# (599f) Uncertainties and Stochastic Optimal Control in Batch Crystallization for Different Types of Objective Functions

Uncertainties
and Stochastic Optimal Control in Batch Crystallization for different types of
Objective Functions

Kirti M. Yenkie1,2 and Urmila Diwekar 1,2

1Department of Bio Engineering, University of
Illinois, Chicago, IL 60607 - USA

2Center for uncertain Systems: Tools for Optimization
& Management (CUSTOM),

Vishwamitra Research Institute, Clarendon Hills, IL
60514 ? USA

Abstract:

Batch crystallization is an industrial
separation and purification process. It is dependent on parameters like temperature,
supersaturation, agitation, solution behavior, etc. The fundamental process
parameters like solubility and crystal structure can have physical
uncertainties, while the process can have uncertainties in engineering design
parameters and operating conditions. These uncertainties were generally ignored
while predicting operating policies and hence, the predicted outcomes were not
fully in accordance with the actual outcomes. To overcome these shifts in the
desired results uncertainties are taken into account while predicting the
crystallization operating policies.

Uncertainties in the system are
characterized as a special class of stochastic processes, known as Ito
processess if they follow certain properties. These uncertainties on
propagation into the process model yields dynamic uncertainties within the
process states. Using the theory of stochastic modeling and Ito's calculus,
these uncertainties in the process states are represented in the mathematical
form. The modified process model involves the state equations with the
stochastic term representing uncertainty. The objective functions in batch
crystallization are dependent on the desired outcome and can be classified into
two categories late growth and early growth functions. Thus, we consider several
objectives to see the effect of uncertainty on both the categories of
functions. This results into stochastic optimal control problems and they are
solved using Ito's lemma, stochastic calculus and stochastic optimal control
theory.

The aim is to study all possible sources of
uncertainty in batch crystallization and model them as Ito processes based on
their behavior. Observe the effects of uncertainty inclusion into the process
model and their effects on the operating policles for the early and late growth
categories of objective functions. Thus, providing operating policies which
shall mimic the actual behavior of the system and yield results with better
predictive value.

Keywords: uncertainty, operating policy,
stochastic processes, Ito's calculus, optimal control

Case
study 1: Optimal control while considering kinetic parameter uncertainty

1.
Kinetic
parameter uncertainty:

The kinetic parameters are
generally empirical constants determined by fitting experimental data to the
model, and hence are a source of uncertainty within the system. In batch
crystallization kinetics, the growth and nucleation expressions have empirical
constants shown in table 3, they can be assumed to follow a Gaussian
distribution, with the fitted values from previous experiments to be the mean.
The values are assumed to deviate around ±5% about the mean.

Table
1: Kinetic parameter uncertainty in batch crystallization model

 Uncertainty Kinetic constants Value from experiments/ model fitting Range of Values G kg 1.44 x 108 µm s-1 1.368 x 108 ? 1.512 x 108 Eg/R 4859 K 4616.05 ? 5101.95 g 1.5 1.425 ? 1.575 B kb 285 (s µm3)-1 270.75 ? 299.25 Eb/R 7517 K 7141.15 ? 7892.85 b 1.45 1.3775 ? 1.5225

We
consider a 95 % confidence interval and hence, the kinetic parameters  lie within two standard deviations of their
values (μ±2σ ). Thus, we evaluate
the standard deviation for all of them using the extreme deviations as minimum
and maximum values.

ü
The sampling operation for
multivariable uncertain parameter domain is performed using the Monte Carlo
sampling technique

ü
100 sample values for each of the
kinetic parameter are generated using inverse transformation over cumulative
probability distribution.

ü
After generating 100 samples for the
kinetic parameter data, the model is simulated using each set.

ü
The resulting dynamic uncertainty in
the state variables due to static uncertainty in the kinetic parameters is
observed in the plots of the dynamic uncertainty as shown in figure 1.

Figure 1 The dynamic uncertainties in
state variables C, μ1s, μ2s, μ3s 2.
Modeling uncertainties as Ito processes

By
studying the dynamic uncertainty plots of the process variables and their
correlation to Ito processes, it has been observed that the uncertainties can
be best modeled with a simple Ito process known as Brownian motion with drift.

where
dz is the increment of the Wiener
process equal to εt√Δt,
and a(y,t) and b(y,t) are known functions. The random value εt has a unit normal distribution with zero mean
and standard deviation of one. In this paper, a simplification of the above
equation is used to represent the time dependent uncertainties in the
concentration, seed and nucleation moments:

Figure 2. State variables modeled as Ito Processes

3.
Stochastic Maximum
Principle:

We
use the stochastic maximum principle similar to the method illustrated in
Ramirez and Diwekar. The objective is to maximize the expected value of mass of
seeded crystals and minimize the expected value of mass of fines, considering
the uncertainties associated with the concentration and moments of batch
crystallization, finding the best operating temperature profile for the
process.

The objective function for the
stochastic formulation can be written as equation 3a or 3b where E is the
expected value.

Subject to seeded batch
crystallization state variables modeled as Ito processes, initial conditions, constraints
for supersaturation condition maintenance. The state equations in general can
be represented as shown in eq 4.

Figure 3. Flowchart for optimal temperature profile evaluation
using stochastic maximum principle (active constraint strategy)

4.
Results

Figure
4 summarizes the optimal temperature profiles evaluated by the deterministic
method and stochastic method. The results for the linear cooling profile, which
is most commonly used is also compared against the optimal trajectories. Figure
5 compares the final size distributions predicted from the moment values, and
it can be seen that the stochastic case yields a much narrower probability
density function for the distribution in case of seed as well as nucleated
crystals.

Table 2: Results of Comparison between
linear, deterministic and stochastic cases

 Case Objective function % Increment in stochastic case Linear 8.53 x 109 5.88 Deterministic 8.79 x 109 2.73 Stochastic 9.03 x 109 N.A.

5.
Conclusion

The
static uncertainty in the kinetics result in dynamic uncertainties within the
state variables. These uncertainties could be modeled as Ito processes, since
they followed certain properties and had a behavioral pattern. The most
important aspect was to solve the stochastic optimal control problem involving
Ito processes and application of stochastic calculus, Ito's Lemma and
stochastic maximum principle.

6.
Future work

We
would like to consider several kinds of objective functions used by researchers
for predicting the best optimal policy. The uncertainties associated with these
functions would be incorporated in the study and a similar strategy to study
and compare the determinstic and stochastic cases will be adopted. Other
sources of uncertainty along with kinetic parameter uncertainty would also be
taken into account to make the operating policies more robust.

Figure 4. Comparison of Temperature profiles

(Linear, Deterministic and Stochastic case)

Figure
5. Predicted particle size distributions (PSD) from moment values for 3 cases.