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Boundary control, stabilization and zero-pole dynamics for a non-linear distributed parameter system
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DOI：10.1002/(SICI)1099-1239(199909)9:11<737::AID-RNC432>3.0.CO;2-3.png)
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Abstract
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In this work we show that the now standard lumped non-linear enhancement of root-locus design still persists for a non-linear distributed parameter boundary control system governed by a scalar viscous Burgers' equation. Namely, we construct a proportional error boundary feedback control law and show that closed-loop trajectories tend to trajectories of the open-loop zero dynamics as the gain parameters are increased to infinity. We also prove a robust version of this result, valid for perturbations by an unknown disturbance with arbitrary L-2 norm. For the controlled Burgers equation forced by a disturbance we prove that, for all initial data in L-2(0, 1), the closed-loop trajectories converge in L-2(0, 1), uniformly in t is an element of [0, T] and in H-1(0, 1), uniformly in t is an element of [t(0), T] for any t(0) > 0, to the trajectories of the corresponding perturbed zero dynamics. We have also extended these results to include the case when additional boundary controls are included in the closed-loop system. This provides a proof of convergence of trajectories in case the zero dynamics is replaced by a non-homogeneous; Dirichlet boundary controlled Burgers' equation. As an application of our convergence of trajectories results, we demonstrate that our boundary feedback control scheme provides a semiglobal exponential stabilizing feedback law in L-2, H-1 and L-infinity for the open-loop system consisting of Burgers' equation with Neumann boundary conditions and zero forcing term. We also show that this result is robust in the sense that if the open-loop system is perturbed by a sufficiently small non-zero disturbance then the resulting closed-loop system is 'practically semiglobally stabilizable' in L-2-norm. Copyright (C) 1999 John Wiley & Sons, Ltd.
Keywords:
Burgers' equation
boundary control
zero dynamics
zero-pole dynamics
robust semiglobal stabilization
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