Mixed H₂/H_∞ fuzzy proportional-spatial integral control design for a class of nonlinear distributed parameter systems

In this paper, a fuzzy feedback control design problem with a mixed H₂/H_∞ performance is addressed 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 with a mixed H₂/H_∞ performance index for the semi-linear parabolic PDIE system. 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 in a given local domain of Hilbert space. Then, based on the T–S fuzzy PDIE model, a distributed fuzzy P-sI state feedback controller is proposed such that the closed-loop PDIE system is locally exponentially stable with a mixed H₂/H_∞ performance. The sufficient condition on the existence of the fuzzy controller is given by using the Lyapunov's direct method, the technique of integration by parts, and vector-valued Wirtinger's inequalities, and presented in terms of standard linear matrix inequalities (LMIs). Moreover, by using the existing LMI optimization techniques, a suboptimal H_∞ fuzzy controller is derived in the sense of minimizing an upper bound of a given H₂ performance function. Finally, the developed design methodology is successfully applied to feedback control of a semi-linear reaction–diffusion system with spatial integral terms.