Likelihood Analysis of Power Spectra and Generalized Moment Problems

We develop an approach to spectral estimation that has been advocated by Ferrante, Masiero and Pavon and, in the context of the scalar-valued covariance extension problem, by Enqvist and Karlsson. The aim is to determine the power spectrum that is consistent with given moments and minimizes the relative entropy between the probability law of the underlying Gaussian stochastic process to that of a prior. The approach is analogous to the framework of earlier work by Byrnes, Georgiou and Lindquist and can also be viewed as a generalization of the classical work by Burg and Jaynes on the maximum entropy method. In the present paper we present a new fast algorithm in the general case (i.e., for general Gaussian priors) and show that for priors with a specific structure the solution can be given in closed form.Comment: 17 pages, 4 figure

Non-Oscillatory Pattern Learning for Non-Stationary Signals

This paper proposes a novel non-oscillatory pattern (NOP) learning scheme for several oscillatory data analysis problems including signal decomposition, super-resolution, and signal sub-sampling. To the best of our knowledge, the proposed NOP is the first algorithm for these problems with fully non-stationary oscillatory data with close and crossover frequencies, and general oscillatory patterns. NOP is capable of handling complicated situations while existing algorithms fail; even in simple cases, e.g., stationary cases with trigonometric patterns, numerical examples show that NOP admits competitive or better performance in terms of accuracy and robustness than several state-of-the-art algorithms

Rational Covariance Extension, Multivariate Spectral Estimation, and Related Moment Problems: Further Results and Applications

This dissertation concerns the problem of spectral estimation subject to moment constraints. Its scalar counterpart is well-known under the name of rational covariance extension which has been extensively studied in past decades. The classical covariance extension problem can be reformulated as a truncated trigonometric moment problem, which in general admits infinitely many solutions. In order to achieve positivity and rationality, optimization with entropy-like functionals has been exploited in the literature to select one solution with a fixed zero structure. Thus spectral zeros serve as an additional degree of freedom and in this way a complete parametrization of rational solutions with bounded degree can be obtained. New theoretical and numerical results are provided in this problem area of systems and control and are summarized in the following. First, a new algorithm for the scalar covariance extension problem formulated in terms of periodic ARMA models is given and its local convergence is demonstrated. The algorithm is formally extended for vector processes and applied to finite-interval model approximation and smoothing problems. Secondly, a general existence result is established for a multivariate spectral estimation problem formulated in a parametric fashion. Efforts are also made to attack the difficult uniqueness question and some preliminary results are obtained. Moreover, well-posedness in a special case is studied throughly, based on which a numerical continuation solver is developed with a provable convergence property. In addition, it is shown that solution to the spectral estimation problem is generally not unique in another parametric family of rational spectra that is advocated in the literature. Thirdly, the problem of image deblurring is formulated and solved in the framework of the multidimensional moment theory with a quadratic penalty as regularization

E-diagnostic de processus physiques à base des méthodes de haute résolution Application : machines éoliennes

The expansion of systems using intelligent sensors has prompted the study of physical processes E-diagnosis based on high resolution methods. The automated control of modern wind machines requires proactive maintenance. We proposed several indicators measuring the performance level of a wireless protocol for routing data packets to the monitoring station. A study to design a diagnosis system entitled IESRCM for local or remote monitoring for the mentioned machines is achieved. A comparison has been realized to appreciate the performance of this system when it is integrated with GPRS or Wi-Max wireless modules. The obtained results by simulation using Proteus ISIS and OPNET software have favored the incorporation of Wi-Max module in the proposed system because its advantages over GPRS. The high resolution spectral estimation methods are effectively used for detecting electromechanical wind turbine faults. In front of the diversity of these methods, an investigation of each algorithm separately has been performed with a composite signal of stator current containing several types of defects and under different noisy environments. It was deduced in therein that the accuracy of the spectral estimation depends on the degree of the signal disturbance, the severity level of the faults, the frequency sampling and the number of data samples. The comparison with simulation in Matlab that we have made between these algorithms has proved the superiority of ESPRIT algorithm. However, this algorithm has a relatively large computing time and requires an important memory size to be executed. To overcome this problem, an improvement of ESPRIT-TLS technique has been proposed to make it applicable in real time. A new version of this method is developed in this thesis entitled Fast-ESPRIT. The proposed development is made by combining pass band recursive filtering technique IIR of Yule-Walker and decimation technique. The evaluation of the proposed technique for wind turbine fault detection of various types is performed. The analysis of the obtained results confirms that the Fast-ESPRIT algorithm provides a very satisfactory spectral accuracy in discriminating the studied faults harmonics. It resulted in a reduced complexity with an eligible ratio, a reduction of the required memory size for its implementation 5 times lower and a decrease of calculation time about 14,25 times less. This method provides better spectral resolution even in presence of a significant number of harmonics of different faults. However, this new method has some limitations because it does not recognize the type and the severity level of a detected fault. Therefore, another real time control approach has been proposed. It combines the developed Fast-ESPRIT method, the fault classification algorithm called CAFH and a fuzzy inference system interconnected with vibration sensors located on various wind turbine components. A new indicator of severity level for each studied fault type was formulated. It allows avoiding unnecessary alarms. Matlab simulation of this approach under four failure types with a noise shows that it provides a good robustness of faults classification.L’expansion des systèmes utilisant des capteurs intelligents a incité l’étude d’E-diagnostic de processus physiques à base des méthodes de haute résolution. Le contrôle automatisé des machines éoliennes modernes nécessite la maintenance proactive. On a proposé plusieurs indicateurs mesurant le niveau de performance d’un protocole d’acheminement sans fil des paquets de données vers la station de supervision. Une étude de conception d’un système de diagnostic IESRCM permettant la surveillance locale ou à distance des machines indiquées est réalisée. Une comparaison a été effectuée pour apprécier les performances de ce système lors de son intégration avec les modules sans fil GPRS ou Wi-Max. Les résultats obtenus avec simulation sous Proteus ISIS et OPNET ont favorisé l’incorporation du module Wi-Max dans le système proposé en raison de ses avantages par rapport au GPRS. Les méthodes d’estimation spectrale à haute résolution sont efficacement utilisées pour la détection de défauts électromécaniques d’éoliennes. Devant la diversité de ces méthodes, une investigation de chaque algorithme à part est réalisée avec un signal composite du courant statorique contenant plusieurs types de défauts et sous un environnement différemment bruité. On a déduit à cet égard que la précision de l’estimation spectrale dépend du degré de perturbation du signal, du niveau de sévérité d’un défaut, de la fréquence d’échantillonnage et du nombre d’échantillons de données. La comparaison avec simulation sous Matlab qu’on a effectuée entre ces algorithmes a prouvé la supériorité de l’algorithme ESPRIT. Cependant, cet algorithme présente un temps de calcul relativement grand et demande une taille mémoire importante pour être exécuté. Pour contourner cet obstacle, on a proposé une amélioration de la technique ESPRIT-TLS pour la rendre applicable en temps réel. Une nouvelle version est développée dans cette thèse intitulée Fast-ESPRIT. L’élaboration envisagée est effectuée en combinant la technique de filtrage passe bande récursif IIR de Yule-Walker et la technique de décimation. L’évaluation de la technique proposée dans la détection de quatre types de défauts d’une éolienne est réalisée. L’analyse des résultats obtenus confirme que l’algorithme Fast-ESPRIT offre une précision spectrale très satisfaisante dans la discrimination des harmoniques des défauts étudiés. On a abouti à une complexité réduite avec un rapport admissible, à une réduction de l’espace mémoire requis pour son exécution 5 fois inférieur et à la diminution du temps de calcul d’environ 14,25 fois moins. Cette méthode offre une meilleure résolution même en présence d’un nombre important d’harmoniques de défauts différents. Cependant, cette nouvelle méthode présente quelques limitations puisqu’elle ne permet pas de reconnaitre le type et le niveau de sévérité d’un défaut détecté. On a donc proposé une autre approche de contrôle en temps réel. Celle-ci combine la méthode Fast-ESPRIT développée, l’algorithme de classification de défauts intitulé CAFH et un système d’inférence flou interconnecté aux capteurs de vibration localisés sur les différentes composantes d’éolienne. Un nouvel indicateur du niveau de sévérité de chaque type de défaut a été formulé. Il permet d’éviter les alarmes inutiles. La simulation sous Matlab de cette approche avec quatre types de défaillances en présence d’un bruit montre qu’elle offre une meilleure robustesse dans la classification des défauts

Novel Results on the Factorization and Estimation of Spectral Densities

This dissertation is divided into two main parts. The first part is concerned with one of the most classical and central problems in Systems and Control Theory, namely the factorization of rational matrix-valued spectral densities, commonly known as the spectral factorization problem. Spectral factorization is a fundamental tool for the solution of a variety of problems involving second-order statistics and quadratic cost functions in control, estimation, signal processing and communications. It can be thought of as the frequency-domain counterpart of the ubiquitous Algebraic Riccati Equation and it is intimately connected with the celebrated Kálmán-Yakubovich-Popov Lemma and, therefore, to passivity theory. Here, we provide a rather in-depth and comprehensive analysis of this problem in the discrete-time setting, a scenario which is becoming increasingly pervasive in control applications. The starting point in our analysis is a general spectral factorization result in the same vein of Dante C. Youla. Building on this fundamental result, we then investigate some key issues related to minimality and parametrization of minimal spectral factors of a given spectral density. To conclude, we show how to extend some of the ideas and results to the more general indefinite or J-spectral factorization problem, a technique of paramount importance in robust control and estimation theory. In the second part of the dissertation, we consider the problem of estimating a spectral density from a finite set of measurements. Following the Byrnes-Georgiou-Lindquist THREE (Tunable High REsolution Estimation) paradigm, we look at spectral estimation as an optimization problem subjected to a generalized moment constraint. In this framework, we examine the global convergence of an efficient algorithm for the estimation of scalar spectral densities that hinges on the Kullback-Leibler criterion. We then move to the multivariate setting by addressing the delicate issue of existence of solutions to a parametric spectral estimation problem. Eventually, we study the geometry of the space of spectral densities by revisiting two natural distances defined in cones for the case of rational spectra. These new distances are used to formulate a "robust" version of THREE-like spectral estimation