[American Institute of Aeronautics and Astronautics AIAA Atmospheric Flight Mechanics Conference - Chicago, Illinois (10 August 2009 - 13 August 2009)] AIAA Atmospheric Flight Mechanics

American Institute of Aeronautics and Astronautics

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An Application of Equation Error Method to Aerodyna mic Model Identification and Parameter Estimation of a Gliding

Flight Vehicle

Ümit Kutluay1 and Dr. Gökmen Mahmutyazıcıoğlu2 TÜBĐTAK-SAGE, Ankara, 06261, TURKEY

and

Prof. Dr. Bülent E. Platin3 Middle East Technical University, Ankara, 06531, TURKEY

This paper describes the application of the equation error method to aerodynamic parameter estimation of a gliding flight vehicle. The flight test data used is not originally designed for estimation, causing data collinearity. A strategy is devised to deal with this situation so that the model identification and parameter estimation can be performed. A computer tool is developed in Matlab® environment for the flight test data processing and aerodynamic identification/estimation. The results of the study show that, although the stepwise regression and equation error method have some potential to be used for flight vehicles other than aircraft, the quality of the flight test data and input design is crucial for aerodynamic parameter identification.

Nomenclature AFCS = automatic flight control system b = span CD = drag coefficient CL = lift coefficient Cl = rolling moment coefficient Cm = pitching moment coefficient Cmq = pitch rate damping stability derivative Cn = yawing moment coefficient CX = axial force coefficient CY = side force coefficient CZ = normal force coefficient d = reference length e = fit error F = F-statistics FTD = flight test data Fv = flight vehicle GFS = global Fourier smoother I = mass moment of inertia J = cost function LS = least squares p = body angular rate about x axis (roll rate) PE = parameter estimation PSD = power spectral density (estimate)

1 Chief Research Engineer, Flight Mechanics Division, [email protected]. 2 Assistant Director, Head of Design Engineering Department, [email protected]. 3 Faculty Member, Mechanical Engineering Department, [email protected].

AIAA Atmospheric Flight Mechanics Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-5724

Copyright © 2009 by TUBITAK-SAGE. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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q = body angular rate about Y axis (pitch rate)

q = combined pitch rate and time rate of change of angle of attack derivative r = body angular rate about z axis (yaw rate) R2 = coefficient of determination S = reference area SI = system identification X = regressor matrix x = regressor y = measurement α = angle of attack β = sideslip angle δ = control deflection angle δa = aileron deflection angle δe = elevator deflection angle δr = rudder deflection angle ε = equation error ρ = density of air ρi = ith correlation coefficient θ = unknown parameter ^ = estimated value ¯ = mean value ˙ = time derivative

I. Problem Definition or over 50 years, engineers working in the field of atmospheric flight mechanics have been practicing and continuously developing methods for identifying the dynamical models and estimating the parameters of these models for flight vehicles (fv). As Hamel and Jategaonkar1 stated, the flight vehicle system

identification and parameter estimation is essential in validating and if necessary updating the mathematical models of aerodynamics and thereof values of aerodynamic parameters of these models, so that simulation, control system design and evaluation, and dynamic analysis can be carried out.

The system identification and parameter estimation of flight vehicles originally started with the need to identify the aerodynamic characteristics of the aircraft; thus, almost all of the methods that are used in practice today were developed for identification of aircraft problems. Although there are plenty of references in the literature on the estimation of aerodynamic coefficients of an aircraft from the flight test data, only a limited number of studies exist for other types of flight vehicles.

The common practice for aircraft system identification starts with the maneuver design. In fact, an optimal input design is considered to be the integral part of the aerodynamic model identification and parameter estimation problem by many engineers. Once an optimal set of inputs are obtained, the aircraft, which is equipped with all the necessary sensors and a data recording system, is flown according to a carefully designed flight test program and the required “high quality” flight test data is obtained.

However, the life is not that easy when it comes to the system identification and parameter estimation of other types of flight vehicles for which the main purpose of the flight tests are not system identification/parameter estimation but rather design validation or flight performance verification. Most of the time, the input design is sacrificed for project schedule and flight test costs. Furthermore, for such vehicles, sometimes only limited measurements can be recorded. Thus, trying to obtain adequate models and good parameter estimates using such flight test data is something like digging a hole with a pin!

Figure 1 shows a typical aircraft flight test maneuver history of Mach number, angle of attack, pitch rate, stabilator, leading edge flap, and trailing edge flap deflections2. As seen, the flight test data (FTD) for the aircraft starts from a trim condition and throughout the maneuver, the flight parameters oscillate about this trim condition. The essential excitation of aircraft dynamic modes is provided by the application of carefully designed inputs.

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Figure 1. Typical aircraft flight test maneuver data2.

On the other hand, Figure 2 shows the important parameters of the flight test data gathered during different test

runs of a flight vehicle in a gliding flight. As seen from the plots, the onboard automatic flight control system (AFCS) assures the vehicle to be always in trimmed flight. However, for this particular case, some unexpected oscillations are encountered during the test runs, most probably due to a combination of factors including atmospheric wind/turbulence and erroneous aerodynamic databases.

Figure 2. Flight test data

In fact, the biggest adverse effect of AFCS to system identification/parameter estimation is caused by its

disturbance rejection feature. For a vehicle with a feedback flight augmentation system, the recorded data is almost “stripped” of the vehicle's inherent dynamics unless some carefully designed maneuvers are flown. This can be seen in Figure 3, which shows the power spectral density (PSD) estimates of the in-flight aerodynamic coefficients for

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the test runs of Figure 2. In spite of the encountered (and in fact, from the performance point of view unwanted) oscillations, it is seen that the rolling and the pitching dynamics of the vehicle are almost indistinguishable from the noise. Also notice very low spectral powers - down to order of 10-7 Watts/Hz for rolling moment and down to 10-4 Watts/Hz for pitching moment. This phenomenon is known as data collinearity.

Figure 3. PSD estimates for the in-flight aerodynamic coefficients

II. Dealing with Data Collinearity The data collinearity is defined as “any situation where regressors are correlated at a high enough level to cause

problems in the parameter estimation”3. The main cause of data collinearity is the near-linear dependence of regressors. There are a number of possible sources of data collinearity:

1. Improper design of flight test maneuvers leads to data collinearity either due to insufficient excitation of fv modes or changing the data of two or more regressors proportionally.

2. Constraints of the fv such as feedback control system lead to data collinearity. As Jategaonkar stated4, “the controller reacts to the motion and suppresses the oscillatory and transient motion….. it is detrimental to parameter estimation, because it drastically reduces the information contents required for estimating parameters”. Also the control allocation algorithm of the flight vehicle may cause the collinearity, since some of the control surfaces are deflected proportionally based on some control mixing strategy hard coded to on board flight computer.

3. Specifying regressors that are small in magnitude can also cause data collinearity. If a regressor is small, then all the higher order regressors derived from it will be very small. Thus, they will be almost the same regressor, which will have an effect on the results as data collinearity.

The FTD of Figure 2 inherently caused data collinearity caused by the AFCS, as explained previously. To tackle this problem, a new approach is devised as “clipping” the FTD to look like an aircraft flight test maneuver data. Since the stepwise regression method, which utilizes equation error method for parameter estimation, does not require the dynamics of the actual maneuver to be matched, that is it treats each data point separately, this approach can be realized with success. Figure 4 shows such an “augmented” FTD, which is obtained by processing the original FTD of Figure 2.

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Figure 4. Augmented FTD

Figure 5 shows the PSD estimate for the in-flight aerodynamic coefficients after the clipping of the flight test

data. Comparing to Figure 3, it is seen that the power is boosted by a factor 10 and the vehicle’s all dynamic modes are now apparent.

Figure 5. PSD estimates for the processed in-flight aerodynamic coefficients

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III. A Brief Overview of Equation Error Method and Stepwise Regression Parameter estimation methods that are not based on probability theory but rely on the laws of statistics are

generally termed as the equation error methods, since they minimize a cost function defined directly in terms of an input-output relationship4. The most widely used subset of equation error methods is the "least squares estimation", which allows the calculation of estimates in a one-shot procedure using matrix algebra. Although least squares (LS) estimators are unbiased, efficient, and consistent in theory, they neither are as efficient, consistent nor unbiased in practice due to the fact that the flight test data always contain some measurement errors. However, LS estimators are still widely preferred mainly because of the simplicity of the algorithm coming from the fact that no iterations are needed to estimate the parameters. Also it is rather easy to combine data from different maneuvers for a single analysis. Furthermore, LS estimates serve as the nominal starting values for the ML methods5. There is an extensive literature available on the least squares estimation problem. Regardless of the previously stated limitations of the method, the equation error approach is still widely preferred in practical applications of aircraft.

Given N discrete data samples of a dependent variable, y, and nq independent variables, xi (i=1,…,nq) a linear combination can be defined at each time step k as4:

�� (1) In Eq.(1), �� are the unknown parameters and is the equation error, representing the model discrepancies and/or

noise in the dependent variable y. The x vector contains the motion variables such as angle of attack and control surface deflections, where �� are the stability and control derivatives. Notice that the values of �� do not depend on time, they are constants.

Eq.(1) can be rewritten in matrix format as follows: � � �� (2) In Eq.(2), the errors are called the "residuals". It is obvious that, the difference between the dependent variable y,

the “observation” and the model consisting of the independent variables x, the “regressors” or the “explanatory variables”, and unknown parameters should be zero for a perfect model. Thus, to obtain the best estimates of the unknown parameters, the residuals should be minimized. This is done via differentiating the following cost function

�� 12 � ��

�

�� 1

2 �� 12 �� (3)

which is the sum of the square errors cost function differentiated with respect to θ to obtain:

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� � �� !��" (4)

Least squares estimates of the unknown parameters, �#, can be found by equating Eq.(4) to zero and solving for

�: �# � !��"$�� (5) Notice that to be able to use the equation error method, the aerodynamic model must be declared in the first

place. If a priori knowledge is available on the model structure, the method can be applied directly on the selected regressors. However, if the model structure is not defined, than a special care must be taken to identify it from the FTD.

A number of methods have been proposed in the past to tackle the problem of finding an adequate model based on some metric rather than pure judgment of an experienced engineer. These methods fall under the classification of regression methods.

The most widely used one among these methods is the stepwise regression due to its advantage of both forward and backward evaluation and selection capabilities. As explained by Klein and Batterson6, the determination of the aerodynamic model structure using stepwise regression includes three steps:

1. Postulation of the terms which might enter the model.


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2. Selection of an adequate model. 3. Verification of the model selected.

A detailed derivation of the method is given by Jategaonkar4. Briefly summarizing his work the four steps of stepwise regression procedure are as follows:

1. A set of possible independent variables (motion variables) are defined and the correlations of each of these independent variables with the dependent variable are sought. The independent variable with the highest correlation is added to the model.

2. The independent variable from the remaining set with the highest partial correlation is added to the model. 3. F-statistics for all the included independent variables are calculated and those found to be below the pre-

specified threshold are excluded from the model. 4. Steps 2-3 are repeated until no other independent variable is left.

IV. Application During the test runs of the flight vehicle, the following data are recorded: • Altitude • Velocities • Body angular rates • Body linear accelerations • Control surface deflections

The following parameters are calculated during post processing of the data using the in-flight measurements: • Angle of attack • Angle of side slip • Mach number

Using six degrees of freedom equations of motion, which can be found in any flight mechanics textbook, in-flight aerodynamic coefficients are obtained as:

%& � '()

*+

%, � '(-*+

%. � '(/*+

%0 � �%& cos�4� � %.sin �4� %7 � %& sin�4� � %.cos �4�

%8 � 9):; � 9)/<; � 9)/:* � !9- � 9/"*<*+=

%> � 9-*; 9)/�:� � <�� 9/ � 9)�:<*+?

%� � 9/<; � 9)/:; 9)/*< � !9) � 9-":**+=

(6)

The angular accelerations in Eq. (6) must be derived from the angular rates that are measured during the test.

This causes a problem when the rate measurements are noisy. Similarly, time derivatives of flight angles must be gathered from the flight angle measurements, and then same problem is faced once again. In fact, regardless of whether the derivative of a signal is required or not, it is always preferred to work with noise free signals.

The common practice to get rid of noise is to utilize digital filters. However, filtering, if not applied correctly, can distort the system identification process since it affects both the phase and magnitude of the data. Among a number of filtering methods, Global Fourier Smoothing (GFS) is mostly preferred for applications of system identification with equation error method.

The GFS method originally proposed by Morelli7 can be summarized as follows3: 1. The end point discontinuities of the signal to be filtered are removed by subtracting a linear trend from the

time series. 2. The obtained time series is reflected about the origin to remove the endpoint slope discontinuities.


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3. Now the signal can be expanded into a Fourier sine series. 4. Since the noise signal is incoherent, in contrast to the signal that contains the actual dynamics of the flight

vehicle, it is assumed to have constant power over a wide frequency range. Thus, the Fourier sine series coefficients associated with the noise signal will be almost constant throughout the frequency of interest where as the Fourier sine series coefficients of the coherent signal (i.e. the actual dynamics of the fv) will rapidly decrease to zero. That is, by plotting the Fourier sine series coefficients versus frequency, it is possible to discriminate between the noise and signal visually (Figure 6).

Figure 6. Fourier sine series coefficients plotted versus frequency

Once the cut-off frequency is determined using the four step procedure described above, a digital filter can be

designed to filter out this noise. For this purpose, the Wiener filter, which is near unity at low frequencies and is near zero at cut-off frequency, is used. Since the Fourier coefficients near the cut-off frequency is small by definition, the Wiener filter tolerates the small errors that might be made during the visual selection of the cut-off frequency.

As the last step of the smoothing procedure, the linear trend which is removed in the first step is restored. The result is a noise free signal, such as the roll rate history shown in Figure 7.

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Figure 7. Rolling rate – smoothed and original signal

One of the decisions that has to be made before the estimation is the sampling rate of the data. Basically, the

sampling rate at which the FTD is recorded is selected based on the a priori information about the dynamics of the flight vehicle. However, since the aerodynamic database contains some discrepancies before the flight test, so does the natural frequencies of the flight vehicle. Thus, a sample rate, which is higher than the one foreseen by the a priori information, is selected for data recording. However, this higher sample rate could lead to large number of samples, which in turn requires more computational power than necessary and sometimes more than available. A direct approach to overcome this “oversampling” is to examine the power spectral density estimates of the FTD, so that a “resampling” can be done before the estimation. For this purpose, the power spectral density estimation, which gives information about how the time series is distributed with frequency, is utilized. Since Mach number and altitude are continuously changing during the flight, the natural frequencies of the modes shift during the test run. So, the raw flight test data is resampled based on the maximum Nyquist frequency of all the coefficients.

A modified stepwise algorithm is implemented for the aerodynamic model identification and parameter estimation. The regressors (Table 1) are grouped as families of primary flight parameters, such as angle of attack, angle of side slip, control deflections, rates and etc. This grouping strategy allows the use of a specific regressor group for the drag/axial force coefficient identification. The specific regressor group is called the “delta family”, which is a non-linear combination of control deflections as given in Eq. (7) . This eliminates the correlation between control deflections, thus a better model is obtained.

@� � @A

� @B� @C

� (7) Note that in Table 1, the regressors that include q can either be built using the pitch rate alone or a linear

combination of pitch rate and time rate of change of angle of attack. A number of stopping rules and exclusion rules are implemented to the algorithm. The most important ones are

the partial correlation and F-ratio limits. During the run, if the partial correlations of the regressors outside the model are below the specified partial correlation lower limit and at the same time the F-ratios of the regressors inside the model are above the F-ratio lower limit then the algorithm stops. Also a limit on goodness of fit (R2) improvement and monotonic increase criteria for adjusted goodness of fit metric are implemented. Although a limit of 1% is recommended for goodness of fit improvement in the literature as a rule of thumb, it is found that a much lower limit can lead to better estimates provided that an exclusion rule is implemented to avoid correlations between the regressors in the model.

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Table 1. Regressors Alpha family

Beta family

Alpha - Beta family

Rates family

Control/Delta family

Mach family

α β αβ p δa M

α2 β2 α2β q δe α3 β3 α3β r δr α4 β4 αβ2 p2 δa

2 αδ2 βδ2 α2β2 q2 δe

2 α2δ2 β2δ2 αβ3 r2 δr

2 αδa βδa αβδ2 p3 δ2 α2δa β2δa αβδa q3 δ4 αδa

2 βδa2 αβδe r3 δa

3 αδe βδe αβδr pq δe

3 α2δe β2δe

p2q δr3

αδe2 βδe

2 pq2

αδr βδr

pr

α2δr β2δr p2r

αδr

2 βδr2

pr2

αp βp qr

α2p β2p

q2r

αp2 βp2 qr2

αq βq α2q β2q αq2 βq2 αr βr α2r β2r

αr2 βr2

αpq βpq

αpr βpr αqr βqr

This exclusion rule is implemented such that, at every selection step of the stepwise regression algorithm, the

parameter correlation matrix is searched for correlated regressors. If there exist correlations between regressors in the model, then this shows up as large elements in the parameter correlation matrix. As a rule of thumb, Klein and Morelli1, suggest that any value larger than 0.9 is a sign of strong correlation between parameters (regressors) of the model. If this is encountered, then the regressor causing the correlation, i.e. the one added latest, is removed from the model and deleted from the pool of candidate regressors. Since the algorithm selects the parameters of the model from the pool of regressors based on their correlation/partial correlation metric, deletion of a regressor does not introduce any complications to estimation algorithm because parameters with higher importance have already been included! Then the algorithm carries on searching for any other regressors which might be included in the model without causing correlation with the parameters already in the model. Also, a convergence time limit is implemented as a safeguard.

In the first step of the stepwise regression algorithm, when there is no regressor included in the model but the static term, correlations between the candidate regressors and FTD are sought. Then the algorithm selects the regressor with the highest correlation coefficient as the first parameter of the model. For a “healthy” FTD, the correlation coefficients in this first step are very high and the algorithm does not encounter any difficulties in


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selecting the parameters. However, in some cases of the flight vehicle under consideration this correlation may be low enough to cause identification of models that are physically meaningless. Then for these cases, it is wise to stop seeking a model by the use of stepwise regression algorithm and switch to the estimation of a default model for that data set. Although the parameter estimates obtained by utilization of equation error method on the default model are still unreliable, at least they are not misleading to some meaningless model.

Table 2 summarizes the proposed default models for the six coefficients of the flight vehicle.

Table 2. Default models Aerodynamic Parameter Default Model Drag Force (D) Coefficient %0 � %0D %0EF4� %0GF @�

Yawing Force (Y) Coefficient %, � %,D %,HI %,GJ @C

Lift Force (L) Coefficient %7 � %7D %7E4 %7GK@B

Rolling Moment (L) Coefficient %8 � %8D %8HI %8GJ@C %8GL @A %8M: N2O

Pitching Moment (M) Coefficient %> � %>D %>E 4 %>GK @B %>�* N2O

Yawing Moment (N) Coefficient %� � %�D %�HI %�GJ @C %�GL @A %�J< N2O

A simple tool is designed in Matlab® for the processing of the flight data and identification of the aerodynamic

model of the flight vehicle. Designed as a modular code, some functions of the tool are based on an open source software package, the SIDPAC of Klein and Morelli3. Figure 8 shows the flowchart of the estimation methodology.

Figure 8. Flowchart of the estimation methodology


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V. Results Figure 9 shows the in-flight aerodynamic coefficients of the processed FTD given in Figure 4. For this FTD, a

stepwise regression is carried out with the following settings: - Re-sampling rate : 12 Hz - Partial correlation lower limit : 0.1 - Correlation lower limit : 0.2 - F-ratio lower limit (automatically decided) : 19.25 - R2 minimum improvement limit : 0.001% - Convergence time limit : 150 seconds

Figure 9. Inflight aerodynamics from FTD

The drag force coefficient is identified with a nonlinear model consisting of five parameters (Table 3). The

maximum error is estimated to be 66% in the normalized Mach derivative. Estimated static term for the drag coefficient is practically zero. The model response seems to agree well with the FTD (Figure 10).

Table 3. Identified model and errors for drag force coefficient

Parameters Percent Error

α 9

αβ 19

α2*P2 9

M 66

CD

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Figure 10. Model vs FTD for drag force coefficient

The yawing force coefficient is identified with a nonlinear model consisting of six parameters (Table 4). The

maximum error is estimated to be 16% in the angle of side slip and rudder deflection squared coupled derivative. The model response seems to agree well with the FTD (Figure 11).

Figure 11. Model vs FTD for yawing force coefficient

Table 4. Identified model and errors for yawing force coefficient


α4 12

β4 8

δr 3

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βδr2 16

α2*P 8

Static term 13 The lift force coefficient is identified with a nonlinear model consisting of four parameters (Table 5). The

maximum error is estimated to be 65% in the static term. The model response seems to agree well with the FTD for most parts (Figure 12).

Table 5. Identified model and errors for lift force coefficient


α 15

δr3 36

α2*P 15

Static term 65

Figure 12. Model vs FTD for lift force coefficient

The identified model for rolling moment coefficient includes a very small, almost negligible static term, and four

other derivatives (Table 6). Although Figure 13 shows that the obtained model is capable of representing most of the actual phenomena, the unmodeled dynamics is obvious as missed peaks.

Table 6. Identified model and errors for rolling moment coefficient


δa 7

αδe2 27

αδr2 19

α2*P 25

Static term 28

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Figure 13. Model vs FTD for rolling moment coefficient

The identified model for pitching moment coefficient includes three terms (Table 7). The obtained model is

capable of representing most of the actual phenomena. However, the mismatch in magnitude between the model response and the data for the second part of the motion is obvious.

Table 7. Identified model and errors for pitching moment coefficient


α 11

αδe 37

Static term 10

Figure 14. Model vs FTD for pitching moment coefficient

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The yawing moment coefficient is identified with a nonlinear model consisting of five parameters (Table 8). The maximum error is estimated to be 20% in the angle of attack squared and aileron deflection coupled derivative. The model response seems to agree well with the FTD (Figure 15).

Table 8. Identified model and errors for yawing moment coefficient


α3β 10

α2δa 20

αδr 3

βδr2 13

Static term 10

Figure 15. Model vs FTD for yawing moment coefficient

VI. Conclusion It is known that input design is a crucial part of system identification/parameter estimation of flight vehicles. In

fact this is the biggest lesson gathered from this study: it cannot be estimated if it is not in the data! Nevertheless, the results of the study are somewhat promising.

When the estimated models are compared, it is seen that utilization of “clipping” strategy yields better results over the unprocessed FTD. In fact, due to very low correlation between the regressors and the unprocessed data, the stepwise regression and equation error estimation sometimes failed to identify a model for moment coefficients. On the other hand, it is shown that representative models are obtained using the clipping strategy.

However, the “quality” of solution is strictly dependent on the selection of the parts of the FTD to be used in identification. In a similar study8, when the FTD from four different test runs of the same vehicle is used, force coefficients and yawing force moment coefficient are obtained, but the algorithm failed to yield models for rolling and pitching moment coefficients. This shows that the clipping strategy is very sensitive to data selection when used in conjunction with equation error method.

As future work, efforts will be concentrated on improvement of the stability of the strategy.

Acknowledgments TÜBĐTAK-SAGE are acknowledged for their support to this study.

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References 1 Hamel, P.G., Jategaonkar, R.V., “Evolution of Flight Vehicle System Identification”, Journal of Aircraft, Vol.33, No.1,

1996, pp.9 – 28. 2 Morelli, E. A., Ward, D.G.., “Automated Simulation Updates Based on Flight Data”, AIAA Atmospheric Flight

Mechanics Conference and Exhibit, Hilton Head, South Carolina, Aug. 20-23, 2007, AIAA 2007-6714 3 Klein, V., Morelli, E. A., “Aircraft System Identification: Theory and Practice”, AIAA Education Series, AIAA, 2006 4 Jategaonkar, R. V., “Flight Vehicle System Identification: A Time Domain Methodology”, Progress in Astronautics

and Aeronautics Volume 216, AIAA, 2006. 5 Morelli, E. A., Klein, V., “Application of System Identification to Aircraft at NASA Langley Research Center”, Journal

of Aircraft, Vol. 42, No.1, 2005, pp. 12 – 25. 6 Klein, V., Batterson, J. G., “Determination of Airplane Model Structure from Flight Data Using Splines and Stepwise

Regression”, NASA TP 2126, NASA, 1983. 7 Morelli, E.A., “Estimating Noise Characteristics From Flight Test Data Using Optimal Fourier Smoothing”, Journal of

Aircraft, Vol.32, No.4, 1995, pp. 689 – 695. 8 Kutluay, U., Mahmutyazicioglu, M., Platin, B.E., “An Application of Equation Error Method to Aerodynamic Model

Identification and Parameter Estimation for Gliding Flight”, to be presented at Ankara International Aerospace Conference, Ankara, Aug. 17-19, 2009, AIAC – 2009 – 050.

Documents

[American Institute of Aeronautics and Astronautics AIAA Atmospheric Flight Mechanics Conference - Chicago, Illinois (10 August 2009 - 13 August 2009)] AIAA Atmospheric Flight Mechanics