Robustness in the Graph Topolgy of a Common Adaptive Controller


French, M., Ilchmann, A. and Ryan, E.P. (2006) Robustness in the Graph Topolgy of a Common Adaptive Controller. SIAM Journal of Control and Optimization, 45, (5), 1736-1757.

Download

[img] PDF
Download (277Kb)

Description/Abstract

For any $m$-input, $m$-output, finite-dimensional, linear, minimum-phase plant $P$ with first Markov parameter having spectrum in the open right-half complex plane, it is well known that the adaptive output feedback control $C$, given by $u=-ky,\ \dot k= \|y\|^2$, yields a closed-loop system $[P,C]$ for which the state converges to zero, the signal $k$ converges to a finite limit, and all other signals are of class $L^2$. It is first shown that these properties continue to hold in the presence of $L^2$-input and $L^2$-output disturbances. By establishing gain function stability of an appropriate closed-loop operator, it is proved that these properties also persist when the plant $P$ is replaced by a stabilizable and detectable linear plant $P_1$ within a sufficiently small neighbourhood of $P$ in the graph topology, provided that the plant initial data and the $L^2$ magnitude of the disturbances are sufficiently small. Example 9 of Georgiou & Smith (IEEE Trans. Autom. Control 42(9) 1200--1221, 1997) is revisited to which the above $L^2$-robustness result applies. Unstable behaviour for large initial conditions and/or large $L^2$ disturbances is shown, demonstrating that the bounds obtained from the $L^2$ theory are qualitatively tight: this contrasts with the $L^\infty$-robustness analysis of Georgiou & Smith which is insufficiently tight to predict the stable behaviour for small initial conditions and zero disturbances.

Item Type: Article
Additional Information: Submitted for publication.
Keywords: adaptive control, gap metric, robust stability
Divisions: Faculty of Physical Sciences and Engineering > Electronics and Computer Science > Comms, Signal Processing & Control
ePrint ID: 261074
Date Deposited: 19 Jul 2005
Last Modified: 27 Mar 2014 20:03
Further Information:Google Scholar
URI: http://eprints.soton.ac.uk/id/eprint/261074

Actions (login required)

View Item View Item

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics