Finite dimensional disturbance observer based control for nonlinear parabolic PDE systems via output feedback

This paper considers the problem of finite dimensional disturbance observer based control (DOBC) via output feedback for a class of nonlinear parabolic partial differential equation (PDE) systems. The external disturbance is generated by an exosystem modeled by ordinary differential equations (ODEs), which enters into the PDE system through the control channel. Motivated by the fact that the dominant dynamic behavior of parabolic PDE systems can be characterized by a finite number of degrees of freedom, the modal decomposition technique is initially applied to the PDE system to derive a slow subsystem of finite dimensional ODEs. Subsequently, based on the slow subsystem and the exosystem, a disturbance observer (DO) and a slow mode observer (SMO) are constructed to estimate the disturbance and the slow modes. Moreover, an observation spillover observer (OSO) is also constructed to cancel approximately the effect of the observation spillover. Then, a finite dimensional DOBC design via output feedback is developed to estimate and compensate the disturbance, such that the closed-loop PDE system is exponentially stable in the presence of the disturbance. The condition for the existence of the proposed controller is given in terms of bilinear matrix inequality. Two algorithms based on the linear matrix inequality (LMI) technique are provided for solving control and observer gain matrices of the proposed controller. Finally, the developed design method is applied to the control of a one-dimensional diffusion-reaction process to illustrate its effectiveness.