LIDS-P-2348
June 1996
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 Astronomics
Massachusetts Institute of Technology
February 5 - May 10 1996
1.4
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~~~~~~~.....Supervisor
Prof.
Prof.
G. Feron
Schweitzer
at
0.Supervisor
at MIT:
ETH:
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30
20
time (s)
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frequency (Hz)
Author: Xavier Paternot
Supervisor at MIT: Prof. E. Feron
Supervisor at ETH: Prof. G. Schweitzer
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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 processed. 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 functions. 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 AA 1 = 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.
1
2
3
4
5
6
7
8
9
10
Sensor Location
left wing forward
left wing aft
left aileron
left vertical tail
left horizontal tail
right vertical tail
right horizontal tail
right wing forward
right wing aft
right aileron
TABLE 1. Accelerometer locations
Right actuator
Left actuator
FIGURE
and2.actuators
Location
ofsensors
on F-18 SRA
Left 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 seconds. 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 possible.
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 considered flight point.
As a result of the required measurements and of the properties of the actuator at disposal the chosen input signal u(t) is a signal of sinusoidal shape with modulated frequency also known as chirp signal. Linear and logarithmic 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 aocan 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 dynamical 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
Noise
load
sesor
Measured
Input
Sensor
Noise
cce
me
Measured
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 purpose 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 methods.
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))
(4)
= X(t,Co)
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 projection of the signal onto functions
w (t-,
a e)
-jwo~
(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 mathematical 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.
Time-Frequency Signal Analysis
7
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 resolution is the same at all location in the time frequency-plane. To overcome this problem functions called wavelets 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 principle will still be satisfied, but now, the time resolution becomes arbitrarily good at high frequencies and the frequency 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)
Time-Frequency Signal Analysis
8
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 generate 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)=
-j
2 -j/2 h2
t kT) .(10)
This results in the wavelet coefficients
cj,
= Jx(t) h
(11)
(t)
In this case the reconstruction problem is unambiguous, as the chosen basis is orthonormal, and could be formulated as follows
x(t) =
j
Cj,
k
k hj,k(t)
k
(12)
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
Time-Frequency Signal Analysis
9
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)
02
2
2
(13)
-e
H(w) = e
0.03
0.02
0.1.
0.8
°0.6
0
0.4 -
-. 01-
0.2
-0.02
0
2
4
4
6
,
B
10
12
frequency(Hz)
14
i
16
i'
18
20
-0.03
4.5
4.6
4.7
4.8
4.9
5
time (s)
5.1
5.2
5.3
5.4
5..
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 composed 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 signals measured on the F- 18 SRA this is not the case, especially for the linear frequency sweeps where the frequency 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 frequencies 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) =
Time-Frequency Signal Analysis
-
(X(o)
H ()
(14)
10
er
t
x 2( t )
tilter2
x(t)
CWT
x (t )
filter n
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 resolution 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
Ao
2
=2
T
co2 -
and
o01
At
2
= 2
02 - (16
(16)
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.
Time-Frequency Signal Analysis
11
Results
Applied to one symmetrical input and output (left wing forward) for the case of a linear sweep the results illustrated 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 dimensional and a three dimensional representation are plotted here.
3s0
25.
4
0.2
0.4
0.6
0.8
time (s)
1
1.2
20
.
time (s)
1.4
0
frequency (Hz)
FIGURE 9. Two and three dimensional representation of a CWT of an input signal in the case of a
linear frequency sweep.
25
Time-Frequency
Analysis
FIGURE10.Signal
To andthrediesoa
rersnainoJ
Wfa otu
linear~~~~ frqcysep
35wi
5
10
t5
12
inli haeo
20
25
30
y
s
time(S)
Time-Frequency
Signal
0
°
frequency (Hz)
time (s)
Anal
124
FIGURE 10. Two and three dimensional representation of a CWT of an output signal in the case of a
linear frequency sweep.
~~~~~~~~~~~~~~~~~~~~~~20~
40
1
0.2
0.4
0.6
0.8
time (s)
1
1.2
1.4
0
0
frequency (Hz)
time (s)
FIGURE 11. Two and three dimensional representation of a CWT of an input signal in the case of a
logarithmic 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.
Time-Frequency Signal Analysis
13
frequency(He)~~~~~~~40
· 40~~
35,
30
10
25
120
15
E20
:5
25
30
30
20
35 -t
4(
5
40
0.2
~~~~~~~~~~~~~~50
10
15
20
25
0.4
0.6
0.8
1
1.2
~~~~~~~
~~time
(s)
3frequency
0
(Hz)
1.4
time (s)
FIGURE 12. Two and three dimensional representation of a cleaned input signal
1.4
to both sides of the
(14).0.8Applying a Fourier transform
using equation
quency domain is first reconstructed
..
.
.
':
(CiT(t)aHf(c)
= X(@)
0(18)
30
40 -
40
50
5
1
10
15
20
A
25
30
time (s)
1
02
0
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 redundant. 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 frequency domain is first reconstructed using equation (14). Applying a Fourier transform to both sides of the
equations one would have
(CWT(t,f))=
Time-Frequency Signal Analysis
X((o)
Hf(O)
.
(18)
14
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
(19)
.
=X (co)
CWT ( ,fo,
To compute the coefficient X (o)
((o)
f
( o0)
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(
l
(oo) =
CWT(w0 ,f)
I
(20)
Hf (%o)
The reconstructed signal in time domain can be obtained by applying (20) at every frequency and by computing 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
time
10.4
10.5
10.6
10.7
10.8
(e)
FIGURE 14. Reconstructed Signal in time domain from a cleaned time-frequency decomposition
Analysis
Time-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
.0.2
.
-
0.1
0
.
-a
.
..
5
10
.
15
20
frequency
25
...
30
35
40
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 dimensional representation relevant or not, this judgement can easily be mislead by wrong interpretation of information 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 frequency 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 corrupted too much.
Time-Frequency Signal Analysis
16
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 chapter 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 performances 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 generally 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 frequency sweep. Thus it is possible to isolate the relevant frequencies in the input and in the output using timefrequency 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 bandpass 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
K
G=PI
(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 window 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 frequency oi:
2io°in
X(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 =
a((Oi)J
u,
2
( Oi) u (oi)J
(23)
((i)
where ys (coi) , ya (Ogi)are the Fourier transform of the outputs for the symmetrical and asymmetrical
dataset at frequency oi, u (i),
uu (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
Application of the time-frequency analysis to the identification problem
18
,_
2(
i)-
L
((i)
U2 (Ci)
(i)]
To obtain a transfer function estimation along the whole considered frequency range the same process is performed 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
TFu2->yl
0.06
0.06
0.05
005
0.04
TF ul ->yl
TF u2 ->yl
0.04
0.05
0.03
0.04
-20.0
004
o0.03
003
appr0.02
0f02.0
0.02
0of01
the time-frequency analysisto0.01
Application0.01
10
.
0...
..
l0
1
20
20
30
0
10
20
30
152000OO00
0
10
20
30
200
-200
1000
1000
100
-400
0
a.a
5O01
-1000
-600
0
-800
0
-100
-1000
-2000
0
10
20
frequency
(Hz)
30
-500
0
10
20
frequency (Hz)
30
-1200
0
-200
10
20
frequency (Hz)
30
0
30
10
20
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
Application of the time-frequency analysis to the identification problem
19
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 maximum 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, localizing 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
s15
10
20
25
30
time (s)
FIGURE 17. Time-frequency trajectory along which the transfer functions have to be estimated
TF ul -> yl
TF u2
0.04
->
yl
0.05
0.03
~0.02
0.01
10
0
-
1
20
30
40
ooo
0o
10
20
30
40
10
20
30
40
-200
-200
100
-400
-800
-1000
-100
0
10
20
30
40
frequency(Hz)
0
frequency (Hz)
FIGURE 18. Estimated transfer functions for the first output
Application of the time-frequency analysis to the identification problem
20
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 Identification (SID) can be improved by using a frequency sweep exciting signal and the above described time-frequency 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
Yk =
where A E
u = [
I
n x n,
BE
n m,
CE
S
, DE
+
CXk
9e
(25)
Duk
, based on the knowledge of specific sequences
... u , 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) transformation.
The following notation is used:
The block Hankel input and output matrices are defined as
Yk+
Yh(k,i,j)
Yk+2l
'
+1 Yk+2
=
Yk+j-]
yk
*
...
... .
Yk+i-
I Yk+i
.'
Yk+j+i-2
and
Uk
Uh (k, i, j)
Uk+l I
=k+2
...
Uk+j-
Uk+j
Uk+i_l Uk+i ... Uk+j+i-_
2
Application of the time-frequency analysis to the identification problem
21
We also introduce the extended observability matrix
CA
[CA' ithe lower block triangular Toeplitz matrix
D
CB
CAB
Htl =
0
D
CB
0
0
D
... 0
... 0
... 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
Application of the time-frequency analysis to the identification problem
22
where S =
S 0
and 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
Therefore
v
=
U 1 iT where v
=
T AT.
is a similar matrix to A and
1/ =
(28)
71U,.
1
* step 4: Compute C using the extended observability matrix. The equality
= CA
2
can be rewritten as
Application of the time-frequency analysis to the identification problem
23
= c[IA ... A'i
Y
72
.
(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 simulated 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
Re
-3
-2.5
-2
-1.5
-1
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
Application of the time-frequency analysis to the identification problem
24
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 functions 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 estimated 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
LG
(jo
B(33)
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)
[1=
I
C(Jco2 1-A)-l
C (joinI-A)-l
Application of the time-frequency analysis to the identification problem
G (j(ol )
G UO02 )
I
(34)
n)
25
The whole principle of this identification method is sketched in figure (20).
u, y
Transfer function
estimation
Signal cleaning
via CWT
u, Ycleaned
jI3Is~~~
~
~
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~~~~~~
0-2010
-100
~10
T
....
.....
............
-400'
10'
frequency (Hz)
FIGURE 21. Comparison of the frequency response of the system to be identified (-) and of the
identified system for a processed output (. _ ) and a non processed output (--)
Application of the time-frequency analysis to the identification problem
26
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 signal 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 system.
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 information 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 timefrequency 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 redundant 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 Macmillan
Company, 1951.
[2] Lura Vernon, In Flight Investigation of rotating Cylinder-Based Structural Excitation System for flutter
Testing. NASA Technical Memorandum 4512. 1993
[3] Leonard S. Voelker, F-18 SRA FlutterAnalysis 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 Representations. 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 signal
Analysis. 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 Filteringand Signal Estimation using
Wigner 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. IEEE
ISCAS, 1989, pp. 1243-1247.
[9] Ning Ma, Didier Vray, Phillippe Delacharte, Albin Dziedzic, Gerard Ginez, Time-Frequency Representation 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, OrthonormalBases 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 SignalAnalysis. 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.Available on the internet: http://www.mathsoft.com/wavelets.html
[15] Martin Vetterli, Cormac Herley, Wavelets andfilter Banks: Theory and design. IEEE Transactions on Signal 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 tutorialoverview offilter banks, wavelets and interrelations.Available on
the internet: http://www.mathsoft.com/wavelets.html.
[18] Gilbert Strang. Wavelet Transforms versus FourierTransforms. CICS-P-332, May 1992.
[19] Gilbert Strang. Wavelets and DilationEquations:A briefIntroduction. 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 of
SPIE 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, Department of Electrical Engineering, Katholieke Universiteit Leuven, Belgium, Feb. 1995.
[23] P. Van Overschee, B. de Moor, Subspace Algorithmfor the Stochastic Identification Problem. Automatica, Vol. 29, No. 3, pp 649-660, 1993.
[24] Y.M Cho, Fast Subspace based System Identification: Theory and Practice.PhD thesis Stanford University, 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 determination 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 measurements. 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 dataset can be changed at all time using the push buttons labelled view.
Appendix A
32
40
fi
20
0.2
0.4
0.6
0.8
1
- g
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 frequency resolution, i.e. a wider filter.
5
$
10
15
time (.)
20
25
30
FIGURE 23. Decomposed signal with a lower frequency resolution
Appendix A
33
3.2 Changing the graphical display of the decomposition
The slider color contrastand the push buttons zoom andfull size act on the window where the time/frequency
decomposition of the signal is represented and allow changes in this representation. For example the decomposition 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/frequency 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 triangleallows 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 decompositions, one where the option rectangle has been used and the other where all information except the fundamental harmonic has been kept.
2O'
-25
5
10
'
'25
15
ne(.)
20
''
25
30
0.2
0.4
0.6
b
08
(s)
1.2
1.4
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
time
10.4
(e)
10.5
10
.6
10.7
10.
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.
TFul ->yt
00:.
5
10
15
7.00
25
30
TF u2->y1
0.04
I
R 20 30 2°
1rqet(Hz)
0
10 20 30 0
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.
3028.
representation
dimensional of a time/frequency decomposition
205Three
700
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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