Designing of Decoupler for a 4 4 Distillation Column Processes with PID Controller

Transcription

1 Dii of Dcouplr for a 4 4 Ditillatio Colum Proc with PD Cotrollr Maah Kumar Sthi Dpt.of Elctroic ad Commuicatio E. Natioal titut of Tcholoy Rourkla Odiha thi.maah@mail.com Abtract Thi papr xtd th cocpt of rlativ ormalizd ai array (RNA) ad a Equivalt Trafr Fuctio (ETF) for ach lmt i th trafr fuctio matrix wa drivd for th clod-loop cotrol ytm ad propo a ytmatic approach to dii dctralizd PD cotrol for ditillatio colum proc cotaii itrator ad/or diffrtiator. Th dcouplr could b aily dtrmid with ach lmt i th Firt-Ordr-plu-Tim-Dlay (FOPTD) form ad rultd i a tabl propr ad caual dcoupld matrix. Th mai advata of thi mthod i it implicity; it rduc xciv calculatio.. NTRODUCTON Dpit th availability of complicat mthod for dii multivariabl cotrol ytmmot of loop ar cotrolld by dctralizd P/PD cotrollr bcau of thir rlativ ffctiv ad thir impl tructur which ca b aily udrtood ad implmtd.th primary tak i th di of dctralizd cotrol ytm i to dtrmi loop cofiuratio i.. pair th maipulatd variabl ad cotrolld variabl.sic cotrollr itract with ach othr i MMO proc th prformac of o loop caot b valuatd without tti iformatio about th cotrollr of othr loop. To olv thi problm a dcoupli cotrol chm ha b propod that itroduc a dcouplr to limiat th prc of itractio. Th th MMO proc ca b tratd a a SSO loop. A ral dcoupli cotrol ytm i rprtd dcouplr matrix ad th cotrollr trafr fuctio matrix rpctivly. () act upo th proc () uch that th trafr fuctio matrix a i Fi. 1 whr ( ) ad C ar th -dimioal proc matrix R ( ) ( ) ( ). (1) Taru Kumar Da. Dpt. of Elctroic ad Commuicatio E. Natioal titut of Tcholoy Rourkla Odiha tkda@itrkl.ac.i Thr ar baically two typ of dcoupli mthod ud for olvi MMO proc uch a tatic ad dyamic dcoupli. f thr i ubtatial diffrc i dyamic charactritic amo th trafr fuctio lmt it i difficult to t th dir prformac. Th dyamic dcouplr provid bttr rult wh th proc dyamic com ito th proc; howvr it i difficult to di practical dyamic dcouplr for hih dimioal MMO ytm. thi papr th ormalizd dcoupli cotrol di mthodoloy wa xtdd from a 4 4 di to a hihdimioal MMO di (A.Wahid ad A.Ahmd (008)Multivariabl cotrol of a (4 4) proc i ditillatio colum). Th di wa bad o th cocpt of Rlativ Normalizd ai Array (RNA) di (Xio & Cai 006; Xio Cai& H007.HCaiNi &Xi009). Ui th ai ad pha iformatio rvald i RNA di th Equivalt Trafr Fuctio (ETF) of ach lmt i th trafr fuctio matrix wa drivd for th clod-loop cotrol ytm. Furthr a quivalt trafr fuctio matrix wa approximatd by th ivr of th proc trafr fuctio matrix ud for calculati dcoupli matrix. Th dcouplr could th b aily dtrmid by multiplyi th ivr of th ETF with a tabl propr ad caual idal-diaoal trafr fuctio. Fially a P/PD cotrollr wa did for th diaoal trafr fuctio to mt th cotrol ytm prformac tart. Both th imulatio ad th ral-tim tti rult of a ditillatio proc dmotratd th ffctiv ad th implicity of th di compot icorporati th applicabl critria that follow.. CASE STUDY Th proc i prtd by Douka ad Luyba(1978).Thy tudid dyamic of a ditillatio colum produci a liquid idtram product.th objctiv i to maitai four compoitio pcificatio o th product tram.th trafr fuctio matrix i of (4 4) modl i iv blow. Th cotrol ad maipulatd variabl ar y1 (tolu impurity i bottom) y (tolu impurity at ditillat) y3 (bz impurity i id tram) y4 (xy impurity at id tram)u1

2 (id tram flow rat) u(rflux ratio) u3 (rboil duty) ad u4 (id draw locatio) (7.91) (36 1)(5 1) (31.6 1)(7 1) (3.71) (91) (351) (31) (81) (17 1) (1 1) (51) (8 1)(50 1) (50 1)(5 1) (43.6 1)(9 1) (48 1)(5 1) A. NPUT-OUTPUT PARN Coidr a ytm with a dctralizd Fdback cotrol tructur a how i Fiur1whr r [ r1r... r ] T u [ u1u... u ] T ar vctor of rfrc iput ad output rpctivly () [ ij ] i th ytm trafr fuctio matrix; ad dia c c1 c c i th dctralizd cotrollr; i j = 1... ar itr idic. B. LOOP PARN FOR PROCESSES CONTANN NTERATORS/DFFERENTATORS Hr it i aumd that K ad T ar ar fiit ad ozro to validat th dfiitio of th RNA. f a MMO proc cotai itrator T ar valu o to ifiity; o th othr had if a MMO proc cotai diffrtiator K qual zro. For uch a proc th RNA caot b computd dirctly ui th dfiitio. To dal with uch proc lt a MMO proc b iv by ( ) [ ij ( )] whr m () k ij () m i a itr ad do ot cotai ij ij ij j ay itrator or diffrtiator ad atifi (0) 1 i. f j m ij 0 th trafr fuctio cotai itrator (); if m ij 0 th trafr fuctio cotai diffrt diffrtiator(); ad if mij 0 it i a ormal trafr fuctio. Suppo that thr xit diaoal output cali matrix 1 () S R ad diaoal iput cali matrix () S R for K ( ) uch that () ca b factorizd a ( ) S1( )[ K ( )] S( ). () whr () [ ij ()]. Bcau th RNA i ivariat to iput/output cali th RNA (dotd by ) for () i th am a that for K ( ). Oc th RNA i obtaid th RA-N-RNA rul for ormal proc ca b applid dirctly to dtrmi th iput output pairi for th MMO proc with itrator/diffrtiator. Th abov loop pairi i limitd to th cla of MMO proc that ca b factorizd a i Eq. (A imilar cotrait occur i th loop pairi ui th RA aalyi) dd may practical MMO proc fall ito thi cla. Typical xampl will b how i ctio KN K Tar kn 11 kn1 KN kn 1 k N.(3) T K N K N.(4) (5) DECOUPLN CONTROL SYSTEM DESN Th di of a idal-diaoal dcouplr problm wa traformd to dtrmi th dcouplr ( ) ( ) ˆ T ( ) R( ).(6) ( ) ˆ ( ) 1 ˆ ( )...1 ˆ ( ) ( ) R11 1 ˆ ( ) 1 ˆ ( )...1 ˆ ( ) 0 ( 0 1 R ˆ ( ) 1 ˆ ( )...1 ˆ ( ) 0 0 ( ) 1 R..7 To obrv how th problm dfiitio ad th di mthod of a ormalizd dcoupli cotrol ytm wa diffrt from th xiti mthod ach lmt of th proc trafr fuctio matrix wa rprtd by Eq(7) (Yuli Sh W-JiaCai ShaoyuaLi (010)) ar from

3 th ETF wa rprtd ad th dird forward trafr fuctio lmt wr of th form Rii Rii () i j 1... Rii 1. (8) whr Rii ad Rii ar th adjutabl tim cotat ad th dad tim of Rii () rpctivly. By ubtituti Eq.()ito()th lmt i th idal dcouplr matrix ha th form ( ˆ Rii ji) ji 1 ij () i j 1... kˆ Rii 1 ji (9) A. CONTROLLER TUNN AND ROBUSTNESS ANALYSS Aftr dtrmiatio ( ) th paramtr i th cotrollr c () could b tud idividually for th corrpodi lmt of c ().Thu th prt SSO PD tui mthod could b dirctly applid to aurac th tability ad prformac of ach loop. thi work th PM (ai ad pha mari) mthod wa implmtd for putti ito practic bcau of it implicity ad robut. Sic ach lmt i R wa rprtd by a FOPDT modl th tadard P cotrollr of th form A diaoal matrix R wa pcifid uch that hold. Coqutly th di of th ormalizd dcouplr tartd from th obtaid ˆ T ( ) dtrmid th diaoal forward trafr fuctio matrix R uch that th dcouplr ( ) from Eq. atifid ralizabl coditio. kiii cii () kpii (10) Th clod-loop forward trafr fuctio wa dfid a kiii - Rii cii ( ) Rii ( ) (11) Th idividual lmt could b drivd a K Accordi to th RA-N-RNA critrio th iput-output Pair ar lctd a 1-1/-/3-3/4-4 Th ETF paramtr wr ˆK= ˆT= ˆ L Which iv ˆ T () Ui th ormalizd dcoupli cotrol ytm di rul th dcoupld forward trafr fuctio wa lctd a () R Which iv a tabl caual ad propr dcouplr () ˆ T () () R.(1)

4 () Th modifid ETF ar dtrmid a To apply th SMC mthod coqutly th PD cotrollr ar obtaid a c ( ) c( ) c33( ) c 44 ( ).04 (Fi. Output rpo of Uit tp iput ar y1 y y3 ad y4)

5 V.CONCLUSON A approach wa propod for dctralizd PD cotrol di of MMO proc with itrator ad/or diffrtiator. Bad o th RA-N-RNA critrio th iput-output pairi wa dtrmid. Th ETF wr drivd for th lctd iput-output pair ui th RA ad RNA iformatio. To maitai itrity th ETF wr modifid for cotrollr tui. Sic th modifid ETF had proprly tak accout of th loop itractio th MMO proc wa prcivd to b dcompod ito a t of idpdt SSO proc o that th PD cotrollr wr did idpdtly. Th ol advata of th propod approach i it implicity i carryi out a ytmatic dctralizd cotrol di which ca aily b udrtood ad implmtd by fild ir. Exampl illutratd th di procdur ad vrifid by ucc of th approach.th output rpo of th cotrollr ar how i th fi.. REFERENCES 1. Yuli Sh W-JiaCai ShaoyuaLi (010) Normalizd dcoupli cotrol for hih-dimioal MMO proc for applicatio i room tmpratur cotrol HVAC ytm. Wuhua Hu W-Jia Cai ad aoxi Xiao(010) Dctralizd Cotrol Sytm Di for MMO Proc with trator/diffrtiator 3. A.Wahid A.Ahmd(008) Multivariabl cotrol of a (4 4) proc i ditillatio colum 4. Xio Q. CaiW.-J.(006) Effctiv trafr fuctio mthod for dctralizd cotrol ytm di of multi-iput multi-output proc. Joural of Proc Cotrol 16(8) Cai W. NiC.-Y.(008) Normalizd dcoupli A w approach for MMO proc cotrol ytm di. dutrial ad Eiri Chmitry Rarch Liu T. Zha W. aof.(007) Aalytical dcoupli traty ui a uity fdback cotrol tructur for MMO proc with tim dlay Joural of Proc Cotrol 17() Xio Q.CaiW.-J.&HM.-J.(007) Equivalt trafr fuctio mthod for P/PD cotrollr di of MMO proc. Joural of Proc Cotrol 17(8) Walltr M. Walltr J.B. WalltrK.V.(003) Dcoupli rviitd. dutrial ad Eiri Chmitry Rarch 4(0)