A Membership-Function-Dependent Approach to Design Fuzzy Pointwise State Feedback Controller for Nonlinear Parabolic Distributed Parameter Systems With Spatially Discrete Actuators

This paper gives a membership-function-dependent approach to solve the design problem of fuzzy pointwise state feedback controller for a class of nonlinear distributed parameter systems modeled by semilinear parabolic partial differential equations (PDEs), where only a few actuators are discretely distributed in space. In the proposed design method, a Takagi–Sugeno (T–S) fuzzy PDE model obtained by using the sector nonlinearity method is first utilized to accurately describe the nonlinear spatiotemporal dynamics of the PDE system. As only the state information at some known specified points in the spatial domain (i.e., the pointwise state information) is available for the controller design, the favorable property offered by sharing all the same premises in the fuzzy PDE plant model and fuzzy controller cannot be employed to develop the fuzzy control design method. To overcome this drawback, a linear matrix inequality (LMI) relaxation technique is developed to enhance the stabilization ability of the fuzzy controller. Based on the T–S fuzzy PDE model, a membership-function-dependent fuzzy pointwise state feedback control design is then proposed by employing the Lyapunov technique, integration by parts, the vector-valued Wirtinger’s inequality and the LMI relaxation technique, and presented in term of standard LMIs. Finally, the satisfactory and better performance of the proposed design method are demonstrated by the extensive numerical simulation results of two numerical examples.