The University of Southampton
University of Southampton Institutional Repository

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), pp. 1736-1757.

Record type: Article


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.

PDF SIAM_FIR_050718.pdf - Other
Download (284kB)

More information

Published date: 2006
Additional Information: Submitted for publication.
Keywords: adaptive control, gap metric, robust stability
Organisations: Southampton Wireless Group


Local EPrints ID: 261074
PURE UUID: 26aeea2f-9666-44b8-87c6-efade957efe1

Catalogue record

Date deposited: 19 Jul 2005
Last modified: 18 Jul 2017 09:05

Export record


Author: M. French
Author: A. Ilchmann
Author: E.P. Ryan

University divisions

Download statistics

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

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton:

ePrints Soton supports OAI 2.0 with a base URL of

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.