A technique is presented for obtaining a control law to regulate the modal dynamics and identify the modal parameters of a flexible structure. The method is based on using a min-max performance index to derive a control law which may be considered to be a best compromise between optimum one step control and identification inputs. Features of the approach are demonstrated by a computer simulation of the controlled modal response of a flexible beam

Thau, F. E.

Catalan 

NASA Technical Reports Server

i 8 8 5 - 3 1 2 0 3  
A NONLINEAR DUAL-ADmiVE 
CONTROL STRATEGY FOR 
IDENTIFICATION AND CONTROL 
OF FLEXIBLE STRUCTURES" 
F. E. Thmo 
City Cokge of The City Univenrity of New Yo& 
New York, NY 10031 
A B S T R A C T  
A t e c h n i q u e  i s  p r e s e n t e d  f o r  o b t a i n i n g  a c o n t r o l  l a w  t o  
r e g u l a t e  t h e  modal d y n a m i c s  a n d  i d e n t i f y  t h e  modal p a r a m e t e r s  
o f  a f l e x i b l e  s t r u c t u r e .  The m e t h o d  i s  b a s e d  o n  u s i n g  a 
min-max p e r f o r m a n c e  i n d e x  t o  d e r i v e  a c o n t r o l  l a w  w h i c h  may 
be c o n s i d e r e d  t o  be a b e s t  c o m p r o m i s e  b e t w e e n  o p t i m u m  
o n e - s t e p  c o n t r o l  a n d  i d e n t i f i c a t i o n  i n p u t s .  F e a t u r e s  o f  t h e  
a p p r o a c h  a r e  d e m o n s t r a t e d  b y  a c o m p u t e r  s i m u l a t i o n  o f  t h e  
c o n t r o l l e d  modal  r e s p o n s e  o f  a f l e x i b l e  beam'. 
I. INTRODUCTION 
A c l a s s  o f  i n d i r e c t  a d a p t i v e  c o n t r o l  s y s t e m s  p r o p o s e d  f o r  t h e  
c o n t r o l  o f  l a r g e  s p a c e  s t r u c t u r e s  [ l ]  i s  b a s e d  on  a modal 
d e c o m p o s i t i o n  o f  t h e  s y s t e m  d y n a m i c s  and may i n c o r p o r a t e  o n e  
o r  more  o n - l i n e  t e s t i n g  schemes [ 2 ]  t o  d e t e r m i n e  when 
s u c c e s s f u l  p a r a m e t e r  i d e n t i f i c a t i o n  h a s  been a c h i e v e .  The 
c o n t r o l  s t r a t e g y  u s e d  i n  c a l c u l a t i n g  t h e  a c t u a t o r  i n p u t s  m u s t  
a c h i e v e  a d e q u a t e  r e g u l a t i o n  o r  t r a c k i n g  p e r f o r m a n c e  and, a t  
t h e  same t i m e ,  prov;de i n p u t s  t o  a l l o w  a d e q u a t e  p a r a m e t e r  
i d e n t i f i c a t i o n .  A on:ro1 s y s t e m  d e s i g n e r  i s  t h u s  f a c e d  w i t h  
t h e  p r o b l e m  o f  ' P v i s i n g  a c o n t r o l  s t r a t e g y  t o  e n s u r e  
a c c e p t a b l e  s y s t e m  p e r f b r m a n c e  e v e n  when o n - l i n e  p a r a m e t e r  
i d e n t i f i a b i l i t y  t e s t s  hasre f a i l e d  b e c a u s e  t h e  s y s t e m  
* T h i s  work  was s u p p o r t e d  b y  N A S A  u n d e r  G r a n t  NAG-1-6. 
117 
https://ntrs.nasa.gov/search.jsp?R=19850022890 2020-03-22T20:01:39+00:00Z
c o n f i g u r a t i o n  has changed o r  t h e  env i ronmen t  i n  w h i c h  t h e  
system o p e r a t e s  has changed. 
I n  t h i s  paper  we f o r m u l a t e  and examine t h e  pe r fo rmance  o f  a 
n o n l i n e a r  d u a l - a d a p t i v e  c o n t r o l  scheme i n  which a 
sampled-data c o n t r o l l e r  i s  d e s i g n e d  t o  s e l e c t  a b e s t  
compromise between an i n p u t  s i g n a l  t h a t  i s  optimum f o r  
mean-square system r e g u l a t i o n  and an i n p u t  s i g n a l  t h a t  i s  
optimum f o r  pa ramete r  i d e n t i f i c a t i o n .  Dual  c o n t r o l  t h e o r y ,  
o r i g i n a l l y  f o r m u l a t e d  by Feldbaum C3,41, has been s t u d i e d  i n  
[ 5 - 7 1  and i n  t h e  r e f e r e n c e s  c i t e d  t h e r e i n .  A key c o n c e p t  
i n t r o d u c e d  by Feldbaum i s  t h e  d u a l  c o n t r o l  s t r a t e g y  based on 
a pe r fo rmance  i n d e x  t h a t  t a k e s  i n t o  a c c o u n t  t h e  f a c t  t h a t  
f u t u r e  o b s e r v a t i o n s  on t h e  p r o c e s s  w i l l  be made. A 
c o n t r o l l e r  m a v  be a b l e  t o  "probe"  t h e  system f o r  s t a t e  and 
pa ramete r  e s t i m a t i o n  improvement, wh ich  t h e n  may improve  
f u t u r e  r e g u l a t i o n  and t r a c k i n g  per formance.  I n  many 
s i ' t u a t i o n s  where t h e  d u a l  n a t u r e  o f  s t o c h a s t i c  c o n t r o l  i s  n o t  
t a k e n  i n t o  a c c o u n t  t h e  c o n t r o l l e r  becomes " c a u t i o u s "  C5,61 
and tends  t o  " t u r n - o f f " .  T h i s  u n d e s i r a b l e  phenomenon i s  
a v o i d e d  by t h e  approach d e s c r i b e d  below. 
11. FORMULATION OF AN A D A P T I V E  PERFORMANCE I N D E X  
The d i s c r e t e - t i m e  dynamics f o r  each mGde i s  assumed t o  be 
d e s c r i b e d  by t h e  ARMA model 
y (  t ) + a l y (  t - l ) + a e y (  t - 2 )  = b l u (  t - l ) + b ~ u (  t - 2 ) + e (  t) 
where y (  t )  denotes modal d i sp lacemen t ,  u( t) denotes modal 
f o r c e ,  and e ( t )  i s  a sequence o f  independent ,  
e q u a l l y - d i s t r i b u t e d ,  normal ( 0 , d )  random v a r i a b l e s .  It i s  
assumed t h a t  e ( t )  i s  i ndependen t  o f  y(t-l),y(:-2),..., 
u ( t - l ) , u ( t - 2 ) , . . .  and t h a t  t h e  pa ramete rs  a ~ , a 2 , b ~ , b 2  
a r e  unknown c o n s t a n t s .  I f  we l e t  Y t  deno te  t h e  i n f o r m a t i o n  
a v a i l a b l e  t o  t h e  c o n t r o l l e r  a t  t i m e  t, 
I Y t  = y ( t )  ,y( t-1) ,. . ., u ( t - 1 )  ,u( t -2) , .  .. { 
x ( t )  denote t h e  modal pa ramete r  v e c t o r  and e ( t )  denote,a 
modal measurement v e c t o r ,  
x T ( t )  = (a1 s a 2  sblsb2) ;  
e T ( t )  f ( - y ( t - l ) , - y ( t - 2 )  , u ( t - l ) , u ( t - e )  
where ( . ) f  denotes v e c t o r  o r  m a t r i x  t ranspose ,  t h e n  ( 1 )  
may be r e w r i t t e n  as 
y (  t )  = e l (  t ) x (  t ) * e (  t )  
( 3 )  
( 4 )  
118 
where t h e  c o n s t a n t  pa ramete r  "dynamics' s a t i s f i e s  
x ( t + l )  = x ( t )  
It can t h a n  be shown, f o l l o w i n g  t h e  a n a l y s i s  o f  L81, t h a t  t h e  
c o n d i t i o n a l  d i s t r i b u t i o n  o f  x ( t + 2 )  g i v e n  Yt+ 
w i t h  mean x ( t + 2 )  and c o v a r i a n c e  m a t r i x  P ( t + 2  t where x ( t )  and 
P ( t )  s a t i s f i e s  t h e  d i f f e r e n c e  e q u a t i o n s  
i s  normal  
( 5 )  
Furthermore,  t h e  c o n t r o l  l a w  t h a t  m i n i m i z e s  t h e  r e g u l a t i o n  
c r i  t e r i  on 
V,(u(t)) = E { y * ( t + l ) I Y t )  ( 9 )  
i s  g i v e n  by 
( 1 0 )  xi ( t+ 1) x3 ( t+ 1 1 +P3i ( t+l 1 ei ( t+ l  1 
x i (  t+ 1 +P 33 ( t+ 1 1 
u ( t )  = - 
T where r )  denotes t h e  sum o v e r  i I 1 t o  4 w i t h  t h e  v a l u e  3 
excluded. 
To p r o v i d e  bounded modal i n  u t s  t h a t  improve pa ramete r  
a m p l i t u d e  w i l l  n o t  become e x c e s s i v e l y  l a r g e ,  t h e  c o n t r o l l e r  i s  
des igned  t o  o p t i m i z e ,  a t  each samp l ing  i n s t a n t  t, t h e  f o l l o w i n g  
per formance c r i t e r i o n :  
i d e n t i f i c a t i o n  accu racy  whi ! e g u a r a n t e e i n g  t h a t  t h e  modal 
s u b j e c t  t o  t h e  c o n s t r a i n t s  
where 
119 
0 
V c  denotes an a c c e p t a b l e  o r  d e s i r e d  l e v e l  o f  r e g u l a t i o n  c o s t .  
V I ( u ( t ) )  denotes and i d e n t i f i c a t i o n  c o s t  f u n c t i o n  o f  u ( t ) ,  
V I ( u ( t ) )  = t r a c e  [ ~ ( t + 2 ; j  ( 1 4 )  
denotes ar,d a c c e p t a b l e  o r  d e s i r e d  l e v e l  o f  i d e n t i f i c a t i o n  
0 
VI 
c o s t .  The m a x i m i z a t i o n  i n d i c a t e d  i n  (11) y i e l d s  a f u n c t i o n  
V ( u ( t ) )  which,  a l t h o u g h  n o t  convex, i s  i n t e r p r e t e d  as 
s p e c i f y i n g ,  f o r  each a d m i s s i b l e  u ( t ) ,  t h e  most c o s t l y  l i n e a r  
c o m b i n a t i o n  o f  r e l a t i v e  r e  u l a t i o n  and r e l a t i v e  i d e n t i f i c a t i o n  
c o s t .  b l i n i m i z a t i o n  o f  V(u3 t h u s  y i e l d s  t h e  modal i n p u t  t h a t  
m i n i m i z e s  t h i s  most  c o s t l y  c o m b i n a t i o n  of  r e l a t i v e  
i d e n t i f i c a t i o n  and r e g u l a t i o n  per formance.  
111. S I M U L A T I O N  RESULTS 
S i n c e  V c ( u ( t ) )  and t r a c e  P ( t + 2 )  a r e  r e l a t i v e l y  s i m p l e  
f u n c t i o n s  o f  u ( t )  t h e  n u m e r i c a l  s o l u t i o n  o f  t h e  one-s tep  
o p t i m i z a t i o n  p rob lem (11 ) - (13 )  a t  each samp l ing  t i m e  i s  q u i t e  
f e a s i b l e .  R e s u l t s  o f  s i m u l a t i o n  s t u d i e s  d e s c r i b e d  below 
i l l u s t r a t e  an i n t e r e s t i n g  f e a t u r e  o f  t h i s  a proach: s i n c e  t h e  
pa ramete rs  i n v o l v e d  i n  t h e  e v a l u a t i o n  o f  V c  P u ( t ) )  and 
V I ( u ( t ) )  depend on system measurements, t h e  optimum 
d i s t r i b u t i o n  o f  r e l a t i v e  c o s t ,  A ( u )  depends on o n - l i n e  
measurement d a t a  and hence, a t  each samp l ing  i n s t a n t ,  t h e  
w e i g h t i n g  between i d e n t i f i c a t i o n  and r e g u l a t i o n  w i l l  change 
depending on t h e  o n - l i n e  system performance. T h i s  i s  i n  
c o n t r a s t  t o C 9 1  i n  wh ich  a f i x e d  w e i g h t i n g  between a b s o l u t e  
c o n t r o l  and i d e n t i f i c a t i o n  c o s t  Is used a t  each sample t i m e .  
I n  t h e  s i m u l a t i o n  s t u d y  we compare t h e  performance o f  t h r e e  
c on t r o l  sy stems : 
a )  A c o n s t r a i n e d  a d a p t i v e  c o n t r o l l e r  t h a t  m i n i m i z e s  (9) 
s u b j e c t  t o  t h e  c o n t r o l  magni tude c o n s t r a i n t .  
b )  An optimum i d e n t i f i c a t i o n  c o n t r o l l e r  t h a t  minirbi i res 
( 1 4 )  s u b j e c t  t o  t h e  c o n t r o l  magni tude c o n s t r a i n t .  
c )  The one-s tep  d u a l - a d a p t i v e  c o n t r o l l e r  based on 
( 1 1 ) - ( 1 3 ) .  
I n  F i g u r e s  1 -3  we p r e s e n t  s i m u l a t e d  modal response d a t a  f o r  t h e  
f i r s t  f l e x i b l e  mode o f  t h e  Lang ley  beam e x p e r i m e n t  d e s c r i b e d  i n  
[lo] where we assume h e r e  t h a t  a s i n g l e  a c t u a t o r  i s  used. The 
accumulated o n - l i n e  r e g u l a t i o n  c o s t ,  VT, shown I n  F i g u r e  1 i s  
d e f i n e d  as 
120 
and t h e  o n - l i n e  i d e n t i f i c a t i o n  c o s t ,  PT ,  i s  d e f i n e d  as 
P T ( N )  = t r a c e  [ P ( N ) l  
where P ( N )  i s  c a l c u l a t e d  o n - l i n e  u s i n g  ( 8 ) .  Note  t h a t  f o r  t h e  
f i r s t  10 t o  15  samp l ing  t i m e s  t h e  r e g u l a t i o n  c o s t  o f  t he  
d u a l - a d a p t i v e  c o n t r o l l e r  i s  c l o s e  t o  t h a t  o f  t h e  c o n s t r a i n e d  
min imum-var iance c o n t r o l l e r  and t h e  f d e n t i f f c a t f o n  c o s t  o f  the  
d u a l - a d a p t i v e  c o n t r o l  system i s  c l o s e  t o  t h a t  o f  t h e  
c o n s t r a i n e d  one-s tep  optimum i d e n t i f i c a t f  on c o n t r o l l e r .  
F i g u r e  2 i n d i c a t e s  t h a t  t h e  d u a l - a d a p t i v e  c o n t r o l l e r ' s  a c t u a t o r  
s i g n a l s  s w i t c h  between i t s  l i n f t s ,  t0 .5,  more f r e q u e n t l y  t h a n  
do t h e  a c t u a t o r  s i g n a l s  o f  t h e  o t h e T  c o n t r o l l e r s .  
due t o  t h e  l a c k  o f  any energy c o n s t r a i . n t  i n  t h e  above p rob lem 
f ormul a t i  on. 
T h i s  may be 
A f u t u r e  s t u d y  w i l l  examine t h e  pe r fo rmance  o f  t h e  
e n e r g y - c o n s t r a i n e d  d u a l - a d a p t i v e  c o n t r o l l e r  i n  compar ison w i t h  
energy- c ons t r a  i ned m i  n i  mum- va r i  ance and one- s t e p  optimum 
i d e n t i f i c a t i o n  c o n t r o l l e r s .  The r e l a t i v e  r e g u l a t i o n  c o s t  and 
r e l a t i v e  i d e n t i f i c a t i o n  c o s t  d e f i n e d  i n  ( 1 3 )  a r e  p l o t t e d  i n  
F i g u r e  3 where 
0 
Vc ( N )  = Q 2 N 
i s  t h e  accumulated c o n t r o l  c o s t  t h a t  wou ld  be a c h i e v e d  i f  t h e  
pa ramete rs  o f  t h e  system where known p r e c i s e 4  and i f  an 
u n c o n s t r a i n e d  c o n t r o l  l a w  were used; ~2 I: 10- i  was used i n  
t h e  s i m u l a t i o n  runs.  A c o n s t a n t  v a l u e  V f  10-4 was chosen 
a s  i n d i c a t i n g  t h e  a c c e p t a b l e  l e v e l  o f  pa ramete r  
i d e n t i f i c a t i o n .  F i g u r e  3 i n d i c a t e s  t h a t ,  dependin on o n - l i n e  
a t  one s a w p l i n g  i n s t a n t  can  have w i d e l y  d i f f e r i n g  shapes f r o m  
t h e i r  r e s p e c t i v e  d i s t r i b u t i o n s  a t  o t h e r  samp l ing  t imes .  T h i s  
l e a d s  t o  t h e  o n - l i n e  v a r i a t i o n s  i n  t h e  d u a l - a d a p t i v e  c o n t r o l  
s t r a t e g y  men t ioned  e a r l i e r .  
measurements, t h e  one-s tep  i d e n t i f i c a t i o n  and r e g u  ! a t i o n  c o s t  
The s i m u l a t i o n  r e s u l t s  i n d i c a t i o n  t h a t  t h e  one-step, 
c o n s t r a i n e d  d u a l  - a d a p t i  ve c o n t r o l  1 e r  has t h e  f e a t u r e  o f  
p r o v i d i n g ,  based on measured data,  sy-ctem i n p u t s  . t h a t  r e s u l t  i n  
pa ramete r  i d e n t f f i c a t i o n  w h f l e  m a i n t a i n i n g  bounded modal 
amp1 i tude  response. 
121 
REFERENCES -
1. 
2. 
3 .  
4. 
5. 
6. 
7 .  
8. 
9. 
Montgomery, R. C. and Thau, F.E., " A d a p t i v e  and L e a r n i n g  
C o n t r o l  o f  L a r g e  Space S t r u c t u r e s , "  A I A A  Guidance and 
C o n t r o l  Conference, June, 1980. 
Sundarara jan,  N. and Montgomery, R. C. , " A d a p t i v e  C o n t r o l  
o f  a F l e x i b l e  Beam U s i n g  L e a s t  Square L a t t i c e  F i l t e r s , "  
1983 L a r g e  Space S t r u c t u r e s  Workshop, B lacksburg ,  Va. 
Feldbaum, A. A., "Dual  C o n t r o l  Theory I - I V ,  Au tomat ion  
and Remote C o n t r o l  1960, v o l .  21, pp. 874, pp. 10~3; 1 ~ b 2 ,  
v o l .  22 , PP. 1, PP. 103. 
Feldbaum, A. A. , O p t i m a l  C o n t r o l  Systems, Academic Press,  
New York,  1965. 
Bar-Shalom, Y;, " S t o c h a s t i c  Dynamic P r o  r a m i n  
and Prob ing ,  
C a u t i o n  
I E E E  Trans., Vol. AC - 28, No. 3; Oct. 
1981, pp, 1184 - 1195. 
U i t t e n m a r k ,  B., " S t o c h a s t i c  Ada t i v e  C o n t r o l  Methods: A 
Survey," In t .  J. C o n t r o l ,  Vol. 2!, No. 5, pp. 705 - 730, 
1975. 
Bar-Shalom, Y.  and Tse, E., "Dual E f f e c t ,  C e r t a i n t y  
Equ iva lence ,  and S e p a r a t i o n  i n  S t o c h a s t i c  C o n t r o l  , 
" I E E E  Trans., Vol .  AC - 19, pp. 494 - 500, Oct. 1974. 
Astrom, K. 3. and Wittenmar!, B o ,  "Pi-obleas o f  
I d e n t i f i c a t i o n  and C o n t r o l ,  3 .  Hath. Anal  and Appl ,  
Wit tenmark,  e . ,  "An A c t i v e  S u b o p t i n f l  Dual  C o t t t r o l l e r  f o r  
Systems w i t h  S t o c h a s t i c  Parameters,  Auto. Con t r .  Theory 
Appl., Canada, Vol. e, pp. 13-19,  1975. 
V O I .  34, pp. 90 - 113, 19-11. 
10. Thau, F. E., Monto ornery, R. C., and Horner,  G.C., 
"On-L i ne S t r u c  tura!  Parameter  I: d e n t i  f f c a t i  qn, 
Guidance and C o n t r o l  Conference, Au u s t ,  1981, 
P r a c  . A I A A  
Albuquerque, New Mexico, pp. 5aO-53 !I . 
122 
500 
450 
400 
350 
300 
250 
200 
150 
100 
VT 
50 
0 
3 - x  to 
40 
PT 
1 I I I I I 
0 10 20 30 4 0 p  
SAMPLE nM 
(a) ONE-STEP OPTIMUM IDENTIFICATION 
(b) MINIMUM-VARIANCE ADAPTIVE 
(e) DUALADAPTIVE 
I 
1_11111 
0 10 20 30 40 50 
SAMPLE TIME 
Fig. 1 .  On-Line Regulation and Identification Cost 
for Three Feedback Control 1 ers 
123 
0.1 
Y O  
-0.1 
0.2r 
U 
-0.2 L 
I 1 I I 1 I 
0 IO 20 30 40 50 
SAMPLE TIME 
(a) ONE-STEP OPTIMUM IDENTIFICATION 
(b) MIN IMUM-VMI ANCE ADAPTIVE 
(c) DUAL-ADAPTIVE 
0.5 
u o  
4 . 5  
0.5 
u o  
-0.5 
0 IG 20 30 40 50 
SAMPLE TIME 
F i g .  2 .  Modal Displacement and Modal Force 
f o r  F i r s t  F l e x i b l e  Mode 
124 
42 
41 
N=13 
1 ! I I I 1 1 I l l  !A 
O f +03 -0.5 
U U. 
F ig 3.  Re la t ive  Control Cost RC 5- V,(u)/Vi apd Re la t ive  
I d e n t i f i c a t i o n  Cost  R I  = V,(u) /Vi  for 
3 Sample Times 
125 


A Nonlinear Dual-Adaptive Control Strategy for Identification and Control of Flexible Structures

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