Metal additive manufacturing (AM) has been intensively advanced due to numerous industrial applications, such as automobiles, aerospace, consumer electronics, and medical devices. The dynamics of the melt pool via laser sintering for metal AM has been studied by means of the thermodynamic phase change model known as the “Stefan problem”. In this article, we develop a control design for the laser power to drive the depth of the melt pool to the desired set point. The governing equation is described by a partial differential equation (PDE) defined on a time-varying spatial domain, which is dependent on the PDE state, and the optical penetration of the laser energy affects the PDE dynamics in the domain as well as at the surface boundary. First, we design the full-state feedback control law utilizing the entire spatial profile of the temperature in the melt pool and the moving interface position. The closed-loop system is proven to satisfy some conditions to validate the physical model, and its origin is shown to be exponentially stable. Next, we propose an observer-based output feedback control law by reconstructing the temperature profile with the availability of only the measured interface position and prove the analogous properties of the closed-loop system. Numerical simulation for a controller designed on a single-phase Stefan model is conducted on a more complex and realistic two-phase Stefan model, which incorporates the cooling effect from the solid phase. In addition, a bias in the interface location measurement is considered. The numerical results illustrate the robustness of the proposed feedback. By lowering the initial temperature in the solid and by increasing the interface sensor bias to more extreme levels, which leads to the controller’s failure (where the failure is exhibited through the entire metal freezing and the melt pool disappearing), we explore the limits of how much uncertainty our control law can handle.

中文翻译：

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PDE反推法用于金属增材制造的激光烧结控制

由于汽车，航空航天，消费电子和医疗设备等众多工业应用，金属增材制造（AM）已得到了高度发展。已经通过被称为“ Stefan问题”的热力学相变模型研究了通过激光烧结金属AM产生的熔池动力学。在本文中，我们开发了一种针对激光功率的控制设计，以将熔池的深度驱动到所需的设定点。该控制方程由在时变空间域上定义的偏微分方程（PDE）来描述，该偏微分方程取决于PDE状态，激光能量的光学穿透会影响该域以及在该区域的PDE动态。表面边界。第一，我们利用熔池中温度的整个空间分布图和运动界面位置来设计全状态反馈控制律。事实证明，该闭环系统满足某些条件可以验证物理模型，并且其起源被证明是指数稳定的。接下来，我们提出了一种基于观察者的输出反馈控制定律，通过仅利用被测接口位置的可用性来重建温度曲线，并证明了闭环系统的类似性质。对单相Stefan模型设计的控制器的数值模拟是在更复杂和更实际的两相Stefan模型上进行的，该模型考虑了固相的冷却效应。另外，考虑接口位置测量中的偏差。数值结果说明了所提出反馈的鲁棒性。通过降低固体中的初始温度并通过将界面传感器偏置增加到更极端的水平，这会导致控制器的故障（故障通过整个金属冻结而显示出来，而熔池消失了），我们探索了如何限制我们的控制法可以处理的不确定性很大。