# On the Group Controllability of Leader-Based Continuous-Time Multiagent Systems

Authors: Bo Liu; Licheng Wu; Rong Li; Housheng Su; Yue Han
Journal: Complexity
 (2020)

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## Abstract

The group controllability of continuous-time multiagent systems (MASs) with multiple leaders is considered in this paper, where the entire group is compartmentalized into a few subgroups. The group controllability concept of continuous-time MASs with multiple leaders is put forward, and the group controllability criteria are obtained for switching and fixed topologies, respectively. Finally, the numerical simulations are given to prove the validity of the theoretical results.


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## Body

## 1. Introduction

In recent decades, distributed coordination control of networked MASs has become a hot and challenging issue in lots of areas, such as applied mathematics, control theory, mechanics, engineering, and neurobiology [1–10]. Studies in this area include several basic problems, such as stability, consensus and synchronization [11], containment [12], controllability [13], and formation control and tracking control [14].The controllability problem is a key essential problem in modern control theory and attracts increasing attention due to its wide applications in engineering. An MAS is controllable if each dynamic follower can attain its desirable configuration from any initial state during finite time by regulating some leaders. In [15], Tanner first put forward the controllability problem of networked systems in a leader-following framework, where a certain agent was acted as the leader (the external control input), and an algebraic feature based on eigenvalues and eigenvectors of such system’s Laplacian matrices was derived by nearest neighbor rules. Based on this, Liu et al. [16] discussed the controllability of discrete-time MASs with a single leader based on nearest neighbor rules and derived a simple controllable condition for such a system on switching topology. Afterwards, further studies on the controllability of MASs have mainly been concentrated from graph-theoretic and algebraic-theoretic points of view, respectively. At present, many works on the controllability of MASs from the perspective of graph theory have concentrated on the basis of partitions of graph topology, such as equitable partition/relaxed equitable partition/external equitable partition in [17], connected component partition in [18], and selection of leaders [19]. Further research studies on the controllability were presented for some different special topology graphs, such as path graphs [20], cycle graphs [21], multichain topologies [22], stars and trees [23], two-time-scale topologies [24], and regular graphs [25]. Lots of algebraic controllable conditions of MASs were characterized in [26, 27].The aforementioned results on the controllability of MASs just contained a single group. However, in engineering practice, a single group can be compartmentalized into some subgroups with the improvement of MASs’ complexity [28]. It is a very challenging work to study the controllability problem of MASs with multiple subgroups and multiple leaders considering the control law, the information topology structure between different subgroups, and the effect of dynamical leaders acting on the follower agents, which will be highlighted in this paper. More recently, the group controllability of continuous-time/discrete-time MASs leaderless with different topologies and communication restrictions in [29, 30] was studied, respectively.Motivated by the results of previous studies, this paper aims at the group controllability of continuous-time MASs consisting of some different subgroups by adjusting the leaders. The main contributions of this paper are summed up as follows:(1)
Different from the group controllability problem of continuous-time MASs under the leaderless framework studied based on the fixed topology in [30], the current work has considered the group controllability of continuous-time MASs under the leader-follower framework with fixed topology and switching topology, respectively, which can be expressed by the system matrices. It is obvious that different models can lead to completely different features for MASs with leaders.(2)
The concepts of the group controllability of continuous-time MASs with multiple leaders are proposed based on switching and fixed topologies, respectively.(3)
Sufficient and/or necessary algebraic- and graph-theoretic group controllable characterizations of continuous-time MASs with multiple leaders under the group consensus protocol are established from the system’s Laplacian matrices.(4)
The effects of subgroups and leaders on the group controllability are discussed.The rest of this work is arranged as follows. The problem formulation is stated in Section2. Section 3 builds the group controllability of MASs with multiple leaders. Numerical example and simulations are given in Section 4. Finally, Section 5 summarises the conclusion.


## 2. Problem Formulation

Consider a continuous-time MAS consisting ofN agents governed by(1)x˙it=uit,i=1,…,N,where xi∈R is the state and ui∈R is the control input, respectively.In engineering practice, the whole group can be compartmentalized into a few subgroups. Without loss of generality, in this paper, such MAS consisting ofm+n+l+k (m,n,l,k>1) agents is compartmentalized into subgroup G1,x1 and subgroup G2,x2, as shown in Figure 1.Figure 1
TopologyG.Denoteℓ1≜1,…,m, ℓ2≜m+1,…,m+n, ℓ1l≜m+n+1,…,m+n+l, ℓ2l≜m+n+l+1,…,m+n+l+k, V1≜v1,…,vm, and V2≜vm+1,…,vm+n; then, ℓ=ℓ1∪ℓ2, ℓl=ℓ1l∪ℓ2l, and V=V1∪V2. Ni is the ith agent’s neighbor set with N1i=vj∈V1:vj,vi∈ℰ and N2i=vj∈V2:vj,vi∈ℰ, where Ni=N1i∪N2i and N1i∩N2i=∅. N1p and N2p, respectively, represent the leaders’ neighbor sets of subgroups 1 and 2.Inspired by [30], the control input ui is designed as follows:(2)ui=∑j∈N1iaijxjt−xit+∑j∈N2iaijxjt+∑q∈N1pbiqyqt−xit,i∈ℓ1,∑j∈N2iaijxjt−xit+∑j∈N1iaijxjt+∑q∈N2pbiqyqt−xit,i∈ℓ2,where aij∈R,biq≥0,∀i,j,q∈ℓ1,ℓ2.Remark 1..
It is noted that thei,jth entry of the system’s adjacency matrix, denoted as aij, in this paper, can be allowed to be negative, which makes it more difficult and complex to discuss the group controllability problem since there are negative factors in the coupling links between different subgroups.
Supposed thatx1≜x1,…,xmT and x2≜xm+1,…,xm+nT are the state vectors of follower agents in G1 and G2 and y1≜y1,…,ylT and y2≜yl+1,…,yl+pT are the state vectors of the leader agents in G1 and G2, respectively. Then, the dynamics of the followers in system (1) becomes(3)x˙1x˙2=−L1−R1C1C2−L2−R2x1x2+B100B2y1y2≜A1C1C2A2x1x2+B100B2y1y2,where L1=lij∈ℝm×m and L2=lij∈ℝn×n are the Laplacian matrices of graphs G1 and G2, respectively, R1=diag∑q∈N1qb1q,…,∑q∈N1qbmq∈ℝm×m, R2=diag∑q∈N2qbm+1,q,…,∑q∈N2qbm+n,q∈ℝn×n,(4)A1≜−L1−R1∈ℝm×m,B1∈ℝm×l,C1∈ℝm×n,A2≜−L2−R2∈ℝn×n,B2∈ℝn×k,C2∈ℝn×m.Remark 2.
Furthermore, becauseaij can be allowed to be negative, nonzero controller gains can be appropriately selected as long as L1 and L2 are Laplacian matrices. In essence, the group controllability of continuous-time leader-based MASs cannot be affected by the controller gains.
In order to discuss the group controllability problem of system (3), its equivalent augmented system can be described by(5)x˙1=A1x1+C1B1x2y1,x˙2=A2x2+C2B2x1y2,where x2,y1 and (x1, y2) are the inputs of subgroup G1,x1 and subgroup G2,x2, respectively.


## 3. Group Controllability Analysis

This section discusses the group controllability of continuous-time leader-based MASs and establishes the group controllability criteria by adjusting appropriate leaders with switching and fixed topologies, respectively.


### 3.1. Group Controllability on Switching Topology

Similar to literature [29], corresponding system (5) with switching topology can be described as(6)x˙1=A1σtx1+C1B1σtx2y1,x˙2=A2σtx2+C2B2σtx1y2,where the switching path σt:ℝ+⟶1,…,K can be described by a piecewise constant scalar function, which presents the coupling links of such time-variant system, and K is the number of probable switching topologies. Moreover, we select Aip,CipBipi=1,2,p=1,2,…,K to achieve the system realizations when σt=p.Some relevant important concepts will be introduced in the following, and more details can be seen in [29].Definition 1.
(see [29], switching sequence). A finite scalars’ set π≜i0,…,iP−1 is said to be a switching sequence, where P∈0,∞ indicates the length of π, and a switching path σp is defined as σp=ip if p∈P¯ (P¯≜0,1,…,P−1) with ip∈1,2,…,P be the index of the pth realization.Definition 2.
(group switching controllability). A nonzero statex of system (6) attains group switching controllability if(1)
There are a time instantP∈0,∞, a switching path σ:P¯⟶1,…,K, and the input (x2, y1) for t∈P¯ such that x10=x1 and x1P=0(2)
There are a time instantP∈0,∞, a switching path σ:P¯⟶1,…,K, and the input (x1, y2) for t∈P¯ such that x20=x2 and x2P=0Definition 3.
(see [29], column space). For a preset matrix Bp×m=b1,…,bm, the column space ℛB is spanned by vectors b1,b2,…,bm, denoted as ℛB≜spanb1,…,bm.Lemma 1.
(see [29]). For matrices Ai∈Rp×mi, i∈r¯≜1,2,…,r and B=A1,A2,…,Ar∈ℝp×m, m=∑i=1rmi, then ℛB=∑i=1rℛAi.Definition 4.
(see [29], cyclic invariant subspace). For A∈RN×N and a linear subspace W⊆ℝN, AW is called as the R-cyclic invariant subspace indicated as AW≜∑i=1KAi−1W.
For notational simplicity, letA1C1B1≜A1ℛC1B1≜A1ℛrij for i=1 and j=1,2,…,l+n, and A2C2B2≜A2ℛrij for i=2 and j=1,2,…,m+k, where CiBi≜rij. For system (6), the subspace sequence is defined as(7)W11=∑i=1KA1ir1i,W12=∑i=1KA1iW11,…,W1m=∑i=1KA1iW1m−1,W21=∑i=1KA2ir2i,W22=∑i=1KA2iW21,…,W2n=∑i=1KA2iW2n−1.Lemma 2.
System (6) attains group switching controllability iff W1m=ℝm and W2n=ℝn.Proof..
Similar proof can be referred from that of Lemma 2 in [29]; here, it is omitted.Theorem 1.
System (6) attains group switching controllability if(8)∑i=1Kℛr1i=ℝm,∑i=1Kℛr2i=ℝn.Proof..
Obviously, fori=1,2,…,K,(9)ℛr1i⊆ℛr1i+ℛA1ir1i+⋯+ℛA1im−1r1i=A1ir1i,we have(10)ℝm=ℛr11+ℛr12+⋯+ℛr1K⊆A11r11+A12r12+⋯+A1Kr1K=W11⊆W12⊆⋯⊆W1m.
On the contrary, it is easy to knowW1m⊆ℝm. Therefore, we can have W1m=ℝm. For the subspace W2n, we can also have the similar result W2n=ℝn. From Lemmas 1 and 2, the assertion holds.Remark 3.
Theorem1 provides an important and simple method to check the controllability of continuous-time MASs with leaders by designing a switching path. At the same time, it is noted that the group controllability of continuous-time MASs with leaders can depend on leader-to-follower information communications (i.e., matrices Bi) and the subgroup-to-subgroup information communications (i.e., matrices Ci) regardless of the internal information communications between subgroups (i.e., matrices Ai) whether the topology of the internal network is fixed or switching, that is, the controllability of the subgroups is not required, which provides an important convenience for designing a switching path to ensure the group controllability for continuous-time MASs with switching topology.
In the following, some special important cases are discussed.


### 3.2. Group Controllability on Fixed Topology

Whenσt=1, system (6) (equivalently, (5)) presents an MAS based on fixed topology.Definition 5.
(see [30] (group controllability)). A nonzero state x of system (5) attains group controllability if(1)
There are a finite timeT∈J and the input (x2,y1) such that x10=x1 and x1T=0(2)
There are a finite timeT∈J and the input (x1,y2) such that x20=x2 and x2T=0Lemma 3.
System (5) attains group controllability iff rankQ1=m and rankQ2=n, where(11)Q1=C1B1⋮A1C1B1⋮A12C1B1⋮⋯⋮A1m−1C1B1,Q2=C2B2⋮A2C2B2⋮A22C2B2⋮⋯⋮A2n−1C2B2.
Here,Q1 is the controllability matrix of G1,x1, and Q2 is the controllability matrix of G2,x2.Proof..
The result is obvious from Definition5.Remark 4.
From Lemma3, it is too complex to compute the controllability matrices of system (5). On this basis, the group controllability of such MAS with leaders is shown by the technique of PBH rank test.Theorem 2.
(PBH rank test). System (5) attains group controllability iff system (5) satisfies(1)
ranksI−A1,C1,B1=m and ranktI−A2,C2,B2=n, ∀s,t∈ℂ, where ℂ is a complex number set(2)
rankλiI−A1,C1,B1=m and rankμiI−A2,C2,B2=n, where λi∀i=1,…,m and μi∀i=1,…,n are, respectively, the eigenvalues of A1 and A2Proof..
Obviously, if condition (1) holds, condition (2) absolutely holds. Therefore, it is only necessary to prove that condition (1) is true.
Necessity: by contradiction, supposed that∃s∈ℂ; then,(12)ranksI−A1,C1,B1<m,and then the rows of sI−A1,C1,B1 are linearly dependent. Thus, ∃α≠0 such that α′sI−A1,C1,B1=0. Therefore,(13)sα′=α′A1,α′C1=0,α′B1=0.
Moreover, we can have(14)α′Q1=α′C1A1C1A12C1⋯A1m−1C1⋮B1A1B1A12B1⋯A1m−1B1=α′C1α′A1C1α′A12C1⋯α′A1m−1C1⋮α′B1α′A1B1α′A12B1⋯α′A1m−1B1=α′B1sα′B1⋯sm−1α′B1⋮α′C1sα′C1⋯sm−1α′C1=0.
Sinceα≠0, then there must be rankQ1<m, which implies that system (5) is uncontrollable, contradicting to the assertion that system (5) attains the group controllability. The necessity of (1) is proved.
Sufficiency: by contradiction, assumed that system (5) is uncontrollable, then ∃λ∈ℂ of A1, which corresponds to the eigenvector β≠0 satisfying(15)β′A1=λβ′,β′C1=0,β′B1=0,and then β′λI−A1,C1,B1=0 so that rankλI−A1,C1,B1<m. This contradicts to ranksI−A1,C1,B1=m for ∀s∈ℂ. The sufficiency of (1) is proved.Theorem 3.
IfLiT=Li (i=1,2), system (5) attains group controllability iff(1)
The eigenvalues ofAi are different(2)
The eigenvectors ofAi are unorthogonal to at least one column of Bi or CiProof..
SinceL1T=L1, then A1T=A1, which can be displayed as A1=U1Λ1U1T, where the columns of U1 and the diagonal matrix Λ1 are made up of orthogonal eigenvectors and eigenvalues of A1, respectively. Moreover, U1U1T=I; then, the controllability matrix of such a system can be expressed as(16)Q1=B1A1B1A12B1⋯A1m−1B1⋮C1A1C1A12C1⋯A1m−1C1=U1U1TB1U1Λ1U1TB1U1Λ12U1TB1⋯U1Λ1m−1U1TB1⋮U1U1TC1U1Λ1U1TC1⋯U1Λ1m−1U1TC1=U1U1TB1Λ1U1TB1Λ12U1TB1⋯Λ1m−1U1TB1⋮U1TC1Λ1U1TC1Λ12U1TC1⋯Λ1m−1U1TC1≜U1Q1˜,where Q1˜=U1TB1Λ1U1TB1Λ12U1TB1⋯Λ1m−1U1TB1⋮U1TC1Λ1U1TC1Λ12U1TC1⋯Λ1m−1U1TC1.
SinceU1 consists of the orthogonal eigenvectors of A1, then U1 is nonsingular, which implies that rankQ1=rankQ1˜. Let λi and ηi be the eigenvalues and their corresponding eigenvectors of A1, respectively, for i=1,2,…,m.
For the convenience of discussion, letU1TB1≜r1,…,rl, U1TC1≜p1,…,pn, ri≜r1i,…,rmiT with rki=ηk,bi,k=1,2,…,m;i=1,2,…,l, and pi=p1i,p2i,…,pmiTi=1,2,…,n; then,(17)Q1˜=r1,r2,…,rl,λ1λ2⋱λmr1,r2,…,rl,…,λ1m−1λ2m−1⋱λmm−1r1,r2,…,rl⋮p1,p2,…,pn,λ1λ2⋱λmp1,p2,…,pn,…,λ1m−1λ2m−1⋱λmm−1p1,p2,…,pn⟶r1,Λ1r1,⋯,Λ1m−1r1,…,rl,Λ1rl,…,Λ1m−1rl⋮p1,Λ1p1,…,Λ1m−1p1,…,pn,Λ1pn,…,Λ1m−1pn⟶r11λ1r11λ12r11⋯λ1m−1r11r21λ2r21λ22r21⋯λ2m−1r21⋮⋮⋮⋮rm1λmrm1λm2rm1⋯λmm−1rm1,…,r1lλ1r1lλ12r1l⋯λ1m−1r1lr2lλ2r2lλ22r2l⋯λ2m−1r2l⋮⋮⋮⋮rmlλmrmlλm2rml⋯λmm−1rml⋮p11λ1p11λ12p11⋯λ1m−1p11p21λ2p21λ22p21⋯λ2m−1p21⋮⋮⋮⋮pm1λmpm1λm2pm1⋯λmm−1pm1,…,p1nλ1p1nλ12p1n⋯λ1m−1p1np2nλ2p2nλ22p2n⋯λ2m−1p2n⋮⋮⋮⋮pmnλmpmnλm2pmn⋯λmm−1pmn⟶r11r21⋱rm11λ1⋯λ1m−11λ2⋯λ2m−1⋮⋮⋮1λm⋯λmm−1,⋯,r1lr2l⋱rml1λ1⋯λ1m−11λ2⋯λ2m−1⋮⋮⋮1λm⋯λmm−1⋮p11p21⋱pm11λ1⋯λ1m−11λ2⋯λ2m−1⋮⋮⋮1λm⋯λmm−1,…,p1np2n⋱pmn1λ1⋯λ1m−11λ2⋯λ2m−1⋮⋮⋮1λm⋯λmm−1⟶η1,b1η2,b1⋱ηm,b1M,…,η1,blη2,bl⋱ηm,blM,…,η1,c11η2,c11⋱ηm,c11M,…,η1,c1nη2,c1n⋱ηm,c1nM,where η1,bl,ηm,c1n is the vector inner product and matrix(18)M=1λ1λ12⋯λ1m−11λ2λ22⋯λ2m−1⋮⋮⋮⋮1λmλm2⋯λmm−1,is a Vandermonde matrix. If Q˜1 has full row rank needing that at least one block of Q˜1 has full row rank, without loss of generality, the first block will be selected to discuss. It is known that Vandermonde matrix M is nonsingular if eigenvalues of A1 are distinct so that the row rank of matrix Q1˜ is decided by matrix diagη1,b1,η2,b1,…,ηm,b1 or matrix diagη1,c11,η2,c11,…,ηm,c11. Since eigenvectors Ai are unorthogonal to at least one column of Bi or Cii=1,2, therefore, matrix diagη1,b1,η2,b1,…,ηm,b1 or matrix diagη1,c11,η2,c11,…,ηm,c11 has full row rank, which means that Q˜1 has full row rank. Similarly, Q˜2 also has full row rank. This completes the proof.
Note that conditionLiT=Li implies that the information weight from agent i to agent j is the same as that from agent j to agent i in the same subgroup, that is, the topological structure is symmetric for the subgroups.Corollary 1.
System (5) is uncontrollable if subgroups G,x1 and G,x2 are both complete graphs (see Figure 2) and aij=biq∀i,j∈ℓ1,ℓ2,∀q∈ℓl, regardless of how to connect G,x1 and G,x2.Figure 2
Complete graph.Proof..
Because subgraphsG,x1 and G,x2 are both complete and aij=biq∀i,j∈ℓ1,ℓ2,∀q∈ℓl, without loss of generality, let aij=biq=1∀i,j∈ℓ1,ℓ2,∀q∈ℓl; then,(19)A1=−m+l−11⋯11−m+l−1⋯1⋮⋮⋮11⋯−m+l−1,A2=−n+k−11⋯11−n+k−1⋯1⋮⋮⋮11⋯−n+k−1.
By simple calculation, we can know thatA1’s eigenvalues are λi=0,−m+l,…,−m+l︸, and A2’s eigenvalues are μi=0,−n+k,…,−n+k︸. Then, A1 has common eigenvalue −m+l, and A2 has common eigenvalue −n+k, which are contrary to the conditions of Theorem 3. Thus, no matter how to connect subgroups G1,x1 and G2,x2, system (5) is uncontrollable.Corollary 2.
IfG,x1 and G,x2 are both star graphs (see Figure 3) as well as aij=biq∀i,j∈ℓ1,ℓ2,∀q∈ℓl regardless of how to connect G,x1 and G,x2, then system (5) is uncontrollable.Figure 3
Star graph.Proof..
Because subgraphsG,x1 and G,x2 are both star groups and aij=biq∀i,j∈ℓ1,ℓ2,∀q∈ℓl, without loss of generality, let aij=biq=1∀i,j∈ℓ1,ℓ2,∀q∈ℓl; therefore,(20)A1=−m−l−11⋯11−m−l−1⋯1⋮⋮⋮11⋯−m−l−1,A2=−n−k−11⋯11−n−k−1⋯1⋮⋮⋮11⋯−n−k−1.
By computing, we can also know that the eigenvalues ofA1 are λi=0,−m+l+2,…,−m+l+2︸, and the eigenvalues of A2 are μi=0,−n+k+2,…,−n+k+2︸. Then, A1 has common eigenvalues −m+l+2, and A2 has common eigenvalues −n+k+2, which contradict to condition (1) of Theorem 3. Thus, no matter how to connect subgroup G1,x1 and subgroup G2,x2, system (5) must be uncontrollable.Remark 5.
It is noted that there must exist a few leaders making the system to reach the desired state from the random initial state if system (5) is controllable. However, how to configure the leaders such that the desired formation can be achieved? That is, how to select the leaders (or design the inputs) with given initial state and desired state?
Here presents an algorithm for designing the leaders.Algorithm 1.
(algorithm for designing leaders). For the given initial and desired statesx0 and xt1, MASs can reach the desired state during 0,t1, where t1>0 is the finial time. Suppose that MAS (5) is controllable; then, its Gram matrix is(21)Wc0,t1≜∫0t1e−AtCBCBTe−ATtdt,where t∈0,t1. Since Wc0,t1 is invertible, we can design a set of inputs (leaders) as(22)ut=−CBTe−ATtWc−10,t1x0.
Then, a solution of system (5) is(23)xt1=eAt1x0+∫0t1eAt1−tCButdt,which can make the system state from x0 to xt1 during 0,t1. Notice that t1 is the longest time to get a set of inputs.


## 3.1. Group Controllability on Switching Topology

Similar to literature [29], corresponding system (5) with switching topology can be described as(6)x˙1=A1σtx1+C1B1σtx2y1,x˙2=A2σtx2+C2B2σtx1y2,where the switching path σt:ℝ+⟶1,…,K can be described by a piecewise constant scalar function, which presents the coupling links of such time-variant system, and K is the number of probable switching topologies. Moreover, we select Aip,CipBipi=1,2,p=1,2,…,K to achieve the system realizations when σt=p.Some relevant important concepts will be introduced in the following, and more details can be seen in [29].Definition 1.
(see [29], switching sequence). A finite scalars’ set π≜i0,…,iP−1 is said to be a switching sequence, where P∈0,∞ indicates the length of π, and a switching path σp is defined as σp=ip if p∈P¯ (P¯≜0,1,…,P−1) with ip∈1,2,…,P be the index of the pth realization.Definition 2.
(group switching controllability). A nonzero statex of system (6) attains group switching controllability if(1)
There are a time instantP∈0,∞, a switching path σ:P¯⟶1,…,K, and the input (x2, y1) for t∈P¯ such that x10=x1 and x1P=0(2)
There are a time instantP∈0,∞, a switching path σ:P¯⟶1,…,K, and the input (x1, y2) for t∈P¯ such that x20=x2 and x2P=0Definition 3.
(see [29], column space). For a preset matrix Bp×m=b1,…,bm, the column space ℛB is spanned by vectors b1,b2,…,bm, denoted as ℛB≜spanb1,…,bm.Lemma 1.
(see [29]). For matrices Ai∈Rp×mi, i∈r¯≜1,2,…,r and B=A1,A2,…,Ar∈ℝp×m, m=∑i=1rmi, then ℛB=∑i=1rℛAi.Definition 4.
(see [29], cyclic invariant subspace). For A∈RN×N and a linear subspace W⊆ℝN, AW is called as the R-cyclic invariant subspace indicated as AW≜∑i=1KAi−1W.
For notational simplicity, letA1C1B1≜A1ℛC1B1≜A1ℛrij for i=1 and j=1,2,…,l+n, and A2C2B2≜A2ℛrij for i=2 and j=1,2,…,m+k, where CiBi≜rij. For system (6), the subspace sequence is defined as(7)W11=∑i=1KA1ir1i,W12=∑i=1KA1iW11,…,W1m=∑i=1KA1iW1m−1,W21=∑i=1KA2ir2i,W22=∑i=1KA2iW21,…,W2n=∑i=1KA2iW2n−1.Lemma 2.
System (6) attains group switching controllability iff W1m=ℝm and W2n=ℝn.Proof..
Similar proof can be referred from that of Lemma 2 in [29]; here, it is omitted.Theorem 1.
System (6) attains group switching controllability if(8)∑i=1Kℛr1i=ℝm,∑i=1Kℛr2i=ℝn.Proof..
Obviously, fori=1,2,…,K,(9)ℛr1i⊆ℛr1i+ℛA1ir1i+⋯+ℛA1im−1r1i=A1ir1i,we have(10)ℝm=ℛr11+ℛr12+⋯+ℛr1K⊆A11r11+A12r12+⋯+A1Kr1K=W11⊆W12⊆⋯⊆W1m.
On the contrary, it is easy to knowW1m⊆ℝm. Therefore, we can have W1m=ℝm. For the subspace W2n, we can also have the similar result W2n=ℝn. From Lemmas 1 and 2, the assertion holds.Remark 3.
Theorem1 provides an important and simple method to check the controllability of continuous-time MASs with leaders by designing a switching path. At the same time, it is noted that the group controllability of continuous-time MASs with leaders can depend on leader-to-follower information communications (i.e., matrices Bi) and the subgroup-to-subgroup information communications (i.e., matrices Ci) regardless of the internal information communications between subgroups (i.e., matrices Ai) whether the topology of the internal network is fixed or switching, that is, the controllability of the subgroups is not required, which provides an important convenience for designing a switching path to ensure the group controllability for continuous-time MASs with switching topology.
In the following, some special important cases are discussed.


## 3.2. Group Controllability on Fixed Topology

Whenσt=1, system (6) (equivalently, (5)) presents an MAS based on fixed topology.Definition 5.
(see [30] (group controllability)). A nonzero state x of system (5) attains group controllability if(1)
There are a finite timeT∈J and the input (x2,y1) such that x10=x1 and x1T=0(2)
There are a finite timeT∈J and the input (x1,y2) such that x20=x2 and x2T=0Lemma 3.
System (5) attains group controllability iff rankQ1=m and rankQ2=n, where(11)Q1=C1B1⋮A1C1B1⋮A12C1B1⋮⋯⋮A1m−1C1B1,Q2=C2B2⋮A2C2B2⋮A22C2B2⋮⋯⋮A2n−1C2B2.
Here,Q1 is the controllability matrix of G1,x1, and Q2 is the controllability matrix of G2,x2.Proof..
The result is obvious from Definition5.Remark 4.
From Lemma3, it is too complex to compute the controllability matrices of system (5). On this basis, the group controllability of such MAS with leaders is shown by the technique of PBH rank test.Theorem 2.
(PBH rank test). System (5) attains group controllability iff system (5) satisfies(1)
ranksI−A1,C1,B1=m and ranktI−A2,C2,B2=n, ∀s,t∈ℂ, where ℂ is a complex number set(2)
rankλiI−A1,C1,B1=m and rankμiI−A2,C2,B2=n, where λi∀i=1,…,m and μi∀i=1,…,n are, respectively, the eigenvalues of A1 and A2Proof..
Obviously, if condition (1) holds, condition (2) absolutely holds. Therefore, it is only necessary to prove that condition (1) is true.
Necessity: by contradiction, supposed that∃s∈ℂ; then,(12)ranksI−A1,C1,B1<m,and then the rows of sI−A1,C1,B1 are linearly dependent. Thus, ∃α≠0 such that α′sI−A1,C1,B1=0. Therefore,(13)sα′=α′A1,α′C1=0,α′B1=0.
Moreover, we can have(14)α′Q1=α′C1A1C1A12C1⋯A1m−1C1⋮B1A1B1A12B1⋯A1m−1B1=α′C1α′A1C1α′A12C1⋯α′A1m−1C1⋮α′B1α′A1B1α′A12B1⋯α′A1m−1B1=α′B1sα′B1⋯sm−1α′B1⋮α′C1sα′C1⋯sm−1α′C1=0.
Sinceα≠0, then there must be rankQ1<m, which implies that system (5) is uncontrollable, contradicting to the assertion that system (5) attains the group controllability. The necessity of (1) is proved.
Sufficiency: by contradiction, assumed that system (5) is uncontrollable, then ∃λ∈ℂ of A1, which corresponds to the eigenvector β≠0 satisfying(15)β′A1=λβ′,β′C1=0,β′B1=0,and then β′λI−A1,C1,B1=0 so that rankλI−A1,C1,B1<m. This contradicts to ranksI−A1,C1,B1=m for ∀s∈ℂ. The sufficiency of (1) is proved.Theorem 3.
IfLiT=Li (i=1,2), system (5) attains group controllability iff(1)
The eigenvalues ofAi are different(2)
The eigenvectors ofAi are unorthogonal to at least one column of Bi or CiProof..
SinceL1T=L1, then A1T=A1, which can be displayed as A1=U1Λ1U1T, where the columns of U1 and the diagonal matrix Λ1 are made up of orthogonal eigenvectors and eigenvalues of A1, respectively. Moreover, U1U1T=I; then, the controllability matrix of such a system can be expressed as(16)Q1=B1A1B1A12B1⋯A1m−1B1⋮C1A1C1A12C1⋯A1m−1C1=U1U1TB1U1Λ1U1TB1U1Λ12U1TB1⋯U1Λ1m−1U1TB1⋮U1U1TC1U1Λ1U1TC1⋯U1Λ1m−1U1TC1=U1U1TB1Λ1U1TB1Λ12U1TB1⋯Λ1m−1U1TB1⋮U1TC1Λ1U1TC1Λ12U1TC1⋯Λ1m−1U1TC1≜U1Q1˜,where Q1˜=U1TB1Λ1U1TB1Λ12U1TB1⋯Λ1m−1U1TB1⋮U1TC1Λ1U1TC1Λ12U1TC1⋯Λ1m−1U1TC1.
SinceU1 consists of the orthogonal eigenvectors of A1, then U1 is nonsingular, which implies that rankQ1=rankQ1˜. Let λi and ηi be the eigenvalues and their corresponding eigenvectors of A1, respectively, for i=1,2,…,m.
For the convenience of discussion, letU1TB1≜r1,…,rl, U1TC1≜p1,…,pn, ri≜r1i,…,rmiT with rki=ηk,bi,k=1,2,…,m;i=1,2,…,l, and pi=p1i,p2i,…,pmiTi=1,2,…,n; then,(17)Q1˜=r1,r2,…,rl,λ1λ2⋱λmr1,r2,…,rl,…,λ1m−1λ2m−1⋱λmm−1r1,r2,…,rl⋮p1,p2,…,pn,λ1λ2⋱λmp1,p2,…,pn,…,λ1m−1λ2m−1⋱λmm−1p1,p2,…,pn⟶r1,Λ1r1,⋯,Λ1m−1r1,…,rl,Λ1rl,…,Λ1m−1rl⋮p1,Λ1p1,…,Λ1m−1p1,…,pn,Λ1pn,…,Λ1m−1pn⟶r11λ1r11λ12r11⋯λ1m−1r11r21λ2r21λ22r21⋯λ2m−1r21⋮⋮⋮⋮rm1λmrm1λm2rm1⋯λmm−1rm1,…,r1lλ1r1lλ12r1l⋯λ1m−1r1lr2lλ2r2lλ22r2l⋯λ2m−1r2l⋮⋮⋮⋮rmlλmrmlλm2rml⋯λmm−1rml⋮p11λ1p11λ12p11⋯λ1m−1p11p21λ2p21λ22p21⋯λ2m−1p21⋮⋮⋮⋮pm1λmpm1λm2pm1⋯λmm−1pm1,…,p1nλ1p1nλ12p1n⋯λ1m−1p1np2nλ2p2nλ22p2n⋯λ2m−1p2n⋮⋮⋮⋮pmnλmpmnλm2pmn⋯λmm−1pmn⟶r11r21⋱rm11λ1⋯λ1m−11λ2⋯λ2m−1⋮⋮⋮1λm⋯λmm−1,⋯,r1lr2l⋱rml1λ1⋯λ1m−11λ2⋯λ2m−1⋮⋮⋮1λm⋯λmm−1⋮p11p21⋱pm11λ1⋯λ1m−11λ2⋯λ2m−1⋮⋮⋮1λm⋯λmm−1,…,p1np2n⋱pmn1λ1⋯λ1m−11λ2⋯λ2m−1⋮⋮⋮1λm⋯λmm−1⟶η1,b1η2,b1⋱ηm,b1M,…,η1,blη2,bl⋱ηm,blM,…,η1,c11η2,c11⋱ηm,c11M,…,η1,c1nη2,c1n⋱ηm,c1nM,where η1,bl,ηm,c1n is the vector inner product and matrix(18)M=1λ1λ12⋯λ1m−11λ2λ22⋯λ2m−1⋮⋮⋮⋮1λmλm2⋯λmm−1,is a Vandermonde matrix. If Q˜1 has full row rank needing that at least one block of Q˜1 has full row rank, without loss of generality, the first block will be selected to discuss. It is known that Vandermonde matrix M is nonsingular if eigenvalues of A1 are distinct so that the row rank of matrix Q1˜ is decided by matrix diagη1,b1,η2,b1,…,ηm,b1 or matrix diagη1,c11,η2,c11,…,ηm,c11. Since eigenvectors Ai are unorthogonal to at least one column of Bi or Cii=1,2, therefore, matrix diagη1,b1,η2,b1,…,ηm,b1 or matrix diagη1,c11,η2,c11,…,ηm,c11 has full row rank, which means that Q˜1 has full row rank. Similarly, Q˜2 also has full row rank. This completes the proof.
Note that conditionLiT=Li implies that the information weight from agent i to agent j is the same as that from agent j to agent i in the same subgroup, that is, the topological structure is symmetric for the subgroups.Corollary 1.
System (5) is uncontrollable if subgroups G,x1 and G,x2 are both complete graphs (see Figure 2) and aij=biq∀i,j∈ℓ1,ℓ2,∀q∈ℓl, regardless of how to connect G,x1 and G,x2.Figure 2
Complete graph.Proof..
Because subgraphsG,x1 and G,x2 are both complete and aij=biq∀i,j∈ℓ1,ℓ2,∀q∈ℓl, without loss of generality, let aij=biq=1∀i,j∈ℓ1,ℓ2,∀q∈ℓl; then,(19)A1=−m+l−11⋯11−m+l−1⋯1⋮⋮⋮11⋯−m+l−1,A2=−n+k−11⋯11−n+k−1⋯1⋮⋮⋮11⋯−n+k−1.
By simple calculation, we can know thatA1’s eigenvalues are λi=0,−m+l,…,−m+l︸, and A2’s eigenvalues are μi=0,−n+k,…,−n+k︸. Then, A1 has common eigenvalue −m+l, and A2 has common eigenvalue −n+k, which are contrary to the conditions of Theorem 3. Thus, no matter how to connect subgroups G1,x1 and G2,x2, system (5) is uncontrollable.Corollary 2.
IfG,x1 and G,x2 are both star graphs (see Figure 3) as well as aij=biq∀i,j∈ℓ1,ℓ2,∀q∈ℓl regardless of how to connect G,x1 and G,x2, then system (5) is uncontrollable.Figure 3
Star graph.Proof..
Because subgraphsG,x1 and G,x2 are both star groups and aij=biq∀i,j∈ℓ1,ℓ2,∀q∈ℓl, without loss of generality, let aij=biq=1∀i,j∈ℓ1,ℓ2,∀q∈ℓl; therefore,(20)A1=−m−l−11⋯11−m−l−1⋯1⋮⋮⋮11⋯−m−l−1,A2=−n−k−11⋯11−n−k−1⋯1⋮⋮⋮11⋯−n−k−1.
By computing, we can also know that the eigenvalues ofA1 are λi=0,−m+l+2,…,−m+l+2︸, and the eigenvalues of A2 are μi=0,−n+k+2,…,−n+k+2︸. Then, A1 has common eigenvalues −m+l+2, and A2 has common eigenvalues −n+k+2, which contradict to condition (1) of Theorem 3. Thus, no matter how to connect subgroup G1,x1 and subgroup G2,x2, system (5) must be uncontrollable.Remark 5.
It is noted that there must exist a few leaders making the system to reach the desired state from the random initial state if system (5) is controllable. However, how to configure the leaders such that the desired formation can be achieved? That is, how to select the leaders (or design the inputs) with given initial state and desired state?
Here presents an algorithm for designing the leaders.Algorithm 1.
(algorithm for designing leaders). For the given initial and desired statesx0 and xt1, MASs can reach the desired state during 0,t1, where t1>0 is the finial time. Suppose that MAS (5) is controllable; then, its Gram matrix is(21)Wc0,t1≜∫0t1e−AtCBCBTe−ATtdt,where t∈0,t1. Since Wc0,t1 is invertible, we can design a set of inputs (leaders) as(22)ut=−CBTe−ATtWc−10,t1x0.
Then, a solution of system (5) is(23)xt1=eAt1x0+∫0t1eAt1−tCButdt,which can make the system state from x0 to xt1 during 0,t1. Notice that t1 is the longest time to get a set of inputs.


## 4. Example and Simulations

A nine-agent system with followers 4 and a leader as subgroup 1 and followers 3 and a leader as subgroup 2 is described by Figure4 with a12=a21=1,a23=a32=2,a34=a43=1,a45=a54=1,a56=a65=1,a67=a76=1; otherwise, aij=0.Figure 4
Topology with 9 agents.From Figure4, the system matrices are as follows:(24)A1=−11001−32002−41001−1,B1=0010,C1=000000000100;A2=−2202−31012,B2=001,C2=000100000000.By computing, we can get the eigenvalues ofA1 and A2 are −5.7711,−2.2430,−0.7904,−0.1955 and −4.6562,−0.5770,2.2332, respectively, and the corresponding eigenvectors are(25)η1=0.1262−0.60230.7714−0.1617,η2=0.4941−0.6141−0.47950.3858,η3=−0.6023−0.12620.16170.7714,η4=−0.6141−0.4941−0.3858−0.4795;μ1=0.5972−0.79320.1192,μ2=0.79490.5656−0.2195,μ3=−0.1067−0.2258−0.9683.We define the first column ofB1 as b11=0010, the first column of C1 as c11=0001, the first column of B2 as b21=001, and the last column of C2 as c24=100, as well as calculate the inner products of every eigenvalue and the corresponding vectors as(26)η1,b11=0.7714,η2,b11=−0.4795,η3,b11=0.1617,η4,b11=−0.3858;η1,c11=−0.1617,η2,c11=0.3858,η1,c11=0.7714,η1,c11=−0.4795,which mean all the eigenvectors of A1 are unorthogonal to the one column of B1 or C1. At the same time,(27)μ1,b21=0.1192,μ2,b21=−0.2195,μ3,b21=−0.9683,μ1,c24=0.5972,μ2,c24=0.7949,μ3,c24=−0.1067,which mean all the eigenvectors of A2 are unorthogonal to any one column of B2 or C2. Those imply that system (5) described by Figure 4 can attain the group controllability.Figures5 and 6 depict the initial states, final states, and moving trajectories of the followers of subgroup 1 and subgroup 2 described by the black star dots and the black circular dots, respectively. Beginning from random initial states, the followers of subgroup 1 and subgroup 2 can be finally governed to a straight-line alignment and a trapezoid alignment, respectively.Figure 5
A straight line.Figure 6
A trapezoid.


## 5. Conclusion

This paper has discussed the group controllability of continuous-time MASs with multiple leaders on switching and fixed topologies, respectively. Some useful and effective results of group controllability are obtained by the rank test and the PBH test. Specially, the group controllability of continuous-time MASs for some special topology graphs has also been studied.



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