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Research Papers

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

[+] Author and Article Information
Miroslav Krstic1

Department of Mechanical & Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411krstic@ucsd.edu

Lionel Magnis

 Ecole des Mines de Paris, 60 Boulevard Saint Michel, 75006 Paris, Francelionel.magnis@ensmp.fr

Rafael Vazquez

Departmento de Ingeniería Aeroespacial, Universidad de Sevilla, Avenida, de los Descubrimientos s.n., 41092 Sevilla, Spainrvazquez@ucsd.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 131(2), 021012 (Feb 09, 2009) (8 pages) doi:10.1115/1.3023128 History: Received January 14, 2008; Revised August 01, 2008; Published February 09, 2009

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.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Solution to the nonlinear trajectory generation problem for a sinusoidal reference with b=0, a=1.2, ω=8

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Figure 2

Trajectory tracking: dashed line: output reference ur(0,t)=1.2sin(8t); and solid line: output response u(0,t) with control gain c1=5

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Figure 3

The Bode plot of the compensator (Eqs. 97,98,99,100) for σ=15

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Figure 4

Simulation of the open-loop system (with constant inputs, Eq. 4) for σ=3, and u0(x)=U(0)+2+(U(1)−U(0)−4)x appears to exhibit a finite-time blow-up

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Figure 5

Convergence of the closed-loop system under the linear output-feedback controller (Eqs. 96,97,98,99,100) for σ=5

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