Dynamic programming and inaccessible state variable theory applied to accelerometer desig

Dyer, P.

Kelly, D. W.

English

NASA Technical Reports Server

' - P  
1 
. 
TECHNICAL MEMORANDUM 
CONTRACT NAS8-20163 
A P P L I C A T I O N  OF CONTINUOUS FORM OF DYNAMIC P R O G W N G  
AND INACCESSIBLE STATE VARIABLE THEORY 
TO THE K-8 ACCEUROMETER 
bY 
P. Dyer 
May 26, 1967 \ 
! GPO PRICE S 
I ff 653 July 65 
i 
https://ntrs.nasa.gov/search.jsp?R=19680019497 2020-03-12T09:51:12+00:00Z
t' - 
I. INTRODUCTION 
Although a l a rge  amount of  i n t e r e s t  has  been shown i n  the  t h e o r e t i c a l  
aspects  of opt imal  con t ro l  over the  l a s t  decade, t h e r e  has only been a 
xinor attempt t o  apply t h i s  somewhat soph i s t i ca t ed  theory.  
t h a t ,  i n  general ,  l a rge  s c a l e  computers would be required.  
s tudy  i s  l imi ted  t o  l i n e a r  systems c e r t a i n  s i m p l i f i c a t i o n s  r e s u l t .  
It is true 
But, i f  t h e  
These s impl i f i ca t ions  lead t o  a simple des ign  technique which has ,  
d2parent ly ,  been overlooked o r  underestimated. The e f fec t iveness  o f  
t h e  rechnique f o r  a l i n e a r  sys t emwi th  access ib l e  s t a t e  v a r i a b l e s  has  
a l r eady  been discussed [l, 21 and t h i s  memorandum r e p o r t s  a pre l iminary  
s tudy  of t he  case'where some of t he  s t a t e  v a r i a b l e s  a r e  inaccess ib le .  
- 
- .  
. 
I .  
11. THEORETICAL BACKGROUND 
The requisite theory is readily available in the modern control 
theory textbooks [3, 41 and only the necessary results are presented 
here. 
- 
Consider a system described by the equations 
(11-1) 
. 
where & is an mth order column vector, 
and 
is an sth order column vector, 
is a disturbance vector. It is desired to minimize the performance 
criterion E, where 
E =  Value Exp e ct of e d LT {Z’Q +I’G} dt , (11-2) 
acd where the primes denote the transpose and A and H are positive 
definite weighting matrices. 
The optimal forcing function has been shown to be, 
where 
(11-3) 
(11-4) 
1 
and where 
. 
$ ( t )  = -A - B'K2 - K B f K CH-IC 'K2 . 2 2 (11-5) 
I f  a l l  of t he  s t a t e  va r i ab le s  5 cannot be measured, then the  opt imal  
conr ro l  has been shown to be, 
A 9C - U -DS , (11-6) 
A 
where 2 i s  the  bes t  es t imate  o f g  i n  a l i n e a r  o r  l e a s t  squares  sense.  
This es t imate  i s  given by 
(II- 7) 
where 2 is  a vec to r  o f  the'measured outputs  contaminated by some n o i s e  
LI) and Q i s  g i v k  by 
Q = A I M  - B f CD . 
The matrix M i s  a measurement matrix, i . e . ,  
- Y = P g + g  
(11-8) 
(11-9) 
aad A i s  given by 1 
2 
1 AI = 1 w w -  , (11- 10) 
where W is formed from the measurement noise vector 2. 
given by the steady state solution of 
Finally K is 
-1 
(11- 11) K = CVC' i. B K f  I(B' - m'w pj-K , 
where CVC' is a matrix formed f rom the disturbances in the system, d. 
It should be noticed that equations (11-5) and (11-11) are very 
similar and the same computer program can be used to compute the K 
and K matrices. 
2 -. 
3 
are 
i 
i 
;. 
f 
t 
111. APPLICATION OF COXTIWOUS FORM OF DYNAMIC PROGMMMING 
i 
The state equations for the K-8 accelerometer, a s  shown i n  Figure 1, t 
I 
(111- 1) 
where 
XI = p (radians) 
X = (radians/sec) 
X3 = a (radians) . 
X = & (radian/sec) 
. 
2 . 
4 
aild 
H = 93.0 gm - em - sec 
2 Ja= 2.66 gm - cm - sec 
J3= 0.123 gm - em - sec 2 
. 
4 
5 
k 
a, 
+I 
a, 
0 
k 
0 
rl 
a, 
0 
E 
4 
s2 
5 
a3 
a, 
W 
0 
ca 
h . 
I 
VI) 
I 
I 
rl 
a n d  
i 
T = (input torque) gm - cm 
, T = (feedback torque) gm - c m  B a 
Limits a r e  imposed upon T and X and a r e  a 1 
ITcl] < 1440.18 gm - cm , 
a ad 
- 
/ X l l  ,< 0.05236 radians . 
The Gbvious choice of a function t o  minimize Ta and X is 1 
Rearranging the  above equations t o  f i t  the form of (11-1) and (I1-2), 
t h e  following matr ices  a r e  obtained. 
0 0 
0 
A =  
0 0 
6 
(111-2) 
(111-3) 
0 
B =  1 
- 
0 
0 
0 
c =  
0 756.097 
0 1.0 
0 
1.0 0 
0 
0 
-34.96 0 
1 
(111-4) 
, H = [A] = [1.0] (111-5) 
. 
Zquations (111-3), (111-4), and (111-5) may be i n s e r t e d  i n t o  (11-5) 
t o  form the R i c a t t i  equation 
I .  
T -1 T I C = - A - B K - K B f K C H  C K  (111-6) 
t , w i t h  zhe f ina l ' condi t ions  K(T) = 0. 
gives the  following R i c a t t i  equation 
Using time r e v e r s a l ,  T = T - 
T -1 T 
K = A + B K + K B - K C H  C K  (111-7) 
wlch i n i t i a l  condi t ions  K(o) = 0. 
s t e a d y - s t a t e  va lue  of  K may be  used to o b t a i n  t h e  feedback necessary  
fo r  the  optimum c o n t r o l  sub jec t  to t h e  c o n s t r a i n t s  of (11-2). 
The s o l u t i o n  of  (111-7) f o r  t h e  
Therefore 
(111-8) 
7 
F i g u r e  2 is the  block diagram of t h e  p l a n t  with t h e  optimal cont ro l .  
Table  1 is a t abu la t ion  of t he  d i f f e r e n t  va lues  of a ( l , l >  and t h e  D 
r n n t r i s  a s  w e l l  a s  the  poles of the cont ro l led  system. 
p l o t  o f  t h e  poles with  a ( 1 , l )  as a parameter. 
Figure 3 i s  a 
2’ x3, and x 4 Unfortunately, i n  this case, the s t a t e  variables X 
a r e  inaccess ib le  and so ‘nave to be estimated. 
. 
8 ’  
t 
9 
I 
 
. I 
?I x 
ri 
0 
k 
U 
C 
0 u 
4 
rb 
‘ri 
U 
PI 
0 
E 
a, s 
E a 
k 
M a 
.ri 
Q 
1 
1 
10 
n 
d 
n 
4 
W 
1 
c7 
n 
d 
n 
l-l 
W 
rc 
I 
0 
0 
0 
0 
0 
0 
rl 
I 
n 
0.3 
0 
rl 
x 
rl 
m 
rl 
0 
rrr - s  
CI 
0 
U 
' m  
I 
0 
rl 
co m 
0 
m 
1- 
+ I  + I  
cv 0 
rl (v 
m a i 
I rl 
1 
I 
I ,  . .  
zf cv I " :  
\o 
\o h 
\o (v 
rl I 
I 
y/9. 
zf 0 
to 
03 I I ? 
co b l x  
fa 0 
N 0 
\o 0 
rl 0 
crl 0 
rl 
I 
- 
I 
rl rl 
to 
h 
I- 
cv 
(v 
\o 
d 
cr) 
I 
0 
0 
0 
0 
0 
0 
0 
0 
d 
I 
n 
n 
X x 
d I 1  
x I :  
11 
I 
i 
a, 
E a 
L4 
rd 
pc 
(d 
a 
a 
n 
r( 
d 
W 
n 
(d 
r -
U 
-rl 
5 
i 
1 
( IV- 1) 
! 
I V .  ATPLICATION OF INACCESSIBLE STATE VARIABLE THEORY 
The s t a t e  equat ions may be rewritten t o  include t h e  e f f e c t s  o f  
co i se ;  i n  t h i s  ca se  a disturbance, da, and measurement noise,WL. 
Refer t o  Figure 4. 
k1 = x2 
2 = H/J -f- T /J 
2 P P B  
x = x  
3 4  
4 - a ! .  a a !  ir = -H/J x2 li- T /J + . 
The output ,  Y1, i s  given by 
Y1 = x  +a1 . 1 ( I V - 2 )  
The coin2utatioA o f  t h e  -D matrix proceeds as before  and t h e  D matrix w i l l  
be  unchanged. The a d d i t i o n a l  matr ices  requi red  for t h e  computation of 
Q and A are ,  1 
x =  [l 0 0 03 
(IV-3) 
12 
13 
x 4 
. 
V = [sd,] , 
where sda  i s  the  spectral  densi ty  of  t h e  d is turbance  dol and ds, i s  tlne 
q e c t r a l  dens i ty  of t he  measurement no i se  w 
I' 
Employing (11-ll), (11-lo), (11-8) and (11-7) enables t h e  formation 
A 
of t h e  bes t  es t imate  of 5, 5. 
A s  t h e r e  t7as l i t t l e  information a v a i l a b l e  on t h e  noise  present  
i n  the system, somewha t a r b i t r a r y  va lues  were used. These corresponded 
t o  noise  of 1 
ainpiitude and 
n-ult iplied by 
-. 
g amplitude and 20 KC/s  bandwidth f o r  du and .005 radians 
a s imi l a r  bandwidth f o r  LU 
0.2,  2 , 5  and 10 t o  g i v e  four  s e t s  of values .  
These f igu res  were then  1' 
The macrices A1 artd Q were  computed f o r  each of these  va lues  and 
The es t imator  (11-7) may now be formed. a r e  given i n  Tables 2 and 3 .  
A l t e r n a t i v e l y  take t h e  Laplace t ransform o f  (11-7) and use  c a p i t a l  
l e t t e r s  t o  denote Laplace*Transforms o f  X and Y: 
i 
i 
[SI + QJX = (IV- 4 )  
where s i s  Laplace's  opera tor  and I, of ,course, t he  u n i t  matrix.  
(IV-5) 
U = -DX , 
14 
(IV-6) 
, 
I 
i 
kist 
Code , 
7 
0. i 
1.0 
5.0 
10.0 
7.87 
77.25 
294.34 
459.01 
31.24 
2,984.16 
43,319.11 
105,344.61 
- .034 
- 3.55 
-21.71 
-27 95 
- 275.33 
- 2,700.83 
-10 , 290.26 
- 16,046.95 
. 
Table 2.--Tabulation of the  al Matrix with the Disturbance, dol, 
and Neasurenlent Noise, "1, as a Parameter. 
Xoise Code Refers co those Noise Values in the Text. 
The 
i 
15 I 
Ya31e 3--Tabula~ion of the Q Matrix wi th  the 
A Iiatrix and Noise Code as a Parameter. 
. 
- ~ ~. 
-4,158.69 108.06 332.48 
.1 x 10 21,547.03 237. OG 552.82 
.1 x LOi2 102,835.67 520.18 856.59 
.1 s 10 l3 359,892.92 1,132.36 1,288.21 
10 .1 x lol1 
.1 x 10 l4 1,172,779.20 2,452 -24 1,911.91 
~~ ~ 
Noise Code 5 
~ - 
1,598.00 108.06 532.48 
27,303.73 237.06 552.82 
108,592.36 520. I8 856.59 
.I x 10 
.1 x 10 365,649.6 1 1,132.36 1,288.21 
.I x lol1- 10 
.1 x l o l3  
.1 x 10 l4 1,178,535.90 2 , 452.24 1,911.91 
12 
Noise Code 1 
9,187k43 108.06 332.48 
34,893.15 237.06 552.82 
.1 x LO 
116,181.79 520.18 856.59 
.1 'c 10 
.1 x 10 
.I '3 73,2 3 9.04 1,132.36 1,288.21 
- 10 - * 
11 
12  
.i x 10 1,186,125.4 2,452.24 1,911.91 
~~ 
Noise Code .1 
.1 x 1010 11,612.93 10s. 06 332.48 
37,3 18.65 237.06 552.82 .i x io 
118,607.29 520.18 856.59 .1 x 10 
.I x 10- 
11 
12 
13 375;664.54 li  132.36 1,288.21 
.i x 10 '' 1,188,550.90 2,452.24 1,911.91 
16 
U = - D ( s I  4- Q)-'Ap . (IV-7) 
Thus the  t r a n s f e r  func t ion  of  t h e  f i l t e r  i s  
U/Y = - D ( s I  4- Q) -1 A Y . 
1 
The po le s  and zeros of t h i s  f i l t e r ,  for d i f f e r e n r  va lues  of t he  D 
x~trix, and d i f f e r e n t  no ise  leve ls ,  a r e  given i n  Table 4.  
c o n p t a L i o n  involved the inversion of a 4 x 4 matrix and i t  i s  f e l z  
t h a :  ::-ere might be some e r r o r  i n  t h e  computation of  t he  zeros  of t h e  
function. 
The 
{ )  
. 
However t h e  poles  given are correct. 
, 
. 
17 
! 
1 
Table 4.--Tabciation of chc Poles  and  Zeros 
of the Transfer Filnc-iion of t h e  
Filter 8s given in (IV-3) with the  
A Kat r ix  arid Noise Code as Parameters. 
c (1,l) Zeros Poles  
0, -I- 110. - 378., -205 j 375. - 55i., -230 5 j 48i. 
0, 4- 38. .. 746., -284 2 j 621. 
0, - so. - 995., -375 2 j 814. 
0, - 115. -1,334., -518 2 j1,088. 
- ,? 0, i IC. 
~ ~~ 
N o i s e  Code 5 
a (1, i) Zeros Poles  
10 
- 11 
12 
14 
.I x 10 0, 4- 172. - 318., -154.2 j 317. 
-1 :i LO 0, 55. - 458., -189.k j 415. 
.1 :< 1g13 0, 2. - 64-8., -251.5 j 547. 
0, - 69. -1,211., -498.i - j1 ,004.  .1 x 10 0, - 40. - 883. ,  -349.2 j 734. .I :i IO 
Noise Code 1 
. 
a (1, I) Zeros Poi& 
0 .  
0, -k 622. - 20S., -100.2 j 241. 
.1 x iG;; 0, + 178. - 332.,  -i49.2 j 322. 
.1 x io-- G, -!- 87. - 492., -220.2 j &43. 
0, i 43. - 713., -326.2 j 622. .1 x 10 
-1,028., -480.i j 887. .1 s 10 14 0, +- 17. 
10 .1 x lo,, 
13 
Noise Code .1 
a (1, i) Zeros P o l e s  
.1 x lo., 10
.I x I C - L  12 .i x IC- 
.1 :< :& 
.I x i0l4 
~ ~- 
0, t 3 , 4 8 2 .  - 170., - 35.2 j 219. 
0, -k 275. - 282., -139.2 j 293. 
0, f 137. - 435., -214.2 j 408. 
0, 52. - 953., -478.2 j 848. 0 3  82 - - 651., -322.2 j 585. 
18 
V. EVALUATION 
The usefu lness  of the  procedure ou t l ined  above depends upon t h e  
e 3 s 2  with  which it m y  be applied and t h e  e f f e c t i v c c e s s  05 t he  o p t i m l  
ccn;i’ol. Once a computer program has been w r i t t e n  t h e  design procedure 
i s  r e l a t i v e l y  sinple so t h a t  the  performance i s  t h e  determining f a c t o r .  
rn. l n e  pcr fo rmnce  of the  system discussed i n  t h i s  memorandurn is now 
b e i q  evaluated a t  Huntsvi l le .  
. 
- .  
1. T. Dyer, D. Rutherford, and A.  R. PI. Nofon, "Application of  
I>ync:aLc Prog~accnLzig to tlic Dcsfgn of Pimarfarit Autostabilixcrs, ' I  
S.  R. AeSoc., Api-ii,  1966. 
2 .  2 .  Dyer, "Application of Var ia t iona l  Nethods t o  A i r c r a f t  Control,  ' I  
l i i  . D . Disse r t a t ion ,  Not t ingham Universi ty ,  Engiand . 
3 .  .4. R .  PI. NoCon, - 1nt rodcc t io ; l  Var ia t ionr ! l  Kethods &I Zonzrol 
EzzFnecrinq, Berganon ?ress, Inc., New York, 1965. 
4 .  A. R. X. Notoa, "The Extension of Cer t a in  i icsu l t s  Ln t he  Trearment 
a; L:xccessLble S t a t e  Variables," IEEE Transactions on huto;.;latic 
Con t ro l ,  Volume AC-11, No. 4 ,  Octcjber, 1966, pp. 569-675. 
- -  
- 
20 - 


Application of continuous form of dynamic programming and inaccessible state variable theory to the K-8 accelerometer

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server