Optimal trajectory approximation by cubic splines on fed-batch control problems

Transcription

1 Opimal rajecory approximaion by cubic splines on fed-bach conrol problems A. ISMAEL F. VAZ Producion and Sysems Deparmen Minho Universiy, Engineering School EUGÉNIO C. FERREIRA Cenro de Engenharia Biológica Universidade do Minho ALZIRA M.T. MOTA Mahemaics Deparmen Poro Polyechnic Insiue, Poro Engineering Insiue Absrac: Opimal conrol problems appear in several engineering fields and in paricular on he conrol of fedbach fermenaion processes. These problems are ofen described by ses of nonlinear differenial and algebraic equaions, usually subjec o consrains in he sae and conrol variables. Tradicional approaches o he opimal feed rajecory compuaion consiss in geing a linear spline ha approximaes he rajecory, which opimizes a given performance of he fed-bach fermenaion process. This approach leads o non-differeniable rajecories ha can pose some problems o implemen in pracice, resuling in a possible discrepancy of he simulaed and real performances. In his paper we develop a echnique o obain a cubic spline for he approximae rajecory, leading o a smooh approximaion funcion. We provide numerical resuls for a se of case sudies where he AMPL modeling language, CVODE ordinary differenial equaions solver and a paricle swarm algorihm were used. Key Words: Fed-bach opimal conrol, Ordinary differenial equaions, Nonlinear programming. 1 Inroducion A grea number of valuable producs, like biopharmaceuicals, are produced using fermenaion processes and hus opimizing such processes is of grea economic imporance. In general highly nonlinear and complex differenial equaions are used o model such fermenaion process. The opimizaion of he fermenaion process resuls in complex opimal conrol problem where an analyical soluion is very hard o obain. A special case of he opimal conrol problem in he fermenaion conex is he feed rajecory deerminaion on a fed-bach bioreacor. A ypical approach o he feed rajecory opimizaion is o approximae i by using linear segmens (linear splines). The resuling rajecory is herefore non-differeniable, making he corresponding opimizaion problem also nondiffereniable. The opimizaion problem belongs o a well known class of programming problems denoed by semi-infinie [5]. By using he semi-infinie opimaliy condiion we can readily ransforme he semiinfinie opimal conrol problem ino a nonlinear opimal conrol problem (e.g. [11]). While using linear splines o approximae he opimal feed rajecory leads o a simpler semi-infinie programming (SIP) problem he resuling rajecory approximaion is no differeniable and herefore he simulaed performance can be significanly differen from he real one. In his work we propose a new approach o he feed rajecory planning by using cubic splines insead of linear ones. The resuling approximaion is now smooh and expeced i o be more realisic han he one obained wih he previous approach. A paricle swarm algorihm was used o obain he numerical resuls. In secion 2 we inroduce he reader o opimal conrol and o he herein used noaion. Secion 3 is used o describe he common approach (he linear

2 splines o approximae he feed rajecory). The new approach is described in secion 4. In secion 5 we provide some implemenaion deails. We leave he numerical resuls o secion 6 and we conclude in secion 7. Some leading research is also presened in he las secion. 2 Opimal conrol rajecory opimizaion Many microorganisms are used for producing valuable biopharmaceuicals producs. During he fermenaion process he biomass and produc concenraions changes considerably. The sysem dynamic behavior moivaes he developmen of opimizaion echniques o find he opimum inpu feeding rajecory of subsrae in order o obain a maximum oucome from he process. The oucome can be, for example, he maximum biomass producion wih a fixed duraion ime or he minimum ime wih a fixed amoun of subsrae. The opimal conrol problem is described by a se of ordinary differenial equaions ẋ = f(x, u, ), x( 0 ) = x 0, 0 f, where x are he sae variables and u he inpu variables ha are a funcion of ime. 0 and f are he inicial and final ime, respecively. The performance index J can be generally saed as J( f ) = ϕ(x( f ), f ) + f 0 φ(x, u, )d, where ϕ is he performance index of he sae variables a final ime f and φ is he inegraed performance index during he operaion. Addiional consrains on he sae and inpu variables can be imposed ha ofen refle some physical limiaion of he sysem. The general maximizaion problem (P ) can be posed as max J( f ) (1) s.. ẋ = f(x, u, ) (2) x x() x, [ 0, f ] (3) u u() u, [ 0, f ] (4) Problem (P ) belong o a well known class of semi-infinie programming problems [5]. These problems are characerized o have a finie number of variables o be opimized subjec o an infinie number of consrains. x() and u() are funcional vecor whose componens are x 1 (), x 2 (),..., u 1 (), u 2 (),..., respecively. Whenever x represens an n dimensional vecor is componens are addressed as x 1, x 2,..., x n and vice-versa. 3 The common approach - Linear splines The opimizaion occurs when deermining he opimal inpu variables u or operaional final ime f. The inpu variables ofen represen feeding (or emperaure, see [9]) rajecories, i.e., in deermining he amoun of subsrae o be fed ino he bioreacor per ime uni. Solving problem (P ) in is original formulaion is no advisable and unpracical, since no of-he-shelf sofware for dealing wih hese ype of problems is available. Insead problem (P ) can be reformulaed as a non-differeniable global opimizaion problem by imposing a penaly for dealing wih consrain (3) and o use linear inerpolaion for dealing wih consrains (4). Imposing he penaly funcion for consrains (3) resuls in redefining he objecive funcion as { J(f ) if x x() x, [ Ĵ( f ) = 0, f ] oherwise An already proposed sraegy o deal wih consrains (4) is o inerpolae he funcion u() by a polynomial (e.g. [9]). A parameer ha makes a grea influence in he final rajecory precision is he number of discreizaion poins (knos) of he domain [ 0, f ]. A well known effec on increasing he number of knos (and consequenly he polynomial degree) is he increasing poor smoohness of he resuling polynomial. Linear spline inerpolaion is herefore beer suied for approximae funcion u(), where increasing he number of knos increases precision wihou a grea affec on smoohness. A linear inerpolaing funcion w() (linear spline) is used o approximae he feeding rajecory funcion u(). Le i, i = 0,..., n, denoe he ime insans (knos) and h i = i i 1, i = 1,..., n, he ime displacemens. Fixed ime inervals are used while he linear spline funcion values u( i ), i = 0,..., n, are o be compued (in fac we may also use i as variables o be opimized, bu keeping in mind ha ill condiion can occur in his case). The linear spline is composed of n linear segmens. The spline segmen w i (), i = 1,..., n, is defined as: w i () =u( i 1 ) + ( i 1 ) (u( i) u( i 1 )), ( i i 1 ) for [ i 1, i ], i = 1,..., n. By using he linear spline w() o approximae he feed rajecory (u()) we obain a SIP problem where consrains (4) are replaced by u w() u. By

3 careful inspecing his consrain and by using he opimaliy condiions for SIP we observe ha candidae poins o make consrains (4) acive, a he soluion, are he spline knos and herefore consrains (4) can be replaced by consrains imposing he limi a knos. Consrain (4) can hen be replaced by u w( i ) u, i = 0,..., n. The opimizaion nonlinear opimizaion problem (NLP) is hen redefined as: max Ĵ( f ) w R n+1 s.. ẋ = f(x, w, ) u w( i ) u, i = 0,..., n. (5) If he iniial dynamic sysem condiions (x 0 ) are o be considered as variable we may also impose some simple bound consrains on is aainable values. We can also consider h R n and f as variables o be opimized increasing he problem dimensional and complexiy. The major moivaion for using derivaive free opimizaion codes has o do wih he fac ha he objecive funcion and he resuling w() rajecory funcion are no differeniable. Recall ha using w() in he dynamic equaion makes hem non-differeniable. By using a sochasic algorihm we can also expec o obain he global opimum for he NLP problem. 4 The new approach - Cubic splines By obaining a non-differeniable rajecory approximaion one expecs bioreacor feed rajecory no o be able o compleely follow he opimal feed. A discrepancy beween he simulaed and real performance is likely o be observed. Using a smooh approximaion o he opimal feed rajecory will resul in a beer simulaed and real performance gap. Meanwhile he rajecory opimal conrol problem will be differeniable if he ordinary differenial equaion and he performance index are also differeniable (please noe ha using an infiniy penaly funcion urns he objecive funcion ino a non-differeniable objecive funcion). Sill obaining he problem derivaive would be a complex and edious ask and herefore he use of a derivaive free opimizaion algorihm is mosly desirable. The new approach consiss herefore in approximaing he opimal feed rajecory by a cubic spline. The penaly funcion Ĵ is again used as he objecive funcion and he feed rajecory u() is approximaed by a cubic spline s(). The cubic segmen i, i = 1,..., n, is defined as s i () = M i 1( i 1 ) 3 + M i( i 1 ) 3 6( i i 1 ) 6( i i 1 ) + [ u(i 1 ) M ] i 1( i i 1 ) ( i )+ i i 1 6 [ u(i ) M ] i( i i 1 ) ( i 1 ) i i 1 6 wih [ i 1, i ], where i, i = 0,..., n, are he ime insans. The semi-infinie programming problem max Ĵ( f ) s R n+1 s.. ẋ = f(x, s, ) u s() u, T [ 0, f ]. is now of a much harder resoluion, since a reducion o a NLP is no sraighforward. Given u( i ), i = 0,..., n, geing he global maximizer of he parameric problem, max s() [ 0, f ] in order o compue which values make he consrain (4) acive is indeed more complex. To avoid he exra complexiy a new redefiniion of he objecive funcion is proposed. The conrol consrain is also included in he infiniy penaly objecive funcion resuling in he new objecive o be opimized, defined as { Ĵ(f ) if u s() u, [ J( f ) = 0, f ] oherwise The opimizaion is hen redefined in he following way max J( f ) s.. ẋ = f(x, s, ) T. (6) In spie of having a differeniable rajecory and an ordinary differeniable sysem of equaions he objecive funcion of problem (6) is sill a nondiffereniable problem. 5 Implemenaion deails We briefly describe he deails regarding he used environmen o address problems (5) and (6).

4 The AMPL [4] modeling language for mahemaical programming was used o code he case sudies (see also [2] for anoher popular modeling language). AMPL provides an easy o use and powerful language, an inerface ha allow communicaion wih a wide variey of solver (see for a lis of available solvers) and he possibiliy o load an exernal dynamic library. By using AMPL as he modeling language modificaions can easily be incorporaed ino he model. If, for example, a consrain in he oal allowed glucose addiion ( G ) is o be imposed in problem (5), he consrain n 1 i=0 h i+1(w( i ) + w( i+1 ))/2 G can easily be considered in he model. The AMPL feaure o load an exernal dynamic library was exploied. The exernal library fed-bach.dll (available from he firs auhor) provides five exernal funcions o AMPL relaed wih each es case considered. These exernal funcions, by is urn, use he CVODE [3] package o solve he ordinary differenial equaions (2). The Newon ieraion wih he CVDiag module was seleced. The MLOCPSOA [10] solver was used o provide he numerical resuls shown in he nex secion. MLOCPSOA sands for Muli-LOCal Paricle Swarm Opimizaion Algorihm. Muli-local opimizaion addresses he finding of all he local and global opima for an opimizaion problem. While MLOCPSOA was developed wih muli-local opimizaion in mind by seing an opion i reverses o he radiional paricle swarm algorihm. MLOCPSOA provides an inerface o AMPL, allowing problems o be easily coded and solved in his modeling language. The MLOCPSOA allows a wide variey of algorihm parameers o be se. The used parameers are size for he populaion size (defauls o min(6 n, 1000)), maxier for he maximum allowed ieraions (defauls o 2000) and mlocal for mulilocal search (defauls o 0 global search insead of muli-local search). All oher parameers were lef by defaul. The reader is poined for he user manual ([10]) for furher deails. 6 Numerical resuls Numerical resuls were obained for five case sudies. The parameers used are presened ogeher wih he numerical resuls in Table 1. Problem column refers o he case sudy (AMPL model file); NT is he number of rajecories in he model; n is he number of ime displacemens (problems wih n + 1 variables) where equal displacemens are considered (h i = f /n, i = 1,..., n). We include also he objecive funcion value obained in he lieraure for he Problem NT n f J( f ) penicillin [1] ehanol [1] chemoherapy [1] hproein [7, 8] rproein [7, 6] Table 1: Tes problems and parameers Cubic Linear Problem f J( f ) J( f ) penicillin ehanol chemoherapy hproein rproein Table 2: Numerical resuls case sudies in column J( f ). Problem penicillin refers o a problem of fed-bach fermenaion process where he opimal feed rajecory is o be compued while he penicillin producion is o be maximized. ehanol refers o a similar opimal conrol problem where he ehanol producion is o be maximized. chemoherapy is he only opimal conrol problem ha does no refers o a fed-bach fermenaion processe. I is a problem of drug adminisraion in chemoherapy. The opimal rajecory o be compued is he quaniy of drug ha mus be presen in order o achieve a specified umor reducion. While hproein opimal conrol problem is o compue a unique rajecory (subsrae o be fed) problem rproein includes also a rajecory for an inducer. Boh problems refer o a maximizaion for proein producion. See Table 1 for he case sudies references. Numerical resuls for he linear and cubic splines are shown in Table 2. Columns have he same meaning as in he previous able. Linear and Cubic columns repors for he J( f ) in he linear and cubic splines rajecory approximaion, respecively. Recall ha J( f ) = Ĵ( f ) = J( f ) for every feasible spline. MLOCPSOA used a populaion size of 60 and a maximum of 1000 ieraions (reaching a maximum of funcion evaluaions). Since MLOCPSOA is a sochasic algorihm we performed 10 solver runs for each problem and he bes soluions obained are repor on Table 2. J( f ) is he objecive funcion value and f is he final ime ( 0 is assumed 0 for all cases). No aemp is made on comparing our resuls wih he previously published ones, since implemenaion deails are no repored in previous papers. The use

5 u Subsrae feed Conrol profile Conrols u Subsrae feed Conrol profile Conrols X1 Cell mass X2 Subsrae X3 Produc X4 Volume Sae profile Saes X1 Cell mass X2 Subsrae X3 Produc X4 Volume Sae profile Saes Figure 1: The ehanol case sudy wih linear spline. u Subsrae feed X1 Cell mass X2 Subsrae X3 Produc X4 Volume Conrol profile Sae profile Figure 2: The ehanol case sudy wih cubic spline. Similar soluion. of differen echniques o solve he differenial equaions, discreizaion sep and precision can influence he objecive value a he soluion found. Meanwhile we can observe ha approximaing he feed rajecory by a linear or cubic spline does no grealy influence he fed-bach performance. ehanol was he case sudy wih he greaes performance difference in he obained numerical resuls. This difference is due o he finding of a new bes feed rajecory for his case. We provide, in figures 1 o 3, plos of he sae and conrol profiles for he ehanol case sudy. In figure 1 we plo he previous resuls wih he feed rajecory approximaed by a linear spline. The similar resul is ploed in figure 2 where he approximaion is a cubic spline. Figure 3 presens he new soluion obained wih he rajecory approximaed by a cubic spline Conrols Saes Figure 3: The ehanol case sudy wih cubic spline. Bes soluion. 7 Conclusions and fuure work The opimal conrol of fed-bach bioprocesses presens challenging nonlinear opimizaion problems where derivaives do no exis or are unpracical. As in a previous work we address he sae consrains by combining i wih he problem objecive funcion in an infinie penaly funcion. In his paper we add an exra difficuly by using cubic splines o approximae he feed rajecory. The conrol consrains wih he rajecory approximaed by cubic splines are handled in a similar way, by adding an infinie violaion o he objecive funcion. The resuling nonlinear opimizaion problem is characerized by possessing a non-convex nondiffereniable objecive funcion subjec o bound consrains in he variables. Paricle swarm opimizaion belongs o a class of sochasic algorihms for global opimizaion and is main advanages are he easily parallelizaion and simpliciy. In his paper we use he MLOCPSOA [10] implemenaion of he paricle swarm paradigm o obain numerical resuls wih some problem formulaions. No comparison wih previous work is provide, since he implemenaion depends on many parameers exernal o he problem and algorihm (ordinary differenial equaion solver, discreizaion sep, ec.). While no aemp is made o compare he previously obained soluions wih he linear spline approach, we provide a comparison among he linear and cubic spline approaches. While similar resuls are obained in erms of objecive funcion values he resuling approximaion o he rajecory is smooh. We expec ha he smoohness obained in he feed rajecory will allow beer resuls in real experi-

6 mens. The MLOCPSOA proved o be able o find he problem soluion wih reasonable accuracy and he paricle swarm paradigm proved o be a valuable ool in solving hese opimal conrol problems. The MLOCPSOA inerface wih AMPL allowed an easy and fas way o code he five case sudies problems. Using he AMPL modeling language ogeher wih a developed exernal dynamic library allows a grea flexibiliy in he problem formulaion. As a fuure research we plan o sudy he E. coli baceria in he same framework. The availabiliy of a lab-scale bioreacor will allow us o confirm ha a cubic approximaion of he feed rajecory produces beer resuls in pracice. Acknowledgemens: The research was suppored by he Algorimi Research Cener and by he Poruguese FCT under gran POCI/MAT/58957/2004. [8] S. Park and W.F. Ramirez. Opimal producion of secreed proein in fed-bach reacors. AIChE Journal, 34(9): , [9] R. Simuis and A. Lübber. A comparaive sudy on random search algorihms for boechnical process opimizaion. Journal of Bioecnology, 52: , [10] A.I.F. Vaz. MuliLOCal paricle swarm opimizaion algorihm v1.1: User s manual. hp:// [11] A.I.F. Vaz and E.C. Ferreira. Opimal conrol of fed-bach process wih paricle swarm opimizaion. In XXIX Congreso Nacional de Esadísica e Invesigación Operaiva, pages , Tenerife, Spain, ISBN X. References: [1] J.R. Banga, E.Balsa-Cano, C.G. Moles, and A.A. Alonso. Dynamic opimizaion of bioprocesses: Efficien and robus numerical sraegies. Journal of Bioechnology, 117: , [2] A. Brooke, D. Kendrick, A. Meeraus, and Ramesh Raman. GAMS: A User s Guide. hp:// December [3] S.D. Cohen and A.C. Hindmarsh. CVODE, a Siff/Nonsiff ODE solver in C. Compuers in Physics, 10(2): , [4] R. Fourer, D.M. Gay, and B.W. Kernighan. A modeling language for mahemaical programming. Managemen Science, 36(5): , [5] R. Heich and K.O. Koranek. Semi-infinie programming: Theory, mehods, and applicaions. SIAM Review, 35(3): , [6] J. Lee and W.F. Ramirez. Opimal fed-bach conrol of induced foreign proein-producion by recombinan baceria. AIChE Journal, 40(5): , [7] R. Oliveira and R. Salcedo. Benchmark esing of simulaed annealing adapaive random search and geneic algorihms for global opimizaion of bioprocesses. In B. Ribeiro, R.F. Albrech, A. Dobnikar, D.W. Pearson, and N.C. Seele, ediors, Adapaive and Naural Compuing Algorihm, pages , Wien, Springer- Verlag.