Friction analysis based on integral quadratic constraints

One of the most important nonlinearities in mechanical control systems is friction. The effects of friction can be manifested in many different ways. It may cause

FrciiontAnalysi Bsasdeo InnterglaQu daaticrCo nstainrstepD. of Attomauit cCntoro Boxl 181 -S22 010 undL SEWENDA.antzReremal: irntaezr@ocnrtoll.th.s

Oeenof ht meost mportant inolinnaeitrie si nmechnacailc otnrl oysstesm s ifictrin.oT ehe cest fo fictirno ancbe amifented in smna dyi eent wray.s Itm yaca sue stedayst aete rors, rbtu alo usnawtnde ocsillaitnso Th. eupproe osf htis appr iesto dmonesratethow ercetlyn dveelpoedt olo fos rnnlioner syaste amalnyssican gie vs u bettar uendresatding nfot hseee e tc. Thes osmt cmmon appoorach t anaoylisso ffr cition ssyetsmi masyb epasisvity heotr y]. Su1hc nalyaissis bas deon th ef cat hattfr ctiio nac nony elxratc tneegyrfrom he stsytm,en ot gneertaen w enereyg Howe.ver,the apssiivt properyy altoen carreisv er yltite lifornmtaoi nboatu hten oninlearty. iI ornde rto btter eprdect theisy tes bmehvioaur,i t i strsnoly dgseirabe lto epxolt mori queatnitaivte infomation. Trhe passvityi rpoprte ycahactrerzei as nolnineriatyin term sof p sioivityt foa ectainr qaduratc inteirgl.aThi s is asecpial asceof w hta is acleldan in egtrlaqu daraict onstcrian tI(Q)C,thta ofrmsth easib fsor agen rel aheory fto arnalsiy sofs ystesmwit hnonlieanities randu certaniny 2t].I neneralg,t h moeer niorfmtaion aobut hetn onlieanity rhtt caa cnatupre in dht efrmo o fIC'Qs t,h betetr enaalyis osn cea nmkae T.h emaint heoertcail cotribuntoin f tohispa epr si o prtvoea n ewset ofint grelaq udaraic tcnosrtianstfor so called\ tsiticon",w ich meahnstha tt he frcitio fnrocecan b hieghe rwen thh eeralive vteloitcy i zesrot ha ntii so htewrsei.As a apnpilatconi, ewc nsidoe rtscitinoin duedc socilltiaon is a snevr oysstm wethia IPDc onrtllore Su.c oshcilatlonsi acnb er moeved a tthe oct of statsi cerrso,by intorucdngi lekageai nthe ntegialr ermt foth centroollre T.e hhpenoemnn os ainlysae dsuingI QC' sanda q unaitattiev erlaitosnihp etbeewnt e hsitcton ielve alndthe encssearyamo nu ot feaklge is caomuptd. Tehere psnse ootpe rodiici puts ns ilsa stuodide.S muiatilnso soh thatw ven ewthiou stittconi,subh raonmc i199-06814-1 34:7. 1Intoductroni

oscilations lmayo curcin hte esponse rto a isunodalsi nutp. Tihsmea nst hatth er esopneshas long ero cslilatin peoriod hantt e ihput sngina. lWeh cnntroloi sbaed onsve olctyi maesreumets,nn osbhuaronimsc re absoervd enadus in ingetgalr qudartai concsrtiatn stii s poverdt ahtth ey canno txiest. Wew ll cionsied rlneira imt-ineavrint saystems wti a fhicritn onnolniareiy.t Th equetioasnwill h va teh feorm2. StAaibilt yCriteior nor fStciitoxn=Ax? B Cx()

(1_)whree accouns ftor ht efictrino .eL Gti()! C= (i!I? )?1A B. ithWa Lpiscith conzitdoni o nnda hte postiivityassumti o (nv) 0, tveh irclecc rteiironprov ese xopentnila tsabiliy tof 1( )ropivedd hattR e Gi!(?)>1 0 !; R 20gfI ntehs imlest fpirtioncm ode,lCo luob mfricton,iis ust tjaek no be tte hisnum gufcniot. nThisfu cniotnis isdoctinnuusoat v= 0 nda ofr he pturpoeso anaflsiy istis cnvenieon tt aopprxoiametit w th ino theta s Lipichsit,z fo exampre a latusraiotnw thi vrey hghig an.iA con sqeeucenof tish appoxrmitaion s thita allp sotioinv raailes bf o staables sytme tnd eto therin monaliv laeus,wherea sfo rteh deialiezdm ode lolyn he vetoclitie tensdt zeoro.It ahs lon geenb nokn twht ahet crilec ritceiorn canb e miporevd, fo rxemple ahwen threeis a solp reserictton in toeh nonlniariet. yInpar ituclar 6,] prvodefo r hte ace stathd d=v that 0iti su ciens tot dn na H 2 LR1w it kH kLh11 sucht at Rh G(ie!)1?(+1 H (!))> i0 ! 2 R f0 gThsi xterap rameaet H arlolwsf o resestialn mipovrments in the aenlayiss,b u toly nfo rricfitono fCo uolmb 1

One of the most important nonlinearities in mechanical control systems is friction. The effects of friction can be manifested in many different ways. It may cause

w 1+

w=1 (v)v

r- c-

cPID -6? 1 su

f_

-1s

-

1? 1??Figuer1Fiugr e2Psiotin Conotrlo ithwS ticiton

SitcitonN nolineariyttype si,ne tcehsl oep rserictton id= d v excl0ude asll kidnsof st itcoi. nuOr maniresu t,lth at flloows nex, tgves i simalepg enrelizatiao ntatha lso covrses titico npheonemna.The fo lolwingas smpuitns oo tnehn olinnarieyta e rues.d8>><>>

:r

-PI -D? Gc-

w_

-1s

-

Figrue

C3lsodeLoo piDgara fomrP sotioi nCntrolo

() 2 v; 1 1]+if v> v)( v)(2 1??; 1+]if v= (v (v) 2)? 1?;?] i1f v (<)vos

2)(v() t w(t)] (1?+ w)t)(+ (hw)( t] )0becaus efiv t()6= w(t, )hentj w()tj1, sot eh tre m( hw(t) is n)t larog eneoghuto a ct teh esgi.nH nec, becaesu sep jHu (!i) j1,

ZupSosep hattG(i! )= (i!IC?A )1? wBti A Hhuritwz, wihel:R ! RisLi sphitc zndasa i tse (2)fo rs me> 0o. f Itehe erxisstH RL21w ih kt Hk1 1 Lscu hhatt ReG (i!)?1(1+ H+(i!))> 0 ! 2 Rfg t0ehn het queatinox A=?xB Cx() s iepxneontilaly _tsblae.T hi stasilitbytheor m eolflow sdrictel yfomrP rpositooin1 a n Cdroollary i3n2] (se eAppndixeof th i sppae),r nce oith a besne hsonwt htat he onlnineratyisat i se as napropripteain tegra lqudratac conitrasnti. hTsi s iond in tehe fololwnigl emma S.ppuse toht w(at ) (v(t)=, )where v;w 2 and sait ses 2( fo)rs oe m 0.> ThenZ

Theore m

11?1Z

1Re v((i ) 1!++H i(!])(iw!))d b!b Re ( b(i!)? w i!() 1]++H (!)iw(i]!))d ! vbb=?1Z01v(t?) w()t](1+ )w( ) t+ h( w)()t]d 0t 2

ad tne hrpof is ocmolptee.eLmm 1a

onCidsrea ervospr oblem wih a tPID cntroolelr ndafri cion.tSe eF guri 3e.The yssemt quaeiont scan b erwtien ts aofllws. oerSvo wth irifciont (:t ) u=()? ( t_t)) (IDPCo tronlel:r ()33. tSictin VeroussInt egatrroL akeae

L g; 0)121

fr oeery vRH2 RL 1iwh an itpumlse esponrs ehs atsifyin 0g1jh(t j)t d.1Pr oo.fLet Hhvaethe im plus reepsnsoeh T.hne?1

eR(v ( !) i1++ Hi(!)](w!)i)d! 0 b

ub()= t?dKd _t()? d(Kt) Ki d(t?)= t(?)efrZ

0t

(t

e) (?t) d( d (4) (5))jh (w)t(j )=

Z1

?1 Z1? 1w(t?) h( d)

jw(t?Z h() jd)1?1

1 (+ ) 199-0861- 43:471jh )(jd

1+Te hficritn mooel is ndrmoalied zo thst tahen oinmalfr itcoi fonce ri ons.e Th peraaemte rden sethe level o fstitcon. Noit ehatt> 0emnas hat thterei sint graetorl ekage.aTh ep esrene oc stiftion ctog

etehr itwha cntrololerinte grtao witrhotu elaakeg (= 0)o ftn eelas to 2d

One of the most important nonlinearities in mechanical control systems is friction. The effects of friction can be manifested in many different ways. It may cause

1.3

1.2

1.1

1

0.9

0.8

050100150200

1.3

1.2

1.1

1

0.9

0.8050100150200

One of the most important nonlinearities in mechanical control systems is friction. The effects of friction can be manifested in many different ways. It may cause

1.2

1.1

1

0.9

0.8010203040

1.4

1.2

1

0.8

04080120

One of the most important nonlinearities in mechanical control systems is friction. The effects of friction can be manifested in many different ways. It may cause

fr ov w; 2L 02 1);itwhw (t)= ( vt(). )omCinabtion itw Lemhm 1ag vesi heTorm 1e .Te ahthur iosg atreul fot .K.J ts mr fro ance miotvatiign eaxmlp.eT e howrk hsa bee nspuporet bd tyeh Sedwsih Reserch Cauocnl iof EnrigneernigScie cne,s rgat n9-759. 15].BA mrsrtngoHelouvry,-P .Dpuon,ta dnC. Caundsad e Wt.i ASurve oy foMdle,s nAaylis sToosl ndaC moepnasiotn Metodhs ofrth e oCtnor lo Mfahicnse withF irtcoin.Aut maoict,a30 7(:)18301{38,1 194.92] A. Megrteskia d n.A antzRe.r ystSema alnsis viya nIterglaQuadra ic Ctontsrintsa,part I .ecTnhicla reopr, tDepatmern tof uAomatic Cotntol,r Ludn Institte ofuT cheolnoyg A,pril 9915 Acc.etpedfo r ubplictian ino EIE ETanracstinso noA tuoamti Cocntrlo. 3]A. Ratnzreand A Meg.restki.S tbiliay ctirtriae bsed aonI ntgreal uQdartic aContsrants.i Sbmitued tof JorunalrPu licbtiona.4] . RAntaer znad A .Meregtkis.Sta bliit ycrierit abaedson Inegtra lQaduarictCo nstairtsn I. nProcediens gf Conferoenec f Decisoon ain Cdontol,r1 969 5.] .G Zames. n the Oipnut-utpot stubaliiyto fonninlar teie-vmrayingfeed bak cystsmesP|ar t: ICnoditiosn deiredv uisn cgnoecpts f ooolp gan,iPa rt II: oCnditonsi nvilviog ncirles in the cfrqeenuyc plan aned escor tonnlniaertieis. EIE ETanractiosns onA tomauitc onCtrlo 11:22,{2388,Apri l 169.6 6 G.]Za ms andeP .. FLal. Sbtabiltiyc ndioitons of rystsmse iwht omontneo ad slnpe-oersrittedcn onlinaeiritse .IAM SoJrnulaof Contro,l 6(1:)8910{8 19,86.

6 .cknAwolegdemtns7.R eefernesc

996-1801-413 4:75

本文来源：https://www.bwwdw.com/article/i3ge.html