Computers Elect. Engng Vol. 17, No. 2, pp. 75-90, 1991 0045-7906/91 $3.00 + 0.00 Printed in Great Britain. All fights reserved Copyright ~]) 1991 Pergamon Press pie

I M P L E M E N T A T I O N O F A C T U A T O R S F O R T H E

I N D E P E N D E N T M O D A L S P A C E C O N T R O L S C H E M E

CHARLES C. NGUYEN

Department of Electrical Engineering, Catholic University of America, Washington, DC 20064, U.S.A.

(Received 19 September 1990; accepted in final revised form 4 January 1991)

Abstract--The Independent Modal Space Control (IMSC) method avoids control spillover generated by conventional control schemes such as Coupled Model Control (CMC) by decoupling the large flexible space structure into independent subsystems of second order and controlling each mode independently. The IMSC implementation requires that the number of actuators be equal to that of modeled modes, which is in general very huge. Consequently the number of required actuators is unrealizable. In this paper, two methods are proposed for the implementation of IMSC with a reduced number of actuators. In the first method, the first m modes are optimized, leaving the last (n - m) modes unchanged. In the second method, generalized inverse matrices are employed to design the feedback controller so that the control scheme is suboptimal with respect to IMSC. The performance of the proposed methods is tested by performing computer simulation on a simply supported beam. Simulation results will be presented and discussed.

1. I N T R O D U C T I O N

The development of the Space Shuttle has opened the possibility of constructing very large space structures (LSS) in space for space explorations• Two control problems for LSS are attitude control and shape control. Complex missions impose many stringent requirements on shape and attitude of the LSS, which lead the control researchers to the concept of distributed active control that places on the structure a number of sensor/actuator pairs in order to optimize the LSS performance and behavior• Active control of LSS has been an active research area in the last several years [1-7, 10, 12, 13]. A large number of control schemes has been developed for LSS, but they represent one form or another of modal control [1]. Two main modal control schemes are the Couples Modal Control (CMC) and Independent Modal Space Control (IMSC). The CMC employs an active controller that consists of a state estimator and a state feedback while the IMSC decouples the LSS into n independent subsystems according to n controlled modes and controls each mode independently by means of a modal filter [5] and an optimal controller• It is well-known that CMC causes control and observation spillover, which together can destabilize the LSS [1]. IMSC eliminates control and observation spillover because each mode is controlled independently. However, the implementation of IMSC requires that the actuator number be equal to the number of modeled modes, which is usually very huge for the modeling of the LSS to be faithful. This fact presents a fundamental limitation of IMSC because the enormous number of modes required for a faithful model of LSS is impractical. The main objective of this paper is to propOse methods implementing the IMSC with a milder requirement of the actuator number. In particular, we will develop two control schemes that use a reduced number of actuators to control all modeled modes in such a way that the closed-loop system modes are as identical as possible to the optimal modes specified by the IMSC scheme. In the first control scheme, the first m modes are optimized, leaving the last (n - m) modes unchanged. In the second scheme, generalized inverse matrices are employed to design the feedback controller so that the control scheme is suboptimal with respect to IMSC.

Matrix notations used in this paper are given below:

Mi 0

block diag (M1 M 2 . . . . . M , ) = .

0

0

M2

0

Om x n = (m x n) null matrix, In = (n x n) identity matrix.

75

76 C-'as, Rt~ C. NOU'tL~

2. S U M M A R Y O F I N D E P E N D E N T M O D A L S P A C E C O N T R O L

A large flexible space structure can be described by the following partial differential equations [4]:

M(P) a2u(p, t)/Ot 2 + Lu(P, t) = f (P, t), (1)

that must be satisfied at every point P of the domain ~ , where u(P, t) is the displacement of point P, L is a linear differential self-adjoint operator of order 2p, expressing the system stiffness, M(P) is the distributed mass a n d f ( P , t) the distributed control force. The displacement u(P, t) is subject to the boundary conditions:

T,u(P, t) = 0, i = 1, 2 . . . . . p, (2)

where T~, i = 1, 2 . . . . , p are linear differential operators of order ranging from 0 to (2p - 1). The associated eigenvalue problem is formulated by:

L~,(P) =),,M(P)O,(P); r = 1, 2 . . . . (3)

with the boundary conditions:

Ti~,(P)=O; i = 1 , 2 . . . . . p; r = l , 2 , . . . , (4)

where )., is the r th eigenvalue and ¢, (P) is the eigenfunction (sometimes also known as mode shape) associated with 2,. Suppose the operator L is self-adjoint and positive-definite, and all eigenvalues are positive and are ordered so that ),~ < )-2 < . . . . Since L is self-adjoint, the eigenfunctions are orthogonal and therefore can be normalized such that:

and

f MO,~, d ~ = 6,,, (5)

f ¢sLO, d~3 =)',6,s; r,s = 1,2 . . . . . (6)

where 6,, is the Kronecker delta. Using the expansion theorem [3], the solution of u(P, t) can be obtained as:

u(P, t)-- ~ ~r(P)u,(t), (7) r = l

where u,(t) is the modal coordinate. Substituting (7) into (1), multiplying both sides of the resulting expression by 0,, integrating over ~ and employing (5) and (6), we obtain:

ii,(t) + 09 2 u,(t) - f ( t ) ; r --- 1, 2 . . . . . (8)

In (8), the mode (or natural frequency) o9, is defined as:

o9, = 0.,)1/2; r = 1, 2 . . . . . (9)

and the modal control force f , ( t ) is computed by:

f,(t) =- fe #,(e) f(P, t) d~3. (10)

In practice, the infinite series in (7) is truncated as:

u(P, t)= ~ *,(P)u,(t), (11) r = l

where n is chosen to be sufficiently large so that u(P, t) can be represented with good fidelity. In this case we are dealing only with the first n modes.

Equation (8) can be transformed into state equation form as follows:

~(t) = A x(t) + W(t), (12)

Implementation of actuators for IMSC 77

where

and

for i -- 1,2 . . . . . n.

x ( t ) = [xr l ( t )x I ( t ) . . . x~(t)] r

W(t ) = [Wit(t) W I ( t ) W~(t)] r

A = block diag (Al, A2 . . . . . A,)

x , ( t ) = [Ur(t)f~r(t)/OJr] T

Wr( t ) = [O f r ( t )/COr] T

(13)

(14)

(15)

(16)

(17)

=[oo, o1, Remark 1

At this point, it is important to recognize the difference between the modes, the eigenvalues of the flexible structure and the eigenvalues of A. If the infinite series given in (11) is truncated to n, then we consider only the first n modes of the flexible system, which are co~, eh . . . . . co,. According to (9), the n eigenvalues of the flexible system are o9~, oJ~ . . . . . o~. According to (15) and (18), the 2n open-loop eigenvalues of the flexible system are the eigenvalues of A, which consist of n pairs of complex conjugate imaginary numbers + o~lj, ___ o~zj, . . . . + co,j. Thus we observe that the rth mode o~, is the magnitude of the eigenvalues of A,. In other words, a mode corresponds to two open-loop eigenvalues of the flexible system.

The system described by (12) consists of n subsystems given by:

Yc,(t) = A ,x , ( t ) + W,(t); r = 1, 2 . . . . . n. (19)

The essence of IMSC is to choose Wr(t) such that it depends on xr(t) alone. Thus:

where

W,(t) = G,x,; r = 1, 2 . . . . . n, (20)

G = F g m gn2]; r = l , 2 . . . . ,n (21) Lg,21 g,22_1

are (2 x 2) gain matrices. Substituting (16) and (17) into (20), we find that G, must assume the following form:

G , - - I ; 0 1, al g,22 " r - - - l ,2 . . . . . n. (22)

For optimal control, g,2m and g,22 should be determined such that the following quadratic cost function is minimized (linear regulator problem):

J = ~ Jr, (23)

where

r ~ l

J, = (xrQ, x, + W~,R, W,) dt, (24)

Q, and R, are positive-semidefinite and positive-definite weighting matrices, respectively, associated with the rth mode.

The form of G, given by (22) requires that R, assume the form given below [7]:

78 CSXgLES C. NotrrElq

Since IV, depends on x, alone as seen in (20), J can be minimized by minimizing each Jr, independently. From optimal control theory [14], the optimal solution for G, is given by:

G , ( t ) = - R T t K , ( t ) ; r = l , 2 , . . . , n , (26)

where K,( t ) is the solution of the Riccati equation:

g r ( t ) = - g r A r - A ? K r -4- g r R f ' l X r - Q,; r = 1, 2 . . . . . n, (27)

with boundary condition K , ( T ) = O. Selecting Qr = 00~I2 and substituting (25) and (18) into (27) yield:

K , = 2oo~Kj2 + r/~ K22 - 00 2, (28)

/~12 = (Dr (K22 - - Kn ) + r ;-~ Kn K22, (29)

K22 ---- - - 200rK,2 + r ; ' K { 2 - 002r, ( 3 0 )

where K 0 ( i , j = 1, 2) are the elements of K,, and Kl2 = K2I. The steady state solutions of (28-30) can be found by letting/~n =/(t2 =/~22 = 0. Introducing R* = (1/00,2)R, where:

:.1 the steady-state solutions of (28-30) are obtained as:

K I 2 K21 2 2 , = = 00r(--00r r, + al/2), (32)

K22 2 , 2 ,2 ---- 00r [rr + -- 200,rr 200~r*~ al/2] 1/2, (33)

Kl I 2 2 w = c%[1/00, + (2/00r r*)a 3/2 2r*r 2 -- r*] I/2, (34)

where

Substituting (32-34) into (26) yields:

G , = [ 0 00, -- b

where

a = _,r,,2r .2_, + r* . (35)

- [200 , ( -00 0 b ) + r*- t] ' / z l ' (36)

b = (002 + r , - l )za. (37)

From (17), (20), (22) and (36), we obtain:

f~(t) = (o , - b )00,u,(t) - [200,( - 00, + b) + r~*-l]l/2f#(t), (38)

for r = 1 , 2 . . . . . n. Substituting (38) into (8) yields:

tT,(t) + [200r(b - co,) + r*- l ] t/2 ti,(t) + o ,b u,(t) = 0. (39)

From (39) the closed-loop poles (eigenvalues) of the system are:

S,l.,~ = -- 1/2 [200,(b - co,) + r*- l ] I/2 _ 1/2 [ - 200,(b + to,) + r*- t ] 1/2, (40)

for r = 1 , 2 , . . . , n . The closed-loop system is stabilized since the poles given in (40) are either complex conjugates

with negative real parts, or both real and negative. Figure 1 illustrates the concept of IMSC in the state equation form. As we observe, each mode is controlled independently by an actuator. Therefore the number of required actuators must be equal to the number of modes that is in general very high. This fact represents a principal disadvantage of IMSC. On the other hand, one main advantage of IMSC is the elimination of control and observation spillover.

Implementation of actuators for IMSC 79

+ X2 X2

Fig. 1. Independent Modal Space Control.

3. I M P L E M E N T A T I O N O F I N D E P E N D E N T M O D A L S P A C E C O N T R O L

The IMSC scheme is represented by equations (19) and (20) that, however, are in the modal form. In order to implement the IMSC, a finite number of discrete sensors and discrete actuators must be employed.

3.1. Discrete sensors and modal filter

Suppose there are 2p discrete sensors consisting ofp displacement sensors and p velocity sensors located at p locations P,, P2 . . . . . Pp. Using (11), the output of the displacement sensor and velocity sensor can be expressed as:

y, = u(P,, t) = ~ ~,(P,)ur( t) , i = 1, 2 . . . . . p r~|

and

~, = ft(Pi, t) = Z ~,(Pi)f t , ( t ) , i = 1, 2 . . . . . p. (41) r ~ l

If we define an output vector y ( t ) containing both displacement and velocity sensor outputs as:

y (t) = [y, p, Y2Y2. . . y,p,]r, (42)

then using (41), we can write:

where

and

y ( t ) = Cx( t ) , (43)

c = [cv], i , j = I, 2 . . . . . p (44)

o ] cu = L 0 coj~j(P,)J" (45)

In order to implement (20), modal state vector x , ( t ) must be extracted from the sensor outputs. A device called the modal filter [5] interpolates the discrete sensor measurements to obtain

i7/2--..C

80 CRates C. NOt~'EN

continuous displacement and velocity profiles ~(P, t) and ~(P, t) and then computes the estimated modal displacement and velocity using the expansion theorem stated by (7):

= f~ M(P)¢,(P)~(P, t) da), (46) ~,(t)

~,(t) = f M(P)O, (P)~(P, t)d~), (47)

for r = 1 , 2 . . . . . n.

3.2. Discrete actuators

Since it is impossible to control force at every point in the domain ~, the distributed control force is realized by m(m < n) discrete point force actuators applied at m points P~, P~ . . . . . Pm in tho domain ~ as given below:

f (P , t) = ~ 6(P - P,)F,(t), (48) i = l

where 6(P - Pi) is a spatial Dirac delta function and Fi(t) is the force applied by the ith actuator on point Pi.

Now substituting (48) into (10) yields:

f,(t) = ~ ¢,(P) ~ 6(P - P,) F~(t) da). (49) d~ i = l

From the property of the Dirac delta function, (49) can be reduced to:

f~(t) = ~ rP,(e,)F,(t). (50) i=1

If we define a force vector F(t) such that:

F(t) = [F~ (t)F2(t) . . . Fm (t)] r, (51)

then using (50) the relation between F(t) and W(t) can be expressed by:

W(t) = BF(t), (52)

where

for i = 1,2 . . . . . n.

B ~ T T [BIB~ . . .B~] T,

B(2i_o=Olxm; i = 1,2 . . . . . n,

B2, = [¢,(P~)/co~ ¢ , ( P 0 / o ~ , . . . ~,(P,)/co,] ,

(53)

(54)

(55)

4. P R O B L E M S T A T E M E N T

Since in this paper, the implementation problem is focused on actuators, we suppose that there exists a modal filter which takes 2p measurements o fp displacement sensors and p velocity sensors and produces a "perfect" estimated state ~(t) of x(t). It is obvious that if the modal displacement u,(t) and modal velocity zi,(t) can be perfectly estimated as shown by (46) and (47), then in view of (16) and (13), state x(t) can also be perfectly estimated. Here state x(t) is said to be perfectly estimated if ~(t) approaches x(t) as t--} ~ . In control theory, the corresponding state estimator is said to be asymptotically stable [10].

Figure 2 illustrates the implementation of IMSC in which the optimal state feedback law is denoted by:

l~'(t) = G~( t ) , (56)

Implementation of actuators for IMSC 81

A i x ~ MODAL 1_ FILTER I ~

Fig. 2. Implementation of IMSC.

y ( t )

where

and

IV(t) = [IVlr(t)IV~(t)... IVT(t)]T (57)

G = block diag (G,, G2 . . . . . G,). (58)

In Section 3, the optimal solution for IMSC is obtained in modal space, namely from (20):

w ( t ) = Gx(t). (59)

Under the assumption of perfect state estimation, we note that IV ~ Gx(t) as t --. oo. Therefore the optimal solution can be achieved if (59) is satisfied. To make W(t) equal to Gx(t), the matrix D in Fig. 2 is chosen such that W(t )= IV(t).

From Fig. 2 we note that:

F(t) = OIV(t). (60)

Substituting (60) into (52) yields:

W(t) = BDIV(t). (61)

Obviously to make W ( t ) = IV(t), D is chosen such that:

BD = 12,. (62)

From the structure of B as given by (53-55), each ( 2 i - 1)th row (odd row) of BD, for i = I, 2 . . . . , n is a row of zeros. We realize that (62) can never be satisfied. However, noting that each odd row of W(t) is also a row of zeros, if we define:

B(t) r r = [B2B4.. . B~,] x (63)

and

O ( t ) = r x [D2D,... Di] T, (64)

where B,. is the ith row of B and Di the ith column of D, then choosing a matrix D such that:

B D = I,, (65)

will ensure that W(t) = IV(t). It is noted that if (65) holds, then BD is a modified identity matrix of order (2n x 2n) whose main diagonal elements are 0 at the (2i - 1, 2i - 1) position and 1 at the (2i, 2i) position for i = 1, 2 . . . . , n.

The solution of (65) for D needs the inverse of B. However, the existence of the inverse of B requires that the number of actuators be equal to the number of modeled modes (m = n). This fact presents the principal limitation of IMSC because the number of modeled modes is usually very high, resulting in an unreasonable number of actuators.

The problem considered in this paper is formulated to develop a control scheme that allows one to use a reduced number of actuators to control all modeled modes such that the closed-loop system is as optimal as possible compared to IMSC.

In the following we will proposed two control schemes that will accomplish the above objective.

82 CHARLES C. N o u n s

5. T H E F I R S T C O N T R O L S C H E M E

The development of the first control scheme is represented in the following theorem:

Theorem 1

Consider a large flexible space structure whose description and solution are given by (1), and (11), respectively. I f the operator L is self-adjoint and the state x( t ) is perfectly estimated, then there exists a control scheme with m actuators, which optimizes the first m modes with respect to the cost function (24) and leaves, the last (n - m ) natural modes unchanged, thus ensuring the system stability. Proof-- I f the hypothesis of Theorem 1 is satisfied, then employing m actuators placed at m distinct points P, , P2 . . . . . Pm in the domain ~ results in a matr ix/~ that can be partitioned as:

F ll, (66)

where Bt is an (m x m) matrix and ~r2 an (n - m) x m matrix. Based on the property of the rows of ~" specified by (55), without loss of generality we can assume that the first m rows of B are linearly independent for the locations of m actuators are distinct. T h u s / ~ is a nonsingular matrix. Now if a matrix D is chosen such that:

L~ = [Bi -~ Om × (n-m)], (67)

~ 2 ~ / t O(n-m) x (n-m)_ J

then it is clear that:

(68)

We proceed by considering the closed-loop feedback system given in Fig. 2 that can be described by:

~(t) = A x( t ) + BDG Yc(t), (69)

under the assumption of perfect state estimation, in the steady state, when t ~ ~ , equation (69) can be rewritten as:

/c(t) = (A + BDG) x(t) . (70)

Now i f /~ is chosen as in (67), then:

A + BDG = block diag(Ai, A : , . . . , An)

I-I* 1 block diag(Gl G2, Gn), (71) O2,, × 2(n

L x 02(n - m) x 2(n - m)_.] ' " " " '

where Xis an 2(n - m) x 2m matrix whose elements are obtained from B2B~ -t and I * is a modified (2m x 2m) identity matrix whose main diagonal elements are 0 at the (2i - 1, 2i - 1) position and 1 at the (2i, 2i) for position for i = 1, 2 . . . . . m.

Recalhng the form of G, for r = 1, 2 . . . . . n as given in (22), we can rewrite (71) as:

A + BDG = [block diag(A1 + Gt, A2 + G2 . . . . . Am + Gin) L X

From (72) the closed-loop ¢igenvalues can be computed as

02m × 2(n - m) 1

block diag(Am + i, A,+2 . . . . . A,)J" (72)

where

#(A + BDG) = ~ ¢r(AI + G,) 0 U i - I l

sl ffi st U s2 O . . . O s= and st O s2 Iml

(73)

Implementation of actuators for IMSC 83

denotes the union operation of two sets s~ and s2. Furthermore, a (A) denotes the set of eigenvalues of the matrix A.

According to Remark 1, it is obvious from (73) that the first m modes are optimized and the last (n - m) modes are unchanged. Since optimal modes correspond to stable closed-loop poles as stated by (40) and natural modes correspond to pairs of imaginary complex conjugate eigenvalues, as pointed out in Remark 1, we conclude that the above control scheme also ensures the system stability. It is noted that the system stability defined here is not asymptotical stability which requires that the real parts of all eigenvalues are negative [15]. We therefore just complete the proof of Theorem 1.

Remark 2

If the control scheme proposed in Theorem 1 is implemented, then there are some unwanted excitations of the ( n - m) modes, which is well-known as control spillover [1]. However the oscillations caused by exciting the (n - m ) modes are insignificant because the mode amplitudes tend to decrease as the number of modes is increased because higher modes require more energy to excite. We also note that in (1), we assume the worst case where no natural damping exists. In practice, existing natural damping in the structure can suppress the unwanted oscillations of the higher modes.

6. T H E S E C O N D C O N T R O L S C H E M E

In this section, another control scheme is developed using the concept of generalized inverse matrices. We first present the following lemma:

Lemma 1

Consider the following equation:

if(t) = ~P(t) , (74)

where i ( t ) and P(t) are matrices consisting of even rows of W(t) and F(t), respectively. If the (n x m) matrix B has rank m, then the solution for (74), which minimizes the weighted norm of error:

[1 e(t) II 2 = II l - - n_~ II 2 = ( l - - B p ) T S ( I - - BF),

where S is the weighted matrix, is given by:

F(t) = B* i ( t )

and the matrix:

(75)

(76)

B* = -(BTS/I)-1BTS,

is the generalized inverse of B. A proof of Lemma 1 can be found in [13]. The following theorem will present the development of the second control scheme:

(77)

Theorem 2

Consider a large flexible space structure whose description and solution are given by (1) and (11), respectively. If the operator L is self-adjoint, then there exists a control scheme with m actuators (m < n) that is suboptimal with respect to (24) in the sense that the closed-loop eigenvalues are assigned as closed as possible to those optimal eigenvalues specified by IMSC. P r o o f - - A control scheme with a reduced number of actuators would be optimal i f /5 could be selected to be a right inverse of B as seen in (65). However, here B has rank m since the discrete actuators apply point forces at m distinct points P~,/'2 . . . . . Pro. It is well-known [13] that an (n x m) matrix (m < n) having rank m does not have any right inverse. Consequently ~ does not possess any right inverse.

84 CHARLES C. NOUYEN

According to Lemma 1, choosing D = B* minimizes:

( i f ' - ~ F ) T S ( f f / - ~F) . (78)

Under perfect estimation and in steady state, we obtain from Fig. 2:

IT/- B• = IT/- B B t ~ = i f ' - BOf f '= (I, - BB)ff ' . (79)

Substituting (79) into (78) yields:

IrgX(I, - ~O)STS([ , - B D )if'. (80)

Based on (80), choosing D = B* minimizes II [ , - / ~ D I12, which makes (65) satisfied with minimal least-square errors. If B/) is exactly equal to 12, then the closed-loop eigenvalues of the flexible system are identical to the optimal eigenvalues of IMSC computed using (40). Here choosing /~ = B* makes/~/~ be as equal to 12 as possible, which in turn makes the closed-loop eigenvalues be as identical as possible to the optimal eigenvalues. Thus the control scheme which uses/5 = B*, with m actuators (m < n), is suboptimal with respect to (24). We just complete the proof of Theorem 2.

7. C O M P U T E R S I M U L A T I O N S T U D Y

In order to evaluate the performance of the proposed control schemes, we consider the control of a simply-supported beam whose dynamics is given by the Euler-BernouUi partial differential equation:

84 82 EI-~x 4 u(x, t )+ m ~ u(x, t ) = f ( x , t ), (81)

where for simplicity we set the mass m, the moment of inertia/, the modulus of elasticity E and the length of the beam to unity. The boundaries of the simply-supported beam are:

u(0, t) = u(1, t) -- 0, (82)

82 8z 0x 2 u(0, t) = ~ x 2 u(1, t) = 0. (83)

The solutions for the eigenvalue problem are given by:

c0r = (rn) 2, •r = (rT~) 4, (84)

~b, = x/~ sin (rnx). (85)

Computer simulation is performed to evaluate the second control scheme and to compare its performance to that of IMSC. Suppose we consider 20 modeled modes and divide the whole length of the beam into 20 sections, specified by x(k) = k/21 for k = 1, 2 . . . . . 21 and select a weighted matrix S = 120 for the second control scheme. Then starting with one actuator we perform computer simulation for different locations of the actuator. After that, computer simulation is performed with an increasing number of actuators. The simulation results are summarized and discussed below:

(a) The location of actuators of the second control scheme--Computer results show that the actuator location affects the closed-loop system dynamics, e.g. the number of asymptot- ically stable closed-loop eigenvalues. Figure 3 illustrates the relationship between the number of asymptotically stable closed-loop eigenvalues and the actuator location for the case of 1 actuator. We note that the optimal actuator locations center around both ends of the beam. The number of the stable closed-loop eigenvalues decreases as the actuator moves toward the center of the beam. Similar results are obtained for the cases of 2, 3 and 4 actuators. (b) The number of actuators of the second control scheme--Simulation results show that the dynamics of the closed-loop system are also affected considerably by the number of actuators. In general it is expected that the number of asymptotically stable closed-loop eigenvalues is

Implementation of actuators for I M S C 85

0J

0 t . .

10

9

8

7

6

5

4

3

2

1

0

one actuator

1 I io

/

15 2'0

a c t u a t o r l o c a t i o n

F ig . 3. Relationship between number of stable eigenvalues and actuator locations for the case of one actuator.

proportional to that of the actuators. According to the computer simulation, the average number of asymptotically stable closed-loop eigenvalues when using 1, 2, 3 and 4 actuators is 10, 36, 38 and 40, respectively, at arbitrary actuator locations. The average number of asymptotically stable eigenvalues is computed by dividing the total number of asymptotically stable eigenvalues by the number of trials, for each of which the computer simulation is performed for a particular location(s) o f actuator(s). It is also noted that for a given number of actuators, the locations of the actuators can be adjusted to increase the number of asymptotically stable closed-loop eigenvalues. For example, in the case of two actuators, first the two actuators are placed near the ends of the beam so that the number of asymptotically stable eigenvalues is 36. Then the relative locations of the actuators are adjusted iteratively

Table I. Open-loop and closed-loop eigenvahes of IMSC and the second control scheme

Closed-loop eigenvalues of

Open-loop IMSC IMSC Second control scheme eigenvalues (20 actuators) (2 actuators) (2 actuators)

( × I0 3) ( × 10 3) ( × I0 ~) ( x 10 3) ~ 0.0099i -0 .0014 + 0.0099i --0.0014 + 0.0099i --0.0014 + 0.0098i ±0.0395i -0 .0020 + 0.0395i -0 .0020 + 0.0395i -0 .0020 ± 0.0398i +0.0888i -0 .0024 + 0.0888i -0 .0000 + 0.0888i -0 .0025 + 0.0884i ±0.1579i -0 .0028 + 0.1579i -0 .0000 + 0.1579i -0 .0028 __+ 0.1580i +0.2467i -0 .0032 -I- 0.2467i - 0.0000 + 0,2467i --0.0032 + 0.2466i -t- 0.3553i - 0.0035 + 0.3553i --0.0000 + 0.3553i -0 .0035 ± 0.3555i =[: 0.4836i --0,0037 ± 0.4836i - 0.0000 ± 0.4836i -0 .0037 ± 0.4834i ±0.6317i -0 .0040 ± 0.6317i - 0.0000 ± 0.6317i --0.0040 ± 0.6316i ± 0.7994i -- 0.0042 ± 0.7994i --0.0000 + 0.7994i --0.0042 :t: 0.7992i ± 0.9870i --0.0045 ± 0.9870i --0.0000 ± 0.9870i -0 .0045 ± 0.9872i ± !.1942i --0.0047 ± 1.1942i -0 .0000 ± 1.1942i -0 .0047 ± 1.1939i ± 1.4212i -0 .0049 ± 1.4212i -0 .0000 :[: 1.4212i -0 .0049 + 1.4213i -t- 1.6680i -0.0051 ± 1.6680i -0 .0000 ± 1.6680i -0.0051 ± 1.6678i ± 1.9344i -0 .0053 ± !.9344i --0.0000 + 1.9344i -0 .0053 ± 1.9345i ± 2.2207i --0.0055 ± 2.2207i - 0.0000 ± 2.2207i -0 .0055 ± 2.2203i ± 2.5266i -0 .0057 + 2.5266i --0.0000 ± 2.5266i --0.0057 ± 2.5266i ±2.8523i --0.0058 ± 2.8523i - 0.0000 ± 2.8523i --0.0058 ± 2.8521i -I- 3.197Si --0.0060 + 3.1978i -- 0.0000 ± 3.1978i --0.0060 ± 3.1979i ± 3.5629i --0.0062 ± 3.5629i --0.0000 ::t: 3.5629i --0.0061 ± 3.5624i -I- 3.9478i --0.0063 ± 3.9478i --0.0000 ± 3.9478i --0.0063 ± 3.9478i

0.025

0.02

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• ~ 0.005 O

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.~-0.005

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-0.015

8 £

IMSC with 20 actuators T

1 015

86 CHARLES C. NGtn,~

"0"020 J 115 ½ 215 3 315 4 415

time in second

Fig. 4. Time history of the vertical displacement of the middle of the beam when IMSC is implemented with 20 actuators.

015

until the number of asymptotically stable eigenvalues reaches 40 when one actuator is located at x = 3/21 and the other at x = 20/21 (see Table 1). (c) The second control scheme vs I M S C - - T h e open-loop eigenvalues of the LSS, the closed-loop eigenvalues of IMSC implemented with 20 actuators and two actuators and the second control scheme implemented with two actuators are presented in Table 1. The table

0.12

0.1

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0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

IMSC with 2 actuators

i 115 ½ 215 3 315 4 415 5

time in second Fig. 5. Time history of the vertical displacement of the middle of the beam when IMSC is implemented

with 2 actuators.

Implementation of actuators for IMSC 87

0.12 Second control scheme with 2 actuators

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Fig. 6. Time history of the vertical displacement of the middle of the beam when the second control scheme is implemented with 2 actuators.

shows that the second control scheme implemented with a reduced number of actuators assigns eigenvalues that are very close to those specified by the optimal IMSC.

Figures 4 and 6 represent the vertical displacements of the beam center when being excited by a unit impulse for IMSC with 20 actuators and the second control scheme with two actuators, respectively, while Fig. 5 represents the case of IMSC with two actuators. We observe that with a reduced number of actuators (two actuators), the second control scheme provides

x10-3 2

IMSC with 20 actuators

h,i o

8

-1

-2

-3

-4

-5 0

..............................................................................................................................................................

................... ~s r ......................................................................................... • .............

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/

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011 012 013 014 015 016 017 018 019 1

beam length in meter

Fig. 7. Beam response when the IMSC is implemented with 20 actuators: after I s ( - - - ) ; after 2 s (... 9; after 3 s ( . . . . ); and after 5 s ( ).

88 CHARLES C. NGUYEN

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IMSC with 2 actuators 0.03

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0

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~ J

-0"040 011 012 013 014 015 016 017 018 019

beam length in meter

Fig. 8. Beam response when the IMSC is implemented with two actuators: after I s ( - - - ) ; after 2 s (. . . .); after 3 s ( . . . . ); and after 5 s ( ).

almost the same performance as that of IMSC when implemented by 20 actuators in the sense that both schemes bring the oscillation down to zero in 3 s. The fact that the use of a reduced number of actuators causes the peak amplitude (0.1 m) of the second scheme to be about four times the peak amplitude (0.0225 m) of the IMSC is not as significant as the amount of time needed to bring the oscillation down to zero, which is the same for both control schemes. As we see

0.01 Second control scheme with 2 actuators

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~ -0.005 8 c~ -0.01

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~ - ~ - . ~ . . . . . . . . . o - -°¸

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beam length in meter Fig. 9. Beam response when the new control scheme is implemented with two actuators: after 1 s ( - - - ) ;

after 2 s ( . . . . ); after 3 s (-.-.); and after 5 s ( ).

Implementation of actuators for IMSC 89

in Fig. 5, when being implemented with a reduced number of actuators (two actuators), the IMSC causes some steady-state oscillation with a nonzero amplitude of about 0.02 m. Beam movements for the case of IMSC with 20 actuators and two actuators are illustrated in Figs 7 and 8, respectively. The case of the second control scheme with two actuators is presented in Fig. 9. As seen in Figs 7 and 9, the IMSC implemented with 20 actuators and the second scheme implemented with two actuators, both are able to bring the beam movement down to zero. In general we conclude that the performance of the second control scheme is suboptimal as compared to the performance of the IMSC implemented with 20 actuators. However, when both schemes are implemented with the same reduced number of actuators (two actuators), the second scheme has better performance as compared to IMSC.

8. C O N C L U S I O N A N D F U T U R E R E S E A R C H

In this paper, we first reviewed the theory of IMSC in the context of optimal control. The requirement that the number of actuators be equal to that of modeled modes for the implementation of IMSC was pointed out as a principal limitation. Assuming that the large space structure is self-adjoint, we developed two control schemes implemented with a reduced number of actuators, which are suboptimal with respect to the IMSC performance. The first control scheme optimizes the first m modes leaving the last ( n - m) modes uncontrolled. The second control scheme employs the method of generalized inverse matrices to assign closed-loop eigenvalues, which are as close as possible to those specified by IMSC. Computer simulation performed on a simply-supported beam showed that the second proposed control scheme performs better than the IMSC when both are implemented by a reduced number of actuators. Future research should focus the attention on the optimization of the number of actuators and the placement of actuators in order to minimize the control spillover. The effect of a modal filter in terms of observation spillover should also be investigated. Finally the method of arbitrary eigenvalue assignment [8, 9] should be considered for the design of control schemes for large space structures.

Acknowledgement--The author wishes to express his appreciation to Goddard Space Flight Center (NASA) for supporting the research presented in this paper under Research Grant NAG 5-949.

R E F E R E N C E S

1. M. J. Balas, Active control of flexible systems. J. Optimiz. Theor. Applic. 25, 415-436 (1978). 2. R. E. Lindberg Jr and R. W. Longman, On the number and placement of actuators for independent modal space

control. J. Guid. Control 7, 215-221 (1984). 3. L. Meirovitch, Analytical Methods in Vibrations. Macmillan, Toronto (1967). 4. L. Meirovitch and H. Baruh, Optimal control of damped flexible gyroscopic systems. J. Guid. Control 4, 157-163

(1981). 5. L. Meirovitch and H. Baruh, The implementation of modal filters for control of structures. J. Guid. Control 8, 107-1 i 6

(1985). 6. L. Meirovitch and H. Oz, Modal space control of distributed gyroscopic systems. J. Guid. Control 8, 707-716

(1985). 7. H. Oz and L. Meirovitch, Optimal modal space control of flexible gyroscopic system. J. Guid. Control 3, 218-226

(1980). 8. C. C. Nguyen, Arbitrary eigenvalu¢ assignments for linear time-varying multivariable control systems. Int. J. Control

45, 1051-1057 (1987). 9, C. C. Ngnyen, Design of reduced-order state estimators for linear time-varying multivariable system. Int. J. Control

46, 2113-2126 (1987). 10. C. C. Ngnyen and A. Baz, Distributed active control of large flexible space structures. NASA Progress Report, Grant

No. NAG 5-749 (1986). 11. C. C. Nguyen and T. N. Lee, Design of state estimator for a class of time-varying multivariable systems. IEEE Trans.

Aurora. Control AC-30, 179-182 (1985). 12. C. C. Nguyen, On the implementation of actuators for control of large flexible space structures. NASA Progress Report,

Grant No. NAG 5-949 (1988). 13. C. C. Nguyen and X. Fang, Optimal control of large space structures via generalized inverse matrix. Proc. 20th

Southeastern Syrup. of System Theory, North Carolina, pp. 652-656 (1988). 14. B. Noble, Applied Linear Algebra. Prentice-Hall, Englewood Cliffs, New Jersey (1969). 15. A. P. Sage and C. C. White, Optimal System Control, 2nd Edn. Prentice Hall, NJ (1977). 16. J. B. Lewis, Linear Dynamic Systems. Matrix Publishing, Champaign, IL (1977).

90 CHARLES C. NGUYEN

A U T H O R ' S B I O G R A P H Y

Charles C. Nguyen---Dr Charles C. Nguyen received the degree of Diplom Ingenieur in Electrical Engineering at Konstanz Technical University, Fed. Rep. Germany in 1978. Later he earned his Master of Science in 1980 and Doctor of Science in 1982 at the George Washington University. He joined the Catholic University of America in 1982 at the rank of Assistant Professor and was promoted to Associate Professor in 1987, a position he still holds. Professor Nguyen has published over 60 technical and scientific papers in the area of control and robotics in major technical journals. He is a member of ISMM, Sigma Xi, and a member and Chief Faculty Advisor of Tau Beta Pi. He is a Senior Member of the IEEE, a senior member of Society of Manufacturing Engineering (SME) and a senior member of Robotics International of SME. He was the recipient of the "Research Initiation Award" from Engineering Foundation in 1986 and was awarded NASA/ASEE Fellowship Awards in 1985 and 1986. He was the recipient of the "Academic Vice President Research Excellence Award," in February 1989 from the Catholic University of America. He was awarded a Senior Research Associateship from the National Research Council/National Academy of Science to conduct robotic research at the Goddard Space Flight Center (NASA) during his sabbatical leave of the academic year 1990-91. His life and achievements are listed in over 22 biographical registers and Who's Who. His research interests lie in the areas of time-varying control systems, control of large space structures, decentralized control, control of robot manipulators, robot vision and neural networks. His research has been continuously supported by several agencies including NASA, JPL, U.S. Air Forces and Engineering Foundation. He has been the principal investigator of 15 research projects funded by NASA and other government agencies totaling $ I million in the last 6 years. He is also a consultant in the area of control and robotics for local companies and government agencies. He is currently the principal investigator of three research grants funded by NASA/Goddard Space Flight Center in the areas of autonomous space robots and telerobotics, one subcontract with The Jet Propulsion Laboratory (JPL) in the area of force-reflecting hand controller and one subcontract with the U.S. Air Forces in the area of Stewart Platform-based high precision manipulators.