For linear and well-defined estimation problems with Gaussian white noise, the Kalman filter (KF) yields the best result in terms of estimation accuracy. However, the KF performance degrades and can fail in cases involving large uncertainties such as modeling errors in the estimation process. The smooth variable structure filter (SVSF) is a relatively new estimation strategy based on sliding mode theory and has been shown to be robust to modeling uncertainties. The SVSF makes use of an existence subspace and of a smoothing boundary layer to keep the estimates bounded within a region of the true state trajectory. Currently, the width of the smoothing boundary layer is chosen based on designer knowledge of the upper bound of modeling uncertainties, such as maximum noise levels and parametric errors. This is a conservative choice, as a more well-defined smoothing boundary layer will yield more accurate results. In this paper, the state error covariance matrix of the SVSF is used for the derivation of an optimal time-varying smoothing boundary layer. The robustness and accuracy of the new form of the SVSF was validated and compared with the KF and the standard SVSF by testing it on a linear electrohydrostatic actuator (EHA).

References

References
1.
Nise
,
N.
, 2004,
Control Systems Engineering
,
4th ed.
,
John Wiley and Sons, Inc.
,
New York
.
2.
Kalman
,
R. E.
, 1960, “
A New Approach to Linear Filtering and Prediction Problems
,”
ASME J. Basic Eng.
,
82
,
pp.
35
45
.10.1115/1.3662552
3.
Anderson
,
B. D. O.
, and
Moore
,
J. B.
, 1979,
Optimal Filtering
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
4.
Ristic
,
B.
,
Arulampalam
,
S.
, and
Gordon
,
N.
, 2004,
Beyond the Kalman Filter: Particle Filters for Tracking Applications
,
Artech House
,
Boston
.
5.
Simon
Haykin
, 2001,
Kalman Filtering and Neural Networks
,
John Wiley and Sons, Inc.
,
New York
.
6.
Gadsden
,
S. A.
,
2011
, “
Smooth Variable Structure Filtering: Theory and Applications
,”
Ph.D. thesis, Department of Mechanical Engineering
,
McMaster University
,
Hamilton, Ontario
.
7.
Gadsden
,
J. A.
,
Al-Shabir
,
M.
, and
Habibi
,
S. R.
,
2011
, “
Estimation Strategies for the Condition Monitoring of a Battery System in a Hybrid Electric Vehicle
,”
ISRN Signal Processing
, 2011, p. 12035110.5402/2011/120351.
8.
Gelb
,
A.
, 1974,
Applied Optimal Estimation
,
MIT Press
,
Cambridge, MA
.
9.
Simon
,
D.
, 2006,
Optimal State Estimation: Kalman, H-Infinity, and Nonlinear Approaches
,
Wiley-Interscience
, New Jersey.
10.
Welch
,
G.
, and
Bishop
,
G.
, 2006, “
An Introduction to the Kalman Filter
,”
Department of Computer Science, University of North Carolina
, Chapel Hill, NC, Report.
11.
Bar-Shalom
,
Y.
,
Li
,
X.-R.
, and
Kirubarajan
,
T.
, 2001,
Estimation With Applications to Tracking and Navigation
,
John Wiley and Sons, Inc.
,
New York
.10.1002/0471221279
12.
Julier
,
S. J.
,
Ulhmann
,
J. K.
, and
Durrant-Whyte
,
H. F.
,
2000
, “
A New Method for Nonlinear Transformation of Means and Covariances in Filters and Estimators
,”
IEEE Trans. Autom. Control
,
45
,
pp.
472
482
.10.1109/9.847726
13.
Grewal
,
M. S.
, and
Andrews
,
A. P.
, 2008,
Kalman Filtering: Theory and Practice Using MATLAB
,
3rd ed.
,
John Wiley and Sons, Inc.
,
New York
.
14.
Kaminski
,
P.
,
Bryson
,
A.
, and
Schmidt
,
S.
,
1971
, “
Discrete Square Root Filtering: A Survey of Current Techniques
,”
IEEE Trans. Autom. Control
,
16
,
pp.
727
786
.10.1109/TAC.1971.1099816
15.
Hammarling
,
S.
,
1977
, “
A Survey of Numerical Aspects of Plane Rotations
,”
Middlesex Polytechnic, Report, Maths
(1). pp.
1
35
.
16.
Wang
,
H.
, and
Gregory
,
R.
,
1964
, “
On the Reduction of an Arbitrary Real Square Matrix to Tridiagonal Form
,”
Math. Comput.
,
18
(
87
),
pp.
501
505
.10.1090/S0025-5718-1964-0165670-0
17.
Chandrasekar
,
J.
,
Kim
,
I. S.
,
Bernstein
,
D. S.
, and
Ridley
,
A. J.
, 2008, “
Cholesky-Based Reduced-Rank Square-Root Kalman Filtering
,” Proceedings of American Control Conference (
ACC
),
pp.
3987
3992
.10.1109/ACC.2008.4587116
18.
Xie
,
L.
,
Soh
,
C.
, and
Souza
,
C. E.
,
1994
, “
Robust Kalman Filtering for Uncertain Discrete-Time Systems
,”
IEEE Trans. Autom Control
,
39
(
6
),
pp.
1310
1314
.10.1109/9.293203
19.
Zhu
,
X.
,
Soh
,
Y. C.
, and
Xie
,
L.
,
2002
, “
Design and Analysis of Discrete-Time Robust Kalman Filters
,”
Automatica
,
38
(
6
),
pp.
1069
1077
.10.1016/S0005-1098(01)00298-9
20.
Berndt
,
B.
,
Evans
,
R.
, and
Williams
,
K.
, 1998,
Gauss and Jacobi Sums
,
John Wiley & Sons, Inc.
,
New York
.
21.
Habibi
,
S. R.
, and
Burton
,
R.
,
2003
, “
The Variable Structure Filter
,”
ASME J. Dyn. Sys., Meas., Control
,
125
,
pp.
287
293
.10.1115/1.1590682
22.
Utkin
,
V. I.
,
1977
, “
Variable Structure Systems With Sliding Mode: A Survey
,”
IEEE Trans. Autom. Control
,
22
,
pp.
212
222
.10.1109/TAC.1977.1101446
23.
Utkin
,
V. I.
,
1978
,
Sliding Mode and Their Application in Variable Structure Systems
,
English Translation ed.
,
Mir Publication
,
Moscow, U.S.S.R
.
24.
Slotine
,
J. J.
, and
Li
,
W.
, 1991,
Applied Nonlinear Control
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
25.
Basin
,
M. V.
,
Ferreira
,
A.
, and
Fridman
,
L.
,
2007
, “
Sliding Mode Identification and Control for Linear Uncertain Stochastic Systems
,”
Int. J. Syst. Sci.
,
pp.
861
869
.10.1080/00207720701409363
26.
Spurgeon
,
S. K.
,
2008
, “
Sliding Mode Observers: A Survey
,”
Int. J. Syst. Sci.
,
pp.
751
764
.10.1080/00207720701847638
27.
Basin
,
M. V.
,
Fridman
,
L.
, and
Skliar
,
M.
,
2002
, “
Optimal and Robust Integral Sliding Mode Filter Design for Systems With Continuous and Delayed Measurements
,” Proceedings of the 41st
IEEE
Conference on Decision and Control,
Las Vegas, NV
,
pp.
2594
2599
.10.1109/CDC.2002.1184229
28.
Habibi
,
S. R.
,
2007
, “
The Smooth Variable Structure Filter
,”
Proc. IEEE
,
95
(
5
),
pp.
1026
1059
.10.1109/JPROC.2007.893255
29.
Al-Shabi
,
M.
,
2011
, “
The General Toeplitz/Observability SVSF
,”
Ph.D. thesis
,
Department of Mechanical Engineering, McMaster University
,
Hamilton, Ontario
.
30.
Gadsden
,
S. A.
, and
Habibi
,
S. R.
,
2010
, “
A New Form of the Smooth Variable Structure Filter With a Covariance Derivation
,”
IEEE
Conference on Decision and Control,
Atlanta, GA
.10.1109/CDC.2010.5717397
31.
Luenberger
,
D. G.
, 1979,
Introduction to Dynamic Systems
,
John Wiley
,
New York.
32.
Gadsden
,
S. A.
,
El Sayed
,
M.
, and
Habibi
,
S. R.
,
2011
, “
Derivation of an Optimal Boundary Layer Width for the Smooth Variable Structure Filter
,”
American Control Conference (ACC)
,
San Francisco, CA
.
33.
Petersen
,
K. B.
, and
Pedersen
,
M. S.
, 2008,
The Matrix Cookbook
,
Technical University of Denmark
,
Copenhagen, Denmark.
34.
Habibi
,
S. R.
, and
Burton
,
R.
,
2007
, “
Parameter Identification for a High Performance Hydrostatic Actuation System Using the Variable Structure Filter Concept
,”
ASME J. Dyn. Sys., Meas., Control
,
129
,
pp.
229–235.10.1115/1.2431816
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