Analysis and experimental validation of a sensor-based event-driven controller 1

Transcription

1 Analysis and ximntal validation of a snso-basd vnt-divn contoll 1 J.H. Sand Eindhovn Univsity of Tchnology Dt. of Elct. Eng. Contol Systms gou W.P.M.H. Hmls Eindhovn Univsity of Tchnology Dt. of Mch. Eng. Contol Systms Tchnology gou S.B.F. Hulsnboom Océ Tchnologis BV Vnlo P.P.J. van dn Bosch Eindhovn Univsity of Tchnology Dt. of Elct. Eng. Contol Systms gou Abstact: A challnging oblm in th dsign of high-tch systms is svo contol at high accuacy and at low cost ic fo its imlmntation. Convntional solutions a oftn not fasibl as high solution ncods a fa too xnsiv and high saml fquncis a ohibitiv as contolls hav to un on low-cost ocssos with ocssing ow that is shad with many oth tass. As a ossibl solution w snt an vnt-divn contoll that is basd on an (xtmly) low solution ncod. Th contol valu is udatd at ach momnt that an ncod uls is dtctd yilding zo masumnt o. Howv as th tim btwn two contol udats is vaying now convntional contoll dsign mthods do not aly as thy nomally assum a constant saml tim. To dal with this oblm th contoll dsign is fomd by tansfoming th systm quations fom th tim domain to th osition (satial) domain in which th ncod ulss and thfo th contoll tigging a quidistant. In this way th contol oblm is wittn as a synchonous oblm fo a nonlina lant. A gain schduld contoll is dsignd and analyzd in th satial domain. This vnt-divn contoll is ximntally validatd on a ototy int in which a 1 uls volution ncod is usd to accuatly contol th motion of imags though th int. By mans of analysis simulation and ximnts w show that th contol fomanc is simila to th initially oosd industial contoll that has fixd (high) saml fquncy and is basd on a much high ncod solution. On to of this th oosd vntdivn contoll involvs a significant low ocsso load. 1. Intoduction On of th challnging oblms in th dsign of a int but also in many oth high-tch systms is th svo contol of sval motos at high accuacy. Convntional solutions a oftn too xnsiv as thy ly on high solution ncods. High saml fquncis a also ohibitiv as contolls hav to un on low-cost ocssos with ocssing ow that is shad with many oth tass. Tyical ncod solutions a 500 ulss volution (PPR) with contolls unning at 500 Hz. By using ths high solution ncods masumnt quantization os can b nglctd. Howv to th systm cost limitd ou aim is to us a low solution ncod to contol a DC-moto. Howv now th quantization os bcom significant. To still achiv satisfactoy contol fomanc this quis an adatation of th convntional contol algoithms. Most alid and sachd solutions that dal with low solution snso data us an obsv-basd aoach to stimat th data at synchonous contoll saml momnts basd on asynchonous masumnt momnts [34789]. In ths solutions th continuous-tim lant is tanslatd into a disct-tim modl which is timvaying as it dnds on th tim btwn succssiv masumnt instants. Th aoachs in [47] us Kalman filting. In [8] a Lunbg-ty obsv is alid to us asynchonous masumnt data in combination with a multi-at contoll schm. In tacing alications a wll-nown tchniqu is th αβtac [3] to stimat osition vlocity and acclation in a tim-disct mann. An ovviw of ths 1 This wo is at of th Ph.D. thsis [1] and has bn caid out as at of th Bodc ojct und th sonsibility of th Embddd Systms Institut. This ojct is atially suotd by th Dutch Ministy of Economic Affais und th Snt TS ogam. Cosonding autho: Tl: / Fax: j.v.d.bosch@tu.nl

2 mthods and a comaison btwn thm is sntd in [9] in th alication of using otical incmntal ncods to masu osition and vlocity. Th main dawbac of ths mthods is that thy gnally qui a high comutational ffot fo comuting th obsv stimats. In this a w will us a diffnt contol aadigm that is siml to imlmnt and dos not suff fom th addd comlxity of an obsv. Th contol stuctu is an asynchonous contol schm in which th contol udats a tiggd by th osition masumnt (ncod uls). Th ida of th asynchonous contoll is basd uon th obsvation that th osition is xactly nown at an ncod uls and thus th is no nd fo an obsv. Howv as th vlocity of th moto vais ov tim both masumnt and contol udats a not quidistant in tim. This quis a comltly nw dsign statgy fo ths vnt-divn contolls of which initial oosals w mad in [6].. Poblm dscition W consid th contol of a DC-moto that can b dscibd by th scond-od modl & θ ( t) = ω( t) 1 (1) & ω( t) = B ω( t) + u( t) d( t) J R R wh θ(t) [ad] is th angula osition of th moto axis ω(t) is its angula vlocity [ad/s] u(t) th moto voltag [V] and d(t) th distubanc toqu [Nm] at tim t. Th moto aamts a obtaind fom data shts of th moto manufactu: th moto intia J m = gm th moto toqu constant = 0.08 Nm/A th moto sistanc R = 1.0 Ω and th moto daming B = Nms/ad. Th intia of th load J l = gm is addd to th moto intia to obtain th total intia J. Th industial quimnts fo thoughut and inting accuacy in th int sult in a fdbac and fdfowad contoll combination such that: Th dviation fom th stady-stat osition o is at most 0.5 ad duing inting. Only dviations fom a constant osition o a visibl in th int quality. All lvant distubancs a jctd sufficintly. 3. Evnt-divn contol Fo th vnt-divn contoll w oos to xcut both th masumnt and th contol udat at th momnt of a ncod uls. Th contol udats a not quidistant in th tim domain in this stting which hams th us of classical tim-divn contol schms. Howv w can aly vaiations of classical dsign mthods if w dfin th modls of th lant and th contoll in th satial (angula osition) domain instad of th tim domain as initially oosd in [6]. This ida is basd on th obsvation that th ncod ulss aiv qually sacd in th satial domain as th ncod ulss hav an quidistant distibution along th axis of th moto. To us this asoning w fist hav to tansfom th moto modl as givn in quation (1) to an quivalnt modl in which th moto angula osition is th indndnt vaiabl. Aft that w will show how th contoll dsign can b fomd using classical contol thoy. 3.1 Tansfomation to satial domain Th tansfomation idas a xlaind in [6] by th authos which stats: d θ dt ( t) = ω( t) ( θ ) = 1 dt ω( θ ) () wh ω(θ) dnots th angula vlocity of th moto and t(θ) dnots th tim sctivly at which th moto achs osition θ. Und th assumtion that ω(t) 0 (th moto dos not chang diction) fo all t>0 a on-toon cosondnc btwn θ and t xists and an intchang of thi ols is ossibl. Using () w obtain th moto modl in th satial domain: dt 1 ( θ ) = ω( θ ) dω 1 d ( θ ) θ = J ω( θ ) R y( θ ) = t( θ ) + B + R ( θ ) u ω θ (3)

3 wh d(θ) and u(θ) dnot th distubanc toqu and th moto voltag sctivly at moto osition θ. Not that tim t is now a function of θ and bcoms a stat vaiabl in this nw dscition. To consid th distubanc d as a function of th angula osition θ is an advantag fo many contoll dsigns as distubanc is oftn could to th angula osition instad of tim. Moov th outut y(θ) is now th tim t(θ) at which th moto achs osition θ. Th o that is inut fo th fdbac contoll is now slctd to b th diffnc btwn th masud tim of an ncod uls (t m (θ)) and th tim at which th ncod uls idally should hav occud basd on th fnc tajctoy (which is dnotd by t (θ)): ( θ ) = t ( θ ) t ( θ ) t m (4) Figu 1: Eos in t and θ In figu 1 it is shown how th tim o can b tanslatd into a osition o. Whn ω is constant and nonzo in th tim intval (t (θ )t m (θ )) whn t m (θ )>t (θ ) o (t m (θ )t (θ )) whn t m (θ )<t (θ ) wh θ is th angula osition at an ncod uls dtction thn it holds that θ ( t ) = ω t ( θ ) Whn ω is not constant quation (5) can b usd as an aoximation with ( θ ) ω o ω. To satisfy th contol objctiv that distubancs a sufficintly jctd w consid th fquncy contnt of ths distubancs in th satial domain: th satial fquncy [ad -1 ]. Th satial fquncy is a chaactistic of any stuctu that is iodic acoss osition in sac. It is a masu of how oftn th stuctu ats unit of distanc (comltly analogous to 'odinay' fquncy with sct to tim). Th conct of satial fquncy is scially usd in wav mchanics and imag ocssing [5]. Fo many iodic movmnts th satial fquncy contnt is almost constant whil th gula fquncy (in Hz) changs with changing sds. Fo th distubancs in th xaml th main comonnt is locatd at a satial fquncy of ad -1. Oiginally fo tim-divn contol this valu was usd in combination with th maximum vlocity of th moto to quantify th bandwidth with sct to tim - in Hz. This claly lads to non-otimal dsigns whn th moto also uns at low vlocitis. 3. Contoll dsign Th sulting modl (3) is non-lina. It can b linaizd aound stady-stat tajctois. Th stady-stat tajctoy is chosn staightfowadly at a constant angula vlocity ω of th moto aft stat-u: 1 ( Bω + d ) R t d u = d + ω θ ω ω (6) ω wh d is chosn as th man valu of th distubanc d. Th vaiations aound this stady-stat tajctoy a dnotd by ( t ω d u). Hnc t=t + t ω=ω + ω tc. Th linaizd dynamics a d t 1 ( θ ) = ω( θ ) ω d ω y 1 Jω ( θ ) = + B ω( θ ) d( θ ) + u( θ ) ( θ ) = t( θ ) R W can now dsign a fdbac PD contoll that comnsats fo th fist-od vaiations u(θ) as dfind in (7). Th contol valu alid to th lant should b u = u + u. As Bω R d R u = ω + + th fd-fowad tm R t (5) (7)

4 tas ca of BR + ω and d has to b comnsatd fo by th PD contoll. Using th modl aamts fom sction w can comut th fd-fowad u = K ff ω : BR K ff = + = W hav chosn to dsign a PD contoll and tund it in th disct osition domain. W aim at using a minimal ncod solution of only 1 PPR. Choosing this ultimatly low solution has th advantag that w do not hav to dal anymo with inaccuat snso distibutions along th moto axis. To tun th contoll quation (7) was fist disctizd in th satial domain using a saml distanc of π ad. ( K + K ) Th PD contoll was chosn as d Kd Hc 1( ) = (8) wh th notation ẑ is usd instad of z to mhasiz that th disctization has bn mad in th satial domain instad of th tim domain. Futhmo this vnt-divn contoll tas th tim o as inut. To find th valus fo th contoll aamts K and K d w usd th oot-locus dsign mthod. Fo ω = and 500 ad/s and choosing K = 1.0 and K d = 1 w obtain th oot-locus as dictd in figu. Th `+' mas indicat th oots fo fdbac gain 1. Th oot-locus shows diffnt oots fo th diffnt sds. Fo ω t = 00 ad/s th oot-locus vn indicats unstabl bhavio. Thfo contoll (9) is suggstd fo which ( K + K ω ) th oot-locus is lottd in figu 3. d d H ( ) = ω (9) c K ω Figu : Root-locus fo contoll (8) Figu 3: Root-locus fo contoll (9) W comutd th snsitivity function (fo th linaizd disctizd lant) as th tansf function fom th distubanc d(θ) to u d (θ). Both signals a indicatd in figu 4 which shows a gahical sntation of quation (7) togth with fdbac contoll (9) fo ω = 388 ad/s. Fo th oth valuatd vlocitis th lot loos simila. Not that ŝ is usd instad of s to mhasiz that th intgation is fomd in th satial domain. Th Bod magnitud lot of this disct-osition tansf is dictd in figu 5. Satial fquncis u th 0.01 ad -1 a attnuatd. This satisfis th quid bandwidth as givn in sction 3.1 bing ad -1. Figu 4: Gahical sntation of linaizd continuous dynamics (7) Figu 5: Bod mag. lot of tansf fom d(θ) to u d (θ) fo ω = 388 ad/s.

5 4. Eximntal sults Eximnts hav bn caid out on a comlt ototy documnt inting systm at Océ Tchnologis BV fo both th oiginally imlmntd obsv-basd contoll and th vnt-divn contoll in which a constant fnc vlocity of 388 ad/s is tacd. Th obsv-basd contoll oats with an ncod having a solution of 1 PPR and at a saml fquncy of 50 Hz [1]. Th ximntal sults fo both contolls a givn in figus 6 and 7. Ths figus show th osition o duing inting ov 5 sconds (aft stat-u). In this iod 5 shts a intd at a sd of 80 ags minut. Comaing th sults in figus 6 and 7 w obsv simila contol fomanc fo th obsv-basd contoll and th vnt-divn contoll both within sc. Th maximum dviation fom th avag osition o is fo both contolls about 0.15 ad which is small than 0.5 ad as quid. Howv th vnt-divn contoll oats with an ncod with a solution that is a facto 1 low than that fo th obsv-basd contoll. Futhmo th obsv-basd contoll uns at a (constant) contol saml fquncy of 50 Hz and th vnt-divn contoll at a much low avag fquncy (aoximatly 6 Hz). Th os causd by th sht assings can b distinguishd in both figus although th a mo distubancs (at diffnt fquncis) acting on th systm as can b sn fom th masumnt data. Figu 6: Eximnt obsv-basd contoll with Figu 7: Eximnt vnt-divn contoll 1 PPR ncod and saml fq. 50 Hz. with 1 PPR ncod. 5. Conclusions In this a w sntd th us of vnt-divn contol to accuatly contol a DC-moto on th basis of an ncod having a solution of 1 PPR. By mans of analysis and industial ximnts w showd that th fomanc of th contoll satisfid th quimnts. Futhmo w showd that simila contol fomanc could b achivd comad to th initially oosd industial obsv-basd contoll. Th advantags of th vnt-divn contoll ov th obsv-basd contoll can b summaizd as follows: Only a cha ncod with a solution of 1 PPR ncssay instad of 1 PPR Low comutational load fo th ocsso (facto 5 duction) Low comlxity of contol algoithm (numb of tuning aamts instad of 4) Eximnts on a ototy int validat th sults and a vy omising. Futh sach is still ndd to fully comhnd such vnt-divn contolls and giv systmatic synthsis mthods. Rfncs 1. Sand J.H. (006). Evnt-divn contol in thoy and actic tad-offs in softwa and contol fomanc. Ph.D. thsis TU Eindhovn ISBN: Åzén Kal-Ei (1999). A siml vnt-basd PID contoll. In: Pocdings of th 14th Wold Congss of IFAC. Bijing P.R. China vol Fidland B. (1973). Otimum stady-stat osition and vlocity stimation using samld osition data. In: IEEE tansactions on Aosac and Elctonic Systms. vol. AES-9 no Glad T. and L. Ljung (1984). Vlocity stimation fom igula noisy osition masumnts. In: Pocdings of th IFAC 9th Tinnial Wold Congss Budast no Goodman J. (005). Intoduction to foui otics. 3d d. Robs & Comany Hmls W.P.M.H. R.J.A. Got A. van Zijl P.P.J. van dn Bosch S. Wiland W. Hndix and M. Vond (1999). Asynchonous masumnt and contol: a cas study on moto synchonization. In: Contol Engining Pactic vol Kucinsi M. Clot C. Tomizua M. and R. Hoowitz (1998). Asynchonous obsv fo a coi a ath. In: Pocdings of th 37th IEEE Confnc on Dcision and Contol vol Phillis A.M. and M. Tomizua (1995). Multiat stimation and contol und tim-vaying data samling with alications to infomation stoag dvics. In: Poc. Amican contol conf. vol Vottis C.V. (003) Extacting mo accuat osition and vlocity infomation fom otical incmntal ncods SAI/y thsis Tchnisch Univsitit Eindhovn ISBN