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-by the system output-of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of nonlinear, single-input, single-output systems modelled by functional differential equations and subject to input saturation. Prespecified is a parameterized performance funnel within which the tracking error is required to evolve; transient and asymptotic behaviour of the tracking error is influenced through choice of parameter values which define the funnel. The control structure is a saturating error feedback with time-varying non-monotone gain designed to evolve in such a way as to preclude contact with the funnel boundary. A feasibility condition-formulated in bounds of the plant data, the saturation bound, the funnel data, the reference signal and the initial data-is presented under which the tracking objective is achieved, whilst maintaining boundedness of the state and gain function.
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