Improving performance using cascade control and a Smith predictor

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Improving performance using cascade control and a Smith predictor

  1. 1. ISA Transactions 40 (2001) 223±234 www.elsevier.com/locate/isatrans Improving performance using cascade control and a Smith predictor Ibrahim Kaya * Inonu University, Engineering Faculty, Electrical and Electronics Department, 44069, Malatya, TurkeyAbstract Many investigations have been done on tuning proportional-integral-derivative (PID) controllers in single-inputsingle-output (SISO) systems. However, only a few investigations have been carried out on tuning PID controllers incascade control systems. In this paper, a new approach, namely the use of a Smith predictor in the outer loop of a cascadecontrol system, is investigated. The method can be used in temperature control problems where the secondary part of theprocess (the inner loop) may have a negligible delay while the primary loop (the outer loop) has a time-delay. Two dif-ferent approaches, including an autotuning method, to ®nd the controller parameters are proposed. It is shown bysome examples that the proposed structure as expected can provide better performance than conventional cascadecontrol, a Smith predictor scheme or single feedback control system. # 2001 Elsevier Science Ltd. All rights reserved.Keywords: PID controller; Cascade control; Smith predictor; Process control1. Introduction control, the corrective action for disturbances does not begin until the controlled variable deviates Many investigations have been done on tuning from the set point. A secondary measurement pointproportional-integral-derivative (PID) controllers and a secondary controller, Gc2, in cascade to thein single-input single-output (SISO) systems, such main controller, Gc1, as shown in Fig. 1, can beas Refs. [1±3]. However, the standard single feed- used to improve the response of the system to loadback control loop does not sometimes provide a changes.good enough performance for processes with long A typical example is the natural draft furnacetime delays and strong disturbances. Cascade con- temperature control problem [10,11], shown introl loops can be used and are a common feature in Fig. 2. When there is a change in hot oil temperature,the process control industries for the control of which may occur due to a change in oil ¯ow rate, thetemperature, ¯ow and pressure loops. conventional single feedback control system, Fig. 2, Cascade control (CC), which was ®rst introduced will immediately take corrective action. However,many years ago by Franks and Workey [4], is one if there is a disturbance in fuel gas pressure noof the strategies that can be used to improve the correction will be made until its e€ect reaches thesystem performance, particularly in the presence of temperature-measuring element. Thus, there is adisturbances. In conventional single feedback considerable lag in correcting for a fuel gas pres- sure change, which subsequently results in a slug- * Corresponding author. Fax: +90-422-3401046. gish response. With the cascade control strategy E-mail address: ikaya@inonu.edu.tr (I. Kaya). shown in Fig. 3, an improved performance can be0019-0578/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.PII: S0019-0578(00)00054-9
  2. 2. 224 I. Kaya / ISA Transactions 40 (2001) 223±234 outer loop of a cascade control. In temperature control, as given above, the process can often be treated as having two transfer functions in cascade with the one in the inner loop, generally, having, no or a small time-delay, while the one in the outer loop has a signi®cant time-delay. It is well known Fig. 1. Block diagram of a cascade control system. [9] that a Smith predictor structure provides good control for processes with a large time delay. Thus, combining a Smith predictor and cascade control strategy can provide a more advantageous struc- ture in the aforementioned situations. The main contribution of the paper is to provide compar- isons, which are fully compatible since optimiza- tion of an integral performance criterion has been used, between the performance with this structure and conventional CC, single feedback loop control and a Smith predictor scheme. The comparisons are carried out by simulations using SIMULINK. The next section outlines some important aspects of the cascade control strategy when compared to a single feedback loop control. Section 3 introducesFig. 2. The natural draft furnace temperature control with the improved cascade control (ICC) strategy andsingle feedback control. also a tuning method to ®nd parameters of the ICC method using an optimization approach which can be used when a mathematical model of the plant is available. Section 4 gives an autotuning method, which makes use of results from relay autotuning where the plant is ®rst approximated by a ®rst order plus dead time (FOPDT) or second order plus dead time (SOPDT), where it is relevant, determined from exact analysis of the limit cycle. The identi®cation procedure is not given here. The interested readers may refer to [16] or [17]. Illus- trative examples are given in Section 5 to show the value of the proposed structure. The paper ends with some conclusions in Section 6.Fig. 3. The natural draft furnace temperature control withcascade control.achieved, since any change in the fuel gas pressure 2. Cascade controlis immediately detected by the pressure-measuringelement and the pressure controller takes corrective Cascade control can improve the response to aaction. set point change by using an intermediate mea- Recent contributions on the tuning of PID con- surement point and two feedback controllers. Thistrollers in cascade loops include [5±8] which have improvement can be shown by examination of theonly concentrated on the conventional cascade transfer functions between the system output andcontrol con®guration. In this paper, an improved the disturbance for the cascade and conventionalcascade control (ICC) structure is proposed. The single feedback loop control con®gurations,idea is to use a Smith predictor structure in the respectively.
  3. 3. I. Kaya / ISA Transactions 40 (2001) 223±234 225 For a cascade control, from Fig. 1, the dis- disturbance. The time constant of designed innerturbance transfer function with d1=0 is given by closed loop should at least be three times larger than the outer loop time constant. The outer loopY1 Gp1 controller is chosen to be a PI or PID. The integral ˆ …I†D2 1 ‡ G™2 Gp2 ‡ G™1 G™2 Gp1 Gp2 action is now required to eliminate the low fre- quency disturbance and o€set in the system. Without the inner feedback loop, and of courseno controller Gc2, the transfer function betweenthe same variables is 3. Improved cascade control (ICC)Y1 Gp1 In many process control systems the plant can ˆ …P†D2 1 ‡ G™1 Gp1 Gp2 often be regarded as a cascade system in which the secondary part of the process (the transfer func-which is quite di€erent from Eq. (1). Therefore, tion in the inner loop) has no or a negligible time-because of the extra degrees of freedom, when delay while the primary part of the process (theappropriate values of the parameters of the two transfer function in the outer loop) has a time-controllers are chosen, the cascade control will delay. A Smith predictor strategy can give a satis-generally result in a better response. factory performance for set point changes, but the A cascade control structure has the following performance for disturbance rejection may not beadvantages over a single feedback loop control satisfactory. In this case, a cascade control can besystem [12]: used to improve the response of the system to dis- turbances. However, a cascade control strategy 1. The secondary controller is used to correct alone may not be enough if a long time delay disturbances arising within the inner loop exists in the outer loop, since it may result in a before they can a€ect the controlled variable. poor response for set point changes. 2. The e€ect of parameter variations in the Thus, this paper reports on the use of a Smith process Gp2 are corrected in the inner loop by predictor strategy, shown in Fig. 4, in cascade the secondary controller. control systems. The general strategy to design 3. The e€ect of any phase lag existing in Gp2 two controllers in CC structure can be followed may be reduced by the secondary loop, thus for the ICC as well. In the next subsection, opti- allowing the speed of response of the primary mization is used to ®nd the tuning parameters for loop to be improved. the two controllers, Gc1 and Gc2, in the ICC Thus, if there are additional measurable vari- structure. Then in section 4, an autotuningables, cascade control can provide better results. method, which makes use of results from the relayHowever, it must be stressed that the inner loop autotuning, is given.should include the major disturbances and be fas-ter reacting than the outer loop in order to achieve 3.1. Tuning ICC using optimizationa signi®cant improved system performance. To complete the design of a cascade control In this section integral performance criteria havesystem, the parameters of two controllers have to been used to determine tuning parameters for thebe determined. The general strategy to ®nd two controllers. There are several integral performancecontroller parameters can be outlined as follows: criteria which may be used to minimize the errorThe inner controller is usually chosen to be a P or signals, e1 and e2, such as the integral of squaredPI controller. The derivative action is not required error, ISE, the integral of absolute error, IAE, andas the inner process disturbance is noisy and its the integral of time-moment weighted squaredfrequency range is higher than the outer process error, ISTE. In this paper, the ISTE criterion hasdisturbance. The inner controller gain should be been chosen to ®nd the tuning parameters for thechosen high to reduce the e€ects of inner loop two controllers, since it normally results in step
  4. 4. 226 I. Kaya / ISA Transactions 40 (2001) 223±234 Fig. 4. Improved cascade control system.responses with a relatively small overshoot and a inner loop process transfer function is usually lowreasonable settling time and the integral can easily order and has no time delay. In this case, eitherbe evaluated in the s-domain [13]. one of the controller parameters, usually the con- In cascade control systems, the auxiliary con- troller gain, can be constrained to a value and thetroller, Gc2, is usually chosen as a PI or P controller remaining controller parameters can be found or aand the main controller, Gc1, as a PI or PID con- small time delay can be added in the optimizationtroller. This is understandable, since the dynamics procedure to ®nd the controller parameters.of the secondary part of the process, Gp2, are Once the parameters of the inner controller areusually low order and derivative action is often obtained, the optimization procedure can be carriednot required. Thus it is assumed that the two con- out to obtain the parameters of the outer or maintrollers have the following ideal transfer functions controller, Gc1, for the resultant plant Gp where 1 G™2 Gp1 Gp2G™1 ˆ Kp1 1 ‡ ‡ Td1 s …Q† Gp ˆ …T† Ti1 s 1 ‡ G™2 Gp2 1 The transfer function of the error signal e1 isG™2 ˆ Kp2 1 ‡ …R† Ti2 s then given by The easiest approach to applying optimization R1 E1 ˆ …U†techniques in a cascade control system to ®nd the 1 ‡ G™1 Gpparameters of the controllers is to optimize theerrors in the loops individually. For this, the inner which is used in the optimization procedure to ®ndloop is considered ®rst. The parameters of the the parameters of the main controller. The maininner loop controller, Gc2, can be obtained by di€erence between the proposed improved cascademinimizing the performance criterion for e2 which control and conventional cascade control is givenis given in the s-domain by by Eq. (7). A similar equation for a conventional cascade control will involve a time-delay in its R2 characteristic equation while here, since idealE2 ˆ …S† 1 ‡ G™2 Gp2 matching has been assumed for the Smith pre- dictor, it is free of the time-delay. It should be noticed that the e€ect of a step dis- Finally, it should be said that, although a Smithturbance d2 on the output y2 gives this same predictor con®guration is used in the outer loop ofequation. Another point which should be pointed the proposed ICC method and it is well knownout is that, the optimization procedure may not that a Smith predictor structure is sensitive to aalways lead to a solution for the inner loop as the mismatch in the plant and model dynamics, the
  5. 5. I. Kaya / ISA Transactions 40 (2001) 223±234 227ICC design gives quite satisfactory results in the For the outer loop a ®rst order plus dead timecase of a mismatch. This is illustrated later by (FOPDT) model may be inadequate, as the overallexamples. transfer function for this loop may be of a higher order. Here, a ®rst order or FOPDT model transfer function is used for the inner loop and a second4. Automatic tuning of ICC order plus dead time (SOPDT) model transfer func- tion for the outer loop. The procedure to ®nd the The controller design given in the previous section controller parameters can be carried out as follows.is based on the assumption that the plant transferfunctions are known. Model based design methods . When the controllers need to be tuned, switchsu€er performance degradation when modelling from the two controllers to the two relays. Seterrors exist and in this respect Smith predictor the heights of the relay in the outer loop tostructures are known to be particularly sensitive. A zero and in the inner one to an appropriaterelay autotuning procedure, which is shown in the value. Measure the limit cycle frequency, wo2,block diagram of Fig. 5, has the advantage that and amplitude, a2, for the inner loop.retuning can be easily accomplished. The Before proceeding further, it is necessaryapproach has been proposed by Hang et al. [14] to comment on tuning of the inner loop. Theand Zhuang and Atherton [5] for a conventional preferred approach is to use an ideal relaycascade system. Hang et al. [14] used limit cycle but if the inner loop plant transfer function isinformation to directly tune loops using Ziegler± of a low order, then one may not get a limitNichols rules [1] or re®ned Ziegler±Nichols rules cycle oscillation. To avoid the waste of time[15]. Zhuang and Atherton [5] stated that the and guarantee a limit cycle oscillation, aloops could be tuned by one of several methods relay with hysteresis and an extra knownwhich use the critical point information, such as circuit, such as an integrator, or a small time Êthe Ziegler±Nichols rules [1] or Astrom and Hag- È È delay can be used in the inner loop. Theglunds phase and gain margin method [2]. In their amount of hysteresis or time delay may havepaper, however, they used a model based tuning to be chosen relatively large, as a smallwhere ®rst an approximate ®rst order plus dead time amount may not yield a limit cycle of low(FOPDT) plant transfer function was found from enough frequency. In most practical caseslimit cycle measurements followed by describing some knowledge of the form of plant transferfunction analysis and then the controller para- function will be available to assist in selectingmeters were obtained using analytic equations the best approach.obtained from optimal responses of a ®rst order . Use the limit cycle information to estimateplus dead time (FOPDT) model [3]. the unknown parameters, based on exact Fig. 5. , Autotuning of improved cascade control (ICC) strategy.
  6. 6. 228 I. Kaya / ISA Transactions 40 (2001) 223±234 analysis, for the inner plant transfer function design method gives a very satisfactory perfor- [16]. mance. Both examples two and three are given to . If the process is modelled by a FOPDT trans- show the use of the autotuning method. Example fer function the tuning rules given by Zhuang two assumes a third order plant transfer function and Atherton [3] can be used, if it is modelled in the inner loop so that the ideal relay feedback by a ®rst order transfer function the tuning test can be carried out. The third example con- rules given by Kaya [17] can be used to ®nd siders a ®rst order plant transfer function in the suitable PI controller parameters. For con- inner loop a situation which was not considered venience, the tables needed in this paper are by either Hang et al. or Zhuang and Atherton. given in the next section, where it is relevent, With a ®rst order plant transfer function in the and formulae are given in the appendix. inner loop, the ideal relay feedback test cannot be . Switch the inner PI controller with the deter- performed as a phase lag of 180 is not achieved. mined parameters into the loop to replace the In the example an ideal relay and time delay are relay. Set the heights of the outer relay to an included in the loop to obtain a limit cycle. In all appropriate value. At the same time open cases, simulations are carried out in SIMULINK switch `S so the loop transfer function is the for comparison of design method performances. CC con®guration. Measure the limit cycle parameters, !o1 and a1, for the outer loop. Example 1. In this example the plant transfer . For the outer loop it is assumed that functions are given by Gpm eÀLm s , where Gpm is the delay free part of the model and Lm is the model time delay, eÀ10s can properly be modelled by a SOPDT Gp1 ˆ …20s ‡ 1†…3s ‡ 1† transfer function. Thus once the limit cycle parameters are measured for the outer loop, 3 the unknown parameters, namely the steady Gp2 ˆ state gain, Km, the two time constants, T1m …10s ‡ 1† and T2m, and the time delay, Lm, for Gpm eÀLm s can easily be evaluated using an Calculations show that no solution exists for exact parameter estimation method based on optimal PI (PID) controller settings for the ISTE asymmetric limit cycle data given by Kaya criterion when the plant has a transfer function and Atherton [16]. Note that one does not with no time delay and a relative order of less than try to ®nd a model for the outer plant three (four). The design problem can be overcome Gp1 eÀLs but rather for the overall plant by constraining one or more controller para- Gpm eÀLm s . Then, the PID tuning parameters meters. Thus, to ®nd optimal PI settings for the for the outer loop can be calculated using inner controller in the CC and ICC designs, the simple tuning formulae given by Kaya [17]. controller gain Kp2 was limited to 3.0 and optimal value of ``I found. Then the outer controller parameters for the cascade control strategy were5. Illustrative examples found. All the parameters are given in Table 1. Again, the aforementioned problem still exists for Three examples are given to illustrate the value calculating the outer controller parameters in theof the proposed method. In the ®rst example improved cascade control strategy and the PIDoptimization is used to ®nd the tuning parameters controller parameters in the Smith predictorfor the controllers. The performance of the ICC structure, since exact matching is assumed. Thus,design is compared to cascade control, Smith pre- the outer controller gain in the improved cascadedictor control and single feedback control systems. control and gain of the PID controller in theIn addition simulation results for mismatching Smith predictor structure were constrained tocases are also given and it is seen that the ICC $2.0$ to provide a fair comparison between the
  7. 7. I. Kaya / ISA Transactions 40 (2001) 223±234 229Table 1PI and PID parameters for Example 1 PID parametersMethod Kp Ti TdCC Gc2 3.0 9.600 ± Gc1 1.622 24.965 6.099 Gc2 3.0 9.600 ±ICC 1.0 15.512 À4.469 Gc1 2.0 19.385 0.363 5.0 22.246 2.773Single 0.545 31.267 10.430Smith 2.0 29.959 8.575 Fig. 7. Step and disturbance responses for +30% variations intwo design methods. A summary of the optimal time delay for Example 1.PID settings are listed in Table 1 for all the cases.Fig. 6 shows the responses to a set point change designs when there is a +30% and À30% varia-and a disturbance d2=-0.2 at t=200 for the dif- tion in the plant time delay. A positive variation inferent controller con®gurations. the plant time delay causes the overshoot to The cascade control gives a good response for increase while a negative variation in the plant timedisturbance rejection, but a poor response for set delay results in less overshoot. In both cases, thepoint change due to the large time delay. In contrast, proposed method gives superior performance tothe Smith predictor results in a good response for a the other design methods.set point change but a poor response for a dis- As the controller gains, Kp1 and Kp2, in the ICCturbance. The proposed method gives good respon- design were constrained, it is appropriate to inves-ses for both a set point change and disturbance tigate the e€ect of their choice on the system per-rejection, since it combines the best features of cas- formance. For this, the inner loop controller gaincade control and the Smith predictor con®guration. Kp2 was kept constant and the outer loop controllerThe disturbance rejection of the CC and ICC con- gain was varied. Control e€ort corresponding to®gurations is so good that it can hardly be seen. the three cases are shown in Fig. 9. As is seen from Figs. 7 and 8, respectively, give the step and the ®gure, the initial control e€ort increases with andisturbance responses for the di€erent controller increase in the gain Kp2. Therefore, the controller Fig. 8. Step and disturbance responses for À30% variations in Fig. 6. Step and disturbance responses for Example 1. time delay for Example 1.
  8. 8. 230 I. Kaya / ISA Transactions 40 (2001) 223±234 Table 2 PI tuning formulae for set-point changes L/T range 0.1±1.0 1.1±2.0 Criterion ISTE IST2E ISTE IST2E a1 0.712 0.569 0.786 0.628 b1 À0.921 À0.951 À0.559 À0.583 a2 0.968 1.023 0.883 1.007 b2 À0.247 À0.179 À0.158 À0.167 T1=7.213, T2=8.178 and L=10.030. Using Table 5 for a PID controller in a Smith predictor scheme gives Ti1=12.641 and Td1=1.397 whenFig. 9. Control signals for the inner loop with three di€erent Kp1 is limited to 1.5. To illustrate the accuracy ofvalues of Kp1 for the ICC design. modelling the overall system by a SOPDT model, Nyquist plots of both the overall system transfergain should not be constrained to very high function and the model obtained are given invalues. In general, the outer loop controller gain Fig. 10 and show good agreement.Kp1 should be chosen smaller than the inner loop To compare the performance of the ICC methodcontroller gain Kp2 so that the inner loop is faster with the single feedback and Smith predictor con-and a better performance can be achieved. trol, the relay feedback test was also performed for the overall plant G=Gp1Gp2 to obtain aExample 2. Assume the outer and inner loop plant FOPDT model to be used in the calculation of thetransfer functions are given by PID controller in a single feedback control system and SOPDT model in the Smith predictor scheme. eÀ9s The FOPDT model was used for the single feed-Gp1 ˆ …8s ‡ 1†…7s ‡ 1† back control since in reference [3], PI and PID tuning parameters were obtained based on a 4 FOPDT model. With the overall plant in the relayGp2 ˆ …s ‡ 1†3 feedback system, limit cycle parameters were obtained as !=0.133, Át1=20.788, amax=2.547 With the height of the outer relay set to zero andinner ideal relay to h1=1 and h2=À0.6 the limitcycle parameters for the inner loop were measuredas 1.685, 1.484, 0.642 and À0.384 for !, Dt1, amaxand amin respectively, see reference [16] for nota-tions. This data was used to ®nd the FOPDTmodel parameters as K=3.994, T=5.194 andL=0.906 using the estimation method given in[16]. These model parameters lead to Kp2=0.889and Ti2=5.616 when Table 2 for the PI controlleris used. Then the inner loop was switched from theinner relay to the inner PI controller with thedetermined tuning parameters and the outer idealrelay was set to h1=1 and h2=À0.6. The outerloop limit cycle parameters were measured as0.146, 18.583, 0.572 and -0.363$ for !, Át1, amax Fig. 10. Nyquist plots for the overall system and SOPDTand amin, respectively, which results in K=0.989, model for Example 2.
  9. 9. I. Kaya / ISA Transactions 40 (2001) 223±234 231and amin=À1.579 which result in K=3.999, T= Example 3. This example shows how the autotun-14.617 and L=15.673 for a FOPDT model and ing method can be used when the plant in theK=3.999, T1=6.015, T2=9.729 and L=11.773 inner loop does not have a phase lag of 180 . Thefor a SOPDT model. Therefore, the tuning para- plant transfer functions are given bymeters for the PID controller in a single feedbackcontrol system are Kp=0.270, Ti=20.028 and 2eÀ8s Gp1 ˆTd=6.080 from Table 3 and in a Smith predictor …10s ‡ 1†…5s ‡ 1†scheme are, from Table 5, Ti=15.682 andTd=3.554 when Kp is limited to 1.5. The step and 5 Gp2 ˆdisturbance responses are given in Fig. 11 for dif- …7:5s ‡ 1†ferent controller design methods. The ICC methodgives better performance both for a set point Since Gp2 is a ®rst order transfer function, achange and disturbance rejection. Fig. 12 illus- limit cycle cannot be obtained for the inner looptrates the case when the plant time delay has using an ideal relay. In this case, a time delay canchanged to 11.7, which corresponds to +30% be added in the relay feedback control loop toincrease. The tuning parameters were kept thesame as before. Fig. 13 shows the responsesobtained after retuning for the changed timedelay. A better closed loop step response, which isalmost the same as for the matching case, isobtained with the retuning.Table 3PID tuning formulae for set-point changesL/T range 0.1±1.0 1.1±2.0 2Criterion ISTE IST E ISTE IST2Ea1 1.042 0.968 1.142 1.061b1 À0.897 À0.904 À0.579 À0.583a2 0.987 0.977 0.919 0.892b2 À0.238 À0.253 À0.172 À0.165 Fig. 12. Step and disturbance responses for +30% change ina3 0.385 0.316 0.384 0.315 time delay for Example 2.b3 0.906 0.892 0.839 0.832 Fig. 13. Step and disturbance responses for +30% change in Fig. 11. Step and disturbance responses for Example 2. time delay for Example 2 after retuning.
  10. 10. 232 I. Kaya / ISA Transactions 40 (2001) 223±234Table 4PI tuning formulae based on a ®rst order modelKKp 1.50±2.50 2.60±5.00 5.10±15.00 2 2Criterion ISTE IST E ISTE IST E ISTE IST2Ea 1.9443 1.3037 1.5556 1.1209 1.2149 1.0196b À0.4722 À0.2426 À0.2173 À0.0666 À0.0679 À0.0075Table 5PID tuning formulae based on a second order modelKKp 1.50±2.50 2.60±5.00 5.10±15.00Criteron ISTE IST2E ISTE IST2E ISTE IST2Ea1 0.6781 0.6338 0.5961 0.5554 0.5169 0.4950b1 À0.2709 À0.2410 À0.1144 À0.0888 À0.0323 À0.0184a2 0.1058 0.1057 0.2446 0.2453 0.4049 0.3978b2 1.3371 1.3041 0.3877 0.3637 0.0767 0.0640obtain a limit cycle and hence estimate theunknown plant transfer function parameters. Inthis example, a time delay of 1.0 was used in theinner loop to obtain a limit cycle oscillation. Withthe chosen time delay and relay parameters ofh1=1, h2=À0.6 and Á=0, a limit cycle oscillationwith !=1.587, Át1=1.542, amax=0.624 andamin=À0.374 were obtained. These limit cycledata result in K=5.003, T=7.506 and L=0.999.Table 4 was used to obtain the inner controllerparameters. Kp2 was limited to 2.5 and then Ti2was calculated as 6.296. The inner controller wasthen switched in with the calculated parameters.To obtain a SOPDT model for Gpm eÀLm s , theouter relay was set to the same parameters used in Fig. 14. Nyquist plots for the overall system and the modelthe inner relay. Limit cycle parameters were mea- obtained for Example 3.sured as !=0.167, Á=16.975, amax=1.058 andamin=À0.759. Then the SOPDT model para-meters were found as K=1.999, T1=10.368, were obtained as !=0.121, Át1=22.364, amax=T2=4.704 and L=8.640. The parameters of the 5.139 and amin=À3.221 which results in K=outer PID controller were calculated using Table 5 10.000, T=20.874 and L=16.066 for a FOPDTas Ti1=14.448, Td1=3.253 when Kp1 is limited to model and K=10.000, T1=10.306, T2=10.306 and4.0. Again Nyquist plots for the overall system and L=10.986 for a SOPDT model. Therefore, the tun-the model obtained are given in Fig. 14 to see the ing parameters for a PID controller using Table 3 inaccuracy of modelling. The ®gure clearly shows a single feedback control system are Kp=0.132,that the model obtained has a good accuracy. Ti=25.968 and Td=6.339 and in a Smith pre- The FOPDT model was used for the single feed- dictor scheme, using Table 5, are Ti=21.477 andback loop as in Example 2. With the overall plant in Td=4.979 when Kp is limited to 1.0. The responsesthe relay feedback system, limit cycle parameters for a set point change and a disturbance of d2=
  11. 11. I. Kaya / ISA Transactions 40 (2001) 223±234 233 Appendix. Tuning formulae and tables This section gives some tuning formulae which are used in the paper to ®nd controller parameters. These formulae were obtained using repeated opti- mizations, using the ISTE or IST2E criteria, on an error signal and then least square ®tting method was used to ®nd coecients in the formulae. A1. Tuning based on a FOPDT The tuning formulae are given [3] by a1 L b1 Fig. 15. Step and disturbance responses for Example 3. Kp ˆ …V† K TÀ0.2 at t=150 are given in Fig. 15. The far superior T Ti ˆ …W†performance of the ICC method is obvious. a2 ‡ b2 …L=T† b3 L6. Conclusions T d ˆ a3 T …IH† T An improved cascade control strategy has been where the (a, b) coecients for a PI and PID con-introduced in this paper. The structure brings troller are given in Tables 2 and 3 respectively.together the best merits of a cascade control andthe Smith predictor structure. The ICC structure A2. Tuning based on a ®rst order modelcan be used in process control problems, such astemperature, ¯ow or pressure control. Two design The tuning formula [17] ismethods have been given to ®nd the tuning para-meters of the PI controller in the inner loop and 1 aÀ Áb ˆ KKp …II†PID controller in the outer loop. Since the plant Ti Ttransfer functions are assumed to be low order,constraining some tuning parameters in the opti- The controller gain is ®rst speci®ed, which is chosenmization may be necessary. This is usually done by so that the normalized gain KKp falls into one oflimiting the gain of the controller to some value the ranges given in Table 4, the controller integralsince this a€ects the maximum magnitude of the time constant Ti is then calculated from Eq. (11).control signal and then performing the optimiza- The coecients in the formula over the varioustion. An autoning method, which is an extention ranges for KKp are listed in Table 4.of the relay feedback tuning for SISO systems, hasalso been given. The autotuning method may be A3. Tuning based on a second order modelmore useful in practice since in the case of a mis-match retuning can be done to obtain new model The tuning formulae are given [17] byparameters and therefore new tuning parameters. The performance of the ICC is compared to single 1 a1 À Áb T1 1=2 ˆ KKp 1 …IP†feedback control, Smith predictor structure and Ti T1 T2cascade control strategy and it is shown by examples À Áb2 T2 1=2that the ICC structure can give a better perfor- Td ˆ T1 a2 KKp …IQ†mance. T1
  12. 12. 234 I. Kaya / ISA Transactions 40 (2001) 223±234 Again the controller gain is ®rst limited to a value Dynamics and Control, John Wiley Sons, New York,and then the remaining controller parameters are 1989.calculated from Eqs. (12) and (13). [12] F.G. Shinskey, Process-Control Systems, McGraw Hill, New York, 1967. The (a, b) coecients are given in Table 5 for [13] M. Zhuang. D.P. Atherton, Tuning PID controllers withdi€erent ranges of KKp. integral performance criteria, in: Matlab Toolboxes and Applications, Peter Peregrinus, 1993, Chapter 8, pp. 131± 144.References [14] C.C. Hang, A.P. Loh, V.U. Vasnani, Relay feedback auto-tuning of cascade controllers, IEEE Trans. Control Systems Technology 2 (1) (1994) 42±45. [1] J.G. Ziegler, N.B. Nichols, Optimum settings for auto- [15] Ê C.C. Hang, K.J. Astrom, W.K. Ho, Re®nements of the È matic controllers, Trans. of ASME 64 (1942) 759±768. Ziegler±Nichols tuning formula, IEE Proc. D, Control Ê [2] K.J. Astrom, T. Hagglund, Automatic tuning of simple È È Theory and Application 138 (2) (1991) 111±118. regulators with speci®cations on phase and gain margins, [16] I. Kaya, D.P. Atherton, An improved parameter estima- Automatica 20 (5) (1984) 645±651. tion method using limit cycle data, UKACC Int. Con- [3] M. Zhuang, D.P. Atherton, Automatic tuning of optimum ference on Control 98, Swansea, UK 1 (1998) 682±687. PID controllers, IEE Proceedings-D 140 (3) (1993) 216± [17] I. Kaya. Relay Feedback Identi®cation and Model Based 224. Controller Design, PhD thesis, University of Sussex, [4] R.G. Franks, C.W. Worley, Quantitive analysis of cascade Brighton, England, 1999. control, Industrial and Engineering Chemistry 48 (6) (1956) 1074±1079. [5] M. Zhuang, D.P. Atherton, Optimum cascade PID con- troller design for SISO systems, UKACC Control 94 Ibrahim Kaya was born in 1971, (1994) 606±611. Diyarbakõr, Turkey. He grad- [6] F.S. Wang, W.S. Juang, C.T. Chan, Optimal tuning of uated from Gaziantep Uni- PID controllers for single and cascade control loops, versity, Electrical Electronics Chem. Eng. Comm. 132 (1995) 15±34. Department, in 1994. In the [7] C.T. Huang, C.J. Chou, L.Z. Chen, An automatic PID same year, he started to work controler tuning method by frequency response techni- as a research assistant at Inonu ques, The Canadian Journal of Chemical Engineering 75 University. In 1996, he started (1997) 596±604. his PhD studies at Sussex Uni- [8] Y. Lee, S. Park, M. Lee, PID controller tuning to obtain versity, Brighton, England. In desired loop responses for cascade control systems, Ind. 2000, he ®nished his PhD and Eng. Chem. Res. 37 (1998) 1859±1865. returned back to Inonu Uni- [9] O.J. Smith, A controller to overcome dead-time, ISA J. 6 versity. He is interested in (2) (1959) 28±33. Relay Feedback Identi®cation,[10] E.F. Johnson, Automatic Process Control, McGraw Hill, Relay Autotuning, PID Controllers, Time Delay Systems, New York, 1967 (pp. 244). Computer-Aided Control System Design and User Interface[11] D.E. Seborg, T.F. Edgar, D.A. Mellichamp, Process Toolkits.

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