INT. J. CONTROL, 1993, VOL. 57, No.3, 537-556 A Luenberger-like observer for nonlinear systems G. CICCARELLAt, M. DALLA MORAt and A. GERMANIt A state observer is proposed for nonlinear continuous time systems which extends the well known Luenberger observer. In particular, on the basis of simple assumptions on the regularity of the system equations (observability and the global Holder condition for suitable functions), which are generally satisfied for physically meaningful dynamic systems, the global asymptotic convergence of the estimated state towards the true state is shown. Finally, some examples of applications are also reported showing the effectiveness of the proposed observer. 1. Introduction The design of state estimators is one of the essential points in control theory whose solution, in the linear case, is the well known Luenberger's observer (1971). A first systematic contribution to the theory of observers for nonlinear systems was a set of conditions under which the dynamics of the observation error is linear (Krener and Isidori 1983, Krener and Respondek 1985, Xia and Gao 1989). Unfortunately the necessary and sufficient conditions for this type of observer to exist are rather restrictive, as in the dual problem of feedback linearization. A nice contribution to the extension of the Luenberger observer for nonlinear systems by a linearization technique has been given by Zeitz (1987). The algorithm proposed by Zeitz, which also uses time derivatives of the input, is easy to implement, but it does not in general guarantee the convergence of the observer. A different approach based on 'high-gain' approximate cancellation of the nonlinearity has been presented by Tornambe (1989). However this approach does not guarantee, with arbitrarily high but finite gain, the asymptotic convergence of the estimated state to the true state. It is easy to demonstrate that, even if the initial state for the observer coincides with that of the system, the error could be in general only bounded and there is no guarantee that it converges asymptotically to zero. Marino (1990) presented an adaptive observer for single-input single-output nonlinear systems that can be transformed to a certain observable canonical form, and Bastin and Gevers (1988) gave necessary and sufficient conditions for a system to be transformed into this canonical form. This observer requires a transformation which is often difficult to find and conditions given by Bastin and Gevers are restrictive from an application point of view. The proposed adaptive Received 20 August 1990. Revised 15 December 1991 and 15 March 1992. Communicated by Professor A. Isidori. t Department of Electrical Engineering, University of L'Aquila, Monteluco, 67100 L'Aquila, Italy. 0020-7179/93 $10.00 © 1993 TaylQr & FrancisLtd