Spatially Piecewise Fuzzy Control Design for Sampled-Data Exponential Stabilization of Semi-linear Parabolic PDE Systems
IEEE transactions on fuzzy systems
This paper employs a Takagi-Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial L ∞ norm ∥ · ∥∞ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the
... near DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov-Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincaré-Wirtinger inequality in 1D spatial domain, as well as a vectorvalued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of ∥·∥∞, if sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally, the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples.