When an airfoil is pitched up rapidly, a dynamic stall vortex forms at the leading edge and produces high transient lift before shedding and stall occur. The aim of this work is to develop low-dimensional models of the dynamics of these leading-edge vortices, which may be used to develop feedback laws to stabilize these vortices using closed-loop control, and maintain high lift. We first perform a numerical study of the two-dimensional incompressible flow past an airfoil at varying angles of attack, finding steady states using a timestepper-based Newton/GMRES scheme, and dominant eigenvectors using ARPACK. These steady states may be either stable or unstable; we develop models linearized about the stable steady states using a method called Balanced Proper Orthogonal Decomposition, an approximation of balanced truncation that is tractable for large systems. The balanced POD models dramatically outperform models using the standard POD/Galerkin procedure, and are used to develop observers that reconstruct the flow state from a single surface pressure measurement.