Robustness of funnel control in the gap metric

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 behaviour of the tracking error $e=y − y_{ref}$ is ensured such that the evolution of $e(t)$ remains in a performance funnel with prespecified boundary $\varphi(t)^{−1}$, where $y_{ref}$ denotes a reference signal bounded with essentially bounded derivative. As opposed to classical adaptive high-gain output feedback, system identification or internal model is not invoked and the gain $k(\cdot)$ is not monotone. We show that the funnel controller is robust by invoking the conceptual framework of the nonlinear gap metric: 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)