A Design of an Improved Anti-Windup Control Using a PI Controller Based on a Pole Placement Method

Transcription

1 KYOHE SAKA e al: A DESGN OF AN MPROVED ANT-WNDUP CONTROL USNG A P CONTROLLER A Deign of an mrove Ani-Winu Conrol Uing a P Conroller Bae on a Pole Placemen Meho Kyohei Saai Grauae School of Science an Technology Meiji univeriy Kanagawa, Jaan ce439@meijiacj Yohihia hia School of Science an Technology Meiji univeriy Kanagawa, Jaan ihia@meijiacj Arac n hi uy, we ecrie an imrove ani-winu conrol uing a Proorional-negral (P) conroller ae on a ole lacemen meho An ani-winu conrol comrie a linear feeac conroller ha aifie he eire non-auraion ecificaion an an ani-winu comenaor ha oerae uring he auraion Thi conrol meho reven large overhoo caue y he winu henomenon However, ince he conrolle lan i no ye-, eay-ae error can occur an he inu o he inegral acion i limie To avoi hee rolem, we ranforme he conrolle yem ino a ye- lan Therefore, no eay-ae error occur when uing a ye- lan an he ouu converge o he arge value n aiion, he conroller gain i erive y he ole lacemen meho Then, i i oile o eign an exene conrolle yem oionally, o ha i i no neceary o reeign he conroller wheher he conrolle lan i a ye- or a ye- Keywor - ani-winu conrol; P conrol; negraor winu; ole lacemen NTRODUCTON n acual yem, we reven amage caue y exceive conrol inu y eing an uer an a lower limi However, if he conrol yem incororae an negraor uch a a P or PD conrol, he inu coninue o increae afer reaching he arge quaniy reuling in an exceive overhoo [] Thi i calle he inegral winu henomenon A numer of ani-winu echnique have een rooe o overcome he winu henomenon []-[4] An aniwinu conrol comrie a linear feeac conroller ha aifie he eire non-auraion ecificaion an an aniwinu comenaor ha oerae uring auraion The ani-winu conrol reven huge overhoo caue y he winu henomenon Thu, i only conier iuaion where he inu limi i urae an o he negraor wihou exceeing he limi an reven exceive overhoo However, if he inu ignal i a e an he conrolle lan i ye-, eay-ae error may occur, an he inu o he inegral acion i limie n hi uy, we rooe an imrove ani-winu conrol meho We ranform he conrolle yem ino an exene conrolle yem calle ye- lan By eigning he ae feeac acion o he conrolle inu, he exene conrolle yem i realize Therefore, even if he conrolle lan i ye-, he ouu of he ani-winu conrol yem follow he arge value wihou reuling in a eay-ae error Thi aer i organize a follow n Secion, we ecrie he influence of he ani-winu conrol uing a ye- lan n Secion 3, we ecrie a echnique o conver a ye- lan ino a ye- lan an erivaion of he conroller gain ae on he ole lacemen meho [5] n Secion 4, we confirm he effecivene of he rooe meho uing variou imulaion Finally, in Secion 5, we reen our concluion ANT-WNDUP COMPENSATOR A a ree winu counermeaure, a meho o o he inegral calculu funcion when an oeraional quaniy reache i uer or lower limi exi [6], [7] The yem i hown in Fig e r y z z ˆ( ) u ˆ( ) u u u Figure Ani-winu comenaor Here, he ea zone (from u o u ) i exree a a linear range of an acuaor an i ecrie y he following equaion [6], [7]: u u if u u, u u, u ˆ( ) ele if u u( ) u Thi yem ecome a P conroller when he inu i wihin he limi f an inu auraion occur, he following equaion i ue: u e z, z e ( u u ) Suiuing ino, we ge he following equaion: DO 53/JSSSTa734 SSN: x online, rin

2 KYOHE SAKA e al: A DESGN OF AN MPROVED ANT-WNDUP CONTROL USNG A P CONTROLLER z e ( e z u ) z u Alying he Lalace ranform, we oain u Z From he final value heorem, we oain u lim z lim Z( ) lim Thu, he inu o he negraor ecome zero Nex, we conier he final value of he negraor ouu, z ˆ( ) We aume ha he inu i a e ignal, r = R For a ye- lan, u ecome zero in eay ae Therefore, we oain he following equaion: (5) (6) u e z ˆ r y z ˆ (7) From he inernal moel rincile, he eay-ae error converge o zero Thu, r = y, an we have z ˆ( ) Meanwhile, for a ye- lan, we have wo coniion erive from he imulaion uy: r u, (8) r u (9) Cae : Afer alying he Lalace ranform, u can e exree a U R Y Zˆ u E E ( U ) u ( R Y ) ( U ) n he eay ae, we have y u Equaion can e wrien a follow: u U ( R U) ( U ) u R From an, we oain Z ˆ( ) U R Y U R U ( ) U( ) R( ) u n he eay ae, we oain he following equaion: u lim z ˆ( ) lim u (3) Cae : The ouu of he ea zone ecome zero, o U can e exree a U E E ( R U ) Equaion can e wrien a follow: U( ) R( ) U( ) R( ) U ( ) R (5) From an (5), we oain Z ˆ( ) ( ) U R ( ) R( ) R( ) ( ) ( )( ) ( ) R ( ) R ( ) (6) n he eay ae, we oain he following equaion: R R lim z ˆ( ) lim ( ) (7) From hee wo coniion, we can efine an uer limi for he ouu of he inegraor a zˆ( ) u (8) Meanwhile, a lower limi for he ouu of he inegraor can e ecrie a DO 53/JSSSTa734 SSN: x online, rin

3 KYOHE SAKA e al: A DESGN OF AN MPROVED ANT-WNDUP CONTROL USNG A P CONTROLLER u zˆ( ) (9) Therefore, he range of he inegraor ouu i ecrie y u zˆ( ) u Hence, for a ye- lan, he ouu of he inegraor i limie y he uer an lower limi of he ea zone Nex, we conier a cloe-loo yem given y Y G ˆ EZ G ( )( R ( ) Y ( ) Z ˆ( )) Y GY ˆ GR GZ, GZ ˆ Y( G) R( G ) R Thu, he cloe-loo yem can e ecrie a Y F R GZ ˆ G G ( ) R ( ) G ˆ Z G ( ) R ( ) Here, he ranfer funcion of a ye- econ-orer lan i G a a Therefore, he cloe-loo yem i a a Z F a a Zˆ( ) ( ) R aa ˆ( ) R Meanwhile, he ae-eay error i ecrie y lim e lim E lim( R Y) (5) lim ( F( )) R( ) R( lim F( )) (6) lim F( ) lim Zˆ( ) R a a ( R u) R( ) Thu, he ae-eay error i wrien a R u lim e ( ) r From he coniion in (ie, zˆ( ) ), we have (7) (8) lim F ( ) (9) Thu, he ae-eay error i given y lim e ( ) (3) Therefore, for a ye- lan, a eay-ae error can occur in he yem DESGN OF AN MPROVED ANT-WNDUP CONTROL A Deign of an exene conrolle yem Thi ecion eign an exene conrolle yem The econ-orer ranfer funcion G can e exree in he form of ae-ace equaion x Ax Bu, (3) a y Cx where x R, A R, B R, C R an ua ( ) are hown a follow: x x,,,, x A a a B C um if u um, ua u ele if um u um, um ele u um Here, he vecor in (33) are alie o (3) a a ae feeac, ie, a a A u (33) Coniering lim F( ) zˆ( ) u ), we have an he coniion in (i) (ie, DO 53/JSSSTa734 3 SSN: x online, rin

4 KYOHE SAKA e al: A DESGN OF AN MPROVED ANT-WNDUP CONTROL USNG A P CONTROLLER Exene Conrolle Plan G Ani-winu Comenaor Conrolle Plan G r e z z ˆ( ) u v u a B A C x y u u ˆ( ) u A u Figure Bloc iagram of he rooe meho Therefore, he maniulae variale ua ( ) i ecrie in (34) Here, v i a new maniulae variale uch ha ua Ax u v (34) Suiuing (34) ino (3), he following equaion i erive: x Ax BAu x v A BAu x Bv Ax B v, (35) y Cx Cx (36) Here, A R, B R an C R are hown a follow A,, B C Therefore, he exene conrolle yem i ecrie a G C A B (37) n hi uy, we eigne an ani-winu conroller uing he exene conrolle yem B Derivaion of gain an Thi ecion ecrie a erivaion of he gain an ae on he ole lacemen meho The conroller in non-aurainon ecome P conroller The conroller i ecrie a C (38) The cloe loo ranfer funcion in non-auraion i erive a (39) For imliciy, N( ) C( ) G ( ) W( ) D ( ) CG ( ) ( ) 3 (39) From (39), he cloe loo characeriic olynomial i exree y D 3 ( ) Here, he eire olynomial i given y a 3a 3 a a From an, he gain an are erive a 3 a, (4a) 3 a, a 3 (4c) Suiuing (4c) ino, he following equaion i erive: 3 (43) a The conroller gain, an he exene conrolle yem arameer, can e ecie uniquely y elecing he arameer a The rooe meho i hown in Fig V SMULATON STUDY n hi ecion, we confirm he effecivene of he rooe meho We imulae he conrol yem uing MATLAB/Simulin We aume ha he inu i a e DO 53/JSSSTa734 4 SSN: x online, rin

5 KYOHE SAKA e al: A DESGN OF AN MPROVED ANT-WNDUP CONTROL USNG A P CONTROLLER ignal The iniial value of he e ignal i r = An inu-ie iurance of magniue - i alie a = 5, an an ouu-ie iurance of magniue i alie a = Our rooe meho i comare o he convenional meho ecrie y Wahyui e al [6] A Tye- lan Thi ecion ecrie he imulaion reul when he lan i ye- Conier he econ-orer lan a G a, where, a = 375, an = 33 ) CASE : r u (44) The limi range of he ea zone i e o u For he ole lacemen meho, he eire olynomial i given y 3 Parameer ha we ue in he imulaion uy are hown in he Tale Fig 3 how he ouu reone of he rooe an convenional meho Fig 4 how he maniulae variale u a Thee reul how ha he ouu reone of he convenional an rooe meho oe no have a eay-ae error m TABLE PARAMETERS N TYPE- PLANT Variale Variale meaning Value u Sauraion range, Conroller gain 8, 45, Exene lan arameer 6, 45 A u Sae feeac vecor 8 Ouu Time[] Figure 3 Se reone of he rooe an convenional meho for Cae (Tye- lan) Targe value, rooe meho, convenional meho wih ani-winu conrol, an convenional meho wihou ani-winu conrol u m Maniulae Variale Time[] Figure 4 Maniulae variale of he rooe an convenional meho for Cae (Tye- lan) Prooe meho, convenional meho wih ani-winu conrol, an convenional meho wihou ani-winu conrol ) CASE : r u The limi range of he ea zone i e o 5 u 5 The eire olynomial i he ame a in ecion A ) The arameer,,, an A u are he ame a he Tale Fig 5 how he ouu reone of he rooe an convenional meho Fig 6 how he maniulae variale u a Thee reul how ha he ouu reone of he convenional an rooe meho oe no have a eay-ae error a in Secion A ) Ouu Time[] Figure 5 Se reone of he rooe an convenional meho for Cae (Tye- lan) Targe value, rooe meho, convenional meho wih ani-winu conrol, an convenional meho wihou ani-winu conrol DO 53/JSSSTa734 5 SSN: x online, rin

6 KYOHE SAKA e al: A DESGN OF AN MPROVED ANT-WNDUP CONTROL USNG A P CONTROLLER Maniulae Variale Ouu Time[] Figure 6 Maniulae variale of he rooe an convenional meho for Cae (Tye- lan) Prooe meho, convenional meho wih ani-winu conrol, an convenional meho wihou ani-winu conrol Time[] Figure 7 Se reone of he rooe an convenional meho for Cae (Tye- lan) Targe value, rooe meho, convenional meho wih ani-winu conrol, an convenional meho wihou ani-winu conrol B Tye- lan Thi ecion ecrie he imulaion reul when he lan i ye- Conier he econ-orer lan a G, a a where, a = 33, a = 375, an = 33 ) CASE : r u (45) Maniulae Variale 5 5 The limi range of he ea zone i e o u The eire olynomial i he ame a in Secion A Parameer ha we ue in he imulaion uy are hown in Tale Fig 7 how he ouu reone of he rooe an convenional meho Fig 8 how maniulae variale u a Thee reul how ha he ouu reone of he convenional meho ha a eay-ae error, wherea he ouu of he rooe meho oe no m TABLE PARAMETERS N TYPE- PLANT Variale Variale meaning Value u Sauraion range, Conroller gain 8, 45, Exene yem arameer 6, 45 A Sae feeac vecor 8 u u m Time[] Figure 8 Maniulae variale of he rooe an convenional meho for Cae (Tye- lan) Prooe meho, convenional meho wih ani-winu conrol, an convenional meho wihou ani-winu conrol ) CASE : r u The limi range of he ea zone i e o 5 u 5 The eire olynomial i he ame a in Secion A The arameer are he ame a in Tale Fig 9 how he ouu reone of he comarion eween he rooe meho an he convenional one Fig how he maniulae variale u a Thee reul how ha he ouu reone of he convenional an rooe meho oe no have a eay-ae error DO 53/JSSSTa734 6 SSN: x online, rin

7 KYOHE SAKA e al: A DESGN OF AN MPROVED ANT-WNDUP CONTROL USNG A P CONTROLLER Ouu Time[] Figure 9 Se reone of he rooe an convenional meho, for Cae (Tye- lan) Targe value, rooe meho, convenional meho wih ani-winu conrol, an convenional meho wihou ani-winu conrol Maniulae Variale Time[] Figure Maniulae variale of he rooe an convenional meho for Cae (Tye- lan) Prooe meho, convenional meho wih ani-winu conrol, an convenional meho wihou ani-winu conrol V CONCLUSONS n hi uy, we have rooe an imrove ani-winu conroller for a ye- lan We alo have analyze he coniion of he arge value larger han u (cae (i)) an he arge value maller han u (cae (ii)) n aiion, he effecivene of he rooe meho ha een confirme uing variou imulaion Our echnique reven eayae error, an i ouu ha no overhoo n aiion, he conroller gain i erive y ole lacemen meho Then, i i oile o eign an exene conrolle yem oionally, o ha i i no neceary o reeign he conroller wheher he conrolle lan i a ye- or a ye- n he cae of acuaor ha hyical limiaion, i i exece o how uerior erformance in inurial fiel REFERENCES [] H Ogawa, K Saai, K Maumoo, an Y hia, A imle aniwinu conrol ae on a P conrol wih an iniial value of he inegral ae variale, Proceeing of he n nernainal Conference on Comuer an Auomaion Engineering, Singaore, Singaore, vol, 49-5, [] N Ban, M Ono, K Saai, K Maumoo, H Shiaai, an Y hia, Aniwinu Conrol Scheme wih Dicree Moifie nenal Moel Conrol, Proceeing of nernaional Conference on Comuaional neligence an Vehicular SyemSeoul, Korea, 6-64, [3] HB Shin, an JG Par, Ani-Winu PD Conroller Wih negral Sae Preicor for Variale-See Moor Drive, EEE Tranacion on nurial Elecronic, vol 5, 59-56, [4] JW Choi, an SC Lee, Aniwinu Sraegy for P-Tyoe See Conroller, EEE Tranacion on nurial Elecronic, vol 56, 39-46, 9 [5] H Shiaai, R Yuof, T Fujio, an Y hia, Simle Moel Following Conrol Deign Meho for a Sale an an Unale Plan, Proceeing of he 4 nernaional Conference on Comuer, Communicaion, an Conrol Technology Langawi, Malayia, 86-9, 4 [6] Wahyui, T Faial, an A Alagul, Ani-Winu Scheme For Pracical Conrol of Poiioning yem, UM Engineering Journal, vol 5, -5, 4 [7] C Bohn, an DP Aheron, An Analyi Pacage Comaring PD Ani-Winu Sraegie, EEE Conrol Syem Magazin, vol 5, DO 53/JSSSTa734 7 SSN: x online, rin