Delay-dependent fuzzy stabilisation design for spatial 2-D time-delay parabolic PDE systems based on mobile sensors and actuators

In the context of actual physical processes over time and space, focussing on a spatial two-dimensional (2-D) framework is more practical. As the number of spatial dimension increases, the complexity of controller design escalates significantly. This study presents a fuzzy stabilisation design aimed at spatial 2-D time-delay nonlinear parabolic partial differential equation (PDE) systems utilising mobile sensor/actuator pairs. Initially, the Takagi-Sugeno (T-S) fuzzy model is employed to accurately characterise the 2-D nonlinear time-delay PDE systems. Following this, a fuzzy controller design scheme is introduced that strategically leverages mobile sensor/actuator pairs placed in various subdomains within the spatial domain based on the T-S fuzzy model. Subsequently, using the developed T-S fuzzy model and the Lyapunov direct method, a membership-dependent fuzzy stabilisation controller design is formulated, along with the guidance laws for mobile sensor/actuator pairs, ensuring that the resulting closed-loop system for the spatial 2-D time-delay scenario achieves the exponential stability. The mobile strategy integrates two directional guidance laws, with each dimension having its own specific guidance form. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed design approach.