Dcntralzd Adaptv Control and th Possblty of Utlzaton of Ntworkd Control Systm MARIÁN ÁRNÍK, JÁN MURGAŠ Slovak Unvrsty of chnology n Bratslava Faculty of Elctrcal Engnrng and Informaton chnology Insttut of Control and Industral Informatcs Ilkovcova 3, 812 19 Bratslava Slovak rpublc maran.tarnk@stuba.sk, jan.murgas@stuba.sk Abstract: h man objctv of ths papr s to dscus a possbl nw concpt of usng communcaton and ntworkd control systm n dcntralzd adaptv control whch guarants zro rsdual trackng rrors n all subsystms. It s studd whr th on-ln coordnaton or communcaton tak plac n th dcntralzd adaptv control schm. hs concpt s an xtnson to th algorthms rportd n th ltratur. Ky Words: Dcntralzd Adaptv Control, Ntworkd Control Systms, zro rsdual trackng rror 1 Introducton For a lmtd amount of nformaton avalabl for th ndvdual subsystms n larg-scal dcntralzd systms, an addtonal communcaton btwn subsystms s usually usd. Such communcaton srvs as a bass for dcson-makng procss and control. h communcaton s provdd by a communcaton ntwork. Undr ths trm w undrstand a st of ntwork dvcs and tchnologs whch ar ndd to mplmnt th communcaton. Whn usng such lmnts n th control systm thn t s classfd as a ntworkd control systm. hus ths trm oftn nvolvs a dcntralzd approach of control systm dsgn. h thortcal problm of dcntralzd adaptv control was frst ntroducd by Ioannou [2], and latr, wth som modfcatons by Gavl and Šljak, for xampl s [1]. Rcntly, th fld of dcntralzd adaptv control s th subjct of ntns rsarch, s [8] and t s rfrncs, and [3, 5, 6, 7]. h bst that can b achvd usng mthods of classcal robust adaptv control n dcntralzd adaptv control n th prsnc of paramtrc uncrtants s th convrgnc of rrors to som boundd rsdual st [5]. Dspt of ths fact t has bn shown that th combnaton of adaptv control and dcntralzd control allows to tak advantag of both flds. Usng addtonal communcaton btwn subsystms s ssntal n achvng of th convrgnc of rrors to zro nstad of som boundd rsdual st. Although w consdr th communcaton btwn subsystms, th dcntralzaton s mantand bcaus th subsystms do not xchang nformaton on thr actual outputs. h fundamntal da has bn proposd n rcnt yars as follows [5, 6]. Each subsystm has th nformaton on rfrnc sgnals of all othr subsystms. In othr words, ach subsystm has th sgnal from th all rfrnc modls. hs allows to achv asymptotc convrgnc of th ovrall rror th trackng rrors of th all subsystms to zro. On possblty s that th subsystms rcv ths nformaton off-ln, bcaus t s oftn avalabl a pror. hrfor thr s no on-ln communcaton and th prncpl of strctly dcntralzd systm s mantand. hs da s rfrrd as th mplct coopraton of subsystms [6]. Slghtly dffrnt pont of vw s that th subsystms ar coordnatd usng outputs of th rfrnc modls. Agan, ths can b th off-ln coordnaton and thrfor th systm s consdrd as strctly dcntralzd. h concpt of coordnaton by mans of rfrnc modls has bn ntroducd by B. M. Mrkn [4]. Indpndntly of th rsults obtand by B. M. Mrkn th smlar rsults has bn rportd by Narndra t al. [6, 7]. h man da n thr work s th compnsaton of ntractons whl th nformaton from th othr subsystms s rplacd by nformaton on thr rfrnc sgnals th rfrnc modl output. It s clar that ths s just a dffrnt pont of vw on th sam concpt. Important n both works s that th control law contans a hgh gan fdback trm. In th work of ISBN: 978-1-6184-57-2 83
Mrkn ar usd th rsults rportd for xampl n [1], and th coordnatng trm s addd as mntond abov. In ths cas t s ssntal that th control law sgnal vctor ncluds an adaptaton rror sgnal [1], whch s not standard n modl rfrnc adaptv control, vn n th spcal applcatons of ths thory, for xampl s [9] or [1]. Authors Narndra and Olng ar closr to th convntons of modl rfrnc adaptv control. hy us th standard sgnal vctor. Howvr, n th control law s ntroducd addtoanal fdback trm whch s a P controllr of trackng rror, assumng ts hgh gan. 2 Utlzaton of Ntworkd Control Systm Our man objctv n ths papr s to dscus a possbl nw concpt of usng communcaton and ntworkd control systm n dcntralzd adaptv control whch guarants zro rsdual trackng rrors n all subsystms. h am of th authors Narndra and Olng s to ncras th control prformanc n th dcntralzd adaptv control systms g unaccptably larg transnt rrors. hy hav proposd to us an addtonal communcaton btwn th subsystms for ths purpos. Rsarch s lookng for an answrs to qustons lk: What nformaton hav to b shard among th subsystm controllrs? At what tm ntrvals? What s an ndcator of ncssty to communcat? It s n ths contxt, whr th rol of th ntworkd control systm obvously appars. In som cass nvolvng th strctly dcntralzd adaptv control, havng nformaton about th ntr trajctors of th othr subsystms may b unncssary and th convrgnc of rrors to zro mght b actually attand wth much lss nformaton [8]. In furthr work w wll consdr th on-ln coordnaton or communcaton nstad of off-ln coordnaton of subsystms. hn w nd to fnd answrs to smlar qustons as mntond abov. Frst stp s to dtrmn whr th on-ln coordnaton or communcaton tak plac n th dcntralzd adaptv control schm. hs ssu s dscussd n th nxt sctons. 2.1 Control algorthm wth rspct to on-ln coordnaton In ths scton w brfly dscrb a dcntralzd adaptv control algorthm wth rspct to on-ln coordnaton whch s basd on th rsults mntond abov. As an xampl w consdr a stat trackng problm. Consdr th larg-scal systm S wth N ntrconnctd subsystms S n th form S : ẋ = A x + b u + b g F x 1 whr x t R n s th stat vctor and u t R s th nput of -th subsystm S. h vctor x = [ x1 xn] s composd of th stat vctors of all subsystms wth lngth n = N =1 n. Matrx F R n n s known, n th form F = blkdag F 1,..., F j,..., F N whr I R n j n j f j F j = f = j 2 h output of functon blkdag s th block dagonal matrx, th argumnts ar placd on dagonal, whl othr lmnts ar zro. Vctor g R n s constant but unknown. Matrx A and vctor b ar constant and unknown. It s assumd that thr xst th vctor k R n and scalar l R, so that A m = A + b k and b m = b l, whr A m s known, asymptotc stabl, and b m s known vctor. A control objctv s to forc x t to track th stat x m t of a gvn rfrnc modl whch s assgnd to ach subsystm n th form S m : ẋ m = A m x m + b m r 3 whr r t s known and boundd rfrnc sgnal. Consdr th control law n th form u = k x γ + l r h F x m 4 whr k t and l t ar th stmats of k and l rspctvly. In th frst two trms n 4 ar usd only local sgnals of subsystm. In th last two trm n 4 ar usd sgnals r t and x m t whch w wll rfr to as an xtrnal sgnals or a coordnatng sgnals, whr xm= [ xm1 xmn]. h adaptd paramtr of th last trm s th vctor h t wth sam sz as g. In ths cas th dal valu of vctor h = g, for mor dtals s [6]. h paramtrγ s th postv ral constant spcfd bllow. h matrx P satsfs Lyapunov quaton n th form A m P +P A m = Q, whr Q s postv dfnt, symmtrc, arbtrary matrx wth corrspondng sz. h subsystm stat trackng rror s dfnd as = x x m 5 W dfn th adaptaton paramtr rrors n th form k = k k ; l = l l ; h = g h 6 ISBN: 978-1-6184-57-2 84
hn th control law 4 can b wrttn n th form u = k x + k x γ + + l r + l r h F 7 x m and th quaton of subsystm 1 n th form ẋ = A x + b u + b g F x m + b g F 8 Substtutng th quaton 7 nto th quaton 8 lads to ẋ = A + b k x + b l r + + b θ ω b γ + 9 + b g F +b g F x m whr th sgnal vctor ω = [ ] x r and th paramtr rror vctorθ = [ k ] l ar ntroducd. Fnally subtractng 3 from 9 lads to th rror quaton n th form ė = A m + b θ ω b γ + + b g F +b h F x m 1 Consdr th adaptaton laws n th form Θ = θ = P b ω 11 ḣ = h = P b F x m 12 h dynamcal systm composd by quatons 1, 11 and 12 s asymptotc stabl wth rspct to rror and stabl wth rspct to paramtr rrors k and l. h outln of th proof s as follows: Consdr th Lyapunov functon canddat n th form V = P +θ θ + h h 13 h tm drvatv V along th trajctory of th consdrd subsystm s n th form V = ė P + P ė + 2θ θ + 2 h h 14 Substtutng quatons 1, 11 and 12 to th quaton 14 lads to V = Q γ 2 γ 2 2 γ 1 Aftr compltng th squars w obtan V = Q γ 2 γ γ 1 g F 2 + g F 2 +γ 1 g F 15 16 h scond and th thrd trms n th rght hand sd of quaton 16 ar always ngatv. hrfor V satsfs an nqualty V λ mn Q 2 +γ 1 g 2 F 2 2 17 V λ mn Q γ 1 g 2 F 2 2 18 whr w hav usd th fact that and λ mn Q s th smallst gnvalu of matrx Q. By slctng a suffcntly larg valu of paramtrγ so that γ >λ 1 mn Q g 2 F 2 19 w hav V along th trajctory of subsystm 1 11 12. h paramtr rror vctorθ and [ h ar boundd, ] thrfor th adaptd paramtrsθ = k l and h ar boundd. h trackng rror wth tm t. In ordr to b abl to choos th valu of paramtrγ, an ucldan norm of vctor g must b known. If th norm of vctor g s not known, w can us th adaptaton law for ths paramtr n th form, s [8] γ = 2 h ovrall Lyapunov functon V of larg-scal systm S s smply th sum of sub-lyapunov functons N V= V 21 =1 hrfor, th ovrall closd-loop systm s stabl bcaus V, and th zro rsdual trackng rrors n all subsystms ar guarantd. 2.2 h control schm wth coordnator W may rmark that th ntroducton of th matrx F n ach subsystm allows to classfy th sgnal x m n th control law 4 as an xtrnal sgnal. h matrx F dtrmns whch componnts of th sgnal x m wll b usd n th control law as an xtrnal sgnal. hs allows to transmt on sgnal to all subsystms. h smlar approach s obvously possbl n th cas of rfrnc sgnals r. h sgnal x m can b gnratd locally n vry tm nstant or t can b rcvd as an xtrnal sgnal not ncssarly n vry tm nstant stablty problms may ars. Introducng th coordnator s straghtforward. h coordnator s formd by th rfrnc modls of all subsystms. h sam concpt s usd n [3, 5]. Consquntly w consdr an on way communcaton from coordnator to th controllrs of ach subsystm. Hr th communcaton ntwork or ntworkd control systm tak plac. h schm of th control algorthm s shown n th Fg. 1. ISBN: 978-1-6184-57-2 85
MUX r 1 RM 1 S 1 x 1 x m1 r... RM... S x x m -th controllr... RM N r N... x mn Coordnator Communcaton ntwork xtrnal sgnals S N x N Larg-scal systm S Intrconnctons Fg. 1: Dcntralzd adaptv control wth on-ln coordnaton 3 Illustratv numrcal xampl As an xampl, to llustrat th abov thory, w consdr th systm wth N = 2 ntrconnctd subsystms n th form 1, whr x 1 t R 2, x 2 t R 2, u 1 t R, u 2 t R and A 1 = A 2 = [ ] 1, b 3 4 1 = [], F 1 1 = [ ] [ ] 1, b 1 3 2 =, F, 5 2 =, 1 1 1 1. h strngth of ntrconnctons s gvn by vctors g 1 =[ 18 3 ] ; g 2 =[ 5 12 ] h rfrnc modls ar n th form 3, whr x m1 t R 2, x m2 t R 2, [ ] [] 1 A m1 =, b 1 2 m1 =, 1 [ ] [] 1 A m2 =, b 4 4 m2 =. 4 h rfrnc sgnals r 1 t and r 2 t hav th form r 1 = 1sgn sn, 4t ; r 2 = 5 sn, 7t h arbtrary paramtrs[ of th ] control algorthm ar 1 chosn to b Q 1 = Q 2 =. h ntal condtons 1 of all dffrntal quatons ar chosn to b zro, ncludng th ntal valus of adaptd paramtrs. h adaptv controllr n ach subsystm has th form 4 wth th adaptaton laws n th form 11 and 12. For th smulaton purposs, t s assumd that th rasonabl stmats of norms g 1 and g 2 ar known g 1 = 18.25; g 2 = 13. hrfor w havγ 1 = 1+λ 1 mn Q 1 g 1 2 F 1 2 = 334 andγ 2 = 1+λ 1 mn Q 2 g 2 2 F 2 2 = 17. h rsults of th smulaton ar n th Fg. 2 and Fg. 3. 4 Concluson and Futur Work hs artcl has dscussd th possblty of usng th on way on-ln communcaton from th coordnator to th subsystms n th dcntralzd adaptv control. hs concpt s an xtnson to th algorthms rportd n th ltratur. h man da s to rplac th off-ln coordnaton by th on-ln coordnaton. h zro rsdual trackng rror s attand usng nformaton from all of th rfrnc modls. If th rfrnc sgnals ar not known a prory, thn th nformaton from all of th rfrnc modls can not b usd off-ln. hs constrant s rlaxd by dscussd concpt. Morovr, ths concpt allows to us much lss nformaton for attanng th zro rsdual rror. In our futur work, w wll study th smlar qustons as mntond n th scton 2. h stablty ssus and th prformanc ssus of th dscussd dcntralzd adaptv control schm wll b studd too. Acknowldgmnts: hs work has bn supportd by Slovak Rsarch and Dvlopmnt Agncy through grant APVV-211-1. It has bn supportd by th projct Rq-48-1 too. ISBN: 978-1-6184-57-2 86
1.2.2.4 1 2 t [sc] 2 2 2 4 1 2 t [sc] Θ 1 6 h 13, h 14 5 1 15 2 a h stat trackng rror of subsystm S 1 1 2 t [sc] b h adaptd paramtrs of subsystm S 1 1 2 t [sc] c h adaptd paramtrs of subsystm S 1 Fg. 2: Smulaton rsults for subsystm S 1 Θ 2 2 2 4 8 h 21, h 22 5 1 15 a h stat trackng rror of subsystm S 2 1 2 t [sc] b h adaptd paramtrs of subsystm S 2 1 2 t [sc] c h adaptd paramtrs of subsystm S 2 Fg. 3: Smulaton rsults for subsystm S 2 Rfrncs: [1] D.. Gavl and D. D. Šljak. Dcntralzd adaptv control: structural condtons for stablty. IEEE ransactons on Automatc Control, 344:413 426, apr 1989. [2] P. Ioannou. Dcntralzd adaptv control of ntrconnctd systms. IEEE ransactons on Automatc Control, 314:291 298, apr 1986. [3] B. M. Mrkn. Dcntralzd adaptv control wth zro rsdual trackng rrors. In Procdngs of th 7th Mdtrranan Confrnc on Control and Automaton MED99 Hafa, Isral, jun 1999. [4] B. M. Mrkn. Commnts on xact output trackng n dcntralzd adaptv control. IEEE ransactons on Automatc Control,, 482:348 35, fb. 23. [5] B. M. Mrkn and P.-O. Gutman. Dcntralzd adaptv control wth mprovd stady stat prformanc. In 15th rnnal World Congrss, Barclona, Span, 22. [6] K. S. Narndra and N. O. Olng. Exact output trackng n dcntralzd adaptv control systms. IEEE ransactons on Automatc Control, 472:39 395, fb 22. [7] K. S. Narndra, N. O. Olng, and S. Mukhopadhyay. Dcntralsd adaptv control wth partal communcaton. IEE Procdngs - Control hory and Applcatons, 1535:546 555, spt. 26. [8] N. O. Olng. Dcntralzd Adaptv Control. PhD thss, Yal Unvrsty, May 24. [9] M. árník and J. Murgaš. Addtonal adaptv controllr for mutual torqu rppl mnmzaton n pmsm drv systms. In 18th IFAC World Congrss, Mlano, August 28 Sptmbr 2, 211. [1] M. árník and J. Murgaš. Modl rfrnc adaptv control of prmannt magnt synchronous motor. Journal of Elctrcal Engnrng, 623:125 133, may 211. ISBN: 978-1-6184-57-2 87