the application of the method of steepest descent.pdf

7/30/2019 THE APPLICATION OF THE METHOD OF STEEPEST DESCENT.pdf

1/8

v NO.68-878

THE APPLICATION OF THE METHOD OF STEEPEST DESCENTTO A PURSUIT-EVASION PROBLEMbyR. F. VACHINO, 1. F. SCHAEFER, D. H. deDOES, B . S. MORGAN, JR.,andG. COOKUnited States Air Force AcademyColorado Springs, Colorado

AlAA PaperNO. 0-070

A l A A Guidance. Control. andfliflht Dvnamics conference~I - -~-PASADENA, CALIFORNIA/AUGUST 12-14, 1968

First QYbliCOtioO r ights reserved by A m e r i c an l n r t i t u t e o f A e r o n o u t i c r ond A i t r o n w l i C I , 1290 A v e n u e of t h e A m e r i c a . N e w Y o rk . N. Y. 1W19.Abs trac ts moy be p u b l i s h e d wi thout Perm i r$ iOn i f c red i t i s g ive n l o a u th o r a n d Io A I A A . (Price: A l A A Member 11.00. N o n m e m b e r 11.501


2/8

THE APPL I CATI ON OF THE METHOD OF STEEPEST DOSCENTTo A PURSUI T- EVASI ON PROBLEMR F. Vachi no, " J . F. Schnef er; * DH deDoes, * BS Mor gan, Jr.,** G Cook"Fr ank J. Sei l cr Resear ch Labor ator y ( OAR)USAF Academy, Col or ado 80840

I. Abst r actPresented ar e the resul t s of the appl i cat i on oft he met hod of st eepest descent t o an engagement bc-t ween a beam r i di ng m ss i l e and an ai rc ra f t . Soughti s the opt i mum f l i ght path of the ai rc r af t f or i tto evade the m ssi l e. The opt i m zat i on cr i t er i oni s t he maxi m zati on of t he di st ance of c l oscst ap-pr oach bet ween t he m ssi l e and the ai r craf t . Asf ormul at ed, t he probl em i s l i near i n t he contr oland has a t wo- s i ded i nequal i t y const r a i nt on t hecont ro l . Two appr oaches ar e t aken to det er m ne t heopt i mum evasi ve f l i ght path. The f i r st start s wi t h

an unconst r ai ned contr ol and i t erat es on t hi s con-t r ol unt i l i t becomes sat ur ated, af t erwards t reat -i ng t he swi t chi ng t i mes as opt i m zati on parameters.The second st art s wi t h a constr ai ned cont r ol andsi mpl y i nvol ves i t erat i ons of t he swi t chi ng t i mes.Bot h appr oaches yi el ded t he Same r esul t s.

11. Pr obl emSt atementThe pr obl em chosen to r epresent t he engagementof a m s s i l e an d an ai r c r af t i s a s i mpl i f i ed vers i onof an act ual m s si l e- a i r c r af t encounter and a s s u m e st h at t h e ai r c r a f t i s f l yi ng i n t he vi c i ni t y o f asur f ace- t o- ai r beam r i di ng m ss i l e and i t s t r acki ngr adar. At some gi ven t i me a m s si l e i s f i r ed a t t heai r c r af t , The opt i mum evasi on probl em i s to det er-m ne the evas i ve f l i ght path that the ai rc r af tstroul d sel ect i n order to maxi m ze i t s m ni mum di s-t ance f r omt he m ssi l e. The si mpl i f i ed t wo di men-si onal model assumes t hat bot h t he ai r cr af t and m s-si l e move wi t h constant speeds; t hat t hei r mot i on i sconf i ned to a pl ane whi ch contai ns bot h t he t r acki ng

wher e R i s t he di st ance al ong the radar beam f r omthe r adar to t he a i r craf t , 0 i s t he angul ar or i en-t at i on of the r adar beam r el at i ve to t he hor i zont al ,a i s the di rec t i on o f t he ai rc raf t ' s vel oc i ty vec-tor wi t h t he hor i zont al , V i s t he vel oc i t y of t heai r c r a f t , V i s t he vel oc i e y of t he m s s i l e, i st he angl e o? the mss i l e' s vel oc i ty vec to r f rom thehor i zont al , and u i s t he cont r ol whi ch produces t hef l i ght path of the ai rc raf t .

The choi ce of t he contr ol l aw, (4), to contr olt he m ssi l e to f ol l ow t he r adar beam docs not con-s i der g- cons t r ai nt s on the tu rni ng abi l i t y of t hem s si l e. llowever, i n or der t o make t he engagcmentmor e r e a l i s t k , i t i s des i r a bl e t o i mpos e a con-s t r ai nt on the t urni ng rat e of t he ai r craf t . Tni sc ons t r a i nt . i s on t he r ate of change of the f l i ghtpat h angl e, a and i s denoted byl a / O f or a11 t

[ t o , t 1, t hen t he m ni mum of occurs at the samet i me as t he m ni mum of D. The t i me t i s assumed tobe a known f i nal t i me, denoti ng the endurance o f t hem s si l e.f f

111. Transf ormati on To Mul t i st age ProcessThe opt i m zat i on c r i t e r i on i s not of the form

( I + )( 5 )

X Resear ch Ass oci at e, Aer ospace Mechani cs Di vi si on, FJ SRL ( OAR)* X Di r ect or, Aerospace Mechani cs Di vi si on, FJ SRL ( OAR)* * * Assi st ant Prof essor of Aer onauti cs and Comput er Sci ence, A C K O S ~ ~ C ~echani cs Di v i s i on, FJSRL ( OAR)

1


3/8

8L8 8 9 z


4/8

j o i n t v a r i a b le s can be w i t t e n asy = vT(xs-xs)cos(xs-xs)3 5 2

+ 7vMx;(x;-x,)4 2 1-x3)(tl)= X2 (tl) M ( t l ) (21) S- 2v ( x 1 - x 3 ) = 0.

( 2 0 )TX = - F Xs, t E T s , s = 1 , 2 .5 "A t t h e t i m e t = t l t h e a d j o i n t v a r i a b l e s are d iscon-

t i n u o u s and s a t i s f y

Wwhere M s s

1 3 . r 25 2 5 2 1 o o o 2vT(xS-x )sin(x'-x )Tsin(xs-xS) -V cos(x -x )/x s ss s s- 2 v T ( x ~ ~ " x ~ - x ~ ) c ~ s ( x5 2x ) / x 1s s s

0 0 0 0 0 0-

1 Us in g t h e p ro p e r t i e s of a d j o i n t s and r e f e l - r i n gr t ,

(22) to Vachino [Z], n e o b t a i i sM ( t l ) = [I + ( f2 - f l ) :lwhere Y i s g iven by (16). In (19) through (22) d~ i T ( t f )m N ( t f ) (,y1 tO + ,J < w d r ( : S Is u b s c r i p t s on t h e vector3 d e n o t e stages.t h e n a t e v e ry s wi t c h i n g t i m e, t ' , t h e s t a t e varia-tion w i l l obey a n e q u a t i o n

0

Ul t i m a t e l y , when t h e c o n t ro l i s a l l b an g-ba ng , whe ne ve r t h e c o n t ro l i s n o t s a t u ra t e d , andTh ( f f ) ?SW,( t f )= ( X 5 % )

( 7 7 )I to6x (ti)= 6x (tl) + ( f - - f+ l t , d t ' , (23) + [h 5 ( f5 - - f5 f ) l d t '+2vT( x3 s ) i n ( x~ - x s ) s i n ( x~ - x~ )4 /

The m a t r i c e s and v e c t o r s t h a t appear i n t h e pre- I t 'c e d i n g e q u a t i o n s a r e g iven by -

r 9vT i n (x 2 -x z ) / (x g )2 o o o 2(vM-vTcos(x'--X~s )

0

0 0 0 0

(74) when the' co n t ro l i s s a t u ra te d , and t he opt imumswit chi ng t imes must be determined. I f o n e choosesl o

TG = [ O 0 0 0 1 01, s = 1 , 2 -t h e t e m i n n l c o n d i ti o n s( f - - f f ) T I t , = [ O 0 0 0 2 2Gmax 01 (: 5 )


5/8

t h e n one can r e l a t e (26) an d ( 2 7 ) t o v a r i a t i o n s o ft h e c o s t f u n c t i o n . There i s np c o n t r i b u t i o n t o d 0caused by changes i n t f s i n c e 0( t , ) i s zero .

v B e fo r e a p p l y i n g t h e pr e c e d i ng r e s u l t s on e mays i mp l i f y Some o f the opera t ion s . For example , hmay he n e g l e c t e d b e c a u s e i t d o e s n o t c o u p l e i n t o 3t h e o t h e r e q u a t i o n s . A l s o , from (2 2 ) one n o t e st h a t M ( t ) i s t h e i d e n t i f y m a tr i x s i n ce f ( t ) =f ( t ). 'Th is means tha t ther e i s no jump in the ad-j h n & v a r i a b l e s . I f c o n d it i o n y = 0 were evers a t i s f i e d ( s e v e r a l minima occurrpng), t h e f ' s wouldn o t b e equal and t he re would be a jump i n the ad-j o i n t s.

2. 1

V . An I l l u s t r a t i v e Exa mp leA s an example, an engagement beginning a t 20,000f e e t s l a n t r an ge w a s c o n s i d e re d . The t a r g e t v el oc -i t y (VA) w a s 1000 f e e t per s ec a nd t h e m i s d i l e v e -

l o c i t y (V ) was 2500 f e e t p e r s e c . The e l e v a t i o no f t h e ragar beam was 30' and t h e t a r g e t v e l o c i t yd i r e c t i o n was 120". The nominal ev asi ve maneuveri s shown by a dashed l i n e i n F i g u re 4 a n d c o n s i s t sof a 4g tu rn i n t h e c l o c k w i s e d i r e c t i o n f o ll o we d b ya 4g t u r n i n a c o u n t er c l o c k w is e d i r e c t i o n .t i a l m i s s i le f l i g h t pa th i s shown by a d o t te d l i n ei n F i g u re 4. A f t e r 110 i t e r a t i o n s o f t h e m eth od o fs t e e p e s t d e sc e n t t h e f l i g h t p a th of t he a i r c r a f th a s be en s i g n i f i c a n t l y a l t e r e d a nd i s shown by t heh ea vy l i n e i n F i g u re 4. T hi s f l i g h t p a t h c o n s i s t sof a 5g t u r n t oward t h e m i s s i l e (a t u r n i n ac o u n t e rc l o c k wi s e d i r e c t i o n ) fo l l owe d by a 59 t u r naway from th e mi ss i l e. The accompanying Table 1p ro v i d e s Some o f t h e S i g n i f i c a n t n u m er i ca l d a t a f o rs e v e r a l i t e r a t i o n s o f t h e co n ve r ge n ce process. The5g c h a r a c t e r i s t i c o f t h e two t u r n s r e f l e c t s t h e e f -c r a f t . The data shown. i n Table 1r e p r e s e n t a b o u t10 minutes o f computer t ime on a Burroughs 85500d i g i t a l c om pu te r. P h y s i c a l l y , t h e r e s u l t i n g f i n a lmaneuver i s r e a d i l y u n d e r s t a n d a b l e an d can be ra -t i o n a l i z e d f r o m a knowledge of t he mathematica lmodel given by (1 ) through ( 5 ) . F r w F i g u re 4 i tca n be seen t h a t t h e i n i t i a l p o r t i o n o f t h e no mi na la i r c r a f t f l i g h t pa th i s n e a r l y p e r p e n d i c u la r t o t h er a d a r beam. This c a u s e s t h e r a d a r beam t o r o t a t er a p i d l y w i th t h e m i s s i l e l a g gi n g t h i s a n gu l ar r o t a -t i o n by a n i n c re a s i n g a mo un t. J u s t b e fo re t h e m i s -s i l e closes i n on t h e a i r c r a f t , t h e n om in al f l i g h tp a th of t h e a i r c r a f t i s v e r y n e a r l y a long t h e r a d a rbeam. Consequently the rad ar beam slows down i t sr o t a t i o n and t h e m i s s i l e r e d u c e s i t s l a g s o t h a t a tt h e p o i nt o f . c l o s e s t ap pr oa ch t h e m i s s i l e l a g i s o ft h e o rd e r o f t h e m i s s d i s t a n c e . F o r an o p t i m a lf l i g h t p a t h t h e a i r c r a f t ch oo se s one which causest h e r a d a r beam t o ro t a t e v e ry r a p i d l y . The m i s s i l ethen lags the r ad ar beam by a gr ea t amount. Thena t t h e a p p r o p r i a t e t i me t h e a i r c r a f t s u dd e nl y re-v e rs e s i t s maneuver a nd t h e m i s s i l e i s u n a b l e t oreduce the lag . F i g u re 5 shows a comparison oft h i s m i s s i l e l a g f o r t h e n om in al a nd o p t i ma l f l i g h tp a t h s . For t h e s e f l i g h t p a th s t h e m i s s d i s t a n c ec o r r e s p o n d s t o t h e d i s t a n c e be twe en t h e p o i n t s Nand n a n d p o i n t s 0 and o i n F i g u r e 5. Figure 6shows t h e a i r c r a f t and m i s s i l e a c c e l e r a t i o n s c a u s e dby these maneuvers. From thes e f i gur es and th ed a t a o f Table 1it! c a n b e seen t h a t f o r t h e n o mi na lf l i g h t p at h t h e m i s s i l e i s p u l l i n g Tg a t t h e p o i n to f c l o s e s t a p p ro ac h . 'This p o i n t occurs a t 11.1

The ini-

-d f e c t of p l a c i n g a maximum g constraint on t h e a i r -

4

seconds wi th a c o r r e s p o n d i n g m i s s d i s t a n c e o f 91f e e t . F o r t h e optimum f l i g h t p a t h ( t h e one gener-a t e d by t h e 40 th i t e r a t i o n ) t h e m i s s i l e i s p u l l i n g9 .7 6 a t t h e p o i n t o f c l o s e s t ap pr oa c h. T ha t p o i n to c c u r s a t 7.3 seconds and the m i s s d i s t a n c e i s 991f e e t .

F i g u r e 7 s ho ws t h e e v o l u t i o n o f t h e c o n t ro lf u n c ti o n d u r i ng t h e i t e r a t i v e process . I t ca n ben ot ed t h a t t h e s u c ce s si v e i t e r a t i o n s r e s u l t i n ag r a du a l ch an ge o f t h e i n i t i a l -4 9 t u r n o f t h e t a r -g e t t o a t 5 g t u r n t ow ar d t h e m i s s i l e . T h i s t u r n i sfol lowed by an unc onst rai ned maneuver, which grad-u a l l y d e v el o ps i n t o a -5g tu rn away from th e m i s -s i l e j u s t p r i o r t o i nt e r c ep t . I t can also be notedt h a t t h e u n c o n s t r a i n e d m an eu ve r fo l l o wi n g t h e t 5 gt u r n becomes g r a d u a l l y s h o r t e r an d s t e e p e r , t e n d in gt o become v e r t i c a l , a nd t e n d i n g t o a c t as n swi tch -i n g t i me . A l t h o rg h Tab le 1 and Figures It, 5 an d 6show r e s u l t s f o r o n l y 40 i t e r a t i o n s , t ile i t e r a t i o np r o c e s s was a l lowed t o c o n t i n u e f o r 80 i t e r a t i o n sa s i s ev idenced b y F i g u r e 7. On t h e 80 t h i t e r a t i o nt h e p r o c es s o f i t e r a t i n g on t h e u n c o n s t r a i n e d p o r -t i o n of t h e c o n t r o l wa s d iscon t inued , and a bang-bang con t r o l wi th a number o f swi t ch in g t imes w a ss u b s t i t u e d f o r u e0 ( t ) .

The i t e r a t i v e p r oc e ss w a s con t inu ed wi th twod i f f e r e n t b an g- ba ng c o n t r o l s ; b o t h o f t h e s e con-s i s t e d o f a l t e r n a t i n g 5g t u r n s s t a r t i n g wi t h ac o u n t e rc l o c k wi s e t u rn . One c o n t r o l h ad f i v e s ub -arcs and t h e s e t o f s w i t c h i n g t i m e s w a s s e l e c t e d a s

= (4 .5 , 6.25, 6.5, 7.0); t h e 0 t h- c o n t r o l con-s i s t e d of f o u r subarcs wi t h t h e s w i t c h i n g t i m e si = (3 .0 , 4.5, 50. he i t e r a t i v e process w a st he n r e s t a r t e d , w i t h t h e aim of f i n d i n g t h e v a r i a -t i o n s i n t h e s w i t c hi n g ti m es t h a t c o n t r i b u t e d t o af u r t h e r i n c r e a s e o f t h e i n t e r c e p t d i s t a n c e . I neach case thes e bang-bang co n t ro l s converged towardt h e t wo -s ub arc o pt im um c o n t r o l d e p i c t e d i n F i g u re6b .t h e f i r s t o f t k s e ba ng-ba ng c o n t r o l s an d t h e s h o r ti t e r a t i v e p r o c e s s l e a d i n g t o t h e o ptimum c o n t r o l .

Table 2 s um ma ri ze s t h e s i g n i f i c a n t d a t a f o r

V I . De t a i l s o f C o m p ut a t i on a l P ro c e du reThe d a t a t a b u l a t e d i n Tab le 1and d i sp layed

g r a p h i c a l l y i n t h e p r e c e d in g f i g u r e s was o b t a i n e dfrom 8 d i g i t a l c om pu te r s i m u l a t i o n o f (2 6 ) an d (27).The v a r i a t i o n cu( t ) was s e l e c t e d as

w ( t ) = K x , ( t )f o r a l l t E [ t o , t f ] when u ( t ) was u n c o n s t r a i n e d ,a nd t h e c o n s t a n t K was e v a l u a t e d e x p e r i m e n t a l l ywi t h t h e a i m t o p ro d u c e p re d i c t a b l e i mp ro ve me nt s i nt h e c o s t f u nc t io n . The scheme that w a s used s u c -c e s s f u l l y i n a l t e r i n g t h e u n c on s tr a i ne d c o n t r o lf u n c t i o n w a s t o u s e an a r b i t r a r i l y s e l e c te d c ha ng e& = 25 f t a nd t o c h oo s e K a s

25K = ,.

The new c o n t r o l fu n c t i o n wa s t h e n d e f i n e d a s".I t i s ne ce ss ar y t o r e i t e r a t e a t t h i s p oi nt t h a t

( t ) = s a t [U i ( t ) + W i ( t ) l , i = 1, 2 , ...f1b e c a us e t h e p rob l e m c o n s i d e r e d i n t h e paper i sl i nea r i n t h e c o n t r o l a nd because t h e c o n t r o l i ss u b j e c t t o a t wo -s id ed i n e q u a l i t y c o n s t r a i n t , one


6/8

can i mmedi at el y concl ude t hat t he opti mumcont r oli s bang- bang, unl ess s i ngul ar cont r o l s ar i se , andt hat t he opt i mumevasi ve maneuver i s composed ofmaxi mum g turns. An appr oach si m l ar to that ofLei t mann [ 3 ] , ai med at deduci ng t he number andorder i ng of subarcs o f t he opti mumcont r ol was at -t empt ed, but wi t h l i t t l e success . From t hi s ap-proach i t i s al so not possi bl e to deduce the pr es-ence of si ngul ar s ubarcs ; consequent l y t he presents ol ut i on of t he evasi on probl em has been obt ai nedst r i c t l y by the appl i cat i on of t he general i zed al -gori t hm The presence of si ngul ar subarcs s e e m svery real f or cer t a i n other sets of i ni t i al c ondi -t i ons , but wer e not experi enced i n the presentst udy. The cl ear conver gence of t he dat a of Tabl es1and 2 l eads to t he concl usi on t hat f or t he pre-sent set of i ni t i al condi t i ons s i ngul ar subarcs donot exi st .

VI I . Refer ences1. Br yson, A E. , et al , " Opt i mal P r o g r a m n gProbl ems W t h I nequal i t y Const r ai nt s" , Par t I ,"Necessary Condi t i ons f or Extr ema1 Sol uti ons",AI AA J ournal , Nov. 1963, p. 2544; Par t 11,"Sol ut i on by St eepest Ascent ", AI AA J ournal ,J an. 1964, p 25.2. Vachi no, R F., " A Gener al i zed St eepest DescentAl gor i t h m f o r Mul t i st age Opt i m zat i on Probl ems" ,Sei l er Resear ch Laborator y Report 68-0002,FJ SRL( OAR) , USAF Academy, Col o.3 Lei t mann, G, Opt i m zat i on Techni ques, Academcpress, N Y., 1962.

Tabl e 1I t e r at i ve Sol ut i on f or an Unconst r ai ned Contr ol-M s s Maxi mum Maxi mumI t era t i on Ti me Di s tance Ai rc r af t M ss i l e

sec f t g ' s g ' Li

Tabl e 2-t e r at i ve Sol ut i on of t he Swi t chi np Ti mestera- I nt ercept I nt ercept Swi t chi ng Ti mcst i on Ti me Di st ance t l . t 2 tsec f t

F i gur e 1 - Evasi on Geomet r y

954.20 4.50 6. 25 6. 50 '1.07.277 716 905.80 4.80 6.37 637 '?.;>

, -1 t fF i gur e 2 - Hypothet i cal Funct i ons

5


7/8

A i r c r a f t N om in alF l i g h t P a th\

\

d-r0000 20000H o r i z o n t a l D i s t a nc e i n F e e tF i g u r e 4 - A i r c r a f t and Missi le F l i g h t P a th s

F i g u re 3 - 'Typical C o n t r o l and Di s t a n c e F u n c t i o n s

Radar L ine-o f-Sigh t Dis ta nce in F e e t0 5000 10000 15000 20000 25000 N- .

Ai rc ra f t No m i n a lF l i g h t P a th.- 50 0

Miss i le Lag fo rF l i g h t P a t h

LEGEM A i r c r a f t O pt im al. . l oo0-Nn Missi le P o s i t i o n f o r Nom ina l F l i g h t P a t h

A i r c r a f t P o s i t i o n f o r Nom ina l F l i g h t P a th

M i s s i l e 0 A i r c r a f t P o s i t i o n f o r O pt im al F l i g h t P a t h~ a gn o Missile P o s i t i o n f o r O pt im al F l i g h t P a t hF e e t ~

F i g u r e 5 - M i s s i l e P o s i t i o n R e l a t i v e t o t h e R ad a r Li n e- o f- S ig h t

6 68 - 818


8/8

rl: / I /B0 in s ec2 - 2 .

Aircra ft Nominal Acceleration- 4 .-

Missile Optimum Acceleration0 18

M 6

- 2

Aircraft Optimal AccelerationF i g u r e 6 - Aircraft and Missile Accelerations

Figure 7 - Trajectories in Control Space7

U

Seconds

the application of the method of steepest descent.pdf

Documents