ESSENTIALS OF
ROBUST CONTROL
Kemin Zhou
May 25, 1999Preface
Robustness of controlsystems to disturbances and uncertainties has always been the
central issue in feedback control. Feedback would not be needed for most
...
ESSENTIALS OF
ROBUST CONTROL
Kemin Zhou
May 25, 1999Preface
Robustness of controlsystems to disturbances and uncertainties has always been the
central issue in feedback control. Feedback would not be needed for most control systems
if there were no disturbances and uncertainties. Developing multivariable robust control
methods has been the focal point in the last two decades in the control community. The
state-of-the-art H ∞ robust control theory is the result of this effort.
This book introduces some essentials of robust and ∞ control theory. H It grew from
another book by this author, John C. Doyle,and Keith Glover,entitled Robust and
OptimalControl,which has been extensively class-tested in many universities around
the world.Unlike that book, which is intended primarily as a comprehensive reference of
robust and H∞ control theory, this book is intended to be a text for a graduate course
in multivariable control.It is also intended to be a reference for practicing control
engineers who are interested in applying the state-of-the-art robust control techniques
in their applications. With this objective in mind, I have streamlined the presentation,
added more than 50 illustrative examples, included many related M R commands atlab 1
and more than 150 exercise problems, and added some recent developments in the area
of robust controlsuch as gap metric, ν-gap metric,modelvalidation,and mixed µ
problem.In addition, many proofs are completely rewritten and some advanced topics
are either deleted completely or do not get an in-depth treatment.
The prerequisite for reading this book is some basic knowledge of classical control
theory and state-space theory. The text contains more material than could be covered in
detail in a one-semester or a one-quarter course. Chapter 1 gives a chapter-by-chapter
summary of the main results presented in the book, which could be used as a guide for
the selection of topics for a specific course. Chapters 2 and 3 can be used as a refresher
for some linear algebra facts and some standard linear system theory. A course focusing
on H∞ control should cover at least most parts of Chapters 4–6, 8, 9, 11–13, and Sections
14.1 and 14.2. An advanced H ∞ control course should also include the rest of Chapter
14, Chapter 16, and possibly Chapters 10, 7, and 15. A course focusing on robustness
and model uncertainty should cover at least Chapters 4, 5, and 8–10. Chapters 17 and
18 can be added to any advanced robust and ∞ control course if time permits. H
I have tried hard to eliminate obvious mistakes. It is, however, impossible for me
to make the book perfect. Readers are encouraged to send corrections, comments, and
1Matlab is a registered trademark of The MathWorks, Inc.
viiviii PREFACE
suggestions to me, preferably by electronic mail, at
[email protected]
I am also planning to put any corrections, modifications, and extensions on the Internet
so that they can be obtained either from the following anonymous ftp:
ftp gate.ee.lsu.edu cd pub/kemin/books/essentials/
or from the author’s home page:
http://kilo.ee.lsu.edu/kemin/books/essentials/
This book would not be possible without the work done jointly for the previous
book with Professor John C. Doyle and Professor Keith Glover. I thank them for their
influence on my research and on this book. Their serious attitudes toward scientific
research have been reference models for me. I am especially grateful to John for having
me as a research fellow in Caltech, where I had two very enjoyable years and had
opportunities to catch a glimpse of his “BIG PICTURE” of control.
I want to thank my editor from Prentice Hall, Tom Robbins, who originally proposed
the idea for this book and has been a constant source of support for me while writing it.
Without his support and encouragement, this project would have been a difficult one.
It has been my great pleasure to work with him.
I would like to express my sincere gratitude to Professor Bruce A. Francis for giving
me many helpful comments and suggestions on this book. Professor Francis has also
kindly provided many exercises in the book. I am also grateful to Professor Kang-Zhi Liu
and Professor Zheng-Hua Luo, who have made many useful comments and suggestions.
I want to thank Professor Glen Vinnicombe for his generous help in the preparation of
Chapters 16 and 17. Special thanks go to Professor Jianqing Mao for providing me the
opportunity to present much of this material in a series of lectures at Beijing University
of Aeronautics and Astronautics in the summer of 1996.
In addition, I would like to thank all those who have helped in many ways in making
this book possible, especially Professor Pramod P. Khargonekar, Professor Andr´eTits,
Professor Andrew Packard, Professor Jie Chen,Professor Jakob Stoustrup, Professor
Hans Henrik Niemann, Professor Malcolm Smith, Professor Tryphon Georgiou, Professor Tongwen Chen, Professor HitayOzbay, ¨ Professor Gary Balas, Professor Carolyn
Beck, Professor Dennis S. Bernstein, Professor Mohamed Darouach, Dr. Bobby Bodenheimer,Professor Guoxiang Gu, Dr. Weimin Lu,Dr. John Morris,Dr. Matt Newlin,
Professor LiQiu, Professor Hector P.Rotstein,Professor Andrew Teel, Professor Jagannathan Ramanujam, Dr. Linda G. Bushnell, Xiang Chen, Greg Salomon, Pablo A.
Parrilo, and many other people.
I would also like to thank the following agencies for supporting my research: National
Science Foundation, Army Research Office (ARO), Air Force of Scientific Research, and
the Board of Regents in the State of Louisiana.
Finally, I would like to thank my wife, Jing, and my son,Eric, for their generous
support, understanding, and patience during the writing of this book.
Kemin ZhouPREFACE ix
Here is how H∞ is pronounced in Chinese:
It means “The joy of love is endless.”Contents
Preface vii
Notation and Symbols xv
List of Acronyms xvii
1 Introduction 1
1.1 What Is This Book About?......................... 1
1.2 Highlights of This Book........................... 3
1.3 Notes and References ............................. 9
1.4 Problems ................................... 10
2 Linear Algebra 11
2.1 Linear Subspaces ............................... 11
2.2 Eigenvalues and Eigenvectors ........................ 12
2.3 Matrix Inversion Formulas .......................... 13
2.4 Invariant Subspaces............................. 15
2.5 Vector Norms and Matrix Norms ...................... 16
2.6 Singular Value Decomposition ........................ 19
2.7 Semidefinite Matrices............................ 23
2.8 Notes and References ............................. 24
2.9 Problems ................................... 24
3LinearSystems 27
3.1 Descriptions of Linear Dynamical Systems ................. 27
3.2 Controllability and Observability ...................... 28
3.3 Observers and Observer-Based Controllers ................. 31
3.4 Operations on Systems ............................ 34
3.5 State-Space Realizations for Transfer Matrices .............. 35
3.6 Multivariable System Poles and Zeros ................... 38
3.7 Notes and References ............................. 41
3.8 Problems ................................... 42
xixii CONTENTS
4 H2 and H∞ Spaces 45
4.1 Hilbert Spaces................................ 45
4.2 H2 and H∞ Spaces .............................. 47
4.3 Computing L2 and H2 Norms ........................ 53
4.4 Computing L∞ and H∞ Norms ....................... 55
4.5 Notes and References ............................. 61
4.6 Problems ................................... 62
5 Internal Stability 65
5.1 Feedback Structure .............................. 65
5.2 Well-Posedness of Feedback Loop ...................... 66
5.3 Internal Stability ............................... 68
5.4 Coprime Factorization over RH ∞ ...................... 71
5.5 Notes and References ............................. 77
5.6 Problems ................................... 77
6 Performance Specifications and Limitations 81
6.1 Feedback Properties............................. 81
6.2 Weighted H2 and H∞ Performance ..................... 85
6.3 Selection of Weighting Functions ...................... 89
6.4 Bode’s Gain and Phase Relation ...................... 94
6.5 Bode’s Sensitivity Integral .......................... 98
6.6 Analyticity Constraints........................... 100
6.7 Notes and References ............................. 102
6.8 Problems ................................... 102
7 Balanced Model Reduction 105
7.1 Lyapunov Equations............................. 106
7.2 Balanced Realizations ............................ 107
7.3 Model Reduction by Balanced Truncation ................. 117
7.4 Frequency-Weighted Balanced Model Reduction .............. 124
7.5 Notes and References ............................. 126
7.6 Problems ................................... 127
8 Uncertainty and Robustness 129
8.1 Model Uncertainty .............................. 129
8.2 Small Gain Theorem ............................. 137
8.3 Stability under Unstructured Uncertainties ................ 141
8.4 Robust Performance............................. 147
8.5 Skewed Specifications ............................ 150
8.6 Classical Control for MIMO Systems .................... 154
8.7 Notes and References ............................. 157
8.8 Problems ................................... 158CONTENTS xiii
9 Linear Fractional Transformation 165
9.1 Linear Fractional Transformations ..................... 165
9.2 Basic Principle................................ 173
9.3 Redheffer Star Products ........................... 178
9.4 Notes and References ............................. 180
9.5 Problems ................................... 181
10 µ and µ Synthesis 183
10.1 General Framework for System Robustness ................ 184
10.2 Structured Singular Value .......................... 187
10.3 Structured Robust Stability and Performance ............... 200
10.4 Overview of µ Synthesis ........................... 213
10.5 Notes and References ............................. 216
10.6 Problems................................... 217
11 Controller Parameterization 221
11.1 Existence of Stabilizing Controllers ..................... 222
11.2 Parameterization of All Stabilizing Controllers .............. 224
11.3 Coprime Factorization Approach ...................... 228
11.4 Notes and References ............................. 231
11.5 Problems................................... 231
12 Algebraic Riccati Equations 233
12.1 Stabilizing Solution and Riccati Operator ................. 234
12.2 Inner Functions ................................ 245
12.3 Notes and References ............................. 246
12.4 Problems................................... 246
13 H2 Optimal Control 253
13.1 Introduction to Regulator Problem ..................... 253
13.2 Standard LQR Problem ........................... 255
13.3 Extended LQR Problem ........................... 258
13.4 Guaranteed Stability Margins of LQR ................... 259
13.5 Standard H2 Problem ............................ 261
13.6 Stability Margins of H 2 Controllers ..................... 265
13.7 Notes and References ............................. 267
13.8 Problems................................... 267
14 H∞ Control 269
14.1 Problem Formulation ............................. 269
14.2 A Simplified H∞ Control Problem ..................... 270
14.3 Optimality and Limiting Behavior ..................... 282
14.4 Minimum Entropy Controller ........................ 286
14.5 An Optimal Controller ............................ 286xiv CONTENTS
14.6 General H∞ Solutions............................ 288
14.7 Relaxing Assumptions ............................ 291
14.8 H2 and H∞ Integral Control........................ 294
14.9 H∞ Filtering ................................. 297
14.10Notes and References ............................. 299
14.11Problems................................... 300
15 Controller Reduction 305
15.1 H∞ Controller Reductions .......................... 306
15.2 Notes and References ............................. 312
15.3 Problems................................... 313
16 H∞ Loop Shaping 315
16.1 Robust Stabilization of Coprime Factors .................. 315
16.2 Loop-Shaping Design ............................. 325
16.3 Justification for H ∞ Loop Shaping ..................... 328
16.4 Further Guidelines for Loop Shaping .................... 334
16.5 Notes and References ............................. 341
16.6 Problems................................... 342
17 Gap Metric and ν-Gap Metric 349
17.1 Gap Metric.................................. 350
17.2 ν-Gap Metric................................. 357
17.3 Geometric Interpretation of ν-Gap Metric ................. 370
17.4 Extended Loop-Shaping Design ....................... 373
17.5 Controller Order Reduction ......................... 375
17.6 Notes and References ............................. 375
17.7 Problems................................... 375
18 Miscellaneous Topics 377
18.1 Model Validation ............................... 377
18.2 Mixed µ Analysis and Synthesis ....................... 381
18.3 Notes and References ............................. 389
18.4 Problems................................... 390
Bibliography 391
Index 407Notation and Symbols
R and C fields of real and complex numbers
F field, either R or C
C
− and C− open and closed left-half plane
C+ and C+ open and closed right-half plane
jR imaginary axis
∈ belong to
⊂ subset
∪ union
∩ intersection
2 end of proof
3 end of remark
:= defined as
' and / asymptotically greater and less than
and much greater and less than
α complex conjugate of α ∈ C
|α| absolute value of α ∈ C
Re(α)realpartofα∈C
In n × n identity matrix
[aij ] a matrix with a ij as its ith row and jth column element
diag(a1,...,a n)ann × n diagonal matrix with ai as its ith diagonal element
AT and A∗ transpose and complex conjugate transpose of A
A−1 and A+ inverse and pseudoinverse of A
A−∗ shorthand for (−A1)∗
det(A) determinant of A
trace(A)traceofA
xvxvi NOTATION AND SYMBOLS
λ(A) eigenvalue of A
ρ(A) spectral radius of A
ρR (A) real spectrum radius of A
σ(A)andσ(A) the largest and the smallest singular values of A
σi(A) ith singular value of A
κ(A) condition number of A
kAk spectral norm of A: kAk = σ(A)
Im(A), R(A) image (or range) space of A
Ker(A), N(A) kernel (or null) space of A
X
−(A) stable invariant subspace of A
Ric(H) the stabilizing solution of an ARE
g ∗ f convolution of g and f
∠ angle
h, i inner product
x ⊥ y orthogonal, hx, yi =0
D⊥ orthogonal complement of D
S⊥ orthogonal complement of subspace S, e.g., 2⊥ H
L2(−∞, ∞) time domain square integrable functions
L2+ := L2[0, ∞) subspace of2L(−∞, ∞) with functions zero for t<0
L2− := L2(−∞, 0] subspace of2L(−∞, ∞) with functions zero for t>0
L2(jR) square integrable functions on 0 including at C ∞
H2 subspace of2L(jR) with functions analytic in Re(s) > 0
H⊥
2 subspace of2L(jR) with functions analytic in Re(s) < 0
L∞ (jR) functions bounded on Re(s) = 0 including at ∞
H∞ the set of L∞ (jR) functions analytic in Re(s) > 0
H−
∞ the set of L∞ (jR) functions analytic in Re(s) < 0
prefix B and Bo closed and open unit ball, e.g. B∆ and Bo∆
prefix R real rational, e.g., RH ∞ and RH2,etc.
R
p(s) rational proper transfer matrices
G∼(s) shorthand for G T (−s)
A B
C D shorthand for state space realization C(sI − −1 A B + ) D
η(G(s)) number of right-half plane poles
η0(G(s)) number of imaginary axis poles
wno(G) winding number
F`(M,Q)lowerLFT
Fu(M,Q) upper LFT
M?N star productList of Acronyms
ARE algebraic Riccati equation
FDLTI finite dimensional linear time invariant
iff if and only if
lcf left coprime factorization
LFT linear fractional transformation
lhp or LHP left-half plane Re(s) < 0
LQG linear quadratic Gaussian
LTI linear time invariant
MIMO multi-input multioutput
nlcf normalized left coprime factorization
NP nominal performance
nrcf normalized right coprime factorization
NS nominal stability
rcf right coprime factorization
rhp or RHP right-half plane Re(s) > 0
RP robust performance
RS robust stability
SISO single-input single-output
SSV structured singular value (µ)
SVD singular value decomposition
xviixviii LIST OF ACRONYMSChapter 1
Introduction
This chapter gives a brief description of the problems considered in this book and the
key results presented in each chapter.
1.1 What Is This Book About?
This book is about basic robust and ∞ Hcontrol theory. We consider a control system
with possibly multiple sources of uncertainties, noises,and disturbances as shown in
Figure 1.1.
controller
reference signals
tracking errors noise
uncertainty uncertainty
other controlled signals
uncertainty
disturbance
System Interconnection
Figure 1.1:General system interconnection
12 INTRODUCTION
We consider mainly two types of problems:
• Analysis problems:Given a controller,determine ifthe controlled signals (including tracking errors, control signals, etc.) satisfy the desired properties for all
admissible noises, disturbances, and model uncertainties.
• Synthesis problems: Design a controller so that the controlled signals satisfy the
desired properties for all admissible noises, disturbances, and model uncertainties.
Most of our analysis and synthesis will be done on a unified linear fractional transformation (LFT) framework. To that end, we shall show that the system shown in Figure 1.1
can be put in the general diagram in Figure 1.2, where P is the interconnection matrix,
K is the controller, ∆ is the set of all possible uncertainty, w is a vector signal including
noises,disturbances, and reference signals, z is a vector signal including all controlled
signals and tracking errors, u is the control signal, and y is the measurement
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