CONTROL OF INTEGRATING PROCESS WITH DEAD TIME USING AUTO-TUNING APPROACH

Transcription

1 Brazilian Journal of Chemical Engineering ISSN Printed in Brazil Vol. 26, No. 0, pp , January - March, 2009 CONROL OF INEGRAING PROCESS WIH DEAD IME USING AUO-UNING APPROACH G. Saravanakumar * and R. S. D. Wahidabanu 2 Department of Intrumentation and Control, Manipal Intitute of echnology, Manipal-57604, Karnataka, India. aravanakumar579@yahoo.com 2 Department of Electronic and Communication Engineering, Govt. College of Engineering, Salem, amilnadu, India. rdwb@yahoo.com (Submitted: April 26, 2007 ; Revied: February, 2008 ; Accepted: February 23, 2008) Abtract - A modification of Smith predictor for controlling higher order procee with integral action and long dead-time i propoed in thi paper. he controller ued in thi Smith predictor i an Integral- Proportional Derivative controller, where the Integrator i in the forward path and the Proportional and Derivative control are in the feedback, acting on the feedback ignal. he main objective of thi paper i to deign a dead time compenator, which ha minimum tuning parameter, imple controller tuning, and robut performance of tuning formulae, and to obtain a critically damped ytem that i a fat a poible in it et point and load diturbance rejection performance. he controller in thi paper i tuned by an adaptive method. hi paper alo preent a urvey of variou dead time compenator and their performance analyi. Keyword: Dead time compenator; Controller; Auto tuning; Integrating proce. INRODUCION Smith predictor are effective tool for compenating the dead-time aociated with the procee. During the lat 20 year, numerou extenion and modification of the Smith predictor, alo called dead-time compenator, DC, (Saravanakumar et al., 2006) have been propoed. he deign of the dead time compenator (Hagglund, 992) require the tuning of more parameter (primary controller and model parameter) than claical PID controller. hi ha been a drawback for the application of DC in indutry, which required 5 parameter. o implify the tuning of the DC controller propoed by Smith, a four parameter controller wa propoed (Coughanowr, 99). NON-SELF REGULAING PROCESSES DC by Watanabe and Ito (98) o overcome the drawback of the DC dicued previouly, Watanabe and Ito (98) propoed the dead time compenator preented in Figure. Figure : DC propoed by Watanabe and Ito (98) *o whom correpondence hould be addreed

2 90 G. Saravanakumar and R.S. D. Wahidabanu Simulation tudie have hown that, with a PI controller the et point and load diturbance were either very ocillatory or highly damped when the proce had a large dead-time. he repone of the ytem alo tend to be low. Much better performance wa obtained when the main controller wa of the PID type. he drawback i that it cannot perform better for load repone for procee with longer dead-time. L Y() = G()e () u() G() = (2) G() G () = (3) + L Here G() i the plant and G () i the model. hi wa the aumption made in thi DC. Here the drawback i that the et point repone i rather low. It too had three controller parameter to be determined, K c, i and d. DC by Atrom et al. (994) o overcome the drawback of the Watanabe method, Atrom et al. (994) propoed a DC for the integrating proce that wa coniderably fater in et point repone and had a better load diturbance rejection. A convenient property of the controller wa that it decouple the et point repone from the load repone. However, the number of tuning parameter, if the velocity gain i unknown, equal ix. DC by Matauek and Micic (996) he controller propoed by Matauek and Micic (996), preented in Figure 2, i a imple and traightforward modification of the Smith predictor for the integrating proce. he high cloed loop ytem performance, i.e., fat et point repone and atifactory load diturbance rejection, can be obtained even in the preence of the unmodelled dynamic. here are only three adjutable parameter that have a clear phyical interpretation and can eaily be tuned manually. Ko = 2Kpτ K r p r (4) = (5) K Figure 2: Modified DC propoed by Matauek and Micic (996). DC by Normey-Rico and Camacho (999) hi tructure, preented in Figure 3, wa the modification of the Watanabe tructure for the integrating proce by including the filter. he tuning criterion wa baed on the definition of the cloed loop performance and conider that the model of the proce i not preciely known. Uing the etimation of the dead-time and the velocity gain of the plant, the propoed control law ha three parameter (o, Kg, L) that can be tuned manually like a PID Controller. hi method, Normey-Rico and Camacho (999) wa better in both et point and load diturbance performance when compared with Matuek and Micic method for longer dead-time. he method preented i applicable to PID Control algorithm in both the Interacting and noninteracting derivative form. Figure 3: DC propoed by Normey-Rico and Camacho (999). DC by Rice and Cooper (2002) Here the work wa baed on the IMC tructure, to derive tuning correlation for integrating procee. One novel contribution of thi work (Rice and Cooper, 2002) wa the extenion of tuning Brazilian Journal of Chemical Engineering

3 Control of Integrating Proce with Dead ime Uing Auto-uning Approach 9 correlation to include the PID with derivative filter form. hi work i applicable where a compromie i ought between ettling time and overhoot, while minimizing the chatter in the controller output due to noie. hi method alo require five parameter to be determined, two model parameter and three controller parameter. DC by Wang et al. (2006) he cheme of thi controller i preented in Figure 4. Here the region of the control parameter to guarantee the ytem tability wa characterized. he control parameter to achieve the given gain and phae margin were determined. Furthermore, the contraint on achievable GPM wa derived. hee reult were obtained on the bai of the normalized ytem that involve only two free parameter and one filter parameter for a total of three parameter. Here (Wang et al., 2006) the et point repone wa fat and the load repone wa rather low for the integrating proce. Figure 4: DC propoed by Wang et al. (2006). So in all the DC dicued above, it wa poible to achieve a perfect repone, either with et point or load repone. Hence it i neceary to deign a method that will be able to achieve a critically damped ytem and i a fat a poible in both et point and load diturbance. open loop by a manipulated or diturbance variable. Such behaviour i characteritic of non-elf regulating procee. If the DC ued for the elfregulating procee are applied for the non-elfregulating procee, thee predictor exhibit a teady-tate error in their load repone. Conequently, there i a need for the modified deign of DC to give atifactory repone for the integrating procee. Several new Smith predictor propoed by variou author were conidered and their repone were analyzed. In Watanabe method (Watanabe and Ito, 98), the repone for the et point and load diturbance rejection were better for the PID controller than the PI controller. he et point repone wa rather low. In Matuek and Micic method, a new DC method wa propoed, which had a very good et point and load repone compared with the previou one, but the drawback wa it cannot produce a better reult with a longer dead time. In Rice and Cooper method, an IMC baed DC wa propoed particularly to reject the chatter in the controller for integrating procee. When affected by noie diturbance, the reult proved that thi method wa better than the tuning formula propoed by Ziegler-Nichol (942) (ZN) and yreu and Luyben (992). Wang et al. (2006) propoed a modified DC with two degree of freedom, a filter and a controller. hi method gave a good et point repone and atifactory load repone. In all the above method, none of the modified DC gave a atifactory performance for both et point and load diturbance. A compromie i obtained between the et point and load diturbance repone with the propoed method in thi paper. Propoed Alternate Method he propoed method i preented in Figure 5. PERFORMANCE ANALYSIS OF MODIFIED DC FOR NON-REGULAING PROCESSES Non-Self Regulating Procee It i not uncommon for ome temperature, level, preure and other meaured proce variable to move in an unbounded manner when perturbed in Figure 5: Modified Smith predictor approach. Brazilian Journal of Chemical Engineering Vol. 26, No. 0, pp , January - March, 2009

4 92 G. Saravanakumar and R.S. D. Wahidabanu Conider the plant with tranfer function L L G p () = e, where e repreent the time delay of L econd. Aume that the time delay L i known and the proce i. Here G p () G 0() = + L (Coughanowr, 99) and L(proce) = L(model). he modification (Saravanakumar et al., 2006) conit of the additional feedback path (K 0 0), from the difference of the plant output Y and the model output Y m to the control input u. When K 0 = 0, the Smith predictor i ued. When K 0 0, then the modified Smith predictor i obtained. he controller ued in thi Smith predictor i an I-PD controller where the Integrator i in the forward path and the proportional and derivative control are in the feedback acting on the feedback ignal. he controller i tuned by both the Ziegler Nichol (ZN) method and the relay method. Since the relay method give the immediate value of ultimate gain and ultimate period, it i preferable to the ZN method. he et point and the load diturbance repone i given by Y() = H ()R() + H ()D() (6) where r d G ()G () p i L + ( e )( G i ()G pd ()) G 0() (7) G i () + G pd () + + L i + pd o H r () = ( e ) G () G () G () ( ) G () + G ()K H d () = G () where, p p o (8) G i ()G pd () + L p + ( e ) G i () + G pd () G o () ( ) Ki G i () =, G pd () = Kp + Kd and G 0() repreent the etimated tranfer function of the plant. From (7) and (8) we have L G 0()G i()e H r () = (9) + G () + G () G () ( ) i pd 0 G p () H d () = + G ()K p L ( ) ( i pd ) o + ( e ) G () + G () G () o i pd o + G () + G () G () (0) for the perfect model of the plant and perfect dead time of the proce. If the cloed loop ytem i table, the et point and the diturbance repone for tep input are a follow: lim H () = lim H r() = r 0 0 lim H () = lim H d() = 0 d 0 0 for K 0 0. () (2) hi how that there will be no teady-tate error for a contant load diturbance. 3 e G 0() =, L=3 (3) ( + 3) he repone for the et point and load diturbance are hown in the imulation reult. Validation of the ime Contant and Dead ime Relationhip In thi paper it i aumed that the model of the G p () proce i G 0() = and L (proce) = L + L (model). hat i, the proce dead time i equal to the time contant of the model. A method i required to validate thi aumption. hu, imulation tudie were undertaken to validate thi aumption. In thi paper, the time contant wa aumed to be 3 ec a that of the deadtime. Simulation tudie are carried out for time contant varying from 0.75 to 6 econd and variou repone are obtained in time domain pecification uch a overhoot, rie time, peak time, delay time and ettling time, etc. From the detailed tudy of thee pecification, it i validated that the time contant of the model hould be equal to the dead-time of the proce. For other time contant a mall teady tate error i obtained for contant load diturbance. able how the variou time contant and dead-time Brazilian Journal of Chemical Engineering

5 Control of Integrating Proce with Dead ime Uing Auto-uning Approach 93 relationhip with value of rie time, peak time, delay time and ettling time. In thee imulation: ) Dead-time i conidered to be contant for all the imulation tudy. 2) he time contant i varied from one fourth of the dead-time to double the dead time. 3) Rie time i the time taken by the proce to reach 90% of the final value. Here the obervation how that the rie i low in the initial tage, then fat with time contant of 3 ec and again lowing down at double the time contant. (4) Peak magnitude repreent the overhoot of the proce and i high at the initial and final value of time contant, but low when the time contant matche the dead time. (5) Settling time i the time taken by the proce to ettle with it et point value, which i alo fat when the time contant matche the dead time, and i low in all other cae. So thee tudie how that when time contant matche the dead time, it i poible to achieve a critically damped ytem. able : Validation through time domain pecification Dead ime ime contant Rie ime A (ec) Peak overhoot magnitude A Settling time A (ec) Peak underhoot Magnitude B Settling ime B(ec) 3 ec A- Repreent the time domain pecification of the et point repone. B-Repreent the time domain pecification of the load repone. Simulation Reult for Validation of ime Contant and Dead ime Relationhip In Figure 6 for a dead time of 3 ec and time contant of 3 ec, the error i of only one polarity (i.e., it never ocillate about the et point). Here the meaure of quality i the duration (tarting point of diturbance to return of the proce variable) of the excurion, for a load change and a maximum error for tranient change. he duration i the time for exceeding the allowable error and to regain the allowable error. he overhoot magnitude i. and, for the tranient change, the proce return to et point at 60 ec. For a load change of unit at 00 ec, the proce variable return at 50 ec. In Figure 7 the ettling time for the et point repone i 58 ec, and ettling time for the load repone i 48 ec, which i comparatively higher than the 45 and 40 ec of the time contant=dead time cae. In Figure 8, when the time contant i 3/4 of the dead time, the ettling time of the et point repone and the load repone are 55 ec and 45 ec, repectively. In Figure 9 for a dead time of 3 ec and a time contant of 6 ec (double the dead time), the ettling time for tranient change i about 80 ec. For a load change of unit at 00ec, the ettling time i around 90 ec. Figure 6: ime contant=/4 of Dead time Figure 7: ime contant=/2 of Dead time Brazilian Journal of Chemical Engineering Vol. 26, No. 0, pp , January - March, 2009

6 94 G. Saravanakumar and R.S. D. Wahidabanu Figure 8: ime contant=3/4 of Dead time Adaptive Controller Gain by Auto-uning hi modified Smith predictor (Figure 5) with I- PD controller, deigned for integrator proce with long dead time, can be ued for auto-tuning the controller gain whenever the plant parameter change due to uncertaintie. here are many autotuning concept, which were formulated earlier (Santacearia and Scattolini, 993). he auto-tuning concept in thi paper firt determine the FF of the plant input for 64 ample and then the FF of the plant output for 64 ample. Now, the output divided by input of the plant give the actual frequency repone of the plant. hen the frequency repone of the model i obtained by giving an impule input to the model. hi model i derived by conidering the plant output, plant input and the plant diturbance. Once the leat quare algorithm i implemented in MALAB, then the theta value (ee equation 22) give the adaptive controller gain, which are the auto-tuned value of the I-PD controller. Now, the determination of I-PD controller gain for auto-tuning in the frequency domain (Saravanakumar et al., 2006) i ummarized a L follow. Let G p () = e. he frequency repone jwl of thi ytem i G p (jw) = e. Let G m (j ω ) jw denote a deired open-loop frequency repone of the reference. hen the I-PD controller gain are determined o a to minimize the cot function J, which i defined by the quare error between the open loop frequency repone of the reference model and that of the actual ytem a M M 2 2 k k k= k= (4) J = ε(j ω ) + ε ω ( j ) where Figure 9: ime contant=2 time Dead time. 2πk ω k = (5) N N i the number of ample, i the ampling time, G(j ω k ) i the dicretized frequency repone of the actual ytem, and G m(j ω k) i the dicretized frequency model G p(jwk)g i(jwk) G( jwk) = (6) + G (jwk)g (jwk) pd jw + (Kp + jw k kk d)g p(jw k)g m(jw k) ε (jwk) = k G (jw ) k = φ(j ω ) θ i p k p (7) where K i, K p, K d are the integral, proportional and derivative gain, repectively. φ (jw k) = [,G p(jw k), jwkg p(jw k)]a(jw k) (8) K K θ= K K K, p, D i i i m k A(jw k) jwk G p (jw k ) (9) G (jw ) = (20) hen the cot function J can be rewritten a J = ( ϕ φθ) ( ϕ φθ ) (2) Brazilian Journal of Chemical Engineering

7 Control of Integrating Proce with Dead ime Uing Auto-uning Approach 95 where φ(j ω), φ(j ω2)... φ(j ωm), φ= φ ω ( j ) φ ω ( j 2)... φ ω ( j m) ϕ= [,,...] Uing the leat quare algorithm, a real valued olution of θ that minimize J i given by θ= φ φ φ ϕ (22) where denote the complex conjugate tranpoe. 3.6 Simulation Reult for Non-Self Regulating Procee he tranfer function conidered i 3 G() = e 3 e and G 0() = for all the method dicued ( + 3) above. he model i derived a per the above dicuion in each method. In the repone hown in Figure 0 it i clear that the alternate method propoed in thi paper give a fat repone for both et point and load diturbance, with fat ettling time. In Figure the repone of the propoed method act a a compromie between both the et point and load diturbance by providing fat rie time and fat ettling time with minimum overhoot and underhoot, wherea the other method were able to produce only either a atifactory et point repone or a load repone for diturbance. Hence, it i clearly proved that the propoed method ha a good diturbance rejection property. hi method i alo verified for the robutne of the controller with unmodeled dynamic (with change in the dead-time). In Figure 2 the et point repone and load repone of the ytem with a PID controller uing the auto-tuning approach i hown. Firt, the repone of analytical PID value i obtained by Ziegler-Nichol tuning. It i clear that the optimum tuning value give a good repone with minimum overhoot and fat ettling time. Next the adaptive PID value are obtained from the Matlab program, and thoe value too reulted in a compromie between minimum overhoot and fat ettling time and the ytem i alo under control. hi wa verified for both et point and load diturbance. hu, the I-PD controller dicued in the propoed method give better repone for both et point and load diturbance than the PI and PID controller conidered in the introductory part. Here the number of tuning parameter i only three, Kp, i and d, which i alo an added advantage of thi method when compared to the previou mode of Smith predictor. Figure 0: Set point and load repone for a non-elf regulating proce by the propoed alternate method. ZN uning: Kc=0.22, Ki=0.09, Ko=0.04 Figure : Set point and load repone of the propoed method in thi paper for a non-elf regulating proce with other method. Brazilian Journal of Chemical Engineering Vol. 26, No. 0, pp , January - March, 2009

8 96 G. Saravanakumar and R.S. D. Wahidabanu Figure 2: Set point repone and load repone of the ytem uing IPD Controller with analytical and adaptive value uing the auto-tuning approach. Robutne of the Propoed Method In thi ection the repone of the propoed method to change in the et point for Benchmark proce are preented. a) Change in the Set-Point Poitive Mimatch erm For the benchmark proce with G () = e 5 the negative mimatch term 0.8 G () = e 4.5 i introduced and oberved for ervo repone of a unit tep change at 0 ec. When compared with other modified Smith predictor, the propoed method give minimum overhoot and fat ettling time without ocillation, a hown in Figure 3. b) Change in the Set-Point Poitive Mimatch erm For the benchmark proce, the poitive mimatch term.2 G () = e 5.2 i introduced and oberved for ervo repone of an unit tep change at 0 ec. When compared with other modified Smith predictor, the propoed method give minimum overhoot and fat ettling time without ocillation, a hown in Figure 4. c) Change in the Load - Negative Mimatch erm For the benchmark proce the negative mimatch term 0.8 G () = e 4.5 i introduced and oberved for load repone of -0. unit at 00 ec. When compared with other modified Smith predictor, the propoed method give minimum underhoot and fat ettling time, a hown in Figure 5. d) Change in the Load - Poitive Mimatch erm For the benchmark proce, the poitive mimatch term.2 G () = e 5.2 i introduced and oberved for load repone of -0. unit at 00 ec. When compared with other modified Smith predictor, the propoed method give minimum underhoot and fat ettling time, a hown in Figure 6. Brazilian Journal of Chemical Engineering

9 Control of Integrating Proce with Dead ime Uing Auto-uning Approach 97 Figure 3: Servo repone for negative mimatch term of BM proce Figure 4: Servo repone for poitive mimatch term of BM proce Figure 5: Load repone for negative mimatch term of BM proce Figure 6: Load repone for poitive mimatch term of BM proce CONCLUSION An I-PD baed modified DC i deigned for Integrator procee with long dead time. It give a atifactory repone for both et point and load change. A critically damped ytem that i a fat a poible i obtained by thi method. It i alo proved that there i no teady-tate error for contant load diturbance. hi method i alo verified for the robutne of the controller with unmodeled dynamic. It i proved that the repone of the proce variable i good when the time contant of the model i equal to the dead time of the proce. hu, a critically damped ytem with fat et point repone and load repone i obtained. K 0 K i K d K p L i r λ κ θ φ ε Diturbance gain Integral gain Derivative gain Proce gain Dead time ime contant Integral time Related time contant of Controller gain K r Sampling time Model uncertainty Normalized gain Adaptive controller gain Unit matrix Error G() G 0 () NOMENCLAURE ranfer function of the Plant ranfer function of the model Acronym DC Dead time compenator IMC Internal model controller PID Proportional, Integral, Brazilian Journal of Chemical Engineering Vol. 26, No. 0, pp , January - March, 2009

10 98 G. Saravanakumar and R.S. D. Wahidabanu PIP GPM ZN FF I-PD Derivative Predictive Proportional Integral Gain phae margin Ziegler Nichol Fat Fourier tranform Integral, proportional, Derivative REFERENCES Atrom K. J., Hang C. C. and Lim B.C., A new Smith predictor for controlling a proce with an integrator and long dead-time IEEE ranaction on Automatic Control Vol 39, no. 2, p (994). Coughanowr D. R., Proce Sytem Analyi and Control, McGraw-Hill International Edition, Chemical Engineering Serie (99). Hagglund., A predictive PI controller for procee with long dead-time IEEE Control Sytem Magazine vol.2, no., p (992). Matauek M. R. and Micic A. D., A modified Smith predictor for controlling a proce with an integrator and long dead-time, IEEE ranaction on Automatic Control, vol. 4, No. 8, p (996). Normey-Rico J. E. and Camacho E. F., Robut tuning of DC for procee with Integrator and longer dead-time, IEEE ranaction on Automatic Control vol.44, no. 8, p (999). Rice R. and Cooper D.J., Deign and tuning of PID controller for integrating (non-elf regulating) procee, Procceding of the ISA Annual Meeting, Chicago (2002). Santacearia C. and Scattolini R., Eay tuning of Smith predictor in preence of delay uncertainty, Automatica, vol. 29, no. 6, p (993). Saravanakumar G., Wahidabanu R. S. D. and Nayak C. G., Deign of modified Smith predictor for table procee with integrator and longer deadtime uing adaptive control, International Journal on Automatic Control and Sytem Engineering, vol 6, no. 4, p. -5 (2006). Wang B., Ree D. and Zhong Q. C., Control of Integral proce with dead-time Part IV: variou iue about PI controller, IEE Proc.-Control heory and Application, vol. 53, no. 3, p (2006). Watanabe K. and Ito M., A proce-model control for linear ytem with delay, IEEE ran. Automatic Control., vol. 26, no. 6, p (98). yreu B. D and Luyben W. L., uning PI Controller for Integrator Dead time procee, Ind. Eng. Chem. Re. vol. 3, p (992). Ziegler J. G. and Nichol N. B., Optimum Setting for Automatic Controller, ran. ASME, 64, p (942). Brazilian Journal of Chemical Engineering