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LMIs in control.pdf
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详细说明：鲁棒控制-线性矩阵不等式处理方法-俞立 线性矩阵不等式的概念等CONTENTS
1.6 Exercises
2 Dissipative dynamical systems
33
2.1 Introductio
33
2.2 Dissipative dynamical systems
.34
2.2.1 Definitions and examples
2.2.2 a classification of sto
functi
37
2.2.3 Dissipation functions
2.3 Linear dissipative syslems with quadratic supply rates
2.3.1 Main results
2.3.2 Dissipation functions and spectral factors
43
2.3.3 The Kalman-Yakubovich-Popov lemma
.45
2.3.4 The positive real lemma
2.3.5 The bounded real lemma
47
2.4 Interconnected dissipative systems
2.5 Further reading
2.6 Exercises
50
3 Stability and nominal performance
55
3.1 Ly apunov stability
3.2 Generalized stability regions for ITI systems
61
3.3 The generalized plant concept
.63
3.4 Nominal performance and lmIs
68
3.4.1 Quadratic nominal performance
3.4.2 Ho nominal performance
69
3.4.3 112 nominal performance
Compilation: November 16. 2004
CONTENTS
3.4.4 Generalized H2 nominal performance
3.4.5 Li or peak-to-peak nominal performance
75
3.5 Further reading
3.6 Exercises
76
thesis
4.1 The setup
..,83
4.2 From analysis to svnthesis - a general procedure
86
4.3 Other performance specilications
92
4.3.1 Ilo Design
4.3.2 Positive real design
..93
4.3.3 H2-problems
4.3.4 Upper bound on peak-to-peak norm
95
4.4 Multi-objective and mixed controller design
96
4.5 Elimination of parameters
4.5.1 Dualization
4.5.2 Special linear matrix inequalities
104
4.5.3 The quadratic performance problem
109
4.5.4 H2-problems
113
4.6 SLate-feedback problems
115
4.7 Discrete-Time Systems
18
4.8 Exercises
.,.,,,,.119
5 Systems with parametric uncertaint
125
5.1 Parametric uncertainties
.125
5.1.1 Affine parameter dependent systems
.126
Compilation: November 16. 2004
ONTENTS
1.2 Polytopic parameter dependent systems
127
5.2 Robust stability for autonomous syster
127
5.2.1 Quadratic stabilit
5.2.2 Quadratic stability
ffi
y of affine
5.2.3 Quadratic stability of polytopic models
130
5.3P
ter dependent lyar
131
5.3.1 Time-invariant uncertainty
5.3.2 Time varyin
134
.135
Compilation: November 16. 2004
List of figures
1.1 Joseph-Louis Lagrange
1.2 Aleksandr Mikhailovich lyapunoy
28
2.1 System interconnections
.49
2.2 Model for suspension system
2.3 Feedback configuration
....53
3.1 A system interconnection
63
3.2 General closed-loop interconnection
3.3 Binary distillation column
80
4. 1 Multi-channel closed-loop system
4.2 Active suspension system
12
VI
LIST OF FIGURES
Compilation: November 16, 2004
Preface
In recent years linear matrix inequalities (LMis) have emerged as a powerful tool to approach con
trol problems that appear hard if not impossible to solve in an analytic fashion although the histor
of linear matrix inequalities goes hack to the fourties with a major emphasis of their role in con
trol in the sixties through the work of Kalman, Yakubovich, Popov and Willems, only during the
last decades powerful numerical interior point techniques have been developed to solve LMIs in a
practically efficient manner(Nesterov, Nemirovskii 1994). Currently, several commercial and non
commercial software packages are available that allow a simple coding of general LMI problems
and provide efficient tools to solve typical control problems in a numerically efficient manner
Boosted by the availability of fast LMI solvers, research in robust control has experienced a sig
nificant paradigm shift. Instead of arriving at an analytical solulion of an optimal control problem
and implementing such a solution in software so as to synthesize optimal controllers, the intention
is to reformulate a given control problem to verifying whether a specific linear matrix inequality is
solvable or, alternatively, to optimizing functionals over linear matrix inequality constraints. This
book aims at providing a state of the art treatment of the theory, the usage and the applications of
linear matrix inequalities in control. The main emphasis of this book is to reveal the basic prin-
ciples and background for formulating desired properties of a control system in the form of linear
matrix inequalities, and to demonstrate the techniques to reduce the corresponding controller syn
thesis problem to an LMI problem. The power of this approach is illustrated by several fundamental
robustness and performance problems in analysis and design of linear control systems
This book has been written for a graduate course on the subject of LMIs in systems and control
Within the graduate program of the dutch Institute of Systems and Control DIsC), this course is
intended to provide up-10-date information on the topic for students involved in either the practical
or theoretical aspects of control system design. DisC courses have the format of two class hours
once per week during a period of eight weeks. Within DISC, the first course on LMIs in control
has been given in 1997, when the first draft of this book has been distributed as lecture notes to the
students. The lecture notes of this course have been evaluating to the present book, thanks to the
many suggestions, criticism and help of many students that followed this course
Readers of this book are supposed to have an academic background in linear algebra, basic calculus,
and possibly in system and control theory
Compilation: November 16. 2004
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