a relative performance monitor for process controllers.pdf

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INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2003; 17: 685708 (DOI: 10.1002/acs.772)

A relative performance monitor for process controllers

Q. Li, J.R. Whiteley and R.R. Rhinehartn

School of Chemical Engineering, Oklahoma State University, 423 Engineering North, Stillwater, OK 74078, USA

SUMMARY

A monitor is developed to automatically detect poor control performance. It provides a measure (relativeperformance index}RPI) of a control-loop performance relative to a reference model of acceptablecontrol. The reference model simulates the controlled variable output of a user-defined, acceptably tunedcontrol loop. The inputs to the reference model are the setpoints (same as the true plant) and the

disturbances (estimated from the measurements). The monitor uses routine plant operation data only.Pending ability to obtain temporally accurate process models, and the validity of process measurements,simulations and experiments show that the monitor can detect the poor control performance caused byimproper controller parameter values or changes in plant characteristics, and can distinguish it from poorperformance caused by external disturbances. Copyright# 2003 John Wiley & Sons, Ltd.

KEY WORDS: control-loop performance; controller performance; process monitoring; performanceassessment

1. INTRODUCTION

The performance of a process controller often changes during plant operation. An initially well-

tuned controller may become undesirably sluggish or aggressive due to many reasons, such aschanges in process gain, process dynamics, valve stiction or constraints. A controller with poor

performance increases manufacturing costs, lowers product quality and even risks process

safety. Therefore, monitoring controller performance is important, and has become a routine

task for process control engineers. However, detecting a poorly performing controller requires

expertise and experience, and is very time consuming. In practice, many poorly performing

controllers often exist in plants unnoticed for a quite long time before being detected.

Therefore, it would be nice to have an automatic monitoring tool to indicate when a control

loop has significant changes in its performance relative to the performance desired by operators.

The monitor should not disturb routine plant operation, and it should use only the routine plant

operation data. The monitor should detect a poorly performing controller, and suggest when

Accepted June 2003Copyright# 2003 John Wiley & Sons, Ltd.

nCorrespondence to: R. Russell Rhinehart, School of Chemical Engineering, Oklahoma State University, 423Engineering North, Stillwater, OK 74078, USA.

Contract/grant sponsor: Measurement and Control Engineering Center (MCEC)


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maintenance is needed for the controller (checking for valve stiction, adjusting process model

and retuning the controller).

Currently, most cited data-based controller performance monitoring techniques are based on

the work of Harris [1], in which the controlled variable variance under minimum variance

control (MVC) is used as a lower bound benchmark to evaluate the performance of single-loopcontrollers. The ratio of the minimum variance to the variance of the controlled variable is

defined as normalized performance index [2]. Other similar measures have also been proposed,

such as the closed-loop potential index [3], the relative variance index that compares actual

control to both MVC and open-loop control [4], and a modified performance index based on the

desired pole locations and MVC [5, 6]. Reviews on the MVC-based techniques can be found in

References [79]. Many methods are also proposed to detect sluggish control [10] or oscillations

[1115]. Applications of the performance assessment schemes in the process industries can be

found in References [1521]. The minimum variance benchmark is widely used in industry for

performance assessment because it can be used to determine the improvement potential in

variance reduction by only requiring an estimate of process delay and routine closed-loop

operation data.

Although the MVC benchmark provides a theoretical lower bound on controlled variablevariances, it is not a practical benchmark that every good controller should try to achieve. It is

because (1) the minimum variance usually cannot be achieved unless the process model and

disturbance model are perfectly known, which is practically impossible and (2) operating too

close to the minimum variance often means excessively large moves of controlled variables,

which is not acceptable in practice. As a consequence, a well-tuned controller in practice

operates with some distance from the minimum variance point. A user has to decide the optimal

distance for good control, which is unique to the balance of issues for each loop. One cannot say

that the closer the actual variance is to the minimum variance, the better the controller

performs.

There are other passive, data-based monitoring techniques. An automated on-line goodness

of control performance monitor was proposed by Rhinehart [2224]. The method uses a

computationally simple, robust statistic, called ther-statistic, which is defined as the ratio of theexpected variance of the deviation of the controlled variable from the setpoint to the expected

variance based on the deviation between two consecutive process measurements. Difficulties in

the r-statistic method are either how to choose the right range of r-values for acceptable and

unacceptable performance or how to select a sampling rate to eliminate auto-correlation.

An online automated control performance monitor based on statistical differences in run-

length (RL) distributions was proposed [25]. The RL index is defined as the time period (number

of sampling periods) between two consecutive zero-crossings (sign changes) of the controller

actuating error signals (setpoint minus controlled variable). The histogram of RL index under

different control performance, such as sluggish control, aggressive control (oscillations) and

good control, are significantly different and therefore are used to detect poor control

performance. The monitor does not require either process knowledge or process model, and it

uses only routine plant operation data. One difficulty in RL-distribution method is how tochoose a representative data to build the reference RL-distribution, which represents acceptable

control performance.

Most control performance monitoring techniques do not differentiate poor control

performance caused by external disturbances from that caused by the control loop itself

(improper controller tuning parameters, constraint or control valve stiction). Obviously, it is

Copyright # 2003 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2003; 17:685708

Q. LI, J. R. WHITELEY AND R. R. RHINEHART686


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important to differentiate these two cases. If the controller or control valve is the reason of the

poor control, maintenance is called for, but if the external disturbance is the cause, loop

maintenance should not be triggered. This work develops a technique that is insensitive to

disturbance behaviour, and identifies problems within the control loop (e.g. tuning, constraints,

valve stiction).The idea of using a reference model has been used in model reference adaptive control [26], in

which the controller parameters are adjusted online to minimize some function of the differences

(errors) between the measured controlled variable output and the output of the reference model,

which represents the desired closed-loop response. This is extended, here, for control

performance monitoring.

The concept is to have a reference model to simulate how a chosen good control system would

respond to the same setpoint sequence and disturbance sequence imposed on an actual plant,

and to compare the actual control performance relative to the performance defined by the

model. In this work, a relative performance monitor is proposed to measure the current control-

loop performance relative to that of a reference model. The reference model represents the

behaviour of an adequately tuned control loop under the same (setpoint and disturbance) inputs

as the actual plant experiences.

2. A RELATIVE PERFORMANCE MONITOR

The basic idea of the proposed relative performance monitor (RPM) is to compare the

performance of a control loop under monitoring to that of a reference model to measure

the relative performance of the actual control loop to the good performance represented by the

reference model. Since the controller actuating error (i.e. setpoint controlled variable) is a

good indication of control-loop performance, we compare the actuating error of a control loop

to that of a reference model to determine the relative performance of the control loop and the

reference model. Figure 1 shows this basic idea. System A in Figure 1 represents a control loop

under monitoring. The inputs to system A are the setpoint SP and disturbances dand the outputis the actuating error eA: UsingeA as plant output is equivalent to using the controlled variableCVA as the plant output because eA SP CVA and the SP is known. The same inputs, SP and

d; are fed into the reference model R; and the output of the reference model is eR eR SP CVR: A comparison of the error sequences of eA and eR through a proposed relativeperformance index gives the relative performance of a control loop and the reference model.

A simplified version of a typical feedback control loop consists of a controller and a process,

as shown in Figure 2. In Figure 2, we arrange the block diagram such that SP and dare the

System A

(controller &

process)

Reference

Model, R

SP

d

eA

eR

RPIRPI

Figure 1. Basic idea of the relative performance monitor.


A RELATIVE PERFORMANCE MONITOR 687


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inputs and the actuating error e is the output. PA represents the actual plant A to be controlled,

CA the controller, CVA the controlled variable, SP the setpoint, MVA the manipulated variable,

and d the effects of disturbances and noise, which are modelled as additive to the controlled

variable CVA:

For the control loop in Figure 2, we have the following Laplace transform relation:

CVAs PAsCAs

1PAsCAsSPs

1

1PAsCAsds 1

Since the actuating error eA SP CVA; we have

eAs 1 PAsCAs

1PAsCAs

SPs

1

1PAsCAsds

eAs 1

1PAsCAsSPs ds 2

The term PAsCAS=1PAsCAs represents the closed-loop response of system A to thesetpoint input. If we use ASPs to represent the closed-loop response of a control loop to the

setpoint input, we have,

ASPs CVAs

SPs

PAsCAs

1PAsCAs 3

or

1ASPs 1

1PAsCAs 4

So, the actuating error of a control loop A can be calculated as

eAs 1ASPsSPs ds 5

In a similar way, by choosing a desired closed-loop setpoint response relation, RSPs, and

usingRSPsas the reference model of good control, we can calculate the actuating error output

of the reference model as

eRs 1RSPsSPs ds 6

The reference model R in Figure 1 represents the desired good relationship between the inputs

(the setpoint and disturbance) and the output (the actuating error or the controlled variable).

Therefore, the CVR output SP2eR of the reference model simulates the good closed-loop

+PACA

SP

System A

d

+

CVAMVA +

eA

Figure 2. Block diagram for a typical feedback control system (system A).




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behaviour under the same input (setpoint and disturbance) conditions as experienced by system

A; the actual control loop. So, a comparison of system A output, eA or CVA; and the referencemodel output, eR or CVR; gives the performance of a control loop A relative to that of thereference model.

A relative performance index (RPI, defined later) is calculated from the actuating errors of acontrol loop and the reference model. The RPI provides a measure of the relative performance

of the control loop and the reference model.

3. REFERENCE MODELS

There are many ways to define a reference model. The reference model can be as simple as a step

response function, or as complex as a simulated adaptive control system.

Here we discuss two simple forms of the reference model: (1) the parametric model form, such

as a first-order or second-order response model, and (2) the non-parametric model form, such as

a finite impulse response (FIR) model, or a step response model.

A parametric model has a small number of parameters that need to be specified. For example,a first-order parametric response model has the following form:

RSPs 1

tRs 1 7

which has only one parameter, the time constant, tR; since the gain must be equal to 1.0 toremove offset. A second-order parametric model has the following form:

RSPs 1

t2s2 2zts 1 8

which has two parameters that need to be specified, the damping factorzand the time constantt

(again, the gain is set to 1.0 to remove offset). When z > 1 (overdamped case) or z 1 (critically

damped case), there is no overshoot in model response to a step setpoint change. When z5

1(underdamped case) for a step change, we have analytical solutions for the following

performance indices:

Overshoot : OS exp pzffiffiffiffiffiffiffiffiffiffiffi ffiffi

1z2p

! 9

Decay ratio : DR OS2 exp 2pzffiffiffiffiffiffiffiffiffiffiffi ffiffi

1z2p

! 10

Period : P 2pz

ffiffiffiffiffiffiffiffiffiffiffiffiffi1z

2p 11By choosing z and t we can determine the desired response to a setpoint change.

The first-order and the second-order models can represent most of the desired response types,

and they require only one or two parameters to be specified, so it is very easy to specify a

parametric reference model. Further, no plant tests are required to obtain a parametric reference

model, so this method is useful when no plant step tests are allowed.




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The other simple form of the reference model is the FIR model, which is usually obtained

through plant tests after the control loop is well tuned by whichever criteria the user wishes.

After the controller tuning, introduce a step changeDSPin setpoint, and record the CV response

samples,s0;s1;s2;. . .;sn;until the CV reaches the new steady state (assuming at sample n). Then,

the reference model RSP; in discrete time, can be represented asRSPq rSP0 rSP1q

1 rSP2q2 rSPnq

n 12

where q is a forward shift operator, and q1 is the backward shift operator, such that for a

sample xk at time k; qxk xk 1 and q1xk xk21; the model coefficients rSPi;i 0; 1; 2;. . .;n; are determined by,

rSPi si si1

DSP; i 1; 2;. . .; n and hSP0 0 13

andn is the number of sampling periods that it takes the CV to reach a new steady state after a

setpoint change.

In discrete time, the actuating error sequence eRkcan be derived in a similar way as deriving

Equation (6), so we haveeRk 1RSPqSPk dk 14

We can prove that 12RSPq is actually a control loops CV response to the disturbance

input. For a control loop as shown in Figure 2, we have

ASPq CVAk

SPk

PAqCAq

1PAqCAq 15

CVAk

dk

1

1PAqCAq 1

PAqCAq

1PAqCAq

Therefore,

CVAk

dk 1 ASPq 16

This also indicates that once the CV response of a feedback control loop to the setpoint input is

fixed, its response to the disturbance input is also fixed.

LetRdqdenote the CV response of the reference model R to the disturbance input, we have,

Rdq rd0 rd1q1 rdnq

n 1RSPq 17

Since

1RSPq 1rSP0 rSP1q1 rSPnq

n

and rSP0 0; we have,

rd0 1

rdi rSPi si1si

DSP; i 1; 2;. . .;n 18




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So, the time-domain-equivalent way to calculate the actuating errors as in Equation (14) is

eRk Xni0

rdiSPk i dki 19

where rdi are defined in Equation (18).The major advantage of using an FIR reference model is that an FIR model can represent the

closed-loop responses of any linear control system with any order or complex dynamics.

4. RELATIVE PERFORMANCE INDEX

The performance of a control loop could be measured by many metrics, such as mean-squared

error (MSE), mean absolute error (MAE), variance, minimum variance-based indices, r-statistic

or the index based on RL distribution differences.

The exponentially weighted-moving-average of squared error (EWMASE) or absolute error

(EWMAAE) could also be used if we want to put more weighting on recent data than old data

or we do not want to take too much computer resources (memory or CPU time) for processingthe data. The EWMASE metric M1 is calculated recursively as follows:

M1k l*M1k 1 1l * e2k 20

whereM1kand M1k 1are the EWMASE metric values calculated at time kand k21;l is a

constant between 0 and 1.0, and thepth sample ofe2 in the past carries a weight of12llp;e is

the difference between the setpoint and the controlled variable, i.e.

eA SP CVA 21

eR SP CVR 22

where subscript A represents the actual control loop, and subscript R represents the reference

model.

From eA; we can calculate a performance metric MeA; such as MSE or EWMASE, for thecontrol loop under monitoring. And similarly, from eR; we can also calculate the sameperformance metric MeR for the reference model.

We define an RPI based on a chosen performance metric M: The RPI based on metric M isdefined as the ratio of the metric value for the reference model MeR; and the metric value forthe control loop MeA;

RPIM MeR

MeA 23

If we choose M1 (i.e. EWMASE) as the performance metric, the RPI based on M1 is

RPIM1 M1R

M1A24

where

M1Rk l *M1Rk 1 1l * e2Rk 25

M1Ak l *M1Ak 1 1l * e2Ak 26

The RPI based on other metrics (such as MSE) could also be defined in a similar way.




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The RPI provides a measure of the relative performance of the control loop and the reference

model, and (1RPI) represents the improvement potential in control-loop performance if

retuning the loop to the reference model level of performance. The RPI value can be interpreted

as follows: (1) An RPI value close to 1.0 means the control-loop performance is close to that

of the reference model; (2) an RPI value5

1.0 indicates the loop performance is much worsethan that of the reference model, and something should be done on the control loop, such as,

re-estimating plant model, retuning controller parameters, checking for valve stiction, etc. In

this case, the chosen performance metric (such as EWMASE or MSE) can be reduced by

(1RPI)n100% under similar input conditions if we retune the control loop to reach the same

performance level represented by the reference model. (3) An RPI value >1:0 means the controlloop has a better performance than the reference model. If the RPI values are much greater than

1.0, we may need to update the reference model.

If we want the monitor to automatically flag poor control, we can specify a critical or

threshold value PRIc, such that when the RPI value exceeds the threshold value, the monitor

automatically flags. A good choice of the threshold value depends not only on the noise level of

the data but also on the users tolerance level on the deviation of a control-loop performance

from that of the reference model. A user can make a choice of the threshold value based on theinterpretation that the variance or MSE or EWMASE can be improved by (1RPI)n100% if

retuning the control loop to the performance level represented by the reference model. For

example, RPI=0.75 means that the variance (or similar measures) can be reduced by 25% if the

controller is retuned to reach the performance level of the reference model. RPI c=0.75 could be

used as an initial threshold value to flag the performance monitor if 25% improvement is worth

the controller-retuning efforts.

Statistical tests could also be used to establish the threshold values for controller monitor

flagging. The authors have used the F-test to test the hypothesis of equal variances

of actual output and reference model output. The critical value to reject the hypothesis was

used as the threshold for monitor flagging. Although statistics-based approaches provide

powerful methods to determine the threshold values, it adds complexity, and will not be

demonstrated here.

5. DISTURBANCE ESTIMATION

The inputs to the reference model, as well as the actual closed control loop, are the setpoint and

disturbances. We know exactly the setpoint input sequence to the actual control system, but we

do not know the disturbance (including noise) sequence, and therefore we have to estimate it.

The disturbance estimation with adequate accuracy is the key to the monitors ability to

distinguish the poor control performance caused by external disturbances and other poor

performance caused by problems within the control loop, such as plant characteristic changes,

poor controller tuning parameters, valve stiction or hitting constraints.

We choose to use the prediction error sequence of the plant model as estimates of the effectivedisturbance and noise, which are added to the true CV. The model prediction error is the

difference between the actual measured CV value and the model predicted CV value.

Let the plant model be fk; uk; y; i.e.

ymodk fk; uk; y 27




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whereymodk is the plant model output at discrete time k; uk the plant input at time k;and ythe set of model parameters. Then the model prediction error sequence is just yk ymodk:

Therefore, the disturbance sequence dk can be estimated by

dk yk fk; uk; y 28

where fk; uk; y can be a linear or non-linear plant model.If we know the plant characteristics do not change much during plant operation, we can use a

fixed plant model fk; uk; y to estimate the disturbances. If no plant models are available, wecan obtain an estimated plant model through a step change in plant input while the controller is

offline. The open-loop response data can be used to build an FIR plant model as described

previously in Equations (12) and (13).

If the plant model parameters change significantly during plant operation and the fixed plant

model cannot reflect the changes, it may be necessary to adaptively estimate and update the

model parameters. If this is the case, we assume a plant model structure, such as ARX,

ARMAX, BoxJenkins, statespace or a non-linear neural networks model and estimate the

model parameters adaptively as the plant changes during operation. For the purpose of

disturbance estimation, we may choose a simple plant model structure and estimate itsparameters under closed-loop condition.

Here, we choose a simple first-order ARX plant model structure

yk b

1aq1uk kd dk 29

where yk; uk and dk are the plant output CV, input MV and effective disturbances (plusnoise), respectively, at discrete time k;kd is the time delay, a and b are constants, andq

1 is the

backward shift operator, such that q1uk uk1: The first-order ARX, which is thediscrete time version of the popular first-order-plus-time-delay (FOPTD) model, is chosen here

because of its simplicity in parameter estimation and its adequacy for our disturbance

estimation for control performance monitoring purpose. There are many system identification

methods to estimate a system order and parameter values [27].

After estimating the plant parameters, a; b and kd; the effective disturbance can be estimatedas

dk yk b

1aq1uk kd 30

We choose the recursive least-square (RLS) estimation method because it works well with

time variant or non-linear processes, which are common in the process industry.

The RLS algorithm has the following form [27]:

#yyk #yyk 1 Kkyk #yyk 31

where #yykis a vector of the model parameters estimated at discrete time k,ykis the observedprocess output at timek, #yykis a prediction ofykbased on observations up to time k1 and

the estimated model at time k1:The gain Kk in Equation (31) has the following form:

Kk Qkck 32




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whereckis the gradient of #yykjywith respect toy; #yykjyis the prediction ofykbased on themodel described by parameters y; and Qk is a matrix that controls the adaptation gain anddirection.

For linear regression model structures, such as AR and ARX, the process output prediction

#

yykjy can be written as#yykjy jTkyk 1

wherejkis the regression vector, which consists of past values of observed inputs and outputs.

Therefore,

ck d

dy #yykjy jk 33

The matrix Qk can be determined by minimizing the following cost function:Xkj1

lkjyj #yyj2 34

where the forgetting factor l is a constant between 0 and 1, and typically between 0.970.995

[27]. The forgetting factor is used to discount old measurements exponentially so that an older

observation will carry a less weight in the cost function than more recent data. The squared

error that ispsamples away in the past from the current time kcarries a weight oflp in the cost

function in Equation (34), which is to be minimized to obtain the model parameters.

For linear regression models, such as ARX, the cost function Equation (34) can be minimized

exactly with the following choice ofQk [27]:

Qk Pk Pk 1

ljkTPk1jk35

Pk 1

lPk1

Pk1jTPk1

ljkTPk1jk 36

For the first-order ARX process model shown in Equation (29), the model parameter vector is

y a bT 37

The regression vector is

jk yk1uk kd T 38

Note that the time delay kd in terms of number of sampling periods must be specified before

we can construct the regression vector y: If we know that the range of change of the processdelay during operation is smaller than one sampling period, we can use a fixed process delay

during monitoring. Otherwise, we need to treat the delay as a variable and estimate it. One way

to estimate the delay is to assume a possible range of change of the process delay during

operation in terms of number of sampling periods, and for each time delay value, construct the

corresponding regression vector and estimate the corresponding model parameters. The delaywith the smallest prediction error is chosen as the estimated delay.

As the new input output data arrive, we can recursively estimate the plant model parameters

(kd; a and b), which can adapt themselves as the plant characteristics change during plantoperation. The closer the estimated model is to the true plant, the more accurate are the

disturbances estimated.




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6. APPLICATION PROCEDURE OF THE RELATIVE PERFORMANCE MONITOR

(1) Specify a reference modelRSP, either through a plant step test or by choosing parameters

for a parametric model.

(a) If plant step tests are allowed, make a step change in SP after tuning the controller atthe plant nominal operating point. Observe the closed-loop CV response. Calculate

the reference model parametersrdi from the CV response data using Equation (18).

(b) If plant step tests are not an option, assume the desired response to setpoint input is

first-order or second-order, and choose the parameter tdas in Equation (7) or the

parameterst and z as in Equation (8).

(2) Choose an approach to estimate the disturbance.

(a) One approach is to use a fixed and predetermined plant model. If no predetermined

model is available, it can be identified by a step test. Make a step change in plant

input at the nominal operating point while the controller is offline, record the plant

output response, and calculate a step response-type model parameter values in the

same way as shown in Equation (13). After a plant model is available, estimate the

effective disturbance sequence using Equation (30).(b) Another approach to estimate the disturbance is to use an adaptable plant model

whose parameters are recursively estimated online. Then, choose a forgetting factor

l; usually between 0.90 and 0.995, according to the desired emphasis of recentmeasurements relative to past ones. After estimating the plant model parameters

using Equations (31)(36), estimate the effective disturbances at each sample time

using Equation (30).

(3) Feed the actual, known setpoint sequence and the estimated effective disturbance

sequence into the reference model to obtain the actuating error output of the reference

model, either using Equation (19) for the non-parametric reference model case (with the

plant step test data) or using Equation (6) for the parametric reference model case (with

chosen model parameters).

(4) Calculate the RPI values using Equations (24)(26), for example.(5) Choose a threshold value RPIc (for example, RPIc=0.75) for flagging poor control

performance based on the users tolerance level on the performance deviation of

the control loop from the reference model. If an RPI value is greater than RPI c, the

relative performance monitor flags poor control performance and suggests loop

maintenance.

7. EVALUATION AND DISCUSSION OF THE RELATIVE PERFORMANCE

MONITOR

The relative performance monitor is demonstrated through computer simulations and

experiments on a water flow control loop.

7.1. Simulation results and discussion

We simulate an SISO first-order-plus-time-delay (FOPTD) plant controlled by a PI controller

using Matlab/Simulink. In all simulations, the plant and the PI controller are simulated using




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continuous-time model, which can be handled by Matlab built-in functions (with variable step

size). The relative performance monitor takes data samples and performs computations at every

discrete time unit.

The plant is

Tpdy

dtyt KputTd 39

whereKp is the process gain, Tp is the process time constant and Td is the time delay. Choose

Kp=1, Tp=10 and Td=1, and then the plant becomes

10dy

dtyt ut1 40

The disturbance, which is added to the plant output, has at least two additive components: (1)

measurement noise represented by a zero-mean Gaussian noise with variance equal to 1, and (2)

a disturbance process driven by a zero-mean Gaussian noise wt:

15dy

dtyt 2wt 41

After tuning, the PI controller has the following parameters: controller gain Kc=5

(dimensionless), and integral time constant Tc=10 (time units), determined from the ITAE

tuning method for setpoint changes and the plant model. Any controller tuning method could

be used to tune the controller to the users satisfaction.

Make a step change in SP, record the CV response and calculate the reference model RSPfor

the setpoint input. Figure 3 shows the plant closed-loop response to the setpoint change, and the

calculated reference model parameters rSPi and rdi:The solid line in the third plot of Figure 3 is the response to the same step setpoint change of a

second-order parametric reference model below:

RSPs 1

1:12s2 2 * 0:7 * 1:1s 1 42

with a damping factor z 0:7 and a time constant t 1:1: We can see that, in this case, theresponse of the second-order reference model Equation (8) with appropriate parameters z and t

is close to the actual control-loop response. In many cases, we can find a parametric reference

model such as Equation (8) or (7) to represent the desired reference model if no plant tests are

allowed.

Here, we choose to use a fixed and predetermined plant model to estimate the disturbance (the

first approach). Since all models have errors, to be realistic, here, we assume the model

parameters are 20% different from the true values. In simulations, we use the following plant

model to estimate the disturbances:

12dy

dt yt 1:2ut1 43

whose gain and time constant are both 20% larger than the true values of plant described by

Equation (40).




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7.2. Simulation of good control performance

First, run a simulation test with the acceptably tuned controller, and with the Gaussian noise

and first-order process disturbance. The results after a warm-up period are shown in Figure 4.

The actual and estimated disturbances are shown in Figure 5.

Figure 4 shows the plant SP, plant output CVA, plant input MVA, reference model

output CVR and RPI index values. A setpoint change from 0 to 20 is introduced at sample

time 150. We can see that the RPI values are overall very close to 1.0 before and after thesetpoint change.

Figure 5 shows the actual disturbance sequence and the estimated disturbance sequence using

the pre-estimated plant model shown in Equation (43). We can see that although the estimated

plant model parameters are 20% larger than the true values, the estimated disturbance is very

close to the true value. Note here, the estimated disturbance is obtained from the difference

Figure 3. Closed-loop step responses to setpoint change, and the reference model parametersrSPi andrdi:The solid line in the third plot is the response of a second-order parametric reference model to the setpoint

change. (Sampling period=1 time unit.)




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between the actual plant output and the plant model output after removing the moving average

mean value from the difference because we assume the disturbance is zero mean. Otherwise, the

estimated disturbances will have an offset from the true disturbances after setpoint changes, and

the offset is due to the plant model errors.

7.3. Simulation of poor control performance due to a too aggressive controller

To see how the monitor detects oscillations caused by a too aggressive controller, change thecontroller gainKcfrom 5 to 10 and the integral time Tifrom 10 to 2, and run the simulator with

other conditions the same as before. The results are shown in Figures 6 and 7.

We can see that except for a short time immediately after the setpoint change, most of the

time the RPI values are smaller than 0.5, which means the squared errors (EWMASE or MSE)

could be reduced by at least 50% if the control loop is retuned to reach the same performance

Figure 4. Plant output CVA, plant input MVA, reference model output CVR and RPI when the controlloop is tuned. (Sampling period=1 time unit.)




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level represented by the reference model. Immediately after the setpoint change, the RPI values

are increased for a short time because a very aggressive controller responds to a setpoint change

much quicker than the reference model, and the quick controller response reduces the errors due

to setpoint change, and improves performance. But, the overshoot due to the aggressive

controller causes the errors to increase, and the RPI values then drop to the previous level.

From Figure 7, we can see that the disturbance estimates are close to the actual values even

when there are oscillations due to an aggressive controller.

7.4. Simulation of poor control performance due to external oscillatory disturbances

To differentiate the oscillations caused internally (by poor controller parameters) and externally

(by oscillatory disturbances), we run a simulation test with a sinusoidal signal below added to

the previous disturbances (Gaussian noise plus a first-order process disturbance)

5 sin2pk=0:4

The amplitude is 5 and the frequency is 0.4 radian/sample, or a period of about 15 samples. The

results are shown in Figures 8 and 9.

From Figure 8, we can see that the RPI values are close to or above 1.0, which means the

relative performance monitor does not indicate poor control when the oscillations are only dueto external oscillatory disturbances. Even a well-tuned controller will oscillate under external

oscillatory disturbance, so does the reference model. Remember an RPI value greater than 1.0

indicates the actual performance is better than the reference model. The disturbances estimated

from a predetermined plant model Equation (43) with a 20% parameter error appear very close

to the true value of the oscillatory disturbances as shown in Figure 9.

Figure 5. The actual and estimated disturbances before and after a setpoint change at time 150 (a well-tuned control loop). (Sampling period=1 time unit.)




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The ability of the relative performance monitor to differentiate oscillations caused internally

and externally is very useful. We do not want to adjust a controller unless the control loop itself

is causing the poor performance. The monitor does not indicate oscillations caused by external

disturbance, and therefore makes it easy to identify the root cause of oscillations.

We can see that the accuracy of the disturbances estimated from a predetermined imperfect

plant model, in the simulations, is good enough for our control performance monitoring

purpose.

7.5. Simulation of control performance under drifting disturbances

To see the effect of the drifting (non-stationary) disturbances on the disturbance estimation and

the performance monitor output, an integral of the Gaussian noise (random walk) is added to

Figure 6. Plant output CVA, plant input MVA, reference model output CVR and RPI when there areoscillations due to a too aggressive controller (poor tuning parameters). (Sampling period=1 time unit.)




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the original disturbances (Gaussian noise plus a first-order process disturbances). The actual

and estimated disturbances are shown in Figure 10.

From Figure 10, we can see that when the disturbances contain a slowly drifting component,

the mean value (more exactly, the zero or very low frequency components) of the actualdisturbances will drift around, and sometimes to a value far away from zero. It is very hard to

estimate this drifting component accurately because an imperfect plant model is used to estimate

the disturbances, and the model errors will accumulate. Note from Figure 10 that except the

drifting mean value (the very low frequency components), the estimated disturbances have a

pattern very similar to the actual disturbance.

Since most well-performing controllers have sufficient integral action and thus can remove

slowly changing drifting (low frequency) disturbance fast enough, these slowly drifting

disturbances have almost no effect on the performance of this good controller. In other words,

the performance measures (such as the variances, MSE or EWMASE) of a well-performing

controller with reasonable integral action should be almost the same no matter whether there

exist the lowly drifting disturbances.

Therefore, although the estimated disturbances, which are fed into the reference model, mayhave different slow-frequency components from the actual disturbances, these slowly changing

errors in the disturbance estimation will not have much effect on the output variances or MSE

of the reference model because the reference model almost always represents a well-performing

controller with sufficient integral action. Figure 11 shows the results of the performance

monitor when there exist the drifting disturbances shown in Figure 10. We can see that the

Figure 7. The actual and estimated disturbances when there are oscillations due to a too aggressivecontroller. (Sampling period=1 time unit.)




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performance monitor outputs with significant drifting is close to the case without the drifting

shown in Figure 4.

7.6. Experimental results and discussion

Experiments on a water flow control loop are also used to evaluate the relative performance

monitor. In the water flow control loop, the controlled variable is the water flow rate, and the

manipulated variable is the signal to the valve. Control is executed by a Camile Tg 2000 system,using 420mA signals}to an i/p device operating a flow control valve in a 1

2inch line, and from an

orifice flow transducer. The inputoutput relation exhibits the first-order plus time-delay dynamics

and non-linear characteristics. A PI controller is used to control the water flow rate at setpoint.

The nominal operating point of the water flow rate was 35 kg/h. After tuning the controller,

make a step change in the setpoint, and record the closed-loop response. Since the controller is

Figure 8. Plant output CVA, plant input MVA, reference model output CVR and RPI when there areoscillations due to external oscillatory disturbances. (Sampling period=1 time unit.)




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Figure 9. The actual and estimated disturbances when there are oscillations due to external oscillatorydisturbances. (Sampling period=1 time unit.)

Figure 10. The actual and estimated disturbances when drifting disturbances are added.(Sampling period=1 time unit.)




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tuned acceptably well as judged by the operator, this response to a setpoint step becomes the

reference response. The experimental data, CVA, and the estimated reference model parameters,

rSPi; are shown in Figure 12.

7.7. Experiments on poor control performance due to plant changes

To see if the monitor can indicate the poor control performance caused by changes in

plant characteristics, we operate the process at different flow rates. Since the process is non-linear, a change in operating point (flow rate) means a change in plant characteristic, or

a change in the parameters in the FOPTD linear plant model, which we use to describe

the process.

To estimate the disturbance, we use the second approach, i.e. use an adaptable plant model

to estimate the disturbance. Since the plant characteristics experience significant changes in

Figure 11. Plant output CVA, plant input MVA, reference model output CVR and RPI when driftingdisturbances are added. (Sampling period=1 time unit.)




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this experiment, an adaptable plant model is appropriate and a fixed plant model will yield

large errors in the disturbance estimation. We choose the FOPTD plant model structure, with

a range of possible delays of 48 samples. The results from the second approach are shown

in Figure 13.

Starting from the nominal operating point, we make several step changes in flow rate setpoint,and observe the changes in control performance due to plant changes while maintaining the

controller parameters unchanged. Figure 13 shows the experimental data CVA, MVA, the

reference model output CVR, the RPI values, and the estimated disturbances.

From Figure 13, we can see that when the plant operates near the nominal operating point

with a water flow rate of 35 kg/h, where the controller is tuned, the RPI values are close to 1.0,

even though there are setpoint changes to 30 kg/h, and then 25 kg/h. As the water flow rate

decreases and moves away from the nominal operating point, the plant characteristics change.

Step tests indicate that the steady-state gain is approximately 0.1 kg/h/% near the nominal

operating point 35 kg/h, but it becomes 0.5 kg/h/% near the operating point 20 kg/h, and 0.9 kg/

h/% near the operating point 5 kg/h.

As the plant changes, the control-loop performance gets worse, and the RPI values drop

far way from 1.0. We can see that the RPI values drops below 0.5 (indicating poor loopperformance) when the flow rate is below 20 kg/h, where the plant steady-state gain is at least

5 times that of the initial model obtained at the nominal operating point, and the poor control

performance with excessive oscillations occur. After the flow rate moves back to the region near

the nominal operating point, the control performance recovers and the RPI values are close to

1.0 again. Figure 13 also shows the estimated disturbances. We can see that, at steady state,

Figure 12. Experimental closed-loop step response data and the reference model RSP parameters rSpi:(Sampling period=0.1 s.)




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the disturbance estimates look very similar to the fluctuations of the CV measurements, but when

there are excessive oscillations, the estimated disturbances have larger errors due to the plant

model errors.

7.8. Discussion summary

From the simulations and experiments, we can see that using a reference model to simulate a

well-tuned controllers response to setpoint and disturbance inputs has the following benefits:(1) it provides a practical, achievable standard for comparing control performances since the

user can specify the desired reference model or build the reference model by a plant step test just

after controller tuning, without the impractical shortcomings of the idealized minimum variance

standard; (2) it provides a relatively fair comparison because the reference model is subject to

exactly the same setpoint sequence and similar (depending on the estimate accuracy) disturbance

Figure 13. Water flow control experimental data CVA, MVA, reference model output CVR, RPI values,and the estimated disturbance. (Sampling period=0.1 s.)




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sequence as the actual control loop, and the comparison standard is constantly adapting to the

plant input levels in setpoint and disturbances rather than comparing to a fixed standard as in

many other methods; (3) the reference model directly simulates the controlled variable output

sequence, which is to be compared with the actual CV output sequence, so a variety of control

performance comparison methods can be used, such as, a simple visual comparison, variancecomparison or other comparisons in overshoot, settling time, mean absolute error, mean square

error or RL distributions.

Although not shown here, our tests reveal the relative performance monitor is able to indicate

poor control performance caused by too sluggish control or constraint-hitting conditions.

Further experiments are needed to claim the monitors ability to indicate poor performance

caused by valve stiction.

We suggest using a fixed, reasonably valid plant model to estimate the disturbance whenever

the plant characteristics do not change much during operation, but when there are significant

changes in plant characteristics during operation, an adaptable plant model should be used.

However, model identification or estimation under closed-loop condition is a very challenging

thing to do.

8. CONCLUSIONS

A relative control-loop performance monitor based on a reference model is developed, and

demonstrated with both simulations and experiments. The simulations demonstrate that the

monitor can distinguish poor control performance caused by external disturbances from those

caused by problems within the control loop, such as improper controller tuning parameters.

This ability is important for the control engineers to identify the root cause of poor control as

well as to decide which control loop needs maintenance. The experimental results demonstrate

that the monitor also can detect poor control performance due to changes in plant

characteristics.

The introduction of the reference model in the proposed relative performance monitor has the

following benefits: (1) it provides a practical, achievable, flexible comparison standard because

the user can specify the reference model, (2) it directly simulates a good control loops behaviour

under the actual setpoint input sequence (the same as the true plant) and estimated disturbance

input sequence, so the comparing standard is adaptive to the changes in the actual input setpoint

or disturbance, (3) the simulated plant output can be visually observed and compared directly

with the actual output using a variety of comparing methods and (4) the monitor uses routine

plant operation data only, and therefore does not disturb plant operations.

Efficacy of the approach requires a reasonably true process model, and is predicated on valid

measurements.

ACKNOWLEDGEMENTS

The authors appreciate both the financial support and guidance from the industrial sponsors of theMeasurement and Control Engineering Center (MCEC).

REFERENCES

1. Harris TJ. Assessment of control loop performance. Canadian Journal of Chemical Engineering 1989; 67:856861.2. Desborough LD, Harris TJ. Performance assessment measures for univariate feedback control. Canadian Journal of

Chemical Engineering 1992; 70:11861197.




24/24

3. Kozub DJ. Controller performance monitoring, diagnosis: experiences and challenges. In Proceedings of the 5thInternational Conference on Chemical Process Control, Kantor JC et al. (eds). AIChE & CACHE: Tahoe, CA, 1996;8396.

4. Bezergianni S, Georgakis C. Controller performance assessment based on minimum and open-loop output variance.Control Engineering Practice 2000; 8:791797.

5. Horch A, Isaksson AJ. A modified index for control performance assessment.Journal of Process Control 1999;

9(6):475483.6. Horch A. Condition monitoring of control loops.Ph.D. Dissertation, Department of signals, sensors and systems,

Royal Institute of Technology: Stockholm, Sweden, 2000.7. Harris TJ, Seppala CT, Desborough LD. A review of performance monitoring and assessment techniques for

univariate and multivariate control systems. Journal of Process Control1999; 9(1):117.8. Qin SJ. Control performance monitoring}a review and assessment. Computers and Chemical Engineering 1999;

23(2):173186.9. Huang B, Shah SL.Performance Assessment of Control Loops: Theory and Applications. Springer: New York, 1999.

10. Hagglund T. Automatic detection of sluggish control loops.Control Engineering Practice 1999; 7(12):15051511.11. Taha O, Dumont GA, Davies MS. Detection and diagnosis of oscillations in control loops. In IEEE International

Conference on Decision and Control, Kobe, Japan, 1996; 24322437.12. Thornhill NF, Hagglund T. Detection and diagnosis of oscillation in control loops. Control Engineering Practice

1997; 5(10):13431354.13. Ettaleb L. Control loop performance assessment, oscillation detection.Ph.D. Dissertation, Department of Electrical

and Computer Engineering, University of British Columbia: Vancouver, Canada, 1999.14. Miao T, Seborg DE. Automatic detection of excessively oscillatory feedback control loops. InProceedings of 1999

IEEE International Conference on Control Applications, Hawaii, 1999; 359364.15. Thornhill NF, Huang B, Zhang H. Detection of multiple oscillations in control loops. Journal of Process Control

2003; 13:91100.16. Perrier M, Roche AA. Towards mill-wide evaluation of control loop performance. In Proceedings of Control

Systems92, Whistler, BC, Canada, 1992; 205209.17. Desborough LD, Harris TJ. Control performance assessment. Pulp and Paper Canada 1994; 94(11):441.18. Jofriet P, Seppala CT, Harvey M, Surgenor B, Harris TJ. An expert system for control loop performance.Pulp and

Paper Canada 1996; 97(6):T207T211.19. Harris TJ, Seppala CT, Jofriet P, Surgenor B. Plant-wide feedback control performance assessment using an expert-

system framework. Control Engineering Practice 1996; 4(9):12971303.20. Thornhill NF, Oettinger M, Fedenczuk P. Performance assessment and diagnosis of refinery control loops.

A.I.Ch.E. Symposium Series 1998; 94(320):373379.21. Thornhill NF, Oettinger M, Fedenczuk P. Refinery-wide control loop performance assessment.Journal of Process

Control1999; 9(2):109124.22. Rhinehart RR. A watch dog for controller performance monitoring. In Proceedings of the 1995 American Control

Conference, Seattle, WA, 1995; 22392240.23. Venkataramanan G, Shukla V, Saini R, Rhinehart RR. An automated on-line monitor of control systemperformance. In Proceedings of the American Control Conference, Albuquerque, NM, 1997; 13551359.

24. Narayanaswamy G. Refinement and experimental verification of a statistical-based goodness-of-control monitor.M.S. Thesis, Department of Chemical Engineering. Texas Tech University: Lubbock, TX, 1998.

25. Li Q, Whiteley JR, Rhinehart RR. An automated performance monitor for process controllers.Control EngineeringPracticein press: accepted June 2003.

26. Butler H. Model Reference Adaptive Control. Prentice-Hall: Hemel Hempstead, UK., 1992.27. Ljung L. System Identification (2 edn). Prentice-Hall PTR: Upper Saddle River, NJ, 1999.



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