Studies were performed to determine location for proof mass actuators, if a significant reduction in the number of sensors would work, and to design a control law to meet requirements for line of sight error and actuators. Conclusions are drawn and briefly discussed

Collins, Emmanuel G.

Norris, Gregory A.

Skelton, Robert E.

English

NASA Technical Reports Server

N 8 T - 1 7 8 3 3 .i
Regulation of the
SCOLE Configuration
by
Gregory A. Norris
Emmanuel G. Collins s
Robert E. Skelton
Purdue U nivers ity
Q "-
https://ntrs.nasa.gov/search.jsp?R=19870008400 2020-03-20T12:42:49+00:00Z
REGULATION OF
THE SCOLE CONFIGURATION
INVESTIGATORS
Gregory A. Norris
Emmanuel G. Collins
Robert E. Skelton
322
.PERF.,.ORMANCE REQUIRE .I_NT$
(I) MAINTAIN RMS OF THE STEADY STATE LINE-OF-SIGHT (LOS)
ERROR WITHIN A SPECIFIED BOUND.
(II) MAINTAIN STEADY STATE ACTUATOR VARIANCES AS CLOSE AS
POSSIBLE TO SPECIFIED BOUNDS.
ORIGINAL SCOLE CONFIGURATION
• LOCATION OF z PROOFMASS ACTUATORS NOT SPECIFIED.
e 42 SENSORS PROVIDED.
, 323
OBJECTIVES
(1) DETERMINE LOCATIONS FOR PROOF MASS ACTUATORS.
(II) DETERMINE A REDUCED SET OF SENSORS.
(III) DESIGN A CONTROL LAW TO MEET PERFORMANCE REQUIREMENTS
FOR LOS ERROR AND ACTUATORS.
• SOLUTIONS TO THE 3 PROBLEMS ARE INTERDEPENDENT.
• CHOICE OF ACTUATORS AND SENSORS INFLUENCES CONTROL LAW.
• CHOICE OF CONTROL LAW INFLUENCES SENSOR AND ACTUATOR
SELECTION.
LINEARIZED
VECTOR SECOND ORDER MODAL FORM
i i
DYNAMICAL MODEL
"" " d _(q + Dq + n =, u+w)
output vector y
Yl = LOSx ' Y2 = LOSy , Y3 = LOSz
E(LOS error) 2 = (Ey21 + Ey 22 + EY3)2 I/2
y=C n
P
measurement vector z
where
Z
p,r
Z
= position & rate measurement vector
= P n+P n+v
p v p,r
a
Z
= acceleration measurement vector
= Qn + va .
Q(__2 _ Dn + _u + _w) + v
n
Z
p,r
Za-QBu
a
=M n+M n+v
p v
=>
[Pp°iM =
P 0 _Q_2_
°1VS --
v 0 -QD
Va+ -
ASSOCIATED SENSOR NOISE (v) & ACTUATOR NOISE (w) ARE
CORRELATED
• MODEL OBTAINED USING CUBIC BEAM ELEMENT SHAPE FUNCTIONS
FOR BEAM BENDING AND LINEAR SHAPE FUNCTION FOR BEAM
TWIST.
• 32 MODES IN ORIGINAL MODEL.
• MODAL COST ANALYSIS USED TO REDUCE TO 23 MODE DESIGN AND
EVALUATION MODEL.
327
MODAL COST ANALYSIS
modal mode
cost rank no. modal cost freq. (hz)
1 1 infinite 0
2 2 infinite 0
3 3 infinite 0
4 5 ,91 I c-+07 .299e400
5 7 .363e+07 . 118e._01
6 4 ,336e+07 .276e_ O0
7 6 . 138o+07 , 811e*00
8 8 ,955e+06 .205e+01
9 10 . 673e+04 .551e+01
10 9 556c-, 04 . 478e_ 01
11 11 .246e-_ 02 . I _.3e- 0_-_
12 14 . 365e+0! .243e+02
13 17 .245e+01 ,395e+OD
14 12 .305e+00 . 129_ _02
_5 18 . 116e+00 . 390e+0_
16 15 . 349e-01 .256e+02
17 26 ,995e-02 . I09e_03
18 25 .377e-02 , IC'3e+03
19 13 376e--02 '-'37 E.-402
20 29 174e-02 . J40e_03
2J _'3-,5 . C..76r_.-03 .215e+03
22 20 . b97e-03 .586e÷0;-:
23 28 . 370e-03 . J 3"-.."_-_ 03
24 23 i "" -• ,_5e-03 . bt7e-*u;-'
25 19 . 310e-04 . 5F_.! _.-t 02
26 34 275c.-04 "_ I _;e +03
• • t"
2.'7 32 .617e-05 . 175e+03
28 31 .294e-05 . 175e+03
29 27 . 131e-05 . 135e+03
30 24 . 140e-07 . lO_e._03
31 30 . J 34e-07 . I&7F+03
32 33 . 4 I 3C-08 . ."'(;Oe4 0"3
33 _ 298e-I0 811 e-t 02
34 18 . 340e--I I . 515E,_ 02
35 21 , 226e-13 .7DZ'e_O2
T'igid
rigid
rigid
bendl
bendl
bendl
bend i
bendl
b c-.ndi
bend_
bendl
bending
twi et
be:,d i:-,9
bendi_ 9
bendln_
b _ndi n(.
b ¢.Tid i n9
b end lhn
b L- lJ d I t'J(J
[ end ! r,_n.
t.C-h C ",n ._
bendir, u
.-,
bend lr, g
b end ing
bending
bend ils9
bending
tuBi st
t. _., i st
t_,i _ t
tu,_ _t.
bending
tu, i =.t
twist
body
body
body
TIg (roll)
ng
ng (pitch)
ng
ng
n9
n9
ng
• FIRST 5 FLEXIBLE MODES DOMINATE MODAL COST
• BEAM BENDING DOMINATES MODAL COST
CONTROL LAW DESIGN VIA
THE OUTPUT VARIANCE ASSIGNMENT ALGORITHM
• ITERATIVE ALGORITHM DEVELOPED BY SKELTON AND DELORENZO
• OBJECTIVE IS TO CHOOSE DIAGONAL Q AND R IN THE LQG COST
FUNCTIONAL
S.T. THE LQG
= E (yTQy + uTRu)
CONTROL LAW SATISFIES
Ej 2 = o2l (or _ o 2 )
WHILE MINIMIZING
V i= I +n
Y
nu Z u2
ill •P.
1
bounds on input variances
..
SENSOR AND ACTUATOR SELECTION
VIA INPUT/OUTPUT COST ANALYSIS
• SUBOPTIMAL APPROACH.
• BASED ON DECOMPOSING COST FUNCTION
= E (yTQy + uTRu)
as
n n
y u
_. v y + Y v u
1 i
i=l i=l
n n
U Z
r vw + E vv
i i "
i=l i=l
a DEFINES ACTUATOR EFFECTIVENESS,
vact=i vl - vl
AND SENSOR EFFECTIVENESS
_sen --_ iV
i i
• DELETES ACTUATOR(S) OR SENSOR(S) WITH LOWEST
EFFECTIVENESS VALUES.
330
SOLUTION PROCEDURE
• BEGIN WITH LARGE SET OF PROOF MASS ACTUATORS AT
FIXED LOCATIONS
DESIGN LQG CONTROLLER
SOME RESULTS
ORIGINAL SCOLE PROPOSAL
rms(los error) _ .02 deg
OUR FINDINGS
if noise through shuttle cmgs only:
rms(los error) ) .045 deg
if equivalent noise through all actuators:
rms(los error) ) .075 deg
CONCLUSIONS
• ORIGINAL SPECS ON LOS ERROR ARE NOT ACHIEVABLE.
• MUST MODIFY LOS SPECS.
332
ACTUATOR SELECTION
DEFINE
n 2
u Eu i dimensionless measure of
USUM = Z
2 total control effort
i=l _i
250
200
USuM
iS0
I D,JO_"
50
• 3Z_"
0
I . i I I I I" , I I I I I t
56 46 36 32 28 24 20 I8 I6 14 12 II I0
N0. C) lu _ 0..TU#_TO_,_
FINDINGS
• BY USING REDUCED SET
SAVINGSIN
OF ACTUATORS THERE IS A 50%
CONTROL EFFORT (AS MEASURED BY USUM).
333
Jm-
Iii I
iii
\
\
MORE E_'FE_.:- UE
Z
334
uSurl,, I00
3.0
2,5
2.0
1.5
1.0
0.5
0.0
42 32
_'_s L_os _-f'r'o_"x,= • o_.. _%
24
+
15
,,,l
I0
• GOOD PERFOR_4ANCE MAY BE ACHIEVED WITH A MUCH SMALLER
SET OF SENSORS.
CONCLUSIONS
(I) RMS(LOS ERROR) _ .02 DEG IS NOT
ACHIEVABLE.
RMS(LOS ERROR) _-- .05 DEG IS
ACHIEVABLE IF NOISE IS ONLY THROUGH
SHUTTLE CMG'S.
RMS(LOS ERROR) <--- .08 DEG
ACHIEVABLE IF (EQUIVALENT)
IS THROUGH ALL ACTUATORS.
IS
NOISE
(II) PROOF NLASS ACTUATORS SHOULD BE
PLACED NEAR TOP OF MAST.
(III) GOOD PERFORMANCE MAY BE ACHIEVED
WITH A (SIGNIFICANTLY) REDUCED
SET OF SENSORS.
33g


Regulation of the SCOLE configuration

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