NASA/TM-2001-210391

Nonstationary Dynamics Data Analysis With

Wavelet-SVD Filtering

Many Brenner

NASA Dryden Flight Research Center

Edwards, California

Dale Groutage

Naval Surface Warfare Center (NSWC)

Bremerton, Washington

April 2001

https://ntrs.nasa.gov/search.jsp?R=20010043991 2018-05-14T14:44:19+00:00Z
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NASA/TM-2001-210391

Nonstationary Dynamics Data Analysis With

Wavelet-SVD Filtering

Marry Brenner

NASA Dryden Flight Research Center

Edwards, California

Dale Groutage

Naval Surface Warfare Center (NSWC)

Bremerton, Washington

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, California 93523-0273

April 2001
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(703) 487-4650
NONSTATIONARY DYNAMICS DATA ANALYSIS

WITH WAVELET-SVD FILTERING

Marty Brenner*

Aerostructures Branch

NASA Dryden Flight Research Center

Edwards, CA 93523-0273

Dale Groutage t

Puget Sound Detachment

Naval Surface Warfare Center (NSWC)

Bremerton, WA 98314-5215

Abstract F_

g

Nonstationary time-frequency analysis is used for identifica- j

tion and classification of aeroelastic and aeroservoelastic dy- i, k, l

namics. Time-frequency multiscale wavelet processing gen- rn, n

erates discrete energy density distributions. The distribu- P,q

tions are processed using the singular value decomposition

r, 8, t

(SVD). Discrete density functions derived from the SVD gen- R

erate moments that detect the principal features in the data.

The SVD standard basis vectors are applied and then corn- SRA

pared with a transformed-SVD, or TSVD, which reduces the SVD

number of features into more compact energy density concen- TSVD

trations. Finally, from the feature extraction, wavelet-based W a

modal parameter estimation is applied.

The primary objective is the automation of time-frequency

analysis with modal system identifcation. The contribution

is a more general approach in which distinct analysis tools are

merged into a unified procedure for linear and nonlinear data

analysis. This method is first applied to aeroelastic pitchplunge wing section models. Instability is detected in the

linear system, and nonlinear dynamics are observed from the

time-frequency map and parameter estimates of the nonlinear

system. Aeroelastic and aeroservoelastie flight data from the

DAST (drone for aerodynamic and structural testing) and

F18 aircraft are also investigated and comparisons made between the SVD and TSVD results. Input-output data is used

to show that this process is an efficient and reliable tool for

automated on-line analysis.

Nomenclature

a wavelet scale

DAST drone for aerodynamic and

structural testing

DWT discrete wavelet transform

*805.276.3793, marty.brenner_dfrc.nasa.gov

f360.476.5927, grout agefdOnswccd.navy.mil

Copyright ©2001 by the American Institute of Aeronautics and

Astronautics, Inc. No copyright is asserted in the United States

under Title 17, U.S. Code. The U.S. Government has a royalty-free

license to exercise all rights under the copyright claimed herein for

government purposes. All other rights axe reserved by the copyright owner.

T

03

OJo

<>

time-frequency feature descriptor

wavelet basis function

imaginary unit

integer indices

number of time and frequency points

moment orders

integer indices

rank of time-frequency map

Systems Research Aircraft

singular value decomposition

transformed-SVD

continuous wavelet transform

wavelet translation time

radian frequency

wavelet peak frequency

complex conjugate transpose

result of moment calculation

Introduction

Many aerospace disciplines face the task of analyzing

nonstationary signals in which the frequency content

changes with time. Most physical phenomena from

acoustics, aerodynamics, thermodynamics, structural

dynamics, propulsion, and controls are naturally timevarying events with frequency variations, transients, and

complex harmonic interactions. Requirements for adaptive nonlinear procedures are now becoming more essential components of modern data analysis of aerospace

systems in concordance with numerous other engineering and scientific applications.

A fundamental objective in data analysis of physical systems is to obtain accurate representation of the dynamics

for a particular analysis. Optimal and relevant representations are desired for efficiency and consistency between

the data decomposition and the physical system [4, 28, 31].

Methods for achieving this goal are often based on entropy or other statistical measures for best-basis decompositions and optimal performance [22,42}.
Parsimonious representation of data has many such implications in denoising, compression, information retrieval, detection, identification, classification, and pattern recognition [sl. This paper emphasizes applications

in dynamics analysis of aeroelastic and aeroservoelastic

systems from test data. Dominant and important features in the data are extracted to provide automated

information retrieval for system identification. The data

analysis becomes an integral part of the system identification procedure for tracking modal stability estimates

in a nonstationary setting.

Time-frequency analysis provides a powerful tool for the

analysis of nonstationaxy signals [18]. Applications have

been demonstrated in music signal analysis [34], machinery diagnosis [2_1, damage detection [40], seismic monitoring {19], medical signal analysis [41], and acoustic and

speech processing [a2]. Transfer functions and modal

parameter estimates derived from time-frequency representations have been applied to estimate state-space

aeroservoelastic models [2, 3, xo].

Time-frequency structure is revealed by quantifying the

distribution of signal energy as a joint function of time

and frequency. Localization of the energy density describes energy density concentrations at specific locations in the time-frequency plane. Dominant and important concentrations need to be accurately discriminated to provide descriptors relating to the location in

time, time duration, frequency location, and local bandwidth of principal energy density {11, 17, 43]. Minimizing

the number of these descriptors while preserving salient

information from the energy distribution for modal estimation is desired for efficient dynamics analysis.

Many approaches have been proposed to extract a signal from noise in the time-frequency representation (33].

One approach performs a masking operation on the

time-frequency map, then separates out desirable features [2, lo]. Often the regions of the signal are wellconcentrated relative to the widely-distributed noise so

this method works well in general. However, it requires

user intervention unless regions are well-defined apriori.

For this reason, an automated filtering procedure is proposed using the singular value decomposition (SVD) of

the time-frequency data [29, 4a].

The singular value decomposition is ubiquitous in the signal processing, identification, and control fields as a tool

for rank-revealing algorithms, model reduction, subspace

detection, and information retrieval [1,201. The SVD is

applied in this paper to derive density functions for generating moments {14]from the joint energy density timefrequency distribution. Moments relate to the principal

signatures of the nonstationary signals [6, 27]. In the context of subspace estimation and information retrieval it

is a data reduction for model identification.

The SVD representation tends to be sparse relative to

an entire time-frequency map. The orthonormality of

the SVD singular vectors for density extraction allows

independent decomposition of time and frequency content in the data, thereby avoiding joint moment considerations. A new transformation of the SVD basis vectors, called the transformed-SVD (TSVD) [15], is also

used to produce more concentrated descriptions of energy density for multicomponent signals containing similar components distributed in the time-frequency plane.

For separation of components requiring very fine time or

frequency resolution, the TSVD may be advantageous.

Multiresolution wavelet signal processing has shown

promise for studying time-frequency characteristics of

signals by decomposing data into cells with properties

of scale and frequency concentrated in time. These cells

form a type of tiling and consist of Gaussian-windowed

sinusoid basis functions (atoms), also known as Morlet

wavelets, creating an atomic multiscale decomposition

in a filter bank structure [44]. Competing requirements

of time and frequency resolution, subject to the uncertainty principle [2s}, is accomplished with a combination

of dyadic multiscale decomposition, compact orthogonality, and harmonic wavelet properties [23, 24].

This paper augments time-frequency multiscale wavelet

processing with SVD filtering and wavelet-based modal

parameter estimation. The contribution is a more general approach in which distinct analysis tools are merged

into a unified procedure:

* multiresolution analysis with wavelet decomposition at multiple scales

, SVD techniques for information retrieval

• feature extraction from moments of densities

• modal parameter estimation from the complex

Morlet basis functions

This automated nonlinear filtering procedure is first

applied to linear and nonlinear two degree-of-freedom

pitch-plunge wing section testbed models. Nonlinear dynamics are detected from the time-frequency map of the

nonlinear system. Aeroelastic dynamics from the experimental flutter flight test drone, DAST (drone for aerodynamic and structural testing), is used to show how the input time-frequency signature can be used for automating

the input-output data analysis. Data from a flutter encounter shows the capability for detecting an approaching instability from actual flight test data. Aeroservoelastic flight data from the F18 Systems Research Aircraft (SRA) is also investigated and comparisons made

between the SVD and TSVD results.
Wavelet-SVD Method

This section describes the method to construct timefrequency signal representations from a Morlet wavelet

decomposition, then to use this energy distribution map

to extract principal features from the SVD. Density functions derived from the SVD singular vectors generate

spectral and temporal moments relating to features of

the original nonstationary time series process. These

moments are used for subsequent wavelet-based modal

parameter estimation and system identification.

Time-Frequency Wavelet Decomposition

Much progress has been made in the development of

time-frequency distributions [5]. For a time-frequency

distribution Q to be interpreted as a joint time-frequency

energy density, it must be nonnegative and satisfy the

correct time and frequency marginals for all time and

frequency

Q(t,f)dt = IX(f)l 2 Q(t,f)df = [x(t)l 2

where X(f) = f___x(t)exp(-j2rcft)dt is the Fourier

transform of the signal x(t). Marginals Ix(t)[ 2 and

[X(f)[ 2 are the energy densities of time and frequency

which are commonly called the instantaneous power and

the energy density spectrum, respectively. Algorithms

for constructing proper positive distributions satisfying

these properties have recently been developed [16, 37].

The joint energy density specifies concentrations of energy in time and frequency.

Feature extraction from joint distributions proceeds from

the joint time-frequency moments of signal x (t) (assumed

to have unit energy) given by

< tPfq > = f_-¢_ f:_o_ tPfqQ(t, f)dtdf

p,q = 1,2,3,... (1)

which defines the temporal and spectral moments

< tp > = tPlx(t)12dt

oo

< fq > = /?fqIx(f)12df. (2)

oo

In the present application, the time-frequency distribution, Q(t, f), is constructed from a multiscale wavelet

transform [3]. Therefore, the distribution is not the strict

form of a joint energy density since the requirement on

marginals is not enforced. The wavelet transform tends

to spread the energy over time and frequency, yet the

wavelet decomposition is essential for identification of

modal dynamics as will be shown. This energy density

approximation is therefore relevant for this problem.

Multiscale transforms are used to exploit multiresolution

analysis, or redundant representations of a signal on multiple frequency bands. Nonorthonormal Morlet wavelets

are approximated with harmonic-like discretizations on

wavelet scales corresponding to subharmonics representing overlapping frequency bands per octave (where each

octave is a doubling of resolution, or a scale twice as

small). These wavelets form a nonorthogonal redundant

basis for the signal space [as]. Adjustment to satisfy the

competing requirements of time and frequency resolution

is accomplished with a combination of compact orthogonal and harmonic wavelet properties [23, 241.

A discrete wavelet transform (DWT) is derived from

the wavelet basis to get a multiresolution analysis of

the sampled continuous Morlet transform. Fast algorithms axe realized with a dyadic multiscale decomposition. Therefore, the DWT is implemented as a dyadic

filter bank covering a pre-defined range of frequencies

with corresponding number of frequency bands per octave (voices/octave) [44], Voices can be viewed as fractional dilations of a single wavelet at a particular scale.

Approximate orthogonality is imposed in the multiresolution representation of the Morlet filters. Multiscale

Morlet DWTs provide efficient and flexible means for

analysis of nonstationary data with adjustable frequency

resolution versus time localization.

The continuous wavelet transform of signal x(t) over the

time-scale (% a) plane is represented as

%(r,a) 1 / t-r --g- = _ x(t)g* ( )at

where scale parameter a is proportional to the duration

and inversely proportional to the peak frequency wo of

the complex Morlet wavelet

1 f2 .

g(t) = e

The spectrum of a dilated and translated Morlet wavelet

G_,r(w) = e-(a_-_°)2eJ_r

reaches a maximum value at a = _-_. The continuous

oJ

Morlet basis functions are discretized for a DWT filter

bank decomposition. Signal power is preserved by enforcing the following identity with the DWT (Cw is an

admissibility constant)

Ix(t)12dt = dr [_(r,a), 2daW- I

a

from which the instantaneous power is expressed in terms

of the wavelet time variable, r, as

This time-scale decomposition of data is often called a

scalogram, or the energy density ]Wg (r, a)] 2 of the signal

over the (r, a) plane [2s]
As anexampleof a bankof discretizedfiniteimpulse

responseMorletfilters,figure1showsanarrayofwavelet

filtersforanumberofvoicesperoctave.Thesefiltersare

adjustedontheplotto lineupthepeakmagnitudesat

thesamelocationsin time. Centerfrequencyincreases

withvoicenumber,andthefiltersarenormalizedtohave

thesamemagnitudein thefrequencydomainat their

respectivepeakfrequencies.

FIR Discrete Wavelet Filters

• . , -- . : . : • .... i 4

2 . - ' " " " ' " " " • '

1

1.5-1 , •

4

g o.5.t.-

04,

600 " /.. ::: " i: ::. . 40

4oo " .--" :i .... ,. 30

Filter Length, se¢ 0 0 Voice Number

Figure 1: Wavelet filters for a number of voices per octave.

Morlet wavelets are being used to create the power distribution and to estimate the modal parameters. Therefore,

an implicit filtering process is being performed independent of the explicit procedure of feature extraction and

noise removal using the SVD and TSVD. The wavelet basis representation of the signal is a projection subspace

from which modal parameters are derived.

SVD Filtering

Numerous wavelet thresholding techniques exist based

on statistical measures of significance for denoising.

These include entropy measures from information and

communication theory. They are generally based on regularity, smoothness, or a low-pass character of the underlying signal relative to noise, and the compromise is

between goodness-of-fit and smoothness. Because this

approach is somewhat subjective by nature, a more deterministic approach is preferable. In aeroelasticity and

aeroservoelasticity, the statistical criteria are generally

not applicable since the dynamics can be of significant

bandwidth and include high-frequency transients.

Assume the time-frequency distribution is a positive distribution in the form of a scalogram, spectrogram, or

any other representation such as the Wigner-Ville distribution. In the present application the basis functions are the discretized complex Morlet wavelets, so

the scalogram magnitude is appropriate. Discretized

wavelets form the multiresolution analysis which can be

exploited at diffferent scales for information extraction

by the SVD.

The SVD spectrum encodes the features of timebandwidth product, frequency-time dependence, and

number and location of signal components. It is invariant to shifts of the signal in time or frequency [29, 43].

While this SVD method is described for the entire timefrequency distribution, it could also be used to analyze

the different scales independently or jointly.

Consider the distribution Q as an m-by-n discrete matrix

A such that A(k,1) = Q(tk,fl), where k = 1,2,...,m

and I = 1,2,... ,n for m time points and n discrete frequencies. This matrix can always be decomposed with

the SVD into a set of basis matrices Ai with corresponding singular values, or weights, ai

R

A = _ aiAi; Ai = uiv_. (3)

i=1

R is the rank, the set of constant weights ai are ordered

such that al _> a: _ ... _> an > 0, and each Ai (or ai)

corresponds to singular vectors ui and vi as shown. Time

and frequency aspects of the A, are independent since

u_ and vi are orthonormal. The maximum value of the

rank R can on!y be as large as the minimum of the rowcolumn (m-n) dimension of A. The objective is to get a

reduced set of energy concentrations defined by the Ai

to generate time-frequency features based on the relative

magnitudes of the associated ai. Note that the rank R

can never exceed the number of discrete frequencies n,

i.e., R < n, since there will always be more time points

than discrete frequencies.

The matrix A is used with the weights to extract highlights from the wavelet map by characterizing the Ai

with the joint distribution of equation 1. For each ai, a

corresponding region in the distribution is desired (Ai)

which is weighted in value by the magnitude of ai relating to its respective information content. From the

original joint moments of equation 1, the SVD separates

the distribution into R primary moments. This decomposition correlates directly with independent time and

spectral moments expressed in equation 2 through independent singular vectors ui and vi in equation 3, so joint

moments need not be considered [141.

A unit-energy positive distribution A matrix does not

guarantee that each A_ will be element-by-element positive, which is necessary for a proper density function.

Matrices ,4_ are constructed from the Ai by squaring

each matrix element to create proper densities for moment calculations. Each Ai is formed from the elementby-element square of the associated singular vectors ui

and vi, and the corresponding squared vectors are denoted by fii and vi. The new basis matrices are

_,_=_ > 0.

Density functions fii and vi compose proper density functions Ai by being positive and orthonormal, and by be-
ingindependent, theyarealsothetimeandfrequency

marginalsof therespective/i.i. Moments of the density functions constructed from the singular time and

frequency vectors are used to formulate features. Temporal and spectral moments corresponding to equation 2

are calculated as

171 n

< >,= < >,= F_,(f,)%(l). tp

k=l 1=1

First moments 09 = q = 1) estimate time-correlated instantaneous frequency, and the second moments (p =

q = 2) determine the time duration and instantaneous

bandwidth for each feature in the distribution. A feature

descriptor Fi is therefore defined by

= -_-_l,<t>i,<f>i, <t2>i-_, <f2>i-j_ .

Each feature descriptor Fi for each Ai is a 5-element vector which includes the normalized singular value weight

b_, time location ti, instantaneous frequency ]i, time

duration ii, and instantaneous bandwidth ]i, of the feature. Features are hi-scaled rectangle regions in the timefrequency map. They are used as windows on the original

scalograrn to extract regions for further analysis.

The number of desired features should preferably be as

small as possible while preserving relevant dynamics.

The separation property of the SVD helps maintain important information with a minimal number of features.

Generalizing the same process to higher-order moments

gives instantaneous skew and kurtosis (6J which may reveal data anomalies, asymmetries, and nonlinearity.

0.8

0.6

0.4

_ 0.2

_ o

_ -0.2

m -0.4

-0.6

-0.8

-1

0

Simulated Four Sine-Burst Signal

1 2 3 4 5 6 7 8 9 10

Time, s6c

Figure 2: Four sine-bursts of equal amplitude.

A simple example of a simulated analysis with SVD filtering uses the signal in figure 2. There are four sine-bursts

at 15, 10, 5, and 25Hz, respectively. A wavelet decomposition and SVD filtering procedure is performed on this

signal to produce the plots in figure 3. The top plot represents an energy density distribution contour (plotted

in two dimensions) of the original wavelet coefficients.

The middle plot represents the sum of all rectangular

features Fi determined from the SVD filter. The bottom plot accentuates the middle plot using a log-scale

(dB) for the out-of-plane contour dimension. A log-scale

region will cover at least as much as the linear scale contour of the middle plot, and all contours are normalized

to their maximum magnitude. The middle and bottom

plots are identical except for the contour scale.

Time-Frequency EnergyDensity Distributions

_4o_

30 _

g 2oI ___

u. 10[ ""----- I I # ! I I I I i

0 1 2 3 4 5 6 7 8 9

30

2O _ -

u-1 --

i i i i i i i i |

0 1 2 3 4 5 6 7 8 9

10

10

3O

20

_-10

0 1 2 3 4 5 6 7 8 9 10

Time. sec

Figure 3: Contour plots from original wavelet decomposition (top), SVD-filtered (middle), and SVD-filtered

(bottom, log-scale) scalograms of four sine-burst signal.

Note the more compact representation from SVD filtering (middle plot) compared to the original decomposition. For an automated analysis, we choose the order

of the distribution to be the full rank R. Of the total 300,000 elements in the distribution, the rank in this

case reduces the number of features to R = 300, resulting

in a dramatic data reduction for model identification.

Transformed-SVD

A potential problem with the standard SVD filter stems

from the information contained in the individual Ai

terms of the decomposition. The principal information

is ordered according to values of ai, but often there are

multiple time-frequency regions with similar energy density magnitudes which will contribute equally to each

Ai. Moments calculated from the singular time and frequency vectors contain simultaneous contributions from

distributed regions since the vectors span the entire

scalogram. Information is not concentrated because the

SVD cannot separate energy density concentration in the

individual Ai so the features consist of linear combinations of many singular vectors.
FeaturextractionwouldbenefitfrommorelocalizedsignalcontentineachAi for two reasons. First, for a fixed

number of features, fewer terms in the SVD would enhance specific localized content. Second, for a fixed number of terms, higher fidelity features would be produced.

Therefore, what is desired is a rotation of the singular

basis to minimize the number of basis vectors to interpret each density concentration. Ideally, each Ai term

should relate to a single localized concentration.

This problem can be posed as follows: find an orthonormal transformation for the u_ and vi vectors such that

the means are maximized in the new basis. The attempt

is to concentrate the densities into smallest regions in the

plane. For unknown transformation C, and orthonormality condition CC* = C*C = I, the means are expressed

in terms of the cik coefficients. Orthonormality implies

rn 2

that _r=l Yi (r) = 1, so the means of the transformed

vectors Yi =- _-,k Ci,kUk are the first moments < yr >

given by

< yr >,= rY, (r)

m R

= Er=l r Es-----I cs,i_s(r) E_R.=I ct,/_t (r)

= c;Mci

In this quadratic form, the unique solution for maximizing the means of the Yi vectors is achieved when the ci are

the eigenvectors of the M matrix. Similarly, an orthonormal matrix D is found independently for the vectors vi by

maximizing the means of vectors xi = _k di,kvk. Compared to the standard SVD, the transformed-SVD is as

follows [151

A = USV*; U = YC*; V = XD*

and the proper substitutions made to get the TSVD

g = (YC*)S(XD*)* = Y(C*SD)X* = YZX*.

Now that instead of just a sequence of singular values a_

from the S matrix of the SVD, there exists a full singular

matrix Z = C*SD of terms to weight the new singular

vectors X and Y in the TSVD. These terms have the

same function as the terms in S, but instead of correspondence to linear combinations of concentrations in

ui and vi, the Z-elements, zik, correlate to transformed

singular vectors y, and xk. Obviously there are significantly more (m-by-n) of the zik than the ai (R < n).

The singular weights in Z are ordered and the new TSVD

decomposition for the first R of the ik-components takes

the form (where now B _ A, compare to equation 3)

R

B = E zikB,; Bi = yix*k. (4)

i=1

Returning to the signal from figure 2, a transformedSVD analysis is performed and plotted in figure 4 for

._. 40

. 30

10

0

4°I

30

2o

L.

u. 10

0

4°f

30

20

_- 10

0

Time-Frequency Energy Density Distributions

- .t

I I I I I I I I

1 2 3 4 5 6 7 8 9 10

........ -7-t

1 2 3 4 5 6 7 8 9 10

I I I I I I _1 I l

1 2 3 4 5 6 7 8 9 10

Time. sec

Figure 4: Contour plots from original wavelet decomposition (top), TSVD-filtered (middle), and TSVD-filtered

(bottom, log-scale) scalograms of four sine-burst signal.

comparison with the SVD result in figure 3. The plots

in figure 4 correspond to those in figure 3, so the top

plot is identical. The middle TSVD contour plot in figure 4 shows an enhanced version of the plot in figure 3

by extension of the time-range estimate of three of the

sine-bursts. Note the longer feature window based on

the computed Fi compared to the original distribution

contour. The burst at 5Hz (7 sec) is not distinguishable

in the TSVD middle plot, but in the remarkably similar

bottom log-scale plot it reappears. This burst at 5Hz

has the least total energy content.

Observe that in the bottom two plots, the TSVD has remarkably more detail with the same number (R = 300)

of singular vectors. Information extraction is devoted to

the regions of the sine-bursts, and not spread over unnecessary areas of the time-frequency plane as in the case of

the SVD in the bottom plot of figure 3. The more compact representation from the TSVD will improve further

analysis by concentrating on the important regions of

interest for modal identification.

It was already shown in a previous study [1_] how the

TSVD separates and improves the quality of the timefrequency features with a small set of descriptors for classification purposes. The present application for flight

data analysis emphasizes the filtering aspects of the

TSVD versus the SVD. The purpose is not to find a minimum number of descriptors, but to get the highest quality filtered output for a fixed number of features determined by the rank, R, of the discretized time-frequency

map. This will allow an automated procedure for preprocessing with a time-frequency filter to augment modal

parameter estimation and system identification.
SVDapproachesareattractivefroma systemsanalysis

perspective.Prevalencein modelreduction,subspace

identification,andoptimalfilteringpointto its significanceasa tool forinformationretrieval.Someof its

powerliesin thegeometryof the singular vectors as a

distance measure between subspaces [7}. Also, for on-line

implementations, popular SVD recursive procedures allow for adaptive signal processing by using fast updatingdowndating schemes [l, 9]. The idea is to perform the

wavelet decompositions, update the time-varying distributions, and apply on-line procedures for the analysis.

Parametric Modal Estimation

Modal parameters can be estimated with wavelets by

analysis of the system impulse response [3, 36] (see appendix). The DWT of a signal using the complex Morlet

wavelet is a complex-valued matrix whose modulus and

phase are related to impulse response parameters. Linear

phase variation of the DWT estimates the instantaneous

frequency. Wavelet modulus decay is used similarly to

derive decay rate for corresponding modal damping estimates assuming the instantaneous frequencies correspond to modal dynamics. In the current application,

this procedure is applied at every time point assuming

at each instant that the response is a sum of multiple

degree-of-freedom impulse responses.

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40

35

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25

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15

10

5

0

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Natural Frequency Dominant Frequencies

50

2 4 6 8 2 4 6

Time. sec Time, sec

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_ 25

$

!f 1

O"

lO o 8 10

Figure 5: Instantaneous frequency estimates (left) and

detected dominant spectrum (right) using the SVD.

Instantaneous frequency estimation of the four sine-burst

signal of figure 2 is illustrated in figures 5 and 6 using

the SVD and TSVD, respectively. The plots show frequency estimates over the corresponding detected period

of time from the DWT. Right plots titled Dominant Frequencies in each figure are subsets of the those titled Natural Frequency. Left plots designate raw estimates from

the respective filtered regions, and right plots refer to the

spectrum detected by using a threshold parameter on the

wavelet coefficients. This threshold is 0.5 of the normalized absolute magnitude of the scalogram. Threshold

estimates are referred to in this report as dominant.

In the SVD analysis of figure 5, numerous frequencies are

detected from the extracted regions of figure 3, but of

those only two dominant frequencies at 15Hz (3 sec) and

25Hz (9 sec) in the right plot are estimated based on this

threshold. The congested left plot indicates that many

detected frequencies are redundant or spurious estimates

from large relatively insignificant regions of the timefrequency map (bottom plot of figure 3). Conversely,

in figure 6 for the TSVD, the left and right plots are

identical and all the frequencies are detected as being

dominant. This demonstrates that the TSVD is detecting only that part of the spectrum that is significant in

the data without any threshold criteria.

LINatural Frequency

30

25

20

15

10

5

0

0 2 4 6 8 10

Time, sec

Dominant Frequencies

30

_-5

_.0

15

iO

5

0

0 2 4 6 8 10

Time. sec

Figure 6: Instantaneous frequency estimates (left) and

detected dominant spectrum (right) using the TSVD.

Aeroelastic Testbed Model

Previous studies have investigated applying wavelets

for analysis of structural and aeroelastic nonlinear systems [25, 30, 45]. SVDs have been used to extract the minimum embedding dimension from noisy nonlinear systems [39]. In this section the wavelet-SVD approaches

are applied to investigate linear and nonlinear reponses

for a simple aeroelastic system. An aeroelastic testbed

has been developed at Texas A&M University for flutter research using a prototypical aeroelastic wing section [12]. This quasisteady pitch-plunge system is described by two-degree-of-freedom aeroelastic equations

of motion. A control surface configuration was used for

closed-loop control [21].

Nonlinearity is introduced to the system dynamics

through the stiffness associated with pitch. This stiffness is described by a nonlinear polynomial function of

the pitch angle. Such structural nonlinearities occur in

physical aeroelastic systems and have been investigated

to determine their effect on inducing limit cycle oscillations. The two parameters that determine the response

of the wing and the onset of limit cycle oscillations, are

the elastic axis location and the freestream velocity. The
"o

e_

_c

"D

¢-

E

E

8

Linear Aeroelastic Testbed Simulation

_i

0 1 2 3 4 5 6 7 8 9 10

200

._ 100

0

_-100

n

-200

-300

0 1 2 3 4 5 6 7 8 9 10

Time, sec

Figure 7: Chirp control command input (top) and pitch

response (bottom) from linearaeroelastictestbed model.

stiffness functions associated with the pitch degree of

freedom are chosen here to represent a linear spring and

a nonlinear hardening spring.

10.

8

g 4

2

10

Damping Ratio 0 Frequency, Hz

Wavelet Coefficients

Damping Ratio 0 Frequency, Hz

lO

8

2.

O_

10

Wavelet Coefficients

2 _ 0.5

Figure 8: Modal parameter estimates from testbed

linearsimulation using SVD (top) and TSVD (bottom).

A classical flutter instability is encountered for the linearized dynamics near a speed of 12 m/s at about

2Hz [25]. Figure 7 represents a control input chirp of

0-5Hz (top) and the unstable pitch response at 12 m/s

(bottom). Both SVD and TSVD results are presented in

figure 8. These plots represent the estimates and corresponding wavelet coefficient magnitudes from which the

estimates were derived. Magnitudes of the coefficients

determine a relative level of significance, or measure of

observability. The upper plot from the SVD shows the

instability detected near 2Hz. The array of zero damping

values at frequencies below the unstable mode indicate

an output tracking of the input (disguised by the magnitude in figure 7) until an instability is detected at 1.5Hz

near the pitch mode natural frequency. TSVD estimates

of the same instability are more accurate near 2Hz I:sJ

with much fewer spurious estimates.

Nonlinear Aeroe|astic Testbed Response

0.5 .....

i ....

0.4 .... : .... i.. :, .i .:........ i i........

o.3 .... i.... i..... ! i..... !.... i- i

0 .... : .... : - •

-0.1 . • :

-0.,2

-0.3 ; ; i ; ; ; ; ; ;

0 1 2 3 4 5 6 7 8 9 10

Time, sec

Figure 9: Pitch response from nonlinear aeroelastic

testbed model chirp command input.

Time-Frequency Energy Density Distributions

11 I

0 1 2 3 4 5 6 7 8 9 10

30

-?-25

20

_,15

glo

', 5

i

0 1 2 3 4 5 6 7 8 9 10

Time. sac

Figure 10: Contour plots from original wavelet decomposition (top) and SVD-filtered (bottom, log-scale)

scalograms of nonlinear aeroelastic testbed simulation.

A nonlinear response from the same chirp input also at

12 m/s with a nonlinear hardening spring in the pitch

degree of freedom is shown in figure 9. The corresponding wavelet scalogram and SVD-filtered contour plot are
shown in figure 10. Note the scattering of the output

spectrum from a well-defined chirp input, thus indicating

the nonlinearity [25]. There is clear indication of higher

harmonics from the primary 2.5Hz mode at higher frequencies near 7.5Hz and 12.5Hz (the third and fifth harmonics). This result is also evident from the corresponding frequency and damping estimates in figure 11. From

the multitude of SVD raw estimates, this plot shows only

the dominant coefficients using the threshold discussed

previously. At any time during the response, at least two

of three of the resonances are evident.

Dominant Wavelet Coefficients

7.

6

¢D 5,

_ 3,

1,

0,

O.08

15

• . . . .

Damping Ratio Frequency, Hz

Figure 11: Modal parameter estimates from nonlinear

aeroelastic testbed simulation using SVD filtering.

30

25

2O

lO

5

o

Featured TSVD Distribution (dB)

; ;

Time. Sac

Figure 12: Contour plot of TSVD-filtered scalograms

(log-scale) of nonlinear testbed simulation response.

More precise tracking of the modes from the TSVD is

evident from the filtered contour plot of figure 12. This

plot corresponds to the previous contours from figure 10.

High definition frequency estimation compared to the

SVD is clearly seen, and results in the raw estimates

plotted in figure 13 being identical to the TSVD dominant estimates (not shown).

Wavelet Coefficients

_ 6.

_ 3.

2.

1.

0=

0.04

Damping Ratio 0 0 Frequency, Hz

Figure 13: Modal parameter estimates from nonlinear

aeroelastic testbed simulation usiag TSVD filtering.

15

DAST Flutter Data Analysis

For an automated data analysis procedure to be effective with a known input signal, it is advantageous to use

the time-varying input spectrum for tracking output responses. Linear dynamics may then be discerned from

unmodeled dynamics and nonlinearity outside of the input frequency range. For this purpose, using actual aircraft data, wingtip accelerometer data is analyzed from

the NASA DAST [13} (drone for aerodynamic and structural testing, figure 14). Part of the DAST program in

1980 was to pursue investigations on a drone equipped

with a flutter suppression system to enable flight beyond

the open-loop stability boundary. Wingtip accelerometer response data was acquired with 10-40Hz logarithmic

chirps and 20Hz sinusoidal doublets into the aileron control surface•

Figure 14: NASA DAST vehicle in flight.

From the time-frequency path of the input signal, a mask

is created by SVD or TSVD filtering to be applied to

the accelerometer response. An aileron chirp input (top)

and wingtip accelerometer response (bottom) are shown

in figure 15 at Mach 0.8 about a minute before the drone

encountered an aeroservoelastic instability [131.
DAST AileronChirpandWingtipResponse

4 6 8 10 12 14

2 4 6 8 10 12 14

Time, sec

Figure 15: Aileron chirp (top) and wingtip accelerometer

response (bottom) from DAST at Mach 0.8, 15,000 feet.

Figure 16 shows contour plots of the aileron chirp input.

The SVD-filtered envelope (bottom plot) bounds the input more than adequately. Also, shown in figure 17,

the estimated modal frequency and damping show the

primary wing bending mode near 20Hz, but also other

minor harmonics of much smaller magnitude near 40Hz

and 60Hz. Note that in the SVD-filtered contour plot of

figure 16, the spread up to 60Hz occurs when the primary

20Hz mode is excited by the chirp input (6-7 sec).

N

T

O

C

O

Or

U_

60

50

40

30

20

10

0

Time-Frequency EnergyDensityDistributions

i ! i i I i

2 4 6 8 10 12 14

N

-1-

>_

U

eu.

60

50

40

30=

20!

10

0 2 4 6 8 10 12

Time, sec

Wavelet Coefficients

Figure 16: Contour plotsfrom original waveletdecomposition(top)and SVD-filtered(bottom,log-scale)

scalogramsofDAST aileronchirpinput,

In contrast, the TSVD-filtered contour of figure 18 discriminates discrete frequency bins directly along the

time-varying chirp input. Dynamics outside of the timefrequency path of the input will not be observed. Estimates will only be derived locally along the input time-

• , • . . . . , . •

• . . . . . . .

• . . . • .

0.4 ' : . ..- ....

• :..-': '_ ... " -.

0.3 _, : 'E'_ " : ....... : ....

o.1 .... : ! :: ........ :......

00.1__''"'""". "''"""" "''''" 80

0__ 40 "_ Dampin Ratio 20

0 Frequency, Hz

Figure 17: Modal parameter estimates from DAST

wingtip acclerometer from chirp using SVD filtering.

frequency window. This property of the TSVD may be

used to separate linear output response from unmodeled

dynamics and nonlinearity.

The SVD filter generally encompasses a much larger

region than the input alone, with the higher contours

more adjacent to the dominant energy spectrum. While

not as precise a filter as the TSVD, it detects a timefrequency envelope that thoroughly bounds the input.

A TSVD filter concentrates more singular vector contributions (moments) into smaller areas than the SVD filter. The TSVD may actually miss some input dynamics

from its moments for a reasonable order approximation

comparable to the SVD. A higher order approximation is

possible with the TSVD as mentioned in the comments

above equation 4, but this paper only addresses equalorder approximations between the SVD and TSVD.

Featured TSVD Distribution (dB)

:?:

;=,.,

60

50

40

30

201

i

14 o 2 4 6 8 10 12 14

Time, sec

Figure 18: Contour plot of TSVD-filtered scalogram

(log-scale) of DAST aileron chirp input•

Wavelet reconstructions are created from the coefficients

of the time-frequency map that are retained by filtering,

masking, or thresholding. In a direct filtering or masking

application, the quality of the wavelet reconstruction of

a time response depends on the time-frequency window

used on the response. Good results are obtained when

10
4

3

2

o

-2

-4

0

Reconstructe¢l Wingtip Response

2 4 6 8 10 12 14

Time. se¢

Reconstructed Wingtlp Response

3 .....................

2 ....................

o°-20 .................... .................... •

0 2 4 6 8 10 12 14

Time, sec

Figure 19: Reconstructed SVD-filtered (top) and TSVD

filtered (bottom) DAST wingtip accelerometer responses

from DAST at Mach 0.8, 15,000 feet (see figure 15).

all the significant dynamics are retained. For instance,

with no filtering or masking, the reconstructed response

is simply a projection onto the wavelet basis functions.

This signal generally resembles the original signal because of the good reconstruction properties from the redundancy in the multiscale wavelet transform.

DAST Wingtip Accelerometer Response

5 ....

4 ........... :...... :...... :.......... : .....

3 ..... i.... i...... i...... i........ i .....

2 . .. :. . •i...... i...... i...... •

i 1 : " . Z . ,

_i

0 1 2 3 4 5 6 7

Time, sec

Figure 20: Wingtip accelerometer response from DAST

aileron-pulsed input at Mach 0.825, 15,000 feet.

For the wingtip accelerometer response of figure 15, comparisons are made between the SVD (top) and TSVDfiltered (bottom) reconstructions in figure 19. The SVD

filter covers more of the time-frequency plane, and this

shows up in the reconstructed response as compared to

the TSVD. Both reconstructions are modulated modal

responses, but the TSVD consists of narrower bandpass

filters in its modulation.

60

N

T 5O

_ 4o

® 30

_r 20

u. 10

Time-Frequency EnergyDensityDistributions

i , , i i , i i

0 1 2 3 4 5 6

N

-1-

O

e-

¢

60-

50

40

30

20----

10

i i i i L i

0 1 2 3 4 5 6

Time, sec

Figure 21: Contour plots from original wavelet decomposition (top) and TSVD-filtered (bottom, log-scale)

scalograms of DAST wingtip accelerometer response.

1.5

Reconstructed Wingtip Accelerometer Response

1

0.5 i

-1

-1.5

0 5 6

Figure 22: Reconstructed TSVD-filtered DAST wingtip

accelerometer response at Mach 0.825, 15,000 feet.

To demonstrate the estimation procedure for an actual

aircraft instability, figure 20 represents the wingtip accelerometer response at Mach 0.825 immediately before

the closed-loop instability. During the last three sine

pulses, the TSVD filter results are shown in the the contour plots of figure 21. The SVD filter has difficulty

making a precise determination of the pulse energy density map (not shown). Tracking the pulse responses is

complete and accurate with the TSVD filter.

Figure 22 represents a reconstructed TSVD-filtered version of the signal in figure 20. From the coefficients of

the filter shown in figure 21, the reconstructed time response resembles an impulse response of the mode near

20Hz. Estimated modal frequency and damping from the

11
TSVDinfigure23showsclearlythe20Hzwingbending

modeandreductionin modaldampingwithtime.

25

20

_- 15

$

_1o

LL

Dominant Frequencies

5

0

0 2 4 6

Time. sec

Dominant Dampings

0.016

0.014

0.012

0.01

0.006 =

0.006

0.004

0.002.

0

0

m

ht

'2 4 6 8

Time. sec

Figure 23: TSVD estimated modal frequencies (left) and

damping ratios (right) from DAST wingtip accelerometer.

F18 Aeroservoelastic Data Analysis

Digital excitation signals were implemented on an the

NASA F18 Systems Research Aircraft (SRA) for a baseline model update and validation effort to support the

F18 Active Aeroelastic Wing [35] program. Multisine signals were added to aileron, stabilator, and rudder actuator commands. In this example, the differential aileron

command is used for excitation at Mach 0.9, 10,000 feet.

Wingtip and lateral fuselage accelerometer responses are

used to demonstrate the SVD and TSVD filtering process

with a discontinuous input in the time-_frequency plane.

Multisine waveforms have a chirp-like quality, but with a

burst of high-frequency chirp at the end of the signal [2].

F16-SRA Wingtip Accelerometer Response

0 5 10 16 20 26 30

Time, sec

Figure 24: F18-SRA wingtip accelerometer response

from a multisine aileron input.

A wingtip accelerometer response to a multisine aileron

input is shown in figure 24. Figure 25 (top plot) shows

the nature of a multisine in the wingtip response for a 35-

to-5Hz sweep. Note the 13-15Hz pulse at the end of the

main sweep (21 sec), characteristic of the multisine. The

_- 4O

30

C

_=20

0"

10

I.L

Time-Frequency EnergyDensityDistributions

A I I I !

5 10 15 20 25 30

4O

30

C

==20

O"

10

LL

i

0 5

_LL-_ , .A> I ,

10 15 20 25

Time. sec

30

Figure 25: Contour plots from original wavelet decomposition (top) and SVD-filtered (bottom, log-scale)

scalograms of F18-SRA wingtip accelerometer response.

Featured TSVD Distribution (dB)

45

40

35

"7"30

g 25

_g20

15

10

0 5 10 15 20 25 30

Time. sec

Figure 26: Contour plot of TSVD-filtered scaiogram of

F18-SRA multisine wingtip accelerometer response.

SVD filter (bottom plot) attempts to bound the entire

time-frequency response, as in the other cases, whereas

the TSVD-filtered output in figure 26 distinguishes fine

frequency divisions over short time intervals. The 13Hz

and 7Hz lines near 20 sec are very distinct and precisely

tuned to the specific frequencies.

A TSVD-filtered reconstruction of the original wingtip

response is in figure 27, which corresponds to the contour

plot of figure 26. Both the SVD (bottom contour plot,

figure 25) and TSVD contour and time history plots show

dominance from the 12-13Hz mode just around 15 sec.

Since the control system is active, this mode corresponds

to the stabilator mode which is excited by the aileron

multisine in closed loop.

12
0.5

0.4

0.3

0.2

0.1

o

_ -0.1

-0.2

-0.3

-0.4

-0.5

0

Reconstructed Wingtip Response

,

10 15 20 25 30

Time, sec

Figure 27: Reconstructed TSVD-filtered F18-SRA wingtip

accelerometer response from multisine aileron input.

Time-Frequency Energy DensityDistributions

_- 40

3o

r20'

| , i i i

5 10 15 20

• i

i

25 3O

_" 4O

30

£:

==20

O"

10

14.

I I i I ¢

0 5 10 15 20 25

Time. sec

3O

Figure 28: Contour plots from original wavelet decomposition (top) and TSVD-filtered (bottom, log-scale),

scaiograms of F18-SRA fuselage lateral accelerometer.

In a final example, figure 28 shows a persistent oscillation of the wing bending mode at 8-9Hz (first fuselage

and wing bending modes) from the fuselage lateral accelerometer for the same multisine aileron input. The

top plot is the original energy density, and the bottom is

the log-scaled TSVD-filtered distribution. The SVD distribution is similar to that in figure 25, so is not shown.

The TSVD defines the persistent mode very well while

also detecting the dominant part of the chirp near 16Hz

(10-15 sec). The final part of the energy density distribution after 20 sec at 20Hz, however, is not detected

since it is relatively less significant than the 8-9Hz modes.

Time-dependent modal frequency and damping ratios estimated with the TSVD filter are displayed in figure 29 to

show the relatively low damping of the steady oscillation

at 8-9Hz compared to modes at 19, 17, and 15Hz (10-15

sec) corresponding to fuselage second bending, wing second bending, and wing first torsion modes, respectively.

Natural Frequency

14

ft.

o, =,=,..

0 10 20 3o

Time. sec

0.03

0.02

0.01

0

0

Damping Ratio

10 20 30

TimA. _,R,c

Figure 29: TSVD estimated modal frequencies (left) and

damping ratios (right) from F18-SRA fuselage accelerometer.

Conclusions

Nonstationary dynamics data analysis is discussed from

the perspective of time-frequency and multiscale resolution analyses. Wavelet decomposition of dynamics data

using a discrete harmonic basis constructed from the

Morlet wavelet establishes a time-frequency representation for modal parameter estimation and system identification procedures. Automated procedures to discern and

extract regions of the time-frequency energy density distribution that are significant for dynamics analysis are

presented. The methods are derived from the singular

value decomposition (SVD) of a time-frequency distribution.

Examples of simulated sinusoids, a testbed aeroelastic

system, aeroelastic flight data from a flutter experiment, and aeroservoelastic data from an F18 are used to

demonstrate and compare the two SVD filtering methods. One method using a standard SVD demonstrates

the ability to discriminate the most significant and other

less dominant dynamics. Its performance may depend

on a time-frequency window determined by the input.

Regions outside of the input time-frequency path relate

to information about noise, unmodeled dynamics, and

nonlinearity.

Another method called the transformed-SVD, or TSVD,

optimally rotates the singular vectors to create a more

precise filter for finer tracking, but may ignore the less

significant dynamics for the same order filter as the standard SVD. Both methods complement each other and

show promise for automatic feature extraction supplemented with noise removal, residual analysis, and uncertainty estimation. These filtering methods contribute to

nonstationary dynamics data analysis at multiple scales

in the time-frequency framework.

13
Appendix

Given a general harmonic signal

x(t) = k(t) cos(¢(t)t)

the WT of x(t) is

W(a, r) = v_k(t)e-(_*(t)-_°)_e_¢(t>.

For fixed dilation parameter ai (equivalently fixed frequency w), the modulus and phase of the WT of x(t)

are

IW(a.r)l = v_k(t)e -(_'¢(t)-_°)=

L[W(a,,_-)] = ¢(t> (5)

Instantaneous frequency of a signal in this case can be

expressed as [3_]

1 d

¢(t) = r)]) (6)

This shows that a general time-varying envelope k(t) or

phase ¢(t) of the signal can be determined from the modulus and phase of the WT for each fixed wavelet frequency.

More specifically, from the impulse response of a single

degree-of-freedom viscous damper

x(t) = Ae -¢"_'t cos(walt + ¢o)

we have from equation 5,

IW( _,,_/I = Ae-¢ _"t

k(t) = ,/aT _(_,,m__o)_

¢(t)t = /[W(ai, r)] = war + ¢o

For a constant wavelet frequency line corresponding to ai

over time 7- in the (a, r) plane, estimation of the WT linear phase variation (or mean value of the instantaneous

frequency over time, from eq. 6) gives ¢(t) _ wd, and the

envelope decay rate is _wn. Natural frequency wn and

modal damping ratio _ are therefore derived. The WT

becomes a complex representation of the original real

signal from which the signal eigenvalues are computed

without any approximation of their range.

Multiple degree-of-freedom systems are analyzed similarly by noting that the dilated Morlet wavelet is a

bandpass filter. With sufficient resolution of dilation ai,

damped modal frequencies Wd_ = _-_ can be discrimiai

nated. To recap, the decay rate of the envelope of each

mode is calculated from the log-slope of the wavelet modulus decay, and damped modal frequency is estimated

as the linear phase variation of the WT as a function

of time. Adequate frequency resolution can be enforced

with the multiscaled compact harmonic Morlet wavelets.

,

,

.

,

,

.

.

.

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16
REPORT DOCUMENTATION PAGE For.,Approved

OMB No. 0704-0188

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

April 2001 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Nonstationary Dynamics Data Analysis With Wavelet-SVD Filtering

6. AUTHOR(S)

Many Brenner and Dale Groutage

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(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

WU 529-50-04-00-RR-00-000

8. PERFORMING ORGANIZATION

REPORT NUMBER

H-2445

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA/TM-2001-210391

11.SUPPLEMENTARYNOTES

Modified from paper presented at 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and

Materials Conference, Seattle, Washington, April 16--19, 2001. Dale Groutage, Naval Surface Warfare Center

(NSWC), Bremerton, Washington.

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Unclassified--Unlimited

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13. ABSTRACT (Maximum 200 words)

Nonstationary time-frequency analysis is used for identification and classification of aeroelastic and

aeroservoelastic dynamics. Time-frequency multiscale wavelet processing generates discrete energy density

distributions. The distributions are processed using the singular value decomposition (SVD). Discrete density

functions derived from the SVD generate moments that detect the principal features in the data. The SVD

standard basis vectors are applied and then compared with a transformed-SVD, or TSVD, which reduces the

number of features into more compact energy density concentrations. Finally, from the feature extraction,

wavelet-based modal parameter estimation is applied.

14.SUBJECTTERMS

Aeroelasticity, Aeroservoelasticity, Singular value decomposition, System

identification, Time-frequency analysis, Wavelet analysis

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