A unified adaptive observer synthesis method for a class of systems with both Lipschitz and monotone nonlinearities

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.