A New Measure To Improve the Reliability of Stiction Detection

Article pubs.acs.org/IECR

A New Measure To Improve the Reliability of Stiction Detection Techniques Babji Srinivasan,*,† Tim Spinner,‡ and Raghunathan Rengaswamy‡,§ †

Department of Chemical Engineering, Indian Institute of Technology-Gandhinagar, Ahmedabad, Gujarat 382424, India Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409, United States § Department of Chemical Engineering, Indian Institute of Technology-Madras, Chennai, Tamil Nadu 600036, India ‡

ABSTRACT: A variety of methods have been developed to identify the presence of stiction in linear closed-loop systems. One of the commonly used approaches is based on identification of a Hammerstein model (linear dynamic model preceded by static nonlinear element) between the controller output and process output. These techniques utilize the fact that control valve stiction introduces nonlinearities in the otherwise linear feedback system. However, the present work shows that these techniques could provide ambiguous results depending on the frequency response of the controller and the process of interest. Therefore, in this work, a reliability measure to validate the results from Hammerstein model-based stiction detection approaches is proposed. This measure of reliability is important from an industrial perspective because (i) it helps in reducing false alarms and (ii) it improves the computational speed by guiding the selection of search space in the linear model identification step. The applicability of this reliability measure in industrial setting is demonstrated using various simulation and industrial case studies.

1. INTRODUCTION In the recent past, researchers in the field of control and optimization have made significant contributions to the areas of oscillation detection, diagnosis, and mitigation in closed-loop systems.1−5 It is well-known that the presence of stiction in control valves is one of the common causes for oscillations in control loops.6−9 Stiction in the control valve introduces delay and nonlinear behavior between the controller signal (OP) and the process output (PV).9,10 There are several methods for stiction detection in linear systems. A brief overview of these techniques is provided here. 1.1. Methods for Stiction Detection. Most of the existing stiction detection techniques for linear closed-loop systems can be classified into one of the following categories: (i) shapebased approaches, (ii) frequency domain-based approaches, and (iii) model-based stiction detection approaches. The methods from all three of these classes utilize the fact that valve stiction introduces nonlinearity into the closed-loop system.7,9,11−17 Horch and Isaksson7 developed stiction detection approaches utilizing the cross-correlation function between the controller output (OP) and the process variable (PV). Later, several techniques based upon the classification of the shape of the periodic components of oscillatory OP and PV data were proposed for stiction detection.11,18,19 Choudhury et al.12 proposed a Higher Order Spectral Analysis (HOSA) technique for detecting and quantifying stiction in control valves. Another widely used stiction detection and quantification approach is based on the identification of models between PV and OP data. Stenman et al.20 proposed a model-based segmentation-based approach for identification of stiction in linear control loops. Srinivasan et al.14 presented a Hammerstein-based model identification approach for the diagnosis and quantification of valve stiction. This latter approach and its variations have been frequently studied for stiction detection in linear closed-loop systems.10,13,21−23 The variants of the © 2015 American Chemical Society

Hammerstein-based method differ by their allowable structures for the nonlinear element and linear dynamics, as well as the optimization approaches used within the identification problem. All of these model-based stiction detection techniques appear to suffer from the important concerns discussed next. 1.2. Motivation and Important Contributions. It has been noted that (i) model-based stiction detection methods like those in Stenman et al.20 and Srinivasan et al.14 have failed to detect stiction in integrating processes as indicated in Ordys et al.,24 and (ii) Hammerstein-based detection techniques rely on the identification of static nonlinearity followed by a linear model with a fixed search space, which plays a crucial role. In this work, it has been shown that the results of the Hammerstein-based approaches change with modifications to this search space. Currently, there are no approaches that provide an optimal search space for model identification and a reliability/confidence measure on the results from model-based stiction detection technique. In general, the issue of the reliability of stiction detection has high practical importance with a recent work analyzing the effect of industrial data quality on the performance of shape-based stiction detection methods.25 This Article addresses the issue of search space determination and reliability measure by analyzing the accuracy of the Hammerstein model-based techniques under different conditions using a frequency domain analysis. In this work, routine operating data (PV and OP) are used to • demonstrate the limitations of Hammerstein modelbased stiction detection methods using simulation and industrial case studies; and Received: Revised: Accepted: Published: 7476

March 11, 2015 June 26, 2015 July 9, 2015 July 9, 2015 DOI: 10.1021/acs.iecr.5b00939 Ind. Eng. Chem. Res. 2015, 54, 7476−7488

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Industrial & Engineering Chemistry Research • develop a reliability measure, independent of the existing model-based approaches, to validate the results from the Hammerstein model-based stiction approaches. This Article is organized as follows: Section 2 presents the problem that is addressed in this work. A brief overview of the Hammerstein-based approach for stiction detection is presented in section 2, wherein two key issues in this approach are highlighted. The summary of frequency domain analysis of the Hammerstein model-based framework for stiction detection in linear control loops is provided in section 3. A reliability measure based on the frequency domain analysis is presented in section 4. Simulation and industrial case studies pertaining to use of the reliability measure for stiction identification in otherwise linear systems are presented in section 5, followed by concluding remarks in section 6.

the value of the valve stiction band. A Hammerstein model (nonlinear one-parameter valve model followed by a linear model) is identified between the controller and process outputs to describe the process with sticky valve. Both the nonlinear and the linear model parameters are estimated using a twostage iterative procedure. As shown in Figure 2, first a grid for

2. ANALYSIS OF HAMMERSTEIN-BASED STICTION DETECTION APPROACHES Hammerstein model-based approaches are widely used for stiction detection in linear closed-loop systems. In this section, the following topics pertaining to the Hammerstein modelbased stiction detection framework are discussed: (i) brief overview, (ii) issues in model order selection, and (iii) failure to detect stiction in an integrating level control loop. 2.1. Brief Overview. A typical closed-loop system with stiction present in the control valve is shown in Figure 1. Here,

Figure 1. Typical control loop with stiction nonlinearity.

Figure 2. Hammerstein model-based stiction detection approach using one-parameter valve model.

yp(t) represents the uncorrupted process output (free of disturbance and noise) of plant function G, er(t) represents the error signal to the controller, v(t) represents the control valve output, Gc represents the linear controller dynamics, e(t) represents the independent and identically distributed (i.i.d.) white noise affecting the process output, and u(t) and y(t) represent the sampled controller output (OP) and process output (PV), respectively (sampled at uniform intervals of time with t = kTs, where Ts is the sampling time (k = 1,2...)). Using the assumption that the closed-loop control system has the structure of Figure 1, the Hammerstein model-based stiction detection approaches attempt to identify the combination of a nonlinear stiction element and a linear dynamical model between the process output (y(t)/PV) and controller output (u(t)/OP). Many data-driven models14,26 have been used to represent the nonlinear behavior of sticky control valves. The data-driven stiction models considered in this work are (i) oneparameter stiction model14 and (ii) two-parameter stiction model.22,27 The one-parameter valve model is14 ⎧ if |u(t ) − x(t − 1)| ≤ d ⎪ x(t − 1) x(t ) = ⎨ ⎪ otherwise ⎩ u(t )

identification of stiction parameter d is created. For a particular d value, one-parameter model is applied to get the valve output v(t). This is followed by identification of best linear model (usually ARMAX) order and its parameters using Akaike’s Information Criterion.28 This procedure (in the inner loop shown in Figure 2) is repeated for all values of stiction parameter d in the grid (outer loop of Figure 2). Now, root mean squared error (RMSE) values are computed between the best models obtained from the inner loop (using AIC) for each value of d and the process output over a period of time. The value of stiction parameter d corresponding to the model with minimum RMSE indicates the presence or absence of stiction; a nonzero value of d indicates stiction, while a zero value implies the absence of stiction. The data-driven stiction model proposed by Choudhury et al.27 has two parameters: (i) S, which provides the information about deadband plus stickband, and (ii) J, which is the slip-jump parameter that describes the jump start of the control valve after it overcomes the stiction and deadband. Identification of the best model follows the same procedure as shown in Figure 2, except that now grid for d is replaced by a two-dimensional grid of S and J. Nonzero values of S and J corresponding to minimum RMSE indicate the presence of stiction, while zero values denote that there is no stiction in the control valve.22

(1)

Here, x(t) and x(t − 1) represent the present and the previous stem positions, u(t) is the present controller output, and d is 7477

DOI: 10.1021/acs.iecr.5b00939 Ind. Eng. Chem. Res. 2015, 54, 7476−7488

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Figure 3. (a) Process output. (b) Controller output. (c) Hammerstein-based stiction detection results for various search spaces.

stiction in the flow control loop. The minimum RMSE value was achieved at a nonzero d = 0.1089, indicating the presence of stiction in the control loop (Figure 3c). In this case, the order of the best model order was na = 4, nb = 1, and nc = 9. The time taken by the method in the first case is 251 s, while that with the larger search space is 595 s, a more than 2-fold increase in computational time. This has to be seen with the fact that a medium/large sized plant has about two/five to five/ ten thousand control loops with stiction detection algorithm to detect oscillations in hundreds (medium sized plant) or thousands (big plants) of control loops where there are no valve positioners, common in old plants. Furthermore, there are several multivariate data analysis algorithms proposed just to identify the control loop, which has the source of oscillations.16 Also, to verify the sensitivity issue, for the same flow control loop, a much smaller search space (na = 2, nb = 2, and nc = 2) was used, which in turn leads to incorrect results indicating the absence of stiction (Figure 3c). This case study serves as a classic example demonstrating the following:

Typically in Hammerstein model-based approaches (as shown in Figure 2), the linear dynamics used to describe the plant and disturbance behavior is the ARMAX (autoregressive moving average with exogenous input) model of the form: y(t ) =

B(q) C(q) u(t ) + e(t ) A(q) A(q)

where q is the forward time shift operator, process model, while

C(q) A(q)

(2) B(q) A(q)

denotes the

represents the noise model.29 The

issues involved in the selection of the order of linear dynamic model are presented next. 2.2. Issues in Model-Based Stiction Detection Approaches. In the Hammerstein model-based stiction detection framework, the search space of the linear dynamic ARMAX model has to be first initialized, and then the order of the best model along with its parameter values in this search space are determined using model selection criterion such as Akaike’s Information Criterion (inner loop of Figure 2). However, it can be noted that choosing a large initial search space (for example, AR (na) order 25, MA (nc) order 25, and exogenous input (nb) order 25) could lead to an increase in computational time, while selecting a small search space (na ∈ {1,2}, nb ∈ {1,2}, and nc ∈ {1,2}) could lead to incorrect results. A simple industrial case study is provided to further illustrate this claim. 2.2.1. Flow Control Loop (FC145). Consider the industrial flow control loop (taken from Horch15) whose PV and OP are shown in Figure 3a and b. It was reported that stiction in the control valve is the cause for oscillations in this loop. Hammerstein model-based stiction detection technique with initial ARMAX search space na ∈ {1···5}, nb ∈ {1···5}, and nc ∈ {1···5} was used to detect stiction in this control loop. However, the method with this initial search space indicated that there was no stiction in the loop (d = 0, has minimum RMSE as shown in Figure 3c), clearly a wrong diagnosis. Furthermore, the best model order identified (using AIC criterion) within this search space was na = 4, nb = 1, and n1c = 4. This means that there is no indication to increase the search space because the best model orders did not reach the search space limits. In the second trial, the search space of the linear ARMAX model in Hammerstein-based approach was increased to na ∈ {1···5}, nb ∈ {1···5}, and nc ∈ {1···10}. Now, the Hammerstein model-based approach detected the presence of

• sensitivity of the Hammerstein model-based stiction detection approach to the choice of search space for the ARMAX model structure; and • significant increase in computational time when the search space is increased. It is now obvious that addressing the search space sensitivity issue in Hammerstein model-based stiction detection approaches is a key element for its applicability to industrial scenarios. Issues with model-based approaches for detection of stiction in a level (integrating) process are discussed next. 2.2.2. Level Control Loop (LC011). The PV and OP data from an integrating level control loop (tag name: LC011) are shown in Figure 4a. It was reported that this control loop had oscillations due to the presence of sticky control valve.30 Furthermore, it is an interesting loop as it has been reported that the model-based segmentation approach for stiction detection proposed by Stenman et al.20 indicates that there is no stiction in this loop. 30 Hammerstein model-based approaches for stiction identification (with search space na ∈ {1···5}, nb ∈ {1···5}, and nc ∈ {1···25}) are used for stiction detection in this loop using both one-parameter and twoparameter stiction models. The method with one-parameter model provided an identified stiction band value of d = 0 (see Figure 4b), indicating that there is no stiction. Also, the 7478


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v(t) is the valve output, e(t) is an independent and identically distributed (i.i.d.) white noise, and G(q) and H(q) are the linear dynamics of the process and disturbance filters, which in the most general case can be represented as ∞

G(q)

= ∑ g (k )q − k k=1 ∞

H(q) =1 +

∑ h(k)q−k (4)

k=1

with ∑k∞= 1|g(k)| ≤ ∞ and ∑k∞= 0|h(k)| ≤ ∞, which ensures stability of each of these filters.29 Henceforth, we consider the case that G(q) and H(q) can be expressed more parsimoniously as rational functions, where each numerator and denominator can be described by a low-order polynomial. Here, we assume that stiction is the only nonlinearity in the otherwise linear system, and further that the presence of stiction acts as a dither, providing enough excitation for identification of the filters G(q) and H(q) from signals v(t) and y(t) using a direct identification approach. These assumptions are the basis of data-driven Hammerstein model-based stiction detection methods.14,22 To proceed with a frequency domain-based analysis, let us consider the following assumptions: • We assume that G(q) and H−1(q) (inverse of the noise model) belong to model set M* used during identification of these models. • It is a standard assumption that terms G(q = 0) = Ĝ (q = 0) = 0 so that there exists at least one unit of time delay between the input and output of filter G(q). Similarly, it is assumed that H(q = 0) = Ĥ (q = 0) = 1. Let model structure M be a differential mapping from open subset DM of RdM to M* such that gradients of prediction function of the process output are stable (chapter 4 of Ljung29). With this definition, the prediction functions of model structure M are

Figure 4. Level loop process − LC011. (a) Plot of process output PV and controller output OP (u). (b) Plot of results from one- parameter model-based Hammerstein approach. (c) Plot of results from twoparameter model-based Hammerstein approach.

−1 −1 y ̂(t |t − 1) = Ĥ (q , θ )Ĝ (q , θ )v(̂ t ) + (1 − Ĥ (q , θ ))

y(t )

Hammerstein model-based approach with two-parameter model provided S = 0 and J = 0 as shown in Figure 4c (minimum RMSE is obtained at these values), indicating that there is no stiction in the control loop. Like the segmentation approach, Hammerstein model-based stiction detection approaches also failed to detect stiction in this integrating level loop process, requiring a detailed analysis of Hammerstein model-based approaches for stiction detection in linear control loops.

where θ represents the vector of parameters used to parametrize model structure M, θ ∈ DM, so that dM = dim(θ). The innovation sequences corresponding to model structure M and parameter vector θ are ε(t , θ ) = y(t ) − y (̂ t |t − 1)

(6)

which is assumed to have a mean zero Gaussian distribution. Akaike’s information criterion (AIC)28 allows for simultaneous selection of the best model structure M and parameter vector θ out of a set of candidate models. The criterion leads to the following decision function to determine the identified model:29

3. DISCUSSION USING FREQUENCY DOMAIN ANALYSIS Let us assume that the true process variable y(t) is generated from the linear dynamical system described by the following equation: y(t ) = G(q)v(t ) + H(q)e(t )

(5)

{M , θ } = arg min min WN (θ , M ) = arg min min M ∈ M * θ ∈ DM

(3)

N

with t = kT, ∀k = 1, 2..., N, where N is the length of sample records u(t) and y(t), and T denotes the sampling period that is considered to be 1 without loss of generality. In the above equation, q is the forward shift operator with qv(t) = v(t + 1),

∑ ε 2 (t , θ ) + t=1

2dM N

M ∈ M * θ ∈ DM

(7)

. From eqs 3, 5, and 6: 7479


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Industrial & Engineering Chemistry Research ε(t , θ ) = y(t ) − y (̂ t |t − 1)

2

N−1

WN (θ , M ) =

−1 −1 = Ĥ (q)y(t ) − Ĥ (q)Ĝ (q)v(̂ t )

∑

Eε(ωk , θ )

k=1

−1

=

−1

= Ĥ (q)(G(q)v(t ) − Ĝ(q)v(̂ t ) + (H(q) − Ĥ (q))e(t )) + e(t )

∑

−1 Ĥ (ωk)(EGv(ωk , θ ) + EH (ωk , θ ))

k=1 2

+ E(ωk)

(8)

+

2dM N (15)

where the last equality is due to the relation:

Suppose that control signal u(t) is generated by linear controller K(q), which has a cutoff frequency ωC, defined by the relation:

−1 −1 Ĥ (q)H(q)e(t ) = Ĥ (q)(H(q) − Ĥ (q))e(t ) + e(t )

(9)

Now, because G(q) and Ĝ (q) both contain at least one unit of delay and because H(q = 0) = Ĥ (q = 0) = 1 implies that the first term of H(q) − Ĥ (q) is zero, then the term e(t) in eq 8 is statistically independent of the remainder of the expression, and changes to estimates Ĝ (q) and Ĥ (q) will not affect this term. It is useful in the foregoing analysis that we rewrite the prediction error as ε(t , θ ) = Ĥ (q)(εGv(t , θ ) + εH (t , θ )) + e(t )

|U(ωk)| = 0, ∀ ωk > ωC

1 √N

|G(ωk)V (ωk)| = 0, ∀ ωk > ωG

(10)

N

(11)

where {ωk}kN−1 = 0 is the set of frequencies at which the discrete Fourier transform can be computed. U(ωk), EGv(ωk,θ), EH(ωk,θ), V(ωk), E(ωk), and V̂ (ωk) represent the discrete Fourier transforms of signals y(t), u(t), εGv(t,θ), εH(t,θ), v(t), e(t), and v̂(t), respectively, and additionally let G(ωk), Ĝ (ωk), H(ωk), Ĥ (ωk), and Ĥ −1(ωk) denote the frequency responses of filters G(q), Ĝ (q), H(q), Ĥ (q), and Ĥ −1(q), respectively. Neglecting finite sample size artifacts in the discrete Fourier transform, which are negligible for large N,29 we obtain EGv(ωk , θ ) = G(q)V (ωk) − Ĝ (q)V̂ (ωk)

(12)

and EH (ωk , θ ) = (H(ωk) − Ĥ (ωk))E(ωk)

(13)

According to Parseval’s theorem: N

N−1

∑ ε2(t , θ) = ∑ |Eε(ωk , θ)|2 t=1

k=1

(17)

To analyze the likelihood of a correct detection result when stiction is present, it is useful to compare the scenario that the correct valve signal is used during identification (v̂(t) = v(t)), against the scenario that no stiction is assumed during identification, wherein v̂(t) = u(t). In practice, it is unlikely to achieve v̂(t) = v(t) exactly, due to the limitations of data-driven stiction model structures to replicate real stiction behavior, as well as due to the inaccuracies caused by the restriction of identified stiction parameters to a predefined grid of values. It is assumed though that a relatively small value of the model selection criteria WN(θ,M) can be achieved using v̂(t) = v(t). We now discuss the possibility of achieving similarly low WN(θ,M) using v̂(t) = u(t), even in the case that stiction is actually present, which could potentially lead to the incorrect determination that stiction is absent. Consider that the controller and measurement signals from systems containing valve stiction often are periodic in nature, consisting of an oscillation at the fundamental frequency along with power at a series of harmonic frequencies. Suppose the signals u(t) and y(t) are of this type. Define ωf as the fundamental frequency of stiction induced oscillation. We then can identify two cases: • Case 1: If ωf ≪ ωG, a large number of stiction harmonics will be available in signal Y(ω). If, additionally, ωC ≪ ωG, much of this harmonic content will not be present in U(ω). Therefore, changes in Ĝ cannot reduce error εGv(ω) at these harmonic frequencies. Attempting to minimize the quantity ε(ω) by adding poles in filter HΔ(ω) or zeros in filter Ĥ −1 will lead to a relatively large penalty dM because of the large number of harmonic frequencies to be canceled. Therefore, in this case, it is unlikely that stiction will be incorrectly identified as absent. • Case 2: If ωG ≈ ωf, then relatively little harmonic content will be available in signal Y(ω). Depending on the value of ωC, errors in ε(ω) at the fundamental and harmonic frequencies may be canceled by appropriately modifying Ĝ or filter HΔ(ω). In this case, a relatively small number of additional poles and zeros will be required, leading to a relatively small penalty dM because of the low number of harmonic frequencies present in Y(ω). Therefore, in this

∑ ε(t , θ) e2jπkt /N t=1

(16)

Additionally, allow ωG to be the cutoff frequency of the plant G(q), which is defined by the relation:

where εGv(t,θ) = G(q)v(t) − Ĝ (q)v̂(t) is the error due to inaccuracies in the estimated plant filtered valve signal and εH(t,θ) = (H(q) − Ĥ (q))e(t) is the error due to the estimated disturbance filter Ĥ (q). It will be useful in the following to define HΔ(q) = H(q)− Ĥ (q). Now we shift focus to the frequency domain properties of these innovation sequences. For signal ε(t), t = 1...N, the discrete Fourier transform denoted Eε(ωk,θ) is calculated via Eε(ωk , θ ) =

2dM N

N−1

= Ĥ (q)(G(q)v(t ) − Ĝ(q)v(̂ t ) + H(q)e(t ))

−1

+

(14)

meaning that the model selection criterion is 7480


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Industrial & Engineering Chemistry Research Table 1. Simulation Example Details and Results from Hammerstein Approach sim no.

process

actual case

prediction

1

3/(100s + 1)

1/(s + 1)

controller

stiction

2

10/(s + 8)

(0.1s + 0.1)/s

stiction

d = 0, S = 0, J = 0 no stiction d = 0.4, S = 0.5, J = 0.5 stiction

Figure 5. Stiction − Simulated using physical valve stiction model. (a) Plot of process output (y) and controller output (u) for closed-loop system I. (b) Plot of process output (y) and controller output (u) for closed-loop system II.

Figure 6. Results from stiction detection approaches. (a) Plot of RMSE versus d value for closed-loop system I. (b) Plot of RMSE versus d value for closed-loop system II. (c) Plot of RMSE versus S value for closed-loop system I. (d) Plot of RMSE versus S value for closed-loop system II.

approach for Hammerstein-based stiction detection approaches with linear output error (OE) models (instead of ARMAX models considered here). Simulation examples and theoretical arguments supporting the above two cases are provided in the following section. 3.1. Simulation Examples. Consider two closed-loop systems whose details are presented in Table 1.The controller transfer functions in these loops (although different from that of regular PI or PID controllers) are utilized to provide a clear illustration of the frequency domain information available for

case, it is much more likely that stiction will be incorrectly identified as absent. To summarize, if the controller cutoff frequency is much smaller than that of the plant (ωC ≪ ωG), and the plant cutoff is also much greater than the oscillation frequency (ωf ≪ ωG), then stiction is likely to be correctly identified, if present. However, if the plant cutoff frequency is not much greater than that of the oscillation frequency (ωG ≈ ωf), the presence of stiction is much less likely to be correctly detected. Spinner31 derived similar conclusions using a rigorous theoretical 7481


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Figure 7. (a) Magnitude response of system I along with the controller. (b) Magnitude response of system II along with the controller. (c) Magnitude spectra of the process (○) and controller output (blue) of system 1. (d) Magnitude spectra of the process (○) and controller output (blue) of system 2.

stiction detection in linear closed-loop systems. Simulation studies presented in this section are free of noise, performed deliberately to show that a linear noise model (even if forced to model data set without noise) cannot capture the harmonic information content if present in the output data, leading to reliable stiction detection results. Applicability of this frequency domain analysis and the reliability measure developed on the basis of this analysis are demonstrated using several simulation and industrial case studies provided in section 5. A physical valve stiction model,32 used as a benchmark for comparison of a number of data-driven parametric valve stiction models,9,14,22 is used for simulating stiction in the two systems. The oscillatory process data after introducing stiction are shown in Figure 5a and b. Results from the Hammerstein-based approach (with search space na = 5, nb = 5, and nc = 25) for the two systems using one-parameter valve model are shown in Figure 6a and b. The d value obtained from the process data of the first system is zero, indicating that there is no stiction. This determination is incorrect, as stiction was introduced in the control loop. However, a d value of 0.4 was identified for the second system, correctly identifying the presence of stiction. Similar results were obtained on application of the Hammerstein-based approach using the two-parameter valve model. These results are presented in Figure 6c and d and summarized in Table 1. For the first system, the approach incorrectly indicates zero stiction, while for the second system, the presence of stiction is identified correctly. Thus, the conclusions for the Hammerstein-based stiction identification techniques having either oneor two-parameter stiction models are the same; each incorrectly indicates the absence of stiction in the first process and correctly identifies the presence of stiction in the second process.

Magnitude plots of the process and controller are computed for both systems and are shown in Figure 7a and b. Fourier transforms of both the controller (U(ωk), ωk = ((2πk)/(N)), k = 1, 2, ..., N) and the process (Y(ωk)) outputs of the two closed-loop systems are computed. The magnitude of these signals at various frequencies obtained is shown in Figure 7c and d. From Figure 7c (along with the inset), OP data of system 1 have appreciable magnitude at all frequencies similar to that of the PV data. In other words, there exists no extra information at various frequencies in the PV data that is not present in the OP data. Contrastingly, it is clear from Figure 7b (along with the inset) that the PV data from system 2 do have appreciable magnitude in the transform of the PV data at frequencies for which the OP data do not display appreciable magnitude (called extra information). Here, appreciable magnitude is defined as magnitudes greater than the small amplitude values, which are generally considered to be spurious artifacts of the Fourier transform computation and disturbances corrupting the plant. System II illustrates the idea provided in case 1 where enough amplitude in the output at various frequencies beyond the controller cutoff frequency (in other words, ωG > ωC, refer to Figure 7b) leads to correct identification of stiction, while system I demonstrates case 2 where there is not enough amplitude in the output at various frequencies beyond the controller cutoff frequency (case where ωG < ωC, refer to Figure 7a) leading to incorrect determination of stiction by Hammerstein model-based approaches.

4. RELIABILITY MEASURE FOR MODEL-BASED STICTION DETECTION APPROACHES The methodology for quantification of the extra information (discussed and visualized in the earlier section), defined as 7482


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Figure 8. (a) |Ysig(ωk)| for simulation system I and (b) |Ysig(ω)| for simulation system II.

⎧ Y (ωk) if|U (ω)| < |S(ωk)|and ⎪ Ysig(ωk) = ⎨ |Y (ωk)| > |S(ωk)| ⎪ ⎩ 0 otherwise

amplitude at various frequencies in the PV data beyond the critical frequency, is provided below: Fourier transforms of process output (Y(ωk)) and controller output (U(ωk)) are computed. For given control loop data, the maximum possible amplitude of oscillation in the controller output, umax = max(u) − min(u), is computed. A signum function given by ⎧−A if u(t ) < 0 ⎪ s(t ) = ⎨ 0 if u(t ) = 0 ⎪ ⎪ A if u(t ) > 0 ⎩

(19)

The constraint |U(ωk)| < |S(ωk)| in eq 19 acts as a noise threshold on u(t), while the condition |Y(ωk)| > |S(ωk)|ensures that Ysig(ωk) is significant. Therefore, Ysig(ωk) represents the extra information (in the PV data), magnitude of the process output at frequencies beyond the critical frequency of the controller output. The plot of Ysig(ωk) with f = ωk/2π for the two closed-loop systems discussed in section 3 are shown in Figure 8a and b. It is clear that in system I, the process output y(t) does not contain any extra information in the PV data, while the second system has amplitude (above the threshold value) at frequencies beyond the cutoff frequency of the controller. Let us define an index Yper, which is an indication of the normalized amplitude present in Ysig(ωk) and is given by

(18)

is used as a threshold to compute the extra information (significant amplitude beyond the critical frequency) present in the amplitude spectrum of PV data. The value of A used in the simulation and industrial case studies is 10% of umax. The Fourier transform of the threshold signal S(ωk) (amplitude spectrum of s(t)) is used as a frequency domain threshold value for the amplitude spectrum of the process (Y(ωk)) and controller (U(ωk)) outputs. The logical reasons for coming up with this particular kind of threshold are (i) the physical valve model equation32 involves a signum function to describe the valve nonlinearity, and (ii) in practice, any oscillation in the controller signal due to valve stiction with amplitude below 10% of umax is considered negligible. Compute a new frequency domain signal Ysig(ωk) given by

Yper =

∑ω |Ysig(ωk)| k

∑ω |Y (ωk)| k

(20)

In the simulation example, the first system has a zero Yper value while the second closed-loop system has a Yper = 8%, indicating the presence of extra information available in the PV data of second system. Thus, the index defined in eq 20 acts as a reliability measure for Hammerstein-based stiction detection 7483


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Industrial & Engineering Chemistry Research approaches. Table 2 provides information on the use of Yper as a reliability measure for Hammerstein-based stiction detection approaches.

loops with the frequency response of these controllers decreasing with increasing frequency right from ω > 0 (due to the presence of integrator). Therefore, it is most likely that the process cutoff frequency will be greater than that of the PI controller cutoff frequency with a typical exception being integrating process. Also, the availability of process knowledge (e.g., the presence of integrating process) could help confirm these results. Another approach that involves fusion of33 results from multiple stiction detection techniques could also enhance the stiction detection results. 4.1. Reliability Measure for Search Space Determination − Demonstration Using Industrial Flow Loop (FC145). The reliability measure algorithm proposed in this section is used to find Ysig(ωk) and Yper of the process data from the flow control loop discussed in section 2.2.1. Figure 9a

Table 2. Use of Yper as Reliability Measure Yper

prediction

reliability

high (≫0) high (≫0) low (∼0) low (∼0)

stiction no stiction stiction no stiction

reliable not reliable (search space issue) not reliable (search space issue) reliable

Remark 4.1: Ideally, the Yper value for PV data with no extra information should be zero. However, for practical purposes, we only consider those Yper values greater than 0.5% as significant. The reason for this is that the spectrum of s(t) (threshold) decreases with increasing frequencies (typical nature of spectrum of the signum function), and, therefore, it is required to neglect the small significant values that might arise due to high frequency noise amplified by the process gain. If a process model/partial knowledge of the process is available, then one can use this knowledge to set the threshold value in computation of the extra information in PV data. Nevertheless, several industrial case studies show that the algorithm with the current threshold value acts as a good reliability measure for Hammerstein-based stiction detection algorithms. Remark 4.2: The proposed measure acts as a measure of reliability (for validating the results from Hammerstein modelbased approaches) and is not intended for use as a new stiction detection technique. This is due to the fact that the complete frequency response of the process is not known, and therefore the proposed measure, used as a stand alone stiction detection technique, could lead to ambiguous results. This is mainly because the proposed reliability measure checks for the presence of significant amplitude (above the threshold) beyond controller cutoff frequency ωC but less than the controller cutoff frequency ωG, which could be due to (i) the presence of stiction leading to significant amplitude at harmonic frequencies, (ii) high amplitude disturbance (above the threshold), and (iii) nonlinearities (like stiction element) in the control loop. Also, the proposed method is completely different from the bicoherence-based stiction detection approach due to the following: (i) Bicoherence-based stiction approach tries to identify the presence of harmonics in the process output or error signal to the controller,16 and (ii) the reliability measure developed in this work is based on the extra frequency domain information between the process output and controller output, which utilizes both PV and OP data unlike bicoherence techniques that utilize only univariate process data. Remark 4.3: In Table 2, if the value of Yper is high and the Hammerstein approach indicates no stiction (row two of the table), row three of the table could result if the search space used in the model identification is not able to capture plant and disturbance dynamics, which could be handled by increasing the search space. Also, in this case, it is assumed that the threshold set during reliability measure calculation is not way off, which could lead to ambiguous results. Finally, in the last row of the Table 2 where both reliability measure and stiction detection approach point to no stiction, a conclusion is determined on the basis of the assumption that the cutoff frequency of the controller is less than that of the process. In general, PI controllers are widely used in industrial control

Figure 9. Reliability measure results on FC525 and LC011. (a) Plot of Ysig(ω) for FC525. (b) Plot of Ysig(ω) for LC011.

shows the plot of |Ysig(ωk)|. It can be noticed that there is significant extra information in the Fourier transform of process output Y(ωk) as compared to that of the controller output U(ωk). The ratio Yper for this loop is 15.28%. This high value (greater than 0.5%) indicates that stiction can be detected in this control loop by Hammerstein-based methods as per the arguments in section 3. In fact, Hammerstein-based stiction detection approaches (for search space na = 5, nb = 5, and nc = 10) on this loop indicated the presence of stiction as discussed in section 2.2.1. The problem of search space determination in Hammerstein-based stiction detection can be effectively addressed using the reliability measure presented in this section. The steps to resolve the search space determination problem are as follows: (1) Compute the Yper for the given process data. Use Hammerstein model-based approach (with nominal fixed search space) to determine the amount of stiction in the control loop. Compare both results using Table 2. (2) If results from both approaches match (as per the table), then the results from Hammerstein model-based approaches are reliable. Otherwise, it is required to increase the linear model identification search space. 7484


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Figure 10. Simulation studies: column I, process output from FOPTD with aggressively tuned controller, stiction, and oscillatory disturbances; column II, results from Hammerstein-based approaches; and column III, |Ysig(ω)| from process data of all three scenarios.

Figure 11. Industrial case studies - I: column I, process output from FC445 and FC525 control with sticky valve; column II, results from Hammerstein-based approaches; and column III, |Ysig(ω)| from process data of the two industrial loops.

4.2. Analysis of Integrating Process − Level Loop LC011. A plot of Ysig(ωk) obtained from the integrating level loop process LC011 data discussed in section 2.2.2 is shown in Figure 9b. The value of Yper for this integrating process is 0.29% (0.5) and 3.05. Interpretation of these values using Table 2 indicates that the findings using the Hammerstein-based approach are valid (based on proposed measure) and also match with the true findings reported in Ordys et al.24 Next, the proposed measure was applied to two different industrial control loops (metal processing industry and pulp and paper industry9). The process data obtained from these two loops are shown in Figure 12a and d. It was reported (by industries) that stiction is not the source for oscillations in the first loop (data from metal processing industry) while the second loop (data from pulp and paper industry) contains a sticky valve. Results from the application of Hammerstein model-based stiction detection approach on these two loops are shown in Figure 12b and e. |Ysig(ω)| obtained from the proposed methodology (shown in Figure 12c and f) are not significant (Yper value is zero) in the first loop, while there is significant Yper (0.74) value indicating the presence of stiction in the second control loop. In Table 2, it is clear that the first loop does not have stiction while the second control has a sticky valve, which coincides with the results from the Hammerstein-based stiction detection approach (shown in Figure 12e). 7486



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algorithm based on the features extracted from the data and resulting from three widely used shape-based algorithms was used to identify the presence of stiction in linear control loops. It was found that none of the individual algorithms were able to provide 100% correct results and also that these methods could not be used in the presence of high nonlinearities and low data resolution.25 With regard to model-based stiction detection approaches, these shape-based approaches can be viewed as imposing further constraints (frequency shaping) on the identified model that could be utilized in improving stiction detection. For instance, there are predefined shapes for PV and OP data in case of the presence/absence of stiction if it is an integrating process. This information can be used in model building as a constraint that along with the proposed measure could further improve the results of model-based stiction detection approaches. Currently, reliability measure is proposed for linear control loops with stiction as the only nonlinear phenomenon. This assumption is true for Hammerstein model-based stiction detection approaches. However, model-based approaches have been extended to identify stiction in control loops with stiction and nonlinear process.34 Reliability measure has to be extended to these methods, which is one of the scope for future work. Furthermore, model-based approaches have also been extended to interacting systems based on approximate dynamic model of the process.35 Although the results are not shown here, the proposed approach fails to provide desired results in case of interacting four tank systems, clearly indicating a modification of the proposed approach for interacting control loops.

REFERENCES

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6. CONCLUSIONS A reliability measure (using the frequency domain analysis of closed-loop systems) is proposed to improve the applicability and fidelity of the Hammerstein model-based stiction detection approaches. This reliability measure independently validates the results provided by the Hammerstein-based stiction detection approach. The proposed algorithm also addresses the problem of determining a search space for the linear model component within Hammerstein model-based stiction detection methods. Using frequency domain analysis and case study results, the issue regarding use of Hammerstein-based stiction detection methods for integrating processes is addressed. Several simulation and industrial case studies show promising results and demonstrate the applicability of the proposed measure. A few guidelines on the use of the reliability measure are also provided. Our current work is directed toward the development of a reliability measure for stiction identification approaches in interacting and nonlinear processes.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from Department of Science and Technology - India (Project: SERB/ET-26/2013) and National Sciene Foundation - United States of America (GOALI Award: 0934348) to pursue this work is gratefully acknowledged. 7487


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Industrial & Engineering Chemistry Research (24) Ordys, A., Uduehi, D., Johnson, M., Eds. Process Control Performance Assessment: From Theory to Implementation; Springer: Berlin, 2007. (25) Zakharov, A.; Zattoni, E.; Xie, L.; Garcia, O. P.; Jämsä-Jounela, S.-L. An autonomous valve stiction detection system based on data characterization. Control Engineering Practice 2013, 21, 1507−1518. (26) Choudhury, M. Detection and Diagnosis of Control Loop Nonlinearities, Valve Stiction and Data Comprression management. Ph.D. thesis, University of Alberta, 2004. (27) Shoukat Choudhury, M. A. A.; Thornhill, N. F.; Shah, S. L. Modeling valve stiction. Control engineering practice 2005, 13 (5), 641− 658. (28) Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control 1974, 19, 716−23. (29) Ljung, L. System Identification: Theory for the User, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1999. (30) Horch, A. Oscillation Diagnosis in Control Loops Stiction and other Causes. Proceedings of the American Control Conference, 2006. (31) Spinner, T. Performance Assessment of Multivariate Control Systems. Ph.D. Thesis, Texas Tech University, 2014. (32) Olsson. Control systems with friction. Ph.D. Thesis, Lund Institute of Technology, Sweden, 1996. (33) Di Capaci, R. B.; Scali, C. A Performance Monitoring Tool To Quantify Valve Stiction in Control Loops; The International Federation of Automatic Control: Cape Town, South Africa, 2014. (34) Nallasivam, U.; Babji, S.; Rengaswamy, R. Stiction identification in nonlinear process control loops. Comput. Chem. Eng. 2010, 34, 1890−1898. (35) Spinner, T.; Srinivasan, B.; Rengaswamy, R. Detection of Stiction in Interacting Systems using a Hammerstein Model Approach. International Symposium on Advanced Control of Chemical Processes, 2014.

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A New Measure To Improve the Reliability of Stiction Detection

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