Some Aspects of Control Loop Performance

TA- 1 -5 Some Aspects of Control Loop Performance Monitoring P-G Eriksson and -41f J. Isaksson Department of Signals, Sensors & Systems Royal Institute of Technology S-100 44 Stockholm, Sweden Fax: +46 8 7907329 Email. pg@elir.e.kth.se, alf@elixir.e.kth.se Keywords- minimum variance control, control loop per- formance, system identification, PID control. Abstract In industries where there are a great number of con- trol loops, it would be convenient to have some tool for monitoring the performance of individual loops. This paper starts from one available approach that evaluates the control performance compared to mini- mum variance control. We show, using an example and also from industrial data, that this technique gives a inadequate measure of the performance if the aim is not stochastic control, but step disturbance rejection. Therefore, some modified criteria are proposed and a monitoring tool utilising all available techniques out- lined. 1 Introduction In industries where there are a great number of con- trol loops it is very time-consuming to ensure that all control loops operate satisfactorily. I t is a sad reality that many of the control loops perform poorly [14]. In- vestigations in the pulp and paper industry [l] show that of the more than two thousand control loops in a typ- ical mill only about 20 per cent work well. Therefore, it would be convenient to have some tool that evaluates the performance of every single loop. Such a tool should be able to work under closed loop conditions and not require extraneous test signals to perturb the system. It should monitor the system and inform the operator when there is a control loop that needs to be retuned. There is, however, not very much published on the subject of control loop performance monitoring. Harris [6] has developed a technique which, under closed loop conditions, can give a performance index for the control systems behaviour for single loops. Stanfelj et al. [12] has extended Harris’s result and developed a hierarchical technique for the analysis of various control schemes, in- cluding a class of feedforward/feedback control schemes. The technique using a performance index to decide the performance of control loops has been tested in several applications [lo], [3], [ll]. In this paper Harris’s technique will be discussed and some drawbacks pointed out. Real data from the pulp and paper industry is examined which indicates that the performance index must be modified for some cases. Fi- nally, a monitoring tool that includes different optional indices is outlined. 2 A n existing approach To obtain a measure of the controller performance Harris suggests comparing its performance with that of a minimum variance controller. A performance index is obtained by dividing the actual output variance, ai, with the minimum achievable output variance, czv, i.e. Consider a linear system described by the discretetime model where u( t ) is the input, and e( t ) a zero mean white noise sequence with variance a:. Here q-’ is the shift opera- tor, and k the time delay of the process. Figure 1 shows the block diagram when a feedback controller (3) controls the system. Figure 1: The process under feedback control. For a system given by (2), it is easy to compute the minimum achievable variance if the polynomials C(q-’) 0-7803-1872-2/94/$4.00 0 1994 IEEE 1029 and D(q-’) are known. See e.g. Astrom [13]. To avoid the problem of estimating C(q-l) and D(q-l)] which may be difficult under closed loop conditions, Harris con- sidered estimating a time series model for the closed loop system instead: T = D ( F R + Bq-kS) A; e ( t ) = -e(t) . (4) C F R Harris pointed out that, given estimates of the two poly- nomials T and A- and the deadtime k, it is possible to estimate the minimum achievable variance. regardless of what controller was used during data collection. The estimated minimum variance is calculated from uLv = (h; + h: + . . . + h’,-,).e’ ( 5 ) where h, are the Markov parameters (or pulse response) of the transfer function Tlh‘. An estimate of uf is ob- tained when the polynomials T and N are identified us- ing a prediction error method as the final value of the loss function. The actual output variance U: can be obtained directly from the measured data. The perfor- mance index is then given by (1). This comparison with the minimum variance controller does not necessarily suggest that such a controller should be used in practice. The minimum variance controller has several well known drawbacks. For example, it can lead to large input signals, and the closed loop quite of- ten has poor robustness properties. However, the mini- mum achievable output variance serves as a theoretical lower bound of the best possible performance. The performance index calculation presented above has some nice properties: If the time delay IC is known? the output y(t) is the only signal that has to be measured to compute the performance index. To estimate k, however, the input u(t) must be measured also. There are algorithms suggested ([4], [j]) for estimation of k under closed loop conditions if u( t ) and y(t) are measured. All signals can be collected under closed loop con- ditions without the need to add extraneous test sig- nals to the process. Neither the plant nor the noise dynamic have to be estimated explicitly. There are. however, also some! more or less serious, To estimate the closed loop polynomials T and :V the degree must be chosen. This choice is not trivial [8]: and with a poor choice the estimation of uLv may fail. To mitigate this problem. it is suggested in [lo] that a Laguerre network may be used to es- timate the closed loop filter. Generally, it is better to overestimate than to underestimate the model order. difficulties involved in the method: 0 Normally, estimation of a time series model (4) re- quires a numerical search with the possibility of l e cal optima. Once again. using a Laguerre network is advantageous. since the output model becomes linear in the parameters. 0 It is not obvious how to interpret the performance index. There is no fixed limit which decides if the performance is poor or acceptable. If. for example, the index is 2.5, is that acceptable or not? Over time we could, of course. always produce a trend plot for the index to see if it is changing or not. but that assumes knowledge that the controller was well tuned in the first place. 0 As mentioned before, the time delay must be known or estimated. There are also some limitations in the method: e If the controller structure is constrained to some specific type, for example a PID-controller, then a comparison to the minimum variance controller may not be all that interesting. The performance index may be very high, when compared t o the minimum variance controller. If, however. the comparison is to the optimal controller within the given structure. the index may well be reasonable. 0 Limitations in the control actions or the possibility to penalise the input are not taken into account. One problem with the above two limitations is that they cannot be dealt with unless we have an explicit model of the plant. In the following sections we wdl illustrate the above remarks; first with a numerical example, then using real data from a couple of paper mills. 3 A numerical example In this example, we highlight some of the remarks of the previous section. The numerical values of model and controller are taken from [IO]. The process is described by a linear, discretetime model of the type (2), where F(q-’) = 1 - 0.67q-’; D(q-’) = 1 - 0.67q-’; B(q-’) = 0.33; C(4-l) = 1 - 0.4q-’. k = 4: The system is controlled by a Dahlin controller, [2]. 0.7 - 0.47q-’ ‘(‘-’) = 0.33 - 0.1Oq-’ - 0.23q-4‘ The noise e ( t ) has U: = 0.36. The minimum variance can now be calculated to be u L y = 0.4033, and the output variance with the Dahlin controller acting on the process uz = 0.6115. Hence this gives a performance index 0.6115 0.4033 I = - = 1.52. If the controller is changed to a P-controller with a gain of 0.1745, the output variance becomes u: = 0.4037. 1030 This yields the performance index 0.4037 0.4033 I = - - - 1.001 indicating that the performance of the P-controller is very close to minimum variance performance. Accord- ing to the index, the P-controller has approximately 50% better control performance than the Dahlin controller. The reason why a P-controller has such a low perfor- mance index in this example is that there is nothing in the noise description that forces the controller to have integral action. Assuming that the system will be influ- enced by step load disturbances, then the P-controller will perform very badly compared to the Dahlin con- troller in this example. This raises the question of the purpose of the controller design. Are there constant or step disturbances acting on the process? Then no sensi- ble control engineer would leave this controller without integral action. Nevertheless, this is what is indicated by the performance index. 4 Alternative indices It would, as pointed out in Section 3, be desirable to have a performance index which is large for controllers without integral action. Some different ways to accom- plish this are discussed below: a) An integrating factor (1 - 4- l ) could be forced into the denominator of the noise model, i.e. changing the right part in (2) to C/D(1 - 4-'). It is in fact possible to calculate a performance index for this case in a way quite similar to that described in Sec- tion 2, using only measurements of the output. b) An alternative would be to omit the polynomials describing the noise dynamics altogether, and re- place them with 1/(1 - 4-l) . This is equivalent to evaluating the performance for step disturbances at the output (or equivalently step setpoint changes). c) Yet another alternative would be to consider step disturbances at the control input instead. This is equivalent to introducing an additive disturbance to u( t ) with the noise model 1/(1- 4-l). We find it hard to motivate the option a), even though it may be an appealing choice. The noise model might as well be omitted completely, and step disturbances be studied in a purely deterministic framework. We will therefore concentrate on the alternatives b) and c). The calculation of these two latter indices is quite similar and will be explained below in the case of b). The b) approach is illustrated in Figure (2): where 6 ( t ) is the unit pulse function. In this case, the closed-loop transfer function becomes b( t ) = Hb(t) . (6) F R = (FR + Bq-kS)(l- 4-1) B I-q-1 h Figure 2: Equivalent scheme for a step disturbance present at y ( t ) . Following the lines from Section 2, the quality of the control is evaluated using 00 U; = y2(t). (7) t = l The current value of crz can, however, no longer be com- puted directly from the collected data. Instead it is use- ful to apply Parseval's identity, yielding The value of this h n d of integral can easily be com- puted, see for example [13]. The disadvantage is that to evaluate (8), we first need to calculate the closed loop transfer function H. This requires knowledge of the cur- rent controller parameters and an explicit model of the plant, hence it is not possible to use only the output for identification as done by Harris. Furthermore, to be able t o estimate the plant it may be necessary to perturb the process with extraneous test signals. The optimal crite- rion value cr,',, can also be computed based on the pulse response of the disturbance model, cf. (5). An index is then formed as (9) Evaluating the modified indices b) and c) for the above example yields for the Dahlin controller 1.02 and 1.5 re- spectively, and for the P controller we, of course, get infinite values. All the results, including the previously presented ones, are summarised in Table 1. It seems that the Dahlin controller was tuned, at least implicitly, t o get the best possible response to step output distur- bances (or step setpoint changes). If that was the in- tention it is clear that the Dahlin controller in this case gives acceptable performance. It is well known that a controller tuned for setpoint changes/output disturbances does not necessarily give a good response for input disturbances. This is also 1031 relatively well illustrated by the results in Table 1: and will be even more evident in the next section. control Dahlin 1.516 1.025 1.521 1.001 0;: Table 1: Three different performance indices. 5 Trials on industr ia l data In this section a couple of different control loops will be examined. The purpose is to illustrate that difficul- ties exist in deriving a meaningful performance index when the primary purpose of the control is unknown. It will also sometimes prove beneficial to restrict the opti- misation to controllers of a particular structure. Data has been collected from some control loops in the pulp and paper industry. 5.1 Dilution water control The first control loop examined is a loop in a wood chip refiner which controls the dilution water flow. A process engineer considered the PI-controller controlhng the process to be improperly tuned. He decided to see if it was possible to improve the performance of the con- trol loop. A standard bump test was used to retune the regulator. The new regulator parameters obtained were implemented and this improved the performance, according to the process engineer. The controller sen- sitivity to load disturbances was decreased significantly. These disturbances occur quite seldom but can affect the process output quite dramatically, and even trip the alarm to cause a plant shut-down. In another experiment a Pseudo Random Binary S e quence (PRBS) was added to the control signal to excite the modes of the process. The input and output signals were measured and used to estimate the process. In the identification the Box-Jenkins model structure (2) was assumed and an estimate obtained using the Sys- tem Identification Toolbox for Matlab 191. Simulations based on the estimated model agreed very well with val- idation data. The following process model was obtained for a sampling period of T, = 0.2 s: F ( q - l ) = 1 - 0.87q-'; B(q-') = 0.22: k = 7; D(q-1) = 1 - 0.9oq-1; C(q-1) = 1 + 0.01q-1. We have calculated the performance indices before and after the tuning of the PI-controller. The results are presented in Table 2. The original performance index (based on stochastic control) increases approximately 25% after retuning of the controller. According to the process engineer, however, the control performance had actually been improved. -1 2.16 8.18 Table 2: Performance indices for a dilution water loop. -1 After tuning 0.136 0.914 Table 3: PI parameters for dilution water control. The controller parameters used are presented in Ta- ble 3. Notice that the integration time was decreased significantly during tuning. Therefore, it is hardly sur- prising that the performance index increased, since there is nothing in the noise model that implies the need for integral action in the controller. On the contrary, the optimal controller within a PI structure will be a P con- troller ( K = 0.43 gives an index of 1.07). Hence, the user has to make up his mind whether minimum vari- ance control is the correct purpose. If not, the other two columns in Table 2 show that the performance was actually improved by tuning. It is, however, quite hard to interpret the two final columns of table 2, especially the last one. Even after tuning, the value 8.18 appears to indicate that the per- formance can be improved quite considerably. We felt it would be interesting to know how much was due to the restriction in the controller structure and how much was due to inadequate tuning. Therefore, we introduce an index that makes a comparison with the optimal PI controller instead, i.e. where u&,pI denotes the minimum value of the integral (8), when the controller structure is restricted to PI. The optimisation can be done numerically using, for example, Matlab. This new index evaluated for the dilution water loop is presented in Table 4. Obviously there is still room for some improvement, in particular if the loop is subject to input disturbances. step at y step at U PI after Table 4: Performance indices for dilution water loop, compared to optimal PI. At this point the reader might ask if minimisation of Cy2(t) really is the proper way of designing a PI 1032 controller, since this is what the current performance is compared with. Perhaps not, because it often leads to quite oscillatory behallour. None the less, the optimal controller can serve as a decent benchmark. There are design methods available that lead to very smooth re- sponses but still come very close to the optimal value. For output disturbances a design based on Internal Model Control normally gives good results ( see e.g. [7]), and for input disturbances the Dominant Pole Placement design by Astrom and coworkers works well (see e.g. [15]). In Figure 3 we show the responses to a step input dis- turbance for the two PI controllers given above, and the optimal one. We have also fine tuned the P I parameters using, in principle, the Dominant Pole Placement design. This is a design that, when compared t o the optimal PI, gave the performance index 1.43 (as compared to 2.79). n stoch. control I 0 2 4 6 8 10 12 14 16 18 20 saonds Conuol sirmai -0.51 ' 3; h ' 1 step step at 'U at U I 0 2 4 6 8 10 12 14 16 18 20 seconds -1.5 Figure 3: Simulated input disturbance steps, for dilution water control. a) Optimal PI controller b) Dominant pole placement P I c) P I after tuning d) P I before tuning. 5.2 A consistency loop As a second example we use data from a consistency loop (see Figure 4) in a paper machine at another Swedish paper mill. In Table 5 the different indices are sum- marised. Notice that once again the optimal P I con- troller for the first column (with index 1.93) is actually a P controller. Also notice that the controller presently in use obviously was tuned for the purpose of minimising the effect of output disturbances. Optimising subject to a P I controller structure clearly enhances the interpre- tation. But also bear in mind that the large d u e in row 1 column 3 is an indication that PI is too restrictive a controller structure for this case. It may therefore be interesting to see what is possible to accomplish with a PID controller (row 3 in Table 5 ) . We see that a change in the controller structure to PID can considerably im- prove the performance. 54 c 521 ' r""l 48 IW 46 I 0 50 100 150 200 250 300 350 400 450 500 seconds Control simd ! I I 20 7.X-J- 'e i lo: 50 100 lS0 200 i 0 300 350 4& 450 5 k seconds Figure 4: Closed loop data from a consistency loop 1 1 1 Restriction 1 trp: I/ Ll; 1.68 1 5.08 1 Restrict ion 2.16 11.15 Table 5: Performance indices for consistency loop. 6 Outline of a monitoring tool The goal of the authors' research is to develop a tool for assessment of the performance of typical control loops in the process industry. In the short term, such a tool probably has t o be a stand alone unit with its own soft- ware that hooks on to and collects data straight from the input of the process computer. In the long term: how- ever, we believe that such a function will be an integral part of any commercial control system. From the