Laser Sintering Control for Metal Additive Manufacturing by PDE Backstepping

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.