A NEW PROPORTIONAL-INTEGRAL-DERIVATIVE (PID) CONTROLLER REALIZATION BY USING CURRENT CONVEYORS AND CALCULATING OPTIMUM PARAMETER TOLERANCES

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1 ISANBUL UNIVESIY ENGINEEING FACULY JOUNAL OF ELECICAL & ELECONICS YEA VOLUME NUMBE : 2001 : 1 : 2 ( ) A NEW POPOIONAL-INEGAL-DEIVAIVE (PID) CONOLLE EALIZAION BY USING CUEN CONVEYOS AND CALCULAING OPIMUM PAAMEE OLEANCES AIM AÞIYICILA ULLANAA YENÝ BÝ OANI- ÝNEGAL-ÜEV (PÝD) ÝPÝ ONOL EDÝCÝ GEÇELEME VE OPÝMUM PAAMEE OLEANSLAININ HESAPLANMASI *Cevat EDAL *Al OE **Cevdet ACA *Ýstanbul echncal Unverst, Facult of Electrcal - Electroncs Engneerng Maslak, ÝSANBUL / UEY Phone: (90) , Fa: (90) , e-mal: [email protected] **YedtepeUnverst, Facult of Engneerng&Archtecture, Electrcal-Electroncs Engneerng Dept. aýþdaðý, Ýstanbul, urke Phone: (90) , Fa: (90) , e-mal : [email protected] ABSAC In process control ndustr, the proportonal-ntegral-dervatve (PID) controllers are one of the most mportant control elements. In practce, operatonal amplfers are generall used n analog controllers. On the other hand, current-mode (CM) crcuts such as second-generaton current conveors (CCII) and current feedback operatonal amplfers (CFOA) have consderable attenton due to ther wder frequenc band, lower power consumpton, better lneart and stablt propertes compared to ther voltage-mode counterparts, operatonal amplfers. he purpose of ths stud s to present a snthess procedure for the realaton of analog PID controller b the use of CCIIs. For the desgned PID crcut, the optmum parameter tolerances are determned. hese tolerances can be used to mprove and to control the senstvt performance of the proposed PID controller. e words: PID controller, current conveor, and optmum parameter tolerances. ÖZE Süreç kontrol endüstrsnde, orantý-türev-ntegral (PÝD) tp kontrolörler en öneml kontrol elemanlarýndan brdr. Ugulamada, analog kontrolörlerde genellkle þlemsel kuvvetlendrcler kullanýlmaktadýr. Öte andan knc nesl akým taþýýcý ve akým ger beslemel þlemsel kuvvetlendrcler gb akým çýkýþlý olarak çalýþan devreler, þlemsel kuvvetlendrclerle karþýlaþtýrýldýklarýnda daha genþ br frekans bandýna, daha düþük güç tüketm le çalýþma, ve daha doðrusallýk ve kararlýlýk öellklerne sahp olduklarýndan, son amanlarda oldukça çok dkkat çekmektedrler. Bu çalýþmanýn amacý akým taþýýcýlar (CCII) kullanarak br analog kontrolör gerçeklemek çn gerekl tasarým öntemn vermektr. asarýmý apýlan PID devres çn optmum parametre toleranslarý hesaplanmýþtýr. Bu toleranslar önerlen PID kontrol edcnn duarlýk davranýþýnýn leþtrlmes ve kontrol edlmesnde kullanýlablrler. Anahtar kelmeler: PÝD kontrolör, akým taþýýcý ve optmum parametre toleranslarý.

2 268 A New Proportonal-Integral-Dervatve (PID) Controller ealaton B Usng Current Conveors And Calculatng Optmum Parameter olerances 1. INODUCION he proportonal-ntegral-dervatve (PID) controllers are one of the most mportant control elements used n process control ndustr [1]. In practce, operatonal amplfers are generall used n analog controllers. On the other hand, current-mode (CM) crcuts such as secondgeneraton current conveors (CCII) and current feedback amplfers (CFOA) have consderable attenton due to ther wder frequenc band, lower power consumpton, better lneart and stablt propertes compared to ther voltagemode counterparts, operatonal amplfers [2-5]. hese CM crcuts have not been used n desgn of analog controllers ecept the one b Erdal et. al. [6] so far. he purpose of ths stud s to present a new PID crcut. In ths crcut, CCIIs are used as current followers n such a wa that voltage trackng non-dealtes are ecluded. he sgnal-flow graph approach s used to facltate the desgn. he crcut presented n ths work emplos also a mnmum number of CCIIs for the general transfer functon realaton of a PID controller. o verf the theor, the smulaton of the proposed crcut s performed and the results are dscussed. hen, the optmum parameter tolerances are determned b the use of parameter senstvtes. hese tolerances keep the relatve error of the output voltage of the controller n tolerance regon and the can also be used to mprove and to control the senstvt performance of the proposed PID controller. 2. CUEN CONVEYO he termnal relatonshps of the postve and negatve tpe CCIIs ( and CCII-) shown n Fg 1. are gven as follows [7]: (t) = 0, v (t) = v (t), (t) = (t) (1) ± (b) Fgure 1.(a) Crcut smbol of second-generaton current conveor (CCII) (b) Crcut smbol of current feedback amplfer (CFOA) Attachng a unt gan voltage buffer at the output of the CCII, whch s actuall a a current feedback operatonal amplfer, CFOA, enables eas cascadng n voltage-mode operaton because of the low output mpedance of the buffer. hus, the proposed PID controller can also be realed b usng of CFOAs. akng the non-dealtes of the CCII nto account Eq. (1) can be rewrtten as follows: (t) = 0, v (t) = β v (t), (t) = ± α (t) (2) Here α = 1 ε denotes the current gan and β = 1 ε v denotes the voltage gan of the current conveor, ε, ( ε <<1) s the current trackng error and ε v, ( ε v <<1) s the voltage trackng error. he non-deal effect of the voltage buffer can smlarl be gven as follows: v v + v + = γ, γ = 1 εo, ε o << 1 (3) o v CCII +v + v o v + v + CCII (a) +v o Here γ s the voltage gan, v o s the output voltage of buffer, and ε o s the voltage trackng error of the voltage buffer. 3. SYNHESÝS POCEDUE Consder the current conveor crcuts shown n Fg. 2. he sgnal-flow graphs of these crcuts can be drawn b usng the defnng equatons of CCII [8,9]. Cevat EDAL, Al OE, Cevdet ACA

3 A New Proportonal-Integral-Dervatve (PID) Controller ealaton B Usng Current Conveors And Calculatng Optmum Parameter olerances 269 V V 2 V V O +V O V 1 V 2-2 / 1 (a) Amplfer crcut 3 V 1 - O / 1 V 2 - O / 2 V O V 1 + I C I +V 2 V 3 - O / 3 (d) Summng crcut 1-s Fgure 2. Basc buldng blocks usng CCIIs together wth correspondng sgnal-flow graphs V 1 + C D V 1-1/ I C I V 2 (b) Integrator crcut (c) Dervatve crcut +V 2 1-1/s V 1 - D C D V 2 D In Fg. 2(a), an amplfer crcut and ts sgnalflow graph are shown. he gan s - 2 / 1 as t can be easl seen from ts sgnal-flow graph. In Fg. 2(b), an ntegrator and ts sgnal-flow graph are shown. he ntegraton tme constant of ths crcut s -1/ I C I. In Fg. 1(c), a CCII based dervatve crcut and correspondng sgnal-flow graph are llustrated. he dervaton tme constant of ths crcut s - D C D. o obtan PID controller the three basc operatons shown n Fgs. 2 (a), (b), and (c) are transmtted to the output b CCII based summng crcut shown n Fg. 2(d). Note that, all the sub-crcuts llustrated n Fg.2 are nvertng crcuts. If a gven transfer functon s represented b a sgnal flow-graph, t can be easl observed from Fg. 2 that the correspondng crcut to the gven transfer functon can be realed b nterconnectng these buldng blocks [10,11]. Note also that, n nondeal case all the transfer functons shown n Fg. 2 should be multpled b the factor of α. If voltage buffers are used ths factor modfes to αγ. he transfer functon of a general analog, proportonal-ntegral-dervatve controller can be wrtten as follows: Cevat EDAL, Al OE, Cevdet ACA

4 270 A New Proportonal-Integral-Dervatve (PID) Controller ealaton B Usng Current Conveors And Calculatng Optmum Parameter olerances = sd (4) s I (s) P + + A sgnal-flow graph of the transfer functon of an analog PID controller can be drawn such as n Fg. 3 [9]. he realaton of the analog CCII based PID controller crcut correspondng to the sgnalflow graph n Fg. 3, whch s realed b the sub-crcuts n Fg. 2, and unt gan voltage buffers s shown n Fg.4. V I 1-s 1- Fgure 3. Sgnal flow graph correspondng to the transfer functon of the general proportonalntegral-dervatve (PID) controller I 1 1 I III V V I 2 C I o + V o 3 C D D Fgure 4. CCII based PID controller realaton correspondng to the sgnal-flow graph n Fg. 3 In Fg. 4, a CCII based PI controller crcut and a CCII based PD controller crcut can also be obtaned b removng path III and path II, respectvel. If the crcut n Fg. 4 s analed the controller coeffcents P, I, and D wll be obtaned as follows: 2 O P = α1γ1α 4 γ 4, (5a) 11 O I = α 2γ 2α 4γ 4, (5b) ICI2 C D D O D = α 3γ 3α4 γ 4. (5c) 3 4. HE ESULS OF SIMULAION In order to confrm the theoretcal results, the PID crcut gven n Fg. 4 s smulated n the SPICE program b usng the macro-model of AD844/AD from Analog Devces. In ths crcut, suppl voltages of ± 12V and n order to llutrate the tme-doman response, the step nput voltage wth 1 Volt ampltude are used. In smulaton procedure, the values of the capactors C I and C D, the resstor values, and the nput voltages also are gven n the capton of Fg. 5. he capactance values are selected such as to have a better llustraton. he smulaton results of the output of the CCII based PID controller are gven n Fgure 5 (a) and (b) respectvel. In both stuatons, the proportonal gans are taken to be ero n order not to confuse the fgures. Cevat EDAL, Al OE, Cevdet ACA

5 A New Proportonal-Integral-Dervatve (PID) Controller ealaton B Usng Current Conveors And Calculatng Optmum Parameter olerances 271 general, t s not known n advance how much each parameter contrbutes to the output error. hat s wh ths defnton s qute reasonable, snce the desgner epects the contrbuton of each parameter varaton on output devaton to be equal to each other. he formulaton of these tolerances was gven b Erdal et. al. [9].As a result, we can defne the optmum parameter tolerances as (a) t = t 0 / n S ( ) ma ω, = 1,..., 18 (6) where t s the th parameter tolerance, t o s the output tolerance of the controller, n s the parameter number,.e. n=18 for the gven confguraton, and ω s the angular frequenc at whch ( ω ) takes ts mamum value,.e. S S { S ( ω) }, ω [ ω ω ] ( ) = ma ma 1, ω (7) 2 (b) Fgure 5. esults of the smulaton of the PID controller n Fg. 4 (a) Output waveform of the PID controller for step response of the nput wth 1V ampltude and I =50kΩ, D =50kΩ, C D =0.1µF and C I as parameter from 0.5µF to 2µF ( O =10kΩ, 1 =2kΩ, 2 =10kΩ, 1 =5MΩ, 2 =40kΩ, 3 =1kΩ) (b) Output waveform of the PID controller for step response of the nput wth 5mV ampltude and I =50kΩ, D =50kΩ, C I =1µF and C D as parameter from 5µF to 20µF ( O =10kΩ, 1 =2kΩ, 2 =10kΩ, 1 =5MΩ, 2 =40kΩ, 3 =1kΩ) From Fgs. 5 (a) and (b), t s clear that the results are n good agreement wth the theoretcal epectatons. 5. CALCULAING OPIMUM PAAMEE OLEANCES he optmum parameter tolerances are defned as the tolerances contrbute equall to the upper bound of the relatve error of the output voltage of the controller ( V o /V o ) gven n Fg. 4. In where ω [ω 1,ω 2 ] descrbes desgner s specfed frequenc band. Hence S ( ω ) S ( ω, ) ma ω [ω 1,ω 2 ]. It should be noted that ω belong to the nterval ω [ω 1,ω 2 ], and ( ω ) has ts S mamum value at ths frequenc. he desgner can easl determne ω b plottng ( ω ) at ths nterval or b usng alread estng mathematcal programs lke Matlab. For eample, assumng that the proportonal gan, P =10, the ntegral gan, I =2 s -1, and the dervatve gan, D =5 s, are gven. hen the parameter values can be selected n Fg. 4 as follows: S o = 10Ω, 1 = 2Ω, 2 = 20Ω, (8a) I = 50 Ω, D = 50 Ω, (8b) 1 = 5Ω, 2 = 40Ω, 3 = 1Ω, (8c) C I = 1 µ F, C D = 10 µ F, (8d) α = 1, = , k 1 (8e) For ths eample, the mamum values of the parameter senstvtes are calculated as follows: S ( ω ) 1, = 1,..., 18 (9) = ma If t s requred that V o /V o 0.1 the parameter tolerances are to be chosen as follows: Cevat EDAL, Al OE, Cevdet ACA

6 272 A New Proportonal-Integral-Dervatve (PID) Controller ealaton B Usng Current Conveors And Calculatng Optmum Parameter olerances t t = t = t = t = t 0.55%, (10a) = 1 2 o CI C D = t = t 0.55 %, (10b) = = t = γ t α 0.55 %, = 1,..., 4. (10c) For ths partcular eample, the optmum tolerances are found to be equal to each other, however the are usuall dfferent n general case. Choosng the parameter tolerances such as above, the desgner can guarantee that the mamum devaton of the output voltage of the controller caused b the parameter varatons due to the envronmental effects wll be less than or equal to 0.1. hs result can also be verfed n SPICE smulaton. If V o /V o 0.01 s requred the parameter tolerances must be chosen ten tmes smaller than the ones n Eq. (10) and so forth. 6. CONCLUSIONS In ths stud, a CCII based PID desgn procedure s gven and a new PID crcut s proposed. hs crcut conssts of onl four CCIIs, two grounded capactors, eght resstors, and four voltage buffers and t s ver convenent for ntegrated crcut mplementaton. It s also ver sutable stable and fast control due to the enhanced propertes of the current-mode crcuts used n the desgn. Consderng that, the controller coeffcents P, I, and D are depended on the tme constants and resstor ratos the can easl be adjusted to an desred value ndependentl from each other. Furthermore, the optmum parameter tolerances are determned to keep the relatve error of the output of the CCIIs based PID controller due to parameter varatons n ts tolerance regon. EFEENCES [1] uo, B.C., Automatc Control Sstems, Prentce-Hall, Upper Saddle ver, N.J., [2] Svoboda, J. A., Current conveors, operatonal amplfers and nullors, IEE Proceedngs, Vol. 136, No. 6, pp , December [3] Wlson, B., ecent developments n current conveor and current-mode crcuts, Proceedng of. IEE pt. G, Vol. 137, (2), 1990, pp [4] oberts, G. W., Sedra, A. S., All current-mode frequenc selectve crcuts, Electroncs Letters, Vol. 25, 1989, pp [5] oumaou, C., Ldge F. J., and Hagh D. G., Analog IC Desgn: he Current-Mode Approach, Peter Peregrnus, [6] Erdal, C., oker, A.,and Acar, C., Current conveor based proportonal-ntegraldervatve (PID) controller and calculatng optmum parameter tolerances, Proceedngs of OPIM 2000, Internatonal Conference on Optmaton of Electrcal and Electronc Equpments, Brasov, Vol. III, pp [7] Smth,. C., Sedra, A., A secondgeneraton current conveor and ts applcatons, IEEE rans. on Crcut heor, Vol. 17, 1970, pp [8] Patranabs D., Ghosh D.., Integrator and dfferentators wth current conveors, IEEE rans. On Crcuts and Sstems, Vol. CAS-31, No. 6, June 1984 [9] Erdal, C., oker, A., and Acar, C., OA-C based proportonal-ntegral-dervatve (PID) controller and calculatng optmum parameter tolerances, urksh Journal Of Electrcal Engneerng & Computer Scences, Elektrk, Vol. 9, No. 1, 2001.(It s accepted and t wll be publshed) [10] Aronhem, P., ransfer functon snthess usng a current conveor, IEEE rans. On Crcuts and Sstems, Vol. CAS-21, pp , March [11] ek, H., Anda, F., Voltage transfer functon snthess usng current conveors, Electroncs Letters, Vol. 25, No 23, November Cevat EDAL, Al OE, Cevdet ACA

7 A New Proportonal-Integral-Dervatve (PID) Controller ealaton B Usng Current Conveors And Calculatng Optmum Parameter olerances 273 Bograph of Cevat Erdal Cevat Erdal was born n Bergama, urke, He receved both the B.Sc. and M.Sc. degrees n electrcal engneerng from the Facult of Electrcal and Electroncs Eng., Istanbul echncal Unverst, urke n He receved the Ph. D. degree n 1987 from the Insttute of Scence and echnolog of the same unverst. He s currentl an assstant professor n Control Engneerng Dept. He s also the author or co-author of about 28 papers publshed n scentfc revews or conference proceedngs. Hs man research nterests are transducers, process control and desgn of new controllers. Bograph of Al oker Al oker was born n Istanbul, urke, He receved the B.Sc. and M.Sc. degrees n electrcal engneerng from the Facult of Electrcal and Electroncs Eng., Istanbul echncal Unverst, urke n 1973 and 1975, respectvel. He receved the Ph. D. degree n 1986 from the Insttute of Scence and echnolog of the same unverst. He s currentl an assocate professor n electroncs, teachng graduate and undergraduate courses. He s also the author or co-author of about 50 papers publshed n scentfc revews or conference proceedngs. Hs man research nterests are desgn of current-mode crcuts and analog sgnal processng applcatons. Bograph of Cevdet Acar Cevdet Acar was born n Hopa, urke, He receved hs B.Sc., M.Sc. and Ph.D. degrees from Istanbul echncal Unverst n 1965, 1967 and 1972, respectvel. In 1970 he joned the Electroncs and Communcaton Engneerng Department of Istanbul echncal Unverst. Snce 1988 he s a professor of crcut and sstems n the same department. Hs research nterests nclude desgn of actve network snthess, analog sgnal processng, and senstvt and tolerance analss. He s the author or the co-author of more than 60 journal papers publshed n nternatonal journals, more than 20 conference papers presented n nternatonal conferences and 2 books related to the above mentoned areas. Cevat EDAL, Al OE, Cevdet ACA