Aircraft Parameter Identification in Frequency Domain with Unsteady Aerodynamic Modelling

AIRCRAFT PARAMETER IDENTIFICATION IN FREQUENCY DOMAIN WITH UNSTEADY AERODYNAMIC MODELLING W. R. Wells* and D. A. Keskar** • Department of Engineering, Wright State University, Day ton, Ohio, U. S. A. "SDC Integrated Services, Inc. , Hampton, Virginia, U. S.A.

Abstract. To assess the effect of unsteady aerodynamic modelling on the estimation of aircraft parameters, numerically simulated data is generated using a math model based on simplified unsteady aerodynamic theory. The data is used in two frequency domain parameter extraction algorithms: one including and the other neglecting unsteady effects. The extracted parameters are compared for any significant difference that may result. Flight data for a light airplane also was used with the two extraction programs for the same purpose. These results indicate that inclusion of unsteady aerodynamics results in significant differences in some of the parameter, particularly in the damping-in-pitch coefficient. Keywords. Unsteady aerodynamics; parameter estimation; data processing; frequency response; modelling. INTRODUCTION

extracted from flight data.

In spite of a wealth of experience over the past two decades in the area of parameter extraction, several problems become apparent when applied to real flight data. Klein (1978) demonstrated the effect of different inputs on the aircraft parameters estimated from flight data by using simple least squares technique and maximum likelihood identification. The data used was preprocessed to remove constant bias errors by using aircraft kinematic equations and an extended Kalman filter. In spite of this degree of accuracy in data, the results indicated there was significant effect of input forms on virtually all estimated lateral stability and control derivatives. In the longitudinal case studied, for two input forms, differences in the estimate of pitching moment derivatives occurred.

MODEL FOR UNSTEADY AERODYNAMICS Unsteady aerodynamic forces on the wing and tail surfaces are estimated through use of the indicia1 lift (lift response due to unit step increase in effective angle of attack) at each surface. Queijo, Wells and Keskar (1978) have shown that the indicia1 lift can be fitted accurately by the equation

where U and c are the freestream velocity and wing root chord respectively. The constants y, z and (CL) are functions of et ss the wing geometry. Queijo, Wells and Keskar (1978) have further shown that the unsteady downwash at the horizontal tail surface due to a unit step change in wing angle of attack can be fitted accurately by the equation

Differences in extracted parameters, and particularly in their variances, have generally been attributed to insufficient excitation of some of the aircraft states and strong correlation between some of the parameters (Wells and Ramachandran, 1976). However, another possible reason is that unsteady aerodynamics associated with load buildup generally have not been modelled. Wells and Queijo (1977) developed a simplified aerodynamic force model based on the physical principle of Prandt1 's lifting line theory and trailing vortex concept to account for unsteady aerodynamic effects in aircraft dynamics. This model is used in the present study to find the effect of unsteady aerodynamic modelling on the aircraft parameters

.,(t) •

(:~)ss 1[1

+

F[l-('-Ut)/o]-l

- G e,p(-2HUt/o

1

(2)

where t is the tail length of the aircraft. The constants F, G, Hand

(~~)ss

functions of the wing geometry. 10 25

are

W. R. Well s and D. A. Kes ka r

1026

The lift and downwash for arbitrary variation in angle of attack can be found from the indicial values by a direct application of Duhamel's integral.

00

f

o

x(t) e- j wt dt

x(t) = (1/2TI ) EQUATIONS OF MOTION

foo

(11 )

x(j w)e j wt dw

Eqs . (3)-(5) are transformed into the following

The perturbed short period longitudinal equations of motion are a (t) = q(t) + (pUS w)/(2m) Cz(t) q(t) = ( u2S c )/(21 ) C (t)

(3)

j w a(j w) = q(j w) - (pUS w)/(2m) {[Nw(j w)

(4)

+ (St/Sw) CLt(j w)] ; (j w)

az(t) = (U/g) [a (t) - q(t)]

(5)

+ (£St/Sw)Nt(jw) q(j w) - C

ww

y

m

where a , q and a z are the angle of attack, pitch rate and vertical acceleration respectively. The functions Cz and Cm are the force and pitching moment coefficients. The unsteady aerodynamic effects are modelled in the Cz(t) and Cm(t) terms as

zo

jw q(j w)

0e(t)

(pUS wCw)/(2I y )

8e (j w)}

(14 )

°e (U/g) [j ~ (j w ) - q(j w)]

where

Cm(t) = -( £St)/(cwS w) CLt(t) + Cm

=

- (£2St )/(C wSwU) Nt(j w) q(j w)

az(jw)

e

(7)

°e

(13)

{- (£St)/(cwSw) CLt(j w) ~ (j w )

(6)

0e(t)

zo

0e(j w)}

e

+ Cm

+C

(12)

_ 00

'lt (jw) • Nt (jw) {I -

(;~t

(1 5)

(l - jwG

• (jw+2HU/c w)-1_ j wcwF/U Ei[( £-c w)jw/U]

where t

CL (t) = f ~ C (t-T) a (T)dT w 0 Lw

(8)

••' p [( , _Cw)jw/U]]

and

t

CL (t) = f ~CL (t-T)[a (T) - £(T) tOt + £/U q( T)] dT t

£(t) = f

o

~£ (t- T ) a (T)dT

(16)

N(j w)

(CL)

[l-jwy

a ss

(9)

(10)

Subscripts "w" and "t" refer to wing and tail respectively. Modification of the longitudinal equations of motion to incorporate unsteady aerodynamics results in integro-differential equations. Numerical integration is time consuming and computation of the sensitivity coefficients in the estimation algorithm is difficult in the time domain. For these reasons, the estimation analysis was performed in the frequency domain. The Fourier transform pair associated with a function x(t) is

• ( j w + 2zU/c w) -1 ]

( 17)

Eqs. (13)-(15) are written in state space form as x(j w) = G(jw, e) u(j w)

(18 )

where X

= [-a , -q, -az]T

u = °e e = [(CL

' (CL ) a sS,w

Cm , Cz ]T oe °e

a ss,t

,

n~~ss

'

1027

Aircraft parameter identifi ca tion

and G{jw,e) is the 3x3 transfer function matrix for the system.

x{jw,e) = x{jw,eo )

The parameters listed above can be related to an alternate set for the aircraft defined as follows:

+ ax{jw,eo)(e-e o) ae

(21)

Substitution of Eq. (21) into Eq. (20) results in L{ e ,R)

- ~ t n IRI

(CL)

+ (St/Sw){C L ) a ss,w a sS,t

(22) 2{ t /C )2 W

(CL) a

where

sS,t

and In the parameter extraction mode, flight data is in the form of time histories of a, q, a z and 0e and are converted into the frequency domain. It is assumed that the measured data is known at equally spaced ins ta nts ill t, i = 0, 1 ... N where N + 1 is the number of data points. The state space form of the frequency spectrum for the measured output is z{jw) = x{j w) + v{jw)

If L{e,R) is maximized with respect to lie and R expressions for the parameter update and R resul t as

(19)

• [ Re N L A* (w.) R-1 v- (w. ~) i =1 1 1

where v is the Fourier transform of the measurement noise assumed to be Gaussian white. The properties of the measurement noise are E{v}

(23)

(24)

=0 A convenient cost function, to judqe the goodness of fit between measured and calculated states is given by

E{vv T} = R MAXIMUM LIKELIHOOD ESTIMATION ALGORITHM

1

J

The parameter estimation algorithm used in this study is the maximum likelihood technique applied in the frequency domain. The algorithm :onsiders noise only in the measurements. The likelihood function to be maximized is

N _

_*

= det [-N L v{w.) v i=l

(w. )]

1

(25)

1

Generally, the fit between measured and calculated states improves with each iteration, and this is reflected in a reducinq cost function. In the present study, iterations were continued until the change in cost J - J function, as defined by k J k l was less

L{ e, R)

k

than 0.01. At that point the parameters were considered to be defined. N

- "2 t n IRI

(20)

This function can be put into a form convenient for maximizing if the state vector is expanded about the nominal parameter e and second-order terms neglected as 0

The parameter-identification alaorithm outlined above requires the computation of the ax (wi ) sensitivity coefficients~. These can J

be determined by taking the indicated

W. R. Well s and D. A. Ke s ka r

102 8

derivatives in Eq. (18) to obtain analytical expressions for each derivative, and then calculating the coefficients based on the measured states and nominal aerodynamic parameters. It was found that a convenient approach was to calculate the sensitivity coefficients using the classical definition of a derivative. First, the frequency response of the various states was calculated for the given control input with the nominal aerodynamic parameters. Each parameter was then incremented individually, and the resulting response characteristics measured. The sensitivity parameters were computed from the equation x(j w,e k+6e k)-x(j w,e k)

TABLE 1 Estimation Simulated {Navion) Parameter

Unsteady

Steady

Cz Cl.

-4 . 99

-4.90 +0.01

C Zq

-4 . 82

-6.28 +0 . 20

C Zo e

-0 . 51

-0.49 +0.01

Cm

-1.00

-0.83 +0.00

Cl.

-

(26)

C mq

-12.77

-14.68 + 0.05

The values of 66k were then reduced, and sensitivity parameters were recomputed. This process was continued until the sensitivity parameters remained very nearly constant for successive reductions in 66 k •

C mc;

-1.42

-1. 36 +0.00

Mk

RESULTS AND DISCUSSION An indication of the effect of modelling the unsteady aerodynamics in the parameter estimation algorithm can be obtained from a numerical simulation. The data simulated is from a light aircraft, the Navion. In this simulation the only unsteady effects were assumed to come from the expression for downwash and not in the indicial lift expressions (i . e. , y=O) since numerical experimentation showed this to be justified .

Fl ight Data The flight data used in this part of the study were for the aircraft configuration shown in Fig. 1. The aircraft geometric and mass characteristics and flight condition are given in Table 2 and the measured flight data is shown in Fig. 2. The constants F, G, and H required to compute frequency response with unsteady aerodynamic modelling were obtained from the geometry of the aircraft. TABLE 2 Characteristics and Flight Conditions

Simulated Data , and ~z generated 0e 0e 0e with the unsteady aerodynamic model were used as "measured" data. A sampling rate of 2/rad was used for data acquisition. An initial guess for the parameters to be estimated was given to start the identification process. In the parameter extraction process, two cases were considered: The responses for

~,~

(1) Estimation of parameters using the extraction algorithm with unsteady aerodynamic terms retained. (2) Estimation using the algorithm with unsteady terms omitted. The results of parameter extraction are shown in Table 1. It is seen that including unsteady aerodynamics in the extraction algorithm has a noticeable effect on the extracted parameters; in particular, C and mq Cz are substantially different for the two q

methods.

e

Item

Wing

Aspect Ratio Taper Ratio Sweep, Deg Root Chord, m Area, m2 F

7.35 1.00 0 1.34 13.56 1.358 0.513 0.065

G

H

Iy

9230 Newtons 2135 Kg_m 2

U

47.5 m/sec

Weight

Horizontal Tail 4.21 1.00 0 0.77 2.51

p

1.076 Kg/m 3

Since the parameter extraction is done in frequency domain, the measured flight data was converted into the frequency domain by Fourier transform techniques. The transformed data for q, and z is shown in Figs. (3)-(5) digitized to 10 points per radian .

a,

a

10 29

Aircraft par ame ter ide ntification

As in the simulated example, the parameters were estimated by two methods; one retaining the unsteady aerodynamic modelling and the other omitting it. Results of each of these two methods are given in Table 3. The fit between measured data and the computed response using estimated parameters with unsteady aerodynamic modelling is shown in Figs. (3)-(5). As is the case with simulated data, the inclusion of unsteady aerodynamics in the extraction algorithm did affect the values of extracted parameters, particularly the damping in pitch parameters C and Cz . mq q TABLE 3 Estimation From Flight (Sundowner) Parameter

Unsteady

Steady

Cz ex

-5.03 +0.04

-4.96 +0.04

Cz q

-0.17 +1.37

-0.25 +1.28

C zo

-0.88 +0.09

-0.75 +0.09

Cm ex

-l. 17 +0.01

-1 . 01 +0.01

C mq

-18.58 + 0.38

-21.93 + 0.38

C mo e

- 3.11 + 0.03

- 3.05 + 0.03

e

dynamics in the extraction model shows significant difference in some of the parameters, particularly in the damping-in-pitch. REFERENCES Klein, V. (1978). Aircraft parameter estimation in frequency domain. Proceedings, AIAA Atmospheric Flight Mechanics Conference, pp . 140-147 . Queijo, M.J., Wells, W.R., and D.A. Keskar (1978). Approximate indicial 1ift function for tapered, swept wings in incompressible flow. NASA Technical Paper 1241. Queijo, M. J., Wells, ~J.R., and D.A. Keskar (1978). The influence of unsteady aerocynamics as extracted aircraft parameters . Proceedings, AIAA Atmospheric Flight Mechanics Conference, pp. 132-139. Wells, W.R., and S. Ramachandran (1976). Flight test design for efficient extraction of aircraft parameters. Proceedings, AIAA Atmospheric Flight Mechanics Conference, pp. 101-107 . Wells, W.R., and M.J. Queijo (1977). Simplified unsteady aerodynamic concepts, with application to parameter estimation. Proceedings, AIAA Atmospheric Flight Mechanics Conference, pp. 39-45.

CONCLUSION Longitudinal equations of motion for lift and drag have been modified to include the effects of unsteady aerodynamics. The result is a pair of complex integro-differential equations, the solution of which is very time consuming even on a modern high speed digital computer. A transformation into the frequency domain using Fourier transform techniques reduced the integro-differential equations to relatively simple algebraic equations, reducing computation time significantly. A parameter extraction program based on the maximum likelihood estimation technique was developed in the frequency domain. The extraction algorithm contains a numerical differentiation method for obtaining sensitivity functions. To assess the effect of unsteady aerodynamic modelling on the extracted aircraft parameters, pseudo data was generated using the unsteady model. Next, parameter extraction was done using two models, one including and the other neglecting unsteady aerodynamic effects. The results indicate that, for the case considered, inclusion of unsteady aero-

· 1. 88

m

2. 50m

Fig. 1. Aircraft used to obtain flight data.

W. R. Wells and D. A. Keskar

10 30

',d"

':r--0

'C:7

-5

c=---

30

20 10 Q.

deg/ sec

O~~~--------~--------~~------~---

-la -20 -30

az 9 units

-;t~~

~--

6e._:~ ~--

deg

-la I

I

I

I

I

I

0

0.5

1.0

1.5

2.0

2.5

I

3.0

Time. t. sec

Fig. 2.

Data obtained from flight test.

, 10 1

++++ Measured

Computed

! I

10

II

0

,.l".~."""" o

2

4

I " " " " de " " " •" 10I

6

Frequency

Fig. 3.

Angle of attack amplitude and phase versus frequency .

103 1

Air c r aft pa r ame t e r ident i f i ca ti on

.. 10 '

++++ Measured Computed

:L" 2

o

/l

+.. + Measured - - - Computed

o~II I I I I IIII I IIIIIIIIIIIIIII~ ' 1 111 1 1

I, " ," " d " " " " d , " " , " d

4

ID'

6

6

('

o

ID

Frequency

l.I

10

B

6

Frequency

.. 10'

/l 10'



az

1II I l!!I! i ! !l !I I I! '[ ! l l ! l l l l li

l.I

6

6

10

Frequency

Fig. 4.

Pitch rate amplitude and phase versus frequency.

Fig. 5.

Acceleration amplitude and phase versus frequency.