# 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.

## 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 Submitted for publication. adaptive control, gap metric, robust stability Faculty of Physical Sciences and Engineering > Electronics and Computer Science > Comms, Signal Processing & Control 261074 19 Jul 2005 27 Mar 2014 20:03 Google Scholar http://eprints.soton.ac.uk/id/eprint/261074