论文标题
二维两阶段Stefan问题的Riccati反馈控制
Riccati-feedback Control of a Two-dimensional Two-phase Stefan Problem
论文作者
论文摘要
我们讨论了二维两阶段Stefan问题的反馈控制问题。在我们的方法中，我们将尖锐的界面表示与网格移动结合使用来跟踪接口位置。为了获得反馈控制，我们将线性季节调节器方法应用于问题的适当线性化。我们讨论有关离散化及其接口表示的详细信息。此外，我们记录了矩阵组件，以生成非自治的广义差异方程。为了在数值上求解riccati方程，我们使用了非自主落后分化公式的低级别和矩阵值的版本，该版本结合了隐式索引还原技术。对于反馈控制的Stefan问题的数值模拟，我们使用时间自适应的分数键盘方案。 我们为开发方法提供了实现，并在几个数值实验中测试了这些方法。通过这些实验，我们表明我们的反馈控制方法适用于Stefan控制问题，并使这个大规模的问题可计算。此外，我们讨论了几个控制器设计参数的影响，例如输入和输出的选择。
We discuss the feedback control problem for a two-dimensional two-phase Stefan problem. In our approach, we use a sharp interface representation in combination with mesh-movement to track the interface position. To attain a feedback control, we apply the linear-quadratic regulator approach to a suitable linearization of the problem. We address details regarding the discretization and the interface representation therein. Further, we document the matrix assembly to generate a non-autonomous generalized differential Riccati equation. To numerically solve the Riccati equation, we use low-rank factored and matrix-valued versions of the non-autonomous backward differentiation formulas, which incorporate implicit index reduction techniques. For the numerical simulation of the feedback controlled Stefan problem, we use a time-adaptive fractional-step-theta scheme. We provide the implementations for the developed methods and test these in several numerical experiments. With these experiments we show that our feedback control approach is applicable to the Stefan control problem and makes this large-scale problem computable. Also, we discuss the influence of several controller design parameters, such as the choice of inputs and outputs.
