Home > Finding the Source of Nonlinearity in a Process With Plant-Wide Oscillation

Finding the Source of Nonlinearity in a Process With Plant-Wide Oscillation

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Finding the Source of Nonlinearity in a Process With Plant-Wide Oscillation
Nina F. Thornhill
Abstract—A plant-wide oscillation in a chemical process often has an impact on product quality and running costs and there is, thus, a motivation for automated diagnosis of the source of such a disturbance. This brief describes a method of analyzing data from routine operation to locate the root cause oscillation in a dynamic system of interacting control loops and to distinguish it from prop- agated secondary oscillations. The novel concept is the application of a nonlinearity index that is strongest at the source. The index is large for the nonsinusoidal oscillating time trends that are typical of the output of a control loop with a limit cycle caused by nonlin- earity. It is sensitive to limit cycles caused both by equipment and by process nonlinearity. The performance of the index is studied in detail and default settings for the parameters in the algorithm are derived so that it can be applied in a large scale setting such as a refinery or petrochemical plant. Issues arising from artifacts in the nonlinearity test when applied to strongly cyclic data have been addressed to provide a robust, reliable and practical method. The technique is demonstrated with three industrial case studies. Index Terms—Fault diagnosis, harmonics, limit cycles, nonlin- earities, spectral analysis, surrogate data, time series.
IT IS important to diagnose and cure oscillations in a con-
trolled process because a system running steadily without oscillation is more profitable and safer [1]. A feedback control loop containing a nonlinearity such as a sticking valve often ex- hibits self-generated and self-sustained limit cycle oscillation [2]–[4], and many surveys have shown that these oscillations are a significant industrial problem [5], [6]. The situation is made worse when the oscillation propagates throughout a dy- namic system such as a chemical plant where it can become widespread due to physical coupling and recycles. Rapid de- termination of the source of a system-wide oscillation allows maintenance activity to focus on the root cause [7]. This article presents a method aimed at that objective and presents three in- dustrial case studies in which the method successfully found the root cause. The time trend of measurements from a limit cycle oscilla- tion is a nonlinear time series, i.e., it cannot be described as the output of a linear system driven by white noise. A nonlinearity test from Kantz and Schreiber [8] has been adapted for the de- tection of limit cycle oscillations and guidelines for its applica- tion to process data have been devised. The underpinning idea in root cause diagnosis is that the nonlinearity is greatest at the
Manuscript received February 18, 2004. Manuscript received in final form June 14, 2004. Recommended by Associate Editor A. T. Vemuri. This work was supported in part by the Royal Academy of Engineering under the Foresight Award and in part by the NSERC-Matrikon-ASRA Industrial Research Chair in Process Control at the University of Alberta. The author is with the Imperial College/UCL Center for Process Systems Engineering, Department of Electronic and Electrical Engineering, University College London, London WC1E 7JE, U.K. (e-mail: n.thornhill@ee.ucl.ac.uk). Digital Object Identifier 10.1109/TCST.2004.839570
source of the problem. By source is meant a measurement as- sociated with the single-input–single-output controller that has been caused to oscillate by a nonlinearity in the loop. Justifica- tion of that assumption is given in the next section. The possibility of a nonlinearity test was outlined in an earlier conference publication [9] and demonstrated in an application at Eastman Chemical Company [10]. The key advance in this brief compared with [9] is the exploitation of the cyclic nature of the measurements to optimize the method. In [10], the focus was on the solution to a particular industrial application and a set of parameters was selected for the algorithm but without a discussion of their optimality. A contribution of this brief is to give fundamental insights into the method and to extend it to additional case studies involving the detection of valve, sensor and process nonlinearities. It gives an in-depth explanation of how the method works when applied to oscillating disturbances and the criteria by which the parameters of the algorithm may be selected. Default parameters are suggested to facilitate routine application of the method to a large-scale plant. II. DIAGNOSIS OF NONLINEARITY A. Nonlinear Time Series Analysis The waveform in a limit cycle is periodic but nonsinusoidal and therefore has harmonics. A distinctive characteristic of a nonlinear time series is the presence of phase coupling which creates coherence between the frequency bands occupied by the harmonics such that the phases are nonrandom and form a reg- ular pattern. Nonlinearity may, thus, be inferred from the pres- ence of harmonics and phase coupling. Methods for nonlinearity detection in the time series include the techniques using surrogate data [8], [11] which have been used in applications ranging from analysis of EEG recordings of people with epilepsy [12] to the analysis of X-rays emitted from a suspected astrophysical black hole [13]. Surrogate data are times series having the same power spectrum as the time series under test but with the phase coupling removed by ran- domization of phases. A key property of the test time series is compared to that of its surrogates and nonlinearity is diagnosed if the property is significantly different in the test time series. Another method of nonlinearity detection uses higher order spectra because these are sensitive to certain types of phase coupling. For instance, the bispectrum [14]–[16] responds to quadratic phase coupling in a signal such as below in which the phase of the frequency component at is , but there is no bispectral response if is a random phase The bispectrum and the related bicoherence have been used to detect the presence of nonlinearity in process data [17], [18].
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A potential disadvantage of the bispectrum for detection of non- linear limit cycle oscillations is that limit cycles may have sym- metrical waveforms (e.g., a square wave or triangular wave) and the bispectrum of a symmetrical waveform is zero. Zang and Howell [19] have investigated the types of limit cycles that are amendable to bispectrum analysis. The presence of harmonics in a time series has also been used successfully for diagnosis of SISO control loop faults [7], [20], [21]. Finding harmonics requires signal processing to isolate the spectral frequencies of interest and inspection to confirm that the frequencies are integer multiples of a fundamental. The inspec- tion is often undertaken by visual examination of the spectra and is therefore unsuitable for a large-scale implementation in- volving several hundred or even a thousand or more plant mea- surements. Moreover, it is possible that components at the har- monic frequencies are not phase coupled in which case the har- monic signature will be a misleading indicator of nonlinearity. B. Propagation of a Nonlinear Limit Cycle Repair of a faulty control loop requires that the engineer knows which control loop should be maintained. In the case of a plant-wide oscillation, it can be a very difficult problem to know which loop to work on because the disturbance from a control loop in a limit cycle typically propagates plant-wide to cause numerous secondary oscillation in other control loops. An automated means is therefore needed to determine which among all the oscillating control loops is the root cause and which are secondary oscillations. Successful studies have used the presence of prominent harmonics to distinguish the source of a limit cycle oscillation from the secondary oscillations in a distillation column in a refinery [22] and in a pulp and paper mill [23]. The reason why secondary oscillations have lower nonlinearity is that as the signal propagates away from its source it passes through physical processes which give linear filtering and which generally add noise. (The filter may be assumed linear if the system is oscillating around a fixed oper- ating point). Such a filter destroys the phase coherence of the time trends and often reduces the magnitudes of the harmonics. Thus, nonlinearity reduces as the disturbance propagates away from the source and the time trend with the highest nonlinearity is the best candidate for the root cause. The nonlinearity statistic to be discussed in Section III can be used for such root-cause diagnosis. C. Surrogate Data Analysis A time series with phase coupling is more structured and more predictable than a similar time series known as a surro- gate having the same power spectrum but with random phases [11]. The spread of values of some statistical property of a group of surrogate data trends provides a reference distribution against which the properties of the test time series can be evaluated. The techniques of surrogate data analysis have been widely applied for detection of nonlinearity in time series [12], [13]. In the process area, Aldrich and Barkhuizen [24] detected nonlin- earity in process data by comparing a singular spectrum analysis of the test data with those from linear surrogate data. Barnard et al. [25] showed that identification of systems is possible by
Fig. 1. Test data and typical surrogate. The time trends are mean centered and scaled to unit standard deviation.
using surrogate methods to classify the data, as well as to vali- date models derived from these data. Issues have been identified with the use of surrogate data with cyclic time series [26], [27]. The surrogate is derived by taking the discrete Fourier transform (DFT) of the test data, randomiza- tion of the arguments followed by an inverse DFT. Nonlinearity testing based on strongly cyclic data can give rise to false de- tection of nonlinearity because when the time trend is strongly cyclic then artifacts in the DFT due to end-matching effects in- fluence the surrogates. A demonstration of the consequences for strongly cyclic data are demonstrated in this article although in practical applications the effect was found to have a minimal impact, as will be discussed later. III. METHOD A. Overview The basis of the test is a comparison of the predictability of the time trend compared to that of its surrogates. Fig. 1 illus- trates the concept. The top panel is an oscillatory time trend of a steam flow measurement from a refinery. It has a clearly defined pattern and a good prediction of where the trend will go after reaching a given position, for example at one of ringed peaks, can be achieved by finding similar peaks in the time trend and observing where the trend went next on those occasions. The lower panel shows a surrogate of the time trend. By con- trast to the original time trend the surrogate lacks structure even though it has the same power spectrum. The removal of phase coherence has upset the regular pattern of peaks. For instance, it is hard to anticipate where the trajectory will go next after emerging from the region highlighted with a circle because there are no other similar peaks. Predictability of the time trend relative to the surrogate gives the basis of a nonlinearity measure. Prediction errors for the sur- rogates define a reference probability distribution under the null hypothesis. A nonlinear time series is more predictable than its surrogates and a prediction error for the test time series smaller than the mean of the reference distribution by more than three standard deviations suggests the time trend is nonlinear.

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B. Construction of the Data Matrix Nonlinear prediction of time series was described by Sug- ihara and May [28] to distinguish determinism from random noise, and the field of nonlinear time series analysis and predic- tion has been reviewed by Schreiber [29]. Rhodes and Morari [30] gave an early process application of nonlinear prediction where the emphasis was on modeling of nonlinear systems when noise corrupts a deterministic signal. Nonlinear prediction uses a data matrix called an embedding having columns each of which is a copy of the original data set delayed by one sampling interval. For instance, a data matrix with 3 is Rows of the matrix Y represent time trajectories that are seg- ments of the original trend. Since the original data formed a con- tinuous time trend the trajectories in adjacent rows are similar. They are called near-in-time neighbors. Also, if the time trend is oscillatory then the trajectories in later rows of Y will be similar to the earlier rows after one or more complete cycles of oscil- lation. For instance, if the period of oscillation is 50 samples per cycle then will be small, where is the 51st row vector of and is the first. Those rows are called near neighbors. C. Calculation of Prediction Error Predictions are generated from near neighbors. Near-in-time neighbors are excluded so that the neighbors are only selected from other cycles in the oscillation. When nearest neighbors have been identified then those near neighbors are used to make an step-ahead prediction. For instance, if row vector were identified as a near neighbor of and if were 3 then would give a prediction of . A sequence of prediction er- rors can, thus, be created by subtracting the average of the pre- dictions of the nearest neighbors from the observed value. The overall prediction error is the rms value of the prediction error sequence. The analysis is noncausal and any element in the time series may be predicted from both earlier and later values. Fig. 2 il- lustrates the principle using a time series from the SE Asia re- finery case study where the embedding dimension is 16 and the prediction is made 16 steps ahead. The upper panel shows the 100th row of the data matrix Y which is a full cycle starting at sample 100, marked with a heavy line. Rows of Y that are nearest neighbors of that cycle begin at samples 67, 133, 166, 199, and 232 and are also shown as a heavy lines in the lower panel. The average of the points marked in the lower panel are then used as a prediction for the value marked . D. Data Preprocessing Detection of plant-wide oscillation is now a solved problem and is starting to be offered by vendors [31], [32]. The periodic nature of the detected oscillation may be exploited in order to
Fig. 2. Illustration of the nearest neighbor concept. The highlighted cycles in the lower plot are the five nearest neighbors of the cycle in the upper plot. The average of the points marked o gives a prediction for the point marked x.
give robust default settings for the parameters. A summary list is presented here and the detailed reasoning behind the recom- mendations will be presented in Section IV. With the data pre- processing steps indicated here the default parameters can be used for any oscillating time trend. 1) The period of the plant-wide oscillation is determined. 2) The number of samples per cycle is adjusted to be no more than 25. The time trends are subsampled if necessary. 3) The number of cycles of oscillation in the data set should be at least 12 for a reliable nonlinearity estimate. 4) The selected data are end-matched to find a subset of the data containing an integer number of full cycles. The algo- rithm and other issues associated with end-matching are explained in detail in Section IV-F. 5) The end-matched data are mean centered and scaled to unit standard deviation. The sequence de- notes end-matched and preprocessed data in the following sections. E. Surrogate Data Surrogate data are derived from the preprocessed and end- matched time trend. Surrogate data have the same power spec- trum as the time trend under test. The magnitudes of the DFT are the same in both cases but the arguments of the DFT of the surrogate data set are randomized. Thus, if the DFT in frequency channel is then the DFT of the surrogate is where is a phase selected from a uniform random distribution in the range . The aliased frequency channels above the Nyquist sampling frequency have the opposite phase added. If the number of samples is even and if the frequency

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channels are indexed as 1 to the Nyquist frequency is in channel and the alias of the th frequency channel is channel . Then and to If is odd and to ceil where ceil is the rounded-up integer value of . Finally, the surrogate data set is created from the inverse Fourier transform of the phase randomized DFT. F. Nonlinearity Test The nonlinearity test requires the determination of mean square prediction errors of surrogates. The statistical dis- tribution of those errors gives a reference distribution. If the test data prediction error lies on the lower tail of the reference distribution then the test signal is more predictable and non- linearity is diagnosed using the following statistic based on a three-sigma test where is the mean square error of the test data, is the mean of the reference distribution and its standard deviation. If then nonlinearity is inferred in the time series. Larger values of are interpreted as meaning the time series has more nonlinearity, those with are taken to be linear. It is possible for the test to give small negative values of . Negative values in the range are not statistically significant and arise from the stochastic nature of the test. Re- sults giving do not arise at all because the surrogate sequences which have no phase coherence are always less pre- dictable than a nonlinear time series with phase coherence. G. Algorithm Summary Step 1) Form the embedded matrix from a preprocessed and end matched subset of the test data Step2) For each row of Y find the indexes of nearest neighbor rows having the smallest values of subject to a near-in-time neighbor exclusion constraint . Step 3) Find the sum of squared prediction errors for the test data Step 4) Create surrogate prediction errors by ap- plying steps 1 through 3 to surrogate data sets.
Step 5) Calculate the nonlinearity from IV. DEFAULT PARAMETER VALUES A. Default Parameter Values Empirical studies have been carried out to ascertain the sen- sitivity of the nonlinearity index to the parameters of the al- gorithm. Reliable results have been achieved using the default values shown in Table I. The floor function in the third row indicates that for noninteger values of then is set to the rounded-down integer value of . The next subsections explore each one of these recommen- dations showing why they were selected. The time trends from Fig. 3. were used for the evaluation (they are from the indus- trial case study in Section V-B). Fig. 3 shows mean centered data normalized to unit standard deviation while the spectra are scaled to the same maximum peak height. The time series of the first three measurements are nonlinear because they are close to the root cause. Their spectra have har- monics and the phase patterns are not random. The last two are far from the root cause and are linear. The data have an os- cillation period of 16.7 sampling intervals and the conditions used were varied around default values of 12 16 8 and 50. B. Number of Samples Per Cycle It is practical to limit the number of samples per cycle . There is a tradeoff between the number of samples needed to properly define the shape of a nonsinusoidal oscillation on the one hand and the speed of the computation on the other. The algorithm requires a distance measure to be ascertained between every pair of rows in the embedded matrix and the time taken for the computation increases as , where is the total number of samples in the time trend. Therefore the number of samples per cycle and the number of cycles cannot be increased arbitrarily. Data sets of 200 samples 8.2 24 and 418 samples ( 16.7 and 25) gave successful results in the industrial case studies reported in Section V. It would be infeasible to operate with fewer than seven sam- ples per cycle because harmonics would not be satisfactorily captured. With 7, any third harmonic present is sampled at

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Fig. 3. Time trends and spectra of the data used for detailed evaluation.
2.33 samples per cycle which just meets the Nyquist criterion of two samples per cycle. The reason for focusing the recommen- dation on the third harmonic is that it is the most prominent har- monic in symmetrical oscillations having square or triangular waveforms. C. Embedding Dimension and Prediction Horizon Fig. 4(a) shows the effect of changing and . They were kept equal to each other and both were varied together. The threshold of nonlinearity 1 is also shown in the plot (horizontal dashed line) as well as the 16 default for the data set (vertical dashed line). Once becomes larger than half a cycle of the oscillation, in this case when , the re- sults for the nonlinearity index become quite steady while for small values of the index falls toward the 1 threshold. An aim of the work presented here is to give reliable default values that are easy to determine. Determination of the period of oscillation is becoming a standard component of controller performance tools [31], [32]. Therefore the recommendation to set floor is robust because it is in the steady region of Fig. 4(a) and is easy to implement because is already known.
Fig. 4. Effects of parameters of the algorithm on the calculated nonlinearity index. The vertical dashed lines in (a)–(c) show the recommended default values.
The poor performance with small values of arises because of the phenomenon of false near neighbors [33], especially when the time trends have high frequency features or noise. The upper panel in Fig. 5 shows an example of what can happen when the embedding dimension is small, in this case 2. The rows of the Y matrix starting at sample 159 comprises just two samples, 159 and 160, shown as small square symbols. Near neighbors are shown in the lower panel, these are two-sample segments of the time trend whose values are similar to samples 159 and 160. However, some of them are false neighbors because they are not from matching parts of the trend. The average of the points marked are used as a prediction for the value marked , but some of them such sample 223 which is based on a false neighbor are not accurate. D. Number of Cycles and Near Neighbors Fig. 4(b) shows a plot of the number of cycles of oscillation presented for analysis versus the nonlinearity value.

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Fig. 5. Illustration of false near neighbors when E = 2. The samples at 221 and 222 are similar to those at 159 and 160 but are from a different part of the cycle. A prediction of sample 161 based on sample 223 will be inaccurate.
The value of the nonlinearity statistic fluctuates up and down but when the number of cycles becomes too few the results start to become unreliable. For instance, when more than twelve cy- cles are used in the analysis then tag 33 is consistently and cor- rectly reported as nonlinear but when fewer than 12 cycles are present its nonlinearity index and that of other tags drops to- ward the 1 threshold. On the basis of these examples it seems necessary to use 12 cycles of oscillation or more. The nonlinearity index becomes consistent if this condition applies because the ranking order of the tags is maintained, for instance Fig. 4(b) shows that tag 34 consistently has the highest nonlin- earity index if . Fig. 4(c) fixes the number of cycles of oscillation at 12 and varies , the number of near neighbors used for prediction. The results are quite steady over a wide range of values of al- though some of the tags with nonlinearity show a drop toward the 1 threshold for large . A recommended value for can be based on the number of cycles of oscillation. From common sense reasoning, it is sensible to make sure that number of near neighbors is smaller than the number of cycles because each near neighbor is one whole cycle if the recommenda- tion is adopted. Given that one or two cycles may be lost during end-matching a conservative choice is 8 when the number of cycles is 12. The same conservative reasoning suggests that for other cases should never be greater than and in prac- tice it has been found quite satisfactory to just set the value to 8. The reason why the nonlinearity tests gives less reliable re- sults for large when the number of cycles is fixed at 12 is that the useful near neighbors run out. For instance, if 20 and if the data set has twelve cycles then any one cycle has eleven near neighbors that are closely matching cycles starting at the same position in the oscillation, like those in Fig. 2. The remaining near neighbors will have to be selected from other rows of the embedded matrix and will not be such good matches. E. Assessment of Variability of the Index Fig. 4(d) shows the variability in the nonlinearity index as the data subset is varied. Each time trend had 512 points. The data set was divided into seven overlapping subsets each having 12 cycles of oscillation. The first subset comprised samples 1 to 202, the second was 50 to 252, and so on. Fig. 4(d) shows that different parts of the data trend exhibit varying amounts of nonlinearity. It is noted, however, that the same conclusion about the tag with the highest nonlinearity applies regardless of the data subset. Tag 1 (34) has the highest nonlinearity right across the board, tags 13 and 33 are next highest and 19 and 25 have little nonlinearity. The white dots in Fig. 4(d) show the effect of varying the number of surrogates. Since it is a statistical test there must be enough surrogates to properly define the reference distribution. The difference between the results from 50 surrogates (black di- amonds) and 250 surrogates (white circles) is about overall and is less than the variability caused by the data subset. It is therefore concluded that 50 surrogates are enough. F. End-Matching Step 1) End-Matching Criterion: Surrogate data analysis re- quires a subset of the data such that the starting gradient and value match well to the final gradient and value. Hegger, et al. [34] recommend finding a subset of the nonmatched data (denoted by samples ) with samples starting at and ending at which minimizes the sum of the normalized discontinuities between the initial and end values and the initial and end gradients, where with being the mean of the sequence . The end matching procedure is modified for use with an oscil- lating signal as recommended in [26] to avoid the artifacts due to spectral leakage described in the next section. End matching of an oscillating time trend as described previously creates a time trend where the last value is the first sample of another cycle. An end matched sequence which contains an exact number of cycles is derived from the sequence by omitting the last sample. 2) False-Positive Results With Strongly Cyclic Data: It is known that calculation of reliable surrogates can be problem- atical for regular and smooth cyclic time series. Unless care is taken with the end matching the test may give false positive re- sults and report nonlinearity for a linear time series. The reason for the false positive results is the phenomenon of spectral leakage in the DFT caused by the use of a finite length data sequence. Fig. 6 illustrates the effect of spectral leakage. The upper panel shows the DFT of a sine wave having eight samples per cycle when the total data length is an exact mul- tiple of the period, in this case exactly eight cycles. The DFT is zero in all frequency channels expect the one at 0.125 corresponding to the frequency of the oscillation. By contrast,

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Fig. 6. Illustration of the importance of end matching. For a strongly cyclic time trend the data set should be an exact number of cycles of oscillation otherwise the Fourier transform will give spectral leakage into adjacent frequency channels.
the lower panel shows the DFT when the total data length is a complete number of cycles minus one sample. It has a nonzero magnitude in frequency channels adjacent to the channel con- taining the main spectral peak. A phase randomized surrogate derived from the DFT in the lower panel therefore contains fre- quencies that were not present in the original signal and will, thus, be less predictable than the original sine wave giving a false indication of nonlinearity. The true surrogate of a sine wave is a phase shifted sine wave at the same frequency and is equally predictable. It is therefore necessary to take special precautions when ana- lyzing cyclic time series. Stam et al. [26] used the end-matching step that ensures the data length of the time series is an exact multiple of the period of the cycle to avoid false nonlinearity de- tection. Small and Tse [27] proposed the calculation of special constrained surrogates that pay particular attention to frequen- cies in the data set having periods longer than the period of the strong cycles. That solution is not applicable to industrial data where the nonlinearity of interest is distortion of the periodic waveform, however. 3) Application to Industrial Data: Experimental laboratory data will give problems of the type outlined previously when the experimental system is driven by a cyclic source such as a labo- ratory signal generator. This is termed in the literature a “strong cyclic component.” Industrial data, however, even when cyclic do not often suffer from the problems described previously be- cause the cyclic behavior is not normally “strong.” Although they might be readily detectable the cycles are normally not completely regular and the spectral power in the test signal is al- ready is spread across several frequency channels. For instance, in Fig. 7 showing oscillatory data from a separation column the time trend labeled AC1 has two shorter cycles at around sam- ples 230 to 250. The effects of spectral leakage in the DFT are less severe in this signal because a wider range of frequency channels are already occupied. Therefore the surrogates more accurately represent the real signals than in the case of strong cyclic data. Table II shows nonlinearity calculations for AC1 and a pure sine wave of the same period of oscillation as the average of AC1. The correct result for the sine wave is , a result
Fig. 7. Time trends from industrial study No 1. Nonlinearity indexes greater than one are shown on the right. TABLE II EFFECT OF END MATCHING ON FALSE-POSITIVE RESULTS
which is achieved when the subset of the data is an exact number of cycles of oscillation. If the subset is longer or shorter, even by one sample, there is a false nonlinearity detection because of spectral leakage contaminating the surrogates. By contrast, the industrial data such as AC1 with its less regular cycles are not sensitive to minor variations in the end-match. No false non- linearity was detected for the industrial data even when mis- matched at the ends by one or two samples.

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Fig. 8. Process schematic for industrial study No 1. Loops PC1 and PC2 (not shown) are the upstream and downstream pressures.
V. INDUSTRIAL CASE STUDIES A. Unit-Wide Oscillation Caused by a Sensor Fault Fig. 8 shows a separation column courtesy of a BP refinery. The sampling interval was 20 s and the controller errors show the presence of a unit-wide oscillation in FC1, TC1, and AC1 with a period of 21 sampling intervals or 7 min. Measurements from upstream and downstream pressure controllers PC1 and PC2 also show evidence of the same oscillation along with other disturbances and noise. It is known that the cause of the oscillation was a faulty steam sensor in the steam flow loop FC1. It was an orifice plate flow meter but there was no weep-hole in the plate. Condensate col- lected on the upstream side until it reached a critical level when the accumulated liquid would periodically clear itself by si- phoning through the orifice. The challenge for nonlinearity de- tection is to identify FC1 as the source of the unit-wide oscilla- tion. The average oscillation period was 21 samples and the set- tings for the algorithm were therefore chosen as: 21 and 8. A data set comprising 500 samples and 24 cycles of oscillation was used. Fig. 7 plots mean centered time trends of controller errors normalized to unit standard deviation. The nonlinearity indexes are presented at the right-hand side. The only nonlinearity index greater than 1 was that of FC1. Therefore the nonlinearity anal- ysis has correctly identified the steam flow control loop causing the unit-wide oscillation. B. Plant-Wide Oscillation Caused by a Valve Fault Fig. 9 shows an outline schematic of a hydrogen reformer from a SE Asian refinery and the mean centered normalized controller errors (1-min samples) are presented in Fig. 10. The measurements from this plant have been discussed before [35] where the main disturbance was shown by spectral principal component analysis to be a 16.7-min oscillation in the reformer and pressure swing absorption (PSA) unit. The exact root cause was not communicated although it has been emphasized that it is a valve fault and pressure cycle swings of the PSA unit were not the cause. The aim of the analysis is to determine which of the oscillating measurements is closest to the root cause. The average oscillation period was 16.7 samples, the settings for the algorithm were therefore
Fig. 9. Process schematic for industrial study No 2. Fig. 10. Time trends from industrial study No 2. Nonlinearity indexes greater than one are shown on the right.
16 and 8. A data set comprising 25 cycles of oscillation was used. The nonlinearity indexes are shown on the right-hand side of Fig. 10 where the largest nonlinearity index is for tag 34. There- fore the flow measured by tag 34 is identified as the physical root cause and the means of propagation of the oscillation is distur- bance to the offgas recycle to the reformer. Tag 25 is upstream yet it was still influenced by the oscilla- tion [35]. The means of propagation from the root of the dis- turbance in the PSA unit to tag 25 is thought to be that tag 25 is waste gas recycled from another unit to which the oscillation has propagated. That Tag 25 is very far from the root cause is clear because its time trend shows no nonlinearity.

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Fig. 11. Process schematic for industrial study No 3. Fig. 12. Time trends from industrial study No 3. Nonlinearity indexes greater than one are shown on the right for samples 261–460 in the oscillatory episode. No nonlinearity was detected before sample 250 in any tag.
The question arises whether multiple sources of nonlinearity can be isolated by the proposed method. It is possible to detect multiple sources if the disturbances have different characteris- tics such as a different oscillation frequency. A cluster of tags oscillating at the same frequency is assumed to be a plant-wide disturbance with a single root cause. For instance, the tags in Fig. 10 share a 16.7-min oscillation. In [35], a second plant-wide disturbance with a different oscillation period was also reported. It was investigated separately and found to be an external dis- turbance originating in another unit. C. Plant-Wide Oscillation Caused by Process Nonlinearity Fig. 11 shows an outline schematic of the solvent recycle in a gas purification system, courtesy of BP Chemicals. The sol- vent absorbs a component from a mixed gas process stream in column 1. The absorbed gas is stripped out in column 2 and the regenerated solvent recycles to column 1. Fig. 12 shows mean centered and normalized time trends from several measurements in the two columns and the chal- lenge was to identify the source of an oscillation that periodi- cally bursts into life, an example of which can be seen starting at sample 250. The prevailing hypothesis was that the oscillation was due to foaming in column 1 because addition of antifoam would stop the oscillation. A successful analysis of this data set should therefore to point to column 1 as the source of oscil- lation and rule out a competing hypothesis that the oscillation was driven by the steam utility system through the steam valve in TC2. The nonlinearity results at the right-hand side of Fig. 12 are from the episode of upset operation, samples 261 to 460. The downward sloping linear trend in DPI1 was removed before analysis. The greatest nonlinearity was in Tag LC1 in column 1. The likely mechanism for generation of the oscillation is a peri- odic buildup and breakdown of foam in column 1 that affects the level sensor LC1, the differential pressure DPI1 and exit temper- ature TI1. The propagation of the oscillatory disturbance is from its root cause in column 1 to the top of column 2. Tag TC2 (the column 2 temperature controller) participates in the disturbance but its nonlinearity is not high so the steam system is ruled out as the root cause. The reason that TC2 has nonlinearity while TI2 (solvent stream temperature) shows no nonlinearity is be- cause TC2 is affected by variations in the unmeasured flow rate of solvent into column 2 as well as by its temperature. Many other measurements in the recycle path are upset by the oscil- lating disturbance but no others had any nonlinearity. This example shows that the nonlinearity index can locate the root causes of a limit cycle caused by process nonlinearity as well as cases where sensors or actuators in control loops cause limit cycling. VI. CONCLUSION Plant-wide oscillations in a system of interacting control loops often originate from a self-sustained limit cycle oscil- lation in just one control loop. Such a disturbance propagates to other parts of the plant and causes secondary oscillation. This brief has presented a method for locating the root cause of a plant-wide oscillation using a nonlinearity test based on the relative predictability of test data and surrogate data. The performance of the procedure was analyzed as key parameters in the algorithm varied, and default parameters were specified so that the test can be applied to new data sets.

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The method was demonstrated using three industrial case studies having a sensor fault, a valve fault and a process non- linearity caused by hydrodynamic instability. The nonlinearity test located the root cause in all three cases. ACKNOWLEDGMENT The author would like to thank A. Meaburn, Z. Rawi, and K. Landells of BP Chemicals, also S. Shah of the University of Alberta and A. Visnubhotla of Matrikon Inc. for providing data and process insights to support the brief. REFERENCES
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