Robustness in the graph topology 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, 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. Working within the conceptual framework of the nonlinear gap metric approach to robust stability, and 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 with a stabilizable and detectable linear plant P-1 within a sufficiently small neighborhood 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 and Smith [IEEE Trans. Automat. Control, 42 (1997), pp. 1200-1221] is revisited. Unstable behavior 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-infinity-robustness analysis of Georgiou and Smith, which is insufficiently tight, to predict the stable behavior for small initial conditions and zero disturbances.