In this paper, the Singular-Polynomial-Fuzzy-Model (SPFM) approach problem and impulse elimination are investigated based on sliding mode control for a class of nonlinear singular system (NSS) with impulses. Considering two numerical examples, the SPFM of the nonlinear singular system is calculated based on the compound function type and simple function type. According to the solvability and the steps of two numerical examples, the method of solving the SPFM form of the nonlinear singular system with (and without) impulse are extended to the more general case. By using the Heine–Borel finite covering theorem, it is proven that a class of nonlinear singular systems with bounded impulse-free item (BIFI) properties and separable impulse item (SII) properties can be approximated by SPFM with arbitrary accuracy. The linear switching function and sliding mode control law are designed to be applied to the impulse elimination of SPFM. Compared with some published works, a human posture inverted pendulum model example and Example 3.2 demonstrate that the approximation error is small enough and that both algorithms are effective. Example 3.3 is to illustrate that sliding mode control can effectively eliminate impulses of SPFM.

中文翻译：

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通过奇异多项式模糊模型方法逼近具有脉冲和滑模控制的非线性奇异系统

在本文中，基于滑模控制的一类具有脉冲的非线性奇异系统（NSS）研究了奇异多项式模糊模型（SPFM）方法问题和脉冲消除问题。考虑两个数值例子，基于复合函数类型和简单函数类型计算非线性奇异系统的SPFM。根据两个数值算例的可解性和步骤，将求解有（和无）冲激非线性奇异系统的SPFM形式的方法推广到更一般的情况。通过使用Heine-Borel有限覆盖定理，证明了一类具有有界无冲项（BIFI）特性和可分离冲量项（SII）特性的非线性奇异系统可以用任意精度的SPFM逼近。设计了线性切换函数和滑模控制律以应用于SPFM的脉冲消除。与一些已发表的作品相比，一个人体姿势倒立摆模型示例和示例3.2表明逼近误差足够小，两种算法都是有效的。例 3.3 是为了说明滑模控制可以有效地消除 SPFM 的脉冲。