Discrete-Time Stochastic Systems: Estimation and Control

ISBN-13: 9781852336493 / Angielski / Miękka / 2002 / 376 str.

Torsten Soderstrom; T. Soderstrom; Torsten Svderstrvm

Opis

Discrete-time Stochastic Systems gives a comprehensive introduction to the estimation and control of dynamic stochastic systems and provides complete derivations of key results such as the basic relations for Wiener filtering. The book covers both state-space methods and those based on the polynomial approach. Similarities and differences between these approaches are highlighted. Some non-linear aspects of stochastic systems (such as the bispectrum and extended Kalman filter) are also introduced and analysed.

Spis treści

1. Introduction.- 1.1 What is a Stochastic 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 Stochastic 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 Stochastic 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 Stochastic 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 Stochastic Systems.- 10.5 Some Stochastic 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.

Blurb

Discrete-time Stochastic Systems gives a comprehensive introduction to the estimation and control of dynamic stochastic systems and provides complete derivations of key results such as the basic relations for Wiener filtering. The book covers both state-space methods and those based on the polynomial approach. Similarities and differences between these approaches are highlighted. Some non-linear aspects of stochastic systems (such as the bispectrum and extended Kalman filter) are also introduced and analysed. The books chief features are as follows:

• inclusion of the polynomial approach provides alternative and simpler computational methods than simple reliance on state-space methods;

• algorithms for analysis and design of stochastic systems allow for ease of implementation and experimentation by the reader;

• the highlighting of spectral factorization gives appropriate emphasis to this key concept often overlooked in the literature;

• explicit solutions of Wiener problems are handy schemes, well suited for computations compared with more commonly available but abstract formulations;

• complex-valued models that are directly applicable to many problems in signal processing and communications.

Changes in the second edition include:

• additional information covering spectral factorisation and the innovations form;

• the chapter on optimal estimation being completely rewritten to focus on a posteriori estimates rather than maximum likelihood;

• new material on fixed lag smoothing and algorithms for solving Riccati equations are improved and more up to date;

• new presentation of polynomial control and new derivation of linear-quadratic-Gaussian control.

Discrete-time Stochastic Systems is primarily of benefit to students taking M.Sc. courses in stochastic estimation and control, electronic engineering and signal processing but may also be of assistance for self study and as a reference.