Modelling and Simulation in Engineering

Volume 2017 (2017), Article ID 2783781, 11 pages

https://doi.org/10.1155/2017/2783781

## A New Sparse Gauss-Hermite Cubature Rule Based on Relative-Weight-Ratios for Bearing-Ranging Target Tracking

College of Science, National University of Defense Technology, Changsha, Hunan 410073, China

Correspondence should be addressed to Lijun Peng

Received 4 January 2017; Revised 15 June 2017; Accepted 24 July 2017; Published 11 September 2017

Academic Editor: Ming-Cong Deng

Copyright © 2017 Lijun Peng 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

A new sparse Gauss-Hermite cubature rule is designed to avoid dimension explosion caused by the traditional full tensor-product based Gauss-Hermite cubature rule. Although Smolyak’s quadrature rule can successfully generate sparse cubature points for high dimensional integral, it has a potential drawback that some cubature points generated by Smolyak’s rule have negative weights, which may result in instability for the computation. A relative-weight-ratio criterion based sparse Gauss-Hermite rule is presented in this paper, in which cubature points are kept symmetric in the input space and corresponding weights are guaranteed to be positive. The generation of the new sparse cubature points set is simple and meaningful for practice. The difference between our new sparse Gauss-Hermite cubature rule and other cubature rules is analysed. Simulation results show that, compared with Kalman filter with those types of full tensor-product based Gauss-Hermite rules, our new sparse Gauss-Hermite cubature rule based Kalman filter can lead to a substantially reduced number of cubature points, more stable computation capability, and maintaining the accuracy of integration at the same time.

#### 1. Introduction

Bayesian recursive estimation is commonly used in target tracking, positioning, and signal processing [1]. Generally, a Bayesian recursive estimation algorithm requires a state model and a measurement model. The a posteriori density function can describe the behaviour of the estimated state [2]. Since a closed form solution to the Bayesian recursive estimation is available only for a few special cases [3], such as the linear Gaussian system (which leads to the classical standard Kalman filter), a suboptimal solution is a preferable choice in the general case [4, 5]. Ito and Xiong [4] suggest that a local Gaussian filter can be used to approximate the general Bayesian recursive estimation suboptimally. The core problem of local Gaussian filters is in fact a high dimensional Gaussian weighted integration, which has been studied in numerical analysis; see [6–9] and the references therein. Lots of researches concentrate on the numerical approximation methods to solve the Gaussian weighted integral problem, and different approaches result in different local Gaussian filters, such as the cubature Kalman filter [10], the quadrature Kalman filter [11], and related variants [12–14]. Gauss-Hermite filter, introduced by [2, 4], makes use of the Gauss-Hermite quadrature rule and has the highest accuracy among all the above filters. However, it suffers from the curse of dimensionality since the number of cubature points increases exponentially with the dimension of the state.

To avoid the curse of dimensionality, several sparse rules are developed. Smolyak rule [15] is one of the useful tools to generate a small number of cubature points for high dimensional integral. The computational cost does not increase exponentially by using this sparse grid method. Reference [16] compares the approximation accuracies of various sparse grid methods, including trapezoidal rule, Crenshaw Curtis rule, and Gauss Patterson rule. Reference [17] combines Smolyak’s rule and Gaussian Quadrature rule for high dimensional likelihood function in economic models. Jia et al. [18, 19] make a tracking comparison among Smolyak’s rule based Gauss-Hermite filter of different levels and the traditional cubature filters in the context of determination of the spacecraft attitude and the lower-earth orbit satellite orbit. Simulations prove the effectiveness of Smolyak’s rule based Gauss-Hermite filter. Reference [20] proposes the multiple sparse grid Gauss-Hermite filter based on sparse grid Gauss-Hermite filter and state-space partitioning. It is claimed that the computational burden is further reduced with respect to the Gauss-Hermite filter and the sparse grid Gauss-Hermite filter.

However, as Heiss and Winschel [17] mentioned, there exists a potential drawback that some of the cubature points have negative weights, which may result in instability for computation. So we come up with the idea to avoid this potential drawback. In this paper, a classification of full tensor-product Gauss-Hermite cubature points is obtained by using equivalence classification by position permutation and signum function. There exist plenty of cubature points with low weights, which can be deleted from the cubature scheme directly considering a relative-weight-ratio threshold. Comparing with the Smolyak-based rule, our construction of the new Gauss-Hermite rule is simple and practically meaningful. The corresponding weights of the cubature points are all positive; meanwhile, the full symmetry property still remains. There also exist some interesting relationships among the Gauss-Hermite filter, the 3rd embedding cubature filter, the sparse Gauss-Hermite filter, and the Unscented Kalman filter.

The remainder of this paper is organized as follows: In Section 2, a brief review of the nonlinear system and its Bayesian recursive estimation framework is presented. In Section 3, a cycle of general local Gaussian filter is presented, which offers six kinds of Gaussian weighted integrals. This leads to various specific local Gaussian filters by various choice of the cubature points set and the corresponding weights. In Section 4, a full tensor-product Gauss-Hermite integral cubature rule is presented. By introducing a simple and elegant operator, the full tensor-product based cubature points can be divided into different categories. A small set of positive weighted cubature points is generated by a threshold of relative-weight-ratio. Relationships between the new sparse Gauss-Hermite filter and the traditional cubature filters are analysed as well. In Section 5, a typical Bearing-Ranging tracking problem with a general 2D manoeuvring target motion is demonstrated to test the performance of our new sparse Gauss-Hermite filter. Some conclusions are given in Section 6.

#### 2. Bayesian Recursive Estimation Algorithm

Consider the following discrete nonlinear system:

where (1) and (2) are the motion model and the measurement model of the system, respectively; is the state of the system and is the measurement; is the process noise and is the measurement noise. is independent with and . By the discrete time Chapman-Kolmogoroff equation,where . By the Bayesian formula,where . Equations (3) and (4) are the time update formula and the measurement update formula. Figure 1 shows the recursive Bayesian estimation algorithm. When is obtained, we can estimate the state and its covariance matrix by minimizing the mean square error. The results are presented as follows:However, when becomes an arbitrary probability density distribution, it is difficult to calculate the high dimensional integrations in (5) directly. A suboptimal way is to place normal distributions on , , and ; thus the first-order moment and second-order moment can be used to describe the posterior probability. In other words, at time epoch , the posterior density isAt time epoch , the predicted probability density of the state isCorrespondingly, the posterior density of the state is