Control lyapunov functions and hybrid zero dynamics

Hybrid zero dynamics extends the Byrnes-Isidori notion of zero dynamics to a class of hybrid models called systems with impulse effects. Specifically, given a smooth submanifold that is contained in the zero set of an output function and is invariant under both the continuous flow of the system with impulse effects as well as its reset map, the restriction dynamics is called the hybrid zero dynamics. Prior results on the stabilization of periodic orbits of the hybrid zero dynamics have relied on input-output linearization of the transverse variables. The principal result of this paper shows how control Lyapunov functions can be used to exponentially stabilize periodic orbits of the hybrid zero dynamics, thereby significantly extending the class of stabilizing controllers. An illustration of this result on a model of a bipedal walking robot is provided.