Systems & Control Letters 58 (2009) 282–288 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle A unified H ∞ adaptive observer synthesis method for a class of systems with both Lipschitz and monotone nonlinearities Ali Zemouche ∗ , Mohamed Boutayeb Centre de Recherche en Automatique de Nancy, CRAN-CNRS UMR 7039, Nancy-Université, France article info Article history: Received 16 January 2008 Received in revised form 4 November 2008 Accepted 12 November 2008 Available online 3 January 2009 Keywords: State observers Nonlinear systems LMI approach The Differential Mean Value Theorem (DMVT) H ∞ adaptive estimator abstract This paper investigates the problem of the H ∞ adaptive observer design for a class of nonlinear dynamical systems. The main contribution consists in providing a unified synthesis method for systems with both Lipschitz and monotone nonlinearities (not necessarily Lipschitz). Thanks to the innovation terms into the nonlinear functions [M. Arcak, P. Kokotovic, Observer-based control of systems with slope-restricted nonlinearities, IEEE Transactions on Automatic Control 46 (7) (2001) 1146–1150] and to the differential mean value theorem [A. Zemouche, M. Boutayeb, G.I. Bara, Observers for a class of Lipschitz systems with extension to H ∞ performance analysis, Systems and Control Letters 57 (1) (2008) 18–27], the stability analysis leads to the solvability of a Linear Matrix Inequality (LMI) with several degrees of freedom. For simplicity, we start by presenting the result in an H ∞ adaptive-free context. Furthermore, we propose an H ∞ adaptive estimator that extends easily the obtained results to systems with unknown parameters in the presence of disturbances. We show, in particular, that the matching condition in terms of an equality constraint required in several works is not necessary and therefore allows reducing the conservatism of the existing conditions. Performances of the proposed approach are shown through a numerical example with a polynomial nonlinearity. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Due to complex behaviors of tremendous natural and artificial processes, observer design for nonlinear dynamic systems has been extensively studied during the last years [1–5]. It remains one of the challenging and open research problems in the area of control theory since used in stabilization, diagnosis or systems supervision. Various approaches have been developed for different types of nonlinear models. One of them is based on a nonlinear change of coordinates to bring the system into a pseudo-linear canonical form easily treated by linear techniques [6–9], however, it requires solving a set of constraints hard to be met for MIMO systems with disturbances. For the latter with Lipschitz nonlinearity, an alternative approach was proposed first by Thau [10]. Since then, significant improvements were established where the stability conditions are expressed in terms of algebraic Riccati equations in connection with the upper bound of the Lipschitz constant [11,12]. The same class of systems is investigated in [13] to construct a state observer, where the convergence of the estimation error has been ∗ Corresponding address: LSIIT-CNRS UMR 7005, Louis Pasteur University, France. Tel.: +33 3 82 39 62 24; fax: +33 3 82 39 62 91. E-mail addresses: Ali.Zemouche@iut-longwy.uhp-nancy.fr (A. Zemouche), Mohamed.Boutayeb@iut-longwy.uhp-nancy.fr (M. Boutayeb). studied by using both Lyapunov functions and functionals, and stability conditions are expressed using LMIs. However, all these stability conditions are difficult to be satisfied for large values of the Lipschitz constant. In a recent work [5], to reduce this conservatism, we introduced the differential mean value theorem in order to represent the dynamics of the estimation error as a Linear Parameter Varying (LPV) system. The observer gain is then obtained by solving a set of LMIs. This methodology (transforming error dynamics into LPV systems) can also be obtained using the contraction theory [14,15]. An alternative and interesting approach has been recently presented in [16,17]. It consists in representing the observer error system as the feedback interconnection of a linear system and time-varying sector nonlinearity. This approach eliminates the global Lipschitz restriction and avoids high gain. The stability conditions expressed in terms of LMIs, under an equality constraint, are non-restrictive and easy to satisfy for monotone systems. Nevertheless, to make this approach much less conservative, it is suitable to avoid the equality constraint that appears in the observer synthesis. This goal was solved in [18] by the same author. Over the last decades, the adaptive observer design problem has become increasingly a subject of research in progress. Several approaches are established in the literature. For an overview, we refer the reader to [19–26]. Nevertheless, all these approaches suffer from some disadvantages, such as the presence of equality constraint in the synthesis conditions, and the difficulty to study 0167-6911/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2008.11.007