A Novel Suboptimal Method for Solving Polynomial Filtering Problems
A Novel Suboptimal Method for Solving Polynomial Filtering Problems
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一种解决多项式滤波问题的新型次优方法
DOI：
10.1016/j.automatica.2015.09.001
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Yau Stephen S.-T.
中科院分区：
文献类型：
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作者：
Luo Xue;Jiao Yang;Chiou Wen-Lin;Yau Stephen S.-T.
本文导出了非线性滤波问题，特别是多项式滤波问题的条件中心矩的随机微分，并通过求解这个发展方程发展了一种新的次优方法。其基本思想是通过包含原非线性系统的条件中心矩来增广原非线性系统的状态，使得增广后的状态截断后形成所谓的双线性系统。在我们的推导过程中，可以清楚地看到，线性滤波问题的条件中心矩的随机微分（即f，g和h都是最多一次多项式）自动形成一个封闭的系统，而无需截断。这给出了线性问题存在最优滤波的一个原因。相反，条件中心矩一般构成一个无穷维系统。为了将其简化为封闭形式，我们让所有足够高的中心矩为零，就像Carleman方法中所做的那样（Germani等人，2007年）。因此，通过处理双线性系统，提出了一种新的次优方法。对三次传感器问题进行了数值模拟，验证了算法的精度和数值稳定性。
In this paper we derive the stochastic differentials of the conditional central moments of the nonlinear filtering problems, especially those of the polynomial filtering problem, and develop a novel suboptimal method by solving this evolution equation. The basic idea is to augment the state of the original nonlinear system by including the original states’ conditional central moments such that the augmented states form a so-called bilinear system after truncating. During our derivation, it is clear to see that the stochastic differentials of the conditional central moments of the linear filtering problem (ie, f, g and h are all at most degree one polynomials) form a closed system automatically without truncation. This gives one reason for the existence of optimal filtering for linear problems. On the contrary, the conditional central moments form an infinite dimensional system, in general. To reduce it to a closed-form, we let all the high enough central moments to be zero, as one did in the Carleman approach (Germani et al., 2007). Consequently, a novel suboptimal method is developed by dealing with the bilinear system. Numerical simulation is performed for the cubic sensor problem to illustrate the accuracy and numerical stability.