A centralized/decentralized control approach for periodic systems with application to chromatographic separation processes

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1 Preprnt, 11t IFAC Symposum on Dynamcs and Control of Process Systems, ncludng Bosystems June 6-8, 16. NTNU, Trondem, Norway A centralzed/decentralzed control approac for perodc systems wt applcaton to cromatograpc separaton processes Mara M. Papatanasou*, **. Muxn Sun*, **. Rcard Oberdeck*, ** Atanasos Mantalars*. Efstratos N. Pstkopoulos** * Dept. of Cemcal Engneerng, Centre for Process Systems Engneerng (CPSE), Imperal College London SW7 AZ, London, U.K ** Arte McFerrn Department of Cemcal Engneerng, Texas A&M Unversty, College Staton TX Abstract: In ts work we demonstrate te applcaton of our recently presented PAROC framework and software platform for te development of a centralzed/decentralzed control strategy for te twn-column Multcolumn Solvent Gradent Purfcaton Process (MCSGP) based on mult-parametrc control polces. Te derved controllers are tested under te presence of dsturbances, wle ter stablty s also assessed. Te proposed control approac s evaluated under an n-slco, closed-loop fason aganst te process model. Te desgned controller captures effcently te process setponts, assurng optmal and stable operaton durng contnuous mode. Keywords: mult-parametrc MPC, control, perodc systems, stablty, contnuous processng 1. INTRODUCTION Te optmzaton and control of perodc systems as been dscussed extensvely n te open lterature, resultng to te development of dfferent strateges and metodologes (Nlcan and Panteldes, 1998, Klatt et al.,, Kawajr and Begler, 6, Grossmann et al., 1, Kajura and Pstkopoulos, 11, Kajura and Pstkopoulos, 13). Suc systems are usually assocated wt gly nonlnear models of large equaton sets, te smulaton of wc requres ncreased computatonal force. In partcular systems tat consder addtonal domans (e.g. spatal) requre specal andlng tat nvolves doman dscretzaton. Te latter can lead to an excessve number of equatons, callengng optmzaton and control studes. Moreover, ter perodc nature poses an extra callenge to te desgn of a olstc control concept and te controller stablty, as tese systems are caracterzed by cyclc steady state (CSS) (Nlcan and Panteldes, 1998, Huang et al., 11, Huang et al., 1). In order for te latter to be establsed, a cyclc nput profle s requred tat wll drve te system to te desred state. Te desgn and soluton of control problems lnked to suc systems, nvolve computatonally demandng smulatons tat are often mpossble to perform onlne usng current solvers. However, te ablty of mult-parametrc Model Predctve Control (mp-mpc) to solve te optmzaton problem offlne decreases sgnfcantly te computatonal effort requred onlne and allows te soluton of more complex problems (Pstkopoulos et al.,, Pstkopoulos et al., 15). In ts work, we present an advanced control strategy of an ndustral cromatograpc separaton process, te Multcolumn Countercurrent Purfcaton Process (MCSGP). MCSGP s a sem-contnuous, perodc, cromatograpc separaton process, used for te purfcaton of varous bomolecules (Aumann and Morbdell, 7). Te control of MCSGP as been vsted n several works (Grossmann et al., 1, Krättl et al., 11, Krättl et al., 13), owever, ere we present a centralzed/decentralzed approac trackng te output ntegral leadng to perodc nput profles and ensurng CSS. Te desgn, soluton and testng of te developed controllers s realzed troug te applcaton of te PAROC framework and software platform (Pstkopoulos et al., 15)..1 Te Process. THEORETICAL BACKGROUND MCSGP s an ndustral, on-excange, cromatograpc separaton process used for te purfcaton of varous bomolecules. Te process separates te mxture n weak mpurtes (W), product (P) and strong mpurtes (S). Te setup comprses two, dentcal (e. same n pyscal propertes and tecncal caracterstcs), on-excange, cromatograpc columns, operatng n countercurrent mode (Fgure 1). Te columns operate n a sem-contnuous fason, alternatng between batc (B-) and nterconnected (I- ) state. At te begnnng of te I1 pase, column starts empty and equlbrated. Durng ts step, te outlet flow of column 1 enters column mxed wt an addtonal fracton of adsorbng eluent (E). Ts elps te recyclng of te mpure fracton of te weak mpurtes and te product. After te completon of I1, te two columns enter B1 pase, were te feed (F) s ntroduced to column and te product s eluted from column 1. In I pase te recyclng stream contanng te mpure fracton of product and strong Copyrgt 16 IFAC 159

2 IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondem, Norway mpurtes exts column 1 and enters column. By te end of I pase, column starts elutng pure W (B pase). B pase fnses wen te overlappng regon of W and P reac te end of column. At ts pont te two columns swtc postons. Terefore, column 1 wll go troug te recyclng and feedng tasks as descrbed above, wle column wll contnue wt te gradent eluton (Aumann and Morbdell, 7, Müller-Spät et al., 8). Te development of advanced optmzaton and control polces for ts process s manly focused on te maxmzaton of product purty and recovery yeld. Fg. 1. Te twn-column MCSGP (Krättl et al., 13).. Te PAROC Framework and Software Platform For te development of te mp-mpc controllers, we follow te PAROC framework and software platform tat can be seamlessly appled to a varety of process systems for te development of advanced mp-mpc controllers. In partcular, te framework allows n-slco valdaton of te desgned controllers aganst te orgnal process model, easng ter performance evaluaton. Fg. llustrates te steps of te framework tat consder: () process model development, () model approxmaton/reducton, () formulaton and soluton of te mp-programmng problem and (v) te n-slco, closed-loop valdaton. For more detals regardng te PAROC framework/software platform te reader s referred to Pstkopoulos et al. (15)..3 Decentralzed control approac durng batc operaton- Sngle-column representaton In our prevously presented work (Papatanasou et al., 15) we develop of a decentralzed control strategy for te batc operaton of te system. In partcular, we decompose te system nto two dentcal sub-systems, for wc te mp- MPC controllers are desgned and tested. Gven te nput pletora (modfer concentraton, flow rate, feed stream) te system s examned under a random nput strategy and te mpact of te nputs on te outputs s evaluated. As a result, te flow rate s excluded from te nput set as t as no sgnfcant mpact on te values of te outputs. Addtonally, te composton of te stream s uncontrollable as t results drectly from te upstream processng and s terefore treated as measured dsturbance wt expermentally defned varaton range. Subsequently, te system consders te modfer as te sole and most sgnfcant nput, te composton of te feed stream as measured dsturbance and te outlet concentratons of te column components as outputs. As mentoned above, te objectve of te separaton s to obtan gly pure product at a suffcently g recovery rate. Grossmann et al. (1) suggested te calculaton of purty and recovery yeld usng te average concentratons over te process cycle (see (1) and ()). Pur, avg cavp s j cavw s j cavp s j c,,, avss j (1) Yj, cavps j cfeedp () were Pur avg and Y j correspond to te average purty and recovery yeld over a process cycle, c avs, j s te average concentraton of te mxture components at te end of eac cycle, c feed p ndcates te feed concentraton of te targeted product, j ndcates te cycle ndex and s te outlet stream. Gven te above presented equatons and amng to derve a control strategy tat wll allow contnuous process montorng, we track te ntegral of te outlet concentratons of te mxture components. In ts fason we can also seamlessly post-calculate purty and yeld tat defne te effcency of te process. Addtonally, snce te ntegral s a contnuous functon of tme t allows tgter process montorng n comparson to cycle-to-cycle control tat montors dscrete outputs..4 Twn-column control concept Fg.. Te PAROC Framework (Pstkopoulos et al., 15). In our prevous work (Papatanasou et al., 15) we focused on te development of decentralzed mp-mpc polces tat are able to montor te two columns ndependently, durng batc operaton (B- pases). In ts work, we extend our metodology to te development of advanced centralzed/decentralzed control strateges tat wll effectvely ensure optmal operaton under contnuous mode (nterconnected (I-) pases). Te decentralzed control strategy presented n Secton.3 s terefore extended for te development of a olstc control concept, were te system 16

3 IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondem, Norway s montored durng contnuous operaton. In ts case, te system comprses two cromatograpc columns, connected n seres (Fg.3). Modfer Feed (W, P, S) C W, C P, C S Modfer C W, C P, C S Fg. 3. Interconnected system based on te prncples of te MCSGP process (two 1x3 systems consderng dsturbances). Te system consders te modfer concentraton as nput, te feed composton as dsturbance and te outlet concentratons of weak mpurtes, product and strong mpurtes as outputs (C W, C P, C S respectvely). Eac column corresponds to a 1x3 sngle nput-multple output (SIMO) confguraton, consderng te modfer concentraton as nput and te outlet concentratons of weak mpurtes, product and strong mpurtes as outputs (C W, C P, C S respectvely). Measured dsturbances are also taken nto consderaton (feed composton). Fg. 4 llustrates te control sceme followed n ts work, were te two columns are connected usng te outlet of column 1 as dsturbance for column. Ts centralzed/decentralzed control sceme consders te nterconnecton of two 1x3 SIMO controllers (as presented n Secton 3). Fg. 4. mp-mpc control sceme for te two-column system. More specfcally, Controller 1 consders te modfer concentraton as nput, te feed composton as measured dsturbance and te ntegrals of te concentratons of te mxture components at te outlet of column 1 as outputs. Smlarly, Controller s operated usng te modfer concentraton njected at te connecton pont as nput, te outlet of column 1 as measured dsturbance and ntegrals of te concentratons of te mxture components at te outlet of column as outputs. calculaton of lqud and te sold pase concentraton respectvely. In total t ncludes 35 parameters, 33 dfferental and algebrac varables and 4119 dfferental and algebrac equatons after spatal dscretzaton usng 5 collocaton ponts (for a more detaled vew on te model te reader s referred to Müller-Spät et al., (8)). Te model was smulated n gproms ModelBulder (v. 4.1.) (Pse, ). c( t D (1 q z t ) (, ) t q( t ax k ( q c(, * z ( c(, Q c( A z col ) q( were c, q and q* correspond to te lqud, sold and equlbrum sold pase concentraton respectvely. A col s te column cross secton, ε s te column porosty, k refers to te lumped mass transfer coeffcent of speces, wle t and z are te tme and space coordnates. Te subscrpts, correspond to te speces and column ndces. 3. Model reducton/system dentfcaton Te orgnal PDAE process model (Müller-Spät et al., 8) s caracterzed by gly nonlnear, complex equatons tat render control studes dffcult or even mpossble to perform (Rvott et al., 1, Lambert et al., 13). We proceed terefore to a system dentfcaton step, were te system s excted wtn te range of nterest and ts beavour s montored. We consder one of te two columns and we apply a pulse nput strategy wtn te operatonal bounds and te effect on te outputs s examned. Te beavour of te oter column wll be dentcal wt tme dfference of alf cycle. Here we coose te applcaton of a pulse strategy over multsne nputs, as t resembles te nput profle followed n te orgnal process. Te results are ten stored n MATLAB, were usng te dentfcaton toolbox we desgn te lnear state space model wt samplng tme of 6 seconds. Te formulaton and te matrces for te model are presented below. x t Ts Axt But Ddt t y Cx t ) (4) (5) (3) 3. APPLICATION OF PAROC ON THE MCSGP 3.1 Te g-fdelty process model Te matematcal model descrbng MCSGP conssts of Partal Dfferental and Algebrac Equatons (PDAEs) and nvolves gly nonlnear terms. Te model s based on lumped knetcs and s usng Eq. (3) and (4) for te were x, u, y are te states, nputs (modfer concentraton) and outputs ( C W, C P, C S ) respectvely, t corresponds to te tme, Ts s te sample tme and A, B, C, D represent te matrces of te state space model. Te state space model was valdated aganst te matematcal, process model and resulted nto: 94.88%, 94.93% and 93.6% ft for te tree 161

4 Concentraton of te strong mpurtes extng column 1 (mg/ml) Concentraton of te product extng column 1 (mg/ml) Concentraton of te weak mpurtes extng column 1 (mg/ml) Concentraton of te strong mpurtes n te feed stream (mg/ml) Concentraton of te product n te feed stream (mg/ml) Concentraton of te weak mpurtes n te feed stream (mg/ml) IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondem, Norway outputs respectvely. Te model as also been cross-valdated under dfferent sets of nputs furter confgurng satsfactory ft of te montored outputs (Pstkopoulos et al., 15, Papatanasou et al., 15). Te state space model for te second column s te same wt te one presented above, as te columns are governed by te same pyscocemcal caracterstcs and prncples. 3.3 Desgn of te mult-parametrc controllers & stablty study Te presented state space model s used for te formulaton of te mult-parametrc programmng problem, usng te tunng parameters sown n Table 1. It sould be noted tat te parameters are te same for bot Controller 1 and Controller. Terefore te two controllers sare te same soluton map. Te problem s solved for varous output and/or control orzon lengts (usng POP toolbox n MATLAB ), owever no sgnfcant cange s observed n te system response wen tested under closed-loop valdaton. Te soluton of te problem sown below s composed by a total of 9 crtcal regons. Table 1. mp-mpc settngs of te SIMO controller (were I s te dentty matrx) Desgn Value Parameters Output 4 Horzon Control Horzon Q 1 QR [1 1 1] R 1 calculated usng te P Rccat equaton Te ssue of stablty s crucal for te adequate desgn of any controller. Based on te works by Mayne et al. () and Huang et al. (11), we ave sown tat our controller fulfls te necessary and suffcent condtons for stablty. Tus, te derved controller s able to reac cyclc steady state. 3.4 Closed-loop valdaton of te controller Te desgned controller s valdated n-slco aganst te orgnal process model and te output profles are montored. Te closed-loop valdaton s performed usng te go:matlab nterface n gproms ModelBulder (v. 4.1.) (Pse, ). For te valdaton experments we assgn ±1% varaton range for te dsturbances appled on Controller 1. In partcular, Fg.5 llustrates te dsturbance profle as mposed on Controller 1. Te strategy follows te exact same pattern followed n te orgnal process (Fg.1), were te feedng s appled once per cycle/per column Tme (mnutes) Fg. 5. Strategy followed for te dsturbances appled on Controller 1 consderng te concentratons of: weak mpurtes, product and (c) strong mpurtes n te feed stream Tme (mnutes) Fg. 6. Strategy followed for te dsturbances appled on Controller consderng te concentratons of: weak mpurtes, product and (c) strong mpurtes at te ext of column1. Smlarly, Fg.6 demonstrates te dsturbance profle appled on Controller. Contrary to Fg.5, te profle s not assgned by te user. It depcts te outlet concentratons of te mxture components at te ext of column 1 tat enter column and are treated as measured dsturbances by Controller as ter varaton range (±1%) can be pre-calculated (see Fg.3 and (c) (c) 16

5 Integral of te outlet concentratons (mg/ml) (Column ) Integral of te outlet concentratons (mg/ml) (Column 1) Modfer Concentraton (mg/ml) Column Modfer Concentraton (mg/ml) Column 1 IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondem, Norway 4) Tme (mnutes) Fg. 7. Input strategy for column 1 and column as ndcated by Controller 1 and Controller respectvely, under closed-loop valdaton Weak mpurtes Product Strong mpurtes Setpont data5 data6 Weak mpurtes Product Strong mpurtes Setpont data5 data Tme (mnutes) Fg. 8. Output beavour aganst te pre-defned setponts for column 1 and column as ndcated by Controller 1 and Controller respectvely, under closed-loop valdaton. Usng te dsturbance profles llustrated n Fg. 5 and 6 we smulate te controllers aganst te g-fdelty process model and we montor te system response. Wle n operaton, te controllers return te nput profles sown n Fg.7. More specfcally, Fg.7a corresponds to te nput (modfer concentraton) appled at te nlet of column 1, wle Fg.7b llustrates te modfer concentraton njected n te system at te connecton pont (see Fg.4). It s observed tat te controllers ndcate nput profles of perodc nature (Fg. 7) tat can effectvely lead to te acevement of cyclc steady state. It sould be noted tat te nput of Controller s dependent on te modfer concentraton tat exts column 1 and enters column, leadng to dfferent nput profles. For example at te 1 t mnute, t s observed tat Controller ncreases te modfer concentraton and mantans t at a g level for approxmately 6 mnutes. Ts s drectly related to te setpont canges appled to Controller for te respectve perod (Fg.8b). In ts case, te modfer concentraton enterng column from column 1 s not adequate and terefore results to addtonal modfer feedng. Fg.8 depcts te comparson between te assgned setponts and te montored outputs (ntegrals of te outlet concentratons of W, P and S) for bot Controller 1 (Fg.8a) and Controller (Fg.8b). It can be observed tat bot controllers manage to drve te system to te desred pont wtout sgnfcant offset (<%). However, partcularly n te case of Controller 1 (Fg.8a) we observe a repettve tme delay n all outputs. In te case of column (Fg. 8b) we observe delay n te extracton of te product and te strong mpurtes, wle te weak mpurtes are caracterzed by early eluton. Ts beavour s n accordance wt te governng laws of on-excange cromatograpy, as te weak mpurtes are te frst to be extracted, followed by te product and lastly by te strong mpurtes (Guocon et al., 1994, Strölen et al., 6, Guocon, 1). Te penomenon of delayed and/or early eluton s a pyscal caracterstc of te system and s nerently correlated to te speed wt wc te components travel troug te column lengt and terefore te flow rate value. In ts work, we mantan te flow rate at a nomnal value for bot columns and terefore te observed tme dfferences can be precomputed. Te latter wll lead to sfted setponts and eventually elmnaton of te delay. 6. CONCLUSIONS Here we demonstrate a detaled study for te development of advanced control strateges for perodc processes, usng te PAROC framework and software platform. In partcular, we focus on te Multcolumn Solvent Gradent Purfcaton (MCSGP) process amng to derve a olstc control strategy for te maxmzaton of purty and yeld. Based on our prevously presented work (Papatanasou et al., 15), ere we successfully demonstrate te operaton of te columns under contnuous mode, were te outputs of column 1 act as dsturbances for column. Te desgned controllers ndcate perodc nput profles tat can effectvely lead to cyclc steady state. Addtonally, we propose trackng of te ntegral of te outlet concentratons tat allows contnuous process montorng and terefore tgter control actons. 7. ACKNOWLEDGEMENTS Te autors would lke to tank te Morbdell Group from ETH Zurc, as well as Dr. Tomas Mueller-Spaet from ComaCon AG for ter valuable nput on te understandng of te MCSGP process. Fnancal support from te European Commsson (OPTICO/G.A. No.8813), EPSRC 163

6 IFAC DYCOPS-CAB, 16 June 6-8, 16. NTNU, Trondem, Norway (EP/M7856/1) & Texas A&M Unversty & ts Energy Insttute are also gratefully acknowledged. REFERENCES Aumann, L. & Morbdell, M. 7. A contnuous multcolumn countercurrent solvent gradent purfcaton (MCSGP) process. Botecnology and Boengneerng, 98, Grossmann, C., Strölen, G., Morar, M. & Morbdell, M. 1. Optmzng model predctve control of te cromatograpc mult-column solvent gradent purfcaton (MCSGP) process. Journal of Process Control,, Guocon, G. 1. Basc prncples of cromatograpy. Ullmann's Encyclopeda of Industral Cemstry. Guocon, G., Golsan-Sraz, S. & Katt, A Fundamentals of nonlnear and preparatve cromatograpy. Academc Press, Boston. Huang, R., Begler, L. T. & Harnat, E. 1. Robust stablty of economcally orented nfnte orzon NMPC tat nclude cyclc processes. Journal of Process Control,, Huang, R., Harnat, E. & Begler, L. T. 11. Lyapunov stablty of economcally orented NMPC for cyclc processes. Journal of Process Control, 1, Kawajr, Y. & Begler, L. T. 6. Optmzaton strateges for smulated movng bed and PowerFeed processes. AICE Journal, 5, Kajura, H. & Pstkopoulos, E. N. 11. Dynamc modelng and explct/mult-parametrc MPC control of pressure swng adsorpton systems. Journal of Process Control, 1, Kajura, H. & Pstkopoulos, E. N. 13. Optmzaton and control of pressure swng adsorpton processes under uncertanty. AICE Journal, 59, Klatt, K.-U., Hansc, F., Dünneber, G. & Engell, S.. Model-based optmzaton and control of cromatograpc processes. Computers & Cemcal Engneerng, 4, Krättl, M., Stenebac, F. & Morbdell, M. 13. Onlne control of te twn-column countercurrent solvent gradent process for bocromatograpy. Journal of Cromatograpy A, 193, Krättl, M., Strölen, G., Aumann, L., Müller-Spät, T. & Morbdell, M. 11. Closed loop control of te mult-column solvent gradent purfcaton process. Journal of Cromatograpy A, 118, Lambert, R. S. C., Rvott, P. & Pstkopoulos, E. N. 13. A Monte-Carlo based model approxmaton tecnque for lnear model predctve control of nonlnear systems. Computers & Cemcal Engneerng, 54, Mayne, D. Q., Rawlngs, J. B., Rao, C. V. & Scokaert, P. O. M.. Constraned model predctve control: Stablty and optmalty. Automatca, 36, Müller-Spät, T., Aumann, L., Melter, L., Strölen, G. & Morbdell, M. 8. Cromatograpc separaton of tree monoclonal antbody varants usng multcolumn countercurrent solvent gradent purfcaton (MCSGP). Botecnology and Boengneerng, 1, Nlcan, S. & Panteldes, C. C On te optmsaton of perodc adsorpton processes. Adsorpton-Journal of te Internatonal Adsorpton Socety, 4, Papatanasou, M. M., Stenebac, F., Stroelen, G., Müller- Spät, T., Nascu, I., Oberdeck, R., Morbdell, M., Mantalars, A. & Pstkopoulos, E. N. 15. A control strategy for perodc systems - applcaton to te twn-column MCSGP. Computer Aded Cemcal Engneerng. Pstkopoulos, E. N., Dangelaks, N. A., Oberdeck, R., Papatanasou, M. M., Nascu, I. & Sun, M. 15. PAROC An ntegrated framework and software platform for te optmsaton and advanced modelbased control of process systems. Cemcal Engneerng Scence, 136, Pstkopoulos, E. N., Dua, V., Bozns, N. A., Bemporad, A. & Morar, M.. On-lne optmzaton va offlne parametrc optmzaton tools. Computers & Cemcal Engneerng, 6, Pse gproms ModelBulder (R). Process Systems Enterpse. Rvott, P., Lambert, R. S. C. & Pstkopoulos, E. N. 1. Combned model approxmaton tecnques and multparametrc programmng for explct nonlnear model predctve control. Computers & Cemcal Engneerng, 4, Strölen, G., Aumann, L., Mazzott, M. & Morbdell, M. 6. A contnuous, counter-current mult-column cromatograpc process ncorporatng modfer gradents for ternary separatons. J Cromatogr A. 164