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DEPARTMENT OF !^SECHANICAL ENGINEERING AND MECHANICSSCHOOL OF E^IGICIEERING

OLD DOMINION UNIVERSITYNCIRFOLK, '; IRGII7IA

CONTROi_ .'YSTEMS DESIGN FOR LAP.GE

FLEXIBLE SPACE STRUCTUR'=S

(N AS A-CR- 162582 ► CONTI+OL SYSTE"IS CESIGN FCRLATIGE PL EXIALR SPACE S?R iICTURfS Pi_nalF^eport, 15 Nov. 1977 - 14 .]an. 19A0 (Gi3Dominion Oniv. Research Foundation) 21 ^F7C A02/MP A01 CSCL 22P 63/18

BY

Suresh M. Joehi, Co-Principal Investigator

and

A. S. Roberts, Jr., Co-Principal Investigator

Final ReportFor the oeriod November 15, 1977 - January 14, 1980

c^repared for the:National Aeronautics and Space AdministrationLangley Research CanterYampton, Virginia

NAO- 11201

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JnderResearch Grant NSG 1473Nelson J. Groom, Technical MonitorFlight Dynamics and Control Division

January 1930

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https://ntrs.nasa.gov/search.jsp?R=19800006942 2018-06-27T12:26:21+00:00Z

DEPARTMENT OF MECHANICAL ENGINEERING AND `MECHANICSSCH()OL OF ENGINEERINGOLD DOMINION UNIVERSITYNORFOLK, VIRGINIA

I

^11^

CONTROL SYSTEMS DESIGN FOR LARGE

FLEXIBLE SPACE STRUCTURES

BY

Suresh M. Joshi, Co-Principal Investigator

and

A. S. Roberts, Jr., Co-Principal Investigator

Final ReportFor the ^eriod November 15, 1977 - January 14, 1980

National .aeronautics and Space AdministrationLangley Research CenterHampton, Virginia 23665

Research Grant NSG 1373Nelson J. room, technical MonitorFlight Dynamics and Control Division

01J Juminion Universit ,.- !-:esearch FoundationP.O. Box 6369Norfolk, Virginia 23508

Januar y 1,)80

'ONTROL SYSTEMS DESIGN FOR LARGE FLE^CIBLE

SPACE STRUCTURES

BY

S. M. Joshi l and A. S'. Roberts, Jr.`

SUMMARY

Large, lightweight space structures represent the basic requirement

of many potentially important, new, space initiatives. This report

contains a description of .he research performed under grant NSG 1473, in

the area of control syster..., design for large space structures (LSS).

Several approaches for the design of reduced-order LAG-type controllers

for LSS were proposed and evaluated using a continuous model of a long

free-free beam. Sufficient conditions were derived for the asymptotic

stability with this type of controller. A finite-element model of a

free-free-free-free square plate was obtained for use in control systems

studies. A method was developed for optimal damping enhancement in

LSS.

INTRODUCTION

Many of the potential)}^ important new space initiatives and mis-

sions which have been identified in reference 1 require large space

structures with dimensions which range from one hundred to several

thousands of meters. Example structures include very lar ge microwave

reflectors, microwave antennas, antenna platrorms, solar energy collectors,

radiators, solar sails, and telescopes. t4her^ the Space Shuttle becomes

operational, development of large space structures such as these will

become feasible.

Research Associate Professor, Old Dominion University ResearchFoundation, P.O. Box 5369, Norfolk, Virginia 23508.

Professor, Department of Mechanical Engineering and Mechanics, Old

Dominion Univers'.'.y, Norfolk, Virginia 23508

Control s}'stems design for structures of the size being contemplated

is a complex and challenging problem because weight and volume constraints

on the structural members will result in extremely low frequency bending

:nodes which are closely spaced in the frequency domain. Because of

stability and pointing requirements, a number of lower frequency modes

will probably fall within the bandwidth of the controller. Thus, the

active control of some of the structure.'. modes appears unavoidable. The

tools of r+^dern control theory, such as Linear-^?uadratic-Gaussian (LQG)

control theory, can be effectively applied to the solution of the problem,

but not using the standard techniques. This is because the order of the

model required to accurately describe the structure, or plant, will be

too high to permit a practical solution using standard LQG theory witt,

full-order estimators and regulators. The feasible solution, therefore,

is almost certain to consist of regulators and estimators of order much

lower than the plant.

Unfortunately, stability of the system with lower order regulators

and estimators is not guaranteed because of the possible excitation of

uncontrolled or "residual" modes. In reference 2, the very descriptive

terms "controller spillover" and "observation spillover" were introduced

to describe the unwanted forcing of the residual modes by the control

inputs and the unwanted contribution of the residual modes to the

observations respectively. If the residual modes have some natural

damping, the closed-loop system would be asymptotically stable in the

absence of either or both spillovers. However, spillover terms are

present in practice and must be considered i.n overall system performance.

This report consists of a summary of the research performed during

the perioa of the grant. For further information, readers are referred

to the papers and reports written by S: H. Joshi during the course of

the study.

REDUCED-ORDER LQG-CONTROLLcR5

The objective of this research was to develop a controller design

methodology for large space structures (LSS1. The first step towards

this objective was the selection of an appropriate model representing

the important 3yrlamic characteristics of LSS while being sufficientl}'

simple to be mathematiczlly tractable. These considerations lad to

the choice of a uniform free-free beam as the basic model for developing

controller design methodology. Reference 3 consists of the development

of a model of a free-free beam. Some interesting properties of the mode

Shapes were also ::erived ir. that report.

In reference J, several approaches to the design of reduced order

controllers fcr large space structures were presented and discussed.

These approaches were based on L^^ control theory and included truncation,

modified truncation regulators and estimators, use of higher order

estimators, and selective modal suppression. .also, the use of direct

sensor feedback, instead of a state estimator, was im^estigated for

some of the approaches. Numerical results were obtained for a long

free-free beam. In addition, sufficient conditions for as}'mptotic

stability were obtained in references 4 and 5. Reference 6 considered

a number of approaches to the LQG controller design for LSS, and also

proposed the use of "polynomial estim^tor5" for explicit estimation of

the observation spillover. Results of reference 6 indicated that tkle

modified truncation regulator and estimator (MTF and }iTE) would be

satisfactory design approaches. The direct sensor feedback (DSF)

implementation was also found to be superior under certain cen^iitions.

The use of a higher order estimator was also foun.i to be a potentially

useful method. However, the "po1^^Inonial estimator" method was not found

to be satisfactory.

FINITE ELEMENT MODEL OF A THIN FLATE

The research described above used a planar model of a free-free

beam. However, a more realistic three-axis model of a large space

structure is more desirable for controller design studies. Since

no such model was available, the task of developing a finite-element

model was undertaken. A 304.3 m < 304.8 m ^ 0.254 cm (100 ft ^ 100

ft X 0.1 in. thick), square aluminum plate was selected for this

purpose. A finite-element model was develope^3 us uiq a 25 Y 25 mesh,

3

by applying the SPAR program for structural anal} • sis. Modal frequency

and mode shape data for 44 structural modes (which represent about 30

kilowords) were stored on a magnetic taFe, and were also printed out in

a book form. This model should be useful to the NASA researchers

working in tt^e LSS control area. Figures 1 to 6 show some of the typical

mode shapes, and table 1 shows the modal frequencies.

MODAL DAMPING ENHANCEMENT IN LSS

It was found in the above described work that the closed-loop stability

of a LSS is heavily dependent on natural damping of the residual modes;

therefore, it is highly desirable to increase the damping of the residual

modes where possible. One method of achieving this is to use "member

dampers' (ref. 7). However, a systematic method is needed for the

selection of member damper gains. Therefore, a method for obtaining

optimal member damper gains wa.s developed (see Appendix). This method

has significant potential wF^^n used in conjunction with the methods

discussed above for the design of the primary controller.

CONCLUDING Re,M.'F.KS

The objective of t:.is project was to develop controller desi gn method-

ologies for LSS. To that effect, several approaches to the cont-oiler

design were proposed and discussed. It appears that tine LQG controller

theory is very well suited as the basic design tool. Of the approaches

considered, the Modified Truncation Regulator and Estimator (MTR and

MTE) design was found to be potentially the best, especiall} • when used

with Direct Sensor Feedback (DSF) implementation. The use of higher

order estimators has also shown promise. The importance of the natural

damping of the LSS to the closed-loop stability cannot be overemphasised.

The natural damping Should be increased wherever possible. To that

effect, a method was developed for the desi gn of optimal member damper

gains. Further investi gation is seeded in this area, and also in the

area of control actuator and sensor analysis and design.

-4

APPENDIX: OPTI^GIL MEMBER D.^1PER I;ONTROLLER DESIGN

FOR LARGE SPACE STUCTURES

INTRODUCTION

Control s y stems design for large flexible space structures is a complex

acrd challenging problem because of their special dvnamir characteristics.

Large space structures tend to have extremel y low-frequency, lightly damned,

bending mules which are close.v sh a;l i„ ^"e fre quency dom.rin. Because of

pointing requirements, a number of lowNr-frequency modes will probably fall

within one bandwiath of the primary controller, thus making active control of

some of t}re medes unavoidable. Since control of all the modes is impractical,

the primary controller will be reduced order. 'Phis introduces stability

problems because of observation rend control spillover irefs. -1, 5, 3). T'}re

stability of a s y stem with a reduced-order controller is heavily dependent

on the natural damping of the residual (uncontrolledl modes. Therefore, it

is desirable to increase the damping of the residual modes where possible.

The member damper approach and the application of multiple member

dampers in an output velocit y feedback cunfiguration were discussed in

reference 7. The member Gamper approach includes local damping elements

which could consist of collocated actuators and velocity sensors. Each

actuator/sensor pair is configured as a single-loop control system and the

member dampers work independently of each other. In the output velocity

feedback configuration, all the sensor signals are distributed b y a gain

matrix to interconnect all the actuators and sensors. This concept waS

further investigated ire references 9 and 10. It has been t^roved in these

references that direct velocity feedback (D^'FBl cannot jestabilice the

sy stem. Such controllers may be used in coniunction with a conventional

is

5

^- ±

modern) active controller, and have the potential to effect significant

improvement in the overall performance.

Selection of velocit y feedback gains for individual member dampers is an

important part of the design. The rout locus technique may be used for this

purpose; however, this could be a complex task, especially if a large number

of actuators are used. IrT this note, ttTe problem of selecting velocity feedback

gains is formulated as aTT optimal output feedback regulator problem, and

necessary conditions are derived for minimising a quadratic performance runct>.on.

The special structure of the gain matrix (i.e., diagonall is taken into account,

aTl d the knowledge of process noise and sensor noise is used to adi•anta^;e.

EQl1ATIONS OF ^tED16ER D,L^IfER CONTROLLERS

The structural model of a large space structure can be (approximately)

^^e^:^•rihed by the equations:

q + Dq + Aq = ^ T f (1)0 0 0 u o 0

y o ^oq o (^)

where qo is the no-dimensional vector of modal amplitudes; ^,^ is the

m X no "mode shape" matrix; f is the m X 1 generalised force vector

(components of f represent :applied forces or torques); }• is t}te m X 10

vector of generalized uisplacements (linear and angular) at the m point= of

application (e l , C ` , gym) of the generalized forces; U o is the> inherent

damping matrix, and /10 is the diagonal matrix of squared natural frequencies.

These equations describe truncated normal-coordinate continuous models, or

finite-element models.

If a member damper is coTinecte^i between two points, eq^ial and opposite

forces (torques), proportional to the sensed relative velocity (relative

b

angular velocity) between the point, are applied at the points. Thus, if

a single-member damper is connected between points t l and ^,,, the

equations are

q o D og o + ^oqo = [m l m,] ^r = [C' 1 - ^ 2 ] f (^)

where the n -vector m. is the ith column of the ^ T T,atrix, and f is theo i o

scalar force.

f = g l v l(^)

yl = (^i - ^2 ) qols)

where gl is ti)e D^'FB gain ( g l and y l are scalars).

Substitution of (J) and (S) into (.i) yie^ds

qo + ( Do - g l ^ l ^ 1 ) v, o + /lo g o = 0

(6)

where

^ I = ^ 1 - ^ 2 (^)

If p member dampers are used, the closed-loop equation becomes

PT

qo * ( Do - ^ g i ^i ^i ) qo + ^1og o = 0 (8)

i=1

where g. is the feedback gain and ^. the effective input matrix [similar1 1

to eq. (7)] for the ith damper. If g i < 0 (i = 1, _', ., p), and Do > 0,

then the effective damping matrix (coefficient matrix of qo in eq. 8) is

positive semidefinite, and the system is stable in the sense of Lyapunov;

if the effective damping matrix is positive definite, the s ystem is

as^^nptotically stable (ref. 9). It should be noted that this is only a

st^.ffic^_ent condition, and the s ystem can be asymptotically stable even thou:.

the effective damping matrix is only positive semidefinite.

OPTI^LaL OUTPUT' -EEDB.aCK FOR^NL>; I0^

=or the purrose oz controller design, n of the n o modes of the structure

are ccnsi.ered. i^ us, the "design model" is of order =:^. .although the member-

damper control s y stem is based on the lower order "design model," it cannot

destabilise :he higher order model. Therefore, this design does not suffer

:rcm the oroble:^s ass,^ciated with the use of reduced-order models in conventional

optimal regulator and estimator design. Let q denote the modal amplitude

vector ror the :nodes in the "design." T'he state equations :or the s;,stem under

consideration, including process noise and sensor noise, ma d• be written as:

x =.fix+ Buv (9)

z = '^T q + w' = Cx + '^

(10)

^..^here

q 0 I 0

x = A = B = (lI)

q _^OnXl !' -D ZnX2n W ZnXp

C = (0 `^T ^pXZn '^ ' ^'^1 '^2 'yn]nXp (1'--)

and where x is the ZnX1 state vector, u is tho pXl vector of effective

inputs, and v and ^a dre zero-mean. white process noise and measurement noise

vectors with covariance intensities V and IV. It should be noted that

C = BT . It is necessary to obtain an input of the type

u = Gz = G (Cx + w) (1^)

where

s

1I^. .

g l 0 0

,; = 0 g, J ll^)

J 0 $p

►.hick g ill minimi_e

.^

dim tl- ^ ^ x T`x ^ (GCx) T R (GCx ; ^ dt 115)._

f J

,.here ^ = QT ? 0, and R = RT > 0 are the state and control •.weighting matrices,

and ^ denotes expected value. The noise-deren^ent part of u is excluded :rom

t`;e perror:,^ance functi.n J since it ^lakes J unbellnded ;ref. 11), the octimal

Output IcPdb3Ck nrol,l^^m IJr 3 gzner3l nondiagonal rectangular G matrix '.:3s

SOlved 111 i'Cftrtn0t 11 . Hoa^ever, .ri thla case, 511101' li 15 d1a^O11;11 , LtIC minimi-

.at ion has to be performed 1^^^ : '1 respect to thr ^^ vari;lblr S 1 . ^_. ..g^^•

(If cross-reedbacks a-^e ai :^^^ee, the prcblem becomes t'le sane as t`le ^ene:'al

optima: output feedba,.:^ problem, with the constraint that the closed-loop syst.T

is stable.) Let g denote the ^^ector (, 1 , g`, ., gp)T.

Let t:^e s^ •^lbol ^ • 5 denote tale el^^ent b^' element aroduct tmatrix} or

matrices ^Y and S. T'lat is,

^ a 3 1 i ) _ ^i J ai)(16)

Define the ^. Y 1 vector-Lunt:. on ^ of a ^. X ^, matrix ,z as

all

1 1^ (^} _ -- (l-^

Yu ^ l

Y

,^._ The ;^e^e^_ary ,.on^di..^r.s for the m:... .._ of J .., equation

X15), with constraints ^f equations (9), (lUl, and (13), are given by

: - fR ,3T ^3) ^^ ( aT P B)^ -1 ^ 13T

-^°) i-^)

, .a + 3G3 r ) TP + P(.a 3GBT ) + ^ + 3GRG3 T = 0 (19)

(.a BG6T ) ^ + ^ (.a + 3GBT ) T « V + 3G^ti'GB T ^ (.0)

^^here P and .. are ?nX_n s'.--^etric matrices.

?roof.- T'he s:.ucture or _i • e proof is very similar to *_hat Uses? .or the ,enersl

optimal output feedbag: problem (ref. 11). (It should be noted that the proof

riven in ref. 11 needs slight modification ?.n light of ref. 1., although the end

result is t}^e same). The only difference is that the derivative of the Hamilto-

nian t.ith respect to the vector g (rather than the matrix G) i^ equated to

_ero. The following easily pre yed pr^^pertics of a matrix trace are used (G is

a diagonal matrix, n an^i 3 are square matrices of com patible dimension):

.g T. [Cif Gc] _-g t gT (^ S ) g ^ ^ ^ i - ^,^ 3)T ^ g ('-1)

3 Tr (G^J = ^(a)og

T,ze :act that C = 3T is also used.

It should be noted that, as in the case of the several opt:ral output

feedbac:c preblem, the t:,eorem does not gusr3ntee the existence or a g that

will maize the s y ste:^ asvmtotica L'^ • stable, although tia necessary conditions

as=.:me the °xistence. Indeed, the rerfor;^ance .`unction of equation (13)

will he meaningful only if such a g exists.

:ne opt.-al gain vector ^ ,^,a^• be computed using .ne algeri^nm si•:en

in reference 11, using equation (1R) or using a m.imerical minimisation

method (such as Davidor.-Fletcher-Powell).

('-=)

1 t)

CONCLUDING REMARKS

:ecessar^ • con^jitions ti:ere obtained for mini^^i=ing a quadratic perfor:^ance

:.:n^::on under the framewor}, of the member damper c^^ncept. Kno^:ledbe of noise

co^^ariancrs is used in the ^3esign. The ,.^etnod presented offers a systematic

approach to the ^:_ign of a class of controllers for en}lancing structural

damping in large space structures. This t^ •Fe of controller has si^nif',...:•^

,^otential if used in coniunction p ith a reduced-order optimal controller

t}lat is designed to control rigiu-body n^o.ies and some selected structural

modes.

11

REFERENCES

1. Outlook for Space: NASA SP - 386 and SP - 387, Jan. 197h.

:. Balas, Mark J.: .active Control of Flexible Systems. Symposium o^Dynamics and Control of Large Flexible Spacecraft. Blacksburg, VA,June 1977.

3. Joshi, S. H.; aiYd Roberts, A. S., .Jr.: Control S ystems Design forLarge Flexibl y Space Structures. Progress Report for NASA grant NSG1 ^3"3 , fay 1978 .

1. Joshi, S. H.; ,T.^i Groom, V. J.: Design of Reduce-Order Controllersfoi^ Large Flexible Space Structures. Proceedings of the loth ,annualAllerton Conference on Communication, Control, and Computing. Oct.4-h, 1978, pp. 780-791.

^. Jos}-i, S. ^1.; and Groom, ^. J.: Stability Bounds for the Control ofLarge Space Structures. AI.aA J. Guidance and Control, Vol. ^, No. •1,Ju' ^^-Aug . 1979.

Josh: S. fit.; and Groom, V. J.: Controller Design Approaches forLarge 'pace Structures Using LQG Control Theory. Second VPI andSU/AIA^. S^Tnposium on Dynamics and Control of Large Flexible Space-craft, (6lacksburg, VA), June _1-^3, 1979.

^. Canavin, J. R.: Control Technology for Large Space Structures.

AIAA Conference on Large Space Platforms (Los Angeles, Calif.),Sept. 1x78.

8. Balas, H. J.: Feedback Control of Flexible S ystems. IEEE Trans. Auto.Contr., Vol. AC-23, \'o. 4, Aug. 1978, pp. 673-679.

9. Balas, ^1. .J.: Direct Velocity Feedback Control of Large SpaceStructures. AI.4A .J. Guidance and Control, Vol. ', No. 3, ^tav - .June1979, pp. X52-253.

10. Strunce, R. R.; and Henderson, T. C.: Application of ^tod yrn ^1oda1

Controller Design to a Large space Structure. Second VPIF,SUSymposium on Dynamics and Control of Large Flexible Spacecraft(Blacksburg, VA), .June 1979.

11. .;^shi, S. ^1.: Design of Optimal Partial State Feedback Controllersfor Linear System: in Stochastic Environments. Proceedings of the

1975 IEEE Southeastcon (Charlotte, NC), Apr. 1975.

1_'. Walsh, F. ^1.: On Symmetric Hatrice5 and the Matrix Minimum Principle.IEEE Trans. .auto. Contr., Vol. AC-^^, No. 6, Dec. 197', pp. 995-996.

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Figure 2. :Node No. S.

15

Figure 3. Mode No. 9.

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'.'i^urc^ •3. Mods Nom. ^^^.

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Figure 5. Mode No. 33.

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Figure 6. Mode No. ^10.

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