Mathematical Problems in Engineering

Volume 2015, Article ID 404986, 8 pages

http://dx.doi.org/10.1155/2015/404986

## The Robust Passive Location Algorithm for Maneuvering Target Tracking

Xi’an Research Institute of High Technology, Xi’an, Shaanxi 710025, China

Received 23 July 2014; Revised 14 September 2014; Accepted 12 October 2014

Academic Editor: Xiao-Sheng Si

Copyright © 2015 Xiaojun Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With the advantages such as high security and far responding distance, the passive location has a broad application in military and civil domains such as radar and aerospace. However, most of the current passive location methods are based on the framework of the probability theory and cannot be used to deal with fuzzy uncertainty in the passive location systems. Though the fuzzy Kalman filter can be used in the uncertainty systems, it could not deal with the abrupt change of state like the maneuvering target which will lead to the filter divergence. Therefore, in order to track the maneuvering target in the fuzzy passive system, we proposed a robust fuzzy extended Kalman filter based on the orthogonality principle and the fuzzy filter in the paper. Conclusion can be made based on the simulation result that this new approach is more precise and more robust than the fuzzy filter.

#### 1. Introduction

With the advantages such as passive receiving and far responding distance, the passive location has been broadly applied in military and civil domains, including radar and geophysics [1, 2]. The current methods of the passive location are the unscented Kalman filter [3], the particle filter [4], and the extended Kalman filter [2]. Extended Kalman filter (EKF) [2] uses the first-order Taylor series expansion to linearize the system where measurement noise is assumed to be Gaussian. The unscented Kalman filter (UKF) [3] has better tracking precision in the nonlinear model, because it transforms the analytic integral operator into an approximate summation operator by a set of deterministic points. But both of the algorithms are not applicable to solve non-Gaussian problems. Particle filter [4] has become popular for nonlinear and/or non-Gaussian filtering and estimation. All of the above methods are based on the certainty model.

But the statistical parameters of the noise may be uncertain due to sensors drift, environmental changes, information incompleteness, and so forth. What is more, the low performance of the noise statistics estimation may lead to poor filter performance or even lead to the divergence of the filter. To solve the problem listed above, some new methods [5–12] are given.

Reference [5] proposes an optimal robust Kalman-type recursive filter for uncertain systems with autocorrelated and cross-correlated noises. Reference [7] uses the H-infinity filter to deal with the uncertain discrete-time systems. Based on the relevance vector machine and gradient descent algorithm, [8] proposes a fuzzy model identification method and [9] gives an off-online fuzzy modeling method. Reference [10] gives a new prediction model for system’s behavior prediction based on belief rule base. The algorithm could use not only numerical data, but human judgmental information with uncertainty as well. Reference [6] proposes a new fuzzy extended Kalman filters (FEKF) method for mobile robots location that uses possibility distributions instead of probability distributions. In the paper, the uncertainty is not necessary to be symmetric and can be described by qualitative knowledge.

Among the above algorithms, [6] is more suitable for passive location and tracking with uncertainty that is based on quantifying uncertainty by trapezoidal possibility distributions. Being motivated by [6], [11] and [12] apply the FEKF in fault prediction and passive location in the nonlinear system with uncertainty, respectively. However, similar to the EKF, the FEKF could not estimate the state in the mismatching model system such as the maneuvering target tracking. The algorithm has poor robustness and is not suitable for maneuvering target tracking. As a result, the passive location may lose tracking ability when the target states change abruptly.

In order to overcome the above disadvantages of the FEKF, combined on the extended orthogonality principle, a robust passive location method is proposed. The algorithm can not only deal with the uncertainty, but also can be suitable for maneuvering target tracking.

We organize this paper as follows. We describe the problem in Section 2 and present the FEKF algorithm in Section 3. Combining the principle of the orthogonality, the robust fuzzy extended Kalman filter (RFEKF) is given to deal with the maneuvering target tracking in Section 4. Section 5 demonstrates the effectiveness of the RFEKF. And we summarize our conclusion in Section 6.

#### 2. The Problem Description

Below is the system equation of the passive location:where stands for the state vector covering velocity and position and so forth, is a measurement vector which can be the time or frequency of arrival, phase-difference rate of change, and so on, is the state function, is the nonlinear measurement function, is input vector, and and are the measurement noise and the process noise, respectively.

The noise model in most passive location algorithm is usually the probability distribution. We will further research the possibility distribution of the noise by using the trapezoidal distributions instead of Gaussian distributions in this paper. The distribution can be asymmetric which is often in the sensor. In addition, the trapezoid with two cuts can be changed to triangle with one cut, line with zero cuts, and so on. The distribution has many merits in the state estimate [6].

Here, is denoted by the fuzzy variable in the universe of discourse and represents a trapezoidal possibility distribution [6] which is shown in Figure 1: