Tracking with Prescribed Transient Behavior for Nonlinear Systems of Known Relative Degree

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.