Matrix Riccati Equations in Control and Systems Theory

584 Pages·2003·39.96 MB·English

by  Hisham Abou-Kandil

Systems & Control: Foundations & Applications
Series Editor
Tamer Ba~ar, University of Illinois at Urbana-Champaign
Editorial Board
Karl Johan Astrom, Lund Institute of Technology, Lund
Han-Fu Chen, Academia Sinica, Bejing
William Helton, University of California, San Diego
Alberto Isidori, University of Rome and
Washington University, St. Louis
Petar V. Kokotovic, University of California, Santa Barbara
Alexander Kurzhanski, Russian Academy of Sciences, Moscow and
University of California, Berkeley
H. Vincent Poor, Princeton University
Mete Soner, K09 University, Istanbul
Hisham Abou-Kandil
Gerhard Freiling
Vlad Ionescu (t)
Gerhard Jank
Matrix Riccati Equations
in Control and Systems Theory
Birkhiiuser Verlag
Basel· Boston· Berlin
Authors:
Hisham Abou-Kandil Gerhard Freiling
Ecole Normale Superieure de Cachan Institute of Mathematics
Laboratoire S.A.T.I.E. (UMR CNRS 8029) University of Duisburg
61, Avenue du President Wilson Lotharstrasse 65
F-94230 Cachan D-47048 Duisburg
France Germany
[email protected] [email protected]
Gerhard Jank Vlad lonescu (t)
Institut fUr Mathematik Department of Mathematics
RWTHAachen University of Bucharest
Templergraben 55 Romania
D-52056 Aachen
[email protected]
2000 Mathematics Subject Classification 49N05, 49NlO, 49N35, 49N40,
49Nff, 90D05, 90D06, 90D10, 90D50, 90D65, 90D25, 93B36, 93B52,
93B35,93E20, l5A24, l5A57, 34Cll, 34K35, 47B35
A CIP catalogue record for this book is available from the Library of
Congress, Washington D.C., USA
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ISBN 3-7643-0085-X Birkhiiuser Verlag, Basel-Boston - Berlin
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© 2003 Birkhiiuser Verlag, p.o. Box 133, CH-4010 Basel, Switzerland
Member of the BertelsmannSpringer Publishing Group
Printed on acid-free paper produced of chlorine-free pulp. TCF 00
Printed in Germany
ISBN 3-7643-0085-X
987654321
To the late V. Ionescu, our esteemed colleague,
and
to our beloved families.
Contents
Preface xi
Notation xvi
1 Basic results for linear equations 1
1.1 Linear differential equations and linear algebraic equations. 1
1.2 Exponential dichotomy and L2 evolutions . . . . . . . . . . 11
2 Hamiltonian Matrices and Algebraic Riccati equations 21
2.1 Solutions of algebraic Riccati equations and graph subspaces 22
2.2 Indefinite scalar products and a canonical form of Hamiltonian
matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Hermitian algebraic Riccati equations ............. 50
2.4 Positive semi-definite solutions of standard algebraic Riccati
equations . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Hermitian discrete-time algebraic Riccati equations. . . 80
3 Global aspects of Riccati differential and difference equations 89
3.1 Riccati differential equations and associated linear systems. 90
3.1.1 Riccati differential equations, Riccati-transformation
and spectral factorization . . . . . . . . . . . . . . . 92
3.1.2 Riccati differential equations and linear boundary value
problems ..................... 95
3.2 A representation formula. . . . . . . . . . . . . . . . . 97
3.3 Flows on GraBmann manifolds: The extended Riccati
differential equation .......................... 110
3.4 General representation formulae for solutions of RDE and PRDE,
the time-continuous and periodic Riccati differential equation, and
dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.4.1 A general representation formula for solutions of RDE . . . 134
3.4.2 A representation formula for solutions of the periodic Riccati
differential equation PRDE . . . . . . . . . . . . . . . . . . 144
viii Contents
3.5 A representation formula for solutions of the discrete time Riccati
equation . . . . . . . . . . . . . . . . . . . . 150
3.5.1 Properties of the solutions to DARE 156
3.5.2 Properties of the solutions to DRDE 158
3.6 Global existence results ..... . 163
4 Hermitian Riccati differential equations 181
4.1 Comparison results for HRDE . 181
4.1.1 Arbitrary coefficients . 181
4.1. 2 Periodic coefficients 188
4.1.3 Constant coefficients . 189
4.1.4 Riccati inequalities . . 193
4.2 Monotonicity and convexity results:
A Frechet derivative based approach 198
4.2.1 Notation and preliminaries 198
4.2.2 Results for HARE . 202
4.2.3 Results for HDARE ... . 206
4.2.4 Results for HRDE .... . 207
4.3 Convergence to the semi-stabilizing solution 209
4.4 Dependence of HRDE on a parameter . 220
4.5 An existence theorem for general HRDE 239
4.6 A special property of HRDE ...... . 247
5 The periodic Riccati equation 257
5.1 Linear periodic differential equations . . . . . . . . . . . . . 257
5.2 Preliminary notation and results for linear periodic systems 262
5.3 Existence results for periodic Hermitian Riccati equations 268
5.4 Positive semi-definite periodic equilibria of PRDE . 279
6 Coupled and generalized Riccati equations 299
6.1 Some basic concepts in dynamic games. . . . . . . . . . . . . . 299
6.2 Non-symmetric Riccati equations in open loop Nash differential
games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
6.3 Discrete-time open loop Nash Riccati equations . . . . . . . 313
6.4 Non-symmetric Riccati equations in open loop Stackelberg
differential games . . . . . . . . . . . . . . . . . . . . . . . . 322
6.5 Discrete-time open loop Stackelberg equations. . . . . . . . 328
6.6 Coupled Riccati equations in closed loop Nash differential games 333
6.7 Rational matrix differential equations arising in stochastic control. 343
6.8 Rational matrix difference equations arising in stochastic control 377
6.9 Coupled Riccati equations in Markovian jump systems . . . . . . . 394
Contents IX
7 Symmetric differential Riccati equations: an operator based approach 411
7.1 Popov triplets: definition and equivalence 412
7.2 Associated objects . . . . . . . . . . 413
7.3 Associated operators . . . . . . . . . 422
7.4 Existence of the stabilizing solution. 428
7.5 Positivity theory and applications. 434
7.6 Differential Riccati inequalities . . . 439
7.7 The signature condition . . . . . . . 444
7.8 Differential Riccati theory: A Hamiltonian descriptor operator
approach. . . . . . . . . . . . . . . 455
7.8.1 Descriptors and dichotomy ........ 455
7.8.2 Hamiltonian descriptors . . . . . . . . . . 461
7.8.3 The stabilizing (anti-stabilizing) solution. 463
8 Applications to Robust Control Systems 467
8.1 The Four Block Nehari Problem . 467
8.1.1 Problem Statement . . . . . 467
8.1.2 A characterization of all solutions. 469
8.1.3 Main Result. . . . 471
8.2 Disturbance Attenuation . . . 479
8.2.1 Problem statement .. 479
8.2.2 A necessary condition 481
8.2.3 The Disturbance Feedforward Problem. 485
8.2.4 The least achievable tolerance of the DF problem . 490
9 Non-symmetric Riccati theory and applications 495
9.1 Non-symmetric Riccati theory ...... . 496
9.1.1 Basic notions and preliminary results ... 496
9.1.2 Toeplitz operators and Riccati equations. 501
9.2 Application to open loop Nash games 507
9.2.1 Definitions and Hilbert space 507
9.2.2 Unique Nash equilibria .... . 509
9.2.3 The general case ....... . 512
9.2.4 If any, then one or infinitely many 513
9.3 Application to open loop Stackelberg games 515
9.3.1 Characterization in Hilbert space 516
9.3.2 Unique Stackelberg equilibria . 517
9.3.3 A value function type approach . 520
A Appendix 527
A.1 Basic facts from control theory 527
A.2 The implicit function theorem. 530
References 533
x Contents
Index 569
List of Figures 572
Preface
In many fields of applied mathematics, engineering and economic sciences there
appear, e.g., as a consequence of variational problems to be solved, matrix (or
operator) Riccati equations. These can be algebraic Riccati equations as well as
Riccati difference or differential equations. Riccati equations are in our opinion
the simplest but most important class of non-linear equations. They show up in
the following domains, just to cite a few:
- linear optimal control and filtering problems with quadratic cost functionals,
- linear dynamic games with quadratic cost functionals,
- decoupling of linear systems of differential and difference equations,
- spectral factorization of operators,
- singular perturbation theory,
- boundary value problems for systems of ODEs,
- invariant embedding and scattering theory,
- differential geometry.
Besides the well-developed theory of symmetric (or Hermitian) Riccati equa
tions there are more and more demands from applications to develop further the
theory for non-symmetric matrix Riccati equations and also for generalized or
perturbed Riccati equation.
In the last decades developments of both algebraic and differential matrix
Riccati equations theory spread out in the scientific literature. In contrast to the
theory of symmetric (or hermitian) matrix algebraic Riccati equations, which has
been presented recently in the book of Lancaster and Rodman [LaRo95], the ba
sic theory of matrix Riccati differential equations is comprised in the monograph
of Reid [Reid72], along with some applications that were developed before 1972.
During the last three decades there was achieved great progress in the math
ematical theory of Riccati equations and in its applications, with emphasis on
control systems and differential games. Whereas symmetric Riccati equations play
a central role in optimal control, non-symmetric matrix Riccati equations show up
for instance in the theory of dynamic games and spectral factorization problems,
while generalized Riccati equations are common in stochastic control problems or
stochastic games.
The aim of this book is to present the state of the art of the theory of sym
metric (Hermitian) matrix Riccati equations and to contribute to the development
of the theory of non-symmetric Riccati equations as well as to certain classes
of coupled and generalized Riccati equations occurring in differential games and
stochastic control. For the major part of the results presented in this book there
exist infinite dimensional counterparts for operator Riccati equations; we do not
address this topic here and refer the reader to the textbooks [BdPDM92] and
[LaTr91]' [LaTrOOa], [LaTrOOb].