The Aerodynamic Inverted Pendulum: Modeling and Stabilization of a Trailing-Edge-Hinged Airfoil

This study investigates the dynamics and control of a fluid-dynamic analogue to the classical inverted pendulum: an airfoil that freely pitches about its trailing edge in a uniform flow. This choice of pitch axis leads to an aerodynamically unstable configuration. The system is modeled using a simplified Theodorsen unsteady aerodynamic model, with a reduced-order linear state-space formulation enabling feedback control design. A full-state feedback controller for stabilizing the airfoil is developed using a linear quadratic regulator, where the input is the acceleration in the streamwise-normal direction of the theoretical ``cart'' to which the airfoil is mounted. After first demonstrating the effectiveness of this control strategy in stabilizing the system with Theodorsen-based models, the robustness of the controller is tested in a vortex element simulation, where effects such as wake vortex roll-up and finite airfoil thickness are accounted for. In all cases, we find that effective stabilization is achieved, with feedback control able to drive the system towards the unstable equilibrium state. We further show that the controller is quite robust to changes in the relative density of the airfoil, and is effective even with large starting angles of attack in the vortex element simulations.