A Developed Method of Tuning PID Controllers with Fuzzy Rules for Integrating Processes

Transcription

1 A Develope Metho of Tunng PID Controllers wth Fuzzy Rules for Integratng Proesses Janmng Zhang, Nng Wang an Shuqng Wang Abstrat The proportonal ntegral ervatve (PID) ontrollers are wely apple n nustral proess, an the tunng metho of PID ontroller parameters s stll a hot researh area. A fuzzy tunng sheme for PID ontroller settngs s evelope for ntegrator plus tme elay proesses n ths paper, n whh a fuzzy rule base reasonng metho are utlze on-lne to etermne a tunng parameter α base on the error an the frst hange of the error of the proess. Then ths tunng parameter α s use to alulate the PID ontroller parameters. Computer smulatons are performe for an example of ntegratng plants. Comparng to some reporte methos, the performane of the presente approah s shown to be satsfyng. I I. INTRODUCTION N nustral proess ontrol area,the proportonal-ntegralervatve (PID) ontrollers are stll wely use beause of ther smple struture, easy unerstanng to operators, robust performane n a we range of operatng ontons, an ther easy mplementaton usng analogue or gtal harware. It has been reporte that more than 95% of the ontrollers n nustral proess ontrol applatons are of the PID type ontroller []. Integratng proesses are frequently enountere n the proess nustres. Many hemal proesses an be moele by a pure ntegrator plus tme elay moel suh as k exp( τs) s. Sne the moel ontans only two parameters (the proportonal oeffent k an the tme elay onstantτ ), t s very onvenent for estmatng the moel parameters by relay feebak metho or lose-loop entfaton [3],[4]. Reent years, more an more PID tunng methos are propose to eal wth varous ntegratng proesses. Chen Manusrpt reeve September 0, 003. Janmng Zhang s wth the Natonal Key Laboratory of Inustral Control Tehnology, Insttute of Avane Proess Control, Zhejang Unversty, Hangzhou, 3007, P. R. of Chna. (Phone: ; Fax: ; e-mal: jmzhang@ p.zju.eu.n). Nng Wang s wth the Natonal Key Laboratory of Inustral Control Tehnology, Insttute of Avane Proess Control, Zhejang Unversty, Hangzhou, (e-mal: nwang@ p.zju.eu.n). Shuqng Wang s wth the Natonal Key Laboratory of Inustral Control Tehnology, Insttute of Avane Proess Control, Zhejang Unversty, Hangzhou, (e-mal: sqwang@ p.zju.eu.n). an Fruehauf [5] propose an nternal moel ontrol (IMC) metho to fn the settngs for a PI ontroller, n a proess onsstng of an ntegrator an a tme elay. Tyreus an Luyben [6] ponte out that the IMC-base PI ontroller oul lea to a poor ontrol performane unless are s taken n seletng the lose-loop tme onstant, an propose an alternatve approah base on lassal frequeny response methos for PI settngs. Ths tunng rule has been extene to esgn PID ontrollers by Luyben n [7]. Wang an Cluett [8] also susse ths ontrol problem an propose a PID ontroller esgnng metho base on spefaton n terms of esre ontrol on sgnal trajetory whh s sale wth respet to the magntue of the oeffent n the seon term of the Taylor s seres expanson. More reently, Vsol [9] propose a tunng metho for ntegratng systems base on mnmzng ISE, ISTE an ITSE wth a genet algorthm. The results are ftte by smple equatons relate to the proportonal oeffent k an the tme elay onstant τ of the proess. However, the optmzaton for servo response results n a PD ontroller an only the results for regulatory response an gve a PID ontroller. Chambaram an Sree [] propose a smple metho for the PI, PD an PID ontroller settngs for suh proess base on mathng the oeffents of orresponng powers of s n the numerator an that n the enomnator of the lose-loop transfer funton. Sne there s only one ajustable parameter α for the PID tuner, are must be taken n seletng the tunng parameter α n orer to obtanng a goo performane. In ths paper, a fuzzy nferene metho s ntroue to fn a proper value α aorng to the response of the lose loop system. The performane of the mofe metho s shown wth an example. The paper s organze as followng. The next seton gves a bref ntrouton of the metho propose n []. Seton III presents the tunng rules wth fuzzy nferene for the tunng parameter α. Computer smulatons an results are gven n seton IV, an the performane of the propose metho s ompare wth that of [], [7]-[9]. Conlusons are summarze n seton V. II. PREVIOUS TUNING METHOD FOR PID CONTROLLERS Chambaram an Sree [] propose a smple metho for

2 the PI, PD an PID ontroller settngs base on mathng the oeffents of orresponng powers of s n the numerator an that n the enomnator of the lose-loop transfer funton for a servo problem. Ther metho s esrbe brefly as follows []. For an ntegrator plus tme elay proess represente by transfer funton k exp( τs) s, onserng the followng PID type ontroller, G = k + + T s. Where, k s T s the proportonal gan of PID ontroller, T an T are known as the ntegral an ervatve tme onstants, respetvely. The lose-loop transfer funton an be gven by y( q) ( Kq + K + K 3q ) exp( q) =. () y ( q) q + ( K q + K + K q ) exp( q) r where, s s the Laplae operator, K T K = kkτ, K =, K 3 = K, q = sτ. () τ τ T 3 Approxmatng the tme elay term n the enomnator by a frst-orer pae approxmaton, 0.5q exp( q). (3) + 0.5q The resultng equaton an be wrtten as y( q) ( Kq + K + K3q )( + 0.5q) exp( q) =. (4) y ( q) q ( + 0.5q) + ( K q + K + K q )( 0.5q) r Removng exp( q) n the numerator an ntroung a parameter α as the oeffents of orresponng powers of q of the numerator to that of the enomnator, one an get the followng set of equatons: α ) K + 0.5( + α) K 0, (5a) ( = 0.5( + α ) + ( α ) K3 = + α ) 3 = α K α, (5b) ( K. (5) By solvng the above equatons, the followng results an be obtane for the PID ontroller parameters, 4α =, (6a) ( + α ) kτ k 0.5τ ( + α) T =, (6b) ( α ) + α T = 0.5τ. (6) α where, α s the tunng parameter an shoul be greater than. Smlar to IMC methos, are shoul be taken n seletng ths tunng parameter. That gves ffult n nustral proess applatons. In orer to avo ths problem, a mofe metho s propose here by a fuzzy tunng metho for fnng the approprate parameter α on-lne aorng to the step response of the proess. 3 a b a b t/se Fg. Proess step response From the equaton (6), one an see that one the parameter α nreases, the proportonal gan k arses whh results n a bg PID ontrol aton; On the ontrary, the smlar results an be mae too. Therefore, aorng to the step response (shown as n Fg.), one an aress the followng onlusons for tunng the parameter α. At the begnnng (pont a ), there s a bg system error, therefore, a bg ontrol aton s esre n orer to aheve a fast rse tme. To proue a bg ontrol sgnal, the PID ontroller shoul have a large proportonal gan, a large ntegral gan an a small ervatve gan. Thus, aorng to the relatons between the tunng the parameter α an the proportonal gan k, ntegral gan T an ervatve gan T, one shoul gve a bg value α ; Aroun the pont b n Fg., a small ontrol aton s expete to avo a large overshoot, an a small value α s requeste; Along wth the ereasng of system error graually, the ontrol aton shoul go to a stea state, therefore, the tunng parameter α shoul not be regulate agan. Form the above analyss, the tunng parameter α shoul be reue from a bg value to a small value graually urng the step response. Sne fuzzy metho has the ablty of reasonng from the system error an the hange of error, t an be utlze on-lne to regulate the parameter α properly. The etal of the approah s to be presente n the next seton. III. THE PROPOSED FUZZY INFERENCE METHOD FOR TUNING THE PARAMETER α Fuzzy sets theory s frst ntroue by Zaeh n 965 [0], an has been apple n the nustral proess ontrol suessfully. Reent years, fuzzy log has been use to tune the parameters of a PID ontroller [], an goo mprovement n the proess response s aheve. Durng the esgn proeure of fuzzy PID ontroller, the human expertse n ontrollng a proess s represente as fuzzy rules or relatons. Ths knowlege base s use by an fuzzy nferene mehansm, n onjunton wth some knowlege

3 Setpont + - Fuzzy Inferene PID Controller Proess Fg.. Fuzzy self-tunng PID ontroller Output of the state of the proess n orer to etermne fuzzy ontrol aton on-lne. The shemat agram of a fuzzy self-tunng PID ontroller use n ths paper s shown n Fg.. Where, the nputs of the fuzzy nferene mehansm are the error e an the hange n error e, an the sutable parameter α for tunng the PID settngs s gven by the output of the fuzzy nferene system. In ths paper, four fuzzy sets (Z, S, M, an B) are onsere for both error an ts hange. Where, Z represents approxmately zero, S, M an B enote small, meum an bg respetvely. Two fuzzy sets (S, B) are use for the output varable α. The error (e) an the hange n error ( e) are normalze wth respet to ther maxmum values. Therefore, ther absolute values an be taken urng the fuzzy nferene. The trangular-type membershp funton s aopte n ths paper for e, e an spefe as n Fg. 3. The sgmo-type membershp funton s use for α an spefe as n Fg. 4. u u SMALL Z S B M e, e Fg. 3. Membershp funton of e an e BIG a Fg. 4. Membershp funton of the output α The grae of the membershp funtons µ an the output varable α has the followng relatonshp: µ B ( α) = ( + exp( 0( α 0.5))) for Bg (7a) or µ S ( α ) = ( + exp( 0( α 0.5))) for Small, (7b) The fuzzy rule base an be extrate from operator s expertse or the step response of the proess. In ths paper, the step response of the proess s use. Aorng to the relatonshp between tunng parameter α an the step response, the followng fuzzy rules an be aresse. At the begnnng (pont a n Fg. ), to proue a bg ontrol aton, the tunng parameter α an be represente by a fuzzy set Bg. An the orresponng rule an be esrbe as IF e s B an e s Z THEN α s BIG. Aroun the pont b n Fg., a small α s requeste for avong a large overshoot an the orresponng fuzzy rule s : IF e s Z an e s B THEN α s SMALL. By the analyss, the whole fuzzy rule base for the tunng parameter α an be summarze n Table. α e Table Fuzzy tunng rules for α e Z S M B Z B S S S S B B S S M B B B S B B B B B In Table, all of the tunng fuzzy rules have the followng esrpton: R : IF e s A an e s B THEN α s C Where, A, B an C are the fuzzy sets. Usng the smplfe fuzzy reasonng metho [], the agreement of the th rule an be alulate by the prout of the member funton values n the anteeent part of the rule: h = µ A ( e) µ B ( e) (8) where µ A s the membershp funton value of the fuzzy set A etermne by the urrent nput value e ; an µ B s the membershp funton value of the fuzzy set B etermne by the urrent nput value e. For eah h, the orresponng output of the fuzzy rule an be alulate from Eq. (7a) or Eq. (7b) as α = 0.05 ln( h ) for α s Bg (8a) or α = 0.05 ln( ( h ) ) for α s Small (8b) Then the entro metho of efuzzfaton s use to get the rsp value of α from the fuzzy varables. the output value of fuzzy nferene system s alulate by

4 α = hα h (9) Here, α s the value of α orresponng to the egree h for the th rule. Aorng to the analyss results n prevous seton, the tunng parameter α shoul be greater than, an shoul be reue from a bg value to a small value slowly urng the transent response proess. Therefore, n ths paper, the true value of α s etermne as followng α ( k) =.+ λ( k) (0) λ( k) = λ( k ) α where λ (0) s a parameter selete by the user, t gves the maxmal value of α. After etermnng the parameter α, the PID ontroller parameters an be alulate by usng the equaton (6). IV. SIMULATION RESULTS Conser the ntegratng plus tme elay proess use n [], whh s esrbe by the followng transfer funton τs e G( s) = k s where, n nomnal ase, k = , an τ = 6. The propose metho wll be evaluate usng ths proess. To ompare the performane to the other prevous work, the orresponng PID ontroller settngs are gven as followngs, respetvely, Vsol [9]: k = 4. 5, T = an T = ; Luyben metho [7]: k =. 5639, T = an T = 3.56 an wth a frst-orer flter tme onstant as 0.38; Chambaram an Sree[]: k = , T = 7 an T =.7 wth α =. 5 ; Wang an Cluett [8]: k =. 03, T = an T =.5674 ; The performane of above PID ontroller an the presente metho for the nomnal proess moel are shown n Fg. 5 an Fg. 8. The servo responses are shown n Fg. 5. The servo response of the propose ontroller s slghtly better than that of Chambaram an Sree [], an s better than that of Vsol [9]. The regulatory responses are shown n Fg. 8. The regulatory response s the best for the metho of Vsol [9], sne the settng was obtane by mnmzng ISE for regulatory problem. In Table, the omparson of ISE values s gven for the above metho an the mofe metho. Obvously, for the regulatory problem, the presente metho s better than that of [7] an [8]. The robustness of the propose ontroller s stue by usng perturbaton n tme elay τ. The ontroller settngs are those alulate for the proess wth nomnal tme elay ( τ = 6 ), whereas the atual values n the smulatons are τ = 7 an τ = 8, respetvely. Fg. 6 an Fg. 7 show the results of servo responses. The responses are obtane for the regulatory problem as shown n Fg. 9 an Fg. 0. Vsol metho gves an osllatory unstable response when τ = 8 both n servo response of regulatory response. The omparson results of ISE values are also gven n Table. For the servo problem, the presente metho gves robust responses an s superor to the metho of [], an the Luyben s metho gves the best robust performane. However, for regulatory problems, the performane of the presente metho s better than that of [7], [7], an s equvalent to that of []. From the vews of ontrol theory, the servo problem s ontratory to the regulatory problem, an ther performane nee to be trae-off at some egrees urng the esgn of a ontroller for the gven proess. From ths pont of vew, the performane of the presente metho n ths paper gves a more effetve ompromse than that of others mentone n ths paper. Table Comparson of ntegral square error for the fve methos No. Methos ISE for servo problem ISE for regulatory problem τ = 6 τ = 7 τ = 8 τ = 6 τ = 7 τ = 8 Present vsol [9] * * 3 Luyben [7] Chambaram an Sree[] Wang an Cluett [8] Remark: PID ontrollers are esgne for τ = 6 ; * means unstable response. Fg. 5. Servo response of the proess for fferent methos ( τ = 6 )

5 Fg. 6. Servo response of the proess for fferent methos ( τ = 7 ) Fg. 9. Regulatory response of the proess for fferent methos ( τ = 7 ) Fg. 7. Servo response of the proess for fferent methos ( τ = 8 ) Fg. 0. Regulatory response of the proess for fferent methos ( τ = 8 ) V. CONCLUSIONS Base on the smple metho propose by Chambaram an Sree n [], a mofe approah s presente n ths paper. The metho apples fuzzy nferene wth smple rules to ompute the tunng parameter α on-lne. That avos the ffulty n seletng a sutable parameter α n Chambaram an Sree s metho []. The omparson of performane for an ntegratng proess s performe. The results valate that the evelopment performane s aheve. REFERENCES Fg. 8. Regulatory response of the proess for fferent methos ( τ = 6 ) [] M. Chambaram an R. P.Sree, A smple metho of tunng PID ontrollers for ntegrator/ea-tme proesses, Computers an Chemal Engneerng, vol. 7, -5, 003. [] C. C. Yu, Auto-tunng of PID Controller. Berln: Sprnger, 990.

6 [3] J. H. Park, S. W. Sung, an I. Lee, Improve relay auto-tunng wth stat loa sturbane, Automata, vol. 33, 7-75, 997. [4] R. Srvya an M. Chambaram, On-lne ontroller tunng for ntegrator plus elay proesses, Proess. Control Qual., vol. 9, [5] I. L. Chen an P. S. Fruehauf, Conser IMC tunng to mprove performane, Chem. Eng. Prog., vol. 0, 33-4, 990. [6] B. D. Tyreus an W. L. Luyben, Tunng PI ontrollers for ntegrator/ea-tme proesses, In. Eng. Chem. Res., vol. 3, 65-68, 99. [7] Luyben, W. L. Tunng Proportonal-Integral-Dervatve Controllers for Integrator/Deatme Proesses. In. Eng. Chem. Res. 996, 35, [8] L. Wang an W. R. Cluett, Tunng PID ontrollers for ntegratng proesses, IEE Pro. CTA, vol. 44, , 997. [9] A. Vsol, Optmal tunng of PID ontrollers for ntegral an unstable proesses, IEE Pro. Control Theory Appl., vol. 48, 80-84, 00. [0] L. A. Zaeh, Fuzzy Sets, Informaton an Control, vol. 8, , 965. [] Z. Y. Zhao, M. Tomzuka an S. Isaka, Fuzzy gan sheulng of PID ontrollers, IEEE Trans. on Systems, Man an Cybernets, vol. 3, , 993.