Robustness in the graph topology of a common adaptive controller

TY - JOUR

T1 - Robustness in the graph topology of a common adaptive controller

AU - French, M.

AU - Ilchmann, A.

AU - Ryan, E.P.

N1 - Submitted for publication.

PY - 2006

Y1 - 2006

N2 - 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, 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 L2. It is first shown that these properties continue to hold in the presence of L2-input and L2-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 P1 within a sufficiently small neighbourhood of P in the graph topology, provided that the plant initial data and the L2 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 L2-robustness result applies. Unstable behaviour for large initial conditions and/or large L2 disturbances is shown, demonstrating that the bounds obtained from the L2 theory are qualitatively tight: this contrasts with the L∞-robustness analysis of Georgiou & Smith which is insufficiently tight to predict the stable behaviour for small initial conditions and zero disturbances.

AB - 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, 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 L2. It is first shown that these properties continue to hold in the presence of L2-input and L2-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 P1 within a sufficiently small neighbourhood of P in the graph topology, provided that the plant initial data and the L2 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 L2-robustness result applies. Unstable behaviour for large initial conditions and/or large L2 disturbances is shown, demonstrating that the bounds obtained from the L2 theory are qualitatively tight: this contrasts with the L∞-robustness analysis of Georgiou & Smith which is insufficiently tight to predict the stable behaviour for small initial conditions and zero disturbances.

KW - adaptive control, gap metric, robust stability

M3 - Article

SN - 0363-0129

VL - 45

SP - 1736

EP - 1757

JO - SIAM JOURNAL ON CONTROL AND OPTIMIZATION

JF - SIAM JOURNAL ON CONTROL AND OPTIMIZATION

IS - 5

ER -