560 Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 Fig. 1. The Smith predictor control scheme.Matausek and Micic ͓5͔ who have shown that by ˇ ´ cesses. Here, it is shown that the structure can alsoadding an additional gain the disturbance response be used to control processes with open loop un-of the Smith predictor can be controlled indepen- stable poles by using a gain only controller in andently of the set point response. The result given inner feedback loop.in Ref. ͓5͔ was later improved by the same authors For autotuning of the proposed Smith predictorby adding a PD controller, instead of a propor- structure, plant model transfer functions are ﬁrsttional only controller, so that a better disturbance obtained using exact limit cycle analysis from arejection can be achieved. More recently, Normey- single relay feedback test. Once the plant modelRico and Camacho ͓6͔ also proposed a new ap- transfer functions are obtained from the relay au-proach to design dead-time compensators for totuning, the parameters of the PI-PD controller instable and integrating processes. the Smith predictor can be calculated from the for- It has been shown that a PI-PD controller can mulas provided. Excellent performance of the pro-give better closed loop responses for processes posed PI-PD Smith predictor over some existingwith large time constants ͓7͔, an integrator ͓8͔, or design methods is illustrated by simulations forunstable open loop poles ͓9͔. In Kaya and Ather- stable, integrating, and unstable process transferton ͓7͔ controller parameters for a PI-PD control- functions.ler in a Smith predictor scheme were obtained us-ing standard forms. The difﬁculty with the designis to involve a trade-off between selected values of 2. Model parameter estimationK p and T i , respectively, the gain and integral timeconstant of the PI controller in the forward path. Recently, the relay feedback test based identiﬁ-The aim of this paper is twofold. First, the control- cation, which is used in this paper, has been veryler design suggested in Kaya and Atherton ͓7͔ is popular. There are several reasons behind the suc-extended by providing simple tuning rules. The cess of the relay feedback method. First, the relayprovided simple tuning formulas have physically feedback method, as normally used, gives impor-meaningful parameters, namely, the damping ratio tant information about the process frequency re- and the natural frequency o to tune a stable sponse at the critical gain and frequency, whichsecond-order plus dead time ͑SOPDT͒ and an are the essential data required for controller de-integrating ﬁrst-order order plus dead-time sign. Second, the relay feedback method is per-͑IFOPDT͒ plant transfer functions and closed loop formed under closed loop control. If appropriatetime constant to tune an unstable ﬁrst-order plus values of the relay parameters are chosen, the pro-dead time ͑UFOPDT͒ plant transfer function. The cess may be kept in the linear region where thevalues of the damping ratio and natural fre- frequency response is of interest. Third, the relayquency o have been obtained based on desired feedback method eliminates the need for a carefulovershoot and the rise time and the value of time choice of frequency, which is the case for tradi-constant has been obtained based on the user tional methods of process identiﬁcation, since anspeciﬁed settling time. Second, the Smith predic- appropriate signal can be generated automatically.tor structure given in Kaya and Atherton ͓7͔ were Finally, the method is so simple that operators un-suggested only for stable and integrating pro- derstand how it works.
Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 561 Fig. 2. The proposed Smith predictor control scheme. Luyben ͓10͔ was one of the ﬁrst to consider es- K m e ϪL m stimating the plant transfer function from limit G 1͑ s ͒ ϭ , ͑1͒ ͑ T 1m sϩ1 ͒͑ T 2m sϩ1 ͒cycle measurements and used the approximate de-scribing function ͑DF͒ method. Several authors K m e ϪL m s͓11,12͔ have presented further approaches, which G 2͑ s ͒ ϭ , ͑2͒ s ͑ T m sϩ1 ͒make use of the approximate DF method. The fact that exact expressions can be found for K m e ϪL m sa limit cycle in a relay feedback system has been G 3͑ s ͒ ϭ . ͑3͒ T m sϪ1known for many years ͓13–15͔. Atherton ͓16͔showed how knowledge of the exact solution for The details of parameter estimation are not givenlimit cycles in relay controlled ﬁrst-order plus here, since interested readers can refer to Kayadead time ͑FOPDT͒ plants could be used to give and Atherton ͑Ref. ͓20͔ or Ref. ͓24͔͒ to determinemore accurate results using the DF method. Re- unknown parameters of the related plant transfercently, several papers ͓17–19͔ have been written function. However, equations for a stable SOPDT,on using exact analysis for parameter estimation in IFOPDT, and UFOPDT plant transfer functions toa relay feedback system, assuming a speciﬁc plant identify its unknown parameters are given in thetransfer function and an odd symmetrical limit Appendix for convenience.cycle. In Kaya and Atherton ͓20͔ asymmetricallimit cycle data is used, however, the effect ofstatic load disturbances is not considered. 3. The new PI-PD Smith predictor Use of expressions based on symmetrical limit conﬁgurationcycles may thus lead to signiﬁcant errors in theestimates under static load disturbances, which In the conventional PID control algorithm, thecause asymmetrical limit cycles. There are only a proportional, integral, and derivative parts arefew works ͓21–23͔ which consider relay autotun- placed in the forward loop, thus acting on the erroring under static load disturbances. All consider between the set point and closed loop response.calculation of the ultimate gain and frequency by This PID controller implementation may lead toﬁrst estimating the disturbance and then injecting an undesirable phenomena, namely the derivativea signal to make the limit cycle odd symmetrical. kick. Also by moving the P͑D͒ part into an innerKaya and Atherton ͓24͔ derived exact expressions feedback loop an unstable or integrating processfor the simple features of asymmetrical limit can be stabilized and the pole locations for a stablecycles in relay controlled loops with both stable process can be modiﬁed. Therefore the new PI-PDand unstable FOPDT and SOPDT plant transfer Smith predictor conﬁguration ͓7͔ is shown in Fig.functions in the presence of static load distur- 2, where G c1 ( s ) is a PI controller, G c2 ( s ) is a PDbances. This enables the parameters to be esti- controller, and G d ( s ) is the disturbance rejectionmated directly, which eliminates the need to try to controller. G c2 ( s ) , as mentioned above, is used toobtain a symmetrical limit cycle. The following stabilize an unstable or integrating process andtransfer functions are used to model a plant trans- modify the pole locations for a stable process. Thefer function by a stable SOPDT, IFOPDT, and other two controllers, G c1 ( s ) and G d ( s ) , are usedUFOPDT, respectively: to take care of servo tracking and regulatory con-
562 Ibrahim Kaya / ISA Transactions 42 (2003) 559–575trol, respectively. When G c2 ( s ) ϭG d ( s ) ϭ0 then the disturbance response of the Smith predictorthe standard Smith predictor is obtained. can be controlled independently of the controllers Assuming exact matching between the process G c1 ( s ) and G c2 ( s ) , which are the controllers usedand the model parameters, that is G ( s ) ϭG m ( s ) for servo tracking and designed by using pole zeroand LϭL m , then the set point and disturbance re- cancellations. The controller G d ( s ) , which is usedsponses are given by to control the disturbance rejection, is designed based on the Nyquist stability criteria. Therefore C ͑ s ͒ ϭT r ͑ s ͒ R ͑ s ͒ ϩT d ͑ s ͒ D ͑ s ͒ , ͑4͒ both the set point response and disturbance rejec- tion of the proposed PI-PD Smith predictor resultwhere in better performance when compared to existing G m ͑ s ͒ G c1 ͑ s ͒ e ϪL m s PI͑D͒ Smith predictor controllers. T r͑ s ͒ ϭ , ͑5͒ 1ϩG m ͑ s ͓͒ G c1 ͑ s ͒ ϩG c2 ͑ s ͔͒ G m ͑ s ͒ ͕ 1ϩG m ͑ s ͓͒ G c2 ͑ s ͒ ϩG c1 ͑ s ͒ ϪG c1 ͑ s ͒ e ϪL m s e ϪL m s 4. Development of autotuning formulas T d͑ s ͒ ϭ . ͕ 1ϩG m ͑ s ͓͒ G c1 ͑ s ͒ ϩG c2 ͑ s ͔͒ ͖ ϫ ͓ 1ϩG d ͑ s ͒ G m ͑ s ͒ e ϪL m s ͔ Case 1: Processes which can be modeled by a ͑6͒ stable SOPDT For this case, the delay free part of the SOPDTThe transfer function for the set point response, model isgiven by Eq. ͑5͒, reveals that the parameters of thetwo controllers, G c1 ( s ) and G c2 ( s ) , may be deter-mined using a model of the delay free part of the Km G m͑ s ͒ ϭ ͑7͒plant. In addition, it is seen that only the distur- ͑ T 1m sϩ1 ͒͑ T 2m sϩ1 ͒bance response T d ( s ) is affected by the controllerG d ( s ) . It has been shown ͓2͔ that the originalSmith predictor gives a steady-state error under and the controllers G c1 ( s ) and G c2 ( s ) have thedisturbances for open loop integrating processes. formsThat is why the controller G d ( s ) has been adoptedin the proposed method, again primarily to im-prove disturbance rejection for integrating and un-stable processes transfer functions. ͩ G c1 ͑ s ͒ ϭK p 1ϩ 1 T isͪ, ͑8͒ The proposed PI-PD Smith predictor controlstructure gives superior performance over classical G c2 ͑ s ͒ ϭK f ͑ 1ϩT f s ͒ . ͑9͒PI or PID Smith predictor control conﬁgurationfor both the set point response and disturbance re- The controller G d ( s ) is not needed in this case asjection. The superior performance of the proposed the plant is a stable and nonintegrating one. Sub-Smith predictor is more evident when the process stituting Eqs. ͑7͒–͑9͒ into Eq. ͑5͒, letting T ihas a large time constant, with or without an inte- ϭT 1m and T f ϭT 1m , the delay free part of thegrator or an unstable pole. This is illustrated later closed loop transfer function for the servo trackingby examples. However, the proposed Smith pre- becomesdictor conﬁguration still suffers from a mismatchbetween the actual process and model dynamics,which is a case also for classical PI͑D͒ Smith pre- K mK pdictor scheme. Another point which must be men- T 1m T 2mtioned about the performance of the proposed T r͑ s ͒ ϭ 1 K mK pPI-PD Smith predictor design is that one can think s 2ϩ ͑ 1ϩK m K f ͒ sϩthat due to pole zero cancellation in the controller T 2m T 1m T 2mdesign procedure, as will be given in the next sec- 2 otion, the load disturbance rejection may be slug- ϭ , ͑10͒gish. However, as is seen from Eqs. ͑5͒ and ͑6͒, s 2 ϩ2 o sϩ 2 o
Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 563where o and are the natural frequency and h 2 ϭϪ0.6 and ⌬ϭ0 was performed. The staticdamping ratio to be speciﬁed in the design. Com- load disturbance was assumed to be dϭ0.05. Thenparing the left-and right-hand side of Eq. ͑10͒, measured limit cycle parameters, ϭ0.088, a max ϭ2.24, a minϭϪ0.64, and ⌬t 1 ϭ35.86, were mea- T 1m T 2m 2 o sured and used in the identiﬁcation procedure K pϭ , ͑11͒ Km given in the Appendix Section 1 to ﬁnd the SOPDT model G m1 ( s ) ϭ4e Ϫ14.69s / ( 21.62sϩ1 ) 2T 2m o Ϫ1 ϫ( 11.34sϩ1 ) . Fig. 3 illustrates the Nyquist plots Kfϭ ͑12͒ ¨ Km for the actual plant proposed, Hang’s and Hag- glund’s obtained models. For this example theare obtained. The other controller parameters are proposed identiﬁcation method and the identiﬁca- T i ϭT 1m ͑13͒ tion method proposed by Hang et al. ͓26͔ gives quite similar estimation results while the identiﬁ-and cation method used by Hagglund ͓27͔ gives poor ¨ estimation results. The reason for the identiﬁcation T f ϭT 1m . ͑14͒ method proposed by Hang et al. ͓26͔ gives betterWhen o and in Eqs. ͑11͒ and ͑12͒ are found the estimations is that the time constants used in thisdesign will be complete. For this, time domain example are larger, which cause the limit cyclespeciﬁcations, namely, the maximum overshoot oscillation to be a good sinusoidal and thereforeand the rise time, will be used. The relation be- eliminating the approximation used in the DFtween the maximum overshoot ͑M͒, the rise time method. The reason for poor estimates obtained by( T r ) , the damping ratio ͑͒, and the natural fre- ¨ Hagglund is that the open loop step becomes inaf-quency ( o ) is given ͓25͔ by fective for processes with large time constants. Speciﬁng a 1% overshoot and 10-s rise time gives ͱ1Ϫ 2 ϭ0.83 and o ϭ0.2664. The PI-PD controller M ϭe Ϫ / ͑15͒ parameters K p and K f were calculated as 4.350and and 1.004, respectively, using Eqs. ͑11͒ and ͑12͒. The other controller parameters are T i ϭT 1m 1Ϫ0.4167 ϩ2.917 2 T rϭ . ͑16͒ ϭ21.620 and T f ϭT 1m ϭ21.620. The controller o ¨ parameters for Hang’s and Hagglund’s designs areRearranging Eqs. ͑15͒ and ͑16͒, the equations K p ϭ0.0625, T i ϭ15.410 and K p ϭ0.250, T i ϭ31, respectively. Fig. 4 shows responses for a unity ϭ ͱ ϩ M M ln 2 ͑ ln ͑ ͒2 ͒2 ͑17͒ step input and disturbance with magnitude of d ϭϪ0.2 at time tϭ200 s for all three design meth- ods. The superior performance of the proposed de-and sign is now clear. Fig. 5 illustrates the good re- sponse of the proposed structure and design in the 1Ϫ0.4167 ϩ2.917 2 case of Ϯ10% change in the plant time delay. oϭ ͑18͒ Tr Case 2: Processes which can be modeled by IFOPDTcan be obtained. Therefore, for the speciﬁed maxi- The delay free part of the IFOPDT model ismum overshoot and rise time, and o can befound from Eqs. ͑17͒ and ͑18͒, respectively. Oncethe value of and o is calculated, K p and K f , Kmrespectively, can be found from Eqs. ͑11͒ and ͑12͒. G m͑ s ͒ ϭ . ͑19͒ s ͑ T m sϩ1 ͒The values of T i and T f are given by Eqs. ͑13͒ and͑14͒, respectively.Example 1 The controllers G c1 ( s ) and G c2 ( s ) are again A fourth-order plant transfer function given given by Eqs. ͑8͒ and ͑9͒. Carrying out the sameby G p1 ( s ) ϭ4e Ϫ10s / ( 20sϩ1 )( 10sϩ1 )( 5sϩ1 )( s procedure as before, the delay free part of theϩ1 ) is considered. In order to ﬁnd the SOPDT closed loop transfer function for servo control, let-transfer function model the relay test with h 1 ϭ1, ting T i ϭT m and T f ϭT m , is obtained as follows:
564 Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 Fig. 3. Nyquist plots for example 1. Fig. 4. Step responses for example 1.
Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 565Fig. 5. Step responses for example 1: ͑a͒ for nominal Lϭ10, ͑b͒ for ϩ10% change in the plant time delay, ͑c͒ for Ϫ10%change in the plant time delay. K mK p The controller G d ( s ) , Tm 2T r͑ s ͒ ϭ ϭ o . G d ͑ s ͒ ϭK d ͑ 1ϩT d s ͒ , ͑25͒ K mK p s 2 ϩ2 o sϩ 2 o s 2 ϩK f K m sϩ is now necessary for a satisfactory load distur- Tm ͑20͒ bance rejection. G d ( s ) is designed based on the stabilization of the second part of the characteris-Then, one can obtain the controller parameters, by tic equation of Eq. ͑6͒,comparing the left- and right-hand sides of Eq.͑20͒, as 1ϩG d ͑ s ͒ G m ͑ s ͒ e ϪL m s ϭ0. ͑26͒ T m 2 o Matausek and Micic ͓28͔ assumed a relation T d ˇ ´ K pϭ , ͑21͒ ϭ ␣ L m to obtain Km 2 o /2Ϫ⌽ pm Kfϭ . ͑22͒ K dϭ Km K m L m ͱ͑ 1Ϫ ␣ ͒ 2 ϩ ͑ /2Ϫ⌽ pm ͒ 2 ␣ 2The other controller parameters are ͑27͒ T i ϭT m ͑23͒ for a speciﬁed phase margin ⌽ pm . It has to be noted that ⌽ pm is not the phase margin corre-and sponding to the system open loop transfer func- T f ϭT m . ͑24͒ tion. The best results can be obtained ͓28͔ with ␣ ϭ0.4 and ⌽ pm ϭ64°. It should be pointed outTherefore ﬁrst and o are, respectively, obtained that Matausek and Micic ͓28͔ use a pure integrator ˇ ´from Eqs. ͑17͒ and ͑18͒ and subsequently K p from plus dead-time process model to ﬁnd K d as givenEq. ͑21͒ and K f from Eq. ͑22͒. T i and T f are given by Eq. ͑27͒. Therefore, since here the IFOPDTby Eqs. ͑23͒ and ͑24͒, respectively. model is used, in Eq. ͑27͒ L m ϭT e ϩL m , where T e
566 Ibrahim Kaya / ISA Transactions 42 (2003) 559–575Fig. 6. Nyquist plots for example 2: ͑a͒ the actual plant, ͑b͒ the proposed model, ͑c͒ the model used by Matausek and Micic. ˇ ´is the sum of the equivalent time constants, must ˇ ´ used in Matausek and Micic. The responses to abe used. unit set point and a disturbance change, which isExample 2 of magnitude Ϫ0.1 at tϭ50 s, are given in Fig. 7. An integrating process given by G p2 ( s ) For this example only a small improvement overϭe Ϫ5s / s ( s ϩ 1 )( 0.5s ϩ 1 )( 0.2s ϩ1 )( 0.1s ϩ1 ) , ˇ Matausek’s approach is obtained. Good step re-which was given in Matausek and Micic ͓28͔, is ˇ ´ sponses for Ϯ10% mismatch in the plant andconsidered. The IFODPT model was obtained as model time delays case, which is the most deterio-G m2 ( s ) ϭe Ϫ5.72s /s ( 1.18sϩ1 ) using the identiﬁca- rative to system performance, are given in Fig. 8.tion method given in Appendix Section 3. The re- Example 3lay parameters were h 1 ϭ1, h 2 ϭϪ0.7 and ⌬ϭ0. Consider G p3 ( s ) ϭe Ϫ6.7s /s ( 10sϩ1 ) , where theThe static load disturbance was assumed to be d plant has both an integrator and a relatively largeϭ0.2. a maxϭ0.99, a minϭϪ0.69, ϭ0.48, and time constant. The relay parameters used in the⌬t 1 ϭ6.61 were the measured limit cycle data. relay feedback test were h 1 ϭ1, h 2 ϭϪ0.7, andFig. 6 shows the Nyquist plots for the actual plant, ⌬ϭ0. The static load disturbance was assumed tothe model obtained by the proposed method, and be dϭ0.2. With the measured limit cycle data, ˇ ´the model used by Matausek and Micic. As is seen a maxϭ0.59, a minϭϪ0.24, ϭ0.29, and ⌬t 1from the ﬁgure both estimation methods result in ϭ11.08, the IFOPDT model was identiﬁed ex-good estimates. Requesting a maximum overshoot actly using the relay estimation method given inof 1% and a rise time of 5 s gives K p ϭ0.335 and the Appendix Section 3. Fig. 9 shows the NyquistK f ϭ0.885, using Eqs. ͑21͒ and ͑22͒. The other plots for the actual plant, the model obtained bycontroller parameters are T i ϭT m ϭ1.18 and T f the proposed method, and the model used by Ma-ϭT m ϭ1.18. K d was calculated from Eq. ͑27͒ as ˇ ´ tausek and Micic. Note that since the model ob-0.1049, for ␣ ϭ0.4 and ⌽ pm ϭ64°. Also, T d tained by the proposed method matches the actualϭ ␣ L m ϭ2.76. Note that here L m ϭT e ϩL m ϭ1.18 plant exactly, its Nyquist plot intersects with theϩ5.72ϭ6.9 was used. In Matausek and Micic ˇ ´ actual plant’s Nyquist plot and hence cannot be͓28͔ K d ϭ0.1065 and T d ϭ2.72 were used. Also, a ˇ seen while the model used by Matausek and Micic ´proportional only controller with gain 0.56 was is quite poor as seen from the ﬁgure. The reason
Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 567 Fig. 7. Step responses for example 2.Fig. 8. Step responses for example 2: ͑a͒ for nominal Lϭ5, ͑b͒ for ϩ10% change in the plant time delay, ͑c͒ for Ϫ10%change in the plant time delay.
568 Ibrahim Kaya / ISA Transactions 42 (2003) 559–575Fig. 9. Nyquist plots for Example 3: ͑a͒ the actual plant, ͑b͒ the proposed model, ͑c͒ the model used by Matausek and Micic. ˇ ´for this is the large time constant. The controller closed loop transfer function for servo control, let-gains K p and K f were obtained as 2.841 and ting T i ϭT m ϩ2T f and K m K f ϭ2, is obtained as0.885, respectively, using Eqs. ͑21͒ and ͑22͒, for a follows:speciﬁed value of 1% overshoot and 5-s rise time.The other controller parameters are T i ϭT m ϭ10 1 1 T r͑ s ͒ ϭ ϭ , ͑29͒and T f ϭT m ϭ10. Using Eq. ͑27͒, K d ϭ0.0434 was ͑ T i /K m K p ͒ sϩ1 sϩ1obtained for ␣ ϭ0.4 and ⌽ pm ϭ64°. Also, T dϭ ␣ L m ϭ6.68. Matausek’s method ͓28͔ has the ˇ where is the closed loop design parameter. Let-same K d and T d values and the main controller ting K m K p ϭ1 results in T i ϭ . Note that it wasgain of 0.1. The responses to a unit set point chosen such that T i ϭT m ϩ2T f . Hence T f ϭ ( change and disturbance of dϭϪ0.1 at tϭ100 s ϪT m ) /2. Therefore the controller parameters areare given in Fig. 10. The far superior performance given byof the proposed design method is now evident. 1Fig. 11 shows the good response of the proposed K pϭ , ͑30͒structure and design in the case of Ϯ10% change Kmin the plant time delay. T iϭ , ͑31͒Case 3: Processes which can be modeled byUFOPDT 2 The delay free part of the unstable FOPDT is Kfϭ , ͑32͒given by Km Km ϪT m G m͑ s ͒ ϭ . ͑28͒ Tfϭ . ͑33͒ ͑ T m sϪ1 ͒ 2Two main controllers G c1 ( s ) and G c2 ( s ) are The value of , which is the desired closed loopagain given by Eqs. ͑8͒ and ͑9͒. Following a simi- time constant, can be found based on the userlar procedure as before the delay free part of the speciﬁed settling time. The settling time is deﬁned
Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 569 Fig. 10. Step responses for example 3.Fig. 11. Step responses for example 3: ͑a͒ for nominal Lϭ6.7, ͑b͒ for ϩ10% change in the plant time delay, ͑c͒ for Ϫ10%change in the plant time delay.
570 Ibrahim Kaya / ISA Transactions 42 (2003) 559–575Fig. 12. Step responses for example 4: ͑a͒ for nominal Lϭ2, ͑b͒ for ϩ10% change in the plant time delay, ͑c͒ for Ϫ10%change in the plant time delay.as the time required for the output to settle within of K d is obtained on the basis of stabilization ofa certain percent of its ﬁnal value. Regardless of second part of characteristic of Eq. ͑6͒,the percentage used, the settling will be directlyproportional to the time constant for a second- 1ϩK d G m ͑ s ͒ e ϪL m s ϭ0. ͑36͒order underdamped system ͓29͔. That is, De Paor and O’Malley ͓31͔ used an optimum T s ϭk , ͑34͒ phase margin criterion to obtainwhere T s is the settling time. In the coefﬁcientdiagram method ͑CDM͒, which is shown to per-form very well for processes with large time con- K dϭ 1 Km ͱT L m m ͑37͒stants, an integrator or unstable poles k is chosen with the constraint L m /T m Ͻ1.between 2.5 and 3.0 ͓30͔. Referring to Eq. ͑34͒ Example 4and using kϭ2.5, An unstable process G p4 ( s ) ϭ4e Ϫ2s / ( 4sϪ1 ) is Ts considered. The plant transfer function was simu- ϭ ͑35͒ lated in SIMULINK with relay parameters of h 1 2.5 ϭ1, h 2 ϭϪ0.9, and ⌬ϭ0.1. The static load dis-is obtained. Therefore once the value of is found turbance was assumed to be dϭ0.1. The fre-from Eq. ͑35͒ with speciﬁcation on the settling quency of the limit cycle , maximum a max , andtime, the controller parameters can then be found minimum a min of the limit cycle amplitude and thefrom Eqs. ͑30͒–͑33͒. pulse duration ⌬t 1 were measured as 0.34, 3.42, The controller G d ( s ) has to be used again for a Ϫ1.82, and 4.94, respectively. The model wassatisfactory load disturbance rejection. De Paor identiﬁed exactly using the identiﬁcation methodand O’Malley ͓31͔ suggested a proportional only given in the Appendix Section 2, since the as-controller, G d ( s ) ϭK d , for the stabilization of an sumed model transfer function matches the actualunstable FOPDT plant transfer function. The value plant transfer function exactly. Letting T s ϭ5s, the
Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 571closed loop time constant is 2.0, from Eq. ͑35͒. 1. Parameter estimation for the SOPDTHence the controller parameters are K p ϭ0.25, T iϭ2, K f ϭ0.5, and T f ϭϪ1, from Eqs. ͑30͒–͑33͒. Assuming a biased relay and load disturbance atThe parameter of the controller G d ( s ) was found the plant input, two equations for the limit cycleusing Eq. ͑35͒, K d ϭ0.354. With these controller frequency and the pulse duration ⌬t 1 can besettings, the response of the closed loop system to obtained and are given ͓24͔ bya unit set point change and a disturbance withmagnitude of Ϫ0.1 at tϭ50 s is given in Fig. 12. ͩThe ﬁgure also shows the response of the closed Ϫ ⌬t 1 T 2m e L m /T 2m ͑ 1Ϫe ⌬t 1 /T 2m ͒loop system in the case of Ϯ10% change in the Km ϩplant time delay. As is seen from the ﬁgure, the 2 ͑ T 1m ϪT 2m ͒ ͑ e 2 / 2 Ϫ1 ͒ ͪproposed Smith predictor structure and design T 1m e L m /T 1m ͑ 1Ϫe ⌬t 1 /T 1m ͒method gives very satisfactory results for unstable Ϫprocesses as well. ͑ T 1m ϪT 2m ͒ ͑ e 2 / 1 Ϫ1 ͒5. Conclusions ϭ Ϫ ͑ h 1 ϩh 2 ͒ ͩ dG ͑ 0 ͒ ϩ⌬ ͪ Simple tuning formulas for a PI-PD Smith pre-dictor conﬁguration have been given. It is shown G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ ϩ ͑38͒by examples that the existing Smith predictor con- Pﬁgurations and design methods for stable and in-tegrating processes may be ineffective when pro-cesses include large time constants. Processes with andhigh orders or large time delays have been mod-eled by lower stable SOPDT, IFOPDT, orUFOPDT models so that the closed loop systemoutput will be a second-order response or a ﬁrst-order response, where it is proper, assuming a per-fect matching. The provided simple tuning formu- Km ͩ ͑ ⌬t 1 Ϫ2 ͒ 2las have physically meaningful parameters, T 2m e L m /T 2m ͑ 1Ϫe ͑ Ϫ ⌬t 1 ϩ2 ͒ / 2 ͒namely the damping ratio and the natural fre- ϩ ͑ T 1m ϪT 2m ͒ ͑ e 2 / 2 Ϫ1 ͒quency o for the SOPDT and IFOPDT and timeconstant for the UFOPDT. The values of thedamping ratio and natural frequency o havebeen obtained based on desired overshoot and the ϩ T 1m ͑ T 1m ϪT 2m ͒ e L m /T 1m ͑ 1Ϫe ͑ Ϫ ⌬t 1 ϩ2 ͒ / 1 ͒ ͑ e 2 / 2 Ϫ1 ͒ ͪrise time and the value of time constant has beenobtained based on the user speciﬁed settling time.The proposed design method has been compared ϭ ͑ h 1 ϩh 2 ͒ ͩ dG ͑ 0 ͒ Ϫ⌬with some existing ones and it is shown by someexamples that the proposed method can be advan-tageous for processes, either stable or integrating, ϩ G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ P , ͪ ͑39͒with large time constants. Also, it is shown thatthe proposed Smith predictor conﬁguration anddesign method can also be used to control pro- where h 1 and h 2 are the relay heights, ⌬ is thecesses with unstable plant transfer function. hysteresis. ⌬t 1 and ⌬t 2 are the pulse durations and Pϭ⌬t 1 ϩ⌬t 2 is the period of the oscillation.Appendix: Parameter estimation 1 ϭ T 1m , 2 ϭ T 2m and d is the magnitude of the disturbance. This section gives equations used to determine Two more equations can be obtained for theunknown parameters of a stable SOPDT, IFOPDT, maximum and minimum of the plant output waveor an UFOPDT plant transfer functions based on form which are given by the following equationsrelay autotuning. ͓24͔:
572 Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ before the disturbance enters the system, where a maxϭdG ͑ 0 ͒ ϩ c ( t ) and y ( t ) are the plant and relay output, re- P spectively, and P is the period of the limit cycle. ϩ 2 ͩ ͑ h 1 ϩh 2 ͒ K m Ϫ ⌬t 1 Once steady-state operation occurs with the dis- turbance existing, the disturbance magnitude can be calculated from T 2m e 1 / 2 ͑ 1Ϫe ⌬t 1 /T 2m ͒ ϩ ͑ T 1m ϪT 2m ͒ ͑ e 2 / 2 Ϫ1 ͒ dϭ 1 G͑ 0 ͒P1 ͵ t tϩ P 1 c ͑ t ͒ dtϪ h 1 ⌬t 1 Ϫh 2 ⌬t 2 P1 , Ϫ T 1m ͑ T 1m ϪT 2m ͒ e 1 / 1 ͑ 1Ϫe ⌬t 1 /T 1m ͒ ͑ e 2 / 1 Ϫ1 ͒ , ͪ where P 1 is the period of the limit cycle. The use ͑43͒ of Eq. ͑42͒ to ﬁnd K m may not be practical. In this ͑40͒ case, the relay test can be performed with its G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ heights set to ͉ h 1 ͉ ϭ ͉ h 2 ͉ ϭh so that the disturbancea minϭdG ͑ 0 ͒ ϩ magnitude can be found using the result given in P Ref. ͓12͔: ϩ ͩ ͑ h 1 ϩh 2 ͒ K m ͑ Ϫ ⌬t 1 ϩ2 ͒ 2 dϭ ⌬a a h, ͑44͒ T 2m e 2 / 2 ͑ e 2 / 2 Ϫe ⌬t 1 /T 2m ͒ where aϭ ( a maxϩ͉amin͉)/2 and ⌬aϭa maxϪa. Once ϩ the magnitude of the disturbance is known, Eq. ͑ T 1m ϪT 2m ͒ ͑ e 2 / 2 Ϫ1 ͒ ͑43͒ can be rearranged to give Ϫ T 1m ͑ T 1m ϪT 2m ͒ e 2 / 1 ͑ e 2 / 1 Ϫe ⌬t 1 /T 1m ͒ ͑ e 2 / 1 Ϫ1 ͒ , ͪ K m ϭG ͑ 0 ͒ ϭ ͵ t tϩ P 1 c ͑ t ͒ dt/ ͑ d P 1 ϩh 1 ⌬t 1 ͑41͒ Ϫh 2 ⌬t 2 ͒ . ͑45͒where For the SOPDT transfer function, Eq. ͑42͒ can be 1ϭ T 1m T 2m ͑ T 2m ϪT 1m ͒ ln ͑ 1Ϫe ͑ 1Ϫe ͩ ⌬t 1 /T 1m ͒͑ e ͒͑ e ⌬t 1 /T 2m 2 / 2 Ϫ1 ͒ Ϫ1 ͒ 2 / 1 ͪ used to ﬁnd the steady-state gain K m and Eq. ͑43͒ to ﬁnd the disturbance magnitude d, or Eq. ͑44͒ can be used to ﬁnd d and then Eq. ͑45͒ to ﬁnd K m . Finally, Eqs. ͑40͒ and ͑41͒ may be used to ﬁnd theand time constants T 1m and T 2m . The only remaining T 1m T 2m unknown, the dead time L m , can then be calcu- 2ϭ lated from either Eqs. ͑38͒ or ͑39͒. ͑ T 2m ϪT 1m ͒ ͩ ͪ 2. Parameter estimation for the UFOPDT ͑ e 2 / 2 Ϫe ⌬t 1 /T 2m ͒͑ e 2 / 1 Ϫ1 ͒ ϫln . ͑ e 2 / 1 Ϫe ⌬t 1 /T 1m ͒͑ e 2 / 2 Ϫ1 ͒ As for the stable SOPDT two equations for the limit cycle frequency and the pulse duration ⌬t 1There are ﬁve unknowns, namely, K m , T 1m , T 2m , ͓24͔ areL m , and d, to be determined. Therefore one moreequation is needed. Fourier analysis can be used toidentify the steady-state gain K and disturbancemagnitude d. It is assumed that the steady-state Km ͩ ⌬t 1 ͑ e Ϫ⌬t 1 /T m Ϫ1 ͒ e ϪL m /T m 2 Ϫ ͑ e Ϫ2 / Ϫ1 ͒ ͪgain can be calculated from ϭ Ϫ ͑ h 1 ϩh 2 ͒ ͩ dG ͑ 0 ͒ ϩ⌬ ͵ tϩ P K ϭG ͑ 0 ͒ ϭ t c ͑ t ͒ dt ͑42͒ ϩ G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ ͪ ͑46͒ ͵ P m tϩ P y ͑ t ͒ dt t and
Ibrahim Kaya / ISA Transactions 42 (2003) 559–575 573 Km ͩ ͑ Ϫ ⌬t 1 ϩ2 ͒ 2 sumed that there is no disturbance to the system ͓32,33͔. If an unbiased relay test with no load dis- turbances is performed, either the gain has to be Ϫ ͑ e ͑ ⌬t 1 Ϫ2 ͒ / Ϫ1 ͒ e ϪL m /T m ͑ e Ϫ2 / Ϫ1 ͒ ͪ assumed known or two relay tests, one with hys- teresis and another without hysteresis, have to be performed. However, the standard relay autotun- ϭ ͑ h 1 ϩh 2 ͒ ͩ dG ͑ 0 ͒ Ϫ⌬ ing can slightly be improved to determine the un- known parameters of the IFOPDT using a biased relay and/or assuming static load disturbances. For ϩ G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ P , ͪ ͑47͒ this, a differentiator is put in front of the IFOPDT plant transfer function to cancel the integrator pole by the zero of the differentiator. In this case, thewhere ϭ T m . overall plant transfer function is a stable FOPDT The other two equations, for the minimum and which has unknown parameters of K m , T m , andmaximum of the plant output, are L m . In theory, differentiating the relay output gives impulses at zero crossings while in practice G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ these impulses can be approximated by pulsesa minϭdG ͑ 0 ͒ ϩ P with short pulse width. Therefore to identify the ͩ ͪ unknown parameters of the IFOPDT, all one needs ͑ h 1 ϩh 2 ͒ K m ⌬t 1 ͑ 1Ϫe Ϫ⌬t 1 /T m ͒ is to derive equations for a stable FOPDT. This is ϩ ϩ 2 ͑ e Ϫ2 / Ϫ1 ͒ the approach used in this paper and the equations required can be found in Refs. ͓32,20͔ and are ͑48͒ given here for convenience.and Equations for the limit cycle frequency and the pulse duration ⌬t 1 ͓32,20͔ are G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ a maxϭdG ͑ 0 ͒ ϩ P Km ͩ Ϫ ⌬t 1 ͑ e ⌬t 1 /T m Ϫ1 ͒ e L m /T m 2 ϩ ͑ e 2 / Ϫ1 ͒ ͪ ϩ ͩ ͑ h 1 ϩh 2 ͒ K m ͑ ⌬t 1 Ϫ2 ͒ 2 ϭ Ϫ ͑ h 1 ϩh 2 ͒ ͩ dG ͑ 0 ͒ ϩ⌬ ϩ e ⌬t 1 /T m ͑e ͑e Ϫ2 / Ϫ2 / Ϫe Ϫ1 ͒ Ϫ⌬t 1 /T m ͒ ͪ . ϩ G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ P ͪ ͑50͒ ͑49͒ andThe steady-state gain K m and the disturbance mag-nitude d are found either using Eqs. ͑42͒ and ͑43͒or Eqs. ͑44͒ and ͑45͒. The time constant T m can beobtained from either Eq. ͑48͒ or Eq. ͑49͒. Finally, Km ͩ ͑ ⌬t 1 Ϫ2 ͒ 2with K m , d, and T m known, the dead time can becalculated from either Eq. ͑46͒ or Eq. ͑47͒. ϩ ͑ e ͑ Ϫ ⌬t 1 ϩ2 ͒ / Ϫ1 ͒ e L m /T m ͑ e 2 / Ϫ1 ͒ ͪ3. Parameter estimation for the IFOPDT Unlike the SOPDT and UFOPDT, the standard ϭ ͑ h 1 ϩh 2 ͒ ͩ dG ͑ 0 ͒ Ϫ⌬relay autotuning under static load disturbances orwith a biased relay cannot be used for parameter ϩ G ͑ 0 ͒͑ h 1 ⌬t 1 Ϫh 2 ⌬t 2 ͒ P , ͪ ͑51͒estimation of the IFOPDT, since in the equationsobtained G ( 0 ) will appear which is inﬁnity for the where ϭ T m .IFOPDT. Therefore for the IFOPDT an unbiased Two more equations can be calculated from therelay has to be used. In addition, it has to be as- plant output, one for the maximum of the plant
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