There has been continuing effort in developing analytical, numerical, and empirical models of laser-based additive manufacturing (AM) processes in the literature. However, advanced physics-based models that can be directly used for feedback control design, i.e., control-oriented models, are severely lacking. In this paper, we develop a physics-based multivariable model for directed energy deposition. One important difference between our model from the existing work lies in a novel parameterization of the material transfer rate in the deposition as a function of the process operating parameters. Such parameterization allows an improved characterization of the steady-state melt-pool geometry compared to the existing lumped-parameter models. Predictions of melt-pool geometry and temperature from our model are validated using experimental data obtained from deposition of Ti-6AL-4V and deposition of Inconel® 718 on a laser engineering net shaping (LENS) AM process and finite-element analysis. Then based on this multivariable model, we design a nonlinear multi-input multi-output (MIMO) control, specifically a feedback linearization (FL) control, to track both melt-pool height and temperature reference trajectories using laser power and laser scan speed.