:or-' The Essentials of Linear State-Space Systems J. Dwight Aplevich GIFT OF THE ASIA FOUNDATION NOT FOR RE-SALE John Wiley & Sons, Inc New York Chichester Weinheim OAI HOC OUOC GIA HA N^l TRUNGTAMTHANCTINTHUVIIN A - 90 /39ii Brisbane Toronto Singapore
Contents Preface, vii WT^ TnfroSiicnon' 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 Solution of state-space equations ^^JHP 27 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
xii Contents 3 Discretization, 50 4 Further study, 54 5 Problems, 54 W Chapter 9 Chapter 4 T1w?5!8fflrfiiethods 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 Discrete-time models, 66 2.1 Free response. 68 2.2 Forced response, 69 Further study, 69 Problems, 69 Writing state-space equations 3 4 5 6 7 8 9 10 11 Graph-based methods: Electric circuits, 74 Energy-based methods: Euler-Lagrange equations, 77 2.1 Quadratic forms. 79 2.2 Standard matrix form, 79 Aggregation. 83 Operational diagrams: Digital filters. 84 4.1 Computer circuits, 85 Continuous-time operational diagrams. 86 High-order equations, 87 6.1 Direct realization of high-order linear equations, 87 Controllable and observable realizations, 89 Factored realizations, 93 Multi-input, multi-output (MIMO) transfer functions. 95 Further study, 96 Problems. 97 H Chapters Matrices over a field 103 Basic definitions, 103 1.1 Field axioms. 103 1.2 f\/latrix definitions and operations, 105 Determinants, 109 2.1 Properties of determinants. /12 Rank, elementary transformations, and equivalence, /13
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 p unapter e vector SPBSSS^'^^^/I/^//I^I/I////I^^ 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 Contents xiil Chapter 7 Similarity transtormat(orfs'^''mhhi ^ 1 Invariance of the external behavior, 172 2 Eigenvalues, eigenvectors, and diagonalization, 173 3 Near-diagonalization: the Jordan canonical form, 182
xiv Contents Functions of square matrices via the Jordan form, 133 General functions of square matrices, 186 Further study. 189 Problems, 190 f Chapter 8 Stability m. 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 p Chapter 9 Minimality via similarity transformations ^1 Step1: Controllability, 2/5 1.1 Construction of the controllability transformation. 214 Step 2: Observability, 218 2.1 Direct transformation, 218 2.2 Observability by constructing the dual system, 219 Minimality, 220 The Kalman canonical decomposition. 228 Further study, 229 Problems, 229 H Chapter 10 Poles and Zeros 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 References
Contents xv Appendix Solutioi^^aii^i^Hiii^^^HM^HiHiHi^i^^MMa^d Chapter 1, 249 Chapter 2, 25) Chapters, 256 Chapter4, 261 Chapters, 266 Chapters, 273 Chapter?, 275 Chapters, 279 Chapters, 283 Chapter 10, 287 Index, 295