Tracking with prescribed transient behaviour

Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains - as a prototype subclass - all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary R^M-valued reference signal r of class W^1,∞ (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that θ(t)∥e(t)∥ < 1 for all t ≥ 0, where θ a prescribed real-valued function of class W^1,∞ with the property that θ(s) > 0 for all s > 0 and lim inf_s→∞ θ(s) > 0. A simple (neither adaptive nor dynamic) error feedback control of the form u(t) = -α(θ(t)∥e(t)∥)e(t) is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain α(θ(·)∥e(·)∥).