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

热量、质量或动量传递方程通常包含梯度， 许多空间方向和可能的积累条件， 依赖时间。 对这样的系统建模涉及到一个 一系列偏微分方程（PDE）， 代表现实，但使分析程序复杂化，因为它们 很少有分析的方法。 这需要使用 数值技术，以获得近似的精确解。 近似解和精确解之间的关系导致： 当这种近似构成时“鲁棒性”问题 反馈控制回路中的元件。 这样的过程控制器起作用 由于它们的设计是基于 近似模型导致系统性能损失， 有时是稳定。 这项工作旨在制定适当的 数学工具来研究模型近似， 分布参数系统在努力引导动态和 对照研究。 PI建议利用 近似模型和严格模型， 对系统特征值的变化，主导系统 模式，模型近似的程度，最重要的是 表征被忽略的系统模式，以判断能力 过程控制器，以保持平稳和有效的操作。 PI一直在使用 一个改进的布里奇曼型炉，并建立一个中试规模 炉组装，以验证结果。 建模和控制 这样一个系统是至关重要的，以确保发展的最佳 晶体和熔体中的温度分布，进而 需要对严格的 模型及其数值近似。 该系统代表一个 过程控制工程师面临的典型问题，是一个理想的 候选人来测试这个项目的结果。

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.