Low dimensional disturbance observer-based control for nonlinear parabolic PDE systems with spatio-temporal disturbances

In this paper, a design problem of low dimensional disturbance observer-based control (DOBC) is considered for a class of nonlinear parabolic partial differential equation (PDE) systems with the spatio-temporal disturbance modeled by an infinite dimensional exosystem of parabolic PDE. Motivated by the fact that the dominant structure of the parabolic PDE is usually characterized by a finite number of degrees of freedom, the modal decomposition method is initially applied to both the PDE system and the PDE exosystem to derive a low dimensional slow system and a low dimensional slow exosystem, which accurately capture the dominant dynamics of the PDE system and the PDE exosystem, respectively. Then, the definition of input-to-state stability for the PDE system with the spatio-temporal disturbance is given to formulate the design objective. Subsequently, based on the derived slow system and slow exosystem, a low dimensional disturbance observer (DO) is constructed to estimate the state of the slow exosystem, and then a low dimensional DOBC is given to compensate the effect of the slow exosystem in order to reject approximately the spatio-temporal disturbance. Then, a design method of low dimensional DOBC is developed in terms of linear matrix inequality to guarantee that not only the closed-loop slow system is exponentially stable in the presence of the slow exosystem but also the closed-loop PDE system is input-to-state stable in the presence of the spatio-temporal disturbance. Finally, simulation results on the control of temperature profile for catalytic rod demonstrate the effectiveness of the proposed method. Copyright (C) 2015 John Wiley & Sons, Ltd.