Elsevier

Neurocomputing

Volume 394, 21 June 2020, Pages 13-26
Neurocomputing

Event-triggered guaranteed cost consensus for uncertain nonlinear multi-agent systems with time delay

https://doi.org/10.1016/j.neucom.2020.02.003Get rights and content

Abstract

The problem of event-triggered guaranteed cost consensus for nonlinear multi-agent systems with time delay and uncertain parameters is solved in this paper. First, a distributed event-triggered strategy is presented, by which control protocol is updated only at the triggering time instants, leading to significant saving of communication resources. Second, based on the Lyapunov method and the proposed event-triggered strategy, some criteria are obtained to ensure that guaranteed cost consensus can be achieved. Moreover, it is proved that the Zeno behavior is excluded under the proposed event-triggered scheme. Third, an algorithm is given to design suitable control gains in terms of linear matrix inequalities. Finally, the validity of the proposed results in this paper is verified via numerical simulation.

Introduction

Multi-agent systems (MASs) have a wide range of practical applications, such as formation control [1], [2], intelligent transportation [3], sensor networks [4], spacecraft rendezvous, and docking problems [5]. Thus, MASs have become a research hotspot in automatic control, artificial intelligence, and other fields [6]. In recent years, the properties of multi-agent systems have been studied extensively [7], [8]. It is well known that consensus is an important dynamic behavior of an MAS under dynamic interaction of each agent. It is a basic requirement for studying cooperation and coordination control of multi-agent systems [9]. Based on algebraic graph theory, matrix theory, together with the Lyapunov method, consensus of linear and nonlinear multi-agent systems is studied in [10]. The second-order consensus problem of multi-agent systems is considered in [11], where it is shown that consensus can be achieved by both position and velocity. Shi and Liu [12], [13] address the problem of leader-follower consensus of multi-agent systems, in which the current state can be adjusted in accordance with feedback information of adjacent followers to reduce the tracking error between leaders and followers.

However, continuous communication among agents may lead to the wastage of communication resources in practical applications. When energy and communication bandwidths are limited, event-triggered strategy is usually adopted to save communication resources [14], [15], [16]. Event-triggered strategy has advantages in reducing unnecessary sampling and transmission. An increasing number of researchers are applying event-triggered strategy to the consensus problem of multi-agent systems [17], [18], and so far, a lot of results have been published on this issue. For example, based on an event-triggered strategy, output feedback control of dynamic multi-agent systems is investigated in [19]. Under a distributed triggered scheme, consensus of nonlinear dynamic multi-agent systems is studied in [20], where the proposed event-triggered scheme can effectively reduce control updates and information transmissions. Recently, by taking network attacks into account, distributed event-triggered consensus of a nonlinear multi-agent system is considered in [21]. Moreover, various event-triggered results on multi-agent systems with uncertainties or time delay have been reported [22]. In [23] and [24], the leader-following consensus problem of multi-agent systems via event-triggered control is studied, but the leader was not controlled by the control protocol and could not receive status samples from other agents. However, in some practical situation, the leader also needs to be controlled. For instance, in the formation of aircraft, the leader aircraft needs to be controlled by the control center. Therefore, we may choose an agent as “Similar Leader”. Different from [23], [24], “Similar Leader” is controlled by a protocol like other agents and also constrained by neighbors. Under the control strategy designed, other agents can achieve consensus with “Similar Leader”. In this paper, we deal with this problem and propose a distributed event-triggered strategy to reduce the number of computations and control updates.

In the aforementioned literature, most results are focused on achieving consensus of multi-agent systems. However, in practical applications of multi-agent systems, each agent may only have limited energy supplies to carry out some tasks, such as perception, communication, and movement. Energy consumption is a real problem that should be considered critically [25]. In [26], [27], a sufficient condition on guaranteed cost and consensus is given through the trade-off between regulating performance and energy consumption. An effective control method for guaranteed cost can provide a performance index with a limited upper bound. In this situation, stability of the system under study is ensured with a bounded performance index [28]. In [29], guaranteed cost consensus has been considered for high-order multi-agent systems with switching topology, while few results have been reported on guaranteed cost of multi-agent systems under event-triggered conditions. For uncertain switched linear systems, both guaranteed cost state-feedback control [30] and event-triggered guaranteed cost problem are investigated in [30] and [31], respectively. It should be mentioned that no time delay is involved in the dynamic model involved in [30] and [31]. However, in practical applications, communication delays among agents are inevitable during information exchange. The problem of guaranteed cost is addressed for a class of high-order multi-agent systems with input delay [32], constant delay [33], and time-varying delay [34], respectively. It is also studied for second-order multi-agent systems with directed topology with or without communication delays [35]. However, those results are based on time-triggered schemes rather than event-triggered ones.

In a word, in recent years, although a number of results on guaranteed cost control have been derived for multi-agent systems with time delay, to the best of our knowledge, the problem of event-triggered guaranteed cost consensus of uncertain multi-agent systems with time delay is rarely studied, which is the main motivation of the paper. Compared with some previous relevant literature, the main contributions of this paper are summarized as follows:

  • (1)

    The problem of guaranteed cost consensus control for a class of nonlinear multi-agent systems with both uncertainties and stated-delays is addressed. To the best of the authors’ knowledge, there is few results on this topic for such systems;

  • (2)

    A new multi-agent consensus control method is proposed. In some practical cases, a multi-agent needs a leader, and this leader should be controlled as other agents so that consensus can be achieved. Bearing that in mind, we choose an agent as the “Similar Leader”, which is controlled by a protocol as other agents and also constrained by its neighbors. With such a strategy, other agents can achieve consensus with the “Similar Leader”. On the other hand, a new distributed event-triggered protocol is proposed, which can reduce the number of control updates effectively. The event-triggered protocol only needs to calculate the triggered threshold based on the estimation error of each agent rather than the sum of the errors of all neighboring agents. The triggered conditions are more accurate, and can avoid the phenomenon where some agents are out of control due to large errors.

  • (3)

    Under the proposed control protocol and distributed event-triggered strategy, sufficient conditions are derived such that the multi-agent system under study can not only achieve consensus but also meet the requirements of stability and suitable performance. Moreover, an algorithm is given to calculate the feedback gain if a number of matrix inequalities are feasible. And it is proved that there is a positive lower bound for the interval of event-triggered execution.

The structure of this paper is as follows. Section 2 illustrates the problem considered in this paper on the basis of graph theory. Section 3 analyzes the stability of multi-agent closed-loop systems via an event-triggered strategy and obtains an upper bound of the guaranteed cost function. The design conditions of the event-triggered guaranteed cost sub-controller are also given. Section 4 provides a positive lower bound on the event-triggered interval to avoid Zeno behavior. Section 5 conducts a numerical simulation to verify the theoretical results. Finally, Section 6 summarizes the paper.

The following standard notations are used in this paper: Rn × m, ‖*‖ and ⊗ represent the set of real matrices, the Euclidean norm of a vector or the derived two-norm of a matrix, and the Kronecker product, respectively. 1N ∈ RN × 1 means the N × 1 column vector of all ones; IN ∈ RN × N is an identity matrix. Υ=(υ,ϑ) denotes a graph composed of N node sets; ϑυ×υ is the set of edges of a graph, and A=(aij)N×N is the weighted collar matrix of γ. The directed edge A of (j, i) ∈ ϑ indicates that agent i can get information from agent j or that agent j can reach agent i. If agent j is a neighbor of agent i, then aij > 0; otherwise, aij=0. A topology is undirected if (i, j) ∈ ϑ↔(j, i) ∈ ϑ. D=diag{d1,d2,,dN} is a diagonal matrix with the ith element di=jNiaij, and Ni={jυ:(i,jϑ)}. Then, the Laplacian matrix L=[lij]Rn×n is usually introduced to describe the topology of the system, where lij=j=1Naij and lij=aij(ij). In addition, the eigenvalues of L are expressed as 0=λ1(L)<λ2(L)···λN(L).

Section snippets

Problem description

Consider a nonlinear multi-agent system model with N agents, in which the ith agent is described as:x˙i(t)=(A+ΔA)xi(t)+(Ad+ΔAd)xi(td)+(B+ΔB)ui(t)+f(xi(t)),where i=1,2,,N, xi(t) ∈ Rn is the system state and xi(td) is the delayed state; ui(t) ∈ Rm is the control input; f(xi(t)) represents the nonlinear dynamic function of the ith agent; A, Ad ∈ Rn × n and B ∈ Rn × m are the known constant matrices with appropriate dimensions; ΔA, ΔAd and ΔB are the uncertain matrices with appropriate

Primary results

In this section, for the multi-agent system (10), the Lyapunov method is used to study the stability of the closed-loop system under event-triggered control. The upper bound J* of the guaranteed performance function is acquired. At the same time, a design method based on LMIs is proposed to solve the control gain matrix.

Theorem 1

For the multi-agent system (10) and any i=2,3,,N, given the positive constants α, β, μ, ε, χ, ρ, if there exit symmetric positive-definite matrices P1, P2 and P3, and an

Zeno behavior exclusion

In this section, the Zeno behavior of the event-triggered strategy (8) is analyzed. It is proved that the event-triggered execution interval is greater than a positive lower bound.

Lemma 4

For the multi-agent closed-loop system (10), all solutions x(t) of the system starting from any initial conditions x(t0) are satisfactory when ϕ → 1, x(t)Zeb(tt0),t>t0, whereZ=max{κx(t0),κb(bg)(g1+g2ebd)(1ebd)bg3},and the constants κ ≥ 1, g > b > 0.

Proof

From the multi-agent system (10), we can acquire thatx˙(t)=H

Numerical simulation

In this section, a numerical example is given to verify the theoretical results. The undirected and connected topology of a multi-agent system with 10 nodes shown in Fig. 1 shows, where the Laplacian matrix L and other system parameters are as follows:L=[4100111000141000110001410001100013000011100021000010001310001100014100011000141000110001410001000012],A=Ad=[101111011110],B=[811],f(xi)=[000.2sin(xi1)],F1=[333333333],F2=[222],F3=[111111111],G1=[120212

Conclusion

Event-triggered guaranteed cost consensus has been studied for multi-agent nonlinear systems with time delay and uncertainty. An event-triggered strategy has been introduced to save communication resources. Only when the event-triggered condition is satisfied can the data be sampled by the system. By employing the Lyapunov method, a sufficient condition on stability and guaranteed cost has been obtained. Based on this condition, event-triggered control protocol can be devised in terms of the

CRediT authorship contribution statement

Yiping Luo: Conceptualization, Formal analysis, Methodology, Project administration, Resources, Funding acquisition, Writing - review & editing, Supervision. Xing Xiao: Data curation, Methodology, Writing - original draft, Software, Validation. Jinde Cao: Visualization, Investigation, Supervision. Anping Li: Investigation, Software, Validation.

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. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Yiping Luo received the Ph.D. degree in control theory and control engineering from South China University of Technology, Guangzhou, China, in 2006. He is Professor with the Hunan Institute of Engineering, Hunan, China. And he is also the academic leader of Hunan Province. He was appointed the New-Century 121 Talent Program of Hunan Province. His current research interests include complex networks and multi-agent systems.

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    Yiping Luo received the Ph.D. degree in control theory and control engineering from South China University of Technology, Guangzhou, China, in 2006. He is Professor with the Hunan Institute of Engineering, Hunan, China. And he is also the academic leader of Hunan Province. He was appointed the New-Century 121 Talent Program of Hunan Province. His current research interests include complex networks and multi-agent systems.

    Xing Xiao was born in Loudi, Hunan, China in 1995. Master student in power engineering, Hunan Institute of Engineering. She received her Bachelor degree in College of Information Science and Engineering from Jishou University in 2018. Her main research interests include multi-agent systems and event-triggered control.

    Jinde Cao is currently a Distinguished Professor, the Dean of the Department of Mathematics, and the Director of the Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing, China. Mr.Cao is a member of the Academy of Europe. He was an Associate Editor of the IEEE Transactions On Neural Networks, the Journal of the Franklin Institute, and the Neurocomputing. He is an Associate Editor of the IEEE Transactions On Cybernetics, the Differential Equations and Dynamical Systems, the Mathematics and Computers in Simulation, and the Neural Networks. He has been named as a Highly Cited Researcher in mathematics, computer science and engineering by Thomson Reuters. He received the National Innovation Award of China (2017).

    Anping Li received the M.S. degree in system theory from Kunming University of Science and Technology, Kunming, China, in 2007 and received the Ph.D. degree in control theory and control engineering from Hunan University, Changsha, China, in 2018. He is currently a lecturer with Hunan Institute of Engineering, Xiangtan, China. His current research interests include multi-agent systems, intelligent control, and robust control.

    This work was jointly supported by the National Natural Science Foundation of China (Grant no. 11972156), the Natural Science Foundation of Hunan Province (Grant no. 2017JJ4004), the Hunan Provincial Innovation Foundation For Postgraduate (CX20190958).

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