Lyapunov-based synchronization of networked systems: From continuous-time to hybrid dynamics

Abstract

Synchronization pertains to the property of interconnected systems according to which their dynamic behavior is coordinated in an appropriate sense. That is to say, some of their state variables, or functions of the latter for that matter, converge to each other. Synchronization may occur naturally or may be induced, controlled, and it may be present between two systems or among a large number. In the latter case, it is convenient to speak of a network of interconnected systems. Understanding synchronization, and how to control it, is an important paradigm as it is present in a variety of scenarios. These involve, e.g., networks of technological systems (robots and vehicles of different kinds), social networks (by which people exchange opinions and agree, or not), networks of biological systems (uni or pluricellular), etc. Owing to the context, the mathematical models to describe such networks and to define synchronization formally, varies dramatically. Ordinary continuous-time or discrete-time models for which modern control theory and Lyapunov stability theory are tailored result inappropriate to incorporate hybrid phenomena that intervene in the network. These may stem from sudden topology changes, the use of digital or intermittent control strategies, the presence of impacts in the intrinsic dynamics of the nodes, etc. In this invited paper, we give an overview of synchronization control problems, mostly of cooperative control of networks of autonomous vehicles (based on continuous-time models). For that matter, the first part of the paper focuses on the main contributions of [1]. Then, we give further perspectives on what we consider significant open problems on synchronization of hybrid systems and hybrid networks.

Introduction

Generally speaking, for a network of dynamical systems, inducing synchronization consists in designing a controller (often one for each node in the network) so that an output or a state variable of each system acquires, asymptotically, a common behavior. For example, the spatial positions of network-interconnected autonomous vehicles (such as drones, mobile robots, platoons of cars, etc.) may be controlled to form a specified geometric pattern, a formation of sorts; the currents and voltages in a network of DC-AC inverters may be controlled to reach the same frequency of oscillation.

Depending on the nature of interaction among the nodes in the network, we distinguish between centralized and decentralized control schemes. In the first case, each node receives global information dictating its desired behavior. This may consist in a desired set-point (a specific position or oscillating frequency), a reference trajectory, or a desired dynamical behavior in more complex cases. In centralized control approaches, each node has an independent task to ensure, which is defined by a central entity. These approaches are often used when the network is composed of a relatively small number of nodes, but abundant computation and communication capacities may be required for the control entity. Decentralized approaches, as we consider in this article, overcome these limitations. There, the control input for each node depends only on local information provided by a given set of nodes called neighbors. The network topology, which determines how the information among the neighbors flows, is typically defined using a communication graph (Ren & Beard, 2005). The communication graph plays a key role in the analysis and design of networked systems.

Decentralized synchronization in networked systems is an active and challenging research topic that has continuously attracted undivided attention from many researchers over the previous two decades —see, for instance, (Jadbabaie, Lin, Morse, 2003, Olfati-Saber, Murray, 2004, Ren, Beard, 2005, Tsitsiklis, Bertsekas, Athans, 1986) in which both continuous and discrete-time networks are considered. Such is the case in engineering disciplines where technological solutions involve groups of systems required to cooperate to complete a common task. Some common examples of these are related, but are not limited, to robotic manipulation and the use of autonomous vehicles (Belleter, Pettersen, 2017, Dong, Farrell, 2008a, Eun, Bang, 2006, Liu, Guo, Lu, 2014, Rodriguez-Angeles, Nijmeijer, 2004), but they are also common in energy networks (Persis, Weitenberg, & Dörfler, 2018), for instance. Other disciplines include neuroscience, in which networks and sub-networks (clusters) formed due to neuronal population activities and interactions are studied (Wendling, Bellanger, Bartolomei, & Chauvel, 2000), social science, in which a subject of study is that of opinion networks formed due to the interactions among individuals (Parsegov, Proskurnikov, Tempo, & Friedkin, 2015), and geography, in which groups of populations that interact when reacting to catastrophes are studied (Cantin, 2017). While the systems models involved in each of these instances are different, a common denominator is the presence of a network composed of nodes having a dynamic behaviors that interact locally with the aim at reaching a common global goal. From a systems-theory viewpoint, at least two primary challenges can be identified when contributing to synchronization problems. The first one arises from recognizing that communication among the nodes is subject to constraints; this includes scenarios where the transfer of information is unreliable, time-dependent, or subjected to delays. The second challenge stems from considering the complexity of the nodes’ dynamics; these may be linear, nonlinear or hybrid in nature and, moreover, may vary throughout the network — such is the case of heterogeneous networks.

Some of these aspects have been addressed in Maghenem (2017), in scenarios pertaining to electromechanical engineering and robotics. The main originality of Maghenem (2017) is to propose advanced Lyapunov-based approaches inspired by techniques for stand-alone nonlinear time-varying systems to analyze the coordination problems after transforming it into global stabilization of a closed, but unbounded, set. Then, Lyapunov-based techniques, via the construction of strict and differentiable Lyapunov functions, are proposed for the network’s model after interconnection. This allows to derive several systematic methods to analyze the coordination task, the network’s performance, and its robustness with respect to perturbations and delays affecting the transfer of information among the nodes. Note that, in related literature, the coordination task is usually analyzed via trajectory-based approaches or using weak Lyapunov functions, in the sense that the Lyapunov function does not systematically allow the verification the coordination task. In that way, for autonomous mobile robots, different coordination tasks are studied in Maghenem (2017), such as consensus and trajectory-tracking formation control or, for networks of oscillators, the frequency synchronization problem has been addressed. In regards to the nature of interaction and transfer of information among the nodes, several scenarios are considered; these include time-varying interconnections as well as communication delays. As far as the intrinsic dynamics of the nodes is considered, only some continuous-time models, such as single and double integrators, time-varying planar oscillators, and nonholonomic unicycles, have been considered.

The results contained in Maghenem (2017) concern exclusively interconnected systems described by continuous-time models. However, the problems solved therein lead to more general open questions that transcend these systems and are, in our opinion, of sufficient significance to awake the interest of a wide readership. The problems that we describe are related to the study of synchronization problems described previously, but in the context of hybrid networked systems. In particular, it is of major importance to handle the presence of discrete phenomena that are coupled with the continuous-time dynamics of the nodes that compose the network. A prominent example that makes this problem relevant is that of digital or intermittent control strategies that interact with continuous-time models, which captures constrained control actions as in power networks (Beneux, Riedinger, Daafouz, Grimaud, 2017, Torquati, Sanfelice, Zaccarian, 2017), in valve-based control systems (Panzani, Colombo, Savaresi, & Zacearan, 2017), or, in networked control systems (Hespanha, Naghshtabrizi, Xu, 2007, Postoyan, de Wouw, Nešić, Heemels, 2014). It also captures the creation (resp. loss) of links and the addition (resp. removal) of nodes as in social networks (Frasca, Tarbouriech, Zaccarian, 2019, Mariano, Morărescu, Postoyan, Zaccarian, 2020, Morărescú, Girard, 2011). Furthermore, in the intrinsic behavior of the nodes, one must also consider systems affected by discontinuities such as impacts and instantaneous jumps like in mechanical systems with impacts (Jiménez-Leudo, Quijano, Rodríguez, 2015, Westervelt, Grizzle, Koditschek, 2003) or in cyber-physical systems (Borgers, Geiselhart, Heemels, 2017, Nowzari, Garcia, Cortés, 2019, Persis, Postoyan, 2017). Hence, such research aims at covering unexplored aspects which, nonetheless, are naturally motivated by intrinsic characteristics of modern systems networks.

The rest of this paper is organized as follows. In the next section, we describe the main contributions of Maghenem (2017). In Section 3, we discuss the relevance and challenges related to synchronization in hybrid networks. Finally, in Section 4, we provide some research directions.