Nonlinear Control of the Viscous Burgers Equation: Trajectory Generation, Tracking, and Observer Design

Abstract

In a companion paper we have solved the basic 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 several advanced problems for this nonlinear partial differential equation (PDE) system. We start with the problems of trajectory generation and tracking. 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. We then 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 output feedback law is illustrated with numerical simulations of the closed-loop system.

The nonlinear heat equation with state-dependent parameters and its connection to the Burgers' and the potential Burgers' equation

“…Balogh and Krstic (2000) introduce H 1 -stability for the Burgers' equation with nonlinear boundary feedback. show results in nonlinear stabilization of shock-like unstable equilibria in the viscous Burgers' equation, whereas Krstic et al (2009) go a step further and present results for the same problem in trajectory generation, tracking and observer design.…”

The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations

“…In more general situations, past efforts in the theory of observers for systems described by PDEs include infinite dimensional observers for wave type equations and reversible systems [8,9], parabolic equations [19], viscous Burgers and shallow water equations [3,13]. Filter and observer design inspired by robust control feedback has been recently developed and studied in a standard (forward) way for medical imaging [16].…”

Recovering the initial state of an infinite-dimensional system using observers

“…Other recent papers on the use of observers for the control of linear DPS are Krstic et al [20] and Guo and Shao [15]. Observers for non-linear systems are studied in Bonnabel et al [7] (in the finite dimensional context), Smyshlyaev and Krstic [32] and Krstic et al [21].…”