Adaptive tracking within prescribed funnels

Output tracking of a reference signal (an absolutely continuous bounded function with essentially bounded derivative) is considered in a context of a class of nonlinear systems described by functional differential equations. The primary control objective is tracking with prescribed accuracy: given λ > 0 (arbitrarily small), ensure that, for every admissible system and reference signal, the tracking error e is ultimately smaller than λ (that is, ||e(t)|| < λ for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F. Adopting the simple feedback control structure u(t) = -k(t)e(t), it is shown that the above objectives can be achieved if the gain k(t) = K_F(t,e(t)) is generated by any continuous function K_F exhibiting two specific properties formulated in terms of the distance of e(t) to the funnel boundary.