This paper investigates the problem of distributed consensus control for multi-agent systems under denial-of-service (DoS) attacks. Different from the existing results where DoS attacks on all the channels are same, in this paper, the adversaries compromise each channel independently. The objective is to design distributed controllers such that the consensus is still achieved in the presence of DoS attacks. Both state-feedback and observer-based controllers are considered. First, the decay rates under different attack modes are obtained by solving a class of linear matrix inequalities. Second, sufficient conditions on the duration of the DoS attacks, under which the consensus is still achieved, are proposed. The difficulty that there is no one-to-one match between the obtained decay rates and DoS duration limitations, is overcome by introducing the equivalent decay rates corresponding to channels. Moreover, the computational complexity is reduced greatly by introducing a novel scaling method. Finally, two examples are presented to illustrate the effectiveness of the proposed approaches.

Introduction

Cyber-physical systems (CPSs) has been intensively studied over the past few years, such as stability analysis [7, 18], sliding-mode observer [33], fault/attack detection [22], and control problems [21, 31], for its immense field of application, such as power grid systems, deep sea exploiting systems, and multi-agent systems (MASs). Compared with the general computing systems where attacks limit their impact to the cyber realm, CPSs where attacks even can impact the physical world for the tight integra- tion of computation, networking, and physical process are more vulnerable [30]. Therefore, security problems for CPSs have attracted considerable attention: the performance degradation under stealthy integrity attacks [23], secure state estimation under sparse sensor attacks [19, 28], security control and distributed filtering under deception attacks [3, 4] and stability analysis under denial-of-service (DoS) attacks [27]. Especially, for the widespread applications of MASs [5], such as wireless sensor networks [1], spacecraft systems [2], service robots [12], and formation control of unmanned vehicles [15], the consensus of MASs has attracted intensive study, such as fault-tolerant consensus control [29], adaptive tracking control problem [17], distributed event-triggered control [13], and synchronization using observer [32]. While most of the previous results focus on leaderless consensus, leader following consensus is considered in [14, 16]. Besides, as a kind of CPSs, the security problems of MASs also have been intensively studied. For example, [26] considers a line consensus network with misbehaving agents which can bee seen as the sparse attacks considered in [19, 28], and [8] investigates distributed secure consensus for MASs under DoS attacks. DoS attacks which is one of the most common attacks in CPSs, compromise the systems through rendering some or all components of a control system inaccessible. Especially, while the agents of MASs always communicate with each other individually, MASs are more vulnerable to DoS attacks since defending DoS attacks for all transmission channels is almost impossible. Although secure consensus for MASs under DoS attacks has been considered in [8, 9, 27], all the channels are seen as one channel from the angle of the adversaries which limits the application of the existing methods. Instead, studying multiple transmission channel case, where the adversaries can attack partial or all channels at any time with DoS duration limitation, is more practical, and this is the major motivation of this study. Although CPSs with multiple channels under DoS have been considered in our previous result [20], the computational complexity has not been addressed well. Especially, while the number of transmission channels grows linearly with the size of common linear systems, as the size of MASs increases, the combinatorial transmission channels lead to explosive growth of the computational complexity. Thus, besides analyzing the stability, more efforts on addressing the computational complexity should be made for MASs. This paper investigates the distributed secure consensus for MASs under DoS attacks. The considered DoS attacks are energy-limited, and on different channels, they are independent of each other. The main contributions of this work are summarized as follows: (i) Both distributed state-feedback and observer-based controllers are proposed.