A Method of Trajectory Optimization by Fast-time Repetitive Computations

7/30/2019 A Method of Trajectory Optimization by Fast-time Repetitive Computations

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N A S A T E C H N IC A L N O T E F_- d -D-3404A S A T N

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A METHOD OF TRAJECTORYOPTIMIZATION BY FAST-TIMEREPETITIVE COMPUTATIONSby Rodney C . Wingrove,James S. Ruby, and D. FrancisAmes Keseurch CenterM offe tt Field, Cali$

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TECH LIBRARY KAFB, NMIIllill11111ullI11111111Ill1110130190 NASA TN D-3404

A METHOD O F TR AJECTOR Y OPTIMIZATION BY FAST-TIMEREPETITIVE COMPUTATIONS

By Rodney C. Wingrove, James S. Raby,and D. F ran cis CraneAmes Research Cente rMoffett Fi eld, Calif.

NA TIONAL AERONAUTICS AND SPACE ADMINISTRATIONFor sale by the Clearingh ouse for Federal Scienti f ic and Techni cal Information

Springfield, Virginia 22151 - Pr ice $0.30


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A METHOD O F TRAJECTORY OPTIMIZATION BY FAST-TIMERFPETITIVE COMmTTATIONS

By Rodney C . Wingrove, James S . Raby,and D. Franc i s CraneAmes Research Center

SUMMARY

T h is r e p o r t p r e s e n t s a p e r t u r b a t i o n method o f co mp utin g o p t i mt r a j e c t o r i e s w he re in c o n t r o l i mp uls e r es po n se f u n c t i o n s a re determined byf a s t - t i m e r e pe t i t i ve c om put at ions o f t h e s t a t e equ at io ns . This method doesn ot r e q u i r e t h e s o l u t i o n of t h e a u x i l i a r y se t o f a d j o i n t e q u at i o n s u s e d w i tho t he r pe r t u r ba t i o n met hods .The mechanization of t h i s computing method on a hybr id computer i s d i s

cussed and an a p p l i c a t i o n t o t h e s t e e p e s t d es ce n t o p ti m i za t i on o f r e e n t r yt r a j e c t o r i e s i s pr e s e n t e d . I n t h i s e xa mple , t he ve h i c l e i s t o a r r i v e a t ad e s i r e d r an g e, t h e h e a t i n p u t t o t h e v e h i c l e i s t o be minimized, and th e cont r o l i s t o r em ain w i t h i n s p e c i f i e d c o n s t r a i n t s . Gptii-mm t r a j e c t o r i e s f o r -t h i s exam ple c ou l d be ob t a i ne d i n a bou t 8 minutes of computing time.INTRODUCTION

It i s i mp o rt an t f o r s pa ce v e h i c l e t r a j e c t o r i e s t o b e n e a r o ptimum i n t h esense tha t some quant i ty i s e i th e r maximized or minimized. For example, i nr e e n t r y t h e t r a i e c t o r y t o d e s i r e d te rm i n al c o nd i ti o ns i s near optimum whent he t o t a l a e r odyna m i c he a t i ng i s a minimum. Th is r e p o r t w i l l cons ide r am ethod f o r f i n d i n g t he Lime h i s t J r i e s o f non l i ne a r c o n t r o l s t h a t c o rr es pondt o optimum t ra . i e c to r ie s . Sev e ra l pe r t ur ba t i on methods , such as t h e c a l c u l u sof v a r i a t b n s , a p p l i c a t i on s o f I he maximum p r i nc i p l e , a nd d i r e c t s t e e pe s tde s ce n t , have be e n c ons i de re d f o r s o l v i ng t h i s c on t r o l op t i m i z a t ion p robl em .Reference 1 c on t a i n s a good review of t he se var i ous methods and ref ere nc e 2gives se ve ra l ana log and d i g i t a l comput ing techniques f o r implement ing them.I n p r i n c i p l e , e ac h o f t h e s e t ec hn iq u es s ho u ld g i ve s a t i s f a c t o r y r e s u l t s ,b u t it has been found th a t f o r many t r a j e c t o r y problems th e computer mechaniz a t i o n s a re cumbersome and r eq u ir e programs t h a t a re d i f f i c u l t f o r e n gi ne er st o formu la te . The comput ing method t o be repo r te d her e i n w a s i n v e s t i g at e d i n

an a tt em pt t o a l l e v i a t e t h e s e d i f f i c u l t i e s and t o p ro vi de a more d i r e c t wayof comput ing opt imiz a t io n so lu t ion s .I n p re v io u s o p t i m iz a t io n s t u d i e s u s i n g p e r t u r b a t i o n t e ch n iq u e s t h ecomputa tions have involv ed th e dynamic sol ut io n of two s e t s of equa t ions :

(1) non l i ne a r s t a t e equa t ions and (2) l i n e a r a d j o i n t e q u a t i o n s . The methodt o b e r e p o rt ed h e r ei n d i f f e r s i n t h a t o n ly t h e s o l u t i o n of t h e n o n li ne ar s t a t e


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equa t ions i s used . The r esponse o f g iven func t ion s ( e .g . , t e rmina l e r ro r o rq u a n t i t y t o b e o p ti m iz e d) t o a con t ro l impu l se i s d e t e r m i n e d a l o n g t h e t r a j e c t o r y b y fas t - t ime r e p e t i t i v e c om pu ta ti on s r a t h e r t h a n by a s o l u t i o n of t h ea d j o i n t e q u at io n s . S in c e a u x i l i a r y a d j o i n t e q u at i on s are no t needed, th ei n v e s t i g a t o r s h ou l d u n d e r s ta n d t h e o p t i m i z a t i o n p r o c e ss m ore e a s i l y ; a l s o t h ecomputer program should be sim ple r . However, t h i s new a l te r n a te computingmethod does re qu i r e many sol u t io ns of th e s t a t e equa t ions . This t a s k of comp u t i n g a la rg e number of dynamic so lu t i on s i s i d e a l l y s u i t e d t o h igh - speedr e p e t i t i v e h y b r i d c o mp ut at io n as w i l l be cons ide red he re in .T h i s r e p o r t w i l l p r e s e n t o n e a p p l i c a t i o n o f t h i s computing t echn ique ; t h a tof t r a j e c t s r y o p t i m i z a ti o n u s in g t h e s t e e p e s t d e s c e n t m ethod ( r e f s . 3 and 4 ) .The mechanization of t h i s me thod on a hybrid computer w i l l be d i scussed andr e s u l t s w i l l b e p r e s e n t e d t o i l l u s t r a t e t he u se of t h i s p ro ce du re i n t h eo p ti m iz a ti o n o f r e e n t r y t r a j e c t o r i e s . For t h e i n t e r e s t e d re a d e r a p pe nd ix Ai l l u s t r a t e s t h e r e l a t i o n s h i p o f t h e i mp ul se r es p o ns e f u n c t io n s c omputed i n t h i sr e p o r t t o t h e s o l u t i o n s o b t a in e d w i t h t h e a d j o i n t e q u a t i o n s a nd t o t h e maximumpr in c i p l e of op t imiza t ion . Th i s appendix a l so p rov ides a background f o r under s t a n d in g t h e s t e e p e s t d e s c e nt o p t i m i z a ti o n e q u a t i o n s.

NOTATION

The fo l lowing no ta t ion i s u s e d i n t h e b od y o f t h e t e x t . Add i t iona lsymbol s a r e d e s c r i b e d as t h e y are i n t roduced i n th e appendixes .-D c o n t r o l value of l i f t - d r a g r a t i on number o f s to r age po i n t s i n co n t r o l t i m e h i s t o r yt t imet f f i n a l t iniet o i n i t i a l t imeAt t i m e increment of c on t r o l impulseU ccritr:1 f u n c t i o nAU c o n t m l i m pu ls e

? c o s t f u n c t i o n a t f i n a l t im e& ( t ) change i n c o s t f u n c t i o n a t f i n a l time due t o con t r a1 impul ses a ttime t$ s t a t e v a l u e a t f i n a l time

2

L


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d e s i r e d s t a t e value a t f i n a l t i m eA\I ' ( t ) change i n s t a t e value a t f i n a l t i m e due t o co n t ro l impul se s a t t i m e t

The method of s t ee pe s t descen t ( r e f s . 3 and 4 ) i s a n i t e r a t i v e p r oc ed ur et h a t has been used f or optimiz ing t r a j e c t o r i e s . The pro ces s commences withany nonoptimal t r a j ec to ry f rom which a s l i g h t l y improved one i s derived. Theimproved t r a j ec to ry i s t hen u sed as a new nominal t ra je c t or y , and th e p rocedurei s r epea ted u n t i l t he optimum or nea r ly optimum t r a j e c t o r y i s found.Genera l Out l ine

T h e i t e r a t i o n i s as fo l low s : (1)E s t i m a t e a reasonable p rogram tha tn e a r l y s a t i s f i e s t h e t e r m i n a l c o n d i ti o n s f or s p e c i f i ed i n i t i a l c on di ti on s;( 2 ) de te rmine impul se r e sponse func t ions t h a t de sc r ibe th e e f f e c t s of s m a l lc ha ng es i n t h e c o n t r o l on t h e t e r m i n a l s t a t e and on t h e c o s t ( t h e q u a n t i t y t obe minimized) . These impulse respons e fun ct i ons , combined wi th s te ep es tdescen t computa tions , in d i ca te the be s t p Js s i b l e way of making s m a l l changest o t h e c o n t r o l t o d e cr ea se t h e c o s t a nd s t i l l a r r i v e a t t h e e nd p o i n t ; (3) addt h i s change i n co n t r o l t o th e prev ious nominal c s n t ro l p rogram. The r e s u l t i sa new t r a j ec to ry .w i th a dec rea sed cos t ; ( 4 ) r e pe a t t h i s p r oc es s u n t i l t h e r ee x i s t s o n l y a very s m a l l ch ange i n c o s t f o r e a ch new t r a j e c t o r y , i n d i c a t i n gt h a t t h e c o n t ro l i s very near a l o c a l optimum. A l i m i t va lue o f t he con t ro lmay be reached be for e th e c o st i s comple te ly min imized . I n t h i s c a se , t h ep roces s i s c on ti nu ed u n t i l t h e c o n s t r a i n t ( c o n t r o l l i m i t ) i s reached, s i n c eno f u r t h e r o p t i m i z at i o n i s p o s s i b l e .

T he p rope r t i e s of st ee pe st descent opt i miz at io n have been documented i nmany prev iaus s tud ies (e .g . , r e f s . 5 - 7 ) . Although t h i s meth.;d has beenregarded as t h e m ost p r a c t i c a l i n many a p p l i c a t i o n s , t h e r e i s no guaranteet h a t it yi e l ds th e abso lu te optimum. That i s , f o r some i n i t i a l c h oi c es of t h en om in al t r a j e c t o r y , t h e f i n a l op t im i z ed t r a j e c t o r y may r e p r e s e n t o n l y a l o c a loptimum path . Also, i n some ap pl ic at io ns , where th e co st fun ct i on may ber e l a t i v e l y i n s e n s i t i v e t o c o n t r o l v a r ia t io n s , a la rg e number of i t e r a t i o n s maybe necessar y t o approach th e optimum so lu t i on .Computation of Impulse RespJnse Fun ctio ns

To i l l u s t r a t e the computat ion o f t h e impul se r esponse func t ions l e t t hequa n t i ty t o be min imized be noted as cp, t h e c s s t e v a l ua te d a t t he f i n a l t i m e .L e t t h e s t a t e var iable a t t h e f i n a l t i m e be noted as \I' and l e t t h e d e s i r e de n d- p oi n t v a l u e f o r t h i s b e d e no te d \I'd.

F igu re 1 i l l u s t r a t e s th e manner i n which the in f l uenc e of s m a l l c o n t r o lchanges on cp and \I' are c a l c u l a t e d i n t h i s r e p o r t . The e q u a ti o n s o f m oti onare f i r s t so lved w i th a co n t ro l change , a p o s i t i v e c o n t r o l impulse a t t i m e t,3


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super imposed upon th e nominal co nt ro l . During the next so lu t io n of the mot ione qua t i ons , a nega t ive cont ro l impulse o f t h e same magnitude i s i n s e r t e d a t t i m et . The change i n co st , 4, and th e change i n t e rm ina l s t a t e , A@ are der ivedf ro m t h e s e t w o s o l u t i o n s . I n a s im i la r manner t he impulse re sponse func t ion scan be pro gress ive ly de termined a t s uc c e s s ive t i m es a long t h e t r a j e c t o r y , andthe t echnique by which they a re de te rmined i s t h e most i m por ta n t f e a t u r e o ft h i s computing method. The computation of t h e f u l l h i s t o r y o f @ ( t )an d A$ ( t )f o r th e same co nt ro l impulse a t d i f f e r e n t t i m e s a l o n g t h e t r a j e c t o r y i s termedone " i t e r a t i o n " s i n c e i t corresponds t o t he p r e v i ous op t i m i z a t i on s t ud i e s w he r eone i t e r a t i o n w it h t h e a d j o i n t e q u a ti o n s w a s us e d t o com pute e s s e n t i a l l y t he s esame fun ct i on s a long th e t r a j ec to ry . Appendix A shows t he r e l a t i o ns h i p of t h i se xpe ri m e n ta l de t e r m i na t i on t o t he s t a nda r d de t e r m i na ti on us i ng a d j o i n t e qua t i o ns . These expe r imen ta l impulse re sponse func t io ns a re shown t o b e d i r e c t l yr e l a t e d t o th e wel l-known "Green' s f un ct i on s . '

Ste epe s t Descent Opt im iza t ion Equat ion sThe impulse re sponse func t ions a re used i n the s te ep es t descent t echnique

t o modify the con t ro l toward th e optimum i n th e fo l lowing manner (see append i x A ) .

The gains Q and Kq a r e c o n s t a n t s f o r e a c h i t e r a t i o n . The ga i n % weightsthe impulse re sponse func t ion f o r t h e c o s t ; i t s s i g n i s n e g at i v e i n o r d e r t odecrease th e co s t . The magni tude of % i s de termined expe r imen ta l ly f o r eachproblem. Too l a r g e a ga i n may cause in s t a b i l i t y i n the convergence procedure ,whi le too s m a l l a ga i n may extend the t ime of convergence.

The gain K q must be c a l c u l a t e d f o r e a ch i t e r a t i o n s o t h a t t h e t e r mKq Aq(t) w i l l account for t e rmina l d i sp lacement due t o the op t imiz ing te rm,Kcp & p ( t ) , and any te rmina l d i sp lacement e r ror f r o m t h e p r ev i ou s i t e r a t i o n w i l lbe cor rec ted . The form ula t io n f o r c a l c u l a t i n g K,J i s as f o l l ow s :S m a l l changes, SJr, i n t h e t e rm i na l s t a t e , $, due t o s m a l l c ha nge s , 6u ( t ) ,i n co nt ro l can be approx.imated by:

64f= 2 A A t J ~ 6u( t )AJr(t)d twhere Au i s th e he igh t of each con t ro l impulse and A t i s t h e ti me i n t e r v a lof e a ch c on t r o l i mpu ls e. Su bs t i t u t i n g % Q(t)+ K,,, A $ ( t ) from (1) f o r 6u,we have :

6$ = 2 nu1A t Jt' ( 3 )t 0

4


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S o l v in g f o r $ and I .et t ing -&$ = $d - $ (prev j .ous t e rmina l e r ror ) we ob t a i n :

, o J < t o JW vSt e e pe s t de s ce n t T e rm i na l e r r o ropt imiza t ion t e r m c o r r e c t i o n t e r mThis g ives t h e ge ne r a l f or m o f t h e s teepes t descent equa t ions . The ac tu a l cal c u l a t i o n s a re next c onsi de re d i n more de t a i l .

HYBRI D COMPUTER MECHANImTIONThe mechaniza t ion of th e o pt im iza t io n procedure on a h i g h - s p e e d r e p e t i t i v eanalog computer i s p r e se n te d i n f i g u r e 2 . F i gu re 2 ( a ) i s t he f low d ia gr am andf i g u r e 2 ( b ) i l l u s t r a t e s t h e l o g i c u s e d i n a u to m a ti c al l y r e g u l a t i n g t h e p ro blem .The mechaniza t ion consis ts of an ana log computer program f o r sol vin g th e t r a j e c t o r y e q u a ti o n s; l o g i c r e q u i r e d t o coord ina te th e procedure ; and a s e r i a lmemory s t orag e un i t f o r s to r in g t he nominal c on t ro l program.The s e r i a l memoryuni t i s cont inuou s ly dr iven by counte r pu l se s (Logicno. 1). The output of the s e r i a l memory i s the nomina l cont ro l t i m e h i s t o r yw i t h n p o i n t s t h a t i s us e d al ong w i t h t he a pp r op r i a t e c o n t r o l i m pu ls e, t os o l ve t he t r a j e c t o r y e qua t i ons . T hese e qua ti ons a re s t a r t e d a t t h e s p e c i f i e di n i t i a l c o n d i t i o n s w i t h L og ic n o. 2 and s topped when th e t r a j ec to ry reaches

t he s pe c i f i e d end po i n t w i t h L og ic no . 3. The f i n a l va l ue s of the cos t quant i t y , cp, and s t a t e qua n t i t y , $', are s t o r e d a t t he e nd o f e a c h r un a s i nd i c a t e dby Logic n38. 4 and 5. The pz s i t i ve o r ne ga ti ve c on t r o l i mpu ls e i s a dded t oth e nominal con t ro l in put w i th Logic nos . 6 and 7 , re s pec t iv e ly . Logic no . 8i n s e r t s th e modifying co nt ro l (Kcp @ +K Aq) i n t o t h e s e r i a l memory. Thisp r sc ed u re r u n s i n e s s e n t i a l l y a continuo us manner; t h a t i s , one po i n t ou t o ft h e n p o i n t s i n t h e no min al c o n t r o l h i s t o r y i s u p da te d a f t e r each two re pe ti t i v e computa tions , and a f t e r 2n r e p e t i t i v e c om pu ta ti on s ( on e i t e r a t i o n ) ,eve ry po in t i n s to rage has been modif ied and the process i s repea ted . For eachi t e r a t i o n t h e g ai ns Kcp and Kq are h e l d as c ons t a n t s . A s pre vio us l y ment i o n e d , t h e value of Kcp de t er m i nes t he r e l a t i v e s pee d and s t a b i l i t y of t h econvergence a n t 3 th e optimum. The corre spo ndi ng val ue of Kq t o b e u s e d w i t he a ch new i t e r a t i o n i s c a l c u l a t e d b y e q u a ti o n ( 4 ) as a f u n c t i o n o f t h e t e r m i n a le r r o r f ro m e ac h p r e vi o u s i t e r a t i o n ($d - $) an d as a f u n c t i o n of t h e followingtwo i n t e g r a t e d q u a n t i t i e s f ro m e ac h p r e v io u s i t e r a t i o n :


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and

The values f o r e q ua ti o ns ( 5 ) and ( 6 ) were computed as i n t e g r a l s a v e r t h et i m e pe r i o d f r om t = to t o t = t f . The t i m e to w a s r e p r e s e n t e d by a l o g i cs i g n a l a t t h e f i r s t r e p e t i t i v e c om pu ta ti on i n a n i t e r a t i o n c y c l e a nd t h e t i m etf w a s r e p r e s e n t e d by a l o g i c s i g n a l a t t h e l a s t computat ion i n am i t e r a t i o nc yc l e . It s ho ul d b e n o t ed t h a t d u ri ng t h o se p u t s of t h e t r a j e c t o r y when t h ec o n t r o l w a s a t a c o n s t r a i n t l i m i t , no f u r t h e r op t i m iz a t i on w a s poss ib le , and thei n t e g r a t i o n of e qua t ions ( 5 ) and (6) w a s t h e r e f o r e n o t c a r r i e d o u t d u r in g t h o s et i m e s .

This type of computer mech aniz ation w i l l b e i l l u s t r a t e d i n more d e t a i l f o rthe following example problem.

APPLICATION TO FEEITTRY TRAJECTORY OPTIMIZATIONStatement of t h e Problem

The p rob lem t o b e i l l u s t r a t e d i n t h i s s e c t i o n i s t h a t o f de te rm i ni ng t hetime h i s t o r y of t h e v a r i a t i o n o f l i f t - d r a g r a t i o ( c o n t ro l L/ D) t ha t m us t beflown f o r a ve h i c l e r e t u r n i ng i n t o t he e a r t h ' s a tm os pher e so t h a t :The t o t a l h e a t i n g l oad t o t h e v e h i c l e i s minimized.The vehicle a r r ives a t a d e s i r e d d e s t i n a t i o n .The c o n t r o l re ma ins w i t h i n s pe c i f i e d c o ns t r a i n t s .

MechanizationThe equat ions of motion, p resen ted i n appendix.B, were f o r a p o i n t mass i nplanar mot ion 3ver a s p h e r i c a l n o n r o ta t in g e a r t h . The v e h i c l e c h a r a c t e r i s t i c sa n d f l i g h t c o n d i t i o n s were t h o s e f o r a manned capsule r e t ur ni ng f rom e a r t ho r b i t .

I n i t i a l c o n d i t i o n s were:A l t i t u d e Veloc i ty F l i g h t - p a t h a n g l e Range t o de s t i na t ion

Fina l s topping condi t ions were :A l t i t u d e

6

76.3 l an7.63 km s-1.81609 Ism

30.48 ~nn

(250,000 f t )(25,000 f p s )(1000 m i )

( i oo, ooo f t )


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-

C ont r o l l i m i t s were:o


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The modifying co nt ro l shown i n th e f ig ur e i s t h e sum Q nCp + Kq A$. Fort h i s s e r i e s of runs a c ons t a n t va l ue o f E;lp = - 2 . 5 ~ 1 0 - ~ [ u n i t sf (L /D )/( J/ m2 )]w a s found to , a l low a f a i r l y r a p i d c onve rgence w h i le m a in t a in i ng p rogra m s t a b i l i t y . The va lue of K$ w a s c a l c u l a t e d by e qua t i on ' ( 4 ) t o be t h a t v a lu e f o re ac h i t e r a t i o n s uc h a s ' t o a l lo w c on ver ge nc e i n t h e s t e e p e s t d e sc e nt m anner.I n t h e lo we r t r a c e o f f i g u r e 3 ( b ) t h e n o min al c o n t r o l i s recorded as it i sread o ut of s e r i a l memory eve ry 128 +1 count er pu lse s (wi t h Logic no. 8 ) . Thisg i ve s a convenien t t ime h i s t o ry t o show th e manner i n which th e co nt r o l hasbeen modif ied dur ing each i t e ra t io n . The co nt r o l i s s e en t o be l i m i t e d w i t h i n

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The r e s u l t s ob ta ined by t h i s computing method appear s a t i s fa c t or y f o rengineer ing purposes; however , th e usu a l d isadvantages o f analog computationare inhe ren t wi t h t h i s method . These d isadvantages are pr imar i ly concernedwi th th e ex t raneous no is e i n the computa t ions and th e abso lu te accuracy (on lyt o w i th in about 1p e r c e n t ) of anal og computer.

CONCLUDING REMARKST his r epo r t ha s p re sen ted a p er tu rb a ti on method of computing optimumtr a j ec to r i e s . The t e chnique u se s fas t - t ime r e p e t i t i v e c o mp ut at io ns i n d e t e r min ing co n t ro l impulse response func t ions and requ i re s on ly the dynamic so l ut i o n o f t he s t a t e equa t ions ; w he reas o the r pe r tu r ba t ion computing t e chn iqueshave r e q u i r e d t h e s o l u t i o n of a d d i t i o n a l a d j o i n t e q u a t io n s .A hybrid computer w a s used i n app ly ing th e method t o t h e s t e e p e s t d e s c e n top t imiza t ion of r ee n t r y t r a j ec to r i e s . Mechaniz ing th e compu ter f o r t h i s t y p eof problem w a s r e l a t i v e l y s im p le , and ne a r Q p t i m m t r a j e c t o r i e s c o u l d b eo b t ai n e d i n a bo ut 3 minutes of computing t i m e and f ul l op t imum t r a j ec to r i e s i nabout 8 minutes .The advantage o f t he t e chn ique ou t l ined he re ove r a l t e rna t ive t e chn iques

i s t h a t t h e i n v e s t i g a t o r n e e d n o t b e fami l i a r w i th or u s e an a u x i l i a r y s e t ofl i ne a r a d jo in t equa t ions f o r t he op t imiza t ion . T h i s te chn ique &ses , how ever,r e q u i r e a large number of dynamic solut ions of t h e s t a t e e q ua t io n s, b u t t h i scomputing t a sk appear s p ra c t i c a l w i th th e h igh -speed r e pe t i t i ve computat ionp ro ce du re p r e se n t ed i n t h i s r e p o r t .. Ames Research CenterNat iona l Aeronau tics and Space Admin is tra t ion

Mof fe t t F i e ld , C a l i f . , Jan . 24 , 1966

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APPENDIX A

RELATIONSHIP OF EXPERI MENTAL I MPULSE RESPONSE FUNCTIONSTO ADJ OI N" SOLUTIONS AND THE MAXIMUM PRINCIPLJTA d j o i n t S o l u t i o n of Impulse Response Functiom

L e t the s t a t e equa t ions be no ted as

where x ( t ) i s a v e c t o r o f s t a t e v a r i a b l e s , u ( t ) i s t h e c o n t r o l v a r i a b l e and fi s a v e c t o r of known func t ions o f x ( t ) and u ( t ) .The au x i l i a r y ad jo in t equ a t ions can be no ted as

where A i s a vec to r o f i n f luence func t ions and (af /ax)T i s t he t r an sposeof the matrix desc r ib ing l i n e a r mo tions about t he nomina l pa th o f x ( t ) andu ( t )

It i s w e l l known t h a t a n y s m a l l change i n t h e c o n t r o l q u a n t i t y 6 u ( t )a long the nomina l pa th w i l l determine a change 6(p i n a ny q u a n t i t y cp a t t h ef i n a l t i m e as fo l lows:

where &+ r e p r e s e n t s a s o l u t i o n o f t h e a d j o i n t e q u a t io n s w i t h t h e b ou nd ar yc o n d i t i o n s a t t he f i n a l t i m e of

TThe quanti ty A ( a f / du ) w i t h i n th e i n t e g r a l i s kn3wn a s Green's fu nc ti 3n .

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Exper imenta l Determinat ion of Z q u l s e Response Functions

The method used i n t h e t e s t .o f t h i s r e p o r t f o r f i n d i n g a change 6cq i s t ope r t u r b t h e c on t r o l e xpe r i m e n t a l l y i n t he f o l l ow i ng manner:

,Nomina l con t ro lLJ! .... =t0 1 t f

With two se qu en t ia l dynamic s ol ut io ns of th e s t a t e e qua t i ons u s i ng f i r s t ap o s i t i v e c o n t r o l i m p u l s e and t he n a n e g a t i v e c o n t r o l impulse, t he f o l l ow i ng i savai la b e :

From equation ( A3) w e can wr i t e the change 26q = Ap, f o r a s m a l l c o n t r o l ,6u = nu, a c t i ng ove r a s m a l l t i m e i n t e r v a l A t , as f o l l ow s :2ep = = (.pT a fz) 2 nu A t

This then re pre sen ts th e correspondence be tween th e impulse responsef u n c ti o n s c a l c u l a t e d i n t h e text a nd t hos e s o l ve d by t he a d j o i n t s o l u t i on .Greens func t ion eva lua ted a t any t i m e , t, a lo n g t h e t r a j e c t o r y c an b e n o t edas

and

Rela t ion ship t o t h e Maximum P r i nc ip le

The m a x i m u m p r i n c i p l e ( r e f . 8) s t a t e s t h a t a n e c es s ar y c o n d i t i o n f o r aminimum ( m a x i m u m ) of t he c os t f unc t i on i s t ha t t he H a m i l t on i a n be maximized( mi ni mi ze d) w i t h r e s pe c t t o t h e c on t r o l a t a l l t i m e s . The Hamilto nian can bew r i t t e n asTH = A f (A91

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where t h e t r a n s v e r s a l i t y c o n d i t i o n must b e s a t i s f i e d a t t h e f i n a l t im e,

an d i s a Lagrange mult ipl ie r cons tant chDsen s3 t h a t t h e t e r m i n a l c o n s t r a i n t i s s a t i s f i e d . T h e boundary c ond i t i ons on t he a d j o i n t e qua t i ons a t t h ef i n a l time are :

Now t o determine i f H i s minimized with respec t t o t h e c o n t r o l w e cant a k e t h e p a r t i a l d e r i v a t iv e of H w i t h r e s p e c t t o u :

Or , no t in g th e correspondence be tween equa t ion s ( Ah) and (Al l ), w e can wri te

T Twhere &,(af /au) i s G re en 's f u n c t i o n f o r t h e c o s t a nd A,(af/au) i s Green'sf u n c t i o n T f o r t h e t e rm i n al c o n s t r a i n t .Reca l l ing the cor re spondence be tween the ad jo in t so lu t ion f or Green' sfun c t io n and th a t de termined expe r imenta l ly , we have t he fo l lowing:

T h i s , t he n , r e p r e s e n t s t he r e l a t i o ns h i p bet we en t he e xpe r i m e n ta l l ydetemnine d impulse r e sponse func t ions and the Hami l ton ian . It i s i n t e r e s t i n gt h a t t h e m a x i m u m p r i n c i p l e can b e a p p l i e d t hr o ug h t h i s r e l a t i o n s h i p w i th o ut anyneed f o r s o l v i n g t h e a d j o i n t e q u at i o ns .St eep es t Descent Equat lons

The gr ea te s t change , @, i n cp f o r a given va lue of 6 u 2 ( t ) d t i st 0obta ined ( r e f . 3) when

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where Kcp and % are c ons t a n t s . T h i s i s t h e s t e e p e s t d e s c e n t ( o r a s c e n t )d i r e c t i o n t o t h e m i n i " ( or m a x i m u m ) cp.When t h e r e are no s t a t e o r c o n t r o l c o n s t r a i n t s , t h e s t e e p e s t d es ce ntprocedure conve rges toward the necessa ry condi t ions for an opt imum solu t ion aspr e v i ous l y no t e d

Q(t>- -q a+>= 0 (A161w here i n t he s t e e pe s t de s c e n t e qua ti ons , Kq/% = 7, on t h e o p t i m s o l ut io nw i t h t h e t er m i n a l c o n s t r a in t s a t i s f i e d .


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APPENDIX BREENTRY T W C T O R Y EQUATIONS

The fo l lo wi ng e qu ati ons were programmed on th e ana log computer f or t h eexample con side red i n t hi s r ep or t . These s im pl i f i ed equa t ions were de r ived f o rf l i g h t w i t h i n t h e a tm os ph er e and t h e p r imary assumptions inc lude : a s p h e r i c a ln o n r o t a ti n g e a r t h , s m a l l f l i g h t - p a t h a n gl es , and a cons tan t g r av i t y t e rm. Thed e r i v a t i o n of t h e s e s i m p l i f i e d e q u a ti o n s and t h e i r a p p l i c a b i l i t y have b ee nc o n s i d e r e d i n a number o f r e p o r t s . S ee f o r i n s t a n c e r e f e r e n c e 9 .The equations are

$ = st'V d tt 0

whereL- c o n t r o l v a l ue of l i f t - d r a g r a t i oDh a l t i t u d e , mV h o r i z o n t a l v e l o c i ty , m /s

f i n a l r an ge , mcp t o t a l h e at i np u t, J/m2P atmosphere densi ty, 1. 225 e -h/7160 %/m3r r a d i u s f r o m e a r t h c e n t e r , 6. 43~10~g l o c a l g r a v i t a t i o n a l a c c e le r a t io n , 9.8 m/s2(?) drag load ing , 0.004 mz/kg14


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REFERENCES 1. Lei tmann, George, ed. : Opt i mzati on Techni ques. Academc Press, 1962. 2. Bal akri shnan, A. V. ; and Neust adt , Luci en W, eds. : Computi ng Methodsi n Opt i mzat i on Probl ems. Academc Press, 1964.3. Bryson, Arthur E. ; and Denham Wal ter F. : A Steepest - Ascent Method f orSol vi ng Opti mumProgrammng Probl ems. Rayt heon Rep. BR 1303, 1961.Al so J . Appl . Mech. , vol . 29, no. 2, J une 1962, pp. 247-257.4. Kel l ey, H J . : Gradi ent Theory of Opt i mal Fl i ght Paths. ARS J . , v o l .30, no. 10, Oct . 1960, pp. 947-954.5. Bryson, A. E. ; Denham W F. ; Carrol l , F. J . ; and Mikami, K. : Determnati on of Li f t or Drag Programs That M ni mze Reent ry Heati ng. J . Aerospace Sei . , vol . 29, no. 4, Apr i l 1962, pp. 420-430.6. Bl anton, H. El more, ed. : Three- Di mensi onal Traj ectory Opt i mzat i on Study.Pt . 1 - Opti mumProgrammng Formul at i on. NASA CR- 57030, 1964.(Supersedes Aero. Sys. Di v. Rep. ASD- TDR- 62- 29?and Raytheon Rep.Br- 1759- 1) .7. Hague, D. S. : Three- Degree- of- FreedomProbl emOpt i mzat i on Formul at i on.FDL- TDR- 64- 1, pt. 1, V O ~ . 3, Oct . 1964.8. Pont ryagi n, L. S. , et al . : The Mathemati cal Theory of Opt i mal Processes. Wl ey and Sons, 1962. 9. Chapman, Dean R. : An Approxi mate Anal yti cal Method f or Studyi ng Ent ry I nto Pl anetary Atmospheres. NASA TR R- 11, 1959.


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-Control, -U -

Stateva Iue

cos

C,on tro I im puIse

71 7

tt0 tf to tf

Trajectory time1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Computer timeFi gu r e 1,- I l l u s t r a t i v e time h i s t o r y of r e pe t i t i ve a na l og c om pu t a t i ons .


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I. c.

91Counter) An a l o g

If 6 I + II I+ impu lse O L III I----I ----Dotted l ines represenl3+2 # 3 log ic s ignals shown-Contro l ( S t a r t ) ( S t o p ) in f i gu re 2 ( b )impu lse(a) Fl ow di agram

Fi gure 2, - Mechani zati on of the opt i mzat i on procedure.


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*- 2 I 1 n nL Solution J#3 I time I(Stop) n n# A ( i t o i e n n

#5 (Store) n n

#7 I I I(- Control)\ i m p u l s e )#8 I I( M o d i f y )

( b ) Problem logic.Figure 2. - Concl uded.


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rc0

/Control impulse.5- n i I NominalControl L .25 - - ~ y - ~ y , - I y control0- '

Final 2 range,$, I -IO 3 km -W -Final heat -Ioad,4, 20--

-i 0.1Second of computer time( a ) D eta i l s of repe t i t i ve t ra j ec to ry compu ta t i ons .

Figure 3.- Recorded t ime h is tor ies f or reen t ry t r a j ec to ry opt imizat ion; minimum heat wi th terminal rangecons t ra in t


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O

? FinalI u range, +I uo IO3 kmPFinal heatload, $107 J oule IO

0- IO Seconds of2 + + computer time

2 -

Modifying -.I %- control .I -- 1( b ) Deta i l s of overa l l conver gence pr ocedure.

F i gur e 3. - Concl uded,


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A Method of Trajectory Optimization by Fast-time Repetitive Computations

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