Global asymptotic stabilization of the inverted equilibrium manifold of the 3-D pendulum by hybrid feedback

Topological obstructions preclude the existence of a continuous state-feedback control law that globally asymptotically stabilizes the inverted equilibrium manifold of the 3D pendulum. Furthermore, memoryless discontinuous feedbacks are either unbounded or not robust to arbitrarily small measurement noise. In this paper, we propose a hybrid feedback that coordinates a “synergistic” family of potential functions and their gradient-based feedbacks to ensure global asymptotic stability of the inverted equilibrium manifold of the 3D pendulum. The hybrid scheme is robust to small perturbations including measurement noise, eliminates performance limitations of smooth state feedback, and obviates the need for large torques created by some non-smooth state-feedback control laws. We provide a brief simulation study to illustrate the efficacy of the method and compare it with a smooth feedback.