SEMI-ANNUAL PROGRESS REPORT 

ON 

DISTRIBUTED ACTIVE CONTROL 

OF 

LARGE FLEXIBLE SPACE STRUCTURES 

(Grant # NAG 5-749) 

Submitted by 

/ 

Dr. Charles C. Nguyen, Principal Investigator 

Dr. Amr Baz, Principal Investigator 

The Catholic University of America 

Washington, D. C. 20064 

to 

Dr. Joseph V. Fedor Code 712 

Space Technology Division 

Goddard Space Flight Center 

Greenbelt, Maryland 20771 

November 1986 
TABLE OF COMENTS 

PAGE 

Sumar y 

Symbols and Notations 

1. Introduction 

2. Updated Research 

3. 

4. Project Research Results 

5. Current and Future Research 

6. Conclusion 

Modeling of The Polar Platform 

References 

Figures and Tables 

4 

5 

16 

16 

20 

21 

-1- 
SMRY 

This progress report summarizes the research work performed at the 

Catholic Universi ty of herica on the research grant entitled 'Distributed 

Active Control of Large Flexible Space Structures,' funded by WWGoddard 

Space Flight Center, under the grant number NAG 5-749, during the period of 

March 19, 1986 to September 19, 1986. 

In this report we first update the research work relevant to the 

project. 

stated. The report is then concluded by a discussion of current and future 

research work 

Then the research accomplished during the above period will be 

-2- 
SYMBOLS MJD NMATl ON 

MI 0 ... 

0 M2... 

.. 

.. 

.. 

00 

= an (mxn) zero matrix 3* Omxn 

4. In = an (nxn) identity matrix 

5. x(t) = 9 x(t) 

dt 

0 

0 

MN 

-3- 
1. Introduction 

The advent of a space transportation system, such as the space shuttle 

makes it possible to conceive of very large satellites and spacecraft which 

could be carried into space and deployed, assembled, or constructed there 

for such diverse purposes as comnunications, surveillance, astronomy, space 

exploration, and electric power generation 121. These large space 

structures (LSS) concepts range from central rigid bodies, to the solar 

electric propulsion spacecraft and the generic polar platform. Two control 

problems for LSS are attitude control and shape control (241. The former 

involves maintaining a given orientation of the spacecraft, e.g., with 

respect to the sun or earth; the latter involves maintaining the shape of 

critical structures of LSS. LSS are distributed parameter systems that 

possess many low resonant frequencies and have very stringent requirements 

for shape, orientation, alignment, vibration suppression and pointing 

accuracy. These requirements lead the control designers to the concept of 

active control of LSS with various sensors and actuators located about the 

structure and operating through on-line computer controllers to tailor the 

performance and behavior of the system. 

There has been considerable interest in the area of active control of 

LSS [11-[243. A number of control schemes were proposed for large flexible 

space structure (LFSS), but they all represent on0 form or another of modal 

control 1113. Two main modal control schemes are the coupled Modal Control 

and Independent Modal Space Control (IMSC). 

controller consisting of a state estimator and a state feedback law; the 

latter controls each mode independently by means of the modal filter [13]. 

The former employs an active 

-4- 
In this report we first update the research in active control of LFSS 

to provide a mathematical framework for the project. 

a generic polar platform is developed using finite element method. 

that some project research results are presented. We then discuss the 

current and future research effort. 

2. UDdated Research 

Then the modelling of 

After 

A conclusion will sumarize the report. 

In this section we first present the mathematical description of large 

space structures (LSS) and then discuss the two main control schemes for 

this type of structures. 

2.1 Mathematical DescriDtion of LSS 

The LSS may be described as a continuum by the following partial 

differential equations 

2 

M(pp u(Pft) + L u(P,t) = f(P,t) 2 at 

(2.1) 

where u(P,t) = displacement of an arbitrary point e 

L= Linear differential self-adjoint operator of 

order 2p, expressing the system stiffness. 

M(P) = distributed mass 

f(P,t) = distributed control force 

Equation (2.1) must be satisfied at every point P in the domain D. The 

displacement of u(P,t) ir subject to the boundary conditions 

ai u(~,t) = 0 for i=1f2f...fp 

where Bi are linear differential operator of order ranging from 

0 to (Pp-1). 

(2.2) 

-5- 
The essocieted eigenualue problem 

L @,(PI = x, M(P) (Pr(P) 

€or r=112,.. . ,w 

with the boundary conditions 

Bi = 0 

for i=112f...1p; r=l12,...,d 

is formulated by 

(2.3) 

(2.4) 

where x is the rth eigenvalues and (0 (p) is the eigenfunction associated 

with 1 . Sometimes @ (p)is called the mode shape. 

r r 

r r 

Equations (2.3) and (2.4) can be solved to obtain the solutions of and 

r 

and in addition, if the operator L is positive definite, then all 

@r 

eigenvalues are positive. The mode frequency (or natural frequency) is 

defined as 

4 

using expansion theorem [lil, the solution of u(P,t) can be obtained as 

r=l 

where ur(t) satisfies 

-6- 
for r=1,2,. . . 

In practice, the infinite series in (2.6) is truncated as 

u(P,t) = 

r=l 

: (2.9) 

where M is chosen to be sufficiently large so that u(P,t) can be represented 

with good fidelity. 

Since M may be quite large, it is not reasonable to control all M 

modes. Hence we select N modes to control. These N modes are called 

Controlled Modes. The remaining R modes (R = M-N) are called Residual Modep 

(uncontrolled modes). 

Equation (2.7) can be transformed into the state equation form as 

follows 

.. T 

where xc(t) = fu, u2 ... u1 u2 ... UNI (2.12) 

WIECEDWQ PAGE BLANK MOT FILMED -7- 
Discrete Actuators: Since it is impossible to implement distributed 

control forces, the distributed control force f(P,t) is realized by k 

discrete point force actuators 

where S(P-Pi) is a spatial Dirac Delta function. 

Substi tuting (2.18) into (2.8) yields 

k 

i=l 

f (t) -= 1 @,(pi) ~~(t); r=1,2,. 

r 

Now using (2.19) we obtain 

B= 

C 

(2.18) 

(2.19) 

(2.20) 

(2.21) 

(2.22) 

(2.23) 

-8- 
T and F(t) = IF1 F2 .... FkI (2.24) 

Discrete Sensors: Suppose there are s discrete sensors consisting of p 

velocity sensors, located at p locations Pvl Pup, ..., pup and 9 

displacement sensors, located at q locations PD1, PD2, ..., PDqe 

(2.9) we can express the velocity sensor output as 

Using 

(2.25) 

for j = 1,2,.. ., p, and the displacement sensor output as 

M 

Fl 

= u(P ,t) = 1 @=(PDj) up) (2.26) 

Dj 

for j = 1,2,..., q. 

if we define the sensor output vector as 

T 

y(t) = [Y, Y2 --. Yp YPl Ype2 -.- Ywsl I 

then using (2.25) and (2.261, we can write 

where 

(2.27) 

(2.29) 

(2.30) 

(2.31) 

(2.32) 

(2.33) 

-9- 
The development in this section can be now summarized by the following modal 

state equation in matrix form 

C 

A [j = 1 - 0 

Y(t) = [ cc 

0 

A R ][:I + (2.34) 

(2.35) 

- 10 - 
2.2 The CouDled Modal Control Scheme 

In this scheme, the active controller consists of two parts: 

a) a state estimator that accepts the sensor output y(t) and produces 

an estimate Xc(t) of the controlled state XC(t) and 

b) a linear state feedback control law that multiplies the estimated 

controlled state XC(t) by constant gain to produce the actuator input F(t). 

The state estimator can be of the Luenberger type that is described by 

E 151 

(2.36) 

The state estimator error is defined as 

eC(t) = xC(t) - xC(t) 

A 

(2.37) 

Theref ore 0 

0 A h 

eC(tl = x(t) - xc(t) = (AC - 0 cc C )e c (t) t GCCRxR(t) (2.38) 

The linear state feedback la is defined by 

A 

F(t) = KC xC(t) (2.39) 

Now substituting (2.39) into (2.34) and (2.381, we obtain the composite 

state equation as follows: 

0 

AC+BCKC BCKC 

GCCR A -G C c cc 0 

A 

'RKC BRKC R 

The objective of the couple modal control scheme ir to select KC and Gc ouch 

that the composite system (2.40) is asymptomatically stable. In (2.40) the 

- 11 - 
control spillover is represented by BR and the observation spillover is 

represented by CR. 

Figure 1. 

2.3 The IndeDendent Modal %ace Control Scheme (IMSC) 

The coupled modal control scheme is illustrated in 

The IMSC scheme was developed by Meirovitch and others [lo] to control 

each mode independently. First an auxiliary variable U (t) is defined by r 

vp = 1/0 qt) (2.41) 

Therefore using (2.7) we can write 

2 

r 

Now using (2.41) and (2.421, we obtain 

irp = l/Ur iir(t) = - w up + fr(t) (2.42) 

(2.43) 

(2.44) 

=[O 'r 

T and xr = [ur vr1 

T 

fr'Wrl (2.45) 

(2.46) 

The essence of the IMSC is to choose Ur such that Wr depends only on 0; 

alone: 

wr = Wr(wr) (2.47) 

The optimal design of Wr was discussed in 1121. 

Modal Filter; To implement the IMSC scheme, the modal displacements 

ur(t) and modal velocities ur(t) are required. 

filter accepts the measurements of displacement u(P,t) and ueiocity u(P,t) 

at every point P at all time and produces the modal displacement ur(t) and 

- 12- 

A device, called modal 
modal velocity ur(t). 

given by 

The mathematical representation of a modal filter is 

(2.48) 

(2.49) 

It is noted that distributed sensors are needed to obtain the 

displacements u(P,t) and velocities u(P,t). 

Implementation of IMSC usins discrete activators and sensors 

The implementation of IMSC requires distributed control, i.e. Control 

forces are applied at every point of the structure, and also requires 

distributed sensors, i.e. the displacements and velocities are measured at 

every point. 

actuators and discrete sensors are used instead. 

This is however not realizable in practice. Thus discrete 

Let us consider the problem of controlling N modes by means of discrete 

actuators. The actuator forces can be treated and distributed by writing 

substituting (2.50) into (2.8) yields 

N 

Introducing the matrix B as 

B= 

we obtain 

... 8 

f(t) = B F(t) 

- 13 - 

(2.50) 

(2.51) 

(2.52) 

(2.53) 
where 

T f(t) = If, f2 .... f,l 

I F(t) = [F, F2 .... FNl 

(2.54) 

(2.55) 

Suppose B is nonsingular, then the actual control vector F(t) can be 

synthesized from the generalized control vector f(t) by 

F(t) = B* f(t) (2.56) 

Next let us assume there are m sensors capable of measuring 

displacements and velocities at the discrete points P=Pk (k=1,2,...,m). 

Then from these measurements the entire displacement pattern u(P,t) and 

velocity pattern u(P,t) can be estimated by using various interpolation 

functions. The structure of the IMSC scheme using discrete actuators and 

sensors is shown in Figure 2. 

2.4 Evaluation of the control schemes 

The design of the coupled modal control scheme is straight forward 

since it is simply a pole placement problem if the observation spillover is 

negligible. 

composite system are the eigenvalues of (Act BCKi; (AC-GCCC)and AR due to 

block triangularity. There are several techniques such as in [15,16 1 that 

Indeed if we set CR=O in (2.4), then the eigenvalues of the 

help the control designer to select KC and Gc such that 6tcC+BcKc and Ac-QcCc 

have desired eigenvalues. The control spillover is represented by 

. (2.57) 

Since eC(t) and x (t) will decay to zero, the control spillover will cause 

unwanted excitations of the residual modes, but it cannot shift the residual 

C 

mode frequencies. Consequently the control spillover results in some 

- 14 - 
unwanted oscillations of the residual modes. So when the observation 

spillover is absent, then the control spillover degrades the system 

responses but cannot destabilize the system. 

When observation spillover exists, then instability may occur when some 

of the eigenvalues of the composite system (2.40) are unstable. 

The control and obsesrvation spillover can be minimized if the 

actuators and sensors are located at (or very near to) the zero of the mode 

shapes of the residual modes. This concept is limited since it creates 

uncontrollable and unobservable triple (AC, Bc, Cc) that hinders the design 

of state estimators and state feedback laws. In addition, the freedom to 

locate sensors and actuators arbitrarily is rarely available to the designer 

since these locations are often already determined by structural 

considerations. 

I 

It is well known 1121 that the IMSC scheme is capable of eliminating 

control and observation spillover. However it requires the implementation 

of distributed actuators and distributed sensors that unfortunately cannot 

be provided by the current state-of-the-art. Consequently, discrete 

actuators and discrete sensors have to be used. A question that is still 

not clearly answered is how many discrete actuators and sensors are 

necessary for a successful implementation of the IMSC scheme. In addition, 

the IMSC scheme when being implemented by discrete actuators also has 

control spillover into the uncontrolled modes. However as pointed out in 

[12], the control spillover effect is not very important. 

Both control schemes, the coupled modal control and the IMSC require 

- that a closed-form solution of the eigenvalue problem exist. Unfortunately 

the vast majority of continuous system% icads to eigenvaiue problems that do 

not have closed-form solutions, owing to nonuniform mass or stiffness 

- 15 - 
distributions, Hence in this case, it is necessary to seek approximate 

solutions of the eigenvalue problem. 

3. Modelinq of the Polar Platform 

One of the project activities is to obtain a model for a generic polar 

platform whose physical parameters are given in Table 1. 

A MSTRAN finite element analysis package (McNeal Schwendler version) 

was employed to model the platform and its appendages by 52 modes. 

model is illustrated in Figure 3. 

The 

The platform itself was modeled by 28 

nodes, the solar panel and the astromast by 16 nodes and the engineering 

module by 8 nodes. 

In the developed model, the payload carried by each module was assumed 

to be distributed uniformly throughout all nodes. Also the engineering 

module was assumed to be very rigid as compared to the platform or the solar 

panel . 

The model considers only those degrees of freedom of the platform that 

contribute to its bending and torsional vibrations. 

Table 2 lists the first ten modes of Vibrations. The first two natural 

frequencies are the first rigid body modes in pitching and rolling. The 

remaining modes are the flexible modes of the platform. 

It can be seen that the flexible modes are very low and highly 

condensed over a narrow band between 0.054Hz and 0.448 Hz. A sample of the 

first three modes of vibrations are shown in Fig. 4-6. 

4. Project Research Resul tt 

In this section we will present some research results obtained by the 

principal investigators during the research period stated in the summary. 

One of the project objectives io to evaluate several candidate control 

- 16 - 
schemes in order to select the most appropriate control scheme for the LFSS. 

The study of coupled-modal control and Independent Modal Space Control 

involves in such problems as pole allocation, eigenvalue assignment, state 

estimator design, canonical transformation etc. ... . During the 

investigation of the above topics, some results have been found and are 

sumar i zed bel ow. 

4.1 Canonical Transformation 

Canonical transformation has been proven to be very useful in the 

design of state estimators and state feedback. In 1141 two canonical 

transformations were deweloped for a class of time-varying multivariable 

control systems. Since most physical systems such as LFSS are time-varying 

in nature, this type of time-varying canonical transformations will be very 

helpful in design of more practical control schemes for LFSS. 

4.2 Eisenvalue Assiqnment and Pole Allocation 

The eigenvalue assignment or pole allocation problem is very essential 

to the control of LFSS since it is related to the system stability. Two 

comon problems in control system design for LFSS are the selection of a 

state feedback gain to shift the natural modes of the LFSS to a set of 

desired damping modes, and the design of a state estimator whose eigenvalues 

can be arbitrarily assigned. These two problems can be treated as 

eigenvalue assignment or pole allocation problems. In E161 it was shown 

that eigenvalues can be arbitrarily assigned to a class of time-varying 

multivariable systems. Canonical transformation developed in [141 was 

employed to design the state feedback gain. The distributed control problem 

was considered in [17] where a control scheme was developed to arbitrarily 

afiocate closed-loop poles to a distributed time-inuariant system that is 

con trolled by several pairs of sensors and actuators. 

- 17 - 
4.3 State Estimator Desisn 

State estimator design is very crucial to the control of LFSS since the 

controlled state vector has to be estimated for the coupled modal control 

case and the modal state for the IMSC case. In 1181 a neu algorith was 

proposed to design reduced-order state estimators for a class of 

time-varying multivariable control systems that are uniformly observable. 

The result in [18] is an extension of that in 1151 where a full-order state 

estimator was designed. 

4.4 Simulation of Control Schemes 

Since the generic polar platform is relatively a complicated structure, 

we start the study of the control scheme by considering the control of a 

simply supported bean with one actuator and one sensor. Complete 

understanding of the control of this simply supported bean will help us to 

I 

I gain some insight of the control of the complete structure. The bean 

dynamics are modelled by the Euler-Bernoulli partial differential equation 

2 

u(x,t) + E1 ---- a4 u(x,t) = f(x,t) a 

at ax 

4 m ---- (4.1) 

For simplicity we set the mass m, the moment of inertia I, the modulus of 

elasticity E and the length of the bean to unity. The boundary conditions 

for this simply supported bean are 

2 

2 

aax 

2 

2 

a ---u(O,t) = ---u(l,t) = 0 

ax 

- ia - 
and 

The eigenvalue solutions for this case are 

2 

w = (kn) k 

Ok = sin (kix) 

The beam is controlled by a single point actuator at x = 1/6: 

.s 

f(x,t) = (1/2) 6(x - 1/61 f(t) (4.6) 

and the displacement is measured by a single point sensor located at x = 

W6: 

y(t) = u(W6,t) (4.7) 

Using the Matlab and the control tool box software package, an active 

I 

controller was designed to control the first three modes of the bean. To 

minimize the following unweighted energy 

J JE(t1 t 0.1f2(t)ldt (4.8) 

the control gain was determined to be 

KC = [-3.87 -3.86 -3.86 -1.01 -6.08 -20.31 (4.10) 

A state estimator was designed to estimate the controlled modes. The 

state estimator gain G given below: C 

Gc = 1711.8 52.34 -7.39 10.06 -5.97 1.981 (4.11) 

- 19 - 
will assign stable eigenvalues to the state estimator. Several simulation 

runs were made to study the performance of the active controllers and the 

effect of the control and observation spillover. 

In all simulation runs, an impulse disturbance was created to excite 

the system. 

spillover. 

reach the zero steady state. 

caused by control spillover into the uncontrolled modes. 

observation exists in this case, the system is till stable. Fig. 10 and 11 

display the sensor output and actuator input when no spillover exists. We 

notice that the displacement war decayed to zero. 

into the 4th and 5th uncontrolled modes is illustrated in Fig. 12 and 13, 

respectively. The eigenvalues of the composite system [Equation (2.4011 for 

all possible cases of spillover are given in Table 2a. 

eigenvalues of the closed-loop system, the state estimator, controlled 

subsystem and uncontrolled subsystem. 

Figures 8 and 9 present the effects of control and observation 

We note that when spillover exists, the displacement did not 

It has an oscillation with small amplitude 

Even though the 

The control spillover 

Table 2b shows the 

In Table 3a we note that when no observation spillover exists, the 

eigenvalues of the composite system consist of the eigenvalues of the 

closed-loop system, the state estimator, and the uncontrolled subsystem. 

This agrees with the discussion made in Section 2.3. 

5. Current and Future Research 

Continuing the research stated in Section 4, we currently focus on the 

study of spillover minimization. 

results showed that the system performance is degraded at the presence of 

control spillover. The LFSS can also become unstable if the observation 

As we have seen, the system simulation 
spillover exists. Therefore spillover minimization study is very crucial to 

the control of LFSS. We are investigating the minimization of the spillover 

by 

(a) Proper locating of actuator and sensors. 

(b) Determining a sufficient number of actuators and sensors needed. 

(c) Applying the IMSC method to the generic polar platform. 

The above study is carried out mostly by simulating the system for 

various situations using system simulation languages in order to set up a 

guideline for selecting and locating actuators and sensors. 

After the control of the simply-supported bean has been studied 

throughout, we will apply the resulting control scheme to the complete 

generic polar platform. 

be addressed in the future research: 

Several control issues such as given below should 

(a) Evaluation of the application of coupled-modal control and IMSP to 

the generic polar platform in terms of feasibility, reliability, spillover, 

and computational effort. 

(b) Implementation of an adaptive control scheme for the system. 

(c) Study of a robust controller when the system parameters are not 

well-known and when unpredictable disturbances are expected. 

6. Conclusion 

In this report we first updated the research related to project. 

modelling of the generic polar platform using finite element method was 

discussed. 

study using computer simulation and some theoretical research results of 

such problems as eigenvalues, state estimator, and state feedback design. 

Current research effort was discussed and future research activities were 

The 

We then presented some preliminary results of the control system 

assessed. 

- 21 - 
References 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

M.J. Balas, 'Active Control of Flexible Systems,' J. Optimization 

Theory and Applications, Uol. 25, No. 3, pp 415-436, July 1978. 

M.J. Balas, 'Trends in Large Space Structure Control Theory: Fondest 

Hopes, Wildest Drems,' IEEE Trans. Automat. Control., AC - 27, No. 3. 

pp. 522-534, June 1982. 

Michel A. Floyd, 'Coment on a Comparison of Control Techniques for 

Large Flexible Systems,' J. 

pp. 634-635, 1984. 

of Guidance and Control, Uol. 7, No. 5, 

A. L. Hale and G. A. Rahn, 'Robust Control of Self - Adjoint 

Distributed - Parameter Structures,' J. 

7, No. 3, pp. 263-273, 1984. 

of Guidance and Control, Uol. 

P. C. Hughes and R. E. Skelton, 'Controllability and Observability for 

Flexible Spacecraft,' J. of Guidance and Control, Uol. 3, pp. 452-459, 

Oct. 1980. 

D. J. Inman, 'Modal Decoupling Conditions for Distributed Control of 

Flexible Structures,' J. of Guidance and Control, Uol. 7, No. 6, pp. 

750-752, 1984. 

T. L. Johnson, 'Progress in Modelling and Control of Flexible 

Spacecraft,' Journal of the Franklin Institute, Uol. 315, No. 516, pp. 

495-520, May/June 1983. 

S. Kumar and J. H. Seinfeld, 'Optimal Location of Measurements for 

Distributed Parameter Estimation,' IEEE Trans. Automat. Control, AC - 

23, No. 4, pp. 690-698, Aug. 1978. 

L. Meirovitch, and H. Baruh, 'On the problem of observation spillover 

in Distributed - Parmeter Systems,' Journal of Optimization Theory and 

Applications, Uol 39, No. 2, pp. 611-620. 

L. Meirouitch and H. Baruh, 'Control of Self-Adjoint Distributed - 

Parameter Systems,' J. of Guidance and Control, Vol. 5, No. 1, pp. 

60-66, Feb. 1982. 

L. Meirovitch and H. Or, 'Modal Space Control of Distributed Gyroscopic 

Systems,' J. of Guidance and Control, Uol. 3, No. 2, pp. 140-150, April 

1380 . L. Wirovitch and others, ' A Comparison of Control Techniques for 

Large Flexible Systems,' J. of Ouidancs, Control, and Dynaaics, Voi. 

6, pp. 302-30, July - Clug. 1983. 

L. Meirovitch and H. Baruh, 'Tho Implementation of nodal Filters for 

- 22 - 
Control of Structures,' J. of Guidance and Control, Vol. 8, No. 6, pp, 

707-716, Dec. 1985. 

14. C. C. Nguyen, 'Canonical Transformation for a class of Time - Varying 

Multivariable Systems,' Int. J. Control, Vol. 43, No. 4, pp. 1061-1074, 

April 1986. 

15. C. C. Nguyen, 'Design of State Estimator for a Class of TimeYarying 

Multivariable Systems,' IEEE Trans. Automat. Control, AC-30, Feb. 1985. 

16. C. C. Nguyen, 'Arbitrary Eigenvalue Assignments for Linear Time-Varying 

Multivariable Control Systems,' to appear in Int. Journal of Control, 

1987. 

17. C. C. Nguyen, 'Pole Allocation for Decentralized-Control Systems,' 

Proc. Int. Conf. Modelling and Simulation, MSF, Williansburg, Sept. 

1986. 

18. C. C. Nguyen, 'Design of Reduced - Order State Estimators for Linear 

Time - Varying Multivariable Systems,' submitted for publication in 

Int. Journal of Control, 1986. 

19. D. Schaechter, 'Distributed Control of Large Space Structures,' Jet 

I Propulsion Lab Report, #81-15, May 1981. 

20. D. Turner and H. M. Chun, 'Optimal Distributed Control of a flexible 

Spacecraft During a Large - Angle Maneuver,' J. of Guidance, 

and Dynamics, Vol. 7, No. 3, pp. 257-264, 1984. 

Control, 

21. Sa hatu, 'Optimization of Sensor and Actuator Locations in a 

Distributed Parameter System,' Journal of the Franklin Institute, Vol. 

315, No. 516, pp. 407-421, May/Juna 1983. 

22. H. 02, and others, 'Some Problems Associated with Digital Control of 

Dynamical Systems,' J. of Guidance and Control, Uol. 3, No. 6, pp. 

523-528, Dec. 1980. 

23. H. E. Vandet Velds, and Ca R. Cariguau, 'Number and Placement of 

Control System Components Considering Possible Failures,' Journal of 

Guidance and Control, Vola 7, No. 6, pp. 703-709, Dec 1984. 

24. fl. Abdel - Rohman and H. .H. Leipholr, 'Automatic Active Control of 

Structures,' Journal of the Structural Division, pp. 663-677, March 

1980 . 25. 6. S. West - Vukouich et al, 'The Decentralized Control of Large 

Flexible Space Structures,' IEEE Trans. Automatic Control, Vol. AC-29, 

No. 10, pp. 866-8799 Octa 1984. 

- 23 - 
I 

I 

i 

I 

I 

i 

i 

i 

i 

i 

I 

l-4 

W 

cr; 

3 

c7 

n 

h. 

L 

- 24 - 
N discrete m discrete 

actuators sensors 

t 

- p1 

P -2 

* 

. 

LARGE FLEXIBLE 

SPACE STRUCTURE 

INTERPOLATION 

FJNCT ION 

plp2 - 

. ur (t) ' A 

A - MODAL 

FILTER 

-'OPTIMAL . 

' CONTROL Gr (t) 5- LAWS - 

T 

'mu(P,t) 1 

w 

G (PI t) 

1 - 

I 

c 

FIGURE 2: The Structure of the Independent Modal Space Control 

- 25 - 
u 

€- 

FIGURE 3: The Finite Element Model of the Generic Polar Platform 

- 26 - 
FIGURE 4: The First Mode of vibration 

- 27 - 
3: 

0 

W 

H 

FIGURE 5 : The Second Mode of vibration 

- 28 - 
FIGURE 6: The Third Mode of Vibration 

- 29 - 
FIGURE 7: The Fourth Mode of vibration 

- 30 - 
-- . .m I I 

- 31 - 
n I ---%en - \- 

--- 

- 

k 

al 

> 

0 

4 

4 

.I4 

a 

rn 

c 

0 

-4 

U 

a 

> 

& 

al 

m 

0 

2 

a 

k 

al 

> 

0 

4 

.4 

-4 

a 

rn 

4 

U 

c 

0 

u 

al 

c 

U 

w 

0 

al 

0 

c 

al 

m 

al 

k nl 

al 

c 

U 

U 

a 

U 

8 

H 5 

& 

0 

U 

a 

5 

U 

u 

4 

m 

w lx 

3 

tc 

.. 

- 34 

i 8 8 

n --- 

a 

c 

1- 

---E 

=--- 1 -4 

-4 

k 

a 

> 

0 

d 

d 

.d 

a 

rn 

C 

0 

4 

U 

ca 

a 

m 

42 

0 

a 

C 

ca 

rl 

0 

k 

U 

C 

0 

u 

0 

C 

G 

U 

.PI 

3 

U 

U 

5 

0 

Li 

C 

e 

cn 

d 

.-I 

w 

p: 

3 

E 

c 

34 

$ 

.. 

2 
--. 

-- 

5 

e 

- 35 - 

s- 

% -- 

a, 

TI 

0 

5: 

c 

JJ 

a, 

c 

JJ 

0 

JJ 

c 

.d 

Li 

a, 

> 

0 

.-4 

rl 

-4 

pc 

v) 

rl 

JJ 

c 

0 

U 

8 

.. 

N 

w 

a 

3 

E 

4 
0, 

a 

0 

c 

E 

c, 

v) 

0, 

L: 

c, 

0 

U 

c 

4 

Lc 

0) 

> 

0 

4 

rl 

-d 

P 

CI) 

I+ 

0 

Lc 

U 

c 

0 

U 

.. 

"I r? 

w 

0: 

3 

ra 

- 36 - 
PARAMETER VALUE 

OVERALL LENGTH OF THE PLATFORM 

PLATFORM WIDTH 

PLATFORM HEIGHT 

LENGTH OF SOLAR ARRAY 

WIDTH OF SOLAR ARRAY 

NUMBER OF MODULES 

MASS/MODULE 

LENGTH OF ENGINEERING MODULE 

HEIGTR OF ENGINEERING MODULE 

MASS OF ENGINEERING MODULE 

LENGTH OF STRUCTURAL MEMBERS 

STRUCTURAL MEMBER OUTER DIAMETER 

STRUCTURAL MEMBER INNER DIAMETER 

STRUCTURAL MEMBER MATERIAL 

DESITY OF THE ASTROMAST SUPPORTING 

THE SOLAR ARRAY 

ASTROMAST FLEXURAL RIGIDITY 

ASTROMAST TORSIONAL RIGIDITY 

45 ft 

6 ft 

6 ft 

60 ft 

15 ft 

6 

600.00 kg 

9 ft 

6 ft 

6 ft 

1 in. 

0.875 in. 

Aluminium 

5000.00 kg 

2 15.20 x10 lb in2 

2.2 kq/gt 

5 Mx10 6lb in 

TABLE 1: Physical Parameters of the Generic Polar Platform 

MODE frequency. in Hz 

0 

5.397 x 

5.457 x 

8.956 x lo-* 

1.404 x 10-1 

3.635 x 10-1 

7 3.837 x 10-1 

8 4.190 x 10-1 

9 4.301 x 10-1 

10 4.488 x 10-1 

TABLE 2: The First Ten Modes of Vibrations 

- 37 - 
TABLE 3a: Eigenvalues of the Composite System 

iio contr. spill. 

TABLE 3b: Eigenvalues of the Closed-Loop System, State Estimator, Controlled 

Subsystem, and Uncontrolled Subsystem 

- 38 -