Time Delay Compensation by A PID Controller

Transcription

1 211 IEEE Internatonal Conerene on Sytem Engneerng and Tehnology (ICSET) Tme Delay Comenaton by A PID Controller Manoureh Emael Det. Downtream Petrohemal Indutre Develoment NPC Tehran, Iran m.emael@n.net Manour Shrvan Det. Chemal Engneerng Iran Unverty o Sene & Tehnology Tehran, Iran Shrvan.m@ut.a.r Abtrat Control o tme delay roee ondered rom a new vew ont. In th vew ont, tme delay ondered a nnte RHP zero that are tranerred to the LHP zero by addng a roer traner unton that hoen baed on domnant gan onet. A a reult, the non mnmum hae behavor o the orgnal oen loo hange to mnmum hae behavor and all o the RHP zero are removed rom the oen loo and the ontrol loo erormane mroved. Keyword: Tme Delay; Comenator; PID Controller I. INTRODUCTION In Control rate many o roee uer rom tme delay. To mrove the tme delay ontrol ytem erormane, ome redtor have been ued [1]. The method whh have ued redtor an be ategorzed nto three laaton. In the rt method, tme delay arameter redted by ung a model and removed rom the oen loo. Smth redtor wa the rt attemt n th eld [2]; however, t ha been lately revealed that Smth' method very entve wth reet to the model mmathe [3-8]. Inaablty o rejetng load dturbane n the ontrol o roee wth ntegraton [9-1] and nablty to ontrol untable roee [11] are the other two roblem mentoned or the Smth redtor. To olve thee roblem, many reearher have reented derent method alled Dead Tme Comenator (DTC) [9-17]. However, the larget det o DTC-baed method lak o t alaton or roe model n whh ther traner unton are rratonal and the tme-delay annot be atored out traghtly rom the traner unton. Ramanathan et al [18-2], nred by the Smth method, ondered the tme delay arameter a nnte Rght Hal Plane (RHP) zero and dvded the model nto two art: : art o the model that nlude RHP zero and : art o the model that doe not nlude RHP zero. By omenatng rom the oen loo traner unton and nut gnal to the ontroller, the behavor o oen loo wa hanged rom non-mnmum hae to mnmum hae. Th method wa alled eneralzed Smth Predtor (SP). In addtonal, Vollmer & Rah [21-22] and Zltek & Hlava [23] ontrolled the roee wth rratonal model by ung H ontrol and removng rom the oen loo. Model redtve aroahe have been ued wdely n reent year a a method or otmal ontrol o nonlnear tmedelay ytem [24, 25]. In the eond method, nut gnal to the ontroller redted by ung a dervatve mode n ther ontroller [26-28]. In thee method, ontroller are ondered a a PID n (1) [26] or a PID n (2) [27], and ome o the arameter are xed at a tyal value. Tyal value ue or T and N were 4 T d and 1, reetvely [26, 27]. K K K d 1 Td K(1 ) T T d 1 N In order to lmt the omlexty o PID ontroller tunng tyal value ued or T and N are 4T d and 1, reetvely [26, 27]. The advantage o thee method ther eay alaton n ndutral ontrol loo [29]. Alo, t hown that tme delay mall, the erormane o a PID ontroller wll be better than a DTC [27]. The thrd method are a onaton o the two above method. In thee method, at rt, the outut gnal o a DTC ontroller dented ung a ytem dentaton method and then the arameter o a PID ontroller are tuned o that t outut gnal wll be the ame a that o the DTC ontroller [3-31].Due to the need o an dentaton te, they are eldom ued n rate. In [32], domnant gan onet ntrodued baed on the requeny reone behavor o an rratonal traner unton. In th aer, a PID degned baed on Domnant an Contrant (DC) to omenate the tme delay. At rt a bre exlanaton o the domnant gan onet reented bellow. II. DOMINANT AIN CONCEPT The onet o domnant gan wa roved n [32]. It tate that In an rratonal traner unton or a QuaRatonal Dtrbuted Sytem (QRDS) model lke: /11/$ IEEE 31

2 211 IEEE Internatonal Conerene on Sytem Engneerng and Tehnology (ICSET) t d 1 2( ) e, n whh the 1( ) the tme-delayree term o the model; n a e requeny range o 1 2, ollow the requeny behavor o domnant gan term n the model. The mot mortant remark n the domnant gan onet ntrodued n the denton bellow: j Denton: I 1( j), ( j) e t d and 2 t j ( j) ( j) ( j) e d are hown a vetor A, B, C 1 2 reetvely a n Fg. 1, then at requene n whh A and B are loated n a ounter dreton to eah other, vetor C wll have mnmum length and wll be loated n the dreton o the larget vetor among A and B. In th tuaton, ( j) have a mnmum gan n the requeny reone lot and t related requeny wll be hown by. demontrate the orreondene between the gan domnaton haratert and the nuer o model RHP zero. Amltude rato Phae (deg.) =1+2 1=1/(+1) 2=.25*ex(-1)/( ) Frequeny (rad/e) wmg1 wmg2 wmg3-3 = =1/(+1) 2=.25*ex(-1)/( ) Frequeny (rad/e) Im Re Fgure 1. Vetor reentaton, when ha mnmum gan Th o rmary mortane n the dynam behavor o the model, beaue the loaton o the zero n a QRDS model or a qua-olynomal deendent on the gan domnaton above. For examle, A B at Or, Z LHPZ Fgure 2. Frequeny reone and the loaton o zero o a QRDS model demontratng an nnte nuer o LHP zero reultng rom all requeny gan domnaton o tme-delay-ree term. Amltude rato =.25/( ) 2=ex(-5)/(+1) = Frequny (rad/e) A B at mg, Z mg In the above equaton, zero to the an be onluded that : RHPZ Z mg an ndator or the related requeny. Alo, rom the above relaton, t Phae (deg.) =.25/( ) -25 2=ex(-5)/(+1) = Frequeny (rad/e) 3 25 A B at, the related zero wll be loated on the magnary ax. In gure 2 through 4, varou requeny reone o a mle QRDS model are hown. The zero are alo drawn to Im Re Fgure 3. Frequeny reone and the loaton o zero o a QRDS model demontratng an nnte nuer o RHP zero reultng rom all requeny gan domnaton o tme-delay-nluded term. 32

3 211 IEEE Internatonal Conerene on Sytem Engneerng and Tehnology (ICSET) In Fg. 2, at all the mnmum hae term (delay ree term) domnant the non-mnmum hae term (delay nluded term). Then, aordng to (3), t exeted that all the zero are loated n LHP. Th exetaton onrmed by zero lot n Fg. 2. Contrary to Fg. 2 n Fg. 3, at all the non-mnmum hae term domnant. Then, aordng to (4), t exeted that all the zero are loated n RHP. In Fg. 4, the mnmum hae term domnant n 4. In th nterval, 3 are ourrng and the remnder o our n loaton where the nonmnmum hae term domnant. Then, t exeted that there are 3 ar o LHP zero and the remnder nnte nuer o zero are loated n RHP. Amltude rato Phae (deg.) =1/(+1) 2 2=.25ex(-5)/(+1) = Frequeny (rad/e) Prooton 1: n a mle qua-ratonal trutured oen td loo traner unton a ol ( ) e n whh doe not have any rght-hal lan ole, a table traner unton, wth an order o 1 or zero, added o that: ( jmg, ) ol ( jmg, ),, the ontrol loo wll be abolutely table. Proo: Aumng the above mentoned ondton, the overall oen loo traner unton wll be: ool ol. I the domnant gan ontrant (eq. (6)) reval, o that the overall oen loo behavor omletely ollow the behavor and beome a mnmum hae traner unton, then there wll not aear any RHP zero n the oen loo. A generally known, a ontrol ytem n whh the oen loo behavor mlar to the behavor o a rt order traner unton abolutely table; thereore, the rooed ontrol ytem abolutely table, too. III. THE PROPOSED METHOD In [18-2], a method wa reented to omenate or the tme-delay. Th method baed on Smth redtor and alled eneralzed Smth Predtor. The ontrol truture o th method hown n Fg =1/(+1) 2 2=.25ex(-5)/(+1) = Frequeny (rad/e) 3 25 Im Re Fgure 4. Frequeny reone and the loaton o zero o a QRDS model demontratng an nnte nuer o RHP zero reultng rom hgh requeny gan domnaton o tme-delay-nluded term. An oote ae o Fg. 4, alo, oble to our, whh not hown here due to lmtaton. In ontrol ytem, the mnmum hae behavor, orreondng to all LHP loated zero (Fg. 2) muh derable. The mot mortant ont that, n an oen loo wth a QRDS truture, a delay-ree term a rt order traner unton, by alyng the Domnant an Contrant (DC), the ontrol ytem wll beome abolutely table, n te o the extene o tme-delay or any hgh order dynam n the roe. Baed on the above ont, tablty o the rooed method roved n the ollowng rooton. Fgure 5. eneralzed Smth Predtor In th method, nred rom Smth method or omenatng the tme-delay eet, the QRDS model dvded nto two art; art o the model whh nlude RHP zero,, and art o the model whh doe not nlude RHP zero,. By omenatng or the eet o, the oen loo behavor wll be hanged rom nonmnmum hae to mnmum hae. I there no model mmath, the oen loo n th method wll be: ( ) SP, oenloo Then, all the RHP zero wll be removed rom the oen loo. For th ytem, the haratert equaton wll be: 1 33

4 211 IEEE Internatonal Conerene on Sytem Engneerng and Tehnology (ICSET) The rooed method here baed on tranerrng the RHP zero o oen loo traner unton to the LHP ung the DC requrement, ntead o requrng a eret model o the roe or elmnatng the tme-delay rom the haratert equaton. The rooed ontrol truture hown n Fg. 6. In th heme, the omenator, whh wll be eleted n a way that t gan beome domnant n omaron to the gan o the mle oen loo traner unton o, ol m. alled here, "Model Bya Phae Lmter" (MBPL) omenator due to the at that t byae all the hgh order a well a the RHP zero dynam n the loo and lmt the hae o the oen loo to t own hae behavor. Thu, by ung th omenator, the requeny reone behavor o the oen loo traner unton wll hange rom delay behavor to non-delay behavor. Proo: In the rooed method, the overall oen loo traner unton o : ool K D ) ( nb) ( ool ol ( mo) K ol N ol ( ( no) Dol ) td e Where, the ubrt ol and ool reer to the mle oen loo (wthout MBPL-omenator) and overall oen loo, reetvely. K and K ol are tat gan o the omenator and mle oen loo traner unton, reetvely. Alo, nb, mo and no are the order o olynomal D, N ol and D ol, reetvely. Then the haratert equaton wll beome: ( no) ( mo) ( nb) K D K N D e t d ol ol ol Fgure 6. The rooed ontrol truture The loed loo traner unton o th ytem wll beome: Y( ) Y 1 Y d( ) 1 m b d d The haratert equaton : m 1 hould be eleted a a mnmum hae traner unton. It addton to the oen loo by onderng the DC requrement (Eq. (6)) reult n the onveron o all the RHP zero n the oen loo traner unton to the LHP zero. IV. CHOOSIN THE ORDER AND AIN OF In reoton 2 and 3, hoong method are reented or the order and gan o. Prooton 2: n the rooed method, ol trtly roer, the mot roer order or rt order and ol roer, the mot roer order or zero order. m m Thu, baed on Domnant an Conet, ol a roer traner unton,.e., no mo, then nb ; and ol a trtly roer traner unton,.e. no mo, then nb 1. In the lat ae, the le value,.e. nb 1, more derable. Th due to the at that, wth a rt order domnant gan o omenator, the ontrol ytem reahe an abolutely table ondton. Prooton 3: The bet value or ( j) the mnmum value whh ate the DC requrement,.e. (6) or the uer bound o ( j) unertanty. ol Proo: Mnmum value whh ate (6) : ( mg, ol j ) ( j ) Sentvty unton or the rooed method are obtaned a (16) and (17): 1 S( ) 1 1 T( ) 1 A een, to whatever level the value nreaed, S nreae and T dereae. Then, the erormane wll beome robut and luggh by nreang the gan o. 34

5 211 IEEE Internatonal Conerene on Sytem Engneerng and Tehnology (ICSET) Alo, to onrm that there not any RHP zero n the oen loo, eq. (15) mut be ated or the uer bound o ( j) unertanty. ol V. COMPARIN THE PROPOSED METHOD WITH A CONVENTIONAL PID CONTROLLER A mentoned beore, beaue o the extene o dervatve mode, PID ontroller are ondered a the redtor. But, tunng the ontroller arameter very omlex due to the ontroller hgh nuer o arameter. Thu, n order to lmt the omlexty o PID ontroller, ome arameter were xed at a tyal value. Sne, n ontrol alaton, more than 93% o the ontroller are o PID tye [29], ome o the reearher [3, 31] have tred to aroxmate Smth redtor erormane to an ordnary eedbak ontrol loo. To aheve th objetve, the onaton o the ontroller and redtor mut be aroxmated a a ontroller. For examle, the ontroller a PI ontroller, a onaton o the ontroller and redtor uually aroxmated a a PID ontroller. In thoe method, PID arameter are dented or aroxmated ung PI ontroller and redtor arameter. On the other hand, the rooed method ealy onverted nto an ordnary eedbak ontrol loo wthout any need or aroxmaton or dentaton. In the rooed method, the ontroller a PI ontroller and t oned wth, a PID ontroller ganed and, by o dong, aordng to Fg. 7 and eq. (12), the rooed ontrol loo onverted to a onventonal eedbak ontrol loo. Thereore, PID ontroller arameter are ealy ound ung mle mathematal orrelaton. Fgure 7. Equvalent truture o the rooed method K (18) PI D [ K ] PID 1 ( ) K VI. SIMULATION In th eton, an examle reented or omenatng tme delay and RHP zero multaneouly. -4 Aume a roe model uh a (2 -.1)e. For 2 ( 1) alyng the rooed method, at rt, the omenator mut be eleted. The way the arameter are hoen derbed a ollow: A. Choong the order, gan and tme ontant o Sne a trtly roer traner unton, eleted a the rt order traner unton. In the rt order traner unton, both the tat gan and the tme ontant are eetve arameter. For mlty n tunng arameter, the tme ontant o eleted a equal to one and the gan o obtaned o that eq. (15) ated or the uer bound o ol ( j) unertanty. Sne that not determned at rt, ol otulated. Thereore, aordng to (15), the gan o ould be equal to the uer bound o ( j) unertanty. Thereore 2 1 eleted. B. Prmary tunng o the ontroller arameter Now, baed on the hoen, the ontroller arameter are ealy determned by ung any knd o tunng roedure. Then, aordng to the ISE rtera, the ontroller are obtaned a:.6658, ( ) ,.2122, PI ( ).8987 and.2246, PID ( ) or the rooed and ordnary eedbak method, reetvely. C. Retunng o the omenator gan to remove all the RHP zero rom the ool Beaue n th ondton, n a e range, ool the non-mnmum hae, the gan o hould be nreaed. In a tral roedure the mnmum value o the gan whh reult n a mnmum hae ool determned a Then, the omenator eleted a. 1 In Fg.8, the rooed and mle eedbak method reone are hown. It hould be noted that a load dturbane equal to. 5 wa nutted n the ontrol ytem at tme 15 e. Fgure 8. The rooed and eedbak method reone 35

6 211 IEEE Internatonal Conerene on Sytem Engneerng and Tehnology (ICSET) In Table 1, two reone a well a underhoot, whh were reented n Fg. 8 are omared by Integral o Abolute Error (IAE) and Integral o Square Error (ISE) meaure. TABLE 1 - IAE AND ISE FOR THE PROPOSED AND FEEDBACK METHODS RESPONSES IN FIURE 9 Method Prooed PI PID IAE ISE Underhoot A an be een n th table, the rooed method reone ha the lower ollaton and lower underhoot. VII. CONCLUSION A new method or ontrollng roee wth large tmedelay rooed. Th baed on the domnant gan onet. The method baed on the dea that a domnant gan redtor wth low order dynam an be ued or hangng the dynam behavor o the oen loo traner unton, rom nonmnmum hae to mnmum hae. I the ontroller ytem oned wth the redtor, a PID ontroller ganed and, by o dong, the rooed ontrol loo onverted to a onventonal eedbak ontrol loo. REFERENCES [1] T. Hägglund, A redtve PI Controller or Proee wth Long Dead- Tme, IEEE Contr. Syt. Mag., vol. 12,. 57 6, [2] O. J. M. Smth, A ontroller to overome dead tme, ISA Journal, vol. 6,.28-33, [3] J. P. Rhard, Tme-delay ytem: an overvew o ome reent advane and oen roblem, Automata, vol. 39, , 23. [4] J.E. Normey-Ro & E. F. Camaho, Dead-tme omenator: A urvey, Control Engneerng Prate, vol.16, , 28. [5] Z. Palmor, Stablty Proerte o Smth Dead-Tme Comenator Controller, Int. J. Control, vol. 32, , 198. [6] J.E. Normey-Ro, C. Bordon, & E. F. Camaho, Imrovng the robutne o Dead-tme omanatng PI ontroller, Control Engneerng Prate, vol. 5, , [7] A. Ingmundaron & T. Hägglund, Robut tunng roedure o Deadtme omenatng ontroller, Control Engneerng Prate,vol. 9, , 21. [8] W. Mhel, & S.. Nuleu, On the delay entvty o Smth redtor, Internatonal Journal o Sytem Sene, vol. 34, , 23. [9] K., Watanabe, and M. Ito. A Proe Model Control or Lnear Sytem wth Tme-Delay, IEEE Tran. Automat. Contr., vol. AC 26, , [1] K. Atrom, C. C. Hang, & B. C. Lm, A New Smth Predtor or Controllng a Proe wth an Integrator and Long Dead-Tme, IEEE Tranaton on Automat Control, vol. 39, , [11] Lu., Xang, Y., Yang, Q. Wang, and W. Zheng, A Double Two - Degree - o -Freedom Control Sheme or Imroved Control o Untable Delay Proee, Journal o Proe Control, vol. 15, , 25. [12] I. Kaya, and D. P. Atherton, A new PI - PD Smth Predtor or Control o Proee wth Long Dead-Tme, n: 14th IFAC World Congre 1999, Vol. C: [13] I. Kaya, Obtanng Controller Parameter or a New PI-PD Smth Predtor Ung Auto tunng, Journal o Proe Control, vol. 13, , 23. [14] S. Majh, and D. P. Atherton. Moded Smth Predtor and Controller or Proee wth Tme-Delay, IEE Pro. Control Theory Al., vol. 146, , [15] S. Majh, and D. P. Atherton. Obtanng Controller Parameter or a New Smth Predtor Ung Auto Tunng, Automata, vol. 36, , 2. [16] M. R. Mataušek, and A. D. Mć. A moded Smth Predtor or Controllng a Proe wth an Integrator and Long Dead-Tme, IEEE Tran. Automat. Control, vol. 41, , [17] M. R Mataušek, and A. D. Mć. On the Moded Smth Predtor or Controllng a Proe wth an Integrator and Long Dead-Tme, IEEE Tran. Automat. Cont., vol. 44, , [18] S. Ramanathan, R. L. Curl And C. Kravar. Dynam and ontrol o the umulatve ma raton o a artle ze dtrbuton, ACC Pro. 1987, Mnneaol.RDS [19] S. Ramanathan, Control o Quaratonal Dtrbuted Sytem wth Examle on the Control o Cumulatve Ma Fraton o Partle Sze Dtrbuton. Ph.D. The, Unverty o Mhgan, Ann Arbor, [2] S. Ramanathan, R. L. Curl and C. Kravar. Dynam and Control o Quaratonal Sytem, AIChE J. 1989; 35(6): [21] U. Vollmer, J. Rah, H -Control o a ontnuou rytallzer, Control Engneerng Prate, vol. 9, , 21. [22] U. Vollmer, J. Rah Poulaton balane modelng and H -Controller degn or a rytallzaton roe, Chemal Engneerng Sene, vol. 57, , 22. [23] P. Zĺtek, J. Hlava, Anohron nternal model ontrol o tme-delay ytem, Control Engneerng Prate, vol. 9, , 21. [24] K.J. Atrom, and T. Hagglund, The uture o PID ontrol, Control Engneerng Prate, vol. 9, , 21. [25] C. Valena,.Enoa, J. ralt, F. ralt, Otmzaton o nvertae roduton n a ed-bath boreator ung mulaton baed dynam rogrammng ouled wth a neural laer, Comuter and Chemal Engneerng, vol.31, , 27. [26] A. Ahoor, B. Mohr, A. Khak-Sedgh and M.R. Bakhtar, Otmal ontrol o a nonlnear ed-bath ermentaton roe ung model redtve aroah, J. Proe Control, vol.19, , 29. [27] A. Ingmundaron, & T. Hägglund, Perormane omaron between PID and Dead-tme omenatng ontroller, Journal o Proe Control, vol. 12, , 22. [28] Adan O'Dwyer. Handbook o PI and PID ontroller tunng rule. Imeral Collage Pre, 2nd edton, 26. [29] K.J. Atrom, and T. Hagglund, n The Control Handbook, Edtor W.S. Levne,(CRC/IEEE Pre), , [3] S. lkman, R. Kuleky, and. Nudelman, PID ontrol degn or ower taton roee baed on tme Delay omenaton Smth redtor, n MMAR 22 IEEE Con., Poland, [31] Ta, K.I., W. Fu-Jen, L. Hao-Chuan and C. Hu-Cheng, A novel ontrol heme or temerature ontrol o mahne tool ooler, J. Aled S., vol. 8, , 28. [32] M. Shrvan, M. Emael., Exlotng dtrbuted arameter roe ytem behavor n requeny doman, 18th nternatonal ongre o hemal and roe engneerng, Cha., Prauge,