MONOPOLI’S MODEL REFERENCE ADAPTIVE CONTROLLER IS INCORRECT Romeo Ortega ∗ Nikita Barabanov ∗∗ Alessandro Astolfi ∗∗∗ ∗ Laboratoire des Signaux et Syst´ emes,Supelec, Plateau du Moulon, 91192 Gif-sur-Yvette, France, ortega@lss.supelec.fr ∗∗ Department of Mathematics, North Dakota State University, Fargo, ND, USA, nikita.barabanov@ndsu.nodak.edu ∗∗∗ Electrical Engineering Department, Imperial College, Exhibition Road, London, SW7 2BT, UK, a.astolfi@ic.ac.uk Abstract: One of the longest standing open questions in adaptive control concerns the correctness of the stability claim of the un–normalized model reference scheme proposed by R. V. Monopoli in 1974. Although provably correct solutions to the problem now abound, in particular, it is well–know that adding a normalization to Monopoli’s original scheme ensures global convergence, it is interesting to know whether this technique– driven modification is really necessary or only required to complete the stability proof in the absence of more elaborate arguments. In this paper we construct a counterexample that provides a definite—unfortunately, negative—answer to the claim. Instrumental for the establishment of this result is a technical lemma that shows that, under some conditions on the regressor that may appear in Monopoli’s scheme, the parameter error freezes as the adaptation gain goes to infinity. On the lighter side, we also prove that the counterexample can be “fixed”, in the sense of achieving semiglobal stability, adjusting some tuning parameters. Keywords: Adaptive control, model reference control, stability of adaptive systems, nonlinear control. 1. INTRODUCTION Model reference adaptive control (MRAC) is unques- tionably the most widely studied problem in the adap- tive literature that has a very long history going back to the 1950’s and extending to the present time. 1 The earliest attempts to solve the MRAC problem followed the classical path of designing an observer, that had 1 The interested reader is referred to [14] for a vivid description of the history as well as to the existing textbooks [21,17,12,9] for further information on MRAC. to be made adaptive because of the unknown plant parameters, and then feeding back the observed state [10]. A first major breakthrough, essentially due to [3,13], was the introduction of the so–called direct control parameterization which revealed that the es- timation of the plant state could be obviated and only a “good” estimation of the controller parameters was needed to achieve the asymptotic model matching ob- jective. A second fundamental development, also reported in [13], was the derivation of a suitable error signal, IFAC Workshop on Adaptation and Learning in Control and Signal Processing, and IFAC Workshop on Periodic Control Systems, Yokohama, Japan, August 30 – September 1, 2004 1