Spring 2019
數位控制系統Digital Control Systems
DCS-22Stability
Feng-Li LianNTU-EE
Feb19 – Jun19
DCS22-Stability-2Feng-Li Lian © 2019
The Research Procedure in Control Science
PhysicalProcess
MathModel
SystemAnalysis
ControlDesign
Differential eqn Laplace transform Transfer function State space form
Root locus Bode diagram Nyquist plot
Stability Robustness Sensitivity Controllability Observability
Estimator Identification Regulation Tracking PID Pole placement Optimal Control
LQR/LQG Adaptive control Robust control Decentralized
(or Multi-person) Control
Difference eqn z transform Transfer function State space form
Plant Sensor Actuator Computer Communication Noise Disturbance
Differential eqn Laplace transform
Introduction: The Design Philosophy of Control Science
DCS22-Stability-3Feng-Li Lian © 2019Introduction: From CT Plant to DT Plant
G(s)u(t) y(t)
C(s)e(t)r(t)
G(z)u[k] y[k]
C(z)e[k]r[k]
Plant (CT): • Input-Output Model:
• State-Space Model:
Plant (DT): • Input-Output Model:
• State-Space Model:
2
3
1
DCS22-Stability-4Feng-Li Lian © 2019Introduction: Model and Analysis
G(s)u(t) y(t)
C(s)e(t)r(t)
G(z)u[k] y[k]
C(z)e[k]r[k]
Plant (CT): • Input-Output Model:
• State-Space Model:
Plant (DT): • Input-Output Model:
• State-Space Model:
System Properties:
Stability
Controllability and Reachability
Observability and Detectability
DCS22-Stability-5Feng-Li Lian © 2019Outline
Solution of a System Stability and Asymptotic Stability Input-Output Stability Stability Tests:
• Jury’s Stability Criterion
• Nyquist and Bode Diagrams
• Nyquist Criterion
• Relative Stability
Lyapunov’s Second Stability
DCS22-Stability-6Feng-Li Lian © 2019Solution of a System
CT:
DT:
Initial Condition:
Solution:
DCS22-Stability-7Feng-Li Lian © 2019Stability
Definition 3.1: Stability
DCS22-Stability-8Feng-Li Lian © 2019Stability
Definition 3.2: Asymptotic Stability
DCS22-Stability-9Feng-Li Lian © 2019Stability of Linear DT Systems
Stability of Linear Discrete-Time Systems
DCS22-Stability-10Feng-Li Lian © 2019Stability of Linear DT Systems
Solution of LTI DT Systems
DCS22-Stability-11Feng-Li Lian © 2019Stability of Linear DT Systems
Theorem 3.1: Asymptotic Stability of Linear Systems
DCS22-Stability-12Feng-Li Lian © 2019Stability of Linear DT Systems
Stability of Linear Continuous-Time Systems
DCS22-Stability-13Feng-Li Lian © 2019Input-Output Stability
Definition 3.3: Bounded-Input-Bounded-Output Stability
Theorem 3.2: Relation between Stability Concept
DCS22-Stability-14Feng-Li Lian © 2019Input-Output Stability
Example 3.1: Harmonic Oscillator
DCS22-Stability-15Feng-Li Lian © 2019Stability Test
DCS22-Stability-16Feng-Li Lian © 2019Stability Test: Jury’s Stability Criterion
(1918) (1922) (1961)
DCS22-Stability-17Feng-Li Lian © 2019Stability Test: Routh’s Stability Criterion (for CT)
Routh in 1874 Hurwitz in 1895
Franklin, Powell, Emami-Naeini 2002
DCS22-Stability-18Feng-Li Lian © 2019Stability Test: Jury’s Stability Criterion
Theorem 3.3: Jury's Stability Test
DCS22-Stability-19Feng-Li Lian © 2019Stability Test: Jury’s Stability Criterion
Example: Jury's Stability Test
DCS22-Stability-20Feng-Li Lian © 2019Stability Test: Nyquist and Bode Diagrams
DCS22-Stability-21Feng-Li Lian © 2019Stability Test: Nyquist and Bode Diagrams
Example 3.3: Frequency Responses
DCS22-Stability-22Feng-Li Lian © 2019Stability Test: Nyquist and Bode Diagrams
CT: - - - DT:
Example 3.3: Frequency Responses
DCS22-Stability-23Feng-Li Lian © 2019Stability Test: Nyquist Criterion
Nyquist Criterion
DCS22-Stability-24Feng-Li Lian © 2019Stability Test: Nyquist Criterion
Franklin, Powell, Emami-Naeini 2002
DCS22-Stability-25Feng-Li Lian © 2019Stability Test: Nyquist Criterion
Example 3.4: A Second-order system
DCS22-Stability-26Feng-Li Lian © 2019Stability Test: Nyquist Criterion
Example 3.4: A Second-order system
DCS22-Stability-27Feng-Li Lian © 2019Stability Test: Relative Stability
Definitions 3.4 & 3.5: Gain & Phase Margins
DCS22-Stability-28Feng-Li Lian © 2019Lyapunov’s Second Stability
Definition 3.6: Lyapunov Function
DCS22-Stability-29Feng-Li Lian © 2019Lyapunov’s Second Stability
Geometric Illustration
DCS22-Stability-30Feng-Li Lian © 2019Lyapunov’s Second Stability
Example 3.6: Lyapunov function
DCS22-Stability-31Feng-Li Lian © 2019Lyapunov’s Second Stability
Example 3.6: Lyapunov function for CT case
Khalil 2002
DCS22-Stability-32Feng-Li Lian © 2019Lyapunov’s Second Stability
Example 3.6: Lyapunov function