Research

Ph.D. Résumé

From smart grids and social and biological networks to fleets of drones, networked systems pervade our daily life. In each one of these systems, we can identify some recurring basic features: elementary dynamical units, called agents, locally mutually interact via a graph topology using local information, and give rise to a globally coherent and collective behavior. In numerous instances, networked systems exhibit continuous-time dynamics that are subject to sudden, instantaneous changes that may naturally arise, as for neurons, or may be enforced by design, as in the case of smart grids, where the control actions happen via switching devices. In both scenarios, it is effective to model the overall system as a so-called  networked hybrid dynamical system. 

The objective of the thesis revolves around demonstrating the strengths of hybrid theoretical tools to model and control in a distributed way important classes of networked systems. We first show how hybrid techniques can be used to model the evolution of opinions in a social network where the interactions between individuals depend on both their past and current opinions. This is a reasonable assumption when every individual knows the identity of the other members of the network. We thus present a model of opinion dynamics where each agent has active or inactive pairwise interactions depending on auxiliary state variables filtering the instantaneous opinions, thereby taking their past values into account. When an interaction is (de)activated, a jump occurs, leading to a networked hybrid dynamical model. The stability properties of this hybrid networked system are then analyzed and we establish that the opinions of the agents  converge to local agreements/clusters as time grows, which is typical in the opinion dynamics literature.

In the second case study,  we demonstrate how hybrid techniques can be used to overcome fundamental limitations of continuous-time coupling to synchronize a network of oscillators. In particular, we envision the engineering scenario where the goal is to design the coupling rules for heterogeneous oscillators to globally and uniformly synchronize to a common phase. Each oscillator has its own time-varying natural frequency taking values in a compact set. This problem is historically addressed in the literature by resorting to the well-known  Kuramoto model whose original formulation comes from biological and physical networks. However, the Kuramoto model exhibits major shortcomings for engineering applications, namely the lack of uniform synchronization and phase-locking outside the synchronization set. To overcome these challenges, the oscillators are designed to be interconnected via a tree-like leaderless network by a class of hybrid coupling rules. The proposed couplings can recover locally the behavior of Kuramoto oscillators while ensuring the uniform global practical or asymptotic stability of the synchronization set, which is impossible with Kuramoto models. We further show that the synchronization set can be made uniformly globally prescribed finite-time stable by selecting the coupling function to be discontinuous at the origin. Novel mathematical tools on non-pathological functions and set-valued Lie derivatives are developed to carry out the stability analysis. 

It appears that this last set of mathematical tools has broader applicability than the considered hybrid network of oscillators. We thus finally exploit these tools to analyze the stability properties of Lur'e systems with piecewise continuous nonlinearities, thanks to the underlying similarities between  Lur'e systems and the continuous-time dynamics describing the synchronizing oscillators. We first extend a result from the literature by establishing the global asymptotic stability of the origin under more general sector conditions.  We then present criteria under which   Lur'e systems with piecewise continuous nonlinearities enjoy output and state finite-time stability properties. Moreover, we provide algebraic proofs of the results, which represents a novelty by itself. We show the relevance of the tools provided, by studying the stability properties of two engineering systems of known interest: cellular neural networks and mechanical systems affected by friction.

Supervisors: Romain Postoyan and Luca Zaccarian

Founding: ANR HANDY Grant