optimal operation and control of simulated moving bed chromatography… · 2017-07-20 · optimal...

Optimal Operation and Control of Simulated Moving Bed Chromatography:A Model-Based Approach

Karsten-U. Klatt∗, Guido Dunnebier, Felix Hanisch and Sebastian EngellDepartment of Chemical Engineering

University of DortmundD-44221 Dortmund, Germany

AbstractChromatographic separations are an expanding technology for the separation of Life Science products, such as pharma-ceuticals, food, and fine chemicals. The simulated moving bed (SMB) process as a continuous chromatographic separationis an interesting alternative to conventional batch chromatography, and gained more and more impact recently. The SMBprocess is realized by connecting several single chromatographic columns in series. A countercurrent movement of thebed is approximated by a cyclic switching of the inlet and outlet ports in the direction of the fluid stream. Becauseof its complex dynamics, the optimal operation and automatic control of SMB processes is a challenging task. Thiscontribution presents an integrated approach to the optimal operation and automatic control of SMB chromatographicseparation processes. It is based on computationally efficient simulation models and combines techniques from mathe-matical optimization, parameter estimation and control theory. The overall concept and the realization of the elementsare explained, and the efficiency of the proposed approach is shown in a simulation study for the separation of fructoseand glucose on an 8-column SMB plant.

KeywordsChromatographic separation, Simulated moving bed, Dynamic models, Optimization, Model-based control

Introduction

The chemical process industry is currently undergoinga substantial restructuring: the classical bulk businessis more and more substituted by Life Science productswith higher profit margins. In this area, particularly inthe development and production of pharmaceuticals, it isof the utmost importance to be ahead of the competitorsin the race to the market. This requires a detailed andintegrated process design already in the product devel-opment phase. In this context, product separation andpurification is the critical element in many cases.

Chromatographic processes provide a versatile toolfor the separation of substances which have differentadsorption affinities. They are especially suitable fortemperature-sensitive compounds and substances withsimilar molecular structure and physico-chemical prop-erties. Chromatography is well established in the fieldof the chemical analysis, but in recent years it gainedmore and more importance on the preparative scaleas a highly efficient, highly selective separation pro-cess. Due to their origin and the close relation to theinstruments from chemical analysis (i.e. HPLC andgas chromatographic analyzers), chromatographic sep-aration processes are mainly operated in the classicalbatch elution mode. To improve the economic viability,a continuous countercurrent operation is often desirable,but the real countercurrent of solids—such as the ad-sorbent in chromatographic processes—leads to seriousoperating problems. Therefore, the simulated movingbed (SMB) process is an interesting alternative since itprovides the advantages of a continuous countercurrent

∗K.-U. Klatt and G. Dunnebier are presently with Bayer AG,D-51368 Leverkusen, Germany.

unit operation while avoiding the technical problems ofa true moving bed.

The SMB process was first realized in the family ofSORBEX processes by UOP (Broughton and Gerhold,1961) and is increasingly used in a wide range of in-dustries. Currently, the main applications of continu-ous chromatographic separations can be divided into twogroups: the large-scale industrial production of relativelycheap specialty products, like xylene production or sugarseparation, and the separation of high-value products insmall amounts, which very often exhibit separation fac-tors near unity (e.g. enantiomer separations in the phar-maceutical industry). The separation costs in both casesare very high in relation to the overall process costs andeasily dominate those. An optimal design and opera-tion might therefore be the only possibility to exploitthe economic potential of the process and to make itsapplication feasible.

In practice, the SMB process is nowadays mainlyrealized by connecting several single chromatographiccolumns in series. The countercurrent movement is thenapproximated by a cyclic switching of the feed streamand the inlet and outlet ports in the direction of thefluid flow. Thus, the process shows mixed continuousand discrete dynamics with complex interactions of thecorresponding process parameters. If the SMB process isoperated close to its economic optimum, high sensitivi-ties to disturbances and changes in the operating param-eters result. Furthermore, concentration measurementsare expensive and can only be installed at the outlet ofthe separation columns. Therefore, the control of SMBchromatographic separation processes in order to ensurea safe and economical operation while guaranteeing theproduct specifications at any time is a challenging task.

239

240 Karsten-U. Klatt, Guido Dunnebier, Felix Hanisch and Sebastian Engell

Currently, most processes are operated at some distancefrom the optimum to avoid off-spec production and to en-sure sufficient robustness margins. In order to exploit thefull economic potential of this increasingly applied tech-nology, model-based optimization and automatic controlof SMB processes are required.

Several publications on both process optimization andfeedback control of SMB processes can be found in theliterature but they predominantly do not treat optimiza-tion and control in an integrated manner. The purposeof this contribution is to propose such an integrated ap-proach based on a rigorous dynamic process model. Theoverall concept and the realization of the elements are ex-plained in the remainder of this paper. In each section,we review the state of the art and refer to related workof other authors. We start from a short description ofchromatographic separations and SMB chromatographyin particular, and then explain the generation and im-plementation of sufficiently accurate and computation-ally efficient process models, which are the essential pre-requisite for model-based optimization and control. Weproceed with the issue of determining the optimal oper-ating regime of the process, followed by the description ofthe overall control concept and its components. The fea-sibility and the capabilities of the proposed approach arethen demonstrated on an application example, the sepa-ration of fructose and glucose on an 8-column SMB lab-oratory plant. We finalize with some conclusions, high-lighting unresolved issues and future research directions.

Process Description

Chromatography is a separation technique which isbased on the preferential adsorption of one component.In adsorption, the solutes are transferred from a liquidor gas mixture to the surface of a solid adsorbent, wherethey are held by intramolecular forces. Desorption is thereverse process whereby the solute, called adsorbate, isremoved from the surface of an adsorbent. By the useof a suitable stationary phase, components that are dif-ficult to separate by other methods can be obtained invery high purities. In comparison to other thermal sepa-ration methods, e.g. distillation, less energy is consumed.Chromatography is particularly useful for the separationof temperature-sensitive components because it can of-ten be performed at room temperature (Hashimoto et al.,1993; Adachi, 1994).

The classical implementations of chromatographic sep-arations are batch processes in elution mode (see Fig-ure 1). A feed pulse, containing the components to beseparated, is injected into a chromatographic columnfilled with a suitable adsorbent, alternating with the sup-ply of pure solvent. On its way along the column, themixture is gradually separated and the products can befractionated at the column outlet. One of the majordrawbacks of this method is the high amount of solvent

Winj

Wcyc

t

4

A,B A,B

CoutCin

t

B A B A

Figure 1: Batch elution chromatography.

��

��

��

��

A

B

Extract A+D Desorbent D

Direction of port switching

Feed A+B Raffinate B+D

Figure 2: Simulated moving bed chromatography.

needed to perform the separation. This also leads to ahigh dilution of the products. During the migration ofthe components along the column, only a small part ofthe adsorbent is used for the separation. Another disad-vantage is the batch operation mode of the process. Inindustrial applications, processes with continuous prod-uct streams are preferred.

These drawbacks led to the development of continuouscountercurrent adsorption processes. The main advan-tage of such an arrangement is the countercurrent flow,as in heat exchangers or distillation columns, that max-imizes the average driving force. Thus, the adsorbentis used more efficiently. However, the movement of thesolid particles is very difficult to realize. One reason isthe inevitable back-mixing of the solid that reduces theseparation efficiency of the columns. Another problemis the abrasion of the particles which is caused by themovement.

The invention of the Simulated Moving Bed processovercame these difficulties, providing a profitable alter-native mainly for the separation of binary mixtures. Thecountercurrent movement of the phases is approximatedby sequentially switching the inlet and outlet valves of in-terconnected columns in the direction of the liquid flow.According to the position of the columns relative to thefeed and the draw-off nodes, the process can be divided

Optimal Operation and Control of Simulated Moving Bed Chromatography: A Model-Based Approach 241

C C

C C

Time [s x 1.0e04] Time [s x 1.0e04]

Elution profile (comp. A)

Axial concentration profile

Elution profile (comp. B)

0.01

2.9 2.93.0 3.03.1 3.12.95 2.953.05 3.05

0.01

0.02 0.02

0.03 0.03

0.04 0.04

Figure 3: Cyclic steady state of the simulated movingbed process. Top: axial concentration profile (end ofperiod), Bottom: elution profiles.

into four different sections (see Figure 2). The flow ratesare different in every section and each section has a spe-cific function in the separation of the mixture. The sep-aration is performed in the two central sections wherecomponent B is desorbed and component A is adsorbed.The desorbent is used to regenerate the adsorbent bydesorption of component A in the first section, and com-ponent B is adsorbed in the fourth section to regeneratethe desorbent. The net flow rates of the componentshave different signs in the central sections II and III,thus component B is transported from the feed inlet up-stream to the raffinate outlet with the fluid stream andcomponent A is transported downstream to the extractoutlet with the “solid stream”.

The stationary operating regime of the SMB processis a cyclic steady state (CSS), in which in each sectionan identical transient takes place during each period be-tween two valve switches. This periodic orbit is practi-cally reached after a certain number of valve switches.The upper part of Figure 3 represents the axial concen-tration profile at the end of a switching period whileoperating in cyclic steady state. The resulting elutionprofiles below represent the time history of the productconcentrations and highlight the periodic nature of theprocess dynamics.

Modeling and Simulation

Modeling of the SMB Process

The modeling and simulation of SMB processes has beena topic of intensive research in recent years. An overviewcan be found e.g. in Ruthven and Ching (1989), Ganet-sos and Barker (1993), Zhong and Guiochon (1998) andKlatt (1999). The modeling approaches can be dividedinto two classes. In the first class, a rigorous SMB model

is assembled from dynamic process models of the singlechromatographic columns under explicit consideration ofthe cyclic switching operation. Alternatively, an equiva-lent solid velocity is deduced from the switching time andthe balance equations for the corresponding true movingbed (TMB) are used.

By neglecting the cyclic port switching, the model isdistinctly simplified, and can be solved very efficiently.It can be shown that the steady state solution of a de-tailed TMB model reproduces the concentration profileof a SMB model reasonably well in case of three or morecolumns per zone and linear adsorption behavior, whichjustifies the use of this type of model for the design ofsuch units (Storti et al., 1988; Lu and Ching, 1997; Paiset al., 1998). However, many of the recent applicationsof the SMB process, especially in the area of fine chemi-cals and pharmaceuticals, are operated with less columnsper zone and at higher concentrations with nonlinearadsorption equilibrium for economic reasons. In thesecases, the accuracy of the TMB approximation becomespoor. Furthermore, only the dynamic SMB model cor-rectly represents the complete process dynamics, whichis essential for an optimization of the operating policyand for model-based control.

The rigorous dynamic SMB model is closely related tothe real process and directly describes the column inter-connection and the switching operation. It mainly con-sists of two parts: the node balances to describe the con-nection of the columns combined with the cyclic switch-ing, and the dynamic simulation models of the singlechromatographic columns. The node balances are usedto calculate the inlet flows and inlet concentrations ofthe four zones of the process based on the mass balancesat the corresponding nodes (Ruthven and Ching, 1989):Desorbent node:

QIV + QD = QI

couti,IV QIV + ci,DQD = cin

i,IQI

(1)

Extract node:QI −QEx = QII

couti,I = cin

i,II = ci,Ex

(2)

Feed node:

QII + QF = QIII

couti,IIQII + ci,F QF = cin

i,IIIQIII

(3)

Raffinate node:

QIII −QRaf = QIV

couti,III = cin

i,IV = ci,Raf

(4)

with Qi being the respective flow rate in each of thefour zones, QD the desorbent flow rate, QF the feed flowrate, QEx the extract flow rate, and QRaf the raffinateflow rate. The switching operation can, from a math-ematical point of view, be represented by shifting the


general rate model

lumped rate modelequilibrium transport

dispersive modelreaction dispersive model

ideal model

Thomas model equilibrium

transport modelequilibrium

dispersive model

convection + ax. dispersion + film mass transfer + particle diffusion + adsorption dynamics

convection + ax. dispersion + rate limiting stepdom. mass transfer resistanceadsorption kinetics

convection + adsorption kineticsconvection + ax. dispersion convection + dom.

mass transfer restc.

convection

Figure 4: Classification of column models.

initial or boundary conditions for the single columns.Using the node model, the dynamic models of the sin-gle columns are interconnected. This modular approachallows the use of different column models which are ap-propriate for the problem at hand.

Modeling of a Chromatographic Column

Modeling and simulation of chromatographic separationcolumns has been a topic of research since the 1950s.An overview can be found in Guiochon et al. (1994), aninteresting presentation of the phenomenological back-ground is given by Tondeur (1995). Mostly, apart fromfew outdated approaches using a stage model, the modelis formulated by a differential mass balance on a crosssection of the chromatographic column. Many differentmodeling approaches can be found in the literature, andthose can be classified by the phenomena which they in-clude and by their level of complexity (see Figure 4).

The simpler modeling approaches in the bottom ofFigure 4 can partly be solved analytically, and there-fore they can be evaluated very efficiently (see e.g. Rheeet al., 1989; Helfferich and Whitley, 1996; Zhong andGuiochon, 1996; Dunnebier and Klatt, 1998; Dunnebieret al., 1998). However, the idealistic assumptions onwhich they are based are very unrealistic for most realsystems. For optimal operation and control, a modelwhich is both accurate and computationally efficient isessential. The more complex process models mainly re-quire an appropriate numerical solution strategy. Themodels consist of a set of partial differential equations ofthe convection-diffusion (or hyperbolic-parabolic) type.Some properties of this type of equations, like shock lay-ers and almost discontinuous solutions, make the appli-cation of many standard discretization procedures dif-ficult. A lot of research has been devoted to the de-velopment of suitable spatial discretization schemes forchromatography column models in order to transformthe PDEs to a set of ODEs (see e.g. Kaczmarski et al.,1997; Kaczmarski and Antos, 1996; Strube and Schmidt-Traub, 1996; Poulain and Finlayson, 1993; Ma and Guio-chon, 1991; Spieker et al., 1998). Common to most of the

known approaches is the need for large computationalpower which makes it difficult, even with modern com-puters, to perform simulations substantially faster thanreal time.

Therefore, the first objective of our research on model-based control of chromatography processes was the for-mulation and implementation of suitable process models.We followed a bottom up strategy, proceeding from theideal model and increasing the complexity as far as nec-essary in order to achieve sufficient accuracy. From amathematical point of view, it is useful to distinguishchromatographic processes by the type of adsorptionisotherms, which describe the thermodynamic equilib-rium of the separation system. Processes with a linearrelation between the fluid phase concentration ci and thesolid phase concentration qi (Henry’s law)

qi = KH,i · ci (5)

lead to systems of decoupled differential equations whichare easier to solve than those with coupled nonlinear ad-sorption behavior, described for instance by competitiveLangmuir isotherms

qi =ai

1 +n∑

j=1

bjcj

· ci . (6)

Van Deemter et al. (1956) have shown that in case of alinear isotherm the effects of axial dispersion and masstransfer resistance are additive and can be incorporatedinto a single parameter, the apparent dispersion coef-ficient Dap. This results in the following quasi-linearparabolic partial differential equation for the fluid phaseconcentration of each component

γi∂ci

∂t+ uL

∂ci

∂x−Dap,i

∂2ci

∂x2= 0 (7)

The parameter γ is defined as

γi = 1 +1− ε

εKH,i (i = A,B)

where ε represents the column void fraction, and the in-terstitial velocity uL is assumed to be constant. Lapidusand Amundsen (1952) proposed a closed form solutionof this type of equation for a set of general initial andboundary equations by double Laplace transform. Fromthis, the dynamic SMB model can be generated by con-necting the solutions for each single column by the re-spective node model (see Dunnebier et al., 1998, fordetails of the implementation). We denote this imple-mentation as the DLI model (dispersive model for linearisotherms). It was shown that simulation times two or-ders of magnitude below real-time can be achieved whilereproducing both the results obtained with more com-plex simulation models and experimental results very ac-curately.


In order to generate both an accurate and computa-tionally efficient dynamic model also in the case of gen-eral nonlinear adsorption isotherms we first followed thesame approach as in the linear case by analyzing a modelwhere the non-idealities are lumped into a single param-eter. However, a closed-form solution is no longer possi-ble in the nonlinear case and the numerical solution usingstandard techniques for the spatial discretization did notimprove the computational efficiency substantially. For-tunately, there exists a very effective numerical solutionfor the detailled general rate model

∂ci

∂t−Dax

∂2ci

∂x2+ uL

∂ci

∂x+

3(1− ε)kl,i

εrp(ci − cp,i(rp)) = 0

(1− εp)∂qi

∂t+ εp

∂cp,i

∂t− εpDp,i

[1r2

∂

∂r

(r2 ∂cp,i

∂r

)]= 0

(8)with complex nonlinear isotherms proposed by (Gu,1995). Here, Dax represents the axial dispersion coef-ficient, cp,i the concentration within the particle pores,and qi the solid phase concentration which is assumedto be in equilibrium with the pore-phase concentration.rp denotes the particle radius, εp the particle poros-ity, kl,i the respective mass transfer coefficient, and Dp,i

the diffusion coefficient within the particle pores. A fi-nite element formulation is used to discretize the fluidphase, and orthogonal collocation for the solid phase.We applied this formulation to SMB processes result-ing in a superb accuracy and simulation times almosttwo orders of magnitude below real time (Dunnebier andKlatt, 2000). Due to the favorable numerical properties,this certain implementation of the complex general ratemodel in terms of computational efficiency even outper-forms the state of the art simulation models for SMB pro-cesses (equilibrium transport dispersive model—secondlayer of complexity in Figure 4) which follow a lineardriving force approach with a lumped mass transfer rate(e.g. Strube and Schmidt-Traub, 1996; Kaczmarski andAntos, 1996; Kaczmarski et al., 1997). Furthermore, interms of physical consistency the general rate model ismore exact, because the lumping of the different masstransfer phenomena is strictly valid only for systems withlinear isotherms and incorrect for substances with largemolecules (as they appear e.g. in bioseparations).

Optimal Operating Regime

State of the Art

Most of the known approaches for the determination ofoperating parameters for simulated moving bed separa-tion processes are not based on mathematical optimiza-tion methods and rigorous dynamic process models. Twomain approaches can be distinguished: The first is to de-rive short-cut design methodologies based on the equiva-lent TMB process. The second type of work uses heuris-tic strategies combined with experiments and dynamic

simulation of the SMB model.By transforming the switching time τ into an equiva-

lent solid flow rate

QS =(1− ε)Vcol

τ(9)

the operating parameters of a SMB process can be ex-pressed in terms of the operating parameters of the cor-responding TMB process. In case of the ideal modeland linear adsorption isotherms according to Equation 5,the TMB model can be solved in closed form. On thebasis of this solution Nicoud (1992) and Ruthven andChing (1989) introduced new operating parameters βi

and stated bounds for which the desired separation canbe achieved:

QF = QS(KH,A/βIII −KH,BβII)QEx = QS(KH,AβI −KH,BβII)QD = QS(KH,AβI −KH,B/βIV )

QIV = QS(KH,B/βIV +ε

1− ε)

1 ≤ βi ≤

√KH,A

KH,B, KH,B < KH,A

(10)

The β-variables were originally intended as slack vari-ables to formulate the conditions for proper operation ofthe separation unit as a set of inequalities. The boundson those variables result from retaining the adsorptionand desorption fronts in the appropriate zone of the unit,i.e. the β’s are safety factors for the ratio of the net massflow rate of the solid and the liquid phase. A detailledexposition of this subject can be found in Zhong andGuiochon (1998).

As the objective function in this framework, the mini-mization of the specific desorbent consumption QD/QF

or the maximization of the throughput QF can be used.Both objective functions lead to the same solution inthe idealistic case, which is QD = QF at the bound-ary of the feasible region with βi = 1 and QD/QF = 1.The maximum feed inflow QF is bounded by the maxi-mum allowed internal flow rates, which are limited by thepressure drop or the efficiency of the adsorbent. Stortiet al. (1993), Mazzotti et al. (1997), and Migliorini et al.(1998) derived a graphical short-cut design methodologybased on these ideas, the so-called triangle theory andextended the theory to systems with nonlinear adsorp-tion isotherms. This methodology is currently state ofthe art and has been applied to a large number of sepa-rations. Due to the unrealistic assumptions of the idealmodel, the triangle theory can only give initial guessesfor a feasible operating point of the process because itdoes not permit a reliable prediction of the product puri-ties which are the most important operating constraints.

To overcome the limitations of the ideal model, Maand Wang (1997) and Wu et al. (1998) presented a stand-


ing wave design approach based on an equilibrium disper-sive transport model of the TMB process. However, incase of systems with dominant mass transfer or diffusioneffects the minimization of the desorbent consumptionand the maximization of the throughput become conflict-ing. Additionally, as already mentioned, the quality ofthe prediction of the TMB model in general is only suffi-cient for a restricted range of applications. It is thereforenecessary to develop design strategies based on rigorousdynamic SMB process models. The operating points de-termined with the triangle theory or the standing wavedesign can be used as initial guesses for the computationof an optimal operating point.

Strube et al. (1999) describe a heuristic design strat-egy based on the dynamic simulation of a SMB processmodel. The optimization objective is formulated in a setof competing and partly contradictory targets which areapproached by parameter variation based on a heuristicstrategy. The major advantage of this method is the useof a realistic process model, the disadvantage is the needfor extensive manual simulation without any guaranteeto determine the optimum, and the need for an experi-enced user.

From the shortcomings and the limitations of the pre-viously described approaches, a list of desired propertiesfor a model-based optimization strategy for simulatedmoving bed chromatographic processes can be stated asfollows:

1. The algorithm has to be based on a realistic andefficient dynamic SMB process model.

2. The objectives for the optimization must be formu-lated in a single objective function avoiding compet-ing and contradictory targets.

3. The product quality has to be included explicitlyin the formulation since this is the most relevantconstraint for the operation.

4. The strategy has to be based on a mathematical op-timization procedure. This is the only way to ensurethat, in combination with suitable initial guesses, asolution at least close to the optimum can be ob-tained in finite time and without requiring too muchcostly expertise.

5. The procedure should be as general as possible, sothat it can be applied to a broad variety of SMBprocesses with linear or nonlinear adsorption equi-librium, any number of columns and any size ofequipment.

A New Model-Based Optimization Strategy

To the best of our knowledge, there are only two ap-proaches documented in the literature which treat thecalculation of the optimal operating regime in a rigor-ous mathematical formulation: the strategy suggestedby Kloppenburg and Gilles (1998), and the approach

proposed in the sequel which is explained in detail inDunnebier and Klatt (1999); Dunnebier et al. (2000).

We here consider the case where the plant design isfixed and the feed inflow is pre-specified, e.g. by theoutflow of an upstream unit. In this case, the desorbentinflow QD constitutes the only variable contribution tothe processing costs. By defining ck as the axial con-centration profile at the end of a switching period, andby expressing both the process dynamics between twoswitching operations and the switching operation itselfby the operator Φ

ck+1 = Φ(ck) (11)

we can write the following condition for the cyclic steadystate:

‖Φ(ck)− ck‖ ≤ δcss (12)

The purity requirements for the products in the extractand raffinate stream are formulated as inequality con-straints. Besides that, the efficiency and functionality ofmost adsorbents is only guaranteed up to a maximuminterstitial velocity, which results in a constraint Qmax

for the flow rate in the first section of the process. Theoptimization problem can then be stated as follows:

minQj ,τ

QD (13)

subject to

‖Φ(ck)− ck‖ ≤ δcss∫ τ

0

cA,Ex(t)cA,Ex(t) + cB,Ex(t)

dt ≥ PurEx,min∫ τ

0

cB,Raf (t)cA,Raf (t) + cB,Raf (t)

dt ≥ PurRaf,min

QI ≤ Qmax

Although the values for the optimization variables areconstant during the switching periods, the optimizationproblem is inherently a dynamic one, because the oper-ating regime of a SMB process is a periodic orbit andnot a steady state. From a mathematical point of view,this is due to the dynamic nature of Equation 12 which isa system of partial differential equations with switchinginitial and boundary conditions, constituting the crucialconstraint of the formulation given in Equation 13.

The natural choices of the degrees of freedom for thecalculation of the optimal operating regime are the desor-bent flow rate QD, the extract flow rate QEx, the switch-ing time τ and the recycle flow rate QIV . Due to thecomplex interactions, this results in a strongly coupledsystem dynamics, since each of the independent variablesaffects every zone of the process. Furthermore, it is im-possible to formulate explicit and independent boundson those variables, and the optimization problem is notwell-conditioned from a numerical point of view.


We therefore exploit the results of the analysis for theideal process model in case of linear isotherms. Equa-tion 10 sets up a transformation where the “natural de-grees of freedom” QD, QEx, QIV , and τ are replaced bythe variables βi. Due to the physical meaning of the βi

(safety margins for stable process operation), the feasi-ble region for the optimization problem is well-defined interms of the bounds for the transformed variables givenin Equation 10. Furthermore, due to the scaling effectand based on the feature, that any of the variables βi

mainly affects one zone of the process, the variable trans-formation results in a more favorably structured opti-mization problem. In case of nonlinear isotherms, theslope of the isotherm is concentration dependent andcan no longer be represented by the constant Henry-coefficient KH . In order to extend Equation 10 to thenonlinear case, we thus chose a reference concentrationto formulate the transformation:

QS =QF

∂gA

∂cA(cF )/βIII −

∂gB

∂cB(cF )βII

=(1− εb)AL

τ

QEx = QS

(∂gA

∂cA(cF )βI −

∂gB

∂cB(cF )βII

)QD = QS

(∂gA

∂cA(cF )βI −

∂gB

∂cB(cF )/βIV

)QIV = QS

(∂gB

∂cB(cF )/βIV +

ε

1− ε

)(14)

where

gi = εpcp,i + (1− εp)qi(cA, cB) (i = A,B)

and describes the equilibrium isotherm. The feed con-centration is a simple and suitable choice to calculate theslopes. The bounds on the variables βi as given in Equa-tion 10 are based on stability considerations for the idealmodel and linear adsorption equilibrium and can there-fore not simply be transferred to more complex systems.In case of nonlinear isotherms the bounds have to be re-laxed suitably, and in terms of the triangle theory, theregion limited by these bounds can be seen as a hull forthe shaped triangle. However, the transformation Equa-tion 14 helps to scale, structure and reduce the searchspace.

The crucial point in the solution of the optimizationproblem (13) is the efficient and reliable computation ofthe cyclic steady state. The solution of the associatedfix-point problem given by Equation 12 corresponds toa mathematically similar formulation for pressure swingadsorption (PSA), the dynamics of which shows somesimilarities to SMB processes. Three main approacheshave been identified and examined in the literature.Firstly, the solution can be achieved by direct dynamicsimulation (Picard iteration). Alternatively, Equation 12can be solved by a Quasi-Newton scheme (Unger et al.,

m

w.r.t.

in ( ) ...

,max

L

4 4

iD

i

ββ

Q

Pur PurPur Pur

Ex Ex, min

Raf Raf, min

=- ≤

≥≥

≤

1 41

1 1

∆

Pur

PurEx

Raf

τ 4 4 4Ex D IV, , ,

Given:- Plant configuration

- Product requirements- Feed properties

Calculation of the naturaloperating parameters

Initializationall columns empty, i=1

Dynamic simulation of the SMB process

for one switching period i

Calculation of the resultingpurities by integration

I=i+1

Feasible minimumfound ?

F Fi iax ax, ,+ − ≥1 δcss

Figure 5: Optimization algorithm.

1997; Kvamsdal and Hertzberg, 1997; Croft and Le Van,1994), or by discretizing the equations in space and timeand solving the resulting system of algebraic equations(Nilchan and Pantelides, 1998; Kloppenburg and Gilles,1998).

Even though there are many analogies between theSMB chromatography and PSA processes, the conver-gence of the direct dynamic simulation approach is muchfaster in the SMB case. For realistic SMB chromatogra-phy processes, the cyclic steady state is reached after afew hundred switching periods at the latest, whereas inthe PSA case possibly several thousand cycles have tobe evaluated. The inherent dynamics of the process aretherefore much faster. On the other hand, especially forsystems with nonlinear adsorption behavior and reactionkinetics, the system becomes very stiff and a very finegrid is needed, which makes the application of a globaldiscretization approach difficult. We therefore choosethe direct dynamic simulation as the most robust andefficient way for the calculation of the cyclic steady statewithin our optimization algorithm, which is schemati-cally shown in Figure 5.


Original Run 1 Run 2PurEx [%] 99.5 99.5 98.0PurRaf [%] 98.3 98.3 98.0rel. QD [%] 100 60.9 41.7QD [cm3/s] 0.8275 0.5043 0.3452QF [cm3/s] 0.2500 0.2500 0.2500QEx [cm3/s] 0.77989 0.4683 0.3489QIV [cm3/s] 0.2695 0.1489 0.1423

τ [s] 1337.4 2096.7 2675.2

Table 1: Optimization results for the separation ofphenylalanine and tryptophan.

We follow a staged sequential approach for the solu-tion of the dynamic optimization problem. The degreesof freedom βi are chosen by the optimizer in an outerloop and are then transformed back into flow rates andswitching times according to Equation 14. In the innerloop, the cyclic steady state is then calculated by directdynamic simulation of the rigorous SMB process model.The purity constraints are evaluated by integration ofthe elution profiles for the cyclic steady state. The non-linear program in the outer loop is small and can besolved by a standard SQP algorithm, while the requiredgradients are evaluated by perturbation methods. Theresults of the optimization are the optimal cyclic oper-ating trajectory (CSS), the minimum desorbent inflowfor the specified product purities, and the correspondingoperating parameters.

The optimization algorithm was tested for a numberof different separation systems with both linear and non-linear adsorption equilibrium. One impressive exampleis shown in Table 1, the separation of phenylalanine andtryptophan. Estimated model parameters and the ref-erence operating conditions were taken from Wu et al.(1998), where the original operating point was optimizedby a standing wave design. Because of the nonlinear ad-sorption equilibrium (Langmuir isotherms), the generalrate model was used within our optimization algorithm.The convergence to the cyclic steady state was achievedafter 75 switching periods in the average, and the SQPsolver in the outer loop converged after 12-15 steps de-pending on the initial point. This resulted in approx. 6-8hours CPU time for each run on a 400 MHz PentiumIIPC. In the first run, the purities for the operating pointobtained by Wu et al. (1998) were taken as constraints.It can be seen, that compared to the operating regimedetermined by the standing wave design technique, thedesorbent requirement was cut down by almost 40% us-ing the proposed optimization approach. Furthermore,the purity requirements can be directly specified. Theresults for run 2 show the economical impact of a reduc-tion of the purity specifications to 98% each.

Control Concept

In real applications, plant/model mismatch and distur-bances will lead to more or less pronounced deviationsfrom the optimal trajectory. However, the online op-timization under real-time requirements is not possi-ble with the computational power currently available.Therefore, a feedback control strategy based on suitabledynamic models and on-line measurement information isrequired in order to keep the process close to the optimaltrajectory.

Only few publications can be found in the open lit-erature which treat the automatic control of simulatedmoving bed chromatographic processes. Ando and Tan-imura (1986), Cohen et al. (1997), Hotier and Nicoud(1996), and Hotier (1998) deal with the basic control ofthe internal flow rates, which itself is a difficult task andforms the basis for the more advanced control strate-gies. The concepts described in Holt (1995), Cansellet al. (1996), and Couenne et al. (1999) propose feed-back control for certain operating variables (e.g. prod-uct purity, system yield) based on some concentrationmeasurements. They are predominantly applied to theseparation of aromatic hydrocarbons where on-line Ra-man spectroscopy (Marteau et al., 1994) can be utilizedto measure the specific concentration of the compoundat the outlet of the chromatographic columns. Thoseas well as the geometric nonlinear control concepts de-scribed in Kloppenburg and Gilles (1999) and Benthabetet al. (1997) are mainly based on a model for the cor-responding true moving bed (TMB) process, where thecyclic port switching is neglected, and thus rely heavilyon the applicability of the TMB model as a simplifiedmodel for the SMB process. This is particularly criticalfor SMB processes with a low number of columns—whichare more and more utilized in industrial applications inorder to reduce the investment costs—where the core-spondence between the SMB dynamics and the TMBapproximation may become poor. In a recent publica-tion, Natarajan and Lee (2000) suggest to apply repet-itive model predictive control on a reduced order linearstate space model of the SMB process. They optimizethe product yield for a given constraint on the puritiesusing the reduced linear model for the control calcula-tions. This approach definitely follows the right objec-tive, i.e. the control of the process in the vicinity of anoptimal operation point. However, in case of the SMBprocess, yield optimization even when considering thepurity constraints does neither generically imply the eco-nomic optimum nor guarantee a stable process operationin the long run. Furthermore, the range of validity forthe linear approximation and its impact on the controlperformance is not explicitly considered.

In order to overcome some of the shortcomings andlimitations mentioned above, we proposed a two-layercontrol architecture which is shown schematically in Fig-


Model-BasedOptimization of

Operating Parameters

Identification ofLocal Models

Model-Based Parameter Estimation

Model-BasedControl

SMB Process Process

On-Line Monitoringand Control

Off-Line Optimization

Opt

imal

Tra

ject

ory

and

Ope

ratin

g Pa

ram

eter

sInputs

Reduced Process Model

Model Parameters

Simulation Data

Measurements c(t)

Figure 6: Control concept for SMB processes.

ure 6 (Klatt et al., 2000). The top layer features theoff-line calculation of the optimal operating trajectoryas described in the previous section, combined with anon-line estimation of the respective model parametersbased on inline concentration measurements. The pur-pose of this estimation algorithm is twofold: It providesactual and reliable values for the model parameters, andtogether with the dynamic simulation model, it enablesthe monitoring of the complete axial concentration pro-file which is not directly measurable in the interior of theseparation columns. If there is a too large discrepancybetween the actual parameters and the parameter valuesused in the trajectory optimization (e.g. caused by ag-ing of the adsorbent material), a new optimization run isinitiated. The remaining control task then is to keep theprocess along the calculated nominal trajectory despitedisturbances and plant/model mismatch, caused e.g. bysmall and non-persistent perturbations of the system pa-rameters. This task is performed by the bottom layerwhere identification models based on simulation data ofthe rigorous process model along the optimal trajectoryare combined with a suitable local controller. The real-ization of the remaining elements of the proposed controlstructure is explained below.

The model-based parameter estimation utilizes on-linemeasurement data from measurement devices located inthe product outlets or in the connecting pipelines be-tween the columns. Because of the high costs of on-line concentration measurements, measurements locatedafter each single column will generally not be feasible.Typically, up to four measurement points can be foundin real plants, if online measurement is employed at all.The devices are system specific and use either spectro-scopic methods or combined measurement techniques todetermine the concentration of each single species.

Starting from a complete set of model parameters de-termined in a priori experiments, the objective of theonline estimation is to fit the model to the real process.The model parameters can be categorized in kinetic pa-rameters (describing mass transfer, diffusion, dispersion)

and adsorption parameters (constituting the adsorptionisotherms). Due to the limited measurement informa-tion available, we solve a reduced parameter estimationproblem, where one crucial parameter per class and com-ponent is adapted by optimizing a quadratic cost func-tional

min J =∫ t2

t1

(ci,meas(t)− ci,sim(t))2dt, i = A,B

(15)where the measured outlet concentrations ci,meas arecompared to the outlet concentrations ci,sim which aredetermined by the solution of the simulation model. Thenumber and arrangement of the measurement devicesdepend on the mixture to be separated. In Zimmeret al. (1999) a measurement setup and estimation algo-rithm was proposed for systems with linear adsorptionisotherms which is briefly sketched in the application ex-ample below.

The rigorous dynamic SMB model, consisting of PDEs(eqs. (7) and (8)) with switching initial and boundaryconditions, is not well suited for a standard controllerdesign. Thus, we base the trajectory control on a lo-cal model. In order to get rid of the hybrid systemdynamics in the bottom layer of the control concept,we consider the reduced model as a discrete time modelwith the sampling interval equal to one switching period.The model predicts one characteristic parameter of theconcentration profile per switching period, and thereforedoes not require the consideration of the discontinuitiesintroduced by the switching operation.

Because of the favorable properties of the nonlineartransformation of the input space (14) which is utilizedin the trajectory optimization, the variables βi, resp. thedeviations from their values on the nominal trajectory,are also chosen as inputs of the reduced model. This al-lows a variable switching time which is essential in case ofinflow disturbances because feed flow-rate changes can becompletely compensated only by a corresponding adap-tation of the virtual solid stream. Furthermore, with anappropriate choice of outputs, the nonlinear transforma-tion helps to create a nearly decoupled system dynamics,particularly for SMB separation processes with linear ad-sorption equilibrium.

One characteristic indicator of the separation perfor-mance of a SMB unit is the axial position of the adsorp-tion and desorption fronts of the two components at acertain moment, e.g. at the end of a switching period (seeFigure 3). They are referred to as front positions pi inthe sequel. Extensive simulation studies showed that thefront positions do not only provide a suitable indicatorfor the long term stability of the system, but addition-ally exhibit an excellent correspondence with the productpurities which are the relevant output variables. How-ever, from a system dynamics point of view, the productconcentrations are ill-suited as controlled variables due


to their strongly delayed response and their lack of sen-sitivity to changes in the manipulated variables. Thus,the deviations of the front positions from their nominalvalues on the optimized trajectory are chosen as outputsof the local model.

The axial concentration profile is not directly mea-surable, but assuming an on-line concentration measure-ment at the end of each column, an approximation ofthe complete axial concentration profile can be obtainedby connecting all measurements of the elution profilesduring one switching period. We call this representationthe Assembled Elution Profile (AEP). If the system is atits cyclic steady state, the AEP can also be obtained byusing the information of a single measurement over ncol

switching periods. In case of a disturbed system, theAEP obtained from a system with less than ncol mea-surements becomes in some respect time variant, sincethe parts of the AEP are recorded at different instantsduring the transition, and the AEP constructed dependson the location of the measurements in the loop. Nev-ertheless, a reduction of the number of measurementsbelow ncol is possible to a certain extent, and the ef-fect on the AEP is comparable to a low pass filter. Therequired number and the exact location of the measure-ment devices depend on the specific system dynamics.

Three different methods for the calculation of the frontpositions from the AEP were evaluated:

I. Functional approximation of the fronts, followed bythe calculation of the inflection point of this func-tion.

II. Surface quadrature to calculate the center of gravityof the fronts.

III. Wavelet analysis to determine the inflection pointsof the fronts.

For method I, a variation of the Gauss error functionproved to supply a good approximation for the frontsof SMB separation processes with linear and moderatelynonlinear adsorption isotherms. The desorption frontsare approximated by

f(t) =12

[1 +

2√π

∫ k(t−z)

0

e−θ2dθ

], (16)

the adsorption fronts by

f(t) =12

[1− 2√

π

∫ k(t−z)

0

e−θ2dθ

]. (17)

The two parameters of the function, k and z are calcu-lated by a least-squares fit to the available measurementswhich is updated in each switching period, and the in-flection point of the function which represents the frontposition is determined analytically afterwards.

In the second method, an upper and lower bound forthe concentration front is assumed first. Then the front

upper bound

lower bound

front position

A1

A2

Figure 7: Evaluation by surface quadrature.

position is defined as the point where A1 = A2 (see Fig-ure 7). The integration is approximated by a summationover the available measurement points.

Wavelet analysis (III) is based on scale and positiondependent coefficients of the form

C(a, k) =∫ ∞

−∞f(t)Ψ(a, k, t)dt (18)

and allows the multi-scale analysis and representationof arbitrary time series. It also enables the direct iden-tification of inflection points from noisy data (see, e.g.Daubechies, 1992). The method of choice mainly de-pends on the characteristics of each different separationprocess, i.e. the shape and the position of the adsorp-tion and desorption fronts. The different methods haveto be evaluated trading off the computational require-ments for the calculation, the robustness against noise,and the physical meaning of the calculated positions.

Having defined the inputs and outputs, the structureof the local model and the input signal for the identifica-tion need to be specified. In our work, we utilize theMatlab System Identification Toolbox (Ljung, 1995)to perform the necessary computations. To obtain thenecessary data for identification, the rigorous simulationmodel is excited around the nominal operation regimewith a random binary signal. For MIMO identification,non-correlated signals in the different channels are to bepreferred, which can be achieved with a slight modifica-tion of the Pseudo Random Binary Signal, the PseudoRandom Multistep Signal (Isermann, 1992). We testedprediction error models as well as subspace state spaceidentification methods and found both approaches suit-able for the SMB separation processes we have investi-gated so far.

After identification of the local linear model any stan-dard linear control design approach or linear model pre-dictive control can be used to realize the controller on


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System Parametersdcol 2.6 cm L 53.6 cmε 0.38 dp 0.0325 cmρ 1 g/cm3 η 0.0058

KA 0.54 KB 0.28Nominal Operating Parameters

Concentration of feed cF 0.5 g/cm3

Feed flow rate QF 0.02 cm3/sDesorbent flow rate QD 0.0414 cm3/sExtract flow rate QEx 0.0348 cm3/sRecycle flow rate QIV 0.0981 cm3/sSwitching time τ 1552 sPurity fructose 99.95%Purity glucose 99.95%β [ 1.1371 1.0993 1.1164 1.1220 ]

Table 2: System and operating parameters for theseparation of fuctose and glucose on an 8-columnSMB laboratory plant.

the bottom layer. At a glance, the structure of the tra-jectory control loop is depicted in Figure 8. The vari-ables represent deviations from the nominal trajectory:four input variables ui = ∆βi, and deviations of thefour front positions (normalized to the scaled length ofall interconnected columns between 0 and 1) as outputsyi = ∆pi.

Application Example

As an application example, we consider the separationof a fructose/glucose mixture on an 8-column laboratoryscale SMB plant. Performing the proposed optimizationalgorithm, the plant was optimized for a product purityof 99.95% both in the extract and the raffinate stream.From a practitioner’s point of view, this is an excep-tionally high purity requirement for a sugar separation.Those are normally operated with a purity specificationof at most 99%. However, the higher the purity specifi-cation the more challenging the control problem, because

Con

cent

ratio

n [g

/cm

]3

Nominal ParametersK -5%, K +5%K -10%, K +10%

A B

A B

Length [cm]

Figure 9: Axial profile (CSS, end of period) of theSMB fructose/glucose separation and its sensitivityto parameter perturbations.

the system is operated much closer to the stability mar-gins. Furthermore, in pharmaceutical separations thepurity requirements are generally very high. Thus wedecided to operate the sugar separation under these highpurity requirements to demonstrate the feasibility of theproposed approach.

The system and operating parameters are shown inTable 2. The plant is operated at 60 ◦C. The liquiddensity can be considered as constant for the given feedconcentration, and the adsorption isotherms are well de-scribed by Henry’s law (5). Thus, the DLI model (7) isutilized within the optimization and control framework.The optimal operating trajectory and parameters weredetermined using the optimization algorithm describedabove. Here, convergence to the cyclic steady state wasachieved after 65 switching periods on the average, andthe SQP solver in the outer loop converged after 10-15steps depending on the initial point. This resulted in ap-prox. 4 hours CPU time on a 400 MHz PentiumII PC.Because of the complex hybrid system dynamics, steadystate multiplicities are possible in principle. We thereforetested different initial points within the feasible region,but for the separation task at hand they all converged tothe optimal solution reported in Table 2. However, thisis of course system dependent and has to be thoroughlyinspected for each individual separation task.

Figure 9 shows the axial concentration profile for theoptimal operation mode and its sensitivity against varia-tions of the adsorption isotherm parameters, caused e.g.by a varying feed quality. We here perturbed the systeminto the direction where the separation becomes moredifficult, i.e. decreasing KA and increasing KB . Table3 depicts the corresponding product purities. It is obvi-ous, that the optimal operating regime is quite sensitiveagainst parameter variations and thus the results of thetrajectory optimization illustrated above essentially de-pend on a reliable determination of the model parame-ters.

In Zimmer et al. (1999) an on-line parameter esti-mation algorithm for systems with linear adsorptionisotherms based on concentration measurements onlyin the product outlets was proposed. A combination


Nominal KA -5% KA -10%KB +5% KB +10%

Extract 99.95% 99.50% 97.04%Raffinate 99.95% 99.61% 96.71%

Table 3: Product purities corresponding to the con-centration profiles shown in Figure 9.

of a polarmeter and a densimeter is used to determinethe specific concentrations of fructose and glucose (Al-tenhohner et al., 1997). Assuming the validity of theDLI model (7), the system has, in the case of a binaryseparation, four parameters describing the physical be-havior, which are γi and Dap,i(i = A,B). γ incorporatesthe column void fraction ε and the respective adsorp-tion parameter, the apparent dispersion coefficient Dap

lumps the kinetic effects. Thus, no further reduction ofthe parameter space is necessary in this case.

A linear adsorption isotherm implies that the com-ponents do not interact with regard to their adsorptionbehavior. Therefore, the two parameter sets (γA, Dap,A)and (γB , Dap,B) can be determined seperately by solv-ing the least-squares problem posed by Equation 15. Thesimulated axial concentration profile ci,sim is determinedby solving the PDE (7) for each column and each compo-nent. Unfortunately, the boundary and initial concentra-tions for each switching period are only partly availablefrom the measurements. Therefore, a special approxima-tion strategy was developed using the concentration offructose and glucose in the extract and raffinate outflowin two consecutive switching periods. This parameter es-timation additionally supplies an approximation of thenon-measurable concentration profiles in the interior ofthe columns prior to the measurement. Because the mea-surement devices are located in the product outlets whichare periodically switched downstream, the complete ax-ial concentration profile can be reconstructed within acomplete cycle (i.e. 8 switching periods).

The major task of the parameter estimation is to sup-ply a set of actual model parameters which can be usedfor the trajectory optimization. Also they indicate ifthere is too large a deviation from the original set ofparameters and thus a new optimization run has to beperformed. However, as online optimization is not possi-ble for the time being, smaller deviations and other dis-turbances have to be adjusted by the trajectory controlloop.

For the fructose/glucose separation, a linear time in-variant model of the ARX type was chosen as the localdynamic model:

A(q)y(t) = B(q)u(t) + e(t) (19)

where A and B are 4 × 4 matrices containing polyno-mial functions of the discrete time shift operator q in

Figure 10: Model validation.

each element. The amplitude of the input signal ∆βi

for excitation of the rigorous DLI model was chosen tobe 10% of the nominal β-value. This input range coversnearly the whole range of practically reasonable operat-ing conditions. A first estimate of the necessary systemorder was obtained by the evaluation of step responsesof the DLI model, which was then adapted based on theperformance of the identified model. For the elementsof the matrices A and B, polynomials of at most sec-ond order proved to be sufficient. The performance ofthe model was tested with a second validation data set.The prediction of the front positions (desorption frontof glucose (component B) as one example) of the rigor-ous and of the reduced model are compared in Figure 10.The prediction horizon is infinite here, i.e. the simulationmodels are reconciled only for initialization and then runindependently. The approximation is very good, only inregions far from the nominal trajectory, where the as-sumption of linearity is violated, slight deviations occur.

As postulated, the system dynamics are diagonallydominant in the vicinity of the optimal operating tra-jectory. As mentioned above, in principle, any standardlinear controller design method could be applied. Forthis example, we transformed the ARX model (19) toa z-domain transfer function representation and appliedinternal model control following the guidelines proposedin Zafiriou and Morari (1985) to design a discrete timeSISO controller for each channel (i.e. the control of eachsingle front position by manipulating the respective β-value). The design of the internal model controllers isvery straightforward for the system at hand and allowsthe direct integration of the system dynamics into thecontroller.

In the conventional feedback representation, the inter-nal model controller is of the form

C(z) =F (z)GC(z)

1− F (z)GC(z)GM (z)(20)


where GC is the internal model controller, GM is thetransfer function of the respective model, and F is afirst order filter

F (z) =z(1− α)z − α

(21)

where α has to be suitably chosen in the range 0 ≤ α <1. After a cancellation of numerically induced pole/zeropairs, we obtained the following transfer functions forthe respective input-output channels

GM1 =

0.0349 · z2

(z − 0.145)(z − 0.8011)

GM2 =

0.0256 · z2

(z − 0.7058 + 0.1106j)(z − 0.7058− 0.1106j)

GM3 =

−0.0365 · z2

(z − 0.6967 + 0.2277j)(z − 0.697− 0.2277j)

GM4 =

−0.0143 · z2

(z − 0.5743)(z − 0.8270)(22)

In this case, direct inversion of GM is possible and theinternal model controller is realized by

GCi =

1z· 1GM

i (z)(23)

The filter parameters were selected from simulation stud-ies as

α1 = 0.4, α2 = 0.4, α3 = 0.6, α4 = 0.6

and the range of the trajectory controller was chosen as a±10% deviation from its value at the optimal operatingregime for each of the βi.

In the following, some simulation studies are presentedin order to demonstrate the capabilities of the proposedtrajectory control. For the simulation scenarios, the gen-eral rate mode according to Equation 8 is used to repre-sent the real plant. This introduces a structural devia-tion between the DLI model on which the control designis based and the simulation model for the validation ex-periments. The scenarios depicted below were performedboth without and with noise added to the concentrationmeasurements. White noise with a standard deviation of1% of the maximum concentration value was assumed.

Flow-Rate Disturbance

In this scenario it is assumed that a step disturbancein each of the internal flow rates occurs one at a timeto analyze the effect on all front positions and possiblecoupling. Under practical considerations, these distur-bances may be caused by an irregularity in the pumpingand piping system or by an offset in the basic controllayer. Each of the subplots in Figure 11 shows the ma-nipulated variable and controlled variable moves result-ing from the disturbance in the respective input channel,

-10

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5x 10 -3 VWHS�GLVWXUEDQFH�X� VWHS�GLVWXUEDQFH�X�

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0

0.05

0.1

Periods60 70 80 90

Periods

0

5

10

x 10 -3 VWHS�GLVWXUEDQFH�X� VWHS�GLVWXUEDQFH�X�

120 130 140 150-0.05

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Periods180 190 200 210

Periods∆p

[-]

∆p[-

]

∆β [-

]∆β

[-]

1 2

21

3

4

43

Figure 11: Closed loop reaction on input disturbancescenario.

where a 10% variation in terms of the β-variable corre-sponds to a 10-20% variation of the internal flow rates.

As can be clearly seen, the closed loop reacts in amostly decoupled manner. The front position deviationis suppressed quickly by the controller, the maximumvalue of the deviation being about 0.01 in terms of scaledlength. Though minor effects of coupling can be seenfor disturbances in channels 2 and 3, the effects remainsmall. As expected, the results of the same sequenceunder noisy measurements show less smooth control ac-tions and a slight wiggling of the front positions due tothe noise’s influence on the calculation of the inflectionpoints. However, the results are principally the samewith no major deterioration as compared to the noise-free setting. The corresponding plot is omitted here forthe sake of brevity.

Feed Batch Change

In real-world applications a separation process followsprior production steps which might be continuous or dis-continuous, a typical example in the Life Science contextbeing a fermenter or a batch reactor. Thus a continuousSMB chromatography has to face changes in the feedbatch quality. In Table 3 and in Figure 9 we alreadyconsidered this scenario, which corresponds to a changeof the characteristic adsorption parameters, and its im-


99.94

99.95

99.96

99.97

purit

ies

[%]

-5

0

5x 10-3

0 5 10 15 20 25 30 35 40-0.1

-0.05

0

0.05

0.1

Periods

Extract

2

2

4

4

1

1

3

3

Raffinate

∆p[-

]∆β

[-]

Figure 12: Closed loop response to change of feedbatch.

99.94

99.95

99.96

99.97

-5

0

5

x 10-3

0 5 10 15 20 25 30 35 40-0.1

-0.05

0

0.05

0.1

Periods

Raffinate

Extract

2

4

1

3

1

4

3

2

purit

ies

[%]

∆p[-

]∆β

[-]

Figure 13: Closed loop response to change of feedbatch in the presence of measurement noise.

pact on the axial concentration profile and the productpurities within the cyclic steady state operating regime.

Starting form the optimal operating regime, after fiveperiods we again perturbed the characteristic adsorptionparameters by ±5%, but now with the feedback controlloop closed. The results are shown in Figure 12 for thecase of noise-free concentration measurements. The tra-jectory controller quickly suppresses the disturbance anddrives the corresponding front positions (controlled vari-ables) back to their nominal values without any offset.The axial concentration profile therefore does not changeits position as in the scenario without feedback control,but is kept in the immediate vicinity of the optimal pro-file. As a result, the product concentrations which arethe essential quality parameters are thus indirectly con-

trolled and kept close to their optimal values. Figure 13shows the same scenario under noisy measurements. Theresults are qualitatively the same and the effect on theproduct concentrations remains very small compared tothe noise-free setting.

Conclusions and Further Research

Simulated moving bed chromatographic separation pro-cesses pose a challenging control problem because of theircomplex hybrid dynamics. In this contribution, we pro-posed an integrated, model-based approach for the op-timal operation and advanced control of a SMB chro-matographic process. To our knowledge, it is the firstapproach which is based solely on a rigorous dynamicprocess model and does not utilize the simplified TMBmodel in any of its elements. Furthermore, the proposedstrategy explicitly aims at controlling the process closeto its economic optimum.

The control architecture features two cascaded lay-ers where in the top layer the optimal operating trajec-tory is calculated off-line by dynamic optimization. Dueto the associated computational complexity, online op-timization based on a rigorous dynamic process modelis currently not possible. The respective model param-eters are determined by an online estimation algorithmwhich, in addition to providing actual and reliable modelparameters for the trajectory optimization, enables theonline monitoring of the non-measurable concentrationprofile within the separation columns. If the discrep-ancy between the actual parameters and the parametervalues on which the trajectory optimization was basedon becomes significant, a new optimization run shouldbe initiated. The task of the second layer of the controlarchitecture is to keep the process along the optimizedtrajectory. This trajectory control is based on a locallinear model which is identified from simulation data ofthe rigorous dynamic process model along the optimizedtrajectory.

The capabilities of the proposed control concept couldbe demonstrated in a set of simulation studies for theseparation of fructose and glucose on an eight columnSMB plant which is currently operated manually in thePlant Design Laboratory at the University of Dortmund.Due to the particular choice of the inputs and outputs,the dynamics of the local ARX model are approximatelydecoupled in the vicinity of the optimal operating tra-jectory, and thus a simple decentralized internal modelcontroller could be used for the trajectory control. Itperforms satisfactorily, rejecting different types of dis-turbances both in a noise-free setting and with noisymeasurements.

The overall control architecture is currently being im-plemented in an industrial standard control system. Itconsists of a decentralized control system (DCS) of theSIEMENS S7-400 series (CPU S7-414-2DP) and the


Windows Control Center (WinCC) as human machineinterface. The algorithms and programs of the compo-nents of the control structure are integrated via the C-script interface, Global Script. Due to the fact, that thetrajectory optimization is performed off-line, this taskcan be swapped out to another machine.

The experimental verification of the proposed ap-proach on the laboratory SMB plant is planned for thenear future, followed by the extension of the conceptto a real pharmaceutical separation. However, for thislatter task, some open issues have to be addressed be-cause pharmaceutical separations may show a very com-plex nonlinear adsorption behavior and not all of thecomponents of the proposed concept are currently com-pletely adapted to this challenge. The off-line trajectoryoptimization can be performed for arbitrary complexisotherms, the successful extension to reactive adsorp-tion processes was recently reported in Dunnebier et al.(2000). The online parameter estimation concept is cur-rently realized only for systems with negligible couplingof the adsorption isotherms. This has to be extended inorder to get a reliable online estimate for the interactionparameters of the adsorption isotherm in case those havea significant impact on the system dynamics. Due to thelimited measurement information, a reduced parameterestimation problem has to be posed in the case of com-plex nonlinear isotherms, restricting the online adapta-tion to the most significant model parameters only. Thedesign strategy for the inner control loop is also subjectto revision because the range of validity for the locallinear model may be too small for very nonlinear andstrongly coupled systems. Currently, the combinationof NARX models based on neural networks and nonlin-ear model predictive control as an alternative to realizethe trajectory control loop for this type of separations isinvestigated. Furthermore, the correspondence betweenthe characteristic points used for control and the productpurities—which is excellent in case of the sugar separa-tion example—has to be validated for different types ofsubstances and the need for alternative calculation meth-ods or even different characteristics to represent the re-spective fronts has to be inspected.

The above mentioned issues are addressed in currentresearch projects and we expect a medium term solu-tion for most of the problems. However, in the long runalso the cascaded control structure is under discussion.A real time optimization based on a dynamic processmodel redundantizing the inner loop would be desirable.To achieve this goal, the computational efficiency of thedynamic optimization has to be enhanced by at least oneorder of magnitude. For this, both the numerical solu-tion of the model equation as well as the calculation ofthe cyclic steady state have to be improved. Adaptivegridding for the solution of the model equations in com-bination with the direct solution of the sensitivity equa-tions seems to be a promising approach and provides an

interesting area for the cooperation between engineersand applied mathematicians.

Acknowledgments

We thank G. Zimmer and M. Turnu for their valu-able contribution to the parameter estimation and re-duced model identification issue, respectively. The finan-cial support of the Bundesministerium fr Bildung undForschung under grant number 03D0062B0 and of theDeutsche Forschungsgemeinschaft under grant numberSCHM 808/5-1 is very gratefully acknowledged.

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