Stability and control of maneuvering high-performance aircraft

(which is described in Appendix R ) . Section 2.3 compares the time responses of linear and nonlinear dynamic aodels.

Section 2.4 is directed at aircraft stability, and S e c t i c ~ ~ 2.5 treats aircraft control. Variations in the aircraft's

-

dynamic characteristics during extreme maneuvering are addressed in Section 2.6, which also introduces rudimentary effects of tLe pilot's control actions while executing a tracking task. The chapter is summarized in Section 2.7.

2.2 PRIOR STUDIES OF AIRCRAFT AT EXTREME FLIGHT CONDITIONS Although published studies of aircraft dynamics shortly followed the Wright L~others' flight (Ref. l), and the concept of stability derivntives was publi-shed in 1913 (Ref. 2 ) ) the dynamics of aircraft which are executing extreme maneuvers received little attention until the late 1940's. (Investikation of the related problem of aircraft spinning had begun a decade earlier.) There are several reasons for this, but the most significant reason is that extreme maneuvers had not presented sufficient problems to merit detailed engineering study. The advent of fighter aircraft with higher speeds, higher roll rates, higher den- sity, lower inherent damping, and higher cost accentuated the importance of understmding extreme maneuvering dynamics.

Furthermore, the improved analytical tools and techniques spawned by World War I1 became available for application to flight dynamic problems.

In addition to the extensive flight testing which high-performance aircraft received, three fundamentally dif- ferent avenues have been followed in the investigation of maneuvering flight. The first approach is the study of rigid-body dynamics of the aircraft, the second is the study of aerodynamics, and the third is the study of control. The

first two areas have a cause-and-effect relationship -- aero-

dynamic forces modify the momentum and energy of the air-

plane -- and there is "feedback," in the sense that the

changing velocity and attitude of the vehicle contribute to changes in the aerodynamic forces. Although dynamic problems result from the interaction of dynamics and aerodynamics, one can distinguish between these two areas in reviewing past work. The third area considers methods of augmenting the natural aircraft stability, of limiting excursions from the normal flight regime, of providing adequate response, and of recovering from fully evolved spins.

2.2.1 Dynamics of the Aircraft The objective of study is the solution of nonlinear and linear equaticns of motion, e.g,, those derived in Appen- dix A . Options for analysis can be classified as explicit - 9 in which a direct solution of motion equations is sought, or implicit, in which the evolution of motions is inferred from characteristics of the system. The solution of these equa- tions describes the aircraft's response to initial conditions

and disturbances, and it provides a basis for identifying - con-

trol policies. The stability of the solution describes its tendency to return to a nominal value. Given an initial dis- turbance, the stable aircraft's solution returns to the nominal solution (or its error is, at least, bounded); the unstable aircraft's solution diverges. These analytical methods can be summarized as follows:

Explicit Analysis - Stability, Response, and Control

a Analog integration of differential equations a Numerical integration of differential equations a Closed-form solution of differential equations a Equilibrium solution of algebraic equations

I m p l i c i t A n a l y s i s - S t a b i l i t y

a G e n e r a l i z e d e n e r g y b a l a n c e ( L i a p u n o v m e t h o d ) a A b s o l u t e s t a b i l i t y b o u n d s ( P o p o v c r i - t e r i o n , c i r c l e c r i t e r i o n , e t c . ) a S t a b i l i t y b o u n d s o f "classical" d i f - f e r e n t i a l e q u a t i o n s ( M a t h i e u l s e q u a - t i o n , e t c . ) E i g e n v a l u e a n a l y s i s ( R o u t h - H u r w i t z c r i t e r i o n , r o o t l o c u s , e t c . ) Q u a s i - l i n e a r E i g e n v a l u e a n a l y s i s

I m p l i c i t A n a l y s i s - R e s p o n s e a n d C o n t r o l

E i g e n v e c t o r a n a l y s i s Time-domain m e t h o d s ( I m p u l s e o r i n d i c i a 1 r e s p o n s e , a u t o - a n d c r o s s - c o r r e l a t i o n f u n c t i o n s , e t c . ) T r a n s f o r m m e t h o d s ( F r e q u e n c y r e s p o n s e , t r a n s f e r f u n c t i o n s , s p e c t r a l d e n s i t y , e t c . ) A p p l i c a t i o n s o f some o f t h e s e t e c h n i q u e s t o t h e m a n e u v e r i n g f l i g h t p r o b l e m a r e d o c u m e n t e d i n t h e l i t e r a - t u r e . Much o f t h e w o r k r e l a t e d t o h a n d l i n g q u a l i t i e s , s t a - b i l i t y , c o u p l i n g , a n d d e p a r t u r e is b a s e d u p o n l i n e a r - t i m e - i n v a r i a n t m o d e l s a n d u s e s e i g e n v a l u e a n d t r a n s f e r f u n c t i o n a n a l y s i s . Work o f t h i s t y p e is r e p o r t e d i n R e f s . 3 t o 10.

I n a d d i t i o n , p a r a m e t e r s o f l i n e a r - t i m e - i n v a r i a n t m o d e l s LCDP, e t c . ) h a v e b e e n c o r r 2 l a t e d w i t h f l i g h t test ( C n ~ , d y n ' or p i l o t e d s i m u l a t i o n d a t a u s i n g l i , ~ t l e o r n o d i r e c t a n a l y s i s o f t h e e q u a t i o n s of m o t i o n ( R e f s . 11 t o 1 4 ) . Q u a s i - l i n e a r i z a t i o n o f a s i g n i f i c a n t s i d e s l i p n o n l i n e a r i t y is a p p l i e d t o t h e w i n g r o c k p r o b l e m i n R e f . 1 5 , a n d c l o s e d - f o r m s o l u t i o n s f o r a c l a s s o f l a r g e m a n e u v e r s are p r e s e n t e d i n R e f . 1 6 . Since t h e early 1 9 6 0 ' s . a l a r g e number o f i n v e s t i - g a t i c n s h n r c ~ u s c d a n a l o g a n d n u m e r i c a l i n t e g r a t i o n i n t h e s t u d y o f d e p a r t u r e , s t a l l , p o s t - s t a l l g y r a t i o n s and s p i n ( R e f s . 17 to 2 7 ) . E q u i l i b r i u m s o l u t i o n s o f 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 h a v e b e e n u s e d t o d e t e r m i n e s p i n c o n d i t i o n s a n d are d i s c u s s e d i n R e f s . 24 a n d 28 t o 3 0 .

A number o f l i n e a r - t i m e - i n v a r i a n t d e p a r t u r e param- --

eters h a v e b e e n i d e n t i f i e d , as r e p o r t e d i n R e f s . 5 , 1 3 , a n d 1 4 . T h e s e p a r a m e t e r s relate t o t r a n s f e r f u n c t i o n n u m e r a t o r s a n d d; n o m i n a t w s a n d are e x p r e s s e d i n terms o f s t a b i l i t y a n d c o n t r o l d e r i v a t i v e s (C " 8 , C l a p Cnsa, C16a, C n g r , C1gr) 9 a n g l e o f a t t a c k ( a 0 ) , moments o f i n e r t i a (Ix, I,), d i r e c - t i o n a l 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 (k, ) , a n d a i l c r r . ? . - r u d d e r - I n t e r c o n n e c t (ARI) g a i n ( k 2 ) : D i r e c t i o n a l S t a b i l i t y P a r a m e t e r A z

= C c o s a - -

'n 0 I X B , d w "0 L a t e r a l C o n t r o l D e p a r t u r e P a r a m e t e r C n, " a

LCDP = C - C1

"6 6 1, Augmented L a t e r a l C o n t r o l D e p a r t u r e P a r a m e t e r ARI L a t e r a l C o n t r a 1 D e p a r t u r e P a r a m e t e r A c c e l e r a t i o c R e s p o n s e D e p a r t u r e P a r a m e t e r ( " 8 p l u s 6 S t a b i l i t y I n d i c a t o r " ) The f i r s t f o u r c r i t e r i a i n d i c a t e r e s i s t a n c e t c d e p a r t u r e when t h e i r m a g n i t u d e s a r e g r e a t e r t h a n z e r o , w h i l e t h e l a s t r e q u i r e s ARDPa t o be g r e a t e r t h a n zero a n d g r e a ; e r The f i r s t c r i t e r i o n r e l a t e s t o t h e o p e n - l o o p t h a n ARDP6.

> 0 is a n s t a t i c s t a b i l i t y o f t h e D u t c h r o l l mode; C n ~ , d y n a p p r o x i m a t e r e q u i r e m e n t f o r s t a b i l i t y . The LCDP's are a p p r o x i - m a t i o n s t o t h e c l o s e d - l o o p s t a t i c s t a b i l i t y o f t h e Du+ch r o l l mode when l a t e r a l c o n t r o l is u s e d t o m a i n t a i n c o n s t a n t r o l l r a t e ; when t h e y a 1 g r e a t e r t h a n z e r o , t h e D u t c h r o l l mode is ~ , s . t i c a l l y s t a b l e , b u t when t h e y a r e less t h a n z e r o , t h e D u t c h r o l l mode is s t a t i c a l l y u n s t a b l e . The "8 p l u s 6 " c r i - t e r i o n is a n a t t e m p t t o combine s t a b i l i t y a n d c o n t r o l c o n - s i d e r a t i o n s i n a s i n g l e d e p a r t u r e i n d i c a t o r .

T h e r e a r e a number o f i n a d e q u a c i e s i n t h e a b o v e p a r a m e t e r s , a l t h o u g h t h e y p r o v i d e i n s i g h t f o r f u t u r e d e v e l o p - m e n t s . They a r e a p p r o x i m a t i o n s t o t h e e x a c t t r a n s f e r f u n c - t i o n c o e f f i c i e n t s a n d d o n 3 t i n d i c a t e a c t u a l p o l e - z e r o loca- t i o n s ; t h e y n e g l e c t dampirig terms e n t i r e l y ; a n d t h e y d o n o t a c c c u n t f o r longitudinal/lateral-directional c o u p l i n g i n d u c e d by l a r g e , s i d e s l i p a n g l e ( B ) a n d a n g u l a r r a t e s ( p , q , r ) -- i n . f a c t , t h e l o n g i t u d i n a l d y n a m i c s a r e i g n o r e d c o m p l e t e l y .

R e f e r e n c e s 8 , 9 , 31, and 3 2 i n t r o d u c e c o u p l i n g e f f e c t s d u e t o 6 , i l l u s t r a t i n g t h e i m p o r t a n c e o f some o f t h e n e g l e c t e d terms, and R e f . 23 t r e a t s t h e d y n a m i c s o f s t e a d y t u r n i n g f l i g h t .

2 . 2 . 2 Aerodvnamics E q u a l l y i m p o r t a n t d e v e l o p m e n t s h a v e b e e n made i n t h e area o f a e r o d y n a m i c s . M e a s u r e m e n t s o f f o r c e s a n d moments a n d v i s u a l i z a t i o n o f f l o w phenomena h a v e i n d i c a t e d t h e large v a r i - a t i o n s i n a e r o d y n a m i c c o n d i t i o n s t o b e e x p e c t e d a t h i g h a n g l e s o f a t t a c k ( a ) a n d s i d e s l i p ( B ) , a n d t h e r e is a n i n c r e a s i n g body o f d b t a r e l a t e d t o t h e e f f e c t s o f l a r g e a n g u l a r rates ( R e f s .

34 t o 4 6 ) . W h i l e t h e s e s e v e r a l r e f e r e n c e s c o v e r a v a r i e t y o f t o p i c s , t h e y p r o v i d e a n i n t r o d u c t i o n t o t h e k i n d s o f a e r o - dynamic p r o b l e m s w h i c h c a n b e e x p e c t e d when a i r c r a f t f l y a t h i g h a n g l e s a n d h i g h r a t e s .

The two d o m i n a n t phenomena w h i c h c o m p l i c a t e t h e c o l l e c t i o n o f v a l i d d a t a a n d t h e f l i g h t o f a c t u a l a i r c r a f t a r e v o r t i c e s a n d s e p a r a t e d f l o w . The v o r t e x is a b y - p r o d u c t o f a e r o d y n a m i c l i f t , a n d e a c h s u r f a c e o r body w h i c h g e n e r a t e s l i f t h a s a c o r r e s p o n d i n g v o r t e x t h a t t r a i l s downstream f r o m t h e l i f t i n g s o u r c e . T h i s s w i r l i n g a i r f l o w a f f e c t s p r e s s u r e d i s t r i b u t i o n s on t h e downstream s u r f a c e s o f t h e a i r c r a f t , a n d it c a n combine w i t h v o r t i c e s g e n e r a t e d o n o t h e r p a r t s o f t h e a i r c r a f t t o p r o d u c e z v e r y complex f l o w f i e l d . A t low a n g l e s o f a t t a c k o r s i d e s l i p , t h e v o r t i c e s f r o m n o s e , w i n g , a n d t a i l u s u a l l y form a u n i f i e d f l o w f i e l d , w h i c h v a r i e s s m o o t h l y as t h e a i r c r a f t ' s a t t i t u d e w i t h r e s p e c t t o t h e wind c h a n g e s .

A s t h e i n c i d e n c e i n c r e a s e s , t h i s smooth v a r i a t i o n may b r e a k down, c a u s i n g t h e f l o w t o become less u n i f i e d .

The d i f f i c u l t i e s i n p r e d i c t i n g t h e a c t u a l f o r c e s a n d moments on f u l l - s c a l e a i r c r a f t f r o m w i n d - t u n n e l d a t a g r e g r e a x e s t f o r l a r g e - a n g l e f l i g h t c o n d i t i o n s , n o t o n l y b e c a u s e t u n n e l c o r r e c t i o n f a c t o r s c a n become s i g n i f i c a n t b u t b e c a u s e s e p a r a t i o n e f f e c t s depend on t h e R e y n o l d s num- b e r o f t h e f l o w ( a n d , t h e r e f o r e , s n t h e s i z e o f t h e a i r - c r a f t ) . H i g h - p e r f o r m a n c e a i r c r a f t a r e most l i k e l y t o p e r f o r m extreme maneuvers at subsonic velocity, in which case Mach number effects may not be significant; however, scaling of the flow to provide representative Reynolds number is re- quired, if model test data is to be applied to the iull-scale aircraft.

Stability problems associated ~ 1 t . h large aerodynamic angles may arise f l o r t l either the nose, wing, or tai!., depend- ing on aircraft configuration. Consequently, it is impossible to identify a single aerodynamic solution to problems of departure (other than to make all aircraft use the same con- figuration). Aerodynamic solutions include wing-root leading- edge extensions, nose strakes, redesign of the nose cross- section and profile, maneuvering (leading-edge) flaps, and adjustment of horizontal tail anhedral.

The aerodynamic forces and moments discussed above are -- static, in that they arise frcm fixed values of a and 6.

These terms establish the static stability and trim points of the aircraft. Forces and moments which result from . . . . . .

angular rates (p,q,r) and accelerations (u,v,w,p,q,r) are dynamic and thus contribute to damping and transient response.

There is indication that assumptions which cocventionally are

made for low-angle flight condjtions, e.g., that the b and

yaw-rate effects are simply additive, break down at high angles. Unfortunately, dynamic forces and moments are dif- ficult to measure in practice, and relatively few facilities are equipped to mzasure dynamic forces, much less to separate k and r effects.

2 . 2 . 3 Control The third subject for study is control of flight motions during rapid maneuvering, and i t is clear that the emphasis of recent studies has shifted away fro^ spin r e c o v e r y t o d e p a r t u r e a n d s p i n p r e v e n t i o n . A t b e s t , sp!n r e c o v e r y is a n emergency p r o c e d u r e , a n d i t is n o t a l w a y r s u c c e s s f u l . S a f e t y is i m p o r t a n t , b u t it is n o t t h e o n l y i s s u e : a n a i r c r a f t w h i c h is p r o n e t o s p i n is less l i k e l y t o c o m p l e t e its m i s s i o n s u c c e s s f u l l y . I t is p r e f e r a b l e , t h e r e f o r e , t o p r e v e n t t h e s p i n b e f o r e it o c c u r s .

N e v e r t h e l e s s , i f a s p i n o c c u r s , it is i m p o r t a n t t o u n d e r s t a n d what c o n t r o l a c t i o n s c a n be u s e d t o r e c o v e r . The m o s t f a v o r e d t e c h n i q u e f o r r e c o v e r y is t o command c o n s t a n t , a n t i - s p i n c o n t r o l s e t t i n g s ( R e f . 4 7 ) . The p r o p e r c o n t r o l s e t t i n g s d e p e n d o n t h e t y p e o f s p i n ( f l a t , s t e e p , o s c i l l a t o r y , or e r r a t i c ) a n d o n t h e a i r c r a f t configuration--particularly t h e t a i l d a m p i n g , a i r c r a f t d e n s i t y , a n d mass d i s t r i b u t i o n .

I n many cases. t h e a v a i l a b l e a n t i - s p i n c o n t r o l moment is less t h a n t h e r e s t o r i n g moments which m a i n t a i n s p i n e q u i - l i b r i u m , i . e . , t h e s p i n c a n n o t be b r o k e n w i t h c o n s t a n t Zon- t r o l s e t t i n g s . The idea o f " r e s o n a t i n g " t h e a i r c r a f t o u t o f t h e s p i n by a p p l y i n g o s c i l l a t o r y c o n t r o l s was p r o p o s e d as e a r l y as 1931 ( R e f . 48) a n d as r e c e n t l y as 1 9 7 4 ( R e f . 4 9 ) .

W h i l e t h i s t a s k may b e d i f f i c u l t f o r t h e p i l o t t o e x e c u t e , simple l o g i c f o r p u l s i n g t h e c o n t r o l s a u t o m a t i c a l l y c a n be d e s i g n e d .

The c o n c e p t o f a u t o m a t i c c o n t r o l s y s t e m s w h i c h p r e - v e n t s t a l l , d e p a r t u r e , a n d s p i n has g a i n e d momentum, and it is now r e c o g n i z e d t h a t d e p a r t u r e p r e v e n t i o n c a n be b u i l t i n t o t h e 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 (SAS), w h i c h v i r - t u a l l y a l l modern h i g h - p e r f o r m a n c e a i r c r a f t c o n t a i n . The basic a p p r o a c h e s t o d e p a r t u r e p r e v e n t i o n t a k e n t o date c a n be c l a s s i f i e d a s limiters ( o r i n h i b i t o r s ) , s t a b i l i t y aug- m e n t e r s , c o n t r o l i n t e r c o n n e c t s , o r some c o m b i n a t i o n of t h e s e t h r e e . A dual-mode s p i n - p r e v e n t i o n s y s t e m is d e v e l o p e d i n R e f . 50. T h i s s y s t e m a p p l i e s c o n s t a n t a n t i s p i n c o n t r o l s when a and r exceed separate threshold values, then switches to a rate-damping mode once the spin is neutralized. Ref- erence 51 presents a departure-prevention system which inhibits a , increases directional stiffness (by stability augmentation), and restricts the aircraft to roll about its flight path. A stall-inhibitor system for a variable-sweep aircraft is described in Ref. 5 7 . This system incorporatss an a limiter, E--dependen< command- and stability-augmentation gains, increased directional stiffness and damping, and aileron-rudder interconnect. A similar philoso~by is adopted il Ref. 53, where speed stability also is augmented to account for a-limiting effects in the landing approach. Departure pre- v~ntion considerations are evident in the designs for two additional high-performance aircraft (Refs. 54 and 55), and the effects of stability augmentation and roll/yaw interconnect are demonstrated in Ref. 56.

While a common thread runs through the designs 1s- ported in Refs. 50 to 56, these reports suggest the need for a unifying control theory to aid the design of future departure prevention systems. These studies have made extensive use of experience, nonlinear simulation, and flight testing to arrive at successful designs, but the underlying concepts of stability, response, and control remain to be identified.

Summary - This section has presented a brief survey

of prior developments related to maneuvering flight, dis- tinguishing between investigations of dynamics, aerodynamics, and control of the aircraft. It is shown that the range of problems, from degraded handling qualities to fully evolved spin, can not be completely solved by focusing on only one area. New developments are required in characterizing the evolution of motions; in the measurement and understanding of forces and moments at extreme flight conditions; and in t h e computation of control s o l u t i o n s . In following s e c t i o n s of t h i s r e p o r t , t h e problems of dynamics and control a r e addressed i n d e t a i l .

2 - 3 COMPARISON O F RESULTS FROM LINEAR AND NONLINEAR SIMULATIONS The use of l i n e a r models i n h i g h l y dynamic s i t u a - t i o n s has been r e s t r i c t e d , i n the p a s t , by a lack of l i n e a r models which include complete dynamic e f f e c t s and by t h e lack of a general method of finding the proper nominal f l i g h t condition. The l i n e a r models developed i n Appendix A include a l l t h e e f f e c t s of a dynamic nominal f l i g h t condition. To verify these models and t o develop methods of using them, t h i s section presents a comparison of l i n e a r and nonlinear r e s u l t s . The nonlinear r e s u l t s are i n _he form of t e s t t r a - j e c t o r i e s generated by a nonlinear a i r c r a f t simulation using aerodynamic and mass data f o r t h e reference a i r c r a f t .

During the early p a r t of t h i s i n v e s t i g a t i o n , l a r g e differences between t h e l i n e a r and no-linear r e s u l t s a p ~ e a r e d along h i g h l y dynamic f l i g h t t r a j e c t o r i e s . These were traced t o the use of an incorrect nominal s t a t e vector. From these observations, the concept of generalized t r i m (Section A . 3 . 2 ) was developed, and a method of f i n d i n g generalized t r i m points was derived. (Section A . 3 . 2 describes the generalized t r i m calculation computer program.)

The generalized t r i m condition i s one i n which the derivatives of the velocity and angular r a t e s t a t e s a r e a s close t o zero a s possible. Dimensionality considerations, a s discussed i n Section A.3.2, lead to the conclusion t h a t t h e generalized t r i m problem involves s i x of the a i r c r a f t s t a t e equations, t h e correspoi~ding s i x s t a t e s , and s i x con-

/

t r o l parameters ( i n t h i s case, four control s e t t i n g s and two Euler angles). The problem becomes a search f o r those values of nominal body-axis v e l o c i t i e s and angular r a t e s t h a t n u l l thc selected nominal s t a t e r a t e s .

The following subsections examine s p e c i f i c r e s u l t s of t h e comparison of l i n e a r and nonlinear t r a j e c t o r i e s t o support these poj n t s .

2 . 3 . 1 - Elevator Control Input

Elevator d e f l e c t i o n produces a change i n p i t c h moment, causing an immediate change i n t h e a i r c r a f t angle of a t t a c k . T h i s causes t h e a i r c r a f t t o climb o r dive. In combination w i t h t h e t h r o t t l e , elevator position e s t a b l i s h e s t h e a i r c r a f t f l i g h t speed, angle of a t t a c k , and f l i g h t path angle. The t e s t s presented here involve small-amplitude elevator inputs when the a i r c r a f t is i n straight-and-level f l i g h t a t slow speed and high angle of attac;. Figure 2.3-1 i l l u s t r a t e s t h e time h i s t o r y of the most important longi- tudinal motion v a r i a b l e s f o r eight seconds following t h e control a p p l i c a t i o n . A l l l a t e r a l v a r i a b l e s a r e approxi- mately zero f o r t h e nonlinear model and exactly zero f o r t h e l i n e a r model.

Comparison of t h e l i n e a r and nonlinear curves indi- c a t e s excellent agreement. I t is important t o note t h a t t h e nonlinear a i r c r a f t responsc v e r i f i e s t h a t the l a t e r a l and iongitudinal modes a r e t r u l y uncoupled i n t h i s f l i g h t con- d i t i o n . The nominal f l i g h t cond~cion is a steady-trim f l i g h t condition and s a t i s f i e s the generalized t r i m condition.

- NONLINEAR

----- LINEAR

COMPARISON FROM 1 r0.0trc NOMINAL FLIGHT CONOlflONS: L) Vo.* dl m/r u, = 2 2 . 0 d e ~ -25 0 1 2 3 4 5 6 7 8 I:........

TIME (=I Figure 2.3-1 Small Amplitude Elevator Input Aileron Control Input The a i l e r o n s p r i n a r i l y provide r o l l moment, and t h e t r a j e c t o r i e s shown i n Fig. 2.3-2 i l l u s t r a t e t h e a i r - c r a f t response t o a small amplitude a i l e r o n doublet. The l i n e a r i z e d t r a j e c t o r y , whose nominal f l i g h t condition is again straight-and-level f l i g h t , d i f f e r s only s l i g h t l y from the t r u e nonlinear response, and t h e l i n e a r and non- l i n e a r t r a j e c t o r i e s e x h i b i t l a t e r a l - l o n g i t u d i n a l s e p a r a t i o n .

2.3.3 Rudder Control Input Large-input, large-response t r a j e c t o r i e s r e s u l t i n g from rudder d e f l e c t i o n a r e examined i n t h i s subsectiou, w i t h the goal of t e s t i n g the t r a j e c t o r y matching c a p a b i l i t i e s af a l i n e a r simulation f o r a h i g h l y dynarnlc f l i g h t condition.

The nor~linear t e s t t r a j e c t o r y l a s t s e i g h t seconds a f t e r the a-K)2?$

- NONLINEAR

----- LINEAR

COMPARISON FROM 110 O u c

- NOMINAL FLIGHT CONDITIONS

I o + = - -

Q V o 194 m h a,@ 15.0 d . 0

L

-2s0 At&,@ *4D&o 100. 1.2.0uo 1 2 3 4 1 6 7 1 TIME IUCI - r ~ a o 120. I S ~ O S U ~

-25 -

0 1 2 3 1 1 b 7 1 TIME IsuJ Figure 2.3-2 Small Amplitude Aileron Input control is applied; l i n e a r t r a j e c t o r i e s s t a r t i n g a t the i n i t i a l time and a t four seconds i n t o t h e t r a j e c t o r y a r e t e s t e d .

Figure 2.3-3 compares t h e nonlinear t r a j e c t o r y t o a l i n e a r t r a j e c t o r y s t a r t i n g a t t h e time of control appli- cation. The nominal t r a j e c t o r y f o r l i n e a r i z a t i o n is t h e 6 r i g i n a l s t a t i c triiil f l i g h t condition of straight-and-level f l i g h t . The t r a j e c t o r y match is acceptable f o r a L w t two seconds, and t h e angle-of-attack p l o t i l l u s t r a t e s the cause of t h e deviation. Because it e x h i b i t s lateral-longitudinal separation, t h e l i n e a r t r a j e c t o r y does not capture the change i n angle of a t t a c k t h a t t h e nonlinear t r a j e c t o r y contains.

T h i s change i n angle of a t t a c k has a l a r g e e f f e c t on the subsequent dynamics which t h e l i n e a r model f a i l s t o d u p l i c a t e .

- NONLINEAR

----- LINEAR

NONLINEAR COMPARISON FROM T:O.Owc RESPCNS NOMINAL FLIGHT CONOlnONk V , 8 94 m/r 0 1 1 3 1 S b 7 8 TlME (wc) - 2 s ? I 2 3 1 5 4 7 6 TlME (we) 0 1 2 3 4 1 6 7 1 TlME (ucl

F i g u r e 2 . 3 - 3 Large A m p l i t u d e R u d d e r I n p u t --

C ~ : , r p n r l s o n of I n i t i a l R e s p o n s e By e x a m i n i n g t h e t r a j e c t o r y b e g i n n i n g f o u r s e c o n d s a f t e r t h e c o n t r o l is a p p l i e d , m e t h o d s of l i n e a r i z a t i o n f o r h i g h l y d y n a m i c t r a j e c t o r i e s c a n b e d e r i v e d . F i g u r e 2.3-4 i l l u s t r a t e s a n e a r l y a t t e m p t . H e r e , t h e p o i n t o f l i n e a r i z a - t i o n is a p p r o x i m a t e , i . e . , i t d o e s n o t s a t i s f y t h e g e n e r a l i z e d trirn c o n d i t i o n d i s c u s s e d b e l o w a n d i n Appendix A . T h e re- s u l t i n g l i n e a r t r a j e c t o r y d i v e r g e s f r o m t h e n o n l i n e a r t r a - j e c t o r y f a i r l y q u i c k l y , and t h e s l o p e s d o n o t m a t c h a t t h e i n i t i a l p o i n t f o r some s t a t e s . F u r t h e r m o r e , t h e f r e q u e n c y o f t h e r e s u l t i n g m o t i o n is c o n s i d e r a b l y d i f f e r e n t f r o m t h a t o f t h e n o n l i n e a r m o t i o n . Due t o i t s d e p e n d e n c e o n ad h o c e s t i m a t i o n c f t h e n o m i n a l f l i g h t c o n d i t i o n , t h e r e s u l t s o f t h i s a p p r o a c h a r e h i g h l y v a r i a b l e i n q u a l i t y .

One o f t h e m o s t s t r i k i n g e r r o r s i n t h e l i n e a r t r a - jectories shown i n F i g . 2.3-4 is t h a t t h e s l o p e s o f t h e s t a t e s do n o t m a t c h a t t h e b e g i n n i n g o f t h e l i n e a r t r a j e c - t o r y . T h i s o b s e r v a t i o n , w h i c h i m p l i e s t h a t t h e n o m i n a l s t a t e -- rstes a r e n o t z e r o , l e d t o t h e d e v e l o p m e n t o f t h e g e n e r a l i z e d t r i m c o n c e p t . I n t h i s c o n t e x t , t h i s c o n c e p t i n d i c a t e s t h a t t o p r o v i d e a n a c c u r a t e r e p r e s e n t a t i o n o f a n o n l i n e a r s y s t e m by a l i n e a r i z e d o n e , it is n e c e s s a r y t o c h o o s e a p o i n t o f l i n e a r i z a t i o n t h a t e x h i b i t s zero n o m i n a l -- s t a t e r a t e s .

A p p l y i n g t h i s g 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 t h e p o i n t f o u r s e c o n d s a f t e r c o n t r o l a p p l i c a t i o n p r o d u c e s t h e r e s u l t s shown i n F i g . 2 . 3 - 5 . Compared t o t h e p r e v i o u s f i g - u r e , t h e g e n e r a l i z e d t r i m p r o c e d u r e p r o d u c e s c l e a r l y s u p e r i o r r e s u l t s . T h e r e a r e no i n i t i a l s l o p e errors e v i d e n t , t h e m a t c h i s e x c e l l e n t f o r two s e c o n d s , a n d it i s r e a s o n a b l y c l o s e f o r much l o n g e r . A d d i t i o n a l l y , t h e f r e q u e n c y o f t h e l i n e a r i z e d m o t i o n s is c l o s e t o t h a t o f t h e n o n l i n e a r Comparison

Figure 2.3-4 Large Amplitude Rudder I n p ~ t --

of Evolved Response With A d Hoc Reference Point for LFnearizat ion NOMINAL ~LIGUT CONDI~ION IS CALCULATED 87 GENERALIZED TRIM.

igure 2.3-5 Large Ampli.tude Rudder Input -- Comparison

o f Evolved Response With Generalized Trim Reference Foint for Linearization t r a j e c t o r y . T h i s , a l o n g w i t h t h e a m p l i t u d e m a t c h , s u p p o r t s t h e u s e o f a p r o p e r l y l i n e a r i z e d m o d e l f o r t h e a n a l y s i s o f a n o n l i n e a r v e h i c l e a l o n g a h i g h l y d y n a m i c t r a j e c t o r y . The s i g n i f i c a n c e o f this r e s u l t is p n t i n p r o p e r p e r s p e c t i v e when i t is r e a l i z e d t h a t t h e v e h i c l e h a s p e r f o r m , d a 360-deg r o l l b e t w e e n t = 0 a n d t = 7 . 2 s e c , a n d t h e p i t c h a n g l e g o e s f r o m 15 d e g t o -45 d e g from t = 0 t o t = 5 sec.

I t s h o u l d b e n o t e d t h a t t h e n o m i n a l f l i g h t c o n d i - t i o n f o r l i n e a r i z a t i o n was f o u n d b y a n a n a l y t i c m e t h o d t h a t d o e s n o t r e q u i r e t h e s o l u t i o n o f a n o n l i n e a r t r a j e c t o r y f r o m w h i c h t o estimate n o m i n a l v a l u e s . T h e g e n e r a l i z e d t r i m p r o c e d u r e is a u s e f u l m e t h o d f o r c a l c u l a t i n g n o m ~ n a l f l i g h t c o n d i t i o n s e v e n a l o n g h i g h l y d y n a m i c t r a j e c t o r i e s .

Summary - T h e s e c o m p a r i s o n s p r e s e n t e d h e r e e s t a b l i s h t h a t n o m i l ~ s l f l i g h t c o n d i t i o n s w h i c h s a t i s f y t h e generalized t r i m c o n d i t i o n p r o d u c e g o o d t r a j e c t o r y ~ a t c h ~ s a n d t h a t t h e c o r r e s p o n d i n g l i n e a r m o d e l s s h o u l d p r o v i d e a c c u r a t e j n f o r - m a t i o n a b o u t t h e n o n l i n e a r s y s t e m d y n a m i c s .

2 . 4 EFFECTS OF ANGULAR MOTION AND FLIGHT C3NDITION ON AIRCRAFT STABILITY T h e e f f e c t s o f a l t i ~ ~ d e a n d v e l o c i t y v a r i a t i o n s , a n g l e - o f - a t t a c k a n d s i d e s l i p a n g l e v a r i a t i o n s , a n d s t e a d y a n g u l a r r a t e s o n a i r c r a f t s t a b i l i t y a r e e x a m i n e d i n t h i s s e c t i o n u s i n g t h e l i n e a r i z e d d y n a m i c m o d e l s a n d e i g e n v a l u e l e i g e n v e c t o r a n a l y s i s t e c h n i q u e p r e s e n t e d I n A p p e n d i x A . T h e p u r p o s e o f t h i s a c a l y s i s is t o show t h e e f f e c t s o f i n d i v i - d u a l f l i g h t v a r i a b l e s , a s w e l l a s ; h e c o m b i n e d e f f e c t s o f f l i g h t v a r i a b l e s w h i c h n o r m a l l y a r e z e r o i n "1-g" s t r a i g h t - a n d - l e v e l f l i g h t . F o r t h i s s t u d y , t h e a i r c r a f t is t r i m m e d i n i t i a l l y f o r "1-g" f l i g h t a t a n a n g l e o f a t t a c k o f 15 d e g a n d a t a n a l t i t u d e o f 6 1 0 0 rn. A s f l i g h t v a r i a b l e s c h a n g e , t h e l c a d f a c t o r may c h a n g e a c c o r d i n g l y ; h o w e v e r , t h e p r i m a r y o b j e c t i v e o f t h i s c h a p t e r is t o i s o l a t e t h e i n d i v i d u a l e f f e c t s o f e a c h s p e c i f i c f l i g h t v a r i a b l e b e i n g e x a m i n e d , so a l l o t h e r v a r i a b l e s a r e h e l d a t t h e i r i n i t i a l v a l u e s .

2 . 4 . 1 A l t i t u d e a n d V e l o c i t y E f f e c t s A l t i t u d e a f f e c t s t h e a i r d e n s i t y a n d , t h e r e f o r e , t h e dynamic p r e s s u r e . T h i s c a u s e s t h e a e r o d y n a m i c f o r c e s a n d moments t o b e r e d u c e d , r e l a t i v e t o t h e i n e r t i a l e f f e c t s , a s a l t i t u d e i n c r e a s e s , a s shown i n F i g . 2 . 4 - 1 . H i g h e r a l t i t u d e c a u s e s b o t h t h e n a t u r a l f r e q u e n c i e s and damping r a t i o s o f t h e D u t c h r o l l a n d s h o r t p e r i o d modes t o d e c r e a s e . T h e r o l l mode a l s o s l o w s down a s a l t i t u d e i n c r e a s e s .

C h a n g e s i n v e l o c i t y a f f e c t t h e d y n a m i c p r e s s u r e , a s w e l l as t h e a n g u l a r r a t e n o r m a l i z a t i o n terms (b/2V a n d c / 2 ~ ) a n d t h e v e l o c i t y - a n g u l a r r a t e c r o s s - p r o d u c t terms. T h e s e c h a n g e s c a u s e s i g n i f i c a n t i n c r e a s e s i n D u t c h i-011 a n d s h o r t p e r i o d i r e q u e n c i e s as v e l o c i t y i n c r e a s e s ( F i g . 2 . 4 - l a ) . T h e damping r a t i o o f t h e s h o r t p e r i o d mode is a f f e c t e d o n l y s l i g h t l y by v e l o c i t y c h a n g e s a v e r t h e r a n g e shown i n F i g .

2 . 4 - l b . The D u t c h r o l l damping d e c r e a s e s as v e l o c i t y d e c r e a s e s , s o t h a t t h e D u t c h r o l l is u n s t a b l e a t t h e l o w e s t v e l o c i t i e s p r e s e n t e d h e r e . The r e f e r e n c e a i r c r z f t ' s r o l l mode ( F i g . 2 . 4 - l c ) is c h a n g e d o n l y s l i g h t l y a s v e l o c i t y v a r i e s , c o n t r a r y t o t h e r k s u l t o b t a i n e d f r o m t h e a p p r o x i m a t e l a t e r a l - l o n g i t u d i n a l e q u a t i o n s d i s c u s s e d b e l o w .

F i g u r e 2 . 4 - 1 i n d i c a t e s o n l y s m a l l i n c r e a s e i n s p i r a l mode s p e e d a n d p h u g o i d f r e q u e n c y a n d damping a t l o w e r a l t i t u d e s . Low v e l o c i t y r e s u l t s i n low p h u g o i d bl REAL PARTS OF COMPLEX ElGENVALUES r 1 IWAGINARARI PARTS OF COMPLEX EIGENVALUES SHORT PERIOD MODE d) COMPARISON OF APPROXIMATE AN0 EXACT EIGENVALUE CALCULATIONS lH-6lOOMl 5 REAL EIGENVALUES

"T

+ ROLL MODE 6, 0.4 APPROXIMATE ROLL 0.1 1 I 1 I I I L VELOCITY. V lm/d 70 80 90 100 110 . 120 VELOCITY. V lrnlrl Figure 2.4-1 A l t i t u d e and V e l o c i t y E f f e c t s on Eigenvalues damping, s o much s o t h a t t h e mode is u n s t a b l e o v e r a s i g n i f i - c a n t p o r t i o n of t h e v e l o c i t y r a n g e examined i n F i g . 2.4-1.

A s n o t e d a b o v e , t h e l a c k of r o l l mode v a r i a t i o n w i t h v e l o c i t y is c o n t r a r y t~ r e s u l t s o b t a i n e d w i t h a c c e p t e d a p p r o x i m a t i o n s . The e x p e c t e d l i n e a r change i n r o i l mode w i t h v e l o c i t y is deduced from a p p r o x i m a t e l a t e r a l - d i r e c t i o n a l e q u a t i o n s , which c a n b e d e r i v e d by n e g l e c t i n g t h e r o l l a n g l e e q u a t i o n ( a n d , t h e r e f o r e , t h e s p i r a ' mode), by assuming t h a t t h e Dutch r o l l mode c o n s i s t s o f wind-axis yawing m o t i o n , and by assuming t h a t t h e r o l l mode c o n s i s t s of w i n d - a x i s r o l l .

The a p p r o x i m a t i o n s t h a t r e s u l t from t h e s e a p p r o x i m a t e l a t e r a l - d i r e c t i o n a l e q u a t i o n s a r e w h e r e A r o l l is t h e r o l l mode e i g e n v a l u e , C n B S d y n rs d e f i n e d as i n E q . (2.2-1) and C = c o s a' C + c o s a . s i n a u C1 I ' I P , dyn O IP is

= c o s a C - s i n a . c o s a C

'n 0 Br 0 n r , d y n P + s i n - s i n a . c o s a . C1 r .

Equations (2.4-1) and (2.4-2) predict values f o r a s shown by the dotted l i n e s i n Fig. 2.4-ld ' r o l l and c u n , ~ ~ f o r the 6100-m case. The actual values, taken from Figs.

2.4-lb and c , a r e shown by s o l i d l i n e s on the same f i g u r e .

The approximate equations do a poor job of predicting mode speed because the subject a i r c r a f t is fuselage-heavy (high I,/Ix r a t i o ) ; hence, the Dutch r o l l contains more r o l l i n g response than is assumed when deriving t h e approximate equations. A s can be seen from Fig. 2.4-ld, t h i s leads t o a damping interchange such t h a t the r o l l mode is f a s t e r than expected and the Dutch r o l l mode is more poorly damped than predicted.

T h i s examination leads t o the following conclusions f o r the reference a i r c r a f t : a Higher a l t i t u d e s r e s u l t i n lower damping and frequency of the Dutch r o l l and short period modes, a s well a s increased r o l l mode time constant.

0 Lower v e l o c i t i e s r e s u l t i n a decrease i n short period frequency a t constant damping r a t i o , as well a s dscreased Dutch r o l l frequency and damping.

0 The approximate l a t e r a l - d i r e c t i o n a l equa- t i o n s l o not p r e a i c t r o l l mode o r Dutch r o l l damping accurately f o r the subject a i r c r a f t . The complete equations should be used for an accurate determination of these parameters.

2 . 4 . 2 Aerodynamic Angle Effects The aerodynamic angles, a and 6 , specify the o r i e n t a - t i o n of the vehicle r e l a t i v e t o the velocity vector, and, t o a large e x t e n t , they d e f i n e the flow f i e l d around the vehicle.

For t h i s reason, the aerodynamic angles a r e prime determi- nants of the aerodynamic forces and moments. Consequently, significant differences in the speeds and shapes of the normal modes occur as o and B o are varied.

Figure 2.4-2 illustrates the boundaries between stability and instability which result from these variations.

These boundaries define the a. and BO for which the real part of one or more eigenvalues migrates from negative (stable) to positive (unstable) sign (see Section A . 4 . 1 ) .

The phugoid mode is a slow mode and is unstable at low a o .

The Dutch roll mode, a fast mode, becomes unstable at high a The dashed line in Fig. 2 . 4 - 2 is an important boundary, 0 ' indicating the transition of a relatively slowly divergent phugoid oscillation into two real roots, one of which is

highly unstable. This transition line occurs at high B O --

about 10 to 15 deg.

10 20 30 ANGLE OF ATTACK, a,(oEG) Figure 2 . 4 - 2 EPfects of Aerodynamic Angles on Aircraft Stability The shape of the Dutch roll stability boundary indicates that moderate values of nominal sideslip angle (two to five deg) stabilize the mode. This is due to lateral-longitudinal coupling; a close examination of the D u t c h r o l l / s h o r t p e r i o d e i g e n v a l u e s i n d i c a t e s t h a t , u p t o a b o u t f i v e d e g o f s i d e s l i p , D u t c h r o l l damping i n c r e a s e s as s h o r t p e r i o d d a m p i n g d e c r e a s e s .

The e i g e n v e c t o r s o f t h e l i n e a r i z e d model p r o v i d e i n f o r m a t i o n a b o u t t h e n o r m a l mode s h a p e s w h i c h i n d i c a t e t h e i n v o l v e m e n t o f e a c h s t a t e i n e a c h mode. F i g u r e 2.4-3 i l l u s - t r a t e s some s p e c i f i c e i g e n v a l u e / e i g e n v e c t o r v a r i a t i o n s w i t h a n g l e o f a t t a c k . R e a l e i g e n v e c t o r s , s u c h a s t h o s e a s s o c i a t e d w i t h t h e r o l l mode, a r e c h a r a c t e r i z e d o n l y b y t h e r e l a t i v e m a g n i t u d e s o f e a c h s t a t e , a s t h e p h a s e a n g l e s a-e e i t h e r 0 o r 180 d e g . A t i m e h i s t o r y o f S h i s mode would show a c o n - s t a n t r a t i o b e t w e e n t h e v a r i o u s s t a t e a m p l i t u d e s . T h e s e a m p l i t u d e s would e v i d e n c e e x p o n e n t i a l d e c a y s w i t h e q ~ a l t i m e c o n s t a n t s , g i v e n b y t h e n e g a t i v e i n v e r s e o f t h e e i g e n v a l u e .

Complex e i g e n v e c t o r s , s u c h as t h o s e o f t h e D u t c h r o l l , a r e c h a r a c t e r i z e d by t h e r e l a t i v e m a g n i t u d e s o f t h e i n v o l v e d s t a t e s a n d by t h e p h a s e a n g l e b e t w e e n them. A t i m e h i s t o r y o f t h i s o s c i l l a t o r y mode is g e n e r a t e d by t h e p r o j e c t i o n s o f t h e e i g e n v e c t o r s on t h e r e a l a x i s as t h e e n t i r e e i g e n v e c t o r set r o t a t e s w i t h a n g u l a r rate g i v e n b y t h e i m a g i n a r y p a r t o f t h e e i g e n v a l u e . The m a g n i t u d e s d e c a y e x p o n e n t i a l l y w i t h t h e time c o n s t a n t g i v e n by t h e n e g a t i v e i n v e r s e o f t h e e i g e n - v a l u e ' s r e a l p a r t .

D e s p i t e t h e l a r g e c h a n g e s i n e i g e n v a l u e s w i t h a n g l e of a t t a c k , F i g . 2.4-3 shows l i t t l e c o r r e s p o n d i n g c h a n g e i n e i g e n v e c t o r s h a p e . T h e o n l y m a j o r c h a n g e s i n v o l v e t h e p r o - p o r t i o n s o f a n g u l a r r a t e s i n t h e f a s t modes, a n d t h e s e c h a n g e s a r e d u e t o mode s p e e d v a r i a t i o n s , a s d e s c r i b e d a b o v e f o r t h e r o l l mode. The s h o r t p e r i o d mode c o n t a i n s i n c r e a s e d p i t c h r a t e a t l a r g e a. f o r t h i s r e a s o n ; i n t h e D u t c h r o l l e i g e n v e c t o r , t h e r o l l r a t e - t o - s i d e s l i p r a t i o i n c r e a s e s a n d d e c r e a s e s w i t h t h e D u t c h r o l l f r e q u e n c y . O v e r a l l , t h e D u t c h r o l l mode of t h i s a i r c r a f t involves a g r e a t deal of r o l l i n g motion, underlining t h e low r o l l i n g i n e r t i a t y p i c a l of modern f i g h t e r s .

The short period eigenvector shows t h a t t h i s o s c i l - l a t i o r typically'involves angle-of-attack perturbations a t constant v e l o c i t y , a s a x i a l and normal v e l o c i t y perturbations a r e approximately 180 deg out of phase w i t h each other and a r e r e l a t e d i n magnitude by tan aO. The short period mode is f a s t e r a t high angle of a t t a c k , and i t includes more p i t c h r a t e than a t low angle of a t t a c k .

The changes i n s p e c i f i c eigecvalues and eigetvectors with s i d e s l i p a ~ g l e a r e i l l u s t r a t e d i n Fig. 2.1-4. Lateral- longitudinal coupling is q u i t e prominent f o r t h e asymmetric f l i g h t conditions portrayed jn t h i s f i g u r e . Modes of com- parable speed couple most r e a d i l y . Roll angle response is found i n t h e phugoid mode, and p i t c h anble becomes a component of the s p i r a l mode, so t h a t both modes involve slow r o l l - p i t c h motion. Angle of a t t a c k appears i n t h e Dutch r o l l eigenvector, and a r o l l - s i d e s l i p combination becomes important i n the short period mode, so t h a t both modes involve an angle of a t t a c k - s i d e s l i p o s c i l l a t i o n . In both cases, Aw and Av ( o r , equivalen- t l y , A a and A B ) a r e almost 180 deg out of phase. Note t h a t the changes i n the speeds of these modes a r e small and gradual a s the s i d e s l i p angle is varied.

To demonstrate some of t h e causes of aerodynamic angle e f f e c t s observed above, the aerodynamic c o e f f i c i e n t s a r e held constant ( a t the values f o r a. = 1 5 deg and B O 0 deg), and t h e body o r i e n t a t i o n w i t h respect t o the velocity vector is varied over the same range of aerodynamic angles used i n Fig. 2 . 4 - 2 . The r e s u l t s , shown i n Fig. 2 . 4 - 5 , d i f f e r s i g n i f i c a n t l y from those shown i n Fig. 2 . 4 - 2 . There is only a s l i ~ h t s i d e s l i p e f f e c t . The Dutch r o l l mode, r a t h e r than oulml ROLL AND MVaOlD STAeLE ANGLE OF ATTACK, C~,(DEG) F i g u r e 2 . 4 - 5 E f f e c t s o f Body O r i e n t a t i o n o n A i r c r a f t S t a b i l i t y becoming u n s t a b l e a t h i g h a o , is d e s t a b i l i z e d by l o w e r ao.

T h e p h u g o i d s t a b i l i t y b o u n d a r y n e a r a. = 1 2 d e g is r o u g h l y similar t o t h a t f o u n d i n F i g . 2 . 4 - 2 , i n d i c a t i n g t h a t t h e l a c k o f p h u g o i d s t a b i l i t y i n t h i s a r e a is n o t d u e t o a e r o d y n a m i c v a r i a t i o n s .

F i g u r e 2 . 4 - 6 assists i n t h e e v a l u a t i o n o f t h e h i g h a n g l e - o f - a t t a c k D u t c h r o l l i n s t a b i l i t y . T h i s f i g u r e com- p a r e s t h e D u t c h r o l l e i g e n v a l u e t o t h e d e p a r t u r e p a r a m e t e r (see S e c t i o n 2 . 2 ) a n d t o Cnr a n d C n r l d y n , t h e l a s t C n O , d y n o f w h i c h is d e f i n e d i n E q . ( 2 . 4 - 5 ) .

The r e s u l t s o f F i g . 2 . 4 4 i n d i c a t e t h a t , a t l e a s t i n t h i s c a s e , C n B l d y n is a good i n d i c a t o r o f t h e D u t c h r o l l p r o v i d e a m o d e ' s i m a g i n a r y p a r t . N e i t h e r Cnr n o r Cn r , dyn p a r t i c u l a r l y u s e f u l i n d i c a t i o n o f D u t c h r o l l s t a b i l i t y .

T h i s e x a m p l e i n d i c a t e s t h a t C n O , d y n h a s o n l y l i m i t e d v a l u e a s a d e p a r t u r e p a r a m e t e r . F o r t h i s a i r c r a f t , D u t c h r o l l F i g u r e 2.4-6 V a r i a t i o n s of D i r e c t i o n a l Aerodynamic.

C o e f f i c i e n t s w i t h Angle of A t t a c k i n s t a b i l i t y is d u e t o n e g a t i v e damping, and Cng.,dgn is i n - a d e q u a t e a s a p r e d i c t o r of d e p a r t u r e .

The f o l l o w i n g c o n c l u s i o n s c o n c e r n i n g aerodynamic a n g l e e f f e c t s on s t a b i l i t y of t h e r e f e r e n c e a i r c r a f t c a n be made: The Dutch r o l l mode becomes u n s t a b l e due t o n e g a t i v e damping a t h i g h ao. T h i s is c a u s e d by c h a n g e s i n t h e aerodynamics a s a0 i n c r e a s e s .

Mean a n g l e of a t t a c k v a r i a t i o n s have s i g - n i f i c a n t effect on t h e e i g e n v a l u e s , b u t mode s h a p e ( e i g e n v e c t o r ) c h a n g e s a r e s m a l l r e l a t i v e t o o t h e r e f f e c t s .

Mean s i d e s l i p a n g l e i n t r o d u c e s l a t e r a l - l o n g i t u d i n a l c o u p l i n g ; t h e r e f o r e it h a s a l a r g e e f f e c t on t h e mode s h a p e s ( e i g e n v e c t o r s ) , w i t h o u t c a u s i n g l a r g e c h a n g e s i n t h e e i g e n v a l u e s . T h i s l a t e r a l - l o n g i t u d i n a l c o u p l i n g p r i m a r i l y o c c u r s between modes o f similar s p e e d and c a n l e a d t o a t r a n s f e r of damping, a s i n t h e s i t u a t i o n where small s i d e s l i p a n g l e s stabilize the Dutch roll mode at the expense of short period damping.

gives a good indi- a The parameter C "6, dyn cation of Dutch roll frequency, but it is not useful as a departure parameter for the subject aircraft.

2.4.3 Angular Rate Effects Non-zsro nominal angular rates have two effects on the linearized aircraft cynamics. The first, an aerodynamic effect, results in a change in the nominal forces and moments due to the steady angular rates. The second is dynamic, and it is due to the cross product of angular rate with velocity (in the force equations) and with angular momentum (in :he moment equations). The specific terms involved (for Ix, = 0) are given in Table 2.4-1. A close examination reveals that mean pitch angular rate, qo, enters both the lateral and longitudinal equations but does not affect lateral-longitu- Mean roll and yaw rates, po and ro, dinal coupling terms.

enter as lateral-longitudinal coupling terms. Steady roll- rate capability of most high-performance aircraft is much higher than pitch- or yaw-rate capability, so roll-rate effects are especially important.

Stability boundaries as functions of pitch rate and yaw rate are illustrated in Fig. 2.4-7. The destabilizing influence of q0 is the major effect, and it has an especially severe effect on the Dutch roll mode. Yaw rate has a mild stabilizing effect on the Dutch roll and spiral modes. This is due partially to lateral-longitudinal coupling, because short period and phugoid damping decrease as Dutch roll damping increases.

TABLE 2.4-1 DYNAMIC EFFECTS OF STEADY ANGULAR RATE A n g u l a r Sate Multiplied By E n t e r s Term DUTCH ROLL STABLE PHUGOlD STABLE PHUGOID STABLE DUTCH ROLL STABLE SPIRAL UNSTABLE 0 10 20 30 PITCH RATE, 40 ( d r g / s u ) Yaw-Rate/Pitch-Rate Effects ( a o = 1 5 deg) Figure 2 . 4 - 7 The e i g e n v e c t o r c h a n g e s t h a t accompany i n c r e a s e s i n q0 are shown i n F i g . 2.4-8. S t e a d y p i t c h r a t e d o e s n o t i n t r o d u c e l a t e r a l - l o n g i t u d i n a l c o u p l i n g , b u t solne c h a n g e s i n mode s h a p e s a p p e a r i n t h e r o l l mode and i n t h e s e p a r a t i o n of t h e complex phugoid mode i n t o two r e a l r o o t s . Both f r e - q u e n c i e s and damping r a t i o s of t h e Dutch r o l l and s h o r t p e r i o d modes c h a n g e , b u t c h a n g e s i n t h e mode s h a p e are minor.

S t e a d y r o l l r a t e is i m p o r t a n t b e c a u s e f i g h t e r a i r - c r e f t a r e c a p a b l e of hig!. p o , and a i r combat maneuvers o f t e n i n c l a d e s u c h m o t i o n s . For t h e a i r c r a f t t o r o l l w i t h con- s t a n t aerodynamic a n g l e s , t h e r o l l r a t e must o c c u r a b o u t t h e wind x - a x i s (which is t h e same as t h e s t a b i l i t y x - a x i s f o r c o n s t a n t nominal aerodynamic a n g l e s ) . S i d e s l i p v a r i a t i o n s a l s o are c o n s i d e r e d , s i n c e p i l o t i n g e r r o r can e a s i l y r e s u l t i n non-zero B O d u r i n g a r o l l i n g maneuver. Both p o s i t i v e and n e g a t i v e po a r e c o n s i d e r e d , t o a c c o u n t f o r r o l l " i n t o " o r " o u t o f " t h e s i d e s l i p .

The s t a b i l i t y b o u n d a ~ i e s t h a t r e s u l t from combined ro': r a t e and s i d e s l i p a r e shown i n F i g . 2.4-9. These b o u n d a r i e s i n d i c a t e t h a t po h a s o n l y a s m a l l e f f e c t on t h e f a s t modes, p r i m a r i l y t h e Dutch r o l l mode. The combination of po and s m a l l v a l u e s of f3' of o p p o s i t e s i g n s e r v e s t o d e s t a b i l i z e t h e Dutch r o l l mode. R o l l rate d e s t a b i l i z e s t h e phugoid mode i n g e n e r a l , b u t t h e r e is a combination of BO and po t h a t m a i n t a i n s phugoid s t a b i l i t y .

High BO r e s u l t s i n a f a s t d i v e r g e n c e f o r a l l v a l u e s of po t e s t e d .

E i g e n v e c t o r v a r i a t i o n s due t o s t e a d y r o l l i n g are i l l u s t r a t z d i n F i g . 2.4-10. E i g e n v a l u e c h a n g e s a r e s i g n i - f i c a n t , c o n s i d e r i n g t h e a n g u l a r r a t e s i n v o l v e d . The mode s h a p e s a l s o c h a n g e , s o t h a t l a t e r a l - l o n g i t u d i n a l c o u p l i n g is i m p o r t a n t . Large r o l l - r a t e l s i d e s l i p p e r t u r b a t i o n s i n ROLL RATE, p ( d e g h c c ) " 0

a PHUGOIO STABLE

F i g u r e 2.4-9 S t a b i l i t y B o u n d a r i e s f o r S i d e s l i p / R o l l - R a t e V a r i a t i o n s ( a 0 = 1 5 d e g ) t h e s h o r t p e r i o d mode and l a r g e a n g l e o f a t t a c k p e r t u r b a - t i o n s i n D u t c h r o l l mode a r e e x a m p l e s of t h i s c o u p l i n g .

C o n c l u s i o n s a b o u t s t e a d y a n g u l a r r a t e e f f e c t s a r e a s f o l l o w s : Mean yaw r a t e and r o l l r a t e c a u s e l a t e r a l - l o n g i t u d i n a l c o u p l i n g and t h e r e f o r e c h a n g e t h e mode s h a p e s s i g n i f i c a n t l y . Roll r a t e is by f a r t h e more s i g n i f i c a n t b e c a u s e o f t h e l a r g e v a l u e s i t c a n e x h i b i t .

Mean p i t c h r a t e c h a n g e s t h e s p e e d s o f t h e normal modes v i t h o u t a f f e c t i n g

t h e i r s h a p e s s i g n i f i c a n t l y . Even low

v a l u e s o f q g ( a b o u t 5 d e g l s e c ) c a n c a u s e t h e Dutch r o l l mode t o b e u n s t a b l e .

2 . 5 EFFECTS OF A??GULAR MOTION AND FLIGHT CO-NDITION ON AIRCRAFT CONTROL The t r a n s f e r f u n c t i o n p r o v i d e s a p r i m a r y m e a s u r e o f t h e q u a l i t y o f a i r c r a f t c o n t r o l , as i t is t h e L a p l a c e t r a n s f o r m o f t h e r a t i o between a s p e c i f i c o u t p u t a n d a s p e - The t r a n s f e r f u n c t i o n g a i n , KF, c i f i c i n p u t (Appendix A ) .

is t h e s t a d y - s t a t e v a l u e o f t h e t r a n s f e r f u n c t i o n a f t e r a l l t r a n s i e n t s damp o u t , a s s m i r i g t h a t a l l t r a n s i e n t s are s t a b l e .

The t r a n s f e r f u n c t i o n g a i n , KI, is ( f o r m o s t a i r c r a f t ) , t h e i n i t i a l s t a t e r a t e r e s p o n s e t o a t r a n s f e r f u n c t i o n ' s c o n t r o l s t e p . T h e p o l e s o f t h e t r a n s f e r f u n c t i o n are t h e e i g e n v a l u e s o f t h e u n f o r c e d s y s t e m , as d e s c r i b e d i n Appendix A .

The z e r o s a f f e c t t h e m a g n i t u d e s o f e x c i t a t i o n o f t h e a o r m a l modes, which are r e l a t e d t o t h e d i s t a n c e between t h e z e r o s and t h e a p p r o p r i a t e e i g e n v a l u e s i n t h e s p l a n e .

I n t h e l i m i t i n g c a s e , a z e r o and p o l e i n t h e same l o c a t i o n c a n c e l , and t h e - c o r r e s p o n d i n g mode d o e s n o t a p p e a r i n t h a t r e s p o n s e . Z e r o s l o c a t e d i n t h e r i g h t h a l f - p l a n e are c a l l e d nonminimum-phase z e r o s ( d u e t o t h e i r e f f e c t s o n t h e p h a s e - s h i f t of s i n u s o i d a l i n p u t s ) , a n d t h e y h a v e m a j o r i m p a c t on t h e a i r c r a f t ' s t r a n s i e n t r e s p o n s e a n d on c o n t r o l l e r d e s i g n .

F o r e x a m p l e , a n u n d e s i r a b l e r e v e r s a l i n t h e i n i t i a l res- p o n s e is c a u s e d by s u c h z e r o s , as i l l u s t r a t e d i n F i g . 2.5-1.

The nonminimum-phase t y p e o f r e s p o n s e is u n d e s i r a b l e b e c a u s e i t makes c l o s e d - l o o p c o n t r o l d i f f i c u l t . The p i l o t c a n b e m i s l e d by t h i s t y p e o f r e s p o n s e , a s t h e m a g n i t u d e a n d s i g n o f t h e m o t i o n a r e u n c e r t a i n . A d d i t i o n o f a h i g h - g a i n f e e d b a c k l o o p a r o u n d a t r a n s f e r f u n c t i o n t h a t e x h i b i t s non- minimum-phase p r o p e r t i e s c a n r e s u l t i n i n s t a b i l i t y o f t h e

- DESIRED RESPONSE - DESIRED RESPONSE

w I- --- UNDESIRED REICONSE --- UNDESIRED REICONSE

\

E

\

5 \

t; \ \ \ F i g u r e 2 . 5 - 1 T y p i c a l S t e p Response Forms c l o s e d - l o o p s y s t e m ( R e f . 9 ) . F i n a l l y , t h i s t y p e of r e s p o n s e c a n make it i m p o s s i b l e t o implement some s i m p l e f o r m s of a d a p t i v e c o n t r o l , a s t h e y c a n s u f f e r from i n s t a b i l i t y f o r an a n a l o g o u s r e a s o n ( R e f . 5 7 ) .

2 . 5 . 1 V e l o c i t y and Aerodynamic Angle E f f e c t s A s h a s b e e n o b s e r v e d p r e v i o u s l y ( S e c t i o n 2 . 4 . 1 ) , v e l o c i t y c h a n g e s t h e dynamic p r e s s u r e , which a f f e c t s t h e T a b l e 2 . 5 - 1 i l l u s t r a t e s some t y p i c a l c o n t r o l e f f e c t i v e n e s s .

v a l u e s of t h e t r a n s f e r f u n c t i o n g a i n , K F , a t d i f f e r e n t s p e e d s , and t h e v a r i a t i o n i s a s e x p e c t e d . A l t i t u d e a l s o a f f e c t s dynamic p r e s s u r e i n t h a t i n c r e a s i n g a l t i t u d e d e c r e a s e s - a t m o s p h e r i c d e n s i t y ; h e n c e , dynamic p r e s s u r e d e c r e a s e s .

TABLE 2.5-1 VELOCITY EFFECTS ON TRANSFER FUNCTION GAIN, KF V e l o c i t y A w / A b h A r / A b a AP/ Ada 70 m / s -3.88 - 0 . 5 0 -4.04 Aerodynamic a n g l e v a r i a t i o n s c a n c a u s e large c h a n g e s i n t h e system e i g e n v a l u e s and c a n b e e x p e c t e d t o h a v e s i g - n i f i c a n t e f f e c t s on t h e numerator o f t h e t r a n s f e r f u n c t i o n as

w e l l . T a b l e 2.5-2 i l l u s t r a t e s v a r i a t i o n s i n KI and % a s

aO and B O v a r y . ( S e e S e c t i o n A . 4 . 3 f o r t h e d e f i n i t i o n o f K I . ) The i n v a r i a b i l i t y of KI w i t h s i d e s l i p i n d i c a t e s t h a t t h e c o n t r o l e f f e c t i v e n e s s d o e s no: depend on s i d e s l i p . T h i s is a f u n c t i o n o f t h e aerodynamic d a t a u s e d h e r e (Appendix B ) , a s t h e d a t a d o e s n o t model t h e e f f e c t s of B O on c o n t r o l e f f e c t i v e n e s s . The s t e a d y - s t a t e g a i n . KF, shows a l a r g e dependence on s i d e s l i p b e c a u s e t h i s g a i n d e p e n d s on t h e p o l e and z e r o l o c a t i o n s , which t h e m s e l v e s v a r y w i t h B O .

The s i g n c h a n g e s i n KF ( a t small a o ) a s B 0 v a r i e s a r e due t o c h a n g e s i n t h e number of u n s t a 3 l e p o l e s and non- minimum-phase z e r o s . From E q . ( 2 . 5 - 6 ) , i t c a n b e s e e n t h a t s u c h a change r e s u l t s i n a KF s i g n change i f t h e s i g n o f K I r e m a i n s t h e same.

One of t h e major e f f e c t s of a n g l e - o f - a t t a c k v a r i a - t i o n s ( b e s t s e e n i n t h e 3 g a i n o i the A p / d S a t r a n s f e r I TABLE 2 . 5 - 2 AERODYNAMIC ANGLE EFFECTS O N TRANSFER FUNCTION GAINS (V0=94rn/s) Av/ACr -0. a:, 0 . 3 7

- 1

-0.89 -1.71 - 0 . 2 7 - 0 . 4 4 -0. c 5

- 3

f u n c t i o n ) i s t h e l o s s o f a i l e r o n r o l l c o n t r o l a t h i g h oo.

T h i s l o s s o f a i l e r o n r o l l e f f e c t i v e n e s s , combined w i t h t h e c o n t i n u e d e f f e c t i v e n e s s o f t h e r u d d e r f o r r o l l a n d yaw c o n t r o l , l e a d s t o t h e c o n c l u s i o n t h a t t h i s a i r c r a f t is r o l l e d m o r e e f f e c t i v e l y w i t h t h e r u d d e r a t h i g h a n g l e s o f a t t a c k .

An e x a m i n a t i o n o f t h e t r a n s f e r f u n c t i o n z e r o s ( T a b l e 2 . 5 - 3 ) i n d i c a t e s t h a t nonminimum-phase zeros a r e q u i t e p r e v a l e n t , a l t h o u g h o f t e n a c c o m p a n i e d by r i g h t - h a l f - p l a n e p o l e s , i . e . , t h e y o f t e n o c c u r i n u n s t a b l e s y s t e m s .

R i g h t - h a l f - p l a n e z e r o s a r e i m p o r t a n t i n c o n t r o l s y s t e m d e s i g n b e c a u s e c l o s e d - l o o p p o l e s o f a s y s t e m w i t h a s i m p l e l o o p c l o s u r e m i g r a t e f r o m t h e o p e n - l o o p p o l e s t o t h e z e r o s as t h e l o o p g a i n is i n c r e a s e d . T h e r e f o r e , i n a s y s t e m w i t h r i g h t - h a l f - p l a n e z e r o s , t o o h i g h a g a i n may r e s u l t i n a n u n s t a b l e c l o s e d - l o o p s y s t e m .

When B O is n o t z e r o , t h e r e is mode c o u p l i n g , a n d a c o n t r o l i n p u t e x c i t e s a l l modes. T h i s is i n d i c a t e d b y T a b l e 2 . 5 - 4 , w h i c h p r e s e n t s t h e p o l e s a n d z e r o s o f t h r e e t r a n s f e r f u n c t i o n s a t a f l i g h t c o n d i t i o n where B o is non- TABLE 2.5-3 E F F E C T S O F ANGLE O F ATTACK ON TRANSFER FUNCTION ZEROS A w / A d h 5 0.0084 ij 0.0824 -35.89.

15 -0.0025 ij 0.1167 -21.07 25 -0.0281 tj 0.1500 -25.56 I Av/Ad, 5 -0.168 -1.465 86.43 15 -0.137 -0.511 1315.0 25 -0.164 0.178 133.7* Ar/Ad, 5 0.350 i 2.914 -1.06 15 2.06 j 9.63 -0.385 25 0.036 i 1.725 -0.2375 AQ/ Ad, 5 -0.141 j 1.399 0. (1270 15 -0.152 j 1.295 0.0278 25 -13.67 22.49 0.0300' A v ~ Ad, S 0.3198 -0.840 40.97 15 0.0956 -0.393 45.74 2 5 0.0795 -0.188 -41. 70* - -- - - - - - - - -p +Accompanied by right-half-plane poles TABLE 2 . 5 - 4 POLE-ZERO C O W A R I S O N AT a 0 = 1 5 DEG, B O = 10 DEG zero. Note t h a t the l a t e r a l mode poles a r e not canceled i n A w / A ~ ~ and t h a t t h e longitudinal mode poles a r e not canceled i n t h e A r l A d , and Ap/A6, t r a n s f e r functions.

The e f f e c t s of velocity and aerodynamic angles on control of t h e example a i r c r a f t can be summarized a s follows: 0 Lower v e l o c i t i e s lead t o decreased control e f f e c t i v e n e s s , a s demonstrated by t r a n s f e r function gains.

Non-zero 00 does not a f f e c t K I , but does change the poles and zeros so t h a t a l l modes a r e excited.

Mean angle of a t t a c k leads t o s i g n i f i c a n t changes i n control e f f e c t i v e n e s s , so much so t h a t t h e rudder is more e f f i c i e n t than t h e a i l e r o n for producing r o l l zit high angles of a t t a c k .

Nonminimum-phase zeros a r e prevalent i n the a i r c r a f t t r a n s f e r functions a t increased angle of a t t a c k .

Angular Rate E f f e c t s Although no e x p l i c i t e f f e c t s of nominal angular r a t e s on t h e control effectiveness a r e included i n the specific aerodynamic data used here, angular r a t e s cause s i g n i f i c a n t changes i n the t r a n s f e r functions due t o pole and zero s h i f t s . T h i s is apparent i n Table 2.5-5, which shows changes i n t r a n s f e r function gains due t o nominal p i t c h r a t e . The i n i t i a l value of the t r a n s f e r function, K I , does not vary w i t h qo because KI depends only on t h e con- t r o l effectiveness. The steady-state g a i n , K F , does vary with q o because of the pole and zero v a r i a t i o n s . A s above, sign v a r i a t i o n s i n KF indicate the appearance of unequal numbers of nonrninimum-phase zeros and r i g h t half-plane poles.

TABLE 2.5-5 EFFECTS OF PITCH RATE O N TRANSFER FUNCTION G A I N S ( a O = 1 5 d e g ) Mean w i n d - a x i s r o l l r a t e , l i k e B O , h a s n o e f f e c t o n c o n t r o l p o w e r b u t d o e s c h a n g e t h e mode s h a p e s s i g n i f i c a n t l y .

T h e s t e a d y - s t a t e t r a n s f e r f u n c t i o n g a l n v a r i e s w i t h p o a s shown i n T a b l e 2.5-6. T h i s v a r i a t i o n is f a i r l y s m o o t h , com- p a r e d t o t h e e f f e c t s o f q o , a n d t h e o n l y s i g n c h a n g e s are i n t h e r o l l - r a t e t ? a n s f e r f u n c t i o n s . B e c a u s e n o n - z e r o r o l l rate creates l a t e r a l - l o n g i t u d i n a l c o u p l i n g , a n y c o n t r o l d i s - p l a c e m e n t e x c i t e s a l l o f t h e n o r m a l m o d e s .

TABLE 2.5-6 EFFECTS OF ROLL RATE O N TRANSFER FUNCTION G A I N , KF ( a 0 = 1 5 d e g ) C o n c l u s i o n s a b o u t a n g u l a r r a t e e f f e c t s o n t h e c o n - t r o l l 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 a r e a s f o l l o w s : Non-zero n o m i n a l a n g u l a r rates d o n o t c h a n g e c c n t r o l e f f e c t i v e n e s s , b u t d o c h a n g e mode s h a p e s a n d / o r s p e e d s , a s w e l l a s z e r o l o c a t i o n s .

Nominal qo has a large effect on KF, primarily due to the creation of non- minimum-phase zeros and unstable modes.

e Nominal po primarily changes mode shapes rather than pole locations, and it causes any control de- flection to excite all response modes.

2.6 DYNAMIC VARIATIONS DURING EXTREME MANEUVERING Aircraft may be especially prone to departure from controlled flight during air combat maneuvering because such maneuvers are executed using the highest possible aircraft performance, and pilot workload during maneuvering flight is high. Although it is possible to fly most combat maneuvers in a smooth, coordinated manner, even small errors can cause difficulty due to instability, unfamiliar coupled mode shapes, or changes in cohtrol effectiveness.

Many air combat maneuvers include periods of high angle-of-attack fligh.~, in order to produce a large normal force for climbing or turning. High angular rates also are typical of many air combat maneuvers. High normal accelera- tion may be accompanied by large qo, and large po may be generated to rapidly orient the lift force in a desired direct ion.

Referring to the earlier sections in this chapter, the difficulties involved in extreme maneuvering become clear. High angles of attack and pitch rate destabilize the normal modes of motion and reduce the available control power, while high roll rate causes lateral-longitudinal coupling and produces mode shapes unfamiliar to the pilot.

The f i r s t two of t h e following s e c t i o n s examine t h e changes i n a i r c r a f t s t a b i l i t y and control along two t y p i c a l a i r combat t r a j e c t o r i e The t h i r d section apprvaches the same problem from a d i f f e r e n t viewpoint, examining t h e e f f e c t s of an elementary target-tracking p i l o t model on a i r c r a f t s t a b i l i t y .

2 . 6 . 1 Wind-Up T u r n

-

In a wind-up turn, t h e a i r c r a f t is r o l l e d and high load f a c t o r is commanded, r e s u l t i n g i n a high p i t c h r a t e . As airspeed bleeds r.ff (which may occur even a t maximum t h r u s t ) , angle of a t t a c k is increased, and t h e a i r c r a f t s t a b i l i t y decreases. Five points taken from a t y p i c a l wind-up turn time history a r e described i n Table 2.6-1, and the corre- sponding eigenvalues a r e given i n Table 2.6-2. O f s p e c i a l i n t e r e s t is the Dutch r o l l damping, which decreases so t h a t the Dutch r o l l mode becomes unstable a s t h e wind-up t u r n progresses. These a r e not synunetric f l i g h t conditions, so i t is expected t h a t the Dutch r o l l eigenvector a l s o con- t a i n s angle-of-attack perturbations.

Ifi addition t o lateral-longitudinal coupling and t h e general reduction i n damping, the control e f f e c t i v e n e s s a l s o decreases, a s i l l u s t r a t e d by the i n i t i a l value of the trans- f e r function, shown i n Table 2.6-3. I t is necessary f o r the p i l o t t o use rudder a s the r o l l control a t high angles of a t t a c k , and t h i s can cause s i d e s l i p perturbations, which can lead to further problems.

These representative points from a wind-up t u r n demonstrate the d e t e r i o r a t i o n of the s t a b i l i t y and contrcl of the example a i r c r a f t a s i t executes one fcrm of a i r combat maneuver.

TABLE 2 . 6 - 1 M'INII-UP T U R N ROHK I N G POT N'I'S ~Vorking Point Description Flight C o n d i t i o n - -..----.----- Roll and T u r n v o = 2 1 7 1;~;:- a 0 = 5 deg t = 0 sec p0 = 5 deg/sec r o = 5 deg/sec I $ ~ = 45 deg P o = 5 deg Rapid Turn Y o = 2 ; : r;ts a O = 11 deg t = 13 sec q0 = 1 0 d e g / s e c r O = 5 d e g l s e c I $ ~ = 85 deg e o = -15 deg Turning Vo = :17 :: I S a 0 = 1 5 deg .; = 30 sec = i O deg/sec r = 5 d e g / s e c u Q i O a 7 0 d e g B 0 = -20 d e g T u r n i n g Vo = I::.; :l :S a0 = 22 deg t = 52 sec = 1 0 d e q / s e c q0 = 15 d e g / s e c P o r o = 1 0 deg/sec 4O = 7 0 deg e 0 = -20 deg Turning lr0 = ~ 1 6 r ' s u , = 27 deg u t = 75 sec po = 12.5 d ~ c / s e c q0 a 12.5 deglsec G O = 6 0 deg B O = -25 deg -- Short Period D u t c h Roll Roll S p l r a l -.---

-

1 -0.935rjl 76 -0.383rj3.43 -1 61 0 . 0 6 4 TABLE 2.6-3 TRANSFER FUNCTION GAIN, K T , ALONG THE WI'XD-UP TURN 2.6.2 Rolling Reversal - .

A rolling reversal combines a rapid pull-up with a rapid rolling maneuver, resulting in a "corkscrew-like" path through space. The combination of a high-accelerationpull- up slid rapid rolling is expected to produce unstable modes with considerable lateral-longitudinal coupling. Table 2.6-4 describes the rolling reversal working points exanined here. The corresponding eigenvalues, shown in Table 2.6-5, illustrate the changes in aircraft stability as rhe rolling reversal progresses. Duo to the high q o involved in this mAneuver, the Dutch roll mode is unstable throughout most of the maneuver.

The initial and final working points of the rolling reversal are symmetric flight conditions so there is no lateral-longitudinal coupling during these phases of the flight. This is demonstrated by the eigsnvectors of the fast modes at the first w r k i n g point, which a r c shown in Fig.

2.6-1. The i n t e r m ~ d i : ~ t ~ work~nq points all o c c l ~ r ciuring the aircraft's roll and invol\~o significant lat~rnl -longitudinal coupling. The e i g c ~ n v e c * t o r s (;t t h . 1 f a z t m ~ . ~ r t t . . - - a t IY~rking Pcint 3 , shown in Fig. 2.6-1, drmonstrnte t h ~ s 'rherc- i:; slgnif'icant ThBLE 2.6-4 ROLLING REVERSAL WORKING POINTS -- -- Working Point D e s c r i p t i o n F l i g h t Condition

V = 217 r;!s a0 - 2 5 deg

H i - G Pull-up t = 0 sec = 1 5 d e g / s e c = 30 deg v0 = 168 n ! s R o l l a 0 = 26 deg = -10 d e g / s e c = 15 d e g / s e c t = 4 sec 0 0 4O = -90 deg e 0 = 5 0 deg I ' = 101 m ! s R o l l a 0 = 26 deg po = -15 d e g / s e c t = 10 sec q 0 = 15 d e g / s e c r = -5 d e g / s e c go = -180 deg g o = -30 deg R o l l and Pull-up V o = 117 111./s a. = 23 deg t = 15 sec po = 10 dcq/sec qO = 1 5 d e g / s e c

r = 5 d e g / s e c eO = 45 deg

e 0 = -55 deg F i n a l Pull-up a. = 2 0 deg t = 22 sec q O = 1 0 d e g / s e c e 0 = -10 deg TABLE 2.6-5 ROLLING REVERSAL ETCENVALUES Working Dutch Roll Short P e r i o d R o l l S p i r a l Phugoid P o i n t b-- + DUTCH ROLL SHORT PERIOD ROLL Figure 2.6-1 Eigenvectors of Rolling Reverskl angie-of-attack (or A w ) motion in the Dutch roll mode and large roll-sideslip perturbation in the short period mode. Even the roll mode contains a significant angle-of-attack excursion.

The aircraft control effectiveness follows trends similar to the aircraft stability, i.e., the control effec- tiveness is degraded throughout the first half of the maneu- w r , but it improves during the second half, as illustrated in Table 2.6-6. The pilot must use the rudder as a roll con- trol during the middle portion of this maneuver. This diffi- culty is complicated by high angular rates, lateral-longi- tudinal coupling, and extreme attitudes.

An aircraft executing a rolling reversal exhibits unstable modes, lateral-longitudinal coupling, and reduced control effectiveness as the maneuver progresses. all of which make the pilot's task more difficult.

TABLE 2.6-6 TRANSFER FUNCTION GAIN, K I , ALONG THE ROLLING REVERSAL / 2.6.3 E f f e c t s of P r o p o r t i o n a l T r a c k i n g Rudimentary p i l o t i n g e f f e c t s c a n be examined by assuming t h a t t h e p i l o t a t t e m p t s t o c o n t r o l t h e a i r c r a f t ' s a t t i t u d e . T h i s c a n b e modeled by a p r o p o r t i o n a l f e e d b a c k o f a n g u l a r d e v i a t i o n ( p i t c h o r r o l l a n g l e ) t o t h e a p p r o p r i a t e c o n t r o l s u r f a c e ( e l e v a t o r o r a i l e r o n ) . P i l o t l a g s o r t i m e d e l a y s are n e g l e c t e d . A s an example, p i t c h a t t i t u d e c o n t r o l is c h o s e n , and t h e f e e d b a c k g a i n s a r e set s o t h a t t h e e f f e c - t i v e p i t c h moment d u e t . ~ p i t c h a n g l e , Me, is a m u l t i p l e o f t h e p i t c h moment due t o a n g l e of a t t a c k , Ma.

S i n c e t h i s is a c h i e v e d by e l e v a t o r f e e d b a c k , t h e r e a r e c h a n g e s t o t h e c o e f f i c i e n t s Xe and Z e a s w e l l . The m u l t i p l y i n g f a c t o r is d e n o t e d by "i" i n t h e f o l l o w i n g t a b l e s , and t h e f e e d b a c k g a i n t h a t p r o d u c e s e q u a l M e and Ma ( i = 1) f o r t h e r e f e r e n c e f l i g h t c o n d i t i o n is 0.64 deg e l e v a t o r p e r deg of p i t c h a n g l e . T h i s s i m p l e model a l s o d i s r e g a r d s p i t c h - r a t e f e e d - b a c k , which t h e p i l o t a l s o might n o r m a l l y p r o v i d e .

For symmetric f l i g h t c o n d i t i o n s , t h i s l o o p c l o s u r e d o e s n o t a f f e c t t h e l a t e r a l e i g e n v a l u e s ; t h e l o n g i t u d i n a l e i g e n v a l u e v a r i a t i o n s are shown i n T a b l e 2.6-7. The s h o r t - p e r i o d mode is b o t h i n c r e a s e d i n f r e q u e n c y and d e c r e a s e d i n TABLE 2.6-7

EIGENVALUE CHANGES DUE TO PROPORTIONAL TRACKING --

SYMMETRIC FLIGHT CONDITIONS a = 15 deg qo = 0 deg/sec

1 VO = 94 m/s

0 I

i I Short Period Phugoid (Pitch

I ArgleISpeed)

I

Phugoid (Pitch

I Short Period

Angle/Speed) damping by the addition of a pitch attitude-to-elevatcr feedback. For straight-and-level flight, attitud -0ntro1 increases the phugoid damping while decreasing th ~tural frequency, resulting in the conversion of the pb~goid mode into two real modes -- a pitching mode and a speed mode.

In steady pitching motion, pitch-attitude control results in increased stability for the pitch angle mode with relatively little effect on the speed mode.

Coupled flight conditions lead to significant effects on the lateral modes due to the longitudinal loop closure. In Ref. 9, a pitch attitude-to-elevator loop closure resulted in an unstable lateral mode when the sub- ject aircraft was in a steady sideslip. As shown in Table 2.6-8 for the example aircraft used in this report, pitch angle-to-elevator feedback generally has a beaeficial influence on the lateral eigenvalues when this aircraft is rolling at zero sideslip. However, the presence of a non- zero nominal sideslip angle results in a mildly diverging speed mode for moderate-to-large feedback gains. There is an inaication that large attitude feedback gains destabilize the Dutch roll mode (for a non-rolling aircraft) or the short period mode (when the aircraft is rolling and slipping).

Regarding the pitch attitude-to-elevator feedback as a simple pilot model, this examination confirms the earlier result that pilot control of the longitudinal motion of an aircraft could result in the destabilization of the aircraft when the vehicle is in a steady sideslip. This would be due to the pilot disregarding the lateral-longitudinal coupling present in asymmetric flight conditions.

This simple attitude feedback underlines the necessity for considering lateral-longitudinal crossfeeds when designing a stability augmentation system for a high- performance aicraft. TG achieve acceptable performance, it may be necessary to design a system that recognizes the aircraft flight condition and adjusts its gains to suit the situation.

2.7 CHAPTER SUMMARY This chapter has presented a study of the dynamic charac ?ristics 3f a high-performance aircraft, with special emphasis on the effects of extreme flight conditions on TABLE 2 . 6 - 8 EJGENVALUE CHANGES DUE TO PROPORTIONAL T R A C K I N G - - SIDESLIP AND ROLL EFFECTS = 0 deg Vo = 94 m / s a = 1 5 deg B O pW0 = 0 d e g / s e c S h o r t P e r i o d Dutch R o l l R o l l S p i r a l I , Phugo i d i ,

--

-0.407kj1.099 0 . 0 7 3 8 k j 2 . 2 5 -0.442 0 -0.017'j0.137 -0.0545 1 - 0 . 3 4 2 t j 1 . 5 5 2 0 . 0 7 3 8 k j 2 . 2 5 -0.442 -0.082'j0.068 -0.0545 2 - 0 . 3 1 9 t j 1 . 9 0 6 0 . 0 7 3 8 k j 2 . 2 5 -0.442 -0.0545 -0.157 -0.0522 4 - 0 . 3 0 2 t j 2 . 4 7 0 . 0 7 3 8 k j 2 . 2 5 -0.442 -0.0545 -0.218 -0.0260 I I Vo = 94 m / s I - L ~ = 1 5 deg B O = 1 0 d s g = 0 d e g / s e c

Pwo

i S h o r t P e r i o d D u t c h R o l l R o l l S p i r a l Phugoid 0 -0.353kj1.363 - 0 . 1 3 4 ' j 2 . 1 1 -0.434 -0.0315 -0.024kjO.146 1 - 0 . 3 0 3 2 j 1 . 7 6 - 0 . 1 5 3 ' j 2 . 1 1 -0.431 - 0 . 0 8 1 - 0 . 0 3 2 t j 0 . 1 2 4 2 - 0 . 2 9 8 t j 2 . 1 5 - 0 . 1 4 3 k j 2 . 0 3 -0.430

-0.021 -0.287 [0.1331

4 - 0 . 3 2 7 + j 2 . 6 3 -0.101'j2.08 - C . 4 2 8 -0.152 - 0 . 0 2 4 7 + j 0 . 1 0 8 .

r V o = 9 4 m / s a O = 1 5 deg = 1 0 deg pW0 = -39 d e g l s e c

Bo

- i S h o r t P e r i o d

Dutch R o l l R o l l S p i r a l I Phugoid

0 - 0 . 5 1 1 + j 1 . 3 2 - 0 . 0 5 7 k j 2 . 2 8 -0.246kj0.199 0 . 0 7 1 k j 0 . 2 2 7 Vo = 94 m / s = 1 5 deg B O = 0 dey pW0 = -39 d e g / s e c

0 . - - - - ---

i S h o r t P e r i o d D u t c h R o l l R o l l S p i r a l Phugoid -- - 0 - 0 . 4 6 4 k j 1 . 2 9 - 0 . 0 3 2 ? . j 2 . 3 4 -0.2932j0.144 0 . 0 4 3 + j 0 . 1 8 9 -0.400?.j1.56 - 0 . 0 4 4 t j 2 . 3 7 - 0 . 1 6 7 i j 0 . 5 0 0 -0.207 -0.062 2 - 0 . 3 3 4 2 j 1 . 8 2 - 0 . 0 7 3 + j 2 . 4 2 - 0 . 2 0 8 + j 0 . 6 0 0 - 0 . 2 2 9 - 0 . 0 3 5 4 -0.169',j2. 10 0 0 6 7 -0.246?.jO. 666 - 0 . 2 3 8 - 0 . 0 2 2 8 A aircraft stability and control. The chapter first examines previous studies of the dynamics, aerodynamics, and control in this flight environment. The difficulty of measuring angular rate and translational acceleration effects leads to limited availability of this aerodynamic data, which has a significant impact on the simulation and analysis of dynamic departures. The survey cf stability and control indicates a need for additional developments in these areas.

EFFECTS OF CONFIGURATION VARIATIONS

-

ON AIRCRAFT DYNAMICS 3.1 OVERVIEW Variations of aircraft configuration lead to changes in the aircraft's eigenvalues, eigenvectors, and control effectiveness. Section 3.2 presents the effects of changes in the most important longitudinal stability derivatives on the mode shapes and speeds. Similar effects caused by changes in lateral stability derivatives are detailed in Section 3.3.

The effects of aircraft mass and rotational inertia variations are given iq - Section 3.4. Section 3.5 presents a general dis- cussion of possible departure modes and illustrates some of the possible departure time histories. Section 3.6 is a summary of the chapter.

3.2 VARIATIONS DUE TO LONGITUDINAL STABILITY DERIVATIVES The longitudinal stability derivatives determine the aerodynamic force and moment contributions to the longitudinal perturbation equations, and these stability derivatives can vary considerably from aircraft to aircraft. This section surveys the changes in normal mode shapes and speeds for different ranges of the most important longitudinal stability derivatives.

Three aerodynamic derivatives dominate the short period mot ion of the, aircraft : The sig- Cmq C m , , and (2%.

nificance of t5ese terms can be seen in reduced-order approxi- mat ions to the d m p l n g ratio, ;, and natural frequency, w of n ' this mode: Over t h e r a n g e o f l i k e l y v a l u e s of t h e c o e f f i c i e n t s , t h e C C Z mq a p r o d u c t u s u a l l y is c o n s i d e r a b l y smaller t h a n C,,, SO i t can a n d C Z a p r i m a r i l y a f f e c t s h o r t p e r i o d b e e x p e c t e a t h a t Cm d a m p i n g , w h i l e C , , c h a n g e s o n l y t h e s h o r t p e r i o d n a t u r a l f r e - q u e n c y .

are The s p e c i f i c e f f e c t s o f v a r y i n g CZ, a n d Cm q I l l u s t r a t e d i n T a b l e 3 . 2 - 1 . The p r i m a r y s h o r t p e r i o d e i g e n v a l u e c h a n g e s o c c u r i n t h e d a m p i n g , a s e x p e c t e d . The p h u g o i d mode d o e s c h a n g e s o m e w h a t , as a n i n c r e a s e i n t h e l i f t - c u r v e s l o p e ( C Z m o r e n e g a t i v e ) i n c r e a s e s t h e p h u g o i d n a t u r a l f r e q u e n c y a a t e s s e n t i a l l y c o n s t a n t damping r a t i o . An i n c r e a s e i n t h e m a g n i t u d e of p i t c h damping (Cm ) d e c r e a s e s p h u g o i d f r e q u e n c y and damping s i g n i f i c a n t l y .

TABLE 3 . 2 - 1 Czc, AND Cm EFFECTS ON EIGENVALUES a . = 15 d e g ) (Vo = 94 m / s , -1

C ~ , ( r a d ) S h o r t P e r i o d P h u g o i d I

I - 1 S h o r t P e r i o d C m q ( r a d ) Phugo i d i -34.4 - 0 , 6 4 2 ? j 1 . 0 5 8 -O.O13+jO.~30 V a r i a t i o n s i n t h e l o n g i t u d i n a l s t a b i l i t y d e r i v a - t i v e s a f f e c t t h e lateral modes o n l y when t h e a i r c r a f t is i n a s y m m e t r i c f l i g h t . I n t h a t case, mode c o u p l i n g occurs-- T a b l e 3.2-2 i l l u s t r a t e s t h e c h a n g e s i n t h e D u t c h r o l l a n d s h o r t p e r i o d modes f o r asymmetric f l i g h t as Cm v a r i e s . The t r a n s f e r o f damping f r o m t h e s h o r t p e r i o d mode t o t h e D u t c h r o l l mode f o r n o n - z e r o B o h a s b e e n o b s e r v e d i n S e c t i o n 2 . 4 .

I n t h e case o f r e d u c e d Cm t h i s t r a n s f e r is e s s e n t i a l l y q ' u n c h a n g e d , i n d i c a t i n g t h a t s h o r t p e r i o d damping d u e t o CZ a ( s e e E q . ( 3 . 2 - 1 ) ) is t r a n s f e r r e d w h e r e a s damping d u e t o C ma is n o t . T h i s is s u p p o r t e d by t h e o b s e r v a t i o n t h a t D u t c h r o l l e i g e n v e c t o r s f o r n o n - z e r o B O ( F i g . 2 . 4 - 4 ) i n c l u d e much more a n g l e - o f - a t t a c k m o t i o n t h a n p i t c h r a t e .

TABLE 3 . 2 - 2 EFFECTS OF Cm ON EIGENVALUES IN ASYMMETRIC FLIGHT = 15 d e g pwO = 0 d e g l s e c

B O - 0 d e g

a.

S h o r t P e r i o d D u t c h Roll = - 1 7 . 8 ( r a d - 0 . 4 0 7 t j 1 . 0 9 9 - 0 . 0 7 4 t j 2 . 2 5 1

I

= 5 d e g pwo = 0 d e g l s e c a = 1 5 d e g B O S h o r t P e r i o d D u t c h Roll

- -

P = - 1 7 . 8 ( r a d - 1 ) - 0 . 3 8 2 k j l . 201 - 0 . 0 9 4 6 + . j 2 . 0 2 4 C"q t I I I pwo = -39 6 d e g l s e c a O = 1 5 d e g B o = 0 deg D u t c h R o l l S h o r t P e r i o d - 0 . 0 3 1 9 k j 2 . 3 4 2 = - 1 7 . 8 ( r a d - l ) - 0 . 4 6 4 2 j 1 . 2 9 4 --------'=I = - 5 . 7 3 ( r a d - 1 ) - 0 . 2 9 1 t j 1 . 2 9 5 - 0 . 0 3 4 3 t j 2 . 3 3 7 i None o f t h e s t a b i l i t y d e r i v a t i v e s d i s c u s s e d h a s any e f f e c t on t h e c o n t r o l e f f e c t i v e n e s s , which d e t e r m i n e s t h e t r a n s f e r f u n c t i o n g a i n , K I ; t h e r e f o r e , t h e i n i t i a l r e s p o n s e t o c o n t r o l i n p u t s d o e s n o t v a r y w i t h CZ, and Cm . Hcwever, s t a b i l i t y d e r i v a t i v e v a r i a t i o n s do a f f e c t t h e t r a n s i e n t res- p o n s e t h r o u g h c h a n g e s i n p o l e s and z e r o s .

V a r i a t i o n s i n t h e a i r c r a f t ' s center o f g r a v i t y c a u s e v a r i a t i o n s i n aerodynamic moment c o e f f i c i e n t s . The c e n t e r o f g r a v i t y ( c . g . ) is t h e r o t a t i o n a l c e n t e r of t h e a i r c r a f t .

F o r f i x e d aerodynamic c e n t e r of p r e s s u r e , c . g . v a r i a t i o n l e a d s t o s t a t i c margin v a r i a t i o n ; h e n c e , t h e moment r e l a - t i o n s h i p s are a l t e r e d .

The s t a t i c margin is t h e d i s t a n c e between t h e c . g .

l o c a t i o n and t h e aerodynamic c e n t e r , and it is u s u a l l y e x p r e s s e d

-

a s a f r a c t i o n o f t h e mean ~ e r o d y n a r n i c c h o r d , c . F i g u r e 3.2-1 d e t a i l s t h e changes i n t h e l o n g i t u d i n a l e i g e n v a l u e s as t h e s t a t i c margin is v a r i e d from 0 . 3 3 t h r o u g h its u s u a l r e f e r e n c e l o c a t i o n o f 0 . 1 7 t o -0.15. T h i s r e s u l t s i n a Cma v a r i a t i o n from -1.17 t o 0.554 r a d - l , a s w e l l a s changes i n C, fro^,' - 2 1 . 1 rad'l t o -11.2 r a d - l .

The e i g e n v e c t o r s ( F i g . 3 . 2 - 2 ) change c o n s i d e r a b l y a s t h e c . g . moves a f t . While t h e s h o r t p e r i o d and phugoid modes still a r e r e c o g n i z a b l e a t a s t a t i c margin of 0 . 0 6 (Cma = -0.17 r a d - l ) , a t r a n s i t i o n r e g i o n is e n t e r e d a s t h e c . g . moves f u r t h e r a f t . A t a s t a t i c margin o f 0 . 0 1 LCma = -0.02 r a d - l ) , a new ( " t h i r d " ) o s c i l l a t o r y mode which d i s ; l a y s s i g n l f i c a n t p e r t u r b a t i o n s i n a l l l o n g i t u d i n a l s t a t e s is e v i d e n t . Two r e a l c o n v e r g e n c e s comprise t h e o t h e r l o n g i - t u d i n a l modes -- one f a s t a t t i t u d e mode o c c u r r i n g a t c o n s t a n t v e l o c i t y and f l i g h t p a t h a n g l e and one s l o w v e l o c i t y mode t h a t i n v o l v e s s i g n i f i c a n t f l i g h t p a t h a n g l e v a r i a t i o n s .

Figure 3.2-1 Longitudinal Eigenvalue Variations with c.g. Location SHORT PtRlOO PHUGZO MODE MOGi STATIC MARGIN; 0.17 - 0 27Sf i0.454 -0.114 fj0.283 ATTITUDE THIRD MODE VELOCITY MODE W O E

J., i

STATIC MARaN: 0.01 Ae * a I -?f$ tLl

're Ad A w A w O w -0.978 0.195 2, 0.284 -0.146 Figure 3.2-2 Longitudinal Eigenvector Variations with c.g. Location C o n t r o l e f f e c t i v e n e s s c h a n g e s o n l y s l i g h t l y w i t h c . g . l o c a t i o n , b n t t h e t r a n s f e r f u n c t i o n g a i n , K F , d o e s v a r y a s shown i n T a b l e 3 . 2 - 3 . The z e r o s o f t h e A w / A S h t r a n s - f e r f u n c t i o n are g i v e n i n T a b l e 3 . 2 - 4 . B o t h o f t h e s e t a b l e s i n d i c a t e t h e w e l l - b e h a v e d n a t u r e o f t h e n u m e r a t o r o f t h e e l e v a t o r - t o - a n g l e o f a t t a c k t r a n s f e r f u n c t i o n .

TABLE 3 . 2 - 3 C.G. LOCATION EFFECTS O N TRANSFER FUNC'A ION G A I N , KF S t a t i c Margin

I

TABLE 3 . 2 - 4 COMPARISON OF ZEROS OF Aw/LSh AT THRFE C.G. LOCATIONS S t a t i c Margin z1 . 2 z 3

-

0 . 1 7 - 0 . 0 0 2 5 t j 0 . 1 1 6 7 -21.0'7 I T h i s s e c t i o n e x a m i n e s t h e e f f e c t s of l o n g i t u d i n a l s t a b i l i t y d e r i v a t i v e s , a n d t h e f o l l o w i n g c o n c l u s i o n s a r e made : R e d u c t i o n s i n p i t c h d a m p i n g (Cm ) a n d q l i f t - c u r v e s l o p e (CZ,) r e d u c e t h e s h o r t p e r i o d d a m p i n g w i t h o u t c h a n g i n g t h e f r e q u e n c y .

L a t e r a l - l o n g i t u d i n a l c o u p l i n g p r o d u c e d b y s i d e s l i p t r a n s f e r s d a m p i n g d u e t o Cxa t o t h e D u t c h r o l l mode f r o m t h e s h o ; t p e r i o d m o d e , b u t d a m p i n g d u e t o Cmq re- m a i n s i n t h e l o n g i t u d i n a l p l a n e .

T h e c . g . l o c a t i o n a f f e c t s Cm, d i r e c t l y , a n d a r e a r w a r d c . g . l o c a t i o n r e s u l t s i n t h e c r e a t i o n o f a new u n s t a b l e o s c i l l a t o r y mode t h a t e x h i b i t s s i g n i f i c a n t p e r c u r b a - t i o n s i n a l l l o n g i t u d i n a l v a r i a b l e s . I n a d d i t i o n , two s t a b l e real mod+s a r e

c r e a t e d -- a f a s t a t t i t u d e mode a n d a

s l o w v e l o c i t y mode.

The n u m e r a t o r o f t h e e l e v a t o r - t o - a n r l e o f a t t a c k t r a n s f e r f u n c t i o n is well b e h a v e d a t r e a r w a r d c g . l o c a t i o n s .

3 . 3 VARIATIONS DUE TO LATERAL-DIRECTIONAL STAEILITY DERIVATIVES I n t h i s s e c t i o n , v a r i a t i o n s i n C l g , C n r , C l p , z n d are s t u d i e d t o d e t e r m i n e t h e i r e f f e c t s o n t h e l a t e r a l - 'nr d i r e c t i o n a i m o d e s . E x p e r i m e n t a l i n f o r m a t i o n on t h e v a l u e o f C n i is l i m i t e d , s o a n i n v e s t i g a t i o n of p o s s i b l e e f f e c t s o f CnB o n e i g e n v a l u e s is i n c l u d e d i n t h i s s e c t i o a .

T h e e Z f e c t s o f C .,, C l , , a n d C1 v a r i a t i o n s are P shown i n F i g . 3.3-1. I n c r e a r r s i n m a g n i t u d e c -. Cng a n d Clg c a u s e a n i n c r e a s e i n t h e f r e q u e n c y u f t h e D u t c h r o l l mode.

A d d i t i o n a l l y , b o t h p a r a m e t e r s c a u s e some c h a n g e i n d a m p i n g , z i t h l a r g e r CnB i n c r e a s i n g t h e d a m p i n g r a t i ~ and l a r g e r C l g m a g n i t u d e d e c r e a s i n g t h e D u t c h r o l l d a m p i n g r a t i o .

L a r g e r C ? . m a g n i t u d e c a u s e s t h e p i r a l mode t o P,e more s t a b l e .

B ****aaC

VARIATION FRSM 0.0 TO -0.Srad '

IP

I

- C VARIATION FROM - 0.095 TO - 0.34 rod-'

'B

- - - C, VARIATION FROM 0.0 TO 0.42 rod"

B e REFERENCE VALUE

I I

'"I

a0.15* I

QOLL 1 SPIRAL

-"'.*..~.'~rl. 8 [(uc") -LO a 13 F i g u r e 3.3-1 E f f e c t s of C l g , CnB, and C1 V a r i a t i o n s P on L a t e r a l - D i r e c t i o n ~ l E i g e n v a l u e s F i g u r e 3.3-1 i n d i c a t e s t h a t a t a. of 15 deg t h e e f f e c t s of C1 a p p e a r i n t h e Dutch r o l l damping r a t i o and P i n t h e r o l l c o r v e r g e n c e mode. The p r e s e n c e of t h e s e e f f e c t s is p r e d i c t e d by t h e a p p r c s i m a t e l a t e r a l - d i r e c t i o n a l equa- t i o n s , Eqs. (2.4-1) t o (2.4-51, b u t t h e a p p r o x i m a t e equa- t i o n s g i v e an i n a c c u r a t e i n d i c a t i o n of t h e s i z e o f t h e s e e f f e c t s . E q u a t i o n s (2.4-1) and (2.4-4) p r e d i c t much l a r g e r v a r i ~ t i o n i n t h e r o l l mode e i g e n v ~ l u e due t o C 1 v a r i a t i o n P t h a n is i n d i c a t e d i n F i g . 3.4-1. C o n v e r s e l y , Eqs. (2.4-2) and ( 2 . 4 - 5 ) p r e d i c t a much s m a l l e r v a r i a t i o n i n Dutch r o l l d a m p i n g , r a t i c t h a n o c c u r s i n t h e complete model o f t h e s u b j e c t a i r c r a f t .

The l a c k o f a c c u r a c y o f t h e a p p r o x i m a t e e q u a - t i o n s r e s u l t s f r o m t h e f a c t t h a t t h e s u b j e c t a i r c r a f t d o e s not conform t o t h e a s s u m p t i o n s upon which t h e a p p r o x i m a t e e q u a t i m s a r e b a s e d . The s u b j e c t a i r c r a f t e x h i b i t s much m o r e r o l l m o t i o n i n D u t c h r o l l mode ( a s shown by t h e e i g e n v e c t o r s i n F i g . 2 . 4 - 3 ) b e c a u s e it r o l l s e a s i l y ( d u e t o low Ix/IZ r a t i o ) a n d b e c a u s e its r o l l - y a w a n d r o l l - k i d e s l i p c r o s s d e r i v a t i v e s ( C l r , CPp, a n d C1 ) a r e no: small when p r o p e r l y compared t o t h e r o l l a n d y a w - s i d e s l i p d e r i v a t i v e s (C l p 7 C n r 9 a n d C ) .

*a

T a b l e 3.3-1 c o m p a r e s t h e l a t e r a l - d i r e c t i o n a l e i g e n - v a l u e s f o r C a n d C v a r i a t i o n s i n t h e p r e s e n c e o f non- B I f 3 z e r o n o m i n a l p i t c h r a t e . A c o m p a r i s o n of t h e e i g e n v a l u e s i n d i c a t e s t h a t t h e e f f e c t s o f ' s t e a d y p i t c h i n g a r e i n d e p e n d e n t o f t h e c h a n g e s i n a e r o d y n a m i c c o e f f i c i e n t s . I n a l l c a s e s , - p o s i t i v e p i t c h r a t e d e s t a b i l i z e s t h e D u t c h r o l l and s ~ i r a l -- modes -- a n d s p e e d s up t h e r o l l mode.

TABLE 3 . 3 - 1 EFFECTS OF C n B AND C l g VARIATIONS IN THE PRESENCE OF STBADY P I T C H RATE D u t c h R o l l R c l l S p i r a l I Ref e r e n ? : -0.07382 j 2 . 2 5 1 - 0 . 4 4 3 =lo = 0 - 0 . 0 5 4 5 V a l u e s q 0 = 1 2 d e g / s e c 0 . 1 1 4 ' j 2 . 3 2 0 - 0 . 8 3 9 0 . 0 2 2 0

cq = 0 . 0 = 0 - 0 . 0 0 4 3 2 j 1 . 8 5 5

- 0 . 1 2 7 - 0 . 4 9 9

q o

( r a d - ' ) q0 = 1 2 d e g / s e c 0 . 2 1 4 k j 1 . 9 9 6 - 1 . 0 1 6 0 . 0 0 6 3 - C l g = - b . 3 3 8 - 0 . 0 5 0 5 r j 1 . 9 9 6 q I 0 ( r a d - I ) qd = 12 d e g / s e l : 0 . 1 5 0 r . j 2 . t i 3 8 The a p p r o x i m a t i o n s o f Eqs. ( 2 . 4 - 2 ) and (2.4-5) i n d i c a t e t h a t t h e c h a n g e i n t h e Dutch r o l l m o d e ' s r e a l p a r t s h o u l d b e a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e c h a n g e i n

. T a b l e 3.3-2 i n d i c a t e s t h a t t h i s is n o t t r u e i n t h i s

C n r e x a m p l e , f o r as Cnr i n c r e a s e s by a f a c t o r o f 2 0 , t h e Dutch r o l l m o d e ' s r e a l p a r t ~ n l y d o u b l e s . The o t h e r s i g n i f i c a n t e f f e c t c a u s e d by a .I.-eduction i n yaw damping m a g n i t u d e is t o d e s t a b i l i z e t h e s p i r a l mode.

TABLE 3.3-2 EFFECTS OF Cnr VARIATIONS D c t c h R o l l R o l l S p i r a l

cnr

- 1 . 2 0 0 - 0 . 0 9 8 3 + j 2 . 2 4 6 8 -0.4612 -0.1396

The d e r i v a t i v e Crib is a n a c c e l e r a t i o n d e r i v a t i v e

( a n a l o g o u s t o C m i ) t h a t a r i s e s b e c a u s e t h e aerodynamic f l o w f i e l d e x h i b i t s some l a g i n r e a r r a n g i n g i t s e l f f o l l o w i n g a A c c e l e r a t i o n terms a r e a p p r o x i - c h a n g e i n aerodynamic a n g l e .

m a t i o n s t o t h e s e f l o w f i e l d d y n a m i c s . E x p e r i m e n t a l l y , a c c e l e r a t i o n d e r i v a t i v e s a r c d i f f i c u l t tc m e a s u r e , and t h e y u s u a l l y a r e combined w i t h t h e r o t a r y d e r i v a t i v e s .

A n a l y t i c a l s t u d i e s o f t e n make t h e a s s u m p t i o n s t h a t ,.

- Cnp - c n p + s i n a. Cn;

- P

e n r - - n r - cos a o Crib

R e f e r e n c e 46 e x a m i n e s some o f t h e p o s s i b l e e f f e c t s o f t h i s a p p r o x i m a t i o n o n a i r c r a f t time h i s t o r i e s a n d param- eter i d e n t i f i c a t i o n . I t is c o n c l u d e d t h a t s i g n i f i c a n t param- eter i d e n t i f i c a t i o n e r r o r s c a n o c c u r i f t h e a c c e l e r a t i o n d e r i v a t i v e s are n o t i n c l u d e d i n t h e m o d e l when n e c e s s a r y .

Two t y p e s o f C e f f e c t s h a v e b e e n e x a m i n e d . I n t h e f i r s t t y p e , n o n - z e r o C n e is a d d e d t o t h e m o d e l w i t h n o o t h e r c h a n g e ; t h i s e f f , ? c l i v e l y d e c r e a s e s t h e t o t a l d a m p i n g .

I n t h e s e c o n d t y p e , a n d Cnr are a d j u s t e d b y s u b t r a c t - C " ~ i n g t h e C terms g i v e n i n E q s . ( 3 . 3 - 8 ) a n d ( 3 . 3 - 9 ) t o

"ii

m a i n t a i n n e a r l y c o n s t a n t d a m p i n g . I n t h e v a r i a b l e d a m p i n g case ( i n w h i c h Cnr a n d Cn are c o n s t a n t ) , t h e o n l y c h a n g e is P i n t h e D u t c h r o l l e i g e n v a l u e s . T h e r e is s i g c i f i c a n t c b m g e i n d a m p i n g a n d a s l i g h t c h a n g e i n f r e q u c n c y . T h i s i n d i c a t e s t h a t Cn* p r i m a r i l y a f f e c t s t h e D u t c h r o l l mode. T h e c o n s t a n t

-

d a m p i n g cases are q u i t e d i f f e r e n t , i n t h a t t h e c h a n g e s i n C n b J C n p J a n d Cnr e f f e c t i v e l y c a n c e l , a s f a r as t h e J u t c h r o l l mode is c o n c e r n e d . I n t h i s case, h o w e v e r , t h e r e a r e s i g n i f i c a n t c h a n g e s i n t h e r o l l a n d s p i r a l m o d e s . T h e s e c h a n g e s are n o t d u e t o C n * ( w h i c h d o e s n o t a f f e c t t h e r o l l B a n d s p i r a l m o d e s ) b u t r a t h e r t o t h e c o r r e s p o n d i n g c h a n g e s i n Cnr a n d Cn T h e r e f o l = , c o m b i n i n g C w i t h Cn a n d Cnr l e a d s

P ' "ti P

t o e r r o n e o u s e i g e n v a l u e c a l c u l a t : . o n s i n modes n o t d i r e c t l y a f f e c t e d by C n i a t a l l .

T a b l e 3 . 3 - 3 i l l u s t r a t e s t h e c o n s t a n t d a r - ? i n g r c s u l t s w i t h a n d w i t h o u t s t e a d y r o l l i n g m o t i o n . N o n - z e r o C n e a f f e c t s t h e f r e q u e n c i e s o f t h e o s c i l l a t i o n s o n l y s l i g h t l y i n t h e p r e s e n c e o f r o l l i n g m o t i o n , b u t t h e r e a r e s i g n i f i c a n t c h a n g e s i n d a m p i n g o f t h e v a r i o u s m o d e s . D u t c h r o l l d a m p i n g r a t i o c h a n g e s f o r n o n - z e r o Cn* when t h e v e h i c l e is r o l l i n g , a n d B TABLE 3 . 3 - 3 EFFECTS OF Cnh I W THE PRESENCE C)F STEADY 3OLLING (Cnp a n d C A d j u s t e d t o i l a i n t a i n C o n s t a n t Damping) n r Short Period D u t c h R c l l 0 1 S p l r z ? Phugo i d - 1

Crib = 0 (rad

p , ~ = 0 d e g l s e c -0.407.jl.099 -0.0738+~2.231 4 . 4 4 2 -0.0545 -0.Oi65rj0.137 -1 Cni = 0.55 ( r a d ) Pw0 = - 3 5 . 6 deg:sca -0.481tj1.290 -0.0411+j2.345 -0.267::G.124 -0.0630?j0.183 t h i s is n o t o b s e r v e d f o r t h e n o n - r o l l i n g , c o n s t a n t damping The c o n t l u s i o n is t h a t a n i m p r o p e r C n i i d e n t i f i c a t i o n case.

may n o t -ear i n t h e D u t c h r o l l mode f o r n o n - r o l l i n g f l i g h t , b u t it Ean c a c s e a s i g n i f i c a n t c h a n g e i n t h e D u t c h r o l l mode

-

d u r i n g a r o l l i n g m a n e u v e r .

T h e f o l l o v . : i n g p o i n t s s u m m a r i z e t h e f i n d i n g s o f t h i s r e p o r t c o n c e r n i n g t h e e f f e c t s o f l a t e r a l s t a b i l i t y d e r i v a t i v e v a r i a t i o n s : T h e l a r g e amount o f r o l l i n g i n t h e s u b j e c t a i r c r a f t means t h a t C1 ! J , d y n a n d C re p o o r i n d i c a t o r s o f "r, dyn t h e r o l l n o d e ' s e i g e n v a l u e and t h e D u t c h r o l l m o d e ' s r e a l p a r t .

and C v a r i a t i o n s p r i m a r i l y a f f e c t C n ~ 16 t h e D u t c h r o l l f r e q u e n c y , a s i n d i c a t , e d

. V a r i a t i o n s d u e t o s t e a d y

by 'n6 , d y n p i t c h i n g motion a r e i n d e p e n d e n t o f t h e e f f e c t s c a u s e d by c h a n g e s i n C and Clg "6 Due t o t h e l a r g e amount of r o l l i ~ g motion i n t h e Dutch r o l l mode, C I D h a s a l a r g e r t h a n e x p e c t e d e f f e c t on ~ u k h r o l l damp- i n g < w h i l e t h e e f f e c t of Cnr is s m a l l e r t h a n might b e e x p e c t e d .

C n g by i t s e l f p r i m a r i l y a f f e c t s t h e Dutch r o l l mode, b u t s u b t r a c t i n g its e f f e c t from CnD and Cnr ( h o l d i n g t h e damping c o n s t a n t ) l e a d s t o l a r g e r o l l mode and s p i r a l mode c h a n g e s and small Dutch r o l l mode v a r i a t i o n s .

0 I n t h e p r e s e n c e of s t e a d y r o l l i n g , t h e r e is a s i g n i f i c a n t C e f f e c t on t h e Dutch r o l l

ni

mode even f o r t h e c o n s t a n t damping case.

Coupled f l i g h t c o n d i t i o n s s h o u l d b e i n v e s t i g ~ t e d when t h e i d e n t i f i c a t i o n o f Cng is d e s i r e d .

3.4 VARIATIONS DUE TO MASS AND INERTIA EFFECTS The v a r i a t i o n s i n t h e a i r c r a f t modes due t o mass and r o t a t i o n a l i n e r t i a changes a r e examined i n t h i s c h a p t e r . The r o t a t i o n a l i n e r t i a s c o n s i d e r e d s p a n t h e r a n g e of a i r c r a f t types (from wing-heavy t o f u s e l a g e - h e a v y ) , and t h e mass v a r i a t i o n s r a n g e from l i g h t t o heavy wing l o a d i n g s .

T a b l e 3.4-1 d e t a i l s t h e a i r c r a f t e i g e n v a l u e t r e n d s a s a i r c r a f t mass v a r i e s . The r o t a t i o n a l i n e r t i a m a t r i x is h e l d c o n s t a n t ( a s i f a p o i n t mass was added o r s u b t r a c t e d a t t h e v e h i c l e ' s c e n t e r of g r a v i t y ) , s o t h e r a t i o s between t h e mass and i n e r t i a a l s o v a r y . The mass change r e p r e s e n t s a change i n r e l a t i v e d e n s i t y , p , which is d e f i n e d as TABLE 3.4-1 EFFECTS OF AIRCRAFT ?,!ASS ON EIGEKVALUE LOCATION ( V o = 94 m!s, aO = 15 d e g ) Short P e r i o d a - - -0.572tj1.07

-

uhl?re c is r e f e r e n c e l e n g t h . T h e r e l a t i v e d e n s i t y r e l a t e s t h e a i r c r a f t mass t o t h e a i r d e n s i t y , a n d t h e r e f o r e i n d i c a t e s t h e r e l a t i v e m a g n i t u d e of a e r o d y n a m i c a n d i n e r t i a l e f f e c t s .

V a r i a t i o n s of t h e a i r c r a f t mass h a v e a l a r g e e f f e c t 011 t h e d a m p i n g o f t h e r o t a t i o n a l o s c i l l a t i o n s -- D u t c h r o l l mode a n d s h o r t p e r i o d mode -- b u t o n l y a n e g l i g i b l e e f f e c t o n t h e i r f r e q u e n c i e s . T h e s p i r a l a n d r o l l m o d e s a l s o are e s s e n t i a l l y ~ l ~ c h a n g e d . '!ass v a r i a t i o n s c h a n g e t h e p h u g o i d m o d e ' s n a t u r a l f r e q u e n c y a n d d a m p i n g r a t i o , b e c a u s e t h i s mode i n v o l v e s t h e i n t e r c h a n g e o f k i n e t i c a n d p o t e n t i a l e n e r g y a n d is h i g h l y m a s s d e p e n d e n t .

T h e r o t x t i ~ n a l i n e r t i ~ o f t h e a i r c r a f t d e s c r i b e s t h e d i s t r i b u t i o n o f t h e a i r c r a f t mass a b o u t t h e c e n t e r o f g r a v i t y .

! h s t o f t h e i n e r t i a i s d u e t o t h e f u s e l a g e a n d t h e w i n g , a n d t h e r e l a t i o n b e t w e e n t h e m l e a d s t o t h e d e s i g n a t i o n o f a s p e c i f i c c o n f i g u r a t i o n as " w i n g - h e a v y ' o r " f u s e l a g e - h e a v y . " T h e y h w i n e r t i a is a p p r o x i m a t e l y 10 t o 1 5 p e r c e n t l n r g e r t h a n t 1 1 ~ p i t c h i - n e r t i a , a n d t h e r o l l i n e r t i a c a n b e from 4 t o 1 2 t lmes smal l p r t h a n t h e yaw i n e r t i a . H i g h p e r - f o r m a n c e f i g h t e r s e m p h a s i z e h i g h r o l l i n g p e r f o r m a n c e , a n d t e n d t o b e f u s e l a g e - h e a t - v . T r a n s p o r t a i r c r a f t are b u i l t f o r c r ~ l i s j n g t , f f ; v i t . n c y a n d tr7nd t o he w i n g - h e a v y .

T a b l e 3.4-2 d e t a i l s t h e r e s u l t s o f two r o t a t i o n a l i n e r t i a i n v e s t i g a t i o n s , o n e w i t h c o n s t a n t r o l l i n g i n e r t i a (I,) a n d v a r y i n g f u s e l a g e i n e r t i a ( I a n d I Z ) , a n d o n e w i t h Y c o n s t a n t f u s e l a g e i n e r t i a a n d v a r y i n g r o l l i n g i n e r t i a . I n n e i t h e r o f t h e s e casc- is t h e p h u g o i d mode s i g n i f i c a n t l y a f f e c t e d , u n d e r l i n i n g t h e c o n c l u s i o n t h a t t h e p h u g o i d is a t r a n s l a t i o n a l mode r a t h e r t h a n a r o t a t i o n a l mode. F o r t h e u n c o u p l e d r e f e r e n c e f l i g h t c o n d i t i o n , t h e s h o r t p e r i o d mode is a f f e c t e d o n l y by a c h a n g e i n p i t c h i n e r t i a ; a n i n c r e a s e i n p i t c h i n e r t i a c a u s e s a m a j o r d e c r e a s e i n s h o r t p e r i o d f r e - q u e n c y a n d a small d e c r e a s e i n damping r a t i o .

TABLE 3.4-2 EFFECT OF RlYTATIONAL INERTIA O N EIGENVALUES

F u s e l a g e I n e r t i a Varied - I, Held Copstant (V0=81 n / s , uo = 2 5 d e g )

Short P e r i o d Dutch Roli R o l l I z / I x I v / I x - - .

4.4 4.0 -0.534tj2.01 0.102tj1.59 -0.483 -0.071 -0.C1.8:,10.156

Roll I n e r t i a Varied - I,, and 1- Held Consrant ! V , = 94 m / s , a , = 15 dey,)

I

I I l / I y I I . . / I Y I Short P e r i o d 1 Dutch R o l l I Roll 1 S p i r a l i Phugoicl 1 The l a t e r a l modes are a f f e c t e d by a n i n c r e a s e in r o l l i n g i n e r t i a , i n t h a t b o t h D u t c h r o l l f r e q u e n c y a n d r o l l T h e r e is a s i g n i - mode r e s p o n s e a r e s l o w e d s i g n i f i c a n t l y .

f i c a n t d e c r e a s e i n Dutch r o l l damping a n d v e r y l i t t l e c h a n g e L a r g e r v a l u e s of yaw i n e r t i a l e a d t o i n t h e s p i r a l mode.

somewhat d i f f e r e n t e f f e c t s . B o t h r o l l a n d s p i r a l modes a r e s i g n i f i c a n t l y s l o w e r , a n d t h e r e is s o r e d e c r e a s e i n D u t c h r o l l damping r a t i o a n d f r e q u e n c y .

The e f f e c t s of a l a r g e r o l l i n g i n e r t i a on t h e r o l l i n g / s l i p p i n g s t a b i l i t y a r e shown i n F i g . 3.4-1. The s t a b i l i t y b o u n d a r i e s a r e g e n e r a 1 , l y s i m i l a r t o t h o s e shown i n F i g . 2.4-10, a l t h o u g h t h e y d i f f e r i n d e t a i l . The Dutch r o l l mode is s t a b l e f o r t h e h i g h e r r o l l i n g i n e r t i a . The p h u g o i d i n s t a b i l i t y combines w i t h an u n s t a b l e s p i r a l mode a t h i g h s i d e s l i p a n g l e s a n d r o l l r a t e s t o form a f a s t , !

h i g h l y u n s t a b l e o s c i 1 l a t i o n .

SIDESLIP. $ Ideel / FAST SPIRAL UNSTABLE UNSTABLE PHUGOlD UNSTABLE OSCILLATION (SPIRAL/PHUGOI~) I ALL MODES (EXCEW WUGolD) STABLE R O L L RATE, Pw0 [dre/ucJ F i g u r e 3.4-1 E f f e c t s o f L a r g e R o l l i n g I n e r t i a on A i r c r a f t S t a b i l i t y ( I, = 14,370 kg-m2) C o n c l u s i o n s c q n c e r n i n g t h e e f f e c t s of mass and i n e r t i a 1 - a r i a t i o n s a r e s w u n a r i z e d a s f o l l o w s : Mass i n c r e a s e s r,?duce Dutch r o l l and s h o r t p e r i o d d m p i n g , b u t mass v a r i a - t i o n s do n c t have l a r g e e f f e c t on t h e s h o r t p e r i o d o r Dutch r o l l mode f r e - q u e n c i e s . They do n o t c a u s e s i g n i f i- c a n t c h a n g e s i n t h e 7011 o r s p i r a l modes.

a\ The p h u g o i d mode is a t r a n s l a t i o n a l mode and is g r e a t l y a f f e c t e d by mass v a r i a - t i o n s . The p h u g o i d e i g e n v a l u e d o e s n o t d e p e n d s t r o n g l y on r o t a t i o n a l i n e r t i a .

I n c r e a s e s i n p i t c h i n e r t i a r e d u c e s h o r t pex i o d f r e q u e n c y a n d d a m p i n g .

I n c r e a s e s i n r o l l i n e r t i a i n c r e a s e t h e r o l l mode time c o n s t a n t , d e c r e a s e t h e D u t c h r o l l f r e q u e 2 c y s i g n i f i c a n t l y , arid m o d i f y t h e e f f e c t s o f mode c o u p l i n g d u e t o a s y m m e t r i c f l i g h t .

I n c r e a s e s i n y a w i n e r t i a p r i m a r i l y slow t h e r o l l a n d s p i r a l modes, a n d t h e r e is some e f f e c t o n D u t c h r o l l damping a n d f r e q u e n c y .

3.5 CLASSIFICATION OF DEPARTURES D e p a r t u r e f r o m c o n t r o l l e d f l i g h t c a n o c c u r i n t w o ways. U n f o r c e d d e p a r t u r e s a r e d u e t o i n s t a b i l i t i e s i n t h e b a s i c a i r c r a f t . Even i f t h e p i l o t d o e s n o t move t h e c o n t r o l s , small p e r t u r b a t i o n s i n t h e a i r c r a f t s t a t e s b u i l d u p u n t i l t h e a i r c r a f t c a n no l o n g e r b e c o n t r o l l e d . I n a f o r c e d d e p a r t u r e , t h e b a s i c a i r c r a f t may o r may n o t b e u n s t a b l e , b u t t h e a d d i - t i o n of a p i l o t l o o p c l o s u r e c r e a t e s an u n s t a b l e v e h i c l e - p i l o t s y s t e m . The two f o l l o w i ~ ~ g s e c t i o n s d i s c u s s t h e s e d e - p a r t u r s c l a s s e s .

3 . 5 . 1 U n f o r c e d D e p a r t u r e !Iudes U n f o r c e d d e p a r t u r e s o c c u r when t h e p i l o t c a n n o t o r dor.:s n o t s t n b j l i z e an u n s t a b l e v e h i c l e . T h e v e h i c l e e i g e n - v a l u e s d i r e c t l y i n d i c a t e t h e o p e n - l o o p s y s t e m s t a b i l i t y i n t h i s c a s e , s o t h n t :.]any o f t h e s t a b i l i t y b o u n d a r i e s t h a t h a v e been shown i n t h i s r e p o r t c a n h i c l a s s e d a s u n f o r c e d d e p z r t u r e -- -- b o u n d a r i e s .

The s p e e d of t h e f a s t modes (Dutch r o l l mode, s h o r t p e r i o d mode and r o l l mode) is s u c h t h a t t h e s t a b i l i t y o f

i

t h e s e modes is c r i t i c a l , and t h e least s t a b l e o f t h e s e u s u a l l y is t h e Dutch r o l l mode. T h i s mode can become u n s t a b l e i n two ways, e i t h e r r e s u l t i n g from n e g a t i v e " s p r i n g terms" o r r e s u l t - i n g from n e g a t i v e damping, b o t h o f which can h e i n f l u e n c e d by aerodynamics and c o u p l i n g e f f e c t s .

The a p p r o x i m a t e e q u a t i o n s f o r t h e Dutch r o l l mode (Eqs.

( 2 . 4 - 1 ) and ( 2 . 4 - 2 ) ) i n d i c a t e a p u r e s t a t i c i n s t a b i l i t y f o r l a r g e , n e g a t i v e C b u t t h e e x a c t res!ilt is somewhat more complex. A s n t 3 ' i n t h e case of l o n g i t u d i n a l mode c o u p l i n g due t o p o s i t i v e Cma ( S e c t i o n 3.2), d i r e c t i o n a l i n s t a b i l i t y c a n c a u s e t h e Dutch r o l l mode t o c o u p l e w i t h t h e c l a s s i c a l r o l l and s p i r a l modes, and it can l e a d t o a new o s c i l l a t o r y mode, a n a l o g o u s t o t h e s o - c a l l e d " r o l l - s p i r a l " o r " l a t e r a l - p h u g o i d " mode. F i p v r e 3.5-1 i l l u s - trates a c a s e i n which n e g a t i v e Cng c a u s c s an o s c i l l a t o r y mode t h a t h a s low n a t u r a l f r e q u e n c y and Is h i g h l y u n s t a b l e .

By comparison t o t h e c o n v e n t i o n a l Dutch r o l l mode ( F i g .

2 . 4 - 3 ) , t h e r e is a s i g n i f i c a n t change i n mode s h a p e . T h e r e SPIRAL ROLL- SPIRAL R O L L F i g u r e 3.5-1 An Example o f L a t e r a l - D i r e c t i o n a l E i g e n v a l u r s f o r N e g a t i v e D i r e c t i o n a l S t a b i l i t y is a 180-deg phase change i n t h e yaw-rate component, a s well as a s u b s t a n t i a l r o l l angle change. In a d d i t i o n , t h e s p i r a l mode has gained s i g n i f i c a n t Av, A r , and Ap components.

The departure caused by t h i s type of i n s t a b i l i t y is shown i n Fig. 3.5-A. Although the l i n e a r model indicates

t h a t t h i s motior. 2s an o s c i i l z t i n n , i t is so unstable t h a t

only p a r t of a period appears on the . L I ~ history p l o t . The f i r s t few seconds of the motion exhibit a rapid roll-yaw angular motion. The p i l o t would sense a rapid r o t a t i o n a l divergence about t h i s a x i s and m i g h t r e f e r t o i t as a r o l l i n g "nose s l i c e " or yaw departure.

TIME. t lsrc) F i g u r e 3.5-2 An Unforced Depa:ture Due t o Negative C

" I 3

Negative yaw damping leads t o a more conventional d e s t a b i l i z i n g of t h e Putch r o l l mbde, which r e t a i n s its c h a r a c t e r i s t i c mode shape. The time hisbory of a departure d ~ e t o dynamic Dutch r o l l i n s t a b i l i t y is shorn i n Fig. 3.5-3, and t h e difference i n shape from t h e departure due t o s t a t i c i n s t a b i l i t y is apparent.

Figure 3.5-3 An Unforced Departure Due t o Negative Dutch Roll Dary~ping The l a r g e amount of r o l l i n g motion i n t h e D u t c h r o i l mode i n d i c a t e s t h a t t h i s may be what p i l o t b r e f e r t o as "wing rock." This is u n c ~ r t a i n , however, s i n c e "wing rock" a l s o could be a r o l l - s p i r a l o s c i l l a t i o n or a l i m i t cycle caused by an aerodynamic non-linearity. In any c a s e , a p i ? z t sensing such an o s c i l l a t i o n probably would unload t ' - e a i r c r a f t by reducing t h e angle of a t t a c k , removing t h e a i r c r a f t from t l - 2 region of i n s t a b i l i t y .

A t h i r d t y p e G f u n f o r c e d d e p a r t u r e can o c c u r a t h l g h s i d e s l i p a n g l e s . The modes o f m o t i o n al, shown i n F i g . 3 . 5 - 4 , a,.' t h e u n s t a b l e r o l l r,ode is s e e n t o e x h i b i t a mixed r c l l i n g - yawing d e p a r t u r e c h a r a c t e r i s t i c . T h i s d i v e r g e n c e is r a p i d and, as shown i n t h e s t a b i l i t y b o u n d a r l 3 o f F i g . 2 . 4 - 2 , c a n a p p e a r w i t h c n l y s m a l l a e r o d y n a m i c a n g l e c h a n g e s from a much more b e n i g n f l i g h t c o n a i t ~ O A I .

F i g u r e 3.5-4 E i g e n v a l u e s and F i g e n v e c t o r s f o r a F l i g h t C o n d i t i o n w i t h Large S i d e s l i p A n g l e T h i s u n s t a b l e r o l l mode is e s s e n t i s l i y a p u r e r o l l a b o u t t h e s t a b i l i t y x - a x i s , b u t , b e c a u s e of t h e l a r g e n o m i n a l a e r o d y n a m i c a n g l e s , i t a p p e a r s a s a r o l l - y ; l w m o t i o n i n b ~ d y a x e s .

I t is p o s s i b l e t h a t a p i l o t f i y i n g a n a i r c r a f t a t these l a r g e a n g l e s h m l d i n t e r p r e t a s i a b i l i t y - a x i s r o l l i n g d e p a ~ t u r e a s a "nose s l i c e . " 3.5.2 F o r c e d D e p a r t u r e Mcdes C o n t r o l i n p u t s f ~ m a ; i I ~ t o r c o n t r o l s y s t e m c a n f o r c e a n a i r c r a f t t o d e p a r t from c u n t r o l l ~ d f l i g h t i n two ways. I n t h e f i r s t way, . A v e r s e rt.spon:-e t o p i l o t i n p u t s moves t h e n o m i n a l f l i g h t c o n d i t i o r . i n t o a n u n s t a h l t . r e g i o n w h e r e a n u n f o r c e d d e p a r t u r e c a n o c c u r . The s e c o n d : - 2 s s i b l c c a u s e o f a f g r c e d d e p a r t u r c is an imprc,per l : m p c l o s u r e t h a t c r e a t e s a n u ! ! s t a b l e c l o s e d - l o o p sy:stem. D e p a r t u r p p r e v e n - t i o ~ p r o c e d u r e s a r e q u i t e d i f f e r e c t for the t w o c a s r s ; i n t h e f o r m e r case, p o s i t i v e c o n t r o l a c t i o n is n e c e s s a r y f o r r e c o v e r y , whi'e a n e u t r a l i z a t i o n of c o n t r o l i n p u t s might allow a r e c o v e r y from t h e l a t t e r d e p a r t u r e .

An example of an improper l o o p c l o s u r e w a s p r e - s e n t e d i n S e c t i o n 2 . 6 . I n t h a t c a s e , t h e t a r g e t - t r a c k i n g p i t c : a t t i t u d e - t o - e l e v a t o r l o o p c l o s u r e c a u s e d l a t e r a l mode s t a b i l i t y problems a t c o u p l e d f l i g h t c o n d i t i o n s . T h i s un- d e r l i n e s t h e n e c e s s i t y o f i n c l u d i n g c o n t r o l c r o s s - c o u p l i n g s i n s i t u a t i o n s r?tere t h e s y s t e m i t s e l f is c o u p l e d .

The t a r g e t - t r a c k i n g example can b e c o n s i d e r e d as a s i t u a t i o n i n which t h e p i l o t l e a r n s t o c o n t r o l t h e air- c r a f t a t o n e f l i g h t c o n d i t i o n b u t d o e s n o t change h i s con- t r o l s t r a t e g y a s t h e f l i g h t c o n d i t i o n changes. T h i s is emphasized by t h e o b s e r v a t i o n t h a t c o u p l e d f l i g h t c o n d i t i o n s o f t e n e x h i b i t d r a s t i c c h a n g e s i n t h e s h a p e s o f t h e normal modes, s o t h a t a p i l o t might a p p l y t h e wrong c o n t r o l a c t i o n .

The i n v e s t i g a t i o n of S e c t i o n 2 . 5 d e m o n s t r a t e s t h a t c o n t r o l e f f e c t i v e n e s s problems may l e a v e t h e p i l o t no alter- n a t i v e b u t t o a p p l y a poor c o n t r o l c o m b i n a t i o n . F o r e x a m p l e , a: 25-deg a n g l e of a t t a c k , t h e r o l l moment due t o a i l e r o n is e s s e n t i a l l y zero. The r u d d e r would c a v e t o be u s e d f o r r o l l c o n t r o l b u t t h i s b r i n g s an u n a v o i d a b l e s i d e s l i p r e s p o n s e w i t h i t . T h i s s i d e s l i p c o u l d d r i v e t h e v e h i c l e i n t o t h e r o l l d i v e r g e n c e r e g i o n c i t e d i n t h e l a s t s e c t i o n .

A s an e x a ~ p l e o f u n e x p e c t e d c o n t r o l r e s p o n s e , F i g .

3.5-5 shows a d e p a r t u r e c a u s e d by an a i l e r o n i n p u t . Nor- m a l l y , t h e r e s u l t would b e a s i g n i f i c a n t n e g a t i v e r o l l r a t e , b u t t h e s i d e s l i p and yaw r a t e b u i l d up s o r a p i d l y t h ~ t t h e i n s t a b i l i t y o f t h e b a s i c a i r c r h ' t r e s u l t s i n a r a p i d r o l l i n g d e p a r t u r e w i t h p o s i t i v e r o l l r a t e .

2 0 0

0 1 2 3 4 5 TIME, t (uc)

M

"Oao I 2 3 4 3 TIME, t Iwcl F i g u r e 3.5-5 - A i l e r o n I n p u t f o r Negative C n B S i t u a t i o n s i n which improper p i l o t i n p u t s a r e l i k e l y a r e d i s c u s s e d i n S e c t i o n 2.5. These s i t u a t i o n s a r e c h a r a c t e r - i z e d by reduced s t a b i l i t y of t h e open-loop system (due t o h i g h a n g l e of a t t a c k o r p i t c h rate) and h i g h l y coupled modes caused by an asymmetric f l i g h t c o n d i t i o n ( s u c h a s non-zero PO o r m o ) .

Nonminirnum-phase z e r o s o f t e n appear and can c a u s e g r e a t d i f - f j c u l t y i f " t i g h t " c o n t r o l is a t t e m p t e d .

The c o n c l u s i o n s r e g a r d i n g d e p a r t u r e modes a r e surnmar- i z e d a s follows: An u n f o r c e d d e p a r t u r e ( o n e d u e t o an u n s t a b l e o p e n - l o o p s y s t e m ) is most l i k e l y t o a p p e a r i n t h e l a t e r a l o s c i l l a - t o r y mode. S t a t i c i n s t a b i l i t y r e s u l t s i n a r a p i d r o l l i n g - y a w i n g d e p a r t u r e , w h i l e dynamic i n s t a b i l i t y c a u s e s an u n s t a b l e "wing r o c k " m o t i o n .

An u n s t a b l e w i n d - a x i s r o l l d i v e r g e n c e c a n a p p e a r a t e x t r e m e a e r o d y n a m i c a n g l e s .

F o r c e d d e p a r t u r e s c a n o c c u r when de- g r a d e d c o n t r o l r e s p o n s e c a u s e s t h e p i l o t t o f l y t h e a i r c r a f t i n t o a f l i g h t con- d i t i o n where u n f o r c e d d e p a r t u r e s are l i k e l y .

Mode c o u p l i n g o r u n e x p e c t e d nonminimum- p h a s e zeros c a n c h a n g e t h e c o n t r o l re- s p o n s e s o t h a t a ''normal" c o n t r o l l o o p c l o s u r e l e a d s t o an u n s t a b l e c l o s e d - l o o p s y s t e m .

3.6 CHAPTER SUMMARY T h i s c h a p t e r h a s p r e s e n t e d e f f e c t s o f c o n f i g u r a t i o n a l v a r i a t i o n s on a i r c r a f t d y n a m i c s . R e l a t i o n s h i p s between mode a p p r o x i m a t i o n s and e x a c t r e s u l t s a r e d i s c u s s e d f o r l o n g i - t u d i n a l , l a t e r a l , and c o u p l e d m o t i o n s , and e x a m p l e s o f v a r i o u s d e p a r t u r e t y p e s a r e p r e s e n t e d . I t is shown t h a t the e f f e c t s o f aerodynamic p a r a m e t e r v a r i a t i o n s are m o d i f i e d by t h e c o u p l i n g which r e s u l t s i n a s y m m e t r i c f l i g h t , p a r - t i c u l a r l y i n r e g a r d t o t h e t r a n s f e r of damping ( d u e t o r o t a r y d e r i v a t i v e s ) from l o n g i t u d i n a l t o l a t e r a l - d i r e c - t i o n a l modes ( a n d v i c e - v e r s a ) . Time h i s t o r i e s o f l i n e a r i z e d - model r e s p o n s e i l l u s t r a t e d e p a r t u r e c h a r a c t e r i s t i c s s i m i l a r t o t h o s e e x p e r i e n c e d i n f l i g h t .

PREVENTION OF DEPARTURE FROM CONTROLLED F L I G H T 4 . 1 OVERVIEW A s i n d i c a t e d by e a r l i e r d e v e l o p m e n t s i n t h i s r e p o r t a n d t h e summary o f p r i o r work i n S e c t i o n 2 . 2 , t h e r e is a m p l e r e a s o n t o c o n s i d e r 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 f o r t h e s p e c i f i c p u r p o s e o f p r e v e n t i n g d e p a r t u r e . A i r c r a f t d e s i g n is d o m i n a t e d b y p e r f o r m a n c e r e q u i r e m e n t s , a n d e v e n u n c o n s t r a i n e d c o n f i g u r a t i o n m o d i f i c a t i o n s may n o t p r o v i d e a d e q u a t e s t a b i l i t y o r c o n t r o l r e s p o n s e ( e s p e c i a l l y d u r i n g e x t r e m e m a n e u v e r i n g ) . A p p e c d i x A a n d C h a p t e r s 2 a n d 3 d e m o n s t r a t e how l i n e a r - t i m e - i n v a r i a n t m o d e l s o f a i r c r a f t d y n a m i c s c a n b e d e r i v e d f o r s t u d y i n g s t a b i l i t y a n d c o n t r o l r e s p o n s e d u r i n g d i f f i c u l t m a n e u v e r s . T h e s e m o d e l s are u s e d t o i l l u s t r a t e 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 c o n c e p t s i n t h e p r e s e n t c h a p t e r .

U n l i k e e a r l i e r s t u d i e s o f d e p a r t u r e p r e v e n t i o n , t h e p o w e r f u l t o o l s o f l i n e a r - o p t i m a l c o n t r o l t h e o r y are a p p l i e d t o t h e p r o b l e m i n t h i s c h a p t e r . S i n c e new g r o u c d is b r o k e n a n d m e t h o d s w h i c h a r e u n f a m i l i a r ( i n t h e d e p a r t u r e p r e v e n t i o n c o n t e x t ) a r e p r e s e n t e d , t h e o b j e c t i v e is t o p r o - v i d e p r e l i m i n a r y g u i d e l i n e s f o r D e p a r t u r e - P r e v e n t i o n 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 (DPSAS) d e v e l o p m e n t . T h e r e f o r e , a s i m p l e o p t i m a l c o n t r o l l e r -- t h e c o n t i n u o u s - t i m e l i n e a r - A is a p p l i e d t o d e p a r t u r e p r e v e n t i o n .

optimal r e g u l a t o r --

l i n e a r - o p t i m a l r e g u l a t o r is a f e e d b a c k c o n t r o l l a w o f t h e f o r m .

where A u ( t ) is t h e v e c t o r of c o n t r o l command p e r t u r b a t i o n s , A x ( t ) - r e p r e s e n t s t h e v e c t ~ ? of t h e a i r c r a f t ' s dynamic s t a t e s , and K is t h e g a i n m a t r i x which s c a l e s t h e s t a t e measurements f o r p r o p e r s t a b i l i z a t i o n a r 4 compensat i o n o f . t h e a i r c r a f t ' s motion. (An e q u i v a l e n t d i s c r e t e - t i m e l i n e a r - o p t i m a l regu- l a t o r , f o r which t h e state is measured and c o n t r o l is com- manded a t a f i x e d sampling i n t e r v a l , c a c be d e r i v e d f o r a d i g i t a l f l i g h t c o n t r o l s y s t e m . ) T h i s c o n t r o l law h a s s e v e r a l q u a l i t i e s which a r e d e s i r a b l e f o r t h e p r e s e n t s t u d y , i n which i is assumed t h a t s y s t e m dynamics are known e x a c t l y and t h a t a l l s t a t e s a r e measured p r e c i s e l y : 0 The c o n t r o l g a i n s g u a r a n t e e s t a b i l i t y of t h e closed-loop system.

a Complete longitudinal/lateral-directionnl c o u p l i n g is assumed and is accounted f o r i n t h e d e s i g n p r o c e s s .

a The c o n t r o l d e s i g n t e c h n i q u e i d e n t i f i e s a l l s i g n i f i c a n t c r o s s f e e d s and i n t e r c o n - n e c t s , a s w e l l as feedback g a i n s .

a T r a d e o f f s between t h e a m p l i t u d e s of s t a t e p e r t u r b a t i o n s and of c o n t r o l motions a r e s p e c i f i e d i n t h e d e s i g n p r o c e s s .

In a d d i t i o n , a gain-scheduling a l g o r i t h m which a c c o u n t s f o r v a r y i n g maneuver c o n d i t i o n s is developed.

The c o n t r o l d e s i g n t e c h n i q u e s a p p l i e d t o t h e DPSAS can be g e n e r a l i z e d t o f u l l command augmentation s y s t e m s f o r a high-performance a i r c r a f t . Reference 58 shows how p r a c t i c a l command-response c o n t r o l laws can be developed f o r a h i g h l y coupled a i r c r a f t , a t a n d e m - r o t o r h e l i c o p t e r , T h e s e c o n t r o l laws s a t i s f y c l a s s i c a l ~ t e p - r e s p o n s e c r i t e r i a , a d a p t t o f l i g h t c o n d i t i o n , honor r a t e - and d i s p l a c e m e n t - l i m i t s on c o n t r o l a c t u a t o r s , and u s e incomplete ( p o s s i b l y n o i s y ) f e e d b a c k measurements. A comrnand-response s y s t e m f o r a h i g h performance f i g h t e r is d e s c r i b e d i n R e f . 59. I t a d a p t s t o f l i g h t c o n d i t i o n t o p r o v i d e u n i f o r m h a n d l i n g q u a l i t i e s t h r o u g h o u t t h e f l i g h t regime. These c o n t r o l laws are de- v e l o p e d f o r d i r e c t i m p l e m e n t a t i o n i n a d i g i t a l computer and u s e low s a m p l i n g rates. T h i s e x t e n s i o n of t h e DPSAS to, a c o m p l e t e f l i g h t c o n t r o l s y s t e m , w h i l e p r o m i s i n g , is a s u b j e c t f o r f u t u r e s t u d y .

The remainder o f t h i s c h a p t e r is directed t o a b r i e f e x p l a n a t i o n o f l i n e a r - o p t i m a l r e g u l a t o r d e s i g n a n d e x t e n s i v e a p p l i c a t i o n of t h i s c o n t r o l d e s i g n a p p r o a c h t o DPSAS examples. S e c t i o n 4.2 p r e s e n t s t h e l i n e a r - o p t i m a l reg- u l a t o r and a d i s c u s s i o n of t h e p a r a m e t e r s u s e d i n computing c o n t r o l g a i n s . C o n t r o l d e s i g n s are d e v e l o p e d f o r a r e f e r - e n c e a i r c r a f t o v e r a wide r a n g e o f a n g l e s o f a t t a c k , p i t c h rates, s i d e s l i p a n g l e s , and r o l l r a t e s a t a s i n g l e a l t i t u d e - v e l o c i t y p o i n t -- t h e c e n t r a l f l i g h t c o n d i t i o n o f 6100 m ,

94 m / s -- i n S e c t i o n 4.3. The symmetric and asymmetric

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 are c o n s i d e r e d s e p a r a t e l y , i n o r d e r t o make t h e d i f f e r e n t i a t i o n betwgen p u l l u p and s i d e s l i p - r o l l i n g e f f e c t s more a p p a r e n t . C o n t r o l g a i n s a r e compGted a t 32 maneuvering c o n d i t i o n s t o o b t a i n t h e r e s u l t s of S e c t i o n 4.3; w i t h e i g h t s t a t e s f e d back t o f o u r c o n t ~ o l e f f e c t o r s , o v e r 1000 g a i n s are g e n e r a t e d . I n S e c t i o n - p 4 4 t h e s e g a i n s a r e c o r r e l a t e d w i t h e a c h o t h e r and w i t h maneuver c o n d i t i o n s t o i d e n t i f y c a n d i d a t e i n t e r c o n n e c t s a n e g a i n - s c h e d u l i n g r e l a t i o n s h i p s . N e g l i g i b l e and c o n s t a n t g a i n s also are i d e n t i f i e d i n t h e p r o c e s s . The c h a p t e r i a summarized i n S e c t i o n 4.5.

4.2 THE LINEAR-OPTIMAL REGULATOR Optimal control theory provides a useful and practical multi-input, multi-output control system design tool. Linear-optimal control methods are based on the differential equations that describe the vehicle in the time domain (Eq. (A.3-3)), and they produce feedback con- trollers which exhibit desirable properties.

The problem is to find a controller for the system described by Eq. (A.3-3), which exhibits a linear feed- back structure (Eq. (4.1-1)) and minimizes a scalar-valued cost functional of the state and the control: This controller is called a linear-optimal regulator, and it is derived in Refs. 60 to 62.

The designer's freedom rests in his choice of the

weighting matrices, Q and n. The design procedure consists

of the choice of Q and R, the computation of the Riccati matrix, an evaluation of closed-loop performance, and the adjustment of Q and R as discussed in Section A.4.4.

The linear-optimal regulator is a tool for design- ing a Departure-Frevention S,tabilitg Augmentation System (DPSAS). It is not a limiter, because no limits are placed on the pilot's control authority, and it is not an auto- matic spin-recovery system, because open-loop anti-spin con- trol settings are not implemented. The DPSAS is intended to augment stability and to minimize the gyrations which pre- cede loss of pilot control. The DPSAS makes full use of

t

t i i a v a i l a b l e c o n t r o l power, a n d , i n t h i s r e s p e c t , c o u l d com- p e t e w i t h t h e p i l o t ' s c o n t r o l commands; however, t h e de- s i g n e r can s p e c i f y t h e amount of c o n t r o l - s u r f a c e d i s p l a c e - ment which n o r m a l l y is a v a i l a b l e t o t h e DPSAS. B a s i n g t h e t s y s t e m on t h e l i n e a r - o p t i m a l r e g u l a t o r , t h e DPSAS c a n be d e s i g n e d t o u s e less t h a n f u l l c o n t r o l a u t h o r i t y f o r e x p e c t e d magnitudes of a i r c r a f t maneuvers, l e a v i n g a p e r c e n t a g e o f c o n t r o l a u t h o r i t y f r e e f o r manual commands.

The p r i m a r y o b j e c t i v e o f t h i s c h a p t e r is t o i d e n - t i f y t h e b a s i c e f f e c t s of v a r y i n g f l i g h t c o n d i t i o n on t h e s t r u c t u r e o f a DPSAS. To keep t h e number o f v a r y i n g param- eters t o a minimum i n t h i s d e m o n s t r a t i o n , t h e s t a t e and c o n t r o l w e i g h t i n g f a c t o r s ( E q , (A.4-21) and (A.4-22)) a r e c h o s e n a t a s i n g l e symmetric f l i g h t c o n d i t i o n and h e l d con- s t a n t t h r o u g h o u t t h e sweep o f 32 maneuvering c o n d i t i o n s . Q and R e l e m e n t s which p r o v i d e s a t i s f a c t o r y e i g e n v a l u e s , a c c e p t a b l e time r e s p o n s e , and r e a s o n a b l e c o n t r o l g a i n s a r e chosen a t t h e c e n t r a l f l i g h t c o n d i t i o n o f t h i s sweep ( a O = 1 5 d e g , V - 9 4 m / s i p s , H = 6 1 0 0 m ) . Thus, it is e x p e c t e d t h a t e i g e n v a l u e s and c o n t r o l g a i n s w i l l v a r y w i t h f l i g h t con- d i t i o n , b u t t h e rms-values of s t a t e and c o n t r o l p e r t u r b a t i o n s s h o u l d remain r e l a t i v e l y c o n s t a n t .

The Q and'R e l e m e n t s are used a s d e s i g n p a r a m e t e r s which can be i n t e r p r e t e d as t h e f o l l o w i n g maximum a l l o w a b l e r m s p e r t u r b a t i o n s : 100% of f u l l scale T h r o t t l e s e t t i n g : E l e v a t o r d e f l e c t i o n : 20 d e g A i l e r o n d e f l e c t i o n : 60 deg Rudder d e f l e c t i o n : 30 deg E u l e r a n g l e : 30 d e g Body a n g u l a r r a t e : 2 5 d e g / s e c B o d y v e l o c i t y : 9 m / s T h e s e v a l u e s i n d i c a t e t h a t t h r o t t l e s e t t i n g , ele- v a t o r , a i l e r o n , a n d r u d d e r a r e a l l o w e d t o v a r y b e t w e e n t h e i r l i m i t s a n d t h a t a n g l e s o f a t t a c k a n d s i d e s l i p m u s t be h e l d w i t h i n 5.6 d e g ( t h e 9 m / s body v e l o c i t i e s c o r r e s p o n d t o a e r o d y n a m i c a n g l e s o f t h i s n u m b e r ) . T a b l e 4.2-1 i n d i c a t e s t h a t t h e p r i m a r y e f f e c t s o f t h e l o o p c l o s u r e s a t t h e c e n - t r a l f l i g h t c o n d i t i o n are t o i n c r e a s e s h o r t p e r i o d , D u t c h r o l l , a n d p h u g o i d damping a n d t o q u i c k e n t h e r o l l a n d s p i r a l modes.

TABLE 4.2-01 EFFECTS OF DPSAS A T THE CENTRAL FLIGHT CONDITION Open-Loo] Characteristics Closed-Loop Characteristlc~ Dynamic Dmpinc Time Natural Dm?ing Time Mode Frequency, Ratio. Ccnstant, Prcquencp, - Ratio, Constant,

- sec rad/scc - sec

rad/sec .I Short Period 1.17

Dutch Roll 1 2.25

Roll I -

Spiral

I -

T h i s i l l u s t r a t e s i m p l i c i t l y t h a t t h e l i n e a r - o p t i m a l r e g u l a t o r d e s i g n c a n p r o d u c e stricter t r a c k t n g t h a n i n d i c a t e d b y t h e c h o i c e o f Q a n d R e l e m e n t s . The A a a n d A$ r e q u i r e - m e n t s c a n be met o n l y by i n c r e a s i n g damping a n d d e c r e a s i n g t i m e c o n s t a n t s . T h i s i n f e r s t h ~ t E u l e r a n g l e s a n d body a n g u l a r r a t e s a l s o are c l o s e l y r e g u l a t e d , e v e n t h o u g h t h e w e i g h t i n g o f t h e c o r r e s p o n d i n g e l e m e n t s i n Q is l i g h t . T a b l e 4.2-1 a l s o i n d i c a t e s t h a t t h e s e l e c t i o n o f e q u a l w e i g h t s o n Av a n d Aw ( a n d , t h e r e f o r e , o n Aa a n d AB) d r i v e s t h e n a t u r a l f r e q u e n c i e s a n d damping r a t i o s o f t h e s h o r t p e r i o d a n d D u t c h r o l l t o similar v a l u e s .

The DPSAS gain matrix for this flight condition is listed in Table 4.2-2. The gain matrix illustrates why damp- ing is increased in the closed-loop system; rate feedbacks are large. The classical longitudinal/lateral-directional partition can be observed in the gains; the control algo- rithm actually computes coupling gains on the order of 10-7 due to the use of single-precision arithmetic. These gains can be ignored. The elevator is seen to be the primary longitudiral controller, as throttle feedback gains are smail (the principal effect of throttle control is to damp the phugoid mode). Lateral-directional control largely parti- tions along the roll and yaw axes. Although the gains shown in Table 4.2-2 have reasonable magnitudes, they could be reduced by reducing the values of q i i (Eq. (A.4-21)). Tran- sient response would be altered, but the system would remain stable.

TABLE 4.2-2 DPSAS GAIN !IATRIX AT TKE CENTRAL FLIGHT CONDITION Pitch R o l l R o l l Angle, Rate. Anglo, Control Wtput a / . deg/loc deg/sec deg Ir8ction of Full Scale Elovator Anglo, deg luddor Anglo, 4.r The performance of the linear-optimal regulator is demonstrated by comparing open- and closed-loop response to perturbations in angle of attack, sideslip angle, and roll rate.

Figure 4.2-1 illustrates that a 1.1-deg ha pertvrbation is moderately damped without the regulator a 1 . J well-damped with OPEN- LOOP RESPONSE

-15 -

o 2.0 co w ao 1ao TlME trrc) CLOSED- LOOP RESPONSE Figure 4.2-1 Longitudinal Response at the Central Flight Condition the regulator. Figure 4 . 2 - 2 shows t h a t the l i g h t l y damped natural motion r e s u l t i n g from a 1-deg A B i n i t i a l condition c r e a t e s a s u b s t a n t i a l amount of r o l l a s well a@ yaw. The regulator damps t h e o s c i l l a t i o n and l i m i t s t h e r o l l angle excursion t o 20 percent of its open-loop value, providing s i g n i f i c a n t decoupling of l a t e r a l and d i r e c t i o n a l motions.

T h i s decoupling e f f e c t is confi.rmed by the a i r c r a f t ' s closed- loop response t o a r o l l - r a t e disturbance of 1 deg/sec (3'ig- ure 4 . 2 - 3 ) . T h i s i n i t i a l condition c r e a t e s a small side- s l i p o s c i l l a t i o n and t r i g g e r s the s p i r a l mode ( i n d i c a t e d by t h e underlying exponential response trend i n r o l l angle).

The regulator damps the o s c i l l a t i o n , reduces t h e s i d e s l i p response by 70 percent, and s t a b i l i z e s the r o l l angle.

Having obtained a representative design point f o r the DPSAS a t the c e n t r a l f l i g h t condition, t h e e f f e c t s of maneuvering on control gains, a i r c r a f t s t a b i l i t y , and t?me response a r e examined i n the next s e c t i o n .

4 . 3 DPSAS CONTROL LAWS The control gains obtained a t the c e n t r a l f l i g h t condition would s t a b i l i z e the a i r c r a f t f o r some range of nominal angles and angular r a t e s ; however, changes i n the a i r c r a f t ' s dynamics ( r e f l e c t e d by v a r i a t i o n s i n F and G) would lead t o less-than-optimal regulation. I t is neces- s a r y , t h e r e f o r e , t o redesign the control gain matrix a t each maneuvering condition i n order t o assess the f u l l p o s s i b i l i t i e s f o r preventing departure w i t h the i i n e a r - o p t ~ m a l control law.

Two separate maneuvering condition sweeps have been conducted, u s i n g the reference a i r c r a f t f l y i n g a t 6100 m OPEN-LOOP RESPONSE TlME (oat) CLOSED-LOOP RESPONSE Directional Response at the CPntral Flight Condition Figure 4.2-2 and 94 m / s i n b o t h cases. The f i r s t is a l o n g i t u d i n a l sweep, i n which a r a n g e of a n g l e s of a t t a c k anti p i t c h r a t e a r e c o n s i d e r e d . A s i n d i c a t e d i n C h a p t e r s 3 a3d 4 , t h e r e is a s i g n i f i c a n t change i n l a t e r a l - d i r e c t i o n a l dynamics d u r i n g p u l l u p maneuvers, a l t h o u g h t h e l o n g i t u d i n a l and l a t e r a l - i d i r e c t i o n a l a x e s remain u n c o u p l e d . The l a t e r a l - d i r e c t i o n a l sweep v a r i e s s i d e s l i p a n g l e and s t a b i l i t y - a x i s r o l l r a t e , i n t r o d u c i n g f u l l c o u p l i n g a b o u t a l l t h r e e a x e s . I n t h e f i r s t sweep, c o n t r o l g a i n s and c l o s e d - l o o p c h a r a c t e r i s t i c s c h a n g e , b u t t h e DPSAS s t r u c t u r e is c o n v e n t i o n a l , i . e . , g a i n s are p a r t i t i o n e d a l o n g u s u a l l i n e s . The s e c o n d sweep g e n e r a t e s u n c o n v e n t i o n a l DPSAS s t r u c t u r e s as w e l l a s g a i n v a r i a t i o n s .

( I n b o t h c a s e s , t h e c o n t r o l l a w is d e s c r i b e d by Eq. (4.1-1).

K c o n t a i n s z e r o s u b - m a t r i c e s i n t h e f i r s t sweep b u t n o t i n t h e s e c o n d . ) 4 . 3 . 1 L o n g i t u d i n a l Sweep T h i s s e c t i o n p r e s e n t s t h e e f f e c t s of a n g l e of a t t a c k and p i t c h r a t e on c l o s e d - l o o p e i g e n v a l u e s , DPSAS c o n t r o l g a i n s , and a i r c r a f t r e s p o n s e . S e c t i o n 2 . 4 showed t h a t t h e r e f e r e n c e a i r c r a f t h a s an u n s t a b l e Dutch r o l l mode a t h i g h a. and u n s t a b l e Dutch r o l l , r o l l , and s p i r a l m ~ d e s a t h i g h qo.

These c o n d i t i o n s a r e s t a b i l i z e d by t h e DPSAS. Using t h e s t a t e and, c o a t r o l w e i g h t i n g f a c t o r s d i s c u s s e d i n t h e p r e v i o u s sec- t i o n , i i n e a r - o p t i m a l r e g u l a t o r s are d e s i g n e d f o r 15 maneuver c o n d i t i o n s ( a o v a r i e s from 5 t o 25 d e g , i n 5-deg i n c ~ e m e n t s , a n d qp is 0 , 1 2 , and 24 d e g l s e c ) . T h i s sweep r e p r e s e n t s r e l a t i v e l y l o w l o a d f a c t o r s ( n Z = 0 . 4 t o 1 . 4 " g ' s M ) , it c o v e r s t h e normal a O r a n g e , and it e x c e e d s t h e normal qo r a n g e .

Cou- s e q u e n t l y , t h e s e f l i g h t c o n d i t i o n s do n o t l i t e r a l l y r e p r e - s e n t c o o r d i n a t e d p u l l u p maneuvers, a l t h o u g h t h e y i n t r o d u c e t h e same symmetric c o u p l i n g terms i n t h e F m a t r i x ( E q . (A.3-4)) t h a t o c c u r i n t h e p u l l u p .

Closed-loop s t a b i l i t y a t t h e 15 c o n d i t i o n s is sum- marized by T a b l e 4.3-1, where i t can b e s e e n t h a t a l l modes are s t a b l e and a t l e a s t m o d e r a t e l y damped. C o n t r o l power d o e s n o t change w i t h q o , b u t it d o e s change w i t h ao; con- s e q u e n t l y , t h e c l o s e d - l o o p s t a b i l i t y a t a g i v e n a. is rela- t i v e l y independent o f qo. There is a g r a d u a l d e c r e a s e i n Dutch r o l l damping a s a . increases, and r o l l r e s p o n s e becomes more s l u g g i s h . T h i s happens b e c a u s e r u d d e r and a i l e r o n a r e less e f f e c t i v e a t t h e h i g h e r a n g l e s , w h i l e t h e e l e m e n t s o f R which weight t h e c o s ~ of u s i n g t h e s e s u r f a c e s remain unchanged. A h e a v i l y damped r o l l - s p i r a l o s c i l l a t o r y mode o c c u r s a t a . of 20 and 25 deg. A coupled r o l l - s p i r a l mode c a n d e g r a d e h a n d l i n g q u a l i t i e s , so a d j u s t m e n t o f TABLE .4.3-1 CLOSED-LOOP STABILITY IN THE LONGITUDINAL SWEEP -

-

Uaneuvcr Dutch Roll Roll Spiral Phupoid Condition Short Period

- --

1 . T .

BCC sec

- -

0.14 1.18 0.14 1.23 0.14 1.32 0.17 1.44 0.17 1.20 0.17 1.30 0.36 1.07 0.37 1.17 0.34 1.30 1.08~ 0 . 8 7 ' 0.82 1.08 1.34 0.66 0.80~ 0 . 9 0 ' 0 . 7 9 ' 0.03~ 0 . 7 8 ' 0. 9 9 '

-

*2 Real Roots +Roll-Spiral u,, and c Q and R c o u l d b e n e c e s s a r y t o e l i m i n a t e t h i s c h a r a c t e r i s t i c .

The n o r m a l l y o s c i l l a t o r y phugoid mode d e g e n e r a t e s i n t o two real modes a t most maneuvering c o n d i t i o n s c o n s i d - e r e d h e r e . The over-damped phugoid mode may r e s u l t from t h e "cost" a s s o c i a t e d w i t h Au p e r t u r b a t i o n s , which c o u l d b e r e l a x e d i n f u t u r e DPSAS d e s i g n s .

T h e r e are 1 6 n o n - t r i v i a l DPSAS g a i n s g e n e r a t e d T o r t h e p u l l u p maneuver. S c h e d u l i n g of t h e s e g a i n s is d i s - c w s e d i n S e c t i o n 4 . 4 , and 1 2 of t h e g a i n s ( f o u r e a c h f o r e l e v a t o r , a i l e r o n , and r u d d e r ) are p r e s e n t e d h e r e . T a b l e 4.3-2 lists t h e s e g a i n s f o r a. of 5 , 1 5 , and 25 deg and q o of 0 , . 1 2 , and 24 d e g l s e c . The f i r s t s u b s c r i p t of k i n d i - c a t e s t h e c o n t r o l e f f e c t o r ( i n t h e o r d e r u s e d i-n T a b l e 4.2-2) and t h e s e c o n d s u b s c r i p t i n d i c a t e s t h e f e e d b a c k v a r i a b l e ( a l s o o r d e r e d i n T a b l e 4 . 2 - 2 ) .

L o n g i t u d i n a l G a i n s (kZ1 t o k,41 - The g a i n s main-

t a i n an o r d e r l y p r o g r e s s i o n w i t h b o t h u o and qo; none change s i g n and most f o l l o w a s i n g l e i n c . r e a s i n g o r d e c r e a s i n g t r e n d w i t h t h e two f l i g h t v a r i a b l e s . T a b l e 4.3-2 shows t h a t t h e p i t c h - r a t e g a i n (k23) is dominant a t a l l maneuver c o n d i t i o n s and h a s a maximum v a r i a t i o n o f less t h a n 2 5 p e r c e n t , which is r e p r e s e n t a t i v e o f t h e v a r i a t i o n s of most g a i n s a t most c o n d i t i o n s .

D i r e c t i o n a l G a i n s ( k t o k ) - S v b s t a n t i a l v a r i a -

15--4 8- t i o n s i n r u d d e r g a i n s c a n b e e x p e c t e d w i t h i n c r e a s i n g a O .

The f u s e l a g e b l o c k s t h e f l o w o v e r t h e v e r t i c a l t a i l a t h i g h and t h e r u d d e r s i d e f o r c e t r a n s f o r m s i n t o s t a b i l i t y a. ' a x i s r o l l and yaw moments d i f f e r e n t l y a t d i f f e r e n t a n g l e s o f a t t a c k . U n l i k e t h e l o n g i t u d i n a l g a i n s , t h e r e is a d r a m a t i c change i n t h e d i r e c t i o n a l g a i n s a s a. i n c r e a s e s from 15 t o TABLE 4.3-2 3PSAS GAINS FOR THE LONGITUDINAL SWEEP - - - - - - - - (THROTTLE GAINS OMITTED) 25 d e g ( T a b l e 4 . 3 - 2 ) . T h e y a w g a l a s ( k 4 5 a n d k46) h a v e T h e r o l l - a n g l a s i g n i f i c a n t c h a n g e s w i t h b o t h aO a n d qo.

g a i n ( k q 8 ) is n o t e d t o c h a n g e s i g n as a . p r o g r e s s e s f r o m 5 t o 15 d e g , w h i l e t h e i n c r e a s e d r o l l - r a t e g a i n ( k d 7 ) a t t e m p t s t o p r o v i d e s t a b i l i t y - a x i s yaw d a m p i n g .

L a t e r a l . G z i n s (k35 t o k g S l - T r e n d s i n t h e a i l e r o n

-.-- g a i n s alsa h a v e l a r g e v a r i a t i o n w i t h a due t o g e o m e t r i c t r a n s f o r m a t i o n , l o s s o f r u d d e r e f f e c t i v e n e s s , a n d a i l e r o n yaw e f f e c t s . More g a i n s c h a n g e s i g n , a n d p i t c h r a t e h a s a g r e a t e r e f f e c t o n g a i n m a g n i t u d e . T h e r e is a n a b r u p t r e d u c - t i o n i n t h e u s e of a i l e r o n f o r r o l l c o n t r o l ( k 3 7 a n d k3g) a t a n a. o f 25 deg, w h i c h i s a c c o m p a n i e d by i n c r e a s e d a i l e r o n u s e f o r y a w c o n t r o l i k a n d k36).

I t was n o t e d e a r l i e r t h a t p i t c h rate d e s t a b i l i z e s t h e D u t c h r o l l , r o l l , a n d s p i r a l m o d e s . T h e c o u p l e d n a t u r e o f t h i s phenomenoc h a s a n i n t e r e s t i n g e f f e c t o n t h e s e c o n d a r y l a t e r a l - d i r e c t i o n a l c o n t r o l p a t h s , i . e . , t h e yaw f e e d b a c k t o t h e r o l l moment c o n t r o l l e r ( a n d t h e c o n v e r s e ) , s u c h as k35, k36, k47, a n d k48 a t t h e l o w e r a n g l e s o f a t t a c k .

T h e s e yains h a v e as g r e a t o r g r e a t e r v a r i a t i o n w i t h qo as w i t h t h e c h a n g e f r o m 5- t o 1 5 - d e g a O , w h i c h is n o t t h e case f o r t h e p r i m a r y c o n t r o l p a t h s ( y a w - t o - r u d d e r a n d r o l l - t o - a i l e r o n ) .

A s i n t h e p r e v i o u s s e c t i o n , t h e p e r f o r m a ~ l c e o f t h e DPSAS i n t h e p u l l u p f l i g h t c o n d i t i o n is a s s e s s e d b y c o m p a r - i n g o p e n - a n d c l o s e d - l o o p time r e s p o n s e s . F i g u y e 4 . 3 - 1 i l l u s t r a t e s t h e a i r c r a f t ' s o p e n - a a d c l o s e d - l o o p r e s p o n s e s t o a n i n i t i a l s i d e s l i p p e r t u r b a t i o n when a . is 15 d e g a n d q0 is 12 d e g / s e c . The o s c i l l a t i o n g r o w s a t a m o d e r a t e r a t e w i t h o u t s t a b i l i t y a u g m e n t a t i o n b u t is damped i n o n e c y c l e w i t h t h e c o n t r o l l o o p s ' c l o s e d . A t h i g h e r a n g l e o f a t t a c k ( 2 5 d e g ) a n d t h e same p i t c h r a t e . t h e o p e n - l o o p o s c i l l a t i o n OPEN- LOOP RESPONSE CLOSED- LOOP RESPONSE u

- O ' : p I { - 2.:p

- 3 as " " 1 i : : p - " - ~ -

- 0 L 0 q - 2 . 0 QI Q4 - 1 . 6 - 4 . 0 o 2 . 0 4 . 0 8 . 0 a0 o 2 . 0 4 . 0 LO ao o 2 . 0 4 . 0 6 a0 o 2.0 4 . 0 6 . 0 0.0 TlME (ml Figure 4.3-1 Pitch-Hate Effect on Directional Response ( a 0 = 1 5 deg, q 0 = 1 2 deglsec) grows a t a f a s t e r r a t e , and t h e c l o s e d - l o o p o s c i l l a t i o n t a k e s two c y c l e s t o d i s a p p e a r ( i n k e e p i n g w i t h t h e r e d u c e d damping r a t i o of t h e Dutch r o l l mode).

4 . 3 . 2 L a t e r a l - D i r e c t i o n a l Sweep Nominal v a l u e s of s i d e s l i p a n g l e and s t a b i l i t y - a x i s r o l l r a t e are v a r i e d i n t h i s s e c t i o n , and t h e i r e f f e c t s on c l o s e d - l o o p e i g e n v a l u e s , DPSAS c o n t r o l g a i n s , a n d a i r c r a f t respoilse a r e p r e s e n t e d . The development of t h i s s e c t i o n f o l l o w s t h e p r e v i o u s s e c t i o n , a l t h o u g h t h e r e s u l t s p r e s e n t e d f o r asymmetric f l i g h t a r e somewhat d i f f e r e n t from t h o s e of t h e l o n g i t u d i n a l sweep. The Q and R m a t r i c e s a r e t h e same as h e f e r e , and l i n e a r - o p t i m a l r e g u l a t o r s a r e d e s i g , ~ e d a t 18 p o i n t s . S i d e s l i p a n g l e s of 0 , 5, and 1 0 d e g a r e con- s i d e r e d i n combination w i t h s t a b i l i t y - a x i s r o l l r a t e s of 0 , f13,f26, and f39 d e g l s e c . ( F o r a g i v e n s i d e s l i p a n g l e , r o l l r a t e s of o p p o s i t e s i g n have d i f f e r e n t d,ynamic e f f e c t s . ) Angle o f a t t a c k , v e l o c i t y , and a l t i t u d e are f i x e d a t 15 d e g , 94 m j s , and G l O O m , r e s p e c t i v e l y .

T a b l e 4.3-3 p r e s e n t s t h e n a t u r a l f r e q u e n c i e s , damping r a t i o s , and time c o n s t a n t s of t ? ~ e a i r c r a f t , w i t h t h e l i n e a r - o p t i m a l r e g u l a t o r l o o p s c l o s e d . The most s t r i k - i n g r e s u l t , i n comparison w i t h T a b l e 4 . 3 - 1 , is t h a t t h e l a t e r a l - d i r e c t i o n a l c l o s e d - l o o p r o o t s e v i d e n c e r e l a t i v e l y l i t t l e v a r i a t i o n w i t h maneuver condition. T h e r e are no r o l l - s p i r a l o r phugoid d e g e n e r a c i e s , and a l l p a r a m e t e r s s t a y w i t h i n 40 p e r c e n t of t h e i r mean v a l u e s . S h o r t p e r i o d , Dutch r o l l , and phugoid n a t u r a l f r e q u e n c i e s d e c r e a s e w i t h i n c r e a s i n g B O magnitude and i n c r e a s e w i t h i n c r e a s i n g p

W o

magnitude. R o l l t i m e c o n s t , a n t and damping of t h e s h o r t p e r i o d and phugoid modes a r e l a r g e l y independent o f 8 magnitude b u t d e c r e a s e w i t h pWO m a g n i t u d e . Dutch r o l l TABLE 4.3-3 CLOSED-LOOP STABILITY I N THE LATERAL-DIRECTIONAL SWEEP damping i n c r e a s e s w i t h Pw0 m a g n i t u d e a n d is l i t t l e a f f e c t e i r b y B O . The s p i r a l mode time c o n s t a n t i n c r e a s e s w i t h PwO m a g n i t u d e , a l t h o u g h its minimum v a l u e o c c u r s a t more n e g a - t i v e Pw3 as B0 i n c r e a s e s .

Examples o f t h e DPSAS g a i n v a r i a t i o n s w i t h s i d e s l i p a n g l e a n d r o l l rate a r e p l o t t e d i n F i g . 4 . 3 - 2 a n d 4 . 3 - 3 .

The m o s t a p p a r e n t t r e n d is t h a t p r i m a r y g a i n s , i . e . , t h o s e which would b e n o n - z e r o i n s y m m e t r i c f l i g h t , c h a c g e v e r y w h i l e c r o s s f e e d g a i n s b.ave s u b - l i t t l e w i t h BO a n d pw - ---- - , -. - - -- --_- _____'_. ..

. . , , * t r - h , s t a n t i a l v a r i a t i o n w i t h maneuver c o n d i t i o n . The s t a n d a r d d e v i a t i o n of e a c h g a i n , computed o v e r t h e 18 l r t e r a l - d i r e c t i o n a l sweep c o n d i t i o n s , is an i n d i c a t i o n of its v a r i a t i o n from a c o n s t a n t v a l u e . The a v e r a g e s t a n d a r d d e v i a t i o n f o r t h e p r i m a r y g a i n s is 1 6 p e r c e n t , and f o r t h e c r o s s f e e d g a i n s it is 422 p e r c e n t . A s d i s c u s s e d i n Sec- t i o n 4 . 6 , t h i s is a f i r s t i n d i c a t i o n of g a i n - s c h e d u l i n g r e q u i r e m e n t s , s u g g e s t i n g t h a t many p r i m a r y g a i n s a r e n e a r l y c o n s t a n t and t h a t most s e c o n d a r y g a i n s must be s c h e d u l e d ( u n l e s s t h e y a r e n e g l i g i b l e ) .

Gain v a r i a t i o n s a r e s e e n t o depend on w h e t h e r t h e v e h i c l e is s i d e s l i p p e d " i n t o " o r " o u t o f " t h e r o l l . (The

v e h i c l e is s i d e s l i p p e d i n t o t h e r o l l when B0 and m0 have

o p p o s i t e s i g n , e . g . , when t h e nose is l e f t and t h e left wing is moving down; it is s i d e s l i p p e d o u t o f t h e r o l l when t h e S i g n s a r e e q u a l . ) F i g u r e s 4.3-2 and 4.3-3 i l l u s t r a t e g a i n v a r i a t i o n s f o r p o s i t i v e B0 o n l y ; f o r n e g a t i v e B O , t h e v a r i a t i o n s w i t h pwo a r e changed. The g r a p h s of p r i m a r y g a i n s f o r n e g a t i v e BO a r e m i r r o r images o f t h o s e f o r p o s i - t i v e B O ( F i g . 4 . 3 - 2 ) . The g r a p h s of c r o s s f e e d g a i n s f o r n e g a t i v e BO s h i f t up o r down, i n o p p o s i t i o n t o t h e BO t r e n d shown i n F i g . 4 . 2 - 3 Primary g a i n s c a n b e monotonic o r convex f u n c t i o n s o f pW ; c r o s s f e e d g a i n s a r e monotonic i n and always p a s s t h r o u g h z e r o when b o t h BO and q, PWO a r e z e r o ( G a i n s f o r symmetril: f l i g h t a r e i n i d c a t e d by "@" i n F i g . 4.3-3). O The c r o s s f e e d g a i n s a r e shown t o b e n o n - t r i v i a l f o r even m o d e r a t e v a l u e s of BO and PW0, and t h o s e shown i n F i g .

4.3-3 c a n b e i n t e r p r e t e d as n o n l i n e a r c o n t r o l e l e m e n t s .

Note t h ~ t e a c h g a i n c o u l d be approximated by a f u n c t i o n o f t h e form where cl and c2 are appropriate constants.

Then the con- trol signals represented by these four graphs would be where the constants are derived b y repression analysis (Section 4.6). The pWOAw and pW A8 terms can be recognized as analogous to so-called "pseudo-k" or "pa" crossfeeds, which have been incorporated in the SAS of modern high-performance aircraft. (An aaditional "pa"-ty~e primary gain is indi- cated in Table 4.3-2. The roll rate-to-rudder gain could be approximated by coo; therefore, the associated r6dder com- mand would be caoAp.) Nonlinearities in the curves of Fig.

4.3-3 suggest that higher-order fits than Eq. (4.3-1) to (4.3-5) are required if design performance is to be obtained over a wide range of B O and wO.

Examples of open- and closed-loop response at two asymmetric flight conditions are shown in the next two fig- ures. Figures 4.3-4 and 4.3-5 show that roll rate intro- duces substantial longitudinal response to a directional input ~.nd that the addition of sideslip angle leads to qualitative changes in response shapes. Roll rate alone introduces regular oscillations in the aircraft's open-loop response (Fig. 4.3-4). The DPSAS damps the oscillation within 13 cycles, although excitation of the phllgoid mode leads to a slow decay in A c i (The effective time constant (-Gun) of the phugoid is 6 sec a t this flight condition).

WEN-LOW RESPONSE TIME (set) CLOSED- L W P RESPONSE Figure 4.3-4 Roll-Rate Effect on Direcciorzl Response ( a 0 = 1 5 deg, B O , = O deg, pw0=39.6 deglsec) C,

0 -

Q) M cn Q) p.l 'tl w arc

; I 1

(110 Q, a a

.d -

!

p*w*:v-x*w*:Tm,, ; . : ~ $ t w w m , ~ ,-,"- "_~,# * , . _,". - _ -._

I ' Z

When t h e a i r c r a f t h a s developed a l a r g e mean s i d e - ! E S s l i p a n g l e a s w e l l a s r o l l r a t e , t h e open-loop p e r t u r b a - % t i o n motions t e n d t o meander, a s s e v e r a l modes a r e involved i n each motion ( F i g . 4.3-5). For example, t h e i n i t i a l A a a p p e a r s t o be damping o u t , b u t a f t e r 5 s e c , it b e g i n s t o wander. P i t c h and r o l l a n g l e develop o f f s e t s which a r e c o n t i n u i n g t o i n c r e a s e a t t h e end of t h e time p e r i o d shown.

The DPSAS r e s t r i c t s t h e maximum i n i t i a l e x c u r s i o n s of A0, A $ , and A0 t o less t h a n h a l f t h e i r open-loop v a l u e s and e l i m i n a t e s t h e meandering c h a r a c t e r i s t i c .

P l o t t i n g B r a t h e r t h a n A 0 i n F i g . 4.3-5 is a reminder t h a t t h e DPSAS p r o v i d e s s t a b - l i t y about a r e f e r - e n c e f l i g h t c o n d i t i o n , i n t h i s c a s e , 10-deg s i d e s l i p a n g l e and -39.6-deg/sec r o l l r a t e . With t h e assumption t h a t t h e s e v a l u e s a r e commanded by t h e p i l o t , it can be seen t h a t t h e DPSAS does n o t l i m i t a i r c r a f t m a n e u v e r a b i l i t y -- i n f a c t , it expands t h e f l i g h t envelope by s t a b i l i z i n g t h e a i r c r a f t i n c o n d i t i o n s w h i c h could n o t b e c o n t r o l l e d by t h e unaided p i l o t . Although non-zero BO is n o t normally d e s i r e d i n maneuvering c u r r e n t high-performance a i r c r a f t , f u t u r e a i r - c r a f t , p a r t i c u l a r l y t h o s e w i t h d i r e c t s i d e f o r c e c o n t r o l , c o u l d u s e t h i s c a p a b i l i t y t o t a c t i c a l advantage.

T h i s s e c t i o n h a s p r e s e n t e d l i n e a r - o p t i m a l DPSAS d e s i g n s f o r t h e r e f e r e n c e a i r c r a f t and f o r a v a r i e t y of maneuvering c o n d i t i o n s . The next s e c t i o n of t h i s c h a p t e r d a o n s t r a t e s how c o n t r o l g a i n s can be adapted t o f l i g h t con- d i t i o n .

4.4 CONTROL-LAW ADAPTATION FOR VARYING FLIGHT CONDITIONS This secticn presents results for a procedure which adapts the control gains of a high-performance aircraft to varyihg flight conditions, including gain correlations for the reference aircraft model. The gains are scheduled by finding functional relationships between aircraft flight variables a ~ d the control gains at the corresponding flight c m d i t ions.

Previous methods for scheduling control gains have been successful and indicate that gain scheduling is a sound approach. The methodology typically is based oa single input/sinsle output concepts, e.g., maintaining constant loop gain. These previous methods, however, pro- vide inadequate insight for scheduling a multivariable system.

The method is a logical extension of previous w o ~ k to multivariable systems. It involves three steps, and it places minimum reliance on past experience and intuition.

The three steps are: The determination of means and stand- ard deviations of the control system gains.

The determination of correlation coefficients between gains and flight variables.

The determination of functional re- lationships (or curve fits) between the chosen flight variables and the gains.

This new gain scheduling procedure, discussed in Section A . 4 . 5 , ;r, bi i s s i m p l e t a u s e , t h e r e s u l t s a r e e a s y t o implement on a -P d i g i t a l c o m p u t e r , and t h e p r o c e d u r e c a n have b r o a d a p p l i - c a t i o n .

The l o n g i t u d i n a l and l a t e r a l - d i r e c t i o n a l sweep . .

c o n t r o l g a i n s d i s c u s s e d i n S e c t i o n 4 . 3 have been c o r r e l a t e d w i t h a number of f l i g h t v a r i a b l e s . I n o r d e r t o i d e n t i f y t h e i n d i v i d u a l e f f e c t s of l o n g i t u d i n a l and l a t e r a l - d i r e c t i o n a l mean m o t i o n s , t h e c o r r e l a t i o n s f o r e a c h sweep a r e done s e p a r a t e l y . I n a f l i g h t s y s t e m , t h e g a i n s s h o u l d be c o r - r e l a t e d j c i n t l y , ;nd a d d i t i o n a l f a c t o r s -- s u c h as w e i g h t , a l t i t u d e , and v e l o c i t y -- must be c o n s i d e r e d .

For e a c h sweep, a list of c a n d i d a t e i n d e p e n d e n t v a r i a b l e s is e s t a b l i s h e d , and v a r i o u s f u n c t i o n s o f t h e i r v a r i a b l e s are c o r r e l a t e d w i t h t h e 32 g a i n s a s s o c i a t e d w i t h e a c h f l i s h t c o n d i t i o n . F u n c t i o n s c o n s i d e r e d i n c l u d e d p o l y - n o m i a l s of o r d e r o n e and t w o , Gain = bo+blm 2 ( 4 . 4 - 2 ) Gain = bo + blm + b2m and l i n e a r r e g r e s s i o n s i n two v a r i a b l e s , Gain = bo+blml+b2m2 Given a f l i g h t v a r i a b l e , y , i n d e p e n u e n t v a r i a b l e s , m , o f

t h e form y , y 2 , l / y , l / y 2 , and y ( y 1 a r e c o n s i d e r e d i n t h e

p o l y n o m i a l r e g r e s s i o n s , E q u a t i o n ( 4 . 4 - 3 ) is u s e d w i t h The o b j e c t i v e o f t h e c o m p u t a t i o n s is t o m l L y l and m 2 = y 2 .

f i n d t h e f u n c t i o n a l a p p r o x i m a t i o n t o e a c h g a i n which h a s t h e g r e a t e s t c o r r e l a t i o n w i t h t h e l i n e a r - o p t i m a l g a i n a t a l l c o n d i t i o n s i n t h e p a r t i c u l a r sweep. I n m a ~ y c a s e s , n l t e r - n a t e f u n c t i o n s have similar c o r r e l a t i o n c o e f f i c i e n t s , so more t h a n one s c h e d u l e c o u l d be c o n s i d e r e d i n i m p l e m e n t a t i o n .

The correlation between gains also is of interest, as it suggests which gains can be scheduled as functions of other gains, and it helps to identify control interconnects.

o This correlation can be computed, using Eq. (A.4-37), by defining the first gain as k and the second as G.

The following results indicate that an aileron-rudder inter- connect could be considered for the stability augmentation system of the reference aircraft.

4.4.1 Longitudinal S w e 9 The procedure followed in establishing gain-sched- uling requirements is to compute means and standard devia- tions (as percentages of the means) of tne gains, to corre- late gains with flight variable functions, and to corre- late gains with other gains. Sixteen crossfeed gains are identically zero, leaving sixteen gains for scheduling.

Table 4.4-1 summarizes the findings for DPSAS gains in the loxgitudinal sweep, presenting the mean and standard deviation of each gain. The independent variables which provide the best gain schedule are listed, along with the correlation between the actual and scheduled gain values.

Also listed is the gain which exhibits the highest cross- correlation, (calculated by applying Eq. (A.4-37) to all pairs of gains) and the value of that cross-correlation.

For example, the gain A6T/A8 exhibits a mean of -0.016 and a standard deviation of 32% over the chosen set of longi- tudinal flight conditions. A gain schedule using normal load factor (nZo) and pitch rate ( q O ) produces a scheduled gain whose correlation factor wlth the actual gain is 0.89.

Finally, A 6 T / A 9 exhibits strongest cross-correlation with U , and the correlation factor is 0.98. The flight variables considered as possible scheduling variables are TABLE 4 . 4 - 1 GAIN CORRELATIONS FOR THE LONGITUDINAL SWEEP L o n g i t u d i n a l Gains Gain Mean of Gain Standard Deviation of Gain, % of Mean Best Scheduling Variables Scheduled/Actual Gain Correlation Gain of Highest Cross Correlation Gain Cross Correlat i o ~ L a t e r a l Gains Gain Mean of Gain Standard Deviation of Gain, % 01 Mean B e s t . Scheduling Variables Scheduled/Actual Gain Correlation Gain of Nighest Cross Correlation Gain Cross Correiat ion * A l l independent v a r i a b l e s e v a l u a t e d a t nominal f l i g h t c o n d i t i o n , ''0" s u b s c r i p t o m i t t e d .

6 I I 1 - .

a n g l e o f a t t a c k ( a n ) . p i t c h r a t e ( a 0 ) . normal l o a d f a c t o r ( n z O ) , s i n a 0 , and c o s a T h i s set of i n d e p e n d e n t v a r i a b l e s i n c l u d e s t h o s e a c t s a l l y v a r i e d ( a o and q o ) a n d some l i k e l y f u n c t i o n s cf them, and it s e r v e s t o i l l u s t r a t e t h e DPSAS g a i n s c h e d u l i n g p r o c e d u r e .

The o n l y g a i n which is r e a s o n a b l y c o n s t a n t i n T a b l e 4 . 4 - 1 is A6h/Aq, t h e p i t c h r a t e - t o - e l e v a t o r g a i n , a s its s t a n d a r d d e v i a t i o n is j u s t 6 p e r c e n t o f t h e mean v a l u e .

A s is t h e c a s e f o r a l l g a i n s which a r e most h i g h l y c o r r e - l a t e d w i t h c o s a o , t h e b e s t f u n c t i o n a l f i t is g i v e n by 2 4 Gain = b + b l / c o s a 0 + b 2 / c o s uo ( 4 . 4 - 4 ) The r o l l r a t e - t o - r u d d e r g a i n , A6r/Ap, is b e s t a p p r o x i m a t e d by a p o l y n o m i a l i n l o a d f a c t o r , Gain = bo + b n2 + b n

=o =o

and t h e r e m a i n i n g g a i n s are " b e s t " f i t by l i n e a r f u n c t i o n s of nZO and qo. T h i s is n o t t o s a y t h a t a l t e r n a t e f u n c t i o n s , e . g . , nzo and p O , would n o t b e b e t t e r . T h e s e c o r r e l a t i o n s a r e " b e s t " f o r t h e i n d e p e n d e n t v a r i a b l e s a n d f u n c t i o n s con- s i d e r e d , and t h e a v e r a g e c o r r e l a t i o n w i t h a c t u a l g a i n v a l u e s is o v e r 0.9.

A l l g a i n s have a c o r r e l a t i o n g r e a t e r t h a n 0 . 8 9 w i t h a t l e a s t o n e o t h e r g a i n e x c e p t Adh/AO, which h a s a c o r r e - l a t i o n above 0 . 8 w i t h s e v e r a l g a i n s . The s u i t a b i l i t y o f a SAS a i l e r o n - r u d d e r interconnect is e v a l u a t e d b y n o t i n g t h t c o r r e l a t i o n between t h e a i l e r o n and r u d d e r g a i n s f o r e a c h f e e d b a c k v a r i a b l e (Av, Ar, Ap, and A$). The c o r r e l a t i o n s ( n o t n e c e s s a r i l y t h e maximums, and n o t n e c e s s a r i l y i n t h e t a b l e ) a r e 3 . 9 3 ( A v ) , 0 . 9 5 ( A r ) , 0 . 8 3 ( A p ) , and O.?O(A$), i n d i - c a t i n g a good p o s s i b i l i t y f o r combining Av and A r f e e d b a c k s w i t h l i t t l e p e r f o r m a n c e d e g r a d a t i o n , a n d a m o d e r a t e p o s s i - b i l i t y f o r c o m b i n i n g Ap and A $ f e e d b a c k s a s w e l l .

4 . 4 . 2 L a t e r a l - D i r e c t i o n a l Sweep C o r r e l a t i o n r e s u l t s f o r t h e l a t e r a l - d i r e c t i o n a l sweep a r e shown i n T a b l e 4 . 4 - 2 w h i c h i n d i c a t e s t h a t n o gain means a r e i d e n t i c a l l y z e r o , a l t h o u g h s e v e r a l a p p e a r n e g l i g - i b l e . The f l i g h t v a r i a b l e s c o n s i d e r e d f o r s c h e d u l i n g are s i d e s l i p a n g l e ( B O ) a l ~ d s t a b i l i t y - a x i s r o l l rate ( p ~ ) , w h i c h are c h o s e n t o i l l u s t r a t e t h e DPSAS g a i n s c h e d u l i n g m e t h o d .

I n a n a c t u a l a p p l i c a t i o n , a d d i t i o n a l i n d e p e n d e n t v a r i a b l e s c o u l d b e i n c l u d e d i n t h e s e a r c h .

U n l i k e t h e l o n g i t u d i n a l s w e e p , i t a p p e a r s t h a t 13 gain:; c c u l d b e c r n s i d e r e d c o n s t a n t , w i t h s t a n d a r d d e v i a t i o n s o f less t h a n 8 p e r c e n t o f t h e mean v a l u e . F i v e g a i n s are i n - a d e q u a t e l y s c h e d u l e d by t h e c h o s e n i n d e p e n d e n t v a r i a b l e s a n d f u n c t i o n s , a s t h e i r c o r r e l a t i o n s a r e below 0 . 7 5 . Two o f t h e s e a r e t h e r u d d e r g a i n s shown i n F i g . 4 . 3 - 2 , w h i c h c a n b e seer t o b e more complex t h a n t h e p o l y n o m i a l s a n d l i n e a r c o m b i n a t i o n c o n s i d e r e d h e r e . H i g h e r - o r d e r c u r v e s would f i t t h e s e g a i n s , a l t h o u g h t h e y a r e c a n d i d a t e s f o r t h e c o n s t a n t - v a l u e a p p r o x i - m a t i o n b e c a u s e t h e i r s t a n d a r d d e v i a t i o n s a r e l o w .

S e v e n t e e n g a i n s a r e most c l o s e l y c o r r e l a t e d w i t h a n d a r e f i t t e d b e s t by s e c o n d - o r d e r p o l y n o m i a l s i n pwO.

Qo

The e l e v e n g a i n s w h i c h a r e m o s t c o r r e l a t e d w i t h B0 a r e f i t t e d a l m o s t as w e l l by s e c o n d - o r d e r p o l y n o m i a l s i n B O , BO, l / B O , Th:?ee o f t h e f o u r l i n e a r BO-pWO f i t s are l / B O , o r B 0 l B g i .

a d e q u a t e , w i t h A b , / i \ r r e q u i r i n g an improved f i t ( a l o n g w i t h A k /Aw, A ~ , / A v , A & , / A r , a n d A6r/Av). ?,lost p a i r s h a v e s t r o n g a.

c o r r e l a t i o n w i t h a t L e a s t o n e o t h e r g a i n . The c o r r e l a t i o n s a s s o c i a t e d w i t h SAS a i l e r o n - r u d d e r i n t e r c o n n e c t a r e 0 , 9 5 ( A v ) , 3.50(Ar), 0.13(Ap), and 0.04(A$), indicating that either rudder or aileron would require additional Ar and Ap feed- backs in parallel with the interconnected control path.

4.4.3 Additional Considerations The longitudinal and lateral-directional sweeps were conducted to illustrate the separate effects of a o , qo, BO, and pwo on DPSAS gains. Furthermore, a limited set of in- dependent variables and scheduling functions were examined.

At a minimum, these sweeps should be combined in a single correlation procedure to obtain a single multi-variable schedule for each gain. Altitude, velocity, and weight effects should be added, principally through indicated air- speed, Mach number, and the ratio of weight-to-dynamic pressure. Permutations of the independent variables, e.g., body-axis rather than stability-axis rates, may provide better correlation or may be easier to implement in a par- ticular system.

The present results suggest that primary gains schedule largely on longitudinal variables and that cross- feed gains schedule primarily on lateral-directional vari- ables. This observation derives from the fact that most

primary DPBAS gains are nearly constant as B O and pwo

change, while crossfeed gains are zero in symmetric flight.

Any approximations made in gain scheduling must be validated by direct simulation, as this is tantamount to changing the gains from their linear-optimal values, thus altering closed-loop response.

An entirely separate issue is the on-board deter- mination (either through measurement or estimation) of the

independent variables to be used for gain scheduling --

d i f f i c u l t are n o t a b l y p a r t i c u l a r l y aO and BO, which t o measure. Two p o t e n t i a l problems are i n a c c u r a t e s t e a d y - s t z t e measurement, which l e a d s t o i n a c c u r a t e c a l c u l a t i o n o f g a i n s , and s u p e r p o s i t i o n o f p e r t u r b a t i o n s on t h e mean v a l u e s , which c o u l d c a u s e l o n g i t u d i n a l . n o t i o n s t o d r i v e lateral- d i r e c t i o n a l motjons ( a n d v i c e v e r s a ) t h r o u g h o s c i l l a t o r y g a i n c h a n g e s . The s o l u t i o n t o b o t h p r o b l e m s , s h o u l d t h e y o c c u r , is found t h r o u g h - s t a t e e s t i m a t i o n , which a l l o w s a l l a v a i l a b l e measurements t o be b l e n d e d i n a u n i f i e d estimate of nominal and p e r t u r b a t i o n motion v a r i a b l e s ( R e f . 6 3 ) . A s a n example, measurements of a , q , nZ, a i r s p e e d , a n d 1 5 ~ Aa, and A q . I f t h e DPSAS c o u l d be used t o estimate a O , q o , i s i n c o r p o r a t e d i n a f u l l command a u g m e n t a t i o n s y s t e m , p i l o t commands c o u l d b e d i r e c t i n d i c a t o r s o f t h e desired ( o r n o m i n a l ) s t a t e ; t h e r e f o r e , t h e y c o u l d be u s e d f o r g a i n s c h e d u l i n g ( R e f . 58). T h i s is a t c p i c f o r f u r t h e r s t u d y .

4 . 5 CHAPTER SUMMARY T h i s c h a p t e r h a s p r e s e n t e d d e s i g n p r i n c i p l e s f o r s t a b i l i t y augmentation s y s t e m s (DPSAS) which p r e v e n t d e p a r - t u r e from c o n t r o l l e d f l i g h t . L i n e a r - o p t i m a l c o n t r o l t h e o r y h a s been used t o develop c o n t r o l s t r u c t u r e s f o r d e p a r t u r e p r e v e n t i o n , and t h e e f f e c t s of maneuvering c o n d i t i o n on o p t i m a l feedback and c r o s s f e e d g a i n s have been e x p l o r e d .

Examples of a i r c r a f t r e s p o n s e t o l o n g i t u d i n a l , l a t e r a l , a n d d i r e c t i o n a l i n i t i a l c o n d i t i o n s i l l u s t r a t e t h e w e l l - c o n t r o l l e d b e h a v i o r which t h e DPSAS p r o v i d e s , a n d c l o s e d - l o o p e i g e n v a l u e s show t h a t v a r i a t i o n i n a i r c r a f t dynamic c h a r a c t e r i s t i c s is minimized f o r a wide r a n g e of maneuvering c o n d i t i o n s .

In many r e s p e c t s , s t a b i l i z i n g t h e r e f e r e n c e a i r c r a f t i n a p u l l u p maneuver is a more c h a l l e n g i n g t a s k t h a n a c c o u n t i n g f o r t h e c o u p l i n g which r e s u l t s from s i d e s l i p and r o l l r a t e ; however, l a t e r a l - d i r e c t i o n a l maneuvering r e s u l t s i n s i g n i f i - I c a n t l i n e a r - o p t i m a l g a i n s which improve a i r c r a f t r e s p o n s e .

I n c o m b i n a t i o n w i t h g a i n - s c h e d u l i n g f u n c t i o n s which depend on mean v a l u e s o f a n g l e s and a n g u l a r r a t e s , t h e DPSAS con- t r o l a l g o r i t h m s are seen t o produce n o n l i n e a r c r o s s f e e d s which are a n a l o g o u s t o c o n t r o l s t r u c t u r e s b e i n g employed i n modern high-performance a i r c r a f t .

aircraft L i n e a r - o p t i m a l c o n t r o l t h e o r y s o l v e s many c o n t r o l problems which have been d i f f i c u l t t o overcome w i t h p a s t d e s i g n t e c h n i q u e s . I t is easy t o u s e , i t g u a r a n t e e s s y s t e m s t a b i l i t y . and it aecomnodates a i r c r a f t w i t h l i m i t e d c o n t r o l a u t h o r i t y .

CONCLUSIONS AND RECOMMENDATIONS This report has illustrated how linear systems analysis can be used to characterize the stability of air- craft during maneuvering flight. I + also presents a design procedure for stability augmentation systems which prevent departure from controlled fllght. The key to linearizing the dynamics of the aircraft is that an accelerated flight condition can be used as a reference path. A linear model can provide a good description of the aircraft's perturbation response (to initial conditions, control inputs, and dis- turbances) even when the aircraft has large aerodynamic angles and angular rates. Contrt>l systems designed for fully coupled linear models and adapted to changing flight con- ditions can provide prote~.tinn agdinst inadvertent depar- ture from controlled flight.

5 . 1 CONCLUSIONS A detailed e x a m i n a t i o n of t h e dynamics of the ref- erence aircraft has led to ::eneral~zations concerning air- craft stability and control. These include the following: The aircraft's stability (as shown by its eigenvalues) is most affected by changes in the nominal longitudinal variables ( V o , a O , and q O ) , while the mode shapes (as described by the aircraft eigenvcctors) are most affected by non-zero nominal values of the lateral variables ( 0 0 and Asymmetric flight leads to iongi- p W O ) tudinal-variable response in typically lateral-directional modes, and vice-versa.

Nonminimum-phase zeros i n t h e a i r c r a f t t r a n s f e r f u n c t i o n s o c c u r f r e q u e n t l y i n a s y m m e t r i c f l i g h t , a n d t h e t r a n s f e r f u n c t i o n n u m e r a t o r s a r e c h a n g e d s u b - s t a n t i a l l y by n o n - z e r o q u .

E x t r e m e m a n e u v e r s a r e o f t e n c h a r a c t e r i z e d by r a p i d c h a n g e s i n b o t h mode s t a p e s a n d s p e e d s d u e t o l a r g e v a l u e s of a o , b,, a n d q 0 . H i g h l y c o u p l e d , u n s t a b l e n a i u r a l modes cbn r e s u l t .

E l e m e n t a r y l o o p c l o s u r e s w p i c h a r e s t a - b i l i z i n g i n s y m m e t r i c f l i g h t c a n l e a d t o u n s t a b l e s y s t e m d y n a m i c s i n a s y m m e t r i c f l i g h t .

0 The d e p a r t u r e p a r a m e t e r , 'nBdyn 7 has l i m i t e d v a l u e i n p r e d i c t i n g a i r c r a f t d e p a r t u r e . I t p r o v i d e s no i n f o r a , a t i on r e g a r d i n g D u t c h r o l l dampi1.g; h e n c e , it d o e s n o t p r e d i c t d e p a r t u r e d u e t o n e g a - t i v e damping ( a s is t h e c a s e f o r t h e s u b j e c t a i r c r a f t ) .

U n f o r c e d d e p a r t u r e s o c c u r when o n e o f t h e f a s t modes ( s h o r t p e r i o d , i l u t c h r o l l , o r r o l l mode) 1s u n s t a b l e . T h e s e d e p a r t - u r e s p r i m a r i l y t a k e t h e form o f a D u t c h r o l l i n s t a b i l i t y , w i t h f a s t r o l l i n g - y a w i n g m o t i o n s o r o s c i l l a t o r y d i v e r g e n c e . The r o l l mode c a n become u n s t a b l e a t e x t r e m e a n g l e s , w h e r e i t e x h i b i t s a f a s t r o l l - yaw d i v e r g e n c e .

a F o r c e d d e p a r t u r e s o c c u r a s a r e s u l t o f p i l o t a c t i o n . T h i s c a n h a p p e n when ae- g r a d e d r e s p o n s e t o c o n t r o l i n p u t s c a u s e s t h e p i l o t t o f l y t h e a i r c r a f t i n t o a f l i g h t r e g i m e w h e r e u n f 3 r c e d d e 9 a r t u r e s a r e l i k e l y o r when p i l o t a c t i ~ n s d e - s t a b i l i z e t h e a i r c r a f t d i r e c t l y .

G u i d e l i n e s f o r t h e d e s i g n o f D e p a r t u r e - P w u e n t i o n S t a b i l i t y Augmentat i o n S y s - tems (DPSAS) h a v e b z e n p r e s e n t e d . An a d a p t i v e - c o n t r o l d e s i g n p r o c e d u r e , u s i n g t h e l i n e a r - o p t i m a l r e g u l a t ~ r f o r fixed-point design followed by gain scheduling, is shown to provide a non- linear control structure containing crossfeeds as well as feedback gains.

The resulting DPSAS has similarities to.the flight control systems of cur- rent high-performance aircr~ft. How- ever, the new design is based on " q ~ a d - ratic synthesis" techniques, which provide a unified set of control gains for all axes from a single set of vector-matrix design equations.

0 The linear-optimal DPSAS prevents departure not by limiting the maneuvering ability of the aircraft but by stabilizing the air- craft in all foreseeable maneuver conditions.

The linear-optimal control law can be readily extended to a full Departure-Prevention Command Augmentation System (DPCAS) which accounts for ~,ontrol actuator rate limits and allows essentially unlimited pilot con- trol authority (within che physical limi- tations of the aircraft).

0 The maneuverability envelope of the subject aircraft could be materially expanded through the incorporation of DPSASIDPCBS concepts, as identift~d in this report.

5.2 RECOMMENDATIONS The following recommendations are made as a result of this study: a Departure prevention studies for high- performance aircraft should be extended to transonic and supersonic flight re- gimes.

The high angle-of-attacklhigh angular rate problems of additional aircraft types, including trailsports, hel'cop- ters, and general aviation aircraft, are amenable to coupled linear analy- sis and bear investigation.

Design requirements for a DPCAS should be investigated. Digital implementation and incorporation of active control conc-pts for improved maneuverabil.ity should be considered.

It is recommended that improvements to the subject aircraft's maneuverability envelope due to DPSAS/DPCAS implementation be explored in ground-based piloted simulation and flight test.

Dynamic coupling is a significant factor in the maneuvers of high-performance aircraft, and a full under- standing of its effects is an important facet of preventing departure from controlled flight. This report has shown how linear models of the aircraft's motions can be used to investigate the stability and control of maneuvering flight, and it has demonstrated the flexibility and ease with which linear-optimal control theory can be used to design departure- preventing control systems.

APPENDIX A ANALYTICAL APPROACH TO AIRCRAFT DYNAMICS A. 1 OVERVIEW T h i s a p p e n d i x d e v e l c p s t h e a n a l y t i c a l a p p r o a c h t a k e n i n t h i s r e p o r t . The g o a l o f t h i s work is t o a n a l y z e t h e air- c r a f t s t a b i l i t y a n d c o n t r o l p r o b l e m s t h a t arise a t h i g h a n g l e s o f a t t a c k a n d d u r i n g r a p i d m a n e u v e r s . The 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 o f a v e h i c l e i n a t m o s p h e r i c f l i g h t are d e v e l o p e d i n S e c t i o n A . 2 . A l z r g e body o f t h e o r y a n d e x p e r i - e n c e r e l a t i n g t o l i n e a r s y s t e m a n a l y s i s a n d c o n t r o l s y n t h e s i s e x i s t s ( R e f . 6 4 t o 6 7 ) , y e t a f u l l a n d c o m p l e t e l i n e a r i z a t i o n o f t h e a i r c r a f t p r o b l e m is n o t r e a d i l y a v a i l a b l e ; t h e r e f o r e , a r i g o r o u s a p p l i c a t i o n o f l i n e a r a n a l y s i s s h o u l d p r o v i d e new i n s i g h t s r e g a r d i n g d e p a r t u r e . S e c t i o n A . 3 p r e s e n t s t h e f u l l l i n e a r i z e d e q u a t i o n s o f m o t i o n i n a g e n e r a l f o r m , a n d it i n c l u d e s a d i s c u s s i o n o f m e t h o d s f o r c h o o s i n g t h e p o i n t o f l i n e a r i z a t i o n . S e c t i o n A.4 p r e s e n t s a n o v e r v i e w o f l i n e a r s y s t e m a n a l y s i s a n d c o n t r o l m e t h o d s .

A . 2 NONLINEAR EQUATIONS OF MOTION The c o m p l e t e n o n l i n e a r r i g i d - b o d y e q u a t i o n s o f m o t i o n a r e d e r i v e d i n t h i s s e c t i o n . They a r e d e v e l o p e d u s i n g " f l a t - e a r t h " a s s u m p t i o n s , i . e . t h e e f f e c t s o f e a r t h c u r v a - t u r e a n d r o t a t i o n a r e assumed n e g l i g i b l ~ . T h i s means t h a t e a r t h - f i x e d and i n e r t i a l r e f e r e n c e f r a m e s a r e e q u i v a l e n t .

T h e r e are f o u r c o o r d i n a t e s y s t e m s o f i n t e r e s t i n t h e s t u d y o f r i g i d - b o d y m o t i o n s o f a e r o d y n a m i c v e h i c l e s .

They are d e s c r i b e d a s f o l l o w s :

a I n e r t i a l - A x i s S j s t e m - T h i s f r a m e is

f i x e d i n i n e r t i a l s p a c e , a n d is t h e f r a m e f r o m w h i c h t h e i n e r t i a l v e l o c i t y a n d a n g u l a r r a : e o f t h e v e h i c l e are m e a s u r e d .

a Body-Axis S y s t e m - The f o r c e s a n d

moments on t h e v e h i c l e , a n d t h e r e - f o r e t h e d y n a m i c e q u a t i o n s , a r e best e x p r e s s e d i n a b o d y - f i x e d r e f e r e n c e f r a m e . T h e s t a b i l i t y - a x i s s y s t e m is a s p e c i a l b o d y - f i x e d r e f e r e n c e f r a m e .

a V e l o c i t y - A x i s S y s t e m - - An e s p e c i a l l y

u s e f u l r e f e r e n c e f r a m e f o r n a v i g a t i o n a n d g u i d a n c e , t h e v e l o c i t y - a x i s s y s t e m r e l a t e s t h e v e h i c l e v e l o c i t y v e c t o r t o i n e r t i a l a x e s .

a Wind-Axis S y s t e m - S i n c e t h e a e r o d y n a m i c

f o r c e s a n d moments d e p e n d l a r g e l y o n t h e b o d y - v e l o c i t y v e c t o r o r i e n t a t i o n , t h e w i n d - a x i s s j s t e m , w h i c h p r o v i d e s s t r a i g h t - f o r w a r d body-wind r e l a t i o n s , is u s e f u l .

F o r t h e m o d e r ~ t e v e l o c i t i e s o f i n t e r e s t i n t h i s r e p o r t ( t y p i c a l l y s u b s o n i c , i . e . , b e l o w a b o u t 340 m / s ) , t h e e q u i v a l e n c e o f e a r t h - f i x e d a n d i n e r t i a l r e f e r e n c e f r a m e s is a good a s s u m p t i o n . The o r i g i n o f t h e i n - e r t i a l r e f e r e n c e f r a m e u s e d h e r e is l o c a t e d on t h e s u r - f a c e o f t h e e a r t h , w i t h t h e x - , y - , a n d z - a x e s i n a n o r t h - e a s t - d o w n o r i e n t a t i o n . S i n c e t h e s i m p l e s t s t a t e m e n t o f N e t w o n ' s S e c o n d Law is g i v e n i n a n i n e r t i a l r e f e r e n c e f r a m e , t h i s f r a m e p l a v s an i m ~ o r t a n t ? a r t i n t h e d e r i v a t i o n o f t h e d y n a m i c e q u a t i o n s .

The v a r i o u s b o d y - f i x e d a x i s s y s t e m s h a v e a common o r i g i n , l o c a t e d a t t h e body c e n t e r o f m a s s , 2nd are f i x e d I " e w . . .". .

I . . .. . - ... . -1

L.

i n o r i e n t a t i o n w i t h respect t o t h e v e h i c l e . G e n e r a l l y , t h e body x - a x i s e x t e n d s f o r w a r d o u t t h e v e h i c l e ' s n o s e , t h e y - a x i s e x t e n d s o u t t h e r i g h t wing, a n d t h e z - a x i s e x t e n d s o u t t h e b o t t o m o f t h e v e h i c l e . T h e x-z p l a n e is u s u a l l y a p l a n e of g e o m e t r i c s y m m e t r y , i f t h e v e h i c l e h a s o n e . T h e r e a r e a number o f p o s s i b l e b o d y - f i x e d r e f e r e n c e f r a m e s , a n d t h e o n e f i x e d b y t h e b u i l d e r is s i m p l y r e f e r r e d t o i n t h i s report a s t h e b o d y - a x i s s y s t e m . F o r a n y n o m i n a l f l i g h t c o n d i t i o n , b o d y - f i x e d a x e s c a n be c h o s e n so t h a t t h e x - a x i s is a l i g n e d w i t h t h e v e l o c i t y v e c t o r , a n d t h e z-axis is i n t h e b o d y a x i s x-z p l a n e . T h i s set o f b o d y - f i x e d a x e s is r e f e r r e d t o as t h e s t a b i l i t y - a x i s s y s t e m .

S i n c e body a x e s are t h e o n l y a x e s i n w h i c h t h e v e h i c l e r o t a t i o n a l i n e r t i a m a t r i x is c o n s t a n t , t h e r o t a t i o n a l d y n a m i c s e q u a t i o n s are u s u a l l y ( t h o u g h n o t e x c l u s i v e l y ) e x p r e s s e d i n t h i s f r a m e . The body f r a m e a l s o is t h e o n e i n w h i c h t h e p i l o t , a n d a l l s e n s o r s a n d c o n t r o l s u r f a c e s a r e located; f o r t h i s r e a s o n , t h e body f r a m e is c o n s i d e r e d t h e b a s i c f r a m e o f r e f e r e n c e i n t h i s r e p o r t . F i g u r e A . 2 - l a i l l u s t r a t e s t h e body f r a m e o r i e n t a t i o n w i t h r e s p e c t t o t h e i n e r t i a l - a x i s s y s t e m .

The v e l o c i t y - a n d wind- a x i s s y s t e m s h a v e a common o r i g i n ( t h e v e h i c l e c e n t e r o f m a s s ) a n d a common x - a x i s , w h i c h is o r i e n t e d a l o n g t h e v e h i c l e ' s i n e r t i a l v e l o c i t y v e c - t o r . The v e l o c i t y r e f e r e n c e f r a m e y - a x i s is p a r a l l e l t o t h e i n e r t i a l x-y p l a n e . T h i s r e s u l t s i n s i m p l e r e l a t i o n s be- tween i n e r t i a l a n d v e l o c i t y a x e s , so t h a t t h e s e a x e s are u s e f u l f o r n a v i g a t i o n a n d p o i n t - m a s s t r a j e c t o r y c a l c u l a - t i o n s . F i g u r e A . 2 - l b i 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 b e t w e e n i n e r t i a l a n d v e l o c i t y a x e s .

The w i n d r e f e r e n c e f r a m e ' s z - a x i s is l o c a t e d i n t h e x-z p l a n e o f t h e body f r a m e ; t h i s r e f e r e n c e f r a m e is a) Inertial-Body Orientation b) Inert ial-Velocity Orientat ion c) Wind-Body Orient at ion Figure A.2-1 Reference Frame Relations very useful in dynamic calculatians because the orientation angles between the wind frame and body frame have large influences on the aerodynamic forces and moments. Figure A.2-lc illustrates the orientation between body and wind axes. Figure A.2-2 summarizes the transformations between reference frames. Any necessary transformation can be identfied from this figure, noting that the Euler angles are given in the order of yaw, pitch, and roll, as speci- fied by the arrows. For example, a transformation from inertial to body-axes is composed of a right-handed yaw through an angle, $ , then a right-handed pitch through an angle, 8, and then a right-handed roll through an angle,

. These three single-angle transformations can be com-

bined to form ac inertial-body transformation as follows: For orthogonal matrices such as these, the matrix inverse, T ( )-I, is equal to the transpose, ( ) .

In the remainder of Section A.2, the vehicle's equation of motion is derived as a single state-vector equation of the form,

where - x is the state vector, 2 is the control vector, - f is

the vector system dynamics equation, and disturbances are neglected. The state vector is a 12-element vector, and the nonlinear state equations are readily derived as four

I INERTIAL \

INERTI.%L-BODY AXIS TRANSFORW ATION VELOCITY

( - 1

W I N P I O D Y " w 8 AXIS

r\ . . A . SF.R.ATl0.

( - )

F i g u r e A.2-2 R e f e r e n c e A x i s T r a n s f o r m a t i o n s (Arrows I n d i c a t e Right-Hand Rotat i o n ) s e t s of t h r e e e q u a t i o n s r e p r e s e n t i n 3 T r a n s l a t i o n a l K i n e m a t i c s R o t a t i o n a l K i n e m a t i c s T r a n s l a t i o n a l Dynamics 0 R o t a t i o n a l Dynamics The k i n e m a t i c e q u a t i o n s r e l a t e t h e v e h i c l e ' s 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 v e l o c i t i e s t o its p o s i t i o n i n i n e r t i a l s p a c e , a n d t h e y i n v o l v e b o d y - a x i s l i n e r t i a l - s x i s r e l a t i o n s h i p s . The dynamic e q u a t i o n s d e s c r i b e t h e c h a n g e s o f t h e v e h i c l e v e l o c - i t i e s c a u s e d b y t h e a p p l i e d f o r c e s a n d moments; t h e y a r e best d e r i v e d i n a b o d y - f i x e d f r a m e o f r e f e r e n c e .

A . 2 . 1 K i n e m a t i c s K i n e m a t i c s is t h e s t u d y o f t h e m o t i o n of a b o d y w i t h o u t r e g a r d t o t h e f o r c e s w h i c h c a u s e t h a t m o t i o n . I n t h i s s e c t i o n , t h e r e l a t i o n s b e t w e e n t h e v e h i c l e ' s p o s i t i o n a n d v e l o c i t y a r e e x a m i n e d . The t r a n s l a t i o n a l a n d a n g u l a r p o s i t i o n o f t h e v e h i c l e a r e g i v e n r e l a t i v e t o i n e r t i a l s p a c e by t h e i n e r t i a l p o s i t i o n v e c t o r , 51, a n d b y t h e i n e r t i a l - body E u l e r a n g l e v e c t o r , xB: ( A . 2 - 3 ) I t is i m p o r t a n t t o n o t e t h a t t h e E u l e r a n g l e " v e c t o r " is n o t a t r u e v e c t o r i n p h y s i c a l s p a c e ; i t is a n o r d e r e d t r i p l e c f r i g h t - h a n d e d r o t a t i o n s mhich o c c u r a b o u t d i f f e r e n t a x e s o f d i f f e r e n t r e f e r e n c e frames.

The t r a n s l a t i o n a l a n d a n g u l a r r a t e v e c t o r s are o f t e n e x p r e s s e d i n body a x e s , a s i n t h e f o l l o w i n g : ( A . 2-6) i The body-axis t r a n s l a t i o n a l r a t e v e c t o r , xB, is an e x p r e s s i o n , i n body a x e s , of t h e d e r i v a t i v e of t h e i n e r t i a l p o s i t i o n vec- t o r . T h i s r e l a t i o n s h i p s u p p l i e s t h e f i r s t p a r t o f t h e non- l i n e a r s t a t e e q u a t i o n s of motion: ( A . 2-7) I where HB is t h e i n v e r s e o f t h e i n e r t i a l - b o d y t r a n s f o r m a t i o n d e r i v e d i n Eq. (A.2-1).

The body a n g u l a r rate v e c t o r a l s o can be r e l a t e d t o t h e d e r i v a t i v e o f t h e E u l e r a n g l e v e c t o r by n o t i n g t h a t t h e E u l e r a n g l e d e r i v a t i v e s o c c u r i n t h r e e d i f f e r e n t r e f e r e n c e frames. The r e s u l t i n g t r a n s f o r m a t i o n is c o n s t r u c t e d i n Eq. (A.2-8), where t h e i n d i v i d u a l t r a n s f o r m a t i o n s a r e t h e same as t h o s e of F i g . A.2-2 and Eq. (A.2-1): ( A . 2-8) T h e non-orthogonal t r a n s f o r m a t i o n , Ig, is T h e o r d e r i n g o f t h e t r a n s f o r m a t i o n s i n E q . ( A . 2 - 8 ) a r i s e s f r o m t h e o r d e r i n g o f t h e E u l e r a n g l e s . A s c a n b e s e e n f r o m F i g . A . 2 - l a , t h e a n g u l a r r a t e , 6 , o c c u r s a b o u t the xg a x i s ,

w h i l e t h e r a t e , 6 , o c c u r s a b o u t t h e y2 a x i s , a n d 4 o c c u r s

a b o u t t h e zl a x i s . T h e i n v e r s e o f E q . ( A . 2 - 8 ) s u p p l i e s t h e r o t a t i o n a l k i n e m a t i c p a r t o f t h e v e h i c l e n o n l i n e a r s t a t e e q u a t i o n s , a n d is g i v e n b y : T h e r e l a t i o n s b e t w e e n t r a n s l a t i o n a l p o s i t i o n a n d v e l o c i t y , Eq. ( A . 2 - 7 ) , a n d b e t w e e n a n g u l a r o r i e n t a t i o n a n d v e l o c i t y , E q . (A.2-101, c o m p r i s e t h e k i n e m a t i c p o r t i o n s o f t h e non- l i n e a r s t a t e e q u a t i o n s .

A . 2 . 2 D y n a m i c s The d y n a m i c s o f t h e v e h i c l e i n v o l v e t h e i n t e r a c t i o n b e t w e e n t h e v e h i c l e m o t i o n a n d t h e f o r c e s t h a t p r o d u c e t h a t n o t i o n . T h i s i n v o l v e s t h e a p p l i c a t i o n o f N e w t o n ' s S e c o n d Law, w h i c h e q u a t e s t h e a p p l i e d f o r c e t o t h e t i m e d e r i v a t i v e o f i n e r t i a l t r a n s l a t i o n a l momentum o f a b o d y . F o r r o t a - t i o n a l m o t i o n , t h i s e q u i v a l e n c e b e c o m e s o n e b e t w e e n t o r q u e a n d t h e d e r i v a t i v e o f a n g u l a r momentum, m e a s u r e d i n a n i n e r t i a l r e f e r e n c e f r a m e .

An e x p r e s s i o n f o r t h e i n e r t i a l t r a n s l a t i o n a l a c c e l - e r a t i o n , e x p r e s s e d i n b o d y - a x i s v a r i a b l e s , is n e e d e d .

T h i s c a n b e d e r i v e d f r o m E q . ( A . 2 - 7 ) by t a k i n g t h e d r i v a t i v e of h o t h s i d e s . I t is i m p o r t a n t t o n o t e t h a t t h e t r a n s f o r m a - t i o n m a t r i x is t i m e - v a r y i n g . T h i s r e s u l t is ( A . 2 - 1 1 ) where (A. 2-12) ."I

and wB is t h e cross-product equivalent matrix f o r gi given by

( A . 2-13) T h i s leads t o t h e body-axis equation; The applied s p e c i f i c forces c o n s i s t of gravita- t i o n a l forces and aerodynamic forces. The gravity f o r c e is especially simple i n i n e r t i a l axes, a s it is confined t o t h e v e r t i c a l a x i s : a ( A . 2-15) The s p e c i f i c contact force can be broken i n t o two components, one of which is due t o aerodynamic forces and one of which is due t o t h r u s t : ( A . 2-17) (Capital l e t t e r s a r e conventionally used i n aerodynamics t o denote the force components.)

The t r a n s l a t i o n a l dynamic equation is formed by equating t h e sum of the aerodynamic and g r a v i t a t i o n a l spe- c i f i c forces t o t h e i n e r t i a l t r a n s l a t i o n a l acceleration of t h e vehicle. When a l l vectors a r e expressed i n body axes and tbe d e r i v a t i v e s of t h e body-axis v e l o c i t i e s a r e iso- l a t e d on t h e left-hand s i d e , the vector equation is To construct t h e r o t a t i o n a l dynamic equation, an expressian f o r the time d e r i v a t i v e of angular momentum mea- sured i n i n e r t i a l axes is necessary. The angular momentum, h is most e a s i l y expressed i n body axes; neglecting r o t a t - -B ' i n g machinery, it is the product of the moment-of-inertia matrix (constant i n body axes) and the angular ra.te vector, where the i n e r t i a matrlx contains a l l products and moments of i n e r t i a : ( A . 2-20) The time d e r i v a t i v e o f t h e a n g u l a r momentum, e x p r e s s e d i n i n e r t i a l a x e s , is e a s i l y d e r i v e d by n o t d g t h a t t h e t r a n s - I f o r m a t i o n HB is t i m e - v a r y i n g : 1-1 I = H ~ I ~ & + H w I g ( A . 2-21) B B B B T h e c o n t a c t moments c o n s i s t o f a e r o d y n a m i c a n d t h r u s t c o m p o n e n t s . T h e s e are d e f i n e d zs ( A . 2-22)

-B - [t]

( A . 2-23) ( C a p i t a l l e t t e r s a r e c o n v e n t i o n a l l y u s e d f o r t h e moment c o m p o n e n t s . ) T h e r o t a t i o n a l d y n a m i c e q u a t i o n c a n be formed by e q u a t i n g t h e a p p l i e d t o r q u e s t o t h e d e r i v a t i v e o f t h e a n g u l a r momentum. A l l v e c t o r s are e x p r e s s e d i : : body a x e s , a n d t h e d e r i v a t i v e o f b o d y - a x i s a n g u l a r r a t e is i s o l a t e d o n t h e l e f t - h a n d side of t h e e q u a t i o n , r e s u l t i n g i n

01 - -1 -1-1 I

( A . 2-24)

I g (gB !B+gB) - I B u B I B-B a

-B A . 2 . 3 Summary o f E q u a t i o n s S e c t i o n A.2 h a s p r e s e n t e d t h e v a r i o u s r e f e r e n c e f r a m e s o f i n t e r e s t a n d h a s d e i i v e d t h e e q u a t i o n s o f m o t i o n o f a n a t m o s p h e r i c v e h i c l e . T h e 1 2 - e l e m e n t s t a t e v e c t o r con- sists o f t h r e e p o s i t i o n s , t h r e e a n g u l a r o r i e n t a t i o n s , t h r e e t r a n s l a t i o n a l rates a n d t h r e e a n g u l a r rates. The s t a t e e q u a t i o n s w e r e f o u n d by e x a m i n i n g t h 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 k i n e m a t i c s a n d d y n a m i c s , a n d a r e r e p e a t e d h e r e : (A. 2-26) (A. 2-27) (A. 2-28) T h e s e e q u a t i o n s f a l l i n t o t h e g e n e r a l s t a t e e q u a - t i o n f o r m , (A. 2-29) by d e f i n i n g t h e s t a t e v e c t o r a s . .

( A . 2-30) a n d n o t i n g t h a t t h e z ~ e r o d y n a m i c f o r c e s a n d moments are f u n c - t i o n s o f t h e s t a t e s , c o n t r o l s , d i s t u r b a n c e s a n d , t o some e x t e n t , t h e s t a t e t i m e h i s t o r y . T h e s e n o n l i n e a r s t a t e e q u a - t i o n s a r e u s e f u l b e c a u s e t h e y are g e n e r a l enough t o a l l o w a t h o r o u g h a n a l y s i s o f t h e d e p a r t u r e p r e v e n t i o n p r o b l e m . They are e x p r e s s e d i n s t a t e - s p a c e f o r m , w h i c h is n o t a t i o n a l l y e f f i c i e n t a n d which makes t h e s u b s e q u e n t l i n e a r i z a t i o n a n e a s i l y f o l l o w e d p r o c e s s .

A. 3 TIINEAR EQUATIONS OF MOTION W h i l e t h e n o n l i n e a r e q u a t i o n s d e r i v e d i n t h e p r e - v i o u s s e c t i o n c a n be s o l v e d o n a d i g i t a l c o m p u t e r , t h e y a r e n o t e a s i l y a n a l y ~ e d by g e n e r a l t e c h n i q u e s , a n d g e n e r a l c l o s e d - f o r m s o l u t i o n s c a n n o t b e o b t a i n e d . Many o f t h e i m - p o r t a n t dynamic a t t r i b u t e s o f t h e a i r c r a f t c a n b e p r e - s e r v e d and t h e a n a l y s i s f a c i l i t a t e d b y d e v e l o p i n g c o r - r e s p o n d i n g l i n e a r i z e d e q u a t i o n s o f m o t i o n , as is d o n e i n R e f s . 64 t o 6 7 .

A . 3 . 1 D e r i v a t i o n f r o m N o n l i n e a r E q u a t i o n s The l i n e a r i z a t i o n p r o c e d u r e b e g i n s w i t h t h e c o n - s t r u c t i o n o f a T a y l o r series e x p a n s i o n r e p r e s e n t i n g t h e n o n l i n e a r e q u a t i o n s a b o u t some n o m i n a l t r a j e c t o r y : A u + H i g h e r O r d e r Terms - -0 u=u

- -0

(A. 3-1) w h e r e t h e s u b s c r i p t "0" i n d i c a t e s t n e n o m i n a l v a l u e a n d t h e p r e f i x " A " d e n o t e s a s m a l l p e r t u r b a t i o n . A l l e x c e p t f i r s t - o r d e r terms a r e t h e n n e g l e c t e d by a r g u i n g t h a t t h e h i g h e r - o r d e r terms a r e s m a l l compared t o l i n e a r t e r m s . The r e s u l t s of t h i s p r o c e d u r e a r e s e p a r a t z d i n t o a n o n l i n e a r e q u a t i o n d e s c r i b i n g t h e n o m i n a l t r a j e c t o r y ( E q . ' ( A . 3 - 2 ) ) a n d a l i n e a r e q u a t i o n d e f i n i n g t h e d y n a m i c s o f t h e p e r t u r b a t i o n s a b o u t

-

t h e n o m i n a l t r a j e c t o r y ( E q . (A.3-3)): where and The l i n e a r i z a t i o n is s t r a i g h t f o r w a r d because t h e n o n l i n e a r s t a t e e q u a t i o n s (Eq. (A.2-25) t o A.2-28))are spe- c i f i e d i n a g e n e r a l s t a t e - s p a c e f o r m a t . E q u a t i o n s f o r t h e p e r t u r b a t i o n s of t h e a x i s t r a n s f o r m a t i o n s a r e e a s i l y d e r i v e d by t a k i n g t h e p a r t i a l d e r i v a t i v e s of each t r a n s f o r m a t i o n with r e s p e c t t o t h e E u l e r a n g l e s o f t h a t t r a n s f o r m a t i o n and mul- t i p l y i n g b y t h e E u l e r a n g l e p e r t u r b a t i o n s . For t h e i n e r t i a l - body t r a n s f o r m a t i o n and its i n v e r s e , t h e s e t r a n s f o r m a t i o n p e r t u r b a t i o n s can be s t a t e d as f o l l o w s : (A. 3-6) where t h e cross-product o p e r a t o r ( - ) is employed, (A. 3-7) and (A.3-8) These r e l a t i o n s h i p s f o r t h e t r a n s f o r m a t i o n p e r t u r b a - t i o n s a r e used t o l i n e a r i z e t h e t r a n s l a t i o n a l kinematic e q u a t i o n , ~ q . ( A . 2 - 2 5 1 , t o g i v e ( A . 3 - 9 ) T h i s e q u a t i o n c l e a r l y shows t h a t t h e p e r t u r b a t i o n i n e r t i a l v e l o c i t y d e p e n d s b o t h o n t h e p e r t u r b a t i o n b o d y - a x i s v e l o c i t y a n d t h e i n e r t i a l - b o d y E u l e r a n g l e p e r t u r b a t i o n s .

The r o t a t i o n a l k i n e m a t i c e q u a t i o n , E q . ( A . 2 - 2 6 ) , r e s u l t s i n t h e f o l l o w i n g l i n e a r p e r t u r b a t i o n e q u a t i o n : (A. 3-10) w h e r e ( A . 3-11] s o t h a t (A. 3-12) The d e f i n i t i o n o f L B ~ t a k e s t h e f o r m it d o e s b e c a u s e t h e l i n e a r r o t a t i o n a l k i n e m a t i c e q u a t i o n was d e r i v e d f r o m t h e o r i g i n a l f o r m o f t h e r o t a t i o n a l k i n e m a t i c s e q u a t i o n , g i v e n i n E q . ( A . 2 - 8 ) .

L i n e a r i z a t i o n o f t h e d y n a m i c e q u a t i o n s r e q u i r e : c o n s i d e r a t i o n o f t h e a e r o d y n a m i c f o r c e a n d moment r e l a t i o n - s h i p s . T h e s e a r e f u n c t i o n s o f t h e s t a t e s , c o n t r o l s , a n d t h e p a s t h i s t o r y of t h e s e v a r i a b l e s . T h i s d e p e n d e n c e on p a s t v a l u e s is c a u s e d by a e r o d y n a m i c f l o w f i e l d e f f e c t s a n d t h e i r p r o p a g a t i o n d e l a y s ; u n s t e a d y a e r o d y n a m i c e f f e c t s c a n be r e p r e s e n t e d a s f u n c t i o n s o f t h e s t a t e rates. The f o r m a l l i n e a r i z a t i o n o f t h e a e r o d y n a m i c f o r c e s a n d moments is a l e n g t h y b u t s t r a i g h t f ~ . r w a r d p r o c e s s w h i c h a m o u n t s t o t a k i n g t h e p a r t i a l d e r i v a t i v e s o f e v e r y a e r o d y n a m i c f o r c e a n d moment v e c t o r w i t h r e s p e c t t o t h e s t a t e s , s t a t e r a t e s , a n d c o n t r o l s . T h e s e p a r t i a l d e r i v a t i v e s a r e c a l l e d s t a b i l i t y d e r i v a t i v e s .

The d i f f i c u l t y r e v o l v e s a r o u n d t h e a c t u a l v a l u e s t o be u s e d f o r o a c h o f t h e s e c o e f f i c i e n t s . T h i s d a t a is p r o - d u c e d p r i m a r i l y by w i n d t u n n e l t e s t i n g , a s d e s c r i p e d , f o r e x a m p l e , i n R e f s . 6 8 a n d 6 9 . T h e r e is a l a r g e amount o f e f f o r t a n d e x p e n s e i n v o l v e d i n g e n e r a t i n g t h i s d a t a , s o o n l y t h e most i m p o r t a n t f u n c t i o n a l r e l a t i o n s h i p s c a n b e e x a m i n e d . T h i s o f t e n r e s u l t s i n d i f f e r e n t d a t a sets f o r e a c h a i r c r a f t .

F o r t h i s r e a s o n , o n l y g e n e r a l terms f o r t h e p e r t u r b a t i o n f o r c e s a n d moments a r e i n c l u d e d i n t h e f o l l o w i n g d i s c u s s i o n .

~ p p e n d i x B c o n t a i n s a d i s c u s s i o n a n d e x a m p l e o f t h e c o n - s t r u c t i o n o f f o r c e a n d moment s t a b i l i t y d e r i v a t i v e m a t r i c e s from r e a l d a t a . Many s t a b i l i t y d e r i v a t i v e matrices are e i t h e r known t o b e z e r o o r a r e s o small a s t o b e n e g l e c t e d i n a l l c a s e s o f i n t e r e s t . Assuming t h a t a l t i t u d e and o r i e n - t a t i o n v a r i a t i o n s h a v e n e g l i g i b l e e f f e c t on c o n t a c t f o l c e a n d moment v a r i a t i o n s , a n d a s s l - m i n g i n s i g n i f i c a n t s t a t e d e r i v a t i v e and a n g u l a r r a t e e f f e c t s on t h r u s t f o r c e s a n d moments, t h e p e r t u r b a t i o n a e r o d y n a m i c f o r c e s a n d moments are a s f o l l o w s : ( A . 3-13) ( A . 3-16) The l i n e a r t r a n s l a t i o n a l dynamic e q u a t i o n is d e r i v e d from Eq. ( A , 2 - 2 7 ) a n d i n c o r p o r a t e s t h e p e r t u r b a t i o n aero- d y n a m i c f o r c e e x p r e s s i o n s p r e s e n t e d a b o v e . Note t h a t t h e state-rate s t a b i l i t y d e r i v a t i v e matrices e n t e r ' t h e e q u a t i o n i n a d i f f e r e n t manner t h a n t h e o t h e r t e r m s ; t h e y must b e moved t o t h e l e f t - h a n d s i d e o f t h e d y n a m i c e q u a t i o n s . The l i n e a r t r a n s l a t i o n a l d y n a m i c e q u a t i o n becomes ( A . 3-17) The l i n e a r r o t a t i o n a l dynamic e q u a t i o n is d e r i v e d s i m i l a r l y , a n d t h e state-rate s t a b i l i t y d e r i v a t i v e s a p p e a r i n t h e same way. The l i n e a r r o t a t i o n a l dynamic e q u a t i o n is ( A . 3-18) The f o u r state e q u a t i o n s ( 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 k i n e m a t i c a n d dynamic e q u a t i o n s ) c a n be w r i t t e n i n s t a n d a r d l i n e a r e q u a t i o n f o r m ( E q . (A.3-3)) by u s i n g t h e f o l l o w i n g state v e c t o r : ( A . 3-19) e The s t a t e e q u a t i o n s t h e n f a l l i n t o t h e f o r m : ( A . 3-20) w h e r e t h e state-rate t r a n s f o r m a t i o n m a t r i x is ( A . 3-21) and t h e three-by-three sub2-matrices a r e ( A . 3-22) ( A . 3-23) ( A . 3-24) ( A . 3-25) The primed s t a t e dynamics matrix is ( A . 3-26) where t h e three-by-three sub-matrices a r e ( A . 3-27) ( A . 3-28) ( A . 3-29) ( A . 3-30) (A. 3-31) (A. 3-32) ( A . 3-33) ( A . 3-34) (A. 3-35) and (A. 3-36) The sub-matrices ( o f t h r e e rows and a s many columns a s con- t r o l s ) a r e ( A . 3-37) (A. 3-38) The complete state e q u a t i o n is produced by pre- m u l t i p l y i n g E q . ( A . 3 - 2 0 ) by t h e i n v e r s e o f t h e s t a t e - r a t e transformation matrix, giving the following r e s u l t T h i s r e s u l t i n g l i n e a r system is analyzed through- out t h i s r e p o r t . I t is important t o note t h a t t h i s s y s - tem s p e c i f i e s t h e s t a t e and control perturbations about t h e i r nominal values. Methods of properly choosing these nominal values a r e examined next.

A . 3 . 2 Generalized T r i m Conditions An a i r c r a f t is i n the trimmed condition when its c o n t r o l s a r e s e t t o produce equilibrium i n t h e equations of motion. Steady t r i m occurs when t h e a i r c r a f t is under no i n e r t i a l a c c e l e r a t i o n , i.e., when t r a n s l a t i o n a l v e l o c i t i e s a r e constant and r o t a t i o n a l r a t e s a r e zero. The t r i m con- cept can be extended t o dynamic f l i g h t conditions b y defin- i n g peneralized t r i m as t h e condition i n which control s e t - t i n g s produce constant v e l o c i t i e s and angular r a t e s . I n t h i s case, t h e vehicle is n o t necessarily i n steady e q u i l i b - rium, due t o changing r o l l and pitch angles.

The importance of these t r i m c l a s s i f i c a t i o n s l i e s i n t h e use of trimmed f l i g h t conditions as nominal t r a j e c - t o r i e s f o r since t r i m implies t h a t t h e nomi- nal v e l o c i t i e s , angular r a t e s , and c o n t r o l s a r e constant, o r , a t most, slowly varying. Thus, the use of the trimmed condition a s a nominal flight condition causes t h e l i n e a r equations t o represent almost a l l of the system dynamics a t t h a t f l i g h t condition. T h i s can be seen by w r i t i n g t h e general equations f o r t h e t o t a l s t a t e r a t e : ( A . 3-40) ( A . 3-41) If a s e t of nominal s t a t e s and controls can be found s o t h a t the generalized t r i m condition .

( A . 3-42) i s s a t i s f i e d , then the perturbation s t a t e r a t e s a r e equal t o the t o t a l s t a t e r a t e s . T h i s can be seen by i n s e r t i n g E y . (A.3-42) i n t o E q s . (A.3-40) and (A.3-41).

Another desirable c h a r a c t e r i s t i c of using t h e trimmed f l i g h t condition a s a nominal f o r l i n e a r i z a t i o n is t h a t the t o t a l s t a t e and control t r a j e c t o r i e s over a s i g n i - f i c a n t i n t e r v a l of time are the sums of the constant nominal values and the l i n e a r perturbation time h i s t o r i e s . T h i s r e l a t i o n is given a s : ( A . 3-43) I I

w ( t ) - w 1 + n g p ( A . 3-44)

-B

-Bo

( A . 3-45) where - vg 0 0 WBO and uo a r e constant.

A s a l u t i o n m e t h o d f o r t h e g e n e r a l i z e d trim p r o b l e m c a n be d e r i v e d by e x a m i n i n g g e n e r a ? f o r m s 01 tile riuii:iiicz~ 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 d y n a m i c e q u - t i o n s , w h i c h a r e d e r i v e d i n S e c t , n A . 2 . 2 as ( A . 3-46) T h e aerodynami: f o r c e s a n d moments a r e a s s u m e d t o b e f u n c - t i o n s of xB, _ w , ~ , a n d - U . One e l e m e n t o f t h e E u l e r a n g l e vec- t o r , + , d o e s n o t a p p e a r i n t h e e q u a t i o n s , a n d a l t i t u d e is a p a r a m e t e r .

Two d i f f e r e n t g e n e r a l i z e d t r i m p r o b l e m s t h e n become p o s s i b l e , t h e f i r s t o f w h i c h i s t h e f o l l o w i n g : F i n d t h e v a l u e s o f v e l o c i t y a n d a n g u l a r

r a t e (% a n d Ei) t h a t p r o d u c e g e n e r a l -

. I

i z e d t r i m (h=iB= 0 ) f o r g i v e n c o n t r o l s

a n d E u l e r a n g l e s ( u , - 3).

T h i s p r o b l e m c o n s i s t s o f s i x e q u a t i o n s ( t h e d y n a m i c e q u a - t i o n s ) i n s i x u1l~;nowns ( t h e v e l o c i t y a n d a n g u l a r r a t e s ) , a n d t h e r e f o r e i t c a c b e e x p e c t e d t o h a v e a s o l u t i o n . T h e c o n t r o l s a n d E u l e r a n g l e s are t h e set p o i n t s t h a t d e t e r m i n e t h e g e n e r a l i z e d t r i m s o l u t i o n , a n d i t s h o u l d b e n o t e d t h a t a l t i t u d e a l s o h a s a n e f f e c t o n t h e s o l u t i o n .

T h e s e c o n d g e n e r a l i z e d t r i m p r o b l e m c a n p r o v i d e t h e t r i m c o n t r o l s t h a t p r o d u c e s p e c i f i c s t a t e v a l u e s : F i n d t h e v a l u e s of t h e c o n t r o l s a n d E u l e r a n g l e s ( u a n d v g ) t l l a t s a t i s f y t h e generalizes t r i m - c o n d i t i o n s

(h= = 0 ) f o r r i v e n v a l ies o f t . h e

-B v e l o c i t y a n d a n g u l a r r a t e v c c t o r s T h e e x i s t e n c e of a s c l u t i o n t o t h e s e c o n d p r o b l e m d e p e n d s o n t h e d e g r e e s o f f r e e d o m a n d power o f t h e c o n t r o l s .

hhny a e r o d y n a m i c v e h i c l e s h a v e a f o u r - e l e m e n t c o n t r o l v e c - t o r ; a l o n g w i t h t h e E u l e r a n l ; l e s , 1 5 a n d 9 , t h i s r e s u l t s i n a p r o b l e m o f s i x e q u a t i o n s ( t h e d y n a m i c e q u a t i o n s ) i n s i x unknowns ( t h r o t t l e , e l e v a t o r , a i - l e r o n , r u d d e r , p i t c h a n g l e , a n d r o l l a n g l e ) .

One a p p r o a c h f o r s o l v i n g e i t h e r of t h e s e g e n e r a l i z e d trim p r o b l e m s is t o u s e f u n c t i o n a l m i n i m i z a t i o n . T h i s -- a p p r o a c h , a l s o c a l l e d p a r a m e t r i c o p t i m i z a t i o n , r e q u i r e s a l l I e l e m e n t s of u - a n d zB ( o ? of yg a n d sR) t o b e s p x i f i e d a s g i v e n o r d e s i r e d , b u t i t ' s n o t t i e d t o a n y p a r t i c u l a r f l i g h t p a t h .

I n t h i s c a s e , E q s . ( A . 3 - 4 6 ) a n d ( A . 3 - 4 7 ) a r e s o l v e d d i r e c t l y u s i n g a n i t e r a t i v e p r o c e s s , e . g . , a s t e e p e s t - d e s c e n t , a c c a l e r a t e d g r a d i e n t a l g o r i t h m , o r d i r e c t n u m e r i c a l s e a r c h ( R e f . 70). A s c a l a r c o s t f u n c t i o n , J , m e a s u r e s trim e r r o r ; a q u a d ~ a t i c f o r m is a p p r o p r i a t e f o r c o m p u t i n g a norm o f t h e v e c t o r e r r o r : I n b o t h g e n e r a l i z e d t r i m p r o b l e m s , t h e t r i m v a l u e is d e t e r m i n e d w h e n J r e a c h e s a min;.-l::, .

A . 3 . 3 Bodv-Axis E a u a t i o n s I t is a d v a n t a g e o u s t o e x p r e s s t h ~ v e h i c l e s t a t e e q u a t i o n s i n o o d y - f i x e d a x e s . T h e s e a r e t h e axes i n w h i c h t h e p i l o t , t h c s e n s o r s , a n d t h e c o n t r o l s u r f a c e s a r e l o c a t e d .

B o d y a x t > y a r e thr. o n l v a x e s ~ n w h i c h t h e m o m e n t - o f - i n e r t i a m a t r i x i s c o n s t a n t . Aerodynamic d a t a c o l l e c t e d f r o m s t i n g - x o u n t e d ;vind t ~ ~ r i r i r - 1 rnod~~1.c: n r frnm t ' l I g h t t ~ s t u s u a 1 1 y is e x p r e s s e d i n body a x e s . C o n s e q u e n t l y , body a x e s are con- s i d e r e d t o b e t h e b a s i c a x e s i n t h i s r e p o r t , a n d a l l o f t h e e q u a t i o n s g i v e n s o f a r h a v e b e e n i n body a x e s .

S t a b i l i t y - A x i s E q u a t i - o n s

-

S t a b i l i t y a x e s a l s o a r e b o d y - f i x e d a x e s , s o t h a t t h e y r e t a i n t h e c o n v e n i e n t c h a r a c t e r i s t i c s m e n t i o n e d a b o v e .

The a x e s a r e f i x e d i n t h e body s o t h a t t h e x - a x i s is a l i g n e d w i t h t h e n o m i n a l v e l o c i t y v e c t o r . A l t h o u g h t h e a x e s are f i x e d i n t h e b o d y , t h e y h a v e d i f f e r e n t o r i e n t a t i o n s a t d i f f e r e n t n o m i n a l f l i g h t c o n d i t i o n s . F u r t h e r , t h e p e r t u r - b a t i o n v e l o c i t y v e c t o r is n o t e x p r e s s e d as t h r e e o r t h o g o n a l v e l o c i t y p e r t u r b a t i o n s b u t a s a v e l o c i t y m a g n i t w p e r t u r - b a t i o n a n d t w o b o d y - v e l o c i t y o r i e n t a t i o n a n g l e s : y e r t u r - b a t i o n a n g l e s o f a t t a c k ( h a ) and s i d e s l i p ( A $ ) . T h i s .?ec- t o r is r e f e r r e d t o h e r e a s A I W .

I t is i m p o r t a n t t o n o t e t h a t s t a b i l i t y a x e s a r e a l s o t h e same a s wind a x e s f o r a s p e c i f i c

. -

n o m i n a l f l i g h t c ' m d i t i o n : iro=6,=0. T h i s is t h e j u s t i f i c a t i o n f o r u s i r i g a "W" s u b s c r i p t t h r o u g h o u t t h i s r e p o r t f o r s t a b i l i t y a x i s v a r i a b l e s .

S t a b i l i t y a x e s s i m p l i f y c e r t a i n a s p e c t s o f t h e l i n e a r e q u a t i o n s . I n l e v e l n o m i n a l f l i g h t , s t a b i l i t y x- and y - a x e s a r e h o r i z o n t a l and t h e z - a x i s is v e r t i c a l . T h e l i f t and d r a g f o r c e s a c t a l o n g s t a b i l i t y a x e s . F i n a l l y , i t o f t e n is t r u e t h a t s t a b i i i t y a x e s a r e c l o s e t o t h e n o r m a l mode a x e s . T h i s means t h a t t h e b a s i c modes o f m o t i o n a p p e a r a s m o t i o n a b o u t o r a l o n g a s i n g l e a x i s o f t h e s t a - b i l i t y r e f e r e n c e f r a m e . The f r e q u e n c y a n d damping o f t h e b a s i c modes t h e n become s i m p l e f u n c t i o n s of t h e a e r o d y n a m i c s t a b i l i t y d e r i v a t i v e s e x p r e s s e d a l o n g s t a b i l i t y a x e s ; t h e r e - f o r e a e r o d y n a m i c s t a b i l i t y d e r i v a t i v e s e x p r e s s e d i n t h e s t a b i l i t y - a x i s f r a m e c a n be u s z d a s a p p r o x i m a t e i n d i c a t i o n s of s t a h i l i t y or i n s t a b i l i t y . F o r e x a m p l e , a s s u m i n g t h a t body a x e s c o i n c i d e w i t h p r i n c i p a l a x e s , t h e s t a b i l i t y - a x i s r o t a t i o n a l dynamic e q u a t i o n s a p p e a r i n p a r t a s : s i n a g C n B

A € + . . . ( A . 3-49)

z

I

I A ? , = -PV 2 0 2 Sb - I, [cosaoCnB-i; 'sins Cl A B + ...

( A . 3-50)

0 B I

where Appendix B d e t a i l s t h e d e f i n i t i o n of t h e i n d i v i d u a l symbols. The d e p a r t u r e p a r a m e t e r C n B P d y n ( S e c t i o n 2 . 2 ) is r e c o g n i z e d a s an e l e m e n t i n t h e yaw e q u a t i o n . T h i s is s i g - n l f i c a n t b e c a u s e t h e s e e q u a t i o n s a r e d e r i v e d by a s i m p l i f i c a t i o n o f t h e c o m p l e t e e q u a t i o n s , and i t is p o s s i b l e t h a t o t h e r u s e - f u l d e p a r t u r e p n r a m e t e r s c a n b e d e r i v e d from t h e same a p p r o a c h .

S i n c e s t a b i l i t y a x e s a r e b o d y - f i x e d a x e s w i t h a p a r - t i c u l a r nominal o r i e n t a t i o n , i t is c o n v e n i e n t t o d e r i v e t h e l i n e a r eqL t i ~ n s i n body a x e s and s i m p l y r o t a t e them t o o b t a i n a s t a b i l i t y - a x i s s e t . T h i s c a n be done by a p p l y i n g t h e t r a : i s f o r m a t i o n m a t r i x , Kg , t o t h e b o d y - a x i s s y s t e m m a t r i c e s , FB and GB, a s f o l l o w s : where

I 1 I . -

and, from Fig. 2.2-2, the wind-body transformation i s ( A . 3-54) while 0 s i n a

=[. 0 0

( A . 3-55) 0 -coaa and JvO is a diagonal matrix whose elements a r e ~ 1 . ~ ~ , ~ ~ c o s ~ ~ ] These r e l a t i o n s (which assume a,=Bo=O) a r e used t o transform body-axis equations t o s t a b i l i t y - a x i s equations f o r f u r t h e r a n a l y s i s .

A . 3 . 5 S t a t e Ordering and Dimension The brder of t h e s t a t e s given by E q . ( A . 2 - 3 0 ) is t h e one t h a t proceeds from considering t r a n s l a t i o n befo1.e r o t a t i o n and kinematics before dynamics. T h i s order does not, however, group r e l a t e d s t a t e s together. For example, many a i r c r a f t demonstrate a n a t u r a l mode t h a t is primarily composed of AU and b e o s c i l l a t i o n s ( t h e phugoid mode), and it is l o g i c a l t o regroup t h e s t a t e s so t h a t ~ u and h e fall next t o each o t h e r .

For some reference f l i g h t conditions, a further major division between longitudinal and l a t e r a l - d i r e c t i o n a l v a r i a b l e s can be made. The former v a r i a b l e s describe motion w i t h i * . t h e vehicle plane of symmetry, while t h e l a t e r a l - d i r e c t i o n a l v a r i a b l e s describe motion out of the plane of symmetry. T h i s division is useful because it allows a quick appraisal 01 t h e extent and nature of cross couplings t h a t a r i s e i n maneuvering f l i g h t .

With t h i s c o n d i t i o n i n mind, t h e o r d e r i n g of s t a t e s given i n Table A.3-1 is suggested f o r a i r c r a f t and s i m i l a r v e h i c l e s . The s i x l o n g i t u d i n a l s t a t e s a r e f i r s t , followed b y t h e s i x l a t e r a l - d i r e c t i o n a l s t a t e s .

TABLE A.3-1 STATE ORDERING Body Axes Ax1 I n any reduced-order approximation, t h i s s t a t e o r d e r - i n g a l s o is u s e f u l . The f o u r " o u t e r n o s t " s t a t e s , f o r example, do not a f f e c t t h e i n n e r e i g h t , but a r e merely i n t e g r a l f u n c t i o n s of them. Thus, AxI, , AyI and A $ can be removed without *=1 changing t h e b a s i c modes of motion of t h e v e h i c l e . The f i r s t two of t h e remaining e i g h t s t a t e s a r e t h e primary s t a t e s involved i n t h e phug3id mode, while t h e next two r e p r e s e n t t h e primary s h o r t p e r i c d l o n g i t u d i n a l mode. The f o u r l a t e r a l - d i r e c t i o n a l s t a t e s o f t e n e x h i b i t a Dutch r o l l o s c i l l a t o r y mode and r o l l and s p i r a l convergence modes.

Except i n s p e c i a l c a s e s , each of t h e s e modes i n v o l v e s most of t h e l a t e r a l - d i r e c t i o n a l s t a . t e s .

The f o r m a t i o n o f m o d e l s o f o r d e r l e s s t h a n e i g h t d e p e n d s on z n e x a m i n a t i o n of t h e i n d i v i d u a l p r o b l e m . F o r e x a m p l e , wher! t h e r e is n o c o u p l i n g b e t w e e n l a t e r a l - d i r e c - t i o n a l a n d l o n g i t u d i n a l m o d e s , t h e e i g h t h - o r d e r model c a n be s p l i t i n t o two i n d e p e n d e n t f o u r t h - o r d e r m o d e l s w i t h n o l&s o f a c c u r a c y . I f t h e t i m e - s p a n o f i n t e r e s t is s h o r t , i t may be p o s s i b l e t o n e g l e c t t h e slower modes ( p h u g o i d mode a n d s p i r a l c o n v e r g e n c e ) w i t h o u t d e c r e a s i n g t h e a c c u - r a c y o f t h e r e s u l t s ; h o w e v e r , a s t h e e i g e n u e c t o r s o f e a r l i e r c h a p t e r s show, o n e r u n s t h e r i s k o f m i s s i n g s i g n i f i c a n t c o u p l i n g e f f e c t s when " i n n e r e i g h t " s t a t e s a r e e l i m i n a t e d .

A d i f f e r e n t o r d e r r e d u c t i o n s u g g e s t s i t s e l f when t h e g e n e r a l i z e d t r i m p r o b l e m is e x a m i n e d . A s d i s c u s s e d i n S e c t i o n A . 3 . 2 , t h e g o a l o f t h e g e n e r a l i z e d t r i m p r o c e d u r e is t o f i n d a n o m i n a l f l i g h t c o n d i t i o n w i t h c o n s t a n t v e l o c i t y and a n g u l a r r a t e s . B e c a u s e t h e y d o n o t a f f e c t t h e v e l o c i t y a n d a n g u l a r r a t e s t a t e s , x I , y Z , zI and $ a r e d r o p p e d i m m e d i a t e l y . To c o m p l e t e l y c o n t r o l t h e s i x d e s i r e d s t a t e s , s i x c o n t r o l s are n e c e s s a r y , b u t m o s t a t m o s p h e r i c f l i g h t v e h i c l e s h a v e less t h a n s i x c o n t r o l e f f e c t o r s . N o t i n g t h a t t h e two E u l e r a n g l e s , 8 and $ , a r e i n v o l v e d p r i m a r i l y i n t h e s l o w modes, t h e s e t w o s t a t e s may be r e g a r d e d a s p a r a m e t e r s .

T h i s r e s u l t s i n a p r o b l e m o f f o u r c o n t r o i Y n r o t t l e , ele- v a t o r , a i l e r o n , r u d d e r ) , two p a r a m e t e r s ( Y , . ) , s i x s t a t e s ( u , v , w , p , q , r ) , a n d s i x s t a t e e q u a t i o n s ( < 1 , + , 6 , 6 , 6 , ? ) t o d e f i n e t r i m .

A . 4 TOOLS FOR LINEAF! ANALYSIS OF AIRCRAFT STABILITY AND CONTROL T h e p r e v i o u s s e c t i o n s o f t h i s c h a p t e r h a v e d e v e l o p e d t h e l i n e a r a i r c r a f t m o d e l a n d d e m o n s t r a t e d i t s v a l i d i t y a l o n g ~ i g h l y d y n a m i c t r a j e c t o r i e s . T h e u s e o f a l i n e a r model is d e s i r a b l e b e c a u s e t h e l a r g e b o d y o f t h e o r y a n d e x p e r i e n c e r e l a t i n g t o t h e a n a l y s i s a n d c o n t r o l o f l i n e a r s y s t e m s t h e n c a n be a p p l i e d t o t h e d e p a r t u r e p r e v e n t i c n p r o b l e m . T h e s e t o o l s a r e d i s c u s s e d i n t h i s s e c t i o n .

A . 4 . 1 E i g e n v a l u e s , E i g e n v e c t o r s , a n d N o r m a l Modes T h e i n i t i a l - c o n d j t i o ~ r e s p o n s e o f a 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 is c o m p o s e d o f a l i n e a r c o m b i n a t i o n o f a l i m i t e d n u m b e r o f n a t u r a l , o r n o r m a l m o d e s . E a c h n o r m a l mode is c h a r a c t e r i z e d b y its time s c a l e , g i v e r ? b y t h e e i g e n - v a l u e of t h a t m o d e , a n d t h e r e l a t i v e i n v o l v e m e n t o f e a c h s t a t e i n t h a t m o d e , i n d i c a t e d b y t h e e i g e n v e c t o r o f t h a t m o d e . I n p h y s i c a l l y r e a l i z a b l e s y s t e m s , t h e m o d e s a r e d e s c r i b e d e i t h e r b y i n d i v i d u a l r e a l e i g e n v a l l i e s o r b y p a i r s o f c o m p l e x e i g e n v a l u e s . A f i r s t - o r d e r mode e x h i b i t s e i t h e r a n e x p o n e n t i a l l y increasing r e s p o n s e ( p o s i t i v e e i g e n v a l u e ) or a n e x p o n e n t i a l l y d e c r e a s i n g r e s p o n s e ( n e g a t i v e e i g e n - v a l u e ) . A c o m p l e x ( s e c o n d - o r d e r ) mode o s c i l l a t e s a t a f r e q u e n c y d e t e r m i r l e d b y t h e i m a g i n a r y p a r t o f t h e e i g e n - v a l u e w i t h i n a n e x p o n e n t i a l e n v e l o p e d e t e r m i n e d b y t h e real p a r t o f t h e 2 i g e n v a l u e . T h e r e f o r e , t h e o s c i l l k t i o n c a n d i v e r g e , d o n v e r g e , o r m a i n t a i n c o n s t a n t a m p l i t u d e .

A s a n e x a m p l e o f t h e e i g e n v a l u e s i n v o l v e d i n t h e n o r m a l m o d e s of f i g h t e r n i r c r a f t , F i g . A . 4 - 1 i l l u s t r a t e s t h o a r e a s i n t h e c o m p l e x p l a n e ..vhich c o n t a i n t h e f i v e m o d e s anL e i g h t b a s i c e i g e n v a l i ~ e s o f a s m a l l h i g h - p e r f o r m a n c e .

Ik'AGlVARY PART OF EIGENVALUE

V, - 90 d l

QO - I S D E G

Co 5 0 DEG Figure A.4-1 Approximate Root Locations of a High-Performance Aircraft aircraft. Because complex eigenvalues occur in pairs of complex conjugates, the lower half of the complex plane is symmetric with the upper half and is not shown.

The eigenvectors indicate the relative involvement of the aircraft states in a given mode. Each of the modes shown in Fig. A.4-1 may involve the motion of every state, but the following generalizations can be made for straight- and-levelmflight. Tbe longitudinal states (60, Au, A q , Aw) exhibit two second-order modes. The phugoid mode is a slow, 1ightly.damped interchange of kinetic energy (speed) and potential energy (altitude) and primarily involves A0 and Au. The short period mode is the rapid, well-damped angular oscillation, and is exhibited primarily by Aq and Aw.

The l a t e r a l - d i r e c t i o n a l modes u s u a l l y c o n s i s t o f a s e c o n d - o r d e r a n d two f i r s t - o r d e r modes. T h e f o r m e r is c a l l e d t h e D u t c h r o l l mode a n d is a f a s t , p o o r l y damped yaw o s c i l l a - t i o n a b o u t t h e s t a b i l i t y z - a x i s . The r o l l c o n v e r g e n c e mode is a f a s t , s t a b l e mode, a n d it r e p r e s e n t s t h e a i r c r a f t r e s p o n s e ( g e n e r a l l y a b o u t t h e s t a b i l i t y x - a x i s ) t o a r o l l moment.

Due t o t h e a n g l e b e t w e e n body a n d s t a b i l i t y a x e s , t h e D u t c h r o l l mode a n d t h e r o l l c o n v e r g e n c e modes a p p e a r i n t h e t h r e e s t a t e s Av, A r , a n d Ap. The s p i r a l mode is s l o w a n d f r e q u e n t l y u n s t a b l e . An u n s t a b l e s p i r a l mode is i m p o r t a n t i n a p i l o t e d a i r c r a f t o n l y i f t h e t i m e c o n s t a n t is s o s h o r t t h a t t h e p i l o t h a s d i f f i c u l t y k e e p i n g t h e w i n g s l e v e l .

T h e s e mode s h a p e s c h a n g e c o n s i d e r a b l y a s t h e f l i g h t c o n d i t i o n v a r i e s from s t r a i g h t - a n d - l e v e l f l i g h t . Asymmetric f l i g h t c o n d i t i o n s r e s u l t I n c o u p l e d l 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 l m o d e s , and a n g u l a r r a t e s c a u s e l a r g e c h a n g e s i n t h e e i g e n v a l u e s . I n c e r t a i n c a s e s , modes c o m b i n e : t h e r o l l a n d s p i r a l modes c a n form a r o l l - s p i r a l o s c i l l a t i o n , f o r e x a m p l e .

The e i g e n v a l u e s o f t h e o u t e r f o u r s t a t e s ( A x I , h y I , and A$) a r e z e r o , i . e . , t h e s e s t a t e s h a v e n o e f f e c t on t h e o t h e r v a r i a b l e s a n d a r e Qure i n t e g r a t i o n s o f t h e o t h e r s t a t e v a r i a b l e s . A s a c o n s e q u e n c e , c o n t r o l - l o o p c l o s u r e s h a v e n o d i r e c t e f f e c t on t h e s e modes u n l e s s t h e o u t e r v a ~ i - a b l e s a r e f e d b a c k d i r e c t l y .

The e i g e n v a l u e s a r e t h e r o o t s o f t h e c h a r z c t e r i s t i c e q u a t i o n o f t h e s y s t e m d y n a m i c s m a t r i x , F , w h e r e I is t h e i d e n t i t y m a t r i x of t h e same o r d e r a s t h e s y s t e m m a t r i x , a n d 1 is a s c a l a r which must e q - l a 1 a n e i g e n - v a l u e f o r E q . (A.4-1) t o be s a t i s f i e d .

A s shown i n Ref. 7 1 , t h e r e is a s e t of vectors associated w i t h t h e eigenvalues which have special.proper- t i e s . The eigenvectors, z i , artb l i n e a r combinations of the elements of the s t a t e vector and a r e s o l u t i o n s t o t h e equa- t i o n s As i n t h e case of eigenvalues, t h e eigenvectors of a second- order mode appear a s complex conjugate p a i r s . The eigen- vectors contain the same information about t h e normal modes t h a t is given by the time vectors of c l a s s i c a l a i r c r a f t s t a - b i l i t y analysis (Ref. 6 5 ) .

The modal matrix is the matrix of eigenvectors arranged columnwise. The inverse of t h i s matrix transforn~s the s t a t e vector i n t o normal mode space, i n which each e l s - rnent of the vector, Ay, represents a normal mode of t h e sys- tem: (Instead of the two complex-conjugate eigenvectors of a second-order mode, I t may be useful t o use two r e a l vectors, one composed of the eigenvector r e a l p a r t and one the imag- inary p a r t . ) The l i n e a r equation 9.f motion, E q . (A.3-3) can be transformed a s well:

b i = M - ~ F M AZ + M-'G A U - (A.4-4)

The normal-mode system matrix, U-'FE.I, is composed of f i r s t - c r second-order diagonal blocks containing the system eigen- values, and the normal mode input matrix, M - ~ G , indicates which i n p u t s a f f e c t which normal modes. T h i s a l t e r n a t e form o f t h e s t a t e e q u a t i o n is u s e f u l b e c a u s e it d e m o n s t r a t e s a method of a n a l y z i n g normal mode e x c i t a t i o n . The e x c i t a t i o n d u e t o t h e s t a t e i n i t i a l c o n d i t i o n c a n be c a l c u l a t e d from E q . (A.4-3), w h i l e t h e e x c i t a t i o n d u e t o c o n t r o l i n p u t s is g i v e n by t h e normal mode i n p u t m a t r i x , M - ~ G .

T h i s s e c t i o n h a s d i s c u s s e d e i g e n v a l u e and e i g e n - v e c t o r c o n c e p t s . Examples o f t h e a p p l i c a t i o n o f t h e s e a n a . l y t i c a 1 t o o l s t o t h e a n a l y s i s o f l i n e a r i z e d a i r c r a f t models a r e c o n t a i n e d i n S e c t i o n s 2 . 4 and 2 . 5 .

A.4.2 C o n t r o l l a b i l i t y I n a m u l t i - i n p u t , m u l t i - o u t p u t s y s t e m , c e r t a i n normal modes may be u n a f f e c t e d by t h e s y s t e m c o n t r o l s w i t h - o u t t h i s b e i n g a p p a r e n t from t h e s y s t e m dynamics and i n p u t m a t r i c e s . T h i s c a n n o t o c c u r i n a n n t h - o r d e r 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 whose c o n t r o l l a b i l i t y t e s t m a t r i x , I', h a s f u l l r a n k : ( A . 4-5) The p r e s e n c e of c o n t r o l l a b i l i t y is n e c e s s a r y f o r t h e c o n s t r u c t i o n of a c o m p l e t e s y s t e m c o n t r o l l e r , and t h i s p r o p e r t y a l m o s t a l w a y s e x i s t s i n p h y s i c a l s y s t e m s of i n t e r - e s t . ( C o n t r o l l a b i l i t y t e s t s show t h a t t h e h i g h - p e r f o r m a a c e f i g h t e r i n v e s t i g a t e d i n t h i s r e p o r t is c o n t r o l l a b l e t h r o u g h - o u t t h e r a n g e o f f l i g h t c o n d i t i o n s . ) Of more i n t e r e s t is t h e i n v e s t i g a t i o n o f c o n t r o l e f f e c t i v e n e s s t h r o u g h o u t t h e f l i g h t r e g i m e . The d i f f i c u l t y is i n d e v i s i n g a s i m p l e m e a s u r e o f c o n t r o l e f f e c t i v e n e s s , b u t t h i s c a n b e overcome, t o some e x t e n t , by u s i n g t h e normal mode c o n t r o l i n p u t m a t r i x , 5 4 - ' 3 , which was i n t r o - duced a b o v e . The rows of t h i s m a t r i x i n d i c s i t e t h e r e l a t i v e i m p o r t a n c e of t h e a i r c r a f t ' s c o n t r o l s i n a f f e c t i n g e a c h o f t h e normal modes.

A.4.3 T r a n s f e r F u n c t i a n s S p e c i f i c i n p u t - o u t p u t r e l a t i o n s h i p s i n l i n e a r - t i m e - i n v a r i a n t dynamic s y s t e m s c a n b e d e s c r i b e d by t r a n s f e r func- tions, which are t y p i c a l l y g i v e n as r a t i o s o f p o l y n o m i a l s i n t h e Lapl.ace o p e r a t o r , s. The L a p l a c e t r a n s f o r m o f t h e o r - d i n a r y d i f f e r e n t i a l e q u a t i o n of motion (Eq. (A.3-3)) is

where I , F, and G have been d e f i n e d , and h x ( s ) - and Au(s)

-

are L a p l a c e t r a n s f o r m s of t h e s t a t e and c o n t r o l r e c t o r s , A t a A t . The i n p u t , A u ( s ) , - and t h e o u t p u t , Ax(s), are r e l a t e d by a t r a n s f e r f u n c t i o n m a t r i x , H ( s ) , which is o b t a i n e d when E q . (A.4-6) is p r e - m u l t i p l i e d by t h e i n v e r s e o f (81-F) : where H ( B ) = ( ~ I - P ; - ~ G ( A . 4-8) bay scalar t r a n s f e r f u n c t i o n of i n t e r t s t ( f o r example, t h e e f f e c t o f t h e ith c o n t r o l on t h e jth motion v a r i a b l e ) , can b e o b t a i n e d from E q . (A.4-8) by e v a l u a t i n g two d e t e r m i n a n t s d e r i v e d from t h e matrices o f Eq. (A.4-6) ( R e f . 7 2 ) , where g i j is an n by n matrix whose elements a r e zero, except f o r the jth column, which c o n ~ a i n s t h e ith column of t h e G matrix. T h i s t r a n s f e r function is a r a t i o of polynomials i n s, and the numerator and denominator can be factored t o iden- t i f y the poles, p, and zeros, z , which describe t h e r e l a t i o n - s h i p between A u i ( s ) and Axj(s): ( A . 4-10a) Alternatively, dividing by the individual poles and zeros, Eq. ( A . 4-10a) becomes The poles of t h e t r a n s f e r function a r e t h e r o o t s of t h e s y s - tem's c h a r a c t e r i s t i c equation, i . e . , they a r e the system's eigenvalues, and they a r e i d e n t i c a l f o r a l l t r a n s f e r func- t i o n s of the system described by F . The zeros depend G n G a s well a s F ; therefore, they vary from one t r a n s f e r function t o t h e next. The t r a n s f e r function g i ~ i n , KF, is t h e steady- s t a t e value of the t r a n s f e r function a f t e r a l l t r a n s i e n t s damp o u t , assuming t h a t a l l t r a n s i e n t s a r e s t a b l e . The trans- , is ( f o r most a i r c r a f t t r a n s f e r func- f e r function gain, K~ t i o n s ) t h e i n i t i a l s t a t e r a t e response t o a control s t e p .

The i n i t i a l value g a i n , K I , is inportant because i t determines the i n i t i a l slope of a given s t a t e v a r i a b l e ' s s t e p response. T h i s can be seen by applying the i n i t i a l value theorem (Ref. 11) t o the s t a t e v a r i a b l e ' s transform, which s t a t e s t h a t l i m x ( t ) = l i m s x ( s ) ( A . + l l ) / t + O s + - i f t h e l i m i t e x i s t s . The L a p l a c e t r a n s f o r m f o r t h e p e r t u r - b a t i o n s t a t e , Axi, g i v e n a u n i t y s t e p i n p u t i n c o n t r o l , , is g i v e n a s S i n c e t h e r e a r e more p o l e s t h a n zeros i n t h e t r a n s f e r f u n c - t i o n s o f i n t e r e s t , t h e i n i t i a l v a l u e o f A x i ( t ) is z e r o .

P h y s i c a l l y , t h i s is a n i n d i c a t i o n t h a t t h e v e h i c l e states do n o t c h a n g e i n s t a n t a n e o u s l y i n r e s p o n s e t o a c o n t r o l i n - p u t . The l o w e s t o r d e r n o n - z e r o s t a t e d e r i v a t i v e is e q u a l t o t h e e x c e s s o f p o l e s o v e r zeros. F o r nust a i r c r a f t t r a n s - f e r f u n c t i o n s o f i n t e r e s t , t h e e x c e s s 1s o n e , a n d T h i s leads t o t h e c a l c u l a t i o n o f t h e i n i t i a l v a l u e o f t h e s t a t e d e r i v a t i v e r e s p o n s e t o a u n i t y s t e p i n p u t as C o u p l e d a i r c r a f t t r a n s f e r f u n c t i o n s t y p ' : a l l y h a v e s e v e n z e r o s a n d e i g h t n o n - z e r o p o l e s , s o K I a n d KF a r e re- l a t e d a s f o l l o w s : A c o m p s . ~ i s o n o f t h e s i g n s o f K I a n d K F l w h l c h are r e l a t e d by t h e s i g n s of t h e p o l e s a n d zeros, a s i n d i c a t e d by E q . ( A . 4 - I S ) , is i m p o r t a n t b e c a u s e t h e s e s i g n s g i v e a n i n d i - c a t i o n o f t h e e x p e c t e d t r a n s i e n t r e s p o n s e . I f t h e t r a n s f e r f u n c t i o n is s t a b l e a n d minimum p h a s e , t h e s i g n s of K I a n d KF a r e t h e same. The r e s u l t i n g r e s p o n s e is s i m ' l a r t o t h e s o l i d l i n e i n F i g . 2.5-1. A nonminiaum-phase z e r o c a u s e s KI a n d K p t o h a v e o p p o s i t e s i g n s , a n d t h e r e s u l t i n g r e s p o n s e r e s e m b l e s o n e o f t h e d a s k e d l i n e s i n F i g . 2 . 5 - 1 . I n t h e s e r e s p o n s e s , t h e i n i t i a l r e s p o n s e d i r e c t i o n is away f r o m t h e d e s i r e d f i n a l v a l u e .

The t r a n s f e r f u n c t i o n h a s b e e n a f u n d a m e u t a l t o o l o f c o n t r o l s y s t e m d e s i g n i n t h e p a s t , a n d , a l t h o ~ g h l i n e a r - o p t i m a l c o n t r o l t h e o r y s e r v e s t h a t p u r p o s e i n t h i s r e p o r t ( C h a p t e r 4 ) , t r a n s f e r f u n c t i o n s c a n b e v z l u a b l e f o r u n d e r - s t a n d i n g d e t a i l s o f t h e a i r c r a f t ' s d y n a m i c s . F o r e x a m p l e , nonminimum-phase z e r o s a n d s i g n c h a n g e s i n KI c a n d e g r a d e h a n d l i n g q u a l i t i e s ( S e c t i o n 2 . 2 ) . When p o l e s a n d zeros are c e a r l y e q u a l , t h e r e is a c a n c e l l i n g e f f e c t which tellds t o remove t h e a s s o c i a t e d n o r m a l mdde f r o m t h e o u t p u t v a r i - a b l e ' s r e s p o n s e t o t h e g i v e n i n p u t . C o n v e r s e l y , f e e d b a c k p a t h s b e t w e e n t h e t r a n s f e r f u n c t i o n ' s o u t p u t a d . i n p u t h a v e n e g l i g i b l e e f f e c t on t h a t norma.1 mode. I n o t h e r w o r d s , t h e t r a n s P e r f u n c t i o n p r o v i d e s t h e i n f o r m a t i o n r e g a r d i n g t h e q u a l i t y o f c o n t r o l l a b i l i t y w h i c h was m i s s i n g i n E q . ( A . 4 - 5 ) .

T h e s e c a p a b i l i t i e s a r e p u t t o m e i n C h a p t e r 2 .

A . 4 . 4 O p t i m a l C o n t r o l T h e o r y A r e g u l a t o z is a f e e d b a c k c o n t r o l l a w w h i c h is d e s i g n e d t o m a i n t a i n a s y m p t o t i c a l l y s t a b l e o u t p u t o f a d y n a m i c s y s t e m , i . e . , i t b o u n d s t h e f l u c t u a t i o n s i n t h e o u t p u t , a n d i t a s s u r e s t h a t t h e o u t p u t g o e s t r ~ z e r o a s time increases. An optimal regulator minimizes a cos; (or penalty) functional of the output and control in stabilizing the dynamic system. A linear-optimal regulator minimizes

a particular cost function -- the time integral of quadratic

functions of the output and controi -- for a linoar dynamic

system, and it takes the form of Eq. (4.1-1) (Hef. 5 0 ) . A linear-optimal regulator can be designed for an aircraft near, at, or beyond its open-loop departure boundary. This dbsign indicates the control loops which must be closed (either automatically or by tho pilot) to prevent departure, ,Jroviding asymptotic stability and minimizing a quadratic cost functional of the output and control.

The basic design objective for the linear-optimal regulator is to define the feedback contrc-1 law which mini- mizes a quadratic cost functional, J , of the perturbation output vector, Ay(t), and the perturbation co-~timl vector,

Au(t): -

The control vector contains all available aircraft control

displacements -- in this case, throttle setting (AtT), ele-

vator (5Clh), aileron (Ada), and rudder (A6r).

The output w vector represents the measured aircraft variables and car be I formulated as a linear combination of the ajrcraft's pertur- bation states, hx(t), - state rates, ~;(t j , and controls, A_u(t):

-

the present development uses the simplifying assumption,

Ay(t) A z ( t ) , where

I

The state-weighting mrtrix is nonnegative-definite and symmetric, a n d t h e c o n t r o l - w s i g h t i n g m a t r i x is p o s i t i v e - d e f i n i t e a n d s y m m e t r i c : ( A . 4 - 1 9 ) E q u a t i o ( A . 4 - 1 5 ) c a n b e w r i t t e n as a n d t h e cost f u n c t i o n a l is s e e n t o b e a w e i g h t e d sum o f t h e i n t e b z a t e d - s q u a r e v a l u e s o f .:he p e r t u r b a t i o n s t a t e a n d c o n t r o l .

I n t h e p r e s a n t c a s e , m i n i m i z i n g t h e w e i g h t e d svm o f i n t e g r a t e d - s q u a r e v a l u e s is e q u i v a l e n t L O m i n i m i z i n g t h e w e i g h t e d sum o f r o o t - m e a n - s q u a r e - ( r m s : v a l u e s o f t h e s t a t e a n d ~ o ~ t r o l .

E q u a t i o ~ . ( A . 4 - 2 0 ) p;"ovides a meaqs o f t r a d i n g o f f t h e - c o s t of o u t p u t e r r o r s a g a i n s t t h e c o s t o f c o n t r o l , a n d it is s i m p l y t h i s : c h o o s e e a c h w e : g h t i n g c o e f f i c i e n t i n Q and R as t h e i n v e r s e o f t h e maximum a l l o w a b l e rnea.c-square v a l u e o f t h e w e i g h t e d v a r i a b l e , i . e . , q i i = l / A x i , i = l t o n msx T h i s n o r m a l i z e s e a c h term i n Eq. (A.4--201, s..: t h a t x r s c o n t r i - b u t i o n t o t h e i n t e g r a i i d is u n i t y when t h e v a r i a b l e e q u a l s its maximum v a l u e . The e l e m e n t s o f R are s p e c i f i e d by t h e c o n - t r o l a u t h o r i t y w h i c h c a n be a s s i g n e d t o t h e DPS.1S. F o r e x a m p l e , i f 1 0 d o g o f e l e v a t o r p e r t u r b a t i o n c a n be a s s i g n e d t o d e p a r t u r e p r e v e n t i o n , t h e c o r r e s p o n d i n g elemerg o f R is r) l / ( l ~ ) ~ = 0 . 0 1 . I f t h e r e is p r i o r i n f o r m a t i o n r e g a r d i n g a l l o w a b l e s t a t e p e r t u r b a t i o n s ( a s i n a t r a c k i n g t a s ~ ) , t h e e l e m e n t s of Q a r e d e t e r m i n e d s i m i l a r l y .

An a l t e r n a t e a p p r o a c h is t o u s e t h e e l e m e n t s o f Q a n d R a s d e s i g n p a r a m e t e r s w h i c h c a n b e v a r i e d u n t i l d e s i r a b l e t r a n s i e n t r e s p o n s e or e i g e n v a l u e s a r e a c h i e v e d . I n s u c h c a s e , t h e e q u i v a l e n c e o f Q e l e m e n t s t o a l l o w a b l e m e a n - s q u a r e v a l u e s is n o t l o s t , and it is p o s s i b l e t o g a i n i n s i g h t r e g a r d i n g t h e c o r r e s p o n d e n c e o f r m s - o u t p u t e r r o r s a n d c l a s s i c a l f i g u r e s o f merit i n e a c h p a r t i c u l a r c a s e .

The m i n i m i z a t i o n o f J m u s t be a c c o n ~ p l i s h e d s u b j e c t t o t h e d y n a m i c c o n s t r a i n t p r o v i d e d by t h e l i n e a r e q u a t i o n o f m o t i o n , ( I t is assumed t h a t F a n d G f o r m a c o n t r o l l a h l e p a i r . ) The m e t h o d o f f i n d i n g t h e c o n t r o l w h i c h m i n i m i z e s J s u b j e c t t o E q . ( A . 4 - 2 3 ) is d e r i v e d i n numerous t e x t s ( e . g . , R e f s . 60 t o 6 2 ) . I n t h e s p e c i a l c a s e o f q u a d r a t i c cost a n d l i n e a r s y s - tem dyr,amics, t h e c o n t r o l s o l u t i o n is a l i n e a r f e e d b a c k l a w ( E q . ( 4 . 1 - 1 ) ) The g a i n m a t r i x o f t h i s c o n t r o l l a w is ( A . 4-24) w h e r e t h e s y m m e t r i c m a t r i x , P , is t h e s t e a d y - s t s t e s o l u t i o n o f t h e m a t r i x R i c c a t i e q u a t i o n I n o t h t r w o r d s , t h e DPSAS g a i n m a t r i x is e a s i l y f o u n d b y two mat. i x m u l t i p 1 i c a t i o : l s o n c e P is f o u n d ( E q . ( A . 4 - 2 4 ) , b u t t h e s o l u t i o n f o r P ( E q . ( A . 4 - 2 5 ) ) a p p e a r s f o r m i d a b l e . T h e r e a r e , h o w e v e r , f o u r r e c o g n i z e d m e t h o d s f o r s t e a d y - s t a t e s o l u t i o n o f E q . ( A . 4 - I s ) ) , a l l o f w h i c h r e q u i r e d i g i t a l c o m p u t a t i o n ( R e f . 6 0 ) : t h e s e a r e d i r e c t i n t e g r a t i o n o f E q . ( A . 4 - 2 5 1 , t h e Newton-Raphson m e t h o d , t h e Kalman-Englar m e t h o d , a n d t h e diagonalization/eigenvalue method. The c h o i c e b e t w e e n t h e s e m e t h o d s must b e b a s e d o n g r o u n d s o f n u m e r i c a l c o n v e n i e n c e a n d e f f i c i e n c y .

The Kalman-Englar method ( R e f . 7 3 ) h a s b e e n u s e d t o g e n e r a t e t h e r e s u l t s which f o l l o w i n l a t e r s e c t i o n s . I n t h i s t e c h n i q u e , P is p r o p a g a t e d t o s t e a d y s t a t e u s i n g t h e r e c u r s i v e e q u a t i o n , T h e m a t r i c e s 0 11, F12, O Z L , a n d 022 a r e t n e a p p r o p r i a t e ( n x n ) s u b - m a t r i c e s o f -ZA t O(At) = e ( A . 4-27) where and t h e p r o p a g a t i o n i n t e r v a l , A t , i s s m a l l compared t o t h e n a t u r a l p e r i o d s of t h e a i r c r a f t m o t i o n .

The DPSAS d e s i g n p r o c e d u r e is summarized and shown t o be a s t r a i g h t f o r w a r d t e c h n i q u e o n c e t h e a p p r o p r i a t e g e n e r a l - p u r p o s e computer r o u t i n e s a r e programmed: D e f i n e c o n t r o l a u t h o r i t y a v a i l a b l e t o t h e DPSAS, t h u s s p e c i f y i n g R .

o D e f i n e a l l o w a b l e s t a t e p e r t u r b a - t i o n s , t h e r e b y s p e c i f y i n g Q .

F o r t h e a i r c r a f t dynamics s p e c i f i e d ?y t h e s t a b i l i t y and c o n t r o l m a t r i c e s , F and G , compute t h e f e e d b a c k g a i n m a t r i x , K , u s i n g E q . ( A . 4 - 2 b ) . ( A . 4 - 2 7 ) , ( A . 4 - 2 6 ) , and ( A . 4 - 2 4 ) .

0 The c o n t r o l l a w f o r t h e DPSAS is g i v e n by E q . ( 4 . 1 - I ) , u s i n g t h e g a i n m a t r i x c a l c u l a t e d i n t h e p r e v i o u s s t e p .

The r e s u l t i n g CPSAS s t a b i l i z e s t h e a i r c r a f t w i t h o u t u s i n g mroe c o n t r o l a u t h o r i t y t h a n t h a t s p e c i f i e d by R f o r s t a t e p e r t u r b a t i o n s d e f i n e d by Q .

A.4.5 Gain S - d u r e -

Means and S i a n d a r d D e v i a t i o n s - Two f e a t u r e s which

s u g g e s t t h a t a g a i n be h e l d c o n s t a n t a r e i t s s t a n d a r d d e v i - a t i o n and mean v a l u e . C e r t a i n g a i n s do n o t e x h i b i t w i d e v a r i a t i o n s a s t h e f l i g h t c o n d i t i o n s c h a n g e . T h i s c a n b e d e t e r m i n e d by c o n s t r u c t i n g a t a b l e of means and s t a n d a r d d e v i a t i o n s f o r t h e g a i n s , as i l l u s t r a t e d by T a b l e A . 4 - 1 .

I n t h e t a b l e , Gain 6 d i s p l a y s a low s t a n d a r d d e v i a t i o n and a l a r g e mean v a l u e . T h i s i n d i c a t e s t h a t t h e g a i n s h o u l d p r o b a b l y n o t b e s c h e d u l e d , i . e . , t h a t its v d n v a l u e c a n b e u s e d a t a l l f l i g h t c o n d i t i o n s .

Another f e a t u r e of each gain is its r e l a t i v e map- nitude compared t o o t h e r gains of a s i m i l a r c l a s s . Gain 4 is small compared t o Gain 6 and a l s o e x h i b i t s a wide varia- t i o n i n magnitude; t h u s it may be d e s i r a b l e t o schedule G a i n 4 , i f its v a r i a t i o n with f l i g h t condition is coherent, o r s e t it t o zero. Simulations should be done t o f u l l y TABLE A.4-I.

EXAXPLE OF MEAN-STA*XDARD U i i V I A T I O N TABLE Standard S.D. Per Cent Mean Deviation of Mean Gain 4 Gain 5 Gain 6 e t c .

determine the zeroed g a i n s ' e f f e c t s on c o n t r o l l i n g t h e a i r - c r a f x . Gain 5 is a l o g i c a l candidate f o r scheduling. The gain magnitude is not n e g l i g i b l e , and it d i s p l a y s enough v a r i a t i o n t o warrant scheduling.

Correlation Between Gains and F l i g h t Veriables -

The a i r c r a f t dynamic model v a r i e s i n a complex but d e t e r - ministic way w i t h f l i g h t conditions, If t h e closed-loop response of t h e a i r c r a f t is maintained e s s e n t i a l l y i n v a r i a n t by automatic c o n t r o l , it is reasonable t o assume t h a t t h e necessary c o n t r o l gains a l s o vary i n a ccmplex but d e t e r - ministic way w i t h f l i g h t conditions; hence the gains and f l i g h t conditions should be c o r r e l a t e d , "he search f w gain!flight v a r i a b l e dependencies begips by determining correlarion c o e f f i c i e n t s between the g e i n s a n d a l l a v a i l a b l e f l i g h t v a r i a b l e s . One m e t h o d o f d e t e r m i n i n g t h e c o r r e l a t i o n c o e f f i c i e n t b e t w e e n a set o f g a i n s ( d e p e n d e n t v a r i a b l e s ) , k i . a n d a f l i g h t v a r i a b l e ( i n d e p e n d e n t v a r i a b l e ) , m , is g i v e n by t h e f o l l o w i n g : I n E q . ( A . 4 - 2 9 ) , I i s t h e number o f f l i g h t c o n d i t i o n s f o r w h i c h t h e g a i n s a r e known, k i is t h e v a l u e o f t h e g a i n o b s e r v e d a t f l i g h t . o i n t i , a n d mi is t h e v a l u e o f t h e f l i g h t v a r i a b l e a t f l i g h t p o i n t i . The v a r i a b l e is t h e mean v a l u e o f t h e g a i n , g i v e n by ( A . 4-30) a n d 6 is t h e mean vzlue o f t h e f l i g h t v a r i a b l e . The c l o s e r t h e m a g n i t u d e o f p i s t o o n e , t h e b e t t e r t h e c o r r e l a t i o n b e t w e e n t h e g a i n a n d t h e f l i g h t c a r t a b l e . I n d e p e n d e n t v a r i a b l e s w h i c h c a n be c o n s i d e r e d . f o r g a i n s c h e d u l i n ~ i n c l u d e i n d i c a t e d a i r s p e e d ( I A S ) , b o d y - a x i s v e l o c i t i e s ( u , v , w ) a n g l e s o f a t t a c k a n d s i d e s l i p ( a , B ) , a n g u l a r rates ( p . q , r ) , a n d c o n t r o l t r i m p o s i t i o n s . T h e s e v a r i a b l e s c a n b e i n v e r t e d , s q u a r e d , a n d s o o n , i n t h e s e a r c h f o r h i g h c o r r e - l a t i o n . An e x a m p l e of a c o r r e l a t i o n c o e f f i c i e n t t a ~ l e is shown i n Ta.ble A.4-2. T:,e c i r c l e d v a l u e s a r e t h e h i g h c o r r e - l a t i o n c o e f f i c i e n t s bet-:,@en g a i n s a n d i n d e p e n d e n t v a r i a b l e f u n c t i o n s .

TABLE A.4-2 EXAMPLE OF CORRELATION COEFFICIENTS TABLE u a l / a IAS I A S ~ p 9 r ... e t c .

.804 .018 .0359 Gain 2 Gain 3 e t c .

Curve F i t t i n g - The t h i r d s t e p i n t h e gain-scheduling

procedure is t o c o n s t r u c t a smooth r e l a t i o n s h i p between t h e g a i n s which a r e t o be scheduled and t h e most h i g h l y c o r r e - l a t e d f l i g h t v a r i a b l e s . M u l t i p l e r e g r e s s i o n and polynomial r e g r e s s i o n , can be used. Equation (A.4-31), t h e m u l t i p l e r e g r e s s i o n , u s e s n d i f f e r e n t independent v a r i a b l e s , w h i l e Eq. (A.4-32)) t h e polynomial r e g r e s s i o n , u s e s powers of t h e h i g h e s t c o r - r e l a t e d independent f l i g h t v a r i a b l e , mi, t o e s t i m a t e t h e g a i n , k. A method f o r determining t h e r e g r e s s i o n c o e f f i - c i e n t s , bi, is shown n e x t .

A m u l t i p l e r e g r e s s i o n a n a l y s i s d e t e r m i n e s t h e r e g r e s s i o n c o e f f i c i e n t s b o p b l , . . . , b n i n Eq. (A.4-31) s o t h a t t h e sum of t h e squared e r r o r between t h e r e g r e s s i o n

e s t i m a t e , 2 , and t h e t l u e v a l u e of k is minimized. For

I f l i g h t c o n d i t i o n s , t h e f u n c t i o n t o be minimized is To m i n i m i z e J , set Then t h e e x p r e s s i o n f o r t h e r e g r e s s i o n c o e f f i c i e n t s b e c o m e s ( A . 4-35)

A b = C -

- w h e r e A a n d - C are d e f i n e d a s i n t h e l e a s t - s q u a r e s f o r m u l a s of R e f . 63.

T h e s e a r e t h e v a l u e s o f bi w h i c h m i n i m i z e t h e s q u a r e d error.

To d e t e r m i n e j u s t how g o o d t h e b i v a l u e s a r e , t h e c o r r e l a t i o n c o e f f i c i e n t f o r t h e m u l t i p l e r e g r e s s i o n f i t c a n be f o u n d , a s i n E q . (A.4-Og): ( A . 4-37) T h e c l o s e r p is t o o n e , t h e b e t t e r t h e f i t of t h e m u l t i p l e r e g r e s s i o n m o d e l . When u s i n g a m u l t i p l e r e g r e s s i o n , t h e m o r e i n d e p e n d e n t v a r i a b l ~ s c h o s e n , t h e h i g h e r t h e v a l u e o f p w i l l b e , u n t i l n = I a n d p = l .

An a l t e r n a t e way of e s t i m a t i n g gain v a l u e s is t o use only one of t h e f l i g h t v a r i a b l e s i n a polynomial r e g r e s - s i o n . A n nth-order polynomial r e g r e s s i o n a n a l y s i s d e t e r - mines t h e r e g r e s s i o n c o e f f i c i e n t s , bo, b l , . . . , b n i n E q .

(A.4-32) s o t h a t t h e sum of t h e squared e r r o r between t h e A r e g r e s s i o n e s t i m a t e , k , o f E q . (A.4-31) and t h e t r u e v a l u e of k is minimum. In t h i s c a s e , t h e r e is only ona kind of independent v a r i a . . l e , m i , f o r each g a i n v a l u e , but it is z r a i s e d t o v a r i o u s powers, i . e . , m l = m m2 = m u n t i l adequate i ' i ' c o r r e l a t i o n is achieved.

The a n a l y s i s f o r t h e polynomial r e g r e s s i o n pro- ceeds a s i n t h e m u l t i 7 l e r e g r e s s i o n , s t a r t i n g a t Eq.

(A.4-33). The polynomial r e g r e s s i o n can be considered a s p e c i a l case of t h e m u l t i p l e r e g r e s s i o n .

Program ALPHA

-

The c o n s t r u c t i o n of t h e complete l i n e a r e q u a t i o n s of motion, '-heir a n a l y s i s , and t h e design of feedback con-

t r o l l e r s has been programmed i n ALPHA--Analysis - - - Program

Figure f o r - High - Angle-of-Attack S t a b i l i t y and Control.

4 . 4 - 2 i l l u s t r a t e s t h e s t r u c t u r e of t h i s program. Input d a t a consist:^ of a i r c r a f t i n e r t i a l and aerodynamic char- a c t e r i s t i c s . The aerodynamic d a t a can be e n t e r e d a s con- v e n t i o n a l s t a b i l i t y d e r i v a t i v e s , dimensionless d e r i v a t i v e s , o r f u l l t a b l e s of nonliiiear f o r c e and moriient c h a r a c t e r i s t i c s .

CONTROL RUNlCASEIPrSS SETUP

STABILITY AUGMENTATION 1 PFFEClS ANALYSIS 1

v DATA I N W T CDNTROLL4BILITY n A N D A R D W WEEP COLITROL . M O T LOCUS.

RETURN m I N T OPTIMAL REGULATOR TIIANSLEI) IUNCTIOUS TRIM OPTIONS

I I rn LIMIT €xCL€OANCLS

u

-- I 9 L L HOOEL MATRICES DEPARTURE PARAMETERS

. TABLE I N M

' %om. %u nmww D ~ V A T I V E S LCOI LINEAR SYSTEM MATRICU d r r U S & IW A*. o* Qdui

C +

TRANSFORMED AND LTI STABILIT) S9iTERIA REDUCED-ORDER MODELS LIGLNVALUES BTUILITV AXIS YOOIL 1. w,. r I1 bh O r r l

. LlGENvL.'IOIW

J I

Figure A.4-2 ALPHA - Analysis Program for High

~ n ~ l e - o f z ~ t t a c k ~tabili ty and-control

-

The program executes a three-step procedure. The first step consists of steady or generalized trim calcula- tion, if desired, and the construction af t\e camplete body- axis system dynamics and control input matrices. During the second step, the linear system is modified, if required, to include any axis transformation, order reduction,or fixed stability augmentation loop closure. The final step consists of the analysis of the resulting system. Eigenvalues, eigen- vectors, transfer functions, linear-optimal stability aug- mentation systems, and time histories can be calculated and plotted.

The executive structure of ALPHA includes logic t o modify the dynamic model on succeeding passes through the program and t o vary the analysis type as cei-tain param- e t e r s are varied over a given range of i n t e r e s t . Program ALPHA provides an e f f i c i e n t t o o l for the thorough analysis of aircraft high angle-of-attack s t a b i l i t y and control.

APPENDIX B AIRCRAFT AERODYNAMIC MODEL The r e f e r e n c e a i r c r a f t is a s m a l l , s u p e r s o n i c f i g h t e r type d e s i g n e d f o r a i r s u p e r i o ~ ~ + y m i s s i o n s . Mass, d i m e n s i o n a l , and i n e r t i a l c h a r a c t e r i s t i c s are l i s t e d i n T a b l e B - 1 . The aerodynamic d a t a set is a c o m p o s i t e of s u b - s c a l e wind t c n n e l measurements f o r t w o c o n f i g u r a t i o n s of t h e r e f e r e n c e a i r c r a f t ; h e n c e , t h e n u m e r i c a l r e s u l t s p r e s e n t e d h e r e d o n o t r e p r e s e n t a s p e c i f i c a i r c r a f t i n d e t a i l .

TABLE B-1 CHARACTERISTICS OF THE REFERENCE AIRCRAFT Mass, kg I,, kg-m2 1x2, kg-m R e f e r e n c e A r e a , m Mean A e r o d y ~ a m i c Chord, m( E ) 2 . 4 6 wing S p a n , m(b) L e n g t h , m R e f e r e n c e C e n t e r o f G r a v i t y ( c . g . ) 0.25: light c . g . 0.17E: The c o n t r o l v a r i a b l e s a r e e l e v a t o r ( o r h o r i ~ o n t a l t a i l , 6 h ) , l e a d i n g l t r a i l i n g edge f l a p s ( 6 f ) , a i l e r o n s ( h a ) , rudder ( d r ) , speed brake ( 6 S B ) , a n d ' m u s t s e r t i n g ( d T ) .

The ranges of t h e s e v a r i a b l e s a r e l i s t e d i n Table B-2.

TABLE B-2 CONTROL VARIABLE RANGES -20 t o +5 deg 6h 0 t o 130 p e r c e n t & f -60 t o +60 deg 6a -30 t o +30 deg 6r 0 t o 45 deg SB 0 t o 100 p e r c e n t &T The aerodynamics of t h z a i r c r a f t a r e r e p r e s e n t e d by 45 coe.:icients which a r e f u n c t i o n s of a c g l e of a t t a c k , s i d e s l i p a n g l e , e l e v a t o r d e f l e c t i o n , and f l a p s e t t i n g .

Using s i m p l i i i e d c o n v e n t i o n a l n o t a t i o n , t h e 6 t c t a l c o e f f i - c i e n t s a r e d e s c r i b e d a s f o l l o w s : The first step in f i n d i n g the individual terms of the perturbat::* forces and moments, Eqs. ( A . 3 - 1 3 ) to ( A . 3 - 1 6 ) , is to evaluate the d e r i v a t i v ~ s o f the f n r c e a.nd moment co- efficjents with respect t o the n ~ n d i r n e n s i ~ : ~ ~ a l states at the nominal flight condition. This r ~ s l ~ l t s in nondimensional stability derivatives, such as t h ~ following: Many of these derivatives contain t h e partial deriva- tives of the nondirnensional wind-axis translational velocities (V/VO, 0 , a) respect t o the nondimensional body-axis translational velocities (u/Vo, v/Vo, w/Vo). This matrix of derivatives, evaluated at the ~ o m i n a l flight condition, is cos a. c o s P O s i n B O sin aO c o s B O T a c v / v o , 0 , a ) -COS aO s i n B O c o s B O -sin a. s i n B O T 3(a/v0, v/vO, w / v O ) -sin ao/cos B O 0 c o s ao/cos B o

I

The dimensional s t a b i l i t y d e r i v a t i v e s a r e formed by t a k i n g t h e d e r i v a t i v e s of t h e dimensional aerodynamic f o r c e s and moments w i t h r e s p e c t t o t h e dimensional s t a t e v a r i a b l e s . These dimensional d e r i v a t i v e s c o n t a i n t h e non- a F ~ dimensional d e r i v a t i v z s ; and a F ~ a r e examples of t h e s e

aa

d e r i v a t i v e s : The complete dimensional s t a b i l i t y d e r i v a t i v e m a t r i c e s a r e These stability derivative matrices are used in Eqs. ( A . 3 - 1 3 ) to ( A . 3 - 1 6 ) to determine the perturbation forces and moments, which themselves determine the aerodynamic terms in the linear perturbation equations of motion.

REFERENCES Lanchester, F.W., Aerodonetics, A. Constable, London, 1908.

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December 1970, pp. 567-569.

McConnell, C.W., "Application of the Lateral and Longi- i tudinal (Coupled) Equations of Motion to an Inertially- Slender Aircraft in Unsymmetric, Rectilinear Flight," M.S. Thesis, Air Force Institute of Technology, December 1973.

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Stengel, R.F., "Effect of Combined Roll Rate and Side- slip Angle on Aircraft Flight Stability," Journal of Aircraft, Vol. 12, No. 8, August 1975, pp. 683-685.

REFERENCES ( C o n t i n u e d l 11. G r e e r , H . D . , "Summary o f Directional D i v e r g e n c e C h a r a c - t e r i s t i c s o f S e v e r a l High-Performance A i r c r a f t Con- f i g u r a t i o n s , " NASA T N D-6993, W a s h i n g t o n , November1972.

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15. R o s s , A . J . , " I n v e s t i g a t i o n o f N o n l i n e a r Motion Exper- i e n c e d on a Slender-Wing R e s e a r c h A i r c r a f t , " J o u r n a l of A i r c r a f t , V o l . 9 , No. 9 , September 1 9 7 2 , pp. 625-631.

1 6 . Haddad, E . K . , "Study o f S t a b i l i t y o f L a r g e Maneuvers of A i r p l a n e s , " NASA CR-2447, W a s h i n g t o n , August 1974.

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2 0 . G r a f t o n , S . B . , "A S t u d y t o D e t e r m i n e E f f e c t s o f Applying T h r u s t on Recovery from I n c i p i e n t a n d Developed S p i n s f o r Four A i r p l a n e C o n f i g u r a t i o n s , " Washington, J u n e 1 9 6 6 .

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Moore, F . L . , A n g l i n , E.L., Adams, M.S., D e a l , P.O.

and P e r s o n , L . H . , J r . , " U t i l i z a t i o n o f a Fixed-Base S i m u l a t o r t o S t u d y t h e S t b l l and S p i n C h a r a c t e r i s t i c s of F i g h t e r A i r p l a n e s , " NASA T N D-6117, W a s h i n g t o n , March 1 9 7 1 .

Champoux, R . L . , "The N u m e r i c a l S o l u t i o n A n a l y s i s of A i r p l a n e S p i n E q u a t i o n s Modeled i n a F i x e d C o o r d i - nate S y s t e m , " M.S. T h e s i s , Naval P o s t g r a d u a t e S c h o o l , December 1 9 7 2 .

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W i l l e n , T.B. and J o h n s o n , K . , " A n t i c i p a t e d S p i n Sus- c e p t i b i l i t y C h a r a c t e r i s t i c s of t h e A-10 A i r c r a f t , " A I A A P a p e r No. 75-33, N e w Y o r k , J a n u a r y 1 9 7 5 .

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J o h n s o n , A.E., "A L i t e r a t u r e S u r v e y o f t h e Problem o f A i r c r a f t S p i n s , " M.S. T h e s i s , Naval P o s t g r a d u a t e S c h o o l , S e p t e m b e r 1 9 7 1 .

McRuer, D . T . , and J o h n s t o n , D . E . , " F l i g h t C o n t r o l S y s t e m s P r o p e r t i e s and P r o b l e m s , Volume I , " NASA CR-2500, Washington, F e b r u a r y 1 9 7 5 .

Hawkins, M.L., "An I n v e s t i g a t i o n o f t h e D e p a r t u r e Modes of a F-4D A i r c r a f t from a S t e a d y S i d e s l i p F l i g h t C o n d i t i o n , " M.S. T h e s i s , A i r F o r c e ? ~ l s t i t u t e of T e c h n o l o g y , December 1 9 7 4 .

REFERENCES (Continued) Rudolph, R.J., "Stability Analysis of an F-4C Air- craft in Steady Level Turning Flight," M . S . Thesis, Air Force Institute of Technology, December 1974.

I Chambers, J.R. and Anglin, E.L., "Analysis of Lateral- Directional Stability Characteristics of a Twin-Jet Fighter Airplane at High Angles of Attack," NASA TN D-5361, Washington, August 1969.

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Coe, P.L., Jr., Chambers, J.R., and Letko, W., "Asymmetric Lateral-Directicnal Characteristics of Pointed Bodies of Revolution at High Angles of Attack," NASA TN D-7095, Washington, November 1972.

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Grafton, S.B., Chambers, J.R., and Coe, P.L., Jr., "Wind-Tunnel Free- Flight Investigation of a Model of a Spin-Resistant Fighter Configuration," NASA TN D-7716, Washington, June 1974.

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Coe, P.L., Jr. and Newsor., W.A., Jr., "Wind-Tunnel Investigation to Determine the Low-Speed Yawing Sta- bility Derivatives of a Twin-Jet Fighter Model at High Angles of Attack," NASA TN D-7721, Washington, August 1974, Hwang, C. and Pi, W.S., "Transonic Buffet Behavior of Northrop F-5k Aircraft," AGARD Report 624, Neuilly- sur-Seine, September 1974.

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44. Kao, H.C.. "Side Forces on Unyawed Slender Inclined ~ e r o d ~ n a m i c Bodies, " Journal of Aircraft, Vol . 12, No. 3, March 1975, pp. 142-150.

45. Tobak, M . and Schiff, L.B., "Generalized Formulation of Nonlinear Pitch-Yaw-Roll Coupling: Part I-Non- axisymmetric Bodies and Part 11-Nonlinear Coning- Rate Dependence," AIAA Journal, Vol. 13, No. 2 , March 1975, pp. 323-332.

46. Nguyen, L.T., "Evaluation of Importance of Lateral Acceleration Derivatives in Extraction of Lateral- Directional Derivatives at High Angles of Attack," NASA TN D-7739, Washington, October 1974; 47. Neihouse, A.I., Klinar, W.J. and Scher, S.H., "Status of Spin Research for Recent. Airplane Designs," NASA TR R-57, Washington, 1960.

48. Fuchs, R. and Schmidt, W., "The Dangerous Flat Spin and the Factors Affecting It," NACA TM 629, Washington, 1931.

49. Young, J.W., "Optimal and Suboptimal Control Technique for Aircraft Spin Recovery," NASA TN D-7714, Washington, October 1974.

50. Gilbert, W.P. and Libbey, C.E., "Investigation of an Automatic Spin-Prevent,ing System for Fighter Air- planes," NASA TN D-6670, Washington, March 1972.

51. Chen, R.T., Newell, F.D., and Schelhorn, A.E., "Develop- ment and Evaluation of an Automatic Departure Prevention System and Stall Inhibitor for Fighter Aircraft," AFFDL TR-73-29, Wright-Patterson AFB, April 1973.

52. Anderson, C , A , and Walker, J. E. , II I , "A Stall Inhi- bitcr for the,F-Ill," NAECON'72 Record, May 1972.

53. Lamars, J.P., "Design for Departure Prevention in the YF-16," AIAA Paper No. 74-794, New York, August 1974.

REFERENCES (Continuedl McDonald, E.H., "F-15 Eagle Flight Control Systemtt in Advances in Control systems, AGARD CP-137, Neuilly-sur-Seine, May 1974.

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Nguyen, L.T., Anglin, E.L., and Gilbert, W.P., "Recent Research Related to Prediction of Stall/Spin Characteristics of Fighter Aircraft," Proceedings of the AIAA 3rd Atmospheric Flight Mechanics Conference, Arlington, Texas, June 1976, pp. 79-91.

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Stengel, R.F., Broussard, J.R. and Berry, P.W., "The Design of Digital-Adaptive Controllers for VTOL Air- craft," The Analytic Sciences Corp., TR-640-1, NASA CR-144912, March 1976.

Kaufman, H., Alag, G., Berry, P. and Kotob, S., "Digital Ad~ptive Flight Controller Development," NASA CR-2466, Washington, December 1974.

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Gelb, A. ed. , Applied Optimal Estimation, !,I. I.T.

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Perkins, C.F. and Hage, R.E., Airplane Performance Stability and Control, J. Wiley and Sons, New York, 1949.

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REFERENCES (Continued) 67. McRuer, D., Ashkenas, I. and Graham, D., Aircraft Ilynamics znd Automatic Control, Princeton Univer- sity Press, Princeton, 1973.

68. Anglin, E.L., "Static Force Tests of a Model of a Twin-Jet Fighter Airplane for Angles of Attack from -100 to 1100 and Sideslip Angles from -400 to 40°," NASA TN D-6425, Washington, August 1971, 69. Grafton, S.B. and Libbey, C.E., "Dynamic Stability Derivatives of a Twin-Jet Fighter Model for Angles of Attack from -100 to 110°," NASA TN D-6091, Washington, September 1970.

70. Jacob, H.G., "An Engineering Optimization Jlethod with Application to STOL-Aircraft Approach and Landing Tra- jectories," NSAS TN D-6978, Washington, September 1972.

DeRusso, P.M., Roy, R.J. and Close, C.M., State Vari- ables for Engineers, J. Wiley and Sons, New York, 1967.

Patel, R.V., "On the Computation of Numerators of

Transfer Functions of Linear Systems," IEEE Transac- -

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*U.S. M E R N U E N T PRIHTIffi OFFICE: 1977 - 735-00412