OVERSHOO FREE PI CONROER UNING BASED ON POE ASSIGNMEN Nevra Bayhan * Mehmet uran Söylemez ** uğba Botan ** e-mail: nevra@itanbul.edu.tr e-mail: oylemez@el.itu.edu.tr e-mail: botan@itu.edu.tr * Itanbul Univerity, Faculty of Engineering, Dertment of Electrical & Electronic Engineering, 385, Avcılar, Itanbul, urey ** Itanbul echnical Univerity, Faculty of Electrical & Electronic Engineering, Dertment of Electrical Engineering, Mala, Itanbul, urey eyword: Pole aignment, PI controller tuning, Padé aroximation, overhoot, critically damed ytem, P36 exeriment et ABSRAC hi er rooe a new formulation for the tuning of roortional-integral (PI) controller. he method i baed on the ole lacement method alied for firt order lu time delay (FOPD) rocee uing Padé aroximation. An imortant roerty of the rooed tuning formulation i that it reult in a fat reone with no overhoot for a wide range of ytem. A tuning rameter i alo rovided to fine-tune a given ytem. he reult are comred with everal well-nown PI tuning formula uing theoretical aroximation, imulation a well a real exeriment. I. INRODUCION Mot of the indutrial ytem are controlled by derivative of PID controller []. When there i a time delay or large amount of no in the roce, the derivative term of PID controller are uually et to zero yielding PI controller. It i imortant for racticing engineer to be able to quicly determine good coefficient for PI controller after identifying ome ey rameter of the ytem with imle exeriment. It i a common ractice to aroximate a given roce to a firt order lu time delay (FOPD) ytem and then calculate the controller coefficient aordingly. here are many well-nown formula derived to tune PI controller among which the following can be counted: Ziegler-Nichol [], Cohen-Coon [3], IMC-PI [], otimum integral abolute error (IAE) [5]-[6], otimum integral time-weighted abolute error (IAE) [5], otimum integral quared error (ISE) [5], and otimum integral time-weighted quared error (ISE) [7]. Mot of thee method are derived baed on time domain erformance. Ziegler-Nichol deign method i a claical method to find a good tarting oint for PI controller tuning. here are two method rooed by Ziegler and Nichol: one i baed on the meaurement of the critical gain and critical frequency of the lant and the other i uing the te reone of the oen-loo ytem. We will be uing the econd method in thi er. Cohen-Coon method i a dominant ole deign method. he ey feature of thi tuning method i that integrated error i minimized. hu, thi method give good load diturbance rejection. IMC-PI tuning method i baed on the idea of cancelling the ole of the roce uing the zero introduced by the PI controller and then tuning the cloedloo ytem reone uing a free rameter, cl (ee able ). he idea behind the other tuning method i to chooe PI controller rameter to minimize an integral cot functional. For the o called etoint formulation the cot function are ued a IAE = r() t y() t dt, and IAE = t r() t y() t dt, () () (t) ISE = t r t y t dt ISE = r() t y() t dt r i the reference inut and y (t) i the outut of the ytem. It i alo oible to ue the diturbance ignal intead of the reference inut in above cot function to obtain the o called load diturbance tuning formula. hi er rooe a new tuning formula that utilize Padé aroximation and the idea of dominant ole aignment. he er i organized a follow. Section and 3 are devoted to ytem rereentation and the derivation of the tuning formula. Section reent imulation reult on two examle ytem and demontrate the advantage of the rooed method. Some exerimental reult carried out on P36 exeriment et are given in Section 5. Finally, Section 6 contain concluive comment and uggetion for future reearch. II. REPRESENAION OF HE FOPD PROCESSES USING PADE APPROXIMAION Conider the feedbac control ytem hown in Figure.
r - F() G ( ) e y G, ( ) = G () = (7) Figure : he FOPD ytem with PI controller A firt-order roce with a time delay can be decribed by the following tranfer function G () G() e () rereent the time delay and G ( ) i the delayfree ytem defined a follow: N () G () = = D() i the teady-tate gain and i the time contant. F () in Figure i a PI controller to control the firt order lu time delay ytem G () decribed by (). he PI controller i given a () NF () i F () = = (3) DF () i P and i rereent the roortional and integral gain, reectively. i i the integral time contant. Padé aroximation of the firt order lu time delay ytem given by () i defined a follow: ˆ ˆ N () G () = G () G () () D ˆ () G () i a rational tranfer function (Padé) aroximation for the term e, and i given by n ( ) h ( ) = e G ( ) (5) n h ( ) h = ( n )! n! = n!!( n )! and n rereent the order of the aroximation. For examle, the firt and the econd order Padé aroximation are given by (6) It hould be noted that the higher the order of aroximation the better the rereentation of G () the time delay term e. It i nown, however, that a firt or a econd order aroximation i enough in mot of ractical alication. In thi er, a econd order Padé aroximation i ued to get a higher auracy in the reulting formulation. Uing () and the PI controller F () decribed in (3), the (aroximate) cloed-loo tranfer function of the unity feedbac ytem in Figure can be written a which can be rewritten a ˆ ˆ GF () () Gc () = (8) GF ˆ ( ) ( ) () ˆ ˆ N () () F N Gc = N () Nˆ() D () Dˆ() F Hence, the (aroximate) cloed-loo ytem characteritic olynomial become F (9) Pˆ () = N () Nˆ() D () Dˆ() () c F F III. PI CONROER UNING BASED ON POE ASSIGNMEN Equation () reveal the fact that the rameter of the controller get into the coefficient of the cloed-loo ytem characteritic olynomial linearly. Actually, it i oible to rewrite () a Pˆ = Dˆ() Nˆ() Nˆ() () c i Uing a econd order Padé aroximation D ˆ () and N ˆ () can be given a ˆ () ( )( D = ) () ˆ () ( ) N= (3) Rearranging ()-(3) to get a monic characteritic olynomial, we have Pˆ c ( ) = [ i ( 6 i) 6 6 () ( i ) ( 6 ) ] 3
We remar that the degree of the cloed-loo ytem characteritic olynomial i. It i oible to how that ole (ay and ) of the characteritic olynomial decribed by () can almot alway be arbitrarily aigned uing the free rameter and i [8]. et the olynomial that correond to the aigned ole be given a P () ( )( ) = ξω ω (5) d n n Here, ξ and ω n are nown a the daming ratio and natural frequency, reectively. Under the aumtion that the remaining two cloed-loo ytem ole are far left of the aigned ole on the comlex-lane, it i oible to dicu that the time domain behavior of the cloed-loo ytem i determined by the daming ratio and the natural frequency defined by (5). In many ractical ituation, no overhoot i allowed, while a fat reone i deired. herefore, chooing the daming ratio ξ = mae ene for many ractical ytem. On the other hand, the cloedloo ytem cannot be arbitrarily fat in comrion to the oen-loo ytem due to hyical contraint. Uually, electing the ettling time of the cloed-loo ytem in the order of that of oen-loo ytem i enible. Conidering the fact that ettling time i inverely roortional to ξω n for the cloed-loo ytem and it i roortional to for the oen-loo ytem, it i oible to chooe the natural frequency ω n a a ωn = (6) a i a free rameter to fine-tune the eed of the cloed-loo ytem reone. In many ractical cae a can be choen between.5 and. It i then oible to exre the cloed-loo characteritic olynomial a Pˆ () = P () P () (7) c d e a a a () = = (8) and P e () i the reidue olynomial formed by the ret of the cloed-loo characteritic olynomial ole, which can be written a Pe () = c c (9) hu, the right ide of (6) i a c ac a c ( ) Pe( ) = [ a ac c 3 a c () Equating the coefficient of the ame ower of in () and (), i, c and c can be found a 3 3 a (a ) ( a ) a = i 3 a (a ) ( a ) a (a ) 3 3 a ( a a ) = a ( a ( ) ) 3 3 a ( a ( )) 3 3 a a a 3 3 ( a ( )) ( ( ) ) c = 3 6a 7( a ) a 7( a ) a c = 3 86a 86 ( 6 a ) () () (3) () = ˆ a (5) Note that equation () and () define a tuning formula for the controller rameter. Although thee equation eem to be comlex, it i traightforward to calculate controller gain once the ytem rameter (, and ) are determined and the free rameter a i choen. It hould be noted that for tability we require that a > and e () i Hurwitz. Since e () i a econd order olynomial, it i Hurwitz, if, and only if, it coefficient ( c and c ) are oitive. Furthermore, for the dominant ole aignment aroach adoted above to be meaningful a neceary condition i that the root of the reidue olynomial e () to be on the left of the dominant ole ( = = a/ ). For thi aim, the olynomial given below required to be Hurwitz. a a a Pe ( ) = ( ) c( ) c a a a = c c c ( ) ( ) (6) herefore when chooing the value of a checing the following condition i a good ractice
c > a a c > c a c and c are a defined in (3) and (). Formula Ziegler-Nichol (ZN) Cohen-Coon (CC) IMC-PI (IMC) ISE for oad Diturbance (ISE) IAE for oad Diturbance (IAE) IAE for Set Point Change (IAESPC) ISE for oad Diturbance (ISE) ISE for Set Point Change (ISESPC) IAE for oad Diturbance (IAE) IAE for Set Point Change (IAESPC) able : Some well-nown PI tuning formula i.9 / 3.9 (7) 3.33 /.33( / ). / ( ) cl.35.98.959.986.9.68.739.77.86.758..33( / ).79.95.535.586.9.7.968.7( / ).859.977.67.68.96.586.3.65( / ) IV. SIMUAION RESUS In thi ection, the rooed ole aignment tuning method (PA) i comred with everal well-nown PI tuning formula (ee able ) on two examle ytem uing comuter imulation. Examle.. Conider a FOPD roce a defined in () and () with =, = and =. he free (finetuning) rameter for the rooed method (PA) i choen to be a =.85 and that of IMC-PI method i choen to be cl =.6 to get near otimal reult. he imulation reult are ummarized in able. A can be een from thi table, all formulation excet PA and ZN reult in a coniderable overhoot. Although delay and r time i long for the rooed PA method the ettling time i very atifactory (coming only after IAE and IAE method). Ste reone obtained uing a Simulin model can be een in Figure. able : he time-domain ecification obtained for Examle..5 Overhoot Settling Delay R PA 6.968.87 3.553 ZN.89.678 6.9366 CC.3853.6.5.8698 IMC.887.99.5795.857 ISE.88593.38.3686.8338 IAE.676 6.66.5573.99 IAESPC.969 7.569.9759.663 ISE.633.776.665.8866 ISESPC.3533 7.7669.6999.89 IAE.79998 6.58986.688.86 IAESPC.3996.385.7555.6353 c.5 c c 3 5 6 7 8 9 Figure : Unit-te reone of PI tuning method for Examle. Examle.. Conider a FOPD roce a defined in () and () with =., =.5 and =. Free rameter a i choen a.3 to obtain a fat reone. he imulation reult are ummarized in able 3. It hould be noted that ince the method CC, IMC, IAESPC, ISE, and ISESPC yield untable cloed-loo ytem they are not included in able 3. Simulin imulation for the te reone are given in Figure 3. able 3: he time-domain ecification Overhoot Settling Delay R PA.5977.9.675 ZN.83355.85.678 ISE.675.957.75868.587 IAE.637 5.673.7578. IAE.6565 9.6678.789.38889 IAESPC.56835 7.9538.7887.73 A can be oberved from able 3, the rooed method (PA) give robably the mot aetable reult eecially when no overhoot i wanted in the outut.
.8.6.5 Outut...8.6.. 3 5 6 [ec] Figure 3: Unit-te reone for examle. V. EXPERIMEN RESUS he rooed tuning formula i alo teted on P36 roce training et that i available in Control aboratory of IU. In thi roce, air in the urrounding atmohere i drawn through a changeable inlet by an axial fan, driven through an electrical heater coil, e through a latic tube and then it i let out to the atmohere. he control roblem in thi roce i to control the temerature of the air going out of the tube. By changing the electrical ower ulied to the heater grid, the temerature i controlled. here are three oition along the latic tube, a thermitor can be laced to meaure the temerature of the air. hi ditance between the thermitor and the heater grid introduce a tranort delay into the ytem. herefore, it i oible to model thi dryer ytem a a FOPD ytem. After a few imle tet, the tranfer function of the exeriment et i determined a G( ) =.875e.3 /(.6 ) Ste reone obtained for different value of the free rameter a, and thoe obtained uing different tuning method are given in Figure and Figure 5, reectively. Figure reveal that a=.3 i a good choice for thi lant. By comring the reult given in Figure 5, it i oible to tate that the tuning method rooed wor very well by roducing a te reone with no overhoot and a very good ettling time (the bet together with IAESPC and IMC) in comrion to the other method. Outut [V].5 3.5 3.5.7.3.6.9..5 a=.7.5 a=. a=.3 a=.6. a=.9.5 a=. a=.5 3 5 6 7 8 9 [ec] Figure : Ste reone obtained for different value of rameter a in P36 exeriment et. Outut [V] 3.5 3.5.5 c c.5.5.5 3 3.5.5 5 [ec] Figure 5: Ste reone obtained for different tuning method in P36 exeriment et. ( a =.3 and cl =.3 ) VI. CONCUSION A new PI tuning method i rooed and comred with other well-nown method. An imortant advantage of the rooed method i the cloed-loo ytem time reone ha uually no overhoot, while a very good ettling time i obtained. Simulation and exeriment reult demontrate thi advantage of the method clearly. R time and delay time characteritic for the rooed method are uually low in comrion to other method. However, thi i exected ince overhoot i avoided. Future reearch will focu on tuning PID and PD controller rameter uing imilar aroache. ACNOWEDGEMEN Part of thi reearch i funded by the IU Scientific Reearch Suort (IU-BAP) Project No. 3685. REFERENCES []. J. Atrom and. Hagglund, PID control'', he Control Handboo (W. S. evine, ed.),. 98 9, Florida: CRC Pre, 996. [] J. G. Ziegler and N. B. Nichol, Otimum Setting for Automatic Controller, ran ASME, Vol. 6,. 759-768, 9. [3] G. H. Cohen and G. A. Coon, heoretical Conideration of Retarded Control, ran ASME, Vol. 5,. 87-83, 953. [] W.. Ho, C. C. Hang and J. H. Zhou, Performance and Gain and Phae Margin of Well-nown PI uning Formula, IEEE ranaction on Control, Sytem echnology, Vol. 3, No.,. 5-8, 995. [5] C. A. Smith and A. B. Corriio, Princile and Practice of Automatic Proce Control, New Yor: Wiley, 985. [6] F. G. Shiney, Proce Control Sytem: Alication, Deign and uning, 3 rd ed. New Yor: McGraw-Hill Boo Co., 985. [7] M. Zhuang, D. P. Atherton, Automatic uning of Otimum PID Controller, IEE Proc-D, Vol., No. 3,. 6-, 993. [8] M.. Söylemez, Pole Aignment for Uncertain Sytem, Reearch Studie Pre, U, 999.