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Game Theoretic Approach to Sensitivity Design of Optimal Systems
GAME THEORETIC APPROACH TO SENSITIVITY DESIGN OF OPTIMAL SYSTEMS
Yoshikazu SAWARAGI and Katsuya OGINO Dept. o f Applied Mathematics & P h y s i c s F a c u l t y o f E n g i n e e r i n g , Kyoto U n i v e r s i t y Kyoto, JAPAN
INTRODUCTION I t is a c h a l l e n g i n g problem t o d e s i g n an o p t i m a l c o n t r o l l e r f o r s y s t e m s w i t h u n c e r t a i n t i e s . Although it is p o s s i b l e t o p r e s e r v e a s a t i s f a c t o r y p e r formance by i d e n t i f y i n g t h e u n c e r t a i n t i e s and cons t r u c t i n g a d a p t i v e c o n t r o l l e r s , it i s u s u a l l y a d i f f i c u l t o r expensive t a s k t o r e a l i z e i t , especial l y when t h e u n c e r t a i n t i e s a r e l a r g e . From t h e p o i n t o f view o f s e n s i t i v i t y d e s i g n a g a i n s t l a r g e u n c e r t a i n t i e s , i t seems r e a s o n a b l e t o assume t h a t t h e s y s t e m u n c e r t a i n t i e s t a k e on v a l u e s which maxmize t h e performance i n d e x . Thus, t h e game t h e o r e t i c approach a p p e a r s t o b e a v e r y n a t u r a l and e f f e c t i v e one f o r t h e s e n s i t i v i t y d e s i g n and h a s t h e advantage t h a t it y i e l d s an optimal c o n t r o l l e r which is e f f e c t i v e u n d e r t h e w o r s t c a s e .
tpg
There have been v a r i o u s a t t e m p t s t o d e v e l o p s e n s it i v i problem a l o n g t h g , + ) i n e . Sworder , indicated the appliMesarovi?Y and D o r a t o e t a l . c a b i l i t y o f t h e g g ~ t h e o r yt o t h e s e n g j t i v i t y problem, w h i l e Salmon and Rohrer e t a l . a d o p t e d t h e min-max approach i n t h e y e g q i t i v i t y d e s i g n . On t h e o t h e r h a n d , Sarma e t a l . ' d i s c u s s e d t h e v a r i o u s d i f f i c u l t i e s inherent i n solving optimal control problems a l o n g t h i s l i n e . T h i s p a p e r p r e s e n t s two d i f f e r e n t game t h e o r e t i c a p p r o a c h e s t o t h e s t a t e r e g u l a t o r problem i n t h e presence o f l a r g e u n c e r t a i n t i e s . In t h e f i r s t p a r t o f t h e p a p e r , t h e s e n s i t i v i t y problem o f t h e s y s t e m w i t h u n c e r t a i n t y is d i s c u s s e d , and t h e o p t i m a l c o n t r o l problem w i t h u n c e r t a i n t y i s p r e c i s e l y f o r mulated and developed a s a game, where t h e c o n t r o l l e r and t h e e q u a t i o n e r r o r caused by t h e s y s t e m uncertainty a r e considered t o be the antagonists. The u p p e r bound o f t h e performance i n d e x p r e s e r v e d by t h e game t h e o r e t i c d e s i g n i s d e r i v e d a s t h e s o l t i o n of a matrix d i f f e r e n t i a l equation. It o f t e n happens i n p r a c t i c a l s i t u a t i o n s t h a t t h e game t h e o r e t i c d e s i g n is t o o p e s s i m i s t i c . S e c o n d l y , t h e n , p r e s e n t e d i s t h e s e n s i t i v e game t h e o r e t i c d e s i g n which i s an i n t e r - m e d i a t e between t h e p r e c e d i n g game t h e o r e t i c d e s i g n and t h e nominal d e s i g n which a d o p t s t h e model s y s t e m as t h e c o n t r o l l e d s y s t e m . The u p p e r bound o f t h e performance i n d e x p r e s e r v e d by t h i s d e s i g n is a l s o d e r i v e d . T h i r d l y , t h e r e l a t i o n between t h e e q u a t i o n e r r o r and t h e performance i n d i c e s c o r r e s p o n d i n g t o t h e t h r e e d e s i g n methods a r e a n a l y z e d and t h e c o n d i t i o n s f o r t h e performance
i n d i c e s t o be p - s e n s i t i v e 9 ) a r e d e r i v e d . Numerical examples a r e p r e s e n t e d t o show t h e e f f e c t i v e n e s s o f t h e p r e s e n t game t h e o r e t i c a p p r o a c h e s t o t h e s e n s i t i v i t y design. OPTIMAL CONTROL PROBLEM IN THE PRESENCE OF UNCERTAINTY L e t t h e model s y s t e m o f a n a c t u a l s y s t e m w i t h uncert a i n t y b e g i v e n by where x , u , A ( t ) and B ( t ) a r e r e s p e c t i v e l y t h e ns t a t e v e c t o r , t h e r - c o n t r o l v e c t o r , t h e nxn- and nxr-nominal m a t r i c e s . The o p t i m a l c o n t r o l problem c o n s i d e r e d i n t h i s p a p e r i s t h e s t a t e r e g u l a t o r problem, t h a t i s , t h e problem o f d e s i g n i n g t h e o p t i m a l c o n t r o l which minimizes t h e f o l l o w i n g performance i n d e x .
where T,S,Q( t ) and R( t ) a r e r e s p e c t i v e l y a f i x e d terminal time, a constant nxn-positive semidefinite m a t r i x , an n x n - p o s i t i v e s e m i d e f i n i t e m a t r i x and a n rxr-positive d e f i n i t e matrix. I t is w e l l known t h a t t h e o p t i m a l c o n t r o l f o r t h e model s y s t e m ( I ) , i n t h e feedback form, i s g i v e n by
u;:
=
R ~ l B t K
(3) 1 and t h e o p t i m a l v a l u e o f performance i n d e x i s g i v e n by 1 J ( u 5 ) = 7 xAKl(0)xo ('4)
where K (t) is t h e s o l u t i o n o f t h e m a t r i x R i c c a t i 1 equation
K1(T) = S
(5)
The d e s i g n which a d o p t s t h e model s y s t e m ( 1 ) a s t h e c o n t r o l l e d system w i l l henceforth be r e f e r r e d t o a s t h e nominal d e s i g n , and u* i n ( 3 ) and J(u'5) i n ( 4 ) w i l l r e s p e c t i v e l y b e c a l l e d nominal o p t i m a l c o n t r o l and nominal o p t i m a l p e r f o r r a n c e . I t is assumed h e r e t h a t t h e a c t u a l s y s t e m w i t h unc e r t a i n t y i s r e p r e s e n t e d by
where AA(t) and AB(t) correspond t o t h e system unc e r t a i n t y . I t i s t o be noted t h a t t h e i n i t i a l cond i t i o n o f t h e system i s assumed t o be e x a c t l y k n m . When t h e nominal o p t i m a l c o n t r o l u* is a p p l i e d , i n t h e feedback form, t o t h e a c t u a l system (61, t h e system is w r i t t e n as
mum energy. Thus t h e d e s i r e d s e n s i t i v i t y design o f t h e system under l a r g e u n c e r t a i n t y can n a t u r a l l y be formulated as a game between t h e c o n t r o l l e r and t h e system u n c e r t a i n t y . By doing t h i s , one can t a k e t h e e f f e c t o f l a r g e system u n c e r t a i n t y i n t o consid e r a t i o n a t t h e i n i t i a l s t a g e o f t h e design. This i s t h e b a s i s of t h e p r e s e n t game t h e o r e t i c design f o r t h e u n c e r t a i n system. The r e g u l a t o r problem can t h u s be r e s t a t e d as a game a s follows: " For t h e system
Here E r e p r e s e n t s t h e e q u a t i o n e r r o r i n t h e model 1 system (l), and i s c a u s e d by t h e v a r i a t i o n i n t h e m a t r i c e s A and B. The value o f t h e a c t u a l p e r f o r mance i n d e x ( 2 ) e v a l u a t e d along t h e s o l u t i o n o f ( 7 ) is given by A 1 J(u*,AA,AB) = 11(0) = x~Vl(0)xo (9) where V ( t ) is t h e s o l u t i o n o f t h e m a t r i x d i f f e r e n 1 t i a l equation
A s t h e nominal design does n o t t a k e t h e system unc e r t a i n t y i n t o account t h e value of t h e performance index I1 is no l o n g e r a c t u a l l y optimal. Thus, when t h e nominal o p t i m a l c o n t r o l is adopted t o t h e a c t u a l system, it can b e s a i d t h a t t h e performance index i s d e t e r i o r a t e d t o t h e v a l u e I1 by t h e equat i o n e r r o r El. E s p e c i a l l y when t h e e q u a t i o n e r r o r i s l a r g e , t h e performance d e t e r i o r a t i o n makes t h e nominal design meaningless. Thus, it i s h i g h l y d e s i r a b l e t o f i n d a s e n s i t i v i t y design which accounts t h e l a r g e u n c e r t a i n t y i n t h e system a t t h e i n i t i a l s t a g e of t h e design. When t h e s e n s i t i v i t y design is viewed a s an o p t i m i z a t i o n problem with u n c e r t a i n t y , t h e game t h e o r e t i c approach appears t o be a very n a t u r a l one.
w i t h payoff
f i n d feedback c o n t r o l s u * ( t ) and v;(t) which a r e o p t i m a l i n t h e s e n s e t h a ? , f o r any feedback c o n t r o l s up( t ) and vg( t ) , t h e r e h o l d s
i . e . , f i n d a s a d d l e p o i n t o f J ( u v ) i f it exists'! The s u b s c r i p t 2 i n (110, ( 1 5 ) & d 2 i 1 9 ) i s u t i l i z e d t o d i s t i n g u i s h t h e above problem from t h e o r i g i n a l r e g u l a t o r problem. I t is t o be n o t e d t h a t t h e m a t r i x R ( t ) i s a s u i t a b l y chosen nxn-positive d e f i n i t e 2 matrix. The l a s t term i n ( 1 5 ) r e p r e s e n t s t h e consuming energy o f t h e system u n c e r t a i n t y . Thus, i n t h e above f o r m u l a t i o n , it is assumed t h a t t h e system unc e r t a i n t y s e e k s an o p t i m a l feedback c o n t r o l which maximizes t h e performance i n d e x ( 2 1 , w h i l e minimiz i n g i t s consuming energy. I n a q u i t e s i m i l a r form t o t h e one-sided r e g u l a t o r probl?m, ~ ~ e l p ~ t i mc oa nl t r o l s o f t h e above problem a r e glven ' by
u* = -R - 1B'K2x
(17)
GAME THEORETIC DESIGN
L e t ' s g i v e a t t e n t i o n t o t h e a c t u a l system ( 6 ) which can be r e w r i t t e n as k=Ax+Bu+v2, v
2
4 AAx +
and t h e value of t h e o p t i m a l payoff is given by
x(0)=xo where K ( t ) i s t h e s o l u t i o n o f t h e m a t r i x R i c c a t i 2 equation
ABu
where t h e n - v e c t o r v ( t ) r e p r e s e n t s t h e system un2 c e r t a i n t y , a n d it i s reasonable t o r e w r i t e v ( t ) i n 2 t h e form v2 = E2x
(13)
Here n m - m a t r i x E ( t ) corresponds t o t h e e q u a t i o n e r r o r i n t h e mode?.
Henceforth t h e o p t i m a l c o n t r o l u * ( t ) i n ( 1 7 ) w i l l be 2 r e f e r r e d t o as t h e GO-control, and t h e above design w i l l be c a l l e d game t h e o r e t i c design.
The unknown v e c t o r v ( t ) , which r e p r e s e n t s t h e 2 l a r g e u n c e r t a i n t y i n t h e system, causes t h e p e r f o r mance d e t e r i o r a t i o n i n t h e design. Then, from t h e p o i n t o f view o f s e n s i t i v i t y d e s i g n , t h e v e c t o r v 2 ( t ) i n t h e form o f (13) s h o u l d be considered t o be under t h e c o n t r o l o f system u n c e r t a i n t y , and/or i n t h e s e n s e o f t h e w o r s t case i t should be consid e r e d as an feedback c o n t r o l by which t h e system u n c e r t a i n t y antagonize t o t h e c o n t r o l l e r and maximize t h e performance index J ( u ) i n ( 2 ) by t h e mini-
I t should be mentioned t h a t a necessary and s u f f i c i e n t c o n d i t i o n f o r t h e s a d d l e p o i n t (u*,vn) t o e x i s t and t o be unique is t h a t the2ma$rix K ( t ) s o l v i n g ( 2 0 ) e x i s t f o r a l l tc[O ,TI. Although ?he game problem ( 1 4 ) and ( 1 5 ) is a game t h e o r e t i c ext e n s i o n of t h e one-sided r e g u l a t o r problem ( T h i s i s t h e main reason of f o r m u l a t i n g t h e s e n s i t i v i t y d e s i g n by ( 1 4 ) and ( 1 5 ) ) , t h e e x i s t e n c e of t h e s o l u t i o n o f t h e m a t r i x R i c c a t i e q u a t i o n ( 2 0 ) i s n o t always assured u n l i k e t h e one-sided r e g u l a t o r problem. T h i s
problem is examined i n Refs.
[lo]
and
e m r i n t h e model.
C111.
When t h e GO-control u $ ( t ) is adopted t o t h e a c t u a l system (11) , t h e o r i m n a l performance index ( 2 ) is, with t h e a i d of t h e s a d d l e p o i n t condition (161, evaluated as
is given by (13) and is evaluated with
where v,(t)
From ( 2 1 ) , (22) and ( 2 3 ) , we o b t a i n t h e upper bound of t h e performance index ( 2 ) corresponding t o t h e GO-control u;(t) as follows: [Theorem 11 When t h e GO-control u$( t ) is adopted t o t h e a c t u a l system i n t h e presence of u n c e r t a i n t y , t h e value of t h e performance index J(u;) is evalua t e d as J(u;)
1 12(0) p
x;V2(0)xo
Thus, t h e performance d e t e r i o r a t i o n is caused by t h e unknown v e c t o r v 3 ( t ) which is un&r t h e i n f l u ence o f t h e l a r g e u n c e r t a i n t y i n t h e system. F ~ w a t h e p o i n t o f view o f t h e s e n s i t i v i t y design, it is again n a t u r a l t o c o n s i d e r t h e v e c t o r v ( t ) as an ant a g o n i s t t o t h e designer and t o t a k e aavantage of t h e worst case design. By doing t h i s , one can account t h e advantage of t h e nominal design t o g e t h e r with t h a t of t h e worst casedesign. This is t h e b a s i c i d e a o f t h e game t h e o r e t i c approach t o t h e s e n s i t i v i t y design i n t h i s s e c t i o n . The design considered i n t h i s section w i l l be called the sensitive game design, s i n c e it tends t o t r a c k t h e new optimum f o r t h e performance i n t h e presence o f system u n c e ~ tainty
.
The r e g u l a t o r problem can t h u s be r e w r i t t e n as a game as follows : " For t h e system
(24)
where nxn-matrix V ( t ) is t h e s o l u t i o n of t h e matrix 2 d i f f e r e n t i a l equation
with payoff
f i n d feedback c o n t r o l s Au ( t ) and v 3 ( t ) which a r e optimal i n t h e sense t h a t , f o r any feedback c o n t r o l s Au( t ) and v3( t ) , t h e r e h o l d s SENSITIVE GAME THEORETIC DESIGN
J3(Au*,v3) ( J3(Au*,v3*)
I t o f t e n happens i n p r a c t i c a l s i t u a t i o n s t h a t t h e game t h e o r e t i c design presented i n t h e preceding s e c t i o n is t o o p e s s i m i s t i c because it is t h e worst case design f o r t h e system with uncertainty. For example, i n some cases when t h e equation e r r o r makes t h e a c t u a l system more s t a b l e than its model system, t h e nominal optimal c o n t r o l preserves a f a i r l y good performance, compared with t h e GO-control. (This f a c t w i l l be seen i n t h e following examples.) Thus, it is d e s i r a b l e t o t a k e t h i s advantage of t h e nomin a l design i n t o consideration a t t h e i n i t i a l s t a g e of t h e worst case design. Suppose t h a t one adopts a new feedback contro1,based on t h e nominal optimal c o n t r o l , a s u
3
= u*
+
(26)
Au
t o t r a c k t h e new optimum f o r t h e performance index whenever t h e r e is t h e u n c e r t a i n t y i n t h e matrices A ( t ) and B( t ) . Then, t h e system under consideration is represented by
k = A x + Bu*
+
BAu+ v3,
x(0)
= xo
(27)
v3
= E3x.
(29
Here nxn-matrix E 3 ( t ) corresponds t o t h e equation
(33)
i . e . , f i n d a s a d d l e p o i n t of J3(Au,v3) i f it e x i s t s . " Here t h e s u b s c r i p t 3 i n (301, (32) and (33) is adopt e d t o d i s t i n g u i s h t h e p r e s e n t design from t h e previous designs. The optimal c o n t r o l s are obtained a s AU* V$
=
-
(3'4)
R-'BIK~X
= R;'K,~
(35)
and t h e value o f t h e optimal payoff is given by 1 J3(Au*,v3) = q xiK3(O)x0 (36) where K3(t) i s t h e s o l u t i o n of t h e matrix R i c c a t i equation i3= 01n3- K ~ D+ K ~ ( B R1B- I R ; ~ ) K~ Q ,
-
-
K3(T)
-
= S
(37)
From ( 3 ) , (26) and (341, t h e optimal c o n t r o l of t h e p r e s e n t design u:(t) is given by u$ =
where t h e n-vector v 3 ( t ) r e p r e s e n t s t h e system unc e r t a i n t y , and it is again reasonable t o express v 3 ( t ) i n t h e form
(J3(Au,9)
- R-hl(K1
+
K3)x.
(38)
I n t h e f o l l w i n g , t h e optimal c o n t r o l u$ is r e f e r r e d t o as t h e SGO-control. With t h e a i d of t h e s a d d l e p o i n t condition (331, t h e o r i g i n a l performance index ( 2 ) corresponding t o t h e SGO-control u $ ( t ) is evaluated as
f o r 11(0)
where v g ( t ) is given by ( 2 8 ) and i s e v a l u a t e d w i t h
+
5 = Dx E3 = AA
+
BAu*
-
E3x,
-1B1(K1+
ABR
x ( 0 ) = xo
5
pJ(u*)
is given by
where W , ( t ) s a t i s f i e s
(40
Kg)
From (391, ( 4 0 ) and (411, we o b t a i n t h e upper bound o f t h e performance i n d e x ( 2 ) corresponding t o t h e SGO-control u$( t ) as follows: [Theorem 21 When t h e SGO-control u:(t) is adopted t o t h e a c t u a l system i n t h e p r e s e n c e o f u n c e r t a i n t y , t h e v a l u e o f t h e performance i n d e x J(u$j) is evalua t e d as
where nxn-matrix V ( t ) is t h e s o l u t i o n o f t h e m a t r i x 3 d i f f e r e n t i a l equation
[Theorem 41 The s u f f i c i e n t c o n d i t i o n f o r p-sensit i v i t y is given by
I n t h e c a s e o f t h e game t h e o r e t i c d e s i g n , t h e upper bound o f t h e performance index i s given by 1 2 ( 0 ) i n ( 2 4 ) and is t o be compared with t h e v a l u e o f t h e nominal o p t i m a l c o n t r o l . I n t h e same way as Mcclamroch, we o b t a i n t h e f o l l w i n g c o n d i t i o n s f o r p-sensitivity. [Theorem 51 The n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r 1 2 ( 0 ) 5 pJ(u*) i s given by
where W ( t ) i s t h e s o l u t i o n o f 2 i2= ( A - B R - ~ I K+? E ~ ) -~w w2 ( ~- B R - ~ B ~ +K E~ ? )
I t is t o be n o t i c e d t h a t compared w i t h t h e precedi n g game t h e o r e t i c d e s i g n , t h e p r e s e n t design i s t h e w o r s t c a s e design b a s i n g on t h e nominal design. I n t h i s s e n s e , i t can be s a i d t h a t t h e p r e s e n t design is an i n t e r mediate between t h e nominal one and t h e game t h e o r e t i c one.
-E2'K1 +C2
,
-
K1E2'
+
(K2
-
1 K ~ ) B RBIM1(K1-
W (T) = 0
K2) (48)
2
[Theorem 6 1 The s u f f i c i e n t c o n d i t i o n f o r p - s e n s i t i v i t y is given by 1 -p(Et2K1 + K1E2) p(K2 - K ~ ) B RB1(K2 - K1)
-
DETERMINATION OF ADMISSIBLE ERROR I n t h e p r e c e d i n g s e c t i o n s , t h e performance d e t e r i o r a t i o n caused by t h e e q u a t i o n e r r o r a r e mainly con-s i d e r e d , and t h e upper bound o f t h e performance i n dex is d e r i v e d as t h e s o l u t i o n o f a m a t r i x d i f f e r e n t i a l e q u a t i o n i n t h e case o f t h e two game t h e o r e t i c d e s i g n s . Conversely, i n t h i s s e c t i o n , t h e problem o f determining t h e a d m i s s i b l e e q u a t i o n e r r o r s o t h a t t h e value o f t h e performance index does n o t exceed some s t a n g ~ r dvalue i s considered by t h e use of p-sensitivity.
+
( p- I ) ( Q
+
K~BR-$IK+ ~ )c2
2 O.
(49)
I n t h e c a s e o f t h e s e n s i t i v e game t h e o r e t i c d e s i g n , t h e upper bound o f t h e performance index i s given by I (0) i n ( 4 2 ) and is t o b e compared with t h e 3 value of nominal o p t i m a l performance. I n t h e same manner, we o b t a i n t h e f o l l o w i n g c o n d i t i o n s f o r psensitivity. [Theorem 71 The necessary and s u f f i c i e n t c o n d i t i o n s f o r 1 3 ( 0 ) 2 pJ(u*) i s given by
F o l l w i n ~McClamroch , t h e p - s e n s i t i v i t y i s d e f i n e d as f o l l w s . [ D e f i n i t i o n ] For some r e a l number p(, 1 1 , some feedback c o n t r o l ;tnd some c l a s s o f e r r o r s E , t h e a c t u a l system i s a i d t o be p - s e n s i t i v e , i f t h e i n equality J(U)
5
pJ(u*)
,
for a l l E
E E
where W3(t) i s t h e s o l u t i o n o f
(44)
i s s a t i s f i e d , where J ( u * ) is t h e nominal o p t i m a l performance v a l u e . When t h e nominal o p t i m a l c o n t r o l i s adopted t o t h e a c t u a l system, t h e performance i n d e x i s given by 11(0) i n (9!. By a p p l y i n g t h e theorem d e r i v e d by Mcclamroch , t h e c o n d i t i o n s f o r p - s e n s i t i v i t y is given as follows: [Theorem 31')
Necessary and s u f f i c i e n t c o n d i t i o n
[Theorem 81 The s u f f i c i e n t c o n d i t i o n f o r p - s e n s i t i v i t y is given by -~(E;K, (Kl
+
+
K 1E 3
Kg)]
-
+
K ~ B R - ~ B+ I K(p-I)[Q ~)
C3,
0,
for a l l
+
( K +~ K ~ ) X
0
(52)
I n t h r e e c a s e s o f d e s i g n , t h e necessary and s u f f i c i -
e n t c o n d i t i o n s and t h e s u f f i c i e n t c o n d i t i o n s f o r t h e a c t u a l system t o be p - s e n s i t i v e a r e d e r i v e d . A s i t is e a s i l y understood, t h e s u f f i c i e n t c o n d i t i o n s a r e more p r a c t i c a l t o apply. From t h e s e c o n d i t i o n s one can o b t a i n t h e information on t h e e f f e c t s o f t h e e q u a t i o n e r r o r on t h e performance index. EXAMPLE For i l l u s t r a t i v e example, c o n s i d e r t h e problem o f f i n d i n g t h e o p t i m a l c o n t r o l which minimizes
f o r t h e f i r s t o r d e r system
k = ax +bu,
x(0) = 1
.
Together w i t h t h e a c t u a l o p t i m a l performance v a l u e , t h e values o f t h e performance index corresponding t o t h e t h r e e design method. versus t h e parameter e m r a r e shown i n Figs. 1,2 and 3 r e s p e c t i v e l y f o r t h e t h r e e cases of uncertainty. The e f f e c t i v e n e s s o f t h e two game t h e o r e t i c designs a r e c l e a r l y understood from t h e s e r e s u l t s . A s i s s e e n i n F i g u r e s , i n t h e case when t h e e q u a t i o n errormakes t h e system mre u n s t a b l e t h a n t h e model, t h e two game t h e o r e t i c design y i e l d s b e t t e r p e r f o r mance t h a n t h e nominal design. And t h e o t h e r way around i n t h e case when t h e e q u a t i o n e r r o r makes t h e system more s t a b l e t h a n t h e model system. This i n f e r i o r i t y o f t h e game designs t o t h e nominal d e s i gn can be understood because t h e game t h e o r e t i c designs a r e t h e worst c a s e design. I t i s a l s o und e r s t o o d from t h e s e Figures t h a t t h e s e n s i t i v e game t h e o r e t i c design i s an i n t e r mediate one between t h e nominal one and game t h e o r e t i c one. CONCLUSIONS Two game t h e o r e t i c approaches t o s e n s i t i v i t y design f o r l i n e a r r e g u l a t o r problem under l a r g e u n c e r t a i n t y a r e developed, where t h e e q u a t i o n e r r o r i n model system and t h e feedback c o n t r o l l e r a r e assumed t o be a n t a g o n i s t s . The s o - c a l l e d game t h e o r e t i c d e s i g n , which i s t h e worst c a s e d e s i g n , a r e f i r s t l y developed, and t h e upper bound o f t h e value of t h e performance index i s derived. Accounting t h e advantage of t h e nomin a l d e s i g n , a s l i g h t l y d i f f e r e n t game design c a l l e d s e n s i t i v e game t h e o r e t i c design is a l s o p r e s e n t e d . The s e n s i t i v e game t h e o r e t i c design i s an i n t e r mediate one between t h e nominal one and t h e preced i n g game t h e o r e t i c design. The problem of determining admissible e r r o r s s o t h a t t h e value o f t h e performance i n d e x does n o t exceed some value i s d i s c u s s e d by t h e use o f p - s e n s i t i v i t y . The c o n d i t i o n s f o r t h e performance index does n o t i n c r e a s e by more t h a n a f a c t o r o f p i n comparison with t h e nominal performance value a r e d e r i v e d f o r t h r e e design c a s e s By g i v i n g simple examples, t h e e f f e c t i v e n e s s o f t h e p r e s e n t game t h e o r e t i c approaches t o t h e s e n s i t i v i t y design a r e shown. A l l numerical computations were performed byFACOM
230-60 a t Data P r o c e s s i n g C e n t e r i n Kyoto University. REFERENCES 1 ) D.D. Sworder: Minimax c o n t r o l o f d i s c r e t e time s t o c h a s t i c systems; J.SIAM on C o n t r o l ser.A, Vol. 2 , No. 3, pp.433-449, 1965. 2) M.D. Mesarovic: S a t i f a c t i o n approach t o t h e synt h e s i s and c o n t r o l o f systems; Proc. 1965 3rd Annual A l l e r t o n Conference on C i r c u i t and s y s tem Theory, pp. 930-942, 3) P. Dorato & R.F.Drenick: O p t i m a l i t y , i n s e n s i t i v i t y and game t h e o r y ; i n S e n s i t i v i t y methods i n c o n t r o l t h e o r y e d i t e d by L. Radanovic, 1966 Pergamon P r e s s . 4) P . Dorato & A. Kestenbaum: A p p l i c a t i o n o f game t h e o r y t o t h e s e n s i t i v i t y design o f optimal system;.IEEE Trans. Vol.AC-12, No.1, pp. 85-87 1968 5 ) D.M. Salmon: minimax c o n t r o l l e r design; IEEE Trans., Vol.AC-13, No. 4 , pp.369-376, 1968 6)R.A. Rohrer & M. S o b r a l Jr.: S e n s i t i v i t y consid e r a t i o n s i n optimal system d e s i g n ; IEEE Trans. Vol.AC-10, No. 1, pp.43-48, 1965. 7) G. I .Sarma & R.K.Ragade: Some c o n s i d e r a t i o n s i n f ? m u l a t i n g o p t i m a l c o n t r o l problem as d i f f e r e n t i a l games; I n t . J. C o n t r . , Vo1.4,No.3, pp. 265-279, 1966. 8 ) R.K.Ragade & I . G . Sarma: A game t h e o r e t i c approach t o optimal c o n t r o l i n t h e presence o f unc e r t a i n t y ; IEEE Trans. , Vol.AC-12, No.4, pp. 395 -401, 1967. 9 ) N.H.Mcclamroch, G.L.ClarJc & J.K.Aggarwa1: Sens i t i v i t y o f l i n e a r c o n t r o l systems t o l a r g e parameter v a r i a t i o n s , Automatics, Vo1.5, pp .257 -263, 1969. 10) H. Kimura: A game t h e o r e t i c approach t o c o n t r o l and s t a b i l i z a t i o n o f systems with d i s t u r b a n c e s ; ( i n J a p a n e s e ) , Trans. SICE, Vo1.6, No.4, pp.366 -371, 1970. 1 1 ) I .B. Rohdes & D.G.Luenberger: D i f f e r e n t i a l games w i t h i m p e r f e c t s t a t e i n f o r m a t i o n ; IEEE Trans. Vol. AC-14, No. 1, pp.29-38,1969.
Yoshikazu SAWARAGI and Katsuya OGINO Dept. o f Applied Mathematics & P h y s i c s F a c u l t y o f E n g i n e e r i n g , Kyoto U n i v e r s i t y Kyoto, JAPAN
INTRODUCTION I t is a c h a l l e n g i n g problem t o d e s i g n an o p t i m a l c o n t r o l l e r f o r s y s t e m s w i t h u n c e r t a i n t i e s . Although it is p o s s i b l e t o p r e s e r v e a s a t i s f a c t o r y p e r formance by i d e n t i f y i n g t h e u n c e r t a i n t i e s and cons t r u c t i n g a d a p t i v e c o n t r o l l e r s , it i s u s u a l l y a d i f f i c u l t o r expensive t a s k t o r e a l i z e i t , especial l y when t h e u n c e r t a i n t i e s a r e l a r g e . From t h e p o i n t o f view o f s e n s i t i v i t y d e s i g n a g a i n s t l a r g e u n c e r t a i n t i e s , i t seems r e a s o n a b l e t o assume t h a t t h e s y s t e m u n c e r t a i n t i e s t a k e on v a l u e s which maxmize t h e performance i n d e x . Thus, t h e game t h e o r e t i c approach a p p e a r s t o b e a v e r y n a t u r a l and e f f e c t i v e one f o r t h e s e n s i t i v i t y d e s i g n and h a s t h e advantage t h a t it y i e l d s an optimal c o n t r o l l e r which is e f f e c t i v e u n d e r t h e w o r s t c a s e .
tpg
There have been v a r i o u s a t t e m p t s t o d e v e l o p s e n s it i v i problem a l o n g t h g , + ) i n e . Sworder , indicated the appliMesarovi?Y and D o r a t o e t a l . c a b i l i t y o f t h e g g ~ t h e o r yt o t h e s e n g j t i v i t y problem, w h i l e Salmon and Rohrer e t a l . a d o p t e d t h e min-max approach i n t h e y e g q i t i v i t y d e s i g n . On t h e o t h e r h a n d , Sarma e t a l . ' d i s c u s s e d t h e v a r i o u s d i f f i c u l t i e s inherent i n solving optimal control problems a l o n g t h i s l i n e . T h i s p a p e r p r e s e n t s two d i f f e r e n t game t h e o r e t i c a p p r o a c h e s t o t h e s t a t e r e g u l a t o r problem i n t h e presence o f l a r g e u n c e r t a i n t i e s . In t h e f i r s t p a r t o f t h e p a p e r , t h e s e n s i t i v i t y problem o f t h e s y s t e m w i t h u n c e r t a i n t y is d i s c u s s e d , and t h e o p t i m a l c o n t r o l problem w i t h u n c e r t a i n t y i s p r e c i s e l y f o r mulated and developed a s a game, where t h e c o n t r o l l e r and t h e e q u a t i o n e r r o r caused by t h e s y s t e m uncertainty a r e considered t o be the antagonists. The u p p e r bound o f t h e performance i n d e x p r e s e r v e d by t h e game t h e o r e t i c d e s i g n i s d e r i v e d a s t h e s o l t i o n of a matrix d i f f e r e n t i a l equation. It o f t e n happens i n p r a c t i c a l s i t u a t i o n s t h a t t h e game t h e o r e t i c d e s i g n is t o o p e s s i m i s t i c . S e c o n d l y , t h e n , p r e s e n t e d i s t h e s e n s i t i v e game t h e o r e t i c d e s i g n which i s an i n t e r - m e d i a t e between t h e p r e c e d i n g game t h e o r e t i c d e s i g n and t h e nominal d e s i g n which a d o p t s t h e model s y s t e m as t h e c o n t r o l l e d s y s t e m . The u p p e r bound o f t h e performance i n d e x p r e s e r v e d by t h i s d e s i g n is a l s o d e r i v e d . T h i r d l y , t h e r e l a t i o n between t h e e q u a t i o n e r r o r and t h e performance i n d i c e s c o r r e s p o n d i n g t o t h e t h r e e d e s i g n methods a r e a n a l y z e d and t h e c o n d i t i o n s f o r t h e performance
i n d i c e s t o be p - s e n s i t i v e 9 ) a r e d e r i v e d . Numerical examples a r e p r e s e n t e d t o show t h e e f f e c t i v e n e s s o f t h e p r e s e n t game t h e o r e t i c a p p r o a c h e s t o t h e s e n s i t i v i t y design. OPTIMAL CONTROL PROBLEM IN THE PRESENCE OF UNCERTAINTY L e t t h e model s y s t e m o f a n a c t u a l s y s t e m w i t h uncert a i n t y b e g i v e n by where x , u , A ( t ) and B ( t ) a r e r e s p e c t i v e l y t h e ns t a t e v e c t o r , t h e r - c o n t r o l v e c t o r , t h e nxn- and nxr-nominal m a t r i c e s . The o p t i m a l c o n t r o l problem c o n s i d e r e d i n t h i s p a p e r i s t h e s t a t e r e g u l a t o r problem, t h a t i s , t h e problem o f d e s i g n i n g t h e o p t i m a l c o n t r o l which minimizes t h e f o l l o w i n g performance i n d e x .
where T,S,Q( t ) and R( t ) a r e r e s p e c t i v e l y a f i x e d terminal time, a constant nxn-positive semidefinite m a t r i x , an n x n - p o s i t i v e s e m i d e f i n i t e m a t r i x and a n rxr-positive d e f i n i t e matrix. I t is w e l l known t h a t t h e o p t i m a l c o n t r o l f o r t h e model s y s t e m ( I ) , i n t h e feedback form, i s g i v e n by
u;:
=
R ~ l B t K
(3) 1 and t h e o p t i m a l v a l u e o f performance i n d e x i s g i v e n by 1 J ( u 5 ) = 7 xAKl(0)xo ('4)
where K (t) is t h e s o l u t i o n o f t h e m a t r i x R i c c a t i 1 equation
K1(T) = S
(5)
The d e s i g n which a d o p t s t h e model s y s t e m ( 1 ) a s t h e c o n t r o l l e d system w i l l henceforth be r e f e r r e d t o a s t h e nominal d e s i g n , and u* i n ( 3 ) and J(u'5) i n ( 4 ) w i l l r e s p e c t i v e l y b e c a l l e d nominal o p t i m a l c o n t r o l and nominal o p t i m a l p e r f o r r a n c e . I t is assumed h e r e t h a t t h e a c t u a l s y s t e m w i t h unc e r t a i n t y i s r e p r e s e n t e d by
where AA(t) and AB(t) correspond t o t h e system unc e r t a i n t y . I t i s t o be noted t h a t t h e i n i t i a l cond i t i o n o f t h e system i s assumed t o be e x a c t l y k n m . When t h e nominal o p t i m a l c o n t r o l u* is a p p l i e d , i n t h e feedback form, t o t h e a c t u a l system (61, t h e system is w r i t t e n as
mum energy. Thus t h e d e s i r e d s e n s i t i v i t y design o f t h e system under l a r g e u n c e r t a i n t y can n a t u r a l l y be formulated as a game between t h e c o n t r o l l e r and t h e system u n c e r t a i n t y . By doing t h i s , one can t a k e t h e e f f e c t o f l a r g e system u n c e r t a i n t y i n t o consid e r a t i o n a t t h e i n i t i a l s t a g e o f t h e design. This i s t h e b a s i s of t h e p r e s e n t game t h e o r e t i c design f o r t h e u n c e r t a i n system. The r e g u l a t o r problem can t h u s be r e s t a t e d as a game a s follows: " For t h e system
Here E r e p r e s e n t s t h e e q u a t i o n e r r o r i n t h e model 1 system (l), and i s c a u s e d by t h e v a r i a t i o n i n t h e m a t r i c e s A and B. The value o f t h e a c t u a l p e r f o r mance i n d e x ( 2 ) e v a l u a t e d along t h e s o l u t i o n o f ( 7 ) is given by A 1 J(u*,AA,AB) = 11(0) = x~Vl(0)xo (9) where V ( t ) is t h e s o l u t i o n o f t h e m a t r i x d i f f e r e n 1 t i a l equation
A s t h e nominal design does n o t t a k e t h e system unc e r t a i n t y i n t o account t h e value of t h e performance index I1 is no l o n g e r a c t u a l l y optimal. Thus, when t h e nominal o p t i m a l c o n t r o l is adopted t o t h e a c t u a l system, it can b e s a i d t h a t t h e performance index i s d e t e r i o r a t e d t o t h e v a l u e I1 by t h e equat i o n e r r o r El. E s p e c i a l l y when t h e e q u a t i o n e r r o r i s l a r g e , t h e performance d e t e r i o r a t i o n makes t h e nominal design meaningless. Thus, it i s h i g h l y d e s i r a b l e t o f i n d a s e n s i t i v i t y design which accounts t h e l a r g e u n c e r t a i n t y i n t h e system a t t h e i n i t i a l s t a g e of t h e design. When t h e s e n s i t i v i t y design is viewed a s an o p t i m i z a t i o n problem with u n c e r t a i n t y , t h e game t h e o r e t i c approach appears t o be a very n a t u r a l one.
w i t h payoff
f i n d feedback c o n t r o l s u * ( t ) and v;(t) which a r e o p t i m a l i n t h e s e n s e t h a ? , f o r any feedback c o n t r o l s up( t ) and vg( t ) , t h e r e h o l d s
i . e . , f i n d a s a d d l e p o i n t o f J ( u v ) i f it exists'! The s u b s c r i p t 2 i n (110, ( 1 5 ) & d 2 i 1 9 ) i s u t i l i z e d t o d i s t i n g u i s h t h e above problem from t h e o r i g i n a l r e g u l a t o r problem. I t is t o be n o t e d t h a t t h e m a t r i x R ( t ) i s a s u i t a b l y chosen nxn-positive d e f i n i t e 2 matrix. The l a s t term i n ( 1 5 ) r e p r e s e n t s t h e consuming energy o f t h e system u n c e r t a i n t y . Thus, i n t h e above f o r m u l a t i o n , it is assumed t h a t t h e system unc e r t a i n t y s e e k s an o p t i m a l feedback c o n t r o l which maximizes t h e performance i n d e x ( 2 1 , w h i l e minimiz i n g i t s consuming energy. I n a q u i t e s i m i l a r form t o t h e one-sided r e g u l a t o r probl?m, ~ ~ e l p ~ t i mc oa nl t r o l s o f t h e above problem a r e glven ' by
u* = -R - 1B'K2x
(17)
GAME THEORETIC DESIGN
L e t ' s g i v e a t t e n t i o n t o t h e a c t u a l system ( 6 ) which can be r e w r i t t e n as k=Ax+Bu+v2, v
2
4 AAx +
and t h e value of t h e o p t i m a l payoff is given by
x(0)=xo where K ( t ) i s t h e s o l u t i o n o f t h e m a t r i x R i c c a t i 2 equation
ABu
where t h e n - v e c t o r v ( t ) r e p r e s e n t s t h e system un2 c e r t a i n t y , a n d it i s reasonable t o r e w r i t e v ( t ) i n 2 t h e form v2 = E2x
(13)
Here n m - m a t r i x E ( t ) corresponds t o t h e e q u a t i o n e r r o r i n t h e mode?.
Henceforth t h e o p t i m a l c o n t r o l u * ( t ) i n ( 1 7 ) w i l l be 2 r e f e r r e d t o as t h e GO-control, and t h e above design w i l l be c a l l e d game t h e o r e t i c design.
The unknown v e c t o r v ( t ) , which r e p r e s e n t s t h e 2 l a r g e u n c e r t a i n t y i n t h e system, causes t h e p e r f o r mance d e t e r i o r a t i o n i n t h e design. Then, from t h e p o i n t o f view o f s e n s i t i v i t y d e s i g n , t h e v e c t o r v 2 ( t ) i n t h e form o f (13) s h o u l d be considered t o be under t h e c o n t r o l o f system u n c e r t a i n t y , and/or i n t h e s e n s e o f t h e w o r s t case i t should be consid e r e d as an feedback c o n t r o l by which t h e system u n c e r t a i n t y antagonize t o t h e c o n t r o l l e r and maximize t h e performance index J ( u ) i n ( 2 ) by t h e mini-
I t should be mentioned t h a t a necessary and s u f f i c i e n t c o n d i t i o n f o r t h e s a d d l e p o i n t (u*,vn) t o e x i s t and t o be unique is t h a t the2ma$rix K ( t ) s o l v i n g ( 2 0 ) e x i s t f o r a l l tc[O ,TI. Although ?he game problem ( 1 4 ) and ( 1 5 ) is a game t h e o r e t i c ext e n s i o n of t h e one-sided r e g u l a t o r problem ( T h i s i s t h e main reason of f o r m u l a t i n g t h e s e n s i t i v i t y d e s i g n by ( 1 4 ) and ( 1 5 ) ) , t h e e x i s t e n c e of t h e s o l u t i o n o f t h e m a t r i x R i c c a t i e q u a t i o n ( 2 0 ) i s n o t always assured u n l i k e t h e one-sided r e g u l a t o r problem. T h i s
problem is examined i n Refs.
[lo]
and
e m r i n t h e model.
C111.
When t h e GO-control u $ ( t ) is adopted t o t h e a c t u a l system (11) , t h e o r i m n a l performance index ( 2 ) is, with t h e a i d of t h e s a d d l e p o i n t condition (161, evaluated as
is given by (13) and is evaluated with
where v,(t)
From ( 2 1 ) , (22) and ( 2 3 ) , we o b t a i n t h e upper bound of t h e performance index ( 2 ) corresponding t o t h e GO-control u;(t) as follows: [Theorem 11 When t h e GO-control u$( t ) is adopted t o t h e a c t u a l system i n t h e presence of u n c e r t a i n t y , t h e value of t h e performance index J(u;) is evalua t e d as J(u;)
1 12(0) p
x;V2(0)xo
Thus, t h e performance d e t e r i o r a t i o n is caused by t h e unknown v e c t o r v 3 ( t ) which is un&r t h e i n f l u ence o f t h e l a r g e u n c e r t a i n t y i n t h e system. F ~ w a t h e p o i n t o f view o f t h e s e n s i t i v i t y design, it is again n a t u r a l t o c o n s i d e r t h e v e c t o r v ( t ) as an ant a g o n i s t t o t h e designer and t o t a k e aavantage of t h e worst case design. By doing t h i s , one can account t h e advantage of t h e nominal design t o g e t h e r with t h a t of t h e worst casedesign. This is t h e b a s i c i d e a o f t h e game t h e o r e t i c approach t o t h e s e n s i t i v i t y design i n t h i s s e c t i o n . The design considered i n t h i s section w i l l be called the sensitive game design, s i n c e it tends t o t r a c k t h e new optimum f o r t h e performance i n t h e presence o f system u n c e ~ tainty
.
The r e g u l a t o r problem can t h u s be r e w r i t t e n as a game as follows : " For t h e system
(24)
where nxn-matrix V ( t ) is t h e s o l u t i o n of t h e matrix 2 d i f f e r e n t i a l equation
with payoff
f i n d feedback c o n t r o l s Au ( t ) and v 3 ( t ) which a r e optimal i n t h e sense t h a t , f o r any feedback c o n t r o l s Au( t ) and v3( t ) , t h e r e h o l d s SENSITIVE GAME THEORETIC DESIGN
J3(Au*,v3) ( J3(Au*,v3*)
I t o f t e n happens i n p r a c t i c a l s i t u a t i o n s t h a t t h e game t h e o r e t i c design presented i n t h e preceding s e c t i o n is t o o p e s s i m i s t i c because it is t h e worst case design f o r t h e system with uncertainty. For example, i n some cases when t h e equation e r r o r makes t h e a c t u a l system more s t a b l e than its model system, t h e nominal optimal c o n t r o l preserves a f a i r l y good performance, compared with t h e GO-control. (This f a c t w i l l be seen i n t h e following examples.) Thus, it is d e s i r a b l e t o t a k e t h i s advantage of t h e nomin a l design i n t o consideration a t t h e i n i t i a l s t a g e of t h e worst case design. Suppose t h a t one adopts a new feedback contro1,based on t h e nominal optimal c o n t r o l , a s u
3
= u*
+
(26)
Au
t o t r a c k t h e new optimum f o r t h e performance index whenever t h e r e is t h e u n c e r t a i n t y i n t h e matrices A ( t ) and B( t ) . Then, t h e system under consideration is represented by
k = A x + Bu*
+
BAu+ v3,
x(0)
= xo
(27)
v3
= E3x.
(29
Here nxn-matrix E 3 ( t ) corresponds t o t h e equation
(33)
i . e . , f i n d a s a d d l e p o i n t of J3(Au,v3) i f it e x i s t s . " Here t h e s u b s c r i p t 3 i n (301, (32) and (33) is adopt e d t o d i s t i n g u i s h t h e p r e s e n t design from t h e previous designs. The optimal c o n t r o l s are obtained a s AU* V$
=
-
(3'4)
R-'BIK~X
= R;'K,~
(35)
and t h e value o f t h e optimal payoff is given by 1 J3(Au*,v3) = q xiK3(O)x0 (36) where K3(t) i s t h e s o l u t i o n of t h e matrix R i c c a t i equation i3= 01n3- K ~ D+ K ~ ( B R1B- I R ; ~ ) K~ Q ,
-
-
K3(T)
-
= S
(37)
From ( 3 ) , (26) and (341, t h e optimal c o n t r o l of t h e p r e s e n t design u:(t) is given by u$ =
where t h e n-vector v 3 ( t ) r e p r e s e n t s t h e system unc e r t a i n t y , and it is again reasonable t o express v 3 ( t ) i n t h e form
(J3(Au,9)
- R-hl(K1
+
K3)x.
(38)
I n t h e f o l l w i n g , t h e optimal c o n t r o l u$ is r e f e r r e d t o as t h e SGO-control. With t h e a i d of t h e s a d d l e p o i n t condition (331, t h e o r i g i n a l performance index ( 2 ) corresponding t o t h e SGO-control u $ ( t ) is evaluated as
f o r 11(0)
where v g ( t ) is given by ( 2 8 ) and i s e v a l u a t e d w i t h
+
5 = Dx E3 = AA
+
BAu*
-
E3x,
-1B1(K1+
ABR
x ( 0 ) = xo
5
pJ(u*)
is given by
where W , ( t ) s a t i s f i e s
(40
Kg)
From (391, ( 4 0 ) and (411, we o b t a i n t h e upper bound o f t h e performance i n d e x ( 2 ) corresponding t o t h e SGO-control u$( t ) as follows: [Theorem 21 When t h e SGO-control u:(t) is adopted t o t h e a c t u a l system i n t h e p r e s e n c e o f u n c e r t a i n t y , t h e v a l u e o f t h e performance i n d e x J(u$j) is evalua t e d as
where nxn-matrix V ( t ) is t h e s o l u t i o n o f t h e m a t r i x 3 d i f f e r e n t i a l equation
[Theorem 41 The s u f f i c i e n t c o n d i t i o n f o r p-sensit i v i t y is given by
I n t h e c a s e o f t h e game t h e o r e t i c d e s i g n , t h e upper bound o f t h e performance index i s given by 1 2 ( 0 ) i n ( 2 4 ) and is t o be compared with t h e v a l u e o f t h e nominal o p t i m a l c o n t r o l . I n t h e same way as Mcclamroch, we o b t a i n t h e f o l l w i n g c o n d i t i o n s f o r p-sensitivity. [Theorem 51 The n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r 1 2 ( 0 ) 5 pJ(u*) i s given by
where W ( t ) i s t h e s o l u t i o n o f 2 i2= ( A - B R - ~ I K+? E ~ ) -~w w2 ( ~- B R - ~ B ~ +K E~ ? )
I t is t o be n o t i c e d t h a t compared w i t h t h e precedi n g game t h e o r e t i c d e s i g n , t h e p r e s e n t design i s t h e w o r s t c a s e design b a s i n g on t h e nominal design. I n t h i s s e n s e , i t can be s a i d t h a t t h e p r e s e n t design is an i n t e r mediate between t h e nominal one and t h e game t h e o r e t i c one.
-E2'K1 +C2
,
-
K1E2'
+
(K2
-
1 K ~ ) B RBIM1(K1-
W (T) = 0
K2) (48)
2
[Theorem 6 1 The s u f f i c i e n t c o n d i t i o n f o r p - s e n s i t i v i t y is given by 1 -p(Et2K1 + K1E2) p(K2 - K ~ ) B RB1(K2 - K1)
-
DETERMINATION OF ADMISSIBLE ERROR I n t h e p r e c e d i n g s e c t i o n s , t h e performance d e t e r i o r a t i o n caused by t h e e q u a t i o n e r r o r a r e mainly con-s i d e r e d , and t h e upper bound o f t h e performance i n dex is d e r i v e d as t h e s o l u t i o n o f a m a t r i x d i f f e r e n t i a l e q u a t i o n i n t h e case o f t h e two game t h e o r e t i c d e s i g n s . Conversely, i n t h i s s e c t i o n , t h e problem o f determining t h e a d m i s s i b l e e q u a t i o n e r r o r s o t h a t t h e value o f t h e performance index does n o t exceed some s t a n g ~ r dvalue i s considered by t h e use of p-sensitivity.
+
( p- I ) ( Q
+
K~BR-$IK+ ~ )c2
2 O.
(49)
I n t h e c a s e o f t h e s e n s i t i v e game t h e o r e t i c d e s i g n , t h e upper bound o f t h e performance index i s given by I (0) i n ( 4 2 ) and is t o b e compared with t h e 3 value of nominal o p t i m a l performance. I n t h e same manner, we o b t a i n t h e f o l l o w i n g c o n d i t i o n s f o r psensitivity. [Theorem 71 The necessary and s u f f i c i e n t c o n d i t i o n s f o r 1 3 ( 0 ) 2 pJ(u*) i s given by
F o l l w i n ~McClamroch , t h e p - s e n s i t i v i t y i s d e f i n e d as f o l l w s . [ D e f i n i t i o n ] For some r e a l number p(, 1 1 , some feedback c o n t r o l ;tnd some c l a s s o f e r r o r s E , t h e a c t u a l system i s a i d t o be p - s e n s i t i v e , i f t h e i n equality J(U)
5
pJ(u*)
,
for a l l E
E E
where W3(t) i s t h e s o l u t i o n o f
(44)
i s s a t i s f i e d , where J ( u * ) is t h e nominal o p t i m a l performance v a l u e . When t h e nominal o p t i m a l c o n t r o l i s adopted t o t h e a c t u a l system, t h e performance i n d e x i s given by 11(0) i n (9!. By a p p l y i n g t h e theorem d e r i v e d by Mcclamroch , t h e c o n d i t i o n s f o r p - s e n s i t i v i t y is given as follows: [Theorem 31')
Necessary and s u f f i c i e n t c o n d i t i o n
[Theorem 81 The s u f f i c i e n t c o n d i t i o n f o r p - s e n s i t i v i t y is given by -~(E;K, (Kl
+
+
K 1E 3
Kg)]
-
+
K ~ B R - ~ B+ I K(p-I)[Q ~)
C3,
0,
for a l l
+
( K +~ K ~ ) X
0
(52)
I n t h r e e c a s e s o f d e s i g n , t h e necessary and s u f f i c i -
e n t c o n d i t i o n s and t h e s u f f i c i e n t c o n d i t i o n s f o r t h e a c t u a l system t o be p - s e n s i t i v e a r e d e r i v e d . A s i t is e a s i l y understood, t h e s u f f i c i e n t c o n d i t i o n s a r e more p r a c t i c a l t o apply. From t h e s e c o n d i t i o n s one can o b t a i n t h e information on t h e e f f e c t s o f t h e e q u a t i o n e r r o r on t h e performance index. EXAMPLE For i l l u s t r a t i v e example, c o n s i d e r t h e problem o f f i n d i n g t h e o p t i m a l c o n t r o l which minimizes
f o r t h e f i r s t o r d e r system
k = ax +bu,
x(0) = 1
.
Together w i t h t h e a c t u a l o p t i m a l performance v a l u e , t h e values o f t h e performance index corresponding t o t h e t h r e e design method. versus t h e parameter e m r a r e shown i n Figs. 1,2 and 3 r e s p e c t i v e l y f o r t h e t h r e e cases of uncertainty. The e f f e c t i v e n e s s o f t h e two game t h e o r e t i c designs a r e c l e a r l y understood from t h e s e r e s u l t s . A s i s s e e n i n F i g u r e s , i n t h e case when t h e e q u a t i o n errormakes t h e system mre u n s t a b l e t h a n t h e model, t h e two game t h e o r e t i c design y i e l d s b e t t e r p e r f o r mance t h a n t h e nominal design. And t h e o t h e r way around i n t h e case when t h e e q u a t i o n e r r o r makes t h e system more s t a b l e t h a n t h e model system. This i n f e r i o r i t y o f t h e game designs t o t h e nominal d e s i gn can be understood because t h e game t h e o r e t i c designs a r e t h e worst c a s e design. I t i s a l s o und e r s t o o d from t h e s e Figures t h a t t h e s e n s i t i v e game t h e o r e t i c design i s an i n t e r mediate one between t h e nominal one and game t h e o r e t i c one. CONCLUSIONS Two game t h e o r e t i c approaches t o s e n s i t i v i t y design f o r l i n e a r r e g u l a t o r problem under l a r g e u n c e r t a i n t y a r e developed, where t h e e q u a t i o n e r r o r i n model system and t h e feedback c o n t r o l l e r a r e assumed t o be a n t a g o n i s t s . The s o - c a l l e d game t h e o r e t i c d e s i g n , which i s t h e worst c a s e d e s i g n , a r e f i r s t l y developed, and t h e upper bound o f t h e value of t h e performance index i s derived. Accounting t h e advantage of t h e nomin a l d e s i g n , a s l i g h t l y d i f f e r e n t game design c a l l e d s e n s i t i v e game t h e o r e t i c design is a l s o p r e s e n t e d . The s e n s i t i v e game t h e o r e t i c design i s an i n t e r mediate one between t h e nominal one and t h e preced i n g game t h e o r e t i c design. The problem of determining admissible e r r o r s s o t h a t t h e value o f t h e performance i n d e x does n o t exceed some value i s d i s c u s s e d by t h e use o f p - s e n s i t i v i t y . The c o n d i t i o n s f o r t h e performance index does n o t i n c r e a s e by more t h a n a f a c t o r o f p i n comparison with t h e nominal performance value a r e d e r i v e d f o r t h r e e design c a s e s By g i v i n g simple examples, t h e e f f e c t i v e n e s s o f t h e p r e s e n t game t h e o r e t i c approaches t o t h e s e n s i t i v i t y design a r e shown. A l l numerical computations were performed byFACOM
230-60 a t Data P r o c e s s i n g C e n t e r i n Kyoto University. REFERENCES 1 ) D.D. Sworder: Minimax c o n t r o l o f d i s c r e t e time s t o c h a s t i c systems; J.SIAM on C o n t r o l ser.A, Vol. 2 , No. 3, pp.433-449, 1965. 2) M.D. Mesarovic: S a t i f a c t i o n approach t o t h e synt h e s i s and c o n t r o l o f systems; Proc. 1965 3rd Annual A l l e r t o n Conference on C i r c u i t and s y s tem Theory, pp. 930-942, 3) P. Dorato & R.F.Drenick: O p t i m a l i t y , i n s e n s i t i v i t y and game t h e o r y ; i n S e n s i t i v i t y methods i n c o n t r o l t h e o r y e d i t e d by L. Radanovic, 1966 Pergamon P r e s s . 4) P . Dorato & A. Kestenbaum: A p p l i c a t i o n o f game t h e o r y t o t h e s e n s i t i v i t y design o f optimal system;.IEEE Trans. Vol.AC-12, No.1, pp. 85-87 1968 5 ) D.M. Salmon: minimax c o n t r o l l e r design; IEEE Trans., Vol.AC-13, No. 4 , pp.369-376, 1968 6)R.A. Rohrer & M. S o b r a l Jr.: S e n s i t i v i t y consid e r a t i o n s i n optimal system d e s i g n ; IEEE Trans. Vol.AC-10, No. 1, pp.43-48, 1965. 7) G. I .Sarma & R.K.Ragade: Some c o n s i d e r a t i o n s i n f ? m u l a t i n g o p t i m a l c o n t r o l problem as d i f f e r e n t i a l games; I n t . J. C o n t r . , Vo1.4,No.3, pp. 265-279, 1966. 8 ) R.K.Ragade & I . G . Sarma: A game t h e o r e t i c approach t o optimal c o n t r o l i n t h e presence o f unc e r t a i n t y ; IEEE Trans. , Vol.AC-12, No.4, pp. 395 -401, 1967. 9 ) N.H.Mcclamroch, G.L.ClarJc & J.K.Aggarwa1: Sens i t i v i t y o f l i n e a r c o n t r o l systems t o l a r g e parameter v a r i a t i o n s , Automatics, Vo1.5, pp .257 -263, 1969. 10) H. Kimura: A game t h e o r e t i c approach t o c o n t r o l and s t a b i l i z a t i o n o f systems with d i s t u r b a n c e s ; ( i n J a p a n e s e ) , Trans. SICE, Vo1.6, No.4, pp.366 -371, 1970. 1 1 ) I .B. Rohdes & D.G.Luenberger: D i f f e r e n t i a l games w i t h i m p e r f e c t s t a t e i n f o r m a t i o n ; IEEE Trans. Vol. AC-14, No. 1, pp.29-38,1969.