The exponential tracking and disturbance rejection for the unstable Burgers' equation with general references and disturbances

TL;DR

This paper presents a novel method for exponential tracking and disturbance rejection of the unstable Burgers' equation without relying on exosystem assumptions, using variable transforms and PDE theory.

Contribution

It introduces a straightforward approach to handle unstable PDE tracking problems without exosystem assumptions, splitting the problem into stabilization and regulation parts.

Findings

Exponential convergence of tracking error is theoretically proved.

Boundary controllers are explicitly constructed without backstepping.

Numerical example confirms the theoretical results.

Abstract

In solving the problem of asymptotic tracking and disturbance rejection, it has been long always assumed that the reference to be tracked and the disturbance to be rejected must be generated by an exosystem such as a finite dimensional exosystem with pure imaginary eigenvalues. The objective of this paper is to solve such a tracking problem for the unstable Burgers' equation without this assumption. Our treatment of this problem is straightforward. Using the method of variable transform, the tracking problem is split into two separate problems: a simple Neumann boundary stabilization problem and a dynamical Neumann boundary regulator problem. Unlike the existing literature where the regulator problem is always kept independent, the stabilization problem here is simplified to an independent linear diffusion equation by moving the instability term and the nonlinear term to the dynamical regulator problem, whereas the dynamical regulator problem does depend on the stabilization problem. Thus we can first easily handle the stabilization problem and then solve the dynamical regulator problem by using the fundamental theory of partial differential equations. The boundary feedforward controller is explicitly constructed by using the reference, the disturbance and the solution of the stabilization problem while the boundary feedback controller is easily designed for the linear diffusion equation without using a complex method such as the backstepping method. It is proved that, under the designed feedback and feedforward controllers, the tracking error converges to zero exponentially. This theoretical result is confirmed by a numerical example.