of a Light Airplane - NASA€¦ · SUMMARY The maximum likelihood parameter estimation technique...

NASA Technical Paper 1043

of a Light Airplane

Robkrt L. Cannaday and William T. Suit

DECEMBER 1977

https://ntrs.nasa.gov/search.jsp?R=19780005128 2020-05-16T18:21:21+00:00Z

TECH LIBRARY KAFB, NM

11II1llIllllllllIllllllllllllllllllllllHllll OL342b5

NASA Technical Paper 1043

Effects of Control Inputs on the Estimation of Stability and Control Parameters of a Light Airplane

Robert L. Cannaday and William T. Suit

Langley Research Center Hampton, Virginia

National Aeronautics and Space Administration

Scientific and Technical information Office

1977

l -

SUMMARY

The maximum l ike l ihood parameter es t imat ion technique was used t o e x t r a c t v a l u e s 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 f r o m f l i g h t test data obta ined from a l i g h t , s ingle-engine, low-wing a i r p l a n e . The f l i g h t t es t s cons i s t ed o f 9 runs i n which the s t a b i l a t o r was used t o e x c i t e the longi tudinal motions and 28 runs i n which the rudde r and a i l e rons were u s e d t o e x c i t e l a t e ra l motions. The va r i - o u s c o n t r o l i n p u t s were i n i t i a t e d f r o m trimmed l e v e l f l i g h t wi th a trimmed airspeed of about 46 m/sec and an i n i t i a l a l t i t u d e o f a b o u t 600 m.

The cons is tency of t h e d e r i v a t i v e estimates as i t relates t o v a r i o u s i n p u t s was i n v e s t i g a t e d t o d e t e r m i n e the i n p u t s which provide the most information for i d e n t i f i c a t i o n . Three cr i ter ia were used i n t h i s i n v e s t i g a t i o n : t he ensemble var iance , t h e estimated Cram&-Rao lower bound, and parameter correlat ions. There were no s i g n i f i c a n t d i f f e r e n c e s i n t h e cons i s t enc ie s ob ta ined by us ing t h e v a r i o u s s t a b i l a t o r i n p u t s . On t h e o the r hand , fo r t h e la teral case, t h e sequen- t i a l inputs ( rudder fo l lowed by a i l e r o n s o r a i l e r o n s f o l l o w e d by rudder) gave no t i ceab le improvement i n parameter va lue cons is tency over t h e rudder or a i l e r o n i n p u t s i n d i v i d u a l l y . Also, some improvement i n c o n s i s t e n c y was no ted fo r bo th a i l e ron and rudde r squa re waves as opposed t o s i n e waves.

The de r iva t ive va lues ob ta ined by us ing the maximum l ike l ihood t echn ique were compared w i t h va lues computed by us ing an empirical approach. They compared f a v o r a b l y i n d i rec t comparison and i n comparison of time h i s t o r i e s o f computed a i rp lane mot ions , a l though the e x t r a c t e d parameters provided the better match.

INTRODUCTION

The maximum l ike l ihood t echn ique (ref. I ) has been used t o estimate va lues of t he ae rodynamic de r iva t ives fo r s eve ra l a i rp l anes r ang ing f rom exps r imen ta l high-performance a i rplanes (refs. 2 and 3) t o s ing le -eng ine l i g h t a i r p l a n e s (ref. 4 ) . The impor tance o f p roper inputs to t h e success o f t h e parameter iden- t i f i c a t i o n p r o c e s s has long been recognized (ref. 51, and s eve ra l i npu t des ign s tudies have been conducted using modern c o n t r o l t h e o r y (refs. 5 t o 8). The inpu t s deve loped i n these s t u d i e s c a n be d i f f i c u l t t o implement i n p r a c t i c e , a n d approximations of these op t ima l i npu t s are o f t en u sed . The a p p r o a c h t a k e n i n t h i s paper to determine the e f f e c t i v e n e s s o f i n p u t s i n parameter i d e n t i f i c a t i o n was t o p e r f o r m r e p e a t e d l y s e v e r a l d i f f e r e n t i n p u t s which were roughly based on some o f these des igned inputs (refs. 6 t o 8 ) . The r e s u l t i n g p a r a m e t e r estimates f o r each type o f i npu t were compared t o i n d i c a t e how well the inputs p rovided t h e in fo rma t ion r equ i r ed t o de f ine each parameter p rec ise ly .

The purpose of t h i s paper is t o p r e s e n t t he eva lua t ion o f t h e e f f e c t i v e n e s s o f d i f f e r e n t i n p u t s i n p r o v i d i n g c o n s i s t e n t p a r a m e t e r estimates. Th i s was done by conduct ing flight tests u s i n g s e v e r a l t y p e s o f i n p u t s , e x t r a c t i n g p a r a m e t e r s

from the f l i g h t test records , and us ing three cr i ter ia to de t e rmine the r e l a t i v e e f f e c t i v e n e s s o f t he v a r i o u s i n p u t s r e l a t i n g t o p a r a m e t e r e s t i m a t i o n . The three cr i ter ia used f o r i n p u t e v a l u a t i o n i n the s tudy were t o compare ( 1 ) t h e v a r i - ances o f the estimated parameter values , (2) t h e estimates o f t h e Cram6r-Rao lower bound (ref. 91 , and ( 3 ) t h e parameter c o r r e l a t i o n s .

This r e p o r t first describes t h e a i rp l ane , i n s t rumen ta t ion , and f l i g h t tests. Then the c o n t r o l i n p u t s u s e d i n the tests are described and t h e three cri teria used for comparing these i n p u t s are d i s c u s s e d b r i e f l y . The r e s u l t s o f these comparisons are presented next . F ina l ly , a comparison of t he parameter estimates wi th those computed by us ing an empir ica l - theore t ica l approach is presented .

SYMBOLS

The aerodynamic parameters are r e f e r e n c e d t o a system of body axes wi th the o r i g i n a t t h e a i r p l a n e c e n t e r o f g r a v i t y , which is loca ted a t 21.25 percent E , and wi th o r i e n t a t i o n o f body axes as shown i n f i g u r e 1 .

accelerat ion measured a long X , Y , and Z body a x i s , r e s p e c t i v e l y , g u n i t s

wing span, m

wing mean geometr ic chord, m

f o r c e a l o n g X , Y , and Z body a x i s , r e s p e c t i v e l y , N

acce l e ra t ion due t o g rav i ty , m/sec2

moment o f i n e r t i a a b o u t X , Y , and Z body a x i s , r e s p e c t i v e l y , kg-m2

p r o d u c t o f i n e r t i a , kg-m2

index

c o s t f u n c t i o n

l i k e l i h o o d f u n c t i o n

d i s t ance f rom a i rp l ane cen te r of g r a v i t y t o c e n t e r o f pressure of h o r i z o n t a l t a i l , m

r o l l i n g , p i t c h i n g , and yawing moments, r e s p e c t i v e l y , N-m

mass, kg

number of data p o i n t s

dimension of system

2

- I

P r o l l rate, r ad / sec

P' number o f p a r a m e t e r s t o b e i d e n t i f i e d

q

zi

R 1

r

S

S

'DCRLB

T

u ' , v ' , w '

p i t c h rate, rad/sec

dynamic p r e s s u r e , N/m2

estimate of e r ro r cova r i ance matrix

yaw rate, rad/sec

wing area, m2

s t anda rd dev ia t ion

s t anda rd dev ia t ion o f estimated Cram&-Rao lower bound

t h r u s t , N

v e l o c i t y a l o n g X , Y , and Z body a x i s , r e s p e c t i v e l y , m/sec

v e l o c i t y component a long X , Y , and Z body a x i s , r e s p e c t i v e l y , a t angle-of-at tack sensor on wing-tip boom, m/sec

a i r p l a n e t o t a l v e l o c i t y , m/sec

body coord ina te axes t h rough a i rp l ane cen te r o f g rav i ty

X , Y , Z body coord ina te s , m

rt vec to r desc r ib ing state o f a i r p l a n e

angle o f a t tack, rad

parameter vec to r

parameter change vector

a n g l e o f s i d e s l i p , rad

l e f t a i l e r o n d e f l e c t i o n minus r i g h t a i l e r o n d e f l e c t i o n , rad

s t a b i l a t o r d e f l e c t i o n , rad

r u d d e r d e f l e c t i o n , rad

a i le ron s ine-wave input

3

a i le ron square-wave input

s t a b i l a t o r r a p i d p u l s e f o l l o w e d by slower decay

s t ab i l a to r s ine -wave i npu t

s tab i la tor square-wave input

rudder sine-wave input

rudder square-wave input

(6a-6r )s i a i le ron s ine-wave input fo l lowed by rudder sine-wave input

(6a-6r)sq a i leron square-wave input fol lowed by rudder square-wave input

(6r-6a)s i rudder s ine-wave input fol lowed by a i le ron s ine-wave input

( 6 4 a ) s q r u d d e r square-wave input followed by ai leron square-wave input

ang le between t h r u s t a x i s and a i r p l a n e X body a x i s , p o s i t i v e f o r t h r u s t u p , rad

measurement no i se vec to r (n x 1)

p i t c h a n g l e , rad

a i r d e n s i t y , kg/m3

ensemble mean es t imated Cram&-Rao lower bound

r o l l a n g l e , rad

ro l l ing-moment coef f ic ien t , MX/?jSb

pi tching-moment coeff ic ient , My/$%

yawing-moment c o e f f i c i e n t , MZ/qSb

t h r u s t c o e f f i c i e n t , T/qS ( u s e d i n some pub l i ca t ions as T c ' )

ax i a l - fo rce coe f f i c i en t , Fx /@

s i d e - f o r c e c o e f f i c i e n t , Fy/qS

normal-force coeff ic ient , Fz/qS

4

CYr - - - acY a ’ b

2v

Subsc r ip t s :

C computed

k index

m measured

0 c o e f f i c i e n t a t trimmed c o n d i t i o n s

t trimmed c o n d i t i o n s

S u p e r s c r i p t s :

T t r anspose ma t r ix

M measured quant i ty

0 nominal evaluat ion

A dot over a symbol s i g n i f i e s a d e r i v a t i v e wi th respect t o time.

5

DESCRIPTION OF AIRPLANE, INSTRUMENTATION, FLIGHT TESTS,

AND DATA REDUCTION

The s u b j e c t a i r p l a n e f o r t h i s s tudy was a four-place, low-wing, s ing le - e n g i n e a i r p l a n e shown i n f i g u r e 2. Its pe r t inen t geomet r i c details are g iven i n t a b l e ’ 1 . The movable con t ro l su r f aces i nc luded the s t a b i l a t o r , r u d d e r , a i le- rons, and f laps; and the a i r p l a n e was ins t rumen ted t o r eco rd con t ro l - su r face movements and a i r p l a n e r e s p o n s e s t o t h e s e movements.

The v a r i a b l e s r e c o r d e d f o r t h i s s tudy were obtained from an onboard instrumentation package. The range of each ins t rument used to record t h e p e r t i n e n t v a r i a b l e s is g i v e n i n table 11. The a c c e l e r a t i o n s , a n g u l a r rates, and angular a t t i t u d e s were recorded cont inuously, whereas the a i r s p e e d , a n g l e of at tack, ang le o f s ides l ip , and con t ro l - su r face pos i t i ons were recorded sequent ia l ly by use o f a commutator which sampled each v a r i a b l e 20 times per second. Airspeed, angle o f at tack, a n d a n g l e o f s i d e s l i p were measured on a boom t h a t was loca ted near t h e wing t i p and extended 3/4 E ahead o f the wing l ead ing edge. (See f ig . 2 . )

The flight tests were d i v i d e d i n t o two g roups : t hose i n which t h e long i tu - d i n a l modes were exc i t ed and t h o s e i n which the la teral modes were exc i t ed . No f l a p s were used during the f l i g h t tests and a l l . t e s t s were i n i t i a t e d from trimmed l e v e l f l i g h t . The test procedure was as fo l lows: The p i l o t t u r n e d on the data recording equipment, made spec i f i c con t ro l i npu t s , a l l owed a i rp l ane r e sponses t o se t t le out , and then turned off the recording equipment . T h i s sequence con- s t i t u t e d a da ta run . The tes ts were conducted in smooth a i r to minimize process noise f rom gus ts . The t h r o t t l e was held f ixed dur ing t h e runs to minimize any c h a n g e s i n t h r u s t so t h a t t h e on ly d i s tu rb ing fo rce on t h e a i r p l a n e was due t o t h e c o n t r o l i n p u t . S i n c e a n g l e - o f - a t t a c k o s c i l l a t i o n s d u r i n g the tests were g e n e r a l l y less than To peak to peak , these v a r i a t i o n s were considered minimal, a l though the reader shou ld no te poss ib l e t h rus t change e f f ec t s on Cxa and Cza as stated in appendix A.

A t o t a l o f 37 flight tes t runs were made, 9 runs concen t r a t ing on long i tu - dinal dynamics and t h e remainder concentrat ing on l a t e ra l dynamics. The l eng th of the runs ranged from about 20 t o 50 sec. S i n c e t h e . o b j e c t o f t h e f l i g h t test was t o e v a l u a t e the r e l a t i v e e f f e c t i v e n e s s o f v a r i o u s i n p u t s f o r p a r a m e t e r es t i - mation, a l l test runs were i n i t i a t e d from approximately the same condi t ion o f trimmed l e v e l f l i g h t . The f l i g h t cond i t ion was between approach and cruise airspeed (approximately 40 percent power). For t h e l o n g i t u d i n a l tests, the mean i n i t i a l trimmed a i r speed was 45.3 m/sec w i t h a s t anda rd dev ia t ion o f 1 .7 m/sec and t h e i n i t i a l a l t i t u d e was 641 m w i t h a s t anda rd dev ia t ion o f 57 m. For t h e la teral tests, t h e mean i n i t i a l trimmed a i r speed was 45.8 m/sec w i t h a s tandard dev ia t ion o f 1.6 m/sec, and t h e i n i t i a l a l t i t u d e was 564 m w i t h a s tandard devi - a t i o n o f 24 m. Effects o f v a r i a t i o n s i n a i r s p e e d a n d a l t i t u d e were considered to be neg l ig ib l e and were ignored i n t h e process ing and ana lys i s o f t h e data.

It was n e c e s s a r y t o a p p l y c o r r e c t i o n s t o some o f the da ta be fo re p rocess ing wi th t he i d e n t i f i c a t i o n a l g o r i t h m . A i r s p e e d was c o r r e c t e d f o r p o s i t i o n e r r o r and a l t i t ude ( a s suming s t anda rd t empera tu re ) t o ob ta in t rue a i r speed . Then airspeed , ang le o f a t t ack , and a n g l e o f s i d e s l i p were c o r r e c t e d f o r upwash and

resolved into velocity components u, v, and w by using the relationships developed in appendix B. The control-surface positions were measured at the respective cables in the vicinity of the cockpit. Because Some of the data were commutated, it was necessary to reconstruct the data between data points by lin- ear interpolation and then sample simultaneously to avoid time-shift errors due to commutation. The continuous data were sampled digitally by using a zero- phase Ormsby filter (ref. IO).

The accelerometers were considered to be located on the airplane center of gravity in the X-Y plane but their z-position relative to thewairplane center of gravity was unknown. The computer program used for identification had the option of identifying the z-position of the Y-accelerometer; so this variable was activated in the algorithm for the appropriate maneuvers. Therefore, the accelerometer data were not corrected to the center-of-gravity position before being examined by use of the identification algorithm. A brief description of the identification algorithm is contained in appendix C.

CONTROL INPUTS

Various control inputs were used to determine dependence of parameter consistency on the inputs. Some of the inputs were suggested by several input design studies (refs. 6, 7, and 8 ) and others were inputs common to flight test- ing. The inputs used in this study were not the optimal inputs of these studies but were simplified forms.

For the identification of the longitudinal parameters, three basic types of inputs were attempted: the stabilator square wave, sine wave, and rapid rise followed by slower decay. The square-wave input (input A , fig. 3) was chosen because it was thought to contain the frequency content necessary to excite the short-period mode. The period and amplitude (stabilator travel) were chosen for ease of pilot implementation, as well as to keep pitch-attitude changes within 5 O to IOo of trim. The stabilator sine-wave input (input B, fig. 3) is often used to characterize a second-order system. The stabilator rapid rise followed by slower decay (input C, fig. 3) was an attempt to approxi- mate an input form suggested by reference 6.

For lateral identification, inputs consisted of rudder or ailerons applied individually or a sequential combination of both. (See fig. 4.) Switching-type (square-wave) inputs were recommended in several references (for example, refs. 7 and 8 ) as an approximation to the optimal input; therefore, square-wave inputs were attempted, although the switching times and amplitudes used for the tests were not optimal. Both sine- and square-wave forms were investigated for each control. The resulting inputs were rudder square wave (input Dl, rudder sine wave (input E), aileron square wave (input F), and aileron sine wave (input GI.

Rudder inputs alone do not provide adequate excitation of the lateral modes for parameter identification (ref. 11). To provide better excitation than single controls (rudder or ailerons individually) can produce, combinations of rudder and ailerons were used sequentially. That is, the aileron inputs were imple- mented followed immediately by rudder inputs or rudder inputs were followed by aileron inputs. The resulting sequential inputs were rudder square wave fol-

7

lowed by a i l e r o n s q u a r e wave ( i n p u t HI, r u d d e r s i n e wave followed by a i l e r o n s i n e wave ( i n p u t I ) , a i l e r o n s q u a r e wave followed by rudder square wave ( i n p u t J), and a i l e r o n s i n e wave followed by r u d d e r s i n e wave ( i n p u t K ) . Typical time h i s t o r i e s o f t h e s e i n p u t s c a n b e s e e n i n f i g u r e 4.

Severa l r epea t runs were made f o r each i n p u t i n a n e f f o r t t o o b t a i n t h e des i red input form. Inspec t ion of t he d a t a f o r each input form showed that none of the inpu t r epea t s exce l l ed ove r t he o the r s , so a l l t he runs were used fo r a n a l y s i s . F o r t h e l o n g i t u d i n a l i n p u t s , t h i s r e s u l t e d i n f i v e r e p e a t s o f i n p u t A and two of B and C. For t h e la teral i n p u t s , D and E each had f o u r r e p e a t s , F had two, G had f o u r , H had f i v e , I had t h r e e , J had four , and K had two.

CRITERIA USED FOR EVALUATING INPUTS

The results from each o f t he s e v e r a l d i f f e r e n t c o n t r o l i n p u t s were examined be fo re p re sen t ing a set 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 v a l u e s which best descr ibed the s u b j e c t a i r p l a n e . I n d e t e r m i n i n g which inputs p rovided the most c o n s i s t e n t i d e n t i f i c a t i o n o f the parameter values , three cr i ter ia were considered: ensemble variance of t h e parameter estimates, es t imated Cram&-Rao lower bound (ref. 9 1 , and pa rame te r co r re l a t ions .

The ensemble variance is t h e var iance o f t h e parameter estimates from t h e same type of input . These var iances o f t h e parameter estimates were compared by us ing the F - t e s t s ( re f . 12). This comparison indicated which inputs produced t h e most c o n s i s t e n t i d e n t i f i c a t i o n o f the unknown parameters .

The n e x t c r i t e r i o n , t he es t imated Cramer-Rao lower bound, was examined a l s o us ing the F - t e s t f o r f u r t h e r i n d i c a t i o n o f t h e e f f e c t i v e n e s s o f the i n p u t s f o r i den t i f i ca t ion . Accord ing t o r e f e rence 9 , t h e estimated Cram&-Rao lower bound is p r o p o r t i o n a l t o the u n c e r t a i n t y l e v e l i n the estimate of an unknown parameter. The pr imary funct ion of these u n c e r t a i n t y l e v e l s u s e d here was t o i n d i c a t e which inputs provided the greatest informat ion and , inverse ly , t h e l eas t unce r t a in ty i n d e f i n i n g a p a r t i c u l a r d e r i v a t i v e v a l u e . The h igher t h e u n c e r t a i n t y l e v e l f o r a p a r t i c u l a r d e r i v a t i v e , t he less information was generated by t h e i n p u t t o d e f i n e t h a t der iva$ive . Therefore , a comparison of the u n c e r t a i n t y l e v e l s ob ta ined fo r a parameter f rom var ious inputs gave some ind ica t ion o f which inputs p rovided the best i d e n t i f i c a t i o n o f t h a t parameter.

Ano the r c r i t e r ion t h a t a ids i n d i s c e r n i n g which inputs p rovide the more r e l i a b l e i d e n t i f i c a t i o n o f t h e unknown parameters is the c o r r e l a t i o n s , o r dependencies , between pairs of parameters ( re f . 6 ) . These c o r r e l a t i o n s between parameters are determined from t h e off-diagonal terms o f the e r ro r cova r i ance ma t r ix for es t imated parameters (parameter covar ' iance matr ix) . Correlat ion values near 21 ind ica te parameter dependencies , tha t is, changes in one parameter value caus- i ng changes i n ano the r . These parameter dependencies do not necessarily arise f rom phys ica l re la t ionships , bu t may a l s o arise f rom inadequate exc i ta t ion o f some o f the states f o r which parameters are t o be i d e n t i f i e d . I n a d e q u a t e e x c i - t a t i on can be ana logous t o the problem of a t tempt ing to ident i fy too many parameters ( re f . 6). High c o r r e l a t i o n s c a n r e s u l t i n a nonuniqueness o f the ident i f i - cation problem which c a n r e s u l t i n p a r a m e t e r r a t i o s or sums of parameters being i d e n t i f i e d rather than the spec i f ic parameters themselves . Al though re l iab le

8

parameter estimates can occur even when correlations are high, in many cases non- realistic parameter values and significant run-to-run scatter have occurred when correlation coefficients are 0.9 o r larger. Therefore, large correlation coefficients are an indication of potential identification problems. One way these parameter dependencies can be reduced is by choosing inputs that adequately excite the modes for which the associated parameters are to be identified. Therefore, examination of the parameter covariance matrix associated with each input revealed which input reduced parameter dependencies o r correlations and, in turn, probably provided the more reliable parameter estimates.

RESULTS AND DISCUSSION

In order to evaluate the uniformity of input repeats, the mean input period and amplitude and the associated standard deviations for each input are presented in table 111. As can be noted, there was considerable variability in the mean period among the lateral inputs (5.4 to 10.4 sec). Generally, this variability did not appear to correspond to the scatter in the parameter estimates.

Effects of Various Longitudinal Control Inputs

The longitudinal parameter values obtained from applying the maximum likelihood technique to the longitudinal flight test data are presented in figure 5 and table IV. In the upper half of figure 5 are the longitudinal parameter values obtained for each run f o r inputs A, B, and C. Note that the run numbers are located beside the extracted values and, in some cases, the extracted values overlap so closely that one symbol may represent two o r three data points. In the lower half of figure 5 are the mean values for each type of input, with the upper and lower bars representing plus and minus values of the standard deviation for each parameter. Also, just to the right of these bars are bars representing the mean plus or minus the mean of the estimated Cram&-Rao lower bound for each input.

Comparing the variances (F-test) of the longitudinal parameter estimates across the inputs indicated no significant differences. Therefore, based on the variance criterion, none of the stabilator inputs showed significant improvement in providing consistent estimates of the parameters.

The next criterion to be considered was the uncertainty level (based on the estimated Cram&-Rao lower bound) in the parameter estimates. A visual inspec- tion of figure 5 shows little difference in longitudinal parameter uncertainty levels among the inputs. The F-test was applied to the square of the uncertainty levels of the parameter estimates for the three inputs. Based on uncertainty levels, none of the longitudinal inputs stood out as significantly improv- ing the consistency of the parameter estimates. This same conclusion was reached by considering the variance, so the conclusions for these two criteria are consistent.

The third criterion, the parameter correlations, was considered next. For the present study, basically the only longitudinal parameters which appeared to be correlated were and Cm&e. For input A , the correlation between these c"q

9

two parameters ranged from 0.90 t o 0.95; f o r i n p u t B, from 0.93 t o 0.96; and for i n p u t C , f rom 0.84 to 0.91. Also, f o r i n p u t B , run 6 e x h i b i t e d r e l a t i v e l y h i g h c o r r e l a t i o n s f o r some o f t he o the r pa rame te r s a l so . Based on parameter correla- t i o n s , i n p u t C a p p e a r e d t o o f f e r s l i g h t l y i m p r o v e d i d e n t i f i c a t i o n . The form of i n p u t C was loose ly pa t t e rned after the op t ima l i npu t o f r e f e rence 6 , which was based on cr i ter ia r e l a t e d t o t h e trace and determinant of t h e F isher in format ion mat r ix . This matrix is the i n v e r s e of the parameter covariance matrix.

Based on the three cr i ter ia considered, none of t h e three l o n g i t u d i n a l i n p u t s c l e a r l y o f f e r e d b e t t e r i d e n t i f i c a t i o n t h a n t h e o t h e r two.

Effects of Various Lateral Cont ro l Inputs

The lateral parameter values obtained from the f l i g h t d a t a are shown i n f i g u r e 6 and table V. The l a t e ra l d a t a are p r e s e n t e d i n t h e same format as tha t used fo r t h e l o n g i t u d i n a l d a t a . Note t h a t no attempt was made t o i d e n t i f y a i le - ron con t ro l de r iva t ives wi th rudder inputs o r r u d d e r c o n t r o l d e r i v a t i v e s with a i l e r o n i n p u t s .

A v i s u a l i n s p e c t i o n o f t h e data r e v e a l s t h a t i n p u t s w i t h rudder a lone (D and E ) genera l ly gave the least c o n s i s t e n t estimates of t h e parameters except f o r t he rudde r con t ro l de r iva t ives . Cor re spond ing ly , a i l e rons a lone ( i npu t s F and G ) genera l ly p rovided more c o n s i s t e n t estimates than t h e rudder a lone . For one o f t he a i l e ron runs ( run 22 of input GI, however, the parameter estimates were c o n s i d e r a b l y o u t o f l i n e f o r the ro l l ing- and yawing-moment parameters. An examination of the time h i s t o r y of run 22 showed t h a t t h e per iod of the i n p u t was about 2 sec longer than the o t h e r runs f o r i n p u t G, a l though the basic form was the same. T h i s l ong i npu t pe r iod may not have properly excited the a i r p l a n e dynamics and thus may h a v e r e s u l t e d i n some iden t i f i ca t ion p rob lems . Parameter c o r r e l a t i o n s f o r run 22 were except ional ly high and are discussed subsequent ly . Due to the problems encountered w i t h run 22, the assoc ia ted parameter va lues were omitted from the computation of t h e s ta t i s t ics shown i n the lower half o f f i g u r e 6.

To i n v e s t i g a t e lateral i n p u t s as they re la te t o c o n s i s t e n t i d e n t i f i c a t i o n , t he three c r i t e r i a - ensemble var iance , uncer ta in ty l eve l , and parameter cor re la - t i o n s - were again used as f o r t h e l o n g i t u d i n a l p a r a m e t e r s .

In o rde r t o de t e rmine whe the r i npu t form affected c o n s i s t e n c y , r e s u l t s from i n p u t s D and E , and F and G were compared by us ing the v a r i a n c e c r i t e r i o n (F-test ) . For rudder input forms, D genera l ly p rovided more c o n s i s t e n t estimates than E bu t on ly CyB, Cygr, and Cn were s i g n i f i c a n t l y more c o n s i s t e n t

(80 -pe rcen t l eve l ) . I n no case d i d inpu t E p r o v i d e s i g n i f i c a n t l y greater cons is - t e n c y i n the parameter estimates than i npu t D. This r e s u l t seems t o i n d i c a t e t ha t , fo r rudde r i npu t s , r udde r squa re waves provided the bet ter i d e n t i f i c a t i o n . F o r a i l e r o n i n p u t s , F provided more c o n s i s t e n t estimates than G ( s ine waves) . Inpu t F d e m o n s t r a t e s s i g n i f i c a n t l y greater c o n s i s t e n c y f o r CY8) CYp, C'B'

and C q a t han i npu t G. Therefore , it appears tha t square-wave inputs p rovided

r

10

the more consistent estimates for ailerons also, but uncertainty levels (estimated Cram&-Rao lower bound) and parameter correlations should be examined before a conclusion is drawn.

Rudder inputs were next compared to aileron inputs. Comparing rudder square-wave inputs (D) with aileron square-wave inputs (F) showed that for CyB, Cyp, Cy,, and C ailerons provided significantly (80-percent level)

more consistent identification and in no case did input D provide more consistent estimates than any of the runs using aileron inputs. 8 similar trend, although less pronounced, held when comparing inputs E and G (rudder and aileron sine waves), with CY and C being significantly more consistently

identified for the aileron inputs. It is interesting to note that for single control inputs, the derivatives which might be assumed to be associated with excitations from the rudder inputs appear to be better defined by using aileron inputs. One explanation is that the correlations between parameters were higher for the rudder inputs, making identification of the correlated parameters difficult. Correlations are discussed in more detail subsequently.

I B '

P "B

Upon comparing the single control inputs (D, E, F, and GI with the sequential inputs (H, I, J, and K) it was found that the sequential inputs provided significantly more consistent estimates of most of the stability derivatives, with the exception of input F. Input F was nearly as consistent as the sequential inputs, but this may be due to the small data sample (2 runs). For the control derivatives, the single inputs generally provided estimates which were equally as consistent as those obtained with sequential inputs.

Inputs H and J, and I and K were compared to determine whether the order of the inputs for the sequential inputs was significant. Input J gave significantly (80-percent level) more consistent estimates of the rolling-moment derivatives except for CIB, and input H gave significantly more consistent estimates

of Cyr and Cygr. These trends did not hold when comparing inputs I and K;

only Cygr was identified as significantly more consistent and this was for

input I. Therefore, based on the information examined thus far, it is difficult to conclude whether order makes any difference.

The next criterion to be considered in evaluating input effectiveness f o r identification purposes was the uncertainty level, o r estimated Cram&-Rao lower bound. Generally, uncertainty levels were less than the standard deviations, especially for the single inputs (D, E, F, and GI. Greater uncertainty levels in many cases corresponded to greater standard deviations, which indicated that uncertainty levels were somewhat analogous to standard deviations, as stated in reference 9. Therefore, the trends determined by using the F-test on the variances should generally hold when considering the square of the uncertainty levels. Looking at input form (comparing data from inputs D and E, and F and G) the distinction between parameter estimates that was noted when comparing ensemble variance was not as obvious when comparing the square of the uncertainty level. Statistically speaking, none of the estimated parameters had signifi-

1 1

and C16a with coefficient values ranging from 0.85 to 0.89 and Cnp and Cnga

with values ranging from 0.815 to 0.869. Inputs I and K were not quite as effective in reducing correlations as H and J but were considerably more effective than D, E, F, and G.

Based on the information contained in the parameter covariance matrix, the sequential inputs clearly reduce parameter correlations, or dependencies, compared to the single control inputs. This implies that the more reliable lateral identification is obtained from the sequential inputs.

The z-position of the Y-accelerometer was also identified for the lateral runs. As for the parameter estimates, the most consistent estimate of the loca- tion was observed for the sequential inputs. These estimates ranged from 0.24 to 0.30 m below the airplane center of gravity.

Use of Estimated Parameters in Mathematical Model

None of the longitudinal inputs attempted offered clear improvement in the consistency of the extracted derivatives. Therefore, the resulting parameters presented for the subject airplane are the arithmetic means of the various parameter values extracted for all 9 longitudinal runs. These mean values along with standard deviations are presented in table VI.

For the lateral data, the sequential inputs generally provided the most consistent estimates of the lateral parameters. Therefore, the arithmetic means of the parameters obtained from the sequential inputs were used to describe the airplane in the subsequent analysis. These mean values are also presented in table VI.

The mean parameter values of table VI were used to generate the computed time histories of figures 7 and 8. The match between the computed airplane responses and the measured responses was considered to be good, although a slightly better fit could be obtained with the actual values for that run.

The last column in table VI contains the standard deviations as a percent of the mean value for each derivative. This quantity, the coefficient of variation, provides a measure of the relative consistency of one derivative with another. For example, CnB (3.6 percent) was the most consistent lateral deriv-

ative identified and CZ, (4.0 percent) was the most consistent longitudinal

derivative. Generally, the static derivatives such as CZ, and C were more "B consistently identified than were. the rotational and control derivatives.

Comparison of the Extracted Parameters

One question which arises whenever derivatives are extracted from flight data is, Are the results reasonable? Good agreement between measured and com-

13

puted time h i s t o r i e s d o e s n o t n e c e s s a r i l y g u a r a n t e e t h a t the parameters are reasonab le . Cor re l a t ions , an i nadequa te mathematical model, o r other problems c o u l d e x i s t which cou ld d r ive the p a r a m e t e r s t o u n r e a l i s t i c v a l u e s t o f i t the data. The re fo re , t o l end some c o n f i d e n c e t o t h e e x t r a c t e d parameter va lues , these va lues were compared wi th r e s u l t s f o r t he same a i r p l a n e , which were determined by another method. The second column of table V I 1 c o n t a i n s t h e estimated parameter va lues g iven in table V I . The t h i r d column o f table V I 1 c o n t a i n s a n estimate of t h e s t a b i l i t y and con t ro l de r iva t ives o f t h e s u b j e c t a i r p l a n e obtained by us ing t h e t h e o r e t i c a l - e m p i r i c a l t e c h n i q u e s p r e s e n t e d i n reference 13. These va lues were t ransformed to body-axis der iva t ives by the t r a n s - fo rma t ions p re sen ted i n r e f e rence 14. Upon comparison, i t can be seen t h a t t h e estimated parameter va lues o f the p resen t s tudy are g e n e r a l l y c o n s i s t e n t wi th those computed from reference 13.

As a matter o f i n t e r e s t , t h e de r iva t ive va lues ob ta ined by us ing t h e maximum l ike l ihood technique , as well as those f rom reference 13, were used i n t h e equat ions o f mot ion to genera te time h i s t o r i e s . The computed time h i s t o r i e s based on the de r iva t ive va lues and the acce le romete r o f f se t ob ta ined by us ing the maximum l ikel ihood technique and the measured f l i g h t data time h i s t o r i e s are shown i n f i g u r e s 7 and 8. The computed time h i s t o r i e s b a s e d on the d e r i v a t i v e s obtained by us ing t h e methods of reference 13 and t h e same acce le romete r o f f se t and the same f l i g h t data time h i s t o r i e s are shown i n f i g u r e s 9 and 10. A s would be expected, t he maximum l ike l ihood de r iva t ives p rov ided t h e be t te r f i t t o t h e f l i g h t data, which was e s p e c i a l l y n o t i c e a b l e f o r t h e l a te ra l data.

CONCLUDING REMARKS

S t a b i l i t y and c o n t r o l d e r i v a t i v e v a l u e s were determined by use of t h e maximum l i ke l ihood a lgo r i thm from flight t e s t data t a k e n i n c r u i s e c o n d i t i o n s f o r a low-wing, s ing le -eng ine a i rp l ane . Var ious con t ro l i npu t s were used to de t e rmine whether c e r t a i n i n p u t s improved the run-to-run consis tency of t h e d e r i v a t i v e estimates. The l o n g i t u d i n a l i n p u t s i n c l u d e d s t a b i l a t o r s q u a r e waves and s i n e waves and a r a p i d rise followed by a slower decay. The l a t e ra l inpu t s cons i s t ed of rudder and a i le ron inputs ( square and s ine waves) ind iv idua l ly and in sequence.

The parameter values obtained f rom the va r ious i npu t s were compared f o r improved consistency by us ing three cr i ter ia : t h e ensemble variance of the parameter estimates, t h e estimated Cram&-Rao lower bound, and parameter cor re- l a t i o n s . The results i n d i c a t e d t h a t none of t he l o n g i t u d i n a l i n p u t s r e s u l t e d i n improved i d e n t i f i c a t i o n . On the other hand, the l a t e ra l i n p u t s which consisted o f rudde r and a i l e rons i n s equence c l ea r ly o f f e red improved consistency over t h e rudder o r a i l e r o n i n p u t s i n d i v i d u a l l y .

For the s i n g l e i n p u t s , t he square-wave inputs for both rudder and a i le rons genera l ly p rovided the more cons is ten t parameter estimates when compared wi th their sine-wave coun te rpa r t s . For t h e sequent ia l inputs , square-wave inputs appeared to provide more precise estimates of the parameters . The o r d e r o f the s e q u e n t i a l i n p u t s made l i t t l e d i f f e r e n c e i n t he cons i s t ency of t h e estimates.

14

I

In order to validate the parameter values determined by use of the maximum likelihood algorithm, parameter values for the subject airplane were also determined by a theoretical-empirical method. These values and those obtained by using the maximum likelihood algorithm were compared both individually and in their ability to describe the airplane dynamics accurately. The results indicated that the values from the maximum likelihood algorithm more nearly described the airplane motions.

Langley Research Center National Aeronautics and Space Administration Hampton, VA 23665 November 17, 1977

15

APPENDIX A

EQUATIONS OF MOTION

The equa t ions u sed i n t h i s program are pe r tu rba t ion equa t ions f rom trimmed l e v e l flight and are w r i t t e n r e l a t i v e t o t h e set of body axes shown i n f i g u r e 1 .

The equat ions used to describe the long i tud ina l mo t ions were

G = -qw + r v - g s i n e + 1 p . ' S [ C X , ~ + c ~ ( c 1 - at)] ( A I ) 2 m

t; = -pv + qu + g cos e cos 4 + + CZc,(a - at1 + cz 1 q a 2 m q 2 v

c1 = tan-1 w U

16

APPENDIX A

The va lues o f t he l a t e ra l states v, p, r , and @ used i n t h e l o n g i t u d i n a l equat ions were t h e f l i gh t -measu red quan t i t i e s .

S ince t h rus t changes are n o t e x p l i c i t l y modeled in t he equa t ions o f mo t ion , C i a o f equa t ion ( A I ) and C i a o f equa t ion (A2) are no t necessa r i ly pu re Cx,

and Cza but may c o n t a i n small c o n t r i b u t i o n s due to changes i n t h rus t . The re -

f o r e , C i a and C i a , as de te rmined i n t h i s s tudy , are given by

S i n c e , i n t h i s s t u d y , t h r u s t was he ld constant and the angle-of-at tack changes were no more than 7 O peak to peak , the c o n t r i b u t i o n s o f t h r u s t t o C i a and C i a were considered minimal.

The equa t ions u sed t o compute t h e la teral motions were

+ = - ru + pw + g cos 8 s i n @ + 1 p + cyB@ + cyr 2v r b 2 m

17

APPENDIX A

v = $2 + v2 + w2 (A161

The va lues o f l ong i tud ina l states u, w, q, and 8 used i n t h e la teral equa- t i o n s were the f l igh t -measured quant i t ies . The equat ions were used t o compute t h e a i r p l a n e state responses . The computed responses were then compared wi th the r eco rded r e sponses f rom the f l i gh t tes ts and the d i f f e r e n c e s were used t o update the p a r a m e t e r s ( 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 ) t o improve the f i t .

The l o n g i t u d i n a l measured and computed responses, or states, used i n the a l g o r i t h m f o r t h i s s tudy were u, w , q , 0 , ax, and az. The la te ra l states used were v , p , r , @, and ay. An abbrevia ted d i scuss ion of t he i d e n t i f i c a - t i on a lgo r i thm is g iven i n append ix C .

18

APPENDIX B

TRANSFORMATION OF MEASURED V , a, AND B

The boom on which t h e dynamic p ressure , angle o f a t tack , and angle of s ide- s l i p were measured was l o c a t e d a t t h e l e f t wing t i p p a r a l l e l t o t h e a i r p l a n e X body axis. The sens ing e lements were loca ted abou t 3/4 E ahead of the lead- i n g edge of t h e wing. These measurements were corrected f o r upwash and t ransformed t o the a i rp l ane cen te r -o f -g rav i ty pos i t i on . The following development d e s c r i b e s t h e procedure used for the data of t h i s s tudy . The fo l lowing ske tches show t h e side view

and t h e top view Y

A

of the schemat ic of t h e l e f t wing t i p . The equa t ions related t o t h e s e s k e t c h e s are

vm = um t a n & (B2 1

The magnitude of t h e v e l o c i t y v e c t o r was measured by t h e p i t o t s t a t i c tube and t h e d i r e c t i o n was g iven by t h e a and f3 vanes. The magnitude of t h e v e l o c i t y v e c t o r is given by t h e equa t ion

19

APPENDIX B

S u b s t i t u t i n g e q u a t i o n s ( B I ) and (B2) i n t o e q u a t i o n (B3) g i v e s

um = Vm \/I + t an2 Bm + t an2 040

It was assumed t h a t t h e v-component was unaf fec ted by t h e upwash and needed no upwash cor rec t ion . Therefore , the p roblem was c o n s i d e r e d t o be c o n t a i n e d i n t h e X-Z plane as shown i n t h e f o l l o w i n g s k e t c h :

x -

-Y, """-

Z

A s can be seen from t h i s s k e t c h , i n o r d e r t o c o r r e c t v e l o c i t y components urn and wm f o r upwash, t h e a f f e c t e d v e l o c i t y v e c t o r n e e d s t o be r o t a t e d a n amount which w i l l r e s u l t i n u ' a n d w' being t h e c o r r e c t components when t h e r o t a t e d vec to r is pro jec ted on to the X- and Z-axes.

Based on experimental data, the angle of at tack c o r r e c t e d f o r upwash ci was de termined to be 0.75%; t h a t is, a = 0.75%. Therefore,

U ' = i n COS (0.75%)

v ' = v,

Subs t i t u t ion o f equa t ions ( B I ) , (B2), and ( B 4 ) i n t o t h e s e e q u a t i o n s y i e l d s

20

APPENDIX B

The components ut, V I , and wt are the upwash corrected components of velocity at the boom. These components must also be corrected to the airplane center-of-gravity position by removing velocities resulting from rotational rates about the center of gravity. This correction results in the following equations :

u = um cos (0.7501,) - qZ + rv (B5 1

w um sin (0.7501,) - py + qX (B7)

where Z, 7, and z denote the position of the sensors relative to the airplane center of gravity.

-

21

APPENDIX C

PARAMETER ESTIMATION PROCEDURE

The pa rame te r e s t ima t ion p rocedure u sed i n t h i s s tudy is known as t h e maximum l ike l ihood technique . This formulat ion of t h e maximum l ike l ihood t echn ique considered only measurement noise. The t e c h n i q u e u t i l i z e s t he d i f f e r e n c e s between t h e measured and computed states ( f i t e r r o r ) t o estimate t h e unknown va lues o f t he parameters necessary to min imize the f i t e r r o r . The measurement no i se

is assumed t o be white w i t h G a u s s i a n d i s t r i b u t i o n . This assumption i s used t o c o n s t r u c t the l i k e l i h o o d f u n c t i o n , t he natural logar i thm of which is

N

where R 1 is the estimate of t he s ta te error covariance matr ix based on t h e nominal solut ion, t i is time, and 20 is t h e nominal parameter vector.

To estimate t h e parameter vector d t h e l i k e l i h o o d f u n c t i o n was maximized with r e s p e c t t o do and R 1 . Because o f the nonl inear na ture o f the e s t ima t ion problem, it was n e c e s s a r y t o u s e a n i t e r a t i v e p r o c e d u r e t o estimate t h e parameters. Therefore , $(d, t i ) , t h e estimated s ta te v e c t o r a t time t i , was expanded about the nominal parameter vector , neglect ing second-order and higher o rde r terms. That is,

expresses the es t imated the change i n the s ta te

s ta te vec to r as a func t ion of t h e nominal so lu t ion p lus vec to r estimate as a func t ion of t h e computed change i n

the parameter vector . This e x p r e s s i o n f o r t h e s ta te vec to r e s t ima t ion is then used i n t h e f i t e r r o r v e c t o r t o o b t a i n

where

and def ines s ta te s e n s i t i v i t y t o t h e cur ren t parameters . Therefore , t he n a t u r a l logar i thm of t he l i ke l ihood func t ion becomes

22

APPENDIX C

J

Maximizing the likelihood function with respect

and solving for Aa yield the parameter change +

1-v- N

to G,

equations,

1

tion. Note that the matrix AT(ti)R1” A(ti) is the estimate of the

error covariance matrix for the estimated parameters (parameter covariance matrix). This matrix contains the parameter dependencies or correlations which are used to construct parameter correlation coefficients.

1 i=l

To estimate R1, the measurement noise covariance matrix, the likelihood function can be maximized with respect to R1, which is approximately

N

according to reference 1.

The effect of maximizing the likelihood function is the same as minimizing the cost function

In using the program, the cost function was computed and displayed on the program operator console (ref. 4 ) . Generally, with each iteration the fit improved, as evidenced by a reduction in the cost function. Once the cost function settled out so that changes from iteration to iteration, as defined by (Jk - &+I)/&, were less than 0.01, the parameters which maximize the likelihood function or minimize the cost function (fit error) were considered identified. The expres- sion (Jk - &+I)/& represents the change in the fit error from successive iterations divided by the fit error of the previous iteration. To avoid‘ possi- ble correlation problems (see section “Criteria Used for Evaluating Inputs1*) between C q and Cmq, C% was held fixed at -4.00 (estimated in a preliminary

study by using ref. 131, so essentially the program was identifying the combination Cmq + Cq with C% held constant.

23

APPENDIX C

Also , to fur ther reduce cor re la t ion p roblems be tween es t imated parameters , CZ6e and Cm6e were assumed t o b e g e o m e t r i c a l l y r e l a t e d as

- c - Cz6e - - Cm6e

I t

as was done i n r e f e r e n c e 4. The program described i n r e f e r e n c e 1 was modified so that CzGe was n o t t r e a t e d as an ac t ive pa rame te r . Then Cm6, was

e x t r a c t e d , and CzGe was c a l c u l a t e d f o r each i t e r a t i o n and its va lue was sub-

s t i t u t e d i n t o t he equa t ions of motion. Although was n o t a c t i v e , i ts

va lue was c h a n g i n g f r o m i t e r a t i o n t o i t e r a t i o n . To accoun t fo r t h i s v a r i a t i o n i n CZGe i n t he s e n s i t i v i t y e q u a t i o n s , Cm6e was s u b s t i t u t e d f o r CzGe i n

I t t h e equat ions and then the p a r t i a l d e r i v a t i v e s were taken of t h e i equa- t i o n wi th r e s p e c t t o C q e and CzGe.

24

t

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Williams, James L.; and Suit, William T.: Extraction From Flight Data of Lateral Aerodynamic Coefficients for F-8 Aircraft With Supercritical Wing. NASA TN D-7749, 1974.

Suit, William T.; and Williams, James L.: Lateral Static and Dynamic Aero- dynamic Parameters of the Kestrel Aircraft (XV-6A) Extracted From Flight Data. NASA TN D-7455, 1974.

Suit, William T.: Aerodynamic Parameters of the Navion Airplane Extracted From Flight Data. NASA TN D-6643, 1972.

Mehra, R. K.; and Gupta, N. K.: Status of Input Design for Aircraft Parame- ter Identification. Methods for Aircraft State and Parameter Identifica- tion, AGARD-CP-172, May 1975, pp. 12-1 - 12-21.

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Gupta, Narendra K.; and Hall, W. Earl, Jr.: Input Design for Identifica- tion of Aircraft Stability and Control Derivatives. NASA CR-2493, 1975.

Reid, Donald B.: Optimal Inputs for System Identification. SUDAAR No. 440 (NASA Grants NGR 05-020-526 and NGL 05-020-007), Stanford Univ., May 1972. (Available as NASA CR- 128 173.

Iliff, Kenneth W.; and Maine, Richard E.: Practical Aspects of Using a Maximum Likelihood Estimation Method To Extract Stability and Control Derivatives From Flight Data. NASA TN D-8209, 1976.

Enochson, Loren D.; and Otnes, Robert K.: Programming and Analysis for Digital Time Series Data. SV"3, U.S. Navy, 1968. (Available from DDC as AD 692 735.)

Klein, V.: Parameter Identification Applied to Aircraft. Cranfield Rep. Aero No. 26, Cranfield Inst. Technol. , [Dec. 19741.

Ullman, Neil R.: Statistics - An Applied Approach. Xerox College Pub., c. 1972.

Smetana, Frederick 0.; Summey, Delbert C.; and Johnson, W. Donald: Riding and Handling Qualities of Light Aircraft - A Review and Analysis. NASA CR-1975, 1972.

Etkin, Bernard: Dynamics of Flight. John Wiley & Sons, Inc., c.1959.

25

TABLE I.- GEOMETRIC DETAILS OF SUBJECT AIRPLANE

Mass. kg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I n e r t i a : Ix. kg-m2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IY. kg-m2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IXZ. kg-m2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iz. kg-m2 . . . . . . . . . . i . . . . . . . . . . . . . . . . . . .

F u s e l a g e 1 e n g t h . m . . . . . . . . . . . . . . . . . . . . . . . . . . . Wing:

Area. m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Span, m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aspect r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean geometric chord, m . . . . . . . . . . . . . . . . . . . . . . .

Vertical tail: Area. m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A s p e c t r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Span, m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rudder area, m2 . . . . . . . . . . . . . . . . . . . . . . . . . . .

Horizonta l t a i l ( s t a b i l a t o r ) : Area, m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A s p e c t r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Span, m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tail l eng th , cen te r of g r a v i t y t o q u a r t e r - c h o r d p o i n t of mean

geomet r i cchord , m . . . . . . . . . . . . . . . . . . . . . . . . . L o c a t i o n r e l a t i v e t o a i r p l a n e c e n t e r of g r a v i t y of sensors to measure

ve loc i ty . ang le of a t tack . and angle of s i d e s l i p : x , m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y , m z , m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

1074.1

1220 1898 2712

68

6.9

14.9 5.625

9.1 1.62

1.07 2.02 1.47 0.38

2.32 4.0

3.05

4.2

1.29 -4.46

0

26

TABLE 11.- INSTRUMENTATION RANGES

Ins t rument func t ion

Airspeed, m/sec

Angle of a t t a c k , d e g

Angle o f s i d e s l i p , d e g

A l t i t u d e , m

Normal a c c e l e r a t i o n , g u n i t s

Long i tud ina l acce l e ra t ion , g u n i t s

Lateral a c c e l e r a t i o n , g u n i t s

E l e v a t o r p o s i t i o n , deg

Ai l e ron pos i t i on , deg

Rudder pos i t i on , deg

T h r o t t l e p o s i t i o n

Engine speed, rpm

P i t c h ra te , deg/sec

Roll ra te , deg/sec

Yaw rate, deg/sec

P i t c h a t t i t u d e , d e g

R o l l a t t i t u d e , d e g

Range

0 t o 61

-30 t o 100

-60 t o 60

0 t o 3050

-3 t o 6

-1 t o 1

-1 t o 1

Ful l t h row (1 .5 t o -19 .0 )

Ful l th row (-43 t o 4 3 )

F u l l t h r o w ( - 2 5 t o 25)

Ful l th row

0 t o 2700

-100 to 100

-180 t o 180

-180 t o 180

-85 t o 8 5

-180 t o 180 -2

27

TABLE 111.- CONSISTENCY OF INPUTS USED I N FLIGHT TEST

:nput

A

B

C

D

E

F

G

H

I

J

K

Control

___- -

S t a b i l a t o r

S t a b i l a t o r

S t a b i l a t o r

Rudder

Rudder

Ai le rons

Ai le rons

Rudder and a i l e r o n s

Rudder and a i l e r o n s

Ailerons and rudder

Ailerons and rudder

-. ~ ".

Input form

( a ) . ~.

Square wave

S i n e wave

Rapid rise, slow dec l ine each d i r e c t i o n

Square wave

S ine wave

Square wave

S ine wave

Rudder square wave followed by a i l e r o n square wave

Rudder s i n e wave followed by a i l e r o n s i n e wave

Ai le ron square wave followed by rudder square wave

A i l e r o n s i n e wave followed by rudde r s ine wave

~ ~ . . .

" "

Mean Ier iod ,

sec

~

4.2

5.5

4 .4

5.5

6.6

5.9

8.0

5.4 5.9

7.9 8.1

5.8 6.0

8 .8 10.4

"" . "

Standard l ev ia t ion of mean per iod ,

I

sec

0.6

1 .o

.8

- .

.7

1.2

.6

1 . 1

.5

.6

.8 1 .2

.5

.4

1 . 1 .8

L

Mean peak-to- leak amplitude ,

de€!

a h p u t s were n o t s t r i c t l y s i n e waves o r square waves a t t empt s a t these forms. See f igures 3 and 4 f o r t y p i c a l

"

2.4

2.5

4.8

11.2

20.8

b15.5

b15.8

10.0 b17. 1

15.7 b13.0

b15.3 12.4

b15.1 17.5

but were

". - - Standard l e v i a t i o n of mean

tmplitude ,

p i l o t

de€!

0.4

.4

.4

". .

1 .o

2.8

2 .1

1.5

1 .o 2.2

2.9 2.6

1.7 1.5

1 . 3 3.5

i n p u t time h i s t o r i e s . bAi le ron def lec t ions are def ined as l e f t a i l e r o n minus r i g h t a i l e r o n .

28

TABLE 1 V . - ENSEMBLE MEAN AND STANDARD DEVIATION FOR EACH LONGITUDINAL

INPUT ALONG WITH ESTIMATED CRAMER-RAO LOWER BOUND

Parameter I Mean va lue I S tandard deviationp-

I I of mean va lue ~

0.0533

.648 1

-. 5533

-4.3828

-1 5.1932

-. 4394

-8.7746

-1 .5265

0.0405

.7424

-. 5829

-4.2332

-20- 9335

-. 4723

-7.6263

- 1 .5265

I n p u t A ( 5 runs)

0.0068

.2275

.0465

.I864

4.1709

.0324

3.7690

.2125 ~~

Inpu t B ( 2 runs )

0.0156

.I573

.0573

.0433

1.0257

.007 1

2.6367

.2850

I I

0.00022

. 0 1830

.00060

.03826

.80940

.00356

.20806

. 0 1830

0.00013

.00815

.00046

.02680

.59945

.00260

.I6445

. 0 1560

0.00010

.00735

.00029

.00947

.22418

. 00 148

.09952

.00870

0.00007

.00233

.00024

.00424

-06074

-00057

. 0 1407

.00042

29

I

TABLE 1 V . - Concluded

Standard devia t ion ] ~ C R L B 1 s ~ C R L B of mean va lue

. ~ .

_ _ _ _ ~

0.0469

.5076

-. 5644

-4.4514

-17.021 1

-. 4925

-8.4654

-1.5786

0.0052

.1623

.0508

.2157

5.2584

.0311

1.1422

.0947

0.00020

.02 120

.00054

.04 105

.72725

.00390

.16715

. 0 1500

0.00006

.01018

.00009

.00615

.04632

.00028

.04674

.00311

30

TABLE V.- ENSEMBLE MEAN AND STANDARD DEVIATION FOR EACH LATERAL INPUT

ALONG WITH ESTIMATED CRAM~R-RAO LOWER BOUND.

(a) Input D (4 runs)

Parameter

cYP

'~6 r

c1 P

'1 r

'16r

'16a

C

C

C "B

"P

cnr

'"6 r

%a C

Mean value ~~

-0.5737

-.0291

.2806

.0342

-. 0631

-. 3400

.0376

.0029

"""_ .052 1

-. 0480

-. 0923

-. 0338

"""_

Standard deviation of mean value

0.0799

.5170

.2652

.oogo

.0229

.I375

.0475

.0038

"""

.0098

.0619

.0212

.002 1

"""

SCRLB

0.0372

.2723

.0730

.0040

.0030

.0179

.0060

.0006

"""

.0018

.0107

.0034

.0003

"""

5 %RLB

0.0229

.1737

.0105

.0007

.002 1

.0127

.0034

.0002

"""_ .0008

.0045

.0012

.00004

"""_

31

TABLE V.- Continued

(b) Inpu t E ( 4 r u n s )

~~ ~~

Mean va lue

~

-0.4639

.8892

.7896

.0764

-. 0438

-.2110

.0647

.0068

"""_ .0492

-. 0802

- . lo42

-. 0368

"""- __ ~~

Standa rd dev ia t ion of mean va lue

0.1868

1.0266

.2648

.0210

.0174

.0907

.0526

.0018

"""

.0203

.I216

.0502

.0014

"""

- OCRLB

0.0332

,1879

.1408

-0074

.0024

.0132

.0076

.0007

"""

.0023

.0130

.0052

.0004

""" I

'*CRLB

0.0100

.0625

.0083

.0008

.0008

.0057

.0034

.0003

"""

.0006

.0038

.0024

.0002

"""

TABLE V.- Continued

(c) Inpu t F ( 2 r u n s )

Mean va lue

-0.5396

.0947

-3182

"""_ -.0413

-. 2538

.0479

"""_ -. 0390

.0598

-. 0462

- . lo72

"""_ ,0058

S tanda rd dev ia t ion of mean v a l u e

~

0.0001

.0079

.0339

"""

.0010

.0262

.0245

"""

.0018

.0038

.0101

.0145

"""

.0002

%RLB

0.0061

.0162

.0265

"""

.0012

.008 1

.0043

"""

.0009

.0004

.0037

.0022

"""

.0004

'DCRLB

0.0008

.0012

.0031

"""

.0003

. 000 1

.0018

"""

. 0001

.0008

.0014

.0004

"""

. 000 1

33

TABLE V.- Continued

( d l Input G (3 r u n s )

Mean va lue

~.

-0.5022

,0328

.1250

"""_ -. 0284 -. 1837 ,007 1

"""_ -. 0288 .0606

-. 0232 -. 1008

"""_ .0083

Standard devia t ion of mean va lue

0.0882

.I789

.1167

- ..

"""

-0147

.0468

.Ob73

"""

.0092

.0011

.0107

-0169

"""

.0022

- %RLB

~.

0.0177

.0432

.OS00

"""

.0020

.0101

.0046

"""

.0011

.ooog

.0064

.0032

"""

.0007 "

S OCRLB

0.0070

.0178

.0137

"""

,0010

-0013

.0020

"""

.0002

.0006

.(lo33

.0008

"""

.0002

34

TABLE V.- Continued

(e) Input H ( 5 runs )

Mean va lue

-0.5442

.I132

.3515

.0307

-. 0475 -. 2374 .0766

.008 1

-. 0397 .0560

-. 0347 -. 0938 -. 0338 .0068


0.0109

.0348

.0315

.0022

.0016

.0128

.0315

.003 1

.0026

.0013

.0046

.0118

.0015

.0012

~

%RLB

~

0.0048

.0152

.0280

.0042

.0012

.0056

.0042

.0008

.0008

.0004

.002 1

. 001 6

.0003

.0068

S ~ C R L B

0.0009

.0030

.0045

.0007

.0002

.0011

.0007

. 000 1

.0002

. 000 1

.0005

.0003

. 000 1

.0012

35

TABLE V.- Continued

(f) Inpu t I ( 3 r u n s )

~

Mean va lue "

~- - - - - - . . . .

-0.5706

.1786

.374 1

.0566

-. 0408

-. 2245

.0678

.0032

-. 0349

.0553

-. 0737

- .09 l l

-. 0386

. 001 8


0.0233

.0274

.0902

.0179

.0040

.0125

.0284

.0046

.0011

.0038

.007 1

.0169

.0028

003 1

". -

.. "~ ..

%RLB -

0.0076

.0242

.0396

.0060

.0016

.0074

.0052

.0009

.0010

.0008

.0043

.0023

.0004

.0006

S %RLB -

0.0007

.0033

.0054

.0006

.0002

.0020

.0008

.0002

.0002

. 000 1

.0006

.0003

.ooo 1

. 000 1

TABLE V.- Continued

( g ) Inpu t J ( 4 r u n s )

Mean va lue

-0.5516

.IO15

.4296

.0457

-.0451

-. 2464

.0750

.0049

-. 0397

.0557

-. 0485

-. 1045

-. 0375

.0056

S tanda rd dev ia t ion of mean va lue

0.0091

.0174

.078 1

.0126

.0018

.0031

.0050

.0013

.0005

.0012

.0058

.0054

.0014

.0009

%RLB

0.0048

.0163

.0273

.00372

.0010

.0048

.0034

.0006

.0007

.0004

.0023

.0016

.0003

.0003

S %RLB

0.0004

.001g

.0024

.0004

.0002

.0008

.0008

. 000 1

. 000 1

.00003

.0002

.0002

. 000 1

.00002

37

Parameter

~

TABLE V.- Concluded

(h) Inpu t K ( 2 r u n s )

Mean valuc

~-

-0.5868

. I 1 4 6

.2929

.0623

-. 0483

-. 2054

.0520

.0106

-. 0364

.0566

-.0451

-. 0948

-. 0380

.0042 "

"" ~ "- ~


0.0035 "

.0267

.0457

.0017

.0032

.0302

.0056

,0060

.0025

.0008

.0113

.0035

.0013

.0008 ___

%RLB - "" ~ . .

0.0078

.0219

.0405

,0058

,0015

.0056

.0049

.0008

.0009

.0006

.0030

.002 1

.0004

.0005

'UCRLB

0.0002

.0012

.0009

. 000 1

.0002

.0004

.0007

. 000 1

.00002

. 0000 1

.0002

.0002

. 0000 1

.00002 _ _ _

TABLE VI.- MAXIMUM LIKELIHOOD ESTIMATES OF STABILITY AND

CONTROL DERIVATIVE VALUES FOR SUBJECT AIRPLANE

Derivative Sample mean, P

0.638

-4.365

-16.875

-. 594

-. 45%

‘-4.000

-8.45 1

-1 .538

-. 558

. I 2 4

.370

.045

-. 046

-. 233

.071

.006

-. 038

.056

-. 048

-. 096

-. 036

.005

Sample standard deviation, s

0.198

.I73

4.264

.072

.035

2.893

.185

.020

.039

.073

.016

.004

.018

.023

.004

.003

.002

.016

.011

.003

.002

Coefficient of variation, s/P x 100

31.0

4.0

25.3

12 .1

7.6

34.2

12.0

3.6

31.4

19.6

34.6

7.9

8.0

32.0

62.3

7.1

3.6

32.8

1 1 . 6

8 .3

40.0

aHeld constant.

39

TABLE V I 1 . - COMPARISON OF DERIVATIVES FROM THE PRESENT STUDY

WITH THOSE FROM METHODS OF REFERENCE 13

-~

P r e s e n t s t u d y ~. . ~

o .638

-4.365

-16.875

-. 594

-. 458

a-4. 000

-8.451

-1.538

-. 558

.124

.370

,045

-. 046

-. 233

.07 1

.006

-. 038

.056

-. 048

-. 096

- .036

.005

Reference

0.285

-4.755

-3.012

-. 535

- .549

-3.052

-6.539

-1.323

-. 285

-.156

.226

.132

-.216

-. 425

.174

.012

b-.071

.058

- .041

- .097

-. 049

b.019

13

bThe derivatives and Cn6a from this source were mul- ‘16a tiplied by -0.5 to make them compatible with the conventions used in present study.

40

6 e

Y

Z

X

Figure 1 . - System of body axes showing pos i t ive sense of v e l o c i t i e s , f o r c e s , and con t ro l de f l ec t ions .

TZ 0.762

I I ns t rumen ta t i on system

""

Accelerometers --

vane

Figure 2.- Three-view drawing of s u b j e c t a i r p l a n e . All dimensions are i n meters.

42

0 10 Time, sec I n p u t A

I-

t 1 ~ 1 1 1

Time, sec

I n p u t B

Figure 3 . - T i m e h i s t o r i e s of s t a b i l a t o r d e f l e c t i o n s i l l u s t r a t i n g t h e t h r e e l o n g i t u d i n a l i n p u t forms used i n t h i s s t u d y .

43

.2

* .1

E O

* .l c m n

c m U .- .-

E O

m m

rQ -.l

- .2

a -.I

- .2

Time, s e t

Time, sec

Time, sec

.25

~n .13 c m U .- E O

ro L -.13

- .25 0 8

- 1 f' 12

Time, sec

Input D Input F

Input E

16 20

Time, sec

8 12 16 Time, sec

Input G

Figure 4.- T ime h i s t o r i e s i l l u s t r a t i n g t h e la teral con t ro l i npu t s u sed i n t h i s s t u d y .

44

a 32 YO

.25

.13 VI c n m L

.m 0

~ " 1 3 'a

- .25 0 e 16 24 32 40

Time, sec Input H

Time, sec

.I3

c n L m

v)

.m 0

'aL - . l3

- .25 0 8 16 32 YO

Time, sec Input J

.2

VI c m

m L

.1

.- n o

o"l

- .2 0 IO 20 30 YO

Time, sec .2

v) c V m

L

._ m O

L - -.l - -2

0 8 16 a 32 YO Time, sec Input I

Time, sec .25

c VI

m

-13

._ w o L

e L - - 1 3

- .2s 0 10 20 30 YO 50

Time, sec Input K

Figure 4.- Concluded.

45

I

0 7

0 6

-1.0 O L

1.0

.8

.6

cxc, .4

.2

0

- T- I Mean 2 Standard deviation "_ I I

Mean f Mean of estimated Cramer-Rao

- 0

-1.0

I -3 .O - 1, - -4.0

Input

lower bound

Figure 5.- Parameter v a l u e s e x t r a c t e d for each l o n g i t u d i n a l i n p u t a l o n g w i t h corresponding run numbers and bars depic t ing ensemble var ia t ions and estimated Cram&-Rao lower bound on v a r i a t i o n s .

46

0

-4

-8

cz -12 q

-16

-20

-24

0 8

0 -

-.1-

-.2 -

- . 3 . a

-.4 -

"5 -

Run

- 0 4

-.6 I I A B C

Input

I Mean f Standard deviation - r-

Mean f Mean of estimated Cram&-Rao lower bound

Or

- T -.2 -T-

I I

I I B C

('e) s i ('e) pul

Input

Input

Q

-.5

I B

('e) si

Input

-T-

t - -1-

Figure 5 .- Continued.

47

- .4

0 9 0 8 cm

6e

I -2.0 L o 4 C A

Input 'T - - 1- I Mean f Mean of estimated Cram&-Rao lower bound

t Mean f Standard deviation

-8 -.8 - I- %

9 -1.2

-1- I J-I

-16

1 A B C A

I I B C

('e) si ('e) pul Input

-T- I I

I I B C

Input Input


0 -

- . 4 -

B

-1.2 "'I

-1.2 - - I 0 1-

- . 4 -

"""" 2: Mean f Standard deviation

I Mean f Mean of estimated Cram&-Rao lower bound

_*-

* Run 22 not included in statistics

"I"l-1"

Input

Figure 6 .- Parameter va lues ex t rac ted f o r each la te ra l input a long w i t h t h e corresponding run numbers and bars depicting ensemble variations and estimated Cram&-Rao lower bound va r i a t ions .

ul 0

P

-2.0

4.0

2 .0

o i -

-2.0

Run

0 10 0 13 8 1'2'

I

0 1 9

""

29

0 30 31 34

"

-T- Mean f Standard deviation -1-

1. Mean t Mean of estimated Cramer-Rao lower bound

-r- I I

I.. Y

"I

1


""I

Input

Figure 6.- Continued.

1.2 -

.8 I-

cy -41 0

Run

0 18 25 " 37 0 34 0 31 8 :O

8 it 0 12

0 11

11" "II -T-

1-

I Mean 2 Mean of estimated Cram&-Rao lower bound

I Mean t Standard deviation 1.2 '

.a

.4 '

0

- 7-

q I

I I I I

-I- -7-

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Figure 9. - Concluded.

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Figure 10.- Lateral f l i g h t data time h i s t o r i e s a n d t h o s e computed by us ing d e r i v a t i v e s o b t a i n e d by methods o f r e f e r e n c e 13 and given i n t a b l e V I I .

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70

~ .

1. Report No. - "___ I 2. Government Accession No.

NASA TP- 1043 .~ "" ~ - . .~ .. - 4. Title and Subtitle

. .

EFFECTS OF CONTROL INPUTS ON THE ESTIMATION OF STABILITY AND CONTROL PARAMETERS OF A LIGHT AIRPLANE

.. -~ ~. . . ." "

7. Author(s)

"

Robert L. Cannaday and William T. Suit ~ ."

9. Performing Organization Name and Address

. " . . "" ~- "-

NASA Langley Research Center Hampton, VA 23665

~ . ~ . . . -. . . - - 12. Sponsoring Agency Name and Address

. "

National Aeronautics and Space Administration Washington, DC 20546

.~~ .. . .

15. Supplementary Notes " "" -

" .

. ..

~ . . 3. Recipient's Catalog No.

5. Report Date December 1977

6. performing Organization Code

8. Performing Organization Report NO. . . ~ -

L-11355 .~

10. Work Unit No.

505-10-11-01 11. Contract or Grant No.

- ~~

13. Type of Report and Period Covered

Technical Paper ~- .. ~

14. Sponsoring Agency Code ~~

. ~- . .~~~ -- - - - . ~~ - ~" - -

16. Abstract

. . . -

The maximum likelihood parameter estimation technique was used to determine the values of stability and control derivatives from flight test data for a low-wing, single-engine, light airplane. Several input forms were used during the tests to investigate the consistency of parameter'estimates as it relates to inputs. These consistencies were compared by using the ensemble variance and estimated Cram&-Rao lower bound. In addition, the relationship between inputs and parameter correlations was investigated. Results from the stabilator inputs are inconclusive but the sequence of rudder input followed by aileron input o r aileron followed by rudder gave more consistent estimates than did rudder or ailerons individually. Also, square-wave inputs appeared to provide slightly improved consistency in the parameter estimates when compared to sine-wave inputs.

17. Key Words (Suggested by Authorb) - .

Parameter extraction Aerodynamic parameters Light aircraft Control inputs

..

19. Security Clanif. (of this report) 20. Security Classif. (of this page) I Unclassified Unclassified

. . . . . . . - . .

18. Distribution Statement ~ -. .

Unclassified - Unlimited

Subject Category 08 ." . . " -

70 * For sale by the Nattonal Technical Information Servlce. Springfield. Vtrglnta 22161

NASA-Langley, 1977

National Aeronautics and Space Administration

Washington, D.C. 20546 Official Business

Penalty for Private Use, $300

THIRD-CLASS BULK RATE Postage and Fees Paid National Aeronautics and Space Administration NASA451

4 1 tu,&, 120577 S00903DS DEPT OF THZ BIB FORCE AF WEAPONS L ABOBBTOBY ATTN: TECHNICAL L I B R A R Y (SUI;) KIRTLAND APB NPl 87177

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