Harmonic tracking
LMS Test.Lab
16A
Copyright Siemens Industry Software NV
Table of Contents
Chapter 1 Introduction ................................................................................................... 5
Section 1.1 Conditions for use ................................................................................ 5
Chapter 2 Theoretical background................................................................................ 7
Section 2.1 Determination of the Rpm ................................................................... 7
Section 2.2 Waveform tracking .............................................................................. 8
Section 2.2.1 The Structural equation ........................................................................ 8
Section 2.2.2 The Data equation ................................................................................ 8
Chapter 3 Practical considerations ............................................................................. 11
Section 3.1 Frequency resolution.......................................................................... 11
Section 3.2 Filter characteristics ........................................................................... 11
Section 3.2.1 Bandwidth characteristics .................................................................. 12
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Chapter 1 Introduction
Chapter 1 Introduction
In This Chapter
Conditions for use ..............................................................5
There are a number of circumstances when it is necessary to track periodic
components (orders) when the signal of interest is buried in noise, or the
rotational speed is changing rapidly. Indeed some effects only manifest
themselves when the rate of change of frequency is high. In these situations,
real time analog and digital filters have limited of resolution due to transients
and excessive processing requirements. The Kalman filter however is able to
accurately track signals of a known structure concealed in a confusion of noise
and other periodic components of unknown structure.
An important characteristic of the Kalman filter is that it is non-stationary. It
functions well at high slew rates, because the system model used does not
presume either fixed time of frequency content, but adapts itself automatically
as the system itself is changing. This ability to derive the system model for each
time sample in the recording (within certain user-defined constraints) frees it
from the usual time/frequency resolution constraint encountered with the
traditional frequency transformations.
Section 1.1 Conditions for use
Some important capabilities of the Kalman filter are -
the ability to track an order with arbitrary fractional order resolution from
signals sampled at a constant rate,
fine spectral resolution of the orders (i.e. 0.01 Hz) obtained after just a few
measurement samples (not even one cycle of the fundamental component),
virtually no slew rate limitations,
the ability to produce an order value for every measurement sample point,
no phase distortion.
In order to use the Kalman filter the following conditions must apply -
The structure of the signal (sine wave) to be tracked must be accurately
known.
The signals must be acquired at a constant sampling rate.
An accurate estimate of the instantaneous Rpm value is required when you
are dealing with signals that vary with rotational speed.
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Chapter 2 Theoretical background
Chapter 2 Theoretical background
In This Chapter
Determination of the Rpm ..................................................7
Waveform tracking .............................................................8
The application of the Kalman filters to track harmonic components involves
two stages.
Step 1
Accurate determination of the Rpm
If you want to track an order, then you must provide the corresponding
Rpm/time trace. Your Rpm may have been determined using a Tacho signal
which results in a pulse train or a swept sine function in which case you will
need to convert it to a Rpm/time function.
Step 2
The tracking of the specified waveform
Section 12.2.2 describes the mathematical background to the operation of the
tracking function.
Some practical considerations are discussed in section 12.3.
Section 2.1 Determination of the Rpm
Since the Kalman filter is highly selective and accurate in tracking a target
signal buried in noise, it is crucial that the instantaneous RPM of the system is
precisely modelled, otherwise the wrong component will be tracked. The rpm
information can be derived from the tachometer channel, which is sampled at
the same rate as the measurement channels to obtain a small statistical
variability in the period estimation. Clearly the tachometer events will occur at a
lower rate and so to reduce the error on the period estimate, resampling is
performed on the original tachometer signal.
The first part of the process therefore is to convert the original tacho signal from
a pulse train to an rpm/time function.
The second step involves obtaining an equidistant function. Since all
mechanical systems have some inertia, it is reasonable to expect the speed to be
a continuous function, so a cubic spline with the appropriate boundary
conditions can be used to obtain the required 'sample-by-sample RPM' estimate
of speed function.
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Chapter 2 Theoretical background
Section 2.2 Waveform tracking
The Kalman filtering method involves setting up and solving a pair of equations
known as the Structural and the Data equations.
Section 2.2.1 The Structural equation
This equation defines the shape or structure of the waveform you wish to track.
A sine wave for example, x(t) of frequency w sampled at time Dt satisfies the
following second order difference equation
by dropping the time increment Dt this can be written more simply as
where c(n) = cos (2 p w Dt)
When the instantaneous frequency w is known, equation 6-2 is a linear
frequency dependent constraint equation on the sine wave which is known as
the structural equation.
When tracking a sine wave which is changing in frequency, and which is
contaminated by noise and other sinusoids, a non homogeneity term e(n) is
introduced. This allows the sine wave to vary in frequency, amplitude and phase
and Equation 6-2 then becomes
e(n) is a deterministic but unknown term which allows for deviations from the
true stationary wave. It is also useful to define Se(n) as the standard deviation of
the non homogeneity of the structural equation.
Section 2.2.2 The Data equation
x(n) is the time history defined by the structural equation, but the measured
signal y(n) contains both the signal that matches the structural equation as well
as noise and other periodic components.
8 LMS Test.Lab Harmonic tracking
Chapter 2 Theoretical background
where h(n) contains noise and periodic components at frequencies other than the
target signal. Once again Sh(n) is defined as the standard deviation of the
nuisance element of the data equation.
Section 2.2.2.1 The Least squares formulation
For any point in time (n), equations 6-3 and 6-4 provide linear equations for
{x(n) x(n-1) x(n-2)}. Rearranging these equations gives an unweighted form of
equation where the structural equation is on the top row and the data equation
on the bottom.
The error in equation 6-5 is made isotropic by applying a weighting factor r(n)
which is defined as the ratio of the standard deviations of the errors in the
structural and data equations.
Equation 6-5 then becomes -
The weighting function r(n) expresses the degree of confidence between the
structural equation and data equation, or, the certainty of the presence of orders
in the data. This function shapes the nature of the Kalman filter and influences
its tracking characteristics. A small value for r(n) leads to a filter that is highly
discriminating in frequency, but which takes time to converge. Conversely, fast
convergence with low frequency resolution is achieved by choosing a large r(n).
When applied to all observed time points Equation 6-7 provides a system of
overdetermined equations which may be solved using standard least squares
techniques.
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Chapter 3 Practical considerations
Chapter 3 Practical considerations
In This Chapter
Frequency resolution ..........................................................11
Filter characteristics ...........................................................11
This section considers some practical characteristics of the Kalman filter and
the parameters that influence them.
Section 3.1 Frequency resolution
In principle the Kalman filter is capable of tracking sinusoidal components of
any frequency up to half the sample frequency. In practice however, it has been
found that the ability to distinguish between two closely spaced sine waves is
inversely proportional to the total observation time. As a consequence, the
observation time should be equal to the inverse of minimum frequency spacing
required between components.
Section 3.2 Filter characteristics
It was mentioned above that the weighting r(n) used in Equation 6-7 can be
used to influence the nature of the tracking filter used. This weighting can be
adjusted through the specification of a harmonic confidence factor which is
defined as the inverse of the weighting factor.
Applying a high value implies confidence in the harmonic (structural data) and
assumes that the error in your measured data is high. In this case the filter will
be narrow so that it is highly discriminating in frequency. This is obtained at
the cost of time to converge in amplitude. Applying a low value implies that the
error in the measured data is low and consequently a wider filter can be used
which while less discriminating in frequency has the advantage that the
amplitude converges more quickly.
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Chapter 3 Practical considerations
The three Kalman filters shown below are characterized by different harmonic
confidence factors which influence the width of the filter.
Figure 6-1 Effect of the Harmonic Confidence Factor
Section 3.2.1 Bandwidth characteristics
Equation 6-7 shows that the weighting function, r(n), which is the inverse of the
harmonic confidence factor, can be different for every time point. This means
that the bandwidth of the filter can vary as a function of the frequency or order
being tracked.
Using a frequency defined band width means that at low Rpm values, a number
of orders will be encompassed by the filter range.
Figure 6-2 Defining the filter bandwidth in terms of frequency and amplitude.
Section 3.2.1.1 Allowable slew rates
The formulation of the Kalman filter assumes that the frequency of the signal to
be tracked remains constant over three consecutive measurement points. When
the frequency is varying, but the variation over these three points is less than the
bandwidth of the filter then no problem arises.
The minimum value of the bandwidth is equal to the inverse of the observation
12 LMS Test.Lab Harmonic tracking
Chapter 3 Practical considerations
time T. If the sample rate is Fs then the slew rate must be less than Fs / 2T.
Tracking closely spaced order signals with a high slew rate requires sampling at
a high frequency over a long period which imposes a heavy computational
effort. However if you consider the significant slew rate encountered during the
deceleration of gas turbines of 75Hz/sec over 5 seconds, from the above this
implies a sample rate of 750Hz. It can be seen therefore that such an extreme
slew rate does not impose any realistic limitation on the sample rate.
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Index
A
Allowable slew rates • 13
B
Bandwidth characteristics • 12
C
Conditions for use • 5
D
Determination of the Rpm • 7
F
Filter characteristics • 11
Frequency resolution • 11
I
Introduction • 5
P
Practical considerations • 11
T
The Data equation • 8
The Least squares formulation • 9
The Structural equation • 8
Theoretical background • 7
W
Waveform tracking • 8
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