Tracking with prescribed transient behaviour

Abstract

Universal tracking control is investigated in the context of aclass S of M-input, M-output dynamical systems modelled byfunctional differential equations. The classencompasses a wide variety of nonlinear and infinite-dimensionalsystems and contains – as a prototype subclass – allfinite-dimensional linear single-input single-output minimum-phasesystems with positive high-frequency gain. The control objectiveis to ensure that, for an arbitrary $\mathbb{R}^M$ -valued reference signalr of class W ^1,∞ (absolutely continuous and boundedwith essentially bounded derivative) and every system of class S, the tracking error e between plant output and referencesignal evolves within a prespecified performance envelope orfunnel in the sense that ${\varphi}(t)\| e(t)\| < 1$ for all t ≥ 0, where φ a prescribed real-valued function of classW ^1,∞ with the property that φ(s) > 0 for all s >0 and $\liminf_{s\rightarrow\infty}{\varphi}(s)>0$ . A simple (neitheradaptive nor dynamic) error feedback control of the form $u(t)=-\alpha ({\varphi}(t)\|e(t)\|)e(t)$ is introduced which achieves theobjective whilst maintaining boundedness of the control and of thescalar gain $\alpha ({\varphi}(\cdot )\|e(\cdot )\|)$ .