Robustness of Periodic Orbits of Impulsive Systems à la Poincaré

Abstract

In this paper, we analyze the robustness of distinct periodic solutions of systems with impulse effects (SIEs) under uncertainty through the method of Poincaré. We work with a class of disturbances that affect both the continuous and discrete update dynamics of the SIE, as well as the geometry of the surface governing state transitions. In particular, we show that in the absence of any disturbances, the fixed point of the corresponding Poincaré map is locally asymptotically stable, if, and only if, in the presence of disturbances the periodic orbit of the SIE is locally input-to-state stable. This result generalizes the method of Poincaré for periodic orbits to explicitly incorporate the effect of disturbances. Although our motivation for this work stems from the need to rigorously and conveniently analyze robust controllers for dynamically moving legged robots, the results presented here are relevant to a much broader class of systems that can be modeled as forced SIEs.