- Email: [email protected]
A fuzzy logic approach for noise signal reconstruction: Application to pressure sensors
ARTICLE IN PRESS
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 19 (2005) 501–508 www.elsevier.com/locate/jnlabr/ymssp
Correspondence
A fuzzy logic approach for noise signal reconstruction: Application to pressure sensors
1. Introduction Surveillance of pressure sensors in nuclear power plants is specified by the technical requirements of the operation [1]. The sensor response time is the quantity determined for assuring a proper behaviour of the transmitters [2]. In order to calculate the response time, noise analysis techniques are preferred to deterministic methods nowadays [3]. Surveillance of about a 100 sensors is periodically performed without perturbing the normal operation of the plant [4], but clearly perturbing the maintenance team. In order to minimise such a perturbation, a pressure sensor surveillance campaign must be planned and some decisions are taken in advance; that is the case of the noise sampling time, the length of the signal record, the signal preconditioning, etc., and of course, one must make sure that the software for the noise analysis is verified. Testing the software is best performed with signals taken from the previous campaign. In many cases, these signals are not available. To do without, a signal reconstruction from a model of the sensor dynamics is a good option, although the coefficients of the corresponding stochastic equation are not accurately known and neither is the driven noise. Fortunately, a plot of the power spectral density (PSD) of the old signals is usually available; accordingly, the noise signal can be reconstructed from its PSD for the software validation purpose. In general terms, a signal cannot be reconstructed from its PSD because the phase information is missing. Although, due to its stochastics character, reconstruction of a noise signal does not mean an ordered reconstruction point by point, but estimating a realisation with the same statistical descriptors. This can be achieved using noise analysis techniques in the time domain, that is, looking for an autoregressive (AR) model driven by a white noise. In order to obtain the PSD, the direct procedure is made following the well-known Box and Jenkins methodology [5,6]: given a noise signal, recorded with a selected sampling time, the autocorrelation function is previously calculated, and from that, by solving the Yule–Walker equations, the AR coefficients. Finally, the PSD is estimated from the AR coefficients. The reverse problem is about estimating a noise signal realisation from the PSD. In that case, the Akaike criterium [7] cannot be used because the sampling time is lacking, so a new criterium is required. Such a criterium should be based on the Information theory. Here, in this work, a criterium for finding the appropriated sampling time is derived. It is based on a quantity also related with the Information theory, the Luca–Termini Fuzzy Entropy [8,9]. 0888-3270/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2004.02.005
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
502
The procedure is illustrated only with a PSD from Rosemount pressure transmitter [3], widely used in actual nuclear power plants, although not showed here, it can be extended to other transmitters or temperature sensors. 2. Noise generation from a PSD Departing from a pressure sensor PSD plot, a noise signal is to be reconstructed having the dynamical features of the sensor and the sensing line. To begin with, the plot is scanned and digitalised choosing a few meaningful points of the PSD. The whole PSD is built by using thirdorder osculating polynomials interpolation. The result is compared with the original PSD plot in order to assure the quality of the digitalisation process. Following, the autocorrelation function is obtained by calculating the inverse Fourier transform of the PSD. The lag time, Dt; is chosen from the Nyquist frequency; therefore, an array Cj ¼ CðjDtÞ is an estimation of the signal autocorrelation function. The variance of the signal, s2 ; can be taken as C0 : In order to generate a corresponding artificial noise, an autoregressive model of order n is fitted from the n þ 1 first values of the autocorrelation. Explicitly, n X ak xjk þ nj ; j ¼ 0; 1; :::; N; ð1Þ xj ¼ k¼1
where xj is the wanted noise signal, nj the driven white noise, ak the AR coefficients, and N the sample size. The coefficients can be calculated solving the Yule–Walker equations, which only require the n þ 1 first values of the autocorrelation [6]. The residual noise variance s2n is estimated from the autocorrelation and the ak coefficients: n X 2 2 a k Ck : ð2Þ sn ¼ s k¼1
It should be remarked that the AR coefficients depend on the sampling time and the model order. In ordinary noise analysis the sampling time is fixed within the recording of the signal, but here, the sampling time can be chosen freely because the autocorrelation is estimated from the inverse Fourier transform of the PSD. So as to remark, ARðn; mDtÞ will denote an AR model of order n and sampling time mDt; being m a natural number. In other words, the procedure for generating noise does not yield a single AR model but a matrix of AR models. A row and a column, i.e. a unique AR model, should be chosen for generating a noise with the same statistical descriptors than the original noise, the corresponding to the original PSD. Once the AR model is chosen, the corresponding PSD is [6]: s2n mDt ð3Þ PSDnm ðf Þ ¼ P 1 n ak expðj2pkfmDtÞ2 k¼1
being the frequency f ofNyquist : This PSD should be compared with the original plot in order to accept the AR model. But for choosing the AR model a criterium is required; unfortunately the Akaike criterium is not applicable because it needs a previous sampling time, so a similar criterium but based on fuzzy logic entropy is sought out.
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
503
3. The selection of the AR model Given the matrix ARðn; mÞ of order n and sampling time Dt ¼ mDtNyq ; we start by fixing Dt and seeking a criterium for selecting an order among the column of AR available models. Following Akaike [7], the residual noise variance s2n ðnÞ is the key quantity. For the purpose, the quantity B is defined as B¼
nDt : ðs=sn Þ2
ð4Þ
Numerator of B is the time length used in the autocorrelation function needed for finding the coefficients of the AR model. We call M ¼ nDt ‘‘Memory’’. On the other hand, the denominator is a kind of ‘‘noise transfer function’’ given by Eq. (2) and depending on the order of the AR model. The transfer function can be regarded as an amplification; in this case, the maximum amplification corresponds to a minimum residual noise variance and consequently to a good AR model. Plotting B against M; for all AR models of the fixed column, the minimum value indicates the order n of the AR model in such a column. However, there is a column for each sampling time, and hence the above criterium is not sufficient to estimate Dt: The qualitative shape of the plot is shown in Fig. 1(a). Next, let us call mDt Bmin =B; see Fig. 1(b). It accomplishes that 0pmp1 for all M; we regard it as a pertenence function [8] for the fuzzy sentence: ‘‘the memory yields a suitable AR model for fitting the noise signal’’. Low memory yields a poor fit; high memory requires many parameters, so according to the parsimony principle there is an optimum M value, corresponding to an order of the AR model. Just the same information as the B plot, but now m is seen as a pertenence function for each sampling time. We shall take advantage of it. To each column of the ARðn; mÞ matrix corresponds a pertenence function m; those functions are alike but with different wideness, see Fig. 1(c). A narrow pertenence function means that only a few set of memory values are suitable for AR representation of the noise; on the other hand, a broad pertenence function means that almost any value of the memory is good enough for the AR representation. Therefore, we can measure the wideness of the pertenence functions. The sampling time is selected by accepting the broadest pertenence function, and the AR order, by choosing the value of the memory corresponding to the maximum of the accepted m:
1
1
∆
B
∆
Broad Narrow
B min 0
(a)
Memory
(b)
0 Memory
(c)
Memory
Fig. 1. Qualitative shapes: (a) the B quantity versus the memory, (b) the pertenence function mDt versus the memory, and (c) several wideness of the pertenence function mDt :
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
504
Wideness of m functions can be assigned calculating the standard deviation as if the m functions were probability distributions. Nevertheless, in order to give some meaning to the criterium, the fuzzy entropy is taken as a measure of its wideness. Departing from the Shannon function [8]: SðmÞ ¼ m ln m þ ð1 mÞln ð1 mÞ:
ð5Þ
Now, given a m function a Shannon value can be calculated for each M: The maximum value of S is ln 2 and corresponds to m ¼ 12; this point stands for the highest ambiguity in the sense that one cannot establish that the AR model order is adequate for fitting the noise. The mean value of all these values is called the Luca–Termini fuzzy entropy [8]: Mx X 1 Si ; ð6Þ g¼ Mx ln 2 i¼1 where Mx is the maximum value of the memory for a given pertenence function m: The criterium for m selection is now: ‘‘among the m functions the selected one corresponds to the maximum fuzzy entropy’’.
4. Results Methodology was tested with the well-known second-order linear system driven by noise. Then, it was applied to a PSD plot, see Fig. 2, corresponding to a Rosemount pressure transmitter, very common in Nuclear Power Plants. The PSD was taken from the literature [10]. The PSD plot was scanned (70 points) and reconstructed by interpolation using third order osculating polynomials (10 000 points). The inverse Fourier transform with a Hanning window was applied to obtain the autocovariance function, see Fig. 3. Other ordinary windows have been tested with similar results. The minimum lag time DtNyq ¼ 2:5 ms. A matrix of AR(14,52) models is calculated solving the Yule–Walker equations using the autocovariance values. In Fig. 4 it is shown the B quantity for Dt ¼ 12:5 ms. According to Fig. 1a,
-5
PSD (dB)
-15
-25
-35
-45
-55 0.1
1
10
10 0
Frequency (Hz)
Fig. 2. Power spectrum density (PSD) for a noise signal taken from the literature. The smooth fitting line corresponds to an autoregressive (AR) model.
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
505
1
Autocovariance
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Fig. 3. Autocovariance function versus the time. Only 560 points are plotted and the minimum lag time is 2.5 ms.
0.01 4
0.01 2
B (s)
0.01
0.00 8
0.00 6
0.00 4 0
0.04
0. 08
0.12
0.16
0.2
Memory (s)
Fig. 4. The B quantity versus the memory for Dt ¼ 12:5 ms. The first value is out of the expected shape.
observe that for the first-order B is out of the expected shape because such an order scarcely represents a PSD with a resonance. It is clear that the third order is the best AR fit for such a sampling time. The pertenence function yields the same order. See Fig. 5. For every sampling time there is a m function and consequently a value for the Luca–Termini fuzzy entropy can be calculated. The result is shown in Fig. 6. In this case, the maximum entropy resulted for Dt ¼ 12:5 ms, and the best AR order was n ¼ 3: With the corresponding AR coefficients the PSD3,5 was calculated and compared with the original. See Fig. 7. Once the order and the sampling time are selected, artificial noise is generated using Eqs. (1) and (2).
ARTICLE IN PRESS 506
Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508 1
Pertenence Function
0.8
0.6
0.4
0.2 0
0.04
0.08
0.12
0.16
0.2
Memory (s)
Fig. 5. The pertenece function mDt versus the memory for Dt ¼ 12:5 ms. The first value is out of the expected shape and the maximum corresponds to the third order, for memory equals 37.5 ms.
0. 9 Maximum ∆t = 0.0125s
Entropy (s-1)
0. 8
0. 7
0. 6
0. 5
0
0.04
0.08
0.12
0.16
Sampling time (s)
Fig. 6. Luca–Termini fuzzy entropy versus the sampling time. Fifty two points are plotted, being the minimum lag time 2.5 ms. The maximum corresponds to the fifth value.
5. Conclusions The solved problem is very specific: simulating artificial noise signal from a given PSD plot. Noise simulation is achieved using AR modelling, but it requires the proper sampling time and the best order of the model. For such a purpose, fuzzy logic criterium has been developed. A pertenence function is designed for each sampling time; the order is selected from the maximum of the function. For each pertenence function the Luca–Termini fuzzy entropy is calculated; the one with the highest entropy is selected.
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
507
0
ARmodel -1 0
PSD (dB)
Original -2 0
-3 0
-4 0
-5 0
0.01
0.1
1
10
100
Frequency (Hz)
Fig. 7. Power spectral density PSD3,5 from an autoregressive AR(3) model whose sampling time equals 5DtNyq : The corresponding original PSD is plotted too.
Applications are designed for the case of Rosemount pressure transmitters. The technique can be applied to any PSD that can be calculated within AR model; it excludes the non-linear noise signals. Besides, the B quantity can be an alternative to the traditional Akaike’s AIC for selecting an AR model, because it fully accomplishes with the parsimony principle.
Acknowledgements We thank Dr P. Vilarroig for critically reading the manuscript and making several useful remarks.
References [1] G.J. Toman, 1986, Inspection surveillance and monitoring of electrical equipment in nuclear power plants. US Nuclear Regulatory Commission, NUREG/CR—4257, Vol. 2. [2] J. Bl!azquez, J. Ballestr!ın, Pressure transmitters surveillance: the dominant real pole, Progress in Nuclear Energy 29 (3) (1995) 139–145. [3] H.M. Hashemian, D.W. Mitchell, R.E. Fain, et al., Long term performance and aging characteristics of nuclear power plant pressure transmitters. US Nuclear Regulatory Commission, NUREG/CR—5851, 1993, pp. 246–252. [4] J. Barbero, J. Bl!azquez, O. Vela, Bubbles in the sensing line of nuclear power plants pressure transmitters: the shift of spectrum resonances, Nuclear Engineering & Design 19 (2000) 327–334. [5] G.E.P. Box, G.M. Jenkins, Time Series Analysis Forecasting and Control, Holder Day, San Francisco, 1970. [6] M.B. Priestly, Spectral Analysis and Time Series, Academic Press Ltd., London, 1981 (Chapter 5). [7] H. Akaike, A new look at the statistical model identification. IEEE Transactions on Automatic Control, AC-19, 1974, pp. 716–723. [8] A. Kaufmann, J. Gil Aluja, Las matem!aticas del azar y de la incertidumbre, Centro de estudios Ramon Areces, Madrid, 1990 (Chapter 8).
ARTICLE IN PRESS 508
Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
[9] A De Luca, S. Termini, A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory, Instrumentation and Control 20 (1972) 301–312. [10] J.M. Chicharro, A. Garc!ıa-Berrocal, J. Bla! zquez, M. Balba! s, Pressure transmitter surveillance using quaternion numbers, Mechanical Systems & Signal Processing 16 (6) (2002) 1083–1091.
A. Garc!ıa-Berrocal, M. Balba! s Applied Physics to Natural Resources Department, ETSIM, Polytechnic University of Madrid, Spain E-mail address: [email protected] J.M. Chicharro Applied Mechanics and Projects Engineering Department, ETSII, University of Castilla-La Mancha, Spain J. Bla! zquez Nuclear Fission Department, CIEMAT, Madrid, Spain
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 19 (2005) 501–508 www.elsevier.com/locate/jnlabr/ymssp
Correspondence
A fuzzy logic approach for noise signal reconstruction: Application to pressure sensors
1. Introduction Surveillance of pressure sensors in nuclear power plants is specified by the technical requirements of the operation [1]. The sensor response time is the quantity determined for assuring a proper behaviour of the transmitters [2]. In order to calculate the response time, noise analysis techniques are preferred to deterministic methods nowadays [3]. Surveillance of about a 100 sensors is periodically performed without perturbing the normal operation of the plant [4], but clearly perturbing the maintenance team. In order to minimise such a perturbation, a pressure sensor surveillance campaign must be planned and some decisions are taken in advance; that is the case of the noise sampling time, the length of the signal record, the signal preconditioning, etc., and of course, one must make sure that the software for the noise analysis is verified. Testing the software is best performed with signals taken from the previous campaign. In many cases, these signals are not available. To do without, a signal reconstruction from a model of the sensor dynamics is a good option, although the coefficients of the corresponding stochastic equation are not accurately known and neither is the driven noise. Fortunately, a plot of the power spectral density (PSD) of the old signals is usually available; accordingly, the noise signal can be reconstructed from its PSD for the software validation purpose. In general terms, a signal cannot be reconstructed from its PSD because the phase information is missing. Although, due to its stochastics character, reconstruction of a noise signal does not mean an ordered reconstruction point by point, but estimating a realisation with the same statistical descriptors. This can be achieved using noise analysis techniques in the time domain, that is, looking for an autoregressive (AR) model driven by a white noise. In order to obtain the PSD, the direct procedure is made following the well-known Box and Jenkins methodology [5,6]: given a noise signal, recorded with a selected sampling time, the autocorrelation function is previously calculated, and from that, by solving the Yule–Walker equations, the AR coefficients. Finally, the PSD is estimated from the AR coefficients. The reverse problem is about estimating a noise signal realisation from the PSD. In that case, the Akaike criterium [7] cannot be used because the sampling time is lacking, so a new criterium is required. Such a criterium should be based on the Information theory. Here, in this work, a criterium for finding the appropriated sampling time is derived. It is based on a quantity also related with the Information theory, the Luca–Termini Fuzzy Entropy [8,9]. 0888-3270/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2004.02.005
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
502
The procedure is illustrated only with a PSD from Rosemount pressure transmitter [3], widely used in actual nuclear power plants, although not showed here, it can be extended to other transmitters or temperature sensors. 2. Noise generation from a PSD Departing from a pressure sensor PSD plot, a noise signal is to be reconstructed having the dynamical features of the sensor and the sensing line. To begin with, the plot is scanned and digitalised choosing a few meaningful points of the PSD. The whole PSD is built by using thirdorder osculating polynomials interpolation. The result is compared with the original PSD plot in order to assure the quality of the digitalisation process. Following, the autocorrelation function is obtained by calculating the inverse Fourier transform of the PSD. The lag time, Dt; is chosen from the Nyquist frequency; therefore, an array Cj ¼ CðjDtÞ is an estimation of the signal autocorrelation function. The variance of the signal, s2 ; can be taken as C0 : In order to generate a corresponding artificial noise, an autoregressive model of order n is fitted from the n þ 1 first values of the autocorrelation. Explicitly, n X ak xjk þ nj ; j ¼ 0; 1; :::; N; ð1Þ xj ¼ k¼1
where xj is the wanted noise signal, nj the driven white noise, ak the AR coefficients, and N the sample size. The coefficients can be calculated solving the Yule–Walker equations, which only require the n þ 1 first values of the autocorrelation [6]. The residual noise variance s2n is estimated from the autocorrelation and the ak coefficients: n X 2 2 a k Ck : ð2Þ sn ¼ s k¼1
It should be remarked that the AR coefficients depend on the sampling time and the model order. In ordinary noise analysis the sampling time is fixed within the recording of the signal, but here, the sampling time can be chosen freely because the autocorrelation is estimated from the inverse Fourier transform of the PSD. So as to remark, ARðn; mDtÞ will denote an AR model of order n and sampling time mDt; being m a natural number. In other words, the procedure for generating noise does not yield a single AR model but a matrix of AR models. A row and a column, i.e. a unique AR model, should be chosen for generating a noise with the same statistical descriptors than the original noise, the corresponding to the original PSD. Once the AR model is chosen, the corresponding PSD is [6]: s2n mDt ð3Þ PSDnm ðf Þ ¼ P 1 n ak expðj2pkfmDtÞ2 k¼1
being the frequency f ofNyquist : This PSD should be compared with the original plot in order to accept the AR model. But for choosing the AR model a criterium is required; unfortunately the Akaike criterium is not applicable because it needs a previous sampling time, so a similar criterium but based on fuzzy logic entropy is sought out.
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
503
3. The selection of the AR model Given the matrix ARðn; mÞ of order n and sampling time Dt ¼ mDtNyq ; we start by fixing Dt and seeking a criterium for selecting an order among the column of AR available models. Following Akaike [7], the residual noise variance s2n ðnÞ is the key quantity. For the purpose, the quantity B is defined as B¼
nDt : ðs=sn Þ2
ð4Þ
Numerator of B is the time length used in the autocorrelation function needed for finding the coefficients of the AR model. We call M ¼ nDt ‘‘Memory’’. On the other hand, the denominator is a kind of ‘‘noise transfer function’’ given by Eq. (2) and depending on the order of the AR model. The transfer function can be regarded as an amplification; in this case, the maximum amplification corresponds to a minimum residual noise variance and consequently to a good AR model. Plotting B against M; for all AR models of the fixed column, the minimum value indicates the order n of the AR model in such a column. However, there is a column for each sampling time, and hence the above criterium is not sufficient to estimate Dt: The qualitative shape of the plot is shown in Fig. 1(a). Next, let us call mDt Bmin =B; see Fig. 1(b). It accomplishes that 0pmp1 for all M; we regard it as a pertenence function [8] for the fuzzy sentence: ‘‘the memory yields a suitable AR model for fitting the noise signal’’. Low memory yields a poor fit; high memory requires many parameters, so according to the parsimony principle there is an optimum M value, corresponding to an order of the AR model. Just the same information as the B plot, but now m is seen as a pertenence function for each sampling time. We shall take advantage of it. To each column of the ARðn; mÞ matrix corresponds a pertenence function m; those functions are alike but with different wideness, see Fig. 1(c). A narrow pertenence function means that only a few set of memory values are suitable for AR representation of the noise; on the other hand, a broad pertenence function means that almost any value of the memory is good enough for the AR representation. Therefore, we can measure the wideness of the pertenence functions. The sampling time is selected by accepting the broadest pertenence function, and the AR order, by choosing the value of the memory corresponding to the maximum of the accepted m:
1
1
∆
B
∆
Broad Narrow
B min 0
(a)
Memory
(b)
0 Memory
(c)
Memory
Fig. 1. Qualitative shapes: (a) the B quantity versus the memory, (b) the pertenence function mDt versus the memory, and (c) several wideness of the pertenence function mDt :
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
504
Wideness of m functions can be assigned calculating the standard deviation as if the m functions were probability distributions. Nevertheless, in order to give some meaning to the criterium, the fuzzy entropy is taken as a measure of its wideness. Departing from the Shannon function [8]: SðmÞ ¼ m ln m þ ð1 mÞln ð1 mÞ:
ð5Þ
Now, given a m function a Shannon value can be calculated for each M: The maximum value of S is ln 2 and corresponds to m ¼ 12; this point stands for the highest ambiguity in the sense that one cannot establish that the AR model order is adequate for fitting the noise. The mean value of all these values is called the Luca–Termini fuzzy entropy [8]: Mx X 1 Si ; ð6Þ g¼ Mx ln 2 i¼1 where Mx is the maximum value of the memory for a given pertenence function m: The criterium for m selection is now: ‘‘among the m functions the selected one corresponds to the maximum fuzzy entropy’’.
4. Results Methodology was tested with the well-known second-order linear system driven by noise. Then, it was applied to a PSD plot, see Fig. 2, corresponding to a Rosemount pressure transmitter, very common in Nuclear Power Plants. The PSD was taken from the literature [10]. The PSD plot was scanned (70 points) and reconstructed by interpolation using third order osculating polynomials (10 000 points). The inverse Fourier transform with a Hanning window was applied to obtain the autocovariance function, see Fig. 3. Other ordinary windows have been tested with similar results. The minimum lag time DtNyq ¼ 2:5 ms. A matrix of AR(14,52) models is calculated solving the Yule–Walker equations using the autocovariance values. In Fig. 4 it is shown the B quantity for Dt ¼ 12:5 ms. According to Fig. 1a,
-5
PSD (dB)
-15
-25
-35
-45
-55 0.1
1
10
10 0
Frequency (Hz)
Fig. 2. Power spectrum density (PSD) for a noise signal taken from the literature. The smooth fitting line corresponds to an autoregressive (AR) model.
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
505
1
Autocovariance
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Fig. 3. Autocovariance function versus the time. Only 560 points are plotted and the minimum lag time is 2.5 ms.
0.01 4
0.01 2
B (s)
0.01
0.00 8
0.00 6
0.00 4 0
0.04
0. 08
0.12
0.16
0.2
Memory (s)
Fig. 4. The B quantity versus the memory for Dt ¼ 12:5 ms. The first value is out of the expected shape.
observe that for the first-order B is out of the expected shape because such an order scarcely represents a PSD with a resonance. It is clear that the third order is the best AR fit for such a sampling time. The pertenence function yields the same order. See Fig. 5. For every sampling time there is a m function and consequently a value for the Luca–Termini fuzzy entropy can be calculated. The result is shown in Fig. 6. In this case, the maximum entropy resulted for Dt ¼ 12:5 ms, and the best AR order was n ¼ 3: With the corresponding AR coefficients the PSD3,5 was calculated and compared with the original. See Fig. 7. Once the order and the sampling time are selected, artificial noise is generated using Eqs. (1) and (2).
ARTICLE IN PRESS 506
Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508 1
Pertenence Function
0.8
0.6
0.4
0.2 0
0.04
0.08
0.12
0.16
0.2
Memory (s)
Fig. 5. The pertenece function mDt versus the memory for Dt ¼ 12:5 ms. The first value is out of the expected shape and the maximum corresponds to the third order, for memory equals 37.5 ms.
0. 9 Maximum ∆t = 0.0125s
Entropy (s-1)
0. 8
0. 7
0. 6
0. 5
0
0.04
0.08
0.12
0.16
Sampling time (s)
Fig. 6. Luca–Termini fuzzy entropy versus the sampling time. Fifty two points are plotted, being the minimum lag time 2.5 ms. The maximum corresponds to the fifth value.
5. Conclusions The solved problem is very specific: simulating artificial noise signal from a given PSD plot. Noise simulation is achieved using AR modelling, but it requires the proper sampling time and the best order of the model. For such a purpose, fuzzy logic criterium has been developed. A pertenence function is designed for each sampling time; the order is selected from the maximum of the function. For each pertenence function the Luca–Termini fuzzy entropy is calculated; the one with the highest entropy is selected.
ARTICLE IN PRESS Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
507
0
ARmodel -1 0
PSD (dB)
Original -2 0
-3 0
-4 0
-5 0
0.01
0.1
1
10
100
Frequency (Hz)
Fig. 7. Power spectral density PSD3,5 from an autoregressive AR(3) model whose sampling time equals 5DtNyq : The corresponding original PSD is plotted too.
Applications are designed for the case of Rosemount pressure transmitters. The technique can be applied to any PSD that can be calculated within AR model; it excludes the non-linear noise signals. Besides, the B quantity can be an alternative to the traditional Akaike’s AIC for selecting an AR model, because it fully accomplishes with the parsimony principle.
Acknowledgements We thank Dr P. Vilarroig for critically reading the manuscript and making several useful remarks.
References [1] G.J. Toman, 1986, Inspection surveillance and monitoring of electrical equipment in nuclear power plants. US Nuclear Regulatory Commission, NUREG/CR—4257, Vol. 2. [2] J. Bl!azquez, J. Ballestr!ın, Pressure transmitters surveillance: the dominant real pole, Progress in Nuclear Energy 29 (3) (1995) 139–145. [3] H.M. Hashemian, D.W. Mitchell, R.E. Fain, et al., Long term performance and aging characteristics of nuclear power plant pressure transmitters. US Nuclear Regulatory Commission, NUREG/CR—5851, 1993, pp. 246–252. [4] J. Barbero, J. Bl!azquez, O. Vela, Bubbles in the sensing line of nuclear power plants pressure transmitters: the shift of spectrum resonances, Nuclear Engineering & Design 19 (2000) 327–334. [5] G.E.P. Box, G.M. Jenkins, Time Series Analysis Forecasting and Control, Holder Day, San Francisco, 1970. [6] M.B. Priestly, Spectral Analysis and Time Series, Academic Press Ltd., London, 1981 (Chapter 5). [7] H. Akaike, A new look at the statistical model identification. IEEE Transactions on Automatic Control, AC-19, 1974, pp. 716–723. [8] A. Kaufmann, J. Gil Aluja, Las matem!aticas del azar y de la incertidumbre, Centro de estudios Ramon Areces, Madrid, 1990 (Chapter 8).
ARTICLE IN PRESS 508
Correspondence / Mechanical Systems and Signal Processing 19 (2005) 501–508
[9] A De Luca, S. Termini, A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory, Instrumentation and Control 20 (1972) 301–312. [10] J.M. Chicharro, A. Garc!ıa-Berrocal, J. Bla! zquez, M. Balba! s, Pressure transmitter surveillance using quaternion numbers, Mechanical Systems & Signal Processing 16 (6) (2002) 1083–1091.
A. Garc!ıa-Berrocal, M. Balba! s Applied Physics to Natural Resources Department, ETSIM, Polytechnic University of Madrid, Spain E-mail address: [email protected] J.M. Chicharro Applied Mechanics and Projects Engineering Department, ETSII, University of Castilla-La Mancha, Spain J. Bla! zquez Nuclear Fission Department, CIEMAT, Madrid, Spain