Vibration Control of Active Structures, An Introduction ...scmero.ulb.ac.be/Teaching/Courses/MECA-H-524/MECA-H-524-Lecture… · Andr¶e PREUMONT Universit¶e Libre de Bruxelles Active

Andre PREUMONT

Universite Libre de Bruxelles

Active Structures Laboratory

Vibration Control of ActiveStructures, An Introduction3rd Edition

Springer

Berlin Heidelberg NewYorkHongKong LondonMilan Paris Tokyo

“. . . le travail eloigne de noustrois grands maux:

l’ennui, le vice et le besoin. ”

Voltaire, Candide (XXX)

Contents

Preface to the third edition . . . . . . . . . . . . . . . . . . . . . . . . . . . .XVII

Preface to the second edition . . . . . . . . . . . . . . . . . . . . . . . . . .XIX

Preface to the first edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XXI

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Active versus passive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Vibration suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Smart materials and structures . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Feedforward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 The various steps of the design . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Plant description, error and control budget . . . . . . . . . . . . . . 141.7 Readership and Organization of the book . . . . . . . . . . . . . . . 161.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Some concepts in structural dynamics . . . . . . . . . . . . . . . . . . 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Equation of motion of a discrete system . . . . . . . . . . . . . . . . . 212.3 Vibration modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Modal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Structure without rigid body modes . . . . . . . . . . . . . . . 242.4.2 Dynamic flexibility matrix . . . . . . . . . . . . . . . . . . . . . . . 262.4.3 Structure with rigid body modes . . . . . . . . . . . . . . . . . 282.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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2.5 Collocated control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.1 Transmission zeros and constrained system. . . . . . . . . 36

2.6 Continuous structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.7 Guyan reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.8 Craig-Bampton reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Electromagnetic and piezoelectric transducers . . . . . . . . . 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Voice coil transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.1 Proof-mass actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.2 Geophone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 General electromechanical transducer . . . . . . . . . . . . . . . . . . . 533.3.1 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.2 Self-sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Reaction wheels and gyrostabilizers . . . . . . . . . . . . . . . . . . . . . 553.5 Smart materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.6 Piezoelectric transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.6.1 Constitutive relations of a discrete transducer . . . . . . 583.6.2 Interpretation of k2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.6.3 Admittance of the piezoelectric transducer . . . . . . . . . 64

3.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Piezoelectric beam, plate and truss . . . . . . . . . . . . . . . . . . . . 694.1 Piezoelectric material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1.2 Coenergy density function . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Hamilton’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Piezoelectric beam actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.1 Hamilton’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.2 Piezoelectric loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4 Laminar sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4.1 Current and charge amplifiers . . . . . . . . . . . . . . . . . . . . 794.4.2 Distributed sensor output . . . . . . . . . . . . . . . . . . . . . . . . 804.4.3 Charge amplifier dynamics . . . . . . . . . . . . . . . . . . . . . . . 81

4.5 Spatial modal filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.5.1 Modal actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.5.2 Modal sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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4.6 Active beam with collocated actuator-sensor . . . . . . . . . . . . . 854.6.1 Frequency response function . . . . . . . . . . . . . . . . . . . . . 854.6.2 Pole-zero pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.6.3 Modal truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.7 Admittance of a beam with a piezoelectric patch . . . . . . . . . 894.8 Piezoelectric laminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.8.1 Two dimensional constitutive equations . . . . . . . . . . . 924.8.2 Kirchhoff theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.8.3 Stiffness matrix of a multi-layer elastic laminate . . . . 944.8.4 Multi-layer laminate with a piezoelectric layer . . . . . . 964.8.5 Equivalent piezoelectric loads . . . . . . . . . . . . . . . . . . . . 974.8.6 Sensor output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.8.7 Beam model vs. plate model . . . . . . . . . . . . . . . . . . . . . 994.8.8 Additional remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.9 Active truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.9.1 Open-loop transfer function . . . . . . . . . . . . . . . . . . . . . . 1064.9.2 Admittance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.10 Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 Passive damping with piezoelectric transducers . . . . . . . . 1135.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2 Resistive shunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.3 Inductive shunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.4 Switched shunt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.4.1 Equivalent damping ratio . . . . . . . . . . . . . . . . . . . . . . . . 1235.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Collocated versus non-collocated control . . . . . . . . . . . . . . . 1276.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Pole-zero flipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.3 The two-mass problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3.1 Collocated control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.3.2 Non-collocated control . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4 Notch filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.5 Effect of pole-zero flipping on the Bode plots . . . . . . . . . . . . 1356.6 Nearly collocated control system . . . . . . . . . . . . . . . . . . . . . . . 1366.7 Non-collocated control systems . . . . . . . . . . . . . . . . . . . . . . . . . 137

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6.8 The role of damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7 Active damping with collocated system . . . . . . . . . . . . . . . . 1437.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.2 Lead control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.3 Direct velocity feedback (DVF) . . . . . . . . . . . . . . . . . . . . . . . . 1477.4 Positive Position Feedback (PPF) . . . . . . . . . . . . . . . . . . . . . . 1507.5 Integral Force Feedback(IFF) . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.6 Duality between the Lead and the IFF controllers . . . . . . . . 158

7.6.1 Root-locus of a single mode . . . . . . . . . . . . . . . . . . . . . . 1587.6.2 Open-loop poles and zeros . . . . . . . . . . . . . . . . . . . . . . . 160

7.7 Actuator and sensor dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 1607.8 Decentralized control with collocated pairs . . . . . . . . . . . . . . 162

7.8.1 Cross talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.8.2 Force actuator and displacement sensor . . . . . . . . . . . . 1627.8.3 Displacement actuator and force sensor . . . . . . . . . . . . 163


8 Vibration isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.2 Relaxation isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

8.2.1 Electromagnetic realization . . . . . . . . . . . . . . . . . . . . . . 1748.3 Active isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.3.1 Sky-hook damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.3.2 Integral Force Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 177

8.4 Flexible body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.4.1 Free-free beam with isolator . . . . . . . . . . . . . . . . . . . . . . 182

8.5 Payload isolation in spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . 1858.5.1 Interaction isolator/attitude control . . . . . . . . . . . . . . . 1868.5.2 Gough-Stewart platform . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.6 Six-axis isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1888.6.1 Relaxation isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898.6.2 Integral Force Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 1908.6.3 Spherical joints, modal spread . . . . . . . . . . . . . . . . . . . . 191

8.7 Active vs. passive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.8 Car suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1958.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Contents XI

8.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9 State space approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079.2 State space description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

9.2.1 Single degree of freedom oscillator . . . . . . . . . . . . . . . . 2099.2.2 Flexible structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2109.2.3 Inverted pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

9.3 System transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139.3.1 Poles and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

9.4 Pole placement by state feedback . . . . . . . . . . . . . . . . . . . . . . . 2169.4.1 Example: oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

9.5 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 2199.5.1 Symmetric root locus . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.5.2 Inverted pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

9.6 Observer design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2229.7 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

9.7.1 Inverted pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2269.8 Reduced order observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

9.8.1 Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2289.8.2 Inverted pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

9.9 Separation principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2309.10 Transfer function of the compensator . . . . . . . . . . . . . . . . . . . 231

9.10.1 The two-mass problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2329.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2359.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

10 Analysis and synthesis in the frequency domain . . . . . . . . 23710.1 Gain and phase margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23710.2 Nyquist criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

10.2.1 Cauchy’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23810.2.2 Nyquist stability criterion . . . . . . . . . . . . . . . . . . . . . . . . 239

10.3 Nichols chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24310.4 Feedback specification for SISO systems . . . . . . . . . . . . . . . . . 244

10.4.1 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24510.4.2 Tracking error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24510.4.3 Performance specification . . . . . . . . . . . . . . . . . . . . . . . . 24610.4.4 Unstructured uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 24710.4.5 Robust performance and robust stability . . . . . . . . . . . 248

10.5 Bode gain-phase relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 250

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10.6 The Bode Ideal Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25410.7 Non-minimum phase systems . . . . . . . . . . . . . . . . . . . . . . . . . . 25610.8 Usual compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

10.8.1 System type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25910.8.2 Lead compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26010.8.3 PI compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26110.8.4 Lag compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26110.8.5 PID compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

10.9 Multivariable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26310.9.1 Performance specification . . . . . . . . . . . . . . . . . . . . . . . . 26310.9.2 Small gain theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26510.9.3 Stability robustness tests . . . . . . . . . . . . . . . . . . . . . . . . 26510.9.4 Residual dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

10.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26810.11Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

11 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27311.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27311.2 Quadratic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27311.3 Deterministic LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27411.4 Stochastic response to a white noise . . . . . . . . . . . . . . . . . . . . 276

11.4.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27811.5 Stochastic LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27811.6 Asymptotic behavior of the closed-loop . . . . . . . . . . . . . . . . . 27911.7 Prescribed degree of stability . . . . . . . . . . . . . . . . . . . . . . . . . . 28111.8 Gain and phase margins of the LQR . . . . . . . . . . . . . . . . . . . . 28211.9 Full state observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

11.9.1 Covariance of the reconstruction error . . . . . . . . . . . . . 28511.10Kalman-Bucy Filter (KBF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28511.11Linear Quadratic Gaussian (LQG). . . . . . . . . . . . . . . . . . . . . . 28611.12Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28711.13Spillover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

11.13.1Spillover reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29111.14Loop Transfer Recovery (LTR) . . . . . . . . . . . . . . . . . . . . . . . . . 29211.15Integral control with state feedback . . . . . . . . . . . . . . . . . . . . . 29311.16Frequency shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

11.16.1Frequency-shaped cost functionals . . . . . . . . . . . . . . . . 29411.16.2Noise model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298


Contents XIII

12 Controllability and Observability . . . . . . . . . . . . . . . . . . . . . . . 30512.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30612.2 Controllability and observability matrices . . . . . . . . . . . . . . . 30612.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

12.3.1 Cart with two inverted pendulums . . . . . . . . . . . . . . . . 30812.3.2 Double inverted pendulum . . . . . . . . . . . . . . . . . . . . . . . 31012.3.3 Two d.o.f. oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

12.4 State transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31212.4.1 Control canonical form . . . . . . . . . . . . . . . . . . . . . . . . . . 31212.4.2 Left and right eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 31412.4.3 Diagonal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

12.5 PBH test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31512.6 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31612.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31712.8 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31812.9 Controllability and observability Gramians . . . . . . . . . . . . . . 31912.10Internally balanced coordinates . . . . . . . . . . . . . . . . . . . . . . . . 32112.11Model reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

12.11.1Transfer equivalent realization . . . . . . . . . . . . . . . . . . . . 32312.11.2Internally balanced realization . . . . . . . . . . . . . . . . . . . . 32312.11.3Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324


13 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

13.1.1 Phase portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33213.2 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

13.2.1 Routh-Hurwitz criterion . . . . . . . . . . . . . . . . . . . . . . . . . 33413.3 Lyapunov’s direct method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

13.3.1 Introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . . 33513.3.2 Stability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33613.3.3 Asymptotic stability theorem . . . . . . . . . . . . . . . . . . . . 33713.3.4 Lasalle’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33813.3.5 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . 33913.3.6 Instability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

13.4 Lyapunov functions for linear systems . . . . . . . . . . . . . . . . . . 34113.5 Lyapunov’s indirect method . . . . . . . . . . . . . . . . . . . . . . . . . . . 34213.6 An application to controller design . . . . . . . . . . . . . . . . . . . . . 343

XIV Contents

13.7 Energy absorbing controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34413.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34613.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

14 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35114.1 Digital implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

14.1.1 Sampling, aliasing and prefiltering . . . . . . . . . . . . . . . . 35214.1.2 Zero-order hold, computational delay . . . . . . . . . . . . . . 35314.1.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35414.1.4 Discretization of a continuous controller . . . . . . . . . . . 355

14.2 Active damping of a truss structure . . . . . . . . . . . . . . . . . . . . 35614.2.1 Actuator placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35814.2.2 Implementation, experimental results . . . . . . . . . . . . . . 359

14.3 Active damping generic interface . . . . . . . . . . . . . . . . . . . . . . . 36114.3.1 Active damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36114.3.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36414.3.3 Pointing and position control . . . . . . . . . . . . . . . . . . . . . 365

14.4 Active damping of a plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36614.4.1 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

14.5 Active damping of a stiff beam . . . . . . . . . . . . . . . . . . . . . . . . . 36914.5.1 System design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

14.6 The HAC/LAC strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37114.6.1 Wide-band position control . . . . . . . . . . . . . . . . . . . . . . 37314.6.2 Compensator design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37514.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

14.7 Vibroacoustics: Volume displacement sensors . . . . . . . . . . . . 37914.7.1 QWSIS sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38014.7.2 Discrete array sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38314.7.3 Spatial aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38714.7.4 Distributed sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393


15 Tendon Control of Cable Structures . . . . . . . . . . . . . . . . . . . . 40115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40115.2 Tendon control of strings and cables . . . . . . . . . . . . . . . . . . . . 40215.3 Active damping strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40415.4 Basic Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40515.5 Linear theory of decentralized active damping . . . . . . . . . . . . 40615.6 Guyed truss experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

Contents XV

15.7 Micro Precision Interferometer testbed . . . . . . . . . . . . . . . . . . 41315.8 Free floating truss experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 41515.9 Application to cable-stayed bridges . . . . . . . . . . . . . . . . . . . . . 41815.10Laboratory experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41815.11Control of parametric resonance . . . . . . . . . . . . . . . . . . . . . . . . 41815.12Large scale experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42115.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

16 Active Control of Large Telescopes . . . . . . . . . . . . . . . . . . . . . 42916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42916.2 Adaptive optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43016.3 Active optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

16.3.1 Monolithic primary mirror . . . . . . . . . . . . . . . . . . . . . . . 43516.3.2 Segmented primary mirror . . . . . . . . . . . . . . . . . . . . . . . 436

16.4 SVD controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43816.4.1 Loop shaping of the SVD controller . . . . . . . . . . . . . . . 439

16.5 Dynamics of a segmented mirror . . . . . . . . . . . . . . . . . . . . . . . 44016.6 Control-structure interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 443

16.6.1 Multiplicative uncertainty . . . . . . . . . . . . . . . . . . . . . . . 44316.6.2 Additive uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44416.6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

16.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

17 Semi-active control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44917.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44917.2 Magneto-rheological fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45017.3 MR devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45217.4 Semi-active suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

17.4.1 Semi-active devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45717.5 Narrow-band disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

17.5.1 Quarter-car semi-active suspension . . . . . . . . . . . . . . . . 45817.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46317.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Preface to the third edition

From the outset, this book was intended to be a bridge between the do-mains of structures and control. This means that both control and struc-tural engineers should feel at home when dealing with their own field (in-cluding familiar notations), while having a chance to become acquaintedwith the other’s discipline and its own specialized vocabulary. That ambi-tion could be summarized by paraphrasing Woody Allen’s movie: Every-thing You Always Wanted to Know About Control-Structure Interaction(But Were Afraid to Ask). Vocabulary and notations are often major ob-stacles in communication between different communities, and this is evenmore so when one deals with smart materials which are multiphysics bynature, forcing us to give up sacrosanct notations.

In the nine years that separate this third edition from the previousone, I have enjoyed a considerable “return on experience” from usersof this book, in academia as well as in industry, and this has guidedme in preparing the present text. Another important lesson has becomeclear: The success of a structural control project relies more on a soundunderstanding of the system than on a sophisticated control algorithm.

This third edition is about 100 pages longer than the second one. Halfof these additional pages constitutes three totally new chapters: Chap-ter 3 is dedicated to electromagnetic and piezoelectric transducers; thedetailed analysis of energy conversion mechanisms is motivated by theincreasing importance of energy harvesting devices and passive damp-ing mechanisms. Chapter 5 is devoted to the passive damping of struc-tures with piezoelectric transducers, including the basic principle of theswitched inductive shunt. Chapter 16 deals with what will become one ofthe most challenging structural control problems of the coming years: theactive control of extremely large segmented telescopes, with a primary

XVIII Contents

mirror of diameter D = 30m and more. This problem is interesting inmany respects: Above all the surface accuracy, because the RMS wave-front error ε cannot exceed a fraction of the wavelength, making the ratioε/D ∼ 10−9 particularly small. The size of the multivariable control sys-tem is also quite unusual: it will involve several thousand sensors andactuators. Finally, control-structure interaction is likely to be critical inthe design; this offers a wonderful example of application of multivariablerobustness tests. Several other chapters have been reorganized to pro-vide the reader with a deeper physical insight, and better tools for designand robustness assessment. In chapter 7 on active damping, the dualitybetween the Direct Velocity Feedback and the Integral Force Feedbackhas been stressed. Chapter 8 on isolation has been expanded to includethe relaxation isolator which has outstanding performance and uses onlypassive components.

I take this opportunity to thank my co-workers and former studentswho have helped me in producing this book. I am particularly indebtedto the following for their work and contributions as listed below: AhmedAbu Hanieh and Bruno de Marneffe for damping and isolation; Abhi-jit Ganguli for machine tool chatter alleviation; Pierre De Man for vi-broacoustics; More Thomas Avraam for MR fluids; Renaud Bastaits andGoncalo Rodrigues for active control of telescopes and adaptive optics;and Christophe Collette for semi-active suspension and many other things.Bilal Mokrani also contributed to several aspects. The quality of the hard-ware involved in the various experimental set-ups is due to the care ofMihaita Horodinca, Iulian Romanescu and Ioan Burda. Special thanks toRenaud who helped me with the figures. The list of colleagues who haveinspired me during my career would be too long to do them justice.

Andre PreumontBrussels, January 2011.

Preface to the second edition

My objective in writing this book was to cross the bridge between thestructural dynamics and control communities, while providing an overviewof the potential of SMART materials for sensing and actuating purposesin active vibration control. I wanted to keep it relatively simple and fo-cused on systems which worked. This resulted in the following: (i) I re-stricted the text to fundamental concepts and left aside most advancedones (i.e. robust control) whose usefulness had not yet clearly been estab-lished for the application at hand. (ii) I promoted the use of collocatedactuator/sensor pairs whose potential, I thought, was strongly underes-timated by the control community. (iii) I emphasized control laws withguaranteed stability for active damping (the wide-ranging applications ofthe IFF are particularly impressive). (iv) I tried to explain why an accu-rate prediction of the transmission zeros (usually called anti-resonancesby the structural dynamicists) is so important in evaluating the perfor-mance of a control system. (v) I emphasized the fact that the open-loopzeros are more difficult to predict than the poles, and that they could bestrongly influenced by the model truncation (high frequency dynamics)or by local effects (such as membrane strains in piezoelectric shells), es-pecially for nearly collocated distributed actuator/sensor pairs; this effectalone explains many disappointments in active control systems. The suc-cess of the first edition confirmed that this approach was useful and it iswith pleasure that I accepted to prepare this second edition in the samespirit as the first one.

The present edition contains three additional chapters: chapter 6 onactive isolation where the celebrated “sky-hook” damper is revisited,chapter 12 on semi-active control, including some material on magneto-rheological fluids whose potential seems enormous, and chapter 14 on the

XX Contents

control of cable-structures. It is somewhat surprising that this last subjectis finding applications for vibration amplitudes which are nine orders ofmagnitude apart (respectively meters for large cable-stayed bridges andnanometers for precision space structures). Some material has also beenadded on the modelling of piezoelectric structures (chapter 3) and on theapplication of distributed sensors in vibroacoustics (chapter 13).

I am deeply indebted to my coworkers, particularly Younes Achkireand Frederic Bossens for the cable-structures, Vincent Piefort for themodelling of piezoelectric structures, Pierre De Man and Arnaud Francoisin vibroacoustics, Ahmed Abu Hanieh and Mihaita Horodinca in activeisolation and, last but not least, Nicolas Loix and Jean-Philippe Ver-schueren who run with enthusiasm and competence our spin-off company,Micromega Dynamics. I greatly enjoyed working with them, exploring notonly the concepts and the modelling techniques, but also the technologyto make these control systems work. I also express my thanks to Davidde Salle who did all the editing, and to the Series Editor, Prof. GrahamGladwell who, once again, improved my English.

Andre PreumontBrussels, November 2001.

Preface to the first edition

I was introduced to structural control by Raphael Haftka and Bill Hal-lauer during a one year stay at the Aerospace and Ocean Engineeringdepartment of Virginia Tech., during the academic year 1985-1986. Atthat time, there was a tremendous interest in large space structures inthe USA, mainly because of the Strategic Defense Initiative and the spacestation program. Most of the work was theoretical or numerical, but BillHallauer was one of the few experimentalists trying to implement controlsystems which worked on actual structures. When I returned to Belgium,I was appointed at the chair of Mechanical Engineering and Robotics atULB, and I decided to start some basic vibration control experiments onmy own. A little later, SMART materials became widely available andoffered completely new possibilities, particularly for precision structures,but also brought new difficulties due to the strong coupling in their consti-tutive equations, which requires a complete reformulation of the classicalmodelling techniques such as finite elements. We started in this new fieldwith the support of the national and regional governments, the EuropeanSpace Agency, and some bilateral collaborations with European aerospacecompanies. Our Active Structures Laboratory was inaugurated in October1995.

In recent years, with the downsizing of the space programs, activestructures seem to have lost some momentum for space applications, butthey gave birth to interesting spin-offs in various fields of engineering,including the car industry, machine tools, consumer products, and evencivil engineering. I believe that the field of SMART materials is still inits infancy; significant improvements can be expected in the next fewyears, that will dramatically improve their recoverable strain and theirload carrying capability.

XXII Contents

This book is the outgrowth of research work carried out at ULB andlecture notes for courses given at the Universities of Brussels and Liege. Itake this opportunity to thank all my coworkers who took part in this re-search, particularly Jean-Paul Dufour, Christian Malekian, Nicolas Loix,Younes Achkire, Paul Alexandre and Pierre De Man; I greatly enjoyedworking with them along the years, and their enthusiasm and creativityhave been a constant stimulus in my work. I particularly thank Pierrewho made almost all the figures.

Finally, I want to thank the Series Editor, Prof. Graham Gladwell who,as he did for my previous book, read the manuscript and corrected manymistakes in my English. His comments have helped to improve the text.

Andre PreumontBruxelles, July 1996.

1

Introduction

1.1 Active versus passive

Consider a precision structure subjected to varying thermal conditions;unless carefully designed, it will distort as a result of the thermal gradi-ents. One way to prevent this is to build the structure from a thermallystable composite material; this is the passive approach. An alternative wayis to use a set of actuators and sensors connected by a feedback loop; sucha structure is active. In this case, we exploit the main virtue of feedback,which is to reduce the sensitivity of the output to parameter variationsand to attenuate the effect of disturbances within the bandwidth of thecontrol system. Depending on the circumstances, active structures maybe cheaper or lighter than passive structures of comparable performances;or they may offer performances that no passive structure could offer, asin the following example.

Until a few years ago, the general belief was that atmospheric turbu-lence would constitute an important limitation to the resolution of earthbased telescopes; this was one of the main reasons for developing theHubble Space Telescope. Nowadays, it is possible to correct in real timethe disturbances produced by atmospheric turbulence on the optical wavefront coming from celestial objects; this allows us to improve the ultimateresolution of the telescope by one order of magnitude, to the limit im-posed by diffraction. The correction is achieved by a deformable mirrorcoupled to a set of actuators (Fig.1.1). A wave front sensor detects thephase difference in the turbulent wave front and the control computersupplies the shape of the deformable mirror which is required to correctthis error. Adaptive optics has become a standard feature in ground-basedastronomy.

2 1 Introduction

Focal plane

Deformablemirror

Degradedimage

Atmosphericturbulence

Imaging camera

Control computer Correctedimage

Wavefrontsensor

Fig. 1.1. Principle of adaptive optics for the compensation of atmospheric turbulence(by courtesy of G.Rousset-ONERA).

The foregoing example is not the only one where active structures haveproved beneficial to astronomy; another example is the primary mirror oflarge telescopes, which can have a diameter of 8 m or more. Large primarymirrors are very difficult to manufacture and assemble. A passive mirrormust be thermally stable and very stiff, in order to keep the right shapein spite of the varying gravity loads during the tracking of a star, andthe dynamic loads from the wind. There are two alternatives to that,both active. The first one, adopted on the Very Large Telescope (VLT)at ESO in Paranal, Chile, consists of having a relatively flexible primarymirror connected at the back to a set of a hundred or so actuators. Asin the previous example, the control system uses an image analyzer toevaluate the amplitude of the perturbation of the optical modes; next,the correction is computed to minimize the effect of the perturbation andis applied to the actuators. The influence matrix J between the actuatorforces f and the optical mode amplitudes w of the wave front errors canbe determined experimentally with the image analyzer:

w = Jf (1.1)

J is a rectangular matrix, because the number of actuators is larger thanthe number of optical modes of interest. Once the modal errors w∗ havebeen evaluated, the correcting forces can be calculated from

f∗ = JT (JJT )−1w∗ (1.2)

1.1 Active versus passive 3

where JT (JJT )−1 is the pseudo-inverse of the rectangular matrix J . Thisis the minimum norm solution to Equ.(1.1) (Problem 1.1).

The second alternative, adopted on the Keck observatory at MaunaKea, Hawaii, consists of using a segmented primary mirror. The potentialadvantages of such a design are lower weight, lower cost, ease of fabrica-tion and assembly. Each segment has a hexagonal shape and is equippedwith three computer controlled degrees of freedom (tilt and piston) andsix edge sensors measuring the relative displacements with respect to theneighboring segments; the control system is used to achieve the opticalquality of a monolithic mirror (by cophasing the segments), to compen-sate for gravity and wind disturbances, and minimize the impact of thetelescope dynamics on the optical performance (Aubrun et al.). Activeand adaptive optics will be discussed more deeply in chapter 16.

As a third example, also related to astronomy, consider the future in-terferometric missions. The aim is to use a number of smaller telescopesas an interferometer to achieve a resolution which could only be achievedwith a much larger monolithic telescope. One possible spacecraft archi-tecture for such an interferometric mission is represented in Fig.1.2; itconsists of a main truss supporting a set of independently pointing tele-scopes. The relative positions of the telescopes are monitored by a sophis-ticated metrology and the optical paths between the individual telescopesand the beam combiner are accurately controlled with optical delay lines,based on the information coming from a wave front sensor. Typically, thedistance between the telescopes could be 50 m or more, and the order

delay line

Large truss Beamcombiner

AttitudeControl

Independentpointing

telescopes

Vibrationisolator

Vibrationisolator

Fig. 1.2. Schematic view of a future interferometric mission.

4 1 Introduction

of magnitude of the error allowed on the optical path length is a fewnanometers; the pointing error of the individual telescopes is as low asa few nanoradians (i.e. one order of magnitude better than the HubbleSpace Telescope). Clearly, such stringent geometrical requirements in theharsh space environment cannot be achieved with a precision monolithicstructure, but rather by active means as suggested in Fig.1.2. The mainrequirement on the supporting truss is not precision but stability, the ac-curacy of the optical path being taken care of by the wide-band vibrationisolation/steering control system of individual telescopes and the opticaldelay lines (described below). Geometric stability includes thermal stabil-ity, vibration damping and prestressing the gaps in deployable structures(this is a critical issue for deployable trusses). In addition to these geomet-ric requirements, this spacecraft would be sent in deep space (e.g. at theLagrange point L2 ) rather than in low earth orbit, to ensure maximumsensitivity; this makes the weight issue particularly important.

Another interesting subsystem necessary to achieve the stringent spec-ifications is the six d.o.f. vibration isolator at the interface between theattitude control module and the supporting truss; this isolator allows thelow frequency attitude control torque to be transmitted, while filteringout the high frequency disturbances generated by the unbalanced cen-trifugal forces in the reaction wheels. Another vibration isolator may beused at the interface between the truss and the independent telescopes,possibly combined with the steering of the telescopes. The third compo-nent relevant to active control is the optical delay line; it consists of ahigh precision single degree of freedom translational mechanism support-ing a mirror, whose function is to control the optical path length betweenevery telescope and the beam combiner, so that these distances are keptidentical to a fraction of the wavelength (e.g. λ/20).

These examples were concerned mainly with performance. However,as technology develops and with the availability of low cost electroniccomponents, it is likely that there will be a growing number of applicationswhere active solutions will become cheaper than passive ones, for the samelevel of performance.

The reader should not conclude that active will always be better andthat a control system can compensate for a bad design. In most cases, abad design will remain bad, active or not, and an active solution shouldnormally be considered only after all other passive means have been ex-hausted. One should always bear in mind that feedback control can com-pensate for external disturbances only in a limited frequency band that

1.2 Vibration suppression 5

is called the bandwidth of the control system. One should never forgetthat outside the bandwidth, the disturbance is actually amplified by thecontrol system.

1.2 Vibration suppression

Mechanical vibrations span amplitudes from meters (civil engineering) tonanometers (precision engineering). Their detrimental effect on systemsmay be of various natures:

Failure: vibration-induced structural failure may occur by excessivestrain during transient events (e.g. building response to earthquake), byinstability due to particular operating conditions (flutter of bridges underwind excitation), or simply by fatigue (mechanical parts in machines).

Comfort: examples where vibrations are detrimental to comfort arenumerous: noise and vibration in helicopters, car suspensions, wind-induced sway of buildings.

Operation of precision devices: numerous systems in precision en-gineering, especially optical systems, put severe restrictions on mechan-ical vibrations. Precision machine tools, wafer steppers1 and telescopesare typical examples. The performances of large interferometers such asthe VLTI are limited by microvibrations affecting the various parts of theoptical path. Lightweight segmented telescopes (space as well as earth-based) will be impossible to build in their final shape with an accuracy ofa fraction of the wavelength, because of the various disturbance sourcessuch as deployment errors and thermal gradients (which dominate thespace environment). Such systems will not exist without the capability tocontrol actively the reflector shape.

Vibration reduction can be achieved in many different ways, dependingon the problem; the most common are stiffening, damping and isolation.Stiffening consists of shifting the resonance frequency of the structurebeyond the frequency band of excitation. Damping consists of reducingthe resonance peaks by dissipating the vibration energy. Isolation consistsof preventing the propagation of disturbances to sensitive parts of thesystems.1 Moore’s law on the number of transistors on an integrated circuit could not hold

without a constant improvement of the accuracy of wafer steppers and other precisionmachines (Taniguchi).

6 1 Introduction

Damping may be achieved passively, with fluid dampers, eddy cur-rents, elastomers or hysteretic elements, or by transferring kinetic energyto Dynamic Vibration Absorbers (DVA). One can also use transducersas energy converters, to transform vibration energy into electrical en-ergy that is dissipated in electrical networks, or stored (energy harvest-ing). Recently, semi-active devices (also called semi-passive) have becomeavailable; they consist of passive devices with controllable properties. TheMagneto-Rheological (MR) fluid damper is a famous example; piezoelec-tric transducers with switched electrical networks is another one. Sincethey behave in a strongly nonlinear way, semi-active devices can transferenergy from one frequency to another, but they are inherently passiveand, unlike active devices, cannot destabilize the system; they are alsoless vulnerable to power failure. When high performance is needed, activecontrol can be used; this involves a set of sensors (strain, acceleration,velocity, force,. . .), a set of actuators (force, inertial, strain,...) and a con-trol algorithm (feedback or feedforward). Active damping is one of themain focuses of this book. The design of an active control system involvesmany issues such as how to configurate the sensors and actuators, howto secure stability and robustness (e.g. collocated actuator/sensor pairs);the power requirements will often determine the size of the actuators, andthe cost of the project.

1.3 Smart materials and structures

An active structure consists of a structure provided with a set of actuatorsand sensors coupled by a controller; if the bandwidth of the controller in-cludes some vibration modes of the structure, its dynamic response mustbe considered. If the set of actuators and sensors are located at discretepoints of the structure, they can be treated separately. The distinctivefeature of smart structures is that the actuators and sensors are often dis-tributed, and have a high degree of integration inside the structure, whichmakes a separate modelling impossible (Fig.1.3). Moreover, in some appli-cations like vibroacoustics, the behaviour of the structure itself is highlycoupled with the surrounding medium; this also requires a coupled mod-elling. From a mechanical point of view, classical structural materials areentirely described by their elastic constants relating stress and strain, andtheir thermal expansion coefficient relating the strain to the temperature.Smart materials are materials where strain can also be generated by dif-ferent mechanisms involving temperature, electric field or magnetic field,

1.3 Smart materials and structures 7

Structure ActuatorsSensors

Controlsystem

SMAPZTMagnetostrictive...

PZTPVDF

Fiber optics...

high degree of integration

Fig. 1.3. Smart structure.

etc... as a result of some coupling in their constitutive equations. Themost celebrated smart materials are briefly described below:

• Shape Memory Alloys (SMA) allow one to recover up to 5 % strainfrom the phase change induced by temperature. Although two-wayapplications are possible after education, SMA are best suited to one-way tasks such as deployment. In any case, they can be used only atlow frequency and for low precision applications, mainly because of thedifficulty of cooling. Fatigue under thermal cycling is also a problem.The best known SMA is called NITINOL; SMA are little used in activevibration control, and will not be discussed in this book.2

• Piezoelectric materials have a recoverable strain of 0.1 % under electricfield; they can be used as actuators as well as sensors. There are twobroad classes of piezoelectric materials used in vibration control: ce-ramics and polymers. The piezopolymers are used mostly as sensors,because they require extremely high voltages and they have a lim-ited control authority; the best known is the polyvinylidene fluoride(PV DF or PV F2). Piezoceramics are used extensively as actuatorsand sensors, for a wide range of frequency including ultrasonic appli-cations; they are well suited for high precision in the nanometer range(1nm = 10−9m). The best known piezoceramic is the Lead ZirconateTitanate (PZT); PZT patches can be glued or co-fired on the support-ing structure.

• Magnetostrictive materials have a recoverable strain of 0.15 % undermagnetic field; the maximum response is obtained when the material

2 The superelastic behavior of SMA may be exploited to achieve damping, for lowfrequency and low cycle applications, such as earthquake protection.

8 1 Introduction

is subjected to compressive loads. Magnetostrictive actuators can beused as load carrying elements (in compression alone) and they havea long lifetime. They can also be used in high precision applications.The best known is the TERFENOL-D; it can be an alternative to PZTin some applications (sonar).

• Magneto-rheological (MR) fluids consist of viscous fluids containingmicron-sized particles of magnetic material. When the fluid is sub-jected to a magnetic field, the particles create columnar structuresrequiring a minimum shear stress to initiate the flow. This effect is re-versible and very fast (response time of the order of millisecond). Somefluids exhibit the same behavior under electrical field; they are calledelectro-rheological (ER) fluids; however, their performances (limited bythe electric field breakdown) are currently inferior to MR fluids. MRand ER fluids are used in semi-active devices.

This brief list of commercially available smart materials is just a flavor ofwhat is to come: phase change materials are currently under developmentand are likely to become available in a few years time; they will offer a re-coverable strain of the order of 1 % under an electric or magnetic field, oneorder of magnitude more than the piezoceramics. Electroactive polymersare also slowly emerging for large strain low stiffness applications.

The range of available devices to measure position, velocity, acceler-ation and strain is extremely wide, and there are more to come, partic-ularly in optomechanics. Displacements can be measured with inductive,capacitive and optical means (laser interferometer); the latter two have aresolution in the nanometer range. Piezoelectric accelerometers are verypopular but they cannot measure a d.c. component. Strain can be mea-sured with strain gages, piezoceramics, piezopolymers and fiber optics.The latter can be embedded in a structure and give a global average mea-sure of the deformation; they offer a great potential for health monitoringas well. Piezopolymers can be shaped to react only to a limited set ofvibration modes (modal filters).

1.4 Control strategies

There are two radically different approaches to disturbance rejection:feedback and feedforward. Although this text is entirely devoted to feed-back control, it is important to point out the salient features of bothapproaches, in order to enable the user to select the most appropriate onefor a given application.

1.4 Control strategies 9

1.4.1 Feedback

The principle of feedback is represented in Fig.1.4; the output y of thesystem is compared to the reference input r, and the error signal, e =r− y, is passed into a compensator H(s) and applied to the system G(s).The design problem consists of finding the appropriate compensator H(s)such that the closed-loop system is stable and behaves in the appropriatemanner.

r ed

yH(s) G(s)

-

Fig. 1.4. Principle of feedback control.

In the control of lightly damped structures, feedback control is usedfor two distinct and somewhat complementary purposes: active dampingand model based feedback.

The objective of active damping is to reduce the effect of the resonantpeaks on the response of the structure. From

y(s)d(s)

=1

1 + GH(1.3)

(Problem 1.2), this requires GH À 1 near the resonances. Active dampingcan generally be achieved with moderate gains; another nice propertyis that it can be achieved without a model of the structure, and withguaranteed stability, provided that the actuator and sensor are collocatedand have perfect dynamics. Of course actuators and sensors always havefinite dynamics and any active damping system has a finite bandwidth.

The control objectives can be more ambitious, and we may wish tokeep a control variable y (a position, or the pointing of an antenna) toa desired value r in spite of external disturbances d in some frequencyrange. From the previous formula and

F (s) =y(s)r(s)

=GH

1 + GH(1.4)

we readily see that this requires large values of GH in the frequency rangewhere y ' r is sought. GH À 1 implies that the closed-loop transfer

10 1 Introduction

ω c

ξ i

Bandwidth

Stability limit

Structural dampingModal dampingof residual modes

k

i

Fig. 1.5. Effect of the control bandwidth on the net damping of the residual modes.

function F (s) is close to 1, which means that the output y tracks theinput r accurately. From Equ.(1.3), this also ensures disturbance rejectionwithin the bandwidth of the control system. In general, to achieve this,we need a more elaborate strategy involving a mathematical model of thesystem which, at best, can only be a low-dimensional approximation ofthe actual system G(s). There are many techniques available to find theappropriate compensator, and only the simplest and the best establishedwill be reviewed in this text. They all have a number of common features:

• The bandwidth ωc of the control system is limited by the accuracy ofthe model; there is always some destabilization of the flexible modesoutside ωc (residual modes). The phenomenon whereby the net damp-ing of the residual modes actually decreases when the bandwidth in-creases is known as spillover (Fig.1.5).

• The disturbance rejection within the bandwidth of the control systemis always compensated by an amplification of the disturbances outsidethe bandwidth.

• When implemented digitally, the sampling frequency ωs must alwaysbe two orders of magnitude larger than ωc to preserve reasonably thebehaviour of the continuous system. This puts some hardware restric-tions on the bandwidth of the control system.

1.4.2 Feedforward

When a signal correlated to the disturbance is available, feedforward adap-tive filtering constitutes an attractive alternative to feedback for distur-bance rejection; it was originally developed for noise control (Nelson &

1.4 Control strategies 11

System

AdaptiveFilter

Error signal

Reference

Primary disturbance source

Secondary source

Fig. 1.6. Principle of feedforward control.

Elliott), but it is very efficient for vibration control too (Fuller et al.).Its principle is explained in Fig.1.6. The method relies on the availabilityof a reference signal correlated to the primary disturbance; this signal ispassed through an adaptive filter, the output of which is applied to thesystem by secondary sources. The filter coefficients are adapted in sucha way that the error signal at one or several critical points is minimized.The idea is to produce a secondary disturbance such that it cancels theeffect of the primary disturbance at the location of the error sensor. Ofcourse, there is no guarantee that the global response is also reduced atother locations and, unless the response is dominated by a single mode,there are places where the response can be amplified; the method cantherefore be considered as a local one, in contrast to feedback which isglobal. Unlike active damping which can only attenuate the disturbancesnear the resonances, feedforward works for any frequency and attemptsto cancel the disturbance completely by generating a secondary signal ofopposite phase.

The method does not need a model of the system, but the adaptionprocedure relies on the measured impulse response. The approach worksbetter for narrow-band disturbances, but wide-band applications havealso been reported. Because it is less sensitive to phase lag than feedback,feedforward control can be used at higher frequency (a good rule of thumbis ωc ' ωs/10); this is why it has been so successful in acoustics.

The main limitation of feedforward adaptive filtering is the availabil-ity of a reference signal correlated to the disturbance. There are manyapplications where such a signal can be readily available from a sensorlocated on the propagation path of the perturbation. For disturbances in-duced by rotating machinery, an impulse train generated by the rotation

12 1 Introduction

Type of control Advantages Disadvantages

Feedback

Active damping • no model needed • effective only near• guaranteed stability resonances

when collocated

Model based • global method • limited bandwidth (ωc ¿ ωs)(LQG,H∞...) • attenuates all • disturbances outside ωc

disturbances within ωc are amplified• spillover

Feedforward

Adaptive filtering • no model necessary • reference neededof reference • wider bandwidth • local method

(x-filtered LMS) (ωc ' ωs/10) (response may be amplifiedin some part of the system)

• works better for • large amount of real timenarrow-band disturb. computations

Table 1.1. Comparison of feedback and feedforward control strategies.

of the main shaft can be used as reference. Table 1.1 summarizes the mainfeatures of the two approaches.

1.5 The various steps of the design

The various steps of the design of a controlled structure are shown inFig.1.7. The starting point is a mechanical system, some performance ob-jectives (e.g. position accuracy) and a specification of the disturbancesapplied to it; the controller cannot be designed without some knowledgeof the disturbance applied to the system. If the frequency distribution ofthe energy of the disturbance (i.e. the power spectral density) is known,the open-loop performances can be evaluated and the need for an activecontrol system can be assessed (see next section). If an active systemis required, its bandwidth can be roughly specified from Equ.(1.3). Thenext step consists of selecting the proper type and location for a set ofsensors to monitor the behavior of the system, and actuators to control

1.5 The various steps of the design 13

System

Evaluation

Closed loopsystem

Controllercontinuous

design

ActuatorSensor

dynamics

Sensor / Actuatorplacement

Disturbancespecification

Performanceobjectives

ControllabilityObservability

Identification Model

Modelreduction

Digitalimplementation

iterate untilperformanceobjectivesare met

Fig. 1.7. The various steps of the design.

it. The concept of controllability measures the capability of an actuatorto interfere with the states of the system. Once the actuators and sen-sors have been selected, a model of the structure is developed, usuallywith finite elements; it can be improved by identification if experimentaltransfer functions are available. Such models generally involve too manydegrees of freedom to be directly useful for design purposes; they must bereduced to produce a control design model involving only a few degreesof freedom, usually the vibration modes of the system, which carry themost important information about the system behavior. At this point, ifthe actuators and sensors can be considered as perfect (in the frequencyband of interest), they can be ignored in the model; their effect on the

14 1 Introduction

control system performance will be tested after the design has been com-pleted. If, on the contrary, the dynamics of the actuators and sensors maysignificantly affect the behavior of the system, they must be included inthe model before the controller design. Even though most controllers areimplemented in a digital manner, nowadays, there are good reasons tocarry out a continuous design and transform the continuous controllerinto a digital one with an appropriate technique. This approach workswell when the sampling frequency is two orders of magnitude faster thanthe bandwidth of the control system, as is generally the case in structuralcontrol.

1.6 Plant description, error and control budget

Consider the block diagram of (Fig.1.8), in which the plant consists of thestructure and its actuator and sensor. w is the disturbance applied to thestructure, z is the controlled variable or performance metrics (that onewants to keep as close as possible to 0), u is the control input and y isthe sensor output (they are all assumed scalar for simplicity). H(s) is thefeedback control law, expressed in the Laplace domain (s is the Laplacevariable). We define the open-loop transfer functions :

Gzw(s): between w and zGzu(s): between u and zGyw(s): between w and yGyu(s): between u and y

From the definition of the open-loop transfer functions,

y = Gyww + GyuHy (1.5)

or

Plant

Disturbance

Control input

Performance metrics

Output measurement

H(s)

w z

yu

Fig. 1.8. Block diagram of the control system.

1.6 Plant description, error and control budget 15

y = (I −GyuH)−1Gyww (1.6)

It follows that

u = Hy = H(I −GyuH)−1Gyww = Tuww (1.7)

On the other handz = Gzww + Gzuu (1.8)

Combining the two foregoing equations, one finds the closed-loop trans-missibility between the disturbance w and the control metrics z :

z = Tzww = [Gzw + GzuH(I −GyuH)−1Gyw]w (1.9)

The frequency content of the disturbance w is usually described byits Power Spectral Density (PSD), Φw(ω) which describes the frequencydistribution of the mean-square (MS) value

σ2w =

∫ ∞

0Φw(ω)dω (1.10)

[the unit of Φw is readily obtained from this equation; it is expressed inunits of w squared per (rad/s)]. From(1.9), the PSD of the control metricsz is given by :

Φz(ω) = |Tzw|2Φw(ω) (1.11)

Φz(ω) gives the frequency distribution of the mean-square value of theperformance metrics. Even more interesting for design is the cumulativeMS response, defined by the integral of the PSD in the frequency range[ω,∞[

σ2z(ω) =

∫ ∞

ωΦz(ν)dν =

∫ ∞

ω|Tzw|2Φw(ν)dν (1.12)

It is a monotonously decreasing function of frequency and describes thecontribution of all the frequencies above ω to the mean-square value ofz. σz(ω) is expressed in the same units as the performance metrics z andσz(0) is the global RMS response; a typical plot is shown in Fig.1.9 foran hypothetical system with 4 modes. For lightly damped structures, thediagram exhibits steps at the natural frequencies of the modes and themagnitude of the steps gives the contribution of each mode to the errorbudget, in the same units as the performance metrics; it is very helpfulto identify the critical modes in a design, at which the effort should be

16 1 Introduction

0

RMS

error

w1 w2 w3 w4

sz

( )w

open-loop

closed-loop H1 ( )g1

H g g2 ( > )2 1

w

Fig. 1.9. Error budget distribution in open-loop and in closed-loop for increasing gains.

targeted. This diagram can be used to assess the control laws and comparedifferent actuator and sensor configurations. In a similar way, the controlbudget can be assessed from

σ2u(ω) =

∫ ∞

ωΦu(ν)dν =

∫ ∞

ω|Tuw|2Φw(ν)dν (1.13)

σu(ω) describes how the RMS control input is distributed over the variousmodes of the structure and plays a critical role in the actuator design.

Clearly, the frequency content of the disturbance w, described byΦw(ω), is essential in the evaluation of the error and control budgetsand it is very difficult, even risky, to attempt to design a control systemwithout prior information on the disturbance.

1.7 Readership and Organization of the book

Structural control and smart structures belong to the general field ofMechatronics; they consist of a mixture of mechanical and electrical en-gineering, structural mechanics, control engineering, material science andcomputer science. This book has been written primarily for structuralengineers willing to acquire some background in structural control, butit will also interest control engineers involved in flexible structures. Ithas been assumed that the reader is familiar with structural dynamicsand has some basic knowledge of linear system theory, including Laplacetransform, root locus, Bode plots, Nyquist plots, etc... Readers who arenot familiar with these concepts are advised to read a basic text on linear

1.8 References 17

system theory (e.g. Cannon, Franklin et al.). Some elementary backgroundin signal processing is also assumed.

Chapter 2 recalls briefly some concepts of structural dynamics; chapter3 to 5 consider the transduction mechanisms, the piezoelectric materialsand structures and the damping via passive networks. Chapter 6 and 7consider collocated (and dual) control systems and their use in activedamping. Chapter 8 is devoted to vibration isolation. Chapter 9 to 13cover classical topics in control: state space modelling, frequency domain,optimal control, controllability and observability, and stability. Variousstructural control applications (active damping, position control of a flex-ible structure, vibroacoustics) are covered in chapter 14; chapter 15 isdevoted to cable-structures and chapter 16 to the wavefront control oflarge optical telescopes. Finally, chapter 17 is devoted to semi-active con-trol. Each chapter is supplemented by a set of problems; it is assumedthat the reader is familiar with MATLAB-SIMULINK or some equivalentcomputer aided control engineering software.

Chapters 1 to 9 plus part of Chapter 10 and some applications ofchapter 14 can constitute a one semester graduate course in structuralcontrol.

1.8 References

AUBRUN, J.N., LORELL, K.R., HAVAS & T.W., HENNINGER, W.C.Performance Analysis of the Segment Alignment Control System for theTen-Meter Telescope, Automatica, Vol.24, No 4, 437-453, 1988.CANNON, R.H. Dynamics of Physical Systems, McGraw-Hill, 1967.FRANKLIN, G.F., POWELL, J.D. & EMAMI-NAEINI, A. FeedbackControl of Dynamic Systems. Addison-Wesley, 1986.FULLER, C.R., ELLIOTT, S.J. & NELSON, P.A. Active Control of Vi-bration, Academic Press, 1996.GANDHI, M.V. & THOMPSON, B.S. Smart Materials and Structures,Chapman & Hall, 1992.NELSON, P.A. & ELLIOTT, S.J. Active Control of Sound, AcademicPress, 1992.TANIGUCHI, N. Current Status in, and Future Trends of, UltraprecisionMachining and Ultrafine Materials Processing, CIRP Annals, Vol.32, No2, 573-582, 1983.UCHINO, K. Ferroelectric Devices, Marcel Dekker, 2000.

18 1 Introduction

General literature on control of flexible structures

CLARK, R.L., SAUNDERS, W.R. & GIBBS, G.P. Adaptive Structures,Dynamics and Control, Wiley, 1998.GAWRONSKI, W.K. Dynamics and Control of Structures - A Modal Ap-proach, Springer, 1998.GAWRONSKI, W.K. Advanced Structural Dynamics and Active Controlof Structures, Springer, 2004.HANSEN, C.H. & SNYDER, S.D, Active Control of Sound and Vibration,E&FN Spon, London, 1996.HYLAND, D.C., JUNKINS, J.L. & LONGMAN, R.W. Active controltechnology for large space structures, J. of Guidance, Control and Dy-namics, Vol.16, No 5, 801-821, Sept.-Oct.1993.INMAN, D.J. Vibration, with Control, Measurement, and Stability. Prentice-Hall, 1989.INMAN, D.J. Vibration with Control, Wiley 2006.JANOCHA, H. (Editor), Adaptronics and Smart Structures (Basics, Ma-terials, Design and Applications), Springer, 1999.JOHSI, S.M. Control of Large Flexible Space Structures, Lecture Notes inControl and Information Sciences, Vol.131, Springer-Verlag, 1989.JUNKINS, J.L. (Editor) Mechanics and Control of Large Flexible Struc-tures, AIAA Progress in Astronautics and Aeronautics, Vol.129, 1990.JUNKINS, J.L. & KIM, Y. Introduction to Dynamics and Control of Flex-ible Structures, AIAA Education Series, 1993.MEIROVITCH, L. Dynamics and Control of Structures, Wiley, 1990.MIU, D.K. Mechatronics - Electromechanics and Contromechanics, Springer-Verlag, 1993.PREUMONT, A. Mechatronics, Dynamics of Electromechanical and Piezo-electric Systems, Springer, 2006.PREUMONT, A. & SETO, K. Active Control of Structures, Wiley, 2008.SKELTON, R.E. Dynamic System Control - Linear System Analysis andSynthesis, Wiley, 1988.SPARKS, D.W. Jr & JUANG, J.N. Survey of experiments and experi-mental facilities for control of flexible structures, AIAA J.of Guidance,Control and Dynamics, Vol.15, No 4, 801-816, July-August 1992.

1.9 Problems 19

1.9 Problems

P.1.1 Consider the underdeterminate system of equations

Jx = w

Show that the minimum norm solution, i.e. the solution of the minimiza-tion problem

minx

(xT x) such that Jx = w

isx = J+w = JT (JJT )−1w

J+ is called the pseudo-inverse of J . [hint: Use Lagrange multipliers toremove the equality constraint.]P.1.2 Consider the feedback control system of Fig.1.4. Show that thetransfer functions from the input r and the disturbance d to the outputy are respectively

y(s)r(s)

=GH

1 + GH

y(s)d(s)

=1

1 + GH

P.1.3 Based on your own experience, describe one application in whichyou feel an active structure may outclass a passive one; outline the systemand suggest a configuration for the actuators and sensors.

2

Some concepts in structural dynamics

2.1 Introduction

This chapter is not intended to be a substitute for a course in structuraldynamics, which is part of the prerequisites to read this book. The goalof this chapter is twofold: (i) recalling some of the notations which will beused throughout this book, and (ii) insisting on some aspects which areparticularly important when dealing with controlled structures and whichmay otherwise be overlooked. As an example, the structural dynamicanalysts are seldom interested in antiresonance frequencies which play acapital role in structural control.

2.2 Equation of motion of a discrete system

Consider the system with three point masses represented in Fig.2.1. Theequations of motion can be established by considering the free body dia-grams of the three masses and applying Newton’s law; one easily gets:

Mx1 + k(x1 − x2) + c(x1 − x2) = f

mx2 + k(2x2 − x1 − x3) + c(2x2 − x1 − x3) = 0

mx3 + k(x3 − x2) + c(x3 − x2) = 0

or, in matrix form,

M 0 00 m 00 0 m

x1

x2

x3

+

c −c 0−c 2c −c0 −c c

x1

x2

x3

+

k −k 0−k 2k −k0 −k k

x1

x2

x3

=

f00

(2.1)

22 2 Some concepts in structural dynamics

x1 x2 x3

c

k

k(x3à x2)k(x2à x1)

c(xç 2à xç 1) c(xç 3à xç 2)

mM m

mmM

c(xç 3à xç 2)c(xç 2à xç 1)

k(x3à x2)k(x2à x1)

k

c

f

f

Fig. 2.1. Three mass system and its free body diagram.

The general form of the equation of motion governing the dynamicequilibrium between the external, elastic, inertia and damping forces act-ing on a non-gyroscopic, discrete, flexible structure with a finite numbern of degrees of freedom (d.o.f.) is

Mx + Cx + Kx = f (2.2)

where x and f are the vectors of generalized displacements (translationsand rotations) and forces (point forces and torques) and M , K and C arerespectively the mass, stiffness and damping matrices; they are symmetricand semi positive definite. M and K arise from the discretization of thestructure, usually with finite elements. A lumped mass system such asthat of Fig.2.1 has a diagonal mass matrix. The finite element methodusually leads to non-diagonal (consistent) mass matrices, but a diagonalmass matrix often provides an acceptable representation of the inertia inthe structure (Problem 2.2).

The damping matrix C represents the various dissipation mechanismsin the structure, which are usually poorly known. To compensate for thislack of knowledge, it is customary to make assumptions on its form. Oneof the most popular hypotheses is the Rayleigh damping:

C = αM + βK (2.3)

The coefficients α and β are selected to fit the structure under consider-ation.

2.3 Vibration modes 23

2.3 Vibration modes

Consider the free response of an undamped (conservative) system of ordern. It is governed by

Mx + Kx = 0 (2.4)

If one tries a solution of the form x = φi ejωit, φi and ωi must satisfy the

eigenvalue problem(K − ω2

i M)φi = 0 (2.5)

Because M and K are symmetric, K is positive semi definite and M ispositive definite, the eigenvalue ω2

i must be real and non negative. ωi is thenatural frequency and φi is the corresponding mode shape; the number ofmodes is equal to the number of degrees of freedom, n. Note that Equ.(2.5)defines only the shape, but not the amplitude of the mode which can bescaled arbitrarily. The modes are usually ordered by increasing frequencies(ω1 ≤ ω2 ≤ ω3 ≤ ...). From Equ.(2.5), one sees that if the structure isreleased from initial conditions x(0) = φi and x(0) = 0, it oscillates atthe frequency ωi according to x(t) = φi cosωit, always keeping the shapeof mode i.

Left multiplying Equ.(2.5) by φTj , one gets the scalar equation

φTj Kφi = ω2

i φTj Mφi

and, upon permuting i and j, one gets similarly,

φTi Kφj = ω2

j φTi Mφj

Substracting these equations, taking into account that a scalar is equalto its transpose, and that K and M are symmetric, one gets

0 = (ω2i − ω2

j )φTj Mφi

which shows that the mode shapes corresponding to distinct natural fre-quencies are orthogonal with respect to the mass matrix.

φTj Mφi = 0 (ωi 6= ωj)

It follows from the foregoing equations that the mode shapes are alsoorthogonal with respect to the stiffness matrix. The orthogonality condi-tions are often written as

φTi Mφj = µi δij (2.6)


φTi Kφj = µi ω

2i δij (2.7)

where δij is the Kronecker delta (δij = 1 if i = j, δij = 0 if i 6= j),µi is the modal mass (also called generalized mass) of mode i. Since themode shapes can be scaled arbitrarily, it is usual to normalize them insuch a way that µi = 1. If one defines the matrix of the mode shapesΦ = (φ1, φ2, ..., φn), the orthogonality relationships read

ΦT MΦ = diag(µi) (2.8)

ΦT KΦ = diag(µiω2i ) (2.9)

To demonstrate the orthogonality conditions, we have used the factthat the natural frequencies were distinct. If several modes have the samenatural frequency (as often occurs in practice because of symmetry), theyform a subspace of dimension equal to the multiplicity of the eigenvalue.Any vector in this subspace is a solution of the eigenvalue problem, andit is always possible to find a set of vectors such that the orthogonalityconditions are satisfied. A rigid body mode is such that there is no strainenergy associated with it (φT

i Kφi = 0). It can be demonstrated that thisimplies that Kφi = 0; the rigid body modes can therefore be regarded assolutions of the eigenvalue problem (2.5) with ωi = 0.

2.4 Modal decomposition

2.4.1 Structure without rigid body modes

Let us perform a change of variables from physical coordinates x to modalcoordinates according to

x = Φz (2.10)

where z is the vector of modal amplitudes. Substituting into Equ.(2.2),we get

MΦz + CΦz + KΦz = f

Left multiplying by ΦT and using the orthogonality relationships (2.8)and (2.9), we obtain

diag(µi)z + ΦT CΦz + diag(µiω2i )z = ΦT f (2.11)

If the matrix ΦT CΦ is diagonal, the damping is said classical or normal.In this case, the modal fraction of critical damping ξi (in short modaldamping) is defined by

2.4 Modal decomposition 25

ΦT CΦ = diag(2ξiµiωi) (2.12)

One can readily check that the Rayleigh damping (2.3) complies with thiscondition and that the corresponding modal damping ratios are

ξi =12(α

ωi+ βωi) (2.13)

The two free parameters α and β can be selected in order to match themodal damping of two modes. Note that the Rayleigh damping tends tooverestimate the damping of the high frequency modes.

Under condition (2.12), the modal equations are decoupled and Equ.(2.11)can be rewritten

z + 2ξ Ω z + Ω2z = µ−1ΦT f (2.14)

with the notationsξ = diag(ξi)

Ω = diag(ωi) (2.15)

µ = diag(µi)

The following values of the modal damping ratio can be regarded astypical: satellites and space structures are generally very lightly damped(ξ ' 0.001− 0.005), because of the extensive use of fiber reinforced com-posites, the absence of aerodynamic damping, and the low strain level.Mechanical engineering applications (steel structures, piping,...) are in therange of ξ ' 0.01−0.02; most dissipation takes place in the joints, and thedamping increases with the strain level. For civil engineering applications,ξ ' 0.05 is typical and, when radiation damping through the ground isinvolved, it may reach ξ ' 0.20, depending on the local soil conditions.The assumption of classical damping is often justified for light damping,but it is questionable when the damping is large, as in problems involvingsoil-structure interaction. Lightly damped structures are usually easier tomodel, but more difficult to control, because their poles are located verynear the imaginary axis and they can be destabilized very easily.

If one accepts the assumption of classical damping, the only differencebetween Equ.(2.2) and (2.14) lies in the change of coordinates (2.10).However, in physical coordinates, the number of degrees of freedom of adiscretized model of the form (2.2) is usually large, especially if the ge-ometry is complicated, because of the difficulty of accurately representingthe stiffness of the structure. This number of degrees of freedom is unnec-essarily large to represent the structural response in a limited bandwidth.


If a structure is excited by a band-limited excitation, its response is dom-inated by the modes whose natural frequencies belong to the bandwidthof the excitation, and the integration of Equ.(2.14) can often be restrictedto these modes. The number of degrees of freedom contributing effectivelyto the response is therefore reduced drastically in modal coordinates.

2.4.2 Dynamic flexibility matrix

Consider the steady state harmonic response of Equ.(2.2) to a vectorexcitation f = Fejωt. The response is also harmonic, x = Xejωt, and theamplitude of F and X are related by

X = [−ω2M + jωC + K]−1F = G(ω)F (2.16)

Where the matrix G(ω) is called the dynamic flexibility matrix ; it is adynamic generalization of the static flexibility matrix, G(0) = K−1. Themodal expansion of G(ω) can be obtained by transforming (2.16) intomodal coordinates x = Φz as we did earlier. The modal response is alsoharmonic, z = Zejωt and one finds easily that

Z = diag 1µi(ω2

i + 2jξiωiω − ω2)ΦT F

leading to

X = ΦZ = Φ diag 1µi(ω2

i + 2jξiωiω − ω2)ΦT F

Comparing with (2.16), one finds the modal expansion of the dynamicflexibility matrix:

G(ω) = [−ω2M + jωC + K]−1 =n∑

i=1

φiφTi

µi(ω2i + 2jξiωiω − ω2)

(2.17)

where the sum extends to all the modes. Glk(ω) expresses the complexamplitude of the structural response of d.o.f. l when a unit harmonic forceejωt is applied at d.o.f. k. G(ω) can be rewritten

G(ω) =n∑

i=1

φiφTi

µiω2i

Di(ω) (2.18)

where


Di(ω) =1

1− ω2/ω2i + 2jξiω/ωi

(2.19)

is the dynamic amplification factor of mode i. Di(ω) is equal to 1 at ω = 0,it exhibits large values in the vicinity of ωi, |Di(ωi)| = (2ξi)−1, and thendecreases beyond ωi (Fig.2.2).1

Excitation bandwidth

!!b

!i !k

1

Di

Mode outside

the bandwidth

0

2øi

1

!!b

F

Fig. 2.2. Fourier spectrum of the excitation F with a limited frequency content ω < ωb

and dynamic amplification Di of mode i such that ωi < ωb and ωk À ωb.

According to the definition of G(ω) the Fourier transform of the re-sponse X(ω) is related to the Fourier transform of the excitation F (ω)by

X(ω) = G(ω)F (ω)

This equation means that all the frequency components work indepen-dently, and if the excitation has no energy at one frequency, there is noenergy in the response at that frequency. From Fig.2.2, one sees that whenthe excitation has a limited bandwidth, ω < ωb, the contribution of all the

1 Qi = 1/2ξi is often called the quality factor of mode i.


high frequency modes (i.e. such that ωk À ωb) to G(ω) can be evaluatedby assuming Dk(ω) ' 1. As a result, if ωm > ωb,

G(ω) 'm∑

i=1

φiφTi

µiω2i

Di(ω) +n∑

i=m+1

φiφTi

µiω2i

(2.20)

This approximation is valid for ω < ωm. The first term in the right handside is the contribution of all the modes which respond dynamically andthe second term is a quasi-static correction for the high frequency modes.Taking into account that

G(0) = K−1 =n∑

i=1

φiφTi

µiω2i

(2.21)

G(ω) can be rewritten in terms of the low frequency modes only:

G(ω) 'm∑

i=1

φiφTi

µiω2i

Di(ω) + K−1 −m∑

i=1

φiφTi

µiω2i

(2.22)

The quasi-static correction of the high frequency modes is often called theresidual mode, denoted by R. Unlike all the terms involving Di(ω) whichreduce to 0 as ω →∞, R is independent of the frequency and introduces afeedthrough (constant) component in the transfer matrix. We will shortlysee that R has a strong influence on the location of the transmission zerosand that neglecting it may lead to substantial errors in the prediction ofthe performance of the control system.

2.4.3 Structure with rigid body modes

The approximation (2.22) applies only at low frequency, ω < ωm. If thestructure has r rigid body modes, the first sum can be split into rigidand flexible modes; however, the residual mode cannot be used any more,because K−1 no longer exists. This problem can be solved in the follow-ing way. The displacements are partitioned into their rigid and flexiblecontributions according to

x = xr + xe = Φrzr + Φeze (2.23)

where Φr and Φe are the matrices whose columns are the rigid bodymodes and the flexible modes, respectively. Assuming no damping, tomake things formally simpler, and taking into account that the rigid bodymodes satisfy KΦr = 0, we obtain the equation of motion


f

f

f

System loaded with f

Self-equilibrated load

System with dummy constraints,loaded with P f

T

− M x r&&

P f f M xT

r= − &&

Fig. 2.3. Structure with rigid body modes.

MΦrzr + MΦeze + KΦeze = f (2.24)

Left multiplying by ΦTr and using the orthogonality relations (2.6) and

(2.7), we see that the rigid body modes are governed by

ΦTr MΦr zr = ΦT

r f

orzr = µ−1

r ΦTr f (2.25)

Substituting this result into Equ.(2.24), we get

MΦeze + KΦeze = f −MΦrzr

= f −MΦrµ−1r ΦT

r f = (I −MΦrµ−1r ΦT

r )f

orMΦeze + KΦeze = P T f (2.26)

where we have defined the projection matrix

P = I − Φrµ−1r ΦT

r M (2.27)

such that P T f is orthogonal to the rigid body modes. In fact, we caneasily check that

PΦr = 0 (2.28)

PΦe = Φe (2.29)


P can therefore be regarded as a filter which leaves unchanged the flexiblemodes and eliminates the rigid body modes.

If we follow the same procedure as in the foregoing section, we needto evaluate the elastic contribution of the static deflection, which is thesolution of

Kxe = P T f (2.30)

Since KΦr = 0, the solution may contain an arbitrary contribution fromthe rigid body modes. On the other hand, P T f = f −Mxr is the super-position of the external forces and the inertia forces associated with themotion as a rigid body; it is self-equilibrated, because it is orthogonal tothe rigid body modes. Since the system is in equilibrium as a rigid body, aparticular solution of Equ.(2.30) can be obtained by adding dummy con-straints to remove the rigid body modes (Fig.2.3). The modified systemis statically determinate and its stiffness matrix can be inverted. If wedenote by Giso the flexibility matrix of the modified system, the generalsolution of (2.30) is

xe = GisoPT f + Φrγ

where γ is a vector of arbitrary constants. The contribution of the rigidbody modes can be eliminated with the projection matrix P , leading to

xe = PGisoPT f (2.31)

PGisoPT is the pseudo-static flexibility matrix of the flexible modes. On

the other hand, left multiplying Equ.(2.24) by ΦTe , we get

ΦTe MΦeze + ΦT

e KΦeze = ΦTe f

where the diagonal matrix ΦTe KΦe is regular. It follows that the pseudo-

static deflection can be written alternatively

xe = Φeze = Φe(ΦTe KΦe)−1ΦT

e f (2.32)

Comparing with Equ.(2.31), we get

PGisoPT = Φe(ΦT

e KΦe)−1ΦTe =

n∑

r+1

φiφTi

µiω2i

(2.33)

This equation is identical to Equ.(2.20) when there are no rigid bodymodes. From this result, we can extend Equ.(2.22) to systems with rigidbody modes:


G(ω) 'r∑

i=1

φiφTi

−µiω2+

m∑

i=r+1

φiφTi

µi(ω2i − ω2 + 2jξiωiω)

+ R (2.34)

where the contribution from the residual mode is

R =n∑

m+1

φiφTi

µiω2i

= PGisoPT −

m∑

r+1

φiφTi

µiω2i

(2.35)

Note that Giso is the flexibility matrix of the system obtained by addingdummy constraints to remove the rigid body modes. Obviously, this canbe achieved in many different ways and it may look surprising that they alllead to the same result (2.35). In fact, different boundary conditions leadto different displacements under the self-equilibrated load P T f , but theydiffer only by a contribution of the rigid body modes, which is destroyedby the projection matrix P , leading to the same PGisoP

T . Let us illustratethe procedure with an example.

2.4.4 Example

Consider the system of three identical masses of Fig.2.4. There is one rigidbody mode and two flexible ones:

Φ = (Φr, Φe) =

1 1 11 0 −21 −1 1

Fig. 2.4. Three mass system: (a) self-equilibrated forces associated with a force fapplied to mass 1; (b) dummy constraints.


andΦT MΦ = diag(3, 2, 6) ΦT KΦ = k.diag(0, 2, 18)

From Equ.(2.27), the projection matrix is

P =

1 0 00 1 00 0 1

−

111

.

13.(1, 1, 1) =

1 0 00 1 00 0 1

− 1

3

1 1 11 1 11 1 1

or

P =13

2 −1 −1−1 2 −1−1 −1 2

We can readily check that

PΦ = P (Φr, Φe) = (0, Φe)

and the self-equilibrated loads associated with a force f applied to mass1 is, Fig.2.4.a

P T f =13

2 −1 −1−1 2 −1−1 −1 2

f00

=

2/3−1/3−1/3

f

If we impose the statically determinate constraint on mass 1, Fig.2.4.b,the resulting flexibility matrix is

Giso =1k

0 0 00 1 10 1 2

leading to

PGisoPT =

19k

5 −1 −4−1 2 −1−4 −1 5

The reader can easily check that other dummy constraints would lead tothe same pseudo-static flexibility matrix (Problem 2.3).

2.5 Collocated control system 33

2.5 Collocated control system

A collocated control system is a control system where the actuator andthe sensor are attached to the same degree of freedom. It is not sufficientto be attached to the same location, but they must also be dual, that isa force actuator must be associated with a translation sensor (measuringdisplacement, velocity or acceleration), and a torque actuator with a ro-tation sensor (measuring an angle or an angular velocity), in such a waythat the product of the actuator signal and the sensor signal represents theenergy (power) exchange between the structure and the control system.Such systems enjoy very interesting properties. The open-loop FrequencyResponse Function (FRF) of a collocated control system corresponds toa diagonal component of the dynamic flexibility matrix. If the actuatorand sensor are attached to d.o.f. k, the open-loop FRF reads

Gkk(ω) =m∑

i=1

φ2i (k)

µiω2i

Di(ω) + Rkk (2.36)

If one assumes that the system is undamped, the FRF is purely real

Gkk(ω) =m∑

i=1

φ2i (k)

µi(ω2i − ω2)

+ Rkk (2.37)

All the residues are positive (square of the modal amplitude) and, as a

static

response

residual

mode

resonance

anti-

resonance

Gkk(!)

zi

Gkk(0) = Kà1kk

Rkk!i+1!i

!

Fig. 2.5. Open-loop FRF of an undamped structure with a collocated actuator/sensorpair (no rigid body modes).


result, Gkk(ω) is a monotonously increasing function of ω, which behavesas illustrated in Fig.2.5. The amplitude of the FRF goes from −∞ at theresonance frequencies ωi (corresponding to a pair of imaginary poles ats = ±jωi in the open-loop transfer function) to +∞ at the next resonancefrequency ωi+1. Since the function is continuous, in every interval, thereis a frequency zi such that ωi < zi < ωi+1 where the amplitude of theFRF vanishes. In structural dynamics, such frequencies are called anti-resonances; they correspond to purely imaginary zeros at ±jzi, in theopen-loop transfer function. Thus, undamped collocated control systemshave alternating poles and zeros on the imaginary axis. The pole / zeropattern is that of Fig.2.6.a. For a lightly damped structure, the polesand zeros are just moved a little in the left-half plane, but they are stillinterlacing, Fig.2.6.b.

Re(s) Re(s)

Im(s) Im(s)

x x

x x

x x

(a) (b)

Fig. 2.6. Pole/Zero pattern of a structure with collocated (dual) actuator and sensor;(a) undamped; (b) lightly damped (only the upper half of the complex plane is shown,the diagram is symmetrical with respect to the real axis).

If the undamped structure is excited harmonically by the actuator atthe frequency of the transmission zero, zi, the amplitude of the response ofthe collocated sensor vanishes. This means that the structure oscillates atthe frequency zi according to the shape shown in dotted line on Fig.2.7.b.We will establish in the next section that this shape, and the frequencyzi, are actually a mode shape and a natural frequency of the systemobtained by constraining the d.o.f. on which the control system acts. Weknow from control theory that the open-loop zeros are asymptotic valuesof the closed-loop poles, when the feedback gain goes to infinity.

The natural frequencies of the constrained system depend on the d.o.f.where the constraint has been added (this is indeed well known in controltheory that the open-loop poles are independent of the actuator and sensor


configuration while the open-loop zeros do depend on it). However, fromthe foregoing discussion, for every actuator/sensor configuration, therewill be one and only one zero between two consecutive poles, and theinterlacing property applies for any location of the collocated pair.

(a)

(b)

(c)

g

u

y

Fig. 2.7. (a) Structure with collocated actuator and sensor; (b) structure with addi-tional constraint; (c) structure with additional stiffness along the controlled d.o.f.

Referring once again to Fig.2.5, one easily sees that neglecting theresidual mode in the modelling amounts to translating the FRF diagramvertically in such a way that its high frequency asymptote becomes tan-gent to the frequency axis. This produces a shift in the location of thetransmission zeros to the right, and the last one even moves to infinityas the feedthrough (constant) component Rkk disappears from the FRF.Thus, neglecting the residual modes tends to overestimate the frequencyof the transmission zeros. As we shall see shortly, the closed-loop poleswhich remain at finite distance move on loops joining the open-loop polesto the open-loop zeros; therefore, altering the open-loop pole/zero patternhas a direct impact on the closed-loop poles.

The open-loop transfer function of a undamped structure with a col-located actuator/sensor pair can be written

G(s) = G0

∏i(s

2/z2i + 1)∏

j(s2/ω2j + 1)

(ωi < zi < ωi+1) (2.38)


For a lightly damped structure, it reads

G(s) = G0

∏i(s

2/z2i + 2ξis/zi + 1)∏

j(s2/ω2j + 2ξjs/ωj + 1)

(2.39)

The corresponding Bode and Nyquist plots are represented in Fig 2.8.Every imaginary pole at ±jωi introduces a 1800 phase lag and everyimaginary zero at±jzi a 1800 phase lead. In this way, the phase diagram isalways contained between 0 and−1800, as a consequence of the interlacingproperty. For the same reason, the Nyquist diagram consists of a setof nearly circles (one per mode), all contained in the third and fourthquadrants. Thus, the entire curve G(ω) is below the real axis (the diameterof every circle is proportional to ξ−1

i ).

Im(G)

Re(G)w = 0G

f

0°

-90°

-180°

w

w

dB

iw!= zi

zi

! = !i

Fig. 2.8. Nyquist diagram and Bode plots of a lightly damped structure with collocatedactuator and sensor.

2.5.1 Transmission zeros and constrained system

We now establish that the transmission zeros of the undamped systemare the poles (natural frequencies) of the constrained system. Considerthe undamped structure of Fig.2.7.a (a displacement sensor is assumedfor simplicity). The governing equations are

Structure:Mx + Kx = b u (2.40)

Output sensor :


y = bT x (2.41)

u is the actuator input (scalar) and y is the sensor output (also scalar).The fact that the same vector b appears in the two equations is due tocollocation. For a stationary harmonic input at the actuator, u = u0e

jω0t;the response is harmonic, x = x0e

jω0t, and the amplitude vector x0 issolution of

(K − ω20M)x0 = b u0 (2.42)

The sensor output is also harmonic, y = y0ejω0t and the output amplitude

is given byy0 = bT x0 = bT (K − ω2

0M)−1b u0 (2.43)

Thus, the transmission zeros (antiresonance frequencies) ω0 are solutionsof

bT (K − ω20M)−1b = 0 (2.44)

Now, consider the system with the additional stiffness g along the samed.o.f. as the actuator/sensor, Fig 2.7.c. The stiffness matrix of the modifiedsystem is K + gbbT . The natural frequencies of the modified system aresolutions of the eigenvalue problem

[K + gbbT − ω2M ]φ = 0 (2.45)

For all g the solution (ω, φ) of the eigenvalue problem is such that

(K − ω2M)φ + gbbT φ = 0 (2.46)

orbT φ = −bT (K − ω2M)−1gbbT φ (2.47)

Since bT φ is a scalar, this implies that

bT (K − ω2M)−1b = −1g

(2.48)

Taking the limit for g →∞, one sees that the eigenvalues ω satisfy

bT (K − ω2M)−1b = 0 (2.49)

which is identical to (2.44). Thus, ω = ω0; the imaginary zeros of theundamped collocated system, solutions of (2.44), are the poles of theconstrained system (2.45) at the limit, when the stiffness g added alongthe actuation d.o.f. increases to ∞:

limg→∞[(K + gbbT )− ω2

0M ]x0 = 0 (2.50)

This is equivalent to placing a kinematic constraint along the control d.o.f.


2.6 Continuous structures

Continuous structures are distributed parameter systems which are gov-erned by partial differential equations. Various discretization techniques,such as the Rayleigh-Ritz method, or finite elements, allow us to ap-proximate the partial differential equation by a finite set of ordinary dif-ferential equations. In this section, we illustrate some of the features ofdistributed parameter systems with continuous beams. This example willbe frequently used in the subsequent chapters.

The plane transverse vibration of a beam is governed by the followingpartial differential equation

(EIw′′)′′ + mw = p (2.51)

This equation is based on the Euler-Bernoulli assumptions that the neu-tral axis undergoes no extension and that the cross section remains per-pendicular to the neutral axis (no shear deformation). EI is the bendingstiffness, m is the mass per unit length and p the distributed external loadper unit length. If the beam is uniform, the free vibration is governed by

wIV +m

EIw = 0 (2.52)

The boundary conditions depend on the support configuration: a simplesupport implies w = 0 and w′′ = 0 (no displacement, no bending moment);for a clamped end, we have w = 0 and w′ = 0 (no displacement, norotation); a free end corresponds to w′′ = 0 and w′′′ = 0 (no bendingmoment, no shear), etc...

A harmonic solution of the form w(x, t) = φ(x) ejωt can be obtained ifφ(x) and ω satisfy

d4φ

dx4− m

EIω2φ = 0 (2.53)

with the appropriate boundary conditions. This equation defines a eigen-value problem; the solution consists of the natural frequencies ωi (infinitein number) and the corresponding mode shapes φi(x). The eigenvaluesare tabulated for various boundary conditions in textbooks on mechani-cal vibrations (e.g. Geradin & Rixen, 1993, p.187). For the pinned-pinnedcase, the natural frequencies and mode shapes are

ω2n = (nπ)4

EI

ml4(2.54)

2.7 Guyan reduction 39

φn(x) = sinnπx

l(2.55)

Just as for discrete systems, the mode shapes are orthogonal with respectto the mass and stiffness distribution:

∫ l

0mφi(x)φj(x) dx = µiδij (2.56)

∫ l

0EI φ′′i (x)φ′′j (x) dx = µiω

2i δij (2.57)

The generalized mass corresponding to Equ.(2.55) is µn = ml/2. As withdiscrete structures, the frequency response function between a point forceactuator at xa and a displacement sensor at xs is

G(ω) =∞∑

i=1

φi(xa)φi(xs)µi(ω2

i − ω2 + 2jξiωiω)(2.58)

where the sum extends to infinity. Exactly as for discrete systems, theexpansion can be limited to a finite set of modes, the high frequency modesbeing included in a quasi-static correction as in Equ.(2.34) (Problem 2.5).

2.7 Guyan reduction

As already mentioned, the size of a discretized model obtained by finiteelements is essentially governed by the representation of the stiffness ofthe structure. For complicated geometries, it may become very large, es-pecially with automated mesh generators. Before solving the eigenvalueproblem (2.5), it may be advisable to reduce the size of the model bycondensing the degrees of freedom with little or no inertia and which arenot excited by external forces, nor involved in the control. The degrees offreedom to be condensed, denoted x2 in what follows, are often referredto as slaves; those kept in the reduced model are called masters and aredenoted x1.

To begin with, consider the undamped forced vibration of a structurewhere the slaves x2 are not excited and have no inertia; the governingequation is

(M11 00 0

) (x1

x2

)+

(K11 K12

K21 K22

) (x1

x2

)=

(f1

0

)(2.59)

or


M11x1 + K11x1 + K12x2 = f1 (2.60)

K21x1 + K22x2 = 0 (2.61)

According to the second equation, the slaves x2 are completely determinedby the masters x1:

x2 = −K−122 K21x1 (2.62)

Substituting into Equ.(2.60), we find the reduced equation

M11x1 + (K11 −K12K−122 K21)x1 = f1 (2.63)

which involves only x1. Note that in this case, the reduced equation hasbeen obtained without approximation.

The idea in the so-called Guyan reduction is to assume that the master-slave relationship (2.62) applies even if the degrees of freedom x2 havesome inertia (i.e. when the sub-matrix M22 6= 0) or applied forces. Thus,one assumes the following transformation

x =

(x1

x2

)=

(I

−K−122 K21

)x1 = Lx1 (2.64)

The reduced mass and stiffness matrices are obtained by substituting theabove transformation into the kinetic and strain energy:

T =12xT Mx =

12xT

1 LT MLx1 =12xT

1 Mx1

U =12xT Kx =

12xT

1 LT KLx1 =12xT

1 Kx1

withM = LT ML K = LT KL (2.65)

The second equation produces K = K11 −K12K−122 K21 as in Equ.(2.63).

If external loads are applied to x2, the reduced loads are obtained byequating the virtual work

δxT f = δxT1 LT f = δxT

1 f1

orf1 = LT f = f1 −K12K

−122 f2 (2.66)

Finally, the reduced equation of motion reads

Mx1 + Kx1 = f1 (2.67)

2.8 Craig-Bampton reduction 41

Usually, it is not necessary to consider the damping matrix in the re-duction, because it is rarely known explicitly at this stage. The Guyanreduction can be performed automatically in commercial finite elementpackages, the selection of masters and slaves being made by the user. Inthe selection process the following should be kept in mind:

• The degrees of freedom without inertia or applied load can be con-densed without affecting the accuracy.

• Translational degrees of freedom carry more information than rota-tional ones. In selecting the masters, preference should be given totranslations, especially if large modal amplitudes are expected (Prob-lem 2.7).

• It can be demonstrated that the error in the mode shape φi associatedwith the Guyan reduction is an increasing function of the ratio

ω2i

ν21

where ωi is the natural frequency of the mode and ν1 is the first naturalfrequency of the constrained system, where all the degrees of freedomx1 (masters) have been blocked [ν1 is the smallest solution of det(K22−ν2M22) = 0]. Therefore, the quality of a Guyan reduction is stronglyrelated to the natural frequencies of the constrained system and ν1

should be kept far above the frequency band ωb where the model isexpected to be accurate. If this is not the case, the model reductioncan be improved as follows.

2.8 Craig-Bampton reduction

Consider the finite element model(

M11 M12

M21 M22

) (x1

x2

)+

(K11 K12

K21 K22

) (x1

x2

)=

(f1

0

)(2.68)

where the degrees of freedom have been partitioned into the masters x1

and the slaves x2. The masters include all the d.o.f. with a specific in-terest in the problem: those where disturbance and control loads are ap-plied, where sensors are located and where the performance is evaluated(controlled d.o.f.). The slaves include all the other d.o.f. which have noparticular interest in the control problem and are ready for elimination.


The Craig-Bampton reduction is conducted in two steps. First, aGuyan reduction is performed according to the static relationship (2.62).In a second step, the constrained system is considered:

M22x2 + K22x2 = 0 (2.69)

(obtained by setting x1 = 0 in the foregoing equation). Let us assumethat the eigen modes of this system constitute the column of the matrixΨ2, and that they are normalized according to ΨT

2 M22Ψ2 = I. We thenperform the change of coordinates

(x1

x2

)=

(I 0

−K−122 K21 Ψ2

) (x1

α

)= T

(x1

α

)(2.70)

Comparing with (2.64), one sees that the solution has been enriched witha set of fixed boundary modes of modal amplitude α. Using the transfor-mation matrix T , the mass and stiffness matrices are obtained as in theprevious section:

M = T T MT K = T T KT (2.71)

leading to(

M11 M12

M12 I

) (x1

α

)+

(K11 00 Ω2

) (x1

α

)=

(f1

0

)(2.72)

In this equation, the stiffness matrix is block diagonal, with K11 = K11−K12K

−122 K21 being the Guyan stiffness matrix and Ω2 = ΨT

2 K22Ψ2 being adiagonal matrix with entries equal to the square of the natural frequenciesof the fixed boundary modes. Similarly, M11 = M11 − M12K

−122 K21 −

K12K−122 M21 + K12K

−122 M22K

−122 K21 is the Guyan mass matrix [the same

as that given by (2.65)]. K11 and M11 are fully populated but do notdepend on the set of constrained modes Ψ2. The off-diagonal term of themass matrix is given by M12 = (M12 − K12K

−122 M22)Ψ2. Since all the

external loads are applied to the master d.o.f., the right hand side of thisequation is unchanged by the transformation. The foregoing equation maybe used with an increasing number of constrained modes (increasing thesize of α), until the model provides an appropriate representation of thesystem in the requested frequency band.

2.9 References 43

2.9 References

BATHE, K.J. & WILSON, E.L. Numerical Methods in Finite ElementAnalysis, Prentice-Hall, 1976.CANNON, R.H. Dynamics of Physical Systems, McGraw-Hill, 1967.CLOUGH, R.W. & PENZIEN, J. Dynamics of Structures, McGraw-Hill,1975.CRAIG, R.R. Structural Dynamics, Wiley, 1981.CRAIG, R.R., BAMPTON, M.C.C. Coupling of Substructures for Dy-namic Analyses, AIAA Journal, Vol.6(7), 1313-1319, 1968.GAWRONSKI, W.K. Advanced Structural Dynamics and Active Controlof Structures, Springer, 2004.GERADIN, M. & RIXEN, D. Mechanical Vibrations, Theory and Appli-cation to Structural Dynamics, Wiley, 1993.HUGHES, P.C. Attitude dynamics of three-axis stabilized satellite witha large flexible solar array, J. Astronautical Sciences, Vol.20, 166-189,Nov.-Dec. 1972.HUGHES, P.C. Dynamics of flexible space vehicles with active attitudecontrol, Celestial Mechanics Journal, Vol.9, 21-39, March 1974.HUGHES, T.J.R. The Finite Element Method, Linear Static and DynamicFinite Element Analysis, Prentice-Hall, 1987.INMAN, D.J. Vibration, with Control, Measurement, and Stability. Prentice-Hall, 1989.MEIROVITCH, L. Computational Methods in Structural Dynamics, Si-jthoff & Noordhoff, 1980.MODI, V.J. Attitude dynamics of satellites with flexible appendages - Abrief review. AIAA J. Spacecraft and Rockets, Vol.11, 743-751, 1974.ZIENKIEWICZ, O.C., & TAYLOR, R.L. The Finite Element Method,Fourth edition (2 vol.), McGraw-Hill, 1989.


2.10 Problems

P.2.1 Using a finite element program, discretize a simply supporteduniform beam with an increasing number of elements (4,8,etc...). Comparethe natural frequencies with those obtained with the continuous beamtheory. Observe that the finite elements tend to overestimate the naturalfrequencies. Why is that so?P.2.2 Using the same stiffness matrix as in the previous example and adiagonal mass matrix obtained by lumping the mass of every element atthe nodes (the entries of the mass matrix for all translational degrees offreedom are ml/nE , where nE is the number of elements; no inertia is at-tributed to the rotations), compute the natural frequencies. Compare theresults with those obtained with a consistent mass matrix in Problem 2.1.Notice that using a diagonal mass matrix usually tends to underestimatethe natural frequencies.P.2.3 Consider the three mass system of section 2.4.4. Show that chang-ing the dummy constraint to mass 2 does not change the pseudo-staticflexibility matrix PGisoP

T .P.2.4 Consider a simply supported beam with the following properties:l = 1m, m = 1kg/m, EI = 10.266 10−3Nm2. It is excited by a pointforce at xa = l/4.(a) Assuming that a displacement sensor is located at xs = l/4 (collo-cated) and that the system is undamped, plot the transfer function for anincreasing number of modes, with and without quasi-static correction forthe high-frequency modes. Comment on the variation of the zeros withthe number of modes and on the absence of mode 4.Note: To evaluate the quasi-static contribution of the high-frequencymodes, it is useful to recall that the static displacement at x = ξ cre-ated by a unit force applied at x = a on a simply supported beam is

δ(ξ, a) =(l − a)ξ6lEI

[a(2l − a)− ξ2] (ξ ≤ a)

δ(ξ, a) =a(l − ξ)6lEI

[ξ(2l − ξ)− a2] (ξ > a)

The symmetric operator δ(ξ, a) is often called “flexibility kernel” orGreen’s function.(b) Including three modes and the quasi-static correction, draw theNyquist and Bode plots and locate the poles and zeros in the complexplane for a uniform modal damping of ξi = 0.01 and ξi = 0.03.

2.10 Problems 45

(c) Do the same as (b) when the sensor location is xs = 3l/4. Notice thatthe interlacing property of the poles and zeros no longer holds.P.2.5 Consider the modal expansion of the transfer function (2.58) andassume that the low frequency amplitude G(0) is available, either fromstatic calculations or from experiments at low frequency. Show that G(ω)can be approximated by the truncated expansion

G(ω) = G(0) +m∑

i=1

φi(xa)φi(xs)µiω2

i

(ω2 − 2jξiωiω)(ω2

i − ω2 + 2jξiωiω)

P.2.6 Show that the impulse response matrix of a damped structure withrigid body modes reads

g(τ) =[ r∑

i=1

φiφTi

µiτ +

n∑

r+1

φiφTi

µiωdie−ξiωiτ sinωdiτ

]1(τ)

where ωdi = ωi

√1− ξ2

i and 1(τ) is the Heaviside step function.P.2.7 Consider a uniform beam clamped at one end and free at theother end; it is discretized with six finite elements of equal size. Thetwelve degrees of freedom are numbered w1, θ1 to w6, θ6 starting from theclamped end. We perform various Guyan reductions in which we selectx1 according to:(a) all wi, θi (12 degrees of freedom, no reduction);(b) all wi (6 d.o.f.);(c) all θi (6 d.o.f.);(d) w2, θ2, w4, θ4, w6, θ6 (6 d.o.f.);(e) w2, w4, w6 (3 d.o.f.);(f) θ2, θ4, θ6 (3 d.o.f.);For each case, compute the natural frequency ωi of the first three modesand the first natural frequency ν1 of the constrained system. Compare theroles of the translations and rotations.P.2.8 Consider a spacecraft consisting of a rigid main body to which oneor several flexible appendages are attached. Assume that there is at leastone axis about which the attitude motion is uncoupled from the otheraxes. Let θ be the (small) angle of rotation about this axis and J be themoment of inertia (of the main body plus the appendages). Show thatthe equations of motion read

Jθ −m∑

i=1

Γizi = T0


µizi + µiΩ2i zi − Γiθ = 0 i = 1, ..., m

where T0 is the torque applied to the main body, µi and Ωi are the modalmasses and the natural frequencies of the constrained modes of the flexibleappendages and Γi are the modal participation factors of the flexiblemodes [i.e. Γi is the work done on mode i of the flexible appendages bythe inertia forces associated with a unit angular acceleration of the mainbody] (Hughes, 1974). [Hint: Decompose the motion into the rigid bodymode and the components of the constrained flexible modes, express thekinetic energy and the strain energy, write the Lagrangian in the form

L = T − V =12Jθ2 −

∑

i

Γiziθ +12

∑

i

µiz2i −

12

∑

i

µiΩ2i zi

and write the Lagrange equations.]

3

Electromagnetic and piezoelectric transducers

3.1 Introduction

Transducers are critical in active structures technology; they can playthe role of actuator, sensor, or simply energy converter, depending on theapplications. In many applications, the actuators are the most critical partof the system; however, the sensors become very important in precisionengineering where sub-micron amplitudes must be detected.

Two broad categories of actuators can be distinguished: “grounded”and “structure borne” actuators. The former react on a fixed support;they include torque motors, force motors (electrodynamic shakers) ortendons. The second category, also called “space realizable”, include jets,reaction wheels, control moment gyros, proof-mass actuators, active mem-bers (capable of both structural functions and generating active controlforces), piezo strips, etc... Active members and all actuating devices in-volving only internal, self-equilibrating forces, cannot influence the rigidbody motion of a structure.

This chapter begins with a description of the voice-coil transducer andits application to the proof-mass actuator and the geophone (absolute ve-locity sensor). Follows a brief discussion of the single axis gyrostabilizer.The remaining of the chapter is devoted to the piezoelectric materials andthe constitutive equations of a discrete piezoelectric transducer. Integrat-ing piezoelectric elements in beams, plates and trusses will be consideredin the following chapter.

48 3 Electromagnetic and piezoelectric transducers

(b)

Fig. 3.1. Voice-coil transducer: (a) Physical principle. (b) Symbolic representation.

3.2 Voice coil transducer

A voice coil transducer is an energy transformer which converts electri-cal power into mechanical power and vice versa. The system consists ofa permanent magnet (Fig.3.1) which produces a uniform magnetic fluxdensity B normal to the gap, and a coil which is free to move axiallywithin the gap. Let v be the velocity of the coil, f the external force act-ing to maintain the coil in equilibrium against the electromagnetic forces,e the voltage difference across the coil and i the current into the coil.In this ideal transducer, we neglect the electrical resistance and the selfinductance of the coil, as well as its mass and damping (if necessary, thesecan be handled by adding R and L to the electrical circuit of the coil,or a mass and damper to its mechanical model). The voice coil actuatoris one of the most popular actuators in mechatronics (e.g. it is used inelectromagnetic loudspeakers), but it is also used as sensor in geophones.

The first constitutive equation of the voice coil transducer follows fromFaraday’s law:

e = 2πnrBv = Tv (3.1)

whereT = 2πnrB (3.2)

is the transducer constant, equal to the product of the length of the coilexposed to the magnetic flux, 2πnr, and the magnetic flux density B. The

3.2 Voice coil transducer 49

second equation follows from the Lorentz force law: The external force frequired to balance the total force of the magnetic field on n turns of theconductor is

f = −i 2πnrB = −Ti (3.3)

where T is again the transducer constant (3.2). Equation (3.1) and (3.3)are the constitutive equations of the voice coil transducer. Notice thatthe transducer constant T appearing in Faraday’s law (3.1), expressedin volt.sec/m, is the same as that appearing in the Lorentz force (3.3),expressed in N/Amp.

The total power delivered to the moving-coil transducer is equal to thesum of the electric power, ei, and the mechanical power, fv. Combiningwith (3.1) and (3.3), one gets

ei + fv = Tvi− Tiv = 0 (3.4)

Thus, at any time, there is an equilibrium between the electrical powerabsorbed by the device and the mechanical power delivered (and viceversa). The moving-coil transducer cannot store energy, and behaves as aperfect electromechanical converter. In practice, however, the transfer isnever perfect due to eddy currents, flux leakage and magnetic hysteresis,leading to slightly different values of T in (3.1) and (3.3).

3.2.1 Proof-mass actuator

A proof-mass actuator (Fig.3.2) is an inertial actuator which is used invarious applications of vibration control. A reaction mass m is connectedto the support structure by a spring k, a damper c and a force actuator fwhich can be either magnetic or hydraulic. In the electromagnetic actuatordiscussed here, the force actuator consists of a voice coil transducer ofconstant T excited by a current generator i; the spring is achieved withmembranes which also guide the linear motion of the moving mass. Thesystem is readily modelled as in Fig.3.2.a. Combining the equation of asingle d.o.f. oscillator with the Lorentz force law (3.3), one finds

mx + cx + kx = Ti (3.5)

or, in the Laplace domain,

x =Ti

ms2 + cs + k(3.6)


S

N

Permanentmagnet

Coil

Membranes

Magneticcircuit

(b)

k c f = à Ti

i = q

x

FSupport

(a)

m

Movingmass

Fig. 3.2. Proof-mass actuator (a) model assuming a current generator; (b) conceptualdesign of an electrodynamic actuator based on a voice coil transducer.

(s is the Laplace variable). The total force applied to the support is equaland opposite to that applied to the mass:

F = −ms2x =−ms2Ti

ms2 + cs + k(3.7)

It follows that the transfer function between the total force F and thecurrent i applied to the coil is

F

i=

−s2T

s2 + 2ξpωps + ω2p

(3.8)

where T is the transducer constant (in N/Amp), ωp = (k/m)1/2 is thenatural frequency of the spring-mass system and ξp is the damping ratio,which in practice is fairly high, typically 20 % or more.1 The Bode plotsof (3.8) are shown in Fig.3.3; one sees that the system behaves like ahigh-pass filter with a high frequency asymptote equal to the transducerconstant T ; above some critical frequency ωc ' 2ωp, the proof-mass actu-ator can be regarded as an ideal force generator. It has no authority overthe rigid body modes and the operation at low frequency requires a largestroke, which is technically difficult. Medium to high frequency actuators(40 Hz and more) are relatively easy to obtain with low cost components(loudspeaker technology).1 the negative sign in (3.8) is irrelevant.

3.2 Voice coil transducer 51

180

T!p !c

!

i

F

Phase

!

Fig. 3.3. Bode plot F/i of an electrodynamic proof-mass actuator.

If the current source is replaced by a voltage source (Fig.3.4), the mod-elling is slightly more complicated and combines the mechanical equation(3.5) and an electrical equation which is readily derived from Faraday’slaw:

T x + Ldi

dt+ Ri = E(t) (3.9)

where L is the inductance and R is the resistance of the electrical circuit.

Fig. 3.4. Model of a proof-mass actuator with a voltage source.


3.2.2 Geophone

The geophone is a transducer which behaves like an absolute velocity sen-sor above some cut-off frequency which depends on its mechanical con-struction. The system of Fig.3.2.a is readily transformed into a geophoneby using the voltage e as the sensor output (Fig.3.5). If x0 is the displace-

x0

e

Fig. 3.5. Model of a geophone based on a voice coil transducer.

ment of the support and if the voice coil is open (i = 0), the governingequations are

mx + c(x− x0) + k(x− x0) = 0

T (x− x0) = e

combining these equations, one readily finds that

x− x0 =−ms2x0

ms2 + cs + k

e = Ts(x− x0) =−s2T

s2 + (c/m)s + k/msx0

e

x0=

−s2T

s2 + 2ξpωps + ω2p

(3.10)

Thus, there is a perfect duality between a proof-mass actuator used witha current source and a geophone (connected to an infinite resistor); abovethe corner frequency, the gain of the geophone is equal to the transducerconstant T . Designing geophones with very low corner frequency is ingeneral difficult, especially if their orientation with respect to the gravityvector is variable; active geophones where the corner frequency is loweredelectronically may constitute a good alternative option.

3.3 General electromechanical transducer 53

3.3 General electromechanical transducer

Tme

Tem

e

ZeiZm v = xç

f

Fig. 3.6. Electrical analog representation of an electromechanical transducer.

3.3.1 Constitutive equations

The constitutive behavior of a wide class of electromechanical transduc-ers can be modelled as in Fig.3.6, where the central box represents theconversion mechanism between electrical energy and mechanical energy,and vice versa. In Laplace form, the constitutive equations read

e = Zei + Temv (3.11)

f = Tmei + Zmv (3.12)

where e is the Laplace transform of the input voltage across the electri-cal terminals, i the input current, f the force applied to the mechanicalterminals, and v the velocity of the mechanical part. Ze is the blockedelectrical impedance, measured for v = 0; Tem is the transduction co-efficient representing the electromotive force (voltage) appearing in theelectrical circuit per unit velocity in the mechanical part (in volt.sec/m).Tme is the transduction coefficient representing the force acting on the me-chanical terminals to balance the electromagnetic force induced per unitcurrent input on the electrical side (in N/Amp), and Zm is the mechanicalimpedance, measured when the electrical side is open (i = 0). As an ex-ample, it is easy to check that the proof-mass with voltage source (Fig.3.4)can be written in this form with Ze = Ls+R, Zm = ms+c+k/s, Tem = Tand Tme = −T . The same representation applies also to the piezoelectrictransducer analyzed below.

In absence of external force (f = 0), v can be eliminated between thetwo foregoing equations, leading to

e = (Ze − TemTme

Zm)i


−TemTme/Zm is called the motional impedance. The total driving pointelectrical impedance is the sum of the blocked and the motional impedances.

3.3.2 Self-sensing

Equation (3.11) shows that the voltage drop across the electrical termi-nals of any electromechanical transducer is the sum of a contributionproportional to the current applied and a contribution proportional tothe velocity of the mechanical terminals. Thus, if Zei can be measuredand subtracted from e, a signal proportional to the velocity is obtained.This suggests the bridge structure of Fig.3.7. The bridge equations are asfollows: for the branch containing the transducer,

Transducer

Fig. 3.7. Bridge circuit for self-sensing actuation.

e = ZeI + Temv + ZbI

I =1

Ze + Zb(e− Temv)

V4 = ZbI =Zb

Ze + Zb(e− Temv)

For the other branch,e = kZei + kZbi

V2 = kZbi =Zb

Ze + Zbe

3.4 Reaction wheels and gyrostabilizers 55

and the bridge output

V4 − V2 = (−Zb Tem

Ze + Zb) v (3.13)

is indeed a linear function of the velocity v of the mechanical termi-nals. Note, however, that −Zb Tem/(Ze + Zb) acts as a filter; the bridgeimpedance Zb must be adapted to the transducer impedance Ze to avoidamplitude distortion and phase shift between the output voltage V4 − V2

and the transducer velocity in the frequency band of interest.

3.4 Reaction wheels and gyrostabilizers

These devices are torque actuators normally used in attitude control ofsatellites. They have authority over the rigid body modes as well as theflexible modes. A reaction wheel consists of a rotating wheel whose axis isfixed with respect to the spacecraft; a torque is generated by increasingor decreasing the angular velocity. If the angular velocity exceeds thespecification, the wheel must be unloaded, using another type of actuator(jets or magnetic).

In control moment gyros (CMG), the rotating wheel is mounted ongimbals, and the gimbal torques are used as control inputs. The principleof a one-axis gyrostabilizer is described in Fig.3.8. Rotating the gimbalabout the x axis with an angular velocity θx produces torques:

Ty = JzΩθx cos θx (3.14)

Tz = JzΩθx sin θx (3.15)

Servo MotorRotor

Gimbal

Ωy

z

x

x

.θ

Fig. 3.8. One-axis gyrostabilizer.


where JzΩ is the angular momentum along the z axis, and θx is thedeviation of the rotor axis with respect to the vertical. The servo motoron the gimbal axis is velocity controlled. The angle θx is measured also,and a small gain feedback maintains the axis of the rotor in the verticalposition (for a deeper discussion of the use of CMG in attitude control,see Jacot & Liska).

3.5 Smart materials

Piezoelectric materials belong to the so-called smart materials, or multi-functional materials, which have the ability to respond significantly tostimuli of different physical natures. Figure 3.9 lists various effects thatare observed in materials in response to various inputs: mechanical, elec-trical, magnetic, thermal, light. The coupling between the physical fieldsof different types is expressed by the non-diagonal cells in the figure; ifits magnitude is sufficient, the coupling can be used to build discrete ordistributed transducers of various types, which can be used as sensors, ac-tuators, or even integrated in structures with various degrees of tailoringand complexity (e.g. as fibers), to make them controllable or responsive

Input

OutputStrain

Electric

charge

Magnetic

fluxTemperature Light

Stress

Electric

field

Elasticity

Heat

Light

Permittivity

Piezo-

electricity

Piezo-

electricity

Magneto-

striction

Magneto-

strictionMagnetic

field

Thermal

expansion

Photostriction

Magneto-

electric

effect

Pyro-

electricity

Photo-

voltaic

effect

Permeability

Specific

heat

Refractive

index

Magneto

-optic

Electro

-optic

effect

Photo-

elasticity

Fig. 3.9. Stimulus-response relations indicating various effects in materials. The smartmaterials correspond to the non-diagonal cells.

3.6 Piezoelectric transducer 57

to their environment (e.g. for shape morphing, precision shape control,damage detection, dynamic response alleviation,...).

3.6 Piezoelectric transducer

The piezoelectric effect was discovered by Pierre and Jacques Curie in1880. The direct piezoelectric effect consists in the ability of certain crys-talline materials to generate an electrical charge in proportion to an exter-nally applied force; the direct effect is used in force transducers. Accordingto the inverse piezoelectric effect, an electric field parallel to the directionof polarization induces an expansion of the material. The piezoelectriceffect is anisotropic; it can be exhibited only by materials whose crystalstructure has no center of symmetry; this is the case for some ceramicsbelow a certain temperature called the Curie temperature; in this phase,the crystal has built-in electric dipoles, but the dipoles are randomly ori-entated and the net electric dipole on a macroscopic scale is zero. Duringthe poling process, when the crystal is cooled in the presence of a highelectric field, the dipoles tend to align, leading to an electric dipole ona macroscopic scale. After cooling and removing of the poling field, thedipoles cannot return to their original position; they remain aligned alongthe poling direction and the material body becomes permanently piezo-electric, with the ability to convert mechanical energy to electrical energyand vice versa; this property will be lost if the temperature exceeds theCurie temperature or if the transducer is subjected to an excessive electricfield in the direction opposed to the poling field.

The most popular piezoelectric materials are Lead-Zirconate-Titanate(PZT) which is a ceramic, and Polyvinylidene fluoride (PVDF) which isa polymer. In addition to the piezoelectric effect, piezoelectric materialsexhibit a pyroelectric effect, according to which electric charges are gen-erated when the material is subjected to temperature; this effect is usedto produce heat detectors; it will not be discussed here.

In this section, we consider a transducer made of a one-dimensionalpiezoelectric material of constitutive equations (we use the notations ofthe IEEE Standard on Piezoelectricity)

D = εT E + d33T (3.16)S = d33E + sET (3.17)


where D is the electric displacement (charge per unit area, expressedin Coulomb/m2), E the electric field (V/m), T the stress (N/m2) andS the strain. εT is the dielectric constant (permittivity) under constantstress, sE is the compliance when the electric field is constant (inverse ofthe Young’s modulus) and d33 is the piezoelectric constant, expressed inm/V or Coulomb/Newton; the reason for the subscript 33 is that, by con-vention, index 3 is always aligned to the poling direction of the material,and we assume that the electric field is parallel to the poling direction.More complicated situations will be considered later. Note that the sameconstant d33 appears in (3.16) and (3.17).

In the absence of an external force, a transducer subjected to a volt-age with the same polarity as that during poling produces an elongation,and a voltage opposed to that during poling makes it shrink (inversepiezoelectric effect). In (3.17), this amounts to a positive d33. Conversely(direct piezoelectric effect), if we consider a transducer with open elec-trodes (D = 0), according to (3.16), E = −(d33/εT )T , which means thata traction stress will produce a voltage with polarity opposed to that dur-ing poling, and a compressive stress will produce a voltage with the samepolarity as that during poling.

3.6.1 Constitutive relations of a discrete transducer

Equations (3.16) and (3.17) can be written in a matrix form

DS

=

[εT d33

d33 sE

] ET

(3.18)

where (E, T ) are the independent variables and (D, S) are the dependentvariables. If (E, S) are taken as the independent variables, they can berewritten

D =d33

sES + εT

(1− d33

2

sEεT

)E

T =1sE

S − d33

sEE

or

DT

=

[εT (1− k2) e33

−e33 cE

] ES

(3.19)


+_

Cross section:Thickness:

# of disks in the stack:

Electric charge:

Capacitance:

At

nl = nt

Q = nAD

Electrode

Free piezoelectric expansion:Voltage driven:

Charge driven:

î = d33nV

î = d33nCQ

t

E = V=t

C = n2"A=l

Fig. 3.10. Piezoelectric linear transducer.

where cE = 1/sE is the Young’s modulus under E = 0 (short circuitedelectrodes), in N/m2 (Pa); e33 = d33/sE , the product of d33 by the Youngmodulus, is the constant relating the electric displacement to the strainfor short-circuited electrodes (in Coulomb/m2), and also that relatingthe compressive stress to the electric field when the transducer is blocked(S = 0).

k2 =d33

2

sEεT=

e332

cEεT(3.20)

k is called the Electromechanical coupling factor of the material; it mea-sures the efficiency of the conversion of mechanical energy into electricalenergy, and vice versa, as discussed below. From (3.19), we note thatεT (1− k2) is the dielectric constant under zero strain.

If one assumes that all the electrical and mechanical quantities areuniformly distributed in a linear transducer formed by a stack of n disksof thickness t and cross section A (Fig.3.10), the global constitutive equa-tions of the transducer are obtained by integrating Equ.(3.18) or (3.19)over the volume of the transducer; one finds (Problem 3.1)

Q∆

=

[C nd33

nd33 1/Ka

] Vf

(3.21)

or

Qf

=

[C(1− k2) nd33Ka

−nd33Ka Ka

] V∆

(3.22)


where Q = nAD is the total electric charge on the electrodes of the trans-ducer, ∆ = Sl is the total extension (l = nt is the length of the trans-ducer), f = AT is the total force and V the voltage applied between theelectrodes of the transducer, resulting in an electric field E = V/t = nV/l.C = εT An2/l is the capacitance of the transducer with no externalload (f = 0), Ka = A/sEl is the stiffness with short-circuited electrodes(V = 0). Note that the electromechanical coupling factor can be writtenalternatively

k2 =d33

2

sEεT=

n2d332Ka

C(3.23)

Equation (3.21) can be inverted

Vf

=

Ka

C(1− k2)

[1/Ka −nd33

−nd33 C

] Q∆

(3.24)

from which we can see that the stiffness with open electrodes (Q = 0) isKa/(1−k2) and the capacitance for a fixed geometry (∆ = 0) is C(1− k2).Note that typical values of k are in the range 0.3− 0.7; for large k, thestiffness changes significantly with the electrical boundary conditions, andsimilarly the capacitance depends on the mechanical boundary conditions.

Next, let us write the total stored electromechanical energy and coen-ergy functions.2 Consider the discrete piezoelectric transducer of Fig.3.11;

Fig. 3.11. Discrete Piezoelectric transducer.

2 Energy and coenergy functions are needed in connection with energy formulationssuch as Hamilton principle, Lagrange equations or finite elements.


the total power delivered to the transducer is the sum of the electric power,V i and the mechanical power, f∆. The net work on the transducer is

dW = V idt + f∆dt = V dQ + fd∆ (3.25)

For a conservative element, this work is converted into stored energy, dWe,and the total stored energy, We(∆,Q) can be obtained by integrating(3.25) from the reference state to the state (∆, Q).3 Upon differentiatingWe(∆, Q),

dWe(∆,Q) =∂We

∂∆d∆ +

∂We

∂QdQ (3.26)

and, comparing with (3.25), we recover the constitutive equations

f =∂We

∂∆V =

∂We

∂Q(3.27)

Substituting f and V from (3.24) into (3.25), one gets

dWe = V dQ + fd∆

=QdQ

C(1− k2)− nd33Ka

C(1− k2)(∆dQ + Q d∆) +

Ka

1− k2∆ d∆

which is the total differential of

We(∆,Q) =Q2

2C(1− k2)− nd33Ka

C(1− k2)Q∆ +

Ka

1− k2

∆2

2(3.28)

This is the analytical expression of the stored electromechanical energyfor the discrete piezoelectric transducer. Equations (3.27) recover the con-stitutive equations (3.24). The first term on the right hand side of (3.28) isthe electrical energy stored in the capacitance C(1− k2) (correspondingto a fixed geometry, 4 = 0); the third term is the elastic strain energystored in a spring of stiffness Ka/(1 − k2) (corresponding to open elec-trodes, Q = 0); the second term is the piezoelectric energy.

The electromechanical energy function uses ∆ and Q as independentstate variables. A coenergy function using ∆ and V as independent vari-ables can be defined by the Legendre transformation

W ∗e (∆,V ) = V Q−We(∆,Q) (3.29)

3 Since the system is conservative, the integration can be done along any path leadingfrom (0, 0) to (∆, Q).


The total differential of the coenergy is

dW ∗e = QdV + V dQ− ∂We

∂∆d∆− ∂We

∂QdQ

dW ∗e = QdV − f d∆ (3.30)

where Equ.(3.27) have been used. It follows that

Q =∂W ∗

e

∂Vand f = −∂W ∗

e

∂∆(3.31)

Introducing the constitutive equations (3.22) into (3.30),

dW ∗e =

[C(1− k2)V + nd33Ka∆

]dV + (nd33KaV −Ka∆) d∆

= C(1− k2)V dV + nd33Ka (∆dV + V d∆)−Ka∆ d∆

which is the total differential of

W ∗e (∆, V ) = C(1− k2)

V 2

2+ nd33KaV ∆−Ka

∆2

2(3.32)

This is the analytical form of the coenergy function for the discrete piezo-electric transducer. The first term on the right hand side of (3.32) isrecognized as the electrical coenergy in the capacitance C(1− k2) (cor-responding to a fixed geometry, ∆ = 0); the third is the strain energystored in a spring of stiffness Ka (corresponding to short-circuited elec-trodes, V = 0). The second term of (3.32) is the piezoelectric coenergy;using the fact that the uniform electric field is E = nV/l and the uniformstrain is S = ∆/l, it can be rewritten

∫

ΩSe33E dΩ (3.33)

where the integral extends to the volume Ω of the transducer.The analytical form (3.28) of the electromechanical energy, together

with the constitutive equations (3.27) can be regarded as an alternativedefinition of a discrete piezoelectric transducer, and similarly for the an-alytical expression of the coenergy (3.32) and the constitutive equations(3.31).


3.6.2 Interpretation of k2

Consider a piezoelectric transducer subjected to the following mechanicalcycle: first, it is loaded with a force F with short-circuited electrodes; theresulting extension is

∆1 =F

Ka

where Ka = A/(sEl) is the stiffness with short-circuited electrodes. Theenergy stored in the system is

W1 =∫ ∆1

0f dx =

F∆1

2=

F 2

2Ka

At this point, the electrodes are open and the transducer is unloaded ac-cording to a path of slope Ka/(1−k2), corresponding to the new electricalboundary conditions,

∆2 =F (1− k2)

Ka

The energy recovered in this way is

W2 =∫ ∆2

0f dx =

F∆2

2=

F 2(1− k2)2Ka

leaving W1−W2 stored in the transducer. The ratio between the remainingstored energy and the initial stored energy is

W1 −W2

W1= k2

Similarly, consider the following electrical cycle: first, a voltage V isapplied to the transducer which is mechanically unconstrained (f = 0).The electric charges appearing on the electrodes are

Q1 = CV

where C = εT An2/l is the unconstrained capacitance, and the energystored in the transducer is

W1 =∫ Q1

0v dq =

V Q1

2=

CV 2

2

At this point, the transducer is blocked mechanically and electrically un-loaded from V to 0. The electrical charges are removed according to


Q2 = C(1− k2)V

where the capacitance for fixed geometry has been used. The energy re-covered in this way is

W2 =∫ Q2

0v dq =

C(1− k2)V 2

2

leaving W1 −W2 stored in the transducer. Here again, the ratio betweenthe remaining stored energy and the initial stored energy is

W1 −W2

W1= k2

Although the foregoing relationships provide a clear physical interpreta-tion of the electromechanical coupling factor, they do not bring a practicalway of measuring k2; the experimental determination of k2 is often basedon impedance (or admittance) measurements.

3.6.3 Admittance of the piezoelectric transducer

(a)

Transducer

dB

(b)

Fig. 3.12. (a) Elementary dynamical model of the piezoelectric transducer. (b) Typicaladmittance FRF of the transducer, in the vicinity of its natural frequency.

Consider the system of Fig.3.12, where the piezoelectric transducer isassumed massless and is connected to a mass M . The force acting on themass is the negative of that acting on the transducer, f = −Mx; using(3.22),

Q−Mx

=

[C(1− k2) nd33Ka

−nd33Ka Ka

] Vx

(3.34)

3.7 References 65

From the second equation, one gets (in Laplace form)

x =nd33Ka

Ms2 + Ka

and, substituting in the first one and using (3.23), one finds

Q

V= C(1− k2)

[Ms2 + Ka/(1− k2)

Ms2 + Ka

](3.35)

It follows that the admittance reads:

I

V=

sQ

V= sC(1− k2)

s2 + z2

s2 + p2(3.36)

where the poles and zeros are respectively

p2 =Ka

Mand z2 =

Ka/(1− k2)M

(3.37)

p is the natural frequency with short-circuited electrodes (V = 0) and zis the natural frequency with open electrodes (I = 0). From the previousequation one sees that

z2 − p2

z2= k2 (3.38)

which constitutes a practical way to determine the electromechanical cou-pling factor from the poles and zeros of admittance (or impedance) FRFmeasurements (Fig.3.12.b).

3.7 References

CADY, W.G. Piezoelectricity: an Introduction to the Theory and Appli-cations of Electromechanical Phenomena in Crystals, McGrawHill, 1946.CRANDALL, S.H., KARNOPP, D.C., KURTZ, E.F, Jr., PRIDMORE-BROWN, D.C. Dynamics of Mechanical and Electromechanical Systems,McGraw-Hill, N-Y, 1968.DE BOER, E., Theory of Motional Feedback, IRE Transactions on Audio,15-21, Jan.-Feb., 1961.HUNT, F.V. Electroacoustics: The Analysis of Transduction, and its His-torical Background, Harvard Monographs in Applied Science, No 5, 1954.Reprinted, Acoustical Society of America, 1982.IEEE Standard on Piezoelectricity. (ANSI/IEEE Std 176-1987).


JACOT, A.D. & LISKA, D.J. Control moment gyros in attitude control,AIAA J. of Spacecraft and Rockets, Vol.3, No 9, 1313-1320, Sept. 1966.Philips Application Book on Piezoelectric Ceramics, (J. Van Randeraat& R.E. Setterington, Edts), Mullard Limited, London, 1974.Physik Intrumente catalogue, Products for Micropositioning (PI GmbH).PRATT, J., FLATAU, A. Development and analysis of self-sensing mag-netostrictive actuator design, SPIE Smart Materials and Structures Con-ference, Vol.1917, 1993.PREUMONT, A. Mechatronics, Dynamics of Electromechanical and Piezo-electric Systems, Springer, 2006.ROSEN, C.A. Ceramic transformers and filters, Proc. Electronic Compo-nent Symposium, p.205-211 (1956).UCHINO, K. Ferroelectric Devices, Marcel Dekker, 2000.WOODSON, H.H., MELCHER, J.R. Electromechanical Dynamics, PartI: Discrete Systems, Wiley, 1968.

3.8 Problems

P.3.1 Consider the piezoelectric linear transducer of Fig.3.10; assumingthat all the electrical and mechanical quantities are uniformly distributed,show that the constitutive equations of the transducer, Equ.(3.22) can bederived from those of the material, Equ.(3.19).P.3.2 A piezoelectric transducer supporting an inertial mass M can beused as an accelerometer or as a proof-mass actuator.

(a) Accelerometer (Fig.3.13.a): the transducer is placed on a surfacesubjected to an acceleration x0 and it is connected to a charge amplifierenforcing the electrical boundary conditions V ' 0 (see section 4.4.1).4

Show that the transfer function between the support acceleration x0 andthe electric charge Q is given by

Q

x0=

−nd33M

1 + 2ξs/ωn + s2/ω2n

(3.39)

where ω2n = Ka/M and c/M = 2ξωn. Comment on what would be the

requirements for a good accelerometer.

4 Some damping is introduced in the system by assuming a mechanical stiffness Ka+csinstead of Ka in the mechanical part of the transducer constitutive equations (3.22)

Qf

=

[C(1− k2) nd33Ka

−nd33Ka Ka + cs

]V∆

3.8 Problems 67

(a)

charge

amplifier

Q

V=0piezoelectric

transducer

0

(b)

V

f

voltage

amplifier

Fig. 3.13. (a) Piezoelectric accelerometer: the input is the support acceleration x0 andthe output is the electric charge Q measured by the charge amplifier. (b) Proof-massactuator: the input is the voltage V applied to the piezo stack and the output is thereaction force f applied to the support.

(b) Proof-mass actuator (Fig.3.13.b): The input is the voltage V ap-plied to the piezo stack and the output is the reaction force f applied tothe support. Show that the transfer function between the voltage V andthe reaction force f is given by

f

V=

s2nd33Ka

s2 + 2ξsωn + ω2n

(3.40)

Compare with the solution based on a voice-coil transducer, Equ.(3.8).P.3.3 Draw the cycle diagrams (f,∆) and (V,Q) of the physical interpre-tations of the electromechanical coupling factor k2, in section 3.6.2.P.3.4 Represent the discrete piezoelectric transducer (3.24) in the elec-trical analog form of Fig.3.6.P.3.5 Electromagnetic damper: Consider a beam with modal propertiesµi, ωi, φi(x) attached to a voice coil transducer of constant T at x = a.Evaluate the modal damping ξi when the coil is shunted on a resistor R.How can this damper be optimized?

4

Piezoelectric beam, plate and truss

4.1 Piezoelectric material

4.1.1 Constitutive relations

The constitutive equations of a general piezoelectric material are

Tij = cEijklSkl − ekijEk (4.1)

Di = eiklSkl + εSikEk (4.2)

where Tij and Skl are the components of the stress and strain tensors,respectively, cE

ijkl are the elastic constants under constant electric field(Hooke’s tensor), eikl the piezoelectric constants (in Coulomb/m2) andεSij the dielectric constant under constant strain. These formulae use clas-

sical tensor notations, where all indices i, j, k, l = 1, 2, 3, and there is asummation on all repeated indices. The above equations are a general-ization of (3.19), with Skl and Ej as independent variables; they can bewritten alternatively with Tkl and Ej as independent variables:

Sij = sEijklTkl + dkijEk (4.3)

Di = diklTkl + εTikEk (4.4)

where sEijkl is the tensor of compliance under constant electric field, dikl

the piezoelectric constants (in Coulomb/Newton) and εTik the dielectric

constant under constant stress. The difference between the properties un-der constant stress and under constant strain has been stressed earlier.As an alternative to the above tensor notations, it is customary to usethe engineering vector notations

70 4 Piezoelectric beam, plate and truss

T =

T11

T22

T33

T23

T31

T12

S =

S11

S22

S33

2S23

2S31

2S12

(4.5)

With these notations, Equ.(4.1) (4.2) can be written in matrix form

T = [c]S − [e]ED = [e]T S+ [ε]E (4.6)

and (4.3), (4.4),

S = [s]T+ [d]ED = [d]T T+ [ε]E (4.7)

where the superscript T stands for the transposed; the other superscriptshave been omitted, but can be guessed from the equation itself. Assum-ing that the coordinate system coincides with the orthotropy axes of thematerial and that the direction of polarization coincides with direction 3,the explicit form of (4.7) is:

Actuation :

S11

S22

S33

2S23

2S31

2S12

=

s11 s12 s13 0 0 0s12 s22 s23 0 0 0s13 s23 s33 0 0 00 0 0 s44 0 00 0 0 0 s55 00 0 0 0 0 s66

T11

T22

T33

T23

T31

T12

+

0 0 d31

0 0 d32

0 0 d33

0 d24 0d15 0 00 0 0

E1

E2

E3

(4.8)

Sensing :

D1

D2

D3

=

0 0 0 0 d15 00 0 0 d24 0 0

d31 d32 d33 0 0 0

T11

T22

T33

T23

T31

T12

+

ε11 0 00 ε22 00 0 ε33

E1

E2

E3

(4.9)

4.1 Piezoelectric material 71

Typical values of the piezoelectric constants for piezoceramics (PZT) andpiezopolymers (PVDF) are given in Table 4.1. Examining the actuatorequation (4.8), we note that when an electric field E3 is applied paral-lel to the direction of polarization, an extension is observed along thesame direction; its amplitude is governed by the piezoelectric coefficientd33. Similarly, a shrinkage is observed along the directions 1 and 2 per-pendicular to the electric field, the amplitude of which is controlled byd31 and d32, respectively (shrinkage, because d31 and d32 are negative).Piezoceramics have an isotropic behaviour in the plane, d31 = d32; on thecontrary, when PVDF is polarized under stress, its piezoelectric proper-ties are highly anisotropic, with d31 ∼ 5d32. Equation (4.8) also indicatesthat an electric field E1 normal to the direction of polarization 3 producesa shear deformation S13, controlled by the piezoelectric constant d15 (sim-ilarly, a shear deformation S23 occurs if an electric field E2 is applied; itis controlled by d24). An interesting feature of this type of actuation is

+

_

ÉL = nd33V

ÉL

ÉL = Ed31L

E = V=t

ÉL

íL0E1

1

3

í = d15E1

ÉL = íL0

V

P

P

V

L

PE t

Supporting structure

P V

d33

d31

d15

Fig. 4.1. Actuation modes of piezoelectric actuators. P indicates the direction ofpolarization.


that d15 is the largest of all piezoelectric coefficients (500 10−12C/N forPZT). The various modes of operation associated with the piezoelectriccoefficients d33, d31 and d15 are illustrated in Fig.4.1.

4.1.2 Coenergy density function

With an approach parallel to that of the discrete transducer, the totalstored energy density in a unit volume of material is the sum of themechanical work and the electrical work,

dWe(S,D) = dST T+ dDT E (4.10)

[compare with (3.25)]. For a conservative system, We(S, D) can be ob-tained by integrating (4.10) from the reference state to the state (S,D);since the system is conservative, the integration can be done along anypath from (0, 0) to (S, D). Upon differentiating We(S,D) and comparingwith (4.10) we recover the constitutive equations

T =

∂We

∂S

and E =

∂We

∂D

(4.11)

which are the distributed counterparts of (3.27). The coenergy densityfunction is defined by the Legendre transformation

W ∗e (S, E) = ET D −We(S, D) (4.12)

[compare with (3.29)]. The total differential is

dW ∗e = dET D+ ET dD − dST

∂We

∂S

− dDT

∂We

∂D

= dET D − dST T (4.13)

where (4.11) have been used. It follows that

D =

∂W ∗e

∂E

and T = −

∂W ∗

e

∂S

(4.14)

Substituting (4.6) into (4.13),

dW ∗e = dET [e]T S+ dET [ε]E − dST [c]S+ dST [e]E

(4.15)which is the total differential of

4.1 Piezoelectric material 73

Material properties PZT PVDF

Piezoelectric constantsd33(10−12C/N or m/V ) 300 -25d31(10−12C/N or m/V ) -150 uni-axial:

d31 = 15d32 = 3bi-axial:

d31 = d32 = 3d15(10−12C/N or m/V ) 500 0

e31 = d31/sE(C/m2) -7.5 0.025Electromechanical coupling factor

k33 0.7k31 0.3 ∼ 0.1k15 0.7

Dielectric constant εT /ε0 1800 10(ε0 = 8.85 10−12F/m)

Max. Electric field (V/mm) 2000 5 105

Max. operating (Curie) T (C) 80 − 150 90

Density (Kg/m3) 7600 1800Young’s modulus 1/sE (GPa) 50 2.5

Maximum stress (MPa)Traction 80 200

Compression 600 200Maximum strain Brittle 50 %

Table 4.1. Typical properties of piezoelectric materials.

W ∗e (S, E) =

12ET [ε]E+ ST [e]E − 1

2ST [c]S (4.16)

[compare with (3.32)]. The first term in the right hand side is the electricalcoenergy stored in the dielectric material (ε is the matrix of permittivityunder constant strain); the third term is the strain energy stored in theelastic material (c is the matrix of elastic constants under constant elec-tric field); the second term is the piezoelectric coenergy, which generalizes(3.33) in three dimensions. Taking the partial derivatives (4.14), one re-covers the constitutive equations (4.6). In that sense, the analytical formof the coenergy density function, (4.16) together with (4.14), can be seenas an alternative definition of the linear piezoelectricity. In the literature,


H(S,E) = −W ∗e (S,E) (4.17)

is known as the electric enthalpy density.

4.2 Hamilton’s principle

According to Hamilton’s principle, the variational indicator

V.I. =∫ t2

t1[δ(T ∗ + W ∗

e ) + δWnc]dt = 0 (4.18)

is stationary for all admissible (virtual) variations δui and δEi of the pathbetween the two fixed configurations at t1 and t2.

T ∗ =12

∫

Ω%uT udΩ (4.19)

is the kinetic (co)energy (% is the density) and

W ∗e =

12

∫

Ω

(ET [ε]E+ 2ST [e]E − ST [c]S

)dΩ (4.20)

has been defined in the previous section. T ∗ + W ∗e is the Lagrangian and

δWnc is the virtual work of nonconservative external forces and appliedcurrents.

4.3 Piezoelectric beam actuator

Consider the piezoelectric beam of Fig.4.2; it is covered with a singlepiezoelectric layer of uniform thickness hp, polarized along the z axis; thesupporting structure is acting as electrode on one side and there is anelectrode of variable width bp(x) on the other side. The voltage differencebetween the electrodes is controlled, so that the part of the piezoelectricmaterial located between the electrodes is subjected to an electric fieldE3 parallel to the polarization (note that the piezoelectric material whichis not covered by the electrode on both sides is useless as active material).We denote by w(x, t) the transverse displacements of the beam; accordingto the Euler-Bernoulli assumption, the stress and strain fields are uniaxial,along Ox; the axial strain S1 is related to the curvature w

′′by

S1 = −zw′′

(4.21)

where z is the distance to the neutral axis. We also assume that the piezo-electric layer is thin enough, so that E3 is constant across the thickness.

4.3 Piezoelectric beam actuator 75

w(x; t)

x

z

Neutral

Axis

Piezoelectric

material

Electrode

hp

h1h2

zm

V

p(x; t)

bp(x)

Fig. 4.2. Piezoelectric beam covered by a single piezoelectric layer with an electrodeprofile of width bp(x).

4.3.1 Hamilton’s principle

The kinetic coenergy reads

T ∗ =12

∫ l

0%Aw2dx (4.22)

where A is the cross-section of the beam. Both the electric field and thestrain vectors have a single non-zero component, respectively E3 and S1;the coenergy function (4.20) is therefore

W ∗e =

12

∫ l

0dx

∫

A

(ε33E

23 + 2S1e31E3 − c11S

21

)dA (4.23)

and, combining with (4.21),

W ∗e =

12

∫ l

0dx

∫

A

(ε33E

23 − 2w

′′ze31E3 − c11w

′′2z2

)dA (4.24)

The first contribution to W ∗e is restricted to the piezoelectric part of the

beam under the electrode area; the integral over the cross section canbe written ε33E

23bphp. The second contribution is also restricted to the

piezoelectric layer; taking into account that∫

AzdA =

∫ h2

h1

bp z dz = bphpzm

where zm is the distance between the mid-plane of the piezoelectric layerand the neutral axis (Fig.4.2), it can be written −2w

′′e31E3bphpzm. The


third term in W ∗e can be rewritten by introducing the bending stiffness

(we give up the classical notation EI familiar to structural engineers toavoid confusion)

D =∫

Ac11z

2dA (4.25)

Thus, W ∗e reads

W ∗e =

12

∫ l

0

(ε33E

23bphp − 2w

′′e31E3bphpzm −Dw

′′2)dx

Next, we can apply Hamilton’s principle, recalling that only the verticaldisplacement is subject to virtual changes, δw, since the electric potentialis fixed (voltage control). Integrating by part the kinetic energy withrespect to time and taking into account that δw(x, t1) = δw(x, t2) = 0,

∫ t2

t1δT ∗dt =

∫ t2

t1dt

∫ l

0%Aw δw dx = −

∫ t2

t1dt

∫ l

0%Aw δw dx

Similarly,

δW ∗e =

∫ l

0[−δw

′′(e31E3bphpzm)−Dw

′′δw

′′]dx

and, integrating by part twice with respect to x,

δW ∗e = − (e31E3bphpzm) δw′

]l

0+(e31E3bphpzm)

′δw

]l

0−

∫ l

0(e31E3bphpzm)

′′δw dx

−Dw′′δw

′]l

0+ (Dw

′′)′δw

]l

0−

∫ l

0(Dw

′′)′′δw dx

The virtual work of nonconservative forces is

δWnc =∫ l

0p(x, t)δw dx

where p(x, t) is the distributed transverse load applied to the beam. In-troducing in Hamilton’s principle (4.18), one gets that

V.I. =∫ t2

t1dt

∫ l

0

[−%Aw − (e31E3bphpzm)

′′ −(Dw

′′)′′+ p

]δw dx

−[(

e31E3bphpzm +Dw′′)

δw′]l

0+

[(e31E3bphpzm)

′+

(Dw

′′)′δw]l

0= 0

for all admissible variations δw compatible with the kinematics of thesystem (i.e. boundary conditions); let us discuss this equation.

4.3 Piezoelectric beam actuator 77

4.3.2 Piezoelectric loads

It follows from the previous equation that the differential equation gov-erning the problem is

%Aw +(Dw

′′)′′= p− (e31E3bphpzm)

′′(4.26)

If one takes into account that only bp depends on the spatial variablex and that E3hp = V , the voltage applied between the electrodes of thepiezoelectric layer, it becomes

%Aw +(Dw

′′)′′= p− e31V zmbp

′′(x) (4.27)

This equation indicates that the effect of the piezoelectric layer is equiva-lent to a distributed load proportional to the second derivative of the widthof the electrode.

Examining the remaining terms, one must also have(e31E3bphpzm +Dw

′′)δw

′= 0

[(e31E3bphpzm)

′+

(Dw

′′)′]δw = 0 at x = 0 and x = l (4.28)

The first condition states that at an end where the rotation is free (wherea virtual rotation is allowed, δw′ 6= 0), one must have

e31V bpzm +Dw′′

= 0 (4.29)

This means that the effect of the piezoelectric layer is that of a bendingmoment proportional to the width of the electrode. Similarly, the secondcondition states that at an end where the displacement is free (where avirtual displacement is allowed, δw 6= 0), one must have

e31V b′pzm +

(Dw

′′)′= 0 (4.30)

(Dw

′′)′represents the transverse shear force along the beam in classical

beam theory, and step changes of the shear distribution occur where pointloads are applied. This means that the effect of a change of slope b′p in thewidth of the electrode is equivalent to a point force proportional to changeof the first derivative of the electrode width. One should always keep inmind that the piezoelectric loading consists of internal forces which arealways self-equilibrated.


Mp

P

Pb

l

bp =l24b x(là x)

P = à e31V zml4b

p = e31V zml28b

(b)

P

P

x

l

p

b

(c)

bzm

V

Mp

Mp

(a)

Mp = à e31V b zm

Mp = à e31V b zm

P = e31V zmlb

Fig. 4.3. Examples of electrode shapes and corresponding piezoelectric loading: (a)rectangular electrode, (b) triangular electrode, (c) parabolic electrode. The piezoelectricloading is always self-equilibrated.

Figure 4.3 shows a few examples of electrode shapes and the cor-responding piezoelectric loading. A rectangular electrode (Fig.4.3.a) isequivalent to a pair of bending moments Mp applied at the ends of theelectrode. A triangular electrode (Fig.4.3.b) is equivalent to a pair of pointforces P and a bending moment Mp; note that if the beam is clamped onthe left side, the corresponding loads will be taken by the support, andthe only remaining force is the point load at the right end. A parabolicelectrode (Fig.4.3.c) is equivalent to a uniform distributed load p and apair of point forces P at the ends.

4.4 Laminar sensor 79

As another example, consider the electrode shape of Fig.4.4. It consistsof a rectangular part of length l1, followed by a part with constant slope,of length l2. According to the foregoing discussion, this is equivalent tobending moments M1 and M2 at the extremities of the electrodes, andpoint forces P at the location where there is a sudden change in the firstderivative b

′(x). Once again, the piezoelectric loading is self-equilibrated.

l1

l2

b1

b2

P

P

M1

M2

M1 = à e31V b1 zm

M2 = à e31V b2 zmP = à e31V (

l2

b2àb1)zm

Fig. 4.4. Self-equilibrated equivalent piezoelectric loading for an electrode with a sud-den change in b′p(x).

4.4 Laminar sensor

4.4.1 Current and charge amplifiers

When used in sensing mode, a piezoelectric transducer is coupled toan operational amplifier (Fig.4.5.a) to form either a current amplifier(Fig.4.5.b), or a charge amplifier (Fig.4.5.c). An operational amplifier isan active electrical circuit working as a high gain linear voltage ampli-fier with infinite input resistance (so that the input currents i− and i+are essentially zero), and zero output resistance, so that the output volt-age e0 is essentially proportional to the voltage difference e+ − e−; theopen loop gain A is usually very high, which means that the allowableinput voltage is very small (millivolt). As a result, when the electrodesof a piezoelectric transducer are connected to an operational amplifier,they can be regarded as short-circuited and the electric field through thepiezo can be considered as E3 = 0. Then, it follows from the constitutive


(a)

(b)

i

i

io

eo = A( )

i

e

e eo

i

i

e

e eo

i

(c)

eà= eo + Ri

i i,e e Very small

' 0

eà= i=sC+ eo

' à i=sC = àQ=C

Piezo

Piezo

e ee

e

R

C

A> 105

e0=àAeà

e0=àAeà

e0= 1+AàARi'àRi

e0 = (1+A)àA

sC

i

Fig. 4.5. (a) Operational amplifier, (b) Current amplifier, (c) Charge amplifier.

equation (4.2) that the electric displacement is proportional to the strain

D3 = e31S1 (4.31)

4.4.2 Distributed sensor output

If one assumes that the piezoelectric sensor is thin with respect to thebeam, the strain can be regarded as uniform over its thickness, S1 =−zmw

′′, and E3 = 0 is enforced by the charge amplifier; integrating over

the electrode area (Fig.4.2), one gets

Q =∫

D3dA = −∫ b

abp(x)zme31w

′′dx = −zme31

∫ b

abp(x)w

′′dx (4.32)

with a constant polarization profile e31. It is assumed that the sensorextends from x = a to x = b along the beam. Thus, the amount of elec-tric charge is proportional to the weighted average of the curvature, theweighing function being the width of the electrode. For an electrode withconstant width,

4.4 Laminar sensor 81

Q = −zme31bp[w′(b)− w

′(a)] (4.33)

The sensor output is proportional to the difference of slopes (i.e. rotations)at the extremities of the sensor strip. We note that this result is dual ofthat of Fig.4.3.a, where the piezoelectric transducer is used in actuationmode.

Equation (4.32) can be integrated by parts, twice, leading to

∫ b

aw′′bp(x)dx = w

′bp

]b

a− wb

′p

]b

a+

∫ b

aw b

′′dx (4.34)

If, as an example, one considers the case of a cantilever beam clamped atx = 0 and covered with a piezoelectric strip and an electrode of triangularshape extending over the whole length as in Fig.4.3.b (a = 0 and b = l),w(0) = w

′(0) = 0 (cantilever beam) and b

′′p = 0, bp(l) = 0, b

′p = −bp(0)/l

(triangular electrode). Substituting into the foregoing equations, one gets

Q = −zme31bp(0)

lw(l) ∼ w(l) (4.35)

Thus, the output signal is proportional to the tip displacement of thecantilever beam. Once again, this result is dual of that obtained in actua-tion mode (the piezoelectric loading is a point force at the tip). Similarly,if one considers a parabolic electrode as in Fig.4.3.c and if the beam issuch that w(0) = w(l) = 0 (this includes pinned-pinned, pinned-clamped,etc), we have bp(0) = bp(l) = 0 and b

′′p(x) = −8b/l2 and, substituting into

(4.34),

Q = zme318b

l2

∫ l

0w(x)dx ∼

∫ l

0w(x)dx (4.36)

Thus, the output signal is proportional to the volume displacement, whichis, once again, dual of the uniform distributed load in actuation mode.All the above results are based on the beam theory which is essentiallyone-dimensional; their accuracy in practical applications will depend verymuch on the relevance of these assumptions for the applications con-cerned. This issue is important in applications, especially in collocatedcontrol systems.

4.4.3 Charge amplifier dynamics

According to Fig.4.5.c, the output voltage is proportional to the amountof electric charge generated on the electrode; the amplifier gain is fixedby the capacitance C. This relation is correct at frequencies beyond some


corner frequency depending on the amplifier construction, but does notapply statically (near ω = 0). If a refined model of the charge amplifieris required, this behavior can be represented by adding a second orderhigh-pass filter

F (s) =s2

s2 + 2ξcωcs + ω2c

(4.37)

with appropriate parameters ωc and ξc. For frequencies well above thecorner frequency ωc, F (s) behaves like a unit gain.

4.5 Spatial modal filters

4.5.1 Modal actuator

According to (4.27), a piezoelectric layer with an electrode of width bp(x)is equivalent to a distributed transverse load proportional to b

′′p(x). Let

w(x, t) =∑

i

zi(t)φi(x) (4.38)

be the modal expansion of the transverse displacements, where zi(t) arethe modal amplitudes, and φi(x) the mode shapes, solutions of the eigen-value problem

[Dφ

′′i (x)

]′′− ω2

i %Aφi = 0 (4.39)

They satisfy the orthogonality conditions∫ l

0%Aφi(x)φj(x)dx = µiδij (4.40)

∫ l

0D φ

′′i (x)φ

′′j (x)dx = µiω

2i δij (4.41)

where µi is the modal mass, ωi the natural frequency of mode i, and δij

is the Kronecker delta (δij = 1 if i = j, δij = 0 if i 6= j). Substituting(4.38) into (4.27) (assuming p = 0), one gets

%A∑

i

ziφi +∑

i

zi(Dφ′′i )′′

= −e31V b′′pzm

or using (4.39),

4.5 Spatial modal filters 83

%A∑

i

ziφi + %A∑

i

ziω2i φi = −e31V b

′′pzm

where the sums extend over all modes. Upon multiplying by φk(x), inte-grating over the length of the beam, and using the orthogonality condition(4.40), one finds easily the equation governing the modal amplitude zk:

µk(zk + ω2kzk) = −e31V zm

∫ l

0b′′p(x)φk(x)dx (4.42)

The right hand side is the modal force pk applied by the piezoelectric stripto mode k. From the first orthogonality condition (4.40), it is readily seenthat if the electrode profile is chosen in such a way that

b′′p ∼ %Aφl(x) (4.43)

all the modal forces pk vanish, except pl:

pk ∼ −e31V zm

∫ l

0%Aφlφkdx ∼ −e31V zmµlδkl (4.44)

such an electrode profile will excite only mode l; it constitutes a modalactuator (for mode l).

4.5.2 Modal sensor

Similarly, if the piezoelectric layer is used as a sensor, the electric chargeappearing on the sensor is given by (4.32). Introducing the modal expan-sion (4.38),

Q = −zme31

∑

i

zi(t)∫ l

0bp(x)φ

′′i (x)dx (4.45)

Comparing this equation with the second orthogonality conditions (4.41),one sees that any specific mode can be made unobservable by choosing theelectrode profile in such a way that the integral vanishes. If the electrodeprofile is chosen according to

bp(x) ∼ Dφ′′l (x) (4.46)

(proportional to the distribution of the bending moment of mode l), theoutput charge becomes


M ode 2odal filter for m

Modal filter for mode 1

Mode 1

Mode 2

+ - +-

Cantilever Simply supported

Fig. 4.6. Electrode profile of modal filters for the first two modes of a uniform beamfor various boundary conditions: left: cantilever, right: simply supported.

Q ∼ −zme31µlω2l zl(t) (4.47)

It contains only a contribution from mode l. This electrode profile consti-tutes a modal sensor. Note that, for a uniform beam, (4.39) implies thatthe mode shapes satisfy φIV

i (x) ∼ φi(x). It follows that the electrodeprofile of a modal sensor also satisfies that of a modal actuator: from(4.46),

b′′p(x) ∼ φIV

l (x) ∼ φl(x) (4.48)

which satisfies (4.43). Figure 4.6 illustrates the electrode profile of modalfilters used for a uniform beam with various boundary conditions; thechange of sign indicates a change in polarity of the piezoelectric strip,which is equivalent to negative values of bp(x). As an alternative, thepart of the sensor with negative polarity can be bonded on the oppositeside of the beam, with the same polarity. The reader will notice that theelectrode shape of the simply supported beam is the same as the modeshape itself, while for the cantilever beam, the electrode shape is that ofthe mode shape of a beam clamped at the opposite end.

Modal filters constitute an attractive option for spillover alleviation,because they allow one to minimize the controllability and observabilityof a known set of modes. In practical applications, however, the beam

4.6 Active beam with collocated actuator-sensor 85

approximation often provides fairly poor modal filters, because the piezo-electric layer reacts as an orthotropic material rather than a unidirectionalone (Preumont et al., 2003).

4.6 Active beam with collocated actuator-sensor

x

M M

F.E. model

Actuator

Sensor

yi

iq1x

2

Fig. 4.7. Active cantilever beam with collocated piezoelectric actuator and sensor.Every node has 2 d.o.f. (yi and θi)

Consider a beam provided with a pair of rectangular piezoelectric ac-tuator and sensor (Fig.4.7). The two patches do not have to be of thesame size, nor have the same material properties, but they are collocatedin the sense of the Euler-Bernoulli beam theory, which means that theyextend over the same length along the beam. The system can, for exam-ple, be modelled by finite elements; the mesh is such that there is a nodeat both ends of the piezo patches (each node has two degrees of freedom,one translation yi and one rotation θi). We seek the open-loop FRF be-tween the voltage V (t) applied to the actuator, and the output voltagev0(t) of the sensor (assumed to be connected to a charge amplifier).

4.6.1 Frequency response function

According to Fig.4.3.a, the rectangular piezoelectric actuator is equivalentto a pair of torques M with opposite signs and proportional to V :

M = −e31zmbpV = gaV (4.49)


where ga is the actuator gain which can be computed from the actuatorsize and the material properties. In the general form of the equation ofmotion, the external force vector in a FE model is

f = bM = bgaV (4.50)

where the influence vector b has the form bT = (.., 0,−1, 0, 1, ...); theonly non-zero components correspond to the rotational degrees of freedomof the nodes located at x = x1 and x = x2 in the model. In modalcoordinates, the system dynamics is governed by a set of independentsecond order equations

zk + 2ξkωkzk + ω2kzk =

φTk f

µk=

pk

µk(4.51)

where ωk is the natural frequency of mode k, ξk the modal damping ratioand µk the modal mass. Using the Laplace variable s, we can write italternatively as

zk =pk

µk(s2 + 2ξkωks + ω2k)

(4.52)

The modal forces pk represent the work of the external loading on thevarious mode shapes:

pk = φTk f = φT

k bgaV = gaV ∆θak (4.53)

where ∆θak = φT

k b is the relative rotation [difference of slope w′(x2) −w′(x1)] between the extremities of the actuator, for mode k. Similarly,according to (4.33), the sensor output is also proportional to the differenceof slopes, that is the relative rotation of the extremities of the sensor,4θs.In modal coordinates,

v0 = gs∆θs = gs

∑

i

zi∆θsi (4.54)

where gs is the sensor gain, depending on the sensor size, material prop-erties and on the charge amplifier gain (which converts the electric chargeinto voltage), and ∆θs

i are the modal components of the relative rotationbetween the extremities of the sensor. Note that if the sensor and the ac-tuator extend over the same length of the beam, they can be consideredas collocated in the sense of the Euler-Bernoulli beam theory, and

∆θsi = ∆θa

i = ∆θi (4.55)

4.6 Active beam with collocated actuator-sensor 87

Combining the foregoing equations, one easily gets the transfer functionbetween the actuator voltage V and the sensor output v0; the FRF followsby substituting s = jω.

v0

V= G(ω) = gags

n∑

i=1

∆θ2i

µi(ω2

i − ω2 + 2jξiωiω) (4.56)

4.6.2 Pole-zero pattern

For an undamped system, the FRF is purely real:

v0

V= G(ω) = gags

n∑

i=1

∆θ2i

µi(ω2

i − ω2) (4.57)

All the residues of the modal expansion are positive and G(ω) is an in-creasing function of ω similar to that represented in Fig.2.5; the pole-zeropattern is that of Fig.2.6.a. As explained in chapter 2, for a lightly dampedstructure, the poles and zeros are slightly moved to the left half plane asin Fig.2.6.b. The position of the zeros in the complex plane depends onthe position of the actuator/sensor pair along the beam, while the polesdo not. The Bode and Nyquist plots of such a system are always similarto those of Fig.2.8. Once again, this interlacing property of the poles and

Fig. 4.8. Experimental open-loop FRF G(ω) of a piezoelectric beam similar to that ofFig.4.7.


zeros is of fundamental importance in control system design for lightlydamped vibrating systems, because it is possible to find a fixed controllerwith guaranteed stability, irrespective to changes in the mass and stiffnessdistribution of the system.

Figure 4.8 shows typical experimental results obtained with a systemsimilar to that of Fig.4.6. Observe that G(ω) does not exhibit any roll-off(decay) at high frequency; this indicates a feedthrough component in thesystem, which is not apparent from the modal expansion (4.56) (accordingto which the high frequency behavior is as ω−2). It will become clearerwhen we consider the modal truncation. 1

4.6.3 Modal truncation

Let us now examine the modal truncation of (4.56) which normally in-cludes all the modes of the system (a finite number n with a discretemodel, or infinite if one looks at the system as a distributed one). Ob-viously, if one wants an accurate model in some frequency band [0, ωc],all the modes (with significant residues) which belong to this frequencyband must be included in the truncated expansion, but the high frequencymodes cannot be completely ignored. To analyze this, one rewrites (4.56)

G(ω) = gags

n∑

i=1

∆θ2i

µiω2i

.Di(ω) (4.58)

where

Di(ω) =n∑

i=1

11− ω2/ω2

i + 2jξiω/ωi(4.59)

is the dynamic amplification of mode i. For any mode with a naturalfrequency ωi substantially larger than ωc, one sees from Fig.2.2 thatDi(ω) ' 1 within [0, ωc] and the sum (4.58) may be replaced by

G(ω) = gags

m∑

i=1

∆θ2i

µiω2i

.Di(ω) + gags

n∑

i=m+1

∆θ2i

µiω2i

(4.60)

where m has been selected in such a way that ωm À ωc. This equationrecognizes the fact that, at low frequency, the high frequency modes re-spond in a quasi-static manner. The sum over the high frequency modescan be eliminated by noting that the static gain satisfies1 Another observation is that a small linear shift appears in the phase diagram, due to

the fact that these results have been obtained digitally (the sampling is responsiblefor a small delay in the system).

4.7 Admittance of a beam with a piezoelectric patch 89

G(0) = gags

n∑

i=1

∆θ2i

µiω2i

(4.61)

leading to

G(ω) = gags

m∑

i=1

∆θ2i

µiω2i

.Di(ω) + [G(0)− gags

m∑

i=1

∆θ2i

µiω2i

] (4.62)

The term between brackets, independent of ω, which corresponds to thehigh frequency modes is often called the residual mode. This equation canbe written alternatively

G(ω) = G(0) + gags

m∑

i=1

∆θ2i

µiω2i

.[Di(ω)− 1]

or

G(ω) = G(0) + gags

m∑

i=1

∆θ2i

µiω2i


i − ω2 + 2jξiωiω)(4.63)

The feedthrough component observed in Fig.4.8 is clearly apparent in(4.62). Note that the above equations require the static gain G(0), but donot require the knowledge of the high frequency modes.

It is important to emphasize the fact that the quasi-static correctionhas a significant impact on the open-loop zeros of G(ω), and consequentlyon the performance of the control system. Referring to Fig.2.5, it is clearthat neglecting the residual mode (quasi-static correction) amounts toshifting the diagram G(ω) along the vertical axis; this operation altersthe location of the zeros which are at the crossing of G(ω) with the hori-zontal axis. Including the quasi-static correction tends to bring the zeroscloser to the poles which, in general, tends to reduce the performanceof the control system. Thus, it is a fairly general statement to say thatneglecting the residual mode (high frequency dynamics) tends to overesti-mate the performance of the control system. Finally, note that since thepiezoelectric loads are self-equilibrated, they would not affect the rigidbody modes if there were any.

4.7 Admittance of a beam with a piezoelectric patch

Let us consider a beam provided with a single piezoelectric patch andestablish the analytical expression of the admittance FRF, or equivalentlyof the dynamic capacitance. Assuming a rectangular patch of length l


(from x1 to x2), width bp, thickness t, and distant zm from the mid-planeof the beam, applying a voltage V generates a pair of self-equilibratedmoments M = −e31bpzmV . As in the previous section, the response ofmode i is governed by (assuming no damping)

µizi + µiω2i zi = M∆θi (4.64)

orzi = −e31bpzmV

∆θi

µi(s2 + ω2i )

(4.65)

where ∆θi = φ′i(x2) − φ′i(x1) is the difference of slope of mode i at theends of the patch. The beam deflection is

w =n∑

i=1

ziφi(x) = −e31bpzmVn∑

i=1

∆θiφi(x)µi(s2 + ω2

i )(4.66)

In the previous section, the charge amplifier cancelled the electric fieldacross the sensor. Here, we must use the second constitutive equation ofa unidirectional piezoelectric material

D = εT (1− k2)E + e31S (4.67)

with the electric field E = V/t; it is assumed that t << zm and thatthe strain level is uniform across the thickness of the patch, leading toS = −zmw′′ according to the Bernoulli assumption. The electric chargeis obtained by integrating over the area of the electrode

Q =∫

AD dA =

∫

AεT (1− k2)

V

tdA− e31zmbp

∫ x2

x1

w′′dx

Q = (1− k2)CV − e31zmbp[w′(x2)− w′(x1)]

Thus,Q

(1− k2)CV= 1 +

(e31zmbp)2

(1− k2)C

n∑

i=1

∆θ2i

µi(s2 + ω2i )

(4.68)

or, after using (3.20) and C = εT bpl/t

Q

(1− k2)CV= 1 +

k2

1− k2

n∑

i=1

cEbptz2m∆θ2

i

l µiω2i

11 + s2/ω2

i

(4.69)

If one notes that the average strain in the piezo patch is S = −zm∆θ/land bplt is the volume of the patch, the strain energy in the patch whenthe structure vibrates according to mode i can be written approximately

4.7 Admittance of a beam with a piezoelectric patch 91

We ' S2.cE .bplt

2=

cEz2m∆θ2

i bpt

2l(4.70)

Since µiω2i /2 is the total strain energy in the beam when it vibrates

according to mode i, the residues in the modal expansion represent thefraction of modal strain energy in the piezo patch, for mode i:

νi =cEbptz

2m∆θ2

i

l µiω2i

= Strain energy in the piezo patchStrain energy in the beam

i (4.71)

and the relation between Q and V reads finally

Q

(1− k2)CV= 1 +

k2

1− k2

n∑

i=1

νi

1 + s2/ω2i

(4.72)

or equivalently, the dynamic capacitance reads

Q

V= C(1− k2)[1 +

n∑

i=1

K2i

1 + s2/ω2i

] (4.73)

where

K2i =

k2νi

1− k2(4.74)

is known as the effective electromechanical coupling factor for mode i.C(1− k2) is the blocked capacitance of the piezo. The static capacitance(at ω = 0) is given by

Cstat = Q

Vω=0 = C(1− k2)(1 +

n∑

i=1

K2i ) (4.75)

The poles of the FRF are the natural frequencies ωi of the beam withshort-circuited electrodes, while the zeros correspond to the natural fre-quencies Ωi with open electrodes (Q = 0). Typically, Ωi is very close toωi, so that in the vicinity of ω = Ωi, the modal expansion is dominatedby the contribution of mode i. It follows that Ωi satisfies the equation

1 +K2

i

1−Ω2i /ω2

i

' 0 (4.76)

or

K2i '

Ω2i − ω2

i

ω2i

(4.77)


Thus, the effective electromechanical coupling factor of all modes can beevaluated from a single admittance FRF test (the poles of which are ωi

and the zeros are Ωi). The foregoing formula is an extension of (3.38).2

Formula (4.73) can be written alternatively

Q

V= Cstat.

∏(1 + s2/Ω2

i )∏(1 + s2/ω2

j )(4.78)

4.8 Piezoelectric laminate

In the first part of this chapter, the partial differential equation governingthe dynamics of a piezoelectric beam, and the equivalent piezoelectricloads were established from Hamilton’s principle. A similar approach canbe used for piezoelectric laminates, but it is lengthy and cumbersome.The analytical expression for the equivalent piezoelectric loads and thesensor output can be obtained alternatively, as in the classical analysisof laminate composites, by using the appropriate constitutive equations;this is essentially the approach followed by (C.K. Lee, 1990).

4.8.1 Two dimensional constitutive equations

Consider a two dimensional piezoelectric laminate in a plane (x, y): thepoling direction z is normal to the laminate and the electric field is alsoapplied along z. In the piezoelectric orthotropy axes, the constitutiveequations (4.1) (4.2) read

T = [c]S −

e31

e32

0

E3 (4.79)

D3 = e31 e32 0S+ εE3 (4.80)

where

T =

T11

T22

T12

S =

S11

S22

2S12

=

∂u/∂x∂v/∂y

∂u/∂y + ∂v/∂x

(4.81)

2 the presence of ωi instead of Ωi at the denominator of (4.77) is insignificant inpractice.

4.8 Piezoelectric laminate 93

are the stress and strain vector, respectively, [c] is the matrix of elasticconstants under constant electric field, E3 is the component of the electricfield along z, D3 is the z component of the electric displacement and εthe dielectric constant under constant strain (εS).

4.8.2 Kirchhoff theory

Following the Kirchhoff theory (e.g. Agarwal and Broutman, 1990), we as-sume that a line originally straight and normal to the midplane remains soin the deformed state. This is equivalent to neglecting the shear deforma-tions S23 and S31. If the midplane undergoes a displacement u0 , v0 , w0 ,a point located on the same normal at a distance z from the midplaneundergoes the displacements (Fig.4.9)

A

B

B

A

wo

ë = @w0=@x

u = uoà ëz

uo

z

w

x

Fig. 4.9. Kinematics of a Kirchhoff plate.

u = u0 − z∂w0

∂x

v = v0 − z∂w0

∂y(4.82)

w = w0

The corresponding strains are

S = S0+ zκ (4.83)

where


S0 =

S011

S022

2S012

=

∂u0/∂x∂v0/∂y

∂u0/∂y + ∂v0/∂x

(4.84)

are the midplane strains and

κ =

κ11

κ22

κ12

= −

∂2w0/∂x2

∂2w0/∂y2

2∂2w0/∂x∂y

(4.85)

are the curvatures (the third component represents twisting). The stressesin the laminate vary from layer to layer (because of varying stiffness prop-erties) and it is convenient to integrate over the thickness to obtain anequivalent system of forces and moments acting on the cross sections:

N =∫ h/2

−h/2Tdz M =

∫ h/2

−h/2Tz dz (4.86)

The positive direction of the resultant forces and moments is given inFig.4.10. N and M are respectively a force per unit length, and amoment per unit length.

z

x

y

Nx

Ny

Nxy

Nxy

Ny

Nxy

NxNxy

z

x

y

MxMxy

MyMxy

MxMxy

My

Fig. 4.10. Resultant forces and moments.

4.8.3 Stiffness matrix of a multi-layer elastic laminate

Before analyzing a piezoelectric laminate, let us recall the stiffness matrixof a multi-layer elastic laminate (Fig.4.11). If [c]k represents the stiffnessmatrix of the material of layer k, expressed in global coordinates, theconstitutive equation within layer k is


T = [c]kS = [c]kS0+ z[c]kκ (4.87)

Upon integrating over the thickness of the laminate, one gets

NM

=

[A BB D

] S0

κ

(4.88)

with

A =n∑

k=1

[c]k(hk − hk−1)

B =12

n∑

k=1

[c]k(h2k − h2

k−1) (4.89)

D =13

n∑

k=1

[c]k(h3k − h3

k−1)

where the sum extends over all the layers of the laminate; this is a clas-sical result in laminate composites. A is the extensional stiffness matrixrelating the in-plane resultant forces to the midplane strains; D is thebending stiffness matrix relating the moments to the curvatures, and B isthe coupling stiffness matrix, which introduces coupling between bendingand extension in a laminated plate; from (4.89), it is readily seen thatB vanishes if the laminate is symmetric, because two symmetric layerscontribute equally, but with opposite signs to the sum.

Mid plane x

Layer khk hkà1

hn

h0

z

Fig. 4.11. Geometry of a multilayered laminate.


4.8.4 Multi-layer laminate with a piezoelectric layer

Next, consider a multi-layer laminate with a single piezoelectric layer(Fig.4.12); the constitutive equations of the piezoelectric layer are (4.79)and (4.80). Upon integrating over the thickness of the laminate as in theprevious section, assuming that the global axes coincide with orthotropyaxes of the piezoelectric layer, one gets

Mid planeh

z

zm

Piezo

hp

Fig. 4.12. Piezoelectric layer.

NM

=

[A BB D

] S0

κ

+

[I3

zmI3

]

e31

e32

0

V (4.90)

D3 = e31 e32 0[I3 zmI3]

S0

κ

− ε V/hp (4.91)

where V is the difference of potential between the electrodes of the piezo-electric layer (E3 = −V/hp), hp the thickness of the piezoelectric layer,zm the distance between the midplane of the piezoelectric layer and themidplane of the laminated; I3 is the unity matrix of rank 3 and A,B,Dare given by (4.89), including the piezoelectric layer.3 In writing (4.91),it has been assumed that the thickness of the piezoelectric layer is smallwith respect to that of the laminate, so that the strain can be regardedas uniform across its thickness.3 The piezoelectric layer contributes to A, B and D with the stiffness properties under

constant electric field.


4.8.5 Equivalent piezoelectric loads

If there is no external load, N and M vanish and (4.90) can berewritten

[A BB D

] S0

κ

= −

[I3

zmI3

]

e31

e32

0

V (4.92)

The right hand side represents the equivalent piezoelectric loads. If thematerial is isotropic, e31 = e32, and the equivalent piezoelectric loadsare hydrostatic (i.e. they are independent of the orientation of the facetwithin the part covered by the electrode). Overall, they consist of an in-plane force normal to the contour of the electrode, and a constant momentacting on the contour of the electrode (Fig.4.13); the force per unit lengthand moment per unit length are respectively

Np = −e31V Mp = −e31zmV (4.93)

M

N

Electrode

piezoelectric patch

p

p

Fig. 4.13. Equivalent piezoelectric loads (per unit length) for an isotropic piezoelectricactuator: Np = −e31V , Mp = −e31zmV .

4.8.6 Sensor output

On the other hand, if the piezoelectric layer is used as a sensor and if itselectrodes are connected to a charge amplifier which enforces V∼0, thesensor equation (4.91) becomes


D3 = e31 e32 0[I3 zmI3]

S0

κ

(4.94)

Upon substituting the midplane strains and curvature from (4.84-85), andintegrating over the electrode area, one gets

Q =∫

ΩD3dΩ =

∫

Ω

[e31

∂u0

∂x+ e32

∂v0

∂y− zm

(e31

∂2w

∂x2+ e32

∂2w

∂y2

)]dΩ

(4.95)The integral extends over the electrode area (the part of the piezo notcovered by the electrode does not contribute to the signal). The first partof the integral is the contribution of the membrane strains, while thesecond is due to bending.

If the piezoelectric properties are isotropic (e31 = e32), the surfaceintegral can be further transformed into a contour integral using the di-vergence theorem; the previous equation is rewritten

Q = e31

∫

Ωdiv ~u0 dΩ − e31zm

∫

Ωdiv. ~gradw dΩ

= e31

∫

C~n.~u0 dl − e31zm

∫

C~n. ~gradw dl

where ~n is the outward normal to the contour of the electrode in its plane.Alternatively,

Q = e31

∫

C(~u0.~n− zm

∂w

∂~n)dl (4.96)

(u .n)n

n

∂ ∂w/ n

C Ω

0

Fig. 4.14. Contributions to the sensor output for an isotropic piezoelectric layer. Ω isthe electrode area.


This integral extends over the contour of the electrode (Fig.4.14); thefirst contribution is the component of the mid-plane, in-plane displace-ment normal to the contour and the second one is associated with theslope along the contour.Once again, the duality between the equivalent piezoelectric loads gen-

erated by the transducer used as actuator, and the sensor output whenthe transducer is connected to a charge amplifier must be pointed out.

4.8.7 Beam model vs. plate model

In this chapter, we have analyzed successively the piezoelectric beamaccording to the assumption of Euler-Bernoulli, and piezoelectric lami-nate according to Kirchhoff’s assumption. The corresponding piezoelectricloads have been illustrated in Fig.4.3 and 4.13, respectively; the sensoroutput, when the transducer is used in sensing mode, can be deducedby duality : a bending moment normal to the contour in actuation modecorresponds to the slope along the contour in sensing mode, and the in-plane force normal to the contour in actuation mode corresponds to thein-plane displacement normal to the contour in sensing mode. Figure 4.15illustrates the equivalent piezoelectric loads according to both theories fora rectangular isotropic piezoceramic patch acting on a structure extend-ing along one dimension: according to the beam theory, the equivalentpiezoelectric loads consist of a pair of torques applied to the ends of theelectrode (Fig.4.15.a), while according to the laminate theory, the torque

(a)

(b)

beam axis

Mp

Mp

Mp

Np

Fig. 4.15. Equivalent piezoelectric loads of a rectangular piezoelectric patch bondedon a beam: (a) beam theory, (b) laminate theory.


is applied along the whole contour of the electrode and it is supplementedby an in-plane force Np normal to the contour (Fig.4.15.b). If the struc-ture extends dominantly along one axis, and if one is interested in thestructural response far away from the actuator (e.g. tip displacement),it is reasonable to think that the piezoelectric loads of the beam theoryare indeed the dominant ones. However, in active vibration control, oneis often interested in configurations where the dual actuator and sensorare close to being collocated, to warrant alternating poles and zeros inthe open-loop FRF, for a wide frequency range (the perfectly dual andcollocated case was considered in section 4.6); in this case, it turns outthat the contributions to the piezoelectric loading and to the sensor outputwhich are ignored in the beam theory are significant, and neglecting themusually leads to substantial errors in the open-loop zeros of the controlsystem. This important issue will be addressed again in later chapters.

50mm

450 mm

Detail ofthe piezos

p4

p3

p2

p1

Test Structure

Fig. 4.16. Cantilever plate with piezoceramics.

To illustrate how a beam model and a plate model can be different fornearly collocated systems, consider the cantilever plate of Fig.4.16; thesteel plate is 0.5 mm thick and 4 piezoceramic strips of 250 µm thicknessare bounded symmetrically as indicated in the figure. The size of thepiezos is respectively 55 mm×25 mm for p1 and p3, and 55 mm×12.5 mmfor p2 and p4. p1 is used as actuator while the sensor is taken successivelyas p2, p3 and p4. Since they cover the same extension along the beam,the various sensor locations cannot be distinguished in the sense of theEuler-Bernoulli beam theory (except for the sign or a constant factor,


GaindB

Freq (Hz)

p3 sensor

p2 sensor

p4 sensor

Fig. 4.17. Cantilever plate with piezoceramics; experimental FRF for the three sensorconfigurations of Fig.4.16.

GaindB

Freq (Hz)

p3 sensor

p2 sensor

p4 sensor

Fig. 4.18. Cantilever plate with piezoceramics; Numerical (finite elements) FRF forthe three sensor configurations of Fig.4.16.

because p2 and p4 are on opposite sides and the size of p3 is twice thatof p2 and p4) and they should lead to the same transfer function. Thisis not the case in practice, as we can see in the experimental resultsshown in Fig.4.17. We see that the transmission zeros vary significantlyfrom one configuration to the other. In fact, because of the nearness of theactuator to the sensor, and the small thickness of the plate, the membranestrains play an important role in the transmission of the strain from theactuator to the sensor and are responsible for the differences between the


three sensor configurations. Figure 4.18 shows computed FRF based onthe plate theory using Mindlin-type finite elements (Piefort, 2001); onesees that the FRF curves behave in a way similar to the experimentalones and that the plate theory accounts for the experimental results. Weshall see later that, because the closed-loop poles are located on branchesgoing from the open-loop poles to the open-loop zeros, overestimatingthe spacing between the poles and the zeros leads to overestimating theclosed-loop performances of the system.

A deeper discussion of the finite element formulation of multi-layerpiezoelectric shells can be found in (Benjeddou, 2000, Garcia Lage et al.,2004, Heyliger et al, 1996, Piefort, 2001) and the literature quoted in thesepapers. The newly available PZT fibers (with interdigitated electrodes ornot), which are usually supplied in a soft polymer cladding, seem to beparticularly difficult to model accurately, due to the stiffness discrepancybetween the supporting structure, the PZT fibers and the soft polymerinterface; this is the subject of on-going research.

4.8.8 Additional remarks

1. Experiments conducted on a cantilever beam excited by a PZT patchon one side and covered with an isotropic PVDF film on the other side,with an electrode shaped as a modal filter for the first mode accordingto the theory of modal sensors developed in section 4.5, have revealedsignificant discrepancies between the measured FRF and that predictedby the beam theory; however, the FRF could be predicted quite accuratelyby the laminated plate theory (Preumont et al., 2003).2. For beams, modal filtering has been achieved by shaping the widthof the electrode. This concept cannot be directly transposed to plates.Spatial filtering of two-dimensional structures will be examined in Chapter14.

4.9 Active truss

Figure 4.19 shows a truss structure where some of the bars have beenreplaced by active struts; each of them consists of a piezoelectric linearactuator colinear with a force transducer. Such an active truss can be usedfor vibration attenuation, or to improve the dimensional stability underthermal gradients. If the stiffness of the active struts matches that of theother bars in the truss, the passive behaviour of the truss is basically

4.9 Active truss 103

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

p K a1 1= δ

p K a2 2= δ

x

y

θ

Detail of anactive member

Forcetransducer

f δ

Piezoelectriclinear actuator

Active member

p2

p1

Fig. 4.19. Active truss. The active struts consist of a piezoelectric linear actuator witha force sensor.

Fig. 4.20. ULB Active truss (1988).


unchanged. Fig.4.20 shows an example of active truss equipped with twoactive struts (built in the late 80’s ).

Consider a structure with a single discrete piezoelectric transducer(Fig.4.21); the transducer is governed by Equ.(3.22)

Qf

=

[C(1− k2) nd33Ka

−nd33Ka Ka

] V

bT x

(4.97)

where ∆ = bT x is the relative displacement at the extremities of thetransducer. The dynamics of the structure is governed by

Mx + K∗x = −bf (4.98)

where K∗ is the stiffness matrix of the structure without the transducerand b is the influence vector of the transducer in the global coordinatesystem of the structure. The non-zero components of b are the directioncosines of the active bar (Problem 4.5). The minus sign on the right handside of the previous equation comes from the fact that the force acting onthe structure is opposed to that acting on the transducer. Note that thesame vector b appears in both equations because the relative displacementis measured along the direction of f .

Substituting f from the constitutive equation into the second equation,one finds

Mx + Kx = bKaδ (4.99)

where K = K∗ + bbT Ka is the global stiffness matrix of the structureincluding the piezoelectric transducer in short-circuited conditions (whichcontributes for bbT Ka); δ = nd33V is the free expansion of the transducer

I

VPiezoelectric

Transducer

Structure

D = b xT

f

M K, *

Fig. 4.21. Structure with a piezoelectric transducer.


induced by a voltage V ; Kaδ is the equivalent piezoelectric loading: theeffect of the piezoelectric transducer on the structure consists of a pair ofself-equilibrating forces applied axially to the ends of the transducer; asfor thermal loads, their magnitude is equal to the product of the stiffnessof the transducer (in short-circuited conditions) by the unconstrainedpiezoelectric expansion; this is known as the thermal analogy.

Let φi be the normal modes, solutions of the eigenvalue problem

(K − ω2i M)φi = 0 (4.100)

They satisfy the usual orthogonality conditions

φTi Mφj = µiδij (4.101)

φTi Kφj = µiω

2i δij (4.102)

where ωi is the natural frequency when the transducer is short-circuited.If the global displacements are expanded into modal coordinates,

x =∑

i

ziφi (4.103)

where zi are the modal amplitudes, (4.99) is easily transformed into

µi(zi + ω2i zi) = φT

i bKaδ (4.104)

Upon taking the Laplace transform, one easily gets

x =n∑

i=1

φiφTi

µi(ω2i + s2)

bKaδ (4.105)

and the transducer extension

∆ = bT x =n∑

i=1

Ka(bT φi)2

µiω2i (1 + s2/ω2

i )δ (4.106)

From Equ.(4.102), µiω2i /2 is clearly the strain energy in the structure

when it vibrates according to mode i, and Ka(bT φi)2/2 represents thestrain energy in the transducer when the structure vibrates according tomode i. Thus, the ratio

νi =Ka(bT φi)2

µiω2i

(4.107)

is readily interpreted as the fraction of modal strain energy in the trans-ducer for mode i. With this notation, the previous equation is rewritten

∆ = bT x =n∑

i=1

νi

(1 + s2/ω2i )

δ (4.108)


Fig. 4.22. (a) Open-loop FRF of the active strut mounted in the structure (un-damped). (b) Admittance of the transducer mounted in the structure.

4.9.1 Open-loop transfer function

From the second constitutive equation (4.97), one readily establish theopen-loop transfer function between the free expansion δ of the transducer(proportional to the applied voltage) and the force f in the active strut

f

δ= Ka[

n∑

i=1

νi

(1 + s2/ω2i )− 1] (4.109)

All the residues being positive, there will be alternating poles and zerosalong the imaginary axis. Note the presence of a feedthrough in the trans-fer function. Figure 4.22.a shows the open-loop FRF in the undampedcase; as expected the poles at ±jωi are interlaced with the zeros at ±zi.As usual, the transfer function can be truncated after m modes:


f

δ= Ka[

m∑

i=1

νi

(1 + s2/ω2i )

+n∑

i=m+1

νi − 1] (4.110)

4.9.2 Admittance function

According to the first constitutive equation (4.97),

Q = C(1− k2)V + nd33KabT x

Using (4.108),

Q = C(1− k2)V + n2d233Ka

n∑

i=1

νi

(1 + s2/ω2i )

V (4.111)

and, taking into account (3.23), one finds the dynamic capacitance

Q

V= C(1− k2)[1 +

k2

1− k2

n∑

i=1

νi

(1 + s2/ω2i )

] (4.112)

(the admittance is related to the dynamic capacitance by I/V = sQ/V ),

Q

V= C(1− k2)[1 +

n∑

i=1

K2i

(1 + s2/ω2i )

] (4.113)

where

K2i =

k2νi

1− k2(4.114)

is once again the effective electromechanical coupling factor for mode i.This equation is identical to (4.73). The corresponding FRF is representedin Fig.4.22(b). The zeros of the admittance (or the dynamic capacitance)function correspond to the natural frequencies Ωi with open electrodesand equations (4.77) and (4.78) apply also for this configuration,

K2i '

Ω2i − ω2

i

ω2i

(4.115)

Q

V= Cstat.

∏ni=1(1 + s2/Ω2

i )∏nj=1(1 + s2/ω2

j )(4.116)


4.10 Finite element formulation

The finite element formulation of a piezoelectric continuum can be derivedfrom Hamilton’s principle (section 4.2). The Lagrangian of a structure in-volving a finite number of discrete piezoelectric transducers can be writtenin the general form

L = T ∗ + W ∗e =

12xT Mx− 1

2xT Kxxx +

12φT Cφφφ + φT Kφxx (4.117)

In this equation, M is the mass matrix, Kxx is the stiffness matrix (includ-ing the mechanical part of the transducers with short circuited electricalboundary conditions), Cφφ is the matrix of capacitance of the transducers(for fixed displacements) and Kφx is the coupling matrix of piezoelectricproperties, relating the mechanical and electrical variables.

The resulting dynamic equations read

Mx + Kxxx−KTφxφ = f (4.118)

Kφxx + Cφφφ = Q (4.119)

where Q is the vector of electric charges appearing on the electrodes.For voltage driven electrodes, the electric potential φ is controlled andEqu.(4.118) can be rewritten (assuming no external load, f = 0)

Mx + Kxxx = KTφxφ (4.120)

where the right hand side represents the self-equilibrated piezoelectricloads associated with the voltage distribution φ. Note that the dynamicsof the system with short-circuited electrodes (φ = 0) is the same as ifthere were no piezoelectric electromechanical coupling.

Conversely, open electrodes correspond to a charge condition Q = 0;in this case, one can substitute φ from Equ.(4.119) into Equ.(4.118)

Mx + (Kxx + KTφxC−1

φφ Kφx)x = f (4.121)

This equation shows that the piezoelectric coupling tends to increase theglobal stiffness of the system if the electrodes are left open. The naturalfrequencies of the system with open electrodes are larger than those withshort-circuited electrodes, as we have discussed extensively earlier in thischapter. If the electrodes are connected to a passive electrical network,the relationship between φ and Q is fixed by the network, as discussed inthe following chapter.

4.11 References 109

4.11 References

AGARWAL, B.D., BROUTMAN, L.J. Analysis and Performance of FiberComposites, Wiley, 2nd Ed., 1990.ALLIK, H., HUGHES T.J.R. Finite Element method for piezoelectricvibration, Int. J. for Numerical Methods in Engineering, Vol.2, 151-157,1970.BENJEDDOU, A. Advances in piezoelectric finite element modeling ofadaptive structural element: a survey, Computers and Structures, Vol.76,347-363, 2000.BURKE, S.E., HUBBARD, J.E. Active vibration control of a simply sup-ported beam using spatially distributed actuator. IEEE Control SystemsMagazine, August, 25-30, 1987.CADY, W.G. Piezoelectricity: an Introduction to the Theory and Appli-cations of Electromechanical Phenomena in Crystals, McGrawHill, 1946.CRAWLEY, E.F., LAZARUS, K.B. Induced strain actuation of isotropicand anisotropic plates, AIAA Journal, Vol.29, No 6, pp.944-951, 1991.de MARNEFFE, B. Active and Passive Vibration Isolation and Dampingvia Shunted Transducers, Ph.D. Thesis, Universite Libre de Bruxelles,Active Structures Laboratory, Dec. 2007.DIMITRIADIS, E.K., FULLER, C.R., ROGERS, C.A. Piezoelectric ac-tuators for distributed vibration excitation of thin plates, Trans. ASME,J. of Vibration and Acoustics, Vol.113, pp.100-107, January 1991.EER NISSE, E.P. Variational method for electrostatic vibration analy-sis, IEEE Trans. on Sonics and Ultrasonics, Vol. SU-14, No 4, 153-160October 1967.GARCIA LAGE, R., MOTA SOARES, C.M., MOTA SOARES, C.A.,REDDY, J.N. Layerwise partial mixed finite element analysis of magneto-electro-elastic plates, Computers an Structures, Vol.82, 1293-1301, 2004.HEYLIGER, P., PEI, K.C., SARAVANOS, D. Layerwise mechanics andfinite element model for laminated piezoelectric shells, AIAA Journal,Vol.34, No 11, 2353-2360, November 1996.HWANG, W.-S., PARK, H.C. Finite element modeling of piezoelectricsensors and actuators, AIAA Journal, Vol.31, No 5, pp.930-937, May 1993.LEE, C.-K. Theory of laminated piezoelectric plates for the design ofdistributed sensors/actuators - Part I: Governing equations and reciprocalrelationships, J. of Acoustical Society of America, Vol.87, No 3, 1144-1158,March 1990.


LEE, C.-K., CHIANG, W.-W., O’SULLIVAN, T.C. Piezoelectric modalsensor/actuator pairs for critical active damping vibration control, J. ofAcoustical Society of America, Vol.90, No 1, 374-384, July 1991.LEE, C.-K., MOON, F.C. Modal sensors/actuators, Trans. ASME, J. ofApplied Mechanics, Vol.57, pp.434-441, June 1990.LERCH, R. Simulation of piezoelectric devices by two and three dimen-sional finite elements, IEEE Transactions on Ultrasonics, Ferroelectrics,and Frequency Control, Vol.37, No 3, May 1990.PIEFORT, V. Finite Element Modeling of Piezoelectric Active Structures,PhD Thesis, Universite Libre de Bruxelles, Active Structures Laboratory,2001.PREUMONT, A., FRANCOIS, A., DE MAN, P., PIEFORT, V. Spatialfilters in structural control, Journal of Sound and Vibration, Vol.265, 61-79, 2003.PREUMONT, A. Mechatronics, Dynamics of Electromechanical and Piezo-electric Systems, Springer, 2006.ROSEN, C.A., Ceramic transformers and filters, Proc. Electronic Com-ponent Symposium, p.205-211 (1956).TIERSTEN, H.F. Hamilton’s principle for linear piezoelectric media, Pro-ceedings of the IEEE, 1523-1524, August 1967.UCHINO, K., Ferroelectric Devices, Marcel Dekker, 2000.

4.12 Problems

P.4.1 Consider a simply supported beam with collocated piezoelectricd31 rectangular actuator and sensor extending longitudinally from x = ato x = b. Using the results of section 2.6, show that the expression ∆θi

appearing at the numerator of Equ.(4.56) can be written analytically

∆θi = 2iπ

lsin iπ

l(a + b

2). sin

iπ

l(a− b

2)

P.4.2 From the result of the previous problem, sketch the actuator (andthe sensor) which maximizes the response of mode 1, mode 2 and mode3, respectively.P.4.3 Consider the active cantilever beam of transfer function (4.56). As-suming that G(0) is available from static calculations or from an experi-ment at low frequency, show that the truncated modal expansion includinga quasi-static correction can be written

4.12 Problems 111

G(ω) = G(0) + gags

m∑

i=1

∆θ2i

µiω2i


i − ω2 + 2jξiωiω)

P.4.4 Consider a free-free beam covered with a piezoelectric layer withthe various electrodes profiles of Fig.4.23. For each of them, sketch theequivalent piezoelectric loading.

parabola

parabola

l

a

a

b

Fig. 4.23. Free-free beam covered with a piezoelectric layer with various electrodeprofiles.

P.4.5 Consider the active truss of Fig.4.19; the coordinates of the finiteelement model are the cartesian coordinates of the nodes (xi, yi).(a) For each active strut, write the influence vector b of the piezoelectricloads in global coordinates.(b) Assuming small displacements, check that the projection of the dif-ferential displacements of the end nodes of the active struts are given bybT x where b is the result of (a).P.4.6 Consider a nine bay planar truss similar to that of Fig.4.19. Eachbay is square with one diagonal; all the bars have the same cross section.For the following boundary conditions, use a finite element program tocalculate the first three flexible modes of the truss. Suggest two reasonablelocations of an active strut to control these modes.(a) Free-free boundary conditions.(b) Clamped-free boundary conditions.


[Hint: Use the modal fraction of strain energy νi as index in the selectionof the active strut location.]P.4.7 Consider a general piezoelectric structure governed by Equ.(4.118-119) (assume f = 0). Write the general form of the admittance matrix Y :sQ = Y φ. Compare it to Equ.(4.113) and interpret the meaning of thevarious terms.P.4.8 The principle of Rosen’s piezoelectric transformer4 is shown inFig.4.24; the left side is the driving section; the input a.c. voltage generatesan axial vibration thanks to the d31 coefficient. The axial vibration istransmitted to the power generating section which is polarized in the axialdirection and generates the output voltage thanks to the d33 coefficient.The system is supposed to work at the second axial resonance of the free-free mechanical system, with a mode shape φ(x) = φ0 cos (πx/l). Showthat the voltage amplification ratio is given by

r =Vout

V=

4π2

k33k31(1− k2

33/2) Qm

l

t(4.122)

where l/t is the length to thickness ratio, k33 and k31 are the respectiveelectromechanical coupling factors of the two sections and Qm = 1/2ξ isthe mechanical quality factor of the axial vibration.

1

2

3

V

Vibration direction

Driving section Power generating section

t

Fig. 4.24. Rosen’s piezoelectric transformer. P indicates the direction of polarization;the grey areas indicate the electrodes. The system vibrates at the second axial resonanceof the free-free mechanical system, with a mode shape φ(x) = φ0 cos (πx/l).

4 Piezoelectric transformers were introduced by Rosen in 1956; they have been verysuccessful for low power applications such as power supply of laptop computers. Dueto the high energy density of piezoelectric materials, the high electromechanical cou-pling factors and the high quality factor of the mechanical resonance (low damping),they tend to be lighter and more efficient than wire wound transformers whose ef-ficiency tends to decrease rapidly as the size is reduced. Besides, they are free fromelectromagnetic interference and the solid-state nature of piezoelectric transformersis the key to mass production.

5

Passive damping with piezoelectric transducers

5.1 Introduction

In this chapter, we examine the basics of the passive damping withshunted piezoelectric transducers. We have seen in the previous chaptersthat a flexible structure (assumed undamped) with embedded piezoelec-tric transducers (Fig.5.1) is characterized by an admittance function ofthe form

I

V=

sQ

V= s.Cstat.

∏ni=1(1 + s2/Ω2

i )∏nj=1(1 + s2/ω2

j )(5.1)

or alternatively

I

V=

sQ

V= sC(1− k2)[1 +

n∑

i=1

K2i

1 + s2/ω2i

] (5.2)

where C(1−k2) is the blocked capacitance1 of the transducer and K2i are

the modal effective electromechanical coupling factors

K2i =

k2νi

1− k2=

Ω2i − ω2

i

ω2i

(5.3)

with ωi being the natural frequencies of the structure when the piezoelec-tric transducer has short-circuited electrodes and Ωi when the electrodesare open. It is assumed that the structure has well separated modes, andthat the transducer produces only a minor perturbation to the originalstructure: the mode shapes are not affected significantly by the electricalboundary conditions and ωi and Ωi are very close to each other.1 comparing the two foregoing equations, one finds that the static capacitance reads

Cstat = C(1− k2)(1 +∑

K2i ).

114 5 Passive damping with piezoelectric transducers

I

VPZT

Transducer

Structure

( )a

I

V

I

V R

I

V

R

L

(c)

(b)

(d)

PZT patch

R-shunt

RL-shunt

Fig. 5.1. Structure with a piezoelectric transducer (a) in d33 mode (b) in d31 mode(c) R shunt (d) RL shunt.

According to the foregoing formula, there are two contributions to K2i :

(i) the electromechanical coupling factor k2 which is a material prop-erty. If the material is used in the extension (d33) mode, the value k33

should be used, and if it is used in shear mode (d13), k13 applies. Typicalvalues are given in Table 4.1; note that k13 is significantly smaller thank33.

(ii) the modal fraction of strain energy νi which depends on the size aswell as the location of the transducer inside the structure. It varies frommode to mode. The transducer should be located where the strain energyis large for the targeted mode(s).

Note that the definition (5.3) of K2i does not imply that K2

i = k2 ifνi = 1. The classical definition is K2

i = (Ω2i − ω2

i )/Ω2i but, Ωi and ωi are

in general very close and the difference is insignificant in most cases.

5.2 Resistive shunting

Using the same positive signs for V and I as for the structure (Fig.5.1.c),the admittance of the shunt is I/V = −1/R. The characteristic equa-tion of the system is obtained by expressing the equality between the

5.2 Resistive shunting 115

admittance of the structure and that of the passive shunt:

− 1R

= sC(1− k2)[1 +n∑

i=1

K2i

1 + s2/ω2i

] (5.4)

or

− 1sRC(1− k2)

= 1 +n∑

i=1

K2i ω2

i

s2 + ω2i

(5.5)

In the vicinity of ±jωi, the sum is dominated by the contribution of modei and the other terms can be neglected; defining γ = [RC(1− k2)]−1, theequation may be simplified as

−γ

s= 1 +

K2i ω2

i

s2 + ω2i

which in turn can be rewritten

1 + γs2 + ω2

i

s(s2 + Ω2i )

= 0 (5.6)

This form of the characteristic equation is that of a root locus in auto-matic control (Fig.5.2);2 where the parameter γ acts as the feedback gainin classical root locus plots. For γ = 0 (R = ∞), the poles are purely

Fig. 5.2. Resistive shunt. Evolution of the poles of the system as γ = [RC(1− k2)]−1

goes from 0 to ∞ (the diagram is symmetrical with respect to the real axis, only theupper half is shown).

2 This root locus will be met many times in the following chapters.


imaginary, ±jΩi, corresponding to the natural frequency of the systemwith open electrodes; the system is undamped. As the resistance decreases(γ increases), the poles move to the left and some damping appears in thesystem; the maximum damping is achieved for γ = Ωi

√Ωi/ωi ' Ωi and

is (Problem 5.1)

ξmaxi =

Ωi − ωi

2ωi' Ω2

i − ω2i

4ω2i

=K2

i

4(5.7)

5.3 Inductive shunting

We proceed in the same way as for the R-shunt (Fig.5.1.d); the admittanceof the shunt is now I/V = −1/(R + Ls). The characteristic equationis obtained by expressing the equality between the admittance of thestructure and that of the passive shunt:

− 1R + Ls

= sC(1− k2)[1 +n∑

i=1

K2i

1 + s2/ω2i

] (5.8)

or

− 1(R + Ls)sC(1− k2)

= 1 +n∑

i=1

K2i ω2

i

s2 + ω2i

(5.9)

Once again, in the vicinity of ±jωi, the sum is dominated by the contri-bution of mode i and the equation is simplified as

− 1(R + Ls)sC(1− k2)

= 1 +K2

i ω2i

s2 + ω2i

(5.10)

Defining the electrical frequency

ω2e =

1LC(1− k2)

(5.11)

and the electrical damping

2ξeωe =R

L(5.12)

Equ.(5.10) is rewritten

− ω2e

2ξeωes + s2= 1 +

K2i ω2

i

s2 + ω2i

=s2 + Ω2

i

s2 + ω2i

(5.13)

5.3 Inductive shunting 117

ors4 + 2ξeωes

3 + (Ω2i + ω2

e)s2 + 2Ω2

i ξeωes + ω2i ω

2e = 0 (5.14)

This can be rewritten in a root locus form

1 + 2ξeωes(s2 + Ω2

i )s4 + (Ω2

i + ω2e)s2 + ω2

i ω2e

= 0 (5.15)

In this formulation, 2ξeωe = R/L plays the role of the gain in a classicalroot locus. Note that, for large R, the poles tend to ±jΩi, as expected. ForR = 0 (i.e. ξe = 0), they are the solutions p1 and p2 of the characteristicequation s4 + (Ω2

i + ω2e)s

2 + ω2i ω

2e = 0 which accounts for the classical

double peak of resonant dampers, with p1 above jΩi and p2 below jωi.Figure 5.3 shows the root locus for a fixed value of ωi/Ωi and variousvalues of the electrical tuning, expressed by the ratio

αe =ωeωi

Ω2i

(5.16)

The locus consists of two loops, starting respectively from p1 and p2; oneof them goes to jΩi and the other goes to the real axis, near −Ωi. Ifαe > 1 (Fig.5.3.a), the upper loop starting from p1 goes to the real axis,and that starting from p2 goes to jΩi, and the upper pole is always moreheavily damped than the lower one (note that, if ωe → ∞, p1 → ∞ andp2 → jωi; the lower branch of the root locus becomes that of the resistiveshunting). The opposite situation occurs if αe < 1 (Fig.5.3.b): the upperloop goes from p1 to jΩi and the lower one goes from p2 to the real axis;the lower pole is always more heavily damped. If αe = 1 (Fig.5.3.c), thetwo poles are always equally damped until the two branches touch eachother in Q. This double root is achieved for

αe =ωeωi

Ω2i

= 1 , ξ2e = 1− ω2

i

Ω2i

' K2i (5.17)

This can be regarded as the optimum tuning of the inductive shunting.The corresponding eigenvalues satisfy

s2 + Ω2i + Ωi(

Ω2i

ω2i

− 1)1/2s = 0 (5.18)

For various values of ωi/Ωi (or Ki), the optimum poles at Q move alonga circle of radius Ωi (Fig.5.3.d). The corresponding damping ratio canbe obtained easily by identifying the previous equation with the classicalform of the damped oscillator, s2 + 2ξiΩis + Ω2

i = 0, leading to


ëe > 1

(a) p1

p2Resistive

shunting

ëe < 1

(b) p1

p2

ëe = 1

(c)

p1

p2

Optimal

Damping

(d)

jÒi

à Òi

Q Q

à Òi

à Òià Òi

Re(s)

Im(s)

jÒi

jÒi

jÒi

j!i j!i

j!i

Fig. 5.3. Root locus plot for inductive shunting (only the upper half is shown). Theoptimum damping at Q is achieved for αe = 1 and ξe = Ki; the maximum modaldamping is ξi ' Ki/2.

ξi =12(Ω2

i

ω2i

− 1)1/2 =Ki

2=

12(

k2νi

1− k2)1/2 (5.19)

This value is significantly higher than that achieved with purely resistiveshunting [it is exactly the square-root of (5.7)]. Note, however, that itis much more sensitive to the tuning of the electrical parameters on thetargeted modes. This is illustrated in Fig.5.4, which displays the evolutionof the damping ratio ξi when the actual natural frequency ω′i moves awayfrom the nominal frequency ωi for which the shunt has been optimized(the damping ratio associated with p1 and p2 is plotted in dotted lines; theratio ω′i/Ω′

i is kept constant in all cases). One sees that the performance ofthe inductive shunting drops rapidly below that of the resistive shunting

5.4 Switched shunt 119

0.1 1 10

0

0.2 0.5 2 5

!0i=!iFrequency ratio

Resistive

shunting0.05

0.1

0.15

0.2

0.25

0.3

Inductive

shunting

øi

p2 p1

Fig. 5.4. Evolution of the damping ratio of the inductive and resistive shunting withthe de-tuning of the structural mode. ωi is the natural frequency for which the shunthas been optimized, ω′i is the actual value (k = 0.5, νi = 0.3).

when the de-tuning increases. Note that, for low frequency modes, theoptimum inductance value can be very large; such large inductors can besynthesized electronically. The multimodal passive damping via resonantshunt has been investigated by (Hollkamp, 1994).

All the dissipation mechanisms considered in this chapter are based onlinear time-invariant filters; an alternative approach based on switchingthe transducer periodically on a small inductor is examined in the nextsection.

5.4 Switched shunt

Consider a piezoelectric transducer connected to a RL shunt with an elec-trical circuit equipped with a switch (Fig.5.5). The governing equationsof the system are obtained by substituting f = −Mx in the constitutiveequations of the transducer:

V

−Mx

=

Ka

C(1− k2)

[1/Ka −nd33

−nd33 C

] Qx

(5.20)

When the switch is closed, the voltage V and the charge Q are relatedthrough the impedance of the RL circuit, V = −(Ls+R)I = −(Ls+R)sQ,leading to:


i

V

R

L

SwitchM

Fig. 5.5. Piezoelectric transducer with a switched RL shunt.

Q + RC(1− k2)Q + LC(1− k2)Q = nd33Kax (5.21)

and, using the definitions (5.11) and (5.12),

Q +2ξe

ωeQ +

1ω2

e

Q = nd33Kax (5.22)

It is assumed that the electrical circuit is such that the electrical resonancefrequency is significantly larger than the mechanical resonance,3 ωe Àωn, in such a way that the displacement may be regarded as constantover one half period π/ωe, when the switch is closed, and the charge Qessentially evolves as the step response of a second order system (Fig.5.6).The overshoot α is related to the electrical damping according to α =e−ξeπ/

√1−ξ2

e .

1

switch closed

1+aQ

33nd aK x

tt = p/we

Q0= 0

Fig. 5.6. Overshoot after closing the RL shunt. α = e−ξeπ/√

1−ξ2e . The switch remains

closed exactly one half of the electrical period, and then opens again.

3 ω2n = Ka/M(1− k2)


0t

V

0

Q1

Q0

Q2

x2

x1

x0

t

Fig. 5.7. Impulse response of the piezoelectric transducer of Fig.5.5. The upper figureshows the displacement x and the electric charge Q; the voltage V between the elec-trodes of the piezoelectric transducer is shown in the lower figure (k = 0.03). For lowk, xi+1 ' −xi.

The control strategy is the following: The electrical switch is closedwhen x = 0 (at the extrema), and remains closed exactly one half of theelectrical period, τ = π/ωe, and then opens again; this technique is knownas Synchronized Switch Damping on Inductor (SSDI).

Assuming that the piezoelectric transducer is initially not charged andthat it starts from non-zero initial conditions, the electric charge after thefirst switch is (Fig.5.7)

Q0 = (1 + α)nd33Kax0

When the switch is open, the charge remains the same until the nextextremum, at x1 ' −x0. Repeating the switching sequence,

Q1 = nd33Kax1 + α(nd33Kax1 −Q0)

The first contribution in the right hand side is the forcing term of thecharge equation (5.22), and the second one is the overshoot associated


with the difference with respect to the previous value. Combining withthe previous equation and taking into account that x1 ' −x0,

Q1 ' nd33Ka(1 + α)(1 + α)x1

Similarly,Q2 = nd33Kax2 + α(nd33Kax2 −Q1)

Q2 ' nd33Ka(1 + α)x2 − nd33Kaα(1 + α)(1 + α)x1

Q2 ' nd33Ka(1 + α)x2[1 + α(1 + α)]

etc...Qn ' nd33Ka(1 + α)xn(1 + α + α2 + . . . + αn)

Since 0 < α < 1, the asymptotic value is

Qn

xn' nd33Ka(

1 + α

1− α) (5.23)

Thus, the ratio between the electric charge and the vibration amplitudetends to stabilize to a constant value which corresponds to the staticresponse of (5.22) amplified by (1 + α)/(1− α), where α is the overshootin the electric charge following the closure of the switch. Note that Qn

has always a sign opposed to that of the velocity in the following half-cycle, which means that it works as dry friction. According to (5.23), thefriction force is maximized when α is close to one; however, small valuesof electrical damping ξe lead to beat (Fig.5.8).

0

øe = 15%øe = 0:4%

x

t

Fig. 5.8. Impulse response of the piezoelectric transducer for two values of the electricaldamping ξe (k = 0.1). For small values of ξe, beat is observed in the damped response.


5.4.1 Equivalent damping ratio

The dynamics of the mass M is governed by the second equation (5.20)

Mx +Ka

1− k2x =

nd33Ka

C(1− k2)Q (5.24)

where the force in the right hand side is opposing the velocity and has aconstant value during every half-cycle separating two extrema (where theswitch occurs); Q is given by Equ.(5.23).

The equivalent damping ratio may be evaluated by comparing theenergy loss in one cycle to that of an equivalent linear viscous damper.The free response of a linear viscous damper is x = x0e

−ξωnt. In one cycle,the amplitude is reduced by x1 = x0e

−2πξ and the strain energy loss4 is

∆VV(x0)

=V(x0)− V(x1)

V(x0)= 1− e−4πξ ' 4πξ (5.25)

Similarly, the energy loss in one cycle associated with the friction damp-ing is the work of the (constant) friction force F0; it is obtained fromEqu.(5.23-24)

∆V = 4x0F0 = 4(nd33Ka)2

C(1− k2)(1 + α

1− α)x2

0 = 4k2 Ka

1− k2(1 + α

1− α)x2

0

after using Equ.(3.23);

∆VV(x0)

= 8k2(1 + α

1− α) (5.26)

Comparing with (5.25), one gets the equivalent viscous damping

ξS =2π

(1 + α

1− α)k2 (5.27)

This result must be compared with ξR = k2/4 for a purely resistiveshunting and ξRL = k/2 for a tuned RL shunt, Fig.5.9. One observesthat the performance of the SSDI depends on the electrical damping ξe;smaller values of ξe lead to larger equivalent mechanical damping, but alower limit exists, corresponding to the appearance of beat. It is indicatedby a triangle in Fig.5.9. Figure 5.10 shows the lower limit of the electricaldamping as a function of the electromechanical coupling factor. Note thatFig. 5.9 and 5.10 have been obtained by considering a discrete transducerwhere the piezoelectric material makes up the entire transducer; for amore complicated system, the modal electromechanical coupling factorKi should be used instead of k.4 the strain energy is proportional to the square of the displacement.


10

-5

10

-4

10

-3

10

-2

0,1

1

10

0,01

0,001

RL shunt

K2

ø%

R shunt

SSDI2%

4%

10%

15% 20%

1%øe=

Fig. 5.9. Equivalent damping ratio ξ as a function of the electromechanical couplingfactor k2. Comparison of the resistive shunting (R), inductive shunting (RL) and syn-chronized switch shunting SSDI, for various values of the electrical damping ξe. Thetriangle indicates the limit value before beat.

10-4

10-3

10-2

Beat

No beat

K2

øe %

1

10

Fig. 5.10. Limit value of the electrical damping ξe under which the beat occurs.

5.5 References

DAVIS, C.L., LESIEUTRE, G.A. A modal strain energy approach to theprediction of resistivity shunted piezoceramic damping, Journal of Soundand Vibration, Vol.184, No 6, 129-139, 1995.de MARNEFFE, B. Active and Passive Vibration Isolation and Dampingvia Shunted Transducers, Ph.D. Thesis, Universite Libre de Bruxelles,Active Structures Laboratory, Dec. 2007.

5.5 References 125

DUCARNE, J. Modelisation et optimisation de dispositifs non-lineairesd’amortissement de structure par systemes piezoelectriques commutes.,PhD thesis, Conservatoire National des Arts et Mtiers, 2009.EDBERG, D.L., BICOS, A.S., FECHTER, J.S. On piezoelectric energyconversion for electronic passive damping enhancement, Proceedings ofDamping’91, San Diego, 1991.FORWARD, R.L. Electronic damping of vibrations in optical structures,Applied Optics, Vol.18, No 5, 690-697, March, 1979.FORWARD, R.L. Electromechanical transducer-coupled mechanical struc-ture with negative capacitance compensation circuit. US Patent 4,158,787,June 1979.GUYOMAR, D., RICHARD, C. Non-linear and hysteretic processingof piezoelement: Application to vibration control, wave control and en-ergy harvesting, Int. Journal of Applied Electromagnetics and Mechanics,Vol.21, 193-207, 2005.GUYOMAR, D., RICHARD, C., MOHAMMADI, S. Semipassive randomvibration control based on statistics., J. of Sound and Vibration Vol.307,818-833, 2007.HAGOOD, N.W., von FLOTOW, A. Damping of structural vibrationswith piezoelectric materials and passive electrical networks, Journal ofSound and Vibration Vol.146, No 2, 243-268, 1991.HOLLKAMP, J.J. Multimodal passive vibration suppression with piezo-electric materials and resonant shunts, J. Intell. Material Syst. Structures,Vol.5, Jan. 1994.LALLART, M., LEFEUVRE, E., RICHARD, C., Self-powered circuit forbroadband, multimodal piezoelectric vibration control, Sensors and Ac-tuators A, Vol. 143, 377-382, 2007.MOHEIMANI, S.O.R. A survey of recent innovations in vibration damp-ing and control using shunted piezoelectric transducers, IEEE Transac-tions on Control Systems Technology, Vol.11, No 4, 482-494, July 2003.NIEDERBERGER, D. Smart Damping Materials Using Shunt Control,PhD thesis, Swiss Federal Institute of Technology - ETHZ, 2005.PREUMONT, A. Mechatronics, Dynamics of Electromechanical and Piezo-electric Systems, Springer, 2006.


5.6 Problems

P.5.1 Show that the maximum damping achievable with a resistive shuntis given by Equ.(5.7). [Hint: The use of a symbolic calculation software isrecommended.]P.5.2 Consider a beam equipped with a rectangular piezoelectric trans-ducer extending from a to b, and a collocated actuator-sensor pair atx = l (Fig.5.11). The natural frequencies and the mode shapes with short-circuited electrodes are respectively ωk and φk(x). This system is intendedto be equipped with various forms of shunt damping or energy harvestingdevices. The input-output relationship of this system can be written inthe form

wi

=

[G11(s) G12(s)G21(s) G22(s)

] fV

Write the analytical form of the various transfer functions Gkj(s) involvedin this expression.

a b

f

w

x

l

i V

shunt

Fig. 5.11. Beam equipped with a piezoelectric transducer extending from a to b, anda collocated actuator sensor pair at x = l.

6

Collocated versus non-collocated control

6.1 Introduction

In the foregoing chapters, we have seen that the use of collocated ac-tuator and sensor pairs, for a lightly damped flexible structure, alwaysleads to alternating poles and zeros near the imaginary axis, Fig.6.1.a. Inthis chapter, using the root locus technique, we show that this propertyguarantees the asymptotic stability of a wide class of single-input single-output (SISO) control systems, even if the system parameters are subjectto large perturbations. This is because the root locus plot keeps the samegeneral shape, and remains entirely within the left half plane when thesystem parameters are changed from their nominal values. Such a controlsystem is said to be robust with respect to stability. The use of collocatedactuator/sensor pairs is recommended whenever it is possible.

Re(s)

Im(s)

xx

x x

x

x

x

x

Pole / Zeroflipping

Fig. 6.1. (a) Alternating pole-zero pattern of a lightly damped flexible structure withcollocated actuator and sensor. (b) Pole-zero flipping for a non-collocated system.

128 6 Collocated versus non-collocated control

This interlacing property of the poles and zeros no longer holds fora non-collocated control, and the root locus plot may experience severealterations for small parameter changes. This is especially true when thesequence of poles and zeros along the imaginary axis is reversed as inFig.6.1.b. This situation is called a pole-zero flipping. It is responsible fora phase uncertainty of 3600, and the only protection against instability isprovided by the damping (systems which are prone to such a huge phaseuncertainty can only be gain-stabilized).

6.2 Pole-zero flipping

ψi

ψiφi

φi

Root locus

Stable StableUnstable Unstable

Root locus

Fig. 6.2. Detail of a root locus showing the effect of the pole-zero flipping on thedeparture angle from a pole. Since the contribution of the far away poles and zeros isunchanged, that of the pole and the nearby zero, φi −ψi must also remain unchanged.

Recall that the root locus shows, in a graphical form, the evolutionof the poles of the closed-loop system as a function of the scalar gain gapplied to the compensator. The open-loop transfer function GH includesthe structure, the compensator, and possibly the actuator and sensordynamics, if necessary. The root locus is the locus of the solution s ofthe closed-loop characteristic equation 1 + gGH(s) = 0 when the realparameter g goes from zero to infinity. If the open-loop transfer functionis written

GH(s) = k

∏mi=1(s− zi)∏ni=1(s− pi)

(6.1)

6.3 The two-mass problem 129

the locus goes from the poles pi (for g = 0) to the zeros zi (as g →∞) ofthe open-loop system, and any point P on the locus is such that

m∑

i=1

φi −n∑

i=1

ψi = 1800 + l 3600 (6.2)

where φi are the phase angles of the vectors ~ai joining the zeros zi toP and ψi are the phase angles of the vector ~bi joining the poles pi to P(see Fig.6.8). Since n ≥ m, there are n −m branches of the locus goingasymptotically to infinity as g increases.

Consider the departure angle from a pole and the arrival angle at thezero when they experience a pole-zero flipping; since the contribution ofthe far away poles and zeros remains essentially unchanged, the differenceφi−ψi must remain constant after flipping. As a result, a nice stabilizingloop before flipping is converted into a destabilizing one after flipping(Fig.6.2). If the system has some damping, the control system is still ableto operate with a small gain after flipping.

Since the root locus technique does not distinguish between the systemand the compensator, the pole-zero flipping may occur in two differentways:

• There are compensator zeros near system poles (this is called a notchfilter). If the actual poles of the system are different from those assumedin the compensator design, the notch filter may become inefficient (ifthe pole moves away from the zero), or worse, a pole-zero flipping mayoccur. This is why notch filters have to be used with extreme care.As we shall see in later chapters, notch filters are generated by opti-mum feedback compensators and this may lead to serious robustnessquestions if the parameter uncertainty is large.

• Some actuator/sensor configurations may produce pole-zero flippingwithin the system alone, for small parameter changes. These situa-tions are often associated with a pole-zero (near) cancellation due toa deficiency in the controllability or the observability of the system(typically, when the actuator or the sensor is close to a nodal point inthe mode shapes). In some cases, however, especially if the dampingis extremely light, instability may occur. No pole-zero flipping can oc-cur within the structure if the actuator and sensor are collocated. Thefollowing sections provide examples illustrating these points.


y d

xf

b

k

M m

Fig. 6.3. Two-mass problem.

6.3 The two-mass problem

Consider the two-mass problem of Fig.6.3. The system has a rigidbody mode along the x axis; it is controlled by a force f applied to themain body M . A flexible appendage m is connected to the main bodyby a spring k and a damper b. First, a position control system will bedesigned, using a sensor placed on the main body (collocated); a sensorattached to the flexible appendage will be considered in the next section.

With f representing the control torque and y and d being the attitudeangles, this problem is representative of the single-axis attitude control ofa satellite, with M representing the main body, and the other inertia rep-resenting either a flexible appendage like a solar panel (in which case thesensor can be on the main body, i.e. collocated), or a scientific instrumentlike a telescope which must be accurately pointed towards a target (nowthe sensor has to be part of the secondary structure; i.e. non-collocated).A more elaborate single-axis model of a spacecraft is considered in Prob-lem 2.8.

The system equations are :

My + (y − d)b + (y − d)k = f (6.3)

md + (d− y)b + (d− y)k = 0 (6.4)

With the notations

ω2o = k/m, µ = m/M, 2ξωo = b/m (6.5)

the transfer functions between the input force f and y and d are respec-tively :

G1(s) =Y (s)F (s)

=s2 + 2ξωos + ω2

o

Ms2 [s2 + (1 + µ) (2ξωos + ω2o)]

(6.6)

6.3 The two-mass problem 131

G2(s) =D(s)F (s)

=2ξω0s + ω2

0

Ms2 [s2 + (1 + µ) (2ξωos + ω2o)]

(6.7)

G2(s) ' ω20

Ms2 [s2 + (1 + µ) (2ξωos + ω2o)]

(6.8)

Approximation (6.8) recognizes the fact that, for low damping (ξ ¿ 1),the far away zero will not influence the closed-loop response. There aretwo poles near the imaginary axis. In G1(s), which refers to the collocatedsensor, there are two zeros also near the imaginary axis, at (−ξω0± jω0).As observed earlier, these zeros are identical to the poles of the modifiedsystem where the main body has been blocked (i.e. constrained mode ofthe flexible appendage). When the mass ratio µ is small, the polynomialsin the numerator and denominator are almost equal, and there is a pole-zero cancellation.

6.3.1 Collocated control

Let us consider a lead compensator

H(s) = gTs + 1αTs + 1

(α < 1) (6.9)

Re(s)

Im(s)

xxx

x

G ss

s s1

2

2 2

0 04 1

0 044 11( )

.

( . . )=

+ +

+ +

s

s

H s( ).

=+10 1

0 04

s

s + 1

Fig. 6.4. Two-mass problem, root locus plot for the collocated control with a leadcompensator (the plot is symmetrical with respect to the real axis, only the upper partis shown).


It includes one pole and one zero located on the negative real axis; thepole is to the left of the zero. Figure 6.4 shows a typical root locus plotfor the collocated case when ω0 = 1, M = 1, ξ = 0.02 and µ = 0.1. Theparameters of the compensator are T = 10 and α = 0.004. Since there aretwo more poles than zeros (n−m = 2), the root locus has two asymptotesat ±900. One observes that the system is stable for every value of the gain,and that the bandwidth of the control system can be a substantial partof ω0. The lead compensator always increases the damping of the flexiblemode.

If there are not one, but several flexible modes, there are as manypole-zero pairs and the number of poles in excess of zeros remains thesame (n − m = 2 in this case), so that the angles of the asymptotesremain ±900 and the root locus never leaves the stable region. The leadcompensator increases the damping ratio of all the flexible modes, butespecially those having their natural frequency between the pole and thezero of the compensator. Of course, we have assumed that the sensor andthe actuator have perfect dynamics; if this is not the case, the foregoingconclusions may be considerably modified, especially for large gains.

6.3.2 Non-collocated control

Figure 6.5 shows the root locus plot for the lead compensator applied tothe non-collocated open-loop system characterized by the transfer func-tion G2(s), Equ.(6.7), with the following numerical data: ω0 = 1, M = 1,µ = 0.1, ξ = 0.02. The excess number of poles is in this case n−m = 3 so

Re(s)

Im(s)

xxx

x

H s( ).

=+

+

10 1

0 04 1

s

s

( )G s

s s2 2 2

0 04 1

0 044 11( )

.

. .=

+

+ +

s

s

Fig. 6.5. Two-mass problem, root locus plot for the non-collocated control with a leadcompensator.

6.4 Notch filter 133

gGH

ω

ω

1

PM = °29

GM =161.

ωc=0.056

-90°

-180°

Bandwidth

0 dB

φ( )gGH

Fig. 6.6. Two-mass problem, Bode plots of the non-collocated control for g = 0.003.

that, for large gains, the flexible modes are heading towards the asymp-totes at ±600, in the right half plane. For a gain g = 0.003, the closed-looppoles are located at −0.0136± 0.0505j and −0.0084± 1.0467j (these lo-cations are not shown in Fig.6.5 for clarity: the poles of the rigid bodymode are close to the origin and those of the flexible mode are locatedbetween the open-loop poles and the imaginary axis). The correspondingBode plots are shown in Fig.6.6; the phase and gain margins are indicated.One observes that even with this small bandwidth (crossover frequencyωc = 0.056), the gain margin is extremely small. A slightly lower value ofthe damping ratio would make the closed-loop system unstable (Problem6.1).

6.4 Notch filter

A classical way of alleviating the effect of the flexible modes in non-collocated control is to supplement the lead compensator with a notchfilter with two zeros located near the flexible poles:


H(s) = g .Ts + 1αTs + 1

.s2/ω2

1 + 1(s/a + 1)2

(6.10)

The zeros of the notch filter, at s = ±jω1, are selected right below theflexible poles. The double pole at −a aims at keeping the compensatorproper (i.e. the degree of the numerator not larger than that of the denom-inator); it can, for example, be selected far enough along the negative realaxis. The corresponding root locus is represented in Fig.6.7.a for ω1 = 0.9and a = 10. This compensator allows a larger bandwidth than the leadcompensator alone (Problem 6.2).

To be effective, a notch filter must be closely tuned to the flexiblemode that we want to attenuate. However, as we already mentioned, thenotch filter suffers from a lack of robustness and should not be used if theuncertainty in the system properties is large. To illustrate this, Fig.6.7.bshows a detail of the root locus near the notch, when the natural frequencyof the system is smaller than expected (in the example, ω0 is reduced from1 rad/s to 0.8 rad/s; the other data are identical to that of Fig.6.5, whilethe notch filter is kept the same (being implemented in the computer, thenotch filter is not subject to parameter uncertainty). The rest of the rootlocus is only slightly affected by the change.

Re(s)

Im(s)

xxxx

x

x

(a) (b)

notch at

= 0.9ω1 j

ω0 = 0.8 rad/s

Fig. 6.7. Two-mass problem, non-collocated control; (a) Lead compensator plus notchfilter (b) Detail of the root locus near the notch for off-nominal open-loop system (ω0

reduced from 1 to 0.8).

Because the open-loop poles of the flexible mode move from aboveto below the zero of the notch filter (from ±j to ±0.8j with the zerosat ±0.9j in the example), there is a pole-zero flipping, with the conse-quence that the branch of the root locus connecting the pole to the zero

6.5 Effect of pole-zero flipping on the Bode plots 135

rapidly becomes unstable. This example emphasizes the fact that notchfilters should be used with extreme care, especially for systems where theuncertainty is large (Problem 6.3).

6.5 Effect of pole-zero flipping on the Bode plots

From Equ.(6.1),

GH(jω) = k

∏mi=1(jω − zi)∏ni=1(jω − pi)

(6.11)

the phase of GH(jω) for a specific value jω is given by

m∑

i=1

φi −n∑

i=1

ψi (6.12)

where φi is the phase angle of the vector ~ai joining the zero zi to jω andψi is the phase angle of the vector ~bi joining the pole pi to jω (Fig.6.8).Accordingly, an imaginary zero at jω0 produces a phase lead of 1800 forω > ω0 and an imaginary pole produces similarly a phase lag of 1800.Therefore, a pole-zero flipping near the imaginary axis produces a phaseuncertainty of 3600 in the frequency range between the pole and the zero.It appears that the only way the closed-loop stability can be guaranteedin the vicinity of a pole-zero flipping is to have the open-loop systemgain-stabilized (i.e. such that |gHG| < 1) in that frequency range.

ψi

φizi

pi

pk

r

bi

rai

r

bk jω

ψ k ψ i ( )−

ψ i ( )−φ i ( )+

φ i ( )+

φ ψi i− = °180 φ ψi i− = − °180

Fig. 6.8. Effect of the pole-zero flipping on the phase diagram.


6.6 Nearly collocated control system

In many cases, the actuator and sensor pair are close to each other withoutbeing strictly collocated. This situation is examined here.

u

a

s

y

Fig. 6.9. Structure with nearly collocated actuator/sensor pair.

Consider the undamped system of Fig.6.9 where the actuator input uis applied at a and the sensor y is located at s. The open-loop FRF of thesystem is given by

G(ω) =y

u=

n∑

i=1

φi(a)φi(s)µi(ω2

i − ω2)(6.13)

where φi(a) and φi(s) are the modal amplitudes at the actuator and thesensor locations, respectively (the sum includes all the normal modes inthis case). The residues of (6.13) are no longer guaranteed to be posi-tive; however, if the actuator location a is close to the sensor location s,the modal amplitudes φi(a) and φi(s) will be close to each other, at leastfor the low frequency modes, and the corresponding residues will again bepositive. The following result can be established in this case : If two neigh-boring modes are such that their residues φi(a)φi(s) and φi+1(a)φi+1(s)have the same sign, there is always an imaginary zero between the twopoles (Martin, 1978).

Since G(ω) is continuous between ωi and ωi+1, this result will be es-tablished if one proves that the sign of G(ω) near ωi is opposite to thatnear ωi+1. At ω = ωi + δω, G(ω) is dominated by the contribution ofmode i and its sign is

sign[φi(a)φi(s)ω2

i − ω2] = −sign[φi(a)φi(s)] (6.14)

At ω = ωi+1 − δω, G(ω) is dominated by the contribution of mode i + 1and its sign is

sign[φi+1(a)φi+1(s)

ω2i+1 − ω2

] = sign[φi+1(a)φi+1(s)] (6.15)

6.7 Non-collocated control systems 137

Thus, if the two residues have the same sign, the sign of G(ω) near ω−i+1 isopposite to that near ω+

i . By continuity, G(ω) must vanishes somewherein between, at zi such that ω+

i < zi < ω−i+1. Note, however, that when theresidues of the expansion (6.13) are not all positive, there is no guaranteethat G(ω) is an increasing function of ω, and one can find situations wherethere are more than one zero between two neighboring poles.

6.7 Non-collocated control systems

Since the low frequency modes vary slowly in space, the sign of φi(a)φi(s)tends to be positive for low frequency modes when the actuator and sen-sor are close to each other, and the interlacing of the poles and zeros ismaintained at low frequency. This is illustrated in the following example:Consider a simply supported uniform beam of mass per unit length m

s > a

y

u

Sensor motion

l=2

l=3 2l=3

l=4 3l=4l=2

Mode 1

Mode 4

Mode 3

Mode 2

a= l=10

sin(ùx=l)

sin(2ùx=l)

sin(3ùx=l)

sin(4ùx=l)

Fig. 6.10. Uniform beam with non-collocated actuator/sensor pair. Mode shapes.


and bending stiffness EI. The natural frequencies and mode shapes arerespectively

ω2i = (iπ)4

EI

ml4(6.16)

andφi(x) = sin

iπx

l(6.17)

(the generalized mass is µi = ml/2). Note that the natural frequencyincreases as the square of the mode order. We assume that a force actuatoris placed at a = 0.1 l and we examine the evolution of the open-loop zerosas a displacement sensor is moved to the right from s = a (collocated),towards the end of the beam (Fig 6.10).

The evolution of the open-loop zeros with the sensor location alongthe beam is shown in Fig 6.11; the plot shows the ratio zi/ω1, so that theopen-loop poles (independent of the actuator/sensor configuration) are at1, 4, 9, 25, etc.... For s = a = 0.1 l, the open-loop zeros are represented by; they alternate with the poles. Another position of the actuator/sensor

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Actuator

25

9

!1

!

ls

16

4

1

0.1 0.80.70.60.50.40.3 0.90.2 1.0

z1z2

z3

z4

0.2=

0.25

0.33

0.5

ls

Fig. 6.11. Evolution of the imaginary zeros when the sensor moves away from theactuator along a simply supported beam (the actuator is at 0.1 l). The abscissa is thesensor location, the ordinate is the frequency of the transmission zero.

6.7 Non-collocated control systems 139

pair along the beam would lead to a different position of the zeros, butalways alternating with the poles.

As the sensor is displaced from the actuator, s > a, the zeros tend toincrease in magnitude as shown in Fig.6.11, but the low frequency onesstill alternate. When s = 0.2 l, z4 becomes equal to ω5 and there is nozero any longer between ω4 and ω5 when s exceeds 0.2 l. Thus, a pole/zeroflipping occurs. Similarly, z3 flips with ω4 for s = l/4, z2 flips with ω3 fors = l/3 and z1 flips with ω2 for s = l/2. Examining the mode shapes, onenotices that the pole-zero flipping always occurs at a node of one of themode shapes, and this corresponds to a change of sign in φi(a)φi(s), asdiscussed above.

This simple example confirms the behavior of the pole/zero pattern fornearly collocated control systems: the poles and zeros are still interlacingat low frequency, but not at higher frequency, and the frequency wherethe interlacing stops decreases as the distance between the actuator andsensor increases. A more accurate analysis (Spector & Flashner, 1989;Miu, 1993) shows that:

For structures such as bars in extension, shafts in torsion or simplyconnected spring-mass systems (non dispersive), when the sensor is dis-placed from the actuator, the zeros migrate along the imaginary axistowards infinity. The imaginary zeros are the resonance frequencies of thetwo substructures formed by constraining the structure at the actuatorand sensor (this generalizes the result of chapter 2).

Imaginary zeros

migrate towards

Real zeros come from

j1

1

Fig. 6.12. Evolution of the zeros of a beam when the sensor moves away from theactuator. Every pair of imaginary zeros which disappears at infinity reappears on thereal axis.


For beams with specific boundary conditions, the imaginary zeros stillmigrate along the imaginary axis, but every pair of zeros that disappearsat infinity reappears symmetrically at infinity on the real axis and movestowards the origin (Fig 6.12). Systems with right half plane zeros arecalled non-minimum phase. Thus, non collocated control systems are al-ways non-minimum phase, but this does not cause difficulties if the righthalf plane zeros lie well outside the desired bandwidth of the closed-loopsystem. When they interfere with the bandwidth, they put severe restric-tions on the control system, by reducing significantly the phase margin;this point will be discussed later in chapter 10.

6.8 The role of damping

To conclude this chapter, we would like to insist on the role of the dampingfor non-collocated control systems. We have seen that the imaginary zerosprovide the necessary phase lead to compensate the undesirable phase lagcaused by the poles. Whenever a flexible pole is not associated with a zero,it produces a net phase lag of 1800. According to the stability criterion,the amplitude of the open-loop transfer function must satisfy |gGH| < 1whenever the phase lag exceeds 1800. Since the amplitude of gGH in theroll-off region is dominated by the resonant peaks of G, it is clear thatthe damping of the flexible modes is essential for non-collocated systems(Problem 6.1).

Damping augmentation can be achieved by passive as well as activemeans. For spacecraft applications, the former often use constrained lay-ers of high damping elastomers placed at appropriate locations in thestructure (e.g. Johnson 1981 or Ikegami, 1986). More varied ways are reg-ularly used in civil engineering applications, such as tuned-mass dampers,tuned liquid dampers, chain dampers, etc... Active damping is the subjectof next chapter.

6.9 References

CANNON, R.H. & ROSENTHAL, D.E. Experiment in control of flexiblestructures with noncolocated sensors and actuators, AIAA Journal ofGuidance, Control and Dynamics, Vol. 7, No 5, Sept-Oct., 546-553, 1984.FRANKLIN, G.F., POWELL, J.D. & EMAMI-NAEINI, A. FeedbackControl of Dynamic Systems. Addison-Wesley, 1986.

6.10 Problems 141

GEVARTER, W.B. Basic relations for control of flexible vehicles. AIAAJournal, Vol.8,No 4, April, 666-672, 1970.HUGHES, P.C. & ABDEL-RAHMAN, T.M. Stability of proportional plusderivative plus integral control of flexible spacecraft, AIAA J. Guidanceand Control, Vol.2, No 6, 499-503, Nov.-Dec. 1979.IKEGAMI, R. & JOHNSON, D.W. The design of viscoelastic passivedamping treatments for satellite equipment support structures, Proceed-ings of DAMPING’86, AFWAL-TR-86-3059, 1986.JOHNSON, C.D., KIENHOLZ, D.A. & ROGERS, L.C. Finite elementprediction of damping in beams with constrained viscoelastic layers, Shockand Vibration Bulletin, No 51, 78-81, May 1981.MARTIN, G.D. On the Control of Flexible Mechanical Systems. Ph.D.Dissertation, Stanford University, 1978.MIU, D.K. Physical interpretation of transfer function zeros for simplecontrol systems with mechanical flexibilities. ASME J. Dynamic SystemsMeasurement and Control, Vol.113, September, 419-424, 1991.SPECTOR, V.A. & FLASHNER, H. Sensitivity of structural models fornoncollocated control systems. ASME J. Dynamic Systems Measurementand Control, Vol.111, No 4, December, 646-655, 1989.SPECTOR, V.A. & FLASHNER, H. Modeling and design implicationsof noncollocated control in flexible systems. ASME J. Dynamic SystemsMeasurement and Control, Vol.112, June, 186-193, 1990.

6.10 Problems

P.6.1 Consider the lead compensator for the non-collocated control of thetwo-mass system.

(a) Determine the value of the damping ratio ξ which would reducethe gain margin to zero.

(b) What would be the gain margin if ξ = 0.04 instead of ξ = 0.02.P.6.2 Consider the lead compensator plus notch filter for the non-collocated control of the two-mass system (section 6.4). Draw the cor-responding Bode plots. Select a reasonable value of the gain g and com-pare the bandwidth, the gain and phase margins with those of the leadcompensator of Fig.6.6.P.6.3 (a) Repeat the previous problem when the frequency of the ap-pendage is lower than that of the notch filter (ω0 =0.8 rad/sec); comparethe Bode plots and comment on the role of the damping.


(b) Same as (a) with the frequency of the appendage moving awayfrom the notch filter (ω0 =1.1 rad/sec). Comment on the importance oftuning the notch filter.P.6.4 Consider the PD regulator

H(s) = g(Ts + 1)

applied to the open-loop structure

G(s) =∞∑

i=1

φi(a)φi(s)s2 + ω2

i

Assuming that the modes are well separated, show that, for small gain g,the closed-loop damping ratio of mode i is

ξi = gTφi(a)φi(s)

2ωi

Conclude on the stability condition (Gevarter, 1970).[Hint: Use a perturbation method, s = ωi[−ξ + j(1 + δ)] in the vicinity ofjωi, and write the closed-loop characteristic equation.]P.6.5 Consider a simply supported uniform beam with a point force ac-tuator and a displacement sensor. Based on the result of the previousproblem, sketch a non-collocated actuator and sensor configuration suchthat a PD regulator is stabilizing for the first three modes.P.6.6 Consider a system of n identical masses M simply connected withn + 1 springs of stiffness k; assume that a point force is applied on massi and a displacement sensor is connected to mass j(> i). Show that thezeros of the transfer function are the resonance frequencies of the twosubstructures (from 1 to i and from j + 1 to n ), formed by constrainingthe masses i and j (Miu, 1991).P.6.7 Consider the non-collocated control of the two-mass problem [thesystem transfer function is given by Equ.(6.7)] with M = 1. For variousvalues of the mass ratio µ = 0.1, 0.01, 0.001, assuming a lead compensator(6.9), draw a diagram of the bandwidth of the control system, ωc/ω0 asa function of the damping ratio ξ for the limit of stability (GM = 0).

7

Active damping with collocated system

7.1 Introduction

The use of collocated (and dual) actuator and sensor pairs, for a lightlydamped flexible structure, always leads to alternating poles and zerosnear the imaginary axis, Fig.7.1. In this chapter, we use this interlacingproperty to develop Single Input-Single Output (SISO) active dampingschemes with guaranteed stability. By active damping, we mean that theprimary objective of the controller is simply to increase the negative realpart of the system poles, while maintaining the natural frequencies es-sentially unchanged. This simply attenuates the resonance peak in thedynamic amplification (Fig 7.2). Recall that the relationship between thedamping ratio ξ and the angle φ with respect to the imaginary axis issinφ = ξ, and that the dynamic amplification at resonance is 1/2ξ. Notethat for typical damping values encountered in practice, the values of φare very close to 0; this is why in most of the root locus plots shown in

Re(s) Re(s)

Im(s) Im(s)

x x

x x

x x

(a) (b)

jzi

j!i

Fig. 7.1. Pole/Zero pattern of a structure with collocated (dual) actuator and sensor;(a) undamped; (b) lightly damped.

144 7 Active damping with collocated system

this chapter, different scales are used for the real and the imaginary axes,leading to a distortion of the angles.

Im(s)

Re(s)

sinþ = ø

Open-loop

(a) Dynamic Amplification (dB)

Damping

(b)

!0

þ

Fig. 7.2. Role of damping (a) System poles. (b) Dynamic amplification (1/2ξ).

Active damping requires relatively little control effort; this is why it isalso called Low Authority Control (LAC), by contrast with other controlstrategies which fully relocate the closed-loop poles (natural frequencyand damping) and are called High Authority control (HAC).

A remarkable feature of the LAC controllers discussed here is thatthe control law requires very little knowledge of the system (at most theknowledge of the natural frequencies). However, guaranteed stability doesnot mean guaranteed performance; good performance does require infor-mation on the system as well as on the disturbance applied to it, forappropriate actuator/sensor placement, actuator sizing, sensor selectionand controller tuning. Actuator placement means good controllability ofthe modes dominating the response (note that controllability and observ-ability go together when the sensor is collocated and dual); this will bereflected by well separated poles and zeros, leading to wide loops in theroot-locus plots.

In order to keep the formal complexity to a minimum, we assume nostructural damping and perfect actuator and sensor dynamics throughoutmost of this chapter. The impact of the actuator and sensor dynamics onstability, and the beneficial effect of passive damping is discussed at theend.

Consider an undamped structure with a collocated, dual actuator/sensorpair (typically, a point force actuator collocated with a displacement sen-sor, or a torque actuator collocated with an angular sensor). The actua-

7.2 Lead control 145

tor/sensor position is defined by the influence vector b.1 We assume thatthe open-loop FRF G(ω) does not have any feedthrough (constant) com-ponent, so that G(ω) decays at high frequency as s−2; the roll-off (highfrequency decay rate) is −40 dB/decade in this case.

The open-loop transfer function of such a system, expressed in modalcoordinates, reads

G(s) =n∑

i=1

(bT φi)2

µi(s2 + ω2i )

(7.1)

where bT φi is the modal amplitude at the actuator/sensor location. Thepole-zero pattern is that of Fig 7.3 (where 3 modes have been assumed);there are two structural poles in excess of zeros, which provide a roll-offrate s−2 (a feedthrough component would introduce an additional pair ofzeros).

7.2 Lead control

This system can be damped with a lead compensator :

Structure

à p à zLeadRe(s)

Im(s)

j!i

jzi

Fig. 7.3. Open-loop pole/zero pattern and root locus of the lead compensator appliedto a structure with collocated actuator/sensor (open-loop transfer function with twopoles in excess of zeros). Different scales are used on the real and imaginary axes andonly the upper half of the plot is shown.

1 Appearing in the equations Mx + Kx = bu and y = bT x.


gs+ps+z+

à

u yP

iöi(s2+!

2i)

(bTþi)2

G(s) =

Fig. 7.4. Block diagram of the lead compensator applied to a structure with collocatedactuator/sensor (open-loop transfer function G(s) with two poles in excess of zeros).

H(s) = gs + z

s + p(p À z) (7.2)

The block diagram of the control system is shown in Fig 7.4. This con-troller takes its name from the fact that it produces a phase lead in thefrequency band between z and p, bringing active damping to all the modesbelonging to z < ωi < p. Figure 7.3 also shows the root locus of the closed-loop poles when the gain g is varied from 0 to ∞. The closed-loop poleswhich remain at finite distance start at the open-loop poles for g = 0 andeventually go to the open-loop zeros for g →∞. Since there are two polesmore than zeros, two branches go to infinity (at ±900). The controllerdoes not have any roll-off, but the roll-off of the structure is enough toguarantee gain stability at high frequency.

Note that the asymptotic values of the closed-loop poles for large gainsbeing the open-loop zeros zi, which are the natural frequencies of theconstrained system, they are therefore independent of the lead controllerparameters z and p. For a structure with well separated modes, the indi-vidual loops in the root-locus (Fig 7.3) are to a large extent independentof each other, and the root-locus for a single mode can be drawn fromthe lead controller and the asymptotic values ωi and zi of that mode only(Fig 7.5). The characteristic equation for this simplified system can bewritten from the pole-zero pattern:

1 + α(s2 + z2

i )(s + z)(s2 + ω2

i )(s + p)= 0 (7.3)

where α is the variable parameter going from α = 0 (open-loop) to infinity.This can be written alternatively

1 +1α

(s2 + ω2i )(s + p)

(s2 + z2i )(s + z)

= 0

If z and p have been chosen in such a way that z ¿ ωi < zi ¿ p, this canbe approximated in the vicinity of jωi by

7.3 Direct velocity feedback (DVF) 147

Im(s)

Re(s)

ømaxj!i

jzi

à p à z

Fig. 7.5. Structure with well separated modes and lead compensator, root-locus of asingle mode.

1 +p

α

(s2 + ω2i )

s(s2 + z2i )

= 0 (7.4)

This characteristic equation turns out to be the same as that of the Inte-gral Force Feedback (IFF) controller discussed a little later in this chap-ter, Equ.(7.31); it follows that the maximum achievable modal dampingis given by

ξmax ' zi − ωi

2ωi(ωi > zi/3) (7.5)

Note that the maximum achievable damping is controlled by the separa-tion between the open-loop pole ωi and the nearby zero zi.

7.3 Direct velocity feedback (DVF)

The Direct Velocity Feedback (DVF) is the particular case of the leadcontroller as z → 0 and p →∞. Returning to the the basic equations:

Structure:Mx + Kx = bu (7.6)

Output (velocity sensor) :y = bT x (7.7)

Control :u = −gy (7.8)

one finds easily the closed-loop equation


Mx + gbbT x + Kx = 0 (7.9)

Upon transforming into modal coordinates, x = Φz and taking into ac-count the orthogonality conditions, one gets

diag(µi)z + gΦT bbT Φ z + diag(µiω2i )z = 0 (7.10)

where z is the vector of modal amplitudes. The matrix ΦT bbT Φ is in gen-eral fully populated. For small gains, one may assume that it is diagonallydominant, ' diag(bT φi)2. This assumption leads to a set of decoupledequations. Mode i is governed by

µizi + g(bT φi)2zi + µiω2i zi = 0 (7.11)

By analogy with a single d.o.f. oscillator, one finds that the active modaldamping ξi is given by

2ξiµiωi = g(bT φi)2 (7.12)

or

ξi =g(bT φi)2

2µiωi(7.13)

Thus, for small gains, the closed-loop poles sensitivity to the gain (i.e.the departure rate from the open-loop poles) is controlled by (bT φi)2, thesquare of the modal amplitude at the actuator/sensor location.

Now, let us examine the asymptotic behavior for large gains. For allg, the closed-loop eigenvalue problem (7.9) is

(Ms2 + gbbT s + K)x = 0 (7.14)

Except for the presence of s in the middle term, this equation is close toEqu.(2.45); proceeding as we did in chapter 2, it follows that

x = −(K + Ms2)−1gsbbT x

orbT x = −gsbT (K + Ms2)−1bbT x (7.15)

Since bT x is a scalar, one must have

sbT (K + Ms2)−1b = −1g

(7.16)

and taking the limit for g →∞

7.3 Direct velocity feedback (DVF) 149

sbT (K + Ms2)−1b = 0 (7.17)

The solutions of this equation are s = 0 and the solutions of (2.49),that are the eigenvalues of the constrained system. The fact that theeigenvalues are purely imaginary, s = ±jω0, stems from the fact that Kand M are symmetric and semi-positive definite. Typical root locus plotsfor a lead controller and a DVF controller are compared in Fig.7.6.

Im(s)

Re(s)

gStructure

Lead

Im(s)

Re(s)

gStructure

DVF

(b)(a)

Fig. 7.6. Collocated control system. (a) Root locus for a lead controller. (b) DVFcontroller.

As for the lead controller, for well separated modes, those which arefar enough from the origin can be analyzed independently of each other.In this way, the characteristic equation for mode i is approximated by

1 + gs(s2 + z2

i )(s2 + ω2

1)(s2 + ω2i )

= 0

(besides the poles at ±jωi and the zeros at ±jzi, we include the zero ats = 0 and the poles at ±jω1) which in turn, if ωi > zi À ω1, may beapproximated by

1 + gs2 + z2

i

s(s2 + ω2i )

= 0 (7.18)

in the vicinity of mode i. This root locus is essentially the same as inthe previous section (with zi appearing in the numerator and ωi in thedenominator), and the formula for the maximum modal damping (7.32)applies

ξmax ' ωi − zi

2zi(zi > ωi/3) (7.19)


7.4 Positive Position Feedback (PPF)

There are frequent situations where the open-loop FRF does not exhibit aroll-off of −40 dB/decade as in the previous section. In fact, a feedthroughcomponent may arise from the truncation of the high frequency dynamics,as in (2.34), or because of the physical nature of the system (e.g. beamsor plates covered with collocated piezoelectric patches, Fig.4.8).

In these situations, the degree of the numerator of G(s) is the sameas that of the denominator and the open-loop pole-zero pattern has anadditional pair of zeros at high frequency. Since the overall degree of thedenominator of H(s)G(s) must exceed the degree of the numerator, thecontroller H(s) must have more poles than zeros. The Positive PositionFeedback was proposed to solve this problem (Goh & Caughey).

The second-order PPF controller consists of a second order filter

H(s) =−g

s2 + 2ξfωfs + ω2f

(7.20)

where the damping ξf is usually rather high (0.5 to 0.7), and the filterfrequency ωf is adapted to target a specific mode. The block diagram ofthe control system is shown in Fig.7.7; the negative sign in H(s), whichproduces a positive feedback, is the origin of the name of this controller.

Fig. 7.7. Block diagram of the second-order PPF controller applied to a structure withcollocated actuator and sensor (the open-loop transfer function has the same numberof poles and zeros).

Figure 7.8 shows typical root loci when the PPF poles are targeted tomode 1 and mode 2, respectively (i.e. ωf close to ω1 or ω2, respectively).One sees that the whole locus is contained in the left half plane, exceptone branch on the positive real axis, but this part of the locus is reachedonly for large values of g, which are not used in practice. Observe that theclosed-loop poles with a natural frequency close to ωf move much fasterthan the others. The stability condition can be established as follows: thecharacteristic equation of the closed-loop system reads

7.4 Positive Position Feedback (PPF) 151

j!i

jzi

Stability limit

PPF

(mode 1)

Structure

Re(s)

Im(s)Im(s)

PPF

(mode 2)

Fig. 7.8. Root locus of the PPF controller applied to a structure with collocatedactuator and sensor (the open-loop transfer function has the same number of poles andzeros). (a) Targeted at mode 1. (b) Targeted at mode 2. (For clarity, different scalesare used for the real and the imaginary axes.)

ψ(s) = 1 + gH(s)G(s) = 1− g


m∑

i=1

bT φiφTi b

µi(s2 + ωi2) = 0

or

ψ(s) = s2 + 2ξfωfs + ω2f − g

m∑

i=1

bT φiφTi b

µi(s2 + ωi2) = 0

According to the Routh-Hurwitz criterion for stability (see chapter 13), ifone of the coefficients of the power expansion of the characteristic equationbecomes negative, the system is unstable. It is not possible to write thepower expansion ψ(s) explicitly for an arbitrary value of m, however, onecan see easily that the constant term (in s0) is

an = ψ(0) = ω2f − g

m∑

i=1

bT φiφTi b

µiωi2

In this case, an becomes negative when the static loop gain gG(0)H(0)becomes larger than 1. The stability condition is therefore

gG(0)H(0) =g

ω2f

m∑

i=1

bT φiφTi b

µiωi2 < 1 (7.21)

Note that it is independent of the structural damping in the system.Since the instability occurs for large gains which are not used in practice,


the PPF can be regarded as unconditionally stable. Unlike the lead con-troller of the previous section which controls all the modes which belongto z < ωi < p and even beyond, the PPF filter must be tuned on thetargeted mode (it is therefore essential to know the natural frequency ac-curately), and the authority on the modes with very different frequenciesis substantially reduced. Several PPF filters can be used in parallel, totarget several modes simultaneously, but they must be tuned with care,because of the cross coupling between the various loops. An applicationwill be considered in section 14.4.

j!i

jzi

PPF

(1st order)

Structure

Im(s)

Re(s)

s = à 1=ü

Fig. 7.9. Root locus of the first-order PPF controller (the scale on the real axis hasbeen magnified for clarity).

The following first-order PPF controller is an alternative to the secondorder controller:

H(s) =−g

1 + τs(7.22)

A typical root locus is shown in Fig.7.9. As compared to the second-ordercontroller, this one does not have to be tuned on targeted modes; theroll-off is reduced to −20 dB/decade instead of −40 dB/decade with thesecond order controller. The part of the locus on the real axis also becomesunstable for large gains. Proceeding exactly as we did for the second ordercontroller, one finds easily that the stability condition is, once again, thatthe static loop gain must be lower than 1:

gG(0) < 1 (7.23)

7.5 Integral Force Feedback(IFF) 153

Thus the gain margin is GM = [gG(0)]−1. The stability condition corre-sponds to the negative stiffness of the controller overcoming that of thestructure. Because of this negative stiffness, the root-locus does not leavethe open-loop poles orthogonally to the imaginary axis; this is responsiblefor larger control efforts, as compared to the other strategies consideredbefore, which may be a serious drawback in applications where the controleffort is an important issue.

7.5 Integral Force Feedback(IFF)

Fig. 7.10. Active truss with an active strut consisting of a displacement actuator and aforce sensor. δ = gau is the unconstrained extension of the strut induced by the controlu. The passive stiffness of the strut is Ka and f/Ka is the elastic extension. ∆ = bT xis the total extension of the active strut.

So far, all the collocated system that we have considered exhibit al-ternating poles and zeros, starting with a pole at low frequency (ω1 <z1 < ω2 < z2 < ω3....). This corresponds to an important class of ac-tuator/sensor pairs, including (force actuator/displacement or velocitysensor), (torque actuator/angular position or angular velocity sensor),(piezoelectric patches used as actuator and sensor). In this section, wediscuss an other interlacing pole-zero configuration but starting with azero (z1 < ω1 < z2 < ω2...). This situation arises, typically, in a actua-tor/sensor pair made of a displacement actuator and a force sensor, such


as that of the piezoelectric active truss of Fig 7.10 already considered insection 4.9. The governing equation has been found to be

Mx + (K∗ + bbT Ka)x = bKaδ (7.24)

where K∗ is the stiffness matrix of the structure without the active strutand K∗ + bbT Ka is the global stiffness matrix, including the active strut.Note that this equation applies to linear actuators of many types, piezo-electric, magnetostrictive, thermal, ball-screw,..., provided that the con-stitutive equation of the actuator is

∆ = δ + f/Ka

or equivalentlyy = f = Ka(bT x− δ) (7.25)

where ∆ = bT x is the total extension of the active strut, δ its free exten-sion (control), f is the force sensor output, and Ka is the strut stiffness.The open-loop transfer function has been found to be

G(s) =y

δ= Ka[

m∑

i=1

νi

s2/ω2i + 1

+n∑

i=m+1

νi − 1] (7.26)

where the residues νi are the fraction of modal strain energy in the activestrut. The FRF exhibit alternating poles and zeros on the imaginary axis,beginning with a zero, Fig.7.11.The feedback control law is in this case a positive Integral Force Feedback

δ =g

Kasy (7.27)

(the Ka at the denominator is for normalization purpose; y/Ka is theelastic extension of the strut). The block diagram of the system is rep-resented in Fig.7.12. The pole-zero pattern of the system is shown inFig 7.13. It consists of interlacing pole-zero pairs on the imaginary axis(z1 < ω1 < z2 < ω2...) and the pole at s = 0 from the controller. The rootlocus plot consists of the negative real axis and a set of loops going fromthe open-loop poles ±jωi to the open-loop zeros ±jzi. All the loops areentirely contained in the left half plane, so that the closed-loop systemis unconditionally stable, for all values of the gain g. Note also that theroot-locus plot does not change significantly if the pole at the origin ismoved slightly in the left half plane, to avoid saturation (which is often


FRF

!1

!z1

!2 !3

z3z2

Fig. 7.11. Open-loop FRF of an active truss. The active strut consists of a piezoelectricactuator and a collocated force sensor.

Ka

P

i=1

n

1+s2=!2i

÷ià 1

ú û+

à

î y

Kas

àg

gD(s) G0(s)

Unconstrained

piezo expansion Force

Fig. 7.12. Block diagram of the IFF control.

associated with integral control). In fact, piezoelectric force sensors havea built-in high-pass filter and cannot measure a static component.2

Equations (7.25) and (7.27) can be combined, leading to

δ =g

s + gbT x (7.28)

and upon substituting into (7.24), one gets the closed-loop characteristicequation2 This issue and the static behavior of the active truss will be reexamined later in in

section 15.5 where the “Beta controller” will be introduced; see also Problem 7.5.


Im(s)

Re(s)

control

j!i

jzi

Structure

Fig. 7.13. Pole-zero pattern of the active strut and root-locus of the IFF.

[Ms2 + (K∗ + bbT Ka)− bbT Kag

s + g]x = 0 (7.29)

For g = 0, the eigenvalues are indeed the open-loop poles, ±jωi. Asymp-totically, for g →∞, the eigenvalue problem becomes

[Ms2 + K∗]x = 0 (7.30)

where K∗ is the stiffness matrix of the structure without the active strut.Thus, the open-loop zeros ±jzi are the natural frequencies of the trussafter removing the active strut. This situation should be compared tothat discussed earlier (section 2.5.1) in connection with a displacementsensor. In that case, the open-loop zeros were found to be the naturalfrequencies of the constrained system where the d.o.f. along which theactuator and sensor operate is blocked (i.e. the sensor output is cancelled).In the present case, if the sensor output is zero, the force carried by theactive strut vanishes and it can be removed.

For well separated modes, the individual loops in the root-locus of Fig7.13 are, to a large extent, independent of each other, and the root locusof a single mode can be drawn from the asymptotic values ±jωi and ±jzi

only (Fig 7.14.a). The corresponding characteristic equation is 3

3 Note the similarity with (7.4) for the lead controller and with (7.18) for the DVF.


IFF

a) b)

Fig. 7.14. (a) IFF root locus of a single mode. (b) Evolution of the root locus as zi

moves away from ωi

1 + g(s2 + z2

i )s(s2 + ω2

i )= 0 (7.31)

The actual root-locus, Fig.7.13, which includes the influence of the othermodes, is only slightly different from that of Fig 7.14.a, with the sameasymptotic values at ±jωi and ±jzi. It can be shown (Problem 7.3) thatthe maximum modal damping for mode i is given by

ξmaxi =

ωi − zi

2zi(zi ≥ ωi/3) (7.32)

It is achieved for g = ωi

√ωi/zi. This result applies only for ξi ≤ 1, that is

for zi ≥ ωi/3. For zi = ωi/3, the locus touches the real axis as indicated inFig 7.14.b, and for even larger differences ωi − zi, the root locus includespart of the real axis, which means that one can achieve enormous dampingvalues. Equation (7.32) relates clearly the maximum achievable dampingand the distance between the pole and the zero. Note, however, thatif the system has several modes, there is a single tuning parameter g,and the various loops are travelled at different speeds. As a result, theoptimal value of g for one mode will not be optimal for another one, anda compromise must be found.

Equation (7.29) can be transformed into modal coordinates. Using theorthogonality conditions, one finds

[diag(µi)s2 + diag(µiω2i )− ΦT bbT ΦKa

g

s + g]z = 0 (7.33)


For small g, the equations are nearly decoupled:

[s2 + ω2i − νiω

2i

g

s + g]zi = 0 (7.34)

after using the definition of νi, Equ.(4.107). Since the root locus plotleaves the open-loop poles orthogonally to the imaginary axis, for smallgain, one can assume a solution of the form s = ωi(−ξi± j). Substitutinginto (7.34), one finds easily

ξi =gνi

2ωi(7.35)

Thus, for small gains, the closed-loop poles sensitivity to the gain, i.e. thedeparture rate from the open-loop poles, is controlled by the fraction ofmodal strain energy in the active element. This result is very useful for thedesign of active trusses. The active strut should be located to maximizeνi for the critical modes of the structure. Note that νi is readily availablefrom finite element softwares; an example is analyzed in section 14.2.

In summary, the active strut placement can be made from the inspec-tion of the map of modal strain energy in the finite element model, oncethe active strut location has been selected, a modal analysis of the trussincluding the active strut gives the open-loop poles ±jωi, and a modalanalysis after removing the active strut gives the open-loop zeros ±jzi.Then, the root-locus plot can be drawn. The case of a truss involvingseveral active struts controlled in a decentralized manner is examined insection 14.3.

7.6 Duality between the Lead and the IFF controllers

In both cases, if the modes are well separated, they behave essentiallyindependently, and their closed-loop behavior may be analyzed as a singlemode, without considering the interaction with the other modes.

7.6.1 Root-locus of a single mode

Figure 7.15 illustrates the duality between the IFF and the Lead con-trollers. If the pole and the zero of the lead controller (7.2) are such thatz ¿ ωi < zi ¿ p, the root-locus plots of every mode turn out to be verysimilar (Fig.7.15.c and d). The DVF can be looked at as the limit of theLead controller as z → 0 and p → ∞; however, depending on the situa-tion, the individual loops starting from an open-loop pole may go eitherto a zero with higher frequency (similar to Fig.7.15.d) or to a zero withlower frequency (similar to Fig.7.15.c).

7.6 Duality between the Lead and the IFF controllers 159

î

IFF

+

Im(s)

Re(s)

j!i

jzi

Im(s)

Re(s)

j!i

jzi

(a) (b)

(d)(c)

Lead

1 +g

p

s(s2+z2i)

s2+!2i = 01 + g

s(s2+!2i)

s2+z2i = 0

É

f

f

f

-

LeadIFF

Fig. 7.15. Duality between the IFF and the Lead (DVF) control configurations. (a)IFF architecture with displacement actuator, force sensor and positive integral forcefeedback. (b) Force actuator and collocated displacement transducer and (negative)Lead controller (DVF is a particular case). (c) IFF control: Root locus for a singlemode. (d) Lead control: Root locus for a single mode (z ¿ ωi < zi ¿ p). The loopsof the DVF can be approximated by one configuration or the other, depending on therelative value of ωi and zi.


7.6.2 Open-loop poles and zeros

For the IFF (Fig.7.15.a), the open-loop poles jωi are the natural fre-quencies of the structure with the active element working passively (con-tributing with its own stiffness Ka), while the open-loop zeros, jzi, arethe natural frequencies when the force f in the active strut is zero, thatis when the active element is removed. On the contrary, in the controlconfiguration of Fig.7.15.b, the open-loop poles are the natural frequen-cies when the active element produces no force (same as the zeros in theprevious case), and the open-loop zeros are the natural frequencies withthe d.o.f. along the actuator blocked (∆ = 0). Note that these are largerthan the open-loop poles in the IFF case, because the stiffness Ka of theactive strut is finite.

7.7 Actuator and sensor dynamics

Re(s)

Im(s)(a)

Re(s)

Im(s)

Actuator

(b)

11

2

3

4

Fig. 7.16. Effect of the actuator dynamics on the Lead compensator. (a) With per-fect actuator. (b) Including the actuator dynamics A(s) given by (7.37) (the cornerfrequency of the actuator is such that ω2 < ωa < ω3).

Throughout this chapter, it has been assumed that the actuator andsensor have perfect dynamics. As a result, the active damping algorithms

7.7 Actuator and sensor dynamics 161

Re(s)

Im(s)

Actuator

Fig. 7.17. Stabilizing effect of the structural damping on the actuator dynamics.

are stable for all gains g (except the PPF). In practice, however, theopen-loop transfer function becomes

g A(s) H(s) G(s) (7.36)

where A(s) includes the sensor, actuator and the digital controller dy-namics. The low frequency behavior of the proof-mass actuator and thecharge amplifier can both be approximated by a second order high-passfilter such as (3.8). The high frequency behavior of the actuators andsensors can often be represented by a second order low-pass filter

A(s) =ω2

a

s2 + 2ξaωas + ω2a

(7.37)

It is easy to see that the two extra poles of A(s) bring the asymptotes ofthe root locus inside the right half plane and substantially alter its shapefor ω > ωa. Figure 7.16 shows the effect of the second order low-pass filteron the root locus plot of the lead compensator (we have assumed ξa = 0.5and ω2 < ωa < ω3). The active damping is no longer unconditionallystable and always has some destabilizing influence on the modes withnatural frequencies beyond ωa. Fortunately, in practice, the modes of thestructure are not exactly on the imaginary axis, because of the structural


damping (Fig.7.17); this allows us to operate the controller with smallgains. The control system becomes insensitive to the actuator dynamicsif ωa is far beyond the cross-over frequency of gH(s)G(s). The effect ofthe low frequency dynamics of a proof-mass actuator is left as a problem(Problem 7.2).

7.8 Decentralized control with collocated pairs

7.8.1 Cross talk

Consider the Multi-Input Multi-Output (MIMO) control of a structurewith two independent control loops using collocated pairs. The input-output relationship for this system can be written in compact form

y1

y2

z

=

G11 G12 G1w

G21 G22 G2w

Gz1 Gz2 Gzw

u1

u2

w

(7.38)

where w is the disturbance and z is the performance metrics. One sees thatthe output y1 responds to u2 through G12 and y2 responds to u1 throughG21, respectively. These terms are called cross-talk, and are responsiblefor interactions between the two loops.

7.8.2 Force actuator and displacement sensor

Consider a control system with m collocated force actuator/displacementsensor pairs. The control is governed by the following equations:

Structure:Mx + Kx = Bu (7.39)

Output:y = BT x (7.40)

where B defines the topology of the actuator/sensor pairs (the size of thevectors u and y is equal to the number m of collocated pairs).

Control:u = −gH(s)y (7.41)

where H(s) is a square matrix and g is a scalar parameter (the discussionis not restricted to decentralized control; H(s) is diagonal if the controlis decentralized). The closed-loop eigenvalue problem is obtained by com-bining the three equations above:

7.8 Decentralized control with collocated pairs 163

[Ms2 + K + gBH(s)BT ]x = 0 (7.42)

One can show that the asymptotic values of the finite eigenvalues of thisequation as g →∞ are independent of H(s) (Davison & Wang); therefore,they can be computed with H(s) = I:

limg→∞[Ms2 + K + gBBT ]x = 0 (7.43)

The asymptotic solutions of this equation are the transmission zeros ofthe MIMO system. The matrix gBBT is the contribution to the globalstiffness matrix of a set of springs of stiffness g connected to all the d.o.f.involved in the control. Asymptotically, when g → ∞, the additionalsprings act as supports restraining the motion along the controlled d.o.f..Thus, the transmission zeros are the poles (natural frequencies) of theconstrained system where the d.o.f. involved in the control are blocked.Since all the matrices involved in (7.43) are symmetrical and positivesemi-definite, the transmission zeros are purely imaginary; since blockingthe controlled d.o.f. reduces the total number of d.o.f. by the number mof control loops, the number of zeros is 2m less than the number of poles.

7.8.3 Displacement actuator and force sensor

The equations are in this case:Structure:

Mx + Kx = BKaδ (7.44)

Output:y = Ka(BT x− δ) (7.45)

Control:δ = gH(s)y (7.46)

where B defines the topology of the active members within the structure,assumed of equal stiffness Ka, H(s) is a square matrix and g is a scalargain (a positive feedback is assumed as in the IFF controller). The closed-loop eigenvalues are solutions of

[Ms2 + K − gBKaH(I + gKaH)−1KaBT ]x = 0 (7.47)

The asymptotic values are respectively, for g = 0, the open-loop poles(natural frequencies of the system including the active members) and, forg →∞,4 they are solution of4 because limg→∞(I + gKaH) ∼ gKaH


(Ms2 + K −BKaBT )x = 0 (7.48)

Thus, asymptotically, as g → ∞, the finite eigenvalues coincide with thetransmission zeros which are the poles (natural frequencies) of the systemwhere the contribution of the active members to the stiffness matrix hasbeen removed. This result applies in particular for independent IFF loops.

7.9 References

AUBRUN, J.N. Theory of the control of structures by low-authority con-trollers. AIAA J. of Guidance, Control and Dynamics, Vol 3, No 5, Sept-Oct.,444-451, 1980.BALAS, M.J. Direct velocity feedback control of large space structures.AIAA J. of Guidance, Control and Dynamics, Vol 2, No 3, 252-253, 1979.BAZ, A., POH, S. & FEDOR, J. Independent modal space control withpositive position feedback. Trans. ASME, J. of Dynamic Systems, Mea-surement, and Control,Vol.114, No 1,March, 96-103, 1992.BENHABIB, R.J., IWENS, R.P. & JACKSON, R.L. Stability of largespace structure control systems using positivity concepts. AIAA J. ofGuidance, Control and Dynamics, Vol 4, No 5, 487-494, Sept.-Oct. 1981.DAVISON, E.J., WANG, S.H. Properties and Calculation of TransmissionZeros of Linear Multivariable Systems, Automatica, Vol.10, 643-658, 1974.de MARNEFFE, B. Active and Passive Vibration Isolation and Dampingvia Shunted Transducers, Ph.D. Thesis, Universite Libre de Bruxelles,Active Structures Laboratory, Dec. 2007.FANSON, J.L. & CAUGHEY, T.K. Positive position feedback control forlarge space structures. AIAA Journal, Vol.28, No 4,April,717-724, 1990.FORWARD, R.L. Electronic damping of orthogonal bending modes in acylindrical mast experiment. AIAA Journal of Spacecraft, Vol.18, No 1,Jan.-Feb., 11-17, 1981.GEVARTER, W.B. Basic relations for control of flexible vehicles. AIAAJournal, Vol.8,No 4, April, 666-672, 1970.GOH, C. & CAUGHEY, T.K. On the stability problem caused by finiteactuator dynamics in the control of large space structures, Int. J. of Con-trol, Vol.41, No 3, 787-802, 1985.PREUMONT, A., DUFOUR, J.P. & MALEKIAN, Ch. Active dampingby a local force feedback with piezoelectric actuators. AIAA J. of Guid-ance, Control and Dynamics, Vol 15, No 2, March-April, 390-395, 1992.

7.10 Problems 165

PREUMONT, A., LOIX, N., MALAISE, D. & LECRENIER, O. Activedamping of optical test benches with acceleration feedback, Machine Vi-bration, Vol.2, 119-124, 1993.PREUMONT, A., ACHKIRE, Y. & BOSSENS, F., Active Tendon Con-trol of Large Trusses, AIAA Journal, Vol.38, No 3, 493-498, March 2000.PREUMONT, A. & BOSSENS, F., Active Tendon Control of Vibrationof Truss Structures: Theory and Experiments, J. of Intelligent MaterialSystems and Structures, Vol.2, No 11, 91-99, Feb. 2000.PREUMONT, A., de MARNEFFE, B., KRENK, S., Transmission Zerosin Structural Control with Collocated MIMO Pairs, AIAA Journal ofGuidance, Control and Dynamics, Vol.31, No 2, 428-431, March-April2008.PREUMONT, A. & SETO, K. Active Control of Structures, Wiley, 2008.SCHAECHTER, D. Optimal local control of flexible structures, AIAA J.of Guidance, Control and Dynamics Vol.4, No 1, 22-26, 1981.SIM, E., & LEE, S.W. Active vibration control of flexible structures withacceleration or combined feedback. AIAA J. of Guidance, Control andDynamics, Vol.16, No 2, 413-415, 1993.

7.10 Problems

P.7.1 Compare the following implementations of the Lead and the DirectVelocity Feedback compensators:

H(s) = s

H(s) = 1 + Ts

H(s) =s

s + a

H(s) =Ts + 1αTs + 1

(α < 1)

H(s) =ω2

fs


Discuss the conditions under which these compensators would be appli-cable for active damping.P.7.2 Consider a vibrating structure with a point force actuator collo-cated with an accelerometer. Consider the two compensators:

H(s) = g/s


H(s) =g


Draw the block diagram of the control system, examine the stability andperformance. What would be the effect of the dynamics of a charge am-plifier, represented as a second order high-pass filter of corner frequencyωp and damping ξp = 0.7, assuming that its corner frequency satisfiesωp ¿ ω1.P.7.3 Show that the maximum damping achievable in an active truss withthe integral force feedback is

ξmaxi =

ωi − zi

2zi(zi ≥ ωi/3)

where ωi is the natural frequency of the truss including the active strutas a passive element and zi is the natural frequency when the active struthas been removed, and that it is achieved for g = ωi

√ωi/zi. [Hint: The

use of a symbolic calculation software is recommended.]P.7.4 Consider the plane truss of Fig.4.19; the coordinates of the finiteelement model are the cartesian coordinates of the nodes (xi, yi). For everyactive strut shown in the figure, write the influence vector b appearing inEqu.(7.24).P.7.5 To avoid the saturation associated with integral control, the IFFcontroller (7.27) may be replaced by the “Beta controller”

δ

y= H(s) =

gs

Ka(s + β)2

where β ¿ ω1 (de Marneffe). See section 15.5.(a) Compare the root locus and the damping performance of the two

controllers.(b) Show that, on the contrary to (7.27), this control law does not

reduce the static stiffness of the structure.P.7.6 Consider the seven-story shear frame of Fig.7.18. It is controlled in adecentralized manner with two independent and identical feedback loops.Every actuator ui applies a pair of forces equal and opposite between floori and floor i − 1,5 while the sensor yi = xi − xi−1 measures the relativedisplacement between the same floors. The mass, stiffness and B matricesare respectively M = mI7,

5 The height of a story is assumed h = 0 in the model, so that the control force maybe assumed to act in the horizontal direction.

7.10 Problems 167

x6

x5

x4

x3

x2

x1u1

u2

!k = 2Ò sin[2(14+1)ù(2kà1)]

k = 1; . . .; 7

zl = 2Ò sin[2(10+1)ù(2là1) ]

l = 1; . . .; 5

y1=x1

y2 = x2 à x1

x7

a) b)

Fig. 7.18. (a) Shear frame with two independent control loops (displacement sensorand force actuator). (b) Configuration corresponding to transmission zeros.

K = k

2 −1 0 . . . 0−1 2 −1 . . . 00 −1 2 . . . 0

. . .0 0 −1 2 −10 . . . 0 −1 1

, B =

1 −10 10 0...

...0 00 0

(7.49)

(m is the mass and k the stiffness of a single floor). The natural frequencyof mode l is given by

ωl = 2

√k

msin[

π

2(2l − 1)(2n + 1)

] , l = 1, · · · , n (7.50)

where n is the number of storeys (for the calculations, one can normalizeaccording to Ω =

√k/m = 1). Consider the lead compensator

H(s) = g1 + Ts

1 + αTs(α < 1) (7.51)

and select the parameters T and α to control properly at least the first 3modes of the system.

(a) Consider a single control loop in the first floor. Compute the trans-mission zeros, draw the root locus and select a reasonable value of the gain;evaluate the active damping obtained in the 3 targeted modes.


(b) Consider the decentralized control with two independent loops withthe same gain. Compute the transmission zeros, draw the root locus andcompare it to the previous one; select a reasonable value of the gain andevaluate the active damping obtained for the 3 targeted modes.

(c) For the single loop controller, evaluate the effect of a band limitedactuator by including a low-pass filter (7.37) with a corner frequencyωa = ω5 and ξa = 0.7. Comment on the feasibility of such a controlsystem.P.7.7 Consider again the seven-story shear frame of Fig.7.18 with one ortwo active struts in the first two floors; the stiffness Ka of the strut issuch that Ka/k = 5. The active struts are equipped with a displacementactuator and a force sensor pair and are controlled in a decentralizedmanner with the IFF controller:

H(s) = g/s (7.52)

(a) Consider a single control loop in the first floor. Compute the open-loop poles and the transmission zeros, draw the root locus and select areasonable value of the gain; evaluate the active damping obtained in the3 targeted modes.

(b) Consider the decentralized control with two independent loops withthe same gain. Compute the open-loop poles and the transmission zeros,draw the root locus and compare it to the previous one; select a reasonablevalue of the gain and evaluate the active damping obtained for the 3targeted modes.

(c) For the single loop controller, evaluate the effect of a band limitedactuator by including a low-pass filter (7.37) with a corner frequencyωa = ω5 and ξa = 0.7. Comment on the feasibility of such a controlsystem and compare to the previous problem.

8

Vibration isolation

8.1 Introduction

There are two broad classes of problems in which vibration isolation isnecessary: (i) Operating equipments generate oscillatory forces which canpropagate into the supporting structure (Fig.8.1.a). This situation corre-sponds to that of an engine in a car. (ii) Sensitive equipments may besupported by a structure which vibrates appreciably (Fig.8.1.b); in thiscase, it is the support motion which constitutes the source of excitation;this situation corresponds to, for example, a telescope in a spacecraft, awafer stepper or a precision machine tool in a workshop, or a passengerseated in a car.

The disturbance may be either deterministic, such as the unbalanceof a motor, or random as in a passenger car riding on a rough road.For deterministic sources of excitation which can be measured, such as

dd

xx x

xx

cc

fkk kc

cc

MM M

a

( c )( b )( a ) f

f

d

s

Fig. 8.1. (a) Operating equipment generating a disturbance force fd. (b) Equipmentsubjected to a support excitation xd. (c) Active isolation device.

170 8 Vibration isolation

a rotating unbalance, feedforward control can be very effective (see forexample chapter 7 of Fuller’s book).1 However, the present chapter isfocused on the feedback strategies for active isolation; they apply to bothdeterministic and random disturbances, and they do not need a directmeasurement of the disturbance.

Let us begin with the system depicted in Fig.8.1.a, excited by a dis-turbance force fd. If the support is fixed, the governing equation is:

Mx + cx + kx = fd (8.1)

The force transmitted to the support is given by

fs = kx + cx (8.2)

In the Laplace domain,

X(s) =Fd(s)

M(s2 + 2ξωns + ω2n)

(8.3)

Fs(s) = M(ω2n + 2ξωns)X(s) (8.4)

where X(s), Fd(s) and Fs(s) stand for the Laplace transform of respec-tively x(t), fd(t) and fs(t), and with the usual notations ω2

n = k/M and2ξωn = c/M . The transmissibility of the support is defined in this caseas the transfer function between the disturbance force fd applied to themass and the force fs transmitted to the support structure; combiningthe foregoing equations, we get

Fs(s)Fd(s)

=1 + 2ξs/ωn

1 + 2ξs/ωn + s2/ω2n

(8.5)

Next, consider the second situation illustrated in Fig.8.1.b; the distur-bance is the displacement xd of the supporting structure and the systemoutput is the displacement xc of the sensitive equipment. Proceeding ina similar way, it is easily established that the transmissibility of this iso-lation system, defined in this case as the transfer function between thesupport displacement and the absolute displacement of the mass M , isgiven by (Problem 8.1)

Xc(s)Xd(s)

=1 + 2ξs/ωn

1 + 2ξs/ωn + s2/ω2n

(8.6)

1 In feedforward control, it is not necessary to measure directly the disturbance force,but rather a signal which is correlated to it, such as the rotation velocity, if thedisturbance results from a rotating unbalance.

8.1 Introduction 171

k

disturbancesource

m

sensitive equipment

M

xcxd

C

0

xc

xd = 0

1

2 1>

1 / s

1 / s²

(j )

(j )

101 2

n

Objective of theactive isolation

10

-10

0.1

-20

dB

Fig. 8.2. Transmissibility of the passive isolator for various values of the damping ratioξ. The performance objectives of active isolation are a high frequency decay like s−2

together with no overshoot at resonance.

which is identical to the previous one; the two isolation problems cantherefore be treated in parallel. The amplitude of the corresponding FRF,for s = jω, is represented in Fig.8.2 for various values of the damping ratioξ. We observe that: (i) All the curves are larger than 1 for ω <

√2 ωn

and become smaller than 1 for ω >√

2 ωn. Thus the critical frequency√2 ωn separates the domains of amplification and attenuation of the

isolator. (ii) When ξ = 0, the high frequency decay rate is s−2, that is -40dB/decade, while very large amplitudes occur near the corner frequencyωn (the natural frequency of the spring-mass system).

Figure 8.2 illustrates the trade-off in passive isolator design: largedamping is desirable at low frequency to reduce the resonant peak whilelow damping is needed at high frequency to maximize the isolation. Onemay already observe that if the disturbance is generated by a rotatingunbalance of a motor, there is an obvious benefit to use a damper withvariable damping characteristics which can be adjusted according to therotation velocity: high when ω <

√2ωn and low when ω >

√2ωn. Such

variable (adaptive) devices will be discussed in chapter 17. Figure 8.2 alsoshows the target of an active isolation system which combines a decayrate of -40 dB/decade with no overshoot at resonance. Before addressing


the active isolation, the following section discusses one way of improvingthe high frequency isolation in a passive way.

8.2 Relaxation isolator

In the relaxation isolator, the viscous damper c is replaced by a Maxwellunit consisting of a damper c and a spring k1 in series (Fig.8.3.a). Thegoverning equations are

Mx + k(x− x0) + c(x− x1) = 0 (8.7)

c(x− x1) = k1(x1 − x0) (8.8)

or, in matrix form using the Laplace variable s,[

Ms2 + cs + k −cs−cs k1 + cs

] xx1

=

kk1

x0 (8.9)

It follows that the transmissibility reads

x

x0=

(k1 + cs)k + k1cs

(Ms2 + cs + k)(k1 + cs)− c2s2=

(k1 + cs)k + k1cs

(Ms2 + k)(k1 + cs) + k1cs(8.10)

One sees that the asymptotic decay rate for large frequencies is in s−2,that is -40 dB/decade. Physically, this corresponds to the fact that, athigh frequency, the viscous damper tends to be blocked, and the system

Fig. 8.3. (a) Relaxation isolator. (b) Electromagnetic realization.

8.2 Relaxation isolator 173

1 10

-40

-20

0

!=!n

xx0dB

c = 0 c!1

copt

A

Fig. 8.4. Transmissibility of the relaxation oscillator for fixed values of k and k1 andvarious values of c. The first peak corresponds to ω = ωn; the second one correspondsto ω = Ωn. All the curves cross each other at A and have an asymptotic decay rate of-40 dB/decade. The curve corresponding to copt is nearly maximum at A.

behaves like an undamped isolator with two springs acting in parallel.Figure 8.4 compares the transmissibility curves for given values of k andk1 and various values of c. For c = 0, the relaxation isolator behaveslike an undamped isolator of natural frequency ωn = (k/M)1/2. Likewise,for c → ∞, it behaves like an undamped isolator of frequency Ωn =[(k + k1)/M ]1/2. In between, the poles of the system are solution of thecharacteristic equation

(Ms2 + k)(k1 + cs) + k1cs = (Ms2 + k)k1 + cs(Ms2 + k + k1) = 0

which can be rewritten in root locus form

1 +k1

c

s2 + ω2n

s(s2 + Ω2n)

= 0 (8.11)

It is very similar to (7.31); it is represented in Fig.8.5 when c varies from0 to ∞; using the results of section 7.5, the maximum damping ratio isachieved for

k1

c=

Ω3/2n

ω1/2n

(8.12)


Fig. 8.5. Root locus of the solutions of Equ.(8.11) as c goes from zero to infinity. The

maximum damping is achieved for k1/c = Ω3/2n ω

−1/2n .

and the corresponding damper constant is

copt =k1

Ωn(ωn

Ωn)1/2 =

k1

Ωn(1 +

k1

k)−1/4 =

k1

ωn(1 +

k1

k)−3/4 (8.13)

The transmissibility corresponding to copt is also represented in Fig.8.4;it is nearly maximum at A.

8.2.1 Electromagnetic realization

The principle of the relaxation isolator is simple and it can be realizedwith viscoelastic materials. However, it may be difficult to integrate inthe system, and also to achieve thermal stability. In some circumstances,especially when thermal stability is critical, it may be more convenientto achieve the same effect through an electromechanical converter whichconsists of a voice coil transducer, an inductor L and a resistor R.

A voice coil transducer is an energy converter transforming mechanicalenergy into electrical energy and vice-versa; its constitutive equations aregiven by Equ.(3.1) to (3.3).

Referring to Fig.8.3.b, the governing equations of the system are

Mx + k(x− x0)− Ti = 0 (8.14)

Ldi

dt+ T (x− x0) + Ri = 0 (8.15)

8.3 Active isolation 175

where T is the transducer constant; in matrix form, using the Laplacevariable, [

Ms2 + k −TTs Ls + R

] xi

=

kTs

x0 (8.16)

It follows that the transmissibility reads

x

x0=

(Ls + R)k + T 2s

(Ms2 + k)(Ls + R) + T 2s(8.17)

Comparing with Equ.(8.10), one sees that the electromechanical isolatorbehaves exactly like a relaxation isolator provided that

Ls + R

T 2=

cs + k1

k1c(8.18)

or

k1 =T 2

Lc =

T 2

R(8.19)

These are the two relationships between the three parameters T , L and Rso that the transmissibility of the electromechanical system of Fig.8.3.bis the same as that of Fig.8.3.a.

8.3 Active isolation

k

Accelerometer

disturbancesource

sensitiveequipment

m

MFd

a

k

F

aF

F

x xcd

Fig. 8.6. Single-axis active isolator.

Consider the single-axis isolator connecting a disturbance source m toa payload M (Fig.8.6). It consists of a soft spring k in parallel with a forceactuator fa; the objective is to isolate the motion xc of the payload M


from the motion xd of m due to the disturbance load fd. The governingequations are

Mxc + k(xc − xd) = fa (8.20)

mxd + k(xd − xc) = fd − fa (8.21)

or, in matrix form using the Laplace variable s,[

Ms2 + k −k−k ms2 + k

] Xc

Xd

=

Fa

Fd − Fa

(8.22)

Upon inverting this equation, one gets

Xc =kFd

s2[Mms2 + (M + m)k]+

mFa

Mms2 + (M + m)k(8.23)

The first term of this expression describes the payload response to thedisturbance load while the second term is the payload response to theactuator. If an accelerometer or a geophone is attached to the payload,measuring the absolute acceleration xc or the absolute velocity xc, theopen-loop transfer function is

G(s) =s2Xc

Fa=

ms2

Mms2 + (M + m)k(8.24)

Consider the closed-loop response to a general feedback law based on theabsolute velocity xc:

Fa = −H(s)sXc(s) (8.25)

Introducing this into (8.22), one gets[

Ms2 + H(s)s + k −k−k −H(s)s ms2 + k

] Xc

Xd

=

0Fd

(8.26)

Upon considering the first line of this equation, one finds easily that theclosed-loop transmissibility is

Xc

Xd=

k

Ms2 + H(s)s + k(8.27)


8.3.1 Sky-hook damper

Equation (8.27) shows the influence of the feedback control law on thetransmissibility; it shows that a simple velocity feedback, H(s) = g leadsto the transmissibility

Xc

Xd=

k

Ms2 + gs + k=

1s2/ω2

n + gs/k + 1(8.28)

which complies with the objectives of active isolation stated in Fig.8.2,because the asymptotic decay rate is in s−2 (i.e. -40 dB/decade) and theovershoot at resonance may be controlled by adjusting the gain g of thecontroller to achieve critical damping. This control law is called sky-hook,because the control force fa = −gxc is identical to that of a viscousdamper of constant g attached to the payload and a fixed point in space(the sky), Fig.8.7.

k

Accelerometerm M

aF

km

M

F = - g s Xa c

(a) (b)“sky”

disturbancesource

sensitiveequipment

g

Fig. 8.7. (a) Isolator based on an acceleration or an absolute velocity sensor (geo-phone). (b) Equivalent sky-hook damper.

The open-loop transfer function between the input force fa and theoutput acceleration xc is given by (8.24); it has a pair of imaginary polesat

pi = ±j

√(M + m)k

Mm(8.29)

and a pair of zeros at the origin. The root-locus of the sky-hook is shownin Fig.8.8; it is entirely contained in the left-half plane, which means thatthe sky-hook damper is unconditionally stable (infinite gain margin).

8.3.2 Integral Force Feedback

We have just seen that the sky-hook damper based on the absolute ve-locity of the payload is unconditionally stable for a rigid body. However,


Im

Re

F s²Xg

sm s

m M s + k (m+M)

H(s) G(s)

(2 zeros + 1 pole)

a c2

2

Fig. 8.8. Root-locus of the sky-hook damper (acceleration sensor).

this is no longer true if the payload cannot be regarded as a rigid body,situation which is frequently met in space applications. Since the abso-lute acceleration of a rigid body is proportional to the force applied toit, F = Mxc, the acceleration feedback of Fig.8.7 may be replaced by aforce feedback as shown in Fig.8.9. Note that, besides the advantage ofachieving alternating poles and zeros discussed here, a force sensor maybe more sensitive than an accelerometer in low frequency applications;for example, a force sensor with a sensitivity of 10−3N is common place;for a mass M of 1000 kg (e.g. a space telescope), this corresponds to anacceleration of 10−6m/s2; such a sensitivity is more difficult to achieve.Force sensing is especially attractive in micro-gravity where one does nothave to consider the dead loads of a structure. The open-loop transferfunction is, in this case

G(s) =F

Fa=

Mms2

Mms2 + (M + m)k(8.30)

k

Fm

F

M

xcxd F = - g/s F

a

a

disturbancesource

sensitiveequipment

Fig. 8.9. Sky-hook based on a force sensor (F is taken positive when it is acting inthe direction of xc on mass M).


disturbancesource

k

Fa

F

x2

x1

sensitiveequipment

h(s)

Fig. 8.10. Arbitrary flexible structures connected by a single-axis isolator with forcefeedback.

which has the same pole/zero pattern and the same root-locus as Fig.8.8.However, when the payload is flexible, the force applied and the acceler-ation are no longer proportional and the pole/zero pattern may differsignificantly. It can be observed that the feedback based on the acceler-ation still leads to alternating poles and zeros in the open-loop transferfunction when the flexible modes are significantly above the suspensionmode, but they do not alternate any more when the flexible modes inter-act with the suspension mode. On the contrary, if two arbitrary undampedflexible bodies are connected by a single-axis isolator with force feedback(Fig.8.10), the poles and zeros of the open-loop transfer function F/Fa

always alternate on the imaginary axis (Fig.8.11).This result is not obvious, because the actuator Fa and the sensor F ,

if collocated, are not dual as requested for alternating poles and zeros (asemphasized in previous chapters); it can be demonstrated as follows: Thesystem with input Fa and output ∆x = x1−x2, the relative displacementbetween the two bodies, is collocated and dual ; therefore, the FRF (whichis purely real in the undamped case) exhibits alternating poles and zeros(full line in Fig.8.11). On the other hand, the control force Fa, the relativedisplacement ∆x and the output (total) force F are related by

F = k∆x− Fa (8.31)

(in this equation, F is assumed positive in traction while Fa is positivewhen it tends to separate the two bodies). It follows that the FRF F/Fa

and ∆X/Fa are related by

F

Fa=

k∆X

Fa− 1 (8.32)

This equation states that the FRF with force sensor, F/Fa, can be ob-tained from that with relative displacement sensor, k∆X/Fa by a simple


1

0

1

Fig. 8.11. FRFs of the single-axis oscillator connecting two arbitrary flexible struc-tures. The full line corresponds to k∆X/Fa and the dotted line to F/Fa; it is obtainedby vertical translation of the full line by -1.

vertical translation bringing the amplitude to 0 at ω = 0 (from the fullline to the dotted line). This changes the locations of the zeros Zi butthe continuity of the FRF curve between two resonances guarantees thatthere is a zero between two consecutive poles (natural frequencies):

ωi < Zi < ωi+1 (8.33)

8.4 Flexible body

When the payload is flexible, the behavior of the acceleration feedbackand the force feedback are no longer the same, due to different poles/zerosconfigurations of the two control strategies. In fact, different sensor con-figurations correspond to different locations of the zeros in the s-plane.To analyze this situation, consider the payload with a flexible appendageof Fig.8.12; the nominal numerical values used in the calculations arem = 1.1kg, M = 1.7kg, k = k1 = 12000N/m, c1 = 0; the mass m1 of theflexible appendage is taken as a parameter to analyze the interaction be-tween the flexible appendage and the isolation system. When m1 is small,the flexible appendage is much more rigid than the isolation system and

8.4 Flexible body 181

k k

F

mF

sensitive equipment

M

xcxd

a

1

1

1

1m

c

x

flexible appendage[disturbance source

Fig. 8.12. Payload with a flexible appendage.

the situation is not much different from that of a rigid body. Figure 8.13shows the root locus plots for m1 = 0.5kg; the acceleration feedback andthe force feedback have similar root locus plots, with a new pole/zero pairappearing higher on the imaginary axis; the poles and zeros still alternateon the imaginary axis and the only difference between the two plots is thedistance between the pole and the zero which is larger for the accelerationfeedback; as a result, the acceleration feedback produces a larger dampingof the higher mode.

By contrast, when m1 is large, the root locus plots are reorganized asshown in Fig.8.14 for m1 = 3.5kg. For force feedback, the poles and zerosstill alternate on the imaginary axis, leading to a stable root locus; thisproperty is lost for the acceleration feedback, leading to an unstable loopfor the lower mode. In practice, however, this loop is moved slightly to theleft by the structural damping, and the control system can still operatewith small gains (conditionally stable).

(b)(a) Im

Re

Im

Re

Fig. 8.13. Root locus of the isolation system with a light flexible appendage (m1 =0.5kg). (a) Force feedback. (b) Acceleration feedback.


(b)(a) Im

Re

Im

Re

Fig. 8.14. Root locus of the isolation system with a heavy flexible appendage (m1 =3.5kg). (a) Force feedback. (b) Acceleration feedback.

8.4.1 Free-free beam with isolator

To analyze a little further the situation when the payload is flexible,consider the vertical isolation of a free-free continuous beam from the dis-turbance of a body of mass m, Fig.8.15.a. This situation is representativeof a large space structure with its attitude control system attached tom (the disturbance is created by the unbalance of the rotating wheels).Note that the rigid body modes are uncontrollable from the internal forceFa. In the numerical example described below, the length of the beamis l = 5m, the mass per unit length is % = 2kg/m, the stiffness of theisolator is k = 1N/m and the mass where the disturbance is applied ism = 1kg; the stiffness EI of the beam is taken as a parameter.

Let Ωi be the natural frequencies of the flexible modes of the free-freebeam alone, Fig.8.15.b and Zi be the transmission zeros corresponding toa force excitation and a collocated displacement sensor (or equivalentlyacceleration). According to what we have seen in the previous chapters,Zi are the natural frequencies of the system with an additional restraintat the connecting degree of freedom of the isolator. Because of the collo-cation, the poles and zeros are alternating on the imaginary axis, so thatΩi and Zi satisfy

Zi < Ωi < Zi+1 (8.34)

Next, consider the complete system (beam + mass m) and let ωi be itsnatural frequencies (flexible mode only, because the rigid body modes arenot controllable from the internal force Fa). If the control system uses aforce sensor, Fig.8.15.d, the transmission zeros, obtained by enforcing a

8.4 Flexible body 183

Fig. 8.15. (a) Free-free beam and single axis isolator. The other figures illustratethe various situations and the boundary conditions corresponding to the transmissionzeros. (b) Free-free beam with displacement sensor and point force actuator. (c) Free-free beam and sky-hook isolator. (d) Free-free beam and isolator with force feedback.

zero force at the connecting d.o.f., are identical to the natural frequenciesof the system when the isolator is disconnected from the beam; whichare identical to the natural frequencies Ωi of the free-free beam. Theconfiguration is exactly that of Fig.8.10 and, accordingly, the open-loopFRF has alternating poles and zeros and the following relation holds:

ωi < Ωi < ωi+1 (8.35)

This condition guarantees the stability of the closed-loop system when aforce feedback is used.

With an acceleration feedback (sky-hook damper, Fig.8.15.c, the polesare still ±jωi while the zeros, obtained by enforcing a zero acceleration at


Fig. 8.16. Flexible beam with an isolator; evolution of ωi,Zi and ωi with the flexibilityof the beam.

the connecting d.o.f. are ±jZi, as for the free-free beam of Fig.8.15.b. Thisactuator/sensor configuration is no longer collocated, so that no conditionsimilar to (8.34) or (8.35) holds between ωi and Zi. When the beam isstiff, the interlacing property ωi < Zi < ωi+1 is satisfied and the stabilityis guaranteed, but as the beam becomes more flexible, the values of ωi andZi decrease at different rates and a pole/zero flipping occurs when theyboth become equal to the natural frequency of the isolator (ω∗ =

√k/m),

Fig.8.16. As a result, the system is no longer unconditionally stable whenthe flexibility is such that ω1 = Z1 = ω∗ =

√k/m, and above.

As a conclusion to this section, it seems that the sky-hook damperimplementation (acceleration feedback) is preferable when the payload isfairly stiff as compared to the isolator corner frequency (e.g. car suspen-sion), to benefit from the better active damping properties of the flexiblemodes (Fig.8.13). On the contrary, the force feedback implementation ispreferable when the payload is very flexible (e.g. space structure), to ben-

8.5 Payload isolation in spacecraft 185

efit from the interlacing of the poles and zeros, leading to guaranteedstability.

8.5 Payload isolation in spacecraft

Space telescopes and precision payloads are subject to jitter due to theunbalanced masses of the attitude control reaction wheels or gyros. Theperformance of the instruments may be improved by inserting one or sev-eral isolators in the transmission path between the disturbance source andthe payload. If the isolator is designed in such a way that its transmis-sibility exhibits a decay rate of −40dB/decade, the jitter can be reducedby a factor 100 by selecting the isolator corner frequency, f0, one decadelower than the first flexible mode of the payload, fn (Fig.8.17). Extremelysensitive payloads may even involve several isolation layers. 2

Fig. 8.17. Effect of the isolator on the transmissibility between the spacecraft bus andthe telescope.

2 The future James Webb Space Telescope, JWST will involve two isolation layers, (i)the wheel isolator supporting six reaction wheels, with corner frequencies at 7 Hzfor rocking and 12 Hz for translation and (ii) a 1 Hz passive isolator at the interfacebetween the telescope deployment tower and the spacecraft bus (Bronowicki).


Fig. 8.18. Spacecraft architecture. (a) Without isolator. (b) Isolator placed betweenthe Reaction Wheel Assembly (RWA) and the spacecraft bus. (c) Isolator between thespacecraft bus and the telescope.

8.5.1 Interaction isolator/attitude control

There are several possible locations for the isolator, depending on thespacecraft architecture (Fig.8.18). If the attitude control wheels arepacked in a single assembly (RWA), the isolator may be placed betweenthe RWA and the spacecraft bus, Fig.8.18.b. Another option consistsin placing the isolator between the spacecraft bus and the instrument,Fig.8.18.c; in this alternative, the rotating wheels are rigidly attached tothe spacecraft bus. The additional compliance introduced by the vibrationisolator has a major impact on the low frequency dynamics of the systemand its interaction with the attitude control system must be taken intoaccount. The most favorable situation is that where the attitude controlactuators and the attitude sensors (star trackers) are both rigidly at-tached to the spacecraft bus (collocated). For non-collocated situations,the stability of the control system requires that the corner frequency f0

of the isolator be one decade above the attitude control bandwidth, fc;altogether,

fc ∼ 0.1f0 ∼ 0.01fn (8.36)

8.6 Six-axis isolator 187

8.5.2 Gough-Stewart platform

To fully isolate two rigid bodies with respect to each other, six single-axisisolators judiciously placed are needed. For a number of space applica-tions, generic multi-purpose isolators have been developed with a standardGough-Stewart platform architecture, in which every leg of the platformconsists of a single-axis active isolator, connected to the base plates byspherical joints. In the cubic architecture (Fig.8.19), the legs are mutu-ally orthogonal, which minimizes the cross coupling between them. Thisconfiguration is particularly attractive, because it also has uniform stiff-ness properties and uniform control capability, and it has been adoptedin most of the projects.

Fig. 8.19. Multi-purpose soft (relaxation) isolator based on a Gough-Stewart platformwith cubic architecture (ULB).


Fig. 8.20. Six-axis isolator (only four legs are represented). The connection between theleg and the support as well as the payload is done with spherical joints. The coordinatesof the payload x = (x, y, z, θx, θy, θz)

T and the leg extensions q = (q1, . . . , q6)T are

related by q = Jx where J is the Jacobian of the isolator.

8.6 Six-axis isolator

Let us consider a payload isolated by six identical isolators (Fig.8.20); ifthe isolators consist of simple springs of stiffness k, the six suspensionmodes are solution of an eigenvalue problem

(Ms2 + K)x = 0 (8.37)

where x is a vector of 6 coordinates describing the position of the payload,e.g. x = (x, y, z, θx, θy, θz)T . The mass matrix M can be obtained bywriting the kinetic energy in terms of x. Similarly, the stiffness matrix isobtained by writing the strain energy in terms of x. The strain energyin the system is V = 1

2k qT q, where q = (q1, . . . , q6)T is the vector ofthe spring extensions in the isolator and k is the stiffness common to allsprings. If J is the Jacobian matrix connecting the spring extensions q tothe coordinates x (J depends on the topology of the isolator),

q = Jx (8.38)

one gets that

V =12k qT q =

12k xT JT Jx (8.39)

which means that the stiffness matrix is

K = kJT J (8.40)


8.6.1 Relaxation isolator

If the linear spring is replaced by a relaxation isolator, the common stiff-ness k must be replaced by the appropriate relationship between thespring force F and the spring extension x − x0. From the constitutiveequations of the isolator, Equ.(8.9), one finds that the dynamic stiffnessof the relaxation isolator is (Problem 8.5):

F

x− x0= k(s) = k[1 +

k1cs

k(k1 + cs)] (8.41)

(the stiffness is k at low frequency and k1 at high frequency). Thus, the(frequency-dependent) stiffness matrix of the six-axis relaxation isolatorreads

K(s) = JT J k[1 +k1cs

k(k1 + cs)] = K[1 +

k1cs

k(k1 + cs)] (8.42)

and the eigenvalue problem (8.37) becomes

Ms2 + K[1 +k1cs

k(k1 + cs)]x = 0 (8.43)

If ωi and Φ = (φ1, . . . , φ6) are the solution of the eigenvalue problem(8.37), normalized according to ΦT MΦ = I, one can transform (8.43)into modal coordinates, x = Φz; using the orthogonality conditions, onefinds a set of decoupled equations

s2 + ω2i [1 +

k1cs

k(k1 + cs)] = 0 (8.44)

Upon introducing

Ω2i = ω2

i (1 +k1

k) (8.45)

the previous equation may be rewritten

k1

c(s2 + ω2

i ) + s(s2 + Ω2i ) = 0

or

1 +k1

c

s2 + ω2i

s(s2 + Ω2i )

= 0 (8.46)

which is identical to (8.11). Thus, according to the foregoing equation,the six suspension modes follow independent root-loci connecting ωi andΩi (Fig.8.5). However, k1/c being a single scalar parameter, the optimaldamping cannot be reached simultaneously in the six modes, because ofthe modal spread (ω1 < ω6).


8.6.2 Integral Force Feedback

If the linear springs are substituted with identical active isolators such asin Fig.8.10, the dynamics of the isolator is governed by

Mx + Kx = Bu (8.47)

where the right hand side is the projection of the six actuator forces,u = (u1, . . . , u6)T in the global coordinate system attached to the payload.The control forces ui act in the direction where the leg extension qi ismeasured; from the principle of virtual work,

(Bu)T δx = uT δq −→ uT BT δx = uT Jδx

which impliesB = JT and K = kBBT (8.48)

The force sensor equation is the same as (8.31):

y = kq − u (8.49)

where y = (y1, . . . , y6)T is the output vector of the six force sensors; theIFF feedback law is

u =g

sy (8.50)

where an equal gain is assumed for the six independent loops. CombiningEqu.(8.47), (8.49) et (8.50), one gets the closed-loop equation

(Ms2 + K)x =g

s + gkBBT x

or(Ms2 + K

s

s + g)x = 0 (8.51)

If we transform into modal coordinates, x = Φz, and take into accountthe orthogonality relationships, the characteristic equation is reduced toa set of uncoupled equations

(s2 + Ω2i

s

s + g)zi = 0 (8.52)

Thus, every mode follows the characteristic equation

s2 + Ω2i

s

s + g= 0


or1 + g

s

s2 + Ω2i

= 0 (8.53)

where Ωi are the natural frequencies of the six suspension modes. Thecorresponding root locus is shown in Fig.8.21.a. It is identical to Fig.8.8for a single-axis isolator; however, unless the 6 natural frequencies areidentical, a given value of the gain g will lead to different pole locations forthe various modes and it will not be possible to achieve the same dampingfor all modes. This is why it is recommended to locate the payload in sucha way that the spread of the modal frequencies is minimized.

(a) (b)

jÒi jÒi

jzi

Im

ReRe

Im

Fig. 8.21. Six-axis active isolator with independent IFF loops: root locus of individualmodes. (a) with perfect spherical joints. (b) with flexible joints.

8.6.3 Spherical joints, modal spread

The foregoing results have been obtained with the assumptions that theconnections are made by perfect spherical joints, so that the only contri-bution to the stiffness matrix come from the axial stiffness of the legs,K = kBBT . However perfect spherical joints do not exist and they havefriction and backlash. Backlash is not acceptable in precision engineeringand the spherical joints are replaced by flexible connections with largelongitudinal and shear stiffness and low bending stiffness, such as the oneshown in Fig.8.22. This is responsible for an additional contribution Ke

to the stiffness matrix. The global stiffness matrix is kBBT + Ke and theclosed-loop equation of the suspension with the IFF controller becomes


F.E. mesh

(Samcef field)

Fig. 8.22. Typical flexible joint for the connections of a six-axis isolator. Its behavioris close to a universal joint, with low bending stiffness while the other d.o.f. are ratherstiff.

(Ms2 + Ke + kBBT s

s + g)x = 0 (8.54)

According to this equation, the transmission zeros, which are the asymp-totic solutions as g → ∞, are no longer at the origin (s = 0), but aresolutions of the eigenvalue problem

[Ms2 + Ke]x = 0 (8.55)

It follows that the zeros are shifted along the imaginary axis, leading to theroot locus of Fig.8.21.b, which reduces the performance of the suspensionsystem.

As mentioned before, the six suspension modes have different natu-ral frequencies and the decentralized IFF controller has a single gain gwhich has to be adjusted to achieve a good compromise in the suspensionperformance for the six modes. The best performance is achieved if thesuspension is designed in such a way that the modal spread, Ω6/Ω1, isminimized. The combined effect of the modal spread and the joint stiff-ness is illustrated in Fig.8.23; there are only 4 different curves becauseof the symmetry of the system. The bullets correspond to the closed-loop poles for a fixed value of g; they illustrate the fact that the variousloops are travelled at different speeds as g increases. How this impactsthe transmissibility is examined below.

8.7 Active vs. passive 193

Im

Re

Ò5;Ò6Ò4

Ò3

Ò1;Ò2

Fig. 8.23. Typical root locus of a complete isolation system with real joints. Thebullets indicate the location of the closed-loop poles for the adopted value of the gaing (from Preumont et al., 2007).

8.7 Active vs. passive

Figure 8.24 compares the components involved in the passive relaxationisolator and the active one. The active isolation requires conditioningelectronics for the force sensor and power electronics for the voice coilactuator. The relaxation isolator requires only a passive RL circuit but

kV

I

fck

Ifc

noisy side

quiet side

fd

Force Sensor

FElectric

circuit

Electromagnetic

transducer noisy side

quiet side

Control-

ler

fd

Fig. 8.24. Comparison of the active isolator (left) with the passive isolator (right);if a RL electrical circuit is used, the passive isolator is a relaxation isolator; a purelyresistive circuit produces a linear viscous isolator.


Fig. 8.25. Leg of a passive relaxation isolator; conceptual design and exploded viewof the transducer showing the membrane, the magnetic circuit, the voice coil and itsconnection with the stinger.

also requires a bigger transducer (with a larger transducer constant T ).Also, it does not have a force sensor, which makes it lighter. In fact,the legs have their own local dynamics which interfere with that of theisolator and impact significantly the transmissibility in the vicinity ofthe resonance frequency of the local modes and beyond. Maximizing thenatural frequency of the local modes of the legs is a major challenge inthe design of a six-axis isolator with broadband isolation capability. Thisis achieved through careful design of all the components of the isolator.Figure 8.25 shows the leg of a passive relaxation isolator; the explodedview of the transducer shows the membrane which acts as an axial springand also plays the role of spherical joint, the magnetic circuit and thevoice coil, and its connection to the stinger, made of CFRP to minimizeits weight.

From the comparison of the root locus plots, of Fig.8.5 and 8.21.b, onewould expect that the active isolator would have less overshoot near theresonance frequencies of the suspension. On the other hand, the passiveisolator does not need a force sensor, which makes the leg a little lighter

8.8 Car suspension 195

and improves the high frequency behavior of the isolator. Fig.8.26 com-pares the transmissibility of the active (IFF) and the passive (relaxation)isolator. The dotted line refer to the transmissibility of the passive isola-tor when the RL circuit is open. As expected, the overshoot of the activeone is a little lower; both have a decay rate of -40 dB/decade in the inter-mediate frequency range, and the high frequency behavior is dominatedby the local modes; the passive isolator behaves better in high frequency,because the local modes have higher frequencies.

Active control

Fig. 8.26. Vertical transmissibility of a six-axis isolator: comparison of the Open-loop(dotted line), closed-loop (IFF) active isolator and passive relaxation isolator with aRL shunt (from de Marneffe et al.).

8.8 Car suspension

Ride comfort requires good vibration isolation; it is usually measured bythe car body acceleration, or sometimes its derivative called jerk. Vehiclehandling requires good road holding, which is measured by the tyre de-flection. In addition to the car body acceleration and the tyre deflection,there are design constraints on the suspension travel, that is the relativedisplacement between the car body and the wheel.


Fig. 8.27. Quarter-car, two-d.o.f. models. (a) Fully active suspension. (b) Passivesuspension with an added sky-hook active control.

Figure 8.27 shows a quarter-car model of a vehicle equipped with a fullyactive suspension (Fig.8.27.a) or a passive suspension with an active, sky-hook damper (Fig.8.27.b); ms refers to the sprung mass, equal to a quarterof the car body mass, and mus is the unsprung mass (wheel); kt is thestiffness of the tyre; k and c are the stiffness and damping of the passivesuspension. The figure also shows the definition of the state variables usedto model the systems: x1 = xs − xus is the relative displacement of thesprung mass with respect to the wheel (suspension travel), x2 = xs is theabsolute velocity of the car body, x3 = xus−w is the tyre deflection, andx4 = xus is the absolute wheel velocity. With this definition of the statevariables, the dynamics of the fully active suspension (Fig.8.27.a) reads

ms x2 = f

mus x4 = −f − ktx3

x1 = x2 − x4

x3 = x4 − v

where v = w is the road velocity. Defining the force per unit sprung massu = f/ms, the unsprung mass ratio µ = mus/ms and the tyre frequencyωt = (kt/mus)1/2, this system is rewritten in matrix form,

x1

x2

x3

x4

=

0 1 0 −10 0 0 00 0 0 10 0 −ω2

t 0

x1

x2

x3

x4

+

010

−1/µ

u +

00−10

v (8.56)


With the same state-variables, the dynamics of the system of Fig.8.27.b(without control) is governed by

ms x2 = f − kx1 + c(x4 − x2)

mus x4 = −f − ktx3 + kx1 + c(x2 − x4)

x1 = x2 − x4

x3 = x4 − v

and, upon defining ω2n = k/ms (ωn is the body resonance), c/ms = 2ξωn,

they are rewritten in matrix form

x1

x2

x3

x4

=

0 1 0 −1−ω2

n −2ξωn 0 2ξωn

0 0 0 1ω2

nµ

2ξωn

µ −ω2t−2ξωn

µ

x1

x2

x3

x4

+

010

−1/µ

u +

00−v0

(8.57)

Passive suspension

The trade-off in the design of a passive suspension can be illustrated withthe following example taken from Chalasani: The nominal values of thepassive suspension are kt = 160000N/m (tyre stiffness), k = 16000N/m(suspension spring stiffness), ms = 240 kg (car body), mus = 36 kg(wheel). Figure 8.28.a shows the transmissibility Txsv between the roadvelocity v = w and the body absolute acceleration x2 = xs for three val-ues of the damping constant, c = 200 Ns/m, 980 Ns/m, 4000 Ns/m. Forthe smallest value of c, one sees clearly the two peaks associated withthe body resonance (sprung mass) and the tyre resonance (unsprungmass); the body resonance is at 7.8 rad/s and the tyre resonance is at69.5 rad/s, respectively very close to ωn =

√k/ms = 8.16 rad/s and

ωt =√

kt/mus = 66.7 rad/s. As the damping increases, the amplitudeof the two peaks is reduced; one sees clearly that the passive dampingcannot control the body resonance without deteriorating the isolation athigher frequency. The cumulative RMS value of the body acceleration isdefined by the integral

σxs(ω) = [∫ ω

0|Txsv|2dν]1/2 (8.58)

it is represented in Fig.8.28.b; since the road velocity is approximately awhite noise, σxs describes how the various frequencies contribute to the


Fig. 8.28. Behavior of the passive suspension for various values of the damping con-stant: c = 200 Ns/m, 980 Ns/m, 4000 Ns/m: (a) Transmissibility Txsv between theroad velocity v = w and the body absolute acceleration x2 = xs. (b) Cumulative RMSvalue of the sprung mass acceleration σxs .

RMS of the body acceleration (in relative terms). When the dampingincreases, the RMS body acceleration initially decreases, and increasesagain for larger values of c.

Active suspension

A partial state feedback is added to the passive suspension; it consists ofa sky-hook damper, f = −gxs as shown in Fig.8.27.b. Figure 8.29.a showsthe impact of the control gain on the transmissibility Txsv between theroad velocity v = w and the body absolute acceleration xs; the cumulativeRMS value of the sprung mass acceleration σxs is shown in Fig.8.29.b.One can see that the active control acts very effectively on the bodyresonance and that the attenuation is achieved without deteriorating thehigh frequency isolation. However, the active control is unable to reducethe wheel resonance. The active control produces a significant reduction ofthe RMS sprung mass acceleration but the control system fails to reducethe wheel resonance, Fig.8.29.c. The RMS tyre deflection is dominatedby the wheel resonance and is not much reduced by the sky-hook control,Fig.8.29.d.


Fig. 8.29. Active suspension for various values of the control gain, g = 0, 1000 and2000. The damping of the shock absorber is c = 200 Ns/m: (a) and (b) Sprung massacceleration (c) and (d) Tyre deflection.


8.9 References

ABU-HANIEH, A. Active Isolation and Damping of Vibrations via Stew-art Platform, PhD Thesis, Universite Libre de Bruxelles, Active Struc-tures Laboratory, 2003.BOURCIER de CARBON, Ch. Perfectionnement a la suspension desvehicules routiers. Amortisseur relaxation. Comptes Rendus de l’Academiedes Sciences de Paris, Vol.225, pp. 722-724, Juillet-Dec. 1947.BRONOWICKI, A.J. Vibration Isolator for Large Space Telescopes,AIAA J. of Spacecraft and Rockets, Vol.43, No 1, 45-53, January-February2006.CHALASANI, R.M. Ride Performance Potential of Active SuspensionSystems, Part 1: Simplified Analysis based on a Quarter-Car Model,ASME Symposium on Simulation and Control of Ground Vehicles andTransportation Systems, Anaheim, CA, Dec. 1984.COBB, R.G., SULLIVAN, J.M., DAS, A., DAVIS, L.P., HYDE, T.T.,DAVIS, T., RAHMAN, Z.H., SPANOS, J.T. Vibration isolation and sup-pression system for precision payloads in space, Smart Mater. Struct.Vol.8, 798-812, 1999.COLLINS, S.A., von FLOTOW, A.H. Active vibration isolation for space-craft, 42nd IAF Congress, paper No IAF-91-289, Montreal, Oct.1991.de MARNEFFE, B. Active and Passive Vibration Isolation and Dampingvia Shunted Transducers, Ph.D. Thesis, Universite Libre de Bruxelles,Active Structures Laboratory, Dec. 2007.de MARNEFFE, B., AVRAAM, M., DERAEMAEKER, A., HOROD-INCA, M. PREUMONT, A. Vibration Isolation of Precision Payloads: ASix-axis Electromagnetic Relaxation Isolator, AIAA Journal of Guidance,Control, and Dynamics, Vol.32, No 2, 395-401, March-April 2009.FULLER, C.R., ELLIOTT, S.J. & NELSON, P.A. Active Control of Vi-bration, Academic Press, 1996.GENG, Z. HAYNES, L. Six degree of freedom active vibration isolationsystem using the Stewart platforms, IEEE Transactions on Control Sys-tems Technology, Vol.2, No 1, 45-53, 1994.HAUGE, G.S., CAMPBELL, M.E. Sensors and control of a spaced-basedsix-axis vibration isolation system, J. of Sound and Vibration, Vol.269,913-931, 2004.HYDE, T.T. & ANDERSON, E.H. Actuator with built-in viscous damp-ing for isolation and structural control, AIAA Journal, Vol.34, No 1, 129-135, January 1996.

8.9 References 201

HROVAT, D. Survey of Advanced Suspension Developments and RelatedOptimal Control Applications, Automatica, Vol.33, No 10, 1781-1817,1997.KAPLOW, C.E., VELMAN, J.R. Active local vibration isolation appliedto a flexible telescope, AIAA J. of Guidance, Control and Dynamics,Vol.3, 227-233, 1980.LASKIN, R.A., SIRLIN, S.W. Future payload isolation and pointing sys-tem technology, AIAA J. of Guidance, Control and Dynamics, Vol.9, 469-477, 1986.KARNOPP, D.C., TRIKHA, A.K. Comparative study of optimizationtechniques for shock and vibration isolation, Trans. ASME, J. of Engi-neering for Industry, Series B, Vol.91, 1128-1132, 1969.McINROY, J.E., O’BRIEN, J.F., NEAT, G.W. Precise, fault-tolerantpointing using a Stewart platform, IEEE/ASME Transactions on Mecha-tronics, Vol.4, No 1, 91-95, March 1999.McINROY, J.E., NEAT, G.W., O’BRIEN, J.F. A robotic approach tofault-tolerant, precision pointing, IEEE Robotics and Automation Maga-zine, 24-37, Dec. 1999.McINROY, J.E., HAMANN, J. Design and control of flexure jointedhexapods, IEEE Transaction on Robotics, Vol.16(4), 372-381, August2000.McINROY, J.E. Modelling and design of flexure jointed Stewart platformsfor control purposes, IEEE/ASME Transaction on Mechatronics, 7(1),March 2002.PREUMONT, A., FRANCOIS, A., BOSSENS, F., ABU-HANIEH, A.Force feedback versus acceleration feedback in active vibration isolation,J. of Sound and Vibration, Vol.257(4), 605-613, 2002.PREUMONT, A., HORODINCA, M., ROMANESCU, I., de MARN-EFFE, B., AVRAAM, M., DERAEMAEKER, A., BOSSENS, F., ABU-HANIEH, A. A six-axis single stage active vibration isolator based onStewart platform, Journal of Sound and Vibration, Vol.300 : 644-661,2007.PREUMONT, A. & SETO, K. Active Control of Structures, Wiley, 2008.RAHMAN, Z.H, SPANOS, J.T, LASKIN, R.A. Multi-axis vibration iso-lation, suppression and steering system for space observational applica-tions, SPIE Symposium on Astronomical Telescopes and Instrumenta-tion, Kona-Hawaii, March 1998.RIVIN, E.I. Passive Vibration Isolation, ASME Press, N-Y, 2003.


SPANOS, J., RAHMAN, Z., BLACKWOOD, G. A soft 6-axis active vi-bration isolator, Proc. of the IEEE American Control Conference, 412-416, 1995.STEWART, D. A platform with six degrees of freedom, Pro. Instn. Mech.Engrs. Vol.180(15), 371-386, 1965-66.THAYER, D., VAGNERS, J., von FLOTOW, A., HARDMAN, C., SCRIB-NER, K. Six-axis vibration isolation system using soft actuators and mul-tiple sensors; AAS 98-064, 497-506, 1998.THAYER, D., CAMPBELL, M., VAGNERS J., von FLOTOW, A. Six-Axis vibration isolation system using soft actuators and multiple sensors,J. of Spacecraft and Rockets, Vol.39, No 2, 206-212, March-April 2002.

8.10 Problems

P.8.1 Consider the passive isolator of Fig.8.1.b. Find the transmissibilityXc(s)/Xd(s) of the isolation system.P.8.2 Consider the active isolator of Fig.8.7.a with a sky-hook controller.Analyze the effect of the passive damping on the transmissibility. Is thedamping beneficial or detrimental ?P.8.3 Write the differential equations governing the system of Fig.8.12 instate variable form. Using the following values of the parameters: m =1.1kg, M = 1.7kg, k = k1 = 1.2 104N/m,m1 = 0.5kg, c1 = 0Ns/m. Writethe open-loop frequency response for the acceleration feedback (xc) andforce feedback (f) configurations and draw the corresponding poles/zerospattern. In both cases draw the root locus for an integral controller. Dothe same for m1 = 3.5kg; investigate the effect of structural damping inthe flexible appendage.P.8.4 Consider the modal expansion of the open-loop FRF (F/Fa) of thesystem of Fig.8.15.d. Show that the residues are all positive and that thisresults in alternating poles and zeros.P.8.5 Show that the dynamic stiffness of the relaxation isolator (Fig.8.3.a)is given by

F

x− x0= k(s) = k[1 +

k1cs

k(k1 + cs)]

P.8.6 Consider the Gough-Stewart platform with cubic architecture ofFig.8.30 (Spanos et al.); the basic frame xb, yb, zb has its origin at node0; the reference (or payload) frame xr, yr, zr has its origin at the geo-metrical center of the hexapod, noted as node 8, and ~zr is perpendicularto the payload plate; the orientation of ~xr and ~yr is shown in the figure.

8.10 Problems 203

23

4

1

1 1

5

6

6

7

85

5

0

33

6

4

4

2

2

zzbb

y

y

b

b

x

x

b

b

x

ry

r

PayloadPlate

+8

1

2

3

4

5

6

r

r

z station of base plate = - L/2 3

z station of payload plate = L/2 3node 0 : (0,0,0) node 4 : (0,L,L)node 1 : (L,0,0) node 5 : (0,0,L)node 2 : (L,L,0) node 6 : (L,0,L)node 3 : (0,L,0) node 7 : (L,L,L)

node 8 : (L/2, L/2, L/2)

BasePlate

Fig. 8.30. Geometry and coordinate systems for the cubic hexapod isolator. Numbersin bold indicate the active struts.

The small displacements of the system are described by the coordinatesx = (xr, yr, zr, θx, θy, θz)T (a) Show that the control influence matrix ap-pearing in Equ.(8.47) reads

B =1√6

1 1 −2 1 1 −2√3 −√3 0

√3 −√3 0√

2√

2√

2√

2√

2√

2−L/2 L/2 L L/2 −L/2 −L

−L√

3/2 −L√

3/2 0 L√

3/2 L√

3/2 0L√

2 −L√

2 L√

2 −L√

2 L√

2 −L√

2

(b) If the base is fixed and the payload is an axisymmetrical rigidbody of mass m with the principal axes of inertia aligned with xr, yr, zr,principal moment of inertia Ix = Iy = mR2

x , Iz = mR2z , and with the

center of mass located at an offset distance Zc from the geometrical center,along the vertical axis zr, show that the mass and stiffness matrices arerespectively


M = m

1 0 0 0 Zc 00 1 0 −Zc 0 00 0 1 0 0 00 −Zc 0 (R2

x + Z2c ) 0 0

Zc 0 0 0 (R2x + Z2

c ) 00 0 0 0 0 R2

z

K = kBBT = k diag(2, 2, 2, 0.5L2, 0.5L2, 2L2)

where k is the stiffness of one strut. Observe that the translational stiff-ness is uniform in all directions and that the two bending stiffnesses areidentical.

(c) Consider the natural frequencies of the isolator, solutions of theeigenvalue problem (8.37). Show that the the z-translation or “bounce”mode and the z-rotation or “torsional” mode are decoupled, with naturalfrequencies given by

Ω3 =√

2 Ω0 Ω6 =√

2ρz

Ω0

where Ω0 =√

k/m and ρz = Rz/L is the z-axis radius of gyration nor-malized to the strut length (for most cases, ρz < 1 and Ω6 > Ω3). Showthat the remaining four modes are lateral bending coupled with shear;their natural frequencies occur in two identical pairs, solutions of thecharacteristic equation

(2− Ω2

Ω20

)(12− ρ2

x

Ω2

Ω20

)− 2ρ2

c

Ω2

Ω20

= 0

where ρx = Rx/L is the x-axis radius of gyration normalized to the strutlength and ρc = Zc/L is the center of mass offset normalized to the strutlength. Show that if the center of mass is at the geometric center (ρc = 0)and if ρx = 1

2 and ρz = 1, the hexapod will have 6 identical naturalfrequencies, all equal to Ω3.P.8.7 Consider the fully active suspension of Fig.8.27.a where v = w isassumed to be a white noise process.

Find the Linear Quadratic Regulator (LQR) minimizing the perfor-mance index

J = E[%1x21 + %2x

22 + u2]

and

8.10 Problems 205

J = E[%1x21 + %2x

22 + %3x

23 + u2]

Discuss the meaning of the various terms in the performance index (Thesolution of this problem requires a prior reading of Chapters 9 an 11).P.8.8 Consider the active suspension of Figure 8.27.b with kt = 160000N/m(tyre stiffness), k = 16000 N/m (suspension spring stiffness), ms = 240 kg(car body), mus = 36 kg (wheel). For the three values of the control gain,g = 0, 1000 and 2000, plot the transmissibility between the road velocityv and the body absolute velocity xs and between the road velocity andthe force in the dash-pot c. Compare their frequency content.

9

State space approach

9.1 Introduction

The methods based on transfer functions are often referred to as ClassicalMethods; they are quite sufficient for the design of single-input single-output (SISO) control systems, but they become difficult to apply tomulti-input multi-output (MIMO) systems.

By contrast, the design methods based on the state space approach,often called Modern Methods, start from a system description using firstorder differential equations governing the state variables. The formalismapplies equally to SISO and MIMO systems, which considerably simplifiesthe design of multivariable control systems. Although it is formally thesame for SISO and MIMO systems, we shall first study the state spacemethods for SISO systems. In this case, it is possible to draw a clearparallel with the frequency domain approach, and it is possible to solve theproblems of the optimum state regulator and the optimum state observeron a purely geometric basis, replacing the solution of the Riccati equationby the use of the symmetric root locus. The optimal control of MIMOsystems will be considered in chapter 11.

In the state variable form, Fig.9.1, a linear system is described by aset of first order linear differential equations

x = Ax + Bu + Ew1 (9.1)

y = Cx + Du + w2 (9.2)

with the following notations:x = state vector,u = input vector,

208 9 State space approach

xu u

y

y +

+C

D

(a) (b)

Fig. 9.1. (a) Transfer function approach; (b) State variable form.

y = output vector,w1 = system noise,w2 = measurement noise,A = system matrix,B = input matrix,C = output matrix,D = feedthrough matrix,E= system noise input matrix.

For SISO systems, both u and y are scalar functions. Equation (9.1) pro-vides a complete description of the internal dynamics of the system, whichmay be partially hidden in the transfer function (the internal dynamicsmay include uncontrollable and unobservable modes which do not appearin the input/output relationship). The feedthrough matrix D is oftenomitted in control textbooks; however, in earlier chapters, we saw sev-eral examples where a feedthrough component does occur as a result ofthe sensor type (e.g. accelerometer) or location (when collocated with theactuator), or as a result of the modal truncation (residual mode); theimpact of a feedthrough component on the compensator design was alsoemphasized.

The system noise w1 may include environmental loads, modelling er-rors, unmodelled dynamics (including that of the actuator and sensor),nonlinearities, and the noise in the input vector. The system noise inputmatrix E is in general different from the input matrix B. The measure-ment noise w2 includes the sensor noise and the modelling errors.

It must be emphasized that the choice of the state variables is notunique. In principle, their number is independent of the specific choice(it is equal to the order of the system), but this is not true in practice,because a model can only be correct over a limited bandwidth; if the statevariables can be selected in such a way that some of them do not respond

9.2 State space description 209

dynamically within the bandwidth of interest, they can be eliminatedfrom the dynamic model and treated as singular perturbations (i.e. quasi-static). In structural control, the modal coordinates often lead to theminimum number of state variables.

9.2 State space description

The state space equations of the car suspension was already considered atthe end of the previous chapter; in this section we review the state variableform of the dynamic equations for a number of mechanical systems thatwill be used later in this chapter.

9.2.1 Single degree of freedom oscillator

First, consider the familiar second order differential equation of a singledegree of freedom oscillator:

x + 2ξωnx + ω2nx =

f

m(9.3)

This second order equation implies that there will be two state variables;they can be selected as

x1 = x, x2 = x (9.4)

With this choice, Equ.(9.3) can be rewritten as a set of two first orderequations

x1 = x2

x2 = −2ξωnx2 − ω2nx1 + f/m (9.5)

or (x1

x2

)=

(0 1

−ω2n −2ξωn

) (x1

x2

)+

(01m

)f (9.6)

This equation explicitly shows the system and input matrices A and B.If one measures the displacement, y = x1, the output matrix is C =(1 0) while if one uses a velocity sensor, y = x2 and C = (0 1). Anaccelerometer can only be accounted for by using a feedthrough matrixD in addition to C. In fact, from Equ.(9.5), one gets


y = x = x2 =(−ω2

n −2ξωn

) (x1

x2

)+

1m

f (9.7)

The output and feedthrough matrices are respectively C = (−ω2n −2ξωn)

and D = 1/m.The choice (9.4) of the state variables is not unique and it may be

convenient to make another choice :

x1 = ωnx, x2 = x (9.8)

The advantages are that both state variables express a velocity, and thatthe free response trajectories in the phase plane (x1, x2) are slowly decay-ing spirals (Problem 9.1). With this choice, Equ.(9.3) can be rewritten

(x1

x2

)=

(0 ωn

−ωn −2ξωn

) (x1

x2

)+

(01m

)f (9.9)

This form is sometimes preferred to (9.6) because the system matrix isdimensionally homogeneous.

9.2.2 Flexible structure

Next, consider the multi degree of freedom vibrating system governed bythe set of second order differential equations

Mx + Cx + Kx = Luf (9.10)

where Lu is the input force influence matrix, indicating the way the in-put forces act on the structure. Equation (9.10) is expressed in physicalcoordinates. If one defines the state vector as z = (xT xT )T , it can berewritten in state space form as

z =

(0 I

−M−1K −M−1C

)z +

(0

M−1Lu

)f (9.11)

The foregoing state variable form is acceptable if M is invertible. Cal-culating M−1 is straightforward for a lumped mass system. However,Equ.(9.11) may not be practical because the size of the state vector (whichis twice the number of degrees of freedom of the system) may be too large.This is why it is customary to transform Equ.(9.10) into modal coordi-nates before defining the state vector. In this way, the state equation canbe restricted to the few structural modes which describe the main part

9.2 State space description 211

of the system dynamics in the frequency range of interest. Neglecting thehigh frequency dynamics of the system in the compensator design, how-ever, must be done with extreme care, because the interaction betweenthe neglected modes and the controller may lead to stability problems(spillover instability). This will be of prime concern in the compensatordesign.

Equation (9.10) can be transformed into modal coordinates followingthe procedure described in chapter 2. If we perform the change of variablesx = Φη, the governing equations in modal coordinates read

η + 2ξΩη + Ω2η = µ−1ΦTu f (9.12)

with the usual notations µ = diag(µi) (modal mass), ξ = diag(ξi) (modaldamping) and Ω = diag(ωi) (modal frequencies). In this equation,

Φu = LTu Φ (9.13)

where Φ is the matrix of the mode shapes and Lu is the input forceinfluence matrix. The columns of Φu contain the modal amplitudes atthe actuators location, so that the vector ΦT

u f represents the work of theinput forces f on the various mode shapes.

Here, exactly as for the single degree of freedom oscillator, one candefine the state variable as either

z =

(ηη

)or z =

(Ωηη

)

In the latter case, the state space equation reads

z =

(0 Ω−Ω −2ξΩ

)z +

(0

µ−1ΦTu

)f (9.14)

This form is similar to Equ.(9.9), except that diagonal matrices are sub-stituted for scalar quantities. Note that, in most cases, the size of thestate vector in Equ(9.14) (which is twice the number of modes includedin the model) is considerably smaller than in Equ.(9.11).

The output equations for a set of sensors distributed according to theinfluence matrix LT

y are as follows

• displacement sensors

y = LTy x = LT

y Φη = Φyη = (ΦyΩ−1 0)z (9.15)


• velocity sensorsy = LT

y x = Φyη = (0 Φy)z (9.16)

• accelerometers

y = LTy x = Φyη = (−ΦyΩ − 2ΦyξΩ)z + Φyµ

−1ΦTu f (9.17)

where the columns of Φy represent the modal amplitudes at the sen-sor locations. As for the single degree of freedom oscillator, there is afeedthrough component in the output equation for acceleration measure-ments. For collocated actuators and sensors, the input and output influ-ence matrices are the same : Lu = Ly and Φu = Φy.

9.2.3 Inverted pendulum

As another example of state space description, consider the inverted pen-dulum with a point mass at a distance l from the support, as representedin Fig.9.2.a. The horizontal displacement u of the support O is the inputof the system. The differential equation governing the motion is

u cos θ + lθ = g sin θ (9.18)

This equation also applies to more complicated situations where thependulum consists of an arbitrary rigid body (Fig.9.2.b) if l is taken asthe equivalent length of the pendulum, defined as

l =J + mL2

mL(9.19)

where J is the moment of inertia about the center of mass and L is thedistance between the center of mass C and the hinge O ( J + mL2 is the

L

J c

u

m(a) (b)

o

x

g

l

θ

o

Fig. 9.2. Inverted pendulum: (a) Point mass. (b) Arbitrary rigid body.

9.3 System transfer function 213

moment of inertia about O). This can be readily established using theLagrange or the Newton/Euler equations (Problem 9.2). Equation (9.18)can be linearized about θ = 0 as

u + lθ = gθ (9.20)

When the displacement u of the hinge is the input, it is convenient tomake a change of variable and introduce the absolute position x of thepoint mass, rather than the angle θ:

x = u + θl (9.21)

With this transformation, Equ.(9.20) becomes

x = ω20x− ω2

0u (9.22)

where ω0 =√

g/l is the natural frequency of the pendulum. Note that,in general, the coordinate x is different from that of the center of mass Cin Fig.9.2.b. Using the state variables x1 = x, x2 = x, we find the statespace equation

(x1

x2

)=

(0 1ω2

0 0

) (x1

x2

)+

(0−ω2

0

)u (9.23)

If the output of the system is the angle of the pendulum, y = θ = x1/l−u/l, the output and feedthrough matrices are

C = (1/l 0), D = −1/l (9.24)

9.3 System transfer function

In this section, the input-output transfer function is derived from thestate space equations. The relationship is formally the same for the scalarand multidimensional cases. The noise terms are deleted in Equ.(9.1) and(9.2), because they have nothing to do with the present discussion. Laplacetransforming the state equation

x = Ax + Bu

givessX(s)− x(0) = AX(s) + BU(s)


and, assuming zero initial conditions,

X(s) = (sI −A)−1B U(s) (9.25)

and, from the output equation (9.2), we have

Y (s) = [C(sI −A)−1B + D] U(s) (9.26)

orY (s) = G(s)U(s) (9.27)

withG(s) = C(sI −A)−1B + D (9.28)

In the scalar (SISO) case, G(s) is the transfer function of the system. ForMIMO systems, it is the transfer matrix. It is rectangular, with as manyrows as there are outputs and as many columns as there are inputs to thesystem.

As an example, consider the single degree of freedom oscillator de-scribed by Equ.(9.6), in which the output is the displacement (y = x1).We readily obtain

sI −A =

(s −1

ω2n s + 2ξωn

)

(sI −A)−1 =1

s2 + 2ξωns + ω2n

(s + 2ξωn 1−ω2

n s

)

G(s) = C(sI −A)−1B =1

m(s2 + 2ξωns + ω2n)

(9.29)

This result could have been obtained more easily by Laplace transformingEqu.(9.3). Similarly, applying Equ.(9.28) to the inverted pendulum, using(9.23) and (9.24), we find

G(s) = C(sI −A)−1B + D = −1l

s2

s2 − ω20

(9.30)

Once again, this result is straightforward from Equ.(9.20).Before discussing the poles and zeros, an important theorem in Matrix

Analysis must be established: If a matrix is partitioned into blocks, thefollowing identity applies

(I 0

−CA−1 I

) (A BC D

)=

(A B0 D − CA−1B

)

9.3 System transfer function 215

The first and third matrices involved in this identity are block triangular,and the determinant of a block triangular matrix is equal to the productof the determinants of the diagonal blocks ; it follows that:

det

(A BC D

)= detA. det[D − CA−1B] (9.31)

Using this theorem with sI − A instead of A, and −B instead of B, weget

det

(sI −A −B

C D

)= det(sI −A).det[D + C(sI −A)−1B]

Comparing with Equ.(9.28), we see that the transfer function of a SISOsystem can be rewritten

G(s) = C(sI −A)−1B + D =1

det(sI −A)det

(sI −A −B

C D

)(9.32)

Now, we demonstrate that, as in the classical pole-zero representationof transfer functions, the denominator and the numerator of Equ.(9.32)explicitly show the poles and zeros of the system.

9.3.1 Poles and zeros

The poles are the values si such that the free response of the system fromsome initial state x0 has the form x = x0e

sit. Substituting this in the freeresponse equation, x = Ax, one gets

six0esit = Ax0e

sit

or(siI −A)x0 = 0 (9.33)

This equation shows that the poles are the eigenvalues of the systemequation A (and the initial conditions are the eigenvectors). The polesare solutions of det(sI −A) = 0, which is the denominator of (9.32).

The zeros of the system are the values of s such that an input

u = u0est (9.34)

applied from appropriate initial conditions x0, produces a zero output,y = 0. The state vector has the form


x = x0est (9.35)

if the following condition is fulfilled:

sx0est = Ax0e

st + Bu0est (9.36)

that is, if(sI −A)x0 = Bu0 (9.37)

Under this condition, the output is

y = Cx + Du = (Cx0 + Du0)est (9.38)

Therefore, having y = 0 requires that

Cx0 + Du0 = 0 (9.39)

The two conditions (9.37) and (9.39), can be cast in compact form as(

sI −A −BC D

) (x0

u0

)= 0 (9.40)

The values of s for which this system of homogeneous equations has anon-trivial solution are the zeros of the system. They are solution of

det

(sI −A −B

C D

)= 0 (9.41)

which is the numerator of (9.32). From Equ.(9.40), we observe that, toachieve a zero output, the system needs to be excited at the frequencyof a transmission zero, and also must start from the appropriate initialconditions x0; other initial conditions would produce a transient outputthat would gradually disappear if the system is asymptotically stable.

9.4 Pole placement by state feedback

The idea in the state space approach is to synthesize a full state feedback

u = −Gx (9.42)

where the gain vector G (we deal with SISO systems first) is selectedto achieve desirable properties of the closed-loop system. The problem isthat, in most practical cases, the state vector is not known. Therefore,

9.4 Pole placement by state feedback 217

xu

y

+

+C

G

D

-

Fig. 9.3. Principle of the state feedback (the state is assumed known).

there must be an additional step of state reconstruction. One of the nicefeatures of the state feedback approach is that the two steps can be per-formed independently; this property is called the separation principle. Weshall address the design of the state feedback first (Fig.9.3), and leave thestate reconstruction until later.

A system of the nth order involves n state variables. Therefore thereare n feedback gains gi that can be adjusted independently. Since thereare n poles in the system, either real or complex conjugate, they can beassigned arbitrarily by proper choice of the gi. This is true, in principle,if the system is controllable, but it may not always be practical, becausethe control effort may be too large for the actuators, or the large valuesof the feedback gains may unduly increase the bandwidth of the controlsystem and lead to noise problems.

Substituting the feedback law (9.42) into the system equation (9.1)yields

x = Ax−BGx + Ew1 = (A−BG)x + Ew1 (9.43)

The closed-loop system matrix is A−BG. Its eigenvalues are the closed-loop poles; they determine the natural behavior of the closed-loop systemand are solutions of the characteristic equation

det[sI − (A−BG)] = 0 (9.44)

The state feedback design consists of selecting the gains gi so that theroots of (9.44) are at desirable locations. If the system is controllable, ar-bitrary pole locations s1, s2, ..., sn (either real or complex conjugate pairs)can be enforced by choosing the gi in such a way that Equ.(9.44) is iden-tical to

αc(s) = (s− s1)(s− s2)...(s− sn) = 0 (9.45)


The gain values gi are achieved by matching the coefficients of (9.44) and(9.45).

Fig. 9.4. Single degree of freedom oscillator: (a)open-loop system, (b)closed-loop poles.

9.4.1 Example: oscillator

As an example, consider the single degree of freedom oscillator of Equ.(9.6)(

x1

x2

)=

(0 1

−ω2n −2ξωn

) (x1

x2

)+

(01m

)f (9.46)

The poles of the open-loop system are represented in Fig.9.4.a. We wantto design a state feedback

u = −g1x1 − g2x2 (9.47)

such that the poles of the closed-loop system are moved to the locationsindicated in Fig.9.4.b. From Equ.(9.43), the closed-loop system matrix,A−BG, reads

(0 1

−ω2n −2ξωn

)−

(01m

) (g1 g2

)=

(0 1

−ω2n − g1

m −2ξωn − g2

m

)

and the characteristic equation

det[sI − (A−BG)] = s2 + s(2ξωn +g2

m) + (ω2

n +g1

m) = 0 (9.48)

9.5 Linear Quadratic Regulator 219

The desired characteristic equation is

αc(s) = s2 + 2ξ0ω0s + ω20 = 0 (9.49)

Comparing Equ.(9.48) and (9.49), one sees that the desired behavior willbe achieved if the gains are selected according to

ω20 = ω2

n +g1

m(9.50)

2ξ0ω0 = 2ξωn +g2

m(9.51)

These equations indicate that the controller will act as a spring (of stiffnessg1), and damper (of damping constant g2). This is a general observationabout the state feedback which consists of a generalization of the PDcontrol.

Matching the coefficients of the characteristic equation to the desiredones rapidly becomes tedious as the size of the system increases. It be-comes straightforward if the state equation is written in control canonicalform. We shall not expand on these aspects because the techniques forpole placement are automated in most control design softwares.

The fact that the poles of the closed-loop system can be located ar-bitrarily in the complex plane is remarkable. Two questions immediatelyarise:

• Can it always be done ?• Is it always practical to do it ?

The answer to the first question is yes if the system is controllable (control-lability is discussed in chapter 12). However, even if the system complieswith the controllability condition, it may not be practical, because thecontrol effort involved would be too large, or because the large valuesof the feedback gains would render the system oversensitive to noise orto modelling errors, when the control law is implemented on the recon-structed state from an observer. These robustness aspects are extremelyimportant and actually dominate the controller design. One reasonableway of selecting the closed-loop poles is discussed in the next section.

9.5 Linear Quadratic Regulator

One effective way of designing a full state feedback in terms of a singlescalar parameter is to use the Optimal Linear Quadratic Regulator (LQR).


In this section, we analyze this problem for SISO systems; a more generalformulation will be examined in chapter 11. We seek a state feedback(9.42) minimizing the performance index

J =∫ ∞

0[z2(t) + %u2(t)]dt (9.52)

where z(t) is the controlled variable, defined by

z = Hx (9.53)

The performance index has two contributions; the first one reflects thedesire of bringing the controlled variable to zero (minimizing the error)while the second one that of keeping the control input as small as possible.% is a scalar weighting factor used as a parameter in the design : largevalues of % correspond to more emphasis being placed on the control costthan on the tracking error. Note that the control variable, z, may or maynot be the actual output of the system. If this is the case, H = C.

9.5.1 Symmetric root locus

The solution of the LQR is independent of the initial conditions; for aSISO system, it can be shown (e.g. Kailath, p.226) that the closed-looppoles are the stable roots (i.e. those in the left half plane) of the charac-teristic equation

1 + %−1G0(s)G0(−s) = 0 (9.54)

whereG0(s) =

Z(s)U(s)

= H(sI −A)−1B (9.55)

is the open-loop transfer function between the input and the controlledvariable. Equation (9.54) defines a root locus problem in terms of thescalar parameter %, the weighting factor of the control cost in the perfor-mance index.

Note that s and −s affect Equ.(9.54) in an identical manner. As aresult, the root locus will be symmetric with respect to the imaginaryaxis, in addition to being symmetric with respect to the real axis. This iswhy it is called a symmetric root locus. Note that, since only the left partof the root locus must be considered, the LQR approach is guaranteed tobe stable.

The regulator design proceeds according to the following steps

9.5 Linear Quadratic Regulator 221

• Select the row vector H which defines the controlled variable appearingin Equ.(9.53).

• Draw the symmetric root locus in terms of the parameter %−1 andconsider only the part of the locus in the left half plane.

• Select a value of %−1 that provides the appropriate balance betweenthe tracking error and the control effort. Once the closed-loop poleshave been selected, the state feedback gains can be computed as forpole placement (a more efficient way of computing the control gainswill be discussed in chapter 11).

• Check with simulations that the corresponding control effort is com-patible with the actuator. If not, return to the previous step.


To illustrate the procedure, consider the inverted pendulum described bythe state space equation (9.23). If we adopt the absolute displacement xas the controlled variable, H = (1 0); the transfer function (9.55) reads

G0(s) = H(sI −A)−1B = − ω20

s2 − ω20

(9.56)

G0(s)G0(−s) =ω4

0

(s2 − ω20)2

(9.57)

The symmetric root locus is represented in Fig.9.5. The closed-loop polesconsist of a complex conjugate pair on the left branch of the locus, s =−ξωn ± jωd. Note that the transfer function G0(s) used in the regulatordesign is different from the open-loop transfer function (9.30) between theinput and the output variable (the latter is the tilt angle θ rather thanthe absolute position x). Using x instead of θ in the performance indexwill not only keep the pendulum vertical, but will also prevent it fromdrifting away from its initial position.

Once appropriate pole locations have been selected, the control gainsG can be calculated as indicated in the previous section, by matching theclosed-loop characteristic equation (9.44) to the desired one (9.45):

det[sI − (A−BG)] = s2 − sω20g2 − ω2

0(1 + g1) = 0 (9.58)

αc(s) = s2 + 2ξωns + ω2n = 0 (9.59)

Equating the coefficients of the various powers in s provides


w0-w0

Fig. 9.5. Symmetric root locus of the inverted pendulum. G0(s)G0(−s) has doublepoles at s = ω0 and s = −ω0.

g1 = −(1 +ω2

n

ω20

) (9.60)

g2 = −2ξωn

ω20

(9.61)

The open-loop system has a stable pole at −ω0 and an unstable one at+ω0. The cheapest optimal state feedback which stabilizes the system,obtained for ρ−1 = 0 on the root locus, simply relocates the unstablepole at −ω0 (thus, the closed-loop system has two poles at −ω0). Thecorresponding characteristic equation (9.59) is

(s + ω0)2 = s2 + 2sω0 + ω20 = 0

and the optimal gains are

g1 = −2 , g2 = − 2ω0

9.6 Observer design

The state feedback (9.42) assumes that the state vector is available at alltimes. This is not the case in general, because it would require too manysensors, and in many applications, some of the states would simply not beavailable, for physical reasons. The aim of the observer is to reconstruct

9.6 Observer design 223

the state vector from a model of the system and the output measurementy. In this way, the state feedback can be implemented on the reconstructedstate, x, rather than on the actual state x. It must be emphasized thatthe state reconstruction requires a model of the system. We shall assumethat an accurate model is available.

To begin with, consider the case where the noise and feedthrough termsare removed from the system and output equations (9.1) and (9.2).

x = Ax + Bu (9.62)

y = Cx (9.63)

The following form is assumed for the full state observer (also called Lu-enberger observer).

˙x = Ax + Bu + K(y − Cx) (9.64)

x(0) = 0

The first part of Equ.(9.64) simulates the system and the second contri-bution (innovation) uses the information contained in the sensor signal;y−Cx is the difference between the actual and the predicted output; thegain matrix K is chosen in such a way that the error between the truestate and the reconstructed one, e = x− x, converges to zero. CombiningEqu.(9.62) to (9.64), we find the error equation

e = (A−KC)e (9.65)

with the initial condition e(0) = x(0). This equation shows that the errorgoes to zero if the eigenvalues of A−KC (the observer poles) have negativereal parts (i.e. if A−KC is asymptotically stable).

In a manner parallel to the state feedback design, if the system isobservable, the n observer poles can be assigned arbitrarily in the complexplane by choosing the values of the n independent gains ki (observabilitywill be discussed in chapter 12).

The procedure for pole assignment is identical to that used for theregulator design : If a set of poles si has been selected, the gains ki canbe computed by matching the coefficients of the characteristic equation

det[sI − (A−KC)] = 0 (9.66)

to that of the desired one


αo(s) = (s− s1)(s− s2)...(s− sn) = 0 (9.67)

(once again, the poles can be only real or complex conjugate pairs). Notethat the nice form of the error equation is due to the fact that the samematrices A, B, C have been used in the system equations (9.62), (9.63)and the observer equation (9.64). This amounts to assuming perfect mod-elling. In practice, however, the actual system will be substituted forEqu.(9.62) and (9.63) while the observer equation (9.64) will be inte-grated numerically in a computer. As a result, there will always be anerror associated with the modelling of the system.

If a feedthrough term appears in the output equation, the innovationterm in Equ.(9.64) must be taken as y − (Cx + Du) instead of y − Cx;this leads to the same error equation (9.65). If noise terms are includedas in Equ.(9.1) and (9.2), the error equation becomes

e = (A−KC)e + Ew1 + Kw2 (9.68)

The plant noise Ew1 appears as an excitation; so does the measurementnoise w2, after being amplified by the observer gain K.

If one wants the regulator poles to dominate the closed-loop response,the observer poles should be faster than (i.e. to the left of) the regulatorpoles. This will ensure that the estimation error decays faster than thedesired dynamics, and the reconstructed state follows closely the actualone (at least without noise and modelling error). As a rule of thumb, theobserver poles should be 2 to 6 times faster than the regulator poles.

With noisy measurements, one may wish to decrease the bandwidthof the observer by having the observer poles closer to those of the regula-tor. This produces some filtering of the measurement noise. In this case,however, the observer poles have a significant influence on the closed-loopresponse. Notice, from Equ.(9.65), that if the open-loop system is stable(and the model accurate), the error will converge to zero even if the gainK is zero.

The observer equation (9.64) is based partly on the system model,Ax + Bu, and partly on the measurement error, y −Cx, the latter beingamplified by the observer gain K. The relative importance which is givento the model and the measurement contributions should depend on theirrespective quality. If the sensor noise is low, more weight can be placedon the measurement error (large gains ki), while noisy measurements dorequire lower gains. The minimum variance observer is that minimizingthe variance of the measurement error. If one assumes that the plantnoise and measurement noise are white noise processes (i.e. with uniform

9.7 Kalman Filter 225

power distribution over the whole frequency range), the minimum varianceobserver is known as the Kalman Filter.

9.7 Kalman Filter

The Kalman Bucy Filter (KBF) will be studied in detail in chapter 11.Here, exactly as we used the LQR as a sensible pole placement strategyfor the regulator design of SISO systems (in terms of a single scalar pa-rameter), we consider the particular case where the plant noise and themeasurement noise are scalar white noise processes. In this case, it is, onceagain, possible to draw a symmetric root locus plot in terms of a singleparameter expressing the relative intensity of the plant and measurementnoise. Let

x = Ax + Bu + Ew (9.69)

y = Cx + v (9.70)

be the system and output equations, where w and v are scalar whitenoise processes. It can be demonstrated that the optimal pole locationscorresponding to the KBF are the stable roots of the symmetric root locusdefined by the equation

1 + qGe(s)Ge(−s) = 0 (9.71)

where Ge(s) is the transfer function between the plant noise w and theoutput y

Ge(s) = C(sI −A)−1E (9.72)

and the parameter q is the ratio between the intensity of the plant noisew and that of the measurement noise v (noisy measurements correspondto small q).

The observer design proceeds exactly as for the regulator: Selecting anoise input matrix E, we draw the symmetric root locus as a functionof q. When proper pole locations have been selected on the left half partof the locus, the gains ki can be computed by matching the coefficientsof the characteristic equation (9.66) to that of the desired one (9.67). Ifnone of the values of q provides a desirable set of poles, another E matrixcan be selected and the procedure repeated.

In the LQR approach, the regulator design involves the symmetricroot locus of the open-loop transfer function between the input u and thecontrolled variable z


Go(s) = H(sI −A)−1B (9.73)

while in the KBF approach, the observer design is based on the transferfunction between the system noise w and the output y.

Ge(s) = C(sI −A)−1E (9.74)

If one assumes that the plant noise w enters the system at the input(E = B) and the controlled variable is the output variable (H = C), thesetransfer functions become identical to the open-loop transfer function

G(s) = C(sI −A)−1B (9.75)

In this case, the regulator and the observer can both be designed from asingle symmetric root locus.

w0-w0

Regulatorpoles

Observerpoles

Im(s)

Re(s)

Fig. 9.6. Inverted pendulum. The same symmetric root locus is used for the design ofthe regulator and the full state observer (E = B).


To illustrate the procedure, consider again the inverted pendulum andassume that the noise enters the system at the input (E = B). FromEqu.(9.23) and (9.24), the transfer function between the noise w and theoutput y, Equ.(9.74), is

9.7 Kalman Filter 227

Ge(s) = −1l

ω20

s2 − ω20

(9.76)

Thus, although the output variable θ is different from the controlledvariable x, Ge(s) is proportional to Go(s) used in the regulator design[Equ.(9.56)]; this is because there is no feedthrough component in Ge(s)as there is in G(s) [Equ.(9.30)]. As a result, the root locus used in theregulator design can also be used for the observer design (Fig.9.6). Howfar the observer poles should be located on the locus really depends onhow noisy the measurements are.

Note that when the regulator poles are near the asymptotes, the KBFobtained here is very close to the design consisting of assigning the ob-server poles by scaling the regulator poles ( ωn, ξ ) to ( αωn, ξ ) where2 < α < 6, according to the rule of thumb mentioned before (Problem9.5).

Finally, assume that the noise enters the system according to E =(1 a)T . From Equ.(9.23) and (9.24), the transfer function between thenoise and the output is readily obtained as

Ge(s) =s + a

l(s2 − ω2o)

(9.77)

The corresponding symmetric root locus is shown in Fig.9.7, assuminga > ω0. One notices that one of the poles goes to the zero at s = −a as qincreases. This is the optimum for the assumed distribution of the plantnoise, but the observer obtained in this way may be sluggish, which maynot be advisable for reasons mentioned before.

-ω0

-a a

ω0

Fig. 9.7. Full state observer of the inverted pendulum. Symmetric root locus when thenoise enters the system according to E = (1 a)T .


9.8 Reduced order observer

With the full state observer, the complete state vector is reconstructedfrom the output variable y. When the number of state variables is smalland the output consists of one of the states, it may be attractive to re-strict the state reconstruction to the missing state variables, so that themeasured state variable is not affected by the modelling error. This isnice, but on the other hand, by building the state feedback partly on theoutput measurement without prior filtering, there is a feedthrough com-ponent from the sensor noise to the control input, which increases thesensitivity to noise.

In this section, we restrict ourselves to the frequent case of reconstruct-ing the velocity from a displacement measurement for a second order me-chanical system. This is probably the most frequent situation where areduced order observer is used. In this case, the reduced observer is of thefirst order while the full state observer is of the second order. Extensionsto more general situations can be found in the literature (e.g. Luenberger,1971).

9.8.1 Oscillator

Consider the single degree of freedom oscillator governed by Equ.(9.6).The output measurement is the displacement (y = x). The velocity equa-tion is (v = x)

v = −2ξωnv − ω2nx + u/m (9.78)

We seek a first order observer governed by the following equations

z = −az − bx + u/m (9.79)

v = z + cx (9.80)

where v is the estimated velocity, z is an internal variable, and a, b andc are free parameters; they are selected in such a way that the errorequation governing the behavior of e = v − v is asymptotically stable,with an appropriate decay rate a:

e = −ae (9.81)

Equations (9.78)-(9.80) give the error equation

e = −av + (c + 2ξωn)v + (ω2n + ac− b)x (9.82)

9.8 Reduced order observer 229

It can be made identical to Equ.(9.81) if the coefficients are selected ac-cording to

a = c + 2ξωn (9.83)

b = ω2n + ac (9.84)

From Equ.(9.81), a is the eigenvalue of the observer; it can be chosenarbitrarily.


The same procedure can be applied to the inverted pendulum describedby Equ.(9.23) and (9.24). The displacement is obtained from the outputmeasurement (y = θ) and the input as

x = ly + u (9.85)

The velocity equation is (v = x)

v = ω20x− ω2

0u (9.86)

The velocity observer has again the form

z = −az − bx− ω20u (9.87)

v = z + cx (9.88)

and the error equation (e = v − v ) is readily written as

e = −av + cv + (−ω20 + ac− b)x (9.89)

It becomes identical to (9.81) if the coefficients satisfy

a = c (9.90)

b = c2 − ω20 (9.91)

The reduced order observer is therefore

z = −cz − (c2 − ω20)x− ω2

0u (9.92)

v = z + cx (9.93)

where the only remaining parameter c is the eigenvalue of the ob-server, which can be selected arbitrarily. Note that c appears also as thefeedthrough component of the measured variable in Equ.(9.93), leading


to a direct effect of the measurement noise on the reconstructed velocity(amplified by cl).

Because of the existence of a feedthrough component from the mea-surement to the reconstructed states, and therefore to the control, thebandwidth of a compensator based on a the reduced observer is muchwider than that of a compensator based on a full state observer. Thesimplicity of the observer structure must be weighed against the highersensitivity to sensor noise. If the latter is significant, the reduced orderobserver becomes less attractive than the full state observer.

9.9 Separation principle

Figure 9.8 shows the complete picture of the state feedback regulator im-plemented on the reconstructed states, obtained from a full state observer(the case without feedthrough component is represented for simplicity).The closed-loop equations are :

x = Ax + Bu

y = Cx

u = −Gx (9.94)˙x = Ax + Bu + K(y − Cx)

The complete system has 2n state variables. If one uses the reconstructionerror, e = x− x, as state variable instead of the reconstructed state vectorx, one can write the closed-loop system equation as

(xe

)=

(A−BG BG

0 A−KC

) (xe

)(9.95)

u

Compensator

yC

G

-

^

^ ^ ^

Fig. 9.8. State feedback with full state observer.

9.10 Transfer function of the compensator 231

It is block triangular and, as a result, the eigenvalues of the closed-loopsystem are those of the diagonal blocks A−BG and A−KC. Thus, thepoles of the closed-loop system consist of the poles of the regulator andthose of the observer. This means that the eigenvalues of the regulator andthe observer are not changed when the two subsystems are put together.Therefore, the design of the regulator and of the observer can be carriedout independently; this is known as the separation principle.

9.10 Transfer function of the compensator

The transfer function of the compensator can be obtained from Equ.(9.94).It reads

H(s) =U(s)Y (s)

= −G(sI −A + KC + BG)−1K (9.96)

The poles of the compensator are solutions of the characteristic equation

det(sI −A + KC + BG) = 0 (9.97)

Note that they have not been specified anywhere in the design, and thatthey are not guaranteed to be stable, even though the closed-loop systemis. Working with an unstable compensator (which fortunately, is stabi-lized by the plant!) may bring practical difficulties as, for example, thatthe open-loop frequency response of the compensator cannot be checkedexperimentally. Figure 9.9 represents the compensator and the system inthe standard unity feedback form used in classical methods (root locus,Bode, Nyquist). There is a major difference between the state feedbackdesign and the classical methods: in classical methods, the structure of thecompensator is selected to achieve desired closed-loop properties; in thestate feedback design, the structure of the compensator is never directlyaddressed because the attention is focused on the closed-loop properties;the compensator is always of the same order as the system.

u y( )G sI A KC BG K− + + −1 ( )C sI A B− −1

System G(s)Compensator H(s)

-

Fig. 9.9. State feedback with full state observer. Unity feedback form of the closed-loopsystem.


9.10.1 The two-mass problem

Let us illustrate this point on the two-mass problem of Fig.6.3. Definingthe state vector x = (d, d, y, y)T , Equ.(6.3) and (6.4) can be written as

x =

0 1 0 0−ω2

0 −2ξω0 ω20 2ξω0

0 0 0 1µω2

0 2µξω0 −µω20 −2µξω0

x +

000

1/M

f (9.98)

For a non-collocated displacement sensor,

d = (1 0 0 0)x

The open-loop transfer function is given by Equ.(6.7). If µ = 0.1, M = 1and the open-loop system has a pair of poles at s = −0.02± j1 (flexiblemode) the open-loop transfer function is

G(s) =D(s)F (s)

=0.036(s + 25)

s2(s + 0.02± j1)(9.99)

where the compact notation s + 0.02± j1 is used for (s + 0.02 + j1)(s +0.02− j1).

Using the output d as the controlled variable and assuming that thesystem noise enters at the input, we can design the regulator and theobserver using the same symmetric root locus based on G(s). It is shownin Fig.9.10. Once the regulator and observer poles have been selected onthe left side of the root locus, the corresponding gains can be calculated bymatching the coefficients of the characteristic equations (9.44) and (9.66)to the desired ones; for the pole locations indicated in the figure, we get

G = (−0.558 0.403 1.364 1.651)

K = (5.78 16.71 34.35 31.26)T

The resulting compensator is

H(s) =−98.4(s + 0.28)(s + 0.056± j0.89)(s + 0.68± j2.90)(s + 3.06± j1.54)

(9.100)

It is of the fourth order, like the system. The numerator consists of a PDplus a pair of zeros near the flexible poles of G(s), to produce a notchfilter. With this nominal compensator, it is possible to draw a conventional

9.10 Transfer function of the compensator 233

Observerpoles

Regulatorpoles

-2

2

-2

-1

Im(s)

Re(s)

Fig. 9.10. Symmetric root locus for the two-mass system.

Im(s)

Re(s)XX

Rigid body mode

Notch filter

Compensator

Flexible mode

Fig. 9.11. Root locus plot, 1 + gHG = 0, for the optimal control of the two-masssystem with nominal values of the parameters. The triangles indicate the location ofthe closed-loop poles for g = 1 (only the upper half is shown).


Im(s)

Re(s)XX

Fig. 9.12. Root locus plot for off-nominal parameters ( ωn = 2 instead of 1).

Im(s)

Re(s)XX

Pole-zero flipping

Fig. 9.13. Root locus of the optimal control when the natural frequency has beenreduced to ωn = 0.8.

root locus plot for gHG (Fig.9.11), which describes the evolution of thepoles of the closed-loop system when the scalar gain g varies from 0 to ∞.The pole locations on this root locus for g = 1 (represented by triangles)coincide with those selected on the symmetric root locus (Fig.9.10).

To assess the robustness of the control system, Fig.9.12 shows the rootlocus plot when the natural frequency is changed from ωn = 1 rad/s to

9.12 Problems 235

ωn = 2 rad/s. One observes that the notch filter does not operate properlyany more (the pair of zeros of the compensator no longer attracts theflexible poles of the structure) and the closed-loop system soon becomesunstable.

The situation is even worse if the natural frequency is reduced to ωn =0.8 rad/s. In this case, a pole-zero flipping occurs between the pole of theflexible mode and the zero of the notch filter (Fig.9.13). As we alreadystressed in chapter 6, the pole-zero flipping changes the departure anglesfrom the poles and the arrival angles at the zeros by 1800, transformingthe stabilizing loop of Fig.9.11 into a destabilizing one as in Fig.9.13. Thislack of robustness is typical of state feedback and notch filters.

9.11 References

BRYSON, A.E., Jr. Some connections between modern and classical con-trol concepts, ASME, Journal of Dynamic Systems, Measurement, andControl, Vol. 101, 91-98, June 1979.CANNON, R.H. & ROSENTHAL, D.E. Experiment in control of flexiblestructures with noncolocated sensors and actuators, AIAA Journal ofGuidance, Control and Dynamics, Vol. 7, No 5, Sept-Oct., 546-553, 1984.FRANKLIN, G.F., POWELL, J.D. & EMAMI-NAEINI, A. FeedbackControl of Dynamic Systems. Addison-Wesley, 1986.KAILATH, T. Linear Systems, Prentice-Hall, 1980.KWAKERNAAK, H. & SIVAN, R. Linear Optimal Control Systems, Wi-ley, 1972.LUENBERGER, D. An introduction to observers, IEEE Trans. Autom.Control, AC-16, pp.596-603, Dec.1971.LUENBERGER, D. Introduction to Dynamic Systems, Wiley, 1979.

9.12 Problems

P.9.1 Consider the s.d.o.f. oscillator described by Equ.(9.9). For non-zeroinitial conditions, sketch the free response in the phase plane (x1, x2).Show that the image point rotates clockwise along a spiral trajectory.Relate the decay rate of the spiral to the damping ratio.P.9.2 Show that the inverted pendulum of Fig.9.2 is governed by Equ.(9.18)to (9.20), where u is the displacement of the support point.P.9.3 Consider an inverted pendulum similar to that of Fig.9.2, butmounted on a cart of mass M and controlled by an horizontal force u


applied to the cart. If x stands for the horizontal displacement of the cart,show that, for small θ, the governing equations can be approximated by

x + lθ = gθ

(M + m)x + mLθ = u

Write the equations in state variable form.P.9.4 For the inverted pendulum of Problem 9.3, assuming that the fullstate is available, find a feedback control that balances the stick and keepsthe cart stationary near x = 0.P.9.5 Consider the inverted pendulum of Fig.9.2. Using the same proce-dure as for the regulator in section 9.5.2, find the analytical expression ofthe observer gains which locates the observer poles at (αωn, ξ).P.9.6 Consider an inverted pendulum with ω0 = 1 (Fig.9.2).(a) Compute the transfer function H(s) of the compensator such that theregulator and the observer poles are respectively at ωn = 2, ξ = 0.5 andαωn = 6, ξ = 0.5.(b) Draw a root locus (1 + gHG = 0) for this compensator. Observethat the closed-loop system is conditionally stable. Find the critical gainscorresponding to the limits of stability. Sketch the Nyquist diagram forthe nominal system.

10

Analysis and synthesis in the frequency domain

10.1 Gain and phase margins

Consider the root locus plot of Fig.6.5; any point s in the locus is solutionof the characteristic equation 1 + gG(s)H(s) = 0. Therefore, we have

|gG(s)H(s)| = 1 φ(gGH) = −π

The locus crosses the imaginary axis at the point of neutral stability. Sinces = jω, the following relations hold at the point of neutral stability:

|gG(jω)H(jω)| = 1 (10.1)

φ(gGH) = −π

Returning to the Bode plot of Fig.6.6, one sees that changing g amountsto moving the amplitude plot along the vertical axis; the point of neutralstability is obtained for g = 0.003 × 1.61 = 0.0048, when the amplitudecurve is tangent to the 0 dB line, near the frequency ω = 1, above whichthe phase exceeds −1800. The system is stable if the algebraic value ofthe phase φ of the open-loop system is larger than −π for all frequencieswhere the amplitude |gGH| is larger than 1. The system is unstable ifthis condition is violated. A measure of the degree of stability is providedby the gain and phase margins:

• The gain margin (GM) indicates the factor by which the gain must beincreased to reach the neutral stability.

• The phase margin (PM) is the amount by which the phase of the open-loop transfer function exceeds −π when |gGH| = 1. The correspondingfrequency is called the crossover frequency.

238 10 Analysis and synthesis in the frequency domain

The situation described above corresponds to a stable open-loop systemG(s); the gain margin is a decreasing function of g, the system becomesunstable when the gain exceeds some critical value. There are more com-plex situations where the system is conditionally stable, when the gainbelongs to some interval g1 < g < g2; it becomes unstable when the gaindecreases below the threshold g1 (see Problem 9.6). These cases can behandled with the Nyquist criterion.

10.2 Nyquist criterion

10.2.1 Cauchy’s principle

Consider the feedback system of Fig.10.1, where G(s) stands for the com-bined open-loop transfer function of the system and the compensator(shorthand for gGH in the previous section). The conformal mappingG(s) transforms the contour C1 in the s plane into a contour C2 in theG(s) plane. One of the properties of the conformal mapping is that twointersecting curves with an angle α in the s plane map into two inter-secting curves with the same angle α in the G(s) plane. Assume G(s) iswritten in the form

G(s) = k


(10.2)

For any s on C1, the phase angle of G(s) is given by

Im(s) Im[G(s)]

Re(s) Re[G(s)]

sG(s)

C1C2

ψi

φi φ

G-

Fig. 10.1. Nyquist contour in the s and G(s) planes.

10.2 Nyquist criterion 239

φ(s) =m∑

i=1

φi −n∑

j=1

ψj (10.3)

where φi and ψj are the phase angles of the vectors connecting respectivelythe zeros zi and the poles pi to s. From Fig.10.1, it can be seen that, if thereare neither poles nor zeros within the contour C1, φ(s) does not changeby 2π when s goes clockwise around C1. The contour C2 will encircle theorigin only if the contour C1 contains one or more singularities of thefunction G(s). Because the contour C1 is travelled clockwise, one pole ofG(s) within C1 produces a phase change of 2π, that is one counterclockwiseencirclement of the origin by C2. Conversely, a zero produces a clockwiseencirclement. Thus, the total number of clockwise encirclements of theorigin by C2 is equal to the number of zeros in excess of poles of G(s),within the contour C1. This is Cauchy’s principle.

10.2.2 Nyquist stability criterion

The foregoing idea provides a simple way of evaluating the number ofsingularities of the closed-loop system in the right half plane, from thepoles and zeros of the open-loop transfer function G(s). The contour C1

is selected in such a way that it encircles the whole right half plane asindicated in Fig.10.2. If there are poles on the imaginary axis, indentationsare made as indicated in the figure, to leave them outside C1. The closed-loop transfer function is

F (s) =G(s)

1 + G(s)(10.4)

R = ∞R = ∞

(a) (b)

C1 C1

ε

Fig. 10.2. Contour encircling the right half plane. (a) With no singularity on theimaginary axis. (b) With three poles on the imaginary axis.


and the closed-loop poles are solutions of the characteristic equation

1 + G(s) = 0

IfG(s) =

n(s)d(s)

1 + G(s) =d(s) + n(s)

d(s)(10.5)

This equation shows that the poles of 1 + G(s) are the same as thoseof the open-loop transfer function G(s). Let P be the [known] numberof unstable poles of the open-loop system. On the other hand, the zerosof 1 + G(s) are the poles of the closed-loop system; we want to evaluatetheir number Z within C1. From Cauchy’s principle, if we consider themapping 1 +G(s), the number N of clockwise encirclements of the originby C2 when s goes clockwise along C1 is

N = Z − P (10.6)

Now, instead of considering the encirclements of the origin by 1 + G(s),it is completely equivalent to consider the encirclements of −1 by G(s).Thus, the number of unstable poles of the closed-loop system is given by

Z = N + P (10.7)

-1

Im

Re

Circle at infinity

ω=0−

ω=0+

Fig. 10.3. Nyquist plot for the non-collocated control of the two-mass problem.

10.2 Nyquist criterion 241

-1

Im

Re

1GM

PM

( )G jω

Unit circle

ωc

Fig. 10.4. Definition of the gain margin GM and phase margin PM on the Nyquistplot.

where N is the number of clockwise encirclements of −1 by G(s), whens follows the contour C1, and P is the number of unstable poles of theopen-loop system.

All physical systems without feedthrough are such that G(s) → 0 ass →∞ (with feedthrough, G(s) goes to a constant value). As a result, onlythe part of the plot corresponding to the imaginary axis (−∞ < jω < ∞)must be considered. The polar plot for positive frequencies can be drawnfrom the Bode plots; that for negative frequencies is the mirror imagewith respect to the real axis, because G(−jω) = G∗(jω).

If there are poles of G(s) on the imaginary axis and if the indenta-tions are made as indicated in Fig.10.2, the poles are outside the contour.According to Equ.(10.2) and (10.3), each pole contributes with an arc atinfinity and a rotation of −π, that is 1800 clockwise.

Returning to the system of Fig.6.5 and 6.6, its Nyquist plot is shownin Fig.10.3: The contribution of the positive frequencies is plotted in fullline, and the circle at infinity corresponds to the indentation of the doublepole at the origin (from ω = 0− to ω = 0+); the contribution of thenegative frequencies (not shown) is the mirror image of that of the positivefrequencies with respect to the real axis. For small gains (as shown in thefigure), the number of encirclements of −1 is zero and the system is stable;for larger gains, there are two encirclements and therefore two unstablepoles. This is readily confirmed by the examination of Fig.6.5.


( )Im G jω

( )Re G jω

−1

2

−1

M = 1

M < 1M > 1

M M p=

P M = °60Unit circle

Amplification

( )G jω

ωc

ωp

PM

Fig. 10.5. Loci of constant magnitude of the closed-loop transfer function (M -circles).The maximum amplification Mp is associated with the smallest circle tangent to theNyquist plot.

Since the instability occurs when G(jω) encircles −1, the distance fromG(jω) to −1 is a measure of the degree of stability of the system. Therelative stability is measured by the gain and phase margins (Fig.10.4).As we shall see, there is a direct relationship between the phase marginand the sensitivity to parameter variations and the disturbance rejectionnear crossover.

Since the closed-loop transfer function F (s) is uniquely determined byG(s) [Equ.(10.4)], loci of constant magnitude |F (jω)| = M and of con-stant phase φ[F (jω)] = N can be drawn in the complex plane G(s); theyhappen to be circles. The M circles are shown in Fig.10.5; the larger mag-nitudes correspond to smaller circles near −1. Thus, the maximum ampli-fication Mp corresponds to the smallest circle, tangent to the Nyquist plotof G(s); it is reached for a frequency ωp close to the crossover frequencyωc (where the Nyquist plot crosses the unit circle). This is why there is adirect relationship between the maximum amplification Mp and the phasemargin PM . In most cases, PM > 600 prevents overshoot of F (Fig.10.6).The bandwidth ωb of the control system is defined as the frequency corre-sponding to an attenuation of -3 dB in the closed-loop transfer function.Since ωb ∼ ωp ∼ ωc, the bandwidth can be approximated by the crossoverfrequency for design purposes.

10.3 Nichols chart 243

PM=22°

PM=45°

PM=90°

dB

log w0

-20

-3

wc

wb

wp

Fig. 10.6. Relation between the overshoot of the closed-loop transfer function andthe phase margin. |F | is maximum at ωp; the bandwidth is the frequency ωb where|F | = −3dB and |G| = 1 at the crossover frequency ωc. In most cases, ωb ∼ ωp ∼ ωc.

10.3 Nichols chart

The Nyquist plot is a convenient tool for evaluating the number of en-circlements and the absolute stability of the system. However, because ofthe linear scale for the magnitude, the Nyquist plot is not always prac-tical in the vicinity of -1. The Nichols chart plot is often more usefulfor evaluating the relative stability when the open-loop system is stable(Fig.10.7). It consists of a plot of the dB magnitude vs. phase angle ofthe open-loop transfer function G(jω). There is a one to one relationshipbetween the Nichols chart and the Nyquist plot, but the former brings aconsiderable amplification to the vicinity of -1, for easy evaluation of thegain and phase margins, and the logarithmic scale allows a much widerrange of magnitude in the graph. Unlike the Nyquist plot, the Nicholschart plot can be obtained from the summation of the individual magni-tude and phase angle contributions of the poles and zeros, and a changeof gain moves the curve along the vertical axis. These two advantages areshared by the Bode plots, but the Nichols chart plot combines the gainand phase information into a single diagram. As for the Nyquist plot, inorder to assist in the design, it is customary to draw the loci of constantamplitude M of the closed-loop system; they are no longer circles.


-200

-180

-100

-50

-20

-10

0

10

20

30

-1 dB

-3 dB

-6 dB

-12 dB

-20 dB

0 dB0.25 dB

6 dB

3 dB

1 dB

0.5 dB

-150

-250

Constant ma gnitude M

PM

GM

|G|dB

fo

(G)

Fig. 10.7. Nichols chart: dB magnitude vs. phase angle of the open-loop transferfunction G(jω) and loci of constant magnitude of the closed-loop transfer function(M -curves).

10.4 Feedback specification for SISO systems

Consider the SISO feedback system of Fig.10.8, where r is the referenceinput, y the output, d the disturbance and n the sensor noise (unlike thedisturbance, the sensor noise does not directly affect the output).

r ed

y

n

G-

Fig. 10.8. Feedback System.

10.4 Feedback specification for SISO systems 245

10.4.1 Sensitivity

In this section, we evaluate the sensitivity of the closed-loop transfer func-tion to the variations of the open-loop transfer function; we assume thatd = n = 0. The closed-loop transfer function is

y

r= F =

G

1 + G

e

r=

11 + G

(10.8)

The sensitivity of F to parameter changes is related to that of the open-loop transfer function G by

∂F

∂p=

1(1 + G)2

∂G

∂p(10.9)

orδF =

1(1 + G)2

δG

orδF

F=

11 + G

δG

G(10.10)

S = (1 + G)−1 is called the sensitivity function and 1 + G the returndifference. Equation (10.10) states that, in the frequency range where|1+G| À 1, the sensitivity of the closed-loop system to parameter changesis much smaller than that of the open-loop system. This is one of theobjectives of feedback. On the other hand, if the phase margin is small,G(jω) goes very near −1 at crossover, and F becomes much more sensitivethan G.

10.4.2 Tracking error

Referring to Fig.10.8, the governing equations are

e = r − y − n

y = Ge + d = G(r − y − n) + d

ory =

G

1 + G(r − n) +

11 + G

d (10.11)

The tracking error is

e∗ = r − y =1

1 + G(r − d) +

G

1 + Gn (10.12)

From Equ.(10.11) and (10.12), we see that good tracking implies that


• The sensitivity function S = (1+G)−1 must be small in the frequencyrange where the command r and the disturbance d are large.

• The closed-loop transfer function F = G(1 + G)−1 must be small inthe frequency range where the sensor noise is large.

Note, however, thatS + F = 1 (10.13)

Therefore, S and F cannot be small simultaneously; this means that dis-turbance rejection and noise rejection cannot be achieved simultaneously.

10.4.3 Performance specification

In the previous section, G referred to the open-loop transfer function,including the system and the compensator. In this section, G will referexplicitly to the system and H to the compensator; the open-loop transferfunction is GH (Fig.10.9).

The general objective of feedback is to achieve sensitivity reduction,good tracking and disturbance rejection at low frequency (ω < ω0) with asensor signal which has been contaminated with noise at high frequency(ω > ω1). From the foregoing section, this is translated into the followingdesign constraints:

|1 + GH(ω)| ≥ ps(ω) ω ≤ ω0 (10.14)

where ps(ω) is a large positive function (performance specification), de-fined in the frequency range where good tracking and disturbance rejec-tion is necessary [zero steady-state error requires that ps(0) = ∞], and

|GH(ω)| ¿ 1 ω > ω1 (10.15)

where ω1 defines the frequency above which the sensor noise becomessignificant. Of course, in addition to that, the closed-loop system must bestable. Thus minimum values for the gain and phase margins must alsobe specified:

GM > µ PM > φ (10.16)

Equivalently, we may require that the GH curve remains outside a par-ticular M circle from the point −1 + j0 (e.g. M =

√2). We shall see later

that stability places constraints on the slope of |GH| near crossover (inthe frequency range near |GH| = 1).


r e u yH G

-

sensornoise

steady state error,disturbance rejection

ps( )ω

ω0 ω1

1

GH

Fig. 10.9. Feedback specification.

10.4.4 Unstructured uncertainty

Unstructured uncertainty When the uncertainty on the physical systemis known to affect some specific physical parameters like, for example,the natural frequencies and the damping, it is called structured. If little isknown about the underlying physical mechanism, it is called unstructured.The unstructured uncertainty will be characterized by an upper boundof the norm of the difference between the transfer function of the actualsystem, G′(ω) and its model G(ω). It can be considered either as additive:

G′(ω) = G(ω) + ∆G(ω), |∆G(ω)| < la(ω), ω > 0 (10.17)

where la(ω) is a positive function defining the upper bound to the ad-ditive uncertainty (including the parameter changes and the neglecteddynamics), or multiplicative:

G′(ω) = G(ω)[1 + L(ω)], |L(ω)| < lm(ω), ω > 0 (10.18)

where lm(ω) is a positive function defining an upper bound to the multi-plicative uncertainty. lm(ω) is usually small (¿ 1) at low frequency wherethe model is accurate, and becomes large (À 1) at high frequency, due tothe neglected dynamics of the system (Fig.10.10). Because lm(ω) definesonly the magnitude of the uncertainty, it can be associated with any phasedistribution; therefore, it clearly defines a worst case situation. Evaluat-ing la(ω) or lm(ω) is not a simple task. For large flexible structures, moremodes are often available than it would be practical to include in a de-sign model; these extra modes can be used to evaluate the associateduncertainty (Problem 10.4).


0 dB

lm ( )ω

<< 1

>> 1

Fig. 10.10. Feedback System.

10.4.5 Robust performance and robust stability

In the face of uncertainties, the design objectives of section 10.4.3 mustbe fulfilled for the perturbed system. Referring to Fig.10.11, the stabilityof the system will be guaranteed in presence of uncertainty if the distanceto instability, |1 + GH| is always larger than the uncertainty ∆GH. Thestability robustness condition is therefore

|1 + GH| ≥ |H|la(ω) (10.19)

This guarantees that the number of encirclements will not be altered byany additive uncertainty ∆G bounded by la(ω), for any phase distribution.Similarly, since la(ω) = lm(ω)|G(ω)|, condition (10.19) can be written interms of the multiplicative uncertainty as

|1 + GH| ≥ |GH|lm(ω) (10.20)

or1

lm(ω)≥ | GH

1 + GH| (10.21)

Normally, this condition will never be violated at low frequency, wherethe uncertainty is small and the loop gain is large. At high frequency,

| GH

1 + GH| ∼ |GH|

and Equ.(10.21) is reduced to

|GH| < 1lm(ω)

, wherever ω is such that lm(ω) À 1 (10.22)

This condition expresses the gain stability. It is not conservative if thephase is totally unknown, as in the case of pole-zero flipping. Multiplica-tion of (10.20) by |GH|−1 gives an alternate form of the stability condition:


∆G

G

G

Additive

Multiplicative

1 + L

perturbedsystem

nominalsystem

Uncertainty circleRadius :

1 + G H GH∆GH

−1

H l GH la m( ) ( )ω ω=

Fig. 10.11. Nyquist plot of the nominal and the perturbed systems.

|1 + (GH)−1| ≥ lm(ω) (10.23)

Now, if we want to achieve the performances with the perturbed system,Equ(10.14) must be replaced by

|1 + (1 + L)GH| ≥ ps(ω)

This equation will be satisfied if

|1 + GH|(1− lm) ≥ ps(ω)

A sufficient condition is

|GH| ≥ ps(ω)1− lm(ω)

(10.24)

The design tradeoff for |GH| is explained in Fig.10.12. The shaded regionat low frequency is excluded for robust performance, and that at highfrequency for robust stability. Considering the vicinity of the crossoverfrequency (Fig.10.13), we can make the following observations:

• At crossover, |GH| = 1, and the stability robustness condition (10.20)becomes

|1 + GH| ≥ lm(ωc) (10.25)

This means that accepting a magnitude error lm(ωc) = 1 at crossoverrequires a phase margin of 600. From this, one can anticipate that thebandwidth of the closed-loop system cannot be much larger than thefrequency where lm = 1.


w1

Robust performanceTracking error &disturbance rejection

Stability robustnesssensor noise rejection

Sensor noise

Fig. 10.12. Design tradeoff for |GH|.

• At crossover, the return difference is related to the phase margin by

|1 + GH| = 2 sin(PM

2) (10.26)

This establishes a direct connection between the phase margin and thesensitivity and the disturbance rejection near crossover.

The following section addresses the conflict between the quality of theloop near crossover (good phase margin) and the attenuation rate of GH.

10.5 Bode gain-phase relationships

Figure 10.12 suggests that good feedback design could be achieved byhaving a large gain at low frequency and a fast enough decay rate athigh frequency. Unfortunately, things are more complicated, because theclosed-loop stability of the nominal system requires that the phase remainslarger than −π as long as the gain is larger than 1. It turns out that forstable, minimum phase systems (i.e. with neither poles nor zeros in theright half plane), the phase angle and the amplitude are uniquely related.This relationship is expressed by the Bode Integrals. In this section, we

10.5 Bode gain-phase relationships 251

Fig. 10.13. Nyquist plot near crossover. Relation between the phase margin and thereturn difference at crossover.

shall state the main results without proof; the interested reader can referto the original work of Bode, or to Horowitz or Lurie.

• Integral # 1

Consider the unity feedback with the stable, minimum phase open-looptransfer function G(jω) = |G|ejφ. If the amplitude diagram has a constantslope corresponding to n poles in a log-log diagram [n×(−20 dB)/decade],the phase is

φ = −n 900 (10.27)

In the general case, the phase at a frequency ω0 is given as a weightedaverage of the gain slope at all frequencies, but with a stronger weight inthe vicinity of ω0:

φ(ω0) =1π

∫ ∞

−∞d ln |G|

duW (u)du (10.28)

where u = ln(ω/ω0) and the weighting function W (u) is defined by

W (u) = ln[coth(|u|/2)]

W (u) is strongly peaked near u = 0 (Problem 10.5); its behavior is nottoo far from that of a Dirac impulse, W (u) ' 1

2π2δ(u), so that

φ(ω0) ' π

2d ln |G|

du]u=0


This relation is very approximate, unless the slope of ln |G| is nearlyconstant in the vicinity of ω = ω0; it applies almost exactly if the slopeis constant over two decades. In this case, one readily sees that

−20 dB/decade ⇒ d ln |G|du

= −1 ⇒ φ = −π/2

−40 dB/decade ⇒ d ln |G|du

= −2 ⇒ φ = −π

The first integral indicates that a large phase can only be achieved if thegain attenuates slowly. It follows that the roll-off rate in the region nearcrossover must not exceed −40 dB/decade and it must often be smallerthan this, in order to keep some phase margin.

• Integral # 2

Assume G(s) is stable and has an asymptotic roll-off corresponding tomore than one pole (n > 1 at infinity). According to section 10.4.2, thesensitivity function S = (1+G)−1 represents the fraction of the commandr, or of the disturbance d, which is transmitted into the tracking error.|S| must be small in the frequency range where r or d are large. However|S| cannot be small everywhere, because the second Bode integral statesthat

∫ ∞

0ln |S| dω = 0 (10.29)

This relation states that if |S| < 1 in one frequency band, there must be|S| > 1 in another frequency band; sensitivity can only be traded fromone frequency band to another, and good disturbance rejection in somefrequency range can be achieved only at the expense of making thingsworse than without feedback outside that frequency range (Problem 10.9).

If the open-loop system has unstable poles pi, the second integral be-comes ∫ ∞

0ln |S|dω = π

∑

i

Re(pi) (10.30)

where the sum extends to the unstable poles. This shows that for a sys-tem unstable in open-loop, the situation is worse, because there is moresensitivity increase than decrease; fast unstable poles are more harmfulthan slow ones because they contribute more to the right hand side ofEqu.(10.30).

10.5 Bode gain-phase relationships 253

• Integral # 3

The third integral states that some reshaping of |G| can be performedwithin the working band (normalized to ω = 1) without affecting thephase outside the working band. This arises from:

∫ ω=1

ω=0(ln |G| − ln |G|∞)d arcsinω = −

∫ ∞

1

φ√ω2 − 1

dω (10.31)

If the open-loop transfer function is altered in the working band, in sucha way that the integral on the left hand side is unchanged (ln |G|∞ is thesame for both transfer functions since only the working band is altered),the weighted phase average is preserved outside the working band. Thissituation is illustrated in Fig.10.14.

G dB

log ω

φ°

G1

G2

φ1

φ 2

working band

ω=1

Fig. 10.14. Effect of reshaping the open-loop transfer function in the working band,which preserves the phase distribution outside the working band.

• Integral # 4

This integral says that the greater the phase lag, the larger will be thefeedback in the working band:

∫ ∞

−∞φ

ωdω = πln |G|∞ − ln |G|0 (10.32)


In particular, if two loop shapes G1 and G2 have the same high frequencybehavior, but G2 has a greater phase lag than G1, G2 has a larger mag-nitude in the working band than G1

∫ ∞

−∞φ1 − φ2

ωdω = π ln |G2

G1|0 (10.33)

The third and fourth integrals tell us that, in order to achieve a largegain within the working band, the phase lag must be as large as possibleoutside the working band. The stability limitations on the phase lag arereflected in feedback limitations in the working band. The next sectionillustrates how these phase-magnitude relationships can be translated intodesign.

10.6 The Bode Ideal Cutoff

The first Bode integral tells us how the phase at one frequency is affectedby the gain slope in the vicinity (about one decade up and down). It alsosays that local phase increase can be achieved by lowering the gain slope(this situation is illustrated at high frequency in Fig.10.14). Figure 10.15shows a Nichols chart of the desired behaviour: One wants to keep thefeedback |G|0 constant and as large as possible within the working band(ω < 1 in reduced frequency), then reduce it while keeping gain and phasemargins of x dB and yπ (shaded rectangle in Fig.10.15). It is thereforelogical to keep the phase constant and compatible with the phase margin:

φ = (y − 1)π (10.34)

The compensator design is thus reduced to that of enforcing an open-loop transfer function (system plus compensator) G(jω) to map the twosegments of Fig.10.15; the result is shown in Fig.10.16: The open-looptransfer function with a constant gain in the working band and a constantphase lag φ outside the working band is

G(jω) =|G|0

[√

1− ω2 + jω]2φπ

(10.35)

This transfer function has an ideal behavior at low frequency. However,the roll-off at high frequency is directly related to the phase margin. Im-proved high frequency noise attenuation requires higher roll-off. Because

10.6 The Bode Ideal Cutoff 255

^

w=1

Fig. 10.15. Nichols chart of the two segment problem.

0.1

0.1

0.01

0.01

-150

-140

-90

-100

-30

-60

-50

-80

0

-40

30

-20

0

-120

-120

1

1

10

10

G (dB)

f° PM = 90°

PM = 60°

Fig. 10.16. Bode plots of the two segment problem, for two values of the phase margin.


working band

x dB

-40(1- )y dB/ decade

-20 n dB/ decade1-2 octaves

^

w=1

wc

Fig. 10.17. The Bode Ideal Cutoff.

of the first integral, we know that if we simply add one segment with ahigher slope to the amplitude diagram of Fig.10.16, it will be reflectedby an additional phase lag near crossover, which is incompatible with thestability margin. The cure to this is to have first a flat segment x dBbelow 0 dB, for approximately one or two octaves, followed by a segmentwith a higher slope, taking care of the sensor noise attenuation. The flatsegment at −x dB provides the extra phase lead near crossover whichcompensates for the extra phase lag associated with the higher rolloff athigh frequency. The final design is sketched in Fig.10.17. The reader cancompare it to the design tradeoff of Fig.10.12. Once the loop transferfunction GH has been obtained, that of the compensator can be deducedand approximated with a finite number of poles and zeros.

10.7 Non-minimum phase systems

In the previous sections, we assumed that G(s) has no singularity in theright half plane; such systems are called minimum phase (this section willexplain why). The effect of right half plane poles was briefly examinedin Equ.(10.30). In this section, we consider the effect of right half planezeros.

To begin with, assume that G(s) has a single right half plane zero ats = a. We can write

10.7 Non-minimum phase systems 257

Im(s) Im[A(s)]

Re(s) Re[A(s)]

-a a

Fig. 10.18. Nyquist diagram of the all-pass function A(s) = − s−as+a

.

G(s) = G0(s).A(s) (10.36)

where A(s) is the all-pass function

A(s) = −s− a

s + a(10.37)

and G0(s) is the minimum phase transfer function obtained by reflectingthe right half plane zero into the left half plane [G0(s) no longer hasa singularity in the right half plane]. The Nyquist diagram of A(s) isrepresented in Fig.10.18; A(jω) follows the unit circle clockwise, fromφ = 0 at ω = 0 to φ = −1800 at ω = ∞. A(s) takes its name from thefact that |A(ω)| = 1 for all ω; G(s) and G0(s) have the same magnitudefor all ω and the same phase at ω = 0, but for ω > 0, their phase differby that of A(ω); at ω = a/2, ∆φ = −530 and at ω = a, ∆φ = −900.

We have seen in section 6.7 that flexible structures with non-collocatedactuators and sensors do have non-minimum phase zeros. If they lie welloutside the bandwidth of the system (ωc ¿ a), they do no harm, becausethe corresponding all-pass function brings only very little phase lag nearcrossover. On the contrary, if a non-minimum phase zero lies at a distancecomparable to the bandwidth, ωc ∼ a and φ[A(a)] = −π/2; this meansthat in the design, the phase angle of G0(ω) cannot exceed −π/2 (with-out any phase margin) and, consequently, its falling rate cannot exceed−20 dB/decade instead of −40 dB/decade for a minimum phase system.

The situation can easily be generalized to an arbitrary number of righthalf plane zeros. If G(s) has k right half plane zeros at ai, it can be writtenin the form (10.36) where G0(s) is minimum phase and A(s) is the all-passfunction


A(s) = (−1)kk∏

i=1

s− ai

s + ai(10.38)

A typical pole-zero pattern of A(s) is shown in Fig.10.19. It is easy tosee that |A(ω)| = 1 for all ω and that the phase angle decreases by kπwhen ω goes from 0 to ∞,or 2kπ when ω goes from −∞ to ∞. This extraclockwise rotation about the origin, associated with the all-pass function,can change the number of encirclements of −1, especially if some zeros areclose to the bandwidth; it is therefore decisive in stability considerations.

Im(s)

Re(s)

Fig. 10.19. Pole-zero pattern of a typical all-pass function.

The phase lag associated with the non-minimum phase transfer func-tion G(jω) is always greater than that of G0(jω), by that of A(jω). Thisis why G0(jω) is called minimum phase, because no other stable functionwith the same amplitude characteristics can have any less phase lag. Thedesign of a compensator for a non-minimum phase system can be doneby considering the minimum phase system G0(s) and modifying the de-sign specifications to account for the extra phase lag generated by theall-pass function. As we have seen, this puts stronger restrictions on theattenuation rate near crossover to avoid instability (Problem 10.7).

A pure time delay T has a transfer function

D(s) = e−sT (10.39)

Just as the all-pass function, it has a constant magnitude and a phaselag φ = ωT which increases linearly with the frequency. A pure delay canalways be approximated as closely as needed by an all-pass network (Padeapproximants), so that the above conclusions also apply (Problem 10.8).

10.8 Usual compensators 259

10.8 Usual compensators

Amongst the desired features of the closed-loop system are the stabil-ity properties, expressed by the gain and phase margins, the transientresponse characteristics, the bandwidth, the high frequency attenuationfor sensor noise rejection and the ability to maintain the output at adesired value with minimum static error. We have seen in the previoussections that these requirements are interdependent and often conflicting.The Ideal Bode Cutoff defines a reasonable compromise. The ease withwhich it can be implemented, however, depends very much on the sys-tem properties. This section briefly reviews the most popular PD, Lead,Lag, PI, PID compensators. Before that, we define the system type whichcontrols the steady state error and tracking ability.

10.8.1 System type

The system type measures the ability of the control system to track poly-nomials. Consider a unity feedback (Fig.10.8) with the open-loop transferfunction

y(s)e(s)

= G(s) =Kn

sn

∏(1− s/zi)∏(1− s/pi)

(10.40)

The transfer function between the reference input r and the error e is

e(s)r(s)

=1

1 + G=

sn ∏(1− s/pi)

sn∏

(1− s/pi) + Kn∏

(1− s/zi)(10.41)

According to the final value theorem, the steady state error, e∞, is givenby

e∞ = lims→0

s e(s) = lims→0

sn+1

sn + Knr(s) (10.42)

The number n of integrators in the open-loop transfer function is calledthe system type. If n = 0, the system is type 0: a constant actuating signale maintains a constant output y.

e∞ = lims→0

s

1 + K0r(s)

If r(t) is a step function, r(s) = r0/s and

e∞ =r0

1 + K0, y∞ =

K0

1 + K0r0 (10.43)


The output follows a step input with a steady state error proportional to(1 + K0)−1.

If there is one integrator in G(s), n = 1 and the system is type 1: aconstant actuating signal e maintains a constant rate of change of theoutput. A step input is tracked with zero steady state error and a rampinput, r = ct, is tracked with a constant error

e∞ = lims→0

s2

s + K1

c

s2 =

c

K1(10.44)

Similarly, a system with two integrators in the open-loop transfer func-tion is type 2: a constant actuating signal e maintains a constant secondderivative of the output (structures with rigid body modes belong to thiscategory); step and ramp inputs are tracked without steady state errorand a parabolic input, r = at2, produces a constant error

e∞ = lims→0

s3

s2 + K2

2a

s3 =

2a

K2(10.45)

Kn is called static error coefficient of the system of type n. FromEqu.(10.40),

Kn = lims→0

snG(s) (10.46)

If a polynomial input tk is applied to a type n system, there is a zerosteady state error if k < n. If k = n, there is a constant steady state errordefined by the static error coefficient Kn; if k > n, the reference signalcannot be tracked.

From the foregoing discussion, we note that long term errors associatedwith persistent disturbances can be zero only if the poles of the distur-bance (one pole at s = 0 for a step, one double pole at s = 0 for a ramp,etc...) are included among the poles of the open-loop transfer functionGH(s); this is called the internal model principle.

10.8.2 Lead compensator

The aim of the cascade compensator, Fig.10.20, is to alter the open-looptransfer function from G(s) to G(s)H(s), in order to improve the charac-teristics of the closed-loop system. The simplest one is the Proportionalplus Derivative (PD):

H(s) = g(Ts + 1) (10.47)

This compensator increases the phase above the breakpoint 1/T . There-fore, selecting 1/T slightly below the crossover frequency of G(ω)H(ω)

10.8 Usual compensators 261

r e yH(s) G(s)

-

Compensator System

Fig. 10.20. Cascade compensator.

produces an increase in the phase margin. The major drawback of thePD compensator is that the compensation increases with frequency andthe high frequency attenuation rate is reduced, which is undesirable if thesensor noise is significant.

The Lead compensator eliminates this drawback by adding a pole athigher frequency

H(s) = gTs + 1αTs + 1

α < 1 (10.48)

The phase lead is still significant but the high frequency amplification isconsiderably reduced, Fig.10.21. The maximum phase lead occurs at

ω =1√αT

(10.49)

Normally, in order to maximize the benefit of the phase lead, α and Tare chosen in such a way that the crossover frequency ωc lies between thezero and the pole of the compensator.

10.8.3 PI compensator

The aim of the Proportional plus Integral (PI) compensator is to increasethe gain at low frequency to eliminate the static error. The compensatoris

H(s) =g

s(s +

1TI

) (10.50)

The system type is increased by 1. The amplification at low frequency isobtained at the expense of a phase lag of −900 below the breakpoint 1/TI .In order to minimize the phase lag at crossover, the breakpoint must besubstantially below the crossover frequency (ωcTI À 1).

10.8.4 Lag compensator

The analytical form of the Lag compensator is the same as that of theLead compensator, except that the pole and zero are placed in reverseorder


H

H H

H

f

f

f

f

1T

1T

1aT

1aT

90°

-90° -90°

90°

0°

0°

-90°

0°

0°

wcwc

wc wc

( )H s gTs

Ts( ) =

+

1

1aa < 1Lead :

Lag : PID :

PI :

g

g

g

1TI

1TI

( )H s gTs

Ts( ) =

+

1

1aa > 1

1TD

H sg

ss( ) ( )= +

H sg

ss T T

ID( ) ( ) ( )= + +1 1s

1

aT

TI

1

Fig. 10.21. Frequency response of the most usual compensators. The range of ωc

indicates the normal location of the crossover frequency of the uncompensated system.

H(s) = gTs + 1αTs + 1

α > 1 (10.51)

Normally, the pole and zero are placed at low frequency, to minimize thephase lag at crossover. The lag compensator can be used

1. to reduce the gain at high frequency without affecting that at lowfrequency, in order to reduce the crossover frequency and increase thephase margin;

10.9 Multivariable systems 263

2. to increase the gain at low frequency without affecting the high fre-quency behavior, in order to improve the steady state error character-istics.

10.8.5 PID compensator

The Proportional plus Integral plus Derivative (PID) compensator con-sists of cascaded PI and PD compensators

H(s) =g

s(s +

1TI

)(TDs + 1) (10.52)

The integral part aims at reducing the static error, while the PD partincreases the phase near crossover. Reasonable breakpoint frequencies are

20TI

∼ 1TD

< ωc (10.53)

If the increasing behavior at high frequency is undesirable, the PD partcan be replaced by a Lead compensator which achieves some phase leadwithout changing the asymptotic decay rate of G(ω)H(ω).

10.9 Multivariable systems

So far, this chapter has been devoted to SISO systems; the analysis anddesign of multi-input multi-output (MIMO) systems is much more com-plicated and a comprehensive treatment of the subject is outside the scopeof this introductory text (e.g. see Maciejowski). However, without enter-ing too much into the details, it is interesting to point out some of thesalient features of the theory which are particularly relevant in structuralcontrol, and their relation to the theory of SISO systems.

10.9.1 Performance specification

Referring to Fig.10.22.a, the governing equations of the closed-loop systemare

y = GHe + d = GH(r − y − n) + d

y = (I + GH)−1GH(r − n) + (I + GH)−1d (10.54)

which generalizes (10.11). Introducing the sensitivity matrix :

S = (I + GH)−1 (10.55)


w0w1

0dB

s( )GH

lower bounds( )GHfor

upper bounds( )GHfor

sensor noiserejection

steady state error,disturbance rejection

r e u yH G

- n

d

(a) (b)

Fig. 10.22. Feedback specification of a MIMO system. (a) Block diagram. (b) Singularvalues of the open-loop transfer matrix GH.

and the closed-loop transfer matrix :

F = (I + GH)−1GH (10.56)

Equ.(10.54) readsy = Sd + F (r − n) (10.57)

It is easy to check thatS + F = I (10.58)

which means that the disturbance d and the noise n cannot be rejectedsimultaneously in the same frequency band.

According to Equ.(10.57), the disturbance rejection is controlled bythe size of the sensitivity matrix, measured by its singular values σ(S).1

If the maximum and minimum singular values, denoted respectively byσ(S) and σ(S) are close to each other, the situation is very similar to thatof a SISO system; otherwise, what matters is the maximum singular value,σ(S), which limits the disturbance rejection capability of the system: gooddisturbance rejection requires that σ(S) be small within the bandwidth.

Similarly, good tracking requires that all the singular values of F beclose to 1, σ(F ) ' 1, σ(F ) ' 1 in the bandwidth. Finally, noise rejectionrequires that the maximum singular value σ(F ) be small at high frequency.These conditions can be translated into conditions on the singular valuesof the open-loop transfer matrix, σ(GH), Fig.10.22.b.

1 About Singular Value Decomposition (SVD), see (Strang).


e

+

+u1

G ( )s1

1

G ( )s2

+

+

u2

e2

Fig. 10.23. Small gain theorem.

10.9.2 Small gain theorem

The small gain theorem plays an important role in the analysis of robuststability of MIMO systems. Referring to Fig.10.23, if G1(s) and G2(s) arestable systems, a sufficient condition for closed-loop stability is the smallgain condition:

σ[G1(jω)G2(jω)] < 1 ∀ ω > 0 (10.59)

An alternative, more conservative condition is

σ[G1(jω)]σ[G2(jω)] < 1 ∀ ω > 0 (10.60)

10.9.3 Stability robustness tests

From the small gain theorem, it is possible to derive sufficient conditionsfor stability which have useful applications:Additive uncertaintyConsider the perturbed unit feedback system with additive uncertaintyof Fig.10.24.a. If one breaks the loop at z and v, one finds that z =−(I +G)−1v and the block diagram can be recast into that of Fig.10.24.b.

According to the small gain theorem, a sufficient condition for stabilityis that

σ(∆G). σ[(I + G)−1] < 1 ω > 0 (10.61)

+

-

(a)DG

z+

+

G

v

z v

DG

- ( + )I G-1

(b)

Fig. 10.24. (a) Additive uncertainty. (b) Seen from the break points z and v.


orσ(∆G).

1σ(I + G)

< 1

which is equivalent to

σ(∆G) < σ(I + G) ω > 0 (10.62)

Multiplicative uncertainty

z v

L

-( + )I G G-1

(c)

(b)

+

-

Lz

+

+

v

G

(a)

+

-

GI L+

Fig. 10.25. (a) and (b) Multiplicative uncertainty. (c) Seen from the break points zand v.

Consider the perturbed unit feedback system with multiplicative uncer-tainty of Fig.10.25.a., or equivalently Fig.10.25.b. Upon breaking the loopat z and v, one finds z = −(I +G)−1Gv and the block diagram can be re-drawn as in Fig.10.25.c. According to the small gain theorem, a sufficientcondition for stability is

σ(L). σ[(I + G)−1G] < 1 ω > 0 (10.63)

orσ(L).

1σ[G−1(I + G)]

< 1

which is equivalent to

σ(L) < σ(I + G−1) ω > 0 (10.64)

10.9.4 Residual dynamics

A frequent form of uncertainty encountered in structural control is theresidual dynamics: the control model includes only the quasi-static re-sponse and the primary modes, G0(s), and the residual (high frequency)


Primary

H

+ +

- +

Residual

Control

OG

RG

(a)

G G H= O

DG = G HR

+ +

- +

(b)

G

DG

G HG= O

L G G= O R-1

+

-

(c)

GI L+

Fig. 10.26. (a) System with residual dynamics. (b) Additive uncertainty. (c) Multi-plicative uncertainty.

modes are considered as uncertainty, GR(s), Fig.10.26. Because many con-trolled structures are lightly damped, the residual modes may be desta-bilized by the controller H(s); this phenomenon is known as spillover ; itwill be studied in detail in the next chapter. In this section, we use theforegoing inequalities to establish a lower bound to the stability margin.

The system of Fig.10.26 may be recast in the standard form of additiveuncertainty by taking G = G0H and ∆G = GRH. It follows that thestability robustness becomes:

σ(GRH) < σ(I + G0H) ω > 0 (10.65)

Alternatively, the system may also be recast in the standard form for mul-tiplicative uncertainty by taking G = HG0 and L = G−1

0 GR. Accordingly,the stability robustness test (10.64) becomes

σ(G−10 GR) < σ[I + (HG0)−1] ω > 0 (10.66)

Note that these tests being sufficient conditions, they are both conser-vative, with different degrees of conservatism. Typical representations ofthese tests are shown in Fig.10.27. In the inequality (10.66), the left handside does not involve the controller at all; it measures the relative sizeof the contribution of the residual dynamics in the global response as afunction of ω. The peaks in the uncertainty curves are at the frequenciesof the residual modes and their amplitude is controlled by the damping.The minimum distance between the two curves may be regarded as alower bound to the gain margin.


10

1

0.1

0.01

GM

s [ + ( ) ]I HG0-1

A

f1

structuraluncertainty

s [ ]G G0 R

-1

10

1

1

0.1

0.01 100.1

[Hz]

0.01

s [ + ( ) ]I G H0

f1

GM

A

s [ ]G HR

uncertainty

(a)

(b)

Fig. 10.27. System with residual dynamics. (a) Robustness test based on multiplicativeuncertainty. (b) Robustness test based on additive uncertainty. The minimum distancebetween the two curves may be regarded as a lower bound to the gain margin.

10.10 References

BODE,H.,W. Relations between attenuation and phase in feedback am-plifier design, Bell System Technical Journal, July 1940.BODE,H.,W. Network Analysis and Feedback Amplifier Design, Van Nos-trand, N-Y, 1945.D’AZZO, J.J., & HOUPIS, C.H. Feedback Control System Analysis &Synthesis, McGraw-Hill (Second Edition), 1966.DISTEFANO, J.J, STUBBERUD, A.R., & WILLIAMS, I.J. Feedback andControl Systems, Shaum’s Outline Series, McGraw-Hill, 1967.

10.11 Problems 269

DOYLE, J.C. & STEIN, G. Multivariable feedback design: concepts fora Classical/Modern synthesis. IEEE Transactions on Automatic Control,Vol.AC-26, No 1, 4-16, February 1981.FRANKLIN, G.F., POWELL, J.D. & EMAMI-NAEINI, A. FeedbackControl of Dynamic Systems. Addison-Wesley, 1986.HOROWITZ, I.M. Synthesis of Feedback Systems, Academic Press, 1963.KISSEL, G.J. The Bode integrals and wave front tilt control. AIAA Guid-ance, Navigation & Control Conference, Portland Oregon, 1990.KOSUT, R.L., SALZWEDEL, H. & EMAMI-NAEINI, A. Robust controlof flexible spacecraft, AIAA J. of Guidance Control and Dynamics, Vol.6,No 2, 104-111, March-April 1983.LURIE, B.J. and ENRIGHT, P.J. Classical Feedback Control, MarcelDekker, 2000.MACIEJOWSKI, J.M. Multivariable Feedback Design, Addison-Wesley,1989.SKELTON, R.E. Model error concepts in control design, Int. J. of Con-trol, Vol.49, No 5, 1725-1753, 1989.STRANG, G. Linear Algebra and its Applications, Harcourt Brace Jo-vanovich, 3rd Edition, 1988.

10.11 Problems

P.10.1 Consider an inverted pendulum with ω0 = 1 (Fig.9.2).(a) Compute the transfer function H(s) of the state feedback compensatorsuch that the regulator and the observer poles are respectively at ωn = 2,ξ = 0.5 and αωn = 6, ξ = 0.5.(b) Draw a root locus and the Nyquist diagram for this compensator.Using the Nyquist criterion, show that the closed-loop system is con-ditionally stable. Find the critical gains corresponding to the limits ofstability.P.10.2 Draw the Nichols chart plot for the two-mass problem of section6.3.2.P.10.3 Consider a direct velocity feedback applied to a structure withthree flexible modes at ω1 = 1 rad/s, ω2 = 3 rad/s and ω3 = 5 rad/s;the modal damping is uniform, ξi = 0.02. Assume that the actuator andsensor are non-collocated and that the uncertainty on the mode shapesis responsible for some uncertainty on the first pair of zeros which varyfrom z1 = ±j2.8 rad/s to z1 = ±j3.2 rad/s. The second pair of zeros arefixed to z2 = ±j4 rad/s. Plot the open-loop transfer function on a Nichols


chart, for various locations of the uncertain zeros. Comment on the effectof the zero on the stability margins.P.10.4 Consider a simply supported beam excited by a point force atxa = 0.2 l and provided with a displacement sensor at xs = 0.3 l. Thenominal system is such that l = 1m, m = 1kg/m, EI = 10.266 10−3Nm2

and the modal damping is uniform ξi = 0.01. Compute an estimate of thetransfer function G′(ω). Next, consider the following modelling errors:(a) Modal truncation: the modal expansion (2.58) is truncated after 4, 6and 10 modes.(b) Quasi-static correction: same as (a), but start from the result of Prob-lem 2.5.(c) Parametric uncertainty: the model is based on EI = 11. 10−3Nm2

and ξi = 0.02.(d) Errors in the mode shapes are simulated by using φ(xs) = φ(0.25 l)in the model.For each situation, compute the transfer function of the model, G(ω),the additive and the multiplicative uncertainty, respectively ∆G(ω) andL(ω) (gain and phase). Comment on the adequacy of the unstructureduncertainty to represent the various situations.P.10.5 Plot the weighting function W (u) appearing in the first Bodeintegral (10.28).P.10.6 Consider a SISO system with the following performance specifi-cations:

|G| > 30 dB ω < 1 rad/s

|G| < −30 dB ω > 10 rad/s

Using the first Bode integral, discuss the feasibility of the design.P.10.7 Consider a non-minimum phase system with a real zero at a = 5ωc.Following the Bode Ideal Cutoff, draw the magnitude and phase diagramsfor the corresponding minimum phase system G0(s) which produce a gainmargin of x dB and a phase margin of yπ for G(s). Do the same fora = 2ωc.P.10.8 Show that at low frequency, the nonrational function e−s can beapproximated by the following all-pass functions of increasing orders:

A1(s) =1− s/21 + s/2

A2(s) =1− s/2 + s2/121 + s/2 + s2/12

10.11 Problems 271

A3(s) =1− s/2 + s2/10− s3/1201 + s/2 + s2/10 + s3/120

Ai(s) are called the Pade approximants of e−s (Ts may be substitutedfor s to allow for any desired delay). For each case, draw the pole-zeropattern and the Bode plots. Compare the phase diagrams and commenton the domain of validity of the various approximations.[Hint: Expand the exponential and the all-pass functions into a McLaurenseries and match the coefficients.]P.10.9 Consider a stable, minimum phase system G(s) with a high fre-quency attenuation rate larger than −20 dB/decade. Using a Nyquistdiagram and geometrical arguments, show that there is always a limitfrequency above which there is an amplification of the disturbances.

11

Optimal control

11.1 Introduction

The optimal control approach for SISO systems has already been intro-duced in chapter 9. In this chapter, we extend it to Multi-Input Multi-Output (MIMO) systems. The general results are stated without demon-stration and the discussion is focused on the aspects which are importantfor the control of lightly damped flexible structures.

11.2 Quadratic integral

Consider the free response of an asymptotically stable linear system

x = Ax (11.1)

The quadratic expression xT Qx is often used to measure the distancefrom the equilibrium, x = 0. Asymptotic stability implies that x → 0 ast →∞, and that the quadratic integral

J =∫ ∞

0xT Qxdt (11.2)

converges, for any semi positive definite weighting matrix Q. Its valuedepends on the initial state x0 alone. To compute J , consider the decayrate of the positive quadratic form V (t) = xT Px, where P is the positivedefinite solution of the matrix Lyapunov equation

PA + AT P + Q = 0 (11.3)

274 11 Optimal control

−V (t) = − d

dt(xT Px) = −xT Px− xT Px

Substituting x = Ax and using Equ(11.3), one gets

− d

dt(xT Px) = −xT (PA + AT P )x = xT Qx (11.4)

Since Q is semi positive definite, this result proves that V (t) is a monoton-ically decaying function [it is indeed a Lyapunov function for the system(11.1)]. Integrating Equ.(11.4) from 0 to ∞ provides the desired result:

J =∫ ∞

0xT Qx dt = xT

0 Px0 (11.5)

where P is solution of the Lyapunov equation (11.3).

11.3 Deterministic LQR

We now formulate the steady state form of the Linear Quadratic Regulator(LQR) problem for MIMO systems. This is the simplest and the mostfrequently used formulation; it can be extended to a cost functional withfinite horizon, and to time varying systems.

Consider the systemx = Ax + Bu (11.6)

where the system matrix A is not necessarily stable, but it is assumedthat the pair (A,B) is controllable (controllability and observability willbe discussed in detail in chapter 12). We seek a linear state feedback withconstant gain

u = −Gx (11.7)

such that the following quadratic cost functional is minimized

J =∫ ∞

0(xT Qx + uT Ru)dt (11.8)

where Q is semi positive definite (Q ≥ 0) and R is strictly positive definite(R > 0). J has two contributions, one from the states x and one from thecontrol u. The fact that R is strictly positive definite expresses that anycontrol has a cost, while Q ≥ 0 implies that some of the states maybe irrelevant for the problem at hand. If a set of controlled variables isdefined,

11.3 Deterministic LQR 275

z = Hx (11.9)

a quadratic functional may be defined as

J =∫ ∞

0(zT z + uT Ru)dt =

∫ ∞

0(xT HT Hx + uT Ru)dt (11.10)

This situation is equivalent to the previous one with Q = HT H. In asimilar way, one often uses the form R = %ST S where S is a scalingmatrix and the scalar % is a design parameter.

It can be shown (see e.g. Kwakernaak & Sivan) that the solution ofthe problem (11.6) to (11.8) is

G = R−1BT P (11.11)

where P is the symmetric positive definite solution of the algebraic Riccatiequation

PA + AT P + Q− PBR−1BT P = 0 (11.12)

The Riccati equation is nonlinear in P ; the existence and uniqueness ofthe solution is guaranteed if (A,B) is a controllable pair (stabilizable isin fact sufficient) and (A,Q1/2) is observable (by Q1/2, we mean a matrixH such that HT H = Q). Under these conditions, the closed-loop system

x = (A−BG)x (11.13)

is asymptotically stable. From Equ.(11.5), the minimum value of the costfunctional (11.8) is given by

J =∫ ∞

0xT (Q + GT RG) x dt = xT

0 P ∗x0 (11.14)

where P ∗ is the solution of the Lyapunov equation of the closed-loopsystem

P ∗(A−BG) + (A−BG)T P ∗ + Q + GT RG = 0 (11.15)

Substituting G = R−1BT P , we get

P ∗A− P ∗BR−1BT P + AT P ∗ − PBR−1BT P ∗ + Q + PBR−1BT P = 0

This equation is identically satisfied by P ∗ = P , where P is solution ofEqu.(11.12). Thus, the minimum value of the cost functional is related tothe solution P of the Riccati equation by


J = min∫ ∞

0(xT Qx + uT Ru)dt = xT

0 Px0 (11.16)

Many techniques are available for solving the Riccati equation; they willnot be discussed in this text because they are automated in most controldesign softwares. In principle, the LQR approach allows the design ofmultivariable state feedbacks which are asymptotically stable. A majordrawback is, of course, that it assumes the knowledge of the full statevector x. Since the latter is, in general, not directly available, it has to bereconstructed.

The poles of the closed-loop system depend on the matrices Q andR. Multiplying both Q and R by a scalar coefficient leads to the samegain matrix G and the same closed-loop poles. In structural control ap-plications, if the controlled variables are not clearly identified, it may besensible to choose Q in such a way that xT Qx represents the total (ki-netic plus strain) energy in the system. Usually R will be chosen R = %R1,where R1 is a constant positive definite matrix and % is an adjustable pa-rameter; its value is selected to achieve reasonably fast closed-loop poleswithout excessive values of the control effort.

11.4 Stochastic response to a white noise

A white noise is a mathematical idealization of a stationary random pro-cess for which there is a total lack of correlation between the values ofthe process at different times. If w is a vector white noise process of zeromean

E[w(t)] = 0,

the covariance matrix reads

E[w(t1)wT (t2)] = Rw(t1 − t2) = W1δ(t1 − t2) = W1δ(τ) (11.17)

where δ(τ) is the Dirac function, τ = t1 − t2 is the delay separating thetwo times t1 and t2, and W1 is symmetric semi positive definite (W1 ≥ 0).W1 is called the covariance intensity matrix, but this is often abbreviatedto covariance matrix. If the components of the random vector are inde-pendent, W1 is diagonal. The corresponding power spectral density matrixis

ΦW (ω) =12π

∫ ∞

−∞Rw(τ)e−jωτ dτ =

12π

W1 (11.18)

This result shows that the power in the signal is uniformly distributed overthe frequency. Although a white noise is not physically realizable [because

11.4 Stochastic response to a white noise 277

its variance would not be finite, see the Dirac function in Equ.(11.17)], it isa convenient approximation which is appropriate whenever the correlationtime of the signal is small compared to the time constant of the systemto which it is applied. In what follows, it is assumed that all processesare Gaussian, so that they are entirely characterized by their second orderstatistics (i.e. the covariance matrix or the power spectral density matrix).

Consider the stationary random response of a linear time invariantsystem to a white noise excitation w of covariance intensity matrix W1.

x = Ax + Dw (11.19)

It can be shown that the covariance matrix of the steady state response,

X = E[x(t)xT (t)] (11.20)

satisfies the Lyapunov equation

AX + XAT + DW1DT = 0 (11.21)

(see e.g. Bryson & Ho). This equation is the dual (adjoint) of Equ.(11.3);it expresses the equilibrium between the damping forces in the systemand the random disturbance acting on it. It has a unique positive definitesolution if the system is stable.

Now, consider the quadratic performance index J = E[xT Qx]. Since

aT b = tr(abT )

we can write

J = E[xT Qx] = Etr[QxxT ] = tr[QExxT ] = tr[QX] (11.22)

where we have used the fact that the expected value applies only to ran-dom quantities. The covariance matrix X is the solution of Equ.(11.21).This result can be written alternatively

J = tr[PDW1DT ] (11.23)

where P is solution of Equ.(11.3); the proof is left as an exercise (Problem11.1).


11.4.1 Remark

The notation used in this section is the most commonly used in the liter-ature; however, for the reader who is not trained in stochastic processes,it deserves some clarification: The definition (11.17) of the covariance ma-trix involves two separate times t1 and t2; for a stationary process, thecovariance matrix is a function of the difference (delay) τ = t1 − t2. Onthe contrary, the covariance matrix involved in Equ.(11.20) and (11.21)is not a function of τ ; X is the particular value of the covariance functionfor τ = 0; it is the generalization of the mean square value for a vectorprocess.

11.5 Stochastic LQR

The linear quadratic regulator can be formulated in a stochastic environ-ment as follows. Consider the linear time invariant system subjected to awhite noise excitation of intensity W1:

x = Ax + Bu + w1, E[w1wT1 ] = W1 (11.24)

Find the control u, function of all information from the past, that mini-mizes the performance index

J = E[xT Qx + uT Ru] (11.25)

The solution of the problem is a constant gain linear state feedback

u = −Gx (11.26)

G = R−1BT P (11.27)

where P is solution of the Riccati equation

PA + AT P + Q− PBR−1BT P = 0 (11.28)

We note that the solution does not depend on the noise intensity matrixW1 and that it is identical to that of the deterministic LQR using the sameweighting matrices Q and R. The closed-loop system behaves accordingto

x = (A−BG)x + w1 (11.29)

and the performance index (11.25) can be rewritten

11.6 Asymptotic behavior of the closed-loop 279

J = E[xT (Q + GT RG)x] (11.30)

From Equ.(11.23),J = tr[P ∗W1]

where P ∗ is solution of the Lyapunov equation of the closed-loop system

P ∗(A−BG) + (A−BG)T P ∗ + Q + GT RG = 0 (11.31)

This equation is identical to (11.15). Substituting the gain matrix from(11.27), we can readily establish that it is identically satisfied by the solu-tion P of the Riccati equation (11.28). Thus, P ∗ = P , and the minimumvalue of the performance index is

J = min E[xT Qx + uT Ru] = tr[PW1] (11.32)

This result is the dual of (11.16) for the deterministic case.

11.6 Asymptotic behavior of the closed-loop

In the particular case where there is a single controlled variable z and asingle input u, the cost functional (11.10) is reduced to (9.52):

J =∫ ∞

0(z2 + %u2)dt (11.33)

We know from section 9.5 that the closed-loop poles are the stable rootsof the characteristic equation

1 +1%G0(s)G0(−s) = 0 (11.34)

whereG0(s) = H(sI −A)−1B (11.35)

is the transfer function between the input u and the controlled variablez = Hx. Only the part in the left half plane of the symmetric root locusis of interest. If

G0(s) = k


Equ.(11.34) becomes

n∏

i=1

(s− pi)(−s− pi) +k2

%

m∏

i=1

(s− zi)(−s− zi) = 0 (11.36)


For large values of %, that is when there is a heavy penalty on the controlcost, the closed-loop poles are identical to the stable poles pi of G0(s),or their mirror image in the left half plane (−pi) if they are unstable.Thus, the cheapest stabilizing control in terms of control amplitude simplyrelocates the unstable poles at their mirror image in the left half plane.

45°22°30'

45°

60° 36°

n=5n=4n=3n=2n=1

Fig. 11.1. Butterworth patterns of increasing order n.

As the penalty % on the control cost decreases, m branches go from thepoles to the left half plane zeros zi, if the system is minimum phase, orto their mirror image, if the zeros are located in the right half plane. Theremaining n−m branches go to infinity. Their asymptotic behavior can beevaluated by ignoring all but the highest power in s in the characteristicequation (11.34). From (11.36), we get

1 +k2

%

sm(−s)m

sn(−s)n= 0

or

s2(n−m) =k2

%(−1)n−m+1 (11.37)

The n − m stable solutions lie on a circle of radius (k2/%)1/2(n−m) in aconfiguration known as a Butterworth pattern of order n−m (Fig.11.1).

For small values of %, the fact that the system is very fast is reflectedby the large distance of the faraway poles to the origin. Some of the poles,however, do not move away but are stuck at the open-loop zeros. Althoughthey remain slow, they will not be noticeable on the controlled variable,because the open-loop zeros are also zeros of the closed-loop system. Ex-tensions to MIMO systems with square open-loop transfer matrix can befound in Kwakernaak & Sivan.

11.7 Prescribed degree of stability 281

11.7 Prescribed degree of stability

In the previous section, we have seen that as % → 0, some of the closed-loop poles go to infinity according to Butterworth patterns, while theothers go to the open-loop zeros. The latter depend on the choice ofthe matrix H defining the controlled variables z. In practice, a singlecontrolled variable cannot always be clearly identified, and the choice of His often more or less arbitrary: The designer picks up a H matrix and findsa root locus, on which reasonable pole locations can be selected. If one failsto find fast enough poles with acceptable control amplitudes, the processis repeated with another H matrix. It may require several iterations toavoid nearby poles. Anderson and Moore’s α−shift procedure allows oneto design a regulator with a prescribed degree of stability. The methodguarantees that the closed-loop eigenvalues (of A−BG) lie on the left ofa vertical line at −α (Fig.11.2).

− α

Fig. 11.2. Prescribed degree of stability.

The idea is to use a modified cost functional

J =∫ ∞

0e2αt(xT Qx + uT Ru)dt (11.38)

where α is a positive constant. If one defines the modified variables x =eαtx and u = eαtu, the system equation in terms of x and u becomes

˙x = (A + αI)x + Bu (11.39)

and the performance index is

J =∫ ∞

0(xT Qx + uT Ru)dt (11.40)

Equations (11.39) and (11.40) define a classical LQR problem in terms ofx and u; the solution is a constant gain feedback


u = −Gx (11.41)

withG = R−1BT P (11.42)

where P is solution of the Riccati equation

P (A + αI) + (AT + αI)P + Q− PBR−1BT P = 0 (11.43)

The closed-loop system

˙x = (A + αI −BR−1BT P )x

is asymptotically stable. If one back substitutes x and u in Equ.(11.41),one gets

u = −Gx (11.44)

Thus, the optimal control is a constant gain linear feedback. Besides,since x(t) = e−αtx(t), and x(t) is asymptotically stable, we know that ast approaches infinity, x(t) → 0 at least as fast as e−αt. The prescribeddegree of stability has been achieved by solving a classical LQR problemfor the modified system with the system matrix A + αI instead of A.

11.8 Gain and phase margins of the LQR

The LQR has been developed as the solution of an optimization problem.The constant gain feedback assumes the perfect knowledge of the state.We momentarily stay with this assumption and postpone the discussionof the state reconstruction until the next section. In the previous chapter,we emphasized the role played by the gain and phase margins in the goodperformance of a control system and in the way the performance will be

xuC G

G

&x A x B u= +- -

(a) (b)

( )sI A B− −1

Fig. 11.3. (a) Classical representation of the state feedback. (b) Unity feedback rep-resentation.

11.8 Gain and phase margins of the LQR 283

degraded if some change occurs in the open-loop system. A good phasemargin improves the disturbance rejection near crossover, and eliminatesthe oscillations in the closed-loop system; it also protects against insta-bility if some delay is introduced in the loop. This section establishesguaranteed margins for the LQR.

Figure 11.3.a shows the classical representation of the state feedback; itis not directly suitable for defining the gain and phase margins. However,the feedback loop can also be represented with a unity feedback as inFig.11.3.b; the corresponding open-loop transfer function is

G0(s) = G(sI −A)−1B (11.45)

It is possible to draw a Nyquist plot of G0(jω), and this allows us todefine the gain and phase margins as in the previous chapter.

The interesting result is that it can be demonstrated (see Anderson &Moore, p.68) that the LQR satisfies the following inequality.

|1 + G(jω −A)−1B| ≥ 1 (11.46)

This implies that the Nyquist plot of G0(jω) always remains outside a diskof unit radius, centered on −1 (Fig.11.4). Simple geometric considerations

-g e

j− φ( )G j I A Bω − −1

Forbiddencircle Unit circle

-1 -1

60°(a)

(1)

(2)

(b)

A

A

Fig. 11.4. Possible Nyquist plots of G0(jω). (a) Stable open-loop transfer function.(b) Open-loop transfer function with two unstable poles.


show that the LQR has a phase margin PM > 600, and a infinite gainmargin. The first result comes from the observation that any open-looptransfer function with a phase margin less than 600 would have to cross theunit circle within the forbidden disk. The second one is the consequence ofthe fact that the number of encirclements of −1 cannot be changed if oneincreases the gain by any factor larger than 1. Note that, for curve (2) inFig.11.4.b, point A could cross −1 if the gain were reduced. However, sinceA is outside the forbidden circle, this cannot occur if the multiplying factorremains larger than 1/2. This result can be extended to multivariablesystems: For each control channel, the LQR has guaranteed margins of

GM = 1/2 to ∞ and PM > 600

For the situation depicted in Fig.11.4.a, the phase is close to 900 abovecrossover; as a result, the gain roll-off rate at high frequency is at most−20 dB/decade.

11.9 Full state observer

The state feedback assumes that the states are known at all times. In mostpractical situations, a direct measurement of all the states would not befeasible. As we already saw earlier, if the system is observable, the statescan be reconstructed from a model of the system and the output measure-ment y. One should never forget, however, that good state reconstructionrequires a good model of the system. If the state feedback is based on thereconstructed states, the separation principle tells us that the design ofthe regulator and that of the observer can be done independently.

Consider the system

x = Ax + Bu + w (11.47)

y = Cx + v (11.48)

where v and w are uncorrelated white noise processes with zero mean andcovariance intensity matrices V and W , respectively. From chapter 9, weknow that without noise, a full state observer of the form

˙x = Ax + Bu + K(y − Cx) (11.49)

x(0) = 0

11.10 Kalman-Bucy Filter (KBF) 285

converges to the actual states, provided that the eigenvalues of (A −KC) are in the left half plane. In fact, the poles of the observer canbe assigned arbitrarily if the system is observable. In presence of plantand measurement noise, w and v, the error equation is the following

e = x− x

e = (A−KC)e + w −Kv (11.50)

It shows that both the system noise w and the measurement noise v act asexcitations on the measurement error. Note that the measurement noiseis amplified by the gain matrix of the observer, which suggests that noisymeasurements will require moderate gains in the observer.

11.9.1 Covariance of the reconstruction error

Comparing Equ.(11.50) with Equ.(11.19) to (11.21), we see that thesteady state reconstruction error has zero mean, and a covariance ma-trix P = E[eeT ] given by the Lyapunov equation:

(A−KC)P + P (A−KC)T + W + KV KT = 0 (11.51)

where we have used the fact that v and w are uncorrelated. It can berewritten

AP + PAT + W − (KCP + PCT KT ) + KV KT = 0 (11.52)

This equation expresses the equilibrium between (as they appear in theequation) the dissipation in the system, the covariance of the disturbanceacting on the system, the reduction of the error covariance due to the useof the measurement, and the measurement noise. The latter two contri-butions depend of the gain matrix K of the observer.

11.10 Kalman-Bucy Filter (KBF)

Since the error covariance matrix depends on the gain matrix K of theobserver, one may look for the optimal choice of K which minimizes aquadratic objective function

J = E[(aT e)2] = aT E[eeT ]a = aT Pa (11.53)


where a is a vector of arbitrary coefficients. There is a universal choice ofK which makes J minimum for all a:

K = PCT V −1 (11.54)

where P is the covariance matrix of the optimal observer, solution of theRiccati equation

AP + PAT + W − PCT V −1CP = 0 (11.55)

This solution has been obtained as a parametric optimization problem forthe assumed structure of the observer given by Equ.(11.49), but it is infact optimal. The minimum variance observer is also called steady stateKalman-Bucy filter (KBF).

11.11 Linear Quadratic Gaussian (LQG)

The so-called Linear Quadratic Gaussian problem is formulated as follows:Consider the completely controllable and observable linear time-invariantsystem

x = Ax + Bu + w (11.56)

y = Cx + v (11.57)

where w and v are uncorrelated white noise processes with intensity ma-trices W ≥ 0 and V > 0. Find the control u such that the cost functional

J = E[xT Qx + uT Ru] Q ≥ 0, R > 0 (11.58)

is minimized.The solution of this problem is a linear, constant gain feedback

u = −Gx (11.59)

where G is the solution of the LQR problem and x is the reconstructedstate obtained from the Kalman-Bucy filter. Combining Equ.(11.56),(11.59) and (11.50), we obtain the closed-loop equation

(xe

)=

(A−BG BG

0 A−KC

) (xe

)+

(I 0I −K

) (wv

)(11.60)

11.12 Duality 287

Its block triangular form implies the separation principle: The eigenvaluesof the closed-loop system consist of two decoupled sets, correspondingto the regulator and the observer. Note that the separation principle isrelated to the state feedback and the structure of the observer, ratherthan to the optimality; it applies to any state feedback and any full stateobserver of the form (11.49).

The compensator equation has exactly the same form as that of Fig.9.9,except that it is no longer restricted to SISO systems. As we alreadymentioned, the stability of the compensator is not guaranteed, becauseonly the closed-loop poles have been considered in the design.

11.12 Duality

Although no obvious relationship exists between the physical problemsof optimal state feedback with a quadratic performance index, and theminimum variance state observer, the algebras of the solution of the twoproblems are closely related, as summarized in Table 11.1.

The duality between the design of the regulator and that of the Kalmanfilter can be expressed as follows. Consider the fictitious dual controlproblem: Find u that minimizes the performance index

J = E[zT Wz + uT V u]

for the systemz = AT z + CT u

The solution isu = −Gz

withG = V −1CP

where P is solution of

PAT + AP + W − PCT V −1CP = 0

It is readily observed that this Riccati equation is that of the Kalmanfilter of the original problem, and that the gain matrix of the minimumvariance observer for the original problem is related to the solution of thefictitious regulator problem by

K = GT


LQR:Gain: G = R−1BT PRiccati: PA + AT P − PBR−1BT P + Q = 0Closed loop: x = (A−BR−1BT P )x

KBF:Gain: K = PCT V −1

Riccati: AP + PAT − PCT V −1CP + W = 0Error equation: e = (A− PCT V −1C)e

Table 11.1. Duality between LQR and KBF.

11.13 Spillover

Flexible structures are distributed parameter systems which, in principle,have an infinite number of degrees of freedom. In practice, they are dis-cretized by a finite number of coordinates (e.g. finite elements) and thisis in general quite sufficient to account for the low frequency dynamicalbehavior in most practical situations. When it comes to control flexiblestructures with state feedback and full state observer, the designer can-not deal directly with the finite element model, which is by far too big.Instead, a reduced model must be developed, which includes the few dom-inant low frequency modes. Due to the inherent low damping of flexiblestructures, particularly in the space environment, there is a danger that astate feedback based on a reduced model destabilizes the residual modes,which are not included in the model of the structure contained in theobserver. The aim of this section is to point out the danger of spilloverinstability. It is assumed that the state variables are the modal amplitudesand the modal velocities (as in section 9.2.2). In what follows, the sub-script c refers to the controlled modes, which are included in the controlmodel, and the subscript r refers to the residual modes which are ignoredin the control design. Although they are not included in the state feed-back, the residual modes are excited by the control input and they alsocontribute to the output measurement (Fig.11.5); it is this closed-loopinteraction, together with the low damping of the residual modes, whichis the origin of the problem. With the foregoing notation, the dynamicsof the open-loop system is

xc = Acxc + Bcu + w (11.61)

11.13 Spillover 289

xr = Arxr + Bru (11.62)

y = Ccxc + Crxr + v (11.63)

A perfect knowledge of the controlled modes is assumed. The full stateobserver is

˙xc = Acxc + Bcu + Kc(y − Ccxc) (11.64)

and the state feedbacku = −Gcxc (11.65)

The interaction between the control system and the residual modes canbe analyzed by considering the composite system formed by the statevariables (xT

c , eTc , xT

r )T , where ec = xc − xc. The governing equation is

xc

ec

xr

=

Ac −BcGc BcGc 00 Ac −KcCc −KcCr

−BrGc BrGc Ar

xc

ec

xr

(11.66)

This equation is the starting point for the analysis of the spillover

Linear observerRegulator

Controlled ModesActuators Sensors

Flexible SystemDynamics

Residual Modes

Bc

Br

Cc

Cr

Ac

Ar

u G xc c= − $

xc

xr

u

u

y

y$xc

+

( )$& $ $x A x B u K y C xc c c c c c c= + + −

Fig. 11.5. Spillover mechanism.


(Fig.11.5). The key terms are KcCr and BrGc. They arise from thesensor output being contaminated by the residual modes via the termCrxr (observation spillover), and the feedback control exciting the resid-ual modes via the term Bru (control spillover). Equ.(11.66) shows thatif either Cr = 0 or Br = 0, the eigenvalues of the system remain decou-pled, that is identical to those of the regulator (Ac −BcGc), the observer(Ac − KcCc) and the residual modes Ar. They are typically located inthe complex plane as indicated in Fig.11.6. The poles of the regulator(controlled modes) have a substantial stability margin, and the poles ofthe observer are located even farther left. On the contrary, the poles cor-responding to the residual modes are barely stable, their only stabilitymargin being provided by the natural damping.

When both Cr 6= 0 and Br 6= 0, i.e. when there is both control andobservation spillover, the eigenvalues of the system shift away from theirdecoupled locations. The magnitude of the shift depends on the couplingterms BrGc and KcCr. Since the stability margin of the residual modes issmall, even a small shift can make them unstable. This is spillover insta-bility. Not all the residual modes are potentially critical from the point ofview of spillover, but only those which are observable, controllable, andare close to the bandwidth of the controller Problem 11.6).

Re(s)

Im(s)

Residual modes

Stability margin

Controlledmodes

Observerpoles

Fig. 11.6. Typical location of the closed-loop poles in the complex plane, showing thesmall stability margin of the residual modes (only the upper half is shown).

11.13 Spillover 291

11.13.1 Spillover reduction

In the previous section, we have seen that the spillover phenomenon arisesfrom the excitation of the residual modes by the control (control spillover,Bru) and the contamination of the sensor output by the residual modes(observation spillover, Crxr). For MIMO systems, both terms can be re-duced by a judicious design of the regulator and the Kalman filter.

Control spillover can be alleviated by minimizing the amount of energyfed into the residual modes. This can be achieved by supplementing thecost functional used in the regulator design by a quadratic term in thecontrol spillover:

J =∫ ∞

0(xT

c Qxc + uT Ru)dt +∫ ∞

0uT BT

r WBru dt (11.67)

where the weighting matrix W allows us to penalize some specific modes.This amounts to using the modified control weighting matrix

R + BTr WBr (11.68)

This control weighting matrix penalizes the excitation u whose shape isfavorable to the residual modes; this tends to produce a control which isorthogonal to the residual modes. Of course, this is achieved more effec-tively when there are many actuators, and it cannot be achieved at allwith a single actuator.

Similarly, a reduction of the observation spillover can be achieved bydesigning the observer as a Kalman filter with a measurement noise in-tensity matrix

V + CrV1CTr (11.69)

The extra contribution to the covariance intensity matrix indicates tothe filter that the measurement noise has the spatial shape of the residualmodes (CrV1C

Tr is the covariance matrix of the observation spillover Crxr

if E[xrxTr ] = V1). This tends to desensitize the reconstructed states to the

residual modes. Here again, the procedure works better if many sensorsare available.

The foregoing methodology for spillover reduction was introducedby Sezak et al. under the name of Model Error Sensitivity Suppression(MESS). It is only one of the many methods for spillover reduction, butit is interesting because it stresses the role of the matrices R and V in theLQG design. Another interesting situation where the plant noise statis-tics have a direct impact on the stability margins is discussed in the nextsection.


11.14 Loop Transfer Recovery (LTR)

In section 11.8, we have seen that the LQR has guaranteed stability mar-gins of GM = 1/2 to ∞ and PM > 600 for each control channel. Thisproperty is lost when the state feedback is based on an observer or aKalman filter. In that case, the margins can become substantially smaller.

The Loop Transfer Recovery (LTR) is a robustness improvement pro-cedure consisting of using a Kalman filter with fictitious noise parameters:If W0 is the nominal plant noise intensity matrix, the KBF is designedwith the following plant noise intensity matrix

W (q) = W0 + q2BW1BT (11.70)

where W1 is an arbitrary symmetric semi positive definite matrix and q isa scalar adjustment parameter. From the presence of the input matrix Bin the second term of (11.70), we see that the extra plant noise is assumedto enter the system at the input. Of course, for q = 0, the resulting KBFis the nominal one. As q → ∞, it can be proved (Doyle & Stein) that,for square, minimum phase open-loop systems G(s), the loop transferfunction H(s)G(s) from the control signal u′ to the compensator outputu (loop broken at the input of the plant, as indicated in Fig.11.7) tendsto that of the LQR:

limq→∞G(sI−A+BG+KC)−1K C(sI−A)−1B = G(sI−A)−1B (11.71)

u'u y

-

-

(a)

(b)

( )G sI A B− −1

( )G sI A BG KC K− + + −1

LQR

LQG

H s( ) G s( )

( )C sI A B− −1

Fig. 11.7. Loop transfer functions of the LQG and the LQR with loop breaking at theinput of the plant.

11.15 Integral control with state feedback 293

As a result, the LQG/LTR recovers asymptotically the margins of theLQR as q →∞.

Note that

• The loop breaking point at the input of the plant, as indicated by ×in Fig.11.7, is a reasonable one, because this is typically one of thelocations where the uncertainty enters the system.

• The KBF gain matrix, K(q) is a function of the scalar parameter q.For q = 0, K(0) is the optimal filter for the true noise parameters. As qincreases, the filter does a poorer job of noise rejection, but the stabilitymargins are improved, with essentially no change in the bandwidth ofthe closed-loop system. Thus, the designer can select q by trading offnoise rejection and stability margins.

• The margins of the LQG/LTR are indeed substantial; they providea good protection against delays and nonlinearities in the actuators.They are not sufficient to guarantee against spillover instability, how-ever, because the phase uncertainty associated with a residual modeoften exceeds 600 (it may reach 1800 if the residual mode belongs tothe bandwidth).

• The LTR procedure is normally applied numerically by solving a set ofRiccati equations for increasing values of q2, until the right compromiseis achieved. For SISO systems, it can also be applied graphically on asymmetric root locus, by assuming that the noise enters the plant atthe input [E = E0 + qB in Equ.(9.72)] (Problem 11.7).

11.15 Integral control with state feedback

Consider a linear time invariant system subject to a constant disturbancew:

x = Ax + Bu + w (11.72)

y = Cx (11.73)

If we use a state feedback u = −Gx to stabilize the system, there willalways be a non-zero steady state error in the output y. Increasing thegain G would reduce the error at the expense of a wider bandwidth anda larger noise sensitivity.

An alternative approach consists of introducing an integral action bysupplementing Equ.(11.72) by

p = y (11.74)


leading to the augmented state vector (xT , pT )T .With the state feedback

u = −Gx−Gpp (11.75)

the closed-loop equation reads(

xp

)=

(A−BG −BGp

C 0

) (xp

)+

(w0

)(11.76)

If G and Gp are chosen in such a way that they stabilize the system, wehave

limt→∞ p = 0 (11.77)

which means that the steady state error will be zero (y∞ = 0), withoutknowledge of the disturbance w.

11.16 Frequency shaping

As we saw in earlier chapters, the desirable features of control systemsinclude some integral action at low frequency to compensate for steadystate errors and very low frequency disturbances, and enough roll-off athigh frequency for noise rejection, and to stabilize the residual dynamics.Moreover, there are special situations where the system is subjected toa narrow-band disturbance at a known frequency. The standard LQGdoes not give the proper answer to these problems (no integral action,and the roll-off rate of the LQR is only −20 dB/decade). We have seen inthe previous section how the state space model can be modified to includesome integral action; in this section, we address the more general questionof frequency shaping.

The weakness of the standard LQG formulation lies in the use of afrequency independent cost functional, and of noise statistics with uniformspectral distribution (white noise). Frequency shaping can be achievedeither by considering a frequency dependent cost functional in the LQRformulation, or by using colored (i.e. non-white) noise statistics in theLQG problem.

11.16.1 Frequency-shaped cost functionals

According to Parseval’s theorem, the cost functional (11.10) of the LQRcan be written in the frequency domain as

11.16 Frequency shaping 295

J =12π

∫ ∞

−∞[x∗(ω)HT Hx(ω) + u∗(ω)Ru(ω)]dω (11.78)

where x(ω) and u(ω) are the Fourier transforms of x and u, and ∗ indicatethe complex conjugate transposed (Hermitian). Equ.(11.78) shows clearlythat the weighing matrices Q = HT H and R = %ST S do not depend onω, meaning that all the frequency components are treated equally.

Next, assume that we select frequency dependent weighing matrices

Q(ω) = H∗(ω)H(ω) and R(ω) = %S∗(ω)S(ω) (11.79)

Clearly, if the shaping objectives are to produce a P+I type of controllerand to increase the roll-off, we must select Q(ω) to put more weight onlow frequency, to achieve some integral action, and R(ω) to put moreweight on high frequency, to attenuate the high frequency contribution ofthe control. Examples of such functions in the scalar case are

Q(ω) =ω2

0 + ω2

ω2R(ω) =

ω2n1 + ω2n

ω2n1

(11.80)

where the corner frequencies ω0 and ω1 and the exponent n are selectedin the appropriate manner. Typical penalty functions are represented inFig.11.8. Likewise, a narrow-band disturbance can be handled by includ-ing a lightly damped oscillator at the appropriate frequency in Q(ω).

Equation (11.78) can be rewritten

( )Q jω

( )R jω

dB

ω (rad / s)

Fig. 11.8. Frequency dependent weighting matrices.


J =12π

∫ ∞

−∞[x∗(ω)H∗(ω)H(ω)x(ω) + %u∗(ω)S∗(ω)S(ω)u(ω)]dω (11.81)

We assume that all the input and output channels are filtered in the sameway, so that the weighing matrices are restricted to the form H(ω) =h(ω)H and S(ω) = s(ω)S, with h(ω) and s(ω) being scalar functions. Ifwe introduce the modified controlled variable

z1 = H(ω)x = h(ω)Hx = h(ω)z (11.82)

and controlu1 = s(ω)u (11.83)

we get the frequency independent cost functional

J =12π

∫ ∞

−∞[z∗1z1 + u∗1Ru1]dω (11.84)

or, in the time domain,

J =∫ ∞

0(zT

1 z1 + uT1 Ru1)dt (11.85)

This cost functional refers to the augmented system of Fig.11.9, includinginput filters s−1(ω) on all input channels and output filters h(ω) on allcontrolled variables. If a state space realization of these filters is avail-able (Problem 11.9), the complete system is governed by the followingequations:

• Structurex = Ax + Bu (11.86)

y = Cx + Du (11.87)

A B

C

i i

i

A B

C D

A B

C D

O O

O O

Input filter Structure Output filter

( )s−1 ω ( )h ω

u1 u z z1

Fig. 11.9. State space realization of the augmented system including frequency shap-ing.

11.16 Frequency shaping 297

• Output filter [state space realization of h(ω)]

x0 = A0x0 + B0z (11.88)

z1 = C0x0 + D0z (11.89)

• Input filter [state space realization of s−1(ω)]

xi = Aixi + Biu1 (11.90)

u = Cixi (11.91)

These equations can be combined together as

x? = A?x? + B?u1 (11.92)

z1 = C?x? (11.93)

with the augmented state vector

x? = (xT , xTi , xT

0 )T

and the notations

A? =

A BCi 00 Ai 0

B0C B0DCi A0

(11.94)

B? =

0Bi

0

(11.95)

C? = (D0C , D0DCi , C0) (11.96)

The state feedback −Gcx? is obtained by solving the LQR problem for the

augmented system with the quadratic performance index (11.85). Noticethat, since the input and output filter equations are solved in the com-puter, the states xi and x0 are known; only the states x of the structuremust be reconstructed with an observer. The overall architecture of thecontroller in shown in Fig.11.10. It can be shown that the poles of theoutput filter (eigenvalues of A0) appear unchanged in the compensator(Problem 11.11); this property can be used to introduce a large gain overa narrow frequency range, by introducing a lightly damped pole in A0

(Problem 11.10).


A B

C

i i

i

A B

C D

O O

O O

Input filter

Structure

Observer

Output filter

u1x0

u

xi

-Gc

LQR

$x

y z=

x*

Fig. 11.10. Architecture of the frequency-shaped LQG controller (y = z).

11.16.2 Noise model

As an alternative to the frequency-shaped cost functionals, loop shapingcan be achieved by assuming that the plant noise w has an appropriatepower spectral density, instead of being a white noise. Thus, we assumethat w is the output of a filter excited by a white noise at the input. Ifthe system is governed by

x = Ax + Bu + Ew (11.97)

y = Cx + Du + v (11.98)

and the plant noise is modelled according to

z = Awz + Bww∗ (11.99)

w = Cwz (11.100)

where Aw is stable and w∗ is a white noise (Problem 11.12), the two setsof equations can be coupled together to form the augmented system

(xz

)=

(A ECw

0 Aw

) (xz

)+

(B0

)u +

(0

Bw

)w∗ (11.101)

y =(

C 0) (

xz

)+ Du + v (11.102)

11.17 References 299

or, with x∗ = (xT , zT )T and the appropriate definitions of A∗, B∗, C∗ andE∗,

x∗ = A∗x∗ + B∗u + E∗w∗ (11.103)

y = C∗x∗ + Du + v (11.104)

Since w∗ and v are white noise processes, the augmented system fits intothe LQG framework and a full state feedback and a full state observercan be constructed by solving the two problems

LQRA∗, B∗, Q∗, R∗

KBFA∗, C∗,W = E∗E∗T , V with the appropriate matrices Q∗, R∗ and V .

In Equ.(11.101), note that the filter dynamics is not controllable fromthe plant input, but this is not a problem provided that Aw is stable, thatis if the augmented system is stabilizable (see next chapter). In principle,a large gain over some frequency range can be obtained by proper selec-tion of the poles of Aw and the input and output matrices Bw and Cw.However, in contrast to the previous section, the poles of Aw do not ap-pear unchanged in the compensator (Problem 11.13) and this techniquemay lead to difficulties for the rejection of narrow-band perturbations(Problem 11.14).

11.17 References

ANDERSON, B.D.O. & MOORE, J.B. Linear Optimal Control, PrenticeHall, Inc. Englewood Cliffs, NJ, 1971.ASTROM, K.J. Introduction to Stochastic Control Theory, AcademicPress, 1970.ATHANS, M. The role and use of the stochastic Linear-Quadratic-Gaussian problem in control system design. IEEE Transactions on Auto-matic Control, Vol.AC-16, No 6, 529-552, December 1971.BALAS, M.J. Active control of flexible systems. Journal of OptimizationTheory and Applications, Vol.25, No 3, 415-436, 1978.DOYLE, J.C. & STEIN, G. Robustness with Observers, IEEE Transac-tions on Automatic Control, Vol.AC-24, No 4, 607-611, August 1979.DOYLE, J.C. & STEIN, G. Multivariable feedback design: concepts fora Classical/Modern synthesis. IEEE Transactions on Automatic Control,Vol.AC-26, No 1, 4-16, February 1981.


GUPTA, N.K. Frequency-shaped cost functionals: extension of linearquadratic Gaussian methods. AIAA J. of Guidance, Control and Dynam-ics, Vol.3, No 6, 529-535, Nov.-Dec. 1980.KWAKERNAAK, H. & SIVAN, R. Linear Optimal Control Systems, Wi-ley, 1972.MACIEJOWSKI, J.M. Multivariable Feedback Design, Addison-Wesley,1989.SEZAK, J.R. , LIKINS, P. & CORADETTI, T. Flexible spacecraft controlby model error sensitivity suppression. Proceedings of the VPI&SU/AIAASymposium on Dynamics & Control of Large Flexible Spacecrafts, Blacks-burg, VA, 1979.WIBERG, D.M. State Space and Linear Systems, McGraw-Hill Schaum’sOutline Series in Engineering, 1971.

11.18 Problems

P.11.1 Consider the linear system (11.19) subjected to a white noiseexcitation with covariance intensity matrix W1. Show that the quadraticperformance index J = E[xT Qx] can be written alternatively

J = tr[PDW1DT ]

where P is the solution of the Lyapunov equation (11.3).P.11.2 Consider the inverted pendulum of section 9.2.3. Using the abso-lute displacement as control variable, design a LQR by solving the Riccatiequation, for various values of the control weight %. Compare the resultto that obtained in section 9.5.2 with the symmetric root locus.P.11.3 Same as Problem 11.2 but with the α−shift procedure of section11.7. Check that for all values of %, the closed-loop poles lie to the left ofthe vertical line at −α (select −α to the left of −ω0). Compare the statefeedback gains to those of the previous problem.P.11.4 For one of the LQR designed at Problem 11.2, draw the Nyquistplot of G0(ω) = G(jωI −A)−1B. Evaluate the gain and phase margins.P.11.5 Consider the state space equation (9.14) of a flexible structure inmodal coordinates and assume that the mode shapes have been normal-ized in such a way that µi = 1. Show that the total energy (kinetic +strain) can be written in the form

T + U = zT Qz with Q =12I

11.18 Problems 301

where z is the state vector defined as z = (ηT Ω, ηT )T .P.11.6 Consider a simply supported uniform beam with a point forceactuator at x = l/6 and a displacement sensor at 5l/6. Assume that thesystem is undamped and that EI = 1Nm2, m = 1kg/m, and l = 1m.(a) Write the equations in state variable form using the state variable zdefined as z = (ηT Ω, ηT )T .(b) Design a LQR for a model truncated after the first three modes,using Q = I (see Problem 11.5); select the control weight in such a waythat the closed-loop poles are (−0.788 ± j9.87), (−1.37 ± j39.48), and(−1.58± j88.83).(c) Check that a full state Luenberger observer with poles located at

−175.39,−20.92,−24.40± j50.87,−7.3± j9.34

shifts the residual mode from p4 = (0± j157.9) to p∗4 = (+0.177± j157.5)(this example was used by Balas to demonstrate the spillover phe-nomenon).(d) Using a model with 3 modes and assuming that the plant noise inten-sity matrix has the form W = wI, design a KBF and plot the evolutionof the residual modes 4 and 5 (in closed-loop) as the noise intensity ratioq = w/v increases (and the observer becomes faster).(e) For the compensator designed in (d), assuming that all the modes havea structural damping of ξi = 0.001, plot the evolution with the parameterq of the open-loop transfer function G5H3 corresponding to 5 structuralmodes (including 2 residual modes).[Hint: Use the result of Problem 2.5 to compute G5(ω).]P.11.7 Reconsider the inverted pendulum of Problem 11.4. Assume thatthe output is the absolute position of the pendulum. Design a Kalman fil-ter assuming that the plant noise enters the system at the input (E = B).Apply the LTR procedure and check that, as q2 increases, the open-looptransfer function GH(ω) tends to that of the LQR (Problem 11.4). Checkthe effect of the procedure on the bandwidth of the control system. [Note:The assumption that the output of the system is the absolute positionx rather than the tilt angle θ may appear as a practical restriction, butit is not, because x can always be obtained indirectly from θ and u byEqu.(9.21). It is necessary to remove the feedthrough component from theoutput before applying the LTR procedure.][Hint: The KBF/LTR is the limit as q →∞ of the symmetric root locus(9.71) based on E = B.]P.11.8 Consider the two-mass problem of section 9.10.1.


(a) Design a LQR by solving the Riccati equation for various values ofthe control weight %. Show that for some %, we obtain the same gains asthose obtained with the symmetric root locus in section 9.10.1.(b) For these gains, draw the Nyquist plot of the LQR,

G0(ω) = G(sI −A)−1B

evaluate the gain and phase margins.(c) Assuming that the plant noise enters at the input, design a KBF bysolving the Riccati equation for various values of the noise intensity ratioq = w/v. Show that for some q, we obtain the same gains as those obtainedwith the symmetric root locus. Calculate the gain and phase margins.(d) Apply the LTR technique with increasing q; draw a set of Nyquistplots of GH(ω) showing the evolution of the gain and phase margins.Check that GH(ω) → G0(ω) as q →∞.P.11.9 Find a state space realization of the input and output filters h(ω)and s−1(ω) corresponding to the weighting matrices (11.80):

|h(ω)|2 =ω2

0 + ω2

ω2

|s−1(ω)|2 =ω2n

1

ω2n1 + ω2n

(n = 2)

The latter is known as Butterworth filter of order n; its poles are locatedon a circle of radius ω1 according to Fig.11.1. P.11.10 Consider the two-mass problem of section 9.10.1. Assume that the system is subjected to asinusoıdal disturbance at ν0 = 0.5 rad/s acting on the main body. Usinga frequency-shaped cost functional, design a LQG controller with gooddisturbance rejection capability. Compare the performance of the newdesign to the nominal one (time response, sensitivity function,...).[Hint: use a lightly damped oscillator as output filter

h(ω) =ν20

ν20 − ω2 + 2jξων0

where ξ is kept as design parameter.]P.11.11 Show that the compensator obtained by the frequency-shapedcost functional has the following state space realization:

ˆxxi

x0

=

A−KfC (B −KfD)Ci 0−BiGcx Ai −BiGci −BiGco

0 0 Ao

xxi

xo

+

Kf

0Bo

y

11.18 Problems 303

u = Cixi

where Kf is the gain of the observer for x and Gc = (Gcx, Gci, Gco) is thegain of the state feedback. Note that, as a result of the structure of thesystem matrix, the poles of the compensator include those of the outputfilter, Ao.P.11.12 Find a state space realization of the noise model (11.99) (11.100)achieving the following power spectral density:

Φw(ω) =ω2

0 + ω2

ω2.

ω2n1

ω2n1 + ω2n

(n = 2)

(this filter combines in cascade the two filters used in Problem 11.9).P.11.13 Show that the compensator obtained by using a noise model inthe loop shaping has the following state space realization:

(˙xz

)=

(A−BGcx −KfxC ECw −BGcw

−KfwC Aw

) (xz

)+ Kfy

u = −Gcxx−Gcwz

where Gc = (Gcx, Gcw) is the gain matrix of the regulator of the aug-mented system and KT

f = (KTfx,KT

fw) is the corresponding observer gainmatrix. Note that the system matrix is no longer block triangular, so thatthe poles of the compensator differ from those of Aw.P.11.14 Repeat Problem 11.10 using a noise model (w is the output ofa second order filter). Compare the results.

12

Controllability and Observability

12.1 Introduction

Controllability measures the ability of a particular actuator configurationto control all the states of the system; conversely, observability measuresthe ability of the particular sensor configuration to supply all the informa-tion necessary to estimate all the states of the system. Classically, controltheory offers controllability and observability tests which are based on therank deficiency of the controllability and observability matrices: The sys-tem is controllable if the controllability matrix is full rank, and observableif the observability matrix is full rank. This answer is often not enoughfor practical engineering problems where we need more quantitative in-formation. Consider for example a simply supported uniform beam; themode shapes are given by (2.55). If the structure is subject to a pointforce acting at the center of the beam, it is obvious that the modes ofeven orders are not controllable because they have a nodal point at thecenter. Similarly, a displacement sensor will be insensitive to the modeshaving a nodal point where it is located. According to the rank tests, assoon as the actuator or the sensor are slightly moved away from the nodalpoint, the rank deficiency disappears, indicating that the correspondingmode becomes controllable or observable. This is too good to be true,and any attempt to control a mode with an actuator located close to anodal point would inevitably lead to difficulties, because this mode is onlyweakly controllable or observable. In this chapter, after having discussedthe basic concepts, we shall turn our attention to the quantitative mea-sures of controllability and observability, and apply the concept to modelreduction.

306 12 Controllability and Observability

12.1.1 Definitions

Consider the linear time-invariant system

x = Ax + Bu (12.1)

y = Cx (12.2)

• The system is completely controllable if the state of the system can betransferred from zero to any final state x∗ within a finite time.

• The system is stabilizable if all the unstable eigenvalues are controllableor, in other words, if the non controllable subspace is stable.

• The system is completely observable if the state x can be determinedfrom the knowledge of u and y over a finite time segment. In thespecialized literature, observability refers to the determination of thecurrent state from future output, while the determination of the statefrom past output is called reconstructibility. For linear, time-invariantsystems, these concepts are equivalent and do not have to be distin-guished.

• The system is detectable if all the unstable eigenvalues are observable,or equivalently, if the unobservable subspace is stable.

12.2 Controllability and observability matrices

The simplest way to introduce the controllability matrix is to consider thesingle input n-dimensional discrete-time system governed by the differenceequation

xk+1 = Axk + buk (12.3)

where A is the n×n system matrix and b the n-dimensional input vector.Assuming that the system starts from rest, x0 = 0, the successive valuesof the state vector resulting from the scalar input uk are

x1 = bu0

x2 = Ax1 + bu1 = Abu0 + bu1

...

xn = An−1bu0 + An−2bu1 + ... + bun−1

or

12.2 Controllability and observability matrices 307

xn = (b, Ab, A2b, ..., An−1b)

un−1

...u1

u0

(12.4)

where n is equal to the order of the system. The n× n matrix

C = (b, Ab, A2b, ..., An−1b) (12.5)

is called the controllability matrix; its columns span the state space whichcan be reached after exactly n samples. If C is full rank, the state vectorcan be transferred to any final value x∗ after only n samples. By solvingEqu.(12.4), one finds

un−1

...u1

u0

= C−1x∗ (12.6)

Next, consider the values of xN for N > n. Once again,

xN = (b, Ab, A2b, ..., AN−1b)

uN−1

...u1

u0

It turns out that the rank of the rectangular matrix

(b, Ab, A2b, ..., AN−1b)

is the same as that of C, and that the columns of the two matrices spanthe same space. This is a consequence of the Cayley-Hamilton theorem,which states that every matrix A satisfies its own characteristic equation.Thus, if the characteristic equation of A is

α(s) = det(sI −A) = sn + a1sn−1 + ... + an−1s + an = 0 (12.7)

A satisfies the matrix equation

An + a1An−1 + ... + an−1A + an = 0 (12.8)

It follows that for any m > n, Amb is linearly dependent on the columns ofthe controllability matrix C; as a result, increasing the number of columnsAmb does not enlarge the space which is spanned (Problem 12.1). In


conclusion, the system (12.3) if completely controllable if and only if (iff)the rank of the controllability matrix C is n.

This result has been established for a single-input discrete-time linearsystem, but it also applies to multi-input discrete as well as continuoustime linear systems. The linear time-invariant system (12.1) with r inputsis completely controllable iff the n× (n× r) controllability matrix

C = (B,AB,A2B, ..., An−1B) (12.9)

is such thatrank(C) = n (12.10)

We then say that the pair (A,B) is controllable. If C is not full rank, thesubspace spanned by its columns defines the controllable subspace of thesystem.

In a similar manner, the system (12.1) (12.2) is observable iff the ob-servability matrix

O =

CCA...

CAn−1

(12.11)

is such thatrank(O) = n (12.12)

In this case, we say that the pair (A,C) is observable.From the fact that

OT = (CT , AT CT , ..., (AT )n−1CT )

we conclude that the pair (A, C) is observable iff the dual system (AT , CT )is controllable. Conversely, the pair (A,B) is controllable iff the dual sys-tem (AT , BT ) is observable. The duality between observability and con-trollability has already been stressed in section 11.12.

12.3 Examples

12.3.1 Cart with two inverted pendulums

Consider two inverted pendulums with the same mass m and lengths l1and l2 placed on a cart of mass M (Fig.12.1.b). Assume that the inputvariable u is the force applied to the cart (in contrast to section 9.2.3,

12.3 Examples 309

u(force)

u

u

x

x

x

m

m

m

m

m(a)

(c)

(b)y

l

l

l

l1

l2

θ

M

M

M

θ1

θ 2

θ1

θ 2

Fig. 12.1. Various configurations of inverted pendulum.

where the input was the displacement of the support). Using the statevariables x = (θ1, θ2, θ1, θ2)T , we can write the linearized equations nearθ1 = θ2 = 0 as

x1

x2

x3

x4

=

0 0 1 00 0 0 1a1 a2 0 0a3 a4 0 0

x1

x2

x3

x4

+

00b1

b2

u (12.13)

where a1 = (g/l1)(1 + m/M), a2 = (g/l1)(m/M), a3 = (g/l2)(m/M),a4 = (g/l2)(1 + m/M), b1 = −1/Ml1 and b2 = −1/Ml2 (Problem 12.3).The controllability matrix is

C =

0 b1 0 a1b1 + a2b2

0 b2 0 a3b1 + a4b2

b1 0 a1b1 + a2b2 0b2 0 a3b1 + a4b2 0

(12.14)

It can be checked easily that this matrix is full rank provided that l1 6= l2.If l1 = l2, the rank of C is reduced to 2. Thus, when the time constants


of the two pendulums are the same, the system is not controllable (inpractical applications, it is likely that the difficulties in controlling thesystem will appear long before reaching l1 = l2).

Next, consider the observability of the system from the measurementof θ1. We have C = (1, 0, 0, 0) and the observability matrix is

O =

1 0 0 00 0 1 0a1 a2 0 00 0 a1 a2

(12.15)

Since det(O) = −a22 6= 0, we conclude that the system is always observable

from a single angle measurement; this result is somewhat surprising, buttrue.

12.3.2 Double inverted pendulum

Next, consider a double inverted pendulum on a cart as in Fig.12.1.c. Tosimplify the equations without losing any generality in the discussion, weassume that the two arms have the same length, and that the two massesare the same. The equations of motion can be written more convenientlyby using the absolute tilt angles of the two arms (Problem 12.4). Usingthe state vector x = (θ1, θ2, θ1, θ2)T , we can write the linearized equationsabout the vertical position as

x =

0 0 1 00 0 0 1

2ω20(1 + a) −ω2

0 0 0−2ω2

0 2ω20 0 0

x +

00

−1/Ml0

u (12.16)

where ω20 = g/l and a = m/M . The controllability matrix reads

C =1

Ml

0 −1 0 −2ω20(1 + a)

0 0 0 −2ω20

−1 0 −2ω20(1 + a) 0

0 0 −2ω20 0

(12.17)

Since det(C) = −4ω40/M

4l4 6= 0, the system is always controllable. Simi-larly, the observability matrix from θ1 reads

O =

1 0 0 00 0 1 0

2ω20(1 + a) −ω2

0 0 00 0 2ω2

0(1 + a) −ω20

(12.18)

12.3 Examples 311

We have det(O) = −ω40 6= 0, which indicates that the system is indeed

observable from θ1 alone.

12.3.3 Two d.o.f. oscillator

Consider the mechanical system of Fig.12.2. It consists of two identicalundamped single d.o.f. oscillators connected with a spring of stiffness εk.The input of the system is the point force applied to mass 1. The massand stiffness matrices are respectively

εk k

u

x1 x2

km m

Fig. 12.2. Two d.o.f. oscillator.

M =

(m 00 m

)K = k

(1 + ε −ε−ε 1 + ε

)(12.19)

Defining the state vector x = (x1, x2, x1, x2)T and using Equ.(9.11), wefind the state space equation

x =

0 0 1 00 0 0 1

−ω2n(1 + ε) ω2

nε 0 0ω2

nε −ω2n(1 + ε) 0 0

x +

00

1/m0

u (12.20)

where ω2n = k/m. The controllability matrix reads

C =1m

0 1 0 −ω2n(1 + ε)

0 0 0 ω2nε

1 0 −ω2n(1 + ε) 0

0 0 ω2nε 0

(12.21)

det(C) = −ω4nε2/m4 indicates that the system is no longer controllable as

ε approaches 0. Indeed, when the stiffness of the coupling spring vanishes,the two masses become uncoupled and mass 2 is uncontrollable from theforce acting on mass 1.


12.4 State transformation

Consider a SISO systemx = Ax + bu

y = cT x

Since A is n × n and b and c are both n × 1, the system has n2 + 2nparameters. If we consider the non singular transformation of the state,

x = Txc (12.22)

the transformed state equation is

xc = T−1ATxc + T−1bu (12.23)

y = cT Txc (12.24)

orxc = Acxc + bcu (12.25)

y = cTc xc (12.26)

with the proper definition of Ac, bc and cc. The non singular transforma-tion matrix T contains n2 free parameters which can be chosen to achievespecial properties for the transformed system; we shall discuss an exam-ple in detail in the next section. It can be shown (Problem 12.5) that thecontrollability matrix of the transformed system, Cc, is related to that ofthe original system by

Cc = T−1C (12.27)

For any non singular transformation T , the rank of Cc is the same as thatof C. Thus, the property of controllability is preserved by any non singulartransformation.

12.4.1 Control canonical form

We have seen in the previous section that the transformation matrix Tcan be selected in such a way that the transformed system has specialproperties. A form which is especially attractive from the state feedbackpoint of view is the control canonical form, where the transformed systemis expressed in terms of the 2n coefficients ai and bj appearing in thesystem transfer function

12.4 State transformation 313

G(s) =y(s)u(s)

=b(s)a(s)

=b1s

n−1 + ... + bn

sn + a1sn−1 + ... + an(12.28)

The transformed matrices are (Problem 12.6)

Ac =

−a1 −a2 ... −an

1 ... 0 00 1 0

0... 0

0 ... 1 0

bc =

10...00

(12.29)

cTc = (b1, ..., bn)

Besides the fact that the transformation between the state space modelin control canonical form and the input-output model is straightforward,it is easy to compute the state feedback gains to achieve a desired closed-loop characteristic equation. Indeed, if the state feedback u = −gT

c xc isapplied, the closed-loop system matrix becomes

Ac − bcgTc =

−a1 − gc1 −a2 − gc2 ... −an − gcn

1 ... 0 00 1 0

0... 0

0 ... 1 0

(12.30)

The corresponding characteristic equation is

αc(s) = sn + (a1 + gc1)sn−1 + ... + (an + gcn) = 0 (12.31)

Thus, in control canonical form, the state feedback gains can be obtaineddirectly from the coefficients of the closed-loop characteristic equation,making pole placement very simple. The state feedback gains in the orig-inal state space system are slightly more difficult to compute, as we nowexamine.

In principle, the linear transformation matrix leading from the originalstate space representation to the control canonical form can be obtainedfrom Equ.(12.27):

T = CC−1c (12.32)

where C and Cc are the controllability matrices of the original system andof the control canonical form (Problem 12.7), respectively. From


u = −gTc xc = −gT

c T−1x = −gT x

it follows that the state feedback gains g of the original model are relatedto those in control canonical form, gc, by

gT = gTc (CcC−1) (12.33)

This formula is not very practical, because it requires the inverse of thecontrollability matrix. However, it can be expressed alternatively by Ack-ermann’s formula

gT = eTnC−1αc(A) (12.34)

where eTn = (0, 0, ..., 1) and αc(A) is the closed-loop characteristic poly-

nomial, expressed in terms of the open-loop system matrix A. Equation(12.34) states that the gain vector is in fact the last row of C−1αc(A). Thedemonstration uses the Cayley-Hamilton theorem; it is left to the reader(Problem 12.8). Note that C−1 does not have to be calculated explicitly;instead, it is more convenient to proceed in two steps, by first solving theequation

bTC = eTn

for b, and then computing

gT = bT αc(A)

12.4.2 Left and right eigenvectors

If the non-symmetric system matrix A has distinct eigenvalues, its eigen-vectors will be linearly independent and can be taken as the columns ofa regular matrix P :

AP = PΛ (12.35)

where Λ = diag(λi) is a diagonal matrix with the eigenvalues of A. Itfollows that

P−1AP = Λ (12.36)

If we define QT = P−1 and right multiply the foregoing equation by QT ,we get

QT A = ΛQT (12.37)

The columns pi of P and qi of Q (i.e. the rows of QT ) are called the rightand left eigenvectors of A, respectively, because

Api = λipi and qTi A = λiq

Ti (12.38)

12.5 PBH test 315

From the definition of QT , the left and right eigenvectors are orthogonal

qTi pj = δij (12.39)

From Equ.(12.36), we have

QT AP = Λ and A = PΛQT (12.40)

12.4.3 Diagonal form

Let us use the right eigenvector matrix P as state transformation matrix

x = Pxd (12.41)

Following the procedure described earlier in this section, we can write thetransformed state equation as

xd = Λxd + QT bu (12.42)

y = cT Pxd (12.43)

Since Λ is a diagonal matrix with entries equal to the poles of the sys-tem, Equ.(12.42) shows that the transformed system behaves like a setof independent first order systems. The diagonal form is also called themodal form, and the states xc are the modes of the system. Note that thisconcept of mode is related only to the matrix A and is different from thevibration modes as defined in section 2.3 (for an undamped structure, theentries of Λ are identical to the natural frequencies of the structure, as il-lustrated in the example of section 12.7). For MIMO systems, Equ.(12.42)and (12.43) become

xd = Λxd + QT Bu (12.44)

y = CPxd (12.45)

12.5 PBH test

It is easy to show (Problem 12.10) that the controllability matrix in di-agonal form reads

Cd = diag(qTi b)

1 λ1 ... λn−11

1 λ2 ... λn−12

...1 λn ... λn−1

n

(12.46)


The second matrix in this expression is called a Vandermonde matrix; itis non-singular if the eigenvalues are distinct. In this case, the rank of Cd

is the same as that of diag(qTi b). As a result, the system is controllable iff

qTi b 6= 0 for all i (12.47)

Thus, any left eigenvector orthogonal to the input vector is uncontrollable.From Equ.(12.42), we see that qT

i b is in fact a measure of the effectiveinput of the control in mode i and can therefore be regarded as a measureof controllability of mode i.

From the duality between controllability and observability, the fore-going results can readily be extended to observability; the observabilitymatrix reads (Problem 12.10)

Od =

1 1 ... 1λ1 λ2 ... λn

...

λn−11 λn−1

2 ... λn−1n

diag(cT pi) (12.48)

Once again, a system with distinct eigenvalues is controllable iff

cT pi 6= 0 for all i (12.49)

Any right eigenvector orthogonal to the output vector is unobservable.From Equ.(12.43), we see that cT pi is a measure of the contribution ofmode i to the output y.

From Equ.(12.46) and (12.48), we conclude that a system with multipleeigenvalues cannot be controlled from a single input, nor observed froma single output. The tests (12.47) and (12.49) are often called the Popov-Belevitch-Hautus (in short PBH) eigenvector tests of controllability andobservability.

For a MIMO system, qTi B is a row vector; its entry k measures the

controllability of mode i from the input k. Similarly, the component j ofCpi measures the observability of mode i from the component j of theoutput vector.

12.6 Residues

Next, consider the open-loop transfer function of the system,

G(s) = cT (sI −A)−1b (12.50)

12.7 Example 317

From Equ.(12.42) and (12.43), it can be written alternatively

G(s) = cT P (sI − Λ)−1QT b (12.51)

Since sI − Λ is diagonal, we easily obtain the following partial fractiondecomposition

G(s) =n∑

i=1

(cT pi)(qTi b)

s− λi=

n∑

i=1

Ri

s− λi(12.52)

where the residue of mode i,

Ri = (cT pi)(qTi b) (12.53)

is the product of the observability and controllability measures of modei. For MIMO systems, the partial fraction decomposition becomes

G(s) = Cn∑

i=1

piqTi

s− λiB =

n∑

i=1

Ri

s− λi(12.54)

with the residue matrixRi = Cpiq

Ti B (12.55)

Its entry (k, l) combines the observability of mode i from output k andthe controllability from input l.

12.7 Example

In order to dissipate any confusion about the eigenvectors of A and themode shapes of the structure (section 2.3), let us consider a flexible struc-ture with one input and one output; we assume that the dynamic equa-tions are written in state variable form (9.14) and, to make things evenclearer, we further assume that the system is undamped (ξ = 0) and thatthe mode shapes are normalized according to µ = 1. We use the nota-tion φ(a) = ΦT

u and φT (s) = Φy to emphasize the fact that φ(a) andφ(s) contain the amplitude of the mode shapes at the actuator and sen-sor locations, respectively. With these notations, the state space equationreads

z =

(0 Ω

−Ω 0

)z +

(0

φ(a)

)f (12.56)


y = (φ(s)T Ω−1, 0)z (12.57)

In this equation, the state vector is

z =

(Ωηη

)(12.58)

where η is the vector of the amplitudes of the structural modes. The non-diagonal system matrix can be brought to diagonal form according toEqu.(12.40); we get

P =1√2

(I I

jI −jI

), QT =

1√2

(I −jII jI

), Λ =

(jΩ 00 −jΩ

)

(12.59)We see that the natural frequencies of the system appear with positiveand negative signs on the diagonal of Λ, but the eigenvectors of A havenothing to do with the mode shapes of the structure. The PBH eigenvectortests read

QT b =1√2

(−jφ(a)jφ(a)

)cT P =

1√2

(φT (s)Ω−1 φT (s)Ω−1

)(12.60)

Thus, the controllability and observability measures qTi b and cT pi are pro-

portional to the modal amplitudes φi(a) and φi(s), respectively. Introduc-ing this in Equ.(12.52) and combining the complex conjugate eigenvalues,the partial fraction decomposition can be reduced to

G(s) =m∑

i=1

φi(a)φi(s)s2 + ω2

i

(12.61)

where the sum extends to all the structural modes (m = n/2). This resultis identical to Equ.(2.58).

To conclude this example, we see that when the state equation is writ-ten in modal coordinates as in Equ.(12.56), the PBH tests and the asso-ciated controllability and observability measures provide no more infor-mation than the amplitude of the mode shapes, φ(a) and φ(s).

12.8 Sensitivity

The ultimate goal of the control system is to relocate the closed-looppoles at desirable locations in the complex plane; this should be done,

12.9 Controllability and observability Gramians 319

preferably, with moderate values of the gain, in order to limit the controleffort and the detrimental effects of noise and modelling errors.

The closed-loop poles sk of a SISO system are solutions of the char-acteristic equation 1 + gH(s)G(s) = 0; they start from the open-looppoles λk for g = 0 and move gradually away as g increases, in a directionwhich is dictated by the compensator H(s). The rate of change of theclosed-loop pole sk near g = 0 is a direct measure of the authority of thecontrol system on this pole; it can be evaluated as follows: for a smallgain g = ∆g, sk = λk + ∆sk; if the open-loop poles are distinct, we canapproximate

sk − λi ' λk − λi (k 6= i)

The partial fraction decomposition (12.52) becomes

G(sk) ' Rk

∆sk+

∑

i6=k

Ri

λk − λi(12.62)

and the characteristic equation

1 + ∆gH(λk) Rk

∆sk+

∑

i6=k

Ri

λk − λi = 0 (12.63)

or∆gH(λk)Rk

∆sk= −1−∆gH(λk)

∑

i6=k

Ri

λk − λi

Upon taking the limit ∆g → 0, we get

(∂sk

∂g)g=0 = lim

∆g−→0

∆sk

∆g= −H(λk)Rk (12.64)

This result shows that the rate of change of the closed-loop poles near g =0 is proportional to the corresponding residue Rk and to the magnitudeof the transfer function of the compensator H(λk). The latter observationexplains why the poles located in the roll-off region of the compensatormove only very slowly for small g.

12.9 Controllability and observability Gramians

Consider the linear time-invariant system (12.1); the controllability mea-sures the ability of the controller to control all the system states from theparticular actuator configuration, or equivalently, the abilty to excite all


the states from the input u. Consider the response of the system to a setof independent white noises of unit intensity:

E[u(t1)uT (t2)] = Iδ(t1 − t2) (12.65)

If the system is asymptotically stable (i.e. if all the poles of A have neg-ative real parts), the response of the system is bounded, and the steadystate covariance matrix is finite; it reads (Problem 12.12.a)

Wc = E[xxT ] =∫ ∞

0eAτBBT eAT τdτ (12.66)

Wc is called the Controllability Gramian. According to section 11.4, it issolution of the Lyapunov equation

AWc + WcAT + BBT = 0 (12.67)

The system is controllable if all the states of the system can be excited;this condition is fulfilled iff Wc is positive definite.

From the duality between the observability and controllability, weknow that the pair (A,C) is observable iff the pair (AT , CT ) is control-lable. It follows that the system is observable iff the observability Gramian

Wo =∫ ∞

0eAT τCT CeAτdτ (12.68)

is positive definite. Substituting (AT , CT ) to (A,B) in Equ.(12.67), wesee that, if A is asymptotically stable, Wo is solution of

AT Wo + WoA + CT C = 0 (12.69)

Just as the controllability Gramian reflects the ability of the input uto perturb the states of the system, the observability Gramian reflectsthe ability of non-zero initial conditions x0 of the state vector to affectthe output y of the system. This can be seen from the following result(Problem 12.12.b): ∫ ∞

0yT y dt = xT

0 Wox0 (12.70)

If we perform a coordinate transformation

x = T x (12.71)

the Gramians are transformed according to

12.10 Internally balanced coordinates 321

Wc(T ) = Wc = T−1WcT−T (12.72)

Wo(T ) = Wo = T T WoT (12.73)

where the notation Wc(T ) refers to the controllability Gramian after thecoordinate transformation (12.71). The proof is left to the reader (Prob-lem 12.13).

12.10 Internally balanced coordinates

As we have just seen, the Gramians depend on the choice of state vari-ables. Since, in most cases, the latter are not dimensionally homogeneous,nor normalized in an appropriate manner, the magnitude of the entriesof the Gramians are not physically meaningful for identifying the leastcontrollable or least observable part of a system. This information wouldbe especially useful for model reduction.

It is possible to perform a coordinate transformation such that thecontrollability and observability Gramians are diagonal and equal; thisunique set of coordinates is called internally balanced (Moore)

Let Wc and Wo be the controllability and observability Gramians of anasymptotically stable time-invariant linear system. We perform a spectraldecomposition of Wc according to

Wc = VcΣ2c V T

c (12.74)

where Vc is a unitary matrix (VcVTc = I) and Σ2

c is the diagonal matrix ofeigenvalues (all positive if Wc is positive definite). If we define Lc = VcΣc,we can write equivalently

Wc = LcLTc (12.75)

(when Lc is a lower triangular matrix, this decomposition is called aCholeski factorization). From Equ.(12.73) and (12.74), if we perform achange of coordinates

x = T1x1 (12.76)

with T1 = Lc, the Gramians become

Wc(T1) = L−1c WcL

−Tc = I (12.77)

Wo(T1) = LTc WoLc (12.78)

Next, we perform the spectral decomposition of Wo(T1) according to


Wo(T1) = UΣ2UT (12.79)

(with UUT = I) and use the transformation matrix T2 = UΣ−1/2 toperform another change of coordinates

x1 = T2x2 (12.80)

Equ.(12.72) and (12.73) show that the Gramians in the new coordinatesystem read

Wc(T1T2) = Σ1/2UT UΣ1/2 = Σ (12.81)

Wo(T1T2) = Σ−1/2UT UΣ2UT UΣ−1/2 = Σ (12.82)

Thus, in the new coordinate system, the controllability and observabilityGramians are equal and diagonal

Wc(T1T2) = Wo(T1T2) = Σ (12.83)

For this reason, the new coordinate system is called internally balanced;it is denotated xb. The global coordinate transformation is

x = T1T2xb (12.84)

and the internally balanced model is readily obtained from Equ.(12.23)and (12.24). From Equ.(12.72) and (12.73), we see that, for any transfor-mation T ,

WcWo = T [Wc(T )Wo(T )]T−1 (12.85)

It follows thatWcWo = (T1T2)Σ2(T1T2)−1 (12.86)

Thus, the eigenvalues of WcWo are the entries of Σ2, and the transforma-tion matrix T1T2 contains the right eigenvectors of WcW0. The eigenval-ues of the Gramians change with the coordinate transformation, but theeigenvalues of the Gramian product WcW0 is invariant (Problem 12.13).The square root of the eigenvalues of the Gramian product, σi, are calledthe Hankel singular values of the system.

12.11 Model reduction

Consider the partition of a state space model according to(

x1

x2

)=

(A11 A12

A21 A22

) (x1

x2

)+

(B1

B2

)u (12.87)

12.11 Model reduction 323

y = C1x1 + C2x2 (12.88)

If, in some coordinate system, the subsystem (A11, B1, C1) has the sameimpulse response as the full order system, it constitutes an exact lowerorder model of the system; the model of minimum order is called theminimum realization.

Model reduction is concerned with approximate models, and involvesa trade-off between the order of the model and its ability to duplicate thebehavior of the full order model within a given frequency range.

12.11.1 Transfer equivalent realization

If we consider the partial fraction decomposition (12.54), one reductionstrategy consists of truncating all the modes with poles far away from thefrequency domain of interest (and possibly including their contributionto the static gains) and also those with small residues Ri, which are onlyweakly controllable or observable (or both).

This procedure produces a realization which approximates the trans-fer function within the frequency band. However, since the uncontrollablepart of the system is deleted, even if it is observable, the reduced modelcannot reproduce the response to disturbances that may excite the sys-tem. This may lead to problems in the state reconstruction. To understandthis, recall that the transfer function Ge(s), which is the relevant one forthe observer design, is that between the plant noise and the output (sec-tion 9.7). If the plant noise does not enter at the input, Ge(s) does havecontributions from all observable modes, including those which are un-controllable from the input. The procedure can be improved by includingall the modes which have a significant contribution to Ge(s) too.

12.11.2 Internally balanced realization

Internally balanced coordinates can be used to extend the concept ofminimum realization. The idea consists of using the entries of the jointGramian Σ to partition the original system into a dominant subsystem,with large entries σi, and a weak one, with small σi. The reduction isachieved by cutting the weak subsystem from the dominant one. The fol-lowing result guarantees that the reduced system remains asymptoticallystable:

If the internally balanced system is partitioned according to (12.87)and if the joint Gramian is


Dominant subsystem ( )Σ1

Weak subsystem ( )Σ2

Reduced modelobtained by cuttingthese connections

A B C11 1 1 +

u y

A12 A2 1

A2 2B2C2

Fig. 12.3. Model reduction.

Wc = Wo =

(Σ1 00 Σ2

)(12.89)

the two subsystems (A11, B1, C1) and (A22, B2, C2) are asymptoticallystable and internally balanced, such that

W 1c = W 1

o = Σ1 = diag(σ1...σk) (12.90)

W 2c = W 2

o = Σ2 = diag(σk+1...σn) (12.91)

The proof is left to the reader (Problem 12.14).Thus, if we order the internally balanced coordinates by decreasing

magnitude of σi and if the subsystems 1 and 2 are selected in such a waythat σk+1 ¿ σk, the global system is clearly dominated by subsystem1. The model reduction consists of severing subsystem 2, as indicated inFig.12.3, which produces the reduced system (A11, B1, C1).

12.11.3 Example

Consider a simply supported uniform beam with a point force actuatorat xa = 0.331 l and a displacement sensor at xs = 0.85 l. We assume thatl = 1 m, EI = 10.266 10−3 Nm2, m = 1kg/m and ξ = 0.005. The naturalfrequencies and the mode shapes are given by (2.54) and (2.55); we findω1 = 1 rad/s, ω2 = 4 rad/s, etc... The system can be written in statevariable form according to (9.14). In a second step, the system can betransformed into internally balanced coordinates following the procedureof section 12.10.1. Two kinds of reduced models have been obtained asfollows:

• Transform into internally balanced coordinates and delete the subsys-tem corresponding to the smallest entries of the joint Gramian.

12.11 Model reduction 325

Reduced models8 states


Full model

Full model

ω (rad / s)

GaindB

GaindB

Fig. 12.4. Input-output frequency response of the full model and the reduced modelsbased on internally balanced coordinates and modal truncation.

• Delete the modal coordinates corresponding to the smallest static gains

φ(a)φ(s)µiω2

i

in the modal expansion of the transfer function

G(s) =m∑

i=1

φi(a)φi(s)µiω2

i

.ω2

i

s2 + 2ξωis + ω2i

(12.92)

Figure 12.4 compares the amplitude plots of the input-output fre-quency response function G(ω) of the reduced models with 8 and 12states, with that of the full model; the internally balanced realization andthe modal truncation based on the static gains are almost identical (theycannot be distinguished on the plot). Figure 12.5 compares the resultsobtained with the same reduced models, for the frequency response func-tion between a disturbance applied at xd = 0.55 l and the output sensor.Once again, the results obtained with the internally balanced realizationand the modal truncation based on the static gains are nearly the same


ω (rad / s)

GaindB

GaindB Full model

Full model



Fig. 12.5. Disturbance-output frequency response of the full model and the reducedmodels based on internally balanced coordinates and modal truncation.

(we can notice a slight difference near ω = 30 rad/s for the reduced mod-els with 12 states); the reduced models with 8 states are substantially inerror in the vicinity of 9 rad/s, because mode 3, which has been elimi-nated during the reduction process (it is almost not controllable from theinput), is excited by the disturbance.

12.12 References

ACKERMANN, J. Sampled-Data Control Systems, Springer-Verlag, 1985.FRANKLIN, G.F., POWELL, J.D. & EMAMI-NAEINI, A. FeedbackControl of Dynamic Systems. Addison-Wesley, 1986.GAWRONSKI, W.K. Advanced Structural Dynamics and Active Controlof Structures, Springer, 2004.HAMDAN, A.M.A. & NAYFEH, A.H. Measure of modal controllabilityand observability for first and second order linear systems, AIAA J. ofGuidance, Control, and Dynamics, Vol.12, No 5, p.768, 1989.HUGHES, P.C. Space structure vibration modes: how many exist? whichones are important? IEEE Control Systems Magazine, February 1987.

12.13 Problems 327

JUNKINS, J.L. & KIM, Y. Introduction to Dynamics and Control of Flex-ible Structures, AIAA Education Series, 1993.KAILATH, T. Linear Systems, Prentice-Hall, 1980.KIM, Y. & JUNKINS, J.L. Measure of controllability for actuator place-ment, AIAA J. of Guidance, Control, and Dynamics, Vol.14, No 5, Sept.-Oct 1991, 895-902.KWAKERNAAK, H. & SIVAN, R. Linear Optimal Control Systems, Wi-ley, 1972.MOORE, B.C. Principal component analysis in linear systems: control-lability, observability and model reduction, IEEE Trans. on AutomaticControl, Vol.AC-26, No 1, 17-32, 1981.SKELTON, R.E. & HUGHES, P.C. Modal cost analysis for linear matrixsecond-order systems, ASME J. of Dynamic Systems, Measurement, andControl, Vol.102, 151-158, Sept. 1980.SKELTON, R.E. Dynamic System Control - Linear System Analysis andSynthesis, Wiley, 1988.WIBERG, D.M. State Space and Linear Systems, McGraw-Hill Schaum’sOutline Series in Engineering, 1971.

12.13 Problems

P.12.1 Show that for a n-dimensional system, the rank of the matrix

(b, Ab, A2b, ..., AN−1b)

is the same as that of the controllability matrix C, for any N > n.P.12.2 Consider the inverted pendulum of Fig.12.1.a, where the inputvariable u is the force applied to the cart. Show that the equation ofmotion near θ = 0 is

θ − g

l(1 +

m

M)θ = − u

Ml

Write the equation in state variable form using x = (θ, θ)T . Compute thecontrollability matrix.[Hint: use Lagrange’s equations]P.12.3 Consider two inverted pendulums on a cart as in Fig.12.1.b. Showthat the equations of motion near θ1 = 0 and θ2 = 0 are

θ1 − g

l1(1 +

m

M)θ1 − g

l1

m

Mθ2 = − u

Ml1


θ2 − g

l2

m

Mθ1 − g

l2(1 +

m

M)θ2 = − u

Ml2

P.12.4 Consider the double inverted pendulum of Fig.12.1.c. Show thatthe equations of motion near θ1 = 0 and θ2 = 0 are

θ1 = 2ω20(1 + a)θ1 − ω2

0θ2 − 1Ml

u

θ2 = −2ω20θ1 + 2ω2

0θ2

where θ1 and θ2 are the absolute angles of the two arms, ω20 = g/l and

a = m/M .P.12.5 Show that for two sets of state variables related by the non singulartransformation x = Txc, the controllability matrices are related by

Cc = T−1C

P.12.6 Show that the control canonical form (12.29) is a state spacerealization of the transfer function (12.28).P.12.7 Show that for n = 3, the controllability matrix of the controlcanonical form reads

1 −a1 a2

1 − a2

0 1 −a1

0 0 1

P.12.8 Demonstrate Ackermann’s formula (12.34) for SISO systems.[Hint: Proceed according to the following steps:

(1) Show that eTi Ac = eT

i−1

(2) Using the Cayley-Hamilton theorem, show that eTnαc(Ac) = gT

c

(3) Show that αc(Ac) = T−1αc(A) = CcC−1αc(A)(4) Using the result of Problem 12.7, show that eT

nCc = eTn .]

P.12.9 Consider the single degree of freedom oscillator of section 9.4.1.Calculate the state feedback gains leading to the characteristic equation(9.49) using Ackermann’s formula. Compare with (9.50) and (9.51).P.12.10 Show that for a system in diagonal form, the controllability andobservability matrices are given by Equ.(12.46) and (12.48).P.12.11 The PBH rank tests state that

• The pair (A, b) is controllable iff

rank[sI −A, b] = n for all s

12.13 Problems 329

• The pair (cT , A) is observable iff

rank

(cT

sI −A

)= n for all s

Show that these tests are equivalent to the eigenvector tests (12.47) and(12.49).P.12.12 Consider an asymptotically stable linear time-invariant system.Show that(a) The steady state covariance matrix due to independent white noiseinputs of unit intensity

E[u(t1)uT (t2)] = Iδ(t1 − t2)

is equal to the controllability Gramian:

Wc = E[xxT ] =∫ ∞

0eAτBBT eAT τdτ

(b) The free response from initial conditions x0 satisfies

∫ ∞

0yT y dτ = xT

0 Wox0

where Wo is the observability Gramian.[Hint: the state impulse response is x(τ) = eAτB and the free outputresponse from non-zero initial conditions x0 is y(τ) = CeAτx0.]P.12.13 Show that for the coordinate transformation x = T x, the Grami-ans are transformed according to

Wc = TWcTT

Wo = T−T WoT−1

Show that the eigenvalues of the Gramian product WcW0 are invariantwith respect to a coordinate transformation.P.12.14 Show that if an internally balanced system is partitioned ac-cording to (12.87), the two subsystems (A11, B1, C1) and (A22, B2, C2)are internally balanced with joint Gramians Σ1 and Σ2.[Hint: Partition the Lyapunov equations governing Wc and Wo.]

13

Stability

13.1 Introduction

A basic knowledge of stability of linear systems has been assumed through-out the previous chapters. Stability was associated with the location ofthe poles of the system in the left half plane. In chapter 9, we saw thatthe poles are the eigenvalues of the system matrix A when the system iswritten in state variable form. In chapter 10, we examined the Nyquistcriterion for closed-loop stability of a SISO system; we concluded on thestability of the closed-loop system G(1 + G)−1 from the number of encir-clements of−1 by the open-loop transfer function G(s). In this chapter, weexamine the salient results of Lyapunov’s theory of stability; it is attrac-tive for mechanical systems because of its exceptional physical meaningand its wide ranging applicability, especially for the analysis of nonlinearsystems, and also in controller design. We will conclude this chapter witha class of collocated controls that are especially useful in practice, becauseof their guaranteed stability, even for nonlinear systems; we will call themenergy absorbing controls. The following discussion will be restricted totime-invariant systems (also called autonomous), but most of the resultscan be extended to time-varying systems. As in the previous chapters,most of the general results are stated without proof and the discussionis focused on vibrating mechanical systems; a deeper discussion can befound in the references.

Consider a time-invariant system, linear or not

• The equilibrium state x = 0 is stable in the sense of Lyapunov if,for every ε > 0 there is some δ > 0 (depending on ε) such that, if||x0|| < δ, then ||x|| < ε for all t > t0.

332 13 Stability

In this statement, ||.|| stands for a norm, measuring the distance to theequilibrium; the Euclidean norm is defined as ||x|| = (xT x)1/2. Stateswhich are not stable in the sense of Lyapunov are unstable. Stability is alocal property; if it is independent of the size of the initial perturbationx0, it is global.

• The equilibrium state x = 0 is asymptotically stable if it is stable inthe sense of Lyapunov and if, for any x0 close to 0, x(t) → 0 as t →∞.

Thus, for a mechanical system, asymptotic stability implies some damp-ing, unlike Lyapunov stability. For a linear time-invariant system, sincex(t) = eAtx0, asymptotic stability is always global; nonlinear systemsexhibit more complicated behaviors and they can have more than oneequilibrium point (Problem 13.1). The stability of an equilibrium pointis related to the behavior of the free trajectories starting in its neighbor-hood; if all the trajectories eventually converge towards the equilibriumpoint, it is asymptotically stable; if the trajectories converge towards alimit cycle, the system is unstable (Problem 13.2).

The above definitions of internal stability refer to the free responsefrom non-zero initial conditions. In some cases, we are more interested inthe input-output response:

• A system is externally stable if every bounded input produces abounded output. For obvious reasons, this is also called BIBO sta-bility.

External stability has no relation to internal (zero-input) stability in gen-eral, except for linear time-invariant systems, where it is equivalent toasymptotic stability (if the system is both controllable and observable).

13.1.1 Phase portrait

As we have already mentioned, the stability of an equilibrium point isrelated to the behavior of the trajectories in its vicinity. If we can alwaysfind a small domain containing the equilibrium point, such that all trajec-tories starting within this domain remain arbitrarily close to the origin,the equilibrium is stable; if all trajectories starting in a small domaineventually converge towards the origin, the equilibrium is asymptoticallystable, and if this occurs for any initial condition, we have global asymp-totic stability. The complete set of trajectories is called the phase portrait;to visualize it, consider the second order system

x + a1(x)x + a2(x) = 0 (13.1)

13.2 Linear systems 333

&x

&x

&x

x

x

x

Limit cycle

(a) (b)

(c)

Fig. 13.1. Phase portrait for various second order systems: (a) x + 2ξx + x = 0,(b) x + x− 2x = 0, (c) x− µ(1− x2)x + x = 0.

Defining the state variables x1 = x and x2 = x, we can easily representthe trajectories in the phase plane (x1, x2); various situations are con-sidered in Fig.13.1. Figure 13.1.a corresponds to a linear oscillator withviscous damping; all the trajectories consist of spirals converging towardsthe origin (the decay rate is governed by the damping); the system isglobally asymptotically stable. Figure 13.1.b shows the phase portrait ofan unstable linear system (poles at -2 and +1); all the trajectories areunbounded. The situation depicted on Fig.13.1.c is that of a Van der Poloscillator (Problem 13.2), all the trajectories converge towards a limitcycle; the system is unstable, although all the trajectories are bounded.

13.2 Linear systems

Since the stability of a system is independent of the state space coor-dinates, it is convenient to consider the diagonal form (12.44), whereΛ = diag(λi) is the diagonal matrix with the eigenvalues of A. The free

334 13 Stability

response from non-zero initial conditions reads

xd(t) = eΛtxd(0) (13.2)

Each state coordinate follows an exponential xi(t) = eλitxi(0). The systemis stable in the sense of Lyapunov if Re(λi) ≤ 0. If Re(λi) < 0 (strictlynegative), the system is globally asymptotically stable (and also externallystable).

If the characteristic equation is available in the form

d(s) = a0sn + a1s

n−1 + ... + an = 0 (13.3)

it is not necessary to compute all the eigenvalues to assess the asymptoticstability of the system; this can be done directly from the coefficients ai

of the characteristic polynomial by the Routh-Hurwitz criterion.

13.2.1 Routh-Hurwitz criterion

Assume that the characteristic polynomial is written in the form (13.3)with a0 > 0.

1. If not all the coefficients ai are positive, that is if ak ≤ 0 for somek, the system is not asymptotically stable (it may still be stable in thesense of Lyapunov if some ak = 0).

2. If all the coefficients ai > 0, a necessary and sufficient conditionfor all the roots λi to have negative real parts is that all the determi-nants ∆1, ∆2, ... ∆n defined below must be positive. The determinantsare constructed as follows:

Step 1. Form the array:

a1 a0 0 0 . . . 0a3 a2 a1 a0 . . . 0a5 a4 a3 a2 0

. . .a2n−1 a2n−2 a2n−3 . . . an

where a1, . . . , an are the coefficients of the characteristic polynomial, andai = 0 (i > n).

Step 2. Compute the determinants:

13.3 Lyapunov’s direct method 335

∆1 = a1

∆2 =

∣∣∣∣∣a1 a0

a3 a2

∣∣∣∣∣

∆3 =

∣∣∣∣∣∣∣

a1 a0 0a3 a2 a1

a5 a4 a3

∣∣∣∣∣∣∣. . .

∆n =

∣∣∣∣∣∣∣∣∣∣∣

a1 a0 0 0 . . . 0a3 a2 a1 a0 . . . 0a5 a4 a3 a2 0

. . .a2n−1 a2n−2 a2n−3 . . . an

∣∣∣∣∣∣∣∣∣∣∣

(13.4)

All the eigenvalues λi have negative real parts iff ∆i > 0 for all i.

13.3 Lyapunov’s direct method

13.3.1 Introductory example

Consider the linear oscillator(

x1

x2

)=

(0 1

−k/m −c/m

) (x1

x2

)(13.5)

We know that it is asymptotically stable for positive damping (c > 0); itsphase portrait is represented in Fig.13.1.a; for any disturbed state x0, thesystem returns to the equilibrium x = 0. The total energy of the systemis the sum of the kinetic energy of the mass and the strain energy in thespring:

E(x) =m

2x2

2 +k

2x2

1 (13.6)

E(x) is positive definite because it satisfies the two conditions

E(0) = 0

E(x) > 0 for all x 6= 0 (13.7)

The time derivative of the total energy during the free response is

E = mx2x2 + kx1x1

336 13 Stability

and, upon substituting x1 and x2 from Equ.(13.5),

E = −cx22 (13.8)

We see that E is always negative for a structure with positive damping.Since E is positive and decreases along all trajectories, it must eventuallygo to E = 0 which, from (13.6), corresponds to the equilibrium statex = 0. This implies that the system is asymptotically stable.

Here, we have proved asymptotic stability by showing that the totalenergy decreases along all trajectories; Lyapunov’s direct method (alsocalled second method, for chronological reasons), generalizes this concept.Unlike other techniques (Eigenvalues, Routh-Hurwitz, Nyquist,...), themethod is also applicable to nonlinear and time-varying systems. In whatfollows, we shall restrict our attention to time-invariant systems for whichthe theorems have a simpler form; more general results can be found inthe literature (e.g. Vidyasagar).

13.3.2 Stability theorem

A time-invariant Lyapunov function candidate V (x) is a continuously dif-ferentiable, locally positive definite function, i.e. satisfying

V (0) = 0

V (x) > 0 for all x 6= 0 in D (13.9)

where D is a certain domain containing the origin.

Theorem: Consider a system governed by the vector differential equation

x = f(x) (13.10)

such that f(0) = 0. The equilibrium state x = 0 is stable (in the sense ofLyapunov) if one can find a Lyapunov function candidate V (x) such that

V (x) ≤ 0 (13.11)

for all trajectories in the neighborhood of the origin. If condition (13.11)is satisfied, V (x) is called a Lyapunov function for the system (13.10).

The Lyapunov function is a generalization of the total energy of thelinear oscillator considered in the introductory example. The foregoingtheorem is only a sufficient condition; the fact that no Lyapunov functioncan be found does not mean that the system is not stable. There is no


general procedure for constructing a Lyapunov function, and this is themain weakness of the method.

As an example, consider the simple pendulum, governed by the equa-tion

θ +g

lsin θ = 0 (13.12)

where l is the length of the pendulum, θ the angle and g the accelerationof gravity. Introducing the state variables x1 = θ and x2 = θ, we rewriteit

x1 = x2

x2 = −g/l sinx1 (13.13)

Let us again use the total energy (kinetic plus potential) as Lyapunovfunction candidate:

V (x) =ml2

2x2

2 + mgl(1− cosx1) (13.14)

It is indeed positive definite in the vicinity of x = 0. We have

V (x) = ml2x2x2 + mgl sinx1x1

and, substituting x1 and x2 from Equ.(13.13), we obtain the time deriva-tive along the trajectories:

V (x) = −mglx2 sinx1 + mglx2 sinx1 = 0

which simply expresses the conservation of energy. Thus, V (x) satisfiescondition (13.11), V (x) is a Lyapunov function for the pendulum and theequilibrium point x = 0 is stable. We now examine a stronger statementfor asymptotic stability.

13.3.3 Asymptotic stability theorem

Theorem: The state x is asymptotically stable if one can find a contin-uously differentiable, positive definite function V (x) such that

V (x) < 0 (13.15)

for all trajectories in some neighborhood of the origin. Besides, if V (x) issuch that there exists a nondecreasing scalar function α(.) of the distanceto the equilibrium (Fig.13.2), such that α(0) = 0 and

0 < α(||x||) ≤ V (x) for all x 6= 0 (13.16)

then the system is globally asymptotically stable under condition (13.15).

338 13 Stability

V(x)

x

x

α ( )

Fig. 13.2. Definition of V (x) and α(||x||) for global stability.

13.3.4 Lasalle’s theorem

Going back to the linear oscillator, we see that Equ.(13.8) does not com-ply with the requirement (13.15) to be strictly negative; indeed, E = 0whenever x2 = 0, even if x1 6= 0 (i.e. whenever the trajectory crosses thex axis in Fig.13.1.a). The answer to that is given by Lasalle’s theorem,which extends the asymptotic stability even if V is not strictly negative.

Theorem: The state x = 0 is asymptotically stable if one can find adifferentiable positive definite function V (x) such that

V (x) ≤ 0 (13.17)

for all trajectories, provided that the set of points where V = 0,

S = x ∈ Rn : V (x) = 0

contains no trajectories other than the trivial one x = 0.As an example, consider the nonlinear spring with friction governed

byx1 = x2

x2 = −f(x2)− g(x1) (13.18)

where g(x1) is the nonlinear restoring force and f(x2) is the friction. Weassume that g and f are continuous functions such that

σ g(σ) > 0, σ f(σ) > 0, σ 6= 0 (13.19)

[f(σ) and g(σ) are entirely contained in the first and third quadrant]. Itis easy to see that the linear oscillator is the particular case with


g(x1) =k

mx1 f(x2) =

c

mx2 (13.20)

and that the simple pendulum corresponds to

g(x1) =g

lsinx1 f(x2) = 0 (13.21)

The total energy is taken as Lyapunov function candidate

V (x1, x2) =12x2

2 +∫ x1

0g(u) du (13.22)

where the first term is the kinetic energy (per unit of mass), and thesecond one, the potential energy stored in the spring. The time derivativeis

V (x1, x2) = x2x2 + g(x1)x1 = −x2f(x2) ≤ 0 (13.23)

Since the set of points where x2 = 0 does not contain trajectories, thesystem is globally asymptotically stable.

13.3.5 Geometric interpretation

To visualize the concept, it is useful to consider, once again, a second ordersystem for which the phase space is a plane. In this case, V (x1, x2) can bevisualized by its contours (Fig.13.3). The stability is associated with thebehavior of the trajectories with respect to the contours of V . If we canfind a locally positive definite function V (x) such that all the trajectoriescross the contours downwards (curve 1), the system is asymptotically

x1

x2

c1

c > c2 1

0

1

2

3

V c=

Fig. 13.3. Contours of V (x1, x2) in the phase plane.

340 13 Stability

stable; if some trajectories follow the contours, V = 0, the system isstable in the sense of Lyapunov (curve 2). The trajectories crossing thecontours upwards (curve 3) correspond to instability, as we now examine.

13.3.6 Instability theorem

In the previous sections, we examined sufficient conditions for stability.We now consider a sufficient condition for instability. Let us start withthe well known example of the Van der Pol oscillator

x1 = x2

x2 = −x1 + µ(1− x21)x2 (13.24)

Taking the Lyapunov function candidate

V (x1, x2) =x2

1

2+

x22

2> 0 (13.25)

we haveV = x1x1 + x2x2 = µ(1− x2

1)x22 (13.26)

We see that, whenever |x1| < 1, V > 0. Thus, V > 0 applies everywherein a small set Ω containing the origin; this allows us to conclude thatthe system is unstable. In this example, V (x) is positive definite; in fact,instability can be concluded with a weaker statement:

Theorem: If there exists a function V (x) continuously differentiable suchthat V > 0 along every trajectory and V (x) > 0 for arbitrarily smallvalues of x, the equilibrium x = 0 is unstable.

It can be further generalized as follows:

Theorem: If there is a continuously differentiable function V (x) suchthat (i) in an arbitrary small neighborhood of the origin, there is a regionΩ1 where V > 0 and V = 0 on its boundaries; (ii) at all points of Ω1,V > 0 along every trajectory and (iii) the origin is on the boundary ofΩ1; then, the system is unstable.

The visual interpretation is shown in Fig.13.4: A trajectory startingat x0 within Ω1 will intersect the contours in the direction of increasingvalues of V , increasing the distance to the origin; it will never cross thelines OA and OB because this would require V < 0.

13.4 Lyapunov functions for linear systems 341

A

0

BΩ

Ω1

V = 0

V = 0

V > 0

V > 0x0

Fig. 13.4. Definition of the domains Ω and Ω1 for the instability theorem.

13.4 Lyapunov functions for linear systems

Consider the linear time-invariant system

x = Ax (13.27)

We select the Lyapunov function candidate

V (x) = xT Px (13.28)

where the matrix P is symmetric positive definite. Its time derivative is

V (x) = xT Px + xT Px = xT (AT P + PA)x

= −xT Qx (13.29)

if P and Q satisfy the matrix equation

AT P + PA = −Q (13.30)

This is the Lyapunov equation, that we already met several times. Thus,if we can find a pair of positive definite matrices P and Q satisfyingEqu.(13.30), both V and−V are positive definite functions and the systemis asymptotically stable.Theorem: The following statements are equivalent for expressing asymp-totic stability:

1. All the eigenvalues of A have negative real parts.2. For some positive definite matrix Q, the Lyapunov equation has a

unique solution P which is positive definite.3. For every positive definite matrix Q, the Lyapunov equation has a

unique solution P which is positive definite.

342 13 Stability

Note that, in view of Lasalle’s theorem, Q can be semi-positive definite,provided that V = −xT Qx 6= 0 on all nontrivial trajectories. The forego-ing theorem states that if the system is asymptotically stable, for everyQ ≥ 0 one can find a solution P > 0 to the Lyapunov equation. Note thatthe converse statement (for every P > 0, the corresponding Q is positivedefinite) is, in general, not true; this means that not every Lyapunov can-didate is a Lyapunov function. The existence of a positive definite solutionof the Lyapunov equation can be compared with the Routh-Hurwitz cri-terion, which allows us to determine whether or not all the eigenvalues ofA have negative real parts without computing them.

13.5 Lyapunov’s indirect method

This method (also known as the first method), allows us to draw conclu-sions about the local stability of a nonlinear system from the analysis ofits linearization about the equilibrium point. Consider the time-invariantnonlinear system

x = f(x) (13.31)

Assume that f(x) is continuously differentiable and that f(0) = 0, so thatx = 0 is an equilibrium point of the system. The Taylor’s series expansionof f(x) near x = 0 reads

f(x) = f(0) + [∂f

∂x]0 x + f1(x) (13.32)

where f1(x) = O(x2). Taking into account that f(0) = 0 and neglectingthe second order term, we obtain the linearization around the equilibriumpoint

x = Ax (13.33)

where A denotes the Jacobian matrix of f , at x = 0:

A = [∂f

∂x]x=0 (13.34)

Lyapunov’s indirect method assesses the local stability of the nonlinearsystem (13.31) from the eigenvalues of its linearization (13.33).

Theorem: The nonlinear system (13.31) is asymptotically stable if theeigenvalues of A have negative real parts. Conversely, the nonlinear systemis unstable if at least one eigenvalue of A has a positive real part. Themethod is inconclusive if some eigenvalues of A are purely imaginary.

13.6 An application to controller design 343

We shall restrict ourselves to the proof of the first part of the theorem.Assume that all the eigenvalues of A have negative real parts; then, wecan find a symmetric positive definite matrix P solution of the Lyapunovequation

AT P + PA = −I (13.35)

Using V = xT Px as Lyapunov function candidate for the nonlinear sys-tem, we have

V = xT Px + xT Px = fT (x)Px + xT Pf(x)

Using the Taylor’s series expansion f(x) = Ax + f1(x), we find

V = xT (AT P + PA)x + 2xT Pf1(x)

Taking into account Equ.(13.35) and the fact that f1(x) = O(x2), weobtain

V = −xT x +O(x3) (13.36)

Sufficiently near x = 0, V is dominated by the quadratic term −xT xwhich is negative; V (x) is therefore a Lyapunov function for the system(13.31) which is asymptotically stable.

We emphasize the fact that the conclusions based on the linearizationare purely local; the global asymptotic stability of the nonlinear systemcan only be established by finding a global Lyapunov function.

13.6 An application to controller design

Consider the asymptotically stable linear system

x = Ax + bu (13.37)

with the scalar input u subject to the saturation constraint

|u| ≤ u∗ (13.38)

If P is solution of the Lyapunov equation

AT P + PA = −Q (13.39)

with Q ≥ 0, V (x) = xT Px is a Lyapunov function of the system withoutcontrol (u = 0). With control, we have

344 13 Stability

V = −xT Qx + 2xT Pbu (13.40)

Any controlu = −ψ(bT Px) (13.41)

where the scalar function ψ(.) is such that σ ψ(σ) > 0 will stabilize thesystem, because V < 0. The following choice of u makes V as negative aspossible:

u = −u∗sign(bT Px) (13.42)

This discontinuous control is often called bang-bang; it is likely to producechattering near the equilibrium. The discontinuity can be removed by

u = −u∗sat(bT Px) (13.43)

where the saturation function is defined as

sat(x) =

1 x > 1x |x| ≤ 1

−1 x < −1(13.44)

However, the practical implementation of this controller requires theknowledge of the full state x, which is usually not available in practice;asymptotic stability is no longer guaranteed if x is reconstructed from astate observer.

13.7 Energy absorbing controls

Consider a vibrating mechanical system, linear or not, stable in open-loop,and such that the conservation of the total energy (kinetic + potential)applies when the damping is neglected. Because there is always somenatural damping in practice, the total energy E of the system actuallydecreases with time during its free response, E < 0. Suppose that we adda control system using a collocated actuator/sensor pair; if we denote byD the power dissipated by damping (D < 0) and by W the power flowfrom the control system to the structure, we have

E = D +W (13.45)

If we can develop a control strategy such that the power actually flowsfrom the structure to the control system, W < 0 (the control systembehaves like an energy sink), the closed-loop system will be asymptoticallystable.

13.7 Energy absorbing controls 345

(a) Velocity feedback (b) Force feedback

Force sensor (T) Piezoactuator (u)

F-g

&u

u

T

F g u= − &

& & &E F u g u= = − <2 0

u g T dt= ∫& &E T u g T= − = − <2 0

Fig. 13.5. Energy absorbing controls.

Now, consider the situation depicted in Fig.13.5.a, where we use a pointforce actuator and a collocated velocity sensor. If a velocity feedback isused,

F = −gu (13.46)

with g > 0, we haveW = Fu = −gu2 < 0 (13.47)

The stability of this Direct Velocity Feedback was already pointed out forlinear system, in section 7.3. Here, it is generalized to nonlinear structures.Even more generally, any nonlinear control

F = −ψ(u) (13.48)

where ψ(.) is such that σ ψ(σ) > 0 will be stabilizing.1

Next, consider the dual situation (Fig.13.5.b) where the actuator con-trols the relative position u of two points inside the structure, and thesensor output is the dynamic force T in the active member (T is collocatedwith u); this situation is that of the active truss considered in section 7.5.

Again, the power flow into the structure is

W = −T u (13.49)

It follows that the positive Integral Force Feedback

1 The above discussion applies also to any collocated dual actuator/sensor pair, as forexample a torque actuator collocated with an angular velocity sensor.

346 13 Stability

u = g

∫ t

0T (τ) dτ (13.50)

with g > 0 will be stabilizing, because

W = −T u = −gT 2 < 0 (13.51)

The stability of the control law (13.50) was established for linear struc-tures using the root locus technique. Here, we extend this result to non-linear structures, assuming perfect actuator and sensor dynamics. Laterin this book, we will apply this control law to the active damping ofcable-structure systems.

Because of their global asymptotic stability for arbitrary nonlinearstructures, we shall refer to the controllers (13.48) and (13.50) as energyabsorbing controllers. Note that, unlike those discussed in the previoussection, these controllers do not require the knowledge of the states, andare ready for implementation; the stability of the closed-loop system reliesvery strongly on the collocation of the sensor and the actuator. Onceagain, we emphasize that we have assumed perfect sensor and actuatordynamics; finite actuator and sensor dynamics always have a detrimentaleffect on stability.

13.8 References

GRAYSON, L.P. The status of synthesis using Lyapunov’s method, Au-tomatica, Vol.3, pp.91-121, 1965.GUILLEMIN, E.A. The Mathematics of Circuit Analysis, Wiley 1949.KALMAN, R.E. & BERTRAM, J.E. Control system analysis and designvia the second method of Lyapunov (1. continuous-time systems), ASMEJ. of Basic Engineering, pp.371-393, June 1960.MEIROVITCH, L. Methods of Analytical Dynamics, McGraw-Hill,1970.NAYFEH, A.H. & MOOK, D.T. Nonlinear Oscillations, Wiley, 1979.PARKS, P.C. & HAHN, V. Stability Theory, Prentice Hall, 1993.VIDYASAGAR, M. Nonlinear Systems Analysis, Prentice-Hall, 1978.WIBERG, D.M. State Space and Linear Systems, Schaum’s Outline Seriesin Engineering, McGraw-Hill, 1971.

13.9 Problems 347

13.9 Problems

P.13.1 Show that the nonlinear oscillator

mx + cx + k1x− k2x3 = 0

with m, c, k1, k2 > 0, has three equilibrium points. Check them for stabil-ity.P.13.2 Consider the Van der Pol oscillator

x− µ(1− x2)x + x = 0

with µ > 0. Show that the trajectories converge towards a limit cycle(Fig.13.1.c) and that the system is unstable.P.13.3 Plot the phase portrait of the simple pendulum

θ + g/l sin θ = 0

P.13.4 Show that a linear system is externally (BIBO) stable if its impulseresponse satisfies the following inequality

∫ t

0|h(τ)|dτ ≤ β < ∞

for all t > 0.P.13.5 Show that a linear time-invariant system is asymptotically stableif its characteristic polynomial can be expanded into elementary polyno-mials (s+ai) and (s2 + bis+ ci) with all the coefficients ai, bi, ci positive.P.13.6 Examine the asymptotic stability of the systems with the followingcharacteristic polynomials:

(i) d1(s) = s6 + 6s5 + 16s4 + 25s3 + 24s2 + 14s + 4(ii) d2(s) = s5 + 3s3 + 2s2 + s + 1(iii) d3(s) = s5 + 2s4 + 3s3 + 3s2 − s + 1

P.13.7 Examine the stability of the Rayleigh equation

x + x = µ(x− x3

3)

with the direct method of Lyapunov.P.13.8 Examine the stability of the following equations:

x + µx2x + x = 0 (µ > 0)

348 13 Stability

x + µ|x|x + x +x3

3= 0 (µ > 0)

P.13.9 (a) Show that, if A is asymptotically stable,

S =∫ t

0eAT τMeAτdτ

where M is a real symmetric matrix, satisfies the matrix differential equa-tion

S = AT S + SA + M [S(0) = 0]

(b) Show that the steady state value

S =∫ ∞

0eAT τMeAτdτ

satisfies the Lyapunov equation

AT S + SA + M = 0

P.13.10 Consider the free response of the asymptotically stable systemx = Ax from the initial state x0. Show that, for any Q ≥ 0, the quadraticintegral

J =∫ ∞

0xT Qxdt

is equal toJ = xT

0 Px0

where P is the solution of the Lyapunov equation

AT P + PA + Q = 0

P.13.11 Consider the linear time invariant system

x = Ax + Bu

Assume that the pair (A,B) is controllable and that the state feedbacku = −Gx has been obtained according to the LQR methodology:

G = R−1BT P

where P is the positive definite solution of the Riccati equation

13.9 Problems 349

AT P + PA + Q− PBR−1BT P = 0

with Q ≥ 0 and R > 0. Prove that the closed-loop system is asymptoti-cally stable by showing that V (x) = xT Px is a Lyapunov function for theclosed-loop system.Note: From section 11.3, we readily see that V (x) is in fact the remainingcost to equilibrium:

V (x) =∫ ∞

t(xT Qx + uT Ru)dτ

P.13.12 Consider the bilinear single-input system

x = Ax + (Nx + b)u

where A is asymptotically stable (the system is linear in x and in u, butit is not jointly linear in x and u, because of the presence of the bilinearmatrix N). Show that the closed-loop system is globally asymptoticallystable for the nonlinear state feedback

u = −(Nx + b)T Px

where P is the solution of the Lyapunov equation

AT P + PA + Q = 0

P.13.13 Consider the free response of a damped vibrating system

Mx + Cx + Kx = 0

The total energy is

E(x) =12xT Mx +

12xT Kx

(a) Show that its decay rate is

E(x) = −xT Cx

(b) Show that if we normalize the mode shapes according to µ = 1 and ifwe use the state space representation (9.14), the total energy reads

E(z) =12zT z

P.13.14 Consider a linear structure with a point force actuator collocatedwith a velocity sensor. Using the state space representation (9.14) andtaking the total energy as Lyapunov function, show that the controller(13.41) is equivalent to (13.48).

14

Applications

After a brief overview of some critical aspects of digital control, this chap-ter applies the concepts developed in the foregoing chapters to a few appli-cations; it is based on the work done at the Active Structures Laboratoryof ULB before 2002. We believe that these early experiments have morethan just an historical value. More applications will be considered in thenext three chapters.

14.1 Digital implementation

In recent years, low cost microprocessors have become widely available,and digital has tended to replace analog implementation. There are manyreasons for this: digital controllers are more flexible (it is easy to changethe coefficients of a programmable digital filter), they have good accuracyand a far better stability than analog devices which are prone to drift dueto temperature and ageing. Digital controllers are available with severalhardware architectures, including microcontrollers, PC boards, and digi-tal signal processors (DSP). It appears that digital signal processors areespecially efficient for structural control applications.

Although most controller implementation is digital, current micropro-cessors are so fast that it is always more convenient, and sometimes wise,to perform a continuous design of the compensator and transform it intoa digital controller as a second step, once a good continuous design hasbeen achieved. This does not mean that the control designer may ignoredigital control theory, because even though the conversion from continu-ous to digital is greatly facilitated by software tools for computer aidedcontrol engineering, there are a number of fundamental issues that haveto be considered with care; they will be briefly mentioned below. For a

352 14 Applications

deeper discussion, the reader may refer to the literature on digital control(e.g. Astrom & Wittenmark; Franklin & Powell).

14.1.1 Sampling, aliasing and prefiltering

Since digital controllers operate on values of the process variables at dis-crete times, it is important to know under what conditions a continuoussignal can be recovered from its discrete values only. The answer to thisquestion is given by Shannon’s theorem (also called sampling theorem),which states that, to recover a band-limited signal with frequency con-tent f < fb from its sampled values, it is necessary to sample at least atfs = 2fb. If a signal is sampled at fs, any frequency component above thelimit frequency fs/2 will appear as a component at a frequency lower thanfs/2. This phenomenon is called aliasing, and the limit frequency that canbe theoretically recovered from a digital signal is often called Nyquist fre-quency, by reference to the exploratory work of Nyquist. Aliasing is ofcourse not acceptable and it is therefore essential to place an analog low-pass filter at a frequency fc < fs/2 before the analog to digital converter(ADC), Fig.14.1.

However analog prefilters have dynamics and, as we know from thefirst Bode integral, a sharp cut-off of the magnitude is always associatedwith a substantial phase lag at the cut-off frequency fc. As fc is related

Fig. 14.1. Prefiltering and A/D conversion.

14.1 Digital implementation 353

to fs, it is always a good idea to sample at a high rate and to make surethat the cut-off frequency of the prefilter is substantially higher than thecrossover frequency of the control system. If the phase lag of the prefilterat crossover is significant, it is necessary to include the prefilter dynamicsin the design (as a rule of thumb, the prefilter dynamics should be includedin the design if the crossover frequency is higher than 0.1fc).

A simple solution to prefiltering is to introduce an analog second orderfilter

Gf (s) =ω2

c

s2 + 2ξωcs + ω2c

(14.1)

which can be built fairly easily with an operational amplifier and a fewpassive components. A second order Butterworth filter corresponds toξ = 0.71. Higher order filters are obtained by cascading first and secondorder systems; for example, a fourth order Butterworth filter is obtainedby cascading two second order filters with the same cut-off frequency andξ = 0.38 and ξ = 0.92, respectively (Problem 14.1).

14.1.2 Zero-order hold, computational delay

Sampling can be viewed as an impulse modulation converting the contin-uous signal x(t) into the impulse train

x∗(t) =∞∑

k=−∞x(t)δ(t− kT ) (14.2)

Zero-orderhold

t t

1

t tT T

T

0 0

0

2T 2T

δ(t)

Fig. 14.2. The zero-order hold transforms an impulse into a rectangle of duration T ,and an impulse train into a staircase function.

354 14 Applications

where T is the sampling period (T = 1/fs). The construction of a processwhich holds the sampled values x(kT ) constant during a sampling periodis made by passing x∗(t) through a zero-order hold which consists of afilter with impulse response (Fig.14.2)

h(t) = 1(t)− 1(t− T )

where 1(t) is the Heaviside step function. It is easy to show that thecorresponding transfer function is (Problem 14.2)

H0(s) =1− e−sT

s(14.3)

and that it introduces a linear phase lag −ωT/2.Another effect of sampling is the computational delay which is always

present between the access to the computer through the ADC and theoutput of the control law at the digital to analog converter DAC. Thisdelay depends on the way the control algorithm is implemented; it maybe fixed, equal to the sampling period T , or variable, depending on thelength of the computations within the sampling period. A time delay T ischaracterized by the transfer function e−Ts; it introduces a linear phaselag −ωT . Rational approximations of the exponential by all-pass functions(Pade approximants) were discussed in Problem 10.8.

The output of the DAC is also a staircase function; in some applica-tions, it may be interesting to smooth the control output, to remove thehigh frequency components of the signal, which could possibly excite highfrequency mechanical resonances. The use of such output filters, however,should be considered with care because they have the same detrimentaleffect on the phase of the control system as the prefilter at the input.

In applications, it is advisable to use a sampling frequency at least 20times, and preferably 100 times higher than the crossover frequency ofthe continuous design, to preserve the behavior of the continuous systemto a reasonable degree.

14.1.3 Quantization

After prefiltering at a frequency fc below the Nyquist frequency fs/2, thesignal is passed into the ADC for sampling and conversion into a digitalsignal of finite word length (typically N=14 or 16 bits) representing thetotal range of the analog signal. Because of the finite number of quanti-zation levels, there is always a roundoff error which represents 2−N times

14.1 Digital implementation 355

the full range of the signal; the quantization error can be regarded as arandom noise. The signal to noise ratio is of the order 2N provided thatthe signal is properly scaled to use the full range of the ADC. Near theequilibrium point, only a small part of the dynamic range is used by thesignal, and the signal to noise ratio drops substantially.

The quantization error is also present at the output of the DAC; thefinite word length of the digital output is responsible for a finite resolutionin the analog output signal; the resolution of the output is δ = R/2M ,where R is the dynamic range of the output and M the number of bits ofthe DAC. To appreciate the limitations associated with this formula, con-sider a positioning problem with a range of R = 10 mm and a DAC of 16bits; the resolution on the output will be limited to δ = 10/216 = 0.15µm.Quantization errors may be responsible for limit cycle oscillations.

Let us briefly mention that the finite word length arithmetic in thedigital controller is another source of error, because finite word lengthoperations are no longer associative or distributive, due to rounding. Weshall not pursue this matter which is closer to the hardware (e.g. Jackson).

14.1.4 Discretization of a continuous controller

Although all the design methods exist in discrete form, it is quite commonto perform a continuous design, and to discretize it in a second step. Thisprocedure works quite well if the sampling rate fs is much higher thanthe crossover frequency fc of the control system (in structural control, itis quite customary to have fs/fc ' 100).

Assume that the compensator transfer function has been obtained inthe form

U(s)Y (s)

= H(s) =b1s

n−1 + . . . + bn

sn + a1sn−1 + . . . + an(14.4)

For digital implementation, it must be transformed to the form of a dif-ference equation

u(k) =n∑

i=1

αiu(k − i) +m∑

j=0

βjy(k − j) (14.5)

The corresponding z-domain transfer function is

U(z)Y (z)

= H(z) =∑m

j=0 βjz−j

1−∑ni=1 αiz−i

(14.6)

356 14 Applications

where z−1 is the delay operator. The coefficients αi and βj of H(z) canbe obtained from those of H(s) following Tustin’s method: H(z) and H(s)are related by the bilinear transform

s =2(z − 1)T (z + 1)

or z =1 + Ts/21− Ts/2

(14.7)

where T is the sampling period. This transformation maps the left half s-plane into the interior of the unit circle in the z-plane, and the imaginaryaxis from ω = 0 to ∞ into the upper half of the unit circle from z = 1 toz = −1 (e.g. see Franklin & Powell or Oppenheim & Schafer).

Tustin’s method can be applied to multivariable systems written instate variables; for the continuous system described by

x = Acx + Bcu (14.8)

y = Cx + Du (14.9)

the corresponding discrete system resulting from the bilinear transform(14.7) reads

x(k + 1) = Ax(k) + B1u(k + 1) + B0u(k) (14.10)

y = Cx + Duk (14.11)

withA = [I −Ac

T

2]−1[I + Ac

T

2] (14.12)

B1 = B0 = [I −AcT

2]−1 T

2Bc (14.13)

The proof is left as an exercise (Problem 14.3).

14.2 Active damping of a truss structure

One of the earliest active damping experiments that we performed atULB is that with the truss of Fig.4.20, built in 1989. It consists of 12bays of 14 cm each, made of steel bars of 4mm diameter; it is clampedat the bottom, and two active struts are located in the lower bay. Similarstudies were performed at other places at about the same time (Fansonet al., Chen et al., Peterson et al.). The distinctive feature of this workwas that the active strut was built with low cost commercial components(Philips linear piezoelectric actuator and Bruel & Kjaer piezoelectric force

14.2 Active damping of a truss structure 357

Mode 1(8.8 Hz)

Mode 2(10.5 Hz)

2

2

98

1

10

9

z

z

x

x

y

y

Element #

98 15.7 0.1

10 2.6 9.4

2 3.1 11.4

1 4.1 11.2

9 3.4 9.4

ν1 (%) ν2 (%)

Fig. 14.3. Finite element model of the truss of Fig.4.20, mode shapes and fractionof modal strain energy in selected elements; the active members have been placed inelements No 2 and 98.

358 14 Applications

sensor). The design was such that the length and the stiffness of the activestrut almost exactly matched that of one bar; in this way, the insertion ofthe active element did not change the stiffness of the structure. Becauseof the high-pass nature of the piezoelectric force sensors (and electronics),only the dynamic component of the force is measured by the force sensor.Other types of active members with built-in viscous damping have beendeveloped (Hyde & Anderson).

The mathematical modelling of an active truss was examined in section4.9 and the active damping with Integral Force Feedback was investigatedin section 7.5. It was found that the closed-loop poles of an active trussprovided with a single active element follow the root locus defined byEqu.(7.31).

14.2.1 Actuator placement

More than any specific control law, the location of the active member isthe most important factor affecting the performance of the control system.The active element should be placed where its authority over the modesit is intended to control is the largest. According to Equ.(7.35), the con-trol authority is proportional to the fraction of modal strain energy in theactive element, νi. It follows that the active struts should be located in or-der to maximize νi in the active members for the critical vibration modes.The search for candidate locations where active struts can be placed isgreatly assisted by the examination of the map of the fraction of strainenergy in the structural elements, which is directly available in commer-cial finite element packages. Such a map is presented in Fig.14.3 ; one seesthat substituting the active member for the bar No 98 provides a strongcontrol on mode 1 [ν1 = 0.157](in-plane bending mode), but no controlon mode 2 (out-of-plane bending mode), which is almost uncontrollablefrom an active member placed in bar No 98 [ν2 = 0.001]. By contrast, anactive member substituted for the bar No 2 offers a reasonable control onmode 1 [ν1 = .031] and excellent control on mode 2 [ν2 = 0.114]; thesetwo locations were selected in the design. The fraction of modal strainenergy is well adapted to optimization techniques for actuator placement.

14.2 Active damping of a truss structure 359

14.2.2 Implementation, experimental results

1Using the bilinear transform (14.7), we can readily transform the integralcontrol law (IFF)

δ =g

sy (14.14)

into the difference equation

δi+1 = δi + gT

2(yi+1 + yi) (14.15)

which we recognize as the trapezoid rule for integration. In order to avoidsaturation, it is wise to slightly modify this relation according to

δi+1 = αδi + gT

2(yi+1 + yi) (14.16)

where α is a forgetting factor slightly lower than 1. α depends on thesampling frequency; it can either be tuned experimentally or obtainedfrom a modified compensator

δ =g

s + ay (14.17)

where the breakpoint frequency a is such that a ¿ ω1 (the first naturalfrequency), to produce a phase of 90o for the first mode and above (Prob-lem 14.5). Note that, for a fast sampling rate, the backward differencerule

δi+1 = αδi + gTyi+1 (14.18)

works just as well as (14.16). In our experiment, the two active membersoperated independently in a decentralized manner with fs =1000 Hz.Figure 14.4 shows the force signal in the active members during the freeresponse after an impulsive load, first without, and then with control.Figure 14.5 shows the frequency response between a point force appliedat A along the truss and an accelerometer located at B, at the top ofthe truss. A damping ratio larger than 0.1 was obtained for the first twomodes. Finally, it is worth pointing out that : (i) The dynamics of thecharge amplifier does not influence the result appreciably, provided thatthe corner frequency of the high-pass filter is significantly lower than the1 The implementation of the IFF controller presented here is that done at the time of

this experiment. Other aspects of the control implementation, particularly concern-ing the recovery of the static stiffness of the uncontrolled structure, will be addressedin section 15.5.

360 14 Applications

Force # 1

Force # 2

Control at this timeon

Free response

Impulse

Fig. 14.4. Force signal from the two active members during the free response after animpulsive load (experimental results).

A

B

with control

without control

Hz8.8 10.5

Fig. 14.5. FRF between A and B, with and without control (experimental results,linear scale).

14.3 Active damping generic interface 361

natural frequency of the targeted mode. (ii) In this application as inall applications involving active damping with piezo struts, no attemptwas made to correct for the large hysteresis of the piezotranslator; it wasfound that the hysteresis does not deteriorate the closed-loop responsesignificantly, as compared to the linear predictions.

14.3 Active damping generic interface

The active strut discussed in the previous section can be integrated intoa generic 6 d.o.f. interface connecting arbitrary substructures. Such aninterface is shown in Fig.14.6.a and b (the diameter of the base platesis 250 mm); it consists of a Stewart platform with a cubic architecture[this provides a uniform control capability and uniform stiffness in all di-rections, and minimizes the cross-coupling thanks to mutually orthogonalactuators (Geng & Haynes)]. However, unlike in section 8.5.2 where eachleg consists of a single d.o.f. soft isolator, every leg consists of an activestrut including a piezoelectric actuator, a force sensor and two flexibletips.

14.3.1 Active damping

The control is a decentralized IFF with the same gain for all loops. Let

Mx + Kx = 0 (14.19)

be the dynamic equation of the passive structure (including the interface).According to section 4.9, the dynamics of the active structure is governedby

Mx + Kx = BKaδ (14.20)

where the right hand side represents the equivalent piezoelectric loads :δ = (δ1, ..., δ6)T is the vector of piezoelectric extensions, Ka is the stiff-ness of one strut and B is the influence matrix of the interface in globalcoordinates. The output y = (y1, ..., y6)T consists of the six force sensorsignals which are proportional to the elastic extension of the active struts

y = Ka(q − δ) (14.21)

where q = (q1, ..., q6)T is the vector of global leg extensions, related to theglobal coordinates by

q = BT x (14.22)

362 14 Applications

Fig. 14.6. Stewart platform with piezoelectric legs as generic active damping interface.(a) General view. (b) With the upper base plate removed. (c) Interface acting as asupport of a truss.

The same matrix appears in Equ.(14.20) and (14.22) because the actua-tors and sensors are collocated. Using a decentralized IFF with constantgain on the elastic extension,

δ =g

Kasy (14.23)


the closed-loop characteristic equation is obtained by combining Equ.(14.20)to (14.23):

[Ms2 + K − g

s + gBKaB

T ]x = 0 (14.24)

In this equation, the stiffness matrix K refers to the complete structure,including the full contribution of the Stewart platform (with the piezo-electric actuators with short-circuited electrodes). The open-loop polesare ±jΩi where Ωi are the natural frequencies of the complete struc-ture. The open-loop zeros are the asymptotic values of the eigenvalues ofEqu.(14.24) when g −→∞; they are solution of

[Ms2 + K −BKaBT ]x = 0 (14.25)

The corresponding stiffness matrix is K−BKaBT where the axial stiffness

of the legs of the Stewart platform has been removed from K. Withoutbending stiffness in the legs, this matrix is singular and the transmissionzeros include the rigid body modes (at s = 0) of the structure wherethe piezo actuators have been removed. However, the flexible tips areresponsible for a non-zero bending stiffness of the legs and the eigenvaluesof Equ.(14.25) are located at ±jωi, at some distance from the origin alongthe imaginary axis.

Upon transforming into modal coordinates, x = Φz and assuming thatthe normal modes are normalized according to ΦT MΦ = I, we get

[s2 + Ω2 − g

s + gΦT BKaB

T Φ]z = 0 (14.26)

whereΩ2 = diag(Ω2

i ) = ΦT KΦ (14.27)

As in section 7.5, the matrix ΦT BKaBT Φ is, in general, fully populated;

assuming it is diagonally dominant and neglecting the off diagonal terms,we can rewrite it

ΦT BKaBT Φ ' diag(νiΩ

2i ) (14.28)

where νi is the fraction of modal strain energy in the active dampinginterface, that is the fraction of the strain energy concentrated in thelegs of the Stewart platform when the structure vibrates according to theglobal mode i. From the definition of the open-loop transmission zeros,±jωi, we also have

ω2 ' diag(ω2i ) = ΦT (K −BT KaB)Φ = diag[Ω2

i (1− νi)] (14.29)

364 14 Applications

and the characteristic equation (14.26) can be rewritten as a set of un-coupled equations

s2 + Ω2i −

g

s + g(Ω2

i − ω2i ) = 0 (14.30)

or

1 + gs2 + ω2

i

s(s2 + Ω2i )

= 0 (14.31)

This equation is identical to Equ.(7.31) and all the results of section 7.5apply. Note that, in this section, the previous results have been extendedto a multi-loop decentralized controller with the same gain for all loops.

14.3.2 Experiment

Fig. 14.7. Impulse response and FRF of the truss mounted on the active interface(experimental results, Abu-Hanieh).

The test set-up is shown in Fig.14.6.c; the interface is used as a supportfor the truss discussed in the previous section (used as a passive truss inthis case). The six independent controllers have been implemented ona DSP board; the feedback gain is the same for all the loops. Figure


Fig. 14.8. Poles, zeros and experimental root locus for the truss mounted on the activeinterface. The continuous lines are the root locus predictions from Equ.(14.31).

14.7 shows some typical experimental results; the time response showsthe signal from one of the force sensors of the Stewart platform whenthe truss is subjected to an impulse at mid height from the base, firstwithout, and then with control. The FRFs (with and without control) areobtained between a disturbance applied to the piezoactuator in one legand its collocated force sensor. One sees that fairly high damping ratioscan be achieved for the low frequency modes (4−5Hz) but also significantdamping in the high frequency modes (40−90Hz). The experimental rootlocus of the first two modes is shown in Fig.14.8; it is compared to theanalytical prediction of Equ.(14.31). In drawing Fig.14.8, the transmissionzeros ±jωi are taken as the asymptotic natural frequencies of the systemas g →∞.

14.3.3 Pointing and position control

As a closing remark, we wish to emphasize the potential of the stiff Stewartplatform described here for precision pointing and precision control. Withpiezoceramic actuators of 50 µm stroke, the overall stroke of the platformis 90, 103 and 95 µm along the x, y and z directions (in the payload

366 14 Applications

plate axis of Fig.8.30) and 1300, 1050 and 700 µrad around the x, y and zaxes, respectively. Embedding active damping in a precision pointing orposition control loop can be done with the HAC/LAC strategy discussedin section 14.6.

14.4 Active damping of a plate

In 1993, at the request of ESA, we developed a laboratory demonstrationmodel of an active plate controlled by PZT piezoceramics; it was latertransformed into a flight model (to be flown in a canister) by our in-dustrial partner SPACEBEL and the experiment (named CFIE: Control-Flexibility Interaction Experiment), was successfully flown by NASA inthe space shuttle in September 1995 (flight STS-69).

Chargeamplifier

DSP

Voltageamplifier

PZT piezoceramic

Support Structure

Laser mirror

Additional Masses

g

Fig. 14.9. Laboratory demonstration model of the CFIE experiment.

According to the specifications, the experiment should fit into a “GAS”canister (cylinder of 50 cm diameter and 80 cm high), demonstrate signif-icant gravity effects, and use the piezoelectric technology. We settled ona very flexible steel plate of 0.5 mm thickness hanging from a support asshown in Fig.14.9; two additional masses were mounted, as indicated inthe figure, to lower the natural frequencies of the system. The first modeis in bending and the second one is in torsion. Because of the additionalmasses, the structure has a significant geometric stiffness due to the grav-ity loads, which is responsible for a rise of the first natural frequency from

14.4 Active damping of a plate 367

f1 =0.5Hz in zero gravity to 0.9Hz with gravity. The finite element modelof the structure in the gravity field could be updated to match the exper-imental results on the ground, but the in-orbit natural frequencies couldonly be predicted numerically and were therefore subject to uncertainties.

As we know from the previous chapters, in order to achieve activedamping, it is preferable to adopt a collocated actuator/sensor configura-tion. In principle, a strictly collocated configuration can be achieved withself-sensing actuators (Dosch et al.), but from our own experience, thesesystems do not work well, mainly because the piezoceramic does not be-have exactly like a capacitance as assumed in the self-sensing electronics.As a result, self-sensing was ruled out and we decided to adopt a nearlycollocated configuration, which is quite sufficient to guarantee alternatingpoles and zeros at low frequencies. However, as we saw in section 4.8.7,nearly collocated piezoelectric plates are not trivial to model, because ofthe importance of the membrane strains in the input-output relationship;this project was at the origin of our work on the finite element modellingof piezoelectric plates and shells (Piefort).

14.4.1 Control design

According to section 7.4, achieving a large active damping with a PositivePosition Feedback (PPF) and strain actuator and sensor pairs relies ontwo conditions: (i) obtaining a precise tuning of the controller naturalfrequency on the targeted mode and (ii) using an actuator/sensor config-uration leading to sufficient spacing between the poles and the zeros, sothat wide loops can be obtained. We will discuss the tuning issue a littlelater; for nearly collocated systems, the distance between the poles andthe zeros depends strongly on local effects in the strain transmission.

In the CFIE experiment, the control system consists of two inde-pendent control loops with actuator/sensor pairs placed as indicated inFig.14.9; finite element calculations confirmed that the spacing betweenthe poles and the zeros was acceptable. The controller consists of twoindependent PPF loops, each of them targeted at modes 1 and 2 of thestructure, respectively at f1 =0.86Hz and f2 =3.01Hz with gravity andf1 =0.47Hz and f2 =2.90Hz in zero gravity (predicted from finite elementcalculations). The compensator reads

D(s) =g1 ω2

f1

s2 + 2ξf1ωf1s + ω2f1

+g2 ω2

f2

s2 + 2ξf2ωf2s + ω2f2

(14.32)

368 14 Applications

The determination of the gains g1 and g2 requires some trial and error; asalready mentioned, it is generally simpler to adjust the gain of the filterof higher frequency first, because the roll-off of the second order filterreduces the influence from the filter tuned on a lower frequency. Notethat, although its stability is guaranteed for moderate values of g1 andg2, the performance of the PPF depends heavily on the tuning of the filterfrequencies ωf1 and ωf2 on the targeted modes ω1 and ω2. It is thereforeessential to predict the natural frequencies accurately.

0

2

4

6

8

10

12

14

-20 -10 0 10 20 30

ξ 1 (%)

∆ω

ω(%)f

f

Fig. 14.10. Sensitivity of the performance to the tuning of the controller.

Time (s)

0 2 4 6 8 10 12 14

control off control on

Fig. 14.11. Free response after some disturbance (laser sensor).

14.5 Active damping of a stiff beam 369

To illustrate the degradation of the performance when the controlleris not tuned properly, Fig.14.10 shows the sensitivity of the performance,taken as the maximum closed-loop modal damping of the first mode, as afunction of the relative error in the frequency of the PPF filter; ∆ωf = 0corresponds to the optimally tuned filter, leading to a modal dampingξ1 over 0.13. We see that even small tuning errors can significantly re-duce the performance, and that a 20 % error makes the control systemalmost ineffective. This problem was particularly important in this exper-iment where the first natural frequency could not be checked from tests.Fig.14.11 illustrates the performance of the control system on the labora-tory demonstration model; it shows the free response measured by laserof one of the additional mass after some disturbance, with and withoutcontrol, when the tuning is optimal.

The laboratory demonstration model shown in Fig.14.9 is very flimsyand would not withstand the environmental loads (static and dynamics)during the launch of the spacecraft; the test plate would even buckle un-der its own weight if turned upside down. As a result, the flight model wassubstantially reinforced with a strong supporting structure, and a latchingmechanism was introduced to hold the plate during the launch. The flightmodel successfully underwent the vibration tests before launch, but thecharge amplifiers were destroyed (!), because the amount of electric chargegenerated during the qualification tests was several orders of magnitudelarger than the level expected during the in-orbit experiment; the prob-lem was solved by changing the electronic design, to include low leakagediodes with appropriate threshold at the input of the charge amplifiers.No problem occurred during the flight.

14.5 Active damping of a stiff beam

We begin with a few words about the background in which this problemwas brought to our attention in the early 90’s. Optical instruments forspace applications require an accuracy on the wave front in the rangeof 10 nm to 50 nm. The ultimate performance of the instruments mustbe evaluated on earth, before launch, in a simulated space environment.This is done on sophisticated test benches resting on huge seismicallyisolated slabs and placed in a thermal vacuum chamber. Because of theconstraint on accuracy, the amplitude of the microvibrations must remainbelow 1 nm or, equivalently, if the first natural frequency of the supportingstructure is around 60Hz, the acceleration must remain below 10−5g. This

370 14 Applications

limit is fairly easy to exceed, even under such apparently harmless exci-tations as the noise generated by the air conditioning of the clean room.Beyond the specific problem that we have just mentioned, the damping ofmicrovibrations is a fairly generic problem which has many applicationsin other fields of precision engineering, such as machine tools, electroniccircuit lithography, etc...

14.5.1 System design

Fig. 14.12. Test structure and impulse response, with and without control.

A simple active damping device has been developed, based on thefollowing premises: (i) The control system should use an accelerometerwhich is more appropriate than a displacement or a velocity sensor forthis problem (an acceleration of 10−5g can be measured with a commercialaccelerometer, while a displacement of 1 nm requires a sophisticated laserinterferometer). (ii) The structures considered here are fairly stiff and wellsuited to the use of a proof-mass actuator without excessive stroke (section3.2.1). (iii) The sensor and the actuator should be collocated, in order tobenefit from guaranteed stability.

14.6 The HAC/LAC strategy 371

The test structure is represented in Fig.14.12; it consists of a 40 kgsteel beam of 4.7 m, mounted on three supports located at the nodes of thesecond free-free mode, to minimize the natural damping. The first naturalfrequency of the beam is f1 = 68Hz. The proof-mass actuator consists ofa standard electrodynamic shaker (Bruel & Kjaer 4810) fitted with anextra mass of 500 gr, to lower its natural frequency to about 20Hz. Inthis way, the amplitude diagram of the frequency response F/i is nearlyconstant for f >40Hz, indicating that the proof-mass actuator behavesnearly as an ideal force generator (section 3.2.1). The phase diagram isalso nearly flat above 40Hz, but contains a linear phase due to the digitalcontroller.

The control law can be either g/s, leading to a Direct Velocity Feed-back, or the set of second order filters as discussed in Problem 7.2. Bothhave guaranteed stability (assuming perfect actuator and sensor dynam-ics). In choosing between the two alternatives, we must take the follow-ing aspects into account: (i) Since the transfer function of the structuredoes not have any roll-off, the roll-off of the open-loop system is en-tirely controlled by the compensator. (ii) The Direct Velocity Feedbackis wide band, while the acceleration feedback, based on second order fil-ters, must be tuned on the targeted modes. (iii) In theory, the phasemargin of the Direct Velocity Feedback is 900 for all modes, but its roll-off is only −20 dB/decade. The acceleration feedback has a roll-off of−40 dB/decade, but the phase margin gradually vanishes for the modeswhich are above the frequency appearing in the filter of the compensator(Problem 14.6). Based on the foregoing facts and depending on the struc-ture considered, one alternative may be more effective than the other innot destabilizing the high frequency dynamics, which is more sensitiveto the finite dynamics of the actuator and sensor, delays, etc... For thetest structure of Fig.14.12, which is fairly simple and does not involveclosely spaced modes, both compensators have been found very effective;the damping ratio of the first mode has been increased from ξ1 =0.002 toξ1 =0.04.

14.6 The HAC/LAC strategy

In active structures for precision engineering applications, the control sys-tem is used to reduce the effect of transient and steady state disturbanceson the controlled variables. Active damping is very effective in reducingthe settling time of transient disturbances and the effect of steady state

372 14 Applications

disturbances near the resonance frequencies of the system; however, awayfrom the resonances, the active damping is completely ineffective andleaves the closed-loop response essentially unchanged. Such low gain con-trollers are often called Low Authority Controllers (LAC), because theymodify the poles of the system only slightly (Aubrun).

To attenuate wide-band disturbances, the controller needs larger gains,in order to cause more substantial modifications to the poles of the open-loop system; this is the reason why they are often called High AuthorityControllers (HAC). Their design requires a model of the structure and, aswe saw in chapter 10, there is a trade-off between the conflicting require-ments of performance-bandwidth and stability in the face of parametricuncertainty and unmodelled dynamics. The parametric uncertainty re-sults from a lack of knowledge of the structure (which could be reducedby identification) or from changing environmental conditions, such as theexposure of a spacecraft to the sun. Unmodelled dynamics include allthe high frequency modes which cannot be predicted properly, but arecandidates for spillover instability.

g D(s)

G (s)o

G(s,g)-

+

-

H(s)r ye u

LAC : collocatedactive damping

StructureHAC compensator

(model-based)

Fig. 14.13. Principle of the dual loop HAC/LAC control.

When collocated actuator/sensor pairs can be used, stability can beachieved using positivity concepts (Benhabib et al.), but in many situa-tions, collocated pairs are not feasible for HAC; we know from chapter6 that such configurations do not have a fixed pole-zero pattern and aremuch more sensitive to parametric uncertainty.

LQG controllers are an example of HAC; their lack of robustness withrespect to the parametric uncertainty was pointed out in section 9.10.The situation is even worse for the unmodelled dynamics, particularly


for very flexible structures which have a high modal density, becausethere are always flexible modes near the crossover frequency. Withoutfrequency shaping, LQG methods often require an accurate modellingfor approximately two decades beyond the bandwidth of the closed-loopsystem, which is unrealistic in most practical situations. The HAC/LACapproach originated at Lockheed in the early 80’s; it consists of combin-ing the two approaches in a dual loop control as shown in Fig.14.13. Theinner loop uses a set of collocated actuator/sensor pairs for decentralizedactive damping with guaranteed stability; the outer loop consists of anon-collocated HAC based on a model of the actively damped structure.This approach has the following advantages:

• The active damping extends outside the bandwidth of the HAC andreduces the settling time of the modes which are outside the band-width.

• The active damping makes it easier to gain-stabilize the modes outsidethe bandwidth of the outer loop (improved gain margin).

• The larger damping of the modes within the controller bandwidthmakes them more robust to the parametric uncertainty (improvedphase margin).

Singular value robustness measures generalize the phase and gain marginfor MIMO systems; some of these tests are discussed in section 10.9 (seealso Kosut et al., or Mukhopadhyay & Newsom).

14.6.1 Wide-band position control

In order to illustrate the HAC/LAC strategy for a non-collocated system,let us consider once again the active truss of Fig.4.20. The objective is todesign a wide-band controller using one of the piezo actuators to controlthe tip displacement y along one coordinate axis (Fig.14.14), measured bya laser interferometer. The compensator should have some integral actionat low frequency, to compensate the thermal perturbations and avoidsteady state errors; the targeted bandwidth of 100 rad/s includes thefirst two vibration modes. Note that the actuator and the displacementsensor are located at opposite ends of the structure, so that the actuatoraction cannot be transmitted to the sensor without exciting the entiretruss.

The LAC consists of the active damping discussed in section 14.2; thetransfer function G(ω, g) between the input voltage of the actuator and

374 14 Applications

the tip displacement y is shown in Fig.14.15 for various values of the gaing of the active damping. One observes that the active damping works verymuch like passive damping, affecting only the frequency range near thenatural frequencies. Below 100 rad/s, the behavior of the system is domi-nated by the first mode; the second mode does not substantially affect theamplitude of G(ω, g), and the phase lag associated with the second modeis compensated by the phase lead of a zero at a frequency slightly lowerthan ω2 (although not shown, the general shape of the phase diagram canbe easily drawn from the amplitude plot). From these observations, weconclude that mode 2, which is close to mode 1, will be phase-stabilizedwith mode 1 and, as a result, the compensator design can be based ona model including a single vibration mode; the active damping can beclosely approximated by passive damping. Thus, the compensator designis based on the very simple model of a damped oscillator.

LaserInterferometer

PositionControl

ActiveDamping

ActiveDamping

+

+

y

HAC

LAC

u

Fig. 14.14. Wide-band position control of the truss. The objective is to control thetip displacement y with one of the piezo actuators; the HAC/LAC controller involvesan inner active damping loop with collocated actuator/sensor pairs.


-30

-20

-10

0

10

20

30

40

10 100

dB

ω (rad / s)

g = 0

g = 0 5.

g = 1

Fig. 14.15. FRF y/u of the structure for various values of the gain g of the activedamping (experimental results).

14.6.2 Compensator design

The compensator should be designed to achieve integral action at lowfrequency and to have enough roll off at high frequency to avoid spilloverinstability. The standard LQG is not well suited to these requirements,because the quadratic performance index puts an equal weight on allfrequencies; the design objectives require larger weights on the controlat high frequency to avoid spillover, and larger weights on the states atlow frequency to achieve integral action; both can be achieved by thefrequency-shaped LQG as explained in section 11.16. The penalty on thehigh frequency components of the control u is obtained by passing thecontrol through a low-pass filter (selected as a second order Butterworthfilter in this case) and the P+I action is achieved by passing the outputy (which is also the control variable z) through a first order system asindicated in Fig.11.9. The state feedback is obtained by solving the LQRproblem for the augmented system with the quadratic performance index

E[zT1 z1 + %uT

1 u1] (14.33)

The structure of the compensator is that of Fig.11.10; the frequency dis-tribution of the weights for the original problem is shown in Fig.11.8 ; thelarge weights Q(ω) on the states at low frequency correspond to the inte-gral action, and the large penalty R(ω) on the control at high frequencyaims at reducing the spillover. The states of the structure (only two in

376 14 Applications

this case) must be reconstructed with an observer; in this case, a Kalmanfilter is used; the noise intensity matrices have been selected to achievethe appropriate dynamics.

14.6.3 Results

The Bode plots of the compensator are shown in Fig.14.16; it behaveslike an integrator at low frequency, provides some phase lead near theflexible mode and crossover, and roll-off at high frequency. The open-looptransfer function of the control system, GH, is shown in Fig.14.17 (G cor-responds to the model); the bandwidth is 100 rad/s and the phase marginis PM = 38o. The effect of this compensator on the actual structure G∗

can be assessed from Fig.14.18. As expected, the second flexible modeis phase stabilized and does not cause any trouble. On the other hand,we observe several peaks corresponding to higher frequency modes in theroll-off region; some of these peaks exceed 0 dB and their stability mustbe assessed from the Nyquist plot, which is also represented in Fig.14.18.We see that the first peak exceeding 1 in the roll-off region (noted 1 inFig.14.18) is indeed stable (it corresponds to the wide loop in the right

ω ( / )rad s

ω ( / )rad s

H

φ

Fig. 14.16. Bode plots of the compensator H(ω).


PM=38.8°

Integral effect

Increased Roll-off

ω ( / )rad s

ω ( / )rad s

GH

φ

dB

Fig. 14.17. Bode plots of the simplified model G(ω, g)H(ω).

side of the Nyquist plot). The second peak in the roll-off region (noted 2)is slightly unstable for the nominal gain of the compensator; some reduc-tion of the gain is necessary to achieve stability (small loop near -1 in theNyquist plot); this reduces the bandwidth to about 70 rad/s. A detailedexamination showed that the potentially unstable mode corresponds to alocal mode of the support of the mirror of the displacement measurementsystem. This local mode is not influenced by the active damping; the sit-uation could be improved by a redesign of the support for more stiffnessand more damping (e.g. passive damping locally applied). This controllerhas been implemented digitally on a DSP processor with a sampling fre-quency of 1000 Hz. Figure 14.19 compares the predicted step responsewith the experimental one. The settling time is reduced to 0.2 s, about10 times faster than what would be achievable with a PID compensator.

378 14 Applications

0

20

-20

-40

0°

-500°

-1000°

-1500°

| |G H*

f

(1)

(2)

w ( )rad/s

10

1

10

2

10

3

w ( )rad/s

10

1

10

2

10

3

(1)

(2)

(-1,0)

Fig. 14.18. Open-loop transfer function G∗(ω, g)H(ω) of the actual control system,Bode plots and Nyquist plot demonstrating the stability (with experimental G∗).

14.7 Vibroacoustics: Volume displacement sensors 379

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4

Time (sec)

Simulation

Experiment

Fig. 14.19. Step response of the control system, comparison between predictions andexperimental results of the top displacement.

14.7 Vibroacoustics: Volume displacement sensors

The general problem of Active Structural Acoustic Control (ASAC) of abaffled plate is represented in Fig.14.20. The performance objective is tominimize the far field radiated noise. The control system consists of one or

Structural

sensor

ActuatorFeedback

controller

Sound power

(at low frequencies)

=~

Transmitted

noise

Baffled plate

Rigid

wall

Acoustic

disturbance

Fig. 14.20. Active Structural Acoustic Control (ASAC) of a baffled plate.

380 14 Applications

several actuators acting on the baffled plate itself and a structural sensormeasuring in real time the sound power radiated by the plate. This sectionis focused on the construction of a volume velocity (or displacement)sensor.

The volume velocity V of a vibrating plate is defined as

V =∫

Sw dS (14.34)

where w is the transverse displacement of the plate and the integral ex-tends over the entire plate area. It is a fairly important quantity in vibroa-coustics, because it is strongly correlated to the sound power radiated bythe plate (Johnson & Elliott), and the modes which do not contribute tothe net volume velocity (anti-symmetric modes for a symmetric plate) arepoor contributors to the sound power radiations at low frequency (Fahy).In this section, we discuss the sensing of the volume velocity with an ar-rangement of piezoelectric sensors; note that the same sensor arrangementcan be used to measure the volume displacement V by using a charge am-plifier instead of a current amplifier as we discussed in Fig.4.5, so that thetwo quantities are fully equivalent from a sensor design viewpoint.

This section examines three totally different concepts for sensing thevolume displacement with piezoelectric sensors; the first one is based ona distributed sensor initially developed for beams, and extended to platesby discretizing them into narrow strips; it is biased, due to the inabilityof the beam theory to represent two-dimensional structures. The secondis based on a discrete array sensor connected to a linear combiner; itis subjected to spatial aliasing. The third concept is based on a porouselectrode design which allows to tailor the effective piezoelectric propertiesof piezoelectric films.

14.7.1 QWSIS sensor

The Quadratically Weighted Strain Integrator Sensor (QWSIS ) is a dis-tributed sensor which applies to any plate without rigid body mode (Rex& Elliott).

BeamConsider a beam fixed at both ends: w(0) = w(a) = 0; it is covered

with a piezoelectric film sensor (e.g. PVDF) with a parabolic electrode,as indicated in Fig.14.21; the profile of the electrode is defined by


Fig. 14.21. Beam covered with a PVDF sensor with a parabolic electrode.

bp(x) = 4∆x

a(1− x

a) (14.35)

According to Equ.(4.32), if the electrode is connected to a charge am-plifier, the sensor output is

v0 ∼∫ a

0bpw

′′dx (14.36)

Upon integrating by parts twice, and taking into account the boundaryconditions bp(0) = bp(a) = 0 and w(0) = w(a) = 0, we get

v0 ∼∫ a

0b′′pw dx (14.37)

Since the width of an electrode of parabolic shape has a constant secondderivative with respect to the space coordinate, the output of the sensoris proportional to the volume displacement:

v0 ∼∫ a

0w dx (14.38)

PlateIn the QWSIS, the plate is discretized in a set of narrow strips

(Fig.14.22) which are provided with parabolic electrodes connected inseries; if we consider the elementary strips as beams, the total amount ofelectric charge is proportional to the volume displacement of the plate.

The QWSIS sensor is based on the beam theory, but the actual be-havior of the plate produces curvatures in two directions; assuming thatthe piezoelectric orthotropy axis 1 of the sensor coincides with the x axis

382 14 Applications

Fig. 14.22. QWSIS sensor.

of the strip, the amount of electric charges generated by the sensor canbe obtained by integrating Equ.(4.80) over the electrode area Ω, with theelectrical boundary condition E = 0 enforced by the charge amplifier:

Q =∫

ΩD dS =

∫

Ω(e31S1 + e32S2) dS (14.39)

where S1 and S2 are the strain components along the orthotropy axes inthe mid-plane of the sensor. If the membrane strains in the plate are smallas compared to the bending strains,

Q = −zm

∫

Ω(e31

∂2w

∂x2+ e32

∂2w

∂y2) dS (14.40)

where zm is the distance between the mid-plane of the sensor and themid-plane of the baffle plate [see Equ.(4.95)]. If e32 = 0, this equation isreduced to that of a beam, which means that the convergence of the sensoris guaranteed when the number of strips increases. However, althoughstrongly anisotropic, PVDF exhibits a piezoelectric coefficient e32 whichis at least 20% of e31, which introduces a bias in the sensor output.


Dual actuatorA piezoelectric strip can be used either as a sensor or as an actuator;

in the latter case, according to the beam theory, a distributed actuator ofwidth bp(x) produces a distributed load proportional to the second deriva-tive of the width of the electrode, b′′p(x). Accordingly, if the QWSIS is usedas an actuator, it is equivalent to a uniform pressure actuator (Fig.4.3.c).This led to the idea of building a collocated active structural acoustic(ASAC) plate with one side covered with a QWSIS volume displacementsensor and the opposite side covered with the dual actuator (Gardonioet al.). Unfortunately, such an arrangement performs poorly, because theinput-output relationship between the strain actuator and the strain sen-sor is dominated by the membrane strains in the plate, which have beenignored in the theory, and are not related to the transverse displacementsw (w is the useful output of the system). The anisotropy of PVDF can beexploited to improve the situation, by placing the strips of the actuatorand the sensor orthogonal to each other (Piefort, p.91).

14.7.2 Discrete array sensor

In this section, we discuss an alternative set-up using a discrete array of nstrain sensors bonded on the plate according to a regular mesh (Fig.14.23).The strain sensors consist of piezo patches connected to individual chargeamplifiers with output Qi; they are connected to a linear combiner, theoutput of which is

y =n∑

i=1

αiQi (14.41)

The coefficients of the linear combiner can be adjusted by software inorder that the sensor output y be as close as possible (in some sense) of adesired quantity such as a modal amplitude, or, in this case, the volumedisplacement.

The electric charges Qi generated by each strain sensor is a linearcombination of the modal amplitude zj :

Qi =∑

j

qijzj (14.42)

where qij is the electric charge generated on sensor i by a unit amplitudeof mode j.

The volume displacement V is also a linear combination of the modalamplitude,

384 14 Applications

Fig. 14.23. Principle of the discrete array sensor of n patches.

V =m∑

j=1

Vjzj (14.43)

where Vj is the modal volume displacement of mode j. At low frequency,V is dominated by the contribution of the first few modes and thereforeonly these modal amplitudes, zj , j = 1, ...,m, have to be reconstructedfrom the electric charges Qi produced by a redundant set of piezoelectricstrain sensors (n > m), leading to

zj =∑

i

ajiQi (14.44)

(where the coefficients aij are unknown at this stage). Combining withEqu.(14.43), we find

V =∑

j

∑

i

VjajiQi =∑

i

αiQi (14.45)

whereαi =

∑

j

Vjaji (14.46)

Equation (14.45) has the form of a linear combiner with constant coeffi-cients αi; it can be rewritten in the frequency domain

V (ω) =n∑

i=1

αiQi(ω) (14.47)


where V (ω) is the FRF between a disturbance applied to the baffled plateand the volume displacement, measured with a laser scanner vibrometer,Qi(ω) is the FRF between the same disturbance and the electric chargeon sensor i in the array. If this equation is written at a set of l discrete fre-quencies (l > n) regularly distributed over the frequency band of interest,it can be transformed into a redundant system of linear equations,

Q1(ω1) ... Qn(ω1)Q1(ω2) ... Qn(ω2)

...Q1(ωl) ... Qn(ωl)

α1

α2

...αn

=

V (ω1)V (ω2)

...V (ωl)

(14.48)

or, in matrix form,Qα = V (14.49)

where Q is a complex valued rectangular matrix (l× n), V is a complex-valued vector and α is the vector of linear combiner coefficients (real).Since the FRFs Q and V are determined experimentally, the solution ofthis redundant system of equations requires some care to eliminate theeffect of noise; the coefficients resulting from the pseudo-inverse in themean-square sense

α = Q+V (14.50)

are highly irregular and highly sensitive to the disturbance source. Thisdifficulty can be overcome by using a singular value decomposition of Q,

Q = U1ΣUH2 (14.51)

where U1 and U2 are unitary matrices containing the eigenvectors of QQH

and QHQ, respectively (the superscript H stands for the Hermitian, thatis the conjugate transpose), and Σ is the rectangular matrix of dimension(l × n) with the singular values σi on the diagonal (equal to the squareroot of the eigenvalues of QQH and QHQ). If ui are the column vectorsof U1 and vi are the column vectors of U2, Equ.(14.51) reads

Q =n∑

i=1

σiuivHi (14.52)

and the pseudo-inverse is

Q+ =n∑

i=1

1σi

viuHi (14.53)

386 14 Applications

This equation shows clearly that, because of the presence of 1/σi, the low-est singular values tend to dominate the pseudo-inverse; this is responsiblefor the high variability of the coefficients αi resulting from Equ.(14.50).The problem can be solved by truncating the singular value expansion(14.53) and deleting the contribution relative to smaller singular valueswhich are dominated by the noise. Without noise, the number of singularvalues which are significant (i.e. the rank of the system) is equal to thenumber of modes responding significantly in the frequency band of inter-est (assuming this number smaller than the number n of sensors in thearray); with noise, the selection is slightly more difficult, because the gapin magnitude between significant and insignificant singular values disap-pears; some trial and error is needed to identify the optimum number ofsingular values in the truncated expansion (Francois et al.). Figure 14.24shows typical results obtained with a glass plate covered with an array of4× 8 PZT patches.

Fig. 14.24. (a) Experimental set-up: glass plate covered with an array of 4 × 8 PZTpatches. (b) Coefficients of the linear combiner to reconstruct the volume displacement.(c) Comparison of the volume displacement FRF obtained with the array sensor and alaser scanner vibrometer.


14.7.3 Spatial aliasing

The volume displacement sensor of Fig.14.24 is intended to be part of acontrol system to reduce the sound transmission through a baffled platein the low frequency range (below 250Hz ), where the correlation betweenthe volume velocity and the sound power radiation is high. Figure 14.24.cshows that the output of the array sensor follows closely the volume dis-placement below 400Hz. However, in order to be included in a feedbackcontrol loop, the quality of the sensor must be guaranteed at least onedecade above the intended bandwidth of the control system. Figure 14.25shows a numerical simulation of the open-loop FRF of a SISO systemwhere the input consists of 4 point force actuators controlled with thesame input current and the output is the volume displacement of the4× 8 array sensor. The comparison of the sensor output with the actualvolume displacement reveals substantial differences at higher frequency,the amplitude of the sensor output being much larger than the actual vol-ume displacement, which is not acceptable from a control point of view,for reasons which have been discussed extensively in chapter 10.

This is due to spatial aliasing, as explained in Fig.14.26. The left partof the figure shows the shape of mode (1,1) and mode (1,15); the diagramson the right show the electric charges Qi generated by the correspondingmode shape on the PZT patches. We observe that the electric chargesgenerated by mode (1,15) have the same shape as those generated bymode (1,1). Thus, at the frequency 1494.4Hz, the plate vibrates accordingto mode (1,15) which contributes only little to the volume displacement;however the output of the array sensor is the same as that of mode (1,1)which contributes a lot to the volume displacement; this explains why thehigh frequency amplitude of the FRF are much larger than expected.

Note that it is a typical property of aliasing that a higher frequencycomponent is aliased into a lower frequency component symmetrical withrespect to the Nyquist frequency. In this case, the number of patches inthe array being 8 along the length of the plate, mode (1,15) is aliased intothe symmetrical one with respect to 8, that is into mode (1,1); similarly,mode (1,13) would be aliased into mode (1,3).

The most obvious way to alleviate aliasing is to increase the samplingrate, that is, in this case, to increase the size of the array; this is illustratedin Fig.14.27 where one can see that an array of 16 × 32 gives a goodagreement up to 5000Hz. However, dealing with such big arrays bringspractical problems with the need for independent conditioning electronics(charge amplifier) prior to the linear combiner.

388 14 Applications

PZT patches

point force actuators

101

102

103

104

-250

-200

-150

-100

-50

dB

101

102

103

104

-3000

-2000

-1000

0

Frequency (Hz)

Phase (deg)

Volume displacement

Sensor output

( )a ( )b

( )c

12

34

0

1

2

3

4

5

6x 10

4

Fig. 14.25. (a) Geometry of the 4 × 8 array sensor and the 4 point force actuators(controlled with the same input current), (b) Weighting coefficients αi of the linearcombiner, (c) Comparison of the FRF between the actuators and the volume displace-ment, and the sensor output (numerical simulation).


If one accepts to give up the programmability of the linear combiner,the coefficients αi can be incorporated into the size of the electrodes,leading to the design of Fig.14.28, which requires only a single chargeamplifier. The shape of this sensor has some similarity with the QWSIS.2

Fig. 14.26. Modes shapes (1,1) and (1,15) and electric charges Qi generated by mode(1,1) and mode (1,15).

2 the electrode shape in Fig.14.28 is nearly that obtained by cutting parabolic stripsin two orthogonal directions.

390 14 Applications

Fig. 14.27. Effect of the size of the array on the open-loop FRF (a) 8×16, (b) 16×32(simulations).

Fig. 14.28. Variable size array with 16× 32 patches interconnected (simulations).


Fig. 14.29. (a) Porous electrode, (b) detail of the pattern with variable porosity, (c)double sided pattern (fraction of electrode area = 50 %), (d) single sided pattern (theother electrode is continuous).

392 14 Applications

Fig. 14.30. Tridimensional finite element analysis; the sample is strained in the di-rection S1, while V = 0 is enforced between the electrodes. The equipotential surfacesshow clearly the edge effect. (a) Two-sided electrode. (b) One-sided electrode.


Fig. 14.31. Effective piezoelectric coefficient vs. fraction of electrode area, for PVDFfilms of 10 µm and 100 µm thickness.

14.7.4 Distributed sensor

For the design of Fig.14.28 involving an electrode connecting 16 × 32patches of variable size, the spatial aliasing still occurs above 10000 Hz;it can be pushed even further by increasing the number of patches. Thissuggests the distributed sensor with a single “porous” electrode shownin Fig.14.29. The electrode is full in the center of the plate and becomesgradually porous as one moves towards the edge of the plate, to achieve an

394 14 Applications

electrode density which produces the desired weighting coefficient α(x, y).This pattern can be placed on one side or on the two sides of the piezomaterial; for manufacturing, it seems simpler to apply the pattern on oneside only, with a continuous electrode on the other side.

The amount of electrical charges generated on the electrode is given byEqu.(14.39) where the integral extends over the area of the electrode; itfollows that tailoring the porosity of the electrode (i.e. Ω) is equivalent totailoring the piezoelectric constants of the material, e31 and e32. Equation(14.39) assumes that the size of the electrode is much larger than itsthickness. However, when the pattern of the electrode becomes small,tridimensional (edge) effects start to appear and the relationship betweenthe porosity and the equivalent piezoelectric property is no longer linear.

The exact relationship between the porosity and the equivalent piezo-electric coefficients can be explored with a tridimensional finite elementanalysis. Figure 14.30 shows the equipotential surfaces for the two elec-trode configurations when a small sample (1 mm × 1 mm × 100 µm) issubjected to a strain along the x axis and a potential difference V = 0is enforced between the electrodes; the material assumed in this study isisotropic PVDF polarized in the direction perpendicular to the electrodes;the edge effects appear clearly in the figures. For this sample, Fig.14.31shows the relationship between the effective piezoelectric coefficient andthe fraction of electrode area; the two electrode configurations are con-sidered for two sample thicknesses (10 µm and 100 µm); we observe thatfor a very thin sensor, the two electrode configurations produce the sameresults and the relationship is almost linear.

The potential of this concept of “porous” electrode for shaping the ef-fective piezoelectric properties of the material for two-dimensional struc-tures is far beyond the design of a volume displacement sensor. Modalfiltering is another obvious application. A transparent sensor has beenrealized for window applications (Preumont et al., 2005).

14.8 References 395

14.8 References

Digital control

ASTROM, K.J. & WITTENMARK, B., Computer-Controlled Systems,Theory and Design, 2nd edition, Prentice-Hall, 1990.ELLIOTT, S.J., Signal Processing for Active Control, Academic Press,2001.FRANKLIN, G.F. & POWELL, J.D., Digital Control of Dynamic Sys-tems, Addison-Wesley, 1980.HANSELMANN, H., Implementation of digital controllers - A survey,Automatica, Vol.23, No 1, pp.7-32, 1987.JACKSON, L.B., Digital Filters and Signal Processing, Kluwer, 1986.KUO, B.C., Digital Control Systems, SRL Pub. Co., 1977.OPPENHEIM A.V. & SCHAFER, R.W., Digital Signal Processing, Prentice-Hall, 1975.

Active damping of a truss

ABU-HANIEH, A. Active Isolation and Damping of Vibrations via Stew-art Platform, PhD Thesis, Universite Libre de Bruxelles, Active Struc-tures Laboratory, 2003.ANDERSON, E.H., MOORE, D.M., FANSON, J.L. & EALEY, M.A.Development of an active member using piezoelectric and electrostrictiveactuation for control of precision structures, SDM conference, AIAA paper90-1085-CP, 1990.CHEN, G.S., LURIE, B.J. & WADA, B.K. Experimental studies of adap-tive structures for precision performance, SDM Conference, AIAA paper89-1327-CP, 1989.FANSON, J.L., BLACKWOOD, G.H. & CHEN, C.C. Active membercontrol of precision structures. SDM Conference, AIAA paper 89-1329-CP, 1989.HYDE, T.T. & ANDERSON, E.H. Actuator with built-in viscous damp-ing for isolation and structural control, AIAA Journal, Vol.34, No 1, 129-135, January 1996.PETERSON, L.D., ALLEN, J.J., LAUFFER, J.P. & MILLER, A.K. Anexperimental and analytical synthesis of controlled structure design. SDMConference, AIAA paper 89-1170-CP, 1989.

396 14 Applications

PREUMONT, A., DUFOUR, J.P. & MALEKIAN, Ch. Active dampingby a local force feedback with piezoelectric actuators. AIAA J. of Guid-ance, Control and Dynamics, Vol 15, No 2, March-April, 390-395, 1992.

Active damping of a plate

DOSCH, J.J., INMAN, D.J. & GARCIA, E. A self-sensing piezoelectricactuator for collocated control, J.of Intelligent Materials, Systems andStructures, Vol.3, 166-185, January 1992.FANSON, J.L. & CAUGHEY, T.K. Positive position feedback control forlarge space structures. AIAA Journal, Vol.28, No 4,April, 717-724, 1990.LOIX, N., CONDE REIS, A., BRAZZALE, P., DETTMAN, J., &PREUMONT, A. CFIE: In-Orbit Active Damping Experiment UsingStrain Actuators, Space Microdynamics and Accurate Control Sympo-sium, Toulouse, May 1997.

Active damping of a stiff beam

PREUMONT, A., LOIX, N., MALAISE, D. & LECRENIER, O. Activedamping of optical test benches with acceleration feedback, Machine Vi-bration, Vol.2, 119-124, 1993.

HAC/LAC

AUBRUN, J.N. Theory of the control of structures by low-authority con-trollers. AIAA J. of Guidance, Vol.3, No 5, Sept-Oct., 444-451, 1980.BENHABIB, R.J., IWENS, R.P. & JACKSON, R.L. Stability of largespace structure control systems using positivity concepts. AIAA J. ofGuidance, Control and Dynamics, Vol 4, No 5, 487-494, Sept.-Oct. 1981.GUPTA, N.K. Frequency-shaped cost functionals: extension of linearquadratic Gaussian methods. AIAA J. of Guidance, Control and Dynam-ics, Vol.3, No 6, 529-535, Nov.-Dec. 1980.KOSUT, R.L., SALZWEDEL, H. & EMAMI-NAEINI, A. Robust controlof flexible spacecraft, AIAA J. of Guidance Control and Dynamics, Vol.6,No 2, 104-111, March-April 1983.MUKHOPADHYAY, V. & NEWSOM, J.R. A multiloop system stabilitymargin study using matrix singular values, AIAA J. of Guidance, Vol.7,No 5, 582-587, Sept.-Oct. 1984.

14.8 References 397

PARSONS, E.K. An experiment demonstrating pointing control on a flex-ible structure, IEEE Control Systems Magazine, 79-86, April 1989.PREUMONT, A. Active structures for vibration suppression and pre-cision pointing, Journal of Structural Control, Vol.2, No 1, 49-63, June1995.

Vibroacoustics

DE MAN, P., FRANCOIS, A. & PREUMONT, A. Vibroacoustic opti-mization of a baffled plate for robust feedback control, ASME Journal ofVibration and Acoustics, Vol.124, pp. 154-157, January 2002.FAHY, F. Sound and Structural Vibration, Academic Press, 1987.FRANCOIS, A., DE MAN, P. & PREUMONT, A. Piezoelectric ArraySensing of Volume Displacement: A Hardware Demonstration J. of Soundand Vibration, Vol.244, No 3, 395-405, July 2001.GARDONIO, P., LEE, Y.S., ELLIOTT, S.J. & DEBOST, S. Active Con-trol of Sound Transmission Through a Panel with a Matched PVDF Sen-sor and Actuator Pair, Active 99, Fort Lauderdale, Fl, Dec 1999.JOHNSON, M. E. & ELLIOTT, S. J. Active Control of Sound Radia-tion Using Volume Velocity Cancellation, J. of the Acoustical Society ofAmerica, Vol.98, 2174-2186, 1995.PIEFORT, V. Finite Element Modeling of Piezoelectric Active Structures,PhD Thesis, Universite Libre de Bruxelles, Active Structures Laboratory,2001.REX, J. & ELLIOTT, S.J. The QWSIS - A New Sensor for StructuralRadiation Control, MOVIC-1, Yokohama, Sept. 1992.PREUMONT, A., FRANCOIS, A., DE MAN, P., PIEFORT, V. Spatialfilters in structural control, Journal of Sound and Vibration, Vol.265, 61-79, 2003.PREUMONT, A., FRANCOIS, A., DE MAN, P., LOIX, N. & HENRI-OULLE, K. Distributed sensors with piezoelectric films in design of spatialfilters for structural control, Journal of Sound and Vibration, Vol.282, No3-5, pp.701-712, April 2005.

398 14 Applications

14.9 Problems

P.14.1 An anti-aliasing filter of bandwidth ωc can be obtained by cas-cading second order filters of the form

ω2

(s/ωc)2 + 2ξω(s/ωc) + ω2

The Butterworth filters correspond to order 2: ω = 1, ξ = 0.71order 4: ω = 1, ξ = 0.38 ω = 1, ξ = 0.92order 6: ω = 1, ξ = 0.26 ω = 1, ξ = 0.71 ω = 1, ξ = 0.97

The Bessel filters correspond to order 2: ω = 1.27, ξ = 0.87order 4: ω = 1.60, ξ = 0.62 ω = 1.43, ξ = 0.96order 6: ω = 1.90, ξ = 0.49 ω = 1.69, ξ = 0.82 ω = 1.61, ξ = 0.98

(a) Compare the Bode plots of the various filters and, for each of them,evaluate the phase lag for 0.1ωc and 0.2ωc.(b) Show that the poles of the Butterworth filter are located on a circleof radius ωc according to the configurations depicted in Fig.11.1.(c) Show that, at low frequency, the Bessel filter has a linear phase, andcan be approximated with a time delay (Astrom & Wittenmark).P.14.2 Show that the transfer function of the zero-order hold is

H0(s) =1− e−sT

s

Show that the frequency response function is

H0(ω) =2ω

sinωT

2.e−jωT/2

Draw the amplitude and phase plots.P.14.3 Using the bilinear transform, show that the discrete equivalent ofEqu.(14.8) (14.9) is given by Equ.(14.10)-(14.13).P.14.4 Consider a truss structure with several identical active memberscontrolled with the same control law (IFF) and the same gain. Makingthe proper assumptions, show that each closed-loop pole follows a rootlocus defined by Equ.(7.31), where the natural frequency ωi is that of theopen-loop structure and the zero zi is that of the structure where theactive members have been removed.P.14.5 For the active truss of section 14.2, show that the compensator

δ =g

s + ay

14.9 Problems 399

is equivalent to δ = g/s provided that the breakpoint frequency a is suchthat a ¿ ω1. Show that its digital counterpart is

δi+1 =2− Ta

2 + Taδi +

gT

2 + Ta(yi+1 + yi)

P.14.6 Consider a simply supported beam with a point force actuatorand a collocated accelerometer at x = l/6. Assume that EI = 1 Nm2,m = 1 kg/m and l = 1 m. Design a compensator to achieve a closed-loop modal damping ξi > 0.1 for i = 1 and 2, using the Direct VelocityFeedback and a second order filter (see Problem 7.2). Draw the Bode plotsfor the two compensators and compare the phase margins. For both cases,check the effect of the delay corresponding to a sampling frequency 100times larger than the first natural frequency of the system (ωs = 100 ω1)and that of the actuator dynamics, assuming that the force actuator is aproof-mass with a natural frequency ωp = ω1/3 (assume ξp = 0.5).

15

Tendon Control of Cable Structures

15.1 Introduction

Cable structures are used extensively in civil engineering: suspendedbridges, cable-stayed bridges, guyed towers, roofs in large public buildingsand stadiums. The main span of current cable-stayed bridges (Fig.15.1)can reach more than 850 m (e.g. Normandy bridge, near Le Havre, inFrance). These structures are very flexible, because the strength of highperformance materials increases faster than their stiffness; as a result,they become more sensitive to wind and traffic induced vibrations. Largebridges are also sensitive to flutter which, in most cases, is associated withthe aeroelastic damping coefficient in torsion becoming negative above acritical velocity (Scanlan, 1974). The situation can be improved either bychanging the aerodynamic shape of the deck, or by increasing the stiff-ness and damping in the system; the difficulty in active damping of cablestructures lies in the strongly nonlinear behavior of the cables, particu-larly when the gravity loads introduce some sag (typical sag to lengthratio is 0.5% for a cable-stayed bridge). The structure and the cables in-teract with linear terms (at the natural frequency of the cable ωi) andquadratic terms resulting from stretching (at 2ωi); the latter may produceparametric resonance if some tuning conditions are satisfied (parametricexcitation has indeed been identified as the source of vibration in severalexisting cable-stayed bridges).

Cable structures are not restricted to civil engineering applications;the use of cables to achieve lightweight spacecrafts was recommended inHerman Oberth’s visionary books on astronautics. Tension truss struc-tures have already been used for large deployable mesh antennas. The useof guy cables is probably the most efficient way to stiffen a structure in

402 15 Tendon Control of Cable Structures

Fig. 15.1. Cable-stayed bridge and conceptual design of an active tendon.

terms of weight; in addition, if the structure is deployable and if the guycables have been properly designed, they may be used to prestress thestructure, to eliminate the geometric uncertainty due to the gaps.

This chapter examines the possibility of connecting guy cables to activetendons to bring active damping into cable structures; the same strategyapplies to large space structures and to cable-stayed bridges and othercivil engineering structures; however, the technology used to implementthe control strategy is vastly different (piezoelectric actuators for spaceand hydraulic actuators for bridges).

15.2 Tendon control of strings and cables

The mechanism by which an active tendon can extract energy from astring or a cable is explained in Fig.15.2 with a simplified model assumingonly one mode (Rayleigh-Ritz) and for situations of increasing complex-ity. The simplest case is that of a linear string with constant tension To

(Fig.15.2.a); the equation becomes nonlinear when the effect of stretching

15.2 Tendon control of strings and cables 403

Fig. 15.2. Mechanism of active tendon control of strings and cables.

is added (cubic nonlinearity). In Fig.15.2.b, a moving support is added;the input u of this active tendon produces a parametric excitation,1 whichis the only way one can control a string with this type of actuator.

The difference between a string and a cable is the effect of gravity,which produces sag (Fig.15.2.c). In this case, the equations of motionin the gravity plane and in the plane orthogonal to it are no longer thesame, and they are coupled. In the gravity plane (z coordinate), the activetendon control u still appears explicitly as a parametric excitation, butalso as an inertia term −αcu whose coefficient αc depends on the sag ofthe cable; even for cables with moderate sag (say sag to length ratio of1% or more), this contribution becomes significant and constitutes thedominant control term of the equation. On the contrary, in the out-of-plane equation (y coordinate), the tendon control u appears explicitlyonly through the parametric excitation, as for the string.

1 the excitation u appears as a parameter in the differential equation.


15.3 Active damping strategy

StructureCable

Activetendon

quM

K

T

z

1s

Fig. 15.3. Cable-structure system with an active tendon.

Figure 15.3 shows a schematic view of a cable-structure system, wherethe control u is the support displacement, T is the tension in the cable, zthe transverse vibration of the cable and q the vibration of the structure;we seek a control strategy for moving the active tendon u to achieveactive damping in the structure and the cable. Any control law based onthe non-collocated measurements of the cable and structure vibration

u = Ψ(z, z, q, q) (15.1)

must, at some stage, rely on a simplified model of the system; as a result,it is sensitive to parametric variations and to spillover. Such control lawshave been investigated by (Chen, 1984) and (Fujino & coworkers) withvery limited success; it turned out that the control laws work in specificconditions, when the vibration is dominated by a single mode, but theybecome unstable when the interaction between the structure and the cableis strong, which removes a lot of their practical value.

By contrast, we saw in section 13.7 that, if a force sensor measuring thetension T in the cable is collocated with the active tendon, the positiveIntegral Force Feedback

u = g

∫ t

0T (τ) dτ (15.2)

produces an energy absorbing controller, which can only extract energyfrom the system. However, for cable-structure applications, a high-passfilter is necessary to eliminate the static tension in the cable. 2

2 To establish the vibration absorbing properties of Equ.(15.2) when T is the dynamiccomponent of the tension in the cable, one can show that the dynamic contribution

15.4 Basic Experiment 405

15.4 Basic Experiment

Figure 15.4.a shows the test structure that was built to represent theideal situation of Fig.15.3; the cable is a 2 m long stainless steel wire of0.196 mm2 cross section provided with additional lumped masses at regu-lar intervals, to achieve a sag to span ratio comparable to actual bridges;the active tendon is materialized by a piezoelectric linear actuator actingon the support point with a lever arm, to amplify the actuator displace-ment by a factor 3.4; this produces a maximum axial displacement of150 µm for the moving support. A piezoelectric force sensor is colinearwith the actuator; because of the high-pass behavior of this type of sen-sor, it measures only the dynamic component of the tension in the cable.The spring-mass system (in black on the figure) has an adjustable masswhereby the natural frequency can be tuned; a shaker and an accelerom-eter are attached to it, to evaluate the performance of the control system.In addition, a non-contact laser measurement system was developed tomeasure the cable vibration (Achkire).

Figure 15.4.b shows the effect of the control system on the structure;we see that the controller brings a substantial amount of damping tothe system. As far as the cable modes are concerned, the out-of-planemodes and the anti-symmetric in-plane modes are not affected by thecontroller (except for large amplitudes where the cable stretching becomessignificant); the amount of active damping brought into the symmetricin-plane modes depends very much on the sag to span ratio. The controlsystem behaves nicely, even at the parametric resonance, when the naturalfrequency of the structure is exactly twice that of the cable.

This experiment was the first demonstration of robust active dampingof a cable-structure; it demonstrates that active damping can be achieved

to the total energy, resulting from the vibration around the static equilibrium posi-tion, is a Lyapunov function. Thus, the stability is guaranteed if we assume perfectsensor and actuator dynamics. Note that the fact that the global stability is guar-anteed does not imply that all the vibration modes are effectively damped. In fact,from a detailed examination of the dynamic equations (e.g. Fujino or Achkire), itappears that not all the cable modes are controllable with this actuator and sen-sor configuration. The odd numbered in-plane modes (in the gravity plane) can bedamped substantially because they are linearly controllable by the active tendon(inertia term in Fig.15.2.c) and linearly observable from the tension in the cable; allthe other cable modes are controllable only through active stiffness variation (para-metric excitation in Fig.15.2), and observable from quadratic terms due to cablestretching. However, these weakly controllable modes are never destabilized by thecontrol system, even at the parametric resonance, when the natural frequency of thestructure is twice that of the cable.


Charge amplifier

Actuator

Lumped

mass

Force

Sensor

Mass for

initial tension

adjustment

Amplifier

DSPMicroprocessor

Current amplifier

Wave generator

Strain gauge

Shaker

Force sensor

Mass for

frequency

adjustment

Accelerometer

Fig. 15.4. (a) Cable-structure laboratory model. (b) Experimental frequency responsebetween the shaker force and the accelerometer, and free response of the structure,with and without control.

without fear of destabilizing the cables, in spite of their complex dynamics;it also suggests that a simple treatment of the cables is acceptable in thedesign of the control system.

15.5 Linear theory of decentralized active damping

In this section, we follow an approach similar to that of section 7.5 topredict the closed-loop poles of the cable-structure system. Each active

15.5 Linear theory of decentralized active damping 407

id

ik

iT

f

Active

tendon

iT

f

Active tendon

id

Forc

e

transducer P

iezoele

ctric

linear a

ctu

ato

r

iT

g.h(s)

Fig. 15.5. Left: Cable-structure system with active tendons and decentralized control.Center: Active tendon. Right: Passive structure. Ti is the tension in the active cable iof axial stiffness ki and free active displacement δi.

tendon consists of a displacement piezoelectric actuator co-linear with aforce sensor. Ti is the tension in the active cable i, measured by the sensorintegrated in the active tendon, and δi is the free extension of the actuator,the variable used to control the system. ki is the combined axial stiffnessof the cable and the active tendon. The control is decentralized, so thateach control loop operates independently. We assume that the dynamicsof the active cables can be neglected and that their interaction with thestructure is restricted to the tension Ti in the active cables (Fig.15.5).Accordingly, the governing equation is

Mx + Kx = −BT + f (15.3)

where x is the vector of global coordinates of the finite element model,M and K are respectively the mass and stiffness matrices of the pas-sive structure (including a linear model of the passive cables, if any, butexcluding the active cables). The right hand side represents the exter-nal forces applied to the system; f is the vector of external disturbances(expressed in global coordinates), T = (T1, . . . , Ti, . . .)T is the vector oftension in the active cables and B is the influence matrix of the cableforces, projecting the cable forces in the global coordinate system (thecolumns of B contain the direction cosines of the various active cables).


If we neglect the cable dynamics, the active cables behave like (mass-less) bars. If δ = (δ1, . . . , δi, . . .)T is the vector of unconstrained activedisplacements of the active tendons acting along the cables, the tensionin the cables are given by

T = Kc(BT x− δ) (15.4)

where Kc = diag(ki) is the stiffness matrix of the cables, BT x are therelative displacements of the end points of the cables projected alongthe chord lines. This equation expresses that the tension in the cable isassociated with the elastic extension of the cable.

Combining Equ.(15.3) and (15.4), we get

Mx + (K + BKcBT )x = BKcδ + f (15.5)

This equation indicates that K + BKcBT is the stiffness matrix of the

structure including all the guy cables (passive + active). Next, we assumethat all the active cables are controlled according to the force feedbacklaw:

δ = gh(s).K−1c T (15.6)

where gh(s) is the scalar control law applied to all control channels. (notethat K−1

c T represents the elastic extension of the active cables). Combin-ing Equ.(15.4) to (15.6), the closed-loop equation is

[Ms2 + K +1

1 + gh(s).BKcB

T ]x = f (15.7)

It is readily observed that the open-loop poles, solutions of the character-istic equation for g = 0, satisfy

[Ms2 + K + BKcBT ]x = 0 (15.8)

(the solutions are the eigenvalues of the structure with all cables), whilethe zeros, solutions of Equ.(15.7) for g −→∞, satisfy

[Ms2 + K]x = 0 (15.9)

which is the eigenvalue problem for the open-loop structure where theactive cables have been removed.

If a IFF controller is used, h(s) = s−1 and the closed-loop equationbecomes

[Ms2 + K +s

s + gBKcB

T ]x = f (15.10)


which indicates that the closed-loop static stiffness matrix

lims=0

[Ms2 + K +s

s + gBKcB

T ] = K (15.11)

This means that the active cables do not contribute to the static stiffnessand this may be problematic in some applications, especially in presenceof gravity loads. However, if the control is slightly changed into the “Betacontroller”

gh(s) =gs

(s + β)2(15.12)

where β is small and positive (the influence of β will be discussed later),the closed-loop equation becomes

[Ms2 + K +(s + β)2

gs + (s + β)2BKcB

T ]x = f (15.13)

and the closed-loop static stiffness matrix becomes

lims=0

[Ms2 + K +(s + β)2

gs + (s + β)2BKcB

T ] = K + BKcBT (15.14)

which indicates that the active cables have a full contribution to the staticstiffness.

Next, let us project the characteristic equation on the normal modes ofthe structure with all the cables, x = Φz, which are normalized accordingto ΦT MΦ = 1. According to the orthogonality condition,

ΦT (K + BKcBT )Φ = Ω2 = diag(Ω2

i ) (15.15)

where Ωi are the natural frequencies of the complete structure. In orderto derive a simple and powerful result about the way each mode evolveswith g, let us assume that the mode shapes are little changed by the activecables, so that we can write

ΦT KΦ = ω2 = diag(ω2i ) (15.16)

where ωi are the natural frequencies of the structure where the activecables have been removed. It follows that the fraction of modal strainenergy is given by

νi =φT

i BKcBT φi

φTi (K + BKcBT )φi

=Ω2

i − ω2i

Ω2i

(15.17)


Considering the IFF controller, the closed-loop characteristic equation(15.10) can be projected into modal coordinates, leading to

(s2 + Ω2i )− g

g + s(Ω2

i − ω2i ) = 0

or

1 + gs2 + ω2

i

s(s2 + Ω2i )

= 0 (15.18)

which is identical to (7.31).This result indicates that the closed-loop poles can be predicted by

performing two modal analyzes (Fig.15.6), one with all the cables, lead-ing to the open-loop poles ±jΩi, and one with only the passive cables,leading to the open-loop zeros ±jωi, and drawing the independent rootloci (15.18). As in section 7.5, the maximum modal damping is given by

natural frequency

with the active cables

active cables removed

Fig. 15.6. Root locus of the closed-loop poles with an IFF controller.


ξmaxi =

Ωi − ωi

2ωi(15.19)

and it is achieved for g = Ωi

√Ωi/ωi. This equation relates directly the

maximum achievable modal damping with the spacing between the poleΩi and the zero ωi, which is essentially controlled by the fraction of modalstrain energy in the active cables, as expressed by Equ.(15.17).

The foregoing results are very easy to use in design. Although they arebased on several assumptions (namely that the dynamics of the activecables can be neglected, the passive cables behave linearly and that themode shapes are unchanged), they are in good agreement with experi-ments as shown below.

If, instead of the IFF controller, the Beta controller is used, Equ.(15.12),the closed-loop characteristic equation projected into modal coordinatesreads

(s2 + Ω2i )− gs

gs + (s + β)2(Ω2

i − ω2i ) = 0

or

1 + gs(s2 + ω2

i )(s + β)2(s2 + Ω2

i )= 0 (15.20)

Thus, as compared to the IFF controller, the pole at the origin has beenreplaced by a zero at the origin and a pair of poles at −β on the real axis.The effect of this change on the root locus is shown in Fig.15.7. Whenβ = 0, there is a pole-zero cancellation and the control is reduced to theIFF. As β increases, the root locus has two branches on the real axis,

Wi

Re s( )

Im s( )

wi

Wi

wi

b/wi= 1

b/wi= 0.5

-b

b/w =i

0

0.250.5

Fig. 15.7. Root locus of the closed-loop poles with the Beta controller gs/(s+β)2, forvarious values of the ratio β/ωi. The IFF controller corresponds to β = 0. The locusis always stable for β < ωi; for β = ωi, it is tangent to the imaginary axis at the zero±jωi.


starting from s = −β in opposite directions; one of the closed-loop polesremains trapped between 0 and −β; the loop still go from ±jΩi to ±jωi,but they tend to be smaller, leading to less active damping; this is the priceto pay for recovering the static stiffness of the active cables. Analyzingthe root locus in detail, one can show that the system is unconditionallystable (for all modes) provided that β < ω1.

15.6 Guyed truss experiment

This experiment aims at comparing the closed-loop predictions of the lin-ear model with experiments. The test structure consists of the active trussof Fig.4.20 equipped with three identical cables made of synthetic fiber“Dynema” of 1 mm diameter (Fig.15.8.a); the tension in the cables is notimportant provided that the effective Young modulus (due to sag) is closeto the actual one; in this experiment, the tension is such that the cablefrequency is above 500 rad/sec. The design of the active tendon is shownin Fig.15.8.b (a better design is shown in Fig.15.13); the amplificationratio of the lever arm is 3, leading to a maximum stroke of 150 µm. The

Fig. 15.8. (a) Guyed truss. (b) Design of the active tendon.

15.7 Micro Precision Interferometer testbed 413

Fig. 15.9. Experimental vs. analytical closed-loop poles.

natural frequencies with and without the active cables are respectivelyΩ1 = 67.9 rad/s, ω2 = 53.8 rad/s, Ω2 = 78.9 rad/s, ω2 = 66 rad/s. Figure15.9 shows the root locus predicted by the linear model together with theexperimental results for various values of the gain; only the upper part ofthe loops is available experimentally because the control gain is limitedby the saturation due to the finite stroke of the actuators. The agreementbetween the experimental results and the linear predictions of Equ.(15.18)is quite good.

15.7 Micro Precision Interferometer testbed

To illustrate further the application of the control strategy to the damp-ing of large space trusses, let us consider a numerical model of the mi-croprecision interferometer (MPI) testbed used at NASA Jet PropulsionLaboratory (JPL) to develop the technology of precision structures for fu-ture interferometric missions (Neat et al.). The first three flexible modesare displayed in Fig.15.10. We investigate the possibility of stiffness aug-mentation and active damping of these modes with a set of three activetendons acting on Kevlar cables of 2 mm diameter, connected as indicatedin Fig.15.11 (Kevlar properties : E = 130 GPa, ρ = 1500 kg/m3, tensile


Fig. 15.10. JPL-MPI testbed, shape of the first three flexible modes (by courtesy ofR. Laskin-JPL).

15.8 Free floating truss experiment 415

strength σy = 2.8 GPa). The global added mass for the three cables isonly 110 gr (not including the active tendons and the control system).The natural frequencies of the first three modes, with and without thecables, are reported in Table 15.1; the root locus of the three global flexi-ble modes as functions of the control gain g are represented in Fig.15.12;for g = 116 rad/s, the modal damping ratios are ξ7 = 0.21, ξ8 = 0.16,ξ9 = 0.14.

Fig. 15.11. Proposed location of the active cables in the JPL-MPI testbed.

i ωi Ωi ξmaxi

7 51.4 74.6 0.238 76.4 101 0.169 83.3 106.4 0.14

Table 15.1. Natural frequencies (rad/s) of the first flexible modes of the JPL-MPItestbed, with and without cables.

15.8 Free floating truss experiment

In order to confirm the spectacular analytical predictions obtained withthe numerical model of the JPL-MPI testbed, a similar structure (al-though smaller) was built and tested (Fig.15.13); the free-floating con-dition was simulated by hanging the structure with soft springs. The


Fig. 15.12. Analytical prediction of the closed-loop poles.

i ωi Ωi

1 119.4 146.12 157.1 173.63 165.7 205.84 208.1 220.7

Table 15.2. Natural frequencies (rad/s) of the free floating truss, with and withoutcables.

active tendon consists of a APA 100 M amplified actuator from CEDRATRecherche together with a B&K 8200 force sensor and flexible tips (thisdesign is much simpler than that used earlier, Fig.15.8). The stroke is110 µm and the total weight of the tendon is 55 gr; the cable is made ofDynema with axial stiffness EA = 19000 N . The natural frequencies ofthe first flexible modes, with and without cables, are reported in Table15.2. Figure 15.14 compares the analytical predictions of the linear modeland the experiments.3

3 All the results discussed above have been obtained for vibrations in a range goingfrom millimeter to micron; in order to apply this technology to future large spaceplatforms for interferometric missions, it is essential that these results be confirmedfor microvibrations. In fact, it could well be that, for very small amplitudes, thebehavior of the control system be dominated by the nonlinearity of the actuator(hysteresis of the piezo) or the noise in the sensor or in the voltage amplifier. Tests

15.8 Free floating truss experiment 417

Fig. 15.13. ULB free floating truss test structure and detail of the active tendon.

Fig. 15.14. ULB free floating truss test structure: Comparison between the analyticalpredictions of the linear model and the experiments. The numbers correspond to equalvalues of the gain.


15.9 Application to cable-stayed bridges

In what follows, we summarize some of the findings of a research projectcalled “ACE” which was funded by the EU in the framework of the Brite-Euram program, between 1997 and 2000, and involved several academicand industrial partners. The overall objective of the project was to demon-strate the use of the active control in civil engineering. Several experimentswere conducted, on different scales; the main results are explained below.

15.10 Laboratory experiment

The test structure is a laboratory model of a cable-stayed bridge dur-ing its construction phase, which is amongst the most critical from thepoint of view of the wind response. The structure consists of two halfdecks mounted symmetrically with respect to a central column of about2 m high (Fig.15.15); each side is supported by 4 cables, two of which areequipped with active piezoelectric tendons identical to those of Fig.15.8.b.The cables are provided with lumped masses at regular intervals, so asto match the sag to length ratio of actual stay cables [a discussion of thesimilarity aspects can be found in (Warnitchai et al.)]. Figure 15.16 com-pares the evolution of the first bending and torsion closed-loop poles ofthe deck with the analytical predictions of the linear theory. The agree-ment is good for small gains, when the modal damping is smaller than20%.

15.11 Control of parametric resonance

In this experiment, the bridge deck is excited harmonically with an elec-trodynamic shaker at a frequency f close to the first torsion mode, andthe tension in the two passive cables on one side is chosen in such a waythat the first in-plane mode of one of them is tuned on the excitationfrequency f , while the other is tuned on f/2, to experience the paramet-ric resonance when the deck vibrates (Fig.15.17). This tuning is achievedby monitoring the cable vibration with a specially developed non-contact

have been conducted for vibrations of decreasing amplitudes, and the influence ofthe various hardware components has been analyzed (Bossens), these tests indicatethat active damping is feasible at the nanometric level, provided that adequatelysensitive components are used.

15.11 Control of parametric resonance 419

Fig. 15.15. Test structure used at ULB to demonstrate the control of parametricresonance. Above left, the Skarnsund cable-stayed bridge during construction (Norway).

100

90

80

70

50

40

30

20

10

0

60

-50 -40 -30 -20 -10 0

x=0.25

x=0.5

Im (s)

Re(s)

deck

1 bending modest

Analyticalprediction

deck torsion mode

Fig. 15.16. Evolution of the first bending and torsion poles of the deck with the controlgain.


optical measurement system (Achkire). Figure 15.18 shows the vibrationamplitude of the deck and the transverse amplitude of the in-plane modeof the two passive cables when the deck is excited at resonance; the exci-tation starts at t = 5 sec and the control is turned on after t = 30 sec.We note that:1. The amplitude of the cable vibration is hundred times larger than thatof the deck vibration.2. The parametric resonance is established after some transient period inwhich the cable vibration changes from frequency f to f/2. The detailof the transition to parametric resonance is shown clearly in the cen-tral part of Fig.15.19 which shows a detail of Fig.15.18 in the range(10 < t < 14 sec).3. The control brings a rapid reduction of the deck amplitude (due toactive damping) and a slower reduction of the amplitude of the cable atresonance f (due only to the reduced excitation from the deck, since noactive damping is applied to this cable).4. The control suppresses entirely the vibration of the cable at parametricresonance f/2. This confirms that a minimum deck amplitude is necessaryto trigger the parametric resonance.

Fig. 15.17. Set-up of the experiment of parametric resonance.

15.12 Large scale experiment 421

-0.04

-0.02

0

0.02

0.04

-4

-2

0

2

4

0 10 20 30 40-4

-2

0

2

4

Deck

Cable at parametric

resonance (f /2)

Cable at

resonance (f)

Time (sec)

Control on

mm

mm

mm

Fig. 15.18. Vibration amplitude under harmonic excitation of the bridge deck at f :deck, passive cable at parametric resonance f/2, passive cable at resonance f ; the activedamping of the four control cables is switched on at t = 30 sec. The control suppressesentirely the vibration of the cable at parametric resonance.

15.12 Large scale experiment

Although appropriate to demonstrate control concepts in labs, the piezo-electric actuators are inadequate for large scale applications. For cable-stayed bridges, the active tendon must simultaneously sustain the highstatic load (up to 400 t) and produce the dynamic load which is at leastone order of magnitude lower than the static one (< ±10%). This hasled to an active tendon design consisting of two cylinders working to-gether: one cylinder pressurized by an accumulator compensates for thestatic load, and a smaller double rod cylinder drives the cable dynamicallyto achieve the control law. The two functions are integrated in a singlecylinder, as illustrated in Fig.15.20; the double rod part of the cylinderis achieved by a “rod in rod” design; this solution saves hydraulic energyand reduces the size of the hydraulic components. The cylinder is positioncontrolled; the long term changes of the static loads as well as the tem-perature differences require adaptation of the hydraulic conditions of theaccumulator. The mock-up (Fig.15.21 and 15.22) was designed and man-


Fig. 15.19. Detail of Fig.15.18 in the range (10 < t < 14 sec) showing the transitionfrom the forced response at f to the parametric resonance at f/2.

Fig. 15.20. Conceptual design of the two-stage hydraulic actuator (by courtesy ofMannesmann Rexroth).


ufactured by Bouygues in the framework of the ACE project; it has beeninstalled on the reaction wall of the ELSA facility at the Joint ResearchCenter in Ispra. It consists of a cantilever beam (l = 30 m) supported by8 stay cables (d = 13 mm); the stay cables are provided with additionalmasses to achieve a representative sag-to-length ratio (the overall massper unit length is 15 kg/m). An intermediate support can also be placedalong the deck to tune the first global mode and the cable frequencies.Because of the actuator dynamics and the presence of a static load, theimplementation of the control requires some alterations from the basicidea of Fig.15.4:(i) A high-pass filter must be included after the force sensor to eliminatethe static load in the active cables.(ii) In hydraulics, the flow rate is directly related to the valve positionwhich is the control element; it is therefore more natural to control theactuator velocity than its position. In addition, a proportional controlleracting on the actuator velocity is equivalent to an integral controller acting

Fig. 15.21. (a) Global view of the large scale mock-up in the ELSA facility (JRCIspra), (b) detail of the hydraulic actuator.


Fig. 15.22. Schematic view of the mock-up and location of the main components.

on the actuator displacement. The actual implementation of the controlis shown in Fig.15.23.

The overall controller includes a high-pass filter with a corner fre-quency at 0.1Hz (to eliminate the static load), an integrator (1/s), anda low-pass filter with corner at 20Hz, to eliminate the internal resonanceof the hydraulic actuator. The overall FRF (u/T ) of the active controldevice is represented in Fig.15.24. The dotted line refers to the digitalcontroller alone (between 1 and 3 in Fig.15.23) while the full line includesthe actuator dynamics (between 1 and 2 in Fig.15.23). Notice that (i)the controller behavior follows closely a pure integrator in the frequencyrange of interest (0.5Hz − 2Hz) and (ii) the actuator dynamics intro-duces a significant phase lag above the dominant modes of the bridge.Figure 15.25 shows the envelope of the time response of the bridge deckdisplacement near the tip when a sweep sine input is applied to a proof-mass actuator (MOOG, max. inertial force 40 kN) located off axis near


Fig. 15.23. Actual control implementation. Both position and velocity constitute theinputs of the local actuator controller.

Fig. 15.24. FRF between T and u (between 1 and 2 in Fig.15.23). The dotted linedoes not include the actuator dynamics (between 1 and 3 in Fig.15.23).


Deckdisplacement

envelope(mm)

Time (sec)

0.5 0.65 0.8 0.95 1.1 1.25 1.4 1.55 1.7 1.85 2

Excitation instantaneous frequency (Hz)

g=0

g=2

g=15

First bendingmode

Mixed lateral& torsion modes

Active cablesresonance

0 100 200 300 400 500 600 700 800 900 1000

0

-50

-150

-100

50

100

150

Fig. 15.25. Envelope of the time response of the bridge deck displacement whena sweep sine input (from 0.5 Hz to 2 Hz in 1000 sec) is applied to the proof-massactuator, for various values of the control gain.

the end of the deck (Fig.15.22). The sweep rate is very slow (from 0.5 Hzto 2 Hz in 1000 sec). The three curves correspond to various values ofthe gain of the decentralized controller when the two active tendons arein operation (g = 0 corresponds to the open-loop response). The instan-taneous frequency of the input signal is also indicated on the time axis,to allow the identification of the main contributions to the response. Thenumerous peaks in the envelope indicate a complex dynamics of the cable-deck system. One sees that the active tendon control brings a substantialreduction in the vibration amplitude of all modes, and especially the firstglobal bending mode. Using a band-limited white noise excitation anda specially developed identification technique based on the spectral mo-ments of the power spectral density of the bridge response, M. Auperinsucceeded in isolating the first global mode of the bridge. Figure 15.26compares the experimental root-locus with the predictions of the linearapproximation; the agreement is surprisingly good, especially if one thinksof the simplifying assumptions leading to Equation (15.18) . The marks

15.13 References 427

Fig. 15.26. Comparison of experimental and analytical root-locus of the first bendingmode.

on the experimental and theoretical curves indicate the fraction of opti-mum gain g/g∗, where g∗ corresponds to the largest modal damping ratio(theoretical value g∗ = Ωi

√Ωi/ωi). Note that the maximum damping

ratio is close to 17%.

15.13 References

ACHKIRE, Y., Active Tendon Control of Cable-Stayed Bridges, Ph.D.Thesis, Universite Libre de Bruxelles, Active Structures Laboratory, May1997.ACHKIRE, Y. & PREUMONT, A., Active tendon control of cable-stayedbridges, Earthquake Engineering and Structural Dynamics, Vol.25, No 6,585-597, 1996.ACHKIRE, Y. & PREUMONT, A., Optical Measurement of Cable andString Vibration, Shock and Vibration, Vol.5, 171-179, 1998.BOSSENS, F., Controle Actif des Structures Cablees: de la Theorie al’Implementation, Ph.D. Thesis, Universite Libre de Bruxelles, ActiveStructures Laboratory, October 2001.


BOSSENS, F. & PREUMONT, A., Active Tendon Control of Cable-Stayed Bridges: A Large-Scale Demonstration, Earthquake Engineeringand Structural Dynamics, Vol.30, 961-979, 2001.CHEN, J.-C., Response of large space structures with stiffness control,AIAA J. of Spacecraft, Vol.21, No 5, 463-467, 1984.FUJINO, Y. & SUSUMPOW, T., An experimental study on active controlof planar cable vibration by axial support motion, Earthquake Engineeringand Structural Dynamics, Vol.23, 1283-1297, 1994.FUJINO, Y., WARNITCHAI, P. & PACHECO, B.M., Active stiffnesscontrol of cable vibration, ASME J. of Applied Mechanics, Vol.60, 948-953, 1993.FUNG, Y.C., An Introduction to the Theory of Aeroelasticity, Dover,1969.NAYFEH, A.H. & MOOK, D.T. Nonlinear Oscillations, Wiley, 1979.PREUMONT, A. & ACHKIRE, Y., Active Damping of Structures withGuy Cables, AIAA J. of Guidance, Control and Dynamics, Vol.20, No 2,320-326, March-April 1997.PREUMONT, A., ACHKIRE, Y. & BOSSENS, F., Active Tendon Con-trol of Large Trusses, AIAA Journal, Vol.38, No 3, 493-498, March 2000.PREUMONT, A. & BOSSENS, F., Active Tendon Control of Vibrationof Truss Structures: Theory and Experiments, J. of Intelligent MaterialSystems and Structures, Vol.2, No 11, 91-99, Feb. 2000.SCANLAN, R.H. & TOMKO, J., Airfoil and bridge deck flutter deriva-tives, ASCE J. Eng. Mech. div., 100, 657-672, August 1974.WARNITCHAI, P., FUJINO, Y., PACHECO, B.M. & AGRET, R., Anexperimental study on active tendon control of cable-stayed bridges,Earthquake Engineering and Structural Dynamics, Vol.22, No 2, 93-111,1993.YANG, J.N. & GIANNOPOULOS, F., Active control and stability ofcable-stayed bridge, ASCE J. Eng. Mech. div., 105, 677-694, August 1979.YANG, J.N. & GIANNOPOULOS, F., Active control of two-cable-stayedbridge, ASCE J. Eng. Mech. div., 105, 795-810, October 1979.

16

Active Control of Large Telescopes

16.1 Introduction

Primary

Mirror

(M1)

...

Image

Plane

wavefront

Spherical

wavefront

(divergent)

Point source

Spherical

wavefront

(convergent)

Secondary

mirror

(M2)

Fig. 16.1. Principle of a reflective telescope with “Alt-Az” mount (only the primaryand the secondary mirrors are considered). The plane wave front is transformed into aconvergent spherical wave front.

A reflective telescope (Fig.16.1) consists of a set of reflectors whichtransform a plane wavefront into a convergent spherical wavefront, insuch a way that a point source at infinity forms a point image in the focalplane. However, because of diffraction, the image of a point object of aperfect (circular) telescope is not a point, but an area of finite size re-sulting from the spreading of the light energy, called Airy disk. Its size isdirectly related to the wavelength λ of the light observed. More generally,

430 16 Active Control of Large Telescopes

the image of a point object is called the Point Spread Function (PSF) ofthe instrument. The theoretical limit resolution of a telescope is propor-tional to the ratio λ/D between the wavelength and the diameter of theprimary mirror M1. Increasing the diameter D has two beneficial effects:(i) gathering more light (more photons), which allows the observation offainter objects, and (ii) increasing the limit resolution of the telescope,which allows to distinguish finer details. Earth based telescopes are sub-jected to two broad classes of aberrations:(i) Atmospheric turbulence produces a temporal and spatial random vari-ation of the refraction index of the air, which distorts the wavefront ofincoming plane waves (Fig.16.2). Atmospheric turbulence can be curedby Adaptive Optics.(ii) Manufacturing errors, thermal gradients and the variations of thegravity loads as the telescope moves to follow a fixed target in the skywhile the earth rotates, are responsible of low frequency but significantstructural deformations, which can be compensated by Active Optics.

An optical system is considered as diffraction limited if the RMS wave-front error is less than λ/14 (0.4 µm < λ < 0.8 µm in the visible spectrumand 0.8 µm < λ < 5 µm in the near infrared); thus, more accuracy is re-quired for shorter wavelength.

16.2 Adaptive optics

The principle of adaptive optics (AO) is shown in Fig.16.2; a small de-formable mirror is inserted in the optical train to counteract in real timethe aberration of the wavefront. The most popular wavefront sensor is theso-called “Shack-Hartmann”; a beam splitter deviates part of the incom-ing light towards an array of micro lenses which measure the normals tothe wavefront of a reference star at an array of discrete points in the tele-scope aperture. The wavefront aberrations are often expressed in a setof orthogonal functions called the Zernike polynomials (Fig.16.3). Thedesign of an AO system depends on many parameters such as the wave-length, the field of view and the size of the primary mirror. Typically, theamplitude of the corrected shape is a few microns, and the bandwidth isin the range 50-100Hz, depending on the wavelength; it is essential thatthe natural frequency of the mirror be significantly higher than the band-width, to operate the deformable mirror in the quasi-static mode. Thenumber of degrees of freedom (of independent actuators) also depends onthe wavelength and on the size of the primary mirror; approaching the

16.2 Adaptive optics 431

Deformable

mirror

Controller

Wavefront analyser

(Shack-Hartmann sensor)

Instrument

8

Atmospheric

turbulence

Deformed

wavefront

Plane

wavefront

Telescope

Fig. 16.2. Principle of adaptive optics to correct atmospheric turbulence. A smalldeformable mirror is controlled in real time to compensate the wavefront aberrations,measured with a wavefront sensor.

diffraction limit in the visible with a large telescope may require thou-sands of independent actuators. Figure 16.4 shows a bimorph deformablemirror made of a silicon wafer of 150 mm diameter and 800 µm thickness,covered with a screen printed thick film of PZT of 70 µm (Rodrigues);the honeycomb actuator array consists of 91 independent electrodes, thevoltage of which can be adjusted independently between 0 and 160V.1

Fig. 16.5 shows examples of experimentally corrected aberrations and thecorresponding voltage distribution.

Let s be the output vector of the wavefront sensor (of dimension n) andv the vector (of dimension m) of input voltages applied to the independentelectrodes. Assuming a linear relationship between s and v, one has

s = Dv (16.1)

1 an offset of 80V is applied to allow corrections of ±80V .


Tilt

Astigmatism Astigmatism

Trefoil Coma Trefoil

Tetrafoil Tetrafoil

Ra

dia

l o

rd

er

Piston

Defocus

Spherical Aberration

Azimuthal Order

0

Tilt

Coma

1 2 3 4-1-2-3-4

0

1

2

3

4

Fig. 16.3. Optical aberrations: low order Zernike polynomials.

where the (n × m) matrix D is the Jacobian of the system, which canbe constructed column by column during the calibration phase of the AOsystem, by applying a given voltage to every input channel, one after theother. n may be larger, equal or smaller than m. Once the matrix D isavailable, the voltage v necessary to correct the wavefront correspondingto a sensor output s is obtained by performing a Singular Value Decom-position (SVD) of the Jacobian matrix according to

D = UΣV T (16.2)

where the columns of U are the orthonormalized sensor modes, thecolumns of V are the orthonormalized actuator modes and Σ is a (n×m)rectangular matrix which contains the singular values σi on its diagonal.The number of non zero singular values is equal to the rank of the matrixD. The control voltage v necessary to correct the sensor error s is given

16.2 Adaptive optics 433

Fig. 16.4. Deformable mirror made of a 150 mm silicon wafer covered on its back sidewith an array of screen printed PZT actuators with honeycomb electrodes (developedjointly with the Fraunhofer IKTS).

byv = D+s = V Σ−1UT s (16.3)

where Σ−1 is (m× n) with σ−1i on its diagonal. The rectangular matrix

D+ is called the pseudo-inverse of the Jacobian matrix. The solutionof the foregoing equation may have different meaning, depending on therelative size of n and m. If the sensor output vector is larger than thevoltage input vector (n > m), the solution (16.3) is that minimizing the

5µm 4.5µm 3µm

Defocus Astigmatism Tetrafoil

Fig. 16.5. Deformable mirror: typical corrected aberrations with the correspondingvoltage distribution within the honeycomb electrodes.


quadratic norm ||s−Dv||; if m > n, the solution is that of minimum norm||v|| satisfying the equality constraint (16.1).2

16.3 Active optics

Adaptive optics is intended to correct the wavefront errors introducedby the atmospheric turbulence, which exist even in a perfect telescopeoperating on earth. However (Fig.16.2), the wavefront sensor cannot sep-arate the wavefront error due to atmospheric turbulence from that dueto the telescope imperfections and the adaptive optics will attempt tocorrect them as well. For large telescopes, however the telescope deforma-tions due to manufacturing errors, thermal gradients and gravity loads areseveral orders of magnitude larger and cannot be fully corrected by theadaptive optics; they are alleviated with a specific control system calledactive optics.

Main axes

M1Shape

M2 RigidBody

Adaptive Optics

Bandwidth [Hz]

0.01 0.1 1 10 100

Spatial

fre

quen

cy

100

3

2

Fig. 16.6. Spatial and temporal frequency distribution of the various active controllayers of extremely large telescopes (adapted from Angeli et al.). Adaptive optics hasamplitudes of a few microns. The shape control of M1 involves much larger amplitudes.

2 The SVD controller will be investigated below in relation with active optics. Notethat the tracking errors involved in adaptive optics are typically of the order of ∼ 10λ.Since an optical system is considered as diffraction limited if the RMS wavefront erroris less than λ/14, the control objectives would be to reduce the RMS surface errorto below λ/28 (the optical path difference is twice the surface error). Overall, thisleads to a gain ∼ 50dB. Another issue is the control-structure interaction when thecontrol bandwidth interferes with the vibration modes of the flexible mirror.

16.3 Active optics 435

Figure 16.6 shows the various control layers of an extremely large tele-scope and their spatial and temporal frequency distribution. Adaptiveoptics covers a wide band of temporal frequencies as well as spatial fre-quencies (Zernike modes of higher orders), but with small amplitudes of afew microns. On the contrary, the control of the rigid body motion of thesecondary mirror M2 and the active shape control of the primary mirrorM1 must counteract disturbances of very low frequency (changes in grav-ity loads take place at one cycle per day, that is 1.16 10−5Hz ), but withmuch larger amplitudes, in millimeters (>> 100λ); this requires largergains than adaptive optics, and a different technology. We will examinesuccessively the shape control of monolithic and segmented primary mir-rors.

16.3.1 Monolithic primary mirror

Figure 16.7 shows the principle of the active optics for a monolithic pri-mary mirror as it was first implemented on the ESO-NTT telescope (Wil-son, 1987). The primary mirror consists of a thin deformable meniscus

M1

M2

Scienceinstrument

Shack-Hartmann

CCDcamera

Controller

( = 0.03Hz)fc

Controller

( ~ 0.1Hz)fc

Alt-az axes

Beamsplitters

Fig. 16.7. Active optics of a monolithic primary mirror (ESO-NTT).


equipped with an array of force actuators on its back3 and the secondarymirror can also be actuated to correct defocus and coma.

The control system uses a Shack-Hartmann sensor and the signal isaveraged over a long period (30 s) to eliminate the effect of atmosphericturbulence. The system is essentially described by Equ.(16.1), where v isthe vector of control inputs (including the forces acting at the back ofM1 and the position of M2) and s is the vector of sensor outputs. TheJacobian is once again determined column by column by analyzing theimpact of every actuator on the wavefront sensor.

16.3.2 Segmented primary mirror

Monolithic mirrors are limited to a maximum size of about 8 m; largermirrors must be segmented. The Keck (located in Hawaı) and the GranTeCan (located in the Cannary Islands) are the largest optical telescopesin operation, with a diameter of the primary mirror (M1) a little largerthan 10 m; their primary mirror consists of 36 segments of hexagonalshape. Extremely Large Telescopes (ELTs), with a primary mirror of di-ameter up to 42 m (E-ELT will have 984 segments!), are currently intheir design phase. Since the sensitivity to disturbance increases with thesize of the telescope and the wavefront error cannot exceed a fraction ofthe wavelength of the incoming light, larger telescopes will rely more and

Controller

Edge sensors ( )yWind

Supporting

trussif

Segments

actuators ( )a

Position

y J a.=1 1 e

Fig. 16.8. Schematic view of a segmented primary mirror, with the supporting truss,the position actuators and the edge sensors. The quasi-static behavior of the reflectorfollows y1 = Jea, where a is the control input (position of the actuators), y1 is the edgesensor output (relative displacement between segments) and Je is the Jacobian of thesegmented mirror.

3 For the NTT (located in La Silla), the diameter of M1 is 3.6 m, the thickness is 0.24m and there are 78 actuators; For the VLT (located in Paranal), the diameter of M1is 8.2 m, the thickness is 0.17 m and there are 150 actuators.

16.3 Active optics 437

more on active control, with higher gains, leading to wider bandwidth. Onthe other hand, since the natural frequencies of the structural system tendto be lower for larger telescopes, control-structure interaction becomes acentral issue in the control system design.

Figure 16.8 shows a schematic view of a segmented primary mirror,with the supporting truss. Every segment can be regarded as a rigid body;it is supported by 3 position actuators via a whiffle tree. A set of sixedge sensors monitor the position of every segment with respect to itsneighbors (overall, there will be 2952 position actuators and 5604 edgesensors for E-ELT); the edge sensors play the key role of co-phasing thevarious segments (i.e. making them work as a single, monolithic mirror).If the supporting truss is assumed infinitely stiff, the quasi-static behaviorof the system is governed by a purely kinematic relationship

y1 = Je.a (16.4)

where Je is the Jacobian of the edge sensors. Since the lower optical modes(piston, tilt, defocus) are not observable from the edge sensors, anotherset of sensors measuring the normal to the segments (e.g. one per segment,or just a few for the entire mirror) may be used, leading to

y2 = Jn.a (16.5)

where Jn is the Jacobian of the normal sensors. Note, however, that theoutput signal y2 is also affected by atmospheric turbulence, which is elimi-nated by time averaging, just as for monolithic mirrors. The active opticscontrol flow is shown in Fig.16.9. The mirror segments are representedby rigid bodies. In order to include the flexibility of the whiffle tree intothe model, the position actuator can be modelled with a force actuatorFa acting in parallel on a spring ka and dash-pot ca; the stiffness ka isselected to account for the local modes of the segments and ca to providethe appropriate damping. the force is related to the unconstrained dis-placement a by Fa = a.ka. The position actuators rest on the supportingtruss carrying the whole mirror. The disturbances d acting on the systemcome from thermal gradients, changing gravity vector with the elevationof the telescope, and wind. Control-structure interaction may arise fromthe force actuator Fa exciting the resonances fi of the supporting truss orthe local modes of the segments. Before addressing the dynamic responseof the mirror, let us discuss the quasi-static shape control of a deformablemirror.


H s( )

Edge Sensorsd

Supportingtruss

m

ka Faca

y1

y2

Normal tosegment +

n Atmosphericturbulence

Normal to wavefront(Shack-Hartmann)

if

aSegment

PositionActuators

(gravity, wind)

Fig. 16.9. Active optics control flow for a large segmented mirror. The axial d.o.f.at both ends of the actuators are retained in the reduced dynamical model. Control-structure interaction may arise from the force actuator Fa exciting the resonances fi

of the supporting truss or the local modes of the segments

16.4 SVD controller

The quasi-static behavior of adaptive optics mirrors, or of deformableprimary mirrors (whether monolithic or segmented) is described by anequation

y =

(Je

Jn

)a = J.a (16.6)

where y is the output of some set of sensors and a is the input of someset of actuators and J is the Jacobian of the system. The block diagramof the SVD controller is shown in Fig.16.10; V Σ−1UT is the inverse ofthe plant; the diagonal gains σ−1

i provide equal authority on all singularvalue modes; only the modes with non-zero singular values are considered

Primarymirror

Positionactuators

Edgesensors

a y

d

-

+

V H )(s S -1 U T

J U= S V T

Fig. 16.10. Block diagram of the SVD controller.

16.4 SVD controller 439

50 250

Coma

Trifoil

Astigmatism

Piston

Tilt

Defocus

0 25 275

Trifoil, Coma

Trifoil, Defocus

Piston

Tilt

Astigmatism

edge sensors

edge + normal

Index

100

10-2

10-15

Fig. 16.11. Distribution of the singular values of a segmented mirror for two sensorconfigurations (edge sensor measurements and actuator displacements are expressed inmeters, tilt angles in radians).

in the control block. H(s) is a diagonal matrix of filters intended to supplyappropriate disturbance rejection and stability margin. The distributionof the singular values depends on the sensor configuration, as illustratedin Fig.16.11 which compares the use of edge sensors alone with the op-tion combining edge sensors and normal sensors (the conditioning of theJacobian is much better in this case).

16.4.1 Loop shaping of the SVD controller

The open loop transfer function of the nominal plant is G0(s) = J =UΣV T and the controller is K(s) = V H(s)Σ−1UT . If one assumes thesame loop shape for all singular value modes, H(s) = h(s)I and

K(s) = h(s)V Σ−1UT (16.7)

essentially inverts the Jacobian of the mirror, leading to the open-looptransfer matrix G0(s)K(s) being diagonal, with all non-zero singular val-ues being equal to

σ(G0K) = |h(jω)| (16.8)

Thus, the loop shaping can be done as for a SISO controller, accordingto the techniques developed in Chapter 10. The control objective is tomaximize the loop gain in the frequency band where the disturbancehas a significant energy content while keeping the roll-off slow enoughnear crossover to achieve a good phase margin. An integral component is


10-4

10-2

100

102

-100

0

100

[Hz]

[dB]

-250

-200

-150

-100

[°]

[dB]

[°]

45°

-180°

f

10 dB0 dB

PM = 70°

GM = 25dB

125dB

10-5

10-3

10-1

101

f = 0.25Hzc

Earth rotation

(1.16 10 Hz)-5

10-4

10-2

100

102

[Hz]

10-5

10-3

10-1

101

Fig. 16.12. Compensator h(s) common to all loops of the SVD controller. Left: Nicholschart. Right: Bode plots showing the gain of 125dB at the earth rotation frequency anda cut off frequency of fc = 0.25Hz.

necessary to eliminate the static error in the mirror shape. The gain at theearth rotation frequency must be large enough to compensate the gravitysag. Figure 16.12 shows typical Bode plots and the corresponding Nicholschart; the compensator consists of an integrator, a lag filter, followed bya lead and a second order Butterworth filter. The crossover frequency isfc = 0.25Hz and the amplitude at the earth rotation frequency is 125dB.The robustness margins of this quasi-static controller are clearly visibleon the Nichols chart [the exclusion zone around the critical point (−1800,0dB) corresponds to (PM= ±450, GM= ±10dB)]. Note that the Nicholschart is invariant with respect to a shift of the Bode plots along thefrequency axis, which gives a simple way to adjust the control bandwidthto achieve the low frequency specification.

However, it is important to point out that the robustness marginsdisplayed by the Nichols chart do not tell anything about the control-structure interaction, since our analysis has been based on a quasi-staticmodel of the plant, ignoring the dynamics of the supporting truss.

16.5 Dynamics of a segmented mirror

The dynamics of the mirror consist of global modes involving the sup-porting truss and the segments, and local modes involving the segments

16.5 Dynamics of a segmented mirror 441

alone. The global modes are critical for the control-structure interaction;the segments are normally designed in such a way that their local modeshave natural frequencies far above the critical frequency range, but theirquasi-static response (to the actuator as well as to gravity and wind dis-turbances) must be dealt with accurately in the model. In order to handlelarge optical configurations, it is important to reduce the model as muchas possible, without losing the features mentioned before. A model of min-imum size can be constructed using a Craig-Bampton reduction (section2.8), where the master d.o.f. consist of the axial d.o.f. at both ends of theposition actuators (represented by circles in Fig.16.9) which are necessaryto describe the kinematics of the system, supplemented by an appropriateset of fixed boundary modes (usually a small number) which take care ofpossible internal modes of the supporting truss.

Figure 16.13 shows the eigenfrequency distribution of a typical seg-mented mirror; the first 20 modes or so are global modes, with modeshapes combining optical aberration modes of low order; they are followedby local modes of the segments (tilt near 75Hz and piston near 100Hz ).If the system is properly designed, only the low frequency modes canpotentially jeopardize the system stability and, provided that the staticbehavior is not altered, the reduced model can be truncated as shownin Fig.16.14 (in the figure, Fm is the quasi-static response of the flexible

50 100 150 200 250 3000

50

100

150

200

Mode index

Eig

en frequency [H

z]

global

modes

PistonTilt

Fig. 16.13. Eigenfrequency distribution of a typical segmented mirror.


GR

GO

+

+

a y

kaSa

Fm

iiii

iifT

s2+2x w s ii

2+w

...

...

Fm

+

-

Sy1

Sy2

y1

y2

Flexible modesi = 1,…,m

(Residual response)

aJ =

JeJn( (

(Primary response)

Jacobian

y1

y2

+ +

y

Fig. 16.14. Input-output relationship of the segmented mirror. The nominal plantG0(s) = J accounts for the quasi-static response (primary response) and the dynamicdeviation GR(s) (residual response) is regarded as an additive uncertainty.

modes included in the residual response, which has already been includedin the Jacobian; the matrices Sy1 and Sy2 describe the sensor topology andSa describes the actuator topology). The global input-output relationshipis written in the form

G(s) = G0(s) + GR(s) (16.9)

where the nominal plant G0(s) = J has been taken into account in thecontroller design (primary response), and GR(s) is the dynamic deviation(residual response), which is considered as an additive uncertainty.

Primary

K

+ +

- +

Residual

Control

a)

G G K= O

G KG= O

L G G= O R-1

L G K= R

OG

RG

+ +

- +

c)

G

L

+

-

b)

GI L+

a y

Fig. 16.15. Block diagram of the control system (a) Mirror represented by its primaryand residual dynamics. (b) Multiplicative uncertainty. (c) Additive uncertainty.

16.6 Control-structure interaction 443

16.6 Control-structure interaction

The controller transfer matrix is essentially the inverse of the quasi-staticresponse of the mirror. However, because the response of the mirror in-cludes a dynamic contribution at the frequency of the lowest structuralmodes and above, the system behaves according to Fig.16.14 and the ro-bustness with respect to control-structure interaction must be examinedwith care. The structure of the control system is that of Fig.16.15.a, wherethe primary response G0(s) corresponds to the quasi-static response de-scribed earlier and the residual response GR(s) is the deviation resultingfrom the dynamic amplification of the flexible modes; K(s) is the con-troller, given by Equ.(16.7).

The control-structure interaction may be addressed with the generalrobustness theory of multivariable feedback systems (section 10.9); theresidual response being considered as uncertainty.

16.6.1 Multiplicative uncertainty

According to section 10.9.3, if one assumes a multiplicative uncertainty,the standard structure of Fig.16.15.b applies, and one can check thatFig.16.15.a and b are equivalent with G(s) = K(s)G0(s) and L =G−1

0 GR.4 A sufficient condition for stability is given by (10.64):

σ[L(jω)] < σ[I + G−1(jω)], ω > 0 (16.10)

(σ and σ stand respectively for the maximum and the minimum singularvalue), which is transformed here into

σ[G−10 GR(jω)] < σ[I + (KG0)−1(jω)], ω > 0 (16.11)

This test is quite meaningful; the left hand side represents an upper boundto the relative magnitude of the uncertainty; it is independent of the con-troller; it starts from 0 at low frequency where the residual dynamicsis negligible and increases gradually when the frequency approaches theflexible modes of the mirror structure, which are not included in the nom-inal model G0; the amplitude is maximum at the resonance frequencieswhere it is only limited by the structural damping of the flexible modes.The right hand side starts from unity at low frequency where | KG0 |À 1(KG0 controls the performance of the control system) and grows larger4 in all this section, the inverse of rectangular matrices should be understood in the

sense of pseudo-inverse.


10.01 100.1 100

[Hz]

10

1

0.1

0.01

GM

s [ + ( ) ]I KG0-1

A

f1

Structuraluncertainty

s [ ]G G0 R

-1

Fig. 16.16. Robustness test assuming multiplicative uncertainty. σ[I+(KG0)−1] refers

to the nominal system used in the controller design. σ[G−10 GR] is an upper bound to

the relative magnitude of the residual dynamics. The critical point A corresponds tothe closest distance between these curves. The vertical distance between A and theupper curve has the meaning of a gain margin.

than 1 outside the bandwidth of the control system where the systemrolls off (| KG0 |¿ 1). A typical robustness test plot is represented inFig.16.16; the critical point A corresponds to the closest distance betweenthese curves. The vertical distance between A and the upper curve hasthe meaning of a gain margin GM (if the gain of all control channels ismultiplied by a scalar g, the high frequency part of the upper curve willbe lowered by g). When the natural frequency of the structure changesfrom f1 to f∗1 , point A moves horizontally according to the ratio f∗1 /f1

(increasing the frequency will move A to the right). Similarly, changingthe damping ratio from ξ1 to ξ∗1 will change the amplitude according toξ1/ξ∗1 (increasing the damping will decrease the amplitude of A).

16.6.2 Additive uncertainty

Alternatively, if one assumes an additive uncertainty, the standard struc-ture of Fig.16.15.c applies, and one can check that Fig.16.15.a and c areequivalent with G = G0K and L = GRK; a sufficient condition for sta-bility is given by (10.62):

σ[L(jω)] < σ[I + G(jω)], ω > 0 (16.12)

which is translated into

σ[GRK(jω)] < σ[I + G0K(jω)], ω > 0 (16.13)

16.6 Control-structure interaction 445

10

1

1

0.1

0.01 100.1 100

[Hz]

0.01

s [ + ( ) ]I KG0

f1

GM

A

s [ ]G KR

Structuraluncertainty

Fig. 16.17. Robustness test assuming additive uncertainty. σ[I + G0K(jω)] refers tothe nominal system used in the controller design. σ[GRK(jω)] is an upper bound tothe effect of the controller on the residual dynamics. The critical point A correspondsto the closest distance between these curves. The vertical distance between A and theupper curve has the meaning of a gain margin.

Again, the smallest distance between these two curves has the meaning ofa gain margin (if the gain of all control channels is multiplied by a scalarg, the lower curve, GRK, will be multiplied by g).

Note that the stability conditions (16.11) and (16.13) come from thesmall gain theorem; being sufficient conditions, they are both conservativeand one may be more conservative than the other.

16.6.3 Discussion

As the telescopes increase in size, so does the gravity sag, requiring highercontrol gains to maintain the right shape, and increasing the control band-width fc; this means that the curve σ[I +(KG0)−1] in Fig.16.16 is movingto the right. At the same time, the natural frequencies fi of the flexiblemodes decrease when the size of the structure increases, which means thatthe curve σ[G−1

0 GR] is moving to the left. The robustness with respectto the control-structure interaction tends to be controlled by the ratiofi/fc between the natural frequency of the critical mode (not necessarilythe first)5 and the control bandwidth. For a given telescope design, usingthe foregoing robustness tests, it is possible to plot the evolution of thegain margin with the frequency ratio f1/fc; this curve depends strongly

5 for the Keck telescope, the critical mode turned out to be a local mode of thesegments with a frequency of ∼ 25Hz.


101

102

-20

0

20

40

60

80

f1/f

c

GM

[dB

]

UNSTABLE

GM=10dB

0.005

0.05

0.02

0.01

x

Fig. 16.18. Evolution of the gain margin with the frequency ratio f1/fc for variousvalues of the damping ratio ξi.

on the structural damping ratio, since the amplitude of the various res-onance peaks is proportional to ξ−1. Figure 16.18 shows a typical plot;one sees that if the critical structural mode of the supporting structurehas a damping ratio of 1%, a gain margin GM=10 requires a frequencyseparation f1/fc significantly larger than one decade. This condition maybe more and more difficult to fulfill as the size of the telescope grows.The situation can be improved by increasing the structural damping ofthe supporting truss, possibly actively.

16.7 References

ANGELI, G.Z., CHO, M.K., WHORTON, M.S. Active optics and archi-tecture for a giant segmented mirror telescope, in Future Giant Telescopes(ANGELI, and GILMOZZI, eds.), 2002. Proc. SPIE 4840, Paper No. 4840-22, pages 129-139.AUBRUN, J.N., LORELL, K.R., MAST, T.S. & NELSON, J.E. DynamicAnalysis of the Actively Controlled Segmented Mirror of the W.M. KeckTen-Meter Telescope, IEEE Control Systems Magazine, 3-10, December1987.AUBRUN, J.N., LORELL, K.R., HAVAS & T.W., HENNINGER, W.C.Performance Analysis of the Segment Alignment Control System for theTen-Meter Telescope, Automatica, Vol.24, No 4, 437-453, 1988.

16.7 References 447

BASTAITS, R. Extremely Large Segmented Mirrors: Dynamics, Controland Scale Effects, PhD Thesis,Universite Libre de Bruxelles, Active Struc-tures Laboratory, 2010.BASTAITS, R., RODRIGUES, G., MOKRANI, B. & PREUMONT, A.Active Optics of Large Segmented Mirrors : Dynamics and Control, AIAAJournal of Guidance, Control, and Dynamics, Vol.32, No 6, 1795-1803,Nov.-Dec. 2009.BELY, P.Y. The Design and Construction of Large Optical Telescopes,Springer, 2003.ENARD, D., MARECHAL, A., & ESPIARD, J. Progress in ground-basedoptical telescopes, Reports on Progress in Physics, Vol. 59, 601-656, 1996.HARDY, J.W. Adaptive Optics for Astronomical Telescopes, Oxford Uni-versity Press, 1998.NOETHE, L. Active Optics in Modern, Large Optical Telescopes. Progressin Optics, 43: 1-70, 2002.PREUMONT, A., BASTAITS, R., RODRIGUES, G. Scale Effects in Ac-tive Optics of Large Segmented Mirrors, Mechatronics, Vol. 19, No 8,1286-1293, December 2009.RODDIER, F. Adaptive Optics in Astronomy, Cambridge UniversityPress, 1999.RODRIGUES, G. Adaptive Optics with Segmented Deformable BimorphMirrors, PhD Thesis, Universite Libre de Bruxelles, Active StructuresLaboratory, 2010.STRANG, G. Linear Algebra and its Applications, Harcourt Brace Jo-vanovich Publishers, 1988.WILSON, R.N., FRANZA, F. & NOETHE, L. Active optics 1. A sys-tem for optimizing the optical quality and reducing the costs of largetelescopes, Journal of Modern Optics, Vol.34, No 4, 485-509, 1987.

17

Semi-active control

17.1 Introduction

Active control systems rely entirely on external power to operate the ac-tuators and supply the control forces. In many applications, such systemsrequire a large power source, which makes them costly (this is why therehas been very few cars equipped with fully active suspensions) and vul-nerable to power failure (this is why the civil engineering community isreluctant to use active control devices for earthquake protection). Semi-active devices require a lot less energy than active devices; and the energycan often be stored locally, in a battery, thus rendering the semi-activedevice independent of any external power supply. Another critical issuewith active control is the stability robustness with respect to sensor fail-ure; this problem is especially difficult when centralized controllers areused.

On the contrary, semi-active control devices are essentially passive de-vices where properties (stiffness, damping,...) can be adjusted in real time,but they cannot input energy directly in the system being controlled. Notehowever that since semi-active devices behave nonlinearly; they can trans-fer energy from one frequency to another. The variable resistance law canbe achieved in a wide variety of forms, as for example position controlledvalves, rheological fluids, or piezoelectrically actuated friction joints.

Over the past few years, semi-active control has found its way in manyvibration control applications, for large and medium amplitudes, (partic-ularly vehicle suspension, but also earthquake protection,...). However,it should be kept in mind that, in most cases, semi-active devices aredesigned to operate in the “post yield” region, when the stress exceedssome controllable threshold; this makes them inappropriate for vibrations

450 17 Semi-active control

of small amplitude where the stress remains below the minimum control-lable threshold in the device. It should also be pointed out that, in manyapplications (e.g. domestic appliances), the cost of the control system isa critical issue (it is much more important than the optimality of theperformances); this often leads to simplified control architectures withextremely simple sensing devices.

Magneto-rheological fluids exhibit very fast switching (of the order ofmillisecond) with a substantial yield strength; this makes them excellentcontenders for semi-active devices, particularly for small and medium-size devices, and justifies their extensive discussion. This chapter beginswith a review of magneto-rheological (MR) fluids and a brief overview oftheir applications to date. Next, some semi-active control strategies arediscussed.

17.2 Magneto-rheological fluids

No fieldApplied field

Fig. 17.1. Chain-like structure formation under the applied external field.

In 1947, W.Winslow observed a large rheological effect (apparentchange of viscosity) induced by the application of an electric field tocolloıdal fluids (insulating oil) containing micron-sized particles; such flu-ids are called electro-rheological (ER) fluids. The discovery of MR fluidwas made in 1951 by J.Rabinow, who observed similar rheological effectsby application of a magnetic field to a fluid containing magnetizable par-ticles. In both cases, the particles create columnar structures parallel tothe applied field (Fig.17.1) and these chain-like structures restrict the flowof the fluid, requiring a minimum shear stress for the flow to be initiated.This phenomenon is reversible, very fast (response time of the order of

17.2 Magneto-rheological fluids 451

milliseconds) and consumes very little energy. When no field is applied,the rheological fluids exhibit a Newtonian behavior.

Typical values of the maximum achievable yield strength τ are givenin Table 17.1. ER fluids performances are generally limited by the electricfield breakdown strength of the fluid while MR fluids performances arelimited by the magnetic saturation of the particles. Iron particles have thehighest saturation magnetization. In Table 17.1, we note that the yieldstress of MR fluids is 20 to 50 times larger than that of ER fluids. Thisjustifies why most practical applications use MR fluids. Typical particlesizes are 0.1 to 10µm and typical particle volume fractions are between0.1 and 0.5; the carrier fluids are selected on the basis of their tribologyproperties and thermal stability (the operable temperature range of MRfluids is −400C< T 0 < 1500C); they also include additives that inhibitsedimentation and aggregation.

Property ER fluid MR fluid

Yield Strength τ 2− 5 kPa 50− 100 kPaMax. field 3− 5 kV/mm 150− 250 kA/mViscosity η(at 25oC under no field)

0.2− 0.3 Pa.s 0.2− 0.3 Pa.s

Density 1− 2 g/cm3 3− 4 g/cm3

Response time ms ms

Table 17.1. Comparison of typical ER and MR fluid properties.

The behavior of MR fluids is often represented as a Bingham plasticmodel with a variable yield strength τy depending on the applied magneticfield H, Fig.17.2. The flow is governed by the equation

τ = τy(H) + η γ , τ > τy(H) (17.1)

where τ is the shear stress, γ is the shear strain and η is the viscosity of thefluid. The operating range is the shaded area in Fig.17.2.c. Below the yieldstress (at strains of order 10−3), the material behaves viscoelastically:

τ = G γ , τ < τy(H) (17.2)

where G is the complex material modulus. Bingham’s plastic model is alsoa good approximation for MR devices (with appropriate definitions for τ ,


(a) (b)

(c) (d)

y (H)

Hmax

H = 0

operatingrange

y (H)

Fig. 17.2. (a) and (b) Bingham plastic model consisting of a constant viscous damperin parallel with a variable friction device. (c) Operating range. (d) Hysteretic behaviorobserved.

γ and η). However, the actual behavior is more complicated and includesstiction and hysteresis such as shown in Fig.17.2.d; more elaborate modelsattempting to account for the hysteresis are available in the literature,Fig.17.3, but Bingham’s model is sufficient for most design work.

17.3 MR devices

Figure 17.4 shows the four operating modes of controllable fluids: valvemode, direct shear mode, squeeze mode and pinch mode. The valve modeis the normal operating mode of MR shock absorbers (Fig.17.5); the con-trol variable is the current through the coil, which controls the magneticfield in the active part of the fluid and, as a result, creates the variableyield force in the device. The direct shear mode is that of clutches andbrakes (Fig.17.6).

17.3 MR devices 453

a)

x

F

c0

F

c0

x1 x2 x3

k1

k 2

b)

k0

Bouc-Wen

x

F

c)

c0

Bouc-Wen

x

F

k1

c1

c1

d)1500

-1500

0

12-12 0

Velocity [cm/s]

Force [N

]

1500

-1500

0

12-12 0

Velocity [cm/s]

Fo

rce

[N

]

1500

-1500

0

12-12 0

Velocity [cm/s]

Force [N

]

1500

-1500

0

12-12 0

Velocity [cm/s]

Force [N

]

model

experiment

model

experiment

model

experiment

model

experiment

k0

c0

Fig. 17.3. MR fluid and MR damper phenomenological models: (a) Bingham model.(b) Gamota and Filisko (c) Bouc-Wen. (d) Spencer et al. Forces-velocity curves areadapted from (Spencer et al.).


applied magnetic field applied magnetic field

applied magnetic field

applied magnetic field

pressure

pressure

flow

force speed

force displacement

flow

MR-fluid MR-fluid

MR-fluidMR-fluid

a) b)

c) d)

A A

gg

w

L

non-magnetic

spacer

Fig. 17.4. Operating modes of controllable fluids: (a) valve mode, (b) direct shearmode, (c) squeeze mode and (d) pinch mode.

a) b)

Fig. 17.5. MR shock absorber (adapted from Carlson, 2007).

17.3 MR devices 455

Fig. 17.6. Various MR brake designs: (a) drum, (b) inverted drum, (c) T-shaped rotor,(d) disk, (e) multiple disks (from Avraam).


17.4 Semi-active suspension

A semi-active suspension consists of a classical suspension provided witha controllable shock absorber, capable of changing its characteristics inreal-time with a small amount of energy. The device remains essentiallypassive and can only dissipate energy, that is to produce a force opposingthe motion applied to the device. In general the term semi-active suspen-sion refers to a suspension provided with a controllable shock absorbercapable of changing its characteristics in wide-band; this requires a fastresponding controllable device. Adaptive suspensions involve controllableshock absorbers with low-frequency capability, allowing the damper char-acteristics to be adapted to optimize ride comfort and road holding for thecurrent road roughness and driving conditions; such a system is availableon many cars, with various degrees of sophistication; they offer new capa-bilities to enhance the vehicle dynamics, in connection with the so-called“ESP” system.

(a) (b)

(u)

max

min

f f

ff

v v

vv

c

c

c

f = c v + (u)

f = c(u)v

max

Fig. 17.7. Semi-active devices and their operating range. (a) Viscous damper withvariable damping coefficient. (b) MR fluid device and its Bingham model.

17.5 Narrow-band disturbance 457

17.4.1 Semi-active devices

Two of the most frequently used semi-active devices are illustrated inFig.17.7, with their respective operating range. The first one (left) consistsof a classical viscous damper with a variable damping coefficient c(u)obtained by controlling the size of the opening of an orifice between thetwo chambers of the damper (e.g. with an electromagnet). The operatingrange is the shaded area between two lines corresponding to the minimumand maximum damping coefficients, cmin < c(u) < cmax. The secondone (right) consists of a MR fluid damper similar to that of Fig.17.5; itsbehavior is represented by its Bingham model (Fig.17.7.b).

17.5 Narrow-band disturbance

Referring to the transmissibility of a passive isolator (Fig.8.2), when thedisturbance frequency is ω <

√2ωn, the overshoot is minimized by set-

ting a high damping constant, while above√

2ωn, the damping should beminimum to enjoy the maximum roll-off rate. This suggests the followingcontrol strategy according to the disturbance frequency ω:

If ω ≤√

2ωn then c = cmax (17.3)

If ω >√

2ωn then c = cmin

Fig. 17.8. (a) MR fluid engine mount and (b) vibration isolation performances(adapted from Carlson, 2007).


If cmax is large enough and cmin is small enough, the transmissibilityachieved in this way fits closely that of the objective of active isolation inFig.8.2. Thus, the semi-active isolation is optimal in this case. One mustbe careful, however, because the behavior a semi-active isolation device,as any nonlinear system, depends strongly on the excitation, and what isoptimal for an harmonic excitation is not for a wide-band excitation.

The MR engine mount of Fig.17.8 is an example of adaptive suspen-sion; the activation of the device allows to go from a low stiffness state(MR valve open, allowing the flow between the upper and the lower sidesof the MR fluid chamber) to a high stiffness state (MR valve closed). Ifthe disturbance is narrow band with a variable frequency (typically therotation speed of the engine) and, if the MR device is activated properly,the overall isolation is the lower bound of the two curves of Fig.17.8.b.

17.5.1 Quarter-car semi-active suspension

The principle of the semi-active suspension is illustrated in Fig.17.9 (com-pare with Fig.8.27.b). The semi-active control unit activates the control-lable device to achieve the variable control force fc subject to the con-straint imposed by the passivity of the device1

fc.(xs − xus) ≤ 0 (17.4)

As a nonlinear device, the response of a controllable shock absorber de-pends on the excitation amplitude and on its frequency content, and ithas the capability to transfer energy from one frequency to another.

The semi-active sky-hook consists of trying to emulate the sky-hookcontrol with the controllable shock absorber, by producing the best pos-sible approximation

fc = −c(u)(xs − xus) ≈ −bxs (17.5)

Because of the passivity constraint (17.4), this is possible only if thesprung mass velocity and the relative velocity have the same sign

xs.(xs − xus) ≥ 0 (17.6)

and if the magnitude of the requested control force belongs to the oper-ating range of the controllable shock absorber,1 fc = −c(u)(xs − xus) is the force applied by the shock absorber to the sprung mass

ms. More complex situations may also be considered, in which the spring stiffnessis also variable.


Fig. 17.9. Principle of the semi-active suspension. One or several sensors monitorthe state of the suspension and a semi-active control unit controls the shock absorberconstant c(u).

cmin ≤ |bxs||xs − xus| ≤ cmax

The damping constant which fits best the requested (sky-hook) controlforce is

c(u) = maxcmin, min[bxs

xs − xus, cmax] (17.7)

However, the sprung mass velocity xs and the suspension relative veloc-ity xs − xus have widely different frequency contents, and the foregoingstrategy tends to produce a fast switching control force fc, as illustratedbelow.

The above strategy requires a fast, calibrated, proportional valve; analternative on/off implementation is

c(u) = cmax If xs.(xs − xus) ≥ 0c(u) = cmin If xs.(xs − xus) < 0

(17.8)

Although simpler, this strategy is likely to produce even sharper changesin the control force. The following example illustrates the energy transferfrom low frequency to high frequency associated with the semi-active sky-hook control.

The system of Fig.17.9 is modelled using the same state variables as forthe passive suspension of Fig.8.27.b, x1 = xs−xus, x2 = xs, x3 = xus−w,


x4 = xus, the set of governing equations is identical to that of the passivesuspension, except that the damping coefficient c(u) of the shock absorberdepends on the control variable u:

ms x2 = −kx1 + c(u)(x4 − x2)mus x4 = −ktx3 + kx1 + c(u)(x2 − x4)

x1 = x2 − x4 (17.9)x3 = x4 − v

where v = w is the road velocity.Time domain simulations have been conducted with the same nu-

merical data as the passive suspension analyzed earlier: ms = 240 kg,mus = 36 kg, k = 16000N/m, kt = 160000N/m, b = 2000Ns/m (gain ofthe sky-hook control). The shock absorber constant is supposed to varybetween cmin = 100Ns/m and cmax = 2000 Ns/m. The body resonanceand the tyre resonance are respectively ωn = (k/ms)1/2 ∼ 8 rad/s andωt = (kt/mus)1/2 ∼ 70 rad/s. The road velocity v is assumed to be awhite noise; the control law is (17.7).

Figure 17.10 shows various time-histories of the quarter-car response,respectively the tyre force ktx3, the body velocity xs = x2, the relativevelocity x1 = xs − xus, the requested (sky-hook) force f = −bxs and theactual control force fc = −c(u)(xs−xus), and finally the damper constantc(u). Note that the relative velocity oscillates much faster (at 70 rad/s)than the body velocity, resulting in sharp changes in the control force fc.

Figure 17.11.a compares the transmissibility between the road veloc-ity and the body acceleration, Txsv of, respectively the passive suspen-sion (c = 200Ns/m), the sky-hook control (c = 200Ns/m and b =2000Ns/m), and the semi-active sky-hook (17.7) with cmin = 100 Ns/mand cmax = 2000Ns/m. The first two curves are the same as in Fig.8.29.a;The semi-active control is successful in reducing the body resonance, andthe transmissibility of the body acceleration is comparable to that of theactive control with b ' 1000Ns/m at low frequency; however, a signif-icant amplification occurs at the wheel resonance, ωt = 70 rad/s, andabove ωt, the transmissibility rolls off much slower than in the previouscases. Besides, one observes peaks at various harmonics of the wheel reso-nance,2 which are likely to excite flexible modes of the vehicle if nothing is2 the two peaks at 132 rad/s and 148 rad/s seem to result from the modulation of

the second harmonic of the wheel mode (2ωt = 140 rad/s) by the car body mode(ωn = 8 rad/s), producing frequency peaks at 2ωt − ωn and 2ωt + ωn.


Fig. 17.10. Quarter-car model with continuous semi-active sky-hook control: (a) tyreforce ktx3, (b) body velocity xs = x2, (c) relative velocity x1 = xs − xus, (d) sky-hookforce f = −bxs and control force fc = −c(u)(xs − xus), (e) damper constant c(u)obtained from (17.7).


done to attenuate them. Figure 17.11.b shows the transmissibility betweenthe road velocity and the tyre deflection; an amplification at the wheelresonance is also observed, but no spurious high frequency componentsappear.

The transmissibility diagrams of Fig.17.11 have been obtained fromtime-histories with cross power spectra and auto power spectra estimates:

Tyx =Φyx

Φxx=

E[Y (ω)X∗(ω)]E[X(ω)X∗(ω)]

(17.10)

Further evidence of the nonlinear energy transfer from low to high fre-quencies can be obtained from the coherence function between the roadvelocity and the body acceleration,

γ2xsv =

|Φxsv|2ΦvvΦxsxs

≤ 1 (17.11)

Fig. 17.11. Quarter-car model with continuous semi-active sky-hook control. (a)Transmissibility between the road velocity and the body acceleration, Txsv; passive sus-pension (c = 200 Ns/m), sky-hook controller (b = 2000 Ns/m), semi-active sky-hook(17.7) with cmin = 100 Ns/m and cmax = 2000 Ns/m. (b) Transmissibility betweenthe road velocity and the tyre deflection, Tx3v.

17.6 References 463

Fig. 17.12. Quarter-car model with continuous semi-active sky-hook control. Coher-ence function γ2

xsv between the road velocity and the body acceleration.

γ2xsv is equal to 1 for a perfect linear system; it measures the causality of

the signal at every frequency; it is a standard tool to detect the presenceof noise and nonlinearities. According to Fig.17.12, the coherence is verygood up to the tyre mode, and falls rapidly to zero above 100 rad/s,which indicates that at those frequencies, the energy content of the bodyacceleration is not due to the road profile.

17.6 References

AVRAAM, M.T. MR-fluid brake design and its application to a portablemuscular rehabilitation device, PhD Thesis, Universite Libre de Bruxelles,Active Structures Laboratory, 2009.CARLSON, J.D. Semi-active vibration suppression. CISM course Semi-Active Vibration Suppression - the Best of Active and Passive Technolo-gies, Udine, Italy, October 2007.CARLSON, J.D. & JOLLY, M. R. MR Fluid, Foam and Elastomer De-vices, Mechatronics, Vol.10, 555-569, 2000.CARLSON, J.D. & SPROSTON, J.L. Controllable Fluids in 2000 - Sta-tus of ER and MR Fluid Technology, Proc. of Actuator 2000, Bremen,Germany, pp. 126-130, June 2000.COLLETTE, C. & PREUMONT, A. High frequency energy transfer insemi-active suspension, Journal of Sound and Vibration, Vol.329, 4604-4616, 2010.COULTER, J.P., WEISS, K.D. & CARLSON, J.D. Engineering Appli-cations of Electrorheological Materials, Journal of Intelligent MaterialSystems and Structures, vol. 4, 248-259, April 1993.


GAUL, L. & NITSCHE, R. Friction Control for Vibration Suppression,(DETC99/ VIB-8191) Movic Symposium, ASME Design EngineeringTechnical Conferences, Las Vegas, 1999.KARNOPP, D. Design Principles for Vibration Control Systems UsingSemi- Active Dampers, Trans. ASME Journal of Dynamic Systems, Mea-surement and Control, Vol. 112, 448-455, Sep. 1990.KARNOPP, D., CROSBY, M. & HARWOOD, R.A. Vibration ControlUsing Semi- Active Suspension Control, Journal of Engineering for In-dustry, Vol. 96, 619-626, 1974.PREUMONT, A. & SETO, K. Active Control of Structures, Wiley, 2008.SIMS, N.D., STANWAY, R. & JOHNSON, A.R. Vibration Control UsingSMART Fluids: A State of the Art Review, The Shock and VibrationDigest, Vol. 31, no. 3, 195-203, May 1999.SPENCER, B.F., DYKE, S.J., SAIN, M.K. & CARLSON, J.D. Phe-nomenological Model of a Magnetorheological Damper, ASCE Journalof Engineering Mechanics, Vol. 123, no. 3, 230-238, 1997.VENHOVENS, P.J.Th. The development and Implementation of Adap-tive Semi-Active Suspension Control, Vehicle System Dynamics, Vol.23,211-235, 1994.

17.7 Problems

P.17.1 Consider a MR device operating according to the direct shearmode, Fig.17.4.b; the electrodes move with respect to each other with arelative velocity U . If A is the active area of the device, g the distancebetween the electrodes, the viscous components Fη and the field-inducedyield stress components Fτ are respectively

Fη = ηU

gA Fτ = τA

From these equations, show that the minimum volume of active fluid toachieve a given control ratio Fτ/Fη for a specified maximum controlledforce Fτ and a maximum relative velocity U reads

V = gA =(

η

τ2

)(Fτ

Fη

)FτU

where η is the viscosity and τ is the maximum yield strength induced bythe magnetic field. From this result, η/τ2 can be regarded as a figure ofmerit of a controllable fluid (Coulter et al.). This explains the superiorityof the MR fluids over ER fluids (Table 17.1).

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Index

AccelerometerPiezoelectric-, 66

Active Optics, 430, 434Active Structural Acoustic Control

(ASAC), 379Active suspension, 195Active truss, 102, 356Adaptive Optics (AO), 1, 430Adaptive suspension, 456Additive uncertainty, 247, 265, 445Admittance, 65, 89, 107, 113Aliasing, 352

spatial-, 387All-pass function, 257Analog to digital converter (ADC), 352Anti-resonance, 34

Bandwidth, 242Bessel filter, 398Beta controller, 166, 409, 411Bilinear transform, 356Bingham model, 451Bode

gain-phase relationships, 250Ideal Cutoff, 254Integrals, 250

Butterworthfilter, 302, 353, 398pattern, 280

Cable Structures, 401Cable-stayed bridge, 401, 418Cauchy’s principle, 238Cayley-Hamilton theorem, 307

Charge amplifier, 79dynamics, 81

Coenergydensity function, 72function, 61

Collocated control, 33, 85, 127, 143, 212Constitutive equations

piezoelectric laminate, 92piezoelectric material, 69piezoelectric transducer, 58

Constrained system, 36Control budget, 16Control canonical form, 312Control-structure interaction, 443, 445Controllability, 305

matrix, 306Covariance intensity matrix, 276Craig-Bampton reduction, 42, 441Cross talk, 162Crossover frequency, 237, 250Cubic architecture, 187, 203, 361Cumulative MS response, 15Curie temperature, 57Current amplifier, 79

Decentralized control, 162, 406Digital to analog converter (DAC), 354Direct piezoelectric effect, 57Direct Velocity Feedback (DVF), 147,

345, 371Discrete array sensor, 383Distributed sensor, 80, 380, 393Duality

478 Index

actuator-sensor, 52, 99, 383Lead-IFF, 158LQR-KBF, 287

Dynamic amplification, 27, 88, 144Dynamic capacitance, 89Dynamic flexibility matrix, 26Dynamic Vibration Absorbers (DVA), 6Dynamics (Actuator and sensor-), 160

E-ELT, 436Electro-rheological (ER) fluid, 450Electrode shape, 77Electromechanical

converter, 49coupling factor, 59, 63, 91, 112, 113transducer, 53

Energy absorbing control, 344Energy density function, 72Energy transformer, 48Error budget, 15

Faraday’s law, 48Feedforward control, 10Feedthrough, 28, 88, 89, 209Flutter, 401Fraction of modal strain energy, 91, 105,

363, 409Frequency shaped LQG, 294, 375Frequency shaping, 294

Gain margin (GM), 237, 246, 282Gain stability, 146, 248Geophone, 52Gough-Stewart platform, 187Gramians

Controllability-, 319Observability-, 319

Guyan reduction, 40Guyed truss, 412Gyrostabilizer, 55

HAC/LAC, 371Hamilton’s principle, 74, 75Hankel singular values, 322High Authority Control (HAC), 144High-pass filter, 81Hydraulic actuator, 421

Impedance (Piezoelectric

transducer), 65Integral control, 293Integral Force Feedback (IFF), 153, 177,

190, 345, 359, 408Interlacing, 34, 143Internally balanced realization, 323Inverse piezoelectric effect, 57Inverted pendulum, 212, 221, 226, 229

double, 308, 310Isolator

Active-, 175, 193Passive-, 170, 193Relaxation-, 172, 189

Kalman Bucy Filter (KBF), 225, 285Kalman Filter, see Kalman Bucy FilterKirchhoff plate theory, 93Kronecker delta, 24, 82

Lag compensator, 261Laminar sensor, 79Lasalle’s theorem, 338Lead compensator, 131, 145, 260Lead-Zirconate-Titanate, 57Legendre transformation, 61, 72Linear Quadratic Gaussian, see LQGLinear Quadratic Regulator, see LQRLoop shaping, 439Loop Transfer Recovery (LTR), 292Lorentz force, 48Low Authority Control (LAC), 144LQG, 286LQR

Deterministic-, 219, 274Stochastic, 278

Luenberger observer, 223Lyapunov

direct method, 335equation, 273, 277, 320, 341function, 274, 336, 339, 341indirect method, 342

Magneto-rheological, see MRMagnetostrictive materials, 7Micro Precision Interferometer, 413MIMO, 207, 263, 273Minimum phase, 256Minimum realization, 323Modal

Index 479

damping, 24filter, 82mass, 24spread, 191truncation, 88

Mode shape, 23, 38MR

clutch and brake, 452engine mount, 458fluid, 450shock absorber, 452

Multi-functional materials, 56Multi-Input Multi-Output, see MIMOMulti-layer laminate, 94Multiplicative uncertainty, 247, 266, 443

Natural frequency, 23Nichols chart, 243Non-minimum phase, 140, 256Notch filter, 133, 235Nyquist

frequency, 352plot, 36, 283stability criterion, 239

Observability, 305matrix, 306

Observer, 222, 284Operational amplifier, 79Orthogonality conditions, 23, 39, 82

Pade approximants, 258, 271Parametric

excitation, 403resonance, 401, 418

Parseval’s theorem, 294Payload isolation, 185PD compensator, 260Performance index, 277, 278Phase margin (PM), 237, 246, 282Phase portrait, 332PI compensator, 261PID compensator, 263Piezoelectric

beam, 74, 99coenergy, 62constants, 69energy, 61laminate, 92, 96

loads, 77, 97material, 56, 69transducer, 58, 64, 104transformer (Rosen’s), 112

Pole, 215Pole placement, 216Pole-zero flipping, 128, 235Pole-zero pattern, 34, 87Poling, 57Polyvinylidene fluoride, 57Popov-Belevitch-Hautus (PBH) test, 316Positive Position Feedback (PPF), 150,

367Power Spectral Density (PSD), 15, 276Prescribed degree of stability, 281Proof-mass actuator, 49, 67PVDF, 57

properties, 72Pyroelectric effect, 57PZT, 57

properties, 72

Quality factor, 27, 112Quantization, 354Quarter-car model, 195Quasi-static correction, 88QWSIS sensor, 380

Rayleigh damping, 22Reaction wheel, 55Reduced order observer, 228Relaxation isolator, see IsolatorResidual dynamics, 266Residual mode, 28, 89Residual modes (spillover), 288Return difference, 245Riccati equation, 275, 278Rigid body mode, 28Robust

performance, 248stability, 248, 265

Robustness test, 265Roll-off, 88, 145Rosen’s piezoelectric transformer, 112Routh-Hurwitz criterion, 151, 334

Sampling, 352Segmented mirror, 436Self-equilibrating forces, 77

480 Index

Self-sensing, 54Semi-active

control, 449sky-hook, 458suspension, 456

Sensitivity function, 245Separation principle, 217, 230, 284Shape Memory Alloys (SMA), 7Shunting

inductive (RL), 116resistive (R), 114switched (SSDI), 119

Singular Value, 264controller, 438Decomposition (SVD), 385, 432

SISO, 207, 244Six-axis isolator, 188Sky-hook damper, 177, 198Small gain theorem, 265Smart materials, 56Sound power, 380Spillover, 10, 84, 211, 288Stability, 331

asymptotic-, 332, 337BIBO-, 332in the sense of Lyapunov, 332

Stability robustness, 248, 265State feedback, 216

State space, 209Stewart platform, see Gough-Stewart,

361Symmetric root locus, 220Synchronized Switch Damping, 121System type, 259

Telescope, 429Tendon Control, 401Thermal analogy, 105Time delay, 258, 271Tracking error, 245Transmissibility, 170Tustin’s method, 356

Unstructured uncertainty, 247

Van der Pol oscillator, 333, 340Vandermonde matrix, 316Vibration isolation, 169Vibroacoustics, 379Voice coil transducer, 48Volume displacement sensor, 81, 379

White noise, 276

Zernike polynomials, 430Zero (transmission-), 36, 163, 215Zero-order hold, 353

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