# Partial Component Consensus of Discrete-Time Multiagent Systems

Authors: Wenjun Hu; Gang Zhang; Zhongjun Ma; Binbin Wu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

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.


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## Body

## 1. Introduction

In recent years, the theory of consistency, as the basis of coordinated control of multiagent system, has attracted extensive attention from many researchers [1–3]. The unified nature of multiagent systems is wildly applied to computing science [4], systems and control [5–7], and distributed sensor networks [8–10].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 [11–14] 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 leader-following 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: Section2 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 to verify the effectiveness of the proposed results; Section 5 offers concluding remarks.


## 2. 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:(1)xk+1=Fxk,where Fx∈CRn,Rn, F0≡0, x=x1,…,xm,xm+1,…,xn∈R. Supposing y=x1,…,xm,z=xm+1,…,xn, (m+p=n), and x=(∑i=1nxi2)1/2, y=(∑i=1mxi2)1/2, z=(∑i=m+1nxi2)1/2, k∈N.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 1.
The trivial solution of (1) is stable for vector y, if ∀ε>0,∀k0∈N,∃δ(k0)>0,∀x0∈Sδ(k0,ε)=x∥<δ(k0,ε), and when k≥k0, there must be y(k,k0,x0)<ε.Definition 2.
The trivial solution of (1) is attracted to vector y, if ∀k0∈N,∃δ(k0)>0,∀ε>0,∀x0∈Sσ(k0)=xx≤δ(k0), ∃T(k0,x0,ε)>0, and when t≥k0+T, there must be y(k,k0,x0)<ε, where Sσ(k0) is the attraction area of the vector y.Definition 3.
The trivial solution of (1) is asymptotically stable for vector y, if it is stable and attracted for vector y.Lemma 4 (see [20]).
LetH=(hij)∈RN×N,B=(bij)∈Rn×n, then there exists a nN-order permutation matrix P=P1⋯Ps, where Pi is the first kind of elementary row transformation matrix, such that the P(H⊗B)P-1=B⊗H can hold, where i=1,…,s,s∈N+,⊗ is the Kronecker product.


## 3. 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(2)xik+1=Axik+uik,i=1,2,…,N,where,xi(k)=(xi1(k),…,xin(k))T∈Rn expresses the state of ith agent; A=diag⁡(A1,…,An) is a diagonal matrix.Supposing the dynamic equation of the leader agent is(3)x0k+1=Ax0k,where x0(k)=(x01(k),…,x0n(k))T∈Rn is the state of leader agent.Next we consider the problem of partial component conformance of discrete leader-following multiagent system under directed network topology. We design the controller as follows:(4)uik=c∑j=1NaijΓxjk-xik-cdiΓxik-x0k,where c>0 denotes coupling strength; Γ=diag⁡(r1,…,rn)∈Rn×n(rk≥0,k=1,…,n) is the intercoupling matrix; if the ith agent can receive the information from the jth agent, then aij>0; otherwise, aij=0. If the ith agent can receive the information from the leader agent, then di>0; otherwise, di>0.LetL=(Lij) satisfy Lij=-sij(i≠j) and Lii=∑i≠j, and let ei(k)=xi(k)-x0(k). We can get the error system from (2), (3), and (4) as follows: (5)eik+1=Aeik-cL+D⊗Γeik.We can write the above equation as a vector form:(6)ek+1=IN⊗AIn-cL+D⊗Γek.where e(k)=(e1(k),…,eN(k))T, ei(k)=(ei1(k),…,ein(k))T, D=diag⁡(d1,…,dN), In represents n-order identity matrix.In order to discuss the asymptotic stability of the trivial solution of the error system (6), we try to do the following transformation. Let e^(k+1)=Pe(k) and εj(k)=(e1j(k),…,eNj(k)), where P is the elementary matrix in Lemma 4 and j=1,…,n. We can get e^(k)=(ε1T(k),…,εnT(k))T through calculation; in this case, (6) will change to the following equation: (7)e^k+1=PIN⊗AIn-cL+D⊗ΓP-1e^k.We can simplify the above equation,(8)e^k+1=AIn⊗IN-cΓ⊗L+De^k.Next we give two useful definitions for this paper.Definition 5.
If there exist1≤l≤n, such that the solution of system (2) and (3) satisfy limk→∞⁡∑j=1lεj(k)=0, then system (2) and (3) achieve consensus for the first l components.Definition 6.
If there existl=1, such that the solution of system (2) and (3) satisfy limk→∞⁡ε1(k)=0, then system (2) and (3) achieve consensus for the first component.Theorem 7.
LetMj=AjIN-crj(L+D),Nj=MjTMj, for the control item, if there exist β∈(0,1), such that system (2) and (3) satisfy(9)Nj<-β-1IN,j=1,2,…,l,where rj>0 is the jth element of the matrix Γ=diag⁡(r1,…,rn), D=diag⁡(d1,…,dN),di≥0; then, system (2) and (3) can achieve consensus for the first l components.Proof.
Define the following Lyapunov function candidate:(10)Vk=∑j=1l∑i=1NeikTkeikk=e^Tk∧e^k,where ∧=diag⁡(1,…,1,︷lN0,…,0), e^(k)=(ε1T(k),…,εnT(k))T. Similar to literature [19], we can obtain the following:(11)ΔVk=Vk+1-Vk=e^TkA⊗IN-cΓ⊗L+DT∧A⊗IN-cΓ⊗L+De^k-e^Tk∧e^k=∑ε^jTkAjIN-crjL+DTAjIN-crjL+D-INε^jk=∑ε^jTkMjTMj-INε^jk=e^TtN1-IN⋱Nl-IN0N×N⋱0N×Ne^t≤-∑j=1lβεjTkεjk=-βVk.In this case, if (9) can be held, then there exist β∈(0,1), such that(12)ΔVk=Vk+1-Vk≤-∑j=1lβεjTkεjk=-βVk.Therefore,(13)Vk+1≤Vk-βVk=1-β2≤1-β2Vk-1≤⋯≤1-βk+1V0,when k→∞, V(k+1)≤(1-β)k+1V(0), i.e., limk→∞⁡εj(k)=0. Therefore, limk→∞⁡∑j=1lεj(k)=0; then the solution of (8) is asymptotically stable for partial vector element; i.e., system (2) and (3) can achieve consensus for the first l components.


## 4. Numerical Examples

In this section, a numerical example has been given to show that our theoretical result obtained above is effective.Example 1.
Given the parametern and N in system (2) as n=3 and N=4, we consider the consensus of the discrete leader-following multiagent system for the first two components (i.e., l=2). Supposing the state of the ith agent (the subscript of the state of the leader denotes 0) is xi(k+1)=Axi(k), A=diag⁡(1.01,1.01,1.03). Designing L and D as follows: (14)L=10-1001-100000-1001,D=0000000000100000,then bring A, L, and D into (4). Through simple calculations, at the same time, we set c=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) throughMatlab software.Figure1 expresses the topology connection of the given agent. Next we give the evolution diagram of the state error of the leader-following agent (Figure 2); Figures 2(a) and 2(b) represent the consensus of system (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.Figure 1
The topology diagram of agent connection.Figure 2
(a) The state error of the leader-following for the first component. (b) The state error of the leader-following for the second component. (c) The state error of the leader-following for the third component.
(a) (b) (c)


## 5. 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.



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