Optimal control of large space structures via generalized inverse matrix

C.C. Nguyen, Fang Xiaowen
[1988] Proceedings. The Twentieth Southeastern Symposium on System Theory  
m: IodepeDdent &YhJ slpam C b n M (lnw is a ant& SabQe that dccwplu tbe space structure into n independent sccmd-opder sutuystarr u m r d i n g to I ) crntrolled modes md cmtmlsescdrpdellldegreadeo tly. I t is d -l h r o r m Uut the l?E r l t e s cmtml and atemation spillover caused wben tbe amaeatiaaal crypled mdal m t r o l scbm? i s employed. rlx independent cmtml of each d e requires tbat the number of .cfwtars k ah is this paper, ~~p r q x s e a amM scheme that U o w s one to ust? a
more » ... ne to ust? a reduced number of actuators to control a l l modelled modes d c~t i m a l l~. J~I particular, the method of generalized inmatrims is en(played to iplement tbe actuetars sud, tbat ule u ' m p a l u e r of the d c s e d -l q system are ar dared as parsihIe to those specified ly the cptimal MFC. Chquter siraulatim of t h e m mntml-Q) a s h q~l y -t e d k is giioen. 1. IllIRoousRM me develcpmt of the space shuttle has the poesibility of constructing very large structures i n space for space aplaratiom. amtrol pmblens for Lss are attitude antrol and sbape rmtrol. cclpplex missions impose many stringent requiremu an shape a d attitude of the Lss, which lead the a n t r o l researchers to the cnrept of distributed active amtrol w!we severdl actuatas ard sasors are placed an the structure to in ader to optimize its performaace ard beharior. %e has been a mmiderable interest in the area of active control of large space structures (US) [11-[131. A number of control sdxmes were stubied, but they represent cne form r n -mother of rodal control t61. d n d a l control schemes are the carpled Ddal antrol Md the Wependent lbdal Space Ccntrol (ItW). '& fmmx uses an active codroller that ansists of a state c s t i m t a and a state feedback; the latter deoxplea tbe 1ss into n ixlepenQnt subsystems aomrdmg to n antrolled .odes md cmtmls ed.l PDde iaQpaden t l y by means of a aodal filter 1 5 I a d an optimal antroller. CaJpled Ip3dal control c a w omtrol .ad abservrrtian spillover that tollether can destabilize the Lss 1101. Msc does not have the spillover prcblap since each d e is cantrolled independently. trowever in order to -1-t the Msc the nmber of actuators is required to be hrse far a faithful Ddelling of the us. mis fact p-esents a h-damental linitatim of M S C since the required number of r h r a h is d i z a b l e . l b e main objective of this paper is to h l e D e n t the Msc with a milder requirement of the actuator nuber. In other words. we w i l l develop a m t r o l sdwne tbat uses a reduaed MIlrber of actuatcrs to antml a l l mo&elled in such a way that the pdes of the closed-locp systm are as closed as possible to the optimal mdes spxified by tbe I)6c rcheae. In particular, the method of generalized inverre matrices is -1 -f t x the hlenentatiar of Msc. equal to tbe armber of m 1 e d EuIks, Rly hi& far d faithful .odeL2q Of 1 -dpaCV StnrCtwPS. fi equal to the rnmrber of amtrolled modes whidl is usually very Watrix notaticns used in this paper is given. belaw: ~~ {NASA-CR-182336) O P T I H A L CONTROL OE' // 9 2 2 si;' g 0 ... 0 0 5 ... 0 . . . . 0 0 6 Tlx! Qsaip?im of a large f l d l e space structure L g i m by thc follcnriig partial diffaentd differential equaticns [31: n w aZ(P,t)/a tZ + Lu(P,t)f(P,t) (1) that rrst be satisfied at e!very point P of t h e ( h a i n D, where u(P,t) is t k displacement of h i n t P, L a linear differential self-adjoint -rotor of order 2p, expressing the system stiffnew, H(P) t h e distributed mss, Md F(P,t) the distributed control fora. Tbe displacement u(P,t) is subject to the bamdary aonditicas: Tju(P,t) = 0: i -I,? ,..., p (2) rkere T . 1=1,2, ...,p are Umar differential operata of order rarNydrm 0 to (2pl). Ibc d t e d eigenvalue pmbla is faculated by: (3) LQ,W = aJl(P) @)PI: ~1 . 2 , ... with thc banbrp coalitiam: = 0 ; i=lV2,...,p; F1,2,... (0 uhere A, ia the rtb eigenvalue and Q PI is the eigenfmcticm (solretimes dm lnown as M e Shape f associated with a, suppoee the o p a t m L is self-adjoint and positive definite, and all dgmvalua .re p s i t h e a d are adered bo tbat A1<$< ... . Since L in W d j o i n t , the eigenfurtiare u e atbogaral and tbcrefarmIbrmxmall& * h t b a t : ubere +(t) is the lDodal amrdiaate. Substitu-(7) into (11, multiplying both sides of tbe resulting expression by gS integratiq onr D ard epoloyirg (5) ard (61, *e &tab .. yet, +(+2 P(t) = fr(t);
doi:10.1109/ssst.1988.17129 fatcat:ouzzqryhs5ho7mszj2fmfts2v4