Robustness of λ-tracking in the gap metric

Form-input, m-output, finite-dimensional, linear systems satisfying the classical assumptions of adaptive control (i.e., (i) minimum phase, (ii) relative degree one and (iii) “positive” highfrequency gain), it is well known that the adaptive λ-tracker ‘u = −k e, k̇ = max{0, |e|−λ}|e|’ achieves λ-tracking of the tracking error e if applied to such a system: all states of the closedloop system are bounded and |e| is ultimately bounded by λ, where λ > 0 is prespecified and may be arbitrarily small. Invoking the conceptual framework of nonlinear gap metric, we show that the λ-tracker is robust. In the present setup this means in particular that the λ-tracker copes with bounded input and output disturbances and, more importantly, it may even be applied to a system not satisfying one of the classical conditions (i)-(iii) as long as the initial conditions and the disturbances are “small” and the system is “close” (in terms of “small” gap) to a system satisfying (i)-(iii).