Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems

In this paper, a finite dimensional guaranteed cost sampled-data fuzzy control (GCSDFC) problem is addressed for a class of nonlinear parabolic partial differential equation (PDE) systems. Initially, applying the Galerkin’s method to the PDE system, a nonlinear ordinary differential equation (ODE) system is derived. The resulting nonlinear ODE system can accurately describe the dominant dynamics of the PDE system, which is subsequently expressed by the Takagi–Sugeno (T–S) fuzzy model. Then, a guaranteed cost sampled-data fuzzy controller is developed to stabilize exponentially the closed-loop slow fuzzy system while providing an upper bound for the quadratic cost function. A novel time-dependent functional is constructed to derive the condition for the existence of the proposed controller which is presented by bilinear matrix inequalities (BMIs). Moreover, a suboptimal GCSDFC problem to minimize the cost bound can be formulated as a BMI optimization problem. A local optimization algorithm that views the BMI as a double linear matrix inequality (LMI) is given to solve this BMI optimization problem, in which a Latin hypercube sampling (LHS) method is proposed to find an initially feasible solution for starting the algorithm. Furthermore, it is shown that the proposed controller can ensure the exponential stability of the closed-loop PDE system. Finally, simulation results on the Fisher equation and the temperature profile of a catalytic rod show that the proposed design strategy is effective.