Bipartite Consensus of Heterogeneous Multiagent Systems Based on Distributed Event-Triggered Control

Wang, Xiaoyu; Liu, Kaien; Ji, Zhijian; Han, Shitao

doi:https://doi.org/10.1155/2020/1867194

Complexity

On this page

Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 1867194 | https://doi.org/10.1155/2020/1867194

Bipartite Consensus of Heterogeneous Multiagent Systems Based on Distributed Event-Triggered Control

Xiaoyu Wang,¹Kaien Liu,¹Zhijian Ji,²and Shitao Han¹

Academic Editor: Thach Ngoc Dinh

Received24 Dec 2019

Revised11 Feb 2020

Accepted21 Feb 2020

Published20 Mar 2020

Abstract

In this paper, the bipartite consensus problem of heterogeneous multiagent systems composed of first-order and second-order agents is considered by utilizing the event-triggered control scheme. Under structurally balanced directed topology, event-triggered bipartite consensus protocol is put forward, and event-triggering functions consisting of measurement error and threshold are designed. To exclude Zeno behavior, an exponential function is introduced in the threshold. The bipartite consensus problem is transformed into the corresponding stability problem by means of gauge transformation and model transformation. By virtue of Lyapunov method, sufficient conditions for systems without input delay are obtained to guarantee bipartite consensus. Furthermore, for the case with input delay, sufficient conditions which include an admissible upper bound of the delay are obtained to guarantee bipartite consensus. Finally, numerical simulations are provided to illustrate the effectiveness of the obtained theoretical results.

1. Introduction

Due to the wide applications on multirobot collaboration, cooperative control of unmanned aerial vehicles, and attitude alignment of satellites, many researchers have devoted to the study of coordination control of multiagent systems [1–4]. As one of the fundamental problems for distributed coordination in multiagent systems, the consensus problem has attracted much attention from multidisciplinary researchers due to its theoretical and practical significance. Consensus means states of all agents can reach a common value under an appropriate control protocol. During the past decade, a vast amount of theoretical achievements have been made on consensus problems [5–15]. Among them, when the single final consensus state cannot satisfy the control requirement with the increase of system size and complexity, group consensus, which means states of different groups of agents may approach to different values, has been put forward [13–15]. In [13], Yu and Wang studied the group consensus problem with both switching topologies and transmission delays. In [14], Altafini first proposed a sufficient and necessary condition for first-order multiagent systems to achieve bipartite consensus. In [15], Jiang et al. extended the results in [14] to general linear multiagent systems with signs.

Due to the complexity of real systems, it is possible for agents to have different dynamics. Hence, it is meaningful to study heterogeneous multiagent systems composed of first-order and second-order agents [16–20]. In [16], based on the properties of nonnegative matrices, sufficient consensus criterion was obtained for the agents with bounded communication delays under fixed topology and switching topologies, respectively. In [17], the consensus problem for continuous-time heterogeneous multiagent systems under directed graphs was investigated. In [18], by designing novel consensus protocols for continuous and discrete heterogeneous multiagent systems under directed topology, Liu et al. proved that the corresponding system asymptotically achieves consensus if and only if the fixed directed topology contains a directed spanning tree. In [19], sufficient and necessary condition for consensus of heterogeneous multiagent systems was given under certain assumptions on control parameters. In [20], some sufficient group consensus conditions which depend on input delays and control parameters were obtained for heterogeneous multiagent systems by using the frequency-domain analysis method and matrix theory.

However, all the control schemes in the aforementioned works depend on continuous or periodic communication among agents. In practical systems, as the communication bandwidth and system energy are usually limited, it is a waste of energy to communicate with others when it is unnecessary. In order to reduce the frequency of communication controller updates, the event-triggered scheme has been developed [21–33]. The basic idea of the event-triggered control is to replace the paradigm of periodic sampling (or continuous control) by aperiodic sampling. In [21], Dimarogonas et al. proposed distributed event-triggered consensus strategies for first-order multiagent systems, whose updates depended on the ratio of a certain measurement error with respect to the norm of a function of the states. Different from [21], event-triggering conditions based on sampled data were designed in [22–25], where energy can be saved by checking the event condition only at periodic sampling instants. In [23], Fan et al. proposed a self-triggered consensus protocol to further reduce the amount of state sampling. In [25], Liu et al. investigated the periodic event-triggered consensus problem of multiagent systems under directed topology which contains a spanning tree. In [26], Xie et al. put forward a novel event-triggered control strategy for second-order multiagent systems. In [27, 28], event-triggered consensus of multiagent systems with general linear dynamics was considered. Event-triggered consensus of multiagent systems with nonlinear dynamics and directed network topology was investigated in [29]. In [30], by using a distributed event-triggered control strategy, Tan et al. addressed the mean square consensus problem of leader-following stochastic multiagent systems with input delay. In [31], the bipartite consensus problem for first-order multiagent systems via the event-triggered control was investigated. In [32], Yin et al. considered the event-triggered consensus of discrete heterogeneous multiagent systems and designed event-triggering functions by using the positions and velocities of agents separately under a directed topology. After that, Yin et al. investigated event-triggered discrete-time heterogeneous multiagent systems with random communication delays represented by a Markov chain in [33].

Motivated by the aforementioned works, the distributed event-triggered scheme is applied to solve the bipartite consensus problem of heterogeneous multiagent systems in this paper. Different from [32], instead of designing the event-triggering function for the second-order agents by using positions and velocities of agents separately, we introduce a variable in the event-triggering function, which has special use in analysis. Moreover, variable y_i is also used in designing control protocol, which can weaken the requirement of control gain. Compared with the threshold designed in [32, 33], where each agent still needs to continuously monitor its neighbors’ states, the threshold in this paper only depends on the latest event-triggered states of agents. To exclude Zeno behavior, we introduce an exponential function in the threshold. As the input delay between the controller and the actuator is ubiquitous in practical systems, the consensus problem of multiagent systems with input delay has been investigated in [10, 20, 30]. To the best of our knowledge, there are no results concerning bipartite consensus of heterogeneous multiagent systems with input delay under the event-triggered scheme. Therefore, we also consider the case with input delay in this study.

The paper is organized as follows. The graph theory and the problem formulation are given in Section 2. Some important definitions and lemmas are also presented in Section 2. The main results of this paper are given in Section 3. Both cases without input delay and with input delay are investigated under topology which contains a directed spanning tree. Numerical examples are given in Section 4 to validate the effectiveness of the obtained results. Finally, conclusions are given in Section 5.

Notations: and represent the set of real and natural numbers, respectively. and denote n-dimensional real vector space and the n × m real matrix space, respectively. 1_n (0_n) denotes a column vector with all 1(0) elements. 0 indicates a zero matrix with a proper order. indicates an index set.

Notation diag{b₁, …, b_N} denotes a diagonal matrix. For a symmetric matrix , means is positive definite. λ (B) represents the eigenvalue of matrix B. Re (λ(B)) represents the real part of λ (B). λ_max (B) and λ_min (B) represent the maximum eigenvalue and the minimum eigenvalue of matrix B, respectively. |⋅| and denote 1-norm and Euclidean norm, respectively, both for vectors and matrices.

2. Preliminaries

In this section, we first give a brief review of graph theory and introduce some lemmas which will be used. Then, the model is formulated.

2.1. Graph Theory

A weighted directed graph (digraph) consists of a node set , an edge set , and a weighted adjacency matrix satisfying a_ij ≠ 0 iff . An edge means that node can receive information from node . We assume that ; hence, a_ii = 0 for all . Define node as a neighbor of node if a_ij ≠ 0. The set of neighbors of node is denoted by . The in-degree of node is defined as , . The Laplacian matrix L of a weighted digraph is defined as L = Λ − A, where Λ = diag{Λ₁₁, Λ₂₂, …, Λ_nn}. A directed path from to is a finite ordered sequence of distinct edges of with the form . A digraph contains a directed spanning tree if there exists a node called the root node such that there exists a directed path from it to every other node.

Definition 1. (structural balance, see [14]). A digraph is said to be structurally balanced if all the nodes of can be partitioned into two nonempty subsets and such that , and the following two conditions hold:(1), where q ∈{1, 2}(2), where q ≠ r and q, r ∈{1, 2}

Definition 2. (gauge transformation, see [14]). Gauge transformation is a change of orthant order in performed by an orthogonal matrix D which is defined as .
Obviously, D = D⁻¹. When the digraph is structurally balanced, the entries of DAD can be guaranteed to be nonnegative by selecting appropriate D. Consequently, the nondiagonal entries of DLD are nonpositive, and the sum of each row is zero.
The following are some lemmas which will be used in Section 3.

Lemma 1 (see [5]). Given a matrix , where b_ii ≥ 0, b_ij ≤ 0 for ∀i ≠ j, and for ∀i, B has at least one zero eigenvalue, and all of the nonzero eigenvalues are in the open right half plane. Furthermore, B has exactly one simple zero eigenvalue if and only if the digraph associated with B contains a directed spanning tree.

Lemma 2 (Young’s inequality, see [34]). Given , for , .

Lemma 3 (comparison principle, see [35]). Consider a differential equation , where f (t, u) is continuous and satisfies local Lipschitz condition in u. Let [t₀, T) be the maximum existence interval of the solution u (t), where T can be infinite. If, for ∀t ∈ [t₀, T), = satisfies , and ≤ u (t₀), then ≤ u (t), t ∈ [t₀, T).

Lemma 4 (see [36]). Assuming that all the eigenvalues of matrix P₁ are in the open left half plane, then for all t ≥ 0, it holds thatwhere β ≥ 1 and r is any positive constant which is smaller than Re (λ_min (−P₁)).

2.2. Problem Formulation

Suppose that the considered heterogeneous multiagent system has a communication topology represented by a digraph , which means each agent is regarded as a node of the interaction digraph. Assume that the system is composed of m (1 ≤ m < n) agents with second-order dynamics and n − m agents with first-order dynamics. Without loss of generality, we assume that agents to are second-order agents, whose index set is denoted as , and agents to are first-order agents, whose index set is denoted as .

Assumption 1. The communication topology is structural balanced with the following groups: the second-order agents constitute and the first-order agents constitute .
The dynamics of the system are described aswhere x_i, , and are the position, velocity, and control input of the ith agent, respectively. Based on the event-triggered control scheme, each agent broadcasts its state information at the event-triggering time instants. The sequence of event-triggering time instants of agent is denoted as , with .
The definition of bipartite consensus of (2) is given as follows.

Definition 3. Bipartite consensus of system (2) can be achieved for any initial conditions if it holds that

3. Consensus Analysis

For system (2), we propose the following bipartite consensus protocol:where , τ ≥ 0 represents the input delay, , α > 0 is the control gain, and and represent the set of neighbors of the ith agent in and , respectively. Note that when τ > 0, the control input u_i (t) is chosen as zero for t ∈ [0, τ).

Remark 1. Compared with [17] where the condition was required to guarantee consensus of heterogeneous multiagent systems, the control gain α is only required to be positive, even though the event-triggered scheme is applied here and it was absent in [17]. More specifically, from the following analysis, we can see that heterogenous system (2) under (4) can be transformed into an equivalent homogeneous system containing n + m agents by virtue of variable y_i, which makes the analysis easier.
For , denote measurement errors as , , and . Event-triggering function for each agent is designed aswhere p_i(t) = , ; is an exponential function with c₁ > 0, c₂ > 0; σ > 0 is the event-triggering parameter which will be further defined later. When f_i1(t) ≥ 0 or f_i2(t) ≥ 0, the event of corresponding agent is triggered immediately.

3.1. Without Input Delay

First, we investigate bipartite consensus of (2) without input delay, i.e., τ = 0. According to the definitions of measurement errors, bipartite consensus protocol (4) can be rewritten as

Substituting (6) into system (2), we can obtain

Based on the grouping of multiagent systems, the adjacency matrix A of the digraph can be described aswhere , , , and . Set , , , and . Thus, (7) can be written in a compact form aswherein which for i = 1, 4, for i = 2, 3,

Obviously, all entries of (i = 1, 2, 3, 4) are nonnegative.

Based on Assumption 1, (9) can be written aswhere with D = diag{I_m, I_m, −I_n−m}. Obviously, the sum of entries in each row of is zero. Meanwhile, the diagonal entries are all nonnegative. Therefore, can be regarded as the Laplacian matrix corresponding to a digraph with n + m nodes.

Lemma 5 (see [18]). The digraph corresponding to the matrix contains a directed spanning tree if and only if digraph contains a directed spanning tree.

It is easy to see that the bipartite consensus problem of system (2) is equivalent to the consensus problem of system (12). Denote and , where E = [−1_n+m−1I_n+m−1]. Correspondingly, and , where . Then, (12) can be rewritten as

Then, from [7], we have the consensus problem of system (12) which can be solved asymptotically if and only if system (13) achieves stability asymptotically. Moreover, when the digraph contains a directed spanning tree, all the eigenvalues of the matrix have negative real parts. So, there exists a positive definite matrix satisfying

For further analysis, we select a positive number ϵ > 0 such that

Theorem 1. Suppose that Assumption 1 holds and the digraph contains a directed spanning tree. The bipartite consensus problem of heterogeneous system (2) can be solved under event-triggered control protocol (4) and event-triggering functions (5) when event-triggering parameter σ satisfies

Proof. Construct the following Lyapunov function:where is the positive definite matrix satisfying (14). The time derivative of V (t) is as follows:Based on event-triggering functions (5), we can get thatAccording to the definitions of in (12), we haveCombining with (16), it is easy to get that . Thus, Therefore, (18) can be further extended to the following inequality:From (16), we have . Then, combining (21) with (17), we can get thatwhere . According to Lemma 3, we have 0 ≤ V(t) ≤ φ(t), where φ(t) satisfies = − ϱφ(t) + with φ(0) = V(0), that is,Obviously, V (t) = 0. Hence, , i.e., system (2) achieves bipartite consensus asymptotically.
The following theorem illustrates that Zeno behavior does not exist, i.e., there is no trajectory with infinite event-triggered time instants in a finite time interval.

Theorem 2. System (2) does not exhibit Zeno behavior under conditions of Theorem 1.

Proof. First, we consider agents in . Assume that and are two adjacent event-triggering time instants of agent , i.e., and . Thus, .
Considering the derivative of over the interval , we haveFrom the proof of Theorem 1, we know that . Thus, is bounded. According to the definition of and gauge transformation, for , , and are all bounded. Thus, = is bounded for . Therefore, is bounded, i.e., there exists a constant M > 0 such that . Then,For , since the event will not be triggered before , by (25), a lower bound on , which is denoted as , satisfies . That is, . Assume that Zeno behavior occurs, which indicates that there exists a positive constant t^∗ such that . Let . According to the definition of sequence limit, for ɛ₀ > 0, there exists a positive integer N₀ such that for m ≥ N₀. Therefore, when . This contradicts with the fact that for m ≥ N₀. Hence, Zeno behavior is strictly excluded.
Next, we consider agents in . Similarly, assume that and are two adjacent event-triggering time instants of agent , i.e., . Since , it holds that, for , . Similar to the above analysis, it is easy to get that u_i(t) is bounded for , i.e., there exists a constant M′ > 0 such that . Thus,The event will not be triggered before for . Combining with (26), the Zeno behavior can be excluded. Since the proof is just as same as that for agents in , we omit it here.
In conclusion, all the agents of system (2) do not exhibit Zeno behavior under conditions of Theorem 1.

Remark 2. From the proof of Theorem 2, we can see that, for second-order agents, to prove that is bounded by virtue of variable y_i which plays a key role to exclude Zeno behavior. This further shows the importance of introducing y_i.

3.2. With Input Delay

In this section, we investigate bipartite consensus of (2) with input delay, i.e., τ > 0. Substituting (4) into system (2) and according to the definitions of measurement errors, we can get that

Then, (27) can be rewritten in a compact form aswhereand , Λ_i, i = {1, 2, 3, 4} and Ψ have the same definitions as in (9). Analogously, according to Assumption 1, system (28) can be written aswhere , and have the same definitions as in (12). By Newton–Leibniz formula, (30) can be rewritten as

Remark 3. Based on Lemma 5, we can get the following result similarly. The digraph corresponding to the matrix has a directed spanning tree if and only if graph has a directed spanning tree. And the bipartite consensus problem of system (2) under control protocol (4) can be solved asymptotically if and only if system (31) achieves consensus asymptotically.
By using vectors and which have been denoted in (13), system (31) can be written as Then, system (30) achieves consensus asymptotically if and only if system (32) achieves stability asymptotically.
The following notations will be used in the next analysis. Set a₁ = ∥−EĤF ∥, , , , and .

Theorem 3. Suppose that Assumption 1 holds and the digraph contains a directed spanning tree. The bipartite consensus problem of system (2) can be solved under event-triggered control protocol (4) and event-triggering functions (5) when σ, τ, and c₂ satisfy 0 < σ < σ₀, 0 ≤ τ < τ₀, and 0 < c₂ < 2r, whereβ and r are the numbers obtained from Lemma 4 with . Moreover, system (2) does not exhibit Zeno behavior.

Proof. By integrating (32), we have According to [7], we know that all the eigenvalues of the matrix have negative real parts. Therefore, based on Lemma 4, it follows thatBased on event-triggering functions (5) and the requirement of σ in Theorem 3, we can get that By replacing (36) into (35), we can derive that Next, we prove that where λ ∈ (0, r) satisfies First, we prove the existence of λ. Define a function h (λ) = βλa₃b₁e^λτ + β [a₁a₂ + a₂ (a₂ + a₃b₁) e^λτ] (e^λτ − 1) − λ (r − λ). It is easy to get that h (0) = 0 and ḣ (0) = β [a₁a₂ + a₂ (a₂ + a₃b₁)]τ + βa₃b₁ − r. Since σ < σ₀, we can get that βa₃b₁ < r. Meanwhile, ḣ (0) < 0 since τ < τ₀. Therefore, there must exists a positive constant λ ∈ (0, r) such that h (λ) < 0. Consequently, (39) holds. Similarly, we can prove that the denominator of χ is positive when 0 < c₂ < 2r, which implies that χ > 0. Next, we prove that (38) holds. Note that for t = 0. If (38) does not hold, there must exist > 0 such that Replacing (38) into (37), we have By some direct calculations, it follows that This contradicts with (40). Hence, (38) holds, and bipartite consensus of system (2) under control protocol (4) and event-triggering function (5) can be solved asymptotically.
The proof of exclusion of Zeno behavior is just as same as that for Theorem 2. The details are omitted here for brevity.

4. Simulation

In this section, we give some numerical simulations to demonstrate the effectiveness of the obtained theoretical results. Consider a heterogeneous multiagent system composed of four agents and under a direct topology shown in Figure 1. Obviously, is structurally balanced with , where is composed of second-order agents and is composed of first-order agents. For simplicity, let a_ij∣ = 1 if a_ij ≠ 0. Let the initial states of system (2) be x (0) = [2,−2,−4,−1]^T, = [−2, 3]^T and choose α = 0.18. According to the topology in Figure 1, we have

Example. 1. This example is for the case without input delay, i.e., τ = 0. By solving the Lyapunov equation in (14), we can get thatBy selecting ϵ = 0.1, we can get γ = 0.39 from (15). By substituting parameters ϵ and γ into (16), we can get that 0 < σ < 0.0088. Therefore, set σ = 0.007 and ζ (t) = e^−0.1t. Figures 2 and 3 show that the heterogeneous multiagent system can achieve bipartite consensus under control protocol (4) and event-triggering functions (5). Figure 4 shows that , which means the introduction of y_i does not change bipartite consensus of the heterogeneous multiagent system. Figure 5 exhibits that all states can achieve consensus after being applied with gauge transformation. Figure 6 shows the event-triggering time instants of all agents. Figure 7 shows the evolution of .

Example 2. This example is for the case with input delay. Since , we can choose r = 0.08, β = 1 and c₂ = 0.1 according to Lemma 4 and conditions of Theorem 3. Therefore, set ζ (t) = e^−0.1t. By substituting the above parameters into (33), we can get that σ₀ = 0.0077, τ₀ = 0.0314. By choosing σ = 0.007 and τ = 0.01, the following numerical simulations are carried out. Figures 8 and 9 show that the heterogeneous multiagent system can achieve bipartite consensus under control protocol (4) and event-triggering functions (5). By comparing Figures 2 and 8 or Figures 3 and 9, we can see that the case with input delay takes longer time to achieve bipartite consensus than the case without input delay. Figure 10 shows the event-triggering time instants of all agents under event-triggered control protocol (4). Figure 11 shows the evolution of .

5. Conclusions and Future Work

In this paper, we have investigated event-triggered bipartite consensus for heterogeneous multiagent systems composed of first-order agents and second-order agents under a structurally balanced topology which contains a directed spanning tree. A special variable is used in designing event-triggered control protocol. By using gauge transformation, we transform a bipartite consensus problem under a structurally balanced signed network into a standard consensus problem under a nonnegative network. Sufficient conditions are obtained to guarantee bipartite consensus for both cases without input delay and with input delay. For the case with input delay, an upper bound of the delay is given to ensure bipartite consensus. It has been shown that no Zeno behavior occurs for heterogeneous multiagent systems under the presented event-triggered control protocol. Investigating bipartite consensus for heterogeneous multiagent systems under adaptive event-triggered control will be our future work.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant nos. 61873136, 61603288, and 61374062, the Science Foundation of Shandong Province for Distinguished Young Scholars under Grant no. JQ201419, and the Natural Science Foundation of Shandong Province under Grant nos. ZR2015FM023 and ZR2017MF055.

References

N. Meskin, K. Khorasani, and C. A. Rabbath, “A hybrid fault detection and isolation strategy for a network of unmanned vehicles in presence of large environmental disturbances,” IEEE Transactions Control Systems Technology, vol. 18, no. 6, pp. 1422–1429, 2010.
View at: Google Scholar
Y. Cao, W. Yu, W. Ren, and G. Chen, “An overview of recent progress in the study of distributed multi-agent coordination,” IEEE Transactions on Industrial Informatics, vol. 9, no. 1, pp. 427–438, 2013.
View at: Publisher Site | Google Scholar
Z. Ji and H. Yu, “A new perspective to graphical characterization of multiagent controllability,” IEEE Transactions on Cybernetics, vol. 47, no. 6, pp. 1471–1483, 2017.
View at: Publisher Site | Google Scholar
Y. Liu, X. Hou, and T. Hou, “Event-triggered consensus control for leader-following multiagent systems using output feedback,” Complexity, vol. 2018, Article ID 8265637, 9 pages, 2018.
View at: Publisher Site | Google Scholar
W. Ren and R. W. Beard, “Consensus seeking in multi-agent systems under dynamically changing interaction topologies,” IEEE Transactions on Automatic Control, vol. 50, no. 5, pp. 655–661, 2005.
View at: Google Scholar
F. Xiao and L. Wang, “Consensus protocols for discrete-time multi-agent systems with time-varying delays,” Automatica, vol. 44, no. 10, pp. 2577–2582, 2008.
View at: Publisher Site | Google Scholar
Y. Sun and L. Wang, “Consensus of multi-agent systems in directed networks with nonuniform time-varying delays,” IEEE Transactions on Automatic Control, vol. 54, no. 7, pp. 1607–1613, 2009.
View at: Google Scholar
K. Liu, G. Xie, W. Ren, and L. Wang, “Consensus for multi-agent systems with inherent nonlinear dynamics under directed topologies,” Systems & Control Letters, vol. 62, no. 2, pp. 152–162, 2013.
View at: Publisher Site | Google Scholar
K. Liu, G. Xie, and L. Wang, “Containment control for second-order multi-agent systems with time-varying delays,” Systems & Control Letters, vol. 67, pp. 24–31, 2014.
View at: Publisher Site | Google Scholar
K. Liu, Z. Ji, and W. Ren, “Necessary and sufficient conditions for consensus of second-order multiagent systems under directed topologies without global gain dependency,” IEEE Transactions on Cybernetics, vol. 47, no. 8, pp. 2089–2098, 2017.
View at: Publisher Site | Google Scholar
J. Qu, Z. Ji, C. Lin, and H. Yu, “Fast consensus seeking on networks with antagonistic interactions,” Complexity, vol. 2018, Article ID 7831317, 15 pages, 2018.
View at: Publisher Site | Google Scholar
J. Xi, C. Wang, X. Yang, and J. Zheng, “Limited-budget output consensus for descriptor multiagent systems with energy constraints,” IEEE Transactions on Cybernetics, 2020, http://arxiv.org/abs/1909.08345.
View at: Google Scholar
J. Yu and L. Wang, “Group consensus in multi-agent systems with switching topologies and communication delays,” Systems & Control Letters, vol. 59, no. 6, pp. 340–348, 2010.
View at: Publisher Site | Google Scholar
C. Altafini, “Consensus problems on networks with antagonistic interactions,” IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 935–946, 2013.
View at: Publisher Site | Google Scholar
Y. Jiang, H. Zhang, and J. Chen, “Sign-consensus of linear multi-agent systems over signed directed graphs,” IEEE Transactions on Industrial Electronics, vol. 64, no. 6, pp. 5075–5083, 2016.
View at: Google Scholar
C.-L. Liu and F. Liu, “Stationary consensus of heterogeneous multi-agent systems with bounded communication delays,” Automatica, vol. 47, no. 9, pp. 2130–2133, 2011.
View at: Publisher Site | Google Scholar
Y. Zheng and L. Wang, “Distributed consensus of heterogeneous multi-agent systems with fixed and switching topologies,” International Journal of Control, vol. 85, no. 12, pp. 1967–1976, 2012.
View at: Google Scholar
K. Liu, Z. Ji, G. Xie, and L. Wang, “Consensus for heterogeneous multi-agent systems under fixed and switching topologies,” Journal of the Franklin Institute, vol. 352, no. 9, pp. 3670–3683, 2015.
View at: Publisher Site | Google Scholar
Y. Feng, S. Xu, F. L. Lewis, and B. Zhang, “Consensus of heterogeneous first- and second-order multi-agent systems with directed communication topologies,” International Journal of Robust and Nonlinear Control, vol. 25, no. 3, pp. 362–375, 2015.
View at: Publisher Site | Google Scholar
G. Wen, Y. Yu, Z. Peng, and H. Wang, “Dynamical group consensus of heterogenous multi-agent systems with input time delays,” Neurocomputing, vol. 175, pp. 278–286, 2016.
View at: Publisher Site | Google Scholar
D. V. Dimarogonas, E. Frazzoli, and K. H. Johansson, “Distributed event-triggered control for multi-agent systems,” IEEE Transactions on Automatic Control, vol. 57, no. 5, pp. 1291–1297, 2012.
View at: Publisher Site | Google Scholar
X. Meng and T. Chen, “Event based agreement protocols for multi-agent networks,” Automatica, vol. 49, no. 7, pp. 2125–2132, 2013.
View at: Publisher Site | Google Scholar
Y. Fan, Y. Yang, and Y. Zhang, “Sampling-based event-triggered consensus for multi-agent systems,” Neurocomputing, vol. 191, pp. 141–147, 2016.
View at: Publisher Site | Google Scholar
A. Wang and Y. Zhao, “Event-triggered consensus control for leader-following multi-agent systems with time-varying delays,” Journal of the Franklin Institute, vol. 353, no. 17, pp. 4754–4771, 2016.
View at: Publisher Site | Google Scholar
K. Liu, Z. Ji, and X. Zhang, “Periodic event-triggered consensus of multi-agent systems under directed topology,” Neurocomputing, vol. 385, pp. 33–41, 2020.
View at: Publisher Site | Google Scholar
D. Xie, S. Xu, Y. Zou, and Z. Li, “Event-triggered consensus control for second-order multi-agent systems,” IET Control Theory & Applications, vol. 9, no. 5, pp. 667–680, 2015.
View at: Publisher Site | Google Scholar
W. Zhu, Z.-P. Jiang, and G. Feng, “Event-based consensus of multi-agent systems with general linear models,” Automatica, vol. 50, no. 2, pp. 552–558, 2014.
View at: Publisher Site | Google Scholar
N. Mu, X. Liao, and T. Huang, “Event-based consensus control for a linear directed multiagent system with time delay,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 62, no. 3, pp. 281–285, 2015.
View at: Publisher Site | Google Scholar
H. Li, G. Chen, T. Huang, W. Zhu, and L. Xiao, “Event-triggered consensus in nonlinear multi-agent systems with nonlinear dynamics and directed network topology,” Neurocomputing, vol. 185, pp. 105–112, 2016.
View at: Publisher Site | Google Scholar
X. Tan, J. Cao, X. Li, and A. Alsaedi, “Leader-following mean square consensus of stochastic multi-agent systems with input delay via event-triggered control,” IET Control Theory & Applications, vol. 12, no. 2, pp. 299–309, 2018.
View at: Publisher Site | Google Scholar
J. Li, X. Chen, F. Hao, and J. Xie, “Event-triggered bipartite consensus for multi-agent systems with antagonistic interactions,” International Journal of Control, Automation and Systems, vol. 17, no. 8, pp. 2046–2058, 2019.
View at: Publisher Site | Google Scholar
X. Yin, D. Yue, and S. Hu, “Distributed event-triggered control of discrete-time heterogeneous multi-agent systems,” Journal of the Franklin Institute, vol. 350, no. 3, pp. 651–669, 2013.
View at: Publisher Site | Google Scholar
X. Yin and D. Yue, “Event-triggered tracking control for heterogeneous multi-agent systems with markov communication delays,” Journal of the Franklin Institute, vol. 350, no. 5, pp. 1312–1334, 2013.
View at: Publisher Site | Google Scholar
H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge, UK, 1952.
Z. Li and Z. Duan, Cooperative Control of Multi-Agent Systems: A Consensus Region Approach, CRC Press, Boca Raton, Florida, US, 2014.
D. Zheng, Linear System Theory, Tsinghua University Press, Beijing, China, 2002.

Copyright

Copyright © 2020 Xiaoyu Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies