HIGH ANGLE-OF-ATTACK STABILITY-AND-CONTROL ANALYSIS* R o b e r tF .S t e n g e l T h eA n a l y t i cS c i e n c e sC o r p o r a t i o n SUMMARY M e t h o d so fl i n e a rs y s t e m sa n a l y s i s are a p p l i e d t o m a t h e m a t i c a l m o d e l s . o f a i r c r a f t f l y i n g a t h i g h a n g l e of attack andmaneuver rate.
F i r s t - o r d e rl o n g i t u d i n a la n dl a t e r a l - d i r e c t i o n a lc o u p l i n g is ob- t a i n e d by l i n e a r i z i n g t h e c o m p l e t e n o n l i n e a r e q u a t i o n s o f m o t i o n a b o u t a g e n e r a l i z e d( q u a s i - s t e a d y ) t r i m p a i n t . "Open-loop" sta- b i l i t y b o u n d a r i e s are d e f i n e d u s i n g t h e l i n e a r d y n a m i c e q u a t i o n s , a n d " p i l o t - i n - t h e - l o o p " e f f e c t s are p r e s e n t e d .S t a b i l i t y augmen- t a t i o n s t r u c t u r e s f o r m a n e u v e r i n g f l i g h t c o n d i t i o n s are shown t o be d e f i n e d r e a d i l y u s i n g o p t i m a l c o n t r o l t h e o r y .
INTRODUCTION H i g h - p e r f o r m a n c ea i r c r a f t are s u s c e p t i b l et od e g r a d e df l y i n g q u a l i t i e sd u r i n gm a n e u v e r i n gf l i g h tf o r a number of reasons. T h e a i r c r a f t f l i e s a t h i g ha n g l eo fa t t a c k , a , whereaerodynamicflow f i e l d s are complexand are s e n s i t i v e t o small v a r i a t i o n s i n f l i g h t c o n d i t i o n .L a t e r a l - d i r e c t i o n a l modes o fm o t i o n are a f f e c t e d by t h e n o s e - h i g ha t t i t u d e ,a n dt h ed e s i r a b l ec o n t r o l moments due t o a i l - e r o n a n d r u d d e r may be o v e r s h a d o w e db ys i g n i f i c a n ta d v e r s e e f f e c t s .
L a r g e r o l l rates may be commanded f o r r a p i d o r i e n t a t i o n o f t h e l i f t v e c t o r , a n d t h e r e s u l t i n g g y r o s c o p i c e f f e c t s c o u p l e t h e l o n g i t u - d i n a la n dl a t e r a l - d i r e c t i o n a l modes of motion. To compound t h e a b o v e d i f f i c u l t i e s , t h e p i l o t m u s t a d a p t h i s c o n t r o l s t r a t e g i e s t o v a r y i n g a i r c r a f t d y n a m i c sa n dc o n t r o lr e s p o n s e s a t h i g h a .
R i g o r o u s s o l u t i o n s t o t h e a i r c r a f t ' s e q u a t i o n s o f m o t i o n are d i f f i c u l tt oo b t a i ni nm a n e u v e r i n gf l i g h t .T h e s ed i f f e r e n t i a l e q u a t i o n s h a v e c o e f f i c i e n t s w h i c h are n o n l i n e a ra n d time v a r y i n g ; h e n c e ,s o l u t i o n so ft h eg e n e r a le q u a t i o n sr e q u i r e direct i n t e g r a - t i o n ,e i t h e rb yn u m e r i c a l or a n a l o gc o m p u t a t i o n .T h er e s u l t i n g time h i s t o r i e s d e s c r i b e t h e e v o l u t i o n of a i r c r a f t m o t i o n s f o r g i v e n c o n - trols, d i s t u r b a n c e s , a n d i n i t i a lc o n d i t i o n s ,a n de a c hc h a n g ei na n y o ft h e s eq u a n t i t i e sl e a d st o a new t i m e h i s t o r y .C o n s e q ' u e n t l y ,t h e are v a l u a b l e f o r d e f i n i n g s p e c i f i c n o n l i n e a r - t i m e - v a r y i n g e q u a t i o n s f l i g h tp a t h s ,b u tt h e i rc o m p l e x i t yc a no b s c u r et h ei d e n t i f i c a t i o n o f t h e u n d e r l y i n g m e c h a n i s m s w h i c h g o v e r n a i r c r a f t r e s p o n s e .
* T h i s work was s u p p o r t e d b y C o n t r a c t s N o . NAS1-13618, NASA Langley R e s e a r c hC e n t e r ,a n d N00014-75-C-0432, O f f i c e o f Naval R e s e a r c h .
I I I 1 ll11lllll11l1l1l T h i s paper p r e s e n t s new r e s u l t s u s i n g l i n e a r - t i m e - i n v a r i a n t d y n a m i ce q u a t i o n s which r e t a i n much of t h e c o u p l i n g of t h e n o n l i n e a r e q u a t i o n s b u t which are amenable t o t h e c o m p r e h e n s i v et e c h n i q u e s of l i n e a rs y s t e m sa n a l y s i s . The s t a b i l i t y characteristics of two con- temporaryhigh-performance a i r c r a f t are compared a t v a r i o u s maneu- v e r i n g f l i g h t c o n d i t i o n s . The effects of i n c r e a s i n g a n g l e of attack o n c o n t r o l r e s p o n s e are d e m o n s t r a t e d ,a n d a d e t a i l e d mathe- matical modelof t h e human p i l o t is a p p l i e d t o t h e p r e d i c t i o n of f l y i n gq u a l i t i e s . The f u l l yc o u p l e dl i n e a r a i r c r a f t m o d e la l s o forms t h e basis for d e s i g n i n g s t a b i l i t y a u g m e n t a t i o n s y s t e m s t h a t i m p r o v eh a n d l i n gq u a l i t i e s and p r e v e n td e p a r t u r ef r o mc o n t r o l l e d f l i g h t .
SYMBOLS F f u n d a m e n t a l m a t r i x o f l i n e a r - t i m e - i n v a r i a n t s y s t e m - f v e c t o r of n o n l i n e a r e q u a t i o n s o f m o t i o n G c o n t r o l effect m a t r i x of l i n e a r - t i m e - i n v a r i a n t s y s t e m I i d e n t i t y m a t r i x K s t a b i l i t y a u g m e n t a t i o n g a i n m a t r i x PYqYr b o d y - a x i s a n g u l a r rates (roll, p i t c h , yaw) t time U - c o n t r o l v a r i a b l e v e c t o r u , v , w b o d y - a x i s v e l o c i t i e s ( a x i a l , l a t e r a l , n o r m a l ) X s t a t e v a r i a b l e v e c t o r - x E , y E , z E t r a n s l a t i o n a l p o s i t i o n ( f o r w a r d , l a t e r a l , v e r t i c a l ) Z e i g e n v e c t o r - a a n g l e o f a t t a c k B s i d e s l i p a n g l e
x e i g e n v a l u e
@ , e , $ E u l e r a n g l e s (roll, p i t c h , yaw) ( > T t r a n s p o s e o f a v e c t o r ( ‘ 1 d e r i v a t i v e w i t h respect t o time A ( 1 p e r t u r b a t i o n q u a n t i t y EQUATIONS OF MOTION T h ef u n d a m e n t a le x p r e s s i o n of r i g i d - b o d ye q u a t i o n s of m o t i o n c o n t a i n sn o n l i n e a ra n dt i m e - v a r y i n gt e r m s ,b u ts i m p l i f i c a t i o n sc a n be c o n s i d e r e di nc e r t a i n cases. F i g u r e 1 i l l u s t r a t e s t h e classes of models which c a n be c o n s i d e r e d f o r a n a l y z i n g f l i g h t m o t i o n s .
If t h e d y n a m i cc o e f f i c i e n t s are c h a n g i n gr a p i d l y w i t h t i m e , i n c o m p a r i s o nw i t ht h e time scale o fm o t i o n s , t h e d y n a m i cm o d e lm u s t b e time v a r y i n g ; i f t h e c o e f f i c i e n t s are r e l a t i v e l y c o n s t a n t , a t i m e - i n v a r i a n t model w i l l s u f f i c e . If f l i g h tm o t i o n se v i d e n c e t h e s u p e r p o s i t i o nc h a r a c t e r i s t i c ;i . e . , i f d o u b l i n g t h e i n p u t d o u b l e st h eo u t p u t ,t h e nl i n e a r models c a n be u s e d ; i f n o t , t h e dynamic model m u s t be n o n l i n e a r .
T h en o n l i n e a re q u a t i o n s of m o t i o nc a n be assembled i n t h e s i n g l e "state-space" ( v e c t o r )e q u a t i o n
- & = "- f ( x , u , t )
where x is a c o l u m nv e c t o r of s t a t e ( o r m o t i o n )v a r i a b l e s , u is a v e c t o r of c o n t r o l v a r i a b l e s , f is t h e v e c t o r of dynamicequa- t i o n s ,a n d t is time. F o r rigid-body m o t i o n , f a n d x each h a v e 1 2 e l e m e n t s r e p r e s e n t i n g t h e d y n a m i cr e l a t i o n s K i p sa n d associated v a r i a b l e s f o r t r a n s l a t i o n a la n dr o t a t i o n a lk i n e m a t i c sa n dd y n a m i c s .
U s i n gc o n v e n t i o n a ln o t a t i o n f o r e a r t h - r e l a t i v e p o s i t i o n , earth- body E u l e ra n g l e s ,a n d body-axis r a t e s , t h e s t a t e v e c t o rc a n be d e f i n e d a s w h e r e ( ) T d e n o t e s t h e column vector t r a n s p o s e , i. e . , a row v e c t o r .T h ec o n t r o lv e c t o r ,u ,c o n t a i n s a t l e a s t t h r e e e l e m e n t s f o r r o t a t i o n a lc o n t r o la b o u t all axes. F u r t h e r d e t a i l s of t h e n o n l i n e a re q u a t i o n s of m o t i o nc a n be f o u n d i n r e f e r e n c e s 1 a n d 2 .
N o n l i n e a r - t i m e - i n v a r i a n t models are u s e f u l i f a m p l i t u d e - d e p e n d e n t e f f e c t s c a n n o t be i g n o r e db u t time v a r i a t i o n s of t h e c o e f f i c i e n t s are n e g l i g i b l e . N o n l i n e a r p h e n o m e n a , s u c h as limit c y c l e s ,s u b h a r m o n i cr e s p o n s e ,j u m pr e s o n a n c e ,a n dn o n l i n e a r cross c o u p l i n g may be r e s p o n s i b l e f o r s u c h a i r c r a f t b e h a v i o r as wing rock, p o r p o i s i n g ( o r b u c k i n g ) ,j u m pr e s p o n s e of r o l l r a t e t o a i l e r o ni n p u t ,a n df o r c i n g of l o n g i t u d i n a l modes by l a t e r a l o s c i l l a t i o n sd u r i n gs y m m e t r i c ,w i n g s - l e v e l f l i g h t ( r e f s . 3 a n d 4 ) .
T h es t a b i l i t ya n dc o n t r o l of small v a r i a t i o n s a b o u t t h e r e f e r e n c e f l i g h t p a t hc a n be i n v e s t i g a t e d u s i n g l i n e a r d y n a m i c models whose c o e f f i c i e n t s may v a r y i n time as t h e f l i g h t c o n d i t i o n v a r i e s .L i n e a r - t i m e - v a r y i n g models c o u l d be r e q u i r e d t o assess a i r c r a f t s t a b i l i t y d u r i n g r a p i d a c c e l e r a t i o n ,d e c e l e r a t i o n , o r f o r s t e e p ,h i g h - s p e e df l i g h tp a t h s ,i nw h i c hc h a n g i n g a i r d e n s i t y a f f e c t s s y s t e m d y n a m i c s ( r e f s . 5 t o 7 ) . R e f e r e n c e 2 i n d i c a t e s t h a t t h e a c c e l e r a t i o n s a s s o c i a t e d w i t h a i r c o m b a tm a n e u v e r i n g i n t r o d u c et i m e - v a r y i n gc o e f f i c i e n t si nl i n e a rm o d e l s ;h o w e v e r , e x p l i c i tt i m e - v a r y i n gd y n a m i c e f f e c t s were n o tf o u n d t o be s i g n i f i c a n t ,a n d a l l major c o n c l u s i o n sr e g a r d i n gt h ef l i g h t s t a b i l i t y o f t h e s u b j e c t a i r c r a f t c o u l d be drawn from t h ee q u i v a - l e n tt i m e - i n v a r i a n tm o d e l .
T h ea i r c r a f te q u a t i o n so fm o t i o nc a n be expressed i n l i n e a r form b yp e r f o r m i n g a T a y l o r s e r i e s e x p a n s i o no fe q u a t i o n (1) and r e t a i n i n go n l yf i r s t - o r d e r terms. T h ee x p a n s i o nr e s u l t si n + f A x + f Au - -X -u - where fx a n d f u a r e p a r t i a l d e r i v a t i v e matrices w i t h respect t o t h e sta€e a n dF o n t r o l ,r e s p e c t i v e l y ,e v a l u a t e da l o n gt h en o m i n a l f l i g h tp a t h .T h en o m i n a lf l i g h tp a t hs a t i s f i e se q u a t i o n (l), and t h ed y n a m i c so f small p e r t u r b a t i o n sf r o mt h ef l i g h tp a t h are described by A; - = F A x + G Au - -
w h e r e fx and f u are d e n o t e db y F a n d G a n d may c o n t a i n time-
v a r y i n g - c o e f f i E i e n t s .
L i n e a r - t i m e - i n v a r i a n t models d e s c r i b e s m a l l - p e r t u r b a t i o n s t a b i l i t y i n t h e v i c i n i t y of a s i n g l e f l i g h t c o n d i t i o n ,a n dc a n b e u s e f u l f o r p r a c t i c a la p p r o x i m a t i o n of s y s t e md y n a m i c s ,f o r s e n s i t i v i t ya n a l y s e s ,a n d f o r c o n t r o l s y s t e m d e s i g n . ( R e f e r e n c e s 8 t o 11 u s e l i n e a r models i nt h ee x a m i n a t i o no fd y n a m i cc o u p l i n g phenomena.) A s shown i nf i g u r e 1, c o e f f i c i e n t so ft h el i n e a r model are d e f i n e db y t h e l o c a l s l o p e s a t a g i v e n f l i g h t c o n d i t i o n , a n d t h e p r i n c i p l e of s u p e r p o s i t i o n a p p l i e s i n c h a r a c t e r i z i n g f l i g h t m o t i o n s .
T h ei m p o r t a n c e of l i n e a r i z i n ga b o u t a g e n e r a l i z e d trim con- d i t i o n is i l l u s t r a t e d b y f i g u r e 2 , w h i c hp r e s e n t sl i n e a ra n d n o n l i n e a rr e s p o n s e s t o a l a r g er u d d e ri n p u t .T h en o n l i n e a r m o d e l ' s t r a c e d e m o n s t r a t e ss i g n i f i c a n tl o n g i t u d i n a l / l a t e r a l - d i r e c t i o n a lc o u p l i n g as well as large r e s p o n s e si n s i d e s l i p , 6 , a n d roll r a t e , p . T h e model which is l i n e a r i z e da b o u tt h ei n i t i a l " w i n g s - l e v e l "f l i g h tc o n d i t i o nd o e sn o tp o s s e s st h i sc o u p l i n g , and t h e m a g n i t u d e s of 6-p o s c i l l a t i o n s are u n d e r e s t i m a t e d( f i g .2 a ) .
U s i n gt h eg e n e r a l i z e d t r i m p r o c e d u r e * t o d e f i n e a l i n e a r i z a t i o n * T h i s trim p r o c e d u r e ,d e s c r i b e di nr e f e r e n c e 1, n u m e r i c a l l yd e f i n e s mean v a l u e s of t r a n s l a t i o n a l a n d r o t a t i o n a l r a t e s w h i c hm i n i m i z e t r a n s l a t i o n a l a n d r o t a t i o n a l a c c e l e r a t i o n s f o r g i v e n E u l e r a n g l e s a n dc o n t r o ls e t t i n g s .
p o i n t 4 sec i n t o t h e m a n e u v e ri n t r o d u c e st h em i s s i n gl o n g i t u d i n a l / l a t e r a l - d i r e c t i o n a lc o u p l i n g ,l e a d i n gt os u b s t a n t i a l l yi m p r o v e d modeling of t h e n o n l i n e a r s y s t e m ' s o s c i l l a t i o n s( f i g .2 b ) . T h e c o u p l e d l i n e a r s y s t e m ' s r e s p o n s ee v e n t u a l l yd i v e r g e sf r o mt h e n o n l i n e a rr e s p o n s e ,b u tt h ef a c tt h a t it s t a y s i n t h e p r o p e r n e i g h b o r h o o df o rs e v e r a ls e c o n d ss u g g e s t s t h e u t i l i t y o f t h e l i n e a r - t i m e - i n v a r i a n tm o d e l i n d e f i n i n g l o c a l s t a b i l i t y , i n p r o - v i d i n g a b a s e l i n e f o r c o n t r o l s y s t e m d e s i g n ,a n d i n a n a l y z i n g p i l o t i n g e f f e c t s .
MANEUVERING EFFECTS ON STABILITY C o n f i g u r a t i o n - d e p e n d e n te f f e c t sa n d common a t t r i b u t e s o f can b es e e n i n c o m p a r i s o n so ft h es t a b i l i t y m a n e u v e r i n gf l i g h t b o u n d a r i e s o f t w o c o n t e m p o r a r y h i g h - p e r f o r m a n c e a i r c r a f t . Air- c r a f t A is a small, s u p e r s o n i c a i r s u p e r i o r i t y f i g h t e r ( r e f . 1); a i r c r a f t B is a l a r g e r s u p e r s o n i c f i g h t e r w i t h similar m i s s i o n ( r e f . 2 ) .
The s t a b i l i t y b o u n d a r i e so f t h e a i r c r a f t are d e f i n e d by t h e c o n d i t i o n s a t w h i c ht h e r e a l p a r t so f t h e e i g e n v a l u e s ( o r r o o t s ) of t h e1 . i n e a r - t i m e - i n v a r i a n t d y n a m i c e q u a t i o n( e q . ( 4 ) ) change s i g n .T h e 1 2 e i g e n v a l u e s , h i ( i = 1 t o1 2 ) ,o ft h ee q u a t i o n are c o m p l e xn u m b e r s ,e a c ho fw h i c hs a t i s f i e s t h e f o l l o w i n ge q u a t i o n : d e t h i I - F = 0 ; i = 1 t o 1 2 w h e r e I is t h e i d e n t i t y m a t r i x .E a c ho s c i l l a t o r y mode is r e p r e - s e n t e d bytwocomplex-conjugate h i , w h i l ee a c hc o n v e r g e n t( o r d i v e r g e n t ) mode is r e p r e s e n t e d by o n e r e a l X i . The X i d e s c r i b e t h e time scales a n ds t a b i l i t yo ft h en o r m a lm o d e so fm o t i o n , w h i c hu s u a l l yp a r t i t i o ni n t o a l o n g i t u d i n a l set ( s h o r tp e r i o d o s c i l l a t i o n ,p h u g o i do s c i l l a t i o n ,a n dp u r ei n t e g r a t i o n s ( X i = O ) f o rr a n g ea n da l t i t u d e )a n d a l a t e r a l - d i r e c t i o n a l s e t (Dutch r o l l o s c i l l a t i o n , r o l l a n d s p i r a l ' c o n v e r g e n c e s ,a n dp u r ei n t e g r a - t i o n s f o r c r o s s r a n g e a n d yaw a n g l e )d u r i n g" w i n g s - l e v e l "f l i g h t .
T h e c o r r e s p o n d i n gc o m p l e x - v a l u e de i g e n v e c t o r s ,z i ,i n d i c a t e t h ei n v o l v e m e n to fe a c hm o t i o nv a r i a b l ei ne a c hm o d e ,s a t i s f y i n g t h ee q u a t i o n
( x i l - F z = o ; i = l t o 12
) -i
Forexample, i n s y m m e t r i c f l i g h t t h e twocomplex-conjugateeigen- v e c t o r s a s s o c i a t e d w i t h t h e s h o r t p e r i o d o s c i l l a t i o n n o r m a l l y i n v o l v e l a r g e p i t c h r a t e a n dn o r m a lv e l o c i t yc o m p o n e n t s (Aq and w ) , small p i t c ha n g l ea n da x i a lv e l o c i t yc o m p o n e n t s (A0 andAu), andnoneof t h e r e m a i n i n gl o n g i t u d i n a la n dl a t e r a l - d i r e c t i o n a l
c o m p o n e n t s .T h u s ,t h e zi are s a i d t o characterize t h e "shape" of
e a c hm o d e ,w h i l et h e X i c h a r a c t e r i z e t h e g r o w t h o r decay of t h e magnitude of zi.
S e v e r a le f f e c t s of m a n e u v e r i n gf l i g h t c a n b e n o t e d . I n - c r e a s i n g mean a n g l e of a t t a c k o r p i t c h r a t e a l t e r s t h e time s c a l e , s t a b i l i t y ,a n ds h a p eo fe a c h mode, b u t it d o e sn o tc o u p l et h e l o n g i t u d i n a la n dl a t e r a l - d i r e c t i o n a l s e t s , as symmetry is main-
t a i n e d ;t h e r e f o r e ,t h e number of non-zero elements i n e a c h zi is
u n c h a n g e d . I n t r o d u c i n g mean s i d e s l i p ,r o l lr a t e , o r yaw r a t e leads t o f u l l c o u p l i n g , a n d it a l t e r s a i r c r a f ts t a b i l i t y ;h e n c e , t h e number of non-zero elements i n e a c h Z i i n c r e a s e s . I t is o f t e nf o u n d t h a t s y m m e t r i c v a r i a t i o n s have g r e a t e s t e f f e c t on s t a b i l i t y( d u et oc h a n g e s i n a e r o d y n a m i cf l o wf i e l d s ) , w h i l e asymmetric v a r i a t i o n sh a v eg r e a t e s te f f e c t on mode s h a p e ( a s a c o n s e q u e n c eo fi n e r t i a lc o u p l i n g ) .
F i g u r e 3 i l l u s t r a t e s t h e e f f e c t s o f a. a n d Bo on t h eo p e n - l o o p s t a b i l i t y b o u n d a r i e so ft h e two a i r c r a f t a t a n a l t i t u d e o f 6100 m . The d i f f e r e n tt r u ea i r s p e e d s , V o , used i n a n a l y s i s have small e f f e c t on t h es t a b i l i t yb o u n d a r i e s ,a l t h o u g h t h e d i f f e r i n g d y n a m i c p r e s s u r e s a f f e c t t h e n a t u r a l f r e q u e n c i e s a n d time c o n s t a n t s of t h e n o r m a lm o d e s .A i r c r a f t A e v i d e n c e s a n u n s t a b l ep h u g o i d mode a t low ct0 and a n u n s t a b l eD u t c h roll mode ( d u e t o n e g a t i v e damping) a t h i g h a , ( f i g . 3 a ) . T h e s e r e s u l t s are a p p a r e n t l y i n s e n s i t i v e t o small s i d e s l i pa n g l e s ;h o w e v e r ,t h e r e i s sub- s t a n t i a l change i n mode s h a p e( n o ts h o w n ) .A l t h o u g hc o n v e n t i o n a l names a r e u s e d , t h e "phugoid" mode c o n t r i b u t e s t o s i g n i f i c a n t r o l l a n g l em o t i o n , a n d t h e " D u t c h r o l l " mode c o n t a i n s n o n - t r i v i a l n o r m a lv e l o c i t yr e s p o n s e . A t h i g h e rs i d e s l i pa n g l e s ,t h e r e are c o u p l e d ,u n s t a b l eo s c i l l a t i o n sa n dd i v e r g e n c e s ,w h i c ha l s o are f o u n di n t h e r e s p o n s eo f Aircraft B ( f i g .3 b ) . The l a t t e r a i r - c r a f t is seen t o p o s s e s s u n s t a b l e D u t c h r o l l a n d r o l l - s p i r a l o s c i l l a t i o n b a n d s i n t h e v i c i n i t i e s o f 20- a n d 30-deg a n g l e of a t t a c k .B o t hi n s t a b i l i t i e s c a n b et r a c e dt o l o s s o fd i r e c t i o n a l r e s t o r i n g moments.
F i g h t e r a i r c r a f t are c a p a b l e o f h i g h r o l l r a t e , a n d a i r c o m b a tm a n e u v e r so f t e ni n c l u d es u c hm o t i o n s .F o r t h e a i r c r a f t t o r o l lw i t hc o n s t a n ta e r o d y n a m i ca n g l e s , t h e roll r a t e , pwo,must o c c u ra b o u tt h e w i n d x - a x i s( w h i c h is t h e same as t h e s t a b i l i t y x - a x i sf o rc o n s t a n tn o m i n a la e r o d y n a m i ca n g l e s ) .S i d e s l i pv a r i a - t i o n s a l s o are c o n s i d e r e d , since p i l o t i n g e r r o r c o u l d e a s i l y r e s u l ti nn o n - z e r o Bo d u r i n g a r o l l i n gm a n e u v e r .B o t hp o s i t i v e a n dn e g a t i v e pw0 are c o n s i d e r e d ,t oa c c o u n tf o rr o l l" i n t o " o r " o u to f " t h e s i d e s l i p .( T h e senses are o p p o s i t e i n t h e f i r s t c a s e ,i d e n t i c a l i n t h es e c o n d . ) The s t a b i l i t y b o u n d a r i e st h a tr e s u l tf r o mc o m b i n e dr o l l r a t e a n ds i d e s l i p are shown i n f i g u r e 4 , and it is i n t e r e s t i n g t o n o t e s t r i k i n g similarities between t h e b o u n d a r i e s o f t h e two a i r c r a f t .
T h e s eb o u n d a r i e s are a n t i s y m m e t r i ca b o u tt h eo r i g i n ( a s i n d i c a t e d i n f i g . 4 b ) b e c a u s e p o s i t i v e s i d e s l i p - p o s i t i v e r o l l r a t e h a st h e same e f f e c t o n a i r c r a f t s t a b i l i t y as n e g a t i v e s i d e s l i p - n e g a t i v e r o l l r a t e . B o t ha i r c r a f th a v es t a b l eb a n d s near f3,=10 d e g when pw0 is l a r g e , a b e n e f i c i a le f f e c to fc o u p l i n g .I n s t a b i l i t y is s u b s t a n t i a l when e a c h a i r c r a f t is s i d e s l i p p e d " i n t o " t h e r o l l .
T h ee i g e n v e c t o r s( n o ts h o w n )i n d i c a t et h a th i g h pw0 c a u s e st h e Ap c o m p o n e n to ft h es h o r t - p e r i o d mode t o b e g r e a t e r t h a n t h e A q component.
R e f e r e n c e s 1 and 2 p r e s e n t r e s u l t s r e g a r d i n g t h e s t a b i l i t y o f symmetric p u l l u p sa n d a maneuver known as a " r o l l i n g r e v e r s a l " ( h i g h - g p u l l u p , roll, i n v e r t e df l i g h t ,r o l l - o u t ,a n dp u l l u p ) .B o t h a i r c r a f t h a v e l a t e r a l - d i r e c t i o n a l i n s t a b i l i t i e s as a r e s u l t o f l a r g ep o s i t i v ep i t c h r a t e . A i r c r a f t A p o s s e s s e s a n u n s t a b l eD u t c h r o l l mode d u r i n gm o s to f t h e 22-secmaneuver. Aircraft B e v i d e n c e s a n u n s t a b l e s p i r a l mode f o r much o ft h em a n e u v e r ,w i t h a Dutch r o l li n s t a b i l i t yd u r i n gt h ef i n a lp u l l u p .T h e s er e s u l t sp r e d i c t i n c r e a s e d p i l o t w o r k l o a d d u r i n g m a n e u v e r i n g f l i g h t as a consequence o ff a c t o r sn o tn o r m a l l yc o n s i d e r e di ns t a b i l i t ya n dc o n t r o l a n a l y s e s .
CONTROL RESPONSE C o n t r o li n p u t time h i s t o r i e sd e m o n s t r a t et h e t r a n s i e n t re- s p o n s eo fA i r c r a f t B w i t hi n c r e a s i n g ao. The l a t e r a l c o n t r o l i n p u t is a p p l i e df o rt w os e c o n d sa n dt h e nr e m o v e d ,a n d a l l re- s p o n s e s a r e c o m p u t e du s i n gt h el i n e a r - t i m e - i n v a r i a n tm o d e l .
F i g u r e 5 shows t h el a t e r a l - d i r e c t i o n a lr e s p o n s e sw h i c hr e s u l t . A s a n g l eo fa t t a c ki n c r e a s e s ,b o t ht h e s y s t e m e i g e n v a l u e sa n dt h e c o n t r o le f f e c t i v e n e s sv a r y . The u n s t a b l eD u t c h roll mode is e x c i t e d a t a . o f 20 d e g ,a n dt h e yaw r a t e r e s p o n s e is r e v e r s e d .
A l t h o u g ht h ea i r c r a f t is a g a i n s t a b l e a t 25 d e g ,t h ec o n t r o l r e s p o n s e is p o o rd u et ot h el a r g ea d v e r s e yaw r e s p o n s e . A t 30 d e g , t h e l a t e r a l c o n t r o l i n p u t e x c i t e s t h e u n s t a b l e r o l l - s p i r a l o s c i l - l a t i o n ,w h i c hd o m i n a t e st h er e s p o n s e .
A d d i t i o n a lr e s u l t sc o n t a i n e d i n r e f e r e n c e 2 i n d i c a t e t h a t l o n g i t u d i n a l r e s p o n s e t o l a t e r a l c o n t r o l is a b o u t 50 p e r c e n to f d i r e c t i o n a lr e s p o n s el e v e l s when a0 and Bo are e a c h 1 0 d e g . When = 75 d e g / s e c , t h e r e s u l t i n g Aq is o n e - t h i r d as l a r g e as Ar, pw0 and A a r e s p o n s e i s t h r e e times g r e a t e r t h a n Af3 r e s p o n s e .
PILOTING EFFECTS T h e e f f e c t sw h i c ht h ep i l o th a so n a i r c r a f t s t a b i l i t y c a n b em o d e l l e d by a c l o s e d - l o o p s y s t e m w h i c h f e e d s b a c k a i r c r a f t m o t i o n st oa v a i l a b l ec o n t r o l s u r f a c e s . An o p t i m a lc o n t r o lp i l o t modelhas been u s e df o r t h i s p u r p o s e i n r e f e r e n c e s 2 a n d 1 2 ,a n d it c o n t a i n st h ef o l l o w i n ge l e m e n t s :a ne s t i m a t o r ,w h i c hp r o - cesses t h e p i l o t ' s o b s e r v a t i o n s t o p r o v i d e a n estimate o f t h e a i r c r a f t s t a t e ; a c o n t r o l l e r ,w h i c hm e c h a n i z e st h ep i l o t ' sr e g u - l a t i n gf u n c t i o n sa n d t r a n s m i t s t h e r e s u l t s t o t h e n e u r o m u s c u l a r dynamics;and a n e u r o m u s c u l a rm o d e l ,w h i c hr e p r e s e n t st h ed y n a m i c s o f t h e p i l o t ' s l i m b s .
I n v e s t i g a t i o n s o f p i l o t - a i r c r a f t i n s t a b i l i t y u s i n gt h ec o n - t r o l - t h e o r e t i cp i l o t model f a l l i n t o t w oc a t e g o r i e s :t h o s e i n which t h e p i l o t f a i l s t o s t a b i l i z e a n u n s t a b l e a i r c r a f t , a n d t h o s e i nw h i c ht h ep i l o td e s t a b i l i z e s a s t a b l e a i r c r a f t . I n t h e f i r s t c a s e , t h e p i l o t ' s time d e l a y ,o b s e r v a t i o nn o i s e ,n e u r o m u s c u l a r time c o n s t a n t s ,a n ds c a n n i n gf a c t o r s are i m p o r t a n tp a r a m e t e r s .
Assuming t h a t t h e a i r c r a f t ' s l i n e a r i z e d d y n a m i c s haveone or more u n s t a b l ee i g e n v a l u e s , t h e a n a l y s i s determines p i l o t p a r a m e t e r s f o r w h i c ht h eo p t i m a lc o n t r o lm o d e lf a i l st oe x i s t . The second c a t e g o r y is r e l a t e d t o t h e p i l o t ' s a b i l i t y t o a d a p t t o c h a n g i n g f l i g h t c o n d i t i o n s . P i l o t - i n d u c e d o s c i l l a t i o n s ( P I O ) a n d d e p a r t u r e s c a n o c c u rb e c a u s e a c o n t r o l s t r a t e g y w h i c h is a p p r o p r i a t e t o o n e f l i g h t c o n d i t i o n is d e s t a b i l i z i n g i n a n o t h e r .
The c o n t r o l - t h e o r e t i c p i l o t m o d e l c a n be u s e d t o a n a l y z e n o n a d a p t i n gp i l o tb e h a v i o r i n a s t r a i g h t f o r w a r d m a n n e r . I n t h e e x a m p l ec o n s i d e r e dh e r e ,t h ep i l o tm o d e l ' sc o n t r o ls t r a t e g y is f i r s td e t e r m i n e d a t a low-a, f l i g h tc o n d i t i o n .T h i ss t r a t e g y is f r o z e n ,a n dt h ea i r c r a f t ' sd y n a m i c s are a l l o w e d t o change. The s t a b i l i t y o f t h e p i l o t - a i r c r a f t s y s t e m is determined b y i t s e i g e n v a l u e s .
N o n a d a p t e d p i l o t i n g e f f e c t s o n p i l o t - a i r c r a f t s t a b i l i t y r e g i o n s c a n be p r e s e n t e d i n t h e a i r c r a f t ' s ao-Bo p l a n e .F i g u r e 6 shows t h e s t a b i l i t y r e g i o n su n d e r t h e a s s u m p t i o n t h a t t h e p i l o t is a d a p t e dt o a. = 1 0 d e g a n d Bo = 0 deg.The i n s t a b i l i t i e s of t h e l o n g i t u d i n a l modes(phugoid a n d s h o r tp e r i o d ) are t h e same i n b o t hc a s e s , s i n c e t h e a v a i l a b l e c o n t r o l is t h e same i n b o t h cases.
F i g u r e 6 s u g g e s t s t h a t i f t h e p i l o t m o d e ld o e sn o ta d a p t , a t some p o i n t low-a. p i l o t i n gp r o c e d u r ec o m b i n e dw i t ha d v e r s e yaw w i l l c a u s ea ni n s t a b i l i t y . I n f i g u r e6 at h i so c c u r s a t a 0 1 1 7 deg, and t h ei n c o r r e c tp r o c e d u r e is c h a r a c t e r i z e d by a nu n s t a b l e( c l o s e d - l o o p )s p i r a l mode. When t h r e ec o n t r o l s a r e u s e d , t h e i n s t a b i l i t y due t o i n c o r r e c t p r o c e d u r e d o e s n o t o c c u r u n t i l a o z 26deg, as shown i n f i g u r e6 b .T h e s er e s u l t s are c o m p a t i b l e w i t h e x p e r i m e n t a l r e s u l t s o b t a i n e d from manned s i m u l a t i o na n df l i g h t t e s t of A i r c r a f t B .
STABILITY AUGMENTATION FOR DEPARTURE PREVENTION F l y i n g q u a l i t i e s c a n be i m p r o v e db yu s i n ga na u t o m a t i c s y s - tem t o c o m p e n s a t ef o rv a r i a t i o n s i n a i r c r a f t dynamic character- istics. C o n t r o ll o g i cf o r a D e p a r t u r e - P r e v e n t i o nS t a b i l i t y AugmentationSystem (DPSAS) c a n be d e v e l o p e du s i n go p t i m a lc o n - t r o lt h e o r y ( r e f s . 1 and 2 ) . The l i n e a r - o p t i m a lr e g u l a t o r is a feedback c o n t r o l l a w of t h e form where K is a g a i n m a t r i x which scales feedback a n dc r o s s f e e d terms f o r p r o p e r s t a b i l i z a t i o n a n d c o m p e n s a t i o n o f t h e a i r c r a f t ' s motion ( r e f . 1 3 ) . K v a r i e st om a i n t a i ng o o df l y i n gq u a l i t i e s f o rl a r g ev a r i a t i o n si nm a n e u v e r i n gc o n d i t i o n s ,g u a r a n t e e i n g c l o s e d - l o o ps t a b i l i t ya n da c c o u n t i n g f o r a l l s i g n i f i c a n t l o n g i - tudinal/lateral-directional c o u p l i n g .
Comparisonsof DPSAS c l o s e d - l o o pr e s p o n s e w i t h open-loop r e s p o n s eo f t h e s u b j e c t a i r c r a f t a r e p r e s e n t e d i n f i g u r e s 7 a n d 8. Each a i r c r a f t is p e r f o r m i n g a c o n s t a n t - r o l l - r a t e maneu- v e r a n d is s u b j e c t e dt o a A B i n i t i a lc o n d i t i o no f 1 d e g . The roll r a t e leads t o s u b s t a n t i a l l o n g i t u d i n a l r e s p o n s e f o r b o t h a i r c r a f t , w i t h Aircraft A e x h i b i t i n g a l i g h t l y damped o s c i l l a - t i o n i n t h e u n a u g m e n t e dc o n d i t i o na n d Aircraft B p o s s e s s i n g a r e a l d i v e r g e n c e .I nb o t hc a s e s , t h e DPSAS q u i c k l y damps a l l p e r t u r b a t i o n s ,u s i n gg a i n sw h i c h are s p e c i f i c t o t h e a i r c r a f t and f l i g h t c o n d i t i o n . L i n e a r - t i m e - i n v a r i a n t d y n a m i c m o d e l s c a n be u s e d t o d e v e l o p t h e n e c e s s a r yv a l u e so f K t h r o u g h o u t t h e f l i g h te n v e l o p e ,a n dg a i n s c a n be s c h e d u l e di n t h e f l i g h t c o n t r o l s y s t e m t o m i n i m i z e t h e p r o b a b i l i t y o f d e p a r t u r e .
CONCLUSION The a n a l y s i so f a i r c r a f t % l y i n g a t h i g ha n g l eo f a t t a c k andmaneuver r a t e c a n be aided b y c o n s i d e r i n g f u l l y c o u p l e d l i n e a r - t i m e - i n v a r i a n t d y n a m i c m o d e l s . The c o u p l e d l i n e a r equa- t i o n s are nomore d i f f i c u l t t o h a n d l e t h a n u n c o u p l e d l o n g i - t u d i n a la n dl a t e r a l - d i r e c t i o n a l sets when "state-space" methods are u s e d ,y e t t h e y c a p t u r e s i g n i f i c a n t aspects o f maneuvering f l i g h t which o t h e r w i s e w o u l d r e q u i r e the s o l u - t i o n of n o n l i n e a re q u a t i o n s of m o t i o n . N e w i n s i g h t s regard- i n go p e n - l o o ps t a b i l i t yb o u n d a r i e s ,h a n d l i n gq u a l i t i e s ,a n d c l o s e d - l o o pc o n t r o lc a n be g a i n e d b y d i r e c t e x t e n s i o n o f w e l l - e s t a b l i s h e dm e t h o d so f l i n e a r s y s t e m sa n a l y s i s .
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21, N O . 11, NOV 1954, pp. 749-762.
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13. Kwakernaak, H . a n dS i v a n ,R . , L i n e a r O p t i m a lC o n t r o lS y s t e m s , W i l e y - I n t e r s c i e n c e , New York, 1972.
P-213 LINEAR.TIME-VARYING EOUATIONS I NONLINEAR, TIME-VARYING EOUATIONS
FLIGHT VARIABLES. FLIGHT m-
VARIABLES.
,' A x ,
4% TIME. I
I TIME i : ! l 5 . Y , l l NUMERICALSOLUTIONSFOREVERYPATH LINEAR. IIME-INVARIANI EOUAIIONS NONLlNEAR,TIME-INVARIANT EOUATIONS Ai:fAz.G4y Ai: I l A : . A y . : , , u d UNIVERSALCLOSED-FORMSOLUTIONSFOR SPECIFICCLOSED-FORMSOLUTIONS PERTURBEDTRAJEClORlES
. 0UASI.LINEARIZATlON FOR ANALYSIS OF
NONLINEAR EFFECTS F i g u r e 1.- Dynamic models for m a n e u v e r i n g f l i g h t .
c - 2 5 I .
a = -50 -75 .'Oo0 1 2 3 1 5 b 7 8 0 1 2 3 4 5 6 7 8 TIME I % c d TIME hecl (a) U n c o u p l e dl i n e a r i z a t i o n .
(b) C o u p l e dl i n e a r i z a t i o n .
F i g u r e 2 . - Comparison o f r e s p o n s e so fn o n l i n e a ra n dl i n e a r models t o a l a r g e r u d d e ri n p u t .( A i r c r a f t A ) .
- t !
- cp & 10 J e l UI 0 IO 20 30 0 10 M 30 40 50 60 ANGLE OF ATTACK. oolOEGI ANGLE OF ATTACK. 00 l e g 1 FAST ROLL PITCH DIVERGENCE UNSTABLE LATERAL-LONGITUDINAL FAST OSCILLATION (a) Aircraft A : Vo = 94 m/s, ( b ) Aircraft B : V o = 2 1 3 m/s.
F i g u r e 3 . - E f f e c t s o f mean a n g l e s of a t t a c k a n d s i d e s l i p o n d y n a m i cs t a b i l i t yb o u n d a r i e s .
-50 -40 -30 -20 -10 0 10 20 30 40 50 ROLL RATE, P ldeg/wcl ' 0 (a) Aircraft A : Vo=94m/s, ao=15 deg. (b) Aircraft B : Vo=213m/s, a,=10 deg.
F i g u r e 4 . - E f f e c t s o f c o m b i n e ds i d e s l i pa n g l ea n dw i n d - a x i s roll r a t e ondynamic s t a b i l i t y b o u n d a r i e s .
o o : l O d c q F i g u r e 5 . - E f f e c t so f a t t a c k on l a t e r a l i n c r e a s i n g a n g l e o f c o n t r o l r e s p o n s e . ( A i r c r a f t B ) .
W 0 10 STABLE 4.
P v) W : 5 0 10 20 Jo 40 50 0 10 20 30 40 50 ANGLE OF ATTACK, a , (dog1 ANGLE OF ATTACK, (10 (degl (b) P i l o tu s e sc o n t r o ls t i c k (a> P i l o tu s e sc o n t r o ls t i c ko n l y .
a n d r u d d e r p e d a l s .
F i g u r e 6 . - S t a b i l i t y b o u n d a r i e s of a i r c r a f t B with p i l o t c o n t r o l .
( P i l o tu s e sc o n t r o ls t r a t e g i e sa p p r o p r i a t e t o a0 = 1 0 d e g . ) OPEN-LOOP RESPONSE fi.20339 2.0
- - 1.0 b{ fb{ {::FJ I $ : : : p d
0 148 . = o = o < -02 l e . 4 -1.0 -5 0 0 2.0 4 0 6.0 8.0 0 2 0 5 0 6.0 8.0 0 2 0 4 0 6.0 8.0 C 2 0 4 . 0 6.0 8 0 TIME lscc) CLOSED-LOOP RESPONSE F i g u r e 7 . - Response o f a i r c r a f t A to s i d e s l i p p e r t u r b a t i o n d u r i n gc o n s t a n t - r o l l - r a t em a n e u v e r . = 3 9 . 6 d e g / s e c ; pWO a. = 15O; Vo = 9 4 m / s .
O P E N - L O O P R L S P O N S E C L O S E D - L O O P W S P O N S E F i g u r e 8.- Response o f a i r c r a f t B t o s i d e s l i p p e r t u r b a t i o n d u r i n gc o n s t a n t - r o l l - r a t em a n e u v e r . = 75 d e g / s e c ; pWO a , = loo; Vp = 213 m/s.