The Kalman filter (KF) is optimal with respect to minimum mean square error (MMSE) if the process noise and measurement noise are Gaussian. However, the KF is suboptimal in the presence of non-Gaussian noise. The maximum correntropy criterion Kalman filter (MCC-KF) is a Kalman-type filter that uses the correntropy measure as its optimality criterion instead of MMSE. In this paper, we modify the correntropy gain in the MCC-KF to obtain a new filter that we call the measurement-specific correntropy filter (MSCF). The MSCF uses a matrix gain rather than a scalar gain to provide better selectivity in the way that it handles the innovation vector. We analytically compare the performance of the KF with that of the MSCF when either the measurement or process noise covariance is unknown. For each of these situations, we analyze two mean square errors (MSEs): the filter-calculated MSE (FMSE) and the true MSE (TMSE). We show that the FMSE of the KF is less than that of the MSCF. However, the TMSE of the KF is greater than that of the MSCF under certain conditions. Illustrative examples are provided to verify the analytical results.

References

References
1.
Reif
,
K.
,
Gunther
,
S.
,
Yaz
,
E.
, and
Unbehauen
,
R.
,
2000
, “
Stochastic Stability of the Continuous-Time Extended Kalman Filter
,”
IEE Proc. Control Theory Appl.
,
147
(
1
), pp.
45
72
.
2.
Sarkka
,
S.
,
2007
, “
On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems
,”
IEEE Trans. Autom. Control
,
52
(
9
), pp.
1631
1641
.
3.
Julier
,
S.
,
Uhlmann
,
K.
, and
Durrant
,
H.
,
1995
, “
A New Approach for Filtering Nonlinear Systems
,” American Control Conference (
ACC
), Seattle, WA, June 21–23, pp.
1628
1632
.
4.
Kulikova
,
M. V.
,
2017
, “
Square-Root Algorithms for Maximum Correntropy Estimation of Linear Discrete-Time Systems in Presence of Non-Gaussian Noise
,”
Syst. Control Lett.
,
108
, pp.
8
15
.
5.
Kulikov
,
G. Y.
, and
Kulikova
,
M.
,
2018
, “
Estimation of Maneuvering Target in the Presence of Non-Gaussian Noise: A Coordinated Turn Case Study
,”
Signal Process.
,
145
, pp.
241
257
.
6.
Yin
,
L.
,
Deng
,
Z.
,
Huo
,
B.
,
Xia
,
Y.
, and
Li
,
C.
,
2018
, “
Robust Derivative Unscented Kalman Filter Under non-Gaussian Noise
,”
IEEE Access
,
6
, pp.
33129
33136
.
7.
Cinar
,
G. T.
, and
Principe
,
J. C.
,
2012
, “
Hidden State Estimation Using the Correntropy Filter With Fixed Point Update and Adaptive Kernel Size
,”
International Joint Conference on Neural Networks
(
IJCNN
), Brisbane, Australia, June 10–15, pp.
1
6
.
8.
Izanloo
,
R.
,
Fakoorian
,
S. A.
,
Sadoghi
,
H.
, and
Simon
,
D.
,
2016
, “
Kalman Filtering Based on the Maximum Correntropy Criterion in the Presence of Non-Gaussian Noise
,”
50th Annual Conference on Information Science and Systems
(
CISS
), Princeton, NJ, Mar. 16–18, pp.
530
535
.
9.
Sarkka
,
S.
, and
Nummenmaa
,
A.
,
2009
, “
Recursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations
,”
IEEE Trans. Autom. Control
,
54
(
3
), pp.
596
600
.
10.
Mehra
,
R.
,
1970
, “
On the Identification of Variances and Adaptive Kalman Filtering
,”
IEEE Trans. Autom. Control
,
15
(
2
), pp.
175
184
.
11.
Haykin
,
S. S.
,
2004
,
Kalman Filtering and Neural Networks
,
Wiley
, Hoboken, NJ.
12.
Li
,
W.
, and
Jia
,
Y.
,
2010
, “
H-Infinity Filtering for a Class of Nonlinear Discrete-Time Systems Based on Unscented Transform
,”
Signal Process.
,
90
(
12
), pp.
3301
3307
.
13.
Sorenson
,
H. W.
, and
Sacks
,
J.
,
1971
, “
Recursive Fading Memory Filtering
,”
Inf. Sci.
,
3
(
2
), pp.
101
119
.
14.
Ge
,
Q.
,
Shao
,
T.
,
Duan
,
Z.
, and
Wen
,
C.
,
2016
, “
Performance Analysis of the Kalman Filter With Mismatched Noise Covariances
,”
IEEE Trans. Autom. Control
,
61
(
12
), pp.
4014
4019
.
15.
Simon
,
D.
,
2006
,
Optimal State Estimation: Kalman, H-Infinity, and Nonlinear Approaches
,
Wiley
, Hoboken, NJ.
16.
Liu
,
W.
,
Pokharel
,
P. P.
, and
Príncipe
,
J. C.
,
2007
, “
Correntropy: Properties and Applications in Non-Gaussian Signal Processing
,”
IEEE Trans. Signal Process.
,
55
(
11
), pp.
5286
5298
.
17.
He
,
R.
,
Zheng
,
W.-S.
, and
Hu
,
B.-G.
,
2011
, “
Maximum Correntropy Criterion for Robust Face Recognition
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
33
(
8
), pp.
1561
1576
.
18.
Higham
,
N. J.
,
2002
,
Accuracy and Stability of Numerical Algorithms
,
SIAM
, Philadelphia, PA.
19.
Daum
,
F.
,
2005
, “
Nonlinear Filters: Beyond the Kalman Filter
,”
IEEE Aerosp. Electron. Syst. Mag.
,
20
(
8
), pp.
57
69
.
20.
Bavdekar
,
V. A.
,
Deshpande
,
A. P.
, and
Patwardhan
,
S. C.
,
2011
, “
Identification of Process and Measurement Noise Covariance for State and Parameter Estimation Using Extended Kalman Filter
,”
J. Process Control
,
21
(
4
), pp.
585
601
.
You do not currently have access to this content.