Partial Component Consensus of Discrete-Time Multiagent Systems

The multiagent system has the advantages of simple structure, strong function, and cost saving, which has received wide attention from different fields. Consensus is the most basic problem in multiagent systems. In this paper, firstly, the problem of partial component consensus in the first-order linear discrete-time multiagent systems with the directed network topology is discussed. Via designing an appropriate pinning control protocol, the corresponding error system is analyzed by using the matrix theory and the partial stability theory. Secondly, a sufficient condition is given to realize partial component consensus in multiagent systems. Finally, the numerical simulations are given to illustrate the theoretical results.


Introduction
In recent years, the theory of consistency, as the basis of coordinated control of multiagent system, has attracted extensive attention from many researchers [1][2][3]. The unified nature of multiagent systems is wildly applied to computing science [4], systems and control [5][6][7], and distributed sensor networks [8][9][10].
The consistency of the discrete multiagent systems is that the state of all agents in a discrete system model can achieve asymptotic convergence under certain conditions. Many researchers have discussed the consistency problem of multiagent systems [11][12][13][14] and have obtained a lot of research results. Xie Dongmei and Wang Shaokun considered the consensus of second-order discrete-time multiagent systems with fixed topology in [15]. In 2016, Gao Yulan et al. studied group consensus for second-order discrete-time multiagent systems with time-varying delays under switching topologies in [16]. At the same year, Cao Yanfen and Sun Yuangong discussed consensus of discrete-time third-order multiagent systems in directed networks in [17]. Furthermore, the consensus of leader-following multiagent has also received a lot of attention. Wang Yunpeng et al. proposed an algorithm to research the consensus of discrete-time linear multiagent systems with communication noises in [18]. Xu Xiaole et al. investigated the leader-following consensus problem of discrete-time multiagent systems through Lyapunov method in [19].
The consistency of discrete multiagent systems has more advantages than continuous multiagent systems. For example, it can reduce a lot of computation, and the speed of convergence is fast and so on. Therefore, the research of discrete consistency has some practical significance. Wu Binbin et al. have studied the partial component consensus of continuous multiagent systems in [20]. In this paper, we discuss the partial component consistency of discrete leaderfollowing multiagent system. Based on the matrix theory and the partial stability theory, together with designing an appropriate pinning control protocol, a sufficient condition is proposed to realize partial component consensus in multiagent systems.
In detail, the remainder of this paper is organized as follows: Section 2 contains the problem statement and preliminaries; Section 3 presents the main result about the partial component consistency of discrete leader-following multiagent system; Section 4 provides a numerical example 2 Mathematical Problems in Engineering to verify the effectiveness of the proposed results; Section 5 offers concluding remarks.

The Problem Statement and Preliminaries
In this section, we will give the basic concept of partial component consistency, basic matrix theory, and some definitions and lemmas. For details, refer to [20,21].
We consider the following n-dimensional discrete system: where Similar to the definition of partial component stability for continuous system in [21], we give the following definition of partial component stability for discrete system.

Definition . The trivial solution of (1) is stable for vector , if
Definition . The trivial solution of (1) is asymptotically stable for vector , if it is stable and attracted for vector .

Main Results
In this section, we consider a first-order discrete multiagent system, which consists of N following agents and a leader, and suppose the equation of state for the following agent is where, ( ) = ( 1 ( ), . . . , ( )) ∈ expresses the state of ℎ agent; = diag( 1 , . . . , ) is a diagonal matrix. Supposing the dynamic equation of the leader agent is where 0 ( ) = ( 01 ( ), . . . , 0 ( )) ∈ is the state of leader agent.

Numerical Examples
In this section, a numerical example has been given to show that our theoretical result obtained above is effective.
then bring , , and into (4). Through simple calculations, at the same time, we set = 0.3 and Γ = diag(1, 1, 0); then, (9) will be held. In this case, it is straightforward to check that all the conditions in Theorem 7 hold. Next we will give the topology diagram of agent connection and the error trajectories of system (2) and (3) (2) and (3) for the first two components and Figure 2(c) represents that system (2) and (3) cannot achieve consensus for the third component.

Conclusion
In this paper, the partial component consensus has been investigated for the first-order discrete leader-following multiagent system. By establishing the suitable control term and using the matrix theory together with the stability theory, the sufficient conditions for the partial component conformance of the discrete system are derived. Furthermore, a numerical example has been given to illustrate the effectiveness of the present results. As an extension to this work, we plan to discuss the partial component consensus for the high-order discrete leader-following multiagent system.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.