Robustness in the graph topology of a common adaptive controller

• Main Authors: French, M. (Author), Ilchmann, A. (Author), Ryan, E.P (Author)
• Format: Article
• Language: English
• Published: 2006.
• Subjects: Article
• Online Access: Get fulltext

 For any <i>m</i>-input, <i>m</i>-output, finite-dimensional, linear, minimum-phase plant <i>P</i> with first Markov parameter having spectrum in the open right-half complex plane, it is well known that the adaptive output feedback control <i>C</i>, given by <i>u</i> = -<i>ky</i>, <i>k</i> = ||<i>y</i>||², yields a closed-loop system [<i>P,C</i>] for which the state converges to zero, the signal <i>k</i> converges to a finite limit, and all other signals are of class <i>L</i>². It is first shown that these properties continue to hold in the presence of <i>L</i>²-input and <i>L</i>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 <i>P</i> is replaced by a stabilizable and detectable linear plant <i>P</i>₁ within a sufficiently small neighbourhood of <i>P</i> in the graph topology, provided that the plant initial data and the <i>L</i>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 <i>L</i>²-robustness result applies. Unstable behaviour for large initial conditions and/or large<i> L</i>² disturbances is shown, demonstrating that the bounds obtained from the<i> L</i>² theory are qualitatively tight: this contrasts with the <i>L</i>^∞-robustness analysis of Georgiou & Smith which is insufficiently tight to predict the stable behaviour for small initial conditions and zero disturbances.