Optimization problems in control theory in terms of criterion function, controller, and information available to controlle

Ho, Y. C.

English

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TOWARDS "GENERALIZED CONTROL THEORY"* 
by 
Y. C .  Ho 
A f a v o r i t e  guessing game popular among symposia, meetings, a f t e r -  
dinner  conversations i n  con t ro l - theo re t i c  circles i n  recent  days seems 
t o  be the  quest ion "What Next?". 
growing mathematical r e s p e c t i b i l i t y  of t h e  d i s c i p l i n e  on one hand and 
t h e  va r ious  "success s t o r i e s ' '  i n  p r a c t i c e  on t h e  o t h e r .  
an at tempt  w i l l  be made i n  e s t a b l i s h i n g  a framework wi th in  which such 
a quest ion might be asked and answered.f 
informal and specu la t ive .  
This is appropr i a t e  i n  view of t h e  
I n  t h i s  note ,  
The discussion w i l l  be  
It seems t o  the  author  t h a t  t h e r e  are t h r e e  main ing red ien t s  i n  
an opt imizat ion problem, whether s t a t i c  o r  dynamic. They are: the  
c r i t e r i o n  func t ion ,  t he  c o n t r o l l e r ,  and the  information a v a i l a b l e  t o  
t h e  c o n t r o l l e r .  For t r a d i t i o n a l  c o n t r o l  theory,  a l l  the  ing red ien t s  
are s i n g u l a r  i n  the sense t h a t  t h e r e  is only one c r i t e r i o n ,  one central 
c o n t r o l l e r  coordinat ing a l l  c o n t r o l  a c t i o n s ,  and one information set 
a v a i l a b l e  t o  t h e  c o n t r o l l e r .  Within t h i s  s e tup ,  some very general  
and deep s ta tements  can and have been made f o r  t h e  problem. On the  
o the r  hand, one can a l s o  argue t h a t  t h i s  viewpoint is unnecessar i ly  
narrow. Surely,  w e  can a l l  v i s u a l i z e  s i t u a t i o n s  o r  problems i n  which 
the re  are more than one performance measure, more than one i n t e l l i g e n t  
c o n t r o l l e r  operat ing with o r  without coordinat ion from o t h e r s ,  and - 
*The research reported i n  t h i s  no te  was made poss ib l e  through sup- 
p o r t  extended by t h e  U. s. O f f i c e  of Naval Research under t h e  
J o i n t  Services  E lec t ron ic s  Program by Contract  N00014-67-A-0298- 
0006 and by t h e  National Aeronautics and Space Administration under 
Grant 1XR 22-007-068. 
-1.1 have d e l i b e r a t e l y  avoided t h e  inc lus ion  here  of r e l a t e d  bu t  never- 
t h e l e s s  important s u b j e c t s  such as,  mathematical system theory,  
s t a b i l i t y  theory,  model theory,  e tc .  where opt imizat ion is not  t h e  
main concern. 
To appear as Letter t o  t h e  Edi to  
December 1969. 
a c t i o n s  on Automatic Control ,  
-23-  
https://ntrs.nasa.gov/search.jsp?R=19700017430 2020-03-12T02:57:00+00:00Z
f i n a l l y  a l l  the  c o n t r o l l e r s  may o r  may not  have t h e  Same information 
set  a v a i l a b l e  t o  them. a start ,  we can d iv ide  each of t he  
ing red ien t s  i n t o  t h r e e  subca tegor ies .  
(i) one J (ii) two J’s with J1 - -J2 and ( i i i )  mu l t ip l e  J ’ s  with  
Ji i= l , . . . , n .  For t h e  c o n t r o l l e r  C, w e  hove ( i )  one C ,  ( i i )  two 
C I S  and (iii) mul t ip l e  C’s. 
t h e  cases  (i) one p e r f e c t  information set  ( i i )  one noisy (or im- 
For the  c r i t e r i o n  J, we have 
F i n a l l y  f o r  t he  information set, we have 
t , ; c ) .4 )  information set and (iii) mul t ip l e  information set e 
s u b c l a s s i f i c a t i o n s ,  t he  fol lowing t a b l e  of organiza t ion  is s e l f -  
With these  
explanatory.  
With t he  except ion of t h e  f i r s t  t w o  ca t egor i e s ,  I have given some 
references  containing r ep resen ta t ive  works where in t roductory  materials, 
r e s u l t s ,  and examples of a p p l i c a t i o n  may be found. One a l s o  no te s  
t h a t  i n  t h i s  framework, t he  names a t tached  t o  these  ca t egor i e s  appear 
inadequate but entrenched For example, a more d e s c r i p t i v e  t i t l e  f o r  
(3) is “2C/2J/lZ problem i n  genera l ized  con t ro l  theory.”  However, 
-24- 
one can hardly expect such a terminology to take hold at this stage. 
Except for the fir 
categories are available only for the static optimization case. Even 
there many basic concepts are subtle and difficult, e.g. the coalition 
problems in (6) and the definition of equilibrium solution of (8). 
Suffice is to paraphrase a famous remark: "they will not be solved 
in the first thousand days e e a but let us begin". 
Y. C. Ho 
Haward University 
Cambridge, Massachusetts 
Ref erencs: 
1. Rufus Isaacs, Differential Games, John Wiley, 1965. 
2.  R. D. Behn and Y. C. Ho, "On a Class of Linear Stochastic Dif- 
ferential Gamess" IEEE Transactions on Automatic Control, AC-13, 
No. 3, June 1968, pp. 227-239. 
3 .  I. B. Rhodes and D. G. Luenberger, "Differential Games with Im- 
perfect State Information," IEEE Transactions on Automatic Con- 
trol, AC-14, No. 1, February 1969, pp. 29-39. 
4 .  W. W. Willman, "Formal Solutions for a Class of Stochastic Pur- 
suit-Evasion Games," IEEE Transactions on Automatic Control, AC- 
14, No. 5, October 1969, to appear. 
5. L. A. Zadeh, "Optimality and Non-Scalar Valued Performance Criteria," 
, IEEE Transactions on Automatic Control, Vol. AC-8, 1963, pp. 59-60. 
6 .  N. 0. da Cunha and E. Pollak, "Constrained Xinimization under 
Vector-Valued Criteria in Linear Topological Spaces," Memorandum 
ERL-M191, November 1966, university of California, Berkeley. 
7. 3. He Case, "Equilibrium Points of N-Person Differential Games," 
University of Michigan, Dept. of Ind. Eng. Tech. Report 1967-1. 
8 .  A .  W. Starr and Y. C. Ho, "Nonzero-Sum Differential Games," 
Journal of Optimization Theory and Applications, 3,  No. 3, 
March 1969. . 
9. A. W. Starr and Y e  C. Ho, "Further Properties of Nonzero-Sum Dif- 
ferential Games," Journal of Optimization Theory and Applicc-;iorls, 
3, No. 4, April 1969. 
10. R. Radner, "Team Decision Problems," Anna1 of Math. Stat. Sept. 
1962. pp. 857-881. 
11. R. Radner, "The Evaluation of Infornation in Organizations," 
Proc. of Fourth Berkely Symp. on Math. Stat. 1961, pp. 491-533. 
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12. H. Witsenhausen, "A Counterexample in Stochastic Optimal Control," 
SIAM Journal on Control 5, No. 1, 1968, pp. 131-147. 
13. J. C. Harsanyi, "Games with Incomplete Information Played By "Bayesian" 
Players ,I1 1-111 Vole 14, No. 3, November 1967, 
pp. 159-182, No. 5, January 1968, pp. 320-334, No. 7, March 1968, pp. 
486-502 e 
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Toward 'generalized control theory'

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