Abstract. 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 Ê 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 ϕ(t) e(t) < 1 for all t ≥ 0, where ϕ a prescribed real-valued function of class W 1,∞ with the property that ϕ(s) > 0 for all s > 0 and lim infs→∞ ϕ(s) > 0. A simple (neither adaptive nor dynamic) error feedback control of the form u(t) = −α(ϕ(t) e(t) )e(t) is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain α(ϕ(·) e(·) ).Mathematics Subject Classification. 93D15, 93C30, 34K20.

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Three related but distinct scenarios for tracking control of uncertain systems are reviewed: asymptotic tracking, approximate tracking with prescribed asymptotic error bound, tracking with prescribed transient behaviour. A variety of system classes are considered, ranging from finite-dimensional linear minimum-phase systems to nonlinear, infinite-dimensional systems described by functional differential equations. These classes are determined only by structural assumptions, such as stable zero dynamics and known relative degree. The objective is a single (and simple) control structure which is effective for every member of the underlying system class: no attempt is made to identify the particular system being controlled.

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of nonlinear systems, with output y, described by functional differential equations (a generalization of the class of linear minimum-phase systems of relative degree one with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given > 0 (arbitrarily small), determine a feedback strategy which ensures that for every admissible system and reference signal, the tracking error e = y − r is ultimately smaller than (that is, e(t) < 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 non-adaptive feedback control structure u(t) = −k(t)e(t), it is shown that the above objectives can be attained if the gain is generated by the nonlinear, memoryless feedback k(t) = K F (t, e(t)), where K F is any continuous function exhibiting two specific properties, the first of which ensures that if (t, e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact, and the second of which obviates the need for large gain values away from the funnel boundary.