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Controlled functional differential equations: approximate and exact asymptotic tracking with prescribed transient performance. (English) Zbl 1175.93188

Summary: A tracking problem is considered in the context of a class \({\mathcal S}\) 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 \({\mathcal S}\): given \(\lambda \geq 0\), construct a feedback strategy which ensures that, for every \(r\) (assumed bounded with 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, in the case \(\lambda=0: \lim_{t\rightarrow\infty}\| e(t)\| = 0\). The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel \({\mathcal S}_\varphi\) (determined by a function \(\varphi)\). 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 \(k(t)=\alpha (\varphi (t)\| e(t)\|)\), 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 \(\lambda \geq 0\).

MSC:

93D15 Stabilization of systems by feedback
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
34K20 Stability theory of functional-differential equations
34A60 Ordinary differential inclusions
93C35 Multivariable systems, multidimensional control systems
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