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.

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered for multi-input, multi-output linear systems satisfying the following structural assumptions: (i) arbitrary-but known-relative degree, (ii) the "high-frequency gain" matrix is sign definite-but of unknown sign, (iii) exponentially stable zero dynamics. The first control objective is tracking, by the output y, with prescribed accuracy: given > 0 (arbitrarily small), determine a feedback strategy which ensures that, for every reference signal r, the tracking error e = y − r is ultimately bounded by (that is, e(t) < for all t sufficiently large). The second objective is guaranteed output transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel F (determined by a function ). Both objectives are achieved by a filter in conjunction with a feedback function of the filter states, the tracking error and a gain parameter. The latter is generated via a feedback function of the tracking error and the funnel parameter . Moreover, the feedback system is robust to nonlinear perturbations bounded by some continuous function of the output. The feedback structure essentially exploits an intrinsic high-gain property of the system/filter interconnection by ensuring that, if (t, e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact.

Abstract.Tracking of reference signals y ref (·) by the output y(·) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e = y − y ref and its derivativeė within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = −k 0 (t) 2 e(t) − k 1 (t)ė(t), where the simple proportional error feedback has gain functions k 0 and k 1 designed in such a way to preclude contact of e andė with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control.

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class Σρ of multi-input, multi-output dynamical systems, modelled by functional differential equations, affine in the control and satisfying the following structural assumptions: (i) arbitrary-but known-relative degree ρ ≥ 1; (ii) the "high-frequency gain" is sign definite-but possibly of unknown sign. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains (as a prototype subclass) all finite-dimensional, linear, m-input, m-output, minimum-phase systems of known strict relative degree. The first control objective is tracking, by the output y, with prescribed accuracy: given λ > 0 (arbitrarily small), determine a feedback strategy which ensures that, for every reference signal r and every system of class Σρ, the tracking error e = y − r is ultimately bounded by λ (that is, e(t) < λ for all t sufficiently large). The second objective is guaranteed output transient performance: the tracking error is required to evolve within a prescribed performance funnel Fϕ (determined by a function ϕ). Both objectives are achieved using a filter in conjunction with a feedback function of the tracking error, the filter states, and the funnel parameter ϕ.

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