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International audience; An Exponential-Polynomial-Trigonometric (EPT) function is defined on [0,∞) by a minimal realization (A, b, c). A stable non-negative EPT function of a fixed degree is fitted to the histogram of a large set of data using an L2 criterion. If we neglect the non-negativity constraint this is shown to be equivalent to a rational approximation problem which is approached using the RARL2 software. We show how, under the additional assumption of the existence of a strictly dominant real pole of the rational function, the non-negativity constraint on the EPT function can be imposed by performing a constraint convex optimization on b at each stage at which an (A, c) pair is determined. In this convex optimization step a recent generalized Budan-Fourier sequence approach to determine non-negativity of an EPT function on a finite interval plays a major role.

International audience; Many normalizations in various classes of linear dynamical state-space systems lead to system representations which are determined up to a state isometry. Here we present a new set of techniques to obtain (local) canonical forms under state isometries, using what we call sub-diagonal pivot structures. These techniques lead to a very flexible, straightforward algorithm to put any system into canonical form under state isometries. The parametrization of these canonical forms is discussed for a number of classes, including lossless systems and input-normal stable systems both in discrete time and in continuous time.

Semi-plenary conference by Ralf L.M. Peeters; International audience; Lossless systems have many applications in systems and control theory, including signal processing, filter design, system identification , system approximation, and the parameterization of classes of linear systems. In this survey paper we address the issue of parameterization of the space of rational lossless matrix functions by successfully combining two approaches. The first approach proceeds in state-space from balanced realizations and triangular pivot structures of reachability matrices. The second approach concerns interpolation theory with linear fractional transformations and the tangential Schur algorithm. We construct balanced realizations (and canonical forms) in terms of the Schur parameters encountered in the tangential Schur algorithm, and conversely, we interpret balanced realizations in discrete-time and in continuous-time in terms of Schur parameters. A number of application areas are discussed to illustrate the importance of this theory for a variety of topics.

International audience; In this paper, the connections are investigated between two different approaches towards the parametrization of multivariable stable all-pass systems in discrete-time. The first approach involves the tangential Schur algorithm, which employs linear fractional transformations. It stems from the theory of reproducing kernel Hilbert spaces and enables the direct construction of overlapping local parametrizations using Schur parameters and interpolation points. The second approach proceeds in terms of state-space realizations. In the scalar case, a balanced canonical form exists that can also be parametrized by Schur parameters. This canonical form can be constructed recursively, using unitary matrix operations. Here, this procedure is generalized to the multivariable case by establishing the connections with the first approach. It gives rise to balanced realizations and overlapping canonical forms directly in terms of the parameters used in the tangential Schur algorithm.

This paper deals with a new and systematic method of approximating exact nonlinear filters with finite dimensional filters. The method used here is based on the differential geometric approach to statistics. The projection filter is derived in the case of exponential families. A characterization of the filters is given in terms of an assumed density principle. An a posteriori measure of the performance of the projection filter is defined. Applications to particular systems, and numerical schemes which can be used to implement the projection filter are given in the final part. The results of simulations for the cubic sensor are discussed.; Cet article propose une méthode nouvelle et systématique pour l'approximation d'un filtre non-linéaire exact par un filtre de dimension finie. La méthode repose sur l'utilisation d'outils de géométrie différentielle en statistique. L'équation du filtre par projection est établie dans le cas des familles exponentielles, et on en donne une caractérisation en tant que filtre de forme donnée. On définit également une mesure a posteriori de la qualité de l'approximation. Dans la dernière partie, on étudie quelques exemples, et on propose un schéma numérique pour la mise en œuvre du filtre par projection. Finalement, on présente des résultats de simulations pour le problème du senseur cubique.