Iterative Learning Control of a Batch Cooling Crystallization Process based on Linear Time- Varying Perturbation Models

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1 Ian Davd Lockart Bogle and Mcael Farweater (Edtor), Proceedng of te nd European Sympoum on Computer Aded Proce Engneerng, 17 - June 1, London. 1 Elever B.V. All rgt reerved. Iteratve Learnng Control of a Batc Coolng Crytallzaton Proce aed on Lnear me- Varyng Perturaton Model Nad Sanzda, Zoltan K. Nagy * Department of Cemcal Engneerng, Lougoroug Unverty, Lougoroug, LE11 U, Unted Kngdom, * z.k.nagy@loro.ac.uk. Atract e paper preent an approac to mprove te product qualty from atc to atc y explotng te repettve nature of atc procee to update te operatng trajectore ung proce knowledge otaned from prevou run. e data-aed optmzaton metodology aed on ung te lnear tme varyng (LV) perturaton model n an teratve learnng control (ILC) framework to provde a convergent atc-to-atc mprovement of te proce performance ndcator. e approac wa evaluated for a atc coolng crytallzaton proce wt te am to control te mean crytal ze y manpulatng te reactor temperature profle. e mulated temperature trajectore reultng from te teratve meaurement-aed optmzaton approac converged to te teoretcally optmal trajectory otaned ung model-aed optmzaton. ee reult demontrate te potental of te ILC approac for controllng atc procee wtout rgorou proce model. Keyword: Batc proce, Iteratve learnng control, LV perturaton model. 1. Introducton Batc cemcal procee are eental for te producton of g value added product, uc a ocemcal, parmaceutcal, mcroelectronc and pecalty cemcal. Generally n te cae of atc procee controllng te operatng condton to mprove te fnal product qualty from atc to atc often te mot practcally acevale control trategy (ong et al., 4), nce t doe not requre wtn atc meaurement and can rely on te generally more frequently avalale reult of laoratory analye at te end of te atce. atc-to-atc control approac explot te repettve nature of atc procee to update te proce operatng trajectore ung proce knowledge otaned from prevou atc run. te man dea of Iteratve Learnng Control (ILC) wc a een uccefully appled from ndutral root to autonomou vecle (Moore, 1998). ILC mprove tranent trackng performance of a ytem tat execute te ame tak repeatedly over a fxed tme nterval (Lee et al., 7), ut t can alo e appled to fnd te tranent etpont to aceve a dered end-pont performance. In practce tere are many procee repeatng te ame tak n a fnte nterval ncludng a atc reactor n te cemcal ndutry. ence t a natural approac to apply ILC n trackng control of product qualty n agle atc manufacturng procee. Addtonally, developng a frt prncple model uually very complcated, tme conumng and ence expenve for ndutral atc procee (ong et al., 1). paper ntroduce a data-drven ILC approac to automate recpe updatng to mprove product qualty from atc to atc n te cae of a atc coolng crytallzaton proce.

N. Sanzda and Z.K. Nagy. Iteratve Learnng Control (ILC) n Batc Cemcal Procee In te atc-to-atc control approac varaton mut e condered on two tmecale.

owever, durng a atc, wtn atc meaurement are often unavalale or adjutment to te operatng condton cannot e made due to lmted onlne enor or actuator.

In t tuaton te wtn atc meaurement (f avalale) are ued to etmate model aed parameter and tate. e updated model ten ued n an ILC framework to mprove te future operatng recpe (ee Fgure ).

, () wc an ILC ceme aed on lnear tme varyng (LV) perturaton model. ypcally te nput/output trajectore n atc procee are nerently nonlnear and tme varyng.

2 N. Sanzda and Z.K. Nagy. Iteratve Learnng Control (ILC) n Batc Cemcal Procee In te atc-to-atc control approac varaton mut e condered on two tmecale. A own on Fgure 1, varaton wtn and etween atce can e condered leadng to an optmzaton prolem on two tme cale (Nagy, 9). owever, durng a atc, wtn atc meaurement are often unavalale or adjutment to te operatng condton cannot e made due to lmted onlne enor or actuator. ence, atc-toatc mprovement tll dependent on learnng from te nformaton otaned uually from after-atc laoratory analye. In t tuaton te wtn atc meaurement (f avalale) are ued to etmate model aed parameter and tate. e updated model ten ued n an ILC framework to mprove te future operatng recpe (ee Fgure ). Fgure 1: Scematc repreentaton of te dynamc two-tme cale varaton n atc control (Nagy, 9). Fgure : Structure of te ILC framework (Nagy, 9)..1. Lnear me Varyng (LV) Perturaton Model e metodology appled aed on te approac propoed y ong et al., () wc an ILC ceme aed on lnear tme varyng (LV) perturaton model. ypcally te nput/output trajectore n atc procee are nerently nonlnear and tme varyng. owever, te concept of ung perturaton varale remove proce nonlnearty y utractng tme varyng nomnal trajectore from te atc operaton trajectore. If we conder te followng nonlnear functon etween nput (t) and output Y (t) n te matrx form, Y Ψ( ) n (1) were, Ψ () te nonlnear tatc functon etween te nput and output and n te vector of meaurement noe. For te ytem (1), an LV perturaton model L developed y lnearzng te nonlnear model along te nomnal trajectore. Intally everal et of torcal proce operatng data are collected and te nput-output data matrce, Ω x [ 1,,..., ] and Ω y [ Y1, Y,..., Y ] repectvely, are contructed, were, te numer of torcal atce. From tee data, te et performng trajectore are elected a te nomnal trajectore (, Y ). At any tme t, te perturaton varale for te t atc are calculated a - and Y Y - Y. Lnearzng te nonlnear model (1) around te nomnal trajectore gve, Ψ( ) Y = Y ( - ) m n ()

3 Iteratve Learnng Control of Batc Cemcal Procee aed on me-varyng Perturaton Model were, m [ m (1), m (),, m ( N )] te equence of model error of N oervaton due to lnearzaton. e LV perturaton model and correpondng aolute model predcton of Equaton () are, Y = L + d () Y = Y + Lˆ (4) e trackng error of te actual proce and of te predcted perturaton model are e = Y - Y d and eˆ = Y - Yˆ, repectvely. e tranfer functon ˆL d predcted accordng to te tep of Equaton (5), for 1,,, N, numer of oervaton per atc and 1,,, te numer of torcal atce, y y - y x x - x, and u [ x1, x,..., x 1( ), x ] (5) y 1 u 1 If, y V and u U y u ten, te ytem, lˆ etmated at eac oervaton y te leat quare metod a, lˆ -1 ( U U ) U V and ˆL otaned a, L ˆ [ˆ l ˆ ˆ 1, l,..., l N ]. After completon of te t atc run, predcton error etween off-lne-meaured or analyzed product qualte and ter model predcton can e calculated a, ε Y Yˆ. e aolute ~ modfed model predcton defned a, Y Yˆ 1 1 ε. e trackng error of te modfed predcton of te perturaton model defned a, ~ ~ e Yd Y. In ILC, t dered tat, lm e mn e. Fnally, te followng quadratc ojectve functon formulated aed on te mnmzaton of te predcted trackng error, 1 ~ ~ J +1 = mn [ e+1 Oe+1 + Δ +1 PΔ + 1] (6) Δ+1 were, O and P are wegtng matrce aed on output performance and nput cange, repectvely. ojectve functon ould e olved upon completon of te t atc to update te nput trajectory for te (+1) t atc a, Kˆ 1 e, were, Kˆ Lˆ OLˆ 1 [ P] Lˆ O, te calculated control acton.. ILC of a Batc Coolng Crytallzaton Proce In t tudy an uneeded atc coolng crytaller wa condered for te eart drug component (Propanolol ae), wt a atc tme of mnute. e knetc of te ytem gven y te followng et of ordnary dfferental equaton, dm / dt = B (7) dm / dt = Gm + Br = 1,, (8) -1 d C / d t =- k r( Gm + Br ) (9) v were, B and G are te nucleaton and growt rate repectvely, S = C - CS ( ) te aolute uperaturaton, C concentraton, C S te olulty a a functon of te temperature, m, m, m, and m are te moment defnng total numer, lengt, area, 1 and volume of crytal n te ytem repectvely, r te ze of te nucle. e ntal

4 4 N. Sanzda and Z.K. Nagy and fnal temperature were 41 o C and 1 o C repectvely. e model parameter are own n ale 1. ale 1: Parameter of te crytallzaton model. 5 Solulty n water ( n K) Growt rate (cm/ec) Nucleaton rate Denty of crytal (g/cm ) = 1.96 C S (S) f S G (S) f S (S) f S B (S) f S Volumetrc ape factor Intal concentraton (g/g olvent) k v =.4 C =.54 A mulaton program wa developed ung MatLa to e ued a te real proce. e ojectve wa to aceve a dered mean crytal ze y manpulatng te reactor temperature. e atc lengt wa dvded nto N = 1 equal tage. Eleven atce conderng dfferent temperature profle were mulated ung te model. From tee eleven data et ten were ued a te torcal data and te data wt te et reult wa elected a te nomnal data et,, Y ued to create te LV model. e frt prncple model wa alo ued to generate te dered product reference trajectory Y d and te teoretcal optmum temperature profle. Ung te torcal data et and te elected nomnal trajectore (, Y ), te parameter of L wa redentfed ung proce meaurement. e wegtng matrce were et a, O = 1 5 dag (1,.5,.5,.5,.5,.5,.5,.5,1.5,1) and P =.5I. Accordng to te ILC algortm, after te completon of eac teraton te new data added to te torcal data et to update L. e forgettng factor wa, β =.8. e temperature profle reultng from eac tep of te ILC ceme wa appled to te mecantc model (.e. real proce) and te correpondng mean crytal ze wa otaned. Mean Lengt dered atc6 atc7 atc1 atc11 fnal atc me (mn) emperature (K) optmum atc6 atc me (mn) atc1 atc11 fnal atc (a) () Fgure : rajectore for te a) mean crytal ze and ) nput temperature at dfferent atce.

5 Iteratve Learnng Control of Batc Cemcal Procee aed on me-varyng Perturaton Model 5 e nput/output data otaned after eac atc durng te ILC wa agan ncluded n te torcal data et and te LV model re-dentfed and ued n a model-aed optmzaton to determne te temperature profle for te next atc. e reult of te ILC approac are own n Fgure, ndcatng tat te mean ze of crytal converged to te dered trajectory and te reultng fnal temperature profle wa very cloe to te teoretcally optmum trajectory. 4.8E-4 4.E-4 RMSE.E-4.4E-4 1.6E-4 8.E-5.E Batc No. Fgure 4: rackng performance Fgure 4 ow te root mean quare error (RMSE) value of te dered and te real mean crytal ze trajectore. It took aout 4 atce to arrve at fnal trajectore wtout te need of any proce model. owever, te reult were almot converged from te 5 t atc onward, and practcally all uequent atce after te 5 t would produce crytal wt very mlar ze to te dered target. 4. Concluon In t paper, an operatng data aed teratve learnng control (ILC) algortm preented. e propoed metod wa evaluated ung a mulated parmaceutcal crytallzaton proce. e reult demontrated tat te approac ale to converge to te teoretcally optmal operatng profle wtout te need of a detaled mecantc proce model. e convergence of te approac can e mproved y ung nonlnear data-drven model n te ILC ceme, uc a artfcal neural network or polynomal cao expanon. Reference J.. Lee and K.S. Lee, 7, Iteratve learnng control appled to atc procee: An Overvew, Control Engneerng Practce, Vol. 15, pp K.L. Moore, 1998, Iteratve learnng control-an expotory overvew, n: B.S. Datta (ed.), Appled & Computatonal Control, Sgnal Proceng, and Crcut, Boton: Kluwer Academc, pp Z. ong, Y. u, J. Dong, and J. Zang, 1, Neural network aed teratve learnng control for product qualte n atc procee, Internatonal Journal of Modellng, Identfcaton and Control, 11(1-), pp Z. ong, and J. Zang, 4, Batc-to-atc optmal control of nonlnear atc procee aed on ncrementally updated model. IEE Proc.-Control eory Appl., 151(), pp Z. ong and J. Zang,, Product qualty trajectory trackng n atc procee ung teratve learnng control aed on tme-varyng perturaton model, Ind. Eng. Cem. Re., 4, pp Z.K. Nagy, 9, Model aed rout atc-to-atc control of partcle ze and ape n parmaceutcal crytallzaton, n S. Engell, Y. Arkun (Ed.) Proc. of te 9t IFAC Symp. on Adv. Control of Cem. Proc. (ADCEM), Itanul, urkey, July 1-15, pp. -5.

Small signal analysis

Small signal analysis Small gnal analy. ntroducton Let u conder the crcut hown n Fg., where the nonlnear retor decrbed by the equaton g v havng graphcal repreentaton hown n Fg.. ( G (t G v(t v Fg. Fg. a D current ource wherea