Foundations of Deterministic and Stochastic Control

"This volume is a textbook on linear control systems with an emphasis on stochastic 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 stochastic 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 stochastic 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

• 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 Stochastic Integrals.- 6.3 Stochastic 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 Stochastic Dynamic Programming.- 7.2 Dynamic Programming Algorithm.- 7.3 Discrete Time Stochastic 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.