Nonlinear Filters

Peyman Setoodeh, PhD, is Visiting Professor with the Centre for Mechatronics and Hybrid Technologies (CMHT) at McMaster University. He is a Senior Member of the IEEE.Saeid Habibi, PhD, is Professor and former Chair of the Department of Mechanical Engineering and the Director of the Centre for Mechatronics and Hybrid Technologies (CMHT) at McMaster University. He is a Fellow of the ASME and the CSME as well as a Canada Research Chair and a Senior NSERC Industrial Research Chair. Simon Haykin, PhD, is Distinguished University Professor with the Department of Electrical and Computer Engineering and the Director of the Cognitive Systems Laboratory (CSL) at McMaster University. He is a Fellow of the IEEE and the Royal Society of Canada. He is a recipient of the Henry Booker Gold Medal from the International Union of Radio Science, the IEEE James H. Mulligan Jr. Education Medal, and the IEEE Denis J. Picard Medal for Radar Technologies and Applications.

• List of Figures xiiiList of Table xvPreface xviiAcknowledgments xixAcronyms xxi1 Introduction 11.1 State of a Dynamic System 11.2 State Estimation 11.3 Construals of Computing 21.4 Statistical Modeling 31.5 Vision for the Book 42 Observability 72.1 Introduction 72.2 State-Space Model 72.3 The Concept of Observability 92.4 Observability of Linear Time-Invariant Systems 102.4.1 Continuous-Time LTI Systems 102.4.2 Discrete-Time LTI Systems 122.4.3 Discretization of LTI Systems 142.5 Observability of Linear Time-Varying Systems 142.5.1 Continuous-Time LTV Systems 142.5.2 Discrete-Time LTV Systems 162.5.3 Discretization of LTV Systems 172.6 Observability of Nonlinear Systems 172.6.1 Continuous-Time Nonlinear Systems 182.6.2 Discrete-Time Nonlinear Systems 212.6.3 Discretization of Nonlinear Systems 222.7 Observability of Stochastic Systems 232.8 Degree of Observability 252.9 Invertibility 262.10 Concluding Remarks 273 Observers 293.1 Introduction 293.2 Luenberger Observer 303.3 Extended Luenberger-Type Observer 313.4 Sliding-Mode Observer 333.5 Unknown-Input Observer 353.6 Concluding Remarks 394 Bayesian Paradigm and Optimal Nonlinear Filtering 414.1 Introduction 414.2 Bayes’ Rule 424.3 Optimal Nonlinear Filtering 424.4 Fisher Information 454.5 Posterior Cramér–Rao Lower Bound 464.6 Concluding Remarks 475 Kalman Filter 495.1 Introduction 495.2 Kalman Filter 505.3 Kalman Smoother 535.4 Information Filter 545.5 Extended Kalman Filter 545.6 Extended Information Filter 545.7 Divided-Difference Filter 545.8 Unscented Kalman Filter 605.9 Cubature Kalman Filter 605.10 Generalized PID Filter 645.11 Gaussian-Sum Filter 655.12 Applications 675.12.1 Information Fusion 675.12.2 Augmented Reality 675.12.3 Urban Traffic Network 675.12.4 Cybersecurity of Power Systems 675.12.5 Incidence of Influenza 685.12.6 COVID-19 Pandemic 685.13 Concluding Remarks 706 Particle Filter 716.1 Introduction 716.2 Monte Carlo Method 726.3 Importance Sampling 726.4 Sequential Importance Sampling 736.5 Resampling 756.6 Sample Impoverishment 766.7 Choosing the Proposal Distribution 776.8 Generic Particle Filter 786.9 Applications 816.9.1 Simultaneous Localization and Mapping 816.10 Concluding Remarks 827 Smooth Variable-Structure Filter 857.1 Introduction 857.2 The Switching Gain 867.3 Stability Analysis 907.4 Smoothing Subspace 937.5 Filter Corrective Term for Linear Systems 967.6 Filter Corrective Term for Nonlinear Systems 1027.7 Bias Compensation 1057.8 The Secondary Performance Indicator 1077.9 Second-Order Smooth Variable Structure Filter 1087.10 Optimal Smoothing Boundary Design 1087.11 Combination of SVSF with Other Filters 1107.12 Applications 1107.12.1 Multiple Target Tracking 1117.12.2 Battery State-of-Charge Estimation 1117.12.3 Robotics 1117.13 Concluding Remarks 1118 Deep Learning 1138.1 Introduction 1138.2 Gradient Descent 1148.3 Stochastic Gradient Descent 1158.4 Natural Gradient Descent 1198.5 Neural Networks 1208.6 Backpropagation 1228.7 Backpropagation Through Time 1228.8 Regularization 1228.9 Initialization 1258.10 Convolutional Neural Network 1258.11 Long Short-Term Memory 1278.12 Hebbian Learning 1298.13 Gibbs Sampling 1318.14 Boltzmann Machine 1318.15 Autoencoder 1358.16 Generative Adversarial Network 1368.17 Transformer 1378.18 Concluding Remarks 1399 Deep Learning-Based Filters 1419.1 Introduction 1419.2 Variational Inference 1429.3 Amortized Variational Inference 1449.4 Deep Kalman Filter 1449.5 Backpropagation Kalman Filter 1469.6 Differentiable Particle Filter 1489.7 Deep Rao–Blackwellized Particle Filter 1529.8 Deep Variational Bayes Filter 1589.9 Kalman Variational Autoencoder 1679.10 Deep Variational Information Bottleneck 1729.11 Wasserstein Distributionally Robust Kalman Filter 1769.12 Hierarchical Invertible Neural Transport 1789.13 Applications 1829.13.1 Prediction of Drug Effect 1829.13.2 Autonomous Driving 1839.14 Concluding Remarks 18310 Expectation Maximization 18510.1 Introduction 18510.2 Expectation Maximization Algorithm 18510.3 Particle Expectation Maximization 18810.4 Expectation Maximization for Gaussian Mixture Models 19010.5 Neural Expectation Maximization 19110.6 Relational Neural Expectation Maximization 19410.7 Variational Filtering Expectation Maximization 19610.8 Amortized Variational Filtering Expectation Maximization 19810.9 Applications 19910.9.1 Stochastic Volatility 19910.9.2 Physical Reasoning 20010.9.3 Speech, Music, and Video Modeling 20010.10 Concluding Remarks 20111 Reinforcement Learning-Based Filter 20311.1 Introduction 20311.2 Reinforcement Learning 20411.3 Variational Inference as Reinforcement Learning 20711.4 Application 21011.4.1 Battery State-of-Charge Estimation 21011.5 Concluding Remarks 21012 Nonparametric Bayesian Models 21312.1 Introduction 21312.2 Parametric vs Nonparametric Models 21312.3 Measure-Theoretic Probability 21412.4 Exchangeability 21912.5 Kolmogorov Extension Theorem 22112.6 Extension of Bayesian Models 22312.7 Conjugacy 22412.8 Construction of Nonparametric Bayesian Models 22612.9 Posterior Computability 22712.10 Algorithmic Sufficiency 22812.11 Applications 23212.11.1 Multiple Object Tracking 23312.11.2 Data-Driven Probabilistic Optimal Power Flow 23312.11.3 Analyzing Single-Molecule Tracks 23312.12 Concluding Remarks 233References 235Index 253