Patent Number: 054596750
Section: description

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS In a method of the invention signals from industrial process sensors can be used to modify or terminate degrading or anomalous processes. The sensor signals are manipulated to provide input data to a statistical analysis technique, such as a process entitled Spectrum Transformed Sequential Testing ("SPRT"). Details of this process and the invention therein are disclosed in U.S. patent application 07/827,776, now U.S. Pat. No. 5,223,207, which is incorporated by reference herein in its entirety. A further illustration of the use of SPRT for analysis of data bases is set forth in the copending application filed contemporaneously, entitled "Processing Data Base Information Having Nonwhite Noise," also incorporated by reference herein in its entirety (Ser. No. 08/068,712). The procedures followed in a preferred method are shown generally in FIG. 8. In performing such a preferred analysis of the sensor signals, a dual transformation method is performed, insofar as it entails both a frequency-domain transformation of the original time-series data and a subsequent time-domain transformation of the resultant data. The data stream that passes through the dual frequency-domain, time-domain transformation is then processed with the SPRT procedure, which uses a log-likelihood ratio test. A computer software appendix is also attached procedure and its implementation in the context of, and modified by, the instant invention. In the preferred embodiment, successive data observations are performed on a discrete process Y, which represents a comparison of the stochastic components of physical processes monitored by a sensor, and most preferably pairs of sensors. In practice, the Y function is obtained by simply differencing the digitized signals from two respective sensors. Let y.sub.k represent a sample from the process Y at time t.sub.k. During normal operation with an undegraded physical system and with sensors that are functioning within specifications the y.sub.k should be normally distributed with mean of zero. Note that if the two signals being compared do not have the same nominal mean values (due, for example, to differences in calibration), then the input signals will be pre-normalized to the same nominal mean values during initial operation. In performing the monitoring of industrial processes, the system's purpose is to declare a first system or a second system degraded if the drift in Y is sufficiently large that the sequence of observations appears to be distributed about a mean +M or -M, where M is our pre-assigned system-disturbance magnitude. We would like to devise a quantitative framework that enables us to decide between two hypotheses, namely: H.sub.1 : Y is drawn from a Gaussian probability distribution function ("PDF") with mean M and variance of .sigma..sup.2. PA1 H.sub.2 : Y is drawn from a Gaussian PDF with mean 0 and variance .sigma..sup.2. We will suppose that if H.sub.1 or H.sub.2 is true, we wish to decide for H.sub.1 or H.sub.2 with probability (1-.beta.) or (1-.alpha.), respectively, where .alpha. and .beta. represent the error (misidentification) probabilities. From the conventional, well known theory of Wald, the test depends on the likelihood ratio 1.sub.n, where ##EQU1## After "n" observations have been made, the sequential probability ratio is just the product of the probability ratios for each step: ##EQU2## where f(y.vertline.H) is the distribution of the random variable y. Wald's theory operates as follows: Continue sampling as long as A&lt;1.sub.n &lt;B. Stop sampling and decide H.sub.1 as soon as 1.sub.n .gtoreq.B, and stop sampling and decide H.sub.2 as soon as 1.sub.n .ltoreq.A. The acceptance thresholds are related to the error (misidentification) probabilities by the following expressions: ##EQU3## The (user specified) value of .alpha. is the probability of accepting H.sub.1 when H.sub.2 is true (false alarm probability). .beta. is the probability of accepting H.sub.2 when H.sub.1 is true (missed alarm probability). If we can assume that the random variable y.sub.k is normally distributed, then the likelihood that H.sub.1 is true (i.e., mean M, variance .sigma..sup.2) is given by: ##EQU4## Similarly for H.sub.2 (mean 0, variance .sigma..sup.2): ##EQU5## The ratio of (5) and (6) gives the likelihood ratio 1.sub.n ##EQU6## Combining (4) and (7), and taking natural logs gives ##EQU7## Our sequential sampling and decision strategy can be concisely represented as: ##EQU8## Following Wald's sequential analysis, it is conventional that a decision test based on the log likelihood ratio has an optimal property; that is, for given probabilities .alpha. and .beta. there is no other procedure with at least as low error probabilities or expected risk and with shorter length average sampling time. A primary limitation that has heretofore precluded the applicability of Wald-type binary hypothesis tests for sensor and equipment surveillance strategies lies in the primary assumption upon which Wald's theory is predicated; i.e., that the original process Y is strictly "white" noise, independently-distributed random data. White noise is thus well known to be a signal which is uncorrelated. Such white noise can, for example, include Gaussian noise. It is, however, very rare to find physical process variables associated with operating machinery that are not contaminated with serially-correlated, deterministic noise components. Serially-correlated noise components are conventionally known to be signal data whose successive time point values are dependent on one another. Noise components includes, for example, auto-correlated (also known as serially correlated) noise and Markov dependent noise. Auto-correlated noise is a known form of noise wherein pairs of correlation coefficients describe the time series correlation of various data signal values along the time series of data. That is, the data U.sub.1, U.sub.2, . . . , U.sub.n have correlation coefficients (U.sub.1, U.sub.2), (U.sub.2, U.sub.3), . . . , (U.sub.n-1, U.sub.n) and likewise have correlation coefficients (U.sub.1, U.sub.3), (U.sub.2, U.sub.4) etc. If these data are auto-correlated, at least some of the coefficients are non-zero. Markov dependent noise on the other hand is a very special form of correlation between past and future data signals. Rather, given the value of U.sub.k, the values of U.sub.n, n&gt;k, do not depend on the values of U.sub.j where j&lt;k. This implies the correlation pairs (U.sub.j, U.sub.n) given the value U.sub.k, are all zero. If, however, the present value is imprecise, then the correlation coefficients may be nonzero. This invention can overcome this limitation to conventional surveillance strategies by integrating the Wald sequential-test approach with a new dual transformation technique. This symbiotic combination of frequency-domain transformations and time-domain transformations produces a tractable solution to a particularly difficult problem that has plagued signal-processing specialists for many years. In the preferred embodiment of the method shown in detail in FIG. 8, serially-correlated data signals from an industrial process can be rendered amenable to the SPRT testing methodology described hereinbefore. This is preferably done by performing a frequency-domain transformation of the original difference function Y. A particularly preferred method of such a frequency transformation is accomplished by generating a Fourier series using a set of highest "1" number of modes. Other procedures for rendering the data amenable to SPRT methods includes, for example, auto regressive techniques, which can accomplish substantially similar results described herein for Fourier analysis. In the preferred approach of Fourier analysis to determine the "1" highest modes (see FIG. 8A): ##EQU9## where a.sub.0 /2 is the mean value of the series, a.sub.m and b.sub.m are the Fourier coefficients corresponding to the Fourier frequency .omega..sub.m, and N is the total number of observations. Using the Fourier coefficients, we next generate a composite function, X.sub.t, using the values of the largest harmonics identified in the Fourier transformation of Y.sub.t. The following numerical approximation to the Fourier transform is useful in determining the Fourier coefficients a.sub.m and b.sub.m. Let x.sub.j be the value of X.sub.t at the jth time increment. Then assuming 2.pi. periodicity and letting .omega..sub.m =2.pi.m/N, the approximation to the Fourier transform yields: ##EQU10## for 0&lt;m&lt;N/2. Furthermore, the power spectral density ("PSD") function for the signal is given by 1.sub.m where ##EQU11## To keep the signal bandwidth as narrow as possible without distorting the PSD, no spectral windows or smoothing are used in our implementation of the frequency-domain transformation. In analysis of a pumping system of the EBR-II reactor of Argonne National Laboratory, the Fourier modes corresponding to the eight highest 1.sub.m provide the amplitudes and frequencies contained in X.sub.t. In our investigations for the particular pumping system data taken, the highest eight 1.sub.m modes were found to give an accurate reconstruction of X.sub.t while reducing most of the serial correlation for the physical variables we have studied. In other industrial processes, the analysis could result in more or fewer modes being needed to accurately construct the functional behavior of a composite curve. Therefore, the number of modes used is a variable which is iterated to minimize the degree of nonwhite noise for any given application. As noted in FIG. 8A a variety of noise tests are applied in order to remove serially correlated noise. The reconstruction of X.sub.t uses the general form of Eqn. (12), where the coefficients and frequencies employed are those associated with the eight highest PSD values. This yields a Fourier composite curve (see end of flowchart in FIG. 8A) with essentially the same correlation structure and the same mean as Y.sub.t. Finally, we generate a discrete residual function R.sub.t by differencing corresponding values of Y.sub.t and X.sub.t. This residual function, which is substantially devoid of serially correlated contamination, is then processed with the SPRT technique described hereinbefore. In a specific example application of the above referenced methodology, certain variables were monitored from the Argonne National Laboratory reactor EBR-II. In particular, EBR-II reactor coolant pumps (RCPs) and delayed neutron (DN) monitoring systems were tested continuously to demonstrate the power and utility of the invention. The RCP and DN systems were chosen for initial application of the approach because SPRT-based techniques have already been under development for both the systems. All data used in this investigation were recorded during full-power, steady state operation at EBR-II. The data have been digitized at a 2-per-second sampling rate using 2.sup.14 (16,384) observations for each signal of interest. FIGS. 1-3 illustrate data associated with the preferred spectral filtering approach as applied to the EBR-II primary pump power signal, which measures the power (in kW) needed to operate the pump. The basic procedure of FIG. 8 was then followed in the analysis. FIG. 1 shows 136 minutes of the original signal as it was digitized at the 2-Hz sampling rate. FIG. 2 shows a Fourier composite constructed from the eight most prominent harmonics identified in the original signal. The residual function, obtained by subtracting the Fourier composite curve from the raw data, is shown in FIG. 3. Periodograms of the raw signal and the residual function have been computed and are plotted in FIG. 4. Note the presence of eight depressions in the periodogram of the residual function in FIG. 4B, corresponding to the most prominent periodicities in the original, unfiltered data. Histograms computed from the raw signal and the residual function are plotted in FIG. 5. For each histogram shown we have superimposed a Gaussian curve (solid line) computed from a purely Gaussian distribution having the same mean and variance. Comparison of FIG. 5A and 5B provide a clear demonstration of the effectiveness of the spectral filtering in reducing asymmetry in the histogram. Quantitatively, this decreased asymmetry is reflected in a decrease in the skewness (or third moment of the noise) from 0.15 (raw signal) to 0.10 (residual function). It should be noted here that selective spectral filtering, which we have designed to reduce the consequences of serial correlation in our sequential testing scheme, does not require that the degree of nonnormality in the data will also be reduced. For many of the signals we have investigated at EBR-II, the reduction in serial correlation is, however, accompanied by a reduction in the absolute value of the skewness for the residual function. To quantitatively evaluate the improvement in whiteness effected by the spectral filtering method, we employ the conventional Fisher Kappa white noise test. For each time series we compute the Fisher Kappa statistic from the defining equation ##EQU12## where 1(.omega..sub.k) is the PSD function (see Eq. 14) at discrete frequencies .omega..sub.k, and 1(L) signifies the largest PSD ordinate identified in the stationary time series. The Kappa statistic is the ratio of the largest PSD ordinate for the signal to the average ordinate for a PSD computed from a signal contaminated with pure white noise. For EBR-II the power signal for the pump used in the present example has a .kappa. of 1940 and 68.7 for the raw signal and the residual function, respectively. Thus, we can say that the spectral filtering procedure has reduced the degree of nonwhiteness in the signal by a factor of 28. Strictly speaking, the residual function is still not a pure white noise process. The 95% critical value for Kappa for a time series with 2.sup.14 observations is 12.6. This means that only for computed Kappa statistics lower than 12.6 could we accept the null hypothesis that the signal is contaminated by pure white noise. The fact that our residual function is not purely white is reasonable on a physical basis because the complex interplay of mechanisms that influence the stochastic components of a physical process would not be expected to have a purely white correlation structure. The important point, however, is that the reduction in nonwhiteness effected by the spectral filtering procedure using only the highest eight harmonics in the raw signal has been found to preserve the pre-specified false alarm and missed alarm probabilities in the SPRT sequential testing procedure (see below). Table I summarizes the computed Fisher Kappa statistics for thirteen EBR-II plant signals that are used in the subject surveillance systems. In every case the table shows a substantial improvement in signal whiteness. The complete SPRT technique integrates the spectral decomposition and filtering process steps described hereinbefore with the known SPRT binary hypothesis procedure. The process can be illustratively demonstrated by application of the SPRT technique to two redundant delayed neutron detectors (designated AND A and DND B) whose signals were archived during long-term normal (i.e., undegraded) operation with a steady DN source in EBR-II. For demonstration purposes a SPRT was designed with a false alarm rate, .alpha., of 0.01. Although this value is higher than we would designate for a production surveillance system, it gives a reasonable frequency of false alarms so that asymptotic values of .alpha. can be obtained with only tens of thousands of discrete observations. According to the theory of the SPRT technique, it can be easily proved that for pure white noise (such as Gaussian), independently distributed processes, .alpha. provides an upper bound to the probability (per observation interval) of obtaining a false alarm--i.e., obtaining a "data disturbance" annunciation when, in fact, the signals under surveillance are undegraded. FIGS. 6 and 7 illustrate sequences of SPRT results for raw DND signals and for spectrally-whitened DND signals, respectively. In FIGS. 6A and 6B, and 7A and 7B, respectively, are shown the DN signals from detectors DND-A and DND-B. The steady-state values of the signals have been normalized to zero. TABLE I ______________________________________ Effectiveness of Spectral Filtering for Measured Plant Signals Fisher Kappa Test Statistic (N = 16,384) Plant Variable I.D. Raw Signal Residual Function ______________________________________ Pump 1 Power 1940 68.7 Pump 2 Power 366 52.2 Pump 1 Speed 181 25.6 Pump 2 Speed 299 30.9 Pump 1 Radial Vibr (top) 123 67.7 Pump 2 Radial Vibr (top) 155 65.4 Pump 1 Radial Vibr (bottom) 1520 290.0 Pump 2 Radial Vibr (bottom) 1694 80.1 DN Monitor A 96 39.4 DN Monitor B 81 44.9 DN Detector 1 86 36.0 DN Detector 2 149 44.1 DN Detector 3 13 8.2 ______________________________________ Normalization to adjust for differences in calibration factor or viewing geometry for redundant sensors does not affect the operability of the SPRT. FIGS. 6C and 7C in each figure show point wise differences of signals DND-A and DND-B. It is this difference function that is input to the SPRT technique. Output from the SPRT method is shown for a 250-second segment in FIGS. 6D and 7D. Interpretation of the SPRT output in FIGS. 6D and 7D is as follows: When the SPRT index reaches a lower threshold, A, one can conclude with a 99% confidence factor that there is no degradation in the sensors. For this demonstration A is equal to 4.60, which corresponds to false-alarm and missed-alarm probabilities of 0.01. As FIGS. 6D and 7D illustrate, each time the SPRT output data reaches A, it is reset to zero and the surveillance continues. If the SPRT index drifts in the positive direction and exceeds a positive threshold, B, of +4.60, then it can be concluded with a 99% confidence factor that there is degradation in at least one of tile sensors. Any triggers of the positive threshold are signified with diamond symbols in FIGS. 6D and 7D. In this case, since we can certify that tile detectors were functioning properly during the time period our signals were being archived, any triggers of the positive threshold are false alarms. If we extend sufficiently the surveillance experiment illustrated in FIG. 6D, we can get an asymptotic estimate of the false alarm probability .alpha.. We have performed this exercise using 1000-observation windows, tracking the frequency of false alarm trips in each window, then repeating the procedure for a total of sixteen independent windows to get an estimate of the variance on this procedure for evaluating .alpha.. The resulting false-alarm frequency for the raw, unfiltered, signals is .alpha.=0.07330 with a variance of 0.000075. The very small variance shows that there would be only a negligible improvement in our estimate by extending the experiment to longer data streams. This value of .alpha. is significantly higher than the design value of .alpha.=0.01, and illustrates the danger of blindly applying a SPRT test technique to signals that may be contaminated by excessive serial correlation. The data output shown in FIG. 7D employs the complete SPRT technique shown schematically in FIG. 8. When we repeat the foregoing exercise using 16 independent 1000-observation windows, we obtain an asymptotic cumulative false-alarm frequency of 0.009142 with a variance of 0.000036. This is less than (i.e., more conservative than) the design value of .alpha.=0.01, as desired. It will be recalled from the description hereinbefore regarding one preferred embodiment, we have used the eight most prominent harmonics in the spectral filtration stage of the SPRT technique. By repeating the foregoing empirical procedure for evaluating the asymptotic values of .alpha., we have found that eight modes are sufficient for the input variables shown in Table I. Furthermore, by simulating subtle degradation in individual signals, we have found that the presence of serial correlation in raw signals gives rise to excessive missed-alarm probabilities as well. In this case spectral whitening is equally effective in ensuring that pre-specified missed-alarm probabilities are not exceeded using the SPRT technique. In another different form of the invention, it is not necessary to have two sensor signals to form a difference function. One sensor can provide a real signal characteristic of an ongoing process and a record artificial signal can be generated to allow formation of a difference function. Techniques such as an auto regressive moving average (ARMA) methodology can be used to provide the appropriate signal, such as a DC level signal, a cyclic signal or other predictable signal. Such an ARMA method is a well-known procedure for generating artificial signal values, and this method can even be used to learn the particular cyclic nature of a process being monitored enabling construction of the artificial signal. The two signals, one a real sensor signal and the other an artificial signal, can thus be used in the same manner as described hereinbefore for two real sensor signals. The difference function Y is then formed, transformations performed and a residual function is determined which is free of serially correlated noise. Fourier techniques are very effective in achieving a whitened signal for analysis, but there are other means to achieve substantially the same results using a different analytical methodology. For example, filtration of serial correlation can be accomplished by using the autoregressive moving average (ARMA) method. This ARMA technique estimates the specific correlation structure existing between sensor points of an industrial process and utilizes this correlation estimate to effectively filter the data sample being evaluated. A technique has therefore been devised which integrates frequency-domain filtering with sequential testing methodology to provide a solution to a problem that is endemic to industrial signal surveillance. The subject invention particularly allows sensing slow degradation that evolves over a long time period (gradual decalibration bias in a sensor, appearance of a new radiation source in the presence of a noisy background signal, wear out or buildup of a radial rub in rotating machinery, etc. ). The system thus can alert the operator of the incipience or onset of the disturbance long before it would be apparent to visual inspection of strip chart or CRT signal traces, and well before conventional threshold limit checks would be tripped. This permits the operator to terminate, modify or avoid events that might otherwise challenge technical specification guidelines or availability goals. Thus, in many cases the operator can schedule corrective actions (sensor replacement or recalibration; component adjustment, alignment, or rebalancing; etc.) to be performed during a scheduled system outage. Another important feature of the technique which distinguishes it from conventional methods is the built-in quantitative false-alarm and missed-alarm probabilities. This is quite important in the context of high-risk industrial processes and applications. The invention makes it possible to apply formal reliability analysis methods to an overall system comprising a network of interacting SPRT modules that are simultaneously monitoring a variety of plan variables. This amenability to formal reliability analysis methodology will, for example, greatly enhance the process of granting approval for nuclear-plant applications of the invention, a system that can potentially save a utility millions of dollars per year per reactor. While preferred embodiments of the invention have been shown and described, it will be clear to those skilled in the art that various changes and modifications can be made without departing from the invention in its broader aspects as set forth in the claims provided hereinafter.