Automatica 82 (2017) 171–180
Contents lists available at ScienceDirect
Automatica
journal homepage: www.elsevier.com/locate/automatica
Brief paper
Filter design based on characteristic functions for one class of
multi-dimensional nonlinear non-Gaussian systems
✩
Chenglin Wen
a
, Xingshuo Cheng
a
, Daxing Xu
b
, Chuanbo Wen
c
a
Institute of Systems Science and Control Engineering, School of Automation, Hangzhou Dianzi University, Hangzhou, 310018, China
b
College of Electrical and Information Engineering, Quzhou University, Quzhou, Zhejiang, 324000, China
c
College of Electrical Engineering, Shanghai Dianji University, Shanghai, 201306, China
a r t i c l e i n f o
Article history:
Received 14 September 2015
Received in revised form
11 December 2016
Accepted 22 February 2017
Available online 18 May 2017
Keywords:
Non-Gaussian system
Characteristic function
Multi-dimensional measurement
Filter gain
Performance index
a b s t r a c t
A filter based on characteristic functions is developed in this paper, to fit to a class of non-Gaussian dy-
namical systems, which state models and measurement models are all nonlinear and multi-dimensional.
The new filter overcomes limitations and expands the application of this kind of filter, which is proved to
just fit to one special kind of systems with multi-dimensional linear state models and one-dimensional
nonlinear measurement models. Firstly, the filter using characteristic function is introduced and its lim-
itation is analysed. Then, we design the new filter to fit to nonlinear states and multi-dimensional mea-
surements. Thirdly, the matrix format of performance index is presented to match to the new filter gain,
and the weighting function vector is given to ensure the uniform boundedness of such a performance
index. Finally, the new filter gain can be obtained by minimizing this performance index, and the process
of filtering design is accomplished. Simulation examples are given to illustrate the effectiveness of the
proposed filter design scheme.
© 2017 Elsevier Ltd. All rights reserved.
1. Introduction
In Guo and Wang (2006); Zhou, Wang, and Zhou (2008); Zhou,
Zhou, Wang, Guo, and Chai (2010), authors have displayed the dis-
tribution function and PDF tracking filters design, which are based
on characteristic functions and fit to the MIMO (Multiple-Input
Multiple-Output) systems. Analytic study shows that this kind of
filter only fits to systems with the one-dimensional measurement,
and incapacitates for multi-dimensional measurement. However,
the systems with multi-dimensional measurements are always en-
countered in practice. Motivated by those reasons, a new filter
based on characteristic function is designed to overcome the limi-
tation in this paper. It fits to the multi-dimensional measurement
models as well as the multi-dimensional nonlinear state models.
There are some existing filters in Gaussian and no-Gaussian sys-
tems, which can be reviewed as follows.
✩
This work was supported in part by National Natural Science Foundation of
China U1509203, 61333011, 61371064, 61271144 and Zhejiang Provincial Nature
Science Foundation of China (Grant LR17F030005). This paper was not presented at
any IFAC meeting. This paper was recommended for publication in revised form by
Associate Editor Antonio Vicino under the direction of Editor Torsten Söderström.
E-mail addresses: wencl@hdu.edu.cn (C. Wen), chengxingshuo@126.com
(X. Cheng), daxingxu@163.com (D. Xu), chuanbowen@163.com (C. Wen).
For Gaussian systems, among a variety of existing filtering
methods, the Kalman filtering approach has been widely adopted
(Anderson & Moore, 1979; Goodwin & Sin, 1984; Jazwinski, 1970).
It is well known that the key of KF is to design an appropriate filter
gain matrix to ensure that the covariance matrix of the estimation
error is minimum (Francois et al., 2013; Kalman, 1960). KF is just
suitable to the linear Gaussian systems, in the case that process
noises and measurement noises are white Gaussian noise (Wen
& Zhou, 2002). In order to extend the scope of application of the
filter, many filters have been designed, such as Extended Kalman
Filter (EKF), Unscented Kalman Filter (UKF), Cubature Kalman Filter
(CKF), Strong Tracking Filter (STF), H
∞
/H
2
filter (Bucy & Renne,
1971; Hou, Jing, & Yang, 2013; Li, Peng, Chen, & Liu, 2014; Li & Sun,
2013; Sunahara, 1970; Wang, Liu, & Liu, 2008). In the derivation
process of EKF, the nonlinear systems are linearized, and fits to
the theory of KF. However, EKF is linearized based on the first-
order Taylor expansion, which just fits to weak nonlinear Gaussian
systems. UKF is a method based on Unscented Transformation (UT)
and the Kalman Filtering theory, which fits to stronger nonlinear
Gaussian systems. By Cubature Transformation (CT), CKF has been
proposed via rigorous derivation to improve the stabilization and
accuracy comparing with UKF for high-dimensional systems. By
introducing the fading factor, STF has been designed according
to EKF to fit to nonlinear systems with uncertainty models (Hu,
Wang, Gao, & Stergioulas, 2012; Li & Sun, 2013). Also polynomial
http://dx.doi.org/10.1016/j.automatica.2017.03.041
0005-1098/© 2017 Elsevier Ltd. All rights reserved.