The generalized polynomial chaos (gPC) mathematical technique, when integrated with the extended Kalman filter (EKF) method, provides a parameter estimation and state tracking method. The truncation of the series expansions degrades the link between parameter convergence and parameter uncertainty which the filter uses to perform the estimations. An empirically derived correction for this problem is implemented, which maintains the original parameter distributions. A comparison is performed to illustrate the improvements of the proposed approach. The method is demonstrated for parameter estimation on a regression system, where it is compared to the recursive least squares (RLS) method.

References

References
1.
Blanchard
,
E. D.
,
Sandu
,
A.
, and
Sandu
,
C.
,
2010
, “
A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
132
(
6
), p.
061404
.
2.
Ljung
,
L.
,
1999
,
System Identification: Theory for the User
,
Prentice Hall
, Upper Saddle River, NJ.
3.
Fathy
,
H. K.
,
Kang
,
D.
, and
Stein
,
J. L.
,
2008
, “
Online Vehicle Mass Estimation Using Recursive Least Squares and Supervisory Data Extraction
,”
American Control Conference
(
ACC
), Seattle, WA, June 11–13, pp. 1842–1848.
4.
Pizarro
,
O.
, and
Sbarbaro
,
D.
,
1998
, “
Parameter Subset Identification by Recursive Least Squares
,”
American Control Conference
(
ACC
), Philadelphia, PA, June 24–26, pp. 3590–3591.
5.
Chensarkar
,
M. M.
, and
Desai
,
U. B.
,
1997
, “
A Robust Recursive Least Squares Algorithm
,”
IEEE Trans. Signal Process.
,
45
(
7
), pp.
1726
1735
.
6.
Chen
,
J.
, and
Ding
,
F.
,
2011
, “
Modified Stochastic Gradient Identification Algorithms With Fast Convergence Rates
,”
J. Vib. Control
,
17
(
9
), pp.
1281
1286
.
7.
Kouritzin
,
M. A.
,
1996
, “
On the Convergence of Linear Stochastic Approximation Procedures
,”
IEEE Trans. Inf. Theory
,
42
(
4
), pp.
1305
1309
.
8.
Gill
,
J.
,
2009
,
Bayesian Methods: A Social and Behavioral Sciences Approach
,
2nd ed.
,
Chapman and Hall/CRC Press
, London/Boca Raton, FL.
9.
Par
,
T. C.
,
Zweiri
,
Y. H.
,
Althoefer
,
K.
, and
Seneviratne
,
L. D.
,
2005
, “
Online Soil Parameter Estimation Scheme Based on Newton-Raphson Method for Autonomous Excavation
,”
IEEE/ASME Trans. Mechatronics
,
10
(
2
), pp.
221
229
.
10.
More
,
J. J.
,
1977
, “
The Levenberg-Marquardt Algorithm: Implementation and Theory
,”
Conference on Numerical Analysis
, Dundee, UK, June 28–July 1, pp.
105
116
.
11.
Astrom
,
K. J.
, and
Eykhoff
,
P.
,
1971
, “
System Identification—A Survey
,”
Automatica
,
7
(
2
), pp.
123
162
.
12.
Sandu
,
A.
,
Sandu
,
C.
, and
Ahmadian
,
M.
,
2006
, “
Modeling Multibody Systems With Uncertainties—Part I: Theoretical and Computational Aspects
,”
Multibody Syst. Dyn.
,
15
(
4
), pp.
369
391
.
13.
Sandu
,
C.
,
Sandu
,
A.
, and
Ahmadian
,
M.
,
2006
, “
Modeling Multibody Systems With Uncertainties—Part II: Numerical Applications
,”
Multibody Syst. Dyn.
,
15
(
3
), pp.
241
262
.
14.
Hays
,
J.
,
2011
, “
Parametric Optimal Design of Uncertain Dynamical Systems
,”
Ph.D. dissertation
, Virginia Tech, Blacksburg, VA.
15.
Blanchard
,
E. D.
,
2010
, “
Polynomial Chaos Approaches to Parameter Estimation and Control Design for Mechanical Systems With Uncertain Parameters
,”
Ph.D. dissertation
, Virginia Tech, Blacksburg, VA.
16.
Blanchard
,
E. D.
,
Sandu
,
A.
, and
Sandu
,
C.
,
2009
, “
Polynomial Chaos-Based Parameter Estimation Methods Applied to Vehicle System
,”
IMechE
,
224
(
1
), pp.
59
81
.
17.
Southward
,
S. C.
,
2007
, “
Real-Time Parameter Id Using Polynomial Chaos Expansions
,”
ASME
Paper No. IMECE2007-43745.
18.
Pence
,
B. L.
,
2011
, “
Recursive Parameter Estimation Using Polynomial Chaos Theory Applied to Vehicle Mass Estimation for Rough Terrain
,”
Ph.D. dissertation
, University of Michigan, Ann Arbor, MI.
19.
Pence
,
B.
,
Hays
,
J.
,
Fathy
,
H.
,
Sandu
,
C.
, and
Stein
,
J.
,
2013
, “
Vehicle Sprung Mass Estimation for Rough Terrain
,”
Int. J. Veh. Des.
,
61
(1/2/3/4), pp.
3
26
.
20.
Shimp
,
S. K.
,
2008
, “
Vehicle Sprung Mass Parameter Estimation Using an Adaptive Polynomial-Chaos Method
,”
Master's thesis
, Virginia Tech, Blacksburg, VA.
21.
Xiu
,
D.
,
2007
, “
Efficient Collocational Approach for Parametric Uncertainty Analysis
,”
Commun. Comput. Phys.
,
2
(
2
), pp.
293
309
.
22.
Cheng
,
H.
, and
Sandu
,
A.
,
2009
, “
Efficient Uncertainty Quantification With the Polynomial Chaos Method for Stiff Systems
,”
Math. Comput. Simul.
,
79
(
11
), pp.
3278
3295
.
23.
Xiu
,
D.
,
2009
, “
Fast Numerical Methods for Stochastic Computations: A Review
,”
Commun. Comput. Phys.
,
5
(2–4), pp.
242
272
.
24.
Xiu
,
D.
, and
Hesthaven
,
J. S.
,
2005
, “
High-Order Collocation Methods for Differential Equations With Random Inputs
,”
SIAM J. Sci. Comput.
,
27
(
3
), pp.
1118
1139
.
25.
Cheng
,
H.
, and
Sandu
,
A.
,
2010
, “
Collocation Least-Squares Polynomial Chaos Method
,” Spring Simulation Multiconference, Orlando, FL, Apr. 11–15, Paper No.
80
.
26.
Welch
,
G.
, and
Bishop
,
G.
,
2006
, “
An Introduction to the Kalman Filter
,” The University of North Carolina at Chapel Hill, Chapel Hill, NC, Report No.
TR 95-041
.
27.
Zarchan
,
P.
, and
Musoff
,
H.
,
2009
,
Fundamentals of Kalman Filtering: A Practical Approach
, Vol.
232
,
American Institute of Aeronautics and Astronautics
, Reston, VA.
28.
Li
,
J.
, and
Xiu
,
D.
,
2009
, “
A Generalized Polynomial Chaos Based Ensemble Kalman Filter With High Accuracy
,”
J. Comput. Phys.
,
228
(
15
), pp.
5454
5469
.
29.
Weisburg
,
S.
,
1980
,
Applied Linear Regression
,
Wiley
, Chichester, UK.
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