SEVERAL NEW DESIGNS FOR PID, IMC, …new method is proposed for the design of multi-variable IMC system aiming at obtaining good loop performance and small loop couplings based on

SEVERAL NEW DESIGNS FOR PID, IMC,DECOUPLING AND FUZZY CONTROL

BY

YANG YONG-SHENG (B.ENG., M.ENG.)

DEPARTMENT OF ELECTRICAL AND

COMPUTER ENGINEERING

A THESIS SUBMITTED

FOR THE DEGREE OF PHILOSOPHY DOCTOR

NATIONAL UNIVERSITY OF SINGAPORE

2003

Acknowledgments

I would like to express my sincere appreciation to my advisor, Professor Wang,

Qing-Guo, for his excellent guidance and gracious encouragement through my

study. His uncompromising research attitude and stimulating advice helped me

in overcoming obstacles in my research. His wealth of knowledge and accurate

foresight benefited me in finding the new ideas. Without him, I would not able

to finish the work here. I am indebted to him for his care and advice not only in

my academic research but also in my daily life. I wish to extend special thanks to

Professor C. C. Hang for his constructive suggestions which benefit my research a

lot. It is also my great pleasure to thank Dr. Chen Ben Mei and Dr. Ge Shuzhi

Sam who have in one way or another give me their kind help.

Also I would like to express my thanks to Dr. Zheng Feng and Dr. Lin Chong,

Dr. Zhang, Yong, Dr. Zhang, Yu, and Dr. Bi, Qiang for their comments, advice,

and inspiration. Special gratitude goes to my friends and colleagues. I would like

to express my thanks to Dr. Yang, Xue-Ping, Mr. Huang Xiaogang, Mr. Huang,

Bin, Ms. He Ru, Mr. Guo Xin, Mr. Zhou Hanqin, Mr. Lu Xiang, Mr. Li Heng

and many others working in the Advanced Control Technology Lab. I enjoyed

very much the time spent with them. I also appreciate the National University of

Singapore for the research facilities and scholarship.

Finally, I wish to express my deepest gratitude to my wife Han Rui. Without

her love, patience, encouragement and sacrifice, I could not have accomplished this.

I also want to thank my parents and brothers for their love and support, It is not

possible to thank them adequately. Instead, I devote this thesis to them and hope

they will find joy in this humble achievement.

i

Contents

Acknowledgements i

List of Figures vii

List of Tables viii

Summary ix

Nomenclature xiii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 10

2 Three New Approaches to PID Controller Design 12

2.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Robust PID Controller Design for Gain and Phase Margins . . . . . 13

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 PID Controller Design Using LMI . . . . . . . . . . . . . . . 14

2.2.3 Tuning Guidelines . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Quantitative Robust Stability Analysis and PID Controller Design . 25

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

ii

Contents iii

2.3.2 Review of Robust Control Theory . . . . . . . . . . . . . . . 26

2.3.3 Quantitative Robust Stability . . . . . . . . . . . . . . . . . 28

2.3.4 Second-Order Uncertain Model . . . . . . . . . . . . . . . . 30

2.3.5 Robust PID controller Design . . . . . . . . . . . . . . . . . 32

2.3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4 PI Controller Design for State Time-Delay Systems via ILMI . . . . 37

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . 38

2.4.3 Stabilizing Control . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.4 Suboptimal H∞ Control . . . . . . . . . . . . . . . . . . . . 42

2.4.5 Control Design with PI Controllers . . . . . . . . . . . . . . 44

2.4.6 A Numerical Example . . . . . . . . . . . . . . . . . . . . . 45

2.4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Advance in Robust IMC Design for Step Input and Smith Con-

troller Design for Unstable Processes 48

3.1 Robust IMC Design via Time Domain Approach . . . . . . . . . . . 49

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1.2 The Robust IMC Design . . . . . . . . . . . . . . . . . . . . 50

3.1.3 LMI Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Robust IMC Controller Design via Frequency Domain Approach . . 60

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.2 IMC Design Review and New Formulation . . . . . . . . . . 60

3.2.3 Controller Design with Fixed Poles . . . . . . . . . . . . . . 62

3.2.4 Controller Design with General Form . . . . . . . . . . . . . 66

3.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3 Modified Smith Predictor Control for Disturbance Rejection with

Unstable Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Contents iv

3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.2 The Proposed New Structure . . . . . . . . . . . . . . . . . 78

3.3.3 Internal Stability . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4 Decoupling with Stability and Decoupling Control Design 87

4.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 Decoupling Problem with Stability via Transfer Function Matrix

Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.2 Minimal C+-Decoupler . . . . . . . . . . . . . . . . . . . . . 89

4.2.3 Decoupling with Stability . . . . . . . . . . . . . . . . . . . 91

4.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3 Decoupling Control Design via LMI Approaches . . . . . . . . . . . 99

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 99

4.3.3 Controller Design via LMI . . . . . . . . . . . . . . . . . . . 101

4.3.4 Stability and Robustness Analysis . . . . . . . . . . . . . . . 103

4.3.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 Fuzzy Modelling and Control for F-16 Aircraft 111

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.1.1 F-16 Aircraft and Control . . . . . . . . . . . . . . . . . . . 111

5.1.2 Objective of the Design . . . . . . . . . . . . . . . . . . . . . 114

5.1.3 Organization of the Chapter . . . . . . . . . . . . . . . . . . 115

5.2 F-16 Aircraft Model . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2.1 Need for Modelling . . . . . . . . . . . . . . . . . . . . . . . 115

Contents v

5.2.2 Modelling Method . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.3 Workable Model . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.3 TS Fuzzy Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.3.1 The Technique . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.3.2 TS Model of F-16 . . . . . . . . . . . . . . . . . . . . . . . . 123

5.3.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . 127

5.4 Lyapunov Based Control . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4.1 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.4.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.5 Gain Scheduling Control . . . . . . . . . . . . . . . . . . . . . . . . 140

5.5.1 Gain Scheduled Linear Quadratic Regulator Design . . . . . 140

5.5.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.6 Discussions and Conclusions . . . . . . . . . . . . . . . . . . . . . . 152

5.6.1 Comparative Studies . . . . . . . . . . . . . . . . . . . . . . 152

5.6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6 Conclusions 160

6.1 Main Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.2 Suggestions for Further Work . . . . . . . . . . . . . . . . . . . . . 162

Bibliography 165

Appendix A TS Fuzzy Basic Model 177

Appendix B TS Fuzzy Augmented Model I 181

Appendix C TS Fuzzy Augmented Model II 188

Appendix D Gain Scheduling Control for Tracking (φ, θ) 195

Appendix E Gain Scheduling Control for Tracking (α, β, φ) 198

Author’s Publications 201

List of Figures

2.1 Unity feedback system . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Step response of proposed method with Am = 3 and φm = 60 . . . . 22

2.3 Step response of proposed method with Am = 2 and φm = 45 . . . . 23

2.4 Step response with Am = 3 and φm = 60 . . . . . . . . . . . . . . . 24

2.5 Step response with Am = 2 and φm = 45 . . . . . . . . . . . . . . . 24

2.6 Controlled uncertain system . . . . . . . . . . . . . . . . . . . . . . 26

2.7 The plot of max|G(jω)| and minargG(jω) . . . . . . . . . . 31

2.8 The plot of max|G(jω)K(jω)| and minargG(jω)K(jω) . . . 37

2.9 Step response of the uncertain system . . . . . . . . . . . . . . . . . 37

3.1 Internal Model Control . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Robust IMC design . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Nominal step response for H2 optimal design . . . . . . . . . . . . . 58

3.4 Nominal step response for robust IMC design . . . . . . . . . . . . 58

3.5 Step response of Robust design for process with mismatch . . . . . 59

3.6 Step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.7 Step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.8 Proposed smith predictor control scheme . . . . . . . . . . . . . . . 78

3.9 Step responses for IPTD process . . . . . . . . . . . . . . . . . . . . 84

3.10 Step responses for unstable FOPTD process . . . . . . . . . . . . . 85

3.11 Step responses for unstable SOPTD process . . . . . . . . . . . . . 85

4.1 Step Tests of the Plant in Example 4.4 . . . . . . . . . . . . . . . . 105

4.2 Step Tests of the plant in Example 4.5 . . . . . . . . . . . . . . . . 107

vi

List of Figures vii

4.3 Robust Stability bound in Example 4.5 . . . . . . . . . . . . . . . . 107

4.4 Step response for perturbed process in Example 4.5 . . . . . . . . . 108

5.1 Definition of aircraft axes and angles. . . . . . . . . . . . . . . . . . 117

5.2 Fuzzy triangle membership functions . . . . . . . . . . . . . . . . . 126

5.3 Approximate error of TS-fuzzy and linear model (different α) . . . . 127

5.4 Approximate error of TS-fuzzy and linear model (different φ) . . . . 127

5.5 Lyapunov based stabilizing control . . . . . . . . . . . . . . . . . . 134

5.6 Linear stabilizing control . . . . . . . . . . . . . . . . . . . . . . . . 135

5.7 Lyapunov based stabilizing control with control signal constraints . 138

5.8 Lyapunov based tracking control . . . . . . . . . . . . . . . . . . . . 141

5.9 Linear tracking control . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.10 The gain scheduled tracking control with φ θ . . . . . . . . . . . . . 146

5.11 The gain scheduled tracking control I with α β and φ . . . . . . . . 147

5.12 The gain scheduled tracking control II with α β and φ . . . . . . . 148

5.13 The gain scheduled tracking control III with α β and φ . . . . . . . 149

5.14 Lee’s Backstepping control . . . . . . . . . . . . . . . . . . . . . . . 150

5.15 Outputs of the two proposed methods . . . . . . . . . . . . . . . . . 153

5.16 Control signals of the two proposed methods . . . . . . . . . . . . . 154

5.17 Initial part of gain scheduling control . . . . . . . . . . . . . . . . . 155

5.18 Outputs of the two proposed methods . . . . . . . . . . . . . . . . . 156

5.19 Control signals of the two proposed methods . . . . . . . . . . . . . 157

5.20 Initial part of gain scheduling control . . . . . . . . . . . . . . . . . 158

List of Tables

2.1 SOF and PI controller and their performance . . . . . . . . . . . . . 46

2.2 P matrix in SOF and PI controller design . . . . . . . . . . . . . . . 47

4.1 Search for Gd,min in Example 4.1 . . . . . . . . . . . . . . . . . . . 92




5.1 The φ tracking specifications of the two control methods . . . . . . 152

5.2 The θ tracking specifications of the two control methods . . . . . . 159

viii

Summary

With the development of industrial competition, the performance requirements of

industrial processes become increasingly stringent. Moreover, it was known that

many controllers is sensitive to model uncertainty. To deal with this problem,

the framework for robustness analysis and design was developed in 1980s and

1990s. Recently, many researchers have developed various approaches for robust

control (Goodwin et al., 1999; Wang, 1999; Wang and Goodwin, 2000). Though the

framework for robust control is available, the method for robust design is usually

very complicated and the resultant controllers are generally of high order. The

implementation of such high order controllers in industrial application is usually

difficult. This thesis is devoted to the development of new control design techniques

for better performance or robustness with relatively simple controller or structure.

Proportional-Integral-Derivative (PID) controllers are the dominant choice in

process control and many researches have been reported in literature. In this thesis,

three schemes are developed to design new PID controllers. The first method is

designed for achieving optimal gain and phase margins for uncertain processes.

Gain and phase margins are typical control loop specifications associated with the

frequency response technique. In the proposed method, the objective is to develop

a scheme such that it can achieve desired gain and phase margins for the uncertain

system. The robust PID controller design problem is converted into a standard

convex optimization problem with linear matrix inequalities (LMI) constraints,

which may be solved effectively using the interior point method. A complete PID

tuning guideline is also presented. Simulation shows that the proposed method

gives good performance. The second proposed scheme is based on the extension of

ix

Summary x

the small gain theorem. The well-known small gain theorem was extensively used in

the analysis of the robust stability and performance robustness of uncertain system.

However, the small gain theorem only constraints the gain of the system, while its

phase may be arbitrary. Thus much conservativeness is introduced. In this thesis, a

new quantitative robust stability criterion is presented. In this criterion, both gain

and phase information is employed to reduce the conservativeness. Examples are

given to show the effectiveness of the proposed criterion. Based on the criterion,

a class of second-order plus dead time uncertain process is discussed and a robust

PID tuning scheme is proposed. Examples are provided to illustrate our analysis

and design. For the processes with state time delay, a new approach is proposed

to design PI controller with iterative LMI optimization. It shows that the problem

of PI controller design may be converted into that of static output feedback (SOF)

controller design after appropriate formulation. The difficulty of SOF synthesis is

that the problem inherently is a bilinear problem which is hard to be solved via an

optimization with LMI constraints. In the thesis, an iterative LMI optimization

method is developed to solve the problem.

For the Internal Model Control (IMC) system, two approaches are developed

to achieve good performance while maintaining the robustness. The first design is

in time domain. A new approach to IMC design is proposed, which aims at obtain-

ing optimal H2 performance under the robust stability constraints. Such a robust

optimal IMC design is formulated into a H2/H∞ multiobjective output-feedback

control problem and solved via a system of LMIs in time domain. The validity of

the approach is illustrated by two examples. The second method is based on the

design in frequency domain, an IMC controller design methodology is presented

to achieve the optimal performance with robust stability. The original problem is

nonlinear and thus difficult to solve. The upper bound and lower bound of the

optimal solution are formulated and converted into LMI or BMI optimization. It

is shown that the optimal solution can be approximated by the upper bound and

lower bound with any accuracy. Examples are given to demonstrate the effective-

ness of the proposed method. The advantage of time domain method is that the

Summary xi

optimization problem encountered is an LMI problem, which is easy to be solved.

However, the method cannot be used for processes with time delay and it intro-

duces some conservativeness in the problem formulation. For frequency domain

method, the global optimal solution may be found without conservativeness and it

can be used for processes with time delay. However, the BMI optimization must

be employed to find the solution. One shortcoming of IMC system is that the

presence of time delay forces the designer to choose lower controller gain to main-

tain stability. To our best knowledge, Smith predictor is the best way to control

the processes with time delay. A new modified Smith predictor control scheme

and its simple control design are proposed for unstable processes. The internal

stability of the proposed structure is analyzed. Simulation results show that the

proposed method yields significant performance improvement with load responses

over existing approaches.

In the decoupling design, we wish to find a systematic scheme to satisfy the

requirements of stability, decoupling, performance and robustness. Firstly, a sim-

ple necessary and sufficient condition for solvability of decoupling with internal

stability for unity output feedback for non-singular plants is proposed. Then, a

new method is proposed for the design of multi-variable IMC system aiming at

obtaining good loop performance and small loop couplings based on LMI opti-

mization. The decoupling design with performance constraint is formulated into

an optimization problem with LMI constraints, thus the problem can be solved

effectively using LMI toolbox. Robust stability is analyzed and simulations show

that good control effects can be achieved.

Takagi-Sugeno (TS) fuzzy modelling and control becomes an effective tool for

nonlinear complex processes. In this thesis, a framework for control of F-16 aircraft

with TS fuzzy systems is developed. First, based on the best-available nonlinear

dynamical model of F-16 aircraft in the open domain, the TS fuzzy model of F-16

aircraft is presented and validated with reasonable accuracy. Then, two control

strategies, namely, Lyapunov based control and gain scheduling control, are pro-

posed using the TS model. Each of them is applied to synthesize a F-16 flight

Summary xii

control system for both stabilizing control and attitude tracking control. Exten-

sive simulation is carried out and comprehensive comparative studies are made

with the normal linear control and among two approaches. It shows that the pro-

posed two control designs are feasible and both of them outperform the linear

control design significantly. In particular, the gain scheduling control has achieved

better performance, which is almost equivalent to the best nonlinear control of

high complexity.

The schemes and results presented in this thesis have both practical values and

theoretical contributions. The results of the simulation show that the proposed

methods are helpful in improving the performance or the robustness of industrial

control systems.

Nomenclature

Abbreviations

BMI Bilinear Matrix Inequalities

DPSUF Decoupling Problem with internal Stability

for Unity output Feedback system

FOPDT First Order Plus Dead Time

HJB Hamilton-Jacobin-Bellman

ILMI Iterative Linear Matrix Inequalities

IMC Internal Model Control

ISE Integral Squared Error

ISTE Integral Squared Time Error

LMI Linear Matrix Inequalities

PI Proportional-Integral

PID Proportional-Integral-Derivative

RHP Right Half Plane

SOF Static Output Feedback

SOPDT Second Order Plus Dead Time

SP Smith Predictor

TS Takagi-Sugeno

xiii

nomenclature xiv

Symbols

a(s) Pole Polynomial in Transfer Function

ai Coefficient of Pole Polynomial in Transfer Function

b(s) Zero Polynomial in Transfer Function

bi Coefficient of Zero Polynomial in Transfer Function

H(s) Closed-loop Transfer Function Without Uncertainty

hi(x) Weights in TS model

G(s) Transfer Function of Plant

G(s) Nominal Model of Plant

G−(s) Stable and Minimal Phase Part of G(s)

G−(s) Unstable and Non-minimal Phase Part of G(s)

Gd Decoupler for G

Gd,min minimal C+- decoupler for G

K(s) Feedback Controller

kp P parameter in PID Controller

ki I parameter in PID controller

kd D parameter in PID controller

L(s) Open-loop Transfer Function

L(s) Open-loop Transfer Function for Nominal Model

L Time delay in Transfer Function

la(s) Transfer Function of Additive Uncertainty Bound

lm(s) Transfer Function of Multiplicative Uncertainty Bound

Q(s) IMC Controller

S(s) Sensitivity Transfer Function

T (s) Complementary Sensitivity Transfer Function

Tr Closed-loop Transfer Function for Set-point Tracking

Td Closed-loop Transfer Function for Disturbance Response

Wi(s) or Vi(s) Weighting Function

nomenclature xv

δp(G) McMillan degree of a transfer function G at the RHP pole p

∆(s) Uncertainty Description Associated with Closed-loop System

¯∆(s) System Determinant in Mason’s Formula

ωg Gain Crossover Frequency

ωp Phase Crossover Frequency

Am Gain Margin

φm Phase Margin

Re Real Part

Im Imaginary Part

Notations

‖ · ‖2 L2 norm

‖ · ‖2 ∞ norm

trace(·) Trace of Matrix

arg(·) Phase Angle of Transfer Function

sup(·) Upper Limit

max(·) Maximal Value

min(·) Minimal Value

diag(·) Diagonal Matrix

Chapter 1

Introduction

1.1 Motivation

Today, the automatic controller can be found in many applications in our lives.

It may range from missile tracking in the military area to water and temperature

control in washing machines. Development of analysis and design of controller

has been a goal for control engineering for a long time. The classical frequency

domain methods were developed during the 1930s and 1940s, the renowned classi-

cal stability theory was proposed by Nyquist (1932) and many methods of system

analysis were found by Black (1977) and Bode (1964). In the fifties of last century,

many analytical design methods were developed, which made it possible to design

a controller for a given model to satisfy the transient performance specifications.

With the development of computer technology, controller design based on the opti-

mization method became the main current from the 1960s. Such methods had the

advantage that many different aspects of the design problem are considered. Both

the analytical and the optimization design are based on the exact model of the pro-

cesses. However, controllers are always designed based on incomplete information.

The accuracy of the model varies but it never perfect. Moreover, the behavior of

the plant may change with time. Thus, the controller should be designed on the

basis of mathematical models with consideration for uncertain description of the

process. Since the early of 1980s, the robust control theory has become a major

1

Chapter 1. Introduction 2

area in control research. For dealing with complex nonlinear processes and using

inaccurate information, the fuzzy control theory and neural network methods are

developed rapidly since the 1980s. Throughout the years, control theory has made

important contributions to our world. As the process industries continue to in-

crease, the performance and robustness requirements of control systems become

more important to ensure strong competitiveness. Thus, it is a strong need to look

for new approaches to increase the performance and guarantee the robustness of

the control systems. This thesis motivated to develop new control techniques for

better performance or robustness.

Among most unity feedback control structures, the majority of regulators used

in the industry are of Proportional-Integral-Derivative (PID) type and a large in-

dustrial plant may have hundreds of such regulators (Astrom et al., 1993). They

have to be tuned individually to match process dynamics for acceptable perfor-

mance. The tuning procedure, if done manually, is very tedious and time consum-

ing, especially for those slow dynamics loops, and the resultant system performance

will mainly depend on the experience and the process knowledge the engineers have.

It is recognized that in practice, many industrial control loops are poorly tuned.

Gain and phase margins are typical control loop specifications associated with

the frequency response technology. Many controller designs about tuning gain and

phase margins have been presented. Ogata (1990) gave solutions using a numerical

method and Franklin et al. (1994) solved the problem using a graphical approach.

Using some approximation. Ho et al. (1995) presented an analytical formulae to

design the PID controller for the first-order and second-order plus dead time pro-

cesses to meet gain and phase margin specifications. However, all these methods

did not consider the uncertainty of the processes. Thus, there is a need to design

a new PID tuning scheme to achieve desired gain and phase margin specifications

for a family of uncertain systems.

One of the significant work on robust stability and performance robustness

of uncertain systems was done by Zames (1981), where he derived a well-known

theorem called the small gain theorem. In recent years, robust stability analysis


of closed-loop system is heavily based on the small gain theorem. Haddad et al.

(2000) discussed the problem of fixed-structure robust controller synthesis using

the small gain theorem. Jiang and Marcels (1997) presented a recursive robust

control scheme using the nonlinear small gain theorem. The usefulness of the

small gain theorem and its variants in addressing a variety of feedback stabilization

problem was clearly established by Praly (1996). However, since small gain analysis

allows uncertainty with arbitrary phase in the frequency domain, it can be overly

conservative, Thus it is desirable to develop a criterion which can employ both

gain and phase information in order to reduce the conservativeness. Moreover, it

is useful to develop improved PID tuning methods for uncertain processes based

on the less conservative criterion.

For processes with state time delay, Niculescu (1998) proposed an approach to

design H∞ state feedback controller via LMI optimization. Later, Mahmoud and

Zribi (1999) developed a scheme for H∞ static output feedback (SOF) control. In

their method, under the strict-positive realness condition, the SOF control design

problem is simplified into a state feedback problem. Obviously, the method cannot

be used for general processes. The difficulty of SOF synthesis is that the problem

inherently is a bilinear problem which is hard to be formulated into an optimization

problem with LMI constraints. Since PID controller is most popular controller

used, a more effective method to cope with the PI/PID controller for general

processes with state time delay is needed.

It is well known that the PID controller is the most popular controller used

in process control. Although PID controller may achieve good performance for

many benign processes, it will lose its effectiveness in more complex environments.

Due to its simple yet effective framework for system design, Internal model con-

trol (IMC) scheme has been under intensive research and development in the last

decades. The main advantage of the IMC system is that if the model is perfect

the IMC system becomes an open-loop system and the stability analysis becomes

trivial. However, the model can never be perfect. Thus, it is of great practical

importance that the controller perform well when the model differs from the real


process. The IMC provides a simple yet effective structure for robust controller

design (Morari and Zafiriou, 1989), thus IMC structure is employed widely in the

robust control system design. Boulet et al. (2002) developed IMC robust tunable

control based on the performance robustness bounds of the system and knowledge

of the plant uncertainty. Chiu et al. (2000) developed IMC for transition control

with uncertain process. Litrico (2002) proposed robust IMC flow control for dam-

river open-channel systems, in which the robustness is estimated with the use of

a bound on multiplicative uncertainty taking into account the model errors, due

to the nonlinear dynamics of the system. However, most of the existing control

design with IMC structure is based on trial and error method. Moreover, the ex-

isting robust controller usually is not optimal for nominal performance. Clearly,

systematic IMC design methods which can achieve optimal nominal performance

and guarantee the robust stability are in demand.

It is well known that a Smith predictor controller, which is an effective dead-

time compensator (Smith, 1959), can be put into an equivalent IMC structure.

However, the original Smith predictor control scheme will be unstable when ap-

plied to an unstable process (Wang et al., 1999b). To overcome this obstacle,

many modifications to the original Smith scheme have been proposed. Astrom

et al. (1994) presented a modified Smith predictor for integrator plus dead-time

process and can achieve faster setpoint response and better load disturbance rejec-

tion with decoupling of the setpoint response from the load disturbance response.

Matausek and Micic (1996) considered the same problem with similar results but

their scheme is easier to tune. Majhi and Atherton (1999) proposed a modified

Smith predictor control scheme which has high performance particularly for unsta-

ble and integrating process. This method achieves optimal integral squared time

error criterion (ISTE) for setpoint response and employs an optimum stability ap-

proach with a proportional controller for an unstable process. Another paper of

Majhi and Atherton (2000) extends their work for better performance and easy

tuning procedure for first order plus dead time (FOPDT) and second order plus

dead time (SOPDT) processes. To our best knowledge, Majhi and Atherton (2000)


achieve best performance for setpoint response with unstable dead time processes

employing modified Smith predictor structure. In this thesis, a new scheme is

motivated to improve the performance of disturbance rejection for the unstable

processes.

As for most of the control systems are of multi-variable characteristics, multi-

variable control design and stability analysis is another important topic of interest

and this is addressed in this thesis. Among the multi-variable design, the problem

of decoupling linear time-invariant multi-variable systems received considerable at-

tention in the literature of system theory for a period of over two decades. Much of

this attention was directed toward decoupling by state feedback (Morgan, 1964).

The problem of block decoupling was investigated by Wood (1986). The neces-

sary and sufficient conditions of decoupling were developed by Wang and Davison

(1975). Decoupling through a combination of pre-compensation and output feed-

back was considered by Pernebo (1981) and Eldem (1996). In contrast to the

above, unity output feedback systems are more widely used in industry. But the

problem of decoupling in such a configuration while maintaining internal stability

of the decoupled system appears to be more difficult. The crucial assumption made

by Gundes (1990) is that the plant does not have unstable pole coinciding with

internal stability. Under this assumption, it has been shown that the problem is

solvable. The condition is, however, not necessary, and it can be relaxed.

For multi-variable systems, interactions usually exist between the control loops.

The goal of controller design to achieve satisfactory loop performance has hence

posed a great challenge in the area of control design. Although multi-variable con-

trollers are capable of providing explicit suppression of interactions, their designs

are usually more complex and their implementation inevitably more costly. More-

over, the real processes cannot be exactly decoupled because of the uncertainty

of the processes model used. Thus, exact decoupling is usually impossible in a

practical environment. Therefore, there is a need to propose a novel method for

general multi-variable processes to achieve good loop performance with relatively

small loop coupling. State space H∞ design has been well established since the


late 1980’s. Ball and Cohen (1987) and Doyle et al. (1989) are the two notable

ones among numerous relevant references. Extensive lists of references and de-

tailed accounts of various approaches are provided in Stoorvogel (1991) and Zhou

et al. (1996). One possible solution of the robust decoupling problem is to adopt a

loop-wise H∞ approach by designing each controller column such that a combined

loop and decoupling performance index is minimized.

For nonlinear system, the linear model is usually not complex enough to de-

scribe the dynamics. Since Takagi and Sugeno (1985) opened a new direction

of research by introducing the Takagi-Sugeno (TS) fuzzy model, there have been

several studies concerning the systematic design of stabilizing fuzzy controllers

(Tanaka and Sugeno, 1992; Tanaka et al., 1996b; Wang et al., 1996; Thathachar

and Viswanath, 1997). In the TS fuzzy model of Takagi and Sugeno (1985), the

overall system is described by several fuzzy IF-THEN rules, each of which rep-

resents a local linear state equation x = Aix + Biu. To derive the stabilizing

controller, the Lyapunov stabilizing theory and LMIs method may be employed.

In the thesis, the complete attitude motion of a rigid spacecraft (Shuster, 1963)

is considered. The TS fuzzy modelling is one of the ways to find a better model

for a complex process, say F-16 aircraft, when the linear model is not enough

to represent the dynamics of the process. Moreover, both stabilizing and attitude

tracking controllers need to be developed based on TS fuzzy model to achieve good

performance.

1.2 Contributions

In this thesis, new control system design issue along with stability, performance

and robustness are addressed for single variable linear processes, multi-variable

linear processes and nonlinear processes. In particular, the thesis has investigated

the following areas:

A. Robust PID Controller Design for Gain and Phase Margins

A new scheme for optimal PID controller is proposed to meet gain and phase


margins for a family of plants. The main contribution is that the uncertainty is

included in the procedure of the optimization. With the new idea, a new method to

design PID controller for uncertain processes is proposed. Using S-procedure and

Schur complement, the PID controller design problem is converted into a standard

convex optimization problem with LMI constraints, which can be solved effectively

using the interior point method. A complete PID tuning method is presented and

simulation examples are provided to show the effectiveness of the proposed method.

B. Quantitative Robust Stability Analysis and Design

A new method quantitative robust stability criterion is proposed and new PID

tuning scheme is developed based on the new criterion. The author begins with a

simple example which shows the conservativeness of the traditional robust stability

theorem, such as small gain theorem. From our analysis, the conservativeness

mainly comes from the unknown sources of uncertainty. In the small gain theorem,

only the gain information of the perturbation is considered, however, the phase

information is discarded. A new quantitative robust stability criterion is in turn

proposed employing both gain and phase information of the uncertain systems. An

example is employed to show that the conservativeness of the new stability criterion

is reduced. Moreover, the gain and phase bounds of the parameter uncertainty for

second order plus dead time models are found and a new PID tuning scheme for

robust performance is also developed.

C. PI Controller Design for State Time-Delay Systems via ILMI Ap-

proach

A new PI controller design method for general processes with state time delay

is proposed. With the augmented state description, we convert the problem of

PI controller design to that of static output feedback controller design. However,

the difficulty is that the problem of static output feedback controller synthesis

inherently is a bilinear problem which is hard to be formulated into an optimization

problem with LMI constraints. In this thesis, An iterative LMI method is proposed

to solved the problem. Both the stabilizing controller and the suboptimal H∞


controller are designed for the processes with state time delay. The sufficient

conditions of existing such controllers are presented and the procedures to find

such controllers are also given. A numerical example is provided to show the

effectiveness of the proposed method.

D. Robust IMC Controller Design via Time Domain Approach

A new approach is proposed to design the IMC controller in order to achieve

the optimal nominal performance under the robust stability constraints. The IMC

design problem is converted into a H2/H∞ multi-objective output feedback control

design problem. With some appropriate manipulations of state space equation of

the closed-lop system, the controller may be obtained via solving a system of LMIs.

Two examples are provided to illustrate the effectiveness of the method.

E. Robust IMC Controller Design via Frequency Domain Approach

A new robust IMC design framework is developed which aims to minimize

the integral square error(ISE) for setpoint step input with the robust stability as a

constraint. The above optimization problem is a nonlinear and nonconvex problem

which is difficult to be solved directly using the description in frequency domain.

For the controller design with fixed poles, it is shown that the optimal solution

is approximated by its upper bound and lower bound which are then solved by

LMI optimization. For the controller design with general form, a branch and

bound method is employed to obtain the upper bound and the lower bound of the

optimal solution. It is shown that the optimal solution can be approximated by

the upper bound and the lower bound with any accuracy. Examples are provided

to show the effectiveness of the proposed method.

F. Modified Smith Predictor Control for Disturbance Rejection with

Unstable Processes

In process control, the Smith predictor (SP) is a well known and very effective

dead-time compensator for stable processes. However, the original Smith predictor

control scheme will be unstable when applied to an unstable process. In this the-

sis, a new modified Smith predictor structure is proposed. A simple but effective


control procedure is designed to improve the performance especially for the distur-

bance rejection. The internal stability of the proposed structure is also analyzed

which is not reported in the previous publications. Simulation is given to illustrate

that the proposed method achieves good performance for both setpoint response

and disturbance rejection. Comparisons show that the proposed scheme has better

performance than the best existing method, especially for disturbance rejection.

G. Stability of Decoupled Systems

A new necessary and sufficient solvability condition is developed for decou-

pling problem with internal stability of unity output feedback system. Firstly,

the definition of the minimal decoupling matrix, which has the minimal set of all

common unstable poles and zeros of the plants, is given. Then, the existence and

uniqueness of the minimal decoupling is analyzed and the procedure of finding the

matrix is provided. Based on the properties of the special matrix, the necessary

and sufficient condition for decoupling with stability is developed. The condition

shows that decoupling with stability is solvable if and only if there exists a mini-

mal decoupling matrix such that it has no unstable pole-zero cancellation with the

controlled process.

H. Decoupling Control Design via LMI

A new method is proposed for the design of multi-variable IMC system aiming

at obtaining good loop performance and small loop couplings based on LMI opti-

mization. The decoupling design with performance constraints is formulated into

an optimization problem with LMI constraints, thus the problem can be solved

effectively using LMI toolbox. Robust stability is analyzed and simulation shows

that good control effects can be achieved.

I. TS Fuzzy Modelling and Control for F-16 Aircraft

In the thesis, the problem to design both stabilizing and tracking controller for

F-16 aircraft systems has been addressed via the TS fuzzy modelling approach.

Both basic and augmented TS fuzzy models of F-16 have been obtained using the

best-available F-16 nonlinear model from the literature and validated to be reason-


ably accurate in the operating range of interests. Two control design methods have

been proposed and shown to be feasible and better than the normal linear control.

The first method, Lyapunov based control, is developed for stabilization of a TS

fuzzy model and extended to handle attitude tracking problem when the model

is augmented with integrator at each output. The most important feature of the

method is guaranteed stability and step-signal tracking of the closed-loop system

if the TS fuzzy model error is small enough. Besides, The solutions are simple to

get since it involves an LMI feasibility problem only. The second method, gain

scheduling control, shows better performance. It has a very simple control struc-

ture and straightforward design from TS model; Yet it can perform as well as the

best nonlinear controller of high complexity and very sophisticated design. How-

ever, the gain scheduling control cannot guarantee the stability of the closed-loop

system.

1.3 Organization of the Thesis

The thesis is organized as follows. Chapter 2 focuses on the development of PID

controller design, where three control strategies are developed. The first work is

to design robust PID controller for gain and phase margins. The second one is to

develop quantitative robust stability criterion and robust PID tuning scheme. The

last one is to design PI controller for state time delay processes via iterative LMI

optimization.

Chapter 3 is devoted to IMC-based feedback system design. A new formulation

of IMC system design in time domain is proposed and the robust IMC controller

may be found via solving an optimization problem with LMI constraints. Another

scheme is developed based on frequency domain approach for the same problem.

A new method is proposed to find the controller via BMI optimization. For the

unstable processes with time delay, a new modified Smith predictor structure is

proposed to achieve better performance for both setpoint response and disturbance

rejection.

Chapter 4 is concerned with the decoupling control. A new necessary and


sufficient condition for solvability of decoupling with internal stability for unity

output feedback for non-singular plants is proposed. A new method is proposed for

the design of multi-variable IMC system aiming at obtaining good loop performance

and small loop couplings based on LMI optimization. The robust stability of the

design is also analyzed.

Chapter 5 investigates the TS fuzzy modelling and controller design for F16

aircraft. The problem to design both stabilizing and tracking controller for F-16

aircraft systems is addressed via the TS fuzzy modelling approach. Two control

designs, namely Lyapunov based control and gain scheduling control, are developed

for the control of the obtained TS fuzzy model.

Finally in Chapter 6, general conclusions are given and suggestions for further

works are presented.

Chapter 2

Three New Approaches to PID

Controller Design

2.1 Preview

The vast majority of the controllers used in industry are of the PID type and

PID control has been an important research topic since 1950’s. According to a

survey paper (Yamamoto and Hashimoto, 1991), more than 90% controllers are of

PID type. Although the PID control is well established in process industries, A

recent survey reports (Cominos and Munro, 2002) that there are still many control

loops poorly tuned. Thus, new tuning schemes for PID controllers are desired and

important for better process operations.

In this chapter, three new PI/PID control schemes are developed. In the first

approach, a new PID control design is proposed to meet gain and phase margins

for a family of processes with norm bounded uncertainty. The robust PID con-

troller design is formulated into an optimization problem with LMI constraints and

solved with interior point method. In the second design, the new robust stability

criterion is developed using both gain and phase information. A new PID tuning

scheme is proposed based on the developed criterion for processed with parameter

uncertainty. The third method aims at designing PI controller for processes with

state time delay. The stabilizing and H∞ suboptimal PI controllers are developed

12

Chapter 2. Three New Approaches to PID Controller Design 13

via iterative LMI optimization.

The chapter is organized as follows. In Section 2.2 a new robust PID method for

gain and phase margins is presented. Section 2.3 proposes a new PID design based

on a new stability criterion. Section 2.4 describes an approach to PI controller

design of stabilizing control and H∞ suboptimal control for processes with state

time delay.

2.2 Robust PID Controller Design for Gain and

Phase Margins

2.2.1 Introduction

PID controllers are the dominant choice in process control and an abundant amount

of research have been reported in the past on the PID controller design (Astrom

and Hagglund, 1995). Gain and phase margins are typical control loop specifi-

cations associated with the frequency response technique (Ho et al., 1995) and

many controller designs have been presented to meet the margins. Ogata (1990)

and Franklin et al. (1994) gave solutions using numerical and graphical methods

respectively. Under some approximations, Ho et al. (1995) presented analytical

formulae to design the PID controller for first-order and second-order plus dead

time plant. A general method to achieve exact gain and phase margin for a general

linear plant is given by Wang et al. (1999a). However, all these methods did not

consider uncertainty of the process. Usually we can only obtain an approximate

linear model because of the complex of the real process. Therefore, it is obvious

the uncertainty of the process should be considered when the controller is designed.

In this section, optimal PID controllers are designed to meet gain and phase

margins for a family of plants. The problem is formulated into an optimization

problem with a set of LMI constraints. The resultant convex optimization can be

solved effectively using interior point method. Simulation is given to demonstrate

the proposed method.


2.2.2 PID Controller Design Using LMI

Suppose an uncertain process G is described as

∣

∣

∣G(jω)− G(jω)∣

∣

∣

2

< ρ(ω) (2.1)

where the nominal model G and the uncertainty bound ρ are known.

K G+

-

r y

Figure 2.1. Unity feedback system

We adopt the unity feedback system showed in Fig. 2.1 and PID controller with

the transfer function:

K(s) = kp +kis+ kds. (2.2)

Assume that the control system specifications are given in terms of gain margin

Am and phase margin φm. Our objective is to determine the controller parameters

kp, ki and kd such that the desired gain margin Am and phase margin φm can be

best achieved for the uncertain process. These specifications mean that

G(jωp)K(jωp) = −1

Am

and

G(jωg)K(jωg) = −ejφm ,

where ωp and ωg are the phase and gain cross over frequencies of the open loop

system respectively. Suppose both ωg and ωp are available, the controller pa-

rameters can be computed if the exact model of the process is available (Ho et

al., 1995; Wang et al., 1999a). However, it will be much more difficult if the un-

certainty of the model is considered. In the sequel, the constraint optimization


methods are used to find the optimal controller in order to make the whole family

of the models meet the gain and phase margin specifications as close as possible.

Denote the open-loop frequency response for the uncertain system and the nominal

system by L(jw) and L(jw) respectively. Set the β as the tuning parameter, the

problem can be formulated as follows

P1 : minK

(γ1 + βγ2) (2.3)

subject to∣

∣

∣L(jωp)− L(jωp)∣

∣

∣

2

< γ1, (2.4)

∣

∣

∣G(jωp)− G(jωp)

∣

∣

∣

2

< ρ1, (2.5)

∣

∣

∣L(jωg)− L(jωg)∣

∣

∣

2

< γ2, (2.6)

∣

∣

∣G(jωg)− G(jωg)

∣

∣

∣

2

< ρ2. (2.7)

Remark 2.1. The formulation of the problem can be explained as follows: The

objective is to design PID controller to achieve desired gain and phase margins

for the open-loop frequency response L(jω). That means we wish L(jω) satisfies

the gain margin index at phase cross-over frequency ωp and the phase margin

index at gain cross-over frequency ωg respectively. So we formulate (2.4) and (2.6).

Considering the uncertainty of the process, we cannot make the inequalities become

equality. Therefore, we add two inequalities (2.5) and (2.7) as the constraints. In

short, we wish the whole family of the systems meet the desired gain and phase

margins as close as possible when the process varies in the bound of the uncertain

model.


Note that

∣

∣

∣L(jωp)− L(jωp)

∣

∣

∣

2

− γ1

=∣

∣

∣G(jωp)K(jωp)− L(jωp)

∣

∣

∣

2

− γ1

=

[real(G(jωp)) + j imag(G(jωp))][kp + j(kiωp− kdwp)]− L(jωp)

2

− γ1

= [kpreal(G(jωp)) + imag(G(jωp))(kiωp− kdωp)− real(L(jωp))]

2

+ [(kdwp −kiωp

)real(G(jωp)) + kpimag(G(jωp))− imag(L0(jwp))]2 − γ1

= real2(L(jωp)) + imag2(L(jωp))− γ1 + 2 real(L(jωp))[

−kp kdωp − kiωp

]

real(G(jωp))

imag(G(jωp))

+ 2 imag(L(jωp))[

kiωp− kdωp −kp

]

real(G(jωp))

imag(G(jωp))

+

real(G(jωp))

imag(G(jωp))

T

k2p 2kp(kdωp − kiωp)

0 ( kiωp− kdωp)

2

real(G(jωp))

imag(G(jωp))

+

real(G(jωp))

imag(G(jωp))

T

( kiωp− kdωp)

2 0

2kp(kdωp − kiωp) k2p

real(G(jωp))

imag(G(jωp))

= real2(L(jωp)) + imag2(L(jωp))− γ1 − 2[real(L(jωp)) imag(L(jωp))]

kp ki/ωp − kdωp

−ki/ωp + kdωp kp

real(G(jωp))

imag(G(jωp))

+

real(G(jωp))

imag(G(jωp))

T

kp ki/ωp − kdωp


T

kp ki/ωp − kdωp


real(G(jωp))

imag(G(jωp))

.

Let

Lp =

Re(L(jωp))

Im(L(jωp))

, L0p =

Re(L(jωp))

Im(L(jωp))

,

Gp =

Re(G(jωp))

Im(G(jωp))

, G0p =

Re(G(jωp))

Im(G(jωp))

,


A(ωp) =

kp ki/ωp − kdωp


, I =

1 0

0 1

.

then equations (2.4) can be rewritten as

1

Gp

T

LT0pL0p − γ1 −LT0pA(ωp)−AT (ωp)L0p AT (wp)A(ωp)

1

Gp

< 0 (2.8)

and from (2.5),we obtain

1

Gp

T

GT0pG0p − ρ1 −GT

0p

−G0p I

1

Gp

< 0. (2.9)

Equation (2.6) and (2.7) can be treated similarly. Therefore, we can convert P1

into the following equivalent problem.

Given the uncertain system (2.1), find the optimal value of the linear weighting

errors of the desired gain and phase margins:

P2 : minK

(γ1 + βγ2)

subject to

1

Gp

T

LT0pL0p − γ1 −LT0pA(ωp)−AT (ωp)L0p AT (ωp)A(ωp)

1

Gp

< 0,

1

Gp

T

GT0pG0p − ρ1 −GT

0p

−G0p I

1

Gp

< 0,

1

Gg

T

LT0gL0g − γ2 −LT0gA(ωg)−AT (ωg)L0g AT (ωg)A(ωg)

1

Gg

< 0,

1

Gg

T

G0gTG0g − ρ2 −GT

0g

−G0g I

1

Gg

T

< 0.

In order to obtain the solution of the problem, we need two lemmas (Boyd et

al., 1994).


Lemma 2.1. (S-procedure) Let F0, ...,Fp be quadratic functions of the variable

ς ∈ Rm:

Fi(ς) = ςTTiς + 2uTi ς + vi, i = 1, ..., p.

where Ti = T Ti . The following condition on F0, ...,Fp: F0(ς) ≥ 0 for all ς such

that Fi(ς) ≥ 0, i = 0, ..., p. holds if there exist τ1 ≥ 0, ..., τp ≥ 0 such that

T0 u0

uT0 v0

−∑p

i=1 τi

Ti ui

uTi vi

≥ 0.

Lemma 2.2. (Schur complement) If Q(x) = Q(x)T , R(s) = R(s)T and S(x)depend affinely on x, the LMI

Q(x) S(x)S(x)T R(x)

> 0

is equivalent to

R(x) > 0, Q(x)− S(x)R(x)−1S(x)T > 0.

Now, we can develop the solution of the original system.

Theorem 2.1. If the solution of problem P2 exists, the solution is unique and it

is equivalent to the solution of the LMI problem

P3 : minK

(γ1 + βγ2) (2.10)

subject to

−γ1 + t1(ρ1 − GTp0Gp0) −t1GT

p0 LTp0

−t1Gp0 −t1I AT (−ωp)Lp0 A(ωp) −I

< 0, (2.11)

and

−γ2 + t2(ρ2 − GTg0Gg0) −t2GT

g0 LTg0

−t2Gg0 −t2I AT (−ωg)Lg0 A(ωg) −I

< 0. (2.12)


Proof. Using the S-procedure, (2.8) and (2.9) can be combined into

LTp0Lp0 − γ1 + t1(ρ1 − GTp0Gp0) LTp0A(ωp)− t1G

Tp0

AT (ωp)Lp0 − t1Gp0 AT (ωp)A(ωp)− t1I

< 0, (2.13)

where t1 is a new variable introduced according to S-procedure Lemma. According

to the Schur complement, (2.13) can be converted into the linear matrix inequality

(LMI) form,

−γ1 + t1(ρ1 − GTp0Gp0) −t1GT

p0 Lp0

−t1Gp0 −t1I AT (−ωp)Lp0 A(ωp) −I

< 0. (2.14)

In a similar way, from (2.6) and (2.7), we have

−γ2 + t2(ρ2 −GTg0Gg0) −t2GT

g0 Lg0

−t2Gg0 −t2I AT (−ωg)Lg0 A(ωg) −I

< 0. (2.15)

Thus, P2 is equivalent to P3. Note that P3 is a standard convex optimization

problem with LMI constraint and it has the unique solution.

Remark 2.2. In this controller design method, the optimal controller design prob-

lem was converted into a standard LMI optimization problem and thus the PID

parameters can be found effectively using interior point method (Boyd et al., 1994).

Remark 2.3. If both ρ1 and ρ2 equal zero, the problem reduce to the nominal design

and the solution will be similar to that of Wang et al. (1999a).

Remark 2.4. Besides finding the optimal PID parameter, we can also obtain the

values of γ1 and γ2 which are the measurement of the errors of the gain and phase

margins for the uncertain system respectively. We can choose the value of β to

make the tradeoff between the error of gain margin and that of the phase margin.

2.2.3 Tuning Guidelines

In order to use the above method to design a PID controller, we must specify gain

margin Am, phase margin φm, phase cross-over frequency ωp and gain cross-over


frequency ωg. Gain and phase margins are typical control loop specifications which

reflect the performance and stability of the system, so we use it as user specified

parameters. Therefore, the remaining work is to choose reasonable phase cross-over

frequency ωp and gain cross-over frequency ωg.

If the bandwidth of a process is the frequency at which the process gain drops

by 3dB below that at the zero frequency, and it is usually approximated by its

phase cross-over frequency, since frequencies below it constitute the most significant

range in controller design. In controller tuning, the closed-loop bandwidth should

be carefully chosen. If it is too large, the control signal will saturate. If it is too

small, sluggish response results. It is well accepted in engineering practice that the

closed-loop bandwidth should be close to its open-loop bandwidth. For example,

we may set

ωp = αωc, α = [0.5, 2], (2.16)

where ωp is the closed-loop bandwidth and ωc is the open-loop bandwidth. The

default value for α is 1. The value of wc is available from the process frequency

response and is the point that satisfies

]G(jωc) = −π. (2.17)

Once ωp is specified, ωg is not free to be chosen. To be this, one obtain from

kpap − (kdωp − ki/ωp)ap = −1/Am,

kpbp + (kdωp − ki/ωp)bp = 0,

kpag − (kdωg − ki/ωg)bg = −cos(φm),

kpbg + (kdωg − ki/ωg)ag = −sin(φm),

where

ap = Re(G(jωp)), bp = Im(G(jωp)),

ag = Re(G(jωg)), bg = Im(G(jωg)).


Eliminating kp, ki and kd from the above four equations yields the relationship

between ωp and ωg:

apa2p + b2p

=Am(cosφmag + sinφmbg)

a2g + b2g, (2.18)

which enable us to find ωg from ωp. The frequency ωg may be determined by

searching down from the frequency ω = ωp towards ω = 0 until (2.18) holds.

In the summary, given the nominal process G(jω), the uncertain bound and

the specified gain margin Am and phase margin φm, the PID parameters can be

found as follows:

i) Obtain the nominal process phase cross-over frequency ωc from G(jω).

ii) Specify the ratio of close-loop from open-loop bandwidth α and set ωp = αωc.

iii) Search from ω = ωp down towards ω = 0 for the frequency ωg that satisfies

(2.18).

iv) Compute PID parameter as the solution of P3.

2.2.4 Simulation

We shall now look at some examples and demonstrate the use of the method.

Example 2.1. Consider a second-order plus dead time process

G(s) =e−0.5s

(1 + s)(1 + 0.5s)

and the uncertain system is described as

G(s) = G(s) + δ∆G(s),

where

|δ| < 1

and

∆G(s) =0.1

s+ 1.


The gain margin and phase margin are set to 3 and 60 degrees. Using the proposed

method, the process phase cross-over frequency wc is obtained as ωc = 3.1416 rad/s.

Choose α = 1, then we have ωp is set as ωp = ωc = 3.1416. Simple search from

ω = ωp towards ω = 0 gives the frequency ωg = 1.0472 at which (2.18) holds. With

the LMI toolbox, we obtain the designed PID controller:

K(s) = 1.5694 +0.8906

s+ 0.7811s.

If the gain and phase margins are specified as 2 and 45 degrees respectively, the

proposed method gives

K(s) = 2.3400 +1.5488

s+ 0.7758s.

The step responses are showed in Fig. 2.2 and Fig. 2.3. The 3 lines correspond to

δ = −1, δ = 0 and δ = 1.

0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

1.2

a1

a2

a3

Figure 2.2. Step response of proposed method with Am = 3 and φm = 60

(a1: δ=-1; a2:δ=0; a3:δ=1)

Example 2.2. Consider a high order process

G(s) =e−0.1s

(s2 + s+ 1)(s+ 2)2.

In order to compare with Ho’ method (Ho et al., 1995), suppose the process can

be approximate by a second-order plus dead time model (Wang and Zhang, 1998b)

G(s) =0.25e−1.420s

(0.7207s+ 1)2


0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a1

a2

a3

Figure 2.3. Step response of proposed method with Am = 2 and φm = 45

(a1: δ=-1; a2:δ=0; a3:δ=1)

and the uncertain system is described as

G(s) = G(s) + δ∆G(s),

where

|δ| < 1

and

∆G(s) =0.2

s+ 5.

For Am = 3 and φm = 60, using the proposed scheme, we obtain the controller

K(s) = 1.9576 +1.5730

s+ 1.2420s.

and Ho’s method gives

K(s) = 2.1239 +1.4749

s+ 1.7646s.

For Am = 2 and φm = 45, the proposed scheme gives

K(s) = 3.4725 +2.1380

s+ 1.4735s.

and Ho’s method gives

K(s) = 3.1858 +2.2124

s+ 1.1469s.

The step responses are showed in Fig. 2.4 and Fig. 2.5. The 3 lines correspond to

δ = −1, δ = 0 and δ = 1.


0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time

a3

a2

a1

Figure 2.4. Step response with Am = 3 and φm = 60

(—— proposed method, - - - Ho’s method)

(a1: δ=-1; a2:δ=0; a3:δ=1)

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2

Time

a2

a1

a3

Figure 2.5. Step response with Am = 2 and φm = 45

(—— proposed method, - - - Ho’s method)

(a1: δ=-1; a2:δ=0; a3:δ=1)

2.2.5 Conclusions

A simple method for the design of PID controllers for uncertain process which

achieves gain and phase margin specifications is proposed. The scheme is based

on the uncertain model of the plant. The design problem was formulated into


a convex optimization problem with quadratic constraint, then it was converted

into an optimization problem with linear matrix inequalities constraint which can

be solved effectively using interior point method. The optimal PID controller is

obtained directly to satisfy the gain and phase margins. Simulation results have

been presented to demonstrate the effectiveness of the method.

2.3 Quantitative Robust Stability Analysis and

PID Controller Design

2.3.1 Introduction

Stability criteria and stabilization methods are important research topics in the

system and control area. Many results have been presented in the area of robust

stability. The small gain theorem is a key result of the robust stability analysis

(Zhou et al., 1996). Essentially, the small gain theorem states that if a feedback

loop consists of stable systems and the loop-gain product is less than unity, then

the closed-loop system is stable. The criterion is widely used in the robust control

area. Unfortunately, it could be very conservative in many cases because only

the gain information of the open-loop system is used and the phase information is

ignored.

In this section, the conservativeness of traditional robust stability results is

studied. With the information on both the gain and phase of the uncertain open-

loop system, a new quantitative robust stability criterion is developed. The gain

and phase bounds of uncertain second-order plus time delay (SOPTD) model are

derived. Based on these new results, a new PID tuning scheme is presented using

the worst gain and phase margins of the open-loop system. Simulation is given to

substantiate the effectiveness of the method.


)(sH r y

+

+

)(s∆

Figure 2.6. Controlled uncertain system

2.3.2 Review of Robust Control Theory

Consider a controlled uncertain system in Fig. 2.6. According to the small gain

theorem (Zhou et al., 1996), if the nominal system H(s) and the perturbation ∆(s)

in Fig. 2.6 are both stable, the feedback system is robustly stable if the inequality

|H(jω)∆(jω)| < 1, ω ∈ [0,∞) (2.19)

holds. Usually, one assumes that ∆(s) is unknown but bounded in magnitude such

that supω

|∆(jω)| = M . Then, the feedback loop given in Fig. 2.6 will be stable

provided that

‖H(s)‖∞ <1

M.

However, in industrial applications, the structure of the uncertainty ∆(s) may be

known. For instance, a large class of the uncertain process can be modeled as a

transfer function with interval parameters. The above criterion for robust stability

may be very conservative when the structure of ∆(s) is specified.

Example 2.3. Consider a second-order nominal plant with transfer function

G(s) =1

s2 + s+ 1

and it is closed in a negative feedback loop with the controller K(s) = 1. Let the

plant uncertainty be structured and given by

G(s) =1 + δ1

s2 + (1 + δ2)s+ (1 + δ3)(2.20)


where δ1 ∈ [−0.5, 0.5], δ2 ∈ [−0.5, 0.5] and δ3 ∈ [−0.5, 0.5]. One may easily see that

the closed-loop system is robustly stable, because its characteristic polynomial,

p(s) = s2 + (1 + δ2)s+ 2 + δ1 + δ3

always has positive coefficients and thus its roots lie in th open left half of the

complex plane.

In order to apply the robust stability criterion (2.19), we express the uncertainty

as an additive one:

∆(s) =1 + δ1

s2 + (1 + δ2)s+ (1 + δ3)− 1

s2 + s+ 1,

and the corresponding H in Fig. 2.6 is

H(s) =−K(s)

1 +K(s)G0(s)= −s

2 + s+ 1

s2 + s+ 2.

We have

supω

|H(jω)∆(jω)| > limω→0|H(jω)∆(jω)| = 1

2× |1 + δ1

1 + δ3− 1|.

If we choose δ1 = 0.5 and δ3 = −0.5, or the plant is perturbed to

G(s) =1.5

s2 + s+ 0.5,

there results in

supω

|H(jω)∆(jω)| > 1,

violating criterion (2.19) though the system is stable. This shows the conservative-

ness of criterion (2.19).

It should be pointed out that criterion (2.19) makes use of only gain information

while any phase lag is permitted including infinity. It should be also pointed

out that what we have said so far on the structured uncertainty is different from

the existing robust stability criterion for structured perturbations, or the theory

of structured singular values, which have been developed for the multivariable

systems and its essence is the same as criterion (2.19). However, we focus our

study here on systems with single uncertainty and attempt to develop a new robust

stability criterion which employs the phase information of the perturbations to

reduce conservativeness.


2.3.3 Quantitative Robust Stability

Let us recall the Nyquist criterion (Zhou et al., 1996): if the controller K(s) and

the plant G(s) have no poles in the right half plane, then the closed-loop system

is stable if and only if the Nyquist curve of G(jω)K(jω) does not encircle the −1point. An uncertain process is actually a set of models. Each model gives rise to

a Nyquist curve and the set thus sweeps a band of Nyquist curves. Then from the

Nyquist criterion, the closed-loop system is robustly stable if and only if the band

of Nyquist cluster for the uncertain process and the controller does not encircle the

−1 point. Conversely, the system is not robustly stable if and only if there exits

a Nyquist curve among the band which encircles −1 point, or there must exist a

specific frequency ω0 whose corresponding uncertain region is such that both

maxG|G(jω0)K(jω0)| > 1

and

minGargG(jω0)K(jω0) 6 −π

hold true.

Note that

maxG|G(jω0)K(jω0)| = |K(jω0)|max

G|G(jω0)|

and

minGargG(jω0)K(jω0) = argK(jω0)+min

GargG(jω0),

thus we establish the following quantitative robust stability theorem.

Theorem 2.2. Supposing that the nominal closed-loop system and the controller

are stable, the uncertain closed-loop system is robustly stable if either of the follow-

ing inequalities holds:

|K(jω)| 6 1

maxG|G(jω)| (2.21)

or

argK(jω) > −π −minGargG(jω). (2.22)


Alternatively, we may transform into the diagram of Fig. 2.6, where H(s) is

the equivalent nominal controlled system, ∆(s) is the uncertainty of the system.

If we modify the inequalities in (2.21) and (2.22) as follows:

|H(jω)| 6 1

max∆|∆(jω)| (2.23)

and

argH(jω) > −π −min∆arg∆(jω), (2.24)

then we have the corollary below.

Corollary 2.3. If the nominal controlled system H(s) and the uncertainty ∆(s)

are all stable, then the uncertain closed-loop system remains stable if either (2.23)

or (2.24) hold for every ω .

Typically, one may use an additive uncertainty model G = G + ∆a or a mul-

tiplicative uncertainty model G = G(1 + ∆m) to describe the uncertain process.

Then corresponding nominal controlled system H is given respectively,

Ha(jω) =−K(jω)

1 +K(jω)G(jω)

or

Hm(jω) =−K(jω)G(jω)

1 +K(jω)G(jω).

For a designed control system, the robust stability criterion in (2.21) and (2.22)

can be tested graphically as follows:

(i) Draw Bode plots for magnitude 1/max|G(jω)| and phase−π−minargG(jω)respectively, according to the given structure of the uncertain plant;

(ii) Draw Bode plots for K(jω) in the same diagram;

(iii) Check if condition (2.21) or (2.22) is satisfied.


2.3.4 Second-Order Uncertain Model

In most practical situations, it is reasonable to approximate high order processes

by low order plus delay models (Pintelon et al., 1994; Glover, 1989). Reduced-order

models are often required for simplifying the design and implementation of control

systems (Seborg et al., 1989). For this reason, model reduction has attracted much

attention in engineering sciences, especially in model-based control. However, real

processes are usually not of first order or second order, and sometimes may be

perturbed by the different operating conditions or other disturbances. Thus the

model error is inevitable in practical applications if only the nominal model is used.

One way to avoid the problems arising from the modeling error is to use a model

set to express the desired closed-loop specifications. First-order plus dead time

(FOPDT) model is relatively simple but may not achieve the accuracy required in

many cases as they carry only real poles, hence are unable to generate peaks in

the frequency response of oscillatory processes. Therefore, it is desirable to use a

second-order plus dead time (SOPDT) model to describe uncertain systems.

Consider an uncertain SOPDT model

G(s) =b

s2 + a1s+ a2e−Ls

where the uncertain parameters are b ∈ [b− b+], a1 ∈ [a−1 , a+1 ], a2 ∈ [a−2 , a

+2 ] and

L ∈ [L−, L+], b− > 0, a−1 > 0, a−2 > 0 and L− > 0. For a fixed ω, the gain of the

process is

|G(jω)| = b/√

(a1ω)2 + (a2 − ω2)2. (2.25)

For this process, max|G(jω)|, min|G(jω))| can be derived as

max|G(jω)| =

b+√(a−1 ω)

2+(a−2 −ω2)2, ω ∈

[

0,√

a−2

]

;

b+

a−1 ω, ω ∈

[

√

a−2 ,√

a+2

]

;

b+√(a−1 ω)

2+(a+2 −ω2)2, ω ∈

[

√

a+2 ,∞]

;

(2.26)

and

min|G(jω)| =

b−√(a+1 ω)

2+(a+2 −ω2)2, ω ∈

[

0,

√

a−2 +a+2

2

]

;

b−√(a+1 ω)

2+(ω2−a−2 )2, ω ∈

[√

a−2 +a+2

2,∞]

.

(2.27)


In a similar way, the phase of the second order system can be obtained as

arg(G(jω)) = −arg(a2 − ω2 + ja1ω)− Lω

Consider the uncertainty of the parameters, we obtain

minarg(G(jω)) =

−arg(a−2 − ω2 + ja+1 ω)− L+ω, ω ∈[

0,√

a−2

]

;

−arg(ja−1 ω + a−2 − ω2)− L+ω, ω ∈[

√

a−2 ,∞]

;

(2.28)

and

maxarg(G(jω)) =

−arg(a+2 − ω2 + ja−1 ω)− L−ω, ω ∈[

0,√

a+2

]

;

−arg(ja+1 ω + a+2 − ω2)− L−ω, ω ∈[

√

a+2 ,∞]

.

(2.29)

10−1

100

101

0

1

2

3

4

5

10−1

100

101

−200

−150

−100

−50

0

Gain

Frequency

Phase

Frequency

Figure 2.7. The plot of max|G(jω)| and minargG(jω)

Example 2.4. In Example 2.3, the small gain theory fails to verify the stability

of the uncertain system. We now use the proposed method to examine if the

uncertain system is stable.

Reconsidering (2.20), we obtain


max|G(jω)| =

1.5√(0.5ω)2+(0.5−ω2)2

, ω ∈[

0,√0.5]

;

1.50.5ω

, ω ∈[√

0.5,√1.5]

;

1.5√(0.5ω)2+(1.5−ω2)2

, ω ∈[√

1.5,∞]

;

and

minarg(G(jω)) =

−arg(0.5− ω2 + j1.5ω), ω ∈[

0,√0.5]

;

−arg(0.5− ω2 + j0.5ω), ω ∈[√

0.5,∞]

.

As in Example 2.3, the unit feedback controller is used, that is K(jω) = 1.

Then we can use Theorem 2.2 to check the stability of the system. Though the

gain condition (2.21) is violated for ω < 1.66 in the gain plot of Fig. 2.7, one sees

that the phase condition (2.22) will be satisfied for all frequencies in the phase plot

of Fig. 2.7. Thus the uncertain system is robustly stable.

2.3.5 Robust PID controller Design

In the context of controller synthesis, we usually get the controller according to

the nominal plant model. If the uncertain range of the plant model is known, then

the above idea of robust stability may be used to design a controller for robust

performance.

Since PID controllers are widely used in the process control industry and easy to

implement (Astrom and Writtenmark, 1984; Zhuang and Atherton, 1993; Wang et

al., 1997), we use the robust quantitative stability criterion developed to tune a PID

controller such that the desired closed-loop specifications are satisfied. Gain and

phase margins are typical control loop specifications associated with the frequency

response technique (Wang et al., 1999a). Here, we shall develop a PID tuning

algorithm which guarantees the required gain and phase margins of the closed-

loop system for the whole family of the uncertain plant. The transfer function of

the PID controller is given by

K(s) = kp +kis+ kds. (2.30)


According to the preceding discussion, we can build a model Gw representing the

worst case of the uncertain process using (2.26) and (2.28), that is

Gw(jω) = max|G(jω)|ejminarg(G(jω)) (2.31)

Let the desired closed-loop gain margin be Am and the phase margin be φm, then

the following two equations follow from the definition of gain and phase margins:

Gw(jωp)

[

kp + j(kdωp −kiωp

)

]

= − 1

Am

(2.32)

and

Gw(jωg)

[

kp + j(kdωg −kiωg

)

]

= −ejφm , (2.33)

where wp and wg are the phase and gain cross-over frequency of the cascaded

process-plus-controller system respectively. The tuning objective is to find param-

eters Kp, Ki and Kd so that the given gain and phase margins are achieved, i.e.

(2.32) and (2.33) are satisfied. It is noted that there are altogether five unknowns,

namely,Kp, Ki, Kd, ωg and ωp, in (2.32) and (2.33). Since the equations are com-

plex, they can be broken down into four real equations. Because the number of the

unknowns exceeds the number of equations, there is no unique solution to (2.32)

and (2.33) unless one more constraint is added.

It is well accepted in engineering practice that the closed-loop bandwidth should

be close to its open-loop value. Thus the value of ωp can be assigned near the plant

phase-crossover frequency ωp, that is,

ωp = αωp.

The default value for α is 1. The solution to (2.32) and (2.33) is then (Wang et

al., 1999a)

kp = Re

[ −1AmGw(jωp)

]

(2.34)

and

ki

kd

=

− 1ωp

ωp

− 1ωg

ωg

−1

Im[

−1AmGw(jωp)

]

Im[

−exp(φm)Gw(jωg)

]

(2.35)


where ωg satisfies

Re

[−exp(jφm)Gw(jωg)

]

= Re

[ −1AmGw(jωp)

]

(2.36)

To test the performance robustness of the closed-loop system for the tuned PID

controller, we simply check both gain and phase margins of the worse case. The

following corollary is from the deviation of Theorem 2.2.

Corollary 2.4. The phase margin of the uncertain process is larger than φm if

argK(jω)+minGargG(jω) > −π + φm

when maxG|G(jω)||K(jω)| > 1 and the gain margin of the uncertain process is

larger than Am if

maxG|G(jω)||K(jω)| 6 1

Am

when argK(jω)+minGargG(jω) < −π.

In summary, given an uncertain plant G(s), the PID controller K(s) in form of

(2.30) can be tuned for the worse case to meet the given gain and phase margins

of Am and φm as follows:

i) Obtain the phase-crossover frequency ωp of the worst model of the uncertain

process from the frequency response.

ii) Set the ωp near that frequency.

iii) Compute kp from (2.34).

iv) Find ωg from (2.36) by searching through the frequency range from ωp down-

wards.

v) Compute ki and kd from (2.35).

Remark 2.5. Although both of the robust PID controllers in this section and last

section are designed for gain and phase margins, there are still much difference

between the approaches. Firstly, the objective of the design in this section is

to satisfy the gain and phase margins for the worst cases of the uncertain model.


While in last section the design purpose is to achieve smallest error with the desired

gain and phase margins for all the uncertain models. In this point of view, the

approach in the second is more conservative. However, it is difficult to compare the

tween approaches with simulation. The reason is that the two approaches cope with

different kinds of uncertain models. In this section design, the uncertain model is

an second order plus dead time model with uncertain parameters. However, the

uncertainty description used in last section is an unstructured uncertainty which

is different with that used in this section.

Example 2.5. Consider a second order plus dead time uncertain process

G(s) =b

s2 + a1s+ a2e−Ls

with the uncertain parameters as b ∈ [0.8, 1.2], a1 ∈ [1.2, 1.8], a2 ∈ [0.8, 1.2] and

L ∈ [4, 6], we obtain from (2.26) ∼ (2.29),

max|G(jω)| =

1.2√(1.2ω)2+(0.8−ω2)2

, ω ∈[

0,√0.8]

1.21.2ω

, ω ∈[√

0.8,√1.2]

1.2√(1.2ω)2+(1.2−ω2)2

, ω ∈[√

1.2,∞]

min|G(jω)| =

0.8√(1.2ω)2+(1.2−ω2)2

, ω ∈[

0,√

0.8+1.22

]

0.8√(1.2ω)2+(ω2−0.8)2

, ω ∈[√

0.8+1.22

,∞]

minarg(G(jω)) =

−arg(0.8− ω2 + j1.8ω)− 6ω, ω ∈[

0,√0.8]

−arg(j1.2ω + 0.8− ω2)− 6ω, ω ∈[√

0.8,∞]

and

maxarg(G(jω)) =

−arg(1.2− ω2 + j1.2ω)− 4ω, ω ∈[

0,√1.2]

−arg(j1.8ω + 1.2− ω2)− 4ω, ω ∈[√

1.2,∞]

The worst case model is then set according to (2.31). The phase cross-over

frequency of Gw(s) is 0.3890rad/s. If we set the gain and phase margins for the


nominal plant to be 3 and 60 degrees respectively, and choose ωp the same as the

plant phase-crossover frequency as ωp = 0.3890. Then the proportional gain Kp

can be readily calculated using (2.33) to be

kp = Re[−1

3×Gw(j0.3890)] = 0.2220

The frequency rage from blow ω = ωp is then searched and it is found that ωg =

0.1047, which satisfied equation (2.36). The value of Ki and Kd are then calculated

using (2.35) to be

ki

kd

=

− 10.3890

0.3890

− 10.1047

0.1047

−1

Im[

−13×Gw(j0.3890)

]

Im[

−exp(j1.05)Gw(j0.1047)

]

=

0.0710

0.4784

The designed PID controller is hence

K(s) = 0.222 +0.071

s+ 0.4784s

Then we check the robustness of the designed controller. In the gain plot of Fig. 2.8,

we can see |Gw(jω)K(jω)| > 1 when ω < 0.1047. In the phase plot of Fig. 2.8,

one sees that at the same frequency interval ω < 0.1047, argGw(jω)K(jω) >

−π + φm = −120. According to Corollary 2.3, the uncertain system satisfies the

phase margin specification. We can check that the gain margin specification is also

satisfied using a similar method. Thus the tuned controller meets the condition

for robust performance of the closed-loop system. With different values for b, a1,

a2 and L, the step responses of the uncertain system are showed in Fig. 2.9.

2.3.6 Conclusions

In this section, a new criterion for robust stability is developed which makes use

of both the gain and phase information of the open-loop system in order to reduce

the conservativeness of the small gain theory. Based on it, a robust PID design is

presented for SOPDT uncertain model. Simulation shows the effectiveness of the

method.


10−2

10−1

100

0

2

4

6

8

10

12

10−2

10−1

100

−400

−350

−300

−250

−200

−150

−100

−50

Gain

Frequency

Phase

Frequency

Figure 2.8. The plot of max|G(jω)K(jω)| and minargG(jω)K(jω)

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 2.9. Step response of the uncertain system

2.4 PI Controller Design for State Time-Delay

Systems via ILMI

2.4.1 Introduction

PID controller is the most popular controller in the industrial world and many ap-

proaches have proposed to design the controller, see Astrom and Hagglund (1995)

and Tan et al. (1999). However, most of the existing methods are developed for sin-

gle input and single output (SISO) systems. Recently, Zheng et al. (2002) proposed


a scheme to design multivariable PID controllers under various performance spec-

ifications and the problem is solved via iterative linear matrix inequalities (ILMI)

approach. However, the method is limited to processes with no time delay.

Niculescu (1998) proposed an approach to design H∞ state feedback controller

for a class of linear systems with delayed state via LMI optimization. Later, Mah-

moud and Zribi (1999) developed a scheme for H∞ static output feedback (SOF)

control. In their method, under the strict-positive realness condition, the SOF

control design problem is simplified into a state feedback problem. Obviously, the

method cannot be used for general processes. The difficulty of SOF synthesis is

that the problem inherently is a bilinear problem which is hard to be formulated

into an optimization problem with LMI constraints.

In this section, we propose a method to design either SOF or PI controller for

general processes with state time delay via iterative LMI approach. Our idea is to

convert the problem of PI controller design to that of static output feedback (SOF)

controller design. In Cao, Lam and Sun (1998), an ILMI algorithm was developed

to solve the problem of SOF stabilization for processes without time delay. Here,

we extend their method to the processes with state time delay.

2.4.2 Problem Description

Consider a class of linear systems with delayed state:

x(t) = Ax(t) + Adx(t− d) +B1w(t) +B2u(t),

ys(t) = Csx(t), yr(t) = Crx(t),(2.37)

where x(t) ∈ Rn is the state, u(t) ∈ Rl1 is the input manipulatable by the controller,

w(t) ∈ Rl2 is the input which stands for some exogenous disturbances and is

not manipulatable by the controller, ys(t) ∈ Rm1 is the measured output and

yr(t) ∈ Rm2 is the controlled output, and A,Ad, B1, B2, CsCr are matrices with

appropriate dimensions. The problem is to find an SOF controller

u = Kys, (2.38)


where K ∈ Rl1×m1 is the control gain or a PI controller

u = K1ys +K2

∫ t

0

ysdt, (2.39)

where K1 ∈ Rl1×m1 and K2 ∈ Rl1×m1 are matrices to be designed, to satisfy some

given specifications.

2.4.3 Stabilizing Control

In this subsection, sufficient conditions are developed for stabilizing control for the

process (2.37) with SOF. The algorithm is also given to find such an SOF controller

(2.38) to stabilizing the process (2.37).

Consider the process described in (2.37). The objective of SOF stabilization

problem is to find an SOF controller in (2.38) such that the closed-loop system

x(t) = Acx(t) + Adx(t− d) +B1w(t),

yr(t) = Crx(t), Ac = A+B2KCs

(2.40)

is asymptotically stable.

Lemma 2.3. The closed-loop system (2.40) is asymptotically stable with w(t) = 0

and d ≥ 0 if one of the following equivalent conditions is satisfied:

(i) There exist matrices 0 < PT = P ∈ Rn×n, 0 < QT = Q ∈ Rn×n,

satisfying the algebraic Riccati inequality (ARI)

P(A+B2KCs) + (A+B2KCs)TP +Q+ PAdQ−1AT

dP < 0. (2.41)

(ii) There exist matrices 0 < PT = P ∈ Rn×n, 0 < QT = Q ∈ Rn×n,

satisfying the matrix inequality

W =

P(A+B2KCs) + (A+B2KCs)TP PAd

ATdP −Q

< 0. (2.42)

Proof. (i) Similar to the method used in Mahmoud and Zribi (1999), we define the

quadratic Lyapunov function candidate

V (x, t) = xT (t)Px(t) +∫ t

t−d

xT (v)Qx(v)dv,


where P ,Q > 0. Note that V (x, t) > 0 when x 6= 0 and V (x, t) = 0 when x = 0.

Suppose that the input signal w(t) = 0, we observe that

V (x, t) =xT (t)Px(t) + xT (t)Px(t) + xT (t)Qx(t)− xT (t− d)Qx(t− d)

=xT (t)(A+B2KCs)TPx(t) + xT (t)P(A+B2KCs)x(t) + xT (t− d)AT

dPx(t)

+ xT (t)PAdx(t− d)− xT (t− d)Qx(t− d)

=ZT (t)WZ(t),

where Z(t) = [xT (t) xT (t− d)]T and W is given by (2.42).

(ii) Using Schur complement, one can obtain (2.41) from (2.42) easily.

Lemma 2.4. The following two matrix inequalities are equivalent:


dP < 0. (2.43)

PA+ ATP +Q−PB2BT2 P + (BT

2 P +KCs)T (BT

2 P +KCs) + PAdQ−1ATdP < 0.

(2.44)

Proof. Sufficiency. Note that


dP

≤(A+B2KCs)TP + P(A+B2KCs) + CT

s KTKCs +Q+ PAdQ−1AT

dP

=ATP + PA+Q−PB2BT2 P + (BT

2 P +KCs)(BT2 P +KCs) + PAdQ−1AT

dP

<0

Necessity. If there exists P > 0 and Q > 0 satisfies inequality (2.43), it is easy to

find there exists a scalar ρ > 0 such that

(A+B2KCs)TP + P(A+B2KCs) +Q+ PAdQ−1AT

dP +1

ρ2CTKTKC < 0.

i.e.

ATP+PA+Q−ρ2PB2BT2 P+(ρBT

2 P+1

ρKCs)(ρB

T2 P+

1

ρKCs)+PAdQ−1AT

dP < 0.

(2.45)

Obviously, equation (2.45) is equivalent to

ρ2ATP+ρ2PA+ρ2Q−ρ4PB2BT2 P+(ρ2BT

2 P+KCs)(ρ2BT2 P+KCs)+ρ

2PAd(ρ2Q)−1AT

dPρ2 < 0.

By substituting ρ2P with P and ρ2Q with Q, we obtain inequality (2.44).


Theorem 2.5. The closed-loop system (2.40) is asymptotically stable with w(t) = 0

and d ≥ 0 if the following inequality is satisfied:

PA + ATP +Q−XB2BT2 P − PB2B

T2 X + XB2B

T2 X (BT

2 P + KCs)T PAd

(BT2 P + KCs) −I 0

ATdP 0 −Q

< 0.

(2.46)

Proof. From Lemmas 2.3 and 2.4, we obtain that the closed-loop system (2.40) is

asymptotically stable with w(t) = 0 and d ≥ 0 if PA + ATP + Q − PB2BT2 P +

(BT2 P+KCs)

T (BT2 P+KCs)+PAdQ−1AT

dP < 0. BecauseXTB2BT2 P+PTB2B

T2 X−

XTB2BT2 X ≤ PTB2B

T2 P for any X and P of the same dimension, we get a suffi-

cient condition for the existence of stabilizing SOF gain matrix K given by

PA+ATP+Q−XB2BT2 P−PB2B

T2 X+XB2B

T2 X+(BT

2 P+KCs)T (BT

2 P+KCs)+PAdQ−1ATdP < 0.

Using the Schur complement, the theorem is proved.

Note that inequality (2.46) is not an LMI. Similar to Cao et. (1998), we may

develop an iterative LMI (ILMI) approach to find K, P > 0 Q > 0 to stabilize the

process (2.37) with SOF.

ILMI algorithm for SOF stabilizing control

Given process (2.37) stabilizable with SOF.

Step 1. Select Y > 0 and solve P from the following algebraic Riccati

equation

ATP + PA− PB2BT2 P + Y = 0.

set i = 1 and Xi = P .Step 2. Solve the following optimization problem for Pi,Qi,F and αi.

OP1: Minimize αi subject to the following LMI constraints

Σ (BT2 Pi +KCs)

T PiAd

(BT2 Pi + kCs) −I 0

ATdPi 0 −Qi

< 0, Pi = PTi > 0, (2.47)

where Σ = PiA+ATPi+Qi−XiB2BT2 Pi−PiB2BT

2 Xi+XiB2BT2 Xi−αiPi. Denote

α∗i as the minimized value of αi.


Step 3. If α∗i < 0, F is the stabilizing SOF control gain, stop. Otherwise go

to Step 4.

Step 4. Solving the following optimization problem for Pi and K with the

fixed Qi and αi obtained from Step 2.

OP2: Minimize trace(Pi) subject to the LMI constraints (2.47). Denote P∗i as

the optimal Pi.Step 5. If ‖XiB2−P∗

i B2‖ < ε, where ε is a prescribed tolerance, go to Step

6; Otherwise set i = i+ 1, Xi = P∗i , and go to Step 2.

Step 6. It cannot be decided by this algorithm whether the question is

solvable. Stop.

2.4.4 Suboptimal H∞ Control

In this subsection, sufficient conditions are developed for suboptimal H∞ control

for the process (2.37) with SOF. That is, the SOF control gain K is design to

simultaneously stabilize (2.37) and guarantee the H∞ norm bound γ of the closed-

loop transfer function Tyrw. An ILMI algorithm is also given to find such a SOF

controller.

Lemma 2.5. The closed-loop system (2.40) is asymptotically stable and ‖Tyrw‖∞ ≤γ, γ > 0 if there exists matrix P = PT > 0, Q = QT > 0 and K such that the

following condition is satisfied:

P(A+B2kCs) + (A+B2KCs)TP +Q+PAdQ−1AT

dP +CTr Cr + γ2PB1BT

1 P < 0.

(2.48)

Proof. Let us construct the matrices U(s) = [−sI − (A+B2KCs)−Ade−ds]−1 and

V (s) = Q+PAdQ−1ATP−edsATdP−e−dsPAd, one has Tyrw(s) = CrU(s)B1. Intro-

ducing the matrixM = −[P(A+B2KCs)+(A+B2KCs)TP+Q+PAdQ−1AT

dP+

CTr Cr + γ2PB1BT

1 P ], one obtains

PU−1(s) + [UT (−s)]−1P − V (s)− CTr Cr − γ2PB1BT

1 P −M = 0.


Pre-multiplying by UT (−s) and post-multiplying by U(s), gives

PU(s) + UT (−s)P − γ2UT (−s)PB1BT1 PU(s) = UT (−s)[V (s) + CT

r Cr +M ]U(s)

(2.49)

Further manipulating (2.49), one gets

BT1 PU(s)B1 +BT

1 UT (−s)PB1 − γ2B1rU

T (−s)PB1BT1 PU(s)B1 − γ2I

=− γ2I +BT1 U

T (−s)[V + CTr Cr +M ]U(s)B1.

It follows that

− [γI − γ−1BT1 PU(−s)B1]T [γI − γ−1BT

1 PU(s)B1]

=[BT1 U

T (−s)CTr CrU(s)B1]− γ2I +BT

1 UT (−s)[V +M ]U(s)B1.

Noting that −[γI − γ−1BT1 PU(−s)B1]T [γI − γ−1BT

1 PU(s)B1] ≤ 0, we obtain

T Tyrw

(−s)Tyrw(s)− γ2I +BT1 U

T (−s)[V +M ]U(s)B1 ≤ 0.

According to Mahmoud and Zribi (1999), we get V ≥ 0, M ≥ 0. Thus we conclude

that ‖Tyrw‖∞ < γ.

Theorem 2.6. The closed-loop system (2.40) is asymptotically stable and ‖Tyrw‖∞ ≤γ, γ > 0 if there exist matrices P = PT > 0, Q = QT > 0 and K such that the

following matrix inequality is satisfied:

Σ (BT2 P +KCs)

T PAd CTr PB1

BT2 P +KCs −I 0 0 0

ATdP 0 −Q 0 0

Cr 0 0 −I 0

BT1 P 0 0 0 −γ2I

< 0, (2.50)

where Σ = PA+ ATP +Q−XB2BT2 P − PB2BT

2 X +XB2BT2 X.

Proof. By Lemmas 2.4 and 2.5 and note the fact that XTB2BT2 P + PB2BT

2 X −XB2B

T2 X ≤ PTB2B

T2 P , the theorem follows.


ILMI algorithm for SOF suboptimal H∞ control

Given process (2.37) stabilizable with SOF and γ > 0.

Step 1. Select Y > 0 and solve P from the following algebraic Riccati

equation

ATP + PA− PB2BT2 P + Y = 0.

set i = 1 and Xi = P .Step 2. Solve the following optimization problem for Pi,Qi,K and αi.

OP1: Minimize αi subject to the following LMI constraints

Σ (BT2 Pi +KCs)

T PiAd CTr PiB1

BT2 Pi +KCs −I 0 0 0

ATdPi 0 −Qi 0 0

Cr 0 0 −I 0

BT1 Pi 0 0 0 −γ2I

< 0, Pi = PTi > 0,

(2.51)

where Σ = PiA+ATPi+Qi−XiB2BT2 Pi−PiB2BT

2 Xi+XiB2BT2 Xi−αiPi. Denote

α∗i as the minimized value of αi.

Step 3. If α∗i < 0, K is the stabilizing SOF control gain, stop. Otherwise

go to Step 4.

Step 4. Solving the following optimization problem for Pi and K with the

fixed Qi and αi obtained from Step 2.

OP2: Minimize trace(Pi) subject to the LMI constraints (2.51). Denote P∗i the

optimal Pi.Step 5. If ‖XiB2−P∗

i B2‖ < ε, where ε is a prescribed tolerance, go to Step

6; Otherwise set i = i+ 1, Xi = P∗i , and go to Step 2.

Step 6. It cannot be decided by this algorithm whether the question is

solvable. Stop.

2.4.5 Control Design with PI Controllers

In this subsection, we investigate the design problem of PI controllers based on the

results in the preceding two subsections.


Consider the process in equation (2.37) and PI controller in equation (2.39),

let z1 = x, z2 =∫ t

0ysdt. Denote z = [zT1 , z

T2 ]

T , the dynamic of augmented process

model is described by

z1 =x = Ax(t) + Adx(t− d) +B2u(t) +B1w(t),

z2 =Csx(t).

The output of the augmented system is written as

yr(t) = [Cr 0]

z1(t)

z2(t)

, ys(t) =

Cs 0

0 I

z1(t)

z2(t)

.

and the PI controller becomes

u(t) = [K1 K2]ys(t).

The close-loop system with PI controller is described as

z(t) =Az(t) + Adz(t− d) + B2u(t) + B1w(t),

yr(t) =B1z(t), ys(t) = Csz(t),

u(t) =Kys(t),

where A =

A 0

C 0

, Ad =

Ad

0

, B2 =

B2

0

, Cr = [Cr 0], Cs =

Cs 0

0 I

and

K = [K1 K2]. Thus, the problem of PID controller design reduces to that of SOF

controller design problem.

2.4.6 A Numerical Example

Consider a system with state time delay is described by

x(t) =Ax(t) + Adx(t− d) +B2u(t) +B1w(t),

yr(t) =Crx(t), ys(t) = Csz(t),

where A =

−3 2 0

1 −2 −13 0 6

, Ad =

1 0 1

1 −1 0

0 0 2

, B2 =

1 1

4 0

2 0

, B1 =

0.1

0.2

0.6

, Cs =

0 1 0

1 0 1

, Cr =[

1 0 1]

and d = 0.1. Using the proposed algorithm, we can


obtain SOF and PI controllers corresponding to different performance specifications

accordingly. The results are summarized in Table 2.1.

Remark 2.6. The disturbance signal w(t) is assumed to be zero for stability proof.

However, for the linear time-delay systems discussing in this section, the distur-

bance signal w(t) will not change the stability of closed-loop system if we may

choose stabilizing control. Moreover, the input signal will also not change the

Hinfty of transfer function between the disturbance input and the output. Thus,

for any inputs, our design may satisfy the objective if we may find such a controller.

Table 2.1. SOF and PI controller and their performance

Problem Feedback matrices Performance required Actual Performance Iteration number α

SOF (stab) K =

22.0090 −65.2154

−0.7559 −2.1137

Stability Stable 3 -0.3170

PI (stab) K1 =

9.6587 −30.8859

37.5367 −28.5601

Stability Stable 3 -0.0004

K2 =

2.5154 −9.8625

14.2639 −10.8638

SOF (H∞) K =

16.8052 −51.2591

−0.3142 −3.4181

H∞ < 2 H∞ = 0.1140 3 -0.0481

PI (H∞) K1 =

16.8748 −56.2466

6.9850 −25.4804

H∞ < 2 H∞ = 0.2484 9 -0.0001

K2 =

65.9504 −49.3472

32.4589 −24.2961

2.4.7 Conclusions

In this subsection, algorithms based on the ILMI technique have been developed

to design SOF and PI controller for linear time invariant systems with state time

delay, which guarantee the stability of closed-loop system or H∞ performance spec-

ification. An example is provided to illustrate the effectiveness of the proposed


Table 2.2. P matrix in SOF and PI controller design

Problem P

SOF (stab)

136.2660 −45.0290 126.2902

−45.0290 15.6715 −42.5981

126.2902 −42.5981 127.7296

PI (stab)

62.8501 −19.9052 53.9336 −82.1373 62.5272

−19.9052 6.9235 −17.4935 26.4612 −20.1282

53.9336 −17.4935 55.0967 −70.6262 53.7021

−82.1373 26.4612 −70.6262 115.4461 −87.8724

62.5272 −20.1282 53.7021 −87.8724 66.9912

SOF (H∞)

97.6294 −31.2270 87.7795

−31.2270 10.7198 −28.8752

87.7795 −28.8752 89.2653

PI (H∞)

93.0279 −28.4674 83.6846 −112.2981 84.0530

−28.4674 9.3379 −25.7822 34.7012 −25.9644

83.6846 −25.7822 85.7134 −102.4361 76.6289

−112.2981 34.7012 −102.4361 142.8461 −106.9121

84.0530 −25.9644 76.6289 −106.9121 80.0845

method.

Chapter 3

Advance in Robust IMC Design

for Step Input and Smith

Controller Design for Unstable

Processes

sectionPreview Most of control designs are based on the model of process. How-

ever, the model is never accurate. Thus, it is of great practical importance that the

controller performs well when the model differs from the real process. The Internal

model control (IMC) provides a simple yet effective structure for robust controller

design (Morari and Zafiriou, 1989), thus IMC structure is employed widely in the

robust control system design. Boulet et al. (2002) developed IMC robust tunable

control based on the performance robustness bounds of the system and knowledge

of the plant uncertainty’s weighting function. Chiu et al. (2000) developed IMC

for transition control with uncertain process. Litrico (2002) proposed robust IMC

flow control for dam-river open-channel systems, in which the robustness is esti-

mated with the use of a bound on multiplicative uncertainty taking into account

the model errors, due to the nonlinear dynamics of the system. However, most of

the existing control design with IMC structure is based on trial and error method.

Moreover, the existing robust controller usually is not optimal for nominal perfor-

48

Chapter 3. Advance in Robust IMC Design for Step Input and Smith ControllerDesign for Unstable Processes 49

mance. It is well known that a Smith predictor can be put into an equivalent IMC

structure. However, the performance of disturbance rejection for existing methods

with unstable processes is not satisfactory.

In this chapter, two new robust IMC control schemes, namely time domain

approach and frequency domain approach, are developed respectively. In the time

domain approach, the IMC design is established through state space description

and LMI optimization. The IMC control design is formulated into an output feed-

back design problem. After appropriate transformation, it is shown that the output

feedback design problem can be solved via available method. In the frequency do-

main method, it is shown that the upper bound and lower bound of the solution

may be found via LMI optimization for the controller with fixed poles and the

upper bound and lower bound of the solution may be found via bilinear matrix

inequalities (BMI) optimization for the controller with general form. Moreover,

the optimal solution can be approximated by the upper bound and lower bound

with arbitrary accuracy. For the modified Smith predictor design, a new struc-

ture is proposed to improve the performance of disturbance rejection for unstable

processes. There are altogether four controllers in the proposed modified Smith

structure, which are designed for disturbance rejection, stability of the time delay

loop, closed-loop dominant pole and setpoint performance.

The chapter is organized as follows. In Section 3.1 a new robust IMC design in

time domain is presented. Section 3.2 proposes an IMC design in frequency domain.

Section 3.3 describes a modified Smith predictor design for unstable processes.

3.1 Robust IMC Design via Time Domain Ap-

proach

3.1.1 Introduction

The IMC is a well-established control scheme in industry control, which is discussed

in the textbook of Morari and Zafiriou (1989) in details. The IMC structure pro-


vides an effective approach in analyzing control system performance and robustness

(Zhu et al., 1995). Moreover, unity feedback control, linear quadratic optimal con-

troller and Smith predictor can be equivalently transformed into the general IMC

form (Garcia and Morari, 1982). Thus, the IMC structure is an important tool

for robust controller design. When the model is perfect, the IMC system becomes

an open-loop system and the problem of controller design and stability analysis

becomes transparent. However, the model usually is not perfect. The model error

gives rise to feedback in the true sense and leads to possible stability problem. It

gives rise to the problem of robust stability (Morari and Zafiriou, 1989). In the

existing IMC design, the procedure usually consists of two step. The first step is

to yield good nominal performance. The second step is to achieve the desired ro-

bustness by detuning the low-pass filter with a trial and error method. It is noted

that with the filter detuning, the performance is no more optimal in any sense.

In this section, we propose a new approach to design the IMC controller in order

to achieve the optimal nominal performance under the robust stability constraints.

The IMC design problem is converted into a H2/H∞ multiobjective output feed-

back control design problem. The controller is obtained via solving a system of

LMIs. Two examples are provided to illustrate the effectiveness of the method.

3.1.2 The Robust IMC Design

y

r +

Q G

G ˆ

u +

+

Figure 3.1. Internal Model Control

Consider an IMC system shown in Figure. 3.1, where G is a stable process to


be controlled, G is the model of the process and Q is the controller to be designed.

Without considering the model uncertainty, the H2 optimal control design is to

determine Q such that the Integral-Square-Error (ISE)

‖e(t)‖22 =∫ ∞

0

e2(t)dt.

is minimized for a particular input r. Using Parseval’s theorem, one obtains the

description of ISE in frequency domain

‖e(t)‖22 = ‖E(jω)‖22 =1

2π

∫ ∞

−∞

|E(jω)|2dω

Note that E(jω) = [1−G(jω)Q(jω)]R(jω), one gets

‖e(t)‖22 = ‖[1−G(jω)Q(jω)]R(jω)‖22

For any practical process, the model is never perfect. In this case, the H2 optimal

control is usually not a desired scheme because it cannot guarantee the stability

of the uncertain system. Thus, we aim to develop a control scheme to develop an

approach to achieve optimal ISE for nominal model and guarantee the stability

of the whole closed-loop system. Suppose that the set of the uncertain model is

described by

Π = G : |G(iω)− G(iω)

G(iω)| < lm(ω). (3.1)

In order to achieve robust stability of the closed-loop system in Fig. 3.1, the fol-

lowing condition should be satisfied (Morari and Zafiriou, 1989):

‖GQlm‖∞ < 1 (3.2)

Thus, the robust IMC design problem becomes

minQ‖[1−G(jω)Q(jω)]R(jω)‖22 (3.3)

subject to (3.2).

Here, for the first time, we proposed a new scheme to convert the robust IMC design

problem (3.3) into a multi-objective optimization problem with dynamic output

feedback. There are methods available to solve the multi-objective optimization


y

r

Gˆ

u

K

+

s

11

2Z

ml

infZ

e

v

Figure 3.2. Robust IMC design

problem with dynamic output feedback via LMI optimization. Suppose the input

of interest r is a unity step and the bound of uncertainty model can be described

by transfer function lm(s), the new system established for robust IMC design is

depicted in Fig. 3.2. We observe that our robust IMC design problem is to find a

dynamic output feedback controller Q in order to minimize the H2 norm between

r and Z2 on the condition that the H∞ norm between r and Zinf is less than 1.

Now let us analysis the proposed new system in Fig. 3.2. Suppose that the state

space description of process model G is

xG = AGxG +BGuG

yG = CGxG +DGuG, (3.4)

and the state space model of lm

xl = Alxl +Blul

yl = Clxl +Dlul. (3.5)

The block 1sin Fig. 3.2 represents the unity step input. However, it makes the

constructed new system unstable, thus it is difficult to compute the controller using

available multiobjective optimization method described in next subsection. Here,

we introduce a small positive number ε and let 1sbecome 1

s+ε. For the element 1

s+ε,

we get the state space description as

xr = εxr + ur

yr = xr. (3.6)


By observing Fig. 3.2, we obtain the following relationship that uG = u, ul = yG,

ur = e, yr = z2, yl = zinf and r = y. Substitute these relationships into (3.4), (3.5)

and (3.6), we get the state equation of the open-loop system without controller Q

in Fig. 3.2 as

xG

xl

xr

=

AG 0 0

BlCG Al 0

−CG 0 ε

xG

xl

xr

+

0 BG

0 BlDG

1 −DG

r

u

, (3.7)

and the output equation as

Z2

Zinf

y

=

0 0 1

DlCG Cl 0

0 0 0

xG

xl

xr

+

0 0

0 DlDG

1 0

r

u

. (3.8)

Let T2 denote the closed-loop transfer function from r to Z2 and Tinf denote the

closed-loop transfer function from r to Zinf . Our goal is to compute a dynamical

output feedback controller

xQ = AQxQ +BQy

u = CQxQ +DQy

such that min ‖T2‖2 subject to ‖Tinf‖∞ < 1. Here we note that the original robust

IMC design problem is converted into a dynamic output feedback control problem.

Remark 3.1. Usually we may only get the amplitude of the uncertain bound at

our interested frequency points. However, we have to obtain the transfer function

description of uncertain bound in order to use our proposed method to solve the

problem. Fortunately, for our propose approach, we just need the amplitude of

the transfer function is same as that of the uncertain bound at each frequency

point. The reason is that the phase of the uncertain bound is irrelevant with

robust stability of the IMC system. In our design, we may only minimal phase

system to describe the uncertain bound. For minimal phase linear system, if we

have the amplitude of frequency response, we may determine the phase at whole

frequency band.


3.1.3 LMI Solution

It is shown (Scherer et al., 1997) that the dynamic output feedback control problem

to meet some specifications can be solved via LMI optimization. In this section,

the specifications under consideration include H∞ and H2 performance. The state

space equation of closed-loop system T is given by

x = Acx+Bcr

z = Ccx+Dcr.

According to Bounded Real Lemma (Boyd et al., 1994), Ac is stable and the H∞

norm of T is smaller than γ if and only if there exists a symmetric P with

ATc P + PAc PBc CT

c

BTc P −γI DT

c

Cc Dc −γI

< 0, P > 0.

Regarding H2 norm, the following result is obtained from Scherer et al. (1997). Ac

is stable and ‖T‖22 < ν if and only if there exist symmetric matrix P and Q such

that

ATc P + PAc PBc

BTc P −I

< 0,

P CTc

Cc Q

> 0,

T race(Q) < ν, Dc = 0,

where Tr(·) is the trace of matrix. In practice, the closed-loop system is usually not

much faster than the dynamic of the process (Morari and Zafiriou, 1989). Thus, we

do not wish the poles of closed-loop system are too fast. So we may limit the closed-

loop eigenvalues in some specified LMI regions. The problem around closed-loop

pole placement in LMI regions is studied by Chilali and Gahinet (1996). The matrix

Ac with all its eigenvalues in the LMI region z ∈ C : [lij +mijz +mjiz]i,j < 0 if

and only if there exists a symmetric P such that

[lijP +mijATc P +mjiPAc]i,j < 0.


In our design, let the Lyapunov matrix P in H∞ norm constraint denote as P1, letthe P in H2 norm optimization as P2 and the P in pole constraints as P3. Accord-ing to the result of Scherer et al. (1997), we must require that all specifications

are forced by a single closed-loop Lyapunov function to satisfy the convexity of

multiobjective optimization problem. Thus, we need to set P1 = P2 = P3 = P in

our computation. Consider (3.7) and (3.8) and denote

A =

AG 0 0

BlCG Al 0

−CG 0 ε

, Bin =

0

0

1

, Bfb =

BG

BlDG

−DG

C2 =[

0 0 1]

, Cinf =[

DlCG Cl 0]

,

Dinf = DlDG,

we obtain the realization of T2 and Tinf as

A =

A BfbCK

0 AK

,B2 =

Bin +BfbDK

BK

C2 =[

C2 0]

,D2 = 0,

and

A =

A BfbCK

0 AK

,B =

Bin +BfbDK

BK

Cinf =[

Cinf 0]

,Dinf = DinfDK ,

respectively. Following the approach in Scherer et al. (1997), partition P and P−1

as

P =

Y N

NT ?

,P−1 =

X M

MT ?

,

and let

A = NAKMT + Y BfbCKM

T + Y AX,

B = NBK + Y BfbDK ,

C = CKMT ,

D = Dk,


the original robust IMC design problem (3.3) becomes the optimization problem

with LMI constraints described by

minQ

ν,

subject to

AX +XAT +Bbf C + (Bbf C)T ? ? ?

A+ AT ATY + Y A ? ?

(Bin +BfbD)T (Y Bin + B)T −γI ?

CinfX +Dinf C Cinf DinfD −γI

< 0, (3.9)

AX +XAT +Bbf C + (Bbf C)T ? ?

A+ AT ATY + Y A ?

(Bin +BfbD)T (Y Bin + B)T −I

< 0,

X ? ?

I Y ?

C2X C2 Q

> 0, T race(Q) < ν, (3.10)

ljk

X I

I Y

+mjk

AX +BfbC A

A Y A

T

+

mkj

AX +BfbC A

A Y A

< 0, (3.11)

where ? replaces blocks that are readily inferred by symmetry.

3.1.4 Simulation

In this subsection, two examples are given to show the validity of the proposed

approach.

Example 3.1. Consider a multi-lag non-minimum phase process with its nominal

model (Wang and zhang, 1998a)

G(s) =1− 0.3s

(s+ 1)(0.8s+ 1)(0.4s+ 1),


and the uncertain bound is given by

lm(s) =2s

s+ 1.

ε in (3.6) is chosen as 0.00001. The fastest open loop pole is 2.5. we limit the

closed-loop poles less than 10 times The fastest open loop pole. In the simulation,

we set the feasible closed-loop pole area as the disk with center at original point

and radius as 25.

We are interested in the H∞ performance from r to zinf and H2 performance

from r to z2. The optimal H2 performance from r to z2 is 0.8191 and the controller

is

Q(s) =100s5 + 8175s4 + 44880s3 + 89420s2 + 76270s + 23660

s5 + 34.03s4 + 748.1s3 + 10290s2 + 33230s + 23660, (3.12)

the step response and control signal for nominal process are given in Fig. 3.3.

When using the controller K(s) in (3.12) for the closed-loop system, the H∞ per-

formance from r to z∞ is 2.5510. It does not satisfy the robust stability condition.

If the real process is G(s) = G(s)+G(s)lm(s), the controller in (3.12) really cannot

stabilize the IMC system.

Now consider the controller design with robust stability constraints. The re-

sulting controller with proposed method is

Q(s) =100s5 + 5917s4 + 31880s3 + 63090s2 + 53680s + 16750

s5 + 45.11s4 + 1085s3 + 14220s2 + 40810s + 16750. (3.13)

The obtained H2 norm is 0.9715 and the step response for the nominal process is

shown in Fig. 3.4. The response is more sluggish than the response in Fig. 3.3.

However, the H∞ norm is 0.9764, which is really less than 1. The IMC system

should be stable if the model uncertainty is in the given bound. The step response

with the process G(s) = G(s) + G(s)lm(s) is given in Fig. 3.5. It is shown that

the controller in (3.13) really stabilize such a process.

Example 3.2. Consider a oscillatory process with dead time

G(s) =1

(s2 + 2s+ 3)(s+ 3)e−0.3s.


0 5 10 15−1

−0.5

0

0.5

1

1.5

time

pla

nt

ou

tpu

t

0 5 10 15−150

−100

−50

0

50

100

150

200

time

co

ntr

ol a

ctio

n

Figure 3.3. Nominal step response for H2 optimal design

0 5 10 15−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time

pla

nt

ou

tpu

t

0 5 10 15−100

−50

0

50

100

150

time

co

ntr

ol a

ctio

n

Figure 3.4. Nominal step response for robust IMC design

Using first order Pade approximation, we can use the proposed method to design

the controller. Suppose the uncertain bound is also

lm(s) =2s

s+ 1,

and ε in (3.6) is chosen as 0.00001. The optimal H2 norm is 0.6621, and the

controller is


0 5 10 15−2

−1

0

1

2

3

time

pla

nt

ou

tpu

t

0 5 10 15−100

−50

0

50

100

150

time

co

ntr

ol a

ctio

n

Figure 3.5. Step response of Robust design for process with mismatch

Q(s) =99.97s6 + 300700s5 + 3798000s4 + 16180000s3

s6 + 53.49s5 + 1844s4 + 41160s3

+33350000s2 + 38580000s + 17920000

+541500s2 + 2492000s + 1991000.

The the H∞ performance from r to z∞ is 2.5816. For robust design, the controller

is

Q(s) =99.97s6 + 150400s5 + 1894000s4 + 8061000s3

s6 + 61.8s5 + 2154s4 + 45940s3

+16610000s2 + 19220000s + 8947000

+552900s2 + 2306000s + 994100.

The relevant H2 norm and H∞ norm are 0.8238 and 0.9563 respectively.

3.1.5 Conclusions

A new approach for robust IMC controller design is proposed. It is shown the

IMC optimal H2 performance design with robust stability constraints is converted

into dynamic output feedback control design problem. It is shown that the robust

IMC controller design is solvable via solving a system of LMI. Two examples are

presented with the demonstrated effectiveness of the proposed method.


3.2 Robust IMC Controller Design via Frequency

Domain Approach

3.2.1 Introduction

In last section, a robust IMC design framework is developed which aims to mini-

mize the integral square error (ISE) for setpoint step input with the robust stability

as a constraint. Using a time domain approach, we have converted the IMC design

problem into output feedback design method and solved using existing LMI opti-

mization. However, the above method cannot cope with the processes with time

delay. Moreover, the obtained control is usually of high order. Thus, we attempt

to solve the IMC design problem in frequency domain.

When the IMC design problem is formulated in frequency domain, it becomes

a nonlinear and nonconvex optimization problem which is difficult to be solved

directly. In this section, we discuss two classes of controllers with fixed pole and

with general form. For the controller design with fixed poles, the optimal solution

is approximated by its upper bound and lower bound which are then solved by

LMI optimization. For the controller design with general form, a branch and

bound method is employed to obtain the upper bound and the lower bound of the

optimal solution. It is shown that the optimal solution can be approximated by

the upper bound and the lower bound with any accuracy. Examples are provided

to show the effectiveness of the proposed method.

3.2.2 IMC Design Review and New Formulation

The schematic of the IMC system is depicted in Fig. 3.1 , where G is the given

stable process to be controlled, G the model of the process, and K(s) the IMC

controller. It follows from Morari and Zafiriou (1989) that the design procedure is

as follows. The model is factorized as

G(s) = G−(s)G+(s), (3.14)


where G−(s) has the stable and minimal phase part of G(s) only and G+(s) has

all the time delay and right half plane zeros of G(s):

G+(s) = e−Ls(∏

i

1− βis

1 + βis), Re(βi) > 0. (3.15)

The controller K(s) takes the form:

Q = G−1− f, (3.16)

where f is a user-specified filter and is usually chosen as

f =1

(τs+ 1)n, (3.17)

where n is large enough to guarantee that K(s) is proper.

The IMC controller design is to achieve the optimal nominal performance

minQ‖(1− GQ)v‖22 (3.18)

subject to the constraints that the IMC system is robustly stable and K is causal

and stable, where v is a set-point input of interest. Assume that the family of the

stable plants Π is described by

Π = G : |G(iω)− G(iω)

G(iω)| < lm(ω). (3.19)

It is shown that the IMC system is robustly stable if and only if

‖GQlm‖∞ = supω

|GQlm| < 1. (3.20)

In short, one solves (3.18) subject to (3.20). However, the problem is difficult

to be solved directly. In practice, (3.18) is solved first and then τ is used as a

tuning parameter to achieve the appropriate compromise between robustness and

performance. A trial and error method is used to find τ .

In the following, we try to find a new framework to design the controller K(s)

to achieve optimal performance with the robust stability constraint for a given

uncertain bound. With the same objective function as in that of Morari’s design,

the robust IMC design problem is now recast into

minQq

γ, (3.21)


subject to∣

∣

∣

∣

∣

∣(1− GQq)v∣

∣

∣

∣

∣

∣

2

2< γ, (3.22)

∣

∣

∣

∣

∣

∣lmGQq∣

∣

∣

∣

∣

∣

2

∞< 1. (3.23)

Suppose that the nominal model of the process is described by

G(s) =β(s)

α(s)=

βnsn + βn−1s

n−1 · · ·+ β1s+ β0αnsn + αn−1sn−1 + · · ·+ α1s+ α0

e−Ls, (3.24)

and the controller takes the general form:

Q(s) =b(s)

a(s)=

bmsm + bm−1s

m−1 · · ·+ b1s+ b0amsm + am−1sm−1 + · · ·+ a1s+ 1

. (3.25)

Our task is to find the optimal controller Q(s) in (3.25) to solve the problem (3.21)-

(3.23). In the subsequent two subsections, two schemes are presented respectively.

The first scheme deals with controller design with fixed poles, while the second

treats controller design without this restriction.

3.2.3 Controller Design with Fixed Poles

In the process control practice, the closed-loop bandwidth ωcb can rarely exceed

ten times of the open loop process bandwidth ωpb (Morari and Zafiriou, 1989).

Usually, the desired closed-loop bandwidth is chosen as ωcb = µωpb, µ ∈ [0.1, 10].

Let the denominator of controller be a(s) = (τs+ 1)n, so that

Q(s) =bms

m + bm−1sm−1 · · ·+ b1s+ b0

(τs+ 1)m. (3.26)

We may relate the closed-loop time constant to the closed-loop bandwidth accord-

ing the approach in (Wang et al., 2001):

τ =

√

m√2− 1

µωpb, µ ∈ [0.1, 10]. (3.27)

To ensure the robust stability will naturally reduce the closed-loop performance or

speed, the bandwidth of controller should be more narrow than the process.

Performance Consider the performance requirements (3.21) and (3.22).

Suppose that v is a unit step.

∣

∣

∣

∣

∣

∣(1− GQ)v∣

∣

∣

∣

∣

∣

2

2=

1

π

∫ ∞

0

∣

∣

∣

∣

∣

G(jω)b(jω)− a(jω)

(jω)a(jω)

∣

∣

∣

∣

∣

2

dω. (3.28)


Write G = Gre + jGim, where Gre and Gim are real and imaginary parts of G

respectively. It follows that

Gb(jω) =Gre[b0 − b2ω2 + b4ω

4 + · · · ]− ωGim[b1 − b3ω2 + b5ω

4 + · · · ]+

jGim[b0 − b2ω2 + b4ω

4 + · · · ] + jωGre[b1 − b3ω2 + b5ω

4 + · · · ],

and

a(jω) = 1 + C1m(jτω) + C2m(jτω)2 + · · ·+ Cm

m (jτω)m,

where Ckm = m!

(m−k)!k!. In order to achieve zero steady-state error for the nominal

IMC system, we set b0 = G−1(0). We thus obtain

∣

∣

∣Gb(jω)− a(jω)

∣

∣

∣

2

=Gre[−b2ω2 + b4ω4 + · · · ]− ωGim[b1 − b3ω

2 + b5ω4 + · · · ] + [C2mτ

2ω2 − C4mτ4ω4 + · · · ]2+

Gim[−b2ω2 + b4ω4 + · · · ] + ωGre[b1 − b3ω

2 + b5ω4 + · · · ] + [−C1mτω + C3mτ

3ω3 + · · · ]2

For simplicity, we let x0 = C2mτ2ω2−C4mτ 4ω4+· · · , y0 = −C1mτω+C3mτ 3ω3+· · · ,

x1 = −ωGim, y1 = ωGre, x2 = ω2Gre, y2 = −ω2Gim, · · · . Thus, we obtain

∣

∣

∣Gb(jω)− a(jω)∣

∣

∣

2

=[x0(ω) + b1x1(ω) + b2x2(ω) + · · · ]2 + [y0(ω) + b1y1(ω) + b2y2(ω) + · · · ]2

=[

1 b1 · · ·]

x20(ω) + y20(ω) x0(ω)x1(ω) + y0(ω)y1(ω) · · ·x0(ω)x1(ω) + y0(ω)y1(ω) x21(ω) + y21(ω) · · ·

......

. . .

1

b1...

.

Let θ = [1 b1 · · · bm]T and

P =

x20(ω) + y20(ω) x0(ω)x1(ω) + y0(ω)y1(ω) · · ·x0(ω)x1(ω) + y0(ω)y1(ω) x21(ω) + y21(ω) · · ·

......

. . .

,

which is symmetric and positive matrix. (3.28) is then written as

∣

∣

∣

∣

∣

∣(1− GQ)v∣

∣

∣

∣

∣

∣

2

2=

1

π

∫ ∞

0

θT P(ω)θω2(τ 2ω2 + 1)m

dω = θT

[

1

π

∫ ∞

0

P(ω)ω2(τ 2ω2 + 1)n

dω

]

θ.

(3.29)


Let

P =1

π

∫ ∞

0

P(ω)ω2(τ 2ω2 + 1)n

dω, (3.30)

(3.22) is converted, by Schur complement (Boyd et al., 1994), into

γ b

bT P−1

> 0. (3.31)

Thus, the optimal performance design is to solve (3.21) subject to (3.31). For a

fixed τ , (3.31) becomes an LMI.

Robustness For the uncertain process described by (3.19), if both G(s)

and G(s) are asymptotically stable, the robust stability condition is (3.23). With

the controller in (3.26) and b0 = G−1(0), we obtain

|b(jω)|2 = |G−1(0) + b1(jω) + b2(jω)2 + · · · |2

= |G−1(0)− b2ω2 + · · · ]2 + ω2[b1 − b3ω

2 + · · · ]2

=[

1 b1 · · ·]

G−2(0) −G−1(0)ω2 · · ·−G−1(0)ω2 ω4 · · ·

......

. . .

1

b1...

.

Let θ = [1 b1 · · · bm]T and

Q(ω) =

G−2(0) −G−1(0)ω2 · · ·−G−1(0)ω2 ω4 · · ·

......

. . .

, (3.32)

where Q(ω) is a symmetric and positive definite matrix. Thus, one has

∣

∣

∣

∣

∣

∣lm(ω)G(jω)Q(jω)∣

∣

∣

∣

∣

∣

2

∞= max

ω∈(0,∞)

∣

∣

∣

∣

∣

lm(ω)G(jω)b(jω)

a(jω)

∣

∣

∣

∣

∣

2

= maxω∈(0,∞)

θT |lm(ω)G(jω)|2Q(ω)θ

ω2(τ 2ω2 + 1)m. (3.33)

Let Q = |lm(ω)G(jω)|2Q(ω)ω2(τ2ω2+1)m

, (3.33) is

maxω∈(0,∞)

θTQ(ω)θ. (3.34)


We notice that (3.34) is a complex nonlinear function with respect to ω and θ even

if τ is fixed. Thus, the exact solution with constraint (3.23) is hard to be found.

However, we can approximate the constraint (3.23) with finite frequency points ωi,

i = 1, · · · , l. Then (3.34) becomes

θTQ(ωi)θ < 1, i = 1, · · · , l. (3.35)

With such a finite point approximation, we notice that the constraint (3.23) is less

strict, thus the lower bound of the optimal solution may be found with the relaxed

constraint (3.35). If we increase l, the lower bound will get closer to the global

optimal solution. However, in this way, we do not know how large the l should be

and if the approximate solution is good enough. Fortunately, we can also find the

upper bound of the optimal solution with some assumption. If bi ≥ 0, i = 1, · · · ,m,

maxω∈(0,∞)

[bTQ(ω)b] < bT [Qmax(ω)]b, where

Qmax(ω) =

maxω∈(0,∞)

Q11 maxω∈(0,∞)

Q12 · · ·

maxω∈(0,∞)

Q21 maxω∈(0,∞)

Q22 · · ·...

.... . .

.

Notice that bT [Qmax(ω)]b < 1 is equivalent to bT [Qmaxi (ω)]b < 1, i = 1, · · · , l + 1,

where

Qmaxi (ω) =

maxω∈(ωi−1,ωi)

Q11 maxω∈(ωi−1,ωi)

Q12 · · ·



Q22 · · ·...

.... . .

,

with ω0 = 0 and ωm+1 =∞. Thus, the constraint (3.23) can be also approximated

by

bT Qmaxi (ω) b < 1. i = 1, · · · , l, l + 1. (3.36)

Since it is a more strict constraint than (3.23). Thus we can obtain the upper

bound of the problem (3.21). Similarly, if we increase l, the the upper bound will

get closer to the global optimal solution. If the error between the upper and lower

bound is small enough, we obtain the good approximate to the optimal solution

with a desired accuracy.


According to Schur complement, (3.35) and (3.45) are converted respectively

into LMIs:

1 b

bT Q(ωi)−1

> 0, i = 1, · · · , l, (3.37)

and

1 b

bT [Qmaxi (ω)]−1

> 0, i = 1, · · · , l, l + 1. (3.38)

Algorithm 3.1. Given G(s) in (3.24), seek Q(s) in (3.26) to minimize (3.21)

subject to (3.22) and (3.23).

Step 1 Compute τ in the denominator of the controller from (3.27).

Step 2 Calculate P and Q(ω) from (3.30) and (3.32) respectively.

Step 3 Select a set of frequency points ωi, i = 1, · · · , l.Step 4 Find the lower bound γ: minimize γ subject to (3.31) and (3.37).

Step 5 Find the upper bound γ: minimize γ subject to (3.31) and (3.38).

Step 6 If γ − γ < ε, where ε is given error bound, stop. Otherwise, increase

l, choose more frequency points and go to Step 4.

3.2.4 Controller Design with General Form

In the preceding subsection, the denominator of controller is chosen based on the

open loop bandwidth. It is obvious that it may yield a good controller but usually

not the optimal one. In this section, we take the controller Q in a general form in

(3.25). Thus if the global optimal solution is found, the controller will be the best

among all the controller of the same order for the IMC system.

Performance In order to achieve optimal performance, we should minimize

γ subject to (3.22). Suppose that v is a unit step, we have

∣

∣

∣

∣

∣

∣(1− GQ)v∣

∣

∣

∣

∣

∣

2

2=

1

π

∫ ∞

0

∣

∣

∣

∣

∣

G(jω)b(jω)− a(jω)

(jω)a(jω)

∣

∣

∣

∣

∣

2

dω. (3.39)


Write G = Gre + jGim, where Gre and Gim are real and imaginary parts of G

respectively. It follows that

Gb(jω) =Gre[b0 − b2ω2 + b4ω

4 + · · · ]− ωGim[b1 − b3ω2 + b5ω

4 + · · · ]+

jGim[b0 − b2ω2 + b4ω

4 + · · · ] + jωGre[b1 − b3ω2 + b5ω

4 + · · · ],

and

a(jω) = 1 + a1(jω) + a2(jω)2 + · · ·

In order to achieve zero steady-state error for the nominal IMC system, we set

b0 = G−1(0). We thus obtain

∣

∣

∣Gb(jω)− a(jω)

∣

∣

∣

2

=Gre[−b2ω2 + b4ω4 + · · · ]− ωGim[b1 − b3ω

2 + b5ω4 + · · · ] + [a2ω

2 − a4ω4 + · · · ]2+

Gim[−b2ω2 + b4ω4 + · · · ] + ωGre[b1 − b3ω

2 + b5ω4 + · · · ] + [−a1ω + a3ω

3 + · · · ]2

=ω2[Gimb1 + Greωb2 − Gimω2b3 − Greω

3b4 + Gimω4b5 + · · · − ωa2 + ω3a4 + · · · ]2+

[Greb1 + Gimωb2 − Greω2b3 − Gimω

3b4 + Greω4b5 + · · ·+ a1 − ω2a3 + · · · ]2

For simplicity, let xb1 = Gim, xb2 = Greω, · · · , xa1 = 0, xa2 = −ω, · · · , yb1 = Gre,

yb2 = Gimω, · · · , ya1 = 1, ya2 = 0, · · · , we obtain

∣

∣

∣Gb(jω)− a(jω)∣

∣

∣

2

ω2

=[a1xa1(ω) + a2x

a2(ω) + · · ·+ b1x

b1(ω) + b2x

b2(ω) + · · · ]2+

[a1ya1(ω) + a2y

a2(ω) + · · ·+ b1y

b1(ω) + b2y

b2(ω) + · · · ]2

=[

a1, a2, · · · , b1, b2 · · ·]

xa12 + ya1

2 xa1xa2 + ya1y

a2 · · · xa1x

b1 + ya1y

b1 xa1x

b2 + ya1y

b2 · · ·

xa2xa1 + ya2y

a1 xa2

2 + ya22 · · · xa2x

b1 + ya2y

b1 xa2x

b2 + ya2y

b2 · · ·

......

. . ....

... · · ·xb1x

a1 + yb1y

a1 xb1x

a2 + yb1y

a2 · · · xb1

2+ yb1

2xb1x

b2 + yb1y

b2 · · ·

xb2xa1 + yb2y

a1 xb2x

a2 + yb2y

a2 · · · xb2x

b1 + yb2y

b1 xb2

2+ yb2

2 · · ·...

.... . .

...... · · ·

a1

a2...

b1

b2...

.


Let φ = [a1, a2, · · · , b1, b2, · · · ]T , ϕ = [1, a1, a2, · · · ]T ,

P(ω) =

xa12 + ya1

2 xa1xa2 + ya1y

a2 · · · xa1x

b1 + ya1y

b1 xa1x

b2 + ya1y

b2 · · ·

xa2xa1 + ya2y

a1 xa2

2 + ya22 · · · xa2x

b1 + ya2y

b1 xa2x

b2 + ya2y

b2 · · ·

......

. . ....

... · · ·xb1x

a1 + yb1y

a1 xb1x

a2 + yb1y

a2 · · · xb1

2+ yb1

2xb1x

b2 + yb1y

b2 · · ·

xb2xa1 + yb2y

a1 xb2x

a2 + yb2y

a2 · · · xb2x

b1 + yb2y

b1 xb2

2+ yb2

2 · · ·...

.... . .

...... · · ·

(3.40)

and

P(ω) =

1 0 −ω2 · · ·0 ω2 0 · · ·−ω2 0 ω4 · · ·...

......

. . .

, (3.41)

where both P(ω) and P(ω) are symmetric and positive definite matrices. (3.29) is

then written as

∣

∣

∣

∣

∣

∣(1− GQ)v

∣

∣

∣

∣

∣

∣

2

2= φT

[

1

π

∫ ∞

0

P(ω)ϕT P(ω)ϕ

dω

]

φ. (3.42)

Let P =[

1π

∫∞

0P(ω)

ϕT P(ω)ϕdω]

, (3.42) is converted, by Schur complement, into

γ φ

φT P−1

> 0. (3.43)

Thus, the optimal performance design is to solve (3.21) subject to (3.43). It is

noticed that (3.43) is an LMI if we set ϕ = ϕ0 = [1, a01, a02, · · · ]T .

Robustness For the uncertain process described by (3.19), if both G(s)

and G(s) are asymptotically stable, the robust stability condition is (3.23). With


the controller in (3.26) and b0 = G−1(0), we obtain

|lm(ω)G(jω)b(jω)|2

=|lm(ω)G(jω)|2|G−1(0) + b1(jω) + b2(jω) + · · · |2

=|lm(ω)G(jω)|2[G−1(0)− b2ω2 + · · · ]2 + ω2[b1 − b3ω

2 + · · · ]2

=[

1 b1 · · ·]

G−2(0)|lm(ω)G(jω)|2 −G−1(0)ω2|lm(ω)G(jω)|2 · · ·−G−1(0)ω2|lm(ω)G(jω)|2 ω4|lm(ω)G(jω)|2 · · ·

......

. . .

1

b1...

.

Let θ = [1, b1, b2, · · · ]T and

Q =

G−2(0)|lm(ω)G(jω)|2 −G−1(0)ω2|lm(ω)G(jω)|2 · · ·−G−1(0)ω2|lm(ω)G(jω)|2 ω4|lm(ω)G(jω)|2 · · ·

......

. . .

,

Q(ω) =

1 ω −ω2 · · ·ω −ω2 ω3 · · ·−ω2 ω3 ω4 · · ·...

......

. . .

,

one has

∣

∣

∣

∣

∣

∣lm(ω)G(jω)Q(jω)

∣

∣

∣

∣

∣

∣

2

∞= max

ω∈(0,∞)

∣

∣

∣

∣

∣

lm(ω)G(jω)b(jω)

a(jω)

∣

∣

∣

∣

∣

2

= maxω∈(0,∞)

θT Q(ω)θ

ϕT Q(ω)ϕ. (3.44)

We notice that (3.44) is a complex nonlinear function respect to ω, θ and ϕ.

Thus, the exact solution with constraint (3.23) is hard to be found. However, we

can approximate the constraint (3.23) with finite frequency point ωi, i = 1, · · · , l.Then (3.44) becomes

θT Q(ωi)θ < ϕT Q(ωi)ϕ, i = 1, · · · , l. (3.45)

With such a finite point approximation, we notice that the constraint (3.23) is less

strict, thus the lower bound of optimal solution may be found with the relaxed


constraint (3.45). If we increase l, the the lower bound will get closer to the global

optimal solution. However, in this way, we do not know how large the l should be

and if the approximate solution is good enough. Fortunately, we can also find the

upper bound of the optimal solution with some assumption. If bi ≥ 0, i = 1, · · · ,m,

maxω∈(0,∞)

[θT Q(ω)θ] ≤ θT [Qmax(ω)]θ, where

Qmax(ω) =

maxω∈(0,∞)

Q11 maxω∈(0,∞)

Q12 · · ·

maxω∈(0,∞)

Q21 maxω∈(0,∞)

Q22 · · ·...

.... . .

.

Moreover, if ai ≥ 0, i = 1, · · · ,m, minω∈(0,∞)

[ϕT Q(ω)ϕ] ≥ ϕT [Qmin(ω)]ϕ, where

Qmin(ω) =

minω∈(0,∞)

Q11 minω∈(0,∞)

Q12 · · ·

minω∈(0,∞)

Q21 minω∈(0,∞)

Q22 · · ·...

.... . .

.

Notice that θT [Qmax(ω)]θ < ϕT [Qmin(ω)]ϕ is equivalent to θT [Qmaxi (ω)]θ < ϕT [Qmin

i (ω)]ϕ, i =

1, · · · , l + 1, where

Qmaxi (ω) =



Q12 · · · maxω∈(ωi−1,ωi)


Q22 · · ·...

.... . .

,

and

Qmini (ω) =

minω∈(ωi−1,ωi)

Q11 minω∈(ωi−1,ωi)

Q12 · · ·

minω∈(ωi−1,ωi)

Q21 minω∈(ωi−1,ωi)

Q22 · · ·...

.... . .

,

with ω0 = 0 and ωl+1 =∞. Thus, the constraint (3.23) can be also approximated

by

θT Qmaxi (ω) θ < ϕT [Qmin

i (ω)]ϕ. i = 1, · · · , k, k + 1, (3.46)

which is more strict constraint than (3.23). Thus we can obtain the upper bound of

the problem (3.21). Similarly, if we increase model order m, the the upper bound

will get closer to the global optimal solution. If the error between the upper and


lower bound is small enough, we obtain the good approximation optimal solution

with a desired accuracy.

According to Schur complement, (3.45) and (3.46) are converted respectively

into

ϕT Q(ωi)ϕ θ

θT Q(ωi)−1

> 0, i = 1, · · · , l, (3.47)

and

ϕT [Qmini (ω)]ϕ θ

θT [Qmaxi (ω)]−1

> 0, i = 1, · · · , l, l + 1. (3.48)

Notice that (3.47) and (3.48) are bilinear matrix inequalities (BMI), which cannot

be solved by LMI optimization directly. Thus, we introduce branch and bound

method to convert the BMI problem into LMI problem.

Branch and Bound

The basic idea of branch and bound is widely used in many forms to handle

difficult optimization problem (Reiner and Tuy, 1992). However, the basic idea is

very simple. Firstly, the lower bound γ(M0) and (if possible) upper bound γ(M0)

of γ(M0) should be found on a feasible set M0 respectively. Secondly, partition

M0 into finite many subsets Mi, i = 1, · · · ,m and find the lower bound γ(Mi) and

upper bound γ(Mi) of each section. Let γ = miniγ(Mi) and γ = min

iγ(Mi) be

overall bounds. If γ is sufficiently close to γ then the algorithm terminates. If not,

then the domain is subdivided and the procedure is repeated for each section. If

the lower bound for a region γ(Mi) is greater than the current upper bound γ then

that region may be discarded. The convergence of the algorithm is guaranteed as

eventually the upper bound and lower bound will be equal to each other within

the specified tolerance.

The most important step to develop an efficient branch and bound algorithm is

to derive tight upper and lower bounds from the objective function over any given

part of the domain. Introducing a set of variables wij, which we wish wij = aiaj.

Thus, the constraint (3.47) is converted into linear matrix constraints as

Q(ωi)wij θ

θT Q(ωi)−1

> 0, i = 1, · · · , l. (3.49)


Suppose that li < ai < ri, i = 1, · · · ,m, wij should satisfy (Fiacco and G.P.McCormick,

1990)

minlilj, lirj, rirj < wij < maxlilj, lirj, rirj

wij 6 liai + rjaj − ljri

wij 6 liai + rjaj − rjli

wij 6 liai + rjaj − lilj

wij 6 liai + rjaj − rirj (3.50)

The lower bound of the optimal problem with the constraint (3.47) can be found

by LMI optimal problem with constraints (3.49) and (3.50). The upper bound can

found by fixing some variables ai = li. Thus the optimal solution can be found by

branch and bounding. Similarly, the optimization problem with constraint (3.48)

can also be solved.

Algorithm 3.2. Given G(s) in (3.24), seek Q(s) in (3.26) to minimize (3.21).

Step 1 Select ϕ0.

Step 2 Calculate P , Q(ω) and Q(ω) from (3.42) and (3.44).

Step 3 Select a set of frequency points ωi, i = 1, · · · , l.Step 4 Find the lower bound of the optimal solution γ: minimize γ subject to

(3.43) and (3.47) using branch and bound method.

Step 5 Find the upper bound of the optimal solution γ: minimize γ subject to

(3.43) and (3.48) using branch and bound method.

Step 6 If γ − γ < ε, where ε is given error bound, go to Step 7. Otherwise,

increase l, choose more frequency points and go to Step 4.

Step 7 If ‖ϕstar −ϕ0‖ < ε1, where ε1 is given error bound, stop. Otherwise, let

ϕ0 = ϕstar, go to Step 2.

3.2.5 Examples

In this subsection, two examples are provided to illustrate the effective of the

proposed design.


0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

plan

t out

put

Figure 3.6. Step response

(- - - Proposed controller with fixed pole,

—- Proposed controller with general form,

...... Morari’s controller.)

Example 3.3. Consider a first order plus dead time model

G =e−s

s+ 1,

with uncertain bound described by∣

∣

∣

∣

∣

G− G

G

∣

∣

∣

∣

∣

< lm(ω) =

√10ω√w2 + 1

.

Let G(0)Q(0) = 1, the first order controller is

Q =bs+ 1

τs+ 1.

Step 1 Following Algorithm 3.1, we calculate τ from (3.27) and obtain

τ =

√

n√2− 1

µωpb=

√2− 1

0.3× 1= 2.894.

Step 2 Set v as unit step input, P is computed as

P =

2.1161 −0.5421−0.5421 0.3201

,


and Q(ω) is

Q(ω) =

4ω4

(ω2+1)2(τ2ω2+1)0

0 4ω2

(ω2+1)2(τ2ω2+1)

.

Step 3 Choose ωi = [0.8, 1.2, 1.5, 2.0, 5], the lower bound of the optimal value

can be computed by

γ = minb

γ

subject to

γ 1 b

1 0.8347 1.4136

b 1.4136 5.5178

> 0,

1 1 b

1 1.0e7 × 2.5006 0

b 0 1.0e7 × 0.0003

> 0,

1 1 b

1 4.0999 0

b 0 1.4760

> 0,

1 1 b

1 2.7808 0

b 0 1.3626

> 0,

1 1 b

1 2.0768 0

b 0 1.3291

> 0,

1 1 b

1 0.9944 0

b 0 1.3291

> 0

1 1 b

1 1.0e3 × 0.0108 0

b 0 1.0e3 × 1.0819

> 0.

The lower bound of the optimal problem is γ = 1.546 and optimal solution b =

1.198.

Step 4 The upper bound of the optimal problem can be computed by

γ = minb

γ

subject to

γ 1 b

1 0.8676 1.2787

b 1.2787 4.6203

> 0,

1 1 b

1 1.0e6 × 1.0002 0

b 0 1.0e6 × 0.0001

> 0,

1 1 b

1 0.1522 0

b 0 0.0548

> 0,

1 1 b

1 0.1008 0

b 0 0.0494

> 0,

1 1 b

1 0.0734 0

b 0 0.0470

> 0,


1 1 b

1 0.0272 0

b 0 0.0465

> 0,

1 1 b

1 0.1988 0

b 0 19.8817

> 0.

The upper bound of the optimal problem is γ = 1.548 and optimal solution b =

1.206.

Step 5 The optimal controller (for upper bound) obtained is

Q =1.206s+ 1

1.s+ 1,

and the optimal H2 norm is γ = 1.548.

With the proposed method for general form controller, we obtain

Q =0.194s+ 1

1.337s+ 1

and the H2 norm is 1.528.

Using Morari’s trial and error method, the controller is given as

Q =s+ 1

2.21s+ 1


The step responses of nominal plant for three methods are given in Fig. 3.6.

Example 3.4. Consider a second order non minimal phase model

G =s− 5

(s+ 1)(s+ 5)

with uncertain bound described by

∣

∣

∣

∣

∣

G− G

G

∣

∣

∣

∣

∣

< lm(ω) =2ω√w2 + 1

.

Let G(0)Q(0) = −1, the first order controller is

Q =bs+ 1

τs+ 1.


0 5 10 15 20 25 30−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time

plan

t out

put

Figure 3.7. Step response

(- - - Proposed controller with fixed pole,

—- Proposed controller with general form,

...... Morari’s controller.)

With the proposed method, the controller with fixed pole is

Q =−2.084s− 1

2.89s+ 1


With the proposed method, the controller with general form is

Q =−0.469s− 1

0.820s+ 1


With Morari’s method, the controller is

Q =−s− 1

2.21s+ 1


The step responses of nominal plant for three methods are given in Fig. 3.7.


3.2.6 Conclusions

In this section, a new IMC controller design method is proposed to achieve the

optimal performance and robust stability. The method is based on LMI/BMI

optimization. It is shown that better optimal performance is achieved compared

with Morari’s method with the same robust stability constraints. It also should be

noted that the proposed scheme is a systemic method while most of existing IMC

tuning schemes use trial and error method.

3.3 Modified Smith Predictor Control for Dis-

turbance Rejection with Unstable Processes

3.3.1 Introduction

In process control, the Smith predictor (SP) (Smith, 1959) is a well known and

very effective dead-time compensator for stable processes. However, the origi-

nal Smith predictor control scheme will be unstable when applied to an unstable

process (Wang et al., 1999b). To overcome this obstacle, many modifications to

the original Smith scheme have been proposed. Astrom et al. (1994) presented

a modified Smith predictor for integrator plus dead-time process and can achieve

faster setpoint response and better load disturbance rejection with decoupling of

the setpoint response from the load disturbance response. Matausek and Micic

(1996) considered the same problem with similar results but their scheme is easier

to tune. Majhi and Atherton (1999) proposed a modified Smith predictor control

scheme which has high performance particularly for unstable and integrating pro-

cess. This method achieves optimal integral squared time error criterion (ISTE)

for setpoint response and employs an optimum stability approach with a propor-

tional controller for an unstable process. Another paper of Majhi and Atherton

(2000) extends their work for better performance and easy tuning procedure for

first order plus dead time (FOPDT) and second order plus dead time (SOPDT)

processes. To our best knowledge, Majhi and Atherton (2000) achieve best the


performance for both setpoint response and disturbance rejection with unstable

dead time processes with modified Smith predictor structure.

In this section, a new modified Smith predictor structure is proposed. A simple

but effective control procedure is designed to improve the performance especially

for the disturbance rejection. The internal stability of the proposed structures is

also analyzed which is not reported in the previous publications. Simulation is

given to illustrate the effectiveness of the proposed method.

3.3.2 The Proposed New Structure

Ls 0 (s)e G -

) ( ˆ s 0 G

Ls e -

u

d

y

y m

r

ˆ

Figure 3.8. Proposed smith predictor control scheme

Let the process and model be represented by G0(s)e−Ls and G0(s)e

−Ls respec-

tively, where G0 and G0 are rational functions with at least one unstable pole. A

new modified Smith predictor control scheme is depicted in Fig.3.8.

Suppose that the model used perfectly matches the plant dynamics, i.e. G0 =

G0 and L = L. Using Mason’s formula (Franklin et al., 2002; Goodwin et al., 2000),

we obtain the system determinant ∆ as:

∆ = 1 +G0K2 +G0K1 +K3G0e−Ls + (1 +G0K2 +G0Q1)K3G0e

−Ls

= (1 +G0K12)(1 +K3G0e−Ls).


The closed-loop response to setpoint and disturbance inputs are given by

Tr =G0K4

1 +G0K12e−Ls,

and

Td =G0e

−Ls(1 +K12G0 −K2G0e−Ls)

(1 +G0K12)(1 +K3G0e−Ls),

where K12 = K1 +K2.

In this new structure, there are four controllers to be designed for different

objectives. We may note that the denominator of the transfer function Tr has

no time delay. Thus, we select controller K12 to place the closed-loop poles to

desired points using pole placement method. Moreover, similar to the two degree

of freedom design, the prefilter K4 may be designed to make the setpoint response

achieve better performance. For the disturbance path, the K3 is employed to

stabilize the time delay part. The controller K2 is chosen to minimize 1+K12G0−K2G0e

−Ls. If we obtain 1 + K12G0 − K2G0e−Ls = 0, the disturbance will be

completely rejected. However, we cannot find such a realizable controller K2 for

most processes. We can only choose appropriate K2 to minimize the disturbance

in critical frequency range.

3.3.3 Internal Stability

Input/output stability does not necessarily acceptable for control system design.

The unstable pole-zero cancellations in the system may cause unbounded signals

in the system and the system may be damaged by such signals. Thus, internal

stability is a prerequisite for any control systems. In our modified Smith predictor

structure, the interconnected system consists of seven subsystems and each of them

is of single input and single output (SISO). It is shown by Wang et al. (1999b) that

an interconnected system consisting only of SISO plants is internally stable if and

only if pc(s) , ∆∏

i pi(s) has all its roots in the open left half of the complex

plane, where pi(s) are the denominators of the subsystem transfer functions and

the ∆ is the system determinant as defined in the Mason’s formula. To apply this


theorem, one sees that the system in Fig.3.8 consists of seven subsystems:

K1(s), K2(s), K3(s), K4(s), G0, G0e−Ls and e−Ls.

Let K1(s) =c1(s)d1(s)

, K2(s) =c2(s)d2(s)

, K3(s) =c3(s)d3(s)

, K4(s) =c4(s)d4(s)

and G0(s) = G0(s) =

α(s)β(s)

. Their respective pi are

p1 = d1(s), p2 = d2(s), p3 = d3(s), p4 = d4(s), p5 = p6 = β(s) and p7 = 1.

Therefore, we have

pc(s) = ¯∆(s)5∏

i=1

pi(s)

= [1 + (K1 +K2)G0][1 +K3G0e−Ls] · d1(s) · d2(s) · d3(s) · d4(s) · β(s) · β(s) · 1

= d4(s)[d1(s)d2(s)β(s) + c1(s)d2(s)α(s) + c2(s)d1(s)α(s)][d3(s)β(s) + c3(s)α(s)e−Ls].

The polynomial, d1(s)d2(s)β(s)+c1(s)d2(s)α(s)+c2(s)d1(s)α(s), reflects stabiliza-

tion of delay-free G0 by the controller K12 = K1 + K2, which is always possible

designed, say, by pole placement. K4 is a stable filter to improve the setpoint re-

sponse. Thus, the overall system is internally stabilizable if and only if the process

with time delay G0e−Ls is stabilizable by controller K3. For a general unstable de-

lay system, an approach to design a stabilizing controller K3 is reported by Bonnet

and Partington (1999).

In view of the above analysis, it can be concluded that unlike the original Smith

system where the characteristic equation is delay-free, the modified Smith scheme

gets no simplification as far as stabilization is concerned. It is indeed the case for

all the existing modified Smith schemes for unstable processes as we found (Majhi

and Atherton, 1999; Astrom et al., 1994; Matausek and Micic, 1996), that is, their

characteristic equations are all delay-dependent.

3.3.4 Controller Design

Most of the industrial processes may be approximated by first order plus time

delay (FOPTD) or second order plus time delay (SOPTD) processes. Moreover,

there are many integral plus time delay (IPTD) processes are reported (Astrom et


al., 1994). In this subsection, controllers are designed for IPTD, unstable FOPTD

and SOPTD processes with the proposed modified Smith predictor structure.

IPTD Processes Consider the following IPTD model:

G(s) = G0(s)e−Ls =

k

se−Ls,

where all coefficients are non-negative. The closed-loop transfer function for set-

point tracking is written as

Tr =kK4

s+ kK12e−Ls =

1

λs+ 1e−Ls,

where λ > 0 is an adjustable closed-loop design parameter. The higher value of λ,

the more robust of the closed-loop system. The small value of λ, the faster of the

response. The controller K12 and K4 are computed as

K12 =1

kλ,

and

K4 = K12.

The closed-loop transfer function for disturbance is given by

Td =ke−Ls(s+ kK12 − kK2e

−Ls)

(λs+ 1)(s+K3ke−Ls).

The controller K3 is designed to stabilize the second part of last equation s +

K3ke−Ls). According to Majhi and Atherton (2000), K3 is computed to satisfy the

optimum phase margin criterion (60o) which gives

K3 =π − 2× π

3

2Lk.

In order to achieve good disturbance rejection performance, K2 is selected to min-

imize s+ kK12 − kK2e−Ls. We approximate e−Ls by 1− Ls and choose

K2 = as+ b.

We wish

s+ kK12 − kK2e−Ls

=s+ kK12 − k[b+ (a− bL)s− aLs2]

=(kK12 − kb) + (1 + kbL− ka)s+ akLs2

=0


Let kK12 − kb = 0 and 1 + kbL − ka=0, we obtains b = K12 and a = 1/k + bL.

Finally, we may get K1 = K12 −K2.

FOPTD Processes Consider the following unstable FOPTD model:

G(s) = G0(s)e−Ls =

k

Ts− 1e−Ls,



Tr =kK4

Ts− 1 + kK12e−Ls =

1

λs+ 1e−Ls,

where λ > 0 is an adjustable closed-loop design parameter. The controller K12 and

K4 are computed as

K12 =1 + T/λ

k,

and

K4 = K12 −1

k.


Td =G0e

−Ls(Ts− 1 + kK12 − kK2e−Ls)

(λs+ 1)(1 +K3G0e−Ls).

The controller K3 is designed to stabilize the second part of last equation 1 +

K3G0e−Ls. De Paor and O’Malley (1989) suggested a proportional controller to

stabilize an unstable FOPTD process based on the optimum phase margin criterion

which gives

K3 =

√

T

Lk2.

In order to achieve good disturbance rejection performance, the K2 is selected to

minimize Ts− 1 + kK12 − kK2e−Ls. We approximate e−Ls by 1− Ls and choose

K2 = as+ b.

We wish

Ts− 1 + kK12 − kK2e−Ls

=Ts− k[b+ (a− bL)s− aLs2] + kK12 − 1

=(kK12 − kb− 1) + (T + kbL− ka)s+ akLs2

=0


Let kK12 − kb − 1 = 0 and T + kbL − ka=0, we obtains b = K12 − 1/k and

a = T/k + bL. Finally, we may get K1 = K12 −K2.

SOPTD Processes Consider the following unstable SOPTD model:

G(s) = G0(s)e−Ls =

k

(T1s− 1)(T2s+ 1)e−Ls,



Tr =kK4/(T2s+ 1)

T1s− 1 + kK12/(T2s+ 1)e−Ls =

1

λs+ 1e−Ls,

where λ > 0 is an adjustable closed-loop design parameter. The controller K12 and

K4 are computed as

K12 = (T2s+ 1)1 + T/λ

k,

and

K4 = (T2s+ 1)(K12

T2s+ 1− 1

k).


Td =G0e

−Ls(Ts− 1 + kK12/(T2s+ 1)− kK2/(T2s+ 1)e−Ls)

(λs+ 1)(1 +K3G0e−Ls).

The controller K3 is designed to stabilize the second part of last equation 1 +

K3G0e−Ls. Using De Paor and O’Malley (1989)’s method

K3 =

√

T1Lk2

(T2s+ 1).

In order to achieve good disturbance rejection performance, the K2 is selected to

minimize Ts− 1 + kK12/(T2s+ 1)− kK2/(T2s+ 1)e−Ls. We approximate e−Ls by

1− Ls and choose

K2 = (as+ b)(T2s+ 1).

We wish

Ts− 1 + kK12/(T2s+ 1)− kK2/(T2s+ 1)e−Ls

=Ts− k[b+ (a− bL)s− aLs2] + kK12/(T2s+ 1)− 1

=(kK12/(T2s+ 1)− kb− 1) + (T + kbL− ka)s+ akLs2

=0


Let kK12/(T2s+ 1)− kb− 1 = 0 and T + kbL− ka=0, we obtains b = K12/(T2s+

1)− 1/k and a = T/k + bL. Finally, we may get K1 = K12 −K2.

3.3.5 Examples

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 3.9. Step responses for IPTD process

(- - - - Majhi’s method; —— proposed method)

Example 3.5. Consider an IPTD process (Majhi and Atherton, 2000)

G(s) =1

se−5s,

we obtain the controllers from the proposed method in last section as K1 = −6s,K2 = 6s + 1, K3 = 0.105 and K4 = 1. The amplitude of setpoint and load

disturbance are set as 1 and 0.1 respectively. Note that K1 and K2 in a form

of PD controller, we implement them with an industrial PD controller with the

form kds+kpkdNs+1

. In the examples, we choose N = 100. The responses of the proposed

method and Majhi’s method are shown in Fig. 3.9. It is illustrated that the

proposed method has better performance for setpoint response and disturbance

rejection.

Example 3.6. Consider an unstable FOPTD process (Majhi and Atherton, 2000)

G(s) =4

10s− 1e−5s,

we obtain the controllers from the proposed method in last section as K1 =

−8.75s + 0.25, K2 = 8.75s + 1.25, K3 = 0.35 and K4 = 1.25. The amplitude


0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 3.10. Step responses for unstable FOPTD process


of setpoint and load disturbance are set as 1 and 0.1 respectively. The responses of

the proposed method and Majhi’s method are shown in Fig. 3.10 . It is illustrated

that the proposed method has better disturbance rejection performance. For this

example, we may check the closed-loop poles with first order Pade approximation

for the time delay. We have the closed-loop poles

pc(s) = [10s− 1 + (−8.75s+ 0.25) ∗ 4 + (8.75s+ 1.25) ∗ 4]× [(10s− 1)(1 + 2.5s) + 4 ∗ 0.35 ∗ (1− 2.5s)]

= 5(2s+ 1)(25s2 + 7.5s− 1 + 1.4− 3.5s)

= 5(2s+ 1)(25s2 + 4s+ 0.4),

thus the closed-loop system is stable.

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 3.11. Step responses for unstable SOPTD process



Example 3.7. Consider an unstable SOPTD process

G(s) =4

(10s− 1)(2s+ 1)e−5s,

we obtain the controllers from the proposed method in last section asK1 = (−15s+0.25)(2s+1), K2 = (15s+2.5)(2s+1), K3 = 0.707s+0.354 and K4 = 2.5(2s+1).

The amplitude of setpoint and load disturbance are set as 1 and 0.1 respectively.

The responses of the proposed method and Majhi’s method are shown in Fig.

3.11. It is illustrated that the proposed method has better disturbance rejection

performance as well as setpoint response.

Remark 3.2. The idea in the proposed method is to use the zero of controller to

compensate the low frequency dynamics of the delay part. The simulations illus-

trate the proposed approach may achieve much better performance for disturbance

rejection especially for the low frequency band, which is the main part of distur-

bance in many cases. Moreover, if we need to achieve better performance, we may

approximate the time delay with higher order zero which results in higher order

controller. Further, we only employ the approximation in the closed-loop zero de-

sign. Thus, the proposed method may also achieve the closed-loop stability for

time delay and maintain the advantage of the original Smith predictor design.

3.3.6 Conclusions

Simple and effective design is proposed based on our new modified Smith predic-

tor structure for unstable process with dead time. Moreover, the internal stabil-

ity of the proposed system is analyzed. Simulations illustrate that the proposed

method yields much improvement of load disturbance responses over the best ex-

isting method.

Chapter 4

Decoupling with Stability and

Decoupling Control Design

4.1 Preview

The problem of decoupling linear time-invariant multi-variable systems received

considerable attention in the system theoretical literature for several decades.

There are many important processes in which multiple control loops interact each

other. The interaction between loops may cause deleterious effects on the sys-

tem stability and performance. Although decentralized control is employed in

most practical multi-variable processes due to its simplicity, the decoupling control

scheme is a desired choice for complex multi-variable system, especially for the

processes which have strongly coupled loops. The trend in the process industry is

towards increased process integration and the highly complex multi-variable sys-

tem is more popular. Thus the decoupling design becomes an important area in

control field.

In this chapter, the solvability of the decoupling problem with internal stability

for unity output feedback systems is discussed. A necessary and sufficient condition

is developed with polynomial approach. Moreover, a robust decoupling control

scheme is developed via LMI optimization. The control aims at gaining good loop

performance and largely removing the coupling simultaneously for multi-variable

87

Chapter 4. Decoupling with Stability and Decoupling Control Design 88

processes. The robust stability of the proposed method is analyzed.

This chapter is organized as follows. In Section 4.2, a new necessary and suffi-

cient condition for the solvability of the decoupling problem with internal stability

for unity output feedback systems is given. In Section 4.3, a decoupling design via

LMI is presented.

4.2 Decoupling Problem with Stability via Trans-

fer Function Matrix Approach

4.2.1 Introduction

In the area of decoupling design, much attention was directed towards decoupling

by state feedback (Morgan, 1964; Falb and Wolovich, 1967; Hautus and Heymann,

1983; Decusse et al., 1988). More general block decoupling was investigated by

Williams and Antsaklis (1986). The necessary and sufficient conditions were given

by Wang and Davison (1975) for static output feedback. Decoupling through a

combination of precompensation and output feedback was considered by Pernebo

(1981) and Eldem and Ozguler (1989). A algebraic theory for decoupling via two-

degree-freedom design was formulated in Desoer and Gundes (1986). Eldem (1996)

presented the solution to the problem of decoupling through specified input-output

channels with internal stability. It is reported that the solvability condition and

the set of all solutions may be obtained. The regular block decoupling problem by

static state feedback was discussed by Camarta using geometric tools (Camarta

and Martnez-Garca, 2001).

In the case of unity output feedback systems, the existence condition of con-

troller for the plants which have full row rank is very simple and well known. But

the problem of decoupling in such a configuration while maintaining internal sta-

bility of the decoupled system appears to be much more difficult. A necessary and

sufficient solvability condition for the diagonal decoupling problem with internal

stability is given by Linnemann and Maier (1990) for 2 × 2 plants, and by Wang


(1992) for square plant. The necessary and sufficient condition for this problem

was also derived based on transfer function matrices and residue by Lin (1997).

Here we will propose another necessary and sufficient solvability condition which

is simpler, more understandable and make use of transfer function matrices only.

4.2.2 Minimal C+-Decoupler

An m×m proper rational transfer function matrix D is said to be decoupled if D

is diagonal and nonsingular; Gd a decoupler for G if GGd is decoupled. Given an

m×m nonsingular proper rational transfer function matrix G, let the set of all the

common right-half plane (RHP) poles and zeros of G be Γ = γ1, γ2, . . . , γn. Letδp(G) denote the McMillan degree of a transfer function G at the RHP pole p, i.e.,

multiplicity of the pole at p. For simplification, we denote δp(G) = δp(G)+δp(G−1).

Definition 4.1. Gd,min is called a minimal C+- decoupler for G if (i) GGd,min

is decoupled; (ii) For any decoupler Gd for G, there holds δγ(Gd,min) 6 δγ(Gd) for

any γ ∈ Γ.

Lemma 4.1. (Existence and uniqueness of Gd,min) For any nonsingular G , (i)

there exists at least a Gd,min; (ii) All Gd,min have the same δγ(Gd,min).

Proof. Obviously, Gd = G−1 is a decoupler for any m×m nonsingular G. One can

see that GdD is still a decoupler for any decoupled D. For each γ ∈ Γ, we perform

the following search to find Gd,min from an initial decoupler Gd. Let lp (resp. lz)

be the greatest multiplicity of the pole of all the elements of Gd (resp. G−1d ) at

γ ∈ Γ. Suppose that lj, j = 1, 2, · · · ,m, are integers, D = diag(s − γ)l1 , (s −γ)l2 , · · · , (s − γ)lm and L = (l1, l2, · · · , lm)| − lz ≤ li ≤ lp, i = 1, 2, · · · ,m. If

(l1, l2, · · · , lm) /∈ L, then there exists at least a l∗j such that (a) l∗j > lp or (b)

l∗j < −lz. Suppose the case (a) (the case (b) can be treated with obvious changes).

Then, Gdiag(s− γ)l1 , · · · , (s− γ)lj−1 , (s− γ)lp , (s− γ)lj+1 , · · · , (s− γ)lm cannot

have any pole-zero cancellation with diag1, · · · , 1, (s− γ)l∗j−lp , 1, · · · , 1 and thus

δr(GD) with (l1, l2, · · · , lm) = (l1, · · · , lj−1, l∗j , lj+1, · · · , lm) is strictly greater than

δr(GD) with (l1, l2, · · · , lm) = (l1, · · · , lj−1, lp, lj+1, · · · , lm). It thus follows that a


minimum δr(GD) will only come from some D with (l1, l2, · · · , lm) ∈ L if it exists.

Further, the combinations of such lj in the range L are finite and the unique

minimum δγ(GdD) must be achieved with one or a few combinations of lj in the

range and the corresponding D will yield Gd,min = GdD. If we start the search with

another initial decoupler Gd, then GGd = D and GGd = D, for some decoupled

D and D, which lead to Gd = G−1D = GdD−1D = GdD for a decoupled D whose

RHP pole and zero structure at γ must correspond to some D above. Thus, the

search will result in the same δγ(Gd,min) as before, in other words, δγ(Gd,min) is

independent of initial Gd and it is the property of G only.

The proof of Lemma 4.1 naturally gives rise to the following constructive pro-

cedure for finding Gd,min from G.

Step 1: Calculate Gd = G−1 and determine the set Γ from pole locations of

elements of G andG−1. If Γ = φ, i.e. empty set, then take Gd,min = Gd;

otherwise, proceed to step 2.

Step 2: For each γ ∈ Γ, find lpγ and lzγ from pole multiplicity of elements of Gd

and G−1d . Let Dγ = diag(s − γ)l1 , (s − γ)l2 , · · · , (s − γ)lm, tabulate

δγ(GdDγ) for all possible combinations of lj such that −lzγ 6 lj 6 lpγ

and determine the minimum δγ(GDγ) and the corresponding D∗γ.

Step 3: Set Gd,min = GdD∗γ1D∗γ2· · ·D∗

γn.

Remark 4.1. Note that the locations of poles of a multivariable system must be

the same as those of its elements (Maciehowski, 1989). Then the zeros of G are the

poles of G−1 and we can find the common RHP zeros and poles of G by examining

the poles of the elements of G and G−1.

Example 4.1. (Wang, 1992) Let

G =

1s+1

1s+2

1(s−1)(s+1)

s(s−1)(s+2)

.

It follows that

G−1 =

s(s+1)s−1

−(s+ 1)

− s+2s+1

(s+ 2)


for which we can easily find that the only common RHP pole and zero of G is

at 1 i.e., Γ = 1. Take Gd = G−1 and obviously we have lpγ = 1 and lzγ = 1.

For D = diag(s − 1)l1 , (s − 1)l2, where −1 6 l1, l2 6 1, the search for Gd,min is

tabulated in Table 4.1, from which two Gd,min are found as

Gd,min1 =

s(s+1)s−1

− s+1s−1

− s+2s−1

s+2s−1

and

Gd,min2 =

s(s+ 1) −(s+ 1)

−(s+ 2) (s+ 2)

.

If we start our procedure with a very different Gd:

Gd =

1 − s+1s+2

− s+2s(s+1)

1

,

with lpγ = 0. And we have

G−1d =

ss−1

s(s+1)(s+2)(s−1)

s+2(s−1)(s+1)

ss−1

,

so that lzγ = 1. For D = diag(s− 1)l1 , (s− 1)l2 with −1 6 l1, l2 6 0, the search

for Gd,min is tabulated in Table 4.2, from which two Gd,min are found as

Gd,min1 =

1 − s+1s+2

− s+2s(s+1)

1

(4.1)

and

Gd,min2 =

1s−1

− s+1(s+2)(s−1)

− s+2s(s+1)(s−1)

1s−1

. (4.2)

Tables 4.1 and 4.2 verify that δγ(Gd,min) is unique for a given G although the Gd,min

could be different for different Gd.

4.2.3 Decoupling with Stability

The problem under consideration is to find the conditions on anm bym nonsingular

proper rational transfer function matrix G under which there is a K such that the


Table 4.1. Search for Gd,min in Example 4.1

(l1, l2) K = Gd

(s− 1)l1 0

0 (s− 1)l2

δ1(K)

(−1,−1)

s(s+1)

(s−1)2− s+1

s−1

− s+2(s+1)2

s+2s−1

3

(−1, 0)

s(s+1)

(s−1)2−(s+ 1)

− s+2(s−1)2

s+ 2

3

(−1, 1)

s(s+1)

(s−1)2−(s+ 1)(s− 1)

− s+2(s−1)2

(s− 1)(s+ 2)

4

(0,−1)

s(s+1)s−1

− s+1s−1

− s+2s−1

s+2s−1

1

(0, 0)

s(s+1)s−1

−(s+ 1)

− s+2s−1

s+ 2

2

(0, 1)

s(s+1)s−1

−(s+ 1)(s− 1)

− s+2s−1

(s− 1)(s+ 2)

3

(1,−1)

s(s+ 1) − s+1s−1

−(s+ 2) s+2s−1

2

(1, 0)

s(s+ 1) −(s+ 1)

−(s+ 2) s+ 2

1

(1, 1)

s(s+ 1) −(s+ 1)(s− 1)

−(s+ 2) (s+ 2)(s− 1)

2

resulting unity output feedback system is internally stable and (I + GK)−1GK

is decoupled. This problem is here abbreviated to the DPSUF (the decoupling

problem with internal stability for unity output feedback system). For later use,

let us recall some notations. For any nonsingular rational matrix G, G = G+G− is

a stability decomposition if G+ is completely unstable and G− is completely stable

(Wang, 1992). The G+ can always be factorized as G+ = N+(D+)−1 where N+



(l1, l2) K = Gd

(s− 1)l1 0

0 (s− 1)l2

δ1(K)

(−1,−1)

1s−1

− s+1(s+2)(s−1)

− s+2s(s+1)(s−1)

1s−1

1

(−1, 0)

1s−1

− s+1s+2

− s+2s(s+1)(s−1)

1

2

(0,−1)

1 − s+1(s−1)(s+2)

− s+2s(s+1)

1s−1

2

(0, 0)

1 − s+1s+2

− s+2s(s+1)

1

1

and D+ are nonsingular polynomial matrices and right coprime. For the pair of

square polynomial matrices N and D, N and D are called externally skew prime

and N and D internally skew prime, if DN = N D with det(N) = αdet(N) with a

real α and N and D left coprime (Wolovich, 1978). Let Pra and Pla be a right and

left strict adjoints of a nonsingular polynomial matrix P respectively (Hammer

and Khargonekar, 1984).

Lemma 4.2. (Wang, 1992) (i) the DPSUF is solvable if and only if there is a

K such that GK is decoupled and has no RHP pole-zero cancellation; (ii) If the

DPSUF is solvable, then all solutions are given by

K = (G−)−1N+

ra(D+

la)−1K2 (4.3)

where K2 is decoupled such that K1 = N+N+ra(D+

laD+)−1 has no RHP pole-zero

cancellations with K2.


Lemma 4.3. If Gd is a solution to DPSUF for G, then there exists a Gd,min such

that (i) Gd = Gd,minD, where D is decoupled; (ii) δγ(Gd) = δγ(Gd,min) + δγ(D),

where γ ∈ Γ.

Proof. Take Gd,min = (G−)−1N+

ra(D+

la)−1 as in (4.3) and the lemma follows from

Lemma 2 (ii).

Theorem 4.1. For a nonsingular G, the DPSUF is solvable if and only if there

exists a Gd,min which has no pole-zero cancellation with G at any γ ∈ Γ ⊂ C+.

Proof. Sufficiency: By the definition of Gd,min, GGd,min is decoupled. The theorem

will follow from Lemma 4.2 (i) if a Gd,min has no RHP pole-zero cancellation with

G at any ρ ∈ C+ but ρ /∈ Γ. Then, such ρ can be a pole or a zero of G but not

both. If ρ be a pole (resp. zero) of G, then ρ must be a zero (resp. pole) of Gd,min

for the cancellation to occur. Form Gd = 1(s−ρ)l

Gd,min (resp. (s − ρ)lGd,min) for

a sufficiently large integer l such that Gd no longer has any zero (resp. pole) at

ρ. This Gd has no pole-zero cancellation with G at ρ. Repeat this elimination of

pole-zero cancellation for all possible ρ and the resultant Gd will have no pole-zero

cancellation with G at ρ ∈ C+ but ρ /∈ Γ.

Necessity: It follows from Lemma 4.2 (i) that if the DPSUF is solvable, then

there is a Gd such that GGd is decoupled and meets

δγ(G) + δγ(Gd) = δγ(GGd), γ ∈ Γ. (4.4)

By Lemma 3, for Gd there exists a Gd,min such that Gd = Gd,minD and

δγ(Gd) = δγ(Gd,min) + δγ(D). (4.5)

Substituting (4.5) into (4.4) yields

δγ(G) + δγ(Gd,min) + δγ(D) = δγ(GGd,minD), (4.6)

implying thatG andGd,min have no pole-zero cancellation at γ ∈ Γ. This completes

the proof.


Remark 4.2. The condition in Theorem 4.1 is much simpler to apply than that in

Wang (1992). The new condition here involves rational matrix manipulations only

and no complicated polynomial operations such as coprime factorization and skew

primeness checking are required.

4.2.4 Examples

In this subsection, several examples are presented to illustrate our results.

Example 4.1. (continued) (Wang, 1992) Consider the same G as in Example

4.1 with Γ = 1. For Gd,min1 in (4.1), one sees

GGd,min1 =

1s+1

1s+2

1(s−1)(s+1)

s(s−1)(s+2)

1 − s+1s+2

− s+2s(s+1)

1

=

s−1s(s+1)

0

0 1s+2

δ1(G) = 1, δ1(Gd,min1) = 0, and δ1(GGd,min1) = 0, δ1(G)+δ1(Gd,min1) 6= δ1(GGd,min1).

There is a pole-zero cancellation between G and Gd,min1 at γ = 1.

For Gd,min2 in (4.2), it follows that

GGd,min2 =

1s+1

1s+2

1(s−1)(s+1)

s(s−1)(s+2)

1s−1

− s+1(s+2)(s−1)

− s+2s(s+1)(s−1)

1s−1

=

1s(s+1)

0

0 1(s+2)(s−1)

,

δ1(G) = 1, δ1(Gd,min2) = 1, and δ1(GGd,min1) = 1, so that δ1(G) + δ1(Gd,min1) 6=δ1(GGd,min1). There is a pole-zero cancellation between G and Gd,min2 at γ = 1 too.

Thus, none of Gd,min can meet the condition in Theorem 4.1 and so the DPSUF

for this plant is not solvable.

Example 4.2. Let

G(s) =

ss+1

1s+2

1s(s+1)

1s(s+2)

.

Choose

Gd = G−1 =

s+1s−1

− s(s+1)s−1

− s+2s−1

s2(s+2)s−1

.

It is readily seen that the only common RHP pole and zero of G(s) is 1, lzγ = 0

and lpγ = 1. From Table 4.3, one can see that Gd,min is not unique. For

Gd,min1 =

s+1s−1

− s(s+1)s−1

− s+2s−1

s2(s+2)s−1

,



(l1, l2) K = Gd

(s− 1)l1 0

0 (s− 1)l2

δ1(K)

(0, 0)

s+1s−1

s(s+1)s−1

− s+2s−1

s2(s+2)s−1

1

(1, 0)

s+ 1 −s(s+1)s−1

−(s+ 2)s2(s+2)s−1

2

(0, 1)

s+1s−1

−s(s− 1)

− s+2s−1

s2(s+ 2)

2

(1, 1)

s+ 1 −s(s+ 1)

−(s+ 2) s2(s+ 2)

1

we have

GGd,min1 =

ss+1

1s+2

1s(s+1)

1s(s+2)

s+1s−1

− s(s+1)s−1

− s+2s−1

s2(s+2)s−1

=

1 0

0 1

,

δ1(G) = 0, δ1(Gd,min1) = 1, and δ1(GGd,min1) = 0, so δ1(G) + δ1(Gd,min1) 6=δ1(GGd,min1). There is a pole-zero cancellation between G and Gd,min1 at γ = 1.

For

Gd,min2 =

s+ 1 −s(s− 1)

−(s+ 2) s2(s+ 2)

,

we have

GGd,min2 =

ss+1

1s+2

1s(s+1)

1s(s+2)

s+ 1 −s(s+ 1)

−(s+ 2) s2(s+ 2)

=

s− 1 0

0 s− 1

,

δ1(G) = 0, δ1(Gd,min2) = 0, and δ1(GGd,min2) = 0, so δ1(G) + δ1(Gd,min2) =

δ1(GGd,min2). There is no pole-zero cancellation between G and Gd,min2 at γ = 1.

So the DPSUF of G(s) is solvable.



(l1, l2, l3) K = Gd

(s− 1)l1 0 0

0 (s− 1)l2 0

0 0 (s− 1)l3

δ1(K)

(−1,−1,−1)

1s−1

1s(s−1)

0

1s−1

1s−1

ss−1

0 0 1s−1

2

(−1,−1, 0)

1s−1

1s(s−1)

0

1s−1

1s−1

s

0 0 1

1

(−1, 0,−1)

− 1s−1

1s

0

1s−1

1 s

0 0 1

3

(−1, 0, 0)

1s−1

1s

0

1s−1

1 s

0 0 1

2

(0,−1,−1)

1 1s(s−1)

0

1 1s−1

ss−1

0 0 1s−1

3

(0,−1, 0)

1 1s(s−1)

0

1 1s−1

s

0 0 1

2

(0, 0,−1)

1 1s

0

1 1 ss−1

0 0 1s−1

2

(0, 0, 0)

1 1s

0

1 1 s

0 0 1

1


Example 4.3. Let

G(s) =

ss−1

− 1s−1

ss−1

−1 1 −s0 0 s− 1

.

From Table 4.4, there are two Gd,min. For

Gd,min1 =

1 1s

0

1 1 s

0 0 1

,

we calculate

GGd,min1 =

ss−1

− 1s−1

ss−1

−1 1 −s0 0 s− 1

1 1s

0

1 1 0

0 0 1

=

1 0 0

0 s−1s

0

0 0 s− 1

,


There is a pole-zero cancellation between G and Gd,min1 at γ = 1.

For

Gd,min2 =

1s−1

1s(s−1)

0

1s−1

1s−1

s

0 0 1

,

we have

GGd,min2 =

ss−1

− 1s−1

ss−1

−1 1 −s0 0 s− 1

1s−1

1s(s−1)

0

1s−1

1s−1

s

0 0 1

=

1s−1

0 0

0 1s

0

0 0 s− 1

,


There is a pole-zero cancellation between G and Gd,min2 at γ = 1, too. So the DP-

SUF for the plant is not solvable.

4.2.5 Conclusions

In this section, a new necessary and sufficient condition for the solvability of the

decoupling problem with internal stability for unity output feedback systems with

square plants is given. The conditions are only concerned with Gd,min of the plant

G, thus it is easy to check.


4.3 Decoupling Control Design via LMI Approaches

4.3.1 Introduction

The problem of decoupling of multivariable systems was first intensively studied by

three researchers, namely, Falb and Wolovich (1967) and Gilbert (1969), by means

of state feedback. These works initiated studies on structure of linear systems in the

1970s. Wang (1992) discussed the internal stability of the decoupled unity output

feedback system. Furthermore, the decoupling control has been tried to apply

to many practical industrial control problems for the last decades. Wang et al.

(1997) proposed a multivariable PID tuning method, where the objectives of control

design, as many applications require, are to make the closed-loop control system

decoupled and the resultant independent loops have good transient and steady state

accuracy in a usual sense of single-variable systems. Therefore, the controllers in

diagonal elements are designed with respect to the “equivalent processes” and

the controllers in off-diagonal elements are designed to achieve decoupling. If

the coupling between the loops is modest, simulations showed that the design

generally gives significant improvement in both system decoupling and diagonal

loop performances compared with conventional multi-loop design. However, if the

coupling is large, PID controller leads to bad decoupling performance and then

yielding poor loop performance. Thus, PID control may fail to yield satisfactory

performance even for simple multivariable models.

In this section, a new control scheme is proposed for the design of general

multivariable system via LMI technique aiming at obtaining good loop performance

and small loop coupling with a fixed order controller. Stability and Robustness are

analyzed and simulation shows that good control effects can be achieved.

4.3.2 Problem Formulation

Consider an IMC system shown in Fig. 3.1, where G = [Gij(s)] is an m×m plant,

G = [Gij(s)] is the model of the plant and Q = [Qij(s)] is the controller to be

designed so as to acquire not only good loop performance but also comparatively


low couplings.

Let S and T be the sensitivity function and complementary sensitivity function

of the IMC system. Then we can obtain the sensitivity transfer matrix S = I−GQand complementary sensitivity transfer matrix T = GQ if the model is perfect. A

particular transfer function, say i-input/j-output, of S and T can be written as

Sij = LiSRj and Tij = LiTRj, where the matrices Li and Rj select the appropriate

channel. If the state space realization of T is (A,B,C,D), then the state space

realization of Tij is (A,BRi, LjC,LjDRi). Similarly, we can obtain the state space

realization of Sij. For decoupling, we wish that ||Tij||∞, i 6= j, is as small as

possible, i.e, the coupling of each loop is minimized. At the same time, we may

consider the loop performance by choosing weighting functions Wi and Vi of Tii

and Sii respectively. One notes that if the single loop is successful in attenuating

the effects of disturbance acting on the plant, Sii should be small, and if the output

is to be insensitive to measurement errors, we need small Tii. Moreover, if we want

to reduce control energy, we need small Tii, and if tracking references is important,

then Sii should be small. In short, if we wish good loop performance, then we need

both Tii and Sii to be small in some sense. However, we know Tii(s) + Sii(s) = 1.

We thus have an unavailable trade-off between different performance specifications.

In practice, the conflict can be resolved by making one small at some frequencies

and the other small at other frequencies. Usually, the solution is to make |Tii(s)|small at high frequency and |Sii(s)| small at low frequency (Maciehowski, 1989).

Using this method, each loop can be tuned to satisfy some low sensitivity and high

noise attenuation performance index. Moreover, we wish the static gain of each

loop is 1, so we set Tii(0) = 1. In short, the design can be described as

P1 : minQij

∑

i,j

aijεij +∑

i

(γTi+ γ

Si) (4.7)

subject to

||WiTii||∞ < γTi, (4.8)

||ViSii||∞ < γSi, (4.9)

||Tij||∞ < εij, i 6= j, (4.10)


Tii(0) = 1. (4.11)

4.3.3 Controller Design via LMI

In order to design controller to achieve the given specification in the last section,

we need the following lemma.

Lemma 4.4. Given any system G = D + C(sI − A)−1B, ||G||∞ < γ and G is

asymptotically stable if and only if the following LMI in X is feasible:

ATX +XA XB CT

BTX −γI DT

C D −γI

< 0, (4.12)

where X > 0.

The above lemma is known as the Bounded Real lemma (Boyd,1994). Note that

ifA andB is fixed, the problem can be easily solved as a semi-definite programming.

Let Qrs = Nrs(s)/Drs(s) be the transfer function of the individual SISO com-

ponent systems in Q, where

Nrs = βn−1rs sn−1 + · · ·+ β1rss+ β0rs,

and

Drs = sn + αn−1rs sn−1 + · · ·+ α1rss+ α0rs.

Then the realization of Qrs can be express as

AQrs=

0 1 · · · 0 0

0 0 · · · 0 0...

.... . .

......

0 0 0 0 1

−α0rs −α1rs · · · −αn−2rs −αn−1rs

, BQrs=

0

0...

0

1

,

CQrs=[

β0rs β1rs · · · βn−1rs

]

, DQrs= 0.

The transfer function Tij can be written as Tij =∑m

l=1QilGlj, where Qil are the

individual SISO entries of Q.


Suppose that the realization of Gij is AGij, BGij

, CGij, DGij

, we obtain the state

space realization of GijQrs in terms of state space matrices of Gij and Qrs.

Aij,rs =

AGij0

BQrsCGij

AQrs

, Bij,rs =

BGij

BQrsDij

,

Cij,rs =[

DQrsCGij

CQrs

]

, Dij,rs = DQrsDGij

.

Finally, we get the desired state space representation for Tij,

Aij =

Ai1,1j 0 · · · 0

0 Ai2,2j · · · 0...

.... . .

...

0 0 · · · Aim,mj

, Bij =

Bi1,1j

Bi2,2j

...

Bim,mj

,

Cij =[

Ci1,1j Ci2,2j · · · Cim,mj

]

, Dij =∑m

l=1Dim,mj .

From the above, we see that if the denominator of controller elements Qij is spec-

ified, all the coefficients of Qij appear in the Cij of Tij. In the same way, we can

also find a similar state space realization of Sij.

As mentioned above, we need to specify the denominator of the controller Q

as a filter, thus the problem can be easily solved using LMI toolbox. Moreover,

we wish the static gain of each loop is 1. Choose Q(0) = G(0)−1, the objective

can be achieved. That means, we can add the following linear constraint into the

optimization.m∑

j=1

gij(0)β0jiα0ji

= 1, i = 1, 2, · · · ,m.

The corresponding algorithm is proposed as follows:

Step 1: According to the performance requirement of each loop, choose appropri-

ate Wi and Vi for each loop.

Step 2: Choose filter

fij =1

(τis+ 1)n

as the denominator of each element in the controller matrix for each i.


Step 3: Form the LMI constraints using the above mentioned method and solve

the the LMI optimization problem using the LMI toolbox.

Remark 4.3. Note that the introduction of the filter fij only aims at maintaining

the properness of controller Cii, so τi is chosen not to influence the decoupling

property of the system. It is only used to tune the performance and the robustness

of each loop.

Remark 4.4. Wi and Vi should be chosen in such a way that W−1i and V −1

i give an

upper bound for the sensitivity and complementary sensitivity functions respec-

tively (Maciehowski, 1989).

4.3.4 Stability and Robustness Analysis

The IMC system in Fig. 3.1 is referred to as the nominal case if G = G. It

follows from the known results (Morari and Zafiriou, 1989) that the IMC system

is nominally stable if both process G and the controller Q are stable. However,

in the real application, the model never represents the process exactly. Thus,

nominal stability is not enough and the robust stability has to be investigated.

As is common in robust analysis of IMC system, it is assumed that the nominal

IMC system is stable (i.e. both the G and Q are stable). Suppose that the actual

process G is also stable and is described by

G ∈∏

= G : σ(G−1(jω)[G(jω)− G(jω)]) = σ(∆m(jω)) ≤ lm(ω),

where lm(ω) is the bound on the multiplicative uncertainty. Suppose that the

following inequalities hold. (This is guaranteed by the control design presented in

last subsection.)

‖WiTii(jω)‖∞ ≤ γTi

i = 1, · · · ,m, (4.13)

and

‖Tij‖∞ ≤ εij i = 1, · · · ,m, j = 1, · · · ,m, j 6= i. (4.14)

Denote

δi(ω) = |γTiW−1

i (jω)|+√

Σmi=1Σ

mj=1,j 6=i|εij|2


We can obtain the following robust stability theorem.

Theorem 4.2. Suppose that Wi(jω) 6= 0,∀ω, the nearly decoupled IMC system isrobustly stable if

δi(ω) ≤ l−1m (ω),∀ω. (4.15)

Proof. The nominal complementary sensitivity function of the IMC system T (jω)

can be written as:

T (jω) = Tdiag(jω) + Toff (jω),

where

Tdiag(jω) =

T11 0 · · · 0

0 T22 · · · 0...

.... . .

...

0 0 · · · Tmm

, Toff (jω) =

0 T12 · · · T1m

T21 0 · · · T2m...

.... . .

...

Tm1 Tm2 · · · 0

.

From the properties of singular value, we have

σ(T (jω)) = σ(Tdiag(jω) + Toff (jω))

≤ σ(Tdiag(jω)) + σ(Toff (jω))

≤ maxi|Tii(jω)|+ [Σm

i=1Σmj=1,j 6=i|Tij(jω)|2]

12 (4.16)

From (4.13) and (4.14), we have

|Tii(jω)| ≤ |γTiW−1

i |,∀ω (4.17)

and

|Tij(jω)| ≤ εij,∀ω. (4.18)

Combining the inequalities (10), (11) and (12), we obtain

σ(T (jω)) ≤ maxi|γ

TiW−1

i (jω)|+√

Σmi=1Σ

mj=1,j 6=i|εij|2,∀ω.

It follows from the known robust stability results (Morari and Zafiriou, 1989) that

the IMC systems is robustly stable if and only if

σ(T (jω)) = σ(G(jω)Q(jω)) < l−1m (jω),∀ω.

Thus, proof is completed.


4.3.5 Simulation

In this section, two examples are given to show the effectiveness of the proposed

design.

0 20 40 60 80 100 120

0

0.5

1

0 20 40 60 80 100 120

−0.5

0

0.5

1

0 20 40 60 80 100 120−1

0

1

2

0 20 40 60 80 100 120−0.2

0

0.2

Time

y1

y2

u2

u1

t

t

t

t

Figure 4.1. Step Tests of the Plant in Example 4.4

( —— proposed method, - - - Luyben’s method)

Example 4.4. Consider the Wood/Berry binary distillation column plant with

first order Pade approximation (Wood and Berry, 1973).

G(s) =

12.8(1−0.5s)(1+16.7s)(1+0.5s)

−18.9(1−1.5s)(1+21s)(1+1.5s)

6.6(1−3.5s)(1+10.9s)(1+3.5s)

−19.4(1−1.5s)(1+14.4s)(1+1.5s)

. (4.19)

For desired loop performance, we choose


W1(s) =s+ 30

s+ 1000,

W2(s) =s+ 10

s+ 1000,

V1(s) =s+ 10

s+ 0.6,

V2(s) =s+ 10

s+ 0.6,

and

f1 =1

(s+ 0.35)3,

f2 =1

(s+ 0.6)3.

Using the proposed method, we obtain the controller as

Q =

q11 q12

q21 q22

,

where

q11 =−0.3856s3 + 0.1115s2 + 0.087s+ 0.0067

(s+ 0.35)3,

q12 =−0.0122s3 + 0.2929s2 − 0.1535s− 0.0314

(s+ 0.6)3,

q21 =−0.1733s3 − 0.0781s2 + 0.0284s+ 0.0023

(s+ 0.35)3,

q22 =0.017s3 + 0.0106s2 − 0.2059s− 0.0218

(s+ 0.6)3.

The step tests is shown in Fig. 4.1 with the solid line. During step tests, two step

reference signals, r1 and r2, each with a step amplitude 1, are introduced at t = 0

and t = 80, respectively. The step tests using Luyben’s method (Luyben, 1986)

is also shown in Fig. 4.1 with dashed line. It can be seen that the response of

the proposed method achieves better performance and less loop interactions than

those of Luyben’s method.


0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60−0.5

0

0.5

1

1.5

Time

Figure 4.2. Step Tests of the plant in Example 4.5

( —— proposed method, - - - Wang’s method)

10−3

10−2

10−1

100

101

102

103

10−4

10−3

10−2

10−1

100

101

102

a1

a2

a3

Figure 4.3. Robust Stability bound in Example 4.5

(a1: δ−11 (ω), a2: δ−12 (ω) a3: lm(ω) )


0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60−0.5

0

0.5

1

1.5

Time

Figure 4.4. Step response for perturbed process in Example 4.5

( —— proposed method, - - - Wang’s method )

Example 4.5. To demonstrate the effectiveness of the proposed design, we

consider the plant adopted in Palmor et al. (1993):

G(s) =

0.5(0.1s+1)2(0.2s+1)2

−1(0.1s+1)(0.2s+1)2

1(0.1s+1)(0.2s+1)2

2.4(0.1s+1)(0.2s+1)2(0.5s+1)

. (4.20)

For desired loop performance, we choose

W1(s) =s+ 0.01

s+ 4,

W2(s) =s+ 0.01

s+ 1,

V1(s) =s+ 3

s+ 0.02,

V2(s) =s+ 3

s+ 0.02,

and

f1 =1

(s+ 0.7)3,


f2 =1

(s+ 0.8)3.

Set a12 = a21 = 100, using the proposed LMI method, we can obtain the controller

as

Q =

q11 q12

q21 q22

,

where

q11 =0.7916s2 + 1.1940s + 0.3742

(s + 0.7)3,

q12 =−0.0345s3 + 0.7996s2 + 0.7429s + 0.2369

(s + 0.8)3,

q21 =−0.1649s3 − 0.5786s− 0.5755s− 0.1559

(s + 0.7)3,

q22 =−0.0388s3 + 0.3386s2 + 0.3736s + 0.1147

(s + 0.8)3.

The step tests is depicted in Fig. 4.2. During step tests, two step reference

signals, r1 and r2, each with a step amplitude 1, are introduced at t = 0 and t = 30.

The step response of Wang’s method (Wang et al., 1997) is also shown in Fig. 4.2.

Next, we will check the robustness of the proposed method. Let the uncertainty

of the process described by:

G ∈∏

= G : σ(G−1(jω)[G(jω)− G(jω)]) = σ(∆m(jω)) ≤ lm(ω),

where

∆m(s) =1.5s

s+ 4I2×2,

and I is an unity matrix. We can obtain

lm(ω) =1.5ω

√16 + ω2

16 + ω2

The resultant γT1

= 0.0262, γT2

= 0.6129, ε21 = 3.7096e − 006 and ε12 = 0.0074.

The frequency curves of lm(ω), δ−11 (ω) and δ−12 (ω) are shown in Fig. 4.3. The step

responses of the perturbed process are also given in Fig. 4.4.

Remark 4.5. From the simulation results, the proposed decoupling design avoids

overshoot but leads to a sluggish response. Moreover, it reduces the coupling of


each loop. The gain of proposed controllers is smaller compared to that of the

Wang’s controller (Wang et al., 1997). The proposed design is more conservative

with the uncertain and disturbance, so it causes sluggish response.

4.3.6 Conclusions

A method for IMC controller design with decoupling has been proposed to gain

good loop performance and largely remove the effect of process coupling simultane-

ously for multi-variable plants. Two examples are presented with the demonstrated

effectiveness of the proposed method.

Chapter 5

Fuzzy Modelling and Control for

F-16 Aircraft

5.1 Introduction

In this chapter, the fuzzy modelling and controller design for F-16 aircraft is dis-

cussed. A TS fuzzy modelling and two control design approaches are developed.

5.1.1 F-16 Aircraft and Control

F-16 aircraft is designed for all electric control in 1970s. The classical flight con-

trol technology can be used via local linearization method. However, the aircraft

model is inherently nonlinear, exact input-output linearization is often used to

control specific output variable sets in nonlinear flight control problems (Lane and

Stengel, 1988; Krsti and Tsiotras, 1999). Feedback linearization is a theoretically

established and widely used method in controlling nonlinear systems. Unfortu-

nately, this direct application of feedback linearization requires the second or third

derivatives of uncertain aerodynamic coefficients and does not guarantee internal

stability for non-minimum phase systems.

Another approach to designing flight control laws with feedback linearization

is to separate the flight dynamics into fast and slow dynamics by using timescale

properties (Menon et al., 1987; Snell et al., 1992). This method allows the con-

111

Chapter 5. Fuzzy Modelling and Control for F-16 Aircraft 112

troller design process to be performed without state transformation. Because each

separated subsystem is square, the number of control inputs is equal to the number

of states. The design process of this method can be divided into two steps. In the

outer loop, the controller for the slow states α, β, and φ is designed to facilitate

tracking of the given commands by assuming that the fast states p, q, and r are

control inputs, which achieve their commanded values instantaneously. With the

slow states controller designed in the outer loop, a separated inner-loop controller

is designed to make the fast states p, q, and r follow the outer loop’s control in-

put trajectories using the real control inputs: aileron, rudder, and elevator. This

method can be justified only if there is sufficient timescale separation between the

inner- and outer-loop dynamics because the fast states p, q, and r are used as

control inputs in the outer-loop system. Hence p, q, and r in the inner loop should

be much faster than the states α, β, and φ in the outer loop. The stability of

this timescale separation approach may be analyzed by the singular perturbation

theory. However, in most nonlinear flight control research, the gain of the inner-

loop controller is set much larger than that of the outer-loop controller, and it is

assumed that the aircraft dynamics satisfies this property. This, therefore, does

not guarantee closed-loop stability.

Schumacher and Khargonekar analyzed theoretically the stability of the flight

control system with the two-timescale separation assumption (Schumacher and

Khargonekar, 1998). Using the Lyapunov theory, they determined the minimal

inner-loop gain guaranteeing closed-loop stability. However, this approach is so

complicated and conservative that the calculated value of minimal inner-loop gain

is too large to be applied in the flight controller. It may excite un-modelled dy-

namics or saturate the control inputs and, therefore, cause robustness problems.

Another difficulty related to the application of feedback linearization to a flight

control system is that a complete and accurate aircraft dynamic model including

aerodynamic coefficients is required. It is difficult to identify accurately aerody-

namic coefficients because they are nonlinear functions of several physical variables.

An inverse optimal design is developed by Sepulchre et al. (1997) to derive the


optimal stabilizing controller. This approach was first proposed by Kalman (1964)

to establish the gain and phase margins of linear quadratic regulators and was

recently revised by Freeman and Kokotovic (1996) to develop a design methodology

of robust nonlinear controllers.

The control problem of a F-16 aircraft model has been addressed by many re-

searchers for the purpose of the control of spacecraft and aircraft (Debs and Athans,

1969; Dabbous and Ahmed, 1982; Porcelli, 1968; Sohoni and Guild, 1970; Vadali et

al., 1984). Also, there have been several works that consider performance indices

such as time and/or fuel in the formulation of the optimal control problems (Athans

et al., 1963; Freeman and Kokotovic, 1970; Junkins et al., 1981; Etter, 1989; Bil-

imoria and Wie, 1991; Scrivener and Thomson, 1994). These studies have mainly

addressed the regulation problem for the angular velocity subsystem and for some

quadratic costs (Athans et al., 1963; Windeknecht, 1963; Kumar, 1965; Tsiotras

et al., 1996). Recently, the attitude control problem of the complete system that

includes the dynamics as well as the kinematics has been investigated by many

researchers. Carrington and Junkins (1986) have used a polynomial expansion ap-

proach to approximate the solution to the Hamilton-Jacobin-Bellman (HJB) equa-

tion. Rotea et al. (1998) have shown that, for some special cases of performance

outputs, Lyapunov functions that include a logarithmic term in the kinematic pa-

rameters result in linear controllers with a finite quadratic cost. For the general

quadratic cost, they have also presented sufficient conditions that guarantee the

existence of a linear, suboptimal, stabilizing controller. Tsiotras et al. (1996) has

derived a new class of globally asymptotically stabilizing feedback control laws for

the complete attitude motion of a nonsymmetric rigid body and has also presented

a family of exponentially stabilizing optimal control laws for the complete system.

Tsiotras (1997), by using the natural decomposition of the complete system into

its kinematics and dynamics subsystems and the inherent passivity properties of

these two subsystems, has presented a partial solution to the optimal regulation

of the symmetry axis of a spinning rigid body. Bharadwaj et al. (1998) have de-

rived a couple of new globally stabilizing attitude control laws using the inverse


optimal approach of Freeman and Kokotovic (1996) where minimal, exponential

coordinates are used to represent the kinematic equations. To design a stabiliz-

ing control law for F-16 aircraft, the method of backstepping (Krsti et al., 1995)

can be considered, which was used by Sontag and Sussmann (1988) for the first

time to design feedback controls laws for an underachieved rigid body. Tsiotras

and Longuski (1994) have employed this method for the attitude stabilization of

an axis symmetric spacecraft with two control torques. In their design, a con-

trol Lyapunov function along with a stabilizing controller is derived and thus the

stabilization of closed-loop system is guaranteed.

5.1.2 Objective of the Design

In this study, we consider the complete attitude motion of a rigid spacecraft

(Shuster, 1963). Moreover, both stabilizing and attitude tracking controller are

needed to be developed based on a nonlinear F-16 model.

Since Takagi and Sugeno (1985) opened a new direction of research in the area

of control by introducing the Takagi-Sugeno (TS) fuzzy model, there have been

several studies concerning the systematic design of stabilizing fuzzy controllers

(Tanaka and Sugeno, 1992; Tanaka et al., 1996b; Wang et al., 1996; Tanaka et

al., 1996a; Wang and Tanaka, 1996; Thathachar and Viswanath, 1997; Tanaka et

al., 1997; Tanaka et al., 1998). In this thesis, we propose two design procedures

yielding the stabilizing controller for the nonlinear system described by a TS fuzzy

model. In the TS fuzzy model (Takagi and Sugeno, 1985), the overall system is

described by several fuzzy IF-THEN rules, each of which represents a local linear

state equation x = Aix+Biu. To derive the first stabilizing controller, we employ

the Lyapunov stabilizing theory and LMIs method. We show that the parameters

of the stabilizing controller can be found by solving an LMI problem. The LMI

formulation of the controller synthesis problems is of great practical value because

it can be solved by using reliable and efficient convex optimization techniques

(Boyd et al., 1994), for example, the LMI Control Toolbox of MATLAB (Gahinet

et al., 1995). Another method is based on gain scheduling, LQR optimal design


and TS fuzzy modelling. The resulting controller is a fuzzy controller.

Our proposed methods are based on the design of the control law for the TS

fuzzy model. To the author’s best knowledge, the proposed approach is the first

attempt to design both stabilizing and tracking control with application to the

complete attitude motion of a rigid spacecraft.

5.1.3 Organization of the Chapter

The rest of this chapter is organized as follows. First, the F-16 aircraft model is

presented. Next, the modelling of TS fuzzy system of F-16 aircraft is described.

Further, synthesis of the stabilizing and tracking controllers for the TS fuzzy sys-

tems is considered. The controller design is studied with two methods, namely,

Lyaponov based control and gain scheduling control. Finally, the discussion and

conclusions are given.

5.2 F-16 Aircraft Model

5.2.1 Need for Modelling

Model building is a fundamental process. An aircraft designer who has a mental

model of the type of aircraft, uses physical models to gather wind-tunnel data,

and designs with mathematical models that incorporate the experimental data

(Taeyoung and Youdan, 2001). The modelling process is often iterative; a math-

ematical model based on the laws of physics will suggest what experimental data

should be taken, and the model may then undergo considerable adjustment in or-

der to fit the data. In building the mathematical model we recognize the onset

of the law of diminishing returns and build a model that is good enough for our

purposes but has known limitations. Some of the limitations of the models involve

uncertainty in the values of their parameters. Later, we attempt to characterize

this uncertainty mathematically and allow for it in control system design.

Because of the high cost of building and flight testing a real aircraft, the impor-

tance of aircraft mathematical models goes far beyond control system design. The


mathematical model is used, in conjunction with computer simulation, to evalu-

ate the performance of the prototype aircraft and hence improve the design. It

can also be used to drive training simulators, to reconstruct the flight conditions

involved in accidents, and to study the effects of modifications to the design. Fur-

thermore, other mathematical models are used in all aspects of the aircraft design

(e.g., structural models for studying stress distribution and predicting fatigue life).

In the aerospace industry it is necessary for a wide range of specialists to work

together, thus flight control engineers must be able to work with the aerodynam-

icists as well as with structural and propulsion engineers. Each have some un-

derstanding of the terms and mathematical models used by the other. This is

becoming increasingly important as designers seek to widen aircraft performance

envelopes by integrating many parts of the whole design process. Furthermore, at

the prototype stage the control system designer must work closely with the test

pilots to make the final adjustments to the control systems. This may take many

hours of simulator time and flight test, and the flight controls engineer involved in

the final stages of the design is also likely to be a qualified pilot.

5.2.2 Modelling Method

The aerodynamic forces and moments on an aircraft are produced by the rela-

tive motion with respect to the air and depend on the orientation of the aircraft

with respect to the airflow. In a uniform airflow these forces and moments are

unchanged after a rotation around the free-stream velocity vector. Therefore, only

two orientation angles (with respect to the relative wind) are needed to specify the

aerodynamic forces and moments. The angles that are used are the angle of attack

(alpha) and the sideslip angle (beta). They are known as the aerodynamic angles

and will now be defined by means of coordinate rotations in three dimensions. Note

that the aerodynamic force and moments are also dependent on angular rates, but

for the moment we are concerned only with their dependence on orientation.

The angle of attack usually specified in the aerodynamic data for an aircraft is

measured with respect to a fuselage reference line and denoted by αflr. We shall


assume that our aircraft body-fixed axes are aligned with this fuselage reference

line; otherwise, the alpha used in the body-axes equations of motion would differ

by a constant from the alpha used in the aerodynamic data base.

Figure 5.1. Definition of aircraft axes and angles.

Fig. 5.1 shows an aircraft with the relative wind on its side (i.e., sideslipping)

and with the conventional right-handed (forward, starboard, and down) set of

body-fixed axes illustrated. The relative wind vector is equal and opposite to

the cg relative velocity vector vB used in the equations of motion. The angles

of attack and sideslip are defined by performing a plane rotation about the body

y-axis, followed by a plane rotation about the new z-axis, such that the final x-axis

is aligned directly into the relative wind. The first rotation defines the stability

axes, and the angle of attack is the angle between the body-fixed x-axis and the

stability x-axis. Alpha is positive if the rotation about the body-fixed y-axis was

negative; thus a positive alpha is shown in the figure. The second rotation leads to

a set of wind axes, and the sideslip angle is the angle between the stability x-axis

and the wind x-axis. Beta is positive if the rotation about the stability z-axis was

positive, and a positive beta is shown in the figure.

An aircraft of conventional shape must fly more or less directly into the appar-


ent wind in order to have low drag; therefore, beta is usually very small in steady

fight. Alpha must be large enough to generate the required lift but is also usually

quite small. Therefore, although the stability axes and wind axes have a variable

orientation depending on the flight condition, they essentially point forward, star-

board, and down, the same as the body-fixed axes. Note that technically all three

sets of axes are “body axes” but only one is body-fixed; we shall drop the term

“fixed” and simply refer to them as body, stability,and wind axes.

A left-handed wind-axes system, aligned backwards, and “up” relative to the

aircraft, has often been used in the past for wind tunnel data. Lift L, drag D, and

cross-wind force C were defined naturally in these axes as the aerodynamic force

components along the respective positive axes. In the right-handed wind axes that

we have defined, we use −L and −D for the z and x force components, and define

a component Y for the aerodynamic side-force measured along the positive y-axis.

The symbol Y is also commonly used for the aerodynamic side-force component

along the body y-axis, but it will be clear from the context which axes are being

used.

5.2.3 Workable Model

The body-fixed axes, nonlinear equations of motion for an aircraft over a flat Earth

are given by (Stevens and Frank, 1992)

V =cosα cos β

m[T + Fx] +

sin β

m[Fy] +

sinα cos β

m[Fz]

+g[− cosα cos β sin θ + sin β sinφ cos θ + sinα cos β cos θ], (5.1a)

α = −p cosα tan β + q − r sinα tan β − sinα

mV cos β[T + Fx]

+cosα

mV cos βFz +

g

V cos β[sinα sin θ + cosα cosφ cos θ], (5.1b)

β = p sinα− r cosα− cosα sin β

mV[T + Fx] +

cos β

mV[Fy]−

sinα cos β

mV[Fz]

+g

V[cosα sin β sin θ + cos β cos θ sinφ− sinα sin β cosφ cos θ], (5.1c)


p = I2pq + I1qr + I3L+ I4N, (5.2a)

q = I5pr − I6(p2 − r2) + I7M, (5.2b)

r = −I2qr + I8pq + I4L+ I9N, (5.2c)

φ = p+ tan θ(q sinφ+ r cosφ), (5.3a)

θ = q cosφ− r sinφ, (5.3b)

ψ =q sinφ+ r cosφ

cos θ, (5.3c)

where the moments of inertia Ii, i = 1, 2, ..., 9, are defined as follows:

I1 =Iz(Iz − Iy) + I2xz

IxIz − I2xz, I2 =

Ixz(Ix − Iy + Iz)

IxIz − I2xz, I3 =

IzIxIz − I2xz

, I4 =Ixz

IxIz − I2xz,

I5 =Iz − IxIy

, I6 =IxzIy, I7 =

1

Iy, I8 =

Ix(Ix − Iy) + I2xzIxIz − I2xz

, I9 =Ix

IxIz − I2xz.

It is assumed that the aerodynamic forces and moments are expressed as functions

of angle of attack, sideslip angle, angular rates, and control surface deflection. For

example, Fx, L and the other symbols are expressed as follows:

Fx = [Cx(α, δe) + (cq/2V )Cxq(α)]qS, (5.4a)

Fy = [Cy(β, δα, δr) + (bp/2V )Cyp(α) + (br/2V )Cyr ]qS, (5.4b)

Fz = [Cz(α, β, δe) + (cq/2V )Czq(α)]qS, (5.4c)

L = [Cl(α, β) + Clδα(α, β)δα + Clδr

(α, β)δr + (p/2V )Clp(α) + (r/2V )Clr(α)]qSb, (5.5a)

M = [Cm(α, δe) + (q/2V )Cmq(α)]qSc (5.5b)

N = [Cn(α, β) + Cnδα(α, β)δα + Cnδr

(α, β)δr + (p/2V )Cnp(α) + (r/2V )Cnr(α)]qSb,(5.5c)

For the F-16 aircraft, the force coefficients Cx, Cxq , Cy, Cyp , Cyr , Cz, Czq and mo-

ment coefficients Cl, Clδα, Clδr

, Clp , Clr , Cm, Cmq, Cn, Cnδα

, Cnδr, Cnp , Cnr and other

necessary parameters can be obtained from Stevens and Frank (1992). The control

input are thrust T , elevator angle (el) δa, aileron angle (ail) δa and rudder angle


(rdr) δr. The nine states are speed V , angle of attack (alpha) α, sideslip angle

(beta) β, roll rate p, pitch rate q, yawing rate r, roll angle (phi) φ, pitch angle

(theta) θ and yaw angle (psi) ψ.

5.3 TS Fuzzy Modelling

From the development of F-16 aircraft nonlinear model in the last section, we

observe that the nonlinear model is rather complex and coefficients can be only

obtained from the look-up tables. Thus, the control design for such model is quite

difficult. We need a simpler model which is easier to be used for control design.

This section reviews the Takagi and Sugeno Fuzzy modelling approach and develop

TS model of F-16 nonlinear model.

5.3.1 The Technique

The fuzzy model proposed by Takagi and Sugeno consists of several fuzzy IF-THEN

rules, each of which represents the local linear state equation of a nonlinear system.

In this report, we consider the fuzzy rules described as follows.

Rule i:

IF x1(t) is Mi1 and · · · and xn(t) is Min,

THEN

x(t) = Aix(t) +Biu(t), i = 1, 2, ..., N, (5.6)

Here, xi(t), i = 1, 2, ..., n, and Mij, i = 1, 2, ..., N , j = 1, 2, ..., n, are state variables

and fuzzy sets, respectively, and N is the number of IF-THEN rules; u ∈ Rm is the

input vector and Ai ∈ Rn×n and Bi ∈ Rn×m, i = 1, 2, ..., N . Following the usual

inference method of the TS fuzzy model, the state equation at time t is represented

in the form of weighted average along the trajectory x(t) ∈ Rn:

x(t) =N∑

i=1

wi(x(t))Aix(t) +Biu(t)/

N∑

i=1

wi(x(t)), (5.7)


In (5.7), the weight functions are defined as

wi(x(t)) =n∏

j=1

Mij(xj(t)),

where Mij(xj(t)) is the grade of xj(t) in the fuzzy set Mij. The weight functions

wi, which are nonnegative and measurable, usually satisfy

N∑

i=1

wi(x(t)) > 0, ∀ t > 0. (5.8)

Throughout this study, it is assumed that (5.8) always holds and that the vector

x(t) can be measured in real time. With the normalization of weight functions

hi(x(t)) = wi(x(t))/

N∑

i=1

wi(x(t)), i = 1, 2, ..., N, (5.9)

the state equation (5.7) can be written in the polytopic form

x(t) =N∑

i=1

hi(x(t))(Aix(t) +Biu(t)), (5.10)

where the normalized weights hi satisfy hi(x(t)) ≥ 0, i = 1, 2, ..., N , and

N∑

i=1

hi(x(t)) = 1, ∀ t ≥ 0.

Suppose that the truth model of a plant has the form

x = F (x, u). (5.11)

Expanding F by means of a Taylor series around (x0, u0) yields

x = F (x0, u0) +∂F

∂x|x=x0,u=u0(x− x0) +

∂F

∂u|x=x0,u=u0(u− u0) (5.12)

The point (x0, u0) is called an equilibrium point of (5.11) if at (x0, u0) we have

x = F (x0, u0) = 0. Let δx = x − x0 and δu = u − u0, the linearized model about

the equilibrium (x0, u0) is obtained by neglecting higher order terms. Observing

that for F (x0, u0) = 0, the linearized model has the form

d

dtδx = Aδx+Bδu, (5.13)


where A = ∂F∂x|x=x0,u=u0 and B = ∂F

∂u|x=x0,u=u0 . We start with generating linear

local models describing plant’s behavior about the equilibrium state (x0, u0) =

(0, 0), we obtain the first linear model by using the above described linearization

technique. The resulting model is given by x = A1x + B1u. Next, we need to

construct local linear models describing plant’s behavior at the remaining operating

points. Suppose that x = xj is the next operating state of interest. The result

of linearization of a nonlinear model using Taylor series expansion operating at a

non-equilibrium point is an affine rather than linear model. Even if the operating

point is an equilibrium point, Taylor series linearization, in general, will not yield

a local model that is linear in x and u. Indeed, suppose that the operating point

(xj, uj) is an equilibrium point, the resulting linearized model is

d

dt(x− xj) = Aj(x− xj) +Bj(u− uj). (5.14)

We can represent model (5.14) in the form

x = Ajx+Bju− (Ajxj +Bjuj). (5.15)

The term (Ajxj + Bjuj) does not have to be equal to zero and, hence, model

(5.15) is not a linear model, but rather an affine model. However, we note that

Taylor’s linearization will yield a linear system in x and u if the equilibrium point

(x0, u0) = (0, 0). Suppose that we are given an operating state x0 that does not

have to be an equilibrium state. Our goal is to construct a linear model in x and u

that approximates the behavior of (5.11) in the vicinity of the operating state x0;

that is, we wish to find constant matrices A and B such that in a neighborhood of

x0

F (x, u) ≈ Ax+Bu (5.16)

and

F (x0, u0) = Ax0 +Bu0. (5.17)

Let A = [a1 a2 · · · an] and B = [b1 b2 · · · bn]. We wish to have

Fi(x0, u0) = aTi x0 + bTi u0 (5.18)


and

bi =∂Fi∂u|x=x0,u=u0 . (5.19)

where Fi is the ith row of F . Moreover, we also wish that

∂Fi∂x|x=x0,u=u0 ≈ ai. (5.20)

Using the method of Marcelo and Stanislaw (1999), we obtain that

ai =∂Fi∂x|x=x0,u=u0 +

Fi(x0, u0)− xT0∂Fi∂x|x=x0,u=u0

‖x0‖2x0, x0 6= 0. (5.21)

ai =∂Fi∂x|x=x0,u=u0 , x0 = 0.

5.3.2 TS Model of F-16

We now apply the proposed fuzzy modelling method to the F-16 aircraft modelling,

where the F-16 nonlinear model is described by (5.1a)-(5.3c). The model can be

written as

x = F (x, u),

y = Cx, (5.22)

where x = [V, α, β, p, q, r, φ, θ, ψ]′ and u = [T, δe, δa, δr]′. In the following, we will

present the details of constructing a F-16 TS model: 1) Selection of equilibrium

state of the nonlinear model. 2) Selection of the operating points around the

equilibrium state. 3) Computation of the linear models at the chosen operating

points. 4) Determination of the weighting function of the TS model.

It has been shown by Stevens and Frank (1992) that one of the equilibrium

states of the F-16 aircraft is

[x0, u0] = [502 0.03936 0 0 0 0 0 0.03936 0.1485, 9.6427− 1.931 0.0 0.0]′

For the convenience of analysis and simulation, we move this point to the original

point x0 = 0. Substituting [x, u] = [x, u] + [x0, u0] into (5.22), we have a new

nonlinear model with the equilibrium point [x0, u0] = [0, 0].


From our extensive simulation, we choose nine operating points to build the TS

fuzzy model. If more points are chosen, the error of the model does not decrease

rapidly and less points result in generally bad approximation. For the states of

the nonlinear F-16 system near the equilibrium point with α ∈ [−0.2, 0.2] andφ ∈ [−0.2, 0.2], the TS model constructed by the proposed method is described by

x =9∑

i=1

hi(α, φ)(Aix+Biu), (5.23)

where hi(α, φ) = MαiMφi . Mαi and Mφi are the fuzzy membership functions of α

and φ respectively. In our simulation, the nine operating points evenly distributed

in the state space. That is, we choose

(α1, φ1) = (−0.2,−0.2),

(α2, φ2) = (−0.2, 0),

(α3, φ3) = (−0.2, 0.2),

(α4, φ4) = (0,−0.2),

(α5, φ5) = (0, 0),

(α6, φ6) = (0, 0.2),

(α7, φ7) = (0.2,−0.2),

(α8, φ8) = (0.2, 0),

(α9, φ9) = (0.2, 0.2)

as the nine operating points.

At each operating point, we can obtain the linear model at these operating

points from (5.19) and (5.21). For example, at the operating point (α, φ) =


(−0.2,−0.2), from (5.21), we get

A1 =

-3.0489 0.0003 -0.9873 0 0 -3.7977 0 0

0 -1.3815 -0.0029 0.0009 -1.1356 0 -0.0335 0

0.0312 0.0025 -0.5181 0 0 9.7538 0 0

0 2.5473 0 -0.0832 191.1245 -6.3862 -80.3479 -31.5328

0 0.9639 0 0.0003 -1.4637 0 0.1846 -0.0127

-0.1609 0 -0.9798 0.0000 0.0004 -0.2779 0.0632 0.0005

1.0000 -0.0078 0.0386 0 0 0 0 0

0 0.9801 0.1987 0 0 0 0 0

,

and from (5.19), we obtain

B1 =

0 0 -0.5983 0.1588

0 -0.1925 0 0

0 0 -0.0337 -0.0675

0 0.4584 0 0

0 -0.0020 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

.

Similarly, we can obtain the linear models (Ai, Bi), i = 2, · · · , 9, at each operating

point.

For weightings, we select triangle function as the fuzzy membership functions,

which are

Mα1 =

1 if α ≤ −0.22.5 ∗ α− 0.5 if − 0.2 < α < 0.2

0 if α ≥ 0.2

Mα2 =

0 if α ≤ −0.25 ∗ α + 1 if − 0.2 < α ≤ 0

−5 ∗ α + 1 if 0 < α < 0.2

0 if α ≥ 0.2


Mα3 =

0 if α ≤ −0.22.5 ∗ α + 0.5 if − 0.2 < α < 0.2

1 if α ≥ 0.2

Mφ1 =

1 if φ ≤ −0.22.5 ∗ φ− 0.5 if − 0.2 < φ < 0.2

0 if φ ≥ 0.2

Mφ2 =

0 if φ ≤ −0.25 ∗ φ+ 1 if − 0.2 < φ ≤ 0

−5 ∗ φ+ 1 if 0 < φ < 0.2

0 if φ ≥ 0.2

Mφ3 =

0 if φ ≤ −0.22.5 ∗ φ+ 0.5 if − 0.2 < φ < 0.2

1 if φ ≥ 0.2

The function is depicted in Fig. 5.2.

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

alpha(theta)

memb

ership

funct

ion

Figure 5.2. Fuzzy triangle membership functions

The complete model with all the data is given in Appendix A.


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

120

140

alpha(rad)

error

( —– δFLS, - - - δFL )

Figure 5.3. Approximate error of TS-fuzzy and linear model (different α)

5.3.3 Model Validation

To show the effectiveness of the obtained TS fuzzy model to approximate the

original nonlinear model, we compare the error between the TS fuzzy model and

linear model. The results are shown in Fig. 5.3 (with different α) and Fig.

5.4 (with different φ), where δFTS = ‖F (x, u) − FTs‖2 and δFL = ‖F (x, u) −

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

phi(rad)

error

( —– δFLS, - - - δFL )

Figure 5.4. Approximate error of TS-fuzzy and linear model (different φ)


FL‖2. In Fig. 5.3, we obtain FTS =∑9

i hi(α, φ0)(Aix + Biu) from TS fuzzy

model, FL = A5x + B5u is from the linear model at equilibrium point, where

x = [p0, q0, r0, V0, α, β0, φ0, θ0, ψ0]′ and u = [T0, δe0, δa0, δr0]

′, α ∈ [−0.2, 0.2]. In

Fig. 5.4, we have FTS =∑9

i hi(α0, φ)(Aix + Biu), FL = A5x + B5u, where

x = [p0, q0, r0, V0, α0, β0, φ, θ0, ψ0]′ and u = [T0, δe0, δa0, δr0]

′, φ ∈ [−0.2, 0.2]. It

is shown that the TS model has smaller error than the linear model around the

equilibrium point. At the interval α ∈ [−0.2, 0.2] , the maximal model error is

δFmaxTS = 2.0214 for TS fuzzy model and the maximal error is δFmax

L = 20.6833 for

linear model in Fig. 5.3. At the interval φ ∈ [−0.2, 0.2], the maximal model error

is δFmaxTS = 0.0001 for TS fuzzy model and the maximal error is δFmax

L = 0.0252

for linear model in Fig. 5.4.

5.4 Lyapunov Based Control

F-16 Aircraft is internally a nonlinear system. The nonlinear system control prob-

lem is generally difficult to solve. Moreover, even the nonlinear controller is found,

it is often very complicated and hard to be implemented. On the other hand, the

linear control is usually designed based on linear model. Because of the severe non-

linearity of the real F-16 aircraft, the linear model cannot represent the dynamics

of the system in large scale. Thus, we try to design controller based on TS fuzzy

model. From the simulation in last subsection, we observe that TS fuzzy model

has much less model error than linear model. In our approaches, Lyaponov based

controller is with linear controller and the gain schedule control is with fuzzy con-

troller. It is obvious that both of the controllers are of simpler structure than most

nonlinear controller. The method discussed in this section is based on Lyapunov

stability theory and called Lyapunov based control.


5.4.1 Stabilization

Suppose that the TS model of the nonlinear systems is described by

x =N∑

i=1

hi(Aix+Biu), (5.24)

y = Cx. (5.25)

The stability conditions for TS fuzzy system can be readily obtained via Lya-

punov stability theory. When u(t) = 0 for all t ≥ 0, the TS fuzzy system becomes

an input-free polytopic system given by

x(t) =N∑

i=1

hiAix(t). (5.26)

It is well known from the stability theory, an autonomous dynamic system

is stable if there exists a positive definite quadratic function V (x) = xTPx that

decreases along every nonzero trajectory of the system. A system having such a

Lyapunov function is called quadratically stable. In the polytopic system (5.26),

the time derivative of V along a nonzero trajectory x(·) is given by

dV

dt=

d

dt[xT (t)Px(t)] = xT (t)

N∑

i=1

hi[x(t)]ATi P + P

N∑

i=1

hi[x(t)]Ai

x(t)

=N∑

i=1

hi[x(t)]xT (t)AT

i P + PAix(t).

Then, we can see that the polytopic system (5.26) is quadratically stable if there

exists a symmetric matrix P satisfying the following inequalities:

P > 0, ATi P + PAi < 0, i = 1, 2, ..., N. (5.27)

Note that the left sides of these inequalities are all linear in the matrix variable P .To find P satisfying Eq.(5.27) or to determine if there does not exist such P is a

convex problem called the LMI feasibility problem. An LMI is any constraint of

the form

A(x) = A0 + x1A1 + · · ·+ xNAN < 0 (5.28)


where x=[x1, ..., xN ]T is the variable and A0, ..., AN are given symmetric matrices.

Since A(y) < 0 and A(z) < 0 implies A[(y + z)/2] < 0, the LMI (5.28) is a convex

constraint on the variable x. It is well known that LMI feasibility problems as well

as LMI-based minimization problems can be solved by interior-point algorithms

with polynomial time, and a toolbox of MATLAB that is dedicated to convex

problems involving LMIs is now available.

For the closed-loop system with state feedback controller

u = Kx, (5.29)

the TS fuzzy system can be described by

x(t) =N∑

i=1

hi(Aix(t) +Biu(t)) (5.30)

=N∑

i=1

hi(Ai +BiK)x(t). (5.31)

Then, we can see that the system (5.30) is quadratically stable if there exists a

symmetric matrix P and a state feedback controller K satisfying the following

inequalities:

P > 0, (Ai +BiK)TP + P(Ai +BiK) < 0, i = 1, 2, ..., N. (5.32)

Now multiplying the inequality on the left and right by P−1 and defining two new

variables Σ = P−1 and Ω = KP−1, we rewrite the conditions as

Σ > 0, ΣTATi + ΩTBT

i + AiΣ +BiΩ < 0, i = 1, 2, ..., N. (5.33)

The above conditions are LMIs with respect to variables Σ and Ω. We can find a

positive definite matrix Σ and Ω satisfying the LMIs or determinate that no such

Σ and Ω exist. A common P and the feedback gains K can be obtained as

P = Σ−1, K = ΩΣ−1 (5.34)

from the solutions of Σ and Ω.

In the control design, we often need to put the constraints on inputs and outputs

to obtain different performance in different conditions. Assume that the constraint


on input is ‖u‖2 ≤ µ, we obtain

uT (t)u(t) = xT (t)KTKx(t) ≤ µ2. (5.35)

Suppose that xT (0)Px(0) = xT (0)Σ−1x(0) ≤ 1 for t ≥ 0, if

1

µ2xT (t)KTKx(t) ≤ xT (t)Σ−1x(t), (5.36)

then (5.35) holds. Therefore, we have

1

µ2KTK − Σ−1 ≤ 0. (5.37)

By defining P = Σ−1 and Ω = KΣ, we can obtain

1

µ2ΩTΩ− Σ ≤ 0. (5.38)

Assume that the initial condition x(0) is known, the constraint ‖u‖2 ≤ µ is enforced

at all times t ≥ 0 if the LMIs

1 xT (0)

x(0) Σ

≥ 0 (5.39)

and

Σ ΩT

Ω µ2I

≥ 0 (5.40)

holds.

In a similar way, we can obtain the following result for output constraints

problem. Suppose the output of the TS system is

y = Cx. (5.41)

Assume that the initial condition x(0) is known, the constraint ‖y‖2 ≤ λ is enforced

as all times t ≥ 0 if the LMIs

1 xT (0)

x(0) Σ

≥ 0 (5.42)


and

Σ ΣCT

CΣ λ2I

≥ 0 (5.43)

holds.

Moreover, the speed of response is related to decay rate. The condition that

V (x(t)) ≤ −2αV (x(t)) for all trajectories is equivalent to

ATi P + PAi + 2αP < 0, i = 1, 2, ..., N. (5.44)

Let P = Σ−1, left multiplied and right multiplied by Σ, the corresponding LMI is

ΣATi + AiΣ + 2αΣ < 0, i = 1, 2, ..., N. (5.45)

5.4.2 Tracking

In control system design, we often need the process to track some reference signals.

In order to obtain zero steady tracking error, we often add a set of new states on

the system, which are the integrals of the error signals. If tracking error is not

zero, the differential of the augmented state, i.e, the error between the specified

signal and the reference signal, will no be zero, that is, the augmented system is

not in steady state or can not be stabilized. Thus, with the augmented states, the

specified state will track the reference signal at the steady state if the augmented

system is stabilized by a controller. The augmented system can be described by

x

xa

=

F (x, u)

x− xr

,

where xa(t) =∫ t

0(xr(τ) − x(τ))dτ , xr are reference signals, and F (x, u) is the

original system dynamics.

In aircraft control, we are often interested not only in regulating the state near

zero, but also in following a nonzero reference command signal. For example, we

may be interested in designing a control system for optimal step-response shaping.

This reference-input tracking or servo design problem is important in the design of

command augmentation system, where the reference signal may be, for instance,


the roll and pitch angle. According to the proposed design, we augment the F-

16 aircraft model with corresponding states in order to obtain zero steady-state

tracking error. Then we can use similar stabilizing control algorithm to stabilize

the augmented system. If the augmented system is stabilized, the differential of

the augmented stats approach to zero when time approaches infinity, that is, the

output is tracking the reference-input. Suppose that we wish the F-16 nonlinear

system to track the step signal of θr and φr, while keeping other states around

zero. The states are augmented for the nonlinear F-16 model. The two augmented

states θa(t) =∫ t

0(θr(τ)− θ(τ))dτ and φa(t) =

∫ t

0(φr(τ)− φ(τ))dτ are incorporated

into the nonlinear F-16 model, where θr and φr are reference signal of θ and φ

respectively. The nonlinear system can be written as

x

θa

φa

=

F (x, u)

θ − θr

φ− φr

Note that the augmented nonlinear model is different from the original F-16 model.

However, with the new model, the similar method can be employed to design the

tracking controller.

5.4.3 Simulation

We now apply the proposed method to F-16 control design. The detail of TS fuzzy

model with F-16 aircraft (5.23) is given in Appendix A. Without any input and

output constraints, we may solve the optimization problem with LMI constraint

(5.33) and find the stabilizing controller

K =

0 0 0 0 0 0 0 0

0 611.2 −0.1 −24.7 −2177.8 0 0.1 7706.3

22.7 −0.1 170.3 0 0.7 387 2276.4 −1.9−15.2 −0.1 338.5 0 0 −201.4 −507.5 −1.1

,


0 1 2 3−50

0

50

100

150

time(s)

rd

r(d

eg

)

0 1 2 3−200

0

200

400

600

time(s)

el(d

eg

)

0 1 2 3−15

−10

−5

0

5

time(s)

alp

ha

(d

eg

)

0 1 2 3−500

0

500

time(s)

ail(d

eg

)

0 1 2 3−15

−10

−5

0

5

time(s)

ph

i(d

eg

)

Figure 5.5. Lyapunov based stabilizing control


0 0.5 1 1.5 2−10

−5

0

5

time(s)

rd

r(d

eg

)

0 0.5 1 1.5 20

50

100

time(s)

el(d

eg

)

0 0.5 1 1.5 2−100

−50

0

time(s)

alp

ha

(d

eg

)

0 0.5 1 1.5 2−6

−4

−2

0

2

time(s)

ail(d

eg

)

0 0.5 1 1.5 2−150

−100

−50

0

time(s)

ph

i(d

eg

)

Figure 5.6. Linear stabilizing control


and the corresponding P matrix is

P =

0.0000 0.0000 −0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0001 −0.0000 −0.0000 −0.0002 −0.0000 0.0000 0.0008

−0.0000 −0.0000 0.0000 0.0000 0.0000 −0.0000 0.0000 −0.00000.0000 −0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −0.00000.0000 −0.0002 0.0000 0.0000 0.0034 0.0000 0.0000 −0.00550.0000 −0.0000 −0.0000 0.0000 0.0000 0.0001 0.0001 −0.00000.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 −0.00000.0000 0.0000 −0.0000 −0.0000 −0.0055 −0.0000 −0.0000 0.0130

.

The initial state is chosen as α = −0.2 rad (−11.46 degree) and φ = −0.2rad (−11.46 degree). Other initial states are all zero. The response of α, φ and

the control signals are shown in Fig. 5.5. For comparison, we also find the con-

troller using linear model on the equilibrium. The obtained controller using same

conditions gives

K =

0 0 0 0 0 0 0 0

−0.0073 5.9730 0.0039 −2.5628 −41.0560 −0.0010 −0.0025 162.8011

−1.6331 0.0046 0.3952 0.0015 0.0002 −16.9832 2.3450 0.0032

11.7954 0.0167 2.5714 0.0185 0.0034 139.4477 3.0502 0.0030

,


P =

0.0021 −0.0000 −0.0000 −0.0000 −0.0000 0.0001 0.0007 0.0000

−0.0000 0.0010 −0.0000 −0.0004 −0.0008 0.0000 0.0000 0.0023

−0.0000 −0.0000 0.0019 −0.0000 −0.0000 −0.0003 −0.0000 0.0000

−0.0000 −0.0004 −0.0000 0.0019 −0.0000 0.0000 −0.0000 −0.0014−0.0000 −0.0008 −0.0000 −0.0000 0.0039 0.0000 −0.0000 −0.00620.0001 0.0000 −0.0003 0.0000 0.0000 0.0018 0.0000 0.0000

0.0007 0.0000 −0.0000 −0.0000 −0.0000 0.0000 0.0021 0.0000

0.0000 0.0023 0.0000 −0.0014 −0.0062 0.0000 0.0000 0.0249

.

We can check that it really gives stabilizing control for the linear system lin-

earized at the equilibrium point. However, it cannot stabilize the nonlinear F-16

model. The response of α, φ and the control signals are shown in Fig. 5.6.


In the above simulation, we may notice that the proposed method can stabilize

the process while linear control fails. However, the control signal of the proposed

method is too large. In the following design, we wish to use constraints to limit

the input signal.

With the same TS fuzzy model of f-16 aircraft, we add the input constraint,

i.e., set µ = 100 and solve the LMIs feasible problem with constraints (5.33), (5.39)

and (5.40). The obtained controller is

K =

0 0 0 0 0 0 0 0

0.0003 4.9962 0.0072 −0.0330 −1.2235 0.0054 0.0031 11.2421

1.8132 0.0012 4.6381 0.0000 0.0001 −9.9594 10.6000 −0.0058−0.3218 0.0023 9.1266 0.0000 0.0032 −8.0655 −0.7151 −0.0071

,


P =

0.0006 0.0000 0.0007 −0.0000 −0.0000 −0.0004 0.0018 0.0000

0.0000 0.0029 0.0000 −0.0000 0.0001 −0.0000 0.0000 0.0048

0.0007 0.0000 0.0113 −0.0000 −0.0000 −0.0093 0.0043 0.0000

−0.0000 −0.0000 −0.0000 0.0000 0.0000 0.0000 −0.0000 −0.0001−0.0000 0.0001 −0.0000 0.0000 0.0135 −0.0000 −0.0000 −0.0104−0.0004 −0.0000 −0.0093 0.0000 −0.0000 0.1018 −0.0058 −0.00000.0018 0.0000 0.0043 −0.0000 −0.0000 −0.0058 0.0125 −0.00000.0000 0.0048 0.0000 −0.0001 −0.0104 −0.0000 −0.0000 0.0213

.

The simulation results are shown in Fig. 5.7. We may observe that the system

gives sluggish response. However, the magnitude of control signals is much smaller.

In attitude tracking design, the TS fuzzy model of the augmented system is

described by

x =9∑

i=1

hi(φ, θ)(Aix+Biu), (5.46)

where hi(φ, θ) = MφiMθi . Mφi and Mθi are the fuzzy membership functions of φ

and θ respectively. In our simulation, the nine operating points evenly distributed



0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

time(s)

rd

r(d

eg

)

0 2 4 6 8 10−0.5

0

0.5

1

time(s)

el(d

eg

)

0 2 4 6 8 10−15

−10

−5

0

5

time(s)

alp

ha

(d

eg

)

0 2 4 6 8 10−3

−2

−1

0

1

time(s)

ail(d

eg

)

0 2 4 6 8 10−15

−10

−5

0

5

time(s)

ph

i(d

eg

)

Figure 5.7. Lyapunov based stabilizing control with control signal constraints


(φ1, θ1) = (−0.15,−0.2),

(φ2, θ2) = (−0.15, 0),

(φ3, θ3) = (−0.15, 0.2),

(φ4, θ4) = (0,−0.2),

(φ5, θ5) = (0, 0),

(φ6, θ6) = (0, 0.2),

(φ7, θ7) = (0.15,−0.2),

(φ8, θ8) = (0.15, 0),

(φ9, θ9) = (0.15, 0.2)

as the nine operating points. At each operating point, we can obtain the lin-

ear model at these operating points from (5.19) and (5.21). For example, at the

operating point (φ, θ) = (−0.2,−0.2), from (5.21), we get

A1 =

-3.6722 0.0003 0.6741 0 0 -30.9113 0 0 0 0 0 0

0 -1.3859 -0.0029 0.0000 -2.4977 0 -0.0003 -0.0002 0 0 0 0

-0.0263 0.0025 -0.4986 0 0 9.4747 0 0 0 0 0 0

0 -0.6501 0 -0.0202 6.8833 -6.3521 0.0559 -31.9571 0 0 0 0

0 0.9048 0 -0.0002 -1.0285 0 0.0064 0.0047 0 0 0 0

0.0389 0 -0.9916 0.0000 -0.0000 -0.3315 0.0636 -0.0005 0 0 0 0

1.000 0.0221 -0.1089 0 0 0 0 0 0 0 0 0

0 0.9801 0.1987 0 0 0 0 0 0 0 0 0

0 -0.1999 0.9861 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1.0000 -0.0000 0 0 0 0

0 0 0 0 0 0 -0.0000 1.0000 0 0 0 0

0 0 0 0 0 0 0 0 1.0000 0 0 0

,

and from (5.19), we obtain

B1 =

0 0 -0.7336 0.1310

0 -0.1824 0 0

0 0 -0.0327 -0.0643

0 0.1704 0 0

0 -0.0022 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

.

The resulting (Ai, Bi), i = 1, 2, · · · , 9, are given in Appendix B. Moreover, we select

triangle function in last section as the fuzzy membership functions, which are the


functions depicted in Fig. 5.2. The complete augmented model with all the data

is given in Appendix B.

In order to obtain better tracking speed, we incorporate the decay rate con-

straint into the controller design. In the simulation, we choose the delay rate

α = 0.0224. The obtained controller is

K = 103

0 0 0 0 0 0 0 0 0 0

0.0000 0.2643 0.0000 0.0002 −0.0516 0.0000 −0.0000 1.4067 −0.0000 0.8095

−0.0004 −0.0000 0.0488 0.0000 0.0000 0.0488 0.0090 −0.0002 −0.0088 −0.0002

0.0016 −0.0003 0.2624 0.0000 0.0000 0.3774 −0.0221 −0.0013 −0.0674 −0.0009

.

The step amplitude for θr and φr are 0.2 rad (11.46 degree) and 0.15 rad (8.59

degree) and the step time is at t = 0s and t = 100s respectively. The result

shows that the tracking problem is solvable by this scheme. For comparison, the

controller ia also designed on the basis of the linear model derived from the same

condition. It is shown that the controller based on the linear model fails to track

the desired trajectory. The simulation results are shown in Fig. 5.8 and Fig. 5.9.

In the simulation, we can observe that the Lyapunov based control can stabilize

the F-16 aircraft as well as tracking the specified altitude signals.

5.5 Gain Scheduling Control

In the last section, we proposed Lyapunov based methods for F-16 control. How-

ever, the method gives slow response in the altitude tracking control. we may

notice that the scheme only employ the linear controller and they inherently linear

control system except that the model is TS fuzzy model. In this section, we try to

use fuzzy controller and gain scheduling method to get faster response of tracking

control.

5.5.1 Gain Scheduled Linear Quadratic Regulator Design

Suppose that the TS model is described by

x(t) =N∑

i=1

hiAix(t) +Biu(t),

y(t) = Cx(t),


0 100 200 300−20

0

20

40

60

time(s)

rd

r(d

eg

re

e)

0 100 200 300−4

−3

−2

−1

0

time(s)e

l(d

eg

re

e)

0 100 200 300−0.1

0

0.1

0.2

time(s)

ph

i(ra

d)

0 100 200 300−30

−20

−10

0

10

time(s)

ail(d

eg

re

e)

0 100 200 3000

0.1

0.2

0.3

0.4

time(s)

the

ta(ra

d)

Figure 5.8. Lyapunov based tracking control


0 50 100 150 200−10

0

10

20

30

time(s)

rd

r(d

eg

re

e)

0 50 100 150 200−40

−20

0

20

40

time(s)

el(d

eg

re

e)

0 50 100 150 200−0.1

0

0.1

0.2

0.3

time(s)

ph

i(ra

d)

0 50 100 150 200−15

−10

−5

0

5

time(s)

ail(d

eg

re

e)

0 50 100 150 2000

0.1

0.2

0.3

0.4

time(s)

the

ta(ra

d)

Figure 5.9. Linear tracking control


and the controller is described by

u =N∑

i=1

hiKix.

Our objective in this section is to develop gain scheduled linear quadratic regulator

(LQR) fuzzy controller for the TS model to obtain desirable closed-loop response

characteristics. In the design, we can select performance criterion at different

design points, which reflect our concern with closed-loop stability and good time

response.

At each design point, we can obtain a linear time-invariant state-variable model

x = Aix+Biu, (5.47)

with x(t) ∈ Rn the state, u(t) ∈ Rm the control input. The controls will be state

feedbacks of the form

u = Kix, (5.48)

whereKi is anm×nmatrix of constant feedback coefficient to be determined by the

design procedure. The objective of state regulation for the aircraft is to drive any

initial condition error to zero, thus maintaining stability. This may be achieved by

selecting the control input u(t) to minimize a quadratic cost or performance index

(PI) of the type

Ji =1

2

∫ ∞

0

(xTQix+ uTRiu)dt, (5.49)

where Qi and Ri are symmetric positive semi-definite weighting matrices. The

definiteness assumption on Qi and Ri guarantee that Ji is nonnegative. The rel-

ative magnitudes of Qi and Ri may be selected to trade off requirements on the

smallness of the state against requirements on the smallness of the input. We can

observe that larger Ri penalizes the control signal, so that they will be smaller in

norm relative to the state vector. On the other hand, to make x(t) go to zero more

quickly with respect to time, we may select a larger Qi.

By substituting the control (5.48) into (5.47), the closed-loop system equations

are found to be

x = (Ai +BiKi)x = Acix, (5.50)


the performance index may be expressed in terms of Ki as

Ji =1

2

∫ ∞

0

xT (Qi +KTi RiKi)xdt, (5.51)

The design problem is now to select the gain Ki so that Ji is minimized subject

to the dynamical constraint (5.49). The dynamical optimization problem may be

converted into an equivalent static one that is easier to solve. Suppose that we can

find a constant, symmetric, positive semi-definite matrix Pi so that

d

dt(xTPix) = −xT (Qi +KT

i RiKi)x, (5.52)

then Ji may be written as

Ji =1

2xT (0)Pix(0)−

1

2limt→∞

xT (t)Pix(t). (5.53)

Assuming that the closed-loop system is asymptotically stable so that x(t) vanishes

with time, this becomes

Ji =1

2xT (0)Pix(0). (5.54)

If P satisfies (5.52), we may use (5.49) to see that

−xT (Qi +KTi RiKi)x =

d

dt(xTPix) = xTPix+ xT ...Pix = xT (AT

ciPi + PiAci)x.(5.55)

Since this must hold for all initial conditions, and hence for all state trajectories

x(t), we may write

ATciPi + PiAci +KT

i RiKi +Qi = 0. (5.56)

If Ki and Qi are given, Pi can be obtained by solving (5.56). We can also obtain

Ki = R−1i BT

i Pi. (5.57)

To obtain the output feedback gain minimizing the performance index, we need

to solve the two coupled equations (5.56) and (5.57). Once the performance index

weighting matrices Qi and Ri have been selected, the determination of the optimal

feedback gain Ki is a formal procedure relying on the solution of nonlinear coupled


matrix equations. For nonlinear control, the process changes with the operating

conditions and the control gains should be gain scheduled. Gain scheduling is a

nonlinear feedback of special type; it has a linear controller whose parameters are

changed as a function of operating conditions in a preprogrammed way. When

scheduling variables have been determined, the controller parameters are calcu-

lated at a number of operating points by using some suitable design method. The

controller is thus tuned for each operating condition. Notice that the stability

and performance of the system are typically evaluated by simulation. In this re-

port, we use gain scheduling method to design tracking control based on TS fuzzy

modelling. In this scheme, we set the fuzzy controller

K =N∑

i=1

hi(x)Ki(t), (5.58)

where each Ki is obtained from the LQR optimal design for each linear model in

TS modelling and hi(x) is got from the coefficients of the TS fuzzy model in section

(5.3).

5.5.2 Simulation

Now, we employed the proposed gain scheduling approach in F-16 aircraft design.

Consider the TS model (5.46) in last section and reference signals are given step

signals on φ and θ. For Qi = diag(1, 1E3, 1, 1, 1, 1, 1, 1, 1, 1E6, 1E7) and Ri =

diag(1, 1, 1, 1), using the gain scheduling optimal design, the fuzzy controller is

given in Appendix D (controller I) and the simulation results are shown in Fig.

5.10.

Moveover, we may try to track α, β and φ, the TS fuzzy model of this system

is described by

x =9∑

i=1

hi(α, φ)(Aix+Biu), (5.59)

where hi(α, φ) = MαiMφi . Mαi and Mφi are the fuzzy membership functions of α

and φ respectively. In our simulation, the nine operating points evenly distributed


0 2 4 6−2

0

2

4

6

time(s)

rd

r(d

eg

)

0 2 4 6−10

−5

0

5

time(s)

el(d

eg

)

0 2 4 60

5

10

time(s)

ph

i(d

eg

)

0 2 4 6−30

−20

−10

0

10

time(s)

ail(d

eg

)

0 2 4 60

5

10

15

time(s)

the

ta(d

eg

)

Figure 5.10. The gain scheduled tracking control with φ θ


0 5 10 15 20−50

0

50

time(s)

rd

r(d

eg

)

0 5 10 15 20−100

0

100

200

time(s)

el(d

eg

)

0 5 10 15 20−5

0

5

10

time(s)

alp

ha

(d

eg

)

0 5 10 15 20−10

0

10

20

time(s)

ail(d

eg

)

0 5 10 15 20−5

0

5

time(s)

be

ta(d

eg

)

0 5 10 15 20−20

0

20

40

60

time(s)

ph

i(d

eg

)

Figure 5.11. The gain scheduled tracking control I with α β and φ


0 5 10 15 20−200

−100

0

100

200

time(s)

rd

r(d

eg

)

0 5 10 15 20−100

0

100

200

time(s)

el(d

eg

)

0 5 10 15 20−5

0

5

10

time(s)

alp

ha

(d

eg

)

0 5 10 15 20−20

0

20

40

60

time(s)

ail(d

eg

)

0 5 10 15 20−0.05

0

0.05

time(s)

be

ta(d

eg

)

0 5 10 15 20−20

0

20

40

60

time(s)

ph

i(d

eg

)

Figure 5.12. The gain scheduled tracking control II with α β and φ


0 5 10 15 20−20

−10

0

10

20

time(s)

rd

r(d

eg

)

0 5 10 15 20−20

−10

0

10

20

time(s)

el(d

eg

)

0 5 10 15 20−5

0

5

10

time(s)

alp

ha

(d

eg

)

0 5 10 15 20−10

−5

0

5

10

time(s)

ail(d

eg

)

0 5 10 15 20−0.4

−0.2

0

0.2

0.4

time(s)

be

ta(d

eg

)

0 5 10 15 20−20

0

20

40

60

time(s)

ph

i(d

eg

)

Figure 5.13. The gain scheduled tracking control III with α β and φ


Figure 5.14. Lee’s Backstepping control



(α1, φ1) = (−0.1,−0.8),

(α2, φ2) = (−0.1, 0),

(α3, φ3) = (−0.1, 0.8),

(α4, φ4) = (0,−0.8),

(α5, φ5) = (0, 0),

(α6, φ6) = (0, 0.8),

(α7, φ7) = (0.1,−0.8),

(α8, φ8) = (0.1, 0),

(α9, φ9) = (0.1, 0.8)

as the nine operating points. At each operating point, we can obtain the linear

model at these operating points from (5.19) and (5.21). The resulting (Ai, Bi), i =

1, 2, · · · , 9, are given in Appendix E. Moreover, we select triangle function in last

section as the fuzzy membership functions, which are the function is depicted in Fig.

5.2. ForRi = diag(1000, 1000, 100, 100 andQi = diag(1, 1e9, 1, 1, 1, 1, 1, 1, 1e10, 1e9, 1e9),

the controller is given in Appendix E (controller I) and simulation results are given

in Fig. 5.11. Compared with the nonlinear control law of Taeyoung and Youdan

(2001) in Fig. 5.14, we achieve similar output performance. However, the ampli-

tude of the beta response is much larger than that of the nonlinear control. We

detune the weightings in our control design. For Ri = diag(1000, 1000, 100, 100

and Qi = diag(1, 1e8, 1, 1, 1, 1, 1, 1, 1e10, 1e15, 1e9), the simulation result is shown

in Fig. 5.12 and controller is given in Appendix E (controller II). We can ob-

serve that the maximal amplitude of beta is less than 0.04 deg which is much

smaller than around 4 deg in the original design in Fig. 5.11. The response of

alpha and theta has little change. Moreover, we redesign our controller to re-

duce the amplitude of the control signal. For Ri = diag(1000, 1000, 1000, 1000)

and Qi = diag(1, 1e9, 1, 1, 1, 1, 1, 1, 1e9, 1e12, 1e8), the simulation result is shown

in Fig. 5.13. We can observe that the amplitude of control signal is smaller and

the response of outputs is more sluggish.


Table 5.1. The φ tracking specifications of the two control methods

RT ST OV |δe|max |δa|max |δr|max

Lyaponuv based control 40 80 0.06 1.4388 1.9203 11.4588

Gain scheduling control I 0.9508 2.5799 0.0523 2.4171 1.6574 14.3380

Gain scheduling control II 0.3562 0.4550 0.0288 2.7179 8.2767 11.5618

5.6 Discussions and Conclusions

5.6.1 Comparative Studies

We now study performance of the altitude tracking control of the proposed two

approaches. The reference signals of φ and θ are chosen as step signals with altitude

0.1 rad (5.73 degree). For the Lypunov based control, the controller is given by

(5.47); for the gain scheduling control I and II, we can find the fuzzy controller

from Appendix D. The comparison of the specification index is given in Table. 5.1

and Table. 5.2 .

In gain scheduling control I, we chooseQi = diag(1, 1E3, 1, 1, 1, 1, 1, 1, 1, 1E6, 1E7)

and Ri = diag(1000, 1000, 1000, 1000), the responses of output and control signals

are shown in Fig. 5.15 and Fig. 5.16. The detail of response in the initial phase is

also given in Fig. 5.17.

In the second design, we try to get even better output performance. For

Qi = diag(1, 1E3, 1, 1, 1, 1, 1, 1, 1, 1E6, 1E7) and Ri = diag(1, 1, 1, 1), the result

of outputs and control signals are given in Fig. 5.18 and Fig. 5.19. The detail

of response in the initial phase is also given in Fig. 5.20. In this design, we can

observe that the output response are speeded up. Inevitably, the control signals

have larger overshoot and (or) undershoot in initial part.


0 5 10 15 20 25 300

1

2

3

4

5

6

7

time(s)

the

ta(d

eg

)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

time(s)

ph

i(d

eg

)

Figure 5.15. Outputs of the two proposed methods

(gain scheduling —, Lyapunov based - - -)

One can see that the gain scheduling controller perform much better than Lya-

punov based controller. This is because that the former is nonlinear and can adapt


0 100 200 300 400 500 600 700−5

0

5

10

15

time(s)

rdr(

de

g)

0 100 200 300 400 500 600 700−3

−2

−1

0

1

time(s)

el(d

eg

)

0 100 200 300 400 500 600 700−2

−1

0

1

time(s)

ail(d

eg

)

Figure 5.16. Control signals of the two proposed methods


to nonlinearity of F-16 aircraft, while the latter is of fixed gain controller and in

fact linear, and they can only ensure stability and tracking by design.


0 2 4 6−0.2

0

0.2

0.4

0.6

time(s)

rdr(

de

g)

0 2 4 6−0.6

−0.4

−0.2

0

0.2

time(s)

el(d

eg

)

0 2 4 60

5

10

time(s)

ph

i(d

eg

)

0 2 4 6−2

−1

0

1

time(s)

ail(d

eg

)

0 2 4 60

2

4

6

8

time(s)

the

ta(d

eg

)

Figure 5.17. Initial part of gain scheduling control

5.6.2 Conclusions

In this chapter, we have carried out and achieved the following:


0 5 10 15 20 25 300

1

2

3

4

5

6

7

time(s)

the

ta(d

eg

)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

time(s)

ph

i(d

eg

)

Figure 5.18. Outputs of the two proposed methods


(i) The problem to design both stabilizing and tracking controller for F-16

aircraft systems has been addressed via the TS fuzzy modelling approach. To our


0 100 200 300 400 500 600 700−5

0

5

10

15

time(s)

rdr(

de

g)

0 100 200 300 400 500 600 700−3

−2

−1

0

1

time(s)

el(d

eg

)

0 100 200 300 400 500 600 700−10

−5

0

5

time(s)

ail(d

eg

)

Figure 5.19. Control signals of the two proposed methods


best knowledge, the TS fuzzy approach has not been used to find the approximated

TS fuzzy model and design both stabilizing and tracking controller for nonlinear


0 2 4 6−1

0

1

2

3

time(s)

rdr(

de

g)

0 2 4 6−3

−2

−1

0

1

time(s)

el(d

eg

)

0 2 4 6−2

0

2

4

6

time(s)

ph

i(d

eg

)

0 2 4 6−10

−5

0

5

time(s)

ail(d

eg

)

0 2 4 60

2

4

6

time(s)

the

ta(d

eg

)

Figure 5.20. Initial part of gain scheduling control

F-16 aircraft systems.

(ii) Both basic and augmented TS fuzzy models of F-16 have been obtained


Table 5.2. The θ tracking specifications of the two control methods

RT ST OV |δe|max |δa|max |δr|max

Lyapunov based control 8 10 0.01 1.4388 1.9203 11.4588

Gain scheduling control I 0.9508 2.0021 0.0653 2.4171 1.6574 14.3380

Gain scheduling control II 0.3562 0.4550 0.0377 2.7179 8.2767 11.5618

using the best-available F-16 nonlinear model the literature and validated to be

reasonable accurate in the operating range of interests.

(iii) Two control methods have been studied through the available TS fuzzy

modelling and fuzzy control design methods (Takagi and Sugeno, 1985) and shown

to be feasible and better than the normal linear control.

(iv) The Lyapunov based control is developed for stabilization of a TS fuzzy

model and extended to handle attitude tracking problem when the model is aug-

mented with integrator at each output. The most important feature of the method

is guaranteed stability and step-signal tracking of the closed-loop system if the TS

fuzzy model error is small enough, while most other nonlinear controllers cannot.

Besides, The solutions are simple to get since it involves an LMI feasibility problem

only.

(v) The gain scheduling control shows the better performance. It has a very

simple control structure and straightforward design from TS model; yet it can

perform as well as the best nonlinear controller of high complexity and very so-

phisticated design. However, the gain scheduling control cannot guarantee the

stability of the closed-loop system.

(vi) These results could be useful to understand and design a flight control

system for rapid maneuver of F-16 aircraft.

Chapter 6

Conclusions

6.1 Main Findings

In this thesis, several new results are obtained around control system design for

better performance and robustness. Briefly, the results are summarized as follows:

A. PID Controller Design

In this thesis, three new PID controller design approaches are developed. In

the first method, a simple scheme for the design of PID controllers for uncertain

processes which achieve gain and phase margin specifications is proposed. The

approach is based on the uncertain model of the plant. The design problem is

formulated into a convex optimization problem with quadratic constraint, then it is

converted into an optimization problem with LMI constraints which can be solved

effectively using interior point method. The optimal PID controller is obtained

directly to satisfy the gain and phase margins. Simulation and comparisons are

presented to demonstrate the effectiveness of the method. For the second approach,

a new criterion for robust stability is developed which made use of both the gain and

phase information of the open-loop system in order to reduce the conservativeness

of the small gain theory. Based on it, a robust PID design approach is developed for

second order plus dead time uncertain processes. Simulation shows the effectiveness

of the method. In the third work, algorithms based on the iterative LMI technique

are developed to design PI controller for linear time invariant processes with state

160

Chapter 6. Conclusions 161

time delay, which guarantee the stability of closed-loop system or H∞ performance

specification. An example is provided to illustrate the applicable of the proposed

method.

B. IMC and Smith Controller Design

Two new approaches for robust IMC controller design, one in time domain

and another in frequency domain, are proposed. For the IMC design in time

domain, it is shown that the IMC optimal H2 performance design with robust

stability constraints can be converted into dynamic output feedback control design

problem. With the new formulation, it is shown that the robust IMC controller

design is solvable via solving a system of LMI. Two examples are presented with

the demonstrated effectiveness of the proposed method. For the IMC design in

frequency domain, a new IMC controller design method is proposed to achieve

the optimal performance and robust stability. The method is based on LMI/BMI

optimization. It is shown that better optimal performance is achieved compared

with Morari’s method with same robust stability constraints. The advantage of

the frequency domain method is that it can cope with the processes with time

delay. Moreover, the controllers obtained through frequency domain approach are

fixed order ones while the resultant controllers via time domain method usually

are of high order. It is also noted that the two proposed schemes are systemic

methods while most of existing IMC tuning schemes use trial and error method.

As we move on to the Smith control with open-loop unstable processes, a simple and

effective design is proposed based on our new modified Smith predictor structure for

unstable process with dead time. Moreover, the internal stability of the proposed

system is analyzed. Simulation illustrates that the proposed method yields much

improvement of load disturbance responses over the best existing method.

C. Decoupling Control Design

In this thesis, a new necessary and sufficient condition for the solvability of the

decoupling problem with internal stability for unity output feedback systems with

square plants is given. The conditions are only concerned with Gd,min of the plant


G. Compared with the existing method, the proposed criterion is simpler, more

understandable and make use of transfer function matrices only. For decoupling

controller design, a new method for IMC system is proposed to gain good loop

performance and largely remove the effect of process coupling simultaneously for

multi-variable plants. The controller may be obtained via LMI optimization. The

robustness of the proposed method is analyzed. Two examples are presented with

the demonstrated effectiveness of the proposed method.

D. TS Fuzzy Control Design

A TS fuzzy modelling method and two fuzzy control strategy are discussed in

the thesis. Both basic and augmented TS fuzzy models of F-16 have been obtained

using the best-available F-16 nonlinear model the literature and validated to be

reasonable accurate in the operating range of interests. Two control methods, Lya-

punov based control and gain scheduling control, have been proposed and shown

to be feasible and better than the normal linear control. The Lyapunov based

control are developed for stabilization of a TS fuzzy model and extended to handle

attitude tracking problem when the model is augmented with integrator at each

output. The most important feature of the method is guaranteed stability and

step-signal tracking of the closed-loop system if the TS fuzzy model error is small

enough, while most other nonlinear controllers cannot. Besides, The solution is

simple to get since it involves an LMI feasibility problem only. The gain schedul-

ing control shows better performance. It has a very simple control structure and

straightforward design from TS model; Yet it can perform as well as the best non-

linear controller of high complexity and very sophisticated design. However, the

gain scheduling control cannot guarantee the stability of the closed-loop system.

The stability of the closed-loop system can only be checked through simulation.

6.2 Suggestions for Further Work

The thesis has taken the full route from initial ideas, via theoretical developments,

to methodologies that can be applied to relevant practical problems. Several new


results have been obtained but some topics remain open and are recommended for

further work.

A. Multi-variable Robust PID Controller Design

Two schemes of single variable robust PID controller design have been proposed

in this thesis. For the single variable case, frequency domain stability margins such

as gain and phase margins are very popular, and used for performance assessment,

design specifications and robustness measure of PID control systems. There are

some attempts to define multi-variable system stability margins in frequency do-

main, but none of them is well known. One may look for better definitions which

are meaningful, useful and easily checkable with clear link to the single variable

case. They can then be used as performance assessment, design specifications and

robustness measure of Multi-variable PID control systems and may lead to a large

branch of tuning rules similar to the single variable case. Basically, there are two

types of Multivariable PID controllers: decentralized (or multi-loop, diagonal) and

centralized (full matrix). The problems should be studied for each type of the

controllers, respectively, as the difference in controller structures calls for different

solution methods in general.

B. Multi-variable Robust IMC Controller Design

In this thesis, the robust IMC designs are developed both in time domain and

frequency domain for SISO systems. Based on the results of single loop design,

more studies may be given to investigate if the ideas is extendible to multi-variable

processes. The robust stability, decoupling and loop performance should be tackled

in the multi-variable IMC system design. Since there are usually more states

and modes in the multi-variable systems, simple design procedures and efficient

computation methods are needed to be investigated.

C. Multi-variable Smith Predictor for Unstable Processes

A modified Smith predictor design for single variable unstable processes has

been proposed. For multi-variable processes, the proposed approach may encounter

problems because of the coupling and different time delay of each element in the


processes. One possible method is to develop decoupling controller to make the

system decoupled. Thus, the scheme used for single variable processes can be used

for the decoupled processed. However, exact decoupling is usually sensitive to

the process model used. Thus, it is desirable to design robust decoupling Smith

predictor such that the interaction of the resultant system is kept within a certain

tolerance for the whole family of the uncertain processes. Another possible scheme

is to find controller with full matrix to make the time delay not appear in the

denominator matrix as well as to maintain the internal stability of the closed-loop

system. Algorithm is needed to be developed to find such controllers.

D. Decoupling Design for Nonlinear Processes

The processes of decoupling design considered in this thesis are assumed to be

linear. However, in practice, all physical systems are nonlinear in nature. Some

complex processes may not be described by linear model with enough accuracy.

Thus, the nonlinear model may be needed in some practical control system design.

In this thesis, the TS fuzzy modelling approach is developed with F-16 aircraft.

However, in the controller design, the decoupling approach is not employed. Works

will be needed to investigate a new method to decouple complex nonlinear control

systems and achieve better performance.

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Appendix A

TS Fuzzy Basic Model

The model with α ∈ [−0.2, 0.2] and φ ∈ [−0.2, 0.2] assumes the form:

x =9∑

i=1

hi(α, φ)(Aix+Biu).

Selection of the operating points and the fuzzy rules

Rule 1: If α = −0.2 and φ = −0.2, then the linearized model is x = A1x+B1u.

Rule 2: If α = 0 and φ = −0.2, then the linearized model is x = A2x+B2u.

Rule 3: If α = 0.2 and φ = −0.2, then the linearized model is x = A3x+B3u.

Rule 4: If α = −0.2 and φ = 0, then the linearized model is x = A4x+B4u.

Rule 5: If α = 0 and φ = 0, then the linearized model is x = A5x+B5u.

Rule 6: If α = 0.2 and φ = 0, then the linearized model is x = A6x+B6u.

Rule 7: If α = −0.2 and φ = 0.2, then the linearized model is x = A7x+B7u.

Rule 8: If α = 0 and φ = 0.2, then the linearized model is x = A8x+B8u.

Rule 9: If α = 0.2 and φ = 0.2, then the linearized model is x = A9x+B9u.

Resulting linear models:

177


A1 =

-3.0489 0.0003 -0.9873 0 0 -3.7977 0 0

0 -1.3815 -0.0029 0.0009 -1.1356 0 -0.0335 0

0.0312 0.0025 -0.5181 0 0 9.7538 0 0

0 2.5473 0 -0.0832 191.1245 -6.3862 -80.3479 -31.5328

0 0.9639 0 0.0003 -1.4637 0 0.1846 -0.0127

-0.1609 0 -0.9798 0.0000 0.0004 -0.2779 0.0632 0.0005

1.0000 -0.0078 0.0386 0 0 0 0 0

0 0.9801 0.1987 0 0 0 0 0

, B1 =

0 0 -0.5983 0.1588

0 -0.1925 0 0

0 0 -0.0337 -0.0675

0 0.4584 0 0

0 -0.0020 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

A2 =

-3.6722 0.0003 0.6741 0 0 -30.9113 0 0

0 -1.3859 -0.0029 0.0000 -2.4977 0 -0.0005 0

-0.0263 0.0025 -0.4986 0 0 9.4747 0 0

0 -0.6501 0 -0.0202 7.2411 -6.3862 0.1259 -32.169

0 0.9048 0 -0.0003 -1.0189 0 0.0064 0.0001

0.0389 0 -0.9916 0.0000 0 -0.3220 0.0636 0.0005

1.0000 -0.0078 0.0386 0 0 0 0 0

0 0.9801 0.1987 0 0 0 0 0

, B2 =

0 0 -0.7336 0.1310

0 -0.1824 0 0

0 0 -0.0327 -0.0643

0 0.1704 0 0

0 -0.0022 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

A3 =

-3.2067 0.0003 1.8464 0 0 -65.6296 0 0

0 -1.6530 -0.0029 -0.0016 -0.7348 0 1.2180 0

-0.0826 0.0025 -0.5456 0 0 8.8395 0 0

0 -7.4582 0 -0.0917 -92.9348 -6.3862 -33.758 -31.5228

0 0.9031 0 -0.0006 -1.0248 0 0.0054 0.0128

0.2390 0 -0.9633 0.0000 -0.0004 -0.2984 0.0632 0.0005

1.0000 -0.0078 0.0386 0 0 0 0 0

0 0.9801 0.1987 0 0 0 0 0

, B3 =

0 0 -0.6839 0.1258

0 -0.2000 0 0

0 0 -0.0254 -0.0641

0 -0.1139 0 0

0 -0.0021 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

A4 =

-3.0489 0.0003 -0.9873 0 0 -3.7977 0 0

0 -1.3815 -0.0029 0.0009 -1.1690 0 0 0

0.0312 0.0025 -0.5181 0 0 9.7538 0 0

0 2.5473 0 -0.0832 111.2891 0 0 -31.5328

0 0.9639 0 0.0003 -1.2854 0 0 -0.0127

-0.1609 0 -0.9798 0.0000 0.000 -0.2779 0.0640 0.00

1.0000 0 0.0394 0 0 0 0 0

0 1.0000 0 0 0 0 0 0

, B4 =

0 0 -0.5983 0.1588

0 -0.1925 0 0

0 0 -0.0337 -0.0675

0 0.4584 0 0

0 -0.0020 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

A5 =

-3.6722 0.0003 0.6741 0 0 -30.9113 0 0

0 -1.3859 -0.0029 0.0000 -2.4977 0 0 0

-0.0263 0.0025 -0.4986 0 0 9.4747 0 0

0 -0.6501 0 -0.0202 7.8814 0 0 -32.169

0 0.9048 0 -0.0003 -1.0190 0 0 0

0.0389 0 -0.9916 0.0000 0 -0.3220 0.064 0

1.0000 0 0.0394 0 0 0 0 0

0 1.0000 0 0 0 0 0 0

, B5 =

0 0 -0.7336 0.1310

0 -0.1824 0 0

0 0 -0.0327 -0.0643

0 0.1704 0 0

0 -0.0022 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

A6 =

-3.2067 0.0003 1.8464 0 0 -65.6296 0 0

0 -1.6530 -0.0029 -0.0016 -1.9527 0 0 0

-0.0826 0.0025 -0.5456 0 0 8.8395 0 0

0 -7.4582 0 -0.0917 -58.4168 0 0 -31.5287

0 0.9031 0 -0.0006 -1.0240 0 0 0.0127

0.2390 0 -0.9633 0.0000 -0.0000 -0.2987 0.0640 0

1.0000 0.00 0.0394 0 0 0 0 0

0 1.0000 0 0 0 0 0 0

, B6 =

0 0 -0.6839 0.1258

0 -0.2000 0 0

0 0 -0.0254 -0.0641

0 -0.1139 0 0

0 -0.0021 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

A7 =

-3.0489 0.0003 -0.9873 0 0 -3.7977 0 0

0 -1.3815 -0.0029 0.0009 -1.1356 0 0.0335 0

0.0312 0.0025 -0.5181 0 0 9.7538 0 0

0 2.5473 0 -0.0832 191.1245 6.3862 80.3479 -31.5328

0 0.9639 0 0.0003 -1.4637 0 0.1846 -0.0127

-0.1609 0 -0.9798 -0.0000 -0.0004 -0.2779 0.0632 -0.0005

1.0000 0.0078 0.0386 0 0 0 0 0

0 0.9801 -0.1987 0 0 0 0 0

, B7 =

0 0 -0.5983 0.1588

0 -0.1925 0 0

0 0 -0.0337 -0.0675

0 0.4584 0 0

0 -0.0020 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0


A8 =

3.6722 0.0003 0.6741 0 0 -30.9113 0 0

0 -1.3859 -0.0029 0.0000 -2.4977 0 0.0005 0

-0.0263 0.0025 -0.4986 0 0 9.4747 0 0

0 -0.6501 0 -0.0202 7.2411 6.3862 -0.1259 -32.169

0 0.9048 0 -0.0003 -1.0189 0 -0.0064 0.0001

0.0389 0 -0.9916 -0.0000 0 -0.3220 0.0636 -0.0005

1.0000 0.0078 0.0386 0 0 0 0 0

0 0.9801 -0.1987 0 0 0 0 0

, B8 =

0 0 -0.7336 0.1310

0 -0.1824 0 0

0 0 -0.0327 -0.0643

0 0.1704 0 0

0 -0.0022 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

A9 =

-3.2067 0.0003 1.8464 0 0 -65.6296 0 0

0 -1.6530 -0.0029 -0.0016 -0.7348 0 -1.2180 0

-0.0826 0.0025 -0.5456 0 0 8.8395 0 0

0 -7.4582 0 -0.0917 -92.9348 6.3862 33.7585 -31.5228

0 0.9031 0 -0.0006 -1.0248 0 -0.0054 0.0128

0.2390 0 -0.9633 -0.0000 0.0004 -0.2984 0.0632 -0.0005

1.0000 0.0078 0.0386 0 0 0 0 0

0 0.9801 -0.1987 0 0 0 0 0

, B9 =

0 0 -0.6839 0.1258

0 -0.2000 0 0

0 0 -0.0254 -0.0641

0 -0.1139 0 0

0 -0.0021 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

Resulting Weighting Functions:

hi(α, φ) =MαiMφi , i = 1, 2, 3,

where

Mα1 =

1 if α ≤ −0.22.5 ∗ α− 0.5 if − 0.2 < α < 0.2

0 if α ≥ 0.2

Mα2 =

0 if α ≤ −0.25 ∗ α + 1 if − 0.2 < α ≤ 0

−5 ∗ α + 1 if 0 < α < 0.2

0 if α ≥ 0.2

Mα3 =

0 if α ≤ −0.22.5 ∗ α + 0.5 if − 0.2 < α < 0.2

1 if α ≥ 0.2

Mφ1 =

1 if φ ≤ −0.22.5 ∗ φ− 0.5 if − 0.2 < φ < 0.2

0 if φ ≥ 0.2


Mφ2 =

0 if φ ≤ −0.25 ∗ φ+ 1 if − 0.2 < φ ≤ 0

−5 ∗ φ+ 1 if 0 < φ < 0.2

0 if φ ≥ 0.2

Mφ3 =

0 if φ ≤ −0.22.5 ∗ φ+ 0.5 if − 0.2 < φ < 0.2

1 if φ ≥ 0.2

Appendix B

TS Fuzzy Augmented Model I

The model with φ ∈ [−0.2, 0.2] and θ ∈ [−0.15, 0.15] assumes the form:

˙x =9∑

i=1

hi(φ, θ)(Aix+ Biu)


Rule 1: If φ = −0.2 and θ = −0.15, then the linearized model is ˙x = A1x+ B1u.

Rule 2: If φ = 0 and θ = −0.15, then the linearized model is ˙x = A2x+ B2u.

Rule 3: If φ = 0.2 and θ = −0.15, then the linearized model is ˙x = A3x+ B3u.

Rule 4: If φ = −0.2 and θ = 0, then the linearized model is ˙x = A4x+ B4u.

Rule 5: If φ = 0 and θ = 0, then the linearized model is ˙x = A5x+ B5u.

Rule 6: If φ = 0.2 and θ = 0, then the linearized model is ˙x = A6x+ B6u.

Rule 7: If φ = −0.2 and θ = 0.15, then the linearized model is ˙x = A7x+ B7u.

Rule 8: If φ = 0 and θ = 0.15, then the linearized model is ˙x = A8x+ B8u.

Rule 9: If φ = 0.2 and θ = 0.15, then the linearized model is ˙x = A9x+ B9u.

181


Resulting linear models:

A1 =

-3.6722 0.0003 0.6741 0 0 -30.9113 0 0 0 0 0 0

0 -1.3859 -0.0029 0.0000 -2.4977 0 -0.0003 -0.0002 0 0 0 0

-0.0263 0.0025 -0.4986 0 0 9.4747 0 0 0 0 0 0

0 -0.6501 0 -0.0202 6.8833 -6.3521 0.0559 -31.9571 0 0 0 0

0 0.9048 0 -0.0002 -1.0285 0 0.0064 0.0047 0 0 0 0

0.0389 0 -0.9916 0.0000 -0.0000 -0.3315 0.0636 -0.0005 0 0 0 0

1.000 0.0221 -0.1089 0 0 0 0 0 0 0 0 0

0 0.9801 0.1987 0 0 0 0 0 0 0 0 0

0 -0.1999 0.9861 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1.0000 -0.0000 0 0 0 0

0 0 0 0 0 0 -0.0000 1.0000 0 0 0 0

0 0 0 0 0 0 0 0 1.0000 0 0 0

B1 =

0 0 -0.7336 0.1310

0 -0.1824 0 0

0 0 -0.0327 -0.0643

0 0.1704 0 0

0 -0.0022 0 0

0 0 0.0003 0.0008

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

A2 =

-3.6722 0.0003 0.6741 0.0000 0.0000 -30.9113 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.3859 -0.0029 0.0000 -2.4977 0.0000 0.0000 -0.0007 0.0000 0.0000 0.0000 0.0000

-0.0263 0.0025 -0.4986 0.0000 0.0000 9.4747 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.6501 0.0000 -0.0202 7.5202 0.0000 0.0000 -32.0497 0.0000 0.0000 0.0000 0.0000

0.0000 0.9048 0.0000 -0.0003 -1.0286 0.0000 0.0000 0.0048 0.0000 0.0000 0.0000 0.0000

0.0389 0.0000 -0.9916 0.0000 0.0000 -0.3316 0.0637 0.0000 0.0000 0.0000 0.0000 0.0000

1.0000 0.0000 -0.1111 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 1.0062 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000

B2 =

0.0000 0.0000 -0.7336 0.1310

0.0000 -0.1824 0.0000 0.0000

0.0000 0.0000 -0.0327 -0.0643

0.0000 0.1704 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


A3 =

-3.6722 0.0003 0.6741 0.0000 0.0000 -30.9113 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.3859 -0.0029 0.0000 -2.4977 0.0000 0.0003 -0.0002 0.0000 0.0000 0.0000 0.0000

-0.0263 0.0025 -0.4986 0.0000 0.0000 9.4747 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.6501 0.0000 -0.0202 6.8833 6.3521 -0.0559 -31.9571 0.0000 0.0000 0.0000 0.0000

0.0000 0.9048 0.0000 -0.0002 -1.0285 0.0000 -0.0064 0.0047 0.0000 0.0000 0.0000 0.0000

0.0389 0.0000 -0.9916 -0.0000 0.0000 -0.3315 0.0636 0.0005 0.0000 0.0000 0.0000 0.0000

1.0000 -0.0221 -0.1089 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.9801 -0.1987 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.1999 0.9861 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000

B3 =

0.0000 0.0000 -0.7336 0.1310

0.0000 -0.1824 0.0000 0.0000

0.0000 0.0000 -0.0327 -0.0643

0.0000 0.1704 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

A4 =

-3.6722 0.0003 0.6741 0.0000 0.0000 -30.9113 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.3859 -0.0029 0.0000 -2.4977 0.0000 -0.0005 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0263 0.0025 -0.4986 0.0000 0.0000 9.4747 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.6501 0.0000 -0.0202 7.2411 -6.3862 0.1259 -32.1690 0.0000 0.0000 0.0000 0.0000

0.0000 0.9048 0.0000 -0.0003 -1.0189 0.0000 0.0064 0.0001 0.0000 0.0000 0.0000 0.0000

0.0389 0.0000 -0.9916 0.0000 0.0000 -0.3220 0.0636 0.0005 0.0000 0.0000 0.0000 0.0000

1.0000 -0.0078 0.0386 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.9801 0.1987 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.1988 0.9808 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000

B4 =

0.0000 0.0000 -0.7336 0.1310

0.0000 -0.1824 0.0000 0.0000

0.0000 0.0000 -0.0327 -0.0643

0.0000 0.1704 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


A5 =

-3.6722 0.0003 0.6741 0.0000 0.0000 -30.9113 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.3859 -0.0029 0.0000 -2.4977 0.0000 -0.0005 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0263 0.0025 -0.4986 0.0000 0.0000 9.4747 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.6501 0.0000 -0.0202 7.2411 -6.3862 0.1259 -32.1690 0.0000 0.0000 0.0000 0.0000

0.0000 0.9048 0.0000 -0.0003 -1.0189 0.0000 0.0064 0.0001 0.0000 0.0000 0.0000 0.0000

0.0389 0.0000 -0.9916 0.0000 0.0000 -0.3220 0.0636 0.0005 0.0000 0.0000 0.0000 0.0000

1.0000 -0.0078 0.0386 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.9801 0.1987 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.1988 0.9808 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000

B5 =

0.0000 0.0000 -0.7336 0.1310

0.0000 -0.1824 0.0000 0.0000

0.0000 0.0000 -0.0327 -0.0643

0.0000 0.1704 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

A6 =

-3.6722 0.0003 0.6741 0.0000 0.0000 -30.9113 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.3859 -0.0029 0.0000 -2.4977 0.0000 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0263 0.0025 -0.4986 0.0000 0.0000 9.4747 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.6501 0.0000 -0.0202 7.2411 6.3862 -0.1259 -32.1690 0.0000 0.0000 0.0000 0.0000

0.0000 0.9048 0.0000 -0.0003 -1.0189 0.0000 -0.0064 0.0001 0.0000 0.0000 0.0000 0.0000

0.0389 0.0000 -0.9916 -0.0000 0.0000 -0.3220 0.0636 -0.0005 0.0000 0.0000 0.0000 0.0000

1.0000 0.0078 0.0386 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.9801 -0.1987 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.1988 0.9808 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000

B6 =

0.0000 0.0000 -0.7336 0.1310

0.0000 -0.1824 0.0000 0.0000

0.0000 0.0000 -0.0327 -0.0643

0.0000 0.1704 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


A7 =

-3.6722 0.0003 0.6741 0.0000 0.0000 -30.9113 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.3859 -0.0029 0.0000 -2.4977 0.0000 -0.0003 0.0002 0.0000 0.0000 0.0000 0.0000

-0.0263 0.0025 -0.4986 0.0000 0.0000 9.4747 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.6501 0.0000 -0.0202 6.8909 -6.2769 0.2860 -31.8332 0.0000 0.0000 0.0000 0.0000

0.0000 0.9048 0.0000 -0.0002 -1.0093 0.0000 0.0063 -0.0047 0.0000 0.0000 0.0000 0.0000

0.0389 0.0000 -0.9916 0.0000 -0.0000 -0.3124 0.0634 0.0011 0.0000 0.0000 0.0000 0.0000

1.0000 -0.0381 0.1878 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.9801 0.1987 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.2023 0.9979 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000

B7 =

0.0000 0.0000 -0.7336 0.1310

0.0000 -0.1824 0.0000 0.0000

0.0000 0.0000 -0.0327 -0.0643

0.0000 0.1704 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

A8 =

-3.6722 0.0003 0.6741 0.0000 0.0000 -30.9113 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.3859 -0.0029 0.0000 -2.4977 0.0000 0.0000 0.0007 0.0000 0.0000 0.0000 0.0000

-0.0263 0.0025 -0.4986 0.0000 0.0000 9.4747 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.6501 0.0000 -0.0202 7.5202 0.0000 0.0000 -32.0493 0.0000 0.0000 0.0000 0.0000

0.0000 0.9048 0.0000 -0.0003 -1.0094 0.0000 0.0000 -0.0048 0.0000 0.0000 0.0000 0.0000

0.0389 0.0000 -0.9916 0.0000 0.0000 -0.3124 0.0629 0.0000 0.0000 0.0000 0.0000 0.0000

1.0000 0.0000 0.1917 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 1.0182 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000

B8 =

0.0000 0.0000 -0.7336 0.1310

0.0000 -0.1824 0.0000 0.0000

0.0000 0.0000 -0.0327 -0.0643

0.0000 0.1704 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


A9 =

-3.6722 0.0003 0.6741 0.0000 0.0000 -30.9113 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.3859 -0.0029 0.0000 -2.4977 0.0000 0.0003 0.0002 0.0000 0.0000 0.0000 0.0000

-0.0263 0.0025 -0.4986 0.0000 0.0000 9.4747 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.6501 0.0000 -0.0202 6.8909 6.2769 -0.2860 -31.8332 0.0000 0.0000 0.0000 0.0000

0.0000 0.9048 0.0000 -0.0002 -1.0093 0.0000 -0.0063 -0.0047 0.0000 0.0000 0.0000 0.0000

0.0389 0.0000 -0.9916 -0.0000 0.0000 -0.3124 0.0634 -0.0011 0.0000 0.0000 0.0000 0.0000

1.0000 0.0381 0.1878 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.9801 -0.1987 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.2023 0.9979 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 -0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000

B9 =

0.0000 0.0000 -0.7336 0.1310

0.0000 -0.1824 0.0000 0.0000

0.0000 0.0000 -0.0327 -0.0643

0.0000 0.1704 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


hi(φ, θ) =MφiMθi , i = 1, 2, 3,

where

Mφ1 =

1 if φ ≤ −0.22.5 ∗ φ− 0.5 if − 0.2 < φ < 0.2

0 if φ ≥ 0.2

Mφ2 =

0 if φ ≤ −0.25 ∗ φ+ 1 if − 0.2 < φ ≤ 0

−5 ∗ φ+ 1 if 0 < φ < 0.2

0 if φ ≥ 0.2

Mφ3 =

0 if φ ≤ −0.22.5 ∗ φ+ 0.5 if − 0.2 < φ < 0.2

1 if φ ≥ 0.2


Mθ1 =

1 if θ ≤ −0.152.5 ∗ θ − 0.5 if − 0.15 < θ < 0.15

0 if θ ≥ 0.15

Mθ2 =

0 if θ ≤ −0.155 ∗ θ + 1 if − 0.15 < θ ≤ 0

−5 ∗ θ + 1 if 0 < θ < 0.15

0 if θ ≥ 0.15

Mθ3 =

0 if θ ≤ −0.152.5 ∗ θ + 0.5 if − 0.15 < θ < 0.15

1 if θ ≥ 0.15

Appendix C

TS Fuzzy Augmented Model II

The model with α ∈ [−0.1, 0.1] and φ ∈ [−0.8, 0.8] assumes the form:

˙x =9∑

i=1

hi(α, φ)(Aix+ Biu),


Rule 1: If α = −0.1 and φ = −0.8, then the linearized model is ˙x = A1x+ B1u.

Rule 2: If α = −0.1 and φ = −0.8, then the linearized model is ˙x = A2x+ B2u.

Rule 3: If α = 0 and φ = −0.8, then the linearized model is ˙x = A3x+ B3u.

Rule 4: If α = 0.1 and φ = −0.8, then the linearized model is ˙x = A4x+ B4u.

Rule 5: If α = −0.1 and φ = 0, then the linearized model is ˙x = A5x+ B5u.

Rule 6: If α = 0 and φ = 0, then the linearized model is ˙x = A6x+ B6u.

Rule 7: If α = 0.1 and φ = 0.8, then the linearized model is ˙x = A7x+ B7u.

Rule 8: If α = 0 and φ = 0.8, then the linearized model is ˙x = A8x+ B8u.

Rule 9: If α = 0.1 and φ = 0.8, then the linearized model is ˙x = A9x+ B9u.

188


Resulting linear models

A1 =

-3.2408 0.0003 -0.0600 0.0000 0.0000 -15.6934 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.3779 -0.0029 -0.0008 2.5382 0.0000 -0.0784 0.0000 0.0000 0.0000 0.0000

0.0159 0.0025 -0.4696 0.0000 0.0000 8.9211 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 2.6825 0.0000 -0.0236 76.3394 -23.0616 -4.9446 -32.0320 0.0000 0.0000 0.0000

0.0000 0.9157 0.0000 -0.0000 -1.0911 0.0000 0.0275 -0.0057 0.0000 0.0000 0.0000

-0.0641 0.0000 -0.9908 0.0001 0.0016 -0.3147 0.0572 0.0017 0.0000 0.0000 0.0000

1.0000 -0.0265 0.0257 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.6967 0.7174 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

B1 =

0.0000 0.0000 -0.7377 0.1355

0.0000 -0.1553 0.0000 0.0000

0.0000 0.0000 -0.0344 -0.0620

0.0000 0.2820 0.0000 0.0000

0.0000 -0.0021 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

A2 =

-3.6776 0.0003 0.6645 0.0000 0.0000 -30.6426 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.0772 -0.0029 0.0000 0.8221 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000

-0.0254 0.0025 -0.4765 0.0000 0.0000 8.5416 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.5750 0.0000 -0.0193 -0.9269 -23.0616 0.4497 -32.1567 0.0000 0.0000 0.0000

0.0000 0.9051 0.0000 -0.0002 -1.0182 0.0000 0.0243 0.0007 0.0000 0.0000 0.0000

0.0364 0.0000 -0.9917 0.0001 -0.0000 -0.3213 0.0574 0.0017 0.0000 0.0000 0.0000

1.0000 -0.0265 0.0257 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.6967 0.7174 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

B2 =

0.0000 0.0000 -0.7331 0.1315

0.0000 -0.1755 0.0000 0.0000

0.0000 0.0000 -0.0319 -0.0620

0.0000 0.1737 0.0000 0.0000

0.0000 -0.0021 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


A3 =

-3.3910 0.0003 1.3693 0.0000 0.0000 -45.5472 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.1801 -0.0029 0.0001 -0.1877 0.0000 -0.0631 0.0000 0.0000 0.0000 0.0000

-0.0688 0.0025 -0.4598 0.0000 0.0000 8.5921 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -4.0169 0.0000 -0.0427 -71.6334 -23.0616 -3.8999 -31.9601 0.0000 0.0000 0.0000

0.0000 0.9010 0.0000 -0.0004 -1.0101 0.0000 0.0251 0.0071 0.0000 0.0000 0.0000

0.1381 0.0000 -0.9826 0.0001 -0.0016 -0.3140 0.0572 0.0017 0.0000 0.0000 0.0000

1.0000 -0.0265 0.0257 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.6967 0.7174 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

B3 =

0.0000 0.0000 -0.7087 0.1261

0.0000 -0.1755 0.0000 0.0000

0.0000 0.0000 -0.0290 -0.0613

0.0000 0.0450 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

A4 =

-3.2408 0.0003 -0.0600 0.0000 0.0000 -15.6934 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.3779 -0.0029 -0.0008 1.9109 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0159 0.0025 -0.4696 0.0000 0.0000 8.9211 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 2.6826 0.0000 -0.0236 42.9300 0.0000 0.0000 -32.0093 0.0000 0.0000 0.0000

0.0000 0.9157 0.0000 -0.0000 -1.0651 0.0000 0.0000 -0.0064 0.0000 0.0000 0.0000

-0.0641 0.0000 -0.9908 0.0000 0.0000 -0.3135 0.0640 0.0000 0.0000 0.0000 0.0000

1.0000 0.0000 0.0369 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

B4 =

0.0000 0.0000 -0.7377 0.1355

0.0000 -0.1553 0.0000 0.0000

0.0000 0.0000 -0.0344 -0.0620

0.0000 0.2820 0.0000 0.0000

0.0000 -0.0021 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


A5 =

-3.6776 0.0003 0.6645 0.0000 0.0000 -30.6426 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.0772 -0.0029 0.0000 0.8221 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0254 0.0025 -0.4765 0.0000 0.0000 8.5416 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.5750 0.0000 -0.0193 8.8168 0.0000 0.0000 -32.1700 0.0000 0.0000 0.0000

0.0000 0.9051 0.0000 -0.0003 -1.0189 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0364 0.0000 -0.9917 0.0000 0.0000 -0.3220 0.0640 0.0000 0.0000 0.0000 0.0000

1.0000 0.0000 0.0369 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

B5 =

0.0000 0.0000 -0.7331 0.1315

0.0000 -0.1755 0.0000 0.0000

0.0000 0.0000 -0.0319 -0.0620

0.0000 0.1737 0.0000 0.0000

0.0000 -0.0021 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

A6 =

-3.3910 0.0003 1.3693 0.0000 0.0000 -45.5472 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.1801 -0.0029 0.0001 0.3169 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0688 0.0025 -0.4598 0.0000 0.0000 8.5921 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -4.0169 0.0000 -0.0427 -27.1270 0.0000 0.0000 -32.0093 0.0000 0.0000 0.0000

0.0000 0.9010 0.0000 -0.0005 -1.0181 0.0000 0.0000 0.0064 0.0000 0.0000 0.0000

0.1381 0.0000 -0.9826 0.0000 0.0000 -0.3166 0.0640 0.0000 0.0000 0.0000 0.0000

1.0000 0.0000 0.0369 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

B6 =

0.0000 0.0000 -0.7087 0.1261

0.0000 -0.1755 0.0000 0.0000

0.0000 0.0000 -0.0290 -0.0613

0.0000 0.0450 0.0000 0.0000

0.0000 -0.0022 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


A7 =

-3.2408 0.0003 -0.0600 0.0000 0.0000 -15.6934 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.3779 -0.0029 -0.0008 2.5382 0.0000 0.0784 0.0000 0.0000 0.0000 0.0000

0.0159 0.0025 -0.4696 0.0000 0.0000 8.9211 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 2.6825 0.0000 -0.0236 76.3394 23.0616 4.9446 -32.0320 0.0000 0.0000 0.0000

0.0000 0.9157 0.0000 -0.0000 -1.0911 0.0000 -0.0275 -0.0057 0.0000 0.0000 0.0000

-0.0641 0.0000 -0.9908 -0.0001 -0.0016 -0.3147 0.0572 -0.0017 0.0000 0.0000 0.0000

1.0000 0.0265 0.0257 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.6967 -0.7174 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

B7 =

0.0000 0.0000 -0.7377 0.1355

0.0000 -0.1553 0.0000 0.0000

0.0000 0.0000 -0.0344 -0.0620

0.0000 0.2820 0.0000 0.0000

0.0000 -0.0021 0.0000 0.0000

0.0000 0.0000 0.0003 0.0008

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

A8 =

-3.6776 0.0003 0.6645 0.0000 0.0000 -30.6426 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.0772 -0.0029 0.0000 0.8221 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0254 0.0025 -0.4765 0.0000 0.0000 8.5416 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -0.5750 0.0000 -0.0193 -0.9269 23.0616 -0.4497 -32.1567 0.0000 0.0000 0.0000

0.0000 0.9051 0.0000 -0.0002 -1.0182 0.0000 -0.0243 0.0007 0.0000 0.0000 0.0000

0.0364 0.0000 -0.9917 -0.0001 0.0000 -0.3213 0.0574 -0.0017 0.0000 0.0000 0.0000

1.0000 0.0265 0.0257 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.6967 -0.7174 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

B8 =

-3.6776 0.0003 0.6645 0.0000

0.0000 -1.0772 -0.0029 0.0000

-0.0254 0.0025 -0.4765 0.0000

0.0000 -0.5750 0.0000 -0.0193

0.0000 0.9051 0.0000 -0.0002

0.0364 0.0000 -0.9917 -0.0001

1.0000 0.0265 0.0257 0.0000

0.0000 0.6967 -0.7174 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


A9 =

-3.3910 0.0003 1.3693 0.0000 0.0000 -45.5472 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -1.1801 -0.0029 0.0001 -0.1877 0.0000 0.0631 0.0000 0.0000 0.0000 0.0000

-0.0688 0.0025 -0.4598 0.0000 0.0000 8.5921 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 -4.0169 0.0000 -0.0427 -71.6334 23.0616 3.8999 -31.9601 0.0000 0.0000 0.0000

0.0000 0.9010 0.0000 -0.0004 -1.0101 0.0000 -0.0251 0.0071 0.0000 0.0000 0.0000

0.1381 0.0000 -0.9826 -0.0001 0.0016 -0.3140 0.0572 -0.0017 0.0000 0.0000 0.0000

1.0000 0.0265 0.0257 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.6967 -0.7174 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000

B9 =

-3.3910 0.0003 1.3693 0.0000

0.0000 -1.1801 -0.0029 0.0001

-0.0688 0.0025 -0.4598 0.0000

0.0000 -4.0169 0.0000 -0.0427

0.0000 0.9010 0.0000 -0.0004

0.1381 0.0000 -0.9826 -0.0001

1.0000 0.0265 0.0257 0.0000

0.0000 0.6967 -0.7174 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000


hi(α, φ) =MαiMφi , i = 1, 2, 3,

where

Mα1 =

1 if α ≤ −0.12.5 ∗ α− 0.5 if − 0.1 < α < 0.1

0 if α ≥ 0.1

Mα2 =

0 if α ≤ −0.15 ∗ α + 1 if − 0.1 < α ≤ 0

−5 ∗ α + 1 if 0 < α < 0.1

0 if α ≥ 0.1

Mα3 =

0 if α ≤ −0.12.5 ∗ α + 0.5 if − 0.1 < α < 0.1

1 if α ≥ 0.1

Mφ1 =

1 if φ ≤ −0.82.5 ∗ φ− 0.5 if − 0.8 < φ < 0.8

0 if φ ≥ 0.8

Mφ2 =

0 if φ ≤ −0.85 ∗ φ+ 1 if − 0.8 < φ ≤ 0

−5 ∗ φ+ 1 if 0 < φ < 0.8

0 if φ ≥ 0.8

Mφ3 =

0 if φ ≤ −0.82.5 ∗ φ+ 0.5 if − 0.8 < φ < 0.8

1 if φ ≥ 0.8

194

Appendix D

Gain Scheduling Control for

Tracking (φ, θ)

The Fuzzy controller I is described by

u = Kx,

K =9∑

i=1

hi(x)Ki,

where hi(x) are the same as those from the TS model in Appendix B and

K1 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.3687 -88.9811 -18.4567 0.0187 13.6331 -10.5508 2.5302 -785.9696 3.5997 6.6243 -3148.3817 0.0934

-19.6926 -1.8319 -0.6296 -0.0295 -0.0409 37.8419 -218.7384 -19.7345 -7.7237 -978.4534 -81.3664 -0.2049

3.8462 -6.8619 -9.9549 -0.1443 -0.2180 -41.8981 46.5754 -84.7639 -36.9614 206.3615 -284.7313 -0.9743

,

K2 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0003 -90.1064 0.0168 0.0050 13.6127 0.0021 -0.0020 -786.3111 -0.0005 -0.0055 -3162.2777 -0.0000

-19.7074 0.0019 0.0217 0.0000 -0.0003 38.8846 -218.8523 0.0107 -6.9331 -978.7325 0.0272 -0.2051

3.7759 0.0062 -7.0133 0.0000 -0.0013 -36.7352 46.0509 0.0407 -33.2557 205.1407 0.0454 -0.9787

,

K3 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.3692 -88.9746 18.4896 0.0187 13.6329 10.5554 -2.5340 -785.9653 -3.6012 -6.6347 -3148.3697 -0.0934

-19.6925 1.8355 -0.6319 0.0295 0.0403 37.8412 -218.7380 19.7584 -7.7239 -978.4526 81.4256 -0.2049

3.8465 6.8739 -9.9658 0.1443 0.2154 -41.9025 46.5769 84.8575 -36.9619 206.3649 284.8479 -0.9743

,

K4 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.4269 -88.9861 -18.4282 0.0193 13.6407 -10.4395 3.3450 -785.9401 3.6987 11.2019 -3148.3571 0.0931

-19.7031 -1.5915 -4.0668 -0.0284 -0.0530 35.9706 -219.1541 -22.6468 -7.3719 -981.4157 -90.9952 -0.1897

3.5044 -6.8992 -8.9633 -0.1507 -0.2986 -42.1319 43.0282 -84.0728 -38.2344 191.5668 -282.0773 -0.9774

,

K5 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0004 -90.1064 0.0168 0.0050 13.6128 0.0020 -0.0030 -786.3134 -0.0005 -0.0100 -3162.2777 -0.0000

-19.7180 0.0015 -3.4165 0.0000 -0.0003 37.0166 -219.2708 0.0133 -6.6003 -981.7266 0.0400 -0.1903

3.4336 0.0062 -6.0098 0.0000 -0.0013 -36.7581 42.4945 0.0391 -34.2506 190.2970 0.0403 -0.9817

,

K6 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.4277 -88.9797 18.4611 0.0193 13.6405 10.4439 -3.3509 -785.9359 -3.7003 -11.2215 -3148.3451 -0.0931

-19.7030 1.5945 -4.0690 0.0284 0.0525 35.9699 -219.1537 22.6759 -7.3721 -981.4146 91.0801 -0.1897

3.5047 6.9113 -8.9741 0.1507 0.2960 -42.1363 43.0298 84.1632 -38.2349 191.5708 282.1837 -0.9774

,

195

K7 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.4849 -88.9933 -18.3769 0.0193 13.6401 -10.3015 4.1588 -785.8823 3.7354 15.8037 -3148.2859 0.0926

-19.7041 -1.3474 -7.5631 -0.0264 -0.0439 34.0512 -219.4880 -25.6273 -6.8772 -984.1375 -100.7489 -0.1745

3.1617 -6.9253 -8.1083 -0.1525 -0.2730 -42.7041 39.4722 -83.6222 -38.9124 176.7022 -279.5447 -0.9803

,

K8 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0004 -90.1065 0.0167 0.0050 13.6147 0.0018 -0.0040 -786.3172 -0.0005 -0.0146 -3162.2777 -0.0000

-19.7197 0.0012 -6.9156 0.0000 -0.0002 35.0816 -219.6131 0.0160 -6.1687 -984.5057 0.0531 -0.1754

3.0907 0.0062 -5.1514 0.0000 -0.0013 -37.2067 38.9350 0.0377 -34.8552 175.3526 0.0359 -0.9845

,

K9 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.4858 -88.9870 18.4096 0.0193 13.6399 10.3057 -4.1668 -785.8781 -3.7369 -15.8325 -3148.2737 -0.0927

-19.7041 1.3497 -7.5653 0.0264 0.0435 34.0507 -219.4876 25.6616 -6.8773 -984.1362 100.8599 -0.1745

3.1620 6.9376 -8.1188 0.1525 0.2704 -42.7083 39.4739 83.7100 -38.9129 176.7069 279.6423 -0.9803

.

The Fuzzy controller II is described by

u = Kx,

K =9∑

i=1

hi(x)Ki,

where hi(x) are the same as those from the TS model in Appendix B and

K1 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0429 -19.9092 -2.6266 0.0003 12.3847 -10.6580 0.3252 -76.2711 0.5021 0.2666 -99.6939 0.0025

-4.2508 -0.2817 -9.0478 -0.0005 0.0622 27.4442 -23.8241 -0.9093 -0.7013 -31.4126 -1.7274 -0.0036

0.0122 -0.9190 -1.7946 -0.0067 0.1970 -7.4881 0.9075 -5.5614 -6.4708 3.6299 -7.6254 -0.0313

,

K2 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0007 -20.1568 0.0089 -0.0002 12.4197 0.0388 -0.0041 -76.0442 -0.0018 -0.0046 -100.0000 -0.0000

-4.2518 -0.0013 -8.9981 -0.0000 -0.0002 27.5275 -23.8278 0.0124 -0.7195 -31.4152 0.0177 -0.0036

0.0071 0.0035 -1.5758 -0.0000 -0.0011 -7.0617 0.8871 0.0184 -6.6230 3.6171 0.0251 -0.0314

,

K3 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0443 -19.9068 2.6437 0.0003 12.3844 10.7364 -0.3335 -76.2721 -0.5056 -0.2760 -99.6893 -0.0025

-4.2508 0.2790 -9.0493 0.0005 -0.0626 27.4402 -23.8240 0.9343 -0.7012 -31.4125 1.7629 -0.0036

0.0122 0.9257 -1.7981 0.0067 -0.1991 -7.4944 0.9076 5.5988 -6.4704 3.6301 7.6764 -0.0313

,

K4 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0563 -19.9074 -2.6121 0.0003 12.4028 -10.7174 0.4147 -76.2638 0.5181 0.4379 -99.6853 0.0025

-4.2525 -0.2249 -9.6707 -0.0004 0.0530 24.9603 -23.8203 -1.3358 -0.5999 -31.4698 -2.1331 -0.0030

-0.0403 -0.9233 -1.7735 -0.0068 0.1907 -7.3174 0.5850 -5.6031 -6.6648 3.0754 -7.6343 -0.0314

,

K5 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0007 -20.1554 0.0089 -0.0002 12.4372 0.0383 -0.0043 -76.0535 -0.0018 -0.0051 -100.0000 -0.0000

-4.2530 -0.0014 -9.6209 -0.0000 -0.0002 25.0637 -23.8211 0.0134 -0.6179 -31.4745 0.0186 -0.0031

-0.0456 0.0035 -1.5526 -0.0000 -0.0010 -6.8960 0.5629 0.0183 -6.8259 3.0588 0.0250 -0.0315

,

K6 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0577 -19.9050 2.6291 0.0003 12.4024 10.7950 -0.4233 -76.2647 -0.5217 -0.4481 -99.6806 -0.0025

-4.2525 0.2220 -9.6721 0.0004 -0.0533 24.9560 -23.8202 1.3627 -0.5997 -31.4697 2.1704 -0.0030

-0.0402 0.9300 -1.7770 0.0068 -0.1928 -7.3237 0.5851 5.6402 -6.6644 3.0756 7.6851 -0.0314

,

K7 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0694 -19.9062 -2.5907 0.0003 12.4204 -10.7558 0.5025 -76.2520 0.5286 0.6100 -99.6747 0.0025

-4.2529 -0.1680 -10.3002 -0.0003 0.0447 22.4445 -23.8097 -1.7683 -0.4872 -31.5161 -2.5413 -0.0025

-0.0924 -0.9251 -1.7787 -0.0070 0.1892 -7.2874 0.2663 -5.6736 -6.7936 2.5226 -7.6487 -0.0314

,

K8 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0008 -20.1540 0.0088 -0.0002 12.4546 0.0379 -0.0045 -76.0626 -0.0018 -0.0055 -100.0000 -0.0000

-4.2545 -0.0016 -10.2535 -0.0000 -0.0001 22.5833 -23.8198 0.0143 -0.5041 -31.5241 0.0195 -0.0025

-0.0979 0.0035 -1.5555 -0.0000 -0.0010 -6.8690 0.2426 0.0183 -6.9596 2.4959 0.0249 -0.0315

,

196

K9 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0710 -19.9037 2.6076 0.0003 12.4201 10.8324 -0.5116 -76.2528 -0.5322 -0.6210 -99.6698 -0.0025

-4.2529 0.1647 -10.3015 0.0003 -0.0449 22.4399 -23.8095 1.7972 -0.4870 -31.5159 2.5804 -0.0025

-0.0924 0.9318 -1.7822 0.0070 -0.1913 -7.2937 0.2664 5.7106 -6.7932 2.5229 7.6993 -0.0314

.

197

Appendix E

Gain Scheduling Control for

Tracking (α, β, φ)

The Fuzzy controller I is described by

u = Kx,

K =9∑

i=1

hi(x)Ki,

where hi(x) are the same as those from the TS model in Appendix C and

K1 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0148 -319.8224 0.0681 0.0035 -1163.5845 0.8604 -0.0505 1.9771 -3161.3123 1.6918 24.6493

-30.4307 0.2319 -26.0497 0.0026 -58.6892 189.3081 -451.9232 0.0475 -234.3342 794.6246 -3059.9151

9.9318 0.0590 -137.7025 0.0137 23.7448 794.3268 177.1732 0.1982 78.3093 3060.8076 794.2567

,

K2 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0169 -315.0360 0.0036 0.0765 -1177.1797 -0.2862 -0.4803 -0.5077 -3161.5351 -0.1421 21.6692

-30.0387 -0.0073 -26.9467 0.0036 -55.9749 183.8534 -470.4688 0.0825 -213.7533 539.8986 -3115.1148

9.2292 0.0584 -138.0877 0.0158 8.9910 797.3002 85.1577 0.1750 35.5965 3115.8481 539.7814

,

K3 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0147 -314.3092 -0.0603 0.1460 -1180.6002 -1.2998 -0.0353 -2.8390 -3161.3692 -1.9262 23.8904

-30.8471 0.1321 -28.2105 0.0042 -65.1202 179.9095 -489.3780 0.2854 -239.6715 279.8238 -3148.9608

8.8470 0.0081 -138.4025 0.0167 -3.9576 796.2902 -9.4219 0.1731 1.9596 3149.8669 279.8894

,

K4 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0001 -319.7585 0.0140 0.0172 -1168.7053 -0.0128 -0.0026 2.1826 -3162.2777 0.0254 -0.0121

-30.4300 0.0267 -26.2300 -0.0000 0.0986 187.7532 -451.9922 -0.0002 0.1805 788.3355 -3062.4381

9.9154 0.0543 -137.6344 -0.0000 0.2210 795.2099 176.9594 -0.0006 0.2163 3062.4381 788.3355

,

K5 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0000 -315.0222 0.0123 0.0911 -1177.2192 -0.0133 -0.0007 -0.3111 -3162.2777 0.0164 -0.0091

-30.0390 0.0196 -27.1345 -0.0000 0.0741 182.1268 -470.4097 0.0000 0.1174 533.5920 -3116.9343

9.2140 0.0422 -138.0574 -0.0000 0.1837 797.5140 84.9014 0.0000 0.1462 3116.9343 533.5920

,

K6 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0001 -314.3445 0.0123 0.1604 -1178.8704 -0.0140 0.0009 -2.7210 -3162.2777 0.0143 -0.0082

-30.8474 0.0176 -28.4079 -0.0000 0.0664 177.8635 -489.1065 0.0002 0.0943 273.5094 -3150.4274

8.8325 0.0417 -138.4122 -0.0000 0.1817 795.7978 -9.5868 0.0006 0.1354 3150.4274 273.5094

,

198

K7 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0146 -319.8224 -0.0400 0.0035 -1163.5841 -0.8852 0.0452 1.9771 -3161.3107 -1.6386 -24.6734

-30.4307 -0.1785 -26.0494 -0.0026 58.8869 189.3112 -451.9230 -0.0479 234.7015 794.6340 -3059.9099

9.9318 0.0495 -137.7025 -0.0137 -23.3008 794.3259 177.1738 -0.1992 -77.8563 3060.8054 794.2694

,

K8 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0169 -315.0360 0.0210 0.0765 -1177.1794 0.2597 0.4789 -0.5077 -3161.5338 0.1750 -21.6875

-30.0387 0.0466 -26.9467 -0.0036 56.1238 183.8562 -470.4689 -0.0824 213.9895 539.9058 -3115.1120

9.2292 0.0261 -138.0877 -0.0158 -8.6237 797.2997 85.1578 -0.1748 -35.3042 3115.8468 539.7907

,

K9 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0146 -314.3092 0.0849 0.1460 -1180.5998 1.2715 0.0372 -2.8390 -3161.3678 1.9538 -23.9068

-30.8471 -0.0969 -28.2103 -0.0042 65.2536 179.9125 -489.3784 -0.2850 239.8591 279.8324 -3148.9587

8.8470 0.0752 -138.4026 -0.0168 4.3192 796.2900 -9.4215 -0.1718 -1.6996 3149.8660 279.9001

.

The Fuzzy controller II is described by

u = Kx,

K =9∑

i=1

hi(x)Ki,


K1 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0160 -319.8224 0.0866 0.0035 -1163.5966 2.8082 -0.1165 1.9771 -3161.3542 113.2555 24.1655

-61.7805 0.1494 -452.2293 0.0416 -51.3543 44345.2652 -411.5411 0.7900 -198.5068 1791246.5206 -2605.2822

-21.1176 -0.0498 -673.2023 0.0624 37.1016 56428.0118 334.3478 1.1332 137.8165 2606038.3294 1790.7163

,

K2 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0167 -315.0360 -0.0045 0.0765 -1177.1799 -0.3313 -0.4806 -0.5077 -3161.5357 -9.5502 21.6613

-24.5328 -0.0135 -155.7431 0.0150 -54.7177 16631.6208 -452.8960 0.3100 -209.1669 823339.3640 -3052.4969

40.6684 0.0220 -879.7431 0.0812 14.8781 73255.1039 178.8092 1.4856 56.3057 3053213.4368 823.1468

,

K3 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0132 -314.3093 -0.0953 0.1460 -1180.6184 -2.8532 -0.1286 -2.8389 -3161.4289 -113.4439 23.1670

-68.9775 0.1435 188.6482 -0.0161 -63.9774 -16783.9465 -556.7211 -0.0941 -230.2015 -340483.2103 -3143.0514

110.8765 0.0146 -819.4763 0.0767 -8.2526 71176.9056 -2.1045 1.4340 -26.0716 3143894.2500 -340.3836

,

K4 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0004 -319.7585 0.0057 0.0172 -1168.7053 0.3638 0.0033 2.1826 -3162.2777 7.9588 0.0257

-61.7686 -0.0103 -451.9914 0.0000 -0.0691 44342.9116 -408.4676 0.0001 -0.1670 1791556.5457 -2605.8252

-21.0894 0.0058 -673.3556 -0.0000 0.0813 56429.7647 339.1063 -0.0001 0.2114 2605825.2326 1791.5565

,

K5 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0003 -315.0222 0.0038 0.0911 -1177.2192 0.3667 -0.0000 -0.3111 -3162.2777 7.2716 -0.0092

-24.5479 0.0123 -155.3781 -0.0000 0.0485 16631.8404 -451.8088 0.0000 0.1074 823883.5124 -3053.0666

40.7040 -0.0057 -879.8141 -0.0000 0.0136 73252.9325 184.7772 0.0000 0.0463 3053066.6481 823.8835

,

K6 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0006 -314.3445 0.0061 0.1604 -1178.8704 0.3068 -0.0026 -2.7210 -3162.2777 8.6844 -0.0350

-69.0080 0.0293 188.8630 -0.0000 0.1528 -16787.9568 -557.7728 0.0003 0.3387 -340247.1483 -3143.9198

110.9017 0.0020 -819.4280 -0.0000 0.0479 71173.4297 3.6736 0.0001 0.1240 3143919.8268 -340.2471

,

K7 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0153 -319.8225 -0.0752 0.0035 -1163.5977 -2.0801 0.1230 1.9771 -3161.3581 -97.3450 -24.1142

-61.7805 -0.1701 -452.2293 -0.0416 51.2175 44345.2677 -411.5408 -0.7897 198.1745 1791246.4609 -2605.2848

-21.1176 0.0615 -673.2023 -0.0624 -36.9405 56428.0101 334.3475 -1.1334 -137.3950 2606038.3768 1790.7195

,

K8 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0172 -315.0360 0.0122 0.0765 -1177.1796 1.0645 0.4805 -0.5077 -3161.5344 24.0847 -21.6799

-24.5328 0.0380 -155.7431 -0.0150 54.8153 16631.6218 -452.8961 -0.3100 209.3843 823339.3165 -3052.4954

40.6684 -0.0335 -879.7431 -0.0812 -14.8511 73255.1036 178.8092 -1.4856 -56.2138 3053213.4489 823.1474

,

199

K9 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0145 -314.3092 0.1075 0.1460 -1180.6170 3.4665 0.1234 -2.8389 -3161.4238 130.8076 -23.2371

-68.9776 -0.0849 188.6482 0.0161 64.2843 -16783.9444 -556.7217 0.0949 230.8797 -340483.2893 -3143.0464

110.8765 -0.0105 -819.4763 -0.0767 8.3488 71176.9058 -2.1046 -1.4338 26.3196 3143894.2346 -340.3818

.

The Fuzzy controller III is described by

u = Kx,

K =9∑

i=1

hi(x)Ki,


K1 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0270 -992.3972 0.2408 0.0012 -706.0301 3.0279 0.3083 6.2426 -998.5364 -0.7241 17.1026

-19.2330 0.1824 -146.2877 0.0136 -17.5950 2820.4547 -67.5476 0.2485 -36.3580 23419.7846 -212.1772

-0.2718 -0.1340 -224.2869 0.0212 20.4004 2774.3344 101.1619 0.3584 40.0386 21248.8515 233.8553

,

K2 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0268 -988.5698 -0.1052 0.2614 -735.5971 -6.7014 -0.1193 -9.8859 -998.7195 -0.2662 15.9980

-10.9429 0.2101 -44.8607 0.0096 -26.1542 567.0138 -101.8420 0.3044 -50.1632 4104.7996 -313.1508

12.6292 -0.0002 -307.5862 0.0284 3.4218 4263.5254 32.0601 0.4727 6.5585 31355.2327 40.9956

,

K3 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0210 -987.8328 -0.4633 0.4812 -760.8022 -11.4936 1.5729 -19.6861 -997.8876 2.2954 20.5433

-24.1816 0.2720 66.2055 0.0039 -33.4559 -1811.9725 -107.0155 1.5549 -56.1619 -15924.4931 -272.6273

23.8109 0.1637 -251.3802 0.0302 -20.2271 3415.6758 -47.6251 1.3891 -32.6516 27320.5145 -158.9098

,

K4 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0001 -992.1874 0.0087 0.0429 -715.6615 0.0864 0.0015 5.4011 -1000.0000 0.7647 0.0078

-19.1997 0.0011 -145.9548 -0.0000 0.0005 2816.7570 -66.5739 -0.0000 0.0014 23406.5985 -212.6338

-0.2262 0.0028 -224.6691 -0.0000 0.0254 2778.8986 102.7283 -0.0002 0.0345 21263.3757 234.0660

,

K5 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0003 -988.5346 0.0049 0.3309 -734.9544 0.1132 0.0012 -11.8979 -1000.0000 0.9023 0.0006

-10.9563 0.0021 -44.1816 -0.0000 0.0036 559.0578 -101.5542 0.0001 0.0019 4077.1723 -313.5884

12.7014 0.0007 -307.6620 -0.0000 0.0209 4264.2442 34.0962 0.0003 0.0285 31358.8371 40.7717

,

K6 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0006 -988.0003 0.0076 0.5436 -747.9396 0.0682 0.0008 -23.5632 -1000.0000 0.6384 -0.0056

-24.2233 0.0034 66.5865 -0.0000 0.0084 -1817.5296 -107.2320 0.0003 0.0051 -15939.8845 -273.1154

23.8456 0.0016 -251.1299 -0.0000 0.0228 3411.5403 -45.7355 0.0007 0.0264 27311.5375 -159.3988

,

K7 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0267 -992.3972 -0.2234 0.0012 -706.0311 -2.8536 -0.3052 6.2424 -998.5390 2.2539 -17.0874

-19.2330 -0.1803 -146.2877 -0.0136 17.5964 2820.4563 -67.5475 -0.2486 36.3615 23419.7864 -212.1772

-0.2718 0.1397 -224.2869 -0.0212 -20.3510 2774.3327 101.1618 -0.3588 -39.9705 21248.8494 233.8565

,

K8 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0275 -988.5698 0.1150 0.2614 -735.5972 6.9282 0.1216 -9.8861 -998.7196 2.0709 -15.9971

-10.9429 -0.2059 -44.8606 -0.0096 26.1620 567.0155 -101.8421 -0.3042 50.1678 4104.8014 -313.1507

12.6292 0.0016 -307.5862 -0.0284 -3.3800 4263.5248 32.0601 -0.4721 -6.5017 31355.2324 40.9965

,

K9 =

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0222 -987.8328 0.4784 0.4812 -760.8010 11.6291 -1.5712 -19.6861 -997.8853 -1.0189 -20.5547

-24.1816 -0.2651 66.2055 -0.0039 33.4732 -1811.9703 -107.0158 -1.5545 56.1726 -15924.4915 -272.6271

23.8109 -0.1604 -251.3802 -0.0302 20.2737 3415.6766 -47.6252 -1.3879 32.7045 27320.5155 -158.9087

.

200

Author’s Publications

Journal publications

[1] Qing-Guo. Wang, C.C Hang, Yongsheng Yang and J.B.He ”Quantitative Ro-

bust Stability Analysis and PID Controller Design”, IEE Proceeding in Control

Theory Application. Vol. 149, No.1, January, 2002.

[2] Yang, Yong-Sheng, Wang, Qing-Guo, Dai, WZ, ”Robust PID controller design

for gain and phase margins”, Journal of Chemical Engineering of Japan, Vol.35,

No.9, pp.874-879, September, 2002.

[3] Wang, Qing-Guo and Yang, Yongsheng, ”Transfer Function Matrix Approach

to Decoupling Problem with Stability”, Systems and Control Letters, Vol. 47, Issue

2, Pages 103-110, October, 2002.

[4] Yongsheng Yang and Qing-Guo Wang, ”Decoupling Control Design via LMI”,

submitted to IEE Proceeding in Control Theory Application.

[5] Yong-Sheng Yang, Qiangguo Pu and Qing-Guo Wang, ” PI Controller Design

for State Time-Delay Systems: An ILMI Approach”, submitted to Automatica.

[6] Yong-Sheng Yang, Qing-Guo Wang and C. C. Hang, ”Modified Smith Predictor

Control for Disturbance Rejection with Unstable Processes”, submitted to Journal

201


of Process Control.

[7] Wang, Qing-Guo, Yong-sheng Yang and Xiang Lu, ”Robust IMC Design via

LMI/BMI Optimization in Frequency Domain”, submitted to Industrial & Engi-

neering Chemistry Research.

[8] Wang, Qing-Guo, Yong-sheng Yang and Yong Zhang ”Robust IMC Design via

LMI optimization”, submitted to Journal of Chemical Engineering of Japan.

Conference Publications

[1] Yongsheng Yang and Q.G. Wang, ”Robust PID Controller Design for Uncertain

Process”, International Conference on Computational Intelligence, Robotics and

Autonomous Systems, Singapore, Nov. 2001.

[2] Bin Huang; Qing-Guo Wang; Yong-Sheng Yang, ”An H∞ approach to decou-

pling control design”, Proceedings of the 4th World Congress on Intelligent Control

and Automation 2002, Volume 4, Page 2878-2882.

[3] Wang, Qing-Guo; Ru, He; Yang, Yongsheng, ”Enhancement of PID control for

complex processes”, IEEE International Symposium on Intelligent Control - Pro-

ceedings, 2002, Pages 592-595.

[4] Yong-Sheng Yang, Xue-Ping Yang, Qing-GuoWang, C. C. Hang and Chong Lin,

”IMC based Control System Design for Unstable Process”, International Sympo-

sium on Design, Operation and Control of Chemical Plants, December 4 - 6, 2002,

Taipei, Taiwan.


[5] Qing-Guo Wang, Yong-Sheng Yang, Xiang Lu, ”Robust IMC Controller Design

In Frequency Domain”, First Humanoid, Nanotechnology, Information Technology,

Communication and Control Environment and Management (HNICEM) Interna-

tional Conference, March 29-31, 2003, Manila, Philippines.

[6] Wang, Qing-Guo, Yong-sheng Yang and Qiangguo Pu, ”Robust IMC Design via

multiobjective optimization”, submitted to the Second International Conference on

Computational Intelligence, Robotics and Autonomous Systems, 2003, Singapore.

Documents

SEVERAL NEW DESIGNS FOR PID, IMC, …new method is proposed for the design of multi-variable IMC system aiming at obtaining good loop performance and small loop couplings based on