Event-triggered bipartite consensus of multi-agent systems in signed networks

1.   Introduction

During the past decade, the cooperative control of multi-agent systems has got compelling attentions owing to intensive applications ^[1,2,3]. As one of the most important topics of cooperative behaviors, the consensus problem of multi-agent systems has attracted much attention, many fruitful and crucial results have been obtained to build the theoretical consensus framework, see ^{[4,5,6,7,8,9,10,11,12,13,14,15]} and the references therein.

Most of the existing results are focused on the cooperative systems, however, the competitive relationship is very common in practical applications^[16,17]. Therefore, more and more attention has been paid to signed networks with both cooperative and antagonistic links^{[18,19,20,21,22]}. In order to achieve bipartite consensus, the structurally balanced condition is essential ^[23]. As an extension to the bipartite consensus, the concept of interval bipartite is introduced in ^[24], and interval bipartite consensus results are derived for structurally balanced networks and structurally unbalanced networks. If the network structure and node function satisfy some given conditions, the synchronization conditions are established for first-order multi-agent systems by utilizing pinning control strategy^[25,26], the above results are extended to second-order nonlinear multi-agent systems^[27,28], where the event-triggered control algorithm and pinning control strategy are used to achieve bipartite consensus. For general linear multi-agent systems, a systematical approach is given in ^[29] for solving the bipartite consensus problem. In ^[30], the bipartite tracking consensus problem is investigated for a dynamic leader. The finite-time bipartite consensus is studied in ^[31]. The adaptive bipartite consensus control is designed for high-order multi-agent systems ^[32]. Additionally, similar to the bipartite consensus problem, the reverse group consensus is firstly studied in the cooperation-competition network ^[33], the results show that the couple group consensus can be achieved if the mirror graph is strongly connected, as an extension, the adaptive robust bipartite consensus is considered for high-order uncertain multi-agent systems over the cooperation-competition network^[34].

In practical multi-agent systems, each agent is equipped with a micro-processor, it can collect information from neighboring agents, therefore, it is desired to implement the controller updates on a digital platform ^[35], the scheduling can be done in a time-triggered scheme or an event-triggered scheme, where the time-triggered scheme has a fixed sampling period, which should be selected to guarantee a desired performance under worst conditions, this kind of triggered method will send many unnecessary sampling data to neighboring agents. Considering the limited network resources, the event-triggered scheme is an effective method to improve the resources utilization while ensuring a satisfactory performance ^[36,37,38].

The design of event-triggered transmission scheme is crucial to determine whether the sampling data should be transmitted. The event-triggered transmission schemes can be classified into two types, the first case is the absolute error-based event-triggered transmission scheme ^[38], which is dependent on a pre-specified positive instant and the state error. The second case is the relative error-based event-triggered transmission scheme, which is related to the system state ^[39,40], these schemes can reduce the data transmission significantly compared with the periodical transmission scheme. More recently, significant considerations have been focused on the event-triggered control for multi-agent systems ^{[41,42,43,44,45,46,47,48,49,50,51]}. Actuator saturation can be observed in many practical systems due to physical or safety constraints, two kinds of distributed event-triggered control schemes are designed for the consensus of nonlinear systems subject to input saturation ^[41], where event-triggered parameters can be adaptively adjusted by using some adaptive laws. In ^[42], the distributed protocol is assumed to be subject to saturation, by utilizing the event-triggered control, the group consensus is discussed for a class of multi-agent systems with non-identical dynamics. The general linear models are considered to investigate event-triggered consensus problems in ^[43], the advantage of the event-based strategy is the significant decrease of the number of controller updates. Three types of event-triggered schemes are proposed for different network topologies ^[44]. Noted that communication topologies face more risks from cyber attacks, some distributed event-triggered schemes are proposed for linear and nonlinear multi-agent systems with Dos attacks^[45,46]. Based on event-triggered schemes, the consensus problems are discussed for nonlinear systems ^[47,48]. Especially, in ^[48], an integrated sampled-data-based event-triggered communication scheme includes advantages of both absolute and relative error-based event-triggered transmission schemes, thus it can lead to a high efficiency of data transmissions. In ^[49], considering quantized information, the periodic event-triggered algorithms are proposed for the consensus of multi-agent systems. The results on observer-based event-triggered consensus are also developed in ^[51,52]. In addition, switching topologies and stochastic sampling have been also considered for the consensus of multi-agent systems by event-triggered schemes^{[53,54,55,56]}, where the stochastic sampling interval randomly switches between two given values. The dynamic event-triggered schemes are developed for the consensus of multi-agent systems ^[57,58], compared with the traditional static event-triggered schemes, the time-varying threshold can ensure less triggered instants. In ^[59], the new event-triggered control law and triggering condition are constructed without continuous inter-neighboring communication, the event-triggered control is designed for the prescribed-time bipartite consensus. In ^[60], a novel observer-based bipartite control scheme is developed on the basis of two event-triggering mechanisms. In ^[61], a mode-dependent event-triggered transmission strategy is proposed for fixed-time bipartite consensus of a class of nonlinear multi-agent systems. In ^[62], distributed event-triggered control strategy is proposed for bipartite consensus for high-order multi-agent systems. However, the above schemes still embrace some room for improvement, in ^{[39,40,45,46]}, the next transmission instant is determined by $t_{k+1}h = t_{k}h+\min \{e^{T}(j_{k}h)\Phi e(j_{k}h) > \sigma x^{T}(t_{k}h)\Phi x(t_{k}h)\}$, where $\Phi$ is a symmetric positive definite matrix, which can be replaced by $t_{k+1}h = t_{k}h+\min \{e^{T}(j_{k}h)\Phi_{1} e(j_{k}h) > \sigma x^{T}(t_{k}h)\Phi_{2} x(t_{k}h)\}$, where $\Phi_{1}$ and $\Phi_{2}$ are two symmetric positive definite matrices, in simulation examples, it can be seen that this scheme can lead to larger sampling period. Compared with ^[27,52,55], in our paper the form of the event-triggered scheme is relatively simple and easy to be applied in practical applications, the lower bound event interval is the sampling period $h$, which is strictly larger than zero, therefore, the Zeno behavior is excluded in our proposed scheme. Moreover, in ^[27,52,55], in order to avoid Zeno behavior, the index term $-\beta e^{-\gamma(t-t_{0})}$ has been introduced in the event-triggered function, which may bring some trouble in the theoretical analysis or practical application. In addition, the bipartite synchronization of Lur'e network or neural network have been investigated in^[25,26], where the controller designs have been based on the continuous or sampled communication information among the agents. In^[63], the consensus problem is discussed for second-order multi-agent systems based on the event-triggered scheme, but the consensus protocol contains the continuous communication information of the leader. To the best of our knowledge, there are few results focusing on bipartite consensus of multi-agent systems with positive and negative communication links via event-triggered strategy, which is the main motivation of this paper.

Motivated by the aforementioned discussion, this paper is devoted to investigating bipartite consensus of Lur'e system in signed networks. The main contributions of this paper can be summarized as follows: (1) Compared with existing communication schemes in the literatures, the proposed event-triggered scheme can ensure the desired performance while further reducing the frequency of data transmission. (2) By using coordinate transform and matrix analysis techniques, sufficient conditions are derived to guarantee that bipartite leaderless consensus and leader-following consensus can be achieved, respectively. (3) The upper bound of the sampling period can be obtained by solving a constrained optimization problem. (4) In the proposed event-triggered communication scheme, the Zeno behavior is excluded owing to the fact that the lower bound of inter event interval is the sampling period.

Notation: ${\mathbb R}^{n}$ denotes the $n$-dimensional Euclidean space and ${\mathbb R}^{n\times m}$ is a set of real $n\times m$ matrices. $diag\{\cdots\}$ stands for a block-diagonal matrix. The notation with the superscript $1_{n}(0_{n})$ indicates the $n$-dimensional column vector with each entry being $1(0)$. $sgn(\cdot)$ is the standard sign function. Notations $\|.\|$ and $\otimes$ denote the Euclidian norm and the Kronecker product, respectively.

2.   Problem formulation and preliminaries

The communication topology of a network can be modeled as a signed graph $g = (v, \varepsilon, A)$, where $v = (v_{1}, v_{2}, \ldots, v_{N})$ denotes the set of nodes, $\varepsilon \subseteq v\times v$ is the set of edges, $A = (a_{ij})_{N\times N}$ denotes a signed adjacency matrix of $g$ representing the communication topology. If $a_{ij} = 0$, it means that agent $i$ cannot receive information from agent $j$. The entry $a_{ij}$ can be allowed to be positive or negative, with $a_{ij} > 0$ indicating that node $i$ and node $j$ are cooperative, $a_{ij} < 0$ indicating the node $i$ and node $j$ are competitive. Based on the weighted adjacency matrix $A$, the Laplacian matrix of the signed graph $g$ can be defined as $L = (l_{ij})_{N\times N}$, where $l_{ii} = \sum_{k = 1, k\neq i}|a_{ik}|$, $l_{ij} = -a_{ij} \; (i\neq j)$. If there exists a rooted node having directed path to all other nodes, the signed graph $g$ contains a directed spanning tree.

Remark 1. For the above definition of Laplacian matrix $L$, the sum of the $i$th row of $L$ is $\sum_{k = 1, k\neq i}^{N}|a_{ik}|-\sum_{k = 1, k\neq i}^{N}a_{ik}$, the element $a_{ij}$ can be allowed to be positive or negative for the signed graph $g$, thus $\sum_{k = 1, k\neq i}^{N}|a_{ik}| = \sum_{k = 1, k\neq i}^{N}a_{ik}$ may not be hold, that is the Laplacian matrix of signed digraph may not be a zero-row-sum matrix, which leads to the consequence that the traditional control protocol design method can not be used directly.

Consider a class of nonlinear multi-agent systems with the Lur'e system form

where $x_{i}(t)\in \mathbb{R}^{n}$ is the state vector of node $i$, $f(Cx_{i}(t)) = [f_{1}(C_{1}^{T}x_{i}(t)), f_{2}(C_{2}^{T}x_{i}(t)), \ldots, f_{m}(C_{m}^{T}x_{i}(t))]^{T}\in \mathbb{R}^{m}$ is a vector-valued nonlinear function. $A\in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}$, $C = [C_{1}, C_{2}, \ldots, C_{i}, \ldots, C_{m}]^{T}\in \mathbb{R}^{m\times n}$ and $C_{i} = [C_{i1}, C_{i2}, \ldots, C_{in}]^{T} \; (i = 1, 2, \ldots, m)$ are some constant matrices. $u_{i}(t)\in \mathbb{R}^{n}$ is the control input.

Assumption 1. Suppose that the nonlinear function $f_{k}(\cdot) \; (k = 1, 2, \ldots, m)$ is an odd functionsatisfying the following sector condition

where $\delta_{k} > 0 \; (k = 1, 2, \ldots, m)$.

Remark 2. The nonlinear function $f(\cdot)$ is assumed to be an odd function due to the fact that the gauge transformation can do work well, which can be found in the proof of Theorem 1. In addition, it is a mild assumption in existing literatures, whether the limitation of nonlinear function can be removed may be discussed in the future.

Definition 1. ^[25] A signed graph $g$ is called structurally balanced if the node set $v$ can be divided into two sub-networks $V_{1}$ and $V_{2}$, which satisfy $V_{1}\cup V_{2} = v$ and $V_{1}\cap V_{2} = \Phi$, and $a_{ij}\geq 0$ holds for node $i$ and node $j$ which are in the same sub-network, and $a_{ij} < 0$ holds for node $i$ and node $j$ which are in different sub-networks. Otherwise, the signed graph $g$ is called structurally unbalanced.

Lemma 1. ^[23]Suppose that the signed network $g$ is structurally balanced, if and only if there is a gauge transformation$W$ such that $W^{T}AW = \bar{A}$ with nonnegative entries, where $W = diag\{w_{1}, w_{2}, \ldots, w_{N}\}$ with $w_{i}\in \{1, -1\}$, $A = (a_{ij})_{N\times N}$ and $\bar{A} = (|a_{ij}|)_{N\times N}$ are the adjacency matrices ofthe signed and unsigned networks, respectively. In addition, $W$ provides a partition, that is $V_{1} = \{v_{i}|w_{i} = 1 \}$ and $V_{2} = \{v_{i}|w_{i} = -1 \}$.

Lemma 2. ^[64]If $x(t):(a, b)\rightarrow \mathbb{R}^{n}$ is an absolutely continuous function satisfying $x(a) = 0$ and $M\in \mathbb{R}^{n\times n}$ is a positive definite matrix, then the following Wirtinger's inequality holds

Lemma 3. ^[65]For any positive definite matrix $M\in \mathbb{R}^{n\times n}$ and a vector function $x(t):(a, b)\rightarrow \mathbb{R}^{n}$, such that the following integrations are well defined, then

Lemma 4. ^[66] For any two real vector functions $y_{1}(t)\in \mathbb{R}^{n}$ and $y_{2}(t)\in \mathbb{R}^{n}$, a positive definite matrix $R\in \mathbb{R}^{n\times n}$ and any matrix $S\in \mathbb{R}^{n\times n}$, then

where $y(t) = \left[ \begin{array}{c} y_{1}(t) \\ y_{2}(t) \\ \end{array} \right]$, $\left[ \begin{array}{cc} R & S^{T} \\ S & R \\ \end{array} \right] > 0$ and $0 < \alpha < 1$.

Lemma 5. ^[67] The following linear matrix inequality

where $Q(x) = Q^{T}(x)$, $R(x) = R^{T}(x)$, is equivalent to either of the following conditions

3.   Main results

3.1.   Bipartite leaderless consensus

In this section, it is assumed that the topology is a directed signed graph and structurally balanced. An event-triggered scheme is designed, and by some transformation, the bipartite leaderless consensus problem is converted to the stability of the error system.

Definition 2. ^[25] The signed Lur'e network (2.1) is said to achieve bipartite leaderless consensus if the following equation holds

where $w_{i} = 1$, $(i\in V_{1})$; $w_{i} = -1$, $(i\in V_{2})$.

Remark 3. If the agents $i, j\in V_{1}$ or $i, j\in V_{2}$, which means they are in the same sub-network, then $w_{i} = w_{j} = 1$ or $w_{i} = w_{j} = -1$, the equation (3.1) reduces to $\lim\limits_{t\rightarrow \infty}\|x_{i}(t)-x_{j}(t)\| = 0$, which is the traditional leaderless consensus of unsigned networks. If the agents $i\in V_{1}, j\in V_{2}$ or $i\in V_{2}, j\in V_{1}$, it means that the agents $i$ and $j$ are in different sub-networks, the equation (3.1) reduces to $\lim\limits_{t\rightarrow \infty}\|x_{i}(t)+x_{j}(t)\| = 0$, which can be rewritten as $\lim\limits_{t\rightarrow \infty}\|x_{i}(t)-(-x_{j}(t))\| = 0$, which represents agent $j$ competes against agent $i$.

For reducing the communication burden among the agents, the following sampled control algorithm is designed

where $c > 0$ is the coupling strength, $t_{k}h \; (k = 0, 1, 2, \ldots)$ is the release times, $t_{0}h = 0$, $\{t_{0}, t_{1}, \ldots\}\subset \{0, 1, 2, \ldots\}$, the data will be updated for $t\in [t_{k}h, t_{k+1}h)$ until $t = t_{k+1}h$, which can effectively reduce the data transmission among the agents. When $a_{ij} > 0$, the coupling information from its neighboring agent can be described as $a_{ij}[x_{i}(t_{k}h)-x_{j}(t_{k}h)]$, which means agent $i$ and agent $j$ are cooperative, when $a_{ij} < 0$, the coupling information from its neighboring agent is $-a_{ij}[x_{i}(t_{k}h)+x_{j}(t_{k}h)]$, which represents that interaction is antagonistic.

Applying (3.2) to the system (2.1) yields

Based on the definition of the Laplacian matrix of signed graph, the equation (3.3) can be rewritten as

where $l_{ii} = \sum_{k = 1, k\neq i}|a_{ik}|$, $l_{ij} = -a_{ij} \; (i\neq j)$.

Considering limited bandwidth of the communication network, to further reduce the data transmission among the agents, an event-triggered communication scheme is proposed for system (2.1), which is utilized to decide whether the current sampled data should be transmitted to neighboring agents or not. Supposed that the latest transmitted data is $x(t_{k}h)$, then the next transmission instant $t_{k+1}h$ can be determined by

where the error $e_{i}(t) = w_{i}x_{i}(t)-w_{1}x_{1}(t)$, $\sigma_{i} > 0$ are event-triggered parameters, $\Phi_{i}^{(1)}$ and $\Phi_{i}^{(2)}$, $(i = 2, 3, \ldots, N)$ are positive definite event-triggered matrices which need to be designed. $t_{k+1}h-t_{k}h$ denotes the release period which corresponds to the sampling period given by the event-triggered scheme in (3.5).

Remark 4. Noted that the above event-triggered communication scheme (3.5) including the following special cases: (I) if $\Phi_{i}^{(1)} = \Phi_{i}^{(2)} = \Phi_{i} \; (i = 2, 3, \ldots, N)$, the scheme (3.5) shrinks to a discrete event-triggered transmission scheme ^[39,40]. (II) if $\sigma_{i} = 0$ and $t_{k+1}h = t_{k}h+h$, then transmission scheme (3.5) shrinks to a periodic transmission scheme. (III) if $\Phi_{i}^{(1)} = \Phi_{i}^{(2)} = I_{n} \; (i = 2, 3, \ldots, N)$, transmission scheme (3.5) becomes a discrete absolute error-based transmission scheme. Hence the proposed event-triggered communication scheme includes some existing schemes. The proposed transmission scheme (3.5) where only the sampled-data is used is different from a continuous absolute error-based scheme that needs to monitor the continuous measurement. Compared with ^[27], the form of the event-triggered scheme is relatively simple and easy to be applied in practical applications, moreover in our paper the lower bound event interval is the sampling period $h$, which is strictly larger than zero, therefore, the Zeno behavior is excluded in our proposed scheme.

Remark 5. The bipartite synchronization of Lur'e network or neural network has been investigated in^[25,26], where the controller designs have been based on the continuous or sampled communication information among the agents. This paper has been based on the event-triggered scheme, the control signals are only updated at a series of triggered instants. In ^[63], the consensus problem is discussed for second-order multi-agent systems based on event-triggered scheme, but the consensus protocol contains the continuous communication information of the leader.

Let $\bar{x}_{i}(t) = w_{i}x_{i}(t) \; (i = 1, 2, \ldots, N)$, here $w_{i} = 1 \; (i\in V_{1})$ and $w_{i} = -1 \; (i\in V_{2})$, it follows from (3.4) that

where $\bar{L} = (\bar{l}_{ij})_{N\times N}$ is the Laplacian matrix of the unsigned graph with zero-row-sum, $\bar{l}_{ii} = \sum_{k = 1, k\neq i}|a_{ik}|$, $\bar{l}_{ij} = w_{i}l_{ij}w_{j} = -|a_{ij}| \; (i\neq j)$.

By using Assumption 1, it is easy to obtain $w_{i}f(Cw_{i}\bar{x}_{i}(t)) = w_{i}^{2}f(C\bar{x}_{i}(t)) = f(C\bar{x}_{i}(t))$ owing to $f(\cdot)$ being an odd function, then (3.6) can be simplified as

In views of (3.3) and $e_{i}(t) = \bar{x}_{i}(t)-\bar{x}_{1}(t) \; (i = 2, 3, \ldots, N)$, one has

Partly inspired by ^[39,40], the interval $[t_{k}h, t_{k+1}h)$ can be expressed as the union of several subintervals

Define $\tau(t) = t-(t_{k}h+lh)$, for $t\in[t_{k}h+lh, t_{k}h+lh+h)$, it can be derived that $0\leq\tau(t) < h$, let $\bar{e}_{i}(t_{k}h+lh) = e_{i}(t_{k}h+lh)-e_{i}(t_{k}h) \; (i = 2, 3, \ldots, N)$, we have

Substituting (3.10) into (3.8) yields

By rewrite (3.11) in compact matrix form, one may further get that

where

Combining (3.5) and (3.10), for $t\in [t_{k}h, t_{k+1}h)$, the current sampled data $e_{i}(t_{k}h+lh)$ will not be sent, then the event-triggered scheme can be rewritten as

By using Kronecher product, from (3.13), one get

where

Theorem 1. Suppose that Assumption 1 holds, for given sampled period $h > 0$, trigger parameters $\sigma_{i} > 0 \; (i = 2, 3, \ldots, N)$ and coupled strength $c > 0$, the bipartite leaderless consensus can be reached in system (2.1)under the control law (3.2) and event-triggered scheme (3.14), if there exist some $(N-1)n\times (N-1)n$ positivedefinite matrices $P > 0$, $Q > 0$, $R > 0$, $\Omega > 0$ and $T = diag\{\tau_{1}, \tau_{2}, \ldots, \tau_{m}\} > 0$, and matrix $U\in \mathbb{R}^{(N-1)n\times (N-1)n}$, such that the following linear matrix inequalities hold

where

Proof. Construct the following Lyapunov function candidate

where

where $P, Q, R, \Omega\in \mathbb{R}^{(N-1)n\times (N-1)n} > 0$.

Next, some necessary explanations are given to show the Lyapunov function $V_{4}(t)$ is valid. By using the Wirtinger's inequality in Lemma 2, one has

it can be easily derived that $V_{4}(t)\geq 0$, thus the Lyapunov function $V_{4}(t)$ is valid.

By using Assumption 1, we can obtain

for $\tau_{k} > 0 \; (k = 1, 2, \ldots, m)$, we have

The above equation (3.20) can be rewritten in the compact matrix form as follows

where

For $t\in [t_{k}h, t_{k+1}h)$, taking the derivative of $V_{i}(t) \; (i = 1, 2, 3, 4)$ along the trajectories (3.12), one gets

For any $\tau(t)\in [0, h)$, it follows from Jensen's inequality in Lemma 3 and the reciprocally convex approach in Lemma 4 that

It can be derived from (3.25) that

From (3.12), (3.22)–(3.27), we can obtain

where

using Schur complement in Lemma 5 with (3.15), we can obtain $\Sigma+h^{2}\Pi^{T}(R+\Omega\Pi) < 0$, then

thus system (3.12) is globally asymptotically stable at the origin, then we have $\lim\limits_{t\rightarrow \infty}\|w_{i}x_{i}(t)-w_{1}x_{1}(t)\| = 0 \; (i = 2, 3, \ldots, N)$, which means $\lim\limits_{t\rightarrow \infty}\|w_{i}x_{i}(t)-w_{j}x_{j}(t)\| = 0 \; (i = 2, 3, \ldots, N; i\neq j)$, by using Definition 1, it can be derived that bipartite leaderless consensus is achieved. This completes the proof.

Remark 6. As a by-product, given the triggered parameters $\sigma_{i} > 0 \; (i = 2, 3, \ldots, N)$ and the coupling strength $c > 0$, the following constrained optimization problem can be employed to find the maximum allowable sampling period $h$, that is

Remark 7. In Theorem 1, the triggered parameters $\sigma_{i} > 0 \; (i = 2, 3, \ldots, N)$ depend on the variation of the system state, if the parameters $\sigma_{i}$ keep a big value, then the network resources can be further saved, for given the coupling strength $c$ and the sampling period $h$, when $\Phi^{(1)} = \Phi^{(2)}$, the maximum values of trigger parameters $\sigma_{i} > 0 \; (i = 2, 3, \ldots, N)$ can be obtained by the following constrained optimization problem

Notice that if $\sigma_{i} = 0 \; (i = 2, 3, \ldots, N)$, the event-triggered scheme will reduce to the time-triggered scheme. Let $\sigma_{i}\rightarrow 0^{+}$ in Theorem 1, we will obtain the bipartite leaderless consensus under the sampled-data control.

Corollary 1. Suppose that Assumption 1 holds, for given sampled period $h > 0$ and coupled strength $c > 0$, the bipartite leaderless consensus under the sampled-data control can be reached in system (2.1)under the control law (3.2), if there exist some $(N-1)n\times (N-1)n$ positivedefinite matrices $P > 0$, $Q > 0$, $R > 0$, $\Omega > 0$ and $T = diag\{\tau_{1}, \tau_{2}, \ldots, \tau_{m}\} > 0$, and matrix $U\in \mathbb{R}^{(N-1)n\times (N-1)n}$, such that the following linear matrix inequalities hold

where

Remark 8. When $\sigma_{i}\rightarrow 0^{+}$, it can be concluded that (3.14) can not be satisfied, the data of each sampling time will be transmitted to the neighbor agents, that is, the event-triggered scheme reduces to the time-triggered scheme, thus bipartite leaderless consensus can be derived based on the sampled control law in ^[25].

3.2.   Bipartite leader-following consensus

In this section, the bipartite leader-following consensus for Lur'e network is considered under the event-triggered communication scheme.

Suppose that there is only one virtual leader, and the dynamics can be described by

where $x_{0}(t)\in R^{n}$ is the position state of the virtual leader, the other notations and the dynamics of the followers are the same as those in (2.1).

Definition 3. ^[25] Under Assumption 1, the signed Lur'e network (2.1) is said to achieve bipartite leader-following consensus if $\lim\limits_{t\rightarrow \infty}\|x_{i}(t)-w_{i}x_{0}(t)\| = 0$ holds for all $i = 1, 2, \ldots, N$, where $w_{i} = 1 \; (i\in V_{1})$ and $w_{i} = -1 \; (i\in V_{2})$.

Remark 9. In Definition 3, if $i\in V_{1}$, we have $w_{i} = 1$, then $\lim\limits_{t\rightarrow \infty}\|x_{i}(t)-w_{i}x_{0}(t)\| = 0$ reduces to $\lim\limits_{t\rightarrow \infty}\|x_{i}(t)-x_{0}(t)\| = 0$. If $i\in v_{2}$, we have $w_{i} = -1$, then $\lim\limits_{t\rightarrow \infty}\|x_{i}(t)-w_{i}x_{0}(t)\| = 0$ reduces to $\lim\limits_{t\rightarrow \infty}\|x_{i}(t)+x_{0}(t)\| = 0$, which means $x_{i}(t)\rightarrow -x_{0}(t) \; (t\rightarrow \infty)$, in other words, one part are synchronized to the state of the leader, the other part is synchronized to the opposite state of the leader, the structural balanced is critical to achieve the bipartite leader-following consensus.

The following sampled control law is proposed to solve the bipartite leader-following consensus problem for Lur'e network (2.1) and (3.32) as follows

where $c > 0$ is the coupling strength, $t_{k}h \; (k = 0, 1, 2, \ldots)$ is the transmitted instants, $t_{0}h = 0$, $\{t_{0}, t_{1}, \ldots\}\subset \{0, 1, 2, \ldots\}$. $d_{i} \; (i = 1, 2, \ldots, N)$ is the pining feedback gain, $d_{i} > 0$ if agent $i$ is pinned, otherwise $d_{i} = 0$.

Applying (3.33) to system (2.1), one has

Let $\bar{x}_{i}(t) = w_{i}x_{i}(t)$ and $e_{i}(t) = \bar{x}_{i}(t)-x_{0}(t) \; (i = 1, 2, \ldots, N)$, it follows from (3.34) that

The event-triggered communication scheme is proposed as follows

where $\sigma_{i} > 0$ are event trigger parameters, $\Phi_{i}^{(1)}$ and $\Phi_{i}^{(2)} \; (i = 1, 2, \ldots, N)$ are positive definite matrices to be designed.

The interval $[t_{k}h, t_{k+1}h)$ can be expressed as follows

Define $\tau(t) = t-(t_{k}h+lh)$, for $t\in[t_{k}h+lh, t_{k}h+lh+h)$, one has $0\leq\tau(t) < h$, let $\bar{e}_{i}(t_{k}h+lh) = e_{i}(t_{k}h+lh)-e_{i}(t_{k}h) \; (i = 2, 3, \ldots, N)$, it can be derived that

then for $t\in [t_{k}h, t_{k+1}h)$, the event-triggered scheme can be rewritten as follows

where

From the above analysis, it can be known from (3.39) that the current sampled data $e_{i}(t_{k}h+lh)$ will not be sent, which will be employed in the following consensus analysis.

For convenience, some notations are given as

Rewrite (3.35) in compact matrix form as

where $\hat{L} = (\hat{l}_{ij})_{N\times N}$, $\hat{l}_{ij} = \bar{l}_{ij} (i\neq j), \hat{l}_{ii} = \bar{l}_{ii}+d_{i}$.

Theorem 2. Suppose that Assumption 1 holds, for given sampled period $h > 0$, triggered parameters $\sigma_{i} > 0 \; (i = 1, 2, \ldots, N)$, pinning feedback gain $d_{i}$ and coupled strength $c > 0$, the bipartite leader-following consensus can be reached in system (2.1)under the control law (39) and event-triggered scheme (3.39), if there exist some $Nn\times Nn$definite matrices $P > 0$, $Q > 0$, $R > 0$, $\Omega > 0$ and $T = diag\{\tau_{1}, \tau_{2}, \ldots, \tau_{m}\} > 0$, and matrix $U\in \mathbb{R}^{Nn\times Nn}$, such that the following matrix inequalities hold

where

Proof. Construct the following Lyapunov function candidate

where

where $P, Q, R, \Omega\in \mathbb{R}^{Nn\times Nn} > 0$.

The proof method is similar to Theorem 1, the detailed process is omitted to save space.

Notice that if $\sigma_{i} = 0 \; (i = 1, 2, \ldots, N)$, the event-triggered scheme will reduce to the time-triggered scheme. In Theorem 2, let $\sigma_{i}\rightarrow 0^{+}$, we will obtain bipartite leader-following consensus under the sampled-data control.

Corollary 2. Suppose that Assumption 1 holds, for given sampled period $h > 0$, pinning feedback gain $d_{i}$ and coupled strength $c > 0$, the bipartite leader-following consensus under the sampled-data control can be reached in system (2.1) under the control law (3.33), if there exist some $Nn\times Nn$definite matrices $P > 0$, $Q > 0$, $R > 0$, $\Omega > 0$ and $T = diag\{\tau_{1}, \tau_{2}, \ldots, \tau_{m}\} > 0$, and matrix $U\in \mathbb{R}^{Nn\times Nn}$, such that the following matrix inequalities hold

where

Remark 10. Similar to Remark 6 and Remark 7, the constrained optimization problem can be employed to find the maximum allowable sampling period $h$ and the maximum values of the trigger parameters $\sigma_{i} \; (i = 1, 2, \ldots, N)$.

Remark 11. For structurally balanced topology, by gauge transformation, the leader-following bipartite consensus of signed network will be converted to the leader-following consensus of unsigned network, then the selected pinning nodes and controller gains can be designed according to the scheme in ^[68].

4.   Numerical examples

This section provides two simulation examples to demonstrate the effectiveness of the proposed event-triggered communication scheme. The first example is for the bipartite leaderless consensus issue, and the second example is for the leader-following consensus issue.

Consider a Lur'e network consisting of seven agents

where

it is easy to verify that the nonlinear function $f(\cdot)$ is an odd function with $\delta_{k} = 1$, which satisfies Assumption 1. The directed communication topology is shown in Figure 1, the solid lines denote the cooperative interactions, the dashed lines denote the antagonistic interactions, obviously, which can be divided into two sub-networks $V_{1} = \{1, 2, 3, 4\}$ and $V_{2} = \{5, 6, 7\}$, the agents are cooperative in $V_{1}$ or $V_{2}$, the agents are competitive between $V_{1}$ and $V_{2}$, by using Lemma 1, we can obtain the gauge transformation $W = diag\{1, 1, 1, 1, -1, -1, -1\}$.

Case 1: Bipartite leaderless consensus

Let the coupling strength $c = 10$, the sampling period $h = 0.01$, the event-triggered parameters are given as $\sigma_{i} = 0.1 \; (i = 2, 3, \ldots, 7)$. By using Theorem 1, the related matrices can be obtained, which means bipartite leaderless consensus can be achieved, the event-triggered matrices are given as follows

taking $t\in [0, 2)$, the simulation results show that only $54$ sampled data is used, which takes $27\%$ of the whole sampled signals. Moreover, it can be computed that the average sampling period is $0.0369$, which is $3.69$ times of the sampling period $h = 0.01$. The release instants and release intervals are illustrated in Figure 2. The curve of $x_{ij}(t) \; (i = 1, 2, \ldots, 7;j = 1, 2, 3)$ are presented in Figure 3, it can be seen the bipartite leaderless consensus is achieved. In order to show the benefits of our proposed event-triggered scheme than ^{[39,40,45,46]}, some comparisons are given in Table 1, it can be seen that our event-triggered scheme can lead to larger sampling period $h$, that is, our results are less conservative than ^{[39,40,45,46]}. In addition, by using Remark 7, the upper bound of $\sigma_{i}$ is shown in Table 2 for given coupling strength $c = 10$ and sampled period $h = 0.01$, which plays an important role owing to a big value can further save the network resources.

If the sampling period takes it's upper bound, that is $h = 0.0227$, taking $t\in [0, 2)$, the simulation results show that only $53$ sampled data is used. The release instants and release intervals are illustrated in Figure 4. The curve of $x_{ij}(t) \; (i = 1, 2, \ldots, 7;j = 1, 2, 3)$ are presented in Figure 5, it can be seen the bipartite leaderless consensus is achieved, the event-triggered matrices are given as follows

Case 2: Bipartite leader-following consensus

Suppose that the dynamics of nonlinear multi-agent systems is the same as Case 1, and the communication topology is shown in Figure 6, which is structurally balanced. Let $V_{1} = \{1, 2, 3\}$ and $V_{2} = \{4, 5, 6\}$, then the agents are cooperative in $V_{1}$ or $V_{2}$, the agents are competitive between $V_{1}$ and $V_{2}$, then we can obtain $W = diag\{1, 1, 1, -1, -1, -1\}$. Take $c = 10$, the sampling period $h = 0.01$, the event-triggered parameters $\sigma_{i} = 0.1 \; (i = 1, 2, \ldots, 6)$, the pinning feedback gain $D = diag\{2, 0, 0, 0, 2, 0\}$, By using Theorem 2, the event-triggered matrices can be obtained as follows

taking $t\in [0, 1)$, the simulation results show that only $33$ sampled data is sent out, which takes $33\%$ of the sampled signals. Moreover, it can be computed that our event-triggered scheme can obtain an average sampling period of $0.03$, the release instants and release intervals are illustrated in Figure 7. The curve of $x_{ij}(t) \; (i = 1, 2, \ldots, 6;j = 1, 2, 3)$ are presented in Figure 8, which can be seen the bipartite leader-following consensus is achieved. In order to show the benefits of our proposed triggered mechanism than ^{[39,40,45,46]}, some comparisons are given in Table 3, it can be seen that our event-triggered scheme can lead to larger sampling period $h$, that is, our results are less conservative than ^{[39,40,45,46]}. In addition, by using Theorem 2 and the similar method of Remark 7, the upper bound of $\sigma_{i}$ is shown in Table 2 given the coupling strength $c = 10$ and the sampled period $h = 0.01$.

5.   Conclusions

The event-triggered communication scheme has been proposed to study the bipartite consensus of Lur'e system in the signed networks. By utilizing matrix transformation techniques and algebraic graph theories, some sufficient conditions in terms of LMIs have been established to ensure that both bipartite leaderless and leader-following consensus can be achieved. Compared with some existing event-triggered results, the simulation examples have shown that the proposed event-triggered communication scheme has the advantage to achieve a better performance. It should be emphasized that the sub-network topology is not required to be connected. Notice that the above results are based on the event-triggered parameters that are given as positive constants, it would be interesting to further investigate adaptive event-triggered communication scheme for bipartite consensus, where the triggered parameters can be adjusted with respect to the dynamic errors. The related results can also be extended to the cases with switching topology and time-delay and so on.

Acknowledgments

This work was jointly supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY17F030020. Jiaxing science and technology project under Grant No. 2016AY13011 and 2016AY13013.

Conflict of interest

The author declare no conflicts of interest in this paper.