OntheGroupControllabilityofLeader-BasedContinuous-Time...

Research ArticleOn the Group Controllability of Leader-Based Continuous-TimeMultiagent Systems

Bo Liu 1 Licheng Wu1 Rong Li2 Housheng Su 3 and Yue Han4

1School of Information Engineering Minzu University of China Beijing 100081 China2School of Statistics and Mathematics Shanghai Lixin University of Accounting and Finance Shanghai 201209 China3Key Laboratory of Imaging Processing and Intelligence Control School of Artificial Intelligence and AutomationHuazhong University of Science and Technology Wuhan 430074 China4College of Science North China University of Technology Beijing 100144 China

Correspondence should be addressed to Bo Liu boliuncuteducn and Housheng Su houshengsuqqcom

Received 11 January 2020 Accepted 7 April 2020 Published 15 May 2020

Academic Editor Cornelio Posadas-Castillo

Copyright copy 2020 Bo Liu et al )is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

In recent decades distributed coordination control of net-workedMASs has become a hot and challenging issue in lotsof areas such as applied mathematics control theory me-chanics engineering and neurobiology [1ndash10] Studies inthis area include several basic problems such as stabilityconsensus and synchronization [11] containment [12]controllability [13] and formation control and trackingcontrol [14]

)e controllability problem is a key essential problemin modern control theory and attracts increasing attentiondue to its wide applications in engineering An MAS iscontrollable if each dynamic follower can attain its de-sirable configuration from any initial state during finitetime by regulating some leaders In [15] Tanner first putforward the controllability problem of networked systemsin a leader-following framework where a certain agentwas acted as the leader (the external control input) and analgebraic feature based on eigenvalues and eigenvectors ofsuch systemrsquos Laplacian matrices was derived by nearestneighbor rules Based on this Liu et al [16] discussed the

controllability of discrete-time MASs with a single leaderbased on nearest neighbor rules and derived a simplecontrollable condition for such a system on switchingtopology Afterwards further studies on the controlla-bility of MASs have mainly been concentrated fromgraph-theoretic and algebraic-theoretic points of viewrespectively At present many works on the controlla-bility of MASs from the perspective of graph theory haveconcentrated on the basis of partitions of graph topologysuch as equitable partitionrelaxed equitable partitionexternal equitable partition in [17] connected componentpartition in [18] and selection of leaders [19] Furtherresearch studies on the controllability were presented forsome different special topology graphs such as pathgraphs [20] cycle graphs [21] multichain topologies [22]stars and trees [23] two-time-scale topologies [24] andregular graphs [25] Lots of algebraic controllable con-ditions of MASs were characterized in [26 27]

)e aforementioned results on the controllability ofMASs just contained a single group However in engi-neering practice a single group can be compartmentalizedinto some subgroups with the improvement of MASsrsquo

HindawiComplexityVolume 2020 Article ID 7892643 11 pageshttpsdoiorg10115520207892643

complexity [28] It is a very challenging work to study thecontrollability problem of MASs with multiple subgroupsand multiple leaders considering the control law the in-formation topology structure between different subgroupsand the effect of dynamical leaders acting on the followeragents which will be highlighted in this paper More re-cently the group controllability of continuous-timedis-crete-time MASs leaderless with different topologies andcommunication restrictions in [29 30] was studiedrespectively

Motivated by the results of previous studies this paperaims at the group controllability of continuous-time MASsconsisting of some different subgroups by adjusting theleaders)emain contributions of this paper are summed upas follows

(1) Different from the group controllability problem ofcontinuous-time MASs under the leaderless frame-work studied based on the fixed topology in [30] thecurrent work has considered the group controlla-bility of continuous-time MASs under the leader-follower framework with fixed topology andswitching topology respectively which can beexpressed by the system matrices It is obvious thatdifferent models can lead to completely differentfeatures for MASs with leaders

(2) )e concepts of the group controllability of con-tinuous-time MASs with multiple leaders are pro-posed based on switching and fixed topologiesrespectively

(3) Sufficient andor necessary algebraic- and graph-theoretic group controllable characterizations ofcontinuous-time MASs with multiple leaders underthe group consensus protocol are established fromthe systemrsquos Laplacian matrices

(4) )e effects of subgroups and leaders on the groupcontrollability are discussed

)e rest of this work is arranged as follows )e problemformulation is stated in Section 2 Section 3 builds the groupcontrollability of MASs with multiple leaders Numericalexample and simulations are given in Section 4 FinallySection 5 summarises the conclusion

2 Problem Formulation

Consider a continuous-time MAS consisting of N agentsgoverned by

_xi(t) ui(t) i 1 N (1)

where xi isin R is the state and ui isin R is the control inputrespectively

In engineering practice the whole group can be com-partmentalized into a few subgroups Without loss ofgenerality in this paper such MAS consisting of m + n +

l + k (m n l kgt 1) agents is compartmentalized into sub-group (G1 x1) and subgroup (G2 x2) as shown in Figure 1

Denote ℓ1 ≜ 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 vm1113864 1113865 and V2 ≜ vm+1 vm+n1113864 1113865 thenℓ ℓ1 cup ℓ2 ℓl ℓ1l

cup ℓ2l and V V1 cup V2 Ni is the ith

agentrsquos neighbor set with N1i vj isin V1 (vj vi) isin E1113966 1113967 andN2i vj isinV2 (vj vi) isin E1113966 1113967 where Ni N1i cup N2i andN1i cap N2i empty N1p and N2p respectively represent theleadersrsquo neighbor sets of subgroups 1 and 2

Inspired by [30] the control input ui is designed asfollows

ui

1113936jisinN1i

aij xj(t) minus xi(t)1113872 1113873 + 1113936jisinN2i

aijxj(t)

+ 1113936qisinN1p

biq yq(t) minus xi(t)1113872 1113873 i isin ℓ1

1113936jisinN2i


aijxj(t)

+ 1113936qisinN2p


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)where aij isin R biq ge 0foralli j q isin ℓ1 ℓ2

Remark 1 It is noted that the (i j)th entry of the systemrsquosadjacency matrix denoted as aij in this paper can beallowed to be negative which makes it more difficult andcomplex to discuss the group controllability problem sincethere are negative factors in the coupling links betweendifferent subgroups

Supposed that x1 ≜ (x1 xm)T andx2 ≜ (xm+1 xm+n)T are the state vectors of followeragents in G1 and G2 and y1 ≜ (y1 yl)

T andy2 ≜ (yl+1 yl+p)T are the state vectors of the leaderagents inG1 andG2 respectively )en the dynamics of thefollowers in system (1) becomes

_x1

_x21113890 1113891 minus L1 minus R1 C1

C2 minus L2 minus R21113890 1113891

x1

x21113890 1113891 +B1 00 B2

1113890 1113891y1

y21113890 1113891

≜A1 C1C2 A2

1113890 1113891x1

x21113890 1113891 +B1 00 B2

1113890 1113891y1

y21113890 1113891

(3)

1

2

3

4

5

Leader 1

6

7

9

8

Leader 2

Group 1 Group 2

Figure 1 Topology G

2 Complexity

where L1 [lij] isin Rmtimesm and L2 [lij] isin Rntimesn are the Lap-lacian matrices of graphs G1 and G2 respectivelyR1 diag(1113936qisinN1q

b1q 1113936qisinN1qbmq) isin Rmtimesm R2 diag

(1113936qisinN2qbm+1q 1113936qisinN2q

bm+nq) isin Rntimesn

A1 ≜ minus L1 minus R1 isin Rmtimesm

B1 isin Rmtimesl

C1 isin Rmtimesn

A2 ≜ minus L2 minus R2 isin Rntimesn

B2 isin Rntimesk

C2 isin Rntimesm

(4)

Remark 2 Furthermore because aij can be allowed to benegative nonzero controller gains can be appropriatelyselected as long as L1 and L2 are Laplacian matrices Inessence the group controllability of continuous-time leader-based MASs cannot be affected by the controller gains

In order to discuss the group controllability problem ofsystem (3) its equivalent augmented system can be de-scribed by

_x1 A1x1 + C1 B11113858 1113859

x2

y11113890 1113891

_x2 A2x2 + C2 B21113858 1113859

x1

y21113890 1113891

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)

where (x2 y1) and (x1 y2) are the inputs of subgroup(G1 x1) and subgroup (G2 x2) respectively

3 Group Controllability Analysis

)is section discusses the group controllability of contin-uous-time leader-based MASs and establishes the groupcontrollability criteria by adjusting appropriate leaders withswitching and fixed topologies respectively

31 Group Controllability on Switching Topology Similar toliterature [29] corresponding system (5) with switchingtopology can be described as

_x1 A1σ(t)x1 + C1 B11113858 1113859σ(t)

x2

y11113890 1113891

_x2 A2σ(t)x2 + C2 B21113858 1113859σ(t)

x1

y21113890 1113891

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)

where the switching path σ(t) R+⟶ 1 K can bedescribed by a piecewise constant scalar function whichpresents the coupling links of such time-variant system andK is the number of probable switching topologies Moreoverwe select (Aip [CipBip])(i 1 2 p 1 2 K) to achievethe system realizations when σ(t) p

Some relevant important concepts will be introduced inthe following and more details can be seen in [29]

Definition 1 (see [29] switching sequence) A finite scalarsrsquoset π ≜ i0 iPminus 11113864 1113865 is said to be a switching sequencewhere P isin (0infin) indicates the length of π and a switchingpath σ(p) is defined as σ(p) ip if p isin P

(P ≜ 0 1 P minus 1) with ip isin 1 2 P be the index ofthe pth realization

Definition 2 (group switching controllability) A nonzerostate x of system (6) attains group switching controllability if

(1) )ere are a time instant P isin (0infin) a switching pathσ P⟶ 1 K and the input (x2 y1) for t isin P

such that x1(0) x1 and x1(P) 0(2) )ere are a time instant P isin (0infin) a switching path

σ P⟶ 1 K and the input (x1 y2) for t isin P

such that x2(0) x2 and x2(P) 0

Definition 3 (see [29] column space) For a preset matrixBptimesm [b1 bm] the column space R(B) is spanned byvectors b1 b2 bm denoted asR(B) ≜ span b1 bm1113864 1113865

Lemma 1 (see [29]) For matrices Ai isin Rptimesmi i isin r ≜ 1 2 r and B [A1 A2 Ar] isin Rptimesmm 1113936

ri1 mi then R(B) 1113936

ri1 R(Ai)

Definition 4 (see [29] cyclic invariant subspace) ForA isin RNtimesN and a linear subspaceW sube RN langA |Wrang is calledas the R-cyclic invariant subspace indicated aslangA |Wrang ≜ 1113936

Ki1 Aiminus 1W

For notational simplicity letlangA1 | (C1 B1)rang ≜ langA1 |R(C1 B1)rang ≜ langA1 |R(rij)rang for i

1 and j 1 2 l + n and langA2 | (C2 B2)rang ≜ langA2 |R(rij)rang

for i 2 and j 1 2 m + k where (CiBi) ≜ (rij) Forsystem (6) the subspace sequence is defined as

W11 1113944K

i1langA1i

1113868111386811138681113868 r1irangW12 1113944K

i1langA1i

1113868111386811138681113868W11rang W1m

1113944K

i1langA1i

1113868111386811138681113868W1(mminus 1)rang

W21 1113944K

i1langA2i

1113868111386811138681113868 r2irangW22 1113944K

i1langA2i

1113868111386811138681113868W21rang W2n

1113944K

i1langA2i

1113868111386811138681113868W2(nminus 1)rang

(7)

Lemma 2 System (6) attains group switching controllabilityiff 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 controlla-bility if

1113944

K

i1R r1i( 1113857 R

m

1113944

K

i1R r2i( 1113857 R

n

(8)

Proof Obviously for i 1 2 K

Complexity 3

R r1i( 1113857 sube R r1i( 1113857 + R A1ir1i( 1113857 + middot middot middot + R Amminus 11i r1i1113872 1113873

langA1i

1113868111386811138681113868 r1irang(9)

we have

Rm

R r11( 1113857 + R r12( 1113857 + middot middot middot + R r1K( 1113857

sube langA111113868111386811138681113868 r11rang +langA12

1113868111386811138681113868 r12rang + middot middot middot +langA1K

1113868111386811138681113868 r1Krang

W11subeW12sube middot middot middotsube W1m

(10)

On the contrary it is easy to know W1m sube Rm)erefore we can have W1m Rm For the subspace W2nwe can also have the similar resultW2n Rn From Lemmas1 and 2 the assertion holds

Remark 3 )eorem 1 provides an important and simplemethod to check the controllability of continuous-timeMASs with leaders by designing a switching path At thesame time it is noted that the group controllability ofcontinuous-time MASs with leaders can depend on leader-to-follower information communications (ie matrices Bi)and the subgroup-to-subgroup information communica-tions (ie matrices Ci) regardless of the internal informationcommunications between subgroups (ie matrices Ai)whether the topology of the internal network is fixed orswitching that is the controllability of the subgroups is notrequired which provides an important convenience fordesigning a switching path to ensure the group controlla-bility for continuous-time MASs with switching topology

In the following some special important cases arediscussed

32GroupControllability onFixedTopology When σ(t) 1system (6) (equivalently (5)) presents an MAS based onfixed topology

Definition 5 (see [30] (group controllability)) A nonzerostate x of system (5) attains group controllability if

(1) )ere are a finite time T isin J and the input (x2 y1)such that x1(0) x1 and x1(T) 0


Lemma 3 System (5) attains group controllability iffrank(Q1) m and rank(Q2) n where

Q1 C1B1( 1113857⋮A1 C1B1( 1113857⋮A21 C1B1( 1113857⋮ middot middot middot⋮A

mminus 11 C1B1( 11138571113960 1113961


nminus 12 C2B2( 11138571113960 1113961

(11)

Here Q1 is the controllability matrix of (G1 x1) and Q2is the controllability matrix of (G2 x2)

Proof )e result is obvious from Definition 5

Remark 4 From Lemma 3 it is too complex to compute thecontrollability matrices of system (5) On this basis thegroup controllability of such MAS with leaders is shown bythe technique of PBH rank test

Theorem 2 (PBH rank test) System (5) attains groupcontrollability iff system (5) satisfies

(1) rank(sI minus A1 C1 B1) m and rank(tI minus A2 C2

B2) n foralls t isin C where C is a complex number set(2) rank(λiI minus A1 C1 B1) m and rank(μiI minus A2 C2

B2) n where λi(foralli 1 m) and μi(foralli 1

n) are respectively the eigenvalues of A1 and A2

Proof Obviously if condition (1) holds condition (2) ab-solutely holds )erefore it is only necessary to prove thatcondition (1) is true

Necessity by contradiction supposed that exists isin C thenrank sI minus A1 C1 B1( 1113857ltm (12)

and then the rows of [sI minus A1 C1 B1] are linearly dependent)us existα(ne 0) such that αprime[sI minus A1 C1 B1] 0 )erefore

sαprime αprimeA1

αprimeC1 0

αprimeB1 0

(13)

Moreover we can have

αprimeQ1 αprime C1 A1C1 A21C1 middot middot middot A

mminus 11 C1⋮B1 A1B1 A

21B1 middot middot middot A

mminus 11 B11113960 1113961

αprimeC1 αprimeA1C1 αprimeA21C1 middot middot middot αprimeAmminus 1

1 C1⋮ αprimeB1 αprimeA1B1 αprimeA21B1 middot middot middot αprimeAmminus 1

1 B11113960 1113961

αprimeB1 sαprimeB1 middot middot middot smminus 1αprimeB1⋮ αprimeC1 sαprimeC1 middot middot middot s

mminus 1αprimeC11113960 1113961 0

(14)

Since αne 0 then there must be rank(Q1)ltm whichimplies that system (5) is uncontrollable contradicting to

the assertion that system (5) attains the group controllability)e necessity of (1) is proved

4 Complexity

Sufficiency by contradiction assumed that system (5) isuncontrollable then existλ isin C of A1 which corresponds to theeigenvector β(ne 0) satisfying

βprimeA1 λβprime

βprimeC1 0

βprimeB1 0

(15)

and then βprime[λI minus A1 C1 B1] 0 so that rank(λIminus

A1 C1 B1)ltm )is contradicts to rank(sI minus A1 C1 B1)

m for foralls isin C )e sufficiency of (1) is proved

Theorem 3 If LTi Li (i 1 2) system (5) attains group

controllability iff

(1) Ge eigenvalues of Ai are different(2) Ge eigenvectors of Ai are unorthogonal to at least one

column of Bi or Ci

Proof Since LT1 L1 then AT

1 A1 which can be displayedas A1 U1Λ1UT

1 where the columns of U1 and the diagonalmatrix Λ1 are made up of orthogonal eigenvectors and ei-genvalues of A1 respectively Moreover U1U1

T I thenthe controllability matrix of such a system can be expressedas

Q1 B1 A1B1 A21B1 middot middot middot A

mminus 11 B1⋮C1 A1C1 A

21C1 middot middot middot A

mminus 11 C11113960 1113961

U1UT1 B1 U1Λ1U

T1 B1 U1Λ

21U

T1 B1 middot middot middot U1Λ

mminus 11 U

T1 B1⋮U1U

T1 C1 U1Λ1U

T1 C1 middot middot middot U1Λ

mminus 11 U

T1 C11113960 1113961

U1 UT1 B1 Λ1U

T1 B1 Λ

21U

T1 B1 middot middot middotΛmminus 1

1 UT1 B1⋮U

T1 C1 Λ1U

T1 C1 Λ

21U

T1 C1 middot middot middotΛmminus 1

1 UT1 C11113960 1113961

≜ U11113958Q1

(16)

where 1113958Q1[UT1 B1Λ1UT

1 B1Λ21UT1 B1 middotmiddotmiddotΛmminus 1

1 UT1 B1⋮UT

1 C1Λ1UT

1 C1Λ21UT1 C1 middotmiddotmiddotΛmminus 1

1 UT1 C1]

SinceU1 consists of the orthogonal eigenvectors ofA1 thenU1 is nonsingular which implies that rank(Q1) rank(1113958Q1)Let λi and ηi be the eigenvalues and their corresponding ei-genvectors of A1 respectively for i 1 2 m

For the convenience of discussion letUT

1 B1 ≜ [r1 rl] UT1 C1 ≜ [p1 pn] ri ≜ [r1i rmi]

T

with rki (ηkbi)(k 12 m i 12 l) and pi [p1i

p2i pmi]T(i 12 n) then

Complexity 5

1113958Q1 r1 r2 rl( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p1 p2 pn( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶ r1Λ1r1 middot middot middot Λmminus 11 r11113872 1113873 rlΛ1rl Λmminus 1

1 rl1113872 1113873⋮ p1Λ1p1 Λmminus 11 p11113872 1113873 pnΛ1pn Λmminus 1

1 pn1113872 11138731113966 1113967

⟶

r11 λ1r11 λ21r11 middot middot middot λmminus 11 r11


⋮ ⋮ ⋮ ⋮

rm1 λmrm1 λ2mrm1 middot middot middot λmminus 1m rm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1l λ1r1l λ21r1l middot middot middot λmminus 11 r1l


⋮ ⋮ ⋮ ⋮

rml λmrml λ2mrml middot middot middot λmminus 1m rml

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p11 λ1p11 λ21p11 middot middot middot λmminus 11 p11


⋮ ⋮ ⋮ ⋮

pm1 λmpm1 λ2mpm1 middot middot middot λmminus 1m pm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1n λ1p1n λ21p1n middot middot middot λmminus 11 p1n


⋮ ⋮ ⋮ ⋮

pmn λmpmn λ2mpmn middot middot middot λmminus 1m pmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

r11

r21

⋱

rm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1 λ1 middot middot middot λmminus 11


⋮ ⋮ ⋮

1 λm middot middot middot λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

middot middot middot

r1l

r2l

⋱

rml

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p11

p21

⋱

pm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1n

p2n

⋱

pmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

η1 b1( 1113857

η2 b1( 1113857

⋱

ηm b1( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 bl( 1113857

η2 bl( 1113857

⋱

ηm bl( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

η1 c11( 1113857

η2 c11( 1113857

⋱

ηm c11( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 c1n( 1113857

η2 c1n( 1113857

⋱

ηm c1n( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(17)

where (η1 bl) (ηm c1n) is the vector inner product andmatrix M

1 λ1 λ21 middot middot middot λmminus 11


⋮ ⋮ ⋮ ⋮

1 λm λ2m middot middot middot λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

6 Complexity

is a Vandermonde matrix If 1113957Q1 has full row rank needingthat at least one block of 1113957Q1 has full row rank without loss ofgenerality the first block will be selected to discuss It isknown that Vandermonde matrix M is nonsingular if ei-genvalues of A1 are distinct so that the row rank of matrix 1113958Q1is decided by matrix diag (η1 b1) (η2 b1) (ηm b1)1113864 1113865 ormatrix diag (η1 c11) (η2 c11) (ηm c11)1113864 1113865 Since eigen-vectors Ai are unorthogonal to at least one column of Bi orCi (i 1 2) therefore matrix diag (η1 b1)1113864

(η2 b1) (ηm b1) or matrix diag (η1 c11) (η2 c11) 1113864

(ηm c11) has full row rank which means that 1113957Q1 has fullrow rank Similarly 1113957Q2 also has full row rank)is completesthe proof

Note that condition LTi 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 topologicalstructure 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(foralli j isin ℓ1 ℓ2forallq isin ℓl) regardless of how toconnect (G x1) and (G x2)

Proof Because subgraphs (G x1) and (G x2) are bothcomplete and aij biq(foralli j isin ℓ1 ℓ2forallq isin ℓl) without loss ofgenerality let aij biq 1(foralli j isin ℓ1 ℓ2forallq isin ℓl) then

A1

minus (m + l minus 1) 1 middot middot middot 1

1 minus (m + l minus 1) middot middot middot 1

⋮ ⋮ ⋮

1 1 middot middot middot minus (m + l minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus (n + k minus 1) 1 middot middot middot 1

1 minus (n + k minus 1) middot middot middot 1

⋮ ⋮ ⋮

1 1 middot middot middot minus (n + k minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

By simple calculation we can know that A1rsquos eigenvaluesare λi 0 minus (m + l) minus (m + l)1113980radicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradic1113981 and A2rsquos eigenvalues areμi 0 minus (n + k) minus (n + k)1113980radicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradic1113981 )en A1 has common ei-genvalue minus (m + l) andA2 has common eigenvalue minus (n + k)which are contrary to the conditions of )eorem 3 )us nomatter how to connect subgroups (G1 x1) and (G2 x2)system (5) is uncontrollable

Corollary 2 If (G x1) and (G x2) are both star graphs (seeFigure 3) as well as aij biq(foralli j isin ℓ1 ℓ2forallq isin ℓl) regardlessof how to connect (G x1) and (G x2) then system (5) isuncontrollable

Proof Because subgraphs (G x1) and (G x2) are both stargroups and aij biq(foralli j isin ℓ1 ℓ2forallq isin ℓl) without loss ofgenerality let aij biq 1(foralli j isin ℓ1 ℓ2forallq isin ℓl) therefore

A1

minus m minus l minus 1 1 middot middot middot 11 minus m minus l minus 1 middot middot middot 1⋮ ⋮ ⋮1 1 middot middot middot minus m minus l minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus n minus k minus 1 1 middot middot middot 11 minus n minus k minus 1 middot middot middot 1⋮ ⋮ ⋮1 1 middot middot middot minus n minus k minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

By computing we can also know that the eigenvalues ofA1 are λi 0 minus (m + l + 2) minus (m + l + 2)1113980radicradicradicradicradicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradicradicradicradicradic1113981 and the ei-genvalues of A2 are μi 0 minus (n + k + 2) minus (n + k + 2)1113980radicradicradicradicradicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradicradicradicradicradic1113981)en A1 has common eigenvalues minus (m + l + 2) and A2 hascommon eigenvalues minus (n + k + 2) which contradict tocondition (1) of )eorem 3 )us no matter how to connectsubgroup (G1 x1) and subgroup (G2 x2) system (5) mustbe uncontrollable

Remark 5 It is noted that there must exist a few leadersmaking the system to reach the desired state from therandom initial state if system (5) is controllable Howeverhow to configure the leaders such that the desired formationcan be achieved )at is how to select the leaders (or designthe inputs) with given initial state and desired state

Here presents an algorithm for designing the leaders

Figure 2 Complete graph

Figure 3 Star graph

Complexity 7

Algorithm 1 (algorithm for designing leaders) For the giveninitial and desired states x(0) and x(t1) MASs can reach thedesired state during [0 t1] where t1 gt 0 is the finial timeSuppose that MAS (5) is controllable then its Gram matrixis

Wc 0 t11113858 1113859 ≜ 1113946t1

0e

minus At[CB][CB]

Te

minus ATtdt (21)

where t isin [0 t1] Since Wc[0 t1] is invertible we can designa set of inputs (leaders) as

u(t) minus [CB]Te

minus ATtW

minus 1c 0 t11113858 1113859x(0) (22)

)en a solution of system (5) is

x t1( 1113857 eAt1x(0) + 1113946

t1

0e

A t1minus t( )[CB]u(t)dt (23)

which can make the system state from x(0) to x(t1) 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 sub-group 1 and followers 3 and a leader as subgroup 2 is de-scribed by Figure 4 with a12 a21 1a23 a32 2a34 a43

1a45 a54 1a56 a65 1a67 a76 1 otherwise aij 0From Figure 4 the system matrices are as follows

A1

minus 1 1 0 0

1 minus 3 2 0

0 2 minus 4 1

0 0 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

0

0

1

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1

0 0 0

0 0 0

0 0 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus 2 2 0

2 minus 3 1

0 1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C2

0 0 0 1

0 0 0 0

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

By computing we can get the eigenvalues of A1 and A2are minus 57711minus 22430minus 07904minus 01955 and minus 46562

minus 0577022332 respectively and the corresponding ei-genvectors are

η1

01262

minus 06023

07714

minus 01617

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η2

04941

minus 06141

minus 04795

03858

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η3

minus 06023

minus 01262

01617

07714

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η4

minus 06141

minus 04941

minus 03858

minus 04795

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ1

05972

minus 07932

01192

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ2

07949

05656

minus 02195

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ3

minus 01067

minus 02258

minus 09683

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(25)

1

2

3

4

5

6

7

1

2

1

1

1

2

Leader 1Leader 2

Group 1 Group 2

Figure 4 Topology with 9 agents

8 Complexity

We define the first column of B1 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 thecorresponding vectors as

η1 b11( 1113857 07714

η2 b11( 1113857 minus 04795

η3 b11( 1113857 01617

η4 b11( 1113857 minus 03858

η1 c11( 1113857 minus 01617

η2 c11( 1113857 03858

η1 c11( 1113857 07714

η1 c11( 1113857 minus 04795

(26)

x position

2

4

6

8

10

12

14

y pos

ition

0 2 4 6 8 10 12 14

Figure 5 A straight line

151050x position

0

1

2

3

4

5

6

7

8

9

y pos

ition

Figure 6 A trapezoid

Complexity 9

which mean all the eigenvectors of A1 are unorthogonal tothe one column of B1 or C1 At the same time

μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)

which mean all the eigenvectors of A2 are unorthogonal toany one column of B2 or C2 )ose imply that system (5)described by Figure 4 can attain the group controllability

Figures 5 and 6 depict the initial states final states andmoving trajectories of the followers of subgroup 1 andsubgroup 2 described by the black star dots and the blackcircular dots respectively Beginning from random initialstates the followers of subgroup 1 and subgroup 2 can befinally governed to a straight-line alignment and a trapezoidalignment respectively

5 Conclusion

)is paper has discussed the group controllability of con-tinuous-time MASs with multiple leaders on switching andfixed topologies respectively Some useful and effectiveresults of group controllability are obtained by the rank testand the PBH test Specially the group controllability ofcontinuous-timeMASs for some special topology graphs hasalso been studied

Data Availability

In this paper no data are needed only mathematical der-ivation is needed

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China under Grant nos 61773023 6199141261773416 and 61873318 the Program for HUST AcademicFrontier Youth Team under Grant no 2018QYTD07 andthe Frontier Research Funds of Applied Foundation ofWuhan under Grant no 2019010701011421

References

[1] X Wang and H Su ldquoConsensus of hybrid multi-agent systemsby event-triggeredself-triggered strategyrdquo Applied Mathe-matics and Computation vol 359 pp 490ndash501 2019

[2] Y Guan L Tian and L Wang ldquoControllability of switchingsigned networksrdquo IEEE Transactions on Circuits and SystemsII Express Briefs 2019

[3] X Liu Z Ji and T Hou ldquoGraph partitions and the con-trollability of directed signed networksrdquo Science China In-formation Sciences vol 62 no 4 Article ID 042202 2019

[4] Y Guan and L Wang ldquoTarget controllability of multiagentsystems under fixed and switching topologiesrdquo InternationalJournal of Robust and Nonlinear Control vol 29 no 9pp 2725ndash2741 2019

[5] X Liu Z Ji T Hou and H Yu ldquoDecentralized stabilizabilityand formation control of multi-agent systems with antago-nistic interactionsrdquo ISA Transactions vol 89 pp 58ndash66 2019

[6] Y Liu and H Su ldquoContainment control of second-ordermulti-agent systems via intermittent sampled position datacommunicationrdquo Applied Mathematics and Computationvol 362 Article ID 124522 2019

[7] B Liu N Xu H Su LWu and J Bai ldquoOn the observability ofleader-based multiagent systems with fixed topologyrdquoComplexity vol 2019 Article ID 9487574 10 pages 2019

[8] Y Sun Z Ji Q Qi and H Ma ldquoBipartite consensus of multi-agent systems with intermittent interactionrdquo IEEE Accessvol 7 pp 130300ndash130311 2019

[9] J Qu Z Ji C Lin and H Yu ldquoFast consensus seeking onnetworks with antagonistic interactionsrdquo Complexityvol 2018 Article ID 7831317 15 pages 2018

[10] Z Ji H Lin S Cao Q Qi and H Ma ldquo)e complexity incomplete graphic characterizations of multiagent controlla-bilityrdquo IEEE Transactions on Cybernetics 2020

[11] X Wang and H Su ldquoSelf-triggered leader-following con-sensus of multi-agent systems with input time delayrdquo Neu-rocomputing vol 330 pp 70ndash77 2019

[12] Y Liu and H Su ldquoSome necessary and sufficient conditions forcontainment of second-order multi-agent systems with sampledposition datardquo Neurocomputing vol 378 pp 228ndash237 2020

[13] Z Lu L Zhang and L Wang ldquoObservability of multi-agentsystems with switching topologyrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 64 no 11pp 1317ndash1321 2017

[14] H Su J Zhang and X Chen ldquoA stochastic samplingmechanism for time-varying formation of multiagent systemswith multiple leaders and communication delaysrdquo IEEETransactions on Neural Networks and Learning Systemsvol 30 no 12 pp 3699ndash3707 2019

[15] H G Tanner ldquoOn the controllability of nearest neighborinterconnectionsrdquo in Proceedings of the 43rd IEEE Conferenceon Decision and Control vol 1 pp 2467ndash2472 NassauBahamas December 2004

[16] B Liu T Chu LWang andG Xie ldquoControllability of a leader-follower dynamic network with switching topologyrdquo IEEETransactions on Automatic Control vol 53 no 4 pp 1009ndash1013 2008

[17] B Liu H Su L Wu and S He ldquoA new perspective to algebraiccharacterization on controllability of multiagent systemsrdquoComplexity vol 2020 Article ID 9703972 12 pages 2020

[18] Z Ji Z Wang H Lin and Z Wang ldquoInterconnection to-pologies for multi-agent coordination under leader-followerframeworkrdquo Automatica vol 45 no 12 pp 2857ndash2863 2009

[19] M Long H Su and B Liu ldquoSecond-order controllability oftwo-time-scale multi-agent systemsrdquo Applied Mathematicsand Computation vol 343 pp 299ndash313 2019

[20] X Liu and Z Ji ldquoControllability of multiagent systems basedon path and cycle graphsrdquo International Journal of Robust andNonlinear Control vol 28 no 1 pp 296ndash309 2018

[21] M Long H Su and B Liu ldquoGroup controllability of two-time-scale multi-agent networksrdquo Journal of the FranklinInstitute vol 355 no 13 pp 6045ndash6061 2018

10 Complexity

[22] M Cao S Zhang and M K Camlibel ldquoA class of uncon-trollable diffusively coupled multiagent systems with multi-chain topologiesrdquo IEEE Transactions on Automatic Controlvol 58 no 2 pp 465ndash469 2013

[23] Z Ji and H Yu ldquoA new perspective to graphical character-ization of multiagent controllabilityrdquo IEEE Transactions onCybernetics vol 47 no 6 pp 1471ndash1483 2017

[24] H Su M Long and Z Zeng ldquoControllability of two-time-scale discrete-time multiagent systemsrdquo IEEE Transactions onCybernetics vol 50 no 4 pp 1440ndash1449 2020

[25] B Liu H Su R Li D Sun and W Hu ldquoSwitching con-trollability of discrete-time multi-agent systems with multipleleaders and time-delaysrdquo Applied Mathematics and Compu-tation vol 228 pp 571ndash588 2014

[26] B Liu T Chu L Wang Z Zuo G Chen and H SuldquoControllability of switching networks of multi-agent sys-temsrdquo International Journal of Robust and Nonlinear Controlvol 22 no 6 pp 630ndash644 2012

[27] Y Guan and L Wang ldquoControllability of multi-agent systemswith directed and weighted signed networksrdquo Systems ampControl Letters vol 116 pp 47ndash55 2018

[28] J Yu and L Wang ldquoGroup consensus in multi-agent systemswith switching topologies and communication delaysrdquo Sys-tems amp Control Letters vol 59 no 6 pp 340ndash348 2010

[29] B Liu Y Han F Jiang H Su and J Zou ldquoGroup con-trollability of discrete-time multi-agent systemsrdquo Journal ofthe Franklin Institute vol 353 no 14 pp 3524ndash3559 2016

[30] B Liu H Su F Jiang Y Gao L Liu and J Qian ldquoGroupcontrollability of continuous-time multi-agent systemsrdquo IETControl Geory amp Applications vol 12 no 11 pp 1665ndash16712018

Complexity 11

complexity [28] It is a very challenging work to study thecontrollability problem of MASs with multiple subgroupsand multiple leaders considering the control law the in-formation topology structure between different subgroupsand the effect of dynamical leaders acting on the followeragents which will be highlighted in this paper More re-cently the group controllability of continuous-timedis-crete-time MASs leaderless with different topologies andcommunication restrictions in [29 30] was studiedrespectively

Motivated by the results of previous studies this paperaims at the group controllability of continuous-time MASsconsisting of some different subgroups by adjusting theleaders)emain contributions of this paper are summed upas follows

(1) Different from the group controllability problem ofcontinuous-time MASs under the leaderless frame-work studied based on the fixed topology in [30] thecurrent work has considered the group controlla-bility of continuous-time MASs under the leader-follower framework with fixed topology andswitching topology respectively which can beexpressed by the system matrices It is obvious thatdifferent models can lead to completely differentfeatures for MASs with leaders

(2) )e concepts of the group controllability of con-tinuous-time MASs with multiple leaders are pro-posed based on switching and fixed topologiesrespectively

(3) Sufficient andor necessary algebraic- and graph-theoretic group controllable characterizations ofcontinuous-time MASs with multiple leaders underthe group consensus protocol are established fromthe systemrsquos Laplacian matrices

(4) )e effects of subgroups and leaders on the groupcontrollability are discussed

)e rest of this work is arranged as follows )e problemformulation is stated in Section 2 Section 3 builds the groupcontrollability of MASs with multiple leaders Numericalexample and simulations are given in Section 4 FinallySection 5 summarises the conclusion

2 Problem Formulation

Consider a continuous-time MAS consisting of N agentsgoverned by

_xi(t) ui(t) i 1 N (1)

where xi isin R is the state and ui isin R is the control inputrespectively

In engineering practice the whole group can be com-partmentalized into a few subgroups Without loss ofgenerality in this paper such MAS consisting of m + n +

l + k (m n l kgt 1) agents is compartmentalized into sub-group (G1 x1) and subgroup (G2 x2) as shown in Figure 1

Denote ℓ1 ≜ 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 vm1113864 1113865 and V2 ≜ vm+1 vm+n1113864 1113865 thenℓ ℓ1 cup ℓ2 ℓl ℓ1l

cup ℓ2l and V V1 cup V2 Ni is the ith

agentrsquos neighbor set with N1i vj isin V1 (vj vi) isin E1113966 1113967 andN2i vj isinV2 (vj vi) isin E1113966 1113967 where Ni N1i cup N2i andN1i cap N2i empty N1p and N2p respectively represent theleadersrsquo neighbor sets of subgroups 1 and 2

Inspired by [30] the control input ui is designed asfollows

ui

1113936jisinN1i


aijxj(t)

+ 1113936qisinN1p


1113936jisinN2i


aijxj(t)

+ 1113936qisinN2p


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)where aij isin R biq ge 0foralli j q isin ℓ1 ℓ2

Remark 1 It is noted that the (i j)th entry of the systemrsquosadjacency matrix denoted as aij in this paper can beallowed to be negative which makes it more difficult andcomplex to discuss the group controllability problem sincethere are negative factors in the coupling links betweendifferent subgroups

Supposed that x1 ≜ (x1 xm)T andx2 ≜ (xm+1 xm+n)T are the state vectors of followeragents in G1 and G2 and y1 ≜ (y1 yl)

T andy2 ≜ (yl+1 yl+p)T are the state vectors of the leaderagents inG1 andG2 respectively )en the dynamics of thefollowers in system (1) becomes

_x1

_x21113890 1113891 minus L1 minus R1 C1

C2 minus L2 minus R21113890 1113891

x1

x21113890 1113891 +B1 00 B2

1113890 1113891y1

y21113890 1113891

≜A1 C1C2 A2

1113890 1113891x1

x21113890 1113891 +B1 00 B2

1113890 1113891y1

y21113890 1113891

(3)

1

2

3

4

5

Leader 1

6

7

9

8

Leader 2

Group 1 Group 2

Figure 1 Topology G

2 Complexity






B1 isin Rmtimesl

C1 isin Rmtimesn


B2 isin Rntimesk

C2 isin Rntimesm

(4)



_x1 A1x1 + C1 B11113858 1113859

x2

y11113890 1113891

_x2 A2x2 + C2 B21113858 1113859

x1

y21113890 1113891

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)





_x1 A1σ(t)x1 + C1 B11113858 1113859σ(t)

x2

y11113890 1113891

_x2 A2σ(t)x2 + C2 B21113858 1113859σ(t)

x1

y21113890 1113891

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)













ri1 R(Ai)


Ki1 Aiminus 1W




W11 1113944K

i1langA1i

1113868111386811138681113868 r1irangW12 1113944K

i1langA1i

1113868111386811138681113868W11rang W1m

1113944K

i1langA1i

1113868111386811138681113868W1(mminus 1)rang

W21 1113944K

i1langA2i

1113868111386811138681113868 r2irangW22 1113944K

i1langA2i

1113868111386811138681113868W21rang W2n

1113944K

i1langA2i

1113868111386811138681113868W2(nminus 1)rang

(7)




1113944

K

i1R r1i( 1113857 R

m

1113944

K

i1R r2i( 1113857 R

n

(8)


Complexity 3


langA1i

1113868111386811138681113868 r1irang(9)

we have

Rm




1113868111386811138681113868 r1Krang


(10)










mminus 11 C1B1( 11138571113960 1113961


nminus 12 C2B2( 11138571113960 1113961

(11)












sαprime αprimeA1

αprimeC1 0

αprimeB1 0

(13)





mminus 11 B11113960 1113961



1 B11113960 1113961


mminus 1αprimeC11113960 1113961 0

(14)



4 Complexity


βprimeA1 λβprime

βprimeC1 0

βprimeB1 0

(15)





controllability iff


column of Bi or Ci








mminus 11 C11113960 1113961

U1UT1 B1 U1Λ1U

T1 B1 U1Λ

21U


mminus 11 U

T1 B1⋮U1U

T1 C1 U1Λ1U


mminus 11 U

T1 C11113960 1113961

U1 UT1 B1 Λ1U

T1 B1 Λ

21U


1 UT1 B1⋮U

T1 C1 Λ1U

T1 C1 Λ

21U


1 UT1 C11113960 1113961

≜ U11113958Q1

(16)



1 UT1 B1⋮UT

1 C1Λ1UT


1 UT1 C1]




T



Complexity 5

1113958Q1 r1 r2 rl( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p1 p2 pn( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭



1 pn1113872 11138731113966 1113967

⟶



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

r11

r21

⋱

rm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


r1l

r2l

⋱

rml

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p11

p21

⋱

pm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1n

p2n

⋱

pmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

η1 b1( 1113857

η2 b1( 1113857

⋱

ηm b1( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 bl( 1113857

η2 bl( 1113857

⋱

ηm bl( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

η1 c11( 1113857

η2 c11( 1113857

⋱

ηm c11( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 c1n( 1113857

η2 c1n( 1113857

⋱

ηm c1n( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(17)




⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

6 Complexity









A1



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)




A1


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)





Figure 3 Star graph

Complexity 7


Wc 0 t11113858 1113859 ≜ 1113946t1

0e

minus At[CB][CB]

Te

minus ATtdt (21)


u(t) minus [CB]Te

minus ATtW

minus 1c 0 t11113858 1113859x(0) (22)


x t1( 1113857 eAt1x(0) + 1113946

t1

0e






A1

minus 1 1 0 0

1 minus 3 2 0

0 2 minus 4 1

0 0 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

0

0

1

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1

0 0 0

0 0 0

0 0 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus 2 2 0

2 minus 3 1

0 1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C2

0 0 0 1

0 0 0 0

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)



η1

01262

minus 06023

07714

minus 01617

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η2

04941

minus 06141

minus 04795

03858

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η3

minus 06023

minus 01262

01617

07714

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η4

minus 06141

minus 04941

minus 03858

minus 04795

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ1

05972

minus 07932

01192

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ2

07949

05656

minus 02195

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ3

minus 01067

minus 02258

minus 09683

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(25)

1

2

3

4

5

6

7

1

2

1

1

1

2

Leader 1Leader 2

Group 1 Group 2


8 Complexity


0010

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ the first

column of C1 as c11

0001

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


b21

001


100

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ as well as


η1 b11( 1113857 07714

η2 b11( 1113857 minus 04795

η3 b11( 1113857 01617

η4 b11( 1113857 minus 03858

η1 c11( 1113857 minus 01617

η2 c11( 1113857 03858

η1 c11( 1113857 07714

η1 c11( 1113857 minus 04795

(26)

x position

2

4

6

8

10

12

14

y pos

ition

0 2 4 6 8 10 12 14


151050x position

0

1

2

3

4

5

6

7

8

9

y pos

ition


Complexity 9


μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)



5 Conclusion


Data Availability




Acknowledgments


References






















10 Complexity










Complexity 11






B1 isin Rmtimesl

C1 isin Rmtimesn


B2 isin Rntimesk

C2 isin Rntimesm

(4)



_x1 A1x1 + C1 B11113858 1113859

x2

y11113890 1113891

_x2 A2x2 + C2 B21113858 1113859

x1

y21113890 1113891

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)





_x1 A1σ(t)x1 + C1 B11113858 1113859σ(t)

x2

y11113890 1113891

_x2 A2σ(t)x2 + C2 B21113858 1113859σ(t)

x1

y21113890 1113891

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)













ri1 R(Ai)


Ki1 Aiminus 1W




W11 1113944K

i1langA1i

1113868111386811138681113868 r1irangW12 1113944K

i1langA1i

1113868111386811138681113868W11rang W1m

1113944K

i1langA1i

1113868111386811138681113868W1(mminus 1)rang

W21 1113944K

i1langA2i

1113868111386811138681113868 r2irangW22 1113944K

i1langA2i

1113868111386811138681113868W21rang W2n

1113944K

i1langA2i

1113868111386811138681113868W2(nminus 1)rang

(7)




1113944

K

i1R r1i( 1113857 R

m

1113944

K

i1R r2i( 1113857 R

n

(8)


Complexity 3


langA1i

1113868111386811138681113868 r1irang(9)

we have

Rm




1113868111386811138681113868 r1Krang


(10)










mminus 11 C1B1( 11138571113960 1113961


nminus 12 C2B2( 11138571113960 1113961

(11)












sαprime αprimeA1

αprimeC1 0

αprimeB1 0

(13)





mminus 11 B11113960 1113961



1 B11113960 1113961


mminus 1αprimeC11113960 1113961 0

(14)



4 Complexity


βprimeA1 λβprime

βprimeC1 0

βprimeB1 0

(15)





controllability iff


column of Bi or Ci








mminus 11 C11113960 1113961

U1UT1 B1 U1Λ1U

T1 B1 U1Λ

21U


mminus 11 U

T1 B1⋮U1U

T1 C1 U1Λ1U


mminus 11 U

T1 C11113960 1113961

U1 UT1 B1 Λ1U

T1 B1 Λ

21U


1 UT1 B1⋮U

T1 C1 Λ1U

T1 C1 Λ

21U


1 UT1 C11113960 1113961

≜ U11113958Q1

(16)



1 UT1 B1⋮UT

1 C1Λ1UT


1 UT1 C1]




T



Complexity 5

1113958Q1 r1 r2 rl( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p1 p2 pn( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭



1 pn1113872 11138731113966 1113967

⟶



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

r11

r21

⋱

rm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


r1l

r2l

⋱

rml

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p11

p21

⋱

pm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1n

p2n

⋱

pmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

η1 b1( 1113857

η2 b1( 1113857

⋱

ηm b1( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 bl( 1113857

η2 bl( 1113857

⋱

ηm bl( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

η1 c11( 1113857

η2 c11( 1113857

⋱

ηm c11( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 c1n( 1113857

η2 c1n( 1113857

⋱

ηm c1n( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(17)




⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

6 Complexity









A1



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)




A1


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)





Figure 3 Star graph

Complexity 7


Wc 0 t11113858 1113859 ≜ 1113946t1

0e

minus At[CB][CB]

Te

minus ATtdt (21)


u(t) minus [CB]Te

minus ATtW

minus 1c 0 t11113858 1113859x(0) (22)


x t1( 1113857 eAt1x(0) + 1113946

t1

0e






A1

minus 1 1 0 0

1 minus 3 2 0

0 2 minus 4 1

0 0 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

0

0

1

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1

0 0 0

0 0 0

0 0 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus 2 2 0

2 minus 3 1

0 1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C2

0 0 0 1

0 0 0 0

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)



η1

01262

minus 06023

07714

minus 01617

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η2

04941

minus 06141

minus 04795

03858

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η3

minus 06023

minus 01262

01617

07714

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η4

minus 06141

minus 04941

minus 03858

minus 04795

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ1

05972

minus 07932

01192

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ2

07949

05656

minus 02195

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ3

minus 01067

minus 02258

minus 09683

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(25)

1

2

3

4

5

6

7

1

2

1

1

1

2

Leader 1Leader 2

Group 1 Group 2


8 Complexity


0010

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ the first

column of C1 as c11

0001

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


b21

001


100

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ as well as


η1 b11( 1113857 07714

η2 b11( 1113857 minus 04795

η3 b11( 1113857 01617

η4 b11( 1113857 minus 03858

η1 c11( 1113857 minus 01617

η2 c11( 1113857 03858

η1 c11( 1113857 07714

η1 c11( 1113857 minus 04795

(26)

x position

2

4

6

8

10

12

14

y pos

ition

0 2 4 6 8 10 12 14


151050x position

0

1

2

3

4

5

6

7

8

9

y pos

ition


Complexity 9


μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)



5 Conclusion


Data Availability




Acknowledgments


References






















10 Complexity










Complexity 11


langA1i

1113868111386811138681113868 r1irang(9)

we have

Rm




1113868111386811138681113868 r1Krang


(10)










mminus 11 C1B1( 11138571113960 1113961


nminus 12 C2B2( 11138571113960 1113961

(11)












sαprime αprimeA1

αprimeC1 0

αprimeB1 0

(13)





mminus 11 B11113960 1113961



1 B11113960 1113961


mminus 1αprimeC11113960 1113961 0

(14)



4 Complexity


βprimeA1 λβprime

βprimeC1 0

βprimeB1 0

(15)





controllability iff


column of Bi or Ci








mminus 11 C11113960 1113961

U1UT1 B1 U1Λ1U

T1 B1 U1Λ

21U


mminus 11 U

T1 B1⋮U1U

T1 C1 U1Λ1U


mminus 11 U

T1 C11113960 1113961

U1 UT1 B1 Λ1U

T1 B1 Λ

21U


1 UT1 B1⋮U

T1 C1 Λ1U

T1 C1 Λ

21U


1 UT1 C11113960 1113961

≜ U11113958Q1

(16)



1 UT1 B1⋮UT

1 C1Λ1UT


1 UT1 C1]




T



Complexity 5

1113958Q1 r1 r2 rl( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p1 p2 pn( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭



1 pn1113872 11138731113966 1113967

⟶



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

r11

r21

⋱

rm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


r1l

r2l

⋱

rml

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p11

p21

⋱

pm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1n

p2n

⋱

pmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

η1 b1( 1113857

η2 b1( 1113857

⋱

ηm b1( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 bl( 1113857

η2 bl( 1113857

⋱

ηm bl( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

η1 c11( 1113857

η2 c11( 1113857

⋱

ηm c11( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 c1n( 1113857

η2 c1n( 1113857

⋱

ηm c1n( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(17)




⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

6 Complexity









A1



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)




A1


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)





Figure 3 Star graph

Complexity 7


Wc 0 t11113858 1113859 ≜ 1113946t1

0e

minus At[CB][CB]

Te

minus ATtdt (21)


u(t) minus [CB]Te

minus ATtW

minus 1c 0 t11113858 1113859x(0) (22)


x t1( 1113857 eAt1x(0) + 1113946

t1

0e






A1

minus 1 1 0 0

1 minus 3 2 0

0 2 minus 4 1

0 0 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

0

0

1

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1

0 0 0

0 0 0

0 0 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus 2 2 0

2 minus 3 1

0 1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C2

0 0 0 1

0 0 0 0

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)



η1

01262

minus 06023

07714

minus 01617

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η2

04941

minus 06141

minus 04795

03858

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η3

minus 06023

minus 01262

01617

07714

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η4

minus 06141

minus 04941

minus 03858

minus 04795

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ1

05972

minus 07932

01192

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ2

07949

05656

minus 02195

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ3

minus 01067

minus 02258

minus 09683

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(25)

1

2

3

4

5

6

7

1

2

1

1

1

2

Leader 1Leader 2

Group 1 Group 2


8 Complexity


0010

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ the first

column of C1 as c11

0001

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


b21

001


100

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ as well as


η1 b11( 1113857 07714

η2 b11( 1113857 minus 04795

η3 b11( 1113857 01617

η4 b11( 1113857 minus 03858

η1 c11( 1113857 minus 01617

η2 c11( 1113857 03858

η1 c11( 1113857 07714

η1 c11( 1113857 minus 04795

(26)

x position

2

4

6

8

10

12

14

y pos

ition

0 2 4 6 8 10 12 14


151050x position

0

1

2

3

4

5

6

7

8

9

y pos

ition


Complexity 9


μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)



5 Conclusion


Data Availability




Acknowledgments


References






















10 Complexity










Complexity 11


βprimeA1 λβprime

βprimeC1 0

βprimeB1 0

(15)





controllability iff


column of Bi or Ci








mminus 11 C11113960 1113961

U1UT1 B1 U1Λ1U

T1 B1 U1Λ

21U


mminus 11 U

T1 B1⋮U1U

T1 C1 U1Λ1U


mminus 11 U

T1 C11113960 1113961

U1 UT1 B1 Λ1U

T1 B1 Λ

21U


1 UT1 B1⋮U

T1 C1 Λ1U

T1 C1 Λ

21U


1 UT1 C11113960 1113961

≜ U11113958Q1

(16)



1 UT1 B1⋮UT

1 C1Λ1UT


1 UT1 C1]




T



Complexity 5

1113958Q1 r1 r2 rl( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p1 p2 pn( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭



1 pn1113872 11138731113966 1113967

⟶



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

r11

r21

⋱

rm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


r1l

r2l

⋱

rml

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p11

p21

⋱

pm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1n

p2n

⋱

pmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

η1 b1( 1113857

η2 b1( 1113857

⋱

ηm b1( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 bl( 1113857

η2 bl( 1113857

⋱

ηm bl( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

η1 c11( 1113857

η2 c11( 1113857

⋱

ηm c11( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 c1n( 1113857

η2 c1n( 1113857

⋱

ηm c1n( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(17)




⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

6 Complexity









A1



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)




A1


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)





Figure 3 Star graph

Complexity 7


Wc 0 t11113858 1113859 ≜ 1113946t1

0e

minus At[CB][CB]

Te

minus ATtdt (21)


u(t) minus [CB]Te

minus ATtW

minus 1c 0 t11113858 1113859x(0) (22)


x t1( 1113857 eAt1x(0) + 1113946

t1

0e






A1

minus 1 1 0 0

1 minus 3 2 0

0 2 minus 4 1

0 0 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

0

0

1

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1

0 0 0

0 0 0

0 0 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus 2 2 0

2 minus 3 1

0 1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C2

0 0 0 1

0 0 0 0

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)



η1

01262

minus 06023

07714

minus 01617

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η2

04941

minus 06141

minus 04795

03858

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η3

minus 06023

minus 01262

01617

07714

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η4

minus 06141

minus 04941

minus 03858

minus 04795

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ1

05972

minus 07932

01192

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ2

07949

05656

minus 02195

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ3

minus 01067

minus 02258

minus 09683

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(25)

1

2

3

4

5

6

7

1

2

1

1

1

2

Leader 1Leader 2

Group 1 Group 2


8 Complexity


0010

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ the first

column of C1 as c11

0001

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


b21

001


100

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ as well as


η1 b11( 1113857 07714

η2 b11( 1113857 minus 04795

η3 b11( 1113857 01617

η4 b11( 1113857 minus 03858

η1 c11( 1113857 minus 01617

η2 c11( 1113857 03858

η1 c11( 1113857 07714

η1 c11( 1113857 minus 04795

(26)

x position

2

4

6

8

10

12

14

y pos

ition

0 2 4 6 8 10 12 14


151050x position

0

1

2

3

4

5

6

7

8

9

y pos

ition


Complexity 9


μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)



5 Conclusion


Data Availability




Acknowledgments


References






















10 Complexity










Complexity 11

1113958Q1 r1 r2 rl( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

r1 r2 rl( 1113857⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p1 p2 pn( 1113857

λ1λ2⋱

λm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

λmminus 11

λmminus 12

⋱

λmminus 1m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1 p2 pn( 1113857

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭



1 pn1113872 11138731113966 1113967

⟶



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

r11

r21

⋱

rm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


r1l

r2l

⋱

rml

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋮

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

p11

p21

⋱

pm1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

p1n

p2n

⋱

pmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

⟶

η1 b1( 1113857

η2 b1( 1113857

⋱

ηm b1( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 bl( 1113857

η2 bl( 1113857

⋱

ηm bl( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

η1 c11( 1113857

η2 c11( 1113857

⋱

ηm c11( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

η1 c1n( 1113857

η2 c1n( 1113857

⋱

ηm c1n( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

M

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(17)




⋮ ⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

6 Complexity









A1



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)




A1


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)





Figure 3 Star graph

Complexity 7


Wc 0 t11113858 1113859 ≜ 1113946t1

0e

minus At[CB][CB]

Te

minus ATtdt (21)


u(t) minus [CB]Te

minus ATtW

minus 1c 0 t11113858 1113859x(0) (22)


x t1( 1113857 eAt1x(0) + 1113946

t1

0e






A1

minus 1 1 0 0

1 minus 3 2 0

0 2 minus 4 1

0 0 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

0

0

1

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1

0 0 0

0 0 0

0 0 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus 2 2 0

2 minus 3 1

0 1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C2

0 0 0 1

0 0 0 0

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)



η1

01262

minus 06023

07714

minus 01617

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η2

04941

minus 06141

minus 04795

03858

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η3

minus 06023

minus 01262

01617

07714

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η4

minus 06141

minus 04941

minus 03858

minus 04795

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ1

05972

minus 07932

01192

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ2

07949

05656

minus 02195

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ3

minus 01067

minus 02258

minus 09683

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(25)

1

2

3

4

5

6

7

1

2

1

1

1

2

Leader 1Leader 2

Group 1 Group 2


8 Complexity


0010

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ the first

column of C1 as c11

0001

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


b21

001


100

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ as well as


η1 b11( 1113857 07714

η2 b11( 1113857 minus 04795

η3 b11( 1113857 01617

η4 b11( 1113857 minus 03858

η1 c11( 1113857 minus 01617

η2 c11( 1113857 03858

η1 c11( 1113857 07714

η1 c11( 1113857 minus 04795

(26)

x position

2

4

6

8

10

12

14

y pos

ition

0 2 4 6 8 10 12 14


151050x position

0

1

2

3

4

5

6

7

8

9

y pos

ition


Complexity 9


μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)



5 Conclusion


Data Availability




Acknowledgments


References






















10 Complexity










Complexity 11









A1



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2



⋮ ⋮ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)




A1


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)





Figure 3 Star graph

Complexity 7


Wc 0 t11113858 1113859 ≜ 1113946t1

0e

minus At[CB][CB]

Te

minus ATtdt (21)


u(t) minus [CB]Te

minus ATtW

minus 1c 0 t11113858 1113859x(0) (22)


x t1( 1113857 eAt1x(0) + 1113946

t1

0e






A1

minus 1 1 0 0

1 minus 3 2 0

0 2 minus 4 1

0 0 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

0

0

1

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1

0 0 0

0 0 0

0 0 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus 2 2 0

2 minus 3 1

0 1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C2

0 0 0 1

0 0 0 0

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)



η1

01262

minus 06023

07714

minus 01617

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η2

04941

minus 06141

minus 04795

03858

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η3

minus 06023

minus 01262

01617

07714

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η4

minus 06141

minus 04941

minus 03858

minus 04795

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ1

05972

minus 07932

01192

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ2

07949

05656

minus 02195

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ3

minus 01067

minus 02258

minus 09683

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(25)

1

2

3

4

5

6

7

1

2

1

1

1

2

Leader 1Leader 2

Group 1 Group 2


8 Complexity


0010

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ the first

column of C1 as c11

0001

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


b21

001


100

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ as well as


η1 b11( 1113857 07714

η2 b11( 1113857 minus 04795

η3 b11( 1113857 01617

η4 b11( 1113857 minus 03858

η1 c11( 1113857 minus 01617

η2 c11( 1113857 03858

η1 c11( 1113857 07714

η1 c11( 1113857 minus 04795

(26)

x position

2

4

6

8

10

12

14

y pos

ition

0 2 4 6 8 10 12 14


151050x position

0

1

2

3

4

5

6

7

8

9

y pos

ition


Complexity 9


μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)



5 Conclusion


Data Availability




Acknowledgments


References






















10 Complexity










Complexity 11


Wc 0 t11113858 1113859 ≜ 1113946t1

0e

minus At[CB][CB]

Te

minus ATtdt (21)


u(t) minus [CB]Te

minus ATtW

minus 1c 0 t11113858 1113859x(0) (22)


x t1( 1113857 eAt1x(0) + 1113946

t1

0e






A1

minus 1 1 0 0

1 minus 3 2 0

0 2 minus 4 1

0 0 1 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

0

0

1

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1

0 0 0

0 0 0

0 0 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus 2 2 0

2 minus 3 1

0 1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

0

0

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C2

0 0 0 1

0 0 0 0

0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)



η1

01262

minus 06023

07714

minus 01617

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η2

04941

minus 06141

minus 04795

03858

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η3

minus 06023

minus 01262

01617

07714

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η4

minus 06141

minus 04941

minus 03858

minus 04795

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ1

05972

minus 07932

01192

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ2

07949

05656

minus 02195

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

μ3

minus 01067

minus 02258

minus 09683

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(25)

1

2

3

4

5

6

7

1

2

1

1

1

2

Leader 1Leader 2

Group 1 Group 2


8 Complexity


0010

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ the first

column of C1 as c11

0001

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


b21

001


100

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ as well as


η1 b11( 1113857 07714

η2 b11( 1113857 minus 04795

η3 b11( 1113857 01617

η4 b11( 1113857 minus 03858

η1 c11( 1113857 minus 01617

η2 c11( 1113857 03858

η1 c11( 1113857 07714

η1 c11( 1113857 minus 04795

(26)

x position

2

4

6

8

10

12

14

y pos

ition

0 2 4 6 8 10 12 14


151050x position

0

1

2

3

4

5

6

7

8

9

y pos

ition


Complexity 9


μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)



5 Conclusion


Data Availability




Acknowledgments


References






















10 Complexity










Complexity 11


0010

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ the first

column of C1 as c11

0001

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


b21

001


100

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ as well as


η1 b11( 1113857 07714

η2 b11( 1113857 minus 04795

η3 b11( 1113857 01617

η4 b11( 1113857 minus 03858

η1 c11( 1113857 minus 01617

η2 c11( 1113857 03858

η1 c11( 1113857 07714

η1 c11( 1113857 minus 04795

(26)

x position

2

4

6

8

10

12

14

y pos

ition

0 2 4 6 8 10 12 14


151050x position

0

1

2

3

4

5

6

7

8

9

y pos

ition


Complexity 9


μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)



5 Conclusion


Data Availability




Acknowledgments


References






















10 Complexity










Complexity 11


μ1 b21( 1113857 01192

μ2 b21( 1113857 minus 02195

μ3 b21( 1113857 minus 09683

μ1 c24( 1113857 05972

μ2 c24( 1113857 07949

μ3 c24( 1113857 minus 01067

(27)



5 Conclusion


Data Availability




Acknowledgments


References






















10 Complexity










Complexity 11










Complexity 11

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