Necessary and sufficient conditions for leader-following consensus of multi-agent systems with random switching topologies

https://doi.org/10.1016/j.nahs.2020.100905Get rights and content

Abstract

This paper is concerned with leader-following consensus of general linear multi-agent systems with random switching topologies, where the dwell time in each topology consists of a fixed part and random part, and the topology switching signal in random part is modeled by a semi-markov process. First, a semi-Markov switched system with state jumps is constructed, and the stochastic stability of the constructed system is shown to be equivalent to that of the original system. Then a necessary and sufficient condition is established by using a Lyapunov approach in terms of linear matrix inequalities. Finally, the effectiveness of our results is illustrated by a numerical example and a practical example.

Introduction

As we all know, with the development of computer technology and networks, an important issue of distributed cooperative control of multi-agent systems has been very popular in the systems and control communities ([1], [2] ) over the past few years owing to its potential applications in the formation flying of unmanned aerial vehicles, cooperative control of mobile robots, sensor networks, etc. Consensus problem plays an important role in addressing the issue of multi-agent systems [3], [4], [5], [6]. By using only local information exchange to design a distributed consensus protocol, the states of all agents converge to an agreement state finally. Generally, in order to make the control objective achieved more effectively, some agents play a role of leaders. When a MAS has one leader, the corresponding problem is called leader-following consensus problem, and finally the followers’ state is the same as the leader’s.

In many applications, communication links between agents may be unreliable due to environmental sudden change, communication range limitations and some disturbances, hence the communication topology among agents may become varying over time [7], [8], [9], [10], [11], [12]. In the last few years, the interaction topology is assumed to be random and Markovian switching. Some good results about consensus in multi-agent systems have been given under Markovian switching topologies. For example, Authors in [9] discussed the linear asymptotic consensus problem for a network of dynamic agents whose communication network is modeled by Markovian switching graphs. In [10], consensus were achieved for MASs with linear time-invariant agent dynamics over Markovian switching topologies. Containment control of general MASs with directed Markovian switching topologies was studied in [11]. Authors in [12] studied L2L containment control of MASs with Markovian switching topologies and non-uniform time-varying delays.

It is known that for Markovian switching, the sojourn time between two consecutive switching obeys exponential distribution in the continuous-time domain (or geometric distribution in discrete time domain, respectively), and owing to the memoryless property of the exponential distribution, the transition rates are constant. However, in practical, the transition rates may be time-varying and the probability distribution of sojourn-time may be Uniform distribution, Gaussian distribution, and Weibull distribution, etc. Based on these characteristics, [13] and [14] proposed the idea of a semi-Markov process in 1954. For the semi-Markov process, there is an embedded Markov chain, and the system makes its transitions from one state i to another state j according to the transition probability matrix of the Markov process, but the waiting time in state i could be an arbitrary random variable depending on the transition. The discrete-time and continuous-time Markov processes are special cases of the semi-Markov process. So the semi-Markov process provided a much more general model for probabilistic systems than Markov process [15], [16].

There are a few results on the synchronization for complex networks with semi-Markov switching topologies [17], [18]. For multi-agent systems, it is worth noting that authors in [19] investigated the exponential consensus of non-linear multi-agent systems with semi-Markov switching topologies. In [20], authors considered leader-following consensus for multi-agent systems with semi-Markov switching topologies and some good results have been obtained by using event-triggered control methods. Authors in [21] studied containment control problem of stochastic MASs with semi-Markovian switching topologies. However, only sufficient conditions are given in above literatures, it seems more challenging and difficult to obtain sufficient and necessary conditions to achieve consensus. If some conditions are changed, can we get sufficient and necessary conditions for the consensus of multi-agent systems with random switching topologies ?

Motivated by above discussion, in this paper, leader-following consensus of general linear multi-agent systems with random switching topologies is considered. The main contribution of this paper is summarized as follows:

(1) Compared with [9], [10], [11], [12] where topologies are Markovian switching and [19], [20], [21] under semi-Markovian switching topologies, in this paper, the dwell time in each topology consists of a fixed part and random part and the topology switching signal in random part is modeled by a semi-markov process, which provides a much more general model.

(2) Compared with [19], [20], [21] where only sufficient conditions are given, a necessary and sufficient condition is established in this paper. Without the fixed part, we can also obtain a necessary and sufficient condition for the consensus of multi-agent systems with semi-Markov switching topologies.

The rest of the paper is organized as follows: in Section 2, some preliminaries and problem formulation are described. Sufficient and necessary conditions to ensure leader-following consensus are proposed in Section 3. Finally, in Sections 4 Simulation results, 5 Conclusion, a numerical example and a practical example to verify the validity of the theory and conclusions are provided, respectively.

Notations: Throughout this paper, denote the Kronecker product by and an identity matrix with proper dimension by I. AT denotes the transpose of a matrix A and denotes the entries of matrices implied by symmetry. Rn and Rm×n denotes the n dimensional Euclidean space and the space of m×n matrices with real entries. P>0(P0) refers to a real symmetric and positive definite (semi-positive definite) matrix.

Section snippets

Preliminaries

The communication topology of a MAS is a weighted directed graph, which is denoted by G=(V,E,A), where V={v1,v2,,vN} is the node set, E={eij=(vi,vj):vi,vjV} is the edge set, and eijE indicates that there is a directed edge from vi to vj, which means that agent j can receive information from agent i (vi and vj are called the parent nodes and child nodes, respectively). Accordingly, agent i is a neighbor of agent j and the neighbor set of node vi is denoted by Ni={j|(vi,vj)V}. A directed

Semi-Markov switched systems with state jumps

In this section, the stochastic stability of system (7) with dwell time (2), (3) will be studied. In order to deal with the system states in the fixed dwell time interval [tn,tn+di), a semi-Markov jump linear system with state jumps is constructed in which the state jumps at switching time and state jumps are used to replace the system states in the fixed dwell time intervals. Moreover, the stochastic stability of the constructed system is shown to be equivalent to that of system (7) with dwell

Simulation results

In this section, a numerical example and a practical example are given to illustrate the effectiveness of the theoretical results obtained in the previous sections.

Consider a MAS with five agents. The leader is labeled as 0, and followers are labeled as 1,2,3,4 respectively. The switching signal r(t)Γ is with three different modes Γ={1,2,3}. The random part is described by a semi-Markov process. The directed interaction topology among agents is shown in Fig. 1. So the corresponding Laplacian m

Conclusion

In this paper, leader-following consensus of general linear multi-agent systems with random switching topologies is considered, where the dwell time in each topology consists of a fixed part and random part, and the topology switching signal in random part is modeled by a semi-markov process, which provides a much more general model. A necessary and sufficient condition is established by using a Lyapunov approach in terms of linear matrix inequalities. Without the fixed part, we can also obtain

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work is supported by The National Natural Science Foundation of China under Grant 11571322 and Grant 11971444 and Foundation of Henan Educational Committee under Grant 16A110023.

References (21)

There are more references available in the full text version of this article.

Cited by (0)

View full text