Systems & Control Letters 79 (2015) 23–29 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Consensus of second-order multi-agent systems in the presence of locally bounded faults ✩ Seyed Mehran Dibaji, Hideaki Ishii ∗ Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259-J2-54, Nagatsuta-cho, Midori-ku Yokohama 226-8502, Japan article info Article history: Received 21 November 2014 Received in revised form 11 February 2015 Accepted 23 February 2015 Available online 27 March 2015 Keywords: Multi-agent systems Cyber-security Consensus abstract We propose an algorithm for consensus of second-order sampled-data multi-agent systems in the pres- ence of misbehaving agents. Each normal agent updates its states following a predetermined control law based on local information while some malicious agents make updates arbitrarily. The normal agents do not know the global topology of the network, but have prior knowledge on the maximum number of ma- licious agents in their neighborhood. Under the assumption that the network has sufficient connectivity in terms of robustness, we develop a resilient algorithm where each agent ignores the neighbors which have large and small position values to avoid being influenced by malicious agents. © 2015 Elsevier B.V. All rights reserved. 1. Introduction Recently, in the area of networked control systems, considera- tion of cyber security has become important since such systems are nowadays often connected to general purpose networks, e.g., the Internet and wireless communication. Malicious attacks can lead the systems to hazardous operations and might cause physical faults or even accidents. Safe distributed algorithms in the pres- ence of faulty behaviors and adversarial agents have been widely studied in computer science [1,2] and control [3–5]. In this paper, we consider networks of agents which interact with each other to accomplish a global objective. In such systems, malicious intruders may take control of some agents and influence other agents to keep them from completing their planned tasks without being noticed. Here, we consider consensus, one of the ba- sic problems in multi-agent systems, where the objective is agree- ment on some state values among the agents [6]. Resilient algorithms for multi-agent systems have appeared in the literature and can be classified into two approaches. One is to achieve consensus among the non-faulty agents by detecting and isolating malicious agents in the network. In the works of [7,8], techniques of observers for systems with unknown inputs are de- veloped for a consensus problem. The papers [4,5] also deal with ✩ This work was supported by Japan Science and Technology Agency under the EMS-CREST program. ∗ Corresponding author. E-mail addresses: dibaji@sc.dis.titech.ac.jp (S.M. Dibaji), ishii@dis.titech.ac.jp (H. Ishii). observer-based methods for fault detection when the agents have second-order dynamics. The other approach aims at consensus by simply ignoring suspicious agents whether or not they are truly faulty. The paper [9] first proposed a consensus algorithm with this idea. The network considered there is however a complete graph. Afterwards, there are some works [10,11] which studied the algorithm for partially connected networks. The term mean sub- sequence reduced (MSR) algorithms coined by [12] for this fam- ily of algorithms has been used in literature (e.g., [13,14]). On the other hand, the papers [14,15] introduce a novel notion of graph robustness to characterize the necessary network structure; other related works include [16,17]. It is noted that these works study only agents whose dynamics is represented as a single integrator. Here, we focus on resilient consensus of sampled-data double- integrator multi-agent systems in the presence of locally bounded malicious agents. Consensus problems for second-order agent dy- namics are motivated by vehicle applications and have been stud- ied, e.g., in [18,19]. Following the second approach mentioned above, we propose a new algorithm to tackle the problem. The dif- ficulty of this problem is two-fold: (i) The presence of malicious agents which might try to deviate the network not to reach con- sensus and (ii) more complicated dynamics due to the double- integrator agents, which requires agreement in both position and velocity values. In our strategy, the non-faulty normal agents are equipped with an algorithm to collect the neighbors’ positions, but to ignore a certain number of them. Specifically, in their updates, they leave out those that take large and small values. In this way, these agents can avoid being affected by the suspicious ones in the course of arriving at consensus. We show that the notion of graph http://dx.doi.org/10.1016/j.sysconle.2015.02.005 0167-6911/© 2015 Elsevier B.V. All rights reserved.