Elsevier

Neurocomputing

Volume 412, 28 October 2020, Pages 276-286
Neurocomputing

Leader-following mean square consensus of stochastic multi-agent systems with ROUs and RONs under periodically variable impulse time windows

https://doi.org/10.1016/j.neucom.2020.05.041Get rights and content

Abstract

In order to bring the theoretical results into harmony or agreement with the reality, this paper copes with the leader-following mean square consensus issue for a class of nonlinear stochastic multi-agent systems with randomly occurring nonlinearities (RONs) and randomly occurring uncertainties (ROUs) under impulse time windows (ITWs). As a majorizing preconfigured mechanism of the windows, the configuration named periodically variable impulse time windows (PVITWs) is put forward. It not only ensures the periodic-based layout of impulse time window (ITW), but also allows the position and width of the window to be arbitrarily adjustable, so that the preset of ITW is more suitable for applications. Based on the Lyapunov stability theory and graph theory, the corresponding sufficient mean square consensus criteria are given. Finally, the feasibility of theoretical analysis is illustrated by numerical simulations.

Introduction

With the rapid development of the computer and communication technology, artificial intelligence and related control theory, the multi-agent systems (MASs) have been widely used in many fields, such as the formation control of robots or aircraft[1], traffic management and control[2], social science[3], finance and business[4], [5], demography[6], etc. Among them, as a hot topic in the research domain of MASs, consensus has been attracted more and more attention from researchers. If one or more real or virtual leaders are introduced into the more general system with nonlinear intrinsic dynamics, the consensus of MASs can be achieved faster and easier[7], [8], [9].

In the engineering application of MASs, the nonlinear intrinsic dynamics and the parameter uncertainties may be subject to random changes in environmental disturbances. Moreover, they may occur in a probabilistic way with certain types and intensity. Very recently, the concepts of randomly occurring uncertainties (ROUs) and randomly occurring nonlinearities (RONs) were proposed to describe this phenomenon in the modeling of controlled system, which have been achieved abundant achievements in many fields, such as synchronization or stability of complex networks [10], [11], [12], consensus of multi-agent systems [13], [14], [15], etc. In fact, the dynamics of an agent may be subject to the stochastic disturbances from its own or external environmental circumstances during the working process. Therefore, taking into account the influence of uncertainties and stochastic disturbances, the study of the leader-following mean square consensus of stochastic MASs (SMASs) with ROUs and RONs is undoubtedly more practical[15], [16], [17], [18].

As a discrete control strategy, the impulsive control has many advantages such as low control cost, fast response and good adaptability, thus has been widely applied to the practical scenarios with limited communication bandwidth [19], [20], [21], [22], [23], [24]. However, it is the fact that the controlled system cannot put impulsive instants at certain time tm precisely. Therefore, the impulse appears randomly in a specific time range tm[tml,tmr] which was defined as ITW in [25], and the window must be known. Generally, the generation of impulse mainly includes two patterns: self-triggering and manually preset. It must be pointed out that the ITWs are unknown under self-triggering mechanism since the impulsive instants generated by triggering function are unpredictable. In other words, we can only fit to investigate the problem of impulsive consensus of MASs under ITWs in the pattern that impulse time sequences are put up beforehand.

Recently, many interesting results for ITWs have emerged [26], [27], [28], [29], [30], [31], [32]. In[29], the asymptotic stability of periodically multiple state-jumps impulsive control systems with ITWs at the origin has been studied. The authors assumed that there exists a period T in which one or more ITWs with fixed width and position are included. In [30], the consensus of MASs with time delay has been studied based on intermittent impulsive control. Similar to the definition of ITW, the impulse is prescribed to appear randomly in a control window with fixed width and position within the period T. Different from [29], [30], for the proposed sandwich control system in [31], the position of left and right endpoints of ITW in each period T is variable. However, this variety is non-arbitrary, which is subject to the constraint of corresponding theorem conditions. In [33], the authors studied the variable impulsive consensus of nonlinear MASs with ITWs from different perspectives, and the windows’ setting has nothing to do with period T. It can be inferred from the results of existing literatures that the allowable upper bound of width of ITW in [33] is smaller than that in [29], [30], [31], which may be conservative. However, the width and position of ITW can be adjusted at will under the configuration without period T, this is different to that in [29], [30], [31], which may be less conservative. Therefore, it is a significant work to construct a new mechanism to ensure that the position and width of ITW are arbitrarily adjustable while it has the characteristics of periodic configuration. Based on the previous work, a majorizing configuration called PVITWs is proposed in this paper. It specifies that the position of left and right endpoints of ITW are arbitrarily adjustable in each designed period T. Although many studies have been published concerning the ITWs, to the best of our knowledge, the problems of leader-following mean square consensus of nonlinear SMASs under ITWs has not been investigated yet.

Motivated by the above discussions, this paper addresses the leader-following mean square consensus of SMASs with ROUs and RONs under PVITWs. The main contributions are as follows:

  • (1) Compared with the presetting strategy of ITW in existing literatures, the configuration of PVITWs may not only makes the windows’ presetting more flexible and practical, but also makes the impulsive control more adaptable.

  • (2) Considering the general situation that ROUs, RONs and stochastic disturbances coexist at the same time, based on the existing results, our work further deepens and expands the research on consensus of MASs under ITW.

  • (3) Compared with the cases of fixed impulse time sequences without ITW and traditional configuration for ITW without period T, the larger impulsive interval can be obtained by PVITWs. Moreover, when the impulsive interval in adjacent ITWs is maximum, our results show that the width and position of corresponding ITWs can be adjusted within a certain range of value under PVITWs. However, the width of corresponding ITWs is fixed and equal to the maximum under ITWs without period T in [33], etc.

  • (4) Actually, the dynamics of system may be stable or unstable, and the impulses can destroy or benefit the stability of error system. Based on the obtained theorem conditions of this paper, the above situations are discussed in detail under the corresponding parameters, and the applicable criteria with low conservatism are given.

The rest of this paper is structured as follows: Section 2 introduces related lemmas, definitions and some basic concepts of graph theory. In Section 3, the relevant results are presented, and the effectiveness is illustrated by numerical simulations in Section 4. Section 5 gives the concise summary of this paper.Nonations: Throughout this paper, R,Rn and Rm×n represent the real numbers, the n-dimensional Euclidean space and the set of matrices with m×n order respectively. N+ is the set of positive integers. Let (Ω,F,{Ft},P) be a complete probability space with filtration {Ft}t0 which satisfy the usual conditions (i.e. F0 contains all P-null sets and Ft is right continuous). |·| and · denote the absolute value and the Euclidean norm (2-norm) respectively. denotes for all sth. is the union. E(·) denotes the expectation operator. denotes the Kronecker product. (·)T represents the transpose for a matrix. λmax(·) is the maximal eigenvalue of specific matrix. L denotes the Kolmogorov operator.

Section snippets

Algebraic graph theory

The topology of MASs is divided into digraph and undirected graph according to the different transmission directions of information exchange among agents. If the exchange between any adjacent agents is bidirectional, the topology graph is undirected. Conversely, it is a digraph, which represented by G=(Φ,ϒ,A). Among them, Φ={Φ1,,ΦN} is the node set, ϒ={(Φj,Φi):i,j=1,,N}Φ×Φ is an edge set, (Φj,Φi) represents the communication path from node Φj to Φi,A=[aij] represents a weighted adjacency

Main results

In this section, the leader-following mean square consensus issue for system (5) is discussed, and the following sufficient criteria are derived.

Theorem 1

Suppose that Assumptions 1–3 hold. If there exists a parameter v>1 such that lnvλY+λT0, then the leader-following mean square consensus of system (5) can be achieved via impulsive control protocol (4), where λ=λmax(ψ1+ψ1T+ε2θ-1MMT+θQTQ+(2Ξϕ+φ2)In) and λY=λmax(ΨTΨ).

Proof

Construct the Lyapunov function as:V(t,e(t))=eT(t)e(t).For t[(m-1)T,tm)(tm,mT),

Numerical simulation

In this section, the following examples are provided to demonstrate the effectiveness of the above theoretical results.

Example 1

Consider a class of system similar to system (8) with four followers and one leader, the following Chua circuit [36] is used as the dynamics of each agent.

ẋi1(t)=-p1xi1(t)+p1xi2(t)-p1gxi1(t),ẋi2(t)=xi1(t)-xi2(t)+xi3(t),ẋi3(t)=-p2xi2(t),where g(xi1(t))=m2xi1(t)+0.5(m1-m2)(|xi1(t)+1|-|xi1(t)-1|) and m1<m2<0. When p1=9.21,p2=15.995,m1=-1.25 and m2=-0.758, the chaotic behavior

Conclusions

The configuration of PVITWs is defined, and on the basis of it, the issue of leader-following mean square consensus of SMASs with ROUs and RONs is addressed. Dealing with the issue from different point of view, we get a series of corresponding results. Compared with the existing literatures, our obtained results are optimized to some extent, which can ensure that the setting of ITWs is more flexible in some circumstances. Therefore, the present study has some better practical significance and

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.

CRediT authorship contribution statement

Zhenhua Zhang: Conceptualization, Methodology, Software, Writing - original draft. Shiguo Peng: Resources, Writing - review & editing, Supervision, Project administration. Tao Chen: Software, Validation, Writing - review & editing.

Acknowledgement

This work was completed with the support of the National Natural Science Foundation of China (Nos. 61973092 and 61374081), the Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515012104).

Zhenhua Zhang received the B.S. degree from West Anhui University, LuAn, China, in 2015, the M.S. degree from Guangdong University of Technology, Guangzhou, China, in 2018. He is currently pursuing a Ph.D. in School of Automation at Guangdong University of Technology, Guangzhou, China. His current research interests include impulsive control theory, event-triggered control theory and consensus of stochastic multi-agent systems.

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Zhenhua Zhang received the B.S. degree from West Anhui University, LuAn, China, in 2015, the M.S. degree from Guangdong University of Technology, Guangzhou, China, in 2018. He is currently pursuing a Ph.D. in School of Automation at Guangdong University of Technology, Guangzhou, China. His current research interests include impulsive control theory, event-triggered control theory and consensus of stochastic multi-agent systems.

Shiguo Peng was born in Hunan, in 1967. He received the B.S. degree from Xiang tan University, Xiangtan, in 1989, the M.S. and Ph.D. degrees from Zhongshan University, Guangzhou, China, in 1992 and 1995, respectively. He is currently a professor in School of Automation, Guangdong University of Technology, Guangzhou. His research interests include nonlinear analysis, robust control, and stochastic systems.

Tao Chen was born in Guangdong, China. He received his B.S. degree from Zhongkai University of Agriculture and Engineering, Guangzhou, China, in 2013, the M.S. degree from Guangdong University of Technology, Guangzhou, China, in 2017. He is currently pursuing the Ph.D degree with the Faculty of Automation, Guangdong University of Technology, Guangzhou, China. His research interests include complex system control and stochastic systems.

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