Foundations of Deterministic and Stochastic Control

by Jon H. Davis

Overview

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 consists of a wide range of topics that include realization problems, linear-quadratic optimal control, stability theory, shastic modeling and recursive estimation algorithms in communications and control, and distributed system modeling. In the early chapters methods based on Wiener—Hopf integral equations are utilized. The fundamentals of both linear control systems as well as shastic 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. 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 the continuous and discrete time cases, followed by a discussion of (dual) filtering problems. Later chapters treat the stationary regulator and filtering problems using 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. "Foundations of Deterministic and Shastic Control" is geared primarily towards advanced mathematics and engineering students in various disciplines.

Product Details

ISBN-13:	9780817642570
Publisher:	Birkhäuser Boston
Publication date:	04/19/2002
Series:	Systems & Control: Foundations & Applications
Edition description:	2002
Pages:	426
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

1 State Space Realizations.- 1.1 Linear Models.- 1.2 Realizations.- 1.3 Constructing Time Invariant Realizations.- 1.4 An Active Suspension Model.- 1.5 A Model Identification Problem.- 1.6 Simulating Recursive Identification.- 1.7 Discrete Time Models.- Problems.- 2 Least Squares Control.- 2.1 Minimum Energy Transfers.- 2.2 The Output Regulator.- 2.3 Linear Regulator Tracking Problems.- 2.4 Dynamic Programming.- Problems.- 3 Stability Theory.- 3.1 Introduction.- 3.2 Introduction to Lyapunov Theory.- 3.3 Definitions.- 3.4 Classical Lyapunov Theorems.- 3.5 The Invariance Approach.- 3.6 Input-Output Stability.- Problems.- 4 Random Variables and Processes.- 4.1 Introduction.- 4.2 Random Variables.- 4.3 Sample Spaces and Probabilities.- 4.4 Densities.- 4.5 Expectations, Inner Products and Variances.- 4.6 Linear Minimum Variance Estimates.- 4.7 Gramians and Covariance Matrices.- 4.8 Random Processes.- 4.9 Gaussian Variables.- Problems.- 5 Kalman-Bucy Filters.- 5.1 The Model.- 5.2 Estimation Criterion.- 5.3 The One Step Predictor.- Problems.- 6 Continuous Time Models.- 6.1 Introduction.- 6.2 Shastic Integrals.- 6.3 Shastic Differential Equations.- 6.4 Linear Models.- 6.5 Second Order Results.- 6.6 Continuous White Noise.- 6.7 Continuous Time Kalman-Bucy Filters.- Problems.- 7 The Separation Theorem.- 7.1 Shastic Dynamic Programming.- 7.2 Dynamic Programming Algorithm.- 7.3 Discrete Time Shastic Regulator.- 7.4 Continuous Time.- 7.5 The Time Invariant Case.- 7.6 Active Suspension.- Problems.- 8 Luenberger Observers.- 8.1 Full State Observers.- 8.2 Reduced Order Observers.- Problems.- 9 Nonlinear and Finite State Problems.- 9.1 Introduction.- 9.2 Finite State Machines.- 9.3 Finite Markov Processes.- 9.4 Hidden Markov Models.- Problems.- 10 Wiener-Hopf Methods.- 10.1Wiener Filters.- 10.2 Spectral Factorization.- 10.3 The Scalar Case - Spectral Factorization.- 10.4 Discrete Time Factorization.- 10.5 Factorization in The Vector Case.- 10.6 Finite Dimensional Symmetric Problems.- 10.7 Spectral Factors and Optimal Gains.- 10.8 Linear Regulators and The Projection Theorem.- Problems.- 11 Distributed System Regulators.- 11.1 Open Loop Unstable Distributed Regulators.- 11.2 The “Wiener-Hopf” Condition.- 11.3 Optimal Feedback Gains.- 11.4 Matched Filter Evasion.- Problems.- 12 Filters Without Riccati Equations.- 12.1 Introduction.- 12.2 Basic Problem Formulation.- 12.3 Spectral Factors.- 12.4 Closed Loop Stability.- 12.5 Realizing The Optimal Filter.- Problems.- 13 Newton’s Method for Riccati Equations.- 13.1 Newton’s Method.- 13.2 Continuous Time Riccati Equations.- 13.3 Discrete Time Riccati Equations.- 13.4 Convergence of Newton’s Method.- 14 Numerical Spectral Factorization.- 14.1 Introduction.- 14.2 An Intuitive Algorithm Derivation.- 14.3 A Convergence Proof for the Continuous Time Algorithm.- 14.4 Implementation.- 14.5 The Discrete Case.- 14.6 Numerical Comments.- A Hilbert and Banach Spaces and Operators.- A.1 Banach and Hilbert Spaces.- A.2 Quotient Spaces.- A.3 Dual Spaces.- A.4 Bounded Linear Operators.- A.5 Induced Norms.- A.6 The Banach Space G(X, Y).- A.7 Adjoint Mappings.- A.8 Orthogonal Complements.- A.9 Projection Theorem.- A.10 Abstract Linear Equations.- A.11 Linear Equations and Adjoints.- A.12 Minimum Miss Distance Problems.- A.13 Minimum Norm Problems.- A.14 Fredholm Operators.- A.15 Banach Algebras.- A.15.1 Inverses and Spectra.- A.15.2 Ideals, Transforms, and Spectra.- A.15.3 Functional Calculus.- B Measure Theoretic Probability.- B.1 Measure Theory.- B.2 Random variables.- B.3 Integrals and Expectation.-B.4 Derivatives and Densities.- B.5 Conditional Probabilities and Expectations.- B.5.1 Conditional Probability.- B.5.2 Conditional Expectations.- References.

Editorial Reviews

"This volume is a textbook on linear control systems with an emphasis on shastic optimal control with solution methods using spectral factorization in line with the original approach of N. Wiener. Continuous-time and discrete-time versions are presented in parallel…. Two appendices introduce functional analytic concepts and probability theory, and there are 77 references and an index. The chapters (except for the last two) end with problems…. [T]he book presents in a clear way important concepts of control theory and can be used for teaching." —Zentralblatt Math

"This is a textbook intended for use in courses on linear control and filtering and estimation on (advanced) levels. Its major purpose is an introduction to both deterministic and shastic control and estimation. Topics are treated in both continuous time and discrete time versions…. Each chapter involves problems and exercises, and the book is supplemented by appendices, where fundamentals on Hilbert and Banach spaces, operator theory, and measure theoretic probability may be found. The book will be very useful for students, but also for a variety of specialists interested in deterministic and shastic control and filtering." —Applications of Mathematics

"The strength of the book under review lies in the choice of specialized topics it contains, which may not be found in this form elsewhere. Also, the first half would make a good standard course in linear control." —Journal of the Indian Institute of Science

From the Publisher