Tracking control: Performance funnels and prescribed transient behaviour

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in a 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 with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given λ > 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 λ (that is, ‖e(t)‖ < λ 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 feedback law k(t) = KF (t, e(t)), where KF 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.