This thesis investigates the average consensus of multi-agent systems with linear dynamics whose interconnections are modelled by balanced digraphs with matrix- weights. The thesis first introduces the notion of balanced digraphs and mirror graphs for matrix weights. Then it proves that the matrix-weight-balanced con- sensus controller is indeed globally asymptotically stable. The Lyapunov stability analysis exploits the properties of the mirror graph of a balanced digraph. Further, the necessary and sufficient condition for the system to achieve average consensus is shown to be positive definiteness of the matrix weights of its balanced digraph. Simulations with robots in SIMULINK verify that positive definite matrix weights on balanced graphs are indeed necessary and sufficient for average consensus. Fi- nally formation control of a multi-robot system is shown to be an application of the matrix-weight-balanced consensus algorithm using real time simulation of Clearpath Ridgeback robots in Gazebo and ROS. / Graduate

Graph signal processing analyzes signals supported on the nodes of a network with respect to a shift operator matrix that conforms to the graph structure. For shift-invariant graph filters, which are polynomial functions of the shift matrix, the filter response is defined by the value of the filter polynomial at the shift matrix eigenvalues. Thus, information regarding the spectral decomposition of the shift matrix plays an important role in filter design. However, under stochastic conditions leading to uncertain network structure, the eigenvalues of the shift matrix become random, complicating the filter design task. In such case, empirical distribution functions built from the random matrix eigenvalues may exhibit deterministic limiting behavior that can be exploited for problems on large-scale random networks. Acceleration filters for distributed average consensus dynamics on random networks provide the application covered in this thesis work. The thesis discusses methods from random matrix theory appropriate for analyzing adjacency matrix spectral asymptotics for both directed and undirected random networks, introducing relevant theorems. Network distribution properties that allow computational simplification of these methods are developed, and the methods are applied to important classes of random network distributions. Subsequently, the thesis presents the main contributions, which consist of optimization problems for consensus acceleration filters based on the obtained asymptotic spectral density information. The presented methods cover several cases for the random network distribution, including both undirected and directed networks as well as both constant and switching random networks. These methods also cover two related optimization objectives, asymptotic convergence rate and graph total variation.

Average consensus is a fundamental concept in distributed computing, where distributed agents exchange messages in order to obtain the average of their ini- tial values without relying on a centralized computing unit. However, achiev- ing average consensus in the presence of communication imperfections, such as quantization and random link or node failures, becomes more challenging. This thesis evaluates various average consensus algorithms regarding their ability to mitigate quantization effects and explores node dropout for reducing communi- cation cost per iteration. It also identifies the conditions required for achieving average consensus in both scenarios.  The first part of this thesis deals with average consensus with quantized up- dates, comparing algorithms such as quantized gossip, average preserving quan- tized gossip, and CHOCO-GOSSIP. CHOCO-GOSSIP stands out as the most ef- fective algorithm, which shows the importance of pre-compensating the quanti- zation error before transmitting the node values. Among other algorithms, aver- age preserving quantized gossip shows slightly better performance. Additionally, graphs with higher connectivity tend to perform better.  The second part of this thesis focuses on energy-efficient average consensus with random node dropout. It compares the optimized node dropout proba- bilities with heuristic designs such as the degree-based method and Metropolis- Hastings method. The degree-based method is shown to give good convergence performance despite its simplicity. Furthermore, in irregular graphs, the perfor- mance difference between optimized probabilities and heuristic designs tends to be more pronounced.

Le consensus des systèmes multi-agents a eu une attention considérable au cours de la dernière décennie. Le consensus est un processus coopératif dans lequel les agents interagissent afin de parvenir à un accord. La plupart des études se sont engagés à l'analyse de l'état d'équilibre du comportement de ce processus. Toutefois, au cours de la transitoire de ce processus une énorme quantité de données est produite. Dans cette thèse, notre objectif est d'exploiter les données produites au cours de la transitoire d'algorithmes de consensus moyenne asymptotique afin de concevoir des protocoles de consensus moyenne en temps fini, évaluer la robustesse du graphique, et éventuellement récupérer la topologie du graphe de manière distribuée. Le consensus de moyenne en temps fini garantit un temps d'exécution minimal qui peut assurer l'efficacité et la précision des algorithmes distribués complexes dans lesquels il est impliqué. Nous nous concentrons d'abord sur l'étape de configuration consacrée à la conception de protocoles de consensus qui garantissent la convergence de la moyenne exacte dans un nombre donné d'étapes. En considérant des réseaux d'agents modélisés avec des graphes non orientés connectés, nous formulons le problème de la factorisation de la matrice de moyenne et étudions des solutions distribuées à ce problème. Puisque, les appareils communicants doivent apprendre leur environnement avant d'établir des liens de communication, nous suggérons l'utilisation de séquences d'apprentissage afin de résoudre le problème de la factorisation. Ensuite, un algorithme semblable à l'algorithme de rétro-propagation du gradient est proposé pour résoudre un problème d'optimisation non convexe sous contrainte. Nous montrons que tout minimum local de la fonction de coût donne une factorisation exacte de la matrice de moyenne. En contraignant les matrices de facteur à être comme les matrices de consensus basées sur la matrice laplacienne, il est maintenant bien connu que la factorisation de la matrice de moyenne est entièrement caractérisé par les valeurs propres non nulles du laplacien. Par conséquent, la résolution de la factorisation de la matrice de la moyenne de manière distribuée avec une telle contrainte sur la matrice laplacienne, permet d'estimer le spectre de la matrice laplacienne. Depuis le spectre peut être utilisé pour calculer des indices de la robustesse (Nombre d'arbres couvrant et la résistance effective du graphe), la deuxième partie de cette thèse est consacrée à l'évaluation de la robustesse du réseau à travers l'estimation distribuée du spectre du Laplacien. Le problème est posé comme un problème de consensus sous contrainte formulé de deux façons différentes. La première formulation (approche directe) cède à un problème d'optimisation non-convexe résolu de manière distribuée au moyen de la méthode des multiplicateurs de Lagrange. La seconde formulation (approche indirecte) est obtenue après une reparamétrisation adéquate. Le problème est alors convexe et résolu en utilisant l'algorithme du sous-gradient distribué et la méthode de direction alternée de multiplicateurs. En outre, trois cas sont considérés: la valeur moyenne finale est parfaitement connue, bruyant, ou complètement inconnue. Nous fournissons également une façon pour calculer les multiplicités des valeurs propres estimées au moyen d'une programmation linéaire en nombres entiers. L'efficacité des solutions proposées est évaluée au moyen de simulations. Cependant, dans plusieurs cas, la convergence des algorithmes proposés est lente et doit être améliorée dans les travaux futurs. En outre, l'approche indirecte n'est pas évolutive pour des graphes de taille importante car elle implique le calcul des racines d'un polynôme de degré égal à la taille du réseau. Cependant, au lieu d'estimer tout le spectre, il peut être possible de récupérer seulement un petit nombre des valeurs propres, puis déduire des limites significatives sur les indices de la robustesse. / Consensus of Multi-agent systems has received tremendous attention during the last decade. Consensus is a cooperative process in which agents interact in order to reach an agreement. Most of studies are committed to analysis of the steady-state behavior of this process. However, during the transient of this process a huge amount of data is produced. In this thesis, our aim is to exploit data produced during the transient of asymptotic average consensus algorithms in order to design finite-time average consensus protocols, assess the robustness of the graph, and eventually recover the topology of the graph in a distributed way. Finite-time Average Consensus guarantees a minimal execution time that can ensure the efficiency and the accuracy of sophisticated distributed algorithms in which it is involved. We first focus on the configuration step devoted to the design of consensus protocols that guarantee convergence to the exact average in a given number of steps. By considering networks of agents modelled with connected undirected graphs, we formulate the problem as the factorization of the averaging matrix and investigate distributed solutions to this problem. Since, communicating devices have to learn their environment before establishing communication links, we suggest the usage of learning sequences in order to solve the factorization problem. Then a gradient backpropagation-like algorithm is proposed to solve a non-convex constrained optimization problem. We show that any local minimum of the cost function provides an accurate factorization of the averaging matrix. By constraining the factor matrices to be as Laplacian-based consensus matrices, it is now well known that the factorization of the averaging matrix is fully characterized by the nonzero Laplacian eigenvalues. Therefore, solving the factorization of the averaging matrix in a distributed way with such Laplacian matrix constraint allows estimating the spectrum of the Laplacian matrix. Since that spectrum can be used to compute some robustness indices (Number of spanning trees and Effective graph Resistance also known as Kirchoff index), the second part of this dissertation is dedicated to Network Robustness Assessment through distributed estimation of the Laplacian spectrum. The problem is posed as a constrained consensus problem formulated in two ways. The first formulation (direct approach) yields a non-convex optimization problem solved in a distributed way by means of the method of Lagrange multipliers. The second formulation (indirect approach) is obtained after an adequate re-parameterization. The problem is then convex and solved by using the distributed subgradient algorithm and the alternating direction method of multipliers. Furthermore, three cases are considered: the final average value is perfectly known, noisy, or completely unknown. We also provide a way for computing the multiplicities of the estimated eigenvalues by means of an Integer programming. In this spectral approach, given the Laplacian spectrum, the network topology can be reconstructed through estimation of Laplacian eigenvector. The efficiency of the proposed solutions is evaluated by means of simulations. However, in several cases, convergence of the proposed algorithms is slow and needs to be improved in future works. In addition, the indirect approach is not scalable to very large graphs since it involves the computation of roots of a polynomial with degree equal to the size of the network. However, instead of estimating all the spectrum, it can be possible to recover only a few number of eigenvalues and then deduce some significant bounds on robustness indices.

A significant progress in the evolution of the computer systems and their interconnection over the past 70 years has allowed replacing the frequently used centralized architectures with the highly distributed ones, formed by independent entities fulfilling specific functionalities as one user-intransparent unit. This has resulted in an intense scientic interest in distributed algorithms and their frequent implementation into real systems. Especially, distributed algorithms for multi-sensor data fusion, ensuring an enhanced QoS of executed applications, find a wide usage. This doctoral thesis addresses an optimization and an analysis of the distributed systems, namely the distributed consensus-based algorithms for an aggregate function estimation (primarily, my attention is focused on a mean estimation). The first section is concerned with a theoretical background of the distributed systems, their evolution, their architectures, and a comparison with the centralized systems (i.e. their advantages/disadvantages). The second chapter deals with multi-sensor data fusion, its application, the classification of the distributed estimation techniques, their mathematical modeling, and frequently quoted algorithms for distributed averaging (e.g. protocol Push-Sum, Metropolis-Hastings weights, Best Constant weights etc.). The practical part is focused on mechanisms for an optimization of the distributed systems, the proposal of novel algorithms and complements for the distributed systems, their analysis, and comparative studies in terms of such as the convergence rate, the estimation precision, the robustness, the applicability to real systems etc.

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We study a scalable approach to information fusion for large sensor networks. The algorithm, field inversion by consensus and compressed sensing (FICCS), is a distributed method for detection, localization, and estimation of a propagating field generated by an unknown number of point sources. The approach combines results in the areas of distributed average consensus and compressed sensing to form low dimensional linear projections of all sensor readings throughout the network, allowing each node to reconstruct a global estimate of the field. Compressed sensing is applied to continuous source localization by quantizing the potential locations of sources, transforming the model of sensor observations to a finite discretized linear model. We study the effects of structured modeling errors induced by spatial quantization and the robustness of ℓ1 penalty methods for field inversion. We develop a perturbations method to analyze the effects of spatial quantization error in compressed sensing and provide a model-robust version of noise-aware basis pursuit with an upperbound on the sparse reconstruction error. Numerical simulations illustrate system design considerations by measuring the performance of decentralized field reconstruction, detection performance of point phenomena, comparing trade-offs of quantization parameters, and studying various sparse estimators. The method is extended to time-varying systems using a recursive sparse estimator that incorporates priors into ℓ1 penalized least squares. This thesis presents the advantages of inter-sensor measurement mixing as a means of efficiently spreading information throughout a network, while identifying sparse estimation as an enabling technology for scalable distributed field reconstruction systems.

Les algorithmes distribués de consensus sont des algorithmes itératifs de faible complexité où les nœuds de capteurs voisins interagissent les uns avec les autres pour parvenir à un accord commun sans unité coordinatrice. Comme les nœuds dans un réseau de capteurs sans fil ont une puissance de calcul et une batterie limitées, ces algorithmes distribués doivent parvenir à un consensus en peu de temps et avec peu d’échange de messages. La première partie de cette thèse s’est basée sur l’étude et la comparaison des différents algorithmes de consensus en mode synchrone et asynchrone en termes de vitesse de convergence et taux de communications. La seconde partie de nos travaux concerne l’application de ces algorithmes de consensus au problème de la détection de trous de couverture dans les réseaux de capteurs sans fil.Ce problème de couverture fournit aussi le contexte de la suite de nos travaux. Il se décrit comme étant la façon dont une région d’intérêt est surveillée par des capteurs. Différentes approches géométriques ont été proposées mais elles sont limitées par la nécessité de connaitre exactement la position des capteurs ; or cette information peut ne pas être disponible si les dispositifs de localisation comme par exemple le GPS ne sont pas sur les capteurs. À partir de l’outil mathématique appelé topologie algébrique, nous avons développé un algorithme distribué de détection de trous de couverture qui recherche une fonction harmonique d’un réseau, c’est-à-dire annulant l’opérateur du Laplacien de dimension 1. Cette fonction harmonique est reliée au groupe d’homologie H1 qui recense les trous de couverture. Une fois une fonction harmonique obtenue, la détection des trous se réalise par une simple marche aléatoire dans le réseau. / Distributed consensus algorithms are iterative algorithms of low complexity where neighboring sensors interact with each other to reach an agreement without coordinating unit. As the nodes in a wireless sensor network have limited computing power and limited battery, these distributed algorithms must reach a consensus in a short time and with little message exchange. The first part of this thesis is based on the study and comparison of different consensus algorithms synchronously and asynchronously in terms of convergence speed and communication rates. The second part of our work concerns the application of these consensus algorithms to the problem of detecting coverage holes in wireless sensor networks.This coverage problem also provides the context for the continuation of our work. This problem is described as how a region of interest is monitored by sensors. Different geometrical approaches have been proposed but are limited by the need to know exactly the position of the sensors; but this information may not be available if the locating devices such as GPS are not on the sensors. From the mathematical tool called algebraic topology, we have developed a distributed algorithm of coverage hole detection searching a harmonic function of a network, that is to say canceling the operator of the 1-dimensional Laplacian. This harmonic function is connected to the homology group H1 which identifies the coverage holes. Once a harmonic function obtained, detection of the holes is realized by a simple random walk in the network.

Avec l’arrivée massive des technologies sans fil, le nombre de terminaux mobiles n’a cessé de croître, pour des usages et des ressources de communication diversifiés. En intégrant les objets du quotidien, nos réseaux de communications sont devenus dynamiques aussi bien en termes de ressources que de topologie physique, offrant accès à des informations de plus en plus riches. La tâche de gestion s’est ainsi complexifiée et requiert des temps de réponse de plus en plus courts difficilement réalisables par un administrateur humain. Il devient indispensable de mettre en œuvre des capacités de gestion autonomes pour les nouveaux réseaux. Dans tous les cas, la gestion d’un système implique une étape essentielle : sa mesure et sa supervision. Peu importe sa nature, c’est cette étape de prise d’information qui permet sa caractérisation, son analyse et son contrôle. Le domaine des réseaux n’échappe pas à cette règle et les objets qui le composent auront besoin d’acquérir des informations sur leur environnement pour mieux s’y adapter. Dans cette thèse, nous nous intéressons au partage efficace de ces informations de mesures à des fins d’auto-analyse et d’évaluation distribuée de la performance. Après avoir formalisé le problème de la mesure distribuée, nous nous consacrons dans un premier temps à l’organisation des échanges de mesures dans les graphes dynamiques. Nous proposons une nouvelle heuristique pour le consensus de la moyenne qui converge plus rapidement que celles de l’état de l’art. Dans un second temps, nous considérons des topologies plus stables pouvant utiliser des flux TCP comme moyen d’échange. Nous proposons un mécanisme d’ordonnancement de ces flux qui conserve le même comportement face à la congestion, tout en réduisant leur latence moyenne. Enfin, nous nous intéressons à l’information de mesure échangée. Nous montrons comment les nœuds peuvent superviser diverses métriques telles que la performance d’un système en se basant sur l’utilité de ses agents, et proposons une méthode pour qu’ils puissent analyser l’évolution de cette performance / With the massive rise of wireless technologies, the number of mobile stations is constantly growing. Both their uses and their communication resources are diversified. By integrating our daily life objects, our communication networks become dynamic in terms of physical topology but also in term of resources. Furthermore, they give access to a richer information. As a result, the management task has become complex and requires shorter response time that a human administrator can not respect. It becomes necessary to develop an autonomic management behavior in next generation networks. In any manner, managing a system requires essential steps which are : its measurement and its supervision. Whatever the nature of a system, this stage of information gathering, allows its characterization and its control. The field of networks is not the exception to the rule and objects that compose them will need to acquire information on their environment for a better adaptation. In this thesis, we focus on the efficient sharing of this information, for self-analysis and distributed performance evaluation purposes. After having formalized the problem of the distributed measurement, we address in a first part the fusion and the diffusion of measures in dynamic graphs. We develop a new heuristic for the average consensus problem offering a better contraction rate than the ones of the state of the art. In a second part, we consider more stable topologies where TCP is used to convey measures. We offer a scheduling mechanism for TCP flows that guaranty the same impact on the network congestion, while reducing the average latency. Finally, we show how nodes can supervise various metrics such as the system performance based on their utilities and suggest a method to allow them to analyze the evolution of this performance