INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING Int. J. Adapt. Control Signal Process. 2003; 17: 635–662 (DOI: 10.1002/acs.770)
Process and controller performance monitoring: overview with industrial applications K. A. Hoo1,n,y, M. J. Piovoso2, P. D. Schnelle3 and D. A. Rowan3 1 2
Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA School of Graduate Professional Studies, Penn State University, Malvern, PA 19355, USA 3 DuPont Engineering Technology, The DuPont Company, Wilmington, DE 19898, USA
SUMMARY Complex processes depend on regulatory control systems to operate. Changes made to the sensor system and the process itself as well as the natural degradation over time of the equipment may cause some controllers’ and process performance to suffer. Since there are hundreds of controllers in most complex processes, it is virtually impossible to monitor their performance manually. This work reviews the state-ofthe-art in controller performance monitoring including both feedforward and feedback control. The experiences of the DuPont Company in the deployment of a system for performance monitoring are presented. Information as to how such a monitoring system is implemented in the operation of a chemical process and examples of its operation are included. Copyright # 2003 John Wiley & Sons, Ltd. KEY WORDS:
compliance metrics; chemical processes; real-time monitoring; controller performance
1. INTRODUCTION Most modern industrial plants have hundreds and even thousands of automatic control loops. These loops can be simple proportional-integral-derivative (PID) or more sophisticated modelbased linear and non-linear control loops. It has been reported that as many as 60% of all industrial controllers have performance problems [1]. Having an automated means of detecting when a loop is not performing well and then diagnosing the root cause is essential because they play a vital role in product quality, safety and ultimately economics. Some of the obstacles that prevent this automatic assessment, from being a part of the day-today maintenance program include a lack of: user-friendly interface, readily understandable report generation, diagnosis information in text form, a single composite index ranking the loop performance and reliable computational software tools. In addition, education of the operations staff is essential to make full use of some of the currently available time and frequency methodologies.
n
Correspondence to: K. A. Hoo, Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA y E-mail:
[email protected]
Copyright # 2003 John Wiley & Sons, Ltd.
Accepted June 2003
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When considering large-scale controller performance assessment, Paulonis and Cox [2] of Eastman Chemical Company, considered the following design issues: * * * * *
*
Ranking loops by performance. Preliminary problem diagnosis for poorly performing loops. Interfaces to all distributed control systems (DCS). Minimal client and server configurations. Reports which could be generated interactively or predefined reports which could be e-mailed to users on a schedule. A friendly user interface, providing accessibility to all company personnel.
They also provided a list of assessment analysis that they expect in any automated process/ controller performance tool. These included: 1. 2. 3. 4. 5. 6.
Time-series trend. Descriptive statistics for controller set point, measurement, output and error. Controller idle index. Cross-correlation between error and output. Harris-style extended horizon performance index. Diagnosis information in text form.
Bialkowski [3] and Kozub [4] provide an industrial perspective on controller performance challenges. Some of the common causes of poor performing loops are [5, 6]: 1. 2. 3. 4. 5. 6. 7.
Incorrect tuning. Changing process dynamics (transitions, unmeasured disturbances). Limited controller output range. Large deadtimes or inaccurate determination of the deadtime. Incorrect sampling interval. Incorrect controlled and manipulated variable pairings. Poor hardware (sensors, actuators) maintenance.
1.1. Univariate measures Performance assessment is concerned with the analysis of available process data against some benchmark. Many researchers have presented different benchmarks to assess univariate feedback controllers [7–10], univariate feedback and feedforward controllers [6, 11], and multiple loop controllers [12, 13]. The most used benchmark is the minimum variance controller (MVC) first introduced by Harris [9] for single loop feedback controllers. It provides a theoretical lower bound on the closed-loop process output variance. The calculation of the MVC assumes that the process can be represented adequately by a linear time-invariant (LTI) transfer function model with additive disturbances. By fitting an autoregressive integrated moving average (ARIMA) time-series model to either open-loop or closed-loop routine output data, this lower bound on the performance can be calculated. Astrom [7] used the measures of bandwidth, rise time and peak errors for servo and regulatory PID controllers to assess achievable performance. Shinskey [8] used the integral of Copyright # 2003 John Wiley & Sons, Ltd.
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the absolute error (IAE) to assess the performance of PID controllers. Desborough and Harris [5] introduced a normalized performance index, which gives the ratio of the variance in excess of that which can be theoretically achieved if the controller was a MVC. They also connected this measure to that of the squared correlation coefficient, r2 ; that is usually calculated in multiple regression analysis. Huang et al. [10] developed an efficient, stable filtering and correlation method (FCOR) to estimate the minimum variance benchmark of Harris [9] and the normalized performance index of Desborough and Harris [5]. Stanfelj et al. [6] and Desborough and Harris [11] addressed the issue of feedback and feedforward control loops. The idea is that if the feedback controller is already providing satisfactory performance but this still exceeds the overall performance then alternatives such as feedforward control can be used to compensate for the disturbances that are measurable Stanfelj et al. [6] developed a cross-correlation test to assess the performance of both types of control loops. Desborough and Harris [11] contributed a procedure to calculate the minimum variance performance when both types of control strategies are present. Their procedure is an extension of the methods in the single loop, feedback case. DeVries and Wu [14] and Kozub and Garcia [15] proposed a closed-loop potential (CLP) factor that quantifies the performance of the controller relative to the minimum variance benchmark. The CLP is based on a normalization of the closed-loop output error variance, which is similar to the normalized performance index of Desborough and Harris [5]. When a model-based feedback control strategy is used, Harris et al. [12] and Kozub [4] propose an extended horizon performance index. Unlike the normalized performance index, this one does not require exact knowledge of the process deadtime and permits an evaluation of long settling times. An interpretation of the extended horizon performance index is that it corresponds to the square of the correlation between the current error and the least-squares estimate of the prediction made in the past. If there is no correlation, then the feedback controller’s performance is deemed satisfactory because the disturbance was rejected after the deadtime elapsed. 1.2. Multivariate measures In the realm of multivariate controllers of multiple-input multiple-output (MIMO) processes, Harris et al. [12] employed a multivariate spectral factorization and the solution of a multivariate Diophantine identity. Huang and Shah [13] extended their single-input singleoutput (SISO) performance measures to address MIMO controller performance by adding the concept of an interactor matrix or time-delay matrix [16]. In both of these contributions, knowledge of all process time delays is a requirement. Shah et al. [17] have also suggested the use of linear quadratic Gaussian (LQG) performance benchmarks and a design performance benchmark. It has been pointed out by many researchers that the implementation of a MVC may not be desirable because (i) it may require excessive controller action, (ii) may not be robust to modelling errors and (iii) the process dynamics may not be invertible. Research related to these issues can be found in References [6, 13, 18, 19]. If the goal is to develop an automated on-line technology then features such as: 1. computational simplicity, 2. reliability, 3. easy interpretability, Copyright # 2003 John Wiley & Sons, Ltd.
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4. prioritization in the case of multivariable process systems, and 5. assessment of controller re-design versus controller re-tuning ought to be a part of the methodology. There are commercial software available to provide automated process loop monitoring, tuning and diagnostics. For instance, the INTUNETM software tools by Control Soft (Cleveland, OH) automatically generate PID parameters, re-tunes loops for optimal performance, and monitors multiple PID loops to determine how complete systems are being controlled. Loop Scout1 by Honeywell Hi Spec Solutions (Minneapolis, MN) provides a noninvasive, systematic identification of controller performance. It makes use of advanced statistics, internet technology and world-wide expertise to diagnose and resolve regulatory control problems. While the promise of these tools do meet a large number of the features in the above list, at least two large chemical companies chose to develop in-house products rather than use a commercial tool. Eastman Chemicals and DuPont evaluated the Loop Scout1 product made by Honeywell in 1999. The former chose not to use this tool for three primary reasons [2]: * *
*
Automated data collection was limited to Honeywell control systems. Substantial amounts of process data would have to be sent to Honeywell, requiring complicated approvals. The cost to assess loops worldwide was prohibitively high given the emphasis on reducing business expenses.
The paper is both a review of the more common existing measures of process and controller performance used in the chemical industries applications and a brief but very informative presentation of a real system, the Performance SurveyorTM ; developed and implemented by the DuPont Company to address these critical issues. The paper is organized as follows. In Section 2, feedback control loop performance measures for univariate systems are reviewed. A case study of an ideal continuous-stirred tank reactor (CSTR) and data from an industrial polymer reactor are used to demonstrate some of the more important concepts. Section 3 introduces model-based feedback control and the issues associated with monitoring loop performance in the presence of plant/model mismatch. A case study is presented to highlight these issues. Section 4 introduces performance monitoring of both feedforward/feedback control loops. Lastly, Section 5 describes the performance monitoring needs of DuPont, the eventual objectives for their in-house performance monitoring tool and a brief overview of an application to illustrate the success of their performance monitoring program.
2. UNIVARIATE SYSTEMS: THE FEEDBACK CONTROLLER 2.1. The minimum variance controller Consider the block diagram of a feedback control system shown in Figure 1. Assume that this single-loop system can be represented adequately by a linear time-invariant, discrete transfer function model and additive disturbance. The process transfer function is given by Gp and Gc Copyright # 2003 John Wiley & Sons, Ltd.
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Gd (q-1) + u(k)
+ Gc (q-1)
r(k)
639
a(k)
d(k)
+ Gp (q-1 )
y(k)
-
Figure 1. Block diagram of a feedback controller. rðkÞ; set point; yðkÞ; output variable; dðkÞ; unmeasured disturbance; uðkÞ; controller output.
and Gd denotes the controller and disturbance transfer functions, respectively, yðkÞ ¼ Gp ðq1 Þuðk bÞ þ dðkÞ ¼
oðq1 Þ uðk bÞ þ dðkÞ dðq1 Þ
ð1Þ
where q1 is the backward shift operator and yðkÞ and uðkÞ are deviations of the measured process output and controller outputs, respectively, from their nominal operating values; dðkÞ is a bounded disturbance; oðq1 Þ and dðq1 Þ are polynomials in the backward shift operator; and b 1 is the number of whole periods of delay in the process. The term dðkÞ is assumed to represent all unmeasured disturbances acting on yðkÞ and it may be deterministic or stochastic. Let dðkÞ be given as a linear function of past values of a statistically independent random sequence of variables, faj g; dðkÞ ¼ Gd ðq1 ÞaðkÞ ¼
yðq1 Þ aðkÞ fðq1 Þ
ð2Þ
The terms yðq1 Þ and fðq1 Þ are assumed to be stable polynomials. For constant reference inputs, the deviations of the outputs from their steady-state values are given by yðkÞ ¼
aðq1 Þ 4 aðkÞ ¼ Cðq1 ÞaðkÞ bðq1 Þ
yðkÞ ¼ ½c0 þ c1 q1 þ cb qb þ þaðkÞ
ð3Þ
where cj is the jth impulse weight [20]. The series in Equation (3) is convergent if the closed loop between yðkÞ and dðkÞ is stable. Because of the delay term, qðb1Þ ; the first b terms in Equation (3) are identical to those computed from the disturbance transfer function Gd ðq1 Þ and can be interpreted as system invariant. Thus, only terms at lag b and beyond are affected by the current controller action. The reason for this is seen as follows. Once a disturbance appears at the output, it is feedback to the controller and the controller makes a correction. However, because of the delay, that corrective action has no effect on the output for b time intervals into the future. No disturbance compensation can occur at the output until the deadtime of the system has expired. Desborough and Harris [5] offer the following interpretation of Equation (3) the first b terms form the b-step ahead forecast errors and the remaining ones are the b-step ahead forecast. The variance of the controlled variable can be calculated by squaring Equation (3) and then applying the expectation operator Efg; s2y ¼ EfyðkÞg ¼ ½c20 þ c21 þ c2b1 þ þs2a Copyright # 2003 John Wiley & Sons, Ltd.
ð4Þ
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where s2a ¼ EfaðkÞ2 g Equations (3) and (4) verify that the variance of the controlled variable is related to the feedback process dynamics, the disturbance model, the controller and the variance of faj g: If the feedback controller is a MVC then the b-step ahead forecast (terms at and beyond b) equals zero and the output variance is given by s2mv ¼ ½c20 þ c21 þ c2b1 s2a
ð5Þ
This controller then rejects the prediction of the disturbance after the deadtime has elapsed leaving only the prediction errors. Thus, the controlled variable under minimum variance control will depend on only the most recent b past disturbances 2 ymv ¼ ½c20 þ c21 þ c2b1 s2a
ð6Þ
The finite stochastic process in the above equation is called a moving average process of order b: It then follows that any controller that is not minimum variance must inflate the variance, that is s2y ¼ s2mv þ s2y#
ð7Þ
where s2y# means the variance of the b-step ahead forecast. Harris [21] has shown that this MVC also minimizes a wide class of symmetric and asymmetric objective functions. 2.2. Performance measures An important consideration in the calculation of the theoretical minimum variance, as presented by Harris [9], is that routine process data, with or without feedback control, are used rather than specially designed tests. The only requirement is that the number of observations is large and representative of the process. Kozub and Garcia [15] and others have commented that the data sequence ought to contain information on important disturbance upsets to avoid using data corrupted by unusual process operations. Furthermore, too short a data record may result in statistical estimates whose variability renders them useless while too lengthy a data set may lead to erroneous results because different response characteristics are contained in the data set. How to select the appropriate data window size remains an open research issue. Thornhill et al. [22] discuss the related issues of sampling interval, model order and data compression on data selection and analysis. The theoretical autocorrelations of the moving average process described by Equation (3) at and beyond lag b are zero when the controller is minimum variance. For real, representative data of size n; the sample autocorrelation value must be compared to some statistical confidence interval because, in practice, the autocorrelations values are never truly zero. If there exist many large autocorrelations beyond lag b; then it can be concluded that the controller’s performance deviates substantially from the minimum variance performance bounds. On the other hand, if only a few values at and beyond lag b are significant, then the performance is close to the lower bound. See Figure 5, which shows the autocorrelation function for an example process. The confidence bands are the 95% or two standard deviation probability limits. Methods to calculate the autocorrelation function can be found in Reference [23]. Copyright # 2003 John Wiley & Sons, Ltd.
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The ratio of the output variance to the theoretical variance under MVC is given by s2y s2mv
s2y s2mv
1
ð8Þ
where s2y is the sampled variance and s2y is the actual variance of the output, where Equation (8) indicates the maximum improvement possible by modification of the controller. This ratio is related to the performance index defined by Desborough and Harris [5] xðbÞ ¼
s2y þ y%2 s2mv
ð9Þ
where y% is the mean of the output. When y%2 term is zero, Equations (8) and (9) have the same meaning and it is clear from Equation (8) that xðbÞ51 but has no upper bound. It is highly possible that the controller is providing minimum variance but its performance exceeds the overall performance (product or process specification). This means that reductions in the output variance can only be realized by modifying the process. In other words, no other modification of the present controller structure will reduce the output variance. The only recourse is to change the system structure. For instance, 1. 2. 3. 4. 5.
Reduce the deadtime. Reduce the variance of the disturbance. Eliminate disturbances through process modification. Introduce feedforward control on measured disturbances. Change manipulated variables.
When the performance is significantly greater than the lower bound, further analysis must be done to ascertain the cause for the variance inflation. This is the diagnosis or third level in the monitoring flowchart given in Stanfelj et al. [6]. Some common causes are: * * * * * * *
De-tuning or poor tuning. Errors in the process model. Wrong sampling interval. Changing process dynamics. Saturation of manipulated variables. Errors in input/output models. Faulty sensors and actuators.
The normalized performance index of Desborough et al. [5] is defined as follows: ZðbÞ ¼ 1
s2mv c20 þ c21 þ þ c2b1 ¼ 1 s2y c20 þ c21 þ c2b1 þ þ
ð10Þ
This index represents the fractional increase in the variance of the output that arises from not implementing an MVC. Further, unlike xðbÞ; ZðbÞ is bounded between ½0; 1: When ZðbÞ ¼ 0; the controller is an MVC. At the other extreme, ZðbÞ approaches a value of one. The closer that ZðbÞ gets to one, the larger the variance of the process output, y; relative to its best possible performance, s2mv : Thus, although theoretically, the controller is capable of eliminating some of the effect of disturbances on the output, it fails to perform as might be expected. This index was shown to be analogous to the multiple coefficient of determination, r2 ; encountered in multiple regression analysis [5]. While r2 indicates the predictability of a times Copyright # 2003 John Wiley & Sons, Ltd.
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series given past information, ZðbÞ provides a measure of the predictable component in the process output, b steps into the future. Calculation of ZðbÞ can be done using linear regression, autocorrelation approach or recursive least squares [24] that are biased (order 1=n), and for systems that are or nearly unstable, the estimates themselves may be unstable. Huang et al. [10] have also introduced a stable FCOR to calculate s2mv and ultimately ZðbÞ: These estimates are also biased. Alternatively, ZðbÞ; might be estimated from the data itself by solving a least-squares problem [5]. First, a set of parameters, faj g; are found using a least-squares method. From these parameters, an estimate of ZðbÞ can be generated y* ¼ X* a 2
y*n
3
7 6 6 y*n1 7 7 6 7 y* ¼ 6 6 . 7; 6 .. 7 5 4 y*bþm
2
y*nb
6 6 y*nb1 6 X* ¼ 6 6 . 6 .. 4 y*m
y*nbs
...
y*nb2
...
.. .
.. .
y*m1
...
y*nbmþ1
3
7 y*nbm 7 7 7 7 .. 7 . 5 y*1
where b is the time delay and m is an estimate of the autoregressive order of the process. Having faj g allows Zb to be calculated using the following equation: ZðbÞ ¼
n b m þ 1 ðy* X* aÞT ðy* X* aÞ n b 2m þ 1 y*T y* þ y*2
ð11Þ
where y% is the mean value of the controlled variable. The extended horizon performance index [4, 12, 13] is defined as Zðb þ hÞ ¼ 1
c20 þ c21 þ c22 þ c2bþh1 c20 þ c21 þ c22 þ c2b þ
ð12Þ
which gives the proportion of the variance arising from non-zero impulse coefficients cj ; j > ðb þ hÞ: The extended horizon performance index is said to correspond to the square of the correlation between the current error and the least-squares estimate of the prediction made at ðb þ hÞ control periods in the past. Similar to ZðbÞ; Zðb þ hÞ 0 means that the controller is performing satisfactory since at current sample k; yðkÞ is not well predicted by the samples older than ðb þ hÞ in the past. Thus, knowledge about the output at times older than ðb þ hÞ gives no indication to discern what is occurring at the present time. A controller regulating a process should be able to eliminate disturbances satisfactorily so that at some point, not too far in the past, the process behaviour is unrelated to the events at the present time. The use of Zðb þ hÞ also implies that minimum variance control may not be feasible. The MVC will often generate large control moves in order to eliminate the disturbance effect at a point in the future after the deadtime. Such large moves are undesirable because of their detrimental effect on actuators. Furthermore, assumptions of approximate linearity that are made in the design of MVC are more likely to be violated significantly by large control efforts. Unlike ZðbÞ; Zðb þ hÞ does not require precise knowledge of the process delay. Ericksson et al. [25] examined the use of ZðbÞ and Zðb þ hÞ for proportional-integral (PI) and PID controllers. Copyright # 2003 John Wiley & Sons, Ltd.
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They concluded that ZðbÞ and Zðb þ hÞ indicate performance deterioration but that Zðb þ hÞ also permits diagnosis of long settling times which is not always possible with ZðbÞ: For such systems, ZðbÞ may be large, and the system is well controlled. Because of constraints around safety and limitations on actuator moves, the closed loop might be far from minimum variance. Using ZðbÞ; the controller may be monitored because h can be chosen according to the desired settling. When the controller performance deteriorates, a greater change will be observed in the value of Zðb þ hÞ as compared to that of ZðbÞ: The closed-loop potential factor given by [14, 15] CLP ¼
s2mv s2y
ð13Þ
is another normalization of the closed-loop output error variance. From Equation (10) CLP is ð1 ZðbÞÞ. Thus, CLP 2 ½0; 1: When CLP is zero, the existing controller can be improved. A value near one means that improvements can only be obtained by process modifications such as the addition of feedforward control, if possible. See the earlier discussion. 2.3. Example: two-state CSTR A reduced order, dimensionless model of a first order, exothermic reaction of A ! B occurring in a jacketed CSTR is given by [26] x’ 1 ¼ fx1 kðx2 Þ þ qðx1f x1 Þ x’ 2 ¼ bfx1 kðx2 Þ ðq þ dÞx2 ðx1f x1 Þ þ dx3 þ qx2f x2 k ¼ exp 1 þ x2 =d with x1 the dimensionless conversion, x2 the dimensionless reactor temperature and x3 the jacket temperature. The controlled variable is x2 ; and x3 is the indirect manipulated variable. The steady operating conditions are xn1 ¼ 0:1039; xn2 ¼ 0:4875; xn3 ¼ 0:6567: From a practical point of view, the jacket temperature is cascaded to the flow rate of the cooling fluid, and with the assumption of fast jacket dynamics, this permits the two-state model representation. In the next section, a modified version of this system will be used to demonstrate model-based feedback control. Table I provides the definition of the parameters and their nominal values. A step test of size 0:1 was imposed on the non-linear system. The results are shown in Figure 2. The positive and negative step responses were first normalized by the magnitude of the changes in the manipulated variable. From the two normalized responses, an average response Table I. Dimensionless parameters in the CSTR example. Parameter
Definition
f ¼ 0:072 b¼8 d ¼ 0:3 g ¼ 20 q¼1 x1f ¼ 1 x2f ¼ 0
Damko. hler number Heat of reaction Heat transfer coefficient Activation energy Flow rate Feed A concentration Feed temperature
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Figure 2. Performance of the two-state CSTR reactor temperature when subjected to step inputs of magnitude 0:1:
Figure 3. Fit of the temperature model to the step response data of the two-state CSTR.
was computed as 0.5((positive step response – original steady state) – (negative step response – original steady state)). This was then used to compute a model for the process. The identified process transfer function model is found to be Gp ðqÞ ¼
0:003739z 0:003739 z 0:9512
Figure 3 shows the fit of this model’s response (labeled by –*–) to an average of the step response data (solid line). From these data, a deadtime of three samples was found. The Copyright # 2003 John Wiley & Sons, Ltd.
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Figure 4. Reactor temperature of the two-state CSTR with added white noise.
response appears to be first order. A PI controller is designed to provide closed-loop poles at 0:5; 0:5: This should yield a closed-loop performance with a settling time of 90 min; a damping coefficient of 1 (critically damped), and a closed-loop time constant of 2 h: The controller has a gain of 6.5232 and integral time constant of 2 h: Note, the PI controller was not optimally tuned. Figure 4 shows the reactor temperature with a filtered white noise disturbance. The disturbance is described by dðkÞ ¼
1 0:9q1 aðkÞ 1 0:99q1
where aðk) is normally distributed white noise with variance of 0.001. The estimated value of Zðb ¼ 3Þ is calculated using a linear regression approach. The estimated value essentially stops declining with a model order of m ¼ 1: This concurs with the model assumption of first order. The estimate of s2y is given by Desborough and Harris [5] as ðy* X* a# ÞT ðy* X* a# Þ ðn b 2m þ 1Þ The value estimated from the data is 0.0011, which is consistent with the variance of the white noise of 0.001. The autocorrelations are shown in Figure 5. The values of for model orders that varied over the range ½1; 4 are shown in Table II. 2.4. Industrial example Figure 6 illustrates data taken from a polymer reactor. It constitutes 15 h of composition data sampled once per minute. This period constitutes normal operation without any major process upset. The process is controlled by a set of PID controllers. For this process, b ¼ 5 and m ¼ 2 provides the minimum. For this situation, ZðbÞ ¼ 0:1007: Figure 7 is the autocorrelation of the output. Copyright # 2003 John Wiley & Sons, Ltd.
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Figure 5. Autocorrelation values of the two-state CSTR example with 95% confidence bands.
Table II. Model orders versus estimate of the normalized performance index. m
1
2
3
4
Z# ðbÞ
0.5894
0.5930
0.5937
0.5962
Controlled output 15.5 15.4 15.3 15.2 15.1 15 14.9 14.8 14.7 14.6 14.5
0
100
200
300
400
500
600
700
800
900
Time
Figure 6. Closed-loop composition data of an industrial polymer reactor. Copyright # 2003 John Wiley & Sons, Ltd.
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Figure 7. Autocorrelation plot of the industrial composition data. The data are within the 95% confidence limit after 33 lags.
3. MODEL-BASED FEEDBACK CONTROL In the preceding discussion, Equation (3) was obtained assuming a constant set point, uncorrelated disturbances and a feedback control structure such as a PID (see Figure 1). The reality is that the disturbances may have significant autocorrelation and the set point may be varying. Stanfelj et al. [6] considered this problem using the model-based feedback concept, shown in Figure 8, to design a feedback controller between the controller output and the model prediction error. They showed that in this situation, it is not possible to attribute poor control performance definitively to modelling errors or to poor tuning using only routine operating data. As such, they suggest the generation of additional data whereby perturbations in the set point are made to facilitate model development. These perturbations must be rich enough to provide sufficient excitation [24]. To observe this, consider the block diagram in Figure 8, where the model prediction error, em ðkÞ; between the process, Gp ðq1 Þ and an approximate model of the process, Gm ðq1 Þ; is used to design the feedback controller, Gc ðq1 Þ; em ðkÞ ¼
Gd ðq1 Þ aðkÞ 1 þ Gc ðq1 Þ½Gp ðq1 Þ Gm ðq1 Þ
ð14Þ
with the other definitions as before with the reference input rðkÞ, held constant. The crosscovariance between uðkÞ and em ðkÞ is given as Gc ðq1 ÞGd ðq1 Þ aðkÞ EfuðkÞem ðk þ hÞg ¼ E 1 þ Gc ðq1 Þ½Gp ðq1 Þ Gm ðq1 Þ Gd ðq1 Þ aðk þ hÞ ð15Þ 1 þ Gc ðq1 Þ½Gp ðq1 Þ Gm ðq1 Þ Copyright # 2003 John Wiley & Sons, Ltd.
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Gd(q-1) + +
u(k) Gc(q-1)
r(k)
a(k)
d(k)
+ Gp(q-1)
y(k)
-
+
em(k)
-1
Gm(q )
-
Figure 8. Block diagram of a model-based feedback controller. eðkÞ; feedback model prediction error.
When the model is perfect, that is, Gm ðq1 Þ ¼ Gp ðq1 Þ; the modelling error is just the disturbance signal dðkÞ: If dðkÞ is white noise, the cross-correlation between the controller output and the modelling error will be zero. More importantly, in this case, any non-zero crosscorrelation values can be attributed to model error. In a more realistic scenario, dðkÞ is not white noise. In this case, it is impossible to determine whether unacceptable controller performance is due to modelling errors or to poor controller settings. Consider set point changes that are independent of future disturbances as a way of deciphering between modelling errors and poor choice of controller parameters. The feedback model prediction error due to set point changes is a function of the disturbance model, the controller and the approximate model of the process, Gm ðq1 Þ em ðkÞ ¼ where rðkÞ correlation correlation tuning and
Gd ðq1 ÞaðkÞ þ Gc ðq1 Þ½Gp ðq1 Þ Gm ðq1 ÞrðkÞ 1 þ Gc ðq1 Þ½Gp ðq1 Þ Gm ðq1 Þ
ð16Þ
is the set point. When there is plant/model mismatch, there will be non-zero values between the prediction error and the set point. When there is no significant present, then other reasons for unsatisfactory performance may be poor controller even poor pairing between the controlled and manipulated variables.
3.1. Example: three-state CSTR The example CSTR, with parameters defined in Table I, was modified to include a third state, the jacket temperature, that can be described by x’ 3 ¼ d1 U ðx3f x3 Þ þ dd1 d2 ðx2 x3 Þ where U is the flow rate ratio, d1 ¼ 1 is the volume ratio of the reactor to the jacket, d2 ¼ 1 is the density-heat capacity ratio and q ¼ 1: A model was identified in a similar fashion as in the twostate CSTR case. Here, the model identified is a third-order transfer function Gm ðsÞ ¼
s3
ð0:1030s þ 0:1149Þ þ 2:924ss þ 2:6810s þ 0:7566
A controller was developed from this model to have all the closed-loop poles at 1: The transfer function of this controller is given by Gc ðsÞ ¼
s3 2:295s2 2:681s 0:7566 0:103s3 þ 0:3209s2 þ 0:3328s þ 0:1149
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Figure 9. Closed-loop performance of the three-state CSTR reactor temperature when model-based feedback control is applied in the presence of an unmeasured disturbance.
The controller and model are implemented digitally. This is done by simulation of the analogue systems. There are a number of techniques for doing this, and in this case it was implemented using Tustin approximation. Thus, the transfer functions for the model and controller are given by Gp ðq1 Þ ¼
103 ð0:2424q3 0:2680q2 þ 0:1912q1 þ 0:2168Þ q3 þ 2:7710q2 þ 2:5669q1 0:7952
Gc ðq1 Þ ¼
9:354q3 þ 25:9213q2 24:0122q1 þ 7:4391 q3 þ 2:7039q2 þ 2:4369q1 0:7321
The same disturbance used in the two-state CSTR case is also applied here. Using the structure in Figure 8, the closed-loop response for this three-state CSTR, with a model given by Gp ðq1 Þ and controller by Gc ðq1 Þ; is shown in Figure 9. Observe that the response reflects the disturbance but appears to remain about the nominal value (0.4875). The auto and cross-correlations for this example are shown in Figures 10 and 11. Note that the autocorrelation for the reactor temperature after lag 14 are within the 95% confidence bands. The estimated values of ZðbÞ using the same procedure and for the same range of the model order yield values shown in the Table III. Note the improvement in the values of ZðbÞ: The estimate of the minimum variance is essentially unchanged from before. It is 0.0012. The spike at 0 lag is due to the fact that the controller has direct feed through.
4. UNIVARIATE CASE: FEEDFORWARD/FEEDBACK Harris et al. [19] and Eriksson et al. [25] partition the disturbance signals into measured and unmeasured ones. The former are said to be deterministic and can be represented by step or Copyright # 2003 John Wiley & Sons, Ltd.
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Figure 10. Autocorrelation of the three-state CSTR reactor temperature with the 95% confidence bands.
Figure 11. The cross-correlations of the three-state CSTR between the reactor temperature and the cooling fluid flow rate. Table III. Model orders versus estimate of the normalized performance index. m
1
2
3
4
Z# ðbÞ
0.0362
0.0928
0.1052
0.0868
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GD(q-1)
D(k) Gff(q-1)
uff(k) +
r(k)
+
+ d(k)
Gd(q-1)
+
a(k)
y(k)
u(k) -1
+ -1
+ ufb(k)
Gc(q )
Gp(q ) + Gm(q-1)
-
+ G^ D(q-1) em(k)
Figure 12. Block diagram of a model-based feedforward/feedback control strategy. DðkÞ; measured disturbance; uff ðkÞ; feedforward controller output; ufb ðkÞ; feedback controller output.
pulse functions, while the latter are stochastic. This separation allows a comparison of the deterministic component to a desired reference trajectory [4]. In fact, Kozub [4] provides an example in which the desired reference trajectory is a random walk. The closed-loop error can be represented by J X aj ðq1 Þ aðq1 Þ em ðkÞ ¼ aðkÞ þ fj ðkÞ ð17Þ bðq1 Þ bj ðq1 Þ j¼1 where fj ðkÞ is the deterministic portion of the disturbance. It is not difficult to compute the performance bounds and the autocorrelation function. Furthermore, the desired trajectory can be used to fit the parameters of the model. When the feedback controller is operating within its performance bound, further reductions in process variability may be achieved by implementing feedforward control [6, 11]. Consider Figure 12 where the controller output consists of two components J X uðkÞ ¼ ufb ðkÞ þ uff;j ðkÞ j¼1
uff;j ðkÞ ¼ Gff;j ðq1 ÞDj ðkÞ
ð18Þ
a feedback and a feedforward part. Dj ðkÞ represents the jth deviation of the measured feedforward disturbance from some average value, and Gff;j ðq1 Þ is the model for the jth feedforward controller. For a single input disturbance, the output is given by yðkÞ ¼ ½Gd ðq1 Þ þ Gff ðq1 ÞGp ðq1 ÞDðkÞ þ dðkÞ þ Gp ðq1 Þufb ðkÞ
ð19Þ
which is driven by the unmeasured and measured disturbances. When the feedforward controller compensates for the measured disturbance then Equation (18) becomes yðkÞ ¼ dðkÞ þ Gp ðq1 Þufb ðkÞ
ð20Þ
Using the cross-correlation test of Stanfelj et al. [6], the expectation is that there is no significant correlation between the future values of the output and the measured disturbances, Copyright # 2003 John Wiley & Sons, Ltd.
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DjðkÞ whenever the measured and unmeasured disturbances themselves are independent of each other. Thus, the feedforward controller is well within acceptable performance bounds. Similar analysis (use of the cross-correlation test) of the prediction error, em ðkÞ indicates that it is a function of DðkÞ; dðkÞ; the controllers, Gc ðq1 Þ and Gff ðq1 Þ and the models Gm ðq1 Þ and G# D ðq1 Þ: If the feedforward controller is performing well, it can be determined that deviation from minimum variance feedback performance is due to controller tuning and modelling errors. On the other hand, when the feedforward controller is not performing well but the feedback controller is adequate, the deviation from minimum variance is due to poor tuning of the feedforward controller. When both are operational and the performance exceeds product or process specifications the cross-correlation test must be supplanted by other methods to arrive at an unequivocal diagnostics. To address this, Desborough and Harris [11] provide a means of estimating the minimum variance performance when both feedforward and feedback control loops are present. Analogous to the earlier discussion, the closed-loop output can be represented by a superposition of ARIMA time-series models, that is yðkÞ ¼ C0 ðq1 Þ þ
J X
Cff ðq1 Þaj ðkÞ
ð21Þ
j¼1
where C0 ðq1 Þ and Cfj ðq1 Þ are the closed-loop transfer functions between the unmeasured and the jth measured disturbances, respectively, and faðkÞg and faf ðkÞg are statistically independent random variates with variance s2a and s2j : Consider only the feedforward forcing function in Equation (21). The variance is found by taking the expectation, EðÞ; of the square of Equation (21) ( )2 J J 1 X X X 2 sy ¼ E Cff aj ðkÞ ¼ C2ff;i s2ff ð22Þ j¼0
j¼0
i¼1
In practice only a finite sum is used. This equation provides an analysis of the process variance, which in turn permits an identification of the significant components of the variance. If the variance exceeds the performance bounds then re-tuning may be used to improve the performance. However, if the variance is acceptable then modification of the feedforward controller will not improve the situation. The design of a minimum variance feedforward/ feedback controller for processes with invertible dynamics can be found in Reference [11]. The process output under MVC is given by the sum of individual errors in forecasting the effects of the disturbances yðkÞ ¼
J X
ej ðkÞ
j¼0
cj;kj qkj þ cj;kj 1 qkj 1 þ þ cj;b1 qb1 aj ðkÞ
tj b
ð23Þ
where tj is the feedforward delay, ej ðkÞ; j > 0 is the contribution of the jth measured disturbance to the total forecast error, and e0 ðkÞ is the effect of the forecast error for the unmeasured disturbance on the closed-loop process output which is assumed independent of ej ðkÞ; j > 0: The first b terms are identical in terms of the expansion given in Equation (2) since k0 ¼ 0: Copyright # 2003 John Wiley & Sons, Ltd.
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The process variance under MVC is given by s2mv ¼
J X
varðej ðkÞÞ
ð24Þ
j¼1
which is similar to Equation (22) but truncated to h ¼ b 1: The term, s2mv ; is a lower bound on the variance of the process output. If s2mv is greater than the maximum allowed process variability then re-tuning or redesigning the controllers will not decrease the process variance to an acceptable level. Rather, process modifications must be considered. When s2y exceeds the lower bound, improvements are possible by re-tuning or implementing alternative control strategies. There are no guarantees and especially in the case of non-invertible unstable dynamics, alternative methods such as those recommended by Tyler and Morari [20] and Sternad and Soderstrom [27] should be investigated. Kozub and Garcia [15], Thornhill et al. [22] provide information on practical experiences in implementing large-scale performance assessment of feedforward/feedback controllers. Their main message is that both formal mathematical analysis and engineering insights are necessary for diagnosis leading to root cause identification. Other examples include: Lynch and Dumont [28], Huang et al. [10], Miller and Huang [29] and Vishnubhotla et al. [30]. An example of the application of feedforward control, consider the three-state CSTR discussed above. In this case, the same PI controller as in the earlier example is applied. The measured disturbance is the cooling fluid flow rate. The response between a step in the disturbance and the normalized controlled variable is shown in Figure 13. The transfer function
Figure 13. Step response from cooling water flow rate to reactor temperature for the three-state CSTR. Copyright # 2003 John Wiley & Sons, Ltd.
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Figure 14. Cross-correlation between the measured disturbance and the reactor temperature under feedforward control.
approximation, Gd ðsÞ; to the above system is given by Gd ðsÞ ¼
0:452 2:8s þ 1
This, together with the earlier defined transfer function between the manipulated variable and the controlled variable, Gp ðsÞ; defines the feedforward compensator as Gcomp ðsÞ ¼
5:901s þ 2:95 2:8s þ 1
Using Tustin approximation for simulation, the discretized compensator is given by Gcomp ðq1 Þ ¼
2:122 2:019q1 1 0:9649q1
Figure 14 shows the cross-correlation between the temperature output and the measured disturbance together with the 95% confidence limits. According to Stanfelj et al. [6], crosscorrelation should have no statistically significant values under perfect feedforward. In this case, the control is almost entirely within the confidence limits.
5. DUPONT PERFORMANCE MONITORING PROGRAM 5.1. Background The DuPont Company began piloting process monitoring techniques in the mid 1990s. Methodologies were developed to assess process performance and associated commercial Copyright # 2003 John Wiley & Sons, Ltd.
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benefits were identified for the various capabilities such a system might provide. The capabilities of these early pilot systems included: 1. Identification and notification of abnormal variability in process variables (largely, but not exclusively controlled variables). 2. Identification and notification of controller saturation conditions. 3. Identification and notification of abnormal controller mode (manual or non-normal mode of operation by the operators). 4. Generating compliance metrics (Cp, Cpk, Six Sigma ‘Z-Scores’, etc). 5. Generating controller performance metrics (minimum variance control index). In practice, the first three capabilities have proven to be of greatest value to the plant operations. Most of the DuPont processes satisfied the desired quality, utility and rate goals during the vast majority of their operation. However, when process conditions, equipment operation or measurement and control hardware degrade, process operation deteriorates quickly and financial losses mount. Control performance assessment was accomplished by auditing the performance of the controller on some set frequency or responding to problems attributed to control and instrumentation. As part of evaluating the controller performance, it is important to understand that in an industrial environment, not all controllers are tuned for minimum variance control. Frequently, controllers are tuned for flow smoothing or tuned for averaging (flow filtering) level control. It is also no unusual to tune controllers to be less tight than the theoretical limit possible for process interaction minimization reasons. The root cause of the degradation is frequently difficult to determine for a number of reasons including the shear complexity of large chemical processes, interaction among process parameters, truly obscure fault conditions, less than adequate technical coverage assigned to the operation, or inexperienced resources. The benefits to the businesses of an early warning system that can identify the existence of degraded process behaviour are quite high. Other performance measures such as compliance and control index are of less initial interest to plant operations but are of value to process control experts at the sites. The early pilot process monitoring efforts established the feasibility and value of this technology as well as clarifying the required functionality for large-scale user acceptance. While the pilot systems generated useful information, they were lacking capability in the area of user interface. Users (typically plant, control or process engineers) were sent an e-mail exception report listing alert conditions; no further diagnostic capability existed. Thus, users were forced to employ other software tools to continue the diagnostic process (graphs of process trends, etc.). This was an obvious barrier to widespread acceptance of the technology. In the late 1990s a process monitoring initiative was put into action by the corporate process control technology network. A team of process control professionals began to explore and plan the necessary steps to make this technology widely available to the corporation. The team reviewed commercial products at that time and found that the number of available products were small and they were largely focused on a different objective. For instance, products such as Loop Scout1 (Honeywell) were targeted to perform occasional (but quite thorough) performance assessment. Thus, it appeared to be more of an audit tool rather than a continuous real-time monitoring tool. Also, Loop Scout1 required transmitting proprietary Copyright # 2003 John Wiley & Sons, Ltd.
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process data over the Internet to Honeywell for processing; this was not acceptable to many businesses within DuPont. Without a viable commercial alternative, the team drew up a system design and estimated the development costs. DuPont did an extensive NPV (net profit value) calculations in order to justify their corporate performance monitoring development and rollout program. A funding proposal was made to corporate management for the internal development and wide deployment of a process performance monitoring system. Funding was approved in early 2001 and development work began immediately; the system was named Performance SurveyorTM : The system was ready for beta test in the first quarter of 2002; broad dissemination began in April 2002. A self-sustaining funding mechanism was established whereby the major internal business units subscribed on an annual basis for Performance SurveyorTM applications. These business-wide subscriptions fund as many applications as desired within that business. In 2002 four business units subscribed with two additional units subscribing in 2003. 5.2. System description The focus of the monitoring work is to keep the controller performance similar to what has been defined as the historical normal. For instance, if the daily average mean squared error (MSE) is 0.25 (whatever that means from a minimum variance standpoint) then the desire is to have the every day average MSE for that loop to be 0:25: Typically, alarms are configured to trigger if a metric, such as the MSE, changes by some prescribed amount. Some metrics like utilization and saturation are easier to set a! priori, but the objective is to find changes relative to accepted historical averages. Making sure the historical averages are satisfactory is very important. Being able to filter out known unusual process operation is critical to successful calibration of these trigger limits. Real-world applications of controller performance monitoring are constrained on several fronts depending on what problem is being addressed. In order to make the tools useful and widely applicable it has to: (i) (ii) (iii) (iv) (v) (vi)
function in many infrastructures, be configurable by a wide variety of end-users, produce useful and easy to understand information available to a general audience, be almost trivial to set-up, provide hassle free operation, and be of low cost to the user.
An internet delivery can provide some of these benefits such as availability, installation and setup ease, and low software maintenance cost. The following assumptions apply to the DuPont Performance SurveyorTM monitoring package. Assumption 1 A web-based system will generate daily reports based on data collected for the last 24-h period. The data will be gathered from different process data systems, from sites located all over the world. The data sets observation length will be 24 h of 1 min data. Metrics will be calculated on pre-specified data array on a daily basis. The data may be compressed, sparse or voluminous. Copyright # 2003 John Wiley & Sons, Ltd.
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Assumption 2 The highest return-on-investment is to provide detection of misbehaviour thus reducing the monitoring task by orders-of-magnitude. The task of actually diagnosing a specific problem is expedited by simple tools for viewing the data in various ways, but the final diagnosis is left to the engineer. Performance SurveyorTM monitors large numbers of process variables/control loops and generates valuable performance metrics. These metrics are used in detecting degrading performance in process conditions, process equipment, instrumentation or control equipment. A daily report alerts plant operating personnel of potential problems. This allows attention to be focused on the highest priority problems. Performance SurveyorTM also provides the means to benchmark performance against the theoretical best performance providing thus a basis for identifying opportunities for improvements under the Six Sigma initiatives as defined by DuPont. Performance SurveyorTM can monitor progress during the Improve phase of a project and sustain the benefits during the Control phase. This real-world monitoring tool focuses primarily on monitoring not diagnostics. The philosophy is that pointing out that there is a problem is the first step in getting it fixed. Examples of problems that have been detected in early applications of this technology include: * * * * * * *
Transmitter zero shift. Transmitter impulse line plugging. Control valve hysteresis. Control valve actuator leaking. Degraded controller tuning. Process block valve leakage causing major flow instability. Reactor agitator malfunction.
These conditions are detected by observing changes in the process variability versus normal operation. Performance SurveyorTM is designed as an Intranet application. It runs on a central computer and extracts data from a plant process historian on a periodic basis (every 24 h). The daily, batch processing of data was a compromise between keeping up with real-time process operation and realizing that plant users would only look at the reports at infrequent intervals. There are a limited number of diagnostic calculations that are done on the daily controller data. Most of these calculation are done and compared to how the loop was running historically for hints as to what is not working properly. The daily processing and reporting has met user needs and batch processing allows the use of high performance mathematical tools such as Matlab1: Performance metrics are calculated from collected process data and stored in an SQL (standard query language) database. Various types of reports are available to plant personnel using a standard web browser. The web-based user interface allows the metrics to be trended along with corresponding process data; this capability allows the engineer to determine the nature and timing of fault events. The user interface also facilitates system configuration and tuning. Data sources for Performance SurveyorTM are the standard process data historians used at the sites including Aspen and other legacy historians. Copyright # 2003 John Wiley & Sons, Ltd.
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Central Server
Plant Site Plant Data Historian
User Interface
Metrics Engine Data Collection Engine
(MatLab)
DB
Display Generator
(Web Display Tools) (Web Browser)
Figure 15. Schematic of the major components of DuPont’s Performance SurveyorTM :
The following list of plots and calculations are done on 24 h 1 m data: 1. 2. 3. 4. 5. 6.
Cross-plots of process variables and outputs (for valve behaviour, and process linearity). Process variable autocorrelations (for opportunity). Deadtime predictions (for process shift). CI as a function of deadtime (for opportunity). Power spectrum (for interaction analysis). Expert system for tuning assistance as a function of type of loop.
A diagram of the major system components is shown in Figure 15. 5.3. A typical real-world example of controller performance monitoring A typical medium to large-scale chemical or material processing plant may have from 1500 to 3000 PID control loops in use. These loops are expected to control variables from the powerhouse to the waste treatment area and everything in between. Not all of these loops are extremely critical to quality operation and only a small fraction of these loops are routinely monitored by Engineering and Instrumentation (E&I) technician. It is very typical for one E&I person to be responsible for hundreds of operating control loops as a small part of his or her overall assignment. It is no wonder control loop performance can degrade and not be recognized until it has an impact on product quality, uptime or yield. The following paragraph discusses a true story about how a hidden control event can seriously affect the process and how a controller performance monitoring application can help to find and fix the problem quickly. One of the, several hundred, loops that the E&I technician is responsible for is the loop labelled 1003TCS.z This loop controls the inlet feed temperature to a critical process analytical measurement that measures final product quality. This is the type of control loop that is supposed to always work. If something goes wrong with this loop, it is very possible that no one z
Fictitious label.
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Figure 16. Malfunctioning temperature loop (1003TCS) affects the analytical loop.
would realize this for many days of operation. They may observe that the product is off spec or degraded and blame the analyser or lab result, but it takes a process expert to track down this kind of problem. Figure 16 shows how the malfunctioning temperature loop affects the analytical loop. The fluctuation in the analytical measurement signal and the failing temperature controller signal are labelled on the figure. The failure was due to a heating utility saturation problem. The controller performance monitoring application pointed to this rather subtle problem (This temperature signal would never have been investigated!) within a 12-h period. This allowed the process engineer to make the necessary utility adjustments and avoid a large quality problem/yield loss.
6. SUMMARY AND CONCLUSIONS Performance monitoring of univariate control systems was extensively reviewed. The MVC was developed and used as reference for the performance of a univariate controller. The ratio of the variance of the controlled output and the expected variance if a MVC is used is a metric for performance. Several variations of this concept are presented. Additionally, it was shown that minimum variance control is usually not desirable because of issues such as model/plant mismatch. Both model based and feedforward/feedback controllers are included with chemical process examples to demonstrate these concepts. The experience of the DuPont Company in the implementation of a controller performance monitoring system, Performance SurveyorTM , was outlined. This system is capable of detecting and reporting unusual variability in process variables, identification and notification of Copyright # 2003 John Wiley & Sons, Ltd.
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saturation conditions in the controller and abnormal controller modes and generating both compliance metrics (Cp, Cpk, etc.) and controller performance metrics. This system is in operation at a number of plant sites.
NOMENCLATURE aðkÞ b dðkÞ DðkÞ em ðkÞ eðkÞ E Gp Gm Gc Gff Gd GD G# D q1
white noise deadtime unmeasured disturbance measured disturbance process/model error feedback error expected value operator process transfer function model transfer function feedback controller transfer function feedforward controller transfer function unmeasured disturbance transfer function measured disturbance transfer function approximate feedforward transfer function model backward shift operator
rðkÞ r2 s uðkÞ uff ðkÞ ufb ðkÞ yðkÞ s2 s2y s2mv s2y ZðbÞ Zðb þ hÞ C
set point multiple regression coefficient Laplace variable controller output feedforward controller output feedback controller output process output population variance population variance in output y theoretical minimum variance sample variance of output y normalized performance index extended horizon performance index impulse weights
List of abbreviations ARIMA API CLP E&I CSTR FCOR
autoregressive integrated moving average absolute performance index closed-loop potential factor engineering and instrumentation continuous-stirred tank reactor filtering and correlation
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LTI LQG MIMO MSE MVC NPV PID SISO
661
linear time invariant linear quadratic Gaussian multiple-input multiple-output mean squared error minimum variance controller net profit value proportional-integral-derivative single-input single-output
ACKNOWLEDGEMENTS
The first two authors wish to acknowledge that Figures 1–14 are reproduced by permission of CRC Press LLC, Boca Raton, FL, USA.
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22. Thornhill N, Oettinger FM, Fedenczuk P. Refinery-wide control loop performance assessment. Journal of Process Control 1999; 9:109–124. 23. Box GEP, Jenkins GM. Time Series Analysis: Forecasting and Control. Holden-Day: Englewood Cliffs, NJ, 1976. 24. Astro. m KJ, Wittenmark B. Adaptive Control, Chapter 3. Addison-Wesley: New York, 1989. 25. Eriksson P-G, Isaksson ZJ. Some aspects of control loop performance monitoring. In Third IEEE Conference On Control Applications, Glasgow, Scotland, 1994; 1029–1034. 26. Kosanovich(Hoo) KA, Charboneau JG, Piovoso MJ. Operating regime-based controller strategy for multi-product processes. Journal of Process Control 1995; 7:43–56. 27. Sternad M, So. derstro. m T. LQG-optimal feedforward regulators. Automatica 1988; 24:557–561. 28. Lynch CB, Dumont GA. Control loop performance monitoring. IEEE Transactions Control Systems Technology 1996; 4:185–192. 29. Miller RM, Huang B. Perspectives on multivariate feedforward/feedback controller performance measures for process diagnosis. In Proceedings IFAC ADCHEM 97, Branff, Alberta, 1997; 495. 30. Vishnubhotla A, Shah S, Huang B. Feedback and feedforward performance analysis of the Shell industrial closedloop data set. In Proceedings IFAC ADCHEM 97, Branff, Alberta, 1997; 295.
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