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The fuzzy matrix equations $A\tilde{X}=\tilde{Y}$ is called a singular fuzzy matrix equations while the coefficients matrix of its equivalent crisp matrix equations be a singular matrix. The singular fuzzy matrix equations are divided into two parts: consistent singular matrix equations and inconsistent fuzzy matrix equations. In this paper, the inconsistent singular fuzzy matrix equations is studied and the effect of generalized inverses in finding minimal solution of an inconsistent singular fuzzy matrix equations are investigated.

Recently, Calvetti et al. have published an interesting paper [Linear Algebra Appl. 316 (2000) 157–169] concerning the least-squares solution of a singular system by using the so-called range restricted GMRES (RRGMRES) method. However, one of the main results (cf. [loc. cit., Theorem 3.3]) seems to be incomplete. As a complement of paper [loc. cit.], in this note we first make an example to show the incompleteness of that theorem, then we give a modified result.

The partial realization problem under consideration consists in finding, for a given sequence s =(s k ) 0 N−1 of blocks, matrices (A,E,B,C) of appropriate size such that s i =CE N−1−i A i B and the identity matrix is a linear combination of A and E. We discuss the question whether there is always a realization of this form for which the state space dimension is equal to the maximal rank of the underlying Hankel matrices. We show that this question has an affirmative answer if the block size is less than or equal to 2 and some other cases but not in general. The paper strengthens results obtained by Manthey et al. [cf. W. Manthey, U. Helmke, D. Hinrichsen, in: U. Helmke et al. (Eds.), Operators, Systems, and Linear Algebra, Teubner, Stuttgart, 1997, pp. 138–156]. The main tools are the results of the authors obtained in connection with Vandermonde factorization of block Hankel matrices. Finally, an interpretation of the problem in periodic discrete-time systems is given.

The Ostrowski–Reich theorem gives the necessary and sufficient condition of convergence of the SOR method for hermitian and positive definite matrices. Ortega and Plemmons have generalized the theorem to non-hermitian matrices. For a general nonsingular matrix A, they have given necessary and sufficient conditions of convergence of splitting methods, and have studied the convergence of the SOR method. This note is a generalization of the Ortega and Plemmons theorems to singular matrices. Some necessary and sufficient conditions of semi-convergence for singular matrices are given.

Krylov subspace methods have been recently considered to solve singular linear systems Ax = b . In this paper, we derive the necessary and sufficient conditions guaranteeing that a Krylov subspace method converges to a vector ADb+Px0, where AD is the Drazin inverse of A and P is the projection P = I−A D A . Let k be the index of A. We further show that A D b+Px 0 , x 0 ∈ R (A k−1 )+ N (A) , is a generalized least-squares solution of Ax = b in R (A k )+ N (A) . Finally, we present the convergence bounds for the quasi-minimal residual algorithm (QMR) and transpose-free quasi-minimal residual algorithm (TFQMR). The index k of A in this paper can be arbitrary, which extends to the main results of Freund and Hochbruck (Numer. Linear Algebra Appl. 1 (1994) 403–420) that only considers the case k = 1 .

International audience; Purpose - The purpose of this paper is to propose a decentralized observer for large-scale singular systems. Design/methodology/approach - In this paper, the authors investigate the problem of observers' design for large-scale singular systems. The proposed decentralized observer is based on a new parameterization of the generalized Sylvester equation solution. The considered system is partitioned into small-sizes interconnected subsystems with unknown interconnections. Findings - A decentralized observer based on new parameterization of generalized Sylvester equation. The performance of the proposed approach is illustrated by a numerical example. Originality/value - The proposed approach unites the full-order, the reduced-order and the minimal order observer design for large-scale system. The conditions of the existence of this observer are given in the linear matrix inequalities (LMI) form.

This paper provides sufficient conditions for both practical stability and finite-time stability of linear singular continuous time-delay systems which can be mathematically described as Ex(t)=Aox(t)+A1x(t-t). Considering a finite-time stability concept, new delay independent and delay dependent conditions have been derived using the approach based on the Lyapunov-like functions and their properties on the subspace of consistent initial conditions. These functions do not need to have the properties of positivity in the whole state space and negative derivatives along the system trajectories. When the practical stability has been analyzed the above mentioned approach was combined and supported by the classical Lyapunov technique to guarantee the attractivity property of the system behavior. Moreover an linear matrix inequality (LMI) approach has been applied in order to get less conservative conditions.

In this dissertation, we investigated the problems of the estimation, $H_{\infty}$ filtering and the controller based-observer design for standard large scale systems and for algebro-differentials ones called also singular large scale systems. So, we began to propose methods for decentralized observer design with unknown interconnections for standard and singular systems. We search here to decouple the unknown interconnections and the dynamics of the observation error. The method is based on LMIs approach to find the gain matrix implemented in the observer matrices. The LMI formulation is based on Lyapunov approach and deduced from various bounded lemmas. We propose then, filters that permits to ensure, in addition to the stability, an $H_{\infty}$ performance criteria; we search to attenuate the perturbations effect, supposed unknown but of bounded energy, on the dynamics of the estimation error. We discussed after the study of interconnected observers for large systems, we have proposed a new method to design a new form of interconnected observers connectively stable. We are interested in the ability of a such observer to be robustly stable towards uncertainties in the interconnections between sub-observers. Finally, we are interested to the application of the proposed estimation methods to the control purpose. Indeed, in a first step, we propose a decentralized control based on a filter $H_{\infty}$ for a class of large scale standard systems with nonlinear interconnections. Then, Then, we focus our attention on the observer based control for singular systems to search for a control law which ensures an $H_{\infty}$ performance criteria. The approach is an extension of recent works of Kalsi and al. to the case of disturbed large scale systems. One of the main contributions of our work was to propose a new method of control design based on a filter $H_ {\infty} $ which generated by using less restrictive conditions than those introduced in the work of Kalsi and al. Thus, we have relaxed the constraint on the distance between the pair of matrices formed by the state matrix and input matrix and the set of pairs of matrices uncontrollable. In addition, we consider the maximization of the bound of the interconnection, which is very important in practice. Our approach is obtained into two steps. The first one consist on calculating the gain state feedback that meets the specifications of the closed loop system. The second one , then we reconstruct this control law using our previous results on the $H_{\infty}$ filtering. The approach has been validated on an example of a system composed of three interconnected electrical machines, Dans ce mémoire, on s'est intéressé aux problèmes d'estimation, de filtrage $H_{\infty}$ et de la commande basée observateur des systèmes de grande dimension. L'étude porte sur les systèmes linéaires standards mais aussi sur les systèmes algèbro-différentiels appelés aussi systèmes singuliers pour couvrir la classe la plus large possible des systèmes de grande dimension. Ainsi, on a commencé notre travail en proposant des méthodes de synthèse d'observateurs décentralisés à interconnexions inconnues pour des systèmes de grande dimension standards et singuliers. On a cherché à éliminer l'effet des interconnections inconnues sur la dynamique de l'erreur d'observation. La synthèse de l'observateur est basée sur des LMIs permettant de déterminer la matrice de gain paramétrant toutes les matrices de l'observateur. La formulation LMI est basée sur l'approche Lyapunov et déduite des différents lemmes bornés. Ensuite, on a proposé des filtres décentralisés qui permettent d'assurer, en plus de la stabilité, un critère de performance $H_{\infty}$, c'est à dire qu'on a cherché à atténuer l'effet des perturbations, supposées être inconnues mais à énergie bornée, sur la dynamique de l'erreur d'estimation. On a abordé après l'étude des observateurs interconnectés pour les systèmes de grande dimension, où on a proposé une nouvelle méthode permettant de synthétiser une nouvelle forme d'observateurs interconnectés connectivement stable. On s'est intéressé à la capacité d'un tel observateur à être stable de manière robuste vis-a-vis des incertitudes sur les interconnexions entre les sous observateurs qui les forment. Enfin, on s'est intéressé à l'application des méthodes d'estimation proposées dans le cadre de la commande. En effet, dans un premier temps, on a proposé une commande décentralisée basée sur un filtre $H_{\infty}$ pour une classe de systèmes de grande dimension standards à interconnections non-linéaires. L'approche est une extension des travaux de Kalsi et al. aux cas des systèmes perturbés standards. En effet, on a commencé par le calcul du gain de retour d'état qui satisfait les spécifications du système bouclé. Puis, on a synthétisé un filtre qui a pour but de fournir en sortie une estimée de ce retour d'état. L'approche a été validée sur un exemple de système composé de trois machines électriques interconnectées. Dans le second volet du chapitre, on a considéré le problème de la commande via un filtre $H_{\infty}$ pour une classe de système singulier de grande dimension soumis à des perturbations à énergie bornée. L'approche est une extension des travaux de Kalsi et al. au cas des systèmes singuliers perturbés. L'un des principaux apports de nos travaux, a été de proposer une nouvelle méthode de synthèse de commande basée sur un filtre $H_{\infty}$ qui générée par des conditions de solvabilité moins restrictives que celles introduites dans les travaux de Kalsi et al. Ainsi, on a relaxé les contraintes qui portait sur la distance entre la paire de matrices formée par la matrice d'état et la matrice d'entrée d'une part et l'ensemble de paires de matrices incontrôlables d'autre part. De plus, on tient compte de la maximisation des bornes de l'interconnexion, ce qui est très important en pratique

In this dissertation, we investigated the problems of the estimation, H-infinity filtering and the controller based-observer design for standard large scale systems and for algebro-differentials ones called also singular large scale systems. So, we began to propose methods for decentralized observer design with unknown interconnections for standard and singular systems. We search here to decouple the unknown interconnections and the dynamics of the observation error. The method is based on LMIs approach to find the gain matrix implemented in the observer matrices. The LMI formulation is based on Lyapunov approach and deduced from various bounded lemmas. We propose then, filters that permits to ensure, in addition to the stability, an H-infinity performance criteria; we search to attenuate the perturbations effect, supposed unknown but of bounded energy, on the dynamics of the estimation error. We discussed after the study of inter- connected observers for large systems, we have proposed a new method to design a new form of interconnected observers connectively stable. We are interested in the ability of observer to be robustly stable towards uncertainties in the interconnections between sub-observers. Finally, we are interested to the application of the proposed estimation methods to the control purpose. Indeed, in a first step, we propose a decentralized control based on a filter H-infinity for a class of large scale standard systems with nonlinear interconnections. Then, we focus our attention on the observer based control for singular systems to search for a control law which ensures an H-infinity performance criteria. The approach is an extension of recent works of Kalsi and al. to the case of disturbed large scale systems. One of the main contributions of our work was to propose a new method of control design based on a filter H-infinity which generated by using less restrictive conditions than those introduced in the work of Kalsi and al. Thus, we have relaxed the constraint on the distance between the pair of matrices formed by the state matrix and input matrix and the set of pairs of matrices uncontrollable. In addition, we consider the maximization of the bound of the interconnection, which is very important in practice. Our approach is obtained into two steps. The first one consists on calculating the gain state feedback that meets the specifications of the closed loop system. The second one, then we reconstruct this control law using our previous results on the H-infinity filtering. The approach has been validated on an example of a system composed of three interconnected electrical machines.; Dans ce mémoire, on s'est intéressé aux problèmes d'estimation, de filtrage H-infini et de la commande basée observateur des systèmes de grande dimension. L'étude porte sur les systèmes linéaires standards mais aussi sur les systèmes algèbro-différentiels appelés aussi systèmes singuliers pour couvrir la classe la plus large possible des systèmes de grande dimension. Ainsi, on a commencé notre travail en proposant des méthodes de synthèse d'observateurs décentralisés à interconnexions inconnues pour des systèmes de grande dimension standards et singuliers. On a cherché à éliminer l'effet des interconnections inconnues sur la dynamique de l'erreur d'observation. La synthèse de l'observateur est basée sur des LMIs permettant de déterminer la matrice de gain paramétrant toutes les matrices de l'observateur. La formulation LMI est basée sur l'approche Lyapunov et déduite des différents lemmes bornés. Ensuite, on a proposé des filtres décentralisés qui permettent d'assurer, en plus de la stabilité, un critère de performance H-infini, c'est à dire qu'on a cherché à atténuer l'effet des perturbations, supposées être inconnues mais à énergie bornée, sur la dynamique de l'erreur d'estimation. On a abordé après l'étude des observateurs interconnectés pour les systèmes de grande dimension, où on a proposé une nouvelle méthode permettant de synthétiser une nouvelle forme d'observateurs interconnectés connectivement stable. On s'est intéressé à la capacité d'un tel observateur à être stable de manière robuste vis-a-vis des incertitudes sur les interconnexions entre les sous observateurs qui les forment. Enfin, on s'est intéressé à l'application des méthodes d'estimation proposées dans le cadre de la commande. En effet, dans un premier temps, on a proposé une commande décentralisée basée sur un filtre H-infini pour une classe de systèmes de grande dimension standards à interconnections non-linéaires. L'approche est une extension des travaux de Kalsi et al. aux cas des systèmes perturbés standards. En effet, on a commencé par le calcul du gain de retour d'état qui satisfait les spécifications du système bouclé. Puis, on a synthétisé un filtre qui a pour but de fournir en sortie une estimée de ce retour d'état. L'approche a été validée sur un exemple de système composé de trois machines électriques interconnectées. Dans le second volet du chapitre, on a considéré le problème de la commande via un filtre H-infini pour une classe de système singulier de grande dimension soumis à des perturbations à énergie bornée. L'approche est une extension des travaux de Kalsi et al. au cas des systèmes singuliers perturbés L'un des principaux apports de nos travaux, a été de proposer une nouvelle méthode de synthèse de commande basée sur un filtre H-infini qui générée par des conditions de solvabilité moins restrictives que celles introduites dans les travaux de Kalsi et al. Ainsi, on a relaxé les contraintes qui portait sur la distance entre la paire de matrices formée par la matrice d'état et la matrice d'entrée d'une part et l'ensemble de paires de matrices incontrôlables d'autre part. De plus, on tient compte de la maximisation des bornes de l'interconnexion, ce qui est très important en pratique.

International audience; Fixed-interval Bayesian smoothing in state-space systems has been addressed for a long time. However, as far as the measurement noise is concerned, only two cases have been addressed so far : the regular case, i.e., with positive definite covariance matrix; and the perfect measurement case, i.e., with zero measurement noise. In this paper we address the smoothing problem in the intermediate case where the measurement noise covariance is positive semi definite with arbitrary rank. We exploit the singularity of the model in order to transform the original state-space system into a pairwise Markov model with reduced state dimension. Finally, the a posteriori Markovianity of the reduced state enables us to propose a family of fixed-interval smoothing algorithms

International audience; In this paper, we propose both discrete-time and z-domain designs of a controller based on function reduced order observer for linear discrete-time multivariable singular systems where measurements are affected by disturbances. The order of this controller is equal to the dimension of the observer state to be estimated, and the control gain is designed using H-infinity techniques. The discrete-time procedure design is based on the unbiasedness of the estimation error using a Sylvester equation; then a new method to avoid the discrete-time advanced disturbance vector w(k+1) at time (k+1) of the disturbance, in the observation error dynamics is proposed; using recently developed bounded real lemma results, this problem is solved by means of Linear Matrix Inequalities (LMIs). The frequency domain procedure design is derived from discrete-time domain results, where we propose some suitable Matrix Fractions Descriptions (MFDs) and mainly, the factorization approach permits to give a polynomial parametrization of the reduced order observer-based controller.

This paper gives a detailed overview of the work and the results of many authors in the area of Lyapunov stability of a particular class of linear systems. This survey covers the period since 1980 up to nowadays and has strong intention to present the main concepts and contributions during the mentioned period throughout the world, published in the respectable international journals or presented at workshops or prestigious conferences.

We consider switched singular linear systems and determine the set of reachable/controllable states. We derive necessary and sufficient conditions for such a system to be reachable/controllable when an “equisingularity condition” holds.

International audience; In this note, we develop a reduced-order observer-based approach for synchronization of time-delayed chaotic systems. Indeed, in a recent work, Wang \textit{et al.} proposed an interesting technique to enhance the complexity of the chaotic behavior of the standard Chua's circuit, through a time-delay feedback. We provide here a specific solution to ensure the synchronization of this class of nonlinear systems, even in the case of an unknown (or time-varying) delay. Our synchronization scheme relies on the design of a reduced-order observer, whose efficiency is tested on the numerical example provided in \cite{Wang01}. This work is the first step in the design of a secure communications scheme.

Ce document fait une synthèse de mon activité professionnelle d'enseignant-chercheur depuis mon recrutement à l'ESISAR en septembre 1999. Je présente ainsi tout d'abord, dans la partie I, mon curriculum vitae détaillé, incluant un CV résumé, une description de mes activités d'enseignement et une présentation générale de mes activités de recherche. Ces recherches sont très largement issues d'encadrement de masters recherche ou de thèses, et s'orientent autour des quatre thèmes principaux : - la commande de véhicules, - l'analyse et le contrôle du trafic routier - le diagnostic des canaux d'irrigation - la commande, l'observation et le diagnostic des systèmes singuliers.La suite est un mémoire scientifique dédié uniquement aux systèmes algébro-différentiels, mettant en évidence mes contributions majeures issues de travaux personnels ou d'encadrements. L'objectif scientifique est de décrire les contributions apportées dans les thématiques présentées, sans entrer dans les justifications des résultats, ceux-ci ayant déjà été publiés.

Continuous and discrete versions of difference operators are introduced, involving special examples of Toeplitz matrices. A spectral analysis of the continuous operator is performed, providing a full characterization of the null space and the singular system. The same analysis is performed for the discrete operator under a simplifying assumption involving the discretization parameter, by showing that the singular value decomposition is preserved in some sense. Many correspondences are shown to be lost in the case where the simplifying assumption is not verified.

The theory of singular lines on the Parameter Plane and in Parameter Space are derived, and applied to control system design. A method for design of compensation of linear feedback control systems using singular line theory is presented, and application of this design method for self-adaptive control systems is considered. General design steps and required procedures for using this design method are summarized and examples are presented. http://archive.org/details/singularlineoryn1094518548 Lieutenant Commander, Greek Navy Approved for public release; distribution is unlimited.