## 1 Introduction  
Laser-based direct energy deposition (DED) is an additive manufacturing (AM) process that is used to fabricate metallic components layer by layer and uses a laser as the heat source to melt additive materials (in either powder or wire form) as they are deposited onto a substrate [1]. The laser-based DED has found broad applications in the aerospace and marine industries due to its capability of making large-scale and customized parts in a cost-effective way. However, the laser-based DED process has poorer stability compared to traditional metal forming processes. It is prone to defects and dimensional inaccuracy due to various factors, including uneven thermal stress, strong melt pool dynamics, localized heat accumulation, inconsistent speed, and other unpredictable disturbances during laser beam delivery and material feeding [2]. Therefore, closed-loop control systems with sensor feedbacks are highly desirable for the laser-based DED process [3]. However, the nonlinearity and varying dynamics of the DED makes the search for robust control algorithms a challenging task. Constant controller parameters may not perform well for all the layers in DED, as the process conditions (e.g., the solidified layer temperature and the associated nominal melt pool size) change over time as the part grows [4]. Moreover, a controller designed for a specific material may not be suitable for another material, since the nominal DED process parameters and material properties (e.g., thermal conductivity, viscosity, and emissivity) are different. Therefore, this research aimed to develop a data-driven adaptive control method in which the controller parameters are variables that can be automatically updated during the laser-based DED process. Melt pool characteristics have a strong correlation with the process stability and part quality in DED, and hence they are frequently used in closed-loop control systems [5]. The effects of DED process parameters on melt pool characteristics have been investigated quantitatively in previous research. It was found that the melt pool size and temperature are both positively influenced by the input energy density and hence the laser power [6; 7]. The proportional-integral-derivative (PID) controller has been widely adopted in the development of melt-pool-based DED control systems due to its simplicity and effectiveness. For example, [PERSON] et al. [8; 9] used a pyrometer to sense the infrared (IR) radiation from the melt pool and sent the reading to a PID controller. Based on the error term calculated as the difference between the real-time IR signal and its nominal value, the PID controller could change the laser power in response to the fluctuation in melt pool temperature. Consistent height of the as-built part was achieved by the above approach. [PERSON] et al. [10] utilized a complementary metal-oxide-semiconductor (CMOS) camera to capture the melt pool image and measure the melt pool width. Then, they applied a PID controller to increase and decrease the laser power to compensate for the rise and fall of the melt pool width, respectively. Similar studies used PID controllers to adjust the laser power based on the variation of melt pool area [11; 12]. Enhancements have also been made to the conventional PID method, aiming to improve the controller performance under the varying dynamics of the DED process. For example, [PERSON] et al. [13] added a feedforward path to a PID controller, which could reduce the overshoot and improve the response speed. The authors also embedded the melt pool size setpoint into the preprogrammed computer numerical control (CNC) code. The position-dependent setpoint allowed the building of changeable geometries using a single-track toolpath. However, extensive experimentation was needed to obtain the correct controller parameters. [PERSON] and [PERSON] [4] implemented an adaptive control strategy that handled the variation of melt pool response across multiple layers.