Discussion on: "Consensus of Second-Order Delayed Multi-Agent Systems with Leader-Following"

a second-order leader-following consensus algorithm in multi-agent systems with directed network topologies and heterogeneous input delays, which generalized the previous work [1] from the case of symmetric coupling weights to that of asymmetric coupling weights. Some sufficient conditions for reaching second-order consensus in a leader-following context were derived by utilizing the Gershgorin disc theorem and the curvature theory. It was shown that to ensure leader-following consensus, the input delays should be less than some critical values determined by the network topology and the control gains. Consensus algorithms in multi-agent systems have been extensively studied in the literature. Existing works have different focuses and might be categorized in a variety of ways. In particular, different consensus algorithms can be designed and analyzed for different system models, among which the most studied models are single-integrator dynamics [2], [3], [4], [5] and double-integrator dynamics [6], [7], [8], [9]. In addition , besides leaderless consensus algorithms, it is often of interest to study leader-following consensus algorithms, i.e., consensus in the presence of a leader or a reference [10], [11]. This paper discussed a leader-following consensus algorithm for double-integrator dynamics when the leader has a constant velocity. The limitation of the algorithm (i.e., its incapability of tracking a leader with a time-varying velocity) has been commented by the authors in Remark 1. There usually exist delays in networked systems due to the finite speed of information transmission and processing. In order to discuss the influence of the delays, one needs to first model the delays, which are usually classified as input delays and communication delays. The input delays can be caused by information processing while the communication delays can be caused by information propagation from one agent to another. Consensus in the presence of input delays and communication delays are respectively. Consensus in the presence of both input and communication delays is considered in [17]. While it is usually assumed that the delays are uniformly fixed [5], the extension to the case of multiple (non-uniform or diverse) time-varying delays is also discussed [18], [19], which take into account a more general case. In this paper, delays were modeled as multiple fixed input delays. When studying the influence of the delays on consensus algorithms, it is desirable to find some sufficient conditions to guarantee the stability of the closed-loop system. Two approaches are generally adopted for the analysis, i.e., the frequency-domain approach and the time-domain approach. …