Obtaining consensus of singular multi-agent linear dynamic systems

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- 1 In e na ional Jou nal o Ma hema ical and Compu a ional Me hods Volume 1, 2016 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 2 In e na ional Jou nal o Ma hema ical and Compu a ional Me hods Volume 1, 2016 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 3 In e na ional Jou nal o Ma hema ical and Compu a ional Me hods Volume 1, 2016 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). 4 In e na ional Jou nal o Ma hema ical and Compu a ional Me hods Volume 1, 2016 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. 5 In e na ional Jou nal o Ma hema ical and Compu a ional Me hods Volume 1, 2016 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. 6 In e na ional Jou nal o Ma hema ical and Compu a ional Me hods 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) 7 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. 8 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 . Re e ences: [1] J.W. B ewe , K onecke p oduc s and ma ix cal- culus in sys em heo y. IEEE T ansac ion on Ci cui s and Sys ems, 25, (9), 1978, pp.772–781. [2] C.T. Chen, In oduc ion o Linea Sys em The- o y. Hol , Rineha and Wins on Inc, New Yo k, 1970. [3] L. 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