This study investigates the bipartite synchronization of heterogeneous signed networks with distributed impulsive control. Leader-follower bipartite synchronization within a nonzero error bound is analyzed when the average impulsive interval is or . Some sufficient conditions to achieve the bipartite quasi-synchronization are presented, and the synchronization error level is estimated by the specific mathematical expression. The correctness of the theoretical results is verified by numerical examples.

1. Introduction

There are some cooperative behaviors of interconnected dynamic systems, such as image processing [1], pattern recognition [2], and secure communication [3], which have been eliciting increasing attention recently. In particular, synchronization, as a typical collective behavior, has been widely studied by many researchers through different control methods, for example, feedback control [4, 5], adaptive control [68], event-triggered control [911], and impulsive control [1218]. Among these methods, impulsive control, as a form of discontinuous control [19], has advantages of low cost and high efficiency and so has been widely used in many fields, such as the population growth model [19] and neural networks [20]. Particularly, a distributed impulsive controller was proposed in [21] and successfully applied to synchronization of a leaderless network. Later, the method of distributed impulsive control was also used to investigate the consensus of multiagent systems [22, 23].

In most early studies, in order to make the system achieve the desired trajectory, the controller is usually applied to all nodes. However, this is difficult to implement or even is inapplicable, especially for large networks. Pinning control is a kind of control strategy which only needs control finite nodes of the system [24, 25]. Considering the advantages of pinning control and impulsive control, the pinning impulsive controller was proposed in [26]. Lu et al. [27] studied the synchronization of complex dynamic networks by using a pinning impulsive controller. In [28], a distributed pinning impulsive controller was developed to investigate the synchronization of heterogeneous networks.

In most of the aforementioned literature, the upper bound and lower bound of the impulsive interval determine the frequency of the impulse occurrence. Lu et al. [29] proposed the concept of average impulsive interval, and then it was used to describe a sparse impulsive sequence with an infinite average impulsive interval in [30]. On the contrary, most studies on interactions among individual systems are assumed to be unsigned, where the weights of the adjacency matrix are nonnegative. In fact, it is more reasonable to consider signed networks since the agents in a network may be cooperative or competitive so that interaction weights between agents can either be positive or negative. Such weights are called signed interaction digraphs. Coordination control on signed networks has become a hot topic in recent years [31, 32]. Proskurnikov et al. [31] investigated opinion dynamics of social networks. Xia et al. [32] studied the separation of opinion in trust-mistrust social networks. Meng et al. [33] achieved bipartite consensus on a signed network. The concept of bipartite synchronization was presented in [34] under a signed network. In [35], bipartite synchronization under a pinning controller was studied on signed and switching networks.

Based on the aforementioned studies, we intend to study the bipartite synchronization of a signed network with distributed impulsive control. The following factors are integrated into our study: (i) interaction graphs are signed; in other words, the weights of the adjacency matrix can be positive or negative; (ii) two error levels are given for bipartite quasi-synchronization of asymmetric networks; (iii) more general impulsive sequences whose average impulsive interval is finite or infinite are considered. By designing a distributed impulsive controller, this study aims to develop the heterogeneous signed network that can be bipartitely quasi-synchronized.

The rest of the paper is organized as follows: in Section 2, we formulate the problem of bipartite quasi-synchronization between the leader and the heterogeneous systems under a distributed impulsive controller and give some necessary preparations. In Section 3, we establish some bipartite quasi-synchronization criteria for the signed network. In Section 4, we design numerical examples to verify the correctness of our theoretical results. In Section 5, we summarize all results in a conclusion.

Notations: throughout the paper, denotes an -dimensional identity matrix; we write if matrix is positive definite; is the maximum singular value for matrix ; is the Euclidean vector norm; is the Kronecker product.

2. Problem Descriptions and Preliminaries

2.1. Problem Formulation

For a heterogeneous system, we use a signed graph to denote its underlying topology, where is the set of nodes, is the set of edges, is the weighted adjacency matrix, and can be positive or negative.

There are some necessary definitions and lemmas in the process of theoretical derivation.

Definition 1. (see [36]). The sighed graph is called structurally balanced if its node set can be partitioned into two disjoint subsets and such that their induced subgraphs and are both unsigned, and the weights of all links between and are negative.

Lemma 1. (see [36]). If a signed graph is structurally balanced, then there exists a diagonal matrix of form , such that all entries of are nonnegative.

Assume that the nodes of the heterogeneous system have the following dynamics:where is the state of the node; is a nonlinear function on ; are constant matrices; and is a control protocol.

The leader node of nonlinear system (1) is described bywhere is bounded.

Assumption 1. The nonlinear function is odd and satisfies the Lipschitz condition. Specifically,and there are nonnegative constants such that, for any ,

Since (1) is heterogeneous, it is difficult to realize full synchronization by a static controller. In recent years, quasi-synchronization about heterogeneous networks received wide attention [28, 35], but this is not applicable for signed heterogeneous networks. According to dynamic properties of network (1) and leader (2), a more general synchronization concept, called bipartite quasi-synchronization, is introduced below.

Definition 2. Suppose that the signed graph of (1) is structurally balanced, , is a diagonal matrix such that all entries of are nonnegative. Signed network (1) is said to achieve bipartite quasi-synchronization if the error converges to some compact set when .

Obviously, when for , bipartite quasi-synchronization becomes quasi-synchronization.

Suppose that the signed graph of (1) is structurally balanced, , is a fixed diagonal matrix such that all entries of are nonnegative. The distributed impulsive protocol with pinning control of (1) is designed aswhere is a constant and if node is pinned; otherwise, . is the Dirac function, and is the impulse sequence satisfying .

Remark 1. Controller (5) indicates that if two nodes and are cooperative, , and the coupling term is expressed as ; if two nodes and are competitive, , and the coupling term is expressed as [36]. On the contrary, if the graph is unsigned, controller (5) degenerates into a traditional distributed impulsive controller defined by previous studies [21, 23, 28].

Based on the aforementioned control protocol (5), the impulsive signed network reads aswhere , , and .

Remark 2. According to Lemma 2, it is necessary that is structurally balanced during investigating the bipartite synchronization. In fact, Meng et al. [33] demonstrated that a network cannot obtain bipartite synchronization when the signed network is structurally unbalanced.

Definition 3. Let be an impulsive sequence on . For any , denotes the impulsive times of on the interval . Then, the average impulsive interval, denoted by , is defined by

Remark 3. Obviously, . For example, let ; then, ; let ; then, . When , it was proposed in [29] and then was applied in [1923, 2628]. The case of was proposed in [30] when impulses occur infinitely but sparsely.

2.2. Synchronization Error Dynamics

Let be the Laplacian matrix of network (6); then, , where with . Let be the unsigned graph corresponding to . Then, , and so, the corresponding Laplacian matrix with for and .

Since is structurally balanced and the diagonal matrix , make sure that . By and , it is easy to see that holds with for and .

Let , , that is, . Then, it follows from (6) that

By Assumption 1, , is odd. Then, holds for any and . Then, from (8), we get

From (2) and (9), based on the fact that is a matrix with zero row sum, the following error system can be obtained:where , , and . Since is bounded and is a Lipschitz function, it is easy to know that is bounded. Suppose , .

Now, we rewrite the error system with a matrix form at the impulse instant :where and .

Lemma 2. (see [37]; Schur complement). Suppose that are both symmetric time-varying matrices; then, for any time-varying matrix ,is equivalent to(1), ,or(2), .

Definition 4. Symmetric time-varying matrix is called to be uniformly bounded and positive definite if there exist constants and such that

Lemma 3. (see [38]). Suppose that , , is nondecreasing in for each fixed , and is nondecreasing in . If satisfythen for implies that for .

3. Main Results

In this section, we derive some bipartite quasi-synchronization criteria for error system (10). As it is known, it is very difficult to achieve bipartite quasi-synchronization when we considered different average impulsive intervals. Here, we focus on two cases of average impulsive intervals. One is the case of , and the other is the case of .

3.1. The Case of

Theorem 1. Suppose that Assumption 1 holds and the average impulsive interval . Let , be a sequence of symmetric time-varying matrices, which are uniformly bounded and positive definite, i.e., there exist positive numbers and satisfying , . If there exist diagonal matrices , , and scalars , , , such thatwhere , , , and , then the bipartite quasi-synchronization can be achieved. Specifically, for , error system (10) between nonlinear system (1) and leader (2) exponentially converges intoand the convergence rate is .

Construct the Lyapunov function

Then, the derivative of is

By Assumption 1,where .

Substituting (21) and (22) into (20) yields

By Lemma 2 and (15), we obtain

When , (11) gives

By Lemma 2 and (16), we have . Thus,

For any , consider the following comparison system:

Let be the unique solution of (27); then, we obtain from Lemma 3 that for all . Using the idea of the variation of parameters, we can write the form of aswhere , , satisfies

Due to the construction of and the definition of , for any , , there is such that, for and ,

Substituting (30) into (28) yields

Let ; one has

Consequently,which implies that (10) converges exponentially into at convergence rate when .

Remark 4. It is noticed that the impulses in Theorem 1 may be desynchronizing , synchronizing , or inactive . However, most of the previous studies are devoted to investigate these impulses separately. The problem of bipartite quasi-synchronization is to find an impulsive interval , a coupling strength , and a pinning matrix so that all stable or unstable isolated nodes in a network synchronize into a bounded region.

Corollary 1. Suppose that Assumption 1 holds and the average impulsive interval . If there exist matrix , diagonal matrices , , and scalars , , , and such thatwhere and , then (10) exponentially converges intoat convergence rate for .

Proof. LetIf , one haswhere .
Using a similar proof as Theorem 1, we haveBy the definition of and (36), for any , , there exists sufficiently large such thatwhen and .
Using Theorem 1, we obtain the conclusion.

3.2. The Case of

This section concerns the bipartite quasi-synchronization of system (1) with average impulsive interval .

Theorem 2. Suppose that Assumption 1 holds and the average impulsive interval . Let be a sequence of time-varying matrices, which are uniformly bounded and positive definite, i.e., there exist positive numbers and satisfying . If there exist diagonal matrices , , and scalars , , and such thatwhere , , , and , then error system (10) exponentially converges intoat convergence rate for .

Proof. Using the similar proof of Theorem 1, we obtainRecalling the construction of and , we can getThus, for any , , there exists sufficiently large such thatwhen .
Then, by (45) and (47), we haveUsing Theorem 1, we obtain the conclusion.

Remark 5. From the conclusion of Theorem 2, we can observe that there is no relationship between the convergence rate and the average impulsive interval when . In other words, the impulsive control, when it occurs infinitely but very sparsely, does not influence the bipartite quasi-synchronization of the networks, but it is necessary to put forward high requirements for their dynamics of the leader and five nodes, which can be seen from Example 2.

Corollary 2. Suppose that Assumption 1 holds and the average impulsive interval . If there exist matrix , diagonal matrices , , and scalars , , , and such thatwhere and , then error system (10) exponentially converges intoat convergence rate for .

4. Simulation Examples

In this section, two numerical examples are used to simulate the theoretical results in the above section.

Example 1. Bipartite quasi-synchronization with .
Suppose that the leader satisfies the following Chua’s circuit:where , , , parameters , , , and , and .
Consider a signed network shown in Figure 1, consisting of five followers described bywhere , , andThe dynamics of the five followers may be chaotic, stable, or periodic. From Figure 1, one sees that the signed network is balanced with and . Lemma 1 holds for , and the Laplacian matrix isIn addition, is the initial condition of the leader node, and is the initial condition of the five nodes in the signed network. By simulation, one gets , , , , and .
Note that the node functions are odd functions satisfying the Lipschitz condition with . Hence, Assumption 1 is satisfied.
Let , i.e., agents 1, 2, and 3 are pinned. Let ; then, it is found that . According to Corollary 1, by solving generalized eigenvalue problems (34)–(36), we can find a feasible solution with , , , , and . Then, . From (43), one obtains that . Consequently, the error level is 0.4305. From Figures 2 and 3, one easily sees that the heterogeneous systems achieve bipartite leader-following quasi-synchronization with the distributed impulsive control, where and .

Example 2. Bipartite quasi-synchronization with .
According to the condition in Corollary 2, the dynamics of the leader and the five followers need to be stable. The leader has the following parameters: , , parameters , , , and , and .
The signed network is the same as that of Example 1, in which five followers have the following parameters: