Systems optimization for motion stability matrix problem - predictive guidance for Thor vehicle trajector

Johnson, R. G.

Lukes, D. L.

English

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fioneywell Report 1546-QR-2 1 June 1965-' 
* PROGRESS REPORT 
For Period 14 February t o  14 May 
0 
ItESEARCH AND STUDY IN 
SYSTEM OPTIMIZATION TECHNIQUES 
. .  
P A  
W 
Office of Astrodynamics and Guidance Theory 
Aero ahd Astrodynamics Division 
George C. Marshall Space P l i g h t  Center 
National Aeronautics and Space"Adminlstrat1on 
Contract No, NAS 8-5222 
GPO PRICE 
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2 c IACCESSION 66-16082 NUMBER) f f H R y l  - 
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(CATEGORY) 
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INASA CR O R  TMX OR AD NUMBER) 
CFSTI PRICE(S1 s 
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ff 853 Julv 65 
Prepared by : 
D. L. Luke8 
R. G. Johnson 
r. e x ,  
Approved by: 
ugo Schuc 
DbkHctor of Reksearch 
. .  
Honeywell, Inc . 
Systems and Research Division 
Research Department 
St. Paul, Minnesota 
https://ntrs.nasa.gov/search.jsp?R=19660006793 2020-03-16T23:13:59+00:00Z
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Honeywell lieport 1546-w-2 1 June 1g@, 
RESEARCH AND STUDY IN 
SYSTEM OPTIMIZATION TECHNIQUES 
SECTION I 
GEMERAL 
This is the second quarterly progress report  submitted i n  
accordance w i t h  the provisions of Modification 4 t o  Contract 
NAS~-5222, 
It covers the period from 14 February 1965 t o  14 May 1965. 
a 
11 I1 Research and Study in  System Optimization Techniques. 
SECTION I1 'a 
SUMMARY OF PRIOR PROGRESS 
S t a b i l i t y  of Motion Study 
\ 
2 
The previous r e s u l t s  that  have been obtained i n  the area of 
the s t a b i l i t y  of motion are presented In Honeywell MPG Report 
1546-QR-1, 22 February 1965. 
of the connection between d i f f e ren t i a l  equations and t h e i r  Lyapunov 
It contains a mathematical analysis  
. functions . 
Guidance Study 
The literature on P-matrix predictive guidance was studied and 
A three the computer sirnulation t o  be programmed was decided upon. 
stage m o r  vehicle was-chosen as  the model with a cross product 
s teer ing  law. 
n 
Data on system parameters were gathered. 
SECTION 111 
PROGRESS DURING REPORTING PERIOD 
Stability of Motion Study 
All of t h e  books, reports and papers collected f o r  the  1JI;eraturc 
survey on methods and techniques for determining regions of s t a b i l i t y  
have been studied. Over t h i r t y  a r t i c l e s  were collected.  Spot 
t rans la t ion  w i t h  an examination of t he  equations was used t o  decide 
the  relevence of the  Russian work t o  the'fundamental problem, Some 
of t he  a r t i c l e s  were found t o  be on unrelated topics ,  
dealt  with Investigations of  very special  problems,, and (th;? 
Other papers 
n 
techniques used could not be applied t o  gur problems, 
of t h e  r e su l t s  on estimates of domains of s tab i l i ty  were so general 
t h a t  they provided l i t t l e  information about spec i f ic  systems. 
Some L. 
Several in te res t ing  resu l t s  concerning the estimate of domains 
of s t a b i l i t y .  have been found and some techniques f o r  estimating 
domains of  s t a b i l i t y  exist for cer ta in  types of' nonlinear systems. 
I n  one of the translated papers, by Dobrotin, estimates of the 
domain of asymptotic s t a b i l i t y  are obtained for systems of the  
form 
b 
where the ai are  constant and f s a t i s f i e s  cer ta in  growth conditions, 
UP estimates of the solutions a re  obtained. The technique used 
is based upon successive approximations and might be generalized 
e 
t o  other systems 
b 
, 
e. 
- . . p  
* .. 
. .. 
I 3 
Although the l i t e r a tu re  survey has not been exhaustive, it 
appeare $ha$ ehe am@ o f  zwmeamh in seabiliCy under canalderation 
is l a rge ly  undeveloped and i s  reveivlng some a t ten t ion  by mathema- 
t i c i a n s  and engineers In both the  U. S. and the U.S.S.R. 
- .  
New r e su l t s  have been obtained for  finding estimates of the  
domain of a t t rac t ion  for systems of the  form 
2 P AX + R(x), 
where A is a s tab i l i ty  matrix and R ( x )  contains higher ordered t e k s .  
The technique I s  based upon obtaining a Laypunov function f o r  the 
system. 
b 
Guidance Study , 
- 
The goal of t h i s  work I s  to  evaluate P-matrix predictive guidance 
- 
by computer simulation. A three-etage Thor vehicle was chosen and 
th rus t  and aerodynamic data were obtained from Douglas Aircraft  
Company. A ro ta t ing  oblate ear th  model and t he  ARDC-59 atmosphere 
were used. A Douglas Thor t ra jec tory  simulation was duplicated, and 
the  t i p -o f f  maneuver and cutoff time was changed t o  get a c i rcu lar  
o r b i t  Inject ion.  The adjoint matrix about t h i s  nominal c i rcu lar  
o r b i t  was generated and calculations using t h i s  adJoint matrix were 
stored on a magnetic tape,  Using these guidance sens i t i v i t i e s  a 
l i nea r  combination of the terminal e r ro r s  was driven to  zero by 
steering and cutoff commands. 
. : ,  . .  
I c 
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* . . e. 4 . '  4 * *  
A standard wind prof i le  was used as a crosswind perturbation. 
Thrust perturbations were made by changing the mass flow rate. 
One percent changes of the mass flow r a t e  plus the crosswind I?oduced 
differences between apogee and perigee of l e s s  than one &le. 
apogee and perigee w e r e  either above o r  below a few miles of the 
c i r cu la r  o r b i t  radius. 
from using a first order scheme In predictting the terminal errors. 
- 
. 
Both 
The apogee-perigee difference resultecl 
S a i l a r  e r ro r s  w i l l  occur from gyro and accelerometer e r ro r s  o r  cutoff 
e r rors .  Thus, the guidance scheme is sat isfactory since the L3rtur-  
bations chosen were extreme. 
- L 
Both velocity and mossproduct s teer ing laws w e r e  t r i e d  K i t h  the ,,' 
- 9  
gain increased or decreased a factor  of ten  from their best Values. 
Guidance was used dur ing  the whole f l ight  or j u s t  the last  20 seconds 
and no difference In apogee or perigee greater thancone tenth o f  a 
mile resulted.  This is because it i s  not necessary t o  s t e e r  c it 
a l l  the onboard e r rors  t o  l e s s  than a hundredth of a foot per 
second when the onboard s t a t e  information is  off' a coupiLe of f e e t  
per second. 
so good that f o r  prac t ica l  purposes programmed steer ing rates are 
actual  r a t e s ,  body dynamios were not included i n  t h i s  simulation. 
Since Douglas reported tha t  the Thor autopilot; was 
. 
The decrease i n  both the apogee and perigee distance occurred 
because the guidance mlssion chosen was t o  produce a cir(cular 
o rb i t  In the same plane as the nominal orb i t .  It was found tha t  
one could guide in to  a c i rcular  o r b i t  using the correct termiral  
s t a t e  even when extreme changes like a 15 per cent decrease i n  t he  
mass flow rate occurred, b u t  the orb i t  was 35 miles low. To 
prevent these low o rb i t s  t he  first stage should perhaps guide t o  
the r ight  a l t i t ude  and then later stages guide t o - t h e  correct 
I c i r cu la r  veloci ty  and zero terminal i n e r t i a l  Pl ight  path angle. 
The r e su l t s  obtained w i l l  be documentgd soon a f t e r  a few 
more runs are  made. 
SECTION IV 
PLANS FOR NEXT QUARTER 
i 
The stability study will be directed toward basic questions 
concerned with domains of a t t rac t ion  of dynamical systems. 
nonlinear system i n  Euclidean space can be studied a s  a l inear  
dynamical system i n  a function space, 
r e su l t s  about how the nonl inear i t ies  a f fec t  the  domain of a t t r  l e t i o n .  
Also a study of  the s t a b i l i t y  o f  smoothly non-stationery sys tem 
A 
The goal is t o  obtain 
w i l l  be car r ied .  on. 
The guidance study is  essent ia l ly  completed and the  r e su l t s  
'wlll b& presented I n  a separate report .  
SECTION V 
EXPENDITURES 
Total expenditures from 7 February through 16 May were $16,262, 
The contract i s  g6 per cent complete. 


Research and study in system optimization techniques Second quarterly progress report, Feb. 14 - May 14, 1965

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