Abstract
Industrial crystallization plants can be operated in basically two different ways: continuous
or batch. These two modes of operation result in two different control problems,
which are both addressed in this dissertation. The thesis is composed of two major parts
corresponding to the two fundamentally different control problems arising in crystallization.
It is the aim of this work to demonstrate that sophisticated up-to-date control synthesis
methods can be applied to crystallization processes on the basis of population balance
models.
Stabilizing Control of Continuous Crystallization
The first part of the thesis comprises Chapters 2 to 4. It deals with continuously operated
crystallization plants. Continuous processes run for very long periods of time. They
are desired to operate at a steady state. Product quality is determined by the steady state
crystal size distribution (CSD). This quantity can be influenced by fines dissolution. Unfortunately,
apart from the desired effect on the CSD this also affects the dynamics of the
system in an undesirable way. Extensive fines dissolution may lead to instability of the
steady state.
In Chapter 2, a detailed population balance model introduced by [Mit02] is presented,
which describes a continuous crystallizer. This model considers nucleation, growth and
attrition of crystals. It comprises two population balances, one for the crystallizer vessel
itself and one for the settling zone used for fines separation. It captures the destabilizing
effect of fines dissolution and predicts the resulting sustained oscillations of CSD and solute
concentration. As a basis for controller design, however, this model is too complex.
Therefore, in Chapter 3, the detailed model is simplified by replacing the rather complex
detailed growth and attrition laws by simpler expressions. Furthermore, the settling zone
is regarded quasi-stationary. Thus, a population balance model is obtained from which a
transcendental transfer function can be derived relating the control input (fines dissolution
rate) and the measured variable (third moment of CSD). In Chapter 4, this transfer
function serves as a basis for the design of stabilizing feedback controllers. Since the
controllers are designed on basis of a simplified model it is essential that they are robust
with respect to plant model mismatch. H-infinity-theory provides a framework for the systematic
consideration of such robustness issues. In this work, a specific version of H-infinity-theory
for infinite-dimensional systems is used [FOT96]. The resulting controllers are tested in
simulations with the simplified and the detailed model.
What distinguishes this approach from most other contributions in the area of continuous
crystallizer control is that controllers are designed on basis of an infinite-dimensional
model. The population balance model is not discretized prior to controller design.
Control of Batch Crystallization Using System Inversion
The second part of the thesis, dealing with batch crystallization, comprises Chapters 5
and 6. In batch cooling crystallization the product quality is determined by the CSD at
the end of the batch, which can be influenced by the cooling profile, i.e. the temperature
trajectory during the batch run. In an experiment or by numeric simulation of a population
balance model, it is obviously possible to determine the product CSD created by
a given temperature trajectory. The solution of the reverse problem, i.e. the design of
a temperature trajectory which produces a specific CSD defined as a function of crystal
length, requires inversion of the system model.
In Chapter 5, a relatively simple standard population balance model [MR94] is presented
which allows the derivation of a finite-dimensional moment model. The notion of differential
flatness [FLMR92] is introduced and it is shown that the moment model is not flat
immediately but can be rendered flat by an appropriate state dependent time scaling. Such
systems are called orbitally flat. In Chapter 6, it is shown that the system model can be inverted,
exploiting the flatness property of the time scaled moment model and the structure
of the time scaled population balance. Thus, for a given desired final CSD it is possible to
check whether it can be produced under the given conditions and, if so, the corresponding
temperature trajectory can be computed analytically. In the literature, the most common
approach to batch crystallizer control is the dynamic optimization of CSD properties. It
is shown that this problem can be simplified considerably taking advantage of the system’s
flatness property. Finally, the design of nonlinear feedback tracking controllers is
presented, which control the system along desired trajectories of the moments.
The distinctive feature of this system inversion approach is the possibility to design a
temperature trajectory achieving one specific CSD defined as a function of crystal length
rather than merely optimizing a scalar property of the CSD. Moreover, temperature trajectories
are determined analytically whereas most other approaches rely on numerical
algorithms.
References
[FLMR92] M. Fliess, J. Levine, P. Martin, and P. Rouchon. On differentially flat nonlinear
systems. In Nonlinear Control Systems Design, pages 408-412. Pergamon Press, 1992.
[FOT96] C. Foias, H. Ozbay, and A. Tannenbaum. Robust Control of Infinite Dimensional
Systems. Springer, 1996.
[Mit02] A. Mitrovic. Population Balance Based Modelling, Simulation, Analysis and
Control of Crystallization Processes. PhD thesis, ISR, Universität Stuttgart,
2002.
[MR94] S.M. Miller and J.B. Rawlings. Model identification and control strategies
for batch cooling crystallizers. AIChE Journal, 40(8):1312-1327, 1994.
