Ob aining Consensus o Singula Mul i-agen Linea Dynamic
Sys ems
M. ISABEL GARC´
IA-PLANAS
Uni e si a Poli `
ecnica de Ca alunya
Depa amen de Ma `
ema ica Aplicada I
Mine ´
ıa 1, Esc. C, 1-3, 08038 Ba celona
SPAIN
[email p o ec ed]
Abs ac : The e is much li e a u e abou he s udy o he consensus p oblem in he case whe e he dynamics o he
agen s a e linea sys ems, bu he p oblem is s ill open o he case whe e he dynamic o he agen s a e singula
linea sys ems. In his pape he consensus p oblem o singula mul i-agen sys ems is conside ed, in which all
agen s ha e an iden ical linea dynamic mode ha can be o any o de . A gene aliza ion o he case all agen s a e
o he same o de bu do no ha e he same linea dynamic is also analyzed.
Key–Wo ds: Singula mul i-agen sys ems, consensus, con ol.
1 In oduc ion
I is well known he g ea in e es c ea ed in many e-
sea ch communi ies abou he s udy o con ol mul i-
agen s sys em, as well as he inc easing in e es in
dis ibu ed con ol and coo dina ion o ne wo ks con-
sis ing o mul iple au onomous (po en ially mobile)
agen s. The e a e an amoun o li e a u e as o exam-
ple [6, 17, 21, 23, 15, 20]. I is due o he mul i-agen s
appea in di e en a eas as o example in consensus
p oblem o communica ion ne wo ks [17], o o ma-
ion con ol o mobile obo s [4].
Jinhuan Wang, Daizhan Cheng and Xiaoming Hu
in [21], s udy he consensus p oblem in he case o
mul iagen sys ems in which all agen s ha e an iden i-
cal linea dynamics and his dynamic is a s able linea
sys em. M.I. Ga c´
ıa-Planas in [6], gene alize his e-
sul o he case whe e he dynamic o he agen s a e
con ollable.
Despi e he o e all p og ess some p oblems o
he consensus heo y s ill emain unexplo ed o he
agen s wi h dynamics de ined as a singula linea sys-
ems. In his pape mul iagen singula sys ems con-
sis ing o k+ 1 agen s wi h dynamics
E1˙x1=A1x1+B1u1
.
.
.
Ek˙xk=Akxk+Bkuk





whe e Ei, Ai∈Mn(IC),Bi∈Mn×1(IC),Ci∈
M1×n(IC), o he cases
i) all agen s ha e an iden ical linea dynamic mode,
(i.e. Ei,Ai=A,Bi=B o all i).
ii) all agen s a e o he same o de bu do no ha e
he same linea dynamic.
a e conside ed.
Wei Ni and Daizhan Cheng in [14], analyze he
s anda d case whe e E1=... =Ek=In,A1=
. . . =Akand B1= 0,B2=...=Bk his pa icula
case has p ac ical scena ios as he ligh o g oups o
bi ds. I is ob ious ha in his case he mechanic o
he i s sys em is independen o he o he s, hen con-
sensus unde a ixed opology can be easily ob ained
and i ollows om he mo ion o he i s equa ion.
This consensus p oblem is known as leade - ollowing
consensus p oblem ([14], [10]).
2 P elimina ies
2.1 Algeb aic G aph heo y
We conside a g aph G= (V,E)o o de kwi h he
se o e ices V={1, . . . , k}and he se o edges
E={(i, j)|i, j ∈ V} ⊂ V ×V.
Gi en an edge (i, j)iis called he pa en node and
jis called he child node and jis in he neighbo o i,
conc e ely we de ine he neighbo o iand we deno e
i by Ni o he se Ni={j∈ V | (i, j)∈ E}.
The g aph is called undi ec ed i e i ies ha
(i, j)∈ E i and only i (j, i)∈ E. The g aph is called
connec ed i he e exis s a pa h be ween any wo e -
ices, o he wise is called disconnec ed.
Associa ed o he g aph we conside he ma ix
G= (gij)called (unweigh ed) adjacency ma ix de-
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ined as ollows gii = 0,gij = 1 i (i, j)∈ E, and
gij = 0 o he wise.
In a mo e gene al case we can conside ha a
weigh ed adjacency ma ix is G= (gij)wi h gii = 0,
gij >0i (i, j)∈ E, and gij = 0 o he wise.
The Laplacian ma ix o he g aph is
L= (lij) = 




|Ni|i i=j
−1i j∈ Ni
0o he wise
Rema k 1 i) I he g aph is undi ec ed we ha e
ha he ma ix Lis symme ic, hen he e exis
an o hogonal ma ix Psuch ha PLP =D.
ii) I he g aph is undi ec ed hen 0 is an eigen-
alue o Land 1k= (1, . . . , 1) is he associa ed
eigen ec o .
iii) I he g aph is undi ec ed and connec ed he
eigen alue 0 is simple.
Figu e 1: Undi ec ed connec ed g aph
Fo mo e de ails abou g aph heo y see [9] and
[22] o example.
2.2 K onecke p oduc
Remembe ha gi en wo ma ices A= (aij)∈
Mn×m(IC) and B= (bij)∈Mp×q(IC) he K onecke
p oduc A⊗Bis de ined as ollows.
De ini ion 2 Le A= (ai
j)∈Mn×m(IC) and B∈
Mp×q(IC) be wo ma ices, he K onecke p oduc o
Aand B, w i e A⊗B, is he ma ix
A⊗B=





a1
1B a1
2B . . . a1
mB
a2
1B a2
2B . . . a2
mB
.
.
..
.
..
.
.
an
1B an
2B . . . an
mB





∈Mnp×mq(IC)
K onecke p oduc e i ies he ollowing p ope -
ies
1) (A+B)⊗C= (A⊗C)+(B⊗C)
2) A⊗(B+C) = (A⊗B) + (A⊗C)
3) (A⊗B)⊗C=A⊗(B⊗C)
4) (A⊗B) =A ⊗B
5) I A∈Gl(n; IC) and B∈Gl(p; IC)), hen A⊗
B∈Gl(np; IC)) and (A⊗B)−1=A−1⊗B−1
6) I he p oduc s AC and BD a e possible, hen
(A⊗B)(C⊗D) = (AC)⊗(BD)
Co olla y 3 The ec o 1k⊗ is an eigen ec o co -
esponding o he ze o eigne alue o L⊗In.
P oo :
(L⊗In)(1k⊗ ) = L1k⊗ = 0 ⊗ = 0
⊓⊔
Consequen ly, i {e1, . . . , en}is a basis o ICn,
hen 1k⊗eiis a basis o he nullspace o L⊗In.
Associa ed o he K onecke p oduc , can be de-
ined he ec o izing ope a o ha ans o ms any ma-
ix Ain o a column ec o , by placing he columns in
he ma ix one a e ano he .
De ini ion 4 Le X= (xi
j)∈Mn×m(IC) be a ma ix,
and we deno e xi= (x1
i, . . . , xn
i) o 1≤i≤m he
i- h column o he ma ix X. We de ine he ec o izing
ope a o ec, as
ec :Mn×m(IC) −→ Mnm×1(IC)
X−→ 





x1
x2
.
.
.
xm






Ob iously, ec is an isomo phism.
Fo mo e in o ma ion see P. Lancas e , M. Tismene -
sky in [11], o J.W. B ewe in [1] o example.
2.3 Con ollabili y and s abili y
De ini ion 5 We ecall ha a sys em is called con ol-
lable (see [3]) i , o any 1>0,x(0) ∈IRnand
w∈IRn, he e exis s a con ol inpu u( )such ha
x( 1) = w.
This de ini ion equi es only ha any ini ial s a e
x(0) can be s ee ed o any inal s a e x1a ime 1.
Howe e , he ajec o y o he dynamical sys em be-
ween 0 and 1is no speci ied. Fu he mo e, he e is
no cons ain s posed on he con ol ec o u( )and he
s a e ec o x( ).
An equi alen de ini ion is gi en by he ollowing
esul
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Theo em 6 ([3]) The sys em E˙x=Ax +Bu is con-
ollable i and only i
ank (E B)=n,
ank (λE −A B)=n, o all λ∈IC.
This esul is a gene aliza ion o a simila one gi en
o linea sys ems, ( o mo e de ails see [5]).
P oposi ion 7 A necessa y condi ion o con ollabil-
i y is ha he sys em be s anda dizable.
Theo em 8 ([7]) The sys em E˙x=Ax +Bu is con-
ollable i and only i he ank o he ma ix










E0 0 . . . 0B0 0 . . . 0 0
A E 0. . . 0 0 B0. . . 0 0
0A E . . . 0 0 0 B . . . 0 0
.........
0 0 0 . . . E 0 0 0 . . . B 0
0 0 0 . . . A 0 0 0 . . . 0B










∈Mn2×((n−1)n+nm)(IC)
is n2
Co olla y 9 Suppose ha Eis an in e ible ma ix
hen, he sys em E˙x=Ax+Bu is con ollable i and
only i , he sys em ˙x=E−1Ax +E−1Bu is con ol-
lable.
P oo :
ank









E0 0 . . . 0B0 0 . . . 0 0
A E 0. . . 0 0 B0. . . 0 0
0A E . . . 0 0 0 B . . . 0 0
.........
0 0 0 . . . E 0 0 0 . . . B 0
0 0 0 . . . A 0 0 0 . . . 0B









=
ank 




In
...
In
(E−1A)n−1B . . . (E−1A)B B




.
⊓⊔
The con ollabili y indices can be compu ed in he
ollowing manne .
We conside he ollowing sequences o anks i
o ma ices
Mi∈M(i+1)n×(in+(i+1)m)(IC).
M0=(B),
M1=(E B 0
A0B),
M2=


E0B0 0
A E 0B0
0A0 0 B

,
.
.
.
Mℓ=






E0 0 ... 0B0. . . 0 0
A E 0... 0 0 B . . . 0 0
.........
0 0 0 . . . E 0 0 . . . B 0
0 0 0 . . . A 0 0 . . . 0B







.
and, we de ine he ollowing collec ion o ρ-numbe s
ha pe mi o deduce he con ollabili y indices o a
con ollable iple.
De ini ion 10 Le ibe he anks o he ma ices Mi,
i= ank Mi
.
Then, we de ine he ρinumbe s as:
ρ0= 0
ρ1= 1− 0−n
ρ2= 2− 1−n
.
.
.
ρs= s−1− s−n.
I is easy o p o e he ollowing p oposi ion.
P oposi ion 11 The con ollabili y indices
[k1, . . . , kp]o a con ollable singula sys em,
a e he conjuga e pa i ion o [ρ0, ρ1, . . . , ρs].
De ini ion 12 The sys em E˙x=Ax +Bu is called
asymp o ically s able i and only i all ini e eigen al-
ues λi,i= 1, . . . ni, o he ma ix pencil (λE −A)
ha e nega i e eal pa s.
De ini ion 13 The sys em E˙x=Ax +Bu is called
asymp o ically s abilizable i and only i all ini e λ
such ha ank (λiE−A B)< n ha e nega i e
eal pa s.
Rema k 14 All con ollable sys ems a e s abilizable
bu he con e se is alse.
I is impo an he ollowing esul
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Theo em 15 a) The sys em E˙x=Ax +Bu is
s abilizable i and only i he e exis some eed-
backs FEand FAsuch ha he close loop sys em
(E−BFE) ˙x= (A−BFA)xis asymp o ically
s able
b) Suppose ank (E B)=n hen he sys em
E˙x=Ax +Bu is s abilizable i and only i
he e exis some eedbacks FEand FAsuch ha
(E−BFE)−1(A−BFA)is s able.
Sensi i i y and s abili y o singula dynamical
linea sys ems had been s udied by M.I. Ga c´
ıa-Planas
in [6].
3 Consensus
Roughly speaking, we can de ine he consensus as a
collec ion o p ocesses such ha each p ocess s a s
wi h an ini ial alue, whe e each one is supposed o
ou pu he same alue and he e is a alidi y condi-
ion ha ela es ou pu s o inpu s. Mo e conc e ely,
he consensus p oblem is a canonical p oblem ha ap-
pea s in he coo dina ion o mul i-agen sys ems. The
objec i e is ha gi en ini ial alues (scala o ec o )
o agen s, es ablish condi ions unde which h ough
local in e ac ions and compu a ions, agen s asymp o -
ically ag ee upon a common alue, ha is o say: o
each a consensus.
The consensus p oblem appea o Example:
- when on y o Con ol mo ing a numbe o
Ae ial Vehicle’s UAVs: alignmen o he head-
ing angles
- when on y o p ocess In o ma ion in senso ne -
wo ks: compu ing a e ages o ini ial local obse -
a ions ( ha is o say consensus on a pa icula
alue)
- also in Design o dis ibu ed op imiza ion algo-
i hms: one needs a mechanism o align es i-
ma es o decision a iables main ained by di e -
en agen s/p ocesso s
3.1 Dynamic o singula mul i-agen ha ing
iden ical dynamical mode
Le us conside a g oup o kiden ical agen s, he dy-
namic o each agen is gi en by he ollowing linea
dynamical sys ems
E˙x1=Ax1+Bu1
.
.
.
E˙xk=Axk+Buk




(1)
xi∈IRn,ui∈IRm,1≤i≤k.
We conside he undi ec ed g aph Gwi h
i) Ve ex se : V={1, . . . , k}
ii) Edge se : E={(i, j)|i, j ∈ V} ⊂ V ×V
de ining he communica ion opology among agen s.
De ini ion 16 Conside he sys em 1, we say ha he
consensus is achie ed using local in o ma ion i he e
is a s a e eedback
ui=K∑
j∈Ni
(xi−xj),1≤i≤k
such ha
lim
→∞∥xi−xj∥= 0,1≤i, j ≤k.
The closed-loop sys em ob ained unde his eed-
back is as ollows
E˙
X=AX +BKZ,
whe e
X=


x1
.
.
.
xk


,˙
X=


˙x1
.
.
.
˙xk


,
E=diagonal(E, . . . , E)
A=diagonal(A, . . . , A)
B=diagonal(B,...,B)
K=diagonal(K, . . . , K)
and
Z=


∑j∈N1x1−xj
.
.
.
∑j∈Nkxk−xj



.
Following his no a ion we can conclude he ol-
lowing.
P oposi ion 17 The closed-loop sys em can be de-
sc ibed as
E˙
X= ((Ik⊗A)+(Ik⊗BK)(L⊗In))X.
Taking in o accoun ha he g aph is undi ec ed,
ollowing ema k 1, we ha e ha he e exis s an o -
hogonal ma ix P∈Gl(k; IR) such ha PLP =
D=diag (λ1, . . . , λk), (λ1≥. . . ≥λk).
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In e na ional Jou nal o Ma hema ical and Compu a ional Me hods
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Co olla y 18 The closed-loop sys em can be de-
sc ibed in e ms o he ma ices E,A,B, he eedback
Kand he eigen alues o Lin he ollowing manne
E˙
b
X=diagonal (A+λ1BK, . . . , A+λkBK)b
X.(2)
P oo :
(Ik⊗BK)(L⊗In) = (Ik⊗BK)(P DP ⊗In) =
(Ik⊗BK)(P ⊗In)(D⊗In)(P⊗In) =
(P ⊗BK)(D⊗In)(P⊗In) =
(P ⊗In)(Ik⊗BK)(D⊗In)(P⊗In) =
(P ⊗In)(D⊗BK)(P⊗In)
(Ik⊗E) = (P ⊗In)(Ik⊗E)(P⊗In)
(Ik⊗A) = (P ⊗In)(Ik⊗A)(P⊗In)
Then,
(P ⊗In)(Ik⊗E)(P⊗In)˙
X=
(P ⊗In)(Ik⊗A)(P⊗In)X+
(P ⊗In)(D⊗BK)(P⊗In)X
so,
(Ik⊗E)(P⊗In)˙
X=
(Ik⊗A)(P⊗In)X+ (D⊗BK)(P⊗In)X
and calling (P⊗In)X=b
Xwe ha e he esul . ⊓⊔
The sys em 2 can be unde s ood as he close loop
sys em co esponding o he sys em



E
...
E


˙
b
X=


A
...
A


b
X+


λ1B
.
.
.
λkB


b
U
(3)
a e o apply he eedback u=Kx.
3.1.1 Consensus p oblem
I would seem ha i he g aph is connec ed he con-
sensus p oblem would be sol able i he e is a Ksuch
ha he sys em 2 is s abilized. Bu aking in o accoun
ha λ1= 0 is necessa y ha E˙x1=Ax1be asymp-
o ically s able.
Suppose now, ha he sys em (E, A, B)is con-
ollable, so he e exis KEand KAsuch ha he close
loop sys em E˙x= (E+BKE) ˙x= (A+BKA)x=
Ax is asymp o ically s able and we apply all esul s
p esen ed in §3.1 o e he g oup o kiden ical agen s,
whe e he dynamic o each agen is gi en by he ol-
lowing linea dynamical sys ems
E˙x1=Ax1+Bu1
.
.
.
E˙xk=Axk+Buk,




(4)
xi∈IRn,ui∈IRm,1≤i≤k.
Lemma 19 Le E˙x=Ax +Bu be a con ollable
singula sys em and we conside he se o k-linea
sys ems
E˙xi=Axi+λiBui,1≤i≤k
wi h λi>0. Then, he e exis eedbacks KEand
KAwhich simul aneously assign he eigen alues o
he sys ems as nega i e as possible.
Mo e conc e ely, o any M > 0, he e exis ui=
KAxi−KE˙xi o 1≤i≤ksuch ha
Re σ(E+BKE, A +λiBKA)<−M, 1≤i≤k.
(σ(E+BKE, A+λiBKA)deno es de spec um
o (E+BKE, A +λiBKA) o each 1≤i≤k).
Rema k 20 We obse e ha i E˙x=Ax +Bu is
con ollable hen, E˙x=Ax +λiBu is con ollable
being λi= 0.
P oo :
Reducing he sys em o he canonical educed
o m
E=PEcQ+P BcFE,A=PAcQ+PBcFA
and B=PBcRwi h Ec=In, and (Ac, Bc)is a pai
in i s B uno sky canonical o m.
de (s(E+BKE)−(A+λiBKA) =
de (s(PEcQ+P BcFE+PBcRKE)−
(PAcQ+P BcFA+λiPBcRKA)) =
de Pde (s(Ec+BcFEQ−1+BcRKEQ−1)−
(Ac+BcFAQ−1+λiBcRKAQ−1)) de Q=
de Pde Qde (s(Ec+Bc
KE)−(Ac+Bc
KA)),
whe e
KE=FEQ−1+RKEQ−1= 0 and
KA=
FAQ−1+λiRKAQ−1.
So, he eigen alues o de (s(E+BKE)−(A+
λiBKA)a e he same han de (sIn−(Ac+Bc
KA)).
Now, i su ices o apply he esul o s anda d
sys ems.
⊓⊔
Rema k 21 The K onecke educed o m o a singu-
la con ollable sys em, can be di ec ly ob ained om
con ollabili y indices de ined in p oposi ion 11.
As a co olla y, we can conside he consensus p ob-
lem.
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Co olla y 22 We conside he sys em 1 wi h a con-
nec adjacen opology. I E˙x=Ax+Bu is a con ol-
lable singula sys em hen, he consensus is achie ed
by means he eedback o lemma 19 and a eedback K
s abilizing E˙x=Ax +Bu.
P oo : Taking in o accoun ha he adjacen opol-
ogy is connec ed we can apply co olla y 3: 0 = λ1<
λ2≤... ≤λkand (1, . . . , 1) =1kis he eigen ec-
o co esponding o he simple eigen alue λ1= 0.
On he o he hand we can ind Ks abilizing
E˙x=Ax +Bu and hen we can ind KEand KA
s abilizing he associa e sys em 4, and we ind b
Xsuch
ha lim →∞ b
X= 0. Consequen ly, we can ind Z
such ha lim →∞ Z= 0.
Using Z= (L⊗In)X= (L⊗In)(P ⊗In)b
Xwe
ha e ha lim →∞ Xis an eigen ec o o L⊗In, ha
is o say lim →∞ X=1k⊗ o some ec o ∈IRn
and he consensus is ob ained. ⊓⊔
Example 1.
We conside h ee singula iden ical agen s wi h
he ollowing dynamics o each agen
E˙x1=Ax1+Bu1
E˙x2=Ax2+Bu2
E˙x3=Ax3+Bu3
(5)
wi h E=(1 0
0 0)A=(0 1
0 0)and B=(0
1).
I is easy o gene a e using he Ma lab ool all pos-
sible g aphs o k= 3, hen selec hose ha a e indi-
ec and connec ed, among o hem, he communica-
ion opology ha we chose in his example is de ined
by he g aph (V,E):
V={1,2,3}
E={(i, j)|i, j ∈ V} ={(1,2),(1,3)} ⊂ V×V
and he adjacency ma ix:
G=


011
100
100

.
The neighbo s o he pa en nodes a e N1=
{2,3},N2={1},N3={1}.
The Laplacian ma ix o he g aph is
L=


2−1−1
−1 1 0
−1 0 1 


wi h eigen alues λ1= 0,λ2= 1,λ3= 3.
ui=K(∑
j∈Ni
(xi−xj)) = Kzi(6)
u1=K((x1−x2)+(x1−x3)) =
=K(2x1−x2−x3),
u2=K(x2−x1),
u3=K(x3−x1).
Fi s o all we obse e ha wi h he de i a i e
eedback KE=(0 1)we ob ain E=Iand he
new mul iagen sys em is ˙xi=Axi+Bui.
Taking in o accoun ha he sys em ˙x1=Ax1
is no s able bu (A, B)is a con ollable sys em, we
conside A=A+BK =(0 1
a b)wi h app op ia e
alues o aand b.
Then, he close loop sys em o 1 wi h con ol 6 is
˙x1=Ax1+BK(2x1−x2−x3) =
= (A+ 2BK)x1−BKx2−BKx3
˙x2=Ax2+BKx2−x1) = (A+BK)x2−BKx1
˙x3=Ax3+BKx3−x1) = (A+BK)x3−BKx1
(7)
O in a ( o mal)-ma ix o m:
˙
X=


A+ 2BK −BK −BK
−BK A +BK 0
−BK 0A+BK

X.
The basis change ma ix diagonalizing he ma ix
Lis
P=


1/√3 0 −2/√6
1/√3 1/√2 1/√6
1/√3−1/√2 1/√6

,
and we ob ain he ollowing equi alen sys em
˙
b
X=


A
A+BK
A+ 3BK

b
X.
The eigen alues a e in onc ion o a, b, c, d, con-
c e ely:
λ1, λ2=b±√b2+4a
2,
λ3, λ4=b+d±√b2+2bd+d2+4a+4c
2,
λ5, λ6=b+3d±√b2+6bd+9d2+4a+12c
2,
Then, he e exis Kand K(de ined by a,b,c,d),
which assign he eigen alues as nega i e as possible.
We will y o each consensus wi h h ee di e en
pa icula eedbacks.
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Volume 1, 2016
i) Fo a=−0.01,b=−0.05,c=−0.05,d=
−0.02 he eigen alues a e
λ1, λ2=−0.0250+0.0968i, −0.0250−0.0968i,
λ3, λ4=−0.0350+0.2424i, −0.0350−0.2424i,
λ5, λ6=−0.0550+0.3962i, −0.0550−0.3962i,
so, he sys em has been s abilized.
Fo ini ial condi ion b
X(0) =
(0,2,2,3,−1,−2) , he ajec o y o each
o he sys ems b
X1=Ab
X1,b
X2= (A+BK)b
X2,
b
X3= (A+ 3BK)b
X3a e showed in igu e 1.
Figu e 1. T ajec o ies 1
The g aphic shows ha he h ee ajec o ies a -
i e a a common poin .
ii) Fo a=−0.1,b=−0.5,c=−0.5,d=−0.2
he eigen alues a e
λ1, λ2=−0.2500+0.1936i, −0.2500−0.1936i,
λ3, λ4=−0.3500+0.6910i, −0.3500−0.6910i,
λ5, λ6=−0.5500+1.1391i, −0.5500−1.1391i,
so, he sys em has been s abilized.
Fo he same ini ial condi ion han he i s case,
i.e. b
X(0) = (0,2,2,3,−1,−2) , he ajec o y
o each o he sys ems b
X1=Ab
X1,b
X2= (A+
BK)b
X2,b
X3= (A+ 3BK)b
X3a e showed in
igu e 2.
I is no ed ha in his second case, he eigen al-
ues ha e a nega i e eal pa smalle han he i s
case, hen consensus is eached as e .
iii) I we conside a=−1,b=−5,c=−5, and
d=−2, he eigen alues a e:
λ1, λ2=−0.2087,−4.7913
λ3, λ4= 1,−6
λ5, λ6=−9.2749,−1.7251,
Figu e 2. T ajec o ies 2
and he sys em is also s abilized.
In his case, he ajec o ies a e showed in igu e
3.
In his hi d case he eigen alues ha e he smalle
eal pa han he second and i s case and he
consensus is eached much as e han he i s
and second case.
Figu e 3. T ajec o ies 3
4 Dynamic o mul i-agen ha ing no
iden ical dynamical mode
Now, we a e going o in oduce in a simila way han
he case whe e he mul iane ha e iden ical mode, we
conside a mul i-agen whe e he dynamic o each
agen is gi en by he ollowing dynamical sys ems:
˙x1=A1x1+B1u1
.
.
.
˙xk=Akxk+Bkuk




(8)
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In e na ional Jou nal o Ma hema ical and Compu a ional Me hods
Volume 1, 2016
xi∈IRn,ui∈IRm,1≤i≤k. Whe e ma ices Ai
and Bia e no necessa ily equal.
The communica ion opology among agen s is de-
ined by means he undi ec ed g aph Gwi h
i) Ve ex se : V={1, . . . , k}
ii) Edge se : E={(i, j)|i, j ∈ V} ⊂ V ×V.
an in a simila way as be o e, we say ha he consen-
sus is achie ed using local in o ma ion i he e exis s
a s a e eedback
ui=Ki∑
j∈Ni
(xi−xj),1≤i≤k
such ha
lim
→∞∥xi−xj∥= 0,1≤i, j ≤k.
The closed-loop sys em ob ained unde his eed-
back is as ollows
˙
X=AX +BKZ
whe e
X=


x1
.
.
.
xk


,˙
X=


˙x1
.
.
.
˙xk



A=diagonal (A1, . . . , Ak)
B=diagonal (B1, . . . , Bk)
K=diagonal (K1, . . . , Kk)
Z=


∑j∈N1x1−xj
.
.
.
∑j∈Nkxk−xj



.
Calling
BK =B ·K
and obse ing ha
Z= (L⊗In)X
we deduce he ollowing p oposi ion
P oposi ion 23 The closed-loop sys em can be de-
duced in e ms o ma ices A,Band Kin he ollow-
ing manne .
˙
X= (A+BK(L⊗In))X(9)
We a e in e es ed in Kisuch ha he consensus is
achie ed.
P oposi ion 24 We conside he sys em 8 which a
connec ed adjacen opology. I he sys em 9 is s a-
ble he consensus p oblem has a solu ion.
Co olla y 25 I he ma ices Aia e s able. Then he
consensus is achie ed.
Rema k 26 The sys em 9 can be w i en as
˙
X=AX +BU wi h U=K(L⊗In)X.
So,
P oposi ion 27 A necessa y (bu no su icien ) con-
di ion o consensus o be eached is ha he sys em
˙
X=AX +BU (10)
is s abilizable.
Co olla y 28 A necessa y condi ion o consensus o
be eached is ha he sys ems
˙xi=Aixi+Biui,∀i= 1, . . . , k
a e s abilizable.
Rema k 29 The eedback Kob ained om he eed-
backs s abilizing he sys ems ˙xi=Aixi+Biuidoes
no necessa ily s abilize he sys em ˙
X= (A+BK(L⊗
In))X.
Example 2.
We conside he ollowing wo one-dimensional
sys ems
˙x1=u1
˙x2=x2+u2
The communica ion opology is de ined by he undi-
ec ed g aph V={1,2},E={(1,2)} ⊂ V ×V. So,
he Laplacian is (1−1
−1 1 ).
Taking as K=(1−1
6−6)we ha e
A+BK =(1−1
6−5)
wi h eigen alues −0.2679, and −3.7321, hen he sys-
em is s able.
Bu aking k1=−1and k2=−2, clea ly hese
eedbacks s abilize he sys ems, bu aking as K=
(k1
k2)=(−1
−2)we ha e
A+BK(L⊗In)) = (−1 1
2−1)
wi h eigen alues 0.4142, and −2.4142, hen he sys-
em is no s able.
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In e na ional Jou nal o Ma hema ical and Compu a ional Me hods
Volume 1, 2016
Tha is o say, we need o s abilize he sys em 10,
mus be s abilized wi h a eedback in he o m K(L⊗
In).
In ou pa icula example, i we conside k1=−2
and k2= 0 he eigen alues o (A+BK(L⊗In)) a e
-2 and -1 and he sys em is s able. Bu , in his case,
he sys em ˙x2=x2+u2wi h k2= 0 is no s able.
Finally, i we conside k1=−5and k2=−3, he
sys ems ˙xi= (Ai+BiKi)xiand (A+BK(L⊗In))
a e s able.
So, o sol e he p oblem we need o ob ain Kin
such a way ha ˙xi= (Ai+BiKi)xiand (A+BK(L⊗
In)) a e s able.
5 Conclusions
In his pape he consensus p oblem o mul i-agen
singula sys ems, o he case whe e all agen s ha e an
iden ical linea dynamic mode, and inally we make a
b ie in oduc ion o he case whe e he agen s a e o
he same o de bu do no ha e he same linea dy-
namic. The solu ion o he consensus p oblem de-
pends on he con ollabili y o he singula sys em,
hen a ank c i e ion o con ollabili y o singula
sys em is in oduced, he eby he wo k is mo e sel -
con ained and unde s andable.
Acknowledgemen s: The au ho wishes o hank
JL Dominguez-Ga cia esea che IREC, wi h i s com-
men s and sugges ions, he a icle has imp o ed i s
con en .
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