# Bipartite Containment Control for Multiagent Systems with Adaptive Quantization Information

Authors: Jie Wu; Heng Lei; Xisheng Zhan
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this study, bipartite containment control for multiagent systems (MASs) with quantitative information is investigated. Communication topology is structurally balanced, and the follower’s trajectory is within the area surrounded by the leader by designing distributed error terms. The leader is an external system, and the matrix information of the leader cannot be accessed by followers. Based on an adaptive quantization information distributed observer, the authors introduce an output feedback protocol to study bipartite containment control. Finally, simulations demonstrate the effectiveness of proposed control algorithms.


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## Body

## 1. Introduction

In recent years, collaborative control has attracted a lot of attention. It is widely used in vehicle formation [1–3], complex dynamic networks [4, 5], and sensor networks [6–8]. With the development of MASs, many interesting results have been obtained, which makes many control problems become hot issues, including consensus of multiagent problem [9], the output regulation problem [10, 11], contains the problem [12, 13], and an adaptive parameter problem [14]. In the field of control, the MASs are a topical research object. It is composed of a group of single multiagent, and the information transmission between them constitutes the communication topology. The consensus of MASs can be divided into two types by whether there is a leader or not, that is, the MASs without leaders who cannot accept the information of other agents and the leader tracking systems with leaders. In [15], the leaderless continuous and discrete time consistency problem of the first-order MASs is studied by the frequency domain method. Based on the leaderless fixed communication topology [16], the finite-time consensus for MASs with external disturbance is studied. In addition, it is also important to consider the existence of leaders in multiagent research. Tang et al. [17] studied the tracking consistency problem, Hong et al. designed a distributed observer for leading tracking MASs [18], and Ni and Cheng solved leader-following consensus of the MASs through switching topology [19]. This study considers multiple leaders, and the leaders are considered to be the exogenous systems, and the leaders’ information cannot be received by the followers.Inspired by the biological behavior of nature, containment control is adopted for systems with multiple leaders in most cases, and the containment control problem is widely used in practice. In containment control, there is more than one leader. The area surrounded by the leader is called the convex shell, and followers will only move within the boundaries defined by the leader. For example, several robots carry equipment through a dangerous stretch of road. Robots without equipment act as leaders. Other robots transmit a safe travel route range to them through sensors or artificial information, while robots with equipment act as followers to enter the movement area surrounded by the leaders so as to avoid dangerous areas in the path. In [20], authors studied the necessary and sufficient conditions to achieve containment control. This paper studies bipartite output containment; that is, some follower curves converge to the region surrounded by the leader and others converge to the region of symmetry. In [21], the authors solved containment control with an output feedback method. In real life, the state is probabilistically unmeasurable, and external disturbance also affects the stability of the system. In [22, 23], by using the output feedback method, containment control of the MASs with exogenous disturbance is solved. This study also considers the influence of quantitative information on information transmission between agents to meet the practical challenges.In the existing distributed controller, the information exchange between agent and neighbor is very accurate, but the accurate information exchange is a simplification, which is unreasonable. In practical applications, agents exchange information through digital communication channels. Due to the limitation of their own information storage capacity and communication width, the quantization effect must be considered [24–26], which is also meaningful and interesting for the study of consensus problems. Therefore, many articles on the consensus issue take into account the quantitative impact. In [27], the problem of quantization information consistency with Markov chains for the first-order MASs is solved. Control input problem: in [28], the author uses the distributed variable adjacency matrix method to deal with the quantized second-order consensus problem. In addition, the problem of the digital digraph of MASs is solved by designing an observer, and the consensus of the system in [29] is realized by using the probabilistic quantization method.Each follower can get the leader’s information, which is not realistic in practical application. Therefore, researchers introduced an adaptive algorithm. In [30], the asymptotic output regulation problem was studied by using the Luenberger observer and introducing an adaptive regulation method. In [31], an adaptive controller was proposed to cope with finite-time bipartite consensus for output feedback in the unknown state information. In [21], containment control for the linear MASs was solved. In addition, multiagent networks with competition and cooperation among agents are more common. Such problems can be called bipartite control problems [32, 33]. In topology graphs, the collar matrix has positive and negative weights. A special example of bipartite control is the control that is only cooperative between agents. The bipartite consensus of the MASs is restudied in [34] by means of output feedback and state feedback. In [35], the leader matrix information is obtained by using an adaptive observer, the solution of the mediation equation is obtained by designing an algorithm, and the consensus of the MASs is solved by output feedback. The bipartite containment problem of the MASs is studied in [36]. To solve some problems in actual life, an adaptive quantization information distributed observer is designed to estimate the leader’s state, and then, a feedback controller is used to reach output bipartite containment.The remaining four sections are as follows. Section 2 is the formulation and preliminary work of the problem. Section 3 is the main theoretical results. Numerical simulation results are shown in Section 4 to verify the effectiveness of theoretical results. In the end, Section 5 gives the conclusion.


## 2. Preliminaries and Model Statement

### 2.1. Algebraic Graph Theory

To achieve the bipartite output containment, communication topology is described in a topology diagram [37]. Let GV,E,A be weighted directed graph and consist of node set V=V1,V2,…,VN which denotes finite nonempty set of N nodes and edge set E⊆V×V describing transfer of information between nodes. We define A=aij∈RN×N as weighted adjacency matrix. Let di=∑j=1Naij be in-degree of vertex i and D=diagd1,d2,…,dN be in-degree matrix of G. Laplace matrix L=lij of weighted directed graph G is defined as L=D−A. The point set is divided into two subgroups, v1 and v2, and v1∪v2=V and v1∩v2=∅, which is different from the general graph theory. Define Ni=Vj|Vj,Vi∈E,i≠j. to represent neighbor points of Vi. If there is an edge Vi,Vj∈E and Vi and Vj are in the same subgroup, it represents cooperative relationship; then, there is aij>0; if Vi and Vj are in different subgroups, it represents antagonistic relationship; then, there is aij<0, otherwise, aij=0. Define τi as a symbolic parameter, where τi=1 is used if Vi∈v1 and τi=−1 is used if Vi∈v2.


### 2.2. Quantizer

Quantizer has been added to many literature studies on multiagent research. In this study, we use the quantizerq:R⟶R as(1)qx=μi,if11+γμi<x≤11−γμi,x>0,0,ifx=0,−q−x,ifx<0,where μi is the quantization level and γ∈0,1 is the quantization accuracy parameter. For the quantization level, one has a set μ¯=±μi:μi=1−γ/1+γiμ0,i=0,±1,±2,…,±n∪0, where μ0>0 is the initial quantization level. qm−m≤γm,∀m∈Rcan be obtained through the definition of quantizer. It is worth noting that, for any set of vectorsy=y1,y2,…,yNT∈Rn, we can writeqy=qy1,qy2,…,qyNT∈Rn. So, obviously there is qy−y=Γy, where Γ=diagΓ1,Γ2,…,ΓN and Γi∈−γ,+γ.


### 2.3. Problem Formulation

In this study, we study the bipartite containment control for a group of MASs composed ofZ leaders and N followers. In addition, Z=N+1,N+2,…,N+Z and N=1,2,…,N are defined as leaders’ set and followers’ set, respectively. The kinetic equation for N followers is as follows:(2)x˙i=Aixi+Biui+∑k=N+1N+ZQikVk,yi=Cixi,i=1,2,…,N,where xi∈Rl,ui∈Rl,andyi∈Rl are state quantity of i th agent, control input quantity of the system, and the control output quantity of the system, respectively. Ai, Bi, Ci, and Qik are parameter matrix with suitable dimensions. Then, the kinetic equation of leaders is in the following form:(3)V˙k=A0Vk,yk=FVk,k∈Z,where Vk∈Rl represents the state of the exogenous system, that is, disturbance quantity or reference input of the system.Definition 1.
In this study, a reasonable controller is designed to solve the problem of bipartite containment control, which makes the followers converge to the positive and negative region defined by the leader, namely, the convex hull. So, for follower systems and leader systems (1) and (2), the following features can be used to solve their bipartite containment problem.(1)
Define the convex hullcoX belonging to set X=x1,x2,…,xN of the form(4)coX=∑i=1Nβixixi∈R,βi≥0,∑i=1Nβi=1.(2)
For followers and leaders with any initial value, the follower can only enter the positive and negative convex hull defined by the leader:(5)limt⟶∞distxi,coVk,k∈Z=0,i∈N1,limt⟶∞distxj,−coVk,k∈Z=0,j∈N2,in whichN1∪N2=N and N1∩N2=∅.
Whenei=0 is used, (2) can be displayed by the following containment error:(6)ei=∑j∈Niaijyi−sgnaijyj+∑k=N+1N+Zaikyi−τiyk,i∈N.
Lete=cole1,e2,…,eN and x=colx1,x2,…,xN; then,(7)e=H⊗INCx−∑k=N+1N+ZA0k⊗FV˜k,where V˜k=G1N⊗VkN with G=diagτ1,τ2,…,τN and 1N is an N-dimensional column vector, H=∑k=N+1N+Z1/ZL+A0k, and A0k=diaga1k,a2k,…,aNk. Then, if limt⟶∞ei=0, we can get the following formula:(8)limt⟶∞Cixi=limt⟶∞∑k=N+1N+ZζikFτiVk,i∈N,where ζik∈Rl is ith row vector of H−1A0k1N.Assumption 1.
GraphG is a signed graph with a balanced structure and a spanning tree.Assumption 2.
Eigenvalues ofA0 are in the right half plane.Assumption 3.
Ci,Ai is observable and Ai,Bi is stabilizable.


## 2.1. Algebraic Graph Theory

To achieve the bipartite output containment, communication topology is described in a topology diagram [37]. Let GV,E,A be weighted directed graph and consist of node set V=V1,V2,…,VN which denotes finite nonempty set of N nodes and edge set E⊆V×V describing transfer of information between nodes. We define A=aij∈RN×N as weighted adjacency matrix. Let di=∑j=1Naij be in-degree of vertex i and D=diagd1,d2,…,dN be in-degree matrix of G. Laplace matrix L=lij of weighted directed graph G is defined as L=D−A. The point set is divided into two subgroups, v1 and v2, and v1∪v2=V and v1∩v2=∅, which is different from the general graph theory. Define Ni=Vj|Vj,Vi∈E,i≠j. to represent neighbor points of Vi. If there is an edge Vi,Vj∈E and Vi and Vj are in the same subgroup, it represents cooperative relationship; then, there is aij>0; if Vi and Vj are in different subgroups, it represents antagonistic relationship; then, there is aij<0, otherwise, aij=0. Define τi as a symbolic parameter, where τi=1 is used if Vi∈v1 and τi=−1 is used if Vi∈v2.


## 2.2. Quantizer

Quantizer has been added to many literature studies on multiagent research. In this study, we use the quantizerq:R⟶R as(1)qx=μi,if11+γμi<x≤11−γμi,x>0,0,ifx=0,−q−x,ifx<0,where μi is the quantization level and γ∈0,1 is the quantization accuracy parameter. For the quantization level, one has a set μ¯=±μi:μi=1−γ/1+γiμ0,i=0,±1,±2,…,±n∪0, where μ0>0 is the initial quantization level. qm−m≤γm,∀m∈Rcan be obtained through the definition of quantizer. It is worth noting that, for any set of vectorsy=y1,y2,…,yNT∈Rn, we can writeqy=qy1,qy2,…,qyNT∈Rn. So, obviously there is qy−y=Γy, where Γ=diagΓ1,Γ2,…,ΓN and Γi∈−γ,+γ.


## 2.3. Problem Formulation

In this study, we study the bipartite containment control for a group of MASs composed ofZ leaders and N followers. In addition, Z=N+1,N+2,…,N+Z and N=1,2,…,N are defined as leaders’ set and followers’ set, respectively. The kinetic equation for N followers is as follows:(2)x˙i=Aixi+Biui+∑k=N+1N+ZQikVk,yi=Cixi,i=1,2,…,N,where xi∈Rl,ui∈Rl,andyi∈Rl are state quantity of i th agent, control input quantity of the system, and the control output quantity of the system, respectively. Ai, Bi, Ci, and Qik are parameter matrix with suitable dimensions. Then, the kinetic equation of leaders is in the following form:(3)V˙k=A0Vk,yk=FVk,k∈Z,where Vk∈Rl represents the state of the exogenous system, that is, disturbance quantity or reference input of the system.Definition 1.
In this study, a reasonable controller is designed to solve the problem of bipartite containment control, which makes the followers converge to the positive and negative region defined by the leader, namely, the convex hull. So, for follower systems and leader systems (1) and (2), the following features can be used to solve their bipartite containment problem.(1)
Define the convex hullcoX belonging to set X=x1,x2,…,xN of the form(4)coX=∑i=1Nβixixi∈R,βi≥0,∑i=1Nβi=1.(2)
For followers and leaders with any initial value, the follower can only enter the positive and negative convex hull defined by the leader:(5)limt⟶∞distxi,coVk,k∈Z=0,i∈N1,limt⟶∞distxj,−coVk,k∈Z=0,j∈N2,in whichN1∪N2=N and N1∩N2=∅.
Whenei=0 is used, (2) can be displayed by the following containment error:(6)ei=∑j∈Niaijyi−sgnaijyj+∑k=N+1N+Zaikyi−τiyk,i∈N.
Lete=cole1,e2,…,eN and x=colx1,x2,…,xN; then,(7)e=H⊗INCx−∑k=N+1N+ZA0k⊗FV˜k,where V˜k=G1N⊗VkN with G=diagτ1,τ2,…,τN and 1N is an N-dimensional column vector, H=∑k=N+1N+Z1/ZL+A0k, and A0k=diaga1k,a2k,…,aNk. Then, if limt⟶∞ei=0, we can get the following formula:(8)limt⟶∞Cixi=limt⟶∞∑k=N+1N+ZζikFτiVk,i∈N,where ζik∈Rl is ith row vector of H−1A0k1N.Assumption 1.
GraphG is a signed graph with a balanced structure and a spanning tree.Assumption 2.
Eigenvalues ofA0 are in the right half plane.Assumption 3.
Ci,Ai is observable and Ai,Bi is stabilizable.


## 3. Main Results

In this section, an adaptive quantitative information observer is designed to observe the information of the leader matrixA0 so as to solve the problem that followers are unknowable of the information of the leader matrix. The Sylvester equation is presented, and an algorithm for solving it is presented. The bipartite containment control is realized by designing an observer and by using the output feedback method.


### 3.1. Bipartite Adaptive Quantization Information Observer

The adaptive quantization information distributed observer of the bipartite containment control is as follows:(9)A˙0i=χ1∑j∈NiaijA0i−sgnaijA0j+∑k=N+1N+ZaikA0i−τiA0,(10)η˙i=τiA0iηi+c1χ2∑j∈Niaijqηi−sgnaijηj+∑k=N+1N+Zaijqηi−τiVk,where ηi∈Rn,i∈N, and χ1,χ2<0. c1>0 is the compensator parameter.Lemma 1.
Consider external systems (2) and adaptive quantized information distributed observer (6) and (7). Let A˜0i=A0i−τiA0 and η˜i=ηi−∑k=N+1N+ZζikτiVk. Then, for χ1,χ2<0 and i∈N, limt⟶∞A˜0it=0 and limt⟶∞η˜it=0.Proof.
According to (6), the derivative of A˜0it can be obtained as follows:(11)A˜˙0i=A˙0i−τiA˙0=χ1∑j∈NiaijA0i−sgnaijA0j+∑k=N+1N+ZaikA0i−τiA0=χ1∑j∈NiaijA0i−τiA0+τiA0−sgnaijA0j−τjA0+τjA0+∑k=N+1N+ZaikA0i−τiA0=χ1∑j∈NiaijA˜0i−sgnaijA˜0j+∑j∈NiaijτiA0−sgnaijτiA0+∑k=N+1N+ZaikA˜0i.
Notice ifi and j are in the same subset i,j∈N1 or i,j∈N2, there are aij>0,τi=τj. So, it is easy to get τi=sgnaijτj. In another case, i and j in different subgroups, we can get aij<0,τi=−τjτi=sgnaijτj. Through the above discussion, τiA0−sgnaijτjA0=0 can be obtained. Then, (8) can be written as(12)A˜˙0i=χ1∑j∈NiaijA˜0i−sgnaijA˜0j+∑k=N+1N+ZaikA˜0i.
LetA˜0I=block diagA˜01,A˜02,…,A˜0N; (9) can be written in concise form:(13)A˜˙0I=χ1H⊗InA˜0I,where H=∑k=N+1N+Z1/ZL+A0k. By assumption 1 and Lemma 2of[29], eigenvalues of matrix L have only positive real part and 1/ZL+A0k is nonsingular. Since A0k has no negative eigenvalues, H=∑k=N+1N+Z1/ZL+A0k also has only nonzero positive eigenvalues. That is to say, λH>0. χ1H⊗In<0 can be obtained by setting the parameter χ1<0, and λχ1H⊗In<0 with all negative eigenvalues can be obtained. And then, we get limt⟶∞A˜0It=0, which means we get limt⟶∞A˜0it=0. Thus, the following formula can be obtained:(14)η˜˙i=η˙i−∑k=N+1N+ZζikτiV˙k=τiA0iηi+c1χ2∑j∈Niaijqηi−sgnaijηj+∑k=N+1N+aijqηi−τiVk−∑k=N+1N+ZζikτiA0Vk.
Since the signature parameterτi=1 or τi=−1, multiply both sides of A˜0i=A0i−τiA0 by τi to obtain τiA˜0i=τiA0i−A0. Then, formula can be written as(15)η˜˙i=τiA˜0i+A0ηi+c1χ2∑j∈Niaijqηi−sgnaijηj+∑k=N+1N+Zaijqηi−τiVk−∑k=N+1N+ZζikτiA0Vk,and (11); from Section 2.2, quantizer definition can be written in the following concise form as(16)η˜˙=G⊗InA˜0I+IN⊗A0η+c11−γχ2H⊗Inη−∑k=N+1N+ZH−1A0k⊗InV˜k−∑k=N+1N+ZH−1A0k⊗InIN⊗A0V˜k=G⊗InA˜0Iη+IN⊗A0η˜+c11−γχ2H⊗Inη˜=G⊗InA˜0Iη˜+∑k=N+1N+ZH−1A0k⊗InV˜k+IN⊗A0η˜+c11−γχ2H⊗Inη˜=IN⊗A0+c11−γχ2H⊗Inη˜+G⊗InA˜0Iη˜+G⊗InA˜0I∑k=N+1N+ZH−1A0k⊗InV˜k.
Under the condition of assumption3, χ2 is negative and small enough; that is, χ2≪0. So, IN⊗A0+χ2H⊗In is Hurwitz. We can get limt⟶∞η˜t=0 and limt⟶∞η˜it=0 through Lemma 1of[30]. The proof is completed.


### 3.2. Solution of Sylvester Equations

In this study, some followers cannot get information from the leader’s matrixA0. A0 is estimated by A0i, and the generalized Lyapunov equation of the Sylvester equation is used to prove the stability of MASs. Therefore, in this section, the Sylvester equations in this paper are solved by the method in [30]. The Sylvester equations are as follows(17)XkIN⊗A0=A^Xk+B^k,(18)0=C^Xk+D^,where Xk is solution to the above equation and A^, B^k, C^, and D^ are given in the simulation examples in Section 4.Lemma 2.
By Lemma3in[30], we give a matrix Pt; limt⟶∞Pt−p=0 can be obtained. For any ε>0,(19)X˙=−εPTtPtX,has and only has a unique solution Xt. There are several X∗ that make pX∗=0 and limt⟶∞Xt−X∗=0.Proof.
The orthogonal matrixM guarantees pM=p¯0 and(20)MTpTpM=p¯Tp¯000,(21)MTpT=p¯0.
LetX∗=MX¯1∗X¯2∗ and p¯X¯1∗=0. Then, the formula is established as follows:(22)pX∗=pMMTX∗=p¯0X¯1∗X¯2∗=0.
LetX¯=MTX; it yields(23)X¯˙=−MTεPTtPtX=−εMTpTpX+εMTpTpX−εMTPTtPtX=−εMTpTpX+εMTpTp−PTtPtX=−εMTpTpMX¯+ft,in which ft=εMTpTp−PTtPtMX¯. Let X¯=X¯1X¯2 and ft=f1tf2t. By Lemma 3 and Remark 2 in [30], limt⟶∞ft=0 and limt⟶∞f1t=0. Then, from (18), we can get the following formula:(24)X¯˙1X¯˙2=−εp¯Tp¯000X¯1X¯2+f1tf2t,and it is concluded that(25)X¯˙1=−εp¯Tp¯X¯1+f1t,X¯˙2=f2t,and there is X¯2∗ to make limt⟶∞X¯2−X¯2∗=0. Let X˜2=X¯1−X¯1∗; derivative of X˜2 is(26)X¯˙1=f1t−εp¯Tp¯X¯1=f1t−εp¯Tp¯X¯1∗−εp¯Tp¯X˜1=f1t−εp¯Tp¯X˜1.
By Lemma1in[30], limt⟶∞X˜1=0 because X¯=MTX has(27)limt⟶∞Xt−X∗=limt⟶∞MX¯−MX¯1∗X¯2∗=limt⟶∞MX¯1−X¯1∗X¯2−X¯2∗=0.
Proof is done.
Now, the feedback protocol is designed:(28)ξ˙i=Aiξi+Biui+∑k=N+1N+ZEikVk+LiCiξi−yiui=K1iξi+ηi,in which K1i is gain matrix. (1), (2), and (19) give us the following formula:(29)x˙i=Aixi+BiK1iξi+Biηi+∑k=N+1N+ZEikVk,ξ˙i=Aiξi+BiK1iξi+Biηi+∑k=N+1N+ZEikVk+LiCiξi−Cixi,
LetX=xξ and Vk=V¯kη, where(30)x=colx1,x2,…,xN,ξ=colξ1,ξ2,…,ξN,V¯k=colVk,Vk,…,Vk∈RNn,η=colη1,η2,…,ηN.
So, (20) can be changed as(31)X˙=A^X+∑k=N+1N+ZB^kVk,where(32)A^=ABK1−LCA+BK1+LC,B^k=Ek1ZBEk1ZB,and(33)A=block diagA1,A2,…,AN,B=block diagB1,B2,…,BN,C=block diagC1,C2,…,BN,Ek=block diagE1k,E2k,…,E3k,K1=block diagK11,K12,…,K1N,K2=block diagK21,K22,…,K2N.
Through (13) and (14), whereD^=0−IN⊗1/ZF and C^=C0. And then, we solve (13) and (14) through the lemma derived from [30].Lemma 3.
Consider equation as(34)Y˙=−εψTtψtY,where ε>0 is large enough and ψt=I2N⊗A0itT⊗I2Nn000−I2Nn⊗A^B^kC^D^. Let Ξt=m3Nn2NnY=XktI2Nn. Then, Xk∗ is the solution of (13) and (14):(35)limt⟶∞Ξt−Xk∗I2Nn=0,which goes to zero no slower than A˜0I.Proof.
(13) and (14) can be written as follows:(36)I2Nn000XkI2NnI2N⊗A0−A^B^kC^D^Xk∗I2Nn=0.
Then, we have(37)ψΘ=0,in which(38)ψ=I2N⊗A0T⊗I2Nn000−I2Nn⊗A^B^kC^D^Θ=Xk∗I2Nn.
limt⟶∞ψt−ψ=0 goes to zero no slower than A˜0I. Based on Lemma 2, limt⟶∞Ξt−Θ=0 can be obtained, where there is a Xk∗ satisfying (21). The proof is completed.


### 3.3. Collaborative Analysis

By the above lemma and proof, we prove (4) and solve the problem of the bipartite containment control. In addition, Xk∗=Xk11∗Xk12∗Xk21∗Xk22∗ is the solution of equations (13) and (14), where Xk11∗,Xk12∗,Xk21∗,Xk22∗∈RNn×Nn.Theorem 1.
For (1) and (2), the MASs composed of the follower and the leader are described, respectively. Under assumption two, the existence of K1i and Li makes Ai+BiK1i and Ai+LiCi are Hurwitz. Then, a matrix T=INn0−INnINn controller (20) can be used to achieve the bipartite containment and the distributed error (4) converges to zero.Proof.
LetA¯=TA^T−1; we obtain(39)A¯=Ai+BK1BK10A+LC.
In assumption2, we can see that A¯ is Hurwitz. Then, we know A^ is Hurwitz. If X˜=X−∑k=N+1N+ZXk∗Vk is set, the derivative of X˜ can be obtained:(40)X˜˙=X˙−∑k=N+1N+ZXk∗V˙k=A^X+∑k=N+1N+ZB^kVk−∑k=N+1N+ZXk∗V¯˙kη˙=A^X+∑k=N+1N+ZB^kVk−∑k=N+1N+ZXk∗IN⊗A0VkIN⊗A0η+∑k=N+1N+ZXk∗0G⊗INA˜0Iη+∑k=N+1N+ZXk∗0c11−γχ2H⊗INη˜=A^X+∑k=N+1N+ZB^kVk−∑k=N+1N+ZXk∗I2N⊗A0Vk−∑k=N+1N+ZXk∗0G⊗INA˜0Iη+∑k=N+1N+ZXk∗0c11−γχ2H⊗INη˜.
Then, formula (13) is used to rewrite (23) into the following form:(41)X˜˙=A^X−∑k=N+1N+ZA^Xk∗Vk−∑k=N+1N+ZXk∗0G⊗INA˜0Iη+∑k=N+1N+ZXk∗0c11−γχ2H⊗INη˜=A^X˜+Λ1η˜+Λ2,in which(42)Λ1=−∑k=N+1N+ZXk∗0c11−γχ2H⊗IN,Λ2=−∑k=N+1N+ZXk∗0G⊗INA˜0Iη,and limt⟶∞Λ2=0 exponentially at least as fast as limt⟶∞A˜0I=0. Therefore, through Lemma 1in[30] and Lemma 1 of this paper, we get limt⟶∞X˜=0. Now, consider distributed error (4), which has the following form:(43)e=H⊗INCx−H⊗ININ⊗Fη+H⊗ININ⊗Fη−∑k=N+1N+ZA0k⊗FV˜k=H⊗Inσ+H⊗Fη˜,where σ=Cx−IN⊗Fη, and σ has this(44)σ=Cx−IN⊗Fη=C0X+∑k=N+1N+Z0−IN⊗1ZFV¯kη=C^X+∑k=N+1N+ZD^Vk=C^X¯+∑k=N+1N+ZC^XVk+∑k=N+1N+ZD^Vk.
Through (14), we have(45)σ=C^X˜.
Because oflimt⟶∞X˜=0, we can get limt⟶∞σ=0. Under Lemma 1, limt⟶∞η˜=0. So, bipartite containment error converges to zero, that is, limt⟶∞e=0.
The proof is completed.


## 3.1. Bipartite Adaptive Quantization Information Observer

The adaptive quantization information distributed observer of the bipartite containment control is as follows:(9)A˙0i=χ1∑j∈NiaijA0i−sgnaijA0j+∑k=N+1N+ZaikA0i−τiA0,(10)η˙i=τiA0iηi+c1χ2∑j∈Niaijqηi−sgnaijηj+∑k=N+1N+Zaijqηi−τiVk,where ηi∈Rn,i∈N, and χ1,χ2<0. c1>0 is the compensator parameter.Lemma 1.
Consider external systems (2) and adaptive quantized information distributed observer (6) and (7). Let A˜0i=A0i−τiA0 and η˜i=ηi−∑k=N+1N+ZζikτiVk. Then, for χ1,χ2<0 and i∈N, limt⟶∞A˜0it=0 and limt⟶∞η˜it=0.Proof.
According to (6), the derivative of A˜0it can be obtained as follows:(11)A˜˙0i=A˙0i−τiA˙0=χ1∑j∈NiaijA0i−sgnaijA0j+∑k=N+1N+ZaikA0i−τiA0=χ1∑j∈NiaijA0i−τiA0+τiA0−sgnaijA0j−τjA0+τjA0+∑k=N+1N+ZaikA0i−τiA0=χ1∑j∈NiaijA˜0i−sgnaijA˜0j+∑j∈NiaijτiA0−sgnaijτiA0+∑k=N+1N+ZaikA˜0i.
Notice ifi and j are in the same subset i,j∈N1 or i,j∈N2, there are aij>0,τi=τj. So, it is easy to get τi=sgnaijτj. In another case, i and j in different subgroups, we can get aij<0,τi=−τjτi=sgnaijτj. Through the above discussion, τiA0−sgnaijτjA0=0 can be obtained. Then, (8) can be written as(12)A˜˙0i=χ1∑j∈NiaijA˜0i−sgnaijA˜0j+∑k=N+1N+ZaikA˜0i.
LetA˜0I=block diagA˜01,A˜02,…,A˜0N; (9) can be written in concise form:(13)A˜˙0I=χ1H⊗InA˜0I,where H=∑k=N+1N+Z1/ZL+A0k. By assumption 1 and Lemma 2of[29], eigenvalues of matrix L have only positive real part and 1/ZL+A0k is nonsingular. Since A0k has no negative eigenvalues, H=∑k=N+1N+Z1/ZL+A0k also has only nonzero positive eigenvalues. That is to say, λH>0. χ1H⊗In<0 can be obtained by setting the parameter χ1<0, and λχ1H⊗In<0 with all negative eigenvalues can be obtained. And then, we get limt⟶∞A˜0It=0, which means we get limt⟶∞A˜0it=0. Thus, the following formula can be obtained:(14)η˜˙i=η˙i−∑k=N+1N+ZζikτiV˙k=τiA0iηi+c1χ2∑j∈Niaijqηi−sgnaijηj+∑k=N+1N+aijqηi−τiVk−∑k=N+1N+ZζikτiA0Vk.
Since the signature parameterτi=1 or τi=−1, multiply both sides of A˜0i=A0i−τiA0 by τi to obtain τiA˜0i=τiA0i−A0. Then, formula can be written as(15)η˜˙i=τiA˜0i+A0ηi+c1χ2∑j∈Niaijqηi−sgnaijηj+∑k=N+1N+Zaijqηi−τiVk−∑k=N+1N+ZζikτiA0Vk,and (11); from Section 2.2, quantizer definition can be written in the following concise form as(16)η˜˙=G⊗InA˜0I+IN⊗A0η+c11−γχ2H⊗Inη−∑k=N+1N+ZH−1A0k⊗InV˜k−∑k=N+1N+ZH−1A0k⊗InIN⊗A0V˜k=G⊗InA˜0Iη+IN⊗A0η˜+c11−γχ2H⊗Inη˜=G⊗InA˜0Iη˜+∑k=N+1N+ZH−1A0k⊗InV˜k+IN⊗A0η˜+c11−γχ2H⊗Inη˜=IN⊗A0+c11−γχ2H⊗Inη˜+G⊗InA˜0Iη˜+G⊗InA˜0I∑k=N+1N+ZH−1A0k⊗InV˜k.
Under the condition of assumption3, χ2 is negative and small enough; that is, χ2≪0. So, IN⊗A0+χ2H⊗In is Hurwitz. We can get limt⟶∞η˜t=0 and limt⟶∞η˜it=0 through Lemma 1of[30]. The proof is completed.


## 3.2. Solution of Sylvester Equations

In this study, some followers cannot get information from the leader’s matrixA0. A0 is estimated by A0i, and the generalized Lyapunov equation of the Sylvester equation is used to prove the stability of MASs. Therefore, in this section, the Sylvester equations in this paper are solved by the method in [30]. The Sylvester equations are as follows(17)XkIN⊗A0=A^Xk+B^k,(18)0=C^Xk+D^,where Xk is solution to the above equation and A^, B^k, C^, and D^ are given in the simulation examples in Section 4.Lemma 2.
By Lemma3in[30], we give a matrix Pt; limt⟶∞Pt−p=0 can be obtained. For any ε>0,(19)X˙=−εPTtPtX,has and only has a unique solution Xt. There are several X∗ that make pX∗=0 and limt⟶∞Xt−X∗=0.Proof.
The orthogonal matrixM guarantees pM=p¯0 and(20)MTpTpM=p¯Tp¯000,(21)MTpT=p¯0.
LetX∗=MX¯1∗X¯2∗ and p¯X¯1∗=0. Then, the formula is established as follows:(22)pX∗=pMMTX∗=p¯0X¯1∗X¯2∗=0.
LetX¯=MTX; it yields(23)X¯˙=−MTεPTtPtX=−εMTpTpX+εMTpTpX−εMTPTtPtX=−εMTpTpX+εMTpTp−PTtPtX=−εMTpTpMX¯+ft,in which ft=εMTpTp−PTtPtMX¯. Let X¯=X¯1X¯2 and ft=f1tf2t. By Lemma 3 and Remark 2 in [30], limt⟶∞ft=0 and limt⟶∞f1t=0. Then, from (18), we can get the following formula:(24)X¯˙1X¯˙2=−εp¯Tp¯000X¯1X¯2+f1tf2t,and it is concluded that(25)X¯˙1=−εp¯Tp¯X¯1+f1t,X¯˙2=f2t,and there is X¯2∗ to make limt⟶∞X¯2−X¯2∗=0. Let X˜2=X¯1−X¯1∗; derivative of X˜2 is(26)X¯˙1=f1t−εp¯Tp¯X¯1=f1t−εp¯Tp¯X¯1∗−εp¯Tp¯X˜1=f1t−εp¯Tp¯X˜1.
By Lemma1in[30], limt⟶∞X˜1=0 because X¯=MTX has(27)limt⟶∞Xt−X∗=limt⟶∞MX¯−MX¯1∗X¯2∗=limt⟶∞MX¯1−X¯1∗X¯2−X¯2∗=0.
Proof is done.
Now, the feedback protocol is designed:(28)ξ˙i=Aiξi+Biui+∑k=N+1N+ZEikVk+LiCiξi−yiui=K1iξi+ηi,in which K1i is gain matrix. (1), (2), and (19) give us the following formula:(29)x˙i=Aixi+BiK1iξi+Biηi+∑k=N+1N+ZEikVk,ξ˙i=Aiξi+BiK1iξi+Biηi+∑k=N+1N+ZEikVk+LiCiξi−Cixi,
LetX=xξ and Vk=V¯kη, where(30)x=colx1,x2,…,xN,ξ=colξ1,ξ2,…,ξN,V¯k=colVk,Vk,…,Vk∈RNn,η=colη1,η2,…,ηN.
So, (20) can be changed as(31)X˙=A^X+∑k=N+1N+ZB^kVk,where(32)A^=ABK1−LCA+BK1+LC,B^k=Ek1ZBEk1ZB,and(33)A=block diagA1,A2,…,AN,B=block diagB1,B2,…,BN,C=block diagC1,C2,…,BN,Ek=block diagE1k,E2k,…,E3k,K1=block diagK11,K12,…,K1N,K2=block diagK21,K22,…,K2N.
Through (13) and (14), whereD^=0−IN⊗1/ZF and C^=C0. And then, we solve (13) and (14) through the lemma derived from [30].Lemma 3.
Consider equation as(34)Y˙=−εψTtψtY,where ε>0 is large enough and ψt=I2N⊗A0itT⊗I2Nn000−I2Nn⊗A^B^kC^D^. Let Ξt=m3Nn2NnY=XktI2Nn. Then, Xk∗ is the solution of (13) and (14):(35)limt⟶∞Ξt−Xk∗I2Nn=0,which goes to zero no slower than A˜0I.Proof.
(13) and (14) can be written as follows:(36)I2Nn000XkI2NnI2N⊗A0−A^B^kC^D^Xk∗I2Nn=0.
Then, we have(37)ψΘ=0,in which(38)ψ=I2N⊗A0T⊗I2Nn000−I2Nn⊗A^B^kC^D^Θ=Xk∗I2Nn.
limt⟶∞ψt−ψ=0 goes to zero no slower than A˜0I. Based on Lemma 2, limt⟶∞Ξt−Θ=0 can be obtained, where there is a Xk∗ satisfying (21). The proof is completed.


## 3.3. Collaborative Analysis

By the above lemma and proof, we prove (4) and solve the problem of the bipartite containment control. In addition, Xk∗=Xk11∗Xk12∗Xk21∗Xk22∗ is the solution of equations (13) and (14), where Xk11∗,Xk12∗,Xk21∗,Xk22∗∈RNn×Nn.Theorem 1.
For (1) and (2), the MASs composed of the follower and the leader are described, respectively. Under assumption two, the existence of K1i and Li makes Ai+BiK1i and Ai+LiCi are Hurwitz. Then, a matrix T=INn0−INnINn controller (20) can be used to achieve the bipartite containment and the distributed error (4) converges to zero.Proof.
LetA¯=TA^T−1; we obtain(39)A¯=Ai+BK1BK10A+LC.
In assumption2, we can see that A¯ is Hurwitz. Then, we know A^ is Hurwitz. If X˜=X−∑k=N+1N+ZXk∗Vk is set, the derivative of X˜ can be obtained:(40)X˜˙=X˙−∑k=N+1N+ZXk∗V˙k=A^X+∑k=N+1N+ZB^kVk−∑k=N+1N+ZXk∗V¯˙kη˙=A^X+∑k=N+1N+ZB^kVk−∑k=N+1N+ZXk∗IN⊗A0VkIN⊗A0η+∑k=N+1N+ZXk∗0G⊗INA˜0Iη+∑k=N+1N+ZXk∗0c11−γχ2H⊗INη˜=A^X+∑k=N+1N+ZB^kVk−∑k=N+1N+ZXk∗I2N⊗A0Vk−∑k=N+1N+ZXk∗0G⊗INA˜0Iη+∑k=N+1N+ZXk∗0c11−γχ2H⊗INη˜.
Then, formula (13) is used to rewrite (23) into the following form:(41)X˜˙=A^X−∑k=N+1N+ZA^Xk∗Vk−∑k=N+1N+ZXk∗0G⊗INA˜0Iη+∑k=N+1N+ZXk∗0c11−γχ2H⊗INη˜=A^X˜+Λ1η˜+Λ2,in which(42)Λ1=−∑k=N+1N+ZXk∗0c11−γχ2H⊗IN,Λ2=−∑k=N+1N+ZXk∗0G⊗INA˜0Iη,and limt⟶∞Λ2=0 exponentially at least as fast as limt⟶∞A˜0I=0. Therefore, through Lemma 1in[30] and Lemma 1 of this paper, we get limt⟶∞X˜=0. Now, consider distributed error (4), which has the following form:(43)e=H⊗INCx−H⊗ININ⊗Fη+H⊗ININ⊗Fη−∑k=N+1N+ZA0k⊗FV˜k=H⊗Inσ+H⊗Fη˜,where σ=Cx−IN⊗Fη, and σ has this(44)σ=Cx−IN⊗Fη=C0X+∑k=N+1N+Z0−IN⊗1ZFV¯kη=C^X+∑k=N+1N+ZD^Vk=C^X¯+∑k=N+1N+ZC^XVk+∑k=N+1N+ZD^Vk.
Through (14), we have(45)σ=C^X˜.
Because oflimt⟶∞X˜=0, we can get limt⟶∞σ=0. Under Lemma 1, limt⟶∞η˜=0. So, bipartite containment error converges to zero, that is, limt⟶∞e=0.
The proof is completed.


## 4. Numerical Simulations

In this section, a numerical example is used to illustrate the effectiveness of the bipartite containment control. Figure1 can be viewed as a topology of (1) and (2), showing the communication relationship between six agents. Since the symbolic graph is considered in this study, we select 5 and 6 as the leader and the other agents as the followers and then divide 1 and 3, 2, and 4 into two different subgroups N1,N2, including N1∪N2=N and N1∩N2=∅. It can be concluded from Figure 1 that the Laplace matrix L and matrix A have the following forms:(46)A=000−10001100000−10,L=1001010−1−10100011.Figure 1
Communication topology of 2 leaders and 4 followers.In addition,A0k=diaga1k,a2k,…,aNk and H=∑k=N+1N+Z1/ZL+A0k have the following forms:(47)A05=1000010000000000,A06=0000000000100001,(48)H=L+A05+A06=2001020−1−10200002.Considering the follower and leader systems (1) and (2), the correlation matrix is given as follows:(49)A0=1−21.5−1,F=4000,Ai=00.5∗i00,Bi=010.5∗i0,C1=8−10.52−7,C2=2−6−3−5.5C3=1.5−8−3−3.8,C4=−4−15−4−2,Ei5=0000.1,Ei6=000.050.,Based on Section3.3, it can be known that Ai+BiK1i and Ai+LiCi are Hurwitz, and then, K1i and Li can be calculated according to the above matrix. c1=3.6 is also given:(50)K11=−0.1604−1.1036−0.9871−0.3207,K12=−0.4142−1.2872−0.9102−0.4142,K13=−0.6074−1.4321−0.7944−0.4049,K14=−0.7310−1.5259−0.6824−0.3655,L1=−0.93210.55860.44130.8974,L2=−0.57060.81810.74340.6689,L3=−0.33070.96810.87980.4754,L4=0.19990.86850.9989−0.0451.Based on the above matrix, Figures2–5 show the bilateral output curve of the agent. The red line represents leader 5, the yellow line represents leader 6, and the remaining colors are followers. Figures 2 and 5 show the first row contained in the leader bilateral output with 5 and 6. Specifically, y51 is the first line of y5=FV5 and y61 is the first line of y6=FV6. y11 is the first line of y1=C1x1, y21 is the first line of y2=C2x2, y31 is the first line of y3=C3x3, and y41 is the first line of y4=C4x4. Similarly, Figures 3 and 4 show the output containing the second row, the second component. To sum up, it can be observed that some followers converge to the region bounded by leaders 5 and 6, while the rest converge to the opposite region. Moreover, follower information transmission takes into account the influence of quantification, which is more interesting and challenging.Figure 2
Output curves of leader 51 and 61 and follower 11 and 31.Figure 3
Output curves of leader 52 and 62 and follower 22 and 42.Figure 4
Output curves of leader 52 and 62 and follower 12 and 32.Figure 5
Output curves of leader 51 and 61 and follower 21 and 41.


## 5. Conclusions

In this study, bipartite containment control is discussed. The matrixA0 of the external leader system is observed by using an adaptive quantization observer, and the adaptive quantization information observer is proved to be effective by the eigenvalue analysis method without M-matrix. A controller with only one gain matrix and an appropriate protocol for the control evolution algorithm is designed. At the same time, considering the influence of quantization on follower information transmission, a quantizer is added. Finally, the bipartite containment control is realized by using the feedback method.In the future, we will further study the bipartite containment control with adaptive quantization information distributed observer under switched topology or in finite time.



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