#### Abstract

This paper investigates the bipartite consensus of linear discrete-time multiagent systems (MASs) with exogenous disturbances. A discrete-time disturbance-observer- (DTDO-) based technology is involved for attenuating the exogenous disturbances. And both the state feedback and observer-based output feedback bipartite consensus protocols are proposed by using the DTDO method. It turned out that bipartite consensus can be realized under the given protocols if the topology is connected and structurally balanced. Finally, numerical simulations are presented to illustrate the theoretical findings.

#### 1. Introduction

In the past two decades, the coordination of MASs has attracted much attention for its wide applications [1–4]. In coordination issues, consensus plays a very important role, which means that the final states of all agents can asymptotically reach a common value. Many works about consensus have been reported in the past few years, including consensus of MASs with different dynamics [5–10] and consensus of MASs via different control methods [11–16].

Many existing works mainly focus on the consensus of cooperative networks. However, in many real networks, there exist competitive relations between the nodes. Bipartite consensus was firstly investigated for MASs with antagonistic links [17], in which the nodes were divided into two parts, one will asymptotically track the leader and the other part will asymptotically converge to the reverse state of the leader. The topology of the discussed network was assumed as structurally balanced. And gauge transformation was used for solving the problems of stability analysis. Then, many efforts were devoted to bipartite consensus of MASs with antagonistic links or competitive topologies. Bipartite consensus was investigated for MASs with time-varying delay [18]. Bipartite edge consensus was studied for MASs with edge dynamics under the corresponding line graph spanned by the nodal graph [19]. Qin et al. [20] investigated global bipartite consensus of MASs with input saturation, and Hu et al. [21] solved bipartite consensus problems of MASs with communication noise. The event-triggered bipartite consensus was investigated for first-order MASs [22]. Under different topologies, event-triggered adaptive bipartite output consensus of heterogenous linear MASs was studied [23]. Considering the rate of convergency, finite-time bipartite consensus was investigated [24].

However, the above papers mainly investigated bipartite consensus of continuous-time MASs without disturbances. Disturbance often exists and is the main resource of poor performance of the controlled systems. In real networks, the subsystem in the network may be sufferred by exogenous disturbances. Therefore, research studies about multiagent systems with exogenous disturbances are very important and significant. By using a novel backstepping method, robust global coordination was investigated for MASs with input saturation and disturbances [25]. In [26], the disturbance-observer- (DO-) based control method was proposed for stabilizing nonlinear systems with exogenous disturbances. Using the continuous DO method proposed in [26], Yang et al. [27] solved the consensus problems of second-order MASs with exogenous disturbances. Containment control of continuous-time MASs with exogenous disturbances was investigated in virtue of the DO technique [28]. Intermittent consensus of MASs with exogenous disturbances was investigated by using both state feedback control and output feedback control protocols [29]. And the DO-based method is used on many other control plants [30, 31].

The existing relative works mainly focus on the consensus of MASs with continuous dynamics and cooperative topology. But there exist few results about bipartite consensus for discrete-time MASs with competitive topology and exogenous disturbances. Motivated by the above literatures, this paper investigates the bipartite consensus of discrete-time linear MASs with exogenous disturbances. The main contributions are as follows. (i) Discrete-time MASs are discussed in this paper, which are more challenging because the analysis of stability is more complex than the continuous-time MASs. (ii) Competitive network is considered in this paper, which can be used to describe more real network. And many results about the bipartite consensus of competitive networks can be applied into the consensus of cooperative networks. (iii) The DTDO-based method is used for attenuating the exogenous disturbances. Both the state feedback and output feedback control protocols are proposed in this paper.

The rest of the paper is organized as follows. Section 2 states the model considered in the paper and gives some basic lemmas and assumptions. In Section 3, discrete-time DO-based state feedback containment protocol is proposed. In Section 4, discrete-time DO-based output feedback containment protocol is given. Numerical examples are included to demonstrate the proposed protocol in Section 5. Finally, Section 6 gives a conclusion for this paper.

#### 2. Preliminaries and Model Description

A network can be described by a graph , which includes a set of nodes , a set of edges , and an adjacent matrix . For an undirected graph, , for . . is a neighbor set of th node. A path between node and node is formed by an array of edges . An undirected graph is connected if there exists a path between any two nodes. is the Laplacian matrix, where , .

The dynamics of the th follower are described as

And the dynamics of the leader are described aswhere , , , and denote the state, control input, exogenous disturbance, and output of the th follower, respectively, and denote the state and output of the leader, and , and are constant matrices. It is assumed that the disturbance is generated by the following exogenous system:where is the state of the exogenous system and and are constant matrices.

The following assumptions and lemmas are necessary for the main results of this paper.

*Definition 1. *(see [24]). A signed graph is said to be structurally balanced if the following hold:(1)It admits a bipartition of nodes as and , where , .(2)The elements of have the following relation:Otherwise, the graph is structurally unbalanced.

*Definition 2. *The leader-following bipartite consensus of system (1) with leader (2) is said to be achieved, if there exists a protocol such thatfor any initial condition , where , if ; , if .

*Remark 1. *According to gauge transformation [17], denoting , one has that is semipositive defined under the assumption that is connected. *Assumption 1*. Suppose the undirected signed graph is connected and structurally balanced. *Assumption 2*. The matrix pair is stabilizable. *Assumption 3*. The matrix pair is detectable.

Lemma 1 (see [32]). *For any matrix with , all eigenvalues of the matrix are positive if Assumption 1 holds, where .*

Lemma 2 (see [33]). *Under Assumption 2, there exists a unique positive definite matrix , satisfying the algebraic Riccati equation*

Lemma 3 (see [34]). *Suppose that are matrices with appropriate dimensions; then, the following inequalities are equivalent:*(1)*.*(2)* and .*(3)* and .*

#### 3. DO-Based State Feedback Bipartite Consensus

In this section, based on the DTDO method, bipartite consensus of MAS with disturbances is solved by using relative state information.

A disturbance observer is designed as follows:where is the internal state variable of the observer, and are the estimated values of and , respectively, and is the gain matrix of the observer.

*Remark 2. *The agents in the network cannot get the information of the disturbances, which leads to that the agents have to estimate the value of the exogenous disturbances. A discrete disturbance observer (7) is proposed for estimating the disturbances.

According to (1) and (7), one hasThen, denoting the state error of exogenous system as , one hasConsider the following distributed bipartite consensus protocol for discrete-time MAS (1):where is the gain matrix to be determined.

Substituting (10) into (1), one has that by (7) and (9),

Theorem 1. *Suppose Assumptions 1 and 2 hold. The bipartite consensus of MAS (1) with leader (2) will be achieved by error system (9) with disturbance observer (7) under bipartite consensus protocol (10) if*(i)*Suppose there exists at least one follower pinned by the leader.*(ii)* is Schur stable.*(iii)*, where is the unique solution of algebraic Riccati equation (6), is the minimum eigenvalue of , and .*

*Proof. *Let . Because , one hasDenote . Then, (12) can be rewritten as follows:Consider the following Lyapunov function candidatewhere , is a matrix designed later, and is a large enough positive constant.

Let and . Then,whereSince is a symmetric matrix, there exists an orthodox matrix such that , where . Moreover, by Assumption 1, (i), and Lemma 1, one has a sequence of eigenvalues . Then, let ; by (iii) and Lemma 2, one hasMeanwhile, by (ii), there exists a positive definite matrix such that the following discrete Lyapunov matrix equation holds:According to (17) and (18), one haswhere , and . By Lemma 3, one has by choosing sufficiently large constant . Then,where if and only if and . Thus, , , , which implies that Theorem 1 holds.

#### 4. DO-Based Output Feedback Bipartite Consensus

In this section, a DO-based output feedback protocol is proposed for disturbed linear MASs. State observer and disturbance observers are given, respectively, and the conditions are obtained.

The state observer of the th follower is designed asand the state observer of the leader is designed aswhere is the observed state of the th follower, is the observed state of the leader, and is the gain matrix to be determined.

The discrete-time disturbance observer based on output information is proposed aswhere is the internal state variable of the observer, and are the estimated values of and , respectively, and is the gain matrix of the observer.

*Remark 3. *For the case that the state of each agent cannot be obtained, the state observer can be used for estimating the state. Moreover, the disturbances exist in the subsystems. One has to design corresponding controller to attenuate the disturbances. Discrete output-based disturbance observer (23) is proposed.

When the state cannot be obtained, state observer can be used for estimating the state. Therefore, the bipartite consensus protocol can be designed aswhere is the gain to be designed.

Substituting (24) into (1), one hasand then, we give the following result.

Theorem 2. *Suppose Assumptions 1–3 hold. The bipartite consensus of MAS (1) with leader (2) will be achieved with state observers (21) and (22) and disturbance observer (23) under bipartite consensus protocol (24) if*(i)*Suppose there exists at least one follower pinned by the leader.*(ii)* is Schur stable.*(iii)*, where is the unique solution of algebraic Riccati equation (6), is the minimum eigenvalue of , and .*

*Proof. *Let , , , and . By , one hasand then, the error system can be written as follows:Denote , , , and (27) can be rewritten as follows:Consider the following Lyapunov candidate function:where , and are positive definite matrices which were designed later, and is a sufficiently large constant.

Let , , and thenwhereAnd, under Assumption 3, the gain matrix can be selected such that is Schur.

Similar to (17), there exists an orthodox matrix such that , where . Let . By (iii) and Lemma 2, one hasand since matrices and are Schur, there exist, respectively, positive definite matrices and such that the discrete Lyapunov matrix equations hold as follows:According to (32)–(34), one haswhereBy choosing sufficiently large constant , according to Lemma 3, one has . Then,where if and only if , , and . Thus, , , . Moreover, , . Thus, Theorem 2 holds.

#### 5. Simulations

In this section, we give two simulation examples to illustrate the theoretical results of Sections 3 and 4. In Figure 1, consider seven agents composed of six followers with one leader in a network with competitive interaction. Moreover, we choose and as the dimensions of the state and output of the seven agents, respectively.

*Example 1. *(the case of state feedback) In Figure 1, note that the signed topology is connected and the bipartition , satisfies Definition 1. Therefore, Assumption 1 is satisfied. Moreover, the matrix can be selected as . Meanwhile, the matrices and can be calculated as follows:and one can obtain .

Then, choose the following system matrices:where matrices and satisfy Assumption 2. And the gain matrix can be selected as follows:and thus one has that the matrix is Schur. By algebraic Riccati equation (6), one hasand thus the gain matrix of bipartite consensus protocol (10) can be obtained:For the MAS (1) with leader (2) and exogenous system (3), the initial values of , , and are given as follows:where . In Figures 2 and 3, the trajectories of , , are displayed. And one can obtain that the bipartite consensus of MAS (1) with leader (2) can be achieved under bipartite consensus protocol (10) by Figures 2 and 3. Thus, the effectiveness of Theorem 1 is verified.

*Example 2. *(the case of output feedback) In this case, consider the same system matrices and the gain matrix as Example 1, and one has that Assumption 1 can be satisfied and the solution of algebraic Riccati equation (6) and the same gain matrix as Example 1 can be calculated. Moreover, we can obtain that the matrix is Schur. Then, choose the output matrix as follows:and thus the matrix pair is detectable satisfying Assumption 3. Meanwhile, the matrix can be selected as follows:and then one has that is Schur.

For MAS (1) with leader (2) and exogenous system (3), we choose the same initial values of , , and as Example 1 and the observed values of , as follows: , , , , , , . In Figures 4 and 5, the trajectories of are presented, and one can note that the bipartite consensus can be achieved for MAS (1) with leader (2) via bipartite consensus protocol (24). Thus, the effectiveness of Theorem 2 is verified.

#### 6. Conclusions

In this paper, bipartite consensus is investigated for discrete-time MASs with exogenous disturbances. With the help of DTDO proposed in this paper, both the state feedback and the output feedback protocols are given, in which the gains can be determined by solving some discrete-time algebraic Riccati equations. Then, using stability theory, some sufficient conditions are obtained. Finally, numerical simulations are presented to illustrate the theoretical findings.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

This study was partially supported by the Natural Science Foundation of Hunan Province (2020JJ6089) and the Key Project of the Department of Education in Hunan Province (19A133).