Optimization-Based PI/PID Control for a Binary Distillation Column

Transcription

1 5 Amercan Control Conference June 8-1, 5. Portland, OR, USA FrA8. Optmzaton-Based PI/PID Control for a Bnary Dstllaton Column Jann-Shou Yang Abstract Ths paper presents the PI/PID control of a bnary dstllaton column va a genetc searchng algorthm (GSA). The tme-doman desgn crteron, expressed as an ntegral of the squared error, s reformulated n the frequency-doman usng the Parseval's relaton and Padé approxmaton. A genetc algorthm s then used to search over the stablty regon n the controller parameter space for the best settngs to mnmze the desgn crteron. Our results ndcate that the genetc algorthm can provde better solutons for the PI/PID control schemes as compared to those usng the sngle-loop and mult-loop Zegler-Nchols tunng methods. We found that the GSA s easer to use than tradtonal optmzaton technques. In addton, no nowledge of complex mathematcs s requred to use the GSA effectvely. I. INTRODUCTION THE dstllaton column s probably one of the most popular and mportant processes studed n chemcal engneerng lterature. Dstllaton s used n many chemcal processes for separatng feed streams and for purfcaton of fnal and ntermedate products. It s nown that hgh-purty dstllaton columns are hghly nonlnear and the composton nteracton between the stages due to the counter flow of vapor and lqud s also large (e.g., [1], []). Thus, the control of columns to gve multple products of constant composton s very dffcult. It s perhaps one of the most challengng problems n process control. Varous methods of controllng dstllaton columns have been reported n the lterature (e.g., nternal model control method [1], rato control [3], [4], non-nteractng control [3]-[5], µ-synthess method [6], lnear-quadratc Gaussan wth loop transfer recovery (LQG/LTR) method [7], fuzzy control [8]). Despte many advanced/modern control desgn technques developed n recent years, PI and PID controllers stll represent the majorty of the controllers used n ndustry. It s, therefore, nterestng to develop an approprate approach to adjust the parameters of the PI and PID controllers for the control of dstllaton columns. In ths paper, the PI/PID control of a bnary dstllaton column s studed. Although most columns handle Fnal manuscrpt submtted February, 5. The author s wth the Department of Electrcal and Computer Engneerng, Unversty of Mnnesota, Duluth, MN 5581 USA (emal: jyang@d.umn.edu). multcomponent feeds, many can be approxmated by bnary (or pseudo bnary) mxtures. However, due to the strong cross couplng and sgnfcant tme delays nherent n the dstllaton column, the smultaneous control of overhead and bottoms composton usng reflux and steam flow as the control varables s stll dffcult. Attemptng to use tunng rules such as the well-nown Zegler-Nchols rule to adjust the PI/PID controller for each ndvdual loop often leads to deterorated control performance for the overall system. Ths s because these tunng rules do not tae the nteracton between the control loops nto consderaton ([3], [9]). In ths paper, nstead of usng a tunng rule we focus on usng a genetc algorthm to determne the optmal PI/PID control settngs for a plot-scale bnary dstllaton column model. A genetc searchng algorthm (GSA) s chosen to test whether such algorthms offer any practcal advantages over tradtonal optmzaton algorthms. For the test example, we select the Wood and Berry bnary dstllaton model [4] because t was derved from an operatonal dstllaton column and s representatve of many chemcal control processes. Determnaton of the optmal gan settngs for the PI/PID controllers wth the Wood and Berry model serves as our optmzaton problem. A parameter space method [1], [11] s used to determne the stablty regon n the controller's parameter plane (space),.e., the search space for the GSA. Lmtng the search of the controller parameters to such a regon guarantees the system stablty. A standard ntegral of the squared error (ISE) s used as our control crteron, whch can be analytcally expressed n terms of the controller gans va the Parseval s Theorem. A second-order Padé approxmaton s also used to approxmate the tme delays nvolved n the model. Based on the crteron, a GSA s mplemented and used to search over the stablty regon for the controller gans that acheved the best performance. The smulaton results are gven, and they are also compared wth those usng the sngle/mult-loop Zegler- Nchols tunng methods. We found that the GSA s easer to use than tradtonal optmzaton technques. II. THE BINARY DISTILLATION COLUMN The bnary dstllaton column we studed bascally separates a mxture of two components havng dfferent /5/$5. 5 AACC 365

2 bolng ponts. Moreover, the process s greatly enhanced by forcng separaton to occur n stages wthn the column. One way to accomplsh ths s by drectng a lqud stream of hgh purty dstllate, or reflux, bac nto the column through an arrangement of seve trays. For example, consder a feed mxture consstng of component A wth a low bolng pont and component B wth a hgher bolng pont. As feed enters the column, some of the mxture wll mmedately vaporze. The remander wll concentrate at the bottom of the column where further bolng s nduced by the re-boler. Consequently, a vapor stream rch n component A s created and rses n opposton to the lqud stream. The seve tray arrangement creates pocets n whch the vapor stream s contact wth the lqud stream s maxmzed. Ths ncreases the lelhood that the lqud stream wll entran the resdual B component from the vapor stream and drect t bac toward the bottom of the column. Most mportantly, the seve tray arrangement provdes the mechansm needed for mult-staged bolng. Each seve tray can be vewed as a separate boler of ncreased effcency as the vapor stream rses toward the top of the column. Whle the benefts of reflux to the column s effcency are certanly to be desred, the resultng dynamcs greatly complcates the smultaneous control of the bottoms and overhead product compostons. Consder the case n whch t s desred to ncrease the overhead product purty by ncreasng reflux flow wth steam flow held constant. Increasng reflux wll certanly remove more of the resdual component B from the vapor stream, but t wll also ncrease the amount of the component A n the bottoms product. It can be seen that there s a dependent relatonshp between steam flow and reflux flow that affects both product compostons. Ths dependence s also apparent n the mathematcal model of the dstllaton column, whch can be expressed as x () s () () () D P s P s 11 1 R s x () s P () s P () s Ss () (1) B 1 where the outputs x D (s) and x B (s) are the overhead and bottom product mole fracton of methanol, respectvely, whle R(s) s the reflux flow rate and S(s) s the re-boler steam flow rate. The transfer functons P j (s) = Ke s / s1 (, j =1, ) have the parameter values shown below: Parameters K P 11 (s) P 1 (s) P 1 (s) P (s) Note that R. K. Wood and M. W. Berry developed the above mathematcal model usng a plot scale replca of an actual bnary dstllaton column [4]. Obvously, the control of the dstllaton column transfer matrx P(s) = [P j (s)] s complcated by the dependent relatonshp that exsts between the steam flow and reflux flow. The nteracton between the two loops maes controller tunng more dffcult. To compensate for undesrable control loop nteractons, consder the bloc dagram of the decouplng scheme [pp , 3] shown n Fg. 1, where D 1 (s) and D 1 (s) are the decouplers used for the decouplng purpose, whle the feedbac controllers C 1 (s) and C (s) are to be desgned to provde a satsfactory set pont tracng for both channels. From Fg. 1, t s easy to see that n order to elmnate the effect of overhead and bottoms composton control actons on the bottoms and overhead composton, respectvely; the two decouplers should be chosen as P1() s P1() s D1() s ; D1() s () P() s P11() s Note that alternatve decouplng technques are possble (e.g., [], [3]). In ths paper, we use the decouplng scheme shown n Fg. 1 before attemptng to desgn feedbac controllers. Usng the two decouplers gven n Eq. (), the system equaton becomes x () s T () s u () s D 11 1 xb () s T() s u() s wheret 11 (s)=p 11 (s)p 1 (s)d 1 (s), T (s)=p (s)p 1 (s)d 1 (s). Note that n realty, the exact cancellatons of couplng can seldom be realzed because of the modelng error and varous uncertantes n the process. However, Seborg et al. clamed that even approxmate cancellatons could be very benefcal n reducng control loop nteractons whle smplfyng the controllers [3]. Besdes that, the approxmated deal decouplers should also wor fne except for hgh-purty dstllaton columns wth large relatve gans [1]. Therefore, we wll use the decouplng scheme descrbed above. Snce the purpose of ths wor s to study the feasblty and advantages of usng a genetc algorthm to tune the controller parameters, a perfect decouplng s assumed so that we can compare the avalable smulaton results wth some other methods dscussed later. After decouplng, we then close the loops and try to desgn the controllers C 1 (s) and C (s) to control x D (s) and x B (s) ndependently. Our objectve s to desgn the PI/PID controllers to mnmze the ntegral of the squared error (ISE) n both channels. That s, our tracng performance s formulated as T e () t e() t dt (4) mn { C1( s), C( s)} (3) 3651

3 where the vector e(t) s defned as e(t)=[e 1 (t) e (t)] T (the a superscrpt T means the transpose) and e (t) represents the tracng error n channel ( =1,). a a1 a a4 a3 a a1 a III. DESIGN CRITERION REFORMULATION a6 a5 a4 a3 a a1 a a After the system s decoupled, the ISE crteron n each 8 a7 a6 a5 a4 a3 a a1 loop can be rewrtten as a8 a7 a6 a5 a4 a3 a8 a7 a6 a J 5 mn ( ) (5) { C ( s)} e tdt a8 a7 From Fg. 1, we also have e (s) = r (s)/(1t (s)c (s)) where T (s) represents the decoupled plant transfer functon n channel ( =1, ). By usng the Parseval s Theorem [13], Eq. (5) can be expressed n the frequency-doman as 1 ISE e t dt e s e s ds j () ( ) () j (6) j 1 and e (-s)e (s) can be further expressed as r( s) r( s) B( s) B( s) e( s) e( s) (1 T ( s) C ( s))(1 T ( s) C ( s)) A( s) A( s) (7) where B(s) and A(s) are Hurwtz polynomals n s. It s clear that the presence of tme delays n T (s) prevents us from obtanng the last expresson n Eq. (7). That s, we cannot fnd the polynomals A(s) and B(s). To overcome ths problem, we can approxmate the tme delay usng a nd -order Padé approxmaton [3], [14] gven below e s s s 1 1 s s 1 1 to replace all the tme delays nvolved n T (s). After such an approxmaton, we can then easly obtan B(s) = 7 8 and A(s) = bs as, where the coeffcents a, b j are functons of the PI/PID gans. Note that the order of the polynomal A(s) s 6 f a frst-order Padé approxmaton s used, and ts order becomes 1 for a thrd-order approxmaton. Usng [15] wth some dervatons, the crteron Eq. (6) can be further expressed as ISE = (-1/ a 8 ) ( 1 / ), where the notaton means the determnant and the matrces 1, are defned, respectvely, as (8) a d a a1 a d 1 a4 a3 a a1 a d a6 a5 a4 a3 a a1 a d3 a8 a7 a6 a5 a4 a3 a d4 a8 a7 a6 a5 a4 d5 a8 a7 a6 d 6 a8 d7 (9) where d b, d b 1 1 b b, dbbb 1 3bb 4, d6b6 bb 57 d3b3 bb 4bb 1 5bb 6, d4 b4 bb 3 5bb 6bb 1 7 d b d b bb bb By substtutng the above two nto - 1 / (a 8 ), we obtan the ISE, whch s a functon of a, b j and these coeffcents are agan functons of the PI gans K p and K, (or the PID gans K p, K, and K d ). Therefore, our objectve s to use a genetc algorthm to search over the set of PI/PID gans n the controller s parameter plane (space) that mantans the system stablty and, at the same tme, mnmzes the ISE crteron. III. THE STABILITY REGION Due to the popularty and wde acceptance of PID control among process ndustres, we use ths control for the dstllaton column. Consder the system shown n Fg. 1 wth the controllers C 1 (s)= P1 I1 /s D1 s and C (s)= P I /s D s where P, I, and D (=1,) are the proportonal gan, ntegral gan, and dervatve gan, respectvely. To acheve the optmal tracng performance (.e., Eq. (4)), we wll fnd the optmal ( P, I, I ) (=1,) by usng a genetc searchng algorthm. Before performng the search, we use the Slja's parameter plane method to 365

4 fnd the set of all PID gans that mantan the system stablty. That s, for each loop of the decoupled system (.e., Eq. (3)), we determne the stablty regon n the P I D -space after nsertng the PID controller n that loop. The search va our genetc algorthm wll then be lmted to the stablty regon for optmalty of the error crteron. For detals about the method to get the stablty regon n the controller parameter space, please refer to [1,11]. IV. THE GENETIC SEARCHING ALGORITHM A genetc algorthm s a functon optmzaton technque based on the deas of evolutonary genetcs and the natural selecton process [16], [17]. Ideally, the algorthm creates new populaton members (chldren) who, wth each successve generaton, are better equpped to succeed n ther present envronment (or, have a hgher degree of ftness). In terms of functon optmzaton, ths process equates to ncorporatng varous weghted operators to randomly select values for the ndependent varable(s) that have a hgh probablty of producng successvely hgher values of the dependent varable untl some global optmum s reached. The mplementaton of a basc genetc algorthm (GA) can be found n the lterature. Our study focuses on usng a genetc approach to determne the optmal settngs for the classcal PID controller. A genetc searchng algorthm s chosen to test whether such algorthms offer any practcal advantages over tradtonal optmzaton algorthms. Desrable propertes of the GSA nclude: (1) No relance on ntegral or dervatve operators to steer ts operaton. Ths allows the GSA to search for a functon s global mnmum or maxmum wthout regard to the contnuty of the functon, () Many ponts wthn the search space are evaluated each generaton. Ths reduces the probablty that the algorthm wll converge to some local mnmum or maxmum wthn the search space, (3) No dependence on the selecton of a startng pont wthn the search space to ntalze the search process, and (4) Ease of use. The GSA does not requre detaled nowledge of complex optmzaton technques. In addton to the basc genetc operators, we choose to ncorporate two ntutve operators of our own. The frst s desgned to elmnate an adverse sde effect of eltsm, that s, the general populaton members are excluded from matng wth the elte populaton set. After some expermentaton, we found that eltsm leads to the average ftness levels of both populatons reachng plateau ftness values for many generatons. Ths s mnmzed by nsertng a copy of the ftness elte populaton member n place of the least ft general populaton member pror to selecton of offsprng producng parents. The second operator s desgned to force the GSA to focus a small amount of ts attenton on the search space regon mmedately surroundng the coordnates of the best elte populaton member for a partcular generaton. It maes sense that f a good soluton s obtaned at a partcular pont that perhaps a better soluton exsts close by. Rather than relyng solely on the crossover and mutaton operators to reach ths concluson randomly, t s much more effcent to strategcally manpulate the lower order bt postons of the best elte populaton member chromosome. Ths manpulaton produces a small number of chldren whose search space coordnates are very close to the best soluton obtaned for the prevous generaton. V. RESULTS ANALYSIS 1. Tunng Based on the GSA Based on the mplemented GSA descrbed n Secton IV, the algorthm searched for the optmal ( P,, d ) n the stablty regon to maxmze the ftness functon. Ths s appled to each loop separately. Results were obtaned after performng fve consecutve searches for each control loop wth the number of generatons set at 5 and populaton sze set at 1. For the PI control scheme (.e., d = ), we found that the GSA has no dffculty convergng to the same soluton durng separate runs of the algorthm. However, the GSA showed less repeatablty for the PID control scheme, gvng three possble solutons for T 11 and two possble solutons for T. Despte the loss of repeatablty, each of these PID settngs yelds better performance than optmally tuned PI controllers as expected. The results we tested can be summarzed as follows: PI Controller Results Parameters Kp K ISE T 11 - loop T - loop PID Controller Results Example 1 Parameters Kp K Kd ISE T 11 - loop T - loop Example Parameters Kp K Kd ISE T 11 - loop T - loop Example 3 Parameters Kp K Kd ISE T 11 - loop T - loop N/A N/A N/A N/A Error response plots for the best gan settng obtaned for each control scheme are shown n Fg. and Fg. 3.. Comparson Study 3653

5 An accurate comparson of the GSA solutons to results obtaned by others s dffcult for a number of reasons. There are a number of crtera that can be used to evaluate a controller s performance. Gan settngs that provde optmal performance for the ISE crteron may not provde the same performance for some other crteron. Dfferences n test methods also mae comparson dffcult. For example, Wood and Berry evaluated the performance of ther PI controllers emprcally usng a dfferent performance crteron [4]. A rough comparson of the GSA PI controller results to the sngle-loop Zegler- Nchols method (sngle-loop/z-n) and mult-loop Zegler- Nchols method (mult-loop/z-n) [3], usng the same ISE crteron, s shown n the followng table. The results are also shown n Fgs. 4 and 5. These comparsons show that our results seem better. Tunng Method ISE (T 11 loop) ISE (T loop) GSA Sngle-loop/Z-N Mult-loop/Z-N VI. CONCLUSION PI and PID control of the Wood and Berry dstllaton column va a genetc searchng algorthm has been presented n ths study. The x multvarable plant s frst decoupled by two decouplers so that the system can be converted nto two ndependent SISO subsystems. We use a PI/PID controller to control each loop separately. The tme-doman tracng error crteron (ISE) s reformulated by usng Parseval s theorem, and the set of the controller gans to mantan each subsystem's stablty s also dentfed n the parameter plane (space). We then search for the optmal PI/PID gans n the stablty regon to mnmze the ISE crteron. A genetc algorthm was mplemented and used n the searchng process. The smulatons show small tracng errors for both channels. Our results also gve a better performance when compared wth the sngle-loop/z-n and multloop/z-n approaches. The proposed method can be easly used to control systems where the controller structure s fxed wth several adjustable parameters to be determned. performance usng PID controllers tuned wth the GSA, we are not able to guarantee that these are the best possble gan settngs. REFERENCES [1] M. Morar and E. Zafrou, Robust Process Control, Prentce Hall, Englewood Clffs, NJ, [] W. L. Luyben, Process Modelng, Smulaton, and Control for Chemcal Engneers, nd ed., McGraw-Hll, New Yor, NY, 199. [3] D. Seborg, T. F. Edgar, and D. A. Mellchamp, Process Dynamcs and Control, John Wley & Sons, New Yor, NY, [4] R. K. Wood and M. W. Berry, Termnal composton control of a bnary dstllaton column, Chemcal Engneerng Scence, vol. 8, pp , [5] W. H. Ray, Advanced Process Control, McGraw-Hll, New Yor, NY, [6] H. E. Musch and M. Stener, Robust PID control for an ndustral dstllaton column, IEEE Control Systems, pp , August [7] C. Zhou, J. R. Whteley, E. A. Msawa, and K. A. M. Gasem, Applcaton of enhanced LQG/LTR for dstllaton control, IEEE Control Systems, pp , August, [8] R. Stenz and U. Kuhn, Automaton of a batch dstllaton column usng fuzzy and conventonal control, IEEE Trans. on Control Systems Technology, vol. 3, no., pp , [9] K. J. Astrom and B. Wttenmar, Adaptve Control, Addson Wesley, Readng, MA, [1] D. D. Slja, Analyss and synthess of feedbac control systems n the parameter plane, part I- lnear contnuous systems, IEEE Trans. Applcatons Industry, vol. 83, pp , [11] D. D. Slja, Generalzaton of the parameter plane method, IEEE Trans. Automatc Control, vol. 11, pp. 63-7, [1] T. McAvoy, Interacton Analyss Theory and Applcaton, Instrum. Soc. of Amerca, Research Trangle Par, NC, [13] A. V. Oppenhem and A. S. Wllsy, Sgnals & Systems, nd ed., Prentce Hall, Englewood Clffs, NJ, [14] G. F. Franln, J. D. Powell, and A. Emam-Naen, Feedbac Control of Dynamc Systems, 4 th ed., Prentce Hall, NJ,. [15] E. I. Jury and A. G. Dewey, A general formulaton of the total square ntegrals for contnuous systems, IEEE Trans. Automatc Control, vol. 1, pp , [16] D. E. Goldberg, Genetc Algorthms n Search, Optmzaton and Machne Learnng, Addson Wesley, Readng, MA, [17] L. Chambers, Practcal Handboo of Genetc Algorthms - New Fronters, vol., CRC Press, Boca Raton, FL, r 1 C 1 U 1 R P 11 X D We are able to show that the GSA s certanly capable of tunng PI and PID controllers for ths partcular applcaton. Our GSA gves good results and s very easy to use. Once stablty regons and a performance crteron are establshed, the operator needs only to ntalze the GSA parameters and perform the search. No nowledge of complex mathematcs s requred to use the GSA effectvely. However, we dd not fnd enough evdence to prove or dsprove the hypothess that GSAs provde superor search results when compared to other optmzaton technques due to the lmted search space n ths example. Whle we are able to obtan better 3654 r - D 1 D 1 P 1 P 1 C U S P Fg. 1 Bloc dagram of the model showng the controllers and decouplers. X B

6 Fg PI versus PID control for overhead loop (PI: curve labeled 1; PID: curve labeled ). Fg. 4 Performance comparson of PI tunng for overhead (GSA-1; Sngle-loop/Z-N: ; Mult-loop/Z-N: 3). Fg. 3 PI versus PID control for bottom loop (PI: curve labeled 1; PID: curve labeled ). Fg. 5 Performance comparson of PI tunng for bottoms loop (GSA-1; Sngle-loop/Z-N: ; Mult-loop/Z-N: 3). 3655