A Substractive Clustering Based Fuzzy Hybrid Reference Control Design for Transient Response Improvement of PID Controller

Transcription

1 IB J. Eng. Sc. Vol. 4, No., 009, A Substrctve lusterng Bsed Fuzzy Hybrd Reference ontrol Desgn for rnsent Response Improvement of PID ontroller Endr Joelnto & Prlndungn H. Stnggng Instrumentton nd ontrol Reserch Group Deprtment of Engneerng Physcs Bndung Insttute of echnology, Bndung 403, Indones E-ml: ejoel@tf.tb.c.d Abstrct. he well nown PID controller hs nherent lmttons n fulfllng smultneously the conflctng control desgn objectves. Prmeters of the tuned PID controller should trde off the requrement of trcng set-pont performnces, dsturbnce rejecton nd stblty robustness. ombnton of hybrd reference control (HR) wth PID controller results n the trnsent response performnces cn be ndependently cheved wthout deterortng the dsturbnce rejecton propertes nd the stblty robustness requrement. hs pper proposes fuzzy bsed HR where the membershp functons of the fuzzy logc system re obtned by usng substrctve clusterng technque. he proposed method gurntees the trnsent response performnces stsfcton whle preservng the stblty robustness of the closed loop system controlled by the PID controller wth effectve nd systemtc procedures n desgnng the fuzzy hybrd reference control system. Keywords: fuzzy systems; hybrd reference control; PID controller; substrctve clusterng; trnsent response. Introducton Hybrd Reference ontrol (HR) hs been developed by Joelnto nd Wllmson [][], nd Joelnto [3] to mprove trnsent response of stblzed closed loop control systems nd to reduce lmttons ntrnsc n one- degreeof-freedom controller by mens of reference sgnl (set-pon mnpulton bsed on lner mpulsve dfferentl equtons. he HR hs been successfully mprovng the trnsent response performnces (rse tme, mmum overshoot, nd settlng tme) or dedbet response t specfed tme of the conventonl controlled systems. By usng HR, the trnsent response performnces cn be desgned ndependently wthout ffectng the stblty propertes of the closed loop systems. he HR system s decson system dded to the conventonl closed loop systems to mnpulte the reference sgnl (set-pon of feedbc control systems bsed on occurrences of events. he bloc dgrm of the HR system s shown n Fgure. In HR method, Receved Februry 4 th, 009, Revsed August 7 th, 009, Accepted for publcton October 0 th, 009.

2 68 Endr Joelnto & Prlndungn H. Stnggng the mn gol s the trnsent response performnce mprovement of the closed loop control systems for gven defult reference sgnl vlues. Durng the trnsent response to rech the defult reference sgnl, the control sgnl s mnpulted s result of pplyng temporry reference sgnls n order to speed up or to slow down the speed of the trnsent response performnces. hs cn be cheved by ncresng or decresng the control sgnl by mens of chngng the reference sgnl pproprtely nd then returnng the reference sgnl to the defult reference sgnl vlue when the output s suffcently close to the defult vlue. Fgure Dgrm bloc of hybrd reference control (HR). As n Joelnto nd Wllmson [], the events ( t,,,, m ) re generted by prttonng the error between the defult reference sgnl nd the output of the controlled system by performnce observer. he ntton to chnge the reference sgnl s ntted by n enbled event when the error s consdered to be suffcently lrge. he event-bsed decson system then sends sequence of temporry reference sgnls to the closed loop system t fed tme ntervl ( t t ). When the output of the controlled system enters the specfed tolernce of error bnd of the defult reference vlue, dsble event ( t m ) s generted tht returns the reference sgnl to ts defult reference vlue. he process s repeted on the occurrences of enble events. In conventonl feedbc desgns, the gol s to see n pproprte gn/control feedbc tht stblzes the closed loop system nd cheves some requred trnsent performnces. On the other hnd, the objectve of the HR desgn s to fnd the

3 rnsent Response Improvement of PID ontroller 69 pproprte temporry reference sgnl tht wll mprove the trnsent response of the stblzed closed loop control. As result, the combnton of HR nd conventonl feedbc systems leds to better control strtegy wth twodegree-of-freedom controller confgurton. Fuzzy logc controllers re nowledge bsed or rule bsed controllers whch re nferred from humn nowledge or eperence on systems. In fuzzy control systems Zdeh [4] nd Wng [5], the gol s to put humn nowledge nto controller n systemtc, effcent nd nlyzble order. In the cse of PID controllers, t s nown tht opertors ply mportnt role n order to cheve good PID controllers by mens of pplyng proper tunng to delt wth conflctng behvour requrements of the closed loop or by mnpultng the nput of the controlled process mnully. Mny publshed ppers mnly focussed on the selecton of the three prmeters of the PID controller s the pplcton of fuzzy system n mmcng the nowledge of the opertors, to llustrte Msr nd Ml [6], nd Mohn nd Snh [7]. However, there hs been lttle ttenton to mplement fuzzy logc to perform smlr wy to n epert opertor who suppresses overshoot by ether ncresng or decresng the nput of the controlled process. Yoogw [8][9] hs proposed fuzzy logc system bsed on the nformton of the process dynmcs to reduce overshoot. hs method hs been one successful emple of the pplcton of fuzzy logc n control system desgn s reported by hu [0]. Joelnto nd nsr [] hs proposed fuzzy logc bsed HR system tht mmcs the wy of opertor n decdng the mgntude of the nput to the controlled process n chevng the specfed trnsent response performnce of the nterest process vrbles. Specfclly, the djustment of the nput of the controlled process by the output of the PID controller cn be produced by the mnpulton of the reference sgnl by mens of fuzzy logc system desgned by usng the well nown fuzzy method g nd Sugeno []. he stblty of the closed control system s mntned by the PID controller wth the three PID prmeters re selected by usng the well nown tunng methods Zegler nd Nchols [3]. Other tunng methods dscussed n Smth nd orrpo [4], oughnowr [5] nd Åström nd Hägglund [6] cn lso be ppled to the PID controller. he dgrm bloc of the fuzzy HR system wth the PID controller developed n Joelnto nd nsr [] s shown n Fgure. he fuzzy logc system processes the error sgnl ( e ( ) whch s the dfference between the output of the controlled system nd the defult reference sgnl nd ts dervtve ( e ( ) n order to decde n pproprte sgnl tht chnges the defult reference sgnl durng the trnsent response.

4 70 Endr Joelnto & Prlndungn H. Stnggng Fuzzy System Defult Set-pont d( - r( e( u( + + y( PID Plnt ontroller - Fgure Bloc dgrm of substrctve clusterng bsed fuzzy HR nd PID controller. he fuzzy bsed HR strts sendng sgnls ( d ( ) tht mnpulte the defult reference sgnl ( r ( ) when n enble event t s detected by the performnce observer embedded n the fuzzy system. hs event nforms the performnce observer tht the devton of the closed loop system output ( y ( ) to the defult reference sgnl ( r ( ) s bgger thn the prescrbed tolernce such tht e ( y( r(. () he fuzzy system sends the reference sgnl ( d ( ) ether contnuously or n predefned tme ntervl ( ) untl the performnce observer detects nother dsble event where the error of the closed loop system s now enterng the llowble tolernce defned by e ( y( r( () When ths event s detected, the fuzzy system then stops sendng reference sgnls ( d ( ) nd the reference sgnl returns to the defult reference sgnl ( r ( ) by sendng the sgnl d ( 0. In developng the fuzzy HR system proposed n [], the mn problem s how to obtn the pproprte membershp functons of the fuzzy system effectvely n shorter tme process desgn s t ws tme consumng n crryng out the trl error method. hs technque s lso ndequte for the requrement of n mplementton usng dptve strteges to del wth plnt prmeters chnges. In ths pper, the substrctve clusterng technque hu [7] s ppled to fnd the membershp

5 rnsent Response Improvement of PID ontroller 7 functons of the fuzzy HR system wth less number of rules nd mnmum mount of computtonl tme. he substrctve clusterng method s ppled to dt consst of error sgnl ( e ( ), dervtve error sgnl ( e ( ) nd the reference sgnl ( d ( ). he proposed method s then referred to s substrctve clusterng bsed fuzzy HR system. Substrctve lusterng Estmton Subtrctve clusterng developed by hu [7] s nown s fst, one-pss lgorthm to estmte the number nd the centers of clusters n gven dt set. It s useful when there s no cler de how mny clusters needed for gven set of dt. he resulted cluster estmtes cn then be used to ntlze tertve optmzton-bsed clusterng methods nd model dentfcton methods. omputtonl problem n subtrctve clusterng s solved by usng dt ponts s the cnddtes for cluster centers. hs leds to the computton s proportonl to the problem sze rther thn the problem dmenson. Although, the ctul cluster centers re not necessrly locted t one of the dt ponts, n most cses, t gves good ppromton. Let us consder collecton of n dt ponts {,,, n} n M dmensonl spce. Wthout loss of generlty, t s lso ssumed tht the dt ponts hve been normlzed n ech dmenson so tht they re bounded by unt hypercube. In substrctve clusterng hu [7], ech dt pont my be consdered s possble cluster center. A mesure of the potentl of dt pont s defned s follows P n j e ( ) j r (3) where denotes Eucldn dstnce nd r 0 s constnt. Equton (3) shows tht the mesure of the potentl for dt pont s functon of ts dstnces to ll other dt ponts. A dt pont wth mny neghborng dt ponts wll hve hgh potentl vlue. he constnt r denotes the rdus defnng neghborhood; where dt ponts outsde ths rdus hve lttle nfluence on the potentl. he potentl of every dt pont hs to be computed usng (3). he frst cluster center s then chosen from the dt pont whch hs the hghest potentl.

6 7 Endr Joelnto & Prlndungn H. Stnggng Let nd P be the locton of the frst cluster center nd ts correspondng potentl vlue respectvely. he potentl of ech dt pont by usng the potentl equton formul s then revsed P ( ) rb e P P (4) where r b 0 s constnt. Equton (4) shows n mount of potentl from ech dt pont s functon of ts dstnce from the frst cluster center. he dt ponts ner the frst cluster center wll hve gretly reduced potentl, nd t wll not be selected s the net cluster center. he constnt r b denotes the rdus defnng the neghborhood whch wll hve mesurble reductons n potentl. o vod obtnng closely spced cluster centers, r b s set to be greter thn r, good choce s r 5 r. 5r hu [8]. b. r hu [7] or b Net, the potentls of ll dt ponts re revsed by usng equton (4), the dt pont wth the hghest remnng potentl s chosen s the second cluster center. th he process s then contnued untl the cluster center hs been chosen, the potentl of ech dt pont s redefned by the potentl equton P ( ) rb e P P (5) where th denotes the locton of the cluster center nd P s ts correspondng potentl vlue. he process of cqurng new cluster center nd revsng potentls s repeted untl the remnng potentl of ll dt ponts flls below some frcton of the potentl of the frst cluster center P. In ddton to ths crteron for endng the clusterng process re crter for cceptng nd rejectng cluster centers tht help to vod mrgnl cluster centers. he lgorthm of substrctve clusterng n fndng new cluster centers, revsng potentls nd repetng the process untl the potentl vlue s below the cceptnce vlue consdered s sgnfcnt effects to the fnl clusterng results cn be found n hu [7][8]. For problem of c clusters nd n dt ponts, Hmmoud nd Krry [9] hs shown the requred number of clcultons s: N clculton n ( c ) n (6) frst. cluster remnder. cluster

7 rnsent Response Improvement of PID ontroller 73 Equton (6) shows tht the number of clcultons s ndependent to the dmenson of the problem. In generl, the number of computtons s n the rnge of few ten thousnds. Although the number of clusters (or rules) s utomtclly determned by the method, t should be noted tht the user specfed prmeter r (the rdus of nfluence of cluster center) strongly ffects the number of clusters tht wll be generted. A lrge r generlly results n fewer clusters wth corser model, whle smll r cn produce ecessve number of clusters nd model tht does not generlze well (by over-fttng the trnng dt). herefore, the constnt r cts s tunng prmeter of the desred resoluton of the model, whch cn be chosen bsed on the resultnt complety nd generlzton blty of the model. It s cler tht choosng r very smll or very lrge wll result n poor ccurcy becuse f r s chosen very smll the densty functon wll gnore the effect of neghborng dt ponts; whle f ten very lrge, the densty functon wll te nto ccount ll the dt ponts n the dt spce. hu [8] suggests the good vlue of r s between 0. nd 0.5, whle Hmmoud nd Krry [9] shows tht vlue of r between 0.4 nd 0.7 s dequte. o etrct fuzzy rules from dt, the dt re seprted nto groups ccordng to ther respectve clsses. In [7][8], subtrctve clusterng s then ppled to the combned nput/output spce of ech group of dt ndvdully for dentfyng ech clss of dt,.e. ech dt pont s vector tht contns both nput nd output vlues. Ech obtned cluster center s n essence prototypcl dt pont tht represents chrcterstc nput/output behvor of the system. onsder set of m cluster centers {,,, } found n M dmensonl m spce nd dvde the spce nto N dmensons correspond to nput vrbles nd the lst M N dmensons correspond to output vrbles. Ech vector s decomposed nto two component vectors y nd z, where y contns the frst N elements of from the coordntes of the cluster center n nput spce nd z contns the lst M N elements from the coordntes of the cluster center n output spce. o dentfy the behvor of the system, ech cluster center my be trnslted nto fuzzy rule tht cn be represented by the rule:

8 74 Endr Joelnto & Prlndungn H. Stnggng Rule : If {nput s ner y } then {output s ner z }. Gven n nput vector y, the degree of fulfllment of y ner follows y s ssgned s j ( ) yy j r e (7) he output vector z s clculted by usng the formul m m z (8) z Equton (8) shows the computtonl model n terms of fuzzy nference system employng trdtonl fuzzy f-then rules. Ech rule cn be wrtten n the more fmlr form: If Y s A & Y s A &... then Z s B & Z s B where Y j s the j th nput vrble nd Z j s the j th output vrble; A j s n eponentl membershp functon n the th rule ssocted wth the j th nput nd B s membershp functon n the th rule ssocted wth the j th output. j For the th rule, whch s represented by cluster center eponentl membershp functon gven by, Aj s selected s n A ( Y j j Yj y ) ep j j (9) nd where j B j s ny symmetrc membershp functon wth the center round y j nd z j re the j th element of y nd z j, z respectvely. he vlue of s set to / 8 [8]. he method of clculton s smlr to n j r nference system tht mplements the AND opertor wth multplcton nd t uses the degree of fulflment of the rule to weght the consequent of ech rule. he vlue of the fnl output s obtned by weghtng verge of ll the consequents. he rules cn be optmzed by usng vlble optmzton

9 rnsent Response Improvement of PID ontroller 75 methods. One smple pproch ppled by hu [8] s by setng the consequent prmeter z s lner functon of the nput vrbles s follows j z j G y h j j (0) where Gjs n N -element vector of coeffcents nd h j s sclr constnt. In ths cse, the f-then rules turn out to be the g-sugeno type. he optmzton of Gj nd h j n ll consequent equtons, gven set of rules wth fed premse membershp functons, s lner lest-squres estmton problem g nd Sugeno []. hs pproch hs been used by hu [7] to optmze the obtned rules from the subtrctve clusterng method. he optmzton s only crred out n the coeffcents of the consequent equtons whch led to less computtonl complety. In ths pper, the substrctve clusterng technque developed by hu [7] whch s vlble n the Fuzzy Logc oolbo of MALAB [0] s used to fnd the number nd the vlue of the membershp functons of fuzzy bsed HR. he cluster estmton s obtned by usng the subclust functon. he genfs functon then genertes Sugeno-type fuzzy nference system tht models the dt behvor. here re three prmeters to be entered to the functons:. Qush fctor ( s f ) s used to multply the rd vlues tht determne the neghborhood of cluster center. It s to qush the potentl for outlyng ponts to be consdered s prt of tht cluster, r s r. Accept rto ( ): A dt pont correspondng potentl P stsfes 3. Reject rto ( ): A dt pont correspondng potentl P stsfes b f wll be ccepted s cluster center f the P P wll be rejected s cluster center f the P P 3 Stblty of losed Loop System o show stblty of the closed loop system, t s necessry to recll the structure of the proposed control desgn s shown n Fgure. he proposed method conssts of three mn prts: the event generton, the fuzzy decson rule nd the PID controller. he PID controller wors on the contnuous ouput sgnl but the fuzzy decson s ctvted when there occur the event generted by the condton n equton () nd (). In ths cse, the equton () nd () hve been selected such tht the equton () genertes n enbled event then

10 76 Endr Joelnto & Prlndungn H. Stnggng followed by tmed events bsed on the smplng tme of the system s shown n Fgure 3. he strtegy hs been selected such tht the modfed reference sgnl wll hve pttern tht wll gurntee the reference sgnl lwys returns to the defult reference sgnl. he reference sgnl sequence s defned s follows. r( r r p p ( t ( t r ( t r ( t 0 ( p ) ) r ( p ) ) r 0 ) r ( t ) r ( t p p3 ( t r ( t ) d 0 ( t d ) d ) d (0 ) ( p ) ) d ( p ) ) d ( ) (( p ) ) (( p ) ) r d ( 0,,, p () where, nd ( ) d denote the enbled event t tme t when the ouput stsfes (), the smplng tme nd the reference sgnl re ntted by the enble event t tme t respectvely nd p s fnte number denotes the number of reference sgnl chnges. he vlue of p s unnown nd s nfluenced by the rel prt of the egenvlues of the closed loop system mtr nd the smplng tme. ; enbled event ( r d... ( r d d (0) d ( ) d ( ) d ( 3 ) d ( 4 ) d (( p ) ) d (( p ) ) Fgure 3 An enbled event followed by tmed events []. he dmssble reference sgnl for the fuzzy HR cn be defned s follows. Defnton he reference sgnls d ( ) re sd to be dmssble reference sgnls f

11 rnsent Response Improvement of PID ontroller 77 d ( ) for 0,,, p where s set of reference sgnls whch fulflls the condton (). In order to nlyze the closed loop system, the PID controller s epressed s full stte feedbc control lw. onsder sngle nput sngle output lner tme nvrnt plnt descrbed by the stte spce equton where the sttes ( A( Bu( y( ( () n R re the soluton of (), the control sgnl ssumed to be the output of PID controller wth nput controller for regultor problem s of the form t u R s y R. he PID d u( K y( d( K y( K3 y( (3) dt 0 whch s n output feedbc control system nd K K 3 p d of whch K p, nd K K p / K K, p, d denote proportonl gn, tme ntegrl nd tme dervtve of the well nown PID controller respectvely. he control lw (3) s epressed s stte feedbc lw usng () by dfferenttng the plnt output y s follows y y A Bu y A ABu Bu. hs mples tht the dervtve of the control sgnl (3) my be wrtten s ( K B) u 3 K A K A K ) ( 3 ( K 3AB KB) u 0 (4) Usng the notton Kˆ s normlzton of K, ths cn be wrtten n more compct form ˆ ˆ K [ K K K3] ˆ ˆ ( K 3B) [ K K K3] (5)

12 78 Endr Joelnto & Prlndungn H. Stnggng or K ck ˆ where c s sclr. hs control lw s then gven by A A K u ] ) ( [ ˆ u A B B K ] [0 ˆ (6) Denote A A K K ] ) ( [ ˆ nd u A B B K K ] [0 ˆ, the bloc dgrm of the control lw (6) s shown n Fgure 4. u u K K u y Bu A Fgure 4 Bloc dgrm of stte spce representton of PID controller. Equton (6) represents n output feedbc lw wth constrned stte feedbc. ht s, the control sgnl (3) my be wrtten s K u (7) where u u, u K A B B A A K ˆ 0 ) ( (8) he ugmented system equton s obtned from () nd (8) s follows u B A (9)

13 rnsent Response Improvement of PID ontroller 79 where A B 0 A ; B 0 0 Equton (7), (8) nd (9) show tht the PID controller cn be vewed s stte vrble feedbc lw for the orgnl system ugmented wth n ntegrtor t ts nput. Once the prmeters of the PID controller re selected, the full stte feedbc gn K cn be clculted usng equton (8) nd the closed loop system s gven by ( A B K ) Acl ; Acl A BK (0) he closed loop system s symptotclly stble f the egenvlues of strctly Hurwtz. Acl re he closed loop system stblty condton cn now be summrzed s follows. heorem onsder the fuzzy hybrd reference control n Fgure. Suppose the prmeters of the PID controller re selected such tht the closed system (0) s symptotclly stble. If the reference sgnls generted by the fuzzy decson re dmssble reference sgnls then the closed loop system s symptotclly stble nd lm e( 0 s r( r ( t ( p ) ) r ( ; 0,,, p t where e( r( y( for ll t t ( p ). Proof p Recll the symptotc stblty n the equton (0) gurntees tht the stte error of the closed loop system goes symptotclly to zero s t rrespectve to the ppled reference sgnl. he convergence of the output to the defult reference sgnl s fulflled by the pttern of the reference sgnl (). he complete proof nvolvng the concept of nternl model prncple (IMP) [] nd the prncples of the hybrd reference control method cn be found n []. It hs been shown n [] tht proper selecton of the mplemented reference sgnl wll mprove the settlng tme to the defult reference sgnl. However, d

14 80 Endr Joelnto & Prlndungn H. Stnggng there s no ssurnce tht the mgntude of the mmum overshoot nd the control sgnl wll be lower compred to the orgnl closed loop system. In ths cse, the generton of the dt used n the substrctve clusterng lgorthm tht produce fster settlng tme nd lower mgntude of mmum overshoot nd control sgnl s very mportnt n order to gurntee the sme result n the PID fuzzy HR. 4 rnsent Response Improvement of PID ontroller Bsed on the conventonl HR n Joelnto nd Wllmson [][][] nd Joelnto [3] nd by usng the stndrd PID, dt re generted by requrng tht the output responses of the closed loop system re wthn specfed tolernces of the desred trnsent response performnces by chngng the reference sgnls durng trnsent response. he plnt whch s frst order system wth dely tme s gven by the followng trnsfer functon G( s) s 30s 5e 9s 0 () he stblty of the closed loop system s controlled by the stndrd PID controller of the form t K p de( u( K pe( e( dt K pd () dt 0 where the prmeters re selected by usng the well nown Zegler-Nchols tunng method developed by Zegler nd Nchols [3]. he obtned PID prmeters re s follow: K 0. 05, 0. 6second, 5second. p In ths pper, the trnsent response specfcton s mmum overshoot ( M ). Dt re collecton of trnsent responses tht stsfy the specfcton s shown n Fgure 3. he dt consst of error ( e ( ), dervtve error ( e ( ) nd the reference sgnl ( d ( ) tht mnpultes the defult reference sgnl ( r ( ). he dt re processed s follows: Reject error sgnl whch hs vlue below stedy stte error ( ) Accept error sgnl whch hs vlue bove for substrctve clusterng dt Reduce zero reference sgnl dt n order to vod hgh concentrton d p

15 rnsent Response Improvement of PID ontroller 8 Equte the number of row between error sgnl, dervtve error sgnl nd reference sgnl Net, rrnge the dt nto mtr X s follows: X X X X 3 where X : error sgnl column mtr, X : dervtve error sgnl column mtr, X 3 : reference sgnl column mtr. he lgorthm of substrctve clusterng bsed fuzzy-hr s summrzed s follow:. Select prmeters of PID controller ( K p,, d ) nd the event tolernce. Generte dt mnully by removng the fuzzy system: error sgnl, dervtve error sgnl nd reference sgnl usng Fgure. 3. Set llowble mmum overshoot ( M p ) nd llowble stedy stte error ( ) 4. Reject dt greter thn M p nd less thn X X nd prmeters of subclust functon 5. Select dt rnge of X 3 ( s f,, ) 6. Arrnge dt nto nput dt (error sgnl nd dervtve error sgnl) nd output dt (reference sgnl) 7. Use the nput dt, output dt nd prmeters of subclust functon nto genfs functon In the emple, the prmeters re selected s follow: rnsent response: M p =5 % nd =% luster rdus: [ ] Dt rnge: Error sgnl [-5 5] Dervtve error sgnl [ ] Reference sgnl [- ] Substrctve clusterng prmeters: [ ]

16 8 Endr Joelnto & Prlndungn H. Stnggng Qush fctor.5 Accept rto 0.5 Reject rto 0. Usng the substrctve clusterng lgorthm, the resulted membershp functons re shown n Fgure 4. he PID controller wth the substrctve clusterng bsed fuzzy HR system s then smulted by usng SIMULINK ccordng to the bloc dgrm n Fgure. In ths pper, the fuzzy HR contnuously sends the reference sgnl ( d ( ) to mnpulte the defult reference sgnl ( r ( ) s s set very smll. he output response, the reference sgnl nd the control sgnl re shown n Fgure 5, t cn be seen the PID fuzzy HR yelds lower mmum overshoot of the output response nd uses lower control sgnl thn the stndrd PID controller s result of pplyng mnpulted reference sgnl (set-pon. Other prmeters selecton leds to dfferent trnsent performnces.

17 rnsent Response Improvement of PID ontroller 83 Fgure 5 Dt nd cluster centers. Fgure 6 Membershp functons.

18 84 Endr Joelnto & Prlndungn H. Stnggng Fgure 7 Output response nd control sgnl of PID nd PID-fuzzy HR. 5 onclusons he pper proposed method clled fuzzy hybrd reference control (fuzzy- HR) whch mmcs the opertor cton n chevng the requred trnsent response performnce by ncresng or decresng the nput of the controlled process by mens of mnpultng the reference sgnl. he combnton of the proposed fuzzy HR wth the PID controller resulted n two-degree-offreedom controller structure whch s useful n solvng conflctng performnce requrements n one degree of freedom controller, such s trnsent performnce nd stblty robustness. In ths method, the stblty property of the closed loop control system ws determned by the prmeter vlues of the PID controller. he fuzzy HR system mproved the trnsent response performnce of the conventonl closed loop system wth PID controller wthout deterortng the closed loop stblty nd t s cheved by usng less control sgnl. he

19 rnsent Response Improvement of PID ontroller 85 pplcton of the substrctve clusterng technque provded fst method to fnd the membershp functons of the fuzzy HR system. he proposed method cn be etended to other stblzng controllers, such tht robust H, robust dssptve. References [] Joelnto, E. & Wllmson, D., Dscrete Event Reference ontrol, Proc. 36 th IEEE onference on Decson ontrol,, , 997. [] Joelnto, E. & Wllmson, D., Optml Full Stte Hybrd Reference ontrol, n Begh, A., Fnesso, L., nd Pcc, G. (Eds.), Mthemtcl heory of Networ Systems, Il Polgrfo, Pdov, Itly, , 998. [3] Joelnto, E., Lner Hybrd Reference ontrol Systems, PhD hess, he Austrln Nttonl Unversty, nberr, Austrl, 000. [4] Zdeh, L.A., Outlne of A New Approch to he Anlyss of omple Systems nd Decson Processes, IEEE rnsctons on Systems, Mn nd ybernetcs, 3, 8-44, 973. [5] Wng, L.X., Adptve Fuzzy Systems nd ontrol: Desgn nd Stblty Anlyss, Prentce Hll, Inc., Upper Sddle Rver, USA, 994. [6] Msr, D. & Ml, H.A., Lpunov Stblty for A Fuzzy PID ontrolled Fleble-Jont Mnpultor, Interntonl Journl of omputer Applctons n echnology, 7, 97-06, 006. [7] Mohn, B.M. & Snh, A., Anlytcl Structure nd Stblty Anlyss of A Fuzzy PID ontroller, Appled Soft omputng Journl, 8, , 008. [8] Yoogw, U4/U4 Dgtl Indctng ontrollers, echncl Informton Publcton I 5B4A7-0E, Yoogw Electrc, Jpn, 990. [9] Yoogw, Green Seres-Dgtl Indctng ontrollers Selecton Gudes, Bulletn 5AA0-E, Yoogw Electrc, Jpn, 998. [0] hu, S., Developng ommercl Applctons of Intellgent ontrol, IEEE ontrol Systems Mgzne, 7, 94-97, 997. [] Joelnto, E. & nsr, O., Fuzzy Logc Bsed Hybrd Reference ontrol for Improvng rnsent Response Performnce of PID ontroller, IB Journl, 39A, 4-45, 007. (In Indonesn) [] g,. & Sugeno, M., Fuzzy Identfcton of Systems nd Its Applcton to Modelng nd ontrol, IEEE rnsctons on Systems, Mn nd ybernetcs, 5, 6-3, 985. [3] Zegler, J.G. & Nchols, N.B., Optmum Settng for Automtc ontrollers, rnsctons ASME, 64, , 94. [4] Smth,.A. & orrpo, A.B., Prncples nd Prctce of Automtc Process ontrol, John Wley & Sons, Inc., nd., 985. [5] oughnowr, D.R., Process Systems Anlyss nd ontrol, nd Ed., McGrw-Hll Int. Edton, Sngpore, 99.

20 86 Endr Joelnto & Prlndungn H. Stnggng [6] Åström, K.J. & Hägglund,., PID ontrollers: heory, Desgn, nd unng, nd Ed., Instrument Socety of Amerc, USA, 995. [7] hu, S.L., Fuzzy Model Identfcton Bsed on luster Estmton, Journl of Intellgent nd Fuzzy Systems,, 67-78, 994. [8] hu, S.L., Etrctng Fuzzy Rules from Dt for Functon Appromton nd Pttern lssfcton, n Dubos, D., Prde, H., nd Yger, R. (Eds.), Fuzzy Informton Engneerng: A Guded our of Applctons. John Wley & Sons, Inc., New Yor, USA, 997. [9] Hmmoud, K. & Krry, F., A omprtve Study of Dt lusterng echnques, SYDE 65: ools of Intellgent Systems Desgn, ourse project, (July 8, 008), 000. [0] Fuzzy Logc oolbo for Use wth MALAB, he MthWors Inc., Ntc, 00. [] Joelnto, E. & Wllmson, D., rnsent Response Improvement of Feedbc ontrol Systems usng Hybrd Reference ontrol, Interntonl Journl of ontrol, 8(0), , 009. [] Frncs, B.A. & Wonhm, W.M., he Internl Model Prncple of ontrol heory, Automtc,, , 976,.