Event-triggered consensus of nonlinear multi-agent systems with stochastic switching topology

Accepted Manuscript

Event-triggered consensus of nonlinear multi-agent systems with stochastic switching topology Lei Liu, Jinjun Shan PII: DOI: Reference:

S0016-0032(17)30276-4 10.1016/j.jfranklin.2017.05.041 FI 3015

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

19 November 2015 25 May 2017 28 May 2017

Please cite this article as: Lei Liu, Jinjun Shan, Event-triggered consensus of nonlinear multiagent systems with stochastic switching topology, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.05.041

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Highlights • Further explanation is included after Eq. (9) to clarify the imperfect communication network.

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• Some typos have been found and corrected.

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Lei Liu∗ Jinjun Shan

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Event-triggered consensus of nonlinear multi-agent systems with stochastic switching topology Department of Earth and Space Science and Engineering, York University

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4700 Keele St., Toronto, Canada, M3J 1P3

Abstract

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This paper is concerned with a leader-follower consensus problem for networked Lipschitz nonlinear multi-agent systems. An event-triggered consensus controller is developed with the consideration of discontinuous state feedback. To further enhance the robustness of the proposed controller, modeling uncertainty and switching topology are also considered in the stability analysis. Meanwhile, a time-delay equivalent approach is adopted to deal with the discrete-time control problem. Particularly, a sufficient condition for the stochastic stabilization of the networked multi-agent systems is proposed based on the Lyapunov functional method. Furthermore, an optimization algorithm is developed to derive the parameters of the controller. Finally, numerical simulation is conducted to demonstrate the effectiveness of the proposed control algorithm.

Keywords: Event-triggered, sampled-data, leader-follower, consensus, Markovian switching topol-

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ogy

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Multi-agent systems have attracted great interests due to their potential applications in a variety

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of areas. In multi-agent systems, consensus seeking algorithm can be considered as one of the most

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Introduction

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crucial issues [1–4]. Consensus has been investigated tremendously in computer science [5], physics [6] and management science [7]. In control area, consensus seeking algorithm has been studied in multiple missions, i.e. formation control [8], rendezvous [9], flocking [10]. ∗ †

Currently a Post-doctoral Fellow at Ryerson University, Email: [email protected] Corresponding author, Professor, Tel: +1-416-7362100 ext. 33854, Email: [email protected]

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In classical single agent system, the system output are expected to converge to a desired tra-

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jectory in the tracking problem. Unlike in single agent system, consensus is the ultimate goal that

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multi-agent systems are supposed to achieve. In multi-agent systems, multiple agents are coupled

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together through the wireless network. To accomplish a common mission, their motion should be

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synchronized to reach the common goal cooperatively. Once their motion is synchronized and the

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networked agents achieve their common mission cooperatively, the consensus is achieved by the

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multi-agent systems. Therefore, compared to the conventional single agent stability, consensus can

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be considered as a specific type of stability for multi-agent systems. In order to achieve the group

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mission, consensus seeking algorithms are applied to each agent in the multi-agent systems.

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Among different types of consensus seeking algorithms, leader-follower consensus algorithm is

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particularly interesting and received broad attention. In previous research on leader-follower con-

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sensus, it is usually assumed that agents exchange information continuously through the coupling

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network [2]. However, it is most likely in practice that information sharing can only take place at

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discrete instants since the bandwidth of the coupling network is limited. With the appearance of

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sampled-data information exchange, the leader-follower consensus problem was investigated in [11].

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In their work, the M-matrix theory is applied to derive the sufficient conditions for system stability,

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while the velocity and acceleration of the leader are unavailable for the controller. Furthermore, the

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stable sampling period can be indicated based on their results. Although the periodically sampling

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strategy can effectively reduce the network consumption, the control output still has to be updated

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periodically. Namely, each agent is still computing the output value periodically though it might

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be unnecessary. To further reduce the computational burden, event-triggered control strategy was

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proposed in [12] for first-order consensus problem. The event-triggered conditions were proposed for

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both centralized and distributed situations in their work. Moreover, the self-triggered multi-agent

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control protocol was proposed to relax the trigger condition. The event-triggered control algorithm was extended to the second-order multi-agent systems in [13]. Particularly, Lipschitz nonlinearity

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was considered in their work because the nonlinear dynamics are almost unavoidable in practice.

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The leader-follower consensus problem for Lipschitz nonlinear multi-agent systems was also consid-

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ered in [14], where the jointly connected topology was assumed for the coupling relationship. The

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leader-follower consensus problem for second-order nonlinear multi-agent systems was investigated

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in [15] with a specific type of nonlinearity. In their work, the stability analysis was conducted on the

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basis of LaSalle’s invariance principle. Further, by taking advantage of M-matrix method and the

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property of nonnegative matrices, the second-order nonlinear multi-agent systems were also investi-

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gated in [16], and it was extensively proven that the leader-follower consensus can be reached easier

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with higher pinning feedback gains. The leader-follower consensus with uncertain Euler-Lagrange

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systems was studied in [17]. In the appearance of the switching communication interaction, the

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convergence of the error systems was guaranteed by their distributed adaptive controller. Moreover,

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the communication topology in their work is not necessarily connected all the time. Specifically,

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the time-varying topology in consensus problem was also widely investigated [4, 18] because it is

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fairly more generic. Ref. [19] conducted the research on Markovian switching topology for second-

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order multi-agent systems, and a necessary and sufficient condition for consensus achievement was

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presented in their work. Markovian switching topology was also considered in [20], where the

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leader-follower consensus problem was investigated with the consideration of nonlinear dynamics

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and communication delay. Furthermore, Ref. [21] discussed the leader-follower consensus with

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switching topology for general linear agent, and the convergence of the closed-loop control system

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was proven along the Riccati-inequality-based approach. With the consideration of the switching

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topology, the leader-follower consensus control was investigated on the basis of the discrete-time

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multi-agent systems in [22]. Both fixed and switched topologies were considered in [23] with a

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globally reachable leader approach. In their work, finite-time convergent leader-follower consensus

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problem was studied and the second-order consensus was successfully reached. Ref. [24] further

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extended the leader-follower consensus algorithm to second-order nonlinear multi-agent systems

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with both fixed and time-varying communication topologies. A large class of nonlinear dynamics was dealt with in their work, and the leader-follower consensus was achieved with the intermittent information exchange. In this work, the sampled-data communication is considered in the stability analysis because

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the networked agents can only share the information intermittently, not continuously. Therefore,

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the entire multi-agent system is essentially a discrete-time dynamical system. To effectively resolve

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the discrete-time control problem, the time-delay equivalent method [25] is adopted in this work to

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convert the discrete-time control problem into a continuous-time issue. Apparently, the sampled-

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data communication can only reduce the network burden. To further reduce the computational load

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of each agent, an event-triggered control strategy is developed and the event-triggered condition is

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proposed in the inequality form. The control input signal is generated only if the event-triggered

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condition is violated, thus the agent actuator does not have to be updated periodically. Namely,

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the computational load of each agent has been effectively reduced because the on-board processor is

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always available for other computational work if the event-triggered condition is satisfied. Further-

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more, the stochastic switching communication topology is considered in this paper, and the finite

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Markov jump process is recruited to describe the interaction switching of the multi-agent systems.

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In order to enhance the robustness of the proposed controller, modeling uncertainty is also consid-

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ered in this work. Since the nonlinear term in the dynamics of single agent might not be precisely

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replicated in the stability analysis, the modeling uncertainty is included in the error dynamics of

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the multi-agent systems. In the stability analysis, the modeling uncertainty is systematically inves-

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tigated, and it is proven that the stability of the networked system can be guaranteed even with

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the appearance of bounded system uncertainty.

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The remainder of this paper is organized as follows. In Section 2, the nonlinear dynamics of the

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multi-agent systems and the error dynamics are formulated. Meantime, the mathematical descrip-

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tion of the interaction relationship between agents is essentially explained using graph theory and

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Markov jump process. Moreover, an event-triggered condition is proposed to reduce the compu-

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tational burden of the multi-agent systems. To further clarify the stability of the error dynamics,

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the stochastic stability is formally defined as well. In Section 3, three assumptions are proposed to clearly claim the communication structure. Based on the three assumptions, the controller de-

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sign and stability analysis are systematically presented with the assistance of Lyapunov functional

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method. Subsequently, the sufficient condition for the convergence of the error dynamics are de-

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rived on the basis of the results in the stability analysis. Moreover, an iterative convex optimization

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algorithm is developed to derive the controller gain. In Section 4, four networked Chua’s circuits

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are used in the numerical simulations. The leader-follower consensus is finally achieved by the four

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Chua’s circuits in the occurrence of stochastic switching interaction. It is shown in the simulation

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that all the tracking errors converge to zero with the appearance of system uncertainties, which

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further demonstrates the effectiveness of the proposed controller. Section 5 concludes this paper.

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Notation: The notations adopted in this work are fairly standard. Rn denotes the n-dimensional

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Euclidean space. Identity matrix and zero matrix with appropriate dimensions are represented using

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I and 0, respectively. The superscript “T ” is used to indicate the matrix transpose, and M > 0

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represents a positive definite matrix. In symmetric matrix, “∗” indicates the entry implied by the

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symmetry.

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Problem formulation

A distributed leader-follower consensus seeking problem is investigated in this work, and k ∈ N

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nonlinear agents are included in the multi-agent systems. The dynamics of each nonlinear agent

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can be described as follows

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x˙ i (t) = Axi (t) + Bf (xi (t)) + ui (tu )

(1)

where xi (t) ∈ Rn is the state vector of agent i, ui (tu ) ∈ Rn represents the control input of agent i,

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A ∈ Rn×n , B ∈ Rn×n are system matrices and nonlinear term f (xi (t)) ∈ Rn satisfies the Lipschitz

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condition, namely, the following inequality is true for any vectors a ∈ Rn and b ∈ Rn

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where α > 0 is the Lipschitz constant. The desired trajectory is generated by a self-driven nonlinear agent with the following dynamics x˙ 0 (t) = Ax0 (t) + Bf (x0 (t))

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[f (a) − f (b)]T [f (a) − f (b)] ≤ α2 (a − b)T (a − b)

where x0 (t) ∈ Rn is the state vector of the desired trajectory. 6

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Remark 1. Linear dynamics are usually studied in previous works [13, 18, 22, 26–28]. In this work,

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Lipschitz nonlinearity is considered in the single agent dynamics, because a large class of nonlinear

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behaviors in mechanical/electrical systems can be described using Lipschitz nonlinearity. Hence,

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once the consensus seeking problem for networked Lipschitz nonlinear systems is solved, a large

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class of consensus seeking problems for networked mechanical/electrical systems have been resolved

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fundamentally.

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Remark 2. Initial condition of Eq. (1) can be freely selected in practice. It can be observed in

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stability analysis (in Section 3) that the Lyapunov functional method is adopted and the initial

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condition of Eq. (1) does not appear explicitly. Therefore, any practical-related initial condition will

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be within the stability domain.

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Remark 3. The control algorithm proposed in this work is characterized by a distributed structure.

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Namely, all the nonlinear agents are coupled locally through a wireless network, and there is not a

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central controller in the multi-agent systems. The decentralized structure has inherently guaranteed

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the low communication burden for each agent. Therefore, neither the communication nor compu-

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tation burden will be increased significantly with the growth of agent number. In this situation, the

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agent number k can be arbitrarily selected without the limitation of upper bound.

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Since the multiple agents are coupled through digital network, it is essential to describe the

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network structure properly. In this work, the communication relationship is mathematically depicted

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using algebraic graph theory. Consequently, each agent is represented by a vertex vi ∈ X , where

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X ∈ X(Rn ) is the set of the vertices. The graph corresponding to a group of agents can be denoted

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by G(X , E), where E(X ) ⊆ X × X is the edge set. The information can be shared between two

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agents if there is a communication channel connecting them, and this information sharing direction

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is indicated by the edge between them. The vertices connected to vi is considered as the neighbors of vi , and the neighbor set can be defined by NG (vi ) = {vj : (vi , vj ) ∈ E(X ∪ {vi })} [29]. The leader

set is defined as L0 , namely, if a vertex vi has access to the desired trajectory, then vi ∈ L0 . The

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graph Laplacian associated with the graph G is defined as (4)

L=H−A

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where H is the degree matrix and A is the adjacency matrix. The following event-triggered control algorithm is considered m(t)

ui (tu ) = Ki

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vj ∈NG (vi ) m(t)

m(t) m(t) pi [xi (tu )

[xi (tu ) − xj (tu )] + Ki

− x0 (tu )]

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(5)

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where Ki

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time instants tu , and m(t) is a finite Markov jump process. The value of m(t) is assigned from a

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finite set.

The transition probability from m(t) = i to m(t) = j is defined as

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∈ Rn×n , tu represents the update instant, i.e. ui (tu ) only updates its value at discrete-

P r {m(t + ) = j|m(t) = i} =



pij  + o() i 6= j 1 + pii  + o() i = j

(6)

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the consideration of controller in Eq. (5), error dynamics of the closed-loop control system can be

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described by the following equation

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Subtracting the leader’s dynamics in Eq. (3) from the dynamics of agent i in Eq. (1) with

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where  is a small positive parameter and o() is a term that is decreasing faster than , i.e. P o()/ → 0. The transition rate pii and pij ≥ 0 satisfy j=1,j6=i pij = −pii .

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e˙ i (t) = Aei (t) + Bf (xi (t), x0 (t)) + ui (tu )

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where ei (t) = xi (t) − x0 (t) and f (xi (t), x0 (t)) = f (xi (t)) − f (x0 (t)). Furthermore, the lumped form

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of the error dynamics of all the agents can be expressed as    e˙ 1 (t) Ae1 (t) + Bf (x1 (t), x0 (t)) + u1 (tu )  e˙ 2 (t)   Ae2 (t) + Bf (x2 (t), x0 (t)) + u2 (tu )     ..  =  ..  .   .

ek (t)



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  = (Ik ⊗ A)  

e1 (t) e2 (t) .. . ek (t)



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     + (Ik ⊗ B)   

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e1 (tu ) e2 (tu ) .. .

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  L ⊗ In )  +Km(t) (L 

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    +  

u1 (tu ) u2 (tu ) .. . uk (tu )

    



ek (tu )



f (x1 (t), x0 (t)) f (x2 (t), x0 (t)) .. .

f (xk (t), x0 (t))

      

    

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     m(t) m(t) D ⊗ I + K  n   

e1 (tu ) e2 (tu ) .. . ek (tu )

    

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n o n o m(t) m(t) m(t) m(t) m(t) m(t) where D m(t) = diag p1 , p2 , . . . , pk and Km(t) = diag K1 , K2 , . . . , Kk . Fur-

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ther details about the derivation with Kronecker product and Laplacian matrix involved can be found in the Appendix.

Ideally, Eq. (8) is the error dynamics of multi-agent systems with respect to the desired tra-

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f (xk (t), x0 (t))

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f (x1 (t), x0 (t)) e1 (t)  f (x2 (t), x0 (t))   e2 (t)      = (Ik ⊗ A)  .  + (Ik ⊗ B)   ..    ..  . f (xk (t), x0 (t)) ek (t)  m(t) m(t) m(t) P p1 [x1 (tu ) − x0 (tu )] K1 vj ∈NG (v1 ) [x1 (tu ) − xj (tu )] + K1  m(t) P m(t) m(t)  K p2 [x2 (tu ) − x0 (tu )]  2 vj ∈NG (v2 ) [x2 (tu ) − xj (tu )] + K2 + .  ..  P m(t) m(t) m(t) pk [xk (tu ) − x0 (tu )] Kk vj ∈NG (vk ) [xk (tu ) − xj (tu )] + Kk

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   

Aek (t) + Bf (xk (t), x0 (t)) + uk (tu )    e1 (t) f (x1 (t), x0 (t))  e2 (t)   f (x2 (t), x0 (t))    = (Ik ⊗ A)  .  + (Ik ⊗ B)  .. .  .   . 

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follows 

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e˙ k (t)

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jectory. However, imperfect communication network is always unavoidable due to time-varying disturbance and other uncertainties. Moreover, the nonlinear term f (xi (t)) and f (x0 (t)) may be in-

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accurate because of perturbations and unmodeling effects. Therefore, the compact form of the error

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dynamics is reformulated in Eq. (9) with the appearance of the stochastic switching communication

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topology and modeling uncertainty. ˙ e(t) = (Ik ⊗ A) e(t) + (Ik ⊗ B) (I + ∆ ) ¯f (x(t), x0 (t)) + Km(t)

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T  eT1 (t) eT2 (t) . . . eTk (t) , ¯f (x(t), x0 (t)) = f T (x1 (t), x0 (t)) , f T (x2 (t), x0 (t)) , T . . . , f T (xk (t), x0 (t)) , L m(t) represents a group of Laplacian matrices, and they are switched

where e(t) =



h  i L m(t) + D m(t) ⊗ In e(tu )(9)

according to the finite Markov jump process m(t). ∆ represents the modeling uncertainty which

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∆k < δ, and δ is an arbitrary positive constant. satisfies k∆

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Remark 4. The consensus seeking problem has been widely investigated in the previous works

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[13, 14, 18–20, 26–28]. However, it is usually assumed in their works that the dynamics of agents

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are precisely derived, thus system uncertainties are usually ignored. In contrast, the system un-

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certainties are essentially considered in this work. Uncertainty always exists in practice because of

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external/internal perturbations, modeling errors or other unmodeled effects. In order to enhance

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the robustness of the proposed control algorithm, system uncertainty is greatly considered in the

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event-triggered consensus algorithm design and the robustness against bounded system uncertainty

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is demonstrated in the stability analysis. This can be considered as one of the main contributions in

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this paper.

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Remark 5. As mentioned above, the modeling uncertainty is assumed to be bounded by a positive

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∆k < δ. Due to the diversity of mechanical/electrical systems, it is impossible to constant, i.e. k∆

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propose a unified method to estimate the boundary of all uncertainties. In practice, this boundary

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can be evaluated based on the characteristics of specific control systems. In particular, trial method

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or frequency domain approach can be utilized in some practical applications [30–33].

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It is commonly assumed in previous works [1, 2] that all the agents exchange information con-

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tinuously. However, it is most likely in practice that agents can only receive data package discontinuously through the digital network. Therefore, the periodically sampling communication is taken

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into account in this work. Meanwhile, to further reduce the computational load, an event-triggered

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manner is investigated for the multi-agent systems. In event-triggered control algorithm, the control

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signal is generated only if the specific event-triggered condition is violated. Obviously, the compu-

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tational burden is dramatically reduced by the event-triggered controller because the control signal

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does not have to be generated in each sampling period. Since the communication is still conducted

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periodically, the event-triggered condition will be verified periodically but the control signal will be

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calculated only if it is necessary. Motivated by [28], the event-triggered condition is designed as

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follows m(t)

σ1 eTi (ts )Pi

m(t)

ei (ts ) > rTi (ts )Pi

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ri (ts )

(10)

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where ts is the periodically sampled time instant, ri (ts ) = ei (ts ) − ei (tu ), σ1 < 1 is a positive

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constant and Pi

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Remark 6. It has been observed in Eq. (5) that the control input ui (tu ) is updated only at instants

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tu . Here, tu is determined by the event-triggered condition in (10). If the event-triggered condition

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is satisfied by ri (ts ) and ei (ts ), the control input ui (tu ) will remain to be the same value; otherwise,

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the the control input ui (tu ) will be updated according to the algorithm in Eq. (5). One of the most

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important advantages of event-triggered algorithm is that it will effectively reduce the burden of the

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on-board processor. For example, in the conventional consensus problem, each agent has to update

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the control input continuously to achieve the consensus with its neighbors, i.e. the computational

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work is being conducted since the mission begins. In contrast, each agent only updates the control

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input when it is necessary in the event-triggered consensus problem. If the control input does not

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need to be calculated, then the on-board processor will be available for other works.

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Remark 7. The consensus seeking problem for networked multi-agent systems has been studied

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in several previous works [11, 19, 20, 27, 34] with the consideration of sampled-data information

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exchange. As explained above, each agent has to update the control signal periodically even when

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is the weight matrix.

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m(t)

it is not necessary in the sampled-data information exchange strategy. In contrast, the control signal in this work will be updated only if the event-triggered condition is violated. Namely, in the

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control strategy proposed in this paper, the computational burden is greatly reduced compared to the

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controllers in the previous works [11, 19, 20, 27, 34]. This is one of the major advantages of the

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controller proposed in this work. 11

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In the consensus seeking mission, the desired trajectory, which is generated by a central work-

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station, is transmitted to the agents in L0 directly. As for any agent vi ∈ X \L0 , it has no direct

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connection with the workstation and they can only exchange information with vj ∈ NG (vi ). Unlike the continuous-time dynamical system, the control system in this paper is a stochastic

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switching system. Hence, the definition of the stability for Markovian jump system in Eq. (9) is

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presented as follows

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Definition 1. [35] Markovian jump system in Eq. (9) is stochastically stable if the following con-

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dition is satisfied lim E

t→∞

Z

0

t

 e (t)e(t)dt < ∞ T

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Based on the definition of the stability of Markovian jump system in Eq. (9), the consensus of

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the networked control system in Eq. (1) can be defined as

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Definition 2. The consensus of the networked control system in Eq. (1) is considered to be achieved

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by the control algorithm in Eq. (5) if Markovian jump system in Eq. (9) is ensured to be stochastically

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stable for any initial condition.

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The main objective of this paper is to develop a consensus seeking algorithm for the networked

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nonlinear multi-agent systems in Eq. (1). Basically, the control algorithm is expected to be in the

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form of Eq. (5), and an iterative algorithm will be proposed to numerically derive the feedback gain

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Km(t) .

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Assumption 1. The communication interaction can be represented by a digraph containing a span-

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ning tree, and each leader is located at the root of the spanning tree.

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Stability analysis

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Assumption 2. The agent information is shared intermittently, and the control signals are updated

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when the event-triggered condition in Eq. (10) is violated.

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Assumption 3. The communication topology is stochastically switched among q structures, where q

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is a finite number. The switching can be mathematically described by a finite Markov jump process. 12

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Lemma 1. [36] Let Y be a symmetric matrix and A, B be matrices with compatible dimensions

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and F satisfying FT F ≤ I. Then, Y + AFB + BT FT AT < 0 holds if and only if there exists a

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scalar ε > 0 such that Y + εAAT + ε−1 BT B < 0.

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Theorem 1. Suppose that the communication topology of the nonlinear multi-agent systems in

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Eq. (1) and the information sharing satisfy Assumptions 1 - 3, then the leader-follower consensus

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of the networked multi-agent systems in Eq. (1) can be achieved by the control algorithm presented

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in Eq. (5) if there exist symmetric matrices Qr > 0, r = 1, . . . , q, Ri > 0, i = 1, 2, Pm(t) = n o m(t) m(t) m(t) diag P1 , P2 , . . . Pk , positive scalars εj , j = 1, . . . , 5 and matrix W such that

and



 Φ 2 hW Ψ 4  ∗ −hR1 Ψ 5  < 0 ∗ ∗ Ψ6

(12)

(13)

CE

PT

ED

M

where h is the step size of the intermittent communication, σ1 , σ2 are arbitrary positive constants,

AC

240

 Φ1 hNT Ψ1  ∗ −hR−1 Ψ 2  < 0 1 ∗ ∗ Ψ3

AN US



239

CR IP T

231

13

ACCEPTED MANUSCRIPT

241

and Φ 1 =MT1

q X i=1

pri Qi M1 + MT1 Qr N + NT Qr M1 − MT1 R2 M1 − MT2 R2 M2 + MT1 R2 M2

+ MT2 R2 M1 + hMT1 R2 N + hNT R2 M1 − hMT2 R2 N − hNT R2 M2 + WM1 + MT1 WT − WM2 − MT2 WT + α2 σ2 MT1 M1 − σ2 MT3 M3 + σ1 MT2 Pm(t) M2 − MT4 Pm(t) M4 q X i=1

pri Qi M1 + MT1 Qr N + NT Qr M1 − MT1 R2 M1 − MT2 R2 M2 + MT1 R2 M2

CR IP T

Φ 2 =MT1

+ MT2 R2 M1 + WM1 + MT1 WT − WM2 − MT2 WT + α2 σ2 MT1 M1 − σ2 MT3 M3 + σ1 MT2 Pm(t) M2 − MT4 Pm(t) M4

Ψ2 =

 

δMT1 Qr NMT3

δε1 MT3

δhMT1 R2 NMT3

δε2 MT3

δhMT2 R2 NMT3 δε3 MT3

0 0 0 0 0 0 δhNMT3

0



AN US

Ψ1 =

0 δε4 MT3



Ψ 3 =diag {−δε1 I, −δε1 I, −δε2 I, −δε2 I, −δε3 I, −δε3 I, −δε4 I, −δε4 I}

Ψ5 =

 

δMT1 Qr NMT3 0 0

δε5 MT3



M3 = M4 = N=

243

244

 

0 I 0 0

 

0 0 I 0



0 0 0 I





Ik ⊗ A Km(t)




 
  L m(t) + D m(t) ⊗ In Ik ⊗ B −Km(t) L m(t) + D m(t) ⊗ In

Remark 8. The time-delay equivalent method is adopted in this work. This method was originally

AC

242



I 0 0 0

PT

M2 =



CE

M1 =

ED

Ψ6 =diag {−δε5 I, −δε5 I}



M

Ψ4 =

developed in [25]. Based on their work, a sufficient condition for sampled-data stabilization of linear systems was proposed in linear matrix inequalities (LMIs) form in [37] along the descriptor approach.

245

To further enhance the theoretical foundation, a discontinuous Lyapunov functional method was

246

presented in [38], based on which the exponential convergence of the sampled-data control system was

247

further investigated in [39] using the discontinuous Lyapunov functional method. The essential part 14

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248

of this method is to recruit an artificial time-delay d(t) so that the sampling time ts is equivalently

249

converted to ts = t − d(t) in each sampling period, which implies that the original discontinuous

250

control problem is transformed to a continuous control problem with a time-varying delay.

251

Proof. Defining the Lyapunov functional T

V (m(t), r) =e (t)Qr e(t) +

Z

t

t−d(t)

˙ )dτ [h − d(t)] e˙ T (τ )R1 e(τ

(14)

T

The weak infinitesimal operator F of the stochastic process {m(t)} is defined as FV (m(t)) = lim

→0+

253

254

E {V (m(t + ))} − V (m(t)) 

To better clarify the derivation, the Lyapunov functional will be separated into three parts and expressed as follows

AN US

252

CR IP T

+ [h − d(t)] [e(t) − e(ts )] R2 [e(t) − e(ts )]

V1 (m(t), r) =eT (t)Qr e(t) Z t ˙ )dτ V2 (m(t), r) = [h − d(t)] e˙ T (τ )R1 e(τ t−d(t)

Consequently,

FV1 (m(t), r) =eT (t)

q X

M

255

V3 (m(t), r) = [h − d(t)] [e(t) − e(ts )]T R2 [e(t) − e(ts )] ˙ pri Qi e(t) + 2eT (t)Qr e(t)

i=1

T

ED

˙ − FV2 (m(t), r) = [h − d(t)] e˙ (t)R1 e(t)

t

˙ )dτ e˙ T (τ )R1 e(τ

t−d(t)

˙ FV3 (m(t), r) = − [e(t) − e(ts )] R2 [e(t) − e(ts )] + 2 [h − d(t)] [e(t) − e(ts )]T R2 e(t)

Therefore,

CE

FV (m(t), r) = FV1 (m(t), r) + FV2 (m(t), r) + FV3 (m(t), r) = eT (t)

AC

256

PT

T

Z

−

Z

q X i=1

t

t−d(t)

˙ ˙ + [h − d(t)] e˙ T (t)R1 e(t) pri Qi e(t) + 2eT (t)Qr e(t)

˙ )dτ − [e(t) − e(ts )]T R2 [e(t) − e(ts )] e˙ T (τ )R1 e(τ

˙ +2 [h − d(t)] [e(t) − e(ts )]T R2 e(t) T

= e (t) −

Z

q X i=1

t

t−d(t)

˙ + [h − d(t)] e˙ T (t)R1 e(t) ˙ pri Qi e(t) + 2eT (t)Qr e(t)

˙ )dτ − eT (t)R2 e(t) − eT (ts )R2 e(ts ) e˙ T (τ )R1 e(τ

˙ − 2 [h − d(t)] eT (ts )R2 e(t) ˙ +2eT (t)R2 e(ts ) + 2 [h − d(t)] eT (t)R2 e(t) (15) 15

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257

258

On the basis of the Newton-Leibniz formula, the following equation is obtained with a free weight matrix W ∈ R4kn×kn 2ξξ T We(t) − 2ξξ T We(ts ) − 2ξξ T W 

eT (t) eT (ts ) ¯f T (x(t), x0 (t)) rT (ts )

T

Z

t

˙ )dτ = 0 e(τ

(16)

ts

where ξ =

260

Remark 9. Reducing the conservativeness is always an important expectation in the investigation

261

on stability condition. In this work, sufficient conditions are derived based on Lyapunov theory.

262

Therefore, the more conservativeness can be reduced, the more generic results can be derived fun-

263

damentally. In order to reduce the conservativeness in the stability analysis, more freedom will be

264

included in the inequality. Motivated by the previous work [34, 37, 38], the free weight matrix is

265

incorporated to increase the freedom of the final results. Consequently, the free weight matrix W

266

will appear in the final sufficient condition to render more flexibilities.

AN US

Eq. (15) can be further manipulated by considering Eq. (16) as follows FV (m(t), r) = e (t) t

i=1

t−d(t)

˙ + [h − d(t)] e˙ T (t)R1 e(t) ˙ pri Qi e(t) + 2eT (t)Qr e(t)

˙ )dτ − eT (t)R2 e(t) − eT (ts )R2 e(ts ) e˙ T (τ )R1 e(τ

ED

−

Z

q X

M

T

˙ − 2 [h − d(t)] eT (ts )R2 e(t) ˙ +2eT (t)R2 e(ts ) + 2 [h − d(t)] eT (t)R2 e(t) Z t ˙ )dτ e(τ +2ξξ T We(t) − 2ξξ T We(ts ) − 2ξξ T W ts

PT = eT (t)

q X i=1

˙ + [h − d(t)] e˙ T (t)R1 e(t) ˙ pri Qi e(t) + 2eT (t)Qr e(t)

CE

−eT (t)R2 e(t) − eT (ts )R2 e(ts ) + 2ξξ T We(t) − 2ξξ T We(ts )

AC

267

.

CR IP T

259

˙ − 2 [h − d(t)] eT (ts )R2 e(t) ˙ +2eT (t)R2 e(ts ) + 2 [h − d(t)] eT (t)R2 e(t) Z t  T   T T T ˙ ) R−1 ˙ ) dτ (17) +d(t)ξξ T WR−1 W ξ + R1 e(τ W ξ + R1 e(τ 1 W ξ − 1 ts

16

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268

Subsequently, the following inequality is equivalent to FV (m(t), r) < 0 ξ T (t)MT1

q X

pri Qi M1ξ (t) + ξ T (t)MT1 Qr Nξξ (t) + ξ T (t)NT Qr M1ξ (t)

i=1

+ξξ T (t)MT1 Qr NMT3 ∆ M3ξ (t) + ξ T (t)MT3 ∆ T M3 NT Qr M1ξ (t) + [h − d(t)] ξ T (t) N + NMT3 ∆ M3


T


R1 N + NMT3 ∆ M3 ξ (t)

CR IP T

−ξξ T (t)MT1 R2 M1ξ − ξ T (t)MT2 R2 M2ξ + ξ T (t)MT1 R2 M2ξ (t) + ξ T (t)MT2 R2 M1ξ (t)


+2 [h − d(t)] ξ T (t)MT1 R2 N + NMT3 ∆ M3 ξ (t) + ξ T (t)WM1ξ (t) + ξ T (t)MT1 WT ξ (t)
−2 [h − d(t)] ξ T (t)MT2 R2 N + NMT3 ∆ M3 ξ (t) − ξ T WM2ξ (t) − ξ T MT2 WT ξ (t) T +d(t)ξξ T WR−1 1 W ξ <0

270

Further taking advantage of inequalities (2) and (10), the following equivalent condition can be

AN US

269

(18)

obtained ξ

T

(t)MT1

pri Qi M1ξ (t) + ξ T (t)MT1 Qr Nξξ (t) + ξ T (t)NT Qr M1ξ (t)

i=1 T (t)M1 Qr NMT3 ∆ M3ξ (t)

+ ξ T (t)MT3 ∆ T M3 NT Qr M1ξ (t)
T


R1 N + NMT3 ∆ M3 ξ (t)

M

+ξξ

T

q X

+ [h − d(t)] ξ T (t) N + NMT3 ∆ M3

ED

−ξξ T (t)MT1 R2 M1ξ − ξ T (t)MT2 R2 M2ξ + ξ T (t)MT1 R2 M2ξ (t) + ξ T (t)MT2 R2 M1ξ (t)


+2 [h − d(t)] ξ T (t)MT1 R2 N + NMT3 ∆ M3 ξ (t) + ξ T (t)WM1ξ (t) + ξ T (t)MT1 WT ξ (t)

PT


−2 [h − d(t)] ξ T (t)MT2 R2 N + NMT3 ∆ M3 ξ (t) − ξ T WM2ξ (t) − ξ T MT2 WT ξ (t) +α2 σ2ξ T (t)MT1 M1ξ (t) − σ2ξ T (t)MT3 M3ξ (t) + σ1ξ T (t)MT2 Pm(t) M2ξ (t)

272

(19)

where σ2 is an arbitrary positive constant.

AC

271

CE

T −ξξ T (t)MT4 Pm(t) M4ξ (t) + d(t)ξξ T (t)WR−1 1 W ξ (t) < 0

Since the left hand side of Eq. (19) is a linear polynomial of d(t), the following inequalities can

17

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273

be derived by setting d(t) = 0 and d(t) = h, respectively. MT1

q X

pri Qi M1 + MT1 Qr N + NT Qr M1 + MT1 Qr NMT3 ∆ M3 + MT3 ∆ T M3 NT Qr M1

i=1

+h N + NMT3 ∆ M3


T


R1 N + NMT3 ∆ M3 − MT1 R2 M1 − MT2 R2 M2



T +MT1 R2 M2 + MT2 R2 M1 + hMT1 R2 N + NMT3 ∆ M3 + h N + NMT3 ∆ M3 R2 M1

CR IP T



T −hMT2 R2 N + NMT3 ∆ M3 − h N + NMT3 ∆ M3 R2 M2 + WM1 + MT1 WT

−WM2 − MT2 WT + α2 σ2 MT1 M1 − σ2 MT3 M3 + σ1 MT2 Pm(t) M2 − MT4 Pm(t) M4 < 0 (20) 274

and MT1

q X

pri Qi M1 + MT1 Qr N + NT Qr M1 + MT1 Qr NMT3 ∆ M3 + MT3 ∆ T M3 NT Qr M1

AN US

i=1

T T T −MT1 R2 M1 − MT2 R2 M2 + MT1 R2 M2 + MT2 R2 M1 + hWR−1 1 W + WM1 + M1 W

−WM2 − MT2 WT + α2 σ2 MT1 M1 − σ2 MT3 M3 + σ1 MT2 Pm(t) M2 − MT4 Pm(t) M4 < 0 (21) Based on the Schur complement lemma, it is obtained from inequality (20) that  where



+



Φ∆ hMT3 ∆ T M3 NT 1 ∗ 0



<0

(22)

ED

276

Φ1 hN ∗ −hR−1 1

M

275

T T T T T T T Φ∆ 1 =M1 Qr NM3 ∆ M3 + M3 ∆ M3 N Qr M1 + hM1 R2 NM3 ∆ M3

ity (13) can be derived from inequality (21).

CE

278

Inequality (12) can be derived from inequality (22) on the basis of Lemma 1. Similarly, inequal-

AC

277

PT

+ hMT3 ∆ T M3 NT R2 M1 − hMT2 R2 NMT3 ∆ M3 − hMT3 ∆ T M3 NT R2 M2

18

ACCEPTED MANUSCRIPT

279

Define ˜ 1 = MT M 1

q X

pri Qi M1 + MT1 Qr N + NT Qr M1 + MT1 Qr NMT3 ∆M3 + MT3 ∆T M3 NT Qr M1

i=1

+h N + NMT3 ∆ M3


T


R1 N + NMT3 ∆ M3 − MT1 R2 M1 − MT2 R2 M2



T +MT1 R2 M2 + MT2 R2 M1 + hMT1 R2 N + NMT3 ∆ M3 + h N + NMT3 ∆ M3 R2 M1

CR IP T



T −hMT2 R2 N + NMT3 ∆ M3 − h N + NMT3 ∆ M3 R2 M2 + WM1 + MT1 WT

−WM2 − MT2 WT + α2 σ2 MT1 M1 − σ2 MT3 M3 + σ1 MT2 Pm(t) M2 − MT4 Pm(t) M4 ˜ 2 = MT M 1

q X

pri Qi M1 + MT1 Qr N + NT Qr M1 + MT1 Qr NMT3 ∆ M3 + MT3 ∆ T M3 NT Qr M1

i=1 T −M1 R2 M1

T T T − MT2 R2 M2 + MT1 R2 M2 + MT2 R2 M1 + hWR−1 1 W + WM1 + M1 W

280

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−WM2 − MT2 WT + α2 σ2 MT1 M1 − σ2 MT3 M3 + σ1 MT2 Pm(t) M2 − MT4 Pm(t) M4 n    o ˜ 1 , λmin M ˜ 2 . According to Eq. (14), it is obtained that and λ1 = min λmin M FV (m(t), r) ≤ −λ1 eT (t)e(t)

On the basis of Dynkin’s formula, it is also obtained that Z t  T E [V (m(t), r) − V (m(t0 ), r)] ≤ −λ1 E e (τ )e(τ )dτ

M

281

and it is further derived that

ED

282

λ1 E

Z

t

T

e (τ )e(τ )dτ

285

≤ V (m(t0 ), r)

CE

Moreover, the following relationship is derived based on Eq. (14)

where λ2 = λmin {Qr }.

 E {V (m(t), r)} ≥ λ2 E eT (t)e(t)

Consequently, following [35], the stochastically stable inequality can be derived as shown below Z t  λ2 T lim E e (t)e(t)dt ≤ 2 < ∞ t→∞ λ1 0

AC

284



PT

0

283

t0

286

According to Definition 1, it is proven that the Markovian jump system in Eq. (9) is stochastically

287

stable, which in turn implies that the leader-follower consensus is achieved by the proposed leader-

288

follower consensus algorithm based on Definition 2. 19

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Remark 10. In this work, the sampled-data vector is equivalently converted to a time-delay vector,

290

thus, the stability analysis can be smoothly conducted without theoretical difficulties. It is also pos-

291

sible to directly deal with the problems with time-varying delays through the similar approach. For

292

˙ = τ < 1, the derivative example, if the artificial time-delay d(t) is a time-varying function with d(t)

293

of the term t − d(t) will be 1 − τ which is always positive. Then, this positive term can be used in

294

the stability analysis. Similarly, other quadratic terms can be included in the Lyapunov functional

295

for the stability proof according to the specific problem.

296

Theorem 2. Suppose that the communication topology of the nonlinear multi-agent systems in

297

Eq. (1) and the information sharing satisfy Assumptions 1 - 3, then the leader-follower consensus

298

problem of the networked multi-agent systems in Eq. (1) is solvable if there exist symmetric matrices

AN US

˜ i > 0, i = 1, 2, 3, such that inequality (13) and the following LMIs are feasible R3 > 0, R   ˜ T Ψ1 Φ 1 hN ¯2  < 0  ∗ −hR3 Ψ ∗ ∗ Ψ3   ˜1 R ˜2 −R < 0 ˜3 ∗ −R    ˜ R1 0 0 R1 0 0 ˜ 2 0   ∗ R2 0  = I  ∗ R ˜3 ∗ ∗ R3 ∗ ∗ R

300

where

¯2 = Ψ

302

˜ m(t) R2 (Ik ⊗ A) K



0

(23) (24) (25)




 L m(t) + D m(t) ⊗ In R2 (Ik ⊗ B) h  ii ˜ m(t) Lm(t) + Dm(t) ⊗ In −K

Proof. By pre- and post-multiplying both side of inequality (12) by diag {I4kn , R2 , I8kn }, the

AC

301



0 0 0 0 0 0 δhR2 NMT3

CE

˜ = N



PT

˜ m(t) Km(t) =R−1 2 K

ED

M

299

CR IP T

289

following inequality can be obtained   Φ1 hNT I4kn 0 0  0   R2 0 ∗ −hR−1 1 0 0 I8kn ∗ ∗

 Ψ1 I4kn 0 R2 Ψ2   0 0 0 Ψ3  ˜T Φ1 hN  ∗ −hR2 R−1 R2 1 ∗ ∗

20

0 0 I8kn



<0

 Ψ1 ¯2  < 0 Ψ Ψ3

(26)

ACCEPTED MANUSCRIPT

If

303

R3 ≤ R2 R−1 1 R2

then the following inequalities are equivalent to inequality (26) based on Schur complement lemma   ˜ T Ψ1 Φ 1 hN ¯2  < 0  ∗ −hR3 Ψ ∗ ∗ Ψ3   −1 −1 −R1 R2 < 0 ∗ −R−1 3 Consequently, the inequalities (23), (24) and Eq. (25) can be derived on the basis of Theorem

305

306

CR IP T

304

(27)

1.

Apparently, the inequalities presented in Theorem 2 cannot be solved linearly due to the inclusion

308

of matrix equality (25). Therefore, the cone complementarity linearization method [40] is employed

309

to derive the feedback gain of the proposed controller.

310

Corollary 1. Suppose that the communication topology of the nonlinear multi-agent systems in

311

Eq. (1) and the information sharing satisfy Assumptions 1 - 3, then the feedback gain Ki

312

Eq. (5) and the matrix parameters in the inequalities (13), (23) and (24) can be derived by solving

313

the following optimization problem

ED min trace

3 X

˜ w Rw R

w=1

!

s.t. LMIs in inequalities (13), (23) and (24)   ˜w R I ≥ 0 w = 1, 2, 3 ∗ Rw

PT CE

m(t)

in

M

AN US

307

(28)

(29)

Remark 11. In classical LMI theory, the feasible values of the matrix variables in LMIs can be

315

derived using well-developed numerical methods [41]. However, Eq. (25) in Theorem 2 is a matrix

316

317

AC

314

equality, not LMI. Thus, the conventional numerical method cannot be applied directly to solve the matrix equality problem. In order to convert the matrix equality problem into a solvable problem,

318

the cone complementarity linearization method is adopted, and a feasible optimization problem is

319

proposed in Corollary 1.

21

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Based on the optimization proposed in Corollary 1, an iterative algorithm is developed to num(t)

321

merically obtain the feedback gain Ki

322

Algorithm 1:

323

324

325

as follows

Step 1 Initialize the maximum number of the iterations imax and the set o ˜ 0 that satisfies inequalities (13), (23), (24) and (29). Q0i , N

Step 2 Solve the following optimization problem:  X ˜ 0 Rw + R ˜ w R0 min trace R w w

n ˜ 0 , R0 , W0 , P0 , R w w

CR IP T

320

s.t. LM Is in inequalities (13), (23), (24) and (29)

326

Step 3 Substitute the feasible solution derived from Step 2 into inequality (12), if it is satisfied,

327

then output the feasible value of the demanded matrices and EXIT.

329

330

Step 4 If i > imax , then EXIT. Otherwise, set i = i + 1. o n o n n ˜ f , where R ˜j = R ˜ fw , ˜ fw , Rfw , Wf , Pf , Qf , N ˜ jw , Rjw , Wj , Pj , Qj , N Step 5 Update R i i o ˜ f is the feasible set derived from Step 2. Rfw , Wf , Pf , Qfi , N

AN US

328

Step 6 Go to Step 2.

332

Remark 12. In Algorithm 1, it has been indicated that the optimization problem in Step 2 is

333

constrained by the inequalities (13), (23), (24) and (29), which implies that the more complicated

334

these inequalities are, the slower the numerical computation will be. Specifically, the dimension of

335

the matrix variables is increased with the growing number of agents. Hence, the more agents the

336

multi-agent systems contain, the slower the feedback gain will be derived. But it should be noticed

337

that the derivation of the feedback gain is conducted before the consensus mission. Once the feedback

338

gain is derived prior to the consensus mission, the consensus will be accomplished successfully using

339

the proposed control algorithm in Eq. (5).

341

ED

PT

CE

AC

340

M

331

Remark 13. It is pointed out in Remark 12 that the growing number of agents will not greatly influence the achievement of consensus. However, it is worth mentioning that the communication

342

topology will have a strong impact on the achievement of consensus. Unsuitable selection of the

343

communication topology may lead to the failure of the consensus mission. It can be observed from

344

the parameter N in inequality (12) that the feedback gain is closely related to the communication 22

ACCEPTED MANUSCRIPT

347

topology. In matrix N, the distribution of the feedback gain Km(t) is largely determined by the matrix

L m(t) + D m(t) ⊗ In , and the structure of the matrix L m(t) + D m(t) ⊗ In is completely determined

348

it may result in an ineffective distribution of the feedback gain, which in turn fails the consensus

349

mission.

350

Remark 14. In this work, the matrix computation has to be conducted to derive the values of

351

eTi (ts )Pi

352

with other event-triggered conditions for the purpose of simplification in the future work. For ex-

353

ample, motivated by the work in [42], the event-triggered condition (10) can hopefully be simplified

354

by just comparing the control output values in different sampling instants. Once the norm of the

355

error vector of the control output is greater than a specific positive scalar, the control output will be

356

updated.

357

Remark 15. It is possible to extend the proposed results to the event-triggered consensus problem

358

with packet dropouts. As motivated by [43], an information indicator can be incorporated in the

359

stability analysis. If the information indicator equals to zero, the information packet will be ignored;

360

otherwise, the information packet is received by the specific agent. It is also possible to further

361

generalize the event-triggered consensus problem with packet dropouts by adopting more generic

362

stochastic model to describe the switching of the information indicator.

363

4

364

Four Chua’s circuits are utilized in the numerical simulation. In the simulated leader-follower

365

mission, a self-driven Chua’s circuit will generate a desired trajectory. At the same time, the desired

366

367

m(t)

ei (ts ) and rTi (ts )Pi

ri (ts ). It is possible to replace the event-triggered condition (10)

PT

ED

M

AN US

m(t)

CR IP T

by the communication topology. Therefore, if the communication topology is inappropriately selected,

Simulation

CE

346

AC

345

trajectory is stochastically broadcast to agent 1 and 2, i.e. they are considered as the leaders of the group. Since the desired trajectory is not available to agent 3 and 4, they can only exchange the

368

local information with their neighbors according to the communication topology, namely, they are

369

the followers in the leader-follower mission.

23

ACCEPTED MANUSCRIPT

370

The dynamics of Chua’s circuit can be described as follows x˙ i (t) = Axi (t) + B (In + ∆ i (t)) f (xi (t)) + ui (t)

and i = 1, 2, 3, 4, a = 9, b = 14.28, c = 1, m0 = 17 , m1 =

2 7

[34].

CR IP T

372

where ∆ i (t) represents the norm bounded uncertainty in agent i,     −am1 a 0 −a(m0 − m1 )  1 −1 1  0 A =  B= 0 −b 0 0  1  xi (t) xi (t) =  x2i (t)  x3i (t)
1 1 xi (t) + c − x1i (t) − c f (x1i (t)) = 2

AN US

371

(30)

Remark 16. System uncertainty is considered in each agent, as shown in Eq. (30). In numerical

374

simulations, bounded random signal is generated to simulate system uncertainty, i.e. ∆ i (t) is a

375

random signal in numerical simulations, and the boundary of the uncertainty in each agent can be

376

observed in Table 1.

M

373

Since the communication relationship is dynamically changing, two communication topologies

378

are considered in the simulation and they are stochastically switched with the evolve of the simula-

379

tion. Figures 2(a) and 2(b) depict these two communication topologies. As shown in Figure 2(a),

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both agent 1 and 2 have access to the desired trajectory, and agent 2 can also get information from

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agent 1. Agent 3 has access to both agent 1 and 2, while agent 4 will only receive information from

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agent 2. In Figure 2(b), agent 1 loses its connection with the self-driven Chua’s circuit, and only

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has access to agent 2. The corresponding Lm(t) and Dm(t) are   1 0 0 0  0 1 0 0   D1 =   0 0 0 0  0 0 0 0   0 0 0 0  −1 1 0 0   L1 =   −1 −1 2 0  0 −1 0 1

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D2

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0 0 0  0 1 0 =   0 0 0 0 0 0  1 −1  0 0 =   −1 −1 0 −1

 0 0   0  0 0 0 2 0

 0 0   0  1

On the basis of the proposed Algorithm 1, the feedback gains are derived as follows   −39.9851 −10.9067 6.5041 6.6656  K11 =  −4.8488 −9.9629 5.7949 10.8603 −35.3108   −15.2637 −4.6030 1.4749 3.8932  K12 =  −2.2798 −4.8693 1.4865 4.9797 −15.3802   −10.4316 −2.7374 1.5970 2.0403  K13 =  −1.5946 −3.8823 1.2688 3.2532 −11.2940   −20.3105 −5.3863 3.7223 4.5170  K14 =  −3.4134 −6.1899 3.1269 5.5011 −22.6398 and

 −10.7246 −2.1144 2.1870 =  −1.2557 −1.9206 1.6094  1.8228 2.2246 −7.5080   −23.8219 −5.7827 4.6505 6.0614  =  −4.1711 −4.8507 4.4891 6.5738 −19.1308   −5.2199 −0.8650 0.8340 =  −0.7885 −1.5873 0.8814  0.7755 0.8893 −4.9528   −9.8336 −1.8675 1.8207 2.1271  =  −1.6449 −2.5436 1.6511 2.0199 −10.0260

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and P21 =

P22 =



11.6449 0.6119  0.6119 11.3196 −0.4361 −0.8545  20.0943 3.0129  3.0129 13.2385 −2.8212 −4.1552  10.9565 −0.0534  −0.0534 11.0480 0.2480 0.0130  10.6029 −0.0930  −0.0930 10.8000 0.2353 0.0607

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The weight matrices in event-triggered condition (10) are derived as follows   16.1566 2.4607 −0.5934 P11 =  2.4607 16.6611 −2.6803  −0.5934 −2.6803 20.0023   15.5444 1.9593 −0.0673 P12 =  1.9593 15.7926 −1.9338  −0.0673 −1.9338 19.2448   11.1575 0.3695 0.6046 P13 =  0.3695 12.8525 −0.6089  0.6046 −0.6089 12.7390   10.8161 0.3009 0.4923 P14 =  0.3009 11.8288 −0.1179  0.4923 −0.1179 12.4691

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The initial values of the self-driven Chua’s circuit and the four agents are           0.1 −0.5 1 −1.5 1.5 x0desired =  0.5  x01 =  2  x02 =  −3  x03 =  −2  x04 =  −3  0.9 1.2 1.5 2 −1

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P24 =

 −0.4361 −0.8545  11.9011  −2.8212 −4.1552  21.1905  0.2480 0.0130  11.2533  0.2353 0.0607  11.0836

The values of other parameters are shown in Table 1

The desired trajectory of the multi-agent systems is shown in Figure 1, and it is generated by the

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self-driven Chua’s circuit. Applying the controller in Eq. (5), the tracking errors of the four agents,

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defined as xdesired (t) − xi (t), are exhibited in Figures 3 and 4. It is clearly observed that all the

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tracking errors along three directions converge to zero with the appearance of system uncertainties in 26

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1 2 3 4

Value 0.01 1 0.1 3 0.1 0.2 0.5 0.3

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Table 1: Parameters Parameter Sampled period, h Lipschitz constant, α σ1 σ2 Boundary of uncertainty in agent Boundary of uncertainty in agent Boundary of uncertainty in agent Boundary of uncertainty in agent

every agent, which firmly demonstrates the effectiveness and robustness of the proposed controller.

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Figure 5 shows the control input signals of agent 1. The solid lines represent the periodically

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sampled signal, while the event-triggered control input signals are accordingly displayed using those

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lines other than solid line. It is further shown in the zoom-in window that the update frequency

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of the event-triggered signal is much lower than that of the periodically sampled signal. Namely,

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the computational burden of each agent is greatly reduced in the proposed event-triggered strategy.

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The switching signal is presented in Figure 6, and the value “1” and “-1” indicates the Topology 1

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and Topology 2, respectively. The topologies are switched stochastically according to the Markov

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jump process.

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Remark 17. Compared with the previous work [14, 16, 18, 19, 26–28], the main characteristics

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can be exhibited from the simulation results. It is displayed in Figure 5 that the update frequency

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of the event-triggered control signal is obviously lower than that of the continuous signal or period-

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ically sampled signal. Moreover, as shown in Figures 3 and 4, the tracking errors converge to zero

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eventually even with the appearance of system uncertainties.

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Conclusion

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A leader-follower consensus algorithm for networked Lipschitz multi-agent systems is systematically

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investigated in this work. A feedback consensus controller is successfully developed with the event-

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triggered condition. In the multi-agent systems, Markov jump process is adopted to describe the

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stochastic switching topologies. Since the information is locally shared through a digital network, a 27

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Figure 1: Desired trajectory

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(a) Communication topology 1

(b) Communication topology 2

Figure 2: Communication topologies

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4

X−axis Y−axis Z−axis

2 0

Tracking error of agent 2

−2 0

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time(s)

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1 0 X−axis Y−axis Z−axis

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time(s)

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Tracking error of agent 4

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Figure 3: Tracking errors of agent 1 and 2

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X−axis Y−axis Z−axis

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Figure 4: Tracking errors of agent 3 and 4

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Event−triggered X−axis Event−triggered Y−axis Event−triggered Z−axis Periodically sampled X−axis Periodically sampled Y−axis Periodically sampled Z−axis

Control input of agent 1

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5 0

−5 −10 −15 0

0.4 0.2 0 −0.2 −0.4

10.1 10.2 10.3 10.4 10.5 5

time(s)

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Figure 5: Control input of agent 1 29

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Switching

1 0.5 0 −0.5

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Figure 6: Stochastic switching of the two topologies

time-delay equivalent approach is utilized to solve the discrete-time control problem caused by the

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discontinuous state feedback. By taking advantage of the Lyapunov functional method, the sufficient

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condition for system stability is derived with the consideration of system uncertainties. Moreover,

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the feedback gain of the proposed controller can be derived by the presented optimization algorithm.

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Furthermore, the effectiveness of the proposed control algorithm is demonstrated by the numerical

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simulation. In the future work, the information sharing and algorithm simplification are expected

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to be improved essentially. It is widely assumed that the information exchange among neighboring

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agents are updated at the same time. However, asynchronous information update is more generic

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and flexible in practice. Hence, the consensus algorithm with asynchronous information update can

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be systematically investigated in the future work. In addition, the consensus algorithm proposed in

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this paper is expected to be simplified to further reduce the computational burden of each agent.

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Appendix

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In order to further clarify the expression in Eqs. (7, 8), Kronecker product is briefly introduced

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with two examples in this section, and details can be found in [44]. Definition 3. Suppose A ∈ Rm×n , B ∈ Rp×q , then the Kronecker product of matrices A and B is

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defined as



 a11 B . . . a1n B   .. .. mp×nq .. A⊗B= ∈R . . . am1 B . . . amn B

Therefore, based on Definition 3, the following expression can be derived as an example to show the derivation with Kronecker product.   Ae1 (t)  Ae2 (t)     =  ..   . Aek (t)

    

A 0 .. .

0 A .. .

0 0 .. .

0

... ...  e1 (t)  e2 (t)  = Ik ⊗ A  .  ..

   

    

ek (t)

(A.2)

Similarly, another example including both Kronecker product and Laplacian matrix is provided as follows      



  =   

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

  =   

− xj (tu )]



 − xj (tu )]    ..  .  m(t) P Kk [x (t ) − x (t )] u j u k vj ∈NG (vk )  P  m(t) [x (t ) − x (t )] K1 0 0 0 1 u j u v ∈N (v )  P j G 1  m(t) 0 K2 0 0  vj ∈NG (v2 ) [x2 (tu ) − xj (tu )]     .. .. .. .. ..   . . . . .  P  m(t) [x (t ) − x (t )] u j u k 0 ... . . . Kk vj ∈NG (vk )   m(t) m(t) m(t) m(t) K1 0 0 0 L11 In L12 In . . . L1k In   m(t) m(t) m(t) m(t) 0 K2 0 0   L21 In L22 In . . . L2k In   .. .. .. .. .. .. .. ..  . . . . . . . . 

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  =   

m(t) P vj ∈NG (v1 ) [x1 (tu ) m(t) P K2 vj ∈NG (v2 ) [x2 (tu )

K1

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A 

   

e1 (t) e2 (t) .. .

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where ei ∈ Rn .



0 0 .. .

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(A.1)

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0

m(t)

K1 0 .. .

0

m(t)

...

. . . Kk

0

0 0 .. .

m(t) K2

.. . ...

0 0 .. .

m(t)

. . . Kk

m(t)

m(t)

Lk1 In Lk2 In   e1 (t)    e2 (t)  m(t)   L ⊗ In  .   .. 

ek (t)

m(t)

. . . Lkk In     

     

e1 (t) e2 (t) .. . ek (t)

     (A.3)

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In Eq. (A.3), Laplacian matrix is involved to simplify the expression with matrix multiplication.

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Details about Laplacian matrix and its application in multi-agent systems can be found in [2, 45, 46] 31

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