International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 4
Number 4
December 2010
423
1 INTRODUCTION
Elements of a typical measuring track include:
measuring transducers, elements adapting the meas-
urement signal to elements of the measurement sys-
tem, analogue-digital transducers, filters, and ele-
ments processing and analysing measurement
signals. A typical measuring track is shown in Fig.1.
Measuring signal
Measuring results
Measuring
transducer
Conditioner
A/D
Filters
Signal analyze
Figure 1. Typical measuring track
Vibration transducers are the initial track ele-
ments. Accelerometers are most commonly em-
ployed in diagnostics. A conditioning and amplify-
ing system adapts the electrical signals from the
transducer (reducing its great internal impedance) to
the input of the analogue-digital transducer. A signal
usually needs to be filtered to achieve its good quali-
ty, necessary for determination of vibration parame-
ters (Szychta 2006). Data are then processed and an-
alysed, compared with available standards, and the
resultant output signal is displayed in the form of,
for instance, a characteristic curve or tabulated re-
sults.
The output signal of a measuring track may be a
signal used to control equipment, e.g. abrupt braking
system, vehicle suspension, etc. In such circum-
stances, the basic criterion for real time controlling
of this device is the correct processing of the input
signal by the measuring transducer and sending of
this signal to a control device over a time-span that
allows the device to respond to the given situation.
This implies correct choice of a transducer adapted
to the input signal (the range of frequencies and sen-
sitivities measured). It must be borne in mind that
each element in a measuring track introduces errors
and delays in processing of the measuring signal, of-
ten amplifying the errors introduced by the function
of upstream elements of the measurement track. The
processing time of a measurement signal by the
track is important in applications that require rapid
measurements. With the number of elements pro-
cessing a signal, the time may be extended till a
point when correct operation of a control system will
no longer be possible.
2 TESTING OF THE MEASURING TRACK IN
LABORATORY CONDITIONS
Figure 2 illustrates a diagrammatic measurement
system used to test characteristics of measuring track
processing. An accelerometer including an inbuilt
DeltaTron preamplifier of sensitivity 10.18 mV/ms
-2
(A2) and the range of measured frequencies 0.3 Hz
to 6 kHz, and a conditioner of the operating range
from 1 Hz to 20 kHz is the sensor under testing. The
reference value of acceleration is obtained from the
accelerometer, whose sensitivity is much greater
than that of the tested accelerometer and whose
measuring signal can be read without the need for
additional equipment. To this end, a piezoelectric
Programmatic Correction of Errors of
Measuring Track Processing
M. Luft, E. Szychta & R. Cioc
Technical University of Radom, Radom, Poland
ABSTRACT: A programmatic method of correcting errors of the measuring track is presented. Methods of
determining transfer function of the measuring track are introduced and measurement results with and without
the correction are compared.
424
accelerometer of sensitivity 317mV/ms
-2
(A1) and
the range of measured frequencies from 50 to 200
Hz is used.
A1 – standard accelerometer
A2 – measuring accelerometer
Signal generator,
amplifier
A1
Oscilloscope
Conditioner
A2
Inductor
Computer,
measuring card,
software
S1
S2
S1
Figure 2. Flow diagram of a measuring track testing system
Figure 3 shows courses of signals from the accel-
erometers A1 and A2 for sinusoidal input function
of frequency 100 Hz and acceleration amplitude of
3.15 m/s
2
, as read from the course of the reference
accelerometer A1. The delay between the signal
from the accelerometer including the preamplifier
A2 and the signal from the piezoelectric accelerome-
ter A1 is 8.9 ms. The determined relative error be-
tween values of computed acceleration amplitudes is
40%. Measuring of the acceleration with the com-
puter, measurement card (of sampling frequency 40
kHz), and the software cause a further delay of 1.1
ms between the measurements. The delay between
measurement of the signal from the transducer A2
and A1 total 10 ms.
Figure 3. The courses of signals from the reference and meas-
urement accelerometer
Figure 4 illustrates the attenuation diagram of the
acceleration of the vibration transducer A2 as deter-
mined in testing. To enhance reliability of the re-
sults, the measurements were made in the frequency
range of the correct operation of the reference trans-
ducer A1: 50 Hz – 200 Hz. Technical specifications
of A1 indicate that, in this range, the amplification
of the output signal's amplitude in relation to the
transducer's input signal is within ± 1 dB. This is
usually the acceptable value for purposes of meas-
urements. Technical specifications of A2 state that,
in the frequency range 0.3 Hz - 6 kHz, the same val-
ue is ± 10 % and corresponds to ± 0.91 dB. It was
assumed that A1 relays the amplitudes closer to the
actual values, as it is designed to operate in a nar-
rower frequency range, at more than 30 times greater
sensitivity, and without additional elements that
would process, and affect, the measurement signal.
Accepting the amplification of ± 1dB in the fre-
quency range 50 Hz - 200 Hz, declared in the speci-
fication of A1, the resulting characteristic curve
shows that the value of amplification of A2 is
achieved in the range of frequencies above 125 Hz.
This is different than the manufacturer's bottom val-
ue of 0.3 Hz.
To avoid problems of correct reading of the val-
ues measured by the transducers, particularly the
reference transducer, assume the amplification
range of correctly measured accelerations ± 2 dB.
The characteristic curve presented in figure 4 indi-
cates this frequency is in the range 125 Hz - 200 Hz.
Figure 4. The attenuation diagram of the measuring sensor
3 CORRECTION ALGORITHM
Determination of a dynamic correction algorithm of
a real measuring transducer requires knowledge of
its dynamics in the form of differential equations
and of the dynamics of the entire measurement sys-
tem. Manufacturers' attempts at maintenance of
transducers' reproduction properties in the manufac-
turing process have been a failure. Depending on the
transducer class, the particular pieces and production
lots suffer from some errors of reproduction, sensi-
tivity, and processing range. Therefore, correction
algorithms must always be determined for specific
measuring transducers and measurement systems,
0,00
1,00
2,00
3,00
4,00
5,00
6,00
7,00
8,00
50 70 90 110 130 150 170 190
Amplification [dB]
Frequency [Hz]
425
and the resultant equations must only be applied to
the particular sensors (Cioc 2006).
In measurements requiring highly accurate re-
sults, other measuring track elements (amplifiers,
measuring cards, signal processing elements, etc.)
should be taken into account when defining dynamic
correction algorithms.
The need to consider elements other than the sen-
sor itself when describing system dynamics is an ini-
tial difficulty with attempts to determine the correc-
tion algorithm. The signal from the reference
transducer and from the measurement system need
to be compared to describe the system's dynamics.
ARX (AutoRegressive with eXternal input) method
(Luft 2005) was employed to identify dynamic pa-
rameters of the measuring track. The voltage signal
from the end of A2's measuring track is the identi-
fied signal, and voltage (S1), which is the response
of the accelerometer A1 to sinusoidal input of 200
Hz, is the comparative (reference) signal. Accelera-
tion values are determined on the basis of voltage
signals from the sensors and the latter's sensitivity.
To limit the calculations and the processing time, the
correction algorithm was assumed to estimate the
voltage signal from the measuring transducer ac-
cording to the algorithm parameters as determined
by comparison of signals of the reference and meas-
uring transducers.
Figure 5. Frequency characteristic curves of a meas-
uring track including an accelerometer
Application of ARX produces the transfer func-
tion G(s) which describes processing dynamics of
the system: accelerometer – conditioner – measuring
card – data acquisition software:
742
62
10309,210678,4
10338,16,131903215,0
)(
⋅+⋅+
⋅++
=
ss
ss
sG
(1)
Magnitude and phase characteristic curves of the
transmittance system (1) are shown in Figure 5. A
great amplification can be seen in the magnitude
characteristic curve as sensors of varying sensitivi-
ties are applied to testing of the voltage signals. The
reference amplification level is expressed as a rela-
tion of the measuring transducer's sensitivity to the
sensitivity of the reference transducer on the loga-
rithmic scale: W
r
= -29.87 dB. Given this value, the
relation of acceleration determined on the basis of
the measuring sensor's voltage signal to the accelera-
tion determined using the reference sensor is a
A1
/a
A2
= 1. The magnitude characteristic curve at the adopt-
ed boundary value of the amplification ± 2dB W
r
is
in the frequency range over 117 Hz.
The dynamic correction algorithm for the transfer
function (1) is expressed in a differential equation:
xx
dt
dt
x
dt
dt
yy
dt
d
y
dt
d
6
2
2
74
2
2
10338,16,131903215,0
10309,210678,4
⋅++
=⋅+⋅+
(2)
where x = input quantity; y = output quantity.
The correction algorithm (Jakubiec 2000) for the
measuring track's transfer function (2) and received
digitisation time Td=0.05 ms becomes a set of equa-
tions:
[ ]
−⋅+−⋅+−⋅−=
−⋅−−⋅⋅+−⋅=
−−−=−
−
)1(
ˆ
376,32)1(
ˆ
9849,0)1(
ˆ
97,441)(
ˆ
)1(
ˆ
0031,0)1(
ˆ
10914,1)1(
ˆ
0893,0)(
ˆ
)1(
ˆ
)1(
03215,0
1
)1(
ˆ
212
2
5
11
1
kxkukuku
kxkukuku
kukykx
(3)
where
)1(
ˆ
−kx
= estimate of input quantity at mo-
ment k-1; y(k-1) = measurement result at moment k-
1; û
1
(k), û
2
(k) = variable at moment k.
The flow diagram of the correction algorithm de-
scribed with (3) and illustrated in Figure 6 suggests
that its correct function depends on correct determi-
nation of the factors of the discrete transducer model
φ and ψ. These are constant for a specific digitisa-
tion time.
delay
y(k-1)
φ
21
1/b
2
ψ
2
φ
11
+
-
+
+
+
x(k-1)
^
u
1(k-1)
^
ψ
1
φ
12
delay
+
+
+
u
1
(k)
^
u
2
(k)
^
u
2
(k-1)
^
φ
22
b
2
=0,03215
φ
11
=0,0893
φ
12
=1,914 10
-5
φ
21
=–441,97
φ
22
=0,9849
ψ
1
=–0,0031
ψ
2
=32,376
.
delay
y(k-1)
φ
21
1/b
2
ψ
2
φ
11
+
-
+
+
+
x(k-1)
^
u
1(k-1)
^
ψ
1
φ
12
delay
+
+
+
u
1
(k)
^
u
2
(k)
^
u
2
(k-1)
^
φ
22
b
2
=0,03215
φ
11
=0,0893
φ
12
=1,914 10
-5
φ
21
=–441,97
φ
22
=0,9849
ψ
1
=–0,0031
ψ
2
=32,376
.
Figure 6. The flow diagram of the correction algorithm of the
measuring accelerometer
In the first step of the algorithm, it is necessary to
adopt an initial value of the variable û
1
(k-1). It was
426
assumed to equal 0. In effect, the algorithm will op-
erate correctly only after a correct variable û
1
(k-1) is
automatically determined. This takes several digiti-
sation steps. Given the accelerometer manufacturers'
recommendations to take measurements for several
dozen seconds to several minutes after the system
start-up, determination of an initial optimum value
of û
1
(k-1) is not necessary
4 CORRECTION RESULTS
Figure 7 shows the waveforms of voltages and the
resultant accelerations from the accelerometers in
the measurement system illustrated in Fig. 4.2. The
measurements were conducted at sinusoidal input of
frequency 200 Hz – which is within the reading
range of both the transducers (according to calibra-
tion cards of their manufacturers) and for which the
input error, according to the magnitude characteristic
curve in Fig. 4.4, becomes the lowest.
In this case, the delay between the waveforms ob-
tained from the measuring (A2) and reference (A1)
accelerometers is 5.3 ms. The acceleration magni-
tude, calculated as the mean value of absolute mag-
nitudes in 2000 measurement samples, equals 7.01
m/s
2
for A1 and 9.05 m/s
2
for A2. This corresponds
to a relative error in measurement of acceleration
magnitude equal to 29.1 %.
Figure 7. Voltage and acceleration waveforms for the reference
and measuring accelerometer
The correction algorithm estimates voltages from
the measurement accelerometer and attempts to ap-
proximate their time waveform to the form obtained
from the reference accelerometer. Correction of
voltage characteristic curves of the measurement
transducer in Figure 7 according to the algorithm
(3), compared to the reference values obtained from
A1, is shown in Figure 8. The delay between the
waveform produced after estimation of the data from
the measurement accelerometer A2 and the wave-
form from the reference accelerometer is 0.05 ms.
This the time of signal sampling. This implies that
the correction algorithm reduced the 5.3 ms delay
between the reference and measurement transducer
to the minimum possible value.
Figure 8. Voltage and acceleration waveforms for the reference
and measuring accelerometer post the correction
The delays resulting from computer mathematical
calculations are negligible. They are not significant
given the current computer capacities and the calcu-
lation simplicity of an algorithm which consists of
five additions and seven multiplications only. An at-
tempted measurement of the time taken for the algo-
rithm calculations in MATLAB environment, Win-
dows XP, the processor Intel Celeron 1.5 Ghz, and
512MB of memory, produced results below 0.1 μs.
Figure 9. Characteristic curves of relative errors of accelera-
tions measured using an accelerometer without and with a cor-
rection
The mean magnitude of estimated acceleration of
A2 for 2000 samples is 7.59 m/s
2
. Compared to the
427
magnitude of the reference accelerometer, the meas-
urement relative error is 8.3 % - which constitutes a
three and half times reduction compared to the same
error without applying the correction algorithm. Fig-
ure 9 illustrates the course of relative error values in
respect of A2 accelerations using and not using the
correction algorithm. The peak values in the charac-
teristic curves result from the reference signal values
approximating zero. Table 1 presents values of the
relative error for an accelerometer employing and
without employing the dynamic correction algorithm
for a selected time range.
Table 1. Relative error values of an accelerometer without and
with the correction
__________________________________________________
Time [s] 0.002 0.0025 0.003 0.0035
__________________________________________________
Relative error -102.13 -98.74 -97,37 -96.37
of accelerometer A2 [%]
Relative error -66.09 -18.61 -2.61 9,03
of accelerometer with
the correction algorithm [%]
__________________________________________________
Time [s] 0.0040 0.0045 0.0050 0.006
__________________________________________________
Relative error -95.40 -93.86 -86.98 -100.08
of accelerometer A2 [%]
Relative error 17.24 32.06 111.86 -35.81
of accelerometer with
the correction algorithm [%]
__________________________________________________
The mean relative errors at the waveform magni-
tudes are: -96.4 % for the signal from the measuring
transducer, and 23.1% for the estimated signal.
Figure 10 presents the characteristic curve of A2's
absolute errors with and without the correction. The
mean absolute error of A2's magnitude, determined
as the absolute mean value of the magnitudes in
2000 samples, is 5.11 m/s
2
. The same error in re-
spect of measurements including the dynamic cor-
rection algorithm diminishes to 1.94 m/s
2
. The great
value of the absolute error, in the case of measure-
ments both with and without the correction, is a re-
sult of the transducer's dynamic properties, i.e. a
phase shift of measurands. The successive measure-
ment values change too fast for the transducer's ca-
pability of reproducing the input magnitude. When
the absolute error is determined, a measurand's
waveform is shifted in relation to the actual value by
a value determined by the transducer's frequency
characteristic curve. The correction reduced the re-
sultant absolute error by more than two and a half
times.
Figure 10. Characteristic curves of acceleration absolute errors
as measured using an accelerometer with and without the cor-
rection
Figure 11 plots the course of the absolute error of
A2's acceleration magnitudes prior to and post the
correction relative to the frequency of the sinusoidal
input signal. At 200 Hz, where parameters of the
correction algorithm were defined, the post-
correction relative error of the acceleration reduces
to a minimum, to rise as it diverges from this value.
The minimum post-correction relative error of the
acceleration is 1.2 %, compared to 4.7 % without the
correction.
Figure 11. Relative error of the acceleration magnitude prior to
and post the correction as dependent on frequency
5 CONCLUSION
Accurate and fast measurements require corrections
to be applied to the measuring track in order to re-
duce the measurement error. The programmatic cor-
rection method proposed by the authors significantly
reduces errors and enables the measurement system
to operate 'on-line'.
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of Technology, Katowice, 2006.
0
5
10
15
20
25
30
120 130 140 150 160 170 180 190 200
Relative error [%]
Frequency [Hz]
A2
A2 po korekcji
with correction
428
Jakubiec J., Roj J.: Pomiarowe przetwarzanie próbkujące
(Measurement sampling processing), Gliwice, Silesian
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