Nonlinear Estimation-Methods and Applications with Deterministic Sample Points

277 Pages·2019·14.928 MB·\277

by  Shovan Bhaumik (Author), Paresh Date (Author)

Nonlinear Estimation
Methods and Applications with
Deterministic Sample Points

Nonlinear Estimation
Methods and Applications with
Deterministic Sample Points
Shovan Bhaumik
Paresh Date
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To my parents
Shovan
To Bhagyashree
Paresh

Contents
Preface xiii
About the Authors xvii
Abbreviations xix
Symbol Description xxi
1 Introduction 1
1.1 Nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Continuous time state space model . . . . . . . . . . . 2
1.1.2 Discrete time state space model . . . . . . . . . . . . . 3
1.2 Discrete time systems with noises . . . . . . . . . . . . . . . 5
1.2.1 Solution of discrete time LTI system . . . . . . . . . . 6
1.2.2 States as a Markov process . . . . . . . . . . . . . . . 6
1.3 Stochastic filtering problem . . . . . . . . . . . . . . . . . . . 7
1.4 Maximum likelihood and maximum a posterori estimate . . . 8
1.4.1 Maximum likelihood (ML) estimator . . . . . . . . . . 8
1.4.2 Maximum a posteriori (MAP) estimate . . . . . . . . 9
1.5 Bayesian framework of filtering . . . . . . . . . . . . . . . . . 9
1.5.1 Bayesian statistics . . . . . . . . . . . . . . . . . . . . 9
1.5.2 RecursiveBayesianfiltering:aconceptualsolution . . . 10
1.6 Particle filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6.1 Importance sampling . . . . . . . . . . . . . . . . . . . 12
1.6.2 Resampling . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Gaussian filter . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7.1 Propagation of mean and covariance of a linear
system . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7.2 Nonlinear filter with Gaussian approximations . . . . 19
1.8 Performance measure . . . . . . . . . . . . . . . . . . . . . . 22
1.8.1 When truth is known . . . . . . . . . . . . . . . . . . 22
1.8.2 When truth is unknown . . . . . . . . . . . . . . . . . 23
1.9 A few applications . . . . . . . . . . . . . . . . . . . . . . . . 23
1.9.1 Target tracking . . . . . . . . . . . . . . . . . . . . . . 23
1.9.2 Navigation . . . . . . . . . . . . . . . . . . . . . . . . 24
vii
viii Contents
1.9.3 Process control . . . . . . . . . . . . . . . . . . . . . . 24
1.9.4 Weather prediction . . . . . . . . . . . . . . . . . . . . 24
1.9.5 Estimating state-of-charge (SoC) . . . . . . . . . . . . 24
1.10 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.11 Organization of chapters . . . . . . . . . . . . . . . . . . . . 25
2 The Kalman filter and the extended Kalman filter 27
2.1 Linear Gaussian case (the Kalman filter) . . . . . . . . . . . 27
2.1.1 Kalman filter: a brief history . . . . . . . . . . . . . . 27
2.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.4 Properties: convergence and stability . . . . . . . . . . 31
2.1.5 Numerical issues . . . . . . . . . . . . . . . . . . . . . 32
2.1.6 The information filter . . . . . . . . . . . . . . . . . . 33
2.1.7 Consistency of state estimators . . . . . . . . . . . . . 34
2.1.8 Simulation example for the Kalman filter . . . . . . . 35
2.1.9 MATLAB(cid:114)-based filtering exercises . . . . . . . . . . 37
2.2 The extended Kalman filter (EKF) . . . . . . . . . . . . . . 38
2.2.1 Simulation example for the EKF . . . . . . . . . . . . 40
2.3 Important variants of the EKF . . . . . . . . . . . . . . . . . 43
2.3.1 The iterated EKF (IEKF) . . . . . . . . . . . . . . . . 43
2.3.2 The second order EKF (SEKF) . . . . . . . . . . . . . 45
2.3.3 Divided difference Kalman filter (DDKF) . . . . . . . 45
2.3.4 MATLAB-based filtering exercises . . . . . . . . . . . 49
2.4 Alternative approaches towards nonlinear filtering . . . . . . 49
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Unscented Kalman filter 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Sigma point generation . . . . . . . . . . . . . . . . . . . . . 52
3.3 Basic UKF algorithm . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Simulation example for the unscented Kalman
filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4 Important variants of the UKF . . . . . . . . . . . . . . . . . 60
3.4.1 Spherical simplex unscented transformation . . . . . . 60
3.4.2 Sigma point filter with 4n+1 points . . . . . . . . . . 61
3.4.3 MATLAB-based filtering exercises . . . . . . . . . . . 64
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Filters based on cubature and quadrature points 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Spherical cubature rule of integration . . . . . . . . . . . . . 66
Contents ix
4.3 Gauss-Laguerre rule of integration . . . . . . . . . . . . . . . 67
4.4 Cubature Kalman filter . . . . . . . . . . . . . . . . . . . . . 68
4.5 Cubature quadrature Kalman filter . . . . . . . . . . . . . . 70
4.5.1 Calculation of cubature quadrature (CQ) points . . . 70
4.5.2 CQKF algorithm . . . . . . . . . . . . . . . . . . . . . 71
4.6 Square root cubature quadrature Kalman filter . . . . . . . . 75
4.7 High-degree (odd) cubature quadrature Kalman filter . . . . 77
4.7.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.7.2 High-degree cubature rule . . . . . . . . . . . . . . . . 77
4.7.3 High-degree cubature quadrature rule . . . . . . . . . 79
4.7.4 Calculation of HDCQ points and weights . . . . . . . 80
4.7.5 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . 80
4.7.6 High-degree cubature quadrature Kalman filter . . . . 86
4.8 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . 87
4.8.1 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.8.2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Gauss-Hermite filter 95
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Gauss-Hermite rule of integration . . . . . . . . . . . . . . . 96
5.2.1 Single dimension . . . . . . . . . . . . . . . . . . . . . 96
5.2.2 Multidimensional integral . . . . . . . . . . . . . . . . 97
5.3 Sparse-grid Gauss-Hermite filter (SGHF) . . . . . . . . . . . 99
5.3.1 Smolyak’s rule . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Generation of points using moment matching method . . . . 104
5.5 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . 105
5.5.1 Tracking an aircraft . . . . . . . . . . . . . . . . . . . 105
5.6 Multiple sparse-grid Gauss-Hermite filter (MSGHF) . . . . . 109
5.6.1 State-space partitioning . . . . . . . . . . . . . . . . . 109
5.6.2 Bayesian filtering formulation for multiple
approach . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6.3 Algorithm of MSGHF . . . . . . . . . . . . . . . . . . 111
5.6.4 Simulation example . . . . . . . . . . . . . . . . . . . 113
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Gaussian sum filters 117
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Gaussian sum approximation . . . . . . . . . . . . . . . . . . 118
6.2.1 Theoretical foundation . . . . . . . . . . . . . . . . . . 118
6.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . 120
6.2.3 Multidimensional systems . . . . . . . . . . . . . . . . 121
6.3 Gaussian sum filter . . . . . . . . . . . . . . . . . . . . . . . 122