MODEL-INVERSE BASED REPETITIVE CONTROL

Transcription

1 MODEL-INVERSE BASED REPETITIVE CONTROL T. Hart*, J. Hätönn*, D.H. Owns* Dparmnt f Autmatic Cntrl and Systms Enginring Th Univrsity f Shffild Mappin Strt S1 3JD Shffild Unitd Kingdm Tl: +44( Fax: +44( J.Hatnn@shffild.ac.uk Kywrds: Rptitiv Cntrl, Rbtics, Rbust Cntrl Abstract This papr xplrs th pssibility f using an invrs plant mdl in th rptitiv cntrl framwrk. Whn n uncrtainty is prsnt in th plant mdl, it is shwn that th algrithm rsults in mntnic cnvrgnc t zr tracking rrr. Furthrmr, th algrithm can tlrat 90 dgrs f phas uncrtainty in th plant mdl, dmnstrating a gd dgr f rbustnss. Simulatin rsults ar usd t dmnstrat th diffrnt thrtical findings in th papr. 1 Intrductin Many signals in nginring ar pridic, r at last thy can b accuratly apprximatd by a pridic signal vr a larg tim intrval. This is tru, fr xampl, f mst signals assciatd with ngins, lctrical mtrs and gnratrs, cnvrtrs, r machins prfrming a task vr and vr again. Hnc it is an imprtant cntrl prblm t try t track a pridic signal with th utput f th plant r try t rjct a pridic disturbanc acting n a cntrl systm.. In rdr t slv this prblm, a rlativly nw rsarch ara calld rptitiv cntrl has mrgd in th cntrl cmmunity. Th ida is t us infrmatin frm prvius prids t mdify th cntrl signal s that th vrall systm wuld `larn' t track prfctly a givn T-pridic rfrnc signal. Th first papr that uss this idlgy sms t b [5], whr th authrs us rptitiv cntrl t btain a dsird prtn acclratin pattrn in a prtn synchrtrn magntic pwr ply. Sinc thn rptitiv cntrl has fund its way t svral practical applicatins, including rbtics [6], mtrs [7], rlling prcsss [4] and rtating mchanisms [3]. Hwvr, mst f th xisting rptitiv cntrl algrithms ar dsignd in cntinuus tim, and thy ithr d nt giv prfct tracking r thy rquir that th riginal prcss is psitiv ral. In rdr t vrcm ths limitatins, this papr prpss a nw mdl-invrs basd rptitiv algrithm fr minimum-phas plants. It is shwn in this papr that nw algrithm rsults in fast cnvrgnc, and th algrithm is rasnably rbust against mdlling uncrtaintis. Th algrithm is applid t an industrial-scal cnvyr blt systm with xcllnt rsults, furthr dmnstrating th applicability f th algrithm t industrial cntrl prblms. Prblm dfinitin As a starting pint in discrt-tim Rptitiv Cntrl (RC it is assumd that a stat-spac mdl x( k + 1 Φx( + Γ, y( Cx( f th plant in qustin xists fr k. Frm nw n it is assumd that th plant is bth cntrllabl and bsrvabl, and that th plant is minimum-phas. Furthrmr, a rfrnc signal is givn, and it is knwn that k+t fr a givn T (in thr wrds th actual shap f is nt ncssarily knwn. Th cntrl dsign bjctiv is t find a fdback cntrllr that causs th utput y( th systm (1 t track th rfrnc signal as accuratly as pssibl. Mr mathmatically, th cntrllr shuld rsult in lim (1 K,, 1,0,1,, K whr :-y( is tracking rrr, whn it is knw that is T-pridic. As was shwn by Francis and Wnhan in [], a ncssary cnditin fr asympttic cnvrgnc is that a cntrllr ( [ Mu ]( [ N]( (3

2 whr M and N ar suitabl pratrs, must hav an intrnal mdl f th rfrnc signal insid th pratr M s that [ ]( Mr (4 Bcaus t is T-pridic, its intrnal mdl is M1-q -T, whr q -1 is th standard lft-shift pratr. This can b sn frm th quatin [ Mr]( [(1 q r]( k T This lads naturally t an rptitiv algrithm structur u ( k T + [ N]( (5 (6 and if N is a causal LTI filtr, th algrithm can b writtn using th q -1 -pratr frmalism as T q + K( (7 and th dsign cnsists f slcting K(. Th fcus f this papr cncntrats n th chic K( β q G 1 ( (8 whr β is a larning gain and G(C(qI-Φ -1 Γ is th transfr functin f (1. Th nxt sctin shws that this algrithm rsults in a fast cnvrgnc rat whn n mdl uncrtainty is prsnt, and, quit surprisingly, that th algrithm can tlrat a rasnabl dgr f multiplicativ uncrtainty. 3 Cnvrgnc analysis 3.1 Nminal cas As xplaind abv, th plant mdl (1 can b writtn using th q -1 -pratr frmalism as y ( G( (9 Cnsidr nw th algrithm (7 whn K( is qual t (8. This rsults in th mdl-basd invrs algrithm 1 q ( + β G ( (10 Nt that this algrithm is causal n th psitiv tim-axis du t th fact that that th tracking rrr t is dlayd T tim stps. Furthrmr, th pratr K( β q G 1 ( is stabl alng th psitiv tim-axis, bcaus G( is assumd t b minimum-phas. Multiplying (9 frm th lft with th plant mdl G( rsults in th rrr vlutin quatin (1 β k T (11 0 < β <, th This quatin shws immdiatly, that if algrithm rsults in ( λ k T, 0 < λ < 1 (1 dmnstrating mntnic cnvrgnc btwn and k- T. This rsult als autmatically implis that lim (13 Furthrmr, th fastst cnvrgnc rat is btaind with β 1, bcaus in th cas fr k>t-1, i.. th algrithm larns th crrct input squnc in n cycl. In th rptitiv cntrl framwrk, hwvr, it is almst always th cas that th plant mdl G( is nt knwn xactly, but mdlling uncrtaintis and nnlinaritis rsult in an uncrtain mdl f th plant. Cnsquntly, th nxt subsctin stablishs th dgr f uncrtainty that th mdl-invrs basd algrithm can tlrat. 3. Multiplicativ uncrtainty Cnsidr nw th cas whn nly a nminal mdl G ( is availabl fr th algrithm dsignr, and th tru plant mdl is rlatd t th nminal mdl thrugh th quatin y( G( G ( U ( (14 whr U( is th multiplicativ uncrtainty f th plant mdl. Applying th algrithm (10 t (14 rsults in th rrr vlutin quatin k q (1 β U ( k Rstrict th tim axis t b psitiv, i.. k 1,,. In this cas (15 rsults in an autnmus systm (15 ( 1 q (1 β U( (16 with th initial cnditins 0 0,, T-1 T-1, whr th initial cnditins ar dpndnt n th initial guss 0, T-1. Accrding t th Nyquist stability tst (s [1], th pls f th systm (16 ar insid th unit circl if th lcus f N z (1 β U ( z (17 z 1

3 ncircls th critical pint (-1,0 N tims, whr N is th numbr f unstabl pls f z N ( 1 ( z. Furthrmr, if th pls ar insid th unit circl, this guarants that k Mα, M > 0,0 < α < 1 and in particular, lim. Assum nw that U( is stabl. In this cas z N ( 1 ( z ds nt hav any unstabl pls, and thrfr N0, and thrfr (16 is nt allwd t ncircl th critical pint (-1,0. A sufficint cnditin fr this is that z N (1 1 < 1 In summary, if th uncrtainty U( is stabl, and it satisfis (19 fr a givn β, th algrithm will cnvrg xpnntially t zr tracking rrr. Hwvr, cnditin (19 ds nt rval any usful infrmatin U( in trms f th tru plant mdl G(. Nt hwvr, that β R{ U (1 * + β U } + β U * (1 which shws that fr (19 t hld it is rquird that [ 0,π ] (18 (19 (0 ω R{ U j } > 0 fr ω (1 and that β is sufficintly small. Nt that R{ U } > 0 fr ω [ 0, is quivalnt t th cnditin that th Nyquist diagram f U( lis strictly in right-half plan. This, n th thr hand, implis that phas f U( shuld lin insid ± 90 fr xpnntial cnvrgnc., shwing a gd dgr rbustnss in th algrithm. In summary, if th phas f th nminal plant G ( lis insid a ±90 dgr tub arund th phas f th tru plant, and β is takn t b sufficintly small, th tracking rrr will cnvrg xpnntially t zr tracking rrr. Small valus f β imply that t t T, shwing that an incrasd cnvrgnc rat ( β 1 rsults in dcrasd rbustnss and incrasd rbustnss ( β 0 in slw cnvrgnc rat. Cnsquntly, th tuning f β plays a crucial rl, but it is nt yt clar hw t autmat this tuning prcss. 4 Filtring Nt that th invrs f th plant G -1 ( has typically a high gain at high frquncis. Thrfr n wuld assum that th algrithm (10 is xtrmly snsitiv t nis y(t. In rdr t analys this situatin, assum that nly a masurd utput y n (t is availabl, whr th riginal utput is crruptd thrugh an additiv nis mdl y n ( t y( t + n( t + ( whr n(t band-limitd whit nis. In this cas th algrithm (10 bcms q 1 ( + β G ( ( t y( t n( t (3 and y(t is rlatd t t and n(t thrugh th quatin ( 1 q (1 β U( y( t q U( β( t n( t (4 Cnsquntly, if U( 1 (i.. n has a prfct mdl f th plant, and th nis is uncrrlatd, th algrithm ds nt amplify high frquncy nis, but mrly dlays it and dcrass its amplitud by β. Hwvr, if U( has high amplitud at a crtain frquncy rang (typically high frquncy rang, th algrithm will start t amplify th masurmnt nis, which will dcras th tracking prfrmanc and pssibly lad t divrgnc. In rdr t vrcm this prblm, cnsidr th fllwing filtrd vrsin f th riginal algrithm (10 1 q ( + β G ( t (5 whr is causal, stabl LTI systm n th psitiv timaxis. Th rrr vlutin quatin in this cas bcms T q ( β U ( + (1 (6 and using a similar argumnt as bfr, it can b shwn that a sufficint cnditin fr mntnic cnvrgnc is (1 1 and th algrithm cnvrgs t th tim squnc < 1 (7 1 (1 β U( (1 (8 This quatin shws that fr ths frquncis, whr th amplitud f F is cls t unity and th phas is 0,

4 th algrithm rsults in prfct tracking f (. On th thr hand, if F 0, th cnvrgnc cnditin (4 is mt trivially. Furthrmr, ths frquncis ar blckd bfr thy ar drivn thrugh 1 j G ω. Thus shuld b mad cls t unity at th frquncis whr ( has significant spctral cntnt. Furthrmr, utsid this frquncy rang, shuld b cls t zr in rdr t nhanc rbustnss against mdlling uncrtainty and masurmnt nis. r r 5 Simulatin xampls Cnsidr th plant mdl s + 1 G ( s (9 + 6 Figur. A typical nis vctr Figur 3 shws hw th algrithm prfrms withut any filtring fr diffrnt valus f β. In this simulatin it is assumd that U(1. which is sampl with T s s using zr-rdr hld. Th systm is psd t track a sinusidal rfrnc π t sin( t (30 6 Figur 1 shws th tracking rrr whn th algrithm (10 is run with β0.5 n th plant (9. Figur 3. y(t fr diffrnt valus f β Th rsults clarly prt th thrtical findings s far, i.. a dcras in β will dcras th cnvrgnc spd but incras th rbustnss f th algrithm against masurmnt nis in th limit. Figur 1. Tracking rrr in th nminal cas Th algrithm larns t track th rfrnc signal in 3 cycls r s, dmnstrating a vry fast cnvrgnc rat. In rdr t tst hw th algrithm (10 can cp with masurmnt nis, y(t is crruptd with additiv nis that has significant spctral cntnt n th frquncy rang abv 40 Hz. A typical nis vctr is shwn in Figur. In rdr t tst th ida f using filtring t mitigat th ffcts f masurmnt nis, a 5 th rdr Buttrwrth filtr was dsign with a cut-ff frquncy f 5 Hz. Th cut-ff frquncy was slctd s that th masurmnt nis wuld b blckd, but at th sam tim nar prfct tracking wuld b btaind in th frquncy rang f t. In this simulatin β is takn t b 0.5. Figur 4 shws th rsult withut any filtring, and it is clar that th tracking rrr has dcrasd th tracking capability. Figur 5, n th thr hand, shws y(t whn th Buttrwrth filtr is bing usd. Th tracking in trms f amplitud is nar prfct, but thr is a slight phas diffrnc btwn th utput and rfrnc. This is du t th fact that th Buttrwrth filtr will intrducs phas lag at vry sing frquncy, and thrfr prfct tracking in trms f phas cannt b achivd.

5 6 Cnclusins This papr has invstigatd th pssibility f using a mdlinvrs basd algrithm in th rptitiv cntrl framwrk. Whn th mdl is a prfct rplicat f th tru plant, it has bn shwn that th algrithm rsults in mntnic cnvrgnc t zr tracking rrr. In th cas f uncrtainty in th mdl, it has bn prvn that that if th phas f mdl lis insid a ± 90 tub arund th phas f th tru plant, th algrithm cnvrgs xpnntially t zr tracking rrr. Figur 4. N filtring Futur wrk cnsists f applying nn-causal filtrs n th windwd tracking rrr [, k-1,,k-t] in rdr t nhanc th rbustnss prprtis f th algrithm at high frquncis. Anthr lin f futur wrk is t adaptivly chang th larning gain β s that a bttr balanc btwn cnvrgnc rat and rbustnss wuld b achivd. Acknwldgmnts Figur 5. Using a Buttrwrth filtr As a final xampl, cnsidr th cas whr th tru plant mdl is givn by s G ( s ( s + 10 but th nminal mdl usd in th dsign f th algrithm is takn t b s + 1 G ( s (3 + 6 whr th high-frquncy pl is nt prprly mdlld. It can b shwn numrically that th rsulting multiplicativ uncrtainty U( is psitiv ral, and thrfr th algrithm shuld cnvrg. Figur 6 shws th rsults fr diffrnt valus f β, as xpctd, th algrithm cnvrgs. J. Hätönn and T. Hart ar prtd by th EPSRC cntract N GR/R7439/10. Rfrncs [1] K. J. Astrm and B. Wittnmark, Cmputr Cntrlld Systms Thry and Dsign, Prntic-Hall, (1984. [] B. A. Francis and W. M. Wnhan, "Th Intrnal Mdl Principl fr linar multivariabl rgulatrs", Appl. Math. Opt., pp , (1975. [3] R. F. Fung and J. S Huang and C. G. Chin and Y. C. Wang, "Dsign and applicatin f cntinuus tim cntrllr fr rtatin mchanisms", Intrnatinal jurnal f mchanical scincs, vl. 4, pp , (000. [4] S. S. Garimlla and K. Srinivasan, "Applicatin f rptitiv cntrl t ccntricity cmpnsatin in rlling", Jurnal f dynamic systms masurmnt and cntrl- Transactins f th ASME, vl. 118, pp , (1996. [5] T. Iny, M. Nakan, S. Kub H. Baba, High accuracy cntrl f a prtn synchtrtn magnt pwr ply, In Prcdings f th 8 th IFAC Wrld Cnfrnc, (1981. [6] K. Kank, R. Hrwitz, "Rptitiv and adaptiv cntrl f rbt manipulatrs with vlcity stimatin", IEEE Trans. On rbtics and autmatin, vl. 13, pp , (1997. [7] Y. Kbayashi, T. Kimuara, S. Yanab, "Rbust Spd Cntrl f ultrasnic mtr basd n H cntrl with rptitiv cmpnsatr", JSME Int. Jurnal Sris C, vl. 4, pp , (1999. Figur 6. Cnvrgnc undr uncrtainty