A New Unified Approach to Valve Stiction Quantification and

Mar 3, 2009 - Our stiction compensation technique is then proposed in section 4. ... When fD is set to zero, pure static friction is experienced. ...
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Ind. Eng. Chem. Res. 2009, 48, 3474–3483

A New Unified Approach to Valve Stiction Quantification and Compensation Lee Zhi Xiang Ivan and S. Lakshminarayanan* Department of Chemical & Biomolecular Engineering, National UniVersity of Singapore, Singapore 117576

Satisfactory control loop performance has always been a vital objective in process industries for economic reasons. However, unsatisfactory performance may prevail, of which valve stiction is a common culprit. In this work, a new unified approach is introduced to quantify valve stiction in control loops and then compensate for it. The newly proposed stiction quantification method uses a one-parameter stiction model and is based on a modified Hammerstein identification approach. In conjunction with the stiction quantification algorithm, a novel compensation technique that adds a constant amplitude signal to the valve input signal is presented. The effectiveness of the proposed scheme is demonstrated using simulation examples. 1. Introduction Safety and profitability of process industries is directly and significantly dependent on satisfactory control loop performance. Central to this issue is the presence of multiple plantwide disturbances, of which oscillatory behavior in control loops is a common reason. Contributions have been made to analyze such disturbances and identify their origins.1 It has been noted that 20-30% of all oscillatory loops are due to control valve stiction.2,3 Other common causes include external disturbances and poor controller tuning.4 Valve stiction essentially comprises two problems: diagnosing its presence in control loops and the subsequent compensation of the variability it causes in the process variable. These two problems have been dealt with separately and together on many occasions. Numerous authors have written about the first problem of stiction diagnosis. Some suggest invasive approaches to identify stiction in control loops,5,6 but in certain situations, this may not be feasible. Other noninvasive methods to detect stiction have also been suggested. Horch7 utilized the crosscorrelation function (CCF) between controller output (OP) and process variable (PV) to detect the presence of stiction. However, the method has several limitations and has been shown to improperly diagnose stiction in some cases.4 A later method by Horch8 was also shown to incorrectly diagnose stiction under significant noise. A curve-fitting method proposed by He et al.4 to detect stiction in a control loop was shown to be suitable for both integrating and nonintegrating processes. However, this development was more concerned about detecting stiction than quantifying the phenomenon or compensating for it. Choudhury9,10 proposed using the bicoherence of the control error signal to ascertain the presence of stiction and subsequently quantify it by fitting an ellipse to OP-PV data. Recently, stiction quantification procedures by means of system identification have been proposed in the literature.2,3,11-14 Srinivasan et al.2,3,13,14 introduced a Hammerstein model approach using separable leastsquares to quantify stiction. In their work, a one-parameter nonlinear element was proposed to represent the “sticking” valve within the Hammerstein model structure. A grid search technique was employed, where for every possible value of this parameter, OP data was first passed through the nonlinear element to obtain estimated valve output, which was then used in conjunction with PV data to give an estimate of the linear section of the system in the form of an ARMAX model. The * To whom correspondence should be addressed. Phone: (65) 65168484. Fax: (65) 67791936. E-mail: [email protected].

loss function of the ARMAX model served as the objective criterion for estimating the correct value of stiction found in the valve. Along similar lines, Lee et al.12 also introduced a closed-loop stiction detection and quantification method that uses ordinary least-squares to identify the process under investigation, which is estimated as a low-order process with possible time delay. The main differences in approach between both system identification approaches lie in the stiction model used as well as the additional explicit assumption on process structure in the latter method. In this paper, we extend Srinivasan et al.’s work by modifying their approach to stiction quantification. First, instead of the nonlinear element put forth by Srinivasan et al. in their papers, we introduce a one-parameter model, motivated by the stiction algorithm proposed in He et al.,4 that mimics stick-slip behavior in valves. This new algorithm for stiction was developed to focus specifically at estimating static friction present in the valve, unlike the original model in the quantification scheme. Other improvements to the original work include an amended preprocessing approach and a refined ARMAX optimization routine so as to better isolate and estimate stiction in industrial data sets. At present, few methods exist to compensate for stiction. Noteworthy are the use of valve positioners in control systems15,16 and also the use of a dither signal pulse in control loops.17 Srinivasan et al.2,3 showed that good reduction in process variability was possible using such a dither signal pulse with their recommended settings based on their Hammerstein quantification approach. We propose a new compensator with constant amplitude added to the controller output signal with the amplitude determined by our modified Hammerstein approach. This has been shown, in our simulations, to produce a greater reduction in process variability than can be achieved with a dither signal pulse operating under the aforementioned recommended settings. Thus, this new compensator, when coupled with our proposed quantification scheme, forms a novel approach to the problem of stiction quantification and compensation. The aim of this paper, then, is to expound to the reader the details and benefits of the new modified quantification algorithm and to highlight the advantages of the proposed compensator. We now state the content of this paper. In section 2, we briefly discuss the stiction model that serves as the basis for our new one-parameter model. Section 3 deals with the subject of stiction quantification, where we detail our new approach. Our stiction compensation technique is then proposed in section 4. The

10.1021/ie800961f CCC: $40.75  2009 American Chemical Society Published on Web 03/03/2009

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Figure 2. Flowchart of He et al.’s stiction model.

Figure 1. Typical OP-MV plot of a “sticking” valve.

framework for combined diagnosis and compensation of stiction in process control loops is shown in section 5. Section 6 presents the results of this new approach on typical simulated examples. Industrial data is considered in section 7 to prove the validity of the proposed quantification technique when compared to other contemporary methods. Conclusions are provided in section 8. 2. Choice of Valve Stiction Model First, we briefly discuss the phenomenon of stiction, the existing stiction models available for describing it, and explain the choice of model used to derive the one-parameter model. The phenomenon of valve stiction has been extensively covered in the literature. A typical OP-MV plot of a “sticking” valve can be seen in Figure 1. When the controller takes action due to deviations of PV from its set point (SP), the valve stem (MV) initially remains stuck until the controller is able to take a large enough action to overcome the static friction (fS) in the valve and move it. fS is the sum of kinetic friction (fD) and slip-“jump” (J). The stem may then (i) jump to the desired position corresponding to the OP input, (ii) overshoot it, or (iii) lag behind it. For a control loop under integral action, the accumulation of the error signal (difference between PV and SP) before the valve starts moving causes the valve position to continue rising beyond the initial desired position at the onset; therefore, the controller will take action in the opposite direction where the valve becomes stuck again and the cycle is repeated. Thus, valve stiction normally results in observed oscillatory behavior in the control loop, commonly referred to as limit cycles. The region labeled S is commonly known as the deadband plus stickband region. Plots such as Figure 1 have become the basis of empirical or data-driven valve stiction models. These models utilize observed characteristics depicted in such plots to simulate the phenomenon of valve stiction, as opposed to physics-based models that employ Newtonian laws to simulate such behavior but have the propensity to include too many parameters. Also, empirical based models require no a priori knowledge of the valve. We therefore choose to use the practically more appealing empirical stiction models to present and illustrate our methodology. Several empirical stiction models exist in the literature. Choudhury et al.9,10,18 developed a two-parameter model based on two parameters, deadband plus stickband, S, and slip-jump, J. However, it has its shortcomings.4 First, it is a deterministic model that fails to properly simulate the behavior of a valve with no stiction. Second, the simulated OP-PV plot was shown to deviate in terms of likeness from a real life plot. Kano’s

Figure 3. Proposed nonlinear element.

Figure 4. Block diagram for CR approach.

model,19 which is stochastic in nature, also fails to capture the characteristics found in industrial OP-PV plots.7 He et al.4 developed a new stochastic stiction model that overcomes the aforementioned inadequacies. Due to such desirable properties, we utilize a derivative of the stiction model of He et al.4 in this work for quantification purposes. Figure 2 shows the flowchart of He et al.’s stiction model and follows closely the earlier description of stiction behavior. Note that when the two parameters in the model, fS and fD, are equal, the valve exhibits pure hysteresis. When fD is set to zero, pure static friction is experienced. For convenience, all variables covered in this work are in units of percentages, unless otherwise stated. We recognize the technical inconsistency in using percentages as the base unit for certain variables such as that for the friction being considered, but we stress that for the simplicity of comparison, we shall presume in this paper that the relevant values have been adjusted by appropriate linear transforms to a common unit. It should be noted that recent literature20 has shown the invalidity of He et al.’s stiction model in producing true stiction behavior. We wish to stress that the one-parameter model derived from their work is merely for use in the quantification and compensation framework, and we are not advocating it for use in simulations due to the limitations as mentioned in the literature. On a related note, to avoid bias when testing our framework, we use Choudhury’s stiction model9,10,18 or Kano’s model19 for simulating the sticky valve in our study and not He et al.’s stiction model due to its underlying ties with our proposed

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of the validity of the stiction quantification method is provided through the analysis of industrial data made available by Lee et al.12 3. A New Valve Stiction Quantification Method

Figure 5. Unified framework for stiction diagnosis and compensation.

Figure 6. Block diagrams of (a) flow control system and (b) level control system. Table 1. Controller Settings for Simulations process

proportional gain

integral time (min)

flow control level control

0.5 3

0.3 30

Table 2. Diagnosis Results for Simulations scenario

f (%)

d (%)

flow control (weak stiction) flow control (strong stiction) level control (weak stiction) level control (strong stiction) level control (sinusoidal disturbance) valve saturation

0.50 2.50 0.50 2.50 0.00 0.00

0.06 4.01 0.75 3.78 0.00 0.04

Table 3. Table of ISE Reduction Ratios for Simulated Stiction Scenarios Scenario

ISE/ISEORG

flow control system (weak stiction) flow control system (strong stiction) level control system (weak stiction) level control system (strong stiction)

1.4 × 10-4 8.7 × 10-3 4.5 × 10-3 3.4 × 10-3

stiction diagnosis method. Our proposed quantification and compensation scheme includes this structural mismatch to demonstrate that the results with our method are robust to alternate realizations of stiction phenomena. Further evidence

In this section, we introduce a new stiction quantification technique based on Srinivasan et al.’s Hammerstein modeling approach after first describing current stiction diagnosis methods. 3.1. Existing Valve Stiction Diagnosis Methods. Horch7 devised a detection method involving the cross-correlation function (CCF) between OP and PV, but the method lacked general applicability as it worked only under certain assumptions about the concerning process. Likewise, it was shown to erroneously diagnose stiction in certain situations.4 A second method was developed by Horch8 to handle the case of stiction in integrating processes, but this technique was also shown to be inadequate in the presence of significant noise. With OP-MV data, stiction can be detected and quantified in a relatively easy manner. The challenge to detect and quantify stiction becomes more formidable when MV data is not available (this is often the case with the data sets available in Singapore chemical plants). Many noteworthy techniques have been proposed to diagnose stiction when MV data is not present. Yamashita21 utilized a simplified shape identification scheme that could be used with both OP-PV and OP-MV data sets. Its performance was investigated by Manum et al.22 and was ascertained to correctly identify the presence of stiction in industry in 50% of industrial cases. An earlier study23 determined that the first scheme proposed by Horch, as well as the algorithms of Choudhury et al.10 and Scali et al.,23 produced uncertainty regions where no decision could be made without further information. The method proposed by Choudhury9 and Choudhury et al.10 involves the calculation of the bicoherence of the control error signal to calculate two metrics, the non-Gaussianity index (NGI) and nonlinearity index (NLI), that determine if the data is nonGaussian and nonlinear. Quantification of stiction comes in the form of apparent stiction (AP). However, for the method to work, certain assumptions must hold true,9 a large number of data samples is also required, and downsampling is necessary when the oscillation frequency is very low.10 He et al.4 implemented a curve-fitting mechanism to ascertain the presence of stiction within a control loop. Their premise is that, whenever stiction is present, the first integrating term after stiction will produce a triangular wave. They developed a metric known as the stiction index (SI) as a measure of confidence of the presence of stiction. The main shortfall of this method includes the requirement of a priori knowledge on whether the process is integrating or not. Furthermore, besides only providing a means of qualifying the existence of stiction, it is also valid only for processes where considerable time delays are not present, otherwise the expected results will be reversed. Srinivasan et al.2,3,13,14 exploited the Hammerstein model structure for stiction diagnosis using a separable least-squares optimization technique. They proposed a stiction model for the nonlinear element as follows: x(t) )

{

x(t - 1) if |u(t) - x(t - 1)| e d u(t) otherwise

(1)

The model is thus memory-based, and the linear component that is cascaded to it is modeled by an ARMAX structure. For a set of model order parameters, the model parameters are first estimated by minimization of the mean squared error (MSE) criterion given by

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V(θN) )

1 N

N

∑ ε (t, θ ) 2

N

(2)

t)1

where θN refers to the model parameters and ε(t, θN) ) y(t) - y˜(t, θN)

(3)

where y˜(t,θN) is the estimated output. The Akaike information criteria (AIC),24 obtained by weighing the MSE and a penalty for model complexity, is used to choose the best model parameters. The particular ARMAX structure and its associated parameters that minimizes AIC is chosen as the optimal estimated model for the linear component. Srinivasan et al. adopt a grid-based search approach to determine the parameters of their model. A grid of d with values ranging from 0 to 100% is created for an appropriate segment of OP data, and valve outputs x˜ are calculated for each value of d. The linear component (ARMAX model) is then estimated with x˜ as input and PV as output. The d value that gives the lowest mean squared error value is consequently chosen as the right estimate of the amount of stiction in the valve. A closer look at Srinivasan et al.’s model shows the equivalency with He et al.’s stiction model with fD set at 0%. We propose an alternative model that specifically mimics pure stick-slip behavior and correctly estimates the static friction fS. 3.2. New Hammerstein Approach to Stiction Diagnosis. Our proposed stiction model, adapted from He et al.’s stiction model, is shown in Figure 3. Here, f is a friction parameter that is assumed to be equal to both static friction fS and kinetic friction fD. However, it differs from He et al.’s model in terms of stiction behavior. In our model, we simulate specifically for the case of pure stick-slip, which, in later sections, will be shown to best suit the compensation scheme we are proposing. Hence, only the S region is present, and after it has been overcome, the valve position immediately jumps to the desired position as specified by controller output. There is thus no lag in MV when the valve is moving and the valve does not stick again as long as valve input (controller output) continues in the prevailing direction. This is in contrast to He et al.’s model, where the MV will always lag the controller output by a value equal to fD. Upon overcoming the static friction f, a residual force equal to it will be present. Thus, if the valve input continues in the same direction as before, static friction will always be overcome and the valve position will move accordingly (since cum will increase beyond f). This continues until valve input changes direction, whereupon deadband plus stickband will be encountered again, since the residual force is no longer larger than static friction (change in valve input acts opposite to residual force to reduce the magnitude of cum to less than static friction). By using this model, the estimated x˜ for each value of f differs from the input OP data only in the S region. Any minimization of the mean squared error criterion is therefore a consequence of better estimation of the static friction in the said region. The possible stiction behavior of the valve between each halfoscillation is ignored. Besides using a different nonlinear element, the identification procedure differs from earlier work in some other aspects. For the identification of the linear element, the ARMAX model order is fixed at 6, and we permutate different combinations of the order parameters to find the right set that minimizes AIC for a given f value during the grid search. Additionally, we calculate fmax, which is the upper bound of possible f values for the valve being considered. It is computed

by taking the means of the local maxima and minima of OP data respectively and then calculating their difference. This follows from the observation that if the value of f is greater than fmax, the controller will always be sticking and the valve output will not change for the entire duration being examined, thus producing a redundant result. An important point to note is that only the maxima and minima corresponding to large peaks, and deep troughs should be used in the calculations and not smaller peaks that may arise from the presence of noise. A moving average filter is used to get rid of such small peaks. However, the smoothed data will only be used for calculating fmax. The original data set will be passed into the modified Hammerstein algorithm, as shown in the next section. 3.2.1. Steps to Undertake in Modified Hammerstein Approach. Our proposed approach is as follows: (i) After oscillatory region has been isolated and denoised (see subsection 3.3), the OP and PV data are expressed in terms of deviation variables. (ii) The difference of the means of the local maxima and minima of OP data (fmax) is computed. A grid of f values from 0 to fmax is then created. (iii) At each value of f, OP data is fed into the stiction model to create estimated stem positions x˜, starting only from the first large minima or maxima in the data, whichever is earlier. This is to resolve the issue of the value for the residual force (ur) at the start of the transformation to x˜. If the first OP data point is a local maxima, ur is equal to f. Otherwise, it is equal to -f. (iv) y(t) is fitted to x˜ using an ARMAX structure with n + m + p ) 6 and with the best combination of n, m, and p, and the parameters of the ARMAX model (ai, bi, ci) are chosen by minimizing the AIC. (v) The value of AIC corresponding to the particular ARMAX structure that minimizes AIC is then stored and the grid moves on to the next value of f. This continues until fmax is reached. (vii) The value of f that gives the least AIC becomes the estimated f for the process. If the estimated f value is less than 0.05, stiction is considered to be absent. 3.3. Preprocessing of Industrial Data. Industrial data is often recorded in terms of units of percentage of the expected span for the variable. Also, the data tends to be noisy. As a result, some amount of preprocessing is required before the data can be used for stiction diagnosis. 3.3.1. Isolating Oscillatory Region. The first important preprocessing activity is the isolation of regions in data where oscillatory behavior is present. We implement the oscillation characterization procedure as proposed by Srinivasan et al.25 but introduce a new test statistic as determinant of the region with the most regular oscillations. An adaptive mean-shifting procedure26 similar to the shifting process of the “Hilbert-Huang transform” is carried out on the OP data. This involves an adaptive mean-shifting procedure.3 Cubic splines are generated over the maxima and minima of OP and then the local means subtracted from each data point. Next, the test statistic r, proposed by Thornhill et al.,1 is used to quantify the regularity of the oscillations: r)

jP 1T 3 σTP

(4)

j P is the mean period of oscillation in the region where jT considered, while σTP is its standard deviation. The region with the highest r value is chosen. Qualitatively, a value of r greater than 1 represents a regularly oscillating segment of data. Note that although we have modified the OP data in order to find the

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Figure 7. ISE/ISEORG plots of (a) flow control system (weak stiction), (b) flow control system (strong stiction), (c) level control system (weak stiction), and (d) level control system (strong stiction).

most oscillatory segment, it is the corresponding portion in the original OP data set that should be passed on to the Hammerstein modeling step. Approximately 1000 isolated data points will suffice to estimate the parameter f. 3.3.2. Denoising Industrial Data. Industrial data is often noisy. While the oscillation characterization procedure can isolate dominant oscillation modes in data, some of the diagnosis approaches suffer in the presence of substantial noise.14 The use of an appropriate wavelet transform denoising technique (to denoise the data) before stiction diagnosis will suffice in most occasions, as has been stated in literature.14 4. A New Valve Stiction Compensation Method In this section, we discuss the problem of stiction compensation. Ideally, a sticking valve should be replaced or repaired. However, replacement or repair is typically done during periods of plant shutdown for which one may have to wait for several months. This is because most continuous processes have fairly long run times, and it is not feasible to replace the faulty valve(s) immediately unless redundant bypass lines exist. The next best solution is to look for ways to compensate for stiction until the next opportunity for repair/replacement arrives. Stiction compensation therefore represents the goal of reducing process output variability caused by stiction behavior in valves. 4.1. Existing Valve Stiction Compensation Methods. Despite being a long-standing problem, only a few stiction

compensation techniques have been published. A common means of stiction compensation is to use valve position control (VPC)15,16 but this is largely not feasible or may prove to be costly as most existing process industry valves do not have positioners.27 Recently, the idea of using dither signals for stiction compensation was put forth. Ha¨gglund17 first proposed the use of a dither signal known as a “knocker” added to OP to compensate for the phenomena of stiction. Srinivasan et al.2,3 followed this up by developing recommendations for knocker settings based on their stiction quantification approach. In our research, we simulated the knocker approach for compensation and found that the best compensation resulting in the lowest ISE reduction ratio [ISE/ISEORG, ratio of integrated squared error (ISE) of the compensated process to the ISE of the original process over the same time span] occurs when the time between subsequent dither signals was insignificant, while Srinivasan et al.’s recommendations for the knocker parameters (as mentioned in their work) only produced a locally optimal solution. 4.2. New Approach to Valve Stiction Compensation. Given the aforementioned observations with the knocker approach, we propose a new compensator, which does away with the periodicity of the dither signal approach and implements a constant amplitude addition to the OP signal (Figure 4). The formula is given as R(t) ) a × sign(∆u)

(5)

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Figure 8. Plot of valve input and PV over time for (a) flow control system (weak stiction), (b) flow control system (strong stiction), (c) level control system (weak stiction), and (d) level control system (strong stiction). The compensator kicks in at time ) 600 and 5000 s, respectively, for the systems.

where R(t) is added to u. The new compensator thus provides constant reinforcement of the OP signal, and the technique is referred to as the “constant reinforcement” (CR) approach. If controller output u is constant, there is no addition to its signal. The recommendation for compensation is to use the estimated stiction parameter f as a. This is to allow for the controller to respond to static friction at every possible control move, particularly in the S region, thereby resulting in a more aggressive controller action, thus reducing variability in the output variable. This recommendation (to be justified in section 6) suggests a strong tie between the new compensation technique and our modified Hammerstein identification approach (subsection 3.2). The unified framework for stiction diagnosis and compensation will be described next.

f is larger than 0.05, stiction is assumed to be significant and a need for its compensation arises. The threshold was determined after observing the results of the quantification algorithm on various processes in the presence and absence of stiction and allows us to say with confidence that significant stiction that requires compensation is present. The “constant reinforcer” is then activated using the estimated value of f to compensate for the process variability caused by stiction. The stiction quantification scheme has been realized in the form of MATLAB codes, which can be made available to interested readers upon request, along with SIMULINK files that incorporate the proposed CR compensation approach.

5. Combined Framework for Stiction Diagnosis and Compensation

In this section we test the effectiveness of the proposed combined framework using closed loop simulations that use Choudhury’s stiction model for the valve. In effect, we allow for a structural mismatch between our stiction model and the “true” stiction phenomena. Two examples are presented here: a flow control system and a level control system as used in two independent papers.4,19 The block diagrams for these processes are shown in Figure 6. The transfer functions are given by

In previous sections, we introduced a new stiction quantification technique and the new CR approach to stiction compensation. Here, we summarize the unified framework for stiction diagnosis and compensation in industrial process control loops. This can be seen in Figure 5. Assume that the OP and PV data can be automatically collected at specific time intervals. The data is then preprocessed and the valve stiction is quantified. If

6. Simulation Results

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PF(S) )

1 1 -s , P (S) ) e 0.2s + 1 L 15s

(6)

where the time constants are in minutes. The controller settings have been adopted from He et al.4 (shown in Table 1). For both systems, the scenarios of weak stiction (fS ) 0.65, fD ) 0.35) as well as high valve stiction (fS ) 3, fD ) 2) in the process are considered. Additionally, a case for sinusoidal disturbance substituting for valve stiction will be shown for the level control system, with the disturbance given by D ) 5 sin(30t)

(7)

where D is the disturbance and t the time in minutes. Limit cycles may also be caused by other factors, such as valve saturation.4 Thus, an additional case of limit cycles caused by valve saturation is also considered. For the case of the flow Figure 10. Plot of valve input and PV over time for level control system (strong stiction) with band-limited white noise power of (a) 0.0001, (b) 0.0002, and (c) 0.0003. The compensator kicks in at time ) 10000 s. Table 5. Table of Influence of Noise on Level Control System

Figure 9. Plot of valve input and PV over time for level control system (strong stiction) with (a) sampling time ) 25 s and (b) 100 s. The compensator kicks in at time ) 5000 s. Table 4. Table of ISE Reduction Ratios for Level Control System (Strong Stiction) at Various Sampling Rates sampling time (s)

ISE/ISEORG

25 50 100

3.3 × 10-3 3.4 × 10-3 3.3 × 10-3

noise power

f (%)

ISE/ISEORG

0.0001 0.0002 0.0003

2.81 4.41 2.09

1.0 × 10-2 1.2 × 10-2 1.7 × 10-2

control, the discrete controller and the compensator implemented both have a sampling time of 1 s, while it is 50 s for the level control system. 6.1. Diagnosis Results. The results of stiction diagnosis by our proposed algorithm are shown in Table 2. The parameter f is seen to estimate closely the static friction force fS and correctly identify the absence of stiction where applicable. More precisely, it is exactly half of the stiction model’s deadband plus stickband region S (equivalently, half the combined value of fS and fD). In fact, while not completely equal to fS, the estimated f values provide a stronger relationship to the best possible compensation using our proposed approach than fS itself, as we will see in the next subsection. It is for this reason that we decided not to identify our parameter as fS per se. Table 2 also shows the results of stiction diagnosis using the method proposed by Srinivasan et al.2,3,13,14 The values obtained for their parameter d suggests that their algorithm is able to correctly distinguish cases where stiction is present from those where it is not. However, it can also be seen that the estimated d values bear no direct correlation to the static friction force fS. 6.2. Compensation Results. The results with the use of the proposed CR compensator in cases where stiction is deemed to exist are summarized in Table 3. In Table 3, we use the ISE reduction ratio (ISE/ISEORG), which is the ratio of ISE of the compensated process to the ISE of the original process over the same time span. The results suggest that the CR compensation approach provides excellent compensation for the presence of valve stiction. The CR compensation is seen to reduce ISE by several orders of magnitude. The next question then is whether the recommended value of f for the “reinforcement” amplitude “a” provides for the best compensation or whether another value will result in an even lower ratio. The ISE reduction ratios versus amplitude for the flow and level control loops are shown in Figure 7. As can be observed, the minima on these plots occur at amplitude values that are close to the estimated f, suggesting that the recommended parameter provides for the

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Figure 11. Plot of valve input and PV over time for (a) flow control system (weak stiction) and (b) level control system (strong stiction), using Kano’s model. The compensator kicks in at time ) 600 and 5000 s, respectively, for the systems. Table 6. Analysis of Industrial Control Loop Data Using Lee’s and the Proposed Quantification Technique loop no.

fS′ (%)

fD′ (%)

f (%)

1 2 3 4 5 6

26.0 41.0 0.0 85.0 5.0 39.0

1.0 9.0 0.0 0.0 2.0 32.0

19.3 33.4 0.0 47.3 0.4 5.0

best compensation in all cases. Also, as can be seen for the case of the level control system, the greatest reduction lies closer to the estimated f as compared to the actual value of fS used in the simulated sticky valve. Further insights can be made on the trends shown in Figure 7. The plots show a huge drop in ISE reduction ratios right before the minimal points on the plots. This is because in this region the compensator provides more aggressive action but is yet not able to overcome static friction immediately; hence, stiction behavior still persists. Only when static friction can be adequately overcome, that is, in the points from and close to f, will there be a significant reduction in ISE reduction ratios, since the compensator is able to prevent the valve from sticking each time the controller takes action. Therefore, the benefit of system identification using our proposed one-parameter stiction model in relation to others found in the literature2,3,11-14 lies in the above results. The estimated value of f used in the compensator achieves stiction compensation that is close to the best possible (if not the best possible). Hence, the estimation of f and its use in the CR compensation scheme makes for an integrated framework. Other stiction diagnosis schemes, in particular those tjat utilize a two-parameter model,11 may present a more comprehensive estimate of the actual stiction phenomena that is present in the process. Our aim here is to have an effective stiction compensation scheme based on the estimation of stiction severity using a one-parameter model. Figure 8 shows plots of valve input and PV versus time for all four control loops with stiction. After the compensator kicks in, it can be seen that the frequency of oscillations in both the valve input (note that the amplitude of the valve input signal does not increase) and process variable PV

increases. However, there is a huge decrease in PV variability once the CR compensator kicks in. Therefore, ISE is reduced at the expense of introducing a high-frequency signal into the valve, which in all likelihood will cause accelerated wear and tear. Hence, we stress once again the short-term nature of our compensation scheme and the need for a long-term objective of repairing or replacing any sticky valve. Srinivasan et al.13 recently introduced two alternate compensation schemes that explicitly consider this tradeoff. However, to apply their approach, one would require a priori knowledge of the plant model (or at least an estimate of it). Notice that, upon compensation, the main trend in the response of PV lacks the distinct shape characteristic of stiction behavior. Observation of Figure 8 brings about another question: will the sampling time for the process and the frequency of control action affect compensation results? As already mentioned, for the case of the level control system, both the discrete controller and the compensator implemented have a sampling time of 50 s. The results shown in Figure 9 indicate that the sampling time does minimally affect the compensation results. The ISE reduction ratios in these cases are presented in Table 4. As can be seen, the numbers differ only marginally from one another. Extensive simulations indicate that the stiction quantification and compensation scheme proposed here is able to reduce the process variability for a wide range of sampling intervals. The next point of contention is the effect of measurement noise on the proposed unified quantification and compensation framework. Although extensive investigation into this area remains to be done, preliminary simulations show that the framework does provide good estimates for stiction, where present, and satisfactory compensation thereafter, even without any form of denoising on the data set. The preliminary results are shown in Table 5 for the level control loop operating under high stiction and sampling time of 50 s, with measurement noise modeled by band-limited white noise. The noise is added to the PV signal leaving the level process block. It is this modified signal that is recorded as well as passed into the feedback loop; accordingly, noise affects the entire process directly, dissimilar to simulations whereby only the recorded signal has an added noise component. Figure 10 shows the degree of compensation that can be achieved for the process under noise. At a noise power of 0.0001, the process is still dominated by valve stiction and produces the characteristic limit cycles in the data. Stiction was adequately compensated for with the proposed framework. However, starting with a noise power of 0.0002 and larger, noise starts to exert a greater influence on the entire process, making the OP-PV data set less regularly oscillatory. Consequently, quantification is affected and thus also the subsequent compensation. We wish to stress that although the noise power used in our preliminary study may appear low, its actual significance has to be viewed relative to the process variable itself. In this case, PV exhibits cyclical movement in a very narrow range of values; hence, a low noise power has considerable influence. More research could be done to characterize, compare, and tune the stiction quantification and compensation approaches when the data is significantly noisy. Recently, it was shown that only a selected number of valve models exhibit behavior that fit with the ISA standards for valve stiction.9 One such model is the Kano16 model. Hence, additional simulations were run using this model to represent the sticky valve. The performance of two of the

3482 Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009

scenarios under our proposed scheme is shown in Figure 11. Once again, it is seen that the process variability is considerably reduced. 7. Industrial Control Loop Data Analysis In section 6, we presented the results of the combined framework for stiction quantification and compensation when tested under simulation. Due to time and resource constraints, the proposed compensation technique has yet to be tested in experimental or industrial settings. In this section, results of stiction quantification using industrial data are presented. The same industrial loop data set as presented in the work by Lee et al.12 is used. Results of the analysis are presented in Table 6, together with the two-parameter estimates (fS′ and fD′) used by Lee et al. in their own work.12 Upon inspection, it can be seen that both methods are able to successfully determine the absence of stiction in loop 3. Also, given the earlier observation that the estimate f was observed to be half of the actual S region when stiction is present, both techniques produce results that correlate well with one another, except for loops 5 and 6, where milder stiction is reported using our proposed algorithm. However, even though the latter results are not directly comparable, both still indicate that loop 6 has stiction that is significantly stronger than that in loop 5. 8. Conclusions In this work, a new unified framework for stiction diagnosis and compensation has been introduced. For the problem of stiction identification, we proposed a modified stiction algorithm that specifically targets stiction behavior in the deadband plus stickband region, where static friction force is most prevalent. The proposed diagnosis was shown to be robust in flagging stiction. A new compensation technique was also presented, and simulations not only provided exemplary results but also showed the intrinsic relationship between our stiction identification and compensation schemes. However, similar to compensation using a knocker, lower variability in PV is achieved at the expense of greater frequency of valve stem oscillations, albeit at a lower amplitude than without compensation. In all practical cases, this will most likely have a detrimental effect on the valve. We thus find it prudent to emphasize that stiction compensation should always be viewed as a temporary solution to stiction behavior in process control loops; priority should always be placed on the repair or replacement of the valve as soon as the situation permits one to do so. Nomenclature OP, u ) controller output MV, x ) valve position PV ) process output SP ) set point S ) deadband plus stickband J ) slip-jump d ) stiction parameter for Srivinisan et al.’s model f ) stiction parameter for proposed model fS ) static friction fD ) kinetic friction r ) test statistic for regularity of oscillations t ) time (s) Tj P ) mean period of oscillation (s) V ) objective function x˜ ) predicted model valve position

y ) actual process output y˜ ) predicted process output ε ) error term σTP ) standard deviation of mean period of oscillation (s) θN ) ARMAX model parameters AP ) apparent stiction NGI ) non-Gaussianity index NLI ) nonlinearity index SI ) stiction index

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ReceiVed for reView June 18, 2008 ReVised manuscript receiVed January 24, 2009 Accepted January 29, 2009 IE800961F