Copyright (c) 2011 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. 1 EKF-like observer with stability for a class of nonlinear systems Lizeth Torres, Gildas Besanc ¸on, and Didier Georges Abstract—An Extended-Kalman-Filter-like observer is derived from a former observer result for a class of nonlinear systems, which can be written as a linear part in the unmeasured states on the one hand, and some additive nonlinearity with a triangular Jacobian on the other hand. It is shown how the previously presented excitation condition for exponential stability of the observer, extends to this EKF version. The observer is illustrated in simulation with two challenging examples, the first one in leak detection, and the second one in chaos synchronization. Index Terms—Nonlinear observer, Extended Kalman Filter, locally regular inputs, high gain, stability. I. I NTRODUCTION Nonlinear observers have been more and more studied in the last years, motivated by various applications, and resulting in various approaches, as presented in [1], [2] for instance. Among those approaches, immersions have more particularly regained interest since early results of [3]. In particular in [4], it was shown how the well-known observability rank condition can allow to turn a system into a particular ’triangular’ form (in the sense of triangular structure of the Jacobian matrix of its dynamics) for which an observer can be designed, and can be shown to converge under appropriate excitation. This observer extends the efficient ’high gain’ technique originally proposed for uniformly observable systems [5], to observers which are not uniformly observable. On the other hand, the well-known Extended Kalman Filter (EKF) is a very popular practical solution for observer design. Even though its stability has been a problem for a long time, it was shown a few years ago how it can be guaranteed for uniformly observable systems, combining it with high gain techniques [1]. In the present paper, the purpose is to take advantage of those results on EKF, to show how the formerly proposed observer for triangular systems which are not necessarily uniformly observable, can be modified as an EKF-like observer, still guaranteeing convergence. In short, this provides the extension of results of [1] for uniformly observable systems to the more general case of systems which are not necessarily so. In other words, the improvement w.r.t. the results of [4] is similar to that of [1] w.r.t. standard high gain observers. In particular this provides a quite general framework for EKF design with stability, via a tuning high gain on top of the EKF parameters. In addition, the proposed observer is illustrated on a couple of The authors are with the Control systems Department of Grenoble Gipsa-lab, Grenoble Univ., BP 46, 38402 Saint-Martin d’H` eres, France. mailto:gildas.besancon@gipsa-lab.grenoble-inp.fr, didier.georges@gipsa-lab.grenoble-inp.fr examples of interest, which have separately motivated a lot of studies, and left open issues in the area of observer results: a first one in the context of leak detection - where the problem of friction estimation is also considered, and a second one in the field of chaos synchronization - in a context of unknown parameters. The remainder of the paper is then organized as follows: section II presents the main observer result, while section III is dedicated to the two proposed illustrative examples. Section IV finally concludes the paper. II. MAIN RESULT The observer which is proposed in this paper is based on the structure below for the system under consideration: ˙ z =A(u, y)z + B(u, z ) y =C(u)z + D(u) (1) where the involved matrices have the following forms: A(u, y) =      0 A 12 (u, y) 0 . . . A q-1q (u, y) 0 0      , B(u, z ) =        B 1 (u, z 1 ) B 2 (u, z 1 ,z 2 ) . . . B q-1 (u, z 1 ,...,z q-1 ) B q (u, z )        , and C(u) = ( C 1 (u) 0 ... 0 ) , with z = col(z 1 ,...,z q ) ∈ IR N , z 1 ∈ IR N1 , C 1 (u) ∈ IR 1×N1 and z i ∈ IR Ni , A i-1 i ∈ IR Ni-1×Ni for i =2 ...q. Although of a special form, such a structure can be reached by state transformations (such as immersions) for systems satisfying the observability rank condition, as emphasized in [4], which means that it can cover many systems in the context of observer design. Notice that from this structure, the system is not necessarily uniformly observable, namely its observability depends on the inputs. With the purpose of high gain observer design, one needs a guarantee of observability at arbitrarily short times. Following former results on such systems, this can be characterized as follows: Definition 1 (Locally regular inputs [6], [7], [4]) An input function u is said to be locally regular for a system (1) if there Limited circulation. For review only Preprint submitted to IEEE Transactions on Automatic Control. Received: October 31, 2011 10:04:41 PST