$H_\infty $ Disturbance Attenuation for Nonlinear Coupled Parabolic PDE–ODE Systems via Fuzzy-Model-Based Control Approach

An <inline-formula> <tex-math notation="LaTeX">${H_\infty }$ </tex-math></inline-formula> fuzzy control design is presented for the disturbance attenuation of a class of coupled systems described by a set of nonlinear ordinary differential equations (ODEs) and a semi-linear parabolic partial differential equation (PDE). The fuzzy control scheme consists of an ODE state feedback fuzzy subcontroller for the ODE subsystem and a PDE static output feedback fuzzy subcontroller for the PDE subsystem by using piecewise uniform actuators and pointwise sensors. Initially, the original nonlinear system is accurately represented by employing a Takagi–Sugeno fuzzy coupled parabolic PDE–ODE model. Then, an <inline-formula> <tex-math notation="LaTeX">${H_\infty }$ </tex-math></inline-formula> fuzzy controller is developed to exponentially stabilize the fuzzy coupled system while satisfying a prescribed <inline-formula> <tex-math notation="LaTeX">${H_\infty }$ </tex-math></inline-formula> performance of disturbance attenuation, whose existence condition is given by linear matrix inequalities. Finally, simulation results on a hypersonic rocket car are given to show the effectiveness of the proposed design method.