Synergistic Lyapunov functions and backstepping hybrid feedbacks

The notion of synergistic potential functions has been introduced recently in the literature and has been used as the basis for the design of hybrid feedback laws that achieve global asymptotic stabilization of a point on a compact manifold (without boundary) such as S¹, S², and SO(3). Here, synergistic potential functions are generalized-to synergistic Lyapunov functions-and are shown to be amenable to backstepping. In particular, if an afflne control system admits a (weak) synergistic Lyapunov function and feedback pair then the system with an integrator added at the input also admits a synergistic Lyapunov and feedback pair. This fact enables "smoothing" hybrid feedbacks, or implementing them through a chain of integrators. In this way, hybrid control designed at a kinematic level can be redesigned for control through forces, torques, or even the derivative of these quantities. We demonstrate the backstepping procedure for attitude stabilization of a rigid body using a quaternion parametrization.