Controlled functional differential equations:approximateand exact asymptotic trackingwith prescribed transient performance

Abstract

A tracking problem is consideredin the context of a class $\mathcal{S}$ of multi-input,multi-output, nonlinear systems modelled by controlled functionaldifferential equations. The class contains, as a prototype, allfinite-dimensional, linear, m-input, m-output, minimum-phasesystems with sign-definite “high-frequency gain". The first controlobjective is tracking of reference signals r by the output y ofany system in $\mathcal{S}$ : given $\lambda \geq 0$ , construct afeedback strategy which ensures that, for every r (assumed boundedwith essentially bounded derivative) and every system of class $\mathcal{S}$ , the tracking error $e = y-r$ is such that, in the case $\lambda >0$ , $\limsup_{t\rightarrow\infty}\|e(t)\|<\lambda$ or, inthe case $\lambda=0$ , $\lim_{t\rightarrow\infty}\|e(t)\| = 0$ . Thesecond objective is guaranteed output transient performance: theerror is required to evolve within a prescribed performance funnel $\mathcal{F}_\varphi$ (determined by a function φ). Forsuitably chosen functions α, ν and θ, bothobjectives are achieved via a control structure of the form $u(t)=-\nu (k(t))\theta (e(t))$ with $k(t)=\alpha (\varphi(t)\|e(t)\|)$ , whilst maintaining boundedness of the control andgain functions u and k. In the case $\lambda=0$ , the feedbackstrategy may be discontinuous: to accommodate this feature, aunifying framework of differential inclusions is adopted in theanalysis of the general case $\lambda \geq 0$ .