# Robust optimal control of polymorphic transformation in batch crystallization

**ABSTRACT** One of the most important problems that can arise in the development of a pharmaceutical crystallization process is the control of polymorphism, in which there exist different crystal forms for the same chemical compound. Different polymorphs can have very different properties, such as bioavailability, which motivates the design of controlled processes to ensure consistent production of the desired polymorph to produce reliable therapeutic benefits upon delivery. The optimal batch control of the polymorphic transformation of L-glutamic acid from the metastable α-form to the stable β-form is studied, with the goal of optimizing batch productivity, while providing robustness to variations in the physicochemical parameters that can occur in practice due to variations in contaminant profiles in the feedstocks. A nonlinear state feedback controller designed to follow an optimal setpoint trajectory defined in the crystallization phase diagram simultaneously provided high-batch productivity and robustness, in contrast to optimal temperature control strategies that were either nonrobust or resulted in long-batch times. The results motivate the incorporation of the proposed approach into the design of operating procedures for polymorphic batch crystallizations. © 2007 American Institute of Chemical Engineers AIChE J, 2007

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**ABSTRACT:**This review discusses important research developments and arising challenges in the field of industrial crystallization with an emphasis on recent problems. The most relevant areas of research have been identified. These are the prediction of phase diagrams; the prediction of effects of impurities and additives; the design of fluid dynamics; the process control with process analytical technologies (PAT) tools; the polymorph and solvate screening; the stabilization of non-stable phases; and the product design. The potential of industrial crystallization in various areas is outlined and discussed with particular reference to the product quality, process design, and control. On this basis, possible future directions for research and development have been pointed out to highlight the importance of crystallization as an outstanding technique for separation, purification as well as for product design.Frontiers of Chemical Science and Engineering. 7(1). - SourceAvailable from: Richard D Braatz
##### Article: Nonlinear model predictive control for the polymorphic transformation of L‐glutamic acid crystals

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**ABSTRACT:**Polymorphism, a phenomenon where a substance can have more than one crystal forms, has recently become a major interest to the food, speciality chemical, and pharmaceutical industries. The different physical properties for polymorphs such as solubility, morphology, and dissolution rate may jeopardize operability or product quality, resulting in significant effort in controlling crystallization processes to ensure consistent production of the desired polymorph. Here, a nonlinear model predictive control (NMPC) strategy is developed for the polymorphic transformation of L-glutamic acid from the metastable α-form to the stable β-form crystals. The robustness of the proposed NMPC strategy to parameter perturbations is compared with temperature control (T-control), concentration control (C-control), and quadratic matrix control with successive linearization (SL-QDMC). Simulation studies show that T-control is the least robust, whereas C-control performs very robustly but long batch times may be required. SL-QDMC performs rather poorly even when there is no plant-model mismatch due to the high process nonlinearity, rendering successive linearization inaccurate. The NMPC strategy shows good overall robustness for two different control objectives, which were both within 7% of their optimal values, while satisfying all constraints on manipulated and state variables within the specified batch time. © 2009 American Institute of Chemical Engineers AIChE J, 2009AIChE Journal 08/2009; 55(10):2631 - 2645. · 2.58 Impact Factor - SourceAvailable from: Richard D Braatz[Show abstract] [Hide abstract]

**ABSTRACT:**Most pharmaceutical manufacturing processes include a series of crystallization processes to increase purity with the last crystallization used to produce crystals of desired size, shape, and crystal form. The fact that different crystal forms (known as polymorphs) can have vastly different characteristics has motivated efforts to understand, simulate, and control polymorphic crystallization processes. This article proposes the use of weighted essentially nonoscillatory (WENO) methods for the numerical simulation of population balance models (PBMs) for crystallization processes, which provide much higher order accuracy than previously considered methods for simulating PBMs, and also excellent accuracy for sharp or discontinuous distributions. Three different WENO methods are shown to provide substantial reductions in numerical diffusion or dispersion compared with the other finite difference and finite volume methods described in the literature for solving PBMs, in an application to the polymorphic crystallization of L-glutamic acid. © 2008 American Institute of Chemical Engineers AIChE J, 2009AIChE Journal 12/2008; 55(1):122 - 131. · 2.58 Impact Factor

Page 1

PROCESS SYSTEMS ENGINEERING

Robust Optimal Control of Polymorphic

Transformation in Batch Crystallization

Martin Wijaya Hermanto and Min-Sen Chiu

Dept. of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576

Xing-Yi Woo

Dept. of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 11576;

and Dept. of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, IL 61801

Richard D. Braatz

Dept. of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, IL 61801

DOI 10.1002/aic.11266

Published online August 27, 2007 in Wiley InterScience (www.interscience.wiley.com).

One of the most important problems that can arise in the development of a pharma-

ceutical crystallization process is the control of polymorphism, in which there exist dif-

ferent crystal forms for the same chemical compound. Different polymorphs can have

very different properties, such as bioavailability, which motivates the design of con-

trolled processes to ensure consistent production of the desired polymorph to produce

reliable therapeutic benefits upon delivery. The optimal batch control of the polymor-

phic transformation of L-glutamic acid from the metastable a-form to the stable b-form

is studied, with the goal of optimizing batch productivity, while providing robustness

to variations in the physicochemical parameters that can occur in practice due to var-

iations in contaminant profiles in the feedstocks. A nonlinear state feedback controller

designed to follow an optimal setpoint trajectory defined in the crystallization phase

diagram simultaneously provided high-batch productivity and robustness, in contrast

to optimal temperature control strategies that were either nonrobust or resulted in

long-batch times. The results motivate the incorporation of the proposed approach into

the design of operating procedures for polymorphic batch crystallizations. ?2007

American Institute of Chemical Engineers AIChE J, 53: 2643–2650, 2007

Keywords: T-control, robust T-control, C-control, polymorphic transformation, phar-

maceutical crystallization

Introduction

Polymorphism, in which multiple crystal forms exist for the

same chemical compound, is of significant interest to the phar-

maceutical industry.1–5According to Ostwald’s Rule of Stages,

in a polymorphic system, the most soluble metastable form

appears first, followed by more stable polymorphs. This rule

holds for most polymorphic systems, which implies that care

must be taken to avoid the formation of metastable crystals

when trying to crystallize the most stable crystal form. Some-

times a relatively small shift in the operating conditions can

result in the appearance of crystals of an undesired polymorph.

Metastable crystals have appeared during the production of

specialty chemicals, such as pharmaceuticals, dyestuffs, and

pesticides. The variation in physical properties, such as crys-

tal shape, solubility, hardness, color, melting point, and

chemical reactivity makes polymorphism an important issue

for the food, specialty chemical, and pharmaceutical indus-

Correspondence concerning this article should be addressed to Min-Sen Chiu at

checms@nus.edu.sg.

?2007 American Institute of Chemical Engineers

AIChE JournalOctober 2007 Vol. 53, No. 102643

Page 2

tries, where products are specified not by chemical com-

position only, but also by their performance.2As a result,

controlling polymorphism to ensure consistent production of

the desired polymorph is important in those industries, includ-

ing the drug manufacturing industry where safety is paramount.

Deliberate isolation of metastable phases is sometimes desired

when they have advantageous processing or application proper-

ties, such as increased dissolution rate. In most cases, however,

the formulation of a product as a metastable phase is undesired

due to potential subsequent phase transformation during drying

or storage, which would change product characteristics.6

Although the crystallization control literature is vast, to

the authors’ knowledge there are no articles on the optimal

control of crystallization processes in which more than one

polymorph occurs. The vast majority of articles on nonpoly-

morph crystallization have considered the optimal control of

only one or two characteristics of the crystal-size distribu-

tion, such as weight mean size. The most widely studied

approach is to determine a temperature profile (T-control)

that optimizes an objective function based on an offline nom-

inal model.7–11Although T-control is simple to implement, it

has become well-known in recent years that T-control can be

very sensitive to variations in the kinetic parameters.12,13

This motivated the development of robust T-control, which

explicitly includes the impact of uncertainties in the objec-

tive, while determining the optimal temperature-time trajec-

tory to be followed during batch operation.14–16With advan-

ces in sensor technologies, another control strategy developed

to provide improved robustness to model uncertainty is C-

control, which follows an optimal or nearly optimal concen-

tration-temperature trajectory.3,13,17–19

Motivated by the industrial need to control polymor-

phism,2,20this article evaluates and compares the perform-

ance of these optimal control strategies for the polymorphic

transformation of L-glutamic acid from the metastable a-

form to the stable b-form. The Process Description section

describes the process model for the polymorphic transforma-

tion of L-glutamic acid. The T-control, robust T-control, and

C-control sections discuss the three control strategies investi-

gated in this article. The simulation results in the Results

section are followed by the conclusions.

Process Description

The population balance equations for the polymorphic

transformation of L-glutamic acid from the metastable a-

form to the stable b-form are21:

@fa

@tþ@ðDafaÞ

@L

¼ 0;

(1)

@fb

@tþ@ðGbfbÞ

@L

¼ BbdðL ? L0Þ;

(2)

where

f0

aðLÞ ¼ faðL;0Þ;

(3)

Da¼ kd;aðC ? Csat;aÞ

100

ða-dissolution rateÞ;

(4)

Gb¼ kg;bLðC ? Csat;bÞ

100

ðb-growth rateÞ;

(5)

Csat;i¼ aiexpðbiTÞ;i 2 fa;bgðsolubilitiesÞ;

(6)

kd;a¼exp

8:98703107

T2

?6:07613105

T

þ1:01413103

??

; (7)

kg;b¼ 0:410exp ?10900

8:314T

??

:

(8)

fiis the number density for the i-form crystals. The initial

crystal-size distributions of polymorph i, Yi(L,0), is described

by the sum of three log-normal distributions, the nthmoment

of the i-form crystals is

Z1

and the nucleation rate for b-form crystals is

li;n¼

0

LnfidL;

(9)

Bb¼ kb;bðC ? Csat;bÞ

100

lb;3;

(10)

kb;b¼ 7 3 107:

(11)

Applying the method of characteristics to Eq. 1, and

method of moments to Eq. 2, gives

?

dlb;n

dt

faðL;tÞ ¼ f0

a

L ?

Zt

0

Dadt

?

;L ? 0 (12)

¼ nG0

blb;nþ BbLn

0;n ¼ 0;1;2;:::

(13)

where G0

mented by the solute mass balance:

b5 Gb/L. The aforementioned equations are aug-

dC

dt¼ ?300qcVslurry

þ kv;aDafaðL0;tÞL3

msolution

1 ?

C

100

???

þ kv;bG0

kv;aDala;2

0

3

blb;3þ kv;bBbL3

0

3

?

;

ð14Þ

where qcis the crystal density, msolutionis the mass of solution,

Vslurryis the total volume of crystals and solution, kv,iis the vol-

umetric shape factor for polymorphic form i, and L0is the size

of the nucleated crystals. The parameter values are in Ref. 21.

In this study, the following uncertain parameters are assumed:

?

k0

g;b¼ kg;bð1 þ h1Þexp ?10900

8:314Th2

?

;

?0:2 ? h1;h2? 0:2

(15)

k0

d;a¼ kd;að1 þ h3Þ;

?0:2 ? h3? 0:2 (16)

C0

sat;a¼ Csat;að1 þ h4Þ;

C0

?0:05 ? h4? 0:05

?0:05 ? h5? 0:05

(17)

sat;b¼ Csat;bð1 þ h5Þ;

where y1and y2are the uncertainties in the growth parame-

ters for the b-form crystals, y3is the uncertainty in the disso-

lution kinetics of the a-form crystals, and y4and y5are the

uncertainties in the solubility curves of the a and b forms,

respectively. The nominal model corresponds to yi5 0, i 5

1,..., 5. While uncertainties in parameters quantified from

(18)

2644 DOI 10.1002/aicPublished on behalf of the AIChE October 2007Vol. 53, No. 10AIChE Journal

Page 3

experimental data are typically correlated,22,23uncorrelated

uncertainties were used here so as to separately assess the

robustness to different types of uncertainties.

T-control

The first journal articles on the control of batch polymor-

phic crystallization implemented temperature control (e.g.,

see21). The most widely studied approach for the optimal

control of nonpolymorphic crystallization processes has uti-

lized T-control, in which the temperature trajectory has been

computed from the optimization of an objective function,

based on an offline model with nominal parameters.9In this

study, the objective function that is maximized is the yield

of the b-form minus a penalty on the time required for the

mass of the a-form to be below some tolerance

Inominal¼ mbðtfÞ ? wntaðcÞ

¼ qckv;bVslurrylb;3ðtfÞ ? wntaðcÞ

where tfis the batch time, mb(tf) is the yield of b-form at the

end of the batch, ta(c) is the time taken to reduce the mass

of a-form below c, and wnis a weighting parameter. The val-

ues of tf, wn, and c are 7 h, 1 3 1023, and 5 3 1024g,

respectively. The second term ta(c) is included to increase

the productivity of the batch crystallizer (shorter batch times

lead to more batches per day).

To implement this strategy, the temperature-time trajectory

is parameterized as a first-order spline with 64 time intervals

(6.56 min). Examination of the temperature trajectories indi-

cated that this temporal resolution is fine enough for this

crystallization process. The temperature trajectory was con-

strained to be within the region where crystals of the a-form

dissolve, while crystals of the b-form grow, that is, the tem-

perature is constrained to be between the saturation tempera-

tures of the a- and b-forms (Tsat,a? T ? Tsat,b).* In addition,

the minimum and maximum temperature can be achieved by

cooling and heating are 258C and 508C. These constraints

were handled by parameterizing the temperature-time trajec-

tory, such that the decision variables were fractions between

0 and 1, with 0 and 1 indicating the lower and upper bounds

on the temperature at each time instance, respectively. A

genetic algorithm was used to determine an initial tempera-

ture trajectory, which was further optimized using sequential

quadratic programming. The resulting concentration-tempera-

ture trajectory for nominal model is shown in Figure 1. There

are two sections in the trajectory (heating followed by cool-

ing) where the supersaturation of b-form is maximized due

to its growth kinetics being the rate-limiting step.

(19)

Robust T-control

The solubility curves and nucleation and growth kinetics

can vary somewhat from batch to batch due to impurities in

the feed. Further, any model parameters obtained from

experiments have uncertainties due to measurement noise

and unmeasured disturbances that occur during the collection

of the experimental data used to estimate parameters. Assum-

ing all the uncertainties are independent from each other,

hmin;i? hi? hmax;i;

(20)

the uncertain parameters may be expressed as:

n

^h ¼hminþ hmax

eh¼

h : h ¼^h þ dh;kWhdhk1? 1

o

;

(21)

2

;

(22)

ðWhÞjj¼

2

hmax;j? hmin;j;

(23)

where y 5 [y1, y2,...,yn]T,^h ¼ ½^h1;^h2;...;^hn?T, ymin 5

[ymin,1, ymin,2,...,ymin,n]T, and ymax 5 [ymax,1, ymax,2,...,

ymax,n]Tare the actual, nominal, minimum, and maximum

values of the uncertain model parameters, respectively, Wyis

a weight matrix quantifying the magnitude of the uncertainty

in each parameter, and k ? k?is the vector ?-norm.

In robust T-control the objective function to be maxi-

mized appends a term to include the impact of uncertain

parameters14–16

Irobust¼ Inominal? wrdIw:c

(24)

where wr[ [0,1] is a weighting parameter and dIw.c.is the

worst-case deviation in the objective due to model uncertain-

ties. If wr5 1, Irobustis the objective function obtained for

the worst-case perturbation at the cost of potential degrada-

tion in nominal performance. The value of wrcan be selected

to be smaller than one depending on the desired trade-off

between the nominal and worst-case performance. In this

paper, wr5 0.6 provided the best balance between nominal

and worst-case performance. In robust T-control the objective

(24) is maximized subject to the condition that the con-

straints in the previous section hold for all parameters within

the uncertainty description.

Figure 1. Solubility curves and concentration-tempera-

ture trajectories of T-control and C-control

with parameters p15 36.56, p25 11.5, p35

0.4340, and p45 0.03656.

*Note that a slight violation of this constraint occurs initially due to the initial

concentration being outside this range. According to Ref. 21 the a-form crystals

grow for a short time before crossing the a solubility curve.

AIChE JournalOctober 2007Vol. 53, No. 10Published on behalf of the AIChEDOI 10.1002/aic2645

Page 4

With the widely-used approximation22,24,25

dI ¼@I

@hdh ¼ Ldh;

(25)

where L is the objective function sensitivity row vector, the

worst-case deviation in the objective function is15,23

max

kWhdhk1?1jdIj ¼ kLW?1

hk1

(26)

and a worst-case parameter uncertainty vector is

dhw:c¼ ?W?1

hv; with vk¼ðLW?1

hÞk

jðLW?1

hÞkj¼ sgn ðLW?1

hÞk;

(27)

where k ? k1is the vector 1-norm and the objective function

sensitivity vector can be computed from

L ¼dI

dh

????

tf

¼dI

dx

????

tf

dx

dh

????

tf

¼

@I

@x1

@I

@x2

...

@I

@xn

??

@x1

@h1

@x1

@h2

...

@x1

@hp

@x2

@hp

...

@xn

@hp

@x2

@h1

...

@xn

@h1

@x2

@h2

...

@xn

@h2

...

..

.

...

2

6666664

3

7777775

(28)

where x 5 [x1, x2,..., xn]Tis the vector of the states

involved in the simulation; for this study,

x ¼ ½la;0;la;1;la;2;la;3;la;4;lb;0;lb;1;lb;2;lb;3;lb;4;C;h?T

(29)

with h 5 $t

The system Eqs. 9, 12, 13, and 14 can be represented as a

system of differential-algebraic equations (DAE):

0Da(s)ds.

M_ x ¼ fðt;x;hÞ

(30)

where M is an n 3 n mass matrix of constant coefficients of

the form

?

and I(s)is the s 3 s identity matrix, and 0(n2s)is the (n 2 s)

3 (n 2 s) matrix of zeroes. In this study, s 5 7 and n 5 12.

Differentiating the system equation with respect to y gives

the sensitivity equation

M ¼

IðsÞ

0

0

0ðn?sÞ

?

(31)

M@_ x

@h¼@f

@x

@x

@hþ@f

@h:

(32)

The following steps were used to compute the robust tem-

perature profile:

(1) The first iteration (j 5 1) was initialized with random

parameters for the temperature profile. For j [ 1, the temper-

ature parameters were determined from a genetic algorithm

applied to the optimization of the robust objective function

subject to all operating constraints for the full range of

uncertain parameters.

(2) The temperature profile from Step 1 was applied to

the nominal model (yi5 0 for i 5 1, ... , 5) by integrating

the system Eq. 30, and the sensitivity Eq. 32. At the final

batch time (tf), the nominal objective function (Inominal) was

obtained from Eq. 19.

(3) The objective function sensitivity vector (L) was com-

puted from Eq. 28, and the worst-case parameter uncertain-

ties obtained from Eq. 27.

(4) The temperature profile from Step 1 was applied to the

model with the worst-case model parameters calculated in

Step 3. Irobustwas calculated from Eq. 24 at the batch time tf.

(5) Steps 1 to 4 were repeated until there was no signifi-

cant change in the temperature profile.

This algorithm uses Eq. 25 only for estimating the worst-

case parameters Eq. 27 with the full dynamic simulation

used to compute Irobust(Step 4).

C-control

In many experimental and simulation studies of nonpolymor-

phic batch crystallizations, the C-control strategy (Figure 2) has

resulted in low-sensitivity of the product quality to most practi-

cal disturbances and variations in kinetic parameters.3,13,17–19,26

In the last two years, the C-control strategy has been applied

experimentally to several polymorphic crystallizations, to pro-

duce large crystals of any selected polymorph.27C-control can

be interpreted as nonlinear state feedback control,26,28in which

the nonlinear master controller acts on the concentration C as a

measured state29to produce the setpoint temperature Tsetas its

manipulated variable.†The difference between the calculated

Tset, and the measured temperature T is used by the slave con-

troller to manipulate the jacket temperature Tj, so that the devia-

tion between Tsetand T is reduced. Because the slave controller

is just temperature control of a mixed tank, and the batch dy-

namics are relatively slow, any reasonably tuned proportional-

integral controller will result inaccurate following ofTset.

Thesetpoint concentration-temperature

parameterized as follows:

(1) Heating section:

(a) This time-dependent relation forces the temperature into

theregion between bothsolubilitycurves in thefirst 5.25mins.

trajectorywas

Figure 2. Prefered implementation of C-control for a

batch cooling crystallizer.26

†For this application, the nonlinear master controller is given by Eqs. 35–38.

2646DOI 10.1002/aic Published on behalf of the AIChEOctober 2007 Vol. 53, No. 10AIChE Journal

Page 5

T0¼ Toþ

p1? To

ð5:25Þð60Þt

(33)

Tset¼ maxfTmin;minfT0;Tmaxgg

where p1(8C) is the first decision variable, To5 258C is the

initial temperature, Tmin5 258C and Tmax5 508C are the

minimum and maximum temperatures achievable by cooling

and heating the water bath, respectively, Tsetis the setpoint

temperature to a lower level controller, and t is time in sec-

onds.

(b) Assuming a linear concentration-temperature trajectory

when heating

(34)

T0¼ Tpþ p2ðC ? CpÞ

Tset¼ maxðTmin;minðT0;TmaxÞÞ

?C

g solute=100 g solution

variable, and Tp(8C) and Cp(g solute/100 g solution) are the

temperature and concentration at time equal to 5.25 mins;

(2) Cooling section:

After the mass of a-form decreases below a certain

value (chosen to be 0.5 g), the assumed trajectory is fol-

lowed

(35)

(36)

where p2

??

is the second decision

T0¼lnðC=p3Þ

p4

(37)

Tset¼ maxðTmin;minðT0;TmaxÞÞ

(38)

where p3(g solute/100 g solution), and p4(1/8C) are the third

and fourth decision variables, respectively.

The rationale behind the structure of Eqs. 33 to 38 is to

obtain the best fit to the concentration-temperature trajectory

obtained by applying the optimal temperature-time trajectory

from T-control to the nominal model. Then, the values for p1

to p4were fitted accordingly. The lower level controllers for

all control strategies are assumed to have very fast response

compared to the overall batch time, which is a good assump-

tion for this process, which has a relatively long batch time.

Results

This section compares the performance and robustness of

the three control strategies to the parameter perturbations

shown in Table 1. The yields and purities of b-form at

the end of batch for all control strategies are tabulated in Ta-

ble 2. The concentration-temperature trajectories in Figure 1

for T-control applied to the nominal model and the corre-

sponding C-control obtained from Eqs. 33 to 38 are coinci-

dent, indicating that the parameterization of Eqs. 33 to 38, is

suitable for representing the C vs. T setpoint used in C-con-

trol. The growth kinetics of the b crystals are relatively slow,

which results in the optimal control trajectories being very

close to the solubility curve for the a-form, to maximize the

supersaturation with respect to the solubility of the b-form

while operating between the two solubility curves.

The nominal temperature trajectory produced by T-control

is highly nonrobust to perturbations in the physicochemical

parameters, as seen in Figure 3b–d, with the temperature

constraints violated for parameter sets 2 and 3. The tempera-

ture trajectories reoptimized for the perturbed model parame-

ters indicate that the optimal temperature trajectory is very

sensitive to shifts in the model parameters. The mass profiles

in Figure 3e,f indicate that the crystals of the a-form com-

pletely dissolve within 2 h for the two feasible parameter

sets, whereas crystals of the b-form continue to grow for 5 h,

which is consistent with the notion that the growth rate of

crystals of the b-form is rate-limiting for the design of the

batch temperature trajectory for this polymorph transforma-

tion.

The temperature trajectories produced by robust T-control

are robust in terms of satisfying the operating constraints for

the whole set of perturbed parameters, but are very conserva-

tive in terms of having very long batch times, and poor pro-

ductivity for all values of the physicochemical parameters

(see Figure 4a–d). Comparing the reoptimized T-control and

robust T-control mass profiles in Figure 4e,h indicates that

robust T-control leads to unnecessarily long batch times for

some values of the physicochemical parameters. Designing a

batch control trajectory to satisfy the operating constraints

for the whole set of potential perturbed model parameters

can result in very sluggish performance irrespective of what

the actual physicochemical parameters happen to be in a par-

ticular batch run. While such approaches have been heavily

studied in the batch design and batch control literature (for

example, see articles cited in16), these approaches can result

in very poor performance when applied to practical batch

processes.

Although C-control does not explicitly include robustness

in its formulation, C-control nearly satisfies all of the operat-

ing constraints for all sets of model parameters (see Figure

5a–d), demonstrating nearly the same robustness as robust T-

control.{Further, C-control results in much faster batch times

and higher productivity than robust T-control for some sets

of physicochemical parameters (see Figure 5a and d). In

addition, C-control results in the batch productivity similar to

that obtained by T-control reoptimized for each parameter

set, as seen by the closeness of the C-control and reoptimized

T-control trajectories in Figure 5a–d. C-control has nearly

the same performance as that of the best T-control trajectory,

with the batch times obtained by C-control are large only

when necessitated by the particular values of the physico-

chemical parameters. This performance is obtained by C-con-

Table 1. Case 1 has no Uncertainties (the Nominal Model),

Case 2 has the Worst-case Parameter Values, Case 3 is the

same as Case 2, but only Includes Variations in the Kinetic

Parameters, and Case 4 are Parameter Variations with Fast

Growth Rate for Crystals of the b-form (Which is the

Rate-limiting Step)

Cases

y1

y2

y3

y4

y5

1

2

3

4

0.00.0

0.2

0.2

0.0

0.2

0.2

0.2

0.0 0.0

0.05

0.0

0.05

20.2

20.2

0.2

20.05

0.0

20.05

20.2

{There is a slight violation of the lower bound on temperature bound for the pa-

rameter set for Case 4, which is due to the shift in the solubility curve of the a-

form. This violation can be removed by shifting the nominal concentration-temper-

ature trajectory slightly away from the a-solubility curve (see Figure 6).

AIChE Journal October 2007Vol. 53, No. 10Published on behalf of the AIChEDOI 10.1002/aic 2647

Page 6

Table 2. The Yields and Purities of b-form at the End of Batch for Different Control Strategies

Cases

OptimalT-controlRobust T-controlC-control

b-yield (g)

b-purity (%)

b-yield (g)

b-purity (%)

b-yield (g)

b-purity (%)

b-yield (g)

b-purity (%)

1

2

3

4

24.15

9.68

13.70

23.88

100.0

97.50

100.0

100.0

24.15

n.a.^

n.a.^

23.88*

100.0

n.a.^

n.a.^

100.0

16.86

9.02

13.21

16.42

100.0

69.80

100.0

100.0

24.15

9.65

13.41

n.a.†

100.0

95.10

100.0

n.a.†

^These values would be meaningless for comparison purposes due to temperature constraint violations.

*From Figure 3f, this value is approached at a much later time (4.67 h) compared to the optimal one (3 h).

†The constraint violation can be removed as mentioned in the text and Figure 6, and the resulting b-yield and purity are 23.86 g and 99.9%, respectively, for

this case.

Figure 3. Temperature profiles applied to: (a) case 1,

(b) case 2, (c) case 3, and (d) case 4, for T-

control (2), the temperature trajectory reop-

timized for the perturbed model parameters

(2 ? 2 ?), and the shaded region showing the

constraints on the temperature for T-control.

Mass profiles for: (e) case 1 and (f) case 4, for T-control

(Ma, 2; Mb, 2 ? 2 ?), and for the temperature trajectory

reoptimized for the perturbed model parameters (Ma, *;

Mb, 1). (The mass profiles for cases 2 and 3 are not shown

since the temperature trajectories do not satisfy the operat-

ing constraints).

Figure 4. Temperature profiles applied to: (a) case 1,

(b) case 2, (c) case 3, and (d) case 4, for ro-

bust T-control (2), the temperature trajectory

reoptimized for the perturbed model parame-

ters (2 ? 2 ?), and the shaded region showing

the constraints on the temperature for robust

T-control.

Mass profiles for: (e) case 1, (f) case 2, (g) case 3, and (h)

case 4, for robust T-control (Ma, 2; Mb, 2 ? 2 ?), and for

the temperature trajectory reoptimized for the perturbed

model parameters (Ma, *; Mb, 1).

2648DOI 10.1002/aicPublished on behalf of the AIChEOctober 2007Vol. 53, No. 10 AIChE Journal

Page 7

trol without having the poor robustness of T-control com-

puted from the nominal model. In summary, C-control has

all the respective advantages of T-control and robust T-con-

trol, without any of their respective disadvantages.

C-control uses nonlinear state feedback control of the con-

centration measurement to follow a desired path in the phase

diagram.26This article, for the first time, demonstrates the

improved robustness of C-control for the solvent-mediated

transformation from one polymorph to another. There is the-

oretical and simulation support that relatively simple nonlin-

ear state feedback controllers can be derived that provide

nearly optimal performance and robustness for batch pro-

cesses.16For a polymorphic crystallization, this paper derives

such a nonlinear state feedback controller, motivated by and

interpreted within the context of the crystallization phase dia-

gram, which has the desired performance and robustness

properties.

Conclusions

The robustness and performance of T-control, robust T-

control, and C-control strategies were compared for maxi-

mizing the batch productivity during the solvent-mediated

polymorphic transformation of L-glutamic acid from the

metastable a-form to the stable b-form crystals. Operating a

batch polymorphiccrystallization

approach based on control along a temperature vs. time tra-

jectory21is shown to be very sensitive to variations in the

nucleation and growth kinetics and shifts in the solubility

curve, resulting in violations of the operating constraints.

For the polymorphic transformation from a-form to b-form

crystals, these constraint violations can result in the nuclea-

tion and re-growth of undesired a-form crystals. Robust T-

control resulted in satisfaction of the operating constraints

for a full range of variations in the physicochemical param-

eters for the kinetics and thermodynamics of the polymor-

phic transformation, but resulted in very poor batch produc-

tivity (long batch times) for parameters in which short batch

times are possible.

A nonlinear state feedback controller designed to follow

an optimal trajectory in the concentration-vs-temperature-

usingtheexisting

Figure 5. Temperature profiles applied to (a) case 1, (b)

case 2, (c) case 3, and (d) case 4, for C-con-

trol (2), the temperature trajectory reopti-

mized for the perturbed model parameters

(2 ? 2 ?), and the shaded region showing the

constraints on the temperature for C-control.

Mass profiles for: (e) case 1, (f) case 2, and (g) case 3, for

C-control (Ma, 2; Mb, 2 ? 2 ?), and for the temperature

trajectory reoptimized for the perturbed model parameters

(Ma, *; Mb, 1). (The mass profiles for case 4 are not

shown since the temperature trajectory (slightly) violates

the T lower limit).

Figure 6. (a) Temperature profile applied to case 4 for

C-control with parameters p15 36.56, p25

11.5, p35 0.4185, and p45 0.03656 (2), the

temperature trajectory reoptimized for the

perturbed model parameters (2 ? 2 ?), and

the shaded region showing the constraints

on the temperature for C-control; (b) the cor-

responding mass profiles for C-control (Ma,

2; Mb, 2 ? 2 ?), and for the temperature tra-

jectory reoptimized for the perturbed model

parameters (Ma, *; Mb, 1).

AIChE JournalOctober 2007 Vol. 53, No. 10 Published on behalf of the AIChEDOI 10.1002/aic 2649

Page 8

phase diagram was highly robust to variations in the kinetic

parameters, while providing batch productivity nearly as high

as optimal control applied to batch crystallization with known

parameters, as illustrated in Figure 5c,g and many simulation

studies (not shown here) with variations in y1to y3(while

maintaining y45 y55 0). Although not explicitly included in

the optimization formulation, the operating constraints were

satisfied for the entire range of physicochemical parameters

(see Figure 5), except for a small constraint violation due to

variation in the solubility of a-form crystals that was removed

by slightly shifting the concentration setpoint trajectory away

from the a-solubility curve (Figure 6). Alternatively, shifts in

any solubility curve can be accounted for by updating meas-

urements of the solubility curve whenever there are significant

changes in feedstocks between batch runs. Automated systems

exist for measuring such solubility curves.17,19,26

Published results,2,20as well as one of the author’s experi-

ence consulting with industry on their polymorphic crystalli-

zations suggest that the solubility curves of most polymorphs

are typically much closer together than for the a and b poly-

morphs of L-glutamic acid (Figure 1).§If this is true, then

the desired operating region for most polymorphic crystalli-

zations is typically much smaller than for the system investi-

gated in this study, making the robustness of batch control

strategies of much greater importance for most polymorphic

crystallizations. The results in this article indicate that the

design of operating procedures for future polymorphic crys-

tallizations should implement C-control.

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2650 DOI 10.1002/aicPublished on behalf of the AIChEOctober 2007 Vol. 53, No. 10 AIChE Journal

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