Controlled functional differential equations: approximate and exact asymptotic tracking with prescribed transient performance

Abstract

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 :g ivenλ ≥ 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 λ> 0, lim supt→∞ � e(t)� <λ or, in the case λ = 0, limt→∞ � e(t)� =0 . The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel Fϕ (determined by a function ϕ). For suitably chosen functions α, ν and θ, both objectives are achieved via a control structure of the form u(t )= −ν(k(t))θ(e(t)) with k(t )= α(ϕ(t)� e(t)� ), whilst maintaining boundedness of the control and gain functions u and k .I n the case λ = 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 λ ≥ 0.