Independent Modal Space Control (IMSC) is a control scheme that decouples the space structure into n independent second-order subsystems according to n controlled modes and controls each mode independently. It is well-known that the IMSC eliminates control and observation spillover caused when the conventional coupled modal control scheme is employed. The independent control of each mode requires that the number of actuators be equal to the number of modelled modes, which is very high for a faithful modeling of large space structures. A control scheme is proposed that allows one to use a reduced number of actuators to control all modeled modes suboptimally. In particular, the method of generalized inverse matrices is employed to implement the actuators such that the eigenvalues of the closed-loop system are as closed as possible to those specified by the optimal IMSC. Computer simulation of the proposed control scheme on a simply supported beam is given

Fang, Xiaowen

Nguyen, Charles C.

English

NASA Technical Reports Server

I 
OPT= CONTROL OF W G B  SPACB STRUCTURES 
VIA GQmRALIZBD I”BRSB IUTRIX 
Charles C. Nguyen, A~mociate Rofessor 
and Xiaoren Fang,  Research Asmiatant 
Department of Electrical Baginaering 
Catholic Univermity of America 
Washington DC. 10064 
m: IodepeDdent &YhJ slpam CbnM (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 mtro l  scbm? is 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 Uows one 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 in- 
matrims is en(played to iplement tbe actuetars sud, tbat ule 
u’mpaluer 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 
an t ro 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 rn- 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 f i l ter  
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 l ini ta t im of M S C  since the required number of 
rhrah is d i z a b l e .  l be  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  s- 
i;’ 
g 0 ... 0 
0 5 ... 0 . .  . .  
0 0  6 
Tlx! Qsaip?im of a large f l d l e  space structure L gim 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 the ( h a i n  D, where 
u(P,t) is tk displacement of h i n t  P, L a linear differential 
self-adjoint -rotor of order 2p, expressing the system 
stiffnew, H(P) the 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) L Q , 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<$< ... . 
Since L in W d j o i n t ,  the eigenfurtiare u e  atbogaral and 
tbcrefarmIbrmxmall& * h t b a t :  
(5) 
(7) 
ubere +(t) is the lDodal amrdiaate. Substitu- (7) into (11, 
multiplying both sides of tbe resulting expression by gS 
integratiq onr D ard epoloyirg (5) ard (61, *e &tab 
(8) 
.. yet, +(+2 P(t) = fr(t); P1,2,... 
L A B G E  N88- 13907 
SPACE STRUCTURES V I A  G E N E R A L X Z E D  INVdRSE 
M A T R I X  ( C a t h o l i c  Uri iv .  of America) S p 
A v a i l :  NTIS HC h 0 2 / ? l P  A01 C S C L  39b Unclas 
C3/63 C 1 1 U 2 2 5  
https://ntrs.nasa.gov/search.jsp?R=19880004525 2020-03-20T09:03:59+00:00Z
I 
are (1x2) gain matrices. 
aJs\rpe the folluuinq form: 
subst i tutb (16) and (17) into (20). we find that C r u s t  
0 
Gr [,,, gr22] : =51,2,...,n (22) 
should be d e t u  such tbat 
axd gP FOP W t M  mtrdl, g a  . the following quadratic cost unction is mininired (linear 
regulator pmblem): 
n 
J x  &r 
where Jr = (.'p,., + Upr$ )dt (24) 
Q and 5 are positive semidefinite and positive definite 
2ightitg matrix, respectively, associated with tbe rtb mie. 
Iha foolof Gr given Ly (22) requilw tbat F$ (u18\1De tbe fadm 
d 
gim brla: 
4 
where $(t) is the solutiocr of the Rimti eqwtiar: 
T 1 irk) +A; A r r  K + I( r d r - Qr :1=1,2,.. .,n (27) 
(19) ud (20) reposent the coacept of E. In 
order to implement (20), the wdal  s ta te  rectors xr for 
~ l , ?  ,..., n llrst be available f a  -t. In [5] a udal 
fi l ter  was develqed to Fpvide an a t i m t e  of thc .oQ1 state 
vectors. Since t h i s  paper focuses an tbe problem of the 
implementation of actuators, we MSW that the mdal state  
rectas x&t) are available fa :  the rtate feedback i n  order to 
avoid the acEpledty d getting involoed in the state cstimaticm 
@le that can wel l  be the subject of a mbequent paper. 
since it is imasible to antral f a r e  at every point in 
the darain D, the distributed m t r o l  force is realized by m 
bcn) discrete point faoe actuatacll applied at I points P1' P2, .... P in tbe &main D as give0 tela: 
fP,t) \6(F" i )  Fi(t1 (30) 
i =  
wbere 6(Fp.) ia a spatial Dirac Delta fmcticrr and Fi(t)  is the 
f a -  Clyplid by tbe ith e t a  cm tbe p i n t  pi. 
Nac m b t i t u t d  (30) into (10) yields 
(31) 
Frca the pmperty of the Dirac Delta function, (31) can be 
rerhredto 
\ q(t) = 1 (Pi) Pi(t) .  (32) 
m 
i .1  
then wing (32) the relatim between Fit)  and W ( t )  CM be 
--=dby 
B(ti-l) = Oh: r=1,2, ..., n 
(35) 
fa ~ 1 , 2  ...., n. 
4. VWBLgl- 
iqleaentaticm of the IIw: schane is illustrated in Figure 1 
*re the optimal state fcedbadr law is defined by 
B ( t )  = G x(t)  (38) 
(39) 
md G = Block diag(G,, G2, . . . , % 1. (40) 
1& optimal soluticm for the W scheme was obtained in 
Sectim 3 as: 
The opt- soluticm is achieved i f  (42) is satisfied. h order 
to .rkc U ( t )  to Qt(t), the matrix D in Fig. 1 i. b i g n d  
Nch thrt W ( t )  - H ( t ) .  Itcr fig. 1 we rlx, have 
Haw sutstituting (42) into (341, we obtain 
In (43) to oake W ( t )  = €i(t), it is obviars that D is designed 
suil tbat 
W ( t )  = G x(t) .  (at) 
F(t)  = D H ( t ) .  (42) 
W ( t )  - B D H ( t ) .  (43 1 
B D = I b  (44) 
holp the structure of B as given by (35)-(37), each (2i-l)t.h 
r ( ~  (c&l row) of BD, for i-1.2 ,..., n is a row of zerce. We 
realize that (44) can rmer be satisfied. tkwwer, not* that 
each odd row of W ( t )  is also a row of zenx, if we define 
and 6(t) = [D2 D q . . .  D2J 
(45) 
(46) 
where B i b  the i t h  d B Md 4 the i t h  m l m  Of D, then 
CaarSM a ~ t r i x - D - ~ ~ ~ b  that 
vi11 ersure that W ( t )  = = B 'p t) .  It IS noted that i f  (47) holds, 
then BD is a modified i b t i t y  matrix of order (2nxZn) whose main 
diwcrd eleuents are 0 a t  the (2i-1,ti-1) positicm and 1 at the 
(2i,2i) positicm for  i 4 . 2  ,..., n. 
(he M o w  solutiop for (47) is to cha;rse D such tbat 
tbuem f a r  the inverse d B to exist, B nust be a nrrsirqular 
stauare mtrix, requiriap that the nlmber of actuators be equal to 
the mnber of xcdelled Ddes ( 1 ~ 1 1 .  Since the nuibex of mu3elled 
.odes is usually very + for a faithful modelling of ISS, the 
W e d  number of actuatas is practicdlly unrealizable. 
5hc pmblem F i d d  in this paper can be formdated as to 
design the matrix D for a rmsqure matrix B (m(n) such that (47) 
is satisfied as well as m i b l e .  h otber words, if this can be 
daw. then the LSS can bc cantrolled by a reduced number of 
q t u a t a s  such that tbe dosed-loop system i s  as optimal as 
passible. 
5. lRIllIlgsu;r 
(47) 
b = K .  (48) 
M : - b i $ g  the follarin;l equaticn: 
u k e  V'(t) and F(t)  are matrices misting of even mrs of W ( t )  
.nd Fit), respectively. If the (NOD) mtrh B has rank m, then 
tbe solutim fa (49) that linimLizes the weighted ~ I Q  of ~ f m  
W ( t )  = aF_(t) (49) 
, 
(51) 
(52) 
T b m m  1 : W i d e r  a large flexible space structure whose 
&scriptian md solutim are given by (1) 4 (111, respectively. 
If the operata t is self-adjaint, thar tbsc Bcisb control 
s&ms with m (r(n) actuators that is subaptiml with respect to 
(24) i n  tbe sense that the closed-loop eigemlues u e  assigned 
as clwed w pcwible to those optimal eiwmalues W i e d  by 
m. 
Proot: A control scheme with a fecuced m n k  d actuators wplld 
be apt- i f  b a u l d  k selected to be a wt inverse of B in 
(47). 
can asnm? that B has rank rn rinca the dircrrte actuators apply 
p i n t  foroes at m distinct points Pl, . Ekcm (12) it is 
w l 1 - k  that an (nm) mtrix (m<n%*r- rn dot8 not 
possess any right inverse. Consequently B does not have any 
right in-. 
(51) min.i~+ (501, selectirg a mtrix D - b w i l l  lini.iLe the 
difference-BD-I ,* rukin~ (47) be satisfied = *ell as parsible. 
selecting D = 1 also make the closed-loop eigenvalues as 
identical as pcssible to thme specified $. Msc. Thus there 
exists a ontrol scheme with a reduced n m k  ab actuators that 
is s u b o p t s  with respect to, (24). Q.E.D. 
6. RUSU 
Ib illustrate the pmpaped antml sdwse II? amsider tbe cmtrol 
of a sinply mpprted beanwbe QMpic is given by the Mer-  
BMleulli partial differential oquatim: 
Ilcrc.rrar hwd QI the tam of 6 gian ir (455) md (31) 4 
Acmrding to Lepma 1, bgcausE*f(t) u gim in 
m(abx')u(x,t) + m(g3at2)u(x,t) = f k t )  (53) 
&ere for shplicity we set tbe mass D, the rrent of inertia I, 
tbae &us of elasticity E md the 1- d the bem to unity. 
Ibe coaditicm fo this sinply bemn are: 
u(O,t) = u ( l , t )  = 0 (U) 
a '/a i w 0 . t )  = a xt(i , t)  = o (55) 
Xk = (kr) (56) 
Ibe mlutiars for tbe "QIF"z" -em 6- by: 
ami = Bsin (kd. (57) 
Faupoed control scheme. Renrzts dml tbat tbe dynalia of the 
closed-lccg SyStm is affected rmsiderably by the rmpber lrnd 
placenent of actuators. suppcse we CarriQ 20 aDdelled modes 
and divide the whole length of the beam into 20 sections 
specified by x(k)=k/2l f o r  k4,2 ,..., Zl. k expected, rimlation 
results sbar tbat in every ase the mnber d rtrble c l d - l o o p  
eipeapalues inerezws with an iacreese in a t o r  nurber. me 
nunben of stable do& loop eiwmalues whm \IS@ 1, 
2, 3, 4 actuators are 5, 18, 19 and 20. respectively. The 
r e m a i h g  eigenvalues are pn-e imginaq -ex amjugate pairs, 
t h  c a w i ~ ~  m instability but unranted add l l a t iw .  It is 
also faard tbat the actuator locaticos E&mizhJ tbe nunber of 
closed locp eigenvalues are centered arorpd both ends of the 
him, nmdy at x-l/2l AMI JPZO/Z~. fig. 2 ~psents the case for 
aw rtuata. As we see, the rwber of stable eigenvalws f a l l s  
W t e r  s a a t i m  was dare to test the performance of the 
4 
d f r t b c r t u r t ~ K J v c ! s t m r d t h e c s r t f f o i * ~ .  similar 
d t a  umre ObtJIWd fa th caaea of 2, 3 md 4 rturtuu. 
W e  1 porrfdcr tbc dcud 1- eigemvdws f a  IKX with 20 
rhytnr ad tha amtrol actme with 4 r t u r t u u .  Am 
tbe table abmm wen with a reduced number of actuator8 the 
gmpc#d m t r o l  retrcr us- e4envalues thnt u e  ocry clam 
to thoa .p#ified by th? optimal m. the 
k e a  lRsBIt f a  the case of D(x: with 20 actuators ud the 
DrcQosed m t m l  rchpe ritb 4 mutars, respectively. *hen the 
b e a i s e x c i t e d b y m i r p i l n .  h t h e f i g w e s s t ~ ~ , t h e p - c p o s e d  
OntIul sdreae with I mbxd Mpber of aCtuata-9 perfar6 .B 
u2u 8s the Insc with 20 actuators, boingirg the beam .wement 
darn t o m a f t e r  5 aecnds. lh time resparres of .ode 1 md 3 
u e  g'im in figures 5 rd 6, respectively. We note that the 
ti.e rupxlse of ude 1 d tbe ontrol schepe is dlmost the sane 
m that of m. 'Item M 
tiD? resrame of .ode 2. me difference is rbailt hlo- . 
figures 3 ud 4 
insignificant differences in !he 
7. canuaas: 
In this paper, we f i r s t  stmans& ' thethecryofMscinthe 
antext of optimal antrol. lben the principal limitaticn of 
USC was pointed out in terms of the required number of 
r t ~ r t a r ,  be* q w l  to that of mdelld oodes. After that, ue 
ptoposed I m t r o l  .drerc which aaployr a reduced n u k r  of 
actuators to  caatrol I large space structure that i!c self- 
adjoint. Cosplter siulatica drowed that the nunter and placement 
of actuators play an -tar& role i n  the stabil i ty of the 
dosed-locp systemr. 'Ibe research of this paper CM be exteaded 
into stuiying the optirinticn problem of actuator placerent Md 
designing a weighting matrix S that naximizes the nmber of 
stable closed-loap eigenvalues. The problem of arbitrary 
assigrrent of e i g e n v a l w  (81 can also be addressed for the 
amtrol of us. 
8 .  Ammummr 
9. - 
1. Balas. H.J., "Actipc Control of Flexible Systcrs," J. 
mtidzaticn Theory &plications, Vol. 25, tb. 3, w. 415- 
4.36, July 1978. 
2. l i i n d b e r g J ~ . , R . E . r d ~ , R . U . , " ( h t h e M . l p b e r a n d  
Placement of Actuators f a  Independent M a l  Space Control," 
rlarrnal of QlidarlCe d Control, Vol. 7, No. 2, 1984, pp. 115 
221 f 
3. neirovitd, L., halyrical kthcds in Vibraticos, Ihc M l a n  
m, Toronto, 1967. 
4. Mmvitch, L. and Bar&, R., "Optimal Control of Damped 
Flexible CYrm0Cpic Syst-.'' Jatrnal of Guidance md Ccntrol, 
5. keirovitch, L. ird %aruh, H., 'Ihe I w l m t a t i c a  of Jicdal 
Filters for Control of Structures," Journal of Guidance urd 
C a o M ,  Vol. 8, WO. 6, 1985, pp. 707-716. 
6. kirovi tch,  L. and ot, 1. "nodal Space Control of 
Vol. 4, No. 2, 1981, pp. l57-163. 
Distributed m c  *t-,* Jarrnal of aidma alxl c€&.ml, 
vol. 3, No. 3, 1980, pp. 218-126. 
7. ca, Il. a d  Heimrritch, L., -optimal nodal space Coatrol of 
F l d b l e  oymeccpic systea," 3annal of Q&mce ami a t r o l ,  
Vol. 3, No. 3, 1980. pp. P8-226. 
8. &yen, C. C., "Arbitruy Eigenvalue Assigumts for Lineat 
Time-VaryinCr I(ultivarirble Cartrol Systems," Int. Journal of 
Ccntrol, Vol. 0, NO. 3, op. l(K1-1057, 1987. 
ONGmAI., PAGE 1s 
9. IlPwQ, c. c., n@84n%-M%- fa 
Unmr *Varying Wltivuiable gylt-," to qqnr kr Int. 
Jcurml d Ctmtml, W. 46, &. 6,1987. 
10. C. C., rd bu, A,, Pttributd ktirr Qntrol of 
Luge Tladble sprea S~KWUCU,~ lRSA Pmmm b t ,  Otmt I 
tw 5749, mvdw 1986. 
11. lbum c. c., kd Lee, T. w., "Design of State B s t i u t a  fa 
a QME of T1.bvuyirp I*lltiwiable Syat-," ?ransactian 
of h a t i c  ccmtrol, K-30, Ib 2, p~ 189-182, 1985. 
12. Noble, B., rppucd LiLoeu Algebra, Prentice Hall, New 
Je2Yey. 1%9. 
13. m, A. P. awl White, C. C., @thm Systea Ccnlml ,  hd 
Riiticn, Rentice Hall, Ner Jersey, 19n. 
F i g .  1 :  I m p l e m e n t a t i o n  o f  I M S C  
. .  I 
F i g .  2 :  R e l a t i o n s h l p  between t h e  n u m b e r  
o f  s t a b l e  c l o s e d - l o o p  modes and 
a n d  t h e  number o f  a c t u a t o r s  
\ 
5 
' '. 
1.0*+881 ' 
-0.0014 + 8.1099i -8.1047 4 1.194Ji -0,0001 + 1.9471i -0.0009 + 0.917Oi 
-0.8014 - 8.1099i -0.8147 - 1.1911i -0.0001 - 1.9171i -0.l009 - L9171i 
-o,ttai + t . 1 1 ~  -1,tou + i.uiJi -O.OOO( t 1.5091 -t.tooa + 1 . 7 ~  
-8.8t18 - 1.11951 -1.t019 - 1.41121 -0.0004 - 1.16191 -0.1001 - 1.19941 
-8.0014 + 1.1IIIi -0.0851 + 1.6610i -0.0007 + l.lY77i -0.0007 t 1.61171 
-0.0014 - 1.8IIIi -0.t051 - 1.6110i -0.0007 - 1.1977i -0.0007 - 1.6111i 
-0.0011 + 8.1579i -0.t051 t 1.9314i -0.0010 + 1.1511i -0.0001 + 0.4116i 
-0.0011 - 8.1579i -0.0051 - 1.9141i -0.0010 - 1.15lli -0.0006 - 0 . 4 0 6 i  
-0.0012 + 8.2467i -0.0055 4 1.2107i -0.0011 t 1.5166i -0.0006 4 0.155Ji 
-0.0011 - 1.1467i -0.1055 - 1.11011 -0.0011 - 1.5166i -0.0006 - 0.1551i 
-0.0015 4 1.J55li -0.0057 + 1.5266i -0.0010 t J.3107i -0.0007 + 0.1417i 
-0.0015 - 1.155Ji -0.0057 - t.5166i -0.0010 - 2.1107i -0.0007 - 0.1467i 
-0.0011 + 1.1116i -0.0051 t 2.I5lJi -0.0001 + l.9n4i -0.0007 + 0.1579i 
-0.0011 - 1.4116i -0.0051 - 1.15JJi -0.oooa - 1.9144i -0.0007 - 11.1579i 
-0.0011 4 8.6117i -0.0060 4 J.1971i -0.0007 t 1.66IOi -0.0011 4 0.0099i 
-0.0040 - 1.6117i -0.0060 - J.1971i -0.0007 - l.66IOi -0.0011 - 0.0099i 
-0.0041 + l.7991i -0.0061 t 1.56JYi -0.0001 t l.4llJi -0.0004 + 0.0395i 
-0.0011 - 0.799ii -0.0061 - J.56J9i -0.0001 - 1.4JlJi -0.0001 - 0.0195i 
-0.0045 4 1.9170i -0.0061 4 J.947Ii -0.0009 4 1.1911i -0.0005 + 0.0111i 
-0.0015 - 0.917Oi -0.0061 - 3.W1i -0.0009 - l.l¶Ui -0.0005 - 0.0111i 
I H S C ( 2 O  A c t u a t o r s )  P r o p o s e d  c o n t r o l  
T a b l e  1: C l o s e d - l o o p  e i g e n v a l u e s  c o r n p a r i s i o n  
* 
s c h e m e ( 4  A c t u a t o r s )  
F i p .  3 :  Beam m o v e m e n t  when t h e  o p t i m a l  
I # S C  i s  a p p l i e d  w i t h  20  a c t u a t o r s .  
I i fJt ' i  VI^. t i l  1: 
F i g .  4 :  Beam movement  when t h e  p r o p o s e d  
c o n t r o l  scheme i s  a p p l i e d  w i t h  
4 a c t u a t o r s .  
1. i III e i. 1.i :; e G 
F i g .  5 :  T ime r e s p o n s e  o f  t h e  f f r s t  mode 
F i g .  6 :  T i m e  r e s p o n s e  o f  t h e  t h i r d  
mode 


Optimal control of large space structures via generalized inverse matrix

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