Robustness of Funnel Control 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), the well-known funnel controller $k(t)=\frac{\varphi(t)}{1-\varphi(t)\|e(t)\|}$, $u(t)=-k(t)e(t)$ achieves output regulation in the following sense: all states of the closed-loop system are bounded, and, most importantly, transient behavior of the tracking error $e=y-y_{\mathrm{ref}}$ is ensured such that the evolution of $e(t)$ remains in a performance funnel with prespecified boundary $1/\varphi(t)$, where $y_{\mathrm{ref}}$ denotes a reference signal bounded with an essentially bounded derivative. As opposed to classical adaptive high-gain output feedback, neither system identification nor the internal model is invoked and the gain $k(\cdot)$ is not monotone. Invoking the conceptual framework of the nonlinear gap metric, we show that the funnel controller is robust in the following sense: the funnel controller 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 a “small” gap) to a system satisfying (i)–(iii).