Valve Stiction Quantification Method Based on a ... - ACS Publications

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Valve Stiction Quantification Method Based on a Semiphysical Valve Stiction Model Q. Peter He*,† and Jin Wang*,‡ †

Department of Chemical Engineering, Tuskegee University, Tuskegee, Alabama 36088, United States Department of Chemical Engineering, Auburn University, Auburn, Alabama 36849, United States

‡

ABSTRACT: Valve stiction is one of the most common equipment problems that can cause poor performance in control loops. Consequently, there is a strong need in the process industry for noninvasive methods that can not only detect but also quantify stiction. In this work, on the basis of a physical and a semiphysical model, a new valve stiction signature is proposed. Industrial evidence is provided to validate the new valve stiction signature. Although valve stiction is a stochastic phenomenon that can not be exactly described by any deterministic model, the revised valve signature provides a better description of sticky valve behavior, particularly when valve stiction is severe. On the basis of the revised valve stiction signature, a noninvasive, simple, and robust valve stiction quantification method is proposed using the routine operation data and limited process knowledge. The proposed quantification method estimates the stiction parameters, namely, static friction and dynamic or kinetic friction, without requiring the valve position signal. Quantification is accomplished by using linear and nonlinear least-squares methods which are robust and easy to implement. The properties of the proposed algorithm are investigated using simulated case studies of first order plus time delay processes, and the performance of the method is compared to other stiction quantification methods using 20 industrial cases.

1. INTRODUCTION

To detect or confirm valve stiction, many methods have been developed in the past decade.1−3,6,9−17 Different detection methods can be roughly categorized into two major groups: shape-based methods and correlation-based methods. Alternatively, valve stiction can be confirmed through the controller gain change method.17,18 The representative methods are collected in Jelali and Huang.19 Once valve stiction is detected, it is desirable to quantify the severity of the stiction so that the appropriate action can be taken: leaving it alone, compensating by control, or requiring immediate valve replacement. Quantification of valve stiction remains a challenging issue despite several approaches having been proposed in recent years. Choudhury et al.20 proposed a quantification method by fitting the filtered process variable (PV) and controller output (OP) using ellipses. Later, Choudhury et al.21 proposed an iterative optimization procedure to identify both the stiction model parameters and the process model simultaneously based on a two parameter model.7 Lee et al.22 proposed a system identification-based method for stiction quantification. In Lee’s method, a suitable model structure of valve stiction is chosen prior to conducting valve stiction detection and quantification; then, given the stiction model structure, a bounded search space of a stiction model is defined and a constrained optimization problem is performed. Later, the same authors proposed stiction estimation using constrained optimization and a contour map.23 Jelali proposed a method to quantify valve stiction in control loops using separable least-squares and global search

Driven by tightened safety regulations and commercial benefits, valve stiction has drawn considerable interest in both industry and academia in the past decade.1−7 The research on valve stiction can be broadly categorized into the following four topics: modeling, detection or confirmation, quantification, and compensation. The four topics are closely related, with stiction modeling serving as the foundation for detection, quantification, and compensation. In this section, existing research on stiction modeling, detection, and quantification is briefly reviewed. To investigate the valve stiction behavior, deterministic models, both physical and empirical (or data-driven), have been developed in the past decade. Physical models8 describe the stiction phenomenon using force balances based on Newton’s second law of motion. They provide a mechanistic description of valve behavior but also have two disadvantages: first, they require knowledge of several parameters such as the mass of the moving parts and different types of friction forces which cannot be easily measured; second, implementation of these models can be tricky due to the stiff ODE system. To address these difficulties, several data-driven models have been proposed.5−7 Compared to physical models, these data-driven models significantly simplify the simulation of valve behavior and have been excessively used in the study of valve stiction. On the other hand, the disadvantages of data-driven models include the following: first, for some data-driven models, there is no direct relationship between the model parameters and the physical properties of a valve; second, different data-driven models are based on different assumptions, and there is a lack of clear understanding on which assumptions agree with the physical valve better. © 2014 American Chemical Society

Received: Revised: Accepted: Published: 12010

March 13, 2014 July 1, 2014 July 10, 2014 July 10, 2014 dx.doi.org/10.1021/ie501069n | Ind. Eng. Chem. Res. 2014, 53, 12010−12022

Industrial & Engineering Chemistry Research

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algorithms.24 The proposed method is based on a Hammerstein model for describing the global system and a separated identification of the linear part, i.e., the transfer function between the manipulated variable (MV) and the process output PV, and the nonlinear part, i.e., the function between the controller output OP and the manipulated variable MV, using available industrial data for OP and PV. Most recently, on the basis of the two-stage procedure,24 Farenzena and Trierweiler proposed a one-stage procedure to estimate stiction parameters.25 In this work, we propose an alternative valve quantification approach using routine process data and limited process knowledge. The main contributions of this work are a revised valve signature derived on the basis of the physical and the semiphysical model we proposed previously,26 as well as a noninvasive, simple, and robust valve stiction quantification method. The rest of the paper is organized as follows. In Section 2, we briefly review the physical model proposed by Kayihan and Doyle8 and the semiphysical model we proposed previously26 and present the revised valve signatures derived on the basis of these models. In Section 3, on the basis of the semiphysical model and the derived valve signature, a noninvasive method is proposed to quantify the severity of valve stiction. Section 4 examines the performance, robustness, and limitations of the proposed method using simulation case studies. An industrial case study is used to further demonstrate the performance of the proposed method. The conclusions are drawn in Section 5.

in the opposite direction to stem motion and is independent of the magnitude of stem velocity v = dx/dt. The combined term vFv is the viscous friction which acts in the opposite direction to stem motion and its magnitude increases linearly with v at a slope of Fv. The term {(Fs − Fc)sign(v)exp[−(v/vs)2]} denotes the Stribeck effect, which is used to account for the valve’s stick−slip (or stiction) behavior, where vs is an empirical Stribeck velocity parameter. The Stribeck term addresses the discontinuity of the friction force going from Fs in magnitude right before the stem starts to move to Fc in magnitude right after the stem starts to move. To facilitate numerical integration, eq 1 is transformed into the state-space representation, (3) ẋ = v mv ̇ = Fa + Fr + Ff + Fp + Fi

and dynamic simulation of a sticky valve can be implemented by integrating eqs 3 and 4 for a given input Fa. However, because of the significant difference between ẋ and v̇ when v → 0, as well as the hash discontinuity caused by the sign function, difficulties in numerical integration exist. These difficulties can be addressed by using a stiff solver and certain approximation. The implementation details of the physical model can be found elsewhere26 with model parameters and their nominal values listed in Table 3. 2.2. He’s Semiphysical Valve Stiction Model. To simplify the simulation of valve stiction, especially for closedloop systems, several data-driven models have been published. Among them, the representative ones are Choudhury’s model,7 Kano’s model27 and He’s model.5 Their similarities and differences have been discussed elsewhere.26 This work focuses on He’s semiphysical model,26 which is an improved version of He’s original model.5 Figure 1 shows the flowchart of He’s semiphysical model with three parameters fs, fd, and K, which are all dimensionless

2. VALVE STICTION MODELING Valve stiction models are the foundation for valve stiction quantification. In this section, we briefly review a physical model8 and a semiphysical model26 for valve stiction. On the basis of the physical and semiphysical models, we derive a revised signature diagram for a sticky valve. Then, supporting evidence is provided to validate the revised valve signature, and its applicability is discussed. 2.1. Physical Model of Valve Stiction. The physical model of a pneumatic valve is derived on the basis of Newton’s second law of motion. Specifically, the force balance equation for the valve stem is the following, M

d 2x = dt 2

∑ Forces = Fa + Fr + Ff

+ Fp + Fi

(1)

where M is the mass of the moving part; x is the relative valve stem position; the pneumatic actuator force Fa = Sa × u where Sa is the area of the diaphragm and u is the actuator air pressure which is also the input to the valve; spring force Fr = −kx where k is the spring constant; Ff is the friction force; Fp is the force caused by fluid pressure drop, Fi is the extra force that maintains the valve in the seat. Among all forces, Fp and Fi are assumed to be zero due to their negligible contributions, and the friction force Ff is the most important component that determines valve dynamics. Ff combines the effects of different forces depending on the status of the valve as detailed below: 2 ⎧ − Fc sign(v) − vFv − (Fs − Fc)sign(v)e−(v / vs) , if v ≠ 0 ⎪ ⎪ Ff = ⎨ − (Fa + Fr ), if v = 0 and |Fa + Fr| ≤ Fs ⎪ ⎪ − Fs sign(Fa + Fr ), if v = 0 and |Fa + Fr| > Fs ⎩

(4)

Figure 1. Flowchart of He’s semiphysical stiction model.

variables as defined in Table 2 and eq 5. The variable e(t) is the current net external force acting on the valve (i.e., the difference between the controller output (or valve input) u′(t) and the previous valve position x′(t − 1)). If the magnitude of e(t) is large enough to overcome the static friction band fs, the valve would jump, causing an overshoot. Otherwise, the valve position will not change. All variables are normalized as dimensionless variables as shown in Table 2, where xm is used as the characteristic length to scale different variables, which denotes the entire valve stroking range, i.e., 0 ≤ x ≤ xm. Other

(2)

where Fc is the Coulomb friction, sometimes also referred to as the kinetic or dynamic friction, which is a constant force acting 12011

dx.doi.org/10.1021/ie501069n | Ind. Eng. Chem. Res. 2014, 53, 12010−12022


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variables and parameters have been listed in Table 1. The dimensionless parameter K is used to account for the overshoot Table 1. Parameters and Their Nominal Values of a Pneumatic Valve8 parameter

description

value/unit (SI)

x v u Sa k m Fc Fs Fv vs xm

stem position stem velocity actuator air pressure diaphragm area spring constant mass of stem and plug coulomb friction static friction viscous friction stribeck velocity valve stroking range

m m/s Pa 6.45 × 10−2 m2 5.25 × 104 N/m 1.36 kg 1423 N 1708 N 1.59 kg/s 2.54 × 10−4 m/s 0.1016 m

observed in the physical model, which has been discussed in detail elsewhere.26 On the basis of the physical model, K can be calculated as follows: ⎛ Fv 2 ⎜ −π 4km K = 1 + OS = 1 + exp⎜ 2 ⎜ 1 − Fv ⎝ 4km

⎞ ⎟ ⎟ ⎟ ⎠

(5)

For the valve with the parameters listed in Table 1, it has been shown that K is insensitive to k, m, and Fv for a wide range of physical parameters.26 For example, a 10-fold difference in Fv around the nominal value in Table 1 will only result in a 2% difference in K, while a 10-fold difference in m or k will result in less than a 1% difference in K. Therefore, if valve physical parameters are not available, K can be assumed to be a value close to 2 (e.g., 1.99) and the three-parameter semiphysical model becomes a two-parameter model. It has been shown in different cases26 that the semiphysical model can accurately reproduce the physical model behavior without involving cumbersome numerical integration. In addition, it was shown that the semiphysical model can satisfactorily simulate several industrial cases.26 2.3. A Revised Valve Signature. The valve input−output characteristic diagram, also called valve signature, is commonly used in industry to characterize a valve.28−30 To generate a valve signature, the valve input (e.g., air pressure) is ramped up and down for a few cycles; then, the valve input and output are plotted to characterize the valve behavior. In this work, we use the valve signature to compare the physical and the semiphysical model. In order to generate the valve signature, the air pressure (i.e., the input to the valve) is cycled from 0 to 12 psi with the rate of 0.02 psi/s. Zero-order-hold (ZOH) is implemented during the sampling interval, which is 1 s. The input signal is shown in Figure 2a. For the physical model, the nominal parameters listed in Table 18 are used, while for the semiphysical model, the parameters are normalized according to Table 2. The corresponding valve signatures obtained on the basis of the physical model and the semiphysical model are shown in Figure 2b. The overlapping trajectories indicate that the two models produce the same valve signature. On the basis of the valve signature shown in Figure 2, if we plot the stable cyclic part only by excluding the initial valve response and scale all variables according to Table 2, we can obtain a normalized valve signature as shown in Figure 3, where

Figure 2. (a) Input signal; (b) valve responses based on the physical and the semiphysical models.

Table 2. Normalized Dimensionless Variables valve input

stem position

static friction

dynamic friction

u′ = (Sau)/(kxm)

x′ = x/xm

fd = Fc/(kxm)

fs = Fs/(kxm)

the bands of fs and fd have been added to the plot to clearly show the overshoot predicted by both the physical and semiphysical models. For comparison, the commonly used valve signature in the literature is shown in Figure 4, where overshoot is not taken into account (i.e., valve jumps (fs − fd), instead of 2( fs − fd)). Comparing Figure 3 with Figure 4, there are two main differences. First, the revised valve signature (Figure 3) shows multiple jumps along the same direction, while the traditional one (Figure 4) does not; second, the revised valve signature cuts through the band of fd due to overshoot, while the traditional one always stays inside of the band of fd. The fundamental reasons for the differences between the two valve signatures boil down to the assumptions adopted by different valve stiction models: (1) whether the valve slips or stops after a jump; (2) whether the valve overshoots when it jumps. It is important to note that there are no definitive answers to these two questions, because friction is a complex phenomena that is determined by the contact surface morphology, material property, and many other factors. It is reasonable to believe that valve stiction is not a deterministic phenomenon but rather a 12012



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expected. However, a “jumpy” response is more likely for a sticky valve if there is zero-order-hold (or during a small step test). It is worth noting that zero-order-hold on an input signal is common in industrial applications for digital controllers. To support our argument on whether the overshoot occurs when the valve jumps, we provide the following industrial evidence. EnTech28 reported a test on a sticky valve, and both the input (air pressure) and output (flow rate) of the valve were measured, which are shown as the top two curves in Figure 5.

Figure 3. New sticky valve signature based on the physical model and the semiphysical model.

Figure 5. Model predictions of an industrial data set. Lines from top to bottom: normalized valve input (%); normalized flow, measured (%); flow predicted by the semiphysical model offset by 1%; flow predicted by He’s original model offset by 2%.

To test whether the overshoot exists when the valve jumps, we compare two different models: He’s original model, which assumes no overshoot when valve jumps, and He’s semiphysical model, which assumes that the overshoot occurs with each jump. Note that both models assume that the valve stops after each jump and the difference between them is whether overshoot occurs with each jump. For both models, we used the valve input and output data to estimate the stiction parameters fs and fd by assuming unit process gain and no delay since the process is just the valve. Then, the valve outputs predicted by different models are compared with measured valve output to evaluate which stiction model agrees with the actual valve better. The prediction results from both models are also plotted in Figure 5 as the bottom two curves. To better visualize the similarities and differences in terms of when and where the valve jumps, we have shifted the model prediction down by 1% for He’s semiphysical model and 2% for He’s original model. As shown in the figure, the trend of both model predictions agreed with the actual measurements; i.e., multiple jumps occur along the same direction. However, the prediction from He’s semiphysical model has the same number of jumps as the actual measurements, while He’s original model5 predicts that the valve jumps twice as often as the actual valve, which is similar to the results we reported before.26 This industrial example confirms that, for the sticky valve under study, it stops (instead of slipping) after each jump and, when it jumps, there is an overshoot of approximately (fs − fd), i.e., K ≈ 2. It is worth noting that, in these reported industrial examples, the levels of valve stiction were relatively severe. It should also be noted that there is no definitive conclusion on whether the revised or the original sticky valve signature is absolutely correct. It is possibly case dependent (e.g., valve type, fluid type, etc.). However, when the level of valve stiction is relatively

Figure 4. Commonly used sticky valve signature with moving or slipping phase and without overshoot.

stochastic process. In other words, for the same sticky valve, it may or may not slip after it jumps when the same driving force is applied in different cycles. In addition, the amount of the overshoot associated with a jump may be different in different cycles as well. Nevertheless, we argue that when valve stiction becomes severe enough to cause a limit cycle in a control loop, it is more likely than not that the valve would stop after each jump, and valve overshoots by ( fs − fd) with each jump (i.e., valve jumps by 2( fs − fd)). To support our argument on whether the valve stops after each jump, we have done a search on industrial valve tests and have found that several industrial practitioners reported the “jumpy” valve responses.28−30 In contrast, we have not found any reported continuous “slippery” response as shown in Figure 4 and elsewhere with multiple jump-slips.21 It is true that, if valve input (e.g., air pressure) changes continuously, a continuous movement or “slipping” of the valve would be 12013



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need to call the valve stiction quantification algorithm. This point is verified in Section 4. Nevertheless, we demonstrate that, under a wide range of controller tunings for a control loop with a sticky valve, only one jump would occur when the limit cycle is reached as illustrated in Section 4 as well. The proposed stiction quantification method is based on a valve’s OP−VP relationship. When a single jump occurs in a limit cycle, the valve signature shown in Figure 3 can be reduced to the OP−VP relationship as shown in Figure 6a, with the corresponding time-series plot of VP shown in Figure 6b. It should be noted that all variables in figures and equations are normalized and dimensionless (according to Table 2) for the rest of this paper. On the basis of the symmetry of AB and BC shown in Figure 6a, we have

severe and causes a limit cycle, we believe that He’s semiphysical model and the revised valve signature provide a better description of the actual sticky valve behavior.

3. VALVE STICTION QUANTIFICATION In this section, we first present the noninvasive valve stiction quantification method derived on the basis of the sticky valve signature generated by the physical and He’s semiphysical model, then we provide two simple illustrative examples to demonstrate the performance of the proposed method. More realistic examples and thorough examination of the proposed method are given in Section 4. 3.1. Quantification Method. The proposed stiction quantification method is based on the revised sticky valve signature as shown in Figure 3. When a stable limit cycle caused by valve stiction is reached, it is assumed that the valve jumps only once before it changes direction and the output of a sticky valve or valve position (VP) resembles a rectangular wave as illustrated in Figure 6b. This assumption is supported by

|BC| = |AB| = fs − fd

(6)

Therefore, the jump band J = |BD| = K (fs − fd )

(7)

with the deadband plus stick S = |GB| = |FC| ≈ 2fd

(8)

In the rest of this section, it is shown that, by making use of OP, PV measurements, and limited knowledge on the process (i.e., process gain Kp and time delay θ), valve stiction parameters can be estimated without requiring measurements of valve position VP. It is worth noting that Kp and θ are not difficult to estimate for most control loops (see, e.g., Bequette31 p.127−147), especially when accurate estimates are not required. 3.1.1. Estimation of fd. In Figure 6a, it can be seen that, because the accumulated input has to be able to overcome fs in order to jump, S or 2fd is approximately (but slightly smaller than) the range of OP under oscillation. Therefore, fd can be estimated directly from the OP signal, which is true for both self-regulating and integrating processes. A self-regulating process is used to illustrate how to estimate fd based on the measurements of OP. Figure 8 shows a segment of the timeseries plot of OP when a sticky valve is present in the control loop. For illustration purposes, time delay is assumed to be zero (i.e., θ = 0), and fd can be estimated as follows.

Figure 6. Valve stiction with a single jump: (a) VP−OP plot; (b) VP− t plot.

various simulation and industrial examples.5,6 However, it is worth noting that there are other reasons that may cause limit cycles, such as poor controller tuning or stable oscillatory disturbance. When the limit cycle is mainly caused by other reasons such poor controller tuning instead of a sticky valve, it is possible that multiple jumps in the same direction (Figure 7) would occur before the valve changes direction. When multiple jumps occur, the assumptions of the proposed valve stiction quantification method are violated and the estimation error would be large. However, because the main cause for multiple jumps to occur before a valve changes direction during a limit cycle is not valve stiction, an effective valve stiction detection algorithm will not classify the valve as sticky and there is no

2fd ≈

(|uB − uA| + |uC − uA|) 2

(9)

The value is calculated on the basis of each half cycle and averaged over multiple cycles to reduce the effect of process and measurement noise. If θ ≠ 0, uB and uC should be moved

Figure 7. Valve stiction with multiple jumps:(a) VP−OP plot; (b) VP−t plot. 12014



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The consideration of time delay is not needed in eq 12 because the fitting segment always starts at the onset of the exponential curve. Also, τ is not needed ahead of time because it is estimated by the curve-fitting process. However, the process gain Kp is needed in order to estimate (fs − fd) as can be seen from eq 12. For a given or estimated Kp, a nonlinear leastsquares method is implemented on the basis of eq 12 to estimate ( fs − fd) using PV of the self-regulating process. For integrating processes, we approximate the process by an integrator, i.e., G (s ) =

Kp (13)

s

For a step input, its corresponding step response is simply Figure 8. OP measurements of a self-regulating process with a sticky valve.

y = y0 + K p

∫ uV dt

(14)

Taking the derivative of the above equation, backward by θ, while uA stays as the peak/valley point of that cycle. For online application, a sliding/rolling window approach or exponentially weighted moving average (EWMA) can be implemented. 3.1.2. Estimation of fs. fs can be estimated after the estimation of J or 2( fs − fd) as shown in Figure 6a. In this case, self-regulating and integrating processes require different approaches. For a self-regulating process, the process output PV can be approximated by the superposition of multiple step responses with different (positive and negative) input magnitudes. The input magnitudes correspond to the rectangular or square wave input resulted from the limit cycle. If we approximate the selfregulating process with a general first order plus time delay (FOPTD) model as shown below,

Gp =

uV =

uV+ =

uV − =

(

)

(16)

1 (Δy)↓ 1 slope (y↓ ) < 0 = K p (Δt )↓ Kp

(17)

On the basis of the valve characteristic shown in Figure 6 and eq 7, 2(fs − fd ) = uV + − uV −

where Kp is the process gain, θ is the time delay, and τ is the time constant, the step response of a single jump in valve input with magnitude J = K(fs − fd) (Figure 6) is y(t ) = K p[K (fs − fd )] 1 −

1 (Δy)↑ 1 slope (y↑ ) > 0 = K p (Δt )↑ Kp

while for PV descending portion y↓,

(10)

t−θ e− τ

(15)

For oscillations caused by valve stiction, the valve position uV follows a rectangular wave; therefore, for PV ascending portion y↑ ,

K pe−θs τs + 1

1 dy K p dt

(18)

plugging into eqs 16 and 17, 2(fs − fd ) =

(11)

=

1 (Δy)↑ 1 (Δy)↓ − K p (Δt )↑ K p (Δt )↓ slope (y↑ ) − slope (y↓ ) Kp

(19)

or (fs − fd ) = Figure 9. A first-order process response (red solid line) to a square wave (black dashed line) valve input.

slope (y↑ ) + slope (y↓ ) 2K p

(20)

For a given or estimated Kp, on the basis of eq 20, ( fs − fd) can be obtained via a simple least-squares method. It is worth noting that, similar to the estimation of fd, multiple cycles are used to estimate (fs − fd) on the basis of eq 12 for selfregulating processes and eq 20 for integrating processes. Once both fd and ( fs − fd) are estimated, fs can be calculated. 3.2. Remarks. If there is a time delay (i.e., θ ≠ 0), the end point of each cycle (for both OP and PV) needs to be shifted backward by θ while the starting point remains as the peak/ valley. This is due to the fact that, if there is a time delay of θ, the switching of OP to the other direction is delayed by θ. The

When the effect of only one previous cycle is considered as shown in Figure 9, the square wave response y(t) can be estimated by superposition of three step responses: t + t1 t + t1+ t 2 ⎞ t ⎛ y(t ) = K p[K (fs − fd )]⎜1 − e− τ + e− τ + e− τ ⎟ ⎝ ⎠

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Figure 10. Cycle switching points detected by PeakFinder: (a) OP; (b) PV.

effect of delay on valve stiction quantification is investigated in Section 4. Stiction has to be detected before utilizing the quantification method. Otherwise, for example, if oscillation is caused by other factors such as controller tuning or external disturbance, misleading results could be obtained by the proposed quantification method as shown in Section 4. To accurately estimate fd and fs, some process knowledge (i.e., Kp and θ) is required. In general, both Kp and θ can be estimated relatively easily through steady-state material and energy balances, or a single step test, or system identification techniques based on good historical data.31,32 In addition, as shown in Section 4, the proposed stiction quantification method is quite robust. Therefore, accurate estimation of Kp and θ is not required to obtain a reasonably accurate estimate of fd and fs. The proposed quantification method requires a good and robust method to detect cycle switching points in the oscillating OP and PV signals. PeakFinder developed by Yoder33 is a good candidate for this purpose, which handles noise much better than the strictly derivative-based peak finding algorithms. In addition, PeakFinder requires little computation time even for large data sets. One example is shown in Figure 10, where PeakFinder correctly identifies all peaks in OP. For PV, if we were to find the range of PV, the algorithm does a very good job in identifying exactly one peak and one valley for each cycle, which is nontrivial considering the noise around the peaks/ valleys. However, the proposed valve quantification algorithm requires the identification of the periods or cycles of PV for least-squares curve fitting. Therefore, PeakFinder alone cannot serve this purpose as shown in Figure 10b. To address this difficulty, we augment the PeakFinder algorithm with a robust spline smoothing method34 and a least-squares curve fitting. The cycle switching points identified by the proposed algorithm are shown in Figure 11. It can be seen that all the cycle switching points are correctly identified. All Matlab codes, including simulations, peak detection, quantification, etc., will be made available online to the public. 3.3. Illustrative Examples. In this subsection, we use two simple simulated examples to illustrate how the proposed valve stiction quantification method works. We consider first order processes, one self-regulating (flow) process, and one integrating (level) process. The transfer functions for the selfregulating and integrating processes are given by Gf (s) =

1 12s + 1

Figure 11. Cycle switching points in PV detected by the proposed method. The light-color thin line is the fitted PV.

G l (s ) =

1 −s e 5s

(22)

PI controllers are used for both control systems and their transfer functions are given by flow control: ⎛ 1⎞ Gc = 1⎜1 + ⎟ ⎝ 6s ⎠

(23)

level control: ⎛ 1 ⎞ ⎟ Gc = 2⎜1 + ⎝ 90s ⎠

(24)

Two cases are examined for both systems: weak stiction and strong stiction. Valve stiction model parameters are summarized in Table 3. Table 3. Valve Stiction Model Parameters case

severity

fd

fs

1 2

weak stiction strong stiction

0.07 0.27

0.08 0.30

Note that fs = 0.3 represents a static friction force that is equivalent to the spring force to open 30% of the valve (i.e., fs = 0.3kxm), which describes a valve with severe stiction. Simulated OP and PV for the flow control are shown in Figure 12, and stiction quantification results are listed in Table 4. Simulated OP and PV for the level control are shown in Figure 13, and stiction quantification results are listed in Table

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Figure 12. Flow control: (a) case 1, weak stiction; (b) case 2, strong stiction.

Table 4. Valve Stiction Quantification for the Flow Control System

Table 5. Valve Stiction Quantification for the Level Control System

case

severity

fd̂ (error %)

fŝ (error %)

case

severity

fd̂ (error %)

fŝ (error %)

1 2


0.070(0.13%) 0.270(0.00%)

0.082(2.26%) 0.303(1.06%)

1 2


0.067(3.71%) 0.262(2.96%)

0.081(0.98%) 0.303(1.06%)

On the basis of the tuning relationships given in Seborg et al.35 (Table 12.1 Case G, on p. 308), with a desired closed-loop time constant of 60s (i.e., (1/2)τ), the obtained controller setting is the following

5. From Tables 4 and 5, it can be seen that the proposed method successfully quantifies the degree or severity of stiction for both self-regulating and integrating processes. The stiction parameters estimated on the basis of OP and PV (i.e., fŝ and fd̂ ) agree well with the true values; all relative errors are less than 5%.

⎛ 1 ⎞⎟ Gc(s) = 2⎜1 + ⎝ 120s ⎠

4. CASE STUDIES In this section, more realistic simulated case studies are used to further investigate the robustness and reliability as well as the limitations of the proposed method. In addition, an industrial case study is provided to further demonstrate the performance of the proposed method. 4.1. Simulated Case Studies. In this subsection, we use a more realistic first order plus time delay (FOPTD) process as the plant model in all simulations, as most industrial processes can be approximated by a FOPTD process. Also a PI controller is applied to control the plant. The nominal process transfer function is G (s ) =

e − 6s 120s + 1

(26)

Below, we examine how the uncertainty of the process model (i.e., error in process time constant τ, process delay θ, or process gain Kp) would affect the quantification performance. 4.1.1. Effect of Process Time Constant. In this case study, we conduct simulations for 14 different processes using the following plant model: G (s ) =

e − 6s τs + 1

(27)

where τ = 60 + i × 30 with i = 1, 2, ..., 14. The same controller as shown in eq 26 is applied to control different processes. Since the controller was tuned for the nominal process eq 25, such configuration would result in different levels of controller tuning (sluggish, optimal, or aggressive) for different processes. For each simulation, valve stiction is introduced with fs = 0.20

(25)

Figure 13. Level control: (a) case 1, weak stiction; (b) case 2, strong stiction. 12017



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Figure 14. Effect of τ: (a) 90s; (b) 210s; (c) 360s; (d) 480s. The thin white lines overlapping the PV signals are the fitted curves based on eq 12, which show excellent fitting.

For two cases where τ is significantly larger than the nominal time constant, i.e., τ = 450 and 480, the quantification performance deteriorates notably. By examining both cases in detail, we found that such deteriorated performance is caused by multiple jumps of the sticky valve before it changes direction as shown in Figure 14d. However, it should be noted that, when τ ≫ τ,̅ the controller, which was tuned on the basis of the nominal model, becomes an overly aggressive controller, which is more likely to cause the loop oscillation. To verify our point, we performed valve stiction detection using the method reported in ref 5 and listed the stiction indices (SI) and detection results in Table 6. It can be seen that for large τ, the oscillation is primarily caused by tuning instead of stiction as they were not detected as stiction. The results confirm that, when a sticky valve is clearly detected, the proposed quantification method works very well. 4.1.2. Effect of Process Delay. In this case study, we conduct simulations for 7 different processes following the process model given below:

and fd = 0.18. Figure 14 shows the simulated process data for 4 different cases: τ = 90, 300, 450, and 480. The valve stiction quantification results for all cases are given in Table 6. Note that the process time constant τ is not needed Table 6. Effect of τ on Stiction Quantification case

τ

fd̂ (error %)

fŝ (error %)

SIa

stiction?a

1 2 3 4 5 6 7 8 9 10 11 12 13 14

90 120 150 180 210 240 270 300 330 360 390 420 450 480

0.177(1.7%) 0.177(1.7%) 0.178(1.1%) 0.178(1.1%) 0.180(0.0%) 0.184(2.2%) 0.181(0.6%) 0.186(3.3%) 0.187(3.9%) 0.189(5.0%) 0.187(3.9%) 0.196(8.9%) 0.210(16.7%) 0.222(23.3%)

0.198(1.0%) 0.195(2.5%) 0.199(0.5%) 0.197(1.5%) 0.196(2.0%) 0.198(1.0%) 0.196(2.0%) 0.198(1.0%) 0.199(0.5%) 0.199(0.5%) 0.198(1.0%) 0.206(3.0%) 0.227(13.5%) 0.241(20.5%)

0.99 1.00 0.99 0.95 0.96 0.98 0.83 0.89 0.84 0.79 0.64 0.51 0.38 0.39

yes yes yes yes yes yes yes yes yes yes yes undetermined no no

G (s ) =

On the basis of He et al.,5 SI ≥ 0.6 indicates stiction, SI ≤ 0.4 indicates no stiction, and it is undetermined if 0.4 < SI < 0.6.

a

e−θis 120s + 1

(28)

where θi = i × 5 with i = 1, 2, ..., 7. Similar to the previous subsection, the same PI controller in eq 26 is used to control the process, and valve stiction is introduced with fs = 0.20 and fd = 0.18. Figure 15 shows the simulated process data for 2 different cases: θ = 10s and 35s. Because information on process delay is needed in the proposed quantification method, we evaluate the effect of the process delay on the quantification performance by comparing two cases: with true delay information (results given in Table 7) and without true delay information (results given in Table 8, where delay is fixed at 17s for all cases). From Table 7, we see that, with true delay information, the quantification results are quite accurate, with all errors less than 2%; while when the

for stiction quantification. From Table 6, we see that for most cases the quantification results are reasonably accurate, with relative errors less than 5%. It is worth noting that, when process time constant is large, PV usually does not reach its steady state before the valve changes direction, as shown in Figure 14b,c. For these cases, PV behaves quite differently from VP. Table 6 shows that the proposed quantification method can provide accurate estimates of fs and fd, regardless of whether a steady state is reached. 12018



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Figure 15. Effect of θ: (a) 10s; (b) 35s.

Table 7. Effect of θ on Stiction Quantificationa

a

case

θ

fd̂ (error %)

fŝ (error %)

1 2 3 4 5 6 7

5 10 15 20 25 30 35

0.177(1.7%) 0.178(1.1%) 0.180(0.0%) 0.178(1.1%) 0.179(0.6%) 0.178(1.1%) 0.179(0.6%)

0.197(1.5%) 0.199(0.5%) 0.201(0.5%) 0.199(0.5%) 0.200(0.0%) 0.199(0.5%) 0.200(0.0%)

Table 9. Effect of Kp on Stiction Quantificationa

True θ is used in quantification.

Table 8. Effect of θ on Stiction Quantificationa

a

case

θ

fd̂ (error %)

fŝ (error %)

1 2 3 4 5 6 7

5 10 15 20 25 30 35

0.173(3.8%) 0.175(3.0%) 0.178(1.2%) 0.179(0.7%) 0.181(0.6%) 0.182(1.1%) 0.184(2.0%)

0.193(3.3%) 0.195(2.3%) 0.199(0.7%) 0.199(0.3%) 0.202(0.9%) 0.203(1.7%) 0.204(2.2%)

a

case

Kp

fd̂ (error %)

fŝ (error %)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%) 0.177(1.4%)

0.213(6.7%) 0.207(3.7%) 0.203(1.6%) 0.200(0.0%) 0.197(1.3%) 0.195(2.3%) 0.194(3.1%) 0.192(3.8%) 0.191(4.3%) 0.190(4.8%) 0.189(5.3%) 0.189(5.6%) 0.188(6.0%) 0.187(6.3%) 0.187(6.5%) 0.186(6.8%)

Kp = 1 is used in simulation.

search and filtering (GSF),36 separable least-squares (SLS),19,24 constrained optimization and contour map (COCM),19,23 and a Hammerstein model identification method proposed by Karra and Karim (KK).19 Table 10 provides stiction quantification results on 20 industrial loops where the real root causes for malfunction are known.19 Results of GSF and COCM were obtained using the codes provided by the proposing authors. Those results were obtained by using the default parameters. Other results taken from the literature19 are marked by asterisks. The cells filled with a short dash (-) indicate that the loop was not analyzed in their original publication. It can be seen that, for the cases where stiction is known to be the rootcause of oscillation, the estimated fd ( fd̂ ) by the proposed method is in good agreement with the published methods. Note that, since the process gain is unknown for all loops, only fd was estimated. However, by comparing with other methods where both fd and fs were estimated, it is confirmed that fd alone provides a good sense of stiction severity. It should also be noted that the proposed method does not differentiate other causes from stiction. In other words, the estimation would be misleading for the case of nonstiction caused oscillation. However, this is also true for other methods as can be seen from Table 10. Therefore, it is not only natural but also necessary to confirm stiction by detection method(s) before trying to quantify the severity of the stiction. Next, an industrial example is used to further demonstrate the performance and robustness of the proposed method.

θ = 17 is used in quantification.

information on delay is not accurate (Table 8), the quantification performance is still reasonably good, with errors around 3% when model plant mismatch is over 100%. 4.1.3. Effect of Process Gain. Equations 9, 11, and 20 show that the estimate of fd does not depend on Kp while the estimate of fs does. In this case study, we examine the effect of the accuracy of Kp on the accuracy of the estimated fs. We use the nominal process model (eq 25) with process gain Kp = 1 and PI controller (eq 26) to simulate process data and then use different Kp values to perform stiction quantification. Totally, 16 different cases are compared, with model-plant mismatch ranging from −50% to 100%. The results are listed in Table 9. When the plant model mismatch is within 100%, the estimate error of fs is less than 7%, which is tolerable for most industrial applications. Such limited effect of inaccurate Kp on the estimated fs can be explained as follows: K̂ p directly affects ( fs − fd) not fs. Because ( fs − fd) is usually small, or in other words, the dominant part of fs is still fd instead of fs − fd, the accuracy of the estimated fs depends more on the accuracy of fd instead of fs − fd. Therefore, even for large model-plant mismatch on Kp, the error of estimated fs is still limited. 4.2. Industrial Case Study. In this subsection, the proposed method is compared with several published stiction quantification methods, including a methodology based on grid 12019



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Table 10. Valve Stiction Quantification for the Industrial Processa

a

loop

stiction?

this work fd̂ (%)

fd̂ (%)

GSF fŝ (%)

fd̂ (%)

COCM fŝ (%)

fd̂ (%)

SLS fŝ (%)

fd̂ (%)

KK fŝ (%)

CHEM 1 CHEM2 CHEM 3 CHEM 6 CHEM 10 CHEM 11 CHEM 12 CHEM 13 CHEM 14 CHEM 16 CHEM 23 CHEM 24 CHEM 29 CHEM 32 PAP 2 PAP 4 PAP 5 PAP 7 PAP 9 MIN 1

yes yes no yes yes yes yes no no no yes yes yes yes yes no yes no no yes

0.4 4.2 0.8 0.2 1.0 0.9 0.9 1.1 1.3 1.5 14 9.7 5.6 11 1.7 7.5 0.2 0.1 1.9 0.8

0.0 1.9 0.0 0.0 0.2 0.2 0.6 0.0 0.6 0.0 11.8 0.0 0.0 0.0 0.6 0.1 0.3 0.3 2.9 0.4

0.0 1.9 0.2 0.0 1.7 0.2 1.1 0.6 1.2 0.1 15.3 0.0 0.0 0.0 3.3 2.5 1.6 0.3 4.6 0.8

0.2 1.4 0.0 0.0b 0.0b 0.0b 0.6b 0.5 0.6 3.3 11.6 9.7b 1.9 6.1 0 2.1b 0.0 0.0 0.0 0b

0.3 1.6 0.0 0.0b 1.7b 0.1b 0.8b 1.7 0.8 5.1 12.2 10.9b 2.6 6.4 1.7 2.1b 0.0 0.1 0.0 1.2b

11.1b 1.1b 0b

11.9b 1.9b 1b

0.9b 0.2b 11.0b -

1.0b 0.3b 12.0b -

A short dash (-) indicates that the loop was not analyzed in the original publication. bValues were obtained from ref 19.

quantification of valve stiction and a proper choice of Kp. The predicted valve position time series is shown in the middle panel of Figure 16. On the basis of the estimated process model, we predict PV using the predicted valve position VP ,̂ which is shown as the green (lighter) line in the bottom panel of Figure 16. We can see that they agree well with each other, which indicates that the proposed method provides satisfactory performance for this challenging case. The estimated fs and fd given in Table 11 indicate significant valve stiction, which is consistent with the high valve stiction detection index (SI) shown in Table 11.

Figure 16 plots the OP (top panel) and PV (blue (dark) line, bottom panel) of the industrial data. The data was collected

Table 11. Valve Stiction Quantification for the Industrial Process fd̂

fŝ

SI

0.139

0.142

0.95

It is worth noting that, because of the likely stochastic nature of valve stiction, the stiction parameters (i.e., fs and fd) estimated on the basis of different cycles are not exactly the same and the estimated valves in this work are all average values across multiple cycles. Therefore, in reproducing/predicting valve positions (VP), the estimated fs and fd were scaled down to compensate for the averaging effect so that the valve jumps whenever OP changes direction. The scaling factor is 0.9 for the industrial case study, and it depends on the variability of fs and fd from different cycles: the higher the variability, the smaller is the scaling factor.

Figure 16. Industrial case study. Top, OP; middle, predicted VP;̂ bottom, PV (blue) and predicted PV or PV ̂ (green).

from a chemical flow control loop (CHEM 23), and no information on the process gain, time constant, or process delay is available.19 Although it is known that the loop has a valve stiction problem, no information on the actual fs or fd is available. As we noted earlier, fd can be obtained with OP alone. Since process delay was not available, it is assumed to be zero for the flow process. However, the estimation of fs requires the process gain. Because none of the actual Kp, fs, and fd is available, we first estimate multiple fs values by using different Kp values; then, we use the identified valve stiction models with the corresponding Kp to predict PV with OP as the input. A good fit between the predicted PV and measured PV would indicate a good

5. SUMMARY AND CONCLUSIONS In this work, we first briefly review a physical model and He’s semiphysical model and derived a revised valve signature on the basis of them. Although it is unclear whether a sticky valve sticks after each move and whether it overshoots when it jumps, industrial evidence was provided to show that, when valve 12020



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(12) Singhal, A.; Salsbury, T. A simple method for detecting valve stiction in oscillating control loops. J. Process Control 2005, 15, 371− 382. (13) Rossi, M.; Scali, C. A comparison of techniques for automatic detection of stiction: Simulation and application to industrial data. J. Process Control 2005, 15, 505−514. (14) Srinivasan, R.; Rengaswamy, R.; Miller, R. Control loop performance assessment. 1. A qualitative approach for stiction diagnosis. Ind. Eng. Chem. Res. 2005, 44, 6708−6718. (15) Srinivasan, R.; Rengaswamy, R.; Narasimhan, S.; Miller, R. Control loop performance assessment. 2. Hammerstein model approach for stiction diagnosis. Ind. Eng. Chem. Res. 2005, 44, 6719−6728. (16) Yamashita, Y. An automatic method for detection of valve stiction in process control loops. Control Eng. Pract. 2006, 14, 503− 510. (17) Haoli, Y.; Lakshminarayanan, S.; Kariwala, V. Confirmation of control valve stiction in interacting systems. Can. J. Chem. Eng. 2009, 87, 632−636. (18) Choudhury, M. S.; Kariwala, V.; Thornhill, N. F.; Douke, H.; Shah, S. L.; Takada, H.; Forbes, J. F. Detection and diagnosis of plantwide oscillations. Can. J. Chem. Eng. 2007, 85, 208−219. (19) Jelali, M., Huang, B., Eds. Detection and Diagnosis of Stiction in Control Loops; Springer: London, 2010. (20) Choudhury, M. S.; Shah, S.; Thornhill, N.; Shook, D. Automatic detection and quantification of stiction in control valves. Control Eng. Pract. 2006, 14, 1395−1412. (21) Choudhury, M. S.; Jain, M.; Shah, S. L. Stiction−definition, modelling, detection and quantification. J. Process Control 2008, 18, 232−243. (22) Lee, K. H.; Ren, Z.; Huang, B. Novel closed-loop stiction detection and quantification method via system identification. Proceedings of Advanced Control of Industrial Processes, Jasper, Canada, 2008; pp 341−346. (23) Lee, K. H.; Ren, Z.; Huang, B. Stiction estimation using constrained optimisation and contour map. In Detection and Diagnosis of Stiction in Control Loops; Jelali, M., Huang, B., Eds.; Advances in Industrial Control; Springer: London, 2010; pp 229−266. (24) Jelali, M. Estimation of valve stiction in control loops using separable least-squares and global search algorithms. J. Process Control 2008, 18, 632−642. (25) Farenzena, M.; Trierweiler, J. Valve stiction estimation using global optimization. Control Eng. Pract. 2012, 20, 379−385. (26) He, Q. P.; Wang, J.; Qin, S. J. An alternative stiction-modelling approach and comparison of different stiction models. In Detection and Diagnosis of Stiction in Control Loops; Jelali, M., Huang, B., Eds.; Advances in Industrial Control; Springer: London, 2010; pp 37−59. (27) Kano, M.; Nagao, K.; Hasebe, S.; Hashimoto, I.; Ohno, H.; Strauss, R.; Bakshi, B. Comparison of statistical process monitoring methods: Application to the Eastman challenge problem. Comput. Chem. Eng. 2000, 24, 175−181. (28) EnTech. Control Valve Dynamic Specification (Version 3.0); Entech: Berwyn, PA, 1998; http://www2.emersonprocess.com/ siteadmincenter/PM PSS Services Documents/Consulting Services/ valvsp30.pdf. (29) Ruel, M. Stiction: The hidden menace. Control Mag. 2000, 69− 75. (30) Gerry, J.; Ruel, M. How to measure and combat valve stiction online. ISA International Fall Conference, Houston, TX, 2001. (31) Bequette, B. W. Process control: modeling, design, and simulation; Prentice Hall Professional: Upper Saddle River, New Jersey, 2003. (32) Bjorklund, S.; Ljung, L. A review of time-delay estimation techniques. Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, 2003; pp 2502−2507. (33) Yoder, N. PeakFinder, 2011; http://www.mathworks.com/ matlabcentral/fileexchange/25500-peakfinder. (34) Garcia, D. Robust smoothing of gridded data in one and higher dimensions with missing values. Comput. Stat. Data Anal. 2010, 54, 1167−1178.

stiction is severe and limit cycles result, it is more likely that the valve would stick (stop) after each jump, and an overshoot exists when the valve jumps. In addition, it is important to note that valve stiction is a stochastic phenomenon that cannot be exactly described by any deterministic model. Next, on the basis of the revised valve signature, a noninvasive valve quantification method is proposed to estimate static and Coulomb frictions fs and fd. The proposed quantification method is a simple curvefitting method, which only requires routine process operation data and limited process knowledge. The proposed method is straightforward to implement with no convergence issue, and the Matlab codes will be made available to the public. Although the proposed method requires some process information (i.e., process gain Kp and process delay θ), simulated case studies show that the proposed method is very robust and is not sensitive to errors in the process gain or process delay. Finally, 20 industrial case studies show that the quantification results of the proposed method are in good agreement with those of the published method.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: (334) 724-4318. Fax: (334) 724-4188. *E-mail: [email protected]. Phone: (334) 844-2020. Fax: (334) 844-2063. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Drs. Huang and Scali and their students for providing their Matlab codes used in this study.

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REFERENCES

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(35) Seborg, D.; Edgar, T.; Mellichamp, D. Process Dynamics and Control; Wiley: New York, 2004. (36) Bacci di Capaci, R.; Scali, C. Stiction quantification: A robust methodology for valve monitoring and maintenance scheduling. Ind. Eng. Chem. Res. 2014, 53, 7507−7516.

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