Chemical Engineering Science 80 (2012) 253–269

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Chemical Engineering Science

0009-25

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n Corr

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journal homepage: www.elsevier.com/locate/ces

Solutions to inversion problems in preferential crystallizationof enantiomers—part I: Batch crystallization in a single vessel

Steffen Hofmann a,n, Jorg Raisch a,b

a Technische Universitat Berlin, Fachgebiet Regelungssysteme, Sekretariat EN 11, Einsteinufer 17, 10587 Berlin, Germanyb Max Planck Institute for Dynamics of Complex Technical Systems, Systems and Control Theory Group, Sandtorstr. 1, 39106 Magdeburg, Germany

H I G H L I G H T S

c Focus on analytical methods for control of preferential crystallization.c Orbital flatness can be excluded for preferential crystallization moment models.c Dynamic inversion for realization of desired final crystal size distributions.c Idealizing assumptions can be justified which greatly simplify dynamic inversion.c Numerical example using a detailed process model.

a r t i c l e i n f o

Article history:

Received 31 January 2012

Received in revised form

29 May 2012

Accepted 10 June 2012Available online 23 June 2012

Keywords:

Enantiomers

Preferential crystallization

Population balance

Moment model

Dynamic inversion

Orbital flatness

09/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ces.2012.06.021

esponding author. Tel.: þ49 30 314 79281; fa

ail address: hofmann@control.tu-berlin.de (S.

a b s t r a c t

In this series of two papers, we investigate inversion techniques for models describing the crystal-

lization of conglomerate forming enantiomers, with application for preferential crystallization. Here, in

Part I, a simple batch process in one vessel is considered. In Part II, the results will be extended to a

technically important setup consisting of two coupled crystallizers. The research presented here builds

on prior work on the use of orbital flatness concepts for trajectory planning and control of single-

substance crystallization. Based on a known necessary condition for flatness, we show that moment

models for crystallization of enantiomers typically do not exhibit the orbital flatness property.

However, we show that a scaling of time introduced in our prior work is still useful. We present

inversion results for the realization of prescribed final-time crystal size distributions (CSDs). Due to the

lack of orbital flatness, these results involve integration of a zero dynamics. A number of idealizations of

the process model are investigated and shown to be suitable for appropriate problem settings. We

present a numerical example which demonstrates the exact as well as the simplified inversion

procedures, and also justifies application of the simplified techniques.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Preferential crystallization is an attractive way for the separationof enantiomers, chiral substances, which share many physical andchemical properties, but may differ, e.g., in their metabolic effects.The crystallization is out of a solution, and its principle is similar tothat of crystallization of a single substance, where mass is transportedfrom the liquid phase, i.e., the solution, to the solid phase, i.e., thecrystalline material. Growth of existing crystals, and nucleation ofnew crystals, are the main phenomena, which influence, over time, apopulation of crystals. We do not consider here other effects suchas conglomeration and breakage. There are a number of ways to

ll rights reserved.

x: þ49 30 314 21137.

Hofmann).

influence, or control, the crystallization process. Frequently, thetemperature of the slurry inside a tank is considered as a manipulatedvariable. This makes sense as often the dynamics of the temperaturetransfer is much faster than that of the crystallization. Also, a low-level controller can help in tracking a temperature reference, providedthat the temperature in the slurry (consisting of solution and crystals)can be measured. A common way to speed up crystallization andmake the process more reproducible consists in the addition of seedcrystals.

Here, in contrast to single substance crystallization, two species ofenantiomers, E1 and E2, are present in the system. Instead of oneconcentration in the liquid phase, two mass fractions are now defined,one for each enantiomer, along with two supersaturations, and so on.The sub-processes corresponding to the crystallization of enantiomersE1 and E2, respectively, may be coupled via their liquid and solidphases.

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269254

Most chiral substances can be classified into one of two types(Lorenz et al., 2006). In this work we focus on conglomerates.‘‘For conglomerates, there exists an attractive possibility to directlycrystallize pure enantiomers from a racemic solution’’ (Lorenzet al., 2006). This means that, under equilibrium conditions,pure E1 crystals and pure E2 crystals can coexist with a racemic(50%/50%) solution. Rather than of mixed crystals, the solidmaterial consists of a mixture of individually pure crystals. Forthe other type, compound forming enantiomers, it is only possible‘‘[y] to obtain pure solid enantiomers, once a certain minimumenrichment is available’’ (Lorenz et al., 2006).

Crystallization of conglomerates becomes preferential by add-ing seed crystals for only one enantiomer, the preferred enantio-mer, into a vessel containing the solution (which may be racemic)and operating in the so-called metastable region. The latter isusually referred to as a certain range of temperature and massfractions, where, qualitatively, growth and secondary nucleationare predominant, and primary nucleation is negligible. At thebeginning of the process, only the seed crystals will grow, andnuclei of the same species as the seed crystals will form bysecondary nucleation. Crystallization will thus be in favor of thepreferred enantiomer. At some time nucleation of the other,counter enantiomer becomes significant, and the process has tobe stopped in order to avoid unacceptable impurification. Thequalitative idea of a metastable region with sharp boundaries isnot sufficient for adequate prediction of this time instant, and,more generally, for control of the system. Rather than that, anaccurate mathematical description of the effects of growth andnucleation is required.

Fig. 1 illustrates a simple batch process for preferential crystal-lization in a stirred tank. Elsner et al. (2005a,b, 2009) havedescribed two modes of operation for the technical realizationof separation of conglomerates by preferential crystallization,with the goal of obtaining pure fractions of both kinds. The cyclic

mode can be thought of as a sequence of simple batch processes.Before each batch, seed crystals of either E1 or E2 are added. Afterthe batch this enantiomer is harvested, and the solution isreplenished (e.g., by adding a racemic mixture of E1 and E2).In the subsequent batch, the roles of E1 and E2 are interchanged,and so on. In a coupled mode, two vessels, A and B, are operatedsimultaneously. In one crystallizer, pure E1 shall be produced, andpure E2 in the other one. Crystal free liquid is exchanged betweenthe vessels. Simply speaking, this measure increases the amountof time during which the two processes can be kept in themetastable region. Of course, the batch can be repeated overand over, as long as enough racemate to replenish the solutionand enough seed crystals are provided.

Control and optimization of (continuous and batch) crystal-lization processes has gained a lot of attention since the 1970s.Modeling and solution methods dealing with the reconstructionof the evolution of crystal size distributions (CSDs), or of CSDproperties, are an essential prerequisite for many of the devel-oped feedforward and feedback schemes. Population balance

Fig. 1. Preferential crystallization in a simple batch.

equations (PBEs) are frequently used to describe the dynamicsof particulate processes (e.g., Randolph and Larson, 1988;Ramkrishna, 2000). Discretization-based methods can be used tocompute directly the evolution of CSDs, such as finite differenceschemes (e.g., Kumar and Ramkrishna, 1996) or high resolutionfinite volume schemes (e.g., Gunawan et al., 2004). On the otherhand, it was recognized early (Hulburt and Katz, 1964) that, forsuitable PBE models, a closed set of a finite number of ordinarydifferential equations (ODEs) can be derived via the so-called(standard) method of moments, which describes the evolution ofthe principal moments of the CSD. It should be noted thatreconstructing a CSD from a finite number of its moments is notunique (John et al., 2007).

Features of more detailed population balance equations, suchas conglomeration and breakage, or growth rates depending onparticle size, prevent the derivation of closed ODE models basedon the (standard) method of moments. A number of methodshave been suggested to overcome this problem, e.g., the quad-rature method of moments (McGraw, 1997). Certain techniquesallow the derivation of approximate low-order ODE models.These include, for example, the method of weighted residuals(Rawlings et al., 1992; Singh and Ramkrishna, 1977; Ramkrishna,2000), or approaches which have recently been suggested byQamar et al. (2009, 2010, 2011) and Bajcinca et al. (2011).

Trajectory planning and optimal control of batch crystallizers,starting with classical contributions such as, e.g., Mullin and Nyvlt(1971), Ajinkya and Ray (1974), Jones (1974), still remains anactive topic (e.g., Ma et al., 2002; Sarkar et al., 2006; Aamir et al.,2009; Bajcinca et al., 2011; Bajcinca and Hofmann, 2011).For example, in Aamir et al. (2009), the quadrature method ofmoments is combined with the method of characteristics tosimulate the evolution of CSDs in a batch crystallization processwith size-dependent growth rate. This feedforward simulationtechnique is subsequently used within a dynamic optimizationscheme aiming at approximately achieving specified target CSDsunder additional constraints, such as process duration; in thisscenario, temperature is considered as the controlled variable.PBEs with growth rate depending on particle size are alsoconsidered in Bajcinca et al. (2011). There, transformation of timeand length coordinates enables efficient computation of tempera-ture profiles for realization of specified CSDs.

With the realization that particulate processes are prone tosignificant uncertainties and disturbances, a lot of recent work putsmuch emphasis on using advanced feedback control schemes. Forexample, based on geometric control theory, a nonlinear outputfeedback controller is designed for a continuous crystallizer in Chiuand Christofides (1999). This approach involves a systematic modelreduction technique based on the combination of the method ofweighted residuals and approximate inertial manifolds. An extendedLuenberger-type observer is used to estimate immeasurable statevariables. In subsequent papers, model uncertainty (Chiu andChristofides, 2000) and actuator constraints (El-Farra et al., 2001)are explicitly considered.

In Vollmer and Raisch (2002), Motz et al. (2003), a continuouscrystallizer model involving size-dependent growth, attrition,fines dissolution and reflux of undersaturated solution is consid-ered, which shows oscillatory behavior. The fines dissolution rateis considered as the manipulated variable. The controller design,which aims at stabilizing an unstable steady state, follows a latelumping approach: an H1 robust performance problem is solvedbased on the linearized infinite-dimensional model, and thecontroller is subsequently approximated by a finite-dimensionaltransfer function.

Inversion of a moment model for batch crystallization isemployed in the feedforward part of a two-degree-of-freedomcontrol scheme suggested in Kleinert et al. (2010). Given a

Fig. 2. Illustration of the inversion problem and the crystal populations involved

in preferential crystallization. In this example, the highlighted final-time CSD shall

be realized by controlling the temperature in the tank. The meaning of tE1and tE2

will be explained in Section 4.1.

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269 255

prescribed temporal evolution of the crystal mean size or super-saturation, the inversion results in trajectories of several vari-ables, including the slurry temperature. Notably, nucleation isneglected in the underlying design model. The computed trajec-tories serve as setpoints for a cascaded feedback part. Anextended Kalman filter is used for observation of the crystal meansize, when only temperature and concentration measurementsare available.

Model predictive control (MPC) and related techniques havealso been actively investigated for crystallizer control (e.g., Nagyand Braatz, 2003; Zhang and Rohani, 2003; Shi et al., 2005, 2006;Nagy, 2009). For example, in the MPC approach in Shi et al.(2005), an optimal control problem for a temperature-controlledbatch crystallizer is repeatedly solved. The aim is the maximiza-tion of the volume-averaged crystal size under additional con-straints regarding, e.g., temperature, supersaturation and alsonucleation rate. An extended Luenberger-type observer generatesthe required estimates of the current state of a standard momentmodel, using only concentration and temperature measurements.In subsequent work (Shi et al., 2006), MPC is investigated for bothcontinuous and batch particulate processes. In the continuouscase, closed-loop asymptotic stabilization of an unstable steadystate can be guaranteed by using a mixed strategy with switchingbetween MPC and bounded Lyapunov-based control. In the batchcase, the optimal control goal now consists in minimizing thetotal volume of fines (i.e., the third moment of the CSD partcorresponding to grown nucleated crystals), while a minimumdesired volume of grown seed crystals poses one of the con-straints. Controller design is again based on standard momentmodels.

Control of preferential crystallization processes, on the otherhand, is a relatively new field. For example, Wang and Ching(2006) have performed trajectory planning with the aim ofcontrolling critical supersaturation in batch preferential crystal-lization. In Angelov et al. (2006), initial conditions of an isother-mally operated, cyclic operation mode (see above) have beenoptimized. Later, in Angelov et al. (2008), the temperature profilewas optimized for batch preferential crystallization. The aimconsisted in maximizing the amount of the preferred enantiomerwhile avoiding inadmissible impurification with the counterenantiomer. Likewise, Bhat and Huang (2009) have performedoptimization of the process in a more complex, multi-objectivesetting. In Hofmann et al. (2011), constant supersaturation con-trol is combined with process stopping based on a worst-caseimpurity estimate.

Work has also been done in fields, which can be seen as relatedto preferential crystallization. For example, for selective crystal-lization of one of the polymorphic forms of L-glutamic acid in Keeet al. (2009), supersaturation profiles are followed by usingconcentration feedback control.

This paper and its companion (Hofmann and Raisch, submitted forpublication) focus on the solution of feedforward control problems.In particular, the problem of realizing prescribed final-time CSDs shallbe extended from single-substance crystallization processes (Vollmerand Raisch, 2006) to preferential crystallization of enantiomers. Here,in Part I, as we restrict ourselves to a simple batch process, this can beillustrated by means of Fig. 2: the final-time CSD of one of the twoenantiomers in the tank is specified, and the temperature profilewhich has to be applied in order to achieve the specification is to bedetermined. Note that the analysis could be repeated a number oftimes for subsequent batches in cyclic mode. Then, in general, everybatch will start with different initial conditions, which result from theprevious batch and some discrete actions, such as product removaland addition of new racemate and seed crystals.

Vollmer and Raisch (2003) have shown that a moment modelfor crystallization of a single substance in a single crystallizer is

orbitally flat, i.e., the model is differentially flat, or just flat, in atransformed time. These concepts will be summarized in Section3. Note that flatness can be seen as a property which makesinversion of the system a straightforward problem, namely apurely algebraic one. Based on an appropriate time scaling ofa partial differential equation (PDE) population balance model, aswell as of a moment model, it is then possible to realizeprescribed forms for the nucleated part of a CSD. Also other tasks,namely the efficient computation of optimal control profiles, aswell as feedback control, can then be solved in a (nearly) analyticway (Vollmer and Raisch, 2003, 2006).

In Raisch et al. (2005), these results were related to controlproblems in preferential crystallization of enantiomers, includingthe cyclic and coupled modes of operation. It was hypothesizedthat, due to different growth rates of the two enantiomer species,transformation of moment models into flat systems is no longerpossible. However, as one general goal of the suggested separa-tion schemes is to keep the crystallized mass of the counterenantiomer as low as possible, neglecting nucleation of thisspecies was suggested as a reasonable idealization; this mayallow transforming a subsystem describing only crystallizationof the preferred enantiomer into a flat system.

The work in this paper and its companion (Hofmann andRaisch, submitted for publication) is twofold: first, we will showhow the inversion results (Vollmer and Raisch, 2003, 2006) can beextended to the case of crystallization of enantiomers, wheremoment models are not orbitally flat. Inversion for these modelsis still possible, but not as efficient as in the orbitally flat case.Consequently, we will investigate a number of idealizing assump-tions. Here, in Part I, we will consider neglecting nucleation terms.As just stated, this especially makes sense if it is done for thecounter enantiomer only. Of course, in this case, inversion canonly be performed w.r.t. the preferred enantiomer (when, forexample, a final-time CSD is specified), but will be greatlysimplified. Alternatively, we will neglect nucleation of the pre-ferred enantiomer only, motivated by the idea that growing seedcrystals outweigh nucleation, both in terms of product propertiesand of feedback inside the model. In Part II, which will be focusedon the coupled mode of operation, we will introduce restrictionsand idealizations pertaining to the growth and nucleation rates,and to the technical realization of the coupling, i.e., the exchangeof liquid between two crystallizer vessels.

This paper is organized as follows: in Section 2, the populationbalance and moment models are given for the process of crystal-lization of two conglomerate forming enantiomers in a singlecrystallizer vessel. Section 3 contains a brief recapitulation ofsome basic concepts and conditions relating to (orbital) flatness.In Sections 4.1 and 4.2, we show how parts of the results byVollmer and Raisch (2003, 2006) can be used in the process ofsystem inversion that is necessary to realize prescribed final-time CSDs.

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269256

In Section 4.3, we show that, normally, moment models forcrystallization of enantiomers are not orbitally flat. Despite this, inSection 4.4, we solve the problem of inverting a moment model,which completes the task of realizing a prescribed final-timeCSD of one of the two enantiomer species. This step involvessolution of differential equations. In Sections 5.1 and 5.2 wepresent simplified solutions for the cases when nucleation of thecounter enantiomer or the preferred enantiomer, respectively, isneglected. In the former case, the resulting simplified systemis orbitally flat.

In a numerical example in Section 6, we compare the idealizedand non-idealized solution methods. We also investigate theresults when temperature trajectories obtained using the idea-lized models are used as inputs to the non-idealized models.

2. Model

We assume only the basic mechanisms of growth and nuclea-tion to take place, no other effects, such as conglomeration andbreakage.

Population balance equations (PBEs) are frequently used todescribe the dynamics of particulate processes (Randolph andLarson, 1988; Ramkrishna, 2000). In the enantiomer context, twoPBEs are required to describe the evolution of the involved crystalpopulations. The PBE for one enantiomer, e.g. E1, is given by

@f E1ðL,tÞ

@t¼�

@ðGE1ðL,tÞf E1

ðL,tÞÞ

@L, ð1aÞ

with boundary and initial conditions

f E1ð0,tÞ ¼

BE1ðtÞ

GE1ð0,tÞ

, ð1bÞ

f E1ðL,0Þ ¼ f E1 ,seedðLÞ, ð1cÞ

when t represents time and the scalar L crystal length, f E1is a

number density function, the so-called crystal size distribution(CSD), and GE1

ðL,tÞ, BE1ðtÞ are growth and nucleation rates of E1,

respectively. Note that newly nucleated crystals are assumed toinitially have size zero (L¼0). We assume size-independentgrowth rates, so that instead of GE1

ðL,tÞ, we can just write GE1ðtÞ.

Note that the time-varying character of GE1and BE1

may resultfrom dependencies on other process variables. This will in thefollowing often be denoted by writing GE1

ð�Þ, BE1ð�Þ. The equations

and definitions are totally analogous for enantiomer E2, and thecomplete model then includes both sets of equations. The rightside of Fig. 2 illustrates the involved crystal populations at finaltime tf. The picture looks qualitatively the same at any time t40.Note that the CSD belonging to E1, f E1

, has been split into twoparts. One represents growing seed crystals, f E1 ,s, and the otherone, f E1 ,n, represents grown nucleated crystals. In the following,these two parts will be referred to as the seed CSD, and thenucleated CSD, respectively. In this scenario, E1 represents thedesired enantiomer, therefore E2 is not seeded, and f E2 ,s isidentically zero. Note that in a cyclic mode of operation, the rolesof E1 and E2 change from batch to batch.

For a single substance crystallization process with size-inde-pendent growth, a closed moment model can be derived (e.g.,Hulburt and Katz, 1964; Myerson, 2002). In the preferentialcrystallization context, more than one population is involved. Inanalogy to a single substance crystallization process, it is possibleto derive a moment model that describes the evolution of theprincipal moments of all relevant CSDs, i.e., that of enantiomers E1

and E2. As will be seen later, a closed model can be obtainedwhich includes only the first four moments of each enantiomer(assuming that nucleation terms do not depend on higher

moments). In total, we then arrive at a model consisting of 8 firstorder ODEs

_m iE1¼ iGE1

ð�Þmði�1ÞE1, i¼ 1 . . .3, ð2aÞ

_m0E1¼ BE1

ð�Þ, ð2bÞ

_m iE2¼ iGE2

ð�Þmði�1ÞE2, i¼ 1 . . .3, ð2cÞ

_m0E2¼ BE2

ð�Þ, ð2dÞ

where the moments are defined as

miE1ðtÞ ¼

Z 10

Lif E1ðL,tÞ dL, i¼ 0 . . .3, ð3aÞ

miE2ðtÞ ¼

Z 10

Lif E2ðL,tÞ dL, i¼ 0 . . .3: ð3bÞ

The associated initial conditions are

miE1ð0Þ ¼ miE1 ,seed ¼

Z 10

Lif E1 ,seedðLÞ dL, i¼ 0 . . .3, ð4aÞ

miE2ð0Þ ¼ miE2 ,seed ¼

Z 10

Lif E2 ,seedðLÞ dL, i¼ 0 . . .3: ð4bÞ

Note that, when the application is preferential crystallization, andE1 is the preferred enantiomer, then miE2 ,seed ¼ 0. As far as amoment model is concerned, the analysis in the remainder ofthis paper, and the numerical example in Section 6, will be basedon (2).

The expressions for the growth and nucleation rates are criticalfor the ability of the model to represent the real crystallizationprocess. In the literature, one can find equations for the processof crystallization of a single substance of various complexity(e.g., Miller and Rawlings, 1994; Mullin, 2001; Mersmann, 2001).For the enantioseparation process, accurate models are expectedto become even more complex, as there are potential cross-couplings between various liquid and solid phases of the respec-tive enantiomers. We have used increasingly detailed models(Elsner et al., 2005a, 2011; Angelov et al., 2008; Eicke et al., 2010;Hofmann et al., 2011).

To account for the current model structure, and for futuremodel developments, investigations in the main part of this paperwill be based on generic functional dependencies of the growthand nucleation rates

GE1ð�Þ ¼ GE1

ðT,wl,E1,wl,E2

Þ, ð5aÞ

GE2ð�Þ ¼ GE2

ðT,wl,E1,wl,E2

Þ, ð5bÞ

BE1ð�Þ ¼ BE1

ðT,wl,E1,wl,E2

,m0E1,m1E1

,m2E1,m3E1

,m0E2,m1E2

,m2E2,m3E2Þ,

ð5cÞ

BE2ð�Þ ¼ BE2

ðT,wl,E1,wl,E2

,m0E1,m1E1

,m2E1,m3E1

,m0E2,m1E2

,m2E2,m3E2Þ:

ð5dÞ

The temperature of the slurry, also called the crystallizationtemperature, is denoted by T. It is sufficient to consider the massfractions wl,E1

, wl,E2in (5), since the mass of solvent (frequently

water) mW in the tank is constant, and thus a unique algebraicrelationship exists between wl,E1

, wl,E2and the dissolved masses

ml,E1, ml,E2

wl,E1¼

ml,E1

ml,E1þml,E2

þmW, ð6aÞ

wl,E2¼

ml,E2

ml,E1þml,E2

þmW: ð6bÞ

The nucleation rates may also depend on the amount and proper-ties of the already existing crystals, expressed in terms of the

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269 257

moments of their size distributions. Note that (5) allows a cross-coupling between the enantiomer species emanating from theirsolid phases.

Additional differential equations describe the evolution of theliquid masses in the single crystallizer

_ml,E1¼�3GE1

ð�Þrskvm2E1, ð7aÞ

_ml,E2¼�3GE2

ð�Þrskvm2E2, ð7bÞ

where rs is the density of the crystals, and kv is a volume shapefactor. Here, as no substance is exchanged with the environmentduring the batch, (7) can be replaced by algebraic mass balanceequations

ml,E1¼ml0,E1

�rskvðm3E1�m3E1

ð0ÞÞ, ð8aÞ

ml,E2¼ml0,E2

�rskvðm3E2�m3E2

ð0ÞÞ: ð8bÞ

Consequently, ml,E1and ml,E2

need not be part of the state.We will now pose basic assumptions on the growth and

nucleation rate terms, i.e., on GE1ð�Þ, GE2

ð�Þ, BE1ð�Þ, BE2

ð�Þ, and alsoon the ratios ðBE1

=GE1Þð�Þ, ðBE2

=GE2Þð�Þ, which will enable us to

establish existence and uniqueness of solutions to the proposedinversion problem, without referring to specific kinetics andparameters.

We require these assumptions to hold for all values of thestate, i.e., the moments, and the input, i.e., the temperature, in acertain physically meaningful domain D, where D is an openconnected set.

We assume that, within D, when GE1ð�Þ, GE2

ð�Þ, BE1ð�Þ and BE2

ð�Þ

are regarded as functions of all their non-constant arguments,they are continuously differentiable, i.e., all partial derivativesw.r.t. the arguments exist and are continuous at every point in D.Also, the functions must be positive in D. The latter implies thatalso any function defined as a ratio of rate expressions will becontinuously differentiable within D. This can be seen by notingthat a function of several variables is continuously differentiableat a point, if all partial derivatives exist and are continuous at thatpoint (Khalil, 2001). If the denominator of the quotient of twofunctions is non-zero, the quotient rule for functions of a singlevariable can be used to compute all the partial derivatives, and itguarantees their existence and continuity.

For the results in Section 4.3, we require GE1ð�Þ, GE2

ð�Þ, BE1ð�Þ and

BE2ð�Þ to be continuously differentiable twice w.r.t. the tempera-

ture input T.Furthermore, we assume that the derivative of at least one of

the ratios ðBE1=GE1Þð�Þ and ðBE2

=GE2Þð�Þ, with respect to the tem-

perature input T, is nowhere equal to zero. From continuity ofthese derivatives, it then follows that the respective one is eithereverywhere positive or negative.

The first part of the above assumptions only poses basicdifferentiability properties, which are not too restrictive. Also,requiring that the growth and nucleation rates be positive is noadditional restriction, since the model itself is not valid otherwise.Only the strict monotonicity assumption between T and ðBE1

=GE1Þ,

respectively ðBE2=GE2Þ, may pose a real restriction. Frequently, in

crystallization systems, there are competing mechanisms govern-ing the rates: for example, supersaturation, as a driving force,decreases with increasing temperature, while an Arrhenius lawcan pose a positive influence of increasing temperature on therespective rate. Consequently, there may be two distinct tem-perature values resulting in the same rate ratio. One possible wayto handle this is to suitably partition D into subsets where therelevant functions are strictly monotonic.

3. Flatness and orbital flatness

In the following, we briefly recapitulate the notions of differ-ential flatness and orbital flatness (e.g., Fliess et al., 1995a, 1999).(Differential) flatness is a property of dynamic systems. A simplecriterion (Rothfuß et al., 1997; Fliess et al., 1995a) states that, if asystem

_xðtÞ ¼ f ðxðtÞ,uðtÞÞ, xðtÞARn, uðtÞARmð9Þ

is differentially flat, this is equivalent to the existence of aso-called flat output,

yðtÞ ¼ hðxðtÞ,uðtÞ, _uðtÞ, . . . ,uðaÞðtÞÞ, yðtÞARm, ð10Þ

which satisfies the following properties:

1.

The output yðtÞ can be expressed as a function of the systemstate xðtÞ and input uðtÞ and finitely many time derivatives ofthe input.

2.

The system state and input can be expressed as functions ofthe output yðtÞ and finitely many of its time derivatives

xðtÞ ¼C1ðyðtÞ, _yðtÞ, . . . ,yðbÞðtÞÞ, ð11Þ

uðtÞ ¼C2ðyðtÞ, _yðtÞ, . . . ,yðbþ1ÞðtÞÞ: ð12Þ

The flat output yðtÞ and its derivatives therefore algebraicallyparametrize the system state and input. Thus, given a sufficientlysmooth output trajectory, inversion (i.e., determination of uðtÞ) ispossible without solving any differential equations. The abovecondition is a way to verify flatness, when a flat output candidateis already known. Note that, as stated in (10), a flat output musthave the same dimension as the system input (Fliess et al., 1999).

For SISO systems in the following control affine form:

_x ¼ f ðxÞþgðxÞu, ð13aÞ

y¼ hðxÞ, ð13bÞ

another way to characterize flatness is to state that the relative

degree n of the system must be equal to n, implying that thesystem must not have an internal (zero) dynamics (Khalil, 2001;Isidori, 1995). The relative degree is defined as the least positiveinteger n, s.t.

LgLn�1f ha0, ð14Þ

where the differential operator Lkg denotes the k-th order

Lie-derivative along a vector field g, defined in the usual way.Related concepts exist for MIMO systems (Isidori, 1995).

Given a suitable output y, verification of flatness may bestraightforward. However, proving that a system is non-flat ismore involved. For SISO systems, a necessary and sufficientcondition can be stated requiring involutivity of a certain dis-tribution of vector fields (Khalil, 2001; Isidori, 1995; Nijmeijerand van der Schaft, 1990). However, the required computation ofrepeated LIE-Brackets may become intense for higher systemdimensions.

For MIMO systems, a necessary condition for flatness can begiven by the ruled manifold criterion (Sluis, 1993; Rouchon, 1994;Fliess et al., 1995b; Martin et al., 1997; Sira-Ramırez, 2004;Levine, 2009). It states (Martin et al., 1997): ‘‘Assume _x ¼ f ðx,uÞis flat. The projection on the p-space of the submanifoldp¼ f ðx,uÞ, where x is considered as a parameter, is a ruledsubmanifold for all x. The criterion just means that eliminatingu from _x ¼ f ðx,uÞ yields a set of equations Fðx, _xÞ ¼ 0 [i.e., animplicit representation of the system] with the following

Fig. 3. Characteristic curves of the population balance equation for enantiomer

E1; (a) without time scaling: f E1ðL,tÞ ¼ const:; (b) with time scaling (18):

f E1ðL,tE1

Þ ¼ const:

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269258

property: for all ðx,pÞ such that Fðx,pÞ ¼ 0, there exists gARn,ga0 such that

8lAR, Fðx,pþlgÞ ¼ 0:’’ ð15Þ

Note that for input-affine systems this condition always holds.A system is orbitally flat (Fliess et al., 1995a; Respondek, 1998;

Guay, 1999) if it can be turned flat by an appropriate state andinput dependent time scaling

½t0,tf �/½t0,tf �, ð16aÞ

dt

dt ¼ sðxðtÞ,uðtÞÞ, tðt0Þ ¼ t0, ð16bÞ

0osðxðtÞ,uðtÞÞo1 8t: ð16cÞ

The last condition ensures that t is strictly increasing in t, andtherefore guarantees that the time scaling is invertible. In scaledtime, the system differential read

d

dtxðtÞ ¼ sðxðtÞ,uðtÞÞ � f ðxðtÞ,uðtÞÞ: ð17Þ

Fig. 4. Subsystem for the crystallization of E1, when using the time transforma-

tion (18).

4. Analysis of the single crystallizer configuration

4.1. Time scaling of the population balance equation

To realize prescribed final-time CSDs, the first step taken byVollmer and Raisch (2003, 2006) is to scale time with the growthrate of the population balance model. This turns the populationbalance PDE into a simple linear transport equation. Using thistransport equation, the trajectory of the output m3ðtÞ, whichrealizes a given final-time CSD f ðL,tf Þ can be directly computed.Moreover, m3ðtÞwas shown to be a flat output of a moment modelin the same transformed time t.

In the enantiomer context, we can scale time either with thegrowth rate of E1 or E2. Without loss of generality, we assume thatthe growth rate of enantiomer E1 is chosen, and denote theresulting scaled time tE1

, i.e.,

dtE1¼ GE1

ð�Þ dt: ð18Þ

Then, the population balance equation for enantiomer E1 becomes

@f E1ðL,tE1

Þ

@tE1

¼�@f E1ðL,tE1

Þ

@L, ð19aÞ

with boundary and initial conditions

f E1ð0,tE1

Þ ¼BE1ðtE1Þ

GE1ðtE1Þ, ð19bÞ

f E1ðL,0Þ ¼ f E1 ,seedðLÞ: ð19cÞ

As in Vollmer and Raisch (2003), the characteristic curves thatdescribe the solutions become straight lines, see Fig. 3, and therequired final-time shape of the nucleated part of the CSDf E1ðL,tE1 ,f Þ, LA ½0,Lmax ¼ tE1 ,f �, can trivially be converted to a

temporal evolution of the boundary condition, i.e.,

BE1

GE1

ðtE1Þ ¼ f E1

ðtE1 ,f�tE1,tE1 ,f Þ, tE1

¼ 0, . . . ,tE1 ,f : ð20Þ

Note that the part of the final-time CSD f E1ðL,tE1 ,f Þ with L4tE1 ,f is

merely a shifted version of the CSD of seed crystals, and cantherefore not be freely prescribed

f E1ðL,tE1 ,f Þ ¼ f E1 ,seedðL�tE1 ,f Þ, L4tE1 ,f : ð21Þ

This shows that tE1corresponds to the gain in length of crystals

belonging to the seed distribution.

Note that (20) and (21) can be immediately deduced fromFig. 3(b). Hence, we can pick one of the enantiomers (here, E1),and transform the corresponding subsystem into the simpletransport equation (19). This subsystem allows computation ofa temporal profile BE1

=GE1ðtE1Þ, which realizes a given final CSD

f E1ðL,tE1 ,f Þ. Note that the profile BE1

=GE1ðtE1Þ represents the bound-

ary condition to this subsystem in scaled time. The ratio BE1=GE1

depends on the state, i.e., some or all moments, and the tem-perature input T. This leaves the task to determine the tempera-ture profile TðtE1

Þ that will generate the required profileBE1

=GE1ðtE1Þ.

However, in the general case, no time scaling exists, whichstraightens out the characteristic curves for both crystallizationsub-processes (E1, E2) at the same time.

Note that prescribing the final-time CSD of E1 and thereforethe temporal evolution of BE1

=GE1determine the control input T,

hence also the final-time CSD of E2, see also Section 4.4.

4.2. Time scaling of the moment model

Following the procedure given by Vollmer and Raisch (2003),the transformation (18) is applied to the moment model (2a) and(2b) for enantiomer E1. The result is

d

dtE1

miE1¼ i mði�1ÞE1

, i¼ 1 . . .3, ð22aÞ

d

dtE1

m0E1¼

BE1

GE1

ðtE1Þ: ð22bÞ

This represents a linear time-invariant system with input signalBE1

=GE1. In fact, it is a simple chain of integrators (see Fig. 4), and it

is immediately clear that the third moment m3E1is a flat output of

this part of the overall system.According to Section 4.1, the profile of BE1

=GE1in scaled time is

determined by the desired final-time CSD for E1. By integratingthis profile, one obtains the temporal evolution of the momentsm0E1

, . . . ,m3E1. For this, the initial conditions of the moments must

be taken into account. Normally, if E1 is the preferred enantiomer,these are the moments of the CSD of the seed crystals, whereas ifE1 is the counter enantiomer they are zero. Note that, when adesired final-time CSD is in a suitable functional form (e.g., a

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269 259

parametrization by splines), then closed form solutions may befound for (22).

To simplify notation, we will define the state vector of theoverall model by

xT ¼4½m3E1

, . . . ,m0E1,m3E2

, . . . ,m0E2�, ð23Þ

and denote the temperature, which serves as the input, u, i.e.,

u¼4

T : ð24Þ

Hence, in original time t, the moment model reads as

_xi ¼ ð4�iÞGE1ðx,uÞxiþ1, i¼ 1 . . .3, ð25aÞ

_x4 ¼ BE1ðx,uÞ, ð25bÞ

_xi ¼ ð8�iÞGE2ðx,uÞxiþ1, i¼ 5 . . .7, ð25cÞ

_x8 ¼ BE2ðx,uÞ: ð25dÞ

For some of the following analysis, we require an input-affinemodel. To achieve this, we consider temperature as another statevariable, and its derivative as a new input u, i.e.,

x9 ¼4

T , ð26Þ

u ¼4 d

dtE1

x9 ¼1

GE1ð�Þ

d

dtx9: ð27Þ

The input u is defined to give simpler equations when the modelis transformed into the time scaled according to GE1

ð�Þ, i.e., tE1.

4.3. Lack of orbital flatness

Based on the ruled manifold criterion (Sluis, 1993; Rouchon,1994; Fliess et al., 1995b; Martin et al., 1997; Sira-Ramırez, 2004;Levine, 2009), see Section 3, we will argue that, typically, themodel (25) is not orbitally flat. Roughly speaking, we show that nostrictly monotonic time scaling can simultaneously ‘‘align’’ morethan two of the right hand sides of (25), which are nonlinear in u.In Appendix A, we derive a general test to exclude orbital flatnessof a given single input system, _x ¼ f ðx,uÞ.

To exclude orbital flatness of a system in the form of (25), thelocal conditions (A.12) have to be checked. More precisely, we canchoose some ðxp,upÞ, and verify, for every one of the eight righthand sides of (25), one after the other, that, when this functiontakes the role of f a, two of the remaining functions can take theroles of f b and f c , so that (A.12) is the case. Recall that, accordingto the basic assumptions in Section 2, the system differentialequations are continuously differentiable twice w.r.t. u.

In the following, the procedure is demonstrated for a verybasic example of growth and nucleation rate functions in crystal-lization of enantiomers, suggesting that practical models for suchprocesses are not orbitally flat. The model is overly simplistic inthat, e.g., only secondary nucleation is considered. It is thereforepresumed that, at time t¼0, crystals of both enantiomer speciesare already present

GE1ð�Þ ¼ kgðwl,E1

v�1Þg , ð28aÞ

GE2ð�Þ ¼ kgðwl,E2

v�1Þg , ð28bÞ

BE1ð�Þ ¼ kbðwl,E1

v�1Þbm3E1, ð28cÞ

BE2ð�Þ ¼ kbðwl,E2

v�1Þbm3E2, ð28dÞ

where

v :¼1

weqðuÞ¼

1

weqðTÞ: ð29Þ

In this simple example, we assume an equilibrium mass fraction,weq(T), which is a static, invertible function of temperature T, andwhich is common for both enantiomers.

This makes it possible to analyze system (25) and (28), with v

as the input, instead of system (25), (28), (29), with u¼4

T as theinput. In fact, noting that ‘‘[a] system _x ¼ f ðx,uÞ is flat if, and onlyif, the closed loop system _x ¼ f ðx,aðx,vÞÞ ¼fðx,vÞ is flat withu¼ aðx,vÞ being an invertible feedback, i.e., v¼ bðx,uÞ withu¼ aðx,bðx,uÞÞ’’ (Sira-Ramırez, 2004), we see that system (25),(28), (29) being orbitally flat would imply that system (25) and(28) also be orbitally flat. Knowing a suitable time scalingfunction sðx,uÞ, one could substitute u by the inverse of (29) inboth sðx,uÞ and f ðx,uÞ and arrive at a system (with input v) whichwould still be flat.

To verify that system (25a) and (28a) is not orbitally flat, wechoose ðxp,vpÞ s:t: ðwl,E1

,wl,E2,vpÞ ¼ ð0:12,0:11,1=0:1Þ, mIE1

a0, i¼

0y3 and miE2a0, i¼0y3. Note that these are reasonable values,

which could be encountered during normal operation of theprocess. Computing the expressions in (A.12), with ð�Þ0 :¼ d=dvð�Þ,ð�Þ00 :¼ d2=dv2

ð�Þ, where, one after the other, each of the eight timederivative functions in (25), with (28) substituted, takes the roleof f a, and each time two of the remaining functions are appro-priately chosen as f b, f c , one can exclude orbital flatness, as longas bag, ba11

6 ðg�1Þ, ba 611g and ba 6

11ðg�1Þ. By conductinganother test, e.g., with ðwl,E1

,wl,E2,vpÞ ¼ ð0:13,0:12,1=0:1Þ, the last

three cases can be ruled out as well, so that only in the case b¼g

orbital flatness cannot be excluded.Note that, when the system would consist only of (25a) and

(25b) (and x5 to x8 are considered as constant parameters), then itwould be easy to see that, regardless of ðxp,upÞ, and which of thefour right hand sides, respectively, are assigned to f a, f b, f c , atleast one equation of (A.12) is zero. This is due to the commonway u enters in the functions (25a), via GE1

ðx,uÞ. The argument isanalogous for a system (25c) and (25d). This is not surprising,since these submodels have the same structure as a momentmodel for single substance crystallization, which was shown to beorbitally flat by Vollmer and Raisch (2003). Following these ideas,in Section 5.1, we will introduce an idealized model with thesame property.

4.4. Dynamic inversion

In this section, we show how inversion of the complete, non-idealized, moment model for enantioseparation in a single crys-tallizer is possible. Although the model is not orbitally flat, thetime scaling proposed in Sections 4.1 and 4.2 still contributesdecisively to the solution.

There exist two growth rates, namely GE1ð�Þ and GE2

ð�Þ, whichare, in general, not equal. Suppose we are primarily interested inone of the enantiomers, say E1. We will therefore use the timetransformation (18)

dtE1¼ GE1

ð�Þdt: ð30Þ

In accordance with Section 4.3, we will show that, if this timetransformation is applied to the complete, non-idealized model,the third moment of enantiomer E1 is not a flat output. However,we will demonstrate that inversion of the system with thisoutput,

y¼4

x1 ð31Þ

is still possible, but involves the solution of differential equations.The existence of a solution then depends also on the zerodynamics.

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269260

When the time transformation (30) is applied to (25)–(29), theresulting set of ODEs is

d

dtE1

xi ¼ ð4�iÞxiþ1, i¼ 1 . . .3, ð32aÞ

d

dtE1

x4 ¼BE1

GE1

ðx1, . . . ,x8,x9Þ, ð32bÞ

d

dtE1

xi ¼ ð8�iÞGE2

GE1

ðxÞxiþ1, i¼ 5 . . .7, ð32cÞ

d

dtE1

x8 ¼BE2

GE1

ðx1, . . . ,x8,x9Þ, ð32dÞ

d

dtE1

x9 ¼ u: ð32eÞ

The expressions BE1=GE1ðxÞ, GE2

=GE1ðxÞ and BE2

=GE1ðxÞ denote

nonlinear functions determined by the respective growth andnucleation rates.

Note that (31) and (32) is in control affine form (13), with

f ðxÞ ¼ 3x2,2x3,x4,BE1

GE1

ðxÞ,3GE2

GE1

ðxÞx6,2GE2

GE1

ðxÞx7,GE2

GE1

ðxÞx8,BE2

GE1

ðxÞ,0

� �T

,

ð33Þ

gðxÞ ¼ ½0;0,0;0,0;0,0;0,1�, ð34Þ

hðxÞ ¼ x1: ð35Þ

Therefore,

L0f h¼ x1; LgL0

f h¼ 0;

L1f h¼ 3x2; LgL1

f h¼ 0;

L2f h¼ 6x3; LgL2

f h¼ 0;

L3f h¼ 6x4; LgL3

f h¼ 0;

L4f h¼ 6

BE1

GE1

ðxÞ; LgL4f h¼ 6

@

@x9

BE1

GE1

ðxÞ

� �: ð36Þ

With regard to the assumptions presented in Section 2, the partialderivative of BE1

=GE1with respect to temperature, i.e., x9, must not

be equal to zero within the physically meaningful domain D.Therefore,

LgL4f ha0: ð37Þ

Consequently, the relative degree of the system is n¼ 5on¼ 9,and y¼

4x1 is not a flat output. Note that the system has strong

relative degree n¼ 5, because (36) and (37) hold in the entiredomain of definition, D.

Nevertheless, inversion of the system is still possible. We cantransform the system into a normal form (Khalil, 2001; Isidori,1995) by introducing a new state z

z1 :¼ y¼ L0f hðxÞ ¼ x1, ð38aÞ

z2 :¼d

dtE1

y¼ L1f hðxÞ ¼ 3x2, ð38bÞ

z3 :¼d2

dt2E1

y¼ L2f hðxÞ ¼ 6x3, ð38cÞ

z4 :¼d3

dt3E1

y¼ L3f hðxÞ ¼ 6x4, ð38dÞ

z5 :¼d4

dt4E1

y¼ L4f hðxÞ ¼ 6

BE1

GE1

ðxÞ, ð38eÞ

zi :¼ xi�1, i¼ 6 . . .9: ð38fÞ

To obtain the inverse transformation, the state variables x1 to x4

and x5 to x8 can be found trivially. For obtaining x9, (38e) must beinverted, i.e., the solution of

6BE1

GE1

ðx1, . . . ,x4,x5, . . . ,x8,x9Þ ¼ z5, ð39Þ

respectively

6BE1

GE1

z1,1

3z2,

1

6z3,

1

6z4,z6,z7,z8,z9,x9

� �¼ z5, ð40Þ

must be computed. Under the assumptions in Section 2, a uniquesolution exists when operation is in the physically meaningfuldomain D, i.e., x9 can be written as a function of z, which, slightlyabusing notation, we denote as

x9 ¼BE1

GE1

� ��1

z1,1

3z2,

1

6z3,

1

6z4,

1

6z5,z6,z7,z8,z9

� �: ð41Þ

The resulting system is

d

dtE1

zi ¼ ziþ1, i¼ 1 . . .4, ð42aÞ

d

dtE1

z5 ¼ L5f hðxÞþLgL4

f hðxÞ � u, ð42bÞ

d

dtE1

z6 ¼ 3GE2

GE1

ðxÞz7, ð42cÞ

d

dtE1

z7 ¼ 2GE2

GE1

ðxÞz8, ð42dÞ

d

dtE1

z8 ¼GE2

GE1

ðxÞz9, ð42eÞ

d

dtE1

z9 ¼BE2

GE1

ðxÞ, ð42fÞ

where x can be substituted by the inverse transform. Note that(42) is in Byrnes–Isidori normal form (Kravaris and Kantor, 1990;Isidori, 1995), since u only enters (in an affine way) in (42b).Using

L5f h¼ 6

@

@x

BE1

GE1

ðxÞ

� �f ðxÞ, ð43Þ

and (36), Eq. (42b) reads

d

dtE1

z5 ¼ 6@

@x

BE1

GE1

ðxÞ

� �f ðxÞþ6

@

@x9

BE1

GE1

ðxÞ

� �� u: ð44Þ

In the Byrnes–Isidori normal form, (42c)–(42f) constitute the zerodynamics. Because z6, . . . ,z9 correspond to x5, . . . ,x8, thisdynamics can be directly identified as the dynamics of thecounter enantiomer subsystem.

When a desired, sufficiently smooth, output trajectory yðtE1Þ,

tE1A ½0,tE1 ,f � is specified, then z1 to z5, respectively x1 to x4 and

BE1=GE1

, are directly determined as functions of tE1, and also their

initial conditions are uniquely determined, i.e., their trajectoriesare determined by the trajectories of the respective derivatives ofthe output, (38a)–(38e). We require that y be continuouslydifferentiable five times with respect to tE1

, so that these func-tions, including BE1

=GE1¼ 1

6ðd4=dt4

E1Þy, are all continuously differ-

entiable w.r.t. tE1. For the realization of a specified final-time CSD

for E1, BE1=GE1

and x1 to x4 are determined according to Sections

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269 261

4.1 and 4.2; the specified CSD must be continuously differentiablew.r.t. L.

For obtaining the trajectory of the remaining part of the state,the zero dynamics, (42c)–(42f), has to be integrated. Therefor, in(42c)–(42f), the explicit time-dependent functions x1ðtE1

Þ tox4ðtE1

Þ can be directly substituted. For x9, the solution (41) canbe substituted (involving the explicit time-dependent functionz5ðtE1

Þ). Under the basic assumptions in Section 2, in the physi-cally meaningful domain D, (41) is continuously differentiablew.r.t. all its arguments. This can be seen using the implicitfunction theorem (e.g., Khalil, 2001), noting in particular that,by assumption, @=@x9ðBE1

=GE1Þð�,x9Þa0. The input uðtE1

Þ can becomputed using (42b). This step is optional, since the ‘‘real’’ inputis uðtE1

Þ ¼ TðtE1Þ ¼ x9ðtE1

Þ.For establishing existence and uniqueness of solutions, it can

be first checked whether the trajectories of the state variables x1

to x4 (known as explicit tE1-time dependent functions) do not

violate D. Then, existence and uniqueness of solutions of(42c)–(42f) (with all substitutions made) has to be checked. Usingthe product rule and the quotient rule for differentiation, andnoting that GE1

ð�Þ40, it can be seen that, inside D, the functions inthese differential equations are continuously differentiable, andtherefore locally Lipschitz, in the state variables, and in the ‘‘time’’tE1

. Thus, a unique local solution exists from any given initialcondition inside D (Khalil, 2001), on a maximal open time interval½0,tE1 ,maxÞ. If tE1 ,max4tE1 ,f (the specified final time), then a uniquesolution has been found. If tE1 ,maxrtE1 ,f , then no solution exists,and one can conclude that the prescribed output profile, respec-tively final-time CSD, is not realizable.

When computing solutions, it can be hard to check whethertE1 ,max4tE1 ,f or not. However, the fact that, if tE1 ,max is finite, asolution must leave any compact subset W of D (Khalil, 2001)shows a practical way to check if an output profile (respectivelyfinal-time CSD) can be realized: one chooses a compact set W,whose boundary keeps a certain safety distance from the bound-ary of D. Then, if a solution can be computed which leaves W,while still remaining in D, one can at least conclude that thespecification cannot be realized while keeping the safety distancefrom the boundary of D.

Fig. 5. Complete inversion scheme for realization of a desired final-time CSD,

f E1 ,des, for enantiomer E1.

Having found the profile x9ðtE1Þ ¼ TðtE1

Þ, the inverse timetransformation,

tðtE1Þ ¼

Z tE1

0

1

GE1ðxðtE1

0ÞÞdt0E1

, ð45Þ

must be performed to arrive at the physically implementable (e.g.,by means of a temperature tracking controller) profileTðtÞ ¼ TðtE1

ÞJtE1ðtÞ, where tE1

ðtÞ denotes the inverse of (45).Fig. 5 illustrates the main steps in the complete inversion scheme,involving results from Sections 4.1, 4.2 and 4.4, in the case when adesired final CSD for enantiomer E1 shall be realized.

To reconstruct the final-time CSD of the other enantiomerspecies, e.g., E2, we can proceed as follows: by applying theinverse time transformation (45) to the known signals GE2

ðtE1Þ

and BE2ðtE1Þ, we obtain the respective temporal growth and

nucleation rate profiles, GE2ðtÞ and BE2

ðtÞ. We can then scale timeaccording to dtE2

¼ GE2ðtÞ dt, and obtain, in particular, the profile

ðBE2=GE2ÞðtE2Þ. The corresponding CSD is then easily determined by

(20) and, if seed crystals were added for the respective enantio-mer species, (21); if necessary, one has to substitute E1 by E2 inthese equations. This argument shows that the non-specifiedfinal-time CSD is completely determined by the specified one.Following these ideas, it is not hard to adapt the results to thepure forward (i.e., simulation) problem: given the temperatureprofile, determine the final-time CSDs of both enantiomer species.

Note that, especially in the enantiomer context, the existenceof solutions for the proposed inversion problem cannot be takenfor granted. Due to different liquid mass fractions of the twoenantiomer species, the temperature required to realize a certainnucleation rate (respectively, nucleation to growth rate ratio) ofone enantiomer may drive supersaturation of the other enantio-mer below one. However, this would imply dissolution of crystalsof that enantiomer, and is not covered by the model introduced inSection 2. Such a situation is indeed relevant in practice, forexample in the cyclic operation mode. There, every batch startswith excess liquid mass fraction of the preferred enantiomer.

5. Analysis using idealizing assumptions

In this section, we discuss the implications on the inversionproblem for a single crystallizer, when nucleation of one enantio-mer species is neglected. Inversion shall of course be performedfor the enantiomer species for which nucleation is not neglected.Otherwise, one could only specify how much the seed crystalsshall grow. This problem would not have a unique solution.

Depending on the actual problem setting, one of the idealiza-tions illustrated in Fig. 6 can make sense. In Fig. 6(a), nucleation ofthe counter enantiomer, E2, is neglected, and the nucleated part ofthe respective CSD, f E2 ,n, vanishes. As no seed crystals are addedfor E2, this means that no crystallization of E2 occurs. In Section5.1, we show how this leads to an idealized system which isorbitally flat (i.e., flat in a scaled time tE1

), and thus easily

Fig. 6. Idealizations; (a) neglecting crystallization of the counter enantiomer;

(b) neglecting nucleation of the preferred enantiomer.

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269262

invertible, effectively describing single substance crystallizationof the preferred enantiomer, E1. An application of this scenariomay arise when a product (preferred enantiomer) with a strictlydefined CSD must be achieved.

In Fig. 6(b), nucleation of the preferred enantiomer, E1, isneglected, so that f E1 ,n vanishes. The CSD of E1 is then fullydescribed by the seed distribution and the gain in length of theseed crystals, tE1

. The final CSD of the counter enantiomer, E2, isspecified, and an appropriate temperature profile is sought. Thiswill be discussed in Section 5.2. Compared to Section 4.4, theinversion will be simplified in that the zero dynamics can bewritten in terms of one state only.

When applying one of these idealizations, one of course has tocheck whether it is justified for the specific problem, possibly a

posteriori, i.e., after a temperature trajectory TðtÞ has beenobtained. This will be demonstrated for a numerical example inSection 6.

5.1. Neglecting nucleation of the counter enantiomer

Here, it is assumed that no crystallization occurs for thecounter enantiomer species, E2, as illustrated in Fig. 6(a). Thismeans that no seed crystals have been added for this species, and,on the other hand, (primary) nucleation is neglected.

When the results for the system inversion in Sections 4.1,4.2 and 4.4 are adopted to this scenario, with the targetedenantiomer being E1, then what remains of Eqs. (42) is

d

dtE1

zi ¼ ziþ1, i¼ 1 . . .4, ð46aÞ

d

dtE1

z5 ¼ L5f hðx1, . . . ,x4,0, . . . ,0,x9Þ

þLgL4f hðx1, . . . ,x4,0, . . . ,0,x9Þ � u, ð46bÞ

where the definitions of x and z follow (23), (26), (38), (41).In particular,

x9 ¼BE1

GE1

� ��1

z1,1

3z2,

1

6z3,

1

6z4,

1

6z5,0;0,0;0

� �: ð47Þ

This system is of dimension n¼5, and it has essentially the samestructure as the one treated by Vollmer and Raisch (2003). Underthe basic assumptions on growth and nucleation rates in Section2, it can be easily verified to be flat: the relative degree isn¼ 5¼ n. Of course, for complicated rate expressions, inversionsolutions may be more demanding than in Vollmer and Raisch(2003). Specifically, it may be necessary to solve (47) numerically.However, unlike in Section 4.4, no differential equations must besolved.

5.2. Neglecting nucleation of the preferred enantiomer

It can be shown in a fashion analogous to Section 4.3 that themoment model resulting from this idealization is not orbitally flat.In this case, in (25), treat the constant x4 as a parameter, substituteit into (25a), (25c), (25d), and remove (25b).

Here, as nucleation of the preferred enantiomer is neglected,the goal is to realize a specified final CSD of the counterenantiomer, E2. The solution of this inversion problem is analo-gous to the procedure described in Sections 4.1, 4.2 and 4.4.However, time must now be scaled with the growth rate of thecounter enantiomer

dtE2¼ GE2

ð�Þ dt: ð48Þ

It still makes sense to define tE1, i.e., the gain in length of the seed

crystals of the preferred enantiomer, as a state variable. In fact, inthe absence of nucleation, the CSD of E1 is merely a shifted

version of the CSD of the seed crystals, where the value of theshift is tE1

. It is easy to show that its moments are fullydetermined by polynomial expressions, in terms of tE1

m0E1ðtE1Þ ¼ m0E1

ð0Þ ¼ const:, ð49aÞ

m1E1ðtE1Þ ¼ m0E1

ð0ÞtE1þm1E1

ð0Þ, ð49bÞ

m2E1ðtE1Þ ¼ m0E1

ð0Þt2E1þ2m1E1

ð0ÞtE1þm2E1

ð0Þ, ð49cÞ

m3E1ðtE1Þ ¼ m0E1

ð0Þt3E1þ3m1E1

ð0Þt2E1þ3m2E1

ð0ÞtE1þm3E1

ð0Þ: ð49dÞ

Note that the final gain in length of the seed crystals, tE1 ,f , is notpart of the specification, and, due to differing growth rates of E1

and E2, it will, in general, be different from the final scaled time,tE2 ,f .

By setting BE1ðxÞ � 0, and substituting the polynomial expres-

sions (49) into the remaining growth and nucleation rate func-tions, the results from Section 4.4 can be adopted for the idealizedsystem of this section, resulting in a simpler zero dynamics, i.e.,one involving less states. When u, x and z are now defined as

u ¼4 d

dtE2

T ¼1

GE2ð�Þ

d

dtT, ð50Þ

xT ¼4½m3E2

,m2E2,m1E2

,m0E2,tE1

,T�, ð51Þ

and

zT ¼4 m3E2

,3m2E2,6m1E2

,6m0E2,6

BE2

GE2

ðxÞ,tE1

� �, ð52Þ

respectively, one arrives at the following system in normal form:

d

dtE2

zi ¼ ziþ1, i¼ 1 . . .4, ð53aÞ

d

dtE2

z5 ¼ L5f hðxÞþLgL4

f hðxÞ � u, ð53bÞ

d

dtE2

z6 ¼GE1

GE2

ðxÞ, ð53cÞ

where

L5f hðxÞ ¼ 6

@

@x

BE2

GE2

ðxÞ

� �f ðxÞ, ð54Þ

LgL4f hðxÞ ¼ 6

@

@x6

BE2

GE2

ðxÞ

� �: ð55Þ

6. Numerical example

6.1. Detailed model

The equations and parameters used in the following case studyare based on ongoing work for modeling the crystallization ofL-/D-threonine in the PCF group (Andreas Seidel-Morgenstern) atthe Max Planck Institute in Magdeburg. For an explanation ofmodeling background, assumptions, equations and determinationof parameters see Elsner et al. (2011). The following model is aresult of further modifications and adjustments of parameters.Notably, a new (empirical) term has been added (e.g., Eicke et al.,2010) to describe the effect of heterogenous primary nucleation,where different enantiomer species in the tank influence thenucleation rates of each other via their solid phases. Essentiallythe same version of the model that we use in the following hasalso been employed by Hofmann et al. (2011). However, asopposed to this contribution, we do not consider size-dependent

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269 263

growth here. Therefore, the model can be obtained from Hofmannet al. (2011) by setting gðLÞ � 1.

In the following, L-threonine corresponds to E1, and D-threo-nine to E2. The supersaturation of an enantiomer, for exampleE1, is given by:

SE1¼

wl,E1

weq,E1

, with weq,E1¼ a1ðT�273:15Þþa0, ð56Þ

where T is the temperature of the slurry, i.e., the so-calledcrystallization temperature. The growth rate expression is

GE1ð�Þ ¼ kg0,E1

exp �EAg,E1

RgT

� �ðSE1ð�Þ�1Þg , ð57Þ

with

kg0,E1¼ kg01,E1

�ong : ð58Þ

Here, o denotes the stirrer speed. For each enantiomer, thenucleation rate is a sum of contributions from primary andsecondary nucleation. For example, for E1,

BE1ð�Þ ¼ Bprim,E1

ð�ÞþBsec,E1ð�Þ: ð59Þ

By primary nucleation, nuclei of an enantiomer species can beformed, even without the presence of existing crystals of thatspecies in the tank. The following expression is based on work byMersmann (2001):

Bprim,E1¼ kb,primT exp �

KT ,visc

T

� �

�exp �wl,E1þwl,E2

Kw,visc

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln

rs

Ceq,E1

� �s� ðSE1

Ceq,E1Þ7=3

Table 1Parameters and initial conditions used in the numerical example.

Variable Symbol

Density of enantiomer crystals rs

Volume shape factor kv

Constants for solution density K1

K2

K3

Solubility constants a1

a0

Universal gas constant Rg

Stirrer speed oCrystal growth

Growth rate coefficient kg01,ðE1 ;E2 Þ

Exponent for stirrer speed (growth) ng

Activation energy for crystal growth EAg,ðE1 ;E2 Þ

Growth exponent g

Primary nucleationNucleation coefficient (primary nucleation) kb,prim

Parameter for temperature dependence KT ,visc

Parameter for mass fraction dependence Kw,visc

Constant for exponential law (primary nucleation) kprim,log

Heterogenous primary nucleation constant A1

Secondary nucleationNucleation coefficient (secondary nucleation) kb,sec01,ðE1

Exponent for stirrer speed (secondary nucleation) nsec,b

Activation energy (secondary nucleation) EAb,sec,ðE1 ;

Nucleation exponent (secondary nucleation) bsec

Nucleation exponent (secondary nucleation) nm3

Initial conditionsMass of solvent (water) mW

Initial liquid (dissolved) masses ml,ðE1 ;E2 Þð0

Initial (seed) moments of the preferred enantiomer CSD m0E1ð0Þ

m1E1ð0Þ

m2E1ð0Þ

m3E1ð0Þ

Initial moments of the counter enantiomer CSD miE2ð0Þ

�expð�kprim,logðlnðrs=Ceq,E1

ÞÞ3

ðlnðSE1ÞÞ

2Þ � ð1þA1m2E2

Þ: ð60Þ

The empirical term ð1þA1m2E2Þ was introduced as a means to

describe the effect of heterogenous primary nucleation. Theequilibrium concentration in the tank for enantiomer E1, Ceq,E1

,is given by

Ceq,E1¼ rsolweq,E1

: ð61Þ

The density of the aqueous solution in the tank is given by

rsol ¼ 1000 � ðrwaterþK3ðwl,E1þwl,E2

ÞÞ, ð62Þ

rwater ¼ 1=ðK1þK2ðT�273:15Þ2Þ: ð63Þ

Secondary nucleation of E1 is induced by existing crystals of E1

(Mersmann, 2001):

Bsec,E1¼ kb,sec0,E1

exp �EAb,sec,E1

RgT

� �� ðSE1�1Þbsec ðm3E1

Þnm3 , ð64Þ

with

kb,sec0,E1¼ kb,sec01,E1

� exp �nsec,b

o

� �: ð65Þ

The kinetics of E2 are analogous to (56)–(65), and are obtainedsimply by exchanging the indexes for E1, E2. Parameters whichhave an index of E1 or E2 may differ between the enantiomerspecies. However, in this case study, these pairs are equal, i.e.,parameters are symmetric w.r.t. the two enantiomer species.

Table 1 lists the values of the parameters and initial condi-tions. Note that in this numerical example stirrer speed o isassumed constant.

Value Unit

1250 (kg m�3)

0.1222 (–)

1.00023 (m3 kg�1)

4.68088�10�6 (m3 kg�1 K�2)

0.3652 (kg m�3)

3.983�10�3 (K�1)

�3.4505�10�2 (–)

8.314472 (J K�1 mol�1)

2p� 250 (min�1)

2.7493�107 (m min�1minng)

0.4573 (–)

75.549�103 (J mol�1)

1.2269 (–)

2.35657�10�5 ((m3 kg�1)7/3 min�1 K�1)

1874.4 (K)

0.29 (–)

4.304�10�3 (–)

6.5608�10�4 (m�2)

;E2 Þ3.2635�1030

ðm�3nm3 min�1Þ

4.9295�103 (min�1)

E2 Þ63.83�103 (J mol�1)

16.2477 (–)

0.65317 (–)

0.35919 (kg)

Þ 0.045405 (kg)

1.4412�104 (–)

7.8331 (m1)

4.7793�10�3 (m2)

3.2733�10�6 (m3)

0 (mi)

Fig. 8. Scenario 1, final MDFs.

Fig. 9. Scenario 1, third moments.

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269264

6.2. Scenario one: final-time CSD of the preferred enantiomer

specified

In this case study, we concentrate on inversion results when thefinal CSD of one of the involved nucleated crystal populations(distinguished by the enantiomer species) is specified. First, weconsider inversion of the model for realization of a prescribed final-time CSD of the preferred enantiomer, i.e., the seeded enantiomer, E1.We compare the exact solution, Sections 4.1, 4.2 and 4.4, to the onebased on orbital flatness of the system resulting from neglecting thenucleation of the counter enantiomer, see Section 5.1.

The specified (desired) final-time CSD for the preferred enan-tiomer, E1, follows an exponential prototype CSD:

f E1 ,desðLÞ ¼

m3f ,n,des � exp �20L

Lmax

� �R Lmax

0 L3 exp �20L

Lmax

� �dL

, LA ½0,Lmax�

f E1 ,seedðL�LmaxÞ, L4Lmax:

8>>>>><>>>>>:

ð66Þ

Recall that the upper CSD part is merely a shifted version of theCSD of the seed crystals. The free parameters m3f ,n,des and Lmax

determine the resulting third moment of the lower (i.e.,nucleated) part of the specified final CSD, and the maximumlength of crystals belonging to that part, respectively. They arechosen as

m3f ,n,des ¼ 10:0 ½cm3�; Lmax ¼ 0:5 ½mm�: ð67Þ

Comparison between exact and simplified solutions is con-ducted in the following manner: the input (temperature) profilescomputed by the exact and simplified solution techniques,respectively, are compared. Furthermore, the temperature profilecorresponding to the simplified solution is applied as an input tothe original model, and the resulting deviations of the final-timeCSD from the specification are shown. In addition to plots of CSDsof final crystal populations, plots of mass density functions(MDFs) are shown, which may be more appropriate to judgeerrors. To obtain these CSDs, respectively MDFs, the forward(i.e., simulation) problem is solved.

Figs. 7–11 show numerical results obtained for scenario one.The plots labeled ‘‘exct. inv.’’ correspond to the exact inversion.The temperature profile obtained using the flatness-based idea-lized solution is used as an input to simulate the original model,

Fig. 7. Scenario 1, final CSDs. Fig. 10. Scenario 1, temperature profiles.

Fig. 11. Scenario 1, supersaturation profiles.

Fig. 12. Scenario 2, final CSDs.

Fig. 13. Scenario 2, final MDFs.

Fig. 14. Scenario 2, third moments.

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269 265

and the resulting trajectories and CSDs (MDFs) are labeled‘‘flat inv.’’. In the plots of the third moments, these momentsare split into parts from nucleation (index n) and growing seeds(index s). For example, m3E1

¼ m3E1 ,nþm3E1 ,s. This splitting is, ofcourse, only relevant for the preferred enantiomer species, forwhich seed crystals are initially added.

Note that the choice of model equations, parameters and specifi-cations for the case study cannot represent all circumstances in whichthe proposed exact as well as idealized solution techniques arepotentially helpful. Nevertheless, the numerical results may be anorientation when using different equations, parameters or problemsettings. We think that, for problems concerning mainly the preferredenantiomer in a single crystallizer, the idealized, flatness-basedapproach is very well suited. Note that an a posteriori justificationof the idealizing assumption can be given as follows: after atemperature profile has been obtained, in can be checked in a singlesimulation of the original (non-idealized) model that crystallization ofthe counter enantiomer is sufficiently low, and, as a consequence, thedegradation of the specified final CSD is acceptably small. Thenumerical results of scenario one, Figs. 7–11, demonstrate how wellthis approach can work, even when the amount of crystallinematerial (third moment) of the preferred enantiomer produced bynucleation is comparable to the amount produced by growing seeds(see Fig. 9), and also the resulting third moment of the counterenantiomer is not considerably lower. The dashed and correspondingsolid lines in these figures virtually coincide.

6.3. Scenario two: final-time CSD of the counter enantiomer

specified

Here, we compute the solution which realizes a prescribedfinal-time CSD of the counter enantiomer, E2. The specificationfollows an exponential prototype CSD:

f E2 ,desðLÞ ¼

m3f ,n,des � exp �20L

Lmax

� �R Lmax

0 L3 exp �20L

Lmax

� �dL

, LA ½0,Lmax�

0, L4Lmax,

8>>>>><>>>>>:

, ð68Þ

with

m3f ,n,des ¼ 0:001 ½cm3�; Lmax ¼ 0:59 ½mm�: ð69Þ

The numerical results are shown in Figs. 12–17. We comparethe exact solution, Sections 4.1, 4.2 and 4.4, to the one involving a

Fig. 15. Scenario 2, third moments, zoom on E2.

Fig. 16. Scenario 2, temperature profiles.

Fig. 17. Scenario 2, supersaturation profiles.

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269266

simplified zero dynamics, resulting from neglecting the nuclea-tion of the preferred enantiomer, see Section 5.2. The temperatureprofile obtained in the latter case is used as an input to simulatethe original model, resulting in the plots labeled ‘‘no p.-nuc. inv.’’.

Scenario two demonstrates that the relation between the finalCSDs of the two enantiomer species may not be straightforward.This time, the specification consists of the counter enantiomerCSD. The same exponential CSD shape as in scenario one is used,but with smaller ‘‘amplitude’’. Fig. 12 shows that a relatively highamount of nucleation of the preferred enantiomer results, i.e., theratio of nucleation of the preferred enantiomer over nucleation ofthe counter enantiomer is much higher than in scenario one. Still,when compared to the growth of the seeds (see Fig. 14) theresulting amount of nucleation of the preferred enantiomer is lowenough for applicability of the idealized solution from Section 5.2.The resulting errors, though noticeable (see Figs. 13 and 15), arereasonably small. In fact, in Figs. 12, 16 and 17, the dashed linesand the corresponding solid lines are virtually indistinguishable.

7. Conclusion

In this paper, we have considered population balance andmoment models describing crystallization of two enantiomerspecies in one vessel. We have summarized the concepts ofdifferential flatness and orbital flatness, which are properties ofnonlinear systems that greatly facilitate analysis and control.Based on a well known necessary condition for flatness, we havederived a test to exclude orbital flatness of typical momentmodels for preferential crystallization. Note that this test couldbe useful for a wider class of single input systems. We then havepresented a technique for inversion of the system, when a given,final crystal size distribution belonging to either one of theenantiomer species must be achieved in a batch process. In atechnical application, one species, the preferred enantiomer, isseeded, while nucleation of the other species, the counter enan-tiomer, shall be kept suitably low.

Exact inversion involves the numerical solution of differentialequations. This computational burden is certainly adequate whenfeedforward control profiles for realization of specified CSDs aresought. The most relevant application in practice may be theachievement of a product (i.e., the preferred enantiomer) with atightly specified CSD. In this context, two general requirementsseem important. First, the nucleated CSD of the preferred enan-tiomer, for which, within certain bounds, any shape can berealized, should represent a significant share of its total CSD(e.g., in terms of the associated third moment). Second, afterobtaining the solution, it has to be checked if the purity require-ment is met, i.e., if the crystalline mass of the counter enantiomeris sufficiently low.

When (only) the preferred enantiomer is targeted by thespecification, a simplified inversion technique, which is basedon orbital flatness of an idealized model (i.e., one where nuclea-tion of the counter enantiomer is neglected), can be applied tosolve the inversion problem. However, the computational advan-tage seems even more important for the solution of optimalcontrol problems concerning properties of the final CSD of thepreferred enantiomer. One could adopt the method proposed byVollmer and Raisch (2003): by parametrizing the final CSD of thepreferred enantiomer the whole evolution of the state is thenalgebraically parametrized, and the dynamic optimization pro-blem is converted to a (nonlinear) static optimization problem.

A different approach, frequently used in practice, to obtain awell defined product CSD, is to rely mainly on the growth of seedcrystals (i.e., the product then mainly consists of a shifted versionof the CSD of the seed crystals). Especially for substances for

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269 267

which nucleation (in particular, secondary nucleation) is gener-ally low relative to growth optimization of properties of thenucleated CSD of the preferred enantiomer is then a less vitaltask. Therefore, entirely neglecting nucleation of the preferredenantiomer makes increased sense in that case, and a second typeof simplified solution presented in this paper can be applied. Inthis case, as nucleation of the counter enantiomer is not

neglected, a dynamic optimization scheme can be most usefulfor the actual separation problem. For example, one could max-imize the purity of the product.

In future work, the extension of the developed ideas to a widerclass of models should be investigated. For example, our analysisis based on closed moment models for preferential crystallization,which cannot be obtained when PBE models are more complex.This can, e.g., be due to size-dependent growth rates. However, itmay be possible to follow the proposed ideas and derive inversionschemes on the basis of approximate ODE models, using conceptsbrought up in the literature review in Section 1. Furthermore, itwas assumed in this work that growth and nucleation ratefunctions are continuously differentiable. In particular, it wasrequired that the process is operated under conditions, where thegrowth (and nucleation) rates of both enantiomer species arepositive; this may not be the case in certain applications. More-over, while not explicitly considered in this paper, the developedmethods may also be useful in areas related to preferentialcrystallization, such as selective crystallization of polymorphs,or fractional crystallization. Transfer of the results should beparticularly simple, if the system at hand is compatible with themodel structure and the basic assumptions given in Section 2.

The developed feedforward control schemes should also becomplemented by appropriate feedback control. Our treatmentnaturally suggests the use of methods from geometric controltheory, such as exact, respectively input–output linearization.The efficient flatness-based inversion schemes enabled by the useof idealized design models, or optimal control solutions based onthis, may also be useful in the context of model predictive control.

In Part II, the inversion results of this paper will be extended toa setup consisting of two coupled crystallizer vessels. This setupenables simultaneous production of pure fractions of two enan-tiomer species, one in each vessel, whereby the exchange ofcrystal free liquid helps in reducing the driving force for nuclea-tion of the counter enantiomers. We will show the effect ofadditional idealizations motivated by the design of the process.In particular, we will show how the above mentioned, simplifiedsolutions can be combined with these new idealizations.

Acknowledgments

We gratefully acknowledge financial support by the GermanResearch Foundation for this project (DFG RA 516/7-1). We alsoacknowledge the productive cooperation with our project part-ners from the PCF group at the Max Planck Institute for Dynamicsof Complex Technical Systems: Matthias Eicke, Martin PeterElsner and Andreas Seidel-Morgenstern.

Appendix A. Test for lack of orbital flatness

Here, based on the ruled manifold criterion (Sluis, 1993;Rouchon, 1994; Fliess et al., 1995b; Martin et al., 1997; Sira-Ramırez, 2004; Levine, 2009), we derive a test to exclude orbitalflatness of a SISO system _x ¼ f ðx,uÞ, with dim x¼ nZ3, anddim u¼m¼ 1. The test requires checking conditions on f and itsderivatives w.r.t. u.

Consider three out of the n system differential equations,combined with a time scaling function sð�Þ; in the following, _xdenotes the derivative of x w.r.t. scaled time

_xa ¼ sðx,uÞ � f aðx,uÞ, ðA:1aÞ

_xb ¼ sðx,uÞ � f bðx,uÞ, ðA:1bÞ

_xc ¼ sðx,uÞ � f cðx,uÞ, ðA:1cÞ

where a,b,cAf1, . . . ,ng. Further, consider a triple ðp,xp,upÞ, whichsatisfies (A.1)

pa ¼ sðxp,upÞ � f aðxp,upÞ, ðA:2aÞ

pb ¼ sðxp,upÞ � f bðxp,upÞ, ðA:2bÞ

pc ¼ sðxp,upÞ � f cðxp,upÞ: ðA:2cÞ

The function f ð�Þ is supposed to be continuous in its arguments,and continuously differentiable twice w.r.t. u, at least in a certainneighborhood of ðxp,upÞ.

The ruled manifold criterion states that, if a system is flat,there must exist, at any xp, a vector gARn, ga0, s.t. a line pþlgis contained in the manifold of all possible values _x for the lefthand sides of the system differential equations; in this regard,x¼ xp is considered a fixed parameter.

Assume that we have chosen an appropriate candidate timescaling function sð�Þ, and want to check if it can turn the systemflat. By appropriate we mean, on the one hand, that the values ofsð�Þ must be positive. On the other hand, we also suppose that, atðxp,upÞ, sð�Þ is continuously differentiable twice w.r.t. u. In thefollowing, we investigate whether, after scaling time according tosð�Þ, the ruled manifold criterion can be satisfied locally, atðp,xp,upÞ. In this context, p, xp and up are constants. Note that pdepends on the value of sðxp,upÞ, so that ðp,xp,upÞ is only fullydetermined once the value of the time scaling function is known.By locally we mean that we allow the domain of u to be restrictedto some small neighborhood of up. By varying u within thatneighborhood, it must then be possible to vary _x on a local (i.e., asmall open) segment of a straight line containing p.

Assume that, for given ðxp,upÞ, u could be eliminated (at leastlocally) from the time-scaled system differential equations, i.e.,_x ¼ sðxp,uÞf ðxp,uÞ. This assumption will be justified at the end ofthis section. More specifically, it will be seen that if the conditionsof the test, which will be developed in the following, are satisfied,this implies that after choosing any appropriate time scalingfunction it is always possible to solve (locally) one of the systemdifferential equations (say, the ith one) for u. In other words, u canbe written as a function of _xi, denoted as u¼ ðsf iÞ

�1ð�, _xiÞ. Knowing

this, it is clear that when trying to verify the ruled manifoldcriterion only straight lines with gia0 have to be taken intoconsideration. Otherwise the projection of the (local segment ofthe) line onto the _xi-space would be a point (pi), which getsmapped by ðsf iÞ

�1ð�Þ onto a point (up) in the input-space. That

point cannot get mapped by sð�Þf ð�Þ to (the local segment of a)straight line. On the other hand, the projection of an opensegment of a straight line containing p, with gia0, onto the_xi-space is an open interval containing pi. The inverse image, w.r.t.the continuous function sð�Þf ið�Þ, of this open set is an open setcontaining up. Furthermore, because sð�Þf ið�Þ does have an inversefunction, this open set is equal to the respective forward image ofthe inverse function ðsf iÞ

�1ð�Þ (see, e.g., Apostol, 1974, Definition

4.21), which is a connected set (because continuous functionsmap connected sets onto connected sets). In conclusion, in orderto vary _x on a local segment of a straight line, u must take on allvalues in an open interval containing up, i.e., a neighborhood of up.

In the following, we investigate input deviations characterizedby Du¼ u�up. The goal is to derive local conditions, at ðp,xp,upÞ,

S. Hofmann, J. Raisch / Chemical Engineering Science 80 (2012) 253–269268

for the value of the time scaling function sð�Þ and its derivativesw.r.t. u. In view of the above considerations, we can now state theconditions of the ruled manifold criterion in the context of theexplicit system representation (A.1): given any ðp,xp,upÞ, whichsatisfy (A.2), there must exist a ga0, s.t.

paþlga ¼ sðxp,upþDuÞ � f aðxp,upþDuÞ, ðA:3aÞ

pbþlgb ¼ sðxp,upþDuÞ � f bðxp,upþDuÞ, ðA:3bÞ

pcþlgc ¼ sðxp,upþDuÞ � f cðxp,upþDuÞ ðA:3cÞ

is satisfiable (a solution lAR exists) for all values of Du, at least ina neighborhood of zero.

According to the ruled manifold criterion, not all componentsof g are allowed to be zero. Suppose therefore that gaa0. Takingtwo of the equations (A.3), (A.3a), (A.3b), l can be eliminated

gb

ga

½sðxp,upþDuÞf aðxp,upþDuÞ�pa�

¼ sðxp,upþDuÞf bðxp,upþDuÞ�pb: ðA:4Þ

Recall that, at ðxp,upÞ, f ið�Þ, iAf1, . . . ,ng, and sð�Þ are supposed to becontinuously differentiable twice w.r.t. u, and that (A.4) must befulfilled for all Du in a neighborhood of zero. It is thereforejustified to take the first and second derivatives of (A.4) w.r.t.Du. In the following, for brevity, we omit the dependencies ðx,uÞof the functions, and we denote ð�Þ0 :¼ d=duð�Þ ¼ d=dðDuÞð�Þ,ð�Þ00 :¼ d2=du2

ð�Þ ¼ d2=dðDuÞ2ð�Þ. The first derivative w.r.t. Du of(A.4) then reads

gb

ga

ðsf 0aþ f as0Þ ¼ sf 0bþ f bs0: ðA:5Þ

In the same way, (A.3a) and (A.3c) can be compared. We nowdenote k1 :¼ gb=ga, k2 :¼ gc=ga. Taking derivatives up to order two,w.r.t. Du, yields the following set of equations:

sðk1f 0a�f 0bÞþs0ðk1f a�f bÞ ¼ 0, ðA:6aÞ

sðk2f 0a�f 0cÞþs0ðk2f a�f cÞ ¼ 0, ðA:6bÞ

sðk1f 00a�f 00bÞþs02ðk1f 0a�f 0bÞþs00ðk1f a�f bÞ ¼ 0, ðA:7aÞ

sðk2f 00a�f 00c Þþs02ðk2f 0a�f 0cÞþs00ðk2f a�f cÞ ¼ 0: ðA:7bÞ

Note that (A.7) are simply the derivatives, w.r.t Du, of (A.6).In order for sð�Þ to be an appropriate time scaling, the solution for s

must be non-trivial. For such a solution to exist in (A.6), therespective determinant must be zero

f 0bf c�f bf 0cþk1ðf af 0c�f 0af cÞ�k2ðf af 0b�f 0af bÞ ¼ 0: ðA:8Þ

Suppose that, at ðxp,upÞ, we have checked that f af 0b�f 0af ba0. Then

k2 ¼f 0bf c�f bf 0cþk1ðf af 0c�f 0af cÞ

f af 0b�f 0af b

: ðA:9Þ

From (A.6a), we also deduce k1f a�f ba0, since, due to theassumption f af 0b�f 0af ba0, the opposite would imply k1f 0a�f 0ba0,immediately leading to the trivial solution s¼0. We can thensolve (A.6a) for s0

s0 ¼ �k1f 0a�f 0bk1f a�f b

s: ðA:10Þ

After substituting the expressions for k2 and s0 into (A.7), we againarrive at a system of equations, which is linear in the twounknowns s and s00, when k1 is considered as an additional(unknown) parameter. We do not show the lengthy equationshere. For a nontrivial solution, i.e., sa0, to exist, the determinantof this two by two system, which takes a rather simple form, must

again be zero

k1f a�f b

f af 0b�f 0af b

� ðf af 0bf 00c�f af 00bf 0c�f 0af bf 00cþ f 0af 00bf cþ f 00af bf 0c�f 00af 0bf cÞ ¼ 0:

ðA:11Þ

Assume we have checked that, at ðxp,upÞ, where also f af 0b�f 0af ba0,the second factor of (A.11) is non-zero. Then the determinantcannot be zero (the condition k1f a�f ba0 was established above).We conclude that, given ðxp,upÞ,

f af 0b�f 0af ba0, ðA:12aÞ

and

f af 0bf 00c�f af 00bf 0c�f 0af bf 00cþ f 0af 00bf cþ f 00af bf 0c�f 00af 0bf c a0, ðA:12bÞ

together, exclude the existence of any appropriate time scalingfunction sðx,uÞ, which would allow a ruling, with gaa0, of themanifold of possible (scaled) time derivatives _x, at the resultingtriple ðp,xp,upÞ. To exclude orbital flatness of a given system_x ¼ f ðx,uÞ, one can select a suitable ðxp,upÞ, and verify (A.12) forevery possible choice of f a among the n components of f ð�Þ.Therefore, each time, two of the remaining components must beappropriately assigned to f b, f c .

Recall that in our argument, it was supposed that, at the pointðxp,upÞ, sð�Þ is continuously differentiable twice w.r.t. u. In otherwords, we have only ruled out all time scaling functions, whichare continuously differentiable twice w.r.t. u at that point, ascandidates for turning the system flat. This condition can berelaxed by noting that, due to continuity of f ð�Þ and its first andsecond derivatives, the verified inequalities (A.12) will hold notonly at ðxp,upÞ, but also in a certain neighborhood of that point.This means that any time scaling function which is continuouslydifferentiable twice w.r.t. u anywhere in that neighborhood canbe ruled out as well. In conclusion, by saying that the system isnot orbitally flat, we mean that no time scaling function which iscontinuously differentiable twice w.r.t. u anywhere in a certainneighborhood of ðxp,upÞ permits the ruled manifold necessarycondition to be locally satisfied.

If the conditions of the test described above are satisfied, thenf af 0b�f 0af ba0 is true for at least one pair (a,b), a,bAf1, . . . ,ng.Using this inequality, it can also be easily seen that no appropriatetime scaling function can cause the derivatives, w.r.t. u, of sð�Þf að�Þ

and sð�Þf bð�Þ to be simultaneously zero. Then, by the implicitfunction theorem, at least one of sð�Þf að�Þ or sð�Þf bð�Þ can (at leastlocally) be solved for u, confirming that u can be eliminated fromthe time-scaled system differential equations. Also, no appropri-ate time scaling function can cause all derivatives of the systemdifferential equations, w.r.t. the transformed time, to besimultaneously zero.

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