341489 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.
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