Input-to-State Stability of Periodic Orbits of Systems With Impulse Effects via Poincar Analysis

In this paper, we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincar 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 zero-input fixed point of the associated forced Poincar map. This result extends the classical Poincar analysis for asymptotic stability of periodic solutions to establish orbital ISS of such solutions under external excitation. In our proof, we define the forced Poincar 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 zero-input (unforced) Poincar map and the ES of the corresponding orbit is recovered. The results can be applied naturally to study the robustness of periodic orbits of continuous-time systems as well. Although our motivation for extending classical Poincar 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.