Iterative Learning Control with Switching Gain PD Feedback for Nonlinear Systems

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1 TIC-STH 9 Ieaive Leaig Coo wih Swichig Gai PD Feedbac o Noiea Sysems P.R. Ouyag.A. Pez F.F. Xi Depame o Aeospace Egieeig Ryeso Uivesiy Tooo ON Caada Absac I his pape we popose a ew ieaive eaig coo caed Swichig Gai PD-PD (SPD-PD) Type Ieaive Leaig Coo o ajecoy acig o ime vayig oiea sysems wih uceaiy ad disubace. I he deveoped coo scheme a PD eedbac coo wih swichig gais o ieaio domais ad a PD ype ieaive eaig coo based o pevious ieaios combies acig eos io he updaed aw. I is pove ha he boudedess o he ia acig eo is guaaeed i he pesece o uceaiy disubace ad iiiaizaio eo. The covegece speed is adjusabe by he adopio o he swichig gais i he ieaio domai. I is show ha a as covegece ad a sma acig eo boud ca be obseved by usig he SPD-PD ype ieaive eaig coo. Keywods- Ieaive eaig coo PD eedbac coo swichig gai ajecoy acig covegece oiea sysem. I. INTRODUCTION Ieaive Leaig Coo (ILC) has daw iceasig aeio because o is simpe coo sucue ad good acig peomace based o he sysem s epeiive opeaios []. The basic picipe behid he ILC is o use iomaio obaied om pevious eecuios o he same as o om he coo acio i ode o impove he peomace om ia o ia. A adiioa cass o ILC agoihms is he eed owad coo sysem ha is heoeicay capabe o educig he acig eo o zeo as he umbe o ieaios iceases owads iiiy. Accodig o he eaig acio ype ILC ca be cassiied as P-ype D- ype I-ype PI-ype PD-ype ad PID-ype. Some deaied suveys o ILC ca be oud i [-4]. A PD-ype aveaged ieaive eaig coo is poposed o iea sysems i [5]. Some eseaches used he cue ieaio acig eo o om he coo siga ha ca be viewed as o-ie ieaio eaig coo [6-]. A adapive P-ype high-gai eaig coo sysem based o he use o he cue ia eedbac was poposed [6]. A simia coo agoihm was used o discee ime sysems [7]. A PID eedbac coo ad high ode PID-ype ILC o he ajecoy acig coo o obos was poposed i [8] whee posiio veociy ad acceeaio iomaio ae eeded o he eaig pocess. Some P-ype ad PD-ype oie ILCs wee deveoped i [9]. As demosaed i ha pape he ieaive eaig coo wih cue cyce siga ca achieve as covegece aes by seecig a high eedbac coo gai. Simia cocusios wee obaied i [ ]. A PD ype eaig coo usig cue cyce sigas was poposed o geea oiea sysems wih uceaiy ad disubace []. O he ohe had eedbac coo is widey used i idusia appicaios such as obo sysems ad pocess coo sysems o achieve good peomace bu cao achieve eacy he desied acig peomace because a o-zeo eo is equied o acivae he eedbac coo. Theeoe he PD/PID coo aoe is o adequae o ajecoy acig especiay whe he sysem has oieaiies ad uceaiies. The combiaio o eedbac coo ad ieaive eaig coo is a pomisig echique o achieve good acig peomace ad speed up he covegece pocess. A P-ype eedbac pus a P-ype ILC coo agoihm (P-P ype) o discee oiea ime-vayig sysems was poposed i [3]. A P-D eaig coo was poposed i [4] o discee iea ime-ivaia sysems. High ode D-D eaig coos wee deveoped i [ 5]. A P-P ype agoihm was deveoped o iemaic pahacig o mobie obos i [6]. I his pape a swichig gai PD ype eedbac coo pus a PD ype ieaive eaig coo (SPD-PD) is poposed o dea wih he uceaiy ad disubace ad o impove he acig eo om cyce o cyce. II. PROLEM STATEMENT I his pape we coside a oiea ime-vayig sysem wih o-epeiive uceaiy ad disubace as oows: ì ( ) = ( ( ) ) + ( ) u( ) + h( ) () î y() = C() () + () whee deoes he ieaive ide. ÎÂ u ÎÂ ad m y ÎÂ ae he sae coo ipu ad oupu o he sysem especivey. The ucio h () ÎÂ epeses he m uceaiy o he sysem ad ucio () ÎÂ is he disubace. Fom he above assumpio oe ca see ha ( ( ) ) ÎÂ () ÎÂ ad C () ÎÂ. I his pape he oowig oaioa coveio is adoped: æ ö = ma i M( ) = ma ç S mi j i i mçèj = ø - h () = sup e h () > Î[ T] /9/$6. 9 IEEE 875

2 whee [ T = ] is a veco M = ém i j ù êë úû ÎÂ is a mai ad h ()( Î [ T]) is a ea ucio whee T is he ime duaio. To esic he discussio he oowig assumpios ae made o he sysem. A) The desied ajecoy yd () is a is-ode coiuiy o Î [ T]. A) The coo ipu mai C () is a is-ode coiuiy o Î [ T]. A3) The ucio ( ( ) ) is gobay uiomy Lipschiz i o Î [ T]. Tha meas ( + ( ) ) - ( ( ) ) c + () -() whee is he ieaio umbe ad c > is a cosa. A4) Uceaiy ad disubace ems h () ad () ae bouded as oows: " Î [ T] ad " h () b h i () b ad i () b. To coo he oiea sysem saed i Eq. () we popose he oowig SPD-PD ype eaig coo aw u+ () = u() + Kp ( + ) e+ () + Kd( + ) e + () () + K () e () + K () e () p d wih eedbac coo gais give o be as: ì Kp ( ) = s( ) Kp ( ) î Kd( ) = s( ) Kd( ) Whee e+ () = yd() - y+ () e() = yd() - y() e+ () = yd() -y+ () ad e() = yd() -y() ae posiio eos ad veociy eos o he oupu veco o he +h ieaio ad h ieaio especivey. K pi ÎÂ ad K di ÎÂ ae he popoioa ad deivaive gai maices especivey. Eq. (3) epeses he swichig gais o he eedbac coo whee s> ( ) is a moooicay iceasig ucio o he ieaio umbe. I ca be see ha he poposed SPD-PD coo aw cosiss o wo coo oops. The is oop is a PD eedbac cooe wih swichig gais i he ieaio domai ad he secod oop is a sadad PD ype ILC. Theeoe SPD-PD ype ieaive eaig coo is eecivey a hybid coo i ha aims o use he advaages oeed by boh eedbac coo ad ILC. The ey pupose o ioducig swichig gais i he eedbac oop is o epedie he covegece o he ieaio ad avoid vibaio o divig acuaos. Aso oe ca see ha he poposed coo agoihm is a eesio o he agoihm deveoped i []. We assume ha o each ieaio he epeaabiiy o he iiia sae seig is saisied wihi a admissibe deviaio eve: (3) ()- () e o =... (4) whee e is a sma posiive cosa ha epeses he accepabe accuacy o he desiged sae veco ad () epeses he desied iiia sae vaue. Fo bieess o discussio he oowig oaios ae used i he oowig secios: ma K () C( ) ma K C( ) d d d d [ T] [ T] pd p d [ T] ma K CK C ma K CK C pd p d [ T] ma K ma K Kp p Kp p [ T] [ T] ma K ma K K Kd d Kd d [ T] [ T] ma ( ) [ T] = + c d pd -c C ma C( ) ma s( ) K [ T] = + c d pd -c - ( ) = ma I + Kd() C = ma I m -K d C Î[ T] Î[ T] = b = + K - K III. s MAIN RESULTS AND CONVERGENCE ANALYSIS Theoem: Fo he oiea ime-vayig sysem () i he SPD-PD ype ieaive eaig coo aw () is used ad he swichig gai agoihm (3) is adoped he he ia sae ad oupu acig eo is bouded give by ì æ F ö im d () + Tb + e h -c çè -b ø æ F ö C im () e h ç -c -b î çè ø e + Tb + + b ( ) ( ) s. (5) F= K + K + + b + + b s s d d h s Kp Kp Wih + + b + K + K e ( ) ( ) s Kd Kd s Povided he coo gai K () d ad he eaig gai K () d ae seeced such ha I () ( ) ( ) m + Kd C is osigua ad - ì ma ( I + K ) d() ( ) C( ) < Î[ T] ma I - Kd ( ) ( ) C( ) < î Î[ T]. (6) 876

3 Aso we popose he oowig iiia sae eaig agoihm: - + () = ( + ( ) Kd( ) C( )) { () + ( ) Kd( ) yd() + ( ) K ( )( y ()- y ())} (7) d d I his pape he om is used o eamie he covegece o he acig eo o he poposed coo agoihm. Fis o a a eaio bewee om ad om is epeseed by Lemma. Lemma : Suppose ha () = [ () () ()] T is deied i Î [ T]. The æ ö - ç ( ) d e ( ) çèò (8) ø Fis we deie he oowig hee vaiabes. d d - e yd - y du ud -u ad. d Fom () we ca cacuae he acig eos as: e C (9) e C The deivaive o he acig eos ca be epeseed as e C C () e C C Fom () we have: du+ = du- s( + ) Kp ( ) e+ ( ) - s( + ) Kd( ) e + ( ) () -K e () -K e () () () p d Submiig (9) ad () io () ges: { } { } du = du -K Cd - - K Cd + Cd - + p d ( ) ( ){ d } - s+ K C - p + + ( ) d( ){ d d } - s+ K C + C - () Aso om Eq. () we ca ge he oowig: u (3) u Submiig () io (3) ad eogaizig ges: I + s( + ) Kd( ) C du+ = ( I-KdC) du - s( + ) ( Kp ( ) C+ Kd( ) C ) d+ -( KpC+ KdC) d - s( + ) Kd( )( Cd+ -Ch+ - + ) -Kd( Cd -Ch - ) + s( + ) Kp ( ) + + Kp (4) ( ) Fom A3) we have d c d ad s o choose a pope d+ c d+. As coo gai K p ( ) ad om (6) we ca esue d d I s K C I K C (5) Theeoe Eq. (4) ca be ewie i he λ-om ad simpiied as: { ( ) ( pd d c ) d ( s d d ) b h ( s Kp Kp) b ( s Kd Kd) b } du du + + c d + s pd d Fo he h ieaio he sae veco ca be wie as () = () + ( ( ( ) ) + ( ) u ( ) ) d+ h () d Fom () we aso have: (6) ò ò (7) d() = d() + ò ( ( d( ) ) + ( ) ud( ) ) d (8) Fom (7) ad (8) we ge: ò d d ò d = ( ( ( ) ) - ( ( ) ) ) d + ( () du- h ) d+ d( ) Appyig A3) o (9) we ge d c d d+ ( () du - h ) d+ d ( ) (9) ò ò () Eq. () ca be wie i he om om as: d c d d+ ( () du + h ) d+ d( ) ò ò () Accodig o he deiiio o λ-om o λ>c appyig Lemma o () obais: d T d + + u b h -c -c -c Fo he + h ieaio we ca ge simia esu: T d du + b h -c -c -c e e () (3) Submiig (~3) io (6) ad simpiyig i ges: ( - s K) du+ ( + K) du + F (4) Eq. (4) ca be simpiied as du + b d + F (5) u Fom (6) we have. I we choose 3 877

4 ( ( + ) + + ) c spd pd sd d > + c he we - ca guaaee. Fiay om (5) we ca ge: im du () F = (6) - b Submiig (6) io () we ca ge: æ F ö im d() = + Tbh + e -c ç b è - ø Fiay om Eq. (9) we ca ge: æ C F ö im e() = + Tbh + e + b -c ç b è - ø (7) (8) Rema : I he iiia sae updaig aw (7) is used we wi esue im () = (). I his mae we ca ge im e =. Theeoe æ C F ö im e () = Tb h b c + + b (9) - çè - ø Rema : I hee is o uceaiy ad disubace i () he he ia acig eo boud becomes: im e () e C = (3) - c Rema 3: I he iiia sae updaig aw (7) is appied ad hee is o uceaiy ad disubace he he ia acig eo wi be im e ( ) =. Such a cocusio ca be deived diecy om Rema ad Rema. IV. EXPERIMENTAL VERIFICATION I his secio we appy he poposed SPD-PD ieaive eaig coo agoihm o a oiea sysem o impove is acig peomace. The simuaio eampe i [] is adoped o he pupose o compaiso. The oiea sysem wih uceaiy ad disubace is descibed by é () ù ési( () ) + si( () ) ùé() ù é ùéu() ù = +ê ê() ú 5 3 ê() ú ê úêu() ú ë û êë úûë û ë ûë û é cos( p ) ù + (.5+ a ) êcos( 4p ) ú ë û éy () ù é4 ùé () ù é si( p ) ù (.5 a ) êy() ú = + + ê úê() ú êsi( 4p ) ú ë û ë ûë û ë û Wih =5Hz. The desied acig ajecoies ae se as y d() = yd() = ( -) o Î[ ] To eam he obusess o he poposed eaig coo agoihm sevea simuaio sudies ae coduced usig diee coo ad eaig gais. Fo a he cases he iiia saes ae se as ()=.3 ad ()=-.3. Aso we assume he mai i he iiia sae eaig schedue (7) is o accuae (The esimaed vaue is.4). I he oowig secios he cassic PD ILC is obaied by seig K p =K d =. A. Eampe : Repeiive Uceaiy ad Disubace I he is simuaio sudy we se α = which meas ha he uceaiy ad disubace is epeiive om ieaio o ieaio. The oowig coo gais ad eaig gais ae chose: Fo cassic ILC: Kp = diag{} Kd = diag{.5.5} Fo SPD-PD: K p( ) = diag{.5.5} Kd( ) = diag{5} K =.6* diag{} K =.6* diag{.5.5} s( ) =. p d Reeig o () we ca obai ha C = diag{4 } o he simuaio eampe. Theeoe he peec ILC eaig gai is Kd = diag{.5.5} ha is used i he cassic ILC. Accodig o he chose gais om (6) we have = o he cassic ILC ad =. ad =.6 o he SPD-PD. Tha meas iaccuae owedge o maices ad C ae cosideed i he eaig coo gai desig o SPD-PD. Figue shows he maimum acig eo boud o hee cases om ieaio o ieaio. Fom his igue we ca see ha he SPD-PD eaig coo agoihm ca obai a vey as covege ae (7 ieaios) ad vey sma ad moooic deceased acig eos. u o he cassic ILC case ahough he bes eaig gai is used (as = ) he acig eos wee i he good-bad-good mode beoe eachig sabe boudedess ad moe ieaios (7 ieaios) wee eeded o obai a eaivey accepabe acig peomace. We ca see ha he acig eos ae si eaivey age compaed wih he SPD-PD. Simia esus wee show i [] whee moe ha ieaios ae eeded o achieve sabe acig peomace. I demosaed ha he swichig gai eedbac coo is moe useu i ems o educig acig eo ad speed up he covegece. Maimum absoue eos e: Cassic ILC e: SPD-PD e: Cassic ILC e: SPD-PD 5 5 Ieaio umbe Figue. Maimum boud eos o eampe

5 Maimum absoue eos e: Cassic ILC e: SPD-PD e: Cassic ILC e: SPD-PD 5 5 Ieaio umbe Figue. Maimum boud eos o eampe.. Eampe : Vayig Uceaiy ad Disubace I his eampe we coside a siuaio whee he uceaiy ad disubace is vaied om ieaio o ieaio by se a =.5. A he coo gais ad eaig gais ae se he same as i eampe. This eampe epeses a moe geea siuaio o ea appicaios. Figue shows he simuaio esus. Fom Fig. we ca see ha a good acig peomace ca be achieved usig he SPD-PD ype eaig coo agoihm eve i he codiios o vayig uceaiy ad disubace om ieaio o ieaio. Tha is aibued o he eedbac coo which ca compesae he disubace. u o he adiioa ILC agoihm hee ae age sabe acig eos. Tha is because he cassic ILC cao compesae he cue ieaio disubace because o he imiaio o he oie eaig saegy. C. Compaiso o Diee Feedbac Coo Gais To adjus he ia acig eo boud by usig diee coo gais diee eedbac coo gais ae used o he SPD-PD ype eaig coo agoihm. I his eampe he eaig coo gais ae se he same as he pevious wo eampes. The oowig eedbac coo gais ae used i he simuaio epeimes. Fo a hee diee gai cases s( ) =. The midde gais ae chose he same as i Eampe. The ow gais ae ha o he midde gais ad he high gais ae oe ad ha imes midde oes. Tabe iss he maimum absoue ia acig eos ude diee eedbac coo gais. Fom his abe we ca see ha wih he icease o he eedbac coo gais he ia acig eos become smae ad he covegece ae is ase. I demosaes ha he coo paamee ρ has sigiica eec o he covegece ae. Tha cocusio is coicide wih he heoeica aaysis show i Secio 3. Tabe shows he ia acig eos (sabe bouday) o SPD-PD eaig coo compaed o ied gai PD-PD eaig coo usig he same eaig gais. Fom his abe i is ceay show ha he SPD-PD eaig coo ca obai much bee acig peomace (a eas 5 imes) eve wih he eisece o he o-epeiive uceaiy ad disubace (Eampe ). Figue 3 shows he acig eo boud esus om ieaio o ieaio based o SPD-PD eaig coo ad he ied gai ( s= ( ) ) PD-PD eaig coo. Coo Gais TALE I. COMPARISON UNDER DIFFERENT CONTROL GAINS Eampe Eampe Ie. ma e ma e Ie. ma e ma e Low Mid High TALE II. COMPARISON FOR SPD-PD AND PD-PD LEARNING CONTROL Eampe Eampe Coo Ie. ma e ma e Ie. ma e ma e PD-PD SPD-PD Maimum absoue eos Maimum absoue eos e: PD-PD e: SPD-PD e: PD-PD e: SPD-PD Ieaio umbe a) Tacig eos o Eampe e: PD-PD e: SPD-PD e: PD-PD e: SPD-PD Ieaio umbe b) Tacig eos o Eampe Figue 3. Compaiso o SPD-PD ad PD-PD eaig coo. Fom Figue 3 ad Tabe we ca see ha he acig peomaces ae impoved i high eedbac coo gais ae used i he deveoped agoihm. I aso ceay shows ha he acig eos moooicay decease om ieaio o ieaio. D. The Eec o Popoioa Gais Fom he covegece aaysis coduced i pevious secio we ca see ha he popoioa eedbac gai ad ieaive eaig gai ae o acos o guaaeeig he covegece as hey ae o icuded i he covegece 5 879

6 codiio. Theeoe i is a good poi o eam he eec o popoioa gais o he acig peomace. Figue 4 shows oe simuaio esu o he acig eos om ieaio o ieaio usig SPD-PD coo SD-PD coo ( K p = ) SPD-D coo ( K p = ) ad SD-D coo ( K p = K p = ). I is ceay show ha K p gais have eec oy o he is ew (3 i his simuaio) ieaios. Ae ha K p gais mae ie coibuio o he covegece speed. The simuaio esus show ha he ia acig eos usig hese ou eaig coo aws ae amos he same. Maimum absoue eos e Maimum absoue eos e e: SPD-PD e: SD-PD e: SPD-D e: SD-D Ieaio umbe e: SPD-PD e: SD-PD e: SPD-D e: SD-D Ieaio umbe Figue 4. The eec o popoioa coo gais o he covegece speed. V. CONCLUSIONS I his pape a ew ieaive eaig coo is poposed ha is a PD eedbac coo wih adjused coo gais i ieaio domai pus a PD-ype ILC wih pevious ieaio iomaio i he updaig coo aw. I aims o mae use o he chaaceisics oeed by boh he eedbac coo ad he ieaive eaig coo. The poposed coo aw aes he advaages o boh he cue ad pevious ieaio iomaio o he sysem o ehace he sabiiy chaaceisics ad quicy dive he acig ajecoies o he desied oes wihi bouds. The ew SPD-PD coo achieves acig accuacy wih vey as covegece speed ad is obus agais upedicabe disubaces ad uceaiies. I addiio i ca povide ea degees o eedom o he choices o he eaig gais. The ia acig eo ad he covegece ae ca be adjused by he swichig gai o he PD eedbac coo ha maes his coo scheme moe pomisig om he pacica viewpoi. Simuaio sudy demosaes he eeciveess o he poposed SPD-PD ieaive eaig coo. Rea appicaio shoud be a uue wo. REFERENCES [] S. Aimoo S. Kawamua ad F. Miyazai eeig opeaio o obos by eaig Joua o Robo Sysems Vo. pp [] K.L. Mooe M. Daheh ad S.P. haachayya Ieaive eaig coo: A suvey ad ew esus Joua o Roboic Sysems Vo. 9 No. 5 pp [3] D.A. isow M. Thaayi ad A.G. Aeye A suvey o ieaive eaig coo: A eaig-based mehod o high-peomace acig coo IEEE Coo Sysems Magazie Vo.6 No.3 pp [4] H.S. Ah Y.Q. Che ad K.L. Mooe Ieaive eaig coo: ie suvey ad caegoizaio IEEE Tasacios o Sysems Ma ad Cybeeics Pa C: Appicaios ad Reviews Vo. 37 No.6 pp [5] K.H. Pa A aveage opeao-based PD-ype ieaive eaig coo o vaiabe iiia sae eo IEEE Tas. o Auomaic Coo Vo. 5 No.6 pp [6] D.H. Owes ad G.S. Mude Uivesa adapive ieaive eaig coo Poc. o he 37 h IEEE Coeece o Decisio & Coo pp [7] N. Ama D.H. Owes ad E. Roges Ieaive eaig coo o discee ime sysems usig opima eedbac ad eedowad acios Poc. o he 34 h Coeece o Decisio & Coo pp [8] Z.H. Qu J. Dosey M. Dawso ad R.W. Johso A ew eaig coo scheme o obos Poceedigs o he 99 IEEE ICRA pp [9] J.X. Xu X.W. Wag ad L.T. Heg Aaysis o coiuous ieaive eaig coo sysems usig cue cyce eedbac Poc. o he Ameica Coo Coeece pp [] Y.Q. Che J.X. Xu ad T.H. Lee Cue ieaio acig eo assised ieaive eaig coo o uceai oiea discee sysem Poc. o 35h IEEE Co. Decisio ad Coo pp [] Y.Q. Che J.X. Xu ad T.H. Lee A ieaive eaig cooe usig cue ieaio acig eo iomaio ad iiia sae eaig Poc. o 35h IEEE Co. Decisio ad Coo pp [] P.R. Ouyag W.J. Zhag ad M.M. Gupa A Adapive Swichig Leaig Coo Mehod o Tajecoy Tacig o Robo Maipuaos Mechaoics Vo. 6 No. pp [3] D.Y. Pi D.Sebog J. Shou Y. Su ad Q. Li Aaysis o cue cyce eo assised ieaive eaig coo o discee oiea imevayig sysems IEEE Ieaioa Coeece o Sysems Ma ad Cybeeics 5 pp [4] S.J. Yu J.H. Wu ad X.W. Ya A PD-ype ope-cosed-oop ieaive eaig coo ad is covegece o discee sysems Poceedigs o he Fis Ieaioa Coeece o Machie Leaig ad Cybeeics pp [5] J.X. Xu L. Su T. Chai ad D. Ta High-ode ope ad cosed oop ieaive eaig coo scheme wih iiia sae eaig 8 h Ieaioa Coeece o Coo Auomaio Roboics ad Visio pp [6] M.K. Kag J.S. Lee ad K.L. Ha Kiemaic pah-acig o mobie obo usig ieaive eaig coo Joua o Roboic Sysems Vo. No