Positive Linear Systems Theory and Applications

by Farina, Lorenzo; Rinaldi, Sergio

• ISBN13: 9780471384564
• ISBN10: 0471384569
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-07-03
• Publisher: Wiley-Interscience

Summary

A complete study on an important class of linear dynamical systems-positive linear systems One of the most often-encountered systems in nearly all areas of science and technology, positive linear systems is a specific but remarkable and fascinating class. Renowned scientists Lorenzo Farina and Sergio Rinaldi introduce readers to the world of positive linear systems in their rigorous but highly accessible book, rich in applications, examples, and figures. This professional 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. Extraordinarily comprehensive, Positive Linear Systems features: * Applications from a variety of backgrounds including modeling, control engineering, computer science, demography, economics, bioengineering, chemistry, and ecology * References and annotated bibliographies throughout the book * Two appendices concerning linear algebra and linear systems theory for readers unfamiliar with the mathematics used Farina and Rinaldi make no effort to hide their enthusiasm for the topics presented, making Positive Linear Systems: Theory and Applications an indispensable resource for researchers and professionals in a broad range of fields.

Author Biography

LORENZO FARINA, PhD, is Associate Professor of Modeling and Simulation, Dipartimento di Informatica e Sistemistica, Università di Roma "La Sapienza," Italy. SERGIO RINALDI, PhD, is Full Professor of Systems Theory, Dipartimento di Elettronica e Informazione, Politecnico di Milano, Italy.

Table of Contents

Preface	ix
PART I DEFINITIONS	1	(32)
Introduction	3	(4)
Definitions and Conditions of Positivity	7	(10)
Influence Graphs	17	(6)
Irreducibility, Excitability, and Transparency	23	(10)
PART II PROPERTIES	33	(74)
Stability	35	(14)
Spectral Characterization of Irreducible Systems	49	(8)
Positivity of Equilibria	57	(8)
Reachability and Observability	65	(16)
Realization	81	(10)
Minimum Phase	91	(10)
Interconnected Systems	101	(6)
PART III APPLICATIONS	107	(80)
Input--Output Analysis	109	(8)
Age-Structured Population Models	117	(14)
Markov Chains	131	(14)
Compartmental Systems	145	(10)
Queueing Systems	155	(32)
Conclusions	167	(2)
Annotated Bibliography	169	(8)
Bibliography	177	(10)
Appendix A: Elements of Linear Algebra and Matrix Theory	187	(38)
A.1 Real Vectors and Matrices	187	(2)
A.2 Vector Spaces	189	(4)
A.3 Dimension of a Vector Space	193	(2)
A.4 Change of Basis	195	(1)
A.5 Linear Transformations and Matrices	196	(2)
A.6 Image and Null Space	198	(3)
A.7 Invariant Subspaces, Eigenvectors, and Eigenvalues	201	(6)
A.8 Jordan Canonical Form	207	(3)
A.9 Annihilating Polynomial and Minimal Polynomial	210	(2)
A.10 Normed Spaces	212	(4)
A.11 Scalar Product and Orthogonality	216	(5)
A.12 Adjoint Transformations	221	(4)
Appendix B: Elements of Linear Systems Theory	225	(78)
B.1 Definition of Linear Systems	225	(3)
B.2 ARMA Model and Transfer Function	228	(3)
B.3 Computation of Transfer Functions and Realization	231	(3)
B.4 Interconnected Subsystems and Mason's Formula	234	(3)
B.5 Change of Coordinates and Equivalent Systems	237	(1)
B.6 Motion, Trajectory, and Equilibrium	238	(3)
B.7 Lagrange's Formula and Transition Matrix	241	(3)
B.8 Reversibility	244	(1)
B.9 Sampled-Data Systems	244	(4)
B.10 Internal Stability: Definitions	248	(1)
B.11 Eigenvalues and Stability	248	(3)
B.12 Tests of Asymptotic Stability	251	(5)
B.13 Energy and Stability	256	(3)
B.14 Dominant Eigenvalue and Eigenvector	259	(1)
B.15 Reachability and Control Law	260	(4)
B.16 Observability and State Reconstruction	264	(4)
B.17 Decomposition Theorem	268	(4)
B.18 Determination of the ARMA Models	272	(7)
B.19 Poles and Zeros of the Transfer Function	279	(3)
B.20 Poles and Zeros of Interconnected Systems	282	(4)
B.21 Impulse Response	286	(2)
B.22 Frequency Response	288	(5)
B.23 Fourier Transform	293	(3)
B.24 Laplace Transform	296	(2)
B.25 Z--Transform	298	(2)
B.26 Laplace and Z--Transforms and Transfer Functions	300	(3)
Index	303