# Intermittent Sampled Data Control for Time-Varying Formation-Containment of the Multiagent System with/without Time Delay

Authors: Ming Chi; Xu-Long Wang; Yangming Dou; Zhi-Wei Liu
Journal: Complexity
 (2021)

---

## Abstract

Time-varying formation-containment problems for a second-order multiagent system (SOMAS) are studied via pulse-modulated intermittent control (PMIC) in this paper. A distributed control framework utilizing the neighbors’ positions and velocities is designed so that leaders in the multiagent system form a formation, and followers move to the convex hull formed by each leader. Different from the traditional formation-containment problems, this paper applies the PMIC framework, which is more common and more in line with the actual control scenarios. Based on the knowledge of matrix theory, algebraic graph theory, and stability theory, some sufficient conditions are given for the time-varying formation-containment problem of the second-order multiagent system. Some numerical simulations are proposed to verify the effectiveness of the results presented in this paper.


---

## Body

## 1. Introduction

Many scholars start to pay attention to the multiagent systems (MASs) with the rapid development of complex network system theory. In recent decades, many major breakthroughs have been made in this field [1], and those results are also widely used in various fields of production and life, such as UAV cruise system [2], smart grid [3, 4], economic dispatching [5], and multiple underactuated surface vessels [6]. The most studied collaboration problems of MASs include swarm, consensus, formation, and distributed optimization. In very recent years, the clustering behavior of MASs has also attracted widespread interest, including but not limited to consensus [7–9], tracking, and formation [10].In these cooperative control problems, many important advances have been made in the field of formation control and containment control. Huang et al. [11] studied the containment control problem of MASs via intermittent control-based sampled data information. Wang et al. [12] investigated the containment control problem of first-order MASs in the noisy communication environments. Rahimi et al. [13] studied time-varying formation control of collaborative heterogeneous MASs. However, most of the existing works are carried out on formation control and containment control separately. In many applications, both these cooperative behaviors often require simultaneous implementation. For example, in the coordination of multiple tanker airplanes and multiple UAVs, the tanker airplanes will form a specific formation in advance and wait for the UAVs to reach the area they surround. In order to solve this kind of problem, Dong et al. designed a continuous control strategy [14] for second-order MASs with multiple leaders and multiple followers, which can ensure the formation control of the leaders and containment of the followers. It is worth noting that the above work [14] adopted a continuous-time control which may be difficult to implement in some cases due to sampled measurement. The traditional zero-order sampling control adopts the same amplitude control input in the whole sampling interval, which can also lead to application difficulties. For instance, in a driverless system, it is difficult for the sensors on the vehicle to work all the time, allowing for fuel economy and other reasons.Aiming at solving the aforementioned problems, the formation-containment problem of the second-order multiagent system (SOMAS) is studied in this paper, and the control framework of pulse-modulated intermittent control (PMIC) is adopted. The main contributions are given as follows. (i) The formation and the containment are achieved simultaneously in second-order MASs, where the leaders form a formation and the followers are contained in this formation. (ii) The PIMC is a framework that can unify impulsive control [8, 15] and zero-order sampling control [16]. It can be applied to a wider range of real-world scenarios. (iii) In this paper, some sufficient conditions are given for the parameters of the control strategy under with/without time delay cases.The remainder sections of this paper are described as follows. Section2 lists the basic preliminary knowledge and a model of the problem to be studied. Section 3 introduces the formation-containment analysis without time delay. Section 4 obtains and provides the results when MASs contain time delay. Section 5 gives several simulations to verify the theorems are correct. Section 6 draws the conclusion.


## 2. Preliminaries

### 2.1. Graph Theory and Some Lemmas

LetG=V,E,A be a weighted digraph, where V=1,…,N is a vertex set, E⊆V×V is a link set, and A=aij∈ℝN×N is a nonnegative weighted adjacency matrix. The information flow from vertex j to vertex i is represented by a directed link eij=j,i. The elements of matrix A are described as follows: aij>0 if eij∈E, and aij=0, otherwise. Furthermore, aii=0 for all i∈V, and aij=aji in an undirected topology. Denote Ni=vj∈V:vj,vi∈E as the set of neighbors of node vi. The Laplacian matrix L is described in terms of L=D−A, where D=diagD1,D2,…,DN is the in-degree matrix with Di=∑j=1Naij. A sequence of edges with different nodes of V as vi,vi1,vi,vi2,…,vi,vj can be obtained if there is a directed path from the ith agent to the jth agent.In a MAS, a SOMAS withN agents is considered, and there are M followers and N−M leaders. Assume that followers can receive messages from the leaders or followers, while the leaders can only receive messages from the leaders. Let F=1,2,…,M and E=M+1,M+2,…,N denote the sets of followers and leaders, respectively. And the Laplacian matrix LN is described as(1)LN=L1L20L3,where L1∈ℝM×M, L2∈ℝM×N−M, and L3∈ℝN−M×N−M.Lemma 1 (see [17]).
A complex characteristic polynomialRz=z2+sz+r is Hurwitz stable if and only if Res>0 and ResImsImr+Re2sRer−Im2r>0.Lemma 2 (see [18]).
If directed graphG contains a spanning tree, then the Laplacian matrix L∈ℝN×N of G has a simple zero eigenvalue with 1⟶N as the associated eigenvector, and all the other N−1 eigenvalues have positive real parts.Assumption 1.
Each follower of graphG has at least one directed path from one leader.Assumption 2.
LetGE be the graph associated with the leaders in the MAS, and GE contains a spanning tree.Lemma 3 (see [19]).
By Assumptions1 and 2, it is obtained that the eigenvalues of L1 have positive real parts, each row of −L1−1L2 has a sum equal, and each entry of −L1−1L2 is nonnegative.


### 2.2. Model Formulation and Some Definitions

The control input of theith agent is denoted by ui∈ℝn, position by pi∈ℝn, and velocity by vi∈ℝn, respectively. Consider a SOMAS as(2)p˙it=vit,v˙it=uit.In the following, for the sake of description, we assume thatn=1. However, more cases such as n>1 can be derived by using the Kronecker product.Let,B=01, and xit=pit,vitTi=1,2,…,N. Then, the dynamic equation of the ith agent can be described in a neat form as follows:(3)x˙it=Axit+Buit.Definition 1.
With the bounded initial state of each agent being chosen arbitrarily, the leaders in MAS (3) are said to realize time-varying formation if there exists a vector function ct∈ℝ2 such that(4)limt⟶∞xit−hit−ct=0i∈E,where hit=hipt,hivtT is a piecewise continuously differentiable vector. Then, we can denote the time-varying formation as hEt=hM+1Tt,hM+2Tt,…,hNTtT∈ℝ2N−M.Definition 2.
Similarly, with the bounded initial state of each agent being chosen arbitrarily, MAS (3) is said to achieve containment if there exist nonnegative constants τk,jk∈F,j∈E such that ∑j=M+1Nτk,j=1 and(5)limt⟶∞xkt−∑j=M+1Nτk,jxjt=0.


## 2.1. Graph Theory and Some Lemmas

LetG=V,E,A be a weighted digraph, where V=1,…,N is a vertex set, E⊆V×V is a link set, and A=aij∈ℝN×N is a nonnegative weighted adjacency matrix. The information flow from vertex j to vertex i is represented by a directed link eij=j,i. The elements of matrix A are described as follows: aij>0 if eij∈E, and aij=0, otherwise. Furthermore, aii=0 for all i∈V, and aij=aji in an undirected topology. Denote Ni=vj∈V:vj,vi∈E as the set of neighbors of node vi. The Laplacian matrix L is described in terms of L=D−A, where D=diagD1,D2,…,DN is the in-degree matrix with Di=∑j=1Naij. A sequence of edges with different nodes of V as vi,vi1,vi,vi2,…,vi,vj can be obtained if there is a directed path from the ith agent to the jth agent.In a MAS, a SOMAS withN agents is considered, and there are M followers and N−M leaders. Assume that followers can receive messages from the leaders or followers, while the leaders can only receive messages from the leaders. Let F=1,2,…,M and E=M+1,M+2,…,N denote the sets of followers and leaders, respectively. And the Laplacian matrix LN is described as(1)LN=L1L20L3,where L1∈ℝM×M, L2∈ℝM×N−M, and L3∈ℝN−M×N−M.Lemma 1 (see [17]).
A complex characteristic polynomialRz=z2+sz+r is Hurwitz stable if and only if Res>0 and ResImsImr+Re2sRer−Im2r>0.Lemma 2 (see [18]).
If directed graphG contains a spanning tree, then the Laplacian matrix L∈ℝN×N of G has a simple zero eigenvalue with 1⟶N as the associated eigenvector, and all the other N−1 eigenvalues have positive real parts.Assumption 1.
Each follower of graphG has at least one directed path from one leader.Assumption 2.
LetGE be the graph associated with the leaders in the MAS, and GE contains a spanning tree.Lemma 3 (see [19]).
By Assumptions1 and 2, it is obtained that the eigenvalues of L1 have positive real parts, each row of −L1−1L2 has a sum equal, and each entry of −L1−1L2 is nonnegative.


## 2.2. Model Formulation and Some Definitions

The control input of theith agent is denoted by ui∈ℝn, position by pi∈ℝn, and velocity by vi∈ℝn, respectively. Consider a SOMAS as(2)p˙it=vit,v˙it=uit.In the following, for the sake of description, we assume thatn=1. However, more cases such as n>1 can be derived by using the Kronecker product.Let,B=01, and xit=pit,vitTi=1,2,…,N. Then, the dynamic equation of the ith agent can be described in a neat form as follows:(3)x˙it=Axit+Buit.Definition 1.
With the bounded initial state of each agent being chosen arbitrarily, the leaders in MAS (3) are said to realize time-varying formation if there exists a vector function ct∈ℝ2 such that(4)limt⟶∞xit−hit−ct=0i∈E,where hit=hipt,hivtT is a piecewise continuously differentiable vector. Then, we can denote the time-varying formation as hEt=hM+1Tt,hM+2Tt,…,hNTtT∈ℝ2N−M.Definition 2.
Similarly, with the bounded initial state of each agent being chosen arbitrarily, MAS (3) is said to achieve containment if there exist nonnegative constants τk,jk∈F,j∈E such that ∑j=M+1Nτk,j=1 and(5)limt⟶∞xkt−∑j=M+1Nτk,jxjt=0.


## 3. Formation-Containment Analysis without Time Delay

In this section, we mainly study how to design the PMIC protocol to make MAS (3) achieve the time-varying formation-containment and propose some sufficient and necessary conditions for parameters. We will investigate the time-varying formation-containment problem in two steps. The first step is to transform the formation-containment problem into a stability problem. The second step is to solve the stability problem according to the related theory.


### 3.1. Problem Transformation

Consider the following PMIC protocols:(6)uit=at−tkK1∑j∈Niaijxitk−hitk−xjtk−hjtk,i∈E,uit=at−tkK2∑j∈Niaijxitk−xjtk,i∈F,where K1=k11,k12∈ℝ1×2 and K2=k21,k22∈ℝ1×2 are feedback gain matrices and tk<t<tk+1. The pulse function at is described as(7)at=a^t,t∈0,d,0,t∉0,d,where a^t is a piecewise continuous function. Let T=tk+1−tk and d<T be the control duration within a complete sampling cycle. tk+d,tk+1 is the rest interval, and tk,tk+d is the control interval [20].Under the control framework (6), MAS (3) can be described in a compact form as follows:(8)x˙Et=IN−M⊗AxEt+at−tkL3⊗BK1xEtk−L3⊗BK1hEtk,x˙Ft=IM⊗AxFt+at−tkL1⊗BK2xFtk+L2⊗BK2xEtk,where xEt=xM+1Tt,xM+2Tt,…,xNTtT and xFt=x1Tt,x2Tt,…,xMTtT.Letθit=xit−hiti∈E and θEt=θM+1Tt,θM+2Tt,…,θNTtT. Then, system (8) can be written as(9)θ.Et=IN−M⊗AθEt+at−tkL3⊗BK1θEtk+IN−M⊗AhEt−IN−M⊗I2h˙Et.The eigenvalues ofL3 relating to GE are denoted by λii∈E, where λM+1=0 with the eigenvector as pM+1=1N−M/N−M, and 0<ReλM+2≤⋯≤ReλN. Let PE−1L3PE=JE, where JE is the Jordan canonical form of L3, PE=pM+1,pM+2,…,pN, and PE−1=p˜M+1,p˜M+2,…,p˜NH. By Lemma 2, one has JE=diag0,J¯E, where J¯E consists of the Jordan blocks relating to λii=M+2,M+3,…,N. Let θ˜Et=PE−1⊗I2θEt; then, system (9) can be rewritten as(10)θ˜Et=IN−M⊗Aθ˜Et+at−tkPE−1L3PE⊗BK1θ˜Etk+PE−1⊗AhEt−PE−1⊗I2h˙Et.LetP˜E=p˜M+2,p˜M+3,…,p˜NH, αEt=p˜M+1H⊗I2θEt, and βEt=P˜E⊗I2θEt. Then, system (10) can be divided as(11)α˙Et=AαEt+p˜M+1H⊗AhEt−p˜M+1H⊗I2h˙Et,β˙Et=IN−M−1⊗AβEt+at−tkJ¯E⊗BK1βEtk+P˜E⊗AhEt−P˜E⊗I2h˙Et.Letδi=∑j∈Niwijxit−xjt,i∈F, and δFt=δ1t,δ2t,…,δMt. Then, the following lemma is used to transform the formation-containment problem.Lemma 4.
MAS (3) under the PMIC framework (6) can achieve time-varying formation-containment if(12)limt⟶∞βEt=02N−M−1,limt⟶∞δFt=02M.Proof.
It is able to be proved by a similar way in [14].Remark 1.
Lemma4 is proved because the following results in [14] are worked out:(13)limt⟶∞xEt−hEt−1N−MN−M⊗I2αEt=02N−M,limt⟶∞xFt−−L1−1L2⊗I2xEt=02M.
Considering Lemma3, we can conclude that equation (13) satisfies Definitions 1 and2, respectively. In other words, Lemma 4 can be proved by the above two equations.


### 3.2. Control Design

By means of Lemma4, the time-varying formation-containment problem can be transformed into the convergence analysis of βEt and δFt. This section presents the conditions for the parameters in control protocol (6) when βEt and δFt converge to 0.Assumption 3.
For∀i∈M+1,M+2,…,N,(14)limt⟶∞Ahit−hjt−h˙it−h˙jt=0,j∈Ni.Lemma 5.
Let(15)β˜˙Et=IN−M−1⊗Aβ˜Et+at−tkJ¯E⊗BK1β˜Etk.Under Assumption3, limt⟶∞βEt=02N−M−1 is satisfied if(16)limt⟶∞β˜Et=02N−M−1.Proof.
If Assumption3 holds, one has(17)limt⟶∞L3⊗AhEt−L3⊗I2h˙Et=0.
SubstituteL3=PEJEPE−1 into (17), and then premultiply both sides of (17) by PE−1⊗I2. One can obtain that(18)limt⟶∞J¯EP˜E⊗AhEt−J¯EP˜⊗I2h˙Et=0.
Under Assumption2 and Lemma 2, J¯E is nonsingular, obviously. By premultiplying both sides of (18) by J¯E−1⊗I2, we have(19)limt⟶∞P˜E⊗AhEt−P˜⊗I2h˙Et=0.
Considering equation (19), Lemma 5 is set up.Lemma 6.
∀i∈M+2,M+3,…,N, equation (16) holds if the following inequalities are satisfied:(20)k12k11>d2d1,(21)2k12k11T−2d2d1T22d2d1T−2k12k11T−4Reλik11d1λi2T−1−16Im2λik112d12λi4T2>0,where d1=∫0da^sds and d2=∫0dsa^sds.Proof.
Equation (16) holds if and only if β˜E in (15) is asymptotically stable. The solution of (15) can be written as(22)β˜Et=eCt−t0β˜Et0+eCt∫t0te−Csas−tkDβ˜Etkds,where C=IN−M−1⊗A and D=J¯E⊗BK1. Let tk=t0 and tk+d=t in (22); then, we have(23)β˜Etk+d=eCdβ˜Etk+eCtk+d∫tktk+de−Csas−tkDβ˜Etkds.
According toC2=0a and the theory of matrix function, one can obtain that(24)β˜Etk+d=I2N−M−1+dC+d1D+dd1−d2CDβ˜Etk.
Notice thatβ˜Etk+1=β˜Etk+T; then, the system can be discretely expressed as(25)β˜Etk+1=I2N−M−1+TC+d1D+Td1−d2CDβ˜Etk.
LetΓ=I2N−M−1+TC+d1D+Td1−d2CD and detsI2N−M−1−Γ=0. It yields(26)s2−2+k12d1λi+k11λiTd1−d2s+1+k12d1λi−k11d2λi=0.
Consideringλi≠0, then it is focused on the conditions that the eigenvalues of Γ are encircled by the unit circle. By applying a bilinear transformation, z=s+1/s−1, an updated polynomial can be found as(27)Rz=z2+2k12d1λi−k11λid2k11λid1Tz+2k11λid2−2k12d1λi−k11d1λiT−4k11λid1T.
s<1 in (26) can be obtained if and only if Rz in (27) is Hurwitz stable. Based on Lemma 1, if (27) is stable, then both equations (20) and (21) hold.Lemma 7.
limt⟶∞δFt=02M holds if and only if ∀i∈1,2,…,M,(28)k22k21>d2d1,(29)2k22k21T−2d2d1T22d2d1T−2k22k21T−4Reλik21d1λi2T−1−16Im2λik212d12λi4T2>0,where d1=∫0da^sds and d2=∫0dsa^sds.Proof.
It is obtained that(30)δFt=L2⊗I2xEt+L1⊗I2xFt.
Substituting (8) into (30), we can obtain that(31)δ˙Ft=IM⊗AδFt+at−tkL1⊗BK2δFtkat−tkL2L3⊗BK1xEtk−hEtk.
By Lemmas4 and 5, MAS (8) can achieve formation hEt. Considering Definition 1, when the leaders’ formation hEt is achieved, then(32)limt⟶∞L2L3⊗BK1xEtk−hEtk=limt⟶∞L2L3⊗BK11⊗ctk.
SinceL31=0, one has(33)limt⟶∞L2L3⊗BK1xEtk−hEtk=0.
Then, consider the following system:(34)δ˙Ft=IM⊗AδFt+at−tkL1⊗BK2δFtk.
limt⟶∞δFt=02M means that system (34) is asymptotically stable. Similar to the analysis of Lemma 5, the conditions that make system (34) asymptotically stable are worked out with (28) and (29).Theorem 1.
MAS (3) under Assumption 3 and PMIC framework (6) realizes formation-containment if the following conditions simultaneously hold:(i)
∀i∈1,2,…,M,(35)k22k21>d2d1,2k22k21T−2d2d1T22d2d1T−2k22k21T−4Reλik21d1λi2T−1−16Im2λik212d12λi4T2>0.(ii)
∀i∈M+2,M+3,…,N,(36)k12k11>d2d1,2k12k11T−2d2d1T22d2d1T−2k12k11T−4Reλik11d1λi2T−1−16Im2λik112d12λi4T2>0,where d1=∫0da^sds and d2=∫0dsa^sds.Proof.
It is proved naturally by Lemmas4, 6, and 7.Remark 2.
When selecting parameters in the control rules, we generally giveK1, K2, d1, and d2 that meet the conditions first. According to the values of these parameters and conditions (35) and (36) in Theorem 1, we can solve parameter T as follows:(i)
∀i∈1,2,…,M,(37)T<minM13λi4+4Reλiλi2M1216k21d1Im2λi+k21d1M12λi4,whereM1=2k21d2−2k22d1.(ii)
∀i∈M+2,M+3,…,N,(38)T<minM13λi4+4Reλiλi2M1216k11d1Im2λi+k11d1M12λi4,where M2=2k11d2−2k12d1.


## 3.1. Problem Transformation

Consider the following PMIC protocols:(6)uit=at−tkK1∑j∈Niaijxitk−hitk−xjtk−hjtk,i∈E,uit=at−tkK2∑j∈Niaijxitk−xjtk,i∈F,where K1=k11,k12∈ℝ1×2 and K2=k21,k22∈ℝ1×2 are feedback gain matrices and tk<t<tk+1. The pulse function at is described as(7)at=a^t,t∈0,d,0,t∉0,d,where a^t is a piecewise continuous function. Let T=tk+1−tk and d<T be the control duration within a complete sampling cycle. tk+d,tk+1 is the rest interval, and tk,tk+d is the control interval [20].Under the control framework (6), MAS (3) can be described in a compact form as follows:(8)x˙Et=IN−M⊗AxEt+at−tkL3⊗BK1xEtk−L3⊗BK1hEtk,x˙Ft=IM⊗AxFt+at−tkL1⊗BK2xFtk+L2⊗BK2xEtk,where xEt=xM+1Tt,xM+2Tt,…,xNTtT and xFt=x1Tt,x2Tt,…,xMTtT.Letθit=xit−hiti∈E and θEt=θM+1Tt,θM+2Tt,…,θNTtT. Then, system (8) can be written as(9)θ.Et=IN−M⊗AθEt+at−tkL3⊗BK1θEtk+IN−M⊗AhEt−IN−M⊗I2h˙Et.The eigenvalues ofL3 relating to GE are denoted by λii∈E, where λM+1=0 with the eigenvector as pM+1=1N−M/N−M, and 0<ReλM+2≤⋯≤ReλN. Let PE−1L3PE=JE, where JE is the Jordan canonical form of L3, PE=pM+1,pM+2,…,pN, and PE−1=p˜M+1,p˜M+2,…,p˜NH. By Lemma 2, one has JE=diag0,J¯E, where J¯E consists of the Jordan blocks relating to λii=M+2,M+3,…,N. Let θ˜Et=PE−1⊗I2θEt; then, system (9) can be rewritten as(10)θ˜Et=IN−M⊗Aθ˜Et+at−tkPE−1L3PE⊗BK1θ˜Etk+PE−1⊗AhEt−PE−1⊗I2h˙Et.LetP˜E=p˜M+2,p˜M+3,…,p˜NH, αEt=p˜M+1H⊗I2θEt, and βEt=P˜E⊗I2θEt. Then, system (10) can be divided as(11)α˙Et=AαEt+p˜M+1H⊗AhEt−p˜M+1H⊗I2h˙Et,β˙Et=IN−M−1⊗AβEt+at−tkJ¯E⊗BK1βEtk+P˜E⊗AhEt−P˜E⊗I2h˙Et.Letδi=∑j∈Niwijxit−xjt,i∈F, and δFt=δ1t,δ2t,…,δMt. Then, the following lemma is used to transform the formation-containment problem.Lemma 4.
MAS (3) under the PMIC framework (6) can achieve time-varying formation-containment if(12)limt⟶∞βEt=02N−M−1,limt⟶∞δFt=02M.Proof.
It is able to be proved by a similar way in [14].Remark 1.
Lemma4 is proved because the following results in [14] are worked out:(13)limt⟶∞xEt−hEt−1N−MN−M⊗I2αEt=02N−M,limt⟶∞xFt−−L1−1L2⊗I2xEt=02M.
Considering Lemma3, we can conclude that equation (13) satisfies Definitions 1 and2, respectively. In other words, Lemma 4 can be proved by the above two equations.


## 3.2. Control Design

By means of Lemma4, the time-varying formation-containment problem can be transformed into the convergence analysis of βEt and δFt. This section presents the conditions for the parameters in control protocol (6) when βEt and δFt converge to 0.Assumption 3.
For∀i∈M+1,M+2,…,N,(14)limt⟶∞Ahit−hjt−h˙it−h˙jt=0,j∈Ni.Lemma 5.
Let(15)β˜˙Et=IN−M−1⊗Aβ˜Et+at−tkJ¯E⊗BK1β˜Etk.Under Assumption3, limt⟶∞βEt=02N−M−1 is satisfied if(16)limt⟶∞β˜Et=02N−M−1.Proof.
If Assumption3 holds, one has(17)limt⟶∞L3⊗AhEt−L3⊗I2h˙Et=0.
SubstituteL3=PEJEPE−1 into (17), and then premultiply both sides of (17) by PE−1⊗I2. One can obtain that(18)limt⟶∞J¯EP˜E⊗AhEt−J¯EP˜⊗I2h˙Et=0.
Under Assumption2 and Lemma 2, J¯E is nonsingular, obviously. By premultiplying both sides of (18) by J¯E−1⊗I2, we have(19)limt⟶∞P˜E⊗AhEt−P˜⊗I2h˙Et=0.
Considering equation (19), Lemma 5 is set up.Lemma 6.
∀i∈M+2,M+3,…,N, equation (16) holds if the following inequalities are satisfied:(20)k12k11>d2d1,(21)2k12k11T−2d2d1T22d2d1T−2k12k11T−4Reλik11d1λi2T−1−16Im2λik112d12λi4T2>0,where d1=∫0da^sds and d2=∫0dsa^sds.Proof.
Equation (16) holds if and only if β˜E in (15) is asymptotically stable. The solution of (15) can be written as(22)β˜Et=eCt−t0β˜Et0+eCt∫t0te−Csas−tkDβ˜Etkds,where C=IN−M−1⊗A and D=J¯E⊗BK1. Let tk=t0 and tk+d=t in (22); then, we have(23)β˜Etk+d=eCdβ˜Etk+eCtk+d∫tktk+de−Csas−tkDβ˜Etkds.
According toC2=0a and the theory of matrix function, one can obtain that(24)β˜Etk+d=I2N−M−1+dC+d1D+dd1−d2CDβ˜Etk.
Notice thatβ˜Etk+1=β˜Etk+T; then, the system can be discretely expressed as(25)β˜Etk+1=I2N−M−1+TC+d1D+Td1−d2CDβ˜Etk.
LetΓ=I2N−M−1+TC+d1D+Td1−d2CD and detsI2N−M−1−Γ=0. It yields(26)s2−2+k12d1λi+k11λiTd1−d2s+1+k12d1λi−k11d2λi=0.
Consideringλi≠0, then it is focused on the conditions that the eigenvalues of Γ are encircled by the unit circle. By applying a bilinear transformation, z=s+1/s−1, an updated polynomial can be found as(27)Rz=z2+2k12d1λi−k11λid2k11λid1Tz+2k11λid2−2k12d1λi−k11d1λiT−4k11λid1T.
s<1 in (26) can be obtained if and only if Rz in (27) is Hurwitz stable. Based on Lemma 1, if (27) is stable, then both equations (20) and (21) hold.Lemma 7.
limt⟶∞δFt=02M holds if and only if ∀i∈1,2,…,M,(28)k22k21>d2d1,(29)2k22k21T−2d2d1T22d2d1T−2k22k21T−4Reλik21d1λi2T−1−16Im2λik212d12λi4T2>0,where d1=∫0da^sds and d2=∫0dsa^sds.Proof.
It is obtained that(30)δFt=L2⊗I2xEt+L1⊗I2xFt.
Substituting (8) into (30), we can obtain that(31)δ˙Ft=IM⊗AδFt+at−tkL1⊗BK2δFtkat−tkL2L3⊗BK1xEtk−hEtk.
By Lemmas4 and 5, MAS (8) can achieve formation hEt. Considering Definition 1, when the leaders’ formation hEt is achieved, then(32)limt⟶∞L2L3⊗BK1xEtk−hEtk=limt⟶∞L2L3⊗BK11⊗ctk.
SinceL31=0, one has(33)limt⟶∞L2L3⊗BK1xEtk−hEtk=0.
Then, consider the following system:(34)δ˙Ft=IM⊗AδFt+at−tkL1⊗BK2δFtk.
limt⟶∞δFt=02M means that system (34) is asymptotically stable. Similar to the analysis of Lemma 5, the conditions that make system (34) asymptotically stable are worked out with (28) and (29).Theorem 1.
MAS (3) under Assumption 3 and PMIC framework (6) realizes formation-containment if the following conditions simultaneously hold:(i)
∀i∈1,2,…,M,(35)k22k21>d2d1,2k22k21T−2d2d1T22d2d1T−2k22k21T−4Reλik21d1λi2T−1−16Im2λik212d12λi4T2>0.(ii)
∀i∈M+2,M+3,…,N,(36)k12k11>d2d1,2k12k11T−2d2d1T22d2d1T−2k12k11T−4Reλik11d1λi2T−1−16Im2λik112d12λi4T2>0,where d1=∫0da^sds and d2=∫0dsa^sds.Proof.
It is proved naturally by Lemmas4, 6, and 7.Remark 2.
When selecting parameters in the control rules, we generally giveK1, K2, d1, and d2 that meet the conditions first. According to the values of these parameters and conditions (35) and (36) in Theorem 1, we can solve parameter T as follows:(i)
∀i∈1,2,…,M,(37)T<minM13λi4+4Reλiλi2M1216k21d1Im2λi+k21d1M12λi4,whereM1=2k21d2−2k22d1.(ii)
∀i∈M+2,M+3,…,N,(38)T<minM13λi4+4Reλiλi2M1216k11d1Im2λi+k11d1M12λi4,where M2=2k11d2−2k12d1.


## 4. Formation-Containment Analysis with Time Delay

We know that the time delay cannot be ignored due to its widely existence in the real world. Then, a control protocol with considering time delay is proposed as follows:(39)uit=at−tkK1∑j∈Niwijxitk−τ−hitk−xjtk−τ−hjtk,i∈E,uit=at−tkK2∑j∈Niwijxitk−τ−xjtk−τ,i∈F.The control protocol (39) is substituted into MAS (3), and we can obtain(40)x˙Et=IN−M⊗AxEt+at−tkL3⊗BK1xEtk−τ−L3⊗BK1hEtk,x˙Ft=IM⊗AxFt+at−tkL1⊗BK2xFtk−τ+L2⊗BK2xEtk−τ.Considering Assumption3, when there is a time delay in the system, similar to Section 3, we have(41)β˜˙Et=IN−M−1⊗Aβ˜Et+at−tkJ¯E⊗BK1β˜Etk−τ,δ˙Ft=IM⊗AδFt+at−tkL1⊗BK2δFtk−τ.It is not hard to see that Lemma4 also applies to cases with time delay. Therefore, we still transform the formation-containment problem of the SOMAS into the stability problem of the system.Remark 3.
Considering that the time delay of the actual system is not long and the intermittent control has the advantage of selectively adjusting the control time intervald, we assume that the time delay τ<mind,T−d.Lemma 8.
limt⟶∞β˜Et=02N−M−1 holds with time delay if ∀i∈M+2,M+3,…,N,(42)−2τT−2d2d1T+2k12k11T>0,(43)−2τT−2d2d1T+2k12k11T2−4Reλiλi2k11d1T−1+2τT+2d2d1T−2k12k11T−16Im2λiλi4k112d12T2>0,where d1=∫0da^sds, d2=∫0dsa^sds, and τ is the time delay.Proof.
Similarly as Lemma6, differential equation (41) is solved. Since τ<mind,T−d, when t∈tk,tk+1, we can obtain that(44)β˜it=Mt−tkβ˜itk+Nt−tkβ˜itk−τ,where Mχ=1t01 and Nχ=td1−d2λik11td1−d2λik12λik11d1λik12d1. To solve the stability problem, we need to consider the iterative problem of the system. When t=tk, one has(45)β˜itk=MTβ˜itk−1+NTβ˜itk−1−τ,β˜itk−τ=MT−τβ˜itk−1+NT−τβ˜itk−1−τ.
Letωit=β˜itβ˜it−τ; combined with (43), it is obtained that(46)ωit=β˜itβ˜it−τ=Γ1Γ2kβ˜it0β˜it0−τ,where Γ1=Mt−tkNt−tkMt−tk−τNt−tk−τ and Γ2=MTNTMT−τNT−τ. It is easy to get that ωit⟶0 if the eigenvalues of Γ2 meet s<1. Let sI4−Γ2=0. It yields(47)s4+a1s3+a0s2=0,where a1=−λik11d1T+λik11d1τ+λik11d2−λik12d1−2 and a0=−λik11d1τ−λik11d2+λik12d1+1. We can obtain that two eigenvalues of Γ2 satisfy s1=s2=0. Then, the remaining proof process has been omitted, which is similar to the proof of Lemma 6.Lemma 9.
limt⟶∞δFt=02M holds with time delay if ∀i∈1,2,…,M,(48)−2τT−2d2d1T+2k22k21T>0,(49)−2τT−2d2d1T+2k22k21T2−4Reλiλi2k21d1T−1+2τT+2d2d1T−2k22k21T−16Im2λiλi4k212d12T2>0,where d1=∫0da^sds, d2=∫0dsa^sds, and τ is the time delay.Proof.
This proof is analogous to Lemma8 and is therefore omitted.Theorem 2.
MAS (3) under Assumption 3 and control framework (39) realizes formation-containment if conditions (42), (43), (48), and (49) hold.Proof.
It is not hard to prove by applying Lemmas4, 8, and 9.


## 5. Simulation

In this section, a two-dimensional formation-containment case of the MAS will be shown by numerical simulations. The simulations illustrate the effectiveness of the results in this paper. In this two-dimensional formation-containment case, suppose there are three leaders and two followers in the MAS. The Laplace matrix of the directed topology is given as(50)L=2−23−13−10−1200−10010−100−110000−11.Note thatpit∈ℝ2, vit∈ℝ2, and xit=piXt,viXt,piYt,viYtT, i∈1,2,…,5. Then, matrices A and B in dynamic equation (3) should be extended as(51)A=I2⊗0100,B=I2⊗01.The formation function for leaders is given by(52)hit=sint+2i−3π3cost+2i−3π3cost+2i−3π3−sint+2i−3π3i=3,4,5.The eigenvalues ofL are λ1=2.8165, λ2=1.1835, λ3=0, λ4=1.5+0.866i, and λ5=1.5−0.866i. The function a^t=t/d is set, and the control duration d=0.2 is chosen; then, d1=0.1 and d2=0.01333.When we do not care about time delays, it follows from (20) and (28) that k12/k11>0.1333 and k22/k21>0.1333. The feedback gains are chosen as K1=I2⊗−1,−2 and K2=I2⊗−1,−2. According to inequalities (21) and (29), T<1.5396 is obtained, and T=0.6 is chosen. When time delays are taken into account, we assume the time delay τ=0.1. According to Lemmas 8 and 9, we also choose the feedback gains as K1=I2⊗−1,−2 and K2=I2⊗−1,−2 and set T=0.6.The simulation results are shown in figures. Among them, Figures1 and 2 are the trajectories of the position and velocity of each agent changing without considering the time delay, in which the initial state is marked with a circle and the final state is marked with a triangular row. Figures 3 and 4 show the trajectories when time delay is taken into account. The same initial state is indicated by a circle, and the final state is indicated by a triangular row. The initial states of all agents are randomly selected.Figure 1
The trajectory of agents’ positions over time without time delay.Figure 2
The trajectory of agents’ velocities over time without time delay.Figure 3
The trajectory of agents’ positions over time with time delay.Figure 4
The trajectory of agents’ velocities over time with time delay.During the simulation, the five agents begin with arbitrary initial states. As time goes on, the three leaders reach a time-varying circular formation, and two followers move into the convex hull formed by leaders. At the same time, it can be seen that the system with time delay converges more slowly with the same sampling period.


## 6. Conclusion

Time-varying formation-containment problems of the SOMAS were studied in this paper. Based on the fundamental theorems of graph theory and matrix theory, the formation-containment problem is transformed into the stability problem of the SOMAS. The PMIC protocols are designed, and the sufficient conditions for the time-varying formation-containment of the SOMAS are given by proving three theorems. The simulations show the correctness of the proposed theoretical results. It is expected that the future work will be carried out on the nonlinear MAS, and the conclusion of this paper will be applied to the actual multi-intelligent vehicle experimental platform.



---

*Source: 9971855-2021-05-07.xml*