Boundary control, stabilization and zero-pole dynamics for a non-linear distributed parameter system
✍ Scribed by Christopher I. Byrnes; David S. Gilliam; Victor I. Shubov
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 238 KB
- Volume
- 9
- Category
- Article
- ISSN
- 1049-8923
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✦ Synopsis
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 in"nity. We also prove a robust version of this result, valid for perturbations by an unknown disturbance with arbitrary ¸ norm. For the controlled Burgers' equation forced by a disturbance we prove that, for all initial data in ¸(0, 1), the closed-loop trajectories converge in ¸(0, 1), uniformly in t3[0, ¹ ] and in H(0, 1), uniformly in t3[t , ¹] for any t '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 ¸, H and ¸ 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 su$ciently small non-zero disturbance then the resulting closed-loop system is &practically semiglobally stabilizable' in ¸-norm.