Nonlinear Filtering: Concepts and Engineering Applications

Cover -- Half Title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Acknowledgements -- Authors -- Introduction -- Section I Mathematical Models, Kalman Filtering and H-Infinity Filters -- Chapter 1. Dynamic System Models and Basic Concepts -- 1.1 Dynamic Systems: The Need for Modelling, Parameter Estimation and Filtering -- 1.2 Mathematical Modelling of Systems -- 1.2.1 Time and Frequency Domain Aspects -- 1.2.2 Differential Equations -- 1.2.3 Difference Equations -- 1.2.4 State Space Models -- 1.2.4.1 Physical Representation -- 1.2.4.2 Controllable Canonical Form -- 1.2.4.3 Observable Canonical Form -- 1.2.4.4 Diagonal Form -- 1.2.4.5 General State Space Models -- 1.2.5 Polynomial Models -- 1.2.6 Time Series Models -- 1.2.6.1 Autoregressive Model -- 1.2.6.2 Least Squares Model -- 1.2.7 Transfer Function Models -- 1.3 Nonlinear Dynamic Systems -- 1.3.1 Nonlinearities in a System -- 1.3.2 Mathematical Models of Nonlinear Systems -- 1.3.2.1 Nonlinear Differential and Difference Equations -- 1.3.2.2 Volterra Series -- 1.3.2.3 Hammerstein Model -- 1.3.2.4 Nonlinear State Space Models -- 1.3.2.5 Nonlinear Time Series Models -- 1.4 Signal and System Norms -- 1.4.1 Signal Norms -- 1.4.2 System Norms -- 1.4.2.1 H2 Norm -- 1.4.2.2 H∞ Norm -- 1.5 Digital Signal Processing, Parameter Estimation and Filtering -- 1.5.1 Signal Processing -- 1.5.2 Parameter Estimation: Recursive Approach -- 1.5.3 Filtering Concepts -- 1.5.4 Simple Recursive Filtering -- Appendix 1A: Mean Square Estimation -- Appendix 1B: Nonlinear Models Based on Artificial Neural Networks and Fuzzy Logic -- Appendix 1C: Illustrative Examples -- Chapter 2. Filtering and Smoothing -- 2.1 Wiener Filtering -- 2.2 Least Squares Parameter Estimation -- 2.3 Recursive Least Squares Filter -- 2.4 State Space Models and Kalman Filtering -- 2.4.1 Discrete Time Filter

2.4.1.1 State and Covariance Matrix Propagation -- 2.4.1.2 Measurement Update -- 2.4.1.3 Kalman Gain -- 2.4.2 Continuous Time Kalman Filter -- 2.4.3 Interpretation of Kalman Filter -- 2.4.3.1 Continuous Time Filter -- 2.4.3.2 Discrete Time Filter -- 2.4.4 Filters for Correlated/Coloured Process and Measurement Noises -- 2.4.4.1 Kalman Filter for the Correlated Process and Measurement Noises -- 2.4.4.2 Handling of Coloured Process Noise and Coloured Measurement Noise in Kalman Filters -- 2.4.5 Time-Varying Linear Kalman Filters -- 2.4.6 Steady State Filtering -- 2.4.7 Kalman Filter Implementation Aspects -- 2.4.8 Parallelization of Kalman Filters -- 2.4.8.1 Measurement Update Parallelization -- 2.4.8.2 Time Propagation Parallelization -- 2.5 Filter Error Methods -- 2.5.1 Output Error Method -- 2.5.2 Process Noise Algorithms for Linear Systems -- 2.5.2.1 Natural Formulation -- 2.5.2.2 Innovations Formulation -- 2.5.2.3 Mixed Formulation -- 2.5.3 Process Noise Algorithms for Nonlinear Systems -- 2.5.3.1 Steady-State Filter -- 2.5.3.2 Time-Varying Filter -- 2.6 Information Filtering -- 2.6.1 Fisher's Information Concept -- 2.6.2 Linear Information Filter -- 2.7 Smoothers -- 2.7.1 Smoothing as a Combination of Forward and Backward Filtering -- 2.7.2 Fixed Interval RTS Smoother -- 2.7.3 Fixed Point Smoother -- 2.7.4 Fixed Lag Smoother -- Appendix 2A: Innovations Approach to Linear Least Squares Estimation -- Appendix 2B: Filtering Algorithms for Delayed State and Missing Measurements - Illustrative Example -- Appendix 2C: Artificial Neural Network Based Filtering -- Appendix 2D: Image Centroid Tracking with Fuzzy Logic in Filtering Algorithms - Illustrative Example -- Appendix 2E: Illustrative Examples -- Chapter 3. H∞ Filtering -- 3.1 H∞ Norm and Robustness -- 3.2 H∞ Filtering Problem -- 3.2.1 H∞ A Posteriori Filter -- 3.2.2 H∞ A Priori Filter

3.3 H∞ Smoother -- 3.4 H∞ Risk-Sensitive Filter -- 3.4.1 A Posteriori-Risk Sensitive Filter -- 3.4.2 A Priori Risk-Sensitive Filter -- 3.4.3 Risk-Sensitive Smoother -- 3.5 Mixed H∞ and Kalman Filtering -- 3.6 Global H∞ Filter -- Appendix 3A: Krein Space and Some Definitions and Theorems -- Appendix 3B: Illustrative Examples -- Chapter 4. Adaptive Filtering -- 4.1 Need of Filter Tuning and Adaptation -- 4.2 Approaches to Adaptive Filtering -- 4.2.1 Heuristic Approach -- 4.2.2 Bayesian Approach -- 4.2.3 Maximum Likelihood-Based Optimal Adaptive Filtering -- 4.2.4 Correlation-Based Adaptation -- 4.2.5 Concept of Covariance Matching -- 4.2.6 Fuzzy Logic-Based Adaptation -- 4.2.6.1 Fuzzy Inference System for R with Known Q -- 4.2.6.2 Fuzzy Inference System for Q with Known R -- 4.3 H∞ Finite Memory Adaptive Filter -- Appendix 4A: Maneuvering Target - Illustrative Examples -- Appendix 4B: Adaptive Kalman Filter - Illustrative Example -- Exercises for Section I (Chapters 1-4) -- References for Section I (Chapters 1-4) -- Section II Factorization and Approximation Filters -- Chapter 5. Factorization Filtering -- 5.1 Divergence of Kalman Filter -- Need of Factorization -- 5.2 UD Factorization Filter -- 5.2.1 Time Propagation -- 5.2.2 Measurement Data Update -- 5.2.3 Filter for Correlated Process Noise and Bias Parameters -- 5.3 Filtering Algorithms Based on Square-Root Arrays -- 5.3.1 H2 Square-Root Arrays -- 5.3.2 Chandrasekhar Recursions -- 5.3.3 H2 Chandrasekhar Recursions -- 5.4 Square-Root Information Filter -- 5.4.1 Inclusion of A Priori Information in the Least Squares Cost Function -- 5.4.2 Measurements Data Update -- 5.4.3 State Propagation of Square- Root Information Filter -- 5.4.4 Measurements Data Update of Square- Root Information Filter -- 5.5 Eigenvalue-Eigenvector Factorization Filtering -- 5.5.1 V-D Discrete Time Measurement Update

5.5.2 V-D Square-Root Filtering -- 5.5.2.1 Continuous Time/Discrete Time Square-Root Filtering Algorithm -- 5.5.2.2 Discrete Time/Discrete Time Square- Root Filtering Algorithm -- 5.6 H-Infinity Square-Root Filters -- 5.6.1 H-Infinity Square-Root Arrays -- 5.6.2 H-Infinity Chandrasekhar Recursions -- Chapter 6. Approximation Filters for Nonlinear Systems -- 6.1 Continuous Extended Kalman-Bucy Filter -- 6.2 Continuous-Discrete Extended Kalman-Bucy Filter -- 6.2.1 Time Propagation Filter -- 6.2.2 Measurement Data Update/Filtering -- 6.3 Continuous Discrete Extended Kalman-Bucy Filter for Joint State Parameter Estimation -- 6.3.1 Time Propagation -- 6.3.2 Measurement Data Update -- 6.4 Iterated Extended Kalman Filter -- 6.5 Linearized Kalman Filter -- 6.6 Continuous Second-Order Minimum Variance Estimator (SOF) -- 6.7 Continuous-Discrete Modified Gaussian Second-Order (CDMGSO) Filter -- 6.7.1 Measurement Update -- 6.7.2 Time Propagation/Prediction Part -- 6.8 Extended Information Filter -- 6.9 Statistically Linearized Filter -- 6.10 Derivative-Free Kalman Filter -- 6.10.1 Derivative-Free Kalman Filter Initialization -- 6.10.2 Sigma Points Computation -- 6.10.3 State and Covariance Propagation -- 6.10.4 State and Covariance Update -- 6.11 Global Approximations Nonlinear Filters -- 6.11.1 Orthogonal Series Expansion Approximations -- 6.11.1.1 Approximation Based on Legendre or Fourier Bases Functions -- 6.11.1.2 Approximation Based on Chebyshev Polynomials -- 6.11.2 Gaussian Sum Approximation -- 6.11.3 Point-Mass Approximation -- 6.11.3.1 Measurement Update -- 6.11.3.2 Time Propagation -- 6.11.3.3 Point Estimates -- 6.11.3.4 Algorithmic Aspects -- 6.11.4 Spline Approximation -- 6.11.4.1 B-Splines -- 6.11.4.2 Spline Filtering -- 6.12 Extended H-Infinity Filters -- 6.12.1 Continuous Time System -- 6.12.2 Discrete Time System

Appendix 6A: Approximate Filters -- Appendix 6B: Basic Numerical Approximation Approaches -- Appendix 6C: Satellite Orbit Determination as a Nonlinear Filtering Problem - Application of the Extended Kalman Filter, Extended UD Filter and Extended UD-RTS Smoother -- Appendix 6D: Application to Planar Tracking Problem - Illustrative Example -- Chapter 7. Generalized Model Error Estimators for Nonlinear Systems -- 7.1 Philosophy of Model Error -- 7.2 Pontryagin's Conditions -- 7.3 Basic Invariant Embedding Approach -- 7.4 Generalized Continuous Time Algorithm -- 7.5 Generalized Discrete Time Algorithm -- 7.6 Conventional Invariance Embedding Estimators -- 7.7 Robust Estimation of Model Error in H-Infinity Setting -- 7.7.1 Performance Norm -- 7.7.2 Constraint on Cost Function -- 7.7.3 Semi-Robust/Adaptive Invariant Embedding Estimators -- 7.8 Model Fitting Procedure to the Discrepancy/Model Error -- 7.9 Features of Model Error Algorithm -- Exercises for Section II (Chapters 5-7) -- References for Section II (Chapters 5-7) -- Section III Nonlinear Filtering, Estimation and Implementation Approaches -- Chapter 8. Nonlinear Estimation and Filtering -- 8.1 The General Estimation Framework -- 8.2 Continuous Time Dynamic Model and Filtering -- 8.2.1 Fokker-Planck Equation -- 8.2.2 Kushner-Stratonovich Equation -- 8.2.3 Minimum Variance Estimation -- 8.2.4 Bayesian Approach to Continuous Time Filtering -- 8.2.4.1 Bayes Formula -- 8.2.4.2 Nonlinear Filtering for Stochastic Differential Equation-Continuous Time Systems -- 8.2.5 Computation of the Filtered Estimates -- 8.3 Bayesian Recursive Estimation-Discrete Time Systems -- 8.3.1 Measurement Data Update/Filtering -- 8.3.2 Prediction-Time Propagation/Evolution -- 8.4 Continuous Time State-Discrete Time Measurement Estimator -- 8.4.1 Filtering/Measurement Data Update -- 8.4.2 Prediction-Time Propagation/Evolution

8.5 Benes Filter