EXPLICIT DAMPING FACTOR SPECIFICATION IN SYMMETRICAL OPTIMUM TUNING OF PI CONTROLLERS

Transcription

1 EXPLICI DAMPING FACOR SPECIFICAION IN SYMMERICAL OPIMUM UNING OF PI CONROLLERS Martin Mahaba and Martin Braae Deartment of Eletrial Engineering, Univerity of Cae own, Rondeboh, Cae own, South Afria Cell: Offie no: Abtrat: he Symmetrial Otimum tuning rooed by Keler (958) and further modified by Voda and Landau (995) enure that maximum hae margin i ahieved for the reulting loed loo ytem. he equation for Symmetrial Otimum tuning a defined by Atrom and agglund (995) have reently been imroved by Preitl and Preu (999). In thi aer the Preitl and Preu equation for Symmetrial Otimum tuning are further refined to allow exliit eifiation of the loed loo daming fator. he reulting tuned ontroller value are alied to oition ontrol of a d ervomotor. Keyword: Daming fator; PI ontroller; Symmetrial Otimum; Phae margin. INRODUCION he adjutable ontrol arameter of PI and PID ontroller for a given roe an be tuned uing a variety of method uh a Ziegler - Nihol, Cohen and Coon, C and Symmetrial Otimum. (Pollard, 97) he Ziegler Nihol method et the ontroller arameter required for reaonably good erformane baed on the te reone of the oen loo ytem. he reone i an exonential urve of a multi-aaitane roe and an be haraterized by two arameter meaured from the reone urve. hee are the delay time, L and the maximum loe, N, a a funtion of the total hange in the variable er unit time. he total hange in the meaured variable K and the maximum loe N are both roortional to the magnitude of the hange in the inut variable M. ( Pollard, 97) Cohen and Coon further extended the Ziegler Nihol method. hey ued the following L K e tranfer funtion to determine the + theoretial value of the ontroller arameter to give reaonable and aetable reone. (Pollard, 97) he Symmetrial Otimum (S.O) tuning method rooed by Keler (958) and further modified by Voda and Landau (995) enure that the tuned ontroller rodue maximum hae margin for the reulting loed loo ytem. he equation for Symmetrial Otimum tuning a defined by Atrom and agglund (995) have reently been imroved by Preitl and Preu (999) to inluded variable daming. In thi aer the Preitl and Preu equation for Symmetrial Otimum tuning are further refined to allow exliit eifiation of the loed loo daming fator, or in other word thi aer rooe a ole laement interretation of the Symmetrial Otimum method in whih the daming fator i exliitly defined a art of the tuning roedure. he aer define Symmetrial Otimum tuning in etion, followed by exliit daming fator eifiation in etion. Setion 4 deal with the aliation of the daming fator aroah to the d ervomotor and reult are given and diued in etion 5. Setion 6 i the onluion. he derivation of equation and table are given in the aendix.

2 . SYMMERICAL OPIMUM UNING he Symmetrial Otimum ontroller tuning method i deigned to enure maximum hae margin. A exreed by Atrom and agglund (995) the otimization ondition are a follow a a a and a a a () Preitl and Preu (999) generalized the equation above by uing the arameter β hene β a a a and β a a a () he additional tuning arameter β that i introdued into the bai Symmetrial Otimum equation effetively et the daming fator of the loed loo ytem, a hown in Fig.. heir reearh indiated that the arameter β hould be hoen to fall in the range 4 to 6, and that three different ituation our: () If β < 9 two of the three ole rodued by the harateriti equation are omlex onjugated. () If β 9 then all ole are real and equal. () If β > 9 all ole are real and ditint. Preitl and Preu tate that if β < 4 the hae margin i very mall, being le than 6, while if β > 6 the hae margin i greater than 6. herefore the domain for β i hoen o a to find the bet trade off between erformane and the minimum value of the deired hae margin hene the domain [4,6]. hu the deign engineer an hange the daming fator by varing the β value.. EXPLICI DAMPING FACOR SPECIFICAION Fig.: he d ervomotor iture he d ervomotor ued in thi wor i hown in Fig., and it dynami an be modeled by the following tranfer funtion ( ) () ( + ) where i the gain and i the time ontant. he PI ontroller i hoen for thi tuning deign and it tranfer funtion i ( + ) ( ) (4) where i the ontroller gain and i the ontroller time ontant. Conider the unity feedba ontrol loo hown in Fig.. It oen loo ytem i then defined by the tranfer funtion ( ) ( ) ( ) or ( + ) ( ) (5) ( + ) he loed loo tranfer funtion i given by ( ) w ( ) + ( ) ( ) Fig. Effet of varying beta β on ole oition. or w ( ) ( + ) ( + ) + ( + ) (6)

3 By omaring the olynomial reulting from the ole oition hown in Fig. and the loed loo harateriti funtion in equation (6) the tuning formulae an be exreed a ( α + ) () Fig.: Feedba oition ontrol By hooing a β < 9 ondition, two of the three ole rodued by the harateriti equation of the loed loo ytem are omlex onjugate and the third i real a how on Fig. Let the daming fator of the omlex mode be defined by oθ (7) + ω Rearranging the equation in term of ω yield ω (8) he real ole from Fig. i defined a α (9) where α > meaning that the real ole i alway fater than the onjugate air. () ( α + ) α () he derivation of thee equation i given in the aendix. 4. APPLICAION O DC SERVOMOOR he tranfer funtion in equation () rereent the d ervomotor in oition ontrol, while equation (4) i the PI ontroller that i alied to it. Exeriment were done on the d ervomotor to find the value of the arameter and. he following are the value found from thee tet. 8.87[ v / v] ().55 e When ued in the tuning equation, and the following ontroller ontant were rodued:. e.55[ v / v] (4) In thi exeriment the daming fator and alha α are eified, to be.77 and reetively. Fig.: Pole oition for β le than 9

4 4. he redited reone 4. he exerimental reult Fig 4.: he redited reone Figure 4. how the reone redited from the loed loo tranfer funtion uing the value for the motor model and the tuned ontant in equation () and (4). It how the following: he reone ha aroximately 5% overhoot and it tae aroximately 8 eond to ettle. Figure 4. how the redited inut ut, alo nown a the ontrol outut, from the loed loo tranfer funtion uing the model value and the tuned ontant in equation () and (4). he inut value are very mall (aroximately.5), and it tae jut under eond to ettle. Fig 4.: he exerimental reone reult Figure 4. how the atual d ervomotor reone to a te inut, and it i a redited. he over hoot i lightly above the redited value of 5%, while the ettling time i almot the ame 8 eond. he direany between the exerimental and redited reone i attributed to the trition effet of the ervomotor ytem. Figure 4.4 how the exerimental inut, the ettling time i fater than exeted beaue of the trition effet of the ervomotor. he value for the inut i aroximately.5 a exeted. Fig 4.4: he exerimental ontroller outut u(t) Fig 4.: he redited ontroller outut u(t)

5 5. CONCLUSION β a a a and β a a a (7) Pole laement interretation of the Symmetrial Otimum method in whih the daming fator i exliitly defined a art of the tuning roedure ha been reented. he main advantage of uing thi method are that he PI ontroller arameter an be tuned by eifying the daming fator, that ha more exliit hyial meaning than the variable β in equation (). he rooed method wa alied to a d ervomotor and the reult indiate that the eifiation imoed on the tuning equation were oberved in ratie when alied to the motor. 6. REFERENCES Pollard, A (97). Proe Control for the hemial and allied fluid-roeing indutrie Keler, C. (958).Da ymmetrihe Otimum. Regelungtehni, 6, 95-4 and 4-46 Preitl, S and R-E Preu (999). An extenion of tuning relation after Symmetrial Otimum method for PI and PID ontroller. Automatia, 5,7-76. Voda, A.A and I.D Landau (995). A method for auto-alibration of PID ontroller, Automatia,,4-5 APPENDIX Derivation of the tuning equation Preitl and Preu (999) invetigated of the loed loo harateriti funtion of the third degree a + a a a + + (4) he otimization ondition aording to the SO method are exreed a: a a a and a a a (6) Subtituting equation 7 into 4 and dividing by the o-effiient of the following moni olynomial funtion reult + β A + β A + A (8) a where A a Chooing a value of β <9 reult in the following ole oition, one real ole and two onjugate air. Let the ole oition be rereented by the following variable ( )( + + jω )( + jω) + (9) ( + ) + ( + + ω ) + ( + ω ) + he real ole an be further defined a α and α > () If the daming fator i defined a () + ω the equation an be rewritten in term of ω a hown below ω () Subtituting equation and into 9 the moni harateriti equation beome + ( α + ) + α () he harateriti equation of the loed loo given in equation 6 an be written a φ ( (4) ) or

6 Comaring the above harateriti equation () and (4) yield the tuning formulae ( α + ) ( α + ) α able 7. how all the arameter in the tuning of the PI ontroller uing daming fator eifiation in Symmetrial Otimum. he arameter in row (4) are the one ued in thi aer, by varying the daming fator only two arameter are affeted namely the ontroller time ontant, and. able 7.: Simulation reult for exeriment : hi table how the tuning of arameter when the daming fator i varied α () () () (4) (5) (6) (7)