Robustness in the Graph Topolgy of a Common Adaptive Controller

Robustness in the Graph Topolgy of a Common Adaptive Controller

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.

adaptive control, gap metric, robust stability

1736-1757

French, M.

22958f0e-d779-4999-adf6-2711e2d910f8

Ilchmann, A.

75c07fbf-9ca5-4765-bfd4-707cf4c42cd3

Ryan, E.P.

d272d821-c68d-4e20-9635-800f7aa07090

2006

French, M.

22958f0e-d779-4999-adf6-2711e2d910f8

Ilchmann, A.

75c07fbf-9ca5-4765-bfd4-707cf4c42cd3

Ryan, E.P.

d272d821-c68d-4e20-9635-800f7aa07090

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), .

## 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.

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## More information

Published date: 2006

Additional Information:
Submitted for publication.

Keywords:
adaptive control, gap metric, robust stability

Organisations:
Southampton Wireless Group

## Identifiers

Local EPrints ID: 261074

URI: https://eprints.soton.ac.uk/id/eprint/261074

PURE UUID: 26aeea2f-9666-44b8-87c6-efade957efe1

## Catalogue record

Date deposited: 19 Jul 2005

Last modified: 18 Jul 2017 09:05

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## Contributors

Author:
M. French

Author:
A. Ilchmann

Author:
E.P. Ryan

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