Discrete-time Stochastic Systems: Estimation and Control

by Torsten Söderström

Overview

This comprehensive introduction to the estimation and control of dynamic shastic systems provides complete derivations of key results. The second edition includes improved and updated material, and a new presentation of polynomial control and new derivation of linear-quadratic-Gaussian control.

Product Details

ISBN-13:	9781852336493
Publisher:	Springer London
Publication date:	07/26/2002
Series:	Advanced Textbooks in Control and Signal Processing
Edition description:	Second Edition 2002
Pages:	376
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1. Introduction.- 1.1 What is a Shastic System?.- Bibhography.- 2. Some Probability Theory.- 2.1 Introduction.- 2.2 Random Variables and Distributions.- 2.2.1 Basic Concepts.- 2.2.2 Gaussian Distributions.- 2.2.3 Correlation and Dependence.- 2.3 Conditional Distributions.- 2.4 The Conditional Mean for Gaussian Variables.- 2.5 Complex-Valued Gaussian Variables.- 2.5.1 The Scalar Case.- 2.5.2 The Multivariable Case.- 2.5.3 The Rayleigh Distribution.- Exercises.- 3. Models.- 3.1 Introduction.- 3.2 Shastic Processes.- 3.3 Markov Processes and the Concept of State.- 3.4 Covariance Function and Spectrum.- 3.5 Bispectrum.- 3.A Appendix. Linear Complex-Valued Signals and Systems.- 3.A.1 Complex-Valued Model of a Narrow-Band Signal.- 3.A.2 Linear Complex-Valued Systems.- 3.B Appendix. Markov Chains.- Exercises.- 4. Analysis.- 4.1 Introduction.- 4.2 Linear Filtering.- 4.2.1 Transfer Function Models.- 4.2.2 State Space Models.- 4.2.3 Yule-Walker Equations.- 4.3 Spectral Factorization.- 4.3.1 Transfer Function Models.- 4.3.2 State Space Models.- 4.3.3 An Example.- 4.4 Continuous-time Models.- 4.4.1 Covariance Function and Spectra.- 4.4.2 Spectral Factorization.- 4.4.3 White Noise.- 4.4.4 Wiener Processes.- 4.4.5 State Space Models.- 4.5 Sampling Shastic Models.- 4.5.1 Introduction.- 4.5.2 State Space Models.- 4.5.3 Aliasing.- 4.6 The Positive Real Part of the Spectrum.- 4.6.1 ARMA Processes.- 4.6.2 State Space Models.- 4.6.3 Continuous-time Processes.- 4.7 Effect of Linear Filtering on the Bispectrum.- 4.8 Algorithms for Covariance Calculations and Sampling.- 4.8.1 ARMA Covariance Function.- 4.8.2 ARMA Cross-Covariance Function.- 4.8.3 Continuous-Time Covariance Function.- 4.8.4 Sampling.- 4.8.5 Solving the Lyapunov Equation.- 4. A Appendix. Auxiliary Lemmas.- Exercises.- 5. Optimal Estimation.- 5.1 Introduction.- 5.2 The Conditional Mean.- 5.3 The Linear Least Mean Square Estimate.- 5.4 Propagation of the Conditional Probability Density Function.- 5.5 Relation to Maximum Likelihood Estimation.- 5.A Appendix. A Lemma for Optimality of the Conditional Mean.- Exercises.- 6. Optimal State Estimation for Linear Systems.- 6.1 Introduction.- 6.2 The Linear Least Mean Square One-Step Prediction and Filter Estimates.- 6.3 The Conditional Mean.- 6.4 Optimal Filtering and Prediction.- 6.5 Smoothing.- 6.5.1 Fixed Point Smoothing.- 6.5.2 Fixed Lag Smoothing.- 6.6 Maximum a posteriori Estimates.- 6.7 The Stationary Case.- 6.8 Algorithms for Solving the Algebraic Riccati Equation.- 6.8.1 Introduction.- 6.8.2 An Algorithm Based on the Euler Matrix.- 6.A Appendix. Proofs.- 6.A.1 The Matrix Inversion Lemma.- 6.A.2 Proof of Theorem 6.1.- 6.A.3 Two Determinant Results.- Exercises.- 7. Optimal Estimation for Linear Systems by Polynomial Methods.- 7.1 Introduction.- 7.2 Optimal Prediction.- 7.2.1 Introduction.- 7.2.2 Optimal Prediction of ARMA Processes.- 7.2.3 A General Case.- 7.2.4 Prediction of Nonstationary Processes.- 7.3 Wiener Filters.- 7.3.1 Statement of the Problem.- 7.3.2 The Unrealizable Wiener Filter.- 7.3.3 The Realizable Wiener Filter.- 7.3.4 Illustration.- 7.3.5 Algorithmic Aspects.- 7.3.6 The Causal Part of a Filter, Partial Fraction Decomposition and a Diophantine Equation.- 7.4 Minimum Variance Filters.- 7.4.1 Introduction.- 7.4.2 Solution.- 7.4.3 The Estimation Error.- 7.4.4 Extensions.- 7.4.5 Illustrations.- 7.5 Robustness Against Modelling Errors.- Exercises.- 8. Illustration of Optimal Linear Estimation.- 8.1 Introduction.- 8.2 Spectral Factorization.- 8.3 Optimal Prediction.- 8.4 Optimal Filtering.- 8.5 Optimal Smoothing.- 8.6 Estimation Error Variance.- 8.7 Weighting Pattern.- 8.8 Frequency Characteristics.- Exercises.- 9. Nonlinear Filtering.- 9.1 Introduction.- 9.2 Extended Kaiman Filters.- 9.2.1 The Basic Algorithm.- 9.2.2 An Iterated Extended Kalman Filter.- 9.2.3 A Second-order Extended Kalman Filter.- 9.2.4 An Example.- 9.3 Gaussian Sum Estimators.- 9.4 The Multiple Model Approach.- 9.4.1 Introduction.- 9.4.2 Fixed Models.- 9.4.3 Switching Models.- 9.4,4 Interacting Multiple Models Algorithm.- 9.5 Monte Carlo Methods for Propagating the Conditional Probability Density Functions.- 9.6 Quantized Measurements.- 9.7 Median Filters.- 9.7.1 Introduction.- 9.7.2 Step Response.- 9.7.3 Response to Sinusoids.- 9.7.4 Effect on Noise.- 9.A Appendix. Auxiliary results.- 9.A.1 Analysis of the Sheppard Correction.- 9.A.2 Some Probability Density Functions.- Exercises.- 10. Introduction to Optimal Shastic Control.- 10.1 Introduction.- 10.2 Some Simple Examples.- 10.2.1 Introduction.- 10.2.2 Deterministic System.- 10 2 3 Random Time Constant.- 10.2.4 Noisy Observations.- 10 2 5 Process Noise.- 10.2.6 Unknown Time Constants and Measurement Noise.- 10 2 7 Unknown Gain.- 10.3 Mathematical Preliminaries.- 10.4 Dynamic Programming.- 10.4.1 Deterministic Systems.- 10.4.2 Shastic Systems.- 10.5 Some Shastic Controllers.- 10.5.1 Dual Control.- 10.5.2 Certainty Equivalence Control.- 10.5.3 Cautious Control.- Exercises.- 11. Linear Quadratic Gaussian Control.- 11.1 Introduction.- 11.2 The Optimal Controllers.- 11.2.1 Optimal Control of Deterministic Systems.- 11.2.2 Optimal Control with Complete State Information.- 11.2.3 Optimal Control with Incomplete State Information.- 11.3 Duality Between Estimation and Control.- 11.4 Closed Loop System Properties.- 114 1 Representations of the Regulator.- 11.4.2 Representations of the Closed Loop System.- 11.4.3 The Closed Loop Poles.- 11.5 Linear Quadratic Gaussian Design by Polynomial Methods.- 11.5.1 Problem Formulation.- 11.5.2 Minimum Variance Control.- 11.5.3 The General Case.- 11.6 Controller Design by Linear Quadratic Gaussian Theory.- 11.6.1 Introduction.- 11.6.2 Choice of Observer Poles.- 11. A Appendix. Derivation of the Optimal Linear Quadratic Gaussian Feedback and the Riccati Equation from the Bellman Equation.- Exercises.- Answers to Selected Exercises.