Signal Processing via Wavelet and IdentificationSignal processing via wavelet and identification To determine the flutter boundary of the F-18 System Research Aircraft at NASA Dryden

June 1996 LIDS-P-2348

Signal Processing via Wavelet and Identification

Final diploma Thesis for the Swiss Federal Institute of Technology(Eidgen6ssische Technische Hochschule Zurich)

written at The Department of Aeronautics and AstronomicsMassachusetts Institute of Technology

February 5 - May 10 1996

1.4 . . .....-.

·'' ' .. .. .......,,-,. .... ....... .. .1 .2 .. ....

~~~~~~~.....Supervisor at MIT: Prof. .... Feron0.Supervisor at ETH: Prof. G. Schweitzer0.4.0.2.

0

30

20

0 10time (s) 0

frequency (Hz)

Author: Xavier PaternotSupervisor at MIT: Prof. E. FeronSupervisor at ETH: Prof. G. Schweitzer

MIT Document Services Room 14-055177 Massachusetts AvenueCambridge, MA 02139ph: 617/253-5668 I fx: 617/253-1690email: docs @ mit.eduhttp://libraries.mit.edu/docs

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Abstract

Signal processing via wavelet and identification

To determine the flutter boundary of the F-18 System Research Aircraft at NASA Dryden Research Centre a

series of in-flight tests have been performed. Based on these measurements a model for the flexible dynamics

of the aircraft structure is to be identified. The available signals are, however, very noisy and must be proc-

essed. The goals of this project are, using a time-frequency approach, to denoise these signals and to find an

estimate of the system's transfer functions. The obtained results should be exploited to improve the presently

used identification methods.

A continuous wavelet decomposition using non orthogonal and non dilated wavelets is performed. Based on

the user's judgement, this decomposition is cleaned manually. From the processed decomposition a denoised

time signal can then be reconstructed. Choosing only relevant areas in the time-frequency decomposition of

the input and output signals transfer functions are estimated. The cleaned signals and the estimated transfer

functions are used to identify a simulated system via a susbspace identification and a frequency fit. A toolbox

written for Matlab 4.2 has been implemented to perform these operations.

The obtained results have shown that the time-frequency approach allows a clear interpretation of the signal's

information content, a sensible elimination of the noise and a smooth estimation of the system's transfer func-

tions. Furthermore, the performance of identification methods can be improved by using these results.

Abstract

Acknowledgement

First of all I wish to thank my supervisor at MIT professor Eric Feron for showing and communicating so

much enthusiasm in my work. I also would like to thank Professor Schweizer at ETH in Ziirich for accepting

to supervise this project. Professor Wehrli at ETH in Zurich and Professor Hollister at MIT in Cambridge have

made my presence here at MIT possible, I would like to express my gratitude for their efforts of bringing the

two schools closer together and for giving me the opportunity to live such an enriching experience.

Special thanks to Laurent Duchesne for numerous helpful discussions and for all his work on this project, and

to all my lab mates for their help and their warm welcome.

Acknowledgement

List of symbols

A state-space A matrix

B state-space B matrix

C state-space C matrix

D sate-space D matrix

At pseudo-inverse of matrix A

A1 A orthogonal (where AA1 = 0)

CWT(ct) Continuous Wavelet Transform

Ff(o) Filter in frequency domain centered around frequency f

G(o) Transfer function at frequency 0o

Hti block triangular Toeplitz Matrix

H(co) Wavelet in frequency domain

U,S,V Singular Value Decomposition matrices

Uh block Hankel input matrix

X(co) Signal in frequency domain

Yh block Hankel output matrix

a scaling factor

h(t) wavelet in time domain

t time

us, ua symmetrical and asymmetrical input

x(t) signal in time domain

yS, ya symmetrical and asymmetrical output

or slope of the frequency sweep

co frequency

F observability matrix

Aco frequency range

At time range

List of symbols

The sensors consist of two load sensors measuring the force applied to the wing and of ten accelerometers

placed along the wings and the tails. These locations are resumed in table I and indicated in figure 2.

Output no. Sensor Location1 left wing forward2 left wing aft3 left aileron4 left vertical tail5 left horizontal tail

6 right vertical tail7 right horizontal tail8 right wing forward9 right wing aft10 right aileron

TABLE 1. Accelerometer locations

Right actuator

Left actuator

FIGURE 2. Location of sensors and actuators on F-18 SRALeft actuator

FIGURE 2. Location of sensors and actuators on F-18 SRA

1.2.2 Input/Output signals

Based on the a priori knowledge of the resonance frequency of the flexible dynamics, the frequency range to

consider is 0 to 30 Hertz. The amount of time at disposal for each measurement during one flight test is 30 sec-

onds. In order to cover the whole flight envelope, the measurements are performed at several different flight

Introduction 3

points. Considering the high cost of a flight the number of required experiments is to be kept as low as possi-

ble.

Since two inputs are considered it is necessary in order to distinguish their influence to run two experiences. In

the case of the F18-SRA a symmetrical measurement, where the inputs are the same, and an asymmetrical

measurement, where the inputs are shifted of 180 degrees in respect to each other, are performed at each con-

sidered flight point.

As a result of the required measurements and of the properties of the actuator at disposal the chosen input sig-

nal u(t) is a signal of sinusoidal shape with modulated frequency also known as chirp signal. Linear and loga-

rithmic frequency sweeps have been tested. In the case of the linear sweep the signal can be written

u(t) = sin(Oct2), (1)

where the coefficient ao can be computed based on the desired frequency range ACO and the amount of time

At at our disposal:

cX = ACO/At. (2)

An example of an actual input signal is given in figure 3 along with its frequency content. In figure 4 an output

signal measured on the forward left wing is plotted. The very high noise content is clearly apparent in both

those signal

FIGURE 3. Exciting signal (force applied on the left wing) and its power spectral density

FIGURE 4. Response of the excited system measured on the left wing and its power spectral density

Introduction 4

1.2.3 The system to identify

The approach adopted by the NASA to locate the flutter boundary of the F-18 SRA is to identify the dynami-

cal behavior of the plant illustrated in figure 5. This system has therefore two inputs, the exciting forces

applied on the wings, and ten outputs, the acceleration of the structure at ten different locations. It is assumed

that it is a linear system.

Noise

Exciting/Signal Actuator

Sensor load Sensor cceNoise sesor Noise me

Measured MeasuredInput output

FIGURE 5. Diagram of the plant to identify

1.3 Goals of the project

The goals of this project are to process the data using a time-frequency approach in order to reduce the noise

and to use these results to improve the performance of the presently used identification methods. For this pur-

pose a time-frequency signal analysis toolbox should be implemented that allows a systematic processing of

the signal within the time-frequency domain and an estimation of the system's transfer functions through a

graphical user interface. These results are to improve the performances of presently used identification meth-

ods.

Introduction 5

2. Time-Frequency Signal Analysis

2.1 Theory

2.1.1 Why a Time-frequency analysis?

To analyse the frequency content of a signal x(t), the well-known Fourier transform

X(O) = fx(t) e dt (3)

is available. This transform is a projection of the signal onto sinewave basis functions of infinite duration and,

as a result, it is very efficient to analyse stationary signals. However if the frequency behavior of x(t) varies

with time, the transform performed in (3) cannot highlight the frequency variation in function of time.

To analyse such signals a transform T of the form

T(x(t)) = X(t,Co) (4)

which locates instantaneous frequencies in x(t) is needed. Such a transform would allow a three dimensional

decomposition of a nonstationary signal where its energy content could be located as a function of time and

frequency simultaneously.

2.1.2 The Resolution Problem

However, such a transform as given in (4) is limited in resolution. Indeed, if one is interested in a frequency

range AC , it will only be possible to perform the analysis within a time window At given by the relation

Am. At>2 . (5)

This relation is known as the Heisenberg uncertainty principle. Any Time/Frequency analysis therefore has for

principal task to handle this resolution problem, and an a priori knowledge of the signal properties is required

in order to minimize its effects.

Time-Frequency Signal Analysis 6

2.1.3 The different Time-Frequency approaches

The Short Time Fourier Transform (STFT)

To localize the instantaneous frequency in a signal it is possible to consider small time windows of the signal

and apply (3) to each of these (see [4] and [5]). The transform is defined as follows

STFT(t,) = fx(=) w(t-')e-l°d~, (6)

where w(t) is the window function centered at time t. The properties of this transform depend strongly on the

shape and size of the window function. Indeed it will determine the frequency and the time resolution which

are constant over all the time-frequency domain.

This decomposition can be interpreted in two different ways. On one hand it can be looked at as a the projec-

tion of the signal onto functions

-jwo~w (t-, a ) e (7)

On the other hand it can be viewed as a filtering of the signal with a bank of filters. At a given frequency to

equation (6) amounts to filtering the signal "at all times" with a bandpass filter having as impulse response the

window function modulated to that frequency.

The Wigner-Ville Decomposition

The Wigner-Ville decomposition (WVD) is defined as

WVD (t, o) = x(t+2x (t- eC_ dt, (8)

where x (t) is the complex conjugate of the signal x (t) . This representation has a number of mathemati-

cal properties considered desirable in a time-frequency representation [5],[6] and[7]. However in practical

applications its use is limited by the presence of cross terms [5] and [10], which makes representations and

interpretations difficult.


The Wavelet Decomposition

In order to handle the resolution problems resulting from the Heisenberg uncertainty principle recent works

from Ingrid Daubechies [11], [12] and Yves Meyer [13], [14] have broaden the principle of the STFT which

lead to a new time-frequency decomposition, the Wavelet Decomposition (WD).

The principle of this decomposition is to perform an orthonormal projection of the signal onto functions that

are "adapted" to the frequency resolution that is needed. Indeed the limitation of the STFT is that the resolu-

tion is the same at all location in the time frequency-plane. To overcome this problem functions called wave-

lets are considered, that are dilated (i.e. extended) and shifted versions of a mother wavelet h(t). The dilation is

chosen so that the ratio

stays constant. The Continuous Wavelet Transform (CWT) can be defined as follow

CWT (, a)= J x (t)h ( )dt, (9)

where a is the scaling factor and h (t) is the complex conjugate of h (t) . The Heisenberg uncertainty princi-

ple will still be satisfied, but now, the time resolution becomes arbitrarily good at high frequencies and the fre-

quency resolution becomes arbitrarily good at low frequencies. The coverage of the time-frequency plane that

was homogenous for the STFT is now logarithmic (figure 6). This kind of analysis works best if the signal is

composed of high frequency components of short duration plus low frequency components of long duration.

a) b) t

FIGURE 6. Coverage of the time-frequency plane for the STFT (a) and the WT (b)


Note that the ha,t(t) are not orthogonal since they are very redundant (they are defined for continuous a and

r). However it is possible to define a Discrete Wavelet Transform (DWT) where the wavelet functions gener-

ate an orthonormal basis, the considered wavelets must then satisfy the so-called dilation equations [19]. The

coefficients a and r are discretized in a dyadic way and the wavelets become

h, k(t)= 2 -j/2 h2 jt- kT) .(10)

This results in the wavelet coefficients

cj, = Jx(t) h (t) (11)

In this case the reconstruction problem is unambiguous, as the chosen basis is orthonormal, and could be for-

mulated as follows

x(t) = Cj, k hj,k(t) k (12)j k

Similarly as for the STFT the wavelet transform can be looked upon in a signal processing point of view as the

filtering of the signal through a bank of filters [17], [15]. The considered filters are then bandpass filters, with

identical shape but with varying frequency width, centered around the frequencies of interest.

2.2 Application to the F-18 SRA Measurements

2.2.1 Time-frequency decomposition of the signal

Choice of transform

As one of the goals of the time-frequency analysis is to process graphically the decomposition it is of high

importance that the performed decomposition appears as "clear" as possible to the naked eye. For this reason

the Discrete Wavelet Transform using orthogonal wavelets as described above is not appropriate since a

dyadic coverage of the time-frequency plane would include too few of the frequencies of interest which range

from 0 to 30 Hertz. Indeed the considered frequencies in this range would only be 2, 4, 8 and 16 Hertz. The

chosen decomposition is therefore a continuous wavelet transform, using as a result non orthogonal wavelets.

For the same reason of clarity the considered wavelets should have no group delay (i.e. no nonzero imaginary


part in the Fourier domain). Several FIR filters with no group delays are tested to compute a CWT. The results

that appears the "clearest" are obtained with morlet wavelets, whose representation H(o) in the Fourier

domain (figure 7) is given by

((O - (2) 0 2 2 2

H(w) = e -e (13)

0.03

0.02

0.8 0.1 .

°0.6 0

0.4 - -. 01-

0.2 -0.02

4 , i i ' -0.030 2 4 6 B 10 12 14 16 18 20 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5..

frequency (Hz) time (s)

FIGURE 7. Morlet Wavelet in Fourier and in time domain

As stated above the wavelet transform that uses dilated wavelets as basis functions is adapted to signals com-

posed of high frequency components of short duration plus low frequency components of long duration, in

other words, for signals where the frequency rate goes through rapid changes in time. For the considered sig-

nals measured on the F- 18 SRA this is not the case, especially for the linear frequency sweeps where the fre-

quency rate of change is constant over the whole signal. Thus a constant resolution is required along the whole

time-frequency domain. For this reason the chosen time-frequency transform uses non dilated wavelets as

functions onto which the signal is projected.

To compute the whole transform the signal x(t) is filtered several times with morlet filters centered around fre-

quencies ranging from 0 to 30 Hertz (figure 8). To simplify the computations the property that the convolution

of two signals in time domain is the multiplication of their Fourier transform in frequency domain is used. The

Continuous Wavelet Transform of the signal considered at time t and frequency f is then

CWT(t,f) = - (X( o) H () (14)


ert

tilter2 x 2( t )

x(t) CWT

filter n x (t )

FIGURE 8. The CWT viewed as a bank of filters

Resolution issues for the linear frequency sweep

Here we try to find, using a priori knowledge on the signal, a good trade-off between time and frequency reso-

lution which can give us an indication of the frequency width of the filter that could be used. Indeed, in the

case of the linear sweep, the frequency varies from () 1 to (02 Hz in T seconds. This means that, assuming

that we want a resolution of AtO in frequency, the time resolution is actually given by the time localisation of

this frequency by

At = T/((o2 -o,)Aco. (15)

Introducing (15) in the Heisenberg uncertainty principle (equation 5) we have

2 T 2 02 - (16Ao =2 and At = 2 (16)co2 - o01 T

For the F-18 SRA the considered linear frequency sweeps range from 0 to 30 Hertz in thirty seconds, which

gives

At = F2 sec and A0O = F2 Hz. (17)

These figures are used to determine an indication for the frequency width of the filters of the CWT.


Results

Applied to one symmetrical input and output (left wing forward) for the case of a linear sweep the results illus-

trated in figures 9 and 10 can be obtained. In figure 11 is plotted the case of a logarithmic frequency sweep. In

order to simplify the representation the absolute value of the decomposition is represented here. A two dimen-

sional and a three dimensional representation are plotted here.

3s0

25.

4 20 .

0.2 0.4 0.6 0.8 1 1.2 1.4 time (s) 0time (s) frequency (Hz)

FIGURE 9. Two and three dimensional representation of a CWT of an input signal in the case of alinear frequency sweep.

25

35wi

5 10 t5 20 25 30 time(S) 0 °

time (s) frequency (Hz)

FIGURE 10. Two and three dimensional representation of a CWT of an output signal in the case of alinear frequency sweep.

Time-Frequency Signal Analysis 12FIGURE10. To andthrediesoa rersnainoJ W fa otu inli h aeolinear~~~~ frqcysep

Time-Frequency Signal Anal y s 124

~~~~~~~~~~~~~~~~~~~~~~20~

40 10.2 0.4 0.6 0.8 1 1.2 1.4 0 0 frequency (Hz)

time (s) time (s)

FIGURE 11. Two and three dimensional representation of a CWT of an input signal in the case of alogarithmic frequency sweep

The observation of these results shows that the relevant signal is clearly visible among the noise. By using

some kind of processing it is therefore conceivable to extract it from the noise. Added to the main component

of the chirp some higher harmonic components appear which can be interpreted as nonlinear effects from the

actuator. From the observation of the decomposition of the output it is also possible to forecast the general

aspect of the transfer function.

2.2.2 Signal processing in the time-frequency domain

Cleaning within the time-frequency domain

Similarly as in [7], the principle of this method consists of eliminating, within the time-frequency domain, the

information that does not "appear relevant" using nothing else than the human eye intuition and an a priori

knowledge of the signal. However one has to be cautious on what information is desired especially when

cleaning input and output signals together for the purpose of identifying the system's dynamics. Indeed the

assumption is made that the system to identify is linear. Thus it is possible to state, if information occurs at a

certain frequency and at a certain time in the output signal when no information had occurred in the input at

the same point, that it has been generated by noise. Therefore by creating a mask around the main harmonic of

the CWT of the input signal and by applying it to the CWT of the output signal the dynamic properties of the

system can still be identified.

By determining graphically the areas that are considered as noise, the results illustrated in figure 12 and 13 can

be obtained.


· 40~~ frequency (He)~~~~~~~40

35,

3010

25

120

15 :5E20

25

3030

20

35 -t 4(

~~~~~~~~~~~~~~50 ~~~~~~~ ~~time (s) 040 5 10 15 20 25 3frequency (Hz)0.2 0.4 0.6 0.8 1 1.2 1.4

time (s)

FIGURE 12. Two and three dimensional representation of a cleaned input signal

1.4

quency domain is first reconstructed using equation (14). Applying a Fourier transform to both sides of the': . . 0.8 ..

(CiT(t) = X(@) aHf(c) 0(18)

30

40 -

40

50 1 A time (s) 1 02 05 10 15 20 25 30 f.eq....y (Hz

time (s)

FIGURE 13. Two and three dimensional representation of a cleaned output signal

Signal reconstruction in time domain

As mentioned above the wavelets used are not orthogonal to one another but are on the contrary very redun-

dant. The idea of this reconstruction is to use this redundancy to estimate a time signal which CWT best

approximates, in a least square sense, the CWT that has to be reconstructed. For this purpose the signal in fre-

quency domain is first reconstructed using equation (14). Applying a Fourier transform to both sides of the

equations one would have

(CWT(t,f))= X((o) .Hf(O) (18)


where ( CWT (t, f) ) could also be written CWT ( o, f) which can be seen as the filtered signal in fre-

quency domain filtered with a filter centered around frequency f. One column of this matrix for a fixed fre-

quency coO can be computed as follows

CWT(woffl) [Hf ((o)

=X (co) . (19)

CWT ( ,fo, f ( o0)

To compute the coefficient X (o) of the fourier transform of the reconstructed signal at frequency co, (19)

can be multiplied from the right by the pseudo-inverse of the right hand side vector, which gives

CWT ( ,fo f (l (

(oo) = l I (20)CWT(w0 ,f) Hf (%o)

The reconstructed signal in time domain can be obtained by applying (20) at every frequency and by comput-

ing the inverse Fourier transform of the obtained signal in frequency domain.

Applying this method to the cleaned CWT illustrated in figure 12 the reconstructed signal shown in figure 14

can be obtained.

s0-.o , o,. J'

, ,, , ,,, ,,0

_-2

-30

9.9 10 1.1 12 10.3 10.4 10.5 10.6 10.7 10.8time (e)

FIGURE 14. Reconstructed Signal in time domain from a cleaned time-frequency decomposition

Time-Frequency Signal Analysis 15Time-Frequency Signal Analysis 15

Figure 15 shows that the cleaning procedure significantly improves the coherence function between the input

and the output signals.

0.9

0.8

0.6 t

0.4 -

0.3 -

0.2 .- -a . ..

0.2

0.1 - .. . ...

0 5 10 15 20 25 30 35 40frequency

FIGURE 15. Coherence function between the first input and the first output (left wing forward) for

the original signals (--) and for the cleaned signals (-)

Limitations

As explained above, this method is based on the user's judgement to consider some areas of the two dimen-

sional representation relevant or not, this judgement can easily be mislead by wrong interpretation of informa-

tion or by ambiguity caused by resolution problems.

Furthermore this signal cleaning method can only deal with noise that occurs at time-frequency points other

than the ones where relevant information is situated. Typically noise occurs at every time and at every fre-

quency which means that a cleaned CWT still contains noise at the selected time-frequency points. At these

precise points it is assumed that the signal to noise ratio is sufficiently high so that the information is not cor-

rupted too much.


3. Application of the time-frequency analysis to the identification problem

In this section we propose to apply the time-frequency approach that has been introduced in the previous chap-

ter to the identification problem. A method to estimate the transfer functions of the considered system will first

be presented. It will then be shown how a time-frequency signal processing can be used to improve the per-

formances of subspace identification methods.

3.1 Transfer functions estimation

An estimation of the transfer function of the system that has to be identified can be useful for several reasons.

In the particular case of flutter boundary estimation for the F- 18 SRA an on-line monitoring of the frequency

behavior of the structure can be performed based on a priori knowledge of the transfer functions. More gener-

ally the estimated transfer functions can be used to identify the parameters of the state-space matrices of the

considered system. An estimation method based on the characteristics of the input signal is introduced here.

Principle

This method, developed by Laurent Duchesne, is based on the a priori knowledge that the input signal is a fre-

quency sweep. Thus it is possible to isolate the relevant frequencies in the input and in the output using time-

frequency considerations. With the same assumption as in chapter 1.2.2, that is the linearity of the system, and

using the same argument it is possible to identify the system's dynamic using only information contained in

the main component of the signal. For this method a set of symmetric and a set of asymmetric measurements

are required. The step by step procedure is explained here:

* step 1: localization of the relevant frequency information by filtering each signal with a very narrow band-

pass filter centered around the frequency coi. Filtering the input and the output of the system does not alter

the identification. Indeed if every output of a linear time invariant (LTI) system P with multiple input and

multiple output (MIMO) is filtered with the same LTI single-input-single-output filter F the total system G

satisfies

Application of the time-frequency analysis to the identification problem 17

G = PI K (21)Therefore if the inputs and the outputs of a system are filtered with the same filter, the filtered data can also

be assumed as the inputs and outputs of the system.

* step 2: localization of the relevant time information by windowing each filtered signals. The chosen win-

dow is the same for every signal and is chosen around the point where the first filtered input has maximum

amplitude. The a priori knowledge that the relevant information at frequency coi is localized in time is used

here. Whenever the same frequency appears in the signal outside the chosen time window it is considered

as noise.

* step 3: computation of the Fourier coefficient for every windowed input and output at the filtered fre-

quency oi:

2io°inX(wi) = Yx[n]e * (22)

n

Steps I till 3 are applied to every symmetrical and asymmetrical input and output.

* step 4: estimation of the coefficients of the transfer functions at frequency coi. Using the symmetrical and

the asymmetrical dataset we have

1 = 2 (23)

a ((Oi)J u, ( Oi) u (oi)J ( (i)

where ys (coi) , ya (Ogi) are the Fourier transform of the outputs for the symmetrical and asymmetrical

dataset at frequency oi, u u (i), u (oi), u' (cOi) and u2 (wi) are the Fourier transform of the first and sec-

ond inputs for the symmetrical and asymmetrical dataset at frequency coi and G1 (Oi) and G2 (C0i) are the

coefficients of the transfer functions for the first input to the output and for the second input to the output at

frequency coi, which results in


,_ 2 ( i)- L ((i) U2 (Ci) (i)]

To obtain a transfer function estimation along the whole considered frequency range the same process is per-

formed for several frequencies.

Results

Transfer functions are estimated with the F-18 SRA measurements using on one hand the method described

above and on the other hand a classical Fourier analysis with no windowing (figure 16). The time-frequency

approach has clearly reduced the influence of the noise on the estimated transfer functions. These transfer

functions have been computed for the first output for the following flight point:

* mach number: 0.85

* altitude: 10000 feet

TFul-,yl TF u2->yl TF ul ->yl TF u2 ->yl0.06 0.06 0.04 0.05

0.05 0050.03 0.04

0.04 004 -20.0

o0.03 003 0.02

appr0.02 0f02.0

Application0.01 0of01 the time-frequency analysisto0.01 0... .. .

10 20 1 l0 20 30 0 10 20 30 0 10 20 30

152000OO00 200

-2001000 1000 100

-400

0 5O01 -600 0

a.a -800-1000 0 -100

-1000

-2000 -500 -1200 -2000 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30

frequency (Hz) frequency (Hz) frequency (Hz) frequency (Hz)

FIGURE 16. Estimated transfer functions with classical Fourier approach and with time-frequency

approach for output 1 in the case of a linear frequency sweep


Definition of the time-frequency trajectory for the transfer function estimation

Instead of localizing the chosen time window around the point where the filtered first input has the maximum

amplitude it is conceivable to define within the CWT of the input signal the time-frequency trajectory along

which the windows should be selected and the transfer functions estimated. By doing this more robustness can

be guaranteed in this method, especially for cases where the signal to noise ratio is small and where the maxi-

mum of the filtered signal might be difficult to identify. Moreover, if the frequency distribution in the signal is

such that the main component of the frequency sweep goes more than once through the same frequency, local-

izing the maximum of the filtered signal would imply neglecting some cases. In figure 17 is illustrated the case

of a logarithmic frequency sweep for the input signal where the considered frequency range is swept back and

forth several times. The chosen time-frequency trajectory along which the transfer functions have to estimated

is defined in the two dimensional time-frequency domain. The estimated transfer functions for the first output

are illustrated in figure 18.

0

i25

5 10 s15 20 25 30time (s)

FIGURE 17. Time-frequency trajectory along which the transfer functions have to be estimated

TF ul -> yl TF u2 -> yl0.04 0.05

0.03

~0.020.01

0 10 20 30 40 0o 10 20 30 40

- 1 ooo -200

-200100

-400

-800

-1000 -1000 10 20 30 40 0 10 20 30 40

frequency (Hz) frequency (Hz)

FIGURE 18. Estimated transfer functions for the first output


3.2 Subspace Identification using signals processed with a time-frequency analysis

In this section we show, based on a fourth simulated SISO system, how results obtained with a Subspace Iden-

tification (SID) can be improved by using a frequency sweep exciting signal and the above described time-fre-

quency analysis.

3.2.1 Identification of the A and C matrices

Principle

The goal of subspace identification, as described in [21],[22] and [24], is to find a linear, time invariant, finite

dimensional state space realization

Xk+l = Axk +Buk(25)

Yk = CXk + Duk

where A E n x n, B E n m, C E S , D E 9e , based on the knowledge of specific sequences

u = [ ... uI , y = [yl ... y]1 where p is the number of samples. However since the states are not directly

observed, the system matrices are identifiable only to within an arbitrary non-singular state (similarity) trans-

formation.

The following notation is used:

The block Hankel input and output matrices are defined as

Yk+ Yk+2l ' Yk+j-]

Yh(k,i,j) = +1 Yk+2 * yk... . ...

Yk+i- I Yk+i .' Yk+j+i-2

and

Uk Uk+l I ... Uk+j-

Uh (k, i, j) =k+2 Uk+j

Uk+i_l Uk+i ... Uk+j+i-_2


We also introduce the extended observability matrix

CA

[CA' i-

the lower block triangular Toeplitz matrix

D 0 0 ... 0

CB D 0 ... 0

Htl = CAB CB D ... 0

CAi-2B CAi-3B CAi-4B ... D

and the state matrix

X = [Xk Xk + ... Xk+j - 1

This notation leads to the following input output history:

Yh(k, i, j) = X + HtlUh. (26)

The step by step procedure of a subspace identification is presented here:

· step 1: find a matrix P that satisfies an equation of the form

P = FQ, (27)

where F is the extended observability matrix and such that rank(P) = rank(F) = n.

In the case of the deterministic system, P can be found by post multiplying equation (26) by a matrix UhI

that satisfies UhUh = 0. We then obtain P = YhUh.

· step 2: perform a singular value decomposition of P

P = USV


where S = S 0 and U = ( U U ) such that U, is the first n columns of U.

Note that S1 is a n x n matrix. With equation (27), we can see that there must exist a full rank matrix T such

that

U1 = FT.

Let us now use the following notation: if M is an m x n matrix, M (resp. M ) will be the matrix with a reduced

number of rows, obtained from M by omitting the first (resp. last) I rows, where I is the number of outputs of

the system.

* step 3: Compute A with the pseudo-inverse of UI. Using the structure of the extended observability

matrix, it is clear that

r = FA

and

UI = _T, U1 = rT,

which leads to

To = 7,T A.

This can also be written as

1= U 1 iT where v = T AT.

Therefore v is a similar matrix to A and

1/ = 711U,. (28)

* step 4: Compute C using the extended observability matrix. The equality

= CA 2

can be rewritten as


72 Y = c[IA ... A'i . (29)

Therefore

C = [71 2 7... [A.. A l] . (30)

Results

To test how signal cleaning as described in chapter 1.2 can improve system identification, the method

described above to identify the A and C matrices of the state-space representation is tested on several simu-

lated measurements. A simple fourth order SISO system is used to generate the simulated datasets. The three

following cases are considered:

* The system is excited with a chirp signal to which unknown white noise is added. Using the input and the

output sequences A and C are identified.

* The simulation is run in an identical way but the identification is performed with an output signal that has

been cleaned using a CWT as described in chapter 2.2.

* The system is excited with colored noise whose Power Spectral Density (PSD) is identical to the chirp's

PSD and to which unknown white noise is added. A and C are then identified.

For each of these identification the identified poles are plotted in figure(19) and compared to the original

poles. The best results are obtained when the exciting signal is a chirp and the output is cleaned via a CWT.

60 -

40 ... - -

20 _ . .... ........ ... ............ .o + . .. . . .

-60 ......... .-

-6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1Re

FIGURE 19. Comparison of the poles (in Hertz) to be identified (+) with the identified poles for a

chirp signal (*), a chirp with cleaned output (o) and colored noise (x) as exciting signal


3.2.2 Identification of the B and D matrices

Principle

As the method proposed in [21] and [24] did not deliver satisfying results for the identification of the B and D

matrices of the state space model a different approach is offered here which uses the estimated transfer func-

tions as described in chapter 3.1. Using this method implies that the exciting signal has to be a frequency

sweep.

The proposed approach tries to match the frequency response of the identified state space model with the esti-

mated transfer functions. For this the well-known formula

G (jo) = C(jtoI -A) B+D (31)

is used. (31) can be rewritten as

G (j0) = IC(joI -A) 1] [(32)

which written for every computed frequency coi gives

G (j(o ) C(j( I- A) I

G(J°)2 ) C(jo,2I-A) -1 I B(33)

LG (jo C(jo _I-A)-I I

where n is the number of frequency points at which the transfer function has been estimated. Therefore, using

the pseudo-inverse, we have

C (J(I1I-A) I G (j(ol )

[1= C(Jco2 1-A)-l G UO02 ) (34)

C (joinI- A)-l I n)


The whole principle of this identification method is sketched in figure (20).

u, y

Signal cleaning Transfer functionvia CWT estimation

u, YcleanedjI3Is~~~ ~ ~ G(jwi)

SID for A and C

Frequency fit for _

D and B

A,B,C,D

FIGURE 20. Sketch of the identification method using time-frequency analysis results

Results

Using a chirp as exciting signal the results obtained for a cleaned output and for a non processed output are

compared in figure (21). The frequency response of the estimated state space representation is clearly more

accurate when a CWT processing has been applied to the output signal.

80 . .

M 60

20-100~~~~~~ ~10

0 -2010-100 T .... ..... ............

-400'10'

frequency (Hz)

FIGURE 21. Comparison of the frequency response of the system to be identified (-) and of theidentified system for a processed output (. _ ) and a non processed output (--)


4. Summary and Conclusion

4.1 Summary

The goals of this project are to bring a computational support for the identification of the flexible dynamics of

the F-18 SRA by performing a time-frequency signal processing of the measured data. This processing is

intended to improve the results of the identification methods presently used.

A wavelet decomposition of the signal is performed using non orthogonal and non dilated morlet wavelets.

This decomposition, applied to measured signals, is processed graphically. The user decides, based on his own

judgement, which time-frequency points should be considered as noise and be eliminated. A cleaned time sig-

nal can then be reconstructed from this processed time-frequency decomposition. It is computed by finding a

signal whose wavelet decomposition best approximates the processed decomposition, in a least square sense.

A similar time-frequency approach can also be used to estimate the transfer functions of the considered sys-

tem.

To execute these operations a toolbox for Matlab 4.2 has been implemented which allows a systematic

processing of the signals through a graphical user interface. A subspace identification can be run using the

results obtained with this toolbox.

4.2 Conclusions

The results obtained have led to the following conclusions:

* The Time-Frequency approach allows a very good visualization and understanding of the signal informa-

tion content.

* The noise content of the signal can be considerably reduced by the time-frequency processing.

* The estimated transfer functions, using a time-frequency considerations, are smoother and less noisy than

the ones estimated with classical Fourier approach.

* Using processed signals and estimated transfer functions with a time-frequency approach, improves the

performances of the presently used subspace identification methods.

Summary and Conclusion 27

4.3 Further work

The first limitation imposed by the signal processing method proposed here is its dependency on the human

intuition to distinguish, within the time-frequency domain, between the noise and the valuable information. A

possible work could concentrate on finding a rigorous and systematic way to perform a cleaning of the time-

frequency decomposition.

Another problem encountered here is the elimination of the noise occurring at the same frequency and at the

same time as the relevant information. A possible approach to handle this problem would be to use the redun-

dant information contained in the computed wavelet coefficients to separate the noise from the signal.

The results obtained with the time-frequency method described here have only been used to identify a simple

Single Input Single Output fourth order simulated system. The next natural step is to apply these results to the

actual real measurements and the real system.

Summary and Conclusion 28

Reference

[1] R.H.Scanlan, R.Rosenbaum, Introduction to the study of Aircraft Vibration and Flutter.The MacmillanCompany, 1951.

[2] Lura Vernon, In Flight Investigation of rotating Cylinder-Based Structural Excitation System for flutterTesting. NASA Technical Memorandum 4512. 1993

[3] Leonard S. Voelker, F-18 SRA Flutter Analysis Results. NASA Technical Memorandum. 1995.

[4] Olivier Rioul and Martin Vetterli, Wavelets and Signal Processing. IEEE Signal Processing Magazine,October 1991, pp. 14 -37 .

[5] Douglas L. Jones, Thomas W. Parks, A resolution Comparison of Several Time-Frequency Representa-tions. IEEE Transactions on Signal Processing, vol. 40 no. 2, February 1992.

[6] A.C.M Claasen, W.F.G. Mecklenbrauker, The Wigner Distribution- A tool for Time-Frequency signalAnalysis. Philips Journal of Research Vol. 35, no. 6, 1980, pp.3 7 2 -3 8 9 .

[7] Gloria Faye Bourdeaux-Bartels, Thomas W. Parks, Time-Varying Filtering and Signal Estimation usingWigner Distribution Synthesis Techniques. IEEE Transaction on Acoustics, Speech and Signal Processing,vol. 34, no. 3, June 1996, pp. 442-450.

[8] William J. Williams, Jechang Jeong, New Time-Frequency Distribution: Theory and Applications. IEEEISCAS, 1989, pp. 1243-1247.

[9] Ning Ma, Didier Vray, Phillippe Delacharte, Albin Dziedzic, Gerard Ginez, Time-Frequency Representa-tion adapted to chirp Signals: Application to Analysis of Sphere Scattering. IEEE Ultrasonics Symposium,1994. pp. 1139-1142.

[10] F. Hlawatsch, W.Krattenhaler. A new Approach to Time-Frequency Decomposition. IEEE ISCAS. 1989.pp. 1248-1251.

[ 11] I. Daubechies, Orthonormal Bases of Compactly Supported Wavelets. Comm. in Pure and Applied Math.,vol. 41, No. 7, pp. 909-996, 1988.

[12] I. Daubechies, The Wavelet Transform, Time Frequency Localization and Signal Analysis. IEEE ICASSP,Vol. 5, pp. 361-364,1992.

[13] Y. Meyer, Ondelettes et Operateurs, Tome I. Ondelettes, Hermann ed., Paris, 1990.

Reference 29

[14] Ronald R. Coifman, Yves Meyer, Victor Wickerhauser. Wavelet Analysis and Signal Processing. Availa-ble on the internet: http://www.mathsoft.com/wavelets.html

[15] Martin Vetterli, Cormac Herley, Wavelets andfilter Banks: Theory and design. IEEE Transactions on Sig-nal Processing, Vol. 40, No. 9, September 1992. pp. 2207-2231.

[16] Jonathan Buckheit, David Denoho, Wavelab and Reproducible Research. Available on the internet: http://playfair.Stanford.EDU:80/-wavelab/

[17] R.A. Gopinath, C.S. Burrus, A tutorial overview offilter banks, wavelets and interrelations. Available onthe internet: http://www.mathsoft.com/wavelets.html.

[18] Gilbert Strang. Wavelet Transforms versus Fourier Transforms. CICS-P-332, May 1992.

[19] Gilbert Strang. Wavelets and Dilation Equations: A brief Introduction. SIAM Review, Vol. 31, NO. 4, pp.614-627, December 1989.

[20] T. Mzaik, J. M. Jagadeesh, Wavelet-based detection of transients in biological signals. Proceedings ofSPIE Vol. 2303, 1994. pp. 105-117.

[21] L. Duschesne, E. Feron, J.D. Pauduano, M. Brenner, Subspace identification with multiple data sets. Dec.1995, to be published.

[221 P. Van Overschee, Subspace Identification: Theory - Implementation - Applications. PhD Thesis, Depart-ment of Electrical Engineering, Katholieke Universiteit Leuven, Belgium, Feb. 1995.

[23] P. Van Overschee, B. de Moor, Subspace Algorithm for the Stochastic Identification Problem. Automat-ica, Vol. 29, No. 3, pp 649-660, 1993.

[24] Y.M Cho, Fast Subspace based System Identification: Theory and Practice. PhD thesis Stanford Univer-sity, Aug. 1993.

Reference 30

Appendix A

Signal analysis Toolbox: installation and operation

A toolbox has been developed in order to perform a systematic time-frequency signal analysis as described in

chapter 2. It has been intended to be applied at NASA Dryden Research centre for the flutter boundary deter-

mination of the F-18 SRA and might be applied on future flights data analysis for the F-18 HARV and the

SR71.

This tool has been written for Matlab 4.2 and allows the following actions:

* time/frequency signal decomposition

* signal cleaning within the time/frequency domain

* signal reconstruction in time domain from a processed time/frequency decomposition

* estimation of a system's transfer functions for a two inputs one output system based on symmetrical and

asymmetrical measurements

All the examples given below are applied on measurements taken on the F18-SRA. In order to be used

properly at least one set of symmetric and asymmetric datas at one flight point are required.

The toolbox is available on anonymous ftp at merlot.mit.edu in directory /pub/feron/wave_tool.

1. Installation

To install this tool in your directory you should:

* create a directory wave_tool with three subdirectories meas, results and tool. In the directory tool put all

the files written for the tool and in the directory meas all the measured data in format .mat.

* Create the required Matlab path so that all three directories are reachable from the current directory

* Compile the C codes (for the moment the only one is reflect.c).

* In the file run_tool.m change the variable passage so that it indicates the path from the current directory to

the directory wave_tool.

Appendix A 31

2. Running the tool

To run the tool the file run_tool.m should be run in Matlab. In this file you should determine:

* which dataset should be considered. The variables datal and data2 correspond to the different measure-

ments. datal is for the symmetric measurements (for example mO85_10k_symm_high.mat for the

F18_SRA measurements) and data2 for the asymmetric ones (for example mO85_1 Ok_asym_high. mat).

* which signal among this you desire to view in time/frequency domain first. This can be done by changing

the variable which_data. The first value of which_data indicates from which dataset datal or data2 the

signal should be taken, the second value indicates which signal from this dataset should be considered (this

value indicates the column in which it appears in the matrix vardat).

· at which sampling frequency the signal was measured.

Type "run_tool" in the Matlab window. You will then be asked, within the Matlab window, the following

questions:

· What is the minimum frequency (Hz) you want to observe?

* What is the maximum frequency (Hz) you want to observe?

* How many voices do you want to compute?

* Which frequency resolution do you want to have?

The first two answers you give determine the frequency range used for the time/frequency decomposition, the

number of voices determines how many frequencies within the considered frequency range will be taken into

consideration. The selected frequency resolution corresponds to the filter width used for the time/frequency

decomposition.

The computation of the decomposition takes approximately 20 seconds. Once it is over two graphical

windows will appear on your screen as shown in figure 22, one containing the decomposed signal, the other

the graphical interface offering a range of possibilities described below. The considered signal from the data-

set can be changed at all time using the push buttons labelled view.

Appendix A 32

40

20 fi - g

0.2 0.4 0.6 0.8 1 1.2 1.4

FIGURE 22. Decomposed signal in time/frequency domain and graphical interface window

3. Working in the time frequency domain using the graphical interface

3.1 Changing the frequency resolution

The slider labelled filter width allows you to change the frequency resolution of the time/frequency

decomposition by changing the width of the filter used for the computation. If this slider is selected the whole

computation for the decomposition will be run again (approximately 20 seconds) with a different resolution.

In figure 23 is illustrated a decomposition of the same signal as in figure 22 recomputed with a lower fre-

quency resolution, i.e. a wider filter.

5 $ 10 15 20 25 30time (.)

FIGURE 23. Decomposed signal with a lower frequency resolution

Appendix A 33

3.2 Changing the graphical display of the decomposition

The slider color contrast and the push buttons zoom and full size act on the window where the time/frequency

decomposition of the signal is represented and allow changes in this representation. For example the decom-

position from figure 22 is illustrated in figure 24 zoomed on a particular area.

ttrnB (8I

FIGURE 24. Zoomed time/frequency decomposition

3.3 Cleaning/recovering information in the time/frequency domain

The push buttons erase and recover allow a cleaning/recovering of graphically defined areas of the time/fre-

quency domain. Both functions' usage are exactly identical. The option box allows you to define a rectangle

(similarly as the one you define with the zoom function) in which the information will be erased or recovered.

The option triangle allows you to determine three points which will define a triangle in which the information

will be processed. According to the chosen option or to the size of the selected area this procedure requires a

computation time of approximately 15 seconds. In figure 25 are illustrated examples of cleaned decomposi-

tions, one where the option rectangle has been used and the other where all information except the fundamen-

tal harmonic has been kept.

2O'

-25 ' '25 ''

5 10 15 20 25 30 0.2 0.4 0.6 08 1.2 1.4ne (.) b (s)

FIGURE 25. Cleaned time/frequency decomposition

Appendix A 34

3.4 Reconstructing the signal in the time domain

From a processed time/frequently decomposition you have the option, using the push button reconstruct to

recover the signal in the time domain. This computation requires approximately one minute. It is important to

note that the reconstructed signal will only contain frequency information up to about 90 percent of the

maximum frequency used for the decomposition. It is therefore wise when the file run_tool is initially run to

select the value of the maximum frequency 10 percent above the highest frequency of interest. In figure 26 are

compared the reconstruction of the decomposition of the signal illustrated in figure 25 with the original signal.

-30

10 10.1 10.2 103 10.4 10.5 10 .6 10.7 10. time (e)

FIGURE 26. Comparison of the processed and reconstructed signal (-) with the original signal (--)

3.5 Estimating the system's transfer functions

The push button estimate TF allows you to compute an estimate of the system's transfer function using

Laurent Duchesne's method which separates the considered signals in time windows and analyses separately

the frequency content of each of these windows. This function is independent of any processing that might

have been executed previously within the time/frequency domain. If you push this button you will be asked

the following questions in the matlab window:

· which output you want to consider to estimate the transfer function?

* How wide you want the considered time window to be? (The required answer is the number of data points

for the desired window).

* do you want to determine graphically the time/frequency trajectory along which the transfer function

should be estimated?

You have the option to compute the two transfer functions from the two inputs to anyone of the outputs (the

outputs appearing in the data matrix vardat from the F18-SRA measurements are considered from 1 to 10, 1

corresponding to the output labelled lwngfwd_nz and 10 to the one labelled raileron_nz). If the answer to the

third question is 'n' the transfer functions are computed and the time windows are centered automatically, if it

is 'y' you will be asked the following:

Appendix A 35

* Do you want to consider a linear trajectory (lin) or do you want to customize its time/freq behavior

(custom)?

If you are considering a linear sweep you should then define two points graphically which will determine the

line along which the transfer function will be estimated. If you want to define any other trajectory you can

determine graphically twenty points within the time/frequency domain, the corresponding spline will then be

plotted as illustrated on figure 27 (for the case of a logarithmic frequency sweep). If this spline satisfies you

the transfer function will be estimated along it, if it doesn't you will be able to define a new one. For the case

of a logarithmic frequency sweep the transfer function for the first output has been estimated and plotted on

figure 27.

TF ul ->yt TF u2->y1

00:. 0.04

5 10 15 7.00 25 30 I R 20 30 2° 0 10 20 30 01rqet(Hz) arfmly (Hz)

FIGURE 27. Trajectory along which the transfer functions should be estimated in the case of

logarithmic frequency sweep and estimated transfer functions

3.6 Plotting a three dimensional representation of the decomposition

The time/frequency decomposition illustrated in figure 22 is a three dimensional picture represented in two

dimensions. By selecting the push button do 3D plot you have the possibility to plot the time/frequency

decomposition of the signal (processed or not processed) in three dimensions as illustrated in figure 28.

40.

30 28. Three dimensional representation of a time/frequency decomposition205700

FIGURE 28. Three dimensional representation of a time/frequency decomposition

Appendix A 36

4. Constraints

Nearly all the functions of the tool require a lot of computation and can take a long time to be run. It is

therefore important first of all to be patient, but also not to click on more than one function at a time in order to

let the computations be executed properly.

Appendix A 37

Appendix B: Matlab Codes

The following codes have been written to implement the signal processing toolbox as described in Appendix A

and to implement the results mentioned in Chapter 3. They heve been written to be run on Matlab 4.2. Their

functionality is described here:

Codes for the signal processing toolbox:

· run_tool.m: Main code of the wavelet toolbbox. Initiates the whole process.

* STFT_morlet.m: Computes the wavelet decomposition of a given signal using morlet wavelets.

* STFT_fir.m: Computes the wavelet decomposition of a given signal using FIR filters as given by the the

matlab signal processing toolbox.

* gui.m: Initiates the graphical User Interface

* modify.m: Computes the modification of the CWT given graphically by the user.

* reflect.c: C code which reflects a vector.

* reconstruct.m: Computes the reconstructed time signal from a processed CWTestim_TF.m

* estimate.m: Initiates the computing of the transfer functions.

* estim_TF.m: Conputes the estimated transfer functions

* plot_2D.m: Plots the two dimensionnal representation of the CWT

* plot_3D.m: Plots the three dimensionnal representation of the CWT

* gord.m: Organizes the figure windows on the screen

Codes for the Subspace identification:

· classic_SID.m: (originally written by Laurent Duchesne modified by X. Paternot) computes the state-space

matrices A and C with a subspace identification.

* BD_estim.m: Computes the state-space matrices B and D with a frequency fit.

Appendix B: Matlab Codes 38

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