An Optimal Fuel-Injection Policy for Performance Enhancement in Internal Combustion Engines

7/30/2019 An Optimal Fuel-Injection Policy for Performance Enhancement in Internal Combustion Engines

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Sddhand, Vol. 22, Part 4, August 1997, pp. 545-552. Printed in India.

A n opt imal f ue l -in jec t ion po l icy f or per f orm anceenh ancem ent in in t erna l com bus t ion eng inesV H G U P T A I a n d S H A L A B H B H A T N A G A R *.*,21 9 , A n a n d N a g a r , R a i p u r 4 9 2 0 0 1 , I n d i a2 D e p a r t m e n t o f E l e c tr i c a l E n g i n e e r i n g , I n d i a n I n s t it u t e o f S c i e n c e ,B a n g a l o r e 5 6 0 0 1 2 , I n d i a* P r e s e n t a d d r e s s: I n s t it u t e o f S y s t e m s R e s e a r c h , 2 2 6 9 A V W i l li a m s B u i l d i n g ,U n i v e r s i ty o f M a r y l a n d , C o l l e g e P a rk , M a r y l a n d , M D 2 0 7 4 2 , U S Ae - m a i l : s h a l a b h @ i s r . u m d . e d uA b s t r a c t . A f u e l - in j e c t i o n in t e r n a l c o m b u s t i o n e n g i n e s y s t e m i s c o n s id e r e d ,w h e r e i n s u p p l y o f f u e l t o t h e e n g i n e i s c o n t r o l l e d w i t h t h e t w i n p u r p o s e s o fm a x i m i z i n g p o w e r o u t p u t a n d m i n i m i z i n g f u e l w as ta g e . T h e s y s t e m i s m o d e l l e da s a c o n t r o l l e d M a r k o v c h a i n a n d a f e e d - b a c k o p t i m a l c o n t r o l p o l i c y is o b t a i n e df o r t h e l o n g - r u n a v e r a g e r e w a r d o p t i m a l i t y c r i t e r i o n u s i n g M a r k o v d e c i s i o nt h e o r y .K e y w o r d s . D y n a m i c p r o g r a m m i n g / o p t i m a l c o n tr o l; i n f in i te s t at e M a r k o vc h a i n ; l o n g r u n a v e r a g e r e w a r d : o p t i m a l p o l i c y ; f u e l i n j e c t i o n c o n t r o l i n I Ce n g i n e s .

1 . I n t r o d u c t i o nI n t e rn a l c o m b u s t i o n ( I C ) e n g i n e s a r e w i d e l y u s e d i n a v a r i e ty o f a p p l i c a t io n s a n d a l m o s tm o s t i m p o r t a n t l y i n a u t o m o b i l e s . A l a r g e n u m b e r o f s u c h m a c h i n e s u s e t h e t w o - s t r o k ep e t r o l e n g i n e s . O n e o f t h e g r a v e s t p r o b l e m s w i t h t w o - s t r o k e p e t ro l e n g i n e s i s th e u n r e l i a b l ec o m b u s t i o n , w h i c h l e a d s to c y c l e s ( a c y c le c o m p r i s e s o n e r e v o l u t i o n o f t h e c r a n k a n d i se q u i v a l e n t to t w o s t ro k e s o f t h e p i s t o n ) t h a t d o n o t p r o d u c e a n y p o w e r a n d a l so r e l e a s eu n i g n i t e d f u e l w h i c h g o e s t o t h e e x h a u s t a n d g e t s w a s t e d . T w e n t y - f i v e p e r c e n t o f t h ec y c l e s b e i n g ' p o w e r l e s s ' i s q u i te c o m m o n e v e n i n t h e l a t e st e n g i n e s . U n t i l re c e n tl y , fu e lw a s s u p p l i e d t o th e s y s t e m u s i n g a f u e l - ai r m i x e r ( t e c h n i c a ll y k n o w n a s c a rb u r e tt o r ) a l m o s ti n a ll c a se s . T h i s c a r b u r e t to r b e i n g i n l i n e w i t h t h e a i r - fl o w o f t h e s y s t e m w o u l d s u p p l y t h ef u e l - a i r m i x t u r e c o n t i n u o u s l y t o th e c o m b u s t i o n c h a m b e r s y s t e m . H o w e v e r , i n re c e n t t im e s ,t h e r e is a s h i f t t o ' d i r e c t p e t r o l i n j e c t i o n ' s y s t e m s , w h e r e a i r s u p p l y i s d o n e i n d e p e n d e n t l ya n d t h e f u e l i s d i r e c t l y i n j e c t e d i n t o t h e s y s t e m , o n c e e v e r y c y c l e . T h i s h a s r e n d e r e d t h e

*Author for correspondence5 4 5


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5 4 6 V H G u p t a a n d S h a l a b h B h a t n a g a rs y s t e m a m e n a b l e t o f u e l i n je c t io n c o n t r o l , s i n c e o n e c o u l d d e c i d e w h e t h e r t h e f u e l w a s t ob e i n j e c t e d o r no t . W e r e f e r t h e re a d e r t o M o s k w a & H e d r i c s ( 1 9 9 2 ) a n d J o n e s e t a l ( 1995 ) ,and r e f e r enc es t he r e in , f o r t he l a t e s t t r ends i n t h i s d i r ec t i on .

I n t h e f o l lo w i n g , w e p r o p o s e a s t o c h a st i c m o d e l o f t h e s y s t e m , a n d a l s o p r o p o s e s t r a te -g i e s f o r c o nt r o l. T h e k e y i d e a i s to p r e d i c t o n - li n e , t h e o c c u r r e n c e s o f ' p o w e r l e s s ' c y c l e ss tochas t i c a l l y , and t o cu t f ue l a t t hose r andom in s t an t s , t hus min imiz ing fue l was t agew i t h o u t c o m p r o m i s i n g o n t h e p o w e r o u t p u t o f t h e e n g in e .

2 . T h e p r o b l e mWe cons ide r an eng ine w i th fue l i n j ec t i on ope ra t i ng i n a s t e ady s t a t e . The two b ina ryva r i ab l e s t ha t a r e o f i n t e r e s t t o u s eve ry i t h cyc l e a r e x i an d Y i , w h i c h a r e d e f i n e d a sf o l l o w s , x i = 0 , i f fue l i s no t in jec ted , and x i = 1 , i f fue l i s in jec ted . S im i la r ly , Y i = 0 , i fc o m b u s t i o n d o e s n o t ta k e p l a c e, a n d Y i = 1 , i f c o m b u s t i o n d o e s t a k e p l a ce . W e n o t e t h a ti f w e d e f in e z i = x i - Y i , t hen z i i s a l so a b ina ry va r i ab l e such t ha t z i = O, i f x i = Y i = 1o r x i = Y i = 0 , a nd z i = 1, if x i = 1 and Y i = 0 . I t i s ev iden t t ha t t he pos s ib i l i t yx i = 0 and Yi = 1 i s i m p o s s i b l e , s in c e t h er e c a n b e n o c o m b u s t i o n w i t h o u t a n y f u e l b e i n gin jec ted . Clear ly , i f z i = 0 , then t he r e i s no w as t age o f f ue l , and i f z i = 1 , then fue l in tha tc y c l e i s w a s t e d . H e n c e , ~ i Z i i s a ' m easu re ' o f t he t o t a l f ue l was t age . A l so no t e t ha t ~],i Yii s a ' m e a s u r e ' o f th e t o ta l p o w e r o u t p u t. I f w e f u r th e r d e n o t e b y n o n - n e g a t i v e c o n s t a n t sB a n d A , t h e c o s t o f f u e l p e r c y c l e a n d t h e p r o fi t o r g a i n f r o m p o w e r o u t p u t p e r c y c l er e s p ec t iv e l y , t h e n t h e p r o b l e m r e d u c e s t o t h e c o m b i n e d t a s k o f m a x i m i z i n g Y ~ i A y i a n dm i n i m i z i n g ~ i B z i . A c c o r d i n g l y , w e c h o o s e a c u m u l a t iv e r e w a r d f u n c t i o n C , d e f i n e d b yC = A ~ i Y i - - B Y ~ i Z i . H e r e , w e a l s o n o t e t h a t x i a nd y i b e i n g r a n d o m v a r i a b l e s , w ec a n n o t d i r e ct l y m a x i m i z e C , t h o u g h w e c o u l d m a x i m i z e t h e e x p e c t at i o n o f C . N o t i n g t h a tY i a r e ra n d o m v a r ia b l e s, w e a l s o m a k e t h e f o l lo w i n g a s s u m p t i o n s .A s s u m p t i o n s A. (1) { Y i } i s s u c h t h a t w h e n e v e r Y i = 0 , ( 2 ) and (3 ) be lo w (o f (A) ) ho ldf r o m i o n w a r d a n d t h i s d o e s n o t d e p e n d o n t h e v a l u e o f i.

( 2) W e d e n o t e b y P l , p 2 . . . . t h e p r o b a b i l it i e s P r o b { y i + l = l l Y i = 0 , x i + 1 = 1} = P l ,and for k > 1 ,

P r o b { y i + k = l [ y i + k - 1 = 1 . . . . . Y i + l = 1 , Y i = O , x i + k = 1} = P k ,where t he ve r t i c a l ba r deno t e s cond i t i ona l p robab i l i t y .

(3) 1 > P l > P2 > P3 > " ' " e tc .W e n o w e x p l a in t h e p h y s i c a l s i g n if i ca n c e o f A ( 1 ) - A ( 3 ) .A ( 1 ) i m p l i e s t h a t w h e n e v e r t h e r e is a ' p o w e r l e s s ' c y c l e , t h e w h o l e s y s t e m b e g i n s a f r e s h

a n d w h a t h a p p e n e d b e f o r e s u c h a c y c l e d o e s n o t a f f e c t w h a t h a p p e n s a f t er w a r d s . T h i s i sq u i t e re a s o n a b l e , s i n c e o t h e r w i s e o n e w i ll h a v e t o a s s u m e e i th e r a n i n fi n it e m e m o r y s y s t e mor an a rb i t r a r i l y a s s igned f i n i t e memory sys t em.

A(2 ) means t ha t combus t i on h i s t o ry p robab i l i s t i c a l l y de t e rmines i t s f u tu r e . One shou ldn o t e h e r e t h a t th i s s t ru c t ur e m u s t b e c o r r o b o r a t e d b y ' e x p e r im e n t a l o b s e r v a t i o n ' .

A ( 3 ) m e a n s t h a t a s t h e s e q u e n c e o f fu e l u ti li z a ti o n ( f u e l -n o n - w a s t a g e ) b e c o m e s l o n g e r,t h e c h a n c e s o f e n c o u n t e ri n g f u e l w a s t a g e i n c r ea s e . T h i s i s p o s s i b l y t h e o u t c o m e o f t h e


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F u e l - in j e c ti o n p o l i c y f o r p e r f o r m a n c e e n h a n c e m e n t i n I C e n g i n e s 5 4 7p e s s i m i s m t h a t t h e lo n g e r t h i n g s g o ' ri g h t ' , t h e c h a n c e s t h a t t h e y g o ' w r o n g ' i n c r e a s e ! T h er e s t ri c t io n P l < 1 i s m i l d a n d m e r e l y a l l o w s f o r c o n s e c u t i v e ' p o w e r l e s s ' c y c l e s . H o w e v e rP l c a n b e a s c l o s e t o 1 a s w e w a n t .

I n w h a t f o l l o w s , w e s h a l l f o r m u l a t e t h i s p r o b l e m a s a c o n t r o l l e d M a r k o v c h a i n a n da p p l y M a r k o v d e c i s i o n t h e o r y t o o b ta i n t h e o p ti m a l c l o s e d l o o p f e e d b a c k c o n t r o lp o l i c y .

W e s h a ll n o w d e s c r i b e t h e m a n n e r i n w h i c h t h is c o n t r o l le d M a r k o v c h a i n i s c o n s t ru c t e d ,i n l i e u o f ( A ) . W e u s e { 0, 1 } a s t h e c o n t r o l s e t w i t h x i a s t h e c o n t r o l a t t h e i t h s t e p . H e n c e ,a t a n y s t a g e w e p i c k x i = 1 a s t h e c o n t r o l i f w e d e c i d e t o s e n d f u e l a n d c h o o s e x i = 0o t h e r w i s e .

L e t {X n } r e p r e s e n t t h e c o n t r o l l e d M a r k o v c h a i n o n t h e s t a te s p a c e o f n o n n e g a t i v e i n t e g e r sw i t h { 0 , 1 } a b o v e a s t h e c o n t r o l s e t . L e t X 0 = j r e p r e s e n t t h e i n i t ia l s t a t e , f o r s o m ei n t e g e r j . T h e n f o r a n y i n t e g e r i > 1 , Xi = j + i i f Y l = Y 2 . . . . . Yi = 1,e l s e i f Yk = 0 f o r s o m e in t e g e r k , X k = 0 a s w e l l . H e n c e f o r i < n , i f Xn = i , t h e nX n - i = 0 . T h e r e f o r e a n y s t a te i in th e s t at e s p a c e o f t h e M a r k o v c h a i n c o r r e s p o n d s t o t h ei t h c o n s e c u t i v e 1 i n t h e Yn s e q u en c e . T h e m o m e n t Yn = 0 f o r s o m e n , t h e M a r k o v c h a i nj u m p s b a c k t o s t a t e 0 . L e t p ( i , j , a ) r e p r e s e n t t h e t ra n s i ti o n p r o b a b i l i t y o f m o v i n g f r o ms t a t e i t o s ta t e j w h e n a c t i o n o r c o n t r o l a 6 { 0 , 1} i s c h o s e n . T h e t r a n s i t i o n p r o b a b i l i t i e st h e n a r e :

p ( i , i + l , 1 ) = p i + l , p (i ,O , 1 ) = l - p i + l , V i > 0 ,a n d p ( i , 0 , 0 ) = 1 , V i > 0 . H e r e Pi are a s i n ( A ) .

L e t R ( i , a ) r e p r e se n t t h e o n e - s te p r e w a r d r a n d o m v a r i a b le s w h e n t h e M a r k o v c h a i n i s ins t a te i a n d a c t i o n a i s c h o s e n . I n li e u o f t h e c u m u l a t i v e r e w a r d f u n c t i o n C d e f i n e d e a r li e r,w e d e f i n e / ~ ( i , a ) a s f o l lo w s :

/ ~ (i , 1 ) = A w . p . P i + l ,= - B w . p . 1 - P i + l ,

V i > 0 a n d R ( i , 0 ) = 0 w . p . 1 , V i > 0 . H e r e a n d e v e r y w h e r e e l s e w . p . s t a n d s f o r ' w i t hp r o b a b i l i t y ' . N o w l e t R ( i , a ) b e t h e o n e - s t e p e x p e c t e d r e w a r d i n s ta t e i w h e n a c t i o n a i sc h o s e n . T h e n ,

R ( i , 1) = A . p i + l - B ( 1 - P i + I ) --- (A + B) .P i+ l - B , Vi > 0 ,a n d R ( i , 0 ) = 0 , V i > 0 .

A p o l i c y i s d e f i n e d a s a r u l e f o r s e l e c t in g a c t i o n s a n d c a n i n g e n e r a l b e r a n d o m i z e d . L e tf o r a n y p o l i c y z r a n d i n i t i a l s t a t e i ,

E ~ [ n ]~ j = 0 R ( X j , a j ) / X o = i4~rr ( i ) = l i ra in f (1)n ~ o e n + 1H e r e E Tr r e p r e s e n t s t h e e x p e c t e d v a l u e u n d e r p o l i c y J r , q~zr i ) r e p r e s e n t s t h e a v e r a g e e x -p e c t e d r e t u rn p e r u n i t t im e w h e n p o l i c y 7 r i s e m p l o y e d a n d i n i ti a l s t a te i s i . W e s h a l l s a yt h a t a p o l i c y r r * is a v e r a g e r e w a r d o p t i m a l i f 4~:r* ( i ) = m a x r r ~brr i ) , V i . O u r o b j e c t i v e s h a l lb e t o f i n d z r * f r o m t h e c l a s s o f a l l p o l i c i e s .


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548 V H Gupta and Sha labh Bha tnagar3 . T h e o p t i m a l p o l i c yLe t Va r ep r e s en t the va l ue f unc t i on f o r t he a - d i s co un t ed r ew ar d c r i te r i on de f i ned a s:

V ~ ( i) = m a x E ~ a n R ( X n , a n ) / X o = i , (2 )7/" n : 0w h e r e th e d i s c o u n t f a c to r a 6 ( 0, 1 ). T h e d y n a m i c p r o g r a m m i n g e q u a t i o n f o r t h e a -d i sco un ted rew ard te l l s us (Ross 1983, p . 31)

V a ( i ) = m a x { R ( i ' a ) + a Z P ( i ' j ' a ) V a ( J ) }

S i n c e w e h a v e o n l y t w o c o n t r o ls 0 a n d 1 , t h e a b o v e e q u a t io n i n o u r c a s e b e c o m e sVot(i) = m a x { R ( / , 1 ) + a p i + l V ~ ( i + 1 ) + o r ( 1 - Pi+l)Vot(O), aVc~(0)}, (3)for i > 0 . In w hat fo l lows , we sha l l f ir s t show tha t V a ( i ) i s non i nc r ea s i ng i n i .Lem ma 1 . Under (A ), R ( i , 1) is nonincreasing in i.P r o o f R ( i + 1, 1) : ( A + B ) . P i+ 2 - B < (A + B ) . P i + l - B = R ( i , 1) , the ineq ual i tyf o l l owi ng becau s e o f A( 3 ) . Th i s i s t r ue f o r eve r y i . Th e c l a i m f o l lows . [ ]P R O P O S I T I O N 1Under (A), V~(i) is nonincreasing in i .P r o o f Co ns i de r t he va l ue i t e r a ti on f o r m o f ( 3 ) ( s ee f o r i n s t ance , chap t e r I I. 3 o f Ros s 1983f o r va l ue i t e r a ti on ) . We have

V n ( i ) = m a x{R ( / , 1) + a p i + l Vn - 1 ( i + 1)+ a ( 1 - P i + l ) V n - l ( O ) , a v n - l ( 0 ) } , ( 4 )

'i > 0 , n > 1 . W e sha l l f i r st show by indu ct ion tha t v n ( i ) i s non i nc r ea s i ng i n i f o r eve r yn , and t hen w e s ha l l s how t ha t vn( i ) - -+ Va( i ) as n ~ oo , t he r eby i m p l y i ng t he c l a i m .

L e t V 2 ( i ) = R ( i , 1) . Then , by t he p r ev i ous l em m a , V 2 ( i + 1) < V d ( i ) . N o w a s su m et ha t v n - 1 ( j ) dec r ea s e s in j . So , a s in ( 4 ), we have f o r i + 1 ,v n ( i + 1) = m a x { R ( / + 1, 1 ) + a p i + 2 v n - l ( i + 2)

+ a ( 1 - P i + 2 ) v n - l ( o ) , a v a n - l ( 0 ) } . ( 5 )W e n e e d t o c o n s i d e r

t h e s e c o n d t e r m s a r eon l y t he f i rs t t e r m s i n t he a r gum en t s on t he RH S o f ( 4 ) and ( 5 ) , s i ncei den t ica l . W e have ,

N o w s i n c e V - 1 ( j )

R ( i , 1) + a p i + l V n - l ( i -q- 1) + a ( 1 - P i + l ) V n - l ( o )= R ( i , 1) + a p i + l ( V n - l ( i + 1 ) - v n - l ( 0 ) ) + a V g - l ( 0 ) . (6)

i s n o n i n c r e a s in g in j , w e h a v ev n - l ( i + 2) < v n - l ( i + 1 ) _ < . . . _< v n - l ( 0 ) ,


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F u e l - i n je c t io n p o l i c y f o r p e r f o r m a n c e e n h a n c e m e n t i n I C e n g i n e s 5 4 9so, Vn - 1 i + 1 ) - V - l ( 0 ) > V ~ - 1 i 2 ) - Vn - 1 0 ). N o w , b y l e m m a 1 a b o v e , w e h a v et h e L H S o f ( 6 ) > R ( i + 1, 1) + o t P i + 2 ( v n - l ( i + 2 ) - v n - 1 (0 )) + o t v n - l ( 0 ) o r , v n ( i ) >v n ( i + 1) , Vi > 0 . H en ce , v n ( i ) i s n o n i n c r e a s i n g i n i , V n b y i n d u c t i o n . S i n c e R ( i , O ) = O ,V i , i t c a n b e e a s il y se e n b y ( 2 ) a n d l e m m a 1 t h a t IV y ( i )] < I g ( 0 , 1 ) 1/ (1 - o r ) . N o w a si n R o ss (1 9 8 3 , p . 3 6 ) V ~ ( i ) = l imn v n ( i ) , u n i f o r m l y in i . H e n c e V ~ ( i ) i s n o n i n c r e a s i n gin i . [ ]

L e t T = m i n { n > O l S n = 0 } r e p r e s e n t t h e f i rs t t i m e t h e M a r k o v c h a i n { X n } hi t s s t a t e0 s t a r ti n g f r o m a n y s t a te . N o t e t h a t T i s a r a n d o m t i m e . L e t f ~ b e t h e u - d i s c o u n t o p t i m a lp o l i c y . F u r t h e r , le t f b e t h e p o l i c y t h a t c h o o s e s a c t i o n 1 f o r a ll n . N o w c l e a r ly , E f~ [ T / X o =i] 1 , T = j w . p .p l p 2 . . . P j - l ( 1 - p j ) . N o w b y A ( 3 ) ,

o~E f [ T / X o = 0 ] = ~ i . p l . . . P i _ l ( 1 - P i )

i =1(x)< Z i . p i l - l ( 1 - p i )i=1

< ~ i .pi1-1i=1

w h e r e , i n t h e f i r s t e q u a t i o n( 1 - P l ) .

1 0 ,a n d i s t h e o n e w h i c h f o r e a c h i , p r e s c r i b e s a n a c t i o n t h a t m a x i m i z e s t h e r i g h t s i d eo f ( 7 ) . A g a i n b y t h e o r e m V . 2. 2 o f R o s s ( 1 9 8 3 ) , 3{C ~n} o f d i s c o u n t f a c t o r s s u c h t h a t0 tn E ( 0 , 1), Yn an d C~n t 1 as n -+ cx~ an d h c~ , ( i ) - + h ( i ) w h i c h i s a b o u n d e df u n c t i o n , a n d w h e r e hc~,( i ) = Vc~,( i ) - Vc~,(O) . B y p r o p o s i t i o n 1 , hc~ ,(i)


6/8

5 5 0 V H G u p t a a n d S h a l a b h B h a t n a g a rDr

or

L e t

( A + B ) p i + l - B + P i + l h ( i + 1) < 0 ,

BPi+l < (9 )- A + B + h ( i H - 1 ) "^i = m i n { i 1 ( 9 ) h o l d s } . ( 1 0 )

^I f t h e s e t i n (1 0 ) i s e m p t y , w e t a k e i = o o .T h e a b o v e i s s u m m a r i z e d i n th e f o l l o w i n g t h e o r e m , w h i c h i s t h e m a i n r e s u l t o f t h is

s e c t i o n .T h e o r e m 1 . U n d e r ( A ) , the re ex ists a s ta te i Van(O) - Van( i ) . T a k i n g l im i t s o n b o t h s i d e s a s n ~ ~ x~ , w e o b t a i n h ( i + 1) < h ( i ) i .

( 2) A s a r e s u l t o f a b o v e , n o t e t h a t t h e R H S o f ( 9) in c r e a s e s w i t h i , w h e r e a s t h e L H S o f ( 9)d e c r e a s e s w i t h i a n d s o , w e w o u l d o n l y b e i n t e r e s te d i n th e f ir s t i a t w h i c h t h e R H So v e r s h o o ts t h e L H S .

( 3 ) R ( 0 , 1 ) > 0 c o r r e s p o n d s t o P l > B / ( A + B ) . F o r t h i s c a s e , c o n s i d e r a p o l i c y f( s a y ) w h i c h w h e n s t a r t i n g i n s t a t e i s e l e c t s a c t i o n 1 i f R ( i, 1 ) > 0 , t il l t h e f i rs t k s u c ht h a t R ( i + k , 1 ) < 0 ( w h i c h o n e e x p e c t s b y l e m m a 1 ) a n d s e l e c t s a c t io n 0 a f t e r t h a t,w i t h t h e s a m e t h i n g r e p e a t e d f r o m s t a t e 0 s u b s e q u e n t l y . I f v f ( i ) i s t h e c u m u l a t i v ea - d i s c o u n t e d r e w a r d f u n c t i o n c o r r e s p o n d i n g t o t h i s p o l i c y w h e n s t a r t i n g i n s t a t e i ,t h e n c l e a r l y V f ( i ) > 0 a n d s o V ~ ( i) > v f ( i ) > O . F u r t h e r i f R ( 0 , 1) < 0 , t h e nR ( i , 1 ) < 0 , V i , a n d h e n c e i t w o u l d b e o p t i m a l t o h a v e a p o l i c y w h i c h s e l e c t s a c ti o n 0f o r e v e r, s t a r ti n g i n a n y s t a te . I t c a n b e f o r m a l l y a r g u e d i n t h i s c a s e t h a t V a ( i ) = O , V i .H e n c e , w e c o n c l u d e h e r e t h a t i r r e s p e c t i v e o f R (0 , 1 ) b e i n g p o s i t i v e o r n e g a t i v e , V,~ ( i )i s a l w a y s n o n n e g a t i v e f o r e v e r y i .

( 4 ) F r o m ( 9 ) , d e f i n e { /3 ~n ( i ) } a s :~ n ( i ) = B / ( A + B + V ~n (i ) - V a ,( O ))

w h e r e {O tn} i s d e f in e d a s b e f o r e . T o c h e c k t h a t it is a p r o b a b il i ty , w e n e e d t o c h e c k t h a t0 < /3 o ~n ( i ) < 1 , i . N o w f o r /3 ~ n ( i ) < 1 , w e n e e d t o c h e c k i f A + V an ( i ) - V ~n ( 0 ) > 0 .F r o m ( 3 ) , n o t e t h a t V~ n(i) > Otn.V~n(O . F u r t h e r a s i n p o i n t ( 3 ) a b o v e , V a n ( 0 ) > 0 , O tn . H e n c e ,


7/8

F u e l - in j e c t i o n p o l i c y f o r p e r f o r m a n c e e n h a n c e m e n t in I C e n g i n e s 5 5 1A + V a , ( i ) - V a n ( O ) > A - ( 1 - O t n ) V ~ , , ( O ) . ( l l )

N o w , a s i n 3 , V un ( 0 ) < ] R ( 0 , 1 ) } / ( 1 - C ~n ). I f R ( 0 , 1 ) < 0 , t h e n V u , ( 0 ) = 0 a n d w ea r e d o n e . S o , l e t R ( 0 , 1 ) _> 0 . T h e n , s i n c e - V u , , ( 0 ) > - R ( 0 , 1 ) / ( 1 - a n ) , w e h a v ef r o m ( 1 1 ) ,

A + V ~ , , ( i ) - V ~ , , ( O ) > A - R ( O , 1)= A - ( A + B ) p l + B= A ( 1 - p L ) + B ( 1 - p l ) > 0 .

F u r t h e r , f o r / 3 ~ ,, ( i ) >_ 0 , w e n e e d d e n o m i n a t o r t o b e p o s i t i v e , a n d t h i s f o l l o w s i m m e -d i a t e l y f r o m a b o v e . N o w , /3 c ~n ( i ) - -+ / 3 ( i ) , i , a s o tn 1" 1 , w h e r e / 5 ( i ) i s t h e R H S o f ( 9 ) .H e n c e , 0 _ < / 3 ( i ) < 1 , V i . I t i s h e n c e , a p r o b a b i l i t y .

( 5 ) L e t u s c o n s i d e r t h e c a s e s B = - 0 a n d A = 0 s e p a r a t e ly . B = 0 c o r r e s p o n d s t o i g n o r i n gf u e l lo s s w h i c h i s r e f l e c t e d f r o m o u r o p t i m a l p o l i c y , s i n c e t h e n , /3 ( i ) = 0 a n d s o ,i t i s o p t i m a l t o k e e p s e n d i n g f u e l f o r e v e r. T h i s i s w h a t o n e e x p e c t s f r o m i n tu i t io n ,f o r m a x i m i z i n g p o w e r . T h e c a s e A = 0 c o r r e s p o n d s t o i g n o r in g p o w e r o u t p u t o ft h e e n g i n e . N o t e th a t /3 u n ( i) = B / ( B + V ~ , ( i ) - V c ~ ,( 0) ), i n t h i s c a s e . H o w e v e r ,R(0 , 1 ) - - - B p l - B = B ( p l - 1 ) < 0 . H e n c e , V u n ( i ) = V c ,, ( 0 ) = 0 , a s m e n t i o n e di n p o i n t ( 3 ) a b o v e . S o , /S u , , ( i ) = 1 , V i , n , a n d h e n c e / 3 ( i ) = 1 , V i , a g a i n as e x p e c t e d ,s i n c e i t t e l l s u s t h a t ' n e v e r t o s e n d f u e l ' p o l i c y i s o p t i m a l .

( 6 ) F i n a l ly , w e l o o k a t t h e p r o b l e m o f e x p l i c i t ly c o m p u t i n g g a n d h ( i ) . B y a s s u m p t i o nA ( 3 ) , w e h a v e 0 < 1 - P l < 1 - p 2 _< . . . . C o n s i d e r a n e w p r o c e s s w i t h i d e n t i c a ls t a te a n d a c t i o n s p a c e s a n d i d e n t i c a l r e w a r d s , b u t w i t h t r a n si t io n p r o b a b i l i t i e s g i v e nb y ,

P i + l p i + l P l - P i + lp ( i , i + 1 , 1) - p ( i , O , 1) : 1 -p l p l p la n d p ( i , 0 , 0 ) = 1 , Y i > 0 . L e t ~ ? (i ) b e th e p l - d i s c o u n t o p t i m a l v a l u e f u n c t i o n f o rt h is n e w p r o c e s s , w h e r e P l i s t r e a t e d a s a d i s c o u n t f a c to r . T h e a n a l y s i s o f (R o s s 1 9 8 3 ) ,p p . 9 8 - 9 9 , c a r r i e s th r o u g h n o w , a n d w e o b t a i n

g - --- ( 1 - p l ) I ? (0 ) , h ( i ) = r e (i ) - V ( O ) .T h e l o n g - r u n a v e r a g e r e t u rn o p t i m a l p o l i c y i s o n e w h i c h s e l e c t s a c t i o n O , t h e fi rs t t i m ei s u c h t h a t P i + l < B / ( A + B + ( ~ '( i + 1 ) - ~ ' ( 0 ) ) ) . T h i s c o m p l e t e s o u r d i s c u s s i o no f t h e r e s u l t s .

5 . C o n c l u d i n g r e m a r k s( 1 ) W e c o n s i d e r e d t h e p r o b l e m o f c o n t r o l l i n g t h e s u p p l y o f f u e l i n a f u e l in j e c t i o n i n te r n a l

c o m b u s t i o n e n g i n e s y s t e m , w i th t h e t w i n p u r p o s e s o f m a x i m i z i n g p o w e r o u t p u t a n dm i n i m i z i n g f u e l w a s t a g e u n d e r v e r y g e n er a l a s s u m p t i o n s .


8/8

55 2 V H G u p t a a n d S h a l a b h B h a t n a g a r( 2 ) We fo rmu la t ed t he p rob l em a s an i n f i n i t e s t a t e con t ro l l ed Markov cha in on t he s t a t e

space o f nonnega t i ve i n t ege r s u s ing b ina ry con t ro l va r i ab l e s, and app l i ed dy nam icp r o g r a m m i n g t e c h n i q u e s t o o b t a in t h e l o n g -r u n a v e r a g e r e w a r d o p t im a l p o l i c y . T h eop t ima l po l i cy t u rned ou t t o be a t h r e sho ld po l i cy wh ich cu t s supp ly o f f ue l a t a c e rt a i ns t a te t m en t i one d i n t heo re m 1 . As a r e su l t i f i < cx~, t he ove ra ll Ma rkov cha in un de rth is po l i cy be co m es f i n it e s t a te w i th s t a t e space { 0, 1 , 2 . . . . . i }.

( 3 ) W e f u l l y a n a l y s e d t h e o p t i m a l p o l i c y . O u r a n a l y s i s s h o w e d t h a t t h e o p t i m a l p o l i c ya c t u a ll y g i v e s t h e t h re s h o l d p r o b a b i l it y . W e a l s o c o n s i d e r e d c a s e s i n w h i c h w e r e d u c et h e p r o b l e m e i th e r to o n e o f m a x i m i z i n g p o w e r o u t p u t a lo n e o r t o o n e in w h i c h j u s tfue l was t age i s min imized , and found t ha t i n e i t he r c a se t he t h r e sho ld po l i cy i s wha tone ex pec t s f r om in tu it i on .

( 4 ) F i n al ly , w e l o o k e d a t th e p r o b l e m o f e x p l ic i t ly c o m p u t i n g t h e l o n g - r u n a v e r a g e r e w a r da n d t h e t h r e s h o ld p r o b a b il it y . W e o b t a i n e d e x p r e s s i o n s f o r b o t h o f th e m i n t e rm so f p l - d i s c o u n t o p t i m a l v a l u e f u n c t i o n o f a m o d i f i e d p r o c e s s , w h i c h c a n b e e a s i l ycom pu ted . Fu r the r i nves t i ga t i on cou ld be ca r r i ed ou t f o r t he expe r ime n ta l ve r i f i c a t i ono f t he a lgo r i t hm.

S B i s in d e b t e d t o P r o f V i v e k S B o r k a r f o r s u g g e s t in g t h e a p p r o a c h t o t h e p r o b l e m a n d f o rm a n y h e l p f u l d i sc u s s i o n s d u r i n g t h e c o u r s e o f th i s w o r k .

Ref e r ence s

Jon es V K, Au lt B A, Franklin G F 199 5 Identification and air-fuel ratio con trol of a spark ignitionengine. IEEE Trans. Control. Syst. Tech. 3 : 1 4 - 2 1M osk w a J J , Hed rics J K 1992 M odeling and validation of autom otive engines for control algorithmdevelopment. AS M E J. Dynam ic Syst. Meas. Control 1 1 4 : 2 7 8 - 2 8 5

Ross S M 1983 Introduction to stochastic dynam ic progra mm ing (New Y ork: Academ ic Press )

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