Table Of ContentMathematical Methods
in Robust Control of
Linear Stochastic Systems
MATHEMATICAL CONCEPTS AND METHODS
IN SCIENCE AND ENGINEERING
Series Editor: Angelo Miele
Mechanical Engineering and Mathematical Sciences
Rice University
Latest Volumes in this series:
MATHEMATICAL METHODS IN ROBUST CONTROL OF LINEAR
STOCHASTIC SYSTEMS • Vasile Dragan, loader Morozan, and
Adrian-Mihail Stoica
PRINCIPLES OF ENGINEERING MECHANICS: VOLUME 2.
DYNAMICS—THE ANALYSIS OF MOTION • Millard F. Beatty, Jr.
CONSTRAINED OPTIMIZATION AND IMAGE SPACE ANALYSIS:
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NONLINEAR EFFECTS IN FLUIDS AND SOLIDS • Editors Michael M. Carroll
and Michael A. Hayes
Mathematical Methods
in Robust Control of
Linear Stochastic Systems
Vasile Dragan
Institute of Mathematics of the Romanian Academy
Bucharest, Romania
Toader Morozan
Institute of Mathematics of the Romanian Academy
Bucharest, Romania
Adrian-Mihail Stoica
University Politehnica of Bucharest
Bucharest, Romania
^Spri nger
Vasile Dragan Toader Morozan
Institute of Mathematics of Institute of Mathematics of
the Romanian Academy the Romanian Academy
RO. Box 1-764, Ro 70700 P.O. Box 1-764, Ro 70700
Bucharest, Romania Bucharest, Romania
[email protected] toader .morozan@ imar. ro
Bucharest, Romania
ainstoica@rdslink. ro
Adrian-Mihail Stoica
University Politehnica of Bucharest
Str. Polizu, No. 1, Ro-77206
Bucharest, Romania
amstoica^rdslink.ro
Mathematics Subject Classification (2000): 93EXX, 34F05
Library of Congress Control Number: 2006927804
ISBN-10: 0-387-30523-8 e-ISBN: 0-387-35924-9
ISBN-13: 978-0-387-30523-8
Printed on acid-free paper.
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Contents
Preface ix
1 Preliminaries to Probability Theory and Stochastic Differential
Equations 1
1.1 Elements of measure theory 1
1.2 Convergence theorems for integrals 5
1.3 Elements of probability theory 7
1.4 Independence 7
1.5 Conditional expectation 8
1.6 Stochastic processes 9
1.7 Stochastic processes with independent increments 11
1.8 Wiener process and Markov chain processes 12
1.9 Stochastic integral 14
1.10 An Ito-type formula 19
1.11 Stochastic differential equations 26
1.12 Stochastic linear differential equations 29
2 Exponential Stability and Lyapunov-Type Linear Equations 33
2.1 Linear positive operators on the Hilbert space of
symmetric matrices 34
2.2 Lyapunov-type differential equations on the space S^ 37
2.3 A class of linear differential equations on the space (R")^ 45
2.4 Exponential stability for Lyapunov-type equations on vS^ 47
2.5 Mean square exponential stability 62
2.6 Numerical examples 75
2.7 Affine systems 79
Notes and references 82
3 Structural Properties of Linear Stochastic Systems 85
3.1 Stabilizability and detectability of stochastic linear systems 85
3.2 Stochastic observability 93
viii Contents
3.3 Stochastic controllability 102
Notes and references 108
4 The Riccati Equations of Stochastic Control 109
4.1 Preliminaries 110
4.2 The maximal solution of SGRDE 114
4.3 Stabilizing solution of the SGRDE 124
4.4 The case 0 e T^ 132
4.5 The filtering Riccati equation 140
4.6 Iterative procedures 142
Notes and references 157
5 Linear Quadratic Control Problem for Linear Stochastic Systems 159
5.1 Formulation of the linear quadratic problem 159
5.2 Solution of the linear quadratic problems 161
5.3 The tracking problem 173
5.4 Stochastic H^ controllers 178
Notes and references 207
6 Stochastic Version of the Bounded Real Lemma and Applications .... 209
6.1 Input-output operators 209
6.2 Stochastic version of the Bounded Real Lemma 218
6.3 Robust stability with respect to linear structured uncertainty 240
Notes and references 256
7 Robust Stabilization of Linear Stochastic Systems 257
7.1 Formulation of the disturbance attenuation problem 257
7.2 Robust stabilization of linear stochastic systems.
The case of full state access 259
7.3 Solution of the DAP in the case of output measurement 273
7.4 DAP for linear stochastic systems with Markovian jumping 292
7.5 An //^-type filtering problem for signals corrupted with
multiplicative white noise 298
Notes and references 303
Bibliography 305
Index 311
Preface
This monograph presents a thorough description of the mathematical theory of robust
Unear stochastic control systems. The interest in this topic is motivated by the variety
of random phenomena arising in physical, engineering, biological, and social pro
cesses. The study of stochastic systems has a long history, but two distinct classes of
such systems drew much attention in the control literature, namely stochastic systems
subjected to white noise perturbations and systems with Markovian jumping. At the
same time, the remarkable progress in recent decades in the control theory of deter
ministic dynamic systems strongly influenced the research effort in the stochastic area.
Thus, the modem treatments of stochastic systems include optimal control, robust sta
bilization, and H^- and //^-type results for both stochastic systems corrupted with
white noise and systems with jump Markov perturbations.
In this context, there are two main objectives of the present book. The first one is
to develop a mathematical theory of linear time-varying stochastic systems including
both white noise and jump Markov perturbations. From the perspective of this gener
alized theory the stochastic systems subjected only to white noise perturbations or to
jump Markov perturbations can be regarded as particular cases. The second objective
is to develop analysis and design methods for advanced control problems of linear
stochastic systems with white noise and Markovian jumping as linear-quadratic con
trol, robust stabilization, and disturbance attenuation problems. Taking into account
the major role played by the Riccati equations in these problems, the book presents this
type of equation in a general framework. Particular attention is paid to the numerical
aspects arising in the control problems of stochastic systems; new numerical algo
rithms to solve coupled matrix algebraic Riccati equations are also proposed and
illustrated by numerical examples.
The book contains seven chapters. Chapter 1 includes some prerequisites con
cerning measure and probability theory that will be used in subsequent developments
in the book. In the second part of this chapter, detailed proofs of some new results,
such as the Ito-type formula in a general case covering the classes of stochastic sys
tems with white noise perturbations and Markovian jumping, are given. The Ito-type
formula plays a crucial role in the proofs of the main results of the book.
X Preface
Chapter 2 is mainly devoted to the exponential stability of linear stochastic sys
tems. It is proved that the exponential stability in the mean square of the considered
class of stochastic systems is equivalent with the exponential stability of an appropri
ate class of deterministic systems over a finite-dimensional Hilbert space. Necessary
and sufficient conditions for exponential stability for such deterministic systems are
derived in terms of some Lyapunov-type equations. Then necessary and sufficient
conditions in terms of Lyapunov functions for mean square exponential stability are
obtained. These results represent a generalization of the known conditions concerning
the exponential stability of stochastic systems subjected to white noise and Markovian
jumping, respectively.
Some structural properties such as controllability, stabilizability, observability,
and detectability of linear stochastic systems subjected to both white noise and jump
Markov perturbations are considered in Chapter 3. These properties play a key role
in the following chapters of the book.
In Chapter 4 differential and algebraic generalized Riccati-type equations arising
in the control problems of stochastic systems are introduced. Our attention turns to the
maximal, minimal, and stabilizing solutions of these equations for which necessary
and sufficient existence conditions are derived. The final part of this chapter provides
an iterative procedure for computing the maximal solution of such equations.
In the fifth chapter of the book, the linear-quadratic problem on the infinite hori
zon for stochastic systems with both white noise and jump Markov perturbations is
considered. The problem refers to a general situation: The considered systems are
subjected to both state and control multiplicative white noise and the optimization is
performed under the class of nonanticipative stochastic controls. The optimal control
is expressed in terms of the stabilizing solution of coupled generalized Riccati equa
tions. As an application of the results deduced in this chapter, we consider the optimal
tracking problem.
Chapter 6 contains corresponding versions of some known results from the deter
ministic case, such as the Bounded Real Lemma, the Small Gain Theorem, and the
stability radius, for the considered class of stochastic systems. Such results have been
obtained separately in the stochastic framework for systems subjected to white noise
and Markov perturbations, respectively. In our book, these results appear as partic
ular situations of a more general class of stochastic systems including both types of
perturbations.
In Chapter 7 the y-attenuation problem of stochastic systems with both white
noise and Markovian jumping is considered. Necessary and sufficient conditions for
the existence of a stabilizing /-attenuating controller are obtained in terms of a system
of coupled game-theoretic Riccati equations and inequalities. These results allow one
to solve various robust stabilization problems of stochastic systems subjected to white
noise and Markov perturbations, as illustrated by numerical examples.
The monograph is based entirely on original recent results of the authors; some
of these results have been recently published in control journals and conferences
proceedings. There are also some other results that appear for the first time in this
book.
