Proeedng of the 3rd WSEAS/IASME Internatonal Conferene on Dynamal Sytem and Control, Arahon, Frane, Otober 3-5, 2007 72 Optmal Degn of Mult-loop PI Controller for Enhaned Dturbane Rejeton n Multvarable Proee TRUON NUYEN LUAN VU AND MOONYON LEE Shool of Chemal Engneerng and Tehnology Yeungnam Unverty 24-, Dae-dong, yeongan, yeongbuk 72-749 KOREA tnluanvu@y http://pd.yu.a.kr Abtrat: - The propoed method preent a new degn for mult-loop PI ontroller n MIMO ytem n two ae: et-pont trakng and dturbane rejeton. It an dentfy the tunng parameter of mult-loop PI ontroller by tartng from the generalzed IMC-PID approah [], whh extended from ngle nput, ngle output (SISO) ytem to multple nput, multple output (MIMO) ytem. Control parameter obtaned from the IMC-PID approah have hown good performane for everal ontrol ytem. However, there not enough robutne n ytem whh ontan a lot of noe and dturbane. A new degn an olve th problem by applyng the magntude of entvty (M theory. A mulaton tudy performed for the well-known proe model and the repone performane ompared favorably wth ome famou tunng method. The reult how that the propoed method uperor to extng tehnque for mult-loop proee. Key-Word: - Mult-loop PI ontroller, MIMO ytem, IMC-PID approah, M Crteron. Introduton The mult-loop PID/PI ontroller ha been tuded for many deade. In the 980, the famou tunng method for alulatng mult-loop PID ontroller parameter wa the Internal Model Control (IMC) [2], t wa publhed by C.. Eonomou and M. Morar. A typal method whh related to the mult-loop IMC degn method wa propoed by M.S. Baualdo and J. L. Marhett [3], whh onder to the nteraton between the ontrol loop. The bgget log modulu (BLT) tunng degn method [4] wa publhed by W. L. Luyben and t tll popular n proe ontrol today. In the 990, Loh et. al [5] tuded the auto-tunng proedure for mprovng the loed-loop frequeny repone n MIMO ytem, and Jung et. al [6] preented the deentralzed lambda tunng (DLT) degn method wth the ame goal of mprovng tablty and robutne. Reently, the generalzed IMC-PID approah degned for the mult-loop PID ontrol ytem by Lee et. al []. Th approah a varaton of Lee et. al [7] whh admtted to SISO ytem. Many mult-loop tunng degn method ext for et-pont trakng problem today. However, there are few method avalable for dturbane rejeton depte the fat that dturbane rejeton a more erou problem n ndutry. Therefore, we propoed a new degn method whh proeed from the generalzed IMC-PID approah and M rteron. The am of th method to degn a mult-loop PI ontroller that enhane dturbane rejeton a well a et-pont trakng. In the mult-loop IMC ontrol ytem, the performane and robutne of the loed-loop ytem largely depend on the loed-loop tme ontant ( λ ). The optmal value for the loed-loop tme ontant an be obtaned by ung M rtera. The propoed method an be ompenated the nfluene of dturbane effetvely by ompenatng the domnant pole n the dagonal element of the proe tranfer funton. r q q q f f2 fn r f - 0 0 0 0 2 0 0 n d 2 n 2 22 2n n n2 nn Fgure. Blok dagram for the mult-loop ontrol ytem y
Proeedng of the 3rd WSEAS/IASME Internatonal Conferene on Dynamal Sytem and Control, Arahon, Frane, Otober 3-5, 2007 73 2 The mult-loop PI ontroller degn In the n n mult-loop feedbak ontrol ytem hown n Fg., the loed-loop repone to the et-pont hange ( I ( ( ) ( ( r( y( = H( r( = () where H( the loed-loop tranfer funton; ( the proe tranfer funton whh open-loop table; ( ) the mult-loop ontroller wth dagonal element only; y( and r( are the ontrolled varable and the et-pont, repetvely. Suppoe that the dered loed-loop repone of the dagonal element n the mult-loop ytem gven by R ( ) = dag [ R, R,..., R ] (2) 2 n Aordng to the degn trategy of the IMC ontroller [], the dered loed-loop repone R of the th loop preented by y( ( ( ) R() β = = (3) n r() f (λ ) where the non-mnmum part of and hoen to be the all pa form; λ an adjutable ontant for ytem performane and robutne; n hoen for the IMC ontroller to be realzable. β degned to anel the domnant pole n the dagonal proe element. ((β ) - ( λ ) n = p =0 (4) Note that λ analogou to the loed-loop tme ontant and thu determne the peed of the loed-loop repone. The mult-loop ontroller ( ) wth ntegral term an be expreed n a Malaurn ere a 2 ( ) [ ( 3 = 0 2 O )] (5) where 0,, 2 an be ondered a the ntegral, proportonal, and dervatve term of the mult-loop PID ontroller, repetvely. A ndated from (5), the mpat of proportonal and dervatve term (.e.,, 2 ) domnate at hgh frequene and thu they hould be degned baed on the proe haratert at hgh frequene. On the other hand, the ntegral term 0 domnatng at low frequene and thu need to be degned baed on the haratert at low frequene. In the mult-loop ytem, the haratert of the loed-loop nteraton hanged aordng to frequeny range. Ung th frequeny-dependent properte of the loed-loop nteraton, analytal degn of the mult-loop PID ontroller an be largely mplfed whle t tll take the nteraton effet fully nto aount a follow []: At hgh frequene, the magntude of open loop gan beome ( jω) ( jω) and thu H( an be approxmated to H( = ( I ( ( ) ( ( ( ( 6) It ndate that 0 and an be degned by onderng only the dagonal element n (, whh mean the generalzed IMC-PID method for the SISO ytem [7] an be appled to the degn of the proportonal and dervatve term n the mult-loop PID ontroller. Therefore, at hgh frequene, the deal mult-loop feedbak ontroller to gve the dered loed-loop repone R ( ) gven by ( = ( ( ( ( ) R I R (7) where ( ) = dag [, 22,..., nn ] Aordngly, the deal mult-loop ontroller of the th loop an be degned by ( ()) ( β ) () = (8) ( ) n λ ()( β) where - the mnmum part of. Sne (0)=, (8) an be rewrtten n a Malaurn ere wth an ntegral term a '' ' f (0) 2 3 ( = ( f(0) f (0) 0( )) (9) 2 where f ( = ( The tandard PID ontrol algorthm gven by () = K( τ D (0) τ I Comparng (9) wth (0) gve the analytal tunng rule for the proportonal gan of the mult-loop PI ontroller a follow: K ' = f (0) () At low frequene, aordng to the degn of the ntegral term 0, the nteraton effet between the ontrol loop an not be negleted. Expanon of ( n a Malaurn ere gve
Proeedng of the 3rd WSEAS/IASME Internatonal Conferene on Dynamal Sytem and Control, Arahon, Frane, Otober 3-5, 2007 74 2 3 ( ) = O( ) (2) 0 2 where 0 = ( 0); = '(0); 2 = "(0)/ 2 By ubttutng (5) and (2) nto (), one an obtan H( a 2 H ( = I ( 0 0) O( ) (3) Furthermore, the dered loed-loop repone R an alo be wrtten n Malaurn ere a R ( ) = R(0) R'(0) O( 2 ) (4) where R (0) = I beaue ( 0) = By omparng the dagonal element of H( n (3) and R ( n (4) for the frt-order term, one an get the analytal tunng rule for the ntegral tme ontant of the mult-loop PID ontroller a follow ' ( (0) nλ β ) τ I = ( (0)) K (5) Tunng formulae by () and (5) provde an mportant advantage to olve the optmzaton problem for fndng the PID parameter value: for a gven proe, all the PID parameter an be expreed by a ngle degn parameter λ and thu the dmenon of the earh pae for optmzaton greatly redued. The lead term by ( β ) n (3) an aue an exeve overhoot n the et-pont repone. The two degree of freedom truture an overome th problem by degnng a et-pont flter q a qf() = (6) ( β ) 3 M rteron for the MIMO ytem M tunng the frequeny-doman method whh relate to the reonant peak M. The M value are related to the reonant peak of the entvty funton. The relatve tablty and robutne of a table loed-loop ytem an be uggeted by the magntude of M. In 996, Skogetad and Potlethwate [8] employed M a a tool for meaurng ytem robutne. In 998, Atrom et. al [9] propoed that the derable value of M for SISO ytem are n the range of.2 to 2. M tunng provde a lmt for the loed-loop tme ontant for a model, and t allow the optmal ontroller parameter to be found. The entvty funton n the mult-loop ontrol ytem an be repreented by - S ( ) = ( I ( () (7) The entvty frequeny repone an be found by ettng = jω n term of ω and λ a follow (jω,λ) = [ (jω,λ ) (jω,λ )] - S I (8) The entvty funton an be expreed by the matrx form a S S2 Sn S2 S22 S 2n (j ωλ, ) { Sj} S = = Sn Sn2 Snn (9) The maxmum entvty M obtaned a the maxmum value of the entvty funton over frequene { Mj} λ,ω 0 M = =max S (jω,λ) (20) j The peak magntude of the entvty funton an be expreed by the matrx form a M M2 Mn M2 M22 M 2n { Mj} M = = (2) Mn Mn2 Mnn The propoed M tunng method amed to mprove the performane and robutne of loed-loop frequeny repone n the mult-loop ontrol ytem by fndng an optmal λ. The mult-loop ontrol ytem an alo be made to meet the tablty bound and all the mult-loop PID parameter an be expreed by a ngle degn parameter λ.th optmzaton problem n the frequeny doman mn ( M ) λ,ω 0.t. j M M low (22) where M low the lower bound of the dagonal M and t alo an be ondered a optmzng value to mnmze the ntegral abolute error (IAE). Fg. 2 how the effet of M low on the overall performane n the OR olumn []. It mple that at mall value of M low, the IAE value are large. However, when M low nreae to hgh value, the IAE value alo nreae. Our extenve mulaton tudy how that the derable value of M low le between.8 and 2. Th range of M low an be ued for the trade-off
Proeedng of the 3rd WSEAS/IASME Internatonal Conferene on Dynamal Sytem and Control, Arahon, Frane, Otober 3-5, 2007 75 between luggh, overhoot, ollaton, and mnmzng IAE. Aordng to (22) t eay to fnd the optmal value of λ whh make mult-loop ontrol ytem table and robut not only for et-pont trakng but alo for dturbane rejeton. Fgure 2. Effet of M low on the IAE: OR olumn. 4 Smulaton tudy In the followng ae tude, we demontrate our tunng rule wth 3x3 ytem from the open lterature, Ogunnake and Ray (OR) olumn, a mult-produt plant dtllaton olumn for eparaton of a bnary ethanol-water mxture, wa modeled expermentally n Ogunnake et al. []. The propoed method alo ompared wth everal well-known tunng method uh a BLT and DLT tunng method. The tranfer funton matrx of the OR olumn gven by 2.6 3.5 0.66e 0.6e 0.0049e 6.7 8.64 9.06 6.5 3 2.e 2.36e 0.0e () = 3.25 5 7.09 9.2 9.4 34.68e 46.2e 0.89(.6 ) e 8.5 0.9 (3.89 )(8.8 ) (23) By ung (22), the optmum λ value were found a 0.07, 8.78, and 2.3 for eah loop, repetvely. Step hange n et-pont and dturbane were equentally made n the ndvdual loop. n n (3) wa hoen a 2 for all loop aordng to the proe model order. The value of.9 wa hoen for M low. All tunng parameter are lted n Table. The et-pont flter an be found a qf,2,3 () = {,, } (5.48 ) (3.43 ) (3.39 ) TABLE Tunng reult by the propoed PI method and varou method: OR olumn Propoed BLT DLT K.62, -0.32, 9.43 τ I 9.32, 7.27, 2.40 λ 0.07, 8.78, 2.3 IAE 8.5, 24.2, 8.48 IAE 2 3.5, 5.76, 24.3 IAE 3 0.07, 0.06, 2.32.5, -0.29, 2.63 6.4, 4.8, 6.6 - - Step hange n et-pont 32.06, 70.22, 49.85 6.9, 45.7, 43.07 0.6, -0.4, 0.39 8.00, 6.50, 6.85 35.86, 55.49, 975.54 6.99, 35.74, 297.52 0.2, 0.5, 3.59 0.2, 0.0, 27.58 IAE t 69.68 350.96 434.96 IAE 6.06, 8.22, 34.37 IAE 2 2.95, 22.22, 34.39 IAE 3 0.05, 0.07,.32 Step hange n dturbane 0.86,.9, 90.96 5.28, 60.93, 94.68 3.2, 7.46, 463.50 8.2, 46.5, 505.40 0.0, 0.02, 2.5 0.08, 0.08, 22.83 IAE t 09.54 277.5 077.0 IAE : IAE for the tep hange n loop. IAE t : um of eah IAE. Fgure 3 and 4 how the loed-loop repone for equental tep hange n et-pont and dturbane, repetvely. Sequental tep hange of magntude,, and 0 were made to eah loop. The BLT method how hgh overhoot and ollaton n the loed-loop repone whle the DLT method lead to very luggh and unbalaned one wth large IAE value. The propoed method provde fat and balaned repone through the llutrated example. The uperorty of the propoed method alo demontrated by omparon of the IAE value.
Proeedng of the 3rd WSEAS/IASME Internatonal Conferene on Dynamal Sytem and Control, Arahon, Frane, Otober 3-5, 2007 76 Fgure 3. Cloed-loop repone to equental tep hange n et-pont for OR olumn. Fgure 4. Cloed-loop repone to equental tep hange n dturbane for OR olumn.
Proeedng of the 3rd WSEAS/IASME Internatonal Conferene on Dynamal Sytem and Control, Arahon, Frane, Otober 3-5, 2007 77 5 Conluon The tunng of mult-loop PI ontroller for MIMO proe uually a omplex problem. In th paper, the generalzed-imc approah ued to develop a mple but effent degn method for the mult-loop PI ontroller. The propoed method ha everal lear advantage. Frtly, the method traghtforward and t an be ealy mplemented n multvarable ontrol ytem. Seondly, M rteron very utable for ahevng good tablty and robutne n mult-loop PI ontrol ytem. Furthermore, t provded the mnmzng of IAE value, whle other ontrol parameter are well-balaned. The mulaton reult how that the propoed method very effetve both et-pont trakng and dturbane rejeton n the mult-loop ontrol ytem. degn, John Wley & Son, New York, 996. [9] K. J. Atrom, H. Panagopoulo and T. Hagglnd, Degn PI ontroller baed on non-onvex optmzaton, Automata, Vol. 34, No. 5, 998, pp. 585-60. [0] R.K. Wood and M. W. Berry, Termnal ompoton ontrol of a bnary dtllaton olumn, Chem. Eng. S., Vol. 28, 973, pp.707-77. [] B. A. Ogunnake, J. P. Lemare, M. Morar, and W. H. Ray, Advaned multvarable ontrol of a plot plant dtllaton olumn, AIChE Journal, Vol. 29, 983,.pp.225-230. Referene: [] M. Y. Lee, K. Lee, C. Km, and J. Lee, Analytal degn of mult-loop PID ontroller for dered loed-loop repone, AIChE Journal, Vol.50, No. 7, 2004, pp. 63-635. [2] C.. Eonomou and M. Morar, Internal model ontrol 6: mult-loop degn, Ind. Eng. Chem. Pro. Proe. De. Dev., Vol. 25, 989, pp. 4-49. [3] M.S. Baualdo and J. L. Marhett, Tunng method for nteratve mult-loop IMC, PI and PID ontroller, Chem. Eng. Commun., Vol. 97, 990, pp.47-55. [4] W. L. Luyben, Smple method for tunng SISO ontroller n multvarable ytem, Ind. Eng. Chem. Proe De. Dev., Vol. 25, No. 3, 986, pp. 654-660. [5] A. P. Loh, C. C. Hang, C. K. Quek, V. N Vanan, Autotunng of multvarable proportonal-ntegral ontroller ung relay feedbak. Ind. Eng. Chem. Re., Vol. 32, 993, pp. 002-007. [6] J. Jung, J. Y. Cho, and J. Lee, One-parameter method for a mult-loop ontrol ytem degn Ind. Eng. Chem. Re., Vol. 38, 999, pp. 580-588. [7] Y. Lee, M. Lee, S. Park, and C. Brolow, PID ontroller tunng for dered loed loop repone for SI/SO ytem, AIChE Journal, Vol.44, No., 998, pp. 06-5. [8] S. Skogetad and I. Potlethwate, Multvarable feedbak ontrol: analy and