REDUCING THE EFFECT OF UNMODELED DYNAMICS BY MRAC CONTROL LAW MODIFICATION. Eva Miklovičová, Ján Murgaš and Michal Gonos

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1 REDUCING HE EFFEC OF UNODELED DYNICS BY RC CONROL LW ODIFICION Eva iklovičová, Já ugaš ad ichal Goos Depatet of utoatic Cotol Systes, Faculty of Electical Egieeig ad Iatio echology, Slovak Uivesity of echology, Ilkovičova, 8 9 Batislava, Slovak Republic tel.: , fa.: , e-ail: eva.iklovicova@stuba.sk, ja.ugas@stuba.sk bstact: Solvig a tackig poble does ot alays give desied esults eve he the adaptive cotol ethods ae used. Soe difficulties ay occu he the apioi assuptios laid do fo the poble solutio ae ot satisfied. Oe of the seious issues is the eistece of uodeled dyaics i the tackig poble. he poposed solutios ae aily based o obustificatio of the adaptatio la. I this pape e popose to educe the effect of uodeled dyaics usig the RC cotol la odificatio so that the stadad adaptatio la esues the sufficietly sall tackig eo. Copyight 5 IFC Keyods: odel efeece adaptive cotol, Lyapuov ethods, tackig systes. INRODUCION he odel efeece cotol stuctues ca be successfully used to solve the tackig poble i case soe ideal coditios egadig the cotolled plat odel ae satisfied (Naeda ad asay, 989). Hoeve, these ideal coditios ae ot ofte satisfied. fequet violatio of the ideal assuptios is a icoect estiatio of the plat odel stuctue that leads to the eistece of uodeled dyaics i the tackig poble. he uodeled dyaics ifluece i adaptive systes epesets a seious poble. he uodeled dyaics ca sigificatly deteioate the cotol paaetes adaptatio o it ca eve destabilize this pocess. Especially i case of RC stuctues the eistece of the uodeled dyaics educes thei applicability to eal pocesses. itesive eseach activity has bee devoted to solve the poble of educig the effect of uodeled dyaics (Goos, et al., 4; Rohs, et al., 985; Sasty ad Bodso, 989) but it still has ot led to satisfactoy esults. he ajoity of the uodeled dyaics poble solutios ae based o the adaptatio la obustificatio (Sasty ad Bodso, 989; Ioaou ad Su, 996). he ai of ou pape is to deostate that the uodeled dyaics poble i stadad RC schee ca also be solved by the cotol la odificatio. he poposed appoach is based o the geeal theoy of stability i vecto (Šiljak, 978).. PROBLE FORULION he tackig poble is alays a oliea task, because the adaptatio eo dyaics is oliea. he let the poble solutio plai the tackig poble ith the liea odel of cotolled syste ad the liea efeece odel has bee chose. Coside that the liea syste ith uodeled dyaics is give by the state space equatios i the

2 + bu y c () hee R p epesets the plat state, y R is the plat output ad u R deotes the cotol sigal. he state vaiables of syste () ca be divided ito to vectos: state vaiables of odeled dyaics R ad state vaiables of uodeled dyaics R, ith +p. he syste () ca the be decoposed so that the odeled ad uodeled dyaics is sepaated b u () hee is a Fobeius ati of the odeled pat of dyaics, is a ati of the uodeled pat of dyaics, () ad () ae atices epesetig the iteactios. Fo the easos of siplicity e coside the class of systes hee oly odeled pat of dyaics is diectly iflueced by the cotol sigal. ssue that the desied closed loop dyaical behavio is descibed by the efeece odel i the + b R () ith the efeece odel state vecto efeece sigal R. R ad I stadad RC schees the cotol la is of the u k k (4) hee k deotes a feedback gai vecto ad k is a feedfoad gai. Hoeve, i case of these cotol stuctues the uodeled dyaics povokes the cosideable deteioatio of adaptive syste tackig peaces. Θ [ ] hee [ κ ρ], ω, * κ ( k ), ( k ) k ρ. k * he stability of the syste (6) is esued by the tackig poble covegece. he stability poof ill be based o the vecto Ljapuov fuctio ethodology (Šiljak, 978). he isolated subsystes of syste (6) ae e& e b Θ ω + b (7) Whe, the equilibiu poit of syste (6) is e,,. o aalyze the equilibiu poit stability the Lyapuov fuctio cadidates fo each isolated subsyste have to be chose as the fuctios of the coespodig subsyste vaiables V V P P V e P e + αθ Θ (8) he coditios of cotiuity ad positive defiiteess ae satisfied fo the fuctios V, V, V. he vecto Lyapuov fuctio ethodology is based o aggegatio, hee it is ecessay to fid the boudaies of the V, V, V tie deivatives alog the elevat subsystes tajectoies. he tie deivatives of the Lyapuov fuctios cadidates (8) alog the subsyste (7) tajectoies ae [ e ω Θb ] P e + e P [ e bθ ω] + Θ Θ e ( P + P ) e ω Θb P e + e& P e + e P e& + Θ Θ (9). REDUCING HE EFFEC OF UNODELLED DYNICS Usig the atchig coditios b bk bk * * (5) the folloig dyaics of adaptatio eo (defied as e ) ca be deived e& e b ( ) & Θ ω + (6) b P + P P ( P + P ) P + P + P () ( + b ) P + P ( + b) ( P + P ) + bp ()

3 Choosig a suitable adaptatio la so that dθ f ( e,,) () ω Θb P e Θ Θ () it is possible to itoduce the folloig bouday fo the V tie deivative e ( G) e ( G) e (4) * * G hee ( b k ) P + P ( bk ). he boudaies fo the V ad V tie deivatives ae ( G ) ( G ) (5) hee P + P G ad ( G ) ( G ) (6) hee P + P G. It is also ecessay to set bouds o the subsyste iteactios e P e P (7) hee ( P P ) / P > (8) P e P e hee P ( ) / P P ad (9) P P e he aggegated syste is of the z & Wz () hee z is the aggegated syste state vecto ad W deotes the aggegatio ati W () ith / / ( G) ( G ) ( G ) ( P P ) ( P P ) () () hee (.) deotes the iial ati eigevalue, (.) is the aial ati eigevalue ad G, G, G ae the solutios of the folloig equatios * * ( b k ) P + P ( b k ) P + P P + P G G he stability coditios ae as follos i) ii) iii) ( > G > ) > (4) (5) Let us o aalyze the possibilities of satisfactio of the coditios (5): he coditio i) is satisfied if the efeece odel is stable. he coditio ii) is essetial ad its satisfactio ill be aalyzed i the folloig. he coditio iii) equies the satisfactio of the coditio ii) as ell as the stability of the efeece odel. Let us defie the stability easue L(,,,, ) i the L(,,,, ) the the coditio ii) of (5) is L (6) (,,, ) > (7) hich afte itoducig () ad () ito(6) ca be eitte ito the folloig ( G ) ( G ) / ( P P ) / ( P P ) > (8) he syetical atices P P ad have positive eigevalues ad the P P poduct ( G ) ( G ) is as ell positive if the

4 uodeled dyaics is stable. he satisfactio of the coditio (8) depeds o the possibility of abitay iceasig the value of ( G ) by eas of the adaptatio. he poble is that the icease of ( G ) povokes also a icease of P ad so a / P P. icease of ( ) he echais of the coditios (5) satisfactio is vey coplicated ad ca ot be aalytically poved. I the cotol systes ith adaptatio the satisfactio of these coditios depeds o the stuctue ad the o of iteactios. Stability easue Fig. he stability easue depedece o the value of ( ) G Usig a siple eaple of the d ode syste cosistig of the secod ode odeled pat ad of the fist ode uodeled pat e ca illustate the depedece of the stability easue o the value of, that epesets the ifluece of state ( ) G (G ) cotolle gais i the pesece of the elatively stog iteactios. I Fig. the gee lie L coespods to the efeece odel stability easue equal to - ad the blue lie L epesets the efeece odel ith the stability easue of -.. It ca be see that the adaptatio eo covegece ca be iflueced by the stability easue of the efeece odel. Hoeve, the efeece odel dyaics is give by the cotol peace equieets, so it is ecessay to esue the icease of the efeece odel stability easue idiectly duig the adaptatio eo tasiet pocesses. his idiect iceasig of the stability easue ca be obtaied by a odificatio of the cotol la (4) to the u + L L k + k k e (9) / ( b k ) P + P ( b k ) ( G ) / ( P P ) ( P P ) > () he syetical atices P P ad have positive eigevalues ad by eas P P of the appopiate adaptatio of k it is possible to esue the satisfactio of the ii) stability coditio i (5). 4. EXPLE Coside the cotolled plat odel i the : y a [ a 4 ] [ ] hee ( i, K,5) a i paaetes. a + + a 5 + u () ae uko sloly vayig he output y(t) is equied to follo as close as possible the output y (t) of the efeece odel y [ ] + () Usig the poposed odificatio of the cotol la (9) ad the stadad adaptatio la (ugaš, et al., 99) dθ αωε () hee α >, ε Pe ad P is the solutio of the Lyapuov ati equatio b P + P I (4) the acceptable tackig eo ca be obtaied, as illustated i Fig. ad Fig.. It ca be see fo Fig. ad Fig. that the uodeled dyaics ifluece eductio by eas of the cotol la odificatio has bee vey efficiet. he geealizatio of the poposed solutio ill be ecessay i the futue. hee the feedback te k e esues the icease of the ati stability easue. fte itoducig () ad () the stability coditio ii) of (5) ca be eitte ito the

5 .5 Output [p.u.] tie [s] Fig. Peaces of the RC ith the poposed cotol la odificatio odif. RC, stad. RC, ef. odel.5 Output [p.u.] tie [s] Fig. Peaces of the RC ith the poposed cotol la odificatio (detail) odif. RC, stad. RC, ef. odel 5. CONCLUSION Τhe ai of the poposed pape has bee to educe the effect of uodeled dyaics i RC tackig pobles. he poposed odificatio of the stadad cotol stuctue iceases the tackig syste obustess eve i case he the stadad adaptatio la is used. CKNOWLEDGEEN his ok has bee suppoted by the Slovak Scietific Gat gecy, Gat No. /58/. REFERENCES Goos,., ugaš, J. ad E. iklovičová (4). RC ith educed state iatio. Poc. of. he IFC Wokshop o daptatio ad Leaig i Cotol ad Sigal Pocessig, Yokohaa, Japa, pp Ioaou, P.. ad J.Su (996). Robust adaptive cotol. Petice Hall, Ne Jesey. ugaš, J., Veselý, V. ad I. Hejda (99). State space stuctues i RC. Poc. of the IFC Wokshop o Stuctues ad Cotol, Pague, Czech Republic, pp.8-8. Naeda, K.S. ad.. asay (989). Stable daptive Systes. Petice Hall, Ne Jesey. Rohs, C.E., Valavai, L., thas,. ad G. Stei (985). Robustess of cotiuous-tie adaptive cotol algoith i the pesece of uodelled dyaics. IEEE as. o uto. Cotol,, pp Sasty, S. ad. Bodso (989). daptive cotol. Stability, covegece ad obustess. Petice Hall, Ne Jesey. Šiljak, D.D. (978). Lage-scale dyaic systes stability ad stuctues. Noth Hollad, steda..