Multi-input–Multi-output (MIMO) Control System Performance

Article pubs.acs.org/IECR

Multi-input−Multi-output (MIMO) Control System Performance Monitoring Based on Dissimilarity Analysis Chen Li,†,‡ Biao Huang,*,‡ Da Zheng,‡ and Feng Qian*,† †

Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai, People’s Republic of China ‡ Department of Chemical and Material Engineering, University of Alberta, Edmonton, Alberta, Canada ABSTRACT: In this paper, a novel dissimilarity-analysis-based method is proposed to monitor the control performance of multi-input−multi-output systems. The proposed approach detects changes in the orientation and volume of hyper-ellipsoids formed by the covariance matrices via analyzing the eigenvalues of transformed covariance matrices. Furthermore, a new performance index is used to quantify performance change of control systems. Simulation results from a numerical example, the Wood Berry distillation column example, and pilot-scale experiment results all demonstrate the effectiveness of the proposed method.

1. INTRODUCTION In industry, there are hundreds of control loops operating under different conditions. Various controllers, from conventional proportional−integral−derivative (PID) controllers to modern model predictive controllers, are utilized to meet various control needs.1 Maintaining hundreds of control loops simultaneously is a formidable task. Although these control loops have satisfying performance at an early stage, their performance will deteriorate as time goes by, because of deteriorating performance of the sensors/actuators, equipment wear, feedstock changes, product changes, and seasonal factors, etc.2−4 Such a challenge in control system maintenance calls for efficient and reliable control performance assessment and monitoring tools.5,6 Moreover, it is desired that the performance assessment tools should be used under closed-loop conditions without disturbing normal operation.7 The field of control performance monitoring has attracted considerable interest from both academia and industry since the seminal work of Harris.8 Harris demonstrated that the lower bound of process output variance is control invariant and can be estimated from closed-loop data. The control performance index was proposed based on this lower bound. The early research focused on the performance assessment of single-input−singleoutput (SISO) processes based on minimum variance benchmark.8,9 Further research involved the extension of the performance assessment to multivariable control systems, especially the model predictive control system. Multivariable control performance assessment requires the knowledge of process time delay, the structure of which is represented by interactor matrix. Huang10 proposed an algorithm using closedloop data to estimate the unitary interactor matrix, which was further used to derive the minimum variance control (MVC) benchmark. Harris et al.11 proposed an approach based on spectrum factorization for multivariate performance assessment. However, interactor matrix or spectrum factorization is conceptually difficult and computationally complex. Since it requires prior knowledge of the process, it is not always implementable in practice. In addition, although the MVC © 2014 American Chemical Society

benchmark sets an absolute lower bound in terms of output variance, it may not be practical to implement in reality, because of reasons such as physical constraints and robustness concern. To avoid the problems stated above, researchers proposed alternative performance benchmarks. Huang et al.12 proposed a pragmatic performance index requiring only the order of the interactor matrix. Considering the control input penalization, Huang and Shah4 proposed a performance benchmark using linear quadratic Gaussian regulator. The proposed approach minimized process output variance while keeping the input variability within a certain range. Patwardhan and Shah13 and Schafer and Cinar14 proposed using a framework based on historical data to form an index for control performance monitoring. However, most methods of multi-input−multioutput (MIMO) control performance assessment only use the trace of the output covariance matrix, i.e., the sum of the diagonal elements in output covariance matrix, without considering the correlations among variables. MacNabb and Qin15 proposed an output covariance-based MIMO performance assessment benchmark by performing principal component analysis (PCA) on both the minimum variance output and the actual output. Qin16 improved this method and proposed a determinant-based performance benchmark to measure the variance−covariance inflation. The ratio of determinants of output covariance matrices was used to develop an index to monitor the MIMO control performance. Yu and Qin17 integrated the determinant based benchmark with the historical data-based benchmark to form a more-informative data-driven index. The proposed data-driven benchmark took into account the volume of the ellipsoid, which represents the distribution of output data and whose volume is defined by the output covariance matrix, based on the assumption that the output data of the control system follows a normal distribution. Exploiting the information contained in Received: June 9, 2014 Accepted: November 2, 2014 Published: November 2, 2014 18226

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minimize the objective function of the original output yt. G̃ l can be expanded into

the ellipsoid defined by the covariance matrix provides an alternative way for performance assessment. However, besides volume, other ellipsoid features, such as the orientation, of the ellipsoid will also provide useful information regarding system performance. By investigating the distribution of process data to monitor the change in control system operating condition has been proposed by Kano et al.18 Since changes in system operating condition will result in changes in system performance, it is intuitive to evaluate system performance based on analysis of process data distribution. In this paper, a novel system performance monitoring approach is proposed based on dissimilarity analysis of process data. The remainder of this paper is organized as follows. Section 2 briefly reviews the minimum variance benchmark. The new dissimilarity performance index is introduced in Section 3. In Section 4, a data-driven and user-defined control system performance monitoring method is developed, based on the proposed performance index. Simulation and pilot-scale experimental results are given in Section 5. Section 6 concludes the paper.

−1 G̃ l = F0 + Fz + ··· + Fd − 1z −d + 1 + z −dR 1

= F + z − dR

Denoting the output under minimum variance control as ymv = Fet, the minimum variance can be calculated from routine operating data as min E(ytT yt ) = trace[Cov(Fet )] = trace[Cov(ymv )]

ηmv =

trace(Cov(ymv )) trace(Cov(y))

3. DISSIMILARITY-BASED PERFORMANCE INDEX 3.1. An Illustrative Example. The conventional MIMO control performance benchmark, as defined by eq 7, is based on the comparison of the sum of diagonal elements of covariance matrices. However, the information conveyed by the off-diagonal elements is ignored.16 To account for the correlations between different process variables, a determinant-based performance index was proposed based on the ratio of the determinant of the minimum covariance matrix and that of the actual output covariance matrix.16 If process output data follows a normal distribution, then the covariance matrix, geometrically, defines a hyper-ellipsoid and the determinant of the covariance matrix provides the volumetric information on the corresponding hyper-ellipsoid. However, neither the trace-based index nor the determinant-based index would be sufficient to reflect the covariance characteristic. To illustrate this, consider the following two examples where Φ1 and Φ2 are two covariance matrices that have the same traces and determinants. Example 1:

where Gp(M × N) and Gl(M × N) are proper, rational transfer function matrices for the plant and noise, respectively, ut is an N × 1 input vector, yt is an M × 1 output vector, and et is a white noise vector with zero mean and a covariance matrix Σe. The system block diagram is shown in Figure 1, where Gc represents the controller transfer function and ysp represents the setpoint of the output.

⎛3 2⎞ ⎟ Φ1 = ⎜ ⎝2 5⎠ Figure 1. Schematic diagram of multi-input−multi-output (MIMO) process under feedback control.

⎛3 2⎞ ⎟ Φ1 = ⎜ ⎝2 5⎠

⎛ 3 −2 ⎞ ⎟ Φ2 = ⎜ ⎝− 2 5 ⎠

Figures 2 and 3 present Monte Carlo simulations with corresponding covariance matrices given by Φ1 and Φ2, respectively. In both cases, the distribution of the data has the shape of an ellipse, whose orientation changes as the multivariate relationship represented by the covariance matrix varies. In Example 1, the variance of one variable increases while that of the other variable decreases. In Example 2, although the value of covariance between two data series remains the same, the sign changes from positive to negative, indicating an orientation change in joint data distribution. Clearly, in both examples, the process data distributions have changed, despite the fact that their covariance matrices have the same traces and determinants. 3.2. Proposed Index. According to Figures 2 and 3, it can be seen that different covariance matrices correspond to different hyper-ellipsoids. Meanwhile, the change in the shape of the hyper-ellipsoids indicates a change in data distribution and, thus,

(2)

−d

Premultiplying both sides of eq 2 by z D, where d is the order of the interactor matrix D and is defined as the maximum power of z in D, gives (3)

Let ỹt = z Dyt and G̃ l = z−dDGl; then, eq 3 becomes −d

yt ̃ = z −dGp̃ ut + G̃ let

⎛5 2⎞ ⎟ Φ2 = ⎜ ⎝2 3⎠

Example 2:

If Gp is a proper, full-rank transfer function matrix, a unitary interactor matrix D can be factored out, such that DGp = G̃ p, where G̃ p is the delay free transfer function matrix of Gp. Therefore, eq 1 can be expressed as

z −dDyt = z −dGp̃ ut + z −dDGlet

(7)

The value of the variance index (ηmv) is between 0 and 1, where the upper bound (1) corresponds to the minimum variance.

(1)

yt = Gput + Glet = D−1Gp̃ ut + Glet

(6)

The minimum-variance-based MIMO control performance index has been defined as

2. MIMO SYSTEM DESCRIPTION AND MINIMUM VARIANCE INDEX A multivariable feedback control system can be modeled as yt = Gput + Glet

(5)

(4)

4

Huang and Shah proved that if D was a unitary interactor matrix, the minimum variance control law that minimized the objective function of the interactor-filtered output ỹt would also 18227

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where P0 is an orthogonal matrix containing the eigenvectors of Φ and Λ is a diagonal matrix with diagonal elements representing the eigenvalues of Φ. Then, define a transformation matrix P as P = P0 Λ−1/2

(11)

and P has the following property: PTΦP = I

(12)

Transfer the minimum covariance matrix and actual covariance matrix through P: Smv = PTΦmv P

Sact = PTΦactP

(13)

It is easy to show that the transformed covariance matrices (Smv and Sact) satisfy

Smv + Sact = I

(14)

For the transformed covariance matrices, their eigenvalues (λmv j mv act and λact j ) and eigenvectors (υj and υj ) are defined by

Figure 2. Two datasets with the same covariance matrix traces and determinants for Example 1.

Smvυjmv = λjmv υjmv

Sactυjact = λjactυjact

(15)

From eqs 14 and 15, the eigenvalues and eigenvectors satisfy Smvυjact = (1 − λjact)υjact

(16)

From eqs 15 and 16, it can be shown that

υjact = υjmv

(17)

and

λjact + λjmv = 1

Equations 17 and 18 show that the transformed covariance matrices of minimum variance output and actual output have the same eigenvectors. Hence, if the eigenvalues of transformed covariance matrices are also close to each other, that is, from eq mv 18, λact j and λj are both ∼0.5, then the distribution of the actual output data is similar to that of the data under MV control, meaning that the actual control performance is close to the minimum variance performance. On the other hand, an eigenvalue that is near 1 or 0 implies that the actual control performance changes significantly, in comparison with the minimum variance control performance. So, the eigenvalues of the transferred covariance matrices completely determined the similarity of the original covariance matrices. Hence, a new control performance assessment index is proposed:

Figure 3. Two datasets with the same covariance matrix traces and determinants for Example 2.

a change in system performance. Hence, the performance change can be determined by analyzing the dissimilarity among the hyper-ellipsoids defined by different covariance matrices. Based on such an idea, in this section, a new performance monitoring method is proposed, based on dissimilarity analysis18 and Karhunen Loeve (KL) transformation19 to monitor changes in system performance. Consider the following two M × M covariance matrices: Φmv = Cov(ymv )

Φact = Cov(yact )

M

ID = 1 −

(8)

M

(19)

4. PERFORMANCE MONITORING BASED ON HISTORICAL DATA BENCHMARK The calculation of minimum variance output ymv, however, requires a priori knowledge of process models, which is not always available in practice. Hence, the historical data benchmark4,16 was proposed. The user-defined reference can be a period of “golden” operation data from the process during which desirable control performance is achieved. In this section, we propose a dissimilarity-based performance monitoring method under this frame. Denoting the reference data as period I and the

(9)

Since Φ is symmetric, it can be factorized as

Φ = P0 ΛP0−1

∑ j = 1 |λjact − λjmv |

where M is the number of variables. ID is between 0 and 1, where the value of 1 corresponds to minimum variance.

where Φmv and Φact are the covariance matrix of minimum variance output and actual output, respectively. Both matrices are positive definite. The joint covariance matrix, which is also positive definite, then is defined as Φ = Φmv + Φact

(18)

(10) 18228

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monitoring data as period II, the historical-data-based performance monitoring index is defined as

It can be seen, from eq 23, that only eigenvalues are used to form the indices. The information contained in the eigenvectors, i.e., the orientation information, are completely ignored. On the other hand, instead of performing eigenvalue decomposition to Cov(yI) and Cov(yII) directly, the KL transformation apply eigenvalue decomposition to the joint covariance matrix Φ as shown in eqs 9 and 10. The eigenvector information carried by P0 is reserved in the transformation matrix P, through which the original covariance matrices are transformed as shown in eqs 11 and 13. The proposed performance index is formed by the eigenvalues of the transformed covariance matrices with the eigenvectors being considered as shown in eqs 17 and 18. Since the eigenvectors represent the directions of the principal components and indicate the direction of the corresponding hyper-ellipsoid, after transformation, any change, either in the orientation or in the volume, of the hyper-ellipsoid defined by the original covariance matrices is reflected by the proposed performance index.

M

IM = 1 −

∑ j = 1 |λjII − λjI|

(20) M where and are eigenvalues of transformed data covariance matrix of reference period I and monitoring period II, respectively. If the control performance of the monitoring period remains unchanged, compared to that of the reference period, the index IM is equal to 1. On the other hand, if the control performance of the monitoring period changes significantly compared to the reference period, then IM will be close to zero. This can be intuitively interpreted by Example 1 and Example 2. Suppose Φ1 and Φ2 represent covariance matrices of reference data and monitoring data, respectively. The calculated IM index and traditional trace-based and determinant-based indices are shown in Table 1. The two traditional performance indices are

λIj

λIIj

Table 1. Comparison of Three Indices for Examples 1 and 2 index

Example 1

Example 2

trace-based index, ηtrace determinant-based index, ηdet dissimilarity-based index, IM

1.0 1.0 0.71

1.0 1.0 0.48

5. CASE STUDIES 5.1. Numerical Simulations. In this section, three cases are considered for a 2 × 2 system. In the first case, the performance of the control system is assessed when the interactions between two loops gradually increase. In the second case, three historical-databased performance indices (i.e., the trace-based index, the determinant-based index, and the proposed dissimilarity-based index) are applied to monitor the performance change when changes in noise model occur. In the last case, the effect of the non-Gaussian noise on the proposed performance index is evaluated. Case 1. The following 2 × 2 multivariable process is taken from ref 7. The process is modeled as

defined in eq 21, where “I” and “II” stand for the reference period and the monitoring period, respectively.16 According to Figure 2, Figure 3 and Table 1, it can be concluded that the two traditional performance indices fail to reflect such changes in data distribution or system operation, indicating their limitation in performance monitoring. ηtrace =

trace(Cov(y I ))

ηdet =

trace(Cov(y II ))

det(Cov(y I )) det(Cov(y II ))

y(k) = T (q−1)u(k) + N (q−1)a(k)

(21)

where the open-loop transfer function matrix T and the disturbance transfer function matrix N are given by

In conclusion, because of the complexity of covariance matrices, there exist situations where the performance of the MIMO control system performance changes significantly while the change of the trace or the determinant of the output covariance matrix is subtle, as illustrated in the above examples. Since the dissimilarity-based performance monitoring index captures the change of the covariance matrices better than conventional trace-based or determinant-based indices, this new index can be considered as an improved performance measure, compared to previously defined indices in the literature. Remark. Performing the eigenvalue decomposition to the covariance matrices Cov(yI) and Cov(yII) leads to Cov(y I ) = Q ΛIQ T

Cov(y II ) = U ΛIIUT

⎡ q −1 K12q−2 ⎤ ⎢ ⎥ −1 −1 − − 1 0.4 1 0.1 q q ⎢ ⎥ T (q−1) = ⎢ ⎥ q −2 ⎢ 0.3q−1 ⎥ ⎢ −1 −1 ⎥ 1 − 0.8q ⎦ ⎣ 1 − 0.1q ⎡ −0.6 ⎤ 1 ⎢ ⎥ −1 1 − 0.5q−1 ⎥ ⎢ 1 − 0.5q −1 N (q ) = ⎢ ⎥ 0.5 1 ⎢ ⎥ −1 −1 ⎢⎣ 1 − 0.5q 1 − 0.5q ⎥⎦

(22)

where M is the number of variables, and Q and U are M × M orthogonal matrices that contain the eigenvectors of Cov(yI) and Cov(yII), respectively. ΛI = [λI1,λI2,...,λIM]T and ΛII = [λII1 ,λII2 ,...,λIIM]T are diagonal matrices with corresponding eigenvalues as diagonal elements. The traditional performance indices shown in eq 21 can be calculated using the expressions ηtrace = ηdet =

trace(Cov(y I )) trace((Cov(y II )) det(Cov(y I )) II

det(Cov(y ))

=

⎡ 0.5 − 0.2q−1 ⎤ ⎢ ⎥ 0 −1 1 − 0.5 q ⎢ ⎥ Q (q−1) = ⎢ ⎥ −1 0.25 − 0.2q ⎢ ⎥ 0 ⎢ −1 −1 ⎥ (1 − 0.5q )(1 + 0.5q ) ⎦ ⎣

∑i = 1 λiI M

∑i = 1 λiII

ΠiM= 1λiI ΠiM= 1λiII

(25)

The noise follows a standard Gaussian distribution, and the sampling time of the system is Ts = 1 s. K12 is a constant, which varies from 0 to 10 to vary the interaction between the two loops. When K12 = 0, two minimum variance controllers are given as

M

=

(24)

(23)

(26) 18229

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the orientation of data distributions, which has been captured by the proposed index. Case 2. In this case, the process model and noise model of the reference period are given by

The minimum variance output of this system can be calculated as20 ⎡ ⎤ −1.1015q−1 − 0.2873q−1 ⎥ a(k ) ymv = ⎢ ⎢−0.2873 + 0.3352q−1 0.9578 + 0.6513q−1⎥ ⎣ ⎦

⎡ ⎤ q −1 q −2 ⎢ ⎥ −1 −1 − − 1 0.4 1 0.1 q q ⎢ ⎥ T (q−1) = ⎢ ⎥ −1 −2 q ⎢ 0.3q ⎥ ⎢ −1 −1 ⎥ 1 − 0.8q ⎦ ⎣ 1 − 0.1q

(27)

As K12 increases, the performance of the closed-loop system will change if the initial controller remains unchanged. Three thousand (3000) data samples are used to calculate the minimum variance-trace-based index and dissimilarity-based performance index (ID). Figure 4 shows the trajectory of both indices, while

⎡ −0.6 ⎤ 1 ⎢ ⎥ −1 1 − 0.5q−1 ⎥ ⎢ 1 − 0.5q −1 N (q ) = ⎢ ⎥ 0.5 1 ⎢ ⎥ ⎢⎣ 1 − 0.5q−1 1 − 0.5q−1 ⎥⎦

(28)

The noise dynamic changes during the monitoring period and is given by ⎡ 0.1 −0.6 ⎤ ⎢ ⎥ −1 1 − 0.5q−1 ⎥ ⎢ 1 − 0.5q −1 N *(q ) = ⎢ ⎥ 0.5 0.9 ⎢ ⎥ −1 −1 ⎢⎣ 1 − 0.5q 1 − 0.5q ⎥⎦

(29)

The two interacting loops are controlled by two PI controllers. One thousand (1000) data points are sampled in each period. The noise is Gaussian white noise with a mean of zero and a variance of 0.0016. Figure 6 shows the three performance Figure 4. Minimum variance index and dissimilarity-based index.

K12 varies from 0 to 10. The dissimilarity-based performance index (ID) considers not only the output variance but also the correlation among output variables. Thus, difference is reflected in the two indices. Figure 5 shows the distribution of the output datasets for K12 = 0, 3, 6 and 10. Figure 5 also demonstrates that the change in the performance of the system is also reflected in

Figure 6. Comparison of three historical-data-based performance indices.

monitoring indices. Figure 7 shows the distribution of reference data and monitoring data, respectively. It can be seen from Figure 7 that, in comparison with the reference period data, the data distribution of the monitoring period has changed remarkably, because of the change of the noise dynamics. Hence, a significant change in system performance should be expected. However, from Figure 6, the trace-based index and the determinant-based index are 1 and 0.94, respectively, which hardly indicated any change in performance. The dissimilarity-based index is 0.32, which indicated a significant change in performance during the monitoring period.

Figure 5. Distribution of reference period data (denoted by open circles, ○) and monitoring period data (denoted by cross symbols, +). 18230

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Figure 7. Data distribution of the reference period and the monitoring period.

Figure 9. Data distribution of the reference period and the monitoring period with non-Gaussian noise.

Case 3. In this simulation, the previous two cases are reevaluated under the circumstance of non-Gaussian noise. The noise is formed by adding the uniformly distributed noise to the normally distributed noise; therefore, the overall distribution deviates slightly from Gaussian. Considering the effect on case 1, the data distributions when K12 = 6 under Gaussian noise and non-Gaussian noise are shown in Figures 8a and 8b, respectively. Because of the effect of the non-Gaussian distributed noise, the distribution of the output data shown in Figure 8b deviates from the shape of an ellipse, compared with the case observed under Gaussian noise. The calculated minimum variance-based and dissimilarity-based indexes are 0.36 and 0.65, respectively. These index values are very close to that calculated in case 1, where the noise follows exactly a Gaussian distribution. For case 2, the added noise follows a uniform distribution bounded between −0.15 and 0.15. Two thousand (2000) data points are sampled in each period. The data distribution of the reference period and the monitoring period is shown in Figure 9. Compared to Figure 7, the data distribution under non-Gaussian noise has clearly diverged from the shape of an ellipse, indicating that the output data does not follow a Gaussian distribution.

Figure 10 further illustrates this fact by presenting the estimated probability distribution of the output data in each period. The traditional indices and the proposed dissimilarity performance index are recalculated and presented in Table 2. According to Table 2, it can be concluded that the dissimilarity-based index can still successfully reflect the performance change, even if the noise is non-Gaussian-distributed. 5.2. Wood-Berry Distillation Column. In this section, the well-known Wood-Berry distillation column process is considered for control performance monitoring with different controller parameter settings. The transfer function of the process is given by21 ⎡ 3.8e−8s ⎤ ⎡ 12.8e−s − 18.9e−3s ⎤ ⎥ ⎥⎡ ⎢ ⎡ y (s)⎤ ⎢ ⎤ u (s) ⎢ 1 ⎥ = ⎢ 16.7s + 1 21.0s + 1 ⎥ ⎢ 1 ⎥ + ⎢ 14.9s + 1 ⎥d(s) ⎥ ⎥ ⎢ ⎢ y (s)⎥ ⎢ 1.6e−7s − 19.4e−3s ⎥ ⎢⎣ u 2(s)⎥⎦ ⎢ 4.9e−3s ⎥ ⎣ 2 ⎦ ⎢ ⎢⎣ 13.2s + 1 ⎥⎦ ⎢⎣ 10.9s + 1 14.4s + 1 ⎥⎦ (30)

The process is controlled by two decentralized PID controllers. The controller parameters for the reference period are P = 0.52, I = 0.076, and D = 0.45 for Loop 1 and P = −0.1, I =

Figure 8. Data distribution of reference period data (denoted by open circles, ○) and monitoring period data (denoted by cross symbols, +): (a) Gaussian noise and (b) non-Gaussian noise. 18231

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Figure 10. Probability distribution of reference period output and monitoring period ouput.

Table 2. Comparison of Three Historical Data Based Indices under Non-Gaussian Noise Case index

value

trace-based index determinant-based index dissimilarity-based index

0.99 0.94 0.33

−0.018, and D = 0.02 for Loop 2. In monitoring period 1, the controller parameters are changed to P = 0.05, I = 0.08, and D = 0.45 for Loop 1 and P = −0.2, I = −0.018, and D = 0.02 for Loop 2. In monitoring period 2, only the proportional gain for Loop 1 is set as 0.02, while the other parameters are the same as period 1. Two thousand (2000) data points are sampled in each period. The calculated trace-based, determinant-based, and dissimilarity-based indexes are shown in Table 3. In monitoring period Table 3. Comparison of Three Indices for Monitoring Periods 1 and 2 index

monitoring period 1

monitoring period 2

trace-based index determinant-based index dissimilarity-based index

1.29 1.0 0.84

1.25 0.82 0.80

Figure 11. Data distribution of reference period and monitoring period 1.

and y2 during the reference period and the two monitoring periods are shown in Figures 13 and 14, respectively.

1, the trace-based index indicates that the performance of the monitoring period is slightly better than the reference period. However, since the trace-based index is not bounded and its upper limit is infinity, the value of 1.29 seems to indicate a very small change and is not sufficiently informative about the actual change. On the other hand, the determinant-based index fails to detect any change in the system’s performance. Figure 11 shows that the distribution of the process output data has slightly changed and the dissimilarity-based index, which is 0.84, indicates such changes in the data distribution. In monitoring period 2, the two traditional indices indicate different changes in direction of the performance: the trace-based index value is larger than 1 and the determinant-based value is smaller than 1. Therefore, because of the complexity of the covariance matrix, the traditional indices are not sufficiently informative, as indicated in this example. The dissimilarity-based index indicates that the performance of the monitoring period has shifted ∼20% in comparison with the reference period. Such changes can also be reflected by the data distribution as shown in Figure 12. The time series plots of the process output variables y1

Figure 12. Data distribution of reference period and monitoring period 2. 18232

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each tank to have water recycle to the reservoir. All the valves are either fully open or fully closed and two identical pumps deliver water to the left and the right tank, respectively. The level in each tank is measured by a DP cell. In this experiment, the measured process outputs are the water levels of the left and the right tank. The water levels in the left tank and the right tank are maintained at 50% and 20%, respectively. To evaluate the proposed method, two scenarios are considered in this experiment to simulate actuator malfunction and process dynamic change, respectively. The results are presented in Table 4. Period I represents the benchmark period and period II Table 4. Comparison of Three Indices for Both Scenarios Scenario 1 Period I valves open trace-based index determinant-based index dissimilarity-based index

Figure 13. Time series of y1 and y2 in the reference period and monitoring period 1.

Period II

5, 6, 7, 8, 9 6, 7, 8, 9 0.94 10.7 0.35

Scenario 2 Period I

Period II

3, 5, 7, 8 5, 9 0.93 0.72 0.38

represents the monitoring period. In this experiment, 300 data are collected during the benchmark and the monitoring period, respectively. Figure 16 shows the time series of the left and the right tank level in Scenario 1. In Figure 16, the fluctuation of the left tank

Figure 14. Time series of y1 and y2 in the reference period and monitoring period 2.

5.3. Pilot-Scale Experiment Using a Hybrid Tank System. The proposed approach is tested on a hybrid tank system to further demonstrate its effectiveness. The hybrid tank system consists of three coupled tanks, as shown in Figure 15. The three tanks are interconnected with each other through top, middle, and bottom pipes. There are three valves at the bottom of

Figure 16. Time series of the left tank level and the right tank level in Scenario 1.

Figure 15. Schematic of a three-tank system.

level decreases because valve 5 is closed during the monitoring period, which subsequently affects the level in the middle tank and causes more fluctuation in the right tank’s level. The change of the data distribution is very obvious, as shown in Figure 17. In this case, the trace-based index is 0.94, which means the performance of the monitoring period is slightly worse than the benchmark period. However, the determinant-based index is 10.7, which indicates that the monitoring period performance is better than that of the benchmark period. Since the upper limit of the determinant-based index is infinity, the value of 10.7 appears to indicate a small change and is not very informative about the actual change. It shows that, because of the complexity of the covariance matrices, the two conventional indices could not be sufficiently informative, as indicated in this experimental study. From Table 4, the dissimilarity index is 0.35, which indicates that 18233

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Figure 17. Data distribution of the reference period and the monitoring period in Scenario 1.

Figure 19. Data distribution of the reference period and the monitoring period in Scenario 2.

the performance of the monitoring period has shifted significantly, compared with the benchmark period as this index is bounded between 0 and 1. In such case, obviously the dissimilarity-based index much better reflects changes in performance than the two conventional indices. In Scenario 2, valves 3, 5, 7, and 8 are open during the benchmark period so that the left tank and the right tank interact with each other. Figure 18 shows the time series plot of the left

6. CONCLUSION In this paper, a new multivariate performance monitoring method is proposed based on analyzing the dissimilarity among different distributions of the process output data. Compared with the minimum variance-based index and the determinant-based performance index, the proposed method considers not only the volume of the hyper-ellipsoid defined by the output data covariance matrix, but also the direction of the hyper-ellipsoid. A new performance index is defined to quantify the performance change. The simulation and experiment results demonstrate the effectiveness of the proposed method and its ability to give a more-informative index for performance monitoring. In addition, the proposed historical data based index can reflect the change in system’s performance but it cannot tell the direction of the change (i.e., better or worse). Future work should be done in this aspect to provide a more-complete performance change profile. Furthermore, methods should be developed when the noise affecting the system does not follow Gaussian distribution or when the process has nonlinearity, such as valve stiction in the control loop.

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AUTHOR INFORMATION

Corresponding Authors

*Tel.: +1 780 492 9016. Fax: +1 780 492 2881. E-mail addresses: [email protected] (B. Huang). *Tel.: +86 21 6425 2060. Fax: +1 780 492 2881. E-mail: fqian@ ecust.edu.cn (F. Qian).

Figure 18. Time series of the left tank level and the right tank level in Scenario 2.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author would like to acknowledge NSERC (Canada), University of Alberta and the financial support of China Scholarship Council (CSC). This work is supported by Major State Basic Research Development Program of China (2012CB720500), National Natural Science Foundation of China (61333010), and the 111 Project (B08021).

tank and right tank level in Scenario 2. During the monitoring period, all the valves are closed except valves 5 and 9. Hence, the left level and the right level control loops become independent. Clearly, the process dynamics changes significantly during the monitoring period. In such case, the trace-based index is 0.93, hardly indicating any change in system performance. The determinant-based index is 0.72, which is also relatively close to 1 and indicates a small change. The dissimilarity index is 0.38, indicating that the performance of the monitoring period has shifted significantly, in comparison with the benchmark period. Such changes can also be reflected by the data distribution as illustrated in Figure 19.

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