AIAA Robert L. Sierakowski Chief Scientist Air Force Research Laboratory AFRL/MNG 101 W. Eglin BLVD, Ste.105, Eglin AFB, Florida

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1 Opmal Rgulaor 4 Dsg of Opmal Rgulaors* Alxadr A. Bolo NRC Sor Rsarch Assoca Ar Forc Rsarch Laoraor 3 Avu R, #6-F, Brool, NY 9, USA TF , abolo@uo.com, aolo@gmal.com, hp:bolo.arod.ru AIAA Ror L. Sraos Chf Scs Ar Forc Rsarch Laoraor AFRLMNG W. Egl BLVD, S.5, Egl AFB, Florda Asrac Curr rsarch suggss h us of a lr quadrac prformac dx for opmal corol of rgulaors varous applcaos. Som xampls clud corrcg h racor of roc ad ar vhcls, vrao supprsso of flxl srucurs, ad arpla sal. I all hs cass, h focus s supprssgdcrasg ssm dvaos rapdl. Hovr, f o compars h Lar Quadrac Rgulaor LQR) soluo h opmal soluos mmum m), s s ha h LQR soluo s lss ha opmal som cass dd 3-6) ms ha oad usg a mmum m soluo. Morovr, h LQR soluo s somms uaccpal pracc du o h fac ha valus of corol xd od admssl lms ad hus h dsgr mus choos coffcs h lar quadrac form, hch ar uo. Th auhors suggs mhods hch allo fdg a quas-opmal LQR soluo h oudd corol hch s closd o h mmum m soluo. Th also rmad h procss of h mmum m dcso. Kords: Opmal rgulaor, mmum m corollr, Lar Quadrac Rgulaor LQR) *Ths papr s dclard a or of h U.S. Govrm ad o suc o coprgh proco h USA. Th mauscrp s accpd as papr AIAA d AIAA Umad Ulmd Ssms, Tchologs, ad Opraos- Arospac, Lad, ad S Cofrc ad Worshop & Exh, Sa Dgo, Calfora, USA, 5-8 Sp 3. Iroduco Th LQR soluo s asl ad covl r usg h Rcca quao as a opmal soluo. Th scs ho accps hs ma acg as a oxcad ma a Russa acdo: o gh a ma s osrvd crpg aroud a srlgh. A passr ass hm, ha ar ou dog? I los mo. Whr dd ou los h mo? Thr a h ohr d of h sr. Th, h ar ou loog hr? Ths s hr h lgh s! Th mmum m soluo s mor complx, hovr, ca covl drmd ma prolms h avalal, grall, of hgh-spd compurs. Also, hs approachs us h a ru mmum m soluo. For a -dmsoal prolm h o corol hs soluo foud gral form rfrc []. For h o-dmsoal cas hs soluo ca prsd graphcall, s rf. []. Mhods for ohr gral opmal soluos ar offrd []-[4]. Th LQR soluo has hr ma ssus: ) Th slco of h marx coffcs h prformac dx ar dsgr slcd ad h soluo s dpd upo h valu of hs coffcs. ) Th rag of corol valus ca larg umr ad hs o admssl for pracc.

2 3) Th opmal LQR soluo ca up o 3-6 ms ors, h h mmum m soluo s h xampl hs papr). If a rsarchr choss o us h LQR soluo, h auhors suggs a mhod for lmg maxmum corol s po ) as ll as for h choc of slcg h coffcs h prformac dx. Ths allos up o a -3 ms mprovm h prformac dx s accompag xampls) ad hus mas h LQR soluo accpal praccal applcaos. Th radoal approach usd h dsg of a corolld srucural ssm s o dsg h srucur frs sasfg gv rqurms ad h o dsg h corol ssm. Th srucur s dsgd h such cosras placd o gh, alloal srsss, dsplacms, uclg, gral sal, frquc dsruos, c. Wh h slco of h gomr, cross-scoal ara of h mmrs, ad maral ar drmd for a spcfd srucur, h h srucural frqucs ad vrao mods com mpora pu h dsg of h corol ssm. Som vsgaors hav r paprs dscussg a grad dsg approach for opmal corol. I mos rfrcs, h corol dsg procdurs usd, do o a o cosdrao h lmaos o h corol forcs dvlopd h acuaors, ad hav o rad as cosras or dsg varals. I hs papr h prolms assocad h h slco of h prformac dx, paramrs, gh coffc h LQR prolm, ad lmao of corol forcs ar addrssd. I h follog scos, hors for h shss of a opmal corol las h a quadrac prformac dx ad oudd corol forcs ar gv. Ths s follod a SISO Sgl Ipu, Sgl Oupu) corol prolm dsgd usg oh approachs for comparso of h d sa racors, h dffr ouds placd o corol forcs. Nx, h corol ssm for a dalzd g-ox s usd o llusra a dsg applcao of h mhod. A dscusso o h applcao of a corol ssm h oudd corol for a grad dsg of a srucur ad corol ssm ca foud rf. [5]. Rlad arcls ar [6]-[]. Th arcl h all fgurs has sz 5.8 M. Th Arxv gvs ol h spac M. Tha a h par of fgurs ar dld.. Opmal Corol Th gral opmal corol prolm ca dscrd h follog quaos I F x, x ) f, x, ) d, dxd = f,x,v), x ) = x, x ) = x -) hr I s h fucoal ocv fuco), s m, x s a -dmsoal vcor of sa, ad v s a p- dmsoal vcor of corol forcs. Th vcor vv hr V ca a oudd doma. Boudar codos,, x, x ar usuall gv,, ) T. Th corol paramr, v s calculad so ha I = m. To fd h soluo o hs prolm, assum h fuco =,x) -) ad r h fucoal J = A + B d [, ], -3) hr A = F o + ) - ), B = f o x)f - ). -4) Hr x) s -dmsoal vcor of paral drvavs. Th gloal mmum s A f A x, x ), B f B, x, v) for T. -5) x, x v, x Dpdg o h aur of h fucos usd for, a dffr s of algorhms for oag h fmum ca dvlopd. For xampl, f Eq.-) as h form = )x, -6) hr ) s a -dmsoal vcor, h gloal mmum fucos ca r as, A o = f F + ) - )), B o = f [f o - f,x,v) dd)x] = f B. -7)

3 3 x,x x,v x,v Usg Bx = ad Eq.-7) gvs dd = - Hx, B = f B, vv, -8) hr H = f,x,u) f o. Eq. -8) ca grad o fd, o oa h opmal corol v ad h opmal racor x). Aohr a s o forc h codo B f f f, vv x -9) vrhr h admssl doma for x. I hs cas, h quao for parcular drvavs ca solvd ad h shss of h opmal corol v = v,x) ad h fld of h opmal racor h admssl doma s oad. Th o corol dsg approachs h cosras o h maxmum corol forcs ar dscussd hs sco. I h frs sco a ocv fuco for salshg of h mmum m o supprss vrao s dscussd ad h scod, h quadrac fuco s mmzd. A. Mmum Tm Sc h ma purpos of h corollr s o supprss vraos mmum m, h m for h ssm o com o rs s a as h ocv fuco. A fucoal xprsso for hs ca r I T d, T = m -) suc o dxd = Ax + f, x) = x o, xt) = -) h corol forc lms f F, =,,,p. -) Ths prolm ca r shor form as T m I d, dxd = Ax + f, x) = x o, xt) =, f F, -3) hr x s h sa vcor of dmso. A s h pla marx, B s p corol marx, f s h corol forc vcor of dmso p, x) s h al sa vcor, ad xt) = s h fal sa of h ssm. B o, Eq.-7), for hs prolm ca r as B o = - x )dx d) ) =.,,, ). -4) Susug = )x =,,,) -5) ad Eq.-) o Eq. -4) gvs p B J a f x. J -6) Tag h paral drvavs of B Bx ) gvs d d = - a =,,,, =,,,. -7) m B o gvs h corol forc f f = F sg ) =,,,p ; =,,,. -8) Usg Eqs.-3),-7) ad -8), h opmal corol forc f ) ad racor x ) ca calculad. Hovr h al ) for our racor h x) = x o s o o. To fd ), a sual

4 4 grad mhod ca usd. For xampl, f h assum som al sa ) ad gra Eqs.- 3),-7) ad 8), ca calcula h fuco I T C x T ), C >, =,,,). -9) Hr C ar gh coffcs. If C x T)< C, hr C o s small, h prolm ca cosdrd as solvd. Tm s opmal ad x ) s h opmal racor hch sasfs h fal codo x T) =. If C x T) > C o ca choos a ) a mhod ad rpa h procss ul sasfs C x T) < C o. I pracc, a dpd varal s roducd as = c; hch ca cludd h Eq.- ) o proudr h addoal quao dd = c. -) Addoall, roducg a fxd rval of grao [, ] a s of quaos com m I cd, dxd = Ax +Bf)c. xo) = x o, x ) =, f F, -) hr c s som cosa, hch s also slcd. Eq.-9 ) hus coms I = C x ), =,,,. -) For h srucural ssm as dfd Eqs.-)--) hs prolm ca solvd for h cas hch h umr of corol pus, p, s qual o h umr of modld srucural dgrs of frdom,. Hovr, umrcal dffculs ould courd h hs codo s o sasfd. Tpcal dffculs ould h occurrc of ma local mmums, poor covrgc, ad h d for smallr sp szs. B. Lar quadrac rgulaor LQR) h oudd corol I hs cas, a prformac dx, J, s dfd as J T T x Qx f R f ) d [,]. -3) Whr Q ad R ar sa ad corol ghg marcs. Th marx Q mus posv sm df x T Qx ), ad R mus posv df f T Rf > ). Th dmsos of Q ad R dpd o h sz of vcors h x ad f, rspcvl. Th marcs Q ad R ca r as Q = Q -4) ad R = )R - -5) hr ad ar h dsg posv varals ad Q ad R - ar cosa d marcs. Th ghg marx R s dfd rms of h vrs of h cosa marx R ordr o maa posv dfss. Th fuco B, Eq. -9) for h prformac dx dfd Eq.-3) ad h cosra quao Eq.-), com T T B f x Qx f Rf ) Ax Bf ) -6) f x If V s a op doma, h fuco ca r If V rprss a op doma, h fuco, ca r h form = x T Px, -7) hr P s a -dmsoal uo marx. Susug Eq. -7) o Eq, -6), oa h quao Q + PA + A T P PBRB T P =. -8) Equao -8) s h Rcca quao. A soluo of hs quao gvs h marx P ad o ca fd h opmal corol forc as

5 5 f = - Gx -9) hr G = RB T P -3) Igrag Eq. -) usg Eq. 9) o oa h opmal racor for h LQR fucoal. Eq. - 9) ma gv uralsc valus of corol dpdg o h slco of. Th magud of corol ca dcrasd crasg, hovr, hs ma caus ohr pruraos of h ssm such as h m as h oscllao o dca) o drora. I ordr o oa mor ralsc rsuls, ouds ca placd o h corol forc. Ths ca r as f F, F = cos, =,,,p -3) hr F s h magud oudg ach corollr. To oa a opmal soluo, h follog rsrcos mus sasfd: ) amog hs opmal shss of h corol mus xs h doma of rs, ) h fuco B Eq. -3) mus covx, ad 3) h lms of F ma cosa or dpd o m ol ad F mus o qual o zro a a m No: f F s vr small a loss sal ca occur). For a soluo, h ssm of Eqs. -) ad -9) mus grad alog h lms mposd quao -3). Th orm for h dsplacms or oal dvao ca dfd R x ) =S = [ x ) ] =,,, -3) = Ths orm s zro a h m h dvao s zro, ad h srucur sops vrag. I h LQR soluo doma hs m quals f. For sudg h havor ad comparso of dffr corol ssms, a masur of prformacs has usd asd upo. Th m rqurd o rduc h orm of h dsplacms o % of hr al valu. Numrcal Exampls. Exampl. SISO prolm. For comparso of ssms h dffr ocv fucos, a vrag srucur h a sgl phscal dgr of frdom as vsgad. Ths ssm s dscrd quao h follog s of dx d = x, dx d = - x x + cf, x ) =, x ) =, f -33) hr = s h frquc, =.3 s h dampg, c =, ad f s h corol. Th prolm s solvd havg a ocv fuco for mmum m as m T T d, x T) =, x T) = -34) Eqs. -7) ad -8) for h ssm dfd Eq. -33) com d d = -, d d = -, f =Fsg. -35) Eqs. -33)--35) ar grad ad h al valus ), ) ar chos such ha h codos x T) = x T) = ar sasfd. Th dals of h soluo schm ar o gv hr caus of spac lmaos. Th prformac for h lar quadrac rgulaor LQR) s J x x d. -36) Usg hs prformac dx ad solvg h Rcca Eq. -9) gvs f = c)c x + c x ), -37) hr c = - [ c o ).5 ]c o, c = { - + [ c )c o ].5 }c o, c o = c. I h cas of = =, h m hsor dpds ol o. Th oal dvao s R x = S = x + x ). -38)

6 6 Eq. -33) s grad h corol gv Eq. -37). Th rsuls of hs vsgao for h cas T = m, =.5 ad ad o corol op-loop ssm) ar sho Fg s,, & 3. Fg. shos h m hsor of dvao of x. As ca s, a LQR h = gvs r rsuls = 4 sc) ha a LQR h =.5 m s mor ha 5 sc) hovr a v r rsul s oad h a ocv fuco of mmum m. I h las cas, oscllaos ar rmad.5 sc. Fgur shos h varao of a oudd corol forc f for h cas of T=m, LQR h =.5 ad =. Th cas LQR =.5) dos o us h full corol forc, h cas LQR =) uss mor of h corol forc, ad cas =m uss h maxmum corol forc all h m. Fg.3 shos h m hsor for h oal dvao R x ) h o corol, h a ocv fuco for mmum m ad h LQR gv corol ouds f. Fg A srucural ssm h a umr of dgrs of frdom ca rasformd o pars of quaos -33)s lar Eq.-4)--48)) hr vr par s dpd from h ohr. If h umr of corols quals h umr of dgrs of frdom h dsg approach asd o mmum m ca usd. Hovr, f h umr of corols s lss ha h umr of pars of quaos, h soluo for h fucoal T = m coms vr complx. I hs cas, h LQR approach s a varal alrav. Fg. dld) Fg.3 dld) Exampl. Wg Box I ordr o llusra h applcao of a approach usg h lar quadrac rgulaor h oudd corol, h g ox prolm rfrc [5] s usd ad sho Fg. 4. Ths srucur has hr-

7 7 o lms ad -four dgrs of frdom. Th srucur s a calvr g ox dalzd h ar lms capal of carrg axal loads ol. Fg.4. Th quaos of moo for a flxl srucur h o xral dsurac ca r as Mu Eu Ku Df, -39) hr M s h mass marx, E s h dampg marx, ad K s h oal sffss marx. Ths marcs ar, hr s h umr of dgrs of frdom of h srucur. I Eq. -39), D s h appld load dsruo marx rlag h corol pu vcor f o h coorda ssm. Th umr of lms vcor f s qual o h umr of acuaors, p. Th vcor u Eq.-39) dfs h srucural rspos. Th coorda rasformao u = [] -4) s roducd hr s h modal coorda ssm ad [] s h modal marx. Usg Eq. - 4), Eq.-39) ca rasformd o ucoupld quaos. Ths ca r as T M E K [] Df -4) hr T M I [ ] M[ ] T E [] [ ] E[ ] -4, -44) T K [ ] [ ] K[ ] Th marcs M, E, ad K ar dagoal squar marcs, s h vcor of srucural frqucs, ad s h vcor of modal dampg facors. Th modal marx [] s ormalzd h rspc o h mass marx. Th corol aalss s prformd rducg h scod-ordr ucoupld quao [Eq.-4)] o a frs-ordr quao. Ol of ucoupld quaos ar usd for h corol ssm dsg. Ths ca achvd usg h rasformao x -45) hr x s h sa varal vcor of sz. Ths gvs x Ax Bf -46) hr A s a marx ad B s a p pu marx. Th A marx ad h pu marcs ar gv

8 8 I A -47) B -48) T D Th sa oupu quao s gv = Cx -49) hr s a q oupu vcor, C s aq oupu marx, ad q s qual o h oal umr of ssors. If h umr of ssors ad acuaors qual ad collocad, h q = p ad C = B T. -5) For hs srucur, Youg's modulus ad gh ds ar assumd o qual o.5 x l 6 ls ad. ls 3, rspcvl. Th acuaors ad ssors ar assumd mddd h srucural lms ad ar collocad. Th acuaors ar assumd o appl forcs alog h axal drcos provdg oh ou of pla, pla ad s corol for h srucur. I s assumd ha all srucural mods hav % srucural dampg ad hus Eq. -9) as.. Th corol ssm ulzs four acuaors ad ssors collocad h four mmrs a h p of h srucur cocg ods -, 3-4, -3 ad -4 rspcvl. No-srucural masss ar locad a ods hrough 8. Thr maguds ar.5 slugs a ods ad ;.5 slugs a ods 3 ad 4;.5 slugs a od 5 ad 7. ad. slugs a ods 6 ad 8 rspcvl. For h 4 srucural dgrs of frdom, h full ordr sa spac marx Eq. -) s 48 x 48. Sc hr ar four acuaors ad ssors, h pu marx B ad oupu marx C ar 48 x 4 ad 4x48, rspcvl. Th cross-scoal aras of h rod lms r qual o.. Th ghg marcs Q ad R Eq. -8), -9) r qual o h d marx. Th four valus of h ghg paramr raos slcd for hs sud ar.,., ad, rspcvl. Th maxmum corol forcs grad h four acuaors ar gv Tal. Tal. Calculad cass Corol oud F = = = = No corol Th al codo usd for dsgg h corollrs s a u dsplacm a od h z- drco. Ths codo s usd for all cass ad also o oa h rspos curvs. Th rspos curvs ar gv for ol a f cass caus of spac lmaos. Th hr lms o h maxmum alloal corol forcs ar s qual o.5,.5, ad.5 rspcvl. Th dffr cass cosdrd ar summarzd Tal. Tal. Maxmum acuaor forcs Acuaor # Valu =

9 9 = = = I h cas of =. h maxmum acuaor forcs ar lss ha.5, ad for =., h ar lss ha.5. Fg. 5 shos h m hsor of h dsplacm orm hou corol oud for h four valus of ad hou corol. Fg.5 dld) Th maxmum valu of h dsplacm orm as a fuco of m s sho Fg. 6. Th m rqurd o dcras h dsplacm orm o % of s al valu. s sho Fg. 7. Fg.6. Fg.7 dld) I h cas of o-corol, h oal m dd o rduc h dsplacm orm o o prc of h al valu s largr ha scods. Th varao h corol forc acuaor as a fuco of m for qual o ad s sho Fg. 8. Fg.8 dld) Fg. 9 shos h m hsor of corol forc acuaor h h uppr oud qual o.5 for =. Th uppr oud s forcd o all h acuaors. Fg.9 8 dld) Th chags h dsplacm orm h m for 8 qual o ar sho Fg. for h cas of corol oud qual o.5 ad hou oud. Fg. dld) Fg. shos h oal m rqurd o rduc h dsplacm orm o. for hr valus of ad four valus of corol oud. As h corol oud dcrass mor m s dd o rduc h dsplacm orm o. for a gv valu of. Th maxmum roo ma squar rspos for dffr cass s sho Fg..

10 Fg. dld) Fg. dld). Soluo of gral lar opmal prolm for o corol No cosdr h gral opmal lar rgulaor prolm h a ocv fuco of mmum m ad o corol paramr. Prolm Sam. Th ssm s dscrd a lar dffral quao vcor form as, -) x Ax Lu hr x x, x,..., x ) s h -dmsoal sa vcor, A a a -dmsoal squar marx of cosa coffcs, L a colum vcor hch coas l, l,..., l ; u a lmd corol, u, ; x) = x o, x ) = x h al ad fal codo, T = rprs h d m of procss, o =. I s o h corol ca hav ol oudar valu lar ssm ad, f gvalus of marx A s ral umrs, h ssm has ol maxmum - schs [7]. Prolm soluo. Th characrsc quao sa - E =, hr E s a u -dmsoal marx, s gvalus of marx A. Cas A. All gvalus ar ral, dffr, ad o qual zro. Usg x =,,,), = cos ca covr h quaos -) o caocal form u; u; u; -) h oudar codos ) = ; ) =. Th opmal corol u = s cosa vrhr. If a varal z = + u s roducd, s possl o r quao -) form z ; z z z; ; z z; -3) A soluo of quao -3) s z c =,,). Rurg o h varal ca r c u =,,); c c ;. -4) Cosdr h valu. Th mom h a corol paramr s chagd s mard a dx lo ad rgh ad lf from po plus ad mus sg o op of maguds. L us suggs, ha h corol has - schs. From couous codo hav From -5). Thrfor hav c u c u =,,). -5) c u u ) c =,,-). -6) Th valu c, c,. From -6), g c c u u ). -7) From h frs quao -4) ad oudar codos for, fd c u ); c u, -8) hr u u. Susug -8) o -7) oa,

11 ) ) u u u u =,,) -9) Ths quao -9) sasfs for all =,,. If o dvd h rgh ad lf pars of quao -9) u o I ), fd, ; ) )... u u ; ; ) )... u u ; ) ; ) )... u u, hr =. Nog ha, quaos -) ca r as ) )... u u, ) )... u u, ) ) )... u u. Equao -) s solvd ordr o fd = o ). Rurg o h orgal varal x, ca r = - l x). hr x o rprss h al po x. Equaos -) ar a s of algrac quaos. From oudar codos o ha.... Tha mpls ha.... For corol u. Ths mpls ha quaos -) mus solvd c. If x ) = hs mas ) = ), h scod soluo s smmrc aou h org. Th soluo of quao -) s asr o valua h h classcal opmal corol soluo. I classcal hor a rsarchr mus solv a oudar prolm for a s of gv dffral quaos ad also fd a s of uo Lagrag mulplrs. I usg quao -) h rsarchr frs salsh h rqurd m crms asd upo oldg of h phscal suao. To fd h sch surfacs, for =, mpls hus h racor s locad o h frs sch surfac. I hs cas quao -). W h s aou solvg h frs - quaos -) for, 3,, ad susu hs soluos o h las quao. Ths lads o a quao ). B susug for ca fd N x). Ths s h frs -)-dmsoal sch surfac. Nx susug =, = h frs - quaos, ad solvg h frs - quaos for 3, 4,, o ca oa soluos ad susu hm o las quao. W hus ca fd a hpr surfac N x) =. Th rsco of hs hpr surfac h N x) cras h scod -)- dmsoal sch surfac. Ohr sch surfacs ca foud as smlar a. Such a opmal corol rsul ca asl foud. I slcg u o, h h sa po rachs h sch surfac N x) =, u = - u o. Wh h sa po rachs h sch surfac N x) =, u = -u ad so o.

12 If m s dld from a o of quao -4), oa a proco of h racor o h surfac c u ) u, c c c. From -) ca fd h oudars of sal, for posv gvalus. For xampl, f I >, >, h ) =. Th cssar ad suffc codo usal soluo s gv ;. W hav ol cosdrd cass h h gvalus ar ral, dffr, ad o-qual o zro. Addoal cass hav cosdrd rfrc [6]. Exampl. Tag a o of quaos -) h gvalus,,,, < ), x ) =, a caocal form of h quaos ca xprssd as, u ; u ; ) = ; ) = ; u. E) Equao -) for u o = ca r as ; For = smplfg for h cas = ), ca oad from E) ad hus For u o = - quaos -) ar ; Tag =, ad usg from E), fd,. E). E3). E4). E5) Usg a cou codo ) = ), h rlaos E3), E5) ca r as o rlao sg sg ). E6) If =, >, h h rlao E6) s grar h zro. From E) s: ll dcras fasr f u = - for > ad u = + for <. Ths mpld ha sg u sg sg sg ). E7) To fd h quaos for opmal racors. Rfrrg quaos -4),-) fd c u ; c u ; c u ) u. E8) Th las quao E8) gvs formao h racors as sho fgur 3. Fg3 dld) Ths racors dpd upo h sgs of,. For h >, < h o-sal rgo s >. For <, > h o-sal rgo s >. I fg.4 also sho opmal racors. Oc aga h dpd up o h sgs of,. Rurg o h varals x, h pcur 4 s affd dform.

13 3 Th offrd mhod allos capur of opmal corol. Th ohr ors rlad o hs opcs ar []-[8]. Fg.4. Summar To opmum corol dsg mhods for supprsso of srucural vrao havg oudd cosras hav compard. Th mmum m ad quadrac prformac dx hav usd as ocv fucos. Th scod approach lads o us of h LQR mhodolog h oudd corol. Th roduco of a mmum m corollr ca usd h h umr of acuaors quals h umr of srucural dgrs of frdom usd h dsg of h corol ssm. Wh h umr of acuaors s lss ha h umr of dgrs of frdom, h mmum m corollr coms mahmacall complcad ad has foud o dffcul o solv du o h prsc of local mmums. Th mmum quadrac fuco corollr, h oudd corol, ca dsgd h a fr umr of acuaors. A SISO srucural corol dsg prolm has solvd usg oh approachs for comparso of racors ad h m dd o supprss vraos. Th fluc of corol lmaos ad h gh coffc of h srucur hav sudd. Rsuls dca ha a opmal slco of h gh coffc ca dcras h supprsso m up o -4 ms. Rcommdaos If possl, h rsarchr should r o dsg h corollr for mmum m. If s vr dffcul, h ca dsg LQR corollr. Hovr hs cas h rsarchr mus:. Cosdr lms o h maxmum valu of h corol forc.. Fd h opmal rao of h gh coffcs. 3. Solv umrcall) a las o m h ral mmum m) prolm ad compar ha ma luc s loss from chagg h T m prolm o h LQR prolm.

14 4 Rfrcs Th radr fds som of hs ors hp:bolo.arod.rup65.hm ad hp:arxv.org). A.A. Bolo, Soluo gral lar opmal prolm h o corol. Joural Prcladaa Mchaca, v.4, #4, 968, pgs. -. Mosco Russa).. Boo: A.A. Bolo, N mhods of opmzao ad hr applcao, Mosco, Tchcal Uvrs amd Bauma, 97, pgs. Russa). 3. A.A. Bolo, Spcal Exrma Opmal Corol Prolms, Aadma Nau, Izvsa, Thchsaa Kra, No., March-Aprl, 969, pp S also Eglsh raslao Eg.Crcs,., March-Aprl,969, pp A.A. Bolo, A N Approach o Fdg a Gloal Opmum, N Amrcas Collcd Scfc Rpors,Vol., 99. h Ba Zo Sa Scss Dvso, N Yor. 5. A.A. Bolo, N.S. Kho, Opmal Srucural Corol Dsg, IAF , 45 h Cogrss of h Iraoal Asroaucal Fdrao, World Spac Cogrss-994. Ocor 9-4,994Jrusalm, Isral. 6. V. Bolas, A. Poza, Lar mul-modl m-opmzao, Joural Opmal Corol Applcaos ad Mhods,Vol.3, Issu 3,, pp Yug Mao, Z Lu, Th opmal fdac corol of h lar-quadrac corol prolm h a corol qual cosra, Joural Opmal Corol Applcaos ad Mhods,Vol., Issu,, pp Hpg Hua, Numrcal soluo of opmal corol prolms, Joural Opmal Corol Applcaos ad Mhods,Vol., Issu 5,, pp H.Sgh, R.H. Brou, D.S. Nadu, Ufd approach o lar quadrac rgulaor h mscal propr, Joural Opmal Corol Applcaos ad Mhods,Vol., Issu,, pp B.J. Drss, N. Sadgh, Mmum-m corol of ssms h Coulom frco: ar gloal va mxd gr lar programmg, Joural Opmal Corol Applcaos ad Mhods,Vol., Issu,, pp Boo: A.A. Bolo, "No-Roc Spac Lauch ad Flgh", Elsvr, Lodo, 6, 488 ps.. A.A. Bolo, Opmal racor of ar ad spac vhcls, AEAT, #, pp.93-4, A.A. Bolo, R. Sraos, Dsg of Opmal Rgulaors, d AIAA Umad Ulmd Ssms, Tchologs, ad Opraos Arospac, Lad, ad sa Cofrc ad Worshop & Exh, Sa Dgo, Calfora, 5-8 Sp 3, AIAA A.A. Bolo, N.S. Kho, Dsg of Srucur corol Ssm usg Boudd LQG,Eg.Op.,997. Vol.9.pp, A.A. Bolo, N.S. Kho, Opmal Boudd Corol Dsg for Vrao Supprsso. Aca Asroaucs, Vol.38, No., pp83-83, A.A., Bolo, Mmum Wgh of Corol Dvcs h Boudd LQG Corol. Th World Spac Cogrss -96, Ju -6, 996, Aluqurqu, MN, USA. 7. A.A. Bolo, N.S. Kho, Dsg of Smar Srucurs h Boudd Corols, Smar Srucurs ad Marals, F. 5-9,996,Sa-Dgo, CA. 8. A.A. Bolo, Opmum Srucural Vrao Corol h Bouds o Corol Forcs, 995 ASME Dsg Tchcal Cofrc, 5h Bal Cofrc o Vrao ad Nos, Spmr 7-,995, Boso, MA, USA. 9. A.A. Bolo, Dsg ad Opmal Corol Smar Srucurs. Cofrc Mahmacs ad Corol Smar Srucurs, 6 F.-3 March 995, Sa Dgo, CA, USA.. A.A. Bolo, Opmal Srucural ad Corol Dsg. 45h Iraoal Acroaucal Cogrss. Jrusalm, Isral. Ocor 9-4, 994, IAF-94-I.4.6.

15 5. A.A. Bolo, Opmzao of Tracors of Mulsag Rocs. Ivsgaos of Flgh Damcs. Mosco, 965, p. -78 Russa). Iraoal Arospac Asrac A # Eglsh).. A.A. Bolo, Spcal xrm opmal corol. Aadma Nau USSR, Izvsa. Thchsaa Kra, No, Mar-Apr.,969, p S also Eglsh raslao Egrg Crcs, #, Mar- Apr.969, p.7-83, Eglsh). 3. A.A. Bolo, Mhods of soluo for oudar-valu prolms of Opmal Corol Thor. Traslad from Prladaa Mhaa, Vol. 7, No 6,97, p , Eglsh). 4. A.A. Bolo, Soluo of dscr prolms of opmal corol o h ass of a gral mmum prcpl Russa. Eglsh summar). Vcsl. Prl. Ma. Kv) Vp ), - 3. Mahmacal Rv A.A. Bolo, Impuls soluo corol prolms. Russa). Izv. Srs. Odl. Aad. Na USSR, 968, No. 3, M.R A.A. Bolo, A cra mhod of solvg opmal prolms. Izv. Srs. Odl. Aad. Nau SSSR. 97, o.8, p M.R. # A.A. Bolo, A mhod for h soluo of opmal prolms Russa). Complx Ssms Corol, pp Nauova Duma, Kv, 965. M.R. # A.A. Bolo, A cra approach o h soluo of opmal prolms. Russa. Eglsh summar). Vcsl. Prl. Ma. Kv). Vp. 97), M.R. # A.A. Bolo, Opmzao of paramrs of varao prolms Uraa. Russa ad Eglsh summars). Dopov Aad. Nau Ura. RSR, 964, M.R. # A.A. Bolo, Th calculus of varaos ad a fucoal quao of Bllma, ad a rprao of Lagrag s udrmd mulplrs. Uraa. Russa ad Eglsh summars). Dopovd Aad. Nau Ura., RSR 964, M.R. # A.A. Bolo, Th xso prcpl ad h Jaco codo of h varao calculus. Uraa. Russa ad Eglsh summars). Dopovd Aad. Nau Ura. RSR M.R. # A.A. Bolo, Elcrosac Solar Wd Propulso Ssm, AIAA Propulso Cofrc, - Jul, 5, Tucso, Arzoa, USA. 33. A.A. Bolo, Elcrosac Ulzao of Asrods for Spac Flgh, AIAA Propulso Cofrc, - Jul, 5, Tucso, Arzoa, USA. 34. A.A. Bolo, Kc A-Gravaor, AIAA Propulso Cofrc, - Jul, 5, Tucso, Arzoa, USA. 35. A.A. Bolo, Slg Roar Spac Lauchr, AIAA Propulso Cofrc, - Jul, 5, Tucso, Arzoa, USA. 36. A.A. Bolo, Radosoop Spac Sal ad Elcrc Graor, AIAA Propulso Cofrc, - Jul, 5, Tucso, Arzoa, USA. 37. A.A. Bolo, Gudd Solar Sal ad Elcrc Graor, AIAA Propulso Cofrc, - Jul, 5, Tucso, Arzoa, USA. 38. A.A. Bolo, Elcrosac Lvao ad Arfcal Grav, AIAA Propulso Cofrc, - Jul, 5, Tucso, Arzoa, USA. 39. A.A. Bolo, Lgh Mul-rflx Eg, JBIS, Vol.57, #9-, A.A. Bolo, Mul-rflx Propulso Ssms for Spac ad Ar Vhcls ad Erg Trasfr for Log Dsac, JBIS Vol. 57, pp , A.A. Bolo, Hgh ffcc rasfr of mchacal rg. Iraoal Erg Covrso Egrg Cofrc a Provdc, RI, Aug.6-9, 4. AIAA A.A. Bolo, Ulzao of Wd Erg a Hgh Alud. Iraoal Erg Covrso Egrg Cofrc a Provdc, RI, Aug.6-9, 4. AIAA , AIAA

16 6 43. A.A. Bolo, Ar Cal Traspor Ssm, Joural of Arcraf, Vol.4, No., March-Aprl A.A. Bolo, Earh Acclraors for Spac Shps ad Mssls, Joural JBIS, Vol.56, pp , A.A. Bolo, No-Roc Trasporao Ssm for Spac Travl, Joural JBIS, Vol.56, pp.3-49, A.A. Bolo, Asrods as propulso Ssms of Spac Shps, JBIS, Vol.56, pp.98-7, A.A. Bolo, Opmal Iflaal Spac Tors h 3- m Hgh, JBIS, Vol.56, pp.87-97, A.A. Bolo, Crfugal Kpr for Spac Saos ad Salls. JBIS, Vol.56, A.A. Bolo, Hprsoc Spac Lauchr of Hgh Capal, Acual prolms of avao ad spac ssm. No.5), vol.8, pp.45-58, A.A. Bolo, J. Clour, Sarch for Em Targs, Tchcal. Rpor AFRL-MN-EG-TR- 3-76, Ju. 49 p. 5. A.A. Bolo, J.Clour, Sarch, Osrvao, ad Aac Prolms, Tchcal Rpor AFRL- MN-EG-TR-3-77, p.. 5. A.A. Bolo, G.Glard, Opmal Pch Thrus Vcor Agl ad Bf for all Flgh Rgms, NASATM A.A. Bolo, G.Glard, Esmad Bfs of Varal-Gomr Wg for Traspor Arcraf,NASATM A.A. Bolo, No-Roc Earh-Moo Traspor Ssm, Joural Advac Spac Rsarch, Vol.3, pp A.A. Bolo, A Hgh Effcc Fuslag propllr Fusfa ) for Susoc Arcraf,!999 World Avao Cogrss, AIAA, # A.A. Bolo, Ar Cal Traspor ad Brdgs, TN 7567, Iraoal Ar & Spac Smposum Th Nx Yars, 4-7 Jul 3, Dao, Oho, USA 57. A.A. Bolo, No-Roc Spac Rop Lauchr for Popl, IAC--V.P.6, 53 rd Iraoal Asroaucal Cogrss. Th World Spac Cogrss, -9 Oc Houso, Txas, USA. 58. A.A. Bolo, No-Roc Mssl Rop Lauchr, IAC--IAA.S.P.4, 53 rd Iraoal Asroaucal Cogrss. Th World Spac Cogrss, -9 Oc Houso, Txas, USA. 59. A.A. Bolo, Ixpsv Cal Spac Lauchr of Hgh Capal, IAC--V.P.7, 53 rd Iraoal Asroaucal Cogrss. Th World Spac Cogrss, -9 Oc. Houso, Txas, USA. 6. A.A. Bolo, Hprsoc Lauch Ssm of Capal up 5 os pr da ad Dlvr Cos $ pr L. IAC--S.P.5, 53 rd Iraoal Asroaucal Cogrss. Th World Spac Cogrss, -9 Oc Houso, Txas, USA. 6. A.A. Bolo, Emplom Asrods for Movm of Spac Shp ad Pros. IAC-- S.6.4, 53 rd Iraoal Asroaucal Cogrss. Th World Spac Cogrss, -9 Oc. Houso, Txas, USA. 6. A.A. Bolo, Opmal Iflaal Spac Tors of Hgh Hgh. COSPAR- C.-35-, 34 h Scfc Assml of h Comm o Spac Rsarch COSPAR). Th World Spac Cogrss, -9 Oc Houso, Txas, USA. 63. A.A. Bolo, No-Roc Earh-Moo Traspor Ssm, COSPAR- B.3-F3.3-3-, -A-6, 34 h Scfc Assml of h Comm o Spac Rsarch COSPAR). Th World Spac Cogrss, -9 Oc Houso, Txas, USA.

17 7 64. A.A. Bolo, No-Roc Earh-Mars Traspor Ssm, COSPAR- B.4-C , 34 h Scfc Assml of h Comm o Spac Rsarch COSPAR). Th World Spac Cogrss, -9 Oc Houso, Txas, USA. 65. A.A. Bolo, Traspor Ssm for dlvr Tourss a Alud 4 m. IAC-- IAA..3.3, 53 rd Iraoal Asroaucal Cogrss. Th World Spac Cogrss, - 9 Oc. Houso, Txas, USA. 66. A.A. Bolo, Spac Cal Lauchrs, TN 857, Iraoal Ar & Spac Smposum Th Nx Yars, 4-7 Jul 3, Dao, Oho, USA. 67. A.A. Bolo, Gomr-Basd Fasl Cosras for Sgl Pursur Mulpl Evadr Prolms, d AIAA Umad Ulmd Ssms, Tchologs, ad Opraos Arospac, Lad, ad sa Cofrc ad Worshop & Exh, Sa Dgo, Calfora, 5-8 Sp 3, AIAA A.A. Bolo, Th Smpls Spac Elcrc Graor ad Moor h Corol Erg ad Thrus, 45h Iraoal Asroaucal Cogrss, Jrusalm. Isral. Oc. 9-4, 994, IAF- 94-R A.A. Bolo, Spac Elcrc Graor, ru Solar Wg. Th World Spac Cogrss, Washgo, DC, USA, 8 Aug.-5 Sp., 99, IAF A.A. Bolo, Smpl Spac Nuclar Racor Moors ad Elcrc Graors Rug o Radoacv Susacs, Th World Spac Cogrss, Washgo, DC, USA, 8 Aug.- 5 Sp., 99, IAF A.A. Bolo, A Spac Moor Usg Solar Wd Erg Magc Parcl Sal). Th World Spac Cogrss, Washgo, DC, USA, 8 Aug.- 5 Sp., 99, IAF A.A. Bolo, Avao, Moor ad Spac Dsgs. Emrgg Tcholog h Sov Uo, 99, Dlphc Ass., Ic., pp Boo: A.A. Bolo, Th Dvlopm of Sov Roc Egs, 99,Dlphc Ass. Ic., p., Washgo,Eg). 74. A.A. Bolo, N Wa of Grao of Elcrcal Erg Spac. Rpor ESTI, 988, 9p. Sov Classfd Proc). 75. A.A. Bolo, N Roor Iral Comuso Eg. Rpor ESTI,988,75p. Sov Classf. proc),rus.) 76. A.A. Bolo, Suprsoc VTOL fghr-hlcopr. Rpor ESTI, 988, p. Sov Clas. Proc), 77. A.A. Bolo, N groud-ffc vhcl. Rpor ESTI, 987, 85p. Sov Classfd Proc), 78. A.A. Bolo, N roor hdraulc rasmsso for cars ad ohr machs. Rpor ESTI, 986. Sov Classfd Proc). 79. A.A. Bolo, Ivsgao of h a off damcs of a VTOL arcraf. Ivsgaos of Flgh Damcs. Mosco, 965, p Russa). Iraoal Arospac Asrac A # Eglsh). 8. A.A. Bolo, Thor of lfg od h corollal radal forc. Ivsgaos of Flgh Damcs. Mosco, 965, p.79-8, Russa).Iraoal Arospac Asrac A #Eg). 8. Boo: A.A. Bolo, Thor of Flgh Modls, Mosco, Assocao of Arm, Ar Forc, ad NAVY, 38p. 96 Russa) Nomclaur A s h pla marx lr prolm a s mmrs of marx A

18 8 B s p corol marx lr prolm s mmrs of marx B C s q ou marx C ar gh coffcs c s cosa F s h magud of h ouds for ach corollr F s fuco of al codos f s h corol forc vcor of dmso p f s h corol lar prolm H s Hamloa I s h fucoal ocv fuco), M, E, K ar dagoal squar marcs P s a -dmsoal uo marx Q s sa ghg marcs R s corol ghg marcs R x ) s orm of dsplacm T s fal m s m varal),, - oudar codo u s h vcor dfs h srucural rspos. v s a p-dmsoal vcor of corol forcs x s a -dmsoal vcor of sa gral prolm, x s h sa vcor of dmso lar prolm. x) s h al sa vcor xt) s h fal sa of h ssm. x, x - oudar codo s h vcor of modal dampg facors ) s a -dmsoal vcor uo coffc s gvalus of marx A =,x) s spcal fuco s h vcor of srucural frqucs