Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements☆
Introduction
Distributed systems and networks have attracted much attention in the last few years because of their flexibility and computational performance. As a fundamental of distributed coordination, the consensus problem which means that a group of autonomous agents reaches an agreement upon some parameters has been widely studied by multi-disciplinary researchers. This is partly due to its broad applications of multi-agent systems in many areas, such as the formation control of robotic systems, the cooperative control of unmanned aerial vehicles, the attitude alignment of satellite clusters, the target tracking of sensor networks, the congestion control of communication networks, and so on [1], [2], [3].
The study of multi-agent consensus is expected to establish the effective consensus protocols with required performance. The convergence speed is an active topic and can reflect the performance of the proposed consensus protocols. Up to now, most of the existing consensus protocols for multi-agent systems are convergent asymptotically, i.e., the systems have at most exponential convergence rates [4], [5]. For such consensus protocols, lots of researchers found that the convergence rates can be influenced by the weights of networks (the second smallest eigenvalue of the interaction graph Laplacian) [6], [7]. One can increase convergence speed with respect to the linear protocols, but the consensus can never be reached in a finite time. However, in many practical situations, it is required that the consensus can be reached in a finite time, such as when high precision performance and stringent convergence time are required. Cortés [8] proposed two normalized and signed gradient flows of a differential function and used these protocols to solve the finite-time consensus problem. Hui [9] used the notion of finite-time semistability to develop the finite-time rendezvous problem. Chen et al. [10] studied the finite-time consensus of a multi-agent system using a binary consensus protocol. However, the aforementioned consensus protocols involved discontinuous dynamical systems, which may lead to chattering or excite high frequency dynamics in applications involving flexible structures [11]. To avoid these negative effects, some continuous consensus protocols are proposed for multi-agent systems which can reach the finite-time consensus [12], [13], [14], [15], [16], [17], [18]. Based on the Lyapunov method, Wang and Xiao [12], [13] showed that the multi-agent systems could solve the finite-time consensus problem for both the bidirectional and unidirectional interaction cases. Jiang and Wang [14], [15] investigated the finite-time consensus of multi-agent systems with respect to a monotonic function under fixed and switching topologies. Zheng et al. [16] studied the finite-time consensus of stochastic multi-agent systems with a general protocol. For second-order multi-agent systems, Wang and Hong [17] gave some protocols and showed that these protocols can reach the finite-time consensus based on homogeneous method. Based on the adding a power integrator method [19], Li et al. [18] designed a protocol and discussed the finite-time consensus of leaderless and leader-following multi-agent systems with external disturbances.
Unfortunately, all the aforementioned multi-agent systems were homogeneous, that is, all the agents have the same dynamics behavior. However, the dynamics of the agents are quite different because of various restrictions or the common goals with mixed agents in the practical systems. Liu and Liu [20] studied the stationary consensus of discrete-time heterogeneous multi-agent systems with communication delays. In [21], [22], the authors considered the consensus of heterogeneous multi-agent systems with and without velocity measurements. To our best knowledge, there is no literature researching the finite-time consensus of heterogeneous multi-agent systems.
Inspired by the recent developments in heterogeneous multi-agent systems, we decide to investigate the finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements under two classes of specific directed networks in this paper, in which the agents are governed by first-order and second-order integrators, respectively. By combining the homogeneous domination method with the adding a power integrator method, we first propose a continuous protocol with velocity measurements for the heterogeneous multi-agent systems and show that these systems can achieve the finite-time consensus. For the second-order integrator agents, the velocity information sometimes is unmeasurable because of technology limitations or environmental disturbances [23], [24], [25]. Therefore, we design a continuous finite-time consensus protocol for the heterogeneous multi-agent systems in which the second-order integrator agents cannot obtain the velocity information. By using the graph theory, Lyapunov theory and the property of a homogeneous function, we prove that the heterogeneous multi-agent systems converge in finite time under two classes of directed networks: (1) strongly connected and satisfies the detailed balance condition, (2) leader-following network and the network among the followers is strongly connected and satisfies the detailed balance condition. Finally, some simulation examples are presented to show the effectiveness of our proposed protocols.
This paper is organized as follows. In Section 2, we present some notions in graph theory and formulate the model to be studied, and assemble some key lemmas. In Sections 3 Finite-time consensus protocol with velocity information, 4 Finite-time consensus protocol without velocity information, we propose the continuous protocols with and without velocity measurements and show that the heterogeneous multi-agent system can achieve the finite-time consensus, respectively. In Section 5, simulation examples are given to illustrate the effectiveness of our proposed protocols. Finally, some conclusions are drawn in Section 6.
Notation: Throughout this paper, we let and be the set of real numbers, positive real numbers and non-negative real numbers, is the -dimensional real vector space, . For a given vector or matrix denotes its transpose. is a vector with elements being all ones. is said to be non-negative (resp. positive) if all entries are non-negative (resp. positive), denoted by (resp. ). , where is the sign function.
Section snippets
Graph theory
The network formed by multi-agent systems can always be represented by a graph. Thus, graph theory is an important tool to analyze the consensus problem for multi-agent systems. In this subsection, some basic concepts and properties are presented in graph theory [26].
Let be a weighted directed graph of order with a vertex set , an edge set and a non-negative asymmetric matrix . agent and can communicate with each
Finite-time consensus protocol with velocity information
In this section, we first propose the protocol with velocity information for the heterogeneous multi-agent system (1)–(2) as follows: where is the aforementioned weighted adjacency matrix associated with the graph , is a ratio of odd integers, , are the feedback gains,
Finite-time consensus protocol without velocity information
In this section, first, we propose the protocol without velocity information for the heterogeneous multi-agent system (1)–(2) as follows: where and is the weighted adjacency matrix associated with the graph is a ratio of odd integers, , are the
Simulations
In this section, we begin with a numerical simulation in Example 1 to illustrate the effectiveness of the theoretical result in Section 3. In Example 2, we provide an illustration of the theoretical result in Section 4.
Fig. 1 shows a strongly connected graph with weight which satisfies the detailed balance condition . Suppose that the vertices 1 and 2 denote the second-order integrator agents and the vertices 3 and 4 denote the first-order integrator agents. In Fig. 2, the agent 4 is a
Conclusions
In this paper, the finite-time consensus problem of the heterogeneous multi-agent system with agents modeled by first-order and second-order integrators was considered. Two kinds of consensus protocols with and without velocity measurements were proposed. Based on the graph theory, the Lyapunov theory and the homogeneous domination method, we proved that the protocols can solve the finite-time consensus under two classes of special directed networks, respectively. Future work may focus on the
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This work was supported by the 973 Program (Grant No. 2012CB821203), NSFC (Grant Nos. 61020106005, 10972002 and 61104212) and the Fundamental Research Funds for the Central Universities (Grant Nos. K50511040005 and K50510040003).