Citation: Wen, C.; Lin, Z. A
Gradually Linearizing Kalman Filter
Bank Designing for Product-Type
Strong Nonlinear Systems. Electronics
2022, 11, 714. https://doi.org/
10.3390/electronics11050714
Academic Editor: Teresa Orlowska-
Kowalska
Received: 12 January 2022
Accepted: 21 February 2022
Published: 25 February 2022
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Article
A Gradually Linearizing Kalman Filter Bank Designing for
Product-Type Strong Nonlinear Systems
Chenglin Wen and Zhipeng Lin *
School of Affiliation, Hangzhou Dianzi University, Hangzhou 310018, China; wencl@hdu.edu.cn
* Correspondence: lin_zp1996@163.com
Abstract:
Our study aimed to improve the poor performance of existing filters, such as EKF, UKF and
CKF, that results from their weak approximation ability to nonlinear systems. This paper proposes
a new extended Kalman filter bank focusing on a class of product-type strong nonlinear systems
composed by system state variables, time-varying parameters and non-linear basic functions. Firstly,
the non-linear basic functions are defined as hidden variables corresponding to system state variables,
and then the strong nonlinear systems are described simplistically. Secondly, we discuss building
two dynamic models between their future values of parameters, as well as hidden variables and their
current values based on the given prior information. Thirdly, we recount how an extended Kalman
filter bank was designed by gradually linearizing the strong nonlinear systems about system state
variables, time-varying parameters and hidden variables, respectively. The first extended Kalman
filter about future hidden variables was designed by using these estimates of the state variables
and parameters, as well as hidden variables at current. The second extended Kalman filter about
future parameters variables was designed by using these estimates of the current state variables
and parameters, as well as future hidden variables. The third extended Kalman filter about future
state variables was designed by using these estimates of the current state variables, as well as future
parameters and hidden variables. Fourthly, we used digital simulation experiments to verify the
effectiveness of this method.
Keywords:
product type; time-varying parameters; hidden variables; strong nonlinear systems;
dynamic models; gradually linearizing
1. Introduction
With the development of the times, filtering theory has played an important role in
various fields at the domestic and international level, especially in national defense, military
and other fields, such as tracking navigation, signal processing, automatic control, target
tracking, etc. [
1
–
6
]. Kalman (R.E. Kalman) proposed a dynamic recursive state estimation
method called the Kalman filter (KF) in 1960 [
7
]. The KF takes into account the statistical
characteristics of the estimated and observed measures, and it designs an optimal filter
based on the minimum mean square error to solve the problem of state estimation and target
tracking in linear Gaussian systems [
8
,
9
]. However, in practical applications, KF is difficult
to use to solve nonlinear models [
10
,
11
]. Bucy proposed the extended Kalman filter (EKF)
method to solve nonlinear problems [
12
]. EKF is a function linear approximation method
for nonlinear system models. It retains the Taylor expansion of the nonlinear function to
the first-order term, thus successfully solving the nonlinear Kalman filter problem [
13
]. The
EKF method has been successfully applied to state estimation and target tracking in the
fields of vehicle control [
14
], traffic dynamics [
15
] and power systems [
16
]. However, since
all the higher-order terms in the Taylor expansion are rounded, a large truncation error
will occur in EKF [
17
]. Moreover, as the nonlinearity of the system increases, the filtering
performance will gradually decrease or even diverge. Since, Zhou et al. proposed a Strong
Tracking Filter (STF)in order to make up for the limitations of EKF. This method introduces
Electronics 2022, 11, 714. https://doi.org/10.3390/electronics11050714 https://www.mdpi.com/journal/electronics
