基于T-S模糊模型的非线性分布参数系统极点配置P-sD控制

Pole Assignment P-sD Control for Nonlinear Distributed Parameter Systems Based on T-S Fuzzy Model

采用Lyapunov直接法讨论分布参数系统的稳定性,建立分布参数系统的Lyapunov函数。为保证分布参数系统稳定,应使Lyapunov函数对时间的微分小于0。由于分布参数系统中存在空间一次微分项与二次微分项,Lyapunov函数对时间的微分中将出现常数项与积分项,针对常数项,引入空间一次微分项来抵消;针对积分项,引入对应的状态反馈来使系统稳定。利用边界条件量化Lyapunov函数对时间的微分中的各项,从而设计控制器,这是一种新的设计P-sD状态控制的方式。其中状态反馈的部分采用极点配置的方法来设计,当分布参数系统中出现状态变量的非线性函数时,采用T-S模糊模型表达,可以通过状态变量的线性组合精确描述非线性项,以便极点配置设计状态反馈。针对两个非线性分布参数数学模型进行仿真,结果证明设计的P-sD状态控制器可以使分布参数系统稳定,并达到期望的效果。

The Lyapunov direct method is used to prove the stability of the distributed parameter system,and the Lyapunov function of distributed parameter system is established.In order to ensure the stability of the distributed parameter system,the time differential of Lyapunov function should be less than 0.Because of the space differential term and the second differential term in the distributed parameter system,the constant term and the integral term appears in the time differential of Lyapunov function.For the constant term,a space differential term is introduced to offset;for the integral term,a suitable state feedback is introduced to make the system stable.By using the boundary condition to quantify the constant term in the differential of Lyapunov function to time and canceling it,the state feedback of pole assignment design is more instructive,and this is a new way of designing P-sD state control.The part of state feedback is designed by the method of pole assignment.When the nonlinear function of state variable appears in the distributed parameter system,it is expressed by the T-S fuzzy model,the nonlinear term can be accurately described by linear combination of state variables,so as to design state feedback of pole assignment.The simulation results of two nonlinear distributed parameter models show that the designed P-sD state controller can make the distributed parameter system stable and achieve the desired effect.