Positive linear systems : theory and applications

✍ Scribed by Lorenzo Farina; S Rinaldi

• Publisher: Wiley
• Year: 2000
• Tongue: English
• Leaves: 315
• Series: Pure and applied mathematics (John Wiley & Sons : Unnumbered)
• Category: Library

✦ Synopsis

"This reference is divided into three main parts: The first part contains the definitions and basic properties of positive linear systems. The second part, following the theoretical exposition, reports the main conceptual results, considering applicable examples taken from a number of widely used models. The third part is devoted to the study of some classes of positive linear systems of particular relevance in applications (such as the Leontief model, the Leslie model, the Markov chains, the compartmental systems, and the queueing systems). Readers familiar with linear algebra and linear systems theory will appreciate the way arguments are treated and presented."--BOOK JACKET. Read more... 2 Definitions and Conditions of Positivity 7 -- 3 Influence Graphs 17 -- 4 Irreducibility, Excitability, and Transparency 23 -- Part II Properties 33 -- 5 Stability 35 -- 6 Spectral Characterization of Irreducible Systems 49 -- 7 Positivity of Equilibria 57 -- 8 Reachability and Observability 65 -- 9 Realization 81 -- 10 Minimum Phase 91 -- 11 Interconnected Systems 101 -- Part III Applications 107 -- 12 Input-Output Analysis 109 -- 13 Age-Structured Population Models 117 -- 14 Markov Chains 131 -- 15 Compartmental Systems 145 -- 16 Queueing Systems 155 -- Appendix A Elements of Linear Algebra and Matrix Theory 187 -- A.1 Real Vectors and Matrices 187 -- A.2 Vector Spaces 189 -- A.3 Dimension of a Vector Space 193 -- A.4 Change of Basis 195 -- A.5 Linear Transformations and Matrices 196 -- A.6 Image and Null Space 198 -- A.7 Invariant Subspaces, Eigenvectors, and Eigenvalues 201 -- A.8 Jordan Canonical Form 207 -- A.9 Annihilating Polynomial and Minimal Polynomial 210 -- A.10 Normed Spaces 212 -- A.11 Scalar Product and Orthogonality 216 -- A.12 Adjoint Transformations 221 -- Appendix B Elements of Linear Systems Theory 225 -- B.1 Definition of Linear Systems 225 -- B.2 ARMA Model and Transfer Function 228 -- B.3 Computation of Transfer Functions and Realization 231 -- B.4 Interconnected Subsystems and Mason's Formula 234 -- B.5 Change of Coordinates and Equivalent Systems 237 -- B.6 Motion, Trajectory, and Equilibrium 238 -- B.7 Lagrange's Formula and Transition Matrix 241 -- B.8 Reversibility 244 -- B.9 Sampled-Data Systems 244 -- B.10 Internal Stability: Definitions 248 -- B.11 Eigenvalues and Stability 248 -- B.12 Tests of Asymptotic Stability 251 -- B.13 Energy and Stability 256 -- B.14 Dominant Eigenvalue and Eigenvector 259 -- B.15 Reachability and Control Law 260 -- B.16 Observability and State Reconstruction 264 -- B.17 Decomposition Theorem 268 -- B.18 Determination of the ARMA Models 272 -- B.19 Poles and Zeros of the Transfer Function 279 -- B.20 Poles and Zeros of Interconnected Systems 282 -- B.21 Impulse Response 286 -- B.22 Frequency Response 288 -- B.23 Fourier Transform 293 -- B.24 Laplace Transform 296 -- B.25 Z-Transform 298 -- B.26 Laplace and Z-Transforms and Transfer Functions 300

✦ Table of Contents

Title......Page 3
ISBN......Page 4
Contents......Page 5
Preface......Page 8
PART I DEFINITIONS......Page 9
1 Introduction......Page 11
2 Definitions and Conditions of Positivity......Page 15
3 Influence Graphs......Page 25
4 Irreducibility, Excitability, and Transparency......Page 31
PART II PROPERTIES......Page 42
5 Stability......Page 43
6 Spectral Characterization of Irreducible Systems......Page 57
7 Positivity of Equilibria......Page 65
8 Reachability and Observability......Page 73
9 Realization......Page 89
10 Minimum Phase......Page 99
11 Interconnected Systems......Page 109
PART III APPLICATIONS......Page 116
12 Input-Output Analysis......Page 117
13 Age-Structured Population Models......Page 125
14 Markov Chains......Page 139
15 Compartmental Systems......Page 153
16 Queueing Systems......Page 163
Conclusions......Page 175
Annotated Bibliography......Page 177
Bibliography......Page 185
A.l Real Vectors and Matrices......Page 195
A.2 Vector Spaces......Page 197
A.3 Dimension of a Vector Space......Page 201
A.4 Change of Basis......Page 203
A.5 Linear Transformations and Matrices......Page 204
A.6 Image and Null Space......Page 206
A.7 Invariant Subspaces, Eigenvectors, and Eigenvalues......Page 209
A.8 Jordan Canonical Form......Page 215
A.9 Annihilating Polynomial and Minimal Polynomial......Page 218
A.10 Normed Spaces......Page 220
A.11 Scalar Product and Orthogonality......Page 224
A.12 Adjoint Transformations......Page 229
B.l Definition of Linear Systems......Page 233
B.2 ARMA Model and Transfer Function......Page 236
B.3 Computation of Transfer Functions and Realization......Page 239
B.4 Interconnected Subsystems and Mason's Formula......Page 242
B.5 Change of Coordinates and Equivalent Systems......Page 245
B.6 Motion, Trajectory, and Equilibrium......Page 246
B.7 Lagrange's Formula and Transition Matrix......Page 249
B.9 Sampled-Data Systems......Page 252
B.ll Eigenvalues and Stability......Page 256
B.12 Tests of Asymptotic Stability......Page 259
B.13 Energy and Stability......Page 264
B.14 Dominant Eigenvalue and Eigenvector......Page 267
B.15 Reachability and Control Law......Page 268
B.16 Observability and State Reconstruction......Page 272
B.17 Decomposition Theorem......Page 276
B.18 Determination of the ARMA Models......Page 280
B.19 Poles and Zeros of the Transfer Function......Page 287
B.20 Poles and Zeros of Interconnected Systems......Page 290
B.21 Impulse Response......Page 294
B.22 Frequency Response......Page 296
B.23 Fourier Transform......Page 301
B.24 Laplace Transform......Page 304
B.25 Z-Transform......Page 306
B.26 Laplace and Z-Transforms and Transfer Functions......Page 308
Index......Page 311