212 Internatonal Truong Journal Nguyen of Control, Luan Vu, Automaton, Jetae Lee, and and Systems, Moonyong vol. Lee 5, no. 2, pp. 212-217, Aprl 2007 Desgn of Mult-loop PID Controllers Based on the Generalzed IMC-PID Method wth Mp Crteron Truong Nguyen Luan Vu, Jetae Lee, and Moonyong Lee* Abstract: A new method of desgnng mult-loop PID controllers s presented n ths paper. By usng the generalzed IMC-PID method for mult-loop systems, the optmzaton problem nvolved n fndng the PID parameters s effcently smplfed to fnd the optmum closed-loop tme constant n a reduced search space. A weghted sum Mp crteron s proposed as a performance cost functon to cope wth both the performance and robustness of a mult-loop control system. Several llustratve examples are ncluded to demonstrate the mproved performance of the mult-loop PID controllers obtaned by the proposed desgn method. Keywords: Generalzed IMC-PID method, MIMO system, Mp crteron, mult-loop PID controller, PID controller tunng, process nteracton. 1. INTRODUCTION Multvarable or mult-nput mult-output (MIMO) systems are frequently encountered n the chemcal and process ndustres. Despte the consderable work that has been done on advanced multvarable controllers for MIMO systems, mult-loop PID controllers (sometmes known as decentralzed PID controllers) are stll much more favored n most commercal process control applcatons, because of ther satsfactory performance along wth ther smple, falure tolerant, and easy to understand structure [1,2]. Mult-loop PID controllers are made up of ndvdual PID controllers actng n a mult-loop fashon and tuned manly on a sngle loop bass. However, due to the process nteractons n MIMO systems, ths approach cannot guarantee stablty when all of the loops are closed smultaneously [3]. Ths s because the closng of one loop affects the dynamcs of the other loops and can make them worse or even unstable. Ths complex nteractve nature of MIMO systems makes the proper tunng of mult-loop PID controllers qute dffcult. Manuscrpt receved September 28, 2005; revsed October 19, 2006; accepted January 30, 2007. Recommended by Edtoral Board member Young Il Lee under the drecton of Edtor Tae-Woong Yoon. Ths work was supported by a Korea Research Foundaton Grant (KRF-2003-002-D00068) and the 2006 Energy Resource and Technology Project. Truong Nguyen Luan Vu and Moonyong Lee are wth the School of Chemcal Engneerng and Technology, Yeungnam Unversty, 214-1, Dae-dong, Gyeongsan, Gyeongbuk 712-749, Korea (e-mals: tnluanvu@yumal.ac.kr, mynlee@yu.ac.kr). Jetae Lee s wth the Department of Chemcal Engneerng, Kyungpook Natonal Unversty, Taegu 702-701, Korea (emal: jtlee@bh.kyungpook.ac.kr). *Correspondng author. For ths reason, despte the wde popularty of mult-loop PID controllers, the number of applcable tunng methods s relatvely lmted. Furthermore, many of the exstng methods of desgnng mult-loop PID controllers are computatonally ntensve and/or requre solvng a large scale optmzaton problem and, therefore, are less appealng to the practtoners. Most exstng tunng methods for mult-loop PID controllers are smlar n that they frst use the sngle loop tunng rules by gnorng process nteractons and then detune the ndvdual loops to preserve stablty [4-7] or adjust them to meet some performance specfcaton [8]. However, snce these approaches are based on sngle loop tunng rules, they often lead to local optmum solutons or too conservatve responses. In ths artcle, we consder the desgn of mult-loop PID controllers for square multvarable systems. The proposed method combnes the nternal model control (IMC) based PID desgn method [9] wth a frequency doman performance crteron, Mp, defned as the maxmum magntude of the closed-loop frequency response. In addton to ts superor performance, the proposed approach has several mportant advantages. Frstly, the optmzaton problem s sgnfcantly smplfed to the fndng of only one desgn parameter,.e., the closed-loop tme constant of each loop. Secondly, there s no restrcton on the process model type. A smulaton study for several well-known dstllaton column models s carred out to llustrate the usefulness of the proposed method. 2. PROPOSED DESIGN METHOD 2.1. Generalzed IMC-PID method for mult-loop PID controller desgn [9] In the n n mult-loop feedback control system shown
Desgn of Mult-loop PID Controllers Based on the Generalzed IMC-PID Method wth Mp Crteron 213 r + n Fg. 1, the closed-loop response to the set-pont change s represented by ( ) 1 c y() s = H()() s r s = I+ G() s G () s G() s G ()(), s r s (1) where H(s) s the closed-loop transfer functon; G(s) s the process transfer functon whch s open-loop stable; G c () s s the mult-loop controller wth dagonal elements only; y(s) and r(s) are the controlled varable and the set-pont, respectvely. Suppose that the desred closed-loop response of the dagonal elements n the mult-loop system s gven by R ( ) [,,..., ]. (2) s = dag R1 R2 R n Accordng to the desgn strategy of the IMC controller [6], the desred closed-loop response R of the th loop s gven by y() s G + () s = R() s =, r( s) n (λ s + 1) _ Gc 1 0 0 0 Gc 2 0 0 0 Gcn G c G G G G G G G G G 11 12 1n 21 22 2n n1 n2 nn Fg. 1. Block dagram for mult-loop control system. c (3) where G + s the non-mnmum part of G and chosen to be the all pass form; λ s an adjustable constant for system performance and robustness; n s chosen for the IMC controller to be realzable. The mult-loop controller G c () s wth ntegral term can be expressed n a Maclaurn seres as 1 2 3 G c() s = c0 c1s c2s O( s ), s G + G + G + (4) where G c0, G c1,g c2 can be consdered as the ntegral, proportonal, and dervatve terms of the mult-loop PID controller, respectvely. As can be gleaned from (4), the mpact of the proportonal and dervatve terms (.e., G c1, G c2) domnates at hgh frequences and, thus, they should be desgned based on the process characterstcs at hgh frequences. On the other hand, the ntegral term G c0 domnates at low frequences and, thus, needs to be desgned based on the process characterstcs at low frequences. At hgh frequences, the magntude of the open loop G y gan becomes G( jω) G c ( jω) 1 and thus H(s) can be approxmated to H() s = ( I+ G() s G ()) s G() s G () s G() s G (). s (5) 1 c c c Ths ndcates that G c0 and G c1 can be desgned by consderng only the dagonal elements n G(s), whch means that the generalzed IMC-PID method for the SISO system [10] can be drectly appled to the desgn of the proportonal and dervatve terms n the mult-loop PID controller. Therefore, at hgh frequences, the deal mult-loop feedback controller gvng the desred closed-loop response R () s s desgned by 1 1 () = () ()( ()), (6) Gc s G s R s I R s where G ( s) = dag[ G11, G22,..., G nn ]. Accordngly, the deal mult-loop controller of the th loop can be desgned by G c 1 ( G ( s)) () s =, n ( λ s + 1) G ( s) + (7) where G - s the mnmum part of G. Snce G + (0)=1, (7) can be rewrtten n a Maclaurn seres wth an ntegral term as follows '' 1 ' f (0) 2 3 c G () s = ( f (0) + f (0) s+ s + 0( s )), (8) s 2 where f (s) = G c (s)s. The standard PID control algorthm s gven by G 1 c () s = K c (1 + s + τ D s ). (9) τ I Comparng (8) wth (9) gves the analytcal tunng rules for the proportonal gan and the dervatve tme constant of the mult-loop PID controller as follows K " ' f (0) = f (0) ; τ =. (10) 2K c D At low frequences, the nteracton effect between the control loops cannot be neglected. The expanson of G(s) n a Maclaurn seres gves 0 1 2 c 2 3 G() s = G + G s+ G s + O( s ), (11) where G0 = G(0); G1 = G'(0); G2 = G "(0) / 2. By substtutng (4) and (11) nto (1), one can obtan H(s) as 1 2 0 c0 H() s = I ( G G ) s+ O( s ). (12)
214 Truong Nguyen Luan Vu, Jetae Lee, and Moonyong Lee Furthermore, the desred closed-loop response R can also be wrtten n a Maclaurn seres as 2 ( s) = (0) + '(0) s+ O( s ), (13) R R R where R (0) = I because G + (0) = 1. By comparng the dagonal element of H(s) n (12) and R (s) n (13) for the frst-order s term, one can obtan the analytcal tunng rule for the ntegral tme constant of the mult-loop PID controller as follows ' ( G+ (0) nλ ) Kc τ I = 1 ( G (0)). (14) The tunng formulae by (10) and (14) provde an mportant advantage n solvng the optmzaton problem used for fndng the PID parameter values: for a gven process, all of the PID parameters can be expressed by a sngle desgn parameter, λ. 2.2. Weghted sum Mp crteron for mult-loop controller tunng The resonant peak Mp n a SISO system s defned as the maxmum magntude of the closed-loop frequency response (.e., the maxmum magntude of the complementary senstvty functon). The Mp s wdely accepted as a good ndex to address both the performance and robustness of a control system. An emprcal study showed that the value of Mp should le between 1.1 and 1.4 n SISO systems [11]. In a MIMO system, the closed-loop transfer functon conssts of the ndvdual closed-loop transfer functons as { } H () s = H j, (15) where H j represents the closed-loop transfer functon of the th loop to the set-pont change n the jth loop. The overall control performance depends not only on the closed-loop transfer functons assocated wth the dagonal elements, but also on those of the offdagonal elements n the closed-loop transfer functon matrx H. Thus, the Mp specfcaton for the dagonal elements does not guarantee good performance by tself. For well-balanced closed-loop performance, the off-dagonal H j should be close to zero as well as the dagonal H beng close to unty. To take these requrements nto account n the desgn of the controller, a weghted sum of the ndvdual Mp values s proposed for the objectve functon used to fnd the optmum PID parameters, as follows. mn (1 w) Mpj + w Mp λ j subject to Mp Mp, (16) low Fg. 2. Effect of Mp low and w on the IAE: WB column example. where Mp j s defned by max ω 0 Hj ( jωλ, ) ; Mp low s the lower bound of the dagonal Mp needed to secure the mnmum speed requred; w s the weghtng factor for the dagonal and the off-dagonal closed-loop responses and ranges from 0 to 1. Mp low and w can also be consdered as optmzng varables to mnmze the ntegral absolute error (IAE) of the closed-loop tme response. However, snce t appears that the effect of these parameters on the IAE s suffcently small over a wde range of values of Mp low and w, they can be fxed as constant values wthn the gven range. Fg. 2 shows an example of the mpact of Mp low and w on the overall performance n the WB column [14] case. As can be seen from ths fgure, the shape of the mnmum IAE surface s qute flat for a wde range of values of Mp low and w. Our extensve smulaton study shows that the desrable value of w les between 0.5 and 0.75 and that of Mp low s from 1.1 to 1.4 for well-balanced closed-loop performance. 3. CASE STUDIES To evaluate the performance of the proposed method, the closed-loop responses by the proposed method were compared wth those by several wellknown exstng desgn methods, such as the bggest log modulus tunng (BLT) method [4], the decentralzed λ tunng (DLT) method [12], and the sequental auto tunng (SAT) method [13]. Example 1: Consder the Wood and Berry (WB) column model for separatng methanol and water [14]. Gs () s 3s 12.8e 18.9e 16.7s+ 1 21s+ 1 6.6e 19.4e 10.9s+ 1 14.4s+ 1 = 7s 3s (17) In the smulaton study, step changes n the set-
Desgn of Mult-loop PID Controllers Based on the Generalzed IMC-PID Method wth Mp Crteron 215 Fg. 3. Closed-loop responses to sequental step changes n set-pont for WB column. pont were sequentally made n the ndvdual loops. n n (3) was chosen as 1 for all loops. The optmum λ values based on (16) were found to be 0.029 and 3.227 for loops 1 and 2, respectvely. All of the control parameters used n the example are lsted n Table 1, ncludng the IAE values of the closed-loop responses. Values of 1.2 and 0.75 were chosen for Mp low and w, respectvely. These Mp low and w values were also appled to all other examples n ths paper. Fg. 3 shows the closedloop responses obtaned usng the varous desgn methods. As seen n ths fgure, the proposed method provdes more well-balanced and faster responses n comparson to the other methods. The superorty of the proposed method s also llustrated by ts IAE values n Table 1. The IAE values for whch all of the parameters n the actual process are assumed to be changed by +10% and -10% are also lsted n Table 1. One can see that the controller obtaned by the proposed method provdes more robust performance than those obtaned usng the other methods. Example 2: Consder the Wardle and Wood (WW) column studed by Luyben [4]. Fg. 4. Closed-loop responses to sequental step changes n set-pont for WW column. -6 s -12 s 0.126 e -0.101 e 60s + 1 (48s + 1)(45s + 1) Gs () = -8 s -8 s 0.094 e -0.12 e 38s + 1 35s + 1 (18) The optmum λ values based on (16) were found to be 5.25 and 0.019 for loops 1 and 2, respectvely. All of the control parameters and the resultng IAE values are also lsted n Table 1. Fg. 4 compares the closedloop response obtaned usng the varous desgn methods. The superor performance of the proposed method s apparent from ths fgure and the IAE values lsted n Table 1. Example 3: Consder the Vnante and Luyben (VL) column studed by Luyben [4]. - s -0.3 s -2.2 e 1.3 e 7s + 1 7s + 1 G(s)= -1.8 s -0.35 s -2.8e 4.3e 9.5s + 1 9.2s + 1 (19) The λ values were calculated to be 0.388 and 0.051
216 Truong Nguyen Luan Vu, Jetae Lee, and Moonyong Lee Table 1. PID parameter and IAE values obtaned usng the varous methods. Process Proposed BLT DLT SAT WB K c 1.30 0.38 0.34 0.87-0.13-0.08-0.14-0.09 τ I 8.55 3.25 17.20 8.29 τ D 0.48-0.49-0.67-1.36 - IAE 16.55 57.99 32.72 23.31 IAE (+10%) 19.83 51.89 30.14 25.19 IAE (-10%) 16.62 61.26 36.07 25.43 WW K c 43.26 27.40 31.25 48.10-40.42-13.30-18.06-25.4 τ I 22.92 41.40 63.00 18.99 14.47 52.90 39.00 26.30 τ D 1.541-2.86-3.718-3.59 - IAE 66.26 212.16 146.97 114.39 IAE (+10%) 77.79 142.08 130.74 158.28 IAE (-10%) 60.89 239.39 165.04 94.78 VL K c -2.43-1.07-3.66-1.35 5.44 1.97 2.94 3.36 τ I 4.53 7.1 12.36 3.00 5.75 2.58 14.36 1.33 τ D 0.35-2.21-0.15-2.20 - IAE 4.79 8.61 14.47 7.18 IAE (+10%) 5.16 8.40 14.72 7.80 IAE (-10%) 4.53 9.03 14.27 6.62 IAE (+10%) and IAE (-10%) denote the IAE values under +10% and -10% parametrc uncertanty, respectvely. The values of the flter tme constant used n the DLT method are (0.38 for loop 1, 0.75 for loop 2), (1.88, 2.22), and (6.54, 6.72) for the WB, WW, and VL processes, respectvely. for loops 1 and 2, respectvely. From Fg. 5 and Table 1, t can be concluded that the proposed method provdes a superor response to that of the other methods. 4. CONCLUSIONS In ths paper, we proposed an effcent method of desgnng mult-loop PID controllers. The proposed method utlzes the one parameter tunng rule to Fg. 5. Closed-loop responses to sequental step changes n set-pont for VL column. effectvely reduce the dmenson of the search space used for fndng the optmum PID parameters. The generalzed IMC-PID method for mult-loop systems was used as the one parameter tunng rule. By usng the frequency-dependent property n the closed-loop nteractons, the analytcal tunng rule can be made to take the nteracton effect fully nto account n a smple but effcent manner. The weghted sum Mp crteron was proposed as a performance measure to cope wth both performance and robustness n multloop systems. The superorty of the proposed method n comparson wth several well-known exstng methods was demonstrated by provdng several llustrated examples. REFERENCES [1] Z. Y. Palmor and N. Krasney, Automatc tunng of decentralzed PID controllers for MIMO processes, Journal of Process Control, vol. 42, pp. 1174-1180, 1996. [2] S. Skogestad and M. Morar, Robust performance of decentralzed control systems by ndependent desgns, Automatca, vol. 25, no. 1,
Desgn of Mult-loop PID Controllers Based on the Generalzed IMC-PID Method wth Mp Crteron 217 pp. 119-125, 1989. [3] A. P. Loh, C. C. Hang, C. K. Quek, and V. N. Vasnan, Autotunng of multvarable proportonal-ntegral controllers usng relay feedback, Ind. Eng. Chem. Res., vol. 32, pp. 1102-1107, 1993. [4] W. L. Luyben, Smple method for tunng SISO controllers n multvarable systems, Ind. Eng. Chem. Process Des. Dev., vol. 25, pp. 654-660, 1986. [5] P. Grosdder and M. Morar, A computer aded methodology for the desgn of decentralzed controllers, Computers Chem. Eng., vol. 5, pp. 309-414, 1987. [6] C. G. Economou and M. Morar, Internal model control 6: Multloop desgn, Ind. Eng. Chem. Proc. Process. Des. Dev., vol. 25, pp. 411-419, 1986. [7] S. Skogestad and M. Morar, Robust performance of decentralzed control systems by ndependent desgn, Automatca, vol. 25, no. 1, pp. 119-125, 1989. [8] D. Y. Lee, M. Lee, Y. Lee, and S. Park, Mp crteron based multloop PID controllers tunng for desred closed-loop responses, Korean. J. Chem. Eng., vol. 20, no. 1, pp. 8-13, 2003. [9] M. Lee, K. Lee, C. Km, and J. Lee, Analytcal desgn of multloop PID controllers for desred closed-loop responses, AIChE Journal, vol. 50, no. 7, pp. 1631-1635, 2004. [10] Y. Lee, M. Lee, S. Park, and C. Broslow, PID controller tunng for desred closed loop responses for SI/SO systems, AIChE Journal, vol. 44, no. 1, pp. 106-115, 1998. [11] S. L. Harrs and D. A. Mellchamp, Controller tunng usng optmzaton to meet multple closed loop crtera, AIChE Journal, vol. 31, no. 3, pp. 484-487, 1985. [12] J. Jung, J. Y. Cho, and J. Lee, A decentralzed controller tunng method wth one desgn parameter of closed-loop tme constant, J. KIChE, vol. 37, no. 6, pp. 844-849, 1998. [13] A. P. Loh, C. C. Hang, C. K. Quek, and V. N Vasnan, Autotunng of multvarable proportonal-ntegral controllers usng relay feedback. Ind. Eng. Chem. Res., vol. 32, pp. 1002-1007, 1993. [14] R. K. Wood and M. W. Berry, Termnal composton control of a bnary dstllaton column, Chem. Eng. Sc., vol. 28, pp. 1707-1717, 1973. Truong Nguyen Luan Vu receved the B.S. degree n Mechancal Engneerng from HCM Cty Unversty of Technology, Vetnam n 2000 and the M.S. degree n the School of Chem. Eng. and Tech. at Yeungnam Unversty n 2005. He was a Lecturer at the Unversty of Techncal Educaton, HCM Cty, Vetnam from 2000 to 2003. He s currently workng on a Ph.D. degree n the School of Chem. Eng. and Tech. at Yeungnam Unversty, Korea. Hs research nterests nclude multvarable control systems, advanced process control and optmzaton. Jtae Lee receved the B.S. degree n Chemcal Engneerng from Seoul Natonal Unversty n 1979, and the M.S. and Ph.D. degrees n Chemcal Engneerng from KAIST n 1981 and 1986, respectvely. Snce 1983, he has been wth Kyungpook Natonal Unversty where he s currently a Professor n the Department of Chemcal Engneerng. Hs research nterests nclude process control and control hardware. Moonyong Lee receved the B.S. degree n Chemcal Engneerng from Seoul Natonal Unversty n 1982, and the M.S. and Ph.D. degrees n Chemcal Engneerng from KAIST n 1984 and 1991, respectvely. From 1984 to 1994, he worked n the refnery and petrochemcal plant of the SK Company, Korea. Snce 1994, he has been wth Yeungnam Unversty where he s currently a Professor n the School of Chem. Eng. and Tech. Hs research nterests nclude process control, onlne montorng, and process desgn.