Repetitive control for systems with uncertain period-time

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1 Automatica 38 (2002) Brif Papr Rptitiv control for systms with uncrtain priod-tim Maartn Stinbuch Control Systms Tchnology Group, Faculty of Mchanical Enginring, Eindhovn Univrsity of Tchnology, P.O. Box 513, 5600 MBEindhovn, Th Nthrlands Rcivd 20 Dcmbr 1999; rcivd in rvisd form 5 Fbruary2002; accptd 14 Jun 2002 Abstract Rptitiv control is usful if priodic disturbancs act on a control systm. Prfct (asymptotic) disturbanc rjction is achivd if th priod-tim is xactlyknown. For thos cass whr th priod-tim changs and cannot b masurd dirctlybyan auxiliarysignal, a robust rptitiv controllr structur is proposd. It uss multipl mmory-loops in a crtain fdback conguration, such that small changs in priod-tim do not diminish th disturbanc rjction proprtis. Th robust rptitiv controllr shows good implmntation rsults for a tracking control problm of a Compact Disc playr.? 2002 Elsvir Scinc Ltd. All rights rsrvd. Kywords: Rptitiv control; Compact disc playr; Priodic disturbancs; Intrnal modl principl 1. Introduction Control systms subjct to priodic disturbancs may wll bnt from th us of rptitiv control. Rptitiv controllrs mployth intrnal modl principl and consist of a priodic signal gnrator, nabling prfct (asymptotic) rjction of priodic disturbancs. In cas of tracking systms whr th task (stpoint) is known to b a prdtrmind (rpatdlysupplid) trajctory, rptitiv control is usd in a smi-opn-loop (fdforward) fashion known as itrativ larning control. In cass whr priodic masurmnt nois is signicant, rptitiv control is applid in th form of digital comb-ltrs for nois rduction. Tomiuka, Tsao, and Chw (1988), Hara, Yamamoto, Omata, and Nakano (1988), Chw and Tomiuka (1990), Hillrstrom (1996), Moor, Dahly, and Bhattacharyya (1992), Yamamoto (1993), Bodson, Sacks, and Khosla (1994), Mssnr and Bodson (1994) and d Roovr, Bosgra, and Stinbuch (2000) covr most of th rlvant dvlopmnts in rptitiv and larning control. A block-diagram of a closd-loop systm including a rptitiv controllr as add-on dvic, is shown in Fig. 1. This papr was not prsntd at anyifac mting. This papr was rcommndd for publication in rvisd form byth Associat Editor Shinji Hara undr th dirction of Editor Robrto Tmpo. addrss: m.stinbuch@tu.nl (M. Stinbuch). Hr th rfrnc signal r, or anydisturbanc acting on th systm, is assumd to b a priodic signal with known and xd priod-tim. Th additional dvic, dnotd as mmory-loop is a dlay lin with lngth qual to th priod-tim of th xtrnal signals and with a positiv fdback. This constituts a priodic signal gnrator. In Sction 2 this standard rptitiv control conguration will b xplaind furthr. On of th drawbacks of rptitiv control is th rquirmnt of xact knowldg of th priod-tim of th xtrnal signals. This mans that in practical applications, ithr th priod-tim is rquird to b constant (±0:1%), or an accurat masurmnt of th priodicityis ncssary. In litratur svral solutions hav bn proposd, most of thm considring a suprvisoryadaptiv schm to stimat th priod-tim from closd-loop on-lin masurmnts, s for instanc Tsao and Nmani (1992), Dotch, Smakman, Van dn Hof, and Stinbuch (1995), and Manayathara, Tsao, Bnstman, and Ross (1996). In contrast to th litratur, in this papr w will propos a nw structur for rptitiv control which is robust for changs in priod-tim (Sction 3). Th robust rptitiv controllr will b xprimntallytstd for th srvo control of a Compact Disc playr (Sction 4). Finally, in Sction 5 w will summari th main contributions in th concluding rmarks /02/$ - s front mattr? 2002 Elsvir Scinc Ltd. All rights rsrvd. PII: S (02)

2 2104 M. Stinbuch / Automatica 38 (2002) r - mmory loop controllr C plant P y Q (s) -st p L (s) Fig. 3. Block diagram of th implmntd mmory-loop. Fig. 1. Block diagram of a control loop including a rptitiv controllr. -st p Fig. 2. Block diagram of a standard mmory-loop. 2. Rptitiv control 2.1. Priodic signal gnrator Th intrnal modl principl (Francis & Wonham, 1975) stats that for asymptotically tracking a rfrnc command byth output of a closd-loop systm, a raliation (modl) of th disturbanc=rfrnc gnrating systm should b includd in th fdback loop. As a wll-known xampl, signals with a DC (! = 0) contnt can b modlld using an intgrator, and inclusion of intgral action in th fdback controllr prvnts stady-stat rrors for constant rfrncs and disturbancs. A discrt tim intgrator is a positiv fdback ovr on dlay, implying that on mmorylocation is usd to stor th intgral valu. With ro input th intgral valu updats itslf byth positiv fdback loop. Similarly, for priodic signals, a mmoryloop can b usd which gnrats an output at frquncis k! p, with k intgr and! p th priod frquncy. In a mmory loop, a signal with priod T p =2=! p is stord in a FIFO bur. Dpnding on T p and th sampl frquncya numbr of mmorylocations is ndd. If a positiv fdback is put ovr this FIFO bur, in stadystat no input is ndd to gnrat an output with priod tim T p. Such a priodic signal has a discrt frquncyspctrum with paks at!=k! p. A block diagram of a mmory-loop with priod tim T p is shown in Fig. 2. Although th common implmntation of mmory-loops is don in discrt tim, in this papr w will adopt a dscription in continuous tim in ordr to driv som insightful rsults. Not that a tim dlayis qual to stp. Th transfr function from to is (s Fig. 2) = G(s)= stp : (1) stp 1 For frquncis! = k! p = k2=t p th magnitud of th dnominator of G(s) is ro, making th gain of th transfr function innit. This, of cours, is th intndd working principl of th rptitiv controllr: to hav innit loop gain at th harmonics of th disturbanc. Howvr, such gains mayasilycaus stabilityproblms in th main (outr) srvo loop. Stabilityof rptitiv controllrs has bn thoroughlyanalyd bytomiuka t al. (1988), Chw and Tomiuka (1990) and Hillrstrom (1996). To guarant stabilitytwo ltrs Q(s) and L(s) ar addd to th rptitiv signal gnrator as shown in Fig. 3. Th ltr L(s) usuallyis calld th larning ltr, and it is usd to compnsat th transfr as sn byth rptitiv controllr. Th ltr Q(s) is usd to rct th mismatch btwn L(s) and th ral systm; it limits th working of th rptitiv controllr to a crtain frquncyband. It is bncial to construct Q(s) such that its phas bhavior is linar with frquncy. In that cas w can compnsat th phas dlayintroducd byq(s) byadjusting th dlayof th priodic signal gnrator. It is not th purpos of this papr to invstigat th dsign of th ltrs L(s) and Q(s). Instad w rfr to th litratur (Tomiuka t al., 1988; Hillrstrom, 1996) Snsitivity analysis Lt us rturn to th basic building block of a rptitiv controllr, s Fig. 2. A frquncyrspons of this ltr G(s) is shown in Fig. 4 for th rst fw harmonics (13 in this cas; T p =1=7:5 s). Clarly, at th harmonic frquncis th gain is innit as prdictd from Eq. (1). Howvr, th additional gain is onlyobtaind in a vry limitd frquncyband cntrd around ach harmonic. In som applications th priod tim is not xactlyknown or cannot b masurd accuratlynough. To furthr analy th snsitivityof th rptitiv controllr for changs in th priod-tim, considr Eq. (1) if th priod tim is prturbd with a multiplicativ rror : T p (1 ). Th transfr function G p (s) for th prturbd priodic signal gnrator thn bcoms = G p(s)= stp(1) 1 stp(1) (2) and at th harmonics w obtain with s =jk2=t p : = jk2(1) : (3) 1 jk2(1) In Fig. 5 th magnitud of Eq. (3), for th rst harmonic (k = 1), is plottd as a function of th prturbation : solid

3 M. Stinbuch / Automatica 38 (2002) magnitud frquncy [H] Fig. 4. Magnitud frquncyrspons G(j!) of a priodic signal gnrator N=2 N=3 magnitud 10 1 N= chang of priodicity α [%] Fig. 5. Magnitud of a priodic signal gnrator as function of th priod-mismatch, valuatd for th rst harmonic (k = 1), for th standard rptitiv controllr (N = 1), and for robust rptitiv controllrs (N = 2 and N = 3). lin (with labl N = 1; th variabl N will b introducd in th nxt sction). As can b sn, alradyfor a prturbation of 1.5% ( =0:015) th gain drops from to 10. For highr harmonics (not shown in th gur) this is vn mor pronouncd: for xampl for th tnth harmonic (k = 10), th gain is only1.1.

4 2106 M. Stinbuch / Automatica 38 (2002) st p -st p -st p W 2 W 1 Fig. 6. Gnralid rptitiv controllr. W 3 In th nxt sction w will prsnt an xtndd rptitiv controllr to improv th robustnss for uncrtaintis in th priod-tim. 3. Robust rptitiv control Considr a gnralid rptitiv controllr, consisting of multipl priodic signal gnrators in a structur as shown in Fig. 6 for an xampl with thr lmnts. In Gotou, Uta, Nakamura, and Matsuo (1991) also multipl mmoryloops ar usd, but hr th wighting factors ar usd to modifyth dynamic rspons in btwn th harmonic frquncis. Th transfr function of a gnralid rptitiv controllr with N priodic signal gnrators can b writtn as = G(s)= H(s) (4) 1 H(s) with th loop transfr function H(s): H(s)= W i istp : (5) Again, w would lik th priodic signal gnrator G(s) to bcom innit at th harmonics, i.. H(s) =1fors = jk2=t p. Substitution of ths valus of s into Eq. (5) givs H(jk2=T p )= W i ijk2 = W i =1: (6) Indd for N =1,wndW 1 =1 which givs th standard rptitiv controllr from th prvious sction. In ordr to improv th robustnss of th rptitiv controllr for variations of th priod-tim, w also impos th =jk2=t p ) = 0 (7) which N W i istp = W i is istp (8) and for s =jk2=t p w p ) = W i ijk2=t p = 0 (9) hnc, for th cas with N 1, W i i = 0 (10) will mt th rquirmnt. Sinc w hav a structur with N dlaylins, and N paramtrs W i, w can impos N constraints. To furthr dcras th snsitivityfor priod-tim variations, w can impos upto (N 1)th drivativs qual to ro: W i i (N 1) =0: (11) In Fig. 7 th magnitud frquncyrsponss ar plottd for a rptitiv controllr with on, two and thr mmory loops, rspctivly. Clarly, for a widr frquncy band around th harmonics th gain is highr, which lads to a smallr snsitivityfor priod-tim changs. This can also b obsrvd in th snsitivityplot Fig. 5 (N =2; 3). As an xampl considr th cas with two mmoryloops (N = 2). Thn w nd from Eqs. (6) and (10) W 1 W 2 =1 and W 1 2W 2 = 0, hnc W 1 =2,W 2 = 1 is th solution pair. Byinduction w can stat that th proposd robust rptitiv controllr with N wights is givn by = 1 (1 ) N stp (1 stp ) N : (12) It should b notd that an quivalnt principl occurs in th tim domain for robustifying th so-calld input shaprs, s Singr (1993) and Singh and Vadali (1993). Th robust rptitiv controllr proposd hr has bn succssfullypatntd (Stinbuch & Schootstra, 1998); an adaptiv vrsion has bn dvlopd for handling largr variations of th priod-tim. This adaptiv vrsion is basd on corrlation analysis of stord information in th FIFO bur, s Schootstra and Stinbuch (1998). In th nxt sction w will show xprimntal rsults for th tracking problm of a Compact Disc mchanism. 4. Application to a compact disc driv 4.1. Standard rptitiv control In Fig. 8 a schmatic viw of a Compact Disc mchanism is shown. Th mchanism is composd of a turn-tabl DC-motor for th rotation of th Compact Disc, and a radial arm for th track-following. An objctiv lns, suspndd bytwo paralll laf springs, can mov in a vrtical dirction to giv a focusing action. In Fig. 9 a block-diagram of th radial (tracking) control loop is shown. Th dirnc btwn th radial track position and th spot position is dtctd byth optical pick-up; it gnrats a radial rror signal (RE) (Stinbuch, van Groos, Schootstra, Wortlbor, & Bosgra, 1998). In currnt systms th srvo controllr is a PID controllr

5 M. Stinbuch / Automatica 38 (2002) magnitud 10 1 N frquncy [H] Fig. 7. Magnitud frquncyrsponss G(j!) of multipl priodic signal gnrators magnitud frquncy [h] Fig. 8. Schmatic viw of a rotating arm Compact Disc mchanism. track RE currnt position srvo actuator - phas frquncy [H] Fig. 10. Opn-loop frquncyrspons of a mmory-loop in sris with a PID controllr and a doubl intgrator. Fig. 9. Conguration of th Compact Disc tracking control loop. (Stinbuch & Norg, 1998). Th tracking control loop has a cross-ovr frquncyof 500 H. Fig. 10 shows th opn-loop frquncyrspons of th Compact Disc control systm modl, in this cas a sris connction of a doubl intgrator as plant (mass to b positiond), a lad ltr with intgral action (PID controllr (Stinbuch & Norg, 1998)) and th rptitiv controllr (with low-pass digital FIR ltrs Q() and L(), cut-o at 150 H, with linar phas charactristics). Using frquncy rspons analysis, stability of th controlld systm can b chckd, using standard Nyquist argumnts. Not that th phas chang at th harmonic frquncis is from 270 back to 90 and thus compnsats for th phas chang btwn th harmonics.

6 2108 M. Stinbuch / Automatica 38 (2002) amplitud Fig. 11. Amplitud spctrum of radial rror (RE) using a standard srvo loop. amplitud frquncy [H] Fig. 14. Spctrum of RE using a robust (N = 2) mmory-loop prturbd by0.5%. amplitud frquncy [H] Fig. 12. Amplitud spctrum of RE whn a mmory-loop is addd. rror rror tim [s] amplitud frquncy [H] Fig. 13. Spctrum of RE using a standard mmory-loop prturbd by 0.5%. A masurd amplitud spctrum of th radial srvo rror (RE) is shown in Fig. 11 for th PID controlld tracking srvo. Th priodic contnts of th tracking rror is clarly visibl in a larg numbr of harmonics of th rotational frquncy(! p =27:5 rad=s) of th disc. If a mmory-loop is corrctly tund and lockd to th disturbanc priod, rror rductions can b vrylarg. As an xampl th amplitud spctrum of th rsulting radial tracking rror is shown in Fig. 12. Up to a frquncyof 150 H (20 harmonics) th priodic componnts ar supprssd. This clarlyshows th rmarkabl improvmnts possibl with rptitiv control. Howvr, as mntiond bfor, th improvmnts ar vrysnsitiv for variations of th priod-tim. To show this, w changd th frquncyof th disc byonly0.5%, whil kping th sam rptitiv controllr. Th rsulting amplitud spctrum of th radial tracking rror is shown in Fig tim [s] Fig. 15. Rmaining rror signals of a standard (top) and a robust mmory-loop (bottom) whn th priod-tim is prturbd by 0.5% Implmntation rsults with robust rptitiv control In ordr to valuat th robust mmory-loop in rality, an implmntation has bn don for th cas N = 2. Th amplitud spctrum of th tracking rror is shown in Fig. 14 for th robust (N = 2) rptitiv controllr with a 0.5% chang in disc rotational frquncy. Compar this with th spctrum plot in Fig. 13 for th standard (N = 1) rptitiv controllr. It is clar that th robust mmory-loop has a much bttr rduction compard to th N = 1 cas. Th improvmnt can b prdictd also for othr prturbation lvls, byth us of Fig. 5. Finally, th rror signals in th tim-domain ar shown in Fig. 15 (th tim-domain signals ar low-pass ltrd at 125 H in ordr to show th dirnc mor clarly). Th robustication of th mmory-loop as xplaind abov, of cours, has th disadvantag of th nd of mor storag capacity. This incrass with th lngth NT p of th FIFO. 5. Conclusions Th us of mmory-loops is bncial in systms with rptitiv disturbancs or tasks. In ordr to improv th ca-

7 M. Stinbuch / Automatica 38 (2002) pabilitis of rptitiv controllrs for thos cass whr th priodicityis hard to masur and is subjct to variation, an xtnsion of rptitiv control is dvlopd. Byusing multipl mmory-loops, and corrct dsign of th cocints, signicant robustnss is achivd for small variations in th priod tim. Th robust rptitiv controllr has bn implmntd succssfullyin a digital control stup of a Compact Disc playr. Futur rsarch will b focusd on possibl application of th sam ida to th so-calld high-ordr itrativ larning control. Acknowldgmnts Th author would lik to thank Grrit Schootstra for his participation in th projct, and Tarunraj Singh, Univrsity of Bualo, for hlpful suggstions and corrctions, spciallyfor pointing out Eq. (12). Rfrncs Bodson, M., Sacks, A., & Khosla, P. (1994). Harmonic gnration in adaptiv fdforward cancllation schms, IEEE Transactions on Automatic Control, 39(9), Chw, K. K., & Tomiuka, M. (1990). Digital control of rptitiv rrors in disk driv systms. Procdings of th Amrican control confrnc (pp ). Dotch, H. G. M., Smakman, H. T., Van dn Hof, P. M. J., & Stinbuch, M. (1995). Adaptiv rptitiv control of a compact disc mchanism. Procdings of th 1995 IEEE confrnc on dcision and control, Nw Orlans (pp ). Francis, B. A., & Wonham, W. M. (1975). Th intrnal modl principl for linar multivariabl rgulators. Applid Mathmatics and Optics, 2, Gotou, M., Uta, E., Nakamura, A., & Matsuo, K. (1991). Dvlopmnt of multirat sampling rptitiv larning srvo systm and its application to a compact camcordr. IEEE=RSJ intrnational workshop on intllignt robots and systms IROS 91 (pp ). Hara, S., Yamamoto, Y., Omata, T., & Nakano, M. (1988). Rptitiv control systm a nw-typ srvo systm. IEEE Transactions on Automatic Control, AC-33, Hillrstrom, G. (1996). Adaptiv supprssion of vibrations a rptitiv control approach. IEEE Transactions on Control Systms Tchnology, 4(1), Manayathara, Th. J., Tsao, T. -C., Bntsman, J., & Ross, D. (1996). Rjction of unknown priodic load disturbancs in continuous stl casting procss using larning rptitiv control approach. IEEE Transactions on Control Systms Tchnology, 4(3), Mssnr, W., & Bodson, M. (1994). Dsign of adaptiv fdforward controllrs using intrnal modl quivalnc. Procdings of th Amrican control confrnc (pp ). Moor, K. L., Dahly, M., & Bhattacharyya, S. P. (1992). Itrativ larning control: A survyand nw rsults. Journal of Robotic Systms, 9(5), d Roovr, D., Bosgra, O. H., & Stinbuch, M. (2000). Intrnal modl basd dsign of rptitiv and itrativ larning controllrs for linar multivariabl systms. Intrnational Journal of Control, 73(10), Schootstra, G., & Stinbuch, M. (1998). Control systm for a procss that xhibits priodic disturbancs. US Patnt No. 5,7842,272, July, 21. Singr, N. C. (1993). Rsidual vibration rduction in computr controlld machins. Ph. D. thsis, Massachussts Institut of Tchnology. Singh, T., & Vadali, S. R. (1993). Robust tim dlaycontrol. ASME Journal of Dynamic Systms, Masurmnt and Control, 115(2A), Stinbuch, M., & Norg, M. L. (1998). Advancd motion control: An industrial prspctiv. Europan Journal of Control, Stinbuch, M., & Schootstra, G. (1998). Filtr, rptitiv control systm and larning control systm both providd with such ltr. US Patnt No. 5,740,090, April 14. Stinbuch, M., van Groos, P. J. M., Schootstra, G., Wortlbor, P. M., & Bosgra, O. H. (1998). -Synthsis of a compact disc playr. Intrnational Journal of Robust and Nonlinar Control, 8, Tomiuka, M., Tsao, T. C., & Chw, K. K. (1988). Discrt-tim domain analysis and synthsis of rptitiv controllrs. Procdings of th 1988 Amrican control confrnc (pp ). Tsao, T. C., & Nmani, M. (1992). Asymptotic rjction of priodic disturbancs with uncrtain priod. Procdings of th Amrican control confrnc (pp ). Yamamoto, Y. (1993). Larning control and rlatd problms in innit-dimnsional systms. in: H. L. Trntlman & J. C. Willms (Eds.), Essays on control: Prspctivs in th thory and its applications (pp ). Boston: Birkhausr. Maartn Stinbuch rcivd M.Sc. (cum laud) in Mchanical Enginring from Dlft Univrsityof Tchnology, Dlft, Th Nthrlands, in From 1984 until 1987 h was a rsarch assistant at Dlft Univrsityof Tchnologyand KEMA (Powr IndustryRsarch Institut), Arnhm, Th Nthrlands. In 1989 h rcivd Ph.D. (Dr.) from Dlft Univrsityof Tchnology on th subjct of Modlling and Control of Wind EnrgyConvrsion Systms. From 1987 to 1998 h was with Philips Rsarch Labs., Eindhovn as a mmbr of th Scintic Sta, working on modlling and control of mchatronic applications. From 1998 to 1999 h was a managr of th Dynamics and Control group at Philips Cntr for Manufacturing Tchnology. Sinc 1999 h is a full profssor of th Control Systms Tchnology group of th Mchanical Enginring Dpartmnt of Eindhovn Univrsityof Tchnology.Prof. Stinbuch has ovr 70 rfrd journal and confrnc publications, and holds two patnts. His rsarch intrsts ar modlling and control of motion systms.h was an associat ditor of th IEEE Transactions on Control Systms Tchnology( ) and of IFAC Control Enginring Practic ( ). H is currntlyassociat ditor of th IEEE Control Systms Magain and ditor-at-larg of th Europan Journal of Control.