AI AA-90-3347-CP MODEL REDUCTION WITH WEIGHTED MODAL COST ANALYSIS A. Hu * Dynacs Engineering, Co., Inc. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347 Palm Harbor, Florida R.E. Skelton t Purdue University, West Lafayette, Indiana Abstract sensor devices can only have finite bandwidth. Furthermore, the drawbacks of the infinite-bandwidth assumption for white noise processes were evident when the norm of the response vector was evaluated as the infinite sum of the modal cost for certain input and output combinations for certain simple continua [4]. For example it mas shown in [4] when the input is a torque and output is rotation rate, the norm of the response, or the infinite s u m of the modal costs was found not t o converge if we choose t o model the disturbances as white noise processes. However, in reality we know that the norm of the response in this case does not increase without bound. This indicates t h a t some appropriate modifications for the MCA theory is necessary, and one direct approach is to model the disturbances as finite bandwidth colored noise processes inqtead of white noise processes. Analytical expressions are developed for the controllability Grammian rnatrix and for the modal costs when the disturbances t o t h e flexible structures a.re modeled as colored noise processes which have only finite bandwidth. This new algorithm is called Weighted Modal Cost Analysis, and is the generalization of the theory of Modal Cost Analysis for which the disturbances are assumed as white noise processes which have infinite bandwidth. Comparing to the conventional numerical procedure, t h e closed form expression derived for controllability Grammian and modal costs are more efficient and more accurate and are thus quite useful in model reduction processes, especially for high order systems since n o interations are required to solve the Lyapunov equation. T h e purpose of this paper is to generalize the theory of Modal Cost Analysis when the disturbances to the flexible structures are modeled as colored noise. Analytical expressions for the controllability Grammian matrix, which is useful in computing the modal costs, are derived. This is particularly efficient for higher order sys tems. 1.O Introduction White noise processes are assumed to have no correlations between any two distinct values of the random processes a n d have equal power spectral densities at all frequencies. It is a convenient model for formulating and solving linear control problems t h a t involve rand o m disturbances a n d noise [l]. Using white noise to model random disturbances acting on space structures, Skelton e t al. have developed the theory of Modal Cost Analysis (MCA) [2,3]. Modal costs represent the contribution of a mode in t h e norm of response vector, and the theory of MCA has been found very useful as mode truncation criteria because it allows t h e control objective to influence model reduction processes. However, such abstract white noise processes do not exist in t h e physical world, since any real actuator and This paper is organized as follows. Section 2.0 briefly reviews the necessary modal cost analysis (MCA) when the linear system is driven by white noise. Section 3.0 provides a mathematical model for Weighted Modal Cost Analysis, t h a t is, a model of a augmented system including the plant and actuators with the disturhances to the plant being modeled as colored noise. This section also derives the closed-form expressions of controllability Grammian, modal costs and other useful theorems. Section 4.0 presents a numerical example with a simply-supported beam and compares the results for modal costs using the new analytical approach with that using Matlab subroutines for solving Lyapunov equations. *Project Engineer, Member AIAA t Professor, School of Aeronautics and Astronautics, Member AIAA Copyright 0 1990 American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 295 2.0 Modal Cost Analysis Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347 A brief review of necessary Modal Cost Analysis (MCA) is included in this section in order to compare with the results obtained using Weighted Modal Cost Analysis to be presented in Section 3.0. The disturbances to the plant model for Modal Cost Analysis are assumed to be white noise processes, whereas the disturbances used in the Weighted Modal Cost Analysis are assumed t o be colored noise processes. In equation (4), W is the noise intensity matrix and Q is the ouput weighting in the cost function (3). If we are interested in lightly damped continua, the second term in equation (4) can be neglected. 3.0 Weighted Modal Cost Analysis Let the flexible space structures driven by actuators with finite-bandwidth be modeled as two interconnected systems. The plant and noise models are assumed in the modal form, and the quantities matrices w, wr represent white and colored noise respectively. Consider an elastic system represented by the following equations of motion. The plant equation is iji Note that w(t) represents a zero-mean white noise with intensity W . T h a t is, + 2CiiwiGi + 4 w , +2Cw,w,,dw, Ew(t) = 0 E{w(t)tuT(T)} = Wb(t - T ) =[p: , . . . l A 2 P 3 { } + R, 4, { } + [ r h ,..., *:l 9, Qw rn t-a, (5) = bZzw, i = 1 , 2 , ...rn (7) +w,,q,, E , = lim E ( V ( t ) ) m (8) In equations (7) and (8), (qw,,dw,)T are the noise model states, w is the zero mean white noise disturbance as in equation (l),and p i and r i are the it* columns of P, and R, respectively. Rearranging equations (5) - (8) we will obtain a state space form for the augmented system The component costs vi, associated with each component x i , are defined by (3) where = b T d , i = 1 , 2 , ...n wt = P, qw where 6 ( t ) is the Dirac delta function. Now let a cost function associated with equation (1) ( 2 ) be given as V, = E,yTQy, W:V~ A x i = ( V i , iilT i = y It was shown in [2,3] that when the system takes the modal term as in equations ( 1 ) and (2), the component cost is called modal cost and can be computed a.s = Az+Bw (9) cz where xT E ( N + M ) with N =A 2n and M The system matrix A is defined as (4) 296 A = 2m between either plant or noise model states respectively, whereas 2 describes the coupling correlations between plant and noise model states. Specifically, X i j , j - i j and X i j are the ( 2 x 2) ( i , j ) t h (, i + n , j + n)'* and ( i ,n j ) t h blocks of the controllability Grammian X , and they are defined as A A A h [ O A ] with + AE (N+M)x(N+M) Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347 where and For example, the correlation between plant states is defined as follows i.' q 211 x= -7112 . .. R XIn "2 s,, Xnl X n z . . . Now that the augmented system in equation (9) is in the standard form of a linear system driven by white noise, its controllability Grammian X can be obtained using various kinds of numerical methods for solving Lynapunov equation (14). But such a numerical approach can be criticized on the grounds that it ignores the special structure of matrices A , A and A exhibited in equations (9) - (13). It can be anticipated that analytical expressions for controllability Grammian A' can be derived for the augmented system. Specifically, we can first solve ,U in terms of noise model parameters, and then determine ,.( in terms of the elements of 3 and finally we can obtain ,. in terms of 2 . These results are summarized in the following theorems. Note t h a t matrix A represents the properties of the plant model, Aq represents the properties of the noise model, and A represents the couplings between the plant and the noise models through the interconnection between actuator ouputs and plant inputs. T h e dimensions of matrices A , A, and A are N x N ,N x M , and Af x M respectively. Also I BT = [ o , O . . . ! 0, b,, . . . O , bwm Theorem 1. Let xij denote the ( 2 x 2) correlation matrix relating noise model states { ( p w , ( t ) , yw,(t))}T and { ( q w J ( t ) , i w J ( t ) ) T }For . the augmented system in equation (9), the elements of %ii depends only on the noise model parameters and can be determined as T h e Lynapunov equation for the augmented system (9) is XAT + A X + BWBT = 0 (14) where the controllability Grammian X has the same dimension of A and is defined as x In equation (15) the matrix partitions X , X and have the same dimensions as those of A,A,and respectively. Note t h a t x ,2 represent the correlations A 297 where Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347 Proof. T h e formulas for plant model correlation matrix elements 2 i j ' s in equation (16) can be directly obtained by writing out the four scalar equations of the ( 2 x 2 ) (i + n , n + j ) t h element of equation (14), i.e., from the following partition. T h e last term in equation (20) can be rewritten as m 2 [ b T ( p i 2 i j l + rigzj) b T ( p i 2 ; : + - 1 0 0 riXzj2) where crz"j' and are defined in equations (19). The resultant four scalar equations of equation (20) are Theorem 2 Let Xij denote the ( 2 x 2) correlation matrix corresponding t o plant state ( q i ( t ) ;7j2(t))*and noise model state ( q w l ( t ) ,i W J ( t ) ) T For . the augmented system in equations (9) - (13), the elements of X'? can be obtained in terms of noise states correlations X,, and plant parameters, i.e., With direct substitution, we can obtain expressions for plant and noise model state coupling correlation kzj's,as provided by equation (18). Theorem 3 Let X i j denote the (2 x 2) correlation matrix relating plant states (77i(t) ,+(t))T and ( q j ( t ) , G j ( t ) ) T . For the augmented system in equation (9), the element of xij can be determined as (1) If i # j (l8) where Proof. In order t o solve for Xij , we need the (2x 2) ( i ,j+n)th partition of equation (14): 298 ( 2 ) If i = j Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347 Now the four scalar equations from equation (24) are where 6 i j ,w i j , pd'j2, and ,f$/ are defined as follows. By using the same procedure as before we can obtain the expressions of X i j ' s in equations (21) and (22) Remark It can be easily verified that 2 and X are symmetric. For example, from equations for X i j ' s and p i j ' s in equation (21), (22) and (23), we can show that $3 = 2 2322 , and X2'j" = X!:, which means X1.l 3I 23 = xr!,X?? X is symmetric. For the lightly damped structures, further simplified expressions for X i j can be obtained. This is summarized in Theorem 4. Proof. The ( 2 x 2 ) ( i , j ) t h element of equation (14) is Theoreiii 4 Let Xij denote the (2 x 2) correlation matrix relating plant states ( q i ( t ) ,i i ( t ) ) Tand ( q j ( t ) ,l i j ( t ) ) T . For the lightly damped structures represented by equation (9)) the element of Xij can be determined as follows (1) If i # j 1 X!? = pa? '3 '3 7n k=l [ 0 -bTpk 0 -bTrh =0 02. (24) The third and the fourth terms in equation (24) can be rewritten as (2) If i = j and Proof and they can be combined as a single ( 2 x 2) matrix 299 '3 The numerical simulation proceeds as follows. First, we do not assume light damping and compare the results using analytical solution for controllability Grammian with that using numerical approach via Matlab subroutines. That is, we use the formulas in Theorems 1-3 to compute the “exact” controllability Grammian and compare the residues of Lyapunov equation (14) using either analytical or numerical method. Secondly, we apply light damping assumption (C -+ 0) and use formulas in Theorem 4 to compute the “approximate” values of controllability Grammian matrix and modal costs based on the analytical method. In this step we will discuss the relative errors of the modal cost based on Theorem 4 normalized by that based on equations (21) - (23) (without assuming light damping). Note that both solutions are obtained using analytical method, but the former is an approximation of the latter when the damping ratio C approaches zero. The above expression can be easily obtained by letting IS 0 in expressions (2 1) - (23). -+ For ease of comparison, the corresponding results in the standard Modal Cost Analysis is cited as follows, that is, for the lightly damped structures when the disturbance w(t) is modeled as white noise processes, the elements of plant state correlation matrix X arc Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347 (1) I f i # x j (2) If i = j 4.1 Analytical Approach vs. Matlab Subroutille ~~~~l~~ x, With X , and X expressed in analytical expressions in Theorems 1-4 we are now in a position to summarize the general steps needed to compute the controllability Grammian X and modal costs for the augmented system (7), that is, Now let us compare the analytical solution of the controlla.bility Grammian with that using Matlab subroutines. In this part of simulation, both the plant and noise models are assumed to contain two modes. The system matrices, frequencies and damping ratios are as follows. Step I. Solve for noise state correlation X in terms of noise model parameters, i.e., determine X i j according t o equations (16) -(17). 0 Step 2. Solve for noise and plant state correlation 3 in terms of 2 a n d other plant parameters according t o equations (18) - (19). 1 A= 1-w1 -2CW1 0 Step 3. Solve for plant state correlation X in terms of 2 and 2 according to equations (21) - (23). w I. - i 2 , Step 4. Form controllability Grammian X according to equation (15). 1 -w; -2Cw2 C = 0.005, i = 1, 2 and Step 5 . Compute the modal cost according to 2 3 The output matrix for the plant is assumed as uw, = tr[XCTQC]2i-i,2i-1 + ty[XCTQC]2i,2i 4.0 A Numerical Example = - (i + 2 ) 2 , Cw = 0.707, i = 1, 2 c = bl, rl Pa, .21, In this section we provide a numerical example t o illustrate the accuracy and efficiency of analytical method for solving controllability Grammian matrix X as derived in the previous section. The plant used is a simply-supported beam, which is driven by colored noise disturbance. The results are presented in Tables 1- 5. ri = Pi = sin (0.35i) icos(0.35i) O I bi = i The output matrix for the disturbance model is assumed as 300 4.2 Approximate Formulas in Theorem 4 for Light Damping We now present numerical results for the approximate formulas (25) - (26) of Theorem 4 for controllability Grammim ,q. We assume small damping for the plant, and investigate the errors of the modal costs. p; = The same beam example is used, but now three modes are retained for both plant and disturbance models. We vary damping ratio C for the plant from 0.001 to 0.01 and compute approximate cost matrix Vu = X,CTQC and “exact” cost matrix Y = XCTQC. Note that X , is obtained using Theorem 4 with small damping assumption, but using Theorems 1 - 3 without small damping assumption. The error in the cost matrix e”, is defined as e u , = (14 - V a , ) / xx 100% and is listed in Table 5. The modal costs and V,, are Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347 bw, = 5i The augmented system matrix has the following structure A A A = [ o A ] and the controllability Grammian is [ x x XT + A A defined as: V , = [ V ( 2 i - 1 , 2 i - 1) V ( 2 i ,Z i ) ] , V,, = [Va(2i - 1 , 2 i - 1) + Ya(2i, 2i)I with Y and Vu being the cost matrices. It should be noted that the plant matrix A and disturbance model matrix are assumed both of dimension (6 x 6) in this case, since three modes are included for both plant and disturbance models. 2.1 a The numerical results of this subsection are summarized in Tables 1-4. Table 1 contains the augmented system matrix, and Table 2 lists the elements of the controllability Grammian X based on the analytical solution. Note that the upper-right (4x4) partition in Table 2 represents the correlations between the plant states. It is clear from Table 5 that the approximate formulas developed in Theorem 4 are adequate for modal cost computation when the damping ratio is small. For example, when ( = 0.01, the errors in VI, V2, and V3 are only 0.62, 0.058, and 0.044 % respectively, and when C = 0.005, the errors in Vll V 2 ,and Vs are only 0.3745, 0.0095, and 0.011% respectively. Next we substitute X in Table 2 obtained using the analytical solution given by Theorems 1-3 into equation (8) to verify whether Lyapunov equation can be satisfied. The resultant residue matrix is listed in Table 3. Note that the largest element in Table 3 is -0.114d-12. Table 1 The Augmented System Matrix A But if we choose t o use subroutines in Matlab t o numerically solve the Lyapunov equation for controllabilty Grammian X ,and then substitute it back into equation (S), the residue matrix can also be obt,ained and is shown in Table 4. Note that the largest element in the residue matrix in this cases is 0.28d-IO, which is greater than that using the analytical method. (columns 1 through 4) 0.00dt00 -. lOd+01 0.00dt00 0.00dt00 0.00dt00 0.00dt00 0.00dt00 0.10dt01 -. IOd-01 0.00dt00 O.OOd+OO 0.00dt00 0.00dt00 0.00dt00 0.00dt00 0.00dt00 It should be noted that same double precision computation has been used in the calculation for both analytical solution and Matlab subroutines so that the comparison is reasonable. Clearly, the analytical method provides more accurate solution than the numerical method. The reason for this is obvious: numerical errors are minimal when closed-form formulas are used. Also, since no iterations are needed, the new analytical method is very fast t o generate results. However, we must admit that the drawback of this new algorithm is that it can be only used when the system takes the special modal form such as in equations (5) - (7). O.OOd+OO 0.00dt00 0.00dt00 -. 16dt02 0.00dt00 O.OOd+OO 0.00dt00 0.00dt00 0.00dt00 0.00dt00 O.lOd+OI - .40d-01 O.OOd+OO 0.00dt00 0.00dt00 O.OOd+OO (columns 5 through 8) 0.00dt00 -. 21dtOi O.OOd+OO - .43d+01 O.OOd+OO -. 36dt02 O.OOd+OO O.OOd+OO 301 0.00dt00 -.45d+01 0.00dt00 - .90d+Ol O.lOd+OO - . 84dt01 O.OOd+OO 0.00dt00 0.00dt00 0.00dt00 - .39d+01 - . 62dt01 0.00dt00 O.OOd+OO -. 78dtOI -. 12dt02 O.OOd+OO O.OOd+OO 0.00dt00 0.00dt00 O.OOd+OO O.lOd+OO - .I ld+03 - . 15d+02 Table 2 T h e System Controllability Grammian Matrix X Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347 (columns 1 through 4) 0.90d+02 O.OOd+OO O.OOd+OO 0.93d+02 -. 17d+00 0.14dtOl -. 14d+01 0.42d+01 - . lld+00 - .40d+00 0.40d+00 - .49d+00 - .39d-0 1 -. 33d+00 0.33d+00 0.56d+00 -. 17d+00 -. 14d+01 O.l4d+OI 0.60d+00 O.OOd+OO - . 17d+00 0.32d+00 -.77d-01 0.58d+00 Table 4 Residue Matrix of the Lyapunov Eqn (Using R4atlab Subroutines) 0.42d+01 O.OOd+OO 0.96d+03 -. 32d+00 -.35d+01 -. 58d+00 -. 58d-01 1.0d-10* (columns 1 through 2) -0.0015- (columns 5 through 8) - . 1 ld+OO 0.40d+00 -. 39d+00 -. 40d+00 -. 49d+00 -. 33d+00 -. 17d+00 0.32d+00 -. 77d-01 -. 32d+00 -. 35d+01 - .58d+00 0.41d-01 O.OOd+OO 0.28d-01 -. 94d-01 O.OOd+OO 0.15d+01 0.94d-01 0.18d+01 0.28d-01 0.94d-01 0.29d-0 I O.OOd+OO -0.0027+ -0.0004+ 0.33d+00 0.56d+00 0.58d+00 - .58d-01 -. 94d-01 0.18d+01 O.OOd+OO 0.33d+01 -0.0009-0.00020.0000-0.0001-0.0031- -0.0003-0.0014+ -0.0039 0.0181+ 0.0002+ -0 .OO 170.0045+ -0.0161- (columns 1 through 4) O.OOd+OO O.OOd+OO O.lld+OO -. 89d-15 0.1 Id-I3 O.OOd+OO 0.lld-13 O.OOd+OO -. Ild-12 O.OOd+OO -.lld-l2 0.16d-14 -. 22d-15 O.OOd+OO - .44d- 15 0.00dt00 - .44d-15 0.71d-I4 0.22d-15 O.OOd+OO 0.17d-15 0.36d-14 -. 13d-I5 0.14d-I3 -. 89d-15 O.OOd+OO - .44d-15 0.71d-14 -. 22d-15 O.OOd+OO -. 22d-15 O.OOd+OO O.OOd+OO 0.22d-I5 O.OOd+OO 0.17d-15 0.00dt00 - .22d-15 O.OOd+OO OfOOd+OO 0.0006i -0.0005+ 0.0006i 0.0056+ 0.0004i 0.01810.0013i 0.1300+ 0.OOOOi -0.0033+ 0.OOOOi -0.0043+ 0.OOOOi 0.0036+ 0.0001i -0.2870+ 0.0018i 0 .OO 13i 0.0023i 0.0076i 0.OOOOi 0.0004i 0.000Oi 0.0001i (columns 5 through 6) -0.0002+ 0.00 12- 0.0002-0.0034- -0.0000- -o.oooo+ -0.00000.0001+ (columns 5 through 8) O.OOd+OO -. 22d-15 O.OOd+OO - .44d-15 O.OOd+OO - .22d-15 O.OOd+OO 0.67d-15 -0.0025- 0 .OO 14i -0.0015+ 0.000Oi -0.00 14- 0.0006i 0.0059- 0 .OO 13i 0.0012+ 0.0011i 0.0011+ 0.0047i 0.0001+ 0.0004i -0.0000- 0.0020i (columns 3 through 4) Table 3 Residue Matrix of the Lyapunov Fqn (Using Analytical Method) O.OOd+OO O.OOd+OO O.OOd+OO 0.lld-13 O.OOd+OO -. 89d-15 O.OOd+OO -. 67d-15 + 0.0001i 0.0013i 0.0007i 0.0024i 0.0001i 0.0037i 0.OOOOi 0.0040i -.67d-15 0.36d-14 -. 13d-15 0.14d-I3 0.67d-15 0.00dt00 0.OOdtOO O.OOd+OO 0.0002i 0.0011i 0.00OOi 0.0OOOi -.OOOOi 0.0001i 0.OOOOi 0.0002i 0.0004+ 0.006-0.00 17+ -0.0046- 0.0036i 0.0049i 0.0001i -.0003i -0.0000- 0.OOOOi 0.0001+ 0.0002i 0.0003+ 0.00OOi 0.0004+ 0.00OOi (columns 7 through 8) -0.0001+ 0.00010.00450.0037- -0.0000+ 0.0003-0.000+ 0.0009- 302 0.OOOOi -0.0031+ 0.0004i -0.0000+ 0.OOOOi -0.0160+ 0.OOOOi -0.28700.00OOi 0.00010.OOOOi 0.0005+ 0.OOOOi 0.0009+ 0.OOOOi -0.0019+ 0.0039i 0.0020i 0.0001i 0.0001i 0.0002i 0.OOOOi 0.OOOOi 0.OOOOi 4. A. Hu and R. Skelton, 1988. Large Space Structures: Dynamics and Control, edited by S.N. Atluvi and A.K. Ames. p.p. 71-94, Chapter 3. M o d a l Cost Analysis for Szmple Continua, Table 5 Error of Modal Cost in Percent Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3347 c e, 1 0.0873 0.1682 0.2427 0.3114 0.3745 0.4325 0.4852 0.5334 0.009 0.5768 0.010 0.6161 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 e,, 0.0012 0.0009 0.0010 0.0045 0.0095 0.0160 0.0241 0.0338 0.0450 0.0577 Springer-Verlag . e, 3 0.0005 0.0018 0.0040 0.0071 0.0111 0.0160 0.0217 0.0284 0.0360 0.0444 vi = [V(2i- 1,2i- 1) + V ( 2 i ,2i)l A v,,=A [Va(2i- 1,2i- 1) + Va(2i,a i ) ] V = XCTQC, Ya = X,CTQC X - Plant state correlation using Theorems 1-3 X, - Plant state correlation using Theorem 4 eu, = (K - Va,)/K x 100% Conclusions In standard Modal Cost Analysis theory, the disturbance is treated as white noise. The Weighted Modal Cost Analysis, developed in this paper, models the disturbance as colored noise. Analytical expressions are developed for the controllability Grammian matrix and for the modal costs. Comparing to the conventional numerical procedure, the closed form expression derived for controllability Grammian and modal costs are very efficient and are thus quite useful in model reduction processes, especially for high order systems since no interations are required to solve the Lyapunov equation. References 1. H. Kwakernaak and R. Sivan, 1972,Linear Optimal System New York: Wiley. and A. Yousuff, 1983, International Journalof Control 37(2), p.p. 285304. Component Cost Analysis of Large Scale Systems. 2. R.E. Skelton 3. R. Skelton 1981 Theory and Applications of Optimal Control in Aerospace Systems (Chapter 8) ed. P. Kant, AGARD publication no.251,KBN 92835-1391-6,July 1981. Distributed in the United States, by NASA Langley Field, Virginia, 23365, attn: Report Distribution and Storage Unit. Control Design of Flexible Spacecraft. 303

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