Iterative Learning Control and Feedforward for LPV Systems: Applied to a Position-Dependent Motion System

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1 7 Amican Contol Confnc Shaton Sattl Hotl May 4 6, 7, Sattl, USA Itativ Laning Contol and Fdfowad fo LPV Systms: Applid to a Position-Dpndnt Motion Systm Robin d Rozaio, Tom Oomn and Maatn Stinbuch Abstact Itativ Laning Contol (ILC) nabls pfomanc impovmnt by laning fom pvious tasks. Th aim of this pap is to dvlop an ILC appoach fo Lina Paamt Vaying (LPV) systms to nabl impovd pfomanc and incasd convgnc spd compad to th lina tim-invaiant appoach. This is achivd though ddicatd analysis and nom-optimal synthsis of LPV laning filts. Application to a position-dpndnt motion systm shows a significant impovmnt in accuacy and convgnc at, thby confiming th potntial of th poposd appoach. I. INTRODUCTION Itativ Laning Contol (ILC) nabls th pfomanc nhancmnt of systms that pfom batch-to-batch tasks [] such as: waf stags [], industial pinting systms [3], and pick-and-plac machins [4]. Among th diffnt ILC dsign famwoks, Lina Tim- Invaiant (LTI) mthods [5] hav gaind populaity bcaus accuat LTI modls can b obtaind using wll-stablishd systm idntification tools [6]. Using ths modls to dsign th ILC laning-filts lads to fast convging schms, wh th at of convgnc is govnd by th modl accuacy [7]. Howv, incasingly stingnt thoughput quimnts sult in v light mchatonic systms [8] with configuation-dpndnt dynamics [9]. Sinc ths systms cannot b asonably appoximatd by an LTI modl, xcssiv obustnss masus a quid in th ILC dsign to achiv convgnc [], o altnativly, th laning spd is dcasd to th point wh it taks many tials to convg to an accptabl lvl of pfomanc []. Ths nonlina dynamic systms can oftn b accuatly modld as Lina Paamt Vaying (LPV) systms [], as is vidncd by th lag numb of succssful applications to mchatonic systms [3], [4], [5]. Extnding th ILC famwok to th class of LPV systms would gatly bnfit ths applications, as is cognizd in [6], [7] and []. In [6], a D-typ laning algoithm is poposd fo Lina Tim-Vaying (LTV) systms by disctizing th tim axis, which sults in a contoll that intpolats local contolls in th tim domain. In [7], th schduling paamt is assumd to b constant duing ach tial, but changs fom tial-to-tial. This lads to a tialvaying laning filt that aims to optimiz th pfomanc dspit tial-vaying dynamics, but is not tating vaying dynamics duing a task. In [], an LTV laning filt Authos a with th Faculty of Mchanical Engining, Contol Systms Tchnology, Eindhovn Univsity of Tchnology, 56 MB Eindhovn, Th Nthlands. of cosponding autho:.d.ozaio@tu.nl is dtmind spaatly fo ach schduling tajctoy of intst, thby not addssing th full LPV cas. Although cnt dvlopmnts hav shown th potntial of byond-lti laning filts in ILC, at psnt no systmatic famwok is availabl yt. Th aim of this pap is to fill this gap though optimal disct-tim LPV filt dsign, using ddicatd citia that a divd basd on ILC poptis. By considing tasks of infinit tim, an appoach sults whin obustnss can b xplicitly addssd and which quis only a singl synthsis stp, thby poviding a significant xtnsion to []. In addition, sinc fdfowad contol can b intptd as a singl ILC itation, th sam appoach is applid to th synthsis of fdfowad contolls. This is achivd though th following contibutions. C In Sction II, th LPV ILC poblm is posd and an updat law is poposd along with conditions und which monotonic convgnc is guaantd. This sults in an optimal lft- and ight-invs matching poblm fo th dsign of laning filts and fdfowad contolls spctivly. C In Sction III, th stuctu of th invs of LPV systms is analyzd and th obtaind insights a usd to xtnd fixd-lag smoothing [8] and pviwbasd contol [9] to th LPV cas, thby ffctivly constituting a complt dsign famwok fo th synthsis of laning filts and fdfowad contolls fo disct-tim LPV systms. C3 In Sction IV, th poposd appoach is applid to a simulatd position-dpndnt motion systm, thby stablishing th pactical lvanc of this mthod. C is povids an xtnsion of optimal invs matching fo LPV systms [] by including pviw to allow fo non-causal filts. This is latd to pviw-basd stabl invsion [] fo Non-Minimum Phas systms (NMP). In this contxt, C xtnds stabl invsion to LPV systms, by combining optimal invs dsign with ILC as in []. A. Concpts and Notation Lt R, Z and Z + dnot th st of als, intgs and nonngativ intgs spctivly. A disct tim signal s dnots th map s Z R ns, and q is dfind as th shift opato, such that s[k] = s[k + τ], with τ, k Z. This pap consids Singl-Input Singl-Output (SISO) disct tim LPV systms, which a psntd in stat-spac as [3], x[k + ] = A(ρ[k])x[k] + B(ρ[k])u[k], (a) Σ(ρ) { y[k] = C(ρ[k])x[k] + D(ρ[k])u[k], (b) /$3. 7 AACC 358

2 wh ρ[k] D R nρ, with D th paamt spac. Fo a givn squnc ρ, th input-output map is dnotd as Σ(ρ) u[k] y[k], i.. y[k] = Σ(ρ[k])u[k], wh th dpndnc on k is omittd whn no ambiguity occus. Fo a constant schduling squnc, i.. ρ[k] = ρ D k, () dscibs an LTI systm, which is calld th fozn bhavio of th LPV systm at ρ [3]. Moov, th l -inducd nom of Σ(ρ) fo a givn squnc ρ, is dfind as, Σ(ρ), sup u l y u, s = k= s [k]s[k]. With slight abus of notation,. L., and. H., indicat th L and H -nom of opatos opating on l (Z) and l (Z + ), spctivly. If Σ(ρ) is such that Σ(ρ), < fo vy possibl squnc ρ, th systm is calld Boundd-Input Boundd-Output (BIBO) stabl in th l -nom []. In th maind of this wok, stability fs to BIBO stability. Moov, lt th plant to b contolld b givn by th l -BIBO stabl LPV systm with static paamt dpndnc, i.. it dpnds only xplicitly on ρ[k] and not on ρ[k + n], n Z /. Moov, dfin th tacking o as y, wh is th dsid output. II. ILC AND FEEDFORWARD CONTROL OF LPV SYSTEMS BY INVERSE MATCHING Itativ Laning Contol is typically mployd to impov th tacking pfomanc of systms that pfom pating tasks. Th aim of this sction is to xtnd th ILC famwok to LPV systms fo tials of infinit lngth. To this nd, an input updat law is poposd and th cosponding convgnc cition is psntd, which is constituts contibution C. Th convgnc condition lads to an optimal lft-invs matching poblm fo th ILC laning filt. In addition, it is shown that fdfowad contol lads to a simila ight-invs matching poblm. A. Itativ Laning Contol fo LPV Systms Fo a systm with input u j and schduling squnc ρ j, th tacking o duing th j-th tial is givn by, j = G(ρ j )u j, () wh th initial condition is assumd to b zo without loss of gnality [7], i.., x[k ] =. ILC aims to dcas j by itativly updating u j, basd on os that w mad duing past tials. In th psnt pap, th following LPV ILC th updat law is considd, u j+ = Q(u j + L(ρ j ) j ), (3) wh Q is a stabl, possibly non-causal LTI obustnss filt and L(ρ) is an LPV laning filt. This contains th LTI vaiant [5] as a spcial cas, which is obtaind by fzing th laning filt L( ρ) fo a spcific valu of th paamt. Th bnfit of th poposd xtnsion with spct to th LTI cas is shown in Sction IV. Th updatd input gnatd by (3) can indd incas th tacking pfomanc itativly as is psntd in th following thom. L(ρ) Fig.. is th plant to b contolld by th fdfowad contoll L(ρ) which th aims to duc, which is inducd by. Thom (LPV ILC Convgnc). Givn, a fnc l, and an admissibl schduling squnc ρ, which a both tial-invaiant, i.., ρ j = ρ j. Thn, if, Q( L(ρ )G(ρ )) L <, (4) th squncs u j and j, govnd by () and (3), convg to th fixd point (u, ), givn by, u = [ Q( G(ρ )L(ρ ))] QL(ρ ), = G(ρ )u, along all possibl paamt tajctois ρ, wh u j u dcass to zo monotonically. Th poof follows by witing u j+ in tms of and u j and dtmining th fixd point (5), which is nabld by assuming tial-invaiant dynamics. Application of th Banach fixd-point thom [7] to u j u thn sults in (4). Th convgnc cition (4) can b intptd as an invs matching poblm, sinc convgnc is nsud if L(ρ) accuatly appoximats G (ρ). In LTI mthods, L is commonly constuctd basd on a modl of G and th obustnss filt Q is usd to nfoc convgnc if th mismatch with th tu invs systm is too lag. Not that fo Q =, it holds that = and pfct tacking is obtaind if (4) is satisfid. In th maind, it is assumd that is xactly known and thfo Q = considd thoughout. Moov, sinc th convgnc spd is invsly popotional to th lft hand sid of (4) [7], fast convgnc is achivd if L(ρ) accuatly appoximats G (ρ). This aim is fomalizd as follows. Poblm (Lft-invs matching). Givn, obtain, L (ρ) = ag min L(ρ) max ρ[k] D L(ρ) L. Sinc L(ρ) is stabl and possibly non-causal, xisting LPV H -synthsis appoachs cannot b applid dictly to solv Poblm. In th nxt sction, an xtnsion of fixd lag smoothing [8] to LPV systms is considd to povid a suitabl appoach to Poblm. Fist, it is shown that fdfowad contol lads to a simila ight-invs modl matching poblm. B. Fdfowad Contol fo LPV systms In fdfowad contol, th systm is pcompnsatd by a filt L(ρ) to duc th tacking as is shown in Figu. A nom-optimal appoach to fdfowad dsign fo LPV systms, which allows fo non-causal contolls, can b fomulatd as follows. Poblm (Right-invs matching). Givn, obtain, L (ρ) = ag min L(ρ) max ρ[k] D L(ρ) L. (5) 359

3 Not that in th SISO LTI cas, Poblm and a idntical. Howv, fo LPV systms this is not th cas, sinc th commutativity popty is lost. Th implication of this diffnc will bcom appant in th nxt sction, wh a solution is povidd by xtnding th concpt of pviw contol [9] to LPV systms. III. OPTIMAL PREVIEW-BASED LPV INVERSES Th aim of this sction is to obtain a mthod fo solving Poblm and. Fist, an analytic solution is tatd, which is only applicabl to a limitd class of systms. Extnding this dict appoach to gnal systms is hampd by two challngs. Ths a addssd by poposing an optimal H -synthsis appoach with fou ky commndations on th stuctu of L(ρ) that follow fom th psntd analysis. Ths fomulations constitut contibution C. A. Invting LPV Systms of Rlativ Dg Zo Not that Poblm and a solvd fo L(ρ) = G (ρ). In this sction, th alization of th invs of LPV systms, with lativ dg zo, is considd to analyz th stuctu of th optimal solution L (ρ). To this nd, th lativ dg is dfind as follows. Dfinition (Rlativ dg). A systm Σ(ρ) is said to hav lativ dg κ N, wh κ is th lowst numb such that u[k] appas xplicitly in y[k + κ] fo som admissibl ρ. In th analysis to com, it is assumd that th lativ dg is constant fo all admissibl ρ. Fo systms with κ =, G (ρ) allows th following stat-spac alization. Lmma. Lt u y b psntd by () with κ =. Thn, G (ρ) y u is givn by, G (ρ) [ A(ρ) B(ρ)D (ρ)c(ρ) D (ρ)c(ρ) B(ρ)D (ρ) D ]. (ρ) Th poof follows fom invting (b) and substituting it in (a). In pactic, this dict appoach povids a solution only fo a limitd class of systms du to th following. I If κ >, G (ρ) is non-causal and cannot b dscibd by a stat-spac alization as givn by (). I This appoach dos not guaant that L(ρ) is stabl. Mo spcifically, if th fozn systm G( ρ) is NMP, G ( ρ) is unstabl, an hnc G (ρ) is unstabl. Challng I is tackld in th nxt subsction by shifting th input o output of th systm fowad in tim, which is also known as pviwing. In th subsction aft that, this pviw-basd appoach is combind with H -optimization to povid a complt solution to Poblm and. B. Towads Rlativ Dg Zo Challng I can b ovcom as follows. I: Apply κ sampls input- o output pviw to to obtain Ḡ(ρ) = o Ḡ(ρ) =, spctivly. II: Dtmin = Ḡ (ρ) by mans of Lmma. III: Apply κ sampls output- o input pviw to spctivly, by composing th non-causal filt as L(ρ) = q κ o L(ρ) = q κ, spctivly. Not that th non-commutativ natu of with LPV systms implis that whn input-pviw is applid to, output-pviw nds to b applid to to maintain quivalnc, and vic vsa. Stp I can b pfomd using stat-spac manipulations, as is psntd in th following Lmma, whas th filt L(ρ) that sults fom stp III dos not allow a stat-spac psntation sinc it is non-causal. Lmma (Input- and output-pviwing). Lt G(ρ[k]) u[k] y[k] b a systm with lativ dg κ. Thn, Ḡ(ρ) u[k +κ] y[k], whos input is pviwd by κ sampls with spct to th input of, is givn by, Ḡ(ρ) [ A(ρ[k]) Aκ (ρ[k])b(ρ[k]) C(ρ[k]) C(ρ[k])A κ (ρ[k])b(ρ[k]) ]. Similaly, th alization of th output-pviwd systm Ḡ(ρ) u[k] y[k + κ], is givn by, Ḡ(ρ) [ A(ρ[k]) B(ρ[k]) ], C(ρ[k],..., ρ[k + κ]) D(ρ[k],..., ρ[k + κ]) κ C(ρ) = C(ρ[k + κ]) j= κ D(ρ) = C(ρ[k + κ]) j= A(ρ[k + j]), A(ρ[k + j])b(ρ[k]). Th poof follows by applying th appopiat stat tansfomation to Ḡ(ρ) in th cas of input-pviwing, as will b psntd lswh. Th output-pviwing cas follows dictly fom cusiv valuation and dfining th output. Only if is stabl will this dict appoach solv Poblm and. In th nxt sction, stability of is nsud though an H -synthsis appoach that includs pviw to nabl non-causal solutions L(ρ) in th sam mann as psntd in this sction. In doing so, Lmma povids insight on th stuctu of th optimal solution L (ρ) = G (ρ). Namly, whn input pviwing is applid, Ḡ(ρ) only dpnds on ρ[k], whas fo output-pviwing, Ḡ(ρ) also dpnds on its tim shifts ρ[k + ],..., ρ[k + κ]. Ths insight a xploitd in nxt sction to povid th stuctu of L(ρ) in th optimization appoach. C. Optimal Pviw-basd Itativ Laning Contol Th dict pviw-basd appoach only povids a solution fo a limitd class of systms, sinc it dos not addss I. Fo Poblm, this is ovcom by cognizing that by using input-pviwing, th fnc can quivalntly b dlayd as is shown in Figu.II. This fomulation is spcially usful sinc both Ḡ(ρ) and a causal, and consquntly, an H poblm can b fomulatd, whos solution sults in a stabl contoll. Poblm 3 (Lft-invs matching with pviw). Givn and a numb of pviw sampls τ, obtain, L (ρ) = ag min max ρ[k] D q τ H. (6) 35

4 I: u j u Ḡ(ρ) u j+ u I: Ḡ(ρ) II: u j u q τ u j+ u II: q τ III: u j u u j+ u L(ρ) III: L(ρ) Fig.. I: Fo with lativ dg κ, th input is pviwd by τ = κ sampls, thby making it bipop. Consquntly, th invs also has lativ dg. II: This is quivalnt to dlaying th upp path, which nabls H -synthsis tchniqus to computing. Additional pviw can b addd to compnsat NMP dynamics, i.., τ κ. III: Th non-causal filt L(ρ) is obtaind by pviwing th output of. This poblm can b solvd using xisting tchniqus [4], [5], [6] and this appoach thby ovcoms I and I latd to th dict appoach. Th complt solution to Poblm is dpictd in Figu, wh all th schms a quivalnt mappings. In Figu.II, th causal filt is obtaind by solving Poblm 3, whas in Figu.III th non-causal filt L(ρ) is obtaind by output-pviwing. Ralizing that th stuctu of is vald by applying Lmma to th alization of Lmma, lads to th following commndations fo synthsizing laning filts fo LPV systms by mans of solving Poblm 3. R It is sufficint fo to dpnd xplicitly on ρ[k] only in od to includ th cas of pfct tacking. R Ca should b takn that allows a sufficintly ich dsciption of th functional dpndnc of ρ to match G (ρ). Fo xampl, whn dpnds on ρ in an affin o polytopic fashion, Ḡ (ρ) gnally dpnds on ρ in a ational way. Consquntly, mthods fo solving Poblm 3 that allow ational dpndnc in a commndd,.g. Lina Factional Rpsntation (LFR) synthsis tools, [4], [6]. R3 Th numb of pviw sampls should b τ κ, wh κ is th lativ dg of to includ th cas of pfct tacking. Taking τ κ may b ncssay whn G( ρ) has NMP dynamics fo som ρ D, in od to guaant a satisfying solution. D. Optimal Pviw-basd Fdfowad Contol Along th sam lins as in th pvious subsction, issu I is ovcom fo Poblm by mans of pviw-contol, which sults in th following fomulation. Poblm 4 (Right-invs matching with pviw). Givn and a numb of pviw sampls τ, obtain, L (ρ) = ag min max ρ[k] D q τ H. (7) In this cas, th fnc can only b quivalntly dlayd by applying output-pviwing as is shown in Figu 3. Consquntly by Lmma, th optimal solution L (ρ) Fig. 3. I: Fo with lativ dg κ, th output is pviwd by τ κ sampls, thby making it bipop. Consquntly, th invs also has lativ dg. II: This is quivalnt to dlaying th upp path, which nabls H -synthsis tchniqus to computing. Additional pviw can b addd to compnsat NMP dynamics, i.., τ κ. III: Th non-causal filt L(ρ) is obtaind by pviwing th input of. dpnds on ρ[k],..., ρ[k + τ]. This lads to th following commndation fo synthsizing Fdfowad contolls of LPV systms by mans of Poblm 4. R4 In addition to commndations R and R3, should dpnd xplicitly on ρ[k],..., ρ[k + τ] in od to includ th cas of pfct tacking. This can b achivd by using paamt-dpndnt Lyapunov function-basd synthsis [7]. In th aim of this sction is to solv Poblm and. This is achivd by th fomulation of Poblm 3 and 4 in combination with commndations R-R4, which followd fom analyzing th invs of LPV systms. Ths appoachs can b viwd as th LPV xtnsions of fixd-lag smoothing and pviw contol spctivly. IV. SIMULATION EXAMPLE In this sction, ILC and fdfowad contol a applid to an xampl systm with position-dpndnt dynamics by applying th pocdu as outlind in Sction III to obtain th non-causal filts. A compaison is mad with an LTI appoach which shows th lvanc of this wok and constituts contibution C3. A. Plant Dsciption Consid th positioning systm as shown in Figu 5. Accuat positioning of th caiag in th y-diction is hampd by a sonanc mod that vais as a function of its x-position as is schmatically shown in Figu 5. A symmtic foc is applid to bam u = u l = u and th output is th avag of th collocatd snsos, y = (y l+y ). A simplifid polytopic LPV modl with κ = is givn by, [ ( ρ[k])a + ρ[k]a B ], D = [, ], C A = , B = A = , C = β

5 G(ρ ) vtx vtx 4 [k] ρ[k].8.6 [-] G(jω, ρ ) [db] ω [Hz] u[k] [k].8 Fig. 4. Th fozn dynamics G(ρ ) display a paamt-vaying sonanc. [-].6 ρ yl ul x y u wh fo β = in th C-matix, th fozn bhavio is shown in Figu 4, which smbls th dynamics of a positiondpndnt motion systm, sampld at khz. Th aim is to accuatly pfom th motion, whil th paamt ρ vais as shown in th upp plot of Figu 6. B. Fdfowad Contol Rcommndations R and R4 as givn in Sction III-D stat that L (ρ) should b adquatly paamtizd in tms of th dpndncy on ρ. Th stuctu of th optimal L (ρ) is vald by applying Lmma to with output pviw fo κ =, followd by applying Lmma. This sults in, ( ρ[k])a + ρ[k]a ( ρ[k])ca + ρ[k]ca B ], CB and subsqunt application of Lmma to G (ρ) yilds, G [ Fig. 6. Upp: th fnc and schduling tajctoy ρ. Low: th optimal fdfowad input u = L (ρ), which sults in pfct tacking as is vidncd by th tacking o. L (ρ) is obtaind as in Sction III. Fig. 5. Th mod shaps of an industial flat bd pint dpndnt on its configuation. A simplifid modl of this bhavio is usd h. G (ρ) [.5 t[s] y ( ρ[k])a + ρ[k]a (CB) C(( ρ[k])a + ρ[k]a ) A = [I B(CB) C]A, B(CB) ], (CB) A = [I B(CB) C]A. It can thus b concludd that G also dpnds on ρ in a polytopic fashion and it only dpnds xplicitly on ρ[k]. Fo this cas, Poblm 4 is solvd as psntd in [5]. Th obtaind solution to poblm 4 fo τ =, yilds an upp bound on th H -nom in (7) of γ = Applying th obtaind fdfowad contoll as shown in Figu, lads to u and, as shown in th low plot of Figu 6. This shows that th computd contoll indd achivs pfct tacking. Altnativly, th analytically dtmind G (ρ) povs to b quadatically stabl [3] and thus th dict appoach yilds th xact solution fo this cas. γ τ Fig. 7. Th achivabl pfomanc bound γ as a function of th numb of pviw sampls τ. This shows that th convgnc at can b incasd by incasing τ, sinc γ is invsly popotional to this at. invs is not stabl and th pocdu dscibd in Sction III-C is ky to computing convgnt laning filts. Along th sam lins, it is concludd that a polytopic paamtization of L (ρ[k]) is sufficint. Following commndation R3, poblm 3 is solvd fo incasing valus of τ. Indd, this sults in an upp bound γ on th H -nom in (6) that dcass as τ incass, as is shown in Figu 7. Th obtaind contoll fo τ = 35 is usd fo ILC by updating th input as givn by (3) with Q =. Th pogssion of is shown as a function of th itations in Figu 8. This shows that that convgs to zo at a stady pac which sults in xcllnt tacking pfomanc aft itations. D. Compaison to ILC fo LTI Systms To highlight th lvanc of this wok, an LTI laning filt is implmntd and compad to th LPV cas. Sinc ρ[k] avags aound, th LPV laning filt L(ρ ) is fozn at ρ = to obtain a snsibl LTI filt. It tuns out that application of this filt dos not lad to a convgnt ILC du to th vaying sonanc. This can b visualizd by considing th LTI convgnc cition [5] fo fozn dynamics, i.., considing (4), ρ D, wh ρ[k] = ρ k, Q(jω )( L(jω, ρ )G(jω, ρ )) <, C. Itativ Laning Contol Nxt, th systm is considd fo β = which sults in fozn NMP dynamics ρ D. Consquntly, th analytic ω [, π). (8) This condition is ncssay, but not sufficint in cas ρ[k] is tim-vaying. Fo th considd LTI filt with Q =, 35

6 [-] L (ρ) L ( ρ), ρ = # itations u[k] [k] t [s] Fig. 8. Upp: convgs fo th LPV laning filt L (ρ), whas fo th LTI filt L ( ρ), with ρ =, additional obustnss is quid as is shown in Figu 9, sulting in limitd pfomanc. Low: u and that sult aft itations with th LPV filt, which show that pfct tacking is achivd.. [db] L( jω, )G(jω, ρ) Q( jω ) ( L( jω, )G(jω, ρ) ) - - ω [Hz] Fig. 9. Visualization of (8) fo th LTI filt L( ρ) with ρ =. This shows that th cition is violatd du to th vaiation in sonant dynamics. To satisfy (8), obustnss nds to b addd, which is th main dawback of applying LTI tchniqus to LPV systms. is (8) is violatd as is shown in Figu 9. To satisfy (8), a obustnss filt is addd. Evn though satisfying (8) is not sufficint, th ILC convgs as is shown in Figu 8. Howv, th achivd pfomanc is not as good as fo th full LPV cas. Th latt did not qui any obustnss filt, sinc th paamt-vaying dynamics a xplicitly taking into account by th laning filt. 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