Launch Point. Ballistic target is launched from different places such as surface of the earth, ship, submarine etc., with different platforms such as mobile launcher, ...
Tracking of Ballistic Target on Re-entry Using Non-linear Filters Thesis Submitted in the partial fulfillment of the requirements for the degree of
MASTER OF CONTROL SYSTEM ENGINEERING
Submitted By NIKHIL KUMAR SINGH Examination Roll number: M4CTL1501 Registration Number: 124858 of 13-14
Under the Guidance of Dr. Samar Bhattacharya & Dr. Shovan Bhaumik
Electrical Engineering Department Faculty Council for UG and PG studies in Engineering and Technology JADAVPUR UNIVERSITY KOLKATA- 700032 May, 2015
Faculty Council for UG and PG studies in Engineering and Technology JADAVPUR UNIVERSITY KOLKATA- 700032 Certificate of Recommendation
This is to certify that Mr. Nikhil Kumar Singh (M4CTL1501) has completed his dissertation entitled, ”Tracking of Ballistic Target on Re-entry Using Non-linear Filters”, under the direct supervision and guidance of Dr. Samar Bhattacharya, Electrical Engineering Department, Jadavpur University and
Dr. Shovan
Bhaumik, Department of Electrical Engineering, IIT Patna. We are satisfied with his work, which is being presented for the partial fulfillment of the degree of Master of Control System Engineering of Jadavpur University, Kolkata-700032.
________________________ Dr. Samar Bhattacharya Professor, Electrical Engineering Department Jadavpur University, Kolkata – 700 032
________________________ Dr. Shovan Bhaumik Asst. Professor, Department of Electrical Engineering IIT Patna
Dr. Swapan Kumar Goswami Head, Electrical Engineering Department Jadavpur University, Kolkata – 700 032
_____________________________
Dr. Sivaji Bandyopadhyay Dean, Faculty of Engineering & Technology Jadavpur University, Kolkata – 700 032
Faculty Council for UG and PG studies in Engineering and Technology JADAVPUR UNIVERSITY KOLKATA- 700032
Certificate of Approval * The foregoing thesis is hereby approved as a creditable study of Master of Control System Engineering and presented in a manner satisfactory to warrant its acceptance as a prerequisite to the degree for which it has been submitted. It is understood that by this approval the undersigned do not necessarily endorse or approve any statement made, opinion expressed or conclusion therein but approve this thesis only for the purpose for which it is submitted.
Final Examination for Evaluation of the Thesis
_____________________________
_____________________________
_____________________________ Signature of Examiners
* Only in case the thesis is approved
Declaration of Originality and Compliance of Academic Ethics
I hereby declare that this thesis contains literature survey and original research work by the undersigned candidate, as part of her Master of Control System Engineering studies. All information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.
Name (Block Letters)
:
NIKHIL KUMAR SINGH
Exam Roll Number
:
M4CTL1501
Thesis Title
:
Tracking of Ballistic Target on Re-entry Using Non-linear Filters
Signature With Date
:
Dedicated to my “Beloved parents”
ACKNOWLEDGEMENTS I heartily thank my supervisor, Prof. Samar Bhattacharya, Department of Electrical Engineering, Jadavpur University, Kolkata, for his invaluable guidance, suggestions and encouragement throughout my project, which helped me in successfully completing it. You gave me freedom to choose a project of my own choice and a chance to get knowledge from two prestigious institutions Jadavpur University Kolkata as well as IIT Patna. I heartily thank my co-supervisor, Asst. Prof. Shovan Bhaumik, Department of Electrical Engineering, IIT Patna, who motivated and helped me in completing the one year project work. Your motivation and valuable suggestions during work, have lifted the thesis to a level I never would have reached on my own. I shall consider this as an inspiration to extend my work in this field. I specially thank Prof. Tapan Kumar Ghoshal, Jadavpur University Kolkata for his innovative classroom discussions which inspired me to do a project work in aerospace field. I am also thankful to Prof. Smita Sadhu, Prof. Gourhari Das, Dr. Madhubanti Maitra and Dr. Rajit Kumar Barai, Jadavpur University Kolkata for giving me a good theoretical and practical knowledge of various control subjects that helped me throughout the thesis work. Now, I would like to thank my IIT Patna friends, specially Abhinoy Kumar Singh, Rahul Radhakrishnan, Sumit Kumar, Suman Kumar Dey and Ajay Kumar Yadav who helped and motivated me throughout my stay in IIT Patna. Finally, I would like to thank my parents and friends for their unconditional support and love. Nikhil Kumar Singh Jadavpur University, Kolkata
Date:
i
ABSTRACT In this thesis work, ballistic target tracking problem has been considered as a subject of study. Position and velocity of a ballistic missile or any freely falling extra terrestrial object are required to be known precisely in order to destroy before it hits the ground. However, estimation of position and velocity of a ballistic target on re-entry becomes challenging due to (i) noise in sensor measurement (ii) nonlinear target and measurement model, (iii) very less engagement time. During the thesis work, literature about different existing nonlinear filters and various process and measurement model of a ballistic target have been studied in detail. Existing nonlinear filters with deterministic sample points, namely the extended Kalman filter, unscented Kalman filter, ensemble Kalman filter, cubature Kalman filter, cubature quadrature Kalman filter, Gauss-Hermite filter and sparse-grid Gauss-Hermite filter have been applied to single and two dimensional ballistic target tracking (during re-entry) problem. The performance of all the above mentioned filters has been compared in terms of (i) estimation accuracy, (ii) computational time, and (iii) missed distance. It has been found that for the single dimensional problem, the ensemble Kalman filter provides the lowest missed distance. However, the computational burden of that filter is highest and may not be suitable for on-board implementation. The Gauss-Hermite filter provides the second lowest missed distance with reasonably low computational cost. For the two dimensional problem, all filters have same estimation accuracy but cubature Kalman filter has lowest computational time. The study carried out in this thesis would help the designers to choose appropriate nonlinear filter which satisfies the mission objective within the allotted computational budget. KEYWORDS:
Target tracking; Ballistic target on re-entry; Cubature Kalman filter ; Cubature quadrature Kalman filter; Gauss-Hermite filter; Sparsegrid Gauss-Hermite filter; Ensemble Kalman filter; Unscented Kalman filter.
ii
Contents ACKNOWLEDGEMENTS
i
ABSTRACT
ii
LIST OF TABLES
vi
LIST OF FIGURES
x
ABBREVIATIONS
xi
NOMENCLATURE
xiii
1
2
INTRODUCTION
1
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.4
Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.5
Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . .
4
LITERATURE SURVEY
6
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Nonlinear Filtering Algorithm . . . . . . . . . . . . . . . . . . . .
6
2.2.1
Extended kalman Filter . . . . . . . . . . . . . . . . . . . .
7
2.2.2
Unscented Kalman Filter . . . . . . . . . . . . . . . . . . .
8
2.2.3
Ensemble Kalman Filter . . . . . . . . . . . . . . . . . . .
10
2.2.4
Cubature Kalman Filter . . . . . . . . . . . . . . . . . . . .
11
2.2.5
Cubature Quadrature Kalman Filter . . . . . . . . . . . . .
13
2.2.6
Gauss-Hermite Filter . . . . . . . . . . . . . . . . . . . . .
14
2.2.7
Sparse-grid Gauss-Hermite Filter . . . . . . . . . . . . . .
15
2.3
Ballistic Target Tracking Problem . . . . . . . . . . . . . . . . . .
16
2.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
iii
3
NONLINEAR FILTERING ALGORITHM
19
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2
Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3
Unscented Kalman Filter . . . . . . . . . . . . . . . . . . . . . . .
24
3.3.1
Unscented Transformation . . . . . . . . . . . . . . . . . .
24
Ensemble Kalman Filter . . . . . . . . . . . . . . . . . . . . . . .
26
3.4.1
Meaning of ensemble in terms of EnKF . . . . . . . . . . .
27
3.4.2
Ensemble Kalman Filter Algorithm . . . . . . . . . . . . .
28
Cubature Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . .
30
3.5.1
Cubature Rule . . . . . . . . . . . . . . . . . . . . . . . .
31
3.5.2
Generation of Cubature Points and Weights . . . . . . . . .
32
Cubature Quadrature Kalman Filter . . . . . . . . . . . . . . . . .
32
3.6.1
Cubature Quadrature Rule . . . . . . . . . . . . . . . . . .
33
3.6.2
Generation of Cubature Quadrature Points and Weights . . .
36
Gauss-Hermite Filter . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.7.1
Gauss-Hermite Quadrature Rule . . . . . . . . . . . . . . .
37
3.7.2
Generation of GHF Points and Weights . . . . . . . . . . .
41
Sparse-grid Gauss-Hermite Filter . . . . . . . . . . . . . . . . . . .
44
3.8.1
Sparse-grid Quadrature Rule . . . . . . . . . . . . . . . . .
44
3.8.2
Generation of SGHF Points and Weights . . . . . . . . . .
46
Filtering Algorithm for UKF, CKF, CQKF, GHF and SGHF . . . . .
48
3.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
BALLISTIC TARGET TRACKING ON RE-ENTRY
52
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4.2
Strategic Defense Against Ballistic Object . . . . . . . . . . . . . .
53
4.2.1
Ballistic Missile . . . . . . . . . . . . . . . . . . . . . . .
53
4.2.2
Ballistic Missile Defense System . . . . . . . . . . . . . . .
56
1-Dimensional Problem on Re-entry Ballistic Target Tracking . . .
58
4.3.1
Problem Formulation . . . . . . . . . . . . . . . . . . . . .
59
4.3.2
Initialization of Filter . . . . . . . . . . . . . . . . . . . . .
62
4.3.3
Simulation Results . . . . . . . . . . . . . . . . . . . . . .
66
2-Dimensional Problem on Re-entry Ballistic Target Tracking . . .
76
3.4
3.5
3.6
3.7
3.8
3.9
4
4.3
4.4
iv
4.5 5
4.4.1
Problem Formulation . . . . . . . . . . . . . . . . . . . . .
77
4.4.2
Initialization of Filter . . . . . . . . . . . . . . . . . . . . .
82
4.4.3
Simulation Results . . . . . . . . . . . . . . . . . . . . . .
83
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
CONCLUSION
97
5.1
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
5.2
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
5.3
Scope of Future Work . . . . . . . . . . . . . . . . . . . . . . . . .
98
Bibliography
100
List of Tables 3.1
CQKF points generation with its order of accuracy level and dimension
35
3.2
GHF points generation with its accuracy level and dimension . . . .
44
3.3
SGHF points generation with its level and dimension . . . . . . . .
48
3.4
Number of points for UKF, CKF, CQKF, GHF and SGHF . . . . . .
49
4.1
Normalized computational time calculation of EnKF, UKF, CKF, CQKF, GHF and SGHF with various accuracy levels for 100 MC runs. . .
75
Average RMSE error and miss distance for RMSE position error calculation of EnKF, UKF, CKF, CQKF, GHF and SGHF with various accuracy levels for 100 MC runs. . . . . . . . . . . . . . . . . . .
76
Normalized computational time calculation of UKF, CKF, CQKF, GHF and SGHF with various accuracy levels for 100 MC runs. . . . . . .
94
Average RMSE error calculation of UKF, CKF, CQKF, GHF and SGHF with various accuracy levels for 100 MC runs. . . . . . . . . . . .
95
Miss distance calculation of UKF, CKF, CQKF, GHF and SGHF with various accuracy levels for 100 MC runs. . . . . . . . . . . . . . .
96
4.2
4.3 4.4 4.5
vi
List of Figures 3.1
Single dimensional third-order cubature quadrature points and its weight.
3.2
2-dimensional third-order cubature quadrature points and its weight.
35
3.3
Single dimensional 3-point Gauss-Hermite quadrature and its weight.
42
3.4
2-dimensional 3-point Gauss-Hermite quadrature and its weight. . .
43
3.5
Sparse-grid points for dimension 2 and accuracy level 3 . . . . . . .
47
3.6
1-dimensional level-2 sparse Gauss-Hermite quadrature point and its weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
2-dimensional level-2 sparse Gauss-Hermite quadrature point and its weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.1
Ballistic missile trajectory . . . . . . . . . . . . . . . . . . . . . .
54
4.2
Block diagram of target tracking, control and guidance . . . . . . .
58
4.3
Ballistic target tracking scenario using stationary ground radar . . .
59
4.4
Typical target trajectory- position vs time . . . . . . . . . . . . . .
61
4.5
Typical target trajectory- velocity vs time . . . . . . . . . . . . . .
61
4.6
Typical target trajectory- acceleration vs time . . . . . . . . . . . .
62
4.7
Mean error of position out of 100 MC runs obtained from EnKF with different ensemble size. . . . . . . . . . . . . . . . . . . . . . . . .
63
Mean error of velocity out of 100 MC runs obtained from EnKF with different ensemble size. . . . . . . . . . . . . . . . . . . . . . . . .
64
Mean error of ballistic coefficient out of 100 MC runs obtained from EnKF with different ensemble size. . . . . . . . . . . . . . . . . . .
64
4.10 Standard deviation of position error out of 100 MC runs obtained from EnKF with different ensemble size. . . . . . . . . . . . . . . . . . .
64
4.11 Standard deviation of velocity error out of 100 MC runs obtained from EnKF with different ensemble size. . . . . . . . . . . . . . . . . . .
65
4.12 Standard deviation of ballistic coefficient error out of 100 MC runs obtained from EnKF with different ensemble size. . . . . . . . . . . .
65
4.13 Mean error of position out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . . . . . . . . . . .
67
4.14 Mean error of velocity out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . . . . . . . . . . .
68
3.7
4.8 4.9
vii
35
4.15 Mean error of ballistic coefficient out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . . . .
68
4.16 Standard deviation of position error out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . . . .
68
4.17 Standard deviation of velocity error out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . . . .
69
4.18 Standard deviation of ballistic coefficient error out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . .
69
4.19 Mean error of position out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . . . . . . .
69
4.20 Mean error of velocity out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . . . . . . .
70
4.21 Mean error of ballistic coefficient out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . .
70
4.22 Standard deviation of position error out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . .
70
4.23 Standard deviation of velocity error out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . .
71
4.24 Standard deviation of position error out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . .
71
4.25 Mean error of position out of 100 MC runs obtained from EnKF-50, UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . . . . .
72
4.26 Mean error of velocity out of 100 MC runs obtained from EnKF-50, UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . . . . .
72
4.27 Mean error of ballistic coefficient out of 100 MC runs obtained from EnKF-50, UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . .
73
4.28 Standard deviation of position error out of 100 MC runs obtained from EnKF-50, UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . .
73
4.29 Standard deviation of velocity error out of 100 MC runs obtained from EnKF-50, UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . .
74
4.30 Standard deviation of ballistic coefficient error out of 100 MC runs obtained from EnKF-50, UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. .
74
4.31 Target tracking coordinate reference system . . . . . . . . . . . . .
77
4.32 Ballistic target trajectory. . . . . . . . . . . . . . . . . . . . . . . .
78
4.33 Velocity of target vs time for ballistic trajectory. . . . . . . . . . . .
79
4.34 Acceleration of target vs time for ballistic trajectory. . . . . . . . .
80
4.35 Mean error of position X out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . . . . . . . . . . .
85
viii
4.36 Mean error of velocity Vx out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . . . . . . . .
85
4.37 Mean error of position Y out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . . . . . . . . . . .
85
4.38 Mean error of velocity Vy out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . . . . . . . .
86
4.39 Standard deviation of position X error out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . .
86
4.40 Standard deviation of velocity Vx error out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . .
86
4.41 Standard deviation of position Y error out of 100 MC runs obtained from CQKF with different order of accuracy level. . . . . . . . . . .
87
4.42 Standard deviation of velocity Vy error out of 100 MC runs obtained from CQKF different order of accuracy level. . . . . . . . . . . . .
87
4.43 Mean error of position X out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . . . . . . .
87
4.44 Mean error of velocity Vx out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . . . . . . .
88
4.45 Mean error of position Y out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . . . . . . .
88
4.46 Mean error of velocity Vy out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . . . . . . . . . .
88
4.47 Standard deviation of position X error out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . .
89
4.48 Standard deviation of velocity Vx error out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . .
89
4.49 Standard deviation of position Y error out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . .
89
4.50 Standard deviation of velocity Vy error out of 100 MC runs obtained from GHF and SGHF with different accuracy level. . . . . . . . . .
90
4.51 Mean error of position X out of 100 MC runs obtained from UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . . . . . . . . . . .
90
4.52 Mean error of velocity Vx out of 100 MC runs obtained from UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . . . . . . . .
91
4.53 Mean error of position Y out of 100 MC runs obtained from UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . . . . . . . . . . .
91
4.54 Mean error of velocity Vy out of 100 MC runs obtained from UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . . . . . . . .
92
ix
4.55 Standard deviation of position X error out of 100 MC runs obtained from UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . .
92
4.56 Standard deviation of velocity Vx error out of 100 MC runs obtained from UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . .
93
4.57 Standard deviation of position Y error out of 100 MC runs obtained from UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . .
93
4.58 Standard deviation of velocity Vy error out of 100 MC runs obtained from UKF, CKF, CQKF-2, GHF-3 and SGHF-L2. . . . . . . . . . .
94
x
ABBREVIATIONS
AAD
Advanced Air Defense
ACQPF
Adaptive Cubature Quadrature Particle Filter
BTT
Bank to Turn
CKF
Cubature Kalman Filter
CLT
Central Limit Theorem
CQKF
Cubature Quadrature Kalman Filter
CQPF
Cubature Quadrature Particle Filter
CRLB
Cramer-Rao Lower Bound
EKF
Extended Kalman Filter
EnKF
Ensemble Kalman Filter
EnOI
Ensemble Optimal Interpolation
EnUPF
Ensemble Unscented Particle Filter
GHF
Gauss-Hermite Filter
gPC
Generalized Polynomial Chaos
GS-QKF
Gaussian Sum Quadrature Kalman Filter
GSCKF
Gaussian Sum Cubature Kalman Filter
ICBM
Intercontinental Ballistic Missile
IEnKF
Iterative Ensemble Kalman Filter
IID
Independent and Identically Distributed
IITP
Indian Institute of Technology Patna
IMM
Interacting Multiple Model
IRBM
Intermediate-range Ballistic Missile
JU
Jadavpur University
MC
Monte Carlo
MRBM
Medium-range Ballistic Missile
MSE
Mean Square Error
PAD
Prithvi Air Defense
PDE
Partial Differential Equation xi
pdf
Probability Density Function
PF
Particle Filter
QKF
Quadrature Kalman Filter
RMSE
Root Mean Square Error
RSCQKF
Risk Based Cubature Quadrature Kalman Filter
SGHF
Sparse-grid Gauss-Hermite Filter
SLBM
Submarine Launched Ballistic Missile
SLR
Statistical Linear Regression
SR-CQKF
Square Root Version of Cubature Quadrature Kalman Filter
SR-QKF
Square Root Version of Quadrature Kalman Filter
SRBM
Short-range Ballistic Missile
TBM
Tactical Ballistic Missile
UKF
Unscented Kalman Filter
UPF
Unscented Particle Filter
UT
Unscented transform
xii
NOMENCLATURE n
State dimension of the process model
n0
Order of accuracy of CQKF
xk
State vector of the system
yk
Measurement vector of the system
wk
Process noise vector
ηk
Measurement noise vector
f (.)
Process model function of the system
h(.)
Measurement model function of the system
Qk
Process noise covariance matrix
Rk
Measurement noise covariance matrix
h
Position of target in ordinate direction
ℵ
Normal distribution
R
Real number
E
Expectation operator
dk
Position in the direction of abscissa coordinate
d˙k
Velocity in the direction of abscissa coordinate
hk
Position in the direction of ordinate coordinate
h˙ k
Velocity in the direction of ordinate coordinate
g
acceleration due to gravity
β
Ballistic coefficient of the target
ρ
air density of atmosphere
φk
Transition matrix of state
T
denotes the transpose of a matrix
τ
Sampling time interval
N
Number of sample considered
v
Velocity of the target trajectory
γ
Angle between the abscissa coordinate and velocity of the target
q, q1 , q2
Process noise parameters
r
range of the target from ground radar
ε
Elevation angle between abscissa coordinate and range of the target trajectory
Γ
Gamma function xiii
βL , βU
Lower and upper limit of ballistic coefficient
λ1 , λ2
Shape parameter of ballistic coefficient
c11 , c22 , c1 , c2 Constant parameters H
Measurement matrix
σd2 , σh2 , σdh
Variance of measurement error in Cartesian coordinates (abscissa and ordinate) and their cross covariance respectively
N
Natural number
x, y, z
Variables
δ(i, j)
Unit impulse function
p(x/y)
Marginal posterior probability
p(x, z)
Joint posterior probability
p(xk |y1:k−1 ) Prior pdf of state vector p(xk |y1:k )
Posterior pdf of state vector
Xk
Stake of ensemble state vector xk
Yk
Stake of ensemble measurement vector yk
Dk
Stake of ensemble state and measurement vector
ˆ k|k−1 X
Prior stake of ensemble state vector
Yˆk|k−1
Prior stake of ensemble measurement vector
ˆ k|k−1 D
Prior stake of ensemble state and measurement vector
ˆ k|k X
Posterior stake of ensemble state vector
Yˆk|k
Posterior stake of ensemble measurement vector
ˆ k|k D
Posterior stake of ensemble state and measurement vector
κ
Scaling parameter
ψ
2n + 1 sigma points
Px,k , Py,k
State covariance matrix and measurement covariance matrix
Pxy,k
state and measurement covariance matrix
ξi , wi
Points and weights of the filters
g(x)
Arbitrary function
Kk
Kalman gain matrix
xˆk|k−1
Prior state
xˆk|k
Posterior state
yˆk|k−1
Prior measurement
yˆk|k
Posterior measurement
xiv
W (x)
Weighted function
Pyy,k|k−1
Prior measurement covariance matrix
Pxy,k|k−1
Prior state and measurement covariance matrix
µ
Mean Gaussian distribution
Σ
Covariance matrix
Fgravity
Gravitational force
Fdrag
Atmospheric force
Cd
Drag coefficient
vT
Terminal velocity
ad
Acceleration due to drag
MT
Target mass
A
Frontal area of the target
xv
Chapter 1
INTRODUCTION
1.1
Background
A ballistic missile is a rocket which follows a predetermined trajectory and deliver warheads to a predetermined location. This missile is guided and powered initial phase of the flight and rest of the flight is unguided and unpowered. Based on the range, ballistic missiles have been classified in different categories; for example, tactical ballistic missile (TBM), short-range ballistic missile (SRBM), medium-range ballistic missile (MRBM), intermediate-range ballistic missile (IRBM) and intercontinental ballistic missile (ICBM) etc. TBM is a ballistic missile which is used in short-range battlefields. The range of TBM is less than 300 km. SRBM and MRBM can destroy the target in the range of 300 − 1000 km and 1000 − 2500 km respectively. IRBM is intermediate-range ballistic missile having range in between 3500 km to 5500 km. ICBM represents intercontinental ballistic missile having range more than 5500 km. Although many countries have short or medium range ballistic missile in their arsenal, few countries like United States of America, Russia, China, France, United Kingdom, Israel, India etc, developed all types of ballistic missiles, including intercontinental one. Till date, only eight countries have developed ballistic missile capable of carrying nuclear warheads. They are United States of America, Russia, China, France, United Kingdom, India, Pakistan and North Korea. India’s indigenous ballistic missile is named as Agni. Based on range, we have Agni-I to Agni-VI. Agni missiles are capable of delivering nuclear warheads to more than 5500 km distance from launching point. Agni-I is a medium range ballistic missile. It is first tested in 1989 from chandipur, It’s range is 700 km to 1250 km. Agni-II (first tested in 1999), is an intermediate range ballistic missile, whose range is from 2000 km to 3000 km. Agni-III and Agni-IV are intermediate ballistic missile, whose range is from 3000 km to 5000 km and 3000 km to 4000 km respectively. Agni-V is an intercontinental ballistic missile, first tested in 2012, can hit the target in a range of
5000 km to 8000 km. These missiles are already in operational phase. Now, India, is working to develop Agni-VI, which will range from 8000 km to 10000 km. Missile defense system used to shield a country or cities from incoming ballistic missile or satellite-debris. Now-a-days, many countries such as United States of America, Russia, France, India, Israel etc. are using ballistic missile defense system to protect their country against enemy’s ballistic target and celestial objects like a meteor or remaining of the satellite. Anti-ballistic missile is a surface to air missile, whose purposes is to destroy the incoming ballistic target. Missile defense system works through detection, tracking and interception of the incoming ballistic target. High-tech RADAR, nonlinear filter and anti-ballistic missile are required for detection, tracking and interception of ballistic target respectively. Initially, RADAR is used to detect the target, and interceptor is fired. Based on the measured value from RADAR, path command is provided to the interceptor. Seeker, located on the interceptor, starts searching for target. When seeker’s S/N ratio is above some threshold, seeker declares that the target is found. Ground RADAR link is terminated. Seeker collects measurement, estimator algorithm estimates the target position, guidance algorithm generates a specified path, and autopilot used controlling surface to follow the path. Ultimately, the interceptor hits the target or missed the target. On-board computer is used to estimate target trajectory, generates guidance command and implement autopilot. Currently, India, uses of antiballistic missile such as Prithvi air defense (PAD) missile for high altitude interception, advanced air defense (AAD) for lower altitude interception etc. The chance of successful interception of target improves, if exact location of the target is known at each time. RADAR measurement data is not sufficient to know target location precisely because the measurement is corrupted with noise. Estimator or filter is required to wipe out the noise and know the target location. If estimator does not provide the location of the target, anti-ballistic missile could not destroy it successfully.
1.2
Motivation
So from the above discussion, it is clear that destroying ballistic target on air is very important. For hitting a ballistic target, we need to know position of the ballistic target at every instant of time. Based on measured value, estimator is used for estimating 2
path of the target position and providing this estimated path to interceptor. The difference between the measured value and estimated value of position are calculated. This difference clarifies that the position error between the target and interceptor. For successful interception of the target, the final error should be within the killing range of the warhead. Tracking a ballistic target during re-entry becomes challenging because nonlinearites of the target is high. Nonlinear filters such as unscented Kalman filter (UKF), ensemble Kalman filter (EnKF), cubature Kalman filter (CKF), cubature quadrature Kalman filter (CQKF), Gauss-Hermite filter (GHF) and sparse-grid Gauss-Hermite filter (SGHF) etc. are required for target’s state estimation. Nonlinear estimator provides the estimated path to interceptor missile for successful interception. In a target tracking problem, a single filter cannot provide the best result for all tracking scenario. This has motivated the author to test the performance of the nonlinear estimators in terms of estimation accuracy, computational time and missed distance. The above named filters use the deterministic sampling point approach to approximate the posterior and prior probability density function of states. The other popular nonlinear filters such as particle filter, grid filter are kept out of scope of this work because they used probabilistic sampling points and computational time is very high which restrict their use on-board.
1.3
Thesis Objective
Objective of the thesis is to study the ballistic target tracking on re-entry problem with the help of available nonlinear filters. To study, we choose the nonlinear filters with deterministic sampling points, namely, UKF, CKF, CQKF, GHF, SGHF etc. We exclude the PF and grid based filters as they take very large time and not suitable for on-board applications. However, we include EnKF, because it is efficient than particle filter and it’s computational time is not as high as particle filter and grid based filter. The objective of this work is to study and compare the Gaussian filters. The study carried out in this thesis would help the designers to choose appropriate nonlinear filter which satisfies the mission objective within the allotted computational budget.
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1.4
Contribution
Literature survey on different type of nonlinear filters such as extended Kalman filter (EKF), unscented Kalman filter (UKF), ensemble Kalman filter (EnKF), cubature Kalman filter (CKF), cubature quadrature Kalman filter (CQKF), Gauss-Hermite filter (GHF), sparse-grid Gauss-Hermite filter (SGHF) etc. and ballistic target have been carried out. The nonlinear filters such as EnKF, UKF, CKF, CQKF, GHF and SGHF with different accuracy levels are applied in 1D and 2D ballistic target tracking problem on re-entry. The performance of all the above mentioned filters has been compared in terms of (i) estimation accuracy, (ii) computational time, and (iii) missed distance. It has been found that for the single dimensional problem, the ensemble Kalman filter provides the lowest missed distance. However, the computational burden of that filter is highest and may not be suitable for on-board implementation. The Gauss-Hermite filter provides the second lowest missed distance with reasonably low computational cost. For the two dimensional problem, all filters have same estimation accuracy but cubature Kalman filter has lowest computational time.
1.5
Organization of Thesis
The organization of the whole thesis consists of five chapters including this present one. The first chapter presents with the introduction to the thesis. This chapter provide motivation behind the selection of research work, thesis objective and the contribution of the author through this research. The second chapter of this thesis is the literature survey. This chapter presents fully dedicated to the previous research made by various personnel in the past and they shows that how it is related to our thesis work. The different works along with the time line, the authors and their contribution is mentioned. The area on which our literature review is presented as follows: nonlinear filtering algorithm and ballistic target tracking problem. In nonlinear filtering algorithm, we present literature survey on various nonlinear filters such as extended Kalman filter, unscented Kalman filter, ensemble Kalman filter, cubature Kalman filter, cubature quadrature Kalman filter, Gauss-Hermite filter and 4
sparse-grid Gauss-Hermite filter. The third chapter of this thesis is nonlinear filtering algorithm. The main focus of this thesis to deal with powerful nonlinear filters, where we study Bayesian approach and algorithm of all nonlinear filters such as UKF, EnKF, cubature and quadrature based filters. The fourth chapter of this thesis is case studies (ballistic target on re-entry). In this chapter, we study the properties of ballistic missile and its defense system. We apply all the above discussed nonlinear filters in two different ballistic target tracking application and comparing the simulation results according to its estimation accuracy and computational time. Finally, in chapter five we conclude our thesis work with the future scope of the work.
5
Chapter 2
LITERATURE SURVEY
2.1
Introduction
Before doing a detailed work on a subject or a topic, a lots of work is required for studying and understanding the topic. This chapter deals with the organized presentation of studying state of art. literature study to find out the gap. The sections of this chapter follows the literature survey of nonlinear filtering algorithm and ballistic target tracking problem. In nonlinear filtering algorithm section summarizes the survey of different advanced nonlinear filter such as extended Kalman filter (EKF), unscented Kalman filter (UKF), ensemble Kalman filter (EnKF), cubature Kalman filter (CKF), cubature quadrature Kalman filter (CQKF), Gauss-Hermite filter (GHF) and sparse-grid GaussHermite filter (SGHF). All the above filter follows the Bayesian approach and Gaussian probability distribution. In ballistic target tracking problem section summarize the survey of highly nonlinear dynamical model of ballistic target in re-entry phase. In this chapter two deals with the literature survey. The section of this chapter presents as follows: nonlinear filtering algorithm, ballistic target tracking problem and finally concludes this chapter. In nonlinear filtering algorithm section, we survey about various nonlinear filters such as extended Kalman filter, unscented Kalman filter, ensemble Kalman filter, cubature Kalman filter, cubature quadrature Kalman filter, GaussHermite filter, sparse-grid Gauss-Hermite filter.
2.2
Nonlinear Filtering Algorithm
Firstly, R. E. Kalman was formulated a Kalman filter in 1960, which works on linear system but actually in real life the system is practically not linear [1, 2, 3]. Kalman filter is not possible to apply in nonlinear system. So a new version of filter was developed, which name was extended Kalman filter. Extended Kalman filter was playing a great
role in target tracking problem. Later, many nonlinear filters were developed such as UKF, EnKF, CKF, CQKF, GHF, SGHF etc. All nonlinear filters are suboptimal in nature and better filter is decided on the basis of its estimation accuracy and computational time.
2.2.1
Extended kalman Filter
Extended Kalman filter (EKF) was the most famous nonlinear filter, which was the extension of the Kalman filter. This filter was created revolution in the nonlinear filtering. This filter has been used in most of the applications from development to till date. This filter works on the first order linearization of the nonlinear system, where it linearizes the current mean and covariance matrix. After linearization of the nonlinear system, directly apply the Kalman filter. In linearization process, a lots of derivation is required for calculating Jacobean matrix, which also gives the nontrivial solutions in many cases as well as accuracy of the system decreases, because neglecting higher order polynomial terms. This is the major drawback of the extended Kalman filter. After development of Kalman filter in 1960 by R. E. Kalman. There were many of them working on extension of Kalman filter. In paper [4], introduced the estimate of the linear system with EKF, where asymptotic behaviour of the EKF was checked and find out its estimate may be diverge or baised in nature. EKF was applied in frequency tracker application for nonlinear system in paper [5, 6], where frequency tracker is considered for weak narrowbands signal for slowly varying frequency. If signal to noise ratio is low then tracking is difficult, but signal to noise increases, then EKF performs accurate tracking. Hence EKF gives better result. In paper [7], convergence analysis of the observer for nonlinear deterministic discretetime systems model have been checked using EKF. Any filter is good only when it converges as fast as possible. Generally, EKF diverges with nonlinearity of the system. Checking the stability of the extended Kalman filter has been done in paper [8]. Tracking observerability critera for EKF has been checked with process and measurement covariance matrix, so take any arbitray value of process and measurement matrix for nonlinear discrete-time systems [9, 10]. In paper [11], compared the performance of several nonlinear filters have been given 7
as, EKF, UKF, particle filter (PF), batch filters and exact recursive filters. These filters faces ’the curse of dimensionality’ problem except PF. It also present that the future research in nonlinear filtering mainly goes on the area such as quasi-Monte Carlo algorithms, ideas borrowed dimension interpolation filter from physics, new mesh-free adjoint methods for solving partial differential equations (PDE) etc. To avoid, the drawback of EKF, recently developed nonlinear filters such as UKF, CKF, CQKF, GHF, SGHF etc. are used. Finally, filter performance depends on its estimation accuracy and computational time.
2.2.2
Unscented Kalman Filter
The unscented Kalman filter (UKF) is a powerful nonlinear filter, which was proposed by Julier et.al. in 1995. In this paper [12], presented a new recursive linear estimator for filtering system with nonlinear process and observation models. It works on the principle of unscented transform. In this transformation, the mean and covariance matrix can be transformed directly by the system equations to give predictions of the transformed mean and covariance. This transformation shows that the UKF is more accurate and far easier to implement than EKF. Firstly, presents the UKF filter in highly nonlinear kinematics of maneuvering vehicles application and tested its result. Another paper of the UKF is presented by Julier et.al. in 1997. In this paper [13], proposed the detailed discussion of the UKF. The performance of the UKF has been compared with Kalman filter as well as extended Kalman filter. Kalman filter gives the optimal result for linear system but it fails for nonlinear system. A new linear estimator, UKF is developed and demonstrated, so a set of discretely sampled points can be used for propagation of mean and covariance, which gives the same result as Kalman filter for linear system and also gives better accuracy than EKF for nonlinear system. This method is not restricted to assuming that the probability distribution of the noise is Gaussian. Another paper of the UKF is presented by Julier et.al. in 2000 [14]. In this paper, a new method for the transformation of mean and covariance matrix is developed in nonlinear filter. This filter gives more accuracy than EKF and Gauss second-order filter. In this paper [15], Wan and Merwe points out the drawback of the EKF and UKF, 8
which is developed by Julier et.al. in 1997. In the EKF, the state distribution is approximated by Gaussian random variable. This approximated state distribution is propagated directly through the first-order linearization of the nonlinear system. So, it introduces a large errors in the posterior mean and covariance matrix of the transformed Gaussian random variable. That clarifies, EKF filter gives a suboptimal result and also diverges sometime. The UKF filter uses the problem in deterministic sampling approach. It uses the approximation of state distribution by Gaussian random variable. A minimum set of sample points are chosen, which is completely captured the ture mean and covariance of the Gaussian random variable, and it is propagated through the nonlinear system. So it gives the exact result upto third-order of Taylor series for any nonlinearity. But, the computational complexity of the UKF is same as EKF. Another paper of the UKF is presented by Julier et.al. in 2004 [16]. In this paper, a new method was developed for the propagation of mean and covariance matrix through nonlinear transformation, which is known as unscented transform (UT). This method is overcome the difficulties arises in EKF due to linearization of nonlinear system. This method is also more accurate, easier to implement and also uses the same order of calculation as linearization due to EKF. In this paper [17], the performance of the modified UKF for nonlinear stochastic discrete-time system with linear measurement has been investigated. It is proved that the estimation error of modified UKF gives the bounded result for certain conditions. The design of noise covariance matrix shows an important role in improving the stability of the modified UKF algorithm. This filter is derived the behavior of the error in terms of mean square error (MSE) and measured the performance of the filter using CramerRao lower bound (CRLB). This filter also verified the results of the two example using Monte Carlo runs. The nonlinear model system problem was not solved by existing linear Kalman filter algorithm. In this paper [18], presented the three extensions to the Kalman filter algorithm: 1) the EKF; 2) the limiting EKF; and 3) the UKF. This new algorithm was applied in the on-line calibration of dynamic traffic assignment models and result shows that the accuracy is comparably best to the previous discussed filter. The robustness approach and varying weather conditions are also discussed. After getting the performance of UKF, It was suitable than existing EKF. The UKF 9
was applied vastly in ballistic target tracking on re-entry application. In paper [19], UKF was applied in ballistic target tracking on re-entry by Farina et.al. in 2002, where position and velocity were considered as state of the system. Also Ristic et.al. in 2003 and Srinivasan et.al. in 2006, applied the UKF in same ballistic target problem in paper [20, 21], where state of the system were position, velocity and unknown ballistic coefficient.
2.2.3
Ensemble Kalman Filter
The ensemble Kalman filter (EnKF) is a grid based filter. It is first developed by Geir Evensen in 1994 [22]. After sixties to till nineties, most of the researchers was working on only EKF for solving nonlinear dynamical problem. But EKF works on first order linearization of nonlinear system, this linarization process is required to solve Jacobean matrix. To overcome this problem, Geir Evenson developed EnKF filter, which is used to forecast the statistical error covariance matrix using Monte Carlo runs. The EnKF is a recursive, suboptimal nonlinear filter, which uses Gaussian distribution and works on Bayesian framework. In this paper, the EnKF filter is applied in ocean dynamic models and checks the accuracy of EnKF, and got better result than EKF. The accuracy of the EnKF depends on the size of the ensemble member. For a high dimensional problem, EnKF performs well but EKF fails because a lot of derivatives exist. Another paper of the EnKF is presented by Geir Evensen in 2003 [23]. In this paper, discussed the theoretical formulation as well as practical implementation of the EnKF upto 2002. This paper shows that the EnKF filter is applied in many applications such as satellite observations, barotropic ocean model, atmospheric transport chemistry model, marine ecosystem model etc. This paper also reviews the idea of the EnKF discussed in different applications and success of the EnKF. This paper is discussed to use of EnKF with nonlinear measurements. This paper is implemented a new algorithm which name is ensemble optimal interpolation (EnOI). The EnOI gives better result than EnKF but computational cost of EnOI is high. The square root version of the EnKF was presented by Geir Evensen in 2004 [24]. The EnKF faces measurement perturbation which introduces sampling errors. This problem can be overcome with square-root version of the EnKF and accuracy of EnKF filter is also maintained.
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In this paper [25], discussed the problem occurred in the EnKF due to randomly generated ensemble member, which create sampling errors. To overcome this problem a different algorithm is used for generation of ensembles i.e generalized polynomial chaos (gPC) expansion. In the new method, firstly solve the stochastic system state equations via gPC to gain efficiency and then sampling the gPC approximation of the stochastic solution with an arbitrary large number of samples. So the sampling errors reduced drastically. The new EnKF filter estimation accuracy increases as well as computation cost is also decreases compared to existing EnKF. The multiple delayed version of EnKF has been developed in paper [26]. In this paper, the ensemble measurement is generated from uncorrelated sensors of the system. It does not use Monte Carlo method for generating ensemble members. This method is also applied in bank to turn (BTT) missile autopilot model. A new filter has been developed, which name is ensemble unscented particle filter (EnUPF) in [27]. It is also a nonlinear suboptimal filter. The EnUPF uses ensembles which is generated from EnKF and applied in the unscented particle filter (UPF). The EnUPF accuracy is same as UPF and computational cost is less compared to UPF. In this paper [28], a new iterative version of filter ia developed, which name is iterative ensemble Kalman filter (IEnKF). The EnKF filter is used, where state model is nonlinear but measurement model is linear. To overcome the problem of EnKF filter, the IEnKF filter is used to develop, where state model as well as measurement model are nonlinear. Filters such as UKF, CKF, CQKF, GHF and SGHF, where generation of points and weights are depended on the dimension of the system but the EnKF ensemble size does not depend on the dimension of the system. The EnKF filter accuracy increases with its ensemble size. First time, the EnKF filter is applied in ballistic target tracking problem in [29]. The EnKF performs better result than EKF for more than 10 ensemble size.
2.2.4
Cubature Kalman Filter
The cubature Kalman filter (CKF) is a cubature based nonlinear filter which is first introduced by Arasaratnam et.al. in 2009 [30]. This filter works on the spherical-radial cubature rule. The multivariate moment integrals are numerically computed in the non-
11
linear Bayesian framework. It was derived from third-degree spherical radical cubature rule that is provided a set of cubature points scaling linearly with the dimension of state vector. The CKF is also provided a symmetric solution for high dimensional nonlinear filtering problem. In this paper, also provided the square version of CKF for its improved numerical stability. In this paper, the CKF filter is tested experimentally with two different problem. In the first problem, the proposed cubature rule is introduced to compute the second-order statistics of a nonlinearly transformed Gaussian random variable. The second problem uses the CKF in maneuvering target tracking problem. In both problem, the CKF performance results is better than other existing conventional nonlinear filters. Another paper of the CKF is presented by Arasaratnam et.al. in 2011 [31]. The CKF is a derivative free approximated nonlinear Bayesian filter built under the Gaussian assumption. In this paper, extended the theory of CKF to nonlinear smoothing problem. Fixed-interval cubature Kalman smoother is used to propagate the square-root error covariances. Fixed- interval cubature Kalman smoother gives better result, so it is applied to track ballistic target on re-entry. In this paper [32], proposed the performance of CKF, which is better than UKF. But 3rd-degree cubature rule fails in many target tracking problem, so higher-degree cubature rule is presented, which accuracy is higher than CKF. The higher-degree CKF performance is grater than EKF, UKF, CKF but just close to GHF with less computational time. After the development of the CKF filter, It is applied mostly on the target tracking problem because it gives better accuracy as well as less computational time than existing conventional nonlinear filter. In paper [33], the CKF filter is applied in ballistic target tracking problem and comparing results with existing EKF and UKF, and find out that CKF result is better than EKF and UKF. The CKF is also applied in bearing-only tracking in [34], where a new different algorithm is presented, which is Gaussian-sum cubature Kalman filter (GSCKF). In bearing only tracking problem has been estimated with several filters such as EKF, PF, UKF, CKF, GSCKF etc and comparing their estimation accuracy and relative computational time to all filters, where finally, we get GSCKF result is better than all of them. In this paper [35], a new nonlinear Gaussian filter is proposed with name 5th-degree 12
cubature Kalman filter rather than existing 3rd-degree CKF for a higher filter accuracy, acceptable computational complexity and stronger numerical stability. The new 5thdegree CKF is also uses 5th degree spherical-radial cubature rule for points and weights calculation. The new CKF is also suboptimal for higher dimensional nonlinear system but higher accuracy than 3rd-degree CKF and also less computational time than GHF. Finally, we got that the CKF estimation accuracy as well as computational time is better than existing conventional nonlinear filters EKF and UKF. The CKF results are closer to GHF but GHF computational time is high. Also GHF faces the ’curse of dimensionality’ but CKF is not. The CKF works easily in high dimensional problem.
2.2.5
Cubature Quadrature Kalman Filter
The cubature quadrature Kalman filter (CQKF) is a cubature based nonlinear filter which is first introduced by Bhaumik et.al. in 2013 [36]. This filter works on the cubature quadrature rule. This is the improved version of CKF. The CQKF works on spherical-radial quadrature rule as well as Gauss-Lagurre quadrature rule for nonlinear state estimation. This CQKF filter had proposed before in the IEEE conference in 2011 [37]. The CKF filter uses third-degree spherical-radial cubature rule for calculating the intractable integrals. The CQKF is used to solve the intractable integrals using multidimensional integral spherical-radial quadrature rule as well as single dimensional GaussLaguerre quadrature rule. The CQKF filter uses the third-degree spherical quadrature rule. The CQKF filter uses Gaussian distribution in Bayesian framework. The performance of CKF filter coincides with CQKF when the evaluation of Gauss-Laguerre quadrature for single points. In CKF, the generation of points and weights depends only on the dimension of the system state model. But in CQKF, the generation of points and weights depends on the dimension of the system state as well as order of the cubature quadrature rule. Total 2nn0 points are generated in CQKF, where n is the dimension and n0 is the order of the Gauss-Lagurre quadrature rule. The CQKF filter estimation accuracy depends on the order of the cubature quadrature rule. Generally, the accuracy increases with its order of accuracy level. The CQKF filter gives better estimation accuracy than EKF, and it is derivative free. The CQKF and
13
GHF filter accuracy depends on the order of the cubature quadrature and Gauss-Hermite quadrature rule. The points and weights of GHF, are increasing exponentially with its dimension, so it faces the ’the curse of dimensionality’ problem, but in CQKF, points and weights are increasing linearly, so CQKF is free from ’the curse of dimensionality’ problem. In this paper [38], a new nonlinear risk based cubature quadrature Kalman filter (RSCQKF) is proposed. It is the improved version of CQKF. The new filter RSCQKF has been used in ballistic target tracking problem and comparing the result with existing CQKF. The result shows that the RSCQKF performs better than CQKF. In this paper [39], presented a square-root version of cubature quadrature Kalman filter (SR-CQKF). It is the improvement of existing CQKF. The CQKF faces Cholesky error in some problems, to overcome this problem, SR-CQKF is used, which is free from Cholesky error. In this paper, compared the results of SR-CQKF with existing EKF and square-root version of cubature Kalman filter. The estimation accuracy of SR-CQKF is better than both of them. In this paper [40], a new method is proposed, whose name is cubature quadrature particle filter (CQPF). The CQPF is used the square-root cubature quadrature Kalman filter (SR-CQKF) to generate the importance proposal distribution. The CQPF is used to avoids the limitation of the particle filter. The CQPF has many drawback, it is used to enhanced the sample size of particle at every time step utilizing the enhanced intervals. To overcome this problem, a new filter is proposed whose name is adaptive cubature quadrature particle filter (ACQPF). The ACQPF uses q-method for initialization of the filter. This ACQPF filter is used in satellite attitude determination for lower earth orbit and compared the accuracy with existing EKF and PF. The estimation accuracy of the ACQPF is better than both EKF and PF.
2.2.6
Gauss-Hermite Filter
The Gauss-Hermite filter (GHF) is a quadrature based nonlinear filter which is first seen in major use by Ito et.al. in 2000 [41]. This filter works on the Gauss-Hermite quadrature rule. The intractable integrals has been approximated numerically on the basis of Gaussian and mixed Gaussian probability distribution function, so the filter gives nu-
14
merically accurate result, like optimal filter. The univariate points and weights is generated according to Gauss-Hermite quadrature rule. To convert this univariate quadrature points and weights, product rule is required. The GHF filter estimation accuracy is better than EKF without any additional cost. In this paper Ito et.al., discussed a new Gaussian sum filter which gives better result than existing Gaussian filters [41, 42]. In this paper [43], a new method is used for generalization of the quadrature weight. This method is easily find out the quadrature weight according to previous method. The performance of the new weight in example is better than the weight used in previous method. In this paper [44], a new version of quadrature Kalman filter (QKF) is developed for nonlinear system. The new quadrature Kalman filter linearizes the process and measurement model using statistical linear regression (SLR). In this paper, the Gaussian sum quadrature Kalman filter (GS-QKF) is developed. The GS-QKF gives the better result than QKF. The square root version of QKF has ben presented in [45]. In many problem, the QKF faces the Cholesky decomposition error. To overcome this error, then square root version of QKF is used i.e. SR-QKF. In higher dimensional system, ’the curse of dimensionality’ problem is occured in GHF. To overcome this problem, many new filter is developed such as cubature Kalman filter (CKF) in 2009 by Arasaratnam et.al. [30], cubature quadrature Kalman filter (CQKF) in 2013 by Bhaumik et.al. [36], sparse-grid Gauss-Hermite filter in 2012 by Jia et.al. [46] etc. The adaptive version of nonlinear Gauss-Hermite filter is proposed in [47]. This paper uses the unknown signal measurement noise statistics. The GHF filter has been used in target tracking problem such as, bearing only tracking in [48] and maneuvering target tracking in [49].
2.2.7
Sparse-grid Gauss-Hermite Filter
The sparse-grid Gauss-Hermite filter (SGHF) is a quadrature based nonlinear filter which is first introduced by Jia et.al. in 2012 [46]. This filter works on sparse-grid quadrature rule. The SGHF filter is the improvement of the GHF filter. The GHF filter faces the curse of dimensionality problem. To overcome from curse of dimensionality 15
problem, SGHF filter is used. The univariate points and weights generation are same for GHF and SGHF filter. But in multivariate case, The SGHF filter uses Smolyak’s rule for generation of points and weights. The estimation accuracy of the SGHF filter is same as GHF filter, but computational cost is less than GHF. The SGHF filter is used by Jia et.al. in spacecraft attitude estimation in [50, 51]. This paper states that the UKF filter is the subset of SGHF filter. The SGHF filter is flexible to use points and wights than UKF. In high dimensional system, SGHF is better than GHF. The numerical integration using sparse-grid theory has been solved in paper [52, 53]. These paper are introduced to solve one dimensional Gaussian quadrature to multidimensional Gauss quadrature using Smolyak’s rule. This paper [52] suggests that to use Gauss-Patterson quadrature rule for calculating the univariate quadrature points and weights. The comparison between sparse-grid quadrature rule and spherical-radial cubature rule has been introduced in paper [54, 55]. These paper shows that both the rules are better than existing EKF and UKF. Also it is better than third-degree CKF. Sphericalradial cubature rule as well as sparse-grid quadrature rule are working well in higher dimensional problem bur GHF fails due to curse of dimensionality problem. These paper introduced that the spherical-radial cubature rule is the projection of the sparsegrid quadrature rule. Also third and fifth degree of spherical-radial cubature rule is calculated from sparse-grid quadrature rule. The SGHF filter is used in target tracking problem in [49, 56].
2.3
Ballistic Target Tracking Problem
Ballistic missile target tracking is a challenge of researcher’s from a decade because nonlinearities of the target is high. In endo-atmospheric region, target is required very less time to reach on the earth surface. So few second is required to converge the estimator. Ballistic missile target tracking is a part of missile defense system. Many countries are working on the missile defense system, our country, India is one of them. Now, our Indian government is working on nuclear shield with two important cities like New Delhi and Mumbai. So these two cities are fully protected with nuclear warheads. Missile defense system has been divided into different parts from launch to impact such 16
as detection, tracking and destruction. High-tech radar is used for detection, advanced nonlinear filter is used for tracking and also high-tech anti-ballistic missile is used for destruction of the enemy target. In target tracking problem, target may be ballistic target, aircraft, satellite, ship, submarine, debris etc. Ballistic target tracking is a difficult task because the target is highly nonlinear dynamical, very less time is required for tracking and also the properties of the target is unknown. All the applied nonlinear estimator in the nonlinear target model are suboptimal in nature. So it estimates close to accurate path. But we required an optimal nonlinear filter for successful tracking. All nonlinear filters are based on approximation and also for drawing a target model. Only consider major states, but neglected few states, so accuracy of the filter decreases. Tracking of ballistic target on re-entry phase is related to the protection and safety of any country. Tracking of re-entry vehicle has been compared in 1971 by R. K. Mehra [57] for several nonlinear filters and classified that extended Kalman filter gives better result. Re-entry ballistic target tracking, where position, velocity and ballistic coefficient as state of the target has been taken in paper [20, 21, 29] and motion of the target is single dimensional. The target has been acting only two forces i.e gravity and drag. These paper has been used such as EKF, UKF, EnKF, PF etc for comparison with one another. In paper [20], shows that the UKF result is better. Paper [21], shows that the result of derivative free filter i.e central difference filter accuracy is better than EKF but close to UKF. Paper [29] shows the EnKF result is better for higher ensemble size but computational time is more. The two dimensional ballistic target tracking on re-entry problem has been shown in [19]. This paper has taken four states of the target i.e position in abscissa, velocity in abscissa, position in ordinate, velocity in ordinate in Cartesian coordinate system. In this paper [58], a new filter was developed to track the ballistic target successfully in real time coding simulation. The target has been tracked in both endo-atmospheric and exo-atmospheric region, where drag profiles are unknown. Different form of reentry ballistic target model has been derived in paper [59]. This paper is also used acceleration and jerk as a state. Tracking of ballistic target on re-entry with on board seeker has been done in paper [60, 61], where the seeker is fixed on anti-ballistic missile. The interacting multiple model (IMM) based filter for real-time tracking of tactical
17
ballistic missile is given in paper [62, 63, 64]. Paper [62] shows that the result of the IIM based filter is better than other conventional filter. When priori information about the missile is not known then other filter fails. To overcome this problem IIM based filter is selecting for tracking the ballistic target.
2.4
Conclusion
In this chapter, we survey the various nonlinear filtering paper as well as ballistic target tracking problem on reentry phase. Nonlinear filters are assumed as suboptimal, follow Gaussian distribution and work on Bayesian framework. Most of the filters are proposed, because one filter doesn’t satisfy all performance criteria for all problems. In nonlinear filtering, all filter has main purpose to increase the estimation accuracy and reduce the computational cost so the new filter gives accurate result and become an optimal filter. Ballistic target tracking problem has many target model is discussed. Some paper considered acceleration and jerks are used as a state. Some paper discussed about the interacting multiple model which is better in tracking of ballistic target.
18
Chapter 3 NONLINEAR FILTERING ALGORITHM
3.1
Introduction
Practically, most of the system are nonlinear, but dealing with nonlinear problem is difficult. Solving a linear system problem is easy and it gives optimal result. Solving a nonlinear problem is difficult. Nonlinear problems provide suboptimal result, so there are many nonlinear filtering algorithm are proposed to get an optimal result. The performance of nonlinear filters have been compared in terms of estimation accuracy and computational time. One filter estimation accuracy is greater than other. Most of the filter uses the noise probability distribution is Gaussian, and works on Bayesian framework. In this thesis, Gaussian noise distribution is used in Bayesian framework because as central limit theorem (CLT) states that if different pdf is mixed an independent and identically distributed (IID) random variables of the noise, the mixture will be of Gaussian in nature with mean zero and variance one. All nonlinear filters are discussed in this thesis were developed from 1960 to 2013. Kalman filter was developed in 1960 by R. E. Kalman for linear filtering estimation, which is an optimal filter. After Kalman filter, a nonlinear filter is developed, which name is extended Kalman filter (EKF). EKF uses first-order linearization of nonlinear system, and then uses Kalman filter algorithm. This linearization, EKF uses Taylor series. In higher dimensional system, EKF is not useful because of a lots of derivatives and also large errors occur due to using only first-order linearization of polynomial. The filters UKF, EnKF, CKF, CQKF, GHF and SGHF give better estimation accuracy than EKF. These filters solve the problem of EKF. This chapter deals with the nonlinear filtering algorithm. The section of this chapter, as follows: Bayesian approach, unscented Kalman filter, ensemble Kalman filter, cubature Kalman filter, cubature quadrature Kalman filter, Gauss-Hermite filter, sparse-grid Gauss-Hermite filter and finally, conclude this chapter.
3.2
Bayesian Approach
Bayesian theory was first introduced by the British researcher Thomas Bayes in 1763 [65]. Bayes theory is the backbone of statistical probability. Stochastic filtering theory is first introduced in 1940, where filtering is an operation that uses the extraction of information about a quantity of interest at time t by using data measured upto and including t. Bayes theory is applied successfully in statistical decision, detection, estimation, pattern recognition and machine learning. The optimal solution is found in Bayes theorem, so the filtering and estimation problem is used in Bayesian approach. Bayes theory allows the uncertainty in the model to provides a prior knowledge and a posterior evidence of measurement in filtering estimation problem. Generally, three types of intractable problems deal with Bayesian statistics: Normalization: The posteriori p(x/y) is obtained by multiplying the prior p(x) and the likelihood p(y/x) is divided by a normalizing factor then, we get p(x/y) = R
p(y/x)p(x) p(y/x)p(x)dx X
(3.1)
Marginalization: The marginal posterior probability p(x/y) is defined as Z p(x, z|y)dz
p(x/y) =
(3.2)
Z
where p(x, z) is the joint posterior probability.The marginalization and factorization plays an important role in Bayesian interface problem. Expectation: The expectation of a function f (x) is defined as Z Ep(x|y) [f (x)] =
f (x)p(x|y)dx
(3.3)
X
According to Bayesian interference property, all the uncertainties such as states, parameters fixed or time varying but unknown, priors etc are treated as random variables. Main objective of the Bayesian interface is to use priors information quantitatively and qualitatively about conditional probability of finite measurements.
20
Let us define a nonlinear target dynamic process models as, xk+1 = f (xk ) + wk
(3.4)
and, its measurement model is defined as yk = h(xk ) + ηk .
(3.5)
Where, xk ∈ Rn and yk ∈ Rp denotes the states and measurements vector of the system respectively at any instant of time k for k = {0, 1, 2, ...}. f (.) and h(.) denotes the process model and measurement model function respectively. wk and ηk are process and measurement noise vector. Where wk ∼ ℵ(0, Qk )
(3.6)
ηk ∼ ℵ(0, Rk ).
(3.7)
and
Where, Qk and Rk are process and measurement covariance matrix. where ℵ(., .) denotes the normal distribution of specified mean and covariance matrix.Also, E[wi wjT ] = Qk δ(i, j), E[ηi ηjT ] = Rk δ(i, j),
f or all integers i and j,
(3.8)
E[wi ηjT ] = 0
where, E denotes the expectation, and δ(i, j) is the unit impulse signal, which is defined as, δ(i, j) =
1 for i = j 0 for i 6= j
In Bayesian framework, the recursive Bayesian estimator estimates the unknown posterior probability density function (pdf) recursively over time using incoming measurement and process model. Two assumptions are used in deriving the recursive Bayesian filter.
21
1. The process model of states follow the first-order Markov rule p(xk |x0:k−1 ) = p(xk |xk−1 )
(3.9)
2. The measurements are the independent of the given states. Under the Bayesian framework, prior and posterior density function are classified as, • The prior probability density function p(xk |y1:k−1 ) is obtained by Chapman-Kolmogorov equation, Z p(xk |y1:k−1 ) =
p(xk |xk−1 )p(xk−1 |y1:k−1 )dxk−1 .
(3.10)
where y1:k−1 = {y1 , y2 ....., yk−1 } is the set of measurements and p(xk |xk−1 ) is the conditional pdf of xk for known xk−1 . • The new measurements yk and the prior probability density function is used to obtain the posterior probability density function using Bayes rule, p(xk |y1:k ) = R
p(yk |xk )p(xk |y1:k−1 ) . p(yk |xk )p(xk |y1:k−1 )dxk
(3.11)
The above equation (3.10) and (3.11) works only for linear and Gaussian function and it fails in nonlinear non-Gaussian system estimation. Non-Gaussian distribution is approximated with its mean as a estimate then we get Gaussian distribution. For this assumption, the prior and posterior probability density can be written as, p(xk |y1:k−1 ) = ℵ(ˆ xk|k−1 , Pk|k−1 )
(3.12)
p(xk |y1:k ) = ℵ(ˆ xk|k , Pk|k ),
(3.13)
and
where ℵ(ˆ xk|k , Pk|k ) denotes the normal distribution with mean xˆk|k and covariance Pk|k . Under the Bayesian framework, the two steps involved in the filtering process are prediction step and update step. Prediction step: The prior estimate is the mean of the prior pdf, which is obtained from
22
prior equation, so xˆk|k−1 = E[xk |y1:k−1 ] = E[(f (xk−1 ) + wk )|y1:k−1 ]
(3.14)
= E[f (xk−1 )|y1:k−1 ] or Z f (xk−1 )p(xk−1 |y1:k−1 )dxk−1
xˆk|k−1 =
(3.15)
Z =
f (xk−1 )ℵ(xk−1 ; xˆk−1|k−1 , Pk−1|k−1 )dxk−1
and Pk|k−1 = E[(xk − xˆk|k−1 )(xk − xˆk|k−1 )T |y1:k−1 ] Z = f (xk−1 )f T (xk−1 )ℵ(xk−1 ; xˆk−1|k−1 , Pk−1|k−1 )dxk−1
(3.16)
− xˆk|k−1 xˆTk|k−1 + Qk Measurement update step: The measurement update under the approximation of the Gaussian posteriori pdf is given as, p(yk |y1:k−1 ) = ℵ(yk ; yˆk|k−1 , Pyy,k|k−1 )
(3.17)
where Z yˆk|k−1 =
Z Pyy,k|k−1 =
h(xk )ℵ(xk ; xˆk|k−1 , Pk|k−1 )dxk
(3.18)
h(xk )hT (xk )ℵ(xk ; xˆk|k−1 , Pk|k−1 )dxk (3.19)
−
T yˆk|k−1 yˆk|k−1
+ Rk
The cross covariance matrix is given by Z Pxy,k|k−1 =
T xk hT (xk )ℵ(xk ; xˆk|k−1 , Pk|k−1 )dxk − xˆk|k−1 yˆk|k−1
23
(3.20)
The new measurement yk of the posterior density function is given by p(xk |y1:k ) = ℵ(xk ; xˆk|k , Pk|k )
(3.21)
xˆk|k = xˆk|k−1 + Kk (yk − yˆk|k−1 )
(3.22)
Pk|k = Pk|k−1 − Kk Pyy,k|k−1 KkT
(3.23)
−1 Kk = Pxy,k|k−1 Pyy,k|k−1
(3.24)
where
3.3
Unscented Kalman Filter
The unscented Kalman filter (UKF) is developed by Julier et.al. in 1995 [12]. The UKF filter is suboptimal nonlinear filter which uses Gaussian distribution and works on Bayesian framework. The UKF is the improvement of the EKF. The EKF filter has two main drawbacks i.e. linearization of the model produces unstable results and the derivation of the Jacobian matrices gives nontrivial solution in most of the problems. The unscented transformation is used for propagating the mean and covariance matrix. This transformation is more accurate than EKF. The UKF filter accuracy is upto thirdorder of polynomial. There are many variants of UKF filter is proposed by Julier et.al. in 1995 [12], 1997 [13], 2004 [16]. The UKF filter is free from curse of dimensionality problem. The UKF filter depends on the states of the system. If the system state has dimension n then total number of points and weights of the unscented transformation is 2n+1. The UKF filter works only when the process model is nonlinear but measurement model is linear. The UKF filter is also applied vastly in target tracking application after its development.
3.3.1
Unscented Transformation
The UKF [15, 18] is a filtering technique which is used in dynamical state space models. It is used to approximate a probability distribution than to approximate an arbitrary 24
non-linear transformation. It uses deterministic sampling approach, called unscented transformation, to represent random variable by using a number of deterministically selected sample points called sigma points. These sigma points capture the mean and covariance of the random variable, when it is propagated through a non-linear system. It captures the posterior mean and covariance accurately upto second order of Taylor series expansion. This filter propagates the mean and covariance accurately only for Gaussian distribution. We denote an n-dimensional random variable xk , state covariance matrix Px,k , measurement vector yk , measurement covariance matrix Py,k , state and measurement covariance matrix Pxy,k , where k denote the sampling step. Now, a matrix Ψ to be generated using 2n + 1 weighted sigma points. We choose a scaling parameter κ ∈ R, p such that ( (n + κ)Px,k )i gives the ith row or column of the square root of matrix (n + κ)Px,k . This square root of the matrix can be done using cholesky decomposition. In the algorithm of UKF, a scaling parameter κ direct effect the sigma points. We use constant weight in the algorithm which is not depending with sampling step k. Unscented transformation can be represented as 1. Generation of sigma points Ψ0,k = xk for i=1 to n Ψi,k = xk + (
q (n + κ)Px,k )i
for i = n + 1 to 2n Ψi,k
q = xk − ( (n + κ)Px,k )i
2. Generation of weights w0 =
κ n+κ
for i = 1 to 2n wi =
1 2(n + κ)
The filtering algorithm of UKF filter has been shown in section 3.9.
25
(3.25)
3.4
Ensemble Kalman Filter
The ensemble Kalman filter (EnKF) is developed by Geir Evensen in 1994 [23]. It has gained popularity because of its formulation and implementation are simple. It requires no derivation and integration of any function to generate ensembles. Those system, where nonlinearity is too high, where EKF fails to give correct result because EKF gives the correct result upto second order Taylor series. So, we use EnKF in highly nonlinear system. This is the improved version of EKF. The EKF filter uses approximation for linearizing the nonlinear model. To overcome the drawback of EKF, the EnKF filter is used, which is derivative free. The EnKF is used to forecast the statistical error covariance matrix using Monte Carlo runs. The EnKF is a recursive suboptimal nonlinear filter, which is used as a Gaussian noise distribution and works on Bayesian framework. It has applied in many applications such as atmospheric and oceanographic models, weather forecasting, target tracking etc. The EnKF is proposed because Kalman filter and EKF doesn’t deal with large scale problems. This filter is used to update all probability distribution function using Bayesian rule which is Gaussian in nature. The mean and covariance matrix is updated easily, every time by Bayesian rule i.e. the benefit of EnKF. It is a recursive filter, where forecasting and updating the ensemble is again and again. If the number of state of the system increases then the EKF fails due to a lot of calculation such as partial differentiation [66]. The EnKF is applied in nonlinear filtering problem, where error statistics of the system are predicted by Monte Carlo or ensemble integration is solved by Fokker-Plank equation. This process is forecasting and updating the ensemble for various Monte Carlo runs. As the number of Monte Carlo runs are increasing then accuracy of the system is automatically increasing. The EnKF is originally developed to estimate extremely higher dimensional nonlinear dynamical atmospheric and oceanographic system model problem, where initial state uncertainty is too high. Later, It has been used in engineering field like target tracking. The EnKF is developed to implement Monte Carlo implementation of Kalman Filter. Kalman filter is an optimal filter. When EnKF is applied in linear problem then 26
it also gives optimal result. But in nonlinear problem, Kalman filter fails to give the correct result then a new filter is developed which is EKF. But EKF gives the correct result upto second order Taylor series. In EKF, we linearized the system with partial differentiation Jacobian equation. This linearization is possible only for lower dimension system. If the system dimension is too high then EKF fails to give correct result. So, a new filter is developed which is EnKF. EnKF generates the ensemble stochastically. All filters use Gaussian probability distribution function and give suboptimal result for nonlinear system. In EnKF, we assume the observational and dynamical noise are Gaussian but practically noise nature is not Gaussian. So EnKF doesn’t give correct result for non-Gaussian noise or system. For non-Gaussian distribution of noise or non-Gaussian filter, we use a new type of filter which is known as particle filter. In UKF [15, 18] and CKF filter, points and weights depend on the dimension of the filter. In quadrature based filter such as CQKF [36], GHF [44, 47, 48, 49], SGHF [46, 50, 51, 56] etc, where quadrature points and weights depend on the order of accuracy level as well as dimension of the system. All filter has point and weight calculated deterministically. But in EnKF, it is required only number of ensemble size for any dimension of the system. We choose the number of ensemble point freely. The EnKF literature shows that the 50 to 100 ensemble size is enough to solve the 100 to 1000 dimensional state of the system.
3.4.1
Meaning of ensemble in terms of EnKF
The EnKF filter uses the word ensemble, which is related to statistical probability distribution. In the filtering problem, the ensemble means a set of random variables. Initially the ensemble size is generated with the help of statistical mean and its covariance, where the size of the ensemble is defined according to the problem. Every time a different set of random variable is generated. After generation of ensemble size, to predict the ensemble points with the help of process and measurement models. After predicting the ensemble size, we update the predicted ensemble points with the help of Kalman gain matrix. This process is repeating again and again, so EnKF filter is also follows recursive property. The EnKF filter uses the Kalman filter algorithm, where process model is nonlinear and measurement model is linear.
27
The implementation of EnKF is based on following steps [29],[28]: • Generation of ensemble points • Propagation of ensemble points through process and measurement model • Update the ensemble points with the help of Kalman gain matrix.
3.4.2
Ensemble Kalman Filter Algorithm
The EnKF is applied in nonlinear system, where error statistics of the system are predicted by Monte Carlo or ensemble integration are solved by Fokker-Plank equation. This process is forecasting and updating the ensemble for various Monte Carlo runs. As the number of Monte Carlo runs increases, the accuracy automatically increases. The nonlinear process and measurement dynamic models are defined in section 3.2. In ensemble Kalman filter, we take an ensemble of size N . Where we define a stake of ensemble state Xk as, Xk =
h
x1k
x2k
x3k
....
xN k
i
(3.26)
where Xk ∈ Rn×N is an ensemble of state vector. The stake of ensemble measurement Yk as, h i Yk = yk1 yk2 yk3 .... ykN
(3.27)
where Yk ∈ Rp×N is an ensemble of measurement vector. The stake of ensemble state and measurement Dk as, 1 2 3 N Xk xk xk xk .... xk Dk = = 1 2 3 N Yk yk yk yk .... yk
(3.28)
where Yk ∈ R(n+p)×N is an ensemble of state and measurement vector, which is also known as stake of ensemble. where ykj = h(xjk )
This EnKF algorithm is defined as following steps:
28
(3.29)
Step 1: Filter initialization Initialize mean x0|0 and covariance P0|0 . Then calculate X0 stochastically with the help of x0|0 and covariance P0|0 . We calculate the measurement ensemble Y0 from (3.29) at instant k = 0. Now we define the ensemble D0 from (3.28) as, N 3 2 1 x0 x0 x0 .... x0 X0 D0 = = y01 y02 y03 .... y0N Y0
(3.30)
Step 2: Predictor / time update / forecast In this step, propagate the each ensemble member by state and measurement equation. The propagating of each ensemble member can be calculated as, ˆ k|k−1 = f (xj ) + wk−1 X k|k
(3.31)
Yˆk|k−1 = h(xjk|k )
(3.32)
ˆ k|k−1 and Yˆk|k−1 are the state and measurement prediction matrix respectively. where X Now, evaluate the prediction error covariance matrix Pk|k−1 as, Pk|k−1 = Lk|k−1 LTk|k−1
(3.33)
where, Lk|k−1
i h 1 2 N 1 ˆ ˆ ˆ =√ (Dk|k−1 − Dk|k−1 ) (Dk|k−1 − Dk|k−1 ) .... (Dk|k−1 − Dk|k−1 ) . N −1 (3.34)
ˆ k|k−1 as, Where, mean of the stake of ensemble D N 1 X j ˆ Dk|k−1 = D N j=1 k|k−1
(3.35)
Step 3: Estimation / measurement update / analysis Now, calculate Kalman gain matrix Kk as, Kk = Pk|k−1 HT (HPk|k−1 HT + Rk )−1
29
(3.36)
h i where H matrix depends on the measurement of the system is given as, H = I1×n I1×p and considered as, Yk = HDk . Now, update the stake of ensmble with the help of Kalman gain matrix as, j ˆj = D ˆj D k|k k|k−1 + Kk (yk,0 − yk|k−1 ).
(3.37)
j Where, measurement noise yk,0 ∼ ℵ(yk , Rk ) are obtained stochastically.
Step 4: EnKF estimate Now, we calculate the mean of the estimated stake of ensemble. N 1 X j ˆ D . Dk|k = N j=1 k|k
(3.38)
We calculate the estimated stake of state ensemble ˆ k|k = GD ˆ k|k . X h
(3.39)
i
Where matrix G is given as G = In×n 0n×p . In the above discussed algorithm, if process model is nonlinear and measurement model is linear then ensemble of the stake can be formed for states only. We should calculate the variance of ensemble by dividing N −1 insteads of N , because the population mean is unknown. This correction is known as Bessel’s correction.
3.5
Cubature Kalman Filter
The cubature Kalman filter (CKF) is developed by Arasaratnam et.al. in 2009 [30]. The CKF filter is nonlinear suboptimal filter which uses Gaussian distribution in Bayesian framework. It works on spherical-radial cubature rule. In this filter, the multiple dimensional intractable integral is converted into surface integral and line integral. Surface integral has been solved using spherical-radial cubature rule over a unit hyper sphere and line integral has been solved using Gauss quadrature rule. The CKF filter points and weights generation depend on the state dimension of the target. This filter is free from curse of dimensionality problem. The performance of the CKF filter is better than 30
EKF and UKF.
3.5.1
Cubature Rule
The CKF filter is a nonlinear filter, which is used as a Gaussian distribution for computing intractable integrals. Let us consider g(x) is an arbitrary function. An integral of this arbitrary function in Gaussian domain is defined as, Z
T
g(x)e−x x dx.
I(g) =
(3.40)
Rn
The variable x ∈ Rn is defined in Cartesian coordinate system. This integral has been solved numerically by transforming this integral into spherical-radial integration form and solved using third-degree spherical-radial cubature rule. First, we transform the Cartesian variable x into spherical-radial variable. Let us consider x = r1 y, where y T y = 1, so xT x = r12 . Where y is the direction vector and r1 is the radius which range is r1 ∈ [0, ∞). The equation (3.40) can be written as, ∞
Z
Z
2
g(r1 y)r1n−1 e−r1 dσ(y)dr1 ,
I(g) = 0
(3.41)
Un
where Un denotes the surface of the sphere, which is defined as Un = {y ∈ Rn |y T y = 1} and σ(.) denotes the spherical surface area element of Un . The radial integral is defined as, Z
∞
I=
2
S(r1 )r1n−1 e−r1 dr1
(3.42)
0
where S(r1 ) is defined as the spherical integral with the unit weighting function w(y) = 1 as, Z S(r1 ) =
g(r1 y)dσ(y).
(3.43)
Un
Spherical Cubature Rule The spherical integral is numerically computed by spherical cubature rule. The thirddegree spherical cubature rule is applied for computing this integral as, Z g(r1 y)dσ(y) ≈ w Un
2n X i=1
31
g[u]i .
(3.44)
Where [u]i denotes the symmetric point set. These cubature point sets are located at the intersection of the unit hyper-sphere and its axes. The constant weight w is defined as, √ 2 πn w= 2nΓ(n/2)
(3.45)
where Γ denotes the Gamma function. For calculation of radial integral, Gaussian quadrature is used. In CKF filter, only considered one point univariate Gaussian quadrature rule. The filtering algorithm of CKF filter has been shown in section 3.9.
3.5.2
Generation of Cubature Points and Weights
The CKF point [u]i is symmetrical in nature and located at the intersection of the unit hyper-sphere and its axes. For example, in single dimensional system, the two cubature point sets are +1 and −1. For two dimensional system, the four cubature point sets are (+1, 0), (−1, 0), (0, +1) and (0, −1). This is free from curse of dimensionality problem.
3.6
Cubature Quadrature Kalman Filter
The cubature quadrature Kalman filter (CQKF) is developed by Bhaumik et.al. in 2013 [36]. The CQKF filter is the improvement of the CKF filter. The CQKF filter is nonlinear suboptimal filter which uses Gaussian distribution in Bayesian framework. It works on spherical-radial cubature quadrature rule. In this filter, the multiple dimensional intractable integral is converted into surface integral and line integral. Surface integral has been solved using spherical-radial cubature rule over a unit hyper sphere and line integral has been solved using Gauss-Laguerre quadrature rule. The CQKF filter points and weights generation depends on the state dimension of the target as well as order of accuracy of the Gauss-Laguerre quadrature rule.
32
3.6.1
Cubature Quadrature Rule
The CQKF [36] filter is a nonlinear filter, which is based on spherical radial quadrature and Gauss-Laguerre quadrature rule. For the first order of accuracy, the CQKF filter gives the same result as CKF filter. The accuracy of the CQKF filter depends on the order of the quadrature rule. As the order of the quadrature increases, the accuracy of the system increases. In Bayesian framework, the prior probability density can be calculated by Chapman-Kolmogorov equation, Z p(xk | y1:k−1 ) =
p(xk | xk−1 )p(xk−1 | y1:k−1 )dxk−1
(3.46)
This equation (3.46) is known as time update equation. Where xk is the state vector of the process model at time k, and y1:k is the measurement equation upto time k. The Bayesian update of posterior density function is given by p(xk | y1:k ) =
p(yk | xk )p(xk | y1:k−1 ) p(yk | y1:k−1 )
(3.47)
where p(yk | y1:k−1 ) is the normalizing constant, given by Z p(yk | y1:k−1 ) =
p(yk | xk )p(xk | y1:k−1 )dxk .
(3.48)
Now, we decompose the integral (3.46) and (3.47) into a surface integral and line integral. The surface integral is calculated using spherical cubature rule over a unit hypersphere of dimension n and a line integral is calculated using Gauss-Laguerre rule. Let us define an integral I(g) of an arbitrary function g(x) is given by Z
1
I(g) = p P | | (2π)n
g(x)e(−1/2)(x−µ)
T
P−1
(x−µ)
dx
(3.49)
Rn
where x ∈ Rn is a variable and (3.49) can be transformed into spherical coordinate system as 1
I(g) = p (2π)n
Z
∞
r1 =0
Z
2
[f (Cr1 y + µ)dσ(y)]r1n−1 e−r1 /2 dr1
(3.50)
Un
where x = Cr1 y + µ, C denotes the Cholesky decomposition of covariance matrix p P , k y k= y T y = 1, µ denotes the mean Gaussian distribution and Un denotes the 33
surface of unit hyper-sphere, r1 denotes the radius of the hyper-sphere. Now, we are approximated the (3.50) with zero mean and unity variance, then we get √ 2n 2 πn X g(r1 y)dσ(y) ' g(r1 [ui ]) 2nΓ(n/2) i=1 Un
Z
(3.51)
where [ui ] denotes the cubature points located at the intersection of unit hyper-sphere and its axes. Now, we approximate the cubature quadrature integral then, we get X 2n X n0 p 1 I(g) = Ai0 g( 2λi0 [ui ]) 2nΓ(n/2) i=1 i0 =1
(3.52)
where λi0 is the Gauss-Laguerre quadrature points, which is determined from ChebyshevLaguerre polynomial equation. i.e. 0
Lαn0
dn α+n0 −λ λ e =0 e dλn0
n0 −α λ
= (−1) λ
(3.53)
n0 !Γ(α + n0 + 1) is the Gauss-Laguerre weights. Where α = (n/2 − 1). λi0 [L˙ αn0 (λi0 )]2 √ Now, from equation (3.52), we get a cubature quadrature point ξj = 2λi0 [ui ] and
and Ai0 =
its corresponding weights wj as, wj =
1 1 n0 !Γ(α + n0 + 1) (Ai0 ) = Ai0 = 2nΓ(n/2) 2nΓ(n/2) λi0 [L˙ αn0 (λi0 )]2
(3.54)
for i = 1, 2, ..., 2n, i0 = 1, 2, ..., n0 and j = 1, 2, ..., 2nn0 . Where n is dimension and n0 is the order of Gauss-Laguerre quadrature points. The filtering algorithm of CQKF filter has been shown in section 3.9.
34
0.45 0.4
Quadrature Weights
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 −4
−3
−2
−1 0 1 Quadrature Points
2
3
4
Figure 3.1: Single dimensional third-order cubature quadrature points and its weight.
Quadrature Weights
0.2
0.15
0.1
0.05
0 4 2
4 2
0
0
−2
−2 −4
Quadrature Points
−4
Quadrature Points
Figure 3.2: 2-dimensional third-order cubature quadrature points and its weight.
Table 3.1: CQKF points generation with its order of accuracy level and dimension
Accuracy level\ Dimension Dim-1 Dim-2 Dim-3 Dim-4 Dim-5 Dim-6 Order-1
2
4
6
8
10
12
Order-2
4
8
12
16
20
24
Order-3
6
12
18
24
30
36
Order-4
8
16
24
32
40
48
Order-5
10
20
30
40
50
60
35
3.6.2
Generation of Cubature Quadrature Points and Weights
The CQKF filter has total 2nn0 points and its corresponding weights, where n is the state dimension of the system and n0 is the order of accuracy level. For example, in single dimension and first order system has total two symmetric points, which are +1 and −1, and its corresponding weights are 0.5 and 0.5. In 2-dimension and first order system has total four points, which are (1.414, 0), (0, 1.414), (−1.414, 0) and (0, −1.414), and its corresponding weights are same i.e. 0.25. The plot of cubature quadrature points and weights for single and 2-dimensional system with accuracy level three i.e. CQKF-3, are shown in Fig. 3.1 and Fig. 3.2 respectively. Table 3.1 shows that the number of points varying linearly with its dimension and order of accuracy level of the system. The CQKF filter is free from the curse of dimensionality problem.
3.7
Gauss-Hermite Filter
The Gauss-Hermite filter (GHF) is proposed by Ito et.al. in 2000 [41]. The GHF filter is a nonlinear suboptimal filter which uses Gaussian distribution and works on Bayesian framework. The GHF filter is used because EKF filter does not provide correct result for highly nonlinear problem. The EKF filter is used to linearize the nonlinear polynomial using Taylor series which produces a large error in the system. A lot of derivation occurs for Jacobean matrices calculation which provides a nontrivial solution for many problem. To overcome this problem, the GHF filter is used. The GHF filter works on the Gauss-Hermite quadrature rule. The GHF is a new version of quadrature Kalman filter which is developed theoretically and tested experimentally [44]. In a nonlinear filtering problem, where both state and measurement models are nonlinear. We use Gaussian filter to approximate the nonlinearities of the system. This filter is suboptimal in nature.
36
3.7.1
Gauss-Hermite Quadrature Rule
In GHF, we approximate the intractable integrals to numerically summation form, with the help of Gauss-Hermite quadrature rule in nonlinear Bayesian filtering framework. Where unknown Gaussian probability density function has been approximated with Gauss-Hermite quadrature rule to form single dimensional quadrature points and their respective weights. For multidimensional system, we convert the single dimensional points to multidimensional points with the help of product rule [44, 47, 48, 49].
Gauss Quadrature rule: In numerical analysis problem, a quadrature rule is defined as an approximation of the definite integral. Where weighted sum is defined for the points in the definite integral domain. In Gaussian quadrature, the integral of the Gaussian density function is 1 with domain of the integral is −∞ to ∞. An m-point Gaussian quadrature rule, a total m point and its corresponding weight is generated exactly upto a polynomial of degree 2m − 1 for univariate quadrature rule. Where the sum of its all weight is equal to 1. We convert this univariate quadrature to multivariate quadrature using product rule. If the univariate quadrature has m points and its dimension is n, then total mn multivariate points and its corresponding weights are generated, where the sum of all the multivariate weights are 1. There are many type of Gauss quadrature rules, where the range of the definite integral defines the interval over which numerical integration is performed and find out the points and non-negative weights. Where W (x) is used as a non-negative weight function. Family of polynomial functions {φn (x)}∞ n=0 i.e φ1 (x), φ2 (x).... each of them are orthogonal with respect to integration over an interval [a, b] with weighted function W (x), as Z
b
φm (x)φn (x)W (x)dx = δmn Cn
(3.55)
a
Legendre polynomials Pk (x): Interval [a, b] = [−1, 1], weight W (x) = 1. Legendre polynomials are represented as, Pk (x) =
k−1 2k − 1 xPk−1 (x) − Pk−2 (x) k k
37
(3.56)
1 where P0 (x) = 1, P1 (x) = x, P2 (x) = (3x2 − 1),.... 2 Laguerre polynomials Lk (x): Interval [a, b] = [0, ∞], weight W (x) = e−x . Leguerre polynomials are represented as, Lk (x) = (2k − x − 1)Lk−1 (x) − (k − 1)2 Lk−2 (x)
(3.57)
where L0 (x) = 1, L1 (x) = 1 − x, L2 (x) = 2 − 4x + x2 ,.... Chebyshev polynomials Tk (x): Interval [a, b] = [−1, 1], weight W (x) = √
1 . 1 − x2
Chebyshev polynomials are represented as, Tk (x) = 2xTk−1 (x) − Tk−2 (x)
(3.58)
where T0 (x) = 1, T1 (x) = x, T2 (x) = 2x2 − 1,.... 2
Hermite polynomials Hk (x): Interval [a, b] = [−∞, ∞], weight W (x) = e−x . Hermite polynomials are represented as, Hk (x) = 2xHk−1 (x) − 2(k − 1)Hk−2 (x)
(3.59)
where H0 (x) = 1, H1 (x) = 2x, H2 (x) = 4x2 − 2,....
Gauss-Hermite quadrature rule We consider any integral of the form Z I=
b
g(x)W (x)dx
(3.60)
a
Where, g(x) is a nonlinear function of x and W (x) is the Gaussian weighted function. Now we use gauss-Hermite quadrature rule to approximate this integral numerically.
I=
m X
g(qi )wi
(3.61)
i=1
where qi and wi are quadrature points and their respective weights. For m-point quadrature rule, it is exact upto polynomial of degree (2m − 1). To calculate the quadrature
38
points, we have taken a symmetric triangular matrix J with zero diagonal element Ji,i+1 = The quadrature points qi =
p i/2,
1 6 i 6 (m − 1).
(3.62)
√ 2ξi , where ξi are the ith eigen value of J. The ith weight
2 wi is chosen as wi = ki1 , where ki1 is the first element of the ith normalized eigenvector
of J. Now, we apply the product rule to convert a single dimensional point to multidimensional point for multidimensional system. Let us consider a multidimensional nonlinear function g(x) with random variable x and weighted function as the standard normal distribution. Z
∞
IN =
g(x)ℵ(x, 0, In )dx
(3.63)
−∞
Apply product rule, then we get
IN =
m X i1 =1
...
m X
g(qi1 , qi2 .., qin )wi1 wi2 ...win .
(3.64)
in =1
In n-dimensional integral system with m-point single dimensional Gauss-Hermite quadrature rule has total mn multidimensional point. In GHF, point will increases exponentially as dimension of the system increases, so this filter suffers from curse of dimensionality problem. Now, we formulate the Gauss-Hermite quadrature points and weights using Golub method which uses the tridiagonal matrix to generate points and weights. The three term recurrence relation polynomial is considered as, Pi (x) = (ai x + bi )Pi−1 (x) − ci Pi−2 (x).
(3.65)
Where, ai > 0, bi > 0, ci > 0, P−1 (x) = 0, P0 (x) = 1 and recurrence relation is written as, P1 (x) = (a1 x + b1 )P0 (x), xP0 (x) =
1 b1 P1 (x) − P0 (x) a1 a1
and P2 (x) = (a2 x + b2 )P1 (x) − c2 P0 (x), 39
xP1 (x) =
b2 c2 1 P2 (x) − P1 (x) + P0 (x) a2 a2 b2
and ...... show on. Now, we derive the above equation in matrix form 1 −b1 P (x) a1 0 ac 1 b2 2 − P1 (x) a2 a2 x = . . . . . . Pn−1 (x) 0 .
0 1 a2 . . .
. . . . cn an
. P (x) 0 0 0 . P1 (x) . + . . . . . 1 bn P (x) P (x) n n−1 − an an
(3.66)
Now, we write the above equation in general form as, xP (x) = M P (x) +
1 Pn (x)en an
(3.67)
iT where T is the tridiagonal matrix and en = 0 0 . . 1 . Now, find the solution of h
the polynomial thus Pn (x) = 0 i.e (λI − T )P (λ) = 0, where λ is the eigen value of matrix T . We write a tridiagonal matrix in general form such as, α β 0 . . 1 1 β1 α2 β2 . . J = . . . . . . . . . . . . . βn−1 αn
(3.68)
1/2 bi ci+1 where αi = − , βi = . For Gauss-Hermite quadrature rule, we use ai ai ai+1 p ai = ai+1 = 2, bi = 0, ci+1 = 2i. Then we get αi = 0 and βi = i/2. The filtering algorithm of GHF filter has been shown in section 3.9.
40
3.7.2
Generation of GHF Points and Weights
Now, first we define a tridiagonal matrix J for accuracy level m.
0 p 1/2 0 J = 0 ... 0
p 1/2
0
0
....
0
1 0 .... 0 p 3/2 .... 0 0 p 3/2 0 .... 0 p ... ... 0 (m − 1)/2 p (m − 1)/2 0 0 0
0 1 0 ... 0
.
(3.69)
m×m
Let us assume, we get m eigenvalues ξ1 , ξ2 , ξ3 , ....., ξm . Then multiply all the eigen √ value by 2 then we get the quadrature points q1 , q2 , q3 , ...., qm . We find the normalized eigenvector for all the eigenvalues. Take the first element of each normalized eigenvector and square them, then we get the corresponding weights w1 , w2 , w3 , ...., wm . Now, apply this quadrature points and weights for n-dimensional multivariate system. Apply product rule, then we get mn quadrature points and its corresponding weight is given as, ξ 1 ξ1 ξ1 : ξ1
ξ2 .... ξm ξ1 ξ2
ξm
... ξm
ξ1 .... ξ1
ξ2 ξ2 .... ξ2
ξ1 .... ξ1
ξ2 ξ2 .... ξ2
:
:
:
:
:
:
:
ξ1
ξ1
ξ1
ξ2 ξ2
ξ2
ξ2
......
ξm ξm : ξm
.
n×mn
Where each column of the matrix is the quadrature points and its mn corresponding weights are given below (w1 .w1 ...w1 ), (w2 .w1 ...w1 ), ...., (wm .w1 ...w1 ), (w1 .w2 ...w2 ), ....(wm .w2 ...w2 ), ..., (wm .wm ...wm ).
Example 1 To find an univariate quadrature points and their respective weights for GHF filter with accuracy level 3.
41
Figure 3.3: Single dimensional 3-point Gauss-Hermite quadrature and its weight.
Now, first we write a tridiagonal matrix J for m = 3, i.e.
p 1/2 0 0 p J = 1/2 0 1 0 1 0 p p Solving the matrix J then we get the eigenvalue ξi = (− 3/2, 0, 3/2). √ √ √ 2ξi = (− 3, 0, 3). h p p p iT Now, we get the normalized eigenvector k11 = − 1/6 1/2 − 1/3 , k21 = h p i h i T T p p p p − 2/3 0 1/3 , k31 = 1/6 1/2 1/3 . The weight of univariate Now, finding the quadrature point qi =
Gauss-Hermite quadrature is defined as, wi = (ki1 )2 . i.e 1 2 1 w1 = , w2 = , w3 = 6 3 6
42
Quadrature Weights
0.5 0.4 0.3 0.2 0.1 0 2 1
2 1
0
0
−1 Quadrature Points
−1 −2
−2
Quadrature Points
Figure 3.4: 2-dimensional 3-point Gauss-Hermite quadrature and its weight.
Example 2 To find an multivariate quadrature points and their respective weights for accuracy level 3 and dimension 2. Use the univariate points and weights discussed in above example 1. Then we get the 32 = 9 quadrature points and its corresponding weights. The quadrature points are expressed as, √ √ √ √ √ √ √ √ √ (− 3, − 3), (0, − 3), ( 3, − 3), (− 3, 0), (0, 0), ( 3, 0), (− 3, 3), √ √ √ (0, 3), ( 3, 3) and its corresponding weights are 1 1 2 1 1 1 1 2 2 2 1 2 1 1 2 1 1 1 ( × ), ( × ), ( × ), ( × ), ( × ), ( × ), ( × ), ( × ), ( × ) 6 6 3 6 6 6 6 3 3 3 6 3 6 6 3 6 6 6 i.e 1 1 1 1 4 1 1 1 1 , , , , , , , , . 36 9 36 9 9 9 36 9 36 The plot of the univariate Gauss-Hermite quadrature points and weights for acccuracy level three i.e. GHF-3, is shown in Fig. 3.3. Two dimensional multivariate GaussHermite quadrature points and weights for accuracy level three i.e. GHF-3, is shown in Fig. 3.4. The GHF filter points has been varying with its dimension and accuracy level which is shown in Table 3.2.
43
Table 3.2: GHF points generation with its accuracy level and dimension
Accuracy Level \Dimension Dim-1 Dim-2 Dim-3 Dim-4 Dim-5 Dim-6
3.8
Level-1
1
1
1
1
1
1
Level-2
2
4
8
16
32
64
Level-3
3
9
27
81
243
729
Level-4
4
16
64
256
1024
4096
Level-5
5
25
125
625
3125
15625
Sparse-grid Gauss-Hermite Filter
The sparse-grid Gauss-Hermite filter (SGHF) is proposed by Jia et.al. in 2012 [46]. It is a powerful nonlinear suboptimal filter which uses Gaussian noise distribution and works on Bayesian framework. The GHF filter is facing curse of dimensionality problem. The SGHF filter is developed to overcome the GHF filter problem. The accuracy of the SGHF filter is better than EKF filter and it is comparable to UKF and CKF filter. The SGHF filter works on sparse-grid quadrature rule. The SGHF filter uses Smolyak’s rule for generation of quadrature points and weights. The SGHF [46, 50, 56] is a nonlinear quadrature based filter which utilizes the sparse-grid quadrature points and weights to approximate the multi-dimensional integral in nonlinear Bayesian algorithm. In GHF, the number of point increases exponentially with dimension, so it arises difficulty in solving a higher dimensional problem. It works on Smolyak’s rule which is used to minimize the quadrature points and weights to alleviate the curse of dimensionality of the system. So, in higher dimensional problem, we prefer SGHF.
3.8.1
Sparse-grid Quadrature Rule
In SGHF, we have taken the same univariate quadrature point as GHF, but in multidimensional system, we approximate the GHF points using Smolyak’s rule which is
44
shown in Fig. 3.5. The spare-grid method is defined as Z g(x)ℵ(x; 0, In )dx ≈ In,L (g)
(3.70)
Rn
In,L (g) =
L−1 X
L−1−q (−1)L−1−q Cn−1
X
(Ii1 ⊗ Ii2 ... ⊗ Iin )(g)
(3.71)
Ξ∈Nqn
q=L−n
where x = [x1 , x2 , ....., xn ]T ∈ Rn , In,L (g) is the approximation of the n-dimensional integral of function g(x) with Gaussian weights ℵ(x, 0, In ) with accuracy level L ∈ N. It gives the exact solution for all polynomial upto degree 2L − 1. C denotes as a binomial coefficient, defined by Ckn = n!/k!(n − k)!, where n! denote as a factorial of n. ⊗ denotes the tensor product. Iij is a single dimensional quadrature rule where ij ∈ Ξ is the accuracy level. It means that Iij gives exact result upto polynomial of degree 2ij − 1. The accuracy level sequence
Nqn =
P {Ξ : n
j=1 ij
= n + q}
∅
f or f or
q>0
(3.72)
q
