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Control Loop Performance Assessment. 1. A Qualitative Approach for Stiction Diagnosis Ranganathan Srinivasan and Raghunathan Rengaswamy* Department of Chemical Engineering, Clarkson University, Potsdam, New York 13699
Randy Miller Honeywell Process Solutions, Thousand Oaks, California 91320
A spate of industrial surveys over the past decade indicate that only about one-third of industrial controllers provide acceptable performance. Since significant commercial benefits exist in diagnosing and improving the remaining two-thirds of the industrial controllers, the past few years have seen an emergence of control loop performance monitoring techniques using routine operating data. About 20-30% of all control loops oscillate due to valve problems caused by static friction or hysteresis. In the first of this two-part paper, a qualitative pattern recognition approach is described for stiction diagnosis. Stiction in control valves leave distinct qualitative shapes in the controller output (OP) and controlled process variable (PV) data. These shapes can be generally categorized as being square, triangular, and saw-toothed. To classify the patterns that evolve due to stiction, a pattern recognition approach using dynamic time warping (DTW) technique is proposed. The success of our proposed approach is built on a new technique for detection and time characterization of oscillations. A robust method for generating a stiction template pattern for each oscillating cycle as opposed to a global pattern for the whole data set is proposed. The qualitative approach was tested on data sets of varying complexity that include nonconstant behavior, intermittent stiction, and external disturbances, and results for eight data sets are presented. 1. Introduction The deployment of distributed control system (DCS), advanced control applications, and information management systems have become commonplace in the process industry. This has led to detailed information about the plant being archived on a daily basis. Competitive pressure and tighter environmental regulations have encouraged control engineers and managers to look at the archived information to identify potential areas of improvement and identify trends and problems in an incipient fashion for preventative maintenance. A spate of surveys on the performance of control loops1-4 reveal that a majority of control loops in processing industries perform poorly. Performance demographics of 26 000 PID controllers collected across a wide variety of processing industries in a 2-year time span indicate that the performance of 16% of the loops can be classified as excellent, 16% as acceptable, 22% as fair, and 10% as poor, and the remaining 36% are in open loop.4 Since PID controllers constitute 97% of all industrial controllers, poorly performing loops pose a significant problem with huge financial implications. Minimum variance as a performance measure for auditing control loops5,6 has been well researched. This benchmark yields a performance index that is defined as a ratio of the minimum output variance achievable to the actual output variance. Since a process engineer is responsible for about 400 loops,4 the availability of a * To whom all correspondence should be addressed. Tel.: (315) 268-4423. Fax: (315) 268-6654. E-mail: raghu@ clarkson.edu. Mailing address: P.O. 5705, Dept of Chemical Engineering, Clarkson University, Potsdam, NY 13699.
set of tools to automatically estimate and diagnose the performance of all control loops is a much-sought after component in the process industry. The primary objectives of any automated performance tool are (i) to detect performance degradation, (ii) to diagnose the cause for performance degradation, and (iii) to suggest corrective action (where applicable). The automated tool must be noninvasive and use routine operating data (controller output (OP); set point (SP); process output (PV)) and should not critically depend on any other information such as the model, controller structure, etc. To meet all three objectives using routine operating data alone will be a daunting task. The first objective has been well researched,7-12 and several commercial packages are available. The second objective, i.e., diagnosing the cause of poor performance has received considerable attention in the recent past, especially in the area of diagnosing oscillations.7,10,12-21 However this objective is only partially met, as there are still no comprehensive procedures available to ascertain the root cause for performance degradation as as one of the following: (1) poor controller tuning, (2) valve nonlinearity, and (3) external disturbance or a combination thereof. The third objective, i.e., suggesting corrective action, is gaining importance. Corrective actions can be, identifying new controller parameters,22 valve maintenance or stiction compensation,23 eliminating upstream disturbances or a combination of these. Performance degradation in control loops manifests itself as (i) poor set point tracking, (ii) oscillations, (iii) poor disturbance rejection, or (iv) high excessive final control element variation. In many cases, performance degradation leads to oscillations in process variables. This leads to increased energy consumption, wastage
10.1021/ie0490280 CCC: $30.25 © 2005 American Chemical Society Published on Web 07/12/2005
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Figure 1. Flow loop in the presence of stiction (industrial data).
of raw material, and products that are off specification. Reducing or removing such oscillations has yielded substantial commercial benefits. Oscillations can happen due to (i) valve nonlinearity (stiction, hysteresis, and backlash), (ii) external oscillation disturbance, (iii) poorly tuned controllers, or (iv) a combination thereof. Since industrial controllers are often conservatively tuned, improper controller tuning is, in general, not the most likely cause of oscillations. The most common reason for oscillation is valve nonlinearity. Twenty percent to 30% of all control loops oscillate due to valve problems caused by static friction or hysteresis1,24 resulting in performance deterioration. An abundance of literature exists for invasive analysis of control valve performance that requires stroking the valve when either in-service or out-of-service.25-31 Installation of very expensive smart positioners that measures stem position in a control valve can also analyze valve performance. Invasive approaches and the use of smart positioners are neither practical nor desirable for routine maintenance of control valves across an entire site. Hence there is a need to develop passive approaches that can classify valve problems. This two-part paper is concerned with the diagnosis of stiction in control loops using routine operating data and controller structure information. Two completely different techniques are proposed for stiction diagnosis. In this first part, a qualitative pattern recognition approach will be described for stiction diagnosis. Stiction in control valves leave distinct qualitative shapes in the controller output (OP) and process variable (PV) data (see Figure 1). These shapes can be generally categorized as being square, triangular, and saw-toothed13,32,33 and depends primarily on the type of controller structure implemented. This is in contrast to the stictionfree case where the oscillations are more sinusoidal in nature. A pattern recognition technique that can exploit this difference in shapes is proposed for diagnosing stiction. In the second part of this two-part paper, a quantitative Hammerstein model approach that estimates a stiction parameter based on the controller output (OP)-process variable (PV) data for diagnosing stiction is proposed. 2. Background An approach to control-loop performance assessment was initially proposed by Harris.6 The present day techniques9,10,12 are variants and extensions of the
approach of Desborough and Harris.7,8 A good review of various performance assessment techniques can be found in Qin34 and Harris et al.35 The main drawback of these performance indices is that they provide very little diagnostics for the performance deterioration. In fact, performance indices might indicate poor performance even if the controller is well tuned. This can happen in situations where the loop is oscillating due to an external disturbance or due to a valve problem like stiction, which results in a temporary breakage of the feedback path during stick-slip conditions, thereby compromising traditional performance assessment techniques. Oscillations in control loops commonly occur due to valve stiction or hysteresis. Although external disturbance, high controller gain, or loop interaction can also make a loop oscillate, industrial surveys in the past decade indicate that a majority of loops oscillate due to valve problems caused by static friction or hysteresis.1,24 Only a few documented techniques exist in the literature for oscillation detection and diagnosis. Ha¨gglund13 proposed a technique to detect oscillating loops “on-line” using the IAE criterion. This method does not assume any particular shape for oscillation and only requires the measurement to deviate significantly from the set point. Ha¨gglund13 also proposed a diagnostic procedure for finding the source of oscillation and eliminating it. The diagnostic procedure is carried out by disconnecting the feedback (i.e. switching the controller to manual mode). This approach is simple and efficient and probably the most comprehensive procedure available for diagnosing root cause for oscillations. However switching the controller to manual mode may not always be allowed, especially if the loop is deemed critical. Further, it will not be possible to apply this approach on thousands of loops in a routine fashion. Thornhill and Ha¨gglund14 presented an offline technique for detecting oscillation using a regularity factor. This method requires the user to specify the root-mean-square value of the noise and a thresholdsa nontrivial task when applied to hundreds of loops. Miao and Seborg36 computed auto-correlation function (ACF) and proposed an oscillation index (0-no oscillation, 1-highly oscillating). Industrial experience appears to be favorable, and oscillations were detected with reasonable reliability for stationary data. Thornhill and Ha¨gglund14 and Thornhill et al.15 proposed a set of procedures to detect and diagnose oscillating loops using off-line data. They combine the techniques of controller performance assessment7 along with operational signatures (OP-PV plots) and spectral analysis37 of the controller error for diagnosis. This technique, though not completely automated, can distinguish the cause of oscillation as one of the following: (i) poor tuning, (ii) nonlinearity, or (iii) external disturbance. However, the downside lies in manually inferring the loop signatures that are based on spectral analysis or on a map of controller output (OP) versus process variable (PV) and isolating the oscillating portion from the entire data. For flow loops the OP-PV map is seen as a set of overlapping tilted trapezoids. This is evident from Figure 2, which shows the OP-PV plot for the industrial data shown in Figure 1. For other loop types (pressure, level, and temperature) based on their process dynamics, the OP-PV map looks more like overlapping tilted ellipses. Recently Paulonis and Cox38 of Eastman Chemical Company “improved” the above technique and
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Figure 2. OP-PV plot for the data given in Figure 1.
Figure 3. Flow loop with varying SP (industrial data).
developed a large-scale system to identify and troubleshoot poorly performing control loops. The algorithm, however, has not been discussed in detail (this may be due to proprietary reasons). Xia and Howell12 proposed various statistics to facilitate the status monitoring of PI/PID loops and isolation of the problem loop. Horch10 presented a simple, practical approach to distinguish oscillating loops that are caused by external disturbances and static friction. This approach is based on cross-correlation between the controller output (OP) and process output (PV). Figures 3 and 4 show industrial examples of stiction. It can be seen that the oscillations are time varying in amplitude and frequency and the set point also varied slowly. Figure 4 shows a loop that exhibited stiction in the presence of a continuously varying set point. It was confirmed by the control engineer that the valve exhibited intermittent stiction between periods [3840-4320]. The cross-correlation technique failed when the data had intermittent oscillations and the set-point was also changing (see Figure 4). Horch and Isakkson39 also proposed a technique to identify stiction using nonlinear filters. The method assumed that information such as mass of stem, diaphragm area, and so on for each valve is readily available. Since in a typical process industry facility there can be thousands of control loops, it may be nearly impossible to build/maintain a knowledge base of control valves, making this technique difficult to implement. Choudhury et al.16 used higher order statistics for detecting nonlinearity in data and have extended the
Figure 4. Intermittent stiction.
method for diagnosing stiction by fitting an ellipse of the OP-PV plot and inferring the stiction from an assumed stiction model.40 The success of this approach lies in correctly identifying the oscillation period and its start and end point in the OP-PV data. Although most of the previously mentioned methods will identify stiction for the data set given in Figure 1, they will have difficulty in diagnosing the presence of stiction for the data sets given in Figures 3 and 4. This difficulty can be attributed to nonconstant behavior in the process data. Although subtracting SP data from PV may work for the data set given in Figure 4, it did not yield the desired results for the data set given in Figure 3. As can be seen, the task of detecting stiction or other nonlinearities in valves from routine operating data is a challenging task. To summarize, techniques based on routine operating data that were presented so far in the literature are useful in (a) assessing the performance of the controller by calculating a figure of meritsgiven that the cause of poor-performance is only due to either an aggressive or sluggishly tuned controller in pure feedback control, (b) detecting oscillating loops with an user-specified parameter, and (c) limited diagnosis of the cause of oscillation based on cross-correlation, power spectral analysis, or OP-PV plots. The current approaches lack (a) the capability to efficiently diagnose sticky loops, (b) a confidence measure while reporting the cause of poorperformance, and (c) an automated means of oscillation diagnosis. In this work, we have attempted to address all three drawbacks. This paper is organized as follows: In section 3, a short description of the functioning of an air operated control valve is given. The stick-slip patterns that evolve in the presence of stiction and different models for stiction are also discussed. In section 4, the idea of pattern comparison is introduced. The use of DTW as a pattern comparison method and the associated issues that need to be addressed when applying DTW for stiction diagnosis are presented. A new algorithm for time characterization of oscillations and the proposed approach for stiction diagnosis are outlined in section 5. The validity of the proposed approach on eight industrial loops is presented in section 6. 3. Control Valve, Stick-Slip BehaviorsIts Effect on a Control Loop and Its Measurements 3.1. Control Valve and Stick-Slip Phenomenon. A control valve consists of two main parts: a valve and
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Figure 5. Control valve and forces acting on the valve stem (Figure taken from ref 61).
an actuator that forces the stem to move. Additionally, it may contain a positioner that controls the valve stem so that it corresponds to the control signal. Since 90% of actuators are air-operated, a pneumatic configuration is considered in this work. Figure 5 displays a simple schematic of a control valve. In process operation, a control valve is subject to the following forces: the valvestem driving force caused by air pressure, spring force associated with the valve travel, seal-friction of the seals sealing the process fluid, and stem thrust originating in the process fluid passing through the valve body. Stiction (also known as stick-slip or static friction) in control valves is thought to occur due to seal degradation, lubricant depletion, inclusion of foreign matter, activation at metal sliding surfaces at high temperatures, and/or tight packing around the stem. The resistance offered from the stem packing is often cited as the main cause of stiction. One other very common cause of stiction is indirectly due to regulations on volatile organic compound (VOC) emissions. In many plants, a team monitors each valve for VOC emissions, usually between the packing and the stem. If any minute leakage is detected, packing in the valve body is tightened, but tightened far more than is necessary. This causes the valve to stick making the process run less efficiently with increased energy consumption. Stiction often varies over time and operating regimes. Since wear is also nonuniform along the body, frictional forces are different at different stem positions. When the control loop is at steady state, and if a valve exhibits this behavior, persistent oscillations in PV on either side of the set point are observed. 3.2. Stiction Models, Adequacy, and Stiction Measure. Several models for stiction have been proposed in open literature.23,40-42 Armstrong-He`louvry et al.41 provide a good collection of friction models. Eborn and Olsson43 have discussed the performance of these stiction models on industrial data. A commonly used stiction model is the classical friction model characterized by three parameters41 is given below:
{
F(v) if v * 0 if v ) 0 & |Fe| < Fs F ) Fe Fs sign(Fe) if v ) 0 & |Fe| g Fs
(1)
where F(v) ) Fcsign(v) + (Fs - Fc)e(v/vs)δ sign(v) Here Fs is the static friction, Fc is the coulomb friction, vs is Stribeck’s constant, Fe is the applied external force,
and v is the stem velocity. Identifying such a detailed stiction model from routine operating data is nearly impossible, as both the stiction model coefficients and the process dynamics have to be estimated simultaneously. Measurements such as stem movement and stem velocity will be required which are typically not measured, and these state variables have to be estimated along with other parameters. Also information such as mass of stem, diaphragm area, etc. have to be known a priori. Similar models given in ref 42 also are specified by several parameters. Since there are several hundred loops in an unit, such detailed information for each valve is not attainable, and computing the friction coefficients specified in eq 1 is a difficult task from closed loop data. A simple yet efficient model was proposed by Ha¨gglund.62 The simple model is given in eq 2:
{
x if |ut - xt-1| e d xt ) ut-1 otherwise t
(2)
Here xt and xt-1 are present and past valve outputs, ut is the present controller output, and ‘d’ is the valve stiction band. With the simple model, it appears that, when the stick condition is overcome, the valve position catches up with the input and instantly sticks again, which is physically unrealistic. However, the use of one parameter model is mathematically attractive when compared to the classical friction model (and other dynamic models for stiction42) for which several parameters such as viscous, coulomb, static friction, etc. have to be estimated. This simple model that has a characteristic of a relay was also considered by other researchers,20,23 and the applicability of this simple model is discussed in ref 44. The adequacy of a two-parameter stiction model reported by Choudhury et al.40 also confirms the applicability of simple models for stiction. This motivated us to use the simple stiction model (eq 2) in all our work for characterizing stiction. Although this stiction model is not directly used in the qualitative approach, the simple model plays a critical role in the quantitative approach for diagnosing stiction. This is presented in part 2 of this series.45 While the stiction phenomenon is well researched in the literature,41,42 measurement of stiction has been reported in several forms.40 In process industries, the feedback is disconnected (i.e. in open loop) before stiction measurement is carried out. A slowly increasing ramp is given as valve input until a noticeable change in the process variable is observed, and this is reported as a stiction measure. The units for stiction is given as a percentage of the valve travel or span of the control signal. Since, in practice, about 90% of pneumatic valves do not transmit the stem’s position (e.g. they lack a digital positioner with linear displacement sensor), in our work, stiction is reported as a percent of the span of the control signal. The stiction model given by eq 2 coincides with a way stiction is measured in industrial control valves. 3.3. Stiction Patterns. As stated before, due to the integral action of the controller, stiction and other nonlinearity leave distinct shapes in process output (PV) and control signal (OP) data. These oscillating shapes can be generally categorized as square, triangular, sawtooth, or sinusoidal.13,18,32 Figure 6 shows a sample industrial data of a flow loop. The control signal (OP), due to integral action, resembles a triangular wave,
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Figure 6. Typical stiction pattern in a flow loop.
whereas the flow (PV) resembles a square wave. If the process lag is high, the process output is generally close to a sinusoidal pattern. When the stiction is higher, the oscillation amplitude of OP and PV increases, causing the patterns in OP and PV to be sharper and more pronounced. Table 1, summarized from the paper of Forsman,18 categorizes the observed stiction patterns of OP and PV for slow and fast dynamic processes. When oscillations are not caused due to stiction, the shapes of OP and PV are more sinusoidal in nature due to the presence of feedback. However, we have to emphasize that the shape library given in Table 1 is not complete by itself as the shapes depend on controller parameters and the process dynamics. This shape library has to be updated as and when newer shapes that occur in the presence of stiction are found. In this work a procedure that discriminates between these kinds of oscillations to provide a stiction diagnostic measure for the plant personnel is proposed. In the following section, use of dynamic time warping (DTW) as the pattern comparison method and related issues that arise are discussed. 4. Pattern Recognition ApproachsIssues and Main Ideas A pattern recognition-based approach to detect stiction was proposed initially by Rengaswamy et al.19 They identified shapes using qualitative shape analysis (QSA).46 The QSA formalism has three components: (1) a set of fundamental units called primitives, (2) a procedure for identifying the primitives from noisy process data, and (3) a grammar for analyzing a combination of primitives. A pattern recognition based diagnosis scheme should ensure the following set of requirements for robust analysis. The diagnosis should (a) be independent of the magnitude of the signals, (b) be robust to time varying oscillations, (c) isolate a combination of faults (like a sticky loop with an external disturbance), (d) handle noisy data, and (e) capture the global trend and be less susceptible to local fluctuations. The approach of Rengaswamy et al.,19 though novel, has
some downsides: (i) one has to choose the basic window size that will allow for the detection of at least increasing and decreasing trends, (ii) one must have a priori knowledge of well-known reference patterns for comparing time series, and (iii) the algorithm is not completely automated. Similar pattern-based approaches16,17 for stiction diagnosis from a two-dimensional plot of OPPV also have the same drawbacks. The proposed work addresses the above limitations and other requirements stated above through the use of dynamic time warping (DTW).47-50 DTW uses the principle of dynamic programming to perform nonlinear temporal warping to achieve a minimum distance between the patterns being compared. In short, DTW does a time alignment and normalization to the data set in order to achieve the best minimum distance possible. The application of DTW in the field of process control has been of research interest.51-53 Colomer and Gamero54 discuss a combination of DTW with episodes based qualitative analysis for diagnosis of a level control system. For a detailed description of DTW, the interested reader is referred to ref 50. 4.1. Issues with Use of Dynamic Time Warping for Stiction Diagnosis. DTW can produce unintuitive alignments where a single point in one time series is mapped on to a large subsection of the other time series. In other instances, DTW can fail to find obvious alignments if a peak, valley, plateau, or other feature is slightly higher or lower than the corresponding feature in the other series. Industrial data (e.g. Figures 3 and 4) reveal that the oscillations at both OP and PV have different time scales and peak amplitudes for each negative and positive half of a full cycle. In addition to that, each full cycle of oscillation has a different overall time period, making the pattern comparison problem complex. When the oscillations are intermittent, a direct pattern matching of process data with a reference template will be meaningless. To summarize, the issues in DTW that need to be addressed for it to be applied for stiction diagnosis are as follows: 1. Amplitude mismatch between two series (test, reference) can lead to misalignment of central peaks. 2. Start and end point of the actual series must match with the reference template for a reasonable inference of the unknown test pattern. 3. Nonuniform positive and negative half time periods and peaks in OP and PV data can result in excessive compression or expansion in the warping path. 4. Nonuniform whole cycle period can increase the distance measure with respect to a reference pattern that is generated from uniform whole periods. 5. Intermittent oscillations in the actual series. 4.2. Main Ideas in the Proposed Approach. The fact that stiction leaves clearly distinguishable patterns in OP-PV data most of the time has been well-known and documented. In our earlier work19 we tried to exploit these patterns for stiction diagnosis. In that work, constant frequency square, triangular, or sawtooth signals were generated for shape matching. As seen, industrial stiction data seldom display constant
Table 1. Stiction Pattern Shapes for Flow, Pressure, Temperature, and Integrating Processes fast process (flow) measurements
dominant (I) action
dominant (P) action
slow process and (pres. and temp)
integrating process (level)
level with PI control
OP PV
triangular (sharp) square
rectangular rectangular
triangular (smooth) sinusoidal
triangular (sharp) triangular (sharp)
triangular (sharp) parabolic
Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6713 Table 2. Distance Measure Obtained Using DTW for Each Reference Pattern with the Test Pattern reference pattern
distance measure
rectangular triangular sinusoidal trapezoidal
2.96 2.94 1.42 0.91
Figure 7. Reference pattern generation.
frequency oscillation, and this lead to the failure of our earlier attempts in a large number of stiction cases. To adjust to this industrial reality, we propose two key ideas in this work: 1. Previous published work (discussed later in section 5.1) in the area of oscillation diagnosis was focused toward detecting oscillation and finding a representative frequency of the oscillation. However, the stiction phenomenon is asymmetric and time varying (see Figure 13, loop 4). In such cases, using a single frequency and generating a reference template will be inappropriate for pattern classification. In our approach, we identify the zero-crossings of oscillation, and the reference template for each cycle (three consecutive zero-crossings constitute one cycle) is generated, thereby accommodating time varying frequency and amplitude changes. 2. Direct pattern matching between the template shape (square, triangular, etc.) and the data for each cycle is still unreliable for stiction diagnosis. Dynamic time warping (DTW) is used to overcome this problem. Each of the issues given in section 4.1 is addressed in our proposed approach. Issues (3), (4), and (5) are addressed by time characterization of oscillating OP and PV data. Once time characterization of OP and PV data are achieved, the reference template can be generated based on the time period of each cycle’s positive and negative half and its respective peak amplitudes. By doing this, issues (1) and (2) are automatically taken care of. 4.2.1. Reference Pattern Generation. A single cycle (between time instants 3600-3800) of loop 4 (flow loop) shown in Figure 13 is considered for demonstrating the pattern generation process. It is seen that the amplitude for the positive and negative half vary significantly (about 50%), and also the time period for each half differs by about 10 sampling instants. Although the valve is stuck due to stiction, the flow drifts and the shape of PV is not square (or trapezoidal) as expected. Reference pattern generation using standard formulas with a fixed frequency and amplitude for square, sinusoidal, triangular, and trapezoidal waveforms will incorrectly capture the test data. Figure 7 shows the reference pattern generation using our approach. Four different reference patterns, namely triangular, sinusoidal, rectangular, and trapezoidal waveforms, are considered for classifying the test pattern. The reference pattern generation is normalized such
Figure 8. Intermittent sticky loop (industrial data).
that their area is matched with the test pattern, and the time periods for each positive and negative half are also matched. Table 2 gives the distance measure obtained using DTW for each reference pattern with the test pattern. It can be seen that the trapezoidal reference pattern gave the least distant measure, and the shape of PV data was classified as trapezoidal. In section 5, the methodology for oscillation characterization is outlined. 5. Proposed Approach 5.1. Oscillation Detection and Characterization. Oscillation characterization is a critical step in our proposed qualitative technique as the success of pattern matching technique depends extensively on the test signal used for pattern classification. It is seen from Figures 3 and 8 that limit cycles caused due to stiction are time varying in both frequency and magnitude. Figure 4 shows another example of stiction in the presence of a continuously varying set point and the valve exhibiting intermittent stiction between periods [3840-4320]. Typical industrial examples show that oscillations caused due to stiction are nonconstant and nonlinear. The use of FFT/spectral, correlation methods55 or computing the area under the signal between successive zero-crossings13,14,56 can reveal oscillations in the signal. However, automating them for several thousand of loops is a difficult task as visual inspection is generally necessary, and the tuning parameters are manually specified. The application of FFT or spectral analysis becomes difficult if the oscillation is intermittent and periods vary every cycle. Although Fourier or spectral analysis is valid under extremely general conditions, there are some crucial restrictions such as the data must be strictly periodic. The analysis of Thornhill et al.15 uses the auto-correlation function (ACF) of the PV data for further diagnosis. The ACF plot of the PV data set depicted in Figure 4 is shown in Figure 9. It can be seen that ACF did not reveal the intermittent oscillation period. The ACF analysis failed
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Figure 9. Auto-correlation function for the PV data shown in Figure 4.
due to its inability to completely remove the nonconstant mean, and only the global oscillation period of the set point was captured. Moreover the ACF analysis enumerates the signal’s dominant periodic component rather than the actual zero-crossings of the signal. More importantly, a component that is missing is time localization of oscillations, i.e., the characterization of the start time of oscillation and its consecutive zerocrossings. Wavelets, Spectrogram, Hilbert-Huang transform57 characterize instantaneous frequencies present in the signal. They do not perform time localization but instead time-frequency localization of a signal. The application of wavelets for time-localization was explored by the authors58 and was concluded that it is difficult to automate the wavelet analysis for characterization of oscillations in industrial data. Wavelet decomposition depends on the type of mother wavelet, its order, and the number of levels. For detailed discussion on application of wavelets to this problem, refer to ref 58. With the available information limited to routine operating data (OP/PV/ SP), PID settings, and loop type (flow, pressure, level, and temperature), time localiza-
tion of an oscillating signal becomes extremely important. Time localization can aid in root cause analysis for plant-wide oscillations; isolation of external disturbances in a loop based on OP and PV phase lags; and diagnosis of stiction and other loop hardware problems. A brief summary of the proposed oscillation characterization technique is given here. A detailed report with validation on more than 50 industrial loop data is given in ref 58. There are three basic steps in the oscillation characterization procedure. The first step removes the nonconstant mean from the data by an adaptive meanshifting procedure. The mean-shifting procedure is very similar to the “sifting process” of Hilbert-Huang57 transform. This step involves finding the local maxima and minima points, respectively. Once the extrema are identified, a cubic spline as the upper envelope connects all the local maxima, and the lower envelope is produced from the local minima. The mean of the data is computed by averaging the upper and the lower envelopes. The shifted signal is obtained by subtracting the calculated mean from the actual signal. The first step is illustrated graphically in Figure 10. The second step involves computing the area of the mean shifted signal and normalizing it. This step is required to remove spurious zero-crossings that occur due to noise. The third step is to find points at which slope changes occur in the area curve, and this identifies the zero-crossings. A clustering technique is employed for highly noisy data to group nearby zero-crossing points.59 The auto-correlation function test55 further confirms the presence of oscillation between the identified zero-crossings. Quiet periods i.e., time periods where oscillations are not present, in an intermittently oscillating signal are identified using a qualitative trend approach.46 Finally the time vector at which zero-crossings occur is reported. This procedure can be completely automated, as it involves parameters that are calculated from the signal being analyzed and operate on the univariate data series. Figure 11 depicts the last two steps of the oscillation characterization algorithm. A detailed discussion of the oscillation characterization technique can be found in ref 58.
Figure 10. Mean shifting process (step 1) of a portion of signal (for clarity) given in Figure 4.
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Figure 11. Steps 2 and 3 of oscillation characterization technique.
Figure 12. Flowchart: Stiction diagnosis - DTW approach.
5.2. Method for Stiction Diagnosis and Estimation using DTW. The flowchart schematic of the proposed approach is shown in Figure 12. It is assumed that the sampling interval for data collection will be adequate enough to preserve the oscillating patterns. Typical sampling intervals for flow, pressure, level, and temperature loops are 1, 5, 30, and 60 s, respectively. Since stiction (except dead-band in non-integrating processes) causes oscillation, first, the signal is tested for the presence of oscillation in OP. Once oscillations are confirmed, based on our oscillation characterization algorithm that identifies the zero-crossings, the amplitude and time period for each cycle of oscillation is calculated from the data. At least 10 cycles are analyzed (where available). Then a template reference pattern
(square, triangular, etc.) with appropriate amplitude and frequency is generated for each of the 10 cycles for both OP and PV. The reference pattern generation procedure was discussed in section 4.2. Four reference patterns, namely rectangular, triangular, sinusoidal, and trapezoidal waveforms, are generated. Although all 10 cycles can be combined to generate one whole data set, this is not considered for two reasons: (a) computational complexity of DTW increases as the square of the signal size,60 and hence, using individual cycles is computationally more efficient and (b) a few of the 10 cycles may be distorted due to the presence of disturbances that may destroy the stiction patterns in OP and PV. A confidence measure calculated as a ratio of the number of cycles identified as sticky against the total number of cycles that were taken for analysis can be obtained. Based on this reference set, classification of test pattern using DTW is done on the basis of the minimum distance achieved between each reference pattern and the actual data for both OP and PV. If the shapes of OP and PV match the shape library given in Table 1, based on the definition of stiction given in section 4.2.1, the maximum peak-to-peak amplitude of the analyzed OP cycles (considered for DTW analysis) is reported as the stiction measure. In cases where oscillations are not caused due to stiction, a further analysis to identify the root cause will be required. This is not addressed here. A detailed stepwise procedure is given below: Step 1: Import OP, PV, and SP data for the chosen loop. Mean shifting of the OP data is done (see Figure 10). Step 2: Confirm presence of oscillations in the mean shifted OP data using Auto-correlation test (Pyror, 1982). Step 3: If oscillations are present, continue to step 4, otherwise the next loop is taken for analysis. Step 4: Characterize the oscillations present in OP as outlined in section 5.1. At least 10 cycles are analyzed (where possible). But each of the 10 cycles is pattern matched separately. Step 5: Since OP and PV oscillate in a similar time range, the time stamps at which OP oscillation occurs is considered for PV data to classify their shapes. Step 6: For the identified zero-crossings of the oscillating OP and PV segment, choose a single full cycle from each of them. Find the peak amplitudes in the positive and negative half and its respective time periods. Step 7: Choose the test pattern as PV. Step 8: Generate reference template: square, triangular, trapezoidal, and sinusoidal signals using the information available from step 6, based on the method discussed in section 4.2. Step 9: Choose one reference pattern from the reference template (for example triangular). Step 10: Perform DTW with the test pattern and the reference pattern. Compute and store the dissimilarity measure. Step 11: Repeat steps 9 and 10 until each reference pattern’s (square, triangular, trapezoidal, and sinusoidal) dissimilarity measure is computed. Step 12: Choose the reference pattern that gives the minimal dissimilarity measure as the closest shape for the PV cycle analyzed. Step 13: Choose the OP test pattern in the same time range as that of PV.
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Table 3. Summary of Results: Industrial Case Study loop no. and type 1 (F) 2 (P) 3 (P) 4 (F) 5 (F) 6 (P) 7 (P) 8 (L)
PV oscillation detected yes yes yes yes yes yes no yes
possible PV oscillation shape
OP oscillation detected
trapezoidal sinusoidal sinusoidal trapezoidal trapezoidal triangular triangular
yes yes yes yes yes yes no yes
possible OP oscillation shape
reporting confidence (0-100%)
(% of OP stiction measure range)
triangular triangular triangular triangular triangular triangular
100 100 100 100 40 100
2.5026 1.6388 0.8244 0.3843 1.6 18.825
triangular
100
5.8
*F, P, and L indicates flow, pressure, and level loops, respectively.
Step 14: Repeat steps 8-12 and identify the shape of OP. Step 15: If none of the shapes in reference template matches the OP/PV data, no decision is taken. This can happen when multiple faults occur simultaneously (e.g. stiction and external disturbance). Step 16: Conclude the loop as sticky if the shapes identified for OP and PV coincides with the shape library given in Table 1. Step 17: If the loop is identified as sticky, based on the definition given for stiction in eq 2, report the stiction measure as a percentage of controller signal. The confidence measure calculated as a ratio of the number of cycles identified as sticky against the total number of cycles that were taken for analysis is also reported. Step 18: In cases where oscillations are not caused due to stiction, i.e., for cases where step 16 is not met or no meaningful shapes are found, a further analysis to identify the root cause will be required. Possible causes can be poor controller tuning, external disturbances, or loop interaction. 6. Results: Application to Industrial Data Eight loops were considered for evaluating the proposed methodology. Figure 13 shows the OP, PV, and SP data for each of the eight loops. Table 3 gives the summary of results analyzed using the proposed methodology. A detailed diagnosis for each loop is given below: Loop 1 (Flow). Plant engineers concluded this loop as a problem loop with a hardware fault in control valve. It has to be noted that, though there was continuous variation in the set point, the proposed technique reported this loop was as sticky with 100 confidence. The stiction parameter was estimated as 2.5% of the OP range. Since it was a flow loop, the PV values almost instantaneously respond to jump changes in stem position (due to stiction) making the PV shape look more rectangular. Nonstationarity in the data can be seen from the Figure 13 as it had oscillations that were varying in amplitude and time period. Loop 2 (Pressure). This loop was diagnosed as a very sticky loop with a large filter on the PV. The estimated stiction value was 1.64. It is worth noting that PV had a sinusoidal shape as expected (see Table 1) and OP had a triangular shape. Loop 3 (Pressure). This loop had a sustained oscillation and was diagnosed as a problem loop with stiction. The stiction estimation was 0.82. Loop 4 (Flow). A case of intermittent stiction can be observed. The loop oscillations were correctly time localized. The oscillation characterization procedure detected a minor amplitude oscillation in OP for the
Figure 13. Data for eight industrial loops.
time range [856 2547], but corresponding OP data had no oscillations. The set of data between [856 2547] was ignored as noise. The DTW algorithm reported that the loop stiction measure of 0.38 though the PV oscillation amplitude was high (27.57). From Figure 13, it is observed that oscillations in OP and PV were varying in amplitude and time period making diagnosis challenging. Loop 5 (Flow). Although the plant engineers concluded the loop as sticky, for the data set given, the DTW algorithm gave a low confidence on the stiction measure. On further analysis the loop had long oscillation periods with only two relevant complete full-cycles in the whole data set. Nevertheless, the algorithm allows the engineer to probe further due to the reported stiction measure. Loop 6 (Pressure). This loop had two simultaneous faults, external disturbance and stiction. Though the shape of PV and OP were distorted due to external disturbance, the algorithm diagnosed OP as oscillating with a triangular shape than to other shapes, and the loop was reported as sticky with a high severity. It is surprising that a high stiction measure of 18.825 was
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reported for this loop. In this case, as can be seen that where external disturbance is dominant, the peak-topeak amplitude is more, and this was considered while reporting stiction measure. However the average peakpeak amplitude is less than 10%. An important point to note is that the conclusion was made after analyzing only two full-cycles, and a final conclusion can be made only after analyzing more data. Loop 7 (Pressure). It was found that PV data was slightly oscillatory due to noise. Stiction analysis was not carried out for this loop as OP had no signs of oscillations. Loop 8 (Level). From Figure 13, it is evident that the level measurement PV and controller output that specifies the flow output (OP) had triangular patterns and matched with the shape library for integrating process. A stiction measure of 5.8 was reported although the average peak-to-peak OP amplitude came to 4.8%. This discrepancy is due to occurrence of a disturbance in inlet flow at the sampling instant around 900 units. Table 3 summarizes the evidence and conclusions for each of the eight loops. The use of qualitative pattern matching approach for stiction diagnosis is demonstrated through these encouraging results. In fact, loop 6 had external disturbances along with stiction. Though the procedure did not attribute both causes in the report, it correctly identified the presence of stiction. The procedure gave a low confidence on stiction measure for loop 5. This is attributed to less number of cycles satisfying the stiction criteria in the data length taken for analysis. Although the proposed approach is attractive and is extendable to other loop types, several improvements may be needed before it can be tested on a larger database. Since the proposed approach is dependent on the zero-crossings obtained from oscillation characterization algorithm to classify the OP and PV patterns, the presence of spurious zero-crossings can lead to wrong pattern classification and lower the overall reporting confidence measure. Though the proposed approach was shown to analyze intermittently sticky loops (e.g. loop 4 in Figure 13), if there are only fewer cycles within the data length (e.g. see Figure 4), choosing the correct cycles for analysis becomes crucial for successful diagnosis. If the oscillation is consistent and there are more than 10 cycles, choosing the subset of cycles for analysis becomes critical, when multiple faults occurs. Analyzing all cycles can improve the reporting confidence. It was seen that the computational load in MATLAB for a single loop in an Intel Pentium 4, 1.5 GHz, 512 MB RAM Windows 2000 system, was about 3 min for analyzing 5000 data set points with 10 cycles considered for pattern classification. However if all cycles have to be analyzed it will increase the computational load significantly. Advance DTW algorithms60 can be implemented to speed up the pattern classification process. Other parametric shape fitting procedures can also be investigated to reduce the computational burden. Also the proposed approach gave higher stiction measure than was observed for two loops (loop 6 and loop 8). This is attributed to the way the stiction is reported i.e., maximum peak-to-peak amplitude obtained from the cycles analyzed. This discrepancy can be avoided by reporting the average peak-to-peak amplitude obtained for those cycles that matched the shape library.
7. Conclusion and Future Work A novel procedure for diagnosis and estimation of stiction was proposed. The success of our proposed approach is built on a new technique for detection and time characterization of oscillations. The oscillation characterization algorithm, instead of just detecting oscillations from the industrial data, also identifies the zero-crossings in the data. This allows the extraction of individual cycles of the oscillation and efficient handling of signals that are nonconstant and nonlinear. A robust method for generating a stiction template pattern for each oscillating cycle as opposed to a global pattern for the whole data set was proposed. Dynamic time warping (DTW) of the template pattern to the actual data in each cycle of the oscillation is used for classification of test patterns. The DTW algorithm works efficiently in discriminating triangular, sinusoidal, rectangular, and trapezoidal patterns. The proposed qualitative procedure makes use of measurements from normal process operations, loop type, and controller structure (where available), which makes it commercially attractive. The overall approach was tested on data of varying complexity such as non-constant in oscillations, intermittent stiction, and external disturbances, and results for eight data sets were presented. Analysis and suggestions from the procedures agreed with the expert opinion of the control engineers. Based on the type of loop (flow, pressure, temperature, or level), the shapes for OP and PV given in step 16 can be modified using the information given in Table 1. In this way, the qualitative approach gives a framework for accommodating different kinds of loops. These patterns can take different shapes in the presence of stiction, when other faults affect the loop at the same time. In such cases, the proposed approach may fail. In cases where oscillations are not diagnosed as stiction, further analysis to identify the root cause will be required. Acknowledgment The authors gratefully acknowledge the Chemical Engineering Department, Clarkson University for fully funding this research work. The authors thank Lane Desborough of Honeywell for providing useful suggestions all along this work. The authors also thank Dr. Pramod Vachhani for providing valuable suggestions on the generation of reference patterns for the DTW algorithm and Ulaganathan (Honeywell India) for providing reviews on this manuscript. Literature Cited (1) Bialkowski, W. L. Dreams versus reality: A view from both sides of the gap. Pulp Pap. Can. 1993, 94 (11), 19. (2) Ender, D. B. Process control performance: Not as good as you think. Control Eng. 1993, 9, 180. (3) Control valve dynamic specification. version 3.0. Entech Control, Division of Emerson Electric Canada Limited, Canada, Toronto, 2005. (4) Desborough, L. D.; Miller, R. M. Increasing customer value of industrial control performance monitoring - honeywell’s experience. CPC-VI, Arizona, U.S.A., 2001. (5) A° stro¨m, K. J. Computer control of a paper machine - an application of linear stochastic control theory. IBM J. 1967, 389. (6) Harris, T. J. Assessment of control loop performance. Can. J. Chem. Eng. 1989, 67, 856. (7) Desborough, L. D.; Harris, T. J. Performance assessment measures for univariate feedback control. Can. J. Chem. Eng. 1992, 70, 1186.
6718
Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005
(8) Desborough, L. D.; Harris, T. J. Performance assessment measures for univariate feedforward/feedback control. Can. J. Chem. Eng. 1993, 71, 605. (9) Stanfelj, N.; Marlin, T. E.; MacGregor, J. F. Monitoring and diagnosing process control performance: The single-loop case. Ind. Eng. Chem. Res. 1993, 22, 301. (10) Horch, A. A simple method for the detection of stiction in control valves. Control Eng. Practice 1999, 7, 1221. (11) Huang, B. Performance assessment of processes with abrubpt changed of disturbances. Can. J. Chem. Eng. 1999, 77, 1044. (12) Xia, C.; Howell, J. Loop status monitoring and fault localisation. J. Process Control 2003, 13 (7), 679. (13) Ha¨gglund, T. A control-loop performance monitor. Control Eng. Practice 1995, 5 (11), 1543. (14) Thornhill, N. F.; Ha¨gglund, T. Detection and diagnosis of oscillation in control loops. Control Eng. Practice 1997, 5, 1343. (15) Thornhill, N. F.; Huang, B.; Zhang, H. Detection of multiple oscillations in control loops. J. Process Control 2003, 13 (1), 91. (16) Choudhury, M. A. A. S.; Shah, S. L.; Thornhill, N. F. Diagnosis of poor control loop performance using higher order statistics. Automatica 2004a, 40 (10), 1719. (17) Kano, M.; Marutha, H.; Kugemoto, H.; Shimzu, K. Practical model and detection algorithm for valve stiction. DYCOPS, Boston, U.S.A., 2004. (18) Forsman, K. On detection and classification of valve stiction. TAPPI Process Control, Electrical and Information Conference, Williamsburg, U.S.A., 2000. (19) Rengaswamy, R.; Ha¨gglund, T.; Venkatasubramanian, V. A qualitative shape analysis formalism for monitoring control loop performance. Eng. Appl. Artif. Intelligence 2001, 14, 23. (20) Stenman, A.; Gustafsson, F.; Forsman, K. A segmentationbased approach for detection of stiction in control valves. Int. J. Adap. Control Signal Process. 2003, 17, 625. (21) Choudhury, M. A. A. S.; Shah, S. L.; Thornhill, N. F., Detection and quantification of control valve stiction. DYCOPS, Boston, U.S.A., 2004. (22) Bezergianni, S.; Georgakis, C., Evaluation of controller performance - use of models derived by subspace identification. Int. J. Adap. Control Signal Process. 2003, 17, 527. (23) Ha¨gglund, T. A friction compensator for pneumatic control valves. J. Process Control 2002, 12, 897. (24) Miller, R. M. Loop scout regulatory control performance study. Honeywell, 2000, unpublished report. (25) Fitzgerald, W. V., Automated troubleshooting of pnematically operated control valves. Proc. ISA 1988, 43. (26) Fitzgerald, W. V. Automated control valve troubleshooting, the key to optimum valve performance. Proc. ISA 1990, 75. (27) Ancrum, R. Control valve diagnostics: Ready for prime time. Control 1996a, 60. (28) Ancrum, R. Control valve diagnostics: Ready for prime time part 2. Control 1996b, 56. (29) Boyle, S. Test systems for evaluating the dynamic performance of final control elements. Proc. ISA 1996, 673. (30) Walle´n, A. Valve diagnostics and automatic tuning. American Control Conference, Albuquerque, New Mexico, 1997. (31) Sharif, M. A.; Grosvenor, R. I. Process plant condition monitoring and fault diagnosis. volume 212 (Part E). Proceedings of the Institute of the Mechanical Engineering, 1998, 13-30. (32) A° stro¨m, K. J. Assessment of achievable performance of simple feedback loops. Int. J. Adap. Control Signal Process. 1991, 5, 3. (33) McMillan, G. K. Improve control valve response. Chem Eng. Prog. 1995, 6, 76. (34) Qin, S. J. Control performance monitoring - a review and assessment. Comput. Chem. Eng. 1998, 23, 173. (35) Harris, T. J.; Seppala, C. T.; Desborough, L. D. A review of performance assessment and process monitoring techniques for univariate and multivariate control systems. J. Process Control 1999, 9 (1). (36) Miao, T.; Seborg, D. E. Automatic detection of excessively oscillating feedback control loops. IEEE Conference on Control Applications, Hawaii, 1999, 359-364. (37) Thornhill, N. F.; Shah, S. L.; Huang, B.; Vishnubhotla, A. Spectral principal component analysis of dynamic process data. Control Eng. Practice 2002, 10 (8), 833.
(38) Paulonis, M. A.; Cox, W. J. A practical approach for largescale controller performance assessment, diagnosis, and improvement. J. Process Control 2003, 13 (2), 155. (39) Horch, A.; Isaksson, A. J. A method for detection of stiction in control valves. IFAC Workshop on On-line fault detection and supervision in the chemical process industry, France, 1998. (40) Choudhury, M. A. A. S.; Thornhill, N. F.; Shah, S. L. Modeling valve stiction. Control Eng. Practice 2005, 13, 641. (41) Armstrong-He`louvry, B.; Dupont, P.; De Wit, C. C. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 1994, 30 (7), 1083. (42) Olsson, H. Control systems with friction. Ph.D. Dissertation, Department Automatic Control, Lund Institute of Technology, Sweden, 1996. (43) Eborn, J.; Olsson, H. Modeling and simulation of an industrial control loop with friction. Proceedings of the 4th IEEE Conference on Control Applications, Albany, New York, 1995, 316-322. (44) Singhal, A.; Salsbury, T. I. A simple method for detecting valve stiction in oscillating control loops. J. Process Control 2005, 15, 371. (45) Srinivasan, R.; Rengaswamy, R.; Narasimhan, S.; Miller, R. M. Control loop performance assessment. 2. Hammerstein model approach for stiction diagnosis. Ind. Eng. Chem. Res. 2005, 44, 6719. (46) Rengaswamy, R.; Venkatasubramanian, V. A syntactic pattern-recognition approach for process monitoring and fault diagnosis. Eng. Appl. Artif. Intelligence 1995, 8 (1), 35. (47) Itakura, F. Minimum prediction residual principle applied to speech recognition. IEEE Trans. Acoust. Speech Signal Process. 1975, 23 (1), 67. (48) Sakoe, H.; Chiba, S. Dynamic programming algorithm for spoken word recognition. IEEE Trans. Acoust. Speech Signal Process. 1978, 26 (1), 43. (49) Myers, C.; Rabiner, L. R. Performance tradeoffs in dynamic time warping algorithms for isolated word recognition. IEEE Trans. Acoust. Speech Signal Process. 1980, 28 (6). (50) Rabiner, L.; Juang, B. Fundamentals of speech recognition; Englewood Cliffs, NJ, Prentice Hall, 1993. (51) Kassidas, A.; MacGregor, J. F.; Taylor, P. A. Synchronization of batch trajectories using dynamic time warping. AIChE J. 1998a, 44 (4), 864. (52) Kassidas, A.; Taylor, P. A.; MacGregor, J. F. Offline diagnosis of deterministic faults in continuous dynamic multivariable processes using speech recognition methods. J. Process Control 1998b, 8, 381. (53) Konstantinov, K. B.; Yoshida, T. Real time qualitative analysis of temporal shapes in bio process variables. AIChE J. 1992, 38 (11), 1703. (54) Colomer, J.; Mele´ndez, J.; Gamero, F. I. Pattern recognition based on episodes and dtw. application to diagnosis of a level control system. Sixteenth International Workshop on Qualitative Reasoning (QR-02), Sitges, Spain, 2002. (55) Pryor, C. Autocovariance and power spectrum analysis -derive new information from process data. Control Eng. 1982, 2, 103. (56) Forsman, K.; Stattin, A. A new criterion for detecting oscillations in control loops. Proceedings of the European Control Conference, Karlsruhe, Germany, 1999. (57) Huang, N. E.; Shen, Z.; Long, S. R.; Wu, M. C.; Shih, E. H.; Zheng, Q.; T. C. C.; Liu, H. H. The empirical mode decomposition and the hilbert spectrum for nonlinear and nonstationary time series. Proc. R. Soc. London, Ser. A 1998, A (454), 903. (58) Srinivasan, R.; Rengaswamy, R.; Miller, R. M. A modified empirical model decomposition (emd) process for oscillation characterization in control loops. Submitted to Control Eng. Practice 2005c. (59) Mitchell, T. M. Machine Learning; McGraw-Hill: 1997. (60) Chu, S.; Keogh, E.; Hart, D.; Passani, M. Iterative deepening dynamic time warping for time series. The second SIAM international conference on Data Mining (SDM-02), Washington, DC, U.S.A., 2002. (61) Kaseda, C.; Ohtsuka, A.; Nagasaka, F. Online control valve diagnostics. http://www.r-t-o-l.com. 2002. (62) Ha¨gglund, T. Private communication, 2005.
Received for review October 8, 2004 Revised manuscript received May 20, 2005 Accepted May 27, 2005 IE0490280
