The Essentials of Linear State-Space Systems

by Aplevich, J. Dwight

• ISBN13: 9780471241331
• ISBN10: 0471241334
• Edition: 1st
• Format: Paperback
• Copyright: 1999-07-23
• Publisher: Wiley

Summary

Provides a thorough introduction to the properties of linear, time-invariant models of dynamical systems, as required for further work in feedback control system design, power system design and analysis, communications, signal processing, robotics, and simulation. The state-space model is used throughout, since it is a fundamental conceptual tool, although the background analysis applies to other models. Modelling and stability of general nonlinear systems is introduced, with the detailed analysis concentrating on LTI systems.

Author Biography

J. Dwight Aplevich is the author of The Essentials of Linear State-Space Systems, published by Wiley.

Table of Contents

Preface	vii
Introduction	1	(26)
The structure of state-space models	1	(8)
The concept of state	7	(2)
Linear models	9	(3)
Time-invariant models	12	(1)
Linear, time-invariant (LTI) models	13	(1)
System properties and model properties	14	(2)
Linearized small-signal models	16	(6)
Further study	22	(1)
Problems	23	(4)
Solution of state-space equations	27	(32)
Solution of discrete-time equations	28	(10)
LTI equations	29	(2)
Free response	31	(1)
Forced response	32	(1)
Weighting sequence	33	(2)
Impulse response	35	(2)
Convolution	37	(1)
Solution of continuous-time equations	38	(12)
Existence and uniqueness	39	(1)
LTI continuous-time equations	40	(1)
Free response of continuous-time LTI systems	40	(4)
Complete response of continuous-time LTI systems	44	(2)
Forced response	46	(1)
Continuous-time impulse response	47	(2)
Continuous-time convolution	49	(1)
Discretization	50	(4)
Further study	54	(1)
Problems	54	(5)
Transform methods	59	(14)
Continuous-time models	61	(5)
Free response	62	(1)
Forced response and transfer matrix	63	(1)
Properties of the transfer matrix	64	(2)
Discrete-time models	66	(3)
Free response	68	(1)
Forced response	69	(1)
Further study	69	(1)
Problems	69	(4)
Writing state-space equations	73	(30)
Graph-based methods: Electric circuits	74	(3)
Energy-based methods: Euler-Lagrange equations	77	(6)
Quadratic forms	79	(1)
Standard matrix form	79	(4)
Aggregation	83	(1)
Operational diagrams: Digital filters	84	(2)
Computer circuits	85	(1)
Continuous-time operational diagrams	86	(1)
High-order equations	87	(2)
Direct realization of high-order linear equations	87	(2)
Controllable and observable realizations	89	(4)
Factored realizations	93	(2)
Multi-input, multi-output (MIMO) transfer functions	95	(1)
Further study	96	(1)
Problems	97	(6)
Matrices over a field	103	(52)
Basic definitions	103	(6)
Field axioms	103	(2)
Matrix definitions and operations	105	(4)
Determinants	109	(4)
Properties of determinants	112	(1)
Rank, elementary transformations, and equivalence	113	(15)
Elementary transformations	113	(2)
Elementary matrices	115	(2)
Echelon forms	117	(2)
Properties of echelon forms	119	(2)
The normal form	121	(2)
The Singular-Value Decomposition (SVD)	123	(5)
Matrix inverses	128	(2)
Left inverse	128	(1)
Right inverse	128	(1)
Inverse	129	(1)
The characteristic equation	130	(3)
The Cayley-Hamilton theorem	131	(2)
The Ho algorithm	133	(11)
The context	133	(1)
Constructive solution	134	(1)
Development of the algorithm	135	(9)
Solution of linear equations	144	(6)
General method	144	(2)
Abbreviated method	146	(2)
Uniqueness and generality of solutions	148	(1)
Special cases	149	(1)
Further study	150	(1)
Problems	150	(5)
Vector spaces	155	(16)
Vector-space axioms	155	(1)
Subspaces	156	(1)
Linear dependence of vectors	157	(1)
Range, basis, dimension, and null space	157	(6)
Bases for the range and null space	160	(2)
Orthogonal bases	162	(1)
Change of basis	163	(4)
Further study	167	(1)
Problems	167	(4)
Similarity transformations	171	(22)
Invariance of the external behavior	172	(1)
Eigenvalues, eigenvectors, and diagonalization	173	(9)
Near-diagonalization: the Jordan canonical form	182	(1)
Functions of square matrices via the Jordan form	183	(3)
General functions of square matrices	186	(3)
Further study	189	(1)
Problems	190	(3)
Stability	193	(18)
Basic definitions	193	(2)
LTI systems	195	(2)
LTI Continuous-time systems	196	(1)
LTI Discrete-time systems	196	(1)
Energy functions and Lyapunov stability	197	(11)
Energy functions for LTI systems	201	(1)
Lyapunov equations for LTI continuous-time systems	201	(2)
Solving continuous Lyapunov equations	203	(3)
Discrete-time Lyapunov equations	206	(2)
Further study	208	(1)
Problems	208	(3)
Minimality via similarity transformations	211	(20)
Step 1: Controllability	213	(5)
Construction of the controllability transformation	214	(4)
Step 2: Observability	218	(2)
Direct transformation	218	(1)
Observability by constructing the dual system	219	(1)
Minimality	220	(8)
The Kalman canonical decomposition	228	(1)
Further study	229	(1)
Problems	229	(2)
Poles and Zeros	231	(14)
The Smith-McMillan form	233	(6)
Construction of the Smith form	237	(2)
Computation of poles and zeros	239	(3)
Further study	242	(1)
Problems	242	(3)
References	245	(4)
Appendix Solutions	249	(46)
Chapter 1	249	(2)
Chapter 2	251	(5)
Chapter 3	256	(5)
Chapter 4	261	(5)
Chapter 5	266	(7)
Chapter 6	273	(2)
Chapter 7	275	(4)
Chapter 8	279	(4)
Chapter 9	283	(4)
Chapter 10	287	(8)
Index	295