Steepest Descent Failure Analysis

7/27/2019 Steepest Descent Failure Analysis

1/15

Steepe s t -D escent Fa i lure Analys isM . M C D E R M O T T , J R . , ~ A N D V ~/. T . F O ' W L E R 2

C o m m u n i c a t e d b y A . M i e le

A b s t r a c t . A n extens ive fa i lu re ana lys i s o f the s t eepe s t -desc ent op t i -m i z a t i on a l go r i t hm ha s be e n m a de . E a c h o f t he w a ys i n w h i c h t hea lgor i thm can fa i l i s d i scussed in t e rms of bo th the mathemat ica l andnumer ica l mani fes ta t ions of a fa i lu re and the in format ion which eacht ype o f f a il u re p r ov i de s a bou t t he f o r m u l a t ion o f t he phys i ca l p r ob l e m .Numer ica l t e s t s fo r each of the va r ious types of fa i lu re a re desc r ibed ;severa l fau l ty pro blem fo rmu la t ions a re presen ted , ea ch of which il lus-t ra t e s a pa r t i cu la r type of fa i lu re . A t ab le i s p resen ted in which a l If a i l u r e m ode s a r e s um m a r i z e d a nd t he c o r r e s pond i ng num e r i c a l t e s t sare exhibited~K e y W or ds . G r a d i e n t m e t hods , il l- pos e d p r ob l e m s , c om p u t i ngmethods , numer ica l me thods , fa i lu re ana lys i s .

1 I n t r o d u c t i o n

T h e s t e e p e s t - d e s c e n t a l g o r i t h m i s w i d e l y u s e d t o o b t a i n n u m e r i c a ls o l u t io n s t o o p t i m i z a t i o n p r o b l e m s . O n e o f t h e m a i n d r a w b a c k s o f t h isa l g o r i t h m is t h a t t h e u s e r is r e q u i r e d t o a r b i t ra r i ly c h o o s e s e v e r a l p r o b l e m -d e p e n d e n t p a r a m e t e r s a n d t i m e f u n c t i o n s i n o r d e r t o i m p l e m e n t t h ea l g o r i t h m , a n d t h e c o n v e r g e n c e c h a r a c te r i st ic s f o r e a c h p a r t i c u l a r p r o b l e ma r e h i g h l y d e p e n d e n t o n t h e s e c h o i c e s . I t is o f t e n d i ff ic u lt f o r a u s e r w h o isn o t i n t i m a t e ly f a m i l ia r w i th t h e a l g o r i t h m t o m a k e a p p r o p r i a t e c h o ic e s o ft h e s e p a r a m e t e r s , t n fa c t , e x p e r i e n c e d u s e r s o f t h e a l g o r i t h m a r e o f t e na c c u s e d o f b e i n g p r a c t i t i o n e r s o f s o m e t y p e o f b l a c k m a g i c . T h e a u t h o r s 'e x p e r i e n c e w i t h th i s a l g o r i t h m a t f ir st t e n d e d t o v a l i d a t e t h is n o t i o n .H o w e v e r , a f t e r c o n s i d e r a b l y a n a l y s i s , m o s t o f t h e s e e m i n g l y m y s t e r i o u s

A s s i s t a n t P r o f e s so r , D e p a r t m e n t o f I n d u s tr i a l E n g i n e e r i n g , T e x a s A & M U n i v e r s i ty , C o l le g eS t a t i o n , T e x a s .

2 A s s o c i a t e P ro f e ss o r , D e p a r t m e n t o f A e r o s p a c e E n g i n e e r i n g a n d E n g i n e e r i n g M e c h a n i c s,T h e U n i v e r s i t y o f T e x a s a t A u s t i n , A u s t i n , T e x a s .

229This journal is copyrighted by Plenum. Each article is available for $7.50 from Plenum PubIishingCorporat ion, 227 WestI7th Street, New York, N.Y 10011.


2/15

230 JOTA: V OL . 23, NO. 2, OCTO BER 1977

p h e n o m e n a w h i ch o c c u r r e d c o u l d be r e a d i ly re l a te d t o p r o b l e m f o r m u l a -t i o n o r n u m e r i c a l l i m i t a t io n s . T h i s p a p e r is a r e s u l t o f t h is a n a l y s is .A l l o f t h e v a r i o u s w a y s i n w h i c h t h e s t e e p e s t - d e s c e n t a l g o r i t h m c a nf a i l a n d t h e r e l a t i o n o f t h e s e f a i l u r e s t o t h e p h y s i c a l p r o b l e m a r e c o n -s i d e r e d . T h e a u t h o r s f e e l t h a t t h e a n a l y s i s p r e s e n t e d h e r e w i ll b e v e r yu s e f u l t o a l l u s e r s o f t h e s t e e p e s t - d e s c e n t a l g o r i t h m i n d e t e r m i n i n g t h ee x a c t c a u s e s o f f a i l u r e s o r e r r a t i c c o n v e r g e n c e c h a r a c t e r i s t i c s o f t h ea l g o r i t h m . T h e i n s i g h t p r o v i d e d b y t h e a n a l y s i s s h o u l d a l s o b e h e l p f u l t ou s e rs o f t h e a l g o r i t h m in c h o o s i n g th e p a r a m e t e r s r e q u i r e d t o o b t a i n an u m e r i c a l s o l u t io n t o a p a r t i c u l a r p r o b l e m .

2. Steepest-Descen t Algorithm~lqae s t e e p e s t - d e s c e n t a l g o r i t h m ( R e f s . 1 - 6 ) i s a n i t e r a t i v e a l g o r i t h m

f o r c o m p u t i n g a t i m e h i s t o r y f o r a n m - v e c t o r o f c o n t r o l v a r i a b l e s u ( t) t om i n i m i z e t h e s c al ar p e r f o r m a n c e i n d e x 3

4 , = 4 , [ x ( t s ) , q ] , ( 1 )w h e r e t h e n - v e c t o r o f s t a t e v a r i a b l e s x i s g o v e r n e d b y t h e d i f f e r e n t i a le q u a t i o n

i = [(x , u, t ) . (2)t h e i n i ti a l s ta t e s a r e f i x e d , a n d t h e t e r m i n a l s t a t e s m u s t s a t i s f y t h e p + 1c o n s t r a i n t s

o = O[x ( q ) , t r ] . ( 3 )T h e a l g o r i th m r e q u ir e s t h a t t h e u s e r c h o o s e o n e c o m p o n e n t o f t h e

v e c t o r f u n c t i o n q~ a s a s t o p p i n g f u n c t i o n . T h i s f u n c t i o n is d e n o t e d b y f ~. O ne a c h i t e r a t i o n , t h e f i n a l t i m e i s d e t e r m i n e d a s t h e t i m e w h e n t h e s t o p p i n gc o n d i t i o n

0 = 1)Ix ( t l ) , t i ] (4)i s s a ti sf ie d . I n w h a t f o l lo w s , t h e r e m a i n i n g p - v e c t o r o f t e r m i n a l c o n s t r a i n tf u n c t i o n s w i l l s ti ll b e d e n o t e d b y ~p.

T o b e g i n t h e c o m p u t a t i o n , t h e u s e r m u s t fi rs t c h o o s e a n o m i n a l c o n t r o lh i s t o r y u ( t) . T h e d i f f e r e n t ia l e q u a t i o n s (2 ) a r e n u m e r i c a l l y i n t e g r a t e d u n t i lt h e s t o p p i n g c o n d i t i o n (4 ) is sa t is f ie d . T h e t r a j e c t o r y w h i c h i s g e n e r a t e d isr e f e r r e d t o a s t h e n o m i n a l t r a j e c to r y , a n d q u a n t i t i e s e v a l u a t e d a l o n g t hi st r a j e c t o r y a r e d e n o t e d b y a su p e r s c r i p t a s t e r is k ; i. e ., ( ) *.3 This paper uses the notation of De nham and Bryson (Refs. 1-4).


3/15

JOTA : VO L. 23, NO. 2, OC TO BER 1977 231

T h e a l g o r i t h m m a k e s u s e o f p r o p e r t i e s o f t h e s y s t e m o f d i f f e r e n t i a le q u a t i o n s a d j o i n t t o E q . ( 2 ) :

J i = - A ~ ' A , ( 5 )w h e r e

A = ( O f / O x ) *a n d A i s t h e v e c t o r o f a d j o i n t v a r i a b l e s . L e t ~ ( t , tr) d e n o t e t h e s t a t et r a n si t io n m a t r i x f o r t h is s y s te m , a n d l e t A e a a n d A C a d e n o t e t h e p a r t i c u la rs o l u t i o n s o f th e s y s t e m w h i c h s a t is f y , r e s p e c t i v e l y , t h e b o u n d a r y c o n d i t i o n s

A ~ . ( t l ) = [ ~ / O x + ( 1 / h ) 4 ; oa/Ox]*=,, ( 6 )A ~ a ( t ) = [ O 0 / O x + ( 1 / f i ) 6 0 f ~ / O x ] * = t , ,w h e r e A ~ a is a n n - v e c t o r a n d A , a i s a n n p m a t ri x . T h e n , a t a n y ti m e t,

A4,a(t) = 'tfft, t~)A~a(tr), (7 )A ,a (t ) = "-Ifft, t f )A ,n(tr) .T h e o p t i m a l v a r i a t i o n i n t h e c o n t r o l i s g i v e n i n t e r m s o f t h e p a r t i c u l a rs o l u ti o n s o f t h e a d j o i n t s y s t e m A + a a n d A , a .

D e n h a m s h o w s th a t t h e o p t i m a l v a r i a t i o n in th e c o n t r o l is6 u = - ( 1 / 2 ~ ) g ,~ -~ B r [ A 4 , a - A , a ( ] , (8 )

w he re W~ = W~ ( t) is a t ime va ry ing , pos i t ive -d e f in i t e we igh t ing m at r ix fo rc o n t r o l v a r i a t i o n s w h i c h m u s t b e c h o s e n b y t h e u s e r o f t h e a l g o r i t h m ,B = [ a f f O u ] * , a n d

= - 1 ; 2 1 1 , 4 , + 2/x 2~4~], (9 )/x = 1 /2[ (~4, - - I , ~ I , + ) / ( R T - ~ 2 _ A o T I f ~ A~0)]~/z (10)

R i s t h e s t e p s i z e in c o n t r o l s p a c e w h i c h m u s t b e c h o s e n b y t h e u s e r o f t h ea l g o ri th m . T o e n s u r e t h a t t h e c o n t r o l v a r i a t i o n i s s m a l l , s o t h a t t h e l i n e a r i t ya s s u m p t i o n s a r e n o t v i o l a te d , t h e c o n t r o l v a r i a t i o n i s r e q u i r e d t o s a t is f y t h ec o n s t r a i n t

I, 9= = ~ u ~ ( ~ - ) w ~ , ( ~ ) a u ( ~ - ) & . ( 1 1 )A O i s t h e r e q u e s t e d c h a n g e i n t h e v a l u e o f t h e t e r m i n a l c o n s t r a i n t f u n c -t i o n s . F o r c o m p u t a t i o n a l c o n v e n i e n c e , u s u a l l y

a 0 = - a ~ * ,w h e r e

O _ < a ~ l .


4/15

232 JOTA: V O L. 23 , NO. 2 , OCTO BER 1977

A l s o ,

f t , y - x T14,4 = A a B W , B A e m d r, ( 1 2 )"J to91 ,4 , = f A ~ f l B W f f l B r A , a d r , ( 1 3 )

a tO

i ty T --1 TI ,o = A ~ a B W u B A , a d . ( 1 4)toN o t e t h a t , if a w e i g h t e d c o n t r o l l a b i li t y m a t r ix C , i s d e f i n e d a s

I [ rC ~ = ~ r ( t y , r ) B ( r ) W 2 ~ ( r ) B r ( r ) ~ ( t r , r ) d r , ( 1 5 )o

t h e nI4,4, = A ~n ( tr ) Cu A4,a( tr ), (16 )I o , = A ~ a ( t i) C ~ A n ( t ~ ) , ( 1 7 )I* * = h ~ n ( t f ) C , A , a ( t r ) . ( 1 8 )

T h e w e i g h t e d c o n t r o ll a b i l i t y m a t r ix C , p r o v i d e s i n fo r m a t i o n a b o u t t h el i n e a ri z e d p r o b l e m ( li n e a ri z e d a b o u t t h e n o m i n a l t r a j e c t o r y ) a n d w ill b ed i s c u s s e d i n t h e f a i l u r e a n a l y s i s w h i c h f o l l o w s .

3 . T y p e s o f F a i l u r e s

T h e c o m p u t a t i o n s i n d i c a t e d i n E q s . ( 6) a n d ( 8 ) - ( i 0 ) a r e n o t p r o p e r l yd e f i n e d if a n y o f t h e f o l lo w i n g c o n d i t i o n s o c c u r :

( A ) 0 = o ,T --1( B ) N = I 4 ,, - 1 , 4 j , , Iq , ~ 0 ,

( c ) D = n 2 - a ~ f ; ~ ~ 4 ,- < 0 ,(D ) I , = s i n g u l a r .

F o r t h e a b o v e f a i l u r e t y p e s , t h e f o l l o w i n g c o m m e n t s a r e p e r t i n e n t .( A ) D i v i s i o n b y ~ i s i n d i c a t e d i n E q . ( 6) .( B ) I f N , t h e n u m e r a t o r i n t h e q u o t i e n t i n d i c a t e d i n E q . ( 10 ) , is z e r o it

f o l l o w s t h a t ~ = 0 , a n d t h e d i v i s i o n b y p i n d i c a t e d i n E q . ( 8 ) i s n o t d e f i n e d .I f N < 0 , p i s i m a g i n a r y , w h i c h i s p h y s i c a l l y m e a n i n g l e s s .


5/15

J O T A : V O L . 2 3 , N O . 2 , O C T O B E R 1 9 77 2 3 3

(C) Division by D, the denominator in the quotient indicated by Eq.(10), is not defined for D = 0. If D < 0, /x is imaginary, which is physicallymeaningless.(D) The computations indicated by Eqs. (9) and (10) require that Io-~exist.Failures caused by ~ = 0 and by D = 0 have been fully discussed in the

literature and are easily avoided. Also, Denham discussed one possiblecause of failure due to N = 0. These results, previously presented by otherauthors, are reviewed here for completeness. In addition, two more pos-sible causes of failure due to N = 0 and two possible causes of failure due toIo ~ being singular are presented.

3.1. Type (A) Failure. Consider first the case in which the timederivative of the stopping function is zero at the final time,

~ [ x ( t ~ ) , t s ] = 0 .The final time is determined as the time at which the stopping function iszero; i.e.,

a [ x ( t ~ ) , t ~ ] = 0 .For the first-order analysis used in the steepest-descent algorithm, thesimultaneous satisfaction of the conditions

~2 = 0 and ~ = 0implies that the stopping function f~[x(t), t] is zero over some finite timeinterval. Note that, for the nonlinear system, an extremum of f~ is possible.Thus, the increment in the final time Atf cannot be uniquely determined.This type of failure is caused by a poor choice of the stopping function andcan be avoided by choosing a stopping function which is monotonic alongany physically realizable trajectory. Methods to construct such a stoppingfunction are well documented in the literature, the most direct methodbeing to make a change in independent variable, so that the final value ofthe new independent variable is fixed. The function

~ = t - t ~is used as the stopping function and

= 1 , o a / o x = O .This transformation was proposed by Long (Ref. 7) for quasilinearizationalgorithms and its application to steepest-descent algorithms is describedby Miele e t a l . (Ref. 8) in the sequential-restoration algorithm (SGRA).


6/15

234 JOTA: VOL. 23, NO. 2, OCT OBE R 1977

3 . 2 . T y p e ( B ) F a i l u re . S e c o n d , c o n s i d e r t h e c a s e w h e r eN =- I,~+ r --I

iS zero .C o n v e r g e n c e . D e n h a m h a s s h o w n t h a t , f o r a p r o p e r l y f o r m u l a t e dp r o b l e m , t h i s e x p r e s s i o n i s p o s i t i v e s e m i d e f i n i t e a n d w i l l b e z e r o o n l y o n

t h e o p t i m a l t r a j e c t o r y . T h u s , f o r a p r o p e r l y f o r m u l a t e d p r o b l e m , N = 0i m p l i e s t h a t c o n v e r g e n c e , r a t h e r t h a n a g r a d i e n t f a i l u r e , h a s o c c u r r e d .P e d o r m a n c e I n d e x C o n s t r a i n e d . T h e q u a n t i t y N c a n b e z e r o if

c e r t ai n t y p e s o f i m p r o p e r p r o b l e m f o r m u l a t i o n o c c u r . C o n s i d e r t h e c o n -s e q u e n c e s o f t h e v e c t o r A 4 , a a n d t h e c o l u m n s o f A 4 ,a b e i n g l in e a rl y d e p e n -den t , so tha t the re ex i s t cons tan t s a i , no t a l l ze ro , such tha t

0 = Y ~ f= 1 A q , , n a i + A 4 , f ~ O L p + I .I f a p + l # O , t h is r e la t i o n c a n b e s o l v e d f o r A , a t o y i e ld

w h e r e

T h e c a s e in w h i c h

A ~ a = A ~ a a , ( 1 9 )

a = ( 1 / a p + l ) [ a l , a 2 , . . . , a p ] T

O~p+l = 0r e s u lt s i n / ~ b e i n g si n g u l a r a n d w ill b e d i s c u s s e d b e l o w . E q u a t i o n ( 1 9) is am a t h e m a t i c a l s t a t e m e n t o f t h e f a c t t h a t t h e p e r f o r m a n c e i n d e x i s c o n -s t r a i n e d i n t h e p h y s i c a l p r o b l e m a n d c a n n o t b e m i n i m i z e d w h i l e s i m u l -t a n e o u s l y s a t i sf y i n g t h e c o n s t r a i n ts . T h a t is , t h e J a c o b i a n m a t r i x0 ( 0 , 4 ) ) / O ( x , t ) i s no t o f fu ll r ank .

I t w il l n o w b e s h o w n t h a t l in e a r d e p e n d e n c e o f A 4 a a n d t h e c o l u m n sof Au,a r esu l t s in N = 0 . Us ing Eqs . (16 ) - (1 9 ) y ie lds

T T T -1 TN = a ( A o a C , A o n ) a - a (Ac, C uAu, a) I4 , , (A4,nC, A oa) a .N o t i n g t h a t

TA~ aC ~ A ~a - I~,~,t h e a b o v e e q u a t i o n r e d u c e s t o

o rN = 0 .


7/15

JOTA: VO L. 23, NO. 2, OCTOB ER 1977 235

T h u s , i f t h e p e r f o r m a n c e i n d e x is c o n s t ra i n e d , t h e n u m e r i c a l m a n i f e s t a ti o nis t h a t N = 0 . F u r t h e r , i n t h is c a s e , N = 0 is c o m p u t e d a s th e d i f f e r e n c eb e t w e e n t w o n o n z e r o t e rm s .

P e r f o rm a n c e I n d e x U n c o n t r o l l a b le . A n o t h e r i m p r o p e r p r o b l e mf o r m u l a t i o n w h i c h w il l c a u s e a f a i l u r e d u e t o N = 0 o c c u r s w h e n t h e v e c t o rA ~ a is i n t h e n u l l s p a c e o f t h e w e i g h t e d c o n t r o l l a b i l i t y m a t r i x C . . T h i s is am a t h e m a t i c a l s t a t e m e n t o f t h e f a c t t h a t t h e p e r f o r m a n c e i n d e x i s u n c o n -t r o l l a b l e in t h e p h y s i c a l p r o b l e m .

I f A ~ a i s i n t h e n u l l s p a c e o f C . , t h e n b y d e f i n i t i o nTI4 ~ = A ~ a C , A ~ = 0,

i q j ~ Q = TA C a C , A + a = 0 ,a n d i t f o l lo w s t h a t

N - 0 .T h u s , a g r a d i e n t f a i l u r e d u e t o N = 0 w i ll o c c u r i f t h e p e r f o r m a n c e i n d e x isu n c o n t r o l l a b l e . I n t h i s c a se , N is c o m p u t e d a s t h e d i f f e r e n c e o f t w o t e r m sw h i ch b o t h a p p r o a c h z e ro .

N u m e r i c a l E r r o r s . I n n u m e r i c a l c o m p u t a t i o n s , N is s o m e t i m e sn e g a t i v e, w h i c h D e n h a m h a s p r o v e d t o b e t h e o r e t ic a l l y i m p o s s ib l e . N e g a -t iv e v a l u e s o f N r e s u l t f r o m n u m e r i c a l e r r o r s a n d i n d i c a t e t h a t t h e t r u ev a l u e o f N is n e a r z e r o ( d u e to c o n v e r g e n c e , c o n s t r a i n e d p e r f o r m a n c e , o ru n c o n t r o l l ab l e p e r f o r m a n c e ) , b u t r o u n d o f f o r t r u n c a t i o n e r r o r s o r b o t hh a v e c a u s e d t h e c o m p u t e d v a l u e t o b e n e g a t i v e . W h e n n u m e r i c a l c o m -p u t a t i o n s a r e p e r f o r m e d , N s h o u l d b e t e s t e d ; a n d , it i t is l es s t h a n s o m es m a ll n u m b e r e, a w a r n i n g s h o u l d b e g i v e n a n d t h e c o m p u t a t i o n s h o u l d b et e r m i n a t e d o r o t h e r a p p r o p r i a t e a c t i o n t a k e n .

3 . 3 . T y p e (C ) F a i l u r e . I n t h e a b o v e d i s c u ss i o n , i t w a s s h o w n t h a t t h eq u a n t i t y N m u s t b e p o s i t i v e . I t f o l l o w s t h a t

D = R 2 - ~ , ~ , ~ I ~ 2 , zX 4 ,m u s t a l so b e p o s i t i v e i f / x is t o b e p r o p e r l y d e f i n e d a n d r e a l . N e g a t i v ev a l u e s o f D m e a n t h a t t h e r e q u e s t e d c h a n g e s i n t h e t e r m i n a l c o n s t r a i n t s 2~0a r e l a r g e r t h a n c a n b e a c h i e v e d w i t h t h e g i v e n s t e p s i z e i n c o n t r o l s p a c e R ;r e c a l l t h a t

i qR 2 = 6 u T W '~ 3 u d ~ '.o


8/15

236 JOTA : VO L. 23, NO. 2, OC TOB ER 1977A n y c o n p u t e r p r o g r a m w h i c h im p l e m e n t s t h e s t e e p e s t -d e s c e n t a lg o r i th mmu s t i nc lude log i c t o choo s e R and A ~ i n such a w ay t ha t D i s a l w ayspos i tive . This log ic wil l p rec lud e the poss ib il i ty o f f a i lu res du e to D -< 0 .

In prac t i ce , s ince no a pr ior i know l edge is ava i l ab l e t o choo s e R , oneusual ly chooses At~ (or 6 u ) t o ach i eve t he g r ea t e s t pos s i b l e pe r f o r mancei n c r e m e n t o n e a c h i t e ra t i o n a n d t h e n , if d e s ir e d , c o m p u t e s t h e c o r r e s p o n d -i ng va l ue f o r R . T he o r e t i ca l and com pu t a t i ona l a s pec ts o f va ri ous s t ra t eg i e sf o r choos i ng t he s t eps ize a r e p r e s en t ed by M i e l e e t a l . (Re f . 8).3 . 4 . T y pe ( D ) Fa i lu r e . T h e fi nal caus e o f f a i lu r e is f o r the ma t r i x I o ,to be s ingular . Before d i scuss ing the spec i f i c causes of f a i lu re , cer ta inp r ope r t i e s o f t he m a t r i x I ~ w ill be i nves ti ga ted . B y de f i n it ion ,

Iti ' A ,a(t f)q ~ (~-,q r 4 t ~ T T t - - 1 Tf ) B ( ~ ) W , (~ ' )B (7)~(~-, f)Aqm(tf) d~'.The fo l lowing fac t s w i l l be used in the ana lys i s which fo l lows . F i r s t no tetha t the m at r ix A~,a(tr ) is cons tan t , so th e f i r st and l as t t e rms in thei n t eg r an d can b e t a ken ou t s i de o f t he i n teg r a l . S i nce W , i s pos i ti ve de f in i t e ,i t fo llows tha t W~ ~ i s a l so pos i t ive def in i t e ; con sequ ent ly , the re ex i s ts anons i ngu l a r m a t r i x H ~ s uch tha t

W 2 ~ = H ~ H ~ .Also , s ince the in tegra nd in the express ion for I ,~ , is a quad ra t i c fo rm , it isa t l eas t p os i t ive sem idef in i t e and i s pos it ive d ef in i t e if f i t i s nons ingular .

B y t ak i ng t he cons t an t m a t r i x A , a ou t s i de o f t he i n teg r a l , I00 can bew r i t t en i n t he f o r m= A T ~ r r T T

T= A , n C ~ A o a,w her e t he i n t eg r a l t e r m C , i s r e f e r r ed t o a s a w e i gh t ed con t r o l l ab i l i t ym at r ix , due to the s imi la r i ty be tw ee n Cu an d the c las s ic cont ro l l ab i l i tymat r ix (Ref . 10)

f o r t he l i nea r s y s t em

qC = fro %~tTBB ~ dT

g.~c = A 6 x + B 6 u.The l inear sys tem i s cont ro l l ab le on the in te rva l [ to , tf ] i f f the matr ix C is


9/15

J O T A : V O L . 2 3, N O . 2 , O C I ' O B E R 1 97 7 2 3 7

p o s i t i v e d e f i n i te . A n e q u i v a l e n t c r i t e r i o n f o r c o n t r o l l a b i l i t y i s t h a t t h em a t r i xr(~-) = B T(~.) ,~(~., t f )

h a s n i n d e p e n d e n t c o l u m n s o n [ to , t~ ]. D e f i n e a m a t r i x Y , (~ ') a sY , ( r ) = H u (~ -)B 7 ( ~ ' ) ~ ( a ts ).

T h e r e l a ti o n s h i p s b e t w e e n t h e m a t r i c e s C ,, a n d Y ~ o f t h e s t e e p e s t - d e s c e n ta l g o r i t h m a n d t h e m a t r i c e s C a n d Y o f t h e c o n t r o l l a b i l i ty p r o b l e m w i lln o w b e c o n s i d e re d .

Constraints Dependent. C o n s i d e r t h e c o n s e q u e n c e s o f t h e c o l u m n so f t h e m a t r ix A , a b e i n g l in e a r l y d e p e n d e n t . T h a t is , t h e r e e x i st s c o n s t a n t sa ~ , n o t a l l z e r o , s u c h t h a t

0 = A or t a .R e c a l l t h a t t h i s i s t h e c a s e r e f e r r e d t o in t h e d i s c u s s i o n o f f a i l u r e s d u e t oN = 0 .T h i s is a m a t h e m a t i c a l s t a t e m e n t o f t h e f a c t t h a t t h e t e r m i n a l c o n -s t ra i n ts ( i n c l u d in g t h e s t o p p i n g c o n d i t i o n ) a r e n o t i n d e p e n d e n t a n d , c o n -s e q u e n t l y , c a n n o t b e s a ti s fi e d si m u l t a n e o u s l y . T h a t i s, t h e J a c o b i a n m a t r i xO ~ / O ( x , t ) i s n o t o f f u ll r a n k . I f t h e r a n k o f A , n i s l e s s t h a n p , i t f o l l o w s t h a tt h e p p m a t r i x

I, = A .GA,m u s t b e o f r a n k l e s s t h a n p ; t h e r e f o r e , i t i s s i n g u l a r.

Constraints Uncontrollable. T h e m a t r i x I o w i l t b e s i n g u l a r i fa n o t h e r i m p r o p e r p r o b l e m f o r m u l a t i o n o c c u r s . A s s u m e t h a t t h e m a t r ix C ,d o e s n o t s p a n t h e c o l u m n s p a c e o f t h e m a t r i x A c a . T h a t is , t h e r e e x i s t s an o n z e r o v e c t o r A w h i c h i s a li n e a r c o m b i n a t i o n o f t h e c o l u m n s o f A c n ( th a ti s, A = A c n / 3 ) a n d 0 = C u A . I t f o l l o w s t h a t

c*J - ~ , t~ u i X O f l pS i n c e / 3 # 0 , i f f o l l o w s t h a t t h e m a t r i x I , i s n o t p o s i t i v e d e f i n i t e ; h e n c e , i t iss in g u l ar . T h i s is a m a t h e m a t i c a l s t a t e m e n t o f t h e f a c t t h a t t h e t e r m i n a lc o n s t r a i n t s c a n n o t b e c o n t r o l l e d i n t h e p h y s i c a l p r o b l e m . I t is e m p h a s i z e dt h a t t h e n o n s i n g u l a r i t y o f 1~ o n l y r e q u i r e s t h a t C ~ s p a n t h e c o l u m n s p a c eo f A , a ; i t d o e s n o t r e q u i r e t h a t t h e l i n e a ri z e d p r o b l e m b e c o m p l e t e l yc o n t r o ll a b l e . T h e p r o b l e m m a y c o n ta i n u n c o n t r o l la b l e f u n c t i o n s , b u t t h ec o n s t r a i n t f u n c t i o n s , a r e c o n t r o l la b l e .


10/15

2 3 8 J O T A : V O L . 2 3 , N O . 2 , O C T O B E R 1 9 77

4 . N u m e r i c a l T e s t s fo r F a i lu r e

A s e r i e s o f n u m e r i c a l t e s t s t o d e t e r m i n e t h e e x i s t e n c e o f a n d t oi d e n t i fy t h e f a i lu r e s d i s c u s se d i n t h e p r e v i o u s s e c t i o n a r e g i v e n in T a b l e 1 .T h e s e q u e n c e i n w h i c h t h e t e s t s a r e t o b e p e r f o r m e d i s i n d i c a t e d i n t h en o t e s a t t h e b o t t o m o f t h e t a b l e. T h e o n l y c o m p u t a t i o n r e q u i r e d t o p e r f o r mt h e t e s ts , b u t n o t r e q u i r e d b y t h e s t e e p e s t - d e s c e n t a l g o r it h m , i s t h e d e t e r -m i n a t i o n o f t h e r a n k o f t h e m a t r ic e s A + a a n d [ A , a ! A ~ a ] . I t is i m p e r a t i v e

T a b l e 1. F a i l u r e t e s ts a n d i n t e r p r e t a t i o n .N u m e r i c a l f a i lu r e s C a u s e s o f f a i l u r e N u m e r i c a l t e s t s

( A ) ~ = 0 ( A 1 ) P o o r c h o ic e o f s to p - t O t < < i f a A l l ~ i t + t a , l .p i n g c o n d i ti o n .

T ~ 1(B ) N = I c g , - I c j 4 4 , 1 ~ , 4 , ( B 1 ) C o n v e r g e n c e .= 0

( 1 3 2 ) P e r f o r m a n c e i n d e x i sc o n s t r a i n e d ,

(c ) R 2 - a O ~ I ~ a ~ - < O

( D ) 1 ~ = s i n g u la r

( B 3 ) P e r f o r m a n c e i n d e x i su n c o n t r o l l a b l e .

( B 4 ) O n e o f t h e c a u s e s o fe r r o r l i s t e d a b o v e p l u sn u m e r i c a l e r r o r s w i l lc a u s e N < 0 .

( C 1 ) I n c o m p a t i b l e c h o i c eo f R a n d A ~ .

( D I ) T e r m i n a l c o n s t r a in t s( i n c l u d i n g t h e s t o p p i n gc o n d i t i o n ) a r e n o t i n d e -p e n d e n t .

( D 2 ) T e r m i n a l c o n s t r a i n t sa r e u n c o n t r o l l a b l e.

][q~lt an d re la t ive ch an ge in q5sma l l .

R a n k [ A c a ! A 4 ,a ] ~ p + 1 .N i s c o m p u t e d a s t h e

d i f f e r e n c e o f t w o n o n z e r on e a r l y e q u a l t e r m s . *[ I , e , [ < E a n d II~,,I#,614,,b[ e,

w h e r e e i s a c r i t e r i o n f o rn u m e r i c a l z e r o . N i s c o m -p u t e d a s t h e d i f f e r e n c e o ft w o t e r m s w h i c h b o t ha p p r o a c h z e r o . : )

L o g i c s h o u l d b e i n c l u d e d i n t h ec o m p u t e r p r o g r a m t o c h oo s eR a n d A s o t h a t t h is e r r o rd o e s n o t o c c u r.

R an k [A+ a] _-.'p .*

R a n k [ I ~ ] :3~p. f* T e s t s D 1 a n d B 2 m u s t b e p e r f o r m e d s e q u e n t ia l ly .t F a i l u r e o n t e s t D 2 i n d i c a t e s u n c o n t r o l l a b i l i t y o n l y a f t e r T e s t D 1 i s p a s s ed .:) F a i l u r e i n T e s t B 3 i n d i c a t e s u n c o n t r o l l a b i l i ty o n l y a f t e r T e s t B 2 i s p a s s e d .


11/15

J O T A : V O L . 2 3 , N O . 2 , O C T O B E R 1 9 77 2 3 9

that the numerical routine used to compute Ie7 have a reliable test todetermine if Iq,+ is singular or ill-conditioned; otherwise, improperly for-mulated problems will not be detected.

5. ExamplesFor alI of the examples of steepest-descent algorithm failure, severaioptimization problems based on a single physical system will be used. The

failures shown will be rather obvious in order that the reader can easily seewhat is happening. However, in a complex optimization problem, thecauses for algorithm failure are anything but obvious. Therefore, thecomputer code should always include checks for the various types ofalgorithm failure to alert the user and inform him of the probable cause offailure.

Consider the two-dimensional equations of motion for a constantlythrusting point mass in a constant gravity field. The mass of the pointdecreases at a uniform rate. The single control variable u is the anglebetween the thrust vector and the coordinate xz. The situation is as shownin Fig. 1. The equations of motion for the system are

JCl ~" X 2,i t2 = ( T / x s ) c o s u ,X3 -~" X 4,2 4 = ( T / x s ) s i n u - g ,

F i g . 1 .

3

X iT h r u s t i n g p o i n t m a s s i n a c o n s t a n t g r a v i t y f i e l d .


12/15

2 4 0 J O T A : V O L . 2 3 , N O , 2 , O C T O B E R 1 9 7 7

w h e r e T a n d C a r e th e c o n s t a n ts su c h t h a tT --> 0 a n d C - 0 .

5 . 1 . T y p e ( A ) F a i l u r e . I t is n o t d i ff ic u lt t o c o n s t r u c t a n e x a m p l e o ft h i s t y p e o f f a i l u r e f o r t h e g i v e n s y s t e m ; h o w e v e r , t h e e x a m p l e w h i c h o n ec a n c o n s t r u c t is t r a n s p a r e n t . L e t C = 0 , s o t h a t

a n d l e t

T h u s ,[ ] = X 5 - - X s [ .

(Z-0,a n d a T y p e ( A ) f a i l u r e o c c u r s . T h e r e a d e r s h o u l d n o t e t h a t , i n m o r ec o m p l e x p r o b l e m s , i t is e a s il y p o s s i b le t o i n a d v e r t e n t l y i n d u c e a f a i l u r e o ft h i s t y p e , e s p e c i a l l y if f ~ i n v o l v e s s e v e r a l s t a t e v a r i a b l e s .

5 . 2 . T y p e ( B ) F a i l u r e . F o r a n e x a m p l e o f t h i s t y p e o f f a i lu r e , a tto = 0 le t

X l = X 2 = X 3 -~- X 4 = 0 an d x5 = Xso.L e t t h e s t o p p i n g c o n d i t io n b e

U l = t - 1 ;a n d , a t t h e t e r m i n a l ti m e t i = 1, l e t x l t h r o u g h x4 b e f re e . L e t C > 0 a n dT > 0 . N o w , l e t u s m a x i m i z e

~b = xs (tf ).I n t h i s c a s e ,

A ca ( t l ) = [0 , 0 , 0 , 0 , 1] T .S i n c e a ll o t h e r v a r i a b l e s a r e f r e e a t ts, A , a d o e s n o t e x is t. A f t e r c a r r y i n g o u tt h e a p p r o p r i a t e m a t r i x o p e r a t i o n s , w e a r r i v e a t t h e c o n c l u s i o n t h a t

N =14, = O.I n t h e c a se d e m o n s t r a t e d , t h e p e r f o r m a n c e i n d e x c o u l d n o t b e c o n tr o l le d .T h i s f a c t c o u l d b e d e t e c t e d b y T e s t B 3 ( se e T a b l e 1 ). W i t h s li g h t v a r i a t i o n s ,a n e x a m p l e i n w h i c h t h e p e r f o r m a n c e i n d e x i s c o n s t r a i n e d c a n b e c o n -s t r u c t e d .


13/15

J O T A : V O L . 2 3, N O . 2 , O C F O B E R 1 97 7 2 4 1

5 . 3 . T y p e ((2 ) F a i l u r e . T h i s t y p e o f f a i lu r e c a n n o t b e e x h i b i t e d e a s il yw i t h o u t u s i ng t h e a c tu a l n u m e r i c a l r es u l ts f r o m a c o m p u t e r p r o g r a m . T h er e a d e r i s r e f e r r e d t o M i e l e e t a l . ( R e f . 8 ) , w h e r e l o g ic t o a v o i d t h is t y p e o fe r ro r i s d i s cus sed in de ta i l .

5 . 4 . T y p e ( D ) F a i lu r e . T o d e m o n s t r a t e t hi s t y p e o f f a il u re , l e tXI-~ 'X2"~-~X3~-X4"~O a nd x s = x s o

a t t = t o. L e t t h e s t o p p i n g c o n d i t i o n b e1 2 = t - 1

a n d t h e p e r f o r m a n c e m a t r i x b e x3 (tr). T h e t e r m i n a l c o n s t r a i n t i st ~ = X 5 - X s f.

For th i s case ,

A l s o ,

a n d

= [ 0 , 0 , 0 , 0 , 1 ] .

[ 0 , 0 , 0 , 0 , 1 1

f 0 .Since I ,~ , s t a r t ed ou t w i th a va lue 0 ,

I ~ - - - 0 .T h u s , Io o i s s ingu la r a s was r equ i red in th i s f a i lu re mode . In th i s case , thet e r m i n a l c o n s t r a i n t w a s u n c o n t r o l l a b l e , s i n c e T e s t D 2 w a s f a i l e d a f t e rs u c c e s s fu l c o m p l e t i o n o f T e s t D 1.

6 . Conclus ions

A l l o f t h e p o s s i b l e w a y s i n w h i c h t h e s t e e p e s t - d e s c e n t n u m e r i c a lo p t i m i z a t i o n a l g o r i t h m c a n f a i l h a v e b e e n e n u m e r a t e d a n d a n a l y z e d .F a i l u r e s o f T y p e s ( A ) a n d ( C ) a r e w e l l d o c u m e n t e d a n d c a n e a s i l y b ea v o i d e d b y c a re f u l i m p l e m e n t a t i o n o f th e s t e e p e s t - d e s c e n t a l g o r it h m .F a i l u r e s o f T y p e s ( B ) a n d ( D ) a r e a t t r i b u t e d t o i m p r o p e r p r o b l e m f o r -m u l a t i o n , n o t t o t h e s t e e p e s t - d e s c e n t a l g o r i t h m . H o w e v e r , t h e n u m e r i c a lt e s t s d e s c r i b e d w i l l d e t e c t t h e s e f o r m u l a t i o n e r r o r s , a n d t h e u s e r i s t h u si n f o r m e d w h i c h t y p e o f e r r o r h a s o c c u r r e d . N u m e r i c a l r e s u lt s o f t h e s e t e s ts


14/15

242 JOTA : V O L. 23, NO. 2, OC TO BER 1977

a r e i n t e r p r e t e d i n te r m s o f t h e f o r m u l a t i o n o f t h e p h y s ic a l p r o b l e m b e i n gc o n s i d e r e d . T h e a u t h o r s h a v e f o u n d t h a t t h e i m p l e m e n t a t i o n o f th e s e t e st s( a s s u m m a r i z e d i n T a b l e 1 ) i n a s t e e p e s t - d e s c e n t c o m p u t e r p r o g r a m i sh i g h l y d e s i r a b le . T h e u s e r i s t h u s p r o v i d e d w i t h c ri ti c al i n f o r m a t i o n a t t h et im e w h e n h e m o s t n e e d s i t.

D e s p i t e t h e f a c t t h a t t h e i n d i c a t e d t e st s a re i m p l e m e n t e d , t h e u s e r o ft h e s t e e p e s t - d e s c e n t a l g o r i t h m m u s t c o n s t a n t ly b e a w a r e t h a t t h e a l g o r i t h mis b a s e d o n a l o c a l, li n e a r a p p r o x i m a t i o n o f a n o n l i n e a r s y s t e m a b o u t s o m en o m i n a l t r a j e c t o r y . T h e r e f o r e , i n f o r m a t i o n a v a i l a b l e f r o m t h e l i n e a ra p p r o x i m a t i o n m a y i n d i c a te a c o n d i t io n w h i c h is n o t t r u e f o r t h e n o n l i n e a rs y s t e m . I n a l m o s t e v e r y c a s e, t h e f a i l u r e s d i s c u s se d h e r e c a n b e c a u s e d b ye i t h e r i n h e r e n t p r o b l e m s i n th e n o n l i n e a r s y s t e m o r b y l o c a l c o n d it io n s t h a tr e s u l t f r o m a p a r t i c u l a r n o m i n a l t r a j e c t o r y . I t i s i m p o s s i b l e t o d i s t i n g u i s hb e t w e e n t h e s e t w o c a s e s n u m e r i c a l l y . T h u s , s t a t e m e n t s t h a t c e r t a i n f u n c -t io n s a r e c o n s t ra i n e d , u n c o n t r o l l a b le , o r l in e a r l y d e p e n d e n t m e a n t h a t t h el o c al , l i n e a r a p p r o x i m a t i o n s o f t h e f u n c t i o n s e x h i b it th e i n d i c a t e d c h a r a c -t e r i s t i c . A s a p r a c t i c a l m a t t e r , i t h a s b e e n t h e a u t h o r s ' e x p e r i e n c e t h a t ,w h e n t h e a l g o r i t h m f a il s, t h e f a i l u r e is m o s t o f t e n i n h e r e n t in t h e p r o b l e mf o r m u l a t i o n r a th e r t h a n t h e p a r t i c u la r n o m i n a l t ra j e c t o r y .

References

1. D E N H A M , W. F. , Steepest-Ascen t Solution of Op timal Programm ing Problems,The R ay theon Com pany Repo r t No . BR-2393 , 1963 .2 . BRY SON, A. E . , JR . , and DEN HAM , W. F., A Steepest-Ascent Method forSolving Optimum Programming problems, Journ al of Appl ied M echanics , Vol .84, No. 2, 1962.3. BRY SON, A . E . , JR., D EN HA M , W . F. , and DREY FUS, S. E. , Optimal pro-

gramm ing P roblems with Inequality Constraints, I: Necessary Cond itions forExtrem al Solutions, A IA A Jou rnal , Vol . 1 , No. 11 , 1963.4 . DENH AM, W . F . , and BRYSO N, A. E . JR. , Optimal Programming problemswith Inequality Constraints, II : Solution by Steepest-Ascent, A I A A J o u r n a l ,Vot. 2, No. 1, 1964.5. KELL EY, H . J., Gradient Theory of Optimal Flight Paths, ARS Journal , Vol .30, No. 10, 1960.6 . K E L L E Y , H . J. , M ethod o f Gradients, Opt imizat ion Techniques , Ed i ted by G.Lei tm ann, Academ ic Press, New Y ork , New Yo rk , 1962.7. LONG, R. S., Newton-Raphson Operator; Problems with U ndetermined En dPoints, AIAA Journal , Vol . 3 , No. 5, 1969.8. MIELE, A . , PRITCHAR D, R. E . , an d DA MO ULA KIS, J . N. , SequentialGradient-Restoration Algorithm fo r O ptimal Con trol Prob lems, Journal ofOp timizat ion The ory and Applications, Vol . 5 , No. 4, 1970.


15/15

JOTA: VOL . 23, NO. 2, OCT OBER 1977 243

9 . MIELE, A . , Recent Advances in Gradient Algorithms for Optimal ControlProblems ( S u r ve y P a pe r ) , J ou r na l o f O p t i m i z a t i on T he o r y a nd A pp l i c a t i ons ,\1ol . 17, Nos . 5/6, 1975.10 . Z A D E H , L . A . , a nd D E S O U R, C . A . , Linear System Theory, M c G r a w - H i l lB o o k C o m p a n y , N e w Y o r k , N e w Y o r k , t 9 6 3 .11 . KALMAN, R. E . , HO , Y . C. , and N ARE ND RA, K. S ., Con trollability o f Linea rDyna mical Systems , Cont r ibu t ions to Di f fe ren t i a l Equa t ions , Vol . 1 , No. 2 ,1962.

Documents

Steepest Descent Failure Analysis