The Essentials of Linear State-Space Systems

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1 :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

2 Contents Preface, vii WT^ TnfroSiicnon' 1 The structure of state-space models, The concept of state, 7 2 Linear models, 9 3 Time-invariant models Linear, time-invariant (LTI) models, 13 5 System properties and model properties, 14 6 Linearized small-signal models, 16 7 Further study Problems, 23 Solution of state-space equations ^^JHP 27 1 Solution of discrete-time equations, LTI equations, Free response, Forced response, Weighting sequence, Impulse response, Convolution, 37 2 Solution of continuous-time equations, Existence and uniqueness, LTI continuous-time equations, Free response of continuous-time LTI systems, Complete response of continuous-time LTI systems, Forced response, Continuous-time impulse response Continuous-time convolution. 49

3 xii Contents 3 Discretization, 50 4 Further study, 54 5 Problems, 54 W Chapter 9 Chapter 4 T1w?5!8fflrfiiethods Continuous-time models, Free response, Forced response and transfer matrix, Properties of the transfer matrix, 64 Discrete-time models, Free response Forced response, 69 Further study, 69 Problems, 69 Writing state-space equations Graph-based methods: Electric circuits, 74 Energy-based methods: Euler-Lagrange equations, Quadratic forms Standard matrix form, 79 Aggregation. 83 Operational diagrams: Digital filters Computer circuits, 85 Continuous-time operational diagrams. 86 High-order equations, 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, Field axioms f\/latrix definitions and operations, 105 Determinants, Properties of determinants. /12 Rank, elementary transformations, and equivalence, /13

4 3.1 Elementary transformations, Elementary matrices, Echelon forms, Properties of echelon forms, The normal form, The Singular-Value Decomposition (SVD), Matrix inverses, Left inverse, Right inverse, Inverse, The characteristic equation, The Cayley-Hamilton theorem, The Ho algorithm, The context, Constructive solution, Development of the algorithm, Solution of linear equations, General method, Abbreviated method, Uniqueness and generality of solutions, Special cases, Further study, Problems. 150 p unapter e vector SPBSSS^'^^^/I/^//I^I/I////I^^ 1 Vector-space axioms, Subspaces, Linear dependence of vectors, Range, basis, dimension, and null space, Bases for the range and null space, Orthogonal bases, Change of basis, Further study, Problems, 167 Contents xiil Chapter 7 Similarity transtormat(orfs'^''mhhi ^ 1 Invariance of the external behavior, Eigenvalues, eigenvectors, and diagonalization, Near-diagonalization: the Jordan canonical form, 182

5 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 LTI systems, LTI Continuous-time systems, LTI Discrete-time systems, Energy functions and Lyapunov stability, Energy functions for LTI systems, Lyapunov equations for LTI continuous-time systems Solving continuous Lyapunov equations, Discrete-time Lyapunov equations, Further study, 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, Direct transformation, 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, Construction of the Smith form Computation of poles and zeros, Further study, Problems, 242 References

6 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

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