Low dimensional anti-disturbance fuzzy control for 2-D spatial parabolic PDE systems with nonlinear ODE exosystem and input dynamics via fuzzy disturbance observer

A design method of low dimensional anti-disturbance fuzzy control via fuzzy disturbance observer (FDO) is proposed for a class of 2-dimensional (2-D) spatial nonlinear parabolic partial differential equation (PDE) systems with dead-zone band input nonlinearities and multiple time-domain disturbances including the disturbance modeled by the nonlinear ordinary differential equation (ODE) exosystem. The modeled disturbance affects the 2-D spatial PDE system through the input channel. First, the modal decomposition technique is applied to the 2-D spatial PDE system to derive a low dimensional nonlinear slow subsystem to capture its dominant dynamics, which is subsequently approximated by a T-S fuzzy model. To deal with dead-zone band input nonlinearities, a dead-zone band compensator is constructed by using its parameters. Next, in order to reconstruct the unknown nonlinearities in the exosystem, a radial basis function (RBF) fuzzy system is employed and then an FDO is developed to estimate the modeled disturbance. Based on T-S fuzzy model, FDO and dead-zone band compensator, a low dimensional anti-disturbance fuzzy control method is developed in terms of linear matrix inequalities (LMIs). The uniformly ultimately boundedness (UUB) of the 2-D spatial PDE system in the presence of the dead-zone band input and multiple disturbances is analyzed via singular perturbation method. Finally, the effectiveness of the proposed design method is demonstrated on the control of 2-D spatial Burgers-KPP-Fisher diffusion-reaction system.