Note: Descriptions are shown in the official language in which they were submitted.
CA 02276571 1999-06-29
BP #11031-001
BERESKIN & PARK CANADA
Title: System and Method for Diagnosing and
Controlling Electric Machines
Inventors: Igor M. Dremine
Victor I. Furletov
Oleg V. Ivanov
Vladimir A. Nechitailo
Vladimir G. Terziev
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Title: System and Method for Diagnosing and Controlling Electric
Machines
FIELD OF THE INVENTION
The present invention relates to a system and method of
monitoring and diagnosing instabilities indicative of impending
malfunctions or failure in electric machinery such as engines, compressors
or turbines. The present invention further relates to a system and method
for controlling such devices to avoid operational malfunction or failure.
BACKGROUND OF THE INVENTION
Many different systems have been developed for analyzing
and monitoring the operation of electric machines such as engines,
motors, turbines, compressors, pumps, and other electromagnetic rotating
machinery. These systems attempt to diagnose or predict impending
failure or malfunction of the electric machine due to instability, and,
where possible, to avert that failure by controlling the device accordingly.
For these purposes, diagnostic systems are usually required to operate in
an on-line manner to provide continuous monitoring of device operation.
Diagnostic systems are generally based on the reading,
recording, and subsequent analysis of physical signals. Sensors may be used
to obtain actual measurements of vibrations, deformations, pressure
variations, temperature, acoustic noise, and other similar physical
phenomenon associated with an operating electric machine. Many failures
in electric machines are accompanied and/or preceded by discernable
changes in such physical signals. For example, the failure of an electric
induction motor may result in, or may be the result of, abnormally high
vibrations in the motor.
For instance, prior art systems have been used to measure the
vibration levels of an electric machine and then continuously monitor the
overall vibration in either the time or the frequency domain. Where
required in these prior art systems, the time domain vibration data are
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converted into the frequency domain by a Fourier transform (typically the
fast Fourier transform or FFT is used for computational purposes). The
measured vibration signal or the transformed vibration signal, often
referred to as the vibration signature of the electric machine, may be
continuously or periodically analyzed and compared to a reference
signature, reference threshold levels, and/or a reference statistical measure
for vibrations of a normally operating machine. See, for example, United
States Patent No. 4,184,205 to Morrow, United States Patent No. 4,366,544 to
Shima et al., United States Patent No. 5,251,151 to Demjanenko et al., and
Max, "Methods and technique of analysis of signals in physical
measurements", M., Mir, 1983; v.l, ch.2, pp 18-35.
Some prior art systems have also been used to diagnose
electric machines based on analysis of the electric current or power
supplied to the machine, since many problems in electric machines may
result in harmonic electric currents or other power variations. For
example in United States Patent No. 5,629,870 to Farag et al., harmonic
current analysis is performed on an operating motor by monitoring the
spectral content of the power signature. Spectral components of the power
or current supplied to the motor are associated with device operating
conditions by referencing a stored knowledge of operational characteristics
of the motor.
Similarly, United States Patent No. 5,587,931 to Jones et al.
describes a tool monitoring system which operates in two distinct modes.
In a learning mode, the system gathers statistical data on the power
consumption of normally operating tools of a selected type. The power
consumption signal of the machine tool is decomposed into time-
frequency components by wavelet transform analysis and then
reconstructed based on certain selected components to reduce the effects of
noise. A statistical power threshold function is then generated based on
the selectively reconstructed power consumption signal. Subsequently, in
a monitoring mode, the power consumption of a tool is compared to the
power threshold. Although the wavelet transformation simultaneously
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provides information on the power consumption signal in both the time
and the frequency domains, the wavelet analysis is for the limited purpose
of establishing power level thresholds for a normally operating power
signature with reduced noise (i.e. in the learning mode). In the
monitoring mode, only a time domain power signature signal is analyzed
for diagnostic purposes.
In prior art diagnostic systems for electrical machines,
statistical or algorithmic analysis of solely the time domain information or
solely the frequency domain information present in a signal generally
provides only a very short precursor or advance warning of impending
device failure. The very short notice provided by such prior art systems is
generally insufficient to provide any opportunity to avoid the predicted
failure. Also, the effectiveness of the statistical or algorithmic analysis in
prior art systems based on Fourier spectral analysis may deteriorate at high
frequencies, because at high frequencies the physical signal is not
measured at sufficiently short time intervals.
Furthermore, these prior art systems require that information
be compiled for a relatively long time, leading to a delayed response in
regulation of the system. For instance, the statistical analysis in some prior
art diagnostic systems is often based on variations in the dispersion
(standard deviation or variance) of the envelope of the spectrum, the
main frequencies of oscillation, and the quasi or harmonic frequencies: see
for example Karasev, Maksimov, and Sidorenko, "Vibrational diagnostics
of aircraft engines", M., Mashinostroenie, 1978, ch.3, pp 60-83, and
Maksimov and Rodov, "Methods and tools for diagnosing unstable flows
in compressors", Turbines and jet devices, N12-1280, M., CIAM, 1990, p.
132. This type of analysis is complex and requires the accumulation of
information or data over a lengthy period time. This delays the time for
regulation of the system. Therefore, for practical purposes these systems
are only applicable to the analysis signals for stationary applications which
are at rest (e.g. in a laboratory), and they are not suitable for performing
on-
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line or real-time analysis for conditions which are not at rest (e.g. in-
flight
analysis for an aircraft) and require rapid on-line analysis.
Systems for automatically controlling electric machines, such
as engines, to avoid potential breakdown may include a sub-system for
diagnosing impending device malfunctions. Ideally, the diagnostic
subsystem recognizes an instability or potential malfunction at definite
stages of device operation, and subsequently the control system
automatically carries out appropriate operations on the device to prevent a
malfunction or failure from developing.
For instance, systems have been designed to use an adaptive
digital system to automatically control a turbine engine in an attempt to
provide full hydrodynamical stability and prevent instabilities in turbine
operation. However, the diagnostics of such systems, similar to the prior
art diagnostic systems described above, are limited to the analysis of the
flight conditions only, and cannot provide sufficient precursors or
warnings of engine failure resulting from internal device problems. As a
result, regulation by prior art control systems leads in most cases to
extremely restricted regimes in which the device or engine is operated far
from its full capacity. While optimal algorithms for entering into high
capacity regimes have been used, these require very extensive analysis and
consideration studies and can, nevertheless, lead to operating decisions
which are far from ideal. Other control systems have been proposed that
react to abruptly cut-off the fuel supply when a malfunction is detected, for
example auto-oscillations arising in an engine (see Waters, "Digital
controller applied to the limitation of reheat combustion roughness", Proc.
of AGARD conference, 1974, N15-1). In attempting to prevent fully
developed oscillations or other malfunctions from negatively influencing
the engine, such abrupt interruptions in fuel supply serve to drastically
reduce the engine's capacity and often lead to unpredictable results.
Thus, there is a need for a diagnostic system which accurately
and reliably predicts instability or impending malfunction in an electrical
device with sufficient warning to allow an associated control system to
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automatically control the operating regime of the electrical device in a safe,
rapid, effective, and efficient manner.
SUMMARY OF THE INVENTION
In a first aspect, the present invention provides a method of
diagnosing a device to determine the presence of an instability in a
physical characteristic of the device, the method comprising the steps of:
(a) receiving a digital input signal representative of the physical
characteristic; (b) performing a wavelet transform on the digital input
signal to provide a set of wavelet coefficients; (c) determining at least a
first
probabilistic measure of at least a portion of the set of wavelet
coefficients;
(d) analyzing at least the first probabilistic measure to identify a precursor
associated with the instability.
Advantageously, steps (a) to (d) may be performed in real
time with the identification of the precursor occurring prior to the
development of a malfunction in the device.
The precursor may be characterized by a significant change in
the probabilistic measure. Preferably, the first probabilistic measure
represents the dispersion of the wavelet coefficients at a particular scale.
In
a preferred embodiment, step (c) further comprises determining at least a
second probabilistic measure of at least a portion of the set of wavelet
coefficients, and step (d) further comprises the step of analyzing at least
the
first and second probabilistic measures to identify a precursor associated
with the instability. In this embodiment, the precursor may be
characterized by a significant change in both the first probabilistic measure
and the second probabilistic measure. Determining a second probabilistic
measure preferably includes the step of determining a high rank
correlation matrix for the wavelet coefficients.
Step (b) may comprise performing a discrete wavelet
transform or a discretized continuous wavelet transform on the digital
input signal. Conveniently, the method also includes the steps of
measuring the physical characteristic of the device with a sensor to
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provide an analog input signal and converting the analog input signal
into the digital input signal.
In a second aspect, the present invention provides a method
of controlling a device driven by a command execution unit comprising
the steps of (i) diagnosing an input signal as described above) and (ii) if a
precursor associated with the instability is identified in step (i), directing
the command execution unit, in accordance with an instability control
approach, to control the device to avert the development of the
malfunction in the device. In one embodiment, the device forms part of a
mufti-functional system and identifying a precursor associated with the
instability further comprises analyzing additional data describing the
operation of the mufti-functional system.
In a third aspect, the present invention provides a system for
diagnosing a device to determine the presence of an instability in a
physical characteristic of the device, comprising a computer system for
receiving a digital input signal representative of the physical
characteristic,
the computer system comprising (a) a wavelet coefficient generation
module for performing a wavelet transform on the digital input signal to
provide a set of wavelet coefficients, (b) a first measure algorithm module
for receiving the wavelet coefficients and determining a first probabilistic
measure of at least a portion of the set of wavelet coefficients, and (c) an
analysis module for receiving and analyzing at least the first probabilistic
measure to identify a precursor associated with the instability prior to the
development of a malfunction in the device.
Preferably, the computer system further comprises a second
measure algorithm module for receiving the wavelet coefficients and
determining a second probabilistic measure of at least a portion of the set
of wavelet coefficients, and wherein the analysis module further receives
and analyzes the second probabilistic measure to identify the precursor.
Also, the first measure algorithm module may determine the dispersion
of the wavelet coefficients at a particular scale, and the second measure
algorithm module may determine a high rank correlation matrix of the
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wavelet coefficients to enable further determination of other probabilistic
measures.
In a fourth aspect, the present invention provides a system
for controlling a device driven by a command execution unit comprising a
system for diagnosing an input signal as described above, wherein the
computer system is connected to the command execution unit and the
computer system further comprises an automatic control module for
directing the command execution unit, in accordance with an instability
control approach, to control the device to avert the development of the
malfunction. In one embodiment, the device forms part of a multi-
functional system including a data unit containing supplemental
information about the multi-functional system, the data unit being
connected to the computer system for providing the supplemental
information to the automatic control module.
The device may be one of the following: an engine, a motor, a
turbine, a compressor, a pump, or other electric machine. The physical
characteristic of the device may also be one of the following: vibrations,
deformations, pressure, acoustic noise, temperature, or power consumed.
The objects and advantages of the present invention will be
more clearly apparent with reference to the remainder of the description
and the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
In the drawings which illustrate, by way of example, preferred
embodiments of the invention:
Figure 1 illustrates the concept of time and frequency
resolution for wavelet transform analysis;
Figure 2 shows an exemplary plot of a discrete wavelet
transform (DWT) scalogram;
Figure 3 illustrates the pyramidal DWT computation
technique;
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Figures 4A-4D show the waveforms of several analyzing or
mother wavelet window functions;
Figure 5 is a block diagram operational overview of the
diagnostic and control system 100 according to a preferred embodiment of
the present invention; and
Figures 6-8 show examples of the diagnostic precursors of
instability which may be provided by the present invention for an aircraft
engine having an axial mufti-stage compressor operating at different
regimes.
DETAILED DESCRIPTION OF THE INVENTION
Prior art diagnostic systems for electric machines analyze a
time varying signal associated with the machine (such as vibration,
pressure, or power consumed) by considering, after spectral conversion,
only the frequency domain information present in the signal or by
considering only the temporal domain information inherently present in
the signal. Thus, prior art systems seek to make conclusions from the
frequency spectrum without taking into account simultaneous
information about the time properties of the signal, or conversely assess
the time content of a signal without concurrently considering frequency
information. In the former case, this limits the analysis of the time
evolution and of the correlation characteristics of the spectrum to the
integral form without any local (in time) information. In the latter case,
the analysis is entirely localized in time but does not reveal the frequencies
involved.
In contrast, the present invention provide a diagnostic system
and method based on mufti-resolution wavelet analysis performed on a
physical signal associated with an electric machine. Sensors positioned at
or near the electrical device may be used to obtain actual time-varying
measurements of vibrations, deformations, pressure variations,
temperature, and/or acoustic noise for example. Alternatively, or in
addition, analysis can be performed on the current or power provided to
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the device. In general, any physical phenomenon that is associated with an
operating electric machine can be used for analysis.
In accordance with the present invention these signals are
digitized and provided to a computer or processor which carries out the
signal analysis by using a wavelet transform that simultaneously provides
both time and frequency (or scale) information about the physical signal.
Wavelet analysis is discussed in detail in: Rioul et al., "Wavelets and
signal processing", IEEE Signal Processing Magazine, October 1991, p. 1-38;
Daubechies, "Ten Lectures on Wavelets", Society for Industrial and
Applied Mathematics Press, vol 61, CBMS-NSF Regional Conference
Series in Applied Mathematics, 1992; Kaiser, A Friendly Guide to Wavelets
(6th), The Virginia Center for Signals and Waves, Birkhauser, Boston
1994; Nievergelt, Wavelets Made Easy, Eastern Washington University,
Birkhauser, Boston, 1999; and Polikar, "The Wavelet Tutorial Parts III and
IV" available in June 1999 at uniform resource locator
http://www.public.iastate.edu /~rpolikar/WAVELETS /WTtutorial.html.
The contents of these references are hereby incorporated into the present
description for background purposes.
Generally, wavelet analysis allows data to be analyzed at
different scales or resolutions (mufti-resolution analysis). Mufti-resolution
wavelet analysis first requires choosing an analyzing or base wavelet
function (also referred to as a "mother wavelet"). The wavelet transform
deconstructs or decomposes or decomposes or decomposes the original
time domain signal into scaled and shifted (or translated) versions or
windows of the base wavelet. The original signal can be represented by a
linear combination of coefficients of these scaled and shifted wavelet
functions. As a result, signal analysis can be carried out on the wavelet
coefficients which conveniently represent a correlation between the
wavelet and a localized part of the original signal. Unlike, the short time
Fourier transform (STFT) or other time-frequency distribution transforms
which have a constant resolution at all times and frequencies, wavelets are
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much more suitable for studying unpredictable (or choppy) signals having
both low frequency components and sharp, high frequency spikes.
Both a continuous wavelet transform (CWT) and a discrete
wavelet transform (DWT) may be performed. While the CWT
theoretically transforms a continuous input signal into a continuous
wavelet coefficient transform, in any practical analysis both the CWT and
the DWT are determined based on a discretized input data stream (and the
CWT will also be discretized in practice).
Generally, for a time domain input signal function f(t)
decomposed with an analyzing or base wavelet function w(t), the wavelet
coefficients W(a,b) for location a and scale b (the scale is effectively
1/frequency) are given as follows:
W(a,b) = f w'(a,b,t)~f(t)~dt
t
The scaled and shifted wavelet functions w(a,b,t) are generated from the
analyzing or mother wavelet w(t) as
w(a,b,t) - ~ w~t_a~
b b
so that the mother wavelet corresponds to the case a = 0 and b = 1. The
function w*(a,b,t) is the complex conjugate of w(a,b,t), and these are equal
when w(a,b,t) is real. Usually, a >_ 0, so that as a increases the wavelet
window is translated along the time axis - the translation a can be
considered the time elapsed since t = 0. The scale variable b is > 0, such
that when 0 < b < 1 the wavelet window is compressed and when b > 1 the
wavelet window is dilated. The factor 1 /~ in w(a,b,t) normalizes the
wavelet so that W(a,b) has the same energy at every scale. Wavelet
windows with a high scale (low frequency) provide a global (non-detailed
or localized) view of the input signal whereas wavelets with a low scale
provide a detailed or localized view of the input. In other words, the
wavelet transform provides good frequency but poor time resolution at
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high scales (low frequencies) and good time but poor frequency resolution
at low scales (high frequencies).
Figure 1 graphically illustrates this trade-off between time and
frequency resolution for wavelet transforms. Each window 10 in the time
frequency plane in Figure 1 corresponds to a value of the wavelet
transform. It may be noted that, in accordance with the Heisenberg
uncertainty principle, the simultaneous mapping of both frequency and
time can never be achieved. Thus, a specific value in the time-frequency
plane can never be known, illustrated by each of the windows 10 in Figure
1 having a non zero area. Each of the windows 10 represents an equivalent
portion of the time-frequency plane (i.e. they have equal areas, but may
have different dimensions), however at higher frequencies (lower scales)
the resolution in time is better or less ambiguous than the resolution in
frequency. This is illustrated by window 12 in Figure 1 which has a narrow
width and large height. Similarly for window 14, the resolution in
frequency is better or less ambiguous than the resolution in time. (Note
that for the short-time Fourier transform, the width and height of the
transform windows does not change since a constant window is used.)
The wavelet coefficients for the continuous transform W(a,b)
can be thought of as a varying surface or landscape over a two dimensional
scale-translation plane whose axes are the location variable a and the scale
variable b. At each point, the CWT is essentially a measure of the
correlation or similarity in frequency between a specific wavelet window
function w(a,b,t) and the input signal f(t). The CWT wavelet coefficients
reflect this similarity of the input to the wavelet at a specific scale and
translation. Therefore, if the input f(t) has a significant frequency
component corresponding to a certain scale and occurring within a certain
translation interval, the CWT coefficient for the corresponding point in
the translation-scale plane will be relatively large.
It should be noted that the wavelet transform can also be
considered a constant relative bandwidth or "constant Q" decomposition
that employs wideband windows at high frequencies and narrowband
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windows at low frequencies. (This notion is considered further when
discussing the DWT below.) For each scale value, the ratio of the frequency
to bandwidth, denoted as Q, remains constant. For the CWT, the analyzing
or mother wavelet can be regarded as the impulse response of a reference
bandpass filter. A set of parallel bandpass filters with constant Q can
therefore be used to realize a wavelet transform.
While W(a,b) is theoretically calculated in a continuous
manner for all values of b > 0 and all values of a >_ 0 (i.e. at every point
in
the translation-scale plane), such a complete transform is generally not
required since a signal is usually bandlimited, as would be the case for
discretized or sampled input data. Also, if necessary, the input signal f(t)
can be reconstructed or synthesized from the wavelet coefficients W(a,b) by
summing, in a linear combination, the products of the wavelet coefficients
with the corresponding wavelets window functions, or equivalently
f(t) = f f W(a,b)~w(a,b,t)~da~db
a b
Reconstruction of the input is generally possible if the wavelet functions
act as a set of orthonormal bases. This requires that
f w(a ,b ,t)~w(a ,b ,t)dt - 1 ifa =a andb =b
1 1 2 2 1 2 1 2
t
0 otherwise
By choosing appropriate wavelet functions as discussed below, these
conditions can be satisfied. In applications where orthonormal wavelet
functions are not available, synthesis may also be possible using
biorthogonal bases functions or the concept of frames.
As mentioned, in practice, the continuous wavelet transform
must be based on data sampled at discrete times or intervals, and only a
sampled or discretized form of the CWT can be computed. Thus W(a,b) is
represented by a matrix having a resolution corresponding to the precision
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of analysis in the computer algorithm. When sampling the CWT, it is
possible to reduce the sampling rate as the scale increases (frequency
decreases), while still remaining at or above a critical sampling rate (i.e.
the
Nyquist rate). This is beneficial since it reduces the number of necessary
computations. If synthesis of the input signal f(t) is required, then the rate
for all scales should be at least equal to the Nyquist sampling rate (i.e.
twice
the frequency at that scale) and preferably somewhat higher to reduce
aliasing. However, if synthesis is not required, computations can be
further reduced, depending on the analysis.
In discretizing the wavelet window functions, the scale
parameter b is generally discretized first on a logarithmic basis. This can
conveniently be accomplished using a base 2 logarithm for computational
simplicity. If so, only wavelet coefficients for b = 2, 4, 8, and so on are
computed (i.e. on a dyadic basis). The translation parameter a is then
discretized with respect to the scale b so that a different sampling rate is
used at each of the scale levels (for dyadic sampling the sampling rate
would be reduced in half each time the scale b jumps to the next level).
The discretized CWT described above, also referred to as a
wavelet series transform, is different from the discrete wavelet transform
(DWT) which is discussed below. In general, the shifting process in the
discretized CWT remains relatively smooth across the sampled data,
unlike for the DWT.
In the discrete wavelet transform (DWT), a and b can be
replaced with m and n respectively where m, n are integers (m > 0, n >_ 0),
so that
M-1
W(m, n) = 2 = ~ f(i)~w(i~2-~' - n)
=o
where f[i] are the discretized samples of the signal f(t) and where M is the
total number of samples. As can be seen from the above equation, for the
DWT the scaling or dilation is stepped exponentially by a power of 2, while
shifting or translation occurs by integer steps. Other scaling and shifting
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bases or step sizes are of course possible, but computationally the DWT as
expressed above is relatively simple and therefore preferable. Technically,
the DWT W(m,n) becomes a two dimensional coefficient matrix evaluated
at the points m,n in the scale location plane; however, as with the
discretized CWT downsampling with increasing scale is usually
performed. An exemplary plot of a DWT, also referred to as a scalogram, is
illustrated in Figure 2.
The scale variable m is also referred to as the level of
resolution, and the number of distinct values which m takes on is the
number of levels in the DWT. The length of the input sample stream
being analyzed determines how many frequency resolutions can possibly
be represented, and these frequency resolutions are referred to as the levels
in the wavelet transform. To illustrate, for an input signal f[i] which has
been discretized into M = 2h samples, there are potentially h levels in the
discrete wavelet transform. After downsampling, the number of
coefficients at a given level or scale m is 2h-m (except for the last computed
resolution level which has 2h-m+1 coefficients).
For the DWT, the analyzing or mother wavelet function w(t)
is generally associated with a high pass filter. For some analyzing wavelets,
the function is given by an explicit formula in time (e.g. the HAAR
wavelet), whereas for others the function is obtained from the coefficients
of the associated high pass filter (e.g. the Daubechies family of wavelets).
In
addition to the analyzing wavelet function w(t), the DWT also uses a
scaling function s(t) associated with a low pass filter. Generally the scaling
function is used to define the averages in a signal decomposition.
The wavelet and scaling functions each satisfy a scale
recursive equation: a scaling function is the weighted sum of translated
and compressed versions of itself; and a wavelet is the weighted sum of
translated and compressed versions of the corresponding scaling function.
The mufti-resolution DWT uses the scaling function and wavelet function
to decompose the input signal into different frequency bands.
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The DWT analysis of an input signal f[i] at different
resolutions (i.e. different frequencies) can be conveniently and efficiently
computed with pyramidal coding techniques. The resolution of the signal
is changed by filtering steps and subsequently the scale is varied by
downsampling the filtered signals. To illustrate, a preferred pyramidal
DWT computation technique is shown in Figure 3.
Referring to Figure 3, the DWT analysis begins by subjecting
the input f[i] to a digital low pass filter LP[i] and a digital high pass
filter
HP[i] at 30 and 32 respectively. The filters 30 and 32 are half band filters
which behave as a quadrature mirror filter (QMF) pair, so that the original
bandwidth of the sequence f[iJ is effectively divided in two. Generally, the
high pass filter provides details about the input signal while the low pass
filter mainly provides approximation information. The filtered outputs
from 30 and 32 are of the same scale (or rate) as f[i] but provide a different
resolution. Since the outputs of the filters 30 and 32 are half the bandwidth
of f[i], they are downsampled by two at 34 and 36 without any resulting loss
of information (assuming f[i] was originally sampled at a rate at or above
the Nyquist rate). Downsampling by 2 is conveniently accomplished by
simply eliminating every second sample. The downsampled sequences 39
and 38 from 34 and 36 have a scale that is twice the scale of f[i].
The sequence 38 and the sequence 39 are defined
mathematically (both as a function of j) as
f[i]~HP[2j-i]
i
f[i]~LP[2j - i]
i
respectively. Also, conveniently, the QMF high pass and low pass filters
are computationally linked to one another by
HP[L- (i+ 1)] _ (-1)'LP[i]
where L is the length of each filter.
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The high-pass filtered and downsampled sequence 38
provides a first level (m=1) of DWT coefficients. Where the input
sequence f[i] has M samples, there are M/2 first level DWT coefficients.
The processing of blocks 30, 32, 34, and 36 is then repeated at blocks 40, 42,
44, and 46 with the low-pass filtered and downsampled sequence 39 as the
input sequence. Consequently a second level (m=2) of DWT coefficients is
provided at 48 (for M original input samples, there would be M/4
coefficients at this level). This is then repeated on the sequence 49 to
generate DWT coefficients for successive levels, as needed. If a full multi-
resolution analysis is performed, the last stage (m = h) will generate a one
element sequence 59 from low pass filter 50 and down-sampler 54 and a
one element sequence 58 from high pass filter 52 and down-sampler 56.
Each of these sequences form the DWT coefficients for the last computed
resolution level, and so the h'th level will have 2 DWT coefficients. Thus,
the DWT will have the same amount of coefficients, M, as in the original
input sequence f[i].
It should be noted that the pyramidal analysis of Figure 3 may
be generalized to provide wavelet packet analysis by decomposing both the
high-pass filtered output sequences as well as the low-pass filtered
sequences. The input is thus transformed into several possible wavelet
decompositions. The wavelet packages are particular linear combinations
of these resulting wavelets usually optimized by recursive algorithms or
the like.
Like the CWT, the DWT localizes time information at high
frequencies better than at low frequencies. Since the information content
of most signals is generally present at higher frequencies, this is usually
advantageous. Each successive wavelet transform level reveals coarser
frequency or change information about a larger part of the original input -
lower scales corresponding to more rapid variations and therefore to
higher frequencies. The information containing frequencies of the original
input map to high amplitude DWT coefficients for the portion of the
DWT that includes those particular frequencies.
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Because of the orthogonality of the DWT, information
represented at a certain scale or level m is disjoint from (i.e. does not
overlap with) information provided by other scales or levels in the DWT.
As for the CWT, reconstruction of the input from the DWT again follows
when the wavelet bases are orthonormal. In particular, with DWT analysis
as in Figure 3, reconstruction simply requires carrying out the analysis in
reverse order, i.e. the signals at every level are upsampled by two, passed
through synthesis filters LP'[i] and HP'[i], and then added. Conveniently,
the analysis (LP[i] and HP[i]) and synthesis filters (LP'[i] and HP'[i]) are
identical except for a time reversal. If the filters are not ideal half-band
filters, reconstruction is more difficult, but an appropriate choice of
wavelet function, e.g. the Daubechies wavelet family, can provide for good
reconstruction.
DWT analysis is computationally much faster than CWT
analysis, including discretized CWT analysis. For many applications, even
a discretized CWT will contain significant redundant information. Also
the DWT is often more suitable for diagnostics or computer solution of
some equations. The CWT, on the other hand, provides for better pattern
recognition: see Afanasyeva, Dremin, Kotelnikov, "Pattern Recognition",
Modern Physics Letters A12 (1997) 1185. As a result, both the CWT and the
DWT can be usefully utilized for many applications.
To complete the discussion on the CWT and DWT, brief
consideration of the types of mother wavelets suitable for use in these
transform algorithms is merited. Four common wavelet window
functions are shown in Figures 4A-4D to illustrate some general principles
regarding the wavelet functions. These are the Daubechies-5 wavelet
(Figure 4A), the Daubechies-8 wavelet (Figure 4B), the HAAR wavelet
(Figure 4C), and the Mexican-Hat wavelet (Figure 4D).
As can be seen from Figures 4A-4D, the mother wavelet
function is generally a small window of finite length and is zero outside a
certain time interval (for example between 0 and a certain constant To) .
The wavelet should be compact, as well as oscillatory to satisfy the zero
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mean requirement. The zero mean requirement for the wavelet is also
referred to as the admissibility condition and requires that the integral of
w(t) over time be zero. When this condition is satisfied, the wavelet
transform will be invertible so that the input can be reconstructed.
Different base wavelet functions make various trade-offs
between how compactly the base function is localized in time and how
smooth the function is (i.e. how well it approximates polynomials).
Referring back to Figure 1, different wavelets can have different window
areas for the windows 10 in Figure 1 - although a lower limit for the area
is set by the uncertainty principle.
As mentioned previously, the wavelet function may be given
by an explicit formula in time (e.g. the HAAR wavelet, the Mexican-Hat
wavelet, the Morlet wavelet) or the function may be obtained from the
coefficients of the associated high pass filter (e.g. the Daubechies family of
wavelets).
Different wavelets have different properties and a specific
wavelet must be selected based on the required analysis. For example, the
HAAR wavelet is a discontinuous function, while the Daubechies
wavelets are specific families of wavelet functions that are particularly
suitable for representing polynomial behaviour (the Daubechies wavelets
are generally sub-categorized by a number indicating the number of
coefficients in the high pass filter associated with the wavelet). The
Mexican-Hat wavelet does not provide an orthogonal analysis, although
this is typically not a concern where reconstruction of the input is not
required. Many other wavelets, including the Morlet, the Coiflets family,
and the Symlets family exist. Indeed, the list of wavelets continues to grow
and to be refined, and potentially this list of possibilities is limitless.
For the purposes of the present invention, the specific
wavelet function used during wavelet analysis, whether continuous or
discrete, is not critical, and the inventors have found that similar results
are attainable with various different wavelet functions, such as with the
HAAR and the Daubechies wavelets.
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As will be understood from the above description of wavelet
analysis, the wavelet coefficients (CWT or DWT) for a physically occurring
time signal f(t) contain a considerable amount of useful temporal and
frequency information about that signal. Since the original signal f(t)
varies in time, a set of wavelet coefficients at any given scale or level of
resolution will also vary in time (although the mean or average of the
coefficients will be zero). While scalograms such as the one in Figure 2 can
be plotted, compared, and analyzed for diagnostic purposes, statistical or
algorithmic analysis which can be carried out quickly and efficiently by a
computer or processor is preferred.
For instance, in Thurner, Feurstein, and Teich,
"Multiresolution Wavelet Analysis of Heartbeat Intervals Discriminates
Healthy Patients from Those with Cardiac Pathology", Phys. Rev. Lett. 80
(1998) 1544-1547, the authors describe a method of diagnosing a human
heart by analyzing the sequence of time intervals between heartbeats. The
time series was transformed into a discrete wavelet transform (DWT) and
the wavelet coefficient standard deviation (or dispersion) as a function of
scale was calculated. The authors determined that, at intermediate scales,
the wavelet coefficients for heart failure patients exhibited substantially
lower variability or dispersion than for normal patients. The analysis
thereby provided a clinically significant measure of the presence of heart
failure with 100% sensitivity and 100% specificity.
While the dispersion analysis performed by Thurner et al. is
effective for diagnosing cardiac pathology under at-rest (non-changing)
conditions, it is generally unsuitable for analyzing the real-time signals or
conditions associated with a "non-stationary" device such as an in-flight
aircraft engine. The method of Thurner et al. is also limited to the analysis
of a single dispersion parameter, and so may be insufficiently reliable for
more complex physical processes.
In accordance with the present invention, a physical time
varying signal associated with the operation of a device such as an engine
or another type of electric machine is analyzed with discrete and/or
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continuous wavelet transforms to determine one or more statistical or
probabilistic measures of the resulting wavelet coefficients. These
coefficient characteristic measures generally include at least the dispersion
of the coefficients (i.e. their standard deviation a or variance a2) at
various
scales. By also determining high order or high rank correlation matrices
for the wavelet coefficients, other statistical or probabilistic measures,
such
as the two dimensional equivalents of the one dimensional measures of
skewness or kurtosis, can be provided. Technically, correlation matrices
with infinite rank form a complete set, and various probabilistic measures
can be expressed as combinations of these matrices. Generally, when the
correlation matrix series is truncated, certain measures become preferable,
depending on the level of truncation.
In a preferred embodiment, more than one probabilistic
wavelet coefficient measure is used to diagnose and assess the physical
signal corresponding to the electric machine so that a more reliable and
accurate diagnosis is achieved with the present invention. Different
computer algorithms can be used to determine different measures of
wavelet coefficient characteristics, and this can also be done at different
resolution levels. Furthermore, this type of diagnostic analysis can also be
carried out simultaneously on more than one physical signal (for example,
for pressure, vibrations, and temperature) to separately provide additional
diagnostic indicators with respect to the operation of a device.
The dispersion of wavelet coefficients is essentially the spread
of the coefficients about their mean, and is generally given by the standard
deviation a. For example, for a set of DWT coefficients W(m,n) the
standard deviation at a given level of resolution m is given by
N-1
N- 1 ~=o
where N is the number of wavelet coefficients at the scale m and
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N=int(Ml
l 2 "'
Correlation is a ratio of the covariance of the two variables to
the product of their standard deviations. (Correlation values are merely
measures of covariances of standardised values. Standardizing a data set
zeros the mean and sets the standard deviation to one, so that the
covariance value is equivalent to the correlation value.) Bivariate
correlation provides a single number which summarises the relationship
between two variables. The correlation coefficient indicates the degree to
which variation in one variable is related to variation in another. The
correlation coefficients which are generally obtained from a least squares
approximation technique, measure the degree of linear correlation
between the two variables.
Typically, a correlation matrix is a variance-covariance matrix
of standardized data. The diagonal of the resulting correlation matrix is the
correlation of one variable with itself and should have a value of 1. In
accordance with the present invention, multi-dimensional correlation
matrices may be calculated from wavelet coefficients by means of a
generating function technique: see De Wolf, Dremin, and Kittel, "Scaling
Laws for Density Correlations and Fluctuations in Multiparticle
Dynamics", Phys. Reports 270 (1996) p. 1. For example, a set of DWT
wavelet coefficients W(m,n) can be used as the coefficients in a polynomial
sequence of two continuous variables, a and v, to provide a generating
function G(u,v). A representative G(u,v) in powers of a and v then
provides the elements of the correlation matrix F(q,p) as follows:
ym~n)u~~" _ ~ (u 1)q(v 1)P F(q~P)
rn,ri q,P q!pl
The correlation matrix F(q,p) is theoretically of infinite dimension, but in
practice it may be truncated at some qmaX and p,naX imposing, for example,
some threshold value on its elements. Generally, F(q,p) indicate the
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correlations between the W(m,n) set of wavelet coefficients with the
parameters q and p defining the rank or order of a particular correlation
Fqp. It should be noted that the second order correlation matrix (q = 1, 2 and
p = 1, 2) provides a measure of the dispersion of the W(m,n) coefficients.
Higher order or higher rank correlation matrices (i.e. q,~,aX > 2 and p,.naX ~
2)
provide distinct probability measures (such as the two dimensional
equivalents of the one dimensional measures of skewness or kurtosis) and
can be used to determine other types of important information about the
wavelet coefficients and hence the original input signal.
Figure 5 shows of block diagram operational overview of the
diagnostic and control system 100 according to a preferred embodiment of
the present invention. The system 100 diagnoses and controls an electrical
device 102 such as an engine. The device or engine 102, in known manner
operates to run a system 108 which may be, for example, an aircraft. The
device 102 may be operating under at rest conditions or under "non-
stationary" conditions which change with time. Furthermore, the device
or engine 102 may be at different operational regimes defined by operation
at a certain percentage of the device's nominal limits. While the device 102
is at or within any of these regimes, the operation may become unstable.
As will be appreciated by those skilled in the art, instability increasingly
becomes a concern as operation of the device approaches its nominal
limits, i.e. full capacity. The operating regime of the device 102 (and any
changes thereto) is generally specified via 136 by the command execution
unit 134.
A sensor or transducer 104 (preferably several like transducers
are used to ensure consistency and accuracy of a physical measurement)
attached to the device 102 provides a time varying signal f(t) corresponding
to the operating characteristic measured by the transducer 104. The
transducer 104 may measure any suitable physical parameters such as
vibrations, deformations, pressure variations, acoustic noise, temperature
etc. Alternatively the signal f(t) may simply represent the power consumed
by the device 102, in which case the sensor 104 is not needed (although the
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signal may need to be converted to an appropriate voltage form). The
signal f(t) which at least partly characterizes the operation of the device
102
is sampled and converted into a digital signal f[i] at 106. The digital stream
f[i] is provided to computer system 110.
Note that while only one signal f(t) is processed in the system
100, the system 100 may separately perform similar analysis on more than
one time varying characteristic of device 102 operation. This allows for
alternative and/or additional diagnostic capability, as will be clear to those
skilled in the art.
The computer system 110 includes a wavelet coefficient
generation module 120, a first measure (M1) algorithm module 122, a
second measure (M2) algorithm module 124, and an automatic control
module 130, as shown. While the computer system 110 is shown as having
these separate modules, this is merely illustrative of system operation, and
it should be noted that software running on computer system 110 may be
organized in a number of different manners. For instance, there may only
be a single measure algorithm module which computes all of the
probabilistic measures, as required. The computer system 110 also has a
CPU or processor and memory resident therein (not shown) for actually
carrying out the operations in modules 120, 122, 124, and 130. The wavelet
coefficient generation module 120 calculates discretized CWT and/or DWT
coefficients based on the input samples f[i]. As the sequence of samples f[i]
is continually updated, so too are the wavelet coefficients calculated by
module 120. Thus the system 100 is capable of running in an on-line or
real-time manner for a device which is not at rest. The module 120 may be
programmed to use a specific analyzing wavelet function (e.g. a
Daubechies wavelet) or it may be programmed to select among various
possible options based on available processing capabilities, the type of
operational characteristic f(t) represents, and any other relevant criteria.
Similarly, the module 120 may be programmed to select between a DWT
and a CWT algorithm.
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Once calculated, the wavelet coefficients from module 120 are
provided to algorithm modules M1 122 and M2 124 which each determine
a probabilistic or statistical measure of the wavelet coefficients, such as
dispersion at a specific scale or a measure provided by a high rank
correlation matrix at a definite, i.e. truncated, correlation rank or level.
Note that although two separate measure algorithms are shown in Figure
5, the system may only determine one measure algorithm (for example
dispersion). Preferably, however, more than one probabilistic wavelet
coefficient measure is used to diagnose and assess the physical signal f(t),
so that a more reliable and accurate diagnosis is achieved. Consequently,
the system 100 may also include more than two measure algorithms.
The computed probabilistic measures from 122 and 124 are
provided to the automatic control module 130 as shown in Figure 5.
Optionally, feedback 126 and 128 from the control module 130 to the
algorithm modules 122 and 124 respectively may be provided, so that the
module 130 can specify, for example, that dispersion be determined at a
particular level or scale or that a correlation matrix of a particular rank be
provided. Changing the level of multi-resolution analysis (or the
correlation matrix rank) in such a manner allows the control module 130
to determine the most appropriate criteria for making a diagnosis and can
also provide the automatic control module 130 with supplementary
diagnostic information. Generally, the use of several criteria and
thresholds in control module 130 provides a reliable means to mutually
validate and corroborate any diagnostic results or conclusions.
It should be noted that a determination of appropriate criteria
(e.g. the resolution level) for analyzing and diagnosing the operation of a
device 102 may initially need to be based, at least to some extent, on
experiments and tests of different criteria with the specific device type.
The automatic control module 130 analyzes the criteria
measures for significant changes, variations, or disruptions within these
measures which vary with time, as the sequence of samples f[i) is
continually updated. Upon doing so, the control module 130 determines
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what feedback response 132, if any, should be supplied to command
execution unit 134 to change the operating regime of the device 102.
Significant variations or disruptions in the probabilistic measures used as
diagnostic criteria are generally indicative of an instability which
(unchecked) will lead to an impending malfunction or failure in the
operation of the device 102. What will qualify as a significant variation in,
for example, the dispersion measure may vary from application to
application (and from one resolution level to another), and an exemplary
illustration of this is provided below. Generally, when stable operation is
detected, i.e. no instabilities or impending malfunctions are diagnosed, the
control module 130 may direct the command execution unit 134 to
maintain the device at the current operating regime. Optionally, the
control module 130 could also direct the command execution unit 134 to
increase the operating regime of the device closer to full capacity if the
device is not already operating sufficiently near full capacity. On the other
hand, when an instability or impending malfunction is detected, the
control module 130 directs the command execution unit 134 to
appropriately reduce the operating regime of the device in accordance with
the instability control approach of the command execution unit 134 -
thereby preventing further development of the instability into a
malfunction or failure in the engine's operation.
The reliability of the system 100 can be further guaranteed by
providing multiple checks of the validity of commands 132 sent by the
control module 130, in addition to the mutual comparison of results
obtained from different probability measures, for example a dispersion
analysis of wavelet coefficients at different scales or a dispersion measure
and a skewness measure from a high rank correlation matrices. As shown
in Figure 5, data unit/recorder 140 of the multi-functional system 108 can
accumulate and provide other information (e.g., the flight data in case of
an aircraft) to the automatic control module 130. Thus, the computer
analysis and control in module 130 via the resulting execution commands
132 can be based on the simultaneous processing of many different
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parameters and on a set of operations prescribed for the engine regulation
(including instability control). The versatility and flexibility of the
control
system of the present invention is clearly apparent from the ability to
analyze several physical parameters, flight or other supplemental data, and
different probability measures, as well as in the selection of an appropriate
approach to engine regulation from among the many available choices.
The specific instability control approach chosen for regulation
of device or engine 102 does not form part of the present invention.
Known approaches include active control, passive control, and avoidance
control, and generally selection of the most suitable approach will depend
on the specific application. See generally, Georgantas, "A Review of
Compressor Aerodynamic Instabilities", National Aeronautical
Establishment (National Research Council Canada), 1994.
Regardless of the instability control approach selected, the
system and method of the present invention can be used to effectively
extend the stable operating range and the scope of regulation of the device
102, allowing the device to be safely and reliably operated closer to full
capacity. The device is therefore operated more efficiently with an
extension of the range of engine regulation not provided by prior art
systems. Conclusions about any corrective adjustment in the operation
regime and its realization are made by a compute program, based on the
time development of these parameters and on the stored algorithms. With
the early detection with precursors of possible failures and by appropriate
and rapid corrective responses, the present invention provides high
reliability and fast feedback both for at rest and in-motion (time varying)
situations and for stable and unstable operating regimes.
The very fast feedback response in the present invention
enables a possible failure to be averted. This ability stems from the highly
reliable and very timely diagnostic conclusions or precursors of
instability/malfunction generated by the invention during operation of
the device. The timely prediction by way of precursors of failure is
primarily due to the wavelet transform properties which have been
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advantageously exploited by the present invention (as discussed in detail
above). Such rapid determination of highly accurate precursors of
impending malfunctions was, hitherto, unavailable in prior art diagnostic
systems.
Furthermore, the system and method of the present
invention differs substantially from prior art systems in that diagnosis of
instability or impending malfunctions can be made under time varying
conditions whether the device is operating within stable or unstable
regimes. The ability to apply the invention not only to systems which are
at rest, but also during the "non-stationary" conditions of device operation
(such as for an in-flight aircraft engine) provides a significant advantage.
Figures 6-8 illustrate, by way of example, the highly accurate
and very timely diagnostic precursors associated with an instability or
impending malfunction in a device, as may be provided by the present
invention. In the experimental examples of Figures 6-8, the device is an
aircraft engine having an axial multi-stage compressor. The rotation of the
rotor is at 76% of its nominal limit in Figure 6, 81% of its nominal limit in
Figure 7, and 100% of its nominal limit in Figure 8. Signals from eight
pressure sensors positioned at various places within the compressor were
recorded and digitized as described above. As is known to those skilled in
the art, an axial multistage compressor is susceptible to the formation of a
rotating stall which may be precipitated by a distorted inlet flow, and this
may also lead to the very serious problem of engine surge. In an attempt to
avoid these problems, the compressor is usually operated below full
capacity at which these instabilities are less likely to occur.
For the experimental test situations in each of Figures 6-8, the
aircraft engine was physically operated under at rest conditions, but an
instability was introduced by increasing the pressure behind the
compressor through a slow injection of extra air into the compressor
intake. After a few minutes, this led to a full blown rotating stall
instability
in the compressor.
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Figures 6-8 show the variation of the pressure within the
compressor for an interval of about 5 seconds. In each case, this interval
includes the occurrence of the fully developed instability, indicated at 200.
Each of Figures 6-8 show the time variation from one of the pressure
sensors at 210 (all pressure sensors provided similar results) and the
dispersion (standard deviation) of a set of DWT coefficients was computed
from the pressure input. The time varying dispersion is shown at 220. In
each case, the DWT was calculated with the HAAR wavelet and dispersion
was measured at the 4-fold resolution interval (i.e. at the 4th level of DWT
resolution) and at a definite (truncated) correlation level .
With reference to Figure 6, it may be seen that the value of
the dispersion during normal operation of the compressor abruptly
decreases at 230. This change is considered to be a precursor of the stall and
the possible destruction of the compressor, reflected in the abrupt, large
increase in the dispersion (at 240) near the end of the time interval. The
smaller values of the dispersion are due to rotational instabilities in the
compressor. Note that the increase in the dispersion measure 220 prior to
the precursor 230 in Figures 6 and 7 is merely a result of a threshold effect
due to the finite length of the wavelet resolution at a given scale.
As indicated, the frequency of the rotation of the rotor was
76% of its nominal limits (n/nli,r, = 0.76) in Figure 6. In this case, the
precursor 230 provided about a 2.0-2.5 second warning before the
beginning of stall 200. This time interval and the significantly large
decrease in the dispersion are more than sufficient for very reliable
diagnostics.
Referring to Figure 7, a similar plot with the frequency of the
rotor rotation equal to 81% of its nominal limit (n/nlim = 0.81) is provided.
Once again the stall precursor 230 provides about 2.2 sec warning before
onset of the rotating stall. Finally, in Figure 8 (at the limiting frequency
of
the rotor rotation (n/num =1)). the precursor 230 starts earlier than in the
previous examples and provides about 1.0 seconds of warning. In each of
CA 02276571 1999-06-29
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Figures 6-8, the precursor was reflected by a drop of about 30-40% in the
dispersion characteristic.
Comparative analysis of Figures 6-8 demonstrates that the
more intensive regimes start developing the instability earlier and possess
shorter precursor warnings. However, all of these time warning intervals
are at least about 1 second. The main delay in an engine control feedback
system such as in the present invention results from the rate of the
physical processes and from the steps used to transfer and process
information about those processes. The inventors' have found that the
measurements from the primary sensors' outputs, the wavelet
transformation analysis of the data accumulated over a period of time, the
mutual comparisons and checks of diagnostic results, and the generation
of appropriate control commands takes approximately up to about 0.2 sec.
Thus, the precursor times in Figures 6-8 are all much longer than the time
generally necessary for reliable computer analysis to be carried out, and
there is enough time for smoothly regulating the operating regime before
a rotating stall starts. In comparison, prior art attempts to predict the
development of a rotating stall with a velocity measuring probe such as a
hot wire anemometer provide a precursor or warning of only about 10
milliseconds. The unambiguous detection of a precursor to instability, its
rapid recognition, and the resulting long time available for preventing the
engine failure are principle advantages of the present invention not found
in the prior art.
In all three cases of Figures 6-8, correlation matrices can also
be used for further and more accurate analysis of the precursors.
Furthermore, measure from correlation matrices can be used to help
reveal the physical origin of a stall.
It will be apparent from Figures 6-8 that no significant
precursors are present in the direct measurements of the highly irregular
pressure variations of the compressor. The diagnostic characteristics of the
wavelet coefficients are clearly much different in principle from those
present in the time-varying signal itself. Furthermore, Fourier
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spectrograms of the pressure variations were also examined, and they
failed to reveal rotational instabilities of sufficiently large amplitude
within the necessary time constraints (i.e. more than 0.2 seconds before the
onset of stall).
Thus it can be seen that the present invention provides a
diagnosis of a possible failure in the engine's operation in a very short
time. High-speed performance stems from the locality properties of the
wavelets and the use of high-speed processing computers. The use of high
rank correlation matrix measures together with the dispersions
advantageously leads to a practically error-free diagnosis. As a result, the
system and method of the present invention can be used for diagnosing
the operating regimes of any regularly (in particular, periodically)
operating systems (aircraft compressors, turbines, auto-motors, electrical
motors, pumps, power plant turbines, etc.) and for preventing their
failure. The invention provides a significant improvement in the
diagnostics of the operating regimes of existing engines, which is
important for preventing their failure and, consequently, for lowering
associated economic losses.
Moreover, in comparison with the prior art, this system and
method make it possible to regulate engine operation quickly and in a
reliable way by taking into account many parameters without necessitating
an abrupt change of the operation regime. This is due to the effectiveness
of diagnostics based on wavelet analysis in detecting precursors of engine
malfunctions/instability and by rapid feedback. Consequently, the present
invention can be used for the automatic regulation of the operating
regimes (to prevent malfunctions and possible failure) of any electric
machine including engines, motors, turbines, compressors, pumps, auto-
motors, other electromagnetic rotating devices, and in general any device
which operates regularly (in particular periodically). Furthermore, the
system and method of the present invention do not require any human
intervention at any stage.
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While preferred embodiments of the present invention have
been described, the embodiments disclosed are illustrative and not
restrictive, and the invention is intended to be defined by the appended
claims.
