Differential flatness applications to industrial machine control

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1 Autotio, Cotrol Itlligt Systs 4; (4: 4-5 Publish oli Sptbr, 4 ( oi:.648/j.cis.44. ISSN: (Prit; ISSN: (Oli Diffrtil fltss pplictios to iustril chi cotrol Ejik C. A, Gsh K. Vygoorthy Elctricl Egirig Progr, Abubkr Tfw Blw Uivrsity, PMB 48, Buchi, Nigri Rl-Ti Powr Itlligt Systs Lbortory, Clso Uivrsity, Clso, USA Eil rss: jik@yhoo.co (E. C. A, gkur@i.org (G. K. Vygoorthy To cit this rticl: Ejik C. A, Gsh K. Vygoorthy. Diffrtil Fltss Applictios to Iustril Mchi Cotrol. Autotio, Cotrol Itlligt Systs. Vol., No. 4, 4, pp oi:.648/j.cis.44. Abstrct: I this rticl th pplictios of iffrtil fltss to so iustril systs r prst. Coputtiol thos of obtiig th flt output th stright forwr tho of costructig th corrspoig cotrol lw r giv. So thorticl iustril systs r us s illustrtio icluig th thir orr sychroous chi ol th o gr of fro gtic lvittio syst ol. Coputtios of th flt output r o usig vrious pprochs. Th Lvi s pproch is prst i such til s to fcilitt quick urstig. Coputtios for th sychroous chi ol yil flt output tht is fuctio of th lo gl whil th gtic lvittio ol yil flt output tht is fuctio of th objcts positio. Rsults showig th stbiliztio of th ppli systs i fult ucrti situtios r iscuss. Kywors: Mgtic Lvittio, Fltss, Fbck Liriztio, Sychroous Mchi. Itrouctio THE cocpt of iffrtil fltss propos by Michl Fliss co-workrs [],[] bout twty yrs go hs volv ito full-flg fil for th stuy of cotrol systs i prcticlly w wy. I this sttig, cotrollbility is lik with syst fltss cotrollbl systs possss this fltss proprty [],[4]. For such systs thr is solutio st cll flt output i th solutio spc cosistig of st of stt vribls tht copltly prtriz th syst without th for solvig iffrtil qutios. Oc this output is show to b flt, it i ffct iplis tht th syst posssss wll chrctriz yics[5]. This is bcus ll syst prtrs cotrol bcos fuctio of th lirizig output tht c bl th grtio of rfrc trjctoris -priori. Th costructio of th fbck lw is o by sipl ivrsio of syst qutios with rspct to th cotrol. Th sch i rivtio is xtsio fro th iputoutput liriztio sch with zro itrl yics. Fliss t-l [] propos th otio of ogous quivlc fi clss of yic fbcks for clssifictio liriztio of systs i th for of Fliss iffrtil lgbric fors. Such clsss of systs r th so-cll iffrtilly flt systs. O of th i cosqucs of thir rsult is costructiv tho of coputig th fbck tht xctly lirizs flt syst. Accorigly cotrol syst M, F is iffrtilly flt rou p if oly if it is quivlt to trivil syst i ighborhoo ofp. A trivil syst c b fi s o which is without yics scrib by collctio of ipt vribls or R, F whr Fs( y, y (, y (,... ( y, y (, y (,..., with y R s y [6]. It is si to b iffrtilly flt if it is iffrtilly flt rou vry p of op s subst ofm. Th st y { y j,..., s} is cll flt or lirizig output of j M scrib by collctio of ipt vribls, th flt output hvig th s ubr of copots s th ubr of cotrol vribls. Th followig uctios r show with proofs i [].. Th ubr of copots of flt output is qul to ubr of iput chls.. A clssic lir syst is flt if oly if it is cotrollbl.. Th cotrollbility of iffrtilly flt systs is rlt to th wll kow strog ccssibility s s

2 4 Ejik C. A Gsh K. Vygoorthy: Diffrtil Fltss Applictios to Iustril Mchi Cotrol proprty of olir systs u to Suss Jurjvic. 4. If clssic olir syst is iffrtilly flt rou p, th it stisfis th strog ccssibility t p. 5. Diffrtil fltss s tht th stt iput y b copltly rcovr fro th flt output without itgrtig th syst iffrtil qutios. Aftr th itrouctio i Sctio, th ppr iscusss th bsic thory of iffrtil fltss i Sctio II. I Sctio III th procur of coputtios of flt output is til. Sctio IV iscusss th xpls for coputig flt outputs for so systs th siultios o o th rsultig cotrollrs of so iustril systs o MATLAB. Coclusios r giv i Sctio V whil i Sctio VI th rfrcs r giv.. Bsic Thory of Fltss A syst vribl is ogous if it c b xprss s lir cobitio of th iput, th output fiit ubr of thir ti rivtivs. Othrwis it is xogous. A sigl iput sigl output (SISO syst is thrfor flt or iffrtilly flt if thr xists ogous vribl cll th flt output, such tht th iput th output c b xprss s lir cobitio of th flt output fiit ubr of its ti rivtivs [7]. Nturlly y othr ogous vribl of th syst joys th s proprty with rspct to th flt output. Thus th flt output iffrtilly prtrizs ll syst vribls. Grlly, th fiitio of syst fltss c b cst i wht follows: Th syst f (ɺ, x x, u ( with x R u R is iffrtilly flt if o c fi st of vribls cll flt output; ( y h( x, uuu,ɺ,ɺɺ,..., u r y R syst vribls, cotrol, ( x α( y,ɺ,ɺɺ,..., y y y q ( u y y y y q β(,ɺ,ɺɺ,..., with q fiit itgr such tht th syst qutio ( ( (4.. Equivlc Fbck Th uthors i [] i thir coprhsiv ppr uifyig thir thory of fltss its ssocit yic fbck, forliz th cocpt tht two systs r quivlt if thr is ivrtibl trsfortio xchgig thir trjctoris, or if y vribl of o syst y b xprss s fuctio of th vribls of th othr syst of fiit ubr of thir ti rivtivs. I or grl ss this trsfortio is si to b Li- Bäcklu isoorphis. If two systs xɺ f ( x, u,( x, u X U R R r yɺ g( y, u,( y, v Y V R R vctor fils ( ( ( ( F( x, uu,, u... ( f ( x, u, uu,, u... ( ( ( ( G( y, v, v, v... ( g( y, v, v, v, v... whr, u α( x, z, w zɺ ( x, z, w, with, z Z R q r iffrtilly quivlt, it bcos possibl to go fro o to othr by yic fbck s show i Figur. Tht is by iffoorphis of th xt stt spc X Z. This yic fbck is ogous if th origil syst is iffrtilly quivlt to th clos loop syst. It is cll ogous bcus th w z vribls c b xprss s fuctios of th stt rivtivs of th iput. Thus fro th work i [9] it c b stt tht, if syst is iffrtilly flt, thr xists ogous yic fbck such tht th clos loop syst is iffoorphic to lir cotrollbl syst. Thrfor for olir syst qutio (, whr rk s (6 (7 (8 f (, (9 f (, ( u its yic fbck lirizbility s th xistc of: yic copstor; α ( q f ( ( y, yɺ, ɺɺ y,..., y, ( q ( α( y, yɺ, ɺɺ y,..., y, β( y, yɺ, ɺɺ y,..., y r iticlly stisfi [8]. q (5 whr zɺ ( x, z, v, z R u b( x, z, v, v R q

3 Autotio, Cotrol Itlligt Systs 4; (4: (,, b(,, iffoorphis; ( xɺ f ( xb, ( x, zv, zɺ β( x, zv, u α( x, z, v ( ξ Ξ( x, z,( ξ R q ( such tht th ( q isiol yics is giv by bcos costt lir cotrollbl syst ɺξ Fξ Gv (4 Figur. Trsfortio of Nolir Syst ito Lir Equivlt. Th copots of u xc b xprss s rllytic fuctios of th copot of qutio (, fiit ubr of thir rivtivs (qutios (, (4. Th yic fbck is si to b ogous if oly if th covrs hols, tht is, if oly if y copot of y c b xprss s rl-lytic fuctio of,x, u fiit ubr of its rivtivs. I fil rrk i [], th flt yics of syst whos output is giv by qutio ( is squr lft right iput-output ivrtibl syst, whr y copot of u or x y, by fiitio b rcovr fro y without itgrtig y iffrtil qutio: It is si to possss trivil zro-yics or trivil rsiul yics. Figur shows th ogous yic fbck liriztio procss cosistig of pol plct liriztio loops.. Grtig Flt Outputs Diffrtil fltss is i tht is turlly ssocit with urtri systs of iffrtil qutios whr syst of lgbric qutios i ukows [4] is writt s: Ax Bf, B, rk[ A, B]. (5 If Ais ivrtibl B is full rk, th x solutios y b writt i trs of fs x A Bf (6 s such k ll solutios prtrizbl i trs of f. I this sttig ogous trsfortio ϕ i which th origil vribls of th syst r trsfor without crtig w xogous vribls is rliz []... Clssicl Mthos Followig [4], cosir SISO syst giv by th trsfr fuctio ( y ( u( (7 ( th syst is cotrollbl if oly if th polyoils ( ( r copri, tht is thy hv o otrivil coo fctors. By Bzout s thor, thr xists polyoils ( b( such tht for ll s C w c writ ( ( b( ( (8. Dfi w vribl f ( u(, (9 ( y ( ( f (, u ( ( f ( ( ultiplyig both sis of (8 by f ( w hv, ( ( f ( b( ( f ( f ( or ( y( b( u( f ( ( which iplis w hv vribl f which is iffrtil fuctio of th syst iput output fiit ubr of thir ti rivtivs. Covrsly ll syst vribls iput r lso iffrtil fuctios of th w vribl. This w vribl qulifis s flt output. Thrfor giv y cotrollbl lir syst i trsfr fuctio

4 45 Ejik C. A Gsh K. Vygoorthy: Diffrtil Fltss Applictios to Iustril Mchi Cotrol for (7, th flt output c b chos s y costt ultipl of th vribl f ( u( or ( k f ( u( for y k, for xpl cosir th ( s, s lir, copri iiu phs fuctio y( u( fro (8, ( ( s b( ( s s( ( sb( b( s, ( b ( stisfis th qutio so tht fro (, f ( y( u(. f thrfor prtrizs ll syst vribls s giv. u( ( f ( ( s f ( sf ( f ( fɺ f siilrly y( ( f ( ( s f ( sf ( f ( fɺ f This trtt c b xt to th stt spc pproch [4]: For giv lir ti-ivrit SISO syst scrib by y( b s b s s s b u(, < ( Figur. Structur of Dyic Fbck Liriztio. with copri polyoils i urtor oitor its flt output κ f ( u(, which i s s trs of iffrtil qutio sclr output qutio givs: f f f u κ f f y b b b f κ ifx f, x fɺ (,., x f x x x th A bu, x x y c x x x x with A b κ c ( b b κ Th flt output of such syst is giv by f (,, ( C x whr C ( b, Ab,, A b is th Kl cotrollbility trix. Expl: Giv DC otor yics [4] LIɺ RI v k ω Jɺ ω Bω k I ( whr IArtur currt, wagulr Vlocity,, r lctricl costts,, r chicl costts. Th stt spc rprsttio is giv by R R k L L xɺ C L L x k L u xɺ k B x JL J J F ( C x JL ω C k k R JL JL L k L (4 Whr,,, r th cotrollbility trix, its ivrs flt output rspctivly. Th cotrol is coput usig s follows: x I JF BF, x ω F ( ɺ k JL ɺɺ LB RJ ɺ RB u v F F kf k k k.. Th Iplicit Rprsttio (Lévi s Mtho (5 Equtio ( c b loclly trsfor ito urtri iplicit syst for F( x, xɺ (6 x X, x, f ( x, u T X ( Tgt spc, for vry u rk f u x. This opts prolog ifol of solutios to th iplicit rprsttio. Th uthor i [] xts th otio of ogous trsfortio (Li- Bäcklu Isoorphis to th iplicit syst, sttig tht if two rgulr iplicit systs of qutio (6 r Li-

5 Autotio, Cotrol Itlligt Systs 4; (4: Bäcklu quivlt th thir lir cotgt pproxitio is loclly Li-Bäcklu quivlt. Th Iplicit syst qutio (6 is flt if oly if it is Li- Bäcklu quivlt. Th syst is flt if thr xists locl ppigs φ stisfyig φ ( y x such tht * f ; i,...,. Φ i * F F Φ f x ɺ x (7 * whr Φ f P( F, which r ctully polyoil trics th iffrtil oprtor is th itrit. Th ivrs of polyoil is ot polyoil th ivrs of squr trix is ot trix. Ths polyoil trics hv th followig chrctristics []:. Thy rquir th us of spcil lgbric ipultios.. P( F M its Sith copositio (or, igol ructio giv by VP F U (. θ (8 (,. A trix is M M is hypr-rgulr if p, q oly if it s Sith copositio ls to I, p, p, q p if p < q; to I P, if p q ; to I p if p > q p q, q 4. A squr trix M M is hypr-rgulr if p, q oly if it is uioulr- ot by u p subgroup of ivrtibl trics M. p, q 5. P(F is hypr-rgulr if oly if th lir cotgt pproxitio of th iplicit syst qutio (6 is cotrollbl iplyig tht th syst is flt. Ths r th copct st of trix ipultios tht l to th tritio of th syst s flt output. 4. Applictio to Sychroous Mchi 4.. Sychroous Mchi Ruc Orr Mol Fro th fourth orr ol of th sychroous chi, th irct xis c b ssu costt rucig it to thir orr o-xis ol [] giv by (9: ɺ ( x x i τ q f q whr i ɺ ω ω i ( ( r ( si R V ( r R ( x x( xq x ( xq x( q V cos q ( ( x ( si x V ( r R ( x x( xq x ( r R( q V cos 4.. Iplicit Mtho Usig Lévi s cssry sufficit coitios for iffrtil fltss [] whr for th syst of qutios (9 th syst orr th ubr of syst iput. Th otio of lir cotgt pproxitio hcforth cll cotgt pproxitio is fi thus. Giv trjctory t x(t of (6of clss C o itrvlj of R, th lir ti-vryig iplicit syst F F ( x( t, xɺ ( t ξ( t ( x( t, xɺ ( t ɺ ξ( t ( x xɺ with ξ ( ξ, ɺ ξ,... ΤΧ, is fi s th lir cotgt pproxitio of qutio (6 rou th trjctoryx. Th syst of qutios (9 is first trsfor to th iplicit quivlt, obti by liitig th yics tht cotis th syst iput f, kig F(, ω, q, ɺ, ɺ ω, ɺ q qul, Such tht H P D( ω ω i i q q ; wr ( ɺ ω ω ( Th cotgt pproxitio to th iplicit qutios ( ( is coput fro: F F F F F F P( F ɺ,, ω ωɺ q ɺ q ( It is otworthy ccorig to th chrctristics bov, tht th cotgt pproxitio of syst of qutios ( ( is hypr-rgulr if oly if it is cotrollbl. A if it is loclly flt roux, its lir cotgt pproxitio rou x is cotrollbl. Thrfor thr ust xist V L Sith ( P( F U R Sith ( P( F such tht H P D( ω ω i i w R q q (9 VP( F U ( I,, (4 Th cotgt pproxitio ftr pplyig qutio ( o qutio ( ( yils:

6 47 Ejik C. A Gsh K. Vygoorthy: Diffrtil Fltss Applictios to Iustril Mchi Cotrol whr: (5 ( ɺ ( R cos xqt si ɺ q( x cosɺ R cos ωv Ht ; ω H D ; P( F of rk rucs to lowr or uppr trigulr polyoil trix to prov its hypr-rgulrity. Th uioulr trics r costruct i such wy to shuffl th lts of th cotgt pproxitio trix chiv lowr trigulr for. Succssiv stps of th ructio r giv s follows []: Stp : Multiplyig qutio (5 with th uioulr trix- givs ω ( xqt x V( x si R cos RV qɺ q ; Ht ω D H t ( r R ( x x( xq x. Stp : Multiplyig qutio (6 with uioulr W ow pply th Sith copositio lgorith to trix- rucs row to [ ] qutio (5 i succssiv polyoil trix ipultios usig uioulr trics of rk util ω ω (7 ω D D D H H H (6 Stp : Multiplyig qutio (7 with uioulr trix - costt, yilig. shuffls row to k try [, ] i (7 ω D ω D ω D ω D (8 H H H H Stp 4: Multiplyig qutio (8 with uioulr trix-4 trigulr trix P (F. ω D H chivs th rquir lowr ω ω ω D ω (9 D D H D H H H Thrfor ( F D H P ω (4 Equtio (4 which is lowr trigulr polyoil trix provs th hypr-rgulrity of qutios (9. By right ultiplyig th uioulr trics to 4 us to grt P( F, th U trix is grt s giv i qutios 4 to 4: Stp b: Uioulr trix- by Uioulr trix-. Stp b: Equtio (4 by uioulr trix- (4 (4

7 Autotio, Cotrol Itlligt Systs 4; (4: Stp Equtio (4 by uioulr trix-4 ω D H ω D H Equtio (4 s U c b rrg copctly U A Thus fro Whr A ω H D ( (4. so givs th flt output y (49 y Vrifictio of th Flt Output of th Thir Orr Sigl- Iput (SMIBS ol is o by showig tht ll th syst stts vribls r fuctio of th flt output its rivtivs. such tht thus ω ɺ ω ɺ ω ɺɺ (5 (5 Rq (( x xqt V( x si R cos q P t t (5 ( t ( D R V ( R si x cos H ω ω ɺɺ qt ω Uɵ Usig th fiitio, ɵ U U I ω ( H D (44 (45 Equtio (5 is qurtic fuctio tht c b vlut for q. Sic th syst stts hv b show to b fuctios of th flt output its rivtivs, it follows tht ll othr syst vribls which r fuctios of th stts r lso fuctios of th flt output its rivtivs. Hc: ζ f, ɺ, ɺ ζ [ i, i, v, v, V ] (5 i i( i q qt t 4.. Copstor Dsig Siultio Rsults QU ɵ (46 it is possibl to coput by furthr trix ipultios Q L Sith( U ɵ which yils Q A (47 whr A is s fi bov. Multiplyig Qby th vctor T (, ω, q, th lst two tris i th rsultig vctor r: ω ω H D ( q which by (5 vishs iticlly ox. Th first try of th vctor is thrfor giv by: ( (, ω, q T y (48 Equtio (48 is trivilly strogly clos such tht It hs b show i th prcig sctio tht th copots of th syst stts othr syst vribls pig o th syst stts c b xprss s rllytic fuctios of th copot of fiit ubr of its rivtivs thus: x A(, ɺ, ɺɺ (54 Th yic fbck is show to b ogous sic th covrs hols, tht is, th flt outputy is xprss s rl-lytic fuctio of o of th stts of th syst. Thus th stt of th SMIBS is fuctio of th lirizig output its rivtivs up to orr α. Th ogous fbck syst to th followig clos loop syst is of orr α, so tht fro th lir syst ɺɺɺ v (55 th copstor follows. Cosirig th systs yicl qutios, prfor th followig stt trsfortios:

8 49 Ejik C. A Gsh K. Vygoorthy: Diffrtil Fltss Applictios to Iustril Mchi Cotrol zɺ z yɺ ɺ ω ω zɺ z ɺɺ y ɺɺ ɺ ω zɺ ɺɺɺ y ɺɺɺ ɺɺ ω v (56 This yils th quivlt orl for for th syst, fro which w c coput th olir cotrollr by ivrtig th xprssios fro ɺɺ ω. Th stt f trsfortios r ivrtibl xist throughout th trsit oprtig zo < < 8 o. Usig th twork prtrs of figur (, th rsultig xcittio cotrol is giv by []: Figur. Fult Loctio o th Sigl Mchi Ifiit Bus Syst (SMIBS f whr, τ Hv ω D ɺ A ɺ B -C q q (x -x i (57 E tω t Tɺ T qt A R -R V si -x V cos ; B x V si( ɺ -R V cos( ɺ ; qt C (x -x ɺ -x V cos( ɺ -R V si( ɺ ; qt T T E (x -x -x V si -R R V cos ; qt T q T Figur 4. Rsposs of Sp Dvitio to -Cycl Fult with without FVDFC. ɺ ((x -x x (x cos( R V si( ɺ q T R q; t T R ( r R ; x ( x x ; x ( x x. T qt q f is hrby prov lso to b fuctio of th flt vribl its rivtivs, tht is f β(, ɺ, ɺɺ (58 Th loop closur is th o to stbiliz th syst. * * * v k ( k ( ɺ ɺ k ( ɺɺ ɺ (59 choos k i ppropritly such tht th lir ti ivrit rror yics ( k k ɺ k ɺɺ (6 whr ( j ( j ( * ( j r stbl. Equtio (57 is th cotrol lw rfrr to s Fil Voltg Dyic Fbck Cotrollr (FVDFC, [] whil (59 is th lir iput tht stbilizs th syst to quilibriu. Siultio of th syst ws o by coctig th sychroous chi s sigl chi ifiit bus syst (SMIBS ur short circuit fult situtio s show i Figur. Figur 5. Rsposs of Tril Voltg to -Cycl Fult with without FVDFC. So siultio rsults with th syst quipp with th sig cotrollr r prst i Figurs 4 5 which r rprsttiv of th syst prforc. Ths figurs clrly show th rsposs of th cotrollr to thr-phs short circuit fult of -cycls urtio. Th syst ws rstor to sty stt oprtig poit s th cotrollr p th fult oscilltios ur thr scos s show by th chi sp vitio th corrspoig tril voltg. Th oscilltios i th ucotroll syst wr ot p withi th s ti urtio.

9 Autotio, Cotrol Itlligt Systs 4; (4: Figur 6. Block igr of INTECO TM glv ol 5. Applictio to Mgtic Lvittio Th ol vlopt of th gtic lvittio is bs o th syst vlop by INTECO TM for th purpos of tchig. Th syst block igr is show i Figur 6. INTECO us piricl lysis to ol th cotrol of th currt tht gos to th lctrogt. Th rsultig lir rltioship is fou to b stright li i ( u u b with zo. Th costts b r tri fro th xpritl t. Th syst yics r scrib i (6 (6. x ɺ x (6 x f _ p _ f p g x f _ p xɺ (6 xɺ ( k u c i x p p x p (6 Whr g is grvittiol forc, is ss of objct, f _ p, f _ p, p, p, ki, c r syst costts. Flt output Th flt output c b tri usig Lvi s tho by pplyig th iplicit fuctio thory liitig th yics with cotrol. Th vritiol qutio is giv by[]: Whr x ɺɺ f _ p f _ p x ( f p x x x f p x x (64 Th polyoil trix will thrfor b or copctly p( f x f _ p x x f _ p x x x x x (65 p ( f [ A b] (66 x f _ p Whr A x - polyoil x f _ p b x. Usig Sith s lgorith for th ipultio of polyoil trics, th followig right Sith stps r prfor. [ A b] A b b [ ], thrfor Uˆ A, such tht b Thrfor, Uˆ A b Q s rquir. (67 x Q x A (68 x b Such tht th first li rs y x which givs y x th flt output, whil th sco li is iticlly qul to zro fro (66 showig th fltss of th syst yics. This follows tht th flt output of this glv ol is th bll positio which is lso syst vribl.

10 5 Ejik C. A Gsh K. Vygoorthy: Diffrtil Fltss Applictios to Iustril Mchi Cotrol 5.. Copstor Dsig Siultio Rsults st poit of.6. Fro th coput flt output th cotrol lw follow fro th followig copstor y x yɺ xɺ x ɺɺ y ɺɺ x xɺ ɺɺɺ y ɺɺɺ x ɺɺ x u L (69 Fro (6, w hv Figur 7. Bll positio for t sco siultio x ( g xɺ f _ p f _ p x f _ P (7 A fro xɺ (69 th cotrol lw is coput s x ɺɺ u x c p p p whr M p x ( ( g xɺ ( ( g xɺ xɺ M p f _ p ki x f _ p f _ P f _ p A th lir cotrol is giv by (7 Figur 8. Appli cotrols for t sco siultio Figs. 9 show th rspos of th syst to crsig st poit lvls lik i scig stircs. This tsk ss to b chllgig cotrol tsk s c b s by th sloppy rspos of th PID cotrollr us o th s syst s s i fig. Th fltss bs cotrollr i ot show th s sloppy bhvior for th scig st poit lvls s s i fig 9. Th PID bhv lik it is hvig ifficulty copig with th shrp trsitios of th bll positio. Stuis of othr systs show tht th fltss cotrollr givs strog first swig cotrol s wll iprovs stbility rgi of th syst. * * * u k ( k ( ɺ ɺ k ( ɺɺ ɺ L (7 Th gis k i r chos such tht th lir ti ivrit rror yics ( k k ɺ k ɺ (7 whr ( j ( j ( * ( j r stbl. To coput th gis, (7 c b rwritt s Hurwitz polyoil by s k s k s k. (74 Th clos loop chrctristic polyoil of thir orr quivlt syst is giv i trs of th turl frqucy pig rtio by ( s such tht coprig (74 (75 givs ξω s ω( s β (75, k ξω β ω, k β k βω ξω Figurs 7 8 shows th bll positio th Fltss cotrol ppli to stbiliz it. Th rsults r for t sco siultio of th glv syst to lvitt bll to Figur 9. Rspos to iput [.5,.4,.,.,.] usig th Fltss bs cotrollr

11 Autotio, Cotrol Itlligt Systs 4; (4: xpcttios prfor wll wh copr with th PID schs. Rfrcs [] Fliss M., Lévi J., Mrti Ph., Roucho P. (999 A Li-Bäcklu pproch to quivlt fltss of olir systs, IEEE Trsctios o Autotic Cotrol, 8: [] Fliss M., Lévi J., Mrti Ph., Roucho P. (99 Fltss fct of olir systs: itrouctory thory xpls, It. J. of Cotrol, 6(6: 7-6. [] Lévi J. ( Rvis (6 O th cssry sufficit coitios for iffrtil fltss Elctroic Prit, Digitl Librry for Physics Astrooy, Hrvr- Sithsoi ctr for Astrophysics, (rxiv: th/6545v. [4] Hbrtt Sir-Rirz, Suil K. Agrwl (4 Diffrtilly flt systs, Mrcl Dkkr, Ic, Nw York Figur. Rspos to iput [.5,.4,.,.,.] usig th PID cotrollr 6. Coclusio This ppr prst so bsic thory of fltss-bs fbck liriztio, vrit of th wll-kow tchiqus of fbck liriztio. Thorticl forultio xpls to hc lrig of th cocpt of fltss how it is coput for crti iustril systs is giv. A ovl tho of coputtio of th flt output vlop by J Lvi is itrouc two iustril systs us to illustrt its fficcy. A pplictio to th sychroous chi gtic lvittio syst ws chiv by costructig cotrol lw rou th flt output. Th tho rquirs th thticl lysis of syst ols for fltss - coitio tht scribs how wll chrctriz th ol is with viw to triig its possssio of virtul (flt output riv by cotributios by th syst stt vribls. This output ws tri for th giv ols us to obti corrspoig fbck lws for th trsfor lir systs quipp with lir cotrollr us to stbiliz th systs to sty stt or p syst oscilltios iuc by fult. For th oxis sigl iput sychroous chi (SMIBS ol thr xists flt output th rotor gl (lt syst vribl whil for th gtic lvittio syst th flt output coput is th bll positio which is lso syst vribl. Th siultio rsults obti gr with th [5] Fliss M., Lévi J., Mrti Ph., Roucho P. (99 Fltss fct of olir systs: itrouctory thory xpls, It. J. of Cotrol, 6(6: 7-6. [6] Roucho P., Fliss M, Lévi J., Mrti Ph. (99 Fltss otio plig: th cr with -trilrs. I Proc. ECC 9, Groig, Pgs [7] Lévi J. (999 Ar thr w iustril prspctivs i th cotrol of chicl systs? I Pul M. Frk, itor, Avcs i Cotrol, pgs Sprigr-Vrlg, Loo. [8] Kiss B., Lévi J., Mullhupt Ph. ( Cotrol of ruc siz ol of us vy cr usig oly otor positio ssors. I A. Isiori, F. Lbhi-Lgrrigu, W. Rspok, itors, Nolir Cotrol i th Yr, volu, pgs. Sprigr. [9] Chrlt B., Lévi J., Mrio R. (99 Sufficit coitios for yic stt fbck liriztio, SIAM J. of Cotrol Optiiztio, 9(:8-57. [] Arso P.M Fou A.A (994 Powr syst cotrol stbility, IEEE sris o Powr Systs. [] E. C. A, J. T. Ag, U. O. Aliyu J. Lvi, (6 A w tchiqu for fbck liristio pplictio i powr syst stbilistio. IASTED Itrtiol Cofrc o POWER, ENERGY APPLICATIONS Gboro Botsw, Sptbr.-, Pgs [] Ejik A, Gsh K. Vygoorthy (, Sior Mbr, IEEE PSO tu fltss bs cotrol of gtic lvittio syst, 45 th IEEE Iustril Autotio Cotrol Aul Cofrc, r 7 th Octobr, Housto Txs.