(613g) Order-Reduction and Optimal Boundary Control of Parabolic PDEs With Time-Varying Domain

In this work, the temperature stabilization problem in
Czochralski crystal growth process, in which domain of interest undergoes
change due to the crystal growth is considered. Being as one of the most
important methods of production of semiconductor crystals, this process is a
representative physical system model governed by the parabolic partial
differential equation (PDE) with time-varying domain. The crystal quality is
contributed to the variations in the temperature distribution and one method to
realize the crystal temperature regulation is by distributed heat input
implemented along the crystal.

It is well established that, the eigenspectrum of the
spatial differential operator in a dissipative PDE model can be partitioned
into a finite number of slow modes and the complement eigenspace of infinite
stable fast modes, therefore, dominant behavior of the model can be
approximated by the finite-dimensional system. The reduced-order model of a
parabolic PDE system with time varying domain can be obtained by Galerkin's
method with the use of the empirical eigenfunctions of the spatial differential
operator obtained by the use of Karhunen-Loeve (KL) decomposition on the
numerical or experimental solutions data of the system [1]. In this method, the
solutions of the PDE system is mapped on a fixed reference configuration while
preserving the invariance of physical properties (energy) of the solutions.
Then, application of KL decomposition results in the dominant modes of the data
on the reference configuration and when these modes mapped on the time-varying
domain, they yield the set of time-varying eigenfunctions that can be used in
Galerkin's method.

The reduced-order ODE model that captures the dominant
dynamics of the PDE system is in the form of a linear time-varying system. For
this system, the LQR control synthesis is considered and the optimization
problem has the solution in the form of the state feedback control law, which
requires resolution of the corresponding differential Riccati equation.

1.  
M. Izadi and S. Dubljevic, Order-Reduction of
Parabolic PDEs with Time-Varying Domain Using Empirical Eigenfunctions,
submitted to AIChE Journal, 2013.