Tracking with prescribed transient behaviour

¹ Institute of Mathematics, Technical University Ilmenau, Weimarer Straße 25, 98693 Ilmenau, Germany; ilchmann@mathematik.tu-ilmenau.de.
² Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK; epr@maths.bath.ac.uk.
³ School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK; C.J.Sangwin@bham.ac.uk.

Received: 7 February 2002
Revised: 17 April 2002

Abstract

Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains – as a prototype subclass – all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary -valued reference signal r of class W^1,∞ (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that for all t ≥ 0, where φ a prescribed real-valued function of class W^1,∞ with the property that φ(s) > 0 for all s > 0 and . A simple (neither adaptive nor dynamic) error feedback control of the form is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain .