NASA Technical Memorandum 4793

Wavelet Analyses of F/A- 18

Aeroelastic and Aeroservoelastic

Flight Test Data

Martin J. Brenner

Dryden Flight Research Center

Edwards, California

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Program

1997

ABSTRACT

Time-frequency signal representations combined with subspace identification methods were used to analyze

aeroelastic flight data from the F/A-18 Systems Research Aircraft (SRA) and aeroservoelastic data from the F/A-18

High Alpha Research Vehicle (HARV). The F/A-18 SRA data were produced from a wingtip excitation system that

generated linear frequency chirps and logarithmic sweeps. HARV data were acquired from digital Schroeder-phased

and sinc pulse excitation signals to actuator commands. Nondilated continuous Morlet wavelets implemented as a

filter bank were chosen for the time-frequency analysis to eliminate phase distortion as it occurs with sliding window

discrete Fourier transform techniques. Wavelet coefficients were filtered to reduce effects of noise and nonlinear

distortions identically in all inputs and outputs. Cleaned reconstructed time domain signals were used to compute

improved transfer functions. Time and frequency domain subspace identification methods were applied to enhanced

reconstructed time domain data and improved transfer functions, respectively. Time domain subspace performed

poorly, even with the enhanced data, compared with frequency domain techniques. A frequency domain subspace

method is shown to produce better results with the data processed using the Morlet time-frequency technique.

NOMENCLATURE

a

[A,B,C,D]

AOA

ASE

CWT

da

DFF

e

f

FIR

G

h

H

H(¢o)

HARV

I

J

K

M

MIMO

n

N

OBES

P

wavelet scaling parameter

system state-space matrices

angle of attack

aeroservoelastic

continuous wavelet coefficient

finite interval of a

discrete Fourier transform

exponential function

center frequency of wavelet filter

Fourier transform

finite impulse response

transfer function

wavelet basis function

Toeplitz matrix of Markov parameters

Fourier transform of wavelet basis function

wavelet filter centered atf

High Alpha Research Vehicle

identify matrix

imaginary ,f_

Kalman gain matrix

number of frequency data

multi-input-multi-output

number of system states

number of time data

on-board excitation system

number system outputs
[P,Q,S]

sISO

SRA

t

T

u

U

V

x

X

x(t)

X(o_)

Y

Y

Z

F

"t

OJ

matrices

single-input-single-output

Systems Research Aircraft

time

total time of wavelet analysis data

system input vector

block Hankel input matrix

block Hankel noise matrix

system state vector

state matrix

analysis data in time domain

analysis data in frequency domain

system output vector

block Hankel output matrix

discrete frequency

extended observability matrix

wavelet time shift parameter

radian frequency

Superscripts

antisymmetric

symmetric

Subscripts

i,k

o

S

integer indices

noise subspace

signal subspace

Symbols

±

1

A

U

orthogonal complement

pseudoinverse

estimate

omission of first rows of matrix U

omission of last rows of matrix U

INTRODUCTION

Envelope expansion of new or modified aircraft often necessitates structural stability testing to verify safety

margins and to prevent catastrophic flutter. Inflight testing allows determination of aeroelastic or aeroservoelastic

effects as a function of flight parameters. Flight data are acquired so stability estimation and system identification

can be compared with analytic predictions. Any anomalies are regarded with care to guarantee maximum flight

2
safety.Becauseatmosphericturbulenceis generallyinsufficiento determinemodalcharacteristics, excitation

systemsareoftenessentialtoresolvestabilitytrendsfromnoisymeasurements.

Flightflutterdataprocessingattemptsto analyzedatacharacteristicsin termsof timing,shape,amplitude,

frequency,anddurationof eventsin thedata.Theseanalyseshavetraditionallybeenaddressedusingclassical

Fouriertechniques.However,Fouriermethodsaresuspectdueto the inherentlytransientnatureof inflight

aeroelasticdynamics.Theinfiniteandatleastlocally-periodicwaveformassumptionsin Fourieranalysiscannot

adequatelydescribethe intermittency,modulation(amplitude,phase,or frequency),nonperiodicity,nonstationariness, time-variance, ornonlinearityinthedata.

Thispaperdescribesanovelfilteringprocedurewhichremovesdistortionsfromaeroelasticoraeroservoelastic

(ASE)flightdata.Thefilteringprocedurebuildsuponandadaptsthemethodsof waveletransformanalysistothe

particulartestdatageneratedfromstructuralexcitationsystems.

Waveletsareversatileharmonicanalysistoolswhichcombinetimeandfrequencyrepresentations intolocalized

waveforms.Givena segmentof experimental data,thewaveletransformconvolvesa selectedseriesof local

waveformswith thedatato identifycorrelatedfeatures,or patterns,in thesignal.Theresultingsetof wavelet

coefficientscanbeinterpretedasmultidimensional correlationcoefficients.Featuresof shape,size,andlocationare

naturallycharacterizedbythesewaveformsandrelatedcoefficients.

Thesalientfeaturesoftheoriginalsignalmaybereconstructedbyexploitingtheredundancyof thewaveletsin

thecontinuouswavelettransform.Time-frequencycomponentscanberemovedfromthesedatabymaskingoffthe

unwantedcomponents(settingthe correspondingwaveletcoefficientsto zero).This methodis usedto filter

unwantedistortionsandextractdesiredfeaturesfromtheinput(excitation)andoutput(structuralresponse)data.

Thisfeature extraction method offers advantages to traditional band-pass filtering or thresholding techniques in that

it removes unwanted features while leaving desirable components intact.

By removing aspects of system inputs and outputs which are detrimental to linear identification methods,

improvement of system identification is realized. Subspace identification methods have recently shown promise as

being efficient, generally applicable, and robust. Experience with these methods will be discussed and comparisons

made between transfer functions estimated with and without feature processing.

AIRCRAFT DESCRIPTIONS AND TEST PROCEDURES

Descriptions of the F/A-18 SRA and HARV follow. Flight flutter testing using a wingtip excitation system on

the SRA is described in detail. High angle-of-attack aeroservoelastic research with the HARV is described using

commands generated from the control system.

Systems Research Aircraft

The F/A-18 Systems Research Aircraft (SRA) is being flight tested at NASA Dryden Flight Research Center,

Edwards, California, for primarily flight systems experiments. These tests include optical sensing, new actuation

concepts, smart structures, and advanced airdata and flight control systems. A major left wing structural modification

was done, however, on the SRA to allow testing of several hydraulic and electromechanical advanced aileron

actuator concepts. Because the test actuators may be larger than the standard one, a fitting called a hinge-half

(supporting the aileron hinge, the actuator, and a fairing) had to be replaced by larger and heavier items.

Approximately 35 lb were added to the wing.

These SRA data were derived from flight flutter testing using a wingtip aerodynamic structural excitation

system I at subsonic, transonic, and supersonic flight conditions. 2 Inputs were linear sinusoidal frequency sweeps, or
chirps,andlogarithmicsweepsof theexcitationsystem.Eachwingtipexciterhadasmallfixedaerodynamicvane

forwardof a rotatingslottedhollowcylinder(fig. 1).Whenthe cylinderrotated,the aerodynamicpressure

distributiononthevanegeneratedaforce.Thisforcechangedattwicethecylinderrotationfrequency.Therotation

directionof thecylindercouldgeneratetwoforcelevels,low-forcelevels(at25percentopen)orhigh-forcelevels

(at75percentopen).Exciterscouldbeplacedinvariouslocations.Threepossibilitiesweremid-wingtip,forward

wingtipatthewingtipleadingedge,oraftwingtipnearthewingtiptrailingedge(aspicturedinfigure1).

Table1summarizesthe matrix of test points for the SRA. Wingtip exciters were mounted on the SRA wingtip

launcher rails in forward and aft positions, independently. They were operated in both low- and high-force modes

for 15-, 30-, or 60-second forward and reverse frequency sweeps. A total of 260 test points were flown with exciters

in one or more of the optional configurations at specified flight conditions. Linear and logarithmic sinusoidal sweeps

of the exciter up to 40 Hz were used to excite the primary modes of interest in the 5 to 30 Hz range (table 2). Some

sweeps were actually multiples of two or four shorter duration sweeps with no interrupt such as double (sweep up,

then down) and quadruple (up-down-up-down) sweeps. Generally sweeps were performed for symmetric and

antisymmetric excitation in each configuration.

Figure 1. Wingtip exciter on F/A- 18 SRA.

Mach number

Altitude, ft

Exciter position

Force

Sweep

Duration, sec

Range, Hz

Multiple

Table 1. SRA aeroelastic flight test matrix.

0.54, 0.65, 0.70, 0.80, 0.85, 0.90, 0.95, 1.05, 1.2, 1.4, 1.6

10k, 30k, and 40k

Both forward, both aft, and left-aft/right-forward

Low and high

Linear and logarithmic

15, 30, and 60

3-12, 25-35, 3-35, 35-3, and 3--40

Single, double, and quadruple contiguous sweeps per maneuver
Table2.F/A-18SRAwithwingtipexciter:Calculatedelasticmodalfrequencies.

SymmetricMode Hz AntisymmetricMode Hz

Wingfirstbending 5.59

Fuselagefirstbending 9.30

Stabilizerfirstbending 13.21

Wingfirsttorsion 13.98

Finfirstbending 16.83

Wingsecondbending 16.95

Wingoutboardtorsion 17.22

Fuselagesecondbending 19.81

Trailing-edgeflaprotation 23.70

Stabilizerforeandaft 28.31

Wingsecondtorsion 29.88

Aileronrotation 33.44

Ailerontorsion 38.60

Stabsecondbending,wingthirdbending 43.17

Fuselagefirstbending 8.15

Wingfirstbending 8.84

Stabilizerfirstbending 12.88

Wingfirsttorsion 14.85

Finfirstbending 15.61

Wingsecondbending 16.79

Fuselagesecondbending 18.62

Trailing-edgeflaprotation 23.47

Fuselagetorsion 24.19

Launcherrail lateral 24.35

Stabilizerforeandaft 28.58

Wingsecondtorsion 29.93

Aft fuselagetorsion 37.80

Wingpitch 39.18

High Alpha Research Vehicle (HARV)

Another F/A-18 aircraft (fig. 2) was modified at the NASA Dryden Flight Research Center to perform flight

research at high angle of attack (AOA) using thrust vectoring and incorporating control law concepts for agility and

performance enhancement, as well as providing data for correlation with computational fluid dynamics solutions.

This vehicle is referred to as the High Alpha Research Vehicle (HARV) 3 throughout this report. As opposed to the

SRA, the HARV was structurally very different from a standard F/A-18 aircraft, as can be seen by comparing the

HARV modal data in table 3 with the SRA data in table 2.

Structural modifications included addition of Inconel ® vanes in each engine exhaust for thrust vectoring.

Corresponding ballast was added in the forward fuselage to maintain the center-of-gravity location. A research

flight control system was incorporated for feedback control of aerodynamic surfaces and the vanes. An inertial

navigation system was installed for AOA and angle of sideslip rate feedbacks, wingtip AOA vanes, and pressure

probes (for airdata research purposes). Additional instrumentation was added for loads, vane temperatures, and

structural dynamics. 4

An important element of the HARV flight system was the onboard excitation system (OBES). This system was

implemented to add programmed digital signals to the control system actuator commands for structural excitation.

Inputs from 5 to 25 Hz were added to the stabilator, aileron, rudder, and pitch- and yaw-vectored thrust commands.

Data were generated with OBES commands at 5° to 70 ° AOA 4 at 1 g. Accelerometers were located in the aircraft

nose, vertical and horizontal tails, and wingtips.

®Inconel is a registered trademark of HuntingtonAlloy Products Div.,International Nickel Co., Huntington, West Virginia.
Figure2.F/A-18HARV.

Table3.F/A-18HARVcalculatedelasticmodalfrequencies.

SymmetricMode Hz AntisymrnetricMode Hz

Wingfirstbending 5.79

Fuselagefirstbending 7.71

Wingfirsttorsion 11.68

Stabilizerfirstbending 13.73

Wingfore-aft 18.53

Finfirstbending 15.92

Wingsecondbending 17.55

Fuselagesecondbending 15.87

Exhaustvanerotation 22.10

Inboardflaprotation 23.60

Wingoutboardtorsion 27.50

Fuselagefirstbending 7.13

Wingfirstbending 8.75

Wingfirsttorsion 12.03

Stabilizerfirstbending 13.63

Wingfore-aft 15.18

Finfirstbending 15.71

Fuselagefirsttorsion 19.13

Fuselagesecondbending 21.42

Exhaustvanerotation 22.10

Inboardflaprotation 23.18

Fore-fuselagetorsion 24.17

.A__atawere _enorat_ fom_i_,_, sinc(_inx)pu_ses an__c.roe_r o_ase_6 inputs intocon_ro, su.ace

commands at generally high AOA. Sinc pulses and Schroeder-phased signals have a theoretically flat frequency

response across a defined frequency band and are optimal for structural excitation because all modes are excited

equally. Furthermore, Schroeder-phased signals have a minimal peak crest factor (difference between upper and

lower bound of the power spectrum) amongst all signals with the same total power and are therefore preferred.
Schroeder-phased signalsaremultifrequencysignalscomposedof alargenumberof harmonicsequallyspaced

infrequency.Eachharmonicisspecifiedbyaphaseshiftsothatwhentheyaresummedthewaveformhasalowpeak

factorandfitsagivenpowerspectrum.It isrepresentedby

x(t) = AZcosIo3t+ _2)

k

(1)

where N is the number of data, and A is a specified amplitude to acquire maximum power under the saturation limits.

TIME-FREQUENCY ANALYSIS METHOD

This section illustrates the limitations of the Fourier transform as a tool for analysis of transient or short time

period phenomena. The wavelet transform is offered as an alternative analysis tool, and the Morlet wavelet is

selected as the wavelet basis function. Criteria for selecting the frequency width of the transform are discussed.

Finally, a method for reconstructing the desirable features of the original signal from the wavelet coefficients and

Morlet wavelet basis function is offered.

Wavelet Transform

Analysis of the frequency content of a signal using the Fourier transform results in a spectral representation that

is a function of frequency. This transform is not localized in time. For a time-varying signal x(t), however, a

transform of the form

T(x(t)) = X(t, o3) (2)

is required to locate instantaneous frequencies. Such a transform is limited in resolution by the Heisenberg

uncertainty principle

Ao3- At > 2 (3)

Resolution problems resulting from the uncertainty principle can be minimized by extending the idea of the

Fourier transform to a new time-frequency decomposition called the wavelet decomposition. 7 This decomposition

performs an orthonormal projection of the signal onto a set of basis functions that are adapted to the required

frequency resolution. The Continuous Wavelet Transform (CWT) is defined as

CWT(% = 1 f._x(t)h,(t-X)dt (4) a)

J_ ka)

where ha, x(t)= h(LYZ2-_] is the wavelet basis function, "_is the local time, anda is a scale or dilation parameter set

\ i._ j

to match the level of resolution desired. Wavelet transforms can be looked upon as the filtering of a signal through

a bank of filters ha,z(t). These filters are band-pass with identical shape but with varying frequency width da centered

around the frequencies of interest. Note that the ha,_(t) are not orthogonal because they are redundant (defined for

continuous a and x). With the CWT the time resolution is arbitrarily good at high frequencies, and the frequency

resolution becomes arbitrarily good at low frequencies (within the limits of the uncertainty principle) because da/a

is constant. Time-frequency analyses attempt to minimize the effects of these resolution problems by using a priori

knowledge of the signal properties or adapting the resolution to the signal. 8,9
Selection of Wavelet Basis

Aeroelastic and aeroservoelastic excitation data are essentially short-time sinusoidal, so the selection of a

wavelet basis function should encompass this characteristic. Therefore, the Morlet wavelet 1°

2

l

h (t) = _ 1 e- "2 e -itot (5)

was chosen as the basis function because of its clear interpretation in the frequency domain (Gaussian window) and

time domain (locally periodic waveform) for the analysis of vibration data. Figures 3(a) and 3(b) show real and

imaginary components of the Morlet wavelet. Figure 3(c) shows a Fourier representation of the real part.

Magnitude 0

.5 m

-.s I

-5 -4 3 4

I I I I J I

-3 -2 -1 0 1 2

Time, sec

5

970001

(a) Real part (top).

Magnitude

.5

-.5

-5 -4 -3 3 4

I 1 _l I

-2 -1 0 1 2

Time, sec

(b) Imaginary part (middle).

5

970002

Magnitude

1.0

.5

I

0 .2 .4 1.4 1.6 1.8 2.0

970003

1 1 i_ [ I I

.6 .8 1.0 1.2

Frequency, Hz

(c) Fourier transform of real part (bottom).

Figure 3. Morlet continuous wavelet transform.
Foranalysisof theSRAdatacomposedof linearfrequencysweepswheretherateof changeisconstantoverthe

wholetimeinterval,aconstantresolutionis requiredalongtheentiretime-frequencydomain.Hence,thechosen

time-frequencytransformusesnondilated(a = constant)Morletwaveletsasfunctionsontowhichthedataare

projected.Fora selectedfrequencyandfrequencyresolution,thewaveletransformconvolvesthecorresponding

Morletfilterwiththesignaltoproducethewaveletcoefficients.

Resolution Criteria

A tradeoff between time and frequency resolution is used to select the appropriate frequency resolution of the

filter. In the case of the sweeps considered here, the frequency varies from 0)1 to 0)2 in T seconds. Assuming a desired

resolution of Ac0 in frequency, the time resolution At is actually given by the time localization of this frequency by

At - T A0) (6)

0)2 - 0)1

From the Heisenberg uncertainty principle (eq. (3)), we have

A0) 0)2 - 0) 1 = T 2 = 2_ and At 2 2_ (7)

T 0)2 - 0)1

For a typical linear sweep from 0 to 30 Hz in 30 sec, these relations give At = _ sec, and Ac0 = ,,/-} Hz. These

values will be used to determine the frequency width of the Morlet filters.

Time-Frequency Filtering and Signal Reconstruction

In the frequency domain, the wavelet coefficient corresponding to a Morlet filter with center frequency fis

CWT(0), f) = ,._"(CWT(t, f)) = X(0)). H f(0)) (8)

where the Fourier transform of the Morlet wavelet is

H f(0)) - H(0), f) (9)

The complex Morlet basis function from equation (5) presents problems because it should have no group delay

(real Fourier representation) and no phase distortion. For example, when reconstructing the original time domain

signal from a bank of complex Morlet filters, phase distortion results. Therefore, only the real part of the Morlet

wavelet is used to ensure that the transform H f(0)) is real, and signal reconstruction is in-phase with the original

signal. The Fourier transform of the real part

_ 2_ f2

Hf(0)) = e -(°)-f)2- e (10)

is shown in figure 3(c). Convolution and inverse Fourier transform operations produce the desired timefrequency coefficients

CWT(t, f) = ._'-1(X(0)) • H f(¢o)) (11)

where X(0)) and H f(0)) are the Fourier transforms of signal and Morlet wavelet, respectively. This real Morlet filter

bank provides a fiflite impulse response (FIR) linear phase filter for each center frequency f and maintains phase

consistency between the original signal and some type of reconstruction.

9
Assumingthatn wavelet basis functions are to be used to perform the signal reconstruction, equation (8) is used

to construct the FIR filter bank. For a given frequency co0, the wavelet coefficients are related to the original signal

spectrum by

-CWT(t°O, fl)] [nfl (_0) 1

i ! /

cwr(<oo,i,,)] Lns(<o0)]

(12)

This equation is solved for X(m 0) using the pseudoinverse at every frequency to0

2(%) = -cwr(%, s_)] Hs,(._o

cwr(%,i.)] HSo(%) ]

(13)

and is inverse transformed to get the time domain reconstruction 2(0.

This procedure will be used subsequently to filter unwanted distortions and extract desired features from the

input excitation and output structural response data from the SRA and HARV aircraft. These feature-filtered data

will then be used to compute cleaned signals for transfer functions and subspace identification algorithms. Figure 4

shows an outline of the entire procedure.

SIGNAL PROCESSING IN THE TIME-FREQUENCY DOMAIN

Transfer functions are routinely used in aeroelastic and aeroservoelastic analyses to acquire state-space

representations of the system modal dynamics and to predict flutter boundaries with damping trends or other

advanced techniques.i 1,12Traditional Fourier transform methods often disguise important features in the data from

I Input excitation data I

i

I Morlet filtering of inputs I

i

i I I

I S,,n.,r.con..uc"*.._,, I I S°'c"""°'"on._<_'I

I ,,--Oo--,n.o_.°.°.,OI I,r.,u.nc, °o_.,n.u_..c.,OI

-I .,.,c,o 1-

970004

Figure 4. Time-frequency Morlet feature-filtered system identification (ID) procedure.

10
averagingandwindowing.Tocircumventtheseproblems,arecipeusingthewavelet-basedfeatureextractionfilter

isemployedtoestimatecleanertransferfunctionsbasedontime-frequencylocalization.Figure4showsaschematic

oftheentireprocedure.

* Featurefilterbymaskingthedominantharmonicbandintheinputsandperformingthesameprocedureto

theoutputs.Theresultinginput--output pairsarefromastrictlylinearsystem.

o Localizetherelevantfrequencycontentby filteringeachinput-outputsignalwith a bankof narrow

band-passFIRfilterscenteredaroundeachtoi"

• Localizetherelevantimeinformationbywindowingeachband-pass-filteredsignalaroundthetimepoint

whereafilteredinputhasmaximumamplitude.Therelevantinformationatfrequencytoiisassumedto be

localizedintime.Energyoutsidethechosentimewindowatidenticalfrequenciesiconsideredasnoise.

• Applyequation(13)foreverywindowedinput--output pairateachto0.

• Reconstruct time signals _(t) for time domain identification or derive transfer functions using the spectral

input-output pairs of ,_((to).

Figure 5 shows an example of what the Morlet filter can do beyond standard Fourier techniques. The top four

plots are derived from standard Fourier techniques, from one exciter (left) to accelerometer output and other exciter

(right) to the same output. The bottom four plots are the corresponding Morlet feature-filtered results. Improvement

is obvious with the Morlet-processed data in identifying modal peaks from the noise and establishing well-defined

phase response.

A comparison is made between the Morlet wavelet filter and signal analysis with a discrete Fourier transform

(DFT) using a sliding window in figure 6. The signal is a linear sweep measured from the SRA exciter vane as the

aeroelastic system input. The sliding DFT is computed for the same frequency resolution as the Morlet filter bank.

A Hanning window is applied, and its real part is plotted as a function of time. The result is a discrete short-time

Fourier transform. A magnitude and time-frequency representation of the sliding DFT is commonly referred to as a

spectrogram. As shown, the results seem comparable, but the sliding DFT suffers from increased smearing in the

resolution of the harmonics (lower left comer) and at the end of the sweep. It is well known that the inverse sliding

transform suffers from loss of resolution, or "time smearing, ''13 and is also computationally inefficient. Furthermore,

the short-time Fourier transform suffers from fixed frequency resolution for all times; whereas, the wavelet filter

bank approach allows adjustments in time and frequency resolution pertaining to the signal characteristics.

APPLICATION TO F/A-18 SRA AEROELASTIC DATA

A procedure for deriving transfer functions from the SRA wingtip excitation data is described. Time-frequency

analyses are applied to this data for enhancing transfer functions and improving system identification methods such

as subspace identification.

Transfer Function Derivation

Estimation of transfer functions from the SRA data was predominantly used to identify state-space matrices for

prediction of flutter boundaries. Because the input was corrupted by exciter vane anomalies and inconsistencies,

extracting information relevant to linear system identification and considering the rest as being detrimental to linear

stability estimation was necessary. Therefore, assuming the system to be identified is linear, the extraneous energy

at frequencies other than the excitation frequency is considered as corruptible unmodeled dynamics, noise, or both.

Along the main harmonic of the input will also be noise (as with output), but the signal-to-noise ratio is assumed to

be sufficiently high to mask the majority of the noise corruption.

11
Magnitude

Phase,

deg

.20

.15

.10

.05

0

200

100

0

-100

- 200

I

i

i i

:1

6 8 10 6

Frequency, Hz

i

L

8 10 12

Frequency, Hz

970005

(a) Classical Fourier transfer functions.

Magnitude

.20

.15

.10

.05

0

Phase,

deg

20O

100

0

-100

- 200

..................................................... [.........................

i

6 8 10 12 6 8 10 12

Frequency, Hz Frequency, Hz

970006

(b) Corresponding time-frequency generated transfer function.

Figure 5. Transfer functions computed with classical Fourier technique and time-frequency method. Left exciter

input to accelerometer output (left column). Right exciter input to accelerometer output (right column).

12
0

5

10

15

Frequency, 20

Hz 25

3O

35

4O

Signal in time/frequency domain

0 5 10 15 20 25 30

Time, sec 970007

(a) Morlet continuous wavelet transform.

Signal in time/frequency domain

0

5

10

15

Frequency, 20

Hz 25

30

35

40

0 5 10 15

Time, sec 970008

(b) Discrete short time Fourier transform (right)

20 25 30

Figure 6. Morlet continuous wavelet transform compared to the discrete short-time Fourier transform of SRA

exciter data.

The procedure for SRA transfer function derivation from the symmetric and antisymmetric exciter inputs

is as follows:

• Compute the Morlet-filtered CWT coefficients of the inputs Ul(t,o i) and u2(toi) from each of the two exciter

vanes (operating simultaneously for symmetric u s and antisymmetric u a sweeps) at each filtered center

frequency o3 i.

• Compute y(to i) for each output yS and ya at each filtered center frequency co i.

• Estimate transfer functions GI((Oi) and G2(t.oi) from each exciter vane input to the output:

• and for each coi

Ys(coi)l = IulS(O_i) u2S(o_i)

ya (fDi)J [ula(f.Oi) u2a(f.Oi)

G1 (°)i)

2 (t, Oi)J

uls((oi ) U2 (03 i

Ula(03i ) u2a(o_i)J

yS ( o_ i)1

ya(o)i)_

(14)

(15)

Time-Frequency Analysis

A demonstration of applications using wavelet time-frequency maps, called scalograms, for data enhancement

in system identification of SRA structural dynamics follows. In figure 7(a), a scalogram showing the original input

signal of a linear frequency sweep from the exciter is displayed. A primary harmonic is clearly seen as well as other

dynamics and noise. The main harmonic is the desired part of the signal because it tracks the programmed input to

the exciter. The actual input to the exciter is much more complicated due to nonlinear aerodynamics and distributed

loading across the exciter vane. By creating a mask around the main harmonic CWT of the input and applying it

to the CWT of the output signal (accelerometer, fig. 8), the dynamic properties of the system can be identified.

Figure 7(b) represents the masked input CWT.

13
0

5

10

15

Frequency, 20

Hz

25

30

35

4O

Signal in time/frequency domain

0

5

10

15

Frequency, 20

Hz

25

3O

35

4O

0 5 10 15 20 25 30 0 5

Time, sac

970009

(a) Scalogram of original SRA exciter signal.

Signal in time/frequency domain

I I

10 15 20 25 30

Time, $ec

970010

(b) Scalogram of cleaned SRA exciter signal by

feature filtering the CWT.

Figure 7. Morlet scalograms of original input and cleaned input.

10

15

Frequency, 20

Hz

25

3O

35

40

O -

5

Signal in time/frequency domain

10

15

Frequency, 20

Hz

25

3O

35

4O

Signal in time/frequency domain

] [ [ 1 I -

I I I I I J 1 I I l

0 5 10 15 20 25 30 0 5 10 15 20 25 30

Time, aec Time, sec

970011 970012

(a) Scalogram of original accelerometer response. (b) Scalogram of cleaned accelerometer response.

Figure 8. Morlet scalograms of raw and cleaned accelerometer output.

Signal reconstruction uses the redundancy of the wavelets to estimate a time signal which the CWT best

approximates. The real transform Hf(o3) assures an in-phase reconstruction compared with the original signal.

Figure 9 shows the original (unprocessed) and reconstructed clean (processed) signals from figure 7. The same

procedure is performed on left and right exciter inputs and an output accelerometer signal to generate transfer

functions in figure 10. Note the extreme contrast between the original results (fig. 10(a)) and the cleaned (fig. 10(b))

transfer functions, especially in phase. These plots show the improved noise removal in the cleaned data for

distinguishing modes below 20 Hz, and the potential for dramatic differences in phase between input and output

responses upon cleaning. Estimation of a state-space system can then be performed using the enhanced data from

the cleaned CWT and an algorithm, such as subspace identification.14-17

14
5O

40

30

20

10

Amplitude 0

-10

- 20

- 30

- 40

- 50

0

(a) Reconstructed

response.

Reconstructed signal in time domain

5 10 15 20 25 30

Time, sec 970013

signal of original accelerometer

4O

3O

20

10

Amplitude 0

-10

-20

- 30

-40

Reconstructed signal in time domain

0 5 10 15 20 25 30

Time, see 970014

(b) Reconstructed signal of cleaned accelerometer

response.

Figure 9. Original signal and cleaned signal reconstruction.

TF ul -> y -

.06

.05

== .04

-_ .03

.02

:_ .01

o

TF u2 -> y - 1

.06 t : i

• .05 l-j- _---,-----_----[ •

__.. .04 _1 _=

= .03 ==

cn

I .o2 ." .01 ::If

0

Left exciter input

0

Right exciter input

m .05

_ .04

--" .03

_ .02

_ .01

0

51111

_ o

_ -500

-looo

L

-1500

0 10 20 30 40

Frequency, Hz

,¢

et

800 _ , _ 500

6oo1_-,-,---1_ ............ i........ i............. 0

400200[_l" __'° - 500

Z" - 1000 0 o.

- 200 - 1500

0 10 20 30 40

Frequency, Hz

970015

(a) Transfer functions using raw time-frequency data.

Left exciter (left column) and right exciter (right

column) to accelerometer output.

\ ,[

0 10 20 30 40

Frequency, Hz

- 200 i i i

_-eoo- 400 ............. i_i

,.-aOOl ! i ! I

- 1000

0 10 20 30 40

Frequency, Hz

970016

(b) Transfer functions using cleaned time-frequency

data. Left exciter (left column) and right exciter (right

column) to accelerometer output.

Figure 10. Transfer functions using raw and cleaned time-frequency data.

An example of filtering the undesirable features of complicated input-output signals using the time-frequency

representation will be a double logarithmic sweep from the SRA excitation system. Figures 11 and 12 show planar

and three-dimensional scalograms of the original and desired sweeps. Harmonics from the strain gauge input

measurement can be readily detected in figures 1l(a) and 12(a). The harmonics indicate nonlinear exciter vane

response from the rotating slotted cylinders at the wingtips, which is deemed undesirable for subsequent linear statespace identification methods. Therefore, the input signal is modified by extracting the desired time-frequency map

from the scalogram (figs. 11 (b) and 12(b)) and reconstructing the time domain input signal.

As exemplified, general maps in the time-frequency plane can be effectively filtered. Also, such signals would

be very difficult to threshold because the harmonic wavelet coefficients are of comparable amplitude. Hence, a

method of automating this feature filtering process will generally not benefit from typical thresholding schemes.18

15
0

5

10

15

Frequency, 20

Hz

25

30

35

4O

Signal in time/frequency domain Signal in time/frequency domain

I I I

i I I

5 10 15 20 25 30 5 10 15 20 25 30

Time, sec Time, sac

970017 970018

0

5

10

15

Frequency, 20

Hz

25

30

35

40

0

(a) Original input. (b) Cleaned input.

Figure 11. Scalograms of original double-sweep logarithmic input and cleaned input.

3D representation of time/frequency decomposition

3D representation of time/frequency decomposition

(a) Original input. (b) Cleaned input.

Figure 12. Three-dimensional scalograms of original double-sweep logarithmic input and cleaned input.

Subspace System Identification

Estimated transfer functions with a classical Fourier method and the time-frequency localization approach are

compared in figure 5. The cleaned transfer functions are clearly superior in illuminating the modal responses relative

to the noise. This distinct difference between the two approaches would considerably influence a system

identification procedure.

Example plots in figure 13 compare SRA transfer functions from a typical aeroelastic data set with transfer

functions generated from the frequency domain algorithm 15 outlined in the appendix. Transfer functions are from

exciter inputs to an accelerometer output and are derived from unprocessed and processed time-frequency data.

Matches between unprocessed data and the subspace estimate are in figure 13(a), and figure 13(b) matches processed

16
Magnitude,

dB

- 20

-30

-40

- 50

- 60

- 70

- 80

I

t

e

J

i

t

I I

ii i

i l •

lj I l t

$ i

,/

I I

Magnitude,

dB

0

- 20

-40

- 60

-80

I I I

t

l J

- ;,,_

I I I

500

0

- 500

- 1000

- 1500

0

I I I

I L I

10 20 30 40

Frequency, Hz

Phase,

deg

1000

500

0

-500

-1000

0

t I I I ,.4

• "..

o

i"

,44

I 0 20 30 40

Frequency, Hz 970021

(a) Transfer functions derived from raw time-frequency data.

Magnitude,

dB

-20 | I i i /

- 40

- 50 ,,

- 60

- 70 _.oF..V, 7

Magnitude,

dB

0

- 20

-40

- 60

- 80

I I I

I I I

Phase,

deg

5OO

0

- 500

-1000

- 1500

0

I I I

10 20 30 40

Frequency, Hz

Phase,

deg

1000

5OO

0

- 500

-1000

I I I

t I 1

10 20 30 40

Frequency, Hz 970022

(b) Transfer functions derived from cleaned time-frequency data.

Figure 13. Discrete-time frequency domain subspace identification results. Solid lines are transfer functions from

exciter input to accelerometer output computed with raw (a) and cleaned (b) time-frequency data. Dashed lines are

transfer function estimates from a frequency domain subspace identification algorithm processing the solid line data.

17
datawithsubspaceestimates.Thesolidlineontheseplotsaretransferfunctionsderivedfromthetime-frequency

procedure,andthedashedlinesaresubspaceidentificationestimates.Agreementis consistentlybetterbetween

processedtransferfunctionsandtheidentification,comparedtotheoriginaltransferfunctions,therebyprovidingan

increaseddegreeof confidenceinthesystemquadruplestimateswhenusingthefeature-filtereddata.

APPLICATION TO F/A-18 HARV AEROSERVOELASTIC DATA

The excitation signals used for generating aeroservoelastic excitation data are described. Morlet decompositions

of data from these maneuvers are used for analysis.

Excitation Mechanisms

Schroeder-phased harmonic signals were compared to sinc-pulses rnr_si__::_because linear and logarithmic

sweeps are deficient at low and high frequencies, respectively. OBES Schroeder-phased and sinc-pulsed signals

were chosen to excite the HARV at the same range of flight conditions of 1 g, 5 ° to 70 ° AOA and 30,000 ft altitude.

Analysis of the Schroeder-phased data compared to linear sweep and sinc-pulse data at the same conditions is

reported in reference 19.

Figure 14 shows a Schroeder wave input to the aileron and the resulting aileron position and roll rate. Figure 15

shows a sinc-pulse input to the aileron with position and roll rate. These signals were input to the surfaces

sequentially in the following order within 70 sec: symmetric stabilator, differential stabilator, ailerons, rudders, pitch

thrust vector vanes, and yaw thrust vector vanes.

Decomposition of Schroeder-Phased Harmonic Data

Figure 16(a) shows an example Morlet scalogram of a 5 to 25 Hz Schroeder input into the pitch thrust-vectoring

command at 10° AOA. The nature of the time-frequency response is the sum of two distinct sweeps, one sweeping

the low frequencies followed by the high frequency sweep with little overlap. All frequencies from 5 to 25 Hz are

being excited equally. The time-frequency-filtering procedure used to analyze the SRA data is applied in

figure 16(b), and subsequent analysis applies accordingly. In contrast to the SRA data, the OBES inputs are

relatively clean because they are generated digitally by the flight system. Figure 17 shows the corresponding transfer

functions of structural response from the OBES inputs of pitch and yaw thrust vectoring. The procedure enhances

the modes near 15 Hz and above relative to the other frequencies, which is justified from figure 18 where the output

accelerometer response scalogram is plotted. The response is dominated by energy around 15 Hz and above relative

to other frequencies, with little response below 15 Hz. Note the order of magnitude difference.

Decomposition of Sinc-Pulse Data

Figure 19 shows scalograms of sinc pulses. In contrast to the Schroeder-phased signal, sinc functions contain

the majority of their energy in a small amount of time. It seems that processing of these data is unnecessary because

nearly all the energy is concentrated in a well-defined area in the time-frequency plane localized in time. Figure 20

shows corresponding transfer functions of structural response from the OBES inputs of pitch and yaw thrust

vectoring. Noise in the spectral response is drastically reduced in figure 20(b) compared to figure 20(a). In

figure 20(a), some difficulty exists in discriminating modal response from spurious noise peaks; in figure 20(b), the

modal response is more distinct. These transfer functions demonstrate that considerable improvement in data quality

is achieved by time-frequency filtering. Time localization of the signal degrades the frequency response unless this

property is exploited in the analysis. In figures 19(b) and 20(b), the time-localized energy is extracted from the

input-output signals to define a much cleaner frequency response from which to perform system identification. Note

the order of magnitude difference in gain between raw and cleaned magnitude responses. Also note that the

responses for the same flight condition at 10° AOA and same input-output signals are very different between the

two types of excitation, Schroeder-phased and sinc-pulsed, either unprocessed or processed.

18
Magnitude

Magnitude 0

-1

Magnitude

0

--.1

°2

_,4

26

_:..................................................................................... i ̧

28 30 32

Time, aec 970023

Figure 14. Example Schroeder-phased harmonic aileron command input (top), aileron position (middle), and roll

rate (bottom).

19
Magnitude

1.1

1.0

.9

.8

.7

.6

.5

.4

.3

.2

.1

0

--,2

--.3

Magnitude,

Magnitude

32.0 32.5 33.0 33.5 34.0

Time, eec 970024

Figure 15. Example sinc-pulse aileron command input (top), aileron position (middle), and roll rate (bottom).

20
10

Frequency, 15

Hz

2O

25

Signal In time/frequency domain

0 2 4 6 8 10 12

6

8

10

12

14

Frequency,

Hz 16 _I_

18

2O

2422 _ I ....

0 2 4 6 8

Signal In time/frequency domain

-1

Time, sec Time, sec 970025

(a) Raw scalogram. (b) Cleaned scalogram.

Figure 16. Morlet scalograms of Schroeder-phased harmonic input.

I I

10 12

970026

101

o

100

C

a_ 10-1

t_

:E

10-2

Pitch input

101

loo 10-1

10-2

_ - 500

0

w

m

• " - 1500

O.

- 2000

0 10 20 30

Frequency, Hz

Yaw Input Pitch input

!!_iii_i_ ;. 101 !_i_i_i_i!t_iiiiiii_

I

10-2

m 2000

._ 1500

1000 50O

_= o

- 500

0 10 20 30

Frequency, Hz

970027

o,i 

- 500 i

• .............. i .............. i..................

= -1500

L

- 2000

0 10 20 30

Frequency, Hz

Yaw input

101 _!_!!!!!i iilii iii-iii :._ !i _!i!_i_!

100

--_10 -1

10-2

3000

2000 _ t

"O

1000

,- 0 1

eL

- 1000

0 10 20 30

Frequency, Hz

970028

(a) Transfer functions derived from raw timefrequency data (left). Pitch thrust vector input (left

column) and yaw thrust vector input (right column).

(b) Transfer functions derived from cleaned timefrequency data (right). Pitch thrust vector input (left

column) and yaw thrust vector input (right column).

Figure 17. Transfer functions from raw and cleaned Schroeder-phased data.

21
Frequency,

Hz

10

15

20

25

Signal in time/frequency domain

I I t i I 1

I I f t t

0 2 4 6 8 10

Time, $ec

Figure 18. Accelerometer response scalogram from Schroeder-phased input.

I

12

970029

5

10

Frequency,

Hz 15

20

25

Signal in time/frequency domain

Frequency,

Hz

0

5

10

15

20

25

Signal in time/frequency domain

1 2 3 4 5 6 7 8 0 1 2 3 4 5

Time, aec Time, aec

970030

(a) Raw scalogram. (b) Cleaned scalogram.

Figure 19. Morlet scalogram of sinc-pulsed input.

6 7 8

970031

22
g

101

lOo

10-1

10-2

Pitch input Yaw input

_!ii::iiiii!i!!i i_ 101 !!_!!!!!!!!!!!"!.!!i__

100 _; 10-1

10-2

Pitch input

101

:_ 10-1lO0 _

10-2

Yaw input

101

100

"_10_ 1

lu

10-2

5O0

m 0

- 500

|

J= - 1000

- 1500

10 20 30

Frequency, Hz

0

_ - 1000 _i I

- - 2000

_. - 3000

- 4000

0 10 20 30

Frequency, Hz

970032

500

_500

: -1000

-1500

0 10 20 30

Frequency, Hz

300o

2000

1000

i o

- 1000

0 10 20 30

Frequency, Hz

970033

(a) Transfer functions derived from raw timefrequency data. Pitch thrust vector input (left column)

and yaw thrust vector input (right column).

(b) Transfer functions derived from cleaned timefrequency data. Pitch thrust vector input (left column)

and yaw thrust vector input (right column).

Figure 20. Transfer functions from raw and cleaned sinc-pulsed data.

CONCLUSIONS

Multiresolution analyses of flight vibration data on two F/A-18 aircraft were performed with nonorthogonal,

nondilated Morlet wavelets. This procedure offered enhanced capabilities for time-frequency feature extraction and

filtering and for subsequent system identification. Such a time-frequency approach allows improved visualization

and understanding of the signal information content. Noise and other distortion dynamics can be identified, reduced,

or eliminated. Automated thresholding would be difficult in the majority of situations when distortion dynamics are

significant relative to the signal content. This proposed approach succeeds when standard threshold techniques are

inappropriate. Identification schemes used to extract modal data, state-space representations, or stability boundaries

demonstrate improved performance with these procedures.

23
APPENDIX

An outline of system identification is presented in time and frequency domains. Proposed advantages and

experiences with algorithms using structural excitation data is described.

Subspace Identification of System Quadruple

Subspace identification methods attempt to find a linear, time-invariant, state-space realization

Xk+ 1 = Axk+BUk+O3k (16)

Yk = CXk + DUk + Vk (17)

based on the input vector sequence u k and output sequence Yk" The state vector x k is not directly observed, so it is

only identifiable to within a similarity transformation. White noise vectors 0Jk and vk are zero-mean process and

measurement noise, respectively. The state-space model identification problem is then as follows:

Given N input-output samples {u k, Yk} of the unknown system, find a consistent estimate of the

state-space quadruple [A,B,C,D] up to a similarity transformation T. Determine the Kalman gain K

such that the output of the estimated system

Xk + l = AXk + Buk + K(yk-Cxk) (18)

Yk = Cxk + DUk (19)

is the minimum variance one-step ahead prediction of the recorded output.

Subspace algorithms are regarded as powerful modem techniques for linear system identification because of

numerous advantages. For example,

• They are noniterative, nonparametric, and linear.

• They are numerically robust algorithms.

• They produce nonparametric state-space models.

• Their MIMO and SISO algorithms are identical.

• They have both time and frequency domain versions.

• They have model insensitivity to matrix perturbations.

• There is good performance for identification of high-order (structural) systems.

For these reasons, the following algorithms are outlined with emphasis on experiences in performance, using data

processed with Morlet wavelet filtering.

Block Hankel matrices that play an important role in subspace methods are

Ul u2 "'" UN-2k+

u2 u3 "'" UN- 2k + 2

: ".. :

Uk Uk + l "'" UN-k

Uk+l Uk+2 ... UN_k+ 1

U2k U2k + l "'" U N

(20)

24
where U_ and Uf designate past and future samples. A similar construction is made for the output matrix Y. The

extendedeobservability matrix is defined as

Icl r = CA (21)

Ai,_

Fundamental to subspace algorithms is the following matrix equation

Y = FX+HU+V (22)

where Y, U, and V are Hankel matrices formed with the output samples, input samples, and noise, and H is the

Toeplitz matrix of Markov parameters

n

D 0 0 ... O"

CB D 0 ... 0

CAB CB D ... 0

;AS-2B cAS-3B cAS-4B ... D

(23)

The following types of subspace identification methods were compared for this study: strictly time domain, 16

combined time and frequency domain, 11 and frequency domain. 14A5'2°Time domain algorithms had the worst

performance, especially in extracting B and D matrices, so a different approach was proposed. The A and C matrices

were derived from the time domain data, then the B and D matrices were computed from matching the state-space

model with estimated transfer functions. This approach gave better results than using a strictly time domain version

in getting the B and D matrices, especially when using the filtered transfer functions from the Morlet time-frequency

method.ll Finally, two algorithms using frequency domain data were tried to take advantage of the improved transfer

functions. All algorithms will be outlined here for completeness•

Time Domain Algorithm

Because the transfer function estimates are based on at least two distinct data sequences from the SRA data

(symmetric and antisymmetric), a method was used to determine system A and C matrices from multiple data sets. 21

Briefly, A and C are derived from the following technique (C is p by n).

• Find P such that P = FQ and rank(P) = rank(F). In the noise-free case (V = 0), P = YU ± because HUU -L= 0

in equation (22) (_1_denotes an orthogonal complement). For two data sets, P = [P1 P2] = F[QI Q2]' The

following steps remain the same.

• Decompose P = USV,, where S = S 1 0 , and U = are rank(P).

0 0 U2

• Evaluate: A - U_U 1 where UI(U1) respectively omit the first (last) p rows of U 1, and 5" denotes

pseudoinverse. C is the first block of U 1.

• Premu_tip_y(2)byF__s_F±F=_andp_st-mu_t_p_yb_Ut_ead_ng__a_eastsq_aress__uti_n__IB 1 •

25
Combined Time and Frequency Domain Algorithm

In the combined time-frequency domain algorithm, the B and D matrices are found from discrete transfer

_k

function G(Zk), where z k = e j% and o_k = _ for M data, k = 0 ..... M

G(Zk) = [C(ZkI-A)-ll][BD]

(24)

for all frequencies cok, the least squares solution separating real and imaginary is

C(ZlI-A) -1

= C(ZE1-A) -1 !

C(ZMI- A) -1 !

t

1

GG(Zl)]

(ZM)J

(25)

Frequency Domain Algorithm

Frequency domain algorithms 14.15,2°are similar in context. In particular, one method 15starts with the premise

that the transfer function G of the system has a real-valued impulse response. As a result, data on [0,r_] can be

extended to [n,2n] by taking the complex conjugate and continuing as follows.

• Extend transfer function samples tothe fullunit circle: GM+ k := GM_k,k- 1.... ,M- 1.

_2gik

1 2M- 1 Gke 2M • Define the real block Hankel matrix: 2M-point inverse DFT hi = _ 2k = 0

hi ... hr

/4 := : ".. " where (q,r) > n and q + r < 2M.

hq ... hq+r-

• C°mputetheSVD: U_f/T'system[/ =n by inspecting singular partitioning: I_".o fq [s_

• Determine order values and // = [(Js(Jo] I-_ -1 9

L0 ol

o

• Determine system matrices ,4 and C as )_ = (O_s)t_ls and C is the first p-block of/_/s.

• Solve least squares, as above, for B and D.

Other algorithms use auto- and cross-spectral data in the same framework 14 or identify a Laplace domain

[A,B,C,D] from discrete transfer function data c0k using numerically well-conditioned Forsythe polynomials. _°

These frequency domain algorithms perform well for high-order structural systems.

To recap, even with processed data from the time-frequency filtering procedure, the subspace time domain

methods perform poorly with rapidly varying inputs. Including some frequency domain information in the subspace

identification method will improve the estimates. Furthermore, enhancements are necessary in data processing for

frequency domain data without assumptions of stationariness, time-invariance, periodicity, and white noise. Timefrequency filtering methods offer a solution. Other examples using a variety of estimation algorithms can be found

in reference 22.

26
REFERENCES

1. Vernon, Lura, In-flight Investigation of a Rotating Cylinder-Based Structural Excitation System for Flutter

Testing, NASA-TM-4512, June 1993.

2. Brenner, M., Rick Lind, and Dave Voracek, "Flight Flutter Testing Research at NASA Dryden" AIAA 97-0000,

Apr. 1997.

3. Bowers, Albion H., Joseph W. Pahle, Joseph R. Wilson, Bradley C. Flick, and Richard L. Rood, An Overview of

the NASA F-18 High Alpha Research Vehicle, NASA-TM-4772, Oct. 1996.

4. Brenner, Martin J., Aeroservoelastic Modeling and Validation of a Thrust-Vectoring F/A-18 Aircraft, NASATP-3647, Sept. 1996.

5. Schroeder, M. R., "Synthesis of Low Peak-Factor Signals and Binary Sequences with Low Auto-correlation,"

IEEE Transactions on Information Theory, vol. 16, pp. 85-89, Jan. 1970.

6. Van der Ouderaa, Edwin, Johan Schoukens, and Jean Renneboog, "Peak Factor Minimization of Input and

Output Signals of Linear Systems," IEEE Transactions on Instrumentation and Measurement, vol. 37, no. 2,

pp. 207-212, June 1988.

7. Strang, Gilbert and Truong Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996.

8. Herley, Cormac, Jelena Kovacevic, Kannan Ramchandran, and Martin Vetterli, "Tilings of the Time-Frequency

Plane: Construction of Arbitrary Orthogonal Bases and Fast Tiling Algorithms" 1EEE Transactions on Signal

Processing, vol. 41, no. 12, Dec. 1993.

9. Jones, Douglas L. and Richard G. Baraniuk, "An Adaptive Optimal-Kernel Time-Frequency Representation,"

IEEE Transactions on Signal Processing, vol. 43, no. 10, Oct. 1995.

10. Daubechies, Ingrid, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia,

PA, 1992.

11. Feron, Eric, and James D. Paduano, Advanced Techniques for Flutter Clearance, Annual report for NASA-MIT

cooperative agreement no. DFRCU-95-025, Oct. 1996.

12. Lind, Rick and Marly Brenner, "Worst-Case Flutter Margins from F/A- 18 Aircraft Aeroelastic Data," 38th AIAA

Structures, Structural Dynamics and Materials Conference, Apr. 1997.

13. Quatieri, Thomas F., Jr., "Short-Time Spectral Analysis with the Conventional and Sliding CZT," IEEE

Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-26, no. 6, Dec. 1978.

14. Liu, Ketao, Robert N. Jacques, and David W. Miller, "Frequency Domain Structural System Identification by

Observability Range Space Extraction," Proceedings of the American Control Conference, Baltimore, MD,

vol. 1, pp. 107-111, 1994.

15. McKelvey, Tomas, Hiiseyin Akqay, and Lennart Ljung, "Subspace-Based Multivariable System Identification

from Frequency Response Data," IEEE Transactions on Automatic Control, vol. 41, no. 7, July 1996.

16. VanOverschee, Peter and Bart DeMoor, Subspace Identification for Linear System Theory, Implementation,

Applications, Kluwer Academic Publishers, 1996.

17. Viberg, Mats, "Subspace-based Methods for the Identification of Linear Time-invariant Systems," Automatica,

vol. 31, no. 12, Dec. 1995.

18. Misiti, Michel, Yves Misiti, Georges Oppenheim, and Jean-Michel Poggi, Wavelet Toolbox User's Guide, For

Use with MATLAB, The Math Works, Inc., 1996.

27
19.Brenner,M., Rick Lind, andDaveVoracek,"AeroservoelasticModelIdentificationwith MultipleInputs,"

AIAA 97-0000,Aug.1997.

20.VanOverschee, PeterandBartDeMoor,"Continuous-TimeFrequencyDomainSubspaceSystemIdentification,"

submittedforpublicationtoIEEE Transactions on Signal Processing, 1995.

21. Duchesne, L., Eric Feron, James D. Paduano, and Marty Brenner, "Subspace Identification with Multiple Data

Sets," Proceedings of the 1996 AIAA Guidance, Navigation, and Control Conference, San Diego, CA,

May 1996.

22. Feron, E., L. Duchesne, A. Turevskiy, J. D. Paduano, and M. Brenner, "Time-Frequency Analysis for Transfer

Function Estimation and Application to F18-SRA," submitted to the Automatica, Jan. 1997.

28

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORTTYPE AND DATES COVERED

April 1997 Technical Memorandum

4.TITLE AND SUBTITLE 5. FUNDING NUMBERS

Wavelet Analyses of F/A- 18 Aeroelastic and Aeroservoelastic

Flight Test Data

6.AUTHOR(S)

Martin J. Brenner

7.PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

9.SPONSORING/MONITORING AGENCYNAME(S)ANDADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

522-32-34-00-RS-00-000

8. PERFORMING ORGANIZATION

REPORT NUMBER

H-2164

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM-4793

11.SUPPLEMENTARYNOTES

Presented at the 38th Structure, Structural Dynamics and Materials Conference Exhibit, AIAA,

Kissimmee, Florida, April 7-10, 1997.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified--Unlimited

Subject Category 08

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

Time-frequency signal representations combined with subspace identification methods were used to

analyze aeroelastic flight data from the F/A-18 Systems Research Aircraft (SRA) and aeroservoelastic data

from the F/A- 18 High Alpha Research Vehicle (HARV). The F/A- 18 SRA data were produced from a wingtip

excitation system that generated linear frequency chirps and logarithmic sweeps. HARV data were acquired

from digital Schroeder-phased and sine pulse excitation signals to actuator commands. Nondilated continuous

Morlet wavelets implemented as a filter bank were chosen for the time-frequency analysis to eliminate phase

distortion as it occurs with sliding window discrete Fourier transform techniques. Wavelet coefficients were

filtered to reduce effects of noise and nonlinear distortions identically in all inputs and outputs. Cleaned

reconstructed time domain signals were used to compute improved transfer functions. Time and frequency

domain subspace identification methods were applied to enhanced reconstructed time domain data and

improved transfer functions, respectively. Time domain subspace performed poorly, even with the enhanced

data, compared with frequency domain techniques. A frequency domain subspace method is shown to produce

better results with the data processed using the Morlet time-frequency technique.

14. SUBJECT TERMS

Aeroelasticity, Aeroservoelasticity, Aircraft vibration, Modal stability,

Structural dynamics, System identification, Wavelet analysis

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION

OF REPORT OF THIS PAGE OF ABSTRACT

Unclassified Unclassified Unclassified

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Linthicum Heights, MD 21090; (301)621-0390

15. NUMBER OF PAGES

32

16. PRICE CODE

A03

20. LIMITATION OF ABSTRACT

Unlimited

Standard Form 298 (Rev. 2-89)

Prera;r_ed by ANSi Sld Z39-18

298-102