# Event-Triggered Bipartite Consensus of Single-Integrator Multi-Agent Systems with Measurement Noise.

1. IntroductionRecent years have witnessed the great achievements in studying the consensus problem of multi-agent systems (MASs) which has broad applications in various fields [18]. 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 G is strongly connected, the MAS is proved to achieve bipartite consensus if and only if G 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. [R.sup.nxm] represents the real matrix of n x m order. 0 denotes vector or matrix whose elements are [0.1.sub.n] represents column vector whose elements are 1. sgn(*) represents the sign function. [cross product] represents Kronecker product. For a given matrix or vector X, [X.sup.T], and [parallel]X[parallel] represent the transpose and European norm of X, respectively. [[parallel]X[parallel].sub.F], [[parallel]X[parallel].sub.1], and [[parallel]X[parallel].sub.[infinity]] represent the Frobenius norm, 1-norm, and [infinity]-norm, respectively. Re([lambda]) is the real part of [lambda].

2. Problem Statement

The communication relations among N agents are described by the signed digraph G = (V, E, A), where V = {1, ..., N} and E [subset not equal to] V x V represent the node set and the edge set, respectively. A = ([a.sub.ij]) [member of] [R.sup.NxN], where [a.sub.ij] > 0 and [a.sub.ij] < 0 represent cooperation and competition between agents i and j, respectively. [a.sub.ij] [not equal to] 0 [left and right arrow] (j,i) [member of] E. We assume that [a.sub.ii] = 0 and [a.sub.ij][a.sub.ji] [greater than or equal to] 0, [for all]i, j [member of] V. L = [C.sub.r] - A is the Laplacian matrix of G, where [C.sub.r] = diag([[summation].sup.N.sub.j=1] [absolute value of [a.sub.1j]], ..., [[summation].sup.N.sub.j=1] [absolute value of [a.sub.Nj]]). A signed digraph G = (V, E, A) is said structurally balanced if V can be divided into two subsets [V.sub.1], [V.sub.2], [V.sub.1] [union] [V.sub.2] = V, [V.sub.1] [intersection] [V.sub.2] = [empty set], such that [a.sub.ij] [greater than or equal to] 0, [for all]i, j [member of] [V.sub.p] (p [member of] {1,2}), and [a.sub.ij] [less than or equal to] 0, [for all]i [member of] [V.sub.p], j [member of] [V.sub.q] (p [not equal to] q,p,q [member of] {1,2}). It is said structurally unbalanced otherwise.

Lemma 1 (see [12]). If G is structurally balanced, Laplacian L of G has at least one zero eigenvalue and all of the nonzero eigenvalues have positive real parts. Furthermore, L has only one zero eigenvalue if and only if G has a spanning tree.

Consider a MAS described by

[[??].sub.i] (t) = [u.sub.i] (t), i=1, ..., N, (1)

where [x.sub.i] (t) [member of] [R.sup.n] is the state of the ith agent and [u.sub.i] (t) [member of] [R.sup.n] is the control input. A signed digraph G = (V, E, A) is used to describe interactions among the N agents.

Since communication is often disturbed by measurement noise, we assume the ith agent receives information from its neighbors with measurement noise [x.sub.j] (t) + [[phi].sub.ij] (t), j [member of] [N.sub.i] i = 1, ..., N. In order to reduce the frequency of controller updates, we design the following event-triggered protocol for the ith agent:

[mathematical expression not reproducible] (2)

where i = 1, ..., N, k = 0,1, ..., b(t) >0 is a piecewise continuous function. {[[phi].sub.ij] (t)} is n 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 b(t) [equivalent to] 1, then (2) is reduced to the protocols in [28, 29] without measurement noise.

Let X(t) = [([x.sup.T.sub.1] (t), ..., [x.sup.T.sub.N] (t)).sup.T] and J = diag[([[zeta].sup.T.sub.1] (t), ..., [[zeta].sup.T.sub.1] (t)).sup.T] be Nx[N.sup.2] dimensional block diagonal matrix, where [[zeta].sup.T.sub.i](t) = ([a.sub.i1], ..., [a.sub.iN]) is the ith row element of matrix A. Then the closed-loop system is

[mathematical expression not reproducible] (3)

where [LAMBDA](t) = [([[LAMBDA].sup.T.sub.1] (t), ..., [[LAMBDA].sup.T.sub.1] (t))).sup.T] and [[lambda].sub.i] (t) = [([[LAMBDA].sup.T.sub.1i] (t), ..., [[LAMBDA].sup.T.sub.Ni] (t))).sup.T], i = 1, ..., N. For i,j = 1, ..., N, [[integral].sup.t.sub.0] [[phi].sub.ij] (s)ds = [[LAMBDA].sub.ij] (t) is n dimensional standard Brownian motion. Let e(t) = [([e.sub.1] (t), ..., [e.sub.N](t)).sup.T] be the measurement error, where [e.sub.i] (t) = [x.sub.i] ([t.sub.k]) - [x.sub.i](t), t [member of] [[t.sub.k], [t.sub.k+1]), k = 0,1, .... Then (3) is changed to

[mathematical expression not reproducible] (4)

We present the following definition of event-triggered bipartite consensus for the stochastic system.

Definition 3. Let U = {[u.sub.i], i = 1, ..., N} be an event-triggered protocol. If for any given X(0) [member of] [R.sup.nN], there exist g = [([g.sub.1]..., [g.sub.N]).sup.T] [member of] [R.sup.N], [g.sub.i] [member of] {[+ or -]1}, i = 1, ..., N and n dimensional random vector [v.sup.*],

[mathematical expression not reproducible], (5)

where E[[parallel][v.sup.*][parallel].sup.2] < [infinity], E[v.sup.*] is dependent on communication relations among agents and X(0), which is deterministic.

Then, event-triggered protocol U is called a mean square bipartite consensus protocol.

We introduce the event-triggered condition

[parallel]e(t)[parallel][less than or equal to][c.sub.1] [e.sup.-[alpha]t], (6)

where [c.sub.1] >0,0 < [alpha] < [min.sub.[lambda](L)[not equal to]0]{Re([lambda](L))}. When the measurement error [parallel]e(t)[parallel] is over the threshold, the controller is triggered and updates itself.

To analyze the closed-loop system in (4), we make the following assumptions:

([Q.sub.1]) G = (V, E, A) is structurally balanced.

([Q.sub.2]) G = (V, E, A) contains a spanning tree.

([Q.sub.3]) [[integral].sup.[infinity].sub.0] b(s)ds = [infinity].

([Q.sub.4]) [[integral].sup.[infinity].sub.0] (s)ds < [infinity].

The following lemma plays an important role in the following section.

Lemma 4 (see [16]). Given linear time-varying system

D[Y.sub.1]/dt = -b(t)[F.sup.[lambda].sub.l] [Y.sub.l], l [member of] N, [lambda] [member of] C, t [greater than or equal to] [t.sub.0] [greater than or equal to] 0, (7)

where [mathematical expression not reproducible] dimensional Jordan block, which [lambda] is the diagonal element. Then the state transition matrix of (7) is [mathematical expression not reproducible]. In addition, we can obtain [mathematical expression not reproducible].

Lemma 5. If the event-triggered protocol (2) is a mean square bipartite consensus protocol, then [there exists]g = [([g.sub.1], ... [g.sub.N]).sup.T] [member of] [R.sup.N] [g.sub.i] [member of] {[+ or -]1}, i = 1, ..., N, and [theta] = [([[theta].sub.j], ..., [[theta].sub.N]).sup.T] [member of] [R.sup.N], such that [mathematical expression not reproducible], where [PSI](t, 0) is the state transition matrix of (4).

Proof. From the above condition, Definition 3 implies that for any given initial state X(0), there exist a vector g and a random vector [mathematical expression not reproducible]. Obviously,

[mathematical expression not reproducible] (8)

Without loss of generality, we assume [[integral].sup.t.sub.0] b(s)[PSI](t, s)(L [cross product] [I.sub.n])e(s)ds and [[integral].sup.t.sub.0] b(s)[PSI](t, s) (J [cross product] [I.sub.n])d[LAMBDA](s) converge to [Y.sup.*] and [Z.sup.*] in mean square sense, respectively. Then,

[mathematical expression not reproducible] (9)

where [mathematical expression not reproducible]. According to Definition 3 and the arbitrariness of X(0), one obtains ([[PSI].sub.[infinity]] [cross product] [I.sub.n])X(0) = g [cross product] [Ed.sup.*], where [Ed.sup.*] [member of] [R.sup.N].

Let [mathematical expression not reproducible]. Then, all elements of [[??].sub.i] have the same absolute value. The same applies for [mathematical expression not reproducible], then by making [theta] = 0, Lemma 5 holds. If [[PSI].sub.[infinity]] has at least one nonzero column, without loss of generality, we assume [mathematical expression not reproducible]. Then [mathematical expression not reproducible]. Without loss of generality, we assume [[theta].sub.1] > 0. For any [alpha], [beta] [member of] [R.sup.n], a, b, [member of] R, a, b [not equal to] 0, a[alpha] + b[beta] [not equal to] [+ or -](a[alpha] - b[beta]). If [mathematical expression not reproducible] for some j [not equal to] 1, then all n dimensional components of [mathematical expression not reproducible] have the same modulus if and only if [mathematical expression not reproducible]. If [mathematical expression not reproducible], we have [mathematical expression not reproducible] by taking [[theta].sub.j] = 0. Then [mathematical expression not reproducible]. In addition, [mathematical expression not reproducible].

Lemma 6. If ([Q.sub.1]-[Q.sub.4]) hold, then for any given initial state X(0), there is a random vector [X.sup.*] such that X(t) converges to [X.sup.*] in mean square sense, i.e., [mathematical expression not reproducible].

Proof. If ([Q.sub.1]) and ([Q.sub.2]) hold, then Laplacian L has exactly one zero eigenvalue and all nonzero eigenvalues have positive real parts by Lemma 1. Thus, there exists an invertible matrix D, such that

[D.sup.-1] LD = F = diag (0, [F.sub.2], ..., [F.sub.[gamma]]), (10)

where [F.sub.i] (i = 2, ..., y) is the [R.sub.i]x[R.sub.i] dimensional Jordan block, which [[lambda].sub.i] is the diagonal element, and [R.sub.2] + ... + [R.sub.[gamma]] = N - 1. Obviously, [[lambda].sub.2], ..., [A.sub.y] are eigenvalues of L and Re([[lambda].sub.i]) > 0, i = 2, ..., [gamma].

Since [PSI](t, [t.sub.0]) ([t.sub.0] [greater than or equal to] 0) is the state transition matrix of (4), [mathematical expression not reproducible]. From Lemma 4,

[mathematical expression not reproducible] (11)

Combining this with ([Q.sub.3]), one has

[mathematical expression not reproducible]. (12)

Thus, there exists T >0 so that for any t [greater than or equal to] [t.sub.0] > 0,

max ([[parallel]Y(t,[t.sub.0])[parallel].sub.1], [[parallel][PSI](t, [t.sub.0])[parallel].sub.[infinity]]) [less than or equal to] T < [infinity]. (13)

By Ito formula, the solution of (4) is given by

[mathematical expression not reproducible] (14)

By (Q4), one obtains that, [mathematical expression not reproducible]. By (12), [mathematical expression not reproducible].

Let [X.sub.2](t) = [[integral].sup.t.sub.0] b(s)[PSI](t, s)(L [cross product] [I.sub.n])e(s)ds, then by (10) and (11), one has

[mathematical expression not reproducible] (15)

By (6), (10), and direct calculation, one has ([D.sup.-1] L[cross product][I.sub.n])e(s) = [(0, [D.sup.T.sub.2] (s), ..., [D.sup.T.sub.N] (s)).sup.T], where [D.sub.i] (s) (i = 2, ..., N) is the linear combination of [e.sub.1](s), ..., [e.sub.N] (s). By L'Hospital and direct calculation, one obtains

[mathematical expression not reproducible] (16)

Noticing that [mathematical expression not reproducible], one has [mathematical expression not reproducible].

Let [X.sub.3] (t) = [[integral].sup.t.sub.0] b(s)[PSI](t,s)(J [cross product] [I.sub.n])d[LAMBDA](s), then

[mathematical expression not reproducible] (17)

Therefore [mathematical expression not reproducible]. It is easy to obtain

[mathematical expression not reproducible] (18)

Noting

[mathematical expression not reproducible] (19)

and

[mathematical expression not reproducible] (20)

one has E[[parallel][X.sub.31][parallel].sup.2] [less than or equal to] [M.sub.4][[epsilon].sup.2] + [M.sub.5][epsilon]. Similarly, one obtains

[mathematical expression not reproducible] (21)

So E[parallel][X.sub.3]([t.sub.2]) - [X.sub.3]([t.sub.1])[parallel].sup.2] [less than or equal to] 2[M.sub.4][[epsilon].sup.2] + (5/2)[M.sub.5] [epsilon]. By Cauchy criterion and the arbitrariness of [epsilon], there exists [X.sup.*.sub.3] such that [X.sub.3] (f) converges to [X-.sub.3] in mean square sense. So there exists [X.sup.*] such that X(t) converges to [X.sup.*] in mean square sense. By (12), [X.sup.*] = [D diag(1,0, ..., 0)[D.sup.-1] [cross product] [I.sub.n]] X(0) + [X.sup.*.sub.3].

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 ([Q.sub.1])-([Q.sub.4]) hold.

Proof (sufficiency).

(5.1) Construct a Bipartition for the MAS. By ([Q.sub.3]), V can be decomposed into two disjoint subsets [mathematical expression not reproducible]. Without loss of generality, we assume [mathematical expression not reproducible]. Let [mathematical expression not reproducible]. By definition, one has [L.sub.g] = 0, where g = [([g.sub.1], ..., [g.sub.n]).sup.T].

(S.2) Prove [mathematical expression not reproducible]. From Lemma 6, [mathematical expression not reproducible]. Without loss of generality, we assume [mathematical expression not reproducible]. Next, we will prove [mathematical expression not reproducible].

Let [phi](f) = [([[phi].sup.T.sub.2](t), ..., [[phi].sup.T.sub.N](t)).sup.T], where [[phi].sub.i] (t) = [g.sub.i][x.sub.1](t) i = 2, ..., N. Now we prove that [mathematical expression not reproducible]. For this purpose, We assume [mathematical expression not reproducible], where

[mathematical expression not reproducible] (22)

Then [mathematical expression not reproducible]. Since

[mathematical expression not reproducible] (23)

by (4), one has

[mathematical expression not reproducible] (24)

By (23), [S.sup.-1] [L.sub.2]S = diag([F.sub.2], ..., [F.sub.[gamma]]), where S is invertible and [F.sub.2], ..., [F.sub.[gamma]] are given in (10). The state transition matrix of the system in (24) is

[mathematical expression not reproducible] (25)

where [mathematical expression not reproducible] are defined as in Lemma 4. Hence, [mathematical expression not reproducible]. Furthermore, [mathematical expression not reproducible], such that, [mathematical expression not reproducible].

By Ito formula, it can be seen that the state of the system in (24) can be described as

[mathematical expression not reproducible] (26)

Therefore,

[mathematical expression not reproducible] (27)

and hence,

[mathematical expression not reproducible] (28)

Since

[mathematical expression not reproducible] (29)

and

[mathematical expression not reproducible] (30)

there exists [mathematical expression not reproducible]. Then

[mathematical expression not reproducible] (31)

Since [mathematical expression not reproducible].

[mathematical expression not reproducible] (32)

From [mathematical expression not reproducible], one gets

[mathematical expression not reproducible] (33)

Combining this with

[mathematical expression not reproducible] (34)

one has

[mathematical expression not reproducible] (35)

By the arbitrariness of [epsilon], one gets [mathematical expression not reproducible].

(S.3) Analyze the Statistical Characteristics of [v.sup.*]. By Lemma 6, g [cross product] [v.sup.*] = [X.sup.*] = [Ddiag(1,0,0, ..., 0)[D.sup.-1] [cross product] [I.sub.n]X(0) + [X.sup.*.sub.3]. So g [cross product] [Ev.sup.*] = [EX.sup.*] = [Ddiag(1,0,0, ..., 0)[D.sup.-1] [cross product] [I.sub.n]X(0).

We assume [m.sub.r], [m.sup.T.sub.l] represent the first column of D and the first row of [D.sup.-1], respectively. Then, g [cross product] [cross product][v.sup.*] = ([m.sub.r] [m.sup.T.sub.l] [cross product] [I.sub.n])X(0). Since [D.sup.-1] LD = F, LD = DF, and [D.sup.-1] L = F[D.sup.-1], L[m.sub.r] = 0 and [m.sup.T.sub.l] L = 0. By (S.1), Lg = 0. Therefore, [m.sub.r] = kg ([kappa] [member of] R and [kappa] [not equal to] 0) and g [cross product] [Ev.sup.*] = g [cross product] [[kappa]([m.sup.T.sub.l] [cross product] [I.sub.n])X(0)]. Then [Ev.sup.*] = [kappa]([m.sup.T.sub.l] [cross product] [I.sub.n])X(0). Clearly, is concerned with communication topology. Thus, [Ev.sup.*] is determined by X(0) and communication topology of MASs.

It is easy to obtain that [PSI](,) is uniformly bounded. Therefore, [for all][epsilon] > 0, [there exist][[GAMMA].sub.3] > [[GAMMA].sub.2],

[mathematical expression not reproducible] (36)

Let [mathematical expression not reproducible].

Then for any [mathematical expression not reproducible]. This together with (36) leads to [mathematical expression not reproducible]. Combining this with [mathematical expression not reproducible], one gets D([v.sup.*]) = [THETA][I.sub.n]. Therefore, E[[parallel]v*[parallel].sup.2] < [infinity]. By Definition 3, the sufficiency is established. Necessity.

(B.1) Prove ([Q.sub.3]), Namely, [[integral].sup.[infinity].sub.0] b(s)ds = [infinity]. By contradiction, we assume that ([Q.sub.3]) does not hold. Then, [mathematical expression not reproducible]. Therefore, [mathematical expression not reproducible]. However, by Lemma 5, [mathematical expression not reproducible]. This is a contradiction. So ([Q.sub.3]) holds.

(B.2) Prove That Laplacian Matrix L Has Exactly One Zero Eigenvalue. By contradiction, we assume that 0 is not an eigenvalue of L. Then all the eigenvalues of L have positive real part and -L is a Hurwitz matrix. By ([Q.sub.3]) and Lemma 4, [mathematical expression not reproducible]. Combining this with Lemma 5, one has g [cross product] [Ev.sup.*] = [EZ.sup.*] - [EY.sup.*]. Since [EZ.sup.*] and [EY.sup.*] are independent of X(0), [Fv.sup.*] is independent of X(0). This contradicts Definition 3. So 0 is an eigenvalue of L.

Let [F.sup.0.sub.1] be a Jordan block with eigenvalue 0. Then it is 1 dimensional. Otherwise, we assume [F.sup.0.sub.1] is dimensional and [R.sub.1] > 1. Then, by ([Q.sub.3]) and the definition of matrix exponent function, one gets that [mathematical expression not reproducible] does not exist, and hence, [mathematical expression not reproducible] does not exist. This contradicts Lemma 5. So [F.sup.0.sub.1] is 1 dimensional.

Let algebra multiplicity of eigenvalue 0 be u>. Then w =1. Otherwise, w >1. Take w = 2 as an example. Since each Jordan block corresponding to eigenvalue 0 is 1 dimensional,

[mathematical expression not reproducible] (37)

Thus, rank([[PSI].sub.[infinity]]) = 2. This contradicts rank([[PSI].sub.[infinity]]) [less than or equal to] 1 from Lemma 6. So Laplacian L has exactly one zero eigenvalue.

(B.3) Prove ([Q.sub.1]) and ([Q.sub.2]). By (B.2) and ([Q.sub.3]), one has (12). By Lemma 5, one gets

D diag (1,0, 0, ..., 0) [D.sup.-1] = g[[theta].sup.T] (38)

Noticing that [m.sub.r] is the first column of D, one has L[m.sub.r] = 0. By (38), one obtains [m.sub.r] = g[kappa]*, where [kappa]* = [[theta].sup.T] [m.sub.r] [member of] R. Then, Lg = 0. By the definition of L, for any j, we obtain [mathematical expression not reproducible]. By definition, G is structurally balanced, that is, ([Q.sub.1]) holds.

By (B.2) and ([Q.sub.1]), Lemma 1 implies that ([Q.sub.2]) holds.

(B.4) Prove ([Q.sub.4]). Assume [[integral].sup.[infinity].sub.0] [b.sup.2] (s)ds = [infinity]. Due to the first row of [D.sup.-1] which is, [m.sup.T.sub.l] L = 0. By (4), we obtain d(([m.sup.T.sub.l] [cross product] [I.sub.n])X(f)) = b(f)([m.sup.T.sub.l] [cross product] [I.sub.n])(J [cross product] [I.sub.n])d[LAMBDA](t), i.e.,

[mathematical expression not reproducible] (39)

From Definition 3, it is known that X(t) converges to g [cross product] [v.sup.*] in mean square sense, where E[[parallel][v.sup.*][parallel].sup.2] < [infinity]. Thus, when t [right arrow] [infinity], ([m.sup.T.sub.l] [cross product] [I.sub.n])(J [cross product] [I.sub.n]) [[integral].sup.T.sub.0] b(s)d[LAMBDA](s) converges to a random variable [X.sub.m] in mean square sense with E[[parallel][X.sub.m][parallel].sup.2] < [infinity]. Then [mathematical expression not reproducible]. This leads to a contradiction. So ([Q.sub.4]) holds.

Remark 8. From Theorem 7 it can be seen that under ([Q.sub.1])-([Q.sub.4]) the event-triggered protocol in (2) ensures agents converging to [v.sup.*] or -[v.sup.*] under measurement noise.

Remark 9. From Theorem 7 one sees that to guarantee the mean square bipartite consensus, ([Q.sub.1])-([Q.sub.2]) are requirements for time-varying gain b(t) while ([Q.sub.3])-([Q.sub.4]) 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, V = {1, ..., 6}, A = ([a.sub.ij]) [a.sub.16] = [a.sub.61] = [a.sub.54] = 1, [a.sub.21] = [a.sub.35] = -1, and [a.sub.42] = [a.sub.63] = 2 in G = (V, E, A). From Figure 1, G satisfies ([Q.sub.1]) and ([Q.sub.2]). Furthermore, all eigenvalues of Laplacian L are [[lambda].sub.1] = 0, [[lambda].sub.2] = 0.6733 + 0.9192j, [A.sub.3] = 0.6733-0.9192j, [A.sub.4] = 2.0887 + 0.7157j, [A.sub.5] = 2.0887 - 0.7157j, and [A.sub.6] = 3.4760 ([j.sup.2] = -1). Obviously, [min.sub.[lambda](L)[not equal to]0] {Re [lambda](L)} = 0.6733. The initial state of the MAS is given by X(0) = (10,-15,20,-25,30,-5). Choose b(t) = ln(f + 1)/(f + 1). By direct calculation we know that b(t) satisfies ([Q.sub.3])-([Q.sub.4]). Assume event-triggered condition (6) is satisfied by taking [c.sub.1] = 1.2 and [alpha] = 0.6. 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.

https://doi.org/10.1155/2018/7186737

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.

References

[1] R. Olfati-Saber and R. M. Murray, "Consensus problems in networks of agents with switching topology and time-delays," IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520-1533, 2004.

[2] W. Ren and R. W. Beard, "Consensus seeking in multiagent systems under dynamically changing interaction topologies," IEEE Transactions on Automatic Control, vol. 50, no. 5, pp. 655-661, 2005.

[3] C. Q. Ma and J. F. Zhang, "Necessary and sufficient conditions for consensusability of linear multi-agent systems," IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1263-1268,2010.

[4] K. You and L. Xie, "Network topology and communication data rate for consensusability of discrete-time multi-agent systems," IEEE Transactions on Automatic Control, vol. 56, no. 10, pp. 2262-2275, 2011.

[5] C. Q. Ma, T. Li, and J. F. Zhang, "Consensus control for leader-following multi-agent systems with measurement noises," Journal of Systems Science and Complexity, vol. 23, no. 1, pp. 35-49, 2010.

[6] Z. Li, Z. Duan, G. Chen, and L. Huang, "Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 57, no. 1, pp. 213-224, 2010.

[7] C. Q. Ma and J. F. Zhang, "On formability of linear continuous multi-agent systems," Journal of Systems Science and Complexity, vol. 25, no. 1, pp. 13-29, 2012.

[8] X. Dong and G. Hu, "Time-varying formation control for general linear multi-agent systems with switching directed topologies," Automatica, vol. 73, pp. 47-55, 2016.

[9] C. Altafini, "Consensus problems on networks with antagonistic interactions," IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 935-946, 2013.

[10] J. Hu and W. X. Zheng, "Emergent collective behaviors on coopetition networks," Physics Letters A, vol. 378, no. 26-27, pp. 1787-1796, 2014.

[11] A. V. Proskurnikov, A. S. Matveev, and M. Cao, "Opinion dynamics in social networks with hostile camps: consensus vs. polarization," Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 61, no. 6, pp. 1524-1536, 2016.

[12] D. Meng, M. Du, and Y. Jia, "Interval bipartite consensus of networked agents associated with signed digraphs," Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 61, no. 12, pp. 3755-3770, 2016.

[13] C. Q. Ma and Z. Y. Qin, "Bipartite consensus on networks of agents with antagonistic interactions and measurement noises," IET Control Theory & Applications, vol. 10, no. 17, pp. 2306-2313, 2016.

[14] J. Hu and Y. Wu, "Interventional bipartite consensus on coopetition networks with unknown dynamics," Journal of The Franklin Institute, vol. 354, no. 11, pp. 4438-4456, 2017.

[15] H. Zhang and J. Chen, "Bipartite consensus of multi-agent systems over signed graphs: state feedback and output feedback control approaches," International Journal of Robust and Nonlinear Control, vol. 27, no. 1, pp. 3-14, 2017.

[16] C. Q. Ma, Z. Y. Qin, and Y. B. Zhao, "Bipartite consensus of integrator multi-agent systems with measurement noise," IET Control Theory & Applications, vol. 11, no. 18, pp. 3313-3320,2017.

[17] C. Q. Ma, W. Zhao, and Y. B. Zhao, "Bipartite linear x-consensus of double-integrator multi-agent systems with measurement noise," Asian Journal of Control, vol. 20, no. 1, pp. 577-584,2018.

[18] C. Q. Ma and L. Xie, "Necessary and sufficient conditions for leader-following bipartite consensus with measurement noise," IEEE Transactions on Systems, Man, and Cybernetics: Systems, pp. 1-6, 2018.

[19] 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.

[20] G. S. Seyboth, D. V Dimarogonas, and K. H. Johansson, "Event-based broadcasting for multi-agent average consensus," Automatica, vol. 49, no. 1, pp. 245-252, 2013.

[21] B. Wang, X. Meng, and T. Chen, "Event based pulse-modulated control of linear stochastic systems," Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 59, no. 8, pp. 2144-2150, 2014.

[22] B. Wang and M. Fu, "Comparison of periodic and event-based sampling for linear state estimation," IFAC Proceedings Volumes, vol. 47, no. 3, pp. 5508-5513, 2014.

[23] S. Liu, D. E. Quevedo, and L. Xie, "Event-triggered distributed constrained consensus," International Journal of Robust and Nonlinear Control, vol. 27, no. 16, pp. 3043-3060, 2017.

[24] E. Garcia, Y. Cao, and D. W. Casbeer, "Decentralised event-triggered consensus of double integrator multi-agent systems with packet losses and communication delays," IET Control Theory & Applications, vol. 10, no. 15, pp. 1835-1843, 2016.

[25] D. Yang, W. Ren, X. Liu, and W. Chen, "Decentralized event-triggered consensus for linear multi-agent systems under general directed graphs," Automatica, vol. 69, pp. 242-249, 2016.

[26] W. Hu and L. Liu, "Cooperative output regulation of heterogeneous linear multi-agent systems by event-triggered control," IEEE Transactions on Cybernetics, vol. 47, no. 1, pp. 105-116,2017.

[27] A. Hu, J. Cao, M. Hu, and L. Guo, "Event-triggered consensus of multi-agent systems with noises," Journal of The Franklin Institute, vol. 352, no. 9, pp. 3489-3503, 2015.

[28] Y. Zhou and J. Hu, "Event-based bipartite consensus on signed networks," in Proceedings of the 2013 IEEE 3rd Annual International Conference on Cyber Technology in Automation, Control, and Intelligent Systems (CYBER), pp. 326-330, China, May 2013.

[29] J. Zeng, F. Li, J. Qin, and W. X. Zheng, "Distributed event-triggered bipartite consensus for multiple agents over signed graph topology," in Proceedings of the 2015 34th Chinese Control Conference (CCC), pp. 6930-6935, Hangzhou, China, July 2015.

Cui-Qin Ma, (1) Yun-Bo Zhao, (2) and Wei-Guo Sun (1)

(1) School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

(2) College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China

Correspondence should be addressed to Cui-Qin Ma; cuiqinma@amss.ac.cn

Received 29 May 2018; Accepted 22 July 2018; Published 6 August 2018

Academic Editor: Yong Chen

Caption: FIGURE 1: Communication graph G among the 6 agents.

Caption: FIGURE 2: State trajectories of six agents.

Caption: FIGURE 3: Control inputs of six agents.

Caption: FIGURE 4: The evolution of error norm.

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Title Annotation: | Research Article |
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Author: | Ma, Cui-Qin; Zhao, Yun-Bo; Sun, Wei-Guo |

Publication: | Journal of Control Science and Engineering |

Article Type: | Report |

Date: | Jan 1, 2018 |

Words: | 5428 |

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