The issue of cyber-security has become ever more prevalent in the analysis and design of cyber-physical systems. In this paper, we investigate self-triggered consensus networks in the presence of communication failures caused by Denialof-Service (DoS) attacks. A general framework is considered in which the network links can fail independent of each other. By introducing a notion of Persistency-of-Communication (PoC), we provide an explicit characterization of DoS frequency and duration under which consensus can be preserved by suitably designing time-varying control and communication policies. An explicit characterization of the effects of DoS on the consensus time is also provided. The considered notion of PoC is compared with classic average connectivity conditions that are found in pure continuous-time consensus networks. Finally, examples are given to substantiate the analysis.

Abstract-The issue of security has become ever more prevalent in the analysis and design of cyber-physical systems. In this paper, we analyze a consensus network in the presence of Denial-of-Service (DoS) attacks, namely attacks that prevent communication among the network agents. By introducing a notion of Persistency-of-Communication (PoC), we provide a characterization of DoS frequency and duration such that consensus is not destroyed. An example is given to substantiate the analysis. I. INTRODUCTIONIn recent years, the issue of security has become ever more prevalent in the analysis and design of cyber-physical systems (CPSs), namely systems that exhibit a tight conjoining of computational resources and physical resources. As argued in [1], [2], security in CPSs drastically differs from security in general-purpose computing systems since attacks can cause disruptions that transcend the cyber realm and affect the physical world. In CPSs, attacks to the communication links can be classified as either deception attacks or Denial-ofService (DoS) attacks. The former affect the trustworthiness of data by manipulating the packets transmitted over the network; see [3]-[5] and the references therein. DoS attacks are instead primarily intended to affect the timeliness of the information exchange, i.e., to cause packet losses; see for instance [6], [7] for an introduction to the topic. This paper is concerned with DoS attacks.In the literature, the problem of securing robustness of CPSs against DoS has been investigated by many research groups [8]- [13]. In these papers, however, the analysis is restricted to a centralized setting, namely to a classical (single-loop) plantcontroller configuration. On the other hand, no quantitative results are available for distributed settings. The purpose of this paper is to explore this topic.We investigate the issues of DoS with respect to consensus networks. Specifically, inspired by [14], we consider a self-triggered consensus network. At each sampling time, a certain subset of active agents poll their neighbors obtaining relative measurements of the consensus variable of interest: the available information is then used by the active agents to update their controls and compute their next update times. The attacker objective is to prevent consensus by denying communication among the agents. Consensus is a prototypical problem in distributed settings with a huge range of applications, spanning from formation and cooperative robotics to

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Network systems are one of the most active research areas in the engineering community as they feature a paradigm shift from centralized to distributed control and computation. When dealing with network systems, a fundamental challenge is to ensure their functioning even when some of the network nodes do not operate as intended due to faults or attacks. The objective of this paper is to address the problem of resilient consensus in a context where the nodes have their own clocks, possibly operating in an asynchronous way, and can make updates at arbitrary time instants. The results represent a first step towards the development of resilient event-triggered and self-triggered coordination protocols. IntroductionNetwork systems are one of the most active research areas in the engineering community as they feature a paradigm shift from centralized to distributed control and computation. When dealing with network systems, a fundamental challenge is to ensure their functioning even when some of the network units (nodes) do not operate as intended due to faults or attacks. The main difficulty originates from the fact that normal (non-misbehaving) nodes can receive, process, and spread erroneous data coming from misbehaving nodes with the consequence that a failure in one point of the network can compromise the whole network functioning.The prototypical problem to study resilience in the presence of misbehaving nodes is the so-called consensus problem [1], which forms the foundation for distributed computing. In resilient consensus, each node is assumed to be aware of only local information available from its neighbors and the goal is to make sure that normal nodes eventually reach a common value despite the presence of misbehaving nodes. The resilient consensus problem has a long history, and it has been investigated first by computer scientists [2,3], usually under the hypothesis that the network graph is complete, that is assuming an allto-all communication structure. More recently, thanks to the widespread of consensus-based applications, this problem has attracted a lot of interest also within the engineering community, mostly in connection with the goal of delineating the minimal connectivity hypotheses that are needed to secure consensus.In [4], the authors consider mean subsequence reduced (MSR) algorithms and define a graph-theoretic property, referred to as network robustness, which charactherizes necessary and sufficient connectivity hypotheses under which normal nodes can reach consensus using only local information available from their neighbors. The results indicate that, while the communication graph should possess a certain degree of redundancy, completeness of the communication graph is not necessary even for very general types of misbehavior. The results of [4] have been extended in many venues. Examples include methods for handling time-varying networks [5], double-integrator systems [6], sparse communication graphs [7] as well as methods for identifying the robustness of specific classes of networks ...

Network systems are one of the most active research areas in the engineering community as they feature a paradigm shift from centralized to distributed control and computation. When dealing with network systems, a fundamental challenge is to ensure their functioning even when some of the network nodes do not operate as intended due to faults or attacks. The objective of this paper is to address the problem of resilient consensus in a context where the nodes have their own clocks, possibly operating in an asynchronous way, and can make updates at arbitrary time instants. The results represent a first step towards the development of resilient event-triggered and self-triggered coordination protocols.