The Essentials of Linear State-Space Systems

Provides a thorough introduction to the properties of linear, time-invariant models of dynamical systems, as required for further work in feedback control system design, power system design and analysis, communications, signal processing, robotics, and simulation. The state-space model is used throughout, since it is a fundamental conceptual tool, although the background analysis applies to other models. Modelling and stability of general nonlinear systems is introduced, with the detailed analysis concentrating on LTI systems.

J. Dwight Aplevich is the author of The Essentials of Linear State-Space Systems, published by Wiley.

Preface

vii

Introduction

1

(26)

The structure of state-space models

1

(8)

The concept of state

7

(2)

Linear models

9

(3)

Time-invariant models

12

(1)

Linear, time-invariant (LTI) models

13

(1)

System properties and model properties

14

(2)

Linearized small-signal models

16

(6)

Further study

22

(1)

Problems

23

(4)

Solution of state-space equations

27

(32)

Solution of discrete-time equations

28

(10)

LTI equations

29

(2)

Free response

31

(1)

Forced response

32

(1)

Weighting sequence

33

(2)

Impulse response

35

(2)

Convolution

37

(1)

Solution of continuous-time equations

38

(12)

Existence and uniqueness

39

(1)

LTI continuous-time equations

40

(1)

Free response of continuous-time LTI systems

40

(4)

Complete response of continuous-time LTI systems

44

(2)

Forced response

46

(1)

Continuous-time impulse response

47

(2)

Continuous-time convolution

49

(1)

Discretization

50

(4)

Further study

54

(1)

Problems

54

(5)

Transform methods

59

(14)

Continuous-time models

61

(5)

Free response

62

(1)

Forced response and transfer matrix

63

(1)

Properties of the transfer matrix

64

(2)

Discrete-time models

66

(3)

Free response

68

(1)

Forced response

69

(1)

Further study

69

(1)

Problems

69

(4)

Writing state-space equations

73

(30)

Graph-based methods: Electric circuits

74

(3)

Energy-based methods: Euler-Lagrange equations

77

(6)

Quadratic forms

79

(1)

Standard matrix form

79

(4)

Aggregation

83

(1)

Operational diagrams: Digital filters

84

(2)

Computer circuits

85

(1)

Continuous-time operational diagrams

86

(1)

High-order equations

87

(2)

Direct realization of high-order linear equations

87

(2)

Controllable and observable realizations

89

(4)

Factored realizations

93

(2)

Multi-input, multi-output (MIMO) transfer functions

95

(1)

Further study

96

(1)

Problems

97

(6)

Matrices over a field

103

(52)

Basic definitions

103

(6)

Field axioms

103

(2)

Matrix definitions and operations

105

(4)

Determinants

109

(4)

Properties of determinants

112

(1)

Rank, elementary transformations, and equivalence

113

(15)

Elementary transformations

113

(2)

Elementary matrices

115

(2)

Echelon forms

117

(2)

Properties of echelon forms

119

(2)

The normal form

121

(2)

The Singular-Value Decomposition (SVD)

123

(5)

Matrix inverses

128

(2)

Left inverse

128

(1)

Right inverse

128

(1)

Inverse

129

(1)

The characteristic equation

130

(3)

The Cayley-Hamilton theorem

131

(2)

The Ho algorithm

133

(11)

The context

133

(1)

Constructive solution

134

(1)

Development of the algorithm

135

(9)

Solution of linear equations

144

(6)

General method

144

(2)

Abbreviated method

146

(2)

Uniqueness and generality of solutions

148

(1)

Special cases

149

(1)

Further study

150

(1)

Problems

150

(5)

Vector spaces

155

(16)

Vector-space axioms

155

(1)

Subspaces

156

(1)

Linear dependence of vectors

157

(1)

Range, basis, dimension, and null space

157

(6)

Bases for the range and null space

160

(2)

Orthogonal bases

162

(1)

Change of basis

163

(4)

Further study

167

(1)

Problems

167

(4)

Similarity transformations

171

(22)

Invariance of the external behavior

172

(1)

Eigenvalues, eigenvectors, and diagonalization

173

(9)

Near-diagonalization: the Jordan canonical form

182

(1)

Functions of square matrices via the Jordan form

183

(3)

General functions of square matrices

186

(3)

Further study

189

(1)

Problems

190

(3)

Stability

193

(18)

Basic definitions

193

(2)

LTI systems

195

(2)

LTI Continuous-time systems

196

(1)

LTI Discrete-time systems

196

(1)

Energy functions and Lyapunov stability

197

(11)

Energy functions for LTI systems

201

(1)

Lyapunov equations for LTI continuous-time systems

201

(2)

Solving continuous Lyapunov equations

203

(3)

Discrete-time Lyapunov equations

206

(2)

Further study

208

(1)

Problems

208

(3)

Minimality via similarity transformations

211

(20)

Step 1: Controllability

213

(5)

Construction of the controllability transformation

214

(4)

Step 2: Observability

218

(2)

Direct transformation

218

(1)

Observability by constructing the dual system

219

(1)

Minimality

220

(8)

The Kalman canonical decomposition

228

(1)

Further study

229

(1)

Problems

229

(2)

Poles and Zeros

231

(14)

The Smith-McMillan form

233

(6)

Construction of the Smith form

237

(2)

Computation of poles and zeros

239

(3)

Further study

242

(1)

Problems

242

(3)

References

245

(4)

Appendix Solutions

249

(46)

Chapter 1

249

(2)

Chapter 2

251

(5)

Chapter 3

256

(5)

Chapter 4

261

(5)

Chapter 5

266

(7)

Chapter 6

273

(2)

Chapter 7

275

(4)

Chapter 8

279

(4)

Chapter 9

283

(4)

Chapter 10

287

(8)

Index

295

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The Essentials of Linear State-Space Systems

Summary

Author Biography

Table of Contents

Supplemental Materials