Henri Bourlès

Bernard Vau, Henri Bourlès

In 1998, A. Karimi and I.D. Landau published in the journal "Systems and Control letters" an article entitled "Comparison of the closed-loop identification methods in terms of bias distribution". One of its main purposes was to provide a bias distribution analysis in the frequency domain of closed-loop output error identification algorithms that had been recently developed. The expressions provided in that paper are only valid for prediction error identification methods (PEM), not for pseudo-linear regression (PLR) ones, for which we give the correct frequency domain bias analysis, both in open- and closed-loop. Although PLR was initially (and is still) considered as an approximation of PEM, we show that it gives better results at high frequencies.

Henri Bourlès

The expression "Algebraic Analysis" was coined by Mikio Sato. It consists of using algebraic notions to solve analytic problem. The origin of Algebraic Analysis is Algebraic Geometry as was developed by Alexander Grothendieck and his school. Mimicking the introduction of Grothendieck's EGA (changing only a few words) one obtains a good definition of the modern theory of linear dynamical systems, as developed by Michel Fliess, Ian Willems, Ulrich Oberst and others.

Bernard Vau, Henri Bourlès

In this paper we generalize three identification recursive algorithms belonging to the pseudo-linear class, by introducing a predictor established on a generalized orthonormal function basis. Contrary to the existing identification schemes that use such functions, no constraint on the model poles is imposed. Not only this predictor parameterization offers the opportunity to relax the convergence conditions of the associated recursive schemes, but it entails a modification of the bias distribution linked to the basis poles. This result is specific to pseudo-linear regression properties, and cannot be transposed to most of prediction error method algorithms. A detailed bias distribution is provided, using the concept of equivalent prediction error, which reveals strong analogies between the three proposed schemes, corresponding to ARMAX, Output Error and a generalization of ARX models. That leads to introduce an indicator of the basis poles location effect on the bias distribution in the frequency domain. As shown by the simulations, the said basis poles play the role of tuning parameters, allowing to manage the model fit in the frequency domain, and allowing efficient identification of fast sampled or stiff discrete-time systems.