Elsevier

Neurocomputing

Volume 223, 5 February 2017, Pages 18-25
Neurocomputing

Static consensus of second-order multi-agent systems with impulsive algorithm and time-delays

https://doi.org/10.1016/j.neucom.2016.10.025Get rights and content

Abstract

This paper studies the consensus problem of second-order multi-agent systems with constant time-delay, fixed topology and impulsive algorithm based on periodic sampling. First, by theory of impulsive differential equations, it is proved that the consensus is achieved if and only if some matrix has a simple 1 eigenvalue and all the other eigenvalues are in the unit circle. Meanwhile, the consensus state of the system is obtained, which indicates that the positions and the velocities of all agents reach, respectively, a constant state and zero. Hence we say a static consensus is achieved for multiple second-order agents. Then, by stability of polynomials, we establish a necessary and sufficient condition from the perspective of topology and protocol parameters, which provides the range of allowable time-delay and the choice of impulse period. Finally, simulation examples are given to illustrate the effectiveness of the theoretical results.

Introduction

In recent years, the study on consensus behavior of multi-agent systems has received a great deal of attention and interested many researchers in various fields. Applications of this research pertain to cooperative control of unmanned air vehicles, autonomous formation flight, control of communication networks, swarm-based computing, rendezvous in space etc. [1], [2], [3]. Systematical framework of consensus problems for first-order multi-agent systems was established in [4] and [5]. In addition, two kinds of consensus protocols for second-order multi-agent systems were proposed in [6] and [7]. For more relevant results, one can refer to the surveys [8], [9] and the recent literature [10], [11], [12].

In many practice, the synthesis of control law can only use the data sampled at discrete sampling instants, although the system itself is a continuous process. Thus sampled-data based consensus were studied in [13], [14], [15], [16], [17], [18], [19] for continuous-time multi-agent systems. [13] and [14] investigated, respectively, the consensus of first-order agents with fixed and switching topologies by using periodic sampling technology and zero-order hold circuit. For multiple second-order agents, [15] proposed two protocols which drove all the agents reach consensus on zero and nonzero constant final velocity, respectively. By assuming that all agents updated their control inputs at their own independently sampling instants, [16] considered the asynchronous consensus of second-order agents with switching topologies. In the case of large sampling periods, [17] proposed a novel protocol when certain position-like states could be only detected over network. [18] and [19] studied the leader-following consensus via sampling control, respectively, for multi-teleoperator systems and linear multi-agent systems with randomly missing data. These sampled-data based protocols are mostly implemented continuously during the sampling intervals, and the dynamics of agents are controlled at all the times.

While different from sampled-data control strategies, impulsive algorithms were presented for continuous-time second-order agents in [20], [21]. The property of impulsive algorithms (with obvious advantages in less energy cost, fast transient, and easier to design) is to instantaneously change the states of agents. There may be no any interactions between agents during sampling intervals. Moreover, the impulsive control is an effective control technique in many practical applications, such as the orbit interception correction of orbiting objects, the population control of a kind of insects, the control of reaction process in a chemical reactor system, and the money supply in a financial system [22], [23], [24], [25], [26]. In [20], two kinds of impulsive algorithms were proposed for the dynamic and static consensus of second-order agents with fixed topology, some necessary and sufficient conditions on sampling period were obtained by using stability of reduced-order system and properties of Laplacian matrix. For the case of aperiodic sampling and switching topology, [21] proved the impulsive consensus of second-order agents where only position measurements were available, by using the property of stochastic matrices. More results on consensus via impulsive control can be found in [27], [28], [29], [30], and the references therein.

Moreover, the disturbance of communication time-delays is unavoidable in real networks due to limited communication capacity of sensing or transmitting equipments. As to the consensus of multi-agent systems with time-delays, recent years have witnessed numerous literature, such as [4], [27], [28], [29], [31], [32], [33], [34], to name a few. Particularly, [27] studied the impulsive consensus of second-order agents with time-delays and undirected topology by analyzing the solutions of impulsive systems. Under leader-following framework, [28] considered the formation tracking of second-order systems via impulsive control with input delays. By using stability theory of discrete-time delayed impulsive linear systems, [29] proposed a strategy for the multi-tracking of discrete-time dynamical networks by mixed impulsive networked control. However, to our best knowledge, there are few results on the consensus of multiple continuous-time second-order agents with directed graph and communication time-delay under the leaderless and impulsive framework. Hence we study the consensus of second-order agents with constant time-delay, directed topology and impulsive control based on periodic sampling. The main contributions of this paper are summarized as follows. First, the condition of consensus is described by the spectrum of matrix, that is, the consensus is achieved if and only if some matrix has a simple 1 eigenvalue and all the other eigenvalues are in the unit circle. Second, the consensus state of the system is provided via theory of impulsive differential equations. Finally, the range of allowable time-delay and the choice of sampling period are given.

The remainder of this paper is organized as follows. Section 2 presents some basic knowledge of graph theory and model formulation. Section 3 gives two necessary and sufficient conditions for consensus achieving. Section 4 carries out numerical example. Section 5 provides some concluding remarks.

Notations: We let R, N and N+ be the sets of real numbers, non-negative integers and positive integers, respectively. Rn is the n-dimensional Euclidean space. Rm×n is the set of m-by-n matrices. InRn×n is an identity matrix. Sometimes we apply 0 to denote the zero matrix with appropriate dimension. 1n=[11]TRn is a column vector with all elements equal to one. For a complex number μ, Re(μ), Im(μ), |μ| and μ¯ are the real part, the imaginary part, the modulus and the complex conjugate of μ, respectively. j is the imaginary unit of complex number field.

Section snippets

Problem formulations and preliminaries

In this section, we present some basic knowledge on graph theory, problem formulations, and some preliminary lemmas.

Main results

In this section, we give the main results on achieving consensus of system (1) with protocol (2).

Theorem 1

Suppose p1>0, p2>0, and Assumption (A1) holds. Then system (1) with protocol (2) achieves consensus if and only if matrix G has a simple 1 eigenvalue and all the other eigenvalues are in the unit circle. In addition, if the consensus is achieved, then limtri(t)=wTr(t0)+hp1wTv(t0), limtvi(t)=0, i=1,2,,N, where wRN satisfies that wTL=0 and wT1N=1; r(t0),v(t0)RN are initial states.

Proof

(Sufficiency.)

Numerical example

Example 1

Consider system (1) of six agents with protocol (2). The topology among agents is described by a directed cycle. The elements of associated Laplacian matrix L are l11=l33=l55=2, l22=l44=l66=1, l21=l43=l65=1 and l32=l54=l16=2, and the others are zero. Then the eigenvalues of L are μ1=0, μ2=2.2541+1.1484j,μ3=2.25411.1484j,μ4=0.7459+1.1484j,μ5=0.74591.1484j,μ6=3. By taking p1=0.9, p2=0.8 and solving (15), then τ¯=0.2950 (which is defined in Remark 4), and the allowable time-delay satisfies 0<τ<

Conclusions

This paper has investigated the impulsive consensus for multiple second-order agents with periodic sampling, constant communication time-delay and fixed topology. By theory of impulsive differential equations and stability of polynomial, two necessary and sufficient conditions of consensus achieving are obtained. The range of allowable time-delay and the choice of sampling period are given explicitly. The consensus state of the system is provided as well, which indicates that the positions and

Fangcui Jiang received the Bachelor and Master degree in applied mathematics from Qufu Normal University, PR China in 2003 and 2006, and the Ph.D. degree in Systems and Control from Peking University in 2011. Now she is an assistant professor at School of Mathematics and Statistics, Shandong University, Weihai, and also studies as a Postdoctoral Research Fellow at School of Control Science and Engineering, Shandong University, Jinan, PR China. Her current research interests focus on networked

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    Fangcui Jiang received the Bachelor and Master degree in applied mathematics from Qufu Normal University, PR China in 2003 and 2006, and the Ph.D. degree in Systems and Control from Peking University in 2011. Now she is an assistant professor at School of Mathematics and Statistics, Shandong University, Weihai, and also studies as a Postdoctoral Research Fellow at School of Control Science and Engineering, Shandong University, Jinan, PR China. Her current research interests focus on networked systems, analysis and cooperative control of multi-agent systems.

    Dongmei Xie received her Bachelor and master degree in applied mathematics from Ludong university, PR China, and Qufu Normal University, PR China in 1998 and 2001, respectively. She received her Ph.D. from Peking University in 2001. From June 2014 to June 2015, she studied as a visiting scholar at the Department of Mathematics, University of California, Irvine, USA. Now she is an associate professor at the Department of Mathematics, Tianjin University, PR China. Her current interests include analysis and control of switched systems, networked control systems and multi-agent systems.

    Bo Liu was born in 1977. She received the Ph.D. degree in Dynamics and Control from Peking University in 2007. She was a visiting research fellow at the City University of Hong Kong in 2009 and is currently an Associate Professor in North China University of Technology. Her research interests include swarm dynamics, networked systems, collective behavior and coordinate control of multi-agent systems.

    This work was supported by National Natural Science Foundation of China (No. 61304163, No. 61473337 and No. 61304049), Natural Science Foundation of Shandong Province (No. ZR2013FQ008), Natural Science Foundation of Tianjin (No. 15JCYBJC19100), and Independent Innovation Foundation of Shandong University (No. 2013ZRQP006).

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