Identification and rational L 2 approximation: a gradient algorithm. Laurent Baratchart, Michel Cardelli, Martine Olivi 1 Abstract: This paper deals with the identification of linear constant dy- namical systems when formalized as a rational approximation problem. The criterion is the l 2 norm of the transfer function, which is of interest in a stochastic context. The problem can be expressed as nonlinear optimization in a Hilbert space, but standard algorithms are usually not well adapted. Here, we present a generic recursive procedure to find a local optimum of the criterion in the case of scalar systems. Our methods are borrowed from differential theory mixed with a bit of classical complex analysis. To our knowledge, the algorithm described in this paper is the first that ensures convergence to a local minimum. Finally, we discuss a number of unsettled issues. 1 Introduction. In this paper, we approach the problem of identification within the framework of Hardy spaces by considering this as a rational approximation problem. We restrict ourselves to linear constant strictly causal single-input single-output dynamical systems (in short systems). We first consider the case of a dis- crete time system. Let f 1 ,f 2 , ..., f m , ... be its impulse response. Identifying the system usually means recognizing the sequence (f m ) as the Taylor coeffi- cients at infinity of a proper rational function whose denominator degree (in irreducible form) is then the order of the system. But since such a sequence might not exist, in practice one has to be content with finding a rational sequence (r m ) that resembles (f m ). This, of course, has no definite meaning, and some criteria has to be chosen. Such criteria can occur in connection with stability. Assume, for instance, that the system is l k -stable for some k ≥ 1, that is ∞  m=1 |f m | k < ∞. 1 Institut National de Recherche en Informatique et Automatique, route des Lucioles, Sophia-Antipolis 06565 Valbonne Cedex (France) 1