1 Introduction 1 -- 1.1 What is System State? 2 -- 1.1.1 Why and How do We Estimate State? 2 -- 1.1.2 What Model to Estimate State? 3 -- 1.1.3 What are Basic State Estimates in Discrete Time? 5 -- 1.2 Properties of State Estimators 6 -- 1.2.1 Structures and Types 6 -- 1.2.2 Optimality 10 -- 1.2.3 Unbiased Optimality (Maximum Likelihood) 11 -- 1.2.4 Suboptimality 14 -- 1.2.5 Unbiasedness 17 -- 1.2.6 Deadbeat 17 -- 1.2.7 Denoising (Noise Power Gain) 17 -- 1.2.8 Stability 18 -- 1.2.9 Robustness 18 -- 1.2.10 Computational Complexity 19 -- 1.2.11 Memory Use 20 -- 1.3 More About FIR State Estimators 20 -- 1.4 Historical Overview and Most Noticeable Works 21 -- 1.5 Summary 26 -- 1.6 Problems 27 -- 2 Probability and Stochastic Processes 31 -- 2.1 Random Variables 31 -- 2.1.1 Moments and Cumulants 33 -- 2.1.2 Product Moments 39 -- 2.1.3 Vector Random Variables 41 -- 2.1.4 Conditional Probability. Bayes' Rule 42 -- 2.1.5 Transformation of Random Variables 45 -- 2.2 Stochastic Processes 47 -- 2.2.1 Correlation Function 48 -- 2.2.2 Power Spectral Density 51 -- 2.2.3 Gaussian Processes 53 -- 2.2.4 White Gaussian Noise 55 -- 2.2.5 Markov Processes 57 -- 2.3 Stochastic Differential Equation 60 -- 2.3.1 Standard Stochastic Differential Equation 61 -- 2.3.2 Itˆo and Stratonovich Stochastic Calculus 61 -- 2.3.3 Diffusion Process Interpretation 62 -- 2.3.4 Fokker-Planck-Kolmogorov Equation 63 -- 2.3.5 Langevin Equation 64 -- 2.4 Summary 65 -- 2.5 Problems 66 -- 3 State Estimation 71 -- 3.1 Lineal Stochastic Process in State Space 71 -- 3.1.1 Continuous-Time Model 73 -- 3.1.2 Discrete-Time Model 77 -- 3.2 Methods of Linear State Estimation 81 -- 3.2.1 Bayesian Estimator 82 -- 3.2.2 Maximum Likelihood Estimator 85 -- 3.2.3 Least Squares Estimator 86 -- 3.2.4 Unbiased Estimator 87 -- 3.2.5 Kalman Filtering Algorithm 88 -- 3.2.6 Backward Kalman Filter 94 -- 3.2.7 Alternative Forms of Kalman Filter 96 -- 3.2.8 General Kalman Filter 98 -- 3.2.9 Kalman-Bucy Filter 110 -- 3.3 Linear Recursive Smoothing 113 -- 3.3.1 Rauch-Tung-Striebel Algorithm 113 -- 3.3.2 Bryson-Frazier Algorithm 114 -- 3.3.3 Two-Filter (Forward-Backward) Smoothing 115 -- 3.4 Nonlinear Models and Estimators 116 -- 3.4.1 Extended Kalman Filter 117 -- 3.4.2 Unscented Kalman Filter 119 -- 3.4.3 Particle Filtering 122 -- 3.5 Robust State Estimation 126 -- 3.5.1 Robustified Kalman Filter 127 -- 3.5.2 Robust Kalman Filter 128 -- 3.5.3 H8 Filtering 131 -- 3.5.4 Game Theory H8 Filter 132 -- 3.6 Summary 133 -- 3.7 Problems 134 -- 4 Optimal FIR and Limited Memory Filtering 139 -- 4.1 Extended State-Space Model 140 -- 4.2 The a posteriori Optimal FIR Filter 142 -- 4.2.1 Batch Estimate and Error Covariance 143 -- 4.2.2 Recursive Forms 145 -- 4.2.3 System Identification 149 -- 4.3 The a posteriori Optimal Unbiased FIR Filter 149 -- 4.3.1 Batch OUFIR-I Estimate and Error Covariance 150 -- 4.3.2 Recursive Forms for OUFIR-I Filter 151 -- 4.3.3 Batch OUFIR-II Estimate and Error Covariance 153 -- 4.3.4 Recursion Forms for OUFIR-II Filter 154 -- 4.4 Maximum Likelihood FIR Estimator 158 -- 4.4.1 ML-I FIR Filtering Estimate 158 -- 4.4.2 Equivalence of ML-I FIR and OUFIR Filters 159 -- 4.4.3 ML-II FIR Filtering Estimate 162 -- 4.4.4 Properties of ML FIR State Estimators 163 -- 4.5 The a priori FIR Filters 164 -- 4.5.1 The a priori Optimal FIR Filter 164 -- 4.5.2 The a priori Optimal Unbiased FIR Filter 165 -- 4.6 Limited Memory Filtering 165 -- 4.6.1 Batch Limited Memory Filter 166 -- 4.6.2 Iterative LMF Algorithm using Recursions 168 -- 4.7 Continuous-Time Optimal FIR Filter 169 -- 4.7.1 Optimal Impulse Response 169 -- 4.7.2 Differential Equation Form 171 -- 4.8 Extended a posteriori OFIR Filtering 172 -- 4.9 Properties of FIR State Estimators 174 -- 4.10 Summary 179 -- 4.11 Problems 182 -- 5 Optimal FIR Smoothing 187 -- 5.1 Introduction 187 -- 5.2 Smoothing Problem 188 -- 5.3 Forward Filter/Forward Model q-lag OFIR Smoothing 189 -- 5.3.1 Batch Smoothing Estimate 190 -- 5.3.2 Error Covariance 193 -- 5.4 Backward OFIR Filtering 195 -- 5.4.1 Backward State-Space Model 195 -- 5.4.2 Batch Estimate 196 -- 5.4.3 Recursive Estimate and Error Covariance 198 -- 5.5 Backward Filter/Backward Model g-lag OFIR Smoother 202 -- 5.5.1 Batch Smoothing Estimate 203 -- 5.5.2 Error Covariance 204 -- 5.6 Forward Filter/Backward Model q-Lag OFIR Smoother 205 -- 5.6.1 Batch Smoothing Estimate 205 -- 5.6.2 Error Covariance 208 -- 5.7 Backward Filter/Forward Model q-Lag OFIR Smoother 208 -- 5.7.1 Batch Smoothing Estimate 208 -- 5.7.2 Error Covariance 211 -- 5.8 Two-Filter q-lag OFIR Smoother 213 -- 5.9 q-Lag ML FIR Smoothing 214 -- 5.9.1 Batch q-lag ML FIR Estimate 215 -- 5.9.2 Error Covariance 216 -- 5.10 Summary 216 -- 5.11 Problems 217 -- 6 Unbiased FIR State Estimation 221 -- 6.1 Introduction 221 -- 6.2 The a posteriori UFIR Filter 222 -- 6.2.1 Batch Form 222 -- 6.2.2 Iterative Algorithm Using Recursions 224 -- 6.2.3 Recursive Error Covariance 226 -- 6.2.4 Optimal Averaging Horizon 228 -- 6.3 Backward a posteriori UFIR Filter 234 -- 6.3.1 Batch Form 235 -- 6.3.2 Recursions and Iterative Algorithm 236 -- 6.3.3 Recursive Error Covariance 239 -- 6.4 The q-lag UFIR Smoother 240 -- 6.4.1 Batch and Recursive Forms 240 -- 6.4.2 Error Covariance 242 -- 6.4.3 Equivalence of UFIR Smoothers 244 -- 6.5 State Estimation using Polynomial Models 245 -- 6.5.1 Problems Solved with UFIR Structures 246 -- 6.5.2 The p-shift UFIR Filter 247 -- 6.5.3 Filtering of Polynomial Models 250 -- 6.5.4 Discrete Shmaliy Moments 252 -- 6.5.5 Smoothing Filtering and Smoothing 252 -- 6.5.6 Generalized Savitzky-Golay Filter 254 -- 6.5.7 Predictive Filtering and Prediction 255 -- 6.6 UFIR State Estimation under Colored Noise 256 -- 6.6.1 Colored Measurement Noise 256 -- 6.6.2 Colored Process Noise 259 -- 6.7 Extended UFIR Filtering 262 -- 6.7.1 First-Order Extended UFIR Filter 263 -- 6.7.2 Second-Order Extended UFIR Filter 263 -- 6.8 Robustness of UFIR Filter 266 -- 6.8.1 Errors in Noise Covariances and Weighted Matrices 268 -- 6.8.2 Model Errors 271 -- 6.8.3 Temporary Uncertainties 274 -- 6.9 Implementation of Polynomial UFIR Filters 276 -- 6.9.1 Filter Structures in z-Domain 276 -- 6.9.2 Transfer Function in DFT Domain 282 -- 6.10 Summary 287 -- 6.11 Problems 288 -- 7 FIR Prediction and Receding Horizon Filtering 295 -- 7.1 Introduction 295 -- 7.2 Prediction Strategies 296 -- 7.2.1 Kalman Predictor 296 -- 7.3 Extended Predictive State-Space Model 298 -- 7.4 UFIR Predictor 298 -- 7.4.1 Batch UFIR Predictor 299 -- 7.4.2 Iterative Algorithm using Recursions 299 -- 7.4.3 Recursive Error Covariance 303 -- 7.5 Optimal FIR Predictor 304 -- 7.5.1 Batch Estimate and Error Covariance 305 -- 7.5.2 Recursive Forms and Iterative Algorithm 306 -- 7.6 Receding Horizon FIR Filtering 308 -- 7.6.1 MVF-I Filter for Stationary Processes 309 -- 7.6.2 MVF-II Filter for Nonstationary Processes 311 -- 7.7 Maximum Likelihood FIR Predictor 313 -- 7.7.1 ML-I FIR Predictor 314 -- 7.7.2 ML-II FIR Predictor 315 -- 7.8 Extended OFIR Prediction 315 -- 7.9 Summary 317 -- 7.10 Problems 318 -- 8 Robust FIR State Estimation under Disturbances 323 -- 8.1 Extended Models under Disturbances 324 -- 8.2 The a posteriori H2 FIR Filtering 326 -- 8.2.1 H2-OFIR Filter 328 -- 8.2.2 Optimal Unbiased H2 FIR Filter 330 -- 8.2.3 Suboptimal H2 FIR Filtering Algorithms 336 -- 8.3 H2 FIR Prediction 338 -- 8.3.1 H2-OFIR Predictor 339 -- 8.3.2 Bias-constrained H2-OUFIR Predictor 341 -- 8.3.3 Suboptimal H2 FIR Predictive Algorithms 341 -- 8.3.4 Receding Horizon H2-MVF Filter 343 -- 8.4 H8 FIR State Estimation 344 -- 8.4.1 The a posteriori H8 FIR Filter 346 -- 8.4.2 H8 FIR Predictor 350 -- 8.5 H2{H8 FIR Filter and Predictor 354 -- 8.6 Generalized H2 FIR State Estimation 355 -- 8.6.1 Energy-to-Peak Lemma 355 -- 8.6.2 L2-to-L8 FIR Filter and Predictor 359 -- 8.7 L1 FIR State Estimation 362 -- 8.7.1 Peak-to-Peak Lemma 363 -- 8.7.2 L8-to-L8 FIR Filtering and Prediction 365 -- 8.8 Game Theory FIR State Estimation 367 -- 8.8.1 The a posteriori Energy-to-Power FIR Filter 368 -- 8.8.2 Energy-to-Power FIR Predictor 370 -- 8.9 Recursive Computation of Robust FIR Estimates 371 -- 8.9.1 Uncontrolled Processes 372 -- 8.9.2 Controlled Processes 372 -- 8.10 FIR Smoothing under Disturbances 374 -- 8.11 Summary 374 -- 8.12 Problems 376 -- 9 Robust FIR State Estimation for Uncertain Systems 379 -- 9.1 Extended Models for Uncertain Systems 380 -- 9.2 The a posteriori H2 FIR Filtering 386 -- 9.2.1 H2-OFIR Filter 387 -- 9.2.2

Bias-constrained H2-OFIR Filter 392 -- 9.3 H2 FIR Prediction 394 -- 9.3.1 Optimal H2 FIR Predictor 395 -- 9.3.2 Bias-constrained H2-OUFIR Predictor 399 -- 9.4 Suboptimal H2 FIR Structures using LMI 400 -- 9.4.1 Suboptimal H2 FIR Filter 401 -- 9.4.2 Bias-Constrained Suboptimal H2 FIR Filter 402 -- 9.4.3 Suboptimal H2 FIR Predictor 403 -- 9.4.4 Bias-Constrained Suboptimal H2 FIR Predictor 404 -- 9.5 H8 FIR State Estimation for Uncertain Systems 405 -- 9.5.1 The a posteriori H8 FIR Filter 405 -- 9.5.2 H8 FIR Predictor 407 -- 9.6 Hybrid H2{H8 FIR Structures 410 -- 9.7 Generalized H2 FIR Structures for Uncertain Systems 411 -- 9.7.1 The a posteriori L2-to-L8 FIR Filter 412 -- 9.7.2 L2-to-L8 FIR Predictor 414 -- 9.8 Robust L1 FIR Structures for Uncertain Systems 416 -- 9.8.1 The a posteriori L8-to-L8 FIR Filter 417 -- 9.8.2 L8-to-L8 FIR Predictor 417 -- 9.9 Summary 418 -- 9.10 Problems 419 -- 10 Advanced Topics in FIR State Estimation 423 -- 10.1 Distributed Filtering over Networks 423 -- 10.1.1 Consensus in Measurements 424 -- 10.1.2 Consensus in Estimates 429 -- 10.2 Optimal Fusion Filtering under Correlated Noise 433 -- 10.2.1 Error Covariances under Cross Correlation 436 -- 10.3 Hybrid Kalman/UFIR Filter Structures 438 -- 10.3.1 Fusing Estimates with Probabilistic Weights 438 -- 10.3.2 Fusing Kalman and ...