Research Initiation: Eigenvalue Inclusion Regions to Study Model Approximations for Distributed Parameter Systems

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Heat, mass or momentum transfer equations often contain gradients in many spatial directions and perhaps accumulation terms to bring about dependence on time. Modeling such systems involves the solution of a series of partial differential equations (PDEs) which accurately represent reality but complicate the analysis procedures since they seldom have analytical solutions. This necessitates the use of numerical techniques to obtain an approximation to the exact solution. The relationship between the approximate and exact solution leads to the problem of "robustness" when such approximations constitute elements in feedback control loops. Such process controllers function with limited process information since their design is based on the aproximate models resulting in the loss of system performance and sometimes, stability. This work is geared towards the development of appropriate mathematical tools to investigate the model approximations for distributed parameter systems in an effort to guide dynamics and control studies. The PI is proposing to exploit the eigenstructure of the approximate and the rigorous models in order to gain some insight towards the variation of system eigenvalues, the dominant system modes, the degree of model aproximation and most importantly the characterization of the neglected system modes to judge the capability of process controllers to maintain smooth and effective operation. The PI has been growing dislocation free single crystals of GaAs using a modified Bridgman type furnace and is setting up a pilot scale furnace assembly to verify the results. The modeling and the control of such a system is critical to ensure the development of optimal temperature profiles in the crystal and the melt and this in turn requires a sound assessment of the relationship between the rigorous model and its numerical approximation. This system represents a typical problem facing the process control engineer and is an ideal candidate to test the results of this project.