Boundary control of nonlinear PDE dynamics with the use of differential flatness theory

Abstract

A boundary control method for nonlinear partial differential equations (PDEs) is developed, with the use of differential flatness theory. First, following the procedure for numerical solution of nonlinear PDEs, it is shown that the PDE is equivalently represented by a set of nonlinear ordinary differential equations (ODEs) and an associated state equations model. For the local subsystems, into which a nonlinear PDE is decomposed, it becomes possible to apply boundary control. Next, the controller design proceeds by showing that the state-space model of the nonlinear PDE stands for a differentially flat system. Moreover, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem's dynamics and can eliminate the subsystem's tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the nonlinear PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally find the control input that should be applied to the nonlinear PDE system so as to assure that all its state variables will converge to the desirable setpoints. By applying the proposed boundary control, it is shown that the nonlinear PDE dynamics can be made to track the designated variation profile.