guidance laws for short range tactical

79-0059

Guidance Laws for Short Range Tactical Missiles

ii

H.L. Pastrick, U.S. Army Missile Research & Development Command, Huntsville, Ala.; and S.M. Seltzer, Control Dynamics Co., Huntsville, Ala.; and M.E. Warren, University of Florida, Gainesville, Fla.

17th AEROSPACE SCIENCES MEETING

New Orleans, La./January 15-17, 1979

For permission lo copy or republish. contact the American institute 01 Aeronautics and Aslronaulics. 1290 Avenue 01 the Americas. New York. N.Y. 10019.

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GUIDANCE LAWS FOR SHORT RANGE TACTICAL MISSILES

H . L . P a s t r i c k " US Army M i s s i l e Research and Development Cornand

H u n t s v i l l e , Alabama

S. M. Sel tzer"* Control Dynamics Company

H u n t s v i l l e , Alabama

M. E . Warrent U n i v e r s i t y o f F l o r i d a G a i n e s v i l l e , F l o r i d a

Abstract_

A comparison of guidance laws a p p l i c a b l e t o s h o r t range t a c t i c a l m i s s i l e s is made. These laws are segmented i n t o several c l a s s e s and t h e p r i n c i - p l e s unde r ly ing each c l a s s are d i s c u s s e d . S p e c i f i c a t t e n t i o n is g iven t o the s t r u c t u r e of t h e guidance technique and t h e requirements f o r i t s implementa- t i o n . E v a l u a t i o n and comparison o f t h e performance of t a c h guidance l a w versus t h e c o s t o f implementing i t are cons ide red . An e x t e n s i v e b ib l iog raphy of r e l e v a n t l i t e r a t u r e i s included.

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

The US Army M i s s i l e Research and Development C o m n d (MIRADCOM) r e c e n t l y began a t a s k t o develop a n advanced guidance and c o n t r o l system f o r f u r t h e r Army Modular Missiles. The i n t e n t i s t o " leapfrog" systems c u r r e n t l y under development. The purpose o f this paper i s eo d e s c r i b e t h e work t h a t has been done wi th in t h i s new t a s k and t o provide an i nd i - c a t i o n of f u t u r e e f f o r t s t h a t are now planned.

T h e reason f o r embarking on this t a s k is now s u m a r i z e d . P r e s e n t weapon systems performance may be s e r i o u s l y degraded i n engagements a g a i n s t t a r g e t s w i th p red ic t ed c h a r a c t e r i s t i c s of t h e 1990s and beyond and i n t h e b a t t l e f i e l d environments of t h a t t ime frame ( s e e , e.g. , p . 1 1 , Avia t ion Week and Space Techno lom, March 20, 1978). I t has been e s t a b l i s h e d t h a t t h e guidance laws c u r r e n t l y i n wide use may not be adequate t o combat t hose t h r e a t s . Thus, i t is p r o j e c t e d t h a t fundamental advances i n guidance and control systems theory are r e q u i r e d t o enhance t h e e f f e c t i v e n e s s of f u t u r e weapon systems. Add i t iona l ly , m i s s i l e a i r f r a m e and p ropu l s ion systems may r e q u i r e advances comen- % r a t e with t h e p red ic t ed t a r g e t s c e n a r i o s . I n p a r t i c u l a r , a i r d e f e n s e weapons c u r r e n t l y i n Research and Development (R&D) may be s e r i o u s l y hampered i n t h e combat s c e n a r i o s env i s ioned . From an overall systems viewpoint , t h i s t a s k s h a l l a d d i e s s t h e i s s u e o f c r e a t i n g new theory i n the guidance and c o n t r o l area t o meet the h i g h performance t h r e a t of t h e f u t u r e as a l ead ing technology item. C lose ly a s s o c i a t e d wi th i t and i n p a r a l l e l w i th t h e guidance and c o n t r o l e f f o r t , weapon system work s h a l l be undertaken t o modify a i r f r a m e and p ropu l s ion modules t o be capab le of engaging t h e 1990s t h r e a t . General suppor t weapons s h a l l be viewed i n i t i a l l y as a s u b s e t of t h e a i r de fense

system. Here to fo re t h e s e two classes of weapons each were developed independent ly . Th i s r e sea rch s h a l l a t tempt t o view them as p o t e n t i a l l y s i m i l a r systems t h a t u t i l i z e d i f f e r e n t modules such as propu l s ion , guidance, warhead, e t c .

The f i r s t scep i n implementing t h i s t a s k was t o conduct a l i t e r a t u r e survey t o e s t a b l i s h a technology base s t a r t i n g p o i n t . C o n s t r a i n t s on l eng th of t h e paper r e s u l t i n summarizing ( a l p h a b e t i c a l l y by au tho r ) t h e l i s t t o inc lude only those r e f e r - ences considered by t h e a u t h o r s t o be most r e l e v a n t .

Following t h i s survey, guidance l a w s were placed i n f i v e c a t e g o r i e s and de f ined mathematical ly . The implementation and p r e d i c t e d performance of each ca t egory was t h t a i n v e s t i g a t e d and compared in l i g h t of c u r r e n t and p r e d i c t e d hardware and so f tware capa- b i l i t i e s . Th i s paper d e s c r i b e s t h e s e r e s u l t s .

11. Guidance Laws

The development o f guidance laws f o r s h o r t range t a c t i c a l m i s s i l e s has become a wel l - researched t o p i c over the p a s t 25 y e a r s . A summary of a d e t a i l e d l i t e r a t u r e survey, how each guidance l aw can be implemented, and guidance l a w p red ic t ed performance are desc r ibed wi th in t h e f i v e guidance law c a t e g o r i e s s t a t e d i n t h e I n t r o d u c t i o n .

L i n e of- S i g h t Guidance (Command- to-Line-of-Sight and B e a m R ide r1

Clemow (1960), i n h i s book M i s s i l e Guidance, provides a d e t a i l e d d i s c u s s i o n o f beam r i d i n g , w h i l e Mahapatra (1976) d i s c u s s e s morphological d e s i g n based on beam r i d i n g , Clemow (1960) d i s - c u s s e s co-nd-to-line-of-sight (CLOS) , while Kain and Yost (1976) u s e t h i s method i n t h e i r s h i p de fense s c e n a r i o , u s i n g Kalman f i l t e r s t o reduce beam j i t t e r .

A t t h e MIRADCOM, CLOS and Beam Rider (BR) con- c e p t s are each cons ide rad a s u b s e t of the l i n e - o f - s i g h t guidance laws. They d i f f e r p r i m a r i l y i n t h e i r mechanizat ion. The CLOS uses a wire f o r t h e t ransponder l i n k , e .g . , TOW o r DRAGON. BR may use an e l e c t r o - o p t i c a l l i n k , e .g . , SHILLELAGH, and f l y i n a d i r e c t e d beam aimed a t t h e t a r g e t . G e n e r i c a l l y they a re s imi la r and w i l l be d i scussed as one.

*Research Aerospace Eng inee r . Member A I A A .

**Senior S c i e n t i s t . Assoc ia t e Fellow A I A A .

t A s s o c i a t e P r o f e s s o r , E l e c t r i c a l Engineer ing Department. Member A I A A .

1 This paper ih declared P work 01 Ihe US. Gorernmcnl and Iherelorei, in thepublicdornsin.

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The l i n e - o f - s i g h t (LOS) guidance scheme (CLOS and BR) i s one i n which t h e missile is guided on an LOS cour se so as t o remain on a l i n e ad jo in ing the t a r g e t and t h e po in t of c o n t r o l . To f l y a long the LOS, t he missile r e q u i r e s a v e l o c i t y component (V )

perpend icu la r t o t h e u)S t h a t i s equal t o t h e MS v e l o c i t y desc r ibed by t h e r e l a t i o n

M 1 i/

'MI = %M 'ST (1)

where RSM r e p r e s e n t s t h e range from t h e m i s s i l e t o

the t r a c k i n g s t a t i o n and yST r e p r e s e n t s t he LOS. I n g e n e r a l , t he missile f l i e s a p u r s u i t guidance cour se a t t h e i n i t i a t i o n o f t he e n t r y i n t o t h e beam a t launch and f l i e s an approximately cons t an t bear - i n g course near impact. This is observed from t h e v e l o c i t y equa t ion

- _ %M 'Mi - RST 'T

where t h e range of t h e t r a c k i n g s t a t i o n t o t h e t a r - g e t i s g iven by R and V r e p r e s e n t s t he t a r g e t

v e l o c i t y r e l a t i v e t o t h e s u r f a c e of t h e e a r t h . ST T

Figures 1 and 2 a r e s i m p l i f i e d c o n t r o l b lock diagrams h i g h l i g h t i n g f e a t u r e s of t h e scheme f o r each of t h e two cases. I t should be noted t h a t t he p r o j e c t o r mount dynamics i n F igu re 1 and t h e t r a c k e r mount dynamics i n F igu re 2 may be cons ide rab ly more

complex than dep ic t ed he re . Also t h e BR missile r e q u i r e s anboard a u t o p i l o t compensation s i n c e t h e p r o j e c t o r does not know where t h e missile i s loca ted once enroute . The CMS scheme, however, does keep t r a c k of t h e m i s s i l e and thus compensates f o r i t s p o s i t i o n p r i o r t o t r a n s m i t t i n g t h e guidance s i g n a l v i a t h e w i r e l i n k .

Performance o f m i s s i l e s f l y i n g t h i s guidance law i s t y p i c a l l y very good. Without t h e man-in- the- loop t r a c k i n g e r r o r , flawless guidance has been t h e r u l e : t a r g e t h i t s can be expected v i r t u a l l y on each sho t . I n t h e more r e a l i s t i c c o n d i t i o n where t h e the t r a c k i n g e r r o r i s t h e major e r r o r sou rce , g iven t h a t a r e l i a b l e round has been f i r e d , t h e p e r f o m - a w e has been found t o be b e t t e r than 1 f t CEP wi th a 90% conf idence l e v e l .

P u r s u i t Guidance

Goodstein (1972) gave a comparison o f t h e q u a l i t i e s and s e n s i t i v i t i e s of LOS, p u r s u i t , and p r o p o r t i o n a l nav iga t ion guidance f o r a i r - to-ground and a i r - t o - a i r missiles. I n ano the r paper i n t h e same r e p o r t , Goodstein d i s c u s s e s t h e guidance and c o n t r o l sys tem t r a d e o f f s i n m i s s i l e des ign .

Two p u r s u i t guidance laws are d i scussed h e r e i n : a t t i t u d e p u r s u i t gu idance and v e l o c i t y p u r s u i t guidance. A t t i t u d e p u r s u i t guidance tries t o keep t h e c e n t e r l i n e of t h e m i s s i l e po in t ed a t t h e t a r s e t . I n a missi le which f l i e s an ang le o f a t t a c k when maneuvering, t h e v e l o c i t y v e c t o r w i l l always l a g

'L ZOOM PROJECTOR

PROJECTOR RANGE ZOOM

ACCELERATION DYNAMICS MAY BE MORE COMPLEX

Figure 1. Beam Rider Scheme$<

TRACKER m! GAIN PROGRAM ACCELERATION AM I

OVNAMICSMAY BE MORE COMPLEX

Figure 2, Command-to-Line-of-Sight Scheme*

. "Implementation scheme provided by R. H. Farmer, Technology Labora tory , US Army Missile Research

and Development Comand.

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t h e v e h i c l e p o i n t i n g d i r e c t i o n . Miss d i s t a n c e i s a s t r o n g f u n c t i o n of the maneuver c a p a b i l i t y o i t h e m i s s i l e and can be reduced by a f a s t responding h igh -g v e h i c l e .

Ve loc i ty p u r s u i t guidance a t t empt s t o keep t h e v e l o c i t y v e c t o r of t h e m i s s i l e po in t ed a t t h e tar- g e t . I t i s mechanized i n sone less s o p h i s t i c a t e d missiles by mounting a target sensor on an a i r vane which i n d i c a t e s r e l a t i v e wind d i r e c t i o n . The d i i - ference between t h i s v e l o c i t y v e c t o r and a t r u e v e l o c i t y v e c t o r is t h e primary error i n t h e scheme. The a t t i t u d e p u r s u i t guidance mechanizat ion decouples t h e ang le o f a t t a c k from t h e t a r g e t s eeke r and improves m i s s d i s t a n c e performancc by an amount p r o p o r t i o n a l t o t h e v e h i c l e ang le of a t t a c k . F igu re 3 d e p i c t s two-dimensional geometry useful f o r d e s c r i b i n g t h e p u r s u i t guidance laws.

t Y

Figure 3 . P u r s u i t Guidance Geometr)

Neglect ing t h e case of a maneuvering t a r g e t , f o r s i m p l i c i t y , one no tes t h a t

i = v - v ( 3 ) T M ' and the LOS r a t e i s

For an i d e a l p u r s u i t , 0 = X . m i s s i l e w i l l always have t o t u r n du r ing t h e a t t a c k except f o r t h e case o f a p e r f e c t head o r t a i l chase.

S ince 6 = A, t h e

F igu res 4 and 5 p r e s e n t examples of s i m p l i f i e d c o n t r o l system diagrams f o r t h e a t t i t u d e and velo- c i t y p u r s u i t guidance laws, r e s p e c t i v e l y . I n t h e former, a wide angle t a r g e t sensor is r equ i r ed s i n c e i t is t y p i c a l l y mechanized t o be body f i x e d . I n t h e l a t t e r , a narrower f i e ld -o f -v i ew (FOV) sen- s o r may be u t i l i z e d as a r e s u l t of t h e decoupl ing o f t h e body from t h e sensor mount as p rev ious ly desc r ibed . I n each c a s e , i s t h e forward guidance

g a i n , and i t w i l l d i f f e r f o r each as w i l l t h e fced- back damping g a i n K R .

F igu re 5 i n d i c a t e s t h a t h i g h e r guidance g a i n i n the v e l o c i t y p u r s u i t law r e q u i r e s some smoothing t o i n h i b i t no i se o f t h e t a r g e t sensor o p t i c s and i ts a s s o c i a t e d e l e c t r o n i c s .

The guidance f i l t e r i n

Figure 4. A t t i t u d e P u r s u i t Guidance

Figure 5. Ve loc i ty P u r s u i t Guidance

The performance t o be expected f rom p u r s u i t 4 guidance i s i n d i c a t e d i n Figures 6 , 7 , and 8. These were obtained f o r a t a c t i c a l weapon of t h e c lass known as close suppor t a n t i t a n k weapons. They are i n d i c a t i v e o f t h e q u a l i t y of performance one may expect f o r t hese guidance laws. Although these are s i m u l a t i o n r e s u l t s , r e c e n t expe r i ence with f l i g h t hardware has v a l i d a t e d t h e s imulated pcriormance t o a h i g h degree of conf idence .

The p r o p o r t i o n a l nav iga t ion guidance (PNG) law performance i s a l s o i n d i c a t e d ; t h e r e f o r e , i t s h a l l be desc r ibed next .

40 I I ATTITUDE PURSUll

VELOCITY PURSUIT

PROPORTIONAL

0 4 8 12 16 READING ERROR Idegl

d Figure 6 , Performance of C l a s s i c a l Guidance Laws

f o r a Given Heading E r r o r a t F i r s t Ta rge t Acqu i s i t i on

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100

80

60

VELOCITY PURSUIT

40

20

0 0 10 20 30 40 60

TARGET VELOCITY lrnph)

Figure 7. Performance o f C l a s s i c a l Guidance Laws f o r a Given Ta rge t Ve loc i ty and as a FUnction of Control Au thor i ty (Units Expressed i n g ' s )

loo 90 7 80 .

70 - CONTROL AUTHORITY = 5 8's

60 - 50 - 40 -

20 -

PROPORTIONAL

0 0.1 0.2 0.3 0.4 0.5 TARGET ACCELERATION (a1

Figure 8. Performance o f C l a s s i c a l Guidance Laws f o r a Given Ta rge t Acce le ra t ion Commencing 5 sec Before Impact

P r o p o r t i o n a l Navigat ion Guidance

Adler (1956) provided one of t h e seminal papers i n t h e area, cons ide r ing PNG i n t h r e e dimensions. Adler provides a r eadab le development of t h e theo ry u s i n g v e c t o r c a l c u l u s , whereas most subsequent a u t h o r s have cons ide red only t h e p l a n a r case f o r s i m p l i f i c a t i o n . Murtaugh and C r i e l ( 1 9 6 6 ) provided ano the r fundamental paper developing three-dimen- . s i o n a l PNG f o r a s a t e l l i t e rendezvous problem.

Clemow (1960) g i v e s ano the r b a s i c d e r i v a t i o n o f PNC, wh i l e Pitman (1972) compiled s e n s i t i v i t y f u n c t i o n s and p r o j e c t e d errors f o r PNG w i t h g a i n s o f 2 , 3, and 4. C a l c u l a t i o n s o f t e rmina l homing parameters were made by Rawlings (1970). I n a 1971 paper , Rawlings cons ide red t h e e f f e c t s o f s a t u r a t i n g aerodynamic s u r f a c e s on t r a j e c t o r i e s flown w i t h PNG. Many a u t h o r s have considered augmenting PNG t o account f o r t a r g e t a c c e l e r a t i o n s . Arbenz (1970) considered making t h e c l o s i n g v e l o c i t y heading r a t e p r o p o r t i o n a l t o LOS r a t e and developed a c l o s e d form expres s ion f o r a modif ied PNG law. Siouris (1974) added an e s t i m a t e of t a r g e t a c c e l e r a t i o n t o t h e missile a c c e l e r a t i o n command t o y i e l d an augmented PNG law. Guelman used geometr ic arguments t o g i v e t h e s t r u c t u r e of t h e m i s s i l e t r a j e c t o r y ; he showed i n 1971 t h a t PNG w i l l a lmost always r e s u l t i n an i n t e r c e p t f o r a c o n s t a n t v e l o c i t y t a r g e t i n 1972. For c o n s t a n t t a r g e t a c c e l e r a t i o n s , q u a l i t a - t i v e t r a j e c t o r i e s were determined and t a r g e t acqu i - s i t i o n boundaries a s ses sed . I n 1976 Cuelman con- s i d e r e d the s t r u c t u r e o f t r a j e c t o r i e s under t m e p r o p o r t i o n a l n a v i g a t i o n guidance (TPNG), where t h e commanded a c c e l e r a t i o n is normal t o t h e LOS r a t h e r t han the m i s s i l e v e l o c i t y . I n this work h e showed t h a t TPNG r e s u l t s i n i n t e r c e p t only if the i n i t i a l c o n d i t i o n s l i e i n a w e l l d e f i n e d Subset of t h e parameter space. Sh ina r (1976) cons ide red PNG f o r a r o l l i n g missile where he cons ide red t h e cross- coup l ing between roll and t h e c o n t r o l system. S l a t e r and Wells (1973) s t u d i e d opt imal e v a s i v e t a c t i c s a g a i n s t a PNG m i s s i l e , i n c o r p o r a t i n g a l a g i n t o t h e missile dynamics, and gene ra t ed two S t r a - t e g i e s based upon d i f f e r e n t o p t i m a l i t y c r i t e r i a . A comprehensive s t u d y of c l a s s i c a l (PNG) and o p t i - mal c o n t r o l techniques f o r t e rmina l homing of CN- c i fo rm and bank- to - tu rn s t e e r i n g m i s s i l e s was per- formed by B a l b i r n i e , S h e p o r a i t i s , and Eferriam (1975).

PNG i s a guidance l a w i n which t h e angu la r r a t e of t h e m i s s i l e f l i g h t p a t h is d i r e c t l y p o r p o r t i o n a l t o t h e angu la r LOS rate o f change. T h i s is shown s i m p l i s t i c a l l y i n F igu re 9 f o r a two-dimensional case.

Y FLIGHTPATH

Figure 9. Geometry of F l i g h t Pa th

The geometry o f F igu re 9 sugges t s

where yH r e p r e s e n t s f l i g h t pa th a n g l e r e l a t i v e t o

a f i x e d reference, h r e p r e s e n t s the LOS r e l a t i v e t o a f i x e d r e f e r e n c e , and N i s t h e n a v i g a t i o n r a t i o . A g e n e r a l expres s ion f o r m i s s i l e a c c e l e r a t i o n may h e w r i t t e n

- - 7 = VMfMR + VMT ( 6)

where q r e p r e s e n t s t o t a l m i s s i l e a c c e l e r a t i o n ; VM, t h e m i s s i l e speed ; T ~ , t h e missile f l i g h t

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- p a t h angle;

miss i le f l i g h t p a t h ; t i a l t o m i s s i l e f l i g h t pa th . The d e f i n i t i o n s o f R and 7 y i e l d t h e r e l a t i o n s h i p , a . may implement p r o p o r t i o n a l nav iga t ion v i a t h e r e l a t i o n s h i p ( i n t h e Laplace domain)

VMl, a u n i t v e c t o r l a t e r a l t o t h e

and 7, a u n i t v e c t o r tangen- -

= 0. One

where 0 r e p r e s e n t s t h e l a t e r a l a c c e l e r a t i o n i n

u n i t s o f g ' s and E r e p r e s e n t s t h e LOS e r r o r which is measured by a seeker hav ing a t ime lag c o n s t a n t of T sec and an error o f d. The symbol k r ep re - s e n t s a guidance g a i n f a c t o r .

M

By no t ing t h a t a c o n s t a n t bea r ing cour se i s determined i f m i s s i l e and t a r g e t are f l y i n g con- s t a n t speed and n e i t h e r i s maneuvering, it may be concluded that the LOS a t each i n s t a n t o f t i m e would be p a r a l l e l t o t h e LOS a t a previous i n s t a n t . L a t e r a l l y p e r t u r b i n g t h e c o l l i s i o n cour se noted by t a r g e t and m i s s i l e p o s i t i o n s X and r e spec t i . ve ly

y i e l d s Z and Z which may be i n t e g r a t e d t o y i e l d

t h e m i s s i l e t r a j e c t o r y u s i n g p r o p o r t i o n a l nav iga t ion :

T

T M

2

- gkTX + ~(3) Td ZM dZM 2 + - = - d t

t=o d t

The pe r fo rmawe and implementation of t h e guidance l a w are b e s t apprec i a t ed i n t h e i r r e l a t i o n - s h i p t o t w o o t h e r guidance laws which are similar i n i n t h a t no requirement i s e s t a b l i s h e d f o r range or range r a t e i n fo rma t ion . Ve loc i ty p u r s u i t and a t t i - t ude p u r s u i t are i n t h i s c a t e g o r y w i t h PNG. The performance i s i n d i c a t e d i n F i g u r e s 6 , 7 , and 6.

O p t i m a l L inea r Guidance

S ince the mid-1960's t h e m i s s i l e guidance l i t e r a t u r e has become i n c r e a s i n g l y permeated by t echn iques based upon opt imal c o n t r o l . The g r e a t s u c c e s s found by l i n e a r - q u a d r a t i c r e g u l a t o r theory a i d i t s dua l ana log , Kalman f i l t e r i n g , p l u s t h e a t t r a c t i v e and e a s i l y determined form of a fced- back s o l u t i o n has l ed t o a lmost a l l work i n t h i s area being based upon l i n e a r model dynamics, w i th q u a d r a t i c c o s t s and a d d i t i v e Gaussian no i se (LQG). Most fo rmula t ions c o n s i d e r t e rmina l m i s s d i s t a n c e and running c o n t r o l e f f o r t only i n t h e c o s t func- t i o n a l . Unlike the Standard r e g u l a t o r format , a running c o s t on t h e s t a t e i s g e n e r a l l y not appro- p r i a t e i n t h i s framework. The general o p t i m i z a t i o n problem then becomes

where E ( . ) d e s i g n a t e s t h e expected va lue o p e r a t i o n , S and R are weight ing m a t r i c e s , x i s t h e s t a t e , u i s t h e c o n t r o l , and w i s a white no i se p rocess . I n most f o m l a t i o n s the dynamics are assumed con- s t a n t t o o b t a i n c losed form s o l u t i o n s .

f

I/ Bryson, Denham, and Dreyfus (1963); Denham

and Bryson (1964) ; and D e n h a m (1964) were among t h e f i r s t t o c o n s i d e r o p t i m i z a t i o n techniques a p p l i e d t o m i s s i l e guidance problems. I n a s e r i e s of papers they formulated and solved t h e opt imal c o n t r o l problem with i n e q u a l i t y c o n s t r a i n t s , and then a p p l i e d t h e r e s u l t t o t h e t r a j e c t o r y shaping o f a s u r f a c e - t o - s u r f a c e missile f o r range maximization.

S r a l l a r d (1968) gave a good t u t o r i a l review of c l a s s i c a l and modern methods f o r homing i n t e r c e p t o r m i s s i l e s . I n an e a r l i e r paper (1966), h e d e a l t w i th an a u t o p i l o t des ign . I n a 1972 paper , he a p p l i e d d i s c r e t e op t ima l control t o a m i s s i l e system wi th u n d e s i r a b l e s t a b i l i t y c h a r a c t e r i s t i c s , and then modified t h e dynamics t o y i e l d a new problem wi th more d e s i r a b l e c h a r a c t e r i s t i c s .

An e x c e l l e n t review of d e t e r m i n i s t i c opt imal c o n t r o l and i ts a p p l i c a t i o n s , w i th an e x t e n s i v e b ib l iog raphy , i s found i n an IEEE paper by Athans (1966). Another e x c e l l e n t review wi th e x t e n s i v e references is i n B 1965 A I A A paper by Paiewonsky. Kokotovic and Rutman (1965) provided a survey of s e n s i t i v i t y methods drawing h e a v i l y upon S o v i e t l i t e r a c u r e .

Some e a r l y a u t h o r s , i n an a t t empt t o j u s t i f y t h e r e s u l t i n g guidance laws achieved v i a l i n e a r opt imal c o n t r o l , showed that t h e i r r e s u l t s were an e x t e n s i o n of PNG. Axelband and Hardy (1969, 1970) used l i n e a r opt imal c o n t r o l t o develop what they c a l l e d quasi-optimum PNG. As w i t h a lmost a l l l i n e a r opt imal schemes, t ime-to-go i s r e q u i r e d f o r implementation.

'4

Lee (1969) desc r ibed s e v e r a l techniques i n non l inea r c o n t r o l which extended t h e l i n e a r rcgu- l a t o r t heo ry and drew h e a v i l y upon r e c e n t d i s s e r - t a t i o n s a t t h e Un ive r s i ty o f Minnesota.

Deyst and P r i c e (1973) used l i n e a r dynamics wi th a p l a n a r engagement t o develop opt imal c o n t r o l laws where the t a r g e t a c c e l e r a t i o n was a f i r s t - o r d e r Markov p rocess , and found t h a t t h e s a t u r a t i o n of conc ro l s u r f a c e s was an important f a c t o r i n modeling a maneuver l i m i t e d m i s s i l e .

Nazaroff (1976) f o m l a t e d an I Q G approach i n which h e assumed extremely s i m p l i f i e d missile dynamics b u t included t a r g e t a c c e l e r a t i o n and j e r k terms. Stockum and Wiener (1976) assumed exponen- t i a l l y c o r r e l a t e d t a r g e t a c c e l e r a t i o n s i n gene ra t - ing an I Q G guidance law. The feedback they obtained on p r o j e c t e d m i s s d i s t a n c e and r a t e was analogous t o t ime-varying PNG. They compared t h e i r guidance scheme a g a i n s t an augumented PNG law.

Asher and Matuseewski (1974) considered t h e opt imal c o n t r o l problem where t a r g e t a c c e l e r a t i o n was accounted f o r as an e x t e r n a l d i s t u r b a n c e , r e s u l t i n g i n a t r a c k i n g f o m l a t i o n of t h e regula- t o r problem. They cons t r a ined t h e f i n a l miss d i s - t ance t o be zero b u t , t o ach ieve t h i s , needed t o know t h e t a r g e t a c c e l e r a t i o n h i s t o r y p r e c i s e l y . '4

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B a l b i m i e , S h e p o r a i t i s , and Merriam (1975) gave a means t o o b t a i n weight ing ma t r i ces which r e s u l t i n " c l a s s i c a l type" c o n t r o l ga ins i n an LQG formula t ion . S h e p o r a i t i s , B a l b i r n i e , and Liebner (1976) cons idered a quadra t i c c o s t on t h e ang le of a t t a c k . They used a second o rde r Newton-Raphson scheme t o s o l v e the r e s u l t i n g op t imiza t ion problem. L

Speyer (1976) app l i ed h i s l i nea r -exponen t i a l - Gaussian (LEG) c o n t r o l l e r t o a t e rmina l guidance problem. Rather than minimizing t h e expec ta t ion o f a quadra t i c form J2 (as given p r e v i o u s l y ) ,

~ p e y e r ' s LEG formula t ion minimizes E ,ex,(lt,J

where p i s a scalar. The dynamical model i s aga in l i n e a r . Speyer used a Kalman f i l t e r t o o b t a i n e s t i m a t e s of S t a t e v a r i a b l e s f o r feedback. H i s c o n t r o l l e r d id n o t en joy a s e p a r a t i o n p r i n c i p a l ; c o n t r o l ga ins depended upon the f i l t e r s t a t e covar iance . Speyer ' s LEG c o s t f u n c t i o n a l had t h e e f f e c t of very h e a v i l y weighing l a r g e excurs ions and thus reduced t h e t a i l s of t h e t e rmina l m i s s d i s t r i b u t i o n .

{ 4

Youngblood (1977) used very s i m p l i f i e d l i n e a r dynamics t o compute i n n e r launch boundaries. Youngblood's r e p o r t c o n t a i n s a d e s c r i p t i o n of t h e F le tcher -Powel l f u n c t i o n a l op t imiza t ion method.

F i ske developed a number of guidance and esti- mat ion schemes based upon LQG formula t ions wi th vary ing degrees of model complexity. form s o l u t i o n s f o r t h e c o n t r o l l e r s are g iven i n t h e r e p o r t , a l lowing one t o examine t h e e f f e c t of model parameters on the ga ins . F i ske also p r e s e n t s a s t o c h a s t i c guidance law, wherein the t a r g e t acce l - e r a t i o n i s assumed t o be a f i r s t o r d e r system d r i v e n by whi te no ise . In a d d i t i o n , a non l inea r law, based upon n u l l i n g p ro jec t ed miss d i s t a n c e over a s i n g l e c o n t r o l i n t e r v a l was given. York and P a s t r i c k (1977) looked a t t h e problem of minimizing t h e t e rmina l m i s s d i s t a n c e and t h e d e v i a t i o n of t he m i s s i l e from a d e s i r e d o r i e n t a t i o n a t t h e f i n a l t i m e . t h a t had f i n i t e t ime de lay . In f a c t , t h e i n c r e a s e and dec rease i n time de lay had i n t e r e s t i n g r ami f i - c a t i o n s on t h e s o l u t i o n . The ang le of a t t a c k assumption was i n v e s t i g a t e d and , a l though not so lved a n a l y t i c a l l y i n c losed form, t h e system was der ived .

The c losed

L/

A f o w l a t i o n was g iven f o r a sys tem

For completeness, t h e fo l lowing example sum- mar izes a t y p i c a l op t imal c o n t r o l law fo- la t ion. The geometry of t h e t a c t i c a l m i s s i l e - t a r g e t pos i - t i o n i s g iven i n F igu re 10 . Assume t h a t t h e ang le o f a t t a c k i s sma l l and thus can be neglec ted ( t h i s assumption w i l l be cons ide red l a t e r ) , and choose the f a l lowing set o f v a r i a b l e s :

where Yd is t h e p o s i t i o n v a r i a b l e from t h e m i s s i l e

t o t h e t a r g e t p ro j ec t ed on t h e ground;

p o s i t i o n v a r i a b l e of t h e t a r g e t ;

Y t i s t h e i Ym is t h e

p o s i t i o n y a r i a b l e of t h e missile p ro jec t ed on t h e ground;

t o t h e t a r g e t v e l o c i t y p r o j e c t e d on t h e ground; AL i s t h e l a t e r a l a c c e l e r a t i o n o f t h e m i s s i l e ; 0 is t h e body a t t i r u d e angle of the missile; and a is t h e ang le of a t t a c k of t h e m i s s i l e shown i n F igu re 10.

Yd is the d e r i v a t i v e o f Yd, t h e missile

F igure 10, Geometry of T a c t i c a l Missile Targe t P o s i t i o n s

Th i s op t imal c o n t r o l problem w i l l have a con- t r o l l e r of t h e form

where Cy, C?, CO,

c i e n t s chosen t o minimize t h e c o s t f u n c t i o n a l

are t ime-varying c o e f f i -

t f

J = Yd( t f ) + y B ( t f ) + B u 2 ( t ) d t 1 (12) 0

For t h e case where t h e ang le o f a t t a c k probably cannot be ignored e . g . , f o r t h e l a r g e r t a c t i c a l missile, t h e system of equa t ions should inc lude t h e a n g l e of a t t a c k a. I n a d d i t i o n , because it i s f e a s i b l e t o achieve only a small a n g l e of a t t a c k a t impact, a r easonab le performance index t o be mini- mized would seem t o he

f t

2 2 u 2 ( t ) d t . 2 J = c lYd( t f ) + c2e ( t f ) + c3a ( t f ) + c4

t o

(13)

The performance o b t a i n a b l e from t h e op t ima l guidance l a w formulated and s imula ted f o r Equat ion (13) was comparabla and even b e t t e r than t h e PNG law i n terms of m i s s d i s t a n c e . Add i t iona l ly , it had t h e added f e a t u r e of meeting a c o n s t r a i n t o r t h e impact ang le which t h e F'NG law could not ach ieve . v i a s i m u l a t i o n on an a l l - d i g i t a l 6-DOF m i s s i l e s i m u l a t i o n t o be w i t h i n 1 deg of t h e d e s i r e d impact ang le and wi th in 1 f t of t h e d e s i r e d m i s s d i s t a n c e . Other r e s e a r c h e r s have co r robora t ed these r e s u l t s .

The performance of any r e a l i s t i c op t imal con-

I n p a r t i c u l a r t h e impact angle was shown

t r o l law i n a m i s s i l e a p p l i c a t i o n i s dependent on the e s t i m a t i o n of f i n a l time o r , e q u i v a l e n t l y , o n

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t ime-to-go. T y p i c a l l y , an e s t i m a t e o f t h e range between t h e t a r g e t and m i s s i l e , and t h e r a t e o f change of t h i s range are ob ta ined from rada r or o t h e r ranging d e v i c e s ; t h e t ime-to-go e s t i m a t e is t h e n c a l c u l a t e d . T h i s e s t i m a t e works q u i t e we l l as long as t h e range and r ange - ra t e i n fo rma t ion a re accu ra t e . I n many i n s t a n c e s , however, t h e d a t a are contaminated by n o i s e e i t h e r c o v e r t l y , as i n t h e case of r a d a r j a m i n g d e v i c e s , or by t h e pro- c e s s i n g e l e c t r o n i c s . T h i s a d v e r s e l y impacts the e s t i m a t e of t ime-to-go and t h e op t ima l c o n t r o l law, and m i s s i l e performance s u f f e r s . P a s t r i c k and Yark (1977) p r e s e n t a d i s c u s s i o n of s e v e r a l a s p e c t s o f t h e problem i n t h e c o n t e x t of a r e a l i s t i c a p p l i c a - t i o n and p rov ide a n a l y t i c computer a lgo r i thms f o r i t s s o l u t i o n , as w e l l as a closed-form r e s u l t . Another more r e c e n t a t t empt t o e s t i m a t e t ime-to-go was made by Fiske. He also addres ses t h e p o s s i b i l i t y o f o b t a i n i n g t h i s v a r i a b l e by an i n t e n s i t y ranging technique.

Another d i f f i c u l t y with op t ima l guidance laws i s t h e i r s e n s i t i v i t y t o i n i t i a l c o n d i t i o n s , as shown by York (1978). The pr imary message t h e r e i n i s t h e s t r o n g need f o r a c c u r a t e modeling of t h e system and the importance of the s e l e c t i o n of numerical q u a n t i t i e s f o r t h e elements of t h e weight- i ng m a t r i c e s i n t h e performance index.

O the r Guidance Schemes

Whiting and Jobe (1972) cons ide red u s i n g a " v i r t u a l t a r g e t " approach t o guidance fo r a s h o r t range a i r - to -g round m i s s i l e .

P o u l t e r and Anderson (1976) cons ide red a non- l i n e a r d i f f e r e n t i a l game framework t o d e r i v e an opt imal s t e e r i n g l a w f o r a t e rmina l homing m i s s i l e . I n a p r i o r pape r , Anderson (1974) gave a method of upda t ing a d i f f e r e n t i a l game s o l u t i o n v i a l i n e a r i z e d two-point boundary v;lue problems.

For background m a t e r i a l , Froning and Gieseking (1973) gave a d e s c r i p t i o n o f a u t o p i l o t s t e e r i n g mechanisms f o r bank-to- turn m i s s i l e s . Gido, J a f f e , and Wilson (1974) provided computer programs t o produce a u t o p i l o t des igns u s i n g b o t h c l a s s i c a l (PNG) and modern (LQG) fo rmula t ions . George (1974) d i s - cussed t r e n d s i n IMU, guidance and c o n t r o l hardware.

Mahmoud (1977) desc r ibed a d u a l l e v e l , h i e r - a r c h i c a l o p t i m i z a t i o n scheme based on i n v a r i a n t imbedding which is used t o o b t a i n approximate solu- t i o n s t o many non l inea r c o n t r o l problems. and Vlach (1977) p re sen ted a new augmented p e n a l t y f u n c t i o n approach t o opt imal c o n t r o l problems. T h e i r fo rmula t ion i s a p p l i c a b l e t o f i n i t e dimen- s i o n a l o p t i m i z a t i o n problems with t e rmina l c o n s t r a i n t s .

Connor

d

Lansing and B a t t i n (1965) p re sen ted m c h background m a t e r i a l on random processes a p p l i e d t o automatic c o n t r o l . R a d b i l l and McCue (1970) provided background m a t e r i a l on q u a s i - l i n e a r i z a t i o n methods i n s o l v i n g coupled n o n l i n e a r two-Point boundary va lue problems.

Ch in ' s book on m i s s i l e d e s i g n (1961) has a s e c t i o n on m i s s i l e t r a n s f e r f u n c t i o n s which i s very u s e f u l i n a s s e s s i n g t h e c o n t r i b u t i o n of aerodynamic f a c t o r s and c o n t r o l s u r f a c e s t o m i s s i l e motion. Goodstein (1972) provided an overview of m i s s i l e c o n t r o l system development and, i n t h e same AGAIU) r e p o r t , Acus (1972) p re sen ted a d e s c r i p t i o n and comparison of i n e r t i a l guidance technology f o r t a c t i c a l m i s s i l e s .

111. Discuss ion

A survey has been made of guidance laws app l i ed t o s h o r t range t a c t i c a l m i s s i l e s , and they have been organized i n t o f i v e c a t e g o r i e s f o r convenience of d i s c u s s i o n .

To b e t t e r p l a c e them i n r e l a t i v e p e r s p e c t i v e , F igu re 11 p r e s e n t s t h e hardware requirements f o r mechanizing them. Though t h e concepts are reason- a b l y s e l f exp lana to ry , i t should be noted t h a t t h e BR and CLOS concep t s are conf igu red wi th a f t sensor and o p t i c s while t h e o t h e r s are forward. Also, t h e opt imal guidance scheme sugges t s t h e need f o r a microprocessor t o hand le t h e more complex guidance a lgo r i thms .

Guidance concept c o n f i g u r a t i o n comparisons dep ic t ed i n t h e Table f e a t u r e t h e r e l a t i v e c o s t and o t h e r items t h a t show complexity and performance. With t h e excep t ion of t h e microprocessor requirement

OPTICS SENSOR AND OPTICS

c MAIN BODY + M A I N BODY SENSOR AND OPTICS

f FINS c

C SENSOR AND r8\ OPTICS r 2 D O F G Y R O

.-MICRO- PROCESSOR

- M A I N BODY

Figure 11. Ins t rumen ta t ion Conf igu ra t ion Concepts

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R. B . Asher and J . H. Matuzewski, "Optimal Guidance E . P. Cunningham, "Lambda Mat r ix Terminal Con t ro l of F i n i t e Bandwidth Systems with Zero Terminal Miss," f o r M i s s i l e Guidance," J. S p a c e c r a f t and Rockets,

pp. 119-121, Jan. 1968. J o i n t Auto. Con t ro l Cbnf. , pp. 4-10, 1974.

R. B. Asher and J. H. Matuzewski, "Optimal Guidance f o r Maneuvering T a r g e t s , " J. Spacec ra f t and Rockets, pp. 204-206, March 1974.

M . Athans, "The S t a t u s o f Optimal Con t ro l Theory and A p p l i c a t i o n s f o r D e t e r m i n i s t i c Systems," IEEE Trans . Auto. C o n t r o l , pp. 580-596, J u l y 1966.

M . Athans, "On Optimal A l l o c a t i o n and Guidance Laws for L i n e a r I n t e r c e p t i o n and Rendeivous Problems," IEEE Trans. Aerospace and E l e c t r o n i c Systems, pp. 843-853, Sep t . 1971.

E . I. Axelband and F. W . Hardy, "Quasi Optimum Pro- p o r t i o n a l Navigat ion," 2nd H a w a i i I n t . Conf. Sys . Sc i ences , pp. 417-421, 1969.

E . I . Axelband and F. W . Hardv. "ODtima1 Feedback _, . Missile Guidance," 3rd H a w a i i 1 C . Conf. Svs. S c i e n c e s , pp. 874-877, 1970.

E . C . B a l b i r n i e , L . P . S h e p o r a i t i s and C. W. Merriam, 'Werging Convent ional and Optimal Con t ro l Techniques f o r P r a c t i c a l M i s s i l e Terminal Guidance," A I A A Guidance and Con t ro l Conf., A I A A paper 75-1127, 1975.

G . Bashein and C. P. Neuman, "Linear Feedback Guid- ance f o r I n t e r c e D t i o n and Rendezvous i n a S t o c h a s t i c Environment," 1st H a w a i i I n t . Conf. Sys. Sc iences , pp. 654-657, 1968.

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