Maximizing Fault Detectability with Closed-Loop Control⁎

Proceedings, Proceedings, 10th 10th IFAC IFAC International International Symposium Symposium on on Proceedings, 10th of IFAC International Symposium on Advanced Chemical Processes Advanced Control Control of Chemical Processes Available online at www.sciencedirect.com Advanced Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018 Proceedings, 10th of IFAC International Symposium on Shenyang,Control Liaoning, China, July 25-27, 2018 Shenyang,Control Liaoning, July 25-27, 2018 Advanced of China, Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018

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IFAC PapersOnLine 51-18 (2018) 696–701

Maximizing Fault Detectability Maximizing Maximizing Fault Fault Detectability Detectability  Closed-Loop Control Maximizing Fault Detectability Closed-Loop Control Closed-Loop Control  Closed-Loop Control Zhijie Sun ∗∗ S. Joe Qin ∗∗

with with with with

Zhijie Sun ∗∗ S. Joe Qin ∗∗ Zhijie Sun S. Joe Qin ∗ ∗ Zhijie Sun S. Joe Qin ∗ ∗ Mork Family Department Family Department of of Chemical Chemical Engineering Engineering and and Materials Materials ∗ ∗ Mork Mork Family Department of Chemical Engineering and CA Materials Science, University of Southern California, Los Angeles, 90089, Science, University of Southern California, Los Angeles, CA 90089, ∗ Mork Family Department of Chemical Engineering and CA Materials Science, University of Southern California, Los Angeles, 90089, USA (e-mail: [email protected], [email protected]). USA (e-mail: [email protected], [email protected]). Science, University of Southern California, [email protected]). Los Angeles, CA 90089, USA (e-mail: [email protected], USA (e-mail: [email protected], [email protected]). Abstract: Abstract: This This paper paper addresses addresses the the fault fault detectability detectability problem problem for for PCA-based PCA-based process process Abstract: methods This paper addresses the control. fault detectability problem for PCA-based process monitoring under closed-loop Unlike previous research monitoring methods under closed-loop control. Unlike previous research assuming assuming the the same same Abstract: This paper addresses theand fault detectability problem for PCA-based process monitoring methods under closed-loop control. Unlike periods, previous research assuming the same process variation for both modeling monitoring the impact of controller process variation for both modeling and monitoring periods, the impact of controller on on monitoring methods control. Unlike previous research assuming the same process variation for under both closed-loop modeling and monitoring periods, the impact ofto controller on data variation is considered. The effective fault direction is given in order describe data variation is considered. The effective fault direction is given in order to describe the the process variation for bothonmodeling and monitoring periods, thecontrollers. impact controller on data variation is variation considered. The effective fault direction is given in order oftoThe describe the effect of process fault detection index under different minimum effect of process variation on fault detection index under different controllers. The minimum data variation is considered. The effective fault direction is given in order to describe the effect of process variation on fault detection index under different controllers. The minimum variance variance along along the the effective effective fault fault direction direction is is developed developed by by introducing introducing the the concept concept of of blockblockeffect of process variation faultdirection detectionis index underbydifferent controllers. The minimum variance along the effectiveonfault developed introducing the concept of blocklower-triangular lower-triangular interactor interactor matrix matrix and and conditional conditional minimum minimum variance variance control. control. The The sufficient sufficient variance alonga the effective fault direction is developed by introducing the concept blocklower-triangular interactor matrix and conditional minimum variance control. The of sufficient condition condition for for a fault fault to to be be detectable detectable under under any any controllers controllers is is provided. provided. The The proposed proposed method method lower-triangular interactor matrix and conditional minimum variance control. The sufficient condition for a fault to besimulation detectable under any controllers is provided. The proposed method is demonstrated with the examples. is demonstrated with the simulation examples. condition for a fault besimulation detectableexamples. under any controllers is provided. The proposed method is demonstrated withto the © demonstrated 2018, IFAC (International Federation ofexamples. Automatic Control) Hosting by Elsevier Ltd. All rights reserved. is with the simulation Keywords: Keywords: fault fault detection, detection, fault fault detectability, detectability, time time delay, delay, minimum minimum variance variance control, control, Keywords: fault detection,time-series fault detectability, time delay, minimum variance control, performance monitoring, analysis performance monitoring, time-series analysis Keywords: fault detection,time-series fault detectability, performance monitoring, analysis time delay, minimum variance control, performance monitoring, time-series analysis 1. INTRODUCTION INTRODUCTION rejection, 1. rejection, and and economics. economics. Improving Improving fault fault detectability detectability unun1. INTRODUCTION rejection, and economics. Improving fault detectabilitysince under abnormal condition is rarely taken into der abnormal condition is rarely taken into account, account, since 1. INTRODUCTION rejection, and economics. Improving fault detectability under abnormal condition is rarely taken into account, since Statistical process process monitoring monitoring has has been been studied studied by by many many the the process process is is designed designed around around aa predetermined predetermined operating operating Statistical der abnormal is rarelya recently taken into account, since the process is condition designed around predetermined operating Statistical process monitoring has been studied by many researchers and successfully applied in the process inpoint. To address this problem, Du et al. (2016) researchers and successfully applied in the process in- point. To address this problem, recently Du et al. (2016) StatisticalMeasurements process monitoring has beenin studied by process is designed around a recently predetermined operating point. To address this problem, Du et aal. (2016) researchers and successfully the process in- the dustries. fromapplied different sensors aremany colpropose dustries. Measurements from different sensors are colpropose an an optimal optimal tuning tuning method method seeking seeking for for a trade-off trade-off researchers and successfully in sensors themodel, process in- point. Toan address this problem, recently Dufor et aal. (2016) dustries. Measurements fromaapplied different arewhich colpropose optimal tuning method seeking trade-off lected and incorporated into data-driven between fault detection and control performance by lected and incorporated into a data-driven model, which between fault detection and control performance by solvsolvdustries. Measurements from different sensors are colpropose an optimal tuning method seeking for a trade-off between fault detection and control performance by solvlected and incorporated into a data-driven model, which is typically typically based based on on principal principal component component analysis analysis (PCA) (PCA) ing is ing an an optimization optimization problem, problem, where where the the fault fault detectability detectability lected and incorporated into acomponent data-driven model, which between faultbydetection and function. control performance byPCAsolving an optimization problem, where the fault detectability is typically based on principal analysis (PCA) (Kresta et al., 1991) or partial least squares (PLS) (Macis expressed a likelihood In the case of (Kresta et al., 1991) or partial least squares (PLS) (Mac- is expressed by a likelihood function. In the case of PCAis typically based onQin principal component analysis (PCA) ing anmonitoring optimization problem, where the fault detectability is expressed by a method, likelihood function. In the case ofequivaPCA(Kresta et al., 1991) or partial least squares (PLS) (MacGregor et al., 1994; and Zheng, 2013). Some statistics based maximizing likelihood is Gregor et al., 1994; Qin and Zheng, 2013). Some statistics based monitoring method, maximizing likelihood is equiva(Kresta etal., al.,1994; 1991) or partial least2013). squares (PLS) (Mac- is expressed by a method, likelihoodmaximizing function. In the effective caseisofequivaPCAGregor et Qin and Zheng, Some statistics based monitoring likelihood are subsequently derived to monitor monitor the process operation lent are subsequently derived to the process operation lent to to reduce reduce the the process process variation variation along along the the effective fault fault Gregor et al., 1994; Qin and Zheng, 2013). Some statistics based monitoring method, maximizing likelihood is equivalent to reduce the process variation along the effective fault are subsequently derived to monitor the process operation faults and and sensor sensor faults. faults. direction faults direction (Dykstra (Dykstra and and Sun, Sun, 2017). 2017). However, However, in in practice practice are subsequently toareduce theofprocess variation along the effective fault direction (Dykstra and Sun, 2017). However, in practice faults and sensor derived faults. to monitor the process operation lent when portion faulty samples are detected by process a portion of faulty samples are detected by process While faultsensor detection and diagnosis diagnosis has has been been researched researched when faults and faults. direction (Dykstra and Sun, 2017). However, in practice when a portion of faulty samples areto detected by process While fault detection and monitoring systems, the user tends tune the controller systems, the user tends to tune the controller While fault analysis detection has based been researched intensively, analysis on and faultdiagnosis detectability based on PCA PCA is is monitoring when a portion of faulty samples aretodetected by process monitoring systems, the user tends tune thethe controller intensively, on fault detectability on such such that that fault fault detectability detectability weighs weighs more more in in the control control While fault analysis detection and diagnosis has based been researched intensively, on fault detectability on PCA is insufficient. Early work by Dunia and Qin (1998c) provides systems, thecan userbetends tomore tune in the controller such that fault detectability weighs the control insufficient. Early work by Dunia and Qin (1998c) provides monitoring objective. Such action detrimental – closed-loop intensively, analysis on for fault detectability based onprovides PCA is objective. Such action can be detrimental – closed-loop insufficient. Early work by Dunia and Qin (1998c) the sufficient condition unidimensional-fault detectabilthat fault more in– the control objective. Such detectability action can beweighs detrimental closed-loop the sufficient condition for unidimensional-fault detectabil- such control performance is sacrificed while faults may performance is sacrificed while faults may never never insufficient. Early work for by unidimensional-fault Dunia Qin statistic. (1998c) provides the with sufficient condition detectability Squared Prediction Errorand (SPE) A more more control objective. Such action be detrimental –illustrated closed-loop control performance isacan sacrificed while faults may never ity with Squared Prediction Error (SPE) statistic. A be fully detected given confidence limit. As be fully detected given a confidence limit. As illustrated in in the with sufficient condition for unidimensional-fault detectability Squared Prediction Error (SPE) statistic. A more rigorous derivation is presented in (Dunia and Qin, 1998a), performance sacrificed while may never be fully detected givenisarate confidence limit.faults As illustrated in rigorous derivation is presented in (Dunia and Qin, 1998a), control Fig. 1, 1, fault fault detection detection rate is is improved improved by by implementing implementing ity withisSquared Prediction Error (SPE) statistic. A more Fig. rigorous derivation is presented in (Dunia and Qin, 1998a), which further extended to the multidimensional-fault fully detected given arate confidence limit. As illustrated by in Fig. 1, fault detection isprocess improved by implementing which is further extended to the multidimensional-fault be a controller that generates data represented controller that generates process data represented by rigorous derivation is presented inand (Dunia and et Qin, whichin isDunia further to the multidimensional-fault case andextended Qin (1998b) Mnassri al.1998a), (2013) aFig. 1, fault that detection ratethis isprocess improved byrepresented implementing a controller generates data by case in Dunia and Qin (1998b) and Mnassri et al. (2013) the magenta ellipse, but ellipse may never collapse magenta ellipse, but this ellipse may never collapse which isDunia further to for the multidimensional-fault case inthe andextended Qin (1998b) and Mnassri al. (2013) where necessary condition SPE is also alsoetderived. derived. To the athe controller that generates process by magenta ellipse, but this ellipsedata may represented never collapse where the necessary condition for SPE is To into into aa line line where where all all faulty faulty samples samples can can be be distinguished. distinguished. case inthe Dunia and Qin (1998b) and Mnassri etderived. al. (2013) where necessary condition for SPE is also To improve fault detectability with PCA, Wachs and Lewin magenta ellipse, but this ellipse may never collapse into a line where all faulty samples can be distinguished. improve fault detectability with PCA, Wachs and Lewin the Therefore, it it calls calls for for aa benchmark benchmark for for fault fault detectability detectability where the necessary condition forPCA, SPEPCA is alsomethod derived. To Therefore, improve fault detectability Wachs and Lewin (1999) have developed an with improved that into athe line where all samples Users canfault be distinguished. Therefore, it calls forfaulty a benchmark for detectability (1999) have developed an improved PCA method that with “best” feedback controller. can with the “best” feedback controller. Users can be be informed informed improve fault detectability with PCA, Wachs and Lewin (1999) have developed an improved PCA method that recursively sums the scores and increases correlations beit calls for amagnitude benchmark for is fault with the “best” feedback controller.that Users candetectability be informed recursively sums the scores and increases correlations be- Therefore, of the minimum fault detectable for of the minimum fault magnitude that is detectable for a a (1999) have developed an improved PCA method that recursively sums the scores and increases correlations between the the input-output input-output data data by by time time shifting. shifting. However, However, with “best” feedback controller.that Users can be informed of thethe minimum fault magnitude is detectable for a tween specific fault direction. recursively sumswork the scores and increases correlations be-a specific fault direction. tween the input-output data by time shifting. However, these research assumes the faulty data contain the minimum fault magnitude that is detectable for a specific fault direction. these research work assumes the faulty data contain a of tween research thebias input-output data by time However, these assumes the faulty data contain constant inwork steady state, which isshifting. parameterized asa This specific faultaddresses direction.the constant bias in steady state, which is parameterized as This paper paper addresses the fault fault detectability detectability problem problem with with these research assumes faulty data contain constant biasofinwork steady state, the which is parameterized asa This paper addresses theunder fault detectability problem with the product fault direction ξ and magnitude f , while PCA-based approaches any feedback the product of fault direction ξ and magnitude f , while PCA-based approaches under any feedback controllers. controllers. constant biasofinfault steady state,of ξwhich is data parameterized as This paper addresses theunder fault detectability with PCA-based approaches any improved feedbackproblem controllers. the product direction and magnitude f , while distribution or variation process are identical identical Since the distribution or variation of process data are Since fault fault detection detection rate rate can can be be improved by by variance variance the product of fault directionof ξprocess and magnitude f ,shown while PCA-based approaches under any feedbackthe controllers. Since faultalong detection rate can bedirection, improved by variance distribution or variation data are identical during modeling and monitoring periods, which are reduction the effective fault minimum during modeling and monitoring periods, which are shown reduction along the effective fault direction, the minimum theblue distribution or of 1. process data are identical Since faultalong detection rate can bedirection, improvedthe byminimum variance reduction the effective fault during andvariation monitoring as and red red ellipses ellipses in Fig. Fig. as blue modeling and in 1.periods, which are shown variance variance of of process process data data along along this this direction direction will will be be exexduring and monitoring the effective fault direction, thewill minimum as blue modeling and red ellipses in Fig. 1.periods, which are shown reduction variance ofalong process data along this direction be explored. While decreasing variation of manipulated plored. While decreasing variation of manipulated varivariIndustrial process systems are usually operated under as blue andprocess red ellipses in Fig. of process data along this is direction will bevariexplored. While decreasing variation of manipulated Industrial systems are1. usually operated under variance ables (MV) (MV) can can be be easily easily achieved, achieved, it it is generally generally unachievunachievIndustrial process systemscontrollers. are usuallyDuring operated under ables the regulation regulation of feedback feedback controllers. During the design design plored. While variation of generally manipulated variables (MV) candecreasing bevariation easily achieved, it is unachievthe of the able to suppress in controlled variables (CV) to Industrial process systems are usually operated under able to suppress variation in controlled variables (CV) to the regulation of feedback controllers. During thedifferent design phase, controller is synthesized synthesized by balancing balancing ablesto (MV) can be easily achieved, it is generally unachievphase, aa controller is by different zero able suppress variation intime controlled variables (CV) to due to the presence of delay. It is shown in the zero due to the presence of time delay. It is shown in the the regulation of feedback controllers. During the design phase, a controller is synthesized by balancing different objectives and and factors, factors, e.g., e.g., setpoint setpoint tracking, tracking, disturbance disturbance able to variation controlled variables (CV) to zero duesuppress towork the presence ofin time delay. It isminimum shown invarithe objectives pioneering by Harris (1989) that the phase, a controller is synthesized bytracking, balancing different pioneering work by Harris (1989) that the minimum variobjectives andwith factors, e.g., setpoint disturbance  zero due to the presence of time delay. It is shown in the pioneering work by Harris (1989) that the minimum variSun is now Halliburton Energy Services, Houston, TX.  Z. Z. Sun is now Halliburton Energy Services, Houston, TX. objectives andwith factors, e.g., setpoint tracking, disturbance  Z. Sun is now with Halliburton Energy Services, Houston, TX. pioneering work by Harris (1989) that the minimum vari-

 Z. Sun is now with Halliburton Energy Services, Houston, TX. 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018, 2018 IFAC 690 Copyright 2018 responsibility IFAC 690Control. Peer review© of International Federation of Automatic Copyright ©under 2018 IFAC 690 10.1016/j.ifacol.2018.09.282 Copyright © 2018 IFAC 690

2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018Zhijie Sun et al. / IFAC PapersOnLine 51-18 (2018) 696–701

(a)

Fig. 1. Motivation. Blue ellipse: variation of normal process data; red ellipse: variation of faulty data with the same controller; magenta ellipse: variation of faulty data with a different controller. ance of CV of single-input-single-output (SISO) systems can be estimated from routine operation data. For multiinput-multi-output (MIMO) systems, unitary interactors are adopted to derive the minimum variance of CV in terms of sum of each CV variance (Huang et al., 1997). When the variance of a CV subspace needs minimization, a block-triangular interactor matrix should be applied to describe the delay structure (Sun et al., 2011), which will be used in the current work. Based on this conditional minimum variance value, the fault detectability radius will be obtained to aid users to evaluate if it is feasible to detect a fault by re-tuning controllers. The remaining part of this paper is organized as follows. Section 2 briefly reviews PCA-based process monitoring techniques and the sufficient condition for fault detectability. In Section 3, the conditional minimum variance is derived first followed by the development of fault detectability radius. Some simulation results are presented in Section 4. Finally conclusions are given in Section 5. 2. FAULT DETECTABILITY FOR PCA-BASED PROCESS MONITORING 2.1 PCA-Based Statistical Process Monitoring The process monitoring method discussed in this paper relies on a PCA model which decomposes normal process data into two parts: ˜ (1) X = TPT + X where the data matrix X = [x1 , x2 , · · · , xN ]T ∈ RN ×m consists of N data samples with each row representing a sample xTi ∈ Rm , which is scaled to zero mean and usually unit variance. P ∈ Rm×l is an orthonormal matrix called loading matrix, where l is the number of principle components (PC) retained. T = XP ∈ RN ×l is the score ˜ is the residual matrix. matrix. X The principal component subspace (PCS) is represented by Sp = span{P} and the residual subspace (RS) is Sr = span{P}⊥ . Therefore, a sample vector x ∈ Rm can be projected onto PCS and RS by ˆ = Cx x (2) ˜ = (I − C)x x (3)

ˆ and where C = PPT is the projection matrix onto PCS. x ˜ are the modeled and residual portion of x. A geometric x interpretation of PCA decomposition of x is shown in Fig. 2.

691

697

(b)

Fig. 2. A sample vector projected onto PCS and RS with (a) SPE detection cylinder, and (b) T 2 detection cylinder. There are many existing work researching the statistics for detecting faults with PCA model. The most popular ones are SPE or Q-statistic, and Hotelling’s T 2 . Assuming process data observe Gaussian distribution, the control limits for either index can be derived. The SPE statistics evaluate the distance between PCS and sample vector x, which is its projection onto RS. SPE ≡ ˜ x2 = (I − C)x2 (4) When correlation among process variables is broken, SPE index increases. Thus, the process is considered abnormal if SPE > δ 2 where the control limits for SPE is denoted by δ 2 . The original expression for δ 2 is developed by Box et al. (1954). Other approximations are intensively studied, e.g., (Dunia et al., 1996). The geometric interpretation of fault detection with SPE is illustrated in Fig. 2. The process is considered to be normal when sample vector is inside the cylinder whose axis is PCS and radius is δ. A sample lies outside the cylinder can be detected by SPE index and will be labeled as faulty data. The Hotelling’s T 2 index mainly detects faults in PCS. One example is normal change of the steady-state operating point. T 2 is defined as T 2 = xT PΛ−1 PT x (5) where matrix Λ is a diagonal matrix with diagonal elements being the first l eigenvalues of Σ = N 1−1 XT X in descending order. Since T 2 statistic can be approximated by a χ2 -distribution with l-degrees of freedom when N is large, the control limits τ 2 becomes τ 2 = χ2l . Similarly, a fault in PCS is detected when T2 > τ2 2.2 Fault Model In this paper, a unidimensional fault is assumed whose exact source may be unknown. However, the fault can be reflected on process data by the following expression: x = x∗ + f ξ (6) m where x is an abnormal sample vector, ξ ∈ R is a known fault direction vector, and f represents the fault magnitude. Variation in process data is denoted by x∗ , which can be different during PCA modeling and monitoring periods. In order to make fair comparison, it is assumed in this paper that the controller does not affect fault magnitude f . This can be generally achieved by setting the steady state of process at E{x} or constraining the steady-state gain of controllers.

2018 IFAC ADCHEM 698 Shenyang, Liaoning, China, July 25-27, 2018Zhijie Sun et al. / IFAC PapersOnLine 51-18 (2018) 696–701

Remark. In other literatures, x∗ is defined as normal process variation. In this work, however, since it is possible to change the controller, the process variation under abnormal condition may be different from that of normal data. Table 1. Summary of control limits Control limit Γ

T2 τ2

SPE δ2

3. CONDITIONAL MINIMUM VARIANCE

2.3 Fault Detectability under Closed-Loop Control Regardless of fault detection subspace, the fault index in quadratic form can always be represented by (7) γ = xT Mx where γ is the index. The semi-positive-definite matrix M equals I − C for SPE and PΛ−1 PT for the T 2 index. 1 Multiplying M 2 on both sides of (6) gives 1

1

1

(8) M 2 x = M 2 x∗ + M 2 f ξ By denoting the control limit as Γ (see Table 1), the fault is sufficiently detectable when 1

1

1

(9) M 2 x  M 2 f ξ − M 2 x∗   Γ Since the fault is not detectable despite the value of f 1 1 when M 2 ξ = 0, it is assumed M 2 ξ > 0 thereafter. Thus, one can define the effective fault direction as 1 M2 ξ ξo ≡ (10) 1 M 2 ξ From (9), to make the fault sufficiently detectable, the 1 equivalent fault magnitude f o = M 2 ξf has to satisfy 1

1

f o = M 2 f ξ  M 2 x∗  + Γ

(11)

1

 M 2 x∗ min + Γ

1

Case 3. Not all nonzero elements of ξ o correspond to either the MV or the CV. The projected process variance x∗T Mx∗ becomes a combination of MV and CV variance, whose minimum value may not be obtained unless the model of true process is known. This case will not be discussed in the current work.

(12) 1

where M 2 x∗ min is the minimum value of M 2 x∗ .

Geometrically, the effective fault direction ξ o is parallel to the radial direction of the cylinder in Fig. 2, while f o represents the distance between the fault vector and longitudinal axis of the cylinder. A fault is considered to be detectable if f o is at least the sum of the control limit Γ and the minimum process variation along the radial axis 1 M 2 x∗ min .

More specifically, for SPE index shown in Fig. 2(a), ξ o ˜ ; a detectable fault should be a least is parallel to x 1 M 2 x∗ min away from the cylindrical surface. Similarly, ˆ for T 2 index illustrated in Fig. 2(b), ξ o aligns with x with the same distance between a detectable fault and the control limit surface. 1

The value of M 2 x∗ min can be determined by the following cases. Case 1. All nonzero elements of ξ o correspond to the controlled variables (CV) of the process. Since digital sam1 pling normally introduces a unit time delay, M 2 x∗ min is generally greater than zero. Its value is computed by the method shown in the next section. Case 2. All nonzero elements of ξ o correspond to the manipulated variables (MV) of the process. In this case, reducing the degree-of-freedom of the controller can force 1 M 2 x∗ min = 0 692

In this section, the conditional minimum variance control is discussed. The corresponding minimum variance of CV in a subspace will be derived, which is used to obtain CV fault detectability. 3.1 Block-Lower-Triangular-Interactor Matrix The time delay of a SISO system can be characterized by q −d where q −1 is the backshift operator and d denotes the delay. However, the time-delay structure of a MIMO system usually takes a more complicated form called interactor. Assume a MIMO process can be modeled as (13) yk = G(q)uk + N (q)ak where y, u, and a represents process output, input, and Gaussian noise vectors with corresponding sizes, respectively. Transfer function matrices G(q) and N (q) are the process and disturbance models. Definition 1. (Sun et al. (2011)). Given a proper and rational transfer-function matrix G(q) with size n × n, D(q) is known as block-lower-triangular interactor matrix if ˜ G(q) = D(q)−1 G(q) ˜ = lim D(q)G(q) = K lim G(q) and

q −1 →0

q −1 →0

 0 ··· 0 D11 (q) 0   D21 (q) D22 (q) · · ·  (14) D(q) =  .. .. .. ..   . . . . Dm,1 (q) Dm,2 (q) · · · Dm,m (q) where K is a full rank finite and non-zero matrix, and Di,i (i = 1, . . . , m) are the unitary interactor matrices. 

In the context of this article, it is sufficient that m = 2 since there are only two types of CVs – the CVs relevant and CVs irrelevant to the fault subspace. 3.2 Derivation of Conditional Minimum Variance 1

To obtain M 2 x∗ min , the coordinate needs to be rotated such that the first CV of the process in the new coordinate aligns with ξ o . One of the solutions is to perform QR decomposition ξ o = QR which provides the rotation matrix QT . Thus, by left multiplying QT on both sides of (13) the new system becomes (15) yk = G (q)uk + N  (q)ak where yk = QT yk , G (q) = QT G(q), and N  (q) = QT N (q). Next, the rotated CV vector yk is divided into two groups:    y  yk = 1,k (16)  y2,k  is the first CV of rotated system which is where y1,k aligned with ξ o while the rest of CVs are included in

2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018Zhijie Sun et al. / IFAC PapersOnLine 51-18 (2018) 696–701

 y2,k . According to (14), the corresponding block-lowertriangular interactor can be expressed by   D11 0 . D= D21 D22

10

5

(17)

˜ = DG where the delay-free transfer function matrix G  ˜ satisfies that limq−1 →0 G is nonzero, finite, and full rank. Note that the time delay d of the MIMO process is defined as the order of matrix D.

y

Hence, the rotated process (15) is rewritten as ˜ uk + N  ak yk = D−1 G

699

0

˜ k = q −d Dyk , further By introducing the filtered output y simplification of (17) leads to  ˜  uk + DN  ak . ˜ t+d =G (18) y

-5 -4

-2

0

2

4

2

4

u

Since N  (q) is a rational model of the delay-free disturbance, DN  can be factorized as

10

DN  = Fd q d + Fd−1 q d−1 + · · · + F1 q +R(q).    F (q)

k+d

5

(19)

Due to causality, it is impossible to eliminate F ak . Therefore, the conditional minimum variance control law can be subsequently derived: ˜ uk = − G

−1

Rak

(1) the variance of first rotated CV; and (2) the sum of variances of the rest CV when 1) is achieved. Proof. See Sun et al. (2011). Based on (20), the conditional minimum variance is obtained. 2 } = tr(eT1 F Σa F T e1 ) (21) min E{y1,k

where e1 = [1, 0, . . . , 0] covariance.

0

(20)

˜  is invertible. We have the following lemma for since G conditional minimum variance control law under blocklower-triangular interactor. Lemma 2. The conditional minimum variance control law (20) minimizes

T

y

Therefore, Equation (18) becomes ˜  uk + F ak + Rak . ˜ =G y

and Σa represents the noise

-5 -4

-2

0

u

Fig. 3. CV fault in Example 4.1. Blue dots are normal data; red crosses are faulty samples. Abnormal data in the upper and bottom figure are regulated by the original controller and the minimum variance controller, respectively. Control limit is represented by the dashed line. 4. SIMULATION EXAMPLES A motivating simulation example is presented first to provide an intuition to the method, which is followed by a more realistic case of MIMO systems. 4.1 A Motivating Example

3.3 Fault Detectability with Conditional Minimum Variance Since the plant and disturbance model of process is linear and noise source is Gaussian, projection of process varia1 tion M 2 x∗ is also multivariate Gaussian. Therefore, given 1 a confidence interval, M 2 x∗ min can be determined from Student’s t-distribution with mean being f ξ and variance calculated by (21). Note that the derivation of conditional minimum variance (21) does not require true process model. Only the delaystructure, i.e., the lower-block-triangular interactor matrix, along with disturbance dynamics N (q) needs to be known. 693

Let a SISO process be yk = G(q)uk + N (q)ak + b (22) −1 −1 −3 where G(q) = q , N (q) = 1 + 0.8q + q , the noise source term ak ∼ N (0, 1) and b is the bias term. Under normal condition, the setpoint is chosen as [ry , ru ] = [0, 0] with zero bias b = 0. The controller is designed to be q −2 (ry − yk ) (23) uk = − 1 + 0.8q −1 Substituting (23) into (22) suggests (24) yk = ak + 0.8ak−1 and (25) uk = −ak−2

2018 IFAC ADCHEM 700 Shenyang, Liaoning, China, July 25-27, 2018Zhijie Sun et al. / IFAC PapersOnLine 51-18 (2018) 696–701

6 4 2

y

Stacking yk and uk gives the sample vector for fault detection: xk = [uk , yk ]T = [−ak−2 , ak + 0.8ak−1 ]T (26) It is obvious that the components of xk are independent. Thus, performing PCA on X results in the first PC being the MV u and the second PC being the CV y as shown in Fig. 3. Setting the number of PC l = 1, the SPE and T 2 indices simply become y 2 and u2 , respectively, with the 95% control limits δ 2 = 6.56 and τ 2 = 4. A total of 1000 samples are collected to build the PCA model.

-2

Faults in CV Suppose there exists a fault on y that shifts the steady-state value of y from 0 to 4.56, which is reflected in the model that ry = b = 4.56. Simple calculation suggests that ξ = ξ o = [0, 1]T , and f = f o = 4.56. If the controller still takes the form of (23), the fault is not fully detectable. In fact, f needs to be at least 2δ = 5.12 to ensure detectability. However, since the process is SISO with unit time-delay, the minimum variance of yk is 2 σMVC = 1 which suggests that f  δ + 2σMVC = 4.56 can be detected. The corresponding minimum variance controller is uk = (0.8 + q −2 )(ry − yk ) (27)

Faults in MV Assume a fault occurs on the MV such that the steady-state value of MV has a bias of 2.05 while the process bias term is b = −2.05. Hence the setpoint becomes [ry , ru ] = [0, −2.05]. If the controller (23) remains unchanged, nearly half of the faulty data cannot be distinguished (Fig. 4). However, when operated open-loop (or equivalently, the controller gain is zero), all abnormal samples are correctly identified as shown in Fig. 4. 4.2 A MIMO Process Example The Wood-Berry distillation column model is simulated. With 1 minute sampling time, the process model is discretized as   0.7665 −0.9000 −4 q −2 q  1 − 0.9419q −1 1 − 0.9535q −1  . G(q) =  0.6055 −1.3472  −8 −4 q q −3 1 − 0.9123q 1 − 0.9329q −1 with the following disturbance model being selected:   1   1 − 0.8q −1 (28) N (q) =  1 − 0.5q −1  1 − q −1 whose associated ak ∼ N (0, I2 ).

An unconstrained model predictive controller (MPC) is designed to control the process under normal condition. The prediction and control horizons are set to 100 and 10, respectively. The MPC has the output weight Wy = diag{100, 1} and input-move weight W∆u = diag{0.01, 0.01}. The setpoint of MPC is selected as ry = 694

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Some visual clues are provided in Fig. 3. A total of 300 faulty samples are generated. Simulation results show that the fault detection rates for faulty samples under same controller and minimum variance controller are 94.3% and 97.7%, respectively.

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Fig. 4. MV fault in Example 4.1. Blue dots are normal data; red crosses are faulty samples. Abnormal data in the upper and bottom figure are regulated by the original controller and the zero-gain controller, respectively. Control limit is represented by the dashed line. [95%, 5%]. Since (28) is nonstationary, the sample vector is composed of CV data only. There are 1000 samples collected for PCA modeling with l = 1. The loading matrix is found to be (29) P = [0.707, −0.707]T Suppose there exists a fault in CV adding a bias term ∆ry = [3.3%, 3.3%] to the setpoint, i.e., ξ = [0.707, 0.707]. Since the fault lies in the RS, SPE index is applied during the monitoring period, and hence ξ o = ξ. The 95% control limit is calculated to be δ 2 = 4.13. In the meanwhile, Eq. (29) suggests that the system needs to be rotated 45◦ so that the first rotated CV aligns with f o . The block-lower-triangular interactor can be obtained  2  q 0 (30) D= 4 4 q q with which the conditional minimum variance along ξ o in the RS space is σMVC = 1.72. Thus, the fault is closed-loop detectable if f o  δ + 2σMVC = 4.655. The lower bound is achieved by conditional minimum variance controller which can be approximated by setting

2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018Zhijie Sun et al. / IFAC PapersOnLine 51-18 (2018) 696–701

701

to reduce uncertainty in the fault direction to enhance fault detectability. REFERENCES

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Fig. 5. Simulation results in Example 4.2. Blue dots represents normal data; red crosses are faulty samples. Abnormal data in the upper and bottom figure are regulated by the original and the modified MPC, respectively. MPC weights Wy = [1, 0.99; 0.99, 1] and W∆u = 0 with no constraints. Simulated results are demonstrated in Fig. 5. It can be seen that with the modified controller variance along fault direction is reduced at the expense of variance along the orthogonal direction. Fault detectability is therefore improved. 5. CONCLUSIONS In this paper, fault detectability of process with arbitrary controller is discussed. It is shown that minimum magnitude for a fault to be sufficiently detectable is the sum of control limit and conditional minimum variance in the equivalent fault direction. The tool of block-lowertriangular interactor is introduced to aid the derivation of minimum variance. The results are supported by simulation results. The results of this paper can be used to enhance fault detectability for processes under closed-loop. For instance, when a fault is detected with only a portion of samples in a time window, one can determine if the fault is detectable under closed-loop conditions and then alter the controller 695

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