MULTI-AXIS MANEUVER DESIGN FOR AIRCRAFTPARAMETER ESTIMATION

Mathias Stefan Roeser, Institute of Flight SystemsDLR (German Aerospace Center)

Lilienthalplatz 7, 38108 Braunschweig, Germany

Abstract

A flight test campaign for system identification is a time and cost consuming task. Mod-els derived from wind tunnel experiments and CFD calculations must be validated and/orupdated with flight data in order to match the real aircraft stability and control character-istics. Classical maneuvers for system identification are mostly one-surface-at-time inputsand need to be performed several times at each flight condition. Multi-surface or multi-axismaneuvers have been designed in the past, however, they are only time or frequency decor-related. Therefore, a new design method based on the wavelet transform was developedthat allows to define multi-axis inputs in the time-frequency domain. Using such inputs,simulated flight test data was generated from a high-quality Airbus A320 dynamic model.System identification was then performed with this data and the results show that accurateaerodynamic parameters can be extracted from these multi-axis maneuvers.

NOMENCLATURE

Symbols

2Nf frequency tiling

2Nt time tiling

A dilation parameter

b wing span, m

B translation parameter

c mean aerodynamic chord, m

CD drag coefficient

CL lift coefficient

Cl, Cm, Cn body-axes nondimensionalmoment coefficients

CX , CY , CZ body-axes nondimensionalforce coefficients

e Oswald factor

E energy spectrum

g acceleration due to gravity, m/s2

iHT horizontal tail deflection, deg

Ix, Iy, Iz, Ixz mass moments of inertia, Nm2

k1 linear drag polar parameter

m aircraft mass, kg

p, q, r body-fixed rotational rates, deg/s

p∗, q∗, r∗ normalized body-fixedrotational rates

q dynamic pressure, N/m2

S wing reference area, m2

SHT horizontal tail reference area, m2

t time, s

∆t time step for multistep input, s

T engine thrust, N

T (A,B) wavelet transform

u, v, w body-fixed translationalvelocities, m/s

V true airspeed, m/s

x(t) continuous signal

xHT , zHT horizontal tail lever arms, m

x′HT distance between horizontal tail

neutral point and wing-bodycenter of gravity, m

Greek Symbols

α angle of attack, deg

αdyn dynamic angle of attack at HT, deg

β angle of sideslip, deg

δε/δα downwash change due tochange of angle of attack

εHT downwash angle at HT, deg

η elevator deflection, deg

Λ wing aspect ratio

ω frequency, 1/s

φ, θ roll and pitch attitude, deg

ψ(A,B)(t) normalized wavelet function

τ time delay, s

ξl, ξr left/right aileron deflection, deg

ζ rudder deflection, deg

Subscripts

0 initial or reference value

HT horizontal tail

WB wing body

Abbreviations

CFD Computational Fluid Dynamics

CWT Continuous Wavelet Transform

DWPT Discrete WaveletPacket Transform

DWT Discrete Wavelet Transform

FFT Fast Fourier Transform

IDWPT Inverse Discrete WaveletPacket Transform

QTG Qualification Test Guide

Sys-ID System Identification

TFP Time-Frequency Plane

TFR Time-Frequency Representation

WPT Wavelet Packet Transform

1 INTRODUCTION

Aerodynamic models used for certification, per-formance and handling qualities evaluations,flight simulators, control law design, etc., mustbe validated and perhaps updated with flight testdata to match the real aircraft stability and con-trol characteristics. For this purpose an exten-sive system identification flight test campaign isusually performed, in which dedicated maneu-vers are conducted at distinct pre-defined flightconditions.

Maneuver design for aircraft parameter esti-mation is done using an a-priori model of theaircraft, typically derived from wind tunnel ex-periments and CFD calculations. It is commonpractice to create inputs that excite the aircraft atits expected eigenmodes. Marchand [1, 2] undPlaetschke et al. [3] showed that evaluating thefrequency response magnitude of the terms ofeach equation of the aircraft’s linear system isone possibility to identify the regions of identifi-ability of each derivative. These regions lie in thevicinity of the natural frequencies of the aircraft’seigenmodes. However, a-priori aircraft modelsare subject to uncertainties and therefore the ma-neuver design must also consider frequenciesslightly above and below the expected eigenfre-quencies.

Classical maneuvers to excite the aircraft atits expected eigenmodes are multistep inputs.Herein the length of the steps is chosen such thatthe natural frequency of the excited mode lies inthe center or upper third of the input spectrum[4]. Multistep inputs like doublets or 3211 signalsare easy to execute manually and are thereforewidely used for system identification purposes.

The 3211-input e.g. has a broad frequency spec-trum, see Figure 1, and suits well for parame-ter estimation maneuvers, because even if thenatural frequency of the aircraft response slightlychanges due to different flight conditions or devi-ations from the a-priori model, this input coversthese uncertainties and must not be redesignedfor every flight condition. The aircraft responseto these multistep inputs is also easy to interpret,which is important when identifying the modelstructure, as will be discussed in Section 6.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

Normalized Frequency, [ω∆t]

Ene

rgy

Spe

ctru

m, [

E/∆

t2 ]

ImpulseDoublet3211

Figure 1: Frequency spectrum comparison of dif-ferent standard inputs

Another type of input for system identificationare frequency sweeps. However, this method istime consuming because each surface has to beexcited separately and the excitation may last upto 2 minutes. The advantage is that a completefrequency response of the system to one singlecontrol input can be evaluated from a single flighttest point.

Using maneuvers that excite multiple controlsurfaces at the same time has a great potential toreduce flight test time and costs. Such multi-axismaneuvers were already investigated in the late70’s, where Ramachandran and Wells [5] inves-tigated the identification of aerodynamic parame-ters for a light aircraft by applying inputs on rud-der and aileron simultaneously to minimize thecorrelation between the model parameters dur-ing identification. Further multi-axis maneuvershave been designed to excite the aircraft in allsix-degrees of freedom.

Morelli introduced a method to create multi-axis input signals based on orthogonal optimizedmulti-sine waves that cover a broad range of fre-quencies [6, 7]. Accurate parameter estimates

using the equation error method in the frequencydomain were obtained from flight test data withthis type of inputs applied continuously during themaneuvers.

Another approach for multi-axis maneuver de-sign has been developed by Lichota [8]. This de-sign method generates a so called D-Optimal sig-nal, based on the Fisher information matrix. Thismethod relies also on a-priori knowledge of theaircraft model, and is only optimal for the flighttest point considered. The a-priori model is as-sumed to be linear and there is no time informa-tion about the frequencies excited.

The goal of this research is to propose newways to design maneuvers in order to reduceflight test time to get very accurate airplane sim-ulation models. These simulation models mustat least fulfill the Qualification Test Guide (QTG)criteria for level-D full flight simulators.

A method is desired that allows to generatemulti-axis input signals with the ability to specifyboth the frequency content and the times whenthe frequencies are excited, with as little parame-ters as possible. A parametrization is performedusing the wavelet transform which has been ap-plied for image processing, seismic signal de-noising and analysis of diverse other physicalphenomena [9]. The wavelet transform yields aTime-Frequency Representation (TFR) of a sig-nal and the signal can be reconstructed from itsTFR.

The idea behind the proposed method is tostart by specifying the desired TFR and to gen-erate the input signals by inverse wavelet trans-form.

This work is structured as follows: the nextSection gives a short overview over the appliedrigid-body equations of motion and the aerody-namic model. Section 3 introduces the wavelettransform and the method used for the signal def-inition is explained in Section 4 together with itsapplicability to multi-axis signals. In Section 5parameter estimation results are presented thatwere obtained from simulated flight test data us-ing the new signal generation method. A com-parative discussion to already existing multi-axismaneuver designs is given in Section 6. Finally,Section 7 sums up the work in this paper giv-ing the next steps and future applicability for themethod introduced.

2 BASIC MODEL FORMULATION

The basic aircraft equations of motion are de-scribed by a six degree-of-freedom kinematicmodel. The translation motion is given by

u = rv − qw +qS

mCX − g sin θ +

T

m

v = pw − ru+qS

mCY + g cos θ sinφ

w = qu− pv +qS

mCZ + g cos θ cosφ

(1)

and the rotational motion is given by

pIx − rIxz = qSbCl − qr(Iz − Iy) + qpIxz

qIy = qScCm − pr(Ix − Iz)− (p2 − r2)Ixz

rIz − pIxz = qSbCn − pq(Iy − Ix)− qrIxz

(2)

The complete set of the aerodynamic equa-tions follows the model in [10]. The coefficientsof the aerodynamic forces and moments for thelateral-directional dynamics (CY , Cl, Cn) are de-rived by simple Taylor series expansion.

CY = CY0 + CYββ + CYζζ + CYpp∗ + CYrr

∗

Cl = Cl0 + Clββ + Clζζ + Clpp∗ + Clrr

∗

+ Clξ1

2(ξr − ξl)

Cn = Cn0 + Cnββ + Cnζζ + Cnpp∗ + Cnrr

∗

with p∗ =pbV

2r∗ =

rbV

2

(3)

where p∗ and r∗ are the normalized roll and yawrates.

The equations for the longitudinal motion arebased on the two-point model described in [11].This model separates the wing and horizontal tailinfluences and allows to account for the down-wash lag effect of the wing to the horizontal tail.

The lift coefficient is separated into a wing-body component and a component for the hor-izontal tail. The drag coefficient is calculatedvia the simple polar equation. The pitching mo-ment coefficient is also separated into a wing-body component and a component calculated viathe body-axis components CXHT and CZHT andthe respective lever-arms.

CL = CLWB+ CLHT

SHTS

cos(αdyn − εHT )

CD = CD0 + k1CL +C2L

eπΛ

Cm = CmWB + CZHTSHTS

xHTc

− CXHTSHTS

zHTc

(4)

The separated influences of wing and tail forthe longitudinal motion are calculated by

CLWB= CL0 + CLα,WB

α+ CLq,WBq∗

CLHT = CL0,HT+ CLα,HTαHT + CLη,HT η

CmWB = Cm0,WB + Cmq,WBq∗

αHT = α+ iHT − εHT + αdyn

αdyn = tan−1(qx′HT /V )

εHT = ε0,HT +δε

δαα(t− τ)

CXHT = CLHT sin(αHT − iHT )

CZHT = −CLHT cos(αHT − iHT )

with q∗ = qcV

(5)

where q∗ is the normalized pitch rate. The pa-rameter τ represents the downwash lag effect ofthe wing to the horizontal tail.

The body-axis force coefficients CX and CZfrom Equations (1) are finally derived from the liftand drag coefficient by

CX = −CD cosα+ CL sinα

CZ = −CL cosα− CD sinα(6)

3 WAVELET TRANSFORM

The wavelet transform is being used for the anal-ysis of diverse physical phenomena, e.g. inthe denoising of seismic signal, climate analysis,heart monitoring, amongst others [9]. The com-plete wavelet transform theory goes far beyondthe scope of this paper and will not be describedin detail. A few essentials are provided to give

the reader some insight into the transform ap-plied in the method. More details can be foundin [9, 12, 13].

In a simple manner, the wavelet transform isa convolution of a signal to be analyzed with awavelet function, also known as wavelet. Thewavelet is a small wave-like function that beginsand ends at zero amplitude. Some commonlyused wavelets are depicted in Figure 2.

(a) Haar (b) Bior

(c) Daubechie (d) Mexican hat

Figure 2: Wavelet examples

The convolution for a signal x(t) in the time do-main is defined as

T (A,B) =

∫ ∞−∞

x(t)ψ∗A,B(t)dt (7)

where ψA,B(t) is the normalized wavelet func-tion1 written as

ψA,B(t) =1√Aψ

(t−BA

)(8)

The parameter A is the dilation parameter andis used to scale (stretch or squeeze) the wavelet.The parameter B is used to translate (shift) thewavelet to various locations, see Figure 3.

The convolution of the signal with a set ofscaled and translated wavelet functions gener-ates a two dimensional transform plane as in-dicated in Figure 4. The x-axis represents thelocation (e.g. time shift) of the wavelet func-tion while the y-axis indicates the current scaleof the wavelet. This two dimensional representa-tion of the signal shows the correlation between

1The symbol ’∗’ indicates that the complex conjugate of awavelet function is used in the transform, when using com-plex wavelet functions.

Figure 3: Wavelet translation and dilation

Current location

Current

scale

Signal

Time [s]

Figure 4: Two dimensional transform planescheme

the wavelet and the signal. If the signal matcheswell with a wavelet at a certain position, a largevalue in the transform plane is expected.

Mallat [12] has shown that a multiresolutionrepresentation of the signal is achieved by apply-ing the Discrete Wavelet Transform (DWT) usinga set of parameters A and B.

The multiresolution representation transformsthe signal into a combination of approximationsand details coefficients, see Figure 5. The ap-proximations contain the low frequency informa-tion and can be interpreted as general trend ofthe signal, whereas the details contain the higherfrequency information. For the inverse opera-tion, an original signal is quickly completely re-constructed, without loosing any information, us-ing its approximation and its detail.

Another method to analyze discrete signalsis the Wavelet Packet Transform (WPT). It is ageneralization of the discrete wavelet transform,

Figure 5: Multiresolution schematic representa-tion from reference [14].

however herein the signal details and the approx-imations are further decomposed at each level.This method is extensively described in reference[13] and allows better resolution at higher fre-quencies creating a decomposition tree structureas seen in Figure 6.

Each level of decomposition in the WPT isrepresented by a Time-Frequency Plane (TFP)which is schematically depicted in the lower partof Figure 6. An example of such a visualization isshown in the lower plot of Figure 7, where a chirpsignal is decomposed and represented by a TFPin natural frequency ordering. As expected, thefrequency of the signal increases with increasingtime (x-axis).

The decomposition coefficients are repre-sented in the TFP’s using so-called Heisenbergboxes. Each of this box defines the center fre-quency and location of the scaled and shiftedwavelet function used to decompose the signal.In Figure 7 these Heisenberg boxes are repre-sented by colored rectangles, where larger co-efficients are plotted darker. This representationis similar to a spectrogram calculated from thetime signal using the Fourier transform, wherethe frequency spectrum of a signal is depictedas it varies with time.

A time domain discrete signal can be recon-structed from the wavelet packet coefficients

Figure 6: Wavelet packet transform

Frequency Sweep

Time−Frequency Plane

Time Localization [s]

Fre

quen

cy

Loca

lizat

ion

[Hz]

Figure 7: Frequency sweep decomposition

stored in the Heisenberg boxes of any admissi-ble TFP. This preserves the time and frequencyinformation for the signal. The methodology pre-sented in the next section relies on this propertyto define control surface inputs with distinct fre-quency excitations at defined time localizationsof the maneuver.

4 METHODOLOGY

The proposed maneuver design method is basedon the idea of creating input signals which havedistinct frequency band excitations at predefinedmaneuver times. An overview of the method isgiven in Figure 8.

Aircraft a-priori information is used to select

Time

Time Localization

Aircraft a-priori Information

Signal Design in the Time-Frequency Domain

Inverse Discrete Wavelet Packet Transform

Input Signal in the Time Domain

Freq

uen

cy

Loca

lizat

ion

A

mp

litu

de

Figure 8: New input design method

the desired frequency band excitations for thesignal. Heisenberg boxes in the TFP are used torepresent this information in the time-frequencydomain. An inverse discrete wavelet packettransform using a selected wavelet yields the de-sired signal in the time domain which can then beassigned to any of the control surfaces.

As will be shown in Section 4.2, this methodalso allows to define a single maneuver thatuses several control surfaces at the same time.Through proper signal definition in the TFP, atime and frequency decorrelation between thedifferent control inputs can be assured.

4.1 Single Input Using the Time-Frequency Plane

As the DWPT uses a dyadic scale the time-frequency plane must be defined as a squarematrix having 2Nt time segments and 2Nf fre-quency segments. Each combination of a timesegment and a frequency segment is a Heisen-berg box. A value can be assigned to any ofthese Heisenberg boxes and the absolute valuerepresents the local energy for the reconstructedsignal, schematically depicted in Figure 9. Fur-thermore, due to the Inverse Discrete WaveletPacket Transform (IDWPT), this value defines theamplitude of the input.

An example of a signal generation startingfrom values specified in a TFP definition is shownin Figure 9. The upper diagram of Figure 9 showsthe TFP with 16 time segments and 16 frequencysegments. The maneuver time was set to 30 sec-onds resulting in Heisenberg boxes with a width

0 5 10 15 20 25 30−4

−2

0

2

4

Def

lect

ion

[deg

]

Time [s]

Rudder

Time−Frequency Plane

Time Localization [s]

Fre

quen

cy L

ocal

izat

ion

[Hz]

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.00.0

0.3

0.6

0.9

1.1

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1 2 100

0.1

0.2

0.3

0.4

Frequency [Hz]

|P1(

f)|

Amplitude Spectrum

Figure 9: Rudder signal generation

of 2 s and a height of 0.2753 Hz 2. In this ex-ample an input with a frequency excitation in the0.3 Hz band at 4 seconds and also a frequencycontent of 0.9 Hz at 18 to 20 seconds was spec-ified by assigning values for the correspondingHeisenberg boxes. The generated signal is thedirect result of the IDWPT and is shown in thecenter diagram of Figure 9. The lower diagram,shows the frequency content of the signal, evalu-ated by a fast Fourier Transform (FFT). This infor-mation can be used to evaluate if the combinationof any wavelet chosen for the IDWPT introducesundesired frequencies for the aircraft excitation,e.g. elastic modes must not be excited duringthe rigid body dynamics identification.

4.2 Multi-Axis Maneuver Design

The multi-axis maneuver design method used inthis work follows the same steps as for a sin-gle input described above. For each control sur-face a TFP is specified. To assess the correla-tion in time and frequency between different con-trol surfaces an overlap diagram was created. Itis a superposition between the TFP’s of the se-lected input signals and gives a visual represen-tation of the information used to define the in-puts. Figure 10 shows the overlap diagram and

2In the time-frequency plane representations within thispaper, the scale was rounded to one decimal place for visu-alization purpose

the resulting time histories for a combined rudderand aileron input maneuver. It is clearly visiblethat there is no concurrent frequency band exci-tation between these two control surfaces, eventhough both control surfaces are deflected simul-taneously.

0 5 10 15 20 25 30−1

−0.5

0

0.5

Aile

ron

[deg

]

0 5 10 15 20 25 30−5

0

5

Rud

der

[deg

]

Time [s]

Figure 10: Multi-input signal generation scheme

5 SIMULATION AND RESULTS

Simulations with a high quality Airbus A320 dy-namic model were used to obtain virtual flight testdata. This model has been derived from an ex-tensive flight test campaign during the DLR in-ternal project OPIAM (Online Parameter Identifi-cation for Integrated Aerodynamic Modeling) andincludes compressibility and ground effects, highalpha characteristics and meets the criteria forsimulator validation and qualification [15].

Maneuvers using the new design method justdescribed were first developed for the longitu-dinal motion, using only elevator and horizontaltail as control inputs. Therefore only the aero-dynamic parameters of the longitudinal motioncould be estimated from a subsequent estimationusing this maneuver.

After this proved to be successful a multi-axismaneuvers using elevator, horizontal tail, aileronand rudder were designed to allow estimation ofthe full set of aerodynamic parameters. Com-pressibility, non-linear lift behavior and ground ef-

fect were neglected for all identification runs andthe parameters were always assumed to be con-stant for the respective maneuver.

5.1 Longitudinal Motion

The method described in Section 4 was used todefine a maneuver to identify the aircraft’s aero-dynamic parameters for the longitudinal motion.Two control surfaces were used for the maneu-ver: the elevator and the horizontal tail.

Inputs for the elevator contain high frequencyexcitations to evaluate the short period motionas well as low frequency excitations to obtain aphugoid-like response. The horizontal tail is be-ing deflected simultaneously in order to estimateits characteristics and the downwash parame-ters. For the elevator a bior wavelet, as depictedin Figure 2, was chosen to generate a signal withno sharp edges.

In order counteract the deviation from the trimpoint, the horizontal tail is being deflected in theopposite way as the elevator. A variation of lessthan 1.5 degrees of angle of attack and of nomore than 4% of true airspeed allows the estima-tion of parameters without any need to accountfor additional compressibility effects. Therefore,the aerodynamic parameters can be assumedconstant throughout the complete maneuver at agiven flight test point.

Figure 11 shows the input signal generated forthe elevator. The upper diagram shows the defi-nition of the time-frequency plane. A 16x16 TFPwas chosen resulting in Heisenberg boxes witha width of 2 s and height of approximately 0.28Hz. The center diagram shows the resulting sig-nal from the inverse wavelet packet transform.The bottom diagram in Figure 11 shows the fre-quency content of the final signal. It can be seenthat no high frequency is excited. This is desiredwhen creating signals for rigid body identification.For the elastic modes, frequencies from 3 Hz andhigher are generally expected. Thus, no elasticmode is expected to be considerably excited dur-ing this maneuver.

Usually some a-priori information for a new air-craft is available from wind tunnel results, CFDcalculations or after the preliminary design. Withthe proposed design method, the expected nat-ural frequencies can easily be excited b usingcorresponding frequency bands. In the current

evaluation, the expected short period natural fre-quency is 0.35 Hz. Therefore three frequencybands, centered at 0.28, 0.55 and 0.83 Hz,are included for the signal generation. For thephugoid mode, the expected natural frequency ismuch lower, in the order of 0.02 Hz. Therefore, aclassical pulse is emulated using the lowest fre-quency band over a few time tilings. For displaypurposes, only relevant portions of the chosentime-frequency plane tilings are shown, meaningthat all other coefficients are zero and do not in-fluence the results of the IDWPT.

0 5 10 15 20 25 30−0.4

−0.2

0

0.2

0.4

0.6

Def

lect

ion

[deg

]

Time [s]

Elevator

Time−Frequency Plane

Time Localization [s]

Fre

quen

cy L

ocal

izat

ion

[Hz]

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.00.0

0.3

0.6

0.8

1.1

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1 2 100

0.01

0.02

0.03

Frequency [Hz]

|P1(

f)|

Amplitude Spectrum

Figure 11: Elevator signal generation

Parameter estimation was performed using theoutput error method in the time domain and theaerodynamic model for a rigid body aircraft forthe longitudinal motion as described by Equa-tions (4) - (5).

The comparison between the simulated flightdata and the identified longitudinal motion modelare shown in Figure 12 and the results of thecomparison of the time histories are given in Ta-ble 1. The maximum absolute difference be-tween the identified model and the simulatedflight test data is shown in the second column.The last column gives the relative difference tothe overall output amplitude during the maneu-ver.

Identified parameters for the longitudinal mo-tion are shown in Table 2. The relative differencebetween the true value from the simulation modeland the estimated value for the reduced model is

0 5 10 15 20 25 30228

230

232

VT

AS

[m/s

]

SimulationIdentified

0 5 10 15 20 25 302

3

4

Alp

ha[d

eg]

0 5 10 15 20 25 30−2

0

2

q[d

eg/s

]

0 5 10 15 20 25 300

5

The

ta[d

eg]

0 5 10 15 20 25 30−0.5

0

0.5

Ele

vato

r [d

eg]

0 5 10 15 20 25 30

−2

−1.5

−1

Hor

izon

tal T

ail

[deg

]

Time [s]

Figure 12: Time history comparison plot of sim-ulated flight data (blue) and identifiedmodel outputs (red), with inputs for el-evator and horizontal tail (black)

Output max. diff. ∆ diff. [%]

V [m/s] 0.011 0.42α [deg] 0.003 0.19q [deg/s] 0.008 0.48θ [deg] 0.019 0.50

Table 1: Comparison between simulated andidentified models outputs for the longi-tudinal maneuver

given in the last column.The lift parameters were obtained with highest

accuracy, whereas the drag parameters are lessaccurate. This could be related to the fact thatthere is little speed variation during the maneu-ver, hence only a small portion of the drag polaris covered. Also CLq,WB

and Cmq,WB are of re-duced accuracy, mainly because these contribu-tions are small in relation to the pitch rate influ-ences derived from the horizontal tail.

These results show that it was possible to de-sign a maneuver using the method in Section 4that allows to successfully identify 10 parametersthat accurately describe the longitudinal motionfor the rigid body aircraft, only using the a-prioriinformation of desired frequencies and a properwavelet function.

Parameter True Value Estimate ∆ [%]

CL0 0.245 0.245 -0.03δε/δα 0.612 0.627 2.44CLq,WB

3.734 3.968 6.28CLα,WB

5.281 5.322 0.76CLα,HT 4.445 4.437 -0.18CLη,HT 1.733 1.736 0.14CD0 0.021 0.018 -13.49e 0.600 0.707 17.90Cm0,WB −0.192 −0.193 0.49Cmq,WB −7.591 −7.176 -5.46

Table 2: Parameter estimates for the longitudinalmotion

5.2 Multi-Axis Maneuver

To be able to estimate the aerodynamic param-eters for a complete aircraft model, a multi-axismaneuver was designed using elevator, aileron,rudder and horizontal tail as control inputs.

For the elevator and horizontal tail, the sameinput as in Section 5.1 were used.

The input signal generation for the other con-trol surfaces is shown for the rudder in Figure 9and for the aileron in Figure 13. The signaldesigns were performed using a 16x16 time-frequency tiling. This results in Heisenberg boxeswith a width of 2 s and height of 0.28 Hz. Thedutch roll of the aircraft is excited in the beginningof the maneuver by a rudder input with low fre-quency at approximately 0.28 Hz. Furthermorethe aileron is deflected to get the roll motion ex-cited. This is the same approach as for the clas-sical maneuver design criteria, described in [4].Additional high-frequency inputs are applied tothe aileron at the beginning of the maneuver andto the rudder in the second half of the maneu-ver, shown in Figure 10. The signals generatedfor the control surfaces do not contain any highfrequencies thus avoiding significant excitation ofelastic modes.

Parameter estimation was again performed us-ing the output error method in the time domainand the aerodynamic model for a rigid body air-craft as described with the complete set of Equa-tions (4) - (5).

Overall, a multi-axis maneuver was success-fully designed and allows the estimation of 21aerodynamic parameters that describe the rigid

0 5 10 15 20 25 30−1

−0.5

0

0.5

Def

lect

ion

[deg

]

Time [s]

Aileron

Time−Frequency Plane

Time Localization [s]

Fre

quen

cy L

ocal

izat

ion

[Hz]

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.00.0

0.3

0.6

0.9

1.1

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1 2 100

0.01

0.02

0.03

0.04

0.05

Frequency [Hz]

|P1(

f)|

Amplitude Spectrum

Figure 13: Aileron signal generation

body aircraft dynamics in six degrees of freedom.Plots for the longitudinal and lateral-directionalmotion of the aircraft can be seen in Figure 14.The comparison results between the outputs ofthe identified model and simulated flight test datais given in Table 3.

Output max. diff. ∆ diff. [%]

V [m/s] 0.015 0.49α [deg] 0.010 0.31q [deg/s] 0.012 0.73θ [deg] 0.027 0.70β [deg] 0.048 0.72p [deg/s] 0.021 0.66r [deg/s] 0.063 0.65φ [deg] 0.149 0.66

Table 3: Comparison between simulated andidentified models outputs for the multi-axis maneuver

Table 4 shows that all main parameters are es-timated with good accuracy. The rolling momentdue to the aileron deflection (Clξ ) has low accu-racy (high deviation to the true parameter) andthe signal could be adjusted for a better estima-tion of this parameter. It should be emphasized,that analogous to the longitudinal maneuver, themodel used for identification is a simplified modelwith some selected parameters to describe the

Parameter True Value Estimate ∆ [%]

CL0 0.245 0.246 0.42δε/δα 0.612 0.635 3.68CLq,WB

3.734 4.911 31.53CLα,WB

5.281 5.310 0.54CLα,HT 4.445 4.445 0.01CLη,HT 1.733 1.736 0.19CD0 0.021 0.018 -15.14e 0.600 0.709 18.24Cm0,WB −0.192 −0.194 1.08Cmq,WB −7.591 −7.166 -5.60CYβ −1.055 −0.998 -5.38CYζ 0.331 0.314 -4.98CYp 0.199 0.299 50.50Clβ −0.419 −0.433 3.31Clξ −0.165 −0.147 -10.69Clζ 0.084 0.084 -0.28Clp −0.901 −0.943 4.66Clr 0.227 0.235 3.51Cnβ 0.432 0.448 3.83Cnζ −0.325 −0.323 -0.54Cnr −0.710 −0.630 -11.22

Table 4: Parameter estimates for the completeaircraft motion

longitudinal and lateral-directional motion. Forinstance there might be an angle-of-attack influ-ence on the (Clξ ), namely (Clξ,α), which is notbeing identified, yielding the higher error in the(Clξ ) parameter. As for the longitudinal motionmaneuver in Section 5.1 the drag parameters areless accurate due to small change in the speedfor a drag polar estimation. Some second orderderivatives such as CYp and Cnr are also esti-mated with low accuracy, however the compari-son plots in Figure 14 show that a good matchbetween simulated flight test data and identifiedmodel is obtained.

6 COMPARISON WITH OTHER MANEU-VER DESIGN METHODS

The proposed methodology reuses an idea whichcan already be found in the literature, namely ap-plying simultaneous uncorrelated inputs for sys-tem identification [6, 7, 16]. The fact that uncor-related inputs are used permits to ensure that theeffects of the respective inputs can be separated

afterward. One possibility is to use sums of puresine signals with disjointed sets of frequencies foreach input. This strategy can be found in manytextbooks, e.g. [16], and has been already usedfor airplane system identification, e.g. by Morelli[6].

Whilst the sum of sines type of signals allowselegant and very efficient parameter estimation inthe frequency-domain, the comparison betweenthe (real) aircraft reaction and the model basedon such signals can hardly be interpreted by sys-tem identification specialists. Indeed, very of-ten the correct model structure is not known be-forehand and the system identification specialistsneed to adjust the model structure and must de-cide which parameters should be estimated.

Usually, a system identification specialist canperform a good diagnostic and know what needsto be changed in the model, just by comparingthe real aircraft and model responses after excit-ing the aircraft with standard well known systemidentification maneuvers (e.g. pulses, doublets,3211-inputs, frequency sweeps, etc.). The per-manent excitations of different frequencies by asum-of-sine signal make this diagnostic impracti-cable and thereby the ”expert-based” model im-provement process impossible.

An automatized search for suitable modelterms can be performed as proposed in [17, 18]but the search is then based on a series of termswhich will not necessarily correspond to physi-cally meaningful effects. For instance, by addinghigher order terms from a polynomial basis anycontinuous nonlinear function can be approxi-mated very well, but no physical interpretationcan be made from a Cm,α3q or Cm,α3η term. Suchhigh-order terms are not physically justified by aparticular phenomenon but just an artificial meanto get a higher goodness of fit.

For this reason, the author and his researchgroup prefer to stay in control of the model struc-ture selection process and add terms only asneeded and wherever adding a new term can bejustified. In this context, having maneuvers towhich the response of the system can be inter-preted by a flight mechanics expert is more ad-vantageous than having maneuvers that enablethe use of a highly efficient parameter estimationformulation in the frequency-domain.

Figure 14: Response time histories from simulated flight test data (blue) and reduced model outputs(red), with multi-surface inputs (black)

7 CONCLUSION AND OUTLOOK

In this work, a new method to design single ormulti-axis maneuvers for aircraft parameter esti-mation was presented and discussed. The pro-posed method permits to design single-axis andmulti-axis excitation maneuvers suited for aircraftparameter estimation. Based on classical de-sign criteria, the methodology developed allowsthe design of complex signals including a-prioriinformation on the system to be identified (herethe aircraft) even though only a quite restrictednumber of parameters is used to describe thesesignals.

The new method for maneuver design hasshown promising results for both, single-axis andmulti-axis excitations. Parameter estimates de-rived from virtual flight test data of maneuversdesigned using the new method have shown thatthis design method can be applied to specify ma-neuvers with the potential to reduce flight testtime, and further on to improve the quality ofaerodynamic models. The proposed method wasdemonstrated based on a 30 second maneuverwhich permitted to successfully identify the air-craft aerodynamic parameters for the longitudinalmotion. It was also successfully demonstrated

that it was possible to design a 30 seconds ma-neuver for which 21 parameters describing thecomplete aircraft rigid body dynamics for a singleflight test point were accurately estimated.

The proposed method can easily lead to verycompact multi-axis maneuvers to which the re-sponse of the system will remain easily inter-pretable by a specialist. However, the result-ing maneuvers may lead to less efficient pa-rameter estimation than some of the alternativeswhen considering frequency-domain parameterestimation. Users will therefore have to choosebased on the requirements of their applicationwhich approach is preferable.

The high level parameters provided, especiallytype of wavelets and scale, seem very well suitedfor further automatic optimization of the maneu-vers aiming at maximizing the quality of the iden-tified parameters. Whilst such an optimizationremains a medium-term goal at this stage ofthe development, it should be noticed that theparametrization used herein will ease the inter-pretation of the parameter values that would beobtained through the optimizer. The other wayaround, the a-priori knowledge available on thesystem can easily be included in the design ofthe signal in order to be certain to excite the rele-

vant dynamic modes of the system.The proposed methodology also permits to de-

sign signals very quickly, not only rich in infor-mation content, but also satisfying very specificconstraints, which can be fully or almost fully un-correlated.

Work on the presented methodology will bepursued for optimizing multi-axis maneuvers ofvirtual flight tests for system identification of flex-ible aircraft (DLR project VicToria [19]) as well asfor optimization of flight test campaigns for sys-tem identification.

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