Robustness in the graph topology of a common adaptive controller

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, 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.

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