Limit cycles and stability of three-axis space vehicle attitude control syste

Burge, W. W.

Russell, D. W.

English

NASA Technical Reports Server

Y) 
I I * 
(I) 
L 
rr 
3 z 
& 
c 
LL 
Technical Memorandum Number 16 
Contract NASS-20104 
Study of Limit Cycles and Stability 
of a Space Vehicle Attitude Control System 
W. W. Burge 
D. W. Russell 
July 24, 1967 
Engineering Experiment Station 
Auburn University 
Auburn, Alabama 
of 
https://ntrs.nasa.gov/search.jsp?R=19680018490 2020-03-12T10:03:42+00:00Z
I INTRODUCTION 
This memorandum extends the work reported i n  the  F i f t een th  
Technical Report (~AS8-20104), Determination of t he  L i m i t  Cycles 
f o r  a Dual-Chaw-el Missile Yaw-Axis Control System [l], to a three- 
a x i s  a t t i t u d e  con t ro l  system f o r  a space vehic le .  The o r i e n t a t i o n  
of the  con t ro l  t h r u s t e r s  about the  center-of-gravi ty  and about the 
three-axes i s  shown i n  Figure 1. 
The objec t ive  of the study was t o  determine the l i m i t  cyc les ,  
the regions of s t a b i l i t y ,  t h e  e f z e c t s  of v a r i a t i o n s  i n  parameters 
on the s t a b i l i t y ,  and methods t o  make the  system more s t a b l e .  
A n a l y t i c a l  and analog computer s imulat ions were employed. 
[ 13 D. W. Russel l ,  e t  a l . ,  "Determination of Limit Cycles f o r  
a Dual-Channel Missile Yaw-Axis Control Sys tern," Auburn University,  
F i f t een th  Technical Report, NAS8-20104, l4arch 31, 1967. 
r 
11. DESCRIPTION OF THE THREE-AXIS 
ATTITUDE CONTROL SYSTEM 
I n  t h i s  system, a t t i t u d e  c o n t r o l  is achieved by modulating 
the force ( t h r u s t )  produced by the four  engines or  t h r u s t e r s ;  
these forces  a r e  modulated about a value set  by the t h r o t t l e  command 
s igna l ,  Tr. 'The d i r e c t i o n  of the  forces  i s  f ixed .  
A second-order a c t u a t o r  pos i t i ons  a valve which con t ro l s  the  
propel len t  flow and the re fo re  the force  produced by an engine. 
The ac tua to r  is  ve loc i ty  l imi ted ,  which l i m i t s  the  rate of change 
of force.  The parameters of the  system are such t h a t  t he  second- 
order ,  ve loc i ty- l imi ted  a c t u a t o r  can be represented adequately as 
an i d e a l  r e l a y ;  t h i s  g r e a t l y  s i m p l i f i e s  the  study and i s  used i n  
a l l  the analyses.  
. 
Angular Pos i t ion ,  4, and angular ve loc i ty ,  0, about each a x i s  
The a re  fed back t o  form a three-axis  closed-loop c o n t r o l  system. 
block diagram i s  given i n  Figure 2.  
A .  Torque Equations 
The torques (T) about t h e  th ree  axes,  i n  terms of t he  engine 
t h r u s t s ,  F1, F2, F3, F4, t he  d i r e c t i o n  cos ines ,  a,  f3, 7 ,  and t h e  
moment arms, a, b, c ,  are: 
TX = c COS (F1 - 3'2 .. F3 + F4) + 
+ b COS 7 (-F1 + F2 + F3 - F4) 
(II- 1) 
2 
3 
Ty = c COS 01 (-F1 + F2 - F3 + F4) + 
a COS 7 (-F1 + F2 - F3 + F4) 
Tz = b COS Q! (F1 + F2 - F3 - F4) + 
9 
a COS f3 (F1 + F2 - F3 - F4) 
(11- 2) 
(11- 3) 
which simplify to : 
Tx = (F1 - F2 - F3 + F4) (C COS 
Ty = (-F1 + F2 -F3 + F4) ( c cos Q! + a cos 7 )  
- b COS 7 )  (11- 4) 
(11- 5) 
(11- 6) Tz = (F1 + F2 - F3 - F4) (b cos (2 + a cos B) 
Thus the angular accelerations about each axis, i n  terms of the 
torques and moments of inertia, I, are: 
0.  
- 
ax - 
$Y - 
.. - 
.. 
o =  
2 
where 
cx = 
- - 
- 
cz - 
1 
Ix ( C  COS @ - b COS y ) - 
- 1 ( c  cos a + Q! cos 7 ) 
IY 
1 
Iz 
- 
(b cos Q + a cos B ) 
(11- 8) 
4 
B.  The State Equations 
The state variables are designated as follows: 
x3 = F3 
x4 = F4 
X 8  = ay 
xg = ay 
x10 = @y 
x11 = @z 
x12 = az 
.. 
.. 
X13 = $2 
The state equations are: 
k1 = FL sgn E1 
' = I?L sgn E2 
$3 = FL sgn E3  
x2 
. (11-9) 
. (11-12) 
5 
Sfj = xg 
ltg = XI0 
;lo - 4 - Cy% (-sgn El + sgn E2 - sgn E3 + sgn E ) (11- 14) 
xll = x12 . 
x13 
- CZ% (sgn El + sgn E2 - sgn E3 - sgn E41 ¶ (11- 15) '13 - 
where 
(11- 16) 
111. STUDY OF TAE STABILITY BOUNDARY AND 
LIMIT CYCItES 
The three-axis  a t t i t u d e  con t ro l  system i s  analyzcd i n  a manner 
s imi l a r  t o  t h a t  reported i n  the F i f t een th  Technical Report. 
A. S t a b i l i t y  i n  the Vic in i ty  of the  Origin 
For t h i s  ana lys i s ,  the  i d e a l  relays are represented a s  ga in  
terms which, i n  the  l i m i t ,  approach i n f i n i t y .  Representing the r e l ays  
as ga ins ,  7 ,  then the s t a t e  equations become: 
# 
x9 = x10 
;'lo = rl[xl - x2 + x3 - x4 - 4aoyx8 - 4aly$] 
(111- 1) 
(111- 2) 
(111- 3) 
6 
7 
Yowever, 
-XI + x2 + x3 - x4 = -x7/c, 
x1 - x2 + x3 - x4 = - X 1 o / C y  
- x2  + x3 + x4 = -x13kz - x1 
Therefore, (111-2), (111-3), and (111-4) become 
(111-5) 
(111- 6) 
(111-7) 
Since the  equat ions for the  th ree  axes are not coupled, t he  
condi t ions f o r  s t a b i l i t y  i n  the  v i c i n i t y  of t he  origin can be 
determined, f o r  each axis, i n  the  same manna as was done i n  the 
Fifteenth Technical Report. Thus, t h e  system is  s t a b l e  i n  the  
v i c i n i t y  of t h e  o r i g i n  if a l l  the parameters, &, Box, alx,  6, 
sly, C, aoz, a12 are g r e a t e r  than zero.  aqr 9 
B. L i m i t  Cycles 
Three l i m i t  cyc les  can be ca l cu la t ed  by the  pp r iod ic i ty  o f  the  
phase space t r a j e c t o s y  for one a x i s ,  with the  other two axes  i n  a 
quiescent  state.  Therefore,  the r e s u l t s  of the  F i f t een th  Technical 
Report can be appl ied t o  c a l c u l a t e  a l i m i t  cyc le  f o r  each a x i s .  
Since the system i s  s t a b l e  a t  the  o r ig in ,  these  are uns tab le  l i m i t  
cycles .  
'=he F o s s i L i l i t y  of o ther  l i m i t  cycles (such as two axes o r  a l l  
th ree  axes being i n  a l i m i t  cyc le  a t  one t i m e )  was s tudied .  To 
i l l u s t r a t e ,  suppose all t h ree  axes a r e  i n  the  same l i m i t  cycle .  The 
r e l a y  switc : .lg sequence i s  shown i n  Figure 3. 
switching, El = E2 = Eg = E4 = 0. 
the 6's as defined i n  Figure 2 ,  
A t  any i n s t a n t  of 
Therefore, a t  t h a t  i n s t a n t ,  with 
Adding the above equat ions ,  
x + x  + X 3 + X 4 ' 0 ,  1 2 
(III- 8) 
which must hold f o r  a l l  switching i n s t a n t s .  However, re fe rence  t o  
Figure 3 r evea l s  t h a t ,  f o r  any a r b i t r a r y  set of d-c ievels f o r  t he  
9 
var i ab le s  xl, x2, x3, and x 4 ,  (111-8) does not hold a t  a l l  switching 
i n s t a n t s .  Therefore,  the assumed l i m i t  cycles cannot e x i s t  i n  all 
t h ree  channels simultaneously.  A similar l i n e  of reasoning can 5e 
appl ied t o  show t h a t  a l i m i t  cyc le  with two axes i p  phase and 0x13 
axis 180' out-of-phase c a m o t  occur. 
The general  case of  two o r  t h r e e  l i m i t  cyc les  of  a r b i t r a r y  
frequency occurring simultaneously has not been thoroughly s tudied .  
Z'urther work needs t o  be done i n  t h i s  area. 
C. S t a b i l i t y  Boundary 
The s t a b i l i t y  bcundary was obtained by t r i a l - and-e r ro r  using 
the analog computer s imulat ion shown i n  Figure 4. The s t a b i l i t y  
boundary f o r  i n i t i a l  condi t ions  of t he  th ree  ve loc i ty  terms (xg, 
x9, XI*), t h e  o the r  s ta te  va r i ab le s  having a zero  i n i t i a l  value, 
i s  shovn i n  Figure 5. It can  be v i sua l i zed  as being similar t o  
two four-sided pyramids whose bases are joined.  The vertices are 
po in t s  on t h e  l i m i t  cyc l e  t r a j e c t o r i e s .  
This work can be extended t o  e s t a b l i s h  t h e  boundary fcjr i n i t i a l  
condi t ions on t h e  p o s i t i o n  anc i cce l e ra t ion  terns. 
IV. C. %LUSIONS AND REC0MMENMTIOK;P 
It was determined t h a t  the  three-axis  con t ro l  system can possess 
a t  l e a s t  three unstable  l i m i t  cycles: a l i m i t  cycle  about each a x i s ,  
conditions about the other  two axis being quiescent .  Pa in ts  on these 
l i m i t  cycle  - t r a j e c t o r i e s  can be ca lcu la ted  using the r e s u l t s  of the 
Fif teenth Technical Report. 
s tud ies  are needed t o  determine i f  o ther  l i m i t  cycles  are possible .  
Addit ional  a n a l y t i c a l  and analog ccmputer 
The condi t ions f o r  s t a b i l i t y  about t he  o r i g i n  were found t o  
be the same as those for a single-axis system: 
gains must be pos i t ive .  
i n  veloci ty  only (see Figure 5) w a s  determined by t r ia l -and-er ror  
using an analog computer qimulation. Addit ional  analog computer 
s tud ies  would y i e l d  information about t he  s t a b i l i t y  boundary f o r  
other i n i t i a l  condi t ions and the e f f e c t s  on .the s t a b i l i t y  boundary 
of variat ion;  i n  system parameters. 
namely, a l l  the  system 
The s t a b i l i t y  boundary for i n i t i a l  condi t ions 
It would be usefu l  t o  extend the  conclusions reached i n  the  
Fif teenth Technical Report f o r  a s ingle-ax is  system t o  t h i s  th ree  
ax i s  system. Here again,  add i t iona l  analog computer s imulat ions 
8re suggested . 
11 
/ N \\ 
// 
// 
a 
m a4 c 
T i  
w 
rl 
0 
k 
3 
0 
E? 
5 
12  
u3 
+ 
cn 
rl 
0 
k 
A 
x-axis 
y-axis 
X 1  
x2 
x3 
t -  
Figure 3. Switching Sequences for Possible Concurrent Limit Cycles 
A 
14 
m 
V 
V 
d 
N 
0 
' I  
6 
d 
3 0
p9 
x 
4 
4 
15 
Figure 5. S t a b i l i t y  Boundary Projected onto the X -X Plane for Vmioua Value3 
9 12 
of X6. A l l  Other Variables Are Assumed To Be In i t ia l ly  Zero. 


Study of limit cycles and stability of a space vehicle attitude control system

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server