# Bipartite Consensus of Linear Discrete-Time Multiagent Systems with Exogenous Disturbances under Competitive Networks

Authors: Yi Yuan; Shamrie Sainin Mohd; Yanhui Zhu
Journal: Complexity
 (2021)

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## 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.


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## Body

## 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. Section2 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 graphG=V,E,A, which includes a set of nodes V=1,2,…,N, a set of edges E⊆V×V, and an adjacent matrix A=aij. For an undirected graph, aij=aji, for i,j=1,2,…,N. aij≠0⇔j,i∈E. Ni=j|j,i⋅E is a neighbor set of ith node. A path between node vi1 and node vin is formed by an array of edges vi1,vi2,vi2,vi3,…,vin−1,vin. An undirected graph is connected if there exists a path between any two nodes. L=lijN×N is the Laplacian matrix, where lii=∑j=1,j≠iNaij, lij=−aij.The dynamics of theith follower are described as(1)xik+1=Axik+Buik+γik,yik=Cxik,i=1,2,…,N.And the dynamics of the leader are described as(2)x0k+1=Ax0k,y0k=Cx0k,where xik∈Rn, uik∈Rp, γik∈Rp, and yik∈Rm denote the state, control input, exogenous disturbance, and output of the ith follower, respectively, x0k∈Rn and y0k∈Rm denote the state and output of the leader, and A∈Rn×n,B∈Rn×p, and C∈Rm×n are constant matrices. It is assumed that the disturbance γik is generated by the following exogenous system:(3)ωik+1=Sωik,γik=Fωik,where ωi∈Rl is the state of the exogenous system and S∈Rl×l and F∈Rp×l are constant matrices.The following assumptions and lemmas are necessary for the main results of this paper.Definition 1.
(see [24]). A signed graph G is said to be structurally balanced if the following hold:(1)
It admits a bipartition of nodes asV1 and V2, where V1∩V2=∅, V1∪V2=V.(2)
The elements ofVq,q∈1,2 have the following relation:(4)aij≥0,∀i,j∈Vq;aij≤0,∀i∈Vq,j∈Vr,q≠r.
Otherwise, the graphG 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 uik such that(5)limt⟶∞xik−dix0k=0,i=1,2,…,N,for any initial condition xl1,l=0,1,…,N, where di=1, if i∈V1; di=−1, if i∈V2.Remark 1.
According to gauge transformation [17], denoting D=diagd1,d2,…,dN, one has that DLD is semipositive defined under the assumption that G is connected.Assumption 1. Suppose the undirected signed graph G is connected and structurally balanced.Assumption 2. The matrix pair A,B is stabilizable.Assumption 3. The matrix pair A,C is detectable.Lemma 1 (see [32]).
For any matrixΛ=diagh,0,…,0 with h>0, all eigenvalues of the matrix DL˜D=DLD+Λ are positive if Assumption 1 holds, where L˜=L+Λ.Lemma 2 (see [33]).
Under Assumption 2, there exists a unique positive definite matrixP, satisfying the algebraic Riccati equation(6)ATPA−P−ATPBBTPB+I−1BTPA+I=0.Lemma 3 (see [34]).
Suppose thatA,B,C are matrices with appropriate dimensions; then, the following inequalities are equivalent:(1)
ABBTC<0.(2)
A<0 and C−BTA−1B<0.(3)
C<0 and A−BC−1BT<0.


## 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:(7)vik+1=S+HBFvik−Hxik+HAxik+Buik,ω^ik=vik−Hxik,γ^ik=Fω^ik,where vik∈Rl×l is the internal state variable of the observer, ω^ik∈Rl and γ^ik∈Rp are the estimated values of ωik and γik, respectively, and H∈Rl×n 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 has(8)ω^ik+1=vik+1−Hxik+1=S+HBFvik−Hxik+HAxik+Buik−HAxik+Buik+γik=S+HBFω^ik−HBFωik.
Then, denoting the state error of exogenous system aseik, one has(9)eik+1=ωik+1−ω^ik+1=Sωik−S+HBFω^ik+HBFωik=S+HBFωik−ω^ik=S+HBFeik.
Consider the following distributed bipartite consensus protocol for discrete-time MAS (1):(10)uik=K∑j∈Niaijxik−sgnaijxjk+hixik−dix0k−γ^ik,where K is the gain matrix to be determined.
Substituting (10) into (1), one has that by (7) and (9),(11)xik+1=Axik+BFeik+BK∑j∈Niaijxik−sgnaijxjk+hixik−dix0k.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)
S+HBF is Schur stable.(iii)
K=−1/λ1DL˜DBTPB+I−1BTPA, where P>0 is the unique solution of algebraic Riccati equation (6), λ1DL˜D is the minimum eigenvalue of DL˜D, and DL˜D=DLD+Λ,Λ=diagh1,h2,…,hN.Proof.
Letx˜ik=xik−dix0k. Because diaijdj=aijsgnaij=aij, one has(12)x˜ik+1=xik+1−dix0k+1=Ax˜ik+BFeik+BK∑j∈Niaijx˜ik−sgnaijx˜jk+hix˜ik,eik+1=S+HBFeik.
Denotex˜k=x˜1Tk,x˜2Tk,…,x˜NTkT,ek=e1Tk,e2Tk,…,eNTkT. Then, (12) can be rewritten as follows:(13)x˜k+1=IN⊗A+L˜⊗BKx˜k+IN⊗BFek,ek+1=IN⊗S+HBFek.
Consider the following Lyapunov function candidate(14)Vk=ηTkIN⊗P00αIN⊗Q1ηk=x˜TkIN⊗Px˜k+αeTkIN⊗Q1ek,where ηk=x˜Tk,eTkT, Q1>0 is a matrix designed later, and α is a large enough positive constant.
Letx¯k=D⊗Inx˜k and η¯k=x¯Tk,eTkT. Then,(15)Vk+1−Vk=x¯TkIN⊗ATPA+2DL˜D⊗ATPBK+DLD2⊗KTBTPBK−IN⊗Px¯k+x¯TkD⊗ATPBF+DL˜⊗KTBTPBFek+eTkD⊗FTBTPA+L˜D⊗FTBTPBKx¯k+eTkIN⊗αS+HBFTQ1S+HBF−αQ1+FTBTPBFek=η¯TkΞ1∗1∗1TΞ2η¯k,where(16)Ξ1=IN⊗ATPA+2DL˜D⊗ATPBK+DLD2⊗KTBTPBK−IN⊗P,Ξ2=IN⊗αS+HBFTQ1S+HBF−αQ1+FTBTPBF,∗1=D⊗ATPBF+DL˜⊗KTBTPBF.
SinceDL˜D is a symmetric matrix, there exists an orthodox matrix ϒ such that ϒTDL˜Dϒ=Λ, where Λ=diagλ1DL˜D,λ2DL˜D,…,λNDL˜D. Moreover, by Assumption 1, (i), and Lemma 1, one has a sequence of eigenvalues 0<λ1DL˜D≤λ2DL˜D≤…≤λNDL˜D. Then, let xˇk=ϒT⊗Inx¯k; by (iii) and Lemma 2, one has(17)x¯TkΞ1x¯k=x^TkIN⊗ATPA+2Λ⊗ATPBK+Λ2⊗KTBTPBK−IN⊗Px^k≤−xˇTkxˇk=−x¯Tkx¯k.
Meanwhile, by (ii), there exists a positive definite matrixQ1>0 such that the following discrete Lyapunov matrix equation holds:(18)S+HBFTQ1S+HBF−Q1+Il=0.
According to (17) and (18), one has(19)Vk+1−Vk≤η¯Tk−INn∗1∗1T−Ψ1η¯k=ηTk−INn∗2∗2T−Ψ1ηk=ηTkΦ1ηk,where Φ1=−INn∗2∗2T−Ψ1, Ψ1=IN⊗αIl−FTBTPBF and ∗2=D⊗In∗1. By Lemma 3, one has Φ1<0 by choosing sufficiently large constant α. Then,(20)Vk+1−Vk≤λmaxΦ1ηTkηk=λmaxΦ1∑i=1Nx˜iTkx˜ik+eiTkeik≤0,where Vk+1−Vk=0 if and only if x˜1=x˜2=⋯=x˜N=0 and e1=e2=⋯=eN=0. Thus, limk⟶∞x˜ik=0, limk⟶∞eik=0, i=1,2,…,N, 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 theith follower is designed as(21)x^ik+1=Ax^ik+Buik+γik+ECx^ik−yik,i=1,2,…,N,and the state observer of the leader is designed as(22)x^0k+1=Ax^0k+ECx^0k−y0k,where x^ik is the observed state of the ith follower, x^0k is the observed state of the leader, and E is the gain matrix to be determined.The discrete-time disturbance observer based on output information is proposed as(23)vik+1=S+HBFvik−Hx^ik+HAx^ik+Buik+ECx^ik−yik,ω^ik=vik−Hx^ik,d^ik=Fω^ik,where vi∈Rl is the internal state variable of the observer, ω^i and γ^i are the estimated values of ωi and γi, respectively, and H∈Rl×n 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 as(24)uik=K∑j∈Niaijx^ik−sgnaijx^jk+hix^ik−dix^0k−γ^ik,where K is the gain to be designed.
Substituting (24) into (1), one has(25)xik+1=Axik+BFωik−ω^ik+BK∑j∈Niaijx^ik−sgnaijx^jk+hix^ik−dix^0k,and 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)
S+HBF is Schur stable.(iii)
K=−1/λ1DL˜DBTPB+I−1BTPA, where P>0 is the unique solution of algebraic Riccati equation (6), λ1DL˜D is the minimum eigenvalue of DL˜D, and DL˜D=DLD+H¯,H¯=diagh1,h2,…,hN.Proof.
Letx˜ik=xik−dix0k, x^˜ik=x^ik−dix^0k, δik=x˜ik−x^˜ik, and eik=ωik−ω^ik. By diaijdj=aijsgnaij=aij, one has(26)aijx^ik−sgnaijx^jk=aijx^˜ik+dix^0k−sgnaijx^jk=aijx^˜ik−sgnaijx^˜jk=aijx˜ik−sgnaijx˜jk+sgnaijδjk−δjk,and then, the error system can be written as follows:(27)x˜ik+1=Ax˜ik+BFeik+BK∑j∈Niaijxik−sgnaijx˜jk+sgnaijδjk−δjk+hix˜ik−δik,x^˜ik+1=Ax^˜ik+BFeik−ECδik+BK∑j∈Niaijx˜ik−sgnaijx˜jk+sgnaijδjk−δjk+hix˜ik−δik,δik+1=A+ECδik,eik+1=S+HBFeik.
Denotex˜k=x˜1Tk,x˜2Tk,…,x˜NTkT, δk=δ1Tk,δ2Tk,…,δNTT, ek=e1Tk,e2Tk,…,eNTkT, and (27) can be rewritten as follows:(28)x˜k+1=IN⊗A+L⊗BKx˜k+IN⊗BFek−L⊗BKδk,δk+1=IN⊗A+ECδk,ek+1=IN⊗S+HBFek.
Consider the following Lyapunov candidate function:(29)Vk=ζTkIN⊗P000βIN⊗Q2000βIN⊗Q3ζk=x˜TkIN⊗Px˜k+βδTkIN⊗Q2δk+βeTkIN⊗Q3ek,where ζk=x˜Tk,δTk,eTkT, Q2 and Q3 are positive definite matrices which were designed later, and β>0 is a sufficiently large constant.
Letx¯k=D⊗Inx˜k, ζ¯k=x¯Tk,δTk,eTkT, and then(30)Vk+1−Vk=x¯TkIN⊗ATPA+2DL˜D⊗ATPBK+DLD2⊗KTBTPBK−IN⊗Px¯k−x¯TkDL˜⊗ATPBK+DL˜2⊗KTBTPBKδk−δTkL˜D⊗KTBTPA+L˜2D⊗KTBTPBKx¯k+x¯TkD⊗ATPBF+DL˜⊗KTBTPBFek+eTkD⊗FTBTPA+L˜D⊗FTBTPBKx¯k−δTkL˜⊗KTBTPBFek−eTkL˜⊗FTBTP×BKδk+δTkβIN⊗A+ECTQ2A+EC−Q2+L˜2⊗KTBTPBKδk+eTkIN⊗βS+HBFTQ3S+HBF−βQ3+FTBTPBFek=ζ¯TkΞ3∗3∗4∗3TΞ4∗5∗4T∗5TΞ5ζ¯k,where(31)Ξ3=IN⊗ATPA+2DL˜D⊗ATPBK+DLD2⊗KTBTPBK−IN⊗P,Ξ4=βIN⊗A+ECTQ2A+EC−Q2+L˜2⊗KTBTPBK,Ξ5=IN⊗βS+HBFTQ3S+HBF−βQ3+FTBTPBF,∗3=DL˜⊗ATPBK+DL˜2⊗KTBTPBK,∗4=D⊗ATPBF+DL˜⊗KTBTPBF,∗5=L˜⊗KTBTPBF.
And, under Assumption 3, the gain matrixE can be selected such that A+EC is Schur.
Similar to (17), there exists an orthodox matrix ϒ such that ϒTDL˜Dϒ=Λ, where Λ=diagλ1DL˜D,λ2DL˜D,…,λNDL˜D. Let xˇk=ϒT⊗Inx¯k. By (iii) and Lemma 2, one has(32)x¯TkΞ3x¯k≤−xˇTkxˇk=−x¯Tkx¯k,and since matrices A+EC and S+HBF are Schur, there exist, respectively, positive definite matrices Q2 and Q3 such that the discrete Lyapunov matrix equations hold as follows:(33)A+ECTQ2A+EC−Q2+In=0,(34)S+HBFTQ3S+HBF−Q3+Il=0.
According to (32)–(34), one has(35)Vk+1−Vk≤ζ¯Tk−INn∗3∗4∗3T−Ψ2∗5∗4T∗5T−Ψ3ζ¯k=ζTk−INn∗6∗7∗6T−Ψ2∗5∗7T∗5T−Ψ3ζk=ζTkΦ2ζk,where(36)Φ2=−INn∗6∗7∗6T−Ψ2∗5∗7T∗5T−Ψ3,Ψ2=βINn−L˜2⊗KTBTPBK,Ψ3=IN⊗βIl−FTBTPBF,∗6=D⊗In∗3,∗7=D⊗In∗4.
By choosing sufficiently large constantβ, according to Lemma 3, one has Φ2<0. Then,(37)Vk+1−Vk≤λmaxΦ2ζTkζk=λmaxΦ2∑i=1Nx˜iTkx˜ik+δiTkδik+eiTkeik≤0,where Vk+1−Vk=0 if and only if x˜1=x˜2=⋯=x˜N=0, δ1k=δ2k=⋯=δNk=0, and e1=e2=⋯=eN=0. Thus, limk⟶∞x˜ik=0, limk⟶∞δik=0, limk⟶∞eik=0. Moreover, limk⟶∞x^˜ik=0, i=1,2,…,N. Thus, Theorem 2 holds.


## 5. Simulations

In this section, we give two simulation examples to illustrate the theoretical results of Sections3 and 4. In Figure 1, consider seven agents composed of six followers with one leader in a network with competitive interaction. Moreover, we choose n=2 and m=3 as the dimensions of the state and output of the seven agents, respectively.Figure 1
A network with competitive interactions containing 7 agents.Example 1.
(the case of state feedback) In Figure1, note that the signed topology is connected and the bipartition V1=1,2,3, V2=4,5,6 satisfies Definition 1. Therefore, Assumption 1 is satisfied. Moreover, the matrix D can be selected as D=diag1,1,1,−1,−1,−1. Meanwhile, the matrices L˜ and DL˜D can be calculated as follows:(38)L˜=3−10100−13001000200110030−101002−1001−1−13,DL˜D=3−10−100−1300−1000200−1−10030−10−1002−100−1−1−13,and one can obtain λ1DL˜D=0.5395.
Then, choose the following system matrices:(39)A=0.9−20−1.01,B=0.02−0.05,F=−0.031.8,S=−1000,where matrices A and B satisfy Assumption 2. And the gain matrix H can be selected as follows:(40)H=−0.200−0.5,and thus one has that the matrix S+HBF is Schur. By algebraic Riccati equation (6), one has(41)P=4.8332−4.7634−4.763440.0959,and thus the gain matrix K of bipartite consensus protocol (10) can be obtained:(42)K=−0.5025−2.4200.
For the MAS (1) with leader (2) and exogenous system (3), the initial values of xik, x0k, and ωik are given as follows:(43)x11=9,10T,x21=−16,−4T,x31=12,−15T,x41=−11,−9T,x51=−9,14T,x61=13,7T,x01=1.5,−1.5T,ω11=2,2T,ω21=−1,−1.2T,ω31=−3,−3T,ω41=5,4T,ω51=−2,−2T,ω61=1,1T,where i=1,2,…,6. In Figures 2 and 3, the trajectories of xik, i=1,2,…,6, 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.Figure 2
The trajectories ofxi1,i=1,2,…,6, under state feedback protocol.Figure 3
The trajectories ofxi2,i=1,2,…,6, under state feedback protocol.Example 2.
(the case of output feedback) In this case, consider the same system matricesA,B,F,S and the gain matrix H as Example 1, and one has that Assumption 1 can be satisfied and the solution P>0 of algebraic Riccati equation (6) and the same gain matrix K as Example 1 can be calculated. Moreover, we can obtain that the matrix S+HBF is Schur. Then, choose the output matrix C as follows:(44)C=−0.5000.500,and thus the matrix pair A,C is detectable satisfying Assumption 3. Meanwhile, the matrix E can be selected as follows:(45)E=100010,and then one has that A+EC is Schur.
For MAS (1) with leader (2) and exogenous system (3), we choose the same initial values of xik, x0k, and ωik as Example 1 and the observed values of x^ik, x^0k as follows: x^11=14,13T, x^21=18,17T, x^31=−10,−11T, x^41=−14,−12T, x^51=−16,−17T, x^61=11,14T, x^01=−0.1,−0.1T. In Figures 4 and 5, the trajectories of xik 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.Figure 4
The trajectories ofxi1,i=1,2,…,6, under output feedback protocol.Figure 5
The trajectories ofxi2,i=1,2,…,6, under output feedback protocol.


## 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.



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