Safety critical and high-integrity systems, such as industrial plants and economic systems can be subject to abrupt changes - for instance due to component or interconnection failure, and sudden environment changes etc. Combining probability and operator theory, Discrete-Time Markov Jump Linear Systems provides a unified and rigorous treatment of recent results for the control theory of discrete jump linear systems, which are used in these areas of application. The book is designed for experts in linear systems with Markov jump parameters, but is also of interest for specialists in stochastic control since it presents stochastic control problems for which an explicit solution is possible - making the book suitable for course use. From the reviews: "This text is very well written...it may prove valuable to those who work in the area, are at home with its mathematics, and are interested in stability of linear systems, optimal control, and filtering." Journal of the American Statistical Association, December 2005

Marcelo Dutra Fragoso is Professor in the Department of Systems and Control at the National Laboratory for Scientific Computing-LNCC/MCT, Rio de Janeiro.

1 Markov Jump Linear Systems

1

(14)

1.1 Introduction

1

(3)

1.2 Some Examples

4

(4)

1.3 Problems Considered in this Book

8

(3)

1.4 Some Motivating Remarks

11

(1)

1.5 A Few Words On Our Approach

12

(1)

1.6 Historical Remarks

13

(2)

2 Background Material

15

(14)

2.1 Some Basics

15

(3)

2.2 Auxiliary Results

18

(2)

2.3 Probabilistic Space

20

(1)

2.4 Linear System Theory

21

(6)

2.4.1 Stability and the Lyapunov Equation

21

(2)

2.4.2 Controllability and Observability

23

(3)

2.4.3 The Algebraic Riccati Equation and the Linear-Quadratic Regulator

26

(1)

2.5 Linear Matrix Inequalities

27

(2)

3 On Stability

29

(42)

3.1 Outline of the Chapter

29

(1)

3.2 Main Operators

30

(6)

3.3 MSS: The Homogeneous Case

36

(12)

3.3.1 Main Result

36

(1)

3.3.2 Examples

37

(4)

3.3.3 Proof of Theorem 3.9

41

(4)

3.3.4 Easy to Check Conditions for Mean Square Stability

45

(3)

3.4 MSS: The Non-homogeneous Case

48

(9)

3.4.1 Main Results

48

(1)

3.4.2 Wide Sense Stationary Input Sequence

49

(6)

3.4.3 The l2-disturbance Case

55

(2)

3.5 Mean Square Stabilizability and Detectability

57

(6)

3.5.1 Definitions and Tests

57

(2)

3.5.2 Stabilizability with Markov Parameter Partially Known

59

(4)

3.6 Stability With Probability One

63

(6)

3.6.1 Main Results

63

(3)

3.6.2 An Application of Almost Sure Convergence Results

66

(3)

3.7 Historical Remarks

69

(2)

4 Optimal Control

71

(30)

4.1 Outline of the Chapter

71

(1)

4.2 The Finite Horizon Quadratic Optimal Control Problem

72

(6)

4.2.1 Problem Statement

72

(2)

4.2.2 The Optimal Control Law

74

(4)

4.3 Infinite Horizon Quadratic Optimal Control Problems

78

(4)

4.3.1 Definition of the Problems

78

(2)

4.3.2 The Markov Jump Linear Quadratic Regulator Problem

80

(1)

4.3.3 The Long Run Average Cost

81

(1)

4.4 The H2-control Problem

82

(8)

4.4.1 Preliminaries and the H2-norm

82

(1)

4.4.2 The H2-norm and the Grammians

83

(3)

4.4.3 An Alternative Definition for the H2-control Problem

86

(1)

4.4.4 Connection Between the CARE and the H2-control Problem

86

(4)

4.5 Quadratic Control with Stochastic l2-input

90

(9)

4.5.1 Preliminaries

90

(1)

4.5.2 Auxiliary Result

91

(3)

4.5.3 The Optimal Control Law

94

(2)

4.5.4 An Application to a Failure Prone Manufacturing System

96

(3)

4.6 Historical Remarks

99

(2)

5 Filtering

101

(30)

5.1 Outline of the Chapter

101

(1)

5.2 Finite Horizon Filtering with theta(kappa) Known

102

(7)

5.3 Infinite Horizon Filtering with theta(kappa) Known

109

(4)

5.4 Optimal Linear Filter with theta(kappa) Unknown

113

(6)

5.4.1 Preliminaries

113

(1)

5.4.2 Optimal Linear Filter

114

(3)

5.4.3 Stationary Linear Filter

117

(2)

5.5 Robust Linear Filter with theta(kappa) Unknown

119

(9)

5.5.1 Preliminaries

119

(1)

5.5.2 Problem Formulation

119

(5)

5.5.3 LMI Formulation of the Filtering Problem

124

(3)

5.5.4 Robust Filter

127

(1)

5.6 Historical Remarks

128

(3)

6 Quadratic Optimal Control with Partial Information

131

(12)

6.1 Outline of the Chapter

131

(1)

6.2 Finite Horizon Case

132

(4)

6.2.1 Preliminaries

132

(1)

6.2.2 A Separation Principle

133

(3)

6.3 Infinite Horizon Case

136

(5)

6.3.1 Preliminaries

136

(1)

6.3.2 Definition of the H2-control Problem

137

(2)

6.3.3 A Separation Principle for the H2-control of MJLS

139

(2)

6.4 Historical Remarks

141

(2)

7 Hinfinity-Control

143

(24)

7.1 Outline of the Chapter

143

(1)

7.2 The MJLS Hinfinity-like Control Problem

144

(4)

7.2.1 The General Problem

144

(1)

7.2.2 Hinfinity Main Result

145

(3)

7.3 Proof of Theorem 7.3

148

(14)

7.3.1 Sufficient Condition

148

(3)

7.3.2 Necessary Condition

151

(11)

7.4 Recursive Algorithm for the Hinfinity-control CARE

162

(4)

7.5 Historical Remarks

166

(1)

8 Design Techniques and Examples

167

(36)

8.1 Some Applications

167

(6)

8.1.1 Optimal Control for a Solar Thermal Receiver

167

(2)

8.1.2 Optimal Policy for the National Income with a Multiplier-Accelerator Model

169

(2)

8.1.3 Adding Noise to the Solar Thermal Receiver problem

171

(2)

8.2 Robust Control via LMI Approximations

173

(15)

8.2.1 Robust H2-control

174

(8)

8.2.2 Robust Mixed H2/Hinfinity-control

182

(5)

8.2.3 Robust Hinfinity-control

187

(1)

8.3 Achieving Optimal Hinfinity-control

188

(9)

8.3.1 Algorithm

188

(1)

8.3.2 Hinfinity-control for the UarmII Manipulator

189

(8)

8.4 Examples of Linear Filtering with theta(kappa) Unknown

197

(4)

8.4.1 Stationary LMMSE Filter

198

(1)

8.4.2 Robust LMMSE Filter

199

(2)

8.5 Historical Remarks

201

(2)

A Coupled Algebraic Riccati Equations

203

(26)

A.1 Duality Between the Control and Filtering CARE

203

(5)

A.2 Maximal Solution for the CARE

208

(8)

A.3 Stabilizing Solution for the CARE

216

(10)

A.3.1 Connection Between Maximal and Stabilizing Solutions

216

(1)

A.3.2 Conditions for the Existence of a Stabilizing Solution

217

(9)

A.4 Asymptotic Convergence

226

(3)

B Auxiliary Results for the Linear Filtering Problem with theta(kappa) Unknown

229

(20)

B.1 Optimal Linear Filter

229

(7)

B.1.1 Proof of Theorem 5.9 and Lemma 5.11

229

(3)

B.1.2 Stationary Filter

232

(4)

B.2 Robust Filter

236

(13)

C Auxiliary Results for the H2-control Problem

249

(8)

References

257

(14)

Notation and Conventions

271

(6)

Index

277

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Discrete-time Markovian Jump Linear Systems

9781852337612

1852337613

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