马尔可夫跳跃非线性系统的高斯和滤波器设计

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本文针对马尔可夫跳跃非线性系统的状态估计问题，提出了Gaussian sum filter (GSF) 的一种新方法。在研究中，作者重点探讨了如何利用高斯和方法来近似条件后验概率密度函数，从而在最小均方误差意义下构建一个通用的GSF框架。这种方法的关键在于利用小的高斯集合设计策略，通过合并初始阶段的高斯成分，有效地降低了计算复杂度。具体而言，作者首先详细介绍了马尔可夫跳跃非线性系统的特性，这种系统在许多实际应用中，如机器人导航、信号处理和控制系统中具有重要意义，其动态模型可能随时间随机变化。传统的滤波算法在处理这类系统的不确定性时可能存在困难，因此引入高斯和滤波器能够更好地捕捉非线性和跳跃性质。在GSF框架中，通过使用一系列独立的高斯分布来逼近状态的概率密度函数，每个高斯分布代表了对系统状态的一种可能性。这种方法允许对复杂的非线性关系进行建模，并且在估计过程中考虑到状态间的跳跃转移。与交互式多模型粒子滤波器（IMMPF）相比，该算法的优势在于能以更低的计算成本实现相近甚至更好的性能。为了验证理论效果，作者进行了数值模拟，结果显示，提出的GSF算法在保持良好估计精度的同时，显著减少了计算量，这使得它在实际应用中具有更高的效率和实用性。此外，文中还讨论了算法的稳定性和鲁棒性，以及可能的扩展和优化方向，这对于理解和改进马尔可夫跳跃非线性系统状态估计的理论和技术具有重要的参考价值。这篇研究论文提供了一种创新的高斯和滤波技术，对于处理马尔可夫跳跃非线性系统中的状态估计问题具有实用价值，不仅提高了估计精度，还优化了计算效率，是当前智能系统估计技术领域的一个重要进展。

Gaussian sum filter of Markov jump non-linear

systems

ISSN 1751-9675

Received on 12th February 2014

Accepted on 28th October 2014

doi: 10.1049/iet-spr.2014.0066

www.ietdl.org

Li Wang

1,2

, Yan Liang

1,2

✉

, Xiaoxu Wang

1,2

, Linfeng Xu

1,2

School of Automation, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China

The Key Laboratory of Information Fusion Technology, Ministry of Education, Xi’an 710072, People’s Republic of China

✉ E-mail: liangyan@nwpu.edu.cn

Abstract: This pap er proposes a Gaussian sum filtering (GSF) framework for the state estimation of Markov jump non-

linear systems. Through presenting the Gaussian sum approximations about the model-con ditioned state posterior

probability density functions, a general GSF framework in the minimum mean square error sense is derived. The

minor Gaussian-set design is utilised to merge the Gaussian components at the beginning, which can effectively limi t

the computational requirements. Simulation result shows that the proposed algorithm demonstrates comparable

performance to the interacting multiple model particle filter with significantly reduced computational cost.

1 Introduction

Markov jump systems (MJSs) involve both time-evolving and

event-driven mechanisms, which have been widely accepted in

many ﬁelds such as target tracking [1, 2], seismic signal

processing [3], process monitoring and fault detection [4]. For the

state estimation problem of MJSs, the recursive optimal estimator

was derived in [5]. However, for most non-linear dynamical and

measurement models, an exact and recursive computation of the

state probability density functions (PDF) is intractable. Deﬁnitely,

one has to pursue suboptimal or approximate non-linear ﬁlters.

Much attention has been paid on the state estimation problem of

non-linear systems. Filtering of the MJSs can be divided into the

following two categories.

The ﬁrst is the state estimation of Markov jump linear systems

(MJLSs), where multiple linear models are stochastically switched

according to a Markov chain. Since one has to match a ﬁlter to

each model, an exponentially increasing number of ﬁlters are

needed, which makes the optimal approach impractical [1]. Hence,

suboptimal techniques must be utilised to avoid the exponentially

increasing number of histories. A simple-minded suboptimal

technique is to keep the N histories with the largest probabilities

and normalise the probabilities. Another effective way is ‘merging’

used in GPB1 (generalised pseudo-Bayesian of order 1), GPB2

and IMM [1, 5, 6], in which the posterior PDFs can be represented

exactly by a sum of Gaussians. Learn from [5], the IMM estimator

can obtain the similar accuracy to GPB2, whereas its computation

burden is similar to that of GPB1.

The second is the state estimation of Markov jump non-linear

systems (MJNLSs), where each model is non-linear. One possible

scheme is to augment the state with mode and then apply the

particle ﬁlter (PF). However, it is computation-intensive by the

fact that the computation burden of PF increases signiﬁcantly as

the state dimension increases. Another ad hoc scheme is that the

extended or unscented Kalman ﬁlter (EKF or UKF) is chosen as

the model-conditioned sub-ﬁlter in the IMM framework for mild

non-linearities [7, 8] or the particle ﬁlter (PF) for strong

non-linearities [9]. In [2], a multiple model particle ﬁlter was

presented for MJNLSs, which obtained superior performance to

the IMM algorithm using EKF. However, the particle cloud of this

ﬁlter can easily degenerate, especially around mode transitions. To

prevent the ﬁlter degenerating, a new particle ﬁlter for MJNLSs

was proposed in [10

], which we refer to as the interacting multiple

model particle ﬁlter (IMMPF). The number of particles per mode

in this ﬁlter is a parameter, which can be selected a priori by the

designer or online based on some adaptive scheme. This particle

ﬁlter for MJNLSs was illustrated more robust than the one in [2]

for the same number of particles. Further research was also carried

out in [11] for the case that the mode switching is state-dependent.

These above algorithms are very promising in accuracy for a class

of problems at the cost of heavy computation load.

For the state estimation of non-linear systems, a new possible

scheme is Gaussian or Gaussian sum approximation to the state

conditional PDF. The advantages are that it can effectively

approximate any PDF as closely as desired (in the L

norm) and

enable a more accurate representation of the non-linearities in the

dynamics and measurement models. The Gaussian sum ﬁlters

(GSF) was developed in [12], in which the conditional probability

density is approximated by a sum of Gaussians. Extensive

researches on GSF have been done in [13–17], which have

become popular in the target tracking community. A

Gaussian-sum based cubature Kalman ﬁlter (CKF) proposed in

[14] showed comparable performance to the PF with reduced

computational cost. However, it was found later that it lacks

robustness where a decrease in accuracy has been observed when

the threshold level for splitting the Gaussian components is very

low. In [15], modiﬁcations were made in the splitting and merging

procedure of the Gaussian components with improved robustness

compared with the algorithm in [14]. An adaptive GSF was

presented in [17] for non-linear tracking problems in high

dimensional data-starved environments such as space surveillance.

These algorithms demonstrate good performance for non-linear

systems.

In fact, Markov jump systems are stochastic non-linear systems. It

will result in a complicated non-linear estimation problem combined

with the non-linearity in system models. To the best of our

knowledge, up to the present, there has no research on GSF design

for the state estimation of MJNLSs. Motivated by the advantages

of the GSF, a GSF framework for MJNLSs based on the Gaussian

sum approximations is developed in this paper to obtain the

compromise of estimation accuracy and computational cost. The

procedure of this ﬁlter design includes the mixing and ﬁltering

steps. In the mixing step, the Minor Gaussian-set design using

moment matching is utilised to control the number of Gaussian

components. The ﬁltering step includes analytical computation and

Gaussian weighted integrals, thus the divided difference ﬁlter

(DDF) [18] based on the second-order Stirling’s interpolation is

chosen as the sub-ﬁlter. Simulation result shows that the proposed

GSF gives comparable performance to the IMMPF [10] with

signiﬁcantly reduced computational cost.

IET Signal Processing

Research Article

IET Signal Process., 2015, Vol. 9, Iss. 4, pp. 335–340

335

The Institution of Engineering and Technology 2015

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