We propose a Lyapunov-based hybrid controller that ensures robust global asymptotic stability of a desired planar orientation, a task that is impossible by continuous or (non-hybrid) discontinuous state feedback due to topological obstructions. The hybrid control structure relies on hysteretically switching between (almost globally) stabilizing control laws which are generated by a family of diffeomorphisms that stretch the space of planar rotations onto itself. A salient feature of the proposed control technique is that it can produce a robustly globally asymptotically stabilizing controller from an existing control law and a corresponding Lyapunov function.
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Mayhew et al. (2010) studied this question.