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ARTICLE IN PRESS
Control Engineering Practice 18 (2010) 279–288
Contents lists available at ScienceDirect
Control Engineering Practice
journal homepage: www.elsevier.com/locate/conengprac
A novel pattern-based approach for diagnostic controller performance monitoring
Rachelle Howard, Douglas Cooper Ã
Department of Chemical, Materials and Biomolecular Engineering, University of Connecticut, 191 Auditorium Road, Unit 3222, Storrs, CT 06269-3222, USA
a r t i c l e in fo
Article history: Receive

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A novel pattern-based approach for diagnostic controllerperformance monitoring
Rachelle Howard, Douglas Cooper
Ã
Department of Chemical, Materials and Biomolecular Engineering, University of Connecticut, 191 Auditorium Road, Unit 3222, Storrs, CT 06269-3222, USA
a r t i c l e i n f o
Article history:
Received 2 June 2009Accepted 10 November 2009Available online 6 January 2010
Keywords:
Performance monitoringPerformance indicesAutocorrelationPattern recognitionProcess control
a b s t r a c t
This paper details a novel method for monitoring the disturbance rejection performance of controllers byapplying a second-order underdamped model as a pattern recognition tool. A controller performance indexbased on the second-order model parameters classiﬁes the patterns into diagnostic categories of sluggish,well-behaved and overly aggressive. The autocorrelation function (ACF) has been used in numerousperformance assessment capacities, and this work builds on these successes by applying the patternrecognition method to automate the ACF assessment across the full range of disturbance rejectionperformance. In addition to the performance diagnostic, a pattern-based visual tuning guide is presented forretuning PI controllers to regain desired performance. The performance assessment method has been testedonnumerouscontrolloopsina25MWcogenerationpowerplantandresultsoftheapplicationarepresented.
&
2009 Elsevier Ltd. All rights reserved.
1. Introduction
Controllers keep measured process variables (PV) at set point(SP)to ensurea safeandproﬁtable plantoperation. Yet controllersare often tuned once and never readdressed, even thoughchanges, for example in operating level, feedstock and mechanicalwear, can cause the process dynamics to change and thecontroller performance to degrade.A mostnotable start to controller performanceassessment wasHarris’ (1989)work that presented a means to establish the limitof achievable performance based upon a minimum variancecontroller assumption. With this method, the quality of controlloops could be assessed without disrupting the process operation.Desborough and Harris (1992)took the work further by proposinga normalized performance index to indicate how far the process isfrom achieving minimum variance. Additional indexes and toolshave been subsequently developed to allow remote identiﬁcationof underperforming control loops. These efforts have beensummarized in several good review articles (Harris, Seppala,&Desborough, 1999;Jelali, 2006;Qin, 1998;Thornhill&Huang,
2008). The work here focuses on single-input single-outputcontrol loops; for a promising approach to performance monitor-ing and diagnosis in multiple-input multiple-output loops, thereader is referred toYu and Qin (2008a, 2008b).Performance indices have been developed to not only indicatethat performance has in general degraded, but also to diagnosecontroller behavior as oscillatory/aggressive (Thornhill, Huang,&Zhang, 2003;Thornhill&Horch, 2007;Xia&Howell, 2005;Zang&
Howell, 2007) or sluggish (Hagglund, 1999, 2005;Kuehl&Horch,
2005). The identiﬁcation of the full range of performance bymeans of a single metric remains an ongoing challenge and anovel approach is detailed in this work.The methods that do diagnose the full range require preproces-sing of the data to isolate a single deterministic load change eventforeach analysis. Salsbury’s
R-
Index method identiﬁes disturbanceevents by tracking the autocorrelation of the error signal(error=SP–PV) using a short moving window (Salsbury, 2005).When a threshold is exceeded, the
R-
index is computed on theassociated data. As shown inFig. 1, peaks in the error signal areidentiﬁed to develop an index based on areas between zerocrossings. This area ratio index is classiﬁed by comparison with anidealized error-to-load change second-order underdamped model.Visioli developed an Area Index that takes a similar approach, butusesthepeakareaaboveandbelowwherethecontrolleroutput(CO)value settles after a disturbance and classiﬁes it with the under-damped model (Visioli, 2006). The work suggests using a high-passﬁlter to distinguish qualifying events on which to compute the AreaIndex. For both methods, only one event is used in each analysis.A challenge for these indexes is that underlying process noisecauses variation in the peak and zero crossing points of the errorsignal, leading to variation in the results. To improve reliability,Salisbury suggested that the
R
-Index can be reported as anexponentially weighted moving average (EWMA) of severaldisturbance events.
1.1. This work
This paper details a method for using stochastic data toproduce a performance index that classiﬁes the full range of
ARTICLE IN PRESS
Contents lists available atScienceDirectjournal homepage:www.elsevier.com/locate/conengprac
Control Engineering Practice
0967-0661/$-see front matter
&
2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.conengprac.2009.11.005
Ã
Corresponding author. Tel.: +8604864092; fax: +8604862959.
E-mail address:
cooper@engr.uconn.edu (D. Cooper).Control Engineering Practice 18 (2010) 279–288
ARTICLE IN PRESS
controller behavior without the need to pre-process the data toisolate speciﬁc events.Inspired by Salsbury and Visioli’s use of the second-orderunderdamped model, a novel direct approach for using the modelas a pattern recognition tool is presented. Like the other methods,it is not necessary to deﬁne a model of the disturbance. Here, thedirect application eliminates the need to identify peaks and zerocrossings in the disturbance rejection pattern.Unique to the method described in this work is that processdata collected over hours and even days can be analyzed. Anexciting contribution is that the resulting controller responsepattern represents the average of all disturbance rejection activityduring the data collection period. Real processes are affected bymany disturbance types and each can present itself differentlyover the course of operation. The composite disturbance rejectionpattern holds allure as a broadly representative snapshot of overall controller performance that cannot be attained with asingle isolated disturbance event identiﬁed by data preprocessing.The autocorrelation function (ACF) is used to generate thiscomposite disturbance rejection pattern. The ACF is related to theclosed-loop process model and thus shares the patterns of theclosed-loop impulse response model (Box&Jenkins, 1976). In
several works (Desborough&Harris, 1992;Harris, 1989;Stanfelj,
Marlin,&MacGregor, 1993), it is shown how the ACF can be used
to determine if a controller is achieving minimum variance byidentifying if the ACF coefﬁcients have settled within the deadtime plus one sample time. The ACF can also be used to detectoscillations in process variables by computing a peak ratio index(Seborg&Miao, 1998). Because of its value, it has been noted that
the ACF can be visually scanned to check for oscillations or overlysluggish behavior, and it should be used as a ﬁrst-pass test beforefurther performance analysis ( Jelali, 2006).Despite its recognized uses, methodology for automating theclassiﬁcation of the patterns of the ACF for the full range of controller performance has not been presented. Here it will beshown how the innovative use of the second-order modelprovides a means to perform the automation. In addition, asolution to the problem of choosing the time window for the ACFfor the analysis is presented. The method does not require
a priori
process information, such as an estimate of dead time or processorder, and it is markedly insensitive to noise and sampling rate,thus making the method directly applicable in industrial practice.Finally, to provide further guidance to users, a tuning mapmethod is presented to direct proportional and integral (PI)controller retuning. As inVisioli (2006), the guide gives a generalindication of how to adjust the parameters, but further, it includesa visual component that enhances the usability. Tuning maps arepresented for both self-regulating and integrating (non-self-regulating) processes, as it is important to distinguish thedifferences in controlling naturally stable and unstable processes.The remainder of this paper is organized as follows. In Section2, the newly proposed pattern recognition method and theRelative Damping Index (
RDI
) algorithm are developed. In Section3, the ACF’s relationship to the closed-loop process is reviewedand an analysis of its application is presented. In Section 4, themethodology for applying the pattern recognition method to theACF is detailed. In Section 5, the results of applying the automatedACF analysis on multiples loops of a cogeneration plant arepresented. In Section 6, the tuning maps and retuning procedureare explained. And in Section 7, concluding remarks are made.
2. Extension of the underdamped model for patternrecognition
Salisbury and Visioli had each deﬁned performance indicesbased on ratios of areas calculated from isolated disturbanceevents in process data. Additionally, both presented categories of the indices to relate to a general underdamped model, deﬁned as
G
CL
ð
s
Þ ¼À
Y
ð
s
Þ
D
ð
s
Þ¼
o
2
n
ss
2
þ
2
z
o
n
s
þ
o
2
n
ð
1
Þ
where
o
n
is thenaturalfrequency and
z
is thedamping factor.Thekey feature of the second-order underdamped model is thedamping factor which deﬁnes process behavior in a continuum of categories as shown inTable 1(Cooper, 2004;Ogunnaike&Ray,
1994).The power of using the underdamped model as a patternrecognition tool emerges from its simplicity. In effect, it providesfor classiﬁcation of disturbance rejection patterns in the full rangeof overdamped (sluggish) to underdamped (aggressive). Here, alook is given to a more direct application of the model forautomating the controller pattern analysis.
2.1. Direct application of the underdamped model
Rewriting Eq. (1) in the time domain provides a clearerapproach to using the underdamped model in a direct patternrecognition application:
t
2
n
d
2
y
ð
t
Þ
dt
2
þ
2
t
n
z
dy
ð
t
Þ
dt
þ
y
ð
t
Þ ¼
Ku
ð
t
Þ ð
2
Þ
where
t
n
is the natural period of oscillation and
K
is a gain. Thenatural period of oscillation has units of time. When the model isapplied for analyzing a pattern of disturbance rejection, its valuecan be used as a characteristic measure. It will be shown to be anessential component of the model ﬁt because it allows theautomated selection of the window size on which to assess thedisturbance rejection pattern.A direct model ﬁt to a composite disturbance rejection patterneliminates the need to precisely deﬁne the response peaks and
Fig. 1.
Disturbance rejection trend features required for the calculation of the
R
-Index.
Table 1
Classiﬁcation of controller response by damping factor values.
f
ValueDynamic closed-loop response
z
4
1Pure exponential decay
—
overdamped
z
=1Fastest exponential decay
—
critically damped
0
o
z
o
1Oscillations and exponential decay
—
underdamped
z
=0Pure oscillation
—
limit of stability
z
o
0Unstable
R. Howard, D. Cooper / Control Engineering Practice 18 (2010) 279–288
280
ARTICLE IN PRESS
zero crossings. And the damping factor from the model ﬁt can bedirectly used in a performance index because of the classiﬁcationsshown inTable 1. With the disturbance dynamics unknown, theperformance index should not be based on an optimal dampingfactor, but rather on a user-deﬁned desired controller perfor-mance.
2.2. User-deﬁned performance index
For every control loop, an operator or engineer has expecta-tions of how that loop should respond to disturbances. Whilesome loops will be more important because of safety, economicsor loop interactions, each response will have a desired behaviorwhich ﬁts a subjective niche of not being ‘‘too slow’’ or ‘‘too fast.’’Therefore, to signal the operator or engineer when the loop is notperforming as desired, a user-deﬁned performance index isdeveloped.The relative damping index,
RDI
, shown in Eq. (3), is based onthe actual damping factor,
z
act
, which is computed in theunderdamped model ﬁt to the disturbance rejection pattern. Theindex can be adjusted relative to the user preference by changing
z
agg
, the limit of acceptable aggressive behavior, and
z
slug
, the limitof acceptable sluggish behavior:
RDI
¼
z
act
À
z
agg
z
slug
À
z
act
ð
3
Þ
The index can be reported in three simple categories as presentedinTable 2. The magnitudes of negative
RDI
values directly relateto the interpretation of the damping factor in that
RDI
valuesgreater than 1 indicate sluggish behavior and
RDI
values less than1 indicate aggressive behavior. Note that an exception isnecessary for the case
z
act
=
z
slug
, for which the
RDI
value shouldbe set to zero.The ﬂexibility of the benchmark allows users to make theassessment stricter for the high importance loops and looserwhere variation can be tolerated.Additionally, the user-deﬁned parameters allow for differencesin loop expectations due to the nature of the process, i.e. self-regulating vs. integrating (non-self-regulating). A self-regulatingprocess has a natural balance point such that in open-loop whendisturbances are quiet, it will settle at a particular operating level.Consider, for example, a heat exchanger. For each particularresidence time, the exchanger exit temperature will steady at aparticular value.An integrating process does not have this natural balancepoint. For example, the level of water in a boiler producing aconstant steam ﬂow will steadily rise or fall if the mass ﬂow of water in does not match the mass of steam out. Without activecontrol to make the mass ﬂow in equal the mass ﬂow out, anintegrating process will not stabilize. Integrating processescontrolled with PI controllers typically display overshoot in theset point tracking response, where as proper tuning can ofteneliminate this in self-regulating processes (Cooper, 2008).The choice of the parameters will be discussed further inSection 6.
2.3. Disturbance dynamics
When the second-order underdamped model is applied as apatternrecognition toolto analyze adisturbancerejection pattern, itis not possible to distinguish whether performance changesoriginate in the process or disturbance, only that the transferfunction, G(
s
), is different. It is stressed, however, that although thechanges cannot be automatically assigned to either the disturbanceor process, the recognition that the transfer function is differentindicates changes in the controller’s disturbance rejection perfor-mance and thus points to the need to reexamine the control loop.
3. Extracting disturbance rejection patterns with the ACF
Using an autoregressive moving average (ARMA) model torepresent the closed-loop process,Box and Jenkins (1976)showedhow the characteristic disturbance rejection response is related tothe ACF coefﬁcients. While the exact process model parametersare not determined by the ACF, it is used to determine the ARMAmodel orders,
p
and
q
. In applyingthe ACF as a process monitoringtool, it has been shown to approximate the settling time of theclosed-loop process (Desborough&Harris, 1992;Harris, 1989;
Stanfelj et al., 1993) and indicate oscillations in the process signal(Qin, 1998;Seborg&Miao, 1998;Thornhill et al., 2003). Until now
though, a pattern recognition method has not been used toautomate the analysis of the ACF to characterize the full range of controller responses.The ACF coefﬁcients are computed for a data set,
y
, according to
r
yy
ð
k
Þ ¼
1
ð
N
À
k
Þ
P
N
À
ki
¼
1
½ð
y
ð
i
ÞÀ
y
Þð
y
ð
i
À
k
ÞÀ
y
Þ
s
2
ð
4
Þ
r
yy
is calculated for
k
=0, 1,
y
,(
N
À
1), where
k
is the number of sampling periods between observations (
k
is often called the lag;Box&Jenkins, 1976),
y
is the average of the series of
y
measurements,
s
2
isthevarianceof
y
and
N
isthetotalnumberofdatasamples.Thereisnowindowingnecessary incomputing theACF, rather thecalculationis performed on several hours to days of data with a resulting ACFcoefﬁcientrelatingtoeachpossibledistancebetweendatapoints,0to(
N
À
1). Because in practical application the data series is discrete andof ﬁnite length, the unbiased expression is applied by limiting thesummation to (
N
À
k
) and dividing by (
N
À
k
).To review the ACF’s relationship to the closed-loop modelfollowing Box and Jenkins, an ARMAX model (ARMA witheXogenous disturbance input) is used to represent a
z
-domaintransfer function of a general closed-loop measured processvariable response to a disturbance change as
Y
ð
z
Þ
D
ð
z
Þ¼ð
c
0
À
c
1
z
À
1
À Á Á Á À
c
q
z
À
q
Þð
1
þ
f
1
z
À
1
þ
f
2
z
À
2
þ Á Á Á þ
f
p
z
À
p
Þð
5
Þ
The linear difference equation of this ARMAX model is
y
ð
i
Þ ¼
f
1
y
ð
i
À
1
Þþ
f
2
y
ð
i
À
2
Þþ Á Á Á þ
f
p
y
ð
i
À
p
ÞÀ
c
0
d
ð
i
ÞÀ
c
1
d
ð
i
À
1
ÞÀ Á Á Á À
c
q
d
ð
i
À
q
Þ ð
6
Þ
Nowthe ACF coefﬁcientsfor this process can be foundby applyingEq. (4) in a step-wise fashion. First, the data set is mean-centeredabout zero by subtracting the average,
y
, from each point in thedata series,
y
. To simplify the expression, substitute
^
y
ð
i
À
x
Þ ¼
y
ð
i
À
x
ÞÀ
y
.Second, multiply the equation by a sample-shifted point,
^
y
ð
i
À
k
Þ
;
to obtain
^
y
ð
i
À
k
Þ
^
y
ð
i
Þ ¼
f
1
^
y
ð
i
À
k
Þ
^
y
ð
i
À
1
Þþ
f
2
^
y
ð
i
À
k
Þ
^
y
ð
i
À
2
Þþ Á Á Áþ
f
p
^
y
ð
i
À
k
Þ
^
y
ð
i
À
p
ÞÀ
c
0
^
y
ð
i
À
k
Þ
d
ð
i
ÞÀ
c
1
^
y
ð
i
À
k
Þ
d
ð
i
À
1
ÞÀ Á Á Á À
c
q
^
y
ð
i
À
k
Þ
d
ð
i
À
q
Þ ð
7
Þ
Table 2
Interpretation of
RDI
results.
RDI
sign
RDI
magnitude Interpretation
+ Any
z
is in range, process running well
À
o
1
z
is out of range, control action is too aggressive
À
4
1
z
is out of range, control action is too sluggish
R. Howard, D. Cooper / Control Engineering Practice 18 (2010) 279–288
281
ARTICLE IN PRESS
Third, sum the product terms from
i
=1 to
N
À
k
and divide by(
N
À
k
) to obtain the theoretical expected value. At this step,taking the expected value yields covariance coefﬁcients. Deﬁnethe autocovariance values as
g
yy
(
x
), for
^
y
ð
i
À
x
Þ
^
y
ð
i
Þ
terms, and crosscovariance values,
g
yd
(
x
), for
^
y
ð
i
À
x
Þ
d
ð
i
Þ
terms. In making thesesubstitutions for the expected values, Eq. (7) becomes
g
yy
ð
k
Þ ¼
f
1
g
yy
ð
k
À
1
Þþ Á Á Á þ
f
p
g
yy
ð
k
À
p
ÞÀ
c
0
g
yd
ð
k
ÞÀ
c
1
g
yd
ð
k
À
1
ÞÀ Á Á Á À
c
q
g
yd
ð
k
À
q
Þ
ð
8
Þ
For the ﬁnal step of the ACF coefﬁcient computation, normalizethe equation by dividing by the autocovariance at
k
=0 i.e. thevariance of
y
,
s
2
. The resulting equation shows how autocorrela-tion coefﬁcients relate to past autocorrelation coefﬁcients andpast cross covariance coefﬁcients:
r
yy
ð
k
Þ ¼
f
1
r
yy
ð
k
À
1
Þþ Á Á Á þ
f
p
r
yy
ð
k
À
p
ÞÀ
c
0
g
yd
ð
k
Þ
s
2
À
c
1
g
yd
ð
k
À
1
Þ
s
2
À Á Á Á À
c
q
g
yd
ð
k
À
q
Þ
s
2
ð
9
Þ
This equation for
r
yy
(
k
) has the same model coefﬁcients as theprocess in Eq. (6). Because they are the same, each ACF coefﬁcientcomputed using Eq. (4),
r
yy
(
k
), contains the same weighting of process and disturbance values as the process point,
y
(
i
), in thelinear difference model. Thus, a plot of the ACF coefﬁcients willreveal the pattern of the closed-loop disturbance rejectionbehavior. While each
r
yy
(
k
) can be computed using Eq. (4), thedisturbances are unmeasured, so the
g
yd
(
k
) terms cannot bedirectly calculated. Therefore, the ACF cannot lead directly to acomplete process model.
3.1. Example ACF implementation
A typical ACF pattern, an example of which is shown to thebottom right inFig. 2, begins at the peak of the disturbancerejection trajectory. This peak corresponds to the maximumcorrelation between points. The normalization by the variance, asin Eq. (4), forces this initial point of the ACF, when
k
=0, to alwayshave a value of one. As each ACF coefﬁcient represents itscorrelation to adjacent points, coefﬁcients at increasing values of
k
will diminish to zero unless components of the signal are notrandom. As established in the theoretical argument in theprevious section, the manner in which the ACF settles to zero isthe same pattern seen when the controller drives the process backto the set point during a disturbance rejection event.Fig. 2shows level data from an industrial boiler steam drumthat is part of the cogeneration facility at the University of Connecticut. Occasionally there are large disruptions due to loadchanges, ﬂow rate changes, and blow down.The ﬁgureshows howa full day’s worth of process data is analyzed to yield an ACFdisturbance rejection pattern that matches the pattern of theprocess during a large disturbance event.The ACF can be computed on a set of data ranging from hoursto many days. As with any data analysis, a shorter window of datawill allow drastic changes to be identiﬁed quickly. But since theimportance of a performance monitoring tool is not to createalarms for fault detection, but rather to survey how well theprocess is maintaining set point and identify degrading controllerperformance. Periodic analysis of large data sets can reveal thevariety of disturbances that occur during operation.
3.2. ACF sensitivity
The ACF was ﬁrst tested with simulations under a wide varietyof conditions to ensure that it serves as a reliable means to extractthe disturbance rejection patterns. Hundreds of simulations wererun for both self-regulating and integrating processes using ﬁrst,second and third order transfer functions controlled with PI andPID controllers. Within each subset of transfer functions, theprocess gain, time constant and dead time were varied to test acomplete range of disturbance rejection behaviors. Additionally,to replicate industrial conditions, the noise level and samplingrate were varied.The ACF results presented inFig. 3highlight the ﬁndings of thestudy. In these representative cases, nine process transferfunctions were each tested under ﬁve conditions: moderateconditions, fast sampling, slow sampling, no noise and highlevel of noise, as will be further deﬁned. As real processes undergochanges while the controller tuning is ﬁxed, here the tuning
Fig. 2.
Steam drum boiler level data captured during a large disturbance event is scaled for direct comparison to the ACF computed from a series of regulatory data withoutsuch signiﬁcant events.
R. Howard, D. Cooper / Control Engineering Practice 18 (2010) 279–288
282

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