Wang Jun-Wei, Wu Huai-Ning, Li Han-Xiong
Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China.
IEEE Trans Syst Man Cybern B Cybern. 2012 Jun;42(3):927-38. doi: 10.1109/TSMCB.2012.2185046. Epub 2012 Feb 7.
In this paper, a distributed fuzzy control design based on Proportional-spatial Derivative (P-sD) is proposed for the exponential stabilization of a class of nonlinear spatially distributed systems described by parabolic partial differential equations (PDEs). Initially, a Takagi-Sugeno (T-S) fuzzy parabolic PDE model is proposed to accurately represent the nonlinear parabolic PDE system. Then, based on the T-S fuzzy PDE model, a novel distributed fuzzy P-sD state feedback controller is developed by combining the PDE theory and the Lyapunov technique, such that the closed-loop PDE system is exponentially stable with a given decay rate. The sufficient condition on the existence of an exponentially stabilizing fuzzy controller is given in terms of a set of spatial differential linear matrix inequalities (SDLMIs). A recursive algorithm based on the finite-difference approximation and the linear matrix inequality (LMI) techniques is also provided to solve these SDLMIs. Finally, the developed design methodology is successfully applied to the feedback control of the Fitz-Hugh-Nagumo equation.
本文针对一类由抛物型偏微分方程（PDEs）描述的非线性空间分布系统的指数镇定问题，提出了一种基于比例 - 空间导数（P - sD）的分布式模糊控制设计方法。首先，提出了一种Takagi - Sugeno（T - S）模糊抛物型PDE模型，以精确表示非线性抛物型PDE系统。然后，基于T - S模糊PDE模型，结合PDE理论和李雅普诺夫技术，开发了一种新型的分布式模糊P - sD状态反馈控制器，使得闭环PDE系统以给定的衰减率指数稳定。根据一组空间微分线性矩阵不等式（SDLMIs）给出了指数镇定模糊控制器存在的充分条件。还提供了一种基于有限差分近似和线性矩阵不等式（LMI）技术的递归算法来求解这些SDLMIs。最后，将所开发的设计方法成功应用于Fitz - Hugh - Nagumo方程的反馈控制。
