Poincaré Analysis for Hybrid Periodic Orbits of Systems with Impulse Effects under External Inputs

In this paper we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincaré maps. In particular, we prove that input-to-state stability (ISS) of a periodic orbit under external excitation in both continuous and discrete time is equivalent to ISS of the corresponding 0-input fixed point of the associated forced Poincaré map. This result extends the classical Poincaré analysis for asymptotic stability of periodic solutions to establish orbital input-to-state stability of such solutions under external excitation. In our proof, we define the forced Poincaré map, and use it to construct ISS estimates for the periodic orbit in terms of ISS estimates of this map under mild assumptions on the input signals. As a consequence of the availability of these estimates, the equivalence between exponential stability (ES) of the fixed point of the 0-input (unforced) Poincaré map and ES of the corresponding orbit is recovered. The results can naturally be applied to continuoustime systems as well. Although our motivation for extending classical Poincaré analysis to address ISS stems from the need to design robust controllers for limit-cycle walking and running robots, the results are applicable to a much broader class of systems that exhibit periodic solutions.