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

热、质量或动量传递方程通常包含许多空间方向上的梯度，可能还包含累加项，使其与时间相关。这类系统的建模需要求解一系列的偏微分方程（PDEs），这些方程准确地反映了实际情况，但由于它们很少有解析解，因此使分析过程复杂化。这就需要使用数值技术来求得精确解的近似值。近似解和精确解之间的关系导致了“鲁棒性”问题，当这种近似构成反馈控制回路中的元素时。这类过程控制器在有限的过程信息下运行，因为它们的设计是基于近似模型，导致系统性能的损失，有时甚至是稳定性的损失。这项工作旨在开发适当的数学工具来研究分布参数系统的模型近似，以指导动力学和控制研究。PI建议利用近似模型和严格模型的特征结构，以了解系统特征值的变化，系统的主导模式，模型的近似程度，最重要的是被忽略的系统模式的表征，以判断过程控制器保持平稳有效运行的能力。PI已经使用改进的Bridgman型炉生长了无位错的GaAs单晶，并正在建立中试规模的炉组件来验证结果。这种系统的建模和控制对于确保晶体和熔体中最佳温度分布的发展至关重要，这反过来又要求对严格模型与其数值近似之间的关系进行合理的评估。该系统代表了过程控制工程师面临的一个典型问题，是测试该项目结果的理想候选者。

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.