Study on Satellite Vibration Control Using Adaptive Algorithm

Transcription

1 ICCAS5 Jue -5, KINTEX, Gyeoggi-Do, Koea Study o Satellite Vibatio Cotol Usig Adaptive Algoithm Choog-Seo Oh *, Se-Boug Oh, ad Hyochoog Bag *Divisio of Aeospace Egieeig, KAIST, Daejo, Republic of Koea (Tel : ; {csoh,sboh,hcbag}@fdcl.aist.ac. Abstact: The picipal idea of vibatio isolatio is to filte out the espose of the system ove the coe fequecy. The isolatio objectives ae to tasmit the attitude cotol toque withi the badwidth of the attitude cotol system ad to filte all the high fequecy compoets comig fom vibatio equipmet above the badwidth. Howeve, whe a eactio wheels o cotol mometum gyos cotol spacecaft attitude, vibatio ievitably occus ad degades the pefomace of sesitive devices. Theefoe, vibatio should be cotolled o isolated fo missios such as Eath obsevig, boadcastig ad telecommuicatio betwee atea ad goud statios. Fo space applicatios, techicias desigig cotolle have to coside a peiodic vibatio ad distubace to esue system pefomace ad obustess completig vaious missios. I geeal, past eseach isolatig vibatio commoly used 6 degee ode feedom isolatos such as Stewat ad Malloc platfoms. I this study, the vibatio isolatio device has 3 degee ode feedom, oe taslatioal ad two otatioal motios. The oigi of the coodiate is located at the cete-of-gavity of the uppe plae. I this pape, adaptive otch filte fids the distubace fequecy ad the efeece sigal i filteed-x least mea squae is geeated by the otch fequecy. The desig paametes of the otch filte ae updated cotiuously usig ecusive least squae algoithm. Theefoe, the adaptive filteed-x least mea squae algoithm is applied to the vibatio suppessig expeimet without efeece seso. This pape shows the expeimetal esults of a active vibatio cotol usig a adaptive filteed-x least mea squaes algoithm. Keywods: Vibatio Isolatio, Jitte, Adaptive Notch Filte, Filteed-x Least Mea Squae. INTRODUCTION These days, may systems eed quiet eviomet fo opeatio without vibatio o oscillatio. Vibatio is udesied distubace souce to the system accuacy, i paticula, of the optics ad atea o spacecaft. Whe a eactio wheels (RW o cotol mometum gyos (CMG cotol spacecaft attitude, vibatio ievitably occus ad degades the pefomace of sesitive devices. Theefoe, vibatio should be cotolled o isolated fo missios such as Eath obsevig, boadcastig ad telecommuicatio betwee atea ad goud statios. Hee we attempt to adopt the adaptive vibatio cotolle. Adaptive cotolle is that its paametes ae o-lie adjustable. A umbe of algoithms have bee developed i implemetatios of adaptive vibatio cotolles. A typical appoach to adaptive vibatio cotol is to feed a eo sigal though a appopiate filte ad apply the esultig sigal to a plat. Coefficiets of the filte ae tued automatically by a adaptive algoithm to achieve best vibatio eductio. Elliott, Stothes ad Nelso peseted a algoithm to adapt coefficiets of a aay of fiite impulse espose (FIR filtes, whose outputs wee liealy coupled to aothe aay of eo detectio poits to miimize mea squae eo sigals.[] Eisso, Allie, ad Geie ivestigated the use of IIR adaptive filtes i adaptive vibatio cotols.[] Bauma studied the potetial of a adaptive feedbac appoach to stuctual vibatio suppessio.[3] The filteed-x least mea squaes (FLMS algoithm was deived by Widow.[4] The adaptatio of the weight coefficiets ivolves use of the filteed efeece sigal vecto. The filteed-x least mea squaes (FLMS algoithm equies the efeece sigal. I this pape, adaptive otch filte (ANF fids the distubace fequecy ad the efeece sigal i FLMS is geeated by the otch fequecy. A adaptive otch file usig the seso output sigal to exactly estimate the actual distubace fequecy could be cosideed. The desig paametes of the otch filte ae updated cotiuously usig ecusive least squae (RLS algoithm. Theefoe, the adaptive FLMS algoithm is applied to the vibatio suppessig expeimet without efeece seso. Adaptive vibatio cotolles ae good cadidates fo poblems whee paametes of a plat ae uow o thee ae ucetaities i a system. Paametes of adaptive cotolles ae adjusted o-lie to achieve the best pefomace. Due to these chaacteistics, the FLMS usig ANF is a good method to atteuate the spacecaft vibatio. Fo the spacecaft, accuacy equiemet i attitude poitig is cotiuously beig heighte. I space thee could be may distubace souces such as sola adiatio pessue, ad aeodyamic foces which ihibit high accuacy. The mai souce of spacecaft vibatio is the RW o CMG. This distubaces ae i the high fequecy domai above Hz ad they degade spacecaft poitig pefomace ad commuicatio accuacy to the goud statio. Theefoe, it is tageted to apply this study to the vibatio isolatio of space applicatios ad shows the expeimetal esults of a adaptive vibatio cotolle.. JITTER SUPPRESSION DEVICE The vibatio isolatio device to which a adaptive vibatio cotol algoithm is adopted has 3-DOF, oe taslatioal ad two otatioal motios as show i Fig., ad associated dyamics equatios ae deived. Figue shows the coodiate defiitio of 3-DOF jitte suppessio device. The oigi of the coodiate is located at the cete-of-gavity (CG of the uppe plae. w deotes z-diectioal taslatio of CG. φ ad θ ae the otatioal motio i the x ad y-axis of the CG. w, w, w 3 which deote z -axis taslatioal motio of each stut ae cotolled by thee actuatos that ae coected paallel to the stut. Thei elatios ae w = w Lsiθ w = w+ Lsiθ siφ ( w = w+ L siθ + L siφ 3 3 4

2 ICCAS5 Jue -5, KINTEX, Gyeoggi-Do, Koea 3-DOF dyamics ae expessed i oliea equatios fom the elatioship amog w, φ, ad θ. 3. ADAPTIVE VIBRATION CONTROLLER I this sectio, the desiged adaptive vibatio cotolle is peseted. The adaptive vibatio cotolle is composed of a adaptive otch filte ad a least mea squae filte. Fig. 3-DOF jitte suppessio device Z θ φ 3. Adaptive otch filte (ANF The multi mode otch filte of a ARMA stuctue is equied to compute matix calculatio. Hece, it is difficult to apply it to eal time system fo multiple siusoids sigal case. To educe the computatio time, the cascade of secod-ode otch filtes is desiged as show i Fig. 3. Whe the umbe of the spacecaft jitte fequecies to be ejected is p, each otch fequecy is detemied by each secod-ode otch filte i the cascade. X Y u ( y ( y ( y ( ( y ( yp( p( H z H ( z p H z W X L W3 b Z L3 L W Fig. Coodiate defiitio fo the 3 DOF jitte suppessio device A spig, a dampe, ad a actuato ae istalled i each stut. The i, c i, ad Fi ( i =,,3 ae spig coefficiet, dampig coefficiet, ad actuato foce, espectively. The foce ad momet equatios ae deived as mw&& =( w Lsi θ ( w + Lsi θ 3( w+ L3si θ c( w& Lsi & θ c( w& + L & θ cos θ c3( w& + L & 3θcos θ + F + F + F 3 I && y θ = ( w Lsi θ L ( w + Lsi θ L cosθ 3( w+ L3si θ L3 c( w& L & θcos θ L c ( w& + L & θcos θ L c ( w& + L & θcos θ L + FL FL FL Y ( (3 I && z φ = ( w si φ 3( w + si φ cosφ +c 3( w& L & 4φcos φ c3( w& + L & 4φcos φ (4 + FL FL Fig. 3 Cascade of secod-ode otch sectios Rao ad Kug[5] poposed the followig fom of a secod ode otch filte i the z-domai jφ jφ ( e z ( e z H ( z = jφ jφ ( ρe z ( ρe z cosφz + = z ρ cos φz + ρ z (5 Howeve, the filte achitectue i Eq. (5 is ot most appopiate to apply the adaptive otch filte algoithm, so the followig otch filte fo adaptatio has bee poposed by Regalia[6] : + a( + b z + bz H ( z = (6 + β ( + β + β z z I ode to match Eq. (5 with Eq. (6, it should be satisfied that β( + β = ρa( + b (7 β = ρ b This ca be educed to ρa( + b β = + ρ b (8 Fo implemetatio of adaptive otch filtes, the umbe of paametes to be adapted should be maitaied miimum. If we assume ρ is equal to, the paametes β, β ca be expessed i tems of ab, such as b β = ρ ρb β ( + β = ρa( + b a( + ρb (9

3 ICCAS5 As a cosequece, β = a ( holds tue ad Eq. (6 ca be ewitte as + a( + b z + bz H ( z = ( + a( + ρb z + ρbz I geeal, bette pefomace of a otch filte is esued whe zeos ae located o the uit cicle. Hece i Eq. (, the paamete b is set equal to uity, ad it ca be simplified as a fuctio of ρ ad a i the followig fom; N ( ( z + az + z H z = = ( D ( ρz + a( + ρ z + ρz I this study, ρ lies betwee.9 ad.99, ad if it is close to the the badwidth becomes aow. Suppose the iput sigal to the otch filte is deoted as u (. Hece the filte output sigal of each sectio ca be expessed as N ( ( ( ( z y = H z y = y, D ( ρz (3 =, L, p, y ( = u( Fo the cascade type otch filte, the Eq. (3 is eplaced by y ( = x ( + x ( + a x ( (4 whee x( epesets a sigal passig though the deomiato of the filte, i.e., x( = y (, =, L, p, y( = u( (5 D ( ρz Now, i ode to desig the adaptatio algoithm a cost fuctio fo optimizatio is poposed as follows; λ = E ( = y ( (6 The ecusive least squae (RLS method as show Eq. (7-(9 is used to estimate the uow paametes θ = [ a L a ] T p. The time aveage coelatio Φ ( ad the time aveage coss-coelatio α ( is give by[7] Φ ( = λφ( + x( (7 { } α ( = λα ( + x ( x ( + x ( (8 Jue -5, KINTEX, Gyeoggi-Do, Koea the weaess of the cascade type otch filte. The -th sectio otch fequecy is estimated by usig the last sectio output i additio to the -th sectio output. Hece, the estimated fequecies ae ot cossig ad do ot covege same fequecy. Fo the cost fuctio of the updated cascade otch filte, the Eq. (4 is eplaced by y( = x( + x( + yp( y( ( + a ( x ( The time-aveage coss-coelatio α ( i Eq. (8 is eplaced by α( = λα( ( + x ( x ( + x ( + y ( y ( { } p At steady state, the estimated fequecy is chatteig, so a smoothig filte is used to emove the chatteig. ω ( = σω ( + ( σ ω ( (3 To impove the covegece, the time vayig paamete is used. A simple way to do this is to let the paametes gow expoetially accodig to ρ( = ρρ( + ( ρ ρ (4 λ( = λλ( + ( λ λ (5 σ ( = σσ( + ( σ σ (6 whee the ρ, λ, σ detemies the ate of chages. u ( y ( y H ( z ( y ( H ( z y ( p H ( y ( p z p Fig. 4 Updated cascade of secod-ode otch sectios 3. Adaptive vibatio algoithm The estimate paamete of -th sectio is give by a( ( α( = Φ (9 Sice the otch poles ae located withi a uit cicle, the paamete a also lies i the age a [,]. Cosequetly, the atual fequecy of the bedig vibatio of the flexible stuctue ca be estimated usig the samplig time ( T i the followig fom: cos [ a ( ] ˆ ( ω = T ( The atual fequecy is updated cotiuously accodig to Eq. (. The paamete a is essetially dive by seso sigal. Figue 4 shows the stuctue desiged to complemet fˆi τ ( v H ( z Fig. 5 Least mea squae algoithm usig ANF

4 ICCAS5 fˆi x ( H ˆ ( z x ( τ v ( H ( z Fig. 6 Filteed-x least mea squae algoithm usig ANF The jitte suppessio of spacecaft is obtaied by the adaptive vibatio cotol algoithm. The vibatio cotolle is the feedfowad cotolle usig the adaptive filteed-x least mea squae (FLMS algoithm.[8] I sectio 3., the jitte fequecies ae estimated usig RLS algoithm. Usig this estimated fequecy, the efeece sigal is p L x( = si( π f ( i = i= (7 whee p is the umbe of the otch fequecy, L is the filte legth. Usig this efeece sigal, the geeal least mea squae (LMS, FLMS filte is modified as Fig. 5 o Fig. 6 espectively. The FLMS usig ANF algoithm is applied to the vibatio suppessig expeimet without efeece seso. The the cotolle update ule i FLMS usig ANF is w( + = w( µ e( H x( (8 whee is the covolutio opeatio ad H is a ( M vecto of filte weight coefficiets. At some time, the vibatio cotol sigal τ v( is simply a weighted combiatio of past efeece samples τ v( = w( x ( (9 whee thee ae L stages i the filte, w is a ( L vecto of filte weight coefficiets ad x is a ( L vecto of efeece sigal. A bloc diagam of the adaptive feedfowad cotol aagemet which will be cosideed i this sectio is show i Fig. 4. The desiged cotolle stability is poved i the followig. Fist, the ANF stability is poved i pevious sectio. The LMS algoithm coveges i mea fom w( to optimal o weight w if ad oly if < µ < (3 λ max whee λmax is the lagest eigevalue of the iput autocoelatio matix. Because computatio of λmax is vey difficult whe L is lage, the Eq. (3 is eplaced with < µ < (3 LP x Jue -5, KINTEX, Gyeoggi-Do, Koea whee Px E[ x ( ]. The filteed-x LMS algoithm will covege withi ealy 9deg of phase eo betwee H ad H.[9] 4. EXPERIMENTAL RESULT The adaptive vibatio cotol algoithm has bee applied to actual expeimetal demostatio. The oveall fuctioal bloc diagam fo the expeimetal set up is peseted i Fig. 7. The acceleatio sigal of the each stut is measued by a acceleomete (CXLP, Cossbow though A/D covete at Hz samplig ate. The cetal pocessig uit is Petium-III.5GHz pocesso which computes cotol sigal with digital filte. The cotolle is commaded by compute output. The compute output sigal of the vibatio cotolle is iceased by amplifie (BOP 7-3M, KEPCO. I this study, a coil actuato is used to geeate the feedbac vibatio cotol foce equied. The coil actuato commad is geeated at Hz. The A/D, ad D/A boad (PCI-65E, Natioal Istumet have -bit esolutio i discetizatio. Vibatio actuato (VTS-, Vibatio Test System geeates a distubig vibatio i ode to simulate spacecaft jitte fequecy. The iput sigal to the vibatio geeato was geeated by fuctio geeatos o S/W pogammed commad ad the sigal is amplified by powe amplifie (CE, Cow. Fig. 7 Expeimetal setup fo the jitte suppessio Expeimetal esults of Hz distubace ae peseted i Fig. 8. At iitial state, the otch filte fequecy is selected as 4Hz. The filte legth of w ad H is that L =, M =, espectively. The otch filte paamete is set to ρ =.95, λ =.98 ad σ =.8, espectively. The covegece coefficiets ae set to µ =., µ =.9 ad µ 3 =., espectively. The cotol begis at 3sec. Afte 3sec, the acceleatio sigal deceased about 9.6dB, 6.5dB, ad 7.5dB, espectively. At steady state, the estimated fequecy is chatteig because the distubace sigal is atteuated. To educe the chatteig of the estimated fequecy, the otch filte paamete is chaged to ρ =.98, λ =.999 ad σ =.999 at steady state. Figue 9 shows the estimated fequecy fom the adaptive

5 ICCAS5 Jue -5, KINTEX, Gyeoggi-Do, Koea Acc.(m/s Acc.(m/s 3 Acc.(m/s Time(sec Fig. 8 Adaptive vibatio cotol esult fo Hz sie distubace 6 Acc.(m /s Acc.(m/s 3 Acc.(m /s Time(sec Fig. Adaptive vibatio cotol esult fo Hz ad Hz sie distubace Estimated distubace fequecy(hz Time(sec Fig. 9 Estimated jitte fequecy fo Hz sie distubace Acc.(m/s Acc.(m/s 3 Acc.(m/s Estimated distubace fequecy(hz Time(sec Fig. Adaptive vibatio cotol esult fo Hz sie distubace Estimated distubace fequecy(hz Time(sec Fig. 3 Estimated jitte fequecies fo Hz ad Hz sie distubace Iput(Volt Iput(Volt Iput3(Volt Time(sec Fig. 4 Cotol iput volts fo Hz ad Hz sie distubace Time(sec Fig. Estimated jitte fequecy fo Hz sie distubace

6 ICCAS5 otch filte. Expeimetal esults of Hz distubace ae peseted i Fig.. The iitial state coditio is equal to the Hz distubace case. The otch filte paamete settig is the same as the Hz distubace case. The cotol begis at 3sec. Afte 3sec, the acceleatio sigal deceased about 9.dB,.9dB, ad 6.9dB, espectively. To educe the chatteig of the estimated fequecy, the otch filte paamete is chaged to ρ =.98, λ =.999 ad σ =.999 at steady state. Figue shows the estimated fequecy fom the adaptive otch filte. The estimated fequecy follows well the eal distubace fequecy. Figue shows the expeimetal esults whe Hz ad Hz distubace excite. The iitial state coditio is equal to the pevious case. The otch filte paamete settig is the same as the Hz distubace case. The cotol begis at 5sec. Afte 5sec, the acceleatio sigal deceased about 8.6dB, 3.5dB, ad 6.dB, espectively. To educe the chatteig of the estimated fequecy, the otch filte paamete is chaged to ρ =.98, λ =.999 ad σ =.999 at steady state. Figue 3 shows the estimated fequecy fom the adaptive otch filte. The estimated fequecies follow the eal distubace fequecy Hz ad Hz. The cotol iput volts ae show i Fig. 4. It is coside the followig ule to select the covegece coefficiet µ. Sice the uppe boud o µ is ivesely popotioal to L i Eq. (3, small µ is used fo lage-ode filtes. Sice µ is made ivesely popotioal to the iput sigal powe, weae sigals ca use a lage µ ad stoge sigals have to use a smalle µ. Istead usig costat ρ, λ ad σ, the otch filte paamete iceases, espectively. These paametes, as metioed aleady, ae so impotat to detemie the distubace fequecies. If the fogettig facto λ is so small, the estimatio pefomace is degaded. But if it is too lage, the pefomace is also degaded fo the case of time vayig paamete estimatio. I the expeimetal ivestigatio, the selectio of a appopiate fogettig facto is impotat fo the esultig pefomace of the vibatio cotolle. Jue -5, KINTEX, Gyeoggi-Do, Koea ACKNOWLEDGMENTS The peset wo was suppoted by Natioal Reseach Lab. (NRL Pogam (M-3--6 by the Miisty of Sciece ad Techology, Koea. Authos fully appeciate the fiacial suppot. REFERENCES [] S. J. Elliott, I. M. Stothes, ad P. A. Nelso, A multiple eo LMS algoithm ad its applicatio to the active cotol of soud ad vibatio, IEEE Tas. of ASSP, Vol. 35, pp , 987. [] L. J. Eisso, M. C. Allie, ad R. A. Geie, The selectio ad applicatio of IIR adaptive filtes fo use i active soud atteuatio, IEEE Tas. of ASSP, Vol. 35, pp , 987. [3] W. T. Bauma, ad H. H. Robetshaw, Active stuctual ascoustic cotol of boadbad distubaces, Joual of the Acoustical Society of Ameica, Vol. 88, pp. 3-38, 99. [4] B. Widow, ad S. D. Steas, Adaptive Sigal Pocessig, Petice Hall, New Jesey, 985. [5] D.V. Rao, ad S.Y. Kug, Adaptive Notch Filteig fo the Retieval of Siusoids i Noise, IEEE Tas. O Acoustic, Speech, Sigal Pocessig, Vol. 3, pp. 79-8, 984. [6] P. A. Regalia, A Impoved Lattice-Based Adaptive IIR Notch Filte, IEEE Tas. o Acoustic Speech, Sigal Pocessig, Vol. 39, pp. 4-8, 99. [7] S. Hayi, Adaptive Filte Theoy, 4 th ed., Petice Hall, New Jesey,, Chaps. 9 [8] C. H. Hase, ad S. D. Syde, Active Cotol of Noise ad Vibatio, E & FN Spo, Lodo, 997. [9] S. M. Kuo, ad D. R. Moga, Active Noise cotol Systems, Joh Wiley & Sos, New Yo, CONCLUSION I the peset study, the jitte isolatio of spacecaft has bee examied. I space thee could be may distubace souces such as sola adiatio pessue, ad aeodyamic foces which ihibit high accuacy. The mai souce of spacecaft vibatio is the RW o CMG. The distubaces ae i the high fequecy domai above Hz ad they degade spacecaft poitig pefomace ad commuicatio accuacy to the goud statio. To atteuate the distubace of the spacecaft, FLMS usig ANF was applied to the jitte suppessio device. Adaptive otch filte (ANF fids the distubace fequecy ad the efeece sigal i FLMS is geeated by this otch fequecy. The desig paametes of the otch filte ae updated cotiuously usig ecusive least squae (RLS algoithm. Theefoe, the adaptive FLMS algoithm is applied to the vibatio suppessig expeimet without efeece seso. Sigificat eductios of vibatio level have bee obseved fo eal-time adaptive cotols. The desiged FLMS usig ANF has a good pefomace without the efeece.