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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e-mail: krstic@ucsd.edu
e-mail: lionel.magnis@ensmp.fr
e-mail: rvazquez@ucsd.edu
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March 2009
Research Papers
Nonlinear Control of the Viscous Burgers Equation: Trajectory Generation, Tracking, and Observer Design
Miroslav Krstic,
Miroslav Krstic
Professor
Department of Mechanical & Aerospace Engineering,
e-mail: krstic@ucsd.edu
University of California, San Diego
, La Jolla, CA 92093-0411
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Lionel Magnis,
e-mail: lionel.magnis@ensmp.fr
Lionel Magnis
Ecole des Mines de Paris
, 60 Boulevard Saint Michel, 75006 Paris, France
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Rafael Vazquez
Rafael Vazquez
Assistant Professor
Departmento de Ingeniería Aeroespacial,
e-mail: rvazquez@ucsd.edu
Universidad de Sevilla
, Avenida, de los Descubrimientos s.n., 41092 Sevilla, Spain
Search for other works by this author on:
Miroslav Krstic
Professor
Department of Mechanical & Aerospace Engineering,
University of California, San Diego
, La Jolla, CA 92093-0411e-mail: krstic@ucsd.edu
Lionel Magnis
Ecole des Mines de Paris
, 60 Boulevard Saint Michel, 75006 Paris, Francee-mail: lionel.magnis@ensmp.fr
Rafael Vazquez
Assistant Professor
Departmento de Ingeniería Aeroespacial,
Universidad de Sevilla
, Avenida, de los Descubrimientos s.n., 41092 Sevilla, Spaine-mail: rvazquez@ucsd.edu
J. Dyn. Sys., Meas., Control. Mar 2009, 131(2): 021012 (8 pages)
Published Online: February 9, 2009
Article history
Received:
January 14, 2008
Revised:
August 1, 2008
Published:
February 9, 2009
Citation
Krstic, M., Magnis, L., and Vazquez, R. (February 9, 2009). "Nonlinear Control of the Viscous Burgers Equation: Trajectory Generation, Tracking, and Observer Design." ASME. J. Dyn. Sys., Meas., Control. March 2009; 131(2): 021012. https://doi.org/10.1115/1.3023128
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