Application of a dead time compensator as a PID controller

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1 Alaton o a dead tme omenator a a PID ontroller M. Emael and M. Shrvan Abtrat In th aer, the alaton o Domnant an Conet or ontrollng tme delay roee ondered rom a new vewont. In th vewont, tme delay ondered a unton wth nnte Rght Hal Plane (RHP) zero whh are tranerred to the Let Hal Plane by addng a roer traner unton to the oen loo unton. Furthermore, by denng a new onet, alled mnmum-gan requeny, the rooton or ontrol ytem tablty and the manner o hoong order and gan o the added traner unton are roved. Alo, the alaton o th method tuded n order to multaneouly omenate or the Rght Hal Plane zero and tme-delay. Fnally, eay alaton o th method hown n an ordnary eedbak loo a a PID ontroller. Index Term Domnant gan, PID ontroller, Rght hal lane zero, Tme delay omenator. I. INTRODUCTION Many o roee n hemal ndutre uer rom tme delay, beaue o whh the nut gnal to ontroller alo ha delay and, thereore, the erormane o ontrol ytem oor. To mrove the ontrol ytem erormane, ome redtor have been ued [1]. The method whh have ued redtor an be ategorzed nto three laaton: 1. The method whh ung a model, redt the tme delay arameter and remove t 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, artularly the error n the tme delay arameter [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 roblem mentoned or the Smth redtor. To olve thee roblem, many reearher have reented derent method alled Dead Tme Comenator (DTC) [9-17]. The larget det o DTC-baed method lak o ther alaton or roe model n whh ther traner unton are rratonal and the tme-delay annot be atored out traghtly rom the traner unton. In order to olve th roblem, 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 rratonal traner unton (Ir-TF) model nto two art: : art o the model that nlude RHP zero and Manurt reeved Marh 28, 211; reved May 3, 211. M.Emael wth the Natonal Petrohemal Comany (orreondng author to rovde hone: ; ax: ; e-mal: m.emael@ n.net). M. hrvan, wth the the Det. o Chem. Eng., Iran Unverty o Sene & Tehnology (e-mal: hrvan.m@ut.a.r). : 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 Zĺtek & Hlava [23] ontrolled the roee wth rratonal model by ung H ontrol and removng rom the oen loo. 2. The method whh redt nut gnal to the ontroller ung a dervatve mode n ther ontroller [24-26]. In thee method, ontroller are ondered ether a equaton (1) [24] or equaton (2) [] and ome o the arameter are xed at a tyal value (a reommended n the textbook on the roe ontrol) n order to lmt the omlexty o PID ontroller. For examle, n [24] and [], T = 4T d and N=1 are rooed. ( ) = d (1) 1 Td ( ) = (1 ) (2) T T d 1 N The advantage o thee method ther eay alaton n ndutral ontrol loo beaue 93% o the ndutral ontrol loo have PID ontroller [27]. Alo, t hown that tme delay mall, the erormane o a PID ontroller wll be better than a DTC []. However, a large tme delay ext, a DTC wll have a better erormane than a PID. 3. Thee method are a onaton o the above two method. In thee method, at rt, the outut gnal o a DTC ontroller dented ung an dentaton ytem 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 [28-29]. Beaue thee method need an dentaton method, they are eldom ued n rate. In [3], domnant gan onet ntrodued baed on the requeny reone behavor o an rratonal traner unton and n [31], th onet wa ued to mrove the erormane o a tme delay ontrol ytem. In th method, a domnant-gan mnmum-hae traner unton, denoted a added to the oen loo. In th way, the non-mnmum hae behavor o the oen loo onverted to the mnmum hae behavor and then the ontrol loo erormane mroved. In th aer, the rooed method n [31] tuded rom a new eretve. In th vewont, tme delay ondered a nnte RHP zero and t hown that how mnmum 7

2 hae and non-mnmum hae behavor o an rratonal traner unton ould relate to the loaton o zero n the RHP or LHP. Furthermore, by denng a new onet, alled mnmum-gan requeny, the rooton o tablty and the manner o hoong order and gan o are roved. Fnally, the rooed method hown to be a onaton o the rt and eond redton method, wth no need or an dentaton method. In th method, wthout any need or the ontroller outut dentaton, a PID ontroller ould be degned. II. DOMINANT AIN CONCEPT The onet o domnant gan wa roved n [3, 31]. It tate that n an rratonal traner unton or a QRDS t model lke ( ) = ( ) ( ) e d, n whh the 1 2 1( ) the tme-delay-ree term o the model, n any requeny range o ω 1 < ω < ω, 2 ollow the requeny behavor o domnant gan term n the model. t Denton 1: 1 ( ), ( ) e d and t d 2 ( ) = 1 ( ) 2 ( ) e are hown a vetor A, B, C reetvely, n the requeny n whh A loated agant B, C wll have mnmum length and wll be loated n the dreton o the larget vetor (Fg.1). In th tuaton, have a mnmum gan n the requeny reone lot and the orreondng requeny wll be hown byω. mg, A = B () at ω = ωmg, zero wll be loated on the magnary ax. In gure 2 through, varou requeny reone o a mle QRDS model are hown. The zero are alo drawn to demontrate the orreondene between the requeny reone and the nuer o model RHP zero. Amltude rato Phae (deg.) =12 1=1/(1) 2=.*ex(-1)/( 2 21) Frequeny (rad/e) wmg1 wmg2 wmg3-3 =12-4 1=1/(1) 2=.*ex(-1)/( 2 21) Frequeny (rad/e) Fg.1. Vetor reentaton, or two ae at mnmum gan ω mg,, n whh ha The mnmum gan requene o are o rmary mortane beaue, n at, the loaton o the zero n a QRDS model or a qua-olynomal deendent on the gan domnaton at thee requene. For examle, A > B Z = LHPZ (3) ω mg, at ω = ωmg, Or, A < B at ω = ω In the above equaton, zero to the ω mg, an be onluded that : mg, Z Z ω mg, ω = mg, RHPZ (4) an ndator or the related requeny. Alo, rom the above relaton, t Fg. 2 Frequeny reone o a QRDS model demontratng an nnte nuer o LHP zero reultng rom all requeny gan domnaton o delay-ree term. In Fg. 2, all ωmg, our n requene where the mnmum hae term domnant omarng to the other term. Then, aordng to the domnant gan onet t exeted that all the zero are loated n LHP. A een n Fg. 2, th exetaton onrmed by the realty. Amltude rato Phae (deg.) =./( 2 21) 2=ex(-)/(1) = Frequny (rad/e) =./( 2 21) - 2=ex(-)/(1) = Frequeny (rad/e) 8

3 Fg. 3 Frequeny reone o a QRDS model demontratng an nnte nuer o RHP zero reultng rom all requeny gan domnaton o delay nluded term. In Fg. 3, all the ω mg, our n loaton where the non-mnmum hae term domnant. Then, aordng to equaton (3), t exeted that all the zero are loated n RHP. Fg. 3, onrm th exetaton. Amltude rato Phae (deg.) =1/(1) 2 2=.ex(-)/(1) = Frequeny (rad/e) =1/(1) 2 2=.ex(-)/(1) = Frequeny (rad/e) Fg. 4 Frequeny reone o a QRDS model demontratng an nnte nuer o RHP zero reultng rom hgh requeny gan domnaton o delay nluded term and a ew nuer o LHP zero reultng rom low requeny gan domnaton o delay ree term. In Fg.4, the mnmum hae term domnant n < ω < 4 rad / e. In th nterval, 3ω are loated n mg, th requeny range and remnder o our n ωmg, requene where the non-mnmum hae term domnant. Then, t exeted that there are 3 LHP zero and the remnder o zero are loated n RHP. Fg. 4 onrm th exetaton. Amltude rato =./(1) =12 2=ex(-)/(1) Frequeny (rad/e) Phae (deg.) =./(1) 2=ex(-)/(1) 2 = Frequeny (rad/e) Fg. Frequeny reone o a QRDS model demontratng an nnte nuer o LHP zero reultng rom low requeny gan domnaton o delay-ree term and a ew nuer o RHP zero reultng rom hgh requeny gan domnaton o delay ree term. In Fg., ontrary to the ae o Fg. 4, the non-mnmum hae term domnant n < ω < 4rad / e. In th nterval, 3 ω and the remnder o ωmg, our n mg, loaton where the mnmum hae term domnant. Then, t exeted that there are 3 RHP zero and the remnder o zero are loated n LHP. Fg., alo, onrm th exetaton. In ontrol ytem, the mnmum hae behavor (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 a Domnant an Contrant (DC) on the oen loo, 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 vewont, tablty o the rooed method roved n the ollowng rooton. Prooton 1: n a mle oen loo traner unton a td ol ( ) = 1 ( ) 2 ( ) e n whh 1( ) and ( ) 2 do not have any rght-hal lane ole, a table traner unton wth an order o 1 or zero, uh a, added n a way that: jω ) ( jω ) (6) ( mg, ol mg, the ontrol loo wll be abolutely table. A an exlanaton, 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 trae the behavor and beome a mnmum hae unton, then there wll not be any RHP zero n the oen loo. A generally known, a ontrol ytem n whh the oen loo a rt order traner unton abolutely table. Thereore, the rooed ontrol ytem abolutely table, rovded that the Domnant an Contrant ated or the non-delay term at all requene. 9

4 III. COMPARIN THE PROPOSED METHOD WITH A ENERALIZED SMITH PREDICTOR A mentoned beore, n [18-2], a method wa reented to omenate or the tme-delay. Th method baed on Smth redtor and beaue t aable o ontrollng QRDS, t alled eneralzed Smth Predtor. The ontrol truture o th method hown n Fg. 6. Fg.6- eneralzed Smth Predtor In th method, nred rom Smth method or omenatng or 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 have RHP zero. By omenatng or the eet o rom the oen loo traner unton, the oen loo behavor wll be hanged rom non-mnmum hae to mnmum hae. I there no model mmath, the oen loo n th method wll beome: SPoenloo = ( ) ( ) (7) Then, all the RHP zero wll be removed rom the oen loo, however, n the oen loo, there are LHP zero yet. For th ytem, the haratert equaton wll be: 1 ( ) ( ) = (8) The rooed method n [31] 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.7. Fg.7.The rooed ontrol truture In th heme, the Domnant an Delay Comenator (DDC), 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 ( ). In th way, all the RHP zero reultng rom the tme-delay wll be onverted to LHP zero. We may alo all the a "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 oen loo unton. Thu, by ung th omenator, the requeny reone behavor o the oen loo traner unton wll hange rom delay behavor to non-delay behavor. The loed loo traner unton o th ytem wll beome: Y ( ) ( ) ( ) ( ) = (9) Y ( ) 1 ( ) ( ) ( ) ( ) ( ) m b Y ( ) d ( ) ( ) d ( ) = (1) d( ) 1 ( ) ( ) ( ) ( ) ( ) The haratert equaton : 1 ( ) ( ) ( ) ( ) m ( ) = (11) The MBPL unton " " hould be eleted a a mnmum hae unton. It addton to the oen loo by onderng the DC requrement (equaton (6)) reult n the onveron o all the RHP zero n the oen loo unton to the LHP zero. IV. COMPARIN THE PROPOSED METHOD WITH A PID CONTROLLER Beaue o the extene o dervatve mode, PID ontroller are known to have redtve roerte. But, tunng the ontroller arameter 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, a reommended n the onventonal textbook on the roe ontrol. On the other hand, ne n roe ontrol alaton, more than 93% o the ontroller are o PID tye [27], ome o the reearher [28, 29] have tred to traner Smth redtor erormane to an ordnary eedbak ontrol loo. To aheve th goal, a onaton o the ontroller and redtor mut be aroxmated a a ontroller. For examle, the orgnal one a PI ontroller, a onaton o the ontroller and redtor uually aroxmated a a PID ontroller. In thee method, PID arameter are dented or aroxmated ung PI ontroller and redtor arameter. However n the rooed method t 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. 8 and equaton (12), the rooed ontrol loo onverted to a onventonal eedbak ontrol loo. Thereore, PID ontroller arameter are ealy ound ung mle mathematal orrelaton. Fg. 8. Equal truture o the rooed method m m 16

5 τ ( ) PI = 1 ( ) 1 1 τ τ D = = [ ] PID τ 1 V. CHOOSIN THE ORDER AND AIN OF (12) In rooton 2 and 3, method or eletng the order and gan o are reented. 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. Proo: n the rooed method, the overall oen loo traner unton o : ool ool ( ) = = D ( ) ( nb) ( ol ) ( ) N ( mo) ol ol ( no) Dol ( ( ) e ) td (13) where, the ubrt ol and ool reer to the mle oen loo (wthout MBPL-omenator) and overall oen loo, reetvely. and 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) D ( ) N ( ) D e t d = (14) ol ol ol Aordng to [32], the requred ondton or elmnatng the RHP zero rom the qua-olynomal, lke the one n (12), : no mo nb nb no mo () ol ol The rt nequalty n () orreond to the hgh requeny gan domnaton requrement whle the eond one ate the low requeny gan domnaton requrement. In the md-requeny range, the gan domnaton requrement an be obtaned by hangng the eond nequalty n () to: Δ > (16) Δ the exeve gan requrement o the omenator to guarantee that there not any remanng RHP zero n the overall oen loo traner unton. Now, the ertnent argument th: wthout atyng the gan domnaton at hgh requene, t moble to elmnate all the RHP zero rom the oen loo traner unton whle the low requeny and md requeny RHP zero are oble to be elmnated eretly by adjutng the omenator gan o. Thereore, there hould be a ou on the order o and ol wth reet to eah other n order to reah to a utable order or the omenator. Thu, ol a roer traner unton,.e., no = mo, then nb = ; ol 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 omenator, " ", the ontrol ytem reahe an abolutely table ondton. Denton 2: The abolute tablty n a mle eedbak ontrol loo reer to the tuaton n whh the oen loo a table delay-ree rt order traner unton. However, by alyng the DC requrement, one an moe the abolutely table ondton when the roe hgh order or nlude tme-delay. Prooton 3: The bet value or the mnmum value whh ate the DC ondton,.e. equaton (6) or the uer bound o ol unertanty. Proo: Mnmum value whh ate equaton (6) : jω ) = ( jω ) (17) ( mg, ol mg, Sentvty unton or the rooed method are obtaned a equaton (18) and (19): 1 S( ) = (18) ( ) ( ) 1 1 ( ) ( ) ( ) T ( ) = 1 ( ) ( ) ( ) (19) A een n the above unton, to whatever level the value nreaed, S nreae and T dereae. Then, the erormane wll be robut and luggh by nreang the value. Thereore, or the roer erormane, equaton (6) mut be ated or the moble mnmum value o. Alo, to onrm that there not any RHP zero n oen loo, (17) mut be ated or the uer bound o unertanty. ol VI. SIMULATION In [31] everal examle o the alaton o the rooed method wa reented or the roee wth model mmathe, varable tme delay, ntegral and QRDS model. In th aer the method ued or omenatng tme delay ytem wth RHP zero multaneouly to how the wdet alablty o the method. In th eton, an examle reented or th ae. Examle: Aume a roe model uh a -4 (2 -.1)e ( ) =. For alyng the rooed method, at 2 ( 1) rt, the omenator mut be eleted. The way the arameter are hoen derbed a the 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 161

6 ontant are eetve arameter. For mlty n tunng the arameter o tme ontant eleted a equal to one and gan obtaned o that equaton () ated or the uer bound o ol model unertanty. Beaue not determned at rt, ol ( ) = ( ). Thereore, aordng to equaton (), the gan o ould be equal to the uer bound o unertanty. 2 Then ( ) = eleted. In Fg. 9, AR lot or the 1 rooed oen loo term are hown or the above ondton. Amltude rato = =2/(1) =(2ex(-4)-.1)/[(1) 2 ] Frequeny (rad/e) ol Fg.9. AR lot or the rooed oen loo term when = ( ) ( ) B. Tunng the ontroller arameter Now, baed on the eleted, the ontroller arameter are ealy determned by ung any knd o tunng roedure. Then, aordng to the ISE rtera, the ontroller are obtaned a.668 ( ) = and.2122 ( ) =.8987 or the rooed and ordnary eedbak method reetvely. C. tunng o the omenator gan to remove all the ool ( ) = ( ) ( ) ( ) RHP zero rom the Beaue n th ondton, n a e range, ool the non-mnmum hae, the gan o hould be nreaed. The mnmum value o the gan whh reult n the ool beomng mnmum hae Then, the omenator eleted a ( ) =. In Fg. 1 1, AR lot or the rooed oen loo term are hown or the above ondton. Amltude rato =1 =2./(1) 1=* Frequeny (rad/e) Fg.1. AR lot or the rooed oen loo term In Fg. 11, the rooed and eedbak method reone are hown. It hould be noted that a load dturbane equal to. wa nutted n the ontrol ytem at tme e. Fg. 11.The rooed and eedbak method reone In Table 1, two reone whh were mentoned n Fg. 9 are omared. TABLE 1- IAE AND ISE FOR THE PROPOSED AND FEEDBAC METHODS Method Prooed Ordnary Feedbak RESPONSES IN FI.9 IAE ISE Underhoot A an be een n th table, the rooed method reone ha the lower ollaton and lower underhoot. VII. CONCLUSIONS In th aer, the rooed method n [31] tuded rom a new ont o vew. In th vewont, tme delay ondered a nnte RHP zero and t hown that how the mnmum hae and non-mnmum hae behavor o an rratonal traner unton ould relate to the loaton o zero n the RHP or LHP. Conderng that n the rooed method, oen loo requeny reone wa ued or determnng arameter, th method doe not need an exat model, whh an advantage or the rooed method wth reet to the delay redton method (.e., Smth redtor, eneralzed Smth redtor and DT). Alo, when the rooed method omared wth a PID ontroller, the tunng omlexty o a PID ontroller not oberved, whh beaue, n the rooed method, dervatve mode arameter were determned ung a domnant gan ontrant. REFERENCES [1] T. Hägglund, A redtve PI Controller or Proee wth Long Dead-Tme, IEEE Contr. Syt. Mag., vol. 12,. 7 6, [2] O. J. M. Smth, A ontroller to overome dead tme, ISA Journal, vol. 6,.28-33, 199. [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. [] 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, rovng the robutne o Dead-tme omanatng PI ontroller, Control Engneerng Prate, vol., , [7] A. Ingmundaron & T. Hägglund, Robut tunng roedure o Dead-tme 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,.43-2, 23. [9]., Watanabe, and M. Ito. A Proe Model Control or Lnear Sytem wth Tme-Delay, IEEE Tran. Automat. Contr., vol. AC 26, ,

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