Foundations of Deterministic and Stochastic Control

by Davis, Jon H.

• ISBN13: 9780817642570
• ISBN10: 0817642579
• Format: Hardcover
• Copyright: 2002-06-01
• Publisher: Birkhauser

Summary

Control theory has applications to a number of areas in engineering and communication theory. This introductory text on the subject is fairly self-contained and aimed primarily at advanced mathematics and engineering students in various disciplines. The topics covered include realization problems, linear-quadratic optimal control, stability theory, stochastic modeling and recursive estimation algorithms in communications and control, and distributed system modeling. These topics have a wide range of applicability, and provide background for further study in the control and communications areas. In the early chapters the basics of linear control systems as well as the fundamentals of stochastic control are presented in a unique way so that the methods generalize to a useful class of distributed parameter and nonlinear system models. The control of distributed parameter systems (systems governed by PDEs) is based on the framework of linear quadratic Gaussian optimization problems. The approach here utilizes methods based on Wiener-Hopf integral equations. Additionally, the important notion of state space modeling of distributed systems is examined. Basic results due to Gohberg and Krein on convolution are given and many results are illustrated with some examples that carry throughout the text. The standard linear regulator problem is studied in both the continuous and discrete time cases, followed by a discussion of the (dual) filtering problems. Later chapters treat the stationary regulator and filtering problems with a Wiener-Hopf approach. This leads to spectral factorization problems and useful iterative algorithms that follow naturally from the methods employed. The interplay between time and frequency domain approaches is emphasized.

Table of Contents

Preface	vii
State Space Realizations	1	(70)
Linear Models	1	(4)
Continuous Time	2	(2)
Discrete Time	4	(1)
Realizations	5	(26)
Controllability	6	(3)
Controllability and Reachability	9	(2)
Observability	11	(3)
Minimal Realizations	14	(8)
Time Invariant Problems	22	(9)
Constructing Time Invariant Realizations	31	(14)
Jordan Form Realizations	32	(4)
Minimality	36	(1)
Companion Realizations	37	(1)
Standard Controllable Realizations	37	(4)
Standard Observable Realizations	41	(4)
An Active Suspension Model	45	(4)
A Model Identification Problem	49	(8)
Measurements and Filtering	49	(2)
Recursive Least Squares	51	(3)
Identifying a Coupled Equation Model	54	(1)
Coupled Problem Formulation	55	(2)
Simulating Recursive Identification	57	(1)
Discrete Time Models	58	(13)
Discrete Realizations	60	(3)
Discrete Time Identification	63	(1)
Problems	64	(7)
Least Squares Control	71	(38)
Minimum Energy Transfers	72	(2)
The Output Regulator	74	(20)
Form of the Optimal Cost	74	(2)
Completing the Square	76	(3)
Riccati Equation Existence	79	(2)
Algebraic Riccati Equation Properties	81	(3)
Riccati Solutions and Canonical Equations	84	(3)
Discrete Time Output Regulator	87	(3)
Stationary Riccati Equations	90	(4)
Linear Regulator Tracking Problems	94	(8)
Continuous Problem Models	95	(1)
The Continuous Optimal Tracking Control	95	(4)
Discrete Time Tracking Model	99	(1)
Discrete Time Optimal Tracking Control	99	(3)
Dynamic Programming	102	(7)
Problems	105	(4)
Stability Theory	109	(42)
Introduction	109	(1)
Introduction to Lyapunov Theory	109	(2)
Definitions	111	(5)
Lyapunov Functions	111	(2)
Notions of Stability	113	(3)
Classical Lyapunov Theorems	116	(7)
Time Invariant Models	119	(2)
Perturbation Results	121	(2)
The Invariance Approach	123	(9)
Discrete Time Systems	124	(6)
Continuous Time Models	130	(2)
Input-Output Stability	132	(19)
Frequency Domain Criteria	132	(3)
Vector Spaces, Norms, and Input-Output Models	135	(3)
Convolution Systems: Nyquist Criterion	138	(4)
The Circle Theorem	142	(5)
Problems	147	(4)
Random Variables and Processes	151	(34)
Introduction	151	(1)
Random Variables	151	(2)
Sample Spaces and Probabilities	153	(1)
Densities	154	(1)
Expectations, Inner Products and Variances	154	(3)
No Densities?	156	(1)
Linear Minimum Variance Estimates	157	(2)
Gramians and Covariance Matrices	159	(1)
Random Processes	160	(14)
Means and Correlations	162	(1)
Vector Processes	163	(2)
Stationarity	165	(2)
Linear Systems	167	(3)
Stationary Processes and LTI Filters	170	(4)
Gaussian Variables	174	(11)
Gaussian Vectors	174	(3)
Conditional Probability	177	(1)
Conditional Expectations	177	(2)
Gaussian Conditional Expectations	179	(2)
Problems	181	(4)
Kalman-Bucy Filters	185	(16)
The Model	185	(1)
Estimation Criterion	186	(1)
Easy Non-Answer	186	(1)
The One Step Predictor	187	(14)
Derivation of the Optimal Filter	188	(5)
Boundedness of the Optimal Filter	193	(2)
Stability of the Optimal Filter	195	(3)
Problems	198	(3)
Continuous Time Models	201	(28)
Introduction	201	(1)
Stochastic Integrals	202	(3)
Stochastic Differential Equations	205	(2)
Linear Models	207	(2)
Second Order Results	209	(3)
Continuous White Noise	212	(2)
Continuous Time Kalman-Bucy Filters	214	(15)
An Intuitive Derivation	214	(3)
Verifying Optimality	217	(6)
Examples	223	(3)
Problems	226	(3)
The Separation Theorem	229	(16)
Stochastic Dynamic Programming	229	(2)
Dynamic Programming Algorithm	231	(1)
Discrete Time Stochastic Regulator	232	(7)
Full State Feedback	232	(2)
Partial State Observations	234	(3)
Separation Theorem	237	(2)
Continuous Time	239	(1)
The Time Invariant Case	240	(1)
Active Suspension	240	(5)
Problems	243	(2)
Luenberger Observers	245	(10)
Full State Observers	247	(2)
Reduced Order Observers	249	(6)
Problems	253	(2)
Nonlinear and Finite State Problems	255	(16)
Introduction	255	(1)
Finite State Machines	256	(4)
Graphical Representations	257	(1)
Transition Graphs	257	(1)
Block Diagrams	258	(1)
Trellis Diagrams	258	(2)
Finite Markov Processes	260	(1)
Hidden Markov Models	261	(10)
BCJR Algorithm	262	(4)
BCJR and Kalman Filters	266	(1)
Hidden Markov Model Smoothing	266	(1)
Computational Requirements	267	(1)
Problems	268	(3)
Wiener-Hopf Methods	271	(26)
Wiener Filters	271	(2)
Spectral Factorization	273	(3)
The Scalar Case - Spectral Factorization	276	(5)
Discrete Time Factorization	281	(2)
Factorization in The Vector Case	283	(3)
Continuous Time Case	283	(1)
Discrete Time Version	284	(1)
Variant Details	285	(1)
Finite Dimensional Symmetric Problems	286	(2)
Spectral Factors and Optimal Gains	288	(3)
Linear Regulators and The Projection Theorem	291	(6)
Problems	295	(2)
Distributed System Regulators	297	(24)
Open Loop Unstable Distributed Regulators	297	(5)
The ``Wiener-Hopf'' Condition	302	(5)
Optimal Feedback Gains	307	(7)
Matched Filter Evasion	314	(7)
Problems	318	(3)
Filters Without Riccati Equations	321	(12)
Introduction	321	(1)
Basic Problem Formulation	322	(2)
Spectral Factors	324	(3)
Closed Loop Stability	327	(3)
Realizing The Optimal Filter	330	(3)
Problems	331	(2)
Newton's Method for Riccati Equations	333	(10)
Newton's Method	333	(1)
Continuous Time Riccati Equations	334	(1)
Discrete Time Riccati Equations	335	(2)
Convergence of Newton's Method	337	(6)
Continuous Time Convergence	337	(3)
Discrete Time Convergence	340	(3)
Numerical Spectral Factorization	343	(14)
Introduction	343	(1)
An Intuitive Algorithm Derivation	344	(2)
A Convergence Proof for the Continuous Time Algorithm	346	(2)
Implementation	348	(4)
The Discrete Case	352	(2)
Numerical Comments	354	(3)
A Hilbert and Banach Spaces and Operators	357	(42)
Banach and Hilbert Spaces	357	(1)
Quotient Spaces	358	(1)
Dual Spaces	359	(2)
Bounded Linear Operators	361	(1)
Induced Norms	362	(2)
The Banach Space L(X, Y)	364	(1)
Adjoint Mappings	364	(1)
Orthogonal Complements	365	(1)
Projection Theorem	366	(3)
Abstract Linear Equations	369	(2)
Linear Equations and Adjoints	371	(2)
Minimum Miss Distance Problems	373	(1)
Minimum Norm Problems	373	(2)
Fredholm Operators	375	(4)
Banach Algebras	379	(20)
Inverses and Spectra	382	(3)
Ideals, Transforms, and Spectra	385	(10)
Functional Calculus	395	(4)
B Measure Theoretic Probability	399	(12)
Measure Theory	399	(2)
Random variables	401	(1)
Integrals and Expectation	402	(1)
Derivatives and Densities	403	(2)
Conditional Probabilities and Expectations	405	(6)
Conditional Probability	405	(1)
Conditional Expectations	406	(5)
References	411	(6)
Index	417