Article
Robust optimal control of polymorphic transformation in batch crystallization
AIChE Journal (Impact Factor: 2.75). 10/2007; 53(10):2643  2650. DOI: 10.1002/aic.11266
Fulltext
Available from: Richard D BraatzPROCESS SYSTEMS ENGINEERING
Robust Optimal Control of Polymorphic
Transformation in Batch Crystallization
Martin Wijaya Hermanto and MinSen Chiu
Dept. of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576
XingYi Woo
Dept. of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 11576;
and Dept. of Chemical and Biomolecular Engineering, University of Illinois at UrbanaChampaign, IL 61801
Richard D. Braatz
Dept. of Chemical and Biomolecular Engineering, University of Illinois at UrbanaChampaign, 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 polymor phism, 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 produc tion of the desired polymorph to produce
reliable therapeutic beneﬁts upon delivery. The optimal batch control of the polymor
phic transformation of Lglutamic acid from the metastable aform to the stable bform
is studied, with the goal of optimizing batch productivity, while providing robustness
to variations in the physicochemical param eters 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 highbatch productivity and robustness, in contrast
to optimal temperature control strategies that were either nonrobust or resulted in
longbatch times. The results motivate the incorporation of the propos ed approach into
the design of operating procedures for polymorphic batch crystallizations.
Ó 2007
American Institute of Chemical Engineers AIChE J, 53: 2643–2650, 2007
Keywords: Tcontrol, robust Tcontrol, Ccontrol, 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.
1–5
According 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 MinSen Chiu at
checms@nus.edu.sg.
Ó 2007 American Institute of Chemical Engineers
AIChE Journal October 2007 Vol. 53, No. 10 2643
Page 1
tries, where products are speciﬁed not by chemical com
position only, but also by their performance.
2
As 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 crystalsize distribu
tion, such as weight mean size. The most widely studied
approach is to determine a temperature proﬁle (Tcontrol)
that optimizes an objective function based on an ofﬂine nom
inal model.
7–11
Although Tcontrol is simple to implement, it
has become wellknown in recent years that Tcontrol can be
very sensitive to variations in the kinetic parameters.
12,13
This motivated the development of robust Tcontrol, which
explicitly includes the impact of uncertainties in the objec
tive, while determining the optimal temperaturetime trajec
tory to be followed during batch operation.
14–16
With 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
trationtemperature trajectory.
3,13,17–19
Motivated by the industrial need to control polymor
phism,
2,20
this article evaluates and compares the perform
ance of these optimal control strategies for the polymorphic
transformation of Lglutamic acid from the metastable a
form to the stable bform. The Process Description section
describes the process model for the polymorphic transforma
tion of Lglutamic acid. The Tcontrol, robust Tcontrol, and
Ccontrol 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 Lglutamic acid from the metastable a
form to the stable bform are
21
:
@f
a
@t
þ
@ðD
a
f
a
Þ
@L
¼ 0; (1)
@f
b
@t
þ
@ðG
b
f
b
Þ
@L
¼ B
b
dðL L
0
Þ; (2)
where
f
0
a
ðLÞ¼f
a
ðL; 0Þ; (3)
D
a
¼ k
d;a
ðC C
sat;a
Þ
100
ðadissolution rateÞ; (4)
G
b
¼ k
g;b
L
ðC C
sat;b
Þ
100
ðbgrowth rateÞ; (5)
C
sat;i
¼ a
i
expðb
i
TÞ; i 2fa; bgðsolubilitiesÞ; (6)
k
d;a
¼ exp
8:98703 10
7
T
2
6:0761 3 10
5
T
þ1:0141310
3
; (7)
k
g;b
¼ 0:410 exp
10900
8:314 T
: (8)
f
i
is the number density for the iform crystals. The initial
crystalsize distributions of polymorph i, Y
i
(L,0), is described
by the sum of three lognormal distributions, the n
th
moment
of the iform crystals is
l
i;n
¼
Z
1
0
L
n
f
i
dL; (9)
and the nucleation rate for bform crystals is
B
b
¼ k
b;b
ðC C
sat;b
Þ
100
l
b;3
; (10)
k
b;b
¼ 7 3 10
7
: (11)
Applying the method of characteristics to Eq. 1, and
method of moments to Eq. 2, gives
f
a
ðL; tÞ¼f
0
a
L
Z
t
0
D
a
dt
; L 0 (12)
d l
b;n
dt
¼ nG
0
b
l
b;n
þ B
b
L
n
0
; n ¼ 0; 1; 2; ::: (13)
where G
0
b
5 G
b
/L. The aforementioned equations are aug
mented by the solute mass balance:
dC
dt
¼300
q
c
V
slurry
m
solution
1
C
100
k
v;a
D
a
l
a;2
þ k
v;a
D
a
f
a
ðL
0
; tÞL
3
0
3
þ k
v;b
G
0
b
l
b;3
þ k
v;b
B
b
L
3
0
3
; ð14Þ
where q
c
is the crystal density, m
solution
is the mass of solution,
V
slurry
is the total volume of crystals and solution, k
v,i
is the vol
umetric shape factor for polymorphic form i,andL
0
is the size
of the nucleated crystals. The parameter values are in Ref. 21.
In this study, the following uncertain parameters are assumed:
k
0
g;b
¼ k
g;b
ð1 þ h
1
Þ exp
10900
8:314T
h
2
; 0:2 h
1
; h
2
0:2
(15)
k
0
d;a
¼ k
d;a
ð1 þ h
3
Þ; 0:2 h
3
0:2 (16)
C
0
sat;a
¼ C
sat;a
ð1 þ h
4
Þ; 0:05 h
4
0:05 (17)
C
0
sat;b
¼ C
sat;b
ð1 þ h
5
Þ; 0:05 h
5
0:05 (18)
where y
1
and y
2
are the uncertainties in the growth parame
ters for the bform crystals, y
3
is the uncertainty in the disso
lution kinetics of the aform crystals, and y
4
and y
5
are the
uncertainties in the solubility curves of the a and b forms,
respectively. The nominal model corresponds to y
i
5 0, i 5
1, ..., 5. While uncertainties in parameters quantiﬁed from
2644 DOI 10.1002/aic Published on behalf of the AIChE October 2007 Vol. 53, No. 10 AIChE Journal
Page 2
experimental data are typically correlated,
22,23
uncorrelated
uncertainties were used here so as to separately assess the
robustness to different types of uncertainties.
Tcontrol
The ﬁrst journal articles on the control of batch polymor
phic crystallization implemented temperature control (e.g.,
see
21
). The most widely studied approach for the optimal
control of nonpolymorphic crystallization processes has uti
lized Tcontrol, in which the temperature trajectory has been
computed from the optimization of an objective function,
based on an ofﬂine model with nominal parameters.
9
In this
study, the objective function that is maximized is the yield
of the bform minus a penalty on the time required for the
mass of the aform to be below some tolerance
I
nominal
¼ m
b
ðt
f
Þw
n
t
a
ðcÞ
¼ q
c
k
v;b
V
slurry
l
b;3
ðt
f
Þw
n
t
a
ðcÞ (19)
where t
f
is the batch time, m
b
(t
f
) is the yield of bform at the
end of the batch, t
a
(c) is the time taken to reduce the mass
of a form below c, and w
n
is a weighting parameter. The val
ues of t
f
, w
n
, and c are 7 h, 1 3 10
23
, and 5 3 10
24
g,
respectively. The second term t
a
(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 temperaturetime trajectory
is parameterized as a ﬁrstorder 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 aform
dissolve, while crystals of the bform grow, that is, the tem
perature is constrained to be between the saturation tempera
tures of the a and bforms (T
sat,a
T T
sat,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 temperaturetime 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 concentrationtempera
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 bform is maximized due
to its growth kinetics being the ratelimiting step.
Robust Tcontrol
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,
h
min;i
h
i
h
max;i
; (20)
the uncertain parameters may be expressed as:
e
h
¼
n
h : h ¼
^
h þ dh; kW
h
dhk
1
1
o
; (21)
^
h ¼
h
min
þ h
max
2
; (22)
ðW
h
Þ
jj
¼
2
h
max;j
h
min;j
; (23)
where y 5 [y
1
, y
2
, ...,y
n
]
T
,
^
h ¼½
^
h
1
;
^
h
2
; ...;
^
h
n
T
, y
min
5
[y
min,1
, y
min,2
, ...,y
min,n
]
T
, and y
max
5 [y
max,1
, y
max,2
, ...,
y
max,n
]
T
are the actual, nominal, minimum, and maximum
values of the uncertain model parameters, respectively, W
y
is
a weight matrix quantifying the magnitude of the uncertainty
in each parameter, and kk
?
is the vector ?norm.
In robust Tcontrol the objective function to be maxi
mized appends a term to include the impact of uncertain
parameters
14–16
I
robust
¼ I
nominal
w
r
dI
w:c
(24)
where w
r
[ [0,1] is a weighting parameter and dI
w.c.
is the
worstcase deviation in the objective due to model uncertain
ties. If w
r
5 1, I
robust
is the objective function obtained for
the worstcase perturbation at the cost of potential degrada
tion in nominal performance. The value of w
r
can be selected
to be smaller than one depending on the desired tradeoff
between the nominal and worstcase performance. In this
paper, w
r
5 0.6 provided the best balance between nominal
and worstcase performance. In robust Tcontrol 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 concentrationtempera
ture trajectories of Tcontrol and Ccontrol
with parameters p
1
5 36.56, p
2
5 11.5, p
3
5
0.4340, and p
4
5 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 aform crystals
grow for a short time before crossing the a solubility curve.
AIChE Journal October 2007 Vol. 53, No. 10 Published on behalf of the AIChE DOI 10.1002/aic 2645
Page 3
With the widelyused approximation
22,24,25
dI ¼
@I
@h
dh ¼ Ldh; (25)
where L is the objective function sensitivity row vector, the
worstcase deviation in the objective function is
15,23
max
kW
h
dhk
1
1
jdIj¼kLW
1
h
k
1
(26)
and a worstcase parameter uncertainty vector is
dh
w:c
¼W
1
h
v; with v
k
¼
ðLW
1
h
Þ
k
jðLW
1
h
Þ
k
j
¼ sgn ðLW
1
h
Þ
k
;
(27)
where kk
1
is the vector 1norm and the objective function
sensitivity vector can be computed from
L ¼
dI
dh
t
f
¼
dI
dx
t
f
dx
dh
t
f
¼
@I
@x
1
@I
@x
2
...
@I
@x
n
@x
1
@h
1
@x
1
@h
2
...
@x
1
@h
p
@x
2
@h
1
@x
2
@h
2
...
@x
2
@h
p
.
.
.
.
.
.
.
.
.
.
.
.
@x
n
@h
1
@x
n
@h
2
...
@x
n
@h
p
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
(28)
where x 5 [x
1
, x
2
, ..., x
n
]
T
is the vector of the states
involved in the simulation; for this study,
x ¼½l
a;0
; l
a;1
; l
a;2
; l
a;3
; l
a;4
; l
b;0
; l
b;1
; l
b;2
; l
b;3
; l
b;4
; C; h
T
(29)
with h 5 $
t
0
D
a
(s)ds.
The system Eqs. 9, 12, 13, and 14 can be represented as a
system of differentialalgebraic equations (DAE):
M
_
x ¼ fðt; x; hÞ (30)
where M is an n 3 n mass matrix of constant coefﬁcients of
the form
M ¼
I
ðsÞ
0
00
ðnsÞ
(31)
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
@
_
x
@h
¼
@f
@x
@x
@h
þ
@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 (y
i
5 0 for i 5 1, ... , 5) by integrating
the system Eq. 30, and the sensitivity Eq. 32. At the ﬁnal
batch time (t
f
), the nominal objective function (I
nominal
) was
obtained from Eq. 19.
(3) The objective function sensitivity vector (L) was com
puted from Eq. 28, and the worstcase parameter uncertain
ties obtained from Eq. 27.
(4) The temperature proﬁle from Step 1 was applied to the
model with the worstcase model parameters calculated in
Step 3. I
robust
was calculated from Eq. 24 at the batch time t
f
.
(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 I
robust
(Step 4).
Ccontrol
In many experimental and simulation studies of nonpolymor
phic batch crystallizations, the Ccontrol strategy (Figure 2) has
resulted in lowsensitivity 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 Ccontrol strategy has been applied
experimentally to several polymorphic crystallizations, to pro
duce large crystals of any selected polymorph.
27
Ccontrol can
be interpreted as nonlinear state feedback control,
26,28
in which
the nonlinear master controller acts on the concentration C as a
measured state
29
to produce the setpoint temperature T
set
as its
manipulated variable.
†
The difference between the calculated
T
set
, and the measured temperature T is used by the slave con
troller to manipulate the jacket temperature T
j
, so that the devia
tion between T
set
and 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 in accurate following of T
set
.
The setpoint concentrationtemperature trajectory was
parameterized as follows:
(1) Heating section:
(a) This timedependent relation forces the temperature into
the region between both solubility curves in the ﬁrst 5.25 mins.
Figure 2. Prefered implementation of Ccontrol for a
batch cooling crystallizer.
26
†
For this application, the nonlinear master controller is given by Eqs. 35–38.
2646 DOI 10.1002/aic Published on behalf of the AIChE October 2007 Vol. 53, No. 10 AIChE Journal
Page 4
T
0
¼ T
o
þ
p
1
T
o
ð5:25Þð60Þ
t (33)
T
set
¼ maxfT
min
; minfT
0
; T
max
gg (34)
where p
1
(8C) is the ﬁrst decision variable, T
o
5 258C is the
initial temperature, T
min
5 258C and T
max
5 508C are the
minimum and maximum temperatures achievable by cooling
and heating the water bath, respectively, T
set
is the setpoint
temperature to a lower level controller, and t is time in sec
onds.
(b) Assuming a linear concentrationtemperature trajectory
when heating
T
0
¼ T
p
þ p
2
ðC C
p
Þ (35)
T
set
¼ maxðT
min
; minðT
0
; T
max
ÞÞ (36)
where p
2
C
g solute
=100 g solution
is the second decision
variable, and T
p
(8C) and C
p
(g solute/100 g solution) are the
temperature and concentration at time equal to 5.25 mins;
(2) Cooling section:
After the mass of aform decreases below a certain
value (chosen to be 0.5 g), the assumed trajectory is fol
lowed
T
0
¼
lnðC=p
3
Þ
p
4
(37)
T
set
¼ maxðT
min
; minðT
0
; T
max
ÞÞ (38)
where p
3
(g solute/100 g solution), and p
4
(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 concentrationtemperature trajectory
obtained by applying the optimal temperaturetime trajectory
from Tcontrol to the nominal model. Then, the values for p
1
to p
4
were 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 bform at
the end of batch for all control strategies are tabulated in Ta
ble 2. The concentrationtemperature trajectories in Figure 1
for Tcontrol applied to the nominal model and the corre
sponding Ccontrol 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 Ccon
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 aform, to maximize the
supersaturation with respect to the solubility of the bform
while operating between the two solubility curves.
The nominal temperature trajectory produced by Tcontrol
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 aform com
pletely dissolve within 2 h for the two feasible parameter
sets, whereas crystals of the bform continue to grow for 5 h,
which is consistent with the notion that the growth rate of
crystals of the bform is ratelimiting for the design of the
batch temperature trajectory for this polymorph transforma
tion.
The temperature trajectories produced by robust Tcontrol
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 Tcontrol and
robust Tcontrol mass proﬁles in Figure 4e,h indicates that
robust Tcontrol 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 in
16
), these approaches can result
in very poor performance when applied to practical batch
processes.
Although Ccontrol does not explicitly include robustness
in its formulation, Ccontrol 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, Ccontrol results in much faster batch times
and higher productivity than robust Tcontrol for some sets
of physicochemical parameters (see Figure 5a and d). In
addition, Ccontrol results in the batch productivity similar to
that obtained by Tcontrol reoptimized for each parameter
set, as seen by the closeness of the Ccontrol and reoptimized
Tcontrol trajectories in Figure 5a–d. Ccontrol has nearly
the same performance as that of the best Tcontrol trajectory,
with the batch times obtained by Ccontrol are large only
when necessitated by the particular values of the physico
chemical parameters. This performance is obtained by Ccon
Table 1. Case 1 has no Uncertainties (the Nominal Model),
Case 2 has the Worstcase 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 bform (Which is the
Ratelimiting Step)
Cases y
1
y
2
y
3
y
4
y
5
1 0.0 0.0 0.0 0.0 0.0
2 20.2 0.2 0.2 20.05 0.05
3 20.2 0.2 0.2 0.0 0.0
4 0.2 20.2 0.2 20.05 0.05
{
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 concentrationtemper
ature trajectory slightly away from the asolubility curve (see Figure 6).
AIChE Journal October 2007 Vol. 53, No. 10 Published on behalf of the AIChE DOI 10.1002/aic 2647
Page 5
Table 2. The Yields and Purities of bform at the End of Batch for Different Control Strategies
Cases
Optimal Tcontrol Robust Tcontrol Ccontrol
byield (g) bpurity (%) byield (g) bpurity (%) byield (g) bpurity (%) byield (g) bpurity (%)
1 24.15 100.0 24.15 100.0 16.86 100.0 24.15 100.0
2 9.68 97.50 n.a.
^
n.a.
^
9.02 69.80 9.65 95.10
3 13.70 100.0 n.a.
^
n.a.
^
13.21 100.0 13.41 100.0
4 23.88 100.0 23.88* 100.0 16.42 100.0 n.a.
†
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 byield 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 Tcontrol.
Mass proﬁles for: (e) case 1 and (f) case 4, for Tcontrol
(M
a
, 2;M
b
, 2 2 ), and for the temperature trajectory
reoptimized for the perturbed model parameters (M
a
, *;
M
b
, 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 Tcontrol (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
Tcontrol.
Mass proﬁles for: (e) case 1, (f) case 2, (g) case 3, and (h)
case 4, for robust Tcontrol (M
a
, 2;M
b
, 2 2 ), and for
the temperature trajectory reoptimized for the perturbed
model parameters (M
a
, *;M
b
, 1).
2648 DOI 10.1002/aic Published on behalf of the AIChE October 2007 Vol. 53, No. 10 AIChE Journal
Page 6
trol without having the poor robustness of Tcontrol com
puted from the nominal model. In summary, Ccontrol has
all the respective advantages of Tcontrol and robust Tcon
trol, without any of their respective disadvantages.
Ccontrol uses nonlinear state feedback control of the con
centration measurement to follow a desired path in the phase
diagram.
26
This article, for the ﬁrst time, demonstrates the
improved robustness of Ccontrol for the solventmediated
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.
16
For 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 Tcontrol, robust T
control, and Ccontrol strategies were compared for maxi
mizing the batch productivity during the solventmediated
polymorphic transformation of Lglutamic acid fr om the
metastable aform to the stable bform crystals. Operating a
batch polymorphic crystallization using the existing
approach based on control along a temp erature vs. time tra
jectory
21
is shown to be very sensitive to variations in the
nucleation and gr owth kinetic s and shifts in the solubility
curve, resulting in violations of the operating constraints.
For the polymorphic transformation fro m aform to bform
crystals, these constraint violations can result in the nuclea
tion and regrowth of undesired aform 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 v ery 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 concentrationvstemperature
Figure 5. Temperature proﬁles applied to (a) case 1, (b)
case 2, (c) case 3, and (d) case 4, for Ccon
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 Ccontrol.
Mass proﬁles for: (e) case 1, (f) case 2, and (g) case 3, for
Ccontrol (M
a
, 2;M
b
, 2 2 ), and for the temperature
trajectory reoptimized for the perturbed model parameters
(M
a
, *;M
b
, 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
Ccontrol with parameters p
1
5 36.56, p
2
5
11.5, p
3
5 0.4185, and p
4
5 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 Ccontrol; (b) the cor
responding mass proﬁles for Ccontrol (M
a
,
2;M
b
, 2 2 ), and for the temperature tra
jectory reoptimized for the perturbed model
parameters (M
a
, *;M
b
, 1).
AIChE Journal October 2007 Vol. 53, No. 10 Published on behalf of the AIChE DOI 10.1002/aic 2649
Page 7
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 y
1
to y
3
(while
maintaining y
4
5 y
5
5 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 aform crystals that was removed
by slightly shifting the concentration setpoint trajectory away
from the asolubility 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,
2,20
as 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 Lglutamic 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 Ccontrol.
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§
This is especially true for enantiomeric polymorphs, in which the solubility
curves intersect.
2650 DOI 10.1002/aic Published on behalf of the AIChE October 2007 Vol. 53, No. 10 AIChE Journal
Page 8
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 "This is the case of microstructured reactors [4]. Unfortunately, channel blocking issues limit, for the moment, the industrial application [4]. From a more fundamental point of view, the complex interaction of the physical chemistry (nucleation, crystal growth rates) and chemical engineering (hydrodynamics, transport processes, scale up), which controls the polymorphic form, crystal stability and CSD, is a key topic. "

 "This is the case of microstructured reactors [4]. Unfortunately, channel blocking issues limit, for the moment, the industrial application [4]. From a more fundamental point of view, the complex interaction of the physical chemistry (nucleation, crystal growth rates) and chemical engineering (hydrodynamics, transport processes, scale up), which controls the polymorphic form, crystal stability and CSD, is a key topic. "
[Show abstract] [Hide abstract] ABSTRACT: Crystallization is one of the major unit operations of chemical process industries and plays a key role for particulate solids production in the pharmaceutical, chemical, electronic, minerals sectors. Most of the current crystallization processes are performed under batch or continuous mode based on a stirred tank process; the need for breakthrough technologies has been highlighted by numerous authors and reports. Membranes are one of the potentially attracting strategies in order to achieve this target. Nevertheless, a relatively limited number of publications have been reported on membranes and crystallization processes, compared to other unit operations. This study intends to provide a stateoftheart review of the different approaches combining membranes and crystallization processes. Hybrid and integrated systems are discussed and the different role and function potentially provided by dedicated membrane materials are analyzed. Based on the results and analyses gained through the different approaches that have been tested, unexplored issues and open questions have been listed. The research efforts which are required in order to make membranes processes for crystallization/precipitation an industrial reality are finally discussed. 
 "The underlying principle is that changing the global properties can change the dynamic pathways of assembly. For example, during crystallization, optimized temperature control can improve the yield of one particular crystal polymorph over another [33,34] . The number of such global variables that can be useful for altering the state of assembly are somewhat limited, which often includes temperature, pressure, concentration, composition, and, for some systems, external fields such as electric and magnetic fields. "
[Show abstract] [Hide abstract] ABSTRACT: Control of selfassembling systems at the micro and nanoscale provides new opportunities for the engineering of novel materials in a bottomup fashion. These systems have several challenges associated with control including highdimensional and stochastic nonlinear dynamics, limited sensors for realtime measurements, limited actuation for control, and kinetic trapping of the system in undesirable configurations. Three main strategies for addressing these challenges are described, which include particle design (active selfassembly), openloop control, and closedloop (feedback) control. The strategies are illustrated using a variety of examples such as the design of patchy and Janus particles, the toggling of magnetic fields to induce the crystallization of paramagnetic colloids, and highthroughput crystallization of organic compounds in nanoliter droplets. An outlook of the future research directions and the necessary technological advancements for control of micro and nanoscale selfassembly is provided.