This article is concerned with the polynomial filtering problem for a class of nonlinear stochastic systems governed by the Itô differential equation. The system under investigation involves polynomial nonlinearities, unknown‐but‐bounded disturbances, and state‐ and disturbance‐dependent noises ((x,d)‐dependent noises for short). By expanding the polynomial nonlinear functions in Taylor series around the state estimate, a new polynomial filter design method is developed with hope to reduce the conservatism of the existing results. In virtue of stochastic analysis and inequality technique, sufficient conditions in terms of parameter‐dependent linear matrix inequalities (PDLMIs) are derived to guarantee that the estimation error system is input‐to‐state stable in probability. Moreover, the desired polynomial matrix can be obtained by solving the PDLMIs via the sum‐of‐squares approach. The effectiveness and applicability of the proposed method are illustrated by two numerical examples with one concerning the permanent magnet synchronous motor.

中文翻译：

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具有状态和扰动相关噪声的非线性随机系统的多项式滤波

本文关注一类由Itô微分方程控制的非线性随机系统的多项式滤波问题。研究中的系统涉及多项式非线性，未知但有界的扰动以及状态和扰动相关的噪声（（x，d）-取决于噪声）。通过围绕状态估计展开泰勒级数中的多项式非线性函数，开发了一种新的多项式滤波器设计方法，以期减少现有结果的保守性。借助随机分析和不等式技术，可以得出足够的条件，取决于参数相关的线性矩阵不等式（PDLMI），以确保估计误差系统的输入状态对概率稳定。此外，可以通过平方和方法求解PDLMI来获得所需的多项式矩阵。通过两个数值示例说明了该方法的有效性和适用性，其中一个涉及永磁同步电动机。