Available online at www.sciencedirect.com Systems & Control Letters 49 (2003) 5–8 www.elsevier.com/locate/sysconle Linear deterministic adaptive control: fundamental limitations? A. Feuer a ; ∗ , G.C. Goodwin b a Department of EE, Technion, Israel Institute of Technology, Haifa, Israel b CIDAC, School of EE&CS, University of Newcastle, Newcastle, Australia Abstract This paper is concerned with the achievable performance of adaptive control algorithms. We show that when the only uncertainty is in the form of xed parameter errors, then there exists an adaptive feedback law whose performance can be made arbitrarily close to that achievable when the system is a priori known. The result is not intended as a practical strategy. Instead, we use it to make the, perhaps obvious, point that meaningful results on performance of adaptive control algorithms must account for non-ideal factors including, at a minimum, noise, parameter time variations and unstructured uncertainty. c  2003 Elsevier Science B.V. All rights reserved. Keywords: Adaptive control; Identication 1. Introduction Many results are available regarding fundamen- tal limitations that apply to linear feedback systems. These results take several dierent forms, e.g. • limitations on time domain transients (see for example [9]); • frequency domain integrals on sensitivity (see for example [3,5]); and • L 2 cheap control results (see for example [11,12]). A summary of some of these results is available in recent texts, see for example [4,5,14]. The above work on fundamental limitations has fo- cused on the case when the model is a priori known. This raises the question, regarding the eect of mod- eling errors on these results. A very preliminary rst step in this direction was discussed in [7]. In that * Corresponding author. E-mail address: feuer@ee.technion.ac.il (A. Feuer). paper, the interaction between a right half plane zero and an uncertain pole was analyzed. The issue of performance limitations in the face of model uncertainty clearly overlaps with the area of adaptive control. Indeed, if we view adaptive con- trol as providing a solution to the control of specic classes of uncertain systems, then the question of per- formance limitations can be rephrased as, “How good can an adaptive controller be?” or “Does there exist a computable lower bound on achievable performance of adaptive control algorithms, and if so, how closely do existing algorithms approach this bound?” Such questions have been posed in the literature. See for example the comparison of Lyapunov-based adaptive controllers and certainty equivalence adap- tive controllers for linear time-invariant systems, dis- cussed in [8]. More recently, Anderson and Gevers [1] have posed several questions regarding fundamental performance of trade-o issues in adaptive control. The later pa- per takes an engineering viewpoint of adaptive con- trol. The authors of the current paper share a similar 0167-6911/03/$-see front matter c  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-6911(02)00338-9