Synthesis of Stable Takagi-Sugeno Fuzzy Systems
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1 Sytess of Stle Tk-Sueo Fuzzy Systems ENATA PYTELKOVÁ AND PET HUŠEK Deptmet of Cotol Eee Fculty of Electcl Eee, Czec Teccl Uvesty Teccká, 66 7 P 6 CZECH EPUBLIC Astct: - Te ppe dels t te polem of sytess of cotolle to te plt desced y Tk-Sueo fuzzy model t le put-output sumodels te coseuets of ules. A pocedue fo des of te cotolle utee stlty of closed-loop system sed o polyoml ppoc s peseted. Te cotolle sses te sme cctestc polyoml fo ll fesle le susystems. Te peseted metod c e used fo ot cotuous-tme d dscete-tme systems. Key-Wods: - fuzzy system, stlty, polyoml, Dopte euto, put-output model, LTV system Itoducto Tk-Sueo fuzzy systems t le sumodels coseuets of ules offe ood posslty to ppoxmte dymcl evou of ole systems. Oe of te dvtes of suc systems s tt cotolles fo Tk-Sueo fuzzy systems c e desed y ell-ko metods used teoy of le systems. Te polem s stlty lyss of suc closed-loop systems. Te m esults coce stlty lyss of suc systems ee ceved y W, Tk d Gff [8]. Ts ok coces Tk-Sueo fuzzy systems t le stte-spce sumodels. Tey poved tt te polem of stlty lyss of suc systems s euvlet to fd commo postve defte mtx to te udtc Lypuov euto d suested some pocedues o to fd t. Some esults ee pulsed lso [], [4] d [6]. It s so tt te polem of stlty lyss of Tk-Sueo fuzzy systems c e solved y test oust stlty of polyomls t polyomc stuctue of te coeffcets. Some esults coce te sytess of cotolles utee stlty of closed-loop system c e foud fo exmple [] fo Tk- Sueo fuzzy systems t stte-spce sumodels d [3] fo Tk-Sueo fuzzy systems t put-output sumodels. I ot cses te Tk- Sueo fuzzy model of plt d Tk-Sueo fuzzy cotolle e supposed. I ts ppe e metod fo des of le tme-vt cotolle to plt desced y Tk-Sueo fuzzy model t le put-output sumodels te coseuets of ules s peseted. Te metod s sed o polyoml ppoc. Te poposed cotolle sses te sme cctestc polyoml fo ll fesle le susystems d utees stlty of closed-loop system. Te peseted esults e vld ude ssumpto of sloly vy pmetes of te model o ets of ules. Te peseted metod c e used fo ot cotuous-tme d dscete-tme systems. Tk-Sueo fuzzy model of te plt Te Tk-Sueo fuzzy model of te plt s cosdeed, ee te ules e tte te follo fom: THEN y () t s M d d v () t : IF v THEN y ( () ) ( t () ) y t + u () t () t s M d d v () t : IF v s M () t y ( t ) + u ( t ) s M,,,, fo cotuous-tme cse d fo dscete-tme cse espectvely, ee v(t)[v (t),, v (t)] M y u () t () t () () vles of te pemse (some mesule plt vles) fuzzy sets te output of te plt te put to te plt
2 Te follo pocedue ll e deved fo dscete-tme cse, ut t c e folloed fo cotuous-tme system loclly. Te totl output of te fuzzy system s: y () t ee y ( t) y ( t) y( t ) ( ) + u t ( v () M v ( t) ( ) (3) (4) M s te de of memesp of v (t) M. It s ssumed tt v, fo,,, d ( v () > fo ( () ll t. Teefoe ( () v. (5) By pply Z-tsfom to (3) ude te ssumpto of sloly vy ets te follo tsfe fucto of te model of plt s oted: G p Y U G ( z) p ( z) ( z) G ( z) z z p (6) ee G p () z deotes te tsfe fucto of te coseuet of te -t ule. Te m de of ts ppe cossts us polyoml metods fo des of cotolle to te Tk-Sueo fuzzy system desced ove c utees stlty of closed-loop system. I ode to ssue stlty of suc systems, metods dested fo le tme-vt (LTI) systems e ofte used ([3],[6]). Hoeve, ts s ot possle eelly. It s to e empszed tt stlty of le tme-vt (LTV) systems us stlty lyss of coespod foze-tme systems c e uteed oly ude ssumpto of sloly vy pmetes (see [7]) c s ofte omtted. Te uesto o te te of te c s lloed ems useed. Neveteless, te expemets evel tt most cses te desed cotolle stlzes te plt (). Ou m s to fd cotolle tt ould ss te sme cctestc polyoml to ll LTI susystems (fo ll dmssle vtos of ets ). Flev [3] poved tt f (z)(z) fo ll,..., ee (z) s Scu polyoml te stlz cotolle c e lys foud s fuzzy pllel dstuted compesto (PDC). It mes tt te cotolle c e desced y Tk- Sueo fuzzy model t put-output sumodels te coseuets of ules s te sme tecedets t te model of te plt. It c e esly so tt ts s ot possle fo te umetos (z) e mutully dffeet eve y us fuzzy o-pdc. Te follo secto dels t te sytess of LTV cotolle stlz plt () closed loop. Let us oce moe emd tt te esults e vld oly ude ssumpto of sloly vy ets,,...,. 3 Des of te cotolle Te polem of fd stlz cotolle to plt desced y te Tk-Sueo fuzzy model s solved ss te sme cctestc polyoml closed-loop system fo ll fesle LTI susystems. Assume tt te plt s desced y Tk- Sueo fuzzy model () d te ets,,..., e sloly vy. Let te cotolle e LTV system expessed y ts tsfe fucto ee G c (7)
3 de ( z de ( z. (8) Te coeffcets, e some fuctos of te vecto of ets of ules. Ou tsk s to fd tese fuctos so tt te closed-loop system s stle. As t s metoed ove ou ol s to ss te sme cctestc polyoml of te closedloop system fo ll H,.e.: ( z ) + ( z), (9) ee (z) s Scu polyoml of ppopte deee. Te ell-ko codto of solvlty of (9) s tt te etest commo dvso of polyomls (z, d (z, dvdes te polyoml (z) fo ll H,.e. ( z ), ) ( z),. () I ode to ceck ete te codto () s stsfed oe c use te ell-ko lotm fo fd te etest commo dvso of to polyomls [5] t espect to te vecto vle. Hoeve te ette soluto cossts solv te Dopte euto (9). If te soluto does ot exst fo some H t mes tt fo tose H te codto () s volted. Te etest commo dvso c e te esly detemed susttut te vecto (6) d us te stdd lotm metoed ove. If (,) s Scu polyoml te e polyoml (z) s to e cose to e le to solve euto (9) (fo exmple (z) (z)* lys leds to solvle euto (9)). If (,) s ot Scu polyoml te cotolle (7) stlz te plt () y ss te sme cctestc polyoml to ll le susystems c ot e foud. As te closed-loop system t lest oe step dely s to e uteed, de (z,<de (z,. Te deees of polyomls (z,, (z, d (z) c e foud comp deees of polyomls of ot sdes of euto (9): de > de de < de de de de. () Susttut (6) d (8) to (9) te follo euto s oted: de de z z + z z () (). z All solutos of () c e expessed s + f ( z) f ( z) (3) ee (z, d (z, e te ptcul solutos of () d f(z) s ty polyoml. Te detemto of polyomls (z, d (z, s euvlet to solv te system of (de +) le eutos s fom () y comp of ll coeffcets z,,...,de. As te solutos, of ts system e fuctos of te eel soluto s eltvely complcted. To ette udestd te desced pocedue ts pcple ll e so o to llusttve exmples. 3. Exmple Fo ske of smplcty ssume plt desced y te follo to-ule Tk-Sueo fuzzy model: IF y(t-) s M THEN y(t)-.y(t-)+u(t) IF y(t-) s M THEN y(t)-.3y(t-)+u(t)+.4u(t-) M M -c c F. Memesp fuctos Te tsfe fucto of te model s: G p ee () z + () z () z + () z () z + ( ) () z () z + ( ) () z
4 () z z, () z z + () z z +., () z [,]..4 z +.3 Fom () te deees of polyomls (z), (z, d (z, c e detemed s follos: de, de, de. Coose te lol cctestc polyoml (z) s () ( z +.) z. By solv te Dopte euto (9) te fom [ ( z +.) + ( )( z +.3) ][ ( ) z + ( )] + [ z + ( )( z +.4) ] ( ) ( z +.) oe c ot ( ) ( ) ( ). It s ot dffcult to see tt fo.5 tee s o soluto of te Dopte euto. elly, f.5 te te etest commo dvso (.5 ),.5) ) ( z +., z +.) z +. does ot dvde te polyoml (z) d te ecessy codto of solvlty of () s volted. Coose te polyoml (z) s () ( z +.)( z +.) z. Oe ptcul soluto of (9) c e foud s ( ) ( ).4. ( ). +. tt coespods to te LTV cotolle t tsfe fucto + G c ). +. z +.4. o euvletly te tme-dom ( k) (..4 ) u( k ) + (.. ) e( k ) u. I F. d F. 3 te esposes o tl codto of te closed-loop system e so fo c. d c espectvely. It s evdet tt te cotolle stlzes te plt fo ty te of c of..8.6 y(t) c t F. espose of closed-loop system fo c. y(t) c t F. 3 espose of closed-loop system fo c 3. Exmple Assume plt desced y te follo to-ule Tk-Sueo fuzzy model composed fom ustle sumodels:
5 IF y(k-) s THEN y(k)-y(k-)+u(k) IF y(k-) s THEN y(k)-3y(k-)+u(k)+u(k-) F M M \W -c c F. 4 Memesp fuctos Coose te lol cctestc polyoml (z) s () ( z +.5) z. Te Dopte euto (9) tus to [ ( z + ) + ( )( z + 3) ][ ( ) z + ( )] + [ z + ( )( z + ) ] ( ) ( z +.5) fom c oe soluto c e oted s + W F. 5 espose of closed-loop system fo c [ F ( ) ( ) ( ) \W Te tsfe fucto of te cotolle s G c ) z o euvletly te tme-dom u () k (.5.5.5) u( k ) + + ( ) e( k ). I F. 5 d F. 6 te esposes o tl codto of te closed-loop system e so fo c d c espectvely. F. 6 espose of closed-loop system fo c Te fues so tt te cotolle stlzes te closed loop oly fo sloly vy et of ule (c). If te te of vce s e (c) te te poposed metod does ot utee te stlty. 4 Cocluso Te e metod fo fd le tme-vt cotolle to plt desced y Tk-Sueo fuzzy model t le put-output sumodels te coseuets of ules utee stlty of te closed-loop system s toduced. Te metod s sed o ss te sme cctestc polyoml fo ll fesle susystems. Te poposed metod utees te stlty oly fo sloly vy vecto of ets of ules. W 5 Ackoledemet Ts ok s ee suppoted y te Msty of Educto of te Czec epulc ude Poect
6 VS97/34 d y te esec pom No. J4/98:33 Decso Mk d Cotol fo Mufctu of te Czec Teccl Uvesty Pue (sposoed y te Msty of Educto of te Czec epulc). efeeces: [] 'YNY 5 DQG +XãHN 3 6WDELOLW\ I Tk-Sueo Fuzzy Systems t Le Iput-Output Sumodels, Poceeds of IEEE MED,, Pts, Geece [] Fe G., Co S. G., ees N. W. d Ck C. K., Des of Fuzzy Cotol Systems t Guteed Stlty, Fuzzy Sets d Systems, Vol. 85, 997, pp. - [3] Flev D., Polyoml Appoc to te Sytess of Stle Fuzzy Systems, Poceeds of NAFIPS 96, 996, pp. -4. [4] +XãHN 3 DQG 'YNY 5 6WDELOLW\ I Tk-Sueo Fuzzy Systems t Le Iput-Output Sumodels: Polyoml Appoc, Poceeds of IFAC CAO,, St. Petesu, uss [5].XþHUD 9 Alyss d Des of Dscete Le Cotol Systems, Petce Hll, 99 [6] Lo J.-C. d Ce Y.-M., Stlty Issues o Tk-Sueo Fuzzy Model - Pmetc Appoc, IEEE Ts. o Fuzzy Systems, Vol. 7, No. 5, 999, pp [7] Skoo. A. d Lu C. G. Y., Istlty of Sloly Vy Systems, IEEE Ts. o Automtc Cotol, Vol. 7, No., 97, pp [8] W H. O., Tk K., d Gff M. F. A Appoc to Fuzzy Cotol of Nole Systems: Stlty d Des Issues, IEEE Tsctos o Fuzzy Systems, Vol. 4, No., 996, pp. 4-3.
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