Interacting T-S fuzzy particle filter algorithm for transfer probability matrix of adaptive online estimation model
Introduction
In the theory of stochastic processes, the filtering problem is a mathematical model for a number of state estimation problems in signal processing and related fields. The general idea is to establish a “best estimate” for the true value of the system state from an incomplete, potentially noisy set of observations on the system. In dealing with linear systems, the Kalman filter [1] theory is the optimal linear Bayesian estimation algorithm. For nonlinear filtering problem, the extended Kalman filtering (EKF) [2], [3] was proposed to further apply the Kalman filtering theory. Subsequently, several second-order generalized Kalman filtering [4] methods were proposed and applied to further improve the estimation performance of Kalman filtering for nonlinear systems. However, with the aggravation of the nonlinearity of the dynamic system, the performance of EKF drops sharply. In order to solve this problem, Julier and Uhlmann proposed unscented Kalman filter (UKF) [5] algorithm. Unlike EKF, the UKF [6], [7], [8] directly uses nonlinear model to approximate the distribution of state random variables by selecting a small number of sample points. For weakly nonlinear Gaussian systems, UKF and EKF have the same estimation performance, but for strongly nonlinear Gaussian systems, the UKF method can obtain better filtering result. However, EKF and UKF have not got rid of the system must be Gaussian constraint, which leads to the poor performance of the non-Gaussian system.
Nowadays, particle filtering [9], as an effective method to deal with nonlinear non-Gaussian problem, is suitable for any nonlinear non-Gaussian complex stochastic system which can be represented by state space model, and has become a hot topic in the field of nonlinear filtering. In fact, this method uses Monte Carlo simulation to complete a recursive Bayesian filtering, whose core is to draw a random set of particles with corresponding weights to represent the desired posterior density. The construction of importance density function in particle filtering is a challenging problem. Therefore, some improved particle filtering methods have emerged, such as the unscented particle filtering (UPF) [10] the Rao-Blackwellized particle filtering (RBPF) [11] and the generalized particle filtering based on Gaussian mixing [12]. In addition, Yang et al. [13] employed K-L divergence (KLD) to measure the difference between the feedback posterior probability distribution and the real posterior distribution, and proposed a multi-variable feedback particle filter.
For obtaining more effective importance density function, the aforementioned methods need to construct accurate target motion model to approach the real model, but in nonlinear target tracking system, there is great uncertainty in target motion model. In view of the uncertainty of maneuvering model, an interacting multiple model particle filter algorithm (IMMPF) was proposed by Boers et al. [14]. Then some improved IMMPF algorithms [15], [16], [17], [18] were proposed to improve the computational efficiency compared with the traditional IMMPF algorithm. However, the IMM-type algorithms have shortcomings such as insufficient model set and lack of prior information. If the spatial-temporal semantic information can be incorporated, the problem of lack of prior information can be solved. It is well known that fuzzy modeling methods can be integrated into prior information of a target by defining different sets of semantic models, thus the research of fuzzy model-based has become one of the research hot spots in complex nonlinear non-Gaussian systems in recent years. For example, the T-S fuzzy modeling method [19], [20], [21], [22] can not only solve the problem of lack of prior information, but also solve the problem of insufficient model set. It divides the nonlinear system into multiple linear subsystems, and the models in T-S method are not set in advance, each linear model is activated according to the motion state of the target, and the motion model of the target is obtained by adaptively adjusting the weight of each model.
There have been many results on the research of T-S fuzzy model. Li et al. [23] proposed a new fuzzy c-regression model clustering algorithm to find the fuzzy structure of the T-S fuzzy model, and used orthogonal least squares to estimate the consequence parameters. Chang et al. [24] proposed a fuzzy c-regression state model (FCRSM) for nonlinear systems represented by equations of state. Catia et al. [25] proposed a T-S fuzzy modeling method based on Hybrid Fuzzy Clustering (MFC) for medical care classification. Li et al. [26] showed that the traditional bell-shaped Gaussian function may not be suitable for matching hyperplane shape clustering algorithm, and proposed a hyperplane shape fuzzy membership function for T-S fuzzy model identification. In order to optimize the fuzzy parameters of the membership function, Simon [27] proposed an optimization method of the membership function through the extended Kalman filter (EKF). However, when there is non-Gaussian noise or high nonlinearity, the EKF method may cause serious problems. In order to overcome the shortcomings of the membership function based on EKF, Chung et al. [28] proposed a membership function adjustment method based on particle filtering. Since Type-2 fuzzy sets have excellent characteristics in dealing with uncertain problems, Zou et al. [29] proposed an improved type-2 fuzzy c-regression model clustering (FCRM) and a new hyperplane Gaussian membership function for T-S fuzzy modeling. Compared with type-1 FCRM, type -2 FCRM can reduce the influence of errors.
The T-S fuzzy model algorithms proposed above were basically used for control systems or medical care classification, and had not been applied to nonlinear maneuvering target tracking systems. Li et al. proposed to use the T-S fuzzy model in a maneuvering target tracking system for the first time, and some related research results [30], [31], [32] had been obtained. In [30], they pointed out that the FCRM algorithm based on correntropy can be effective captured the higher moments of error probability. However, semantics is often obtained through training on large scale dataset, the identification of parameters of the T-S fuzzy model with a small amount of data is difficult using this method. Considering the advantages of the T-S fuzzy model particle filtering [31], [32] in modeling fuzzy semantic information, as in the IMM-type methods, we focus on research on interacting T-S fuzzy model particle filtering for maneuvering target tracking, by introducing the interacting process, which has ability to effectively adjust the probability of each model.
In this paper, a novel interacting T-S fuzzy modeling particle filtering algorithm (ITS-PF) is proposed to solve the modeling problem of nonlinear non-Gaussian dynamic system in maneuvering target tracking. The main contributions are as follows: 1) The proposed algorithm can be regarded as a switching dynamical model. But unlike popular interacting multiple model-based method, it derives the probabilistic switching model by fuzzy sets through using their closeness degree to incorporate the spatial-temporal features. 2) A fuzzy C-regression clustering method based on maximum correntropy principle and spatial-temporal information is proposed for parameter identification in T-S fuzzy model. Meanwhile, the consequence parameters are identified based on the modified strong tracking method. And we use the Linear Matrix Inequalities (LMI) to analyze the stability of the proposed algorithm. 3) The importance density function is constructed by using the interacting T-S fuzzy model, which contains a wealth of prior knowledge and the latest observation information, effectively improving the diversity of particles. 4) The proposed algorithm is used to deal with the state estimation problem in the maneuvering target tracking system.
The rest of this paper is organized as follows. The proposed interacting T-S fuzzy modeling particle filtering algorithm is presented in the Section 2. The simulation results that compare the performances of all algorithms are described in the Section 3. Finally, some conclusions of the proposed algorithm are given in Section 4.
Section snippets
The proposed algorithm
Consider the nonlinear discrete-time dynamic system: where denotes discrete time, and denote some appropriate nonlinear functions respectively. is dimensional state vector. is dimensional observation vector. is the process noise, with zero mean value and covariance , and is the non-Gaussian observation noise.
Simulation results and analysis
In order to compare the performance of the ITS-PF with that of Interacting Multiple Model (IMM) [36], Interacting Multiple Model Unscented Kalman Filtering (IMMUKF) [37], Interacting Multiple Model Particle Filtering (IMMPF) [14] and Interacting Multiple Model Rao-Blackwellized Particle Filtering (IMMRBPF) [38]. In Section 3.1, multiple simulation radar data maneuvering target tracking problem are analyzed, which make sense in defense application. In Section 3.2, the tracking results of real
Conclusion
In this paper, a novel interacting T-S fuzzy modeling particle filtering algorithm is proposed. Based on the closeness degree between fuzzy sets, the transition probabilities between fuzzy sets are derived, which replaces the mode transition probabilities, and a general interacting T-S fuzzy model framework is constructed. The premise parameters of T-S fuzzy model are identified by kernel fuzzy C regression clustering algorithm, which combines the spatial-temporal information and correntropy,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (61773267), Science & Technology Program of Shenzhen (Grant No JCYJ20190808120417257).
Xiao-li Wang was born in 1992. She is a doctoral student of school of information engineering, Shenzhen University. Her research interests include multi-sensor information fusion, target tracking, etc.
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Xiao-li Wang was born in 1992. She is a doctoral student of school of information engineering, Shenzhen University. Her research interests include multi-sensor information fusion, target tracking, etc.
Wei-xin Xie was born in 1941. He is a professor and doctoral tutor of School of information engineering, Shenzhen University. His research interests include radar target recognition, multi-sensor information fusion, fuzzy information processing, and image processing, etc.
Liang-qun Li was born in 1979. He is a professor of school of information engineering, Shenzhen University. His research interests include multi-sensor information fusion, target tracking, etc.