Nonlinear control of the Burgers PDE—Part II: Observer design, trajectory generation, and tracking

In a companion paper we have solved the problem of full-state stabilization of unstable "shock-like" equilibrium profiles of the viscous Burgers equation with actuation at the boundaries. In this paper we consider the problem of output feedback stabilization. We design a nonlinear observer for the Burgers equation that employs only boundary sensing. We employ its state estimates in an output feedback control law which we prove to be locally stabilizing. The main idea is to use a nonlinear spatially- scaled transformation (that employs three ingredients, of which one is the Hopf-Cole nonlinear integral transformation) and then employ the linear backstepping observer design method. The stabilization properties of the output feedback law are illustrated with numerical simulations of the closed-loop system. We also consider the problems of trajectory generation and tracking for the fully nonlinear Burgers equation. Our algorithm is applicable to a large class of functions of time as reference trajectories of the boundary output, though we focus in more detail on the special case of sinusoidal references. Since the Burgers equation is not globally controllable, the reference amplitudes cannot be arbitrarily large. We provide a sufficient condition that characterizes the allowable amplitudes and frequencies, under which the state trajectory is bounded and tracking is achieved.