Finite-time consensus of second-order multi-agent systems via auxiliary system approach

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Abstract

This paper investigates the problem of finite-time consensus protocols design for second-order multi-agent systems without velocity measurements. Based on the global finite-time stability theory of continuous systems and homogeneity with dilation, a new class of finite-time consensus (FTC) protocols are proposed for the multi-agent systems by the aid of an auxiliary system. Under the leader-following topological structure, the goal of FTC can be achieved when the velocity information is unavailable. Finally, two examples are given to verify the efficiency of the proposed method.

Introduction

Consensus, which means that a team of subsystems achieve an agreement on a common value by interacting with their local neighbors. Recently, consensus of multi-agent systems has been an emerging research topic in the systems and control community, due to its broad applications in various areas such as formation control of mobile robots, sensor networks, and cooperative surveillance [1], [2], [3], [4], [5], [6]. Generally, existing consensus algorithms can be divided into two types: leaderless consensus and leader–follower consensus [7], [8]. Independent of any global information of the communication graph and based on only the agent dynamics and the relative states of neighboring agents, Li et al. [9] designed a new distributed adaptive consensus protocol to achieve leader–follower consensus in the presence of a leader with a zero input for any communication graph containing a directed spanning tree.

In the past decade, a lot of results have been reported on consensus of multi-agent systems with first-order integrator dynamics [10], [11], and it is well known that such consensus can be realized if and only if the time-varying network topology contains a directed spanning tree jointly as the network evolves over time [12].

In contrast to first-order multi-agent systems, second-order multi-agent systems are more interesting because it can describe a large class of real networked systems. Recently some interesting research results have been obtained for second-order multi-agent systems [13], [14], [15], [16], [17], [18]. In [17], to avoid fragmentation, Su proposed a connectivity-preserving algorithm to ensure the second-order consensus of multiple nonlinear dynamical mobile agents when only a small fraction of agents in the group have access to the information of the virtual leader. In [18], by defining a new concept about the generalized algebraic connectivity for strongly connected networks, Yu derived some sufficient conditions for guaranteeing the second-order consensus in multi-agent systems based on algebraic graph theory, matrix theory, and Lyapunov control approach.

When studying consensus problems, the convergence rate is an important index to evaluate the designed protocols. Many of the existing protocols, including those aforementioned literatures, only result in consensus asymptotically, which means that convergence rates of the considered systems are asymptotic with infinite settling time and the goal of consensus cannot be achieved in finite time. However, in practical application, finite-time consensus is more desirable which holds the advantage of better disturbance rejection properties and better robustness against uncertainties [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. Recently, finite-time consensus problems have attracted much attention and some related works can be briefly reviewed as follows. Guan et al. [32] discussed the finite-time consensus problem of leader-following second-order multi-agent systems by the graph theory, matrix theory, homogeneity with dilation and LaSalle׳s invariance principle. Li and Wang [33] addressed some position consensus algorithms without/with collision avoidance for leaderless and leader–follower multi-AUV systems. Cao and Ren [34] studied the finite-time consensus of multi-agent networks with unknown inherent nonlinear dynamics by the proposed distributed nonlinear consensus algorithm based on a comparison based approach.

It is worth noting that most consensus algorithms for multi-agent systems mainly depend on the availability of the full state for feedback, very few works have been done when velocity information is not available. Actually, such situation is often happened when vehicles are not equipped with velocity sensors, or velocity is not precisely measured. In [35], the consensus strategy was designed for double-integrator dynamics in the case that the control inputs are required to be a priori bounded and the velocity is not available for feedback. In [36], the finite-time consensus problems of second-order multi-agent systems were investigated by proposing two bounded control laws, which are independent of velocity information. In [37], [38], the finite-time consensus problem of the heterogeneous multi-agent system with agents modeled by first-order and second-order integrators was considered, where 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.

Compared to full state feedback cases [39], [40], [41], [42], there are fewer results dealing with consensus protocols without velocity measurement and there still have many challenges remaining. In this paper, we design a new class of consensus protocols for second-order multi-agent systems without velocity measurements, and the innovation lies in three aspects: Firstly, based on the auxiliary system method, a new class of consensus protocols without velocity measurements are presented for second-order multi-agent systems. Secondly, utilizing the graph theory, Lyapunov stability theory and the homogeneity with dilation, the new designed consensus protocols can be proved to solve the finite-time consensus of second-order multi-agent systems under leader–follower network structure. Finally, the proposed consensus protocols are further successfully applied into the finite-time consensus issue of multi-agent systems with the directed network topology.

The notations in this paper are quite standard. Rn and Rn×m denote, respectively, the n dimensional Euclidean space and the set of all n×m real matrices. The superscript “T” denotes the transpose and the notation XY (respectively, X>Y) where X and Y are symmetric matrices, means that XY is positive semi-definite (respectively, positive definite). I and 0 represent the identity matrix and a zero matrix, respectively; diag(···) stands for a block-diagonal matrix; λmin(·) denotes the minimum eigenvalue of a matrix. sign(·) is the sign function. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

The rest of this paper is organized as follows. Preliminaries are given in Section 2. Section 3 presents the main results with proofs. Simulation results aiming at verifying the theoretical analysis are reported in Section 4. Finally, conclusions are drawn in Section 5.

Section snippets

Graph theory

Let G=(S,E,A) be a weighted directed or undirected graph of order n+1, with the set of nodes S={s0,s1,s2,,sn}, the set of edges ES×S, and a weighted adjacency matrix A=[aij](n+1)×(n+1) with weights aij>0 if (sj,si)E and aij=0 otherwise. Assume that all the graphs under consideration are simple, that is, (si,si)E, with no self-loops, thus aii=0. For undirected graphs, aij=aji. The Laplacian matrix L(A)=[lij]n×n of graph G is defined by lij=aij for ij, i,jN={1,2,,n} and lii=dini=j=1,jina

FTC of multi-agent systems

In this section, we will establish the main results about FTC of leader–follower multi-agent systems.

Consider a distributed dynamic system consisting of n followers in In and one leader labeled as s0, where each agent׳s dynamic is described by a second-order dynamic in the form of{ẋi=vi,v̇i=ui,iIn+1,where xi(t), vi(t) are the agent position and velocity, respectively. ui(t) is a control protocol to be designed.

From the viewpoint of mathematics, the finite-time consensus (FTC) means that there

Two numerical examples

To show the effectiveness of the designed protocols, two examples about finite-time consensus are presented for illustration, where the function ϕ(x) is chosen as the following two forms:ϕ1(x)=tanh(x),ϕ2(x)=x.

Example 1

Under the Topology B, consider the multi-agent systems (4) with six follower agents and one leader agent, where communication topology G is shown in Fig. 1 and the weight of each edge is 1. That is, a51=a42=a32=a64=a63=b1=b2=1.

The initial conditions of the leader and six followers are

Conclusions

In this paper, based on the auxiliary system method, a new class of consensus protocols without velocity measurements were presented for second-order multi-agent systems. Under leader–follower network structure, the new designed consensus protocols could be proved to solve the finite-time consensus of second-order multi-agent systems by utilizing the graph theory, Lyapunov stability theory and the homogeneity with dilation. Then, the proposed consensus protocols were further successfully

Acknowledgments

The authors would like to thank the Associated Editor and all the reviewers for their constructive suggestions and to acknowledge Dr. Xin He in City University of Hong Kong and Dr. Xiuxian Li in The University of Hong Kong for inspiring discussions on the topic. Research is partially supported by the National Natural Science Foundation of China under Grant nos. 61304174, 61573096, 61322302, 61272530, 61272297, 61402207, and the Jiangsu Provincial Natural Science Foundation under Grant nos.

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