In this paper, a new robust Kalman filter is proposed for discrete-time time-varying linear stochastic systems. The system under consideration is subject to stochastic and norm-bounded uncertainties in all matrices of the system model. In the proposed approach, the filter is first achieved by solving a stochastic min–max optimization problem. Next, we find an upper bound on the estimation error covariance, and then, by using a linear matrix inequality (LMI) optimization problem, unknown parameters of the filter are determined such that the obtained upper bound is minimized. Finally, two numerical examples are given to demonstrate the effectiveness and performance of the proposed filtering approach compared to the existing robust filters.

References

References
1.
Kailath
,
T.
,
Sayed
,
A. H.
, and
Hassibi
,
B.
,
2000
,
Linear Estimation
, Vol.
1
,
Prentice Hall
,
Upper Saddle River, NJ
.
2.
Simon
,
D.
,
2006
,
Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches
,
Wiley
,
New York
.
3.
D'Appolito
,
J.
, and
Hutchinson
,
C.
,
1969
, “
Low Sensitivity Filters for State Estimation in the Presence of Large Parameter Uncertainties
,”
IEEE Trans. Autom. Control
,
14
(
3
), pp.
310
312
.
4.
Leondes
,
C.
, and
Pearson
,
J.
,
1972
, “
A Minimax Filter for Systems With Large Plant Uncertainties
,”
IEEE Trans. Autom. Control
,
17
(
2
), pp.
266
268
.
5.
Theodor
,
Y.
, and
Shaked
,
U.
,
1996
, “
Robust Discrete-Time Minimum-Variance Filtering
,”
IEEE Trans. Signal Process.
,
44
(
2
), pp.
181
189
.
6.
Zhu
,
X.
,
Soh
,
Y. C.
, and
Xie
,
L.
,
2002
, “
Design and Analysis of Discrete-Time Robust Kalman Filters
,”
Automatica
,
38
(
6
), pp.
1069
1077
.
7.
Yang
,
F.
,
Wang
,
Z.
, and
Hung
,
Y.
,
2002
, “
Robust Kalman Filtering for Discrete Time-Varying Uncertain Systems With Multiplicative Noises
,”
IEEE Trans. Autom. Control
,
47
(
7
), pp.
1179
1183
.
8.
Dong
,
Z.
, and
You
,
Z.
,
2006
, “
Finite-Horizon Robust Kalmanfiltering for Uncertain Discrete Time-Varying Systems With Uncertain-Covariance White Noises
,”
IEEE Signal Process. Lett.
,
8
(
13
), pp.
493
496
.
9.
Gershon
,
E.
,
Shaked
,
U.
, and
Yaesh
,
I.
,
2001
, “
H∞ Control and Filtering of Discrete-Time Stochastic Systems With Multiplicative Noise
,”
Automatica
,
37
(
3
), pp.
409
417
.
10.
Ding
,
D.
,
Wang
,
Z.
,
Shen
,
B.
, and
Dong
,
H.
,
2015
, “
Envelope-Constrained H∞ Filtering With Fading Measurements and Randomly Occurring Nonlinearities: The Finite Horizon Case
,”
Automatica
,
55
, pp.
37
45
.
11.
Wang
,
Z.
,
Yang
,
F.
,
Ho
,
D. W.
, and
Liu
,
X.
,
2006
, “
Robust H∞ Filtering for Stochastic Time-Delay Systems With Missing Measurements
,”
IEEE Trans. Signal Process.
,
54
(
7
), pp.
2579
2587
.
12.
Liu
,
Y.
,
Alsaadi
,
F. E.
,
Yin
,
X.
, and
Wang
,
Y.
,
2015
, “
Robust H∞ Filtering for Discrete Nonlinear Delayed Stochastic Systems With Missing Measurements and Randomly Occurring Nonlinearities
,”
Int. J. General Syst.
,
44
(
2
), pp.
169
181
.
13.
Li
,
X.
,
Lam
,
J.
,
Gao
,
H.
, and
Xiong
,
J.
,
2016
, “
H∞ and H2 Filtering for Linear Systems With Uncertain Markov Transitions
,”
Automatica
,
67
, pp.
252
266
.
14.
Li
,
X.
, and
Gao
,
H.
,
2012
, “
Robust Finite Frequency H∞ Filtering for Uncertain 2-D Roesser Systems
,”
Automatica
,
48
(
6
), pp.
1163
1170
.
15.
Wang
,
T.
,
Qiu
,
J.
, and
Gao
,
H.
,
2016
, “
Adaptive Neural Control of Stochastic Nonlinear Time-Delay Systems With Multiple Constraints
,”
IEEE Trans. Syst., Man, Cybern.: Syst.
,
47
(8), pp. 1875–1883.
16.
Wang
,
T.
,
Zhang
,
Y.
,
Qiu
,
J.
, and
Gao
,
H.
,
2015
, “
Adaptive Fuzzy Backstepping Control for a Class of Nonlinear Systems With Sampled and Delayed Measurements
,”
IEEE Trans. Fuzzy Syst.
,
23
(
2
), pp.
302
312
.
17.
Wang
,
T.
,
Qiu
,
J.
,
Fu
,
S.
, and
Ji
,
W.
,
2016
, “
Distributed Fuzzy H∞ Filtering for Nonlinear Multirate Networked Double-Layer Industrial Processes
,”
IEEE Trans. Ind. Electron.
,
64
(
6
), pp.
5203
5211
.
18.
Wang
,
T.
,
Qiu
,
J.
,
Gao
,
H.
, and
Wang
,
C.
,
2016
, “
Network-Based Fuzzy Control for Nonlinear Industrial Processes With Predictive Compensation Strategy
,”
IEEE Trans. Syst., Man, Cybern.: Syst.
,
47
(8), pp. 2137–2147.
19.
Petersen
, I
. R.
, and
McFarlane
,
D. C.
,
1991
, “
Robust State Estimation for Uncertain Systems
,”
30th IEEE Conference on Decision and Control
(
CDC
), Brighton, UK, Dec. 11–13, pp.
2630
2631
.
20.
Xie
,
L.
,
Soh
,
Y. C.
, and
de Souza
,
C. E.
,
1994
, “
Robust Kalman Filtering for Uncertain Discrete-Time Systems
,”
IEEE Trans. Autom. Control
,
39
(
6
), pp.
1310
1314
.
21.
Petersen
, I
. R.
, and
McFarlane
,
D. C.
,
1996
, “
Optimal Guaranteed Cost Filtering for Uncertain Discrete-Time Linear Systems
,”
Int. J. Robust Nonlinear Control
,
6
(
4
), pp.
267
280
.
22.
Bolzern
,
P.
,
Colaneri
,
P.
, and
De Nicolao
,
G.
,
1996
, “
Optimal Robust Filtering With Time-Varying Parameter Uncertainty
,”
Int. J. Control
,
63
(
3
), pp.
557
576
.
23.
Sayed
,
A. H.
,
2001
, “
A Framework for State-Space Estimation With Uncertain Models
,”
IEEE Trans. Autom. Control
,
46
(
7
), pp.
998
1013
.
24.
Terra
,
M. H.
,
Ishihara
,
J. Y.
, and
Inoue
,
R. S.
,
2011
, “
A New Approach to Robust Linear Filtering Problems
,”
IFAC Proc. Vol.
,
44
(
1
), pp.
1174
1179
.
25.
Ishihara
,
J. Y.
,
Terra
,
M. H.
, and
Cerri
,
J. P.
,
2015
, “
Optimal Robust Filtering for Systems Subject to Uncertainties
,”
Automatica
,
52
, pp.
111
117
.
26.
Terra
,
M. H.
,
Ishihara
,
J. Y.
,
Jesus
,
G.
, and
Cerri
,
J. P.
,
2013
, “
Robust Estimation for Discrete-Time Markovian Jump Linear Systems
,”
IEEE Trans. Autom. Control
,
58
(
8
), pp.
2065
2071
.
27.
de Jesus
,
G. Q.
,
Inoue
,
R. S.
, and
Terra
,
M. H.
,
2016
, “
Information Filtering and Array Algorithms for Discrete-Time Markovian Jump Linear Systems Subject to Parametric Uncertainties
,”
Inf. Sci.
,
369
, pp.
287
303
.
28.
Terra
,
M. H.
,
Ishihara
,
J. Y.
, and
Padoan
,
A. C.
,
2007
, “
Information Filtering and Array Algorithms for Descriptor Systems Subject to Parameter Uncertainties
,”
IEEE Trans. Signal Process.
,
55
(
1
), pp.
1
9
.
29.
Hsieh
,
C.-S.
, 2013 “
Robust State Estimation Via the Descriptor Kalman Filtering Method
,”
9th Asian Control Conference
(
ASCC
), Istanbul, Turkey, June 23–26, pp.
1
6
.
30.
Subramanian
,
A.
, and
Sayed
,
A. H.
,
2004
, “
Regularized Robust Filters for Time-Varying Uncertain Discrete-Time Systems
,”
IEEE Trans. Autom. Control
,
49
(
6
), pp.
970
976
.
31.
Abolhasani
,
M.
, and
Rahmani
,
M.
,
2017
, “
Robust Kalman Filtering for Discrete-Time Systems With Stochastic Uncertain Time-Varying Parameters
,”
Electron. Lett.
,
53
(
3
), pp.
146
148
.
32.
Gershon
,
E.
,
Shaked
,
U.
, and
Yaesh
,
I.
,
2005
, H∞
Control and Estimation of State-Multiplicative Linear Systems
, Vol.
318
,
Springer Science & Business Media
,
New York
.
33.
Gershon
,
E.
,
Limebeer
,
D. J.
,
Shaked
,
U.
, and
Yaesh
,
I.
,
2001
, “
Robust H∞ Filtering of Stationary Continuous-Time Linear Systems With Stochastic Uncertainties
,”
IEEE Trans. Autom. Control
,
46
(
11
), pp.
1788
1793
.
34.
Wang
,
F.
, and
Balakrishnan
,
V.
,
2002
, “
Robust Kalman Filters for Linear Time-Varying Systems With Stochastic Parametric Uncertainties
,”
IEEE Trans. Signal Process.
,
50
(
4
), pp.
803
813
.
35.
Halabi
,
S.
,
Ali
,
H. S.
,
Rafaralahy
,
H.
, and
Zasadzinski
,
M.
,
2009
, “
H∞ Functional Filtering for Stochastic Bilinear Systems With Multiplicative Noises
,”
Automatica
,
45
(
4
), pp.
1038
1045
.
36.
Pang
,
C.
, and
Sun
,
S.
,
2015
, “
Fusion Predictors for Multisensor Stochastic Uncertain Systems With Missing Measurements and Unknown Measurement Disturbances
,”
IEEE Sens. J.
,
15
(
8
), pp.
4346
4354
.
37.
Kai
,
X.
,
Wei
,
C.
, and
Liu
,
L.
,
2010
, “
Robust Extended Kalman Filtering for Nonlinear Systems With Stochastic Uncertainties
,”
IEEE Trans. Syst., Man, Cybern. Part A: Syst. Hum.
,
40
(
2
), pp.
399
405
.
38.
Sayed
,
A. H.
, and
Nascimento
,
V. H.
,
1999
, “
Design Criteria for Uncertain Models With Structured and Unstructured Uncertainties
,”
Robustness in Identification and Control
,
Springer
,
London
, pp.
159
173
.
39.
Boyd
,
S.
, and
Vandenberghe
,
L.
,
2004
,
Convex Optimization
,
Cambridge University Press
,
New York
.
You do not currently have access to this content.