A Chebyshev Approximation for Solving Optimal Control Problems

7/30/2019 A Chebyshev Approximation for Solving Optimal Control Problems

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P e r g a m o n Computers Math. A pplic. Vol. 29, No. 6, pp. 35-45, 1995Copyright(~)1995 Elsevier S cience L tdPrinte d in Gre at Britain. All r ights reserved0898-1221/95 $9.50 + 0.000898-1221 (95)0 0005 .4

A C h e b y s h e v A p p r o x i m a t io nf o r S o l v i n g O p t i m a l C o n t r o l P r o b l e m sW . M . E L - G I N D Y A ND H . M . E L - H A W A R YD e p a r t m e n t o f M a t h e m a t ic sFac ul ty of Sc ience , Ass iu t U niver s i tyAss iu t , Egypt

M . S . S A L IMD e p a r t m e n t o f M a t h e m a t i c sFac ul ty of Sc ience , Azha r U niver s i tyA s s iu t , E g y p tM . E L - K A D YD e p a r t m e n t o f M a t h e m a t ic sFac ul ty of Sc ience , Ass iu t Un iver s i tyAss iu t , Egypt

(Received November 1993; revised and accepted June 1994)A b s t r a c t - - T h i s pape r presents a numerical solut ion for solving opt imal control problems, and thecontrolled Du ffing oscillator. A new Ch ebyshev spectral p rocedure is introduced . Con trol variablesand state variables are approximated by Chebyshev series. Then the system dynamics is trans-formed into systems of algebraic equations. Th e optimal control problem is reduced to a constrainedoptimization problem. Results and com parisons are give n at th e end o f the paper.K e y w o rd s - -C h e b y sh e v approximation ; Opt imal control p roblem; Ordinary and par t i a l d if fe ren-tial equ ations.

1 . I N T R O D U C T I O N

B e l l m a n ' s d y n a m i c p r o g r a m m i n g [1] a n d P o n t r y a g i n ' s m a x i m u m p r in c ip l e m e t h o d [2] r e p r e s e n tt h e m o s t k n o w n m e t h o d s f o r s o l vi n g o p t i m a l c o n t r o l p r o b l e m s . I n t h i s p a p e r , a n a l t e r n a t i v ea l g o r i t h m f o r s o l v in g s u c h p ro b l e m s i s p r e s e n t e d . T h i s a p p r o a c h is b a s e d o n t h e e x p a n s i o n o ft h e c o n t r o l v a r ia b l e i n C h e b y s h e v s e ri e s w i t h u n k n o w n c o e f f ic i en t s. I n t h e s y s t e m d y n a m i c s , t h es t a t e v a r i a b le s c a n b e o b t a i n e d b y t r a n s f o r m i n g t h e b o u n d a r y v a l u e p r o b le m f or o r d i n a r y a n dp a r t i a l d i f f e r e n t ia l e q u a t i o n s t o i n t e g ra l f o r m u l a e . U s i n g E 1 - G e n d i' s m e t h o d [ 3] , C h e b y s h e v s p e c -t r a l a p p r o x i m a t i o n s f o r t h e s e i n t e g r a ls [4] c a n b e o b t a i n e d . T h i s i s a c c o m p l i s h e d b y s t a r t i n g w i t ha C h e b y s h e v s p e c t r a l a p p r o x i m a t i o n f o r t h e h i g h e s t o r d e r d e r i v a t iv e a n d g e n e r a t i n g a p p r o x i m a -t i o n s t o t h e l o w e r o r d e r d e r i v a t i v e s t h r o u g h s u c c es s iv e i n te g r a t i o n . T h e r e f o r e , t h e d i f f e r e n ti a l a n di n t e g r a l e x p r e s si o n s w h i c h a ri s e fo r t h e s y s t e m d y n a m i c s a n d t h e p e r f o r m a n c e i n d e x , t h e i n i ti a lo r b o u n d a r y c o n d i t i o n s , o r e v e n fo r g e n e r a l m u l t i p o i n t b o u n d a r y c o n d i t i o n s a r e c o n v e r t e d i n t o al -g e b r a i c e q u a t i o n s w i t h u n k n o w n c o e ff ic i en t s. I n t h i s w a y , t h e o p t i m a l c o n t r o l p r o b l e m i s r e p l a c e db y a p a r a m e t e r o p t i m i z a t i o n p r o b l e m , w h i c h c o n s is ts o f t h e m i n i m i z a t io n o r m a x i m i z a t i o n o f t h ep e r f o r m a n c e in d e x , s u b j e c t t o a l ge b r ai c c on s t ra i n ts . T h e n , t h e c o n s t r a in e d e x t r e m u m p r o b l e mc a n b e r e p l a c e d b y a n u n c o n s t r a i n e d e x t r e m u m p r o b l e m b y a p p l y i n g t h e m e t h o d o f L a g r a n g e [5]

T y pe se t b y A ~ - T E X35


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3 6 T . M . E L -G IN D Y e t a l .o r t h e p e n a l t y f u n c t i o n t e c h n i q u e [ 6 ] . T h e s a m e c o m p u t a t i o n a l t e c h n i q u e c a n b e e x t e n d e d t oso l v e t h e co n t ro l l ed D u f f i n g o sc i l l a t o r .

2 . M A T H E M A T I C A L F O R M U L A T I O NT h e b e h a v i o u r o f a d y n a m i c s y s t e m c a n b e r e p r e s e n t e d b y t h e f o ll ow i n g s e t o f o r d i n a r y d i ff er -

e n t i a l e q u a t i o n s :d x i = f ~ ( x l , x 2 , . . . , x n , U y , U 2 , . . . , u r , T ), i = 1 , 2 , . . . , N ,d T

o r i n t h e v e c t o r f o r m

w i t h i n i ti a l c o n d i t i o nd Xd T f ( X , U , ~-) , 0 < T < T , (2 .1)

X (0 ) = X 0 , (2 .2 )w h e r e X a n d U a r e v e c t o r fu n c t i o n s o f T , ( X l, X 2 , . . . , X n) a re t h e s t a t e v a r i a b l e s , a n d( U l, u 2 , . . . , u r ) a r e t h e c o n t r o l v a r ia b l e s .

T h e p r o b l e m o f o p t i m a l c o n t r o l i s t h e n t o f i n d t h e c o n t r o l u ~, i = 1 , . . . , N , t r a n s f e r r i n g t h es y s t e m ( 2 .1 ) f r o m t h e p o s i t io n x i = x i ( r 0 ) t o t h e p o s i t i o n x i = X~(T ) w i t h i n t h e t i m e (T - T O),a n d y i e ld i n g th e o p t i m u m o f p e rf o r m a n c e i n d e x I , g iv e n b y [7]

I = h [ X ( T ) , T ] + g ( X , U , 7 , T ) d T . (2.3)T h e v e c t o r f u n c t i o n f a n d t h e s c a l a r f u n c t i o n s h a n d g a r e g e n e r a l l y n o n l i n e a r , a n d a r e a s s u m e dt o b e c o n t i n u o u s l y d i ff e r e n t ia b l e w i t h r e s p e c t t o t h e i r a r g u m e n t s . W i t h o u t l o ss o f g e n e r a l i ty , w ew i ll a s s u m e t h a t n = r = 1 . T h e t i m e tr a n s f o r m a t i o n

T = T (1 + t ) (2.4)is i n t r o d u c e d i n o r d e r t o u s e C h e b y s h e v p o l y n o m i a l s o f t h e f i rs t k i n d , d e f i n e d o n t h e i n t e r v a l[ -1 , 1 ]. I t f o ll o w s t h a t eq u a t i o n s (2 . 1 ) - (2 . 3 ) a r e r ep l aced b y :

d xd--- /= F (x , u , t ) , - 1 < t < 1 , (2 .5)

x ( - 1 ) = x 0 , ( 2 .6 )/ j .= H [ x ( 1 ) , T ) + G ( x , u , t , T ) d r. (2.7)12 .1 . A p p r o x i m a t i o n o f t h e S y s t e m D y n a m i c s

T o so l v e eq u a t i o n (2 . 5 ) , w e p u t [4 ] d x ( t )d t - ( t ) . (2 .8)F r o m t h e i n i t ia l c o n d i t i o n ( 2 .6 ) , a n d b y i n t e g r a t i n g e q u a t i o n ( 2 .8 ) , w e g e t/ :( t ) = ( t ) d t + x0. (2.9)1H e r e , w e c a n g i v e C h e b y s h e v s p e c t r a l a p p r o x i m a t i o n s a s f ol lo w s :

NX i = X ( t i ) = ~ b i j ( t j ) + X o, i = 1 , . . . , N , ( 2 . 1 0 )

j = 0


3/11

C h e b y s h e v A p p r o x i m a t i o n 3 7w h e r e t~ - - - c o s ~ a r e t h e C h e b y s h e v p o i n t s a n d b it a r e th e e l e m e n t s o f t h e m a t r i x B a s g i v e ni n [ 3 ] .

B y e x p a n d i n g t h e c o n t r o l v a r ia b l e s i n a C h e b y s h e v s e r ie s o f o r d e r m , w e h a v e [8]m

m ( t) = ) - - : ' c , ( 2 . 1 1 )i----0

h e r e T i ( t) is th e i th C h e b y s h e v p o l y n o m i a l . A s u m m a t i o n s y m b o l w i t h t h e p r i m e d e n o t e s a s u mw i t h f i r s t t e r m h a l v e d . H e n c e , t h e s y s t e m o f e q u a t i o n s ( 2 .5 ) c a n b e a p p r o x i m a t e d a s f o ll ow s :s u b s t i t u t i n g f r o m e q u a t i o n s ( 2 .8 ) , ( 2 .1 0 ), a n d ( 2 .1 1 ) i n t o e q u a t i o n ( 2 .5 ) , w e h a v e

( t i ) = F b i t ( t t ) + x 0 , ck T k ( t i ) , t i , i = 1 , . . . , N , (2 .1 2 )k=Ow h i c h c a n b e w r i t t e n i n t h e f o r m :

F ( a , / 3 ) : 0 , ( 2 .1 3 )w h e re cg - ( ( t 0 ) , ( t 1 ) , . . - , ( t N ) ) , /~ - - (C O , C l , . . . , Cm).2 .2 . A p p r o x i m a t i o n o f t h e P e r f o r m a n c e I n d e x

T h e p e r f o r m a n c e i n d e x ( 2 . 7) c a n b e a p p r o x i m a t e d a s f ol lo w s : u s in g t h e E 1 - H a w a r y t e c h n i q u e[4] , an d su b s t i t u t i n g f ro m (2 .1 0 ) an d (2 .1 1 ) i n t o (2 . 7 ), w e h av e

NJ = H [ x ( T ) , T ] + ~ b N j G ( x ( t j ) , u ( t j ) , T ) ( 2 .1 4 )j = 0

= g bNj ( t N ) , T + Z bNj G bjs ( t s ) + xo, ~ cr T~( t j ) , T = g (~ , f~) .\ j = 0 t = 0 r = 0 ( 2 .1 5 )

G e n e r a l l y , J i s n o n l i n e a r i n a , /3 .T h e o p t i m a l c o n t r o l p r o b l e m h a s b ee n r ed u c e d t o a p a r a m e t e r o p t i m i z a t i o n p r o b le m . T h e

p r o b l e m n o w i s t o f i n d t h e m i n i m u m v a l u e o f J = J ( t ~ , f~ ) g i v e n b y ( 2. 15 ) , s u b j e c t t o t h e e q u a l i t yco n s t r a i n t s (2 . 1 3 ) , i . e . ,

M i n i m i z e J = J ( a , / ~ ) ,s u b j e c t t o F ( t x , / ~ ) = 0 .

M a n y t e c h n i q u e s a r e a v a i l a b l e i n s u c h c a s e , s u c h a s L a g r a n g e m u l t i p l i e r s , p e n a l t y f u n c t i o n ,e t c . W e p r e fe r t h e p e n a l t y f u n c t io n a p p r o a c h w i t h p a r t i a l q u a d r a t i c i n t e r p o l a ti o n m e t h o d w h i c his ca l le d P e n a l t y P a r t i a l Q u a d r a t i c I n t e r p o l a t io n ( P P Q I ) [9 ].

U r a b e [ 10 ,1 1 ] h a s d e s c r i b e d a m e t h o d t o d e t e r m i n e v e r y a c c u r a t e l y t h e n u m e r i c a l s o l u t io n o fn o n l i n e a r o r d i n a r y d i f fe r e n ti a l e q u a t io n s , a n d h a s s h o w n h o w t o s t u d y t h e e x i s t e n ce a n d u n i q u e -n e s s p r o b le m o f a n e x a c t s o l u t io n n e a r t h e c a l c u l a t e d C h e b y s h e v a p p r o x i m a t i o n , a n d h o w t oe s t i m a t e t h e e r r o r o f t h e a p p r o x i m a t i o n . H o w e v e r , w e b e li e v e i t w i ll b e v e r y h a r d t o a p p l yU r a b e ' s r e s u l t s t o t h e o p t i m a l c o n t r o l p r o b l e m . J a c q u e s a n d R e n ~ [7] s u g g e s t , fo r e n g i n e e r i n gp u r p o s e s , a s o m e w h a t d i f f e r e n t v i e w o f t h e e r r o r e s t i m a t i o n p r o b l e m [ 1 2] . W e th e r e f o r e u s e e i t h e r

IJ(aiV+l,~3g+l) - - J(aN,~N)l < ~1, o rF < ~2,

o r b o t h , t o d e c i d e w h e t h e r t h e c o m p u t e d s o l u ti o n is clo s e e n o u g h t o t h e o p t i m a l s o l u t io n .


4/11


5/11

m

3579

C h e b y s h e v A p p r o x i m a t i o nT a b l e i. T h e F e l d b a u m p r o b l e m .

N =5 N =7 N =9 N =I I0.1929074640.192881804

0.192909305 0.1929093060.192909292 0.1929092990.192906918 0.192909299

- - O .192909306

0.1929093060.1929092980.1929092980.192909298

39

3 . 2 . M i n i m u m T i m e O r b i t T r a n s f e r P r o b l e mO n e o f t h e b e s t k n o w n t r a j e c t o r y o p t i m i z a t i o n e x a m p l e s i s t h e p r o b l e m o f m i n i m i z i n g t h e

t r a n s f e r t i m e o f a c o n s t a n t l o w - t h r u s t io n ro c k e t b e t w e e n t h e o r b i t s o f E a r t h a n d M a r s . T h i si n v o lv e s t h e d e t e r m i n a t i o n o f t h e t h r u s t a n g l e h i st o r y , f o r w h i c h n o e x a c t s o l u t i o n i s k n o w n [ 7].T h e p e r f o r m a n c e i n d e x o f t h e p r o b l e m c a n b e s t a t e d a s fo ll ow s :

M i n i m i z es u b j e c t t o

d X td rdX2d r

a x 3

I=T ,t h e f o l lo w i n g t i m e - v a r y i n g e q u a t i o n s [1 3]:

aTw i t h t h e b o u n d a r y c o n d i t i o n s

Xl( O) = 1.0,X I ( T ) = 1 .5 2 5,

- - ----X 2 ,X 2 ~ R o s i n u

- X 1 X 2 4 - m 0 4 - m r 'X 2 X 3 R o c o s u

- - - - - - 4 "X l to o+ mr' 0


6/11

40 T .M . EI,-GINDYet al .F r o m e q u a t i o n ( 3 .1 2 ) a n d b y i n t e g r a t i n g e q u a t i o n ( 3 .1 3 ), w e g e t

fl ( t ) = ( t ) d t + 1 ,1 i2 ( t ) = ( t ) d r ,1N ~ ( t ,) = ~ b ~ 5 C j + 1 ,j= O

Nj= O

N 3(t~ ) = ~ b~ j o j + 1,j=O

i3( t ) = 9( t ) d t + 1 ,1

i = 1 , . . . , N ,

(3.14)

w h e r e t i = - c o s ~ a r e t h e C h e b y s h e v p o i n t si n [ 3 ] . T h e b o u n d a r y c o n d i t i o n s g i v e C 1 =t e c h n i q u e , w e g e t

a n d b i5 a r e t h e e l e m e n t s o f t h e m a t r i x B a s g iv e n1 .0 , C2 = 0 .0 , C3 = 1 .0 , an d us ing E1- Ge ndi ' s

Nx 1 ( 1 ) = ~ -: ~ b ~ 5 5 + 1 = 1 . 5 2 5 ,

5=0N

x 2 ( l ) = ~ b ~ j 5 = 0 . 0 ,j= O

N 3 (1 1 = ~ b ~ j oj + 1 = 0 . 8 0 9 8

5=0

(3.15)

B y e x p a n d i n g t h e c o n t ro l v a r ia b le s i n C h e b y s h e v s e ri es o f o r d e r m ,m

u r n (t) = ~ ' ck ~ ( t ) ,k=O

( 3 . 1 6 )

a n d s u b s t i t u t i n g f r o m ( 3 .1 3 ), ( 3 .1 4 ), a n d ( 3 .1 6 ) i n t o t h e s y s t e m d y n a m i c s ( 3 .1 1 ) , w e h a v e t h ef o l lo w i n g a p p r o x i m a t i o n s :

g l s ( i , ~bi , Oi , Ck , T ) = 0,g 2 p ( i , 4 , O i , C k , T ) = O ,g3q(~, i , 0~,C k , T ) = O ,

s = 1 , . . . , N ,p = 1 , . . . , N ,q - - 1 , . . . , N ,

( 3 . 1 7 )

w h e r e i = 1 , . . . , N , k = 1 , . . . , m .E q u a t i o n s ( 3 . 1 5 ) a n d ( 3 . 1 7 ) g i v e ( 3 N + 6 ) e q u a t i o n s i n ( 3 N + m + 5 ) u n k n o w n s . H e n c e , t h e

o p t i m a l c o n t r o l p r o b l e m c a n b e s t a t e d a s f o l l o w s .F i n d i , i , 8 i, c k , T , s o t h a t t h e p e r f o r m a n c e i n d e x J --- T i s t o b e m i n i m i z e d , s u b j e c t t o t h e

c o n s t r a i n t e q u a t i o n s i n ( 3 . 1 7 ) , w i t h t h e b o u n d a r y c o n d i t i o n s ( 3 . 1 5 ) .T h e s t a t e a n d c o n t r o l v a r ia b l e s o b t a i n e d f o r m = 1 1 a n d N = 9 a r e l is t e d in T a b l e 2. F o r t h i s

c a s e , w e h a v e t h e f i n a l t i m e T = 3 .3 11 71 . T h e b o u n d a r y c o n d i t i o n s ( 3.9 ) a r e a c c u r a t e l y s a t i s fi e d( s ee T a b l e 2 ) , a n d t h e e s t i m a t e d e r r o r s 1 a n d 62 a r e t h e n o f o r d e r 1 0 - 7 .

T h i s p r o b l e m h a s a l so b e e n s u c c e s s f u ll y s o l v ed b y m a n y a u t h o r s [ 7 ] . I n T a b l e 3 , th e r e a r ec o m p a r i s o n s b e t w e e n t h e s e m e t h o d s a n d t h e p r e s e n t m e t h o d .


7/11

C h e b y s h e v A p p r o x i m a t i o nT a b l e 2 . C o n v e r g e d f u n c t i o n s f o r E x a m p l e 3 . 2 ; N --- 9 , m - - 1 1 .

t X 1 ( t ) X 2 ( t ) X 3 ( t ) U ( T )- 1 . 0 0 0- 0 . 9 4 0- 0 . 7 6 6- 0 . 5 0 0- 0 . 1 7 4

0 . 1 7 40 . 5 0 00 . 7 6 60 . 9 4 01 . 0 0 0

1 . 0 0 0 0 01 . 0 0 0 2 91 . 0 0 7 6 81 . 0 4 5 8 41 . 1 6 2 4 31 . 3 4 1 6 51 . 4 6 4 2 01 . 5 1 2 4 81 . 5 2 4 3 51 . 5 2 5 0 0

0 . 0 0 0 0 00 . 0 0 7 1 50 . 0 4 6 0 80 . 1 3 6 5 70 . 2 8 9 5 80 . 2 9 3 4 20 . 1 5 6 0 60 . 0 6 8 1 30 . 0 1 4 2 70 . 0 0 0 0 0

1 . 0 0 0 0 01 . 0 1 2 4 61 . 0 4 0 1 21 . 0 5 1 7 00 . 9 6 9 8 30 . 8 2 8 9 70 . 7 6 8 6 90 . 7 7 5 3 20 . 7 9 7 8 90 . 8 0 9 8 0

6 . 6 9 2 0 46 . 7 5 1 8 50 . 6 0 1 1 0

- 5 . 4 5 7 4 71 . 6 3 7 5 04 . 5 5 4 1 3

- 1 . 2 0 0 4 7- 1 . 0 9 2 1 5- 0 . 9 3 6 9 31 8 . 0 9 9 9 0

4 1

T a b l e 3 . R e s u l t s fo r t h e m i n i m u m - t i m e o r b i t t ra n s f e r p r o b l e m .M a x . E r r o r B o u n d a r yM e t h o d s T C o n d i t i o n s

M o y e r , P i n k h a m [14]- g r a d i e n t

f i r s ts e c o n d

- g e n e r. N e w t o n - R a p h s o nF a l b , d e J o n g [ 1 5 ]H o n t o i r , C r u z [ 1 3 ]T a y l o r , C o n s t a n t i n i d e s [ 1 2 ]J a c q u e s , R e n ~ [ 7 ]- m = 7- m = 9- m = l l

P r e s e n t m e t h o d

3 . 3 1 73 . 3 1 73 . 3 2 0 73 . 3 1 9 33 . 3 1 9 43 . 3 8 1 9

3 . 3 3 0 6 93 . 3 2 2 6 33 . 3 1 8 7 43 . 3 1 1 7 1

0 . 1 %0.05%

0

< 10 -13< 10 -13< 10 -13

0

4 . T H E C O N T R O L L E D L I N E A R O S C I L L A T O RW e w i ll c o n s i d e r t h e o p t i m a l c o n t r o l o f a l i n e a r o s c i ll a to r g o v e r n e d b y t h e d i f fe r e n t ia l e q u a t i o n

~: + w 2x = u , (4 .1)i n w h i c h a d o t ( . ) m e a n s d i f f e r e n t i a t i o n w i t h r e s p e c t t o T , w h e r e - T < T < 0 a n d T i s s p e c i f ie d .E q u a t i o n ( 4 .1 ) is e q u i v a l e n t t o t h e d y n a m i c s t a t e e q u a t i o n s

:~I ~ X2~

w i t h t h e b o u n d a r y c o n d i t i o n sz I ( - T ) : x l o ,

x l ( 0 ) = 0 ,x 2 ( - T ) = x 2 0 ,

x 2 ( 0 ) = 0 .a n d

O n e w i s h es t o c o n t r o l t h e s t a t e o f t h i s p l a n t s u c h t h a t t h e p e r f o r m a n c e i n d e x

i f u 2 d rI = - ~ 1i s m i n i m i z e d o v e r a l l a d m i s s i b l e c o n t r o l f u n c t i o n s U ( T ) .

( 4 . 2 )

(4.3)

(4 .4)


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4 2 T . M . E L -G IN D Y et al.

P o n t r y a g i n ' s m a x i m u m p r i nc i p le m e t h o d [2] a p p l ie d t o t h is o p t i m a l c o n t r o l p r o b l e m y i e ld s th ef o l lo w i n g e x a c t a n a l y t i c a l s o l u t i o n r e p r e s e n t a t i o n [7]:

1x l = ~ [ A w T s i n w T + B ( s i n w T - w r C O S W T ) ] ,

1 [ A ( s i n w r + wT c o s WT) + B WT s in w ~ ']2~--- ~U(T) = A c o s w r + B s i n w T ,

1 [ 2 w T ( A s + B s ) + ( A s - B 2 ) s i n 2 w T - 4 A B s i n s w T ] ,

(4 . 5 )

w h e r e 2 w [ x l o w s T s i n w T - X 2o ( w T c o s w T - s i n w T ) ]A = w s T 2 - s i n 2 w T2 w s [x s o T s i n w T + x l o ( s i n w T + w T c o s w T ]B = w s T 2 - s i n 2 w T

( 4 . 6 )

5 . S O L U T I O N O F T H E P R O B L E M A N D I T S R E S U L T ST h e o p t i m a l c o n t r o l p r o b l e m d e s c r i b e d i n (4 . 1 ), ( 4 .3 ) , a n d ( 4 .4 ) c a n b e r e s t a t e d a s fo l lo w s :

M i n i m i z es u b j e c t t o

T f lI = ] u s d t ,J - 1= 1 T 2 ( _ w 2 x + u ) ,

( 5 . 1 )(5 . 2 )

w i t hz ( - 1 ) = x - l ,

T o s o l v e e q u a t i o n ( 5 .2 ) , w e p u t5 : ( - 1 ) = X - l , z ( 1 ) = 0 , ~ ( 1 ) = 0 . (5.3)

d S x ( t )d t 2 = ( t ) , t h e nd x ( t ) / ' _= ( t ' ) d t ' + C 1,

1t t" (t')dede'+cl +cs.1 1

(5 . 4 )(5 . 5 )

(5 . 6 )F r o m t h e b o u n d a r y c o n d i t i o n ( 5 .3 ) , w e g e t

2 ( - 1 ) = 2 - 1 = C 1 , z ( - 1 ) = - C 1 + C s = x - l ,h e n c e

C 2 ---- X - 1 5 - 1 -

N O W w e c a n g i v e t h e f o ll o w i n g a p p r o x i m a t i o n s :Nx~ z(td ~ (s) 1) + x_ 1,= b ~ ( t~ ) + ~ - l ( t ~ +

5=0N

2 ( t i ) = E b i j ( t j ) + 2 - 1 , f or i = 1 , . . . , N ,j=O

(5 . 7 )

( 5 . s )

w h e r e b ! 2 ) = (t~ - t j ) b i j , t i = - c o s - ~ , a n d b i j a r e t h e e l e m e n t s o f t h e m a t r i x B a s g i v e n i n [3 ].93


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C h e b y s h e v A p p r o x i m a t i o n 4 3

T h e b o u n d a r y c o n d i t i o n s ( 5 . 3 ) a r e a p p r o x i m a t e d b y :

A s a b o v e , l e t t h e a p p r o x i m a t i o n

?i(1 ) = ( t ' ) d t ' d t + 2 ~ - 1 + x - 1 = O,1 1N

x ( 1 ) = ~ b ( ~ ( t j ) + 2 5 - 1 + x - 1 = 0,j = 0

5 (1 ) = ( t ) d t + 5 _ 1 = O ,1

N5 ( 1 1 = b N , ( t , ) + = 0 .

j = 0

o f t h e c o n t r o l v a r i a b le b e g i v e n b ym

u r n ( t ) =i = 0

(5.9)

(5.10)

(5.11)

H e n c e , t h e s y s t e m d y n a m i c s (5 .2 ) c a n b e a p p r o x i m a t e d a s fo ll ow s . S u b s t i t u t i n g f r o m e q u a t i o n s(5 . 4 ), ( 5 . 7 ) , an d (5 .1 1 ) i n t o eq u a t i o n (5 .2 ) , w e g e t

( t~ ) I T 2 - w 2 h !2 1 ) E ' c k T k ( t ) = 0 ,- ( t j ) + . ~ _ l ( t ~ + + +j = 0 k = O

w h i c h c a n b e w r i t t e n i n t h e f o r m :

i -- 1 , . . . , N ,(5 .12)

F i ( a , / 9 ) = 0 , i = 1 , . . . , N , ( 5 .1 3 )w h e r e

( :I t - -~ ( ( ~ ( t 0 ) , ( ~ ( t l ) , - , ( tN) ) , / 9 ---- ( C 0 , C I , . - - , a m ) ,w h i c h i s t h e n a s y s t e m o f ( N + 3 ) l i n e a r e q u a t i o n s i n ( N + m + 2 ) u n k n o w n s c ~ , / 9 .

T h e p e r f o r m a n c e i n d e x ( 5 . 1 ) c a n b e a p p r o x i m a t e d a s f o l l o w s . S u b s t i t u t i n g f r o m ( 5 . 1 1 ) i n t o(5 .1 ) an d u s i n g [3] , w e g e t ) '= - - W ~ - "b N j C r T r ( t j ) - - J ( / 9 ) . (5 .14)G e n e r a l l y , J i s n o n l i n e a r i n a , /9 .

T h e o p t i m a l c o n tr o l p r o b l e m h a s b ee n r e d u c ed to a p a r a m e t e r o p t i m i z a t i o n p r o bl e m . T h ep r o b l e m n o w i s t o f i n d t h e m i n i m u m v a l u e o f J = J ( f~ ) g iv e n b y ( 5 .1 4 ) s u b j e c t t o t h e e q u a l i t yco n s t r a i n t s (5 . 1 3 ) , i . e . ,

M i n i m i z e J - - J ( / 9 ) ,s u b j e c t t o F ( a , / 9 ) = 0 .

B y u s i n g th e P e n a l t y P a r t i a l Q u a d r a t i c I n t e r p o l a t i o n ( P P Q I ) [ 9] , w e g e t t h e r e s u l t s o f t h e s t a t ea n d c o n t r o l v a ri a b l e s w i t h N = 1 5 a n d m = 7 l i st e d i n T a b l e 4. T h e o p t i m a l v a l u e o f t h e c o s tf u n c t i o n a l J f o r d i f f e r e n t v a l u e s o f m a n d N i s g i v e n in T a b l e 5 .

T h e e s t i m a t e d e r r o r s o n t h e b o u n d a r y c o n d i t io n s ar e 0 . 2 7 9 4 E - 0 3 a n d 0 . 2 6 2 7 E - 0 8 , r e s p e c t i v e ly .The exac t so lu t ion as g iven in [7] i s J = 0 .184858542.


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44 T. M. EL-GINDYe t a l .Table 4. w -- 1, T -- 2, 2:-1 ~-- 0.5, and ~:-1 ---- --0.5.

t i- 1 . 0 0 0

-0.978-0.914-0.809-0.669-0.500-0.309-0.105

0.1050.3090.5000.6690.8090.9140.9781.000

x ( t ~ ) ~ ( t~ ) u ( t ~ )0.5000E+ 000.4891E + 000.4568E+ 000.4052E+ 000.3382E+ 000.2628E+ 000.1875E+ 000.1210E + 000.6921E - 010.3431E - 010.1430E - 01

-0.4846E - 020.1350E - 020.4283E- 030.2861E- 030.2794E- 03

-0.5000E+ 00-0.5001E+ 00

0.4892E+ 000.4946E+ 000.5090E+ 000.5278E+ 000.5440E+ 000.5493E+ 000.5365E+ 000.5012E+ 000.4436E+ 000.3685E+ 000.2844E+ 000.2012E+ 000.1280E+ 000.7153E- 010.3620E- 010.2421E- 01

-0.4 982E+ O0-0.4890E+00-0.4658E+00-0.4238E+00-0.3626E+ 00-0.2874E+ 00-0.2078E+ 00-0.1348E+ 00-0.7677E- 01-0.37 13E- 01-0.14 48E- 01-0.41 22E- 02-0.65 85E- 03-0.2627E - 08

Table 5. The optimal cost functional J.N : 9 N : 11 N = 15

0.184851218 0.184851229 0.184851242- - 0.184851228 0.184851242- - -- 0.184851242

m7911

6 . T H E C O N T R O L L E D D U F F I N G O S C I L L A T O RLet us now invest igate the optimal control of the Duffing oscil lator, described by the nonli near

differential e quatio n5 : + w 2 x + ~ x 3 = u , (6.1)

subjec t to the same b oun dar y condit ions as before and tak ing the same perfor mance index expres-sion. Of course, the exact solut ion in this case is not known. The approach sy stem dynamics ,boundary condit ions, and performance index take the same expressions, at least formally, asequations (5.13), (5.14), (5.9), and (5.10).

Table 6 lists the opt imal value of the cost fu nction al J for different values of N with m = 7.Table 6. The optimal cost functional J* in case of m = 7.N= 9 N = 11 N---- 13 N---- 15

0.187433708 0.187433709 0.18743 3709 0.187433791The appr oxi mate soluti on (m = 11) for the perfor mance in dex as given in [5] is 0.187444856.

7 . C O N C L U S I O N STables 1-3 give a comparison between the proposed technique and other methods. The results

show tha t the suggested method is quite rel iable. Compa ring our method with the one of Jacquesand Ren~ (which is bet ter tha n other methods) [7], we notice tha t we have 33 equatio ns and 43unk now ns in our method, against 39 equations and 49 unkno wns in the Jacques and Ren~ method,which gives some superior ity to our method; particularly, our mi ni mu m time J* = 3.31171 isbetter t han that of Jacques and Ren~ (J* = 3.31874). The m ajor advantag e of this method isthat we can deal direct ly with the highest-order derivatives in the differential equation without


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Chebyshev Approximation 45t rans forming i t to a sys tem of fi rs t order , and that wi l l reduce the num ber of the un know ns . Th isfact has been shown in applying the sugges ted t echnique on the control l ed Duf f ing osc i l lator .Final ly , we no t i ce that our t echnique i s much eas ier than th e num erical integrat ion of the nonl inearTPBV P der i ved f rom Pont ryag i n ' s m ax i m um pr i nc i p l e m et hod .

R E F E R E N C E S1. R. Bellman, Dynamic Programming, University Press, Princeton, NJ, (1957).2. L.S. Pontryagin, The Mathemat ical Theory of Optimal Processes, In Interscience, John Wiley & Sons,(1962).3. S.E. E1-Gendi, Chebyshev solution of differential, integral and integro-differential equations, Computer Y.12, 282-287 (1969).4. H. EloHawary, Numerical treatment of differential equations by spectral methods, Ph.D. Thesis, Faculty ofScience, Assiut University (1990).5. H.J. Kelley, Methods of gradients, In Optimization Techniques, (Edited by G. Leitmann), pp. 205-254,

Academic Press, London, (1962).6. H.J. Walsh, Methods of Optimization, Wiley, London, (1975).7. R. Van Dooren and J. Vlassenbroeck, A Chebyshev technique for solving nonl inear optimal control problems,IEEE Trans. Automat. Contr. 33 (4), 333-339 (April 1988).8. L. Fox and I.B. Parker, Chebyshev Polynomials in Numerical Analysis, University Press, Oxford, (1972).9. T.M. E1-Gindy and M.S. Salim, Penalty function with partial quadratic interpolation technique in theconstrained optimization problems, Journal of Insti tute of Math. ~ Computer Sci. 3 (1), 85-90 (1990).10. M. Urabe, Numerical solut ion of mult i-point boundary value problems in Chebyshev series, Theory of themethod, Numer. Math. 9, 341-366 (1967).11. M. Urabe, Numerical solut ion of boundary value problems in Chebyshev series, A method of computationand error estimation, Lecture Notes Math. 109, 40-86 (1969).12. J.M. Taylor and C.T. Constantinides, Optimization: Application of the epsilon method, IEEE Trans.Automat. Contr. AC-17, 128-131 (Feb. 1972).13. Y. Hontoir and J.B. Cruz, A manifold imbedding algorithm for optimizat ion problems, Automatica 8,581-588 (1972).14. H.G. Moyer and G. Pinkham, Several trajectory optimization techniques, Par t II: Application, In Comput-ing Methods in Optimization Problems, (Edited by A.V. Balakrishnan and L.W. Neustadt), pp. 91-109,Academic Press, New York (1964).15. P.L. Falb and J.L. de Jong, Some Successive Approximation Methods in Control and Oscillation Theory,Academic Press, New York, (1969).

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A Chebyshev Approximation for Solving Optimal Control Problems