On the adequate model for aircraft parameter estimation

6 Ü3

1975

CRANFIELD

INSTITUTE OF TECHNOLOGY

Bibliotheek T U Delft

Faculteit der Luchtvaart- en Ruimtevaarttechniek Kluyverv^/eg 1

2629 HS Delft

ON THE ADEQUATE MODEL FOR

AIRCRAFT PARAMETER ESTIMATION

by

Cranfield Institute of Technology

ON THE ADEQUATE MODEL FOR AIRCRAFT PARAMETER ESTIMATION

by

SUMMARY

The problem of the selection from measured data of an aircraft of

an adequate model which v.ould he the simplest and sufficient

approximation to the correct model and which would facilitate the

successful determination of the unknown parameters is discussed.

Two v/ays for the proper model structure verification are recommended,

namely sensitivity analysis and/or testing of a hypothesis as to the

significance of unknovm parameters in the model proposed, and the

analysis of residuals. Finally some approaches towards the assessment

of parameter and adequate model accuracies are proposed. The procedures

mentioned are demonstrated in an example.

CONTENTS

Page

1 . INTRODUCTION "•

2. CORRECT AND ADEQUATE iMODEL 3

3. SIGNIFICANT PARAMETERS 5

4. ANALYSIS OF RESIDUALS 9

5. ACCURACY OF THE MODEL AND PARAMETER ESTIMATES 11

6. EXAMPLE 12

7 . CONCLUSION ^^

8 . REFERENCES 17

TABLES

LIST OF TABLES

Table

1. Estimates, Sensitivities and t- Statistics for Different

Models Used («^ = 20.5 deg).

2. Comparison of Parameter Determined from Repeated Measurements.

(«g = 7.2 deg).

LIST OF FIGURES

Fig.

1. Comparison of Time Histories Measured with those Computed.

(a = 20.5 deg).

2. Time Histories of Normalized Residuals (a = 20.5 deg).

3. Sample Autocovariance Functions of Residuals (a = 20.5 deg).

4. Cumulative Frequency of Residuals. Adequate Model (a = 20.5 deg).

5. Input Forms Used in the Excitation of the Aircraft Motion

(ag = 7 . 2 d e g ) .

6. Comparison of Time Histories Measured with those Computed. Run 1

(og = 7.2 deg).

7. Comparison of Time Histories Measured with those Predicted. Run 1

(cg = 7.2 deg).

8. Time Histories of Differences betv/een Measured and Predicted

Output Variables. Run 1 (« = 7.2 deg).

NOTATION

g acceleration due to gravity, (m/s^)

H sensitivity matrix

M a) pitching moment, (Nm)

b) information matrix

m number of outputs

H number of data points

n reading of the vertical accelerometer, (g units)

* g

q a) pitching velocity, (rad/s)

b) number of unknown parameters

R measurement noise covariance matrix

R autocovariance function

r correlation lag number

s(.) standard error

2

s (•) variance estimate

t a) time, (s)

b) Student's t - v a r i a b l e

u a) longitudinal airspeed component, (m/s)

b) input vector

Z vertical force, (M)

X state vector

y output vector

z measurement vector

o a) angle of attack

b) level of confidence

0 vector of unknown parameters

n eleven d e f l e c t i o n , ' ( r a d )

0 pitch angle, (rad)

V residual

o standard deviation ^ ,

a^-

standard error

Matrix exponents:

T indicates transpose matrix operation

-1 indicates inverse natrix operation

Subscripts:

E measured quantity

P predicted quantity

e steady-state value

Additional notation:

cov( •) covariance

|.| determinant of a matrix enclosed

over symbol indicates an estimate

INTRODUCTION

The e s t i m a t i o n o f s t a b i l i t y and c o n t r o l parameters from f l i g h t data** has Lecome the standard procedure f o r a i r c r a f t and f l i g h t c o n d i t i o n s where the: aerodynamic c h a r a c t e r i s t i c s can be described i n l i n e a r terms o n l y , and where no s i g n i f i c a n t e x t e r n a l disturbances are presented. New t h e r e i s a tendency t o widen the parameter e s t i m a t i o n technique i n t o f l i g h t regimes w i t h n o n - l i n e a r aerodynamic e f f e c t s and i n t o f l i g h t manoeuvres f o r which the non-linear functions o f the a i r c r a f t s t a t e s expressing the i n e r t i a forces and niotrents, and g r a v i t y terms must i n general be used.

The problem of modelling a complicated system raises the fundamental question o f how complex the model should bo. Although a more complex model can be j u s t i f i e d f o r proper d e s c r i p t i o n o f a i r c r a f t m o t i o n , i t has not been c l e a r i n the case of parameter e s t i m a t i o n what would be the best r e l a t i o n s h i p between model complexity and measurement

i n f o r m a t i o n . I f too many unknown parameters are sought f o r a l i m i t e d amount o f d a t a , then a reduced confidence i n evaluated parameters can be expected ( l a r g e covariance and/or non-physical values of some parariie"".ers), or attempts t o i d e n t i f y a l l parameters niiglit f a i l .

I n the f i e l d of system i d e n t i f i c a t i o n w i t h general a p p l i c a t i o n a number of d i f f e r e n t methods f o r determining an adequate model have' been developed. Most o f these nietnods introduced i n R e f . l are

connected w i t h the determination of model order in parameter e s t i m a t i o n f o r the s i n g l e i n p u t - s i n g l e output syste.-ii.

One of the f i r s t attempts t o t e s t the correctness of the model r e p r e s e n t i n g an a i r c r a f t was introduced in K e f . 2 , The appropriate

s t a t i s t i c was formed by the r a t i o between variance estiirates of the neasureJ frequency response c u r v e . Gne o f these cstiiiiates was outdined from repeated measurements under the same c o n d i t i o n s , the other from the r e s i d u a l s .

In Ref ,3 the analysis of r e s i d u a l s v/as recommended f o r checking

for finding the parameters in the model proposed whose effect on the

measured responses v/as neglinible. Neither approach, however, has

been brought to a conclusion as to the adequate structure of the

model used.

More comprehensive treatment of the model structure determination is

proposed and used by Hall and others in Ref.5. It is based on the

testing of model parameters in the regression analysis. A criterion

for retaining significant parameters is proposed. An adequate model

is developed either from simulated data using a priori knowledge from

wind-tunnel measurements or other sources, or from measured flight

data which are then analysed by the more advanced technique, e.g. the

maximum likelihood method. In the first case the results can be used

for the proper design of an experiment, mainly an optimal input. In the

second case the expected adequate form of the model enters the final part

of the analysis providing even better estimates of unknov/n parameters.

The procedure mentioned assumes, without any explicit statement, the

same adequate model for both the regression analysis and the maximum

likelihood method and no effect of bias errors in the least-squares

para-meter estimates on the F-statisties used for testing of the parapara-meter

significance.

Stephner and Mehra in Ref.6 proposed a new criterion for fit error.

This criterion does not improve monotonically with the increased number

of unknown parameters, but has some minimum value which could determine

the optimal number of parameters included in a m.odel.

Finally Taylor in Ref.7 developed another criterion for finding the

optimal number of unknown parameters which is based on the expected model

response error. The criterion selects the most promising model from

various candidate ciodels.

In the report presented an attempt is made in the generalisation of

experiences fron: Rof.3 and Ref;.4 to achieve an objective approach towards

a decision as to an adequate model for parameter estimation from a given

set of flight data.

CORRECT AND ADEQUATE MODEL

To represent any f l y i n g v e h i c l e completely v/ould be a task of immense d i f f i c u l t y . The c o r r e c t model of an a i r c r a f t i s i n general unknown and unknowable. In system i d e n t i f i c a t i o n the problem i s t h e r e f o r e the s e l e c t i o n from m.easured data o f an adequate model which would be the • simplest and s u f f i c i e n t approximation t o the c o r r e c t model and v/hich v/ould f a c i l i t a t e the successful determination of the unknov/n parameters. An adequate model should include only these terms which have physical meaning and which s i g n i f i c a n t l y i n f l u e n c e the output o f the system.

The i d e n t i f i c a t i o n procedure w i t h an adequate model w i l l wery l i k e l y r e s u l t i n greater f i t e r r o r than t h a t w i t h the over-parameterized one. On the other hand the use of an adequate model should provide more accurate parameter e s t i m a t e s .

The determination of an adequate model involves tv/o s t e p s , namely c h a r a c t e r i z a t i o n and v e r i f i c a t i o n . For an a i r c r a f t of orthodox design i n a s m a l l - d i s t u r b a n c e l o n g i t u d i n a l or l a t e r a l motion around some steady-state e q u i l i b r i u m c o n d i t i o n s , the equations of motion are y/ery w e l l known and no problems are expected in the f o r m u l a t i o n o f an adequate model. In g e n e r a l , hov/ever, the c h a r a c t e r i z a t i o n could be more d i f f i c u l t and must take i n t o account the known physics of the i n v e s t i g a t e d manoeuvre and the a p r i o r i knowledge of the aerodynamic c h a r a c t e r i s t i c s of the a i r c r a f t , mainly obtained from wind-tunnel m.easurements. These considerations w i l l then be r e f l e c t e d i n model c o m p l e x i t y , i . e . the number o f s t a t e variables and the form o f expressions f o r the aerodynamic forces and moments.

The models f o r l a r g o - d i s t u r b a n c e manoeuvres or coupled motions of an

a i r c r a f t can i n some cases be s i m p l i f i e d by i n t r o d u c i n g the less s i g n i f i c a n t s t a t e v a r i a b l e s as the a d d i t i o n a l i n p u t to the system. As examples t h e r e are the s h o r t - p e r i o d l o n g i t u d i n a l motion w i t h small airspeed f l u c t u a t i o n s and the r a p i d - r o l l i n g r.anocuvro w i t h induced l o n g i t u d i n a l motion.

A m.ore d i f f i c u l t procedure, however, w i l l be i n deciding which aero-dynamic parameters adequately describe the i n p u t - o u t p u t data f o r a given

mianoeuvre. I f the l i n e a r approximation seems to be i n s u f f i c i e n t , then the quadratic f u n c t i o n or the e n t i r e polynomial must be used. Both i n t e r p r e t a t i o n s r e s u l t from s e r i e s expansions around some t r i m c o n d i t i o n s assuming continuous d e r i v a t i v e s .

The n o n - l i n e a r r e p r e s e n t a t i o n of the aerodynamic forces and moments has been s u c c e s s f u l l y used i n some cases, as reported in R e f . 4 , by Wells i n Ref.8 (quadratic approximation) and by Hall and others i n Ref.5 (polynomial, f o r m ) . However, even i n those cases where non-linear aerodynamic terms are s u b s t a n t i a t e d , the l i n e a r model can s t i l l be good approximation and the estimated parameters from i t could provide a d d i t i o n a l a p r i o r i

i n f o r m a t i o n .

When the parameter e s t i m a t i o n i s completed, the accuracy of the model s t r u c t u r e should be v e r i f i e d . In t h i s r e p o r t tv/o ways are recommended. The f i r s t i s connected w i t h the e f f e c t o f a l l unknown parameters included i n the model proposed. S e n s i t i v i t y analysis ( s u b j e c t i v e approach) and the t e s t i n g o f a c e r t a i n hypothesis ( o b j e c t i v e approach) can be employed.

The second assessment i s based on the analysis of r e s i d u a l s , and i s considered as the i n v e s t i g a t i o n of the model's o v e r a l l adequacy. When both checks are p o s i t i v e , the f i n a l accuracy of the model and parameters can be e s t a b l i s h e d . I f the opposite r e s u l t s are o b t a i n e d , then the whole i d e n t i f i c a t i o n must be repeated on the c h a r a c t e r i z a t i o n l e v e l .

SIGNIFICANT PARAMETERS

The state and output equations of a system with no process noise can

be v/ritten as

f(x, u, 6 ) , x(0)

(3.1)

g(x, u, e)

(3.2)

where x i s an (n x 1) s t a t e v e c t o r , u i s an (P x 1) i n p u t v e c t o r , B i s a (q X 1) vector o f unknown parameters and y i s an (m x 1) output v e c t o r . I t i s assumed v/ithout any f u r t h e r loss o f g e n e r a l i t y t h a t a l l the i n i t i a l c o n d i t i o n s are equal t o zero.

The concept of s e n s i t i v i t y can be t r e a t e d as the s e n s i t i v i t y o f the f u n c t i o n y = g ( x , u , 8 ) w i t h respect to the parameters. Therefore the parameter s e n s i t i v i t y f u n c t i o n s are

'^w

ae,

/ k = 1 , 2 , . . . ,m

' ' ' ^ j - 1 . 2 . . . . , q

At each time i n t e r v a l t - the values of the parameter s e n s i t i v i t y f u n c t i o n s form the elements of the (m x q) s e n s i t i v i t y matrix H . , i = 1 , 2 , . . . , N , where N i s the number of data p o i n t s . I f the modified Newton-Raphson computing technique f o r parameter e s t i m a t i o n i s used, then the s e n s i t i v i t y matrix enters the i n f o r m a t i o n matrix v/hich f o r the maximum l i k e l i h o o d (ML) e s t i m a t i o n i s given as

N M = z i = l ( — - ) ^-^ f • ' \ N i = l r T 1 1 H. R H . ( 3 . 3 )

v/here R i s the measurement noise covariance m a t r i x ,

For the comparison o f a l l s e n s i t i v i t y functions r e l a t e d t o d i f f e r e n t parameters, the r e l a t i v e s e n s i t i v i t y f u n c t i o n

1 9 y k ( t ) B- 9 y k ( t )

Oy a i n p . a^ aPj

of the measureP'.ent noise.

Nevertheless, the assessment of the sensitivity based on (3.4) is not

very practical. It is therefore preferable to replace the time functions in (3.4) by constant terms defining the sensitivity for the time interval considered.

One o f the possibilities v/ould be the formulation of the term for the sensitivity as

0,^

1 = 1 I

\

J "•''

taking into account each output variable separately, and the formulation of the term for the sensitivity of the v/hole system as

k J

Comparing (3.G) with the form of tht information matrix in (3.3) yields the relationship between the relative sensitivity and the information matrix in the form

2 m , N ray^(ti) ,2

J k = l OyJ- 1=1 93j J "J JJ

= B ' M , . (3.7)

where M.- i s tfie main diagonal element i n M corresponding t o &• JJ J •

The o v e r a l l s e n s i t i v i t y ( 3 . 6 ) w i l l depend on the c h a r a c t e r i s t i c s o f the system, tiie design o f the experiment, and the accuracy of measurement. The nuhierical values o f ( 3 . 6 ) can detect c r i t i c a l parai„eters, i . e . parameters -which have a n e g l i g i b l e e f f e c t on the response o f the system and -which

t h e r e f o r e w i l l yery l i k e l y be estimated w i t h an unacceptable accuracy.

For the o b j e c t i v e d e c i s i o n as to which terms should be deleted from the

model proposed f o r paranieter e s t i i n a t i o n , a t e s t s t a t i s t i c i.-ay be used, This s t a t i s t i c has the fonr.

4. _ ! i - ( 3 . 8 )

'' ~ s - .

BJ

where B i s the parameter estimate and S j - i s the standard e r r o r o f the

J p j

e s t i i r a t e , see e . g . Tcef.S. Using the ML estimation the s t a t i s t i c i s changed as

t = —i- (3.9)

JJ

wliere M.' is the main diagonal element in the error covariance matrix for the estimated parameters.

The null hypothesis thcit 3- = ü is accepted if

'^' ^ ''^a, N-q (3.1Ü)

where a is the level of confidence (usually a = Ü.35) and N-q is the

number of statistical degrees of freedom. The value of t , ., ,. ^ . ^ a/2, N-q IS found

from the tables of Student's distribution. For a = 0.95 and N-q-x», t=1.96.

It| < t^/2,M-q, tliG alternative hypothesis that B- ={= 0 is then valid.

J

The a p p l i c a t i o n of the t e s t mentioned i s as f u n o \ / s : suppose a confidence l e v e l of 0.95 i s cliosen. The t - s t u t i s t i c is then computed f o r each estimated paraiLC'ter and the parameter v;ith n.iniiiium | t | i s found. I f I t l . >t , , ^ ' ' ' 'min a/2 a l l terms are ccncluoed to be s i g n i f i c a n t at the 0,u5 l e v e l . I f

U L - < t , , the parameter corresponding t o the Itl,.,- i s dropped from the ' 'min a/2 liiin ' ^

model and the e s t i m a t i o n i s recomputed w i t h the new model. The process i s ' rcpeatea t i l l the adequate form of the model i s reached.

the t e s t i n g procedure, the a p r i o r i weighting can be used. With a decreasing number of parameters, the c o n s t r a i n t due t o a p r i o r i v/eighting can gradually be l i f t e d and possibly abandoned f o r an

adequate model,

The adequate miodel r e s u l t i n g from the t e s t i n g of paramieters i s not n e c e s s a r i l y unique. The proper form of t h i s model w i l l depend on the i n i t i a l m o d e l l i n g , where tfie a p r i o r i knowledge as to the a e r o -dynandc c h a r a c t e r i s t i c s of the a i r c r a f t and the physics of the motion are considered.

4. ANALYSIS OF RESIDUALS

For the ML estimation, the residuals v. = z. - y. should form a sequence

of uncorrelated random variables with gaussian distribution and zero mean,

see e.g.Ref.6.

The simplest way to check some of the assumptions mentioned is the time

sequence plot of residuals. From this time history trends can be apparent

and possible deterministic components discovered. If the normalized residuals

^t,(^-;)/s(yi,), k= 1.2 ...,m are plotted, then any values greater than three

can be considered as outliers.

The check of normality uses the calculated percentage cumulative distribution,

P-, for each of N samples of the residuals V|,(t.). Because of the large

number of data points, the grouped residuals are usually used in the

calculation. The probability of the residuals v. less than or equal to the

jth class 1imit is

P. = P , v . i « j . ^ , , . (4.1)

wherev^are the midpoints of the class intervals and Av is the length of the

J

class intervals.

It is advantageous to plot v. as abcissas and the corresponding P. values

J J

as ordinates on the probability paper, on which the ordinate scale is

graduated according to the area under a normal distribution function. Then

the fitted line of all values of P,. m.ust, in the case of a normal distribution,

be a straight 1ine.

The mean value can also be checked and is zero if the condition

P{v. < v.. = 0} = 50%

is satisfied. The difference in abcissas for P^. = 50% and P. = 15.9% is

o J

an estimate of

a.

The established value of the standard error can be

therefore used for fitting the straight line to the plotted (P,v.) data.

-J -J

The assumption of uncorrelated residuals is checked by the autocorrelation

function of the residuals. Its estimate is found from the expression

T N-r

^ ( 0 = - ^ , ^ ^ - i . r ' ^ = ^ ' » 1 - - - ' W ('-^^

i - 1

I f the r e s i d u a l s are to be u n c o r r e l a t e d , and i n the normal case also independent, then the c o n d i t i o n K ( r f 0) = O must be s a t i s f i e d .

In p r a c t i c e even f o r an adequate model R ( r f 0) i s never e x a c t l y z e r o , but v a r i e s s l i g h t l y around t h i s value. Hov/ever, these non-zero values should be v/ell w i t h i n the 2a-bounds. The r e l a t i v e standard e r r o r f o r the a u t o c o r r e l a t i o n f u n c t i o n estimate i s approximated as

— = — ( 4 . 3 ) R ( r = 0) ^

ACCURACY OF THE MODEL AND PARAMETER ESTIMATES

When the parameter and adequate p-odel estimations are completed, the accuracy of the results obtained ought to be checked. In this process presumably the most important point is the comparison between the estimated parameters and those for which either a priori values or at least limits on their values are known. This comparison must also take into account the standard errors and correlation coefficients of the estimates.

The standard errors define the confidence limits for each parameter

regardless of the remaining parameters. The correlation coefficient is the measure of stochastic dependence. There has been no effort to develop a criterion for testing the significant correlation. Usually a value of the correlation coefficient greater than 0.85 is considered significant.

In the next step of the accuracy assessment the output time history match between the actual and estimated responses found from the identified model is checked. A small fit error is a necessary but not sufficient condition for accurate result.

Increased confidence in the estimated parameters and adequate form of a model can be obtained from repeated measurements under the same conditions. The problem of applying either similar or different inputs must be carefully considered. A decision should be taken separately for each group of

repeated measurements. In principle, the differences in input forms should not substantially change the sensitivities with respect to unknown parameters If these changes occur, then their effect must be taken into account in

comparing repeated parameter estimates.

From a large number of repeated measurements the ensemble mean values and ensemble standard errors of estimated parameters can be obtained. For accurate results the two quantities are expected to be close to the corresponding ones from each individual measurement.

The ability of the estimates from one test to predict the response of another flight can also be checked. For comparing the predicted and measured output time histories, the confidence limits for the predictions may be found. These limits follov/ from the covariance matrix of prediction

error, which has the form

/ • •

cov {y(t. B)} = H.M

H T

+ R (5.1)

The prediction error is defined as

Spi = 2^(3) -

y^ih

(5.2)

EXAMPLE

As an example, the parameter e s t i m a t i o n and adequate model determination f o r a slender d e l t a - w i n g research a i r c r a f t are presented. The l o n g i t u d i n a l responses o f the a i r c r a f t were e x c i t e d from the h o r i z o n t a l s t e a d y - s t a t e f l i g h t s by elevon d e f l e c t i o n . Because o f the input form used and the a i r c r a f t c h a r a c t e r i s t i c s , the airspeed changes during the t r a n s i e n t motion v/ere negl i g i b l e .

Taking i n t o account the physics o f the motion, the v/ind-tunnel measurements and p r e l i m i n a r y f l i g h t t e s t r e s u l t s , the equation of motion were formulated as a = Z a + Z q + Z 2^ + ^„a Q" + Z „ na + _{a q^ a / q" ' n} + Z n + Z'n + Z _{n n o} q = M a + M q + M y'^^ + Ka^°^ + ''•'' na + ( 6 . 1 ) ' a q a'^ q ^ na M n ^ M-n + M n + n 0 0 = q

In these equations only the pitch rate, q, was measured. The second output

variable n is related to the state variables by the equation

n* = Z a + Zlq + Z 9 + Z 23^ + Z„ 1^ +

Z a q^ a n^ qa '

+ Z not + Z n + Z 'n + Z ^

(f, o\

The s t a t e and out[;'jt oouations are developed i n R e f . 4 , v/hore the parameters contained i n these equations are also d e f i n e d . The ML e s t i m a t i o n techni^'jc without process noise described i n Ref.3 vas used i n the data a n a l y s i s .

The r e s u l t s from one ter-t run at h-;nh angle of attack (a = 20,5 deg.) are summarized in Table 1 . These r e s u l t s include the estimated parameters,

standard e r r o r s of the estimates (lower bounds), variance estimates of measurement n o i s e , the logarithm of the determinant of measurement noise covariance m a t r i x ( o v e r a l l f i t ) , s e n s i t i v i t i e s and t - s t a t i s t i c s . The tabulated t - v a l u e f o r a = 0.05 v/as equal to 1.98. ^

The f i r s t paraireter ostimatt.'S v.'ero obtained from the l i n e a r model

a = Z CÏ + Z 'i + Z n + Z q = M a + M t, + I'! n + M ^ a q T) : 0 e = q _(6.3) Z ni + Z / ' i + Z o + Z n + Z^ o; 'i 0 r, o

where Z„ was known and Z„, Z,, and 7. v/ere fixed on the wind-tunnel values. 0 T f] n

The r e s u l t e d estimates agreed reasonably w i t h those from wind-tunnel d a t a , but the parameters f1 and M v/ere s t r o n g l y c o r r e l a t e d and the f i t between measured and computed v e r t i c a l a c c e l e r a t i o n time h i s t o r i e s was r a t h e r poor.

The next e s t i m a t i o n was based on the .complete equations ( 6 . 1 ) and (6.2) w i t h Z , Z and Z. treated as knovn values. Because of inconsistency i n

the number of unknown parameters and measured o u t p u t s , v/eighting of the basic l i n e a r parameters had t o be used. The previous estimates of Z , M ,

' ' a a

M and M from the l i n e a r model and t h e i r standard e r r o r s formed the a p r i o r i q n

data.

The e s t i n ; 3 t i o n r e s u l t e d in s i g n i f i c a n t l y changed damping parameter M , b e t t e r accuracy of the estimates, and b e t t o r f i t i n both output v a r i a b l e s

The sensitivity analysis revealed small effects of all non-linear terms

in the lift-force equation on measured responses. This v/as also confirmed

by com.paring the computed and tabulated t-statistics.

In the following estimation the insignificant param.eters Z 2> Z a and Z

were successively dropped from the model. Then the remaining parameters

v/ere proved to be significant. The adequate model obtained had increased

sensitivities, lower standard errors in some estimates, and only slightly

higher fit errors. Hov/ever, the main advantage of the adequate model in

comparison with the complete one v/as a considerable improvement in the

convergence of the iterative procedure.

Even for the adequate model, it was still necessary to use a priori

v/eighting. Nevertheless the effect of the gradual decrease of v/eights v/as

investigated. As a result, no substantial changes in the estimates v/ere

observed, but the convergence v/as slowing down and the parameter covariances

were increasing.

The measured and computed outputs are plotted in Fig.l. The time histories

of normalized residuals for the linear and adequate model are compared in

Fig.2, the corresponding autocovariance functions in Fig.3. A substantial

improvement in the shape of the autocovariance functions for the adequate

model is apparent. However, their forms are still different from those

for v/hite noise. This might be due to poor quality of measured data, which

is demonstrated by the residuals for both inodels used and also by the

cumulative frequencies for the adequate m.odel plotted in Fig.4.

The comparison of results from two repeated measurements with some of the

pararrieters obtained from wind-tunnel and steady-state measurements

(parameter M ) is made in Table 2. The inputs used in both runs are

presented in Fig.5, the measured and computed outputs for Run 1 in Fig.6.

The repeatibility of results from the tv/o runs is

very

good, as follows

from Table 2 and also from Figs. 7 and 8. In Fig.7 the measured outputs

are plotted, together with the predicted ones based on the estimates from

Run 2. The resulting differences between the measured and predicted

responses are shown in Fig.8. These differences are within the 2a bounds

for the prediction error defined b^- equation (5.2).

CONCLUSION

Because in general the correct model of an aircraft is unknown and

unknowable the problem of identification also encompasses the selection

from measured data of an adequate model. This model should be the

simplest and sufficient approximation to the correct model and should

facilitate the successful determination of the unknown parameters. The

determination of an adequate model involves two steps, namely

character-ization and verification.

The characterization must take into account the known physics of the

investigated manoeuvre and the a priori knowledge of the aerodynamic

characteristics of the aircraft. These considerations will then be

reflected in model complexity, i.e. the number of state variables and

the form of expressions for the aerodynamic forces and moments.

For the verification of the model proposed two steps have been developed.

The first one is based on the sensitivity analysis and/or testing a certain

hypothesis. The sensitivity analysis is the subjective approach which can

detect critical parameters which have a negligible effect on the response

of the aircraft under test and which therefore will very likely be

estimated with an unacceptable accuracy.

For the objective decision as to which terms should be deleted from the

model proposed the t-test may be used. In this test the computed t-statisties

for each estimated parameter are compared with the tabulated value from

Student's distribution and the insignificant terms successively dropped

from the model till its adequate form is found.

Both approaches require a very limited amount of additional computing when

the parameter estimation is completed. Their disadvantage could be in

using an over-parameterized model in the first step of the procedure.

This drawback can be overcome, however, by applying a priori weighting

during the parameter estimation,

The second step in the model verification is based on the analysis of

residuals and is considered as the investigation of the model's overall

adequacy.

Where both steps

are

positive, the final accuracy of the model and

parameters can be established. If the opposite results are obtained,

then the whole identification must be repeated on the characterization

level.

With the tendency to widen the aircraft parameter estimation into flight

regimes with non-linear aerodynamic effects and into flight manoeuvres

with other non-linear functions the increasing importance.of the proper

model structure can be expected. For this reason more experience with

the procedures developed for the determination of an adequate model and

the further research in this area are highly recommended,

REFERENCES

1. UNBEHAUEN, H, GORING, B,

Test for Determining Model Order in Parameter Estimation.

Automatica, Vol. 10, pp.233-244, Perganon Press, 1974

2. KLEIN, V. TOSOVSKY, J,

General Theory of Complex Random Variable and its A|)plication to the Curve Fitting a Frequency Response

(Summary Report).

Zprava VZLU, 2-11 December 1967.

KLEIN, V, Parameter I d e n t i f i c a t i o n Applied t o A i r c r a f t .

Report CIT-FI-73-018, October 1973 or

Cranfield Report Aero No.26. December 1974.

KLEIN, V, Longitudinal Aerodynamic D e r i v a t i v e s o f a Slender Delta-Wing Research A i r c r a f t Extracted from F l i g h t Data. Report CIT-FI-74-023, J u l y 1974 or

C r a n f i e l d Report Aero No.27, January 1975,

5. HALL, W.E., J r . GUPTA, N.K. TYLER, J . S , , J r .

Model S t r u c t u r e Determination and

Parameter I d e n t i f i c a t i o n f o r Non-Linear Aerodynamic F l i g h t Regimes.

Paper presented at AGARD S p e c i a l i s t s Meeting on Methods f o r A i r c r a f t State and Parameter E s t i m a t i o n , NASA Langley Research Center, Hampton, V i r g i n i a , November 5 - 8 , 1974.

STEPHNER, D.E MEHRA, R.K.

Maximum L i k e l i h o o d I d e n t i f i c a t i o n

and Optimal Input Design f o r I d e n t i f y i n g A i r c r a f t S t a b i l i t y and Control D e r i v a t i v e s ,

NASA CR-22Ü0 March 1973.

TAYLOR L,W., Jr. Application of a New Criterion for Model1 ing Systems.

Paper presented at AGARD Specialists Meeting on Methods for Aircraft State and Parameter Estimation, NASA Langley Research Center, Hampton, Virginia, November 5-8, 1974.

RAMACHANDRAN, S, WELLS, W.R.

Estimation of Non-linear Aerodynamic Derivatives of a Variable Geometry Fighter Aircraft from Flight Data. AlAA Paper No. 74-790, AlAA.

Mechanics and Control of Flight

Conference, Anaheim, Cal., August 5-9, 1974.

HIMMELBLAU, D.M. Process Analysis by Statistical Methods John Wiley & Sons., Inc., 1970.

Za Z„2 \ a \ . \ Zn Ma M q Ma^

V

Mria ^ >^fl

1 s^q)

s 2 ( n * ) I H I R I

h

- 1 . 4 3 5 -- 0 , 2 7 -- 1 . 7 6 - 0 . 7 8 -- 3 . 5 4 s ( 6 j ) 0 . 0 8 0 -0 . 1 5 0 . 1 0 -0 . 1 1 -4 . 5 8 X 1 0 - 5 1 6 . 5 X 1 0 " ^ - 2 1 , 0 0 <%

'i

- 1 . 4 3 0 - 1 . 7 1.4 3 , 2 - 0 . 2 0 4 - 0 , 0 0 5 6 - 1 . 5 8 - 0 . 3 8 7 - 5 5 18 164 - 3 . 4 0 6 - 0 . 3 3 7 s ( 0 j ) 0 . 0 5 1 5 . 9 2 . 7 2 . 6 0 . 0 1 1 0 . 0 0 1 6 0 . 1 3 0 . 0 5 6 13 9 . 2 44 0 . 0 9 3 0 . 0 2 5 B?M.. J J J 7286 2 2 ^ 646 29 3636 1153 1005 184 312 5 4 9 5 6 1374 1 . 0 9 X 1 0 - 5 5 . 2 0 X 1 0 - 6 - 2 3 . 5 9 x)

11|

2 8 . 2 0 . 3 0 . 5 1.2 1 9 . 2 3 , 4 1 1 . 9 6 . 9 4 . 2 2 . 0 3 . 7 3 6 . 7 1 3 . 3 6 . J - 1 . 4 3 2 -- 0 . 2 0 4 - 0 . 0 0 4 9 - 1 . 5 4 - 0 . 4 7 - 7 0 23 179 - 3 . 2 2 2 - 0 . 3 0 7 s ( 6 . ) J 0 . 0 4 0 -0 . -0 -0 8 9 0 . 0 0 1 0 0 . 1 3 0 . 0 5 8 11 6 . 9 48 0 . 0 0 8 7 0 . 0 2 5 8 ? M . . J J J 13063 -1726 48 4485 1819 2536 436 363 79594 2973 1 . 2 1 X 1 0 - 5 4 . 8 7 X 1 0 - 6 - 2 3 . 5 5 x )

Itl

3 5 . 9 -2 -2 . 9 4 . 9 1 1 . 9 8 . 0 6 . 2 3 . 4 3 . 7 3 7 . 1

12.5

1

TABLE 1. ESTIMATES, SENSITIVITIES AND T - STATISTICS FOR DIFFERENT MODELS USED (ttg = 20,5 deg).

ITEM Z a Z na

z

n

z.

n

M a M q M qa M

n

M.

n

s2(q) s2(n*) In R| PRED - 1,7 - 0.38 - 6.8 - 1.6 -13

h

- 1.525 - 0.38 - 5,366 - 1,993 -14,08

s(êj)

0.031 0.091 0.062 0,23 2.28 X 10~^ _6 4,80 X 10 - 20.68 B. J - 1.516 7.8 - 0.292 - 0.0115 - 5.450 - 2.197 -17.6 -13.8 - 0.083

s(ê.)

0.021 2.5 0.019 0,0012 0.061 0.056 2.9 0.15 0.032 2.08 X 10 ^ _6 2.57 X 10 - 23.65 6. J - 1.544 - 0.38 - 5,96 - 1,743 -13.31

s(ê.)

0.041 0.12 0.077 0,29 -5 3,96 X 10 _6 7,70 X 10 - 21.91 B. J - 1,505 3.3 - 0.402 - 0.0096 - 5.678 - 1.754 -18.9 -12,32 - 0.495 s(B^) 0.024 3.0 0.030 0,0018 0.077 0,070 3.8 0.16 0.031 -5 1.87 X 10 _6 4.45 X 10 - 23.21

TABLE 2. COMPARISON OF PARAMETER DETERMINED FROM REPEATED MEASUREMENTS, (a- = 7,2deg).

q [deg/s]

O

- 2 - 3 - 4 - 5 ^

1

r

V'

/ /

f

y^

\/WVv Measured ouni|Jutt;u,uueL|uuLt: model z [g units] • 2

• 15

• I

•05

O - • 0 5 - .1

f\

\

' ' j

-V

_V

t\^

r\

\ \

vJ

V

\ ^

/VW-WV

4 5

'[O

FIG. 1 COMPARISON OF TIME HISTORIES MEASURED WITH THOSE COMPUTED. («g = 20.5 deg).

n 6

O

- 2

O

4

t[s]

FIG. 1 COMPARISON OF TIME HISTORIES MEASURED WITH THOSE COMPUTED. (o;g= 20.5 deg). _ C o n c l u d e d .

s(q) O - 4 4

. r

\

Y

'^

hh-J

Vyy*"^

- V W M A / V " (f Y

M^

\ _{^ A A / ^ ^ • ' ' ^} .' nz O - 4 sCq) •4 4

II

H/^A^

v^

/^yvM»

huH

\ *«„

_WV|/VV

Adequate Model

rtJA.

^ V

# • nz - 4

R

_qO

Adequate Model 0 - • 5 1 0 nzO - . 5

K

\

.__ riN_

r s •^ ^ —

^\ 1

^ ^ ^ ^ : r ^ ^ > ^

10 15

r

J

m

80

50

20

y

yo / "

r

/

o/°

o /

p

y"

- • 6

- 3

• 3

[deg/s]

6

99

P. J

{i\

80

50

20

n ^ O O ^ o ^ ^ . ^ ' o ^ ^ o -{•50

-75

•75 1 5 0 10x1)^^ [q units]

FIG, 4 CUMULATIVE FREQUENCY OF RESIDUALS, ADEQUATE MODEL («g = 20,5 deg).

[deg]

•'^•V^v^.^s/^ v v v » . . » mtm,

_"VV^

/fc^«w>r>rV*^

8

[deg]

O

- 2

A

1

V

- \ /

\ r^-^—

Run 2

i 6

FIG. 5 INPUT FORMS USED IN THE EXCITATION OF THE AIRCRAFT MOTION (ag,= 7.2deg).

q

[deg/s]

O

- 2

- 4

8

10

ir^

/ /

V

[ V

. . y ^ " ^'^^^^^^

/ /

' . - > . - / i >^A

/vr^^rs^/ruv\

S-yVÖi / \ i ^ 1

•v^v'v r V r

Measured

computea,aaequate

model

i.

[g units]

• 5

4

• 3

•2

-•2

- • 3

ft

n

f

\ \ \ \ \

" V s ^

•** —""^ 'VWü^A-t^^g^VA

v*-^--^J

5 6

t [s]

FIG, 6 COMPARISON OF TIME HISTORIES MEASURED WITH THOSE COMPUTED. RUN 1 («g = 7.2 deg).

j

/

\j

f

Measured,Run 1 Predict edjparametcrs from Run 2 0

r

1

, A . .

1

V

0

\ \ ; ^ . ) 1 • • • • " \ A \ ..:

^.A'V^'VVA/

> 6

t[s]

G.7 COMPARISON OF TIME HISTORIES MEASURED WITH THOSE PREDICTED. RUN 1 (ag=7.2deg).

n ^ n

-[g units]

• 08

04

O

- . 0 4

08

» 1

• v

11 ^

f

y

I l i

UK

yn

Sivjw

\ .

X

VAMKAT

Al

AA

%

f

5 6

t[0

FIG. 8 TIME HISTORIES OF DIFFERENCES BETWEEN MEASURED AND PREDICTED OUTPUT VARIABLES. Run 1 (0^ = 7.2 deg).