Leader-following consensus of multi-agent systems under fixed and switching topologies☆
Introduction
A new feature of the modern control theory is that the environment is an information-rich world and the control involves networked communications. Under this circumstance, systems are considered as agents forming a network and information is exchanged between these agents through the network. Although each individual agent has limited processing power, the interconnected system as a whole can perform complex tasks in a coordinated fashion. Therefore, comparing with conventional control systems, multi-agent systems have many advantages such as reducing cost, improving system efficiency, flexibility, reliability, and providing new capability. Applications of multi-agent systems cover a wide range, including unmanned air vehicles [1], formation control [2], rendezvous [3], flocking [4], and sensor networks [5].
A central problem for multi-agent systems is to design simple control law for each agent, using local information from its neighbors, such that the networked system can achieve prescribed collective behaviors such as maintaining a formation, swarming, rendezvousing, or reaching a consensus. The control of multi-agent systems is a very active area of research. Jadbabie [6] successfully explained the consensus on the heading angles observed by Vicsek [7]; Savkin [8] gave another qualitative analysis for the Vicsek model. Motivated by the simulation results in [7], Tanner et al. [9] considered a group of mobile agents moving on the plane with double-integrator dynamics, and they proposed a set of control laws which render the group to generate a stable flocking motion. Some less restrictive conditions for consensus were obtained in [10], [11], where consensus is ensured provided there exists a spanning tree. The systematical framework of consensus problem in networks of dynamic agents with switching topology and time-delays was proposed in [12], [13]. Some other relevant research topics have also been addressed, such as agreement over random networks [14], coordination and consensus of networked agents with noisy measurements [15], networks with time-delays [16], [17], consensus filters [5]. In an endeavor to better tackle the control problems, tools from graph theory [18], stochastic matrix analysis [6], system theory such as Lyapunov functions [19], [20] and set-valued Lyapunov theory [16], contraction theory [21], [22], [23], passivity theory [24], [25], [26] etc. have been proved useful.
A particularly interesting topic is the consensus of a group of agents with a leader, where the leader is a special agent whose motion is independent of all the other agents and thus is followed by all the other ones. Such a problem is commonly called leader-following consensus problem. It was reported that the leader-following configuration is an energy saving mechanism [27] which was found in many biological systems, and it can also enhance the communication and orientation of the flock [28]. The leader-following consensus has been an active area of research. Jadbabaie et al. [6] considered such a leader-following consensus problem and proved that if all the agents were jointly connected with their leader, their states would converge to that of the leader as time goes on. The controllability of a Leader–follower dynamic network was studied in [29], where the leader is a particular agent acting as an external input to steer the other agents. The problem was also considered by Ji et al. [30] and Rahmani et al. [31] from a graph-theoretic perspective. When an agent is a particle moving under Newton’s law, consensus of leader-following multi-agent systems is considered in [19], [32]. [19] proposed a distributed control law using local information and [32] provided a rigorous proof for the consensus using an extension of LaSalle’s invariance principle. Distributed observer design for leader-following control of multi-agent networks was considered in [33]. Leader–follower cooperative attitude control of multiple rigid bodies was considered in [34]. Leader-following formation control of nonholonomic mobile robots was considered in [35]. Peng et al. [36] studied the leader-following consensus for a multi-agent system with a varying-velocity leader and time-varying delays. Wang et al. [37] studied different leader roles in networks.
In this paper, we consider the leader-following consensus problem in a more general case, where the dynamics of each agent is an th order linear control system, rather than integrators or double integrators in most existing literature. Achieving consensus under fixed interaction topology is relatively easy. However when the interaction topology is time-varying, this problem becomes challenging. How to design control of multi-agent system with variable communication topologies? When the agent dynamics is integrator, a useful method to handle this problem is to use products of stochastic matrices; see [6] for details. Since the system matrices considered in this paper are not stochastic matrices, the analysis from [6] cannot be extended straightforward to our system. Therefore, we propose a Riccati-inequality-based approach, together with graph theory. This paper is partly motivated by Wang etc. [38]. The major differences between this work and [38] are as follows:
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[38] considered the case without a leader, the case we considered is with a leader.
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In [38] it is required that the graph is frequently connected, which means within a certain duration the graph is connected at least one time with a dwell time . In our case, only joint connectedness is assumed. Hence our restriction is much weaker than the one in [38].
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In [38] the computation of feedback matrix is a heavy job. For example, when there are five agents, one has to compute matrices. Unlike [38], we compute the feedback gain matrix by exploiting the Riccati inequality, which tremendously reduces the computation load.
From the practical point of view, it is hoped that our consensus schemes are robust with respect to communication noises. The robust analysis of consensus was discussed in [39] for fixed interaction topology, and in [40] for variable interaction topology with first order agent model. For general model, we show by simulations that our consensus schemes have certain robustness.
The rest of this paper is organized as follows. Section 2 contains the problem formulation and some preliminary results from graph theory. The leader-following consensus problem under fixed interconnection topologies is discussed in Section 3. Section 4 generalizes the result obtained in Section 3 to the case of switching interconnection topologies. Illustrative examples are presented in Section 4. Section 5 is a brief conclusion.
Section snippets
Problem formulation and preliminaries
We first introduce some notations and definitions. For a symmetric matrix , by we mean that is positive definite (positive semidefinite, negative definite, or negative semidefinite). Letting , denotes the smallest nonzero eigenvalue of . For a finite set , denotes the number of its elements. A matrix is said to be diagonally dominant if for all .
Consider a multi-agent system consisting of agents and a leader. The
Leader-following consensus under fixed interaction topology
In this section, we consider the leader-following consensus problem when the graph is fixed. We use the following control law for agent : where is a feedback matrix to be designed and is defined to be 1 whenever the agent is a neighbor of the leader and 0 otherwise. In order to analyze the leader-following consensus problem, we denote the state error between the agent and the leader as . Then the dynamics of is
Leader-following consensus under switching interaction topology
In this section, we extend the result in last section to the case when the interaction topology is switching. In order to achieve leader-following consensus under switching topology, we must impose a constraint on the dynamics. We assume Assumption 3 The matrix has no positive real part eigenvalues.
In switching topology case, the control is designed as Comparing with that of fixed graph, the difference is that the neighbors of each agent and the index
Simulation results and extended discussions
In this section, we give three examples to illustrate the validity of the results. The first example considers the case when the interaction topology is fixed and the second one considers the case when the interaction topology is switching. We also discuss the robustness properties of the leader-following consensus in the third example. Example 3 Consider a multi-agent system consisting of a leader and four agents. Assume the system matrices are
Conclusions
In this paper we consider the leader-following consensus problem by designing distributed controllers using local information to ensure that all the agents follow the leader. Graph theory is used to describe the interconnection topology. Riccati inequality and Lyapunov inequality are implemented to control design and stability analysis. The connectivity of the graph in fixed topology and the joint connectivity of the graph in switching topology are assumed as key conditions to ensure the
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This work is supported by the National Nature Science Foundation of China under Grants 60674022, 60736022 and 60821091.