Systems & Control Letters 47 (2002) 365–374 www.elsevier.com/locate/sysconle Controlled functional dierential equations and adaptive tracking  E.P. Ryan a , C.J. Sangwin b; ∗ a Department of Mathematical Sciences, University of Bath, Bath, UK BA2 7AY b School of Mathematics and Statistics, University of Birmingham, Birmingham, UK B15 2TT Received 24 August 2001; received in revised form 28 May 2002; accepted 26 June 2002 Abstract Adaptive tracking control of a class N of single-input, single-output systems described by nonlinear functional dierential equations is considered: the control objective is that of tracking, by the system output, of reference signals of class R (absolutely continuous and bounded with essentially bounded derivative). A (N; R)-universal servomechanism, in the form of an adaptive error feedback strategy incorporating gains of Nussbaum type, is developed which, for every system of class N and every reference signal of class R, ensures either (i) practical tracking (in the sense that prespecied asymptotic tracking accuracy, quantied by ¿ 0, is assured), or (ii) asymptotic tracking (in the sense that the tracking error approaches zero). The rst case (i) is achievable by continuous feedback; the second case (ii) necessitates discontinuous feedback. Both cases are developed within a framework of functional dierential inclusions. c  2002 Elsevier Science B.V. All rights reserved. Keywords: Adaptive control; Nonlinear systems; Functional dierential equations; Tracking; Universal servomechanism 1. Introduction In [11], a universal stabilizer (adaptive output feed- back control) was developed for a class of systems, modelled by nonlinear functional dierential equa- tions, having structure represented by Fig. 1 in the particular case wherein r =0= v. The dynamic block  1 (a forced nonlinear ordinary dierential equation of the form ˙ y = g(p;y)+ w + bu,  Based on work supported, in part, by the UK Science & Engineering Research Council (Grant GR/L78086). * Corresponding author. Tel.: +44-121-414-6197; fax: +44- 121-414-3389. E-mail addresses: epr@maths.bath.ac.uk (E.P. Ryan), c.j.sangwin@bham.ac.uk (C.J. Sangwin). where b is a nonzero constant and p ∈ L ∞ (R) is a bounded perturbation term), which can be inuenced directly by the control u, is also driven by the output w from the dynamic block  2 . Viewed abstractly, the block  2 can be regarded as a causal operator ˆ T which maps y to w. In the present paper, we revisit systems of the above form, viz. ˙ y = g(p;y)+ ˆ Ty + bu, but now in the con- text of a tracking problem wherein r = 0 represents a reference signal and v represents an additive measure- ment disturbance signal on the system output y. Un- der stronger hypotheses (vis  a vis those posited in the case of the stabilization problem (r =0=v) considered in [11]) on the operator ˆ T representing the subsystem  2 , we construct an adaptive error feedback strategy (parameterized by  ¿ 0) which ensures that, in the 0167-6911/02/$ - see front matter c  2002 Elsevier Science B.V. All rights reserved. PII:S0167-6911(02)00232-3