Lecture Notesin Control and Information Sciences Edited by M.Thoma and A.Wyner 131 S. M. Joshi Control of Large Flexible Space Structures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong S e r i e s Ed i t o r sM.T h o ma A.Wy n e rAd v i s o r y B o a r dL. D. Davi sson A.G.J. Ma c Fa r l a ne - H.Kwaker naak J. L. Ma s s e y YaZ.Tsypki n A.J. Vi t er biA u t h o rSu r e s h M. Joshi510Bl ue Ri d g e Hunt Rd.Ha mp t o n , V A 2 3 6 6 6U S AI S B N 3 - 5 4 0 - 5 1 4 6 7 - 8 S p r i n g e r - V e r l a g B e r l i n H e i d e l b e r g N e w Y o r kI S B N 0 - 3 8 7 - 5 1 4 6 7 - 8 S p r i n g e r - V e r l a g N e w Y o r k B e r l i n H e i d e l b e r gThi s wor k is subj ectt o copyri ght .Al l ri ghts are reserved, whet hert he whol e orpart oft he materi ali s concerned,speci f i cal l y t he ri ght s oftransl ati on, repri nt i ng,re-use ofi l l ustrati ons, reci tati on, broadcasti ng, r epr oduct i on onmi crof i l ms orin ot herways, and st orage in dat a banks. Dupl i cat i on oft hi spubl i cat i on orparts t her eofis onl y permi t t ed undert heprovi si ons oft he German Copyr i ghtLaw ofSept ember9, 1965,ini ts versi on ofJune 24,1985, and' a copyr i ghtf eemustal ways be paid. Vi ol at i ons fal l under t heprosecut i on actoft he German Copyr i ghtLaw. Spdnger - VedagBerl i n, Hei del ber g 1989 Pri nt edi nGer many The use of regi stered names, trademarks, etc.in t hi spubl i cat i on does noti mpl y, even in t he absence ofa speci f i cstatement, t hatsuchnames are exemptf rom t he r el evantpr ot ect i ve l aws and regul at i ons and t heref ore f r ee f orgeneraluse. Of f set pri nt i ng: Mercedes-Druck, Berl i n Bi ndi ng: B. Helm, Berl i n 216113020-543210Pr i nt edona c i d - f r e e paper .TO Shyamala,to myparent s,andto myparents-in-law Theviewsandcont ent soft hi sbookaresolelyt hoseoft heaut hor andnotoft heNat i onal Aer onaut i csandSpaceAdmi ni st r at i on.Pref ace Inthehistoryof scienceandtechnology, theoreticaladvancementsandpractical applicationshavealwayshadamutuallystimulatingeffect.Theaerospacesciences areaisnoexceptiontothis.Asnew,moreadvancedaerospacesystemsemerge,the requirements for better pefformance~ higher reliability and less cost,are becoming more stringent, requiring further theoretical developments in many different disciplines.One sucharea,whichisbecomingincreasingly important,isthedesign,constructionand operation of very large satellites in Earth orbit.This upcoming class of satellites, which would requirelargestructurestobedeployed, assembled,andmaintainedinspace,is expectedtohaveaprofoundimpactonthequalityoflifehereonEarth,bymaking quantumimprovementsincommunications,astronomy,andEarthobservation;byes- tablishing permanent human presencein space;andby facilitating mannedexploration of thesolar system. In order toachieve the required performance, itis of utmostimportance to beable to control such structuresin space with high precision in attitude and shape.Control of such structures is a challenging problem because of their special dynamic characteristics, whichresult from theirlarge sizeandlight weight. Thisbookisintended tobearesearchmonographthatpresentstheproblemsen- countered in controlling large, flexiblespace structures(LFSS), and some of the control synthesismethodsdevelopedbytheauthor,aswellasresultsoftheirapplicationto realisticspacecraftmodels.Intheinterestof economy of space,wehavenotincluded alltheexhaustiveresearchdoneinthisareabyseveralresearchers,butratherhave concentratedonsomeof ourownresults.However, wehaveattemptedtoprovidean adequatebibliographyforthosewishingtorefertoworksofotherresearchers.Itis hopedt hat thebook will stimulate researchactivity incontroltheory withapplication toflexible spacecraft~andt hat itwill bealso beuseful topractitionersworking inthe VI ar eaof sat el l i t econt rol . The bookwillalsoserveasani nt r oduct or yt ext f or persons i nt endi ngt os t ar t worki ngint hear eaofflexible spacecr af t cont r ol .The or gani zat i onoft hebookisas follows:The basi cmat hemat i cal model sofLFSS ar epr esent edint heChapt er 1,andt hedifficultiesandchallengesi nvol vedindesi gni ng cont r ol lawsareexpl ai ned. Weconcent r at eonusi ngfi ni t e-di mensi onal model s[i.e., i nvol vi ngor di nar ydi fferent i alequat i ons(ODE)], al t houghi nfi ni t e-di mensi onal model s [i.e.,descr i bedbypar t i al di fferent i al equat i ons( PDE) ] arebri efl ydi scussedandt hei rr el at i onshi pt oODEmodel sispoi nt edout . The r easonfor following t heODEappr oach ist ha t model sformos t real i st i cspacecr af t canonl ybegener at edusi ngfinite-el ementcomput er pr ogr amssuchasNASTRAN, andareint heODEf or m.Chapt er s 2and3addresst hepr obl emofpreci si ona t t i t ude cont r ol s ys t emsynt he-sisforLFSS. Inmost appl i cat i onsitisessentialt ocont r ol t hea t t i t ude oft heLFSSt o specifiedpreci si oninor der fort heLFSSt of unct i onasr equi r ed. For i nst ance, acom- muni cat i onsa nt e nna mus t bepoi nt edwi t hext r emel yhi ghaccur acyt owar dst het ar getonEar t h. I t mus t alsor et ai naccur at eshapeof itsrefl ect or, andaccur at er el at i vepc* si t i onbet weent hefeedsandt herefl ect or. The at t i t udecont r ol pr obl emweconsi der t husi ncl udest hecont r ol ofnot onl y poi nt i ng, but alsoshapedi st or t i onsr esul t i ngf r om flexibility.Twoappr oachesareconsi der edforat t i t udecont r ol s ys t emdesign.Thefirstappr oach( pr esent edinChapt er 2)ist he~dissipative"cont rol l er, whi chutilizes anumber of act uat or s andsensorsat same(orclose)l ocat i ons, di st r i but edt hr ough-out t hes t r uct ur e. Usingt hedi ssi pat i vi t yt ypepr oper t i esofLFSS, wepr ovet ha t such cont rol l ersarer obus t t omodel i nger r or saswellasact uat or andsensor nonl i neari t i es andphaselags.Wealsopr esent resul t sof appl i cat i onof suchcont rol l erst oalarge,flexible spaceant enna. Apar t i cul ar t ypeof act uat or , t heAnnul ar Mome nt umCont r olDevi ce( AMCD) , whi chhast hei nher ent char act er i st i cof a c t ua t or / s e ns or col l ocat i on,isdescri bed, anditsst abi l i t yandr obust nesspr oper t i esareanal yzed.Cha pt e r 3consi derst heuseof l i near - quadr at i c- Gaussi an{LQG)-t ype cont r ol l er s fora t t i t ude cont r ol of LFS$. The t i me- domai ndesi gnappr oachisconsi der edfirst, and ma t r i xnor mboundsonmodel i ngerrors, whi chensurest abi l i t y, areobt ai ned. Mul t i -vari abl ef r equency domai nt echni quesfordesigningcont rol l ers whi char er obus t t orood- VII clinger r or sarenext consi dered. Inpar t i cul ar , t heLQG/ l oopt r ansf er r ecover y( LTR)met hodis modi fi ed for appl i cat i ont oLFSS, andresul t sforal argespaceant ennamodelare pr esent ed. The st abi l i t yofLQG- t ype cont rol l ersint hepr esence ofreal i st i cact uat ornonl i neari t i esisi nvest i gat ed, andexpressi onsfort heregi onsofat t r act i onandul t i mat e boundednessar eobt ai ned.Chapt er 4addressessomer el at edt opi cs, namel y, par amet er i dent i fi abi l i t yst udi es,andt hel argeat t i t ude- angl emaneuver i ngpr obl em. Areasforf ut ur er esear charealso discussed. To demons t r at et hecont r ol l ersynt hesi smet hods, we havea t t e mpt e dt omakeuseofnumeri calexampl es basedon real i st i cLFSS. Inpar t i cul ar , t heLFSS model we haveused most of t enist ha t ofal argespaceant enna, namel y, t he122mdi amet er ~hoop/ col umn" ant enna. The ant ennamodel servesasat hr eadwhi chisc ommont oallt hechapt er s,andhopef ul l ysimplifiest heexpl anat i onof vari ousmet hods discussed.I woul dlike t ot hankmy wifeShyamal a for her pat i ent s uppor t andencour agement .Iwoul dalsoliket ot hankNASALangl eyResearchCent er forgr ant i ngmeper mi ssi on t owr i t et hi sbook. Manyt hanksarealsoduet oMs.Ba r ba r a Jef f r eyforher exper tt ypi ngoft hefinalmanuscr i pt .Hampt on, Vi rgi ni a December 1988 Sur eshM. JoshiCoes CHAPTER1.I NTRODUCTI ON. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1Mat hemat i calModel sof LargeFl exi bl e SpaceSt ruct ures . . . . . . . . . . . . . . . . . . . . 3 1.1.1Infi ni t e-Di mensi onalModel s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2Appr oxi mat e Met hodsforFi ni t e-Di mensi onalModel s . . . . . . . . . . . . . . . . . 8 1.1.3Cont rol l abi l i t y andObservabi l i t y ofFi ni t e-Di mensi onalModel s. . . . . 21 1.2Pr obl emsinCont rol l erDesignforLargeSpaceSt r uct ur es . . . . . . . . . . . . . . . . . . 24 CHAPTER2.ACLASSOFROBUST DI SSI PATI VECONTROLLERS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1Descri pt i on ofDi ssi pat i veCont rol l ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2St abi l i t y Pr oper t i esof Di ssi pat i ve Cont rol l ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3DesignofDi ssi pat i ve Cont rol l ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4Robust nessPr oper t i esofDi ssi pat i ve Cont rol l ers . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.1Robust nesstoImpreci seCol l ocat i on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.2Robust nesst oAct uat or / SensorNonl i neari t i es andDynami cs. . . . . . . 48 2.4.3Summar yofRobust nessResul t s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.5.Exampl e: Di ssi pat i ve Cont rol l erDesignforaLargeSpaceAnt enna. . . . . . . 71 2.6TheAnnul ar Moment umCont rol Device(AMCD): AnAct uat or ConceptforDi ssi pat i ve Cont rol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.6.1Mat hemat i calModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.6.2Dampi ngEnhancement UsingAMCD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.6.3Dampi ngEnhancement UsingSeveralAMCD' s. . . . . . . . . . . . . . . . . . . . . 89 2.6.4Numeri cal Exampl e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.6.5At t i t udeCont rol forAMCD- Act uat edLFSS. . . . . . . . . . . . . . . . . . . . . . . . 98 2.7Remar ksonDi ssi pat i ve Cont rol l ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 IX CHAPTER3. LQG- BASEDCONTROLLERS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.1Ti me - Doma i n LQGDesi gnAppr oaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.1.1St abi l i t yBoundsonSpillover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.1.2Cont r ol l er Desi gnforHoop/ Col umnAnt e nna. . . . . . . . . . . . . . . . . . . . . . 116 3.2Mul t i var i abl e Fr equency- Domai n Met hods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.2.1TheLQG/ LTRMet hod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.2.2Appl i cat i on t oHoop/ Col umnAnt e nna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.3EffectofAct uat or Nonl i near i t i es. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.3.1Type- I Nonl i neari t i es: Regi onofAt t r act i on . . . . . . . . . . . . . . . . . . . . . . . . 142 3.3.2Type- I INonl i near i t i es: Regi onofUl t i mat eBoundedness. . . . . . . . . . . 150 3.3.3Ext ens i on t oSt at e- Es t i mat eFeedback. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 CHAPTER4. RELATEDTOPI CS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.1Par amet er I dent i f i abi l i t y St udi es. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.1.1Cr amdr - RaoLowerBounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.2TheManeuver i ng Pr obl em . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.2.1Opt i mal Cont r ol Pr obl emFor mul at i on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.3Fut ur eResearch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 BI BLI OGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 I NDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Cha pt e r 1 I nt r o duc t i o n Future utilization of space is expected to require large space structures in low-Earth aswellasgeosynchronous orbits.Examples of such future missions include:electronic mail systems, Earth observation systems, mobile satellite communication systems, solar powersatellites,largeopticalreflectors,andspacestations.Suchmissionstypically requirelargeantennas,platformsandsolararrays.Thesemissionswouldbefeasible becauseofthelaunchingcapabilityof theSpaceShuttle,whichcanbeexpandedby augmentation with OrbitTransfer Vehicles(OTV)for placement in the geosynchronous orbit. Thedimensionsofsuchstructureswouldtypically rangefrom50meters(m)to possiblyseveralkilometers(km).Forexample,onemobilepersonalcommunication system concept, for the entire continental United States,would requireaspaceantenna with adiameter of 122 m.To establish such structures in space at minimum cost would requiret hat their weight beminimized.Itwill also benecessary tocompactly package theminsub-assemblies,eachofwhichisdeployableandcanfitintheShuttlecargo bay.Someof thestructures(e.g.,spacestation)willrequireon-orbitassemblyusing components suchasdeployable beams. Because of their light weight and expansive sizes,these structures will tend to have extremely low-frequency, lightly damped structural(elastic)modes.Natural frequencies oftheelasticmodeswouldbegenerally closely spaced,andsomenaturalfrequencies may belowerthanthecontroller bandwidth.Inaddition, theelastic modeparameters (natural frequencies, mode shapesand damping ratios)would not be known accurately. 2F o rt h e s er e a s o n s ,c o n t r o ls y s t e m s d e s i g nfor l a r g eflexible s p a c es t r u c t u r e s( L F S S ) is adifficult a n dc h a l l e n g i n gp r o b l e m .T w o o ft h em o s ti m p o r t a n tc o n t r o lp r o b l e m sfor L F S S a r e :i) F i n e - p o i n t i n go fL F S Sin s p a c ew i t ht h er e q u i r e dprecision in a t t i t u d e( t h a tis, t h et h r e ep o i n t i n ga n g l e s )a n ds h a p e , a n d ii) L a r g e a n g l em a n e u v e r i n g ( " s l e w i n g " )t h e L F S S to o r i e n tto a different target. T h ep e r f o r m a n c e r e q u i r e m e n t s for b o t ho ft h e s ep r o b l e m s a r eu s u a l l yv e r yhigh. F o r e x a m p l e , in s o m e applications, it is n e c e s s a r y t om a n e u v e r t h e L F S S q u i c k l yto a c q u i r e a n e w t a r g e to n t h e E a r t h . T h i s h a s t o b e d o n e in t h e m i n i m u m possible t i m e , a n d w i t h t h e m i n i m u m f u e le x p e n d i t u r e , w h i l e k e e p i n g t h e elastic m o t i o n a n da c c o m p a n y i n g stresses w i t h i n a c c e p t a b l e limits.I n g e o s y n c h r o n o u s applications, t h ea n t i c i p a t e dE a r t h - p o i n t i n g s l e ww o u l d r o u g h l y t r a n s l a t ei n t oa2 0d e g r e e m a n e u v e r in 1 0 s e c o n d s o r less.(20d e g r e e sw o u l d c o v e r t h e e n t i r ed i a m e t e r o ft h e E a r t h att h eg e o s y n c h r o n o u s altitude). T h e s l e wa n g l e sa r em u c h larger for l o w - E a r t h orbits. O n c et h e t a r g e tis a c q u i r e d , t h e L F S S m u s t p o i n t t o it w i t h t h e r e q u i r e d precision.F o re x a m p l e , for t h e m o b i l e c o m m u n i c a t i o n s y s t e m m i s s i o n ,t h e a n t e n n a w i l lh a v e t o b ep o i n t e d to w i t h i n 0 . 0 3d e g r e e r o o tm e a n s q u a r e ( R M S ) . T h e r e q u i r e m e n t s for o t h e rm i s s i o n sv a r y ,b u t s o m e a r ee x p e c t e d to b e e v e n m o r e s t r i n g e n t - o nt h e o r d e r o f0 . 0 1a r c - s e c o n d .T h i sb o o k m a i n l y a d d r e s s e st h ep r o b l e m o ffine-pointing. T h e b a s i cp r o b l e m s in p o i n t i n gc o n t r o lof L F S S h a v eb e e n k n o w n for s e v e r a ly e a r sin t h ec o n t e x to fc o n t r o lof c o n v e n t i o n a lspacecraft, w h i c ha r erelatively rigid, b u tw h i c hh a v esufficient flexibility t onecessitate c o n s i d e r a t i o nin t h e d e s i g np r o c e s s . E x a m p l e so ft h e e a r l ys t u d i e sin t h i sa r e a i n c l u d eS a t u r n Ia n d I B [Kie.64], S a t u r n V A p o l l o[Pin.69],andSkylab[Har.76].[Gev.70]alsoincludesananalysis of t he probl emsarising from lightlydampedelastic modes.However, t hedistinguishingcharacteristicof LFSS is theirhighly prominentst ruct ural flexibility, which makes LFSS anew class of spacecraft. (A detailed literaturesurvey on dynamics andcontrol of LFSS may be found in[Nur.84]). Inthischapter, t hebasicmat hemat i cal modelsofLFSSarepresented.Theyin- cludeinfinite-dimensionalmodelsdescribedbypartialdifferentialequations(PDE' s),andfinitedimensionalapproximations,i.e.,involvingordinarydifferentialequations (ODE' s). Therelationshipbet weenPDEandODEmodelsisbrieflydiscussed,al- 3 though weconcentrate onusing onlyODEmodelsintheremainderof thebook.The reasonfor following theODEapproachisthatmostengineering modelsaregenerated using finite element computer programs such asNASTRAN,andareintheODEform. Two examples of finite element models are presented, and the procedure for using finite elementdatatogeneratestatespacemodelsisexplained.Controllability andobserv- abilityofODEmodelsarediscussed,andthedifficultiesandchallengesinvolvedin control systems synthesis areexplained. 1.1.Ma t he ma t i c a l Model sof Lar geFl exi bl eSpaceSt r uct ur esTwotypesofdynamicmodelshavebeenbeenusedintheliteratureforLFSS. Thefirsttypeconsistsofcontinuousmodels,whichresultindistrlbuted-parameter (infinite-dimensional) systems representedbypartialdifferential equationswithappro- priateboundaryconditions.Thesecondtypeof modelsconsistoffinite-dimensional systems, representedby ordinary differential equations. 1.1.1I nf i ni t e- Di mensi onalModel s AclassofLFSScanbegenericallydescribedbyasystemofpartialdifferential equations(PDEs)suchas[Bal.82]: as c3 m ( s ) f f f i u ( s , t ) + D ~ i u ( s , t ) + A u ( s , t ) = F ( s , t ) (1) whereu ( s , t ) isadisplacementCtranslational orrotational)of thestructurefrom itsequilibriumposition,asafunctionofspacevariablesandtimet . r n ( s ) isthe massdensity, Aisatime-invariantdifferential operator,whosedomainconsistsof all smooth functions satisfying (1)with appropriate boundary conditions, and is thus dense intheinfinite dimensionalHilbertspaceL 2 ( f t ) , whereftdenotesthestructure.The operatorAisgenerally self-adjoint andnon-negative.F ( s , t ) isthedistribution of the applied generalized force (i.e.,forces and moments).Drepresentsthe inherent damping operator,whichisapropertyof thestructure(materials, jointdesign, etc). In most cases, thecharacterization of thedamping operatoris notstraightforward. Thestandardmethodistoassume"proportionaldamping", whichisnotwell-defined evenforsimplestructureslikeauniformbeam.Modelingofthedampingtermis 4 t r e a t e din[Hug.82],[Bal.86],[Bal.87],andisstillanar eaofact i veresearch. Because t hedampi ngis usual l y ver y small, andbecausearel i abl e dampi ngmodel is not available,Di s usual l yassumedt obezero(anul l oper at or ) . Incaseswher efi ni t e-di mensi onalappr oxi mat i onsar eused, pr opor t i onaI (orvi scous)dampi ngisaddedl at er , af t er such amodel hasbeenobt ai nedassumi ngzerodampi ng.Inmos t cases, t heope r a t or Ahasadi scret es pect r um, anditsei genval ueequat i on canbewr i t t enas: AkCs)=mCs)kCs)k=1 , 2 , 3 , . . . ( 2 )wher e~k(s)andw~aret heei genvect orandt heei genr~l ue. ~kiscalledt henat ur al (or st r uct ur al ) frequency, and~k(~)iscalledt he~modeshape"f unct i onoft hek t hmode.Inmos t real appl i cat i ons, t heappl i edforcesandmoment s arepr oducedbylocal- izedor "poi nt devi ces", t hat is,t heyareappl i edat di scret epoi nt sont hes t r uct ur e.Assumi ngt ha t forcesf ~( t ) areappl i edat poi nt ss i(i=1, 2, .., m) , t heforcedi st r i but i on canbeexpr essedas: V f ti --1 wher e 6(.)denot es t he Di r ac del t a f unct i on. At or que Tat a poi ntrcanbeappr oxi mat ed ast woequal andopposi t eforces, Fand- F , appl i edat : s=r-~,ands=r-i- respect i vel y, wher eF=T / 2 ~ . Theseforceshavet obei ncl udedint heappr opr i at e t r ans l at i oncomponent of t heforcedi st r i but i onF ( s , t ) . As~--*0,t heforcesappr oach t hedoubl et : T a S ( s -r ) / a a .Int hi sf or mul at i onwehaveomi t t edt hegr avi t at i onal field,or bi t al effects,at mo-spherei cdr ag, sol arpressure, et c. Thes ecanbet r eat edasext er nal di st ur banceswhi ch ar eusual l ypr edi ct abl eandr epeat abl e, or havesmal l magni t udes, andt heyandarenotconsi der edher ei n. Exampl esof appl i edforcesandmoment s i ncl uder eact i oncont r olj et s, r eact i onwheel s, mome nt umwheel s, cont r ol moment gyr os( CMG) etc.The sensorsaboar danLFSSmayi ncl udest ar sensorsorSunsensors( at t i t ude meas ur ement ) , r at egyr os( at t i t uder at emeasur ement s) , accel er omet er s, l as er / opt i calr angi ngdevi ces(t omeasur er el at i vedefl ect i on), etc. The yareusual l y"poi nt "devices,andt hemeas ur ement s ( t r ansl at i onal or r ot at i onal di spl acement s)at l ocat i ons Icanbe expressedas: y j ( t ) =u ( s j , t ) (4) Ther at eor vel oci t y measur ement sar emer el y t het i me- der i vat i ves oft hemeas ur ement s( 4 ) .Usi ngt hef act t ha t Aissel f-adj oi nt , itcanbeshownt ha t t heei genvect orsCkar e or t hogonal int hesenset ha t = ~ = 0f o r i#j( 5 )wher er epr esent st heusual L2( f l ) - i nnerpr oduct . If i ar enor mal i zedbydi vi di ng eachi by: , wehave = ~ i j ( 6 )wher e5i j r epr esent st heKr onecker del t af unct i on. Thi si nfi ni t e-di mensi onal model can bet r ans f or medi nt oa"modal "model consi st i ngofi nfi ni t enumber of second- or derODg' s . Toi l l ust r at et hi sl et usconsi derauni f or m free-free be a mwhosepl anar mot i on (i.e.,bendi ngal onganaxispependi cul ar t ot hebe a ml ongi duni al axis)isdescr i bedby t heEul er - Ber noul l i PDE:_0 2 u 0 4 ump ' =I j : l( 7 )wher efit andE 1 denot et hemassper uni t l engt handt hebendi ngstiffness oft hebeam,anduist het r ansver sedi spl acement . The bounda r ycondi t i onsforbot hendst obe compl et el yfreeare:8 2 u ( s , t ) / O sz=a 3 u ( s , t ) / S sa=0at s=0ands=L, wher eLis 6 t hel engt hof t hebeam. (Such boundar ycondi t i ons,whichinvolve second ort hi r dorder spat i al derivativesof u, arecalled"dynami c"or"nat ur al "boundar ycondi t i ons because t heyareobt ai nedbyappl yi ngforceormoment bal anceont heboundar y. Boundar y condi t i onsinvolving spat i al derivativesof order0or1 arecalled"geomet ri c"boundar y condi t i ons; e.g.,i f t hebeamiscl ampedat oneend, u ( s , t ) =O u ( s , t ) / c g s =0at s=0). SupposeeigenvectorsCk (~)arenormal i zedsot ha t = i(s) t heyf or manor t honor mal basisoft heHi l bert spaceL2(fl ). Thusu ( s , t)hasauni que represent at i on:oo u(~, t)=~qk (t)k C~)(9) k=lwhereqkCt)iscalledt he"modal ampl i t ude"fort hek t h mode. Subst i t ut i ngforu ( s , t )f r om(9)i nt o(7)andt aki ngi nnerpr oduct of bot hsidesof(7)wi t hi (s)yields[using (2)]: rnp 4, +~ q ~ =~ , ( ~ j ) / i Ct ) +~~(r;)TiCt)i=1 , 2 , 3, .... i=1i =I(io) whereapr i medenot est hespat i al der i wt i ve("sl ope"). Inwri t i ng(10)we haveused(8) andt hefact t ha t ~bkareort hogonal . Wehavealsomadeuseof t hefact t hat t hei nner pr oudct ofiwi t ht hedoubl et t er mO~(s-r i ) / 0 8 in(7)yields:~( r i ) Ti ( t ) .Thedi spl acement at l ocat i ons iisgivenby: o ok=la~nd t heangul ar di spl acement at s iisgivenby: Cil) oo k=l(12) 7 The translational and rotational velocities are simply thetime- derivatives of these terms(Eqs.11and12),which would involve theqk,themodal velocities. In the above discussion we have illustrated the transformation of an infinite-d|mens- ionalsystemintoaninfinitesetofODE'sforasimplefree-freebeam.Forgeneral three-dimensional structures, however, mere formulation of theproblemintheinfinite- dimensional settingisquiteformidable.(Forexample,see[Bal.85]whereinarealistic LFSSmodelisformulatedasanabstractwaveequation).Intheory,theprocedure used above for the simple beamcan also beused for more complex spacecraft([Rob.85], [ J o s . 8 4 ] ) . However, the eigenvalue problem is generally very difficult to solve for complex non-symmetrlcstructures,anditisusuallynecessarytouseapproximatemethods, which resultinafinite number of ODE's. Theoperatorequation(1)describesthelinear motion of theLFSS, which consists of elasticandrigid-body(zerofrequency)modes.Therigid-bodymodesinclude three translationsandthreerotations(withrespecttothreeorthogonalaxes).Thesemodes axeveryimportantbecausetheprimaryobjectiveistocontrolthebasicrigid-body attitude(threepointing angles)and positionof thespacecraft.Inmany casestherigid motion is too large to be in the linear range, although the elastic motion is small.In some such casesanapproximate model may beobtained by super-imposing thesmall elastic motion on thenonlinear rigid motion.For theprecision attitudecontrol problem,both rigidandelasticmotionaresmall(aboutthezeroequilibriumposition)andmutually uncoupled.Itisusually simplerto writetherigidand elasticmodelsseparatelyand to combine them.Thelineaxized rigid-body equationsare: r n=: , ( 13)i =lr np s: , =R, x:,+ i - - 1 y = lwherezdenotesthe31 vector representingtheposition of thecenterof mass(c.m.) inaninertialcoordinatesystem,MsandJsdenotethemassandthe3x3moment of inertiamatrix(aboutthec.m.),a=(, 0, )Trepresentsthe(rigid-body)attitude 8 anglesaboutX,YandZ axes,andRidenotesthe31 coordinate vector of thepoint of application of force fiwithrespecttothec.m(ab denotethecross-productof 3- vectors aandb).The elasticODEmodelisgiven by(10),(wherein only elasticmodes areincluded).Thusthecomplete linearized ODErepresentationisgiven byEqs.(10), ( 1 3 ) a n d ( 1 4 ) .The sensors mentioned previously arephysical devices t hat measurethetotalmo- tion, which includes the rigid and elastic motion.Theinertial position of apoint on the LFSSis: co =a c t } -d s a c t ) + ( a j ) q k ( t ) ( 1 5 3k=lwheredjisthepositionvector of thepoint,andCkisthe(31)translational mode shapeforthek thmode.Thetotalattitudeatt hat point(whichismeasured,for example, byastarsensor)isgiven by: co fly=aCt) +ECk(dj)qk(t)(16) k=lwherekdenotesthe3x1 rotational mode shapefor thekthmode.Anattitude ratesensoratpointd iwould measurethetime-derivative of (16). Inprecisionattitudecontrolproblems,therigid-bodytranslationisusuallynot important.Itoccursonly whenforceactuationdevicessuchasreaction jetsareused for attitude control and orbitcorrection.Application of torques or moments isabetter wayofcontrollingtheattitude,sinceitleavesthec.m.position,andthereforethe orbitalparameters(altitude,velocity, etc.)unaffected.Inordertostudytheattitude controlproblem,we only need toconsider(10) and(14),andthesensorequations. 1.1.2Appr oxi mat eMet hods forFi ni t e- Di mensi onalModel s LFSSdesigned forrealmissionsareexpectedtobequitecomplex and wouldlead toeigenvalue problemsforwhichdosed-form solutionsarenotpossible.Therearea numberofapproximatemethodswhichcanbeusedforsuchcases.Themethodsare essentiallyschemesfordiscretizationofcontinuoussystems,andcanbedividedinto t w o c l a s s e s .T h e first c l a s sr e p r e s e n t s t h es o l u t i o n a s a finite s u m m a t i o n o ff u n c t i o n so fs p a c ev a r i a b l e s ,m u l t i p l i e d b y g e n e r a l i z e d c o o r d i n a t e s , w h i c h a r ef u n c t i o n s o ft i m e .T h es e c o n dc l a s so fm e t h o d s u s e sal u m p e d - m a s s a p p r o x i m a t i o n o fc o n t i n u o u ss y s t e m s .( S e e[ M e i . 8 0 ]o ro t h e rs t a n d a r dt e x t so nv i b r a t i o n a la n a l y s i sf o rad e t a i l e dd i s c u s s i o no fa p p r o x i m a t e m e t h o d s ) . H e r e i nw e b r i e f l yd i s c u s so n l yt h efirst c l a s so fm e t h o d s , w h i c his u s e d m o r e w i d e l y . I np a r t i c u l a r ,t h e b e s t k n o w n m e t h o d s i nt h e first c l a s sa r e :t h eR a y l e i g h - R i t zm e t h o d , t h em e t h o d o fw e i g h t e dr e s i d u a l s ,a n dt h efinite e l e m e n tm e t h o d .Ra yl e i gh- Ri t z Me t h o dInthismet hod, whichisapplicableonlyt oself-adjointsystems, anapproxi mat e solution(eigenvector)of theeigenvalueprobl emisexpressedas; i - - - - 1whereviarerealcoefficient,anduiareknownsmoot h, independentfunctionswhich satisfyt hegeometricboundar yconditions,but notnecessarilythedifferentialequation of thesystem.Suppose ~b andA areaneigenvector andthecorrespondingeigenvalue, whichsatisfy theoperat oreigenvalueequaiton: A~=mA~(18) Takinginnerproduct wi t h~b, = A = A (19) o r = ( s o ) 10 Si nceui n(17)isanappr oxi mat i onandnot anei genvect or, (20)willnot hol di fis r epl acedbyu. Definet heRayl ei ghquot i ent as: I E ' ; " , " , , E? ,,,A,.) =E ' : , , a. , " , " , 1( 2 1 )wher e~=( vl ,v2, ..., vn) TIt canbeshownt ha t V whi chgivest hebest appr oxi mat i on fort he ei genvect or isobt ai nedbymaki ngc g R/ O~=O,whi chyields: ( K-) , M) v=0(22) wher e K=KT~0andM=MT>0ar ennmat r i ces, whi chare cal l edt he"stiffness mat r i x" andt he"massmat r i x" respect i vel y. Eq. (22)r epr esent sanal gebrai c( r at hert ha noper at or ) ei genval ue pr obl em, for whi cht heneigenvalueshiandt hecor r espondi ng ei genvect orsVi, (i=1, 2, . . . , n) , canber eadi l ycomput ed. The appr oxi mat es t r uct ur alfrequenci esaret hengi venbywi=~ , andt heappr oxi mat eei genfunct i onsar egi ven byusi ngt hecoefficient vect or sviin(17). Th e "assumed- modes"me t hodiscloselyr el at edt ot heRayl ei gh- Ri t zmet hod, and maybeconsi dered t o beavar i at i on of t hel at t er . Int hi smet hod, t hesol ut i onis assumed t obeoft heform:n i =1 ( 2 3 )wher e~ar et heknownadmi ssi bl ef unct i ons(i.e.,sat i sfyi ngt hegeomet r i cbounda r y condi t i ons)whi chr epr esent t hebest guessoft hemode- shapef unct i ons, andr/iar et he general i zedcoor di nat es, whi charef unct i onsoft i me, r at her t ha nconst ant sasin(17). The subsequent st epsint heassumed- modesme t hodi ncl udewr i t i ngt heexpressi ons fort heki net i candpot ent i al ener gyasf unct i onsof ' s andr/' s,andappl i cat i onof t he Lagr angi anf or mul at i on, whi chresul t sanequat i onoft hefollowing f or m:M~+K~=0(24) 11 whereM=MT~.0a n d K=KT>0.TheMandKmat r i cesobt ai nedint hi s manner areidenticalt ot hosein(22).Fur t her mor e, t hi ssetof differential equat i onscan berecast [Mei.80]asanalgebraiceigenvalueprobl emoft hef or mofEq.(22).Fort he eigenvalue probl emin(22),supposei and~(i=1, 2, ..., n)represent t heeigenvectors andt heeigenvalues.Theeigenvectorscanbeshownt obeM- or t hogonal , t ha t is, CTM~b j =0 ifi # jSupposei arenormal i zedsot ha t f ) T M i =1 . Denot i ng = [ 1 , 2 , . . . , ~ 1 , a n d A = d i a g [ ~ , ~ 2 ~ , - - - , ~ . lit c a nb es h o w n that ~TM~~-- Iandq~r K~=A(25) Usingt het r ansf or mat i on:r/Ct)=~ q ( t ) C26) whereq=( q l , q2, . . . , q n )T,Eq.(24)ist r ansf or medintot hefollowing setof uncoupl ed second-orderODE' s:+ A q = 0 ( 2 ~ )q i ( t ) a n d f ~ i ( s ) , ( i =1, 2, . . . , n ) arecallednat ur al modal ampl i t udesandmodeshapesof t hesyst em, respectively. Me t h o d o f We i g h t e d Re s i d u a l sThi smet hodismoregeneral t hant heRal ei gh-Ri t zmet hodbecauseitsapplica- t i onisnot l i mi t edt oself-adjointsyst ems. Int hi smet hod, anappr oxi mat esol ut i onis assumedasin(17),andt he"residual"of t heeigenvalueequat i onisdefi nedas: 12 R ( u , s ) =A u ( s ) -A m ( s ) u ( 8 ) (28) Let , ~( i =1, 2, ..., n)representani ndependent setof funct i onsinL2(D). Foragiven, ,wea t t e mpt t oobt ai nwei ght i ngfunct i onsi ( i =1, 2, ..., n), allof whi chareor t hogonalt oR. Asn~oo,Rwillbeort hogonal t ot heent i respaceL2(f~},whi chimplies: R--+0.Byi mposi ngt hecondi t i ont hat all~biareort hogonal t oR, onecanobt ai n t healgebraiceigenvalue pr obl emof t heform(22),wherei nMandKaregeneral l ynotsymmet r i c. Dependi ngont hechoiceof{~b}, avari et yof met hodsforsol ut i oncanbe obt ai ned. Thebest knownofthesemet hodsisperhapst heGal erki nmet hod, inwhich ~=u ~ ( i =1, 2,..., n). If Ais self-adjoint,t heresul t i ngalgebraiceigenvalue pr obl emis identicalt ot hat obt ai nedbyt heRayl ei gh-Ri t zmet hod.F i n i t e E l e me n t Me t h o dThefinite el ement met hodisbyfart hemost commonl y usedmet hodformodel i ng LFSS,because of its versat al i t y in handl i ng hi ghl y compl ex st r uct ur es, anditsamenabi l - i t y t oaut omat i onbydi gi t alcomput er . Thebasicphi l osophy of t hi smet hodist odivide acont i nuouss uys t emi nt oanumberofelementsusi ngfictitiousdi vi di nglines.The poi nt sofi nt ersect i onof t hedi vi di nglinesarereferredt oas"nodes"or"j oi nt s". Each j oi nt hasacert ai nnumber of degreesof freedom(DOF), upt oamaxi mumof sixDOF (t hreet r ansl at i onal andt hreer ot at i onal ) . Forexampl e, forpl anar mot i onofasl ender beamel ement , t hedi spl acement (bending)perpendi cul artot heaxis,isexpressedasa funct i onoft hespacevariablesas: 4 i =1 ( 2 9 )whereu l ( t ) a n d u 2 ( t ) axet het ransl at i onal andr ot at i onal di spl acement sat onej oi nt[oneend)oft heel ement , andu 3 ( t ) a n d u 4 ( t ) aret hoseat t heot her end. el ( s) are t heshapefunct i ons whi ch areusual l y obt ai nedusing t hedifferential equat i ongoverning st at i cbeambendi ng. Incont r ast tot hepreviousmet hods, t hecoefficientsu~( t ) in (29)aret heact ual physi cal coordi nat es, whichisanicef eat ur eoft hefi ni t eel ement13 met hod. Lagrangianformulationissubsequentlyappliedwhereint hej oi nt forcesare determinedasfunctionsoft heappliedforcedistributionandforcesproducedbyt he adjacentelements.Theproblemfinallyreducesto: M f i +K u =0(30) whereu ( t ) i s ofdimensionn,givenby n i i =1 ( 3 1 )nj beingt henumberof joints, andn D O F ( i ) isthenumberofDOFforthei t h j oi nt .Eq.(30)hast hesameformas(24).Forgivent ypesofelements(e.g.,beam, plate, shell,membrane, etc.),t hest andardshapefunctionsareknown;therefore,dividing anLFSSintosuchelements,andusingtheinterconnectioninformation,makesthe generationof KandMhighlyamenabletomachinecomputation. Subsequentsolution of t heeigenvalueprobl emusingst andardnumericaltechniquesisalsostraightforward. If point-forcesandtorquesareappliedateachj oi nt , thefinal"modal"modelhast he form(afterusingt hetransformation:u ( t ) =* q ( t ) a s in(26)): +Aq(t)=r r f ( t ) (32) y ( t ) = r q ( t ) ( 3 3 )whereqist hen1 modalampl i t udevector,I" isthen ngeneralizedmode-shapema- trLx,fis the vect or of generalizedapplied forces(consistingof forces andtorques),andy is t hen 1 displacementvector,consisting of translationalandrotationaldisplacements at each joint. Thefiniteelementmet hodgives bot htherigidandtheelasticmodes.That is,q ( t )inEq.(32)alsoincludeszero-frequencymodes.However,itisusuallymoreconvenient touse only t heelastic-modeportionof t hefiniteelementmodel,andt oaugmentitwith t herigid-bodyequations[Eqs.(13)and(14)]obt ai nedindependently.Itisalsomore 14 accur at et odot hi sbecausezero-frequencymodesareusual l ycomput edbyt hefinite el ement met hodassomecombi nat i onsof t hebasict r ansl at i onal andr ot at i onal modes,andwi t h smal lnon-zerofrequenciesduet onumeri cal errors,Thusweassumef r om here ont hat q(t)in(32)cont ai nsonl y elasticmodal ampl i t udes.Modelsobt ai nedint hi smanner cannot predi ct t hei nherent s t r uct ur al dampi ng.It iscus t omar yt oadddampi ngt ermst oEq.(32),i.e.,aft ert heeigenvalueprobl em hasbeensolved.Themost commonl yuseddampi ngt er mispr opor t i onal (orviscous) dampi ng, whi chmodifies(32)asfollows: where 4(t)+D4(t )+Aq(t}=r r s ( t )D=2 di ag ( p l wl ,P22,- . . , PnV~n) (34) ( 3 5 )wherepiandwldenot et hei nherent dampi ngrat i oandt henat ur al frequencyoft he i thelasticmode. ForLFSS, pi ' saret ypi cal l y ont heorderof 0.001-0.01.Thecompl et e model ist hengivenby(13),(14)and(34),andcanbeeasilyrepresent edint hest at e variableform.E x a mp l e s o f F i n l t e - E l e me n t Mo d e l sE x a mp l e 1:Lar ge~t h i n , c o mp l e t e l y free~f i at p l a t e -Consi dera100ft x100ft0.1in.,free-free-free-free(i.e.,freeat allt heboundari es)al umi numpl at e. Suppose itisdi vi dedi nt o2424equal squarepl at eel ement s. Eachcornerofeachel ementrepresentsa"j oi nt "; t hust hereare625 j oi nt s. Supposet hepl at eisrigidabout t heaxis per pendi cul ar t oitsplane. Thent heelasticmot i onconsistsofonl yt heout -of-pl ane elastic di spl acementand two t ransverserot at i ons. That is, each j oi nthas t hreedegrees of freedom(nDOF=3 forall j oi nt s). Thedi mensi on of t heeigenvalue probl emis:n=625 3-~1875.However,weusual l yneedt ocomput efewerelgenvalues(frequencies)and eigenvectors(generalizedmode-shapes). Table1showst hefirst44nat ur al frequencies of t hepl at e, comput edusingt heSPARcomput er package[Whe.78].Thust hermat r i x is187544.Thei t/= modeshapecanbepl ot t edbygivingqiareasonabl evalue(while maki ng allot her qi ' szero),andcomput i ng t heout -of-pl anedi spl acement sat each j oi nt .15 Tabl e1. Elastic modefrequencies for large,thin, flat pl at e MODEF R E Q ( R A D / S E C )10 . 0 5 4 9 920 . 0 8 0 0 230 . 0 9 9 1 140 . 1 4 2 1 150 . 1 4 2 1 160 . 2 4 9 4 870 . 2 4 9 4 880.26008 90 . 2 8 2 8 6100 . 3 1 5 1 5110 . 4 3 0 6 8120 . 4 3 0 6 8130. 47824 140 . 5 0 0 0 3150 . 5 3 6 8 9160.53689 170 . 6 2 4 2 2180 . 6 5 9 5 8190 . 6 8 8 0 8200 . 8 0 9 7 3210 . 8 0 9 7 3220.83371 230 . 8 7 3 7 7240 . 8 7 9 8 2250 . 8 7 9 8 2260 . 9 9 2 1 6270 . 9 9 2 1 6281 1483 291.1922 301.1996 311 . 2 1 9 432~ . 2 2 5 0331.2532 341 . 2 5 3 235Io3742 361 . 4 0 8 2371.4871 381. 4871 391 . 6 0 5 9401 . 6 0 5 9411 . 6 9 7 2421 . 6 9 7 2431. 7111 441 . 7 5 2 316 ..','~2 '~- , . . ' . ' : ' "" - tMode3(0,0158 Hz)Hode4(0.0226 Hz) Mode5( 0 . 0 2 2 6 H z ) Mode6( 0 . 0 3 9 7 Hz )Fi gur e1. Mo d e - s h a p e pl ot sf orl arge, thin,f l atpl at e 17 S T R U C T U R A L M O D E NO.23,F R E Q =. 8 3 7 7 E + 0 0 R A D / S E C ( . 1 3 9 6 E + 0 0 HZ) J O I N T C O O R D I N A T E S P H I - Z T H E T A - X T H E T A - YNO.FT.I N / I N R A D / I N R A D / I N5 0 1502 503 504 5 O 55 0 65 0 75 O 85 O 95 1 05 1 1512 513 514 5 1 55 1 6517 5 1 85 1 95 2 O5 2 1522 523 5 2 45 2 5( 1 6 . 6 7( 1 6 . 6 7( 1 6 . 6 7( 1 6 . 6 7(16.67 (16.67 (16,67 ( 16. 67 ( 1 6 . 6 7( 1 6 . 6 7( 1 6 . 6 7( 16. 67 ( 16. 67 ( 1 6 . 6 7( 1 6 . 6 7( 16. 67 ( 16. 67 ( 1 6 . 6 7(16. 67 ( 16. 67 ( 1 6 . 6 7( 16. 67 ( 1 6 . 6 7( 1 6 . 6 7( ( 1 6 . 6 7 ,0 . 0 0 ) 4 . 1 7 ) 8 . 3 3 ) 1 2 . 5 0 )1 6 . 6 7 )20.83) 2 5 . 0 0 )2 9 . 1 7 )33. 33)37. 50)41. 67)4 5 . 8 3 )50.00) 5 4 . 1 7 )58.33) 6 2 . 5 0 )66. 77)7 0 . 8 3 )75.00) 7 9 . 1 7 )8 3 . 3 3 )87.50) 9 1 . 6 7 )95.83) 100. 0)- . 3 1 9 9 E - 0 11 0 2 8 E + 0 02 3 8 9 E + 0 03 4 1 1 E + 0 03 7 9 9 E + 0 03 4 2 8 E + 0 02 4 0 6 E + 0 0I 0 4 4 E + 0 0-2 4 7 3 E - 0 1-i 0 9 2 E + 0 0-1 2 7 8 E + 0 0-8 3 1 5 E - 0 11 2 3 9 E - 0 58 3 1 6 E - 0 1. 1 2 7 8 E + 0 0. I 0 9 2 E + 0 02 4 7 3 E - 0 1-I 0 4 4 E + 0 0-2 4 0 6 E + 0 0-3 4 2 8 E + 0 0-3 9 8 0 E + 0 0-3 4 1 1 E + 0 0- . 2 3 8 9 E + 0 0- . I 0 2 8 E + 0 0. 3 2 0 0 E - 0 1. 2 4 6 0 E - 0 2 - . I I 0 3 E - 0 2. 2 8 1 1 E - 0 2 - . 2 0 1 3 E - 0 3. 2 4 9 4 E - 0 2 - . 4 7 6 4 E - 0 3. 1 4 7 9 E - 0 2 - . 2 0 1 3 E - 0 3. I 1 6 0 E - 0 4 - . I I 0 9 E - 0 4- . 1 4 7 9 E - 0 2 . 6 2 4 3 E - 0 4- . 2 5 2 8 E - 0 2 . 2 2 7 1 E - 0 4- . 2 8 1 1 E - 0 2 - . 9 0 2 7 E - 0 4- . 2 2 5 8 E - 0 2 - . 2 1 4 2 E - 0 3- . I 0 8 2 E - 0 2 - . 2 8 7 4 E - 0 3. 2 9 1 2 E - 0 3 - . 2 7 1 4 E - 0 3. 1 3 7 4 E - 0 2 - . 1 6 4 1 E - 0 3. 1 7 8 3 E - 0 2 . 1 2 8 1 E - 0 7. 1 3 7 4 E - 0 2 . 1 6 4 1 E - 0 3. 2 9 1 2 E - 0 3 . 2 7 1 4 E - 0 3- . I 0 8 2 E - 0 2 . 2 8 7 5 E - 0 3- . 2 2 5 8 E - 0 2 . 2 1 4 2 E - 0 3- . 2 8 1 1 E - 0 2 . 9 0 2 9 E - 0 4- . 2 5 2 8 E - 0 2 - . 2 2 6 9 E - 0 4- . 1 4 7 9 E - 0 2 - . 6 2 4 1 E - 0 4. I I 0 9 E - 0 4 . 1 1 l I E - 0 4. 1 4 7 9 E - 0 2 . 2 0 1 3 E - 0 3. 2 4 9 4 E - 0 2 . 4 7 6 4 E - 0 3. 2 8 1 1 E - 0 2 . 7 8 7 5 E - 0 3. 2 4 6 0 E - 0 2 . i 1 0 3 E - 0 2Fi gure2. Typi cal finiteel ement model data 18 Theplotsof t hefirstsixmode-shapesareshowninFigureI.Figure2showsatypical out put dat apagefort hefinite-elementmodel,whichcanbeusedtoobt ai nthevalues of elementsof t hegeneralizedmode-shapematrix.Forexample,therowcorresponding t o jointnumber501givesitscoordinates, followed byr(s00xs+l),23,F(s0ox2+l),23,and r(so0xs+3),2~.Thecomplete model, which consistsof all the1875 44(--82,500)entries of r , wouldoccupyabout 1200pagessimilartoFigure2.However,wenormallyneed theelementsofronlycorrespondingtothesensorandact uat orlocations,whichare farfewer t han82,500.If therearemactuators, t hermat ri xin(34)isreplacedbyan mnqmat ri xr ! (assumingonlynqmodesareincludedint hemodel).Ifthereare sensors,t hesensorout put isgivenby(33)whereinrisreplacedbythelnqmat ri x r s . If =m, andif t heact uat orsandsensorsarecollocatedandcompat i bl e(i.e.,force act uat orandpositionsensor,ort orqueact uat orandat t i t udesensor),t henr a--rI.Manyfinite-elementcomput erout put scustomarilyexpress t hedat aininch-pound- second(in.-lb.-sec.)units.That is,forces,torques, displacementsandrot at i onsare expressedin:lb.,in.-lb.,in.,andradiansrespectively.If these unitsareused,r8andr /arenot equalfort hecollocatedact uat or/ sensorcase.Conversiontofundamentalunits (ft.-lb.-sec.)canbeaccomplishedby[Jos.80c]i)dividingt herowsofrcorresponding t odisplacement(translations)byv ~ , andii)multiplyingtherowsofrcorresponding t orot at i onsbyV~ . Afterthismodification,r~=rIforthecollocatedcase. E x a m p l e 2:L a r g e s p a c e a n t e n n a - Consider the122 mdiameterh o o p / c o l u m nant ennaconcept([Rus.80],[Sul.82]),shownschematicallyinFigure3.Theant enna conceptconsistsofadeployablecentralmast at t achedt oadeployablehoopbycables heldintension.Asecondarydrawingsurfaceisusedt oproducet hedesiredcont ourof t he radio- frequency(RF)reflective mesh.The shapingof the RFsurface is accomplished by meshshapingties.Thedeployable mast consistsof anumberof tclescoplng sections whicharedeployedbymeansofacabledrivesystem. Thehoopconsistsof48rigid segments,andis deployed by four mot ordrive units.Thereflective mesh,whichis made ofknitgold-platedmol ybdenumwire,isat t achedt othehoopbyquartzorgraphite fibers.Precision shapingof the RFmesh(e.g., spherical,parabolic,etc.}is accomplished bycontrolcordsaat t achedtot hemesht hrought hesecondarydrawingsurface. 19 Z ; c a b l e sY X b l e sFigure 3. Hoop/column ant enna schematic Table 2. Hoop/column ant enna parameters M a s s = 4 5 4 4 . 3 K g .I n e r t i a a b o u t a x e s t h r o u g h c e n t e r o f m a s S ( K g - m2) I=5 . 7 2 4 x1 0 6 I=5 . 7 4 7 x1 0 6x x y yI=4 . 3 8 3 x1 0 6 I=3 . 9 0 6 x1 0 4z z X ZI=I=0 x y y zIMo d e n o . 1 F r e q . I r a d / s e e i M o d e no. F r e q ,r a d / s e e5 0.75 5i 5 0 . 8 5q 1 . 3 5 1 . 7 0 3 . 1 8 4 . 5 3 5 . 5 9 5 . 7 8 6 . 8 4 7 . 4 8 . 7 81 1 . 2 4 1 5 5 . 0 5 1 1 7 . 4 1 5 5 . 7 5 1 1 6 . 0 4 1 _ 1 . 8 ~20 M o d e 2 : 1 . 3 5 r a d / s e c( f i r s t b e n d i n g , X - Z plane) x M o d e 3:I .70r a d / s e cf i r s t b e n d i n g , Y - Z p l a n e )xx~ M o d e 4 : 3 . 1 8 r a d / s e c( s u r f a c e t o r s i o n )z M o d e 5 : 4 . 5 3 r a d / s e e M o d e 7 : 5 . 7 8 r a d / s e cg l Y-Zpl a be ndi ng, X-Zplanel r xx ~M o d e 8 : 6 . 8 4 r a d / s e c~C3x z Figure4.Typicalantenna modeshapes 21 A 20-elastic-mode finite-elementmodel of t heant ennawas obt ai ned[Sui.82]andt he resultingelasticmodefrequenciesareshowninTable2,alongwi t hotherparamet ers oftheantenna. Usingthemode-shapedat afor112joints(eachwi t hDOF=6)on themast , t hefeeds,t hefeedpanels,thesolarpanels,andtheRFmesh,afewtypical mode-shapesarepl ot t edinFigure4.Theformat fort hefinite-elementmodeldat ais very similart ot hat fort heplate(Figure2),but each jointhassixdegreesof freedom, comparedt ot hreeDOFforeach jointfortheplate. 1. 1. 3Cont r ol l a bi l i t ya n d Ob s e r v a b i l i t y o f Fi n i t e - Di me n s i o n a l Mo d e l sAfinitedimensionalmodelof anLFSScanbeexpressedinthest at e-spaceform as: :~=A z +B u (36) where T T z=(qrb, qrb,q l , q l , q2, q2, . . . , q,~q,(l,~q)T u is t h em 1c o n t r o lv e c t o rc o n s i s t i n go fa p p l i e d f o r c e sa n d t o r q u e s , qrbis t h e6 1v e c t o ro fr i g i dt r a n s l a t i o n sa n d r o t a t i o n s ,a n d qi d e n o t e s t h eith m o d a l a m p l i t u d e .A=diag(Arb, A , , A 2 , . . . ., A n q ) ( 3 s )[ 0 6 / 6 ] ( 3 9 )Arb=0606 (OkandIkdenot ethekknullandidentitymatricesrespectively). Ai[0I](40) =2- - 2 p i wi--co i B , b ]B = B2( 4 1 )22 B i ' : b Asanexampl e, consi dert heat t i t udecont r ol pr obl emwi t honl yt or queact uat or s , and a t t i t ude andr at esensors. Int ha t case, t heri gi dt r ansl at i onal mot i onisi gnorabl e, and qrb=(, 0, ) Twhi char et het hr eer i gi d- bodyr ot at i on(Eul er)angles, and Arb =0303 Consi der asingle3-axk~t or queact uat or , whi chyields: 03, B,b=Lj F ] (44) wher e,Isist he3 x 3moment ofi ner t i amat r i x. If asingle3-axis a t t i t ude sensor isused,t hesensor out put is: =c ~ (45) wher e C = [OrbCl C~ c , 6 ={z8o 3 ]C i = [ C / ( 3 l ) 0 3 I ]Ifat hr ee- axi sat t i t uder at esensorisused,. . . C . q ]c ~ b = [ o z 3 1oi - - [oc ~ ]If bot ha t t i t ude andr at esensorsareused(at t hesamel ocat i on),(46) ( 4 7 )( d S )(4 9 ) ( 5 0 )23 =I o ( 5 1 )Ci =di ag(ci ,ei)(52) Thefollowingt he or e mgivescondi t i onsforcont rol l abi l i t y.T h e o r e m. Thes ys t emgivenbyEq.(36)iscontrollablei f andonl yi f (iff )allof the followingcondi t i onsar esatisfied: i)Brbisoff ul l rank ii)Eachbi[see(42)] correspondingtodi st i nct eigenvaluesofAhasatleastone nonzero ent ry iii)Foreachmul t i pl eeigenvalueAiofmul t i pl i ci t yv, themat r i x B j = bE isofrankv, wher ebTkist herowof t heBmat r i xcorrespondingtot hesecondrow of t hek ~h2x2blockf or Aj. P r o o f . Th e pr oof canbeobt ai nedbyst r ai ght f or war dappl i cat i onoft hePopov-Bel evi t ch- Hant us( PBH) r ankt est [Kai.80]. (Thesymbol ""isusedt odenot et heendof aproof, or, whennopr oofisgiven, t he endofat heor ems t at ement . )Asanexampl e, fort heat t i t udecont r ol pr obl emwi t honl yt or queact uat or s , con- di t i on(i)woul dbesat i sfi ediff t her eisat l east onet or quea c t ua t or per axis.Condi t i on (ii)woul dbesat i sfi ediff t her ot at i onal modeshape(or"mode- sl ope") foreachmodeis nonzer oat t hel ocat i onof at l east oneact uat or . Condi t i on(iii)needst obet est edonl y 24 whent her ear emor et ha noneelasticmodeswi t ht hesames t r uct ur al frequency. Thi s isnot anunc ommonoccur r enceint hecaseof s ymmet r i cs t r uct ur es (forexampl e, see Tabl e1). Si mi l arnecessar yandsufficientcondi t i onscanbeobt ai nedf or obser vabi l i t yin anent i r el yanal ogousmanner , andt hat discussionisomi t t ed. However , itiswor t h not i ngfort hea t t i t ude cont r ol pr obl emt ha t t her i gi d- bodymodesarenot observabl e usi ngat t i t ude- r at esensorsal one(i.e.,wi t hnoa t t i t ude sensor). The s ys t emcanbe obser vabl eifat l east t hr eeat t i t udesensors(oneper axis)areused, evenwhenr at e sensorsar eabsent .1. 2P r o b l e ms i nCo n t r o l l e r De s i g n f o r L a r g e S p a c e S t r u c t u r e sThe mai npr obl emconsi der edint hi sbookist hea t t i t ude (orfi ne-poi nt i ng)con- t r ol pr obl em, whi chi mpl i escont r ol l i ngt heri gi dr ot at i onal modesa n d suppr essi ngt he el ast i cvi br at i on. The obj ect i veoft hecont r ol l er ist o 1)qui ckl y da mpout t hepoi nt i nger r or sr esul t i ngf r om st epdi st ur bances(suchast her -mal di st or t i onr esul t i ngf r om ent er i ngor l eavi ngEa r t h' s shadow) , or nonzer o i ni t i alcondi t i ons(e.g.,r esul t i ngf r omt hecompl et i onofal arge-angl ea t t i t ude maneuver ) ,and 2)mai nt ai nt hea t t i t ude ascloseaspossiblet ot hedesi redat t i t udeint hepr esenceof noi seanddi st ur bances.The firstobj ect i vet r ansl at esi nt ot hecl osed-l oopbandwi dt hr equi r ement , while t hesecondonet r ansl at esi nt omi ni mi zi ngt her oot meansquar e(RMS)poi nt i nger r or .Inaddi t i on, itisr equi r edt ha t t heel ast i cmot i onbever ysmall;i.e.,t heRMSshape di st or t i onsmust bebel owpr escr i bedlimits. Forappl i cat i onsuchast hel argecommu-ni cat i onsant enna, t het ypi cal ba ndwi dt hwoul dbe0.1r ad/ s ec, wi t hat mos t 4second t i mecons t ant f or allt heel ast i cmodes(cl osed-l oop). Typi calal l owabl eRMSer r or sare: 0.03deg.poi nt i nger r or , and6mmsurfacedi st or t i on.The pr obl emsencount er edindesi gni ngana t t i t ude cont r ol l er are:1)Anadequat emodel ofanLFSSisofhi ghor der becauseitcont ai nsal argenumberof el ast i cmodes. Ther ef or e, apr act i cal l yi mpl ement abl econt r ol l er hast obeof 25 7 / / / / I I Co n t r o l l e d ' Mo d e s f I1\ \ \ Process . _~ Noise\ f t Input _ ~ F e e d b a c k 1,t Controller \ Mode1\Sensor Noi se I ( I ) o 1 ~ - I Mode2 1 ModeN,1 - ~ ModeN+I I-I ' - " ' 1 ~-Observation ControlSpUlover Spillover O 0Sensor 11 Output F i g u r e 5. E f f e c t o f m o d a l t r u n c a t i o n26 reduced order. 2)Theinherent energy dissipation(damping)isvery small. 3)Theelasticfrequencies arelowandclosely-spaced. 4)Theparameters(frequencies,dampingratiosandmodeshapes)arenotknown accurately apriori. The simplest controller design approach would be to truncate the model ata certain mode,and tousethe truncated"design model"todesign acontroller, which, of course, wouldheof reducedorder.Thisapproachisroutinely usedforcontrollingrelatively rigidconventional spacecraft,whereinonlytherigidmodesareretainedinthedesign model.Second-orderfiltersaxeincludedinthelooptoattenuatethecontributionof theelasticmodes.Thisapproachdoesnotgenerally work forLFSS because theelastic modesaremuch more prominent.Figure 5shows the effect of using atruncated design model.When building a control loop around the~controlled" modes(which are included inthedesignmodel,onealsounintentionally buildsafeedbacklooparoundthetrun- cated(or~residual')modes.Theresultingcontrolsystemmaymaketheclosed-loop system unstable.Theinadvertentexcitationof theresidualmodesandtheunwanted contribution of theresidual modestothesensed outputweretermedbyBalas[Bal.77] respectively as"control spillover"and"observation spillover'.Thespillover termsmay cause reduction inperformance,andeveninstability leading tocatastrophicfailure. Inaddition tothetruncationproblem,thedesignerusuallylacksaccurateknowl- edgeof theparameters.Approximatemethods(suchasfiniteelement)areknownto givereasonablyaccurateestimatesofthefrequenciesandmodeshapesonlyforthe firstfewmodes,andcanprovidenoestimatesof inherentdampingratios.Pre-mis- sion ground-testing for parameter estimation would beinfeasible becauseLFSS arenot designed towithstandthegravitationalforce,andbecausethetestfacilitiesrequired (e.g.,vacuum chamber)would beexcessively large.Another consideration in controller designisthattheactuatorsandsensorshavenonlinearitiesandfiniteresponsetimes. Inviewof theseproblems,theattitudecontrollermustbea"robust~one,thatis,it mustmaintainatleaststability,andperhapsperformance,despitemodelingerrors, uncertainties, nonlinearities andcomponent failures. Chapt e r 2 ACl as s o f Ro b u s tDi s s i pa t i v e Co n t r o l l e r sThemai npr obl emindesi gni ngacont r ol l er foranLFSSist hemodel i nger r or("spi l l overs' )ari si ngf r omt r uncat i ngt hemodel , andf r omt helackofaccur at eknowl- edge of t hepar amet er s . Int hi schapt er , weconsi deraclassof cont r ol l er swhi chci r cum-ventt hesepr obl ems. Suchcont rol l ers, t er med"di ssi pat i ve"or "col l ocat ed"cont rol l ers,consistofcompat i bl epai rsofact uat or s andsensorswhi chmaybedi s t r i but edt hr ough-outt heLFSS. For exampl e, anat t i t udeandanat t i t uder at esensor isl ocat edat t he same poi nt asat or queact uat or ; di spl acement andvel oci t y sensorsar ecol l ocat edwi t ha force act uat or . We consi der t wot ypesdi ssi pat i vecont rol l ers: 1)col l ocat eda t t i t ude con- troller( CAC) , and2)cont rol l ersusi ng vel oci t y f eedback. The secondt ypeofcont r ol l er s includesa)col l ocat eddampi ngenhancement cont r ol l er s( CDEC) , andb)t ot al vel oci t y feedbackcont r ol l er s( TVFC) . ACACisdesi gnedt ocont r ol bot ht heri gi dmodesand the el ast i cmodes, wher east hef unct i onof aCDECandTVF Cist oda mpout t heel ast i c mot i on. The CDECisusedt oenhancet hes t r uct ur al dampi ngwi t hout affect i ngt he rigidmodes, whi l et heTVFCaddi t i onal l yst abi l i zest heri gi dmot i onint hesenset ha tallr i gi d- bodyr at esalsot endt ozero. Bot ht ypesof cont r ol l er suseout put f eedback,andt her ef or ear esi mpl et oi mpl ement .2. 1De s c r i p t i o n o f Di s s i p a t i v e Co n t r o l l e r sThe l i neari zedmat hemat i cal model of t hesyst emisasfollows: Ri gi dmodes: Cent er ofmass(c. m. )t r ansl at i on:28 m!=y. f ,( , )i = Iwherern 8 ist heLFSSmass, ~ist hedi spl acement of t hec. m. , andf i (i=1, 2, ..., mi )aret heappl i edforces,eachbei nga3-vectorconsistingofX,YandZcomponent s.Rot at i on:m y r n TJ . a =~r,r, +~T;(2) i-----I. 7 ' = 1whereJsist he3x3moment ofi nert i amat r i x, a=(, 0, ) Tist herigid-bodyEuler anglevect orabout X,YandZaxes,riist he3x1 coor di nat evect orof t hel ocat i onof t heappl i edforcef i ,andT j ( j =1,2,. . . , roT)ist he3x1t orqueat l ocat i onj .Elasticmot i on[fromeq.(34),Chapt er 1]: +D~+hq=~ T F +e~TT ( 3 )whereqist henq1 modal ampl i t udevect or, andk~ andiI~ are(respectively)3 ml xnq and3rn Txnqmode- shapeandmode-sl ope(i.e.,r ot at i onal modeshape)mat ri ces. F andTdenot et heappl i edforceandt orquevectors.Wefirstconsi dert hecasewheret heact uat or sandsensorsare"ideal"(i.e.,linear andi nst ant aneous) . Eqs.(1)-(3)canbecombi nedas: Asf i +Bs ' ) +C~p=FT u( 4 )where [ o 0 0 ]A~ = J~0 0I, ~ (Thesymbol Ikisusedt odenot et hekxki dent i t ymat r i x) .(5) B, = d i a g ( 0 a , 03,Dn,xn, ) (6) 2 9whereDist henqxIZqs ymme t r i c ma t r i xr epr esent i ngt hei nher ent s t r uc t ur a l dampi ng,and0hdenot est hekxknullma t r i x. Sinces omedampi ng, noma t t e r howsmal l , is alwayspr es ent , wea s s ume D>0. C. =di ag( Oz, Oa,An. xn. ) (7) wher eAist hedi agonal ma t r i xofs quar edel ast i cmode frequenci es, and rT~ _1 3 , I 3 , . . . , 1 3 , 0 3 , 0 3 , . . . , 0 3~ 1 , ~ 2 , . . . , F , , ,I , 1 3 , 1 3. . . . ,I ~"'''rnf,""'rtlT denot est hecrosspr oduc t ma t r i xoft he3- vect or r=( r z,ry,r z )T 0- - rzrz ]~ =r z 0- r u - - rzry0 ( 8 )p = ( ( T , a T, q T) T(9) u( f T , f ~2, TTq~Tq~T~T =" " , f mi , T l , ~2, ' " , ~ mr )( 1 o )In(8),koyand4 idenot et he3nqmode - s ha pe andmode- sl opemat r i ces fort hej t hact uat or l ocat i on. Supposemy 3-axi st r ans l at i onal posi t i onandvel oci t ysensor sar e pl acedat t hes a me l ocat i onsast her n f forceact uat or s , andmT3-axi sa t t i t ude andr at e sensorsar epl acedat t hes a me l ocat i onsast hemTt or queact uat or s . The nt he3x1 t r ansl at i onal di s pl acement vect or at t hek t hs e ns or l ocat i onwillbe:Ylk=~- rk~- t- ~kq Similarly,t he3x1a t t i t ude vect or at sensor l ocat i onkwillbe:( 1 1 )YTk=a+C k a ( 1 2 )30 Denoting thesensor outputvectorby yp, TT YP(YT,TTTYT,,T ) :, U I ~ , ' " ' Y f ~ f ' Y T I ' " "Itcanbereadily verified that ( 1 3 )y p :Fp(14) ItcanbeseenthatthematricesAs,Ba,andCaaresymmetric.Inaddition,As ispositivedefinite,andBaandC8arepositivesemidefinite.Notet hat the"F"in (14)isthesame"r"whichappearsin(4).Itisthispropertyofthesystem(which isduetocollocationof theactuatorsandsensors),togethterwiththesymmetry and non-negativity of As,Bs,andCa,t hat enablesthedesign of aclassof robustly stable outputfeedback contrtollers. Theoutputof theratesenorsisgiven by: y , =r p 0 5 )Inreality,thesensoroutputwillbecontaminatedbyadditiveobservationnoise. However, sincethenoiseisof noconsequence inthestabilityanalysis, weignoreitfor themoment.Consider thecontrol law: = - o , , y p - c , . y , . ( 1 6 )whereGpandGrdenotethe3 ( r n f -t-mr ) x3 ( m . f -]- r n T ) proportionalandrategain matrices. TheCollocateddampingenhancementcontroller(CDEC)issimilartotheCAC described above, except t hat its purpose is to enhance structural damping using velocity feedback, withoutaffecting therigid modes.IntheCDECmode,therigid components ofthetranslationalandrotationalvelocitiesareremovedfromthevelocityfeedback signal.Forexample,supposetherearetwotorqueactuatorsandcollocatedattitude 31 ratesensorsat t wodi st i nct l ocat i onsont hes t r uct ur e. The nt heel ast i cmot i oncanbe describedby:+D O +Aq=(I)tTT1 +~ T 2 (17) If weput t hecons t r ai nt t ha t T~=- T I =T, t hent heri ght ha ndsideoft heabove equat i onbecomes: (~2-O ) I ) T T .Inaddi t i on, if wes ubt r a c t t heout put s oft het wor at e gyrosignals,weget: y=- ( 1 8 )By following t hi spr ocedur eusi ngpai r edact uat or andsensorl ocat i ons, t heri gi dmot i on canber emovedf r omt heequat i ons. The el ast i cmot i oncanbeexpr essedint heform:+D4 + Aq=r r u(19) whereFdenot est hema t r i xobt ai nedbys ubt r act i ngt hemodeslopes. The modi fi ed rat esensorout put vect or is: # r =t 0 ( 2 o )Itshoul dbenot edt ha t t heuseofforceact uat or s int hi smanner woul dgener al l yaffect the ri gi d r ot at i onal modes; t her ef or e,t or queact uat or s are mor e sui t abl efor t hi spur pose.Th e CDECcont r ol lawisgi venby:u=- Gr OrIfGr isdi agonal , t her esul t i ngCDECconf i gur at i oniscal l ed" member da mpe r " con-t rol l er[Can.78]. 32 Thet ot al velocity feedback controller(TVFC)isidenticaltot heCAC,exceptthat thepositiongainGp=0ineq.(16).Thusthiscontrollerincludest hefeedback of both t herigidandt heelasticcomponent sof t herates.TheTVFCcanaccomplishdamping enhancementandat t i t udestabilization(but not control),int hesenset hat therigid bodyratest endtozero. 2. 2St a b i l i t y P r o p e r t i e s of Di s s i pa t i ve Cont r ol l e r sInthissectionweinvestigatethestabilitypropertiesof dissipitavecontrollerswith t heassumpt i ont hat thesensorsandact uat orsareperfect,i.e.,linearandinstantaneous. Wefirstconsidert heCAC.From(4),(14),(15)and(16),theclosed-loopequations become: (21) where ~s=B,+r r a r r , andO,=C, +r r c pr (22) Thefollowingt heoremprovesthestabilityoftheclosed-loopsyst emfortheCAC. Th e o r e m1.SupposeGpandGraresymmet ri c, andGp>O.Thentheclosed-loop system givenbyeqs.(21)and(22)is stableinthesenseo[ Lyapunovi fCr>O,andis asymptoticallystablei fGr>O. Pr o o f . Wefirstprovet hat Csispositivedefinite.Considerthequadrat i cform: W( p) =pT~'sp-----qT hq-4- pTFTGpFp SinceA >0,W( p) canbe0onlyif q=0,andpr Fr CpFp=0;i.e.,onlyif ( ~ 5 , q * ) r * a ~ rF.l=o t ; o j3 3wherexrb=( ~r , a ~' ) T- Thus W(p)canbe0onl yif TT xr b( I GpI)x,.b=0,( 2 3 )where [ I 3 x 3 , . . . , z 8 o 3 , 0 8 , . . . , o 3 ]Z = '. . . , "/ 3 , I s , l s Lr l , r2,rmy, ..., JClearly,si ncet hecoefficientmat r i xin(23)isposi t i vedefi ni t e, W(p)canbezero onlyif XTb =0.Thus weconcl udet ha t W(p)=0onl yif p=0,andt ha t C, isposi t i ve definite. Nowconsi dert heLyapunovf unct i on:V (p, ~)=p rCsp +~T As~ (24) Visposi t i vedef ni t e. Using(21), "2:2pTC'sp -4- 2pT Asp ( 2 5 )IfGr_>0,' 2isnegat i vesemi defi ni t e. Ther ef or e, t hes ys t emisst abl eint hesenseofLyapunov.SupposeGr >0.Using(22)in(25), ' 2=- 2 f f r r Gr r p - 2dTTD{7 Ther ef or e, for' 2=O,itisnecessar yt hat ~=O,andhencet ha t ~Tb=O.Consi deri ng t hecl osed-l oopequat i on, t hi simplies ~ p = o ( 2 6 )34 which can happen(since C,>0)only when p=0. Thus V0.Th a t is,adi r ect t r ansmi ssi ont er m, no mat t er howsmall, waspr esent . Fr omThe or e m1,t hecl osed-l oops ys t emisa.s.for anye>0.Ther ef or e, t hecl osed-l oopei genval uesareallint heopenlefthal f-pl ane.Becauseof cont i nui t y, itisobvi oust hat , whenc=0,t heei genval ueswillnot crosst he i magi naryaxis. Th a t is,t heei genval ueswillbeint heclosedlefthal f-pl ane. The or e m9 given bel owconsi derst hecasewhen6=0.It essent i al l yshowst hat , if t hecl osed-l oop system wi t hnoel ast i cmodesint heloopisa.s.,t hensoist hes ys t emwi t hel ast i cmodes,provi dedt ha t (100)issatisfiedwi t hHrepl aci ngH.T h e o r e m9. Suppose~1isanon-anticipative,asympoticallystable,observable,LTIoperatorwithrationalt r ansf er mat ri xH(s)whichis properandminimum-phase.I f the closed-loops ys t emfortherigidbodymodelal one(/.e.,Eqs.(27),(34),(35)(88)-(91) withnq=O)isa.s.,thentheentireclosed-loopsyst em(Le.,withnq7~ O)isa.s.i fH( j w) (w2Gr-j apw) +(w2ar-j apw) H* (jw)>0forallreal w.(114) 58 P r o o L Consi deri ngt her i gi d- bodyequat i ons,J , a :~u~=X( uo+%)(115) wher eu s =- G p a -Gr&anduq:- G p q -G r ( ~ Thust het r ansf er f unct i onfrom t o& isgi venby M ( s ) = [I + H ( s ) { G p + s a ~ } ] - I H ( s ) { G p + s G ~ } # (116) Since t hecl osed-l oop r i gi d- bodysyst emis st r i ct l yst abl ebyassumpt i on, M( s ) isstrictly st abl eandfi ni t e-gai n, whi chimplies II~llT~rll~lIT+IIh,~llT (117) wher erist hegai nofMandhmisitsfreeresponse. Pr oceedi ngasint hepr oof of The or e m8,wecanarri veat Eq. (109)wher ei nc, =0andh0isr epl acedbyhm. Since uer=- Gr ( &+~) , wehavef r om(117), Iluc,[IT~cI[[~I[T +c211hmilT (118) wher ecl ande2areposi t i veconst ant s. Compl et i ngsquaresasin(110)andnoting t ha t IIh,nllisfinite, i t canbepr ovedt ha t IlqllTisboundedforallT>0,andthat Iim~_.oo ~(~)---- 0.Fr om(118),ucralsot endst ozeroast--~oo.The r emai nder of the pr oof issi mi l art ot ha t ofThe or e m9. Co r o l l a r y 9. 1. Undert he condi t i ons o f Th e o r e m9,i f Gp,Gr , Haredi agonal , then t he cl osed-l oops y s t e misa.s.i f (112)issat i sf i edwi t hHrepl aci ng[ I . Fr omCor ol l ar y9.1,fort hecasewher eHi ( s ) =k i / ( s +hi )wi t hk i , a l >O,the cl osed-l oopas ympt ot i cst abi l i t yisassuredifGpi-o I I I 0. ~;.50 (P~) d 3 4 c , J JlI X0 - 5- - - - "- - - - - DD- I l l - - I l l0iIII_I O. 2 5 , 50 (P=)d x10-3 2( 10 _ o25, 50 (P~)d 1 .(rnm) "8 . t t~- - ~1II 2~, 50 ( o ~ )d . 6E , 2,L ( r n r n )0. 25o50 (pro)d F i g u r e 6 . P e r f o r m a n c e o f C A CO - O - O0-- O.,1a=0. 02r a d/ s e cO)a=0. 1t a d/ s e eO)m=0. 25r a d/ s e c77 106 I 0 ~ 6pp I 0~iL 1 0~2 0 , 2~ (P~)d 7so tO+ I 0~6po I Ot2 10 o O, 25 ( P~) d . 5 0i 0~1 0 6lO s 2 6 q 2 0 .2S (Pro) d .50 10 n 1 0 1 oI OB I1 0 7 ~. . . .i 0e ~~ X~-~ lO st 0 l Om "~ r l OB~, ,-~ i 0~ .~~ " ~~ ' ~I Os t,25..sO0,25 (P~) d(p~) d F i g u r e 7.P e r f o r m a n c e c o e f f i c i e n t 6p .50 O S "4 p l u e p g j e o o o o u e u ~ o j ~ d " S ~ n g ! - dP ( m o )S g "fj , , ~~ l l~ ! ~ I ] % ,z ~ " %I0 O S "t m O I' I n O I' SI ~ r O ISI e i O IP ( ~ d )S 8 " 0/ \ ,i x / ' \ Si Ir , / ~ -II c ~ O IP ( r o d )0 " S 8 ' OL O !i~ 98 0 I' ~ 1 9 G O IP ( ~ o )O S " S 8 "0 0 Ih. _ . . - - 4 ~ ~ - - - - - - . ~ 1 8 _ ~ O Ia e 4998s O 1V ( r o d )O S " S 8 "PO9 0 I89gt O [o 0 Io a s l p ~ 1 T ' O = = 00 - 0 - . . 00 4 3 - C 1o - o - o8 /~ D a = 0 . 0 2 r a d l s e cU l ~ C O = = 0 . i t a d / s e e(.4)o. = 0 . 2 5 r a d / s e c79 I0~ e S 10~8 6 10"6 ~4 0.25.SO (Pm) d l O a e 10.+~ 2 .25.,50 ( pm) d 1 0-3 II t O-~ 2 tO'+ t l O 4 0 k \ \\ \.25 (poO d BO 1 0 : I510o 10 4t 5 10 41 0 ~x x\ x% \ \ \(ore) d 1001 5 ~ s10 41 5 1 0- ~I.r~oo . ~ . , 50 (P~) d Fi gur e9.Pe r f or ma nc e coef f i ci ent 6a. 80 threer esponsespeeds. Gener i cda t a suchast hesecanpr ovi deuseful guidelineslot sel ect i ngt hecont r ol s ys t emspeci fi cat i ons.I nor der t oeval uat et hecont r ol l er mor ecompl et el y, t hefollowinginvestigations weremade:Ef f ect o f i mpr e c i s e c o l l o c a t i o n: Allsensorsweredi spl acedf r omt hecorre- spondi ngact uat or l ocat i onsby4-60cmal ongt hemas t . Fort henomi nal case[wa= 0.1r ad/ s ec, (pw)d--0.25]t hecl osed-l oopei genval uesr emai nedvi r t ual l yunchanged,andt heRMSer r or sshowedIesst ha n1%i ncrease.Ef f ect o f i mp r e c i s e k n o wl e d g e o f t h e p a r a me t e r s : The par amet er s (cJi,pi, and~) wer echangedby+10%f r omt heval uesusedint hedesign;t hi sr esul t edina ma xi mumof 2.5%det er i or at i onint heper f or mance.Ins ummar y, acol l ocat edat t i t udecont r ol l er canbedesi gnedt omeettheperfor- mancer equi r ement sint er msofbot ht hebandwi dt handt heRMSer r or . For this ant enna, aba ndwi dt hof 0.1r ad/ s ecand(0w)~of0.25appear t obesat i sf act or y. The cont r ol l er per f or mancewasf oundt ober el at i vel yi nsensi t i vet opar amet er inaccura- cies.Ame t hodof par amet r i zi ngt heda t a waspr esent ed, whi chprovi desguidelinesfor sel ect i ngt hedesi gnpar amet er s .2. 6T h e An n u l a r Mo me n t u mCo n t r o l De v i c e ( AMCD) : An Ac t u a t o rCo n c e p t f o r Di s s i p a t i v e Co n t r o lAsdi scussedint hepr evi oussect i on, di ssi pat i vecont rol l erscanbereal i zedinprac- t i ce by usi ng t or queact uat or s andat t i t udeandr at esensors.Int hi ssect i onwei nt roduce aspecifica c t ua t or concept whi chhast hei nher ent f eat ur eofcol l ocat i onof act uat ors andsensors. Thi sact uat or concept , cal l edt he"Annul ar Mome nt umCont r ol Device ( AMCD) ' , wasori gi nal l ydevel opedat NASA- Langl eyResear chCent er [And.75]for a t t i t ude cont r ol of convent i onal (rel at i vel yrigid)spacecr af t . However , becauseofits i nher ent design, itisalsowellsui t edfort heact uat i onof di ssi pat i vecont r ol l er s(with allt heaccompanyi ngr obust nesspr oper t i es) , aswillbeshownint hi ssect i on.The concept ual devel opmentof t heAMCDwasmot i vat edbyt heneedt omaxi mi ze t het or queout put t oact uat or weightrat i o. The ma xi mumout put t or queof ast ored- Magnetic7 actuators motor- ~ ~81 Figure10. AMCD/LFSSconfiguration 82 mome nt um device(e.g.,mome nt um wheels,CMG' s, etc. )ispr opor t i onal t ot heangular mome nt umwhi chcanbest oredinitsrot or, whichisequaltot hemoment ofinertia t i mest heangul ar velocity.Theopt i mumrot orshapeformaxi mi zi ngt hemoment 0f i ner t i aforagivenmassisanannul ar r i mwi t hnocent ral hub, i.e.,wi t hallitsmass concent r at edat t heri m. Thi sispreciselyt heshapeofanAMCD. AsshowninFigure 10,t heAMCDconsistsofar ot at i ngri msuspendedbyt hreeormorenoncontacting el ect romagnet i cact uat or st at i onsandspunbyanoncont act i nglinearelectromagnetic mot or . Eachel ect romagnet i cact uat or st at i onconsistsof anaxialandar adi al actuator whi chcanappl yt hecommandedforceint hecorrespondi ngdi rect i on. Axialandradial r i mpr oxi mi t ysensorsarealsopl acedat t heact uat or l ocat i ons; hencet heinherent col l ocat i onpropert y. Thethicknessoft her i misbasicallydet er mi nedbyt helimiton t het i pspeed, whi chispr opor t i onal t ot hetensilestress. Increasi ngt her i mradius increasest hetipspeedandt hetensilestress.Inaddi t i on, excessivelylarger i mradius woul dmaket her i mflexible,t huscompl i cat i ngt heprobl em. Therefore, welimitthe AMCDri msint hi sanal ysi st oabout 2-3met ersdi amet er size,andassumet hat they arerigid.2. 6. 1Ma t h e ma t i c a l Mo d e lSupposeasingleAMCDissuspendedusing! (>3)magnet i cact uat or st at i ons. We arepri mari l yconcernedwi t hr ot at i onsabout twoaxes(XandY,int hepl aneofthe rim}.Ther ot at i onabout t heZ-axiscanbecont rol l edbyt hespi nmot or {Z-axistorque canbeappl i edsi mpl y byaccel erat i ng ordecel erat i ngt her i musingt hespi nmotor};and is not i ncl uded int hi sanal ysi s. Thelinearized two-axisAMCDr ot at i onal equat i onsare givenby:[Jos.80a] Ja&,~ +W&a=Cl f (148) whereJ~ist het ransverse-axi sr i mi ner t i amat r i x(2x2),~a---(Ca, 0a) a" r ot at i onanglesabout t heXandYaxes, aret herim Yl,Y2,--.,Yt] c , = ( 1 4 9 1--2~1,--Z2~, . . , - - ~ 83 (zi ,yi ) beingt hel ocat i onof t hei t hact uat or int heLFSS-fixed coor di nat esyst emwi t h originat t henomi nal posi t i onof t her i mcenter.whereHist her i mangul ar moment umabout t heZ-axis.( / - / =J ,zwz,wherewzist he rim spinvelocity, whi chisassumedtobeconst ant ) .f=( F 1 , F 2 , . . . , F t ) T(151) whereFi ist heaxial(Z-direction)forceat act uat or i.Theri mt r ansl at i onisgivenby: m a z a =C 2 f (152) wherema andz~aret her i mmassandt heZ-axisdi spl acement oft her i mcent er,respectively,and C2=(1, 1,..., 1) 1xl (153) Thetwo-axisri gi d-bodyequat i onsof mot i onfort helarge,flexiblespacest r uct ur e (LFSS)aregivenby: J , &, =- ~C2f -Cl f(154) where =( y c , - x c ) T( 1 5 5 )(xc,Yc)bei ngt hel ocat i onoft henomi nal posi t i onoft her i mcent erint heLFSS-fl xed coordinate syst em wi t h originat t heLFSSc. m. Jaist he2 2LFSSi nert i amat r i x, and aa=(s, 0a)ist hetwo-axisLFSSri gi d-bodyat t i t udevector. TheZ-axisri gi d-body translationalequat i onof t heLFSSc.m.isgivenby: rn,%=- - C2 f(156) 84 wher em 8ist heLFSSmas s andzsist heZ-axi sc. m. t r ans l at i on. Th e fl exi bl emotion of t heLFSSisgi venby:+D4 + Aq=_ ~ T f( 1 5 7 1wher eq,DandA havebeendefi nedpr evi ous l y(See.2.1),andkoist heXnqmatrix of t heZ-axi smode s hapes at t hea c t ua t or l ocat i ons. Asint hepr evi ousanal ysi s, we as s umet h a t s omeda mpi ngispr esent ineachflexiblemode; i.e.,D--D T>0. Let zsadenot et heposi t i onof t hel ocat i onont heLFSSwhi chcor r es ponds t othe nomi nal r i mcent er posi t i on. The abs ol ut et r ans l at i ons zs~andzaar enot i mpor t ant in t hi sanal ysi s; r a t he r t her el at i vet r ans l at i oneisi mpor t a nt , wher e (158) Fr omeqs. (158),(152)and(156),wehave =m- l C2f _~T&(159) wher e , . , - , = . - , , , , . , - , , , / ( , ? - , , , + , ' , , . )(160) The equat i ons ofmot i oncanbecombi nedas: wher e A~ +B~+Cx=/ ~f( 1 6 1 )85 A= J i 00 r n r n ~T rn~J .+ m ~ r 00 0 o 0 I nq (163) B = d i a g ( W, 0s,D) (164) C = d i a g(0s3_. , . . ) (165) =[c, c L - e L - q, ] ~" (166) Using t het ransformat i on: x=Th, where h : (O~aT, g,Or , - - n' a, q rT (167) T= Io -/3 _ o 10 0 ,~o___ : o 1 1O' O [o1 ; o 1 0', I . , J (168) we have t heequat i onsint heform: ~+ BA + Ch= ~r f(169) where =T T A T(17o) ")': [ O I ~ x 2 , 0 2 T ,- - CT ,--%111(171) Thelx1act uat or (axial)cent eri ngerrorvect or6isgiven by: 86 =c T - . +Vq-c , ~ - ~ -c ~ ~=- ~ h (172) 2. 6. 2Da mpi ngEnhancement Us i ngAMCD WefirstconsidertheuseofanAMCDfordampingenhancementonly,without attempting tocontrol theattitude.In thismode, thecontrol objective andtheanalysis arevery similartothetotalvelocity feedback controller(TVFC)discussedinSec.2.1 and2.2;t hat is,weareinterestedmainlyinenhancingthestructuralmodedamping andstabilizing theattitude(inthesensethatattituderateas~0). Considerthemagneticactuatorcontrol law: f=Gp~ +Grt (173) where GpandGrare lx symmetric positive definite matrices.Using eqs.(169),(172) and(173),theclosedloopequation becomes: Ah +]~h +(~h =0(174) where [ : 0 ]=B+U r a r ~ =( 175),TGr,+ (o~) [oooo)](176) 0=C-4- " l T a p ? =rITGP ~+( o =[C T, - C~, - ~l (177) Inthisconfiguration, theclosed-loop system will always have two zeroelgenvalues (corresponding toAMCDattitude aaor equivalently, LFSS rigid-body attitudeas),as shown below.Denoting ~ q ~ ) ~ (17s)p=@,O~a -- O~8, 87 Eq. (174)canber ewr i t t enint heform:&a =Dll&a+D12f) +E12P(179) =D21&,~ +D22~ +E22p(180) whereDiiandEi5areappr opr i at el ydi mensi onedmat r i ces. Si ncet her i ght handsides of eqs.(179)and(180)donot cont ai naa, t her earet wofreei nt egr at or sint hesyst em.Denoting&, =w~,wei nvest i gat et hest abi l i t yof t heAMCD/ LFSSs ys t em(174)af t erremovinga , , whi chisr ewr i t t enbel owas: =E12I,DHD, 2(181) LE22I,D21D22J It shoul dbenot edt ha t t heremoval ofaaf r omt hecl osed-l oopequat i onspr esent s nopr obl embecauseour mai nobj ect i veher eist oachi evedampi ngenhancement . Theasympt ot i cst abi l i t yof (181)woul di mpl yt hat , ast--,oo,wa,e, (a~-aa) , andq allt endt o0,andt ha t aa, aat endt oconst ant s(i.e.,t heLFSSwoul dbest abi l i zed).Forcont rol l i ngt heat t i t udefornor mal oper at i on, itwoul dbenecessar yt oempl oyan addi t i onal out er loopforat t i t udefeedback. Fornow, wel i mi t our at t ent i ont ot he asympt ot i cst abi l i t yoft hes ys t em(181). In or dert oi nvest i gat et hest abi l i t yofeq.(181),we firstprove t hefollowing l emmas:L e mma 1. For ~ _> 3,t hema t r i xLisalwaysof fullrank, wher e [ 1 1 1] L==- Yl -Y2.-.- Ye(182) - C1XlX2....Tl P r o o f . The fi rst t wocol umnsof L( denot edbyL1andL2)ar el i near l yi ndependent .Supposet hei thcol umnLi(i>2)isl i nearl ydependent onL1andL2. The nt her eexi stconst ant so~1anda2sucht ha t88 al L1+a2L2-:-Li (183) Th e t opequat i onin(183)i mpl i est ha t ot I-{- ot 2----1.Ther ef or e, f r om(182)and ( 1 8 3 ) ,cqLt +(1-al )L2=Li(184) whi chi mpl i est ha t t hea c t ua t or iisl ocat edont hes t r ai ght line j oi ni nga c t ua t or s 1and 2.Thi s is,of cour se, not t r ue si ncet hea c t ua t or s ar el ocat edonacircle. Thus t herank of Lis3.[ ]L e mma 2. f fGp>0,thematrixC22>O,wher e P r o o f . Let y=(yr,yT)T, wher eYlandy~ar ereal 31andnq1vect or s. Because A>O, yTC22ycanbezer oonl yifY2=0,andy~LGpLTyl =0;i. e. , onl yifYl=0 (sincer a nkof Lis3). T h e o r e m17. Theclosed-loopAMCD/ LFSSsystemgivenbyEq.(181)isLyapunov- s t abl ei f Gp>0a ndGr >__0.Thesystemisasymptoticallystablei fGp>O,Gr>0, a n d H~O.P r o o f . Cons i der t heLya punovf unct i on:=prC2p +kr2h (186) wh e r e h = (~T, pT)T. S i n c e A>0,Ai s al so>0.C22waspr ovedt obe>0 i nLe mma 2.Ther ef or e, V>0.Di f f er ent i at i ngVwi t hr espect t otandusi ng(174),we get t hefol l owi ngaf t er si mpl i f i cat i on:=- / , T ( B +B r ) h=--2(~TrtTGrrl~.q- ~TDO) (187) 89 Inwr i t i ngt heaboveequat i onwehaveut i l i zedt hef act t ha t Wisskew- symmet r i c foranyH. I f Gr >_ 0,t hi si mpl i eslJ"0andH~0.The nr~canbe0onl yif~=0and15 =0(since rankof Lis3), whi chimpliesp=const ant . Fr omeq.(174),t hi si mpl i est ha t II=0 onlyif AAllDa+ Wwa=0(188) AA21~a 4- C22dl=0(189) wher edlisa21const ant vect or , andAqareappr opr i at es ubmat r i ces ofA,Al lbeing22.Usi ngEq. (170),i t canbeverifiedt ha t ATl=[rn~, J8 + rn~ T, 0],whi ch isof fullr ank. Ther ef or e, V=0onl yif d:a=d2,aconst ant , orif wa=d3+d2t ( d 3 = c o n s t a n t )i.e.,onl yi f Alld2+ W(d3 + d2t) = 0 Si nceH#0,Wisnonsi ngul ar , andVcanbezeroonl ywhend2=d3=0.Thus I2#0al ongallt r aj ect or i es, andt hesyst emisa.s. 2. 6. 3Da mp i n g E n h a n c e me n t Us i n g S e v e r a l AMCD' sInt hi ssubsect i on, weext endt hemat hemat i cal model andst abi l i t yanal ysi spre- sent edabove, t ot hecasewher ema nyAMCD' s areused. Asint hepr evi oussubsect i on,weassumet heLFSSt obealarge, fiat,pl at f or m- t ypes t r uct ur e, wi t ht hespi naxisofeachAMCDbei ngper pendi cul ar t ot hepl aneoft hes t r uc t ur e (Fig. 10).Thus , onl y XandYaxisr ot at i ons , andZ-axist r ansl at i onsar econsi der edint hi sconf i gur at i on,whichsufficest opr esent t hepri nci pl es. It woul dbepossiblet oext endt hi sanal ysi sin ast r ai ght f or war dbut cumber s omemanner , t ogeneral LFSSwi t ht heAMCDspi naxes poi nt i ngindi fferent di r ect i ons.9 0Weassumet ha t ~AMCD' s aredi st r i but edt hr oughout t hes t r uct ur e, andt hat the r i mmassandt he(22)t r ansver se- axisr i mi nt er t i ama t r i xof t hei tuAMCDare denot edbymai andJai respect i vel y. Assumi ngt ha t t hei thAMCDusesl i actuator st at i ons( t i _> 3), l et Cl i andC2idenot et he2x&and1x&mat r i cesdefi nedsimilar t ot hosefort hesingleAMCDcase(Eqs.149and153).Let a~i andeidenot et he2 r i ma t t i t ude vect or andt heZ-axisr i mcent er di spl acement of t hei *uAMCD, andletfl denot et het i 1axi al act uat or forcevect or . The equat i onsofmot i onfort hemultiple AMCDcasecant henbederi vedint heform:A:~ +B:~ +C x =,.tT I(i9o) wher e TTTTT__T( 1 9 1 )=( ~, " , , -" . , ~o~-~, , . . . , ~, , ~, ~, , e2 . . . . . ~, , qr ) r: =( f f, f f. . . . , : ~7( 1 9 2 )A=di ag[AI,3,,+~),,(3,,+,},I%]( 1 9 3 )wher e A~-x= z _ j ; - , ~ j ~- r _ j ~- I ..._ _ j ~- I , ' I_j ~- x, ( j , - t +j ~ , ) J~-' ...J~- ' I J~- ' ~ I !' I' I_ j ~- I ,j ; - 1 . . . . ( j ~ 1 + a v, , "- ,"'"j - I V J - I ~[- _ ~ T j ~ l lI~ Tj ; - 1 ~ T j f - 1 . . . ~ T j ~ - I I ( ~ T j ~ - I ~ +rn-;li~,+M~- I( l o 4 )h(194) 9 1= ( ~ 1 , ~ , . . . , ~ v ) , ~ , = ( y ~ , - = i ) r ,(195) - ' V i a =di ag ( r eal ,ma2 . . . . .m a t , ) ( 1 9 6 )B= I tW1t I W2 I Wt W2W~ W10 W20 IIIWv'Wt, r 0I0 I0,0 1 I~0tO L- - I - -,0~0 I Ii O I Oi Ii II II iI II' 0 1 0LI II IO, D( 1 9 7 )where [ o : ]w~=-H, ( 1 9 8 )Hibeingt heZ-axlsangul ar mome nt umof t heit?`AMCDr i m.C =diag[0(av+~), A% xn,](199) , . , / T =-02xtd i a g ( Cn , C n , ..., Clt,) diag(C21, C22, ..., C2t,) . - , r L - , r ~. . . . . . . . - ~ , ~ ,n 2 Xg z~[0 2 x l] ( 2 o o )92 ~di ( i =1, 2, . . . , u) r epr esent sinamodeshapemat r i xfora c t ua t or l ocat i onsofthe i t hAMCD, -=-~ = 1 i, n l =nq+3u, andn2=nl +2.Let ~ikdenot et heaxialrim cent er i nger r or at act uat or s t at i onkoft hei thAMCD. Asint hesingleAMCDcase, t her ankof [ C~, C~] is3,andt her ankofW ist hen3u. The ~ x1r i mcent er i ngerror vect or isgivenby: (2oi) Asint hesingleAMCDcase, consi dert hecont r ol law: f=Gp6+Gr t (202) wher eGpandGr ar e ~lreal s ymmet r i cmat r i ces.The mat hemat i cal model pr esent edabovefor t hemul t i pl eAMCDcaseisderi ved in ama nne r ver y si mi l art ot hesingleAMCDcase.Th e onl y di fference ist ha t t hexvector issel ect edinasl i ght l ydi fferentma nne r for conveni ence. It isobt ai nedf r om t heoriginal vect or (consi st i ngofan, e, as andq)byusi nganappr opr i at esi mi l ar i t yt r ansf or mat i on.The cl osed-l oopequat i onsbecome:A~+B$ +Cx=0(203) wher e / ~=B+~ T G r ~ ( 204)wher e C':C+"~TGp7 = d i a g [02,C22(3~+2)~3v+~)] C22=~TGp~?+diag[0n2, A] (2o5) ( 2 0 6 )Asint hepr evi ouscase, t her earealwayst wozeroeigenvaluescor r espondi ngt ot hetwo ri gi dLFSSr ot at i onal modes. Defining 93 ( T TTT) - ~- ~al--~s,"',OCav --~s~61,'",~u~q TT ( 2 o 7 )we exami net hest abi l i t yoft hecl osed-l oopsyst emconsi st i ngoft hes t at evect or :(pT, ~T) r , whi chhast heform:00' 1=~=E1 . 2 ~"' E 2 . 2 x . 2The fi rst st epist opr ovet ha t C22gi venbyeq.(206)isposi t i vedefi ni t e. Thi scan beaccompl i shedinama nne r si mi l art ot hesingleAMCDcase.The followingt heor em generalizest hest abi l i t yr esul t t ot hemul t i pl eAMCDcase. T h e o r e m18. Thedosed-loopmultipleAMCD/ LFSSs ys t emgi venbyEq.(208)is stableinthesenseofLyapunovif Gp>0andGr>_ O.Thesystemisasymptotically v stablei fGp>0,Gr >0,andEHi ~O. i =1 Thepr oof isent i r el yanal ogoust ot ha t ofThe or e m17,andwillt her ef or ebeomi t t ed.Thest abi l i t yof t heAMCD/ LFSSs yt emis guar ant eedregardl essof t hemodel or derort helackof knowl edgeof t hepar amet er s . It shoul dbenot edt ha t t hes ys t emwoul d stillbeLyapunovst abl eforEHi =0;however, t hedampi ngenhancement , whi chwill onlybeduet ot her i mmasses, willbemi ni mal .2 . 6 . 4 Nu me r i c a l : Ex a mp l eFort hepur pos eofdemons t r at i onoft heeffectivenessof t heAMCDfordampi ng enhancement , weconsi dert he44-flexiblemodefiniteel ement model ofa30. 48mx 30. 48m x2.54rnm(lOOft. x100f t . x 0. 1i n. )flexible al umi numpl at e, whi chwasdescr i bed inChapt er 1.The i nher ent dampi ngwasassumedt obezeroforallt hemodes. Letusfirsti nvest i gat et heeffectofasingleAMCD, wi t hr i mdi amet er of 1.79m(5.88ft . ),ri mmass34Kg(wei ght 75lb. ), andwi t hf our act uat or st at i ons, l ocat edont hepl at e asshowninFi gur e11as"AMCDNo.i ' . The nomi nal spi nspeedof t heAMCDis assumedt obe5000RPM. Thesepar amet er s arechosent obewi t hi nt helimitsof t he present dayt echnol ogy. Let t i ngGp--gpIandGr =grI,gpwasfixedat 146N/ m, and 94 AMODNo.3 AMCDNo.1 Rimdiameter-5.88ftO 100ft AMCDNo.2 C Figure 11. AMCD locations on LFSS 9 5. 3 0- 2 7- 2 1> _ . 1 8[If c E7 _, - - 4 . 1 5 D~-~ - 1 2. 0 6. 0 30 - . I 0_=\( 0 : 0 . 2 ,f = o . ~=\\ =\\ __- \~ ZT . ~\ -M O D E 7\M O D E 6 -\\ - - % . \ -%.\ _\ \p=o.~ - - \ _=-%.\ _--2 "- -N O M I N A L D E S I G N %. - - %.\\ k \ MODE 3~ \\ \ \ \ ~ 1 - . 0 ~ - . 0 8 - . 0 7 - . 0 6 - . 0 5 - - OLt - - . 03- - 0 2 - - 0 1 0 RERL Fi gur e12.Ro o t l ocuswi t hr e s pe c t t og r96 .30 ,27 . 2 4. 2 t, 1 8O fC KZ ~-~ .t5 C DCIZ ~-~ . t 2. 0 9. 0 6. 0 3-- =-\- L\N O M I N A L\ \ \ \ D E S I G N\ / ' 0 = 0 . 1X\ \p=O, 2\ \\ x \ \\ \\ \\ \\ . " , . . t f " \\ Z __=- =\\ ~--\\ __-\\ ~-\ \ 3 o~. . . . l l . , . . ~h . I t t t l t h l t t . l t d , t . , , , , , l L ~, . ~, l l , , . , , , , l l . ~. ~ - . 10- . 0 9 - . 0 8 - . 0 7 - . 0 6 - . 0 5 --.Oq- . 0 3 - . 0 2 - . 0 1 0 RERL Fi gur e 13.Ro o t l ocuswi t hr e s pe c t t oH . 30 . 27 . 24 . 2 ].18 ~i . 15 ~. 12 . 0 9. 0 6 - -. 0 3 - -- . ]97 t \,t" . - ~\ . . . ~ -" . ... . q 4 t... / I,I I I . . I I I I l-.09-.08-.07-.06-.05-.04-. 03-.02-.01 R E A LSi ngl e A MC DSe c o ndA MC D adde d . . . . . . . . . . . Thi rdA MC D adde d Figure14.Rootloci with three AMCD' s 98 grwasvar i edf r om0t o20,000N-see~re.The r esul t i ngr oot lociareshowninFigure 12fort hefi rst sevenmodes. Dampi ngformodes3,5,and7i mpr ovedsignificantly,but t ha t formodes 1,4and6i mpr ovedonl ymargi nal l y, becauset hea c t ua t or l ocat i onwas mor ef avor abl et ot hef or merset ofmodes. Inpr act i ce, t hel ocat i onsmus t bechosen to maxi mi zet heeffectont hedesi redmodes.Inor der t oi nvest i gat et heeffect ofvar i at i onoft hemome nt um, Hwasnext varied f r om0t of our t i mesitsnomi nal val ue, wi t ht hegai nsfixedat gp--146N/ mand gr--5 6 1 4 N- ~ec/ m. At zer o mome nt um, t her eisasmal l dampi ngbecauset hemassof t her i mact slikeapr oof - massact uat or . AsshowninFi gur e13,t hedampi ngimproved si gni fi cant l yf or modes1,3,5and7. Inor der t os t udyt heeffectof manyAMCD' s, t womor ei dent i cal AMCD' s (No. 2andNo. 3)werenext addedat di fferent l ocat i onsasshowninFi gur e11.Keeping t heposi t i ongai nsoft heAMCD' s at : gpl=146N/ r nandgp2~-gp3:14. 6N/ rn, the r at egai nswer ei ncr eased{st art i ngf r om0)for t hefirst, t hent hesecond, andfinally the t hi r dAMCD. Th e r esul t i ngr oot lociar eshowninFi gur e14.The fi gureshowst hatsubst ant i al dampi ngenhancement ispossi bl eusi ngmul t i pl eAMCD' s .2. 6. 5At t i t u d e Co n t r o l f o r AMCD- Ac t u a t e d L F S SSof ar wehavediscusseddampi ngenhancement usi ngAMCD' s . Wehaveproved t ha t t heas ympt ot i cst abi l i t yof t hes ys t emcosi st i ngof&s,(aa~-as ) , ei ( i =1, 2, . . . , u),andq,isguar ant eed; t ha t is,t hesevari abl eswillt e ndt ozer oast-~co. Inot her wor ds, t hes t r uc t ur e willt endt oas t eadys t at east~or , al t houghitsa t t i t ude will t endt osome(non-zero)const ant . Thi spr oper t y(whi chwasalsodi scussedinSee. 2.2inconnect i onwi t hTVFC}isext r emel yusefulforst abi l i zi ngas t r uc t ur e whi chis vi br at i ngor begi nni ngt ot umbl e. Forexampl e, suchacont r ol l er woul dbever yuseful formai nt ai ni ngt hest abi l i t yandi nt egr i t yof t heLFSSdur i ngdepl oyment , assembly, or i ni t i al post - assembl yphase, whent hepar amet er s areyet t obees t i mat ed, anda hi gh- per f or mancecont r ol l er isyet t obedesi gned.Dur i ngact ual oper at i on, however, inaddi t i ont odampi ngenhancement , t heat- t i t udeof t heLFSSmus t becont r ol l ed. Thi s canbeaccompl i shedint woways:1) 99 usingaddi t i onal t orqueact uat or sandat t i t udeandr at esensors, or2)usi ngt hesame AMCD' s. Wecallt hefirstmet hoda"twolevel"controller, consi st i ngofasecondary controllerfordampi ngenhancement , andapr i mar ycont rol l erforat t i t udecont rol .At t i t u d e Co n t r o l Us i n g aTwo - Le v e l Co n t r o l l e rWi t ht hesecondarycont rol l erconsi st i ngofoneormor eAMCD' s , supposet her e aremr , 2-axist orqueact uat or s( mr _> 1),col l ocat edwi t hmT , 2-axisat t i t udesensors andr at esensors. Webegi nwi t hequat i on(203),wherei nt hesecondar ycont rol loopis al readyclosed.Becauseof t headdi t i onal t orqueact uat or s, (203)ismodi fi edas: (209) whereA,BandC'aredefinedin(193),(204),and(205),and u=( Tr , " r r - r r ,~T - L2~.,,~. L r n T) (210) I2...I21F r:0sv, . T(211) T ~T...CmT where~i representst he2nqmode-sl opemat r i xat t hei tht or queact uat or l ocat i on.Theat t i t udeandr at evect orsat t heact uat or locationsaregivenby(ignoringnoise): y p :( 2 1 2 )and Yr--F~(213) Equat i ons( 209) - (213)havet hesameformaseqs.(4),(14),and(15).Themai n differenceist ha t Bin(209)isnot symmet r i c. Thet op(2v+2)(2v+2)pri nci palsubmat r i xofB(Eq.197)isskew-symmet ri c. Consi dert hecont rol law: 100 u=- O ~, u . ~, . -(214) Wenowinvestigatet hest abi l i t yof t heentiresyst emusingt hetwo-levelcontroller. T h e o r e m19. Theclosed-loopsyst emgivenby eqs.(209)-(214)isa.s.i fGp=G~> 0 and O~= Or>O. Pr o o f . Wi t ht hecontrollaw(214),(209)becomes: where A~ + / ~ +Cx=0(215) =B+r r O, r (216) 0=o+r r o p r (217) Wefirstshowt hat ~7 ispositivedefiniteifGp>O.Indeed, [ : 0 ]0=( 2 1 8 )C22 whereC22waspreviouslyprovedt obepositivedefinite.Sincet herankofFisatleast 2,itfollows from(217)t hat Cispositiveallfinite. Considert heLyapunovfunction ( 2 1 9 )Asint hepreviouscases,wehave =_ ~ r ( b +h r ) ~ =_ ~ r ( ~ +~ r +2 r r 0 ~ r ) ~( 2 2 o )Usingt hedefinitionof / ~from(204)andt heskew-symmet ryof W~(i=1, 2,..., w), 101 ~-- - 2 4 T D 4 - - 2~. T( . T T Gr , 7 +rTO, r) where~/isdefi nedin(200). SinceD>0, IY =0implies~=0,and ~ T t l r G r ~ l ~ =~ J T C r I ~ a =0 (221) ( 2 2 2 )where ----- - a s ,- - . , O c a v - - a a ,~ ' 1 , ~ 2 , . . . , ~ ' v( 2 3 )I ( I 2 , r. . . . , I2)2x2mr(224) and~/isdefi nedin(200).Sincer/andIareoffullrank, 1~"---0implies/~--0and &a=0;i.e.,~--0.Thefact t hat C>0t henimplies[(from215)]t ha t V=0onl yif x--0,andt hes ys t emisa.s. It wasprovedprevi ousl y(Sec.2.2)t hat t heclosed-loopasympt ot i cst abi l i t y canbe obt ai nedwi t honl yt or queact uat or s, wi t hout usi ngAMCD' s. However,t heAMCD' s significantly enhancet hest r uct ur al dampi ngwi t hrel at i vel ysmal l wei ght penal t y. One canaccompl i sh t hesamedampi ngenhancement usi ngt or queact uat or s; but t heywould addconsi derabl yt ot hewei ght , andt herefore, t ot hemissioncost. Becauseoft hei r opt i mi zedshape, AMCD' s arei deal l y sui t edfordampi ngenhancement .Wenext i nvest i gat et hepr obl emofat t i t udecont rol usi ngt hesameAMCD' s; i.e., wi t hout usi ngaddi t i onal t or queact uat or s.At t i t u d e Co n t r o l Us i n g AMCD' sInt hefi ne-poi nt i ngmode, itispossibletocont rol t heat t i t udeusi ngt hesame AMCD' swhi charesi mul t aneousl ybeingusedfordampi ngenhancement . Thi sisac- compl i shedbyt or qui ngagai nst t heAMCDangul ar moment a. However, int hi sdualcontrolmode, we cannot si mul t aneousl y cont rol t herel at i ver ot at i onal angles(~ai -a , )102 andr i gi d- bodya t t i t ude ae; t hes ys t emconsi st i ngof bot hof t heseinitss t at evect orcan beeasilyshownt obeuncont r ol l abl e. The dual cont rol isaccompl i shedint wost eps, as follows:i)uset hemagnet i cact uat or cont r ol law(202)onl yforel ast i cmot i ondamping andr i m cent er i ng, andii)super i mposet heLFSSa t t i t ude cont r olsi gnal ont hemagnetic act uat or s basedont her equi r edat t i t udecont rol t orque, i . e. , asin(214)].Inst ep(ii),we assumet ha t a t t i t ude andr at esensorsar epl acedont heLFSSnear eachAMCDcenter, anduset hedesi redt or quec omma ndt ogener at eappr opr i at emagnet i ca c t ua t or forces. Si ncet heAMCD' s arerel at i vel ysmall, t hi scont r ol lawappr oxi mat est hetwo-level con- t r ol l er di scussedint hepr evi oussect i on, wher ei naddi t i onal t or queact uat or s wer eused forpr i ma r ya t t i t ude cont rol .Let usconsi der t hefirstst ep. The "posi t i on"feedbackgai nGpin(202)must be r edesi gnedt oexcl udet hefeedbackof ( aai -a , ) . However, t her at es( ~ a i - ~ , s ) must bezeroins t eadyst at e, andmust befedback. Thus Gpmust beredesi gnedt ocontrol onl yeisoast okeept her i mcent er snear t hei r nomi nal posi t i ons, andt heAMCDrim t r ansver ser ot at i onangl es(aai) ar eallowedt obenonzer o const ant sins t eadyst at e. In t hefi ne-poi nt i ngmode, ( " a i -a , ) areexpect edt obever ysmal l (ont heor der of one degree)anditisr easonabl et oexpect t hat t heel ect r omagnet i cact uat or gapl i mi t swill not beexceeded.The col umnsof t hema t r i xCu aregi venby: (yq, - x q )T, j=1, 2, ..., Ii . Sincethe act uat or s ofeachAMCDarel ocat edal ongacircle,t hefirstt wocol umnsof Cl i are l i nearl yi ndependent , andcol umns3t hr oughl i canbeexpr essedasl i nearcombi nat i ons of t hefi rst t wocol umns. Th a t is,Clicanbeexpressedas: =c d h ! r d (225) wher eC~ist he2x2ma t r i xconsi st i ngof t hefirstt wocol umnsof Cl i andI' iisa 2x(i-2)mat i r x. Let t heposi t i ongai nbegi venby: wher eCp=~pr>0isan(e-2t,)(-2v)mat r i x. Usingt hepr opor t i onal - pl us-der i vat i vecont r ol lawofEq. (202),itcanbeverifiedbydi r ect s ubs t i t ut i ont ha t this 103 choice ofGpr esul t sint heel i mi nat i onof t hefeedbackof ( a T --a T , ...,a T --aTs) Tf r om thecont rol i nput forcejr.