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

Performance Evaluation of two Industrial MPC

Controllers

Jianping Gao1, Rohit-Patwardhan1,

K. Akamatsu2, Y. Hashimoto2, G.Emoto2,

Sirish. L. Shah3∗, Biao Huang3

1 Matrikon Inc. # 1800, 10405 Jasper Avenue, Edmonton, AB, Canada T5J 3N42 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 mul-

tivariate 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 controller-

based 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 as-

sessment 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 1996-

1999, 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 perfor-

mance 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:sirish.shah@ualberta.ca

1

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 con-

trollers (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 evalua-

tion 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 perfor-

mance 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 monitor-

ing. Since MPC is a multivariate, constrained, optimizing, model-based controller, methods that con-

sider 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 objec-

tive 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 = TUt + 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→0DT = limq−1→0T = 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 = D0qd +D1q

d−1 +D2qd−2 + . . .+Dd−1q (3)

where d is denoted as the delay order and is unique for a given transfer function matrix, and Dis are

constant coefficient matrices(Huang et al., 1997). The interactor matrix D can be of any form: If

D = qdI, then the transfer function matrix T is regarded as having a simple interactor matrix. If D is

a diagonal matrix, i.e., D = diag(qd1, qd2 , . . . , 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

DT (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.

E[YtTYt] = E[Yt

TYt] (5)

where

Yt = q−dDYt (6)

Therefore, in terms of the quadratic measure of the variance, performance assessment of Yt is the same

as performance assessment of Yt.

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

3.1 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 Corpo-

ration’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 Xy-

lene 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’ con-troller was implemented on the PX unit. ARX models were identified from the plant input/outputdata and control adjustments are calculated based on the generalized predictive control algorithm tominimize the following cost function:

J = κ

N2Xj=N1

(y(t+ j)− w(t+ j))2 + λNuXk=0

∆u(t+ k)2 (7)

4

LightEnd

Column

OrthoXyleneColumn

OrthoXylene

PurificationColumn

NewXylene

LC

CV5

OrthoXylene

HeavyComp.

LightComp.

HeatFurnace

CV2

IsomerizationProcess

CV3R/D

FF4FF4

FF2FF2

MV6MV6Fuel

Fuel

MV4MV4 MV5MV5

MV3MV3CV6

MV2

Steam

MV1MV1

RecycleXylene

CrystallizationProcess

FF1FF1

FF3

CV4Hold Up

FF5

FF6Cooling Water Temp.

Ambient Temp.

CV1

ParaXylene

R/D

Steam

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

where y(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 MPC performance monitoring

3.3.1 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 parameters for the MPC-based multivariate objective function

CV No. Tag Weight Horizon Range

CV1 Xylene feed 5.0 20 0-25

CV2 Tray #5 temperature 2.5 100 0-200

CV3 Internal reflux ratio 0.05 100 0-45

CV4 OX Hold up 6.0 100 0-100

CV5 OX reflux drum level 0.3 100 0-100

CV6 OX reflux ratio 1.4 50 0-10

MV No. Tag Weight Horizon Range

MV1 Xylene feed 35 3 0-25

MV2 Reboiler steam 200 3 0-15000

MV3 Internal reflux 15 3 0-45

MV4 OX reflux 20 3 0-180

MV5 OX distillate 100 3 0-125

MV6 Fuel heat calorie 25 3 0-200

Table 2: Disturbance VariablesFF No. Tag

FF1 Isomerization feed

FF2 OX column feed

FF3 Heat furnace fuel heat calorie residual

FF4 OX purification column feed

FF5 Cooling water temperature

FF6 Ambient temperature

3.3.2 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(withtwo MVs and one CV out of service this time) is listed below. The model is used to factor out theopen loop interactor matrix:

z−1 0 0 0 0 0

s21 s22 s23 0 s25 0

s31 s32 s33 0 s35 0

0 0 0 0 s45 0

s51 s52 0 s54 s55 s56

0 0 0 0.2z−1 −0.1679z−1 0

6

0 1 2 3 4 5

x 104

15

20MPC was ON

CV1

0 1 2 3 4 5

x 104

120

140

160

CV2

0 1 2 3 4 5

x 104

6

8

10CV3

0 1 2 3 4 5

x 104

30

40

50

CV4

0 1 2 3 4 5

x 104

40

50

60

CV5

0 1 2 3 4 5

x 104

0.5

1

1.5

Samples

CV6

0 1 2 3 4 5

x 104

15

20MPC was OFF

CV1

0 1 2 3 4 5

x 104

120

140

160

CV2

0 1 2 3 4 5

x 104

5

10

CV3

0 1 2 3 4 5

x 104

30

40

50

CV4

0 1 2 3 4 5

x 104

20

40

60

CV5

0 1 2 3 4 5

x 104

0.5

1

1.5

Samples

CV6

Figure 2: The result of MPC application to the distillation units of the para-xylene plant

where

s21 = (−0.06809z−1 − 0.1408z−2)/(1− 0.2845z−1 − 0.6746z−2)

s22 = (−0.06038z−1 + 0.2498z−2)/(1− 1.69z−1 + 0.6978z−2)

s23 = (−0.02414z−1 − 0.266z−2)/(1− 0.8188z−1 − 0.153z−2)

s25 = (−0.1297z−1 + 0.4256z−2)/(1− 1.301z−1 + 0.3159z−2)

s31 = (−0.02498z−1 + 0.03939z−2)/(1− 1.868z−1 + 0.8764z−2)

s32 = (0.1092z−1 − 0.1551z−2)/(1− 1.873z−1 + 0.8796z−2)

s33 = (0.01148z−1 + 0.01025z−2)/(1− 1.876z−1 + 0.8823z−2)

s35 = (−0.0408z−1 − 0.008103z−2)/(1− 1.814z−1 + 0.8235z−2)

s45 = (−0.1968z−1)/(1− 0.99z−1) s51 = (−0.1464z−2)/(1− 0.99z−1)

s52 = (0.34z−2)/(1− 0.99z−1) s54 = (−1.399z−2)/(1− 0.99z−1)

s55 = (−0.4911z−2)/(1− 0.99z−1) s56 = (3.766z−2)/(1− 0.99z−1)

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

0.05925z 0.7234z 0.4z −0.5596z 0 0

0.006454z 0.4251z −0.9003z −0.0934z 0 0

−0.02356z −0.3103z −0.09755z −0.4734z 0 0.8183z

−0.03354z −0.4417z −0.1389z −0.6739z 0 −0.5748z−0.9974z 0.06791z 0.02491z 0 0 0

0 0 0 0 z2 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 as-

sumed 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 overlap-

ping 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 im-

provement 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).

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6Performance when MPC was OFF

Perfo

rman

ce in

dex

Days

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6Performance when MPC was ON

Perform

ance

inde

x

Days

Figure 3: Multivariate performance assessment, with minimum variance as a benchmark, of the Para-

Xylene unit over 30 days(each bar indicates performance measure for 1 day)

1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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

Figure 4: Individual loop performance assessment of the Para-Xylene control loops over 30 days (with

multivariate minimum variance as a benchmark)

8

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.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

250

260

270

280

290

300

310

320

330

340

350

Samples

Qua

dratic c

ost fu

nctio

n

MPC was OFFMPC was ON

Figure 5: Overall quadratic cost function is plotted versus time (based on the moving horizons and

weights of CV and MV data)

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.

1The diagonal term in coherency plot is always 1.

10

0 1 2 3 4 5

x 104

0

20

40

60

80Comparison of T-112 FX feed

0 1 2 3 4 5

x 104

0

50

100

150Comparison of T-112 #5 Temp

0 1 2 3 4 5

x 104

0

0.05

0.1

0.15

0.2Comparison of T-112 Internal Reflux Ratio

0 1 2 3 4 5

x 104

0

50

100

150Comparison of T-111 Hold up

0 1 2 3 4 5

x 104

0

5

10Comparison of T-111 Reflux Drum Level

Samples0 1 2 3 4 5

x 104

0

0.5

1Comparison of T-111 Reflux Ratio

Samples

MPC was offMPC was on

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 = E0at + E1at−1 + . . .+ Ed−1at−d+1 + Edat−d + . . . (8)

and the filtered output variance is:

Yt = q−dD(F0at + F1at−1 + . . .+ Fd−1at−d+1 + Fdat−d + . . .) (9)

we have:

E(Y Tt Yt) = E(Y

Tt Yt) = trace(F0

Pa F

T0 ) + trace(F1

Pa F

T1 ) + . . . , (10)

where the first NMIR coefficient is given byqtrace(F0

P0 F

T0 ) and the second NMIR coefficient is

given byqtrace(F1

Pa F

T1 ), 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

0 0.5 10

0.5

1

0 0.5 10

0.05

0.1

0 0.5 10

0.02

0.04

0 0.5 10

0.02

0.04

0 0.5 10

0.01

0.02

0 0.5 10

0.02

0.04

0 0.5 10

0.05

0.1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.1

0.2

0 0.5 10

0.1

0.2

0 0.5 10

0.05

0.1

0 0.5 10

0.02

0.04

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.1

0.2

0 0.5 10

0.05

0.1

0 0.5 10

0.2

0.4

0 0.5 10

0.02

0.04

0 0.5 10

0.1

0.2

0 0.5 10

0.1

0.2

0 0.5 10

0.5

1

0 0.5 10

0.5

0 0.5 10

0.1

0.2

0 0.5 10

0.01

0.02

0 0.5 10

0.1

0.2

0 0.5 10

0.05

0.1

0 0.5 10

0.5

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.02

0.04

0 0.5 10

0.05

0.1

0 0.5 10

0.2

0.4

0 0.5 10

0.1

0.2

0 0.5 10

0.5

1

0 0.5 10

0.5

1

MPC on MPC off

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):

limt→∞(trace(E0P

aET0 ) + trace(E1

PaE

T1 ) + . . .) =

(trace(F0P

a FT0 ) + trace(F1

Pa F

T1 ) + . . .)

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.

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8Normalized Multivariate Impulse Response when MPC was off

NMIR NMIR-wof

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5Normalized Multivariate Impulse Response when MPC was on

NMIR NMIR-wof

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:

RPIhis =(Pn

i=1 κivar(CVi) +Pm

i=1 λivar(∆MVi))benchmark

(Pn

i=1 κivar(CVi) +Pm

i=1 λivar(∆MVi))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, RPIhis for different j, j = 1 : n can be calculated as:

RPIj =(κjvar(CVj) +

Pmi=1 λivar(∆MVi))benchmark

(κjvar(CVj) +Pm

i=1 λivar(∆MVi))act

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

• RPI = 1 implies that the control system performance is meeting benchmark specifications.

• RPI > 1 implies that the control system is performing better than the benchmark.

• RPI < 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:

CER =(Pm

i=1 λimean(MVi))ben

(Pm

i=1 λimean(MVi))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 im-

pulse 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:

RPIst =(1 + g2

1 + g22 + . . .)

(1 + f21 + f

22 + . . .)

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:

RPIst =trace(E0

PaE

T0 ) ∗ (1 + g2

1 + g22 + . . .)

((trace(E0P

aET0 ) + trace(E1

PaE

T1 ) + . . .))

Note that term trace(E0P

aET0 ) 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 perfor-

mance 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 (C3F) 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 per-

formance assessment:(1) multivariable settling time benchmarking, and (2) historical benchmarking.

The results of the performance assessment step are presented below.

14

Streams

Reflux Distillate(D)

Condenser

Reboiler FC

FC

Bottoms

Steam

FC

LC

GConce

/ 20 min

GConce/ 20 min

D2

D1

Y2

Y1

U1

U2

FC

Figure 9: A schematic diagram of the distillation column showing the relevant controlled and manip-

ulated variables

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

Days: July 1 to Oct. 28

Propylene production rate

Normal Load

HighLoad

LowLoad

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

0 50 100 1502.58

2.6

2.62

2.64

2.66

y1

0 50 100 1506.9

6.95

7

7.05

7.1

7.15

y2

0 50 100 150137

138

139

140

141

142

u1

0 50 100 150

0.74

0.75

0.76

u2

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 Diagnostics

4.2.1 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 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Poor

Poor

Satisfactory

Good

1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

His

tori

cal B

ench

mar

k

(a) (b)

Settl

ing

Tim

e Be

nchm

ark

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)

0 20 40 60 8036

38

40

42

44Achieved objective function

0 20 40 60 8034

36

38

40

42Achieved output variance

0 20 40 60 801.5

2

2.5

3Achieved Input variance

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5Achieved output bias/offset

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

0 500 1000 1500 20002.4

2.45

2.5

2.55

2.6

2.65

y1

0 500 1000 1500 20006.8

7

7.2

7.4

7.6

7.8

y2

0 500 1000 1500 2000108

110

112

114

116

u1

0 500 1000 1500 20000.56

0.58

0.6

0.62

0.64

0.66

u2

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:

d(k/k − 1) = y(k − 1)− y(k − 1)

where y is the measured value and y 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∆)−1w +N(I +Q∆)−1a+Q∆(I +Q∆)−1r (11)

where Q is MPC controller, ∆ = P − P is the model plant mismatch(MPM), a is disturbance, r is thesetpoint and w is the dither signal or external excitation. If w = 0, we have,

d = N(I +Q∆)−1a+Q∆(I +Q∆)−1r = (I +∆Q)−1{∆Qr +Na}.

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

0 100 200 300 400 500 600 700 800 900 1000105

110

115

120

125

130

u1

0 100 200 300 400 500 600 700 800 900 10002.4

2.45

2.5

2.55

2.6

2.65

y1

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.

Q P

yr +

+-+

N

a

P

-+

++

w

d

u

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) pro-

posed the use of minimum variance benchmarking on the prediction errors to obtain a prediction

index. We applied this idea for the C3F 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

0 20 40 60 80 100 120 140-0.04

-0.02

0

0.02

0.04

y1

predicted vs. actual output

0 20 40 60 80 100 120 140-0.02

-0.01

0

0.01

0.02

y2

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 en-

sure 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 as-

sessment 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

0 50 100 150 200 250-0.15

-0.1

-0.05

0

0.05

0.1

y1

predicted vs. actual output

0 50 100 150 200 250-0.05

0

0.05

0.1

0.15

y2

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 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0-0 .1 5

-0 .1

-0 .0 5

0

0 .0 5

0 .1

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0-0 .1

-0 .0 5

0

0 .0 5

0 .1

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.

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Prentice Hall.

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0 100 200 300 400 500 600 700 800 900 1000-0.1

-0.05

0

0.05

0.1

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0.5

1

Prediction..

Actual Values

Figure 20: The prediction errors for low load data set B.

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22

Normal Load High Load Low Load 1 Load Load 2

Performance

Measures

Settling time

Benchmark0.8572 0.4026 0.2423 0.6790

Historical

Benchmark1.000 0.2637 0.2544 0.4181

Achieved

Objective- 42.24 105.07 25.2

Constraints

Input

ConstraintsActive Active

Output

ConstraintsViolations Violations

Prediction

Measures

Settling time

based index0.6841 0.1440 0.1196 0.1647

Prediction

error index

(Historical)

1.000 0.0254 0.0620 0.0803

Overall

PerformanceGood Poor Poor Satisfactory

Table 4: Performance measures and the prediction measures for the propylene splitter MPC appli-

cation.

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ods, Applications and Challenges, Presented at CPC-6, Tuscon, AZ, Jan. 2001, pp 187-219. Also

in AIChE Symposium series 326, Volume 98, 2002.

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23