Distributed Fuzzy Proportional-spatial Integral Control Design for a Class of Nonlinear Distributed Parameter Systems

The fuzzy feedback control design problem is addressed in this paper by using the distributed proportional-spatial integral (P-sI) control approach for a class of nonlinear distributed parameter systems represented by semi-linear parabolic partial differential-integral equations (PDIEs). The objective of this paper is to develop a fuzzy distributed P-sI controller for the semi-linear parabolic PDIE system such that the resulting closed-loop system is exponentially stable. To do this, the semi-linear parabolic PDIE system is first assumed to be exactly represented by a Takagi-Sugeno (T-S) fuzzy parabolic PDIE model. A new vector-valued integral inequality is established via the vector-valued Wirtinger's inequality. Then, based on the T-S fuzzy PDIE model and this new integral inequality, a distributed fuzzy P-sI state feedback controller is proposed such that the closed-loop PDIE system is exponentially stable. The sufficient condition on the existence of this fuzzy controller is given in terms of a set of standard linear matrix inequalities (LMIs), which can be effectively solved by using the existing convex optimization techniques. Finally, the developed design methodology is successfully applied to solve the feedback control design of a semi-linear reaction-diffusion system with a spatial integral term.