Exact Linearization and Fuzzy Logic Applied to the Control of a Magnetic Levitation System

Transcription

1 WCCI IEEE World Cogress o Coputatioal Itelligee July, 8-3, - CCIB, Bareloa, Spai UZZ-IEEE Exat Liearizatio ad uzzy Logi Applied to the Cotrol o a Mageti Levitatio Syste Luiz H. S. orres, Carlos A. V. Vasoelos, Jr., Leizer Shita, ad J. A. M elippe de Souza Abstrat I reet years the area o oliear otrol systes has bee the subet o ay studies. Coputatioal developets have eabled ore oplex appliatios to provide solutios to oliear probles. he purpose o this paper is to use a obiatio o two tehiques to otrol a oliear syste: the Mageti Levitatio Syste. irst, the exat liearizatio tehique with state eedbak is applied to obtai a liear syste. Seod, the liearizatio is ade via diret aellatio o oliear utios, whih represet the pheoeologial odel o the syste. ially, to deal with the presee o uertaity i the syste odel, a adaptive otroller is used. he otroller is based o uzzy logi to estiate the utios that otai the oliearities o the syste. he uzzy syste is a zero-order akagi-sugeo-kag struture ad the adaptive otroller is ipleeted i a siulated eviroet (Matlab Siulik ). he ethodology guaratees the overgee o the estiates to their optial values, ad i tur the overall stability o the syste. he results show the otroller output sigal traks a reeree iput sigal. or uture work this adaptive otroller should be ipleeted i a real physial syste. I I. INODUCION N reet years the area o oliear otrol systes has bee the subet o ay studies. Coputatioal developets have eabled ore oplex appliatios to provide solutios to oliear probles. his paper shows a obiatio o a liearizatio tehique ad Artiiial Itelligee (AI) to otrol a Mageti Levitatio Syste (MLS). his syste was hose sie it has oliear dyais ad a didati kit o the physial syste is available to peror tests ad to otiue with uture work. he MLS used is auatured by ECP Eduatioal Cotrol Produts ( ad will be desribed i ore detail i setio II. he ai o this work is to otrol a ageti dis oveet over a glass olu as a result o the appliatio o a eletrial urret o a oil [],[]. Luiz H. S. orres is with the Cotrol Lab, Post-graduate Progra i Idustrial Egieerig, ederal Uiversity o Bahia, ua Aristides Novis, º, Esola Politéia, º adar,.-63, Salvador, Bahia, Brazil; (eail: luizhstorres@gail.o). Carlos A. V. Vasoelos, Jr. ad Leizer Shita are with Cotrol Lab, Post-graduate Progra i Mehatrois, ederal Uiversity o Bahia, ua Aristides Novis, º, Esola Politéia, º adar,.-63, Salvador, Bahia, Brazil (e-ails: arlosvasoelos@uba.br ad leizer@uba.br). J.A.M elippe de Souza is with the Eletroehaial Egieerig Departet, Uiversity o Beira Iterior, Covilhã, Portugal (phoe: ; e-ail: elippe@ubi.pt). he relatioship betwee the eletrial low ad the ageti dis oveet is give by a seod order oliear ordiary dieretial equatio o the type x & = φ( t, x) = Ax + B[ ( x) + G( x) u]. Several oliear otrol strategies a be used to otrol the dis positio, or exaple, uzzy, eural etwork, adaptive otrol, exat liearizatio [3],[]. I this paper both the exat liearizatio tehique ad uzzy will be used. Exat liearizatio eables a trasoratio ro a oliear syste to a liear syste through the additio o oliear opesators i the syste otrol loops [5],[6]. However, the exat liearizatio tehique with state eedbak requires a atheati odel that represets the dyais o the real plat well. urtherore, the uertaities i the pheoeologial odel aot guaratee good results a oit better results. o deal with soe uertaities i the syste odel, a adaptive otroller is used [7]. he otroller is based o uzzy logi x G x that otai the to estiate the utios ( ) ad ( ) oliearities o the syste. he uzzy syste is a zero-order akagi-sugeo-kag (SK) struture ad the adaptive otroller is ipleeted i a siulated eviroet [8],[9]. he ethodology adopted guaratees the overgee o the estiates to their optial values, ad i tur the overall stability o the syste []. II. HE MODEL A. Mageti Levitatio Syste I this paper the MLS ade by ECP was used ad is show i ig.. It oprises two ageti diss, a glass olu, two laser sesors ad two oils. he sesors are used to obtai the syste respose assoiated with the dis positios. he syste iput is give by the appliatio o a eletrial urret to the oils. he physial syste ouiates with a oputer via Digital Sigal Proessig (DSP) ad a blak box is resposible or the eletrial urret drivers ad the eergy supply. his MLS a be lassiied aordig to two odes, SISO (Sigle Iput Sigle Output) or MIMO (Multiple Iput Multiple Output) ad this depeds o the desired syste oiguratio. I the SISO ode oly oe dis is used whereas i the MIMO ode two diss are used. Here the //$6. IEEE 37

2 MLS was oigured to operate i the SISO ode. A. Paraeter Estiatio here are ive paraeters i (3): g,,, a ad b. he paraeters g = 9.8 [/s ], =. [Kg] ad =.5 [Ns/] (ro []). he paraeters a ad b are ostats related with ageti oil properties ad ust be estiated. I [3], the least square ad Mote Carlo ethods were used to estiate a ad b. Aordigly with [3], based o a ost utio oept, the Mote Carlo ethod preseted the best values or these paraeters. he values are a =.95 ad b = 6.8. hese values will be used i this paper. III. HE CONOL ECHNIQUE he MLS aual [] shows the atheati odel, based o the physial laws, that allows us obtaiig its dieretial equatio odel. he developet o the atheati odel is beyod the sope o this paper. hrough the balae o ores, the equatio is give by (see []): y+ y = where, y - ageti dis positio g y - irst derivative ageti dis positio y - seod derivative ageti dis positio - air visosity oeiiet - ageti dis ass - ageti ore applied to the ageti dis. he ageti ore a be writte i the ollowig way (see []) i = () a(y +b) where, i - eletrial urret applied o the oil a ad b - are ostats related with the oil properties. By substitutig () i (), a oliear relatioship betwee the ageti dis positio ad eletrial urret applied to the oil gives y = g ig.. Mageti Levitatio Syste ade by ECP. y+ a( y b) i () (3) A. Exat Liearizatio with State eedbak Exat liearizatio with state eedbak a be applied to a variety o oliear systes, iludig the MLS studied. he otrol shee uses the exat liearizatio tehique based o the aellatio o oliear utios. However, to eable the appliatio o the tehique, the syste dyai ust be represeted by (see []) X = ( X ) + G( X ) u where the utios (X) ad G(X) represet the oliearities o the states, u is the otrol syste iput ad X is the state vetor. urtherore, two oditios ust be satisied. he irst oe is that the syste ust be otrollable. or this irst oditio the atrix ored by vetorial ields i (5) ust otai order, where is the syste order (see [5]) [ ad G ad G... ad G] where ad G is the otatio o Lie braket. he seod oe is that the syste be ivolutive. It eas that the distributio expressed i (6) also be ivolutive (see [6]) D = spa ad { G ad G ad G} () (5)... (6) where D is the ivolutive distributio o G(X) expaded i aylor s series (represeted here by the otatio spa{.}) o a equilibriu state X. he order o D is give by. I order to the distributio i (6) to be ivolutive, it is eessary that the order o the expressio i (7) be equal to di(d) i (6) [ ad G,ad G]. (7) Oe the oditios are satisied it is possible to deterie a dieoorphis Z = (X). Ater this, the dyai o the syste give by () a be trasored ito the or (see [7]) 38

3 Z = AZ + Bβ ( Z)[ u α( Z)]. (8) A eedbak otrol sigal u or the oliear syste is hose i the or i (9) u = α ( Z) + β v (9) where α (Z ) ad β (Z) represet the states eedbaks. hus, the liear syste a be writte i the or i () Z = AZ + Bv where v is the iput sigal or the syste ater liearizatio. he deteriatio o v will be disussed i ext setio. B. Liearizatio o the MLS Made by ECP () he odel o the MLS ade by ECP was preseted i the (3) ad the two oditios or appliatio o the exat liearizatio were preseted i the last subsetio. he variables o states ad the eedbak otrol sigal u a be set as ollows i () u = i x = y x = y. () )( Z ) α = ( ga + az + b ) (6) β = a( Z + b. (7) ially, the eedbak otrol sigal u ould be rewritte by usig (6) ad (7) u = ( ga + az )( Z + b) + a( Z + b v. (8) ) he appliatio o the eedbak otrol sigal u over the syste give by () will ael the oliearities ad the syste will be trasored ito a liear syste give by (). I the sae way but by usig (5) x x Z = x = v. (9) A blok diagra was ipleeted i Matlab Siulik whih siulates the exat liearizatio tehique applied i the MLS, see. ig. below he dyai o the syste give by (3) a be rewritte i the or give i () x x x = + u. g x a( x + b) () ig.. Blok diagra ipleeted i Matlab/Siulik or the exat liearizatio i the MLS. he utios (X) ad G (X ) that otai the oliearities o the syste a be set as ollows x ( X ) = g x ( ). ( ) X = a x + b (3) G () he trasoratio Z = (X ) a be set i the or give by (see [5]) Z x Z = = ( X ) =. (5) Z x he utios α (Z ) ad β (Z) a be alulated i the or give by A. uzzy Estiators IV. HE UZZY SUCUE A zero-order SK syste with rules (ig. 3) is used i this paper ad has the ollowig or I x is A... ad x is x = x,..., x where [ ] A the y is B, () is the iput vetor, { } A,..., A / B are the uzzy set o iput ad output, respetively, assoiated =,..., ad y is the output o the uzzy with a th rule ( ) syste. he output axiu value ( ) y is the poit whih ( =) B y i the or θ = [ y,..., y ] is expressed by ( x) B is the µ ad θ is the paraeter vetor, so the output o the uzzy syste y = θ W () 39

4 where W ( x) [ W ( x),... W ( x) ] W or ( x) = = ad, ( xk ) ( x ) µ = () k= Ak ( k= µ k ) Ak =,..., ad W ( x) [,], where µ is the ebership utio Z = AZ ˆ ( )[ ( ˆ + Bβ Z θ β u α Z θ α )] + (6) + k C( Z Z ). A error o the estiatio is set as [ e e e ] e = Z Z =,,...,. (7) ig. 3. uzzy struture diagra, zero-order SK. he eedbak otrol sigal u expressed i (9) ould ot be ipleeted beause the utios α(z) ad β (Z ) are ukow ad eed to be estiated. However, [8] ad [9] used uzzy strutures to estiate soe utios. he ai idea here is to ostrut a uzzy struture able to geerate the estiates α( Z ˆ θα ) ad β ( Z ˆ θα ), where θˆ α ad θˆ β are paraeter vetors. So, a adaptive shee is used here to obtai these paraeters vetors ad (9) a be expressed i ters o uzzy strutures i the or below ig.. Blok diagra i Siulik α( Z ˆ θ ˆ α ) = θ α W (3) β ( Z ˆ θ ˆ β ) = θ β W. () B. Adaptive Cotrol Shee he adaptive otrol shee is based o state observers. However, the utios α (Z ) ad β (Z) are substituted by the uzzy estiates, respetively Z = AZ + k C ( Z Z + Bβ ( Z θˆ )[ u ( Z ˆ β α θ α )] + (5) ), k = k,..., k ad where k is a gai vetor i the or [ ] Z is state estiated. here are optial paraeters θ α ad β θ whih are able to estiate the utios α (Z) ad β (Z), there will also be estiates or optial states he adaptive laws i the or (see []) ˆ& θα = γ α e PBW (8) ˆ& θβ = γ β e PBW u, (9) where γ α ad γ β are positive ostats. As regards Lyapuov s equatio expressed i (3), to obtai the value o atrix P ig. 5. Syste respose with the adaptive proposed otroller ad step P + P = Q (3) 35

5 where = A + k C ad Q is deied a positive atrix. egardig stable, so there is a uique deied positive atrix P that satisy (3). ially, the hoie o a deied Lyapuov sei-egative or the syste, guaratees that the error o the estiate e is ollowed. he appliatio o the Barbalat s lea ause e whe t. ro (7), e iplies that Z Z ad the overgee o the estiated paraeters to their optial values is ahieved. V. SIMULAION ESULS he Matlab Siulik was used to siulate a proposed otroller or the MLS i the preset work. he blok diagra desiged i Siulik is show i igure. he odel paraeters used here were preseted i subsetio II.B. he siulatios were perored regardig r as a step reeree sigal, with values ragig ro a iiu o to a axiu o 3, respetively. I ig. 5, a siulatio o the otroller respose is show. his was obtaied by the obiatio o exat liearizatio with the uzzy estiates ad the step reeree sigal. CONCLUSION I this paper the obiatio o two tehiques to otrol a MLS were preseted: exat liearizatio with states eedbak ad uzzy logi. It was possible to veriy that the siulated results show a overshoot i the respose o the otroller. However, these sae results show that the otroller output sigal traks a reeree iput sigal. or uture work this adaptive otroller should be ipleeted i a real physial syste. ACKNOWLEDGMEN he authors wish to akowledge the support with ailities ad irastruture ro the Idustrial Autoatio ad ehology Ceter at the ederal Uiversity o Bahia. EEENCES [] Maual or Model 73: Mageti Levitatio Syste. Bell Cayo : ECP, 999. [] E.. Laithwaite, Eletroageti Levitatio, i Pro. Istitutio o Eletrial Egieers, 965, v.,., pp [3] E. B. Silva, Modelig ad Cotrollig a Mageti Levitatio Syste, B.S. thesis, Dept. Elet. Eg., UBA Uiv., Salvador Bahia, Brazil, 9. [] N. Bedrossia, Noliear Cotrol Usig Liearizig rasoratios, Ph.D. thesis, Dept. Meha.. Eg., MI, Massahusetts, 993. [5] H. Khalil, Noliear Cotrol, d ed. Mihiga: Pretie Hall,. [6] G. Guadarbassi, ad S. Savaresi, Approxiate liearizatio via eedbak A overview, Autoatia,, v. 37, pp. -5. [7] A. Isidori, Noliear Cotrol Systes, 3rd ed. Berli: Spriger- Verlag, 995. [8] L.-X. Wag, Stable adaptive uzzy otrol o oliear systes, IEEE ras. o uzzy Syste, vol, º, pp. 6-55, 993. [9] L.-X. Wag, Adaptive uzzy Systes ad Cotrol. Pretie Hall, 99, h. 8,9. [] L. Shita, Itelliget Cotrollers or Dyais Syste with State ad Iput Costraits ad Subeted to Model Uertaities, Ph.D. thesis, Dept. Elet. Eg., IA, São Paulo,. 35