Low Design-Cost Fuzzy Controllers for Robust Stabilization of Nonlinear Partial Differential Systems

Abstract

In general, the design cost of a fuzzy controller for nonlinear partial differential systems (PDSs) is very expensive. In this paper, a new approach for robust fuzzy $H_{\infty }$ stabilization design is developed for a class of $N$ -dimensional nonlinear parabolic PDSs. Further, two low design-cost robust fuzzy controllers, called the robust fuzzy area-controller and point-controller, as well as a normal design-cost robust fuzzy full-controller are proposed for this problem. The difference between the three control designs lies in their controller placement in the spatial domain. First, we present the $N$ -dimensional parabolic Takagi–Sugeno (T–S) fuzzy PDS based on the knowledge-based fuzzy system technique. Next, these three robust fuzzy controllers are constructed via solving diffusion matrix inequality (DMI) problems. With the proposed simple but general method using the Poincare inequality, the linear matrix inequality problems are provided to replace DMI problems for the robust fuzzy $H_{\infty }$ stabilization designs for computational simplicity. Further, the comparison of these three robust fuzzy controllers are demonstrated to enable a designer to select a low-cost option. Finally, a simulation example is provided to illustrate the design procedure and verify the performance of the proposed designs.