Journal of Process Control

Transcription

1 Journal of Process Control ) Contents lsts avalable at ScenceDrect Journal of Process Control journal homepage: wwwelsevercom/locate/jprocont Independent desgn of mult-loop PI/PID controllers for nteractng multvarable processes Truong Nguyen Luan Vu, Moonyong Lee School of Chemcal Engneerng and Technology, Yeungnam Unversty, Daedong 214-1, Kyongsan, Kyongbuk , South Korea artcle nfo abstract Artcle hstory: Receved 2 January 2010 Receved n revsed form 26 Aprl 2010 Accepted 14 June 2010 Keywords: Mult-loop PI/PID controller tunng Effectve open-loop transfer functon EOTF) Model reducton Internal model control IMC) Dynamc relatve gan array The nteractons between nput/output varables are a common phenomenon and the man obstacle encountered n the desgn of mult-loop controllers for nteractng multvarable processes In ths study, a novel method for the ndependent desgn of mult-loop PI/PID controllers s proposed The dea of an effectve open-loop transfer functon EOTF) s frst ntroduced to decompose a mult-loop control system nto a set of equvalent ndependent sngle loops Usng a model reducton technque, the EOTF s further approxmated to the reduced-order form Based on the correspondng EOTF model, the ndvdual controller of each sngle loop s then ndependently desgned by applyng the nternal model control IMC)-based PID tunng approach for sngle-nput/sngle-output SISO) systems, whle the man effects of the dynamc nteractons are properly taken nto account Several llustratve examples are employed to demonstrate the effectveness of the proposed method 2010 Elsever Ltd All rghts reserved 1 Introducton Most chemcal processes are bascally mult-nput/mult-output MIMO) systems Despte the development of advanced multvarable controllers, the mult-loop PI/PID control usng multple sngle-nput/sngle-output SISO) PI/PID controllers remans the standard for controllng MIMO systems wth modest nteracton because of ts smple and falure tolerant structure and adequate performance 1,2 However, due to process and loop nteractons, the desgn and tunng of mult-loop controllers s much more dffcult compared wth that of sngle-loop controllers Snce the controllers nteract wth each other, the tunng of one loop cannot be done ndependently Applyng the tunng methods for a SISO system to mult-loop systems often leads to poor performance and stablty Much research has been focused on how to effcently take loop nteractons nto account n the mult-loop controller desgn Many methods have been proposed, ncludng the detunng method, sequental loop closng SLC) method, relay auto-tunng method, and ndependent loop method The bggest log modulus tunng BLT) method proposed by Luyben 3 s a typcal example of the detunng method, wheren each ndvdual controller s frst desgned based on the Zegler Nchols Z N) tunng rules 4 by gnorng process nteractons from other loops Then, the nteractons are taken nto account by detunng Correspondng author Tel: ; fax: E-mal address: mynlee@yuackr M Lee) each controller untl the multvarable Nyqust stablty s satsfed The attractveness of ths method s due to the smplcty n mplementaton and comprehensbleness for control engneers However, a dsadvantage s that the controller settngs are made more conservatve The well-known SLC method for the desgn of mult-loop controllers was frst ntroduced by Mayne 5 and later studed by Hovd and Skogestad 6 In ths method, the controllers are tuned sequentally, wheren the controller of the fastest loops should be tuned frst by consderng a selected nput output par; ths loop s then closed and then the controller of the lower loops s tuned for a second par whle the frst control loop remans closed and so on The SLC method s smpler than the detunng method as each controller s desgned usng SISO desgn methods In relay auto-tunng for the mult-loop control system 7 10, the relay feedback technque s appled to the desgn of each correspondng SISO controller The control loops are tuned sequentally or smultaneously Furthermore, on the bass of sequental algorthm, the mult-loop control system s desgned n a sequence of SISO desgn problems and the nteracton taken nto account n a sequental fashon In ths way, Loh et al 8 and Shen and Yu 9 have drectly combned the effcency of a sngle-loop relay and SLC method to desgn mult-loop controllers, whch s sometmes called as the auto-tunng SLC method These methods requre mnmal process nformaton but tunng sequence has to be repeated for the correct sequence f the desgn sequence s not approprate The ndependent loop method s used to surmount the restrcton of the relay auto-tunng As dscussed by a number of authors /$ see front matter 2010 Elsever Ltd All rghts reserved do:101016/jjprocont

2 TNL Vu, M Lee / Journal of Process Control ) Fg 1 Block dagram for the concept of the effectve open-loop transfer functon n a n n mult-loop system: loop s open whle all other loops are closed 11 14, the ndependent loop method has a potental advantage n that the falure tolerance of the overall control system s automatcally guaranteed, wheren each controller s ndependently desgned based on the correspondng open-loop and closed-loop transfer functons, thereby satsfyng the nequalty constrants on the process nteractons 11 A potental dsadvantage of the method s one of the conservatsm due to the nherent assumpton of the ndependent desgn, whch does not explot the nformaton regardng controllers n other loops 12 For ths, the ndependent loop method for IMC type mult-loop controllers 13,14 s used to reduce the conservatsm Recently, several researchers have ntroduced the concept of an effectve open-loop transfer functon EOTF) to take nto account the loop nteractons n the novel desgn of a mult-loop controller Usng ths concept, the desgn of a mult-loop controller can be reasonably converted to the desgn of a sngle-loop controller On the bass of structure decomposton, the mult-loop control system s completely separated nto equvalent ndvdual SISO loops, and thus the effects of the process and controller on the loop nteracton and subsequent system propertes, such as rght half plane RHP) zeros and poles, ntegrty, and stablty, are elucdated 15 Moreover, He et al 16 have suggested the dynamc relatve nteracton to derve the multplcatve model factor MMF) for an ndvdual control loop; the equvalent transfer functon s then obtaned by multplyng the orgnal loop transfer functon wth the approxmated MMF wthn the neghborhood of the ndvdual control loop crtcal frequency Huang et al 17 found that the EOTF s formulated wthout pror knowledge of controller dynamcs n other loops and that the controller s ndependently desgned for equvalent sngle loops In an alternate manner, Xong and Ca 18 suggested that the EOTF provdes both gan and phase nformaton for mult-loop controller desgn n four ways The advantages of the EOTF nvolve reduced modelng requrements and ease of mplementaton whle the potental dsadvantage s reducton n achevable control performance due to restrcted controller structure 19 The control performance of the mult-loop systems s also closely related to the control loop parng Gven ts clear and useful defnton, the well-known RGA 20 has been wdely used for the mult-loop structure desgn, such as a rato of an open-loop Fg 2 Mult-loop system and equvalent ndependent SISO systems wth the correspondng EOTFs

3 924 TNL Vu, M Lee / Journal of Process Control ) gan to a closed-loop gan The defnton of RGA was extended to dynamc RGA DRGA), wth frequency-dependent terms, by replacng the steady-state gans wth the correspondng transfer functons Xong et al 25 proposed the relatve effectve gan array REGA) by employng the steady-state gan and bandwdth of the process transfer functon element that combnes the benefts of both the RGA and DRGA He et al 26 suggested the relatve normalzed gan array RNGA) for loop nteracton measurements Snce ths takes nto account both the steady-state and transent nformaton of the process transfer functon, t provdes more accurate nteracton assessment than the conventonal RGAbased loop parng crteron In ths paper, an effcent approach to ncorporate the loop nteracton effect nto the desgn of mult-loop PI/PID controllers s proposed by utlzng the concept of the EOTF Based on the characterstcs of the closed-loop dynamcs, the effect of other loops upon the partcular loop s effectvely expressed n terms of the DRGA wth no need of a pror nformaton of other controllers A multloop control system s then decomposed nto a set of ndependent SISO loops represented by correspondng EOTFs; the tunng of the mult-loop PI/PID controller s thus converted to the desgn of ndependent sngle-loop PI/PID controllers Several llustrated smulaton examples are employed to demonstrate the effectveness of the proposed method for varous nteractng multvarable processes 2 Effectve open-loop transfer functon and dynamc relatve gan Consder the open-loop stable mult-loop system n Fg 1, where r, ū, and ȳ are the set-pont, manpulated, and controlled varable vectors, where r, u, and y are dscarded from r, u, and y, respectvely Let the EOTF of loop be defned as the transfer functon relatng u wth y where loop s open whle all other loops are closed Fg 1 shows the block dagram for the concept of the EOTF of loop The EOTF dffers from the orgnal open-loop transfer functon OTF) by transmsson nteracton through a path ncludng other loops It s clear that the EOTF corresponds to the actual openloop transfer functon under mult-loop stuatons and thus, tunng of the controller of loop should be done based on the EOTF,, rather than the orgnal OTF, g Furthermore, as shown n Fg 2, a mult-loop MIMO system can be consdered a set of n ndependent SISO systems wth correspondng EOTFs From the block dagram n Fg 1 wth r = 0, ū s obtaned by ū = G cȳ = G c ḡc u + Ḡ ū ) 1) where G c denotes a mult-loop controller matrx n whch g c s dropped from G c Ḡ denotes a transfer functon matrx where both the th row and column are removed from G The terms ḡ r and ḡ c denote the th row and column vector of matrx G where g s dscarded, respectvely Rearrangng 1) yelds ū = G c I + Ḡ G c ) 1 ḡ c u 2) Therefore, the relaton between y and u s wrtten as y = g u + ḡ r ū = g ḡ r G c I + Ḡ G c ) 1 ḡ c u 3) The complcaton of the dynamc nteracton s clear from 3) The open-loop dynamcs between y and u depend not only upon the sngle transfer functon, g, but also on the process and controller terms n all other loops Ths also mples that n prncple, tunng of one controller cannot be done ndependently and depends on other controllers However, ths complcated relaton due to the dynamc nteracton can be reasonably smplfed by ntroducng several assumptons Assumng that the controllers nclude ntegral actons to avod offset and that the closed-loop dynamcs, by properly tuned controllers, are suffcently faster than the open-loop dynamcs, then a perfect control approxmaton, Ḡ G c I + Ḡ G c ) 1 = I, can be consdered at frequences lower than the cross-over frequency Therefore, 3) s reasonably smplfed as follows y = g ḡ r Ḡ ) 1 Ḡ G c I + Ḡ G c ) 1 ḡ c u = g ḡ r Ḡ ) 1 ḡ c u = u 4) Note that the EOTF of loop,, conssts of a process dynamcs term only and does not nclude knowledge of other controllers Furthermore, the EOTF can be compactly expressed n terms of the DRGA, as follows see detals of dervaton n Appendx A) = g 5) where denotes the th dagonal element of the DRGA and s calculated by = G G 1 ) T 6) where the symbol denotes the element by element multplcaton Hadamard or Schur product) and the superscrpt T desgnates the transpose of a matrx The physcal meanng of DRGA s clearly ndcated from 5): the th dagonal element of the DRGA mples a rato of the open-loop transfer functon to the effectve open-loop transfer functons of loop, under the assumpton of perfect control of other loops In ths paper, the dervaton of the EOTF and ts physcal meanng have been dscussed n a dfferent transparent way from the prevous works Reduced EOTF for controller desgn One of the most common approaches for controller desgn s to use a reduced-order model that smplfes the process dynamcs Snce the EOTF s lkely to show a complcated dynamc model form, the approach usng a reduced-order model s generally requred Any conventonal model reducton technque can be appled for ths purpose In ths secton, a smple model reducton technque s appled to approxmate the EOTF to a reduced-order model, such as the frst-order plus dead tme FOPDT) and the second-order plus dead tme SOPDT) models A two-nput, two-output TITO) mult-delay process s one of the most commonly encountered multvarable processes n the process ndustry A large number of prevous studes focused on desgnng mult-loop control system of TITO processes For a 2 2 system, the general stable square transfer functon matrx s represented as g11 s) g 12 s) Gs) = 7) g 21 s) g 22 s) The DRGA obtaned from 6) s ) g 11 s)g 22 s) 11 s) = 22 s) = 8) g 11 s)g 22 s) g 12 s)g 21 s) Therefore, the EOTFs for the frst and second loops are found usng 5), respectvely: 11 s) = g 11s) g 12s)g 21 s) 9) g 22 s) 22 s) = g 22s) g 12s)g 21 s) g 11 s) 10)

4 TNL Vu, M Lee / Journal of Process Control ) Fora3 3 system, the dervaton of EOTF s substantated n a smlar manner as that of the 2 2 system The dagonal elements of the DRGA are establshed from 6) as follows 11 = 22 = 33 = g 11g 22g 33 g 23g 32) g 11g 22g 33 + g 21g 13g 32 + g 31g 12g 23 g 11g 23g 32 g 21g 12g 33 g 31g 13g 22 11) g 22g 11g 33 g 13g 31) g 11g 22g 33 + g 21g 13g 32 + g 31g 12g 23 g 11g 23g 32 g 21g 12g 33 g 31g 13g 22 12) g 33g 11g 22 g 12g 21) g 11g 22g 33 + g 21g 13g 32 + g 31g 12g 23 g 11g 23g 32 g 21g 12g 33 g 31g 13g 22 13) Thus, the EOTFs for the frst, second, and thrd loops are consttuted as 11 = g g12 g 21 g 33 + g 12 g 23 g 31 ) g 13 g 31 g 22 + g 13 g 32 g 21 ) 11 g 22 g 33 g 23 g 32 14) 22 = g g12 g 21 g 33 + g 13 g 32 g 21 ) g 23 g 32 g 11 + g 23 g 31 g 12 ) 22 g 11 g 33 g 13 g 31 15) 33 = g g13 g 31 g 22 + g 12 g 23 g 31 ) g 23 g 32 g 11 + g 13 g 32 g 21 ) 33 g 11 g 22 g 12 g 21 16) As seen from the equatons above, the resultng EOTFs are usually too complcate to be drectly utlzed for the controller desgn To overcome ths dffculty, the EOTFs have to be smplfed to low-order models, such as FOPDT and SOPDT A lot of model reducton technques are avalable, ncludng, but not lmted to, the least squares algorthm 30 32, polynomal approxmaton 33, Laguerre expanson 34 36, and the Gaussan frequency doman approach 37 Any technque can be appled toward fttng the EOTFs nto a low-order model In ths work, for the purpose of evaluatng the proposed EOTF, a smple model reducton technque was proposed based on the coeffcent matchng method Expandng n a Maclaurn seres n s gves s) = a + b s + c s 2 + d s 3 + e s 4 + Os 5 ) 17) where the coeffcents of ths polynomal are a = 0) 18a) b = dgeff s) ds 18b) s=0 c = 1 d 2 s) 2 ds 2 18c) s=0 d = 1 d 3 s) 6 ds 3 18d) s=0 e = 1 d 4 s) 24 ds 4 18e) s=0 Snce the FOPDT model s most wdely used for modelng monotonc stable dynamcs, gven ts smplcty wth reasonable performance, the FOPDT dynamcs as a reduced-order model must be consdered frst g r eff = Ke s 19) s + 1 Expandng the reduced EOTF gven by 19) n a Maclaurn seres n s also gves 1 g r eff s) = K K + )s + K ) s 2 + Os 3 ) 20) Table 1 Relatons between process parameters and polynomal coeffcents for typcal process models Model Relatons ) c = Ke s K = a s+1 ); + ) = b ; a ±as+1)ke s K = a 1 s+1) 2 s+1) ); X 0 a) = b ; X a 1 + X 0 2 a) = a ) c ) a ; X 2 + X 3 2 a) = ) d e ) ; X a 4 + X 5 2 a) = a where X 0 = , X 1 = 2 /2)+ + 1) 1, X 2 = 3 /6)+ 2 /2)+ + 1) 1) 1, X 3 = 2 /2)+ + 1) ) , X 4 = 4 /24) + 3 /6)+ 1 2 /2) ) 1, X 5 = 3 /6)+ 1 2 /2) /2) where K,, and should be dentfed to approxmate as close as possble over control relevant frequency ranges Comparng the frst, second, and thrd terms of 20) wth those of 17) leads to the followng explct equatons for K,, and : K = a = 2c b ) 2 a a = b a 2c b ) 2 a a 21a) 21b) 21c) In order for the resultng FOPDT model to be feasble, and should be real and postve It s clear from 21) that the followng condton should be satsfed for feasble and values: 2c > b ) > a a 2c b ) 2 22) a a When the EOTF dynamcs s too complcated to be properly expressed by a smple FOPDT model, reducton to the FOPDT model results n nfeasble and/or values that are negatve or complex As such, the FOPDT model s not vald for approxmatng the EOTF dynamcs and as such, more general dynamcs such as a SOPDT model has to be consdered It has been shown from prevous works 17,38 40 that a SOPDT model wth a negatve/postve zero satsfactorly represents most of the complcated process dynamcs for the control relevant purpose g r eff = ± as + 1)Ke s 23) 1 s + 1) 2 s + 1) The relatons between the polynomal coeffcents and the process parameters for the SOPDT model are lsted n Table 1 The SOPDT model parameter values of 1, 2, a, and can also be obtaned from those relatons It s mportant to predct when and/or of the FOPDT model becomes nfeasble n the proposed model reducton method Suppose that the FOPDT model s attempted whle the actual EOTF has the SOPDT dynamcs gven by 23) Accordngly, mathematcal manpulaton from the relatons for the SOPDT process n Table 1 and the nequalty constrant gven by 22) provdes the followng condtons for when reducton to the FOPDT model leads to the nfeasble parameter values: For the case of SOPDT wth a negatve zero, a > > a > 1 2 or X X ) X ) X2 0 24a)

5 926 TNL Vu, M Lee / Journal of Process Control ) For the case of SOPDT wth a postve zero, a > b) where X 0 = For example, n the proposed model reducton method, ether a large overshoot or a strong nverse response such as: a > causes nfeasble and/or values n the FOPDT model If 2 the EOTF s monotonc and has no lead term, the proposed method always gves feasble and values It should be noted that feasble parameter values do not guarantee accuracy of model reducton Identfcaton performance by any reduced-order model requres fnal confrmaton by comparng the EOTF wth the reduced EOTF 4 Mult-loop PID controller desgn Once a reduced EOTF s obtaned, any PID tunng method for a SISO system can be appled for the desgn of each ndvdual PID controller In ths study, the IMC-PID tunng rules suggested by Lee et al 41 were chosen The IMC-PID desgn approach s commonly used for the PID controller tunng n the process ndustry because of ts many advantages, ncludng smplcty, robust performance, and analytcal form The overall procedure for drvng the tunng rules of loop s as follows: Frst, the reduced EOTF, g r eff, s decomposed to g r eff = p A p M, where p A and p M are the non-mnmum porton wth an all-pass form and the mnmum phase porton, respectvely The conventonal IMC flter, f, s selected as: f s)=1/ s +1)m, n whch s a desgn parameter that provdes the tradeoff between performance and robustness It s the desred closed-loop tme constant for the set-pont trackng The flter order m s selected as a postve nteger so that the controller s proper and realzable Then, the deal feedback controller to yeld the desred closedloop response perfectly s gven by g c = 1 q 1 g r eff q ) = p M s) s + 1) m p A s) 25) where q s the IMC controller and s desgned by: q = p 1 M f Snce the above resultng controller does not have a standard PID controller form, t s requred to approxmate the deal feedback controller G c to the equvalent PID controller forms Expandng g c n a Maclaurn seres n s yelds g c f s) s = 1 s f 0) + f f 0)s + 0) 2! s2 + f 0) 3! s3 + 26) The controller gven by 25) s nterpreted as the standard PID controller by usng the frst three terms and truncatng the hgher order terms, gven by g c s) = K c ) I s + Ds 27) where K c = f 0) I = f 0) f 0) D = f 0) 2f 0) 28a) 29b) 28c) The dervatve and ntegral tme constants computed from 27) could have negatve values when the reduced EOTF model has a strong lead term In ths case, a PID controller n seres wth the frst-order lag flter structure s recommended for use g c s) = K c ) I s + 1 Ds 29) F s + 1 where K c = f 0) I = f 0) 3f 0) + D = 1 2 F = 1 3 2f f f f 0) 3f 0) + f 0) f 0) f 0) 0)f 0) 3f 0))2 0)f 0) 3f 0)f 0) 0) f 0) 5 Smulaton study 30a) 30b) 30c) 30d) In ths secton, three examples are consdered to demonstrate the performance of the proposed method n comparson wth those of other well-known methods To ensure a far comparson, the performance and robustness of the control system are measured by the followng evaluaton crtera 51 Performance ndex To evaluate closed-loop performance, the ntegral absolute error IAE) crteron s consdered, whch s defned as: IAE = et) dt 31) 0 where et) = rt) yt) 52 Robustness ndex In ths study, a well-known method for robust stablty 2,24,42,43 s utlzed for a far comparson wth other comparatve methods The multple sources of uncertanty are lumped nto a sngle complex perturbaton multplcatve nput/output form) Snce the output uncertanty s often less restrctve than nput uncertanty n terms of control performance 24, the robust stablty of mult-loop control systems s examned under output multplcaton uncertanty For a process wth an output uncertanty of I + 0 s)gs), the upper bound of the robust stablty s gven as 43: = 0 ) < 1/ I + Gjω) G c jω)) 1 Gjω) G c jω) < - I +Gjω) G c jω)) 1, ω 0 32) where represents the degree of robust stablty, 0 perturbaton as a multplcatve output, and and - maxmum and mnmum sngular values, respectvely It should be noted that: Gjω) G c jω) s nvertble For a far comparson, all of the controllers beng compared were desgned to have the same degree of robust stablty n terms of the value throughout all smulaton examples For the proposed control system, the value was kept the same as or larger than those of the other methods Note that a control system wth a larger value mples more robust stablty Example 1 Vnante and Luyben VL) Column) A 24-tray tower separatng a mxture of methanol and water, examned by Luyben 3,

6 TNL Vu, M Lee / Journal of Process Control ) Fg 3 Bode magntude plots of the reduced EOTFs, EOTFs, g, and actual EOTFs for the VL column Fg 4 The tme responses of the reduced EOTFs, EOTFs, g, and actual EOTFs for the VL column

7 928 TNL Vu, M Lee / Journal of Process Control ) Fg 5 Closed-loop responses to the sequental step changes n the set-pont for the VL column has the followng transfer functon matrx For ths TITO system, t follows from 9) and 10) that the EOTFs for the frst and second loops are obtaned as Gs) = 22e s 7s e 18s 95s e 03 s 7s e 035s 33) 92s e s 11 s) = 7s s + 1)e 175s + 7s + 1)95s + 1) = 43e 035s 92s e 11s 95s + 1 ; 22 s) Fg 6 Output responses of the VL column where loop 1 s closed and loop 2 s open

8 TNL Vu, M Lee / Journal of Process Control ) Table 2 Controller parameters and resultng performance ndces for the VL column Tunng method Loop K c I D IAE t Nomnal VL +40%) VL 40%) Proposed PID Proposed PI J Lee et al Lee et al He et al IAE t: total sum of each loop s IAE VL +40%) and VL 40%) denote the plant-model msmatch cases under +40 and 40% gan uncertanty, respectvely The reduced EOTFs for the correspondng EOTFs are consttuted usng 21a) 21c) as follows g r eff 11 = 1354e 0682s eff ; gr 6661s = 2646e 0052s 8841s + 1 To evaluate how closely the proposed reduced EOTF approxmates the actual EOTF, the Bode dagrams and the tme responses are drawn for several cases Fgs 3 and 4 compare the Bode dagrams and the tme responses of the reduced EOTF, EOTF, orgnal OTF, and actual EOTF From Fg 3, both the reduced EOTF and EOTF show a farly good concdence wth the actual EOTFs over the control relevant low and mddle frequency ranges, becomng more conservatve as the frequency ncreases Note that the response of the actual EOTF s the actual response based on 3), and thus depends on the controller of other loops Fg 4 compares the tme responses of the reduced EOTF, EOTF, orgnal OTF, and actual EOTF As seen from Fg 4, the tme response of the reduced EOTF s closely approxmated to the actual EOTF These close approxmatons to the actual EOTF n the frequency and tme responses llustrate the valdty of the EOTF and the reduced EOTF, and also essentally lead to satsfactory control performance of the mult-loop controller desgned based on the EOTF Furthermore, the sgnfcant dfference between the orgnal OTF, g, and the other EOTFs confrm that the tunng of the ndvdual mult-loop controller should be done based on the EOTF rather than the orgnal OTF Fg 5 shows the closed-loop responses by several tunng methods In the smulaton study, the unt step set-pont changes were sequentally ntroduced nto the ndvdual loops For both the proposed method and Lee et al s 44 method, was adjusted to have a degree of robust stablty as = 053, whch s the same as that obtaned by Lee et al s 2 method and smaller than that of He et al 16 Note that snce the method of He et al 16 utlzed the SIMC-PID tunng rule 45 by settng equal to, ther respectve tunng values were employed n the smulaton wthout adjustng the value Fg 6 compares the output responses afforded by the proposed method wth those gven by others where loop 1 s closed and loop 2 s open The resultng controller parameters, together wth the performance ndces calculated usng the abovementoned methods, are summarzed n Table 2 It s apparent from Fgs 5 and 6 that the proposed PI/PID controller provdes a good performance wth fast and well-balanced responses n comparson wth those of the exstng methods The effectveness of the proposed PI/PID controller s also confrmed by ts smallest IAE value n Table 2 The robustness of the controller s evaluated by nsertng a perturbaton uncertanty of ±40% n the process gan nto the actual process, whereas the controller settngs are those provded for the nomnal process The resultng performance ndex for the plantmodel msmatch s tabulated n Table 2 As seen from the table, t s obvous that the proposed controller affords a good robust performance consstently Example 2 Wood and Berry WB) Column) Wood and Berry 46 ntroduced the followng model of a plot-scale dstllaton column consstng of an eght-tray plus reboler separatng methanol and water Gs) = 128e s 167s e 7s 109s e 3s 21s e 3s 144s ) Table 3 Controller parameters and resultng performance ndces for the WB column Tunng method Loop K c I D IAE t Nomnal WB +40%) WB 40%) Proposed PID Proposed PI Ho et al Lee et al Loh et al

9 930 TNL Vu, M Lee / Journal of Process Control ) Fg 7 Closed-loop responses to the sequental step changes n the set-pont for the WB column The EOTFs for the frst and second loops are found as 128e s s + 1)e 7s 11 s) = 167s s + 1)109s + 1) ; geff 22 s) = 194e 3s 144s s + 1)e 9s 21s + 1)109s + 1) The EOTFs are approxmated to the reduced EOTFs by usng the proposed model reducton method as follows g r eff 11 = 6370e 0308s eff ; gr 10529s = 9655e 4265s 6271s + 1 The mult-loop PI controller desgn methods suggested by Ho et al 47, Lee et al 44, and Loh et al 8 were employed for the comparson as they have demonstrated effectveness over other exstng methods The controller parameters used are lsted n Table 3 For a far comparson, the values for both the proposed method and that of Lee et al 44 were adjusted for = 047, the same as the method of Ho et al 47 and larger than that of Loh et al s 8 method Fg 7 compares the closed-loop tme responses by the proposed method and the above-mentoned desgn methods, where the unt step changes n the set-pont were sequentally made at t = 0 and t = 80 to the 1st and 2nd loops, respectvely The performance ndces are also tabulated n Table 3 The good performance of the proposed method s readly apparent To demonstrate the robust performance of the proposed method, the smulaton study was also done by nsertng a perturbaton uncertanty of ±40% n process gan As shown n Table 3, the controller settngs of the proposed method provde a good robust performance for the set-pont change Example 3 Ogunnake and Ray OR) Column) The multproduct plant dstllaton column for the separaton of a bnary ethanol water mxture was modeled expermentally by Ogunnake Table 4 Controller parameters and resultng performance ndces for the OR column Tunng method Loop K c I D F IAE t Proposed PID BLT DLT Halev et al Jung et al

10 TNL Vu, M Lee / Journal of Process Control ) Fg 8 Closed-loop responses to the sequental step changes n the set-pont for the OR column et al 48, the open-loop transfer functon matrx s gven by 066e 26s 061e 35s 00049e s 67s s s e Gs) = 65s 236e 3s 001e 12s 325s + 1 5s s e 92s 462e 94s s + 1)e s 815s s s + 1)188s + 1) 35) In ths example, snce a smple FOPDT model dd not take major process nteractons nto account, the SOPDT model wth a lead term was chosen as the reduced EOTF model The reduced EOTFs are consttuted as follows g r eff 11 = = s + 1)e 21s 1262s + 1)373s + 1) ; s + 1)e 262s 1489s + 1)284s + 1) ; eff gr 22 eff s + 1)e 108s gr 33 = 1051s + 1)380s + 1) For a SOPDT model wth negatve zero, the PID controller n seres wth the frst-order lag flter structure was recommended to enhance the closed-loop performance In the smulaton study, the values of the proposed method were chosen to obtan the = 006, whch gves a hgher robustness level than those of the BLT 3, Halev et al s 10, decentralzed lambda tunng DLT) 14, and Jung et al s 14 methods The controller parameters used are gven n Table 4 Fg 8 shows the closed-loop tme response of the proposed method n comparson wth those by BLT 3, DLT 14, and Jung et al s 14 methods, where the magntudes of the sequental step changes n the set-pont of loops 1, 2, and 3 are 1, 1, and 5, respectvely A mult-loop PID controller n seres wth a frst-order lag flter structure was used for the proposed, DLT 14, and Jung et al s 14 methods, whereas the standard PI controller structure was utlzed for the BLT 3 method The order of the IMC flter was chosen as 1 for all loops It s clear from Fg 8 that the PID controller obtaned by the proposed method yelded fast and wellbalanced responses The performance ndces of all the comparatve methods are also tabulated n Table 4, of whch the smallest total IAE value was found for the closed-loop response provded by the proposed method n terms of hgher robustness level than other methods

11 932 TNL Vu, M Lee / Journal of Process Control ) Concluson A novel method for the ndependent desgn of a mult-loop PID controller s proposed, based on the concept of EOTF The EOTF concept was successfully appled to decompose the complex mult-loop control systems nto a number of ndependent SISO loops n whch dynamc nteracton s taken nto account Therefore, the mult-loop PID controller was accomplshed by desgnng the SISO PI/PID controllers for each loop based on the correspondng EOTF model It was also shown that the well-known DRGA was nterpreted as a rato of the OTF to the EOTF A model reducton technque was proposed to further smplfy the EOTF to the reduced-order form The IMC-PID approach 41 was appled to the reduced EOTF to desgn the ndvdual PI/PID controller n each loop Results from the frequency and tme response analyss confrm that the EOTF and the reduced EOTF closely approxmate the dynamc nteracton among loops and the actual EOTF A smulaton study was also carred out to evaluate the proposed approach In order to nsure a far comparson, the maxmum upper bound on the sngular values of the output multplcatve uncertanty for the robust stablty was utlzed as a measure of the robustness level The smulaton results ndcate that the proposed method consstently affords a good performance wth a fast and well-balanced closed-loop tme response Robustness study was also conducted by nsertng a perturbaton uncertanty of ±40% n the process gan The results showed that the proposed control systems held robust stablty well n the plant-model msmatch case Acknowledgment Ths research was supported by a grant from the Gas Plant R&D Center funded by the Mnstry of Land, Transportaton and Martme Affars MLTM) of the Korean government Appendx A From the propertes of matrx GadjG) = G I A1) where I denotes the dentty matrx, G the determnant of G, and adjg the adjont of G, and the transpose of the matrx of cofactors correspondng to the entres of G C 11 C 21 C n1 C 12 C 22 C n2 adjg = A2) C 1n C 2n C nn From A1), t s mpled that g 1 C 1 + +g, 1 C, 1 + g C + g,+1 C,+1 + +g n C n = G g 11 g 1, 1 g 1, g 1,+1 g 1,n g 1,1 g 1, 1 g 1, g 1,+1 g 1,n g +1,1 g +1, 1 g +1, g +1,+1 g +1,n g n1 g n, 1 g n, g n,+1 g nn C 1 C, 1 C, C,+1 C,n = A3) A4) where g j and C j denote the jth element of G and ts cofactor, respectvely From A3), the th dagonal element of G s wrtten as g = 1 C G g 1 C 1 + g 2 C 2 + +g, 1 C, 1 + g,+1 C,+1 + +g n C n ) = G ḡr c C C C where C c = C 1 C 2 C, 1 C,+1 C n T A5) Note that C c corresponds to the th column vector of adjg droppng the element C A4) s rearranged as Ḡ C c + ḡ c C = 0 A6) Thus, = Ḡ ḡ c C c A7) C Substtutng A5) and A7) nto 4), the EOTF s expressed as ) ) G = ḡr c C ḡ rḡ ) 1 Ḡ C c = G A8) C C C C Furthermore, each dagonal element of the DRGA matrx s calculated as C = g A9) G Therefore, rearrangng A9) for G and substtutng t nto A8) leads to = g References A10) 1 PJ Campo, M Morar, Achevable closed-loop propertes of systems under decentralzed control: condtons nvolvng the steady-state gan, IEEE Trans Automat Control ) J Lee, W Cho, TF Edgar, Mult-loop PI controller tunng for nteractng multvarable processes, Comput Chem Eng ) WL Luyben, Smple method for tunng SISO controllers n multvarable systems, Ind Eng Chem Process Des Dev ) JG Zegler, NB Nchols, Optmum settngs for automatc controllers, Trans ASME ) DQ Mayne, The desgn of lnear multvarable systems, Automatca ) M Hovd, S Skogestad, Sequental desgn of decentralzed controllers, Automatca ) KJ Åström, T Hägglund, Automatc Tunng of PID Controllers, Instrument Socety of Amerca, Research Trangle Park, NC, AP Loh, CC Hang, CK Quek, VU Vasnan, Auto-tunng of mult-loop proportonal-ntegral controllers usng relay feedback, Ind Eng Chem Res ) SH Shen, CC Yu, Use of relay-feedback test for automatc tunng of multvarable systems, AIChE J ) Y Halev, ZJ Palmor, T Efrat, Automatc tunng of decentralzed PID controllers for MIMO processes, J Process Control ) P Grosdder, M Morar, Interacton measures for systems under decentralzed control, Automatca ) S Skogestad, M Morar, Robust performance of decentralzed control system by ndependent desgn, Automatca ) M Hovd, S Skogestad, Improved ndependent desgn of robust decentralzed controllers, J Process Control 3 1) 1993) J Jung, JY Cho, J Lee, One parameter method for a mult-loop control system desgn, Ind Eng Chem Res ) ZX Zhu, Structural analyss and stablty condtons of decentralzed control systems, Ind Eng Chem Res ) M-J He, WJ Ca, BF Wu, M He, Smple decentralzed PID controller desgn method based on dynamc relatve nteracton analyss, Ind Eng Chem Res ) HP Huang, JC Jeng, CH Chang, W Pan, A drect method for mult-loop PI/PID controller desgn, J Process Control ) Q Xong, W-J Ca, Effectve transfer functon method for decentralzed control system desgn of mult-nput mult-output processes, J Process Control )

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