THE DESIGN METHOD OF ROBUST CONTROL BY FLEXIBLE SPACECRAFT V.Yu. Rukosky, S.D. Zelyko, V.M. Sukhno, V.M. Gluo Insiue of Conol Sciences, Russin Acey of Sciences, Pofsoyuzny 65, 7997, Moscow, Russi Tel:+7(95)334 87 3; e-il : uko@ipu.ssi.u Absc: The poble of ngul oion sbilizion by echnicl syses wih nonigi consucion is consiee. The conol is elize in he clss of iscee syses wih piecewise consn conol cions which cn le o ppence n gowh of he elsic oscillions of he obec consucion. The noion of he influence funcion of bse conol on he elsic oscillions is inouce. Using his funcion he sk of he subsyse foion of he inellecul ignosics n uning of he bse lgoih is sole. Soe esuls of igil siulion of he suggese eho of conol by ulifequency lge spce sucue e gien. Copyigh 5 IFAC Keywos: Lge spce sucues, oscillion, obus conol. INTRODUCTION AND STATEMENT OF THE PROBLEM Thee e ny ypes of copoun echnicl obecs h equie oienion of hei posiion ino spce. The equieens fo ecese of he el expeniue fo hei nufcue le o elsiciy of he consucion. As he esul hey e flexible ulifequency obecs. The ypicl exples of such obecs e he lge spce sucues (LSS) (Nue, e l., 985), (Kik, (e.), 99, 993, 996, 999), spce n subine oboic oules h he long nipulos links n long pylos. Such exoic obecs s ciil ehquke-poof uli-soie builings on oing founion belong o his clss of he obecs oo (Spence n Soong, 999). The peculiiies of consiee obecs e he exisence of in igi boy conining sensos n cuos of he conol syse. Tnslionl n oionl oions e efine by he cooines q n q of he boy-fixe fe Oxyz in he ineil spce. Ache o he in boy iionl eleens o blocks cn be flexible. Thei posiions e efine by finie nube n of he genelize cooines q. Fis of ll i us be noe h fuhe he specific peculiiies of conol by spce ehicles e king ino ccoun. Bu he os esuls e lso ue fo ll foeenione obecs. Dynics of he consiee obecs usully is escibe by Lgnge equions h cn be euce o he following fo of he finie-eleens oel (Nue, e l., 985) A( q) q + Hq + Bq= Q+ R, () T T T whee q= ( q is he ( ' eco of genelize cooines h e efine he posiion n, q) n ) configuion of he flexible obec ( n' = n+ 6, = T T q ; A, B e he syeic ( q, q ) T ) ( n' n') ices of he sses n igiiies; H is he ( n ' n') -ix of ping; Q= K( q ) µ is he eco of genelize foces; Kq ( ) is he ( n ' 6) - T ix of he cuos effecieness; µ = ( µ, µ ) T T T is he eco of conol cions; µ = ( µ, µ, µ 3) is he subeco of nslionl oion conol n T µ = ( µ, µ, µ ) is he subeco of oionl o- 3
ion conol by he in boy; R = Rqq (, ) is he eco of Coiolis n cenifugl foces. As ule he coponens of he (6 ) eco e he conolle cooines. Consiee in his ppe poble is eenion of he eco cooines q in he oin of sll isplceens fo hei esie lues q. Sll isplceens of he eco co- ponens q llow o use he lineize equion A A x x Kµ + =, (3) A q A x x B x = x, x = x, µ = µ K = o, µ = µ o A = A, i, =,, n K K = K e he sub-ices of he ices A* n K*. i Hee x i e he cooines of he in boy oion in he cse if ll consucion woul be igi; x i e iionl isplceens of he cooines x i ue o he elsic oscillions of he che flexible eleens. The cooines of he eco x= ( x i ), i=,,3, e consiee s Eule ngles. Sepion (4) of he cooines x i peis o epesen equion (3) in he following fo * A x+ Hx + Bx= K µ, () x = N ; µ s + ω s = G µ ; x = Ls; x = x + x, (5) T T T whee x = x = q A * * = ( x, x), x q, ;, K e x = x + x, (6) consn ices now. Vey ofen i is possible o ssue h in equion () he coponens of he ices efining nslionl n oionl ine-oion coupling e sll. I is li in he oion of he obec ino spce o in he oion unewe low elociies. In his cse he e Hx is sll oo. Then equion () is iie ino wo nlogous syses h in he fis ppoxiion e inepenen. Ech syse fe soe nsfoions hs he following fo whee oion n x = x, x = x fo oionl oion. A i = Ai i fo nslionl whee s= ( s ) is n -eco of nol cooi- nes; ω = ig( ω ), =, n; ω e funenl fequencies of he elsic oscillions of he obec; J µ = Kµ ; J = ig( J, J,J 33) ; N = A J ; L n G e he ices efine in (Gluo, e l., 998). If gs-e engines o hn-wheels e use s conol eices he syse of equions (5) y be iie ino hee inepenen ol-physicl oels (MPM) (Gluo, e l., 998) ech of h epesens he obec oion x i, i =, 3, wih espec o ny of hee ohogonl xes of he ineil cooine syse (fuhe subscip i will be oie). In he scl fo MPM is wien s follows: M : x =, x + ωx = k, x= x+ x, x = x. (7) n = Fo efinieness in his ppe oionl oion of he obec is consiee. A soluion of he sk of he lgoihs synhesis of enegy econoic n obus conol by nonigi obecs escibe by equion (3) is suggese. The equieen o he econoic conol is efine by liie soe of he enegy fo conol une coniion of long-uion opeion of spcecfs on he obi. The equieen o he obus conol is efine by he fc h i is ipossible o clcule he exc lues of he flexible spcecf pees une ehly coniions.. TRANSFORMATION OF THE OBJECT S MODEL TO THE MODAL-PHYSICAL FORM As he fis sep of he suggese eho of esigning conol syse is he nsfoion of he equion (3) o he ol-physicl fo (Gluo, e l., 998). Fo h he cooines x i of he eco x e epesene s he su of wo coponens n xi = xi + x i, x i = x i, i =,, 3. (4) = The obec oupu x = x + x is esue by ngule senso wih igil oupu z= z+ z whee z = xk [ ], z = xk [ ], k is iscee ie. 3. THE CHARACTERISTICS OF THE REGULATORS INFLUENCE ON THE ELASTIC OSCILLATIONS As isinc fo (Kuo, ) le us inouce he noion oel of he egie's se of he oinn oe inse of he se of uonoous oels of he isole elsic oes. I is efine s MPM M,( =, n) in scl fo (7) n i is isinguishe by picul cobinion of he eco coponens iniil coniions lues x = ( x ), when x x n x, x. This oel M fo beiy is clle s he oel of he obec's -egie ( =, n). Fuhe he iscee conol syses h e wiely pplie fo spce obecs e consiee. In such syses he sey-se oion is epesene by s-
ble lii cycle Γ= Γ[(, uz )] wih peio τ Γ n issible pliue. The influence of he conol cion u [ ( z, )] on he oscilling coponen x () of he oel M cn be eeine wih he help of qusi-enelope ρ (, ) = Enx [ ( uz (, ),)] of he nsien pocess of he coponen x ( ) on he ie inel τ (cie inel) belonging o he lii cycle peio τ Γ when u [ ]. This enelope cn be efine by he funcion ( ) ρ (, λ ( )) = e λ. (8) In esigning conol syse enelope (8) is clcule wihou ouble using equions (7). Bu he fequecies ω cn no be known excly n oeoe hey cn chnge uing he cie life of he obec. So i is necessy o he he eho of enelope (8) clculion in he pocess of he obec opeion. In his cse he following eho cn be suggese. Accoing o his eho se Zs = {[ zk ]} is use. I cn be obine fo exple by pocessing of he cooine zk [ ] by he eho of Kln filion (Eilo, e l., 4). The oupus of he Kln file e z [ k ] n z [ k]. Bu i is ppopie o obin he coponen zk [ ] s he iffeence z [ k] = z[ k] z[ k] since he coponen z[ k ] on he Kln file oupu coneges oe quickly n oe ccuely in cope wih he coponen zk [ ]. Moeoe he coponen zk [ ] on he Kln file oupu conins only soe oes of he elsic oscillions. Fuhe he opeion of ecificion, z [ ] [ ] k = z k is use wih subsequen selecion of ll xiu lues z [] l. The se Z s is nsfoe ino he se Z = { z [ l] } oinly wih he se of he coesponen insns T = {}. l Define by he ses Z, T he funcion f = {( lz, ) T Z z = fl ( )} is subece o wo-sge ppoxiion. A he fis sge his funcion is eplce by he ppoxiing polynoil P () = q ν pν ( l ) ν=, whee e consn coefficiens, l is he fis eleen of he se T. Appoxiion is elize by he eho of les sques n funcion Polifi in MATLAB. The secon sge is he poceue of iniizion of he funcionl τ λ ( ) τ Jλ = P( τ ) e P( τ) τ. Clcule he lue τ λ ( ) efines he e of he coponen x pliue chnging. The sign λ efines he chce of he oscilling pocess: λ < i coneges ( x (, ) x ); λ > i ieges ( x (, x + ). p ν ) If λ ( ) ε hen fo sign[ λ ( )] =± he egulo ffecs on he elsic coponen wekly ( x (, ) x ). So fo ny consn lue of he pee he influence of he egulo on he oscilling coponen x () of he oel cn be efine by he single nube λ. Vying he pee, epeing fo ny new lue ν * = λ ( ) copue siulion of he obec sbilizion egie wih he oel n clculing he inex M * M of he qusi-enelope λ soe influence funcion λ = λ ( ) will be obine. This funcion eflecs he influence of he bse lgoih uz (, ) on he elsic oscillions fo -egie. The oliy of he influence funcions Λ = { λ ( )} ( =, n) is use fuhe s n infoionl inex fo he subsyse of he inellecul ignosis (Duboin, e l., 3) of he oscilling coponen coniion. The exple of liie nube of he influence funcions λ ( ), =,3, fo he LSS n iscee nlog of PD-lgoih sbilizion wih isceeness peio Т (Kuo, ) (wihin peio T conol cion = cons) is shown in Fig.. The coefficiens of oel (7) fo n = 6, I=5 kg, I =67 kg e inice in Tble. i= i= i=3 i=4 i=5 i=6 i f i [ Hz].3.6.9..5 3. ω i 8.7.5.94 3.8 5.7. x k i Tble. 8.5 5.6 4.8.5.4. Anlysis of he influence funcions kes i possible o selec wo oins of cie ( T,5 s) n Fig.. Toliy of he influence funcions. neul ( >, 5 s) influence of he egulo on he elsic oscillions. An i is cle in wh oin of
ying pee one of he oinn oes ( =,, 3 ) is he cuse of he LSS oion insbiliy. The sk of he influence funcions λ ( ) nlysis cn be consiee s peicion of possible ciicl egies of he LSS ynics (huning phenoenon of he egulo by he elsic oscillions n ohes (Rukosky n Sukhno, 973)). These funcions cn be use boh in esigning conol syse n fo elizion of obus conol by he obec (uning of he pee uing he fligh). The os ineesing sk is he ls one. Le in he syse une influence of he conol cion u [ ( z, )], ( [ in, x] is he noinl lue of he une pee) -egie occu which is esie using he se Z = { z[ l]}. I is equie:. To efine he nube "" of he oinn oe on he bsis of he fequency ω, =, n, ienificion poceue;. Using he nube "" i is necessy o choose fo he oliy Λ = { λ ( )} h e soe in he copue he influence funcion λ ( ) n o efine new lue [, wih eg o necessy in x ] ( sign [ λ ( )] = ) n sufficien ( λ [, + ] ) coniions which he () = ( λ ) in conol cion u [ ( z, )] gunees xiu + e of he oinn oe ping. Hee, e he bounies of he oin in which necessy coniion ( λ ( ) ) of he coponen x () ping is fulfille. The soluion of his sk elizes he synhesis poceue of he subsyse of he bse lgoih pie uning. The gol of his uning is he oinn oe ping wihou iionl enegy consupion. 4. DESIGNING SUBSYSTEM OF THE BASE ALGORITHM PARAMETER TUNING In he synhesis of he subsyse of he ying pee uning he ses Z n T e use. The lues λ (, ) e obine wih he help of escibe eho of he oscilling coponen z qusi-enelope consucion. This poceue is elize ech cie inel τ of he lii cycle. In he egie of he oinn oe he lues of he os iffeences [ ] = { [ l] l ]} of he [ cen eleens of he se T = {} l e coincies ppoxiely wih sei-peio, 5τ of he oscilling coponen h hs he xiu pliue. L Afe he opeion of eging τ = [ ], L = L = it, he fequency of he oinn oe ω = πτ is efine. Fo ienificion of he oinn oe nube "" he iffeences ω = ω ω ( =, n) e inesige n i is ccepe h = fo he coniion ω = ω ω = in. Fuhe using he nube "" of he oinn oe i is necessy o choose he coesponing influence funcion λ ( ). If λ ( ) > i is equie o selec he new lue of he pee s i ws wien elie. In he cse when = he inex λ ( ) he inensiy of he oscilling pocess x () cn be high o low (he pliue of he oinn oe cn be lge o sll). High inensiy cn le o insbiliy of he obec oion. Becuse of his, inensiy of he oscilling pocess is esie by he en lue z of he se Z eleens. The en lue z is cope wih he issible lue z * n if * z > z he uning (selecion) of he pee us be elize. If * z z he uning of he pee is no = equie., - 4 Fig.. Bse egie of he LSS sbilizion n oupus of he infoion oule. Discee nlog of he PD-lgoih T = s ws use. I is cle h signls λ () n () inice he pesence of he oinn oe () = n is iegence ( λ () > ). Signls, () λ z () n z u [ (, )] = s, x() λ (, ) =, s = 5 5 5 3 35 4 In Fig. he exple of he bse conol elizion (n = 6) is shown. (signl in Fig. is no epice) wee obine by pocessing of he cooine z[ k] ( ) of he lii cycle. A > τ he signls τ = s λ () n z e eine fixe unil he en of he cuen lii cycle. on he cie ps
Obec x Anlogue-igil conee Digil senso k [ ] Kln file z = x[ k] z - z Infoion oule of he subsyse of ignosis n uning of he pee ( ) λ ν () z Conol eice u Bse lgoih u= u(, z ) Sech lgoih of op (pecise uning of ) D bse Λ = { λ ( )}, z n subsyse of ough uning Fig. 3. Block schee of he conol syse. Descibe poceue of he pee uning is elize on-line using logicl opeions n he influence funcions. This uning is quick bu ough. As ule he new lue is no equl o is opil one + fo h ( ) becuse op λ op = λin [, ] he influence funcions e clcule in he pocess of he conol syse esigning on he bsis of esigne pees of he elsic oscillions. So i is ppopie o he n opil subsyse fo pecise uning of he pee. Fo h one of he well-known lgoihs of n exeu sech (Ksosky (e.), 987) y be use. In ou cse he iniu of he inex λ ( ) in he + oin [, ] is ken o be he exeu. In he opil subsyse he es signl is genee n el inex λ ( is clcule in he infoion ) oule. The sech is coplee when λ ( ) = λ. Afe h he lue ν in is eine consn n he oinn oe pliue is ecesing wih xiu e. Bu he pliues of ohe oes cn incese s long s one of he will becoe s he oinn one. In his cse he pocess of he pee ough uning is ecue. ν op By his ens he suggese syse hs hee loops. Is block-schee is shown in Fig. 3. The fis, in, loop consiss of he obec, he nlogue-igil conee, he conol eice n copue h elize he bse lgoih. The in loop gunees equie quliy of conol ll lues of he pee [, ] poie h he elsic oscillions x () e sll. Infoion oule of in x he subsyse of ignosis n uning of he pee esies he inex λ, nube of he oi- nn oe ''" n he inex z using he elsic coponen zk [ ]. The secon loop elizes he ough uning of he pee n he hi one elizes he pecise uning of his pee. 5. COMPUTER SIMULATION OF THE SUGGESTED SYSTEM Suggese syse ws epouce in MATLAB- Siulink king ino ccoun nonline chceisics of he iue senso n conol eice. The lues of he obec pees e epice in he Tble (n = 6). An i ws ssue h he funenl fequencies n coefficiens of excibiliy cn be gien wih he =, =, = =,85 =3 3 =, = λ(, ) 658 687 9 9 545 675 9 [s] Fig. 4. Dynics of he LSS sbilizion pocess.
eos (bou %). The iscee PD-lgoih (Kuo, ) wih ying pee T ws chosen s he in lgoih of he sbilizion. The iniil coniions of he sbilizion egie wih espec o he «igi» oion wee: x =,5, x = n T = s. A h he fis oe ws s he oinn one. Ohe fie oes he eliely sll pliues. In Fig. 4 soe esuls of igil siulion e shown. Becuse of high uion of he sbilizion pocess obseion (T obs = 3 s, τ Γ = 5s, τ = s ) he oupus λ n e wien () () only on ypicl ps of he pocess h e connece wih chnging in he coniion of he LSS ynics (fo exple, chnge of he oinn oe nube). The Fig. 4 shows h he fis cie p of conol i ws efine ou insbiliy of he fis oe λ( T =, ) =,5 (Fig ) n wih eg o he = λ T he signl n he influence funcion ( ) lue of ( ) pee T ws chnge o =, s T =, s ( λ ( T =,) =,35< n i is he iniu λ T ). So he fis oinn oe bece sble. A 4s he fis oe pliue is bou zeo. Bu he hi oe bece unsble (Fig ) n fo he insn 545 s i ws s he oinn oe h is = 3. This fc le o he nex chnging T fo T =, s o T =,8 s (Fig. ). Theefe he hi oe begn o conege. A 675 s is pliue ws bou zeo. Now he secon oe ws excie (he signl = ws ppee) n his le o he nex chnging T fo T =,8 s o T =, s (Fig ) n o conegence of his oinn oe. Fuhe hese chnges of he pee T e ecue uing ll uion of he LSS sbilizion pocess. A ohe lues of he obec pees he oe of he oinn oes ppeing will be iffeen. 7. CONCLUSION The esuls of his ppe cooboe possibiliy of he high efficien conol by lge iensionl flexible echnicl syses in he cse when i is no possible o choose he consn pees of he bse conol lgoih which coul gunee he conegence of ll elsic oes of he obec consucion. In suggese eho he eleens of he inellecul ignosis e use fo elsic oscillions ping. This ping is elize wih he help of he pie uning of he bse lgoih pee n wihou iionl consupion of he enegy fo conol. This eho cn be pplie o ny ypes of he spce obecs. Bu i shoul be noe h he cse when he fequencies of he elsic oes e close o he funenl fequency of he igi consucion soe oubles cn occu. ACKNOWLEDGEMENT The wok epoe in he ppe is conibuion o Poec 3--6, fune by Russin Founion fo Bsic Resech fo which uhos e geful. REFERENCES Nue G.S., R.S. Ryn, H.N. Skofiel, J.I. Sis (985). Dynics n conol of lge spce sucues // Aeospce Technique. V. 3. 6. С. 7-8. Kik, C. (Eio) (99, 993, 996, 999). Dynics n conol of sucues in spce. I-IV. Poceeings of he Inenionl confeences on ynics n conol of sucues in spce, Copuionl echnics publicions, Souhpon Boson. Spence B.F., T.T. Soong (999). "New Applicions n Deelopen of Acie, Sei-Acie n Hybi Conol Techniques fo Seisic n Non-Seisic Vibion in he Vibion in he USA", Poceeings of Inenionl Pos SMIRT Confeence on Seisic Isolion, Pssie Enegy Dissipion n cie Conol of Vibion of Sucues, Koe. Gluo V.M., S.D. Zelyko, V.Yu. Rukosky, V.M. Sukhno (998). Spil Angul Moion of Flexible Spcecf. The Mol Physicl Moel n is Chceisics. // Auoion n Reoe Conol. Vol 59. N, P. P. 78-738. Kuo I.N., (). Suying Sbiliy of he Flexible Spcecf wih iscee Conol Syse. Auoion n Reoe Conol. Vol. 6,. pp. 964-977. Eilo T.V., V.M. Sukhno, A.S. Eilo, V.G. Boiso (4). Recuen esiion of he ngle oion cooines of flexible obecs of eospce echniques. Aeospce Insuenion. V. 6, P. 63-68. (In Russin). Rukosky V.Yu., V.M. Sukhno (973). Specific ely conol of flexible sellies. Poceeings of he 5-h IFAC Syposiu on Auoic Conol in Spce, Geno. Duboin V., S. Subboin, A. Boguslye, V. Yzenko (3). Inelligen Mens of Dignosics n Peicion of Relibiliy of Ai engines. Join-Sock Copny "Moo Sich", 3.-79 p. (ISBN 966-78-59-7). (In Russin). Ksosky A.A. (e.) (987). Refeence book on he uoic conol heoy. M.: Nuk. (In Russin).