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), .
Description/Abstract
For any $m$input, $m$output, finitedimensional, linear, minimumphase plant $P$ with first Markov parameter having spectrum in the open righthalf complex plane, it is well known that the adaptive output feedback control $C$, given by $u=ky,\ \dot k= \y\^2$, yields a closedloop 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 closedloop 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) 12001221, 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 
Organisations: 
Southampton Wireless Group 
ePrint ID: 
261074 
Date : 

Date Deposited: 
19 Jul 2005 
Last Modified: 
17 Apr 2017 22:03 
Further Information:  Google Scholar 
URI: 
http://eprints.soton.ac.uk/id/eprint/261074 
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