Data-Driven Plant-Wide Control Performance Monitoring - Industrial

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Process Systems Engineering

Data-driven plant-wide control performance monitoring David Alejandro R. Zumoffen, Lautaro Braccia, and Patricio A. Luppi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b06293 • Publication Date (Web): 01 Apr 2019 Downloaded from http://pubs.acs.org on April 7, 2019

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Industrial & Engineering Chemistry Research

Data-driven plant-wide control performance monitoring David Zumoffen,∗,†,‡ Lautaro Braccia,† and Patricio Luppi†,¶ †Grupo de Ingenier´ıa de Sistemas de Procesos (GISP), Centro Internacional Franco-Argentino de Ciencias de la Informaci´on y de Sistemas (CIFASIS), CONICET-UNR, 27 de Febrero 210 bis, (S2000EZP) Rosario, Argentina. ‡Universidad Tecnol´ogica Nacional – FRRo, Rosario, Argentina ¶Universidad Nacional de Rosario – FCEIA, Rosario, Argentina E-mail: [email protected] Phone: +54-341-4237248 int. 332. Fax: +54-341-482-1772 Abstract In this work a new data-driven plant-wide control performance monitoring methodology is proposed. The main constitutive parts of the suggested method are based on three well-known research areas from process systems engineering (PSE): 1- the sum of squared deviations (SSD) concepts from the plant-wide control design topic, 2- the partial least squares (PLS) modeling technique from the multivariate statistics area, and 3- the covariance-based performance index and diagnosis (CID) from the control performance monitoring field. All these approaches are integrated and reformulated in the current work to perform a MIMO control structure performance/feasibility assessment, an open-loop steady-state model identification by using close-loop normal data, and a covariance-based procedure for diagnosis purposes. This strategy requires minimum interference with the industrial process operation and generates valuable information (off-line as well as on-line) to evaluate the already installed control policy

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and suggest potential control structure modifications and/or potential controller retuning. Two typical case studies are proposed to analyze the scope of the suggested approach.

1

Introduction

Control performance monitoring (CPM) has acquired great relevance and popularity the last decades either from the academy as well as the industrial perspective. The CPM, also called control performance assessment (CPA), is devoted to audit the health of the control system installed in some industrial plant and try to identify/diagnose the root causes related with its performance degradation. The main objective of the CPM is to provide an on-line automated procedure to deliver information to the plant personnel 1 . Several approaches have emerged in the last years to address the CPM either for the SISO as well as MIMO point of view, which have been reviewed and summarized in Qin 2 , Qin and Yu 3 and Jelali 1 . Generally, the main objective of the CPM procedure is to compare the current monitored closed-loop data against some kind of benchmark. The most common benchmark is the well-known “minimum variance benchmark” popularized by Harris 2,4 , which is based on the minimum variance control (MVC) theory. Jelali(2006) 1 compares and summarizes several benchmarks methodologies such as linear quadratic gausian (LQG), optimal PID (OPID), generalized minimum variance (GMV), and historical data (HIS), among others. Alternative developments in CPM include the data-driven covariance benchmark developed by Yu and Qin 5 where a generalized eigenvalue problem is solved to identify the directions related to the degraded or improved performance in MIMO control. The minimum achievable output variance (MAOV) benchmark is addressed in Fu et al. 6 based on iterative convex programming for a SISO PID control loop. Gao et al. 7 suggest a data-driven method for simultaneous performance assessment and PID retuning for a SISO control loop application. This approach is based on a reference model as a benchmark and a convex approximation problem to solve the PID tuning optimization. A recent methodology proposed 2


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by Yan et al. 8 suggest a multiobjective scheme for CPA and control system monitoring via hypothesis test on output covariance matrices. This new approach extends the F-test to covariance matrices and improves the information content compared with the data-driven covariance method of Yu and Qin 5 . Anyway, this promising strategy needs to be tested in large-scale processes and integrated to plant-wide control concepts. Various authors identified the main limitations of the MVC for MIMO control structures, which become this approach not attractive for practical implementations 1,3 . Therefore, these authors suggest some variants called “user-defined benchmarks”, where these benchmarks are usually related to some historical data period in which the desirable MIMO control performance was achieved. Although, the CPM is a relatively mature field, to the best of our knowledge, there are no contributions from a plant-wide control (PWC) perspective. Furthermore, all the mentioned methodologies address the control performance monitoring and retuning in some extend but there are no contributions addressing the performance evaluation of the PWC structure and the recommendations of what it is necessary to do when the performance degradation cannot be improved by retuning the controller. From the statistical process monitoring (SPM) area several approaches are suggested to quantify the current status of the industrial processes from a data-driven point of view. The SPM generally is based on multivariate statistics and machine learning methods to fault detection and diagnosis 9 . Principal components analysis (PCA) and partial least squares (PLS) appear to be the main multivariate used tools in multivariate quality control (MQC) to characterize the normal operation from data. Several modifications have been appear in the last decades, which include adaptive/recursive approaches 10–13 , non-linear strategies 9,14 , and dynamic extensions as summarized in Qin (2012) 9 and Ge (2017) 15 . Recently, a hierarchical multiblock total PLS based monitoring is presented in Liu et al. (2018) 16 . This work suggests a methodology for on-line process operation monitoring based on the user-defined benchmark approach. In addition, a contribution analysis is suggested for abnormal operation diagnosis. Although, the SPM research field seems to be related to the CPM area there are no direct

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connections or contributions relating these two research lines. Before explicitly defining the major contributions of the current work, the authors want literally re-write the main conclusions and thoughts from Qin 2 about the required features for any practical MIMO CPM system: ... it must have the following features similar to the SISO control assessment. (a) It requires minimum interference to process operation. (b) It provides achievable performance bound for the existing control structure, e.g. multi-loop control. (c) It should diagnose whether the existing control structure is adequate or not. For example, it could recommend alternative control structure that will improve the performance significantly. The third item is more difficult to achieve but is more desirable in practice. It would be of tremendous value to practitioners if, by assessing the performance, one can determine whether alternative control strategies are needed to achieve the control objectives...

1.1

Contribution of the work

The present work suggests a new methodology for control performance monitoring, which addresses the Qin’s 2 conclusions stated above. To the best of our knowledge, the proposed PWC performance monitoring approach is the first data-driven methodology addressing the MIMO control structure performance/feasibility quantification in a plant-wide sense. The procedure requires minimum interference with the industrial process operation and generates valuable information (off-line as well as on-line) to evaluate the already installed control policy and suggest potential control structure modifications (MVs, CVs, pairing) and/or potential controller retuning (revamping/redesign). All this information gives more reliability to the decision making process. The main concepts behind the proposed method are: 1- the integration between the sum of squared deviations (SSD) strategy and the block-wise recursive partial least squares (BWRPLS) 10,17 , and 2- the block-wise recursive formulation of the covariance-based performance 4


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index and diagnosis (BW-RCID). Both concepts working together allow to detect modifications in the PWC performance due to multivariable static gains changes and/or dynamic implications. In the next section the proposed method is discussed in detail. The specific contributions of the suggested CPM system can be summarized in the following list: • it allows to identify the open-loop steady-state gains of the process by using close-loop normal data, so the procedure is minimally invasive from the process operation point of view, • it performs a PWC performance monitoring (off-line/on-line), therefore it allows to reevaluate the feasibility of the MIMO control policy, • it integrates the sum of squared deviations (SSD) concept 17,18 with a batch-wise as well as block-wise recursive PLS strategy 10 , so the data-driven monitoring index development and the model estimations become direct and explicit, • it extends the covariance-based performance index and diagnosis (CID) 3 with a blockwise recursive formulation 12 , so the dynamic implications and the diagnosis features are considered with this proposal. The work is organized as follows. The proposed methodology is presented in Section 2. Section 2.1 and 2.2 discuss in detail the main constitutive parts such as the SSD index based on the block-wise recursive PLS approach and the block-wise recursive covariancebased methodology, respectively. The overall procedure, the main tasks (off-line/on-line), and the monitoring indexes are presented in Section 2.3. The suggested approach is applied and analyzed on two typical case studies such as the Shell evaporator plant and Tennessee Eastman Process in Section 3. Finally, the major conclusions are stated in Section 4.

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NO

SS MODEL ID PWC Design SSD MIQP

CONTROLLED PROCESS?

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YES

SS MODEL ID SSD-BASED PLS APPROACH

OFF-LINE CLOSED-LOOP APPROACH

* EVALUATE CURRENT PWC * DEFINE BENCHMARK

CLOSED-LOOP PROCESS PWC MONITORING * SSD BW-RPLS * BW-RCID

DEFINE BENCHMARK OFF-LINE OPEN-LOOP APPROACH

NO

DEVIATION?

IMPLEMENT MODIFICATIONS

YES * EVALUATE PLS MODEL * PWC DESIGN: SSD MIQP * EVALUATE PAIRINGS * WORST DIRECTIONS ON-LINE PWC PERFORMANCE MONITORING AND REDESIGN

Figure 1: The data-driven PWC performance monitoring approach

2

The data-driven PWC performance monitoring approach

In this section the PWC performance monitoring approach is explained and its constitutive parts are discussed in detail. The overall layout is shown in Fig. 1 and clearly two principal branches can be identified depending on if the actual process is already controlled or not. If the industrial process is not controlled, the first task is the PWC design to fulfill with some operational objectives. This is the left branch in Fig. 1 and it is called the “offline open-loop approach” (green background). For example, the methodology suggested and studied by the authors in Braccia et al.(2017) 17 can be a good selection. This approach needs a steady-state model of the process, so any identification methodology could be used in this regard. Once the PWC policy is defined, the process works at closed-loop and the relevant

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information (historical data) is saved to develop the corresponding user-defined benchmarks, which will be used by the on-line monitoring counterpart. It is important to note that the current work is not focused in this “off-line open-loop approach” since several researchers, included the current authors, have been proposed numerous methodologies to address this PWC design problem. We will concentrate in the second branch of Fig. 1, i.e. when the process is already controlled. The first task is the “off-line closed-loop approach” (blue background), which is based on the integration between the SSD concept and the batch-wise PLS tool. This connection allows to identify an open-loop steady-state of the process by using closed-loop historical data (normal operation data). In this procedure is not required any kind of information about the installed controllers. The only relevant information at this stage is to know which are the MVs and CVs. This SSD-PLS closed-loop model identification gives the foundations to perform the first PWC assessment: 1- compute the SSD index for the installed control policy and 2- compute the optimal SSD index and its corresponding PWC structure by using the SSD-MIQP approach 17 . At this stage the user-defined benchmarks (UDB), to be used in the on-line monitoring tasks, are also selected. These UDB are usually related to some historical data period in which the desirable MIMO control performance was achieved. The next activities are grouped into the “on-line PWC performance monitoring and redesign” tasks (orange background). In this section the normalized indexes from the sum of squared deviations based block-wise recursive PLS (SSD-based BW-RPLS) and the blockwise recursive formulation of the covariance-based performance index and diagnosis (BWRCID) are used to monitor the overall process performance. On the one hand, the SSD-based BW-RPLS indicates potential degradation problems in the PWC performance (operability/controlability) due to static gains modification on the multivariable process and, on the other hand, the BW-RCID evidences modifications in the variance of the variables involved in the control structure basically due to controller detuning problems (dynamic issues). In

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the later case, a retuning of the controller could be enough to return the process operation near to the stipulated benchmark. In contrast, in the former case, depending on the magnitude and direction of the changes the complete installed PWC could become obsolete. If the normalized indexes deviate: 1- the PLS model is analyzed and the new SSD index computed, 2- the potential new control structures are obtained based on the SSD-MIQP 17 approach, 3the feasibility of the input-output pairing for the installed control policy is evaluated, and 4- the worst performance direction is obtained from the BW-RCID for diagnosis purposes. All this information can be reported to the plant personnel for the decision making process and the potential modifications could be implemented. In the next sections all the constitutive parts of the proposed PWC performance monitoring approach are discussed in detail.

2.1

The SSD-based BW-RPLS methodology

In this section the integration between the sum of squared deviations (SSD) strategy and the block-wise recursive partial least squares (BW-RPLS) is proposed. In the following subsections the main concepts of both approaches are revisited.

2.1.1

The SSD approach

Let us consider the following transfer functions matrix (TFM) representation (Laplace domain) of some stable (or stabilized) process,

y(s) = G(s)u(s) + D(s)d∗ (s),

(1)

which can be partitioned as: 





G∗s (s)









 ys (s)   Gs (s)   us (s)   Ds (s)  ∗  +  d (s)  = ∗ yr (s) Gr (s) Gr (s) ur (s) Dr (s)

8


(2)

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where y(s) are the potential output measurements, u(s) are the available MVs, and d∗ (s) are the disturbance variables (DVs) with size (ny × 1), (nu × 1), and (nd × 1), respectively. For some particular partitioning of G(s), as shown in eq. (2), ys (s) and yr (s) are the selected controlled variables (CVs) and uncontrolled variables (UVs) with size (ncv × 1) and ((ny − ncv )×1), respectively. Besides, us (s) are the selected manipulated variables (MVs) to control the subprocess Gs (s) and ur (s) are the remaining potential MVs, with size (ncv × 1) and ((nu − ncv ) × 1), respectively. Let us assume that the subprocess Gs (s) in Eq. (2) is controlled with some structure based on integral action, then at steady-state (s = 0) we have ys = yssp and,  







yssp



  −1 −1 ∗  (G∗r − Gr G−1 s Gs ) (Dr − Gr Gs Ds )    yr   Gr Gs    ur   =  ∗   G−1 −G−1 −G−1 us s s Gs s Ds d∗

(3)

where yssp is the set point vector. The sum of squared deviations (SSD) index is defined as 2 2 + Dr − Gr G−1 SSD = Gr G−1 D s s s F F

(4)

and it quantifies the deviations of the uncontrolled variables yr from their nominal operating points when set points and disturbances occur in the plant individually. The current authors 17–19 have been studied the minimization of the SSD index to obtain an optimal partitioning in eq. (2) and the corresponding impacts on the PWC design. These works suggest that the SSD minimization tends to maximize the minimum singular value of the subprocess Gs to be controlled, i.e. a well-conditioned1 subprocess is obtained (easy to control). 1

A well-conditioned process means that there are no directions in which the control could lose effectiveness.

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2.1.2

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Recursive PLS

The partial least squares (PLS) methodology has been widely applied in different engineering fields such as chemometrics, steady-state process modeling, dynamic process modeling, and process monitoring in the last three decades 10,14 . In most of the PLS applications the batch-wise modeling approach is used, i.e. the process data are stored and the PLS regression is carried out on the whole batch. The PLS approach allows to deal with data highly correlated (collinearity problem) by orthogonal projections onto latent variables. Let us consider the input and output process data as Xp and Yp , respectively. If ny , nu , and ns represent the number of output variables, input variables, and samples, then these matrices have dimension (ns × nu ) and (ns × ny ), respectively. Let us assume that X and Y are the normalized version to zero mean and unit variance of those original data matrices. The basic concepts behind the classical PLS regression assume a linear relationship between input and output process data matrices:

Y = XC + V

(5)

where C is the matrix of coefficients and V is the noise matrix. This linear model is developed by decomposing matrices X and Y into bilinear terms by using the so-called outer model: X = TPT + E (6) T

Y = UQ + F where T and U are the latent score matrices both with dimension (ns × nc ) and P (nu × nc ) and Q (ny × nc ) the corresponding loading matrices. Where nc is the number of components

10


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selected. The latent score matrices are related by a linear inner model:

U = BT + R

(7)

where the coefficients of the diagonal matrix B (nc × nc ) are determined by minimizing the residual matrix R. Considering Lemma 1 and Lemma 2 from Qin (1998) 10 (nc = rank(X)) the coefficients matrix in eq. (5) can be computed as:

C = PPT

−1

PBQT .

(8)

The traditional batch-wise PLS method is an iterative process and it is shown in the Appendix section. Let us assume that the following new data Xk (ns1 ×nu ) and Yk (ns1 ×ny ) is available, i.e. the k-th incoming data block. Where ns1 are the samples of the new block of data and the corresponding PLS model for this data base results Pbk , Bbk , Qbk . Note that the superscript “b” indicates that this PLS model was computed from the current block of data. Let us consider that {Xk−1 , Yk−1 } is the data pair resulting to combine all the previous data blocks and its PLS model was opportunely computed as Pk−1 , Bk−1 , Qk−1 . The combined data up to the current k-th block results: 





 Xk−1  Xck =   Xk

and

(ns +ns1 )×nu



 Yk−1  Ykc =   Yk

(9)

(ns +ns1 )×ny

according to Qin (1998) 10 the following expressions can be stated

C = XT X

+

XT Y,

XT X = PPT ,

where (·)+ denotes the generalized inverse.

11


XT Y = PBQT

(10)


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Algorithm 1: Block-wise recursive PLS (BW-RPLS) 1

2

3

4

5

Initialization: set k = 0, use the original block of data {Xp , Yp } and normalize it to zero mean and unit variance. Derive the PLS model by using the batch-wise PLS algorithm to compute P0 , Q0 , and B0 . Set k = k + 1. When a new block of data Xpk , Ykp is available, normalize it by using the normalization factor computed in step 1. Apply the batch-wise PLS algorithm to compute Pbk , Qbk , and Bbk . Combine both models: " # " # T T P B Q k−1 k−1 k−1 T , Ykc = T Xck = Pbk Bbk Qbk Apply the batch-wise PLS algorithm with combined data from step 3 to obtain the actualized matrices Pk , Qk , Bk , and compute the PLS model by using eq. (8): −1 Ck = Pk PT Pk Bk QT k k Return to step 2.

Thus the combined data in eq. (9) results

 



 

 PT k−1

Bk−1 QT k−1

Pk

Bbk

  ,  b T

 , b T

(11)

Qk

and the combined PLS model at the k-th incoming data block is

 C k = 

+ 

T 



T 



PT k−1

PT k−1

PT k−1

Bk−1 QT k−1

Pk

Pk

Pk

Bbk Qk

    T b

    T b

    T b

 , T b

(12)

and this implies that the regular PLS modeling approach on the data base in eq. (9) with a run-size of (ns + ns1 ) can be reduced to a PLS regression on the data base in eq. (11) with a run-size of 2nc . It is important to note that ns > ns1 >> nc . In this context, the block-wise recursive PLS (BW-RPLS) approach shown in Algorithm 1 is proposed for on-line steady-state model adaptation 10 . Note that this approach is based on the batch-wise PLS algorithm summarized in the Supporting Information File. The block-wise recursive PLS approach can be easily extended to take into account the relative importance between the new incoming data blocks and the older ones. In this case, we use a forgetting factor-based strategy, which is based on an exponential decaying 12


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weight, ξ, to progressively forget the effect of the old model into the new adaptation. A small ξ implies fast forgetting effect. The BW-RPLS approach shown in Algorithm 1 can be modified to include the forgetting factor strategy by weighting the old model by ξ before combining the models in step 3. It is worth mentioning that some improved versions of the Algorithm 1 are suggested by Vijaysai et al. (2003) 11 and Ni et al. (2012) 13 for both block-wise and sample-wise approaches, respectively. These proposals define a variable/adaptive forgetting factor and the block-wise case uses a new merging data block philosophy integrated with a condition number testing. We select the Qin’s approach 10 (Algorithm 1) because of its simplicity, effectiveness, and low computational burden. This give to us the basis for a proper integration with the SSD concepts. Finally, let us consider that σx = [σx1 , σx2 , . . . , σxnu ] and σy = [σy1 , σy2 , . . . , σyny ] are the row vectors of standard deviations used in the normalization at step 1 in Algorithm 1, then the coefficients matrix Ck can be unscaled as: C*k = diag σx−1 Ck diag (σy )

(13)

Comparing eqs. (3) and (5) and considering eq. (13) the following connection between the BW-RPLS and the SSD-based PWC methodology can be stated:   C*T k = 

Gr G−1 s G−1 s

(G∗r

−

∗ Gr G−1 s Gs )

(Dr −

∗ −G−1 s Gs

Gr G−1 s Ds )

−G−1 s Ds

  ,

(14)

if each row xp (l) and yp (l) from Xp and Yp , respectively, at sample instant l are given by: xp (l) = yssp T (l) ur T (l) d∗ T (l) , yp (l) = yr T (l) us T (l) .

(15)

It is worth noting that eq. (14) is a direct data-driven estimation of the SSD index. 13



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Indeed, considering the eq. (4), an on-line PLS-based SSD index evaluation can be suggested by: 2 2 *T SSDk = T1 C*T k T2 F + T1 Ck T3 F

(16)

where T1 , T2 , and T3 are binary selection matrices with the following structure: T T T1 = I(ny −nc ) 0((ny −nc )×nc ) , T2 = I(nc ) 0(nc ×(nu −nc +nd )) , T3 = 0(nd ×nu ) I(nd ) . (17)

where I(i) means the identity matrix with dimension (i × i) and 0(i×j) the (i × j) zero matrix. It is worth mentioning that these selection matrices are designed off-line and depend on the current PWC installed. A PWC performance monitoring can be developed by comparing the instantaneous SSDk index in eq. (16) against a user-defined benchmark based on historical data. This benchmark is called SSDn and it is related to the normal process operation:

ISSDk =

SSDk SSDn

(18)

the value of SSDn can be estimated by the SSD-based batch-wise PLS approach applied off line and using the normal data base, i.e. period in which the desirable MIMO control performance was achieved. The ISSDk index is a positive value, which can be greater or lower than one indicating a degradation or improvement in the PWC performance, respectively. Some modifications in the process gains can produce that the current PWC strategy loses performance and, depending on the direction of these changes, the current PWC could become totally obsolete or unstable. This fact happens because of the controllers detuning or the loss of validity of the control structure implemented. Moreover, due to the multivariable nature of the process, some modifications in the process gains can produce that the current PWC strategy improves its operability/controlability properties. This fact is due to an improvement in

14


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the numerical conditioning of the subprocess to be controlled, and then the corresponding input-output pairing gains. Anyway, in this last case if the controller results detuned, the dynamic behavior could be poor. Remark 1: It is important to note that the SSD index and the open loop steady-state gain matrices of the process can be estimated off-line and/or on-line without requiring any kind of information about the controller installed on the plant. The only relevant information is how many servo/regulatory control loops exist on the process and what variables are involved in the current PWC. Remark 2: If other sub-matrices need to be retrieved from C*T k similar selection matrices can be developed as well. For example, if all degrees of freedom are used in the installed PWC (nu = ncv ) and if we define the selection binary matrix T4 = 0(nc )×(ny −nc ) I(nc ) the open-loop steady-state process model can be retrieved as: Gs = T4 C*T k T2

−1

−1 Ds = − T4 C*T T T4 C*T 2 k k T3 −1 *T Gr = T1 C*T k T2 T4 Ck T2 h −1 i *T Dr = I − T1 C*T T T C T T4 C*T 2 4 k 2 k k T3

2.2

(19)

The BW-RCID strategy

The second approach proposed here, which complements the SSD-based RPLS one, is the block-wise recursive covariance-based performance index and diagnosis (BW-RCID). The main idea is to integrate the capabilities for MIMO controller performance monitoring and diagnosis of the covariance-based strategy with a block-wise recursive covariance adaptation for on-line supervision.

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2.2.1

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The CID method

This procedure is based on comparing the actual process outputs (CVs) variance against some specific benchmark. The most popular benchmark is the well-known MIMO minimum variance benchmark from the minimum variance control (MVC) theory. The methodology behind the MVC requires a priori dynamic as well as structural knowledge of the process, which is not attractive for practical implementations. Qin and Yu (2007) 3 suggest the covariance based user-defined benchmark from historical data as an alternative methodology. Let us consider the CVs covariance matrix Sn for some particular operation period (desired normal operation). Generally, this data base is related with a period in which the desirable MIMO control performance was achieved. Similarly, Sm represents the current covariance matrix of the CVs which need to be monitored (period under analysis). The direction along which the largest variance inflation occurs is given by

r = arg max

rT Sm r rT Sn r

(20)

where the solution is a generalized eigenvector problem

Sm r = µSn r

(21)

with µ the generalized eigenvalue and r the corresponding eigenvector. The direction r for the largest generalized eigenvalue is called the worst performance direction (WPD). The overall variability of the monitored period Sm can be supervised by defining a volumelike performance index as: Iv =

det (Sm ) det (Sn )

(22)

where Iv is a positive value which can be greater than or less than one. Values of the index above one indicate that the monitored period has a poor performance respect to the benchmark. If the index is significantly less than one, the monitored data present an

16


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improved performance.

2.2.2

The covariance adaptation

When the process operating conditions change (gradually or abruptly) the covariance matrix will not be constant and will need to be updated. Depending on the intrinsic dynamic of the process, this adaptation could be done based on a single sample as well as a block of data 12,20 . Generally, the industrial processes change slowly and the sample time is small compared with the time constant of the process, so a block-wise adaptation is more efficient. Let us consider that Xk is the normalized k-th incoming block of data and Sbk is its corresponding covariance matrix, then the recursive adaptation of the covariance can be defined as 12 :

Sk = (1 − β)Sbk + βSk−1

(23)

where β is the forgetting factor and Sk−1 the covariance updating for the previous (k − 1) block of data. Considering the volume-like index in eq. (22) and the covariance adaptation in eq. (23), the process variability can be block-wise monitored by de following updated index:

Ivk =

2.3 2.3.1

det (Sk ) det (Sn )

(24)

Procedure and monitoring indexes Off-line procedure

Let us suppose that some industrial process is controlled with some kind of integral feedback control strategy. With ny potential CVs, nu available MVs, nd disturbances, and ncv installed control loops. Under normal operation, where typical set points and disturbances changes happen and the desirable MIMO control performance was achieved, the historical 17



Page 18 of 42

data base defined in eq. (15) is collected according to the structure given in eq. (3) with ur = 0 (remaining potential MVs do not change). In this context, the classical batch-wise PLS procedure shown in the Supporting Information File can be applied to the data pair {X, Y} to identify the nominal/normal (notice the superscript “n”) PLS model C*n and nT T therefore the corresponding open-loop steady-state process model Gn = GnT and s Gr nT T Dn = DnT , via proper binary selection matrices (eqs. (14) and (17)). So, two SSD s Dr indexes can be computed at this stage: • The current SSD value (user-defined benchmark): according to the installed control structure, 2 2 SSDn = Gnr (Gns )−1 F + Dnr − Gnr (Gns )−1 Dns F 2 2 = T1 C∗nT T2 F + T1 C∗nT T3 F

(25)

which is a fixed value and quantifies the current PWC performance. • The optimal SSD value (notice the superscript “op”): re-evaluation of the current control structure, op op −1 2 op op −1 op 2 SSDop = Gop (G ) + D − G (G ) D r s r r s s F F

(26)

op op op where the optimal partitioning Gop s , Gr , Ds , and Dr is the solution of a MIQP

optimization problem as stated in Braccia et al. (2017) 17 by using the normal steadystate process matrices Gn and Dn . The normal data base can be used in this stage to tune the forgetting factors ξ and β for the SSD-BW-RPLS and BW-RCID strategies, respectively. The main objective is to obtain a robust recursive adaptation avoiding excessive variability for the ISSDk and Ivk indexes. Furthermore, these monitoring indexes can be characterized according to their n n mean, µnISSD and µnIv , and standard deviation, σISSD and σIv , values for this normal data base.

18


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These parameters will be used later in the on-line monitoring to normalize the instantaneous indexes.

2.3.2

On-line procedure

From the monitoring perspective it is important to have a quick and effective detection of any change in the performance of the current PWC. In this context, we propose to normalize the instantaneous indexes in eqs. (18) and (24) by using their corresponding statistics from the normal operation case as it is shown in eq. (27),

NISSDk =

|ISSDk − µnISSD | , n 3σISSD

NIvk =

|Ivk − µnIv | n 3σIv

(27)

n n are the mean and standard deviation computed for each index , and σIv where µnISSD , µnIv , σISSD

as commented in previous section. So, the PWC performance does not have modification if NISSDk ≤ 1 and NIvk ≤ 1, i.e. the process operation is close to the user-defined benchmarks. In contrast, if NISSDk > 1 or NIvk > 1 a PWC performance modification can be assumed and the estimated matrices need to be analyzed. This monitoring approach represents two univariate control charts (Shewhart charts) by using the so-called “3-σ” rule as control limit (99% confidence). Once an abnormal situation is detected at the kdb -th incoming block, the final PLS model estimation is performed by averaging the instantaneous PLS models for the next kav consecutive data blocks, C∗av =

1 kav

kdb +kav X

C∗k

(28)

k=kdb +1

this is done because of the time response of the process as well as robustness issues. Considering eq. (14) and eq. (19), the open loop steady-state gain matrices of the process Gs , Ds , Gr , and Dr can be retrieved and the relative gain array (RGA) can be evaluated T as Λ = Gs ⊗ (G−1 s ) . These matrices help us to diagnose the abnormal operation by direct

comparison with the corresponding matrices for the normal case (user-defined benchmark). 19



Page 20 of 42

In the same sense, the generalized eigenvector problem defined in eq. (21) can be used in this case to identify the problematic directions, which are directly associated with the degraded control loops. An alternative index can be implemented to monitor the current performance of the installed input-output pairing. This can be made based on the instantaneous estimation of the RGA matrix when Gs is updated via eq. (19) and it is compared against the off-line estimation for normal operation case. Let us assume a diagonal input-output pairing, so this index can be defined based on the mean squared difference as:

Λmsd = k

ncv 1 X [Λn (i, i) − Λk (i, i)]2 ncv i=1

(29)

where Λn (i, i) and Λk (i, i) are the main diagonal of the RGA matrix given by the model identified off-line (for the user-defined benchmark period) and the current recursive model, respectively. The Λmsd evaluation could be useful to complement the diagnosis given by k NISSDk and NIvk indexes.

3

Case studies

In this section two case studies are proposed to evaluate the data-driven plant-wide control performance monitoring system described in previous sections. Initially, the academic Shell fractionator process is used for showing, step-by-step, the complete procedure shown in Fig. 1 under several scenarios. Finally, the Tennessee Eastman Process is proposed for testing the methodology on a complex large-scale non linear benchmark.

3.1

The Shell Process

The Shell oil fractionator process presented in Maciejowski 21 and Zumoffen and Basualdo 22 is considered here. The overall process layout and the installed decentralized control structure

20


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y3 Top Composition y1

d1

CC u1

y4 sp1 d2 y5

y6

Side Composition

u3

y2 CC

TC sp7

u2 y7

sp2

Feed

Figure 2: Shell process can be observed in Fig. 2 with the following description: • Output variables: y1 and y2 top and side composition, y3 to y7 top, upper reflux, side draw, intermediate reflux and bottom temperatures, • Manipulated variables : u1 and u2 top and side draw flows, u3 bottom reflux duty, • Disturbance variables: d1 and d2 intermediate and upper reflux duty, • Control structure : decentralized control loops with pairing y1 −u1 , y2 −u2 , and y7 −u3 . According to the Section 2.1.1 then ny = 7, nu = 3, nd = 2, and ncv = 3. It is important to note that in this case the process disturbances are considered to be measured variables. In the next case study (TEP) we will consider the unmeasured disturbances scenario. The steady-state model of the controlled subprocess, its corresponding relative gain array (RGA), and the current input-output pairing (highlighted with gray background) are

21



Page 22 of 42

presented in eq. (30), 





 4.05 1.77 5.88      Grn s =  5.39 5.72 6.90    4.38 4.42 7.20

  rn Λ =  

2.08 −0.73 −0.35   3.42 0.93 −3.36    4.71 −4.50 0.79



(30)

where the superscript “rn” means that this is the “real under normal operation” process inforrn rn rn , θ22 , θ33 ] = mation2 . In addition, the delays associated with each control loop are Θrn = [θ11

[27, 14, 0] minutes. This information will be useful to evaluate the performance of the recursive estimation and to diagnose some abnormal events. Let us consider that Gs (i, j) is the ij-entry of the gain matrix Gs for the subprocess under control. Three different scenarios are proposed to represent potential process modifications and their corresponding PWC performance degradations: Scenario I- Change in the static gain Gs (1, 1) (-11%): this modification impacts notably on the numerical properties of the subprocess under control. This fact will be clearly identified by a drastic SSD index deviation. The RGA pairings could become obsolete depending on the ill-conditioning degree of Gs . Moreover, this change produces also the detuning of the controller in loop no. 1, so the BW-RCID procedure could identify this fact, depending on the detuning degree. Scenario II- Change in the static gains Gs (1, 1) and Gs (2, 2) (-15%): This is a more drastic change of the process. It impacts severally on both the SSD index and the RGA pairings. In addition, the BW-RCID procedure will identify clearly the detuned control loops in this case. rn Scenario III- Change in the delay θ11 (×3): in this case no static changes are suggested,

in contrast, this modification only affect the dynamic behavior evidenced by a typical detuning in the controller no. 1. The BW-RCID procedure will detect this degraded 2

Real information means that some rigorous model of the process is available for comparison purposes.

22


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performance in variance easily. In contrast, from the SSD perspective, the PWC performance will remain unchanged with the same numerical properties and the RGA pairings still valid. In all cases the process modifications occur at the sample time l = 1 × 104 .

3.1.1

Off-line procedure: batch-wise PLS and user-defined benchmark

Remembering the model partitioning stated in Section 2.1.1, the input and output matrices definition in Section 2.1.2 and the controlled Shell process description given in the previous paragraph, the l-th row in X and Y is given by: x(l) = [sp1 (l), sp2 (l), sp7 (l), d1 (l), d2 (l)] y(l) = [y3 (l), y4 (l), y5 (l), y6 (l), u1 (l), u2 (l), u3 (l)] respectively, where spj (l) is the j-th set point variable for the sample time l, with j = 1, 2, 7. The normal operation is collected by using an sample time of Ts = 2 min into a batch run size of ns = 1 × 104 samples, with the controlled process working under common/typical variations in the set points and disturbances variables. The off-line procedure is applied based on the batch-wise PLS approach suggested in the Supporting Information File with the number of components given by nc = rank(X) = 5 and a tolerance of convergence of ρ = 1 × 10−3 . These parameters are summarized in Table 1 and the quality of the obtained PLS model, C∗n , is evaluated against the real one, C∗rn , by using the Frobenius norm as:

∗

∗rn

||∆C ||F = ||C

∗n

− C ||F =

sX X i

|C∗rn (i, j) − C∗n (i, j)|2

(31)

j

where C∗rn (i, j) means the ij-entry of the matrix C∗rn . All the estimated matrices can be evaluated according to this metric. In this context, the Table 1 also shows the estimation n error of Gs by evaluating ||∆Gs ||F = ||Grn s − Gs ||F . The open-loop steady-state model of

the process can be retrieved by defining specific selection matrices as suggested in eq. (17). 23



Page 24 of 42

Table 1: Shell process - Bath-wise PLS parameters - Normal operation nc

||∆C∗ ||F

||∆Gs ||F

5

0.30

0.45

SSDrn 4.83

SSD index SSDn SSDop 4.64 2.37

ρ 1 × 10−3

Ts [min] 2

ns [samples] 4 × 104

Table 2: Shell process - Block-wise RPLS parameters µnISSD

n σISSD

ξ

µnIv

n σIv

β

0.93

0.08

0.85

1.00

0.11

0.97

ns1 [samples] 1000

Control Limit 1

ns [samples] 4 × 104

In this context, two different SSD indexes can be computed: 1- the actual PWC performance called SSDn for the current control structure implemented, i.e. the user-defined benchmark and 2- the optimal SSD index named SSDop by applying the procedure described in Braccia et al. 17 . Both indexes are based on the current off-line PLS model estimation. Table 1 shows these indexes against the real value SSDrn . The important conclusion at this stage is that the implemented PWC structure in Fig. 2 with SSDn = 4.64 is not the best control policy. In fact, there is another decentralized control structure with better performance, SSDop = 2.37, which suggest the following control loops: y2 − u2 , y4 − u1 , and y7 − u3 . This conclusion agrees with the opportunely suggested one in Zumoffen and Basualdo 22 . Therefore, the off-line procedure suggested here can be used to re-evaluate if the currently implemented PWC structure is the best selection or not, i.e. the off-line PWC performance assessment. Obviously, this procedure relies on the MIQP strategy proposed by Braccia et al. 17 for designing PWC structures. All this information can be used to evaluate some kind of revamping/redesign of the control policy. The next step is to compute the normalization statistics to be used in eq. (27). The normal data base (user-defined benchmark) with ns = 4 × 104 samples is used initially to define some important parameters for the on-line performance monitoring. The block-wise RPLS in Algorithm 1 is run for the normal data base with a block size of ns1 = 1000 samples and a forgetting factor of ξ = 0.85 as it is shown in Table 2. This setting was selected by considering the slower response of the controlled process ≈1000 samples. It is important to note that if the length of the block ns1 increases the updating is less frequent and the forgetting factor 24


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needs to be low for allowing a quick adaptation of the PLS model. In contrast, if the block length is small, the block update rate is high and the forgetting factor needs to be high enough to made robust the on-line PLS model estimation. A similar discussion can be made for the selection of the forgetting factor β for the RCID case. Processing all the ns samples n n in the normal data base the mean µnISSD and µnIv and the variance σISSD and σIv statistics

of the index ISSDk and Ivk can be computed, respectively. The normalized version of these indexes, NISSDk and NIvk , and its respective control limits represent an univariate control chart. All these parameters are summarized in Table 2.

3.1.2

On-line procedure: PWC performance monitoring

In this section an on-line PWC performance monitoring is implemented on the controlled process shown in Fig. 2 by using the methodology proposed in Fig. 1 based on SSD-BWRPLS and BW-RCID displayed in previous sections. The first simulation case, called Scenario I, is shown in Fig. 3 where a total simulation time of 4 × 104 samples is used. The Figure 3(a) shows the temporal evolution of the normalized ISSD index, NISSDk , and its control limit. An abnormal situation is detected when NISSD > 1 and this happens at sample ld = 14001 (detection sample time). In this case, ld belongs to the fifteenth processed data block, kdb = 15. The normalized recursive covariance index is shown in Fig. 3(b). It is clear that NIvk does not detect any abnormal situation in this case. The RGA mean squared difference Λmsd in Fig. 3(c) shows some minor k modifications in the main diagonal pairings, which could indicate some kind of problem in the current control structure. The worst performance direction (WPD) in Fig. 3(d), tied to the largest eigenvalue, indicates that the control loops no. 1 and no. 2 are the main contributors to the variance performance degradation. In this context, the static gains related to these loops need to be analyzed first to evaluate some inconsistencies from the normal operation case. In the Supporting Information file all the estimated matrices are shown and those clearly indicate that there is a modification in Gs (1, 1) of ≈ −16%, which

25



Page 26 of 42

Table 3: Shell process - Block-wise RPLS with forgetting factor - Abnormal operation Scenario I II III

||∆Gs ||F

||∆Λ||F

0.86 1.26 0.73

0.89 0.91 0.83

SSD index SSDrf SSDav 10.57 10.43 10.18 9.79 4.83 5.27

ld [samples] 14001 14001 30001

Table 4: Shell process - RGA estimation - Normal/Abnormal operation Λn Normal Operation 2.35 −0.71 −0.64 1.12 −3.01 2.89 −4.24 0.60 4.65

Λav Scenario II 1.03 −1.01 0.98 4.17 −0.07 −3.10 −4.20 2.07 3.12

Scenario I 2.51 −0.91 −0.59 0.23 −3.12 3.89 −5.40 1.68 4.72

Scenario III 2.09 −0.68 −0.41 0.97 −2.95 2.98 −4.07 0.71 4.36

allows us to corroborate this testing scenario. The norm-based estimation errors of the open-loop process model Gs and the RGA matrix Λ given by the identified model are shown in Table 3. In addition, this table summarizes the real, SSDrf , as well as the estimated, SSDav , sum of squared deviations values. This scenario produces a drastic degradation of the PWC performance. This means that the current sets of CVs and MVs are not the best selection to guarantee operability/controlability of the process. Furthermore, if the average estimated RGA matrix, for this scenario, is compared with the opportunely estimated off-line in the normal operation (benchmark) case, the final conclusion is that there is no feasible input-output pairing in this case3 . This fact can be observed from Table 4 (Scenario I) and it is important to note that the current input-output pairing is highlighted with gray background. So, this scenario produces both the PWC degradation as well as an obsolete input-out pairing. In this case, the control problems cannot be solved by retuning the controller only, it needs a complete PWC redesign (new CVs, MVs, and pairing). If the SSD-MIQP approach, suggested by Braccia et al. 17 , is used in this scenario the new PWC results: y2 − u2 , y4 − u1 , and y7 − u3 with SSDop = 2.35. This control structure is the same as the optimal one opportunely proposed in the off-line procedure in Section 3.1.1. The second set of simulation corresponds to the Scenario II, where both Gs (1, 1) and 3

The pairing between output i and input j is called feasible if δmin ≤ Λ(i, j) ≤ δmax . In this case we select [δmin , δmax ] = [0.4, 10]

26


Page 27 of 42

6 NISSD Control Limit

1

NIv Control Limit

5 0.8

0.6

NIv

NISSD

4

3

0.4 2

0.2

1

0 0

0.5

1

1.5

2 Samples

2.5

3

3.5

0 0

4

0.5

1

4

x 10

(a) NISSDk index

1.5

2 Samples

2.5

3

3.5

4 4

x 10

(b) NIvk index

1

0.8

0.9

0.6

0.8

0.4

Eigenvector direction

0.7 0.6

Λmsd

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5 0.4 0.3

0.2 0 −0.2 −0.4

0.2

−0.6

0.1

−0.8

0 0

0.5

1

1.5

2 Samples

2.5

3

3.5

4 4

−1

1

x 10

(c) Λmsd index k

2 Control loop

3

(d) Eigenvector - WPD

Figure 3: Performance monitoring - Shell process - Scenario I Gs (2, 2) change −15% from their nominal values. The temporal profiles of the normalized PWC performance index NISSDk and the covariance-based index NIvk used for monitoring purposes are presented in Fig.4(a) and Fig.4(b), respectively. Similar monitoring conclusions to the previous scenario can be stated here for the NISSDk index, but in addition the NIvk index also presents a considerable deviation indicating that the controllers are detuned compared with the user-defined benchmark case. The norm-based estimation performance, the SSD average value estimation, and the detection sample can be observed in Table 3. The Figure 4(c) shows the mean squared pairing difference Λmsd which evidences potential drastic k modifications in the RGA-based pairing respect to the nominal case. Figure 4(d) summarizes the WPD. This direction indicates that control loops no. 1 and no. 2 present, again here, the major contribution in the overall variance performance degradation. Analyzing the estimated

27



steady-state matrix of the process, see the Supporting Information file, modifications of ≈ −20% and ≈ −11% can be identified for the components Gs (1, 1) and Gs (2, 2), respectively, which allows us to corroborate this testing scenario. Comparing in Table 4 the estimated average RGA matrix for this scenario and the normal one it can be observed that the current pairing results infeasible, i.e. the pairing y2 − u2 presents a negative value near to zero. Furthermore, there is another potential pairing to resolve this issue, i.e. y1 − u3 , y2 − u1 , and y7 − u2 . In this scenario two effect are superimposed: on the one hand, the PWC performance degrades indicating that the current sets of CVs and MVs are not the best selection to guarantee operability/controlability of the process and, on the other hand, the RGA matrix suggest another input-output pairing to ensure closed-loop stability. Again here, the highlighted entries in the RGA represents the installed pairing. In this scenario, the control problems cannot be solved by retuning the controller only, it needs a new input-ouput pairing selection and/or a complete PWC redesign. Finally, the simulation results for Scenario III are displayed in Fig. 5. In this case, no static modifications of the process are proposed. In contrast, a modification of the delay rn , which impacts only dynamically is suggested. In this case, in theory, no deviations of θ11

the SSD index or pairing problems should be observed, but the covariance index will modify to indicate variance performance degradation. This fact shows the complementary behavior of the suggested monitoring indexes. The Figures 5(a) and 5(c) corroborate the hypothesis argued previously, and no deviation exist for NISSDk and Λmsd indexes. Conversely, the k recursive covariance index NIvk deviates from its normal operation state and at ld = 30001 samples (detection time) the abnormal behavior is detected. In this case, ld belongs to the data block, kdb = 31. The worst performance direction (WPD) in Fig. 5(d), tied to the largest eigenvalue, indicates that the control loop no. 1 is the major contributor to the variance performance degradation. Analyzing the estimated steady-state matrices of the process, shown in the Supporting Information file, can be conclude that no gain modifications occur in this scenario. The norm-based estimation errors of the process model Gs and the RGA

28


Page 28 of 42

Page 29 of 42

6

2.5 NISSD Control Limit

NIv Control Limit

5 2

4

NIv

NISSD

1.5 3

1 2

0.5 1

0 0

0.5

1

1.5

2 Samples

2.5

3

3.5

0 0

4

0.5

1

4

x 10

(a) NISSD index 1.8

0.8

1.6

0.6

1.4

0.4

1.2

0.2

1 0.8 0.6

−0.8

1.5

2 Samples

2.5

3

3.5

4 4

x 10

−0.4 −0.6

1

2.5

−0.2

0.2

0.5

2 Samples

0

0.4

0 0

1.5

(b) NIv index


Λmsd

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3

3.5

4

−1

4

1

x 10

(c) Λmsd index k

2 Control loop

3


Figure 4: Performance monitoring - Shell process - Scenario II matrix Λ are shown in Table 3. In addition, this table summarizes the real SSDrf value and the average estimated one SSDav . This scenario III does not produces the PWC performance degradation, which indicates that the current sets of CVs and MVs are good enough to guarantee the operability/controlability of the process, i.e. the same PWC performance than the benchmark case. A similar conclusion can be obtained for the current input-output pairing and this can be observed from Table 4 where the RGA average estimation for this monitored scenario is very close to the off-line estimation with the benchmark data. So, this scenario only produces variance performance degradation due to the detuning of the controller no. 1. In this case, the control problems can be solved by retuning the controller only, a complete PWC redesign it is not needed.

29



3

NISSD Control Limit

1

NIv Control Limit 2.5

0.8

0.6

NIv

NISSD

2

1.5

0.4 1

0.2

0.5

0 0

0.5

1

1.5

2 Samples

2.5

3

3.5

0 0

4

0.5

1

4

x 10

(a) NISSD index

1.5

2 Samples

2.5

3

3.5

4 4

x 10

(b) NIv index

1

0.6

0.9

0.4

0.8 0.2


0.7 0.6

Λmsd

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 42

0.5 0.4

0

−0.2

−0.4

0.3 −0.6 0.2 −0.8

0.1 0 0

0.5

1

1.5

2 Samples

2.5

3

3.5

4

−1

4

1

x 10

(c) Λmsd index k

2 Control loop

3


Figure 5: Performance monitoring - Shell process - Scenario III

3.2

The Tennessee Eastman Process

The Tennessee Eastman process (TEP), proposed in Downs and Vogel 23 as a multivariable nonlinear benchmark, is used here to test the suggested data-driven methodology for PWC performance monitoring. Figure 6 shows the TEP layout and the corresponding decentralized control structure (CS) installed on the process. The stabilizing as well as the servo/regulator control loops are identified with gray and white background, respectively. The TEP variables description is given in Table 5 and the 8 × 8 CS is defined by the following input-output pairing: y3 − u7 , y4 − u8 , y7 − u6 , y8 − u4 , y9 − u2 , y10 − u3 , y11 − u5 , and y12 − u1 . Note its corresponding RGA matrix in Table 6. The off-line procedure proposed in Section 2.3 is applied here to the TEP. The normal operation data base is processed by considering a sampling time of Ts = 5/60 hours and a 30


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u4 u5

u2

1

A

CWS

CC

PC

u1 2

D FI

FC

y8

JI

u8

PI

13 TC

7

JC

Compressor

CWR

Condenser

LC 3

E

9

y1 8

LC

LI

FC

FI

TI y4, y5

LI 5

CWS A N A L Y Z E R

6

TI

y2 y3, y9

Reactor

TI

TC

TC

u6 LI

Stm Cond

C

XB y11

y7

FC

u3

CC

y6 CWR

12 TI

Purge

Vap/Liq Separator

10

TC u7

A N A L Y Z E R

FI

FC

LC

A N A L Y Z E R

XG

XH

Stripper 4

11

Product y10, y12

Figure 6: Controlled Tennessee Eastman Process batch run size of ns = 1 × 104 samples. The complete parameters settings for the batch-wise PLS algorithms and the obtained open-loop steady-state model are shown in the Supporting Information file. In this context, two different SSD indexes were computed: 1- the actual PWC performance, SSDn = 3.90, for the current control structure implemented and 2- the optimal SSD index, SSDop = 2.06, by applying the procedure described in Braccia et al. 17 . Both indexes are based on the current off-line PLS model estimation and these results clearly indicate that the current PWC is not the best selection. Furthermore, the control structure corresponding to the optimal SSD is a 7 × 7 decentralized policy with the following pairing: y4 − u8 , y7 − u6 , y8 − u4 , y9 − u2 , y10 − u3 , y11 − u5 , and y12 − u1 . Summarizing, the installed PWC has an extra control loop which does not give any kind of improvement on the PWC performance, increasing the investment and maintenance costs. Therefore, the off-line procedure suggested here can be used to re-evaluate if the current implemented PWC structure is the best selection or not. All this information can be used to evaluate some kind of revamping/redesign of the control policy.

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Table 5: TEP variables u1 u2 u3 u4 u5 u6 u7 u8 y1 y2 y3 y4 y5 y6 y7 y8 y9 (*) y10 (*) y11 (*) y12 (*)

Inputs D flow [kg/h] A flow [kg/h] A/C flow [kscmh] Compressor rec. valve [%] Purge valve [%] Stripper steam valve [%] RCWO temp. setpoint [o C] CCW Flow [m3/h] Outputs Recycle flow [kscmh] Reactor flow [kscmh] Reactor temp. [o C] Separator temp. [o C] Separator pressure [kPa] Stripper pressure [kPa] Stripper temp. [o C] Compressor work [kW] Reactor pressure [kPa] Production rate [m3/h] B comp. purge [mol%] G/H comp. ratio

Variable XM V (1) XM V (3) XM V (4) XM V (5) XM V (6) XM V (9) XM E(21)sp XM V (11) XM E(5) XM E(6) XM E(9) XM E(11) XM E(13) XM E(16) XM E(18) XM E(20) XM E(7) XM E(17) XM E(30) XM EG/H

It is important to note that in the TEP case we cannot know the “normal” steadystate model with certainty (we assume the TEP as the real industrial process), therefore we compare the performance of the methodology based on PLS against other methods such R as a system identification experiment (SIE) and the linear analysis from Simulink . In

the Supporting Information file this comparison is done. It is worth mentioning that most R based approach needs a SIE methods needs open-loop historical data and the Simulink

complete non linear model, while the SSD-based PLS procedure suggested in the current work only needs the closed-loop normal operation data from the process. So, the later procedure seems to be minimally invasive from the process operation point of view. The methodology proposed in Fig. 1 based on SSD-BW-RPLS and BW-RCID is tested on the TEP by suggesting a steady-state gain modification of −40% in the Gs (1, 7) component (the input-output pairing y3 − u7 ). This gain modification was implemented by using an additive uncertainty structure (Gs + ∆Gs ). Figure 7 shows this scenario where the total 32


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simulation time was fixed to 4 × 104 samples. Figure 7(a) shows the temporal evolution of the normalized ISSD index, NISSDk , and its control limit. An abnormal situation is detected when NISSD > 1 and this happens at sample ld = 15001 (detection time). The averaging SSD index results SSDav = 7.52 indicating a clear degradation of the PWC performance (remember the normal case of SSDn = 3.90). This means that the current sets of CVs and MVs are not the best selection to guarantee operability/controlability of the process. The normalized recursive covariance index, NIvk , is shown in Fig. 7(b) and this statistic also displays a divergence respect to the normal operation case, indicating a degraded control performance. in Fig. 7(c) shows strong modifications in Similarly, the RGA mean squared difference Λmsd k the installed input-output pairing. The worst performance direction (WPD) in Fig. 7(d), computed for CVs and MVs, indicates that the control pairing called y3 − u7 (Gs (1, 7)) is the main responsible of the variance performance degradation. The normalization statistics to be used in eq. (27) and all the parameters required for the on-line BW-RPLS approach are shown in the Supporting Information file. Furthermore, the average estimated RGA matrix, for this scenario, is compared with the opportunely estimated off-line in the normal operation (benchmark) case in Table 6. It is clear here that the original input-output pairing (highlighted with gray background) has a feasibility problem, mainly given by the RGA entries: Λav (2, 8), Λav (4, 4), and Λav (5, 2) which suffer a considerable degradation. So, this scenario produces both a high SSD index as well as an obsolete input-out pairing. In this case, the control problems cannot be solved by retuning the controller only. Some potential solutions to this scenario are listed in the following: 1. Change the input-output pairing of Λav (2, 8) and Λav (5, 2) by Λav (2, 2) and Λav (5, 8), respectively. This is indicated in Table 6 with an asterisk. The pairing is more feasible, but the SSD index remains with a value of 7.52 indicating potential conditioning problems of Gs . 2. Run the MIQP approach suggested by Braccia et al. 17 with the steady-state model 33



9

100 NISSD Control Limit

8

NIv

90

Control Limit

80

7

70 6

NIv

NISSD

60 5 50

4 40 3

30

2

20

1

10

0 0

0.5

1

1.5

2 Samples

2.5

3

3.5

0 0

4

0.5

1

4

x 10

(a) NISSD index

1.5

2 Samples

2.5

3

3.5

4 4

x 10

(b) NIv index

5 10 4.5 4 5


3.5 3

Λmsd

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2.5 2 1.5

0

−5

1 0.5 0 0

−10 0.5

1

1.5

2 Samples

2.5

3

3.5

4 4

y3

y4

y7

y8

x 10

(c) Λmsd index k

y9 y10 y11 y12 u1 u2 CVs and MVs

u3

u4

u5

u6

u7

u8


Figure 7: Performance monitoring - TEP - Gs (1, 7) change %40 identified on-line by the BW-RPLS methodology. This solution proposes the following 7 × 7 decentralized control structure: y4 − u2 , y7 − u6 , y8 − u4 , y9 − u8 , y10 − u3 , y11 − u5 , and y12 − u1 with SSD = 2.31. The solution no. 2 seems to be the best choice because implies a reduced PWC (one less loop), the problematic loop (y3 − u7 ) is not included in the solution, all the involved measurements and manipulated variables were already considered in the original PWC, and the final SSD index of 2.31 reports an improvement from the original PWC structure (for both normal (3.90) and faulty (7.52) cases).

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Table 6: TEP - RGA estimation - Normal/Abnormal operation

4

y3 y4 y7 y8 y9 y10 y11 y12

u1 0.00 0.04 -0.02 -0.01 -0.01 -0.07 -0.01 1.08

y3 y4 y7 y8 y9 y10 y11 y12

0.00 0.03 -0.02 0.04 -0.01 0.07 -0.15 1.05

u2 -0.07 0.23 0.00 0.03 0.67 0.10 0.02 0.03 -0.14 0.66 0.07 -0.01 0.11 0.10 0.12 0.09

∗

Normal Operation – Λn u3 u4 u5 u6 0.02 0.11 -0.01 0.00 -0.02 0.24 -0.01 0.01 0.03 0.00 0.00 0.99 0.00 0.64 0.06 0.00 0.12 -0.08 0.05 -0.01 0.96 0.06 -0.01 0.01 0.01 0.04 0.92 0.00 -0.13 0.00 0.00 0.00 Gs (1, 7) change %40 – Λav 0.03 -0.03 -0.07 -0.02 0.06 0.09 -0.06 0.08 0.04 0.06 -0.02 0.91 0.04 0.31 0.31 0.00 0.06 0.32 -0.11 0.02 0.77 0.22 -0.01 0.01 0.08 0.07 0.97 0.00 -0.09 -0.04 -0.02 0.00

u7 1.02 -0.13 0.02 0.12 0.00 -0.04 0.01 0.01

u8 -0.07 0.65 -0.01 0.16 0.26 0.00 0.02 0.00

1.30 -0.21 -0.15 0.14 0.04 -0.15 0.01 0.02

-0.08 0.36 0.10 0.17 ∗ 0.57 -0.01 -0.11 0.00

Conclusions

In this work a new data-driven plant-wide control (PWC) performance monitoring system is proposed. The main philosophy behind the method relies on: 1- the integration between the sum of squared deviations (SSD) approach and the partial least squares strategy (batch-wise as well as recursive) and 2- the block-wise recursive covariance-based methodology for monitoring and diagnosis purposes. Both procedures allow to monitor the PWC performance detecting potential steady-state gains modification, controller detuning problems (static and dynamic), and input-output pairing drawbacks. The proposed approach generates valuable information (off-line as well as on-line) to the plant personnel for evaluating the already installed control policy and suggesting potential control structure modifications and/or potential controller retuning (revamping/redesign). All this information gives more reliability to the decision making process. The overall procedure is minimally invasive from the process operation point of view, since both the off-line as well as the on-line procedures only require closed-loop normal data to perform the corresponding monitoring/estimation tasks. This is an important difference from the classical identification approaches, which require open-loop

35



data rich enough. The simulation results shown that all the information related to the SSD index, the CID index, the pairing deviation, the WPD, and the steady-state model estimation need to be evaluated in a whole to take some reliable conclusion about the current status of the process. It is important to note that the SSD approach is scaling dependent, so the normalization criterion needs to maintained along the overall monitoring procedure. The length of the incoming block and the forgetting factors play an important role in the block-wise recursive algorithms. If the block length increases the updating is less frequent and the forgetting factor needs to be low for allowing a quick adaptation. In contrast, if the block length is small, the block update rate is high and the forgetting factor needs to be high enough to made robust the on-line estimation. This conclusion applies to both the BW-RPLS as well as the BW-RCID approaches. Although the proposed data-driven plant-wide control performance monitoring system gives an approximated estimation about the steady-state gain modifications of the process, no estimations are given related to dynamic implications such as delays or time constants, which need to be evaluated by some other methodology. Future works in this line will complement this deficiency by suggesting a dynamic SSD approach. Furthermore, we handle the main hypothesis that: –adaptive forgetting factors, –adaptive means, –persistence to excitation test, –variable data block length, and –indexes to enable or not the model update, could be very useful to improve the overall PLS recursive modeling performance.

Acknowledgement The authors thank the financial support of CIFASIS-CONICET, ANPCYT, UNR-FCEIA and UTN-FRRo from Argentina.

Nomenclature 36


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Variables

Acronyms

B: Diagonal coefficients matrix Bbk : B for the k-th block of data C: Coefficients matrix (PLS) BW-RCID: Block-wise RCID

Ck : Coefficients matrix for the k-th block of

BW-RPLS: Block-wise RPLS

data

CPA: Control performance assessment

C∗av : Average C∗k along tav blocks

CPM: Control performance monitoring

C∗k : Unscaled Ck

CID: Covariance-based index and diagnosis

C∗rn : Real C∗ - normal operation

CS: Control structure

d∗ : Disturbance vector

CVs: Controlled variables

D: Steady-state disturbance process model

DVs: Disturbance variables

Dn : D - normal operation

MIQP: Mixed-integer quadratic programming

Ds : Controlled disturbance sub-process model

MIMO: Multi-input/multi-output

Dns : Ds - base-case design

MQC: Multivariate quality control

Dop s : Ds - optimal design

MVs: Manipulated variables

E: Residual matrix

MVC: Minimum variance control

F: Residual matrix

PCA: Principal components analysis

G: Steady-state process model

PID: Proportional-integarl-derivative

Gn : G - normal operation

PLS: Partial least squares

Gs : Controlled sub-process model

PWC: Plant-wide control

Gns : Gs - base-case design

RCID: Recursive CID

Grn s : Real Gs - normal operation

RGA: Relative gain array

Gop s : Gs - optimal design

RPLS: Recursive PLS

ISSDk : SSD PWC monitoring index for the k-th

SISO: Single-input/single-output

block of data

SPM: Statistical process monitoring

Iv: CID monitoring index

SSD: Sum of squared deviations

Ivk : CID monitoring index for the k-th block of

UVs: Uncontrolled variables

data

WPD: Worst performance direction

k: Incoming data block

37



kdb : Detection data block

SSDop : Optimal SSD index

kav : Number of blocks for averaging

T: Latent score matrix

l: Sample time

T1 , T2 : Binary selection matrices

ld : Detection sample time

T3 , T4 : Binary selection matrices

nc : Number of components

u: Input vector of the process

ncv : Number of CVs

ur : Fixed input vector

nd : Number of disturbances

us : MVs vector

ns : Number of samples

U: Latent score matrix

ns1 : Number of samples of block no. 1

V: Noise matrix

nu : Number of inputs

X: Normalized input process data

ny : Number of outputs

Xk : k-th block for the k-th block of data

NISSDk : Normalized ISSD for the k-th block of Xp : Input process data base data

Xpk : k-th block - input process data

NIvk : Normalized Iv for the k-th block of data

Xck : k-th block - combined PLS models

P: Loading matrix

y: Output vector of the process

Pbk : P for the k-th block of data

yr : UVs vector

Q: Loading matrix

ys : CVs vector

Qbk : Q for the k-th block of data

yssp : Set point vector

r: Eigenvector

Y: Normalized output process data

R: Residual matrix

Yk : k-th block of output data

Sk : Monitored covariance matrix for the k-th Yp : Output process data base Ykp : k-th block - output process data

block of data

Sbk : Block covariance matrix for the k-th block Ykc : k-th block - combined PLS models of data Sn : Benchmark covariance matrix Sm : Monitored covariance matrix

Greek letters

SSDk : SSD index for the k-th block of data SSDrn : Real SSD - normal operation SSDn : Benchmark SSD index

rn : Real delay for pairing ij - normal operation θij

µ: Eigenvalue vector µnISSD : ISSD mean - normal operation

38


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µnIv : Iv mean - normal operation

Λ: RGA matrix

n σISSD : ISSD standard deviation - normal opera- Λav : Average RGA matrix

tion

Λn : RGA matrix - normal operation

n : Iv standard deviation - normal operation σIv

Λrn : Real RGA matrix - normal operation

σx : Input standard deviation vector

Λk : RGA matrix for the k-th block of data

σy : Output standard deviation vector

n Λmsd k : Mean squeared difference between Λ and

ξ: Forgetting factor RPLS

Λk

β: Forgetting factor RCID

Supporting Information This material is available free of charge via the Internet at http://pubs.acs.org/. The steady-state model estimation (all scenarios) for the Shell plant is summarized. The complete parameters setting for the TEP case is reported and different steady-state model identification strategies compared.

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(5) Yu, J.; Qin, S. J. Statistical MIMO controller performance monitoring. Part I: Datadriven covariance benchmark. Journal of Process Control 2008, 18, 277–296. (6) Fu, R.; Xie, L.; Song, Z.; Cheng, Y. PID control performance assessment using iterative convex programming. Journal of Process Control 2012, 22, 1793–1799. (7) Gao, X.; Yang, F.; Shang, C.; Huang, D. A Novel Data-Driven Method for Simultaneous Performance Assessment and Retuning of PID Controllers. Industrial & Engineering Chemistry Research 2017, 56, 2127–2139. (8) Yan, Z.; Chan, C.; Yao, Y. Multivariate Control Performance Assessment and Control System Monitoring via Hypothesis Test on Output Covariance Matrices. Industrial & Engineering Chemistry Research 2015, 54, 5261–5272. (9) Qin, S. J. Survey on data-driven industrial process monitoring and diagnosis. Annual Reviews in Control 2012, 36, 220–234. (10) Qin, S. J. Recursive PLS algorithms for adaptive data modeling. Computers & Chemical Engineering 1998, 22, 503–514. (11) Vijaysai, P.; Gudi, R.; Lakshminarayanan, S. Identification on demand using a blockwise partial least-squares technique. Industrial & Engineering Chemistry Research 2003, 42, 540–554. (12) Choi, S. W.; Martin, E. B.; Morris, A. J.; Lee, I. B. Adaptive multivariate statistical process control for monitoring time-varying processes. Industrial & Engineering Chemistry Research 2006, 45, 3108–3118. (13) Ni, W.; Tan, S. K.; Ng, W. J.; Brown, S. D. Localized, adaptive recursive partial least squares regression for dynamic system modeling. Industrial & Engineering Chemistry Research 2012, 51, 8025–8039.

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(14) Godoy, J.; Zumoffen, D. A.; Vega, J. R.; Marchetti, J. L. New contributions to nonlinear process monitoring through kernel partial least squares. Chemometrics and Intelligent Laboratory Systems 2014, 135, 76–89. (15) Ge, Z. Review on data-driven modeling and monitoring for plant-wide industrial processes. Chemometrics and Intelligent Laboratory Systems 2017, 171, 16–25. (16) Liu, Y.; Wang, F.; Gao, F.; Cui, H. Hierarchical multiblock T-PLS based operating performance assessment for plant-wide processes. Industrial & Engineering Chemistry Research 2018, 57, 14617–14627. (17) Braccia, L.; Marchetti, P. A.; Luppi, P.; Zumoffen, D. Multivariable Control Structure Design Based on Mixed-Integer Quadratic Programming. Industrial & Engineering Chemistry Research 2017, 56, 11228–11244. (18) Zumoffen, D. Plant-wide control design based on steady-state combined indexes. ISA Transactions 2016, 60, 191–205. (19) Zumoffen, D. Oversizing analysis in plant-wide control design for industrial processes. Computers & Chemical Engineering 2013, 59, 145–155. (20) Liu, X.; Chen, T.; Thornton, S. M. Eigenspace updating for non-stationary process and its application to face recognition. Pattern Recognition 2003, 36, 1945–1959. (21) Maciejowski, J. Predictive Control With Constraints; Prentice Hall, 2002. (22) Zumoffen, D.; Basualdo, M. Improvements on multiloop control design via net load evaluation. Computers & Chemical Engineering 2013, 50, 54–70. (23) Downs, J.; Vogel, E. A plant-wide industrial process control problem. Comput. Chem. Eng. 1992, 17, 245–255.

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TOC Graphic OFF-LINE APPROACH DATA BASE

ON-LINE APPROACH

PWC

SSD BATCH-WISE PLS BENCHMARKS SELECTION INSTALLED PWC PERFORMANCE EVALUATION

MONITORING AND DIAGNOSIS SSD BW-RPLS + BW-RCID DEFINE NEW PWC DEFINE MODIFICATIONS

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Data-Driven Plant-Wide Control Performance Monitoring - Industrial

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