This article presents the design

Transcription

1 Dgnng Rbut Frc Cntrl f Dt Sytm and Envrnmntal Uncrtant By Navd Nkfat and Narman Shr Th artcl rnt th dgn f a rbut frc cntrllr fr a hydraulc actuatr ntractng wth an uncrtan nvrnmnt va quanttatv fdback thry (QFT). Aftr th drvatn f a raltc nnlnar dffrntal quatn mdl, a lnarzd lant tranfr functn dvld. Th ffct f nnlnart ar accuntd fr by dcrbng th lnarzd mdl aramtr a tructurd uncrtanty. Th mact f nvrnmntal varablty a wll a varatn n hydraulc cmnnt aramtr ar al ncludd a uncrtanty n th mdl. Th QFT dgn rcdur carrd ut t dgn a rbut cntrllr that atf rfrmanc cfcatn fr trackng and dturbanc rjctn. Th dgnd cntrllr njy th mlcty f fxd-gan cntrllr, ay t mlmnt, and at th am tm rbut t th varatn f hydraulc functn a wll a nvrnmntal tffn. Th cntrllr mlmntd n an ndutral hydraulc actuatr qud wth a lw-ct rrtnal valv. Th xrmntal rult hw that rbut tablty agant ytm uncrtant and undr varyng cndtn achvd and th rfrmanc gal ar atfd. Intrductn Many ndutral alcatn, uch a manufacturng autmatn and matral handlng, nvlv ntractn wth th nvrnmnt. In th alcatn, frc cntrl rqurd. Hydraulc actuatr ar advantagu fr uch a- ABB Cntrl Valv. Nkfat and Shr (narman@cc.umantba.ca) ar wth th Dartmnt f Mchancal Engnrng, Th Unvrty f Mantba, Wnng, Mantba, RT 5V6 Canada. 7-78//$. IEEE 66 IEEE Cntrl Sytm Magazn Arl

2 lcatn bcau f thr hgh frc-t-wght rat and fat rn tm. Addtnally, thy ar abl t mantan thr ladng caacty ndfntly, whch wuld uually cau xcv hat gnratn n lctrcal cmnnt []. Unlk n lctrc actuatr, hwvr, frc cntrl n hydraulc actuatr a dffcult rblm [], []. In a hydraulc actuatr, th cntrl gnal actvat th l valv that cntrl th flw f hydraulc flud nt and ut f th actuatr. Th flw n turn cau a rur dffrntal buldu that rrtnal t th actuatr frc. Evn f th l valv dynamc ar gnrd, th cntrl gnal fundamntally cntrl th drvatv f th actuatr frc and nt th frc tlf. Furthrmr, hydraulc ytm ar hghly nnlnar and ubjct t aramtr uncrtanty; aramtr chang wth tm a a rult f varatn n ratng cndtn and cmnnt dgradatn. Fr xaml, th uly rur ubjct t varatn that may b gnratd by th ratn f thr actuatr n a multur nvrnmnt []. Th flw and rur cffcnt, charactrzng flud flw nt and ut f th valv, ar functn f lad and uly rur and can vary undr dffrnt ratng cndtn [5]. Al, th ffctv bulk mdulu n hydraulc ytm can gnfcantly chang undr varu lad cndtn, l tmratur, and ar cntnt n th l [6]. Dgn f a cntrllr n th fac f uch a rang f aramtr varatn and dturbanc challngng. Th artcl rnt th alcatn f QFT t th dgn f a rbut frc cntrllr fr hydraulc actuatr. In th ltratur, vral frc cntrl tratg hav bn rd fr hydraulc actuatr. Cnrad and Jnn [] ud cmbnatn f vlcty fdfrward, utut fdback, and a Lunbrgr brvr wth tat tmat fdback fr frc cntrl f a dubl-rd hydraulc actuatr. Th mulatn and xrmntal rult fr a cntant tnt frc hwd urr rfrmanc f th rd mthd vr cnvntnal (P r PI) frc fdback cntrllr. Hwvr, th varatn f lad and uly rur wr nt cndrd n thr tudy. Chn t al. [7] dgnd a ldng-md cntrllr fr a ngl-rd hydraulc actuatr ntractng wth a rng a an nvrnmnt. Ung tn, vlcty, acclratn, frc, and rur fdback, th varabl-tructur cntrllr rvd t b caabl n bth tatc and dynamc frc cntrl tak. Th ffct f rv-amlfr gan varatn wa al xamnd; hwvr, th ffct f varatn n nvrnmntal tffn wa nt tudd. Sun t al. [8] mlyd a ldng-md cntrllr wth a rturbatn brvr fr a ngl-rd lctrhydraulc ytm. Th ffct f cylndr tn and vlcty n th rur dynamc wa cndrd a rturbatn t th cntrl mdl, whch wa tmatd by an brvr. Th xrmntal rult vrfd mrvd tady-tat and trannt rfrmanc a cmard wth tradtnal rrtnal-ntgral-dffrntal (PID) cntrllr. Adatv cntrl tratg hav al bn cndrd fr hydraulc frc cntrl. Lu and Allyn [9] dvld a wtchng cntrl chm ung a Lyaunv-bad adatv law t rduc aramtrc uncrtanty. Th mlmntatn f th cntrllr, whch bad n th maurmnt f tn, vlcty, acclratn, rur, and l dlacmnt, hwd gd rfrmanc fr hgh-frquncy frc/ rur trackng. Wu t al. [] ald a gnralzd rdctv cntrl algrthm t a hydraulc frc cntrl ytm. Th cntrllr wa xrmntally valuatd fr varu nvrnmntal tffn and tnt. Th mthd, hwvr, rl havly n nln aramtr tmatn and cnquntly cmutatnally xnv. Laval t al. [] ud an H arach t rbutly cntrl th frc xrtd by a dubl-actng ymmtrc hydraulc cylndr wth a rvvalv. Th mrtanc f uncrtant n th nvrnmnt, maurmnt, and nnlnart n th rfrmanc f hydraulc frc cntrl ytm wa hghlghtd. Lmtd tt rult, dmntratng th achvmnt f a tablty/rfrmanc trad-ff utlzng an H arach, wr rntd. Dt th xtnc f a grat numbr f frc cntrl cnct, mthd, and algrthm, thr tll a larg ga btwn thry and ndutral ractc. Th ran hav bn acrbd t th r ndutral cntrl archtctur, whch d nt allw th mlmntatn f htcatd algrthm []. In th artcl, w mly th QFT tchnqu t dgn an xlct frc cntrllr fr an ndutral hydraulc actuatr. Th gal t arrv at a fxd-gan cntrllr that: ) f lw rdr and ay t mlmnt, ) rbut agant uncrtant n bth nvrnmntal tffn and actuatr functn, and ) d nt rqur xact knwldg f th ytm aramtr. QFT a rbut cntrllr dgn mthdlgy amd at lant wth aramtrc and untructurd uncrtant. Th cnct wa frt ntrducd by Hrwtz n th arly xt and wa latr rfnd by hm and thr nt a tchnqu []-[5]. QFT mhaz th fact that fdback nly ncary bcau f uncrtanty and that th amunt f fdback huld thrfr b drctly rlatd t th xtnt f lant uncrtanty and unknwn xtrnal dturbanc. Mnmzng th ct f fdback, a maurd by th amunt f cntrllr bandwdth, th man bjctv f QFT [6]. Thrfr, th lant uncrtanty and th cld-l tlranc ar frmulatd quanttatvly that th ct f fdback can b ad at ach tag f th dgn rc [7]. (S th dbar An Ovrvw f QFT n. 68.) Th mthd ha bn ald t a wd rang f ngnrng rblm, ncludng flght cntrl [8], rbt tn cntrl [9], and manufacturng ytm []. Rgardng th alcatn f QFT t hydraulc ytm, th tchnqu wa ud t cntrl an lctrhydrtatc actuatr a art f a flght cntrl ytm by Pachtr t al. []. Th cntrllr ffcacy wa valdatd by mulatn rult. Thmn and Krmr [] dvld a QFT cntrllr fr a varabl-dlacmnt um ytm bad n a lnarzd mdl wth Arl IEEE Cntrl Sytm Magazn 67

3 Quanttatv fdback thry (QFT) a frquncy dman mthd ntndd fr ractcal cntrl ytm dgn gvn rbut rfrmanc cfcatn. It gal t dgn a lw-bandwdth cntrllr that atf rfrmanc cfcatn,dt ytm uncrtant and dturbanc. A lw-bandwdth cntrllr a ky u n any ractcal dgn t avd rblm wth n amlfcatn,rnanc, and unmdld dynamc. A brf utln f th clacal QFT rblm rntd hr. A full tratmnt gvn n [] and [5]. Shrt vrvw can al b fund n [8] and [9]. QFT Cnfguratn A tw dgr-f-frdm ytm tycally aumd fr th QFT tchnqu whn nly utut Y and cmmand nut R can b maurd ndndntly. Plant P cntan ytm uncrtant and xd t unknwn dturbanc D. G th cntrllr, whch can hl rduc th varatn f th lant utut du t uncrtant and dturbanc,whl F ud mrly a a fltr t talr th rn t mt th cntrl ytm cfcatn. Th uncrtan lant can b dcrbd a P(, α) m n ( α) q ( α) whr α Ω R an uncrtan aramtr vctr. Ω a cmact t f aramtr varatn and may b gvn a { αα : [ α, α],,, } Ω whr ach uncrtan aramtrα (,..., ) var ndndntly wthn an ntrval. Chng a rfrnc aramtr vctr, An Ovrvw f QFT α,yld th nmnal lant P( ) P(, α ). Many rblm f ractcal ntrt can b xrd by th abv lant mdl wth aramtrc uncrtanty. QFT Dgn Prblm Th QFT dgn rblm t ynthz a ar f trctly rr, ratnal,tabl functn G ( ) and F ( ) uch that th fllwng cfcatn ar atfd whl th bandwdth f th cntrllr kt a lw a bl. Rbut Stablty: Th cld-l ytm T(, α) L (, α) L (, α) mut b tabl α Ω. L (, α) P (, α) G ( ) th n-l tranfr functn. Rbut Trackng: A trackng cfcatn fr th rfrnc nut gvn a T( ω) F( ω) T( ω, α) T ( ω) ω [, ) l whr th frquncy rn bund, T l ( ω) and T u ( ω ),ar cfd by th dgnr. Dturbanc Attnuatn: Th rqurmnt fr dturbanc rjctn at lant utut xrd a whr max T ( jωα, ) M ( ω) α Ω D u D aramtrc uncrtanty. Th cld-l frquncy dman cfcatn wr atfd, and th crrndng t rn wr fund t b ranabl. Th uccful alcatn f th mthd t a wd rang f ngnrng rblm rvd vdnc f t tntal ffctvn fr hydraulc ytm. On bjctv f th tudy t valuat and dmntrat th bnft f QFT whn ald t hydraulc frc cntrl ytm. Th artcl rganzd a fllw. In th nxt ctn, w drv a mathmatcal mdl f a hydraulc actuatr n cntact wth th nvrnmnt. Th mdl thn ud t cntruct a aramtrcally uncrtan lnar tranfr functn t dcrb th ytm. Th uncrtanty rflct varatn n ratng-nt-dndnt aramtr, chang n th nvrnmnt, and chang n hydraulc actuatr functn. Thn, w u th QFT mthd t dgn a ngl fxd-gan rbut frc cntrllr fr th abv uncrtan lant. Fllwng that, w dmntrat th cntrllr rfrmanc n an xrmntal hydraulc actuatr tt rg. Fnally, w rnt cnclun. Elctrhydraulc Actuatr Mdlng Prntatn f Nnlnar Dynamc Equatn A chmatc dagram f a hydraulc actuatr n cntact wth th nvrnmnt hwn n Fg.. In th tudy, th wll-knwn mdl f manulatr-nr-nvrnmnt [], [] culd wth th nnlnar hydraulc actuatr dynamc. Th nr wth tffn k and damng d cnnct th actuatr tn, rrntd by th ma m a, t th nvrnmnt. Th nvrnmnt rrntd by ma m, damng d, and tffn k. Th dynamc quatn ar mx a fa d( x x ) dx k( x x) () mx d( x x ) dx k( x x) kx () f k( x x) () 68 IEEE Cntrl Sytm Magazn Arl

4 T (, ) α D L (, α ) th tranfr functn frm th dturbanc t th utut and M D ( ω) th magntud f dturbanc rjctn. QFT Dgn Prcdur Th dgn rcdur nvlv th fllwng t. ) Gnratng th lant tmlat: Th rgn n th Nchl chart ccud by th cmlx valu f P( jωα, ) at ach frquncy calld th lant tmlat ( th fgur at rght). Th ntally charactrz th lant uncrtanty by caturng th gan and ha varatn f th lant at a gvn frquncy. Th tmlat ar ud t gnrat bund n th Nchl chart fr th cntrllr dgn. ) Gnratng rfrmanc bund: Th dgn cfcatn utlnd abv huld b tranlatd nt crtan bund n th nmnal n-l tranfr functn, L( ) P( ) G( ),t rval th trad-ff btwn rfrmanc cfcatn and rbutn at ach frquncy. Th bund ar drvd thr by mvng th lant tmlat btwn th cld-l magntud cntur n th Nchl chart r by a cmutr arch. Examl f bund ar hwn n th fgur. Th bund ar ud n th nxt t a a gud fr l hang th nmnal n-l tranfr functn. ) L hang: Onc th QFT bund ar dtrmnd,th nmnal l tranfr functn, L( ),huld b dgnd,by addng rr l and zr,t yld a tabl nmnal cld l,whl at th am tm atfyng all bund. A tycal lt f L( ) gvn n th fgur. An tmum L( ) n that atf th bund and dcra a radly a bl wth frquncy t k th cntrllr bandwdth mall. Durng th tag,th dgnr huld ffct a trad-ff btwn cnflctng cfcatn,cntrllr cmlxty,and th ct f th fdback n th bandwdth. Onc a atfactry L( ) arrvd at,th cntrllr can b xtractd frm L( )by dvdng t by th nmnal lant tranfr functn P( ). ) Dgn f rfltr: Dgn f a rr L( ) nly guarant that th varatn n th cld-l tranfr functn, T(, α ), l than r qual t that allwd. Thrfr,a rfltr rqurd t brng th rn wthn th ur and lwr tlranc, T l ( ω) and T u ( ω ). Th rfltr magntud bund at ach frquncy can b calculatd frm th rbut trackng cfcatn. On-L Gan (db) 6 5 Plant Tmlat at ω ω QFT Bund Nmnal L, L( jω) L( jω ) On-L Pha (dg) Tycal bund and nmnal l functn n th Nchl chart. ω ω ω ω ω whr f th nd frc and x, x, and d ar th actuatr dlacmnt, th nvrnmntal dlacmnt, and th quvalnt vcu damng cffcnt f th actuatr, rctvly. Th frc rgnatd frm th hydraulc actuatr, f a, fa A A () whr A and A ar th tn ffctv ara and and ar th nut and utut ln rur, rctvly. Th gvrnng nnlnar quatn dcrbng th flud flw dtrbutn n th valv ar wrttn, n thr mlt frm, a [5] x (xtnn) q cdwx ρ ( ) q c wx d ρ ( ) (5a) q cdwx ρ ( ) q c wx d ρ ( ) (5b) whr q and q rrnt flud flw nt and ut f th valv, rctvly. c d th rfc cffcnt f dcharg, ρ th ma dnty f th flud, th um rur, and th rturn (xt) rur. w th ara gradnt that rlat th l dlacmnt( x ) t th rfc ara. Cntnuty quatn fr l flw thrugh th cylndr, nglctng th lakag flw acr th actuatr tn, ar q A dx V d dt β dt (6a) x < (rtractn) q A dx V d. dt β dt (6b) Arl IEEE Cntrl Sytm Magazn 69

5 Hr β th ffctv bulk mdulu f th hydraulc flud, andv andv ar th vlum f flud trad at th d f th actuatr. Thy can b xrd a functn f actuatr dlacmnt a uncrtanty. In th ctn, th drvatn rcdur wll b fllwd t-by-t t hghlght th aumtn that ar mad durng th drvatn and mlfcatn. Th lnarzd flw rlatnh frm (5) ar Hydraulc actuatr ar advantagu bcau f thr hgh frc-t-wght rat and fat rn tm. V( x) V xa (7a) q K x K (9a) q K x K (9b) whr K (K ) and K (K ) rrnt flw and rur ntvty gan f th valv, rctvly. Surcrt and tand fr nut and utut, rctvly. Thy ar gvn a fllw: fr xtnn ( x ) fr rtractn ( x < ) V( x) V xa (7b) whr V and V ar th ntal vlum trad n th blnd and rd d f th actuatr. Th rlatnh f th l dlacmnt, x, and th nut vltag, u, t th rrtnal valv can b xrd a a frt-rdr ytm that accurat nugh t mdl lw-ct rrtnal valv [6] τ dx u k dt k x whr τ and k ar gan dcrbng th valv dynamc. Equatn ()-(8) xr th rlatnh btwn th cntact frc, f, and th nut cntrl vltag, u. Drvatn f Lnar Tranfr Functn Th nnlnar quatn drvd abv ar nw lnarzd abut an ratng nt t btan a lnar tranfr functn. Th bjctv t utlz th mdl wthn th QFT arach. A wll b n latr, th varatn f th ratngnt-dndnt aramtr wll b ncludd n th mdl (8) K cdw ( ) K cdw ( ) ρ ρ K cdw ( ) K cw d ( ) ρ ρ K K cwx cwx K ( ) ( ) d ρ d ρ cwx cwx d K ρ( ) ρ( ). d A can b n, K (K ) and K (K ) ar lad- and rur-dndnt varabl. Hnc, n th tudy, thy ar cndrd uncrtan but bundd aramtr. Aumng mall tn dlacmnt wthn th vcnty f th mdtrk, th fllwng arxmatn mad: V( x) V( x) V V C. β β β Thu, (6) can b wrttn n th Lalac dman a () x x Q AX CP (a) k k q A A q d m a d m d Q AX CP. (b) Subttutng (9) nt () and rarrangng fr ln rur, w hav x A P C K X K C K X (a) P P P Fgur. Schmatc mdl f th hydraulc actuatr ntractng wth th nvrnmnt. A P C K X K C K X. (b) 7 IEEE Cntrl Sytm Magazn Arl

6 Incrratng () nt () w hav F a K A C K K A C K X A C K A C K X. () Equatn ()-() ar nw cmbnd t frm th fllwng tranfr functn: ( m a d)( m ( d d) k k) Fa( ) F( ) k m d k ( d ( ) k) m d k m d k () Tabl. Oratng valu and aramtr rang rtanng t th lnar tranfr functn. Paramtr Nmnal Valu Rang k 75 (kn/m) 5- K.75 (m /a.).5-.5 K.5() (m /) -5() C.5() (m /a) () -() d 7 (N/m/) 6-8 m a (kg) A. (m ).9-. A.5 (m ).-.6 X( ) m ( d d) k k F( ) k m d k. (5) k. (m/v).-. τ 5 (m) - Subttutng F ( a )and X ( )frm () and (5) nt () and rarrangng t, th tranfr functn dcrbng th rlatn btwn cntact frc and l dlacmnt drvd a (6), hwn blw, whr ϕ( ) m d k, ψ( ) ma d, and ζ( ) d k. T mlfy th abv tranfr functn, K (K ) and K ( K ) ar rlacd by K and K. Hnc tranfr functn (6) rducd t (7), hwn blw. Any varatn r arxmatn n K (K ) and K (K ) ar nw ncludd a uncrtanty n K and K, rctvly. Th abv rlatn can b furthr mlfd f th tffn f th frc nr and th tn rd ar hgh cmard wth th nvrnmntal tffn and th hydraulc cmlanc. Hnc thr dynamc ar nt xctd durng th cntact and can b lumd tgthr a a rgd bdy. Bad n th aumtn, (7) rwrttn a F( ) K( A A)( m d k). X( ) ( K C)( ma ( d d) k) ( A A ) (8) Furthr, w aum th dynamc f th nvrnmnt dmnatd by a ur tffn, k. Th ty f nvrnmnt ha njyd ularty amng many rarchr [7]. Th ytm tranfr functn fnally wrttn n t mlt frm a F( ) Kk ( A A) X( ) ( K C)( ma d k) ( A A ). (9) Includng th dynamc f th valv frm (8), th tranfr functn btwn maurd cntact frc, F( ), and cntrl vltag, U( ), wrttn a F( ) k Kk ( A A) U( ) ( τ ) ( K C)( ma d k) ( A A ). () Equatn () nw cndrd a a aramtrcally uncrtan ytm. Fr xaml, th uncrtanty rang n K and K rflct varatn n th ratng nt, uly rur, and, n art, th rfc ara gradnt. Th varatn n th nvrnmntal tffn and damng f th ytm ar ncludd n aramtr k and d, rctvly. Furthrmr, th uncrtanty n C rrnt th chang n th flud bulk mdul and th vlum f flud trad at th d f th actuatr. Th uncrtanty n th valv charactrtc caturd n varatn f τ and k. All th aramtr ar knwn t hav an ffct n th ytm tablty. Tabl lt th nmnal valu f all aramtr n () and thr crrndng uncrtanty rang. Th valu, whch ar rrntatv f th hydraulc actuatr undr nvtgatn ( th Exrmntal Stu ctn), wr btand frm avalabl rfrnc (ncludng manufacturr cfcatn), drct maurmnt f m gmtrc aramtr, and ndrct dntfcatn bad n xrmntal brvatn f th rn [6]. Th uncrtan aram- F( ) X ( ) ( K ( )( ( ) ( ) k m d k K A K C K A K C C)( K C)( ψ( ) ϕ( ) ζ( ) ϕ( ) ψ( ) ζ( )) ( A ( K ( ) C) A( K C))( ϕ( ) ζ ( )) (6) F( ) kk ( A A) m d k. (7) X ( ) ( K C)(( ψ( ) ϕ ( ) ζ( ) ϕ( ) ψ( ) ζ( )) ( A A)( ϕ( ) ζ( )) Arl IEEE Cntrl Sytm Magazn 7

7 R () Prfltr F () Cntrllr tr ar grud nt a vctr, dntd a α. Thn th ytm n-l tranfr functn can b dfnd a F( ) P(, α ). U( ) Uncrtan Plant G () P (, α) Fgur. Tw-dgr-f-frdm QFT cntrl ytm. () Th lant valuatd at t nmnal ratng nt rfrrd t hr a th nmnal lant and hwn a P( ) P(, α ) () whr α th vctr cntanng all nmnal valu f th lant tranfr functn. D () Y () Cntrllr Synth Th gal f th ctn t dgn a rbut frc cntrllr fr th hydraulc actuatr that rrntd by th uncrtan tranfr functn (). A tycal tw-dgr-f-frdm fdback ytm cnfguratn n QFT hwn n Fg.. A trctly rr cntrllr, G ( ), and a trctly rr rfltr, F( ), ar t b dgnd uch that th fllwng cndtn ar atfd (radr ar rfrrd t [8] fr a dcrtn f th QFT dgn rcdur): ) Cld-l rbut tablty. Th acatd QFT rbutn cntrant n trm f th nmnal l tranfr functn, L ( ) P ( ) G( ), gvn by [8]: L ( ω) M. [, ) ( ω) ω L ( ω) ( ωα, ) () whch ml an arxmatly db gan margn fr th cld-l ytm. Magntud (db) Frc (Nrmalzd) 6 8 Ur Bund Lwr Bund Tu( ) T l () Frquncy (rad/) (a) Ur Bund Lwr Bund Tm () (b) Fgur. Frquncy and tm dman trackng bund. Magntud (db) Magntud (db) ω7 ω 5 Pha (dg) (a) Pha (dg) (b) ω5 ω.5 ω. 5 5 ω. ω.5 ω ω5 ω ω Fgur. QFT bund n Nchl chart wth: (a) uncmnatd lant and (b) nmnal l. 5 7 IEEE Cntrl Sytm Magazn Arl

8 ) Rbut rfrnc nut. T mt trackng rfrmanc rqurmnt, th cntrllr huld atfy th fllwng nqualty: T( ω) F( ω) l L ( ω) Tu( ω) ω [, ) ( ω) L ( ω) ( ωα, ) () whr th ur and lwr trackng bund ar dfnd a 8. Tu( ) 7 8 (5a) T ( ) l (5b) Th bund ar bult frm th tm dman fgur f mrt fr t rn uch a ak vrht, ak tm, and ttlng tm []. In th wrk, th drd lwr trackng bund, Tl ( ), bult t hav an vrdamd rn wth ttlng tm. Fr th ur, a mdl wth a ral l at 8. and a ar f cmlx l ar chn. Th ral l mut b mr dmnant than th cmlx l []. Mrvr, a hgh-frquncy l at 8 nrtd n Tl ( ), whch d nt affct th drd rfrmanc cfcatn but wdn th rang btwn T ( u )and T( ) l n th hgh-frquncy band. Th fgur f mrt fr ur bund, Tl ( ), ar a 5% ak vrht and ttlng tm. Tu ( ) lctd wth thr ral l and a zr. Th zr clr t th rgn than th l t hav an undrdamd rn. Th frquncy and tm dman lt f th bund ar hwn n Fg. (a) and (b), rctvly. ) Cld-l dturbanc attnuatn (ntvty rductn). Fr dturbanc rjctn at th lant utut, an ur tlranc md n th ntvty functn. Hr w cndr nly a cntant ur bund t lmt th ak valu f dturbanc amlfcatn a fllw: S ( ωα, ) max ( ω) ( ωα, ) ( ω) L ( ω) ( ωα, ) max M D ( ω). ω [, ]. (6) Th abv dgn cfcatn m cntrant n th allwabl l gan, L. Th cntrant can b hwn n a Nchl chart a a curv bundary at ach dgn frquncy. Onc th bund ar drvd, th nmnal n-l tranfr functn, L, huld b dtrmnd uch that t l n r abv th crrndng bund at vry frquncy. Th can b dn ung a rcdur that calld l hang n QFT. Fnally, th cntrllr can b xtractd frm L by dvdng by th nmnal lant tranfr functn, P. Dgn frqunc ar lctd a ω {.,.5,.,.5,,5,,5,7,} rad/. Th bund gnratd by cntrant (), (), and (6) ar hwn n Fg. (a) fr th uncmnatd lant. Th nmnal n-l tranfr func- Magntud (db) Magntud (db) Frc (N) Ur Bund Ur Bund Lwr Bund Frquncy (rad/) (a) Ur Bund Lwr Bund Frquncy (rad/) (b) Lwr Bund δ δ all Tm () (c) Fgur 5. Sytm rn vr rang f aramtrc uncrtanty (mulatn): (a) cld-l frquncy rn wthut rfltr, (b) cld-l frquncy rn wth rfltr, and (c) tm rn. Arl IEEE Cntrl Sytm Magazn 7

9 tn, L, mut b had t atfy all th bund at ach frquncy. Fr lw frqunc, L huld l n r abv th bund t atfy th cntrant. Hwvr, fr hghr frqunc n th rang,.g., ω 5, 7, and rad/, th cntrant gnrat cld bundar whch L huld nt ntr. Fr th ndutral hydraulc actuatr undr nvtgatn, th valv dad band rduc a tady-tat rrr n th ytm rn n th abnc f an ntgratr. Fr a zr Mantanng a lw-rdr cntrllr wa an mrtant factr n th dgn rc. tady-tat rrr, L mut cntan an ntgratr. Thu, a bl l can b btand by cacadng an ntgratr. Furthr, a gan f.65 rqurd t brng th n-l tranfr functn wthn th cfcatn bund. Tw zr huld b addd t th tructur f th cntrllr t atfy th bund and t frc th nmnal l, L,tth rght-hand d f th cld bundar. Aftr vral tratn th cntrllr zr wr chn at and. At th tag, L clly fllw th bund u t ω 7 rad/. 5 Th nxt t t add tw hgh- frquncy l t allw fr a quck dcnd f th nmnal l t dcra th ct f fdback. Fg. (b) hw th fnal l hang f th ytm. Th cntrllr that atf th cfcatn. 65 G ( ). (7) T atfy th trackng cfcatn, a rfltr, F( ), rqurd t lac th cldl frquncy rn n th cfd rang, btwnt ( u )andt l ( ). Fg 5(a) hw th frquncy rn f th cld-l ytm wthut a rfltr whr th dgnd cntrllr (7) nly guarant that th varatn n th ytm magntud wll b l than r qual t that allwd. Th rfltr ynthzd by calculatng t magntud bund at ach frquncy. Fr xaml, n Fg. 5(a), cndr th frquncy rn f th cntrl ytm at ω 5 rad/. Th maxmum frquncy varatn, δ, abut 7 db ( 5), whch l than th amunt allwd fr th frquncy,.., δ all 5 db Tu( 5) Tl ( 5) ( 55). Th acctabl magntud f th rfltr at th frquncy thn fund t b 55 ( 5) F( 5) ( ). By alyng th abv rcdur fr th dgn frquncy rang, a utabl rfltr fund t b Analg and Dgtal Intrfac Bard u 6 5 F( ) (8) Fgur 6. Exrmntal tt tatn. : rrtnal valu, : um wth rur rgulatr, : rur tranducr, : ncrmntal ncdr, 5: frc nr, 6: nvrnmnt. Th cld-l Bd lt f th lant wth rfltr ar gvn n Fg. 5(b). Nt that all th trackng cfcatn ar mt n th frquncy dman. Th crrndng cld-l mulatd tm rn t a t nut f N ar hwn n Fg. 5(c). A can b n, fr all ca rtanng t xtrm art f th ratng nvl, th cfcatn ar atfd. Exrmnt Fgur 7. Cl-u htgrah f th nvrnmnt. Exrmntal Stu Th tt tatn cnt f a hydraulc unt, a 86/66-bad PC qud wth a Mtrabyt M5 quadratur ncrmntal ncdr card, and a DAS-6 analg-t-dgtal (A/D) cnvrn card (Fg. 6). Th um rvd cntant ratnal uly rur u t. Th hydraulc valv a lw-ct cld-cntr fur-way rrtnal valv. Th tnng f th valv l bad n th ul-wdth mdulatn rncl. A rng ud t r- 7 IEEE Cntrl Sytm Magazn Arl

10 rnt th nvrnmnt (Fg. 7); rlacng th rng chang th tffn f th nvrnmnt. A frc tranducr wth a caacty f lb (.5 kn) nrtd btwn th nvrnmnt and th tn rd. Th -bt A/D cnvrn allw a frc rlutn f N whn th full rang ud. Thr rur tranmttr rad um, uly ln, and rturn ln rur wth ±% accuracy. An ncrmntal ncdr wth nng rlutn f.6 mm rad th dlacmnt f th cylndr tn. Th cntrl gnal, gnratd by th cntrl algrthm, cnvrtd t an analg gnal by th A/D card and thn tranmttd t th hydraulc valv amlfr. Th lw-ct valv rat wthn th rang ±.8 V and ha a dad band f ±% (±.5 V) wthn whch th actuatr d nt mv. N attmt wa mad t lmnat r bya th nndal ffct du t th ntndd alcatn. Rult Th cntrllr dcrbd by tranfr functn (7) wa dcrtzd and mlmntd n th xrmntal tt tand. Th amlng frquncy fr th cntrllr wa Hz. Svral xrmnt wr rfrmd t tudy th ffct f varatn f nvrnmntal tffn, uly rur, and frc tnt. Frt, th varatn f nvrnmntal tffn wa tudd by ung dffrnt rng. Tw dffrnt rng wth tffn f 5 and kn/m wr ud fr th ur. Th rult ar hwn n Fg. 8. A can b n, th cntrllr caabl f handlng th chang n nvrnmntal tffn and th tady-tat rrr ar mall. Wth rfrnc t Fg. 8(a), rn xhbt ntal dlay, manly du t th dad band (±%) and dry frctn n th lw-ct valv ud n th xrmnt. Cnquntly, th rn d nt rcly ft wthn th dgn cfcatn bund hwn wth dttd ln. Indd, th ffct f valv dad band wa nt ncrratd nt th QFT dgn rcdur and wa nly handld by cacadng an ntgratr n th cntrllr. Th actuatr dlacmnt and th cntrl gnal fr uch a tral ar hwn n Fg. 8(b) and (c), rctvly. Th cntrl gnal mth, but t cntan hgh-frquncy cllatn that rgnatd frm th n n th frc nr. Fg. 9 cmar th tt rult fr tw dffrnt rfrnc frc (5 N and N) and wth mlar nvrnmntal tffn (5 kn/m). A hwn n th fgur, dt changng th ladng cndtn, th ytm r tm dd nt chang cndrably. Th ablty f th cntrllr t c wth um rur varatn wa al ttd. Tycal rult ar hwn n Fg., whr th um rur wa vard %. A can b n, th cntrl ffrt rducd fr th hghr um rur. Fnally, a tt wa arrangd t nvtgat th ratablty f th cntrllr and th ffct f lng-trm ytm ratn n th rn. A -N t rn wa rfrmd nc and thn agan aftr th machn had bn n cntnuu ratn fr mr than n hur. A hwn n Fg., thr n ntcabl dffrnc btwn th tw rn. Cnclun Th artcl ha dcrbd th alcatn f th QFT tchnqu t th dvlmnt f a frc cntrllr fr hydraulc actuatr. A lnar aramtrcally uncrtan furth-rdr mdl wa dvld t rrnt th rlatn btwn th cntrl gnal and th frc actng n th nvrnmnt. Plant Frc (N) Cntrl Sgnal (V) Dlacmnt (m) T u T l Rfrnc Frc 5 N K 5 kn/m K kn/m.5.5 Tm () (a).5.5 Tm () (b).5.5 Tm () (c).5 K 5 kn/m K kn/m.5 K 5 kn/m K kn/m Fgur 8. St rn wth dffrnt nvrnmntal tffn (xrmnt): (a) frc, (b) cntrl gnal, and (c) dlacmnt..5 Arl IEEE Cntrl Sytm Magazn 75

11 Frc (N) 8 6 K 5 kn/m Tm () Fgur 9. St rn fr dffrnt rfrnc frc (xrmnt). uncrtanty wa quantfd wthn th lnar mdl t cmnat fr tm-varyng dynamc and t allw fr varatn n th nvrnmntal tffn, hydraulc flud flw and rur gan, actuatr cmlanc, and many thr aramtr cfc t th hydraulc actuatr cmnnt. A rbut cntrllr wa thn dgnd that, alng wth a rfltr, mantan a atfactry frc cntrl rfrmanc agant th nvrnmnt dt a wd rang f uncrtanty. Mantanng a lw-rdr cntrllr wa an mrtant factr n th dgn rc t nur ay mlmntatn and dmntrat t utablty fr ndutral alcatn. Th QFT cntrllr dgnd hr wa mlmntd uccfully n an ndutral hydraulc actuatr qud wth a lw-ct rrtnal valv. Th tranarncy f th dgn rc facltatd th quck adjutmnt f th cntrllr fr xrmntal mlmntatn. Svral tt wr rfrmd t hw th frc-rgulatng ablty f th dvld cntrllr. In artcular, th xrmntal rult dmntratd th rbutn f th QFT cntrllr fr u t % varatn n nvrnmntal tffn, uly rur, and rfrnc frc. Gd rfrmanc wa al btand dt gnfcant varatn n actuatr dynamc. Th ngl-l fxd-gan cntrllr wa ay t mlmnt, rqurd vry lttl cmutatnal ffrt, ndd n nln tunng r gan chdulng, and rultd n gd rfrmanc n bth trannt and tady-tat rd. Knwldg f th lwr and ur bund f uncrtan aramtr wa th nly rqurmnt fr th cntrllr dgn. In ummary, th rult f th wrk, whch rrnt a cntrbutn t th ublhd wrk n th alcatn f QFT, rvd nght nt th tntal and ffctvn f th tchnqu fr th dgn f rbut frc cntrllr fr hydraulc actuatr. In artcular, w hav hwn that t fabl t mly a ngl fxd-gan frc cntrllr fr a Frc (N) 8 6 Stffn 75 kn/m Pum Prur 5 Pum Prur Frc (N) 8 6 Stffn 75 kn/m Pum Prur 75 Cntrl Sgnal (v) Tm () (a).5 Pum Prur 5 Pum Prur Cntrl Sgnal (v) Tm () (a) Tm () (b) Fgur. St frc rn fr dffrnt uly rur (xrmnt) Tm () (b) Fgur. Ratablty tt (xrmnt) IEEE Cntrl Sytm Magazn Arl

12 hydraulc actuatr, and that cntrl rfrmanc can b nntv t tructurd lant varatn. It al bcam aarnt that th uncrtan lnar furth-rdr mdl adquatly charactrz th majr fatur f th cla f hydraulc ytm undr nvtgatn. In futur wrk, w lan t u th QFT mthd t dgn autmatc frc cntrllr fr havy-duty hydraulc qumnt uch a xcavatr. Th machn trngly ntract wth nvrnmnt, and accurat mdl fr th ntractn ar dffcult t btan. Currntly, w ar nvtgatng th fablty f aumng th nvrnmntal frc a a cmbnatn f dturbanc and untructurd uncrtanty t b handld by th QFT cntrllr. Rfrnc [] A. Allyn, Nnlnar frc cntrl f an lctrhydraulc actuatr, n Prc. Jaan/USA Sym. Flxbl Autmatn, Btn, MA, 996,. 9-. [] F. Cnrad and C.J.D. Jnn, Dgn f hydraulc frc cntrl ytm wth tat tmat fdback, n Prc. IFAC th Trnnal Cngr., Munch, Grmany, 987,. 7-. [] N. Nkfat and N. Shr, Rbut frc cntrllr dgn fr an lctrhydraulc actuatr bad n nnlnar mdl, n Prc. IEEE Cnf. Rbtc and Autmatn, Dtrt, MI, 999,. -6. [] R.F. Panntt, P.K. Chawdhry, and C.R. Burrw, Altrnatv rbut cntrl tratg fr dturbanc rjctn n flud wr ytm, n Prc. ACC, San Dg, CA, 999, [5] J. Wattn, On lnarzd cffcnt fr an undrlad rv-valv culd t a ngl-rd cylndr, ASME J. Dynam. Syt., Maur., Cntr., vl., n., , 99. [6] J. Yu, Z. Chn, and Y. Lu, Th varatn f l ffctv bulk mdulu wth rur n hydraulc ytm, ASME J. Dynam. Syt., Maur., Cntr., vl. 6, n.,. 6-5, 99. [7] Y.N. Chn, C.B. L, and C.H. Tng, A varabl-tructur cntrllr dgn fr an lctrhydraulc frc cntrl rv ytm, J. Chn Sc. Mch. Eng., vl., n. 6,. 5-56, 99. [8] H. Sun and G.T.C. Chu, Nnlnar brvr bad frc cntrl f lctrhydraulc actuatr, n Prc. ACC, San Dg, CA, 999, [9] R. Lu and A. Allyn, Nnlnar frc/rur trackng f an lctrhydraulc actuatr, n Prc. IFAC th Trnnal Cngr., Bjng, Chna, 999, [] G. Wu, N. Shr, and K. Za, Dgn f a hydraulc frc cntrl ytm ung a gnralzd rdctv cntrl algrthm, Prc. Int. Elc. Eng.-Cntr. Thry Alcat., vl. 5, n. 5,. 8-6, 998. [] L. Laval, N.K. M Srd, and J.C Cadu, H frc cntrl f a hydraulc rv-actuatr wth nvrnmntal uncrtant, n Prc. IEEE Cnf. Rbtc and Autmatn, Mnnal, MN, 996, [] G. Frrtt, G. Magnan, and P. Rcc, On th tablty f ntgral frc cntrl n ca f cntact wth tff urfac, ASME J. Dynam. Syt., Maur., Cntr., vl. 7, n., , 995. [] I.M. Hrwtz, Quanttatv Fdback Dgn Thry (QFT). Buldr, CO: QFT Publcatn, 99. [] J.J. D Azz and C.H. Hu, Lnar Cntrl Sytm Analy and Dgn. Nw Yrk: McGraw-Hll, 988. [5] O. Yanv, Quanttatv Fdback Dgn f Lnar and Nnlnar Cntrl Sytm. Nrwll, MA: Kluwr, 999. [6] S. Jayaurya, O. Yanv, O.D.I. Nwkah, and Y. Chat, Bnchmark rblm lutn by quanttatv fdback thry, J. Gudanc, Cntr., Dynam., vl. 5, n. 5,. 87-9, 99. [7] S.G. Brln and M.J. Grmbl, Lngtudnal cntrl f an advancd cmbat arcraft ung quanttatv fdback thry, n Prc. ACC, Albuqurqu, NM,997,. -7. [8] S.N. Shldn and S.J. Ramun, Dvlmnt and frt uccful flght tt f a QFT flght cntrl ytm, n Prc. IEEE Natnal Arac and Elctrnc Cnf., NAECON, Daytn, OH, 99, [9] M.B. Lahy, D.E. Brt, and P.V. Whaln, Rbut mdl-bad cntrl: An xrmntal ca tudy, n Prc. IEEE Cnf. n Rbtc and Autmatn, Sacramnt, CA, 99, [] S.J. Rbr, Y.C. Shn, and O.D.I. Nwkah, A dgtal rbut cntrllr fr cuttng frc cntrl n th nd mllng rc, ASME J. Dynam. Syt., Maur., Cntr., vl. 9, n.,. 6-5, 997. [] M. Pachtr, C.H. Hu, and K. Kang, Mdlng and cntrl f an lctrhydrtatc actuatr, Int. J. Rbut Nnlnar Cntr., vl. 7, n. 6, , 997. [] D.F. Thmn and G.G. Krmr, Quanttatv fdback dgn fr a varabl-dlacmnt hydraulc van um, n Prc. ACC, Albuqurqu, NM, 997, [] R. Vl and P. Khla, Thrtcal analy and xrmntal vrfcatn f a manulatr/nr/nvrnmnt mdl fr frc cntrl, n Prc. IEEE Cnf. Sytm, Man, Cybrntc, L Angl, CA, 99, [] S.D. Engr and W.P Srng, On dynamc mdl f rbt frc cntrl, Prc. IEEE Cnf. Rbtc and Autmatn, San Francc, CA, 986,. 9-. [5] H.E. Mrrtt, Hydraulc Cntrl Sytm. Nw Yrk: Wly, 967. [6] K. Za, Dvlmnt f a nnlnar adatv cntrl chm fr hydraulc actuatr, M.Sc. th, Dt. Mch. Eng., Unv. f Mantba, Wnng, Canada, 998. [7] H. Sraj and R. Clbaugh, Frc trackng n mdanc cntrl, Int. J. Rbt. R., vl. 6, n.,. 97-7, 997. [8] S. Jayaurya, Frquncy dman dgn fr rbut rfrmanc undr aramtrc, untructurd, r mxd uncrtant, ASME J. Dynam. Syt., Maur., Cntr., vl. 5, n.,. 9-5, 99. [9] I.M. Hrwtz, Survy f quanttatv fdback thry (QFT), Int. J. Cntr., vl. 5, n.,. 55-9, 99. Navd Nkfat rcvd th B.Sc. dgr frm Sharf Unvrty f Tchnlgy, Iran, and th M.Sc. dgr frm Ifahan Unvrty f Tchnlgy, Iran, n 99 and 997, rctvly, bth n mchancal ngnrng. Snc January 998, h ha bn wth th Exrmntal Rbtc and Tlratn Labratry, Unvrty f Mantba, Canada, whr h ha hld tachng and rarch atanth tn. H currnt rarch ntrt nclud rbut cntrl and cntrl f dcntnuu and nnlnar ytm, wth alcatn t rbtc and hydraulc ytm. H rcvd vral award and a fllwh frm th Unvrty f Mantba durng Narman Shr jnd th Dartmnt f Mchancal and Indutral Engnrng at th Unvrty f Mantba n 99, whr h currntly a Prfr. H rcvd an M.Sc. n cmutr-add rc lannng n 986 and a Ph.D. n rbtc and cntrl n 99, bth frm th Unvrty f Brth Clumba. H ara f ntrt ar mathmatcal mdlng and ral-tm mulatn, rbut cntrl dgn, and flud wr. An ara f rarch ntrt th autmatn f havy-duty ndutral (frt and mnng) machnry ung rbtc and advancd cntrl tchnlgy. H a mmbr f th Amrcan Scty f Mchancal Engnr and th IEEE. Arl IEEE Cntrl Sytm Magazn 77