Robustness of $\lambda$-Tracking in the Gap Metric

Abstract

For $m$-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” high-frequency gain), it is well known that the adaptive $\lambda$-tracker “$u=-k\,e$, $\dot k=\max\{0,|e|-\lambda\}|e|$” achieves $\lambda$-tracking of the tracking error $e$ if applied to such a system: all states of the closed-loop system are bounded, and $|e|$ is ultimately bounded by $\lambda$, where $\lambda>0$ is prespecified and may be arbitrarily small. Invoking the conceptual framework of the nonlinear gap metric, we show that the $\lambda$-tracker is robust. In the present setup this means in particular that the $\lambda$-tracker copes with bounded input and output disturbances, and, more importantly, it may even be applied to a system not satisfying any 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).