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引用次数: 3
摘要
Stefan PDE系统是许多科学和工程过程中出现的熔化和凝固等热相变现象的代表性模型。用偏微分方程(PDE)给出了温度分布的数学描述,该温度分布定义在具有运动边界的空间区间上,其中边界代表液固界面,其动力学由常微分方程(ODE)控制。边界处的PDE-ODE耦合是非线性的,对具有可证明的收敛性和鲁棒性的状态估计提出了重大挑战。本文介绍了一种基于PDE反演的Stefan系统状态估计方法,该方法仅在移动边界处使用测量值。PDE反演观测器设计通过将观测器误差状态Volterra变换为理想的目标系统,求解变换核的goursat形式PDE,并对目标观测器误差系统进行Lyapunov分析,从而产生观测器增益。观测者应用于由气候变化和可再生能源存储需求引起的问题模型:极地冰动力学模型和锂离子电池充放电模型。极地冰的数值结果表明,所设计的估计器对于海冰中未模拟的盐度效应具有良好的鲁棒性。使用相变材料的锂离子电池的电化学PDE模型的结果表明,即使在存在传感器噪声的情况下,也可以在5分钟内消除超过15%的充电状态估计误差。
State estimation of the Stefan PDE: A tutorial on design and applications to polar ice and batteries
The Stefan PDE system is a representative model for thermal phase change phenomena, such as melting and solidification, arising in numerous science and engineering processes. The mathematical description is given by a Partial Differential Equation (PDE) of the temperature distribution defined on a spatial interval with a moving boundary, where the boundary represents the liquid–solid interface and its dynamics are governed by an Ordinary Differential Equation (ODE). The PDE–ODE coupling at the boundary is nonlinear and creates a significant challenge for state estimation with provable convergence and robustness.
This tutorial article presents a state estimation method based on PDE backstepping for the Stefan system, using measurements only at the moving boundary. PDE backstepping observer design generates an observer gain by employing a Volterra transformation of the observer error state into a desirable target system, solving a Goursat-form PDE for the transformation’s kernel, and performing a Lyapunov analysis of the target observer error system.
The observer is applied to models of problems motivated by climate change and the need for renewable energy storage: a model of polar ice dynamics and a model of charging and discharging in lithium-ion batteries. The numerical results for polar ice demonstrate a robust performance of the designed estimator with respect to the unmodeled salinity effect in sea ice. The results for an electrochemical PDE model of a lithium-ion battery with a phase transition material show the elimination of more than 15 % error in State-of-Charge estimate within 5 min even in the presence of sensor noise.
期刊介绍:
The field of Control is changing very fast now with technology-driven “societal grand challenges” and with the deployment of new digital technologies. The aim of Annual Reviews in Control is to provide comprehensive and visionary views of the field of Control, by publishing the following types of review articles:
Survey Article: Review papers on main methodologies or technical advances adding considerable technical value to the state of the art. Note that papers which purely rely on mechanistic searches and lack comprehensive analysis providing a clear contribution to the field will be rejected.
Vision Article: Cutting-edge and emerging topics with visionary perspective on the future of the field or how it will bridge multiple disciplines, and
Tutorial research Article: Fundamental guides for future studies.