2. Higher Order Consensus

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1 Prepared by F.L. Lews Updated: Wedesday, February 3, 0. Hgher Order Cosesus I Seto we dsussed ooperatve otrol o graphs for dyamal systems that have frstorder dyams, that s, a sgle tegrator or shft regster at eah ode. ow we dsuss ooperatve otrol for etworked systems wth hgher-order dyams. Frst we osder lear systems, the systems havg seod-order posto/veloty type dyams the form of ewto s thrd law. I ths seto we use A to deote the ode dyams system matrx, ad a j to deote the graph edge weghts. hese should ot be ofused. Lear Systems o Graphs Cosder the systems dstrbuted o ommuato graph G wth detal lear tmevarat ode dyams x Ax Bu (. R m where x ( t s the state of ode ad u ( t R ts otrol put. Selet the ode otrols as the loal state varable feedbak u K aj( xj x (. j wth salar ouplg ga 0 ad feedbak otrol matrx K m R. Here, a j are the elemets of the graph adjaey matrx, ot the ode system matrx. hat s, they are the edge weghts of graph G. hese otrols are based o the loal votg protool struture where the otrol put of eah ode depeds o the dfferee betwee ts state ad those of all ts eghbors. he ouplg gas ad feedbak gas K of all odes are the same. he ode losed-loop dyams are x Ax Bu Ax BK a ( x x (.3 j j j he overall losed-loop graph dyams s x ( I A LBK x (.4 where the overall (global state vetor s x x x x R, L s the graph Laplaa matrx, ad s the Kroeker produt (Appedx A. We use the Kroeker produt heavly ths seto, maly the property ( M ( PQ MP Q for ommesurate matres M,,P,Q. Defe the ooperatve feedbak losed-loop system matrx o ommuato graph G as A ( I A L BK (.5

2 hs matrx reflets the loal losed-loop system matrx ( A BK as modfed by dstrbuted otrol o the graph struture L. Its stablty depeds o both the stablty propertes of ( A BK ad the graph topology propertes. he ext key result from [Fax ad Murray 004] provdes a method for aalyss of the dyams (.3/(.4 ad relates ther stablty propertes to the graph egevalues. Lemma. Let,, be the egevalues of the graph Laplaa matrx L. he the stablty propertes of dyams (.3/(.4 are equvalet to the stablty propertes of the systems A BK,, (.6 that they have the same egevalues. Proof: Defe a trasformato M suh that J M LM s upper tragular wth the egevalues,, of L o the dagoal. Apply the state-spae trasformato ( M I ( I A LBK( M I (.7 ( I A J BK ad defe ew state ( M I. he the trasformed system s blok tragular form wth dagoal bloks of the form ( A BK (.8 Hee, stablty of (.3/(.4 s equvalet to stablty of all of these systems, se a state spae trasformato does ot hage the egevalues.. hs proof shows that A ( I A L BK s equvalet to the system dag{( A BK, ( A BK, ( A BK} (.9 wth,, the egevalues of the graph Laplaa matrx L, the sese that they have the same egevalues. Seod-Order Cosesus Oe applato of osesus ooperatve otrol s the otrol of formatos of vehles. herefore, we ow wsh to study osesus for oupled systems that satsfy ewto s law x u, whh yelds the seod-order ode dyams x v (.0 v u wth posto x R, veloty v R, ad aelerato put u R. Cosder the dstrbuted posto/veloty feedbak at eah ode gve by the seod-order loal eghborhood protools u a ( x x a ( v v a ( x x ( v v (. j j j j j j j j j j where 0 s a stffess ga ad 0 s a dampg ga. hs s based o loal votg protools both posto ad veloty, so that eah ode seeks to math all ts eghbors

3 postos as well as ther velotes. We would lke to determe whe ths protool delvers osesus, ad to fd the osesus values of posto ad veloty. We aalyze ths protool by two methods. Aalyss of Seod-Order Cosesus Usg Posto/Veloty Loal ode States he frst method for aalyzg seod-order osesus follows [Xe ad Wag 007]. Defe the ode states as z x v ad wrte the posto/veloty ode dyams ad protool at eah ode as 0 0 z z u Az Bu 0 0 (. xj x u aj K aj( zj z j v j v (.3 j K. ow detfy ths wth (.3, defe where the feedbak ga matrx s z z z z R, ad wrte the global losed-loop dyams (.4 as z ( I A LBK z A z (.4 for the spef A,B,K gve. Assume the graph has a spag tree. he, L has a smple egevalue at 0, ad hee has rak -, ad the rest of ts egevalues are strtly the rght-half of the s-plae. Uder these odtos we wat to study osesus propertes of the protool (.. Frst, we eed to look at the stablty propertes of (.4, ad seod to fd the posto ad veloty osesus values. Stablty of Seod-Order Cosesus Protool. o exame stablty of the protool, aordg to the proof of Lemma, A (.4 s equvalet to dag{ A,( A BK, ( A BK} (.5 wth Re{ } 0,,,. Matrx A has two egevalues at 0 wth geometr multplty equal to oe ad a Jorda blok of order two (Appedx A. he haraterst polyomals of the other bloks are si ( A BK s s (.6 he the haraterst polyomal of ( A s si A s s (.7 herefore, the egevalues of A are gve by ( 4 s (.8 3

4 here are two egevalues of A for eah egevalue of L. he stablty of the egevalues of A a easly be studed usg the Routh test. Suppose s real. he the Routh test shows that (.6 s asymptotally stable (AS for all 0. Suppose s omplex. he * * ( s s( s s ( s s ( s s where * deotes omplex ojugate, Re{ },. Performg the Routh test shows that ths s AS f ad oly f ( / (.0 where Im{ }. herefore, all the systems (.5 for,, are AS f ad oly f Im { } Re { } max,,, Re{ } (. Cosesus Values for Posto ad Veloty. he ext result tells whe the odes reah osesus uder the seod-order protool (., ad spefes the osesus values reahed. heorem. Cosesus of Seod-Order Dyams Cosder the ewto s law systems at eah ode gve by (.0 wth the loal eghborhood protools (.. he odes reah osesus both posto ad veloty f ad oly f the graph has a spag tree ad the gas, are seleted to satsfy odto (., wth the egevalues of the graph Laplaa L. he, the posto ad veloty osesus values are gve respetvely by pv (0 (. x p x (0 t pv (0 v (.3 Proof: Frst we eed to fd the left ad rght egevetors of matrx A (.4, whh s equvalet to (.5 the sese that they have the same egevalues. Matrx A (. has a egevalue at 0 of geometr multplty oe ad algebra multplty. he rak- rght egevetor s 0 se A 0 0. he rak- rght egevetor s 0 se 0 A 0. he rak- left egevetor s 0 se 0 A 0. he rak- left egevetor s 0 se 0 A 0. he rght egevetor of L for 0 s w p p be R. Let the orrespodg left egevetor, ormalzed so that w. he we lam a rak- rght egevetor 4

5 of A (.4 for 0 y 0. hs s easly verfed by hekg that Ay 0. It s easy to verfy that a rak- rght egevetor of A s y 0 by hekg that Ay y s. Smlarly, t s foud that a rak- left egevetor of A for 0 s gve by w ad a rak- left egevetor by w w w 0 0. ow we wll study the Jorda ormal form of A ad perform a modal deomposto o (.4 to fd the steady-state osesus values for posto ad veloty. Brg A to Jorda ormal form 0 0 ( w A MJM y y ( w 0 0 stable (.4 where M s the matrx havg rght egevetors as olums, ad M s the matrx havg left egevetors as rows. he t 0 ( w At Jt zt ( e z(0 Me M z(0 y y 0 0 ( w z(0 0 0 stable (.5 yelds the modal deomposto ( w z(0 t( w z(0 zt ( y y ( w z(0 (.6 ( w z(0 t( w z(0 y ( w z(0 y stable terms ote ow that 0 zt ( xt ( vt ( 0 (.7 wth global posto ad veloty vetors x x x x R, v v v v R. Aordg to the deftos of the left ad rght egevetors of A for 0, oe sees that the posto ad veloty osesus values dued by the protool (. are (., (.3. If the graph s strogly oeted the t has a spag tree ad the theorem holds. Moreover, f the tal velotes are all equal to zero, the the osesus veloty s equal to zero ad all odes ed up at rest at the weghted eter of gravty of the tal postos, that s, the frst term (.. he ext result has also bee establshed. Corollary. Let the graph have a spag tree ad all egevalues of the Laplaa L be real. he the osesus values (., (.3.are reahed for ay postve otrol gas, (.. 5

6 Proof: he Routh test reveals that (.6 s asymptotally stable uder the hypotheses. If the graph s udreted, the the Laplaa egevalues are real ad ths orollary holds. Aalyss of Seod-Order Cosesus Usg Posto/Veloty Global State he seod method for aalyzg seod-order osesus follows [Re ad Atks 007], [Re ad Beard 008]. It hges o aother way to wrte the global dyams tha (.4. ake ewto s law ode dyams (.0 ad the seod-order protools (.. Defe the global posto ad veloty vetors as x x x x R, v v v v R. Defe the global state vetor posto/veloty form as z x v R. he the global dyams a be wrtte as 0 I z z A z L L (.8 hs s posto/veloty form, whereas (.4 has the global state defed terms of the terleaved ode postos ad velotes aordg to the loal state deftos (.. he haraterst polyomal of ths system s si I si A s I Ls L (.9 L si L As the proof of Lemma, defe a trasformato M suh that J M LM s upper tragular wth the egevalues,, of L o the dagoal. Wrte si Ls L M M si Ls L M ( si Ls LM si Js J (.30 Matrx ( sijs J s blok tragular form wth the egevalues of L o the dagoal. Cosequetly silsl ( s s (.3 herefore, the egevalues of A are gve by ( 4 s (.3 ad there are two egevalues of A for eah egevalue of L. hese results are the same as (.7. herefore all the subsequet developmet there follows here, evetually provdg aother path to heorem. Example : Seod-Order Cosesus 6

7 Example : Destruto of Cosesus By Ireasg Iformato Exhage I ths example we show that gve a ommuato graph topology ad a seod-order protool that yelds osesus, t s possble to destroy the osesus behavor by addg extra lks betwee them. hus, addg formato exhage may be detrmetal to the performae of etworked teams. Formato Cotrol Seod-Order Protool I the otrol of formatos t s desred for all vehles to reah the same osesus veloty, but also for the steady-state postos to be some presrbed formato relatve to eah other. he protool just gve auses all odes to move to the same fal posto. Moreover, formato otrol the postos are -D or 3-D. herefore, let the formato move a -dmesoal spae ad osder the ode dyams x v (.33 v u wth vetor posto x R, veloty v R, ad aelerato put u R. For moto the -D plae, for stae, oe would take x [ p q ] where ( p ( t, q ( t s the posto of ode the (x,y-plae. he ode states are z x v R ad the loal ode dyams are gve by 0 I 0 z z u Az Bu 0 0 I (.34 Defe the ostat desred posto of ode relatve to the movg formato eter x 0 as R. Suppose t s desred for the odes to follow a leader or otrol ode wth posto ad veloty gve by x 0 v0. Cosder the dstrbuted seod-order leader-followg protools at eah ode gve by u v 0 kv( v0 v kp ( x0 x aj ( xj j ( x aj ( vj v j j (.35 v k ( v v k ( x x a ( x ( x ( v v 0 v 0 p 0 j j j j j where 0 s a stffess ga ad 0 s a dampg ga. hs protool ludes leader s veloty feedbak wth ga of kv 0, leader s posto feedbak wth ga of k p 0, ad leader s aelerato feedforward. he am of ths protool s to esure that all odes reah osesus posto ad veloty ad that x( t x0 ( t, v ( t v0 (, t asymptotally. Defe the moto errors x x x0, v v v0. he the losed-loop ode dyams are 0 I 0 z z aj( xj j ( x ( vj v ki p ki v I (.36 j 7

8 whh yelds the global error dyams z ( I A LBK z Az (.37 wth z z z z R A ki, K I I 0 I ki p v, ad (.38 Cosesus Stablty of the Formato Protool. We wat to exame stablty of the protool ad determe whe x( t x0( t, v( t v0(, t. Assume the graph has a spag tree. o exame stablty, aordg to the proof of Lemma, A (.37 s equvalet to dag{ A,( A BK, ( A BK} (.39 wth the egevalues of Laplaa matrx L ad Re{ } 0,,,. Wthout loss of geeralty, assume moto -D ad set. Moto other oordate dretos wll be the same as the moto alog a le as aalyzed here. Matrx A has the haraterst polyomal s kvs kp, whh s stable for all kp, kv 0. he haraterst polyomals of the other bloks are si ( A BK s ( k s ( k (.40 v p he the haraterst polyomal of A s ( ( v ( v si A s k s k (.4 herefore, the egevalues of A are gve by ( ( v 4( p s k k k (.4 here are two egevalues of A ( eah dmeso of moto for eah egevalue of L. Se the graph has a spag tree we have 0 ad 0,,. For real the Routh test shows that (.40 s asymptotally stable (AS for all 0, 0. If s omplex t s dffult to get a odto for stablzg gas. Routh test reveals that all the systems (.39 are AS f 0 ad * * ( s s( s s ( s s ( s s where * deotes omplex ojugate, Re{ },. Performg the Routh test shows that ths s AS f ad oly f ( / (.44 where Im{ }. herefore, all the systems (.5 for,, are AS f ad oly f 8

9 Im { } Re { } max,,, Re{ } (.45 heorem. Cosesus of Formato Protool Cosder the ewto s law moto dyams at eah ode gve by (.33. he leader-followg protools (.35 guaratee that x( t x0( t, v( t v0(, t asymptotally f the graph has a spag tree ad the gas, are seleted to satsfy odto (., wth the egevalues of the graph Laplaa L. Proof: Uder the hypotheses, the polyomals (.4 are asymptotally stable so that the error dyams (.37 are AS. herefore, x x x0 ad v v v0 go to zero ad the theorem follows. he followg result has also bee establshed. Corollary. Let the graph have a spag tree ad all egevalues of the Laplaa L be real. he x( t x0( t, v( t v0(, t asymptotally for ay gas, 0 kp, kv 0. Proof: he Routh test reveals that (.40 s asymptotally stable uder the hypotheses. If the graph s udreted, the the Laplaa egevalues are real ad ths orollary holds. A dsadvatage of protool (.35 s that all odes must kow the leader ode s posto x ( t ad veloty 0 v ( 0 t. I realst formatos, eah ode oly kows posto ad veloty formato about a few eghbors. Oly a few mmedate followers of the leader, hs wgme, a sese ts moto, ad other odes follow these wgme, ad so o. We shall show how to orret ths defey later whe we dsuss the ooperatve trakg problem, also kow as pg otrol. Example : Formato Cotrol Protool 9

10 Referees J.A. Fax ad R. M. Murray. 004, Iformato Flow ad Cooperatve Cotrol of Vehle Formatos, IEEE ras. Automat Cotrol 49, o. 9: W. Re, Cosesus strateges for ooperatve otrol of vehle formatos, IE Cotrol heory ad Applatos, vo., o., pp , 007. W. Re ad E. Atks, Dstrbuted mult-vehle oordated otrol va loal formato exhage, It. J. Robust ad olear Cotrol, vol. 7, pp , 007. W. Re ad R.W. Beard, Dstrbuted Cosesus Mult-Vehle Cooperatve Cotrol, Sprger- Verlag, Lodo, 008. G. Xe ad L. Wag, Cosesus otrol for a lass of etworks of dyam agets, It. J. Robust ad olear Cotrol, vol. 7, pp ,