Delay-dependent fuzzy stabilisation design of nonlinear time-delay parabolic PDE-ODE coupled systems

This article addresses the challenging problem of fuzzy stabilisation for nonlinear time-delay parabolic partial differential equation (PDE) and ordinary differential equation (ODE) coupled systems, which are widely encountered in complex industrial processes but remain underexplored in the control literature. To overcome the complexity of nonlinear coupled dynamics, we propose an advanced Takagi-Sugeno (T-S) fuzzy modelling approach that accurately captures the spatio-temporal interactions between PDE and ODE components. By developing a time-delay-dependent Lyapunov functional that simultaneously accounts for time delays, parabolic PDE dynamics, and cross-coupling effects, we establish sufficient conditions for the exponential stability of closed-loop coupled PDE-ODE systems in the form of linear matrix inequalities (LMIs). The proposed control framework significantly extends existing results by providing a systematic design methodology for time-delay coupled PDE-ODE systems. Finally, numerical simulation studies demonstrate the validity of the presented method.