Abstract
Event-triggered bipartite consensus of single-integrator multi-agent systems is investigated in the presence of measurement noise. A time-varying gain function is proposed in the event-triggered bipartite consensus protocol to reduce the negative effects of the noise corrupted information processed by the agents. Using the state transition matrix, It formula, and the algebraic graph theory, necessary and sufficient conditions are given for the proposed protocol to yield mean square bipartite consensus. We find that the weakest communication requirement to ensure the mean square bipartite consensus under event-triggered protocol is that the signed digraph is structurally balanced and contains a spanning tree. Numerical examples validated the theoretical findings where the system shows no Zeno behavior.
1. Introduction
Recent years have witnessed the great achievements in studying the consensus problem of multi-agent systems (MASs) which has broad applications in various fields [1–8]. We notice that in these mentioned works interactions among agents are all assumed to be cooperative to achieve consensus. However, it is very natural to see, in many real examples, that in MASs some agents cooperate while others compete, and MASs with competitive interactions can introduce more complex behaviors. To quantitatively model such a scenario, the concept of bipartite consensus, i.e., agents agree on a certain quantity with the equal modulus but different signs, has been proposed [9], and many achievements have been made [9–18]. In [9], for single-integrator MASs, a linear feedback protocol is designed and under the assumption that the communication topology is strongly connected, the MAS is proved to achieve bipartite consensus if and only if is structurally balanced. Then, in [10], the communication condition in [9] is relaxed to containing a spanning tree. In [11], the communication topology in [9] is extended to the time-varying case.
It is worth noting that the above literatures mainly focus on continuous feedback protocols, where the agent state is monitored continuously and its controller is updated all the time. However, updating the controller in real-time easily increases the computational burden. Therefore, reducing the update frequency for a trade-off between the system performance and the resource usage is usually desired. This requirement then naturally brings event-triggered schemes into consideration, which updates only at some predetermined discrete time instants. Event-triggered techniques have already been widely used in traditional consensus problems of MASs [19–27]. For example, a self-triggered protocol is proposed in [19] and a decentralized event-triggered protocol ensuring average consensus is proposed in [20] for single-integrator MASs, time-dependent triggering functions are investigated in [24] for second-order MASs, and event-triggered consensus problems are considered in [25, 26] for general linear systems, just name a few. Despite these achievements, event-triggered protocols have not been well studied for bipartite consensus [28, 29], which thus motivates the present study.
In another parallel line, measurement noise is unavoidable in practice, making the investigation on the event-triggered bipartite consensus of MASs with noise even interesting. In fact, studies on bipartite consensus with measurement noise can be found in [13, 16–18], which are however all with time-triggered controllers. Event-triggered bipartite consensus for MASs with measurement noise still remains to tackle.
In this paper, we investigate event-triggered bipartite consensus for single-integrator MASs with measurement noise. A time-varying control gain is introduced into the event-triggered protocols, leading to a time-varying closed-loop system. With the help of the state transition matrix and stochastic analysis theory, the closed-loop system is analyzed. Necessary and sufficient conditions for the system to achieve mean square bipartite consensus based on event-triggered protocols are given. We find that the communication topology being structurally balanced and containing a spanning tree are necessary and sufficient for ensuring a mean square bipartite consensus based on event-triggered protocols.
Organization. Section 2 gives the algebraic graph preliminaries and the problem in question. Section 3 contains the main results of the paper. Section 4 applies the results to examples of MASs with six agents. Section 5 closes this paper.
Notations. represents the real matrix of order. 0 denotes vector or matrix whose elements are 0. represents column vector whose elements are 1. represents the sign function. represents Kronecker product. For a given matrix or vector , , and represent the transpose and European norm of , respectively. , , and represent the Frobenius norm, 1-norm, and -norm, respectively. is the real part of .
2. Problem Statement
The communication relations among agents are described by the signed digraph , where and represent the node set and the edge set, respectively. , where and represent cooperation and competition between agents and , respectively. . We assume that and , . is the Laplacian matrix of , where . A signed digraph is said structurally balanced if can be divided into two subsets , such that , , and , . It is said structurally unbalanced otherwise.
Lemma 1 (see [12]). If is structurally balanced, Laplacian of has at least one zero eigenvalue and all of the nonzero eigenvalues have positive real parts. Furthermore, has only one zero eigenvalue if and only if has a spanning tree.
Consider a MAS described bywhere is the state of the th agent and is the control input. A signed digraph is used to describe interactions among the agents.
Since communication is often disturbed by measurement noise, we assume the th agent receives information from its neighbors with measurement noise In order to reduce the frequency of controller updates, we design the following event-triggered protocol for the th agent:where is a piecewise continuous function. is dimensional independent standard white noise.
Remark 2. As far as we know the existing results [28, 29] for event-triggered bipartite consensus did not consider measurement noise. Here, we take noise into consideration. If we take , then (2) is reduced to the protocols in [28, 29] without measurement noise.
Let and be dimensional block diagonal matrix, where is the th row element of matrix . Then the closed-loop system iswhere and , . For , is dimensional standard Brownian motion. Let be the measurement error, where , . Then (3) is changed toWe present the following definition of event-triggered bipartite consensus for the stochastic system.
Definition 3. Let be an event-triggered protocol. If for any given , there exist , , and dimensional random vector , where , is dependent on communication relations among agents and , which is deterministic. Then, event-triggered protocol is called a mean square bipartite consensus protocol.
We introduce the event-triggered conditionwhere . When the measurement error is over the threshold, the controller is triggered and updates itself.
To analyze the closed-loop system in (4), we make the following assumptions: is structurally balanced. contains a spanning tree. .
The following lemma plays an important role in the following section.
Lemma 4 (see [16]). Given linear time-varying systemwhere and is the dimensional Jordan block, which is the diagonal element. Then the state transition matrix of (7) is . In addition, we can obtain if and .
Lemma 5. If the event-triggered protocol (2) is a mean square bipartite consensus protocol, then , , , and , such that , where is the state transition matrix of (4).
Proof. From the above condition, Definition 3 implies that for any given initial state , there exist a vector and a random vector so that . Obviously, Without loss of generality, we assume and converge to and in mean square sense, respectively. Then, where . According to Definition 3 and the arbitrariness of , one obtains , where .
Let . Then, all elements of have the same absolute value. The same applies for , where . If , then by making , Lemma 5 holds. If has at least one nonzero column, without loss of generality, we assume . Then . Without loss of generality, we assume . For any , . If for some , then all dimensional components of have the same modulus if and only if . If , we have by taking . Then . In addition, , so .
Lemma 6. If hold, then for any given initial state , there is a random vector such that converges to in mean square sense, i.e., .
Proof. If and hold, then Laplacian has exactly one zero eigenvalue and all nonzero eigenvalues have positive real parts by Lemma 1. Thus, there exists an invertible matrix , such thatwhere is the dimensional Jordan block, which is the diagonal element, and . Obviously, are eigenvalues of and
Since is the state transition matrix of (4), . From Lemma 4,Combining this with , one hasThus, there exists so that for any ,By It formula, the solution of (4) is given by By , one obtains that, , , . By (12), , such that , , .
Let , then by (10) and (11), one hasBy (6), (10), and direct calculation, one has , where is the linear combination of . By L’Hospital and direct calculation, one obtains Noticing that , one has .
Let , then Therefore It is easy to obtain Notingand one has Similarly, one obtains So . By Cauchy criterion and the arbitrariness of , there exists such that converges to in mean square sense. So there exists such that converges to in mean square sense. By (12), .
3. Main Results
In this section, we give necessary and sufficient conditions for the proposed event-triggered protocols to guarantee a mean square bipartite consensus.
Theorem 7. The event-triggered protocol in (2) is a mean square bipartite consensus protocol for the system in (1) if and only if - hold.
Proof (sufficiency). (S.1) Construct a Bipartition for the MAS. By (, can be decomposed into two disjoint subsets , , , and for , and for . Without loss of generality, we assume , . Let for and for . By definition, one has , where .
(S.2) Prove . From Lemma 6, , . Without loss of generality, we assume . Next, we will prove
Let , where . Now we prove that . For this purpose, We assume , whereThen , where . Sinceby (4), one hasBy (23), , where is invertible and are given in (10). The state transition matrix of the system in (24) is where are defined as in Lemma 4. Hence, , i.e., , , such that , . Furthermore, , such that, , .
By It formula, it can be seen that the state of the system in (24) can be described as Therefore, and hence, Since and there exists such that . Then Since , where and , one has Therefore, From , one gets Combining this with one has By the arbitrariness of , one gets . Hence, .
(S.3) Analyze the Statistical Characteristics of . By Lemma 6, So
We assume , represent the first column of and the first row of , respectively. Then, . Since , , and , and . By , . Therefore, and and . Then . Clearly, is concerned with communication topology. Thus, is determined by and communication topology of MASs.
It is easy to obtain that is uniformly bounded. Therefore, , ,Let . Then for any , . This together with (36) leads to , where . Combining this with , one gets . Therefore, . By Definition 3, the sufficiency is established.
Necessity.
(B.1) Prove , Namely, . By contradiction, we assume that does not hold. Then, , , and . Therefore, However, by Lemma 5, and . This is a contradiction. So holds.
(B.2) Prove That Laplacian Matrix Has Exactly One Zero Eigenvalue. By contradiction, we assume that is not an eigenvalue of . Then all the eigenvalues of have positive real part and is a Hurwitz matrix. By and Lemma 4, . Combining this with Lemma 5, one has . Since and are independent of , is independent of . This contradicts Definition 3. So is an eigenvalue of .
Let be a Jordan block with eigenvalue 0. Then it is 1 dimensional. Otherwise, we assume is dimensional and . Then, by and the definition of matrix exponent function, one gets that does not exist, and hence, does not exist. This contradicts Lemma 5. So is 1 dimensional.
Let algebra multiplicity of eigenvalue 0 be . Then . Otherwise, . Take as an example. Since each Jordan block corresponding to eigenvalue 0 is 1 dimensional, Thus, . This contradicts from Lemma 6. So Laplacian has exactly one zero eigenvalue.
(B.3) Prove and . By and , one has (12). By Lemma 5, one getsNoticing that is the first column of , one has . By (38), one obtains , where . Then, . By the definition of , for any , we obtain Since and , . So . Let and , then , . If , then or . By definition, is structurally balanced, that is, holds.
By and , Lemma 1 implies that holds.
(B.4) Prove . Assume . Due to the first row of which is , . By (4), we obtain , , From Definition 3, it is known that converges to in mean square sense, where . Thus, when , converges to a random variable in mean square sense with . Then . This leads to a contradiction. So holds.
Remark 8. From Theorem 7 it can be seen that under - the event-triggered protocol in (2) ensures agents converging to or under measurement noise.
Remark 9. From Theorem 7 one sees that to guarantee the mean square bipartite consensus, - are requirements for time-varying gain while - are the weakest connectivity assumptions.
4. Numerical Simulation
To demonstrate the developed result in the preceding, we consider an MAS of six agents, whose dynamics satisfy the system in (1). The communication graph that connects the six agents is illustrated in Figure 1. Clearly, , , , , and in . From Figure 1, satisfies and . Furthermore, all eigenvalues of Laplacian are , , , , , and . Obviously, . The initial state of the MAS is given by . Choose . By direct calculation we know that satisfies -. Assume event-triggered condition (6) is satisfied by taking and . Applying protocol (2) to the system in (1), we get the six agents’ state trajectories. As shown in Figure 2 one can see that the states of agents 1, 3, and 6 converge to 5 in mean square sense while the states of agents 2, 4, and 5 converge to -5 in mean square sense. Thus, mean square bipartite consensus is achieved with event-triggered protocol (2). On the other hand, from Figure 3 we know that the inputs are constants between the event triggering time interval. Moreover, from Figure 4, it can be seen that the absolute value of the measurement error of each agent converges to zero. This means that the MAS does not exhibit Zeno behavior.
5. Conclusion
Mean square bipartite consensus problem of single-integrator MASs is investigated in the context of event-triggered control and measurement noise. By using time-varying gain, an event-triggered bipartite consensus protocol is proposed under measurement noise, with which the controller update frequency is reduced. With given necessary and sufficient conditions on protocol gain and communication topology, the MAS is proved to achieve event-triggered bipartite consensus. The simulation shows that the system will not show Zeno behavior.
Data Availability
The Matlab based models used to support the findings of this study are available from the corresponding author upon request.
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 Grants 61104136 and 61673350, Postgraduate Education Innovation Program of Shandong Province under Grant no. SDYY16088, and the Young Teacher Capability Enhancement Program for Domestic Study, Qufu Normal University.