Performance Evaluation of two Industrial MPC ...

Paper to appear in the Special Issue of “Prize Winning Application Papers”, Control Engineering Practice, IFAC,2003

Performance Evaluation of two Industrial MPC Controllers

1

Jianping Gao1 , Rohit-Patwardhan1 , K. Akamatsu2 , Y. Hashimoto2 , G.Emoto2 , Sirish. L. Shah3∗, Biao Huang3 Matrikon Inc. # 1800, 10405 Jasper Avenue, Edmonton, AB, Canada T5J 3N4 2 Mitsuibishi Chemical Corp., Mizushima, Japan 3 Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada T6G 2G6

Abstract This paper presents industrial case studies of the performance evaluation of two industrial multivariate MPC-based controllers at the Mitsubishi chemical complex in Mizushima, Japan: 1) a 6-output, 6-input Para-Xylene(PX) production process with 6 measured disturbance variables that are used for feedforward control; and 2) a multivariate MPC controller for a 6-output, 5 input poly-propylene splitter column with 2 measured disturbances. A generalized predictive controllerbased MPC algorithm has been implemented on the PX process. Data from the PX unit before and after the MPC implementation are analyzed to obtain and compare several different measures of multivariate controller performance. The second case study is concerned with performance assessment of a commercial MPC controller on a propylene splitter. A discussion on the diagnosis of poor performance for the second MPC application suggests significant model-plant-mismatch under varying load conditions and highlights the role of constraints.

1

Introduction

The last decade has witnessed a growing interest by practitioners and academics alike in the field of controller performance monitoring (Harris, Harris and co-workers 1989-1999, Huang and Shah 19961999, Kozub 1996). The basic idea in performance monitoring is to obtain a measure of ‘performance’ of a closed loop system from routine closed-loop output and input data. In short, the role of performance evaluation is to see if the controller is doing its job satisfactorily and if not, further analyze closed loop data with process information to diagnose the causes of poor performance. Routine monitoring of controller performance ensures optimal operation of the regulatory control layers and the higher level advanced process control (APC) applications. Model predictive control (MPC) is currently the main vehicle for implementing the higher layer APC. The APC algorithms include a class of model based controllers which compute future control actions by minimizing a performance objective function over a finite prediction horizon. This family of controllers is truly ∗

The author to whom all correspondence should be addressed. Email:[email protected]

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multivariate in nature and has the ability to run the process close to its limits. It is for these reasons that MPC has been widely accepted by the process industry. Various commercial versions of MPC have become the norm in industry for processes where interactions are of foremost importance and constraints have to be taken into account. Most commercial MPC controllers also include a linear programming stage that deals with steady-state optimization and constraint management. Several authors have proposed approaches for evaluation of the performance of multivariate controllers (see Harris et al 1996., Huang and Shah 1996-1999, Shah et al 2001., Ko and Edgar 2000). This paper is concerned with the application of some of these methods towards performance evaluation of two industrial MPC controllers. In this paper a graphical measure of multivariate controller performance is adopted. This is a generalization of the univariate impulse response plot (between the process output and the whitened disturbance variable) to the multivariate case, and defined as the ‘Normalized Multivariate Impulse Response’ plot. A particular form of this plot, that does not require knowledge of the process time-delay matrix, is used here. Such a plot provides a graphical measure of the multivariate controller performance in terms of settling time, decay rates etc. This graphical measure is compared with the multivariate minimum variance benchmark in which the interactor or the time delay matrix is first computed from the step-response model required for the design of the MPC controller. Other measures of multivariate performance are also explored in detail in this study: the use of the design or historical performance as a benchmark, and the use of the settling time as benchmark. The design objective function based approach can be applied to constrained MPC type controllers and is therefore a practical measure. However, it does not tell you how close the performance is relative to the lowest achievable limits. In this respect the minimum variance-benchmarking index complements the objective function measure very well. In this paper, several MPC specific monitoring methods are investigated for performance monitoring. Since MPC is a multivariate, constrained, optimizing, model-based controller, methods that consider all or some of these features of MPC are applied. For example, constraint analysis is performed to evaluate constraints violation, manipulated variables (MV) saturation properties and capability indices of controlled variables (CV) and MVs; optimization analysis is performed in terms of objective function to be minimized by MPC; interaction analysis is performed in the frequency domain to evaluate the interaction reduction properties of MPC; a new performance assessment method based on NMIR (normalized multivariate impulse response) is proposed to look at performance of MPC in a graphical form; evaluation of MPC performance in terms of user-specified benchmarks is also discussed.

2

Introduction to MPC performance monitoring

Performance assessment of univariate control loops is carried out by comparing the actual output variance with the minimum output variance. The latter term is estimated by simple time series analysis of routine closed-loop operating data. This idea has been extended to multivariate control loop performance assessment resulting in the Multivariate Filtering and Correlation (MFCOR) Analysis algorithm [11]. The concept of a delay term is important in minimum variance control. This idea obviously carries over to the multivariate minimum variance control case as well. The generalization of the univariate delay term to the multivariate case is known as the interactor matrix or delay matrix. The difficulty in the multivariate case is the factorization of the delay matrix to design a multivariate minimum variance controller. The interactor matrix as proposed by (Wolovich and Falb, 1976), had 2

a lower triangular form. However, the form of an interactor matrix is application-dependent, i.e. it may take an upper triangular form or a full matrix form. Huang and Shah (1999) has found that a unitary interactor matrix is an optimal factorization of time-delays for multivariate systems in terms of minimum variance control and control loop performance assessment. This unitary interactor is an all-pass term, and factorization of such a unitary interactor matrix does not change the spectral property of the underlying system. If a multivariate process can be represented as Yt = T Ut + Vt

(1)

where T (n × m) is a proper, rational transfer functions matrix in the backshift operator q −1 ; Yt , Ut and Vt are output, input and disturbance vectors of appropriate dimensions, there exists a unique, n × n lower left triangular polynomial matrix D, such that |D| = qr and limq−1 →0 DT = limq−1 →0 T˜ = K

(2)

where K is a full rank matrix and r is defined as the number of infinite zeros of T , and T˜ is the delay free (or Hermite canonical form) transfer function matrix of T (Wolovich and Falb, 1976, Goodwin and Sin, 1984). The matrix D is defined as the interactor matrix and can be written as D = D0 qd + D1 qd−1 + D2 q d−2 + . . . + Dd−1 q

(3)

where d is denoted as the delay order and is unique for a given transfer function matrix, and Di s are constant coefficient matrices(Huang et al., 1997). The interactor matrix D can be of any form: If D = q d I, then the transfer function matrix T is regarded as having a simple interactor matrix. If D is a diagonal matrix, i.e., D = diag(q d1 , q d2 , . . . , qdn ), the T is regarded as having a diagonal interactor matrix. Otherwise T is considered to have a general interactor matrix, which may be a full, lower triangular or upper triangular matrix. If the interactor matrix satisfies D T (q −1 )D(q) = I

(4)

then this interactor matrix is denoted as the unitary interactor matrix (Huang et al., 1997). It have been shown in Huang et al.(1997) that multiplying a unitary interactor matrix to the output variables does not change the quadratic measure of the variance of the corresponding output variables, i.e. T E[Y˜t Y˜t ] = E[Yt T Yt ]

(5)

Y˜t = q −d DYt

(6)

where Therefore, in terms of the quadratic measure of the variance, performance assessment of Yt is the same as performance assessment of Y˜t . In short, the interactor matrix is an equivalent form of the time delay in multivariate system. It needs the open loop model (at least the first few Markov Matrices) and an algorithm (Shah et al. 1987, Peng and Kinnaert 1992) to factor out the fewest number of Markov matrices to capture the delay terms. A key to performance assessment of multivariate processes using minimum variance control as a benchmark, is to estimate the benchmark performance from routine operating data with a priori knowledge of time-delays or interactor-matrices. The expression for the feedback controller-invariant (minimum variance) term is then derived by using the interactor matrices. It is shown that this term 3

can be estimated from routine operating data using multivariate time series analysis [10]. Minimum variance characterizes the most fundamental performance limitation of a system due to existence of time-delays. Practically there are many limitations on the achievable control loop performance. For example, a feedback controller that indicates poor performance relative to minimum variance control is not necessarily a poor controller. Further analysis of other performance limitations with more realistic benchmarks is usually required. For example, performance assessment in a more practical context such as a user-defined benchmark may be desirable. We will show the application of two practical user-defined benchmarks that do not require knowledge of the process on two industrial applications. Nevertheless, performance assessment using minimum variance control as benchmark does provide useful information such as what is the absolute lower bound of process variance, and whether the controller needs re-tuning or the process needs re-engineering.

3 3.1

Case Study 1: MPC control of the Para-Xylene Unit Process Description

In order to achieve operational efficiency in terms of production costs, stable and reduced variance product composition and automated versus manual process operation, a multivariable model predictive controller was designed and implemented on the PX distillation unit at Mitsubishi Chemical Corporation’s, Mizushima plant in Japan. The distillation unit of the PX plant consists of three columns where raw Xylene is separated into the main product, Ortho-Xylene, OX, and other byproducts. A schematic of the process flow sheet is shown in Figure 1. Xylene feed and recycled Xylene from the isomerization section are mixed and fed to the light end column. In the light end column, the light components (C1-C5, toluene and benzene) are separated and the bottom products, composed of Xylene and ‘heavies’, (more than C9), are fed to the OX column. In the OX column, the OX and heavy components show up as the bottom products whereas mixed PX and Meta-Xylene are distillated to the overhead. The mixed Xylenes in the overhead are fed to the crystallization section. The OX and heavy components are fed to the OX purification column where the heavy components end up at the bottom and the pure OX, as a product, is distillated to the overhead. The heat furnace is used as a reboiler for the OX column, because high temperatures and significantly large heat duties are necessary for OX distillation. The separated light and heavy components are used as fuel to the heat furnace. In the conventional operation of this unit, reboiler steam at the light end column, heat furnace fuel and the OX column were all operated manually to keep the reflux ratio and column operational. Ambient temperature and fuel composition changes are the main disturbances to this unit. These disturbances forced manual operation of the column, one of the consequences of which resulted in high reflux ratio to keep product specification.

3.2

Multivariable model predictive controller strategy

A multivariable model predictive control software package based on the ‘Hitachi PS21 IMPACT’ controller was implemented on the PX unit. ARX models were identified from the plant input/output data and control adjustments are calculated based on the generalized predictive control algorithm to minimize the following cost function: J =κ

N2 X

j=N1

(ˆ y(t + j) − w(t + j))2 + λ

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∆u(t + k)2

(7)

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Figure 1: Process flow sheet of the Para-Xylene unit

where yˆ(t + j) is j-step ahead prediction of the system output on data up to time t, N1 and N2 are the minimum and maximum costing horizons, Nu is the control horizon, κ and λ are the weights and w(t) is the future reference. A list of controlled, manipulated and disturbance variables and the corresponding weights(κ, λ) with the prediction horizons are given in Table 1. Note that in this table, CV1 and MV1 are identical. This is frequently done under MPC either to square the system or alternately to maximize the feed as is the case here, and yet also be able to manipulate it, should circumstances dictate so, e.g., to prevent flooding. The implementation results of this GPC-based MPC control algorithm are shown in the left hand column in Figure 2. The right column shows the control results before the MPC was installed.

3.3 3.3.1

MPC performance monitoring Data Preprocessing

The first and foremost step in data driven analysis is to prepare the working data set for the analysis. It is always important to spend time previewing and processing the data set before proceeding to the analysis. The quality of the results obtained during the analysis is directly related to the quality of the data used in the analysis. In general, certain procedures are required when performing data preprocessing relevant to MPC performance monitoring. Bad, null and non-numerical data in the dataset need to be replaced; if the data is not evenly or regularly sampled, it needs interpolation to make it evenly sampled before any numerical analysis can proceed; before proceeding with time-series analysis to calculate performance metrics, the data needs to be mean-centered and all outliers should be removed. 5

Table 1: Controller design CV No. CV1 CV2 CV3 CV4 CV5 CV6 MV No. MV1 MV2 MV3 MV4 MV5 MV6

parameters for the MPC-based multivariate objective function Tag Weight Horizon Range Xylene feed 5.0 20 0-25 Tray #5 temperature 2.5 100 0-200 Internal reflux ratio 0.05 100 0-45 OX Hold up 6.0 100 0-100 OX reflux drum level 0.3 100 0-100 OX reflux ratio 1.4 50 0-10 Tag Weight Horizon Range Xylene feed 35 3 0-25 Reboiler steam 200 3 0-15000 Internal reflux 15 3 0-45 OX reflux 20 3 0-180 OX distillate 100 3 0-125 Fuel heat calorie 25 3 0-200

FF No. FF1 FF2 FF3 FF4 FF5 FF6

3.3.2

Table 2: Disturbance Variables Tag Isomerization feed OX column feed Heat furnace fuel heat calorie residual OX purification column feed Cooling water temperature Ambient temperature

Performance evaluation in terms of minimum variance benchmark

Calculation of the unitary interactor matrix based on open loop model

The current open loop process model for the PX process as used in the Hitachi IMPACT MPC(with two MVs and one CV out of service this time) is listed below. The model is used to factor out the open loop interactor matrix:         

z −1 s21 s31 0 s51 0

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Figure 2: The result of MPC application to the distillation units of the para-xylene plant where s21 s22 s23 s25 s31 s32 s33 s35 s45 s52 s55

= (−0.06809z −1 − 0.1408z −2 )/(1 − 0.2845z −1 − 0.6746z −2 ) = (−0.06038z −1 + 0.2498z −2 )/(1 − 1.69z −1 + 0.6978z −2 ) = (−0.02414z −1 − 0.266z −2 )/(1 − 0.8188z −1 − 0.153z −2 ) = (−0.1297z −1 + 0.4256z −2 )/(1 − 1.301z −1 + 0.3159z −2 ) = (−0.02498z −1 + 0.03939z −2 )/(1 − 1.868z −1 + 0.8764z −2 ) = (0.1092z −1 − 0.1551z −2 )/(1 − 1.873z −1 + 0.8796z −2 ) = (0.01148z −1 + 0.01025z −2 )/(1 − 1.876z −1 + 0.8823z −2 ) = (−0.0408z −1 − 0.008103z −2 )/(1 − 1.814z −1 + 0.8235z −2 ) = (−0.1968z −1 )/(1 − 0.99z −1 ) s51 = (−0.1464z −2 )/(1 − 0.99z −1 ) −2 −1 = (0.34z )/(1 − 0.99z ) s54 = (−1.399z −2 )/(1 − 0.99z −1 ) −2 −1 = (−0.4911z )/(1 − 0.99z ) s56 = (3.766z −2 )/(1 − 0.99z −1 )

An unitary interactor matrix is obtained (For the algorithm, see Huang and Shah, 1999):         

0.05925z 0.006454z −0.02356z −0.03354z −0.9974z 0

0.7234z 0.4251z −0.3103z −0.4417z 0.06791z 0

0.4z −0.9003z −0.09755z −0.1389z 0.02491z 0

−0.5596z −0.0934z −0.4734z −0.6739z 0 0

0 0 0 0 0 z2

0 0 0.8183z −0.5748z 0 0

        

Time series analysis of closed loop data via the MFCOR algorithm The closed loop data used for performance monitoring corresponding to PX MPC(IMPACT) ON and OFF was available at 1 min sampling rate with a length of 43200 samples(over 30 days). It is assumed that the two data sets with and without IMPACT correspond to similar periods of operating conditions. To evaluate the daily performance, a moving window of 1440 data points without overlapping corresponding to each day was used to calculate the daily multivariate performance index (see Figure 3) as well as individual performance indices at different channels or for different outputs. The definition of multi-variate performance index and detailed MFCOR algorithm with interactor matrix 7

filtering is provided in the book by Huang and Shah 1999. Bar charts corresponding to the ‘ON’ and ‘OFF’ status of the IMPACT controller were generated. It is clearly seen that a significant improvement in performance resulted after the IMPACT controller was implemented. The multivariate performance index, for the same data, with MPC ‘ON’ is 0.3549 versus 0.1398 when MPC was ‘OFF’. This is clearly a significant improvement(over 200%). In Figure 4, individual performance indices over a 30-day period are displayed. It is noticed as well that the performance of a few controlled variables improved significantly while the performance of others degraded as a result of different controller weightings. It is to be expected that performance in some loops would improve at the expense of reduced performance in other loops. (Note the significant weightings in Table 1 for CVs:1,4 and 6 and the corresponding improvement in performance for these 3 important CVs). Performance when MPC was OFF P erform anceindex

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Figure 3: Multivariate performance assessment, with minimum variance as a benchmark, of the ParaXylene unit over 30 days(each bar indicates performance measure for 1 day)

1 Performance of each output, MPC is ON Performance of each output, MPC is OFF 0.9

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3.3.3

Performance evaluation via alternative methods

Constraints analysis An important feature of the MPC controller is that it can handle CV and MV constraints explictly. By examining the CVs and MVs against their upper and lower constraints listed in Table 1, it is found that there is no MV saturation or CV violation of their constraints limits. This indicates that both MPC-ON and MPC-OFF are able to handle constraints effectively. Another method to analyze the variability of a variable against its upper and lower specification or constraints is the capability analysis in terms of Cp/Cpk and Pp/Ppk as articulated by Shunta [4]. If the values for Cp and Cpk are the same, this indicates a mean-centered process. If the Cp and Cpk values are not equal, this suggests that the process is not centered, and that a mean shift may have occurred. In this case, the Cpk value, which is adjusted to reflect an off-center distribution, should be used for analyzing the Cp/Cpk results. Pp and Ppk are process performance indices that represent the current process variability based on user-defined specifications. The Pp and Ppk values indicate the current level of performance for a specific variable. If the values for Pp and Ppk are the same, this indicates a mean-centered process. If the Pp and Ppk values are not equal, this suggests that the process is not centered. Therefore, the Ppk value should be used for analyzing the Pp/Ppk results. If Cp/Cpk of a variable is higher, it indicates that this variable has more room or capacity to perform better, while if its Pp/Ppk is higher, it indicates that such variable has lower variability relative to the specified range. For example, the Cp/Cpk element can be used to determine which variables have the most freedom or room to move to carry out the regulatory task. Given the engineering range of CVs and MVs listed in Table 1, Capability analysis in terms of Cp/Cpk and Pp/Ppk was performed and the results are summarized in Table 3.: Table 3: Capability analysis of CVs/MVs when MPC is ON and OFF No.CV Cpk (MPC is ON) Cpk (MPC is OFF) Ppk(MPC is ON) Ppk(MPC is OFF) CV1 807.17 473.45 4.48 3.89 CV2 117.48 79.53 19.16 18.89 CV3 351.21 71.03 5.5 7.60 CV4 565.86 86.60 81.23 13.91 CV5 362.69 48.61 11.25 9.42 CV6 259.22 60.10 57.03 2.86 No.MV Cpk (MPC is ON) Cpk (MPC is OFF) Ppk(MPC is ON) Ppk(MPC is OFF) MV1 807.2 251.30 4.48 3.88 MV2 2123.2 677.31 4.37 4.07 MV3 321.4 166.41 3.56 6.69 MV4 262.7 60.24 33.85 2.84 MV5 328.9 404.65 12.65 12.07 MV6 812.0 420.16 21.45 5.95

It is reasonable to have all Cpk and Ppk of CVs and MVs greater than 1 because engineering ranges of CVs/MVs were applied to perform the analysis and the engineering range is much wider than the practical range. The results in Table 3 indicate that after MPC was ON, the Cpk and Ppk of all CVs and MVs increased significantly compared with those when MPC was OFF except CV3 9

or MV3 as a result of lower weightings. This result confirms that with MPC in “ON” mode, the capability or room for control improvement increased significantly while actual variability decreased within the engineering range. Optimization analysis The objective of this MPC controller is to minimize the quadratic objective function, as specified in eqn 7 and Table.1. In order to demonstrate that the current MPC controller is working effectively towards this objective, the values of this objective function corresponding to the state when MPC was ‘ON’ and ‘OFF’ were calculated. As shown in Figure 5, the objective function corresponding to MPC-ON shows a significant decrease (ie. a lower cost) in the quadratic cost function compared with that corresponding to MPC-OFF. If contributions by different controlled variables were computed and compared as shown in Figure 6, decreases in channels corresponding to CV1, CV4 and CV6 are obvious as a result of higher weighting for these variables as indicated in Table 1.

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Interaction analysis Another objective of MPC is to reduce the interaction between different loops. Coherency analysis or the normalized cross spectrum between different controlled variables is a useful tool to assess the interaction between different control variables. In this industrial application, the coherency of six control variables during the MPC-ON versus MPC-OFF periods were computed and compared with each other. Figure 7 shows this plot where off-diagonal terms represent the coherency between the corresponding control variables. As shown in Figure 7, all off-diagonal terms1 in the frequency domain are far below 1 which indicates that interactions between different loops are small. In particular, the magnitudes of some off-diagonal terms (see coherency plot of “CV2 versus CV3” and “CV4 versus CV5” ) decrease significantly as a result of MPC application, which indicates that the MPC controller is effective in interaction reduction.

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Figure 6: Contributions towards the objective function(individual quadratic cost are plotted versus sample period) 3.3.4

Normalized multivariate impulse response as a performance measure

An impulse response curve represents the dynamic relationship between the whitened disturbance and the process output. In the univariate case, the first “d” impulse response coefficients are feedback control invariant, where “d” is the process time delay. Therefore, if the loop is under minimum variance control, the impulse response coefficients should be zero after “d” lags. The Normalized Multivariate Impulse Response (NMIR) curve reflects this idea for a multivariate controlled system. The first “d” NMIR coefficients are feedback controller invariant, where “d” is the maximum order of the interactor matrix. If the loop is under multivariate minimum variance control, then the NMIR coefficients should decay to zero after “d” lags. The sum of squares under the NMIR curve is equivalent to the sum of the trace of the covariance matrix of the data. If the output variance is Yt = E0 at + E1 at−1 + . . . + Ed−1 at−d+1 + Ed at−d + . . .

(8)

and the filtered output variance is: Y˜t = q −d D(F0 at + F1 at−1 + . . . + Fd−1 at−d+1 + Fd at−d + . . .) we have:

P P E(YtT Yt ) = E(Y˜tT Y˜t ) = trace(F0 a F0T ) + trace(F1 a F1T ) + . . . ,

where the first NMIR coefficient is given by q

P

q

trace(F0

P

T 0 F0 )

(9)

(10)

and the second NMIR coefficient is

given by trace(F1 a F1T ), and so on. The multivariate performance index is then equal to the ratio of the sum of the squares of the first “d” NMIR coefficients to the sum of all NMIR coefficients. The NMIR outlined above requires apriori knowledge of the interactor matrix. In this specific application, the NMIR of the interactor filtered output from knowledge of the computed interactor matrix can be computed. As a complement, a similar normalized multivariate impulse curve without interactor filtering is also computed to serve a similar purpose. In this calculation, the NMIRwof coefficients are given by the E-matrices(E0 , E1 , . . .) instead of the F-matrices(F0 , F1 , . . .). The rationale for using the NMIRwof is that the two calculations are asymptotically equal (see Eqn.10 and Shah et 11

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Figure 7: Coherency analysis of control variables. The x-axis is the normalized frequency whereas the y-axis is the cohereny value. al. 2001 for further details): P

P

limt→∞ (trace(E0 a E0T ) + trace(E1 a E1T ) + . . .) = P P (trace(F0 a F0T ) + trace(F1 a F1T ) + . . .)

The result is clearly shown in Figure 8, which complies with the theoretical derivation for this specific industrial application. The NMIRwof curve corresponding to the MPC “ON” case decays quickly, which again leads to the conclusion that the MPC controller improves the performance of the multivariate controlled system significantly. Normalized Multivariate Impulse Response when MPC was off 0.8 NMIR NMIR-wof 0.6

0.4

0.2

0

0

5

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30

Normalized Multivariate Impulse Response when MPC was on 2.5 NMIR NMIR-wof

2 1.5 1 0.5 0

0

5

10

15

20

25

30

Figure 8: Plots of the Normalized Multivariate Impulse Response versus time-lags

12

3.3.5

Performance evaluation in terms of user-specified benchmark

Historical benchmark Historical performance benchmark is the actual cost function for the closed loop system over a specific period when it is considered to have a good response. A relative performance index in terms of historical benchmark can be calculated to measure the current actual response speed in comparison with a historical benchmark. A relative performance index(RPI) relative to the historical benchmark can be defined as follows: Pn Pm i=1 κi var(CVi ) + i=1 λi var(∆M Vi ))benchmark Pn Pm RP Ihis =

(

(

i=1 κi var(CVi ) +

i=1 λi var(∆M Vi ))act

where n and m is the number of CV and MV respectively, and κi , λi are the weights of CVs and MVs. Similarly, for each CV, RP Ihis for different j, j = 1 : n can be calculated as: RP Ij =

P

(κj var(CVj ) + m i=1 λi var(∆M Vi ))benchmark P (κj var(CVj ) + m i=1 λi var(∆M Vi ))act

Such a relative measure of performance varies between 0 and infinity.

• RP I = 1 implies that the control system performance is meeting benchmark specifications. • RP I > 1 implies that the control system is performing better than the benchmark. • RP I < 1 implies that the control system is performing poorly relative to the benchmark. The ratio between the benchmark controller energy consumption (CER) and energy consumed by the actual system is also a good metric to evaluate the performance of MPC since it gives a measure of the energy saving after MPC was ON. It is defined as: P

( m λi mean(M Vi ))ben CER = Pi=1 ( m i=1 λi mean(M Vi ))act

If controller performance with MPC ‘OFF’ is chosen as a historical benchmark and it is assumed that the two data sets with and without IMPACT corresponded to similar periods of operation, then the relative improvements in performance are indicated as RPI to be 4.5 and CER to be 1.18 respectively. It is noticed that both RPI and CER are greater than 1, which indicates that the relative performance of MPC is much better than the benchmark and MPC is very effective in reducing controller energy and output variability. Settling Time Benchmark A settling time benchmark is a user-defined time period over which the controller should return to its set point after a disturbance. This idea has been widely applied in single loop performance assessment as a compliment to the minimum variance benchmark. In this paper, a relative performance index in terms of settling time benchmark is proposed for multivariable case and can be calculated to measure the actual response speed in comparison with the target benchmark in multivariate controlled system(Patwardhan, 1999). In single loop performance assessment, an ideal first-order system with expected settling time is chosen to be the target benchmark. Its impulse response is compared with the closed loop impulse response to calculate the relative performance index, for example, if the benchmark impulse 13

response coefficients are {1, g1 , g2 , . . . , } and the actual closed loop impulse response coefficients are {1, f1 , f2 , . . . , } respectively, then: RP Ist =

(1 + g12 + g22 + . . .) (1 + f12 + f22 + . . .)

This idea can be extended to multivariable controller performance evaluation if NMIR without filtering is calculated. The squared sum of NMIRwof can be compared with the squared sum of settling time benchmark to calculate the RPI. Therefore, the modified formula is: P

trace(E0 a E0T ) ∗ (1 + g12 + g22 + . . .) RP Ist = P P ((trace(E0 a E0T ) + trace(E1 a E1T ) + . . .)) P

Note that term trace(E0 a E0T ) is a normalization factor to make these two curves start at the same point. If 10 samples are chosen to be the expected settling time as a benchmark, the RPI in terms settling time benchmark can be calculated. The RPI is 1.6 when MPC was ‘ON’ versus 1.2 when MPC was ‘OFF’. The improvement when MPC was ON also leads to the conclusion that the performance has improved in comparison with the operation when MPC was OFF. The advantage of these two proposed benchmarks is that the process model is no longer required. Without the process model, the NMIR without filter is able to provide a graphical measure of performance and the relative performance assessments vs settling time and historical benchmark metrics can also be calculated. This idea is promising for online application of performance assessment technology to be feasible without knowing the process model. Such tools should therefore promote more extensive industrial applications of performance assessment technology.

4

Case Study 2: MPC Control of a Propylene Splitter Column

The process under consideration is a propylene splitter (C3 F) column. A schematic of the process is shown in Figure 9. A commercial MPC controller is currently used to control the key variables of interest which, in this case, are the top and the bottoms impurities. The natural logarithm of the top impurity is controlled as it shows a more linear response. There are a total of 6 controlled variables (CVs), 5 manipulated variables and 2 measured disturbances. The model was available in a step response form obtained from the identification algorithm. The measurements of the main CVs were available at the rate of 20 minutes each. The main CVs and MV considered in this analysis are highlighted in Figure 9, i.e. the performance assessment is considered only for a subset of inputs and outputs, i.e. 2-inputs and 2-outputs problem with constraints. The controller in place is a multivariable MPC controller. The column is operated under normal, low and high load (feed flow rate) conditions depending upon the production requirements (see Figure 10). The step response models were originally identified under normal load conditions.

4.1

Assessment Results

Performance assessment was carried out for different load conditions. Two methods were used for performance assessment:(1) multivariable settling time benchmarking, and (2) historical benchmarking. The results of the performance assessment step are presented below. 14

GC

Conde nse r

Y1

once / 20 min Re flux

Distillate (D)

FC

U1

FC

LC

D2

FC

Stre ams

Ste am

D1

Re boile r

U2

FC

Bottoms once / 20 min

GC

Y2

Figure 9: A schematic diagram of the distillation column showing the relevant controlled and manipulated variables Propylene production rate 8

High Load

7

Normal Load 6 5 4 3 2

Low Load

1 0

0

20

40 60 80 Days : July 1 to Oct. 28

100

120

Figure 10: The Production requirements in normalized units from July 1 to Oct. 28. Normal Load Data The operating data for the two main input and output variables under normal load conditions is shown in Figure 11. The top part of the graph shows the outputs along with the setpoints and the bottom part shows the manipulated variables. The sampling time for control is 6 minutes and the data shown here correspond to each control sampling interval. A desired settling time of 5 samples was chosen for each of the outputs based on the open loop settling time and the recommendation of the plant personnel. Since the settling time index was 0.86 for this data set, the normal load data as our benchmark data for historical benchmarking was chosen. For this MPC application the output weighting matrix is given by: Γ = diag(200, 2). The performance indices for the cases where: (1) the weightings were taken into account and (2) the raw data was used are 0.8 and remain the same irrespective of whether the data is weighted or not. The top impurity is the economically important variable and is weighted by a factor of 100 times 15

2.66

7.15 7.1 y2

y1

2.64 2.62

7.05 7

2.6

6.95

2.58

6.9 0

50

100

150

142

0

50

100

150

0

50

100

150

0.76

140

u 2

u 1

141 0.75

139 138

0.74

137 0

50

100

150

Figure 11: Input-output data for the normal load conditions. The solid line in the top two figures represents the setpoint. more than the bottom. It is important to consider these weightings in any benchmarking scheme. For the normal data set the settling time measures show good performance irrespective of the weightings. The historical benchmark measure equals 1 since this data set itself is the benchmark data set. The achieved objective function could not be calculated due to insufficient data. For the remaining data sets we do not show the data trends but the assessment results are summarized below. The results for the normal high and low load data are reported below. The settling time and the historical measures are shown in Figure 12. Both the measures indicate a significant drop in performance compared to the normal load conditions. In the later sections we attempt to isolate the cause for this drop in performance. The achieved objective function for the last 60 samples is shown in Figure 13. The achieved objective function shows a decreasing trend, which implies improving performance. The output variance term is dominant compared to the offset and the input variance terms. For the first low load data set, both the settling time and the historical measures indicate a drop in performance for the low load case. For this case the achieved objective function was dominated by the offset term. The contribution of the input variance term is negligible. The overall performance assessment results are shown in Figure 12. The settling time and historical benchmarks indicate good performance in the normal load region and poor performance in the high and low load regions. The first low load data set showed poor performance due to the fact that constraints were active throughout this run. The second low load data set showed satisfactory performance because constraints were active only for the initial part of this run (see Figures 14 and Figure 15). The historical benchmarking method confirmed the settling time approach based results except for the second low load data set. For this data, the historical benchmark indicated no significant improvement in performance. This can be attributed to an increase in input variance or output offset, since the historical benchmark takes them into account explicitly.

4.2 4.2.1

Diagnostics Analysis of constraints

During the low load conditions the lower constraint on reflux flow rate was activated. Material balance considerations can be used to explain this phenomenona. The lower limit on the input constraint 16

1

0.9

Good 0.9

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0.7

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0.6

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Poor 0.5 0.4

Poor

0.3 0.2 0.1

0 1

2

3

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1

2

(a)

3

4

(b)

Figure 12: (a) The settling time performance indices and (b) the historical performance measures for different data sets (1-Normal, 2-High, 3-Low A, 4-Low-B) Achieved objective function

Achieved output variance

44

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Figure 13: The achieved objective and the contributing factors for the high load data.

was an operator specified quantity. Since the reflux flow rate was saturated at the constraint, the control system was effectively reduced to a single-input-two-output system. Theoretically this is an uncontrollable system. Economically, the top impurity is more important than the bottom impurity, and due to the emphasis placed on this output, the bottom flow rate was devoted entirely to controlling output 1 during this period. An offset was observed in output 2 for a part of the low load region(See Figure 14).

4.2.2

Prediction Errors in the closed loop

In order to investigate the reasons for poor performance, the prediction errors for the MPC controller were investigated. MPC uses a step response model of the process combined with an estimated disturbance to predict the process outputs over the prediction horizon as shown in Figure 16. The 17

7.6

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114 u2

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2000

Figure 14: The input-output data for the low load data (A). The dotted line in the top two figures represents the setpoint change. disturbance is estimated through feedback: ˆ d(k/k − 1) = y(k − 1) − yˆ(k − 1) where y is the measured value and yˆ is the predicted output. The open loop predictions are updated with this disturbance estimate. This step in MPC allows us to rewrite the MPC controller in the IMC form. For the propylene splitter MPC, the step response model is also used to predict the compositions between two measurements. This is equivalent to an observer in control terminology. From Figure 16, prediction errors in the closed loop can be expressed as, d = ∆(I + Q∆)−1 w + N(I + Q∆)−1 a + Q∆(I + Q∆)−1 r

(11)

where Q is MPC controller, ∆ = P − Pˆ is the model plant mismatch(MPM), a is disturbance, r is the setpoint and w is the dither signal or external excitation. If w = 0, we have, d = N (I + Q∆)−1 a + Q∆(I + Q∆)−1 r = (I + ∆Q)−1 {∆Qr + N a}. Thus, the prediction errors, under closed loop, are influenced by the controller, Q and the extent of (MPM). However, notice that if there is no MPM and no external signals (r and a), d = 0 holds irrespective of the controller. For example, with r = 0, the prediction errors could be compared under several different circumstances to determine if the changes in N and ∆ are significant. This observation forms the basis for deciding if the model is satisfactory. If not, several factors, other than mismatch, could be held responsible - Q, N, r, a. In an effort to isolate the reasons of poor performance, the infinite horizon prediction errors of the step response model under closed loop conditions were examined. The following Figures 17 and 18 show the prediction errors for the normal and high load conditions respectively. The model seemed to worsen on a relative basis under high load condition. It should be noted here that we are dealing with closed loop data. Hence, the controller and/or unmeasured disturbances can affect the predictive capabilities significantly, under closed loop. 18

2. 65 2. 6

y1

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Figure 15: The input-output data for the low load (B) data set. The reflux flow rate is at the lower operating limit as specified by the solid horizontal line.

a

N

w +

r Q

+

y

+

u + P

-

+

+ P d

Figure 16: The internal model control structure for MPC

Low Load Data Set 1 The prediction errors for the low load data are shown in Figures 19 and Figure 20. For the second low load data set the predictions for output 2 appear to have worsened considerably. Quantification of prediction errors is important in establishing model quality. Kozub (1997) proposed the use of minimum variance benchmarking on the prediction errors to obtain a prediction index. We applied this idea for the C3 F splitter MPC. This measure can be treated as a necessary test for goodness of model quality. If it equals unity, then the model quality is perfect; if it is less than unity, several reasons can be listed for its deviation from unity and no statement can be made about the model quality. For our data sets, the MVC measure indicated satisfactory model quality for the normal load case relative to the other data sets. However, MPC performance deteriorates under significant low-load conditions. 19

predicted vs. actual output 0.04

y1

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Figure 17: Prediction errors for the MPC model: normal load. The dashed line and solid line represent predicted and actual values respectively.

Discussion The statistics for assessment of control errors and the prediction errors are reported in Table 4. It should be noted that the different factors are often correlated. For example, during the high load conditions there was a disturbance in the feed flow rate which caused a change in plant dynamics and the controller tunings could be inappropriate due to these changes. Constraints were shown to play an important role in the analysis of performance. Some evidence of MPM was found through prediction analysis at different loading conditions.

5

Concluding Remarks

Several measures of multivariate controller performance monitoring have been introduced and applied to performance evaluation of two MPC controllers. It is shown that routine monitoring of MPC application can ensure that corrective measures will be taken when control degrades to finally ensure good and optimal control. The authors hope that these two applications of the new multivariate performance assessment technology will advocate more extensive and intelligent use of performance assessment technology and eventually lead to automated monitoring of the design, tuning and upgrading of the control loops. In the first application, MPC control was achieved and operator interventions were reduced by 87%. As a result, the OX column operation was stabilized as well as reflux ratio was reduced. The latter improvement resulted in significant reduction of fuel consumption. Finally stable operation of OX product composition was achieved. In the second application, the diagnosis of poor performance for the MPC controller suggests significant model-plant-mismatch under varying load conditions as well as poor choices of constraints.

Acknowledgments: The project has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), Matrikon Inc.(Edmonton, Alberta) and the Alberta Science and Research Authority (ASRA), 20

predicted vs. actual output 0.1 0.05

y1

0 -0.05 -0.1 -0.15 0

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y2

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Figure 18: Prediction errors for the MPC model under high load conditions. The dashed line and solid line represent predicted and actual values respectively. 0 .1 0 .0 5 0 -0 .0 5 -0 .1 -0 .1 5

0

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Prediction.

Actual Values.

Figure 19: The prediction errors for low load data set A. through the NSERC-Matrikon-ASRA Senior Industrial Research Chair program at the University of Alberta.

References [1] Desborough, L. and Harris, T.(1992), Performance Assessment Measures for Univariate Feedback Control, Can. J. Chem. Eng., 70, 1186-1197. [2] Wolovich, W.A. and Falb, P.L. 1976. Invariants and Canonical Forms under Dynamic Compensation. SIAM J. Control, Vol 14. 996-1008. [3] Goodwin, G.C., and Sin, K.S. 1984. Adaptive Filtering Prediction and Control. Englewood Cliffs: Prentice Hall. 21

0.1 0.05 0 -0.05 -0.1 0

100

200

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500

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100

200

300

400

500

600

700

800

900

1000

800

900

1000

1

0.5

0

-0.5 0

Prediction. .

Actual Values

Figure 20: The prediction errors for low load data set B. [4] Shunta, Joseph P. Achieving World Class Manufacturing Through Process Control. Prentice Hall PTR, Englewood Cliffs, New Jersey 07632. [5] Harris, T (1989), Assessment of Closed-loop Performance, Can. J. Chem. Eng. 67, 856-861 [6] Harris, T., Seppala, C. and Desborough, L. (1999), A Review of Performance Monitoring and assessment Techniques for Univariate and Multivariate Control Systems, J. of Process Control, 9, 1-17. [7] Harris, T. Boudreau, F. and MacGregor (1996), Performance Assessment of Multivariate Feedback Controllers, Automatica 32(11), 1505-1518 [8] Huang, B., Shah, S.L. and Kwok K. (1996), How Good Is Your Controller? Application of Control Loop Performance Assessment Techniques to MIMO Processes, In Proc. 13th IFAC World Congress, Vol M, San Francisco, 229-234 [9] Huang, B., Shah, S.L. and Kwok K. (1997), Good, Bad or Optimal? Performance Assessment of Multivariable Processes, 33, 1175-1183, Automatica [10] Huang,B, Shah,S.L. and Fujii,H., The Unitary Interactor Matrix and Its Estimation from Closedloop Data, Journal of Process Control, 7, 6, 195-207, 1997 [11] Huang, B. and Shah, S.L. (1999), Performance Assessment of Control Loops: Theory and Applications, Springer Verlag. [12] Ko, B.S. and Edgar, T.F., Performance Assessment of Multivariable Feedback Control Systems, In the Proc.of American Control Conference, 2000. [13] Kozub, D. (1996), Controller Performance Monitoring and Diagnosis Experiences and Challenges, In Proceedings of CPC-V, Lake Tahoe, CA [14] Patwardhan, R. S., Studies in the Synthesis and Analysis of Model Predictive Controllers, PhD Thesis, Department of Chemical and Materials Engineering, University of Alberta, 1999 22

Performance Measures Settling time Benchmark Historical Benchmark Achieved Objective Constraints Input Constraints Output Constraints Prediction Measures Settling time based index Prediction error index (Historical) Overall Performance Table 4: cation.

Normal Load

High Load

Low Load 1

Load Load 2

0.8572

0.4026

0.2423

0.6790

1.000

0.2637

0.2544

0.4181

-

42.24

105.07

25.2

Active

Active

Violations

Violations

0.6841

0.1440

0.1196

0.1647

1.000

0.0254

0.0620

0.0803

Good

Poor

Poor

Satisfactory

Performance measures and the prediction measures for the propylene splitter MPC appli-

[15] Shah, S. L., Patwardhan, R. and Huang, B. Multivariate Controller Performance Analysis: Methods, Applications and Challenges, Presented at CPC-6, Tuscon, AZ, Jan. 2001, pp 187-219. Also in AIChE Symposium series 326, Volume 98, 2002. [16] Thornhill, N.F., Oettinger, M., and Fedenczuk, P., Refinery-wide control loop performance assessment, Journal of Process Control, 9(2), 109-124, 1999. [17] Patwardhan, Rohit.S., Shah, Sirish L., Emoto, G., and Fujii, H., Performance Analysis of Modelbased Predictive Controllers: An Industrial Case Study , AICHE Annual Meeting, Miami, Nov. 15-19, 1998 [18] Shah, S.L., Mohtadi, C. and Clarke, D.W., Multivariable adaptive control without a prior knowledge of the delay matrix, Systems & Control Letters, 9, 295-306, 1987. [19] Peng, Y., and Kinnaert, M., Explicit Solution to the Singular LQ regulation Problem. IEEE Trans AC, 37(May), 633-636, 1992.

23