Controlled Functional Differential Equations: Approximate and Exact Asymptotic Tracking With Prescribed Transient Performance

A tracking problem is considered in the context of a class S of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, m-input, m-output, minimum-phase systems with sign-definite "high-frequency gain". The first control objective is tracking of reference signals r by the output y of any system in S: given lambda >= 0, construct a feedback strategy which ensures that, for every r (assumed bounded with essentially bounded derivative) and every system of class S, the tracking error e = y - r is such that, in the case lambda > 0, lim sup(t ->infinity) parallel to e(t)parallel to <lambda or, in the case lambda = 0, lim(t ->infinity) parallel to e(t)parallel to = 0. The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel F phi (determined by a function phi). For suitably chosen functions alpha, nu and theta, both objectives are achieved via a control structure of the form u(t) = nu(k(t))theta(e(t)) with k(t) = alpha(phi(t) parallel to e(t)parallel to), whilst maintaining boundedness of the control and gain functions u and k. In the case lambda = 0, the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case lambda >= 0.