Enhanced descriptions from Syndetics:

This comprehensive introduction to the estimation and control of dynamic stochastic 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.

Table of contents provided by Syndetics

• 1 Introduction (p. 1)
• 1.1 What is a Stochastic System? (p. 1)
• Bibliography (p. 5)
• 2 Some Probability Theory (p. 7)
• 2.1 Introduction (p. 7)
• 2.2 Random Variables and Distributions (p. 7)
• 2.2.1 Basic Concepts (p. 7)
• 2.2.2 Gaussian Distributions (p. 11)
• 2.2.3 Correlation and Dependence (p. 12)
• 2.3 Conditional Distributions (p. 12)
• 2.4 The Conditional Mean for Gaussian Variables (p. 14)
• 2.5 Complex-Valued Gaussian Variables (p. 17)
• 2.5.1 The Scalar Case (p. 17)
• 2.5.2 The Multivariate Case (p. 19)
• 2.5.3 The Rayleigh Distribution (p. 24)
• Exercises (p. 26)
• Bibliography (p. 27)
• 3 Models (p. 29)
• 3.1 Introduction (p. 29)
• 3.2 Stochastic Processes (p. 30)
• 3.3 Markov Processes and the Concept of State (p. 33)
• 3.4 Covariance Function and Spectrum (p. 36)
• 3.5 Bispectrum (p. 46)
• 3.A Appendix. Linear Complex-Valued Signals and Systems (p. 48)
• 3.A.1 Complex-Valued Model of a Narrow-Band Signal (p. 48)
• 3.A.2 Linear Complex-Valued Systems (p. 49)
• 3.B Appendix. Markov Chains (p. 51)
• Exercises (p. 55)
• Bibliography (p. 58)
• 4 Analysis (p. 59)
• 4.1 Introduction (p. 59)
• 4.2 Linear Filtering (p. 59)
• 4.2.1 Transfer Function Models (p. 59)
• 4.2.2 State Space Models (p. 61)
• 4.2.3 Yule-Walker Equations (p. 67)
• 4.3 Spectral Factorization (p. 71)
• 4.3.1 Transfer Function Models (p. 71)
• 4.3.2 State Space Models (p. 74)
• 4.3.3 An Example (p. 78)
• 4.4 Continuous-time Models (p. 80)
• 4.4.1 Covariance Function and Spectra (p. 80)
• 4.4.2 Spectral Factorization (p. 81)
• 4.4.3 White Noise (p. 82)
• 4.4.4 Wiener Processes (p. 83)
• 4.4.5 State Space Models (p. 84)
• 4.5 Sampling Stochastic Models (p. 86)
• 4.5.1 Introduction (p. 86)
• 4.5.2 State Space Models (p. 87)
• 4.5.3 Aliasing (p. 88)
• 4.6 The Positive Real Part of the Spectrum (p. 90)
• 4.6.1 ARM A Processes (p. 90)
• 4.6.2 State Space Models (p. 95)
• 4.6.3 Continuous-time Processes (p. 98)
• 4.7 Effect of Linear Filtering on the Bispectrum (p. 99)
• 4.8 Algorithms for Covariance Calculations and Sampling (p. 103)
• 4.8.1 ARMA Covariance Function (p. 103)
• 4.8.2 ARMA Cross-Covariance Function (p. 105)
• 4.8.3 Continuous-Time Covariance Function (p. 107)
• 4.8.4 Sampling (p. 108)
• 4.8.5 Solving the Lyapunov Equation (p. 110)
• 4.A Appendix. Auxiliary Lemmas (p. 111)
• Exercises (p. 114)
• Bibliography (p. 121)
• 5 Optimal Estimation (p. 123)
• 5.1 Introduction (p. 123)
• 5.2 The Conditional Mean (p. 123)
• 5.3 The Linear Least Mean Square Estimate (p. 126)
• 5.4 Propagation of the Conditional Probability Density Function (p. 128)
• 5.5 Relation to Maximum Likelihood Estimation (p. 130)
• 5.A Appendix. A Lemma for Optimality of the Conditional Mean (p. 133)
• Exercises (p. 134)
• Bibliography (p. 135)
• 6 Optimal State Estimation for Linear Systems (p. 137)
• 6.1 Introduction (p. 137)
• 6.2 The Linear Least Mean Square One-Step Prediction and Filter Estimates (p. 138)
• 6.3 The Conditional Mean (p. 145)
• 6.4 Optimal Filtering and Prediction (p. 147)
• 6.5 Smoothing (p. 148)
• 6.5.1 Fixed Point Smoothing (p. 149)
• 6.5.2 Fixed Lag Smoothing (p. 150)
• 6.6 Maximum a posteriori Estimates (p. 153)
• 6.7 The Stationary Case (p. 155)
• 6.8 Algorithms for Solving the Algebraic Riccati Equation (p. 159)
• 6.8.1 Introduction (p. 159)
• 6.8.2 An Algorithm Based on the Euler Matrix (p. 161)
• 6.A Appendix. Proofs (p. 165)
• 6.A.1 The Matrix Inversion Lemma (p. 165)
• 6.A.2 Proof of Theorem 6.1 (p. 166)
• 6.A.3 Two Determinant Results (p. 171)
• Exercises (p. 171)
• Bibliography (p. 182)
• 7 Optimal Estimation for Linear Systems by Polynomial Methods (p. 185)
• 7.1 Introduction (p. 185)
• 7.2 Optimal Prediction (p. 185)
• 7.2.1 Introduction (p. 185)
• 7.2.2 Optimal Prediction of ARMA Processes (p. 187)
• 7.2.3 A General Case (p. 191)
• 7.2.4 Prediction of Nonstationary Processes (p. 193)
• 7.3 Wiener Filters (p. 194)
• 7.3.1 Statement of the Problem (p. 194)
• 7.3.2 The Unrealizable Wiener Filter (p. 196)
• 7.3.3 The Realizable Wiener Filter (p. 197)
• 7.3.4 Illustration (p. 199)
• 7.3.5 Algorithmic Aspects (p. 200)
• 7.3.6 The Causal Part of a Filter, Partial Fraction Decomposition and a Diophantine Equation (p. 202)
• 7.4 Minimum Variance Filters (p. 205)
• 7.4.1 Introduction (p. 205)
• 7.4.2 Solution (p. 206)
• 7.4.3 The Estimation Error (p. 207)
• 7.4.4 Extensions (p. 209)
• 7.4.5 Illustrations (p. 211)
• 7.5 Robustness Against Modelling Errors (p. 215)
• Exercises (p. 218)
• Bibliography (p. 220)
• 8 Illustration of Optimal Linear Estimation (p. 223)
• 8.1 Introduction (p. 223)
• 8.2 Spectral Factorization (p. 223)
• 8.3 Optimal Prediction (p. 225)
• 8.4 Optimal Filtering (p. 227)
• 8.5 Optimal Smoothing (p. 228)
• 8.6 Estimation Error Variance (p. 232)
• 8.7 Weighting Pattern (p. 234)
• 8.8 Frequency Characteristics (p. 235)
• Exercises (p. 242)
• 9 Nonlinear Filtering (p. 245)
• 9.1 Introduction (p. 245)
• 9.2 Extended Kaiman Filters (p. 245)
• 9.2.1 The Basic Algorithm (p. 245)
• 9.2.2 An Iterated Extended Kaiman Filter (p. 247)
• 9.2.3 A Second-order Extended Kaiman Filter (p. 248)
• 9.2.4 An Example (p. 250)
• 9.3 Gaussian Sum Estimators (p. 254)
• 9.4 The Multiple Model Approach (p. 257)
• 9.4.1 Introduction (p. 257)
• 9.4.2 Fixed Models (p. 257)
• 9.4.3 Switching Models (p. 259)
• 9.4.4 Interacting Multiple Models Algorithm (p. 260)
• 9.5 Monte Carlo Methods for Propagating the Conditional Probability Density Functions (p. 265)
• 9.6 Quantized Measurements (p. 269)
• 9.7 Median Filters (p. 270)
• 9.7.1 Introduction (p. 270)
• 9.7.2 Step Response (p. 271)
• 9.7.3 Response to Sinusoids (p. 272)
• 9.7.4 Effect on Noise (p. 273)
• 9.A Appendix. Auxiliary results (p. 277)
• 9.A.1 Analysis of the Sheppard Correction (p. 277)
• 9.A.2 Some Probability Density Functions (p. 280)
• Exercises (p. 281)
• Bibliography (p. 294)
• 10 Introduction to Optimal Stochastic Control (p. 297)
• 10.1 Introduction (p. 297)
• 10.2 Some Simple Examples (p. 297)
• 10.2.1 Introduction (p. 297)
• 10.2.2 Deterministic System (p. 297)
• 10.2.3 Random Time Constant (p. 298)
• 10.2.4 Noisy Observations (p. 299)
• 10.2.5 Process Noise (p. 300)
• 10.2.6 Unknown Time Constants and Measurement Noise (p. 300)
• 10.2.7 Unknown Gain (p. 301)
• 10.3 Mathematical Preliminaries (p. 303)
• 10.4 Dynamic Programming (p. 304)
• 10.4.1 Deterministic Systems (p. 305)
• 10.4.2 Stochastic Systems (p. 306)
• 10.5 Some Stochastic Controllers (p. 311)
• 10.5.1 Dual Control (p. 313)
• 10.5.2 Certainty Equivalence Control (p. 313)
• 10.5.3 Cautious Control (p. 314)
• Exercises (p. 316)
• Bibliography (p. 317)
• 11 Linear Quadratic Gaussian Control (p. 319)
• 11.1 Introduction (p. 319)
• 11.2 The Optimal Controllers (p. 320)
• 11.2.1 Optimal Control of Deterministic Systems (p. 320)
• 11.2.2 Optimal Control with Complete State Information (p. 323)
• 11.2.3 Optimal Control with Incomplete State Information (p. 324)
• 11.3 Duality Between Estimation and Control (p. 326)
• 11.4 Closed Loop System Properties (p. 328)
• 11.4.1 Representations of the Regulator (p. 328)
• 11.4.2 Representations of the Closed Loop System (p. 329)
• 11.4.3 The Closed Loop Poles (p. 331)
• 11.5 Linear Quadratic Gaussian Design by Polynomial Methods (p. 332)
• 11.5.1 Problem Formulation (p. 332)
• 11.5.2 Minimum Variance Control (p. 333)
• 11.5.3 The General Case (p. 337)
• 11.6 Controller Design by Linear Quadratic Gaussian Theory (p. 344)
• 11.6.1 Introduction (p. 344)
• 11.6.2 Choice of Observer Poles (p. 351)
• 11.A Appendix. Derivation of the Optimal Linear Quadratic Gaus-sian Feedback and the Riccati Equation from the Bellman Equation (p. 360)
• Exercises (p. 362)
• Bibliography (p. 364)
• Answers to Selected Exercises (p. 367)
• Index (p. 373)