Control Engineering Practice 8 (2000) 791}797
Controller performance assessment based on minimum and open-loop output variance Stella Bezergianni, Christos Georgakis* Chemical Process Modeling and Control Research Center, Department of Chemical Engineering, Lehigh University, 111 Research Drive, Bethlehem, PA 18015, USA Accepted 4 February 2000
Abstract The need to achieve reliable and e!ective automatic control systems has directed numerous research e!orts towards the development of monitoring and assessment methods of the closed-loop performance. A new controller assessment index, called relative variance index (RVI) is introduced in this work, according to which the closed-loop performance is compared with the best theoretical control action (minimum variance control) and no control action. For the estimation of RVI, an identi"cation method is employed to extract the system models and an estimate of the process unit delays. The proposed index, RVI, is in agreement with the classical deterministic closed-loop assessment measures and thus can be utilized for the controller performance evaluation more e!ectively. 2000 Elsevier Science ¸td. All rights reserved. Keywords: Controller performance assessment; Minimum variance control; Structure target factor analysis
1. Introduction The desire to monitor and assess the performance of controllers has emerged due to the increased need to achieve reliable and e!ective automatic control systems. In the industrial world, there are numerous automatic control systems that aim to maintain constant product quality standards and satisfactory plant performance. However, controller performance deterioration is a phenomenon commonly observed in practice. Most of the control systems fail to perform as they were initially designed. In many cases, controllers have even been found to increase the process variability due to poor tuning, according to a recent study by Ender (1993). Several techniques developing e!ective tools for controller performance monitoring and assessment, have been proposed in the literature during the last decade. These techniques can be divided into two major categories, deterministic and stochastic controller performance assessment methods. The most prevailing stochastic
* Corresponding author. Tel.: #1-610-758-5432; fax: #1-610-7585297. E-mail address:
[email protected] (C. Georgakis).
performance assessment method was proposed by Desborough and Harris (1992), who suggested the comparison of the output variance of the current control action with the minimum variance that could be achieved via minimum variance control (MVC) as a measure of closed-loop performance. Thornhill, Oettinger and Fedenczuk (1999) also adopted and applied the principles of MVC performance monitoring to single-input}single-output (SISO) controllers of a re"nery with positive results. As stroK m (1991) combined the classical performance monitoring measures for qualitative and quantitative assessment of the performance that can be achieved under a simple feedback loop, forming the "rst deterministic controller performance assessment technique. Eriksson and Isaksson (1994), motivated by the fact that the MVC-based controller performance assessment index was di$cult to interpret and could not evaluate the e!ect of deterministic changes to a closed-loop system, presented some alternative indices, which required the exact models of the process and the controller. In the present paper, a new controller performance assessment index is presented. The index compares the currently achieved closed-loop variance with both the achievable minimum variance as well as the open-loop variance of the process. It provides a more realistic
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comparison than previous indexes. The proposed index, although stochastic in nature, is in agreement with the classical deterministic closed-loop assessment measures and can, therefore, evaluate the controller performance under both stochastic and deterministic changes. The methodology of estimating the proposed index is also presented. This methodology estimates the number of process unit delays and identi"es the approximate models for the controller, process, and disturbance using closed-loop data of the output, y, the input, u, and the set-point, yQN. In the section that follows, the motivation and description of the proposed new controller performance assessment index will be introduced. Some case studies where the new index has been used, as well as some comparisons with the MVC-based index, will be presented later.
2. A new generalized controller performance assessment index The minimum variance controller assessment index, g, has been extensively presented in the literature (Desborough & Harris, 1992, 1993), and can be de"ned as p g"1! +4! , (1) p W where p is the output variance under the current control W action and p is the output variance under MVC, +4! which is treated as the best possible theoretical control action. This normalized MVC-based controller performance index, g, is equal to zero for the best possible controller performance (p"p ) and one for the W +4! worst controller performance (pPR). However, the W minimum variance-based index is very di$cult to interpret since it compares the present variance of the closedloop system with only the theoretical minimum variance which can be achieved under a minimum variance controller (MVC). Even though the MVC has a signi"cant theoretical value, it is not often possible to implement it in practice because it leads to excessive input moves. Therefore, it might be misleading to compare the current controller performance with that of just the MVC one. To compensate for the above limitations, a new closedloop performance assessment index, to be called relative variance index (RVI), is proposed. The proposed new index compares the current output variance, under closed-loop conditions, with the variance under the best and worst possible control action. The minimum variance controller represents the best possible closed-loop conditions, while the open-loop case is considered as the least desirable closed-loop conditions. The new controller performance assessment index is de"ned by the following equation: p !p W , RVI" -* p !p -* +4!
(2)
where p is the output variance under current control W action, p is the theoretically possible minimum vari+4! ance and p is the open-loop output variance. The RVI -* is equal to zero when the controller performance is equal to the open-loop performance (p"p ), and equal to W -* one when the controller performs as well as a MVC (p"p ). The main characteristic of this index is that W +4! it evaluates the present control action in a more judicious manner by using two bases of comparison, assuring a more realistic and practical assessment of the closedloop process. In later sections, the MVC-based index will be represented by 1!g in order to facilitate the comparison of the new index with the RVI one.
3. Methodology The estimation of the proposed RVI requires knowledge of the process, controller, and disturbance models as well as the number of process unit delays. In this section, a new approach for the approximate identi"cation of these models and the process delay from closedloop operating data is presented. 3.1. Data requirements The estimation of the relative variance index requires operating data of the output variable (y), the input or manipulated variable (u), and the set-point (yQN). The systems examined here are SISO control systems consisting of a process model, G , of order n with d unit delays, N N a disturbance model, G , of order n , and a linear controlB B ler, G . The classical cases of P, PI, and PID are A considered in detail. The three measured variables are collected at equidistant time intervals, with sampling times small enough to adequately capture the system dynamics. For identi"cation purposes, at least one setpoint change is required in order to use the transient region data for extracting the controller, process, and disturbance models. Process data that include more than one set-point change will increase the accuracy of the models identi"ed. The data samples of y, u, and yQN form a data matrix X as K X "[X(k) X(k!1) 2 X(k!m)]. K
(3)
The X(k) is a matrix that contains the variables y(k), u(k), and yQN(k) in a column format, having maximum dimensions of N;3 where N is the number of data points collected. The data matrix X then becomes a K (N!m);3(m#1) matrix, where m is the number of lags that the data is shifted to. The matrix X is called K dynamic data matrix of order m. If m"0, then the matrix is de"ned as a static matrix. As will be shown in the following sections, these static and dynamic matrices
S. Bezergianni, C. Georgakis / Control Engineering Practice 8 (2000) 791}797
are used to identify dynamic controller, process, and disturbance models. 3.2. Identixcation of controller model In most cases, the controller model is known and available. However, there are many cases where the controller model is not readily available and it is di$cult and time consuming to extract it from the distributed control system (DCS) system on the plant. Moreover, when a great number of control loops have to be assessed, it is an unwieldy task to ask the operators to provide the control models of all the loops under examination. For these cases, the following methodology for identifying the controller model has been developed. The "rst model that is extracted from the closed-loop data using the proposed approach is the controller model. The methodology is not only able to identify the controller's parameters but also its structure, provided the present controller is linear and time-invariant. The "rst step is to set all the possible control structures to be considered (target candidates), for example, P, PI, and PID controllers, and express them in a vector form, as shown in Table 1. According to this table, if the controller is proportional, then the controller relationships are contained in the static data, since the P controller relationship is zeroth order (static). Similarly, if the controller is PI or PID, dynamic data of "rst and second order, respectively, are required to identify the corresponding "rst and second order controller models. In order to identify the controller structure and parameters, the following procedure is followed: E Compose the dynamic data matrix of order m, equal to the candidate controller model order, n . If it is A a P controller, then use a static matrix (m"0). Use transient region data only. E Apply singular value decomposition (SVD) on the covariance of a dynamic matrix of order n : A R 0 V2 X2A X A . L L P[ U U ] (4) 0 m!1 0 V2
793
E Find singular vectors corresponding to the zero singular values, V , which form the basis of the vector space to which the controller relation belongs. This subspace is represented by a projection operator, P, which is calculated as P"V V2 .
(5)
E Target a controller model of order n , as represented A by the target vector t A (see Table 1). The candidate L model structure is tested by projecting it through P as (6)
r A "Pt A L L
If t A is a true model contained in the range of P, L then the response vector, r A , should be identical to t A . L L By using structured target factor analysis (STFA) (Fotopoulos, Georgakis & Stenger, 1994), one can calculate the unknown controller model parameters as a least-squares solution of Eq. (6). E Calculate the percent projection error E A as L ""t !r A "" L 100. E A " LA L ""t A "" L
(7)
If E A 0, then the controller model relationship of L order n and its parameters are given by either the A r A or the t A vectors (since r A t A ). Otherwise, repeat L L L L all the above using dynamic data corresponding to a value of n , increased by one unit. A In the above algorithm, it is essential to use dynamic data of orders equal to the candidate controller model order to avoid the presence of multiple copies of the controller relationship, shifted by one or more time steps, that are present when using high-order dynamic data sets. It is also worth noting that the identi"ed model will be very close to the actual one because the values of the three measurements y, u, and yQN, are directly involved in the controller relationship. For cases where the controller relationship is known, the above algorithm can still be used for examining it. The available controller model can be represented as
Table 1 Controller transfer functions and corresponding target vectors for the identi"cation of the controller model using STFA Controller
Transfer function G (z\) A
Target vector t
P
K A
[c 1 !c ]2
0
PI
K A
(*t/q #1)!z\ ' 1!z\
[c 1 !c c !1 !c ]2
1
PID
K A
(*t/q #q /*t#1)!(1#2q /*t)z\#(q /*t)z\ ' " " " 1!z\
[c 1 !c c !1 !c c 0 !c ]2
2
A
Order (m)
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a target vector, t A which can be projected through the L projection operator P constructed from the corresponding closed-loop data as in Eq. (6). If the projection error, as calculated from Eq. (7), is small, the available controller model is correct. If it is large, then either the controller model is not accurate, or the closed-loop data utilized are not informative enough or not related to the control loop under examination.
3.3. Identixcation of process model The "rst step towards the identi"cation of the process model, is to estimate the number of process unit delays. The process delay is a very important factor for the calculation of the RVI, as it also is for the MVC index. In many cases, the number of process unit delays changes as the process changes operating points. The number of process delays should therefore be calculated frequently from the current data, so that the controller performance assessment index has a more realistic value. Elnaggar, Dumont and Elshafei (1991) suggested the variable regression delay estimation technique (VRE), which is utilized in the overall approach of determining the process model. The next step towards the process model identi"cation is the estimation of the process model order, n . The N deterministic-stochastic subspace identi"cation algorithm Van Overschee (1995) on [ y(k#d ), u(k)] data is used for the estimation of n . The parameter d denotes the numN ber of unit delays. The approach for the identi"cation of the model parameters is summarized by the following steps: E Compose a dynamic data matrix of only input (u) and output (y) data of order n #d, X N , where n is the N N L >B model order estimated. E Apply SVD on X N and calculate singular vectors L >B corresponding to the small singular values, V (as in the case of controller model identi"cation, Eq. (4)). E Compose the corresponding projection operator P"V V2 . E Target a process-model of n order with d unit delays: N b #2#b N z\LN L G (z\)" z\B (8) N 1#2#a N z\LN L represented by the target vector t N "[0, 1, 0, a ,2, 0, a ,b ,a , L >B B\ B b ,a , ,b ,a , B> 2 LN \B LN bN , 0, , b , 0]2 L \B> 2 LN
(9)
indicating the following arrangement of variables: t N "[u(k), y(k),2, u(k!n !d), L >B N y(k!n !d)]2. N
(10)
Then, calculate the response vector r r
LN >B
"Pt
LN >B
.
LN >B
using STFA: (11)
E Calculate projection error E N as L >B ""t !r N "" L >B 100. E N " LN >B L >B ""t N "" L >B
(12)
If E N 0, then the process model relationship is of L >B order n with d unit delays and its parameters are N given by the vector r N . Else, repeat the above proL >B cedure using dynamic matrix of order n #d#1 for N identifying a process model of order n #1 with d unit N delays. It is more di$cult to extract the process model from the closed-loop data than to extract the controller model, because of the existence of the measurement and input noise. Nevertheless, for monitoring purposes and for the cases considered, the identi"ed process model is quite close to the actual one, considering that the data collected is not as informative as it would have been if a substantial external excitation had been introduced in the process. 3.4. Identixcation of disturbance model To identify the disturbance model, one needs to calculate the vector of di!erence, e , between the measured LMGQC output and the one predicted by the process and controller model identi"ed above in the closed-loop formation. This di!erence is attributed to the presence of disturbances, and is used for the extraction of the disturbance model. The disturbance model order, n , will be extracted B using the stochastic subspace identi"cation algorithm, Van Overschee (1995). The identi"cation of the parameters of the disturbance model is outlined as follows: E Compose a dynamic matrix of order n , using B e data only. LMGQC E Apply SVD on the covariance of the dynamic matrix of order n and "nd the singular vectors, V , that B correspond to the zero singular values, and compose the projection operator P"V V2 . E Target a disturbance model of order n , using a target B vector t B "[1 d d 2 d B ]2 and calculate the reL L sponse r B using STFA: L r B "Pt B . L L
(13)
E Calculate projection error E B as L ""t !r B "" L 100. E B " LB L ""t B "" L
(14)
S. Bezergianni, C. Georgakis / Control Engineering Practice 8 (2000) 791}797
If E B 0, then the disturbance model relationship is of L order n and its parameters are given by the vector r B . B L Otherwise, repeat the above procedure using dynamic matrix of order n #1. B 3.5. Calculation of RVI Once the three models of the process, controller, and disturbance are identi"ed, the variance of the current control action, p, the minimum variance, p , and the W +4! open-loop output variance, p can be calculated. The -* most accurate estimation of the three variances is performed by using the sum of the squares of the impulse response coe$cients of the corresponding close- and open-loop models. For the closed-loop output variance under the current control action, p is calculated as W the impulse response coe$cients, , of the closedG loop transfer function between e and y, G (z\)/ B (1#G (z\)G (z\)). The variance of the current control A N action is given by p"(1# #2# #2)p, W B\ C
(15)
where p is the variance of the white noise signal e. The C minimum achievable variance, p , can then be ob+4! tained from Eq. (15) by setting " "2"0: B B> p "(1# #2# )p, +4! B\ C
(16)
where d is the number of process unit delays estimated earlier. The open-loop output variance can be calculated from the impulse response coe$cients, t , of the open-loop G transfer function from e to y, G (z\), as shown in the B equation p "(1#t #2#t #2)p. -* @\ C
795
The models were discretized using the sampling time *t"0.1 min and the discrete models of the process and disturbance were G (z\)"0.04877z\/(1!0.9512z\) N and G (z\)"0.00396z\/(1!0.9802z\), respectively. B Using the above system, seven di!erent processes were considered by adding to the above model a time delay equal to 1, 5, 10, 15, 20, 25, and 30. This corresponds to a time delay equal to 0.1, 0.5, 1, 1.5, 2, 2.5, and 3 min. For each of the above systems, a PI controller was used with its tuning parameters calculated through the Ziegler}Nichols criterion. The controller parameter values used in the above-mentioned seven cases were equal to 9.320, 3.445, 1.801, 1.291, 1.045, 0.901, and 0.808 for K , while the values for q were equal to 0.595, 1.419, A ' 2.743, 3.946, 5.063, 6.120, and 7.132 min, respectively. Data from the closed-loop response of each process to a unit step set-point change over a 10 min (100 samples) interval were collected. For the aforementioned seven process examples, the RVI and the MVC-based controller performance indexes were calculated and compared, as shown in Fig. 1. From Fig. 1, it can be seen that the MVC-based controller performance assessment index, 1!g, increases as the number of process delays increase. This increase in 1!g prompts the authors to conclude that the performance of the closed-loop system improves as the dead-time increases. This obviously is an incorrect conclusion. On the other hand, the RVI shows that the controller performance decreases with an increasing number of process unit delays. This is in accordance with the intuitive expectation on the controller performance. Furthermore, this is also in agreement with some additional classical controller assessment indices, such as the rise time t , 0 settling time t , and integral square error ISE, given in 1 Table 2. Therefore, the RVI index shows a more realistic evaluation of the control system performance, while the
(17)
Having the three variances, p, p , and p , the relaW +4! -* tive variance index can be estimated using Eq. (2). In the analysis section that follows, the e$ciency and the estimation accuracy of the RVI will be presented through some case studies.
4. Analysis * results Values for the RVI have been examined extensively using di!erent simulation models. One of the models used is a "rst-order process, G (z\), with a disturbance N added onto it, G (z\), described by the following transB fer functions: 1 , G (s)" N qs#1
q"2 min,
0.2 G (s)" . B 5s#1
(18)
Fig. 1. Comparison of estimated relative variance index, RVI (), and the MVC-based index, 1!g (*).
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Table 2 Comparison of expected, 1!g and estimated, 1!g , MVC-based controller performance index, and expected, RVI , and estimated, RVI , relative ? ? variance index, with rise time, t , settling time t and ISE for systems with di!erent process unit delays, b 0 1 b
1!g
1 5 10 15 20 25 30
0.330 0.506 0.626 0.690 0.732 0.761 0.782
1!g 0.338 0.296 0.267 0.252 0.209 0.165 0.094
?
RVI
RVI ?
t (min) 0
t (min) 1
ISE
0.917 0.831 0.741 0.663 0.583 0.494 0.389
0.772 0.765 0.680 0.645 0.570 0.483 0.448
10.5 11.1 12.3 13.6 14.9 42.4 50.8
11.0 12.3 16.1 21.3 34.7 39.1 48.5
0.1031 0.1076 0.1144 0.1205 0.1296 0.1395 0.1459
MVC focuses on a very idealistic comparison with minimum variance. The above problem of the MVC-based methodology has also been reported by other investigators (Eriksson & Isaksson, 1994). The problem is caused by the ability of the MVC index to assess the control behavior only under stochastic changes, thus classifying it as a stochastic controller performance assessment technique, while the classical non-statistical assessment indices are part of the deterministic controller performance assessment methodologies. On the other hand, the relative variance index, though it is a stochastic-in-nature performance assessment index, is also in agreement with the deterministic measures of closed-loop dynamics. The RVI can be estimated from closed-loop operating data as described in Section 3. The more accurate the identi"ed models, the better the estimate of the relative variance index. For the simulation cases that are studied, the models are in very good agreement with the original systems. The proximity of the estimation of the RVI to its actual value (which is known for the simulation cases) was also examined by conducting a series of Monte Carlo experiments. In Figs. 2 and 3, the results of one thousand Monte Carlo simulations for the "rst-order process with "ve unit delays are displayed. The accuracy of the RVI estimation from one-step set-point change data using the proposed identi"cation methodology are depicted in Fig. 2. The accuracy of the RVI estimation appears adequate as is also the case for the accuracy of the MVC-based index, presented in Fig. 3, and estimated from closed-loop output data using linear-regression analysis, as suggested by Desborough and Harris (1992). The negative and positive bias in the RVI and MVI estimation, respectively, is most probably due to the limited of information contained in the data utilized. Moreover, for all the aforementioned "rst-order processes with di!erent dead times, similar Monte Carlo simulations were performed, showing that the larger the data sample, the more accurate the estimate is for the RVI, which is expected. On the other hand, the larger the process delay, the less accurate is the estimate. Nevertheless, the estimated RVI was very satisfactory for all the
Fig. 2. Accuracy of relative variance index (RVI) estimation for a "rstorder process with "ve delays, using the proposed estimation methodology. (Solid line: Estimated Values of RVI, Dashed line: Actual RVI).
cases studied, if one takes into consideration that it is an assessment tool rather than a design one.
5. Conclusions A new controller performance assessment index called relative variance index (RVI) was introduced. The RVI compares the closed-loop output variance under the current control action with the minimum and open-loop output variances. There are several issues to be noted about the e!ectiveness of the relative variance index. The estimation methodology of RVI is a systematic and non-parametric methodology. It also employs a technique for identifying the number of process unit delays. On the other hand, the
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used o!-line or intermittently on-line provided that the closed-loop data employed are informative enough. Overall, it is a non-parametric systematic methodology that gives valuable information about the controller tuning parameters and performance.
References
Fig. 3. Accuracy of MVC-based index, 1!g, estimation for a "rstorder process with "ve delays, using linear regression technique (Solid line: Estimated Values of 1!g, Dashed line: Actual 1!g).
existing MVC-based methodology utilizes parametric identi"cation techniques, and requires prior knowledge of the process unit delays. Although stochastic in nature, RVI is in agreement with the classical deterministic measures (i.e. settling time, raise time, ISE to a step response). The proposed methodology could either be
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