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) =phi(t)/1-phi(t)parallel to e(t)parallel to, 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(ref) is ensured such that the evolution of e(t) remains in a performance funnel with prespecified boundary 1/phi(t),where y(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(.) 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).