Feedback control of unsteady flow structures is a challenging problem that is of interest for the creation of agile bio-inspired micro aerial vehicles. This thesis presents two separate results in the estimation and control of unsteady flow structures: the application of a principled estimation method that generates full flowfield estimates using data obtained from a limited number of pressure sensors, and the analysis of a nonlinear control system consisting of a single vortex in a freestream near an actuated cylinder and an airfoil. The estimation method is based on Dynamic Mode Decompositions (DMD), a data-driven algorithm that approximates a time series of data as a sum of modes that evolve linearly. DMD is used here to create a linear system that approximates the flow dynamics and pressure sensor output from Particle Image Velocimetry (PIV) and pressure measurements of the flowfield around the airfoil. A DMD Kalman Filter (DMD-KF) uses the pressure measurements to estimate the evolution of this linear system, and thus produce an approximation of the flowfield from the pressure data alone. The DMD-KF is implemented for experimental data from two different setups: a pitching cambered ellipse airfoil and a surging NACA 0012 airfoil. Filter performance is evaluated using the original flowfield PIV data, and compared with a DMD reconstruction. For control analysis, heaving and/or surging are used as input to stabilize the vortex position relative to the body. The closed-loop system utilizes a linear state-feedback control law. Conditions on the control gains to stabilize any of the equilibrium points are determined analytically for the cylinder case and numerically for the airfoil. Simulations of the open- and closed-loop systems illustrate the bifurcations that arise from varying the vortex strength, bound circulation and/or control gains.