Dragan / Morozan / Stoica Mathematical Methods in Robust Control of Discrete-Time Linear Stochastic Systems

1. Auflage 2009
ISBN: 978-1-4419-0630-4
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 346 Seiten

In this monograph the authors develop a theory for the robust control of discrete-time stochastic systems, subjected to both independent random perturbations and to Markov chains. Such systems are widely used to provide mathematical models for real processes in fields such as aerospace engineering, communications, manufacturing, finance and economy. The theory is a continuation of the authors' work presented in their previous book entitled 'Mathematical Methods in Robust Control of Linear Stochastic Systems' published by Springer in 2006. Key features: - Provides a common unifying framework for discrete-time stochastic systems corrupted with both independent random perturbations and with Markovian jumps which are usually treated separately in the control literature; - Covers preliminary material on probability theory, independent random variables, conditional expectation and Markov chains; - Proposes new numerical algorithms to solve coupled matrix algebraic Riccati equations; - Leads the reader in a natural way to the original results through a systematic presentation; - Presents new theoretical results with detailed numerical examples. The monograph is geared to researchers and graduate students in advanced control engineering, applied mathematics, mathematical systems theory and finance. It is also accessible to undergraduate students with a fundamental knowledge in the theory of stochastic systems.

Autoren/Hrsg.

Dragan, Vasile

Morozan, Toader

Stoica, Adrian-Mihail

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;1 Elements of probability theory;11
3.1;1.1 Probability spaces;11
3.2;1.2 Random variables;13
3.2.1;1.2.1 Definitions and basic results;13
3.2.2;1.2.2 Integrable random variables. Expectation;13
3.2.3;1.2.3 Independent random variables;15
3.3;1.3 Conditional expectation;15
3.4;1.4 Markov chains;17
3.4.1;1.4.1 Stochastic matrices;17
3.4.2;1.4.2 Markov chains;17
3.5;1.5 Some remarkable sequences of random variables;20
3.6;1.6 Discrete-time controlled stochastic linear systems;22
3.7;1.7 The outline of the book;27
3.8;1.8 Notes and references;29
4;2 Discrete-time linear equations defined by positive operators;30
4.1;2.1 Some preliminaries;30
4.1.1;2.1.1 Convex cones;30
4.1.2;2.1.2 Minkovski seminorms and Minkovski norms;32
4.2;2.2 Discrete-time equations defined by positive linear operators on ordered Hilbert spaces;36
4.2.1;2.2.1 Positive linear operators on ordered Hilbert spaces;36
4.2.2;2.2.2 Discrete-time affine equations;40
4.3;2.3 Exponential stability;42
4.4;2.4 Some robustness results;51
4.5;2.5 Lyapunov-type operators;54
4.5.1;2.5.1 Sequences of Lyapunov-type operators;54
4.5.2;2.5.2 Exponential stability;57
4.5.3;2.5.3 Several special cases;64
4.5.4;2.5.4 A class of generalized Lyapunov-type operators;66
4.6;2.6 Notes and references;67
5;3 Mean square exponential stability;68
5.1;3.1 Some representation theorems;69
5.2;3.2 Mean square exponential stability. The general case;77
5.3;3.3 Lyapunov-type criteria;85
5.4;3.4 The case of homogeneous Markov chain;86
5.5;3.5 Some special cases;88
5.5.1;3.5.1 The periodic case;88
5.5.2;3.5.2 The time-invariant case;93
5.5.3;3.5.3 Another particular case;95
5.6;3.6 The case of the systems with coefficients depending upon .t and .t-1;98
5.7;3.7 Discrete-time affine systems;104
5.8;3.8 Notes and references;110
6;4 Structural properties of linear stochastic systems;111
6.1;4.1 Stochastic stabilizability and stochastic detectability;111
6.1.1;4.1.1 Definitions and criteria for stochastic stabilizability and stochastic detectability;111
6.1.2;4.1.2 A stability criterion;115
6.2;4.2 Stochastic observability;119
6.3;4.3 Some illustrative examples;129
6.4;4.4 A generalization of the concept of uniform observability;131
6.5;4.5 The case of the systems with coefficients depending upon .t, .t-1;134
6.6;4.6 A generalization of the concept of stabilizability;136
6.7;4.7 Notes and references;137
7;5 Discrete-time Riccati equations of stochastic control;138
7.1;5.1 An overview on discrete-time Riccati-type equations of stochastic control;138
7.2;5.2 A class of discrete-time backward nonlinear equations;142
7.2.1;5.2.1 Several notations;142
7.2.2;5.2.2 A class of discrete-time generalized Riccati equations;143
7.3;5.3 A comparison theorem and several consequences;147
7.4;5.4 The maximal solution;148
7.5;5.5 The stabilizing solution;155
7.6;5.6 The Minimal Solution;161
7.7;5.7 An iterative procedure to compute the maximal solution and the stabilizing solution of DTSGRE;165
7.8;5.8 Discrete-time Riccati equations of stochastic control;173
7.8.1;5.8.1 The maximal solution and the stabilizing solution of DTSRE-C;173
7.8.2;5.8.2 The case of DTSRE-C with definite sign of weighting matrices;177
7.8.3;5.8.3 The case of the systems with coefficients depending upon .t and .t-1;180
7.9;5.9 Discrete-time Riccati filtering equations;184
7.10;5.10 A numerical example;188
7.11;5.11 Notes and references;190
8;6 Linear quadratic optimization problems;191
8.1;6.1 Some preliminaries;191
8.1.1;6.1.1 A brief discussion on the linear quadratic optimization problems;191
8.1.2;6.1.2 A usual class of stochastic processes;193
8.1.3;6.1.3 Several auxiliary results;193
8.2;6.2 The problem of the linear quadratic regulator;199
8.3;6.3 The linear quadratic optimization problem;202
8.4;6.4 The linear quadratic problem. The affine case;210
8.4.1;6.4.1 The problem setting;211
8.4.2;6.4.2 Solution of the problem OP 1;212
8.4.3;6.4.3 On the global bounded solution of (6.11);214
8.4.4;6.4.4 The solution of the problem OP 2;217
8.5;6.5 Tracking problems;222
8.6;6.6 Notes and references;227
9;7 Discrete-time stochastic H2 optimal control;228
9.1;7.1 H2 norms of discrete-time linear stochastic systems;229
9.1.1;7.1.1 Model setting;229
9.1.2;7.1.2 H2-type norms;230
9.1.3;7.1.3 Systems with coefficients depending upon .t and .t-1;231
9.2;7.2 The computation of H2-type norms;231
9.2.1;7.2.1 The computations of the norm G 2 and the norm ˜ G˜ 2;232
9.2.2;7.2.2 The computation of the norm |||G|||2;241
9.2.3;7.2.3 The computation of the H2 norms for the system of type (7.1);246
9.3;7.3 Some robustness issues;250
9.4;7.4 H2 optimal controllers. The case with full access to measurements;252
9.4.1;7.4.1 H2 optimization;252
9.4.2;7.4.2 The case of systems with coefficients depending upon .t and .t-1;258
9.5;7.5 The H2 optimal control. The case with partial access to measurements;260
9.5.1;7.5.1 Problem formulation;260
9.5.2;7.5.2 Some preliminaries;262
9.5.3;7.5.3 The solution of the H2 optimization problems;265
9.6;7.6 H2 suboptimal controllers in a state estimator form;270
9.7;7.7 An H2 filtering problem;279
9.8;7.8 A case study;287
9.9;7.9 Notes and references;288
10;8 Robust stability and robust stabilization of discrete-time linear stochastic systems;291
10.1;8.1 A brief motivation;291
10.2;8.2 Input–output operators;293
10.3;8.3 Stochastic version of bounded real lemma;302
10.3.1;8.3.1 Stochastic bounded real lemma. The finite horizon time case;303
10.3.2;8.3.2 The bounded real lemma. The infinite time horizon case;306
10.3.3;8.3.3 An H8-type filtering problem;316
10.4;8.4 Robust stability. An estimate of the stability radius;322
10.4.1;8.4.1 The small gain theorems;322
10.4.2;8.4.2 An estimate of the stability radius;328
10.5;8.5 The disturbance attenuation problem;331
10.5.1;8.5.1 The problem formulation;331
10.5.2;8.5.2 The solution of the disturbance attenuation problem. The case of full state measurements;333
10.5.3;8.5.3 Solution of a robust stabilization problem;339
10.6;8.6 Notes and references;340
11;Bibliography;341
12;Abbreviations;347
13;Index;348