Design of Suboptimal Local Piecewise Fuzzy Controller With Multiple Constraints for Quasi-Linear Spatiotemporal Dynamic Systems

This paper discusses the problem of suboptimal local piecewise <inline-formula> <tex-math notation="LaTeX">${H} _{\infty }$ </tex-math></inline-formula> fuzzy control of quasi-linear spatiotemporal dynamic systems with control magnitude constraints. A Takagi–Sugeno fuzzy partial differential equation (PDE) model with space-varying coefficient matrices is first assumed to be derived for exactly describing nonlinear system dynamics. In the light of the fuzzy model, a local piecewise fuzzy feedback controller is then constructed to guarantee the exponential stability with a prescribed <inline-formula> <tex-math notation="LaTeX">${H} _{\infty }$ </tex-math></inline-formula> disturbance attenuation level for the resulting closed-loop system, while the control constraints are also ensured. A sufficient condition on the existence of such fuzzy controller is developed by the Lyapunov direct method and an integral inequality and presented in terms of space algebraic linear matrix inequalities (LMIs) coupled with LMIs. By virtue of extreme value theorem, a suboptimal-constrained local piecewise <inline-formula> <tex-math notation="LaTeX">${H} _{\infty }$ </tex-math></inline-formula> fuzzy control design in the sense of minimizing the disturbance attenuation level is formulated as a minimization optimization problem with LMI constraints. Finally, the proposed method is applied to solve the feedback control of a quasi-linear FitzHugh–Nagumo equation with space-varying coefficients, and simulation results show its effectiveness and merit.