An Improved Wavefront Control Algorithm for Large Space Telescopes

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1 A Impoved Wvefot Cotol Algothm fo Lge Spce Telescopes Ek Sdck*, Scott A. Bsge, d Dvd C. Reddg Jet Populso Lbotoy, Clfo Isttute of Techology, 48 Ok Gove Dve, Psde, CA, USA 99 ABSTRACT Wvefot sesg d cotol s equed thoughout the msso lfecycle of lge spce telescopes such s Jmes Webb Spce Telescope (JWST). Whe optc of such telescope s cotolled th both sufce-defomg d gdbody ctutos, the sestvty-mtx obted fom the ext pupl vefot vecto dvded by the coespodg ctuto commd vlue c sometmes become sgul due to dffeece ctuto types d ctuto commd vlues. I ths ppe, e popose smple ppoch fo pevetg sestvty-mtx fom sgulty. We lso toduce e mmum-vefot d optml cotol compesto. It uses optml cotol g mtx obted by feedg bck the ctuto commds log th the mesued o estmted vefot phse fomto to the estmto, thus elmtg the ctuto modes tht e ot obsevble the vefot sesg pocess. Keyods: Wve-fot sesg d cotol, dptve optcs, defomble mos, spce telescopes.. INTRODUCTION Wvefot sesg d cotol (WFS&C) s equed thoughout the msso lfecycle of lge spce telescopes such s JWST []. The most commoly used cotol techque s the mmum-vefot compesto. I ths ppoch, the vefot s computed t the ext pupl d s used to deteme e ctuto commds fo the cotollble optcs usg le mtx model of the optcs. The mtces defg system e ethe obted theoetclly fom modelg o mesued dectly fom the optcs. The telescope typclly uses combto of to kds of ctutos: Sufce-defomg (SD) ctutos d gd-body (RB) ctutos. The fome s used to cotol the sufce shpe of defomble mo, d the ltte s used to move cotollble optc up to 6 degees of feedom, cludg tsltos d ottos. The cotol put fo SD ctutos e usully tcked ethe voltge o stoke, d the puts fo RB degees of feedom c be bookkept s ottos d tsltos of the optc. I the le mtx model, the omlzed fluece fuctos e obted, hch mesue the chge the vefot, y-by-y, by pplyg cotol degees of feedom oe t tme, d dvdg the chge vefot by the mgtude of the ctuto. These fluece fuctos e equvlet to optcl sestvty mtx hose pseudo-vese s the lest-sques cotol g-mtx. Hoeve, the sestvty d the g mtces c esly become ll-codtoed becuse of the vst dffeeces cotol put pmetes. I such cse, the WFC pocess becomes ethe umeclly ustble o opetolly effcet. I ths ppe, e popose method to ovecome ths poblem. We sho tht by usg set of ppoptely chose sclg fctos fo ctuto commds d optmzg the umbe of egemodes o the theshold of the sgul vlues eted, oe c cheve hgh umecl stblty d hgh opetol effcecy the WFC of lge spce optcs. We lso toduce to e compestos. Oe s mmum-vefot d optml cotol compesto, d the othe s ts sptlly-flteed d geelzed veso. I these ppoches, the cost fucto to be mmzed s the squed-sum of the vefot phse eo vlues plus the squed-sum of ctuto commds. Ths s equvlet to feedg bck the ctuto commds log th the mesued vefot phse fomto to the estmto, d the compesto, usg the optml cotol gs tht do ot ctute uobsevble ctuto modes. Such ppoch elmtes the ctuto modes tht e ot obsevble the vefot sesg pocess. Some spects of ths ppoch hve bee descbed the cotol l of Shck-Htm seso bsed dptve optcs system fo the Plom Mout Hle Telescope [2]. Hee e cosde oly tefeometc-bsed WFS system, d ll peset the e methods log th modelg d smulto esults. *Ek.Sdck@jpl.s.gov; Phoe ; Fx ;.jpl.s.gov Adptve Optcs Systems, edted by Nobet Hub, Cle E. Mx, Pete L. Wzoch, Poc. of SPIE Vol. 75, 754P, (28) X/8/$8 do:.7/ Poc. of SPIE Vol P-

2 2. BRIEF DESCRIPTION OF THE OPITCAL SYSTEM Ths ppe s ot bout ptcul spce telescope. Hoeve, to expl ou method fo mpovg the sestvty mtx d ou e compestos, e ll sho some umecl exmples. The optcl system used to geete ou smulto esults s bsed o the Hgh-Cotst Imgg Testbed (HCIT) t Jet Populso Lbotoy (JPL), Clfo Isttute of Techology [3-4]. I ths secto, e befly descbe the HCIT s optcl system. The schemtc dgm of the HCIT lyout the xz-ple s sho Fgue. Atfcl stlght s ceted by 5µm phole llumted by optcl fbe. A off-xs pbolc mo (OAP) collmtes the lght fom the phole d dects t to hgh-desty, 32x32 ctuto defomble mo (DM), hch pefoms vefot cotol. A ccul petue msk o the DM defes the system pupl of the HCIT, d hs dmete of D=3mm. Afte the DM, the collmted lght s e-mged oto the focl ple of occultg msk by OAP2 d fold-mo (FM). A occultg msk tteutes the stlght, d lmost hs o effect o off-xs lght, potetlly eflected fom obtg plet. The bck-ed of the system, fom the occultg msk to the bck focus ple, suppots expemetto th dvese coogph cofgutos d podztos. A fold-mo (FM2) d OAP3 e-collmte the lght pssg though the occulte msk d fom smesze shp mge of the DM pupl t the Lyot ple. A Lyot Stop blocks the g-lke esdul lght dffcted off the occultg msk hle lettg most of the off-xs plet lght though. Afte OAP4 foms mge fom the emg stell d plet lghts, t s the mgfed (M 3) by the OPA5-OAP6 p fo pope smplg o the CCD scece cme locted t the bck focl ple. Moe fomto o the HCIT d the DM c be foud Refs. [3-4]. I the peset cse, e use ths system s stdd mgg system thout cludg the Occulte Msk d the Lyot Stop ou optcl model. The ssumed velegth of the llumtg bem ou umecl smultos s λ = 785m. DM Lght Souce OAP FM OAP2 Occulte OAP3 FM2 Lyot Stop OAP5 OAP4 Bck Focus OAP6 Fgue. Schemtc dgm of the Hgh Cotst Imgg Testbed lyout. The lght souce ( stlght ) s 5µm phole llumted by optcl fbe, d CCD scece cme s locted t the bck focl ple fo detectg the mge of the stlght. Poc. of SPIE Vol P-2

3 3. MATHEMATICAL NOTATIONS The vefot phse eo t the ext pupl of optcl system c be epeseted by 2D, m mtx hch e deote s W. I the fomultos tht follos, e lso use vefot phse colum-vecto, otted s. s fomed by stckg the elemets of W fllg sde the ext pupl cle petue cet ode, see the follog p s exmple: W = , 2 T = [ ], () hee the supescpt T deotes mtx-tspose. The zeo vlues th W epeset the gdded dt pots outsde the cle petue of the ext pupl, d hece e ot ecessy fo computtos egdg the vefot. The sestvty vecto s (lso clled Ifluece Fucto ) of the th degee of feedom (DOF) s defed s the dffeece of the oml vefot d the vefot esultg fom dsplcemet of the th DOF, epeseted by vectos, d, dvded by the mgtude of the dsplcemet, u. Tht s, s =. (2) u s s computed fo ech cotollble DOF the system though umecl dffeetto, by ecodg the tl vefot, d the dsplcg dvdul DOFs oe t tme d ecodg the e. Ths c be doe usg the model o mesued dectly th tefeomete. The coespodg sestvty-mtx fo ll N ctutos s S = s s... s, (3) [ ] 2 N hee N s the totl umbe of DOFs. I geel, S hs to pts: The fst N colums coespod to the SD ctutos, d ts lst N b colums coespod to the RB ctutos, hee the subscpts d b std fo sufce-defomg ctuto d gd-body ctuto, espectvely. Tht s, S = [ S S b ]. (4) Thus, N = N + N b d S s N N mtx. I geel, cludes the oml vefot om peset by desg he the telescope s ubeted, the vefot b cused by betos to the system, d the vefot u poduced by pplyg set of ctuto commds. Tht s, om + b + u =. (5) c be obted by settg the commds of ll ctutos to zeo ( u =), o th set of u to cheve desed stte of the telescope. The oml vefot om s ot usully zeo, but the hs some smll stgmtsm s esult of blcg betos mog vous optcs to meet the feld pefomce objectves of the telescope. A full cycle WFC pocess cludes the follog steps: Poc. of SPIE Vol P-3

4 () Apply set of ctuto commds, u, d mesue o estmte the combed tl vefot,. Fo smll motos of the ctutos, u s le fucto of u. Theefoe, ths c be expessed mthemtclly s u = Su. (6) Smll efes to the ge of petubtos ove hch the smll-gle ppoxmto s vld fo y-tce though the optcl system. (2) Geete e set of ctuto commds bsed o. I ts smplest fom, ou cotol objectve s to select the dffeetl commds u such tht they ll dve the post-cotol WF eo (WFE), f = + S u, (7) to ts mmum. Tht s, f s the vefot fte oe WFC cycle. Oe c cheve ths by mmzg the cost fucto J: I mtx fom, ths becomes N 2 f k 2 k = J =, (8) T J = f f. (9) 2 If e tke the devtve of J th espect to the ctuto commds, the optml soluto occus he dj s zeo, tht s, N T f T T T T dj u f = u S f = u S ( + S u) =. () = u Equto (7) hs bee utlzed to obt the lst expesso Eq. (). The o-tvl soluto to ths equto s: u = T T + ( S S) S = S = G, () hee G s the optcl cotol g mtx, d the supescpt + deotes the pseudo-vese of mtx. It must be poted out tht the ctuto commds u obted fom Eq. () e dffeetl vlues d must be dded to the pevous vlues u he cyg out the ext WFC cycle. The ucosted lest-sques compesto Eq. () s efeed to s vefot cotol Method ths ppe. The G s N N mtx d s usully obted usg sgul-vlue decomposto pocedue. The bove soluto povdes the del stteous cotol put tht ould esult globl mmum fo the WFE, povded tht the sestvty mtx s exct. The g mtx G hs othe foms, epesetg dffeet types of compestos. The oe defed Eq. () coespods to smple lest-sques compesto. Othe foms of compestos c be foud Refs. [5-8]. We ll toduce to e WF compestos Secto 5 of ths ppe. 4. METHOD FOR IMPROVING A SENSITIVITY MATRIX Whe dptve o ctve optcl system utlzes both SD d RB ctutos, thee e to ppoches to pefom WFC. I the fst ppoch, oe ces out RB ctuto WFC fst, folloed by SD ctuto WFC. I ths cse, to dffeet g mtces e geeted septely fom the S d the S b defed Eq. (4). I the secod ppoch, oe full WFC cycle s completed sgle-step, usg the S lso defed Eq. (4). The dsdvtge of the fst ppoch s Poc. of SPIE Vol P-4

5 tht the WF eeds to be estmted o mesued tce to complete sgle WFC cycle, equg loge tme to complete oe WFC teto s comped to the secod ppoch. I both ppoches descbed bove, oe eeds to compute g mtx fom mesued o smulted sestvty mtx to pefom WFC. To obt the smulto esults to be peseted belo, e used detled d elstc optcl model of the HCIT [9]. The 2D OPD mp t the ext pupl s obted by ug MACOS (Modelg d Alyss fo Cotolled Optcl Systems) [] th gve set of pmetes. MACOS s vestle optcl modelg tool developed t JPL d used my flght pojects. It llos us to pply set of ctuto commds to the optcl model dectly, d cy out ed-to-ed full dffctve popgto lyss to estmte the vefot t the ext pupl. As metoed ele, sometmes optcl system utlzes mxed types of ctutos, such s SD d RB ctutos. The esposes of dffeet SD ctutos c be dffeet, d the RB ctutos c lso clude those tht pefom RB ottos d those tht pefom RB tsltos. Such stuto leds to lge dffeeces mog the ege-mode o sgul vlues of the fluece fuctos of dffeet ctutos. Fo exmple, fo the HCIT sho Fg., e obted tl smulted S fo the 24 SD ctutos of the DM, the tp/tlt (otto) ctutos of the DM d the psto (tslto) ctuto of the detecto. Fgue 2 shos the dffeetl optcl pth dffeece (OPD) mps, W W om, of oe SD ctuto d the thee RB ctutos. The SD ctutos of the DM utlzed o the HCIT hve poke lmt of () SDA #256 (b) RBA - Ox (c) RBA - Oy /(d) RBA N- L\z S PV = 55.2m PV = 59.3m PV = 58.7m PV = 2.6m Fgue 2. Dffeetl optcl pth-dffeece (OPD) W W of 4 dffeet types of ctutos. Fo the SD ctutos, oly oe exmple s cluded Pt (). Pts (b) d (c) coespod to the RB tp/tlt (x- d y-ottos) of the DM, d Pt (d) coespods to the psto of the detecto locl coodte systems. These fluece fuctos e the oes used ll of the umecl exmples ths ppe. The ctuto pmetes used e: Stoke s = 3 m fo the SD ctutos, θx = θy = µd fo the DM gd-body ctutos, d z = mm fo the detecto. The sme colo-mp scle s used ll of the OPD mps. Stdd Sestvty Mtx Nomlzed Sestvty Mtx Tolece Vlue, Tol = E Ege-Mode Numbe Fgue 3. Dstbutos of sgul vlues of the ogl S (ed-cuve) s ell s the omlzed oe (blue-cuve), S. The gee cuve epesets pseudo-vese tolece vlue used oe of the umecl exmples. Poc. of SPIE Vol P-5

6 u sd m. Theefoe, e used poke vlue of usd = 3m fo ll of the SD ctutos of the DM, u θ = µd fo the DM tp/tlt ctutos, d u z = mm fo the detecto psto ctuto Fg. 2. The sme vlues of ctuto commds ee used he obtg the sestvty mtx used ou umecl exmples. The ogl S tued out to be sgul mtx. Ths s cused by the fct the SD ctutos locted the 4 coes of the DM e f outsde the cle petue of the system d do ot ffect the ext pupl WF. Theefoe, e fst elmted those SD ctutos hose om hve pek-to-vlley (PV) vlues less th.m. Ths educed the umbe of SD ctutos fom 24 to 93 d esulted e, smlle S. The ed-cuve Fg. 3 shos the dstbuto of the sgul vlues of ths e S th =93. The codto umbe (the to of the lgest sgul vlue to the smllest) of S s 8 N 9.6 d the sgul-vlue tolece s defed Mtlb (see the Help fomto fo pv ) s.95. Tht s, the e S s o loge sgul. I the exmple descbed bove, the ogl S becomes sgul becuse the SD ctutos locted completely outsde the cle petue of the ext pupl do ot hve effect o the WF. Hoeve, oe sometmes ecoutes stuto, especlly lge spce telescope th fly complex optcl desg such s the JWST, hee the S becomes sgul o ely sgul due to the lge dffeeces mog the fluece fuctos of SD, otto d tslto ctutos. Oe ppoch to solve ths poblem s to use omlzed s, o s, defed s s om =, (2) u / u hee u s the sclg fcto. I the follog, e use the p, ( S G ) to deote the coespodg (omlzed) sestvty d g mtces, espectvely. I ths cse, Eq. () becomes s T U = G, u = [ Uu U 2u 2... U N u N ], (3) hee U e the omlzed ctuto commds. Ths s oe of the e ppoches toduced ths ppe. Ths ppoch s especlly useful fo beges ths feld. I pctce, oe c choose the vlues of u fo dffeet types of ctutos such tht the esultg sufce fgue mps hve compble RMS o PV vlues, fo exmple, betee d 2 m. I ou exmple sho Fg. 2, the PV vlues ge fom 2 m to 6 m fo the SD ctutos ot locted e the edges of the ext pupl cle petue d the othe thee RB ctutos. Afte choosg the vlues of u such y, oe c set u = u, tht s, the sclg fctos of ll ctutos e equl to the petubed vlues. The blue-cuve Fg. 3 shos the dstbuto of the sgul vlues of S obted ths y. The codto umbe of ths S s , d ts tolece s 2.. Tht s, the veso chctestc of the sestvty mtx hs o mpoved dmtclly by ths sclg method. We o sho sevel WFC exmples. Fgue 4 shos the ext pupl OPD mp of the HCIT befoe WFC s pefomed. It s obted by cludg the mesued fgue eos of vous optcs employed o the HCIT [9], plus some ddtol lo-ode Zeke-mode eos toduced by OAP. Pt () of Fg. 5 shos the OPD mp fte sgle teto of WFC usg the G obted th tolece vlue of Tol = o usg smple lest-sques compesto. Ths s equvlet to cludg ll of the ege-modes the WFC. The colo-scle of ths fgue s compessed to ± 3m to mke the ek detls vsble. I elty, thee e dt pots th vlues outsde tht ge s evdeced by the PV vlue gve s the x-lbel of the fgue. The udesble, post-cotol, betos ths fgue e composed of to pts. The fst pt cludes some hot pxels t the edges of the cle petue d s cused by the stuto of the ctutos those loctos. The secod pt coespods to the hgh sptl-fequecy compoets of the OPD mp sho Fg. 4 tht fll outsde the cotollble bd of the DM. I el stuto, the eo emg the vefot ould hve othe ddtol compoets. These clude, but ot lmted to, the eo the vefot 9 Poc. of SPIE Vol P-6

7 RMS = 38. m, PV = 7.8 m Fgue 4. Ext pupl OPD mp of the HCIT befoe WFC. It s obted by cludg the mesued sufce fgue mps of the vous optcs o the HCIT d by toducg some ddtol lo-ode Zeke-mode eos to the OAP. The lbel udeeth the plot dsplys the RMS d PV vlues of the OPD mp. The colo-b s m. () Tol = (b) Tol = le-5 (c) \OPD -...,,.:.63, 9.24 m.626, 9.26 m.23, 6.2 m Fgue 5. Ext pupl OPD mps of the HCIT fte WFC. The g mtx s obted fom the S th () Tol = d (b) Tol =E-5, espectvely. Pt (c) shos the dffeece of the OPD mps sho Pts () d (b). The lbel udeeth ech plot dsplys the RMS d PV vlues fo ech plot. The colo-b s m Acuto Idex Fgue 6. Soted vlues of u used to obt the esults Fg. 5. The ctuto commds obted th Tol = e fst soted descedg ode d plotted s fucto of ctuto dex. The ctuto commds obted th Tol = e lso plotted the sme ode th blue ccles. Poc. of SPIE Vol P-7

8 estmte, ctuto eo, d the effects of lmted sptl esoluto the mge ple. Fo lge tl vefot eo the olety of ctuto espose to the ppled commds lso becomes fcto d WFC typclly eques moe th oe tetos to covege. Oe ell-ko ppoch fequetly used WFC pocess s to geete the g mtx fom the coespodg sestvty mtx th o-zeo tolece vlue. We cll ths WFC ppoch lest-sques compesto th tolece d smply efe to t s the vefot cotol Method 2 ths ppe. I Mtlb, the g mtx of ths ppoch s obted s G = pv( ddu, Tol). (4) hee ddu = S. Ths opeto e-dstbutes the ctuto tsks of ek ctutos, o ctutos hose flueces to the WF e much eke th the oml oes due to some stuctul dffeece, stuctul defcecy o locto t the ext pupl, to the eghbog stog ctutos d pevets the ek ctutos fom stutg. Fo ou secod exmple, e used Tol = to get e G, d coducted sgle-teto WFC smulto. The esdul OPD mp of ths cse s sho Fg. 5(b), d the dffeece of the OPD mps Fgs. 5() d 5(b) s sho Fg. 5(c). The RMS d the PV vlues of the OPD mps pts () d (b) e lmost detcl. Hoeve, e c see fom pt (c) tht the m dffeece of these to mps e e the edges of ext pupl, d most of the stog esdul WF vlues pt () o loge exst pt (b). Fgue 6 shos ll of the SD ctuto commds used to obt the esults Fgs. 5() d 5(b), espectvely. I ths fgue, the u vlues of ll the SD ctutos obted th Tol = e soted descedg ode d plotted s fucto of ctuto umbe dex the soted ode usg ed-cuve. The u vlues obted th Tol = e lso plotted th blue ccles the sme ode. A totl of 25 ctutos e stuted d totl of 54 ctutos hve u 2m the cse of Tol =. We foud tht ll of these 54 ctutos e locted t the edges of the ext pupl. Tht s, these ctutos hve ek flueces o the WFC pocess becuse they e oly ptlly vsble sde the cle petue of the ext pupl. Whees o ctuto s stuted d u 2m fo ll the ctutos the cse hee Tol =. Also, my ctutos ll be foced to hve lge stoke vlues he Tol = th he Tol =. Fo exmple, ote the ctutos th dces gete th 6 Fg MINIMUM-WAVEFRONT AND OPTIMAL CONTROL COMPENSATOR I ths secto, e toduce e mmum-vefot d optml cotol compesto. It cludes both the WFE d the ctuto commds the cost fucto to be mmzed. Ths compesto c be used to educe the ovell mgtudes of the ctuto commds, d c lso be used to cotol u-y ctuto stutos. Some cotol ctutos, such s the SD ctutos t the edges of system s cle petue o mo s ell s RB otto d tslto ctutos, oly ekly mpct WFE. Dug WFC pocess, these ctutos get dve hd to poduce smll WF chge, usully hde th s desble d sometmes to the ctuto lmt. I such cse, some ctutos get stuted (ech the ctuto lmts) d stll cot povde the ecessy WF chge. Ths poblem c be ovecome th e g mtx obted fom the follog jot cost fucto: T T ( + γ u u) J = f f u, (5) 2 d by djustg the γ u pmete. The cotol soluto s deved the sme y s Eq. (). The esult s: u = T T [( S S) + γu I ] S = G, (6) hee I s the detty mtx. Ths cesed vestlty comes t the cost of moe complex mplemetto the computtos, hoeve. We efe to the mmum vefot d optcl cotol compesto Eq. (6) s vefot cotol Method 3 ths ppe. Poc. of SPIE Vol P-8

9 Hee e toduce othe, geelzed ppoch to the cotol tht c be qute useful hee ose o lsg s poblem. I ths ppoch, the cost fucto to be mmzed s ot the WFE tself, but ts sptlly-flteed veso [8]. Ths ppoch my be pefeed he scetfc objectve clls fo sptlly-flteed metc such s eccled eegy. Sptl fltes c be mplemeted usg mtx-fom Foue flte. Tht s, F = F, (7) Whee F s the sptlly-flteed WF d F s the sptl flte mplemeted mtx fom. A sml flte F u c be dded to the ctuto commd tem Eq. (5). I ts most geel fom, cludg M vefot d M ctuto commd sptl fltes, the cost fucto becomes fom hch e obt the cotol commds u = M m= M M T T J = γ m F m F m + f f γ 2 m= = T u F () =. (b) =. (c) =. u T u F u u, (8) M M T T T T T γ S F F S γu Fu Fu γ S F F m m m + m m m = G. (9) = m= We cll the bove ppoch the sptlly-flteed, mmum vefot d optml cotol compesto. I; I;.6, 9.27 m.64, 9.27 m.637, 9.3 m Fgue 7. Dffeece of ext pupl OPD mp of the HCIT fte WFC obted th Tol =E-5 mus d OPD mps obted th γ u =.,. d., espectvely. The lbel udeeth ech plot shos the RMS d the PV vlues of the fte-wfc OPD mp obted usg the g mtx defed Eq. (6). The colo-b s m. To sho the dffeece of the g mtces defed Eqs. (4) d (6), espectvely, e ced out some WFC smultos o the optcl system Fg.. We used vlues of.,. d. fo γ u, d comped the OPD mps fte WFC th tht obted th Tol =, see Fg. 5(b). Fgues 7()-7(c) sho the dffeece of the OPD mp Fg. 5(b) d those obted th γ u =.,. d., espectvely. Just lke the cse of Fg. 5(b), the WFC s ced out fo oly oe teto the cuet cse. As see fom Fgs. 7, the OPD mps fte WFC obted fom the bove to ppoches e mly dffeet t the edges of the esulted OPD mps; the WFE vlues t the edges get less suppessed th cesed γ u s comped the cse of Tol =. We soted the cotol vlues coespodg to Tol = descedg ode, d plotted them s ell s the cotol vlues coespodg to γ u =. the sme ode Fg. 8(). I Fg. 8(b), e peseted the hstogms of the cotol vlues obted th Tol = s ell s th the thee dffeet vlues of γ u. These fgues sho tht the Method 3 compesto Poc. of SPIE Vol P-9

10 exteds the ge of u s comped to Method 2. Also, Fg. 8(b) shos tht cesg the vlue of γ u educes the umbe ctutos th lge cotol vlues, s expected. 25 Tol= E-5 Yu=. Yu =. 2 Y (b) Acuto Idex l Il R Cotol Vlues (m) Fgue 8. Dstbuto of cotol vlues u obted sgle-cycle WFC smulto. () The cotol vlues of the cse Tol = E-5 e soted descedg ode d plotted th ed-cuve. The u of the cse γ u =. e lso plotted th blue-ccles the sme ode. (b) Hstogms of the cotol vlues obted th to dffeet compestos. Fo the cse of Eq. (6), the vlues of u coespodg to thee dffeet γ u vlues e cluded. All cotol vlues e fo SD ctutos oly; the cotol vlues of the RB ctutos e ot cluded these fgues. 6. CONCLUSION We hve toduced e method fo pevetg the sestvty mtx of lge spce telescope fom becomg sgul. I ths method, the e-codtog of the sestvty mtx to ell-behved oe tems of vetblty s ccomplshed by usg set of ppoptely selected sclg fctos fo ll ctuto commds. Ths method s vey useful he the ext pupl vefot of the telescope s cotolled th both sufce-defomg d gd-body ctutos tht dffe getly fom oe othe the flueces o the vefot. We hve expled ths e ppoch th some umecl exmples obted though modelg d smultos coducted o the optcl model of the Hgh- Cotst Imgg Testbed (HCIT) t JPL. We hve lso descbed sevel commoly used o vlble vefot compestos o cotol methods. These methods clude: Method : Smple lest-sques compesto, Eq. () Method 2: Smple lest-sques compesto th tolece, Eq. (4) Method 3: Mmum vefot d optcl cotol compesto, Eq. (6) Method 4: Geelzed, sptlly-flteed mmum vefot d optcl cotol compesto, Eq. (9) Amog them, Methods 3 d 4 hve bee epoted fo the fst tme ths ppe. I these to methods, the WFS sgls s ell s the ctuto commds e cluded the feedbck loop. The ctuto commd tems povde dect obsevblty of ctuto modes tht e ot obsevble the WFS sgls, thus elmtg susceptblty to ffle d othe poblemtc modes. We hve sho usg some HCIT smulto esults tht Method 3 s compble to Method 2 tems of cotollg u-y ctuto stutos o pevetg the ek ctutos fom gettg stuted dug vefot cotol opeto. We hve used these to methods successfully sevel dffeet expemets d pojects. They hve bee foud to be useful stutos th ucet pmetes, ole ctutos, fluece fucto eos, d/o hgh ose. They my hve pooe pedcted pefomce le models of the mo cotol poblem, yet yeld bette ovell pefomce, becuse they e less depedet o the bsolute ccucy of the cotol models. They vod poblems hee cotolle ll dve ctutos opposto to cete smll theoetcl Poc. of SPIE Vol P-

11 beefcl effect -- smll dffeece of lge umbes -- oly to fd tht the eos the lge umbes cete lge eo th the expected smll beeft. We expect tht both the sestvty mtx omlzto method d the mmum vefot d optml cotol compesto peseted ths ppe ll ply mpott oles lge spce telescopes. REFERENCES. Del S. Acto, Pul D. Atcheso, Mck Cemk, L K. Kgsbuy, Fg Sh d Dvd C. Reddg, Jmes Webb Spce Telescope vefot sesg d cotol lgothms, Poc. SPIE, 5487, 887, D. Reddg, S. Bsge, G. Bck, d R. Deky, Adptve optcs ecostucto utlzg supe-smpled defomble mo fluece fuctos, Poc. SPIE, 3353, 543, Ade E. Lom, Joh T. Tuge, B Godo, Joseph J. Gee, Dght Moody, Albet F. Nesse, d Fg Sh, Hgh-cotst mgg testbed fo the Teestl Plet Fde coogph, Poc. SPIE, 5487, 246, Joh T. Tuge, Chs Buos, B Godo, Joseph J. Gee, Ade E. Lom, Dght Moody, Albet F. Nesse, Fg Sh, d Del Wlso, Coogph cotst demosttos th the hgh-cotst mgg testbed, Poc. SPIE, 5487, 33, W. Wld, Iovtve vefot estmtos fo zol dptve optcs, Poc. SPIE, 326, 278, D. Reddg, S. Bsge, A. E. Lom, d A. Kssl, Wvefot sesg d cotol fo ext geeto spce telescope, Poc. SPIE, 3356, 758, D. Reddg, S. Bsge, G. Bck, d R. Deky, Adptve optcs ecostucto utlzg supe-smpled defomble mo fluece fuctos, Poc. SPIE, 3356, 543, D. Reddg, F. Sh, S. Bsge, D. Cohe, J. J. Gee, A. E. Lom, d C. M. Oh, Wvefot sesg d cotol fo lge spce optcs, IEEEAC Poc. Vol , -6, Ek Sdck, Fg Sh, Scott Bsge, Dght Moody, Ade E. Lom, Ades C. Kuhet, d Joh T. Tuge, Pefomce of TPF s Hgh-Cotst Imgg Testbed: Modelg d smultos, Poc. SPIE, 6265, 62653L, 26.. Modelg d Alyss fo Cotolled Optcl Systems (MACOS) Use s Mul, Jet Populso Lbotoy, Clfo Isttute of Techology, Psde, CA. Poc. of SPIE Vol P-