Elsevier

Information Sciences

Volumes 370–371, 20 November 2016, Pages 667-679
Information Sciences

Asynchronous impulsive containment control in switched multi-agent systems

https://doi.org/10.1016/j.ins.2016.01.072Get rights and content

Abstract

A novel impulsive containment control is derived for multi-agent systems with the switching behavior and time delay, where an asynchronous impulsive control formulation is introduced in a distributed synthesis. Containment control criterion is established by using multiple Lyapunov functions and the Razumikhin technique in terms of mathematical induction. The relationship between impulsive control and the average dwell time is presented and thus can serve as a tool to design an impulsive containment controller. Unlike traditional impulsive control strategies, the proposed methodology does not require time instants of impulsive control to simultaneously occur with switching signals, which significantly leads to a less conservativeness of the designed impulsive controller to handle the switched system. Finally, the effectiveness of the proposed method is verified by a simulation example.

Introduction

Consensus of multi-agent systems has regained popularity due to its wide applications in distributed computing. When the computational issues are sophisticated for an individual agent, the collective computation of multi-agent systems is an alternative to solve these challenging problems in a networked environment, e.g., surveillance and monitoring of power networks [18], [28], [38], collaborative information processing [1], and distributed sensor fusion scheme from spatially distributed sensor networks [41]. Indeed, consensus based algorithms can be considered as an effective method for performing distributed computation problems over a network environment. Based on consensus of multi-agent systems, a large number of control schemes have been investigated for various scenarios, e.g., synchronization of networks [10], [36], distributed optimization [33], [44], distributed filtering [4], [43], and formation control [22].

The consensus problem is derived from distributed and parallel computation, where all of agents are required to reach the agreement of states. Inspired by biological collective behaviors [32], Jadbabaie [14] provides a theoretical explanation for the collective behavior. After this study, insightful researches on the consensus [8], [19], [20], [34] and the synchronization problems [9], [45] are published. Regarding the different distributed protocols, analytic approaches via eigenvalue analysis [42], nonnegative matrix [26] and Lyapunov functions [13], [40] have been successfully employed. Using the eigenvalue analysis, some necessary and sufficient conditions are obtained for the second-order consensus in [42]. Moreover, high-order consensus problem of the multi-agent dynamical network with switching topology is studied in [35] by Lyapunov functions. By consideration of the limited communication channel, distributed event-triggered based strategies [7] are designed by Lyapunov functions for consensus of multi-agent systems. In addition, finite-time control algorithms [3] are developed for the second-order multi-agent system via nonsmooth analysis.

As known, the switching topology widely exists in the multi-agent systems. Concerning the synchronization problem of the multi-agent systems with switching topology, a multiple-Lyapunov function approach [37] is used for pinning the network into a synchronous state. Leader-following consensus of multi-agent system is studied in [2], [31] and [29]. Recently, discontinuous control method has attracted much attentions, i.e. impulsive control [5], [15], [39] and [24]. Impulsive control based consensus problems have been studied for the multi-agent system with sampled information in [6]. Impulsive synchronization problems of the network have been investigated in [11]. A further study on the impulsive synchronization schemes of complex network with switching topology can be found in [17].

However, the impulsive containment control algorithm has merely been developed for multi-agent systems with ordinary nonlinearities, such as nonlinear dynamics and time delays. Especially, the multi-agent system involving nonlinearity with topology switching behaviors brings about several challenges for deriving a containment control strategy. Furthermore, for impulsive control in a switched system, the existing algorithms rigorously require each time instant of the impulsive control coinciding with the time instant when the switching behavior is triggered [21], [25], [27]. If the impulsive controller fails to perform precisely at the switching instant, the system may not be stabilized. Meanwhile, the time delays from a networked environment may also jeopardize the effectiveness of an impulsive controller through which there is a time difference between the impulsive control and the switching signal. These issues makes it impossible to efficiently implement the impulsive control scheme for the multi-agent systems. To better handle these problems, we propose an impulsive containment control for multi-agent systems with time delay and switching behavior, which asynchronously performs impulsive control for the whole time horizon. Undoubtedly, this strategy will enhance the flexibility of impulsive control for a switched system. In this work, the main contributions are twofold. First, an impulsive containment control algorithm is built for a multi-agent system with switching behavior and time delays. To ensure stability of the containment control scheme, we use an asynchronous impulsive control formulation derived from a distributed synthesis, where local information exchange is only allowed. Second, multiple Lyapunov functions and the Razumikhin technique are employed to derive impulsive containment control criterion. Both analytical and numerical conditions of impulsive control algorithms are proposed. In fact, the relationship between impulsive control and the average dwell time is established for the multi-agent system.

The rest of the paper is organized as follows. First, basic mathematical descriptions of the switched multi-agent system with the time delay are defined in Section 2. Then, impulsive containment control is designed for the multiple leader-following based multi-agent system. Following on, the stability analysis of the impulsive controlled system is considered via the Lyapunov–Razumikhin technique. Moreover, the discussion on the difference between the asynchronous and the synchronous impulsive control is presented. Section 4 is devoted verifying the effectiveness of the impulsive containment control algorithm by a numerical simulation. The last section concludes the paper and briefly discusses future research.

Let Rn denote the n-dimensional Euclidean space and ‖•‖ be the Euclidean norm in Rn. 1n=[1,1,,1]T. Let In be the n × n identity matrix (or simply I if no confusion arises), and 0n × m is the n × m matrix with all elements zero (or simply 0 if no confusion arises). Define the finite set S={1,2,,S} with a finite positive integer S, K={0,1,2,}. Given a matrix X, X > 0(<, ≥, ≤) means that X is a symmetric positive definite matrix (negative definite, positive semi-definite, negative semi-definite, respectively). Denote the maximum and minimum eigenvalue of the matrix by λmax (•) and λmin (•), respectively. The superscript T stands for matrix transposition. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The notation ⌈x⌉ stands for the minimal integer not less than x.

Consider a switched multi-agent system consists of N agents with nonlinear dynamics. Assume that there are m leaders and Nm followers. The interaction for all agents can be naturally modeled by a directed graph G=(V,E,A), where V={1,2,,N} is a set of agents, EV×V is a set of edges, A=[aij],i,j=1,2,,n is a weighted adjacency matrix. If {j,i}E, then aij > 0, otherwise aij=0. A directed path from node i to node j is a sequence of edges (i,s1),(s1,s2),,(sk,j), where (i,s1),(s1,s2),,(sk,j)E. Let M={1,2,,m} denote the set of followers, and N={m+1,,N} denote the set of leaders. The Laplacian matrix L=[lij]RN×N, where lii=j=1naij and lij=aij for ij. The Laplacian matrix L can be partitioned as: L=[L11L1200]where L11Rm × m and L12Rm×(Nm) denote the communication among the followers and the communication among the followers and leaders, respectively.

Assumption 1

In the directed graph G, for each follower, there exists at least one leader that has a directed path to that follower.

From [23], the following lemma can be obtained.

Lemma 1

If Assumption 1 holds, then all the eigenvalues of L11 have positive real parts and all the elements ofL111 are nonnegative and(L111L12)1Nm=1m, whereNm is the number of leaders.

Generally, the dynamics of the multi-agent system is described by x˙i(t)={gσ(t)(t,xi(t),xi(td(t)))+ui(t),iM,gσ(t)(t,xi(t),xi(td(t))),iN,where xi(t) ∈ Rn is the position state of the ith agent. ui(t) ∈ Rn is its control input. The time delay d(t) may be unknown (constant or time-varying) but bounded by a known constant 0<d(t)<d¯, where d¯R+ is upper bound of the time delay. σ(t) is switching signal. σ(t):R+S with a constant s being the total number of the switching modes. For each time instant t[t0,+), the switching signal σ establishes the state change among different subsystems with respect to the multi-agent system. The logical rules that generate the switching signals constitute the switching logic, and the index σ(t) is called the active mode at the time instant t. Let t0=0 be the starting mode.

Definition 1 see [12]

For the switching signal σ and any ta > tb > t0, let Nσ(ta, tb) be the switching numbers of σ over the interval [tb, ta). We have Nσ(ta,tb)tatbT+N0,where T and N0 are called average dwell time and the chatter bound, respectively.

gi(·, ·, ·): R × Rn × RnRn, i=1,2,,l are the nonlinear functions which satisfy the following assumption.

Assumption 2

The nonlinear function gs(t,xi(t),xi(td(t))),s=1,2,,l satisfies the convex Lipschitz condition. That is, for any given nonnegative constant θi0,i=1,2,,Nm, there exist two positive numbers ks and ls, such that for y, z, yi, ziRn, gs(t,y,z)j=1Nmθigs(t,yj,zj)ksyj=1Nmθjyj+lszj=1Nmθjzj.

Definition 2 [30]

A subset C of Rn is said to be convex if (1λ)x+λyC whenever xC, yC and 0 < λ < 1. The convex hull of a finite set of points x1,x2,,xqRn (q is a positive integer) is the minimal convex set containing all points in {x1,x2,,xq}. We use co{x1,x2,,xq} to denote it, i.e. co{x1,x2,,xq}={j=1qθjxj|θjR,θj0,j=1qθj=1}.

Definition 3

The containment control is achieved in the multi-agent system (2) if for any initial values xi(0) Rn(i=1,2,,N),xi(t)co{xm+1(t),,xN(t)} as t+,iM. That is, each state of the followers will converge into the convex hull formed by the states of the leaders as t+.

To achieve the containment control, the impulsive control is designed as ui(t)=αk=1[jVaij(xi(t)xj(t))]δ(ttk),iM,where α is the control gain to be determined, A=[aij] is the adjacency matrix associated with the directed graph G, and δ( · ) is the Dirac impulsive function. tk, k=1,2, are the impulsive instants and limk+tk=+. That is, t1 < t2 < ⋅⋅⋅tk < ⋅⋅⋅ are the impulsive instant sequences, where t1 > t0, and t0=0 is the initial time.

The multi-agent system (2) under impulsive control (4) can be reformulated by the following impulsive differential equation. {{x˙i(t)=gσ(t)(t,xi(t),xi(td(t))),ttkΔxi(tk)=αj=1maij(xi(tk)xj(tk)),iMx˙i(t)=gσ(t)(t,xi(t),xi(td(t))),t>0,iNwhere Δxi(tk)=xi(tk)xi(tk),k=1,2,, and xi(tk)=limttkxi(t).

We assume that xi(t), iM, are right-hand continuous at t=tk, that is, xi(tk)=xi(tk+),k=1,2,, where xi(tk+)=limttk+xi(t).

Let xF(t)=[x1(t),,xm(t)]T,xL(t)=[xm+1(t),,xN(t)]T,Gσ(t)F(t)=[gσ(t)(t,x1(t),x1(td(t))),,gσ(t)(t,xm(t),xm(td(t)))]T,Gσ(t)L(t)=[gσ(t)(t,xm+1(t),xm+1(td(t))),,gσ(t)(t,xN(t),xN(td(t)))]T.

Then, according to (1), the system (5) can be written in a vector form: {{x˙F(t)=Gσ(t)F(t),ttkΔxF(tk)=αL11[xF(tk)+L111L12xL(tk)]x˙L(t)=Gσ(t)L(t),t>0

By defining the error w(t)=xF(t)+(L111L12)xL(t), then we can derive the following error system {w˙(t)=Gσ(t)F(t)+(L111L12)Gσ(t)L(t)Δw(tk)=αL11w(tk)

Remark 1

It is noticeable that the impulsive instants of the proposed control strategy are able to force the system reach the containment control in an arbitrary way as long as the convergence criterion can be satisfied. In other words, the time instants of impulsive control unnecessarily coincide with the triggered switching signal, for details please refer to the section of the comparison.

Section snippets

Asynchronous impulsive containment control

In this section, we consider the global exponential stability of the aforementioned error system (7) by employing the multiple Lyapunov functions and the Razumikhin technique. Precisely, the multiple Lyapunov functions of the error system are used to estimate the destabilizing effect of each subsystem which is usually associated with the switched multi-agent system. The time delays are handled by using the Razumikhin technique. By mathematical induction method, the criterion of global

Simulation

In this section, we will give an illustrative example to demonstrate the validity of the previous theoretical results. Suppose that the interaction graph with two leaders and ten followers are chosen as illustrated in Fig. 1, where the graph connection matrix is given as follow L=(101000000001131000000100113010000000000310100010001141000010000011000000000100100000001001020000000000102100000000001201000000000000000000000000).

Note that, by the network topology L, there is a

Conclusion

This paper introduced a novel impulsive containment control for the multi-agent system with the state switching and time delay, where an asynchronous distributed strategy is utilized. In terms of Lyapunov–Razumikhin technique, the global exponential stability criterion of the impulsive controlled system is established, which indicates the multi-agent system is able to achieve the containment control. More specifically, the proposed impulsive control algorithm takes an asynchronous strategy that

References (46)

  • W. Ni et al.

    Leader-following consensus of multi-agent systems under fixed and switching topologies

    Syst. Control Lett.

    (2010)
  • Q. Song et al.

    Second-order leader-following consensus of nonlinear multi-agent systems via pinning control

    Syst. Control Lett.

    (2010)
  • W. Yu et al.

    Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems

    Automatica

    (2010)
  • D. Yuan et al.

    Inexact dual averaging method for distributed multi-agent optimization

    Syst. Control Lett.

    (2014)
  • S. Boyd et al.

    Gossip algorithms: Design, analysis and applications

    Proceedings of the 24th Annual Joint Conference of the IEEE Computer and Communications Societies, IEEE

    (2005)
  • H. Du et al.

    Finite-time synchronization of a class of second-order nonlinear multi-agent systems using output feedback control

    IEEE Trans. Circuits Syst. I: Regular Papers

    (2014)
  • Z.-H. Guan et al.

    Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control

    IEEE Trans. Circuits Syst. I: Regular Papers,

    (2010)
  • W. He et al.

    Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control

    Inf. Sci.

    (2015)
  • W. He et al.

    Lag quasi-synchronization of coupled delayed systems with parameter mismatch

    IEEE Trans. Circuits Syst. I: Regular Papers

    (2011)
  • W. He et al.

    Synchronization error estimation and controller design for delayed lur’e systems with parameter mismatches

    IEEE Trans. Neural Netw. Learn. Syst.

    (2012)
  • J.P. Hespanha

    Stability of switched systems with average dwell-time

    Proceedings of the 38th IEEE Conference on Decision and Control, IEEE

    (1999)
  • Y. Hong et al.

    Lyapunov-based approach to multiagent systems with switching jointly connected interconnection

    IEEE Trans. Autom. Control

    (2007)
  • A. Jadbabaie et al.

    Coordination of groups of mobile autonomous agents using nearest neighbor rules

    IEEE Trans. Autom. Control

    (2003)
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    This paper was not presented at any IFAC meeting. This work was supported by the Australian Research Council (Nos. DP130104765, DP140100544), the National Natural Science Foundation of China under grants (No. 61304152) and the open fund of Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education (No. MCCSE2015A02).

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