Some remarks on the bias distribution analysis of discrete-time identification algorithms based on pseudo-linear regressions

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

Identification algorithms based on pseudo-linear regression with predictors parametrized on generalized bases of orthonormal transfer functions

Cette thèse porte sur l’identification des systèmes linéaires stationnaires, représentés par des fonctions de transfert en temps discret. Pour un ordre donné, contrairement aux méthodes d'identification visant explicitement à minimiser la variance de l'erreur de prédiction, les algorithmes basés sur la régression pseudo-linéaire induisent des modèles dont la distribution des biais est dépendante de la paramétrisation du prédicteur. Ceci a été démontré grâce au concept innovant d'erreur de prédiction équivalente, signal en général non mesurable, dont la variance est effectivement minimisée dans le cadre de la régression pseudo-linéaire.Dans un second temps, sont proposées des versions revisitées des algorithmes récursifs de l'erreur de sortie et des moindres carrés étendus (ainsi que de leurs équivalents en boucle fermée), dont les prédicteurs sont exprimés sur les bases généralisées de fonctions de transfert orthonormales, introduites par Heuberger et al. dans les années 1990 et 2000. La sélection des pôles de la base revient à imposer le noyau reproduisant de l'espace de Hilbert auquel appartiennent ces fonctions de transfert, et à spécifier la manière dont l'approximation est réalisée par les algorithmes. Nous utilisons une expression particulière de ce noyau reproduisant pour introduire un indicateur de l'effet des pôles de la base sur la qualité de l'ajustement du modèle dans le domaine fréquentiel. Cet indicateur joue un grand rôle d'un point de vue heuristique. Enfin, un test de validation en adéquation avec ces algorithmes d'identification est proposé, dont les propriétés statistiques sont explicitées. Les retombées concrètes de ces travaux résident dans la mise à disposition de paramètres de réglages simples et peu nombreux (les pôles de la base), utilisables en fonction du but implicite assigné à l'identification. L'obtention de modèles d'ordre réduit s'en trouve facilitée. De plus l'identification des systèmes raides - comportant des modes dont les fréquences sont séparées de plusieurs décades- jusqu'alors impossible en temps discret, est rendue accessible.This thesis deals with identification of linear time invariant systems described by discrete-time transfer functions. For a given order, contrary to identification methods minimizing explicitly the prediction error variance, algorithms based on pseudo-linear regression produce models with a bias distribution dependent on the predictor parametrization. This has been demonstrated by the innovating concept of equivalent prediction error, a signal in general non-measurable, whose variance is effectively minimized by the pseudo-linear regression.In a second step, revisited versions of recursive algorithms are proposed (Output Error, extended least squares, and their equivalents in closed-loop), whose predictors are expressed on generalized bases of transfer functions introduced by Heuberger et al. in the 1990s and 2000s. The selection of the basis poles is equivalent to define the reproducing kernel of the Hilbert space associated to these functions, and to impose how approximation is achieved by the algorithms. A particular expression of this reproducing kernel is employed to introduce an indicator of the basis poles effect on the model fit in the frequency domain. This indicator plays a great role from a heuristic point of view.At last, a validation test in accordance with these algorithms is proposed. Its statistical properties are given. This set of algorithms provides to the user some simple tuning parameters (the basis poles) that can be selected in function of the implicit purpose assigned to the identification procedure. Obtaining reduced order models is made easier, while identification of stiff systems –impossible until now in discrete-time- becomes accessible

Generalized convergence conditions of the parameter adaptation algorithm in discrete-time recursive identification and adaptive control

International audienceIn this paper, we extend convergence conditions for the parameter adaptation algorithm, used in discrete-time recursive identification schemes, or in adaptive control. Whereas the classical stability analysis of this algorithm consists in checking the strictly real positiveness of an associated transfer function, we demonstrate that convergence can be obtained even when this condition is not fulfilled, under some assumptions on the algorithm forgetting factors. These results regarding both deterministic and stochastic contexts are obtained by analyzing convergence with a prescribed degree of stability

Closed-loop output error identification algorithms with predictors based on generalized orthonormal transfer functions: Convergence conditions and bias distribution

International audienceThis paper proposes an improved version of closed-loop output-error identification algorithms, where the predictor is established on a generalized basis of orthonormal transfer functions. It is shown that the selection of the basis poles impacts the convergence conditions and the bias distribution of the schemes.These algorithms present several advantages: They are able to identify in closed-loop fast sampled systems, stiff systems (with modes spread over three decades or more), and reduced order models. Moreover, they are suitable for unstable systems or controllers. A simulation example shows the effectiveness of this approach. These algorithms can be employed in an open-loop context by using a straightforward simplification

Some remarks on the bias distribution analysis of discrete-time identification algorithms based on pseudo-linear regressions

International audienceIn 1998, A. Karimi and I.D. Landau published in this journal an article entitled ''Comparison of the closed-loop identication 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 identication 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

Laguerre based predictors in discrete-time recursive algorithms: A solution for open-loop identification under oversampling

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Adaptive Rejection of Narrow-band Disturbances in the Presence of Plant Uncertainties -A Dual Youla-Kucera Approach

International audienceThe stability of adaptive disturbance rejection schemes using Youla-Kucera (YK) parametrization and the internal model principle (IMP) in the presence of plant model uncertainties is investigated. The problem is approached by using the dual Youla Kucera parametrization for the description of the plant model uncertainties. The known disturbance case is discussed first, emphasizing the need of over parametrization of the Youla Kucera filter used for control in order to both solving the IMP and the stability problems. Then this solution is extended for the case of unknown disturbances leading to the use of a parameter adaptation algorithm with projection. A stability analysis of the adaptive scheme is provided. Simulation results on relevant examples and experimental evaluation on an active noise attenuation system illustrate the possibilities of this approach for handling significant plant-model mismatch

A robust whitness test for the identification of discrete-time linear models: Use of orthonormal transfer functions

International audienceA novel whiteness test of residuals is proposed, which makes use of generalized bases of orthonormal transfer functions. It can be viewed as a robustified version of the classical whiteness test in the sense that it reduces the risk of type II errors, by introducing a frequency weighting in the assessment of the flatness in the residual power spectrum density. This frequency weighting, which depends on the basis poles, can be employed for the validation of reduced order models, when the flatness of the residual power spectrum density is evaluated over a limited frequency band