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Since 1960-1961, when Rudolf E. Kalman has published his seminal work on discrete-time recursive optimal filtering and, together with Richard S. Bucy, on continuous-time optimal filtering of linear nonstationary stochastic processes with white Gaussian noise, recursive filtering and smoothing algorithm have become and still remain a key tool for real-time state estimation. This is despite the fact that Bayesian and convolution-based approaches say that optimal recursions are only available for white Gaussian and colored Gauss-Markov noise. Otherwise, since non-Gaussian noise has high-order statistics, the hypothetical recursive forms seem to be so complex that it is hardly practical to use them instead of batch forms. Therefore, recursive forms are widely used, although this often calls into question their accuracy in harsh environments.
When solving state estimation problems for signal processing and control using recursive algorithms, researchers traditionally associate them with Kalman filtering, even when not using all its recursive forms. This is even though some solutions, such as the robust iterative UFIR filter, as well as the transfer function-based Hâ filter, generalized H2 filter, L1 filter, etc., have nothing to do with Kalman filtering. Moreover, data-driven and AI-aided model-based filtering algorithms also lose connection to it. This leads to the idea that instead of thinking of recursive algorithms as Kalman-like, it is worth focusing on the general recursive form and cover all available recursive state estimators under one umbrella, treating Kalman filter is a special case.
This book attempts to do this by describing 53 pseudo codes and other forms of optimal, suboptimal, and robust recursive state estimation algorithms.
Contents
Foreword xv
Preface xvii
Acronims xix
1 Introduction 1
1.1 Brief pre-Kalman history . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Kalman filtering approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Recursive filtering estimate . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Dynamic process in state space . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 What is system state? . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 What do we need to estimate state? . . . . . . . . . . . . . . . . . . 7
1.3.3 What model to estimate state? . . . . . . . . . . . . . . . . . . . . . 8
1.3.4 What is state estimation problem? . . . . . . . . . . . . . . . . . . . 10
1.3.4.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4.2 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.4.3 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5 What types of linear state estimators? . . . . . . . . . . . . . . . . . 13
1.3.5.1 Unbiased estimator . . . . . . . . . . . . . . . . . . . . . . 13
1.3.5.2 Optimal estimator . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.5.3 Optimal unbiased (maximum likelihood) estimator . . . . . 15
1.3.6 What criteria to evaluate estimator? . . . . . . . . . . . . . . . . . . 16
1.4 Properties of Kalman filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.2 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.4 Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.5 General functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
I Optimal and Suboptimal Estimates 25
2 Kalman Filter for Beginners 27
2.1 Continuous-time stochastic system . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.1 Representation in state space . . . . . . . . . . . . . . . . . . . . . . 28
2.1.2 General state space model . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Discrete-time state-space model . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Eulerâs methods of system discretization . . . . . . . . . . . . . . . . 30
vii
viii Contents
2.2.2 LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2.1 Forward Euler method . . . . . . . . . . . . . . . . . . . . 31
2.2.2.2 Backward Euler method . . . . . . . . . . . . . . . . . . . . 33
2.2.3 LTV systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3.1 Forward Euler method . . . . . . . . . . . . . . . . . . . . 35
2.2.3.2 Backward Euler method . . . . . . . . . . . . . . . . . . . . 35
2.3 Intuitive derivation of the Kalman filter . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Basic a posteriori Kalman filter . . . . . . . . . . . . . . . . . . . . . 40
2.3.2 Alternate a posteriori Kalman filter . . . . . . . . . . . . . . . . . . 40
2.3.3 The a priori Kalman filter . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.4 Stationary Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Algorithmic variants of the Kalman filter . . . . . . . . . . . . . . . . . . . 43
2.4.1 Information Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.1.1 Information Kalman filtering algorithm . . . . . . . . . . . 45
2.4.2 Backward Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.2.1 Backward Kalman filtering algorithm . . . . . . . . . . . . 46
2.4.3 Forward-backward (two-filter) smoothing . . . . . . . . . . . . . . . 47
2.5 Kalman-Bucy filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Unbiasedness and stability of Kalman filter . . . . . . . . . . . . . . . . . . 51
2.6.1 Unbiasedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Bayesian Approach 57
3.1 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 Bayesâ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.2 Conditional probability density . . . . . . . . . . . . . . . . . . . . . 60
3.1.2.1 Two random variables . . . . . . . . . . . . . . . . . . . . . 60
3.1.2.2 Multiple random variables . . . . . . . . . . . . . . . . . . 61
3.2 Bayesian estimator (filter) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.1 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.1.1 Linear model (scalar case) . . . . . . . . . . . . . . . . . . 63
3.2.1.2 Linear model (vector case) . . . . . . . . . . . . . . . . . . 63
3.2.2 Nonlinear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Bayesian filtering of Gaussian models . . . . . . . . . . . . . . . . . . . . . . 64
3.3.1 Time update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.2 Measurement update . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.3 Recursive Gaussian filter (nonlinear case) . . . . . . . . . . . . . . . 66
3.3.3.1 Non-additive noise case . . . . . . . . . . . . . . . . . . . . 67
3.3.4 Recursive Gaussian filter (linear case) . . . . . . . . . . . . . . . . . 68
3.4 Kalman filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.1 Alternate Kalman filter recursions (scalar case) . . . . . . . . . . . . 70
3.4.2 Alternate Kalman filter recursions (vector case) . . . . . . . . . . . . 73
3.4.3 Basic Kalman filter recursions . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Kalman smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5.1 Kalman smoother recursions . . . . . . . . . . . . . . . . . . . . . . 78
3.5.2 Rauch-Tung-Striebel Smoother . . . . . . . . . . . . . . . . . . . . . 82
3.5.2.1 Kalman-RTS smoothing algorithm . . . . . . . . . . . . . . 83
3.6 Sequential Monte Carlo methods . . . . . . . . . . . . . . . . . . . . . . . . 84
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Contents ix
3.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Convolution-based approach 89
4.1 Convolution forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1.1 Problems solved with convolution . . . . . . . . . . . . . . . . . . . . 91
4.2 The a posteriori optimal FIR filter . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.1 Extended state-space model . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.2 Batch estimate and error covariance . . . . . . . . . . . . . . . . . . 93
4.2.2.1 Batch a posteriori optimal FIR filtering algorithm . . . . . 95
4.2.3 Recursive forms for OFIR filter . . . . . . . . . . . . . . . . . . . . . 97
4.2.3.1 Iterative a posteriori OFIR filtering algorithm . . . . . . . 101
4.3 The a posteriori optimal unbiased FIR filter . . . . . . . . . . . . . . . . . . 102
4.3.1 Batch estimate and error covariance . . . . . . . . . . . . . . . . . . 102
4.3.1.1 Batch a posteriori OUFIR filtering algorithm . . . . . . . . 103
4.3.2 Batch maximum likelihood filter . . . . . . . . . . . . . . . . . . . . 103
4.3.2.1 Batch ML filtering algorithm . . . . . . . . . . . . . . . . . 104
4.3.3 Recursive forms for OUFIR filter . . . . . . . . . . . . . . . . . . . . 104
4.3.3.1 Special case: infinite horizon . . . . . . . . . . . . . . . . . 106
4.3.3.2 Special case: constant state . . . . . . . . . . . . . . . . . . 107
4.3.3.3 Recursive ML filtering of constant state . . . . . . . . . . . 108
4.3.4 Properties of bias-constrained filters . . . . . . . . . . . . . . . . . . 111
4.4 The a posteriori optimal IIR filter . . . . . . . . . . . . . . . . . . . . . . . 111
4.4.1 Extended state-space model . . . . . . . . . . . . . . . . . . . . . . . 112
4.4.2 Batch a posteriori optimal IIR filter . . . . . . . . . . . . . . . . . . 112
4.4.3 Recursive forms for OIIR filter . . . . . . . . . . . . . . . . . . . . . 113
4.5 Kalman filter properties from convolution . . . . . . . . . . . . . . . . . . . 116
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5 General Kalman Filter 121
5.1 Time-Correlated Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.1 Noise de-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.1.1 GKF algorithm for de-correlated noise . . . . . . . . . . . . 123
5.1.2 New Kalman gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1.2.1 GKF algorithm for time-correlated noise . . . . . . . . . . 124
5.2 Colored Measurement Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2.1 Augmented state vector . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.2 Measurement differencing . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.2.1 Time-correlated noise . . . . . . . . . . . . . . . . . . . . . 127
5.2.2.2 De-correlated noise . . . . . . . . . . . . . . . . . . . . . . 129
5.2.3 Equivalence of GKF algorithms for CMN . . . . . . . . . . . . . . . 131
5.3 Colored process noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.1 Augmented state vector . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.2 State differencing (LTV case) . . . . . . . . . . . . . . . . . . . . . . 134
5.3.2.1 GKF algorithm for LTV systems with CPN . . . . . . . . . 135
5.3.3 State differencing (LTI case) . . . . . . . . . . . . . . . . . . . . . . 136
5.3.3.1 GKF algorithm for LTI systems with CPN . . . . . . . . . 138
5.4 Colored process and measurement noise . . . . . . . . . . . . . . . . . . . . 139
5.4.1 Augmented state vector . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4.2 State and measurement differencing . . . . . . . . . . . . . . . . . . 140
5.4.2.1 State differencing . . . . . . . . . . . . . . . . . . . . . . . 140
x Contents
5.4.2.2 Measurement differencing . . . . . . . . . . . . . . . . . . . 140
5.4.2.3 GKF algorithm for LTI systems with CPN and CMN . . . 141
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6 Suboptimal Kalman filtering 147
6.1 Extended Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.1.1 The first-order extension . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1.1.1 The first-order a posteriori EKF algorithm . . . . . . . . . 151
6.1.2 Iterated extended Kalman filter . . . . . . . . . . . . . . . . . . . . . 152
6.1.2.1 The a posteriori iterated EKF algorithm . . . . . . . . . . 153
6.1.3 The second-order extension . . . . . . . . . . . . . . . . . . . . . . . 154
6.1.3.1 The a priori state estimate . . . . . . . . . . . . . . . . . . 155
6.1.3.2 The a priori error covariance . . . . . . . . . . . . . . . . . 156
6.1.3.3 The a posteriori state estimate . . . . . . . . . . . . . . . . 157
6.1.3.4 The a posteriori error covariance . . . . . . . . . . . . . . . 157
6.2 Sigma-points approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2.1 Statistical linearization . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.2.2 Gaussian quadrature and cubature . . . . . . . . . . . . . . . . . . . 164
6.3 Unscented Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.3.1 The unscented transformation . . . . . . . . . . . . . . . . . . . . . . 167
6.3.2 The unscented Kalman filter . . . . . . . . . . . . . . . . . . . . . . 168
6.3.2.1 The UKF algorithm . . . . . . . . . . . . . . . . . . . . . . 169
6.4 Gauss-Hermite Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.4.1 Gauss-Hermite approximate integration . . . . . . . . . . . . . . . . 170
6.4.2 Gauss-Hermite cubature recursions . . . . . . . . . . . . . . . . . . . 172
6.5 Cubature Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.5.1 Cubature rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.5.1.1 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . 175
6.5.1.2 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . 176
6.5.1.3 CKF algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.6 Adaptive Kalman filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.6.1 Adaptation through covariance mismatch . . . . . . . . . . . . . . . 179
6.6.1.1 Measurement noise covariance adaptation . . . . . . . . . . 181
6.6.1.2 System noise covariance adaptation . . . . . . . . . . . . . 182
6.6.1.3 Adaptive Kalman filtering algorithm . . . . . . . . . . . . . 184
6.6.1.4 Strong tracking Kalman filter . . . . . . . . . . . . . . . . . 187
6.6.2 Adaptation using fuzzy inference . . . . . . . . . . . . . . . . . . . . 190
6.6.2.1 Fuzzy strong tracking Kalman filter . . . . . . . . . . . . . 194
6.6.3 Neural network aided adaptation . . . . . . . . . . . . . . . . . . . . 197
6.6.3.1 Correcting output estimate . . . . . . . . . . . . . . . . . . 200
6.6.3.2 Computing Kalman gain . . . . . . . . . . . . . . . . . . . 202
6.7 Fault detection and diagnosis using Kalman filtering . . . . . . . . . . . . . 203
6.7.1 Fault detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.7.2 Fault isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6.7.2.1 Measurement fault isolation . . . . . . . . . . . . . . . . . . 208
6.7.3 Fault estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.8 Some notable applied problems . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.8.1 Data association uncertainty . . . . . . . . . . . . . . . . . . . . . . 212
6.8.2 Random jitter in sampling interval . . . . . . . . . . . . . . . . . . . 215
6.8.2.1 Noise covariances under timing jitter . . . . . . . . . . . . 217
Contents xi
6.8.2.2 Jitter Kalman filtering algorithm . . . . . . . . . . . . . . . 221
6.8.3 Fast Kalman filter variants . . . . . . . . . . . . . . . . . . . . . . . 223
6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7 Kalman filtering for networks 229
7.1 Ensemble Kalman filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.1.1 High-dimensional state vectors . . . . . . . . . . . . . . . . . . . . . 230
7.1.1.1 The ensemble Kalman filtering algorithm . . . . . . . . . . 232
7.2 Consensus distributed Kalman filtering . . . . . . . . . . . . . . . . . . . . . 233
7.2.1 Consensus on measurements . . . . . . . . . . . . . . . . . . . . . . . 235
7.2.2 Consensus on estimates . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.2.2.1 Disagreement in estimates . . . . . . . . . . . . . . . . . . 237
7.2.2.2 Consensus on a posteriori estimates . . . . . . . . . . . . . 238
7.2.2.3 Consensus on a priori estimates . . . . . . . . . . . . . . . 242
7.2.3 Consensus on information . . . . . . . . . . . . . . . . . . . . . . . . 243
7.2.3.1 Distributed information Kalman filtering . . . . . . . . . . 244
7.2.4 Stability of consensus filtering . . . . . . . . . . . . . . . . . . . . . . 246
7.3 Fusion Kalman filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
7.3.1 Optimal fusion of estimates . . . . . . . . . . . . . . . . . . . . . . . 248
7.3.1.1 Kalman filter-based algorithm for optimal fusion of estimates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
7.3.2 Optimal data fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7.3.2.1 Kalman filter-based data fusion algorithms . . . . . . . . . 253
7.4 Filtering with intermittent observations . . . . . . . . . . . . . . . . . . . . 254
7.4.1 Intermittent Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . 257
7.4.2 Optimal fusion with intermittent observations . . . . . . . . . . . . . 258
7.4.2.1 Intermittent Kalman filter for optimal fusion . . . . . . . . 260
7.5 Delayed and lost data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.5.1 Timestamp one-step delays and packet dropouts . . . . . . . . . . . 262
7.5.1.1 Kalman filter for timestamp data with one-step delays and
packet dropouts . . . . . . . . . . . . . . . . . . . . . . . . 264
7.5.2 Random one-step delays and packet dropouts . . . . . . . . . . . . . 264
7.5.2.1 Kalman filter for data with random binary one-step delays
and packet dropouts . . . . . . . . . . . . . . . . . . . . . . 267
7.5.3 Optimal data fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
7.5.3.1 Fusion Kalman filter for timestamp one-step delays and
packet dropouts . . . . . . . . . . . . . . . . . . . . . . . . 272
7.6 Event-triggered data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
7.6.1 Send-on-delta event-triggering . . . . . . . . . . . . . . . . . . . . . . 274
7.6.2 Send-on-delta event-triggered Kalman filter . . . . . . . . . . . . . . 275
7.6.2.1 Event-triggered Kalman filtering algorithm . . . . . . . . . 277
7.6.3 Optimal fusion of event-triggered data . . . . . . . . . . . . . . . . . 278
7.6.3.1 Data received from a single sensor . . . . . . . . . . . . . . 278
7.6.3.2 Data received from multiple sensors . . . . . . . . . . . . . 281
7.6.3.3 Fusion of event-triggered data . . . . . . . . . . . . . . . . 282
7.6.3.4 Fusion event-triggered Kalman filtering algorithm for timestamp
data . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
7.7 Fault tolerant Kalman filtering . . . . . . . . . . . . . . . . . . . . . . . . . 286
7.7.1 Fault tolerant optimal fusion of estimates . . . . . . . . . . . . . . . 287
7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
xii Contents
7.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
II Robust Estimates 295
8 Robust approaches to recursive filtering 297
8.1 Robust state estimation problem . . . . . . . . . . . . . . . . . . . . . . . . 297
8.2 Noticeable early solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
8.3 Robust performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
8.4 Error models for robust filtering . . . . . . . . . . . . . . . . . . . . . . . . 304
8.4.1 Disturbance models . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
8.4.1.1 Standard error model . . . . . . . . . . . . . . . . . . . . . 305
8.4.1.2 Augmented error model . . . . . . . . . . . . . . . . . . . . 307
8.4.2 Uncertainty models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
8.4.2.1 Forward Euler method-based error model . . . . . . . . . . 309
8.4.2.2 Backward Euler method-based error model . . . . . . . . . 311
8.4.3 Disturbance and uncertainty models . . . . . . . . . . . . . . . . . . 312
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
8.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
9 Unbiased filtering 315
9.1 The a posteriori UFIR filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.1.1 Iterative computation using recursions . . . . . . . . . . . . . . . . . 317
9.1.1.1 Iterative a posteriori UFIR filtering algorithm . . . . . . . 320
9.1.1.2 UFIR filter with improved performance . . . . . . . . . . . 323
9.1.2 Recursive form for error covariance . . . . . . . . . . . . . . . . . . . 323
9.1.3 Minimizing error covariance . . . . . . . . . . . . . . . . . . . . . . . 325
9.1.3.1 Available ground truth . . . . . . . . . . . . . . . . . . . . 325
9.1.3.2 Unavailable ground truth . . . . . . . . . . . . . . . . . . . 326
9.2 General UFIR filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
9.2.1 Gauss-Markov colored measurement noise . . . . . . . . . . . . . . . 328
9.2.1.1 General UFIR filtering algorithm for CMN . . . . . . . . . 329
9.2.2 Gauss-Markov colored process noise . . . . . . . . . . . . . . . . . . 330
9.2.2.1 General UFIR filtering algorithm for LTI systems with CPN 331
9.3 Unbiased smoothing and prediction . . . . . . . . . . . . . . . . . . . . . . . 332
9.3.1 Recursive error covariance . . . . . . . . . . . . . . . . . . . . . . . . 335
9.3.1.1 Time-varying case . . . . . . . . . . . . . . . . . . . . . . . 336
9.3.1.2 Time-invariant case . . . . . . . . . . . . . . . . . . . . . . 337
9.4 Extended UFIR filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
9.4.1 First-order extended UFIR filter . . . . . . . . . . . . . . . . . . . . 338
9.4.2 Second-order extended UFIR filter . . . . . . . . . . . . . . . . . . . 340
9.5 Robustness of UFIR filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 343
9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
10 H2 filtering 349
10.1 Transfer function approach for H2 filtering . . . . . . . . . . . . . . . . . . . 349
10.2 Recursive H2 filtering under disturbances . . . . . . . . . . . . . . . . . . . 350
10.2.1 Forward Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . 350
10.2.2 Backward Euler method . . . . . . . . . . . . . . . . . . . . . . . . . 352
10.2.3 Recursive H2 filtering algorithm for disturbed models . . . . . . . . 354
Contents xiii
10.2.4 Computing H2 filter gain using LMI . . . . . . . . . . . . . . . . . . 357
10.3 Filtering of uncertain models . . . . . . . . . . . . . . . . . . . . . . . . . . 359
10.3.1 Covariances of uncertain terms . . . . . . . . . . . . . . . . . . . . . 360
10.3.1.1 H2 filtering algorithm for uncertain models . . . . . . . . . 363
10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
10.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
11 Hâ filtering 367
11.1 The Hâ filtering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
11.2 Disturbed model based on forward Euler method . . . . . . . . . . . . . . . 369
11.2.1 Bounded real lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
11.2.2 Hâ filterâ"option-(I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
11.2.2.1 Recursive Hâ filtering algorithm . . . . . . . . . . . . . . . 374
11.2.3 Hâ filterâ"option-(II) . . . . . . . . . . . . . . . . . . . . . . . . . . 374
11.2.3.1 Recursive Hâ filtering algorithm . . . . . . . . . . . . . . . 375
11.2.4 Iterative Hâ filtering algorithm . . . . . . . . . . . . . . . . . . . . . 376
11.3 Disturbed model based on backward Euler method . . . . . . . . . . . . . . 378
11.3.1 Bounded real lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
11.3.2 Hâ filterâ"option-(I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
11.3.2.1 Recursive Hâ filtering algorithm . . . . . . . . . . . . . . . 380
11.3.3 Hâ filterâ"option-(II) . . . . . . . . . . . . . . . . . . . . . . . . . . 381
11.3.3.1 Recursive Hâ filtering algorithm . . . . . . . . . . . . . . . 382
11.3.4 Iterative Hâ filtering algorithm . . . . . . . . . . . . . . . . . . . . . 382
11.4 Filtering of uncertain models . . . . . . . . . . . . . . . . . . . . . . . . . . 383
11.4.1 Recursive Hâ filtering algorithm for uncertain models . . . . . . . . 383
11.5 Hybrid filtering structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
11.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
12 Generalized H2 filtering 389
12.1 The energy-to-peak filtering problem . . . . . . . . . . . . . . . . . . . . . . 389
12.2 Energy-to-peak lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
12.2.1 Eulerâs forward method-based model . . . . . . . . . . . . . . . . . . 391
12.2.2 Eulerâs backward method-based model . . . . . . . . . . . . . . . . . 394
12.3 GH2 filter for disturbed modelsâ"option-(I) . . . . . . . . . . . . . . . . . . 395
12.3.1 Recursive GH2 filtering algorithm . . . . . . . . . . . . . . . . . . . 397
12.4 GH2 filter for disturbed modelsâ"option-(II) . . . . . . . . . . . . . . . . . . 397
12.4.1 Recursive GH2 filtering algorithm . . . . . . . . . . . . . . . . . . . 400
12.5 Iterative GH2 filtering algorithm . . . . . . . . . . . . . . . . . . . . . . . . 403
12.6 Filtering of uncertain models . . . . . . . . . . . . . . . . . . . . . . . . . . 403
12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
12.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
13 L1 filtering 407
13.1 The L1 filtering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
13.2 Peak-to-peak lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
13.2.1 Eulerâs forward method-based model . . . . . . . . . . . . . . . . . . 409
13.2.2 Eulerâs backward method-based model . . . . . . . . . . . . . . . . . 411
13.3 L1 filtering of disturbed models . . . . . . . . . . . . . . . . . . . . . . . . . 413
13.3.1 L1 filterâ"option-(I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
13.3.2 L1 filterâ"option-(II) . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
xiv Contents
13.3.2.1 Recursive L1 filtering algorithm . . . . . . . . . . . . . . . 416
13.3.3 Iterative L1 filtering algorithm . . . . . . . . . . . . . . . . . . . . . 420
13.4 Filtering of uncertain models . . . . . . . . . . . . . . . . . . . . . . . . . . 421
13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
13.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
14 Where recursive state estimation meets artificial intelligence 425
14.1 Model-based vs. data-driven state estimation . . . . . . . . . . . . . . . . . 426
14.1.1 Data-driven Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . 427
14.1.2 Maximum likelihood estimate . . . . . . . . . . . . . . . . . . . . . . 428
14.2 Machine learning-aided Kalman filtering . . . . . . . . . . . . . . . . . . . . 430
14.2.1 Tuning parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
14.2.1.1 Measurement noise covariance . . . . . . . . . . . . . . . . 432
14.2.1.2 Process noise covariance . . . . . . . . . . . . . . . . . . . . 432
14.2.1.3 Process and measurement noise covariance . . . . . . . . . 433
14.2.1.4 State space model parameters . . . . . . . . . . . . . . . . 433
14.2.1.5 Bias correction gain . . . . . . . . . . . . . . . . . . . . . . 433
14.2.2 Compensation for estimation errors . . . . . . . . . . . . . . . . . . . 434
14.3 Kalman filter-aided machine learning . . . . . . . . . . . . . . . . . . . . . . 435
14.3.1 Network training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
14.3.2 Hybrid schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
14.3.2.1 Improving network performance . . . . . . . . . . . . . . . 436
14.3.2.2 Long short-term memory Kalman filter . . . . . . . . . . . 437
14.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
14.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
15 Applications 441
15.1 GPS navigation of a moving vehicle . . . . . . . . . . . . . . . . . . . . . . 441
15.2 UWB-based localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
15.2.1 Linear robotic dog localization . . . . . . . . . . . . . . . . . . . . . 445
15.2.2 Nonlinear ad hoc localization . . . . . . . . . . . . . . . . . . . . . . 451
15.3 Pedestrian visual object tracking . . . . . . . . . . . . . . . . . . . . . . . . 455
15.3.1 Eight-state space model . . . . . . . . . . . . . . . . . . . . . . . . . 456
15.3.2 Separate coordinate filtering . . . . . . . . . . . . . . . . . . . . . . . 458
15.4 Air pollution estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
15.4.1 CO concentration model . . . . . . . . . . . . . . . . . . . . . . . . . 461
15.5 EMG amplitude estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
15.5.1 Typical EMG signaling . . . . . . . . . . . . . . . . . . . . . . . . . 465
15.6 Person activity estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
15.6.1 Walking activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
15.6.2 Standing up from sitting position . . . . . . . . . . . . . . . . . . . . 470
15.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
15.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
Bibliography 475
Index 497
ISBN: 9781041286134
ISBN-10: 1041286139
Available: 11th November 2026
Format: Hardcover
Language: English
Audience: General Adult
Country of Publication: GB
Edition Number: 1
Dimensions (cm): 23.4 x 15.6
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