Le comportement dynamique de structures complexes, comme les alternateurs, doit être maîtrisé afin d’en garantir un fonctionnement fiable. Cependant, la modélisation comporte de nombreuses incertitudes rendant délicates la prédiction du comportement vibratoire. Ces travaux de recherche ont pour objectif de fournir des outils d’aide à la décision afin de faciliter la prise de décision rapide suite au redémarrage d’alternateurs. Basé sur la théorie info-gap, un premier outil d’aide à la décision est proposé : il a pour objectif d’évaluer la robustesse de réponses dynamiques vis-à-vis d’un modèle modal incertain. Nous avons également étudié comment de l’information nouvelle peut être intégrée au modèle d’incertitude pour améliorer sa représentativité à la réalité.Une expansion par l’erreur en relation de comportement étendue de modes propres identifiés permet ensuite d’enrichir la représentativité du modèle numérique fournissant ainsi un modèle qualifié d’hybride et permettant d’évaluer les niveaux de réponse. Comme la modélisation comporte de nombreuses méconnaissances, nous avons proposé le procédé d’expansion robuste dont l’objectif est d’obtenir des vecteurs étendus robustes. En présence de méconnaissances sévères, nous montrons enfin qu’il est préférable de calibrer un modèle en maximisant la robustesse vis-à-vis des incertitudes plutôt qu’en maximisant uniquement la fidélité vis-à-vis des données. Couplée à des techniques de réduction de modèle et de construction de méta modèles,nous appliquons cette démarche à une structure de complexité industrielle représentative du contexte industriel. / The complex structural dynamic behavior of alternator must be well understood in order to insuretheir reliable and safe operation. The numerical model is however difficult to construct mainlydue to the presence of a high level of uncertainty. The objective of this work is to providedecision support tools in order to assess the vibratory levels in operation before to restart thealternator. Based on info-gap theory, a first decision support tool is proposed: the objective hereis to assess the robustness of the dynamical response to the uncertain modal model. Based on realdata, the calibration of an info-gap model of uncertainty is also proposed in order to enhance itsfidelity to reality. Then, the extended constitutive relation error is used to expand identified modeshapes which are used to assess the vibratory levels. The robust expansion process is proposed inorder to obtain robust expanded mode shapes to parametric uncertainties. In presence of lack-ofknowledge,the trade-off between fidelity-to-data and robustness-to-uncertainties which expressesthat robustness improves as fidelity deteriorates is emphasized on an industrial structure by usingboth reduced order model and surrogate model techniques.

This thesis is divided into two parts. The first part deals with theNevanlinna-Pick interpolation problem, a problem which occursnaturally in several applications such as robust control, signalprocessing and circuit theory. We consider the problem of shaping andapproximating solutions to the Nevanlinna-Pick problem in a systematicway. In the second part, we study distance measures between powerspectra for spectral estimation. We postulate a situation where wewant to quantify robustness based on a finite set of covariances, andthis leads naturally to considering the weak*-topology. Severalweak*-continuous metrics are proposed and studied in this context.In the first paper we consider the correspondence between weighted entropyfunctionals and minimizing interpolants in order to find appropriateinterpolants for, e.g., control synthesis. There are two basic issues that weaddress: we first characterize admissible shapes of minimizers bystudying the corresponding inverse problem, and then we developeffective ways of shaping minimizers via suitable choices of weights.These results are used in order to systematize feedback controlsynthesis to obtain frequency dependent robustness bounds with aconstraint on the controller degree.The second paper studies contractive interpolants obtained as minimizersof a weighted entropy functional and analyzes the role of weights andinterpolation conditions as design parameters for shaping theinterpolants. We first show that, if, for a sequence of interpolants,the values of the corresponding entropy gains converge to theoptimum, then the interpolants converge in H_2, but not necessarily inH-infinity. This result is then used to describe the asymptoticbehaviour of the interpolant as an interpolation point approaches theboundary of the domain of analyticity.A quite comprehensive theory of analytic interpolation with degreeconstraint, dealing with rational analytic interpolants with an apriori bound, has been developed in recent years. In the third paper,we consider the limit case when this bound is removed, and only stableinterpolants with a prescribed maximum degree are sought. This leadsto weighted H_2 minimization, where the interpolants areparameterized by the weights. The inverse problem of determining theweight given a desired interpolant profile is considered, and arational approximation procedure based on the theory is proposed. Thisprovides a tool for tuning the solution for attaining designspecifications. The purpose of the fourth paper is to study the topology and develop metricsthat allow for localization of power spectra, based on second-orderstatistics. We show that the appropriate topology is theweak*-topology and give several examples on how to construct suchmetrics. This allows us to quantify uncertainty of spectra in anatural way and to calculate a priori bounds on spectral uncertainty,based on second-order statistics. Finally, we study identification ofspectral densities and relate this to the trade-off between resolutionand variance of spectral estimates.In the fifth paper, we present an axiomatic framework for seekingdistances between power spectra. The axioms requirethat the sought metric respects the effects of additive andmultiplicative noise in reducing our ability to discriminate spectra.They also require continuity of statistical quantities withrespect to perturbations measured in the metric. We then present aparticular metric which abides by these requirements. The metric isbased on the Monge-Kantorovich transportation problem and iscontrasted to an earlier Riemannian metric based on theminimum-variance prediction geometry of the underlying time-series. Itis also being compared with the more traditional Itakura-Saitodistance measure, as well as the aforementioned prediction metric, ontwo representative examples. / QC 20100817