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(1)

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(2)

F-

BIMONTHLY REPORT

r ^)4rlr,) ^!`T T. ^'^Y'^n t, 1.^f' T'r iTGN

".*T41 Y""; `^•n-1"p `'i1T $PL''r'iIO^1^ PnI!r^T^1(i

°Y IT...,

"imonthly ^ ,. pnrt (^y:ttrn: - , ,irch

T^h., 'himpai jn, T11.) '4 1) H:' AQ1/MF i ^A l ^G1

CS) C'L :" 'i t /1 h

X179-140 13

r ^{incla :,}

99473

RESEARCH STUDY ON STABILIZATION AND CONTROL r

MODERN SAMPLED-DATA CONTROL THEORY

. ^^ ti213141 ^{^} ^{1^^ry}

SYSTEMS RESEARCH LABORATORY 3q,Y^9/

P.O. BOX 2277, STATION A `^'F,,

⁸ ^{^V}

3206 VALLEY BROOK DRIVE'F

CHAMPAIGN, ILLINOIS 61820

ei

^

^,c

1^ti

PREPARED FOR GEORGE C. MARSHALL SPACE FLIGHT CENTER HUNTSVILLE, ALABAMA

I

(3)

BIMONTHLY REPORT

ICIC ITAL CONTROLLER DESIGN SLIbIiIIt':

ANALYSIS OF THE ANNULAR SUSPENSION POINTING SYSTEM

December 1, 1

977

Contract No. NAS8

-32358 1-7-

EO-07418(IF)

PREPARED FOR GEORGE C. MARSHALL SPACE FLIGHT CENTER

HUNTSVILLE, ALABAMA

SYSTEMS RESEARCH LABORATORY

P.O. BOX 2277, STATION A CHAMPAIGN, ILLINOIS 61820

(4)

r

115

VII. SIMPLIFIED MODELS OF THE ANNULAR i

SUSPENSION AND POINTING SYSTEM (ASPS)

1

),I Introduction III

The Annular Suspension and Pointing System (ASPS) is a payload .,uxiliary pointing device of the Space Shuttle. The ASPS is comprised of two major subassemblies, a vernier and a coarse pointing subsystem.

The experiment i ,, attached to a mounting plate/rim combination which is suspended on magnetic bearing/actuators (MBA's) strategically located about the rim. Fine pointing is achieved by "(J mbalIing" the plate/rim within the MBA claps.

Control about the experiment line-of -sight is obtained through the use of a non- contacting rim drive and positioning torquer. All Sensors used to close the servo loops on the vernier system are noncontactinq elements. Therefore, the experiment is a free-flyer constrained only by the magnetic forces generated by the control loops.

The configuration of thr ASPS is ,hown in Fig. 7-l. The payload/plate/rim combination is mounted on a set of coarse gimbals; an elevation and a lateral coarse girnbal, which providt • the slewing and coarse pointing capability of the•

system. The pointing system concept is unique in that the vernier and coarse pointing subsystem are separate entities. This approach allows for sub-aresecond pointing of the payload at any coarse gimbal position.

The three functions provided by the ASPS are: (1) pointing the payload, (2) centering the payload in the magnetic actuator assembly, and

(3)

tracking the payload mc.inting plate and shuttle motions by the coarse gimbals. Rate and position errors sensed by gyros and celestial sensors located on the payload are processed by a controller which subsequently commands appropriate actuator forces, to point the payload. Proximeter sensors associated with the actuator clusters detect the payload translation errors which are subsequently processed by the

(5)

ION

1 1

1Z

OAD

VTING SURFACE AXIAL ACTUATOR AND SENSOR

-- RADIAL ACTUATOR AND SENSOR PALLET

ORIGNAI, PAGE IS OF POOR QUALITY SHUTTLE

MAA

Figur e J-I. ASPS configuration.

Figure 1-2. Payload and magnetic actuator assembly.

(6)

controller and used to ascertain the appropriate centering forces.

Fiqure 7-2 shows the payload and its mounting surface which is controlled by the magnetic actuator a ,,,embly (MAA). The cables shown are for the purpose of connecting electric power from the shuttle to the payload and the MAA on the pallet.

7.2 The Planar Model of the ASPS 121

In this section the equations of motion of a simplified planar model of the ASPS are derived.

The small-angle, small-displacement model Shown in Fig. 7-3 is planar with four degrees of freedom and is composed of amount, a gimbal assembly (elevation), a pallet with magnetic actuators, and a payload. The pallet has one rotational degree of freedom relative to the mount, and the payload has two t ran sIation.)I and one rotational degrees of freedom relative to the pallet.

Let the four degrees of freedom be

I = attitude degree of freedom of the pallet relative to the mount y 2 = attitude degree of freedom ^ofthe payload relative to the pallet x = tr anslation degree of freedom of the payload relative to the pallet

z translation degree of freedom of the payload relative to the pallet The following coordinates are defined:

(x O ,z 0 ) _ inertial axes

(x G ,z G ) = inertial axes rotated through an angle of ¢ M relative to the (x G ,r p ) axes, _(PM is defined as the gimb. angle).

(x m ,zm ) - axes fixed at the pallet center of gravity (CG) (x l ,z l ) = static axes of thu payload

(x i

1z.

r ) = axes fixed at the payload center of gravity (CG) (x ) ,z i ) = axes fixed at the center of the base of the payload.

117

(7)

A

x G PALLET -~" Z,

r b ^^ QnN8AL

ASSEMBLY

10 w^--~---e~x-

MOUNT BASE v

/|M

I

" °

Z o

F|qure 7-3. P|onar ASPS model.

ORIGIN

AL ^ PAgI", 18

OF PWQRQt;ALJTY.

(8)

The following system parameters are defined:

M.r = mass of the payload - 600 Kg M

m

⁼mass of tht• pallet - 82 Kg

J Ot the pallet About its mass center 3.1 Kg-m2 J. = inertia of the payload about its mass center 503 Kg-m2

L = radius of the payload = 0.406 to

La = distance from the gimbal to the pallet CG = 0.2064 m

Lb w distance from the pallet center to the payload CG = 1.4$6 m r - distance from the gimbal assembly to the pallet center - 0.47 m rb = distance from the gimbal assembly to the payload CG = 1.956 m r0 = distance from the mount base to the gimbal assembly = 0.75 rn

Since the payload is suspended with respect to the pallet, there are many ways of fixing its coordinates for the motion of rotation. in other words, the angle 1 2 can be defined in a number of ways. Figure 7-4 illustrates the small- angle rotation of the pallet and the payload with 12 measured as the angle between

the coordinate axes of (x 1 ,7 1 ) and (Aj ,z

i). This configuration is defined as Model 1 of the ASPS. Figure 7-5 illustrates the model of the ASPS with `Z2

measured at the CG of the payload; i.e., between the axes of (x 1 ,71) and (xi,zi) 119

The following coordinate transformations are identified:

Transformation from the static pellet axes to the mount axes:

cos(,)

M -s i n(pM rG =

s i nJ M cos'l,M

(7-1)

Transformation from the dynamic pallet axes to the static pallet axis:

cos,( I -s i n^ l

I ;

_I

TI = (7-2)

sing, I cosQ,l

^'I

3

le

(9)

zi

xo z

Izo

Figure ]-b. Planar ASPS Model I.

(10)

121

Fiyurc 7-5. Planer ASPS Model 2.

ORIGIN AI. PAGE 1S

OF po ()It (ZUAL11'Y

(11)

IIransformation from the dynamic psi/load axes to the dynamic pellet axes:

T

1

^{= (}

cos'y

2

-sin4r2

I_` 2

^

(7-3)

s

111;

2

cosrp2

^2 I

The force vector~ applied to the payload by the magnctic actuator assembly

are defined is:

fl

F = I -

magnet

is fr,rcc a p p I ied it !ht! po^,i it ive x

III

displacerlient (7-4) f3

fI

F 2 = = nrignetic force

app Iiud at the negative x

M

displacement (7-5)

IF

The forces F ,Ind F 2 .are i I Iustrated as shown in Fig. 7 - 5.

The torque applied by the: gimbal assembly 1s designated as T

c

, as shown in F

i,j. J-5.

The following vector distances are defined for the pallet and the payload.

R

I

= vector di°.stance fror» the gimbal assembly to the payload point of application of FI

R

1

= vector distance from the gimbal assembly to the 1-ayload point of application of F2

R

3

= vector distance from the gimbal assembly to the pallet paint of application of

F

R

I#

= vector distance from the gimbal assembly to the pallet point of application of F2.

Ecivations of Motion o f Model I

Usinq the degrees of freedom defined in the preceding sections, the kinetic energy of the System in Fiq. 7-4 is

T = K.E. = 2 mMr ► Rn ► + 2 f, i m i + 2 J m`''I + Z J i ^' I + $

, 2

)

2

(7-6)

where the primes denote the transpose of a matrix, and

122

(12)

-L

J Rill

-

^I

Y1 (7-7)

^ o

I b

-r'b

R i

⁺

y2

^♦ ^I ^(7-9)

z

^

^_ 0

O

SLrb-.tI tut ion

of Eqs. (7-7) and ( 7 -8) into Eq. (7-6) yiv.-5

T = K. E. IM L2v1 +

1

M.z

l

⁺

I

M .(x - r ;- - L )

2

⁺

I

J

'?

^{+ 1-J.('} ²

2 ni a l 2 r 1 2 r I b l t. : 2 m' I 2 r' i ^2)

(7-9)

Lei. the sprViq farce applied to the system payload due to the cable be

designated as I- t

fsx(x1)

f

s

- (7-l0)

^ f sz 1/ )

and the spring torque applied to the payload clue to the cable be T

s (,

^y

? ).

The sprinq torque applied to the pallet dui to the cable is denoted as T1)(;.I).

The rel.ition between the force F and the potential of the system, U, is

F = -VU (,-I I)

Thu-,

U = U - jF

•

dx (7-12)

where U

D

= constant.

The potential energy of Model 1 is

/ ^y 2 z1 xi

U = U

U -

^j(f

I

- f

1r )

x

I

+

(

f

2

⁺

F 3I

+

jo T,(;')di'

+

j o

fsz(z)dz + l ⁰fyx W dx

+ (t 3 - f 2 )L; 2 _{+ ( +1 r} c ( ,y ) dy + f

o ^T ^I, (^J)d y (7-13)

0

The Lagrangian is defined as

of -- K.E. - U (7-14)

Then f rom Eq5 . (7-9) and (7-13) , we get

ORIGINAL pAG6 15 OF Pwit QUALITY

113

(13)

124

1, 2

.

2 1 -2 1 2 1

•

2 1 2

M

L

a I

⁴

1

M

i

^I

I +

^ZM

i

^(x

1

^-^{rh; 1 - L}

I

^;

2

) + 2^m,.I +

2

^J

i (;

.

1

f 2)

1

^2

z1

(

+ (t

- f4)xl + ( f 2 _j lz I

+

T,'(y)c4, + JO f5z(z)dz + O f tix (x)dx O

1 21

+ ( f 3 - f Z )W + Tc(i )4 + T^C^)dy - U O ( 7

-

15)

0 0

r

The Lagrange c(itiat i t,n .)f

uiolt iun

is

I/ 1

^i

`—'_l!r /

0

I 1 x2 .3,4

(7-16)

dt x.

where x - x

1 ,

x

2

^r^z

1 ,

x

3 - ,

r

l

and x

4 -

^2.

Fcir

i = I ,

we have

JA f - f + 1 (x

a

:

^ I 4

sx

I ) (7-11)

^A - M

i

(x - , ;

I -

^{L^^ )}

Thus,

^, _ cl a

Jx

l

dt^;)xl, _ -M

ixl + +Mr' I MiLb

2

⁺

(

1

1 -

f

1e )

_{+ ftix(x1} 0 (7-18)

For i - 2, we ha

ve.

f2 + f

3

_{+ fSz(z1)}

J^

i)z1

=

Mi/i

Then, f

^z

- Jt^ if-J = -Mi1i ⁺

(

^f

2

^{+ f}

3 )

^{+ f}

Sz (

^z

I -

⁰ (1-^9)

For i = 3,

we

have

;) I =A _{1 ` (;,} _{I )} _{+ T} _{h (,^,} ₁₎

); 1

= M^ n L? :'1 + M i (

-

r h x

_{+ rbv}

I

+ r b L ^'Z ) +

^ni^ I

^{+ J i +}

1

+ J

i`t2

(14)

125

71 - 1 - ^^^r

^{1 ,}

T,(^'

I

) +

T

p

(

^I' • Mirbxl

(gym + Ji + MnrL2 + Mi12),I

(Ji • ^Mⁱ^L^b^r^b⁾4, 0 (7-20) 11

For i - 4, we have

< ► ` (t

³

- t )L + Ts(^L,2) 2

'^2

MiLbx1 +

rbLbMi'I + Milb^? + ji'I + Ji'2

JA-

^d ^I

f (I,.

) + (t - f )L + Ml x i :,;

2

dt

(^{^lt2l}

2 3 2 b l

(M

i

t

b

rb

+ J

ⁱ

)

^'

y

l -

(Pi

i

L2 + J

I )

Z

2

0

(7-21)

The Laqrange equations in Eys. (7-1$), (7-19), (7-20) and (7-21) are written in matrix form as follows:

M i 0 -Mirk

-MA

f1

0 Mi 0 0 rl t2+f3+fsx(zl)

(7-22) - M.rb 0

Jm+Ji+MrnLa+Mirl Ji+MiLbrb yl ^Tc('I,I)+Tp(II)

-MA 0 J1+MiLt^rb Ji+MiL2 y2

(t3-f2)L+Ts(1L2)^

Equations of Motion of Model 2

For thy• ASPS system Model 2, the k ine

•

t is energy of the system i It i I I

given by Eq. (7-6), and R rrI is as defined ii Eq. (7-7), except that

x1 +

- rb

r

t ^

^l

1

⁰

1 ,1

Substitution of Eys. (7-7) and (7-23) into Eq. (7-6) (lives

1 2

^.

2 1 2

K.E.

2 mLa'y 1 + 2rtilI +

1

2Mi (x1 -

2

r b A I ) +

1 2

2 J rri ; I

1 2

+

2Ji (' I + Z2)

The potential energy of the Model 2 is

()RIc',ItiA1. UAIS1'IYS OF POUR

(7-23)

(7-24)

ML

(15)

T_. _T

126

f

r+2

U - ^UO (fl4 )^xl + 0•>x(x)dx +

(f2 +tj)rl«JOfsz(z)dr + JO Ty(1)d4^

+ "

I

T

c

(Od, +

I` (t')^I; , (f 3 -

f

2

)Ly

2

+

(

f

l -

f

4 )

L

h

y

2 1

(7-25)

f

_o

⁰

The Lagrangiin 4 1s given by

K. E. - U =

^I^M

L

2 2

+ ^M.z

I

+

²^{Im (x -}

r )

2

+ ^ J

1

+ 1J. I ^{t )2}

2

nr

a'

^f

I 2 r l

^{2 r}

I b

^f

I 2 .^^ 1 2 i + "2

+ (fI - ^f4 )xI ^{+ f01 f}5X^{(x)dx +}(f2 ^{+ f}3 )zl +

f0I fsz(z)dz

+ I

2

T

5

(+)do + (

1

T

^c

Q)dj + fo)TI^(^;.)do + 1(f3-f2)L

0 0

+ (fI - t4)Lb1:2 - U (1-26;

The Lagrange equation of motion is given by Eq. (7-16).

Fullowinq the same procedure as for Model 1, the Laqrange equations of Model 2 are derived by use of Eqs. (7-16) -6 (1-26), and the result is

Mi 0 -Mirb 0 xI^

0 Mi 0 0 zl

0 Jr'r+` !Mm L z +M. r2 -M i r b

0 0

Ji

fI-f4+fsx(xl) f2^+f3

+fsz(zl)

Tc(LZI)+T,)(,,#, )

(f3-f 2)L+(fI-f4)Lb+TS42)^

(7-27) In the analysis conducted I" the ensuing sections the equations of motions of Model 2 will be used. One reason for this selection is that the mass matrix of Eq. (7-21) is Simpler than that of Model I in Eq. (7-22). Another reason for using Model 2 is that the model uses the center of gravity of the payload as the reference point of rotation, which is more logical.

SLbS titution of the system parameters into Eq. (7-27), we have

I`

i

(16)

600 0 1173.6 0 xI fI-f4+^Sx(xI)

0 600 0 0

z f2+f3+fsz(zI

.S

-1173.6 0 1805.15 503 I TCGI)+TP(Vl)

0 0 503 503

1Y2

(f3-f2L+(fI-t4)Lb+Ts2)

(7-28)

7.3 Control of they Z Dynamics of the Payload

Equation (7-28) indicates that the z dynamics of the ASPS are riot coupled to the other three degrees of freedom. The z dynamics are described by

M i z I = f 2 + f 3 + f Sz (z l ) (7-29)

The magnetic actuator force ,. f 2 + f 3 are controlled by feeding back the variables z and z I . The control equation is

f2 ^{+ f}3 = -Kpzl - Kr_^I

(7-30)

where K p = 37.861 N/m and K r = 211.01 N^/m/sec.

Substitution of Eq. (7-30) into Eq. (7-29), we have.

M i z l = -K p z l - K r_z t + f SZ (z I ) (7-31)

Figure 7-6 shows the state diagram of the z dynamics of the ASPS with the continuous-data position-plus-rate controller. The notation N sz (z I ) rn the state dia(Iram represents the functional relation of the mire cable which is attached to the center of the payload mounting surface.

If the wire cable is modelling by a linear spring, N Sz (z I ) is simply a constant, -K

s (W/m); that is, f sz ( z I )

= -KszI

(7-32)

A nonlinear spring characteristic for N sz (z I ) is shown in Firl. 7-7.

However, since the mass of the payload is 600 Ky, and the spring constant is

Ur ^I ^Y()k)YL ^Tx

127

it

(17)

Its

7,

4 HWF

t

)PE

^{= K,S} 10. Z f^ ,( Z')

fst(Z,)

- _{K /MI}

Figure 7-6. State diagram of the z dynamics of the ASPS with positron-plus-rate controller

(continuous-data System).

Figure 7-7. Nonlinear spring characteristic for the wire-cable torque of the ASPS.

(18)

129

only 0.35 N/iii, the effect of the wire cable on thv payload dynamics is not going to be substantial.

The characteristic equation of the continuous-data ASPS dynamic system with the linear wire cable spring characteristic is

M ,2 + K

r

s + K

p

+ K

S

= 0 ( 7-33)

l

6005

2

+ 211.OIs + 38.211 = 0 (7-34)

The damping ratio of the system is

t., = 0.6968 (7-35)

and the undamped natural frequency is

w = 0.2524 rad/sec (7-36)

Analysis of the Digital ASPS z Dynamics

When the z dynamics of the ASPS are controlled by a digital position-plus- rase controller, the dynamic equation is

M

i

z I + K

s

z

I =

f

2 (

t )

+ f3(t)

(7-37)

where

f 2 (t) + f 3 (t) = f 2 Q) + f 3 Q) kT < t < (k+l)T (7-38) Then the control equation i,,

f 2 (kT) + f 3 (kt) _ -K p z i (U) - K r z i Q) (7-39)

Figure 7-8 shows the block diagram of the linear digital ASPS payload (z I dynamics).

Since all the system parameters are given, except the sampling period T, we shall Study the maximum value of T for asymptotic stability.

The characteristic equation of the digital system in Fig. 7-8 i%

A(z) = I + M. (1 - i Iq

K A 2 K

r

_{+ ___}E

/s 3 ` (

= 0

I

(7-40)

i

-2

K

I+

^M

r

s -2

I+

i

Ms s I

J or

OItI('=ItiAI, P,^IiI^" I OV POOR QUAIU1'Yi

ti

(19)

130

The z-tr.rnsforms in th

y

• last vg

g

ation are: eval"ated as follows:

M.K KS zsin Mi T

(7_4l)

K

S2 + Ks

z2

-

2zc^ My T + 1

r r

I _ Mi

z z(z ^_coslKy/Mi T)

_ - (7-42)

s(s2 } K

s) Ks z 1

z 2 2zcos^K s /M i T + I

l Mi

Substitution of the lost two equations into Eq. (7-40) and simplifying, we have

K

r

K

^

Y. K K K K K K

z2 + M.Y. sin

^M'

T - ^^ cos

My

^{T +}

K - -

2c<^ .

My

^T ^{r I} KL - cos

rty

^T

r s r

r 5

r

5 s i

K r I"

// sin T = 0 (7-43)

^Mi

Substituting the system parameters into the last equation, yielding,

z 2 + (14.55975in0.02415T - 110.1688cos0.02415T + 108.1688)z r I - 14.5597sin0.02415T -108.1688cos0.02415T 1 108.1688 = 0 (7-44) The roots of the last equation as a function of T are tabulated below and the root

locus diagram with T as a variable parameter i5 shown in Fig. 7-9. The erilical value of T for anymptotie stability is approximately 5./ sec.

Sampling period

Characteristic Equation Roots

T (sec)

0.1 z2 - 1.962 + 0.965 = o 0.98 + j0.069 0.5 z2 - 1.8162 + 0.832 = 0 0.908 + j0.0864

1.0 z2 - 1.6163z + ME = 0 0.808 + jo.164

2.0 z2 - 1.16862 + o.4232 = 0 0.584 + jo.286 3.0 z2 - o.657z + 0.2298 = 0 0.328 + j0.349

(20)

sec Rez

1?I

LM'

Figure 7-8. Block diagram of the linear digital LISPS payload z dynamics).

Figure 7-9. Root loci of z. dynamics of the digital ASPS payload as

thu

sampling period T varies.

pRlcrrn+

AL P AG 1; 1.

OF pWlt QUALM

(21)

132

4.o z2 - 0.082lz + 0.1 . 0 0.041 + j 0.3135 5.0 z2 + 0.556z + 0.0338 = 0 -0.4865, -0.0695

5.7 12 { 1.04z + 0.02533 - 0 -1.01, -0.025

6.0 z2 + 1.257z + 0 03124 = 0 -1.23, -0.0254

The time responses of* the digital system in Fiq. 1^{-8 for} ^various ^sampling periods are Shown in Fig. 7-10. The initial value of e I (t) is chosen to be 0.002 m, sine the static bearing gap of I is only 0.0076 m, so that ttic maximum

constraints on the magnitude of

,II are +0.0038 m. The time responses in Fiy. 7-10 substantiates th. root locus findings; when T = 6 sec, the closed-loop system

is unstable. Th,- time responses are quite good for T less than or equal to 3 Seconds.

Effects of Quantization

The block diagram of the digital ASPS payload z dynamics with the quantization effect is shown in ^Fic;.7-11. The input-output characteristics of the quantizer are illustrated in Fiq. 7-12. The quantization levt•I is denoted by h in meter.

The effects of quantization can be classified into three categories: (1) stable system with steady-state error, (2) unstable system with sustained oscillation, and (3) unstable system with unbounded responses. The last case is possible since no saturation is assumed in the system model.

The steady-state error due to quatization can be determined by using the least-upper bound method 131 and the condition of sustained oscillations is found by use of the discrete describing function.

The "characteristic equ.ition" of the system shown in Fiy. 1-II is written

as K _ + Kf

^ /s

A(z) = 1 +

Q(z)

(I - z- I _{M.Is^ r} ₀ t I + t1s s

i

where Q(z) denotes the di ,,crete describing function of the quantizer.

4 .+... _mow

le

(22)

0. 00?

0.001 t 1 (t (m)

I

0.001 - -6.002

F

(23)

INPUT

134

Figure 7-II. BIock dia(lram of ^thodi(I jt,rl ASPS payload , I Jyn.unics with yuantizati Oil .

I iyurc % - I.'. 1r i r I ,tt - .,utI I ut -h ir,r(.t,-r i'.l i C S „I d yu.uil iZCI'.

(24)

o, c

L u

v

Lv

u

vN

C to 7 Q 4-

O

N

c E

311,

Oz

C M O O

61 II

L

— Y

V U_

' L 'E

V C T Tc m u ^N N

_O

• T m

v

CL

LA Q M 4- i O h C O v •- L u 7 V O. C

•— 7 LL 4-

135

-

o

^ r

o ^

N N

(V ca M

M

.-, cv

F- CJ

V \ci

tD I M ^l ^

It •

n N

z

ti z v v

J) ^

M)

v

m

lD U-) ^)

OD Ul)

N

(D

N F- i - V

i^ Z

Z Z ^)

N

m

I ti

u

a! i d

^D 'n

V Y

1 i

aD

Q

11 z

r

M

v

of t

c

M I

u

^?

z

J^

a

n

Z

^

tD

V

I N

m

ORIGIN N^- ^U ^.^y 0f' PUQI

(25)

L

u v v v v u

v

N

G

^D7 Q w

O bC

^ EO ^

C um

O

Q1 oj it

L N

Y u .. u

u mC v r

c v o N a+

O Z7

a o

^^ TN M

Q U/f

C) a

Ln

? v- O

1 fr c

O

v •- :3u

U, c

i^ w

1 3b

r'

(26)

T^

131

C0

•v^

d L

.v

v

.V I,

^►- E V ^C C

^ T

C V

E N 7e v Un O

M -

T c n

^o u. a

M

2

u, O

M

• C O O

11 ^ J V^ C ]L 7

w

- 'O

G (T

^U G

11 • L

u

L ^a^

0 v

^ v

«.

L

O u

CL •-

N L

^' NUN J-

C7 C

Ln v

^ w

r• O

0! LA

L ^

7 C

Q. 7 - O

D

(27)

A(z) n I + Q(z)G (z) a 0

Vq ^(7-46)

138

The i-transform of the last equation Is evaluated usinq the results in Eqs. (7-41) and (7-42). Fquation (7-45) becomes

where Kr KS K

s K^1 K Ks K Ks K^

(41 N

₁

KK in ^; T

^KP

cos

^M,

T + K-/ft - .sin- T - (

M- T + K

G (z)^s---^r ^s ^-

r_ s r s --- r s

^_ _s

( 7-47) eq

z1 - 2zcos

My T t I r

For K

p a 37.861, K1. 111.01, M. = 600, and K .1' . 0. 35, the last. equation is

r

simplified to

G (z) . L4--5597%In0.02415T-108.1688cos0.02415T+108.1688)z - 14.5597sin0.02415T

`q z2 - 2zcos0.0241;T + 1

(7-48) -108.1688cos0.02415T + 108.1688

Figure 7-13 shows the plots of G

eq (z) for various periods cif sustained oscillations T = NT, N 1 2,3,4,... . The sampling period T varies along the curves. The square block in the figure Which i ,, centered at -I it-present'.' the bounds on the critical regions of -11Q(z) 1 1+1 . Theoretically, the intersects between the critical regions of -I/Q(z) and G eq(i) represent conditions of

pelf -sustained osciIIA ions. It is clear from Fig. 7-13 that the system should be free from sustained

osci Il,rtions fur all sampling

period,, le-,s

than 2 seconds.

Figure 7-I 1+ 'sllustrates the Gc

q (z) plot~ for K s = 3._: N/m, 10 times the ,

nomin.0 value. As po:nted out earlier, since the mass of the payload is so large, the light spring effect of the wire cable does not materially affect the performance of the system. Fitlure /-15 further illustrates that evt l n with K % = 35 N/m, 100 times the nom ; -r,rl value, the characteristics of the system for sampl ing periods less than 2 seconds are not significantly affected.

The least-upper bound error analysis of the quantization effect is performed by reft • rrinq to the system block diagram shown in fig. 7-16. The quantizer i,.

0RIGP.-AT• PA(7 1' OF POUR QUALITY

I — -- -- — —

--.=MMA

(28)

139

I ,

+h

+ -EE

_-1

Figure 1-16. Block diagram of the diij i ta I ASPS payload / I if Ynibill i CS

t-t) v t t i c I ej-,t -upper bound an,i I ys is of cItianti z.ition effects,.

i, imFis- irri FF-14T ,

Ys

F: rFPr4-1 ^P- 17 ^{T 91711}ITP

i.

^—

imt-inti

^{Hii i •t:}⁴ⁱ^,^sit^m4^f r- • " it f. • i I` •H •W%AH • P-.AH9 *-A HI

FILL TIME.

I r+

^{111 -} ^{H r1I}^{ILD f+}

I .

iTF I-, P HL i l

l

+

I

1 1

F

t F: ^{nF T INT.}

H=--_-.00-4 T ' P= I . T 7 1 11-4 1-1 T'L P T = il . T INT=11 I F-

PrDT

F-5

tZ4 P

T

FiLlure 7-11. Computer program of the s him la -, ior, of the ASPS pay I ().)d dynamics with quanti zation.

(29)

140

P. T I OF I F*PT

M T I T 7-1-1 T J , =Ulirl

P '-'P-1 T 7, T I 14 T

P" 1-1 T 4 F F-'Pr. C⁷, rjT

14 M

f ir) T I'll

F'F' 1- 1 T T,Fp,(,•

I ri

I

m I HL F F T D.1 1 "UX F,'1 L+

P n 1 1 T I I i F F I

- T T I M E 4Y [IF

rl I IMFN - I Dri

'Ilt •'.'1H

-WrIN ^'DCT•T - t4 M I •^01 I ti'v' 1 9 TPFT•I F

I

i r, 4 F*T I MF 9V 1 'DMI-IrIll Ht•' I •I'l► H1• P .ri1 P • T P H •H%,-' I H • I'111.1 1H+.•

7 P 11 T 2

-rinT

Eli T ^IMF — T T T + T

41-1 1 H

=.'1

F H T D T F1 2i1 H= I ^V1 H H +

Ii7 TO -:11 --H MiT I HUF

V.1 H = I F I:-:, 0:i',! I H I Hl', =;:,I, qH T , _IH OH

721qT=—i4if -I'll IHI-I

=

7

f ^InT F. T _T_1._{1 PN}

F 11.4 [1

IPP,r PUTP# T I MF ripve

I

MUT T tjE HL-F otill 11-1 FF'MT

1`110FIC TGH 'r' ,:2, orl^{FVO. "}R.) qPPP1T('5 I

Fjr-1 I ti., - F PT 11-1 E o'... I

,imt :,^it4 ^:⁷ ^r,^{CT T}^{* P^} ^{M 1} ^{'-,1 ^}TPPT TFr4Ti H

rimrinri ^HO^{I -} ⁿ¹^{P -}^q^I-&^I ^ , r - F •

H

q 1^4 ^{V I}^H^{q I I}^{H "} I H'^-

I': T I t-It- — FP'T I OF L T L-PT - PE TUP 11

PR T I ME RT I II-T + TPF; T

TE I T I NF 4'v'l H HI-I q (Y' I• i ^IHL F

rrIPTIPT f' 1: i=71 • I .• F 1 5 9 1:-.: - E I IX E 12 V:' -- 2 ^ I X 10 E 12' 5 1•1

T I I F'I 4 F11D

Figure

7-17• Computer program of the s irmi I at ion of the ASPS payload z d y ri;) i i I _i

(_ s W i I [ i , I U a r I

I i za t i () I i .

ORIGJ,\j1j,J^

3F POOR (4u ^{7, ,} I

(30)

141

j

W

y

O.007

i

O.001 • ..-. . .

With j uanti at ion

I

o,0000

3^

¹

30 46

, ....

i

I IIML sec)

f^4 ^{Yuan 1.}

i

^{za t}

ⁱ

^can

h = 2'4

o.000444

With d^anti7otion

• I

-O.00 ^l ^: . l .... ....

-

^{0.0 0 2} ^. ^{. . . ...} ^.. ^{. ..} ^..

Fi tire 7-18. Time resporns s of

the

^{I I i}

itaI ASPS p.yI a(I ^ I dyr amies;

wi h JIIJ Without.

cauantizatlon-

t = I IL-cond.

(31)

0

i

^ T`^

- 0.002 1 _.

-0.001

A

Wi^h quantization

h 2-4

I ..

I

i

30 0 0

'

Without q6Lant(zation ^Tl

^(sec^)

W th

quaht

z t i ,i^

-0.0003 i r h

0

^2_g

i

I

Ii I

u

0.001

I

_0.002 1

rigure 7-19.

vli th and wi th

Time responses of the Jigit31 ASPS 1 payload z 1 dynamics;

out quantization.

i

T seconds.

ORIGNAL PAGE IS

OF POOR QUALITY

(32)

1

143

replaced by the noise input with an amplitude of +h/2.

The z-transform of the displacement z duce to the noise• input is

-I

s

Z (z) _ (I - r )

_ I I(+ h/2) z (7-49)

I 11^z^ ^ ^^M ((1 + K s /^1 z- I

i s r

where AQ) is as given in Eq. (7-40).

In Eq. (1-49),

(I - z

^-1

)^ 3 - _

¹

_ _ 22- _ (I - ^z ^- I ) 1

M

i

^{s (I + K}

s

s /M

i

⁾ ^M.s(s ^{+ IN /M}

I

(z + I)(I - cos K

s

/M

i

^T)

--- - (7-50)

Ks z2 - 2zcosK

s

^/M

i

^{T + I}

Thus,

(z K

S

+ 1)(I - cos Ks/Mi

T) (+2)(z z )

r _ K

Z1 z

z2 K Z)

sin 5 T - K P co, -

^s

T + K ^ - 2c

+ os

⁵

TIz + I + ^'

.K M, K ^r1. ^K ^M. ¹ ^K

s 5 r (7-51)

K K

Kco' M. T M.K ^sin

M.

^t

s i i s .

The MY steady-state value of ^z

I

^(kT)is given by

IS z

I

(kT) = IN (I - z - I)

k-

^I

K K (1 - cos

MS ^{T) (+ 2)} ^{± 2}

s r

(7-52)

K K K + K

,I +K ) (1 - cos

MS ^T) ^s P

s i

This result shows that the least-upper- bound on the steady-state value of zI(kT) due to quantization is inversely proportional to K

s

^{and K}

For the given values of K

s

^{and K}

P ,

^{we have}

ORIGIN F

it ^L ^^`IY

0^' Y

(33)

14 + h

tin ► zl(LT) 38.211 + 0.013085237h (7-53)

Thus, for a quantization level of 2 - tr , the final error in z is +0.000811827 in, whereas it is +0.000051114 rn for a qu.►ntization level of

2-8.

7.4 Computer Simulation of the ASPS Payload r. I Dynamics with Quantization In this section the z dynamics of the ASPS payload are simulated to study the effects of quantization. The computer program

is

(liven in Fig.

7-17.

Figure 7-18 illustrates the time responses of z 1 (t) of the ASPS payload with and without quantization, for the sampling period of T = I second. The initial

value of

z W

was chosen to be 0.002 m. As predicted by the discrete describing function analysis, the system does not exhibit any sustained oscillations when T = I sec-. However, the nonzero quantization levels did produce steady-state errors in z I (t). The computer simulated results and the result obtained by the least-upper bound method are tabuiated below for comparison. It is expected that the errors predicted by the least-upper bound method will be greater, since it is a worst-case study.

Sampling Period T = I sec

Quantization level h (m) zI(-,) least-up er bound (m)

zl

(°)) simulation (m)

2 -4 + 0.00081/8 -0.000444

2 -8

+. 0.000051114 0.000036

Figure 7-18 also shows that with the quantization level of 2 -8 (8 bits), .he time response of z 1 (t) is very close to that of z 1 (t) without quantization, so th.►t a larder word length seems unnecessary unless a sm,jIler steady-state error is required.

Figure 7-19 illustrates the time responses of z l (t) for T = 2 sec. For h = 2 4, the error is -0.0003 at t = 50 sec and -till increasing. For h =

2-8,

(34)

145

the response actually exhibited a sustained oscillation with a peak-to-peak amplitude of 0.000066 m. AS Shown in Fiy. 7-13, when T - 2 Sec, the system is mar,jinal in generating sustained osci'latiuns. It should be noted that the digital computer is riot the must suitable for c-imulatmy digit.il systems with quantizers, Since th-: computer itself is a digital system with its own quantization

love k . However, the results obtained here are conclusive enough to indicate the quantization effects in the ASPS payload, and are useful in the selection of the sampling period and the quantization IeveI.

For sampling periods greater than 2 seconds, the computer simulation results showed that sustained oscillations always existed.

ORIGM1, PAGE IS

OF PWR QU:1 i "

^IT

(35)

146

References

I. Technical Report, Space Shuttle Experiment I l ointin(I Mount UPM) Systems, An Evaluation of Concepts and Technologies, Jet Propulsion Laboratory

101-I, April I, 1917•

2. H. B. WAites, Equations of Motion Derivation for a S imp Iitied ASPS Midi' I, Memorandum, NASA, OLtober 1977.

3. B. C. Kuo, DIGITAL CONTROL SYSTEMS, SRL Publishinq Company, 1977.

• 4. Final Report, Stability Analysis of

the

Low-Cost Larye Space Telescope System, Systems Research Laboratory, June

General Disclaimer. One or more of the Following Statements may affect this Document