Influence of structural flexibility on a single-axis linear attitude controller

February 1976

INFLUENCE OF STRUCTURAL FLEXIBILITY ON A SINGLE-AXIS LINEAR ATTITUDE CONTROLLER

. 1 SEP.

197e

by

Tarek Abde1-Rahman

UTIAS Technica1 Note No. 198 CN ISSN 0082-5263

..

INFLUENCE OF STRUCTURAL FLEXIBILITY

ON A

SINGLE~AXIS

LINEAR ATTITUDE CONTROLLER

Februa,ry,

1975

by

Tarek Abde1-Rahman

UTIAS Technical Note No.

198

•

..

Acknowledgements

This investigation was carried out as a research thesis in ·partial fulfilment of the requirements for an M.A.Sc. degree. The author wishes to thank his thesis supervisor, Professor P. C. Hughes, for suggesting this research topiç and for his assistance during its execution.

This work was sponsored by the Departmentfof Communications (Commun-ication~ Research Centre) under Contract No. OSR3-0017 •

- - - - _ . _ - - - -- - - .

Abstract

The effect of structural flexibility on a linear attitude control

syst"'em employing areaction wheel is investigated. The parameters of a

compensator in the form of (proportional + integral + derivative) feedback

are chosen in an optimal way: to minimize the real part of right-most r00t

of the system characteristic equation. This is done assuming the satellite to be rigide Then the effects of flexibility are investigated through

stability diagrams drawn by inspection of the real parts of the roots of the characteristic equation, showing stable and unstable regions • Root locus plots for different parameter values are also used to illustratethe effect

of flexibility on the roots. Increasing the structural damping and designing

the appendages as rigidly as possible are important means of insuring stability

of the control system. otherwise, the controller should be designed with

structural flexibility eXplicitly included.

The flexibility investigations have been carried out for two general types 0f flexiblevibration, namely 'rod-like' behaviour and 'membrane-like'

behaviour. A method to courrteract the destabilizirig effect that may arise due

TABLE OF CONTENTS

Acknow1edgements

Abstract

Notation

1.

INTRODUCTION

2.

CONTROL SYSTEM COMPONENTS

3.

2.1 Reaction Whee1

2.2 Compensator

2.3

An

Optimum Design Criterion

2.4 Control System Characteristics

2.5 Another Possible Compensator

INFLUENCE OF STRUCTURAL FLEXIBILITY ON ATTITUDE STABILITY

3.1 General Appendage Model

3.2 Formulation of the Characteristic P01ynomial

3.3 Stabili ty Diagrams

3.4 Root Locus Plots

3.4.1 Effect of Structural Flexibility on Root Loci

3.4.2 Effect of Structural Damping on Root Loci

3.4 .3 Effect of Appendage Inertia on Root Loci

4

~

ATTITUDE CONTROL SYSTEM IMPROVEMENT

5.

CONCLUDING REMARKS

REFERENCES

TABLES

iv

ii

iii v 1 1 1

5

5

8

9

10 10

12

12

15

15

35

35

35

38

39

w'

I

k kIl km

Kn

Zn

~=If/I w m w s

n

n NGrATION

Moment of inertia of spacecraf't about the axis (flexible

+

rigid)

=

I _f "" I _r

The inertia associated with the flexible part of the structure about the control axis

The inertia associated wi th the rigid portion of the struC'ture about the control axi s

Proportional gain factor of the controller Rate gain factor of the controller

Integrator gain factor of the controller

Double integrator gain factor of the controller Motor gain factor

Appendage modal gains Modal damping ratios

Relati ve inertia of the appendages

=

1/,-_{m m m}

(T

is the m::>tor time constant),.

38.40

se cs for CTS

=

1/'-_s

(T

_s is the sensor time constant) Natural frequencies of appendage vibration

1. INTRODUCTION

The dynamical interaction between spacecraf't atti tude control systems and flexible appendages has received considerable attention because many append-ages carmot reasonably be designed with sufficient rigidity to justify the optimistic assl.llIq)tion that the dynamic response to attitude control devices can be uncoupled from vehicle vibrations. When coupling is present, the result may vary from slight performance degradation to, instability •

This study was motivated by the Connnunication Technology Satellite (CTS). Reference 1 gives an overall description of CTS.

An

artist's in;pres-sion of the satellite appears in Fig. 1, and a conceptual diagram of the subsystems in Fig. 2. CTS is a high-power connnunication satelli'te a..n.d is scheduled for launch early in

1976.

The power needswill be supplied by two arrays of solar cells which generate about 1.2 kilowatts 0 These arrays make CTS interesting from a control standpoint, since they contribute the predominant structural flexibility.

The pitch axis (parallel to the orbit normal) of the satellite coincides

with the axis of the array. I t is evident that twisting motions of the array will excite, and will be excited by, the pitch motion of the satelli te and this fact has ramifications for the pitch attitude control system (Ref. 2). Assumingthe ideal symmetry which is suggested by the overall spacecraft design (Fig. 1), pitch/twist oscillations are not coupled to any of the other modes of oscil-lation. The principal dynamical element in the control system is a

nongim-balled :momentum reaction wheel whose axis coincides wi th the pitch axis. Thus roll and yaw are mutually coupled but not (ideally) to pitch. The twisting modes of the syrnmetrically deployed array may be further sub-classified into symmetric ones (in which both panels rotate equally one way, and the main body rotates the other) and skew-symmetric ones (in which the panels rotate equally in opposite directions, and the main body does not rotate). Only the former are directly of interest since only they involve the pitch atti tude motion of the body, where the sensors and antenna are located. References 2-5 have treated this satellite in a relatively straightforward manner by modelling the array blanket as a sin;ple membrane • A distinction is made between 'constrained' and

'unconstrained' modes of flexible vibration. The former concern the array only (wi th a fixed base); the latter one for the whole sp ace craf't • The constrained

modes will be used here since they are inherently characterizations of the appendages alone (Ref. 2), and they are thus easier to deal wi th when we come to choosing 'general ' appendage models. Mot i vated by the pitch attitude control

system of CTS then, a general 'class of single-axis attitude control systems is shown in Fig.

3.

The flexible appendage dynamics appears on the right-hand side of the figure •

2. CONTROL SYSTEM COMPONENTS 2.1 Reaction Wheel

The control actuator used in the control system shown in Fig.

3

is a reaction wheel. This device produces torques on the satellite by accelerating an inertia wheel (using the principa:).. of conservation of angular momentum) and , can be represented for most purposes as a linear system (Refs.

6,7).

Us basic

function is to counteract environmental disturbing torques which are cyclical

FIG. 1: Artist's Impression of the Communications Techno1ogy

FIG. 2:

'XTlNeU Ilr: lOL". AIU'AY

AUAY T~ACk''''' _ IE.OR

ARRAY UUNO'" 10011

CONTROL THRUSTl'!:S

____ -•• ", ... , . MOTOft ( NOT SHO'tWN)

TUN"1~ ORIIT tOUR CILLS

ATT.TUDf: CONT~OI. TH~USTl~S

HYOR:AZ IN( ",[TS

ts Ir., CATAll'TIC T"RUSTE."

CU AH)

TRACk.NO SUN UN_

Communications Technology Satellite as Deployed in Synchronous Orbit (from Ref. 1 )

8

S2 Kn S2+ 2Z

n

nn

s+n~ I S2 K~ I Ir--

1--'

S2+2Z2.Q2S +.Q~

.

I - _{Td(S) .} S2KI r-- ₂

_.0.

_.0.

2 S +2Z₁ IS+ I dIS) + E(s) ks Es (S) k+ k

i

+kos V (S) kms Tc(S) _{+ I} +

,

'"-

ws+s _Wm+S

~

_T

~

_S"2

-Sensor Compensator Reaction Wheel Dynamics

FIG. 3 BLOCK DIAGRAM REPRESENTATION OF SINGLE-AXIS CONTROL SYSTEM OF A FLEXIBLE VEHICLE EMPLOYING PROPORTIONAL + INTEGRAL + DERIVATIVE CONTROLLER

I

(Le., have zero average) • The adion of secular components of disturbing torque leads to a buildup 'in wheel speed until a saturation is met. Tt then becomes necessary to 'unload' or 'dump' the excess angular momentum to prevent the speeds of the reaction wheel exceeding its limit. The mst obvious neans for unloading is through the use of a mass expulsion. system. This system would be energized when the speed of the wheel exceeds a preset threshold. There are two principal methods of mechanizing such an operation:

(i) The appropriate jet is connnanded ON and remains ON until the wheel speed falls below a lower threshold. The reduction of wheel speed is accom-plished through the normal action of the wheel control loop, which treats the jet torque as a disturbing torque.

(ii) A change of control mode is initiated, the new mode controlling the attitude of the spacecraft p.l rely by use of jet torque. Simultaneously,

maximum deceleration torque is applied to the wheel. When the wheel speed falls below an appropriate threshold, thenormal control mode is re-enabled.

Reaction wheel unloading mayalso be accomplished through the use of magnetic torquing to save fuel (Ref.

7).

Recently proposals have been made

(Ref. 9) for unloading systems which make use of knowledge or estimation of disturbing torque time variations in order to operate both economically and with minimum frequency.

2.2 Compensator

The output voltage of the compensator is assumed to be the combina-tion of three signals. The first is proporcombina-tional to the error and, remenbering that this voltage goes into driving the wheel, Hs basic fundion is to improve the transient response. The second is proportional to the rate of change of

error; its merit lies in that it is a measure of how fast the error signal is changing and thus corresponds to anticipation. The third is the integral type, and is provided to get a constant steady-state attitude error in response to a constant torque.

A rationale must now be found for the selection of the compensator parameters k, kr and kD (see Fig.

3).

The criterion of minimi~ing the right-most root of the charaderistic equation of the control system will be applied

(Ref. 10). The control system charaderistics will then be obtained by using a Nyquist plot. The steady-state error in response to a constant disturbing torque is mentioned. AIlOther possible choice for the compensator will also be discussed.

2.3 An Optimum Design Criterion

The criterion chosen for optimization is the transient response, a measure of which is the real part of the right-most system root. The time constant T may be defined as follows: the distance from the imaginary axis to the right-most root is a

=

l/T. The optimization process then reduces to choosing the system gains so as to minimize T (i. e., to maximize a).

The attitude control system bloek diagram shown in Fig. 3 eau be reduced to that shown in Fig. 4, assuming the satellite to be rigid. The relationship between the old and new constants is:

k' ::;: _{k k kI/I} I s m k' ::;: k km

~/I

s

kiJ

=

k km _s

~/I

:E

::;: w

₊

w s m TI

=

w w s m

The eharacteristic equation of this system eau be written as follows:

(2.1)

(2.2)

Since k', k

n

and kj are free parameters, then to maximize the magnitude of the real part of the right ... moE;t root, the four roots have to be all located together at a distanee

a ::;:

:E/4

1

(2.3)

from the imaginaryaxis. This occurs when the controller gains are seleeted as follows:

kJJ

k'

kj

For the values (typical of CTS)

::;: 6a~ I - TI = 4a3 I = a4 I

w

=

1/3840 sec~~ In

(2.4)

+ k Ir + kiS + kiD S 2

FIG. 4 REDUCED SINGLE-AXIS ATTITUDE CONTROL SYSTEM BLOCK

DIAGRAM ASSUMING THE SATELLITE TO BE RIGID

I

/

/

I

-. / .... , / ~-_/

\

\

"'"

---

~

/

/

8

\

\

/

/

FIG. 5 NYQUIST PLOT FOR THE OPEN LOOP TRANSFER FUNCTION OF SINGLE -AXIS

in particular,

(2.3)

and

(2.4)

give

al

=

0.125

(2.6)

k'

=

7.82

X

10-

3

2.4

Control System Characteristics

Figure 5 shows a Nyquist plot for the open loop control function shown

in Fig.

4,

(2.7)

The characteristics of the control system for the gains given in

(2.6)

can be summarized as follows:

Gain margin Phase margin

Phase crossover frequency

(w )

c

Gain crossover frequency

(W

_g)

Band width Resonance peak Resonant frequency

=

0.199

=

14.0

db

=

0.761

rad

=

43.6°

=

0.557

sec-~

=

0.181

sec-l.

_(2.8)

=

0.3~6 sec -~

=

1.61

=

0.113

sec-;t

Applying the final value theorem, the steady-state error in response to constant disturbance torque T

d is: 8

(t)

lim T d lLI S2 Tdll

=

s -+ ~s2',

=

k k

k

I ss S-70 s

+..L.

k k (kl + ks 1 s m s m I S2 (S2

+

L: s +

TI)

- -- - - -- - -

-2.5 Another Possible Compensator

e

This section is copcerned with the discussion of the possibility of

adding a double integrator to the controller. Thus °the controller bloek diagram

will be as shown in Fig. 6.

The characteristic equation will be, in this case,

(2.10)

where,

(2.11)

Applying the same definition of optimality given in Section 2.3,the five roots

of the polynomial can ,again be located altogether by choosing all

=

l/T as

follows: ~ aU

= -

₅

k];

=

10a2 II - TI kt

₌

_10a~I

_(2.12) kJ:

=

5a4 ₁₁ 5 kJ:I=,oa II

For the values of ~ and TI given in (2.6) ,the numerical values for

(2.12) are, approximately, all

=

0.100

kiJ

=

0.100 kt '"

=

10-2 _(2.13) kt

₌

5.01 X 10-4 I kIl

=

10- 5 9

The steady-state attitude error in response to constant disturba.nce~torque is zero. Thus the addition of a double integrator iII!Proves the steady-state response bu,t deteriorates the transient response, since all

<

al'

3.

INFLUENCE OF STRUCTURAL FLEXIBILITY ON ATTITUDE STABILIT:(

The effe cts of structural flexibility are now investigated by consid-ering the interaction of the control system with a general flexible appendage. In an effort to limit the number of appendage parameters to as few as possible while at the same time retaining the greatest possible generality, the sugges-tions made in Ref. 2 will be adopted.

3.1

General Avpendage Model

Firstly, concerning the natural frequencies of vibration (n~, n2 , ••• ) it is noted (Ref.

2)

that generally

(3.1)

The bounds in

(3.1)

are not always satisfied for all structures. However they seem reasonable general estimates of the magnitude of nn. Since nn

=

nnJ. is satisfied by strings, membranes, etc., the lower bound in

(3.1)

will be called

'm.embrane-like' behaviour. Similarly, since nn

=

n2nJ. for a rod (at least for large n), the upper bound in

(3.1)

will be termed 'rod-like' behaviour.

Reference 2 goes on to suggest that the modal gains(K;I.., 1\2, ••• ; see Fig. 3) might reasonably be expected to be estimated by

K

=

(

I f ) P

/n2

n I M for membrane-like behaviour, where

while, for rod-like behaviour

where

(3.2)

Es(S)

bL__

~ ~

- _

k

+

k

oS

+

S

+

$'

v

(S)

FIG. 6 PROPORTIONAL + DERIVATIVE + INTEGRAL + DOUBLE INTEGRAL

CON.TROLLER BLOCK DIAGRAM

kll+kIS+kIOS2

5

2

(S2.,. IS

+1T')

--"

FIG. 7 THE REDUCED SINGLE-AXIS ATTITUDE CONTROL SYSTEM BLOCK DIAGRAM FOR THE FLEXIBLE SATELLITE

In

(3.2)

and

(3.4),

If is the inertia of the flexible portions of the spacecraft, and I is the inertia of the entire spacecraft. Clearly,

1

O<-!<l

- I -

(3.6)

Finally, as to damping, in the absence of any better ass~tion, it will be assumed that

z~

:

= Z2 = Z3

= ••. =

Z

The advantage of the model described in the preceding paragraph is that only three appendage parameters emerge (DJ., If/I and Z) plus a decision

on rod-like vs mernbrane-like behaviour.

3.2

Formulation of the Characteristic Polynomial that

The block diagram shown in Fig.

3

can be reduced, for Td(s) shown in Fig.

7,

where 00

\ ' {S2K

.n

(S2

+ 2Z.D. s +

D~)}

() L

n J.=l J. J. J. Gfn s _n=l irn = ""'"Gf--d-.C-s ... )

TI

(S2

+

2Z

n

s

+

D2) n=l n n n

=

0, to

(3.8)

An

approximate value for Gf(s} can be obtained by employing only N modes of constrained flexible vibrations. Thus

3.3

stability Diagrams

Two groups of stability diagrams have been generated both for rod-like behaviour and for mernbrane-like behaviour. In the first group, two flexible modes have been incorporated. These stability diagrams, preparedUsing If/I

(=

f3)

and

n

jw

_c as parameters, have been generated by a direct investigation of the real parts of the roots of the characteristic polynomial. The subroutine DPRBM was used which ~mploys Bairstow' s me'thod for finding the roots of a polynomial.

Figures 8-10 show stability diagrams for Z = 0.001, .01 and 0.05, respecti vely, and rod-like behaviour. Figures 11-13 show stability diagrams for Z = 0.001, 0.01 and O.OB:j and membrane-like behaviour.

10.00 1.00 Ron-LIKE OEHAVIOUR ++++++++++++++t+++++++++1++++++~~+++++++++++++++++ ++++++++++++1+~+++++++++++++++++++++++++++++++++++ +++++++-+-+++.+.+.++-+-.++.+++++++.+++.+.++++++-+-++++++ +T+++.+++++++.++++++.+++++.-+-+-+-++++·.+·++-+-·.++·++.+ ++++++++++++++++++++++++++++++++++++++++++++++++++ ++.++++++++++.++.+.+-+-+-+-+.++++++++.++.++.++-+-+.+.++. ++++.++ .... ++ +++ ... +++ .... t ++'+++1 ++++++++-+-+++++ • • -+-+-+-+++ ++++++++++++++++++++++1+++++++++++++++++++++++++++ +++++-+-++++++.++-+-.++-+-++++-+-++++++++." • • ++i.+-+--+--+-.++-+-+ +++++++++~++++++++++++++++++++++++++++++++++++++++ ++++++++++++******.*********§k~*****.**.********** ++********~*.********************~*****~********** ++***~*****(~~J*******«~**~******~~***~~********** +++*****~**.~****~**~**~.~***~««~~******«.*.**«.** ++++++***~*****~~C**XX~«~~:~~*~~*):~***«~*~******* +++++++++++****«********~)**~**.~*~».**~*.»****.** ++.+++++++++++~++++++*************~***«~*~**~***** ++++++++++++++++++++++++~++++++++++++++++********* +++++++++++++++~+++++++++++t++++++++.+++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ O.10-+---~---r---~---r---, O.U 0.2 0.4 0.6 BETA

FIG.8 STABILITY DIAGRMl

IZETA.0.001.NU~·.BER OF CONSTRAINEll ~;OllES INCLUDW=21

* = UNSTAHLE + STA ALE

B POINT ON THE BOUNDARY

.al Wc Ron-LIKt REHAVIOUR u.5574t 00 0.442tlE 00 0.3517E 00 U.27Y4t 00 U.221Y~ 00 J.1763E !.JO U.14UOE. UU 0.!112E 00 0.U835t-vl U.7018E-Ul 0.,574l-01 U.44ldE-U1 0.3517t-Ol U.n~4E-Ol 0.221~E-Ol U 01763E-0 1 0014001:-01 0.1112E-Ul U.8835.-02 U.7U1UE-02 OMt('A 1 10.00 ++++++++++++++++++++++++++++++++++++++++++++++++++ U.5574E ou ++++++++++++++++++t+++++++++++++++++++++++++++++++ 0.442~E uo ++++++++++++1+++++++++++t++++++++++++++1++++++++++ O.3517E 00 +++++++++11++++++++++++~++++++++++++++++++~+++++++ O.l7~4E 00 +++++++++1+++++++++++++~1++++++++++++++++~++++++++ 0.221~E 00 ++++++++++++++++++++~+++++++++++++++++++++++++++++ O.17b3E 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.14UOE 00 +++++++++++++++++++4+1++++++++++++++++++++++++++++ u.1112E OU +++++++++++++++++++++.++++++++++++++++++++++++++++ 0.d835E-01 + +++++++++++ +++4 + + +++"H ~++++++ t.,. ++++++ 1 +++++++++++ ü. 1018 E-O 1 1.00 +++++++++++++++++++1++++++++++I+++t+++++++++++++++ U.~~74E-01 ++++++++++++++++++++++4+++************************ 0.4428E-01 +++++++++++++.++++++++++++++**~******************* U.3~17E-U1 ++++++++++++++++++++++++++++++++++~++++++********* 0.2794E-01 ++++++++++++++++++++++++++++++++++++++++++++++++++ O.l219E-01 +++++++++++++++++++++++++++T++++++++++++++++++++++ O.1763E-01 ++++++++++++++++++I++lt++++~++~+++++++++++++++++++ U.14UUE-U1 +++++++++++++++++++4+1++++1+++++++++++++++++++++++ u.1112E-Ol ++++++++++++++++++++++ I ++1+1"++ I I ++t-+1 1+++ +-+ t++++++ U.tl83SC:-02 ++++++++++++++++++++++++++++++++++t++1+++t1+++++++ U.7U1~E-02 O.10-t---~---~---~---~---, 0.0 0.2 0.4 0.6 1.0 BElA

FIG. 9 STMILITY OIAGI<Mol

IZETA.0.Ol0.NU~OER OF CONSTRAI~En ~UDES INCLUllEll-ZI UNSTABLE

STAALt

A POINT ON THE BOU~DARY

Rcn-LIKE REf~VlnUR OMEuA 1 10.00 ++++++++++++++++++++++++++++++++++++++++++++++++++ ~.5574E 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ J.442öE OU ++++++++++++1+++++++++1+++++++++++++++++++++++++++ O.3S17E UU ++++++++++++++++++++++++++++++++++++++++++++++++++ u.2794E 00 ++++++++++++++++++11++++++++++++++++++++++++++++++ O.2219E 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.1763E 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ Q.1400E 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.1112E 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.d835E-01 ++++++++++++t+t+++++T++++++++++++++~++++++++++++++ u.701df-Ol 1.no ++++++++++++++++++++++++++++++++++++++++++++++++++ U.S574E-Ul ++++++++++++1+++++++++++++++++++++++++++++++++++++ 0.4428E-01 ++++++++++++++++++++++++++++++++++++++++++++++++++ U.351·(~-Ol +++++++++1++++++++++++++++++++++++++++++++++++++++ U.2794E-Ol ++++++++++++++++++++++++++++++++++++++++++++++++++ O.221Y~-Ul +++++++++++++++++++++++t+++++++++~++++++~++++++~++ 0.1763E-01 +++++++++++++++++++++++++++++++++++++++++++++.++++ O.14UOE-Ol +++++t++++++++++++++++++++++++++++++++++++++++++++ 0.1112E-Ol +++++++++++++++++++++++++++~++++++++++++++++++++++ O.ö835E-02 +++++++++++++++++.++++++++++++++++++++++++++++++++ u.7~ldE-U2 0.10 0.0 0.2 0.4 0.6

FIG.IO STAOILITY DIAGRA:"

0.8 1.0

BETA

IZETA"0.D5a.NU~HER OF CO~STRAI~ED MODES INCLUDEll"21 U.'ISTAHLf

STARLE

~EMBRANE-liKE HEHAVIOUR U;·H,GA 1 10.00 ++++++++++++++++++++++++++++++++~+++++++++++++++++ ++++++++++++++++++++~+++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ ~.14vu~ Ju ++++++++++++++~+++++++++++++++++++++++++++++++++++ u.1112e vU +++++++++++++++++++.++++++++++++++++++++++++++++++ U.bb~5E-Ul ++++++++++++++++++++++++++++++++++++++++++++++++++ O.7ül~~-vl u.5574E )JU U.4428~ 00 v035l'1E vO J.2794~ ou U.2219~ Uu 0.1763. ue 1.00 +++++++++++++++++++**.************«*~*.~.~***.**** U.5~74E-Ul ++++*******~**~.**.**c«r*~*~*»*o******~~«~~*****.* O.442B~-Ol ++++**.************.~**~9 •• *******.*.*~~* •• *.***** U.3517E-Ol ++++++*.********~~**.**~~***.*****~***.~~.***.**~* O.21~4~-01 ++++++++++*********~*******«*~*«c**~**~~~**.*****' O.221~E-Ol +++++++++++++++++**.*.*.***************.****.***** O.1763~-Ol ++++++++++++++++++++++++***************~***.****** 0.1400~-01 ++++++++++++++++++++++++++++++++++.+++++********** O.1112~-~1 ++++++++++++++++++++++++++++++++++++++++++++++++++ u.U~35~-02 ++++++++++++++++++++++++++++.+.++++++~++++++++++++ U.7U1Hf-U2 O.IO~---T---r---~---r---' 0.0 0.2 0.4 0.6 0.8 1.0 ~ETA

FIG. 11 STABILlTY DIAGRA~:

IZETAcO.001.NU,',BER OF CO~STRAINED !.ODES INClUUtD=2.1

* = UNSTABLE

+ • STABlE

• POINT ON THE AOU~DARY

~E~A~A~E-lIK~ "EHAVIOU~

10.00 ++++++++++++++++T++++T++++++++++++++++++++++++++++ 0.5574~ 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ O.442b~ ua ++++++++++++1++++++++~++~+~+++++++++++++++++++++++ U.351/E vU +++++++++++++++++++++++++T.+++++~+++++++++++++++++ O.2194~ OU +++++++++++++++++++++++T++++++++++++++++++++++++++ 0.2219E 00 +++++++++++++++++++++++++++++++++++++++~++++++++++ ü.17ó3E ua ++++++++++++~++++++++++1+++T++++++++++T+++++++++++ 0.1400~ 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ O.1112E 00 ++++++++++++++++++++++~+++++++++++++++~+~+rT++++++ o.aij35~-Ol +++++++++++++++++++++++++++++++++++++++++.++++++++ O.701tlE-Ol 1.00 ++++++++++++++++++++++++++++++++++T+++++++++++++++ O.5574~-U! ++++++++++++++++++++++++++++++++++++++++********** 0.44lB~-01 +++++++++++++++++++++++++++++++++++++++++++******* O.351/~-vl ++++++++++++++++++++++++++++++++++++++++++++++++++ ü.27~4E-Ol ++++++++++++++++++++++++++++++++++++++++++++++++++ O.2219~-OI ++++++++++++++++++++++++++++++++++++++++++++++++++ O.1763E-Ol ++++++++++++++++++++++++++++++++++T++++++T++++++++ 0.1400E-Ol ++++++++++++++++++++++++++++++++++++++++++++++++++ O.1112E-Ol ++++++++++++++~+++++++++++++++++++T+++++++++++++++ O.ijB35~-02 ++++++++++++t+++++++++++++++++++++++++++++++++++++ a.701SE-a2 O.lC~---~~---r---~~---r---' 0.0 0.6 BETA

FIG. ~ STABIliTY DIAGRAM

IZETA=0.010.NU~BER OF CONSTRAINED MODES I~ClUDED=4)

UNSTABlE STARlE

• PO INT O,~ THE BOU~DARY

~;EMRRANf:-Ll KE ~~IlAV IOUH

10.00 ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.5574~ 00 +++++++++++++++++++++1++++++++++++++++++++++++++++ O.442~~ uu ++++++++++++++~+++++++++++++++++++++++++++++++++++ 0.3511~ 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.~794E OU ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.Z219E uu ++++++++++++++~+++++++++++++++++++++++++++++++++++ O.1763E 00 ++++++++++++1+++++++++++++++++++++++++++++++++++++ 0.140~E 00 ++++++++++++++++++++++++++T++++++++++++T++++++++++ u.l11~t 00 ++++++++++++++++++++++++++++++++++++++++++++++++++ J.~U35ë-Ul +++++++++~++++++++++++++++++++++++++++++++++++++++ U.7018~-Ul l.no +++++++++.+++++++++++++++++++++++++++++++++++++++++ U.SS74ë-Ol +++++++++++++++++++++T++++++++++++++++++++++++++++ O.442H~-Ul ++++++++~++.+++++++++++++++~.+++++++++++++++++++++++ U.3517E-Ol ++++++++i+++1+~++++++i++1i++++'I+++T++++~T+++++++++ O.2794~-01 +++++++++1+ 1 ++I+i++++++++T++++++++++++++++++++++++ U.221~~-Ol +++++++1++++++++++++++++++++++++++++++++++++++++++ O.1763E-01 ++++++++++i+1+i++++++++++++i++++++++~+TT++++++++++ v.14UUE-Ul +++++++++++T~~++++T+++++++++++++++++++++++++++++++ O.lll~~-Ol

+++++++++i·++i+++++++T+i+++++++++++++++++++++++++++ U.HH3SE-02

++++++++++++++++++++++++++++++++++++++++++++++++++ U.7018ê-U2

0.4

SElA FIG. a STASIliTY DIAG~A~

(ZETA=C.05C,!!U··'AEf~ OF C(I,\~sn'(~\I,'I;:G '<iOI>ES lr"ClUDED=2)

---~---

-In the second group four flexible modes have been incorporated. The resulting stability diagrams coincide exactly with those shown in Figs. 8-13. This implies that the third and fourth modes, at least under the current assumptions, do not contribute materially to the stability question. Figures 8-10 were also checked using the Routh-Hurwitz criteria. The resulting figures

coincided exactly, as expected.

A second check, using Nyquist's stability criterion, was then carried out for Z

=

0.001, 0.01, 0.05 and for both rod-like and membrane-like behaviour. The results for ~

=

0.1 are shown in Tables 1 and 2. I t is clear that these results agree wi-th those obtained with ~

=

0.1 in the stability diagrams. Of course they give much more information, gain and phase margin, for the stable cases. Figures 14 and 15 show sketches of the open-loop transfer function for a typical stable case, and a typica~ unstable case, respectively.

From the stability diagrams it is clear tbat the unstable region _

shrinks as the structural damping ratio Zincreases until the unstable region disappears completely for a sUfficiently high Z (perhaps as low as 0.05). Secondly, for a gi ven structural damping ratio, the unstable region becomes more wider as ~ increases (i.e., as a greater fraction of the satellite is flexible) •

even can, ages

3.4

Of interest is the possibili ty of a stable region in which n~ < wc, for low Z. In this region the satellite is so flexible that the controller in effect, control in a quasi-steady marmer. The torques from the append-on the main body change append-only very slowly.

Root Locus Plots ( ~

Deeper insight into the effect of flexibility on the transient response is now obtained by studying the effect on the roots of the characteristic equation of varying n~. The latter indicates the degree of flexibility of the appendages. Root loci for both rOd-like and membrane-like behaviour employi.ng both two and four modes are shown in Figs. 16-51. These figures show that the roots obtained by using two constrained modes agree very well with the corresponding roots obtained by using four modes. The largest deviation (still small) oecurs in the higher frequency roots for membrane-like behaviour, very low structural damping ratio, and ~

=

0.9 (as in Figs. 38-39). The instabili ty for a certain interval of n~ is seen to occur dueto the crossing by three roots of the

imaginaryaxis (in the case of four modes), and by two roots when two modes are used.

In each root locus the effect of flexibili ty (obtained by varying n~) on the roots is displayed for specific values of the parameters Z and~. The values of the structural damping ratio parameter Z have been chosen, as before, to be 0.001, 0.01, 0.05 to represent very low, low and moderate structural

damping, respectively. The values for the parameter ~ (namely 0.1, 0.5 and 0.9) have been chosen to represent small, moderate and large appendages, respectivelJl:. A discussion of the significant trends in these root loci follows.

3.4.1 Effect of Structural Flexibility on Root Loci

For gi ven (Z, ~),

nJ.

may be considered as a measure of flexibility;the rigidity increases as n~ increases. If the appendages are designed with sufficient

lil 25 >-~

g.

~ < ::;: H

F--IG· 14 NYQUIST PLOT

·25

( N=4 • 8ETA=0.i 9 ZETA=0·010 • OMEGA 1=0.04428 • 'ROO-LIKE 8EHAVIOUR'

lil 25

-70 50

RE,A,L 5

-25

FIG,15 NYOUIST PLOT

-... _ .... -.. _ - - - -- - - -- -- - - ---,:,.,.--- - -- - - , .1 -lC 1C' 01 '>- . ~ J Cl." .,. ~ H X. ~ _{~ .,} ~. 10

..

10 0 ~O ·1 10 r.MfIjA 1 sc X 10 .~ 10 1-00 ~EAL 5

FICj, 16 ~OOT LCCUS ( N=2 , BUA=Od , ZUA=O.OOl '~CC -LIKE BEHAVICUR' )

l + = BAr--r; WICTI-i ) ~1 >- 1 Cl.". iO :!;; "'-. H L~ ..: ~

,

H 10 , 10 iO 0 ·1 10 lJMUjA 1 50 X 10 ..,. .J 10 l-OO ..> ~)-10 ·1 o"a~t) r --1(' REAL 5

I="IG, 17 ~OClT LOCUS l N=4 8ETA=O·i • ZETA=O·DD1 '~OO'-LIKE 8EHAVIOUR' )

, + = BANC WICTH )

·3

3 ·-10 U1 >- ei Cl:: 10 ~ H ~ X. _{~ 2} H ₁₀

,

10 10°

-

,

10 OMEG'" 1 50 X 10 '7 ₁ Ö-1·00 A 0·10 ~ 10. noc; ~ RS'IL 5

FIG- ~ ROOT LOCUS ( N=2 9 BETA=O.5 • ZETA=O-OOl 'ROO-LIKE BEHAVIOUR' )

( + = 8"'1\0 WIOTH ) Ul >- 3 Cl:: 10 ~ H ~ < ::::> ₂ H ₁₀ 1 10 0 10

-

,

10 OMEG'" 1 50 X 10 ..." 10 1·00 A 0·10 <l 10.005 ~ REAL 5

F IG, 19 ROOT LOCUS N=4 • BETA=O,5 9 ZETA=O·OOl 'ROO-LIKE BEHAVIOUR' )

-3

·._--_._- - - . " . . , - . - - - , :';1 >- ,~ .. (l:' ~ H x _~ ::i'. ., ..., 10' 10~ kO ") ·1 10 " )

V-.• r-, .L.' . .;

...

, {'Ir-, .L. ~ ...J .... J

..

G~ iO 'J (' ~ r-r.ç.~ ~ RE ... ·L 5

nel' 20 ~(JOT LOCUS ( N=2 , aETA=O,'~ ~ ZETA=O"OOl '~OO-LIKE 8EHAVIDUR' )

( + = aAi'[] wroTH ) tn ~. 1D 3 ..:: z H l~ <. :::> 10' H 10' 0 iO

r

-

·1 10 ~,r-. ~ OM':::;" 1 -. SO x 10

.,.

.:. • ~"'IC' ~ c-iO .;! ';""";"'~'Ol- ~

ÇI~· ~ ~ODT LOCUS ( N=4

REi'L 5

2ETA=O,g • ZETA=O,ClOi '~OO-LIKE 2EH.WIOL.!R' )

l + = BANC WICTH )

·3 ~o

3 '-10 L1 >- 3 CY, ::"0 ~ H 1éJ <. ::E ₂ H ₁₀ i 10 10 0 -1 10 C».1EGA 1 50 X 10 v 1Ö2 1·00 L>. 0·10 <I 10·005 ~ REAL 5

FIG. 22 ROOT LOCUS ( N=2 , BETA=O.l • ZETA=O·010 'ROO-LIKE BEHAVIOUR' ) ( + = BAI\O WIDTH ) U1 >- 3 (Y. 10 ~ H 1éJ <. ::E 10~ H 1 :1.0 0 '10 .. 1 10 OMEG." 1 :;0 X W '7 2 10 1·00 _"" 0-10 <I 10.005 t· 3 ·10 REAL 5

. FIS, 23 ~O!JT LOCUS ( N=4 , BETA=O.i • ZETA=O,010 'ROO'-LIKE BEHAVIOUR'

-3 10

CM:Gf.. 1

50 X

10 v

1·00 L>-0-10

FIG·24 ROOT LOCUS

().EGA 1 ~ ' X 10 '7 1.00 A 0-10 IJl

t

103 H

~

10" H 101 100 1Ö1 -:> 10 REAL 5

N=2 , BETA=O.S , ZETA=O.010 'ROO-LIKE BEHAVIOUR' )

( +

=

SM[) wroTH ) IJl 3 10 10" 1 10 o 10 -1 iO -:> 10 ~3~~~~~~~~~~~~~~~~~~~~~~~~~~~--~~~~~~~~~~~~~~~10~-3 ~iO REAL 5

FIG· 25 ROOT LOCUS ( N=4 • BETA=O·S , ZE:TA=O·010 'ROO--LIKE BEHAVIOUR' )

~---,r---' x 'jO X 10 .1,QQ ,,::., O,lQ '1 (1, (n'; ~ RE:AL 5 l + -:: BAND WIDTH ) U1 -1 10

__

---~ r---Ul ft 1.;) -, <C Z H L'J

..-::;> 2 H ₁₀ l ii) c iO 1 10 -_I 10 --; 1.D ~,~~--~~~L-~~~J-~~~~~~~~~~~~--~~~~~~~L-~S~· ~~ __ J~-L~~~~~~WL~~~~ 3 'l~"l -:h' -1,)'" -10"'; -10 -iC 1D lC'l RF.'\L 5

-

-

---

-

-

---

-

--

_ . _ - - - -

- - - . , . , . . . , . - - -

_ . _

-.i. .. ' :l ·iO :":;0 .iO ~ 'Jo;; l~) c: ,. ,:-.:r.t.: I x 5C X 10 "" 0-10 <I ". "0<; ~ '. REAL 5 .~ ... 10,.' '1 10 -2 10

Fm,28 RCCT LCCUS ( N=2 , Br_TA=O,i ~ ZETA=Oo.050 'ROO-UKE. BEHAVIDUR' 1

X ':" u ·1 ~ l + = 8AND WIDTH ) UI ~ 10 ~ H L'J -< ::::; ₂ H ₁₀ ::. 10 c ... 0 -1 10

-

,

10 RE.AL 5

BETA=On:i , ZETA=O·050 'ROO··LlKE BEHAVICLJR' )

( +: 8A!\C WICTH )

·3

J 10 Ul ~ 0'. ~v ~. H ' L~ X. <. ::;; 2 "- H 10 " 10 W° -1 10 C. ... fIiA 1 50 X 10 "7 ··2 10 1·00 L:. Q-iC ~ Irj·['Cf.i ~

FIG,30 ROOT LOCUS ( N=2 , BETA=O,':i , ZETA=O·O':iO 'ROD -LIKE BEHAVIOUR' )

l +

=

BANC WICTH ) UI ~ 3 fr: ~D « Z H 1.9 <. ::ï'. '" H ₁₀ 1 10 0 iD -l 10 50 X 10 ..,.. 10

,

1-00 .:. O·lJ -J n ~ r;1"'\r.~ REAL 5

FIG,31 ~OCT LOC~S ( N=4 BETA=O·5 , ZETA=O.050 'ROO-LIKE BEHAVIOUR'

- j

::'0

Ul >- _'" a:: ... :.... <: z H t.:J <: ::;: _-, H ₁₀ :.. 1D :..0 0 -1 10 50 X .10 ..,. 10 -.1 :"·00

.,

a-iQ ~ {J -"{J':i ~ REAL 5

f--IG, 32 ROJT LOCUS ( N=2 , BETA=O,9 • Z[TA=O,O"iO 'ROD-LIKt: BEHAVIOUR' )

\ + = BAND WIDTH ) Ul iÇ. 10 :\ ~ H lél <: ::;: " H ₁₀ '" 10 G :LO -i 10 50 X 10 -:-O-lD ·1

~

-;J 10 o ~ r'C~j c· REAL 5

Fl[;, 33 ~OOT LOCUS ( N=1l 8ETA=O"1 , ZETA=O,050 'ROO--LIKE 8EHAVIOUR' )

\ + = 81\ND WICTH )

-3

:0

-3

J ··10 Lr) >- ~ cr. ~O ~ H LJ ~ 1 I-' ₁₀

,

10 .. 0 0

V

10 .1 1Ö" ~ OMEfjA 1 50 X 10 "7 1·00 A 0-10 <l lo.IJO" C-REAL 5

FIG.34 ROOT LOCUS ( N=2 , BETA=O.i • ZETA=O.OOl 'MEMBRANE-LIKE BEHAVIOUR' 1

OMEf",A 1 50 X 10 1·00 0·10 O·':JN ( + = BANJ WroTH ) UI >-cr ~ H l'J ...:. ~ H REAl 5 3 1;) 2 10

,

10 c iO '1 10 -J 10

FIG· 35 ~OOT LOCUS ( N=4 9 BETA=O·i • ZETA=O·OOl 'MEMBRANE-LIKE BEHAVIOUR' )

-3

.

,

·10 ] '10 ~1 >- ,~ ~ .,' « z f-' l.:J <". ::.ö J X H ₁₀

><-,

10 .i.0 0

-

.

10

,

;

'2

ti

10

*

O!IIErJA 1 50 X iO .,. 1-00

.,

0-10 ~ o _ rlQt~ _~ RCI\L 5

FIG· 36 ROOT LOCUS ( N=2 • BETA=O.S • ZlTA=O.001 '~E~BRANE-LIKE 8EHAVIOUR" )

OMEf,A 1 ~o .1.0 i"OO SolO 0·· CGt· X l + -= BAr--o WroTH ) ,,[AL 5 Ul ~ ~û « z H l~ <" ::> " H ₁₀

,

10 :iC (, , ij)

.

,

10

3 1.Cl :-3C X ." .Lv .,-1"00 A O'lC 1 IJ. r,r><~ _~ OMf.fjA i ~o X 10 1":)0 -'" ':;"1,::: .J 0 .~.~r..:: ~\. x REAL 5 I + = 8i\~ WIOTH ~

-,

-iO REAL 5 ;.i 1 " '.., " 1.0' (> ~O iOl.

:,1 >- ; (Y ~o ~ H l~ < ::::;; ~ X H 10 10' :1.0 0

1;

10 -1 -1 10 IJ'..1t.fjA 1 50 X 10

.,.

1-00 .c..

*

0-10 <l 0-r)()<j ~ Rt:AL 5

FI[3,40 ROOT LOCUS l N=2 • 8ETA=O,i • ZlTA=O,010 'MEM8RANE-LIKE BEHAVIOUR' )

( + = 8AI\O WroTH ) lf) >- :l (Y ~D ~ H ~ ..:: ::;:

"

H _1D

,

10 G 10 -1 10 OMfJ; .... 1 50 X 10 .,. -~ l-Oe: ~ 10 -0·10 <I !1.cv-:'-- ~ REAL 5

FIG,41 ROOT LOCUS ( N=4 • BETA=O·i • ZETA=O·010 'MEM8RANE-LIKE BEHAVIOUR' l_

e +

=

8AND WIOTH )

-3

Ln ~ 3 Ct: ia ~ H 1.'J ~ ~ H ₁₀ 1 10 iO 0 -1 10 OMEGA 1 50 X 10 v -1 10 1·00 L> 0·10 4 In.me:; ~ ~3UL~~~WU-L~~~~~~~--~UL~--~WU-L-llWU~~-W~~L-~~~ __ L-~~UW~LLLUWL~~~_3 -10 -10 10 10 3 -10 REAL 5

FIG,42 ROOT LOCUS ( N=2 9 BETA=O,5 • ZETA=O,010 'MEMBRANE-LIKE BEHAVIOUR' )

( +

=

BAl\!] WIDTH ) Ln ~ 3 10 ~ H 1.'J <. :::;: " H ₁₀ 1 10 10 0 -1 10 OMEGA 1 SO X 10 ..,. 1·00 L> -2 10 O,iO <I In,noe:; ~ REAL 5

FIG,43 ROOT LOCUS ( N=4 9 BETA=O,5 • ZETA=O,010 'MEMBRANE-LIKE BEHAVIOUR' )

-3

J "i~ ._._-- - - - -- - - -- - - -- - - " " . - - - , x 10 Rf. AL ::;

FIf3, 44 RO~JT LOCUS l !\I=2 , Br_ TA=O··(~ , ZETA=O,C;10 '''-AE.MBRA''~r~-LIKE BEHi'lVIOUR' 1

j.:} X . .., ~ '-1,00 .:. ::>,10 ·1 O···~O'"~

"

~=4 l + ~ BAr-"o w roTH ) RFAL :; + = 8.~NC WIOTH } :.1\ 1; ..: z H L::J < ~ H _10:l (. oiO 1 10 '2 10

r---~~---,

I

I I I I

i

O\,1C~;A 50 10 1-00 a-iQ n~0(!t:; x i X "7 A ~ ~ J1 >-~ ~v <' Z H ~ :> " H ₁₀ 10 iOO

V

-1 10 -1 10 , lJ( ~~~--~~~~~~~~~~-L~~~~~~~-L~~~-L~~~-L-ll~~-L~L~~~~~-L~~~-L~~_3 .l "10 10 REAL 5

FIG,46 ROOT LOCUS ( N=2 • BETA=O.l p ZETA=O,O~O '~EMBRANE-LIKE BEHAVIOUR' )

( + = BAND wroTH ) UJ >- 3 [Y 1.0 ~ H L'l « :> _:1 H ₁₀ i 10 10 (. i 10 ::50 x .:? 10 10 --:-l~OO ,~ 0,,10 <1 0··(\01:; ~ ~EAL :;

8ETA=O·1 • ZETA=O·O~O ·~EMBRANE-LIKE BEHAVIOUR'

~o

J -10 .1 .. 10 50 X 10 ~·IJO 0-10 -10 X REAL ::; ~l

ft

_<. Z f-' LJ <. ::;: H

,

,~ ~-' 10 10 :) ~O -2 10

FIG,48 ROOT LOCUS l N=2 9 8ETA=O,~ , ZETA=O,O~O 'MEMBRANE-LIKE BEHAVIOUR' )

fJMtJ;A 1 SO :i..0 l·OG 0·-10 O·'SO~; x l + = BAl\[] WIDTH ) IJ) >- ~D .. [Y ~ H L'J <. ~ 2 H ₁₀

,

10 " 'lO > 10 2 10 Rt:AL S

aETA=O,~ , ZETA=O·O~O 'MEMaRA~E-LIKE 8EHAVIOUR'

+ = BAND WICTH

-3

'10

-J

3 -10 m >- 3 IY. 10 ~ H l'l ~ 2 H ₁₀ 10' :l0 0 -1 10 OMEG'" 1 50 X 10 'Q' -2 10 1·00 .t::. 0·10 <l 10'005 ~ RE"'L 5

FIG.50 ROOT LOCUS ( N=2 • BETA=O,9 • ZETA=O,050 'MEMBRANE-LIKE BEHAVIOUR' )

( + = 8."NO wroTH ) m ~ 10 3 <. z H (!) <. :;E 2 H ₁₀ 1 10 0 iO -, 10 OMEGA 1 50 X 10 .,. 10 -2 1·00 _'" 0,10 <l la· oae:; ~ -3 10 ~3~U-L-~~~~llll~~~~~-J~~~-W~LL~~~LL~~~~~~~-JL-~~~~~~~U-LUllW_3 --10 10 10 RE"'L 5

"

rigidity, the roots associated with the attitude control system (the lower frequency roots in this case) deviate very little from those obtained in

Section

2.4

for the rigid satellite, and the roots associated with the flexible appendages deviate very little from the appendage modal charac·teristics. However, this deviation increases as n~ decreases.

An i~rovement in the damping ratio of the roots associated with

flexibility cau be noted when n~ becOJIles within the bandwidth of the control system. In other words, the attitude control helps to damp the appendage oscillations. As the appendages become even more flexible, the roots travel nearer to the imaginary axis, bringing about a marked decrease in stability margin, or even instability. Also it can be noticed that 'rod-like behaviour' tends to affect the control system roots more than 'membraue-like behaviour' does.

3.4.2

Effect of structural D~ing on Root Loci

Instabili ty tends to occur for low structural damping ratio. For all the cases studied, increasing the structural damping reduces the instability

region. In fact, for sufficiently high damping ratio (Z = 0.05), stability is assured.

3.4.3

Effect of Appendage Inertia on Root Loci

As the appendage inertia increases (relatively), the region of

instability is increased. Also, for low Z, greater deterioration in the right-most root occurs.

4.

ATTITUDE CONTROL SYSTEM IMPROVEMENT

Section

3

shows a great need to improve control system parameters in order to stabilize a spacecraft with a large, flexible appendage whose damping is low.

A straightforward approach is to vary the gains of the compensator through multiplication of each of the compensator parameters (k

D,

k', kt) by au "improving gain factor", g. Thus the control system block diagram will be as shown in Fig. 52. Values of kj, k' and k

n

remainthose calculated in Eq.

(2.4)

for the rigid satelli te. An unstable case has been chosen to illustrate

the effect on the roots of varying g. Figure

53

shows that as g increases it stabilizes the (flexibility) rootsthat caused insta1:>ility. The notation (+)

shown in Fig.

53

determines the roots at g

=

4.4.

The Nyquist stability criterion requires g

>

4.319

which agrees with the root locus shown. Further increases in g above

4.319

improve the minimum darnping ratio for the system until, when g :::::: 100, the minimum damping ratio reaches the value of the struc-tural damping ratio. No further improvement in the minimum damping ratio could be obtained by further increasing g. This f'urther increase in g is seen to

deteriorate the damping ratio of the higher frequency root. It should also be noted that as g grows greater than unity, the steady- state atti tude error in response to a constant disturbing torque is reducedo Tt will be recalled that

i t is proportional to the inverse of the integrator gaine

8

_d

-g

( klr+kls + kIOS2) _I

~ _{S2(S2+~S+-n-) I-Gf (5)}

FIG. 52 SINGLE - AXIS ATTITUDE CONTROL SYSTEM WITH IMPROVING

GAIN FACTOR

9

8

.

~---~

o ,~ :.

/

o ~ I .. , I .. " I" " 1 .... I 1 I S AèJ'vNI9VV'Ïl

/

/

/

/

-: -:. -'"0 \3 _o -< '0 -~ .,-j ~ -I .. , .. 1 I 1 I .. " 11 1 1 1 .. , ! I I'"'' I I fT .. " , I I 1,11" 1 ~xt> ~'V6 - ("'\! ...:.. ' 0 ~ 1 . .,., ...:. ' 0 ~ 1 ·.1 -" ~ 1 • ':ol Z r - - - J ...:. Cl ~ 0 0 0 ~ L'J ..--l 0 0 0 0 ..--l..--l00 ~ [\j :::

~---

----

---

---~======~~;~

;~ 0 ~ Ul 0: =:J 0 H

>

<-I W m

w

:::sC H -.J I W Z <:( 0: m ~

w

::?

..

,:-j 0 0 ~ 0 11

<- f-UJ N "\J 11 Z U1 :=J U

o

-.J

1--o

o

Q'.

tB

a Cl H LL ~. 0 0 N N 0

.

0 11

5.

CONCLUDING REMARKS

Great care has to be taken wh en designing the control system of a flexible satelli te, since even for a control system that is optimally designed considering the satellite to be rigid, instability may arise due to structural flexibili ty. This is especially true for very low structural damping ratios, and large, very flexible appendages.

stability margins are improved markedly as the lowest natural frequency of the appendage goes higher than Wil" (the frequency at which the imaginary part of the open loop of the rigid satellite equals zero). It has also been shown that two modes of vibration approximatethe appendage reasonably well, especially for 'rod-like' appendages.

Increasing the compensator gains improves the transient performance and also reduces the steady- state error but other control design considerations beyond the scope of this work would limit these gains.

1. Franklin, C. A. Davison, E. H. 2. P. C. Hugh~s S. C. Garg 3. Hughes, P. C. 4. Hughes, P. C.

5.

Hughes, P. C. REFERENCES

"A High Power Communications Techno1ogy Sate1litE;l f'or 12 and 14 GHz Bands", AIM Paper No. 72-580, Presented at the 4th AJ.AA Communications Sate11ite System Conf'erence, Washington, D.C., April 1972. "Dynanrics of' Large F1exib1e Solar Arrays and App1i-cation to Spacecraf't Attitude Control System Design",

urIAS Report No. 179, February 1973.

"F1exibi1ity Considerations f'or the ·Pitch Attitude Control of' the Communications Techno1ogy Sate11i te", CASI Transactions, Vol. 5, No. 1, March 1972, pp. 1-4.

"Attitude Dynanrics of a Three-Axis Stabilized Sate1-1ite with a Large F1exib1e Solar Array", Journa1 of' the Astronautica1 Sciences, Vol. XX, No. 3, November-December, 1972, pp. 166-189.

"Attitude Control of Sate11ites with Large F1exib1e Solar Arrays", AIM Paper No. 72-733, Presented at the CASI/AIAA Meeting, ottawa, Canada, Ju1y 1972.

6. Cannon, Robert H. Jr."Some Basic Response Re1ations for Reaction-Whee1 Attitude Control" , ARS Journa1, January 1962, pp. 61-74. 7. Frick, Martin A. 8. Auc1air, G. F. 9. Johnson, C.

D.

Ske1ton, R •. E. 1.0. Hughes, P. C.

"Atti tude Stabi1ization of' Sate11i te in Orbit" , "Rotationa1 Dynanrics", AGARD-LS-45-71

"Advanced Reaction Whee1 Controller f'or Spacecraf't Attitude Control", AIM Guidance, Control

&

F1ight Mechanics Conf'erence, Princeton, New Jersey, U.S.A., Paper No. 69-855, August 1969.

"Optimal Desaturation of' Momentum Exchange Control Systems", AIM J. Vol. 9, No. 1, January 1971, pp. 12-22.

"Optimized Reaction Whee1 Attitude Control Systems", CRC Memo 6665-10-1 (NSTL), August 10, 1970.

"

TABLE 1 -

ROD-LlKE BEHAVIOUR GAIN MARGIN (GM) AND FRASE MARGIN

(FM)

FOR If'/I

=

0.1

[2J.

Z

=

0.001

- .

Z

=

0.01

Z

=

0.05

radlsec

GM

PM

GM

PM

GM

\

0.55740

14.03

43.50

14.03

43.50

14.03

0.44280

14.02

43.45

14.02

43.46

14.02

0.35170

14.02

43.35

14.02

43.37

14.02

0.27940

14.00

43.14

14.00

43.19

14.01

0.22190

13.98

42.63

13.98

42.81

13·99

0.17630

13.95

41031

13.95

41.95

13

.

95

0.14000

13.89

38.30

13.89

40.35

13.90

0.11120

13.77

32.97

13.78

38.14

13.80

0.08835

13.52

25.14

13.54

36.29

13.62

0.07018

12.77

14.99

12.88

38.20

13.28

0.05574

5.321

3.497

10.45

44.21

13.32

0.04428

Unstab1e

Unstab1e

9.563

49.03

15.48

0.03517

Unstab1e

Unstab1e

11

.76

44.24

19.64

0.02794

Unstable

Unstab1e

15.65

44.20

25.26

0.02219

2.325

47.16

20.52

44.59

32.51

0.01763

14.89

50.40

14.9

44.20

14.98

0.01400

13.47

44.17

14.59

44.19

15.27

0.01112

11.74

44.18

17.92

44.20

15.15

0.008835

14.32

44.17

14.99

44.18

15.03

0.007018

18.55

44

.

17

14.94

,

44.17

14.98

PM

43.52

43.49

43.46

43.44

43.61

44.90

51.84

44.96

44.63

44.49

44.68

44.94

44.41

44.32

44.95

44.29

44.26

44.27

44.21

44.20

TABLE 2 - MEMBRANE-LlKE BEHAVIOUR

GAIN MARGIN (GM) AND PRASE MARGIN

(PM)

FOR If'/I

=

0.1

nJ.

z

=

.001

Z

=.01

Z =

.05

rad/sec

GM

PM

GM

PM

GM

PM

0.5574

14.03

43.52

14.03

43.53

14.03

43.54

0.4428

14.03

43.49

14.03

43049

14.03

43.52

0.3517

14.02

43.42

14.02

43.43

14.02

43

.

50

0.2794

14.01

43.27

14.01

43.31

14.01

43

0'49

0.2219

14.00

42.89

14.00

43.04

14.00

43.69

0.1763

13.97

41.76

13.98

42042

13.98

45.49

0.1400

13.93

38.88

13.93

41.50

13.94

45.38

0.1112

13.85

33.69

13086

41.10

13.87

45.12

0.08835

13.68

26.10

13.69

43.67

13.74

46.00

0.07018

13.16

16.40

13.23

57.97

13.50

45

.

48

0.05574

7.115

5.822

11.35

44.37

13.70

45.13

0.04428

Unstab1e

Unstable

11.22

44.36

16.29

44.86

0.03517

Unstab1e

Unstab1e

13.94

44.20

21.21

44.38

0.02794

1.558

28.05

18.21

44.16

15.07

44.28

0.02219

5.813

44.12

23.36

44-..14

15.22

44.22

0.01763

8.724

44.12

20.28

44.13

15.80

44.19

0.01400

12

..

21

44.11

26

.71

44.12

15.31

44.17

0.01112

16.82

44.11

14

.

99

44.12

15.06

44.15

'"

11rIAS TECIUlICAL NarE NO. 198

Institut~ for Aerospace Studies, University of Toronto

IIIi'LUENCE OF STRUCTURAL FLEIUBILITY ON A SINOLE-AXIS LINEAR ATTITUDE CONrROLLER

Abdel-Rahnnn, Tacek 40 pages 53 flgures 2 tables

1.

I.

!:;11UCCCTuft Atti t.ude Contro1 2. Spacecraft Structura1 F1exibili ty

AWel-Rutunun, Tarck Il. UTIAS Technica1 Not-e No. 198

~

Thc ctTcct of 5tl'uct.uronl flexibility on a linear attltude contral system employing a reaction wheel la

inv~:;til1atc(l. The pa.rameter:. of a compensator in the form of (proportional + integral + derivative)

fcctl.Lack :1I"C chosell in OJl o}ltinnl way: ta minimize the r eal part of right-roost root of t he syst·em

chal'af;tcd:;tir: cqulltion. This is done assum1.ng the satellite ta be rigid. 'l'hen the effects of

flcxllJility 0.'·0 invo,!Gtlgated through stability diagrams drawn by inspectian of the real parts of the

root:; or thc cho.ractcristic equa.tion, showing stabie and Wlstable regions. Root locus plots for

û1rrcl'cnt parametel' valuc,O are also used to illustrate tbe effect of flexibility on the roots.

Incrc(\3in~ the stl'uctural damping o.nd designing the appendages as rigldly as passible are important

mcans of insuring :::ta.bility of thc control system. Otherwise, the controller shauld be designed with

structuroJ. flcxibili ty cxplici tly included.

The flexibility investigations have been carried out for two general types of flexib1e vibration, name1y

'rod-like' behaviour and 'menbrane-like' behaviour. A method to counteract the destabllizlng effect that may arise due to structural flexibility is also given.

11rIAS TECHNICAL NarE NO. 198

~

Institute for Aerospace Studies, l1niverslty of Taronto

INFLUENCE OF STRUCTURAL FLEXIBILITY ON A SINOLE-AXIS LINEAR A..'"TITUDE CONrROLLER

Abdel-Rahma.n, Tarek 40 pages 53 figures 2 tables

1. Spacecratt Attitude Control 2. Spacecraft Structural Flexibili ty

I. Abde1-Rahman. Tarek 11. I1l'IAS Technlcal Note No. 198

The effect of structural nexibility on a linear attitude control system employing a reaction wbeel ia

investigated. The parameters of a compensator in tbe for,m of (proportianal + integral + derivative)

feedback are chosen in an optimal way: ta minimize the real part of right-most root of t he system characteristic equatian. Thls is done asswning the satellite ta be rigid. Then the effects of

flexlbillty are investigated through stabl1ity diagrams drawn by inspection of the real pal'ts aC the

roots of the characteristic equatian, showing stabie snd unstable regions . Root locus plots for

different parameter values are also used to illustrate the effect of flexibili ty on the roots. Increasing the structural d.amping and designing the appendages as rigidly as possible are ilJ!)ortant

means of insuring stabili ty of tbe control system. otherwise, the controller shauld be designed wi th

structural flexibllity explicit1y inc1uded.

The fiexibillty investigations have been carried out for two general types of flexib1e vibration, name1y

'rod-like' behaviour and I menbrane-1ike I behaviour. A method to cOWlteract the destabilizing effect

tbat may arise due to structural flexibil1ty la also given.

Available copies of th is report are limited. Return this card to UTIAS, if you require a copy. Available copies of th is report are limited. Return this card to UTIAS, if you require a copy.

I

11rIAS TECllNICfIL NarE NO. 198

~

Instltute for Acrospace Studies, Unlversity of Toronta

INFLUENCE OF STRUCTURAL FLEXIBILITY ON A SINOLE-AXIS LINEAR ATTITUDE CONrROLLER

Abdel-Rahmil.n, Tarck 40 pages 53 figures 2 tables

1. SJXlcecra.ft Attilude Contr01 2. Spacecraft Structural F1exibi1ity

J. Abdel-Rahman, Tarek IJ. trrIAS Technical Note No. 198

The effect of stJ'uctUl'al flexibility on a linear attitude control system employing areaction wheel is

invc!;tieated. The 1,aramcters of a cOUJlensatar in the farm of (propartional + integral + derivative)

fcedback ::1I'C chosen in 31l. optim::ll way: ta minimize the real part of right-lOOst root of t he system

cha,ractcristir. equation. This is dane assuming the satellite ta be r1gid. Then the effects of

flexlbility o,,'c investigated through stability diagrams drawn by inspectian of the real parts of the

root::: of' the chnractcristic equation, showing stable and Wlstable reglans . Root locus plots far

different pnrnJOOter vnlues are alsa used ta illustrate the effect of flexibility on the raets.

Incrcasing tbc stl'uctural dampin,; and designing the appendages as rigld.ly as possible are inpartant

mc~~ns of insw'irlg 5tnbility of the control system. Otherwise, the controller shoUld be designed with

structural flexibility explicit1y inc1uded.

The flexibl1i ty lnvestigations have been carried out far two general types of flexible vibration, namely

I rod-like' behaviour and 'meni:lrane-like ' behaviaur. A method to cOWlteract tbe destabilizlng effect

that may arise due to structural flexibility is also given.

~

Available co pies of this report are limited. Return this card to UTIAS, if you require a copy.

11rIAS TECHNICAL NarE NO. 198

Instltute for Aeraspace Studies, l1niversity of Toronto

INFLUENCE OF STRUCTURAL FLEXIBILITY ON A SINOLE-AXIS LINEAR ATTITUDE CONrROLLER

Abdel-Rahman, Tarek 40 pages 53 figures 2 tables

1. Spacecraft Attitude Control 2. Spacecraft Structura1 F1exibility

r. Abde1-Rahman, Tarek II. 11rIAS Technical Note No. 198

~

_~

The effect of structural flexibility on a linear attitude control system cmploying areaction wheel is

investigated. The parameters of a compensator in the farm of (proportional + integral + derivo.tive)

feedback are chosen in sn optimal way: ta minimize the real part of right-most root of t he system

characteristic equation. This is done assuming the satellite ta be rigid. Then the effects of

flexibility are lnvestigated through stability diagrams drawn by inspectian of the real parts of the roots of the characteristic equatian, showing stable and unstable regions . Root locus plots for

different parameter values are alsa used ta illustrate tbe effect of flexibility on the roets.

Increasing the structural damplng and designing the appendages as rigidly as passible are important

means of insuring stability of the control system. otherwise, the control',er should be designed with

structural flexibllity explicitly inc1uded.

The fiexibility investigations have been carried out for two general types of flexible vibration, namely

'rod-like' behavioW' and 'menilrane-like' behaviour. A method to counteract tbe destabilizing effect

that may arise due to structW'al flexibillty la also given.