Performance funnels and tracking control

Tracking of an absolutely continuous reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of nonlinear, single-input, single-output, dynamical systems modelled by functional differential equations satisfying certain structural hypotheses (which, interpreted in the particular case of linear systems, translate into assumptions – ubiquitous in the adaptive control literature – of (i) relative degree one, (ii) positive high-frequency gain and (iii) stable zero dynamics). The control objective is evolution of the tracking error within a prespecified funnel, thereby guaranteeing prescribed transient performance and prescribed asymptotic tracking accuracy. This objective is achieved by a so-called funnel controller, which takes the form of linear error feedback with time-varying gain. The gain is generated by a nonlinear feedback law in which the reciprocal of the distance of the instantaneous tracking error to the funnel boundary plays a central role. In common with many established high-gain adaptive control methodologies, the overall feedback structure exploits an intrinsic high-gain property of the system, but differs from these approaches in two fundamental respects: the funnel control gain is not dynamically generated and is not necessarily monotone. The main distinguishing feature of the present paper vis à vis previous contributions on funnel control is twofold: (a) a larger system class can be accommodated – in particular, nonlinearities of a general nature can be tolerated in the input channel; (b) a wider choice of formulations of prescribed transient behaviour (including, for example, practical (M,μ)-stability wherein, for prescribed parameter values M > 1, μ > 0 and λ > 0, the tracking error e(·) is required to satisfy |e(t)| < max{Me|e(0)| , λ} for all t ≥ 0) is encompassed.