Robustness of λ-tracking and 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 definite high-frequency gain matrix), two control strategies are considered: the well-known λ-tracking and funnel control. An application of the λ-tracker to systems satisfying (i)-(iii) yields that all states of the closed-loop system are bounded and |e| is ultimately bounded by some prespecified λ > 0. An application of the funnel controller achieves tracking of the error e within a prescribed performance funnel if applied to linear systems satisfying (i)-(iii). Moreover, all states of the closed-loop system are bounded. The funnel boundary can be chosen from a large set of functions. Invoking the conceptual framework of the nonlinear gap metric, we show that the λ-tracker and the funnel controller are robust. In the present setup this means in particular that λ-tracking and funnel control copes with bounded input and output disturbances and, more importantly, may be applied to any system which is "close" (in terms of a "small" gap) to a system satisfying (i)-(iii), and which may not satisfy any of the classical conditions (i)-(iii), as long as the initial conditions and the disturbances are "small".