Information on the identification and control of spacecraft is given. Maximum likelihood estimation, identification accuracy issues, steady state identifiability analysis and stochastic error with process noise are among the topics addressed

Barrett, M. F.

Greene, C. S.

English

NASA Technical Reports Server

IDENTIFICATION AND CONTROL OF SPACECRAFT 
C. S. Greene 
M. F. Barrett 
Honeywell Systems and Research 
Minneapolis, Minnesota 
79 
https://ntrs.nasa.gov/search.jsp?R=19830013991 2020-03-21T03:44:54+00:00Z
IDENTIFICATION & CONTROL OF SPACECRAFT 
* THE PROBLEM 
+ CONTROL 
- CLASS I CAL 
- MODERN 
* IDENTIFICATION 
RIGID PAYLOAD 
MANIPULATOR 
80 
CLASSICAL CONTROL 
* SCALAR FEEDBACK DESIGN 
* LEAD-LAG CONTROLLER 
K(s) = 
k(s+a) 
(s2+2<b+b2) 
* BANDWIDTH AROUND 1 RAD/SEC 
SHUTTLE ElND PAYLOAD: FROM ALPHA TO Y-ATTITUDE 
FREQUENCY <radocc> 
81 
SHUTTLE hND PfiYLOhD: FROll ALPIIA TO Y-ATTITUDE 
\ 
- 
- 
- 
FREQUEHCY (rad/secj 
MODERN CONTROL 
7-l 
_1 
1e2 183 
* START WITh FULL-STATE DESIGN 
- GOAL: MINIMIZE PAYLOAD ATTITUDE ERRORS 
- ITERATE ON CONTROL PENALTY TO ACHIEVE BW OF 5 R/S 
* DESIGN FILTER TO RECOVER LQ RESPONSE 
- USE STEIN/DOYLE ROBUSTNESS RECOVERY RESULTS 
+ TEST ROBUSTNESS 
STATE FEEDBACK LQ 
1 OS freuuencu 
u1unt.r: Singular values of K<s>tGXf;> w=lEB 
10s frawency 
83 
I + KG 
1 og frequent 9 
-I 1’ r 
e 
a 
1 
d _ 
- 
log frequency 
84 
IDENTIFICATION 
k4XlttUM LIKELIHOOD ESTM4TION (MLE) 
-.-__ ~- 
IDENTIFICATION 
PROCESS 
1 E[wT] - H E[VVT] = R 
o STATE SPACE 
n* NODES (2x2 8LocKs) 
m INPUTS 
P OUTPUTS 
MODEL STRUCTURE 
-_ . 
Ij 
-Fi (t), 
xi ;= 
: 
(U= [I [ I Y . . . ; ci 0 :... : 1 
l FREQUENCY DOMAIN 
Y(S) = (G~s)1uk.)+w(s)1 + v(s) 
G*(s) = 
ci tl; 
i- 1 s~+2QpiL 
: . -_ 
I 
“1 (t)+ Xi -_. . 
t)+ ” (t) [1 
o PARAMETER VECTOR 
85 
. . . . . . . . . . .-.- . ..-__._-_. -_-.--- 
MLE IDENTIFICATION SETUP 
l RESIDUAL DEFINITION 
'k 8 y(kT) - $(kT) 
o LIKELIHOOD FUNCTION (NEGATIVE LOG) 
L(a) f! 
N 
k& 
LkW 
A ;l = 
k=O 
7 LOG DET Sk + r: Sk rk 1 
WHERE 
a b Iw2 i, 23ioi, bi, ci; i = 1, . . . . nj = UNKNOWN PARAMETERS 
'k 
A 
= Eatrk+ = PREDICTED RESIDUAL COVARIANCE 
MLE IDENTIFICATION SOLUTION 
m PARAMETER ESTIMATE (THEORETICAL) 
G t ARG[~:~ L(.)1 = PARAMETER ESTIMATE 
o ITERATIVE ALGORITHMS 
,,. .I, > GRADIENT: iJ+l = iJ - cJ VL(a,) 
NEWTON-RHAPSON: iJ+l = iJ - [ 02LQ] -lvLGJ, 
L’ : ;!: ,:;- 
WHERE 
86 
IDENTlFICATlON ACCURACY ISSUES 
0 SYSTEMATIC ERRORS: E,(8) - aI 
- MODEL-ORDER MISMATCH 
- TEST SIGNAL AMPLITUDE AND SHAPING 
- SYSTEMATIC DISTURBANCES 
-SENSOR/ACTUATOR MODEL ERRORS 
l STOCHASTIC ERRORS: 0 -1 
“CN 
- RANDOM DISTURBANCES ANU SENSOR NOISE 
- TEST SIGNAL AMPLITUDE AND SHAPING 
- IDENTIFICATION TIME INTERVAL 
STEADY-STATE IDENTIFIABILITY ANALYSIS (YARED) 
l EXPECTED LIKELIHOOD FUNCTION 
= ; [,,, DET s + TR ts-‘s.,] 
a RESIDUAL COVARIANCES 
KALHAN FILTER PREDICTED RESIDUAL COVARIANCE HATRIX 
= ACTUAL RESIDUAL COVARIANCE HATRIX 
NOTE: S. AND l’(a) CAN ONLY BE COHPUTED WHEN THE TRUE PLANT PARUIETERS ARE 
KNOWN I 
87 
EXPECTED ALE IDENTIFICATION SOLUTION 
o EXPECTED PARAMETER ESTIMATE (THEORETICAL) 
a. p E. (3 = ARG {"AN I'w] 
o ITERATIVE ALGORITHMS 
GRADIENT: ieJ+l = iaJ - oJVI*(~.,) 
NEWTON-RHAPSON: kJ+l = AcJ - [021%.,,] -l VI%.J 
WHERE VI*(a) i g (a) 
v*1*(4 x 2i 
aa (al 
STEADY STATE IDENTIFICATION ACCURACY 
a SYSTEMATIC ERRORS (BIASES) 
- PARAMETER RORS 
A, agIAS = a, - a* = 0 WHEN NO MODEL MISMATCH 
- INFORMATION KASURE (YARED) 
A 
I(a, ; ;,, 0 I*kl*) - I*&) 2 0 
o STOCHASTIC ERRORS 
E (vm CL,] -l f--g) WHEN NO MODEL MISMATCH 
88 
STOCHASTIC ERROR WITH PROCESS NOISE 
KAL"hN FILTER 
E(uuT) = UI 
E(wwT) = WI 
E{VvTl - RI 
Gi(joi) = 
j2Fiui 
i 
2R 
a0 ; 
c- 
IGi(juill 2 % <cl IG. (jw.) I2 E '~1 1 I R 
U = TEST SIGNAL COVARIANCE 
I 
DOT,, CONSTANT 
R = "EASUREHENT NOISE COVA"IANCE 
-- 
0 
w (PROCESS NOISE COVARIANCE) 
SIMPLIFIED IDENTIFICATION ACCURACY ANALYSIS 
(FOR EACH MODE) 
w Y 
E[uuT] = UI E[wwT] = WI E[VvT] = RI 
cb T 
s2+2zws+w2 
I I 
a p (w’, 2.3~. hl,... h,,,, cl,..., c,1 - (bL or cM) 
02 2 
wz8F2 
(w2) 2 
1; 
SNR (ti+l)~ 
p 4 max ( 
e,m 
2 
1; (WAS. NOISE1 
sNR (N+~)T 
(PROC. NOISE) 
Obe 
2 
he2 = ii& 
l [ l$ + p-l] 1 
(N+~)T 
!!XE: THIS ANALYSIS ASSUMES THAT UT<<1, E cc 1 AND APPLIES FOR EACH MODE 
89 
- 
NUMERICAL RESULTS 
o PROBLEM SIZE 
12 MODES (=22 - 8R.B, - 2 SMALL) 
2 INPUTS (a, 8 GIMBAL ANGLES) 
2 OUTPUTS (x, Y ATTITUDES) 
60 PARAMETERS (=l2 MODES x 5 PARAKTERS/MODE) 
o TEST SIGNAL, NOISE STATISTICS 
SAMPLE TIME: T = 0.1 SEC 
TEST SIGNAL: U = 4000(IN-LB)* 
PROCESS NOISE: W = 4O(IN-LB)2 
MEASUREMENT NOISE: R = 4 x lo-l2 RAD2 
o WORST-CASE RELATIVE ERRORS AT TIME (MODE 9): 
PARAMETERLSL(u14 
.* 0.0265 0.007 0.0007 
*w 3.75 1.0 0.1 
bl 13.6 3.6 0.36 
Cl 2.7 0.7 0.07 
=2 11.7 3.1 0.31 
SUMMARY 
o CONTROL PROBLEM 
- MODERN LQ CONTROL DESIGN WITH ROBUSTNESS RECOVERY 
PRODUCES ROBUST CONTROLLERS FOR LSS 
- ACCURATE ID ALLOWS A FIVE-FOLD INCREASE IN LOOP BW 
o IDENTIFICATION PROBLEM 
- STRUCTURAL MODES MAY BE IDENTIFIED ONE AT A TIME FOR 
SMALL DAMPING 
- LSS ID W/O KF 
--GREATLY REDUCES PARAMETER BIASES 
--GIVES ONLY MODEST INCREASE IN STOCHASTIC ERRORS 
- RELATIVE ERRORS IN PARAMETERS AFTER ID ARE SMALLER FOR 
FREQUENCY THAN FOR DAMPING OR MODE SHAPES 
o OPEN ISSUE: HOW ACCURATE MUST ID BE FOR ROBUST CONTROL DESIGN? 
90 


Identification and control of spacecraft

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