The Essentials of Linear State Space Systems

Contents

Preface, vii

• 1 The structure of state-space models, 1
• 1.1 The concept of state, 7
• 2 Linear models, 9
• 3 Time-invariant models, 12
• 4 Linear, time-invariant (LTI) models, 13
• 5 System properties and model properties, 14
• 6 Linearized small-signal models, 16
• 7 Further study, 22
• 8 Problems, 23

• 1 Solution of discrete-time equations, 28
• 1.1 LTI equations, 29
• 1.2 Free response, 31
• 1.3 Forced response, 32
• 1.4 Weighting sequence, 33
• 1.5 Impulse response, 35
• 1.6 Convolution, 37
• 2 Solution of continuous-time equations, 38
• 2.1 Existence and uniqueness, 39
• 2.2 LTI continuous-time equations, 40
• 2.3 Free response of continuous-time LTI systems, 40
• 2.4 Complete response of continuous-time LTI systems, 44
• 2.5 Forced response, 46
• 2.6 Continuous-time impulse response, 47
• 2.7 Continuous-time convolution, 49
• 3 Discretization, 50
• 4 Further study, 54
• 5 Problems, 54

• 1 Continuous-time models, 61
• 1.1 Free response, 62
• 1.2 Forced response and transfer matrix, 63
• 1.3 Properties of the transfer matrix, 64
• 2 Discrete-time models, 66
• 2.1 Free response, 68
• 2.2 Forced response, 69
• 3 Further study, 69
• 4 Problems, 69

• 1 Graph-based methods: Electric circuits, 74
• 2 Energy-based methods: Euler-Lagrange equations, 77
• 2.1 Quadratic forms, 79
• 2.2 Standard matrix form, 79
• 3 Aggregation, 83
• 4 Operational diagrams: Digital filters, 84
• 4.1 Computer circuits, 85
• 5 Continuous-time operational diagrams, 86
• 6 High-order equations, 87
• 6.1 Direct realization of high-order linear equations, 87
• 7 Controllable and observable realizations, 89
• 8 Factored realizations, 93
• 9 Multi-input, multi-output (MIMO) transfer functions, 95
• 10 Further study, 96
• 11 Problems, 97

• 1 Basic definitions, 103
• 1.1 Field axioms, 103
• 1.2 Matrix definitions and operations, 105
• 2 Determinants, 109
• 2.1 Properties of determinants, 112
• 3 Rank, elementary transformations, and equivalence, 113
• 3.1 Elementary transformations, 113
• 3.2 Elementary matrices, 115
• 3.3 Echelon forms, 117
• 3.4 Properties of echelon forms, 119
• 3.5 The normal form, 121
• 3.6 The Singular-Value Decomposition (SVD), 123
• 4 Matrix inverses, 128
• 4.1 Left inverse, 128
• 4.2 Right inverse, 128
• 4.3 Inverse, 129
• 5 The characteristic equation, 130
• 5.1 The Cayley-Hamilton theorem, 131
• 6 The Ho algorithm, 133
• 6.1 The context, 133
• 6.2 Constructive solution, 134
• 6.3 Development of the algorithm, 135
• 7 Solution of linear equations, 144
• 7.1 General method, 144
• 7.2 Abbreviated method, 146
• 7.3 Uniqueness and generality of solutions, 148
• 7.4 Special cases, 149
• 8 Further study, 150
• 9 Problems, 150

• 1 Vector-space axioms, 155
• 2 Subspaces, 156
• 3 Linear dependence of vectors, 157
• 4 Range, basis, dimension, and null space, 157
• 4.1 Bases for the range and null space, 160
• 4.2 Orthogonal bases, 162
• 5 Change of basis, 163
• 6 Further study, 167
• 7 Problems, 167

• 1 Invariance of the external behavior, 172
• 2 Eigenvalues, eigenvectors, and diagonalization, 173
• 3 Near-diagonalization: the Jordan canonical form, 182
• 4 Functions of square matrices via the Jordan form, 183
• 5 General functions of square matrices, 186
• 6 Further study, 189
• 7 Problems, 190

• 1 Basic definitions, 193
• 2 LTI systems, 195
• 2.1 LTI Continuous-time systems, 196
• 2.2 LTI Discrete-time systems, 196
• 3 Energy functions and Lyapunov stability, 197
• 3.1 Energy functions for LTI systems, 201
• 3.2 Lyapunov equations for LTI continuous-time systems, 201
• 3.3 Solving continuous Lyapunov equations, 203
• 3.4 Discrete-time Lyapunov equations, 206
• 4 Further study, 208
• 5 Problems, 208

• 1 Step 1: Controllability, 213
• 1.1 Construction of the controllability transformation, 214
• 2 Step 2: Observability, 218
• 2.1 Direct transformation, 218
• 2.2 Observability by constructing the dual system, 219
• 3 Minimality, 220
• 4 The Kalman canonical decomposition, 228
• 5 Further study, 229
• 6 Problems, 229

• 1 The Smith-McMillan form, 233
• 1.1 Construction of the Smith form, 237
• 2 Computation of poles and zeros, 239
• 3 Further study, 242
• 4 Problems, 242

• Chapter 1, 249
• Chapter 2, 251
• Chapter 3, 256
• Chapter 4, 261
• Chapter 5, 266
• Chapter 6, 273
• Chapter 7, 275
• Chapter 8, 279
• Chapter 9, 283
• Chapter 10, 287

Index, 295