Input-to-State Stability of Periodic Orbits of Systems with Impulse Effects via Poincar\'e Analysis

TL;DR

This paper establishes a theoretical link between the robustness of periodic orbits in impulsive systems and their Poincaré maps, extending classical stability analysis to include input-to-state stability under external disturbances.

Contribution

It proves that input-to-state stability of periodic orbits is equivalent to ISS of the fixed point of the forced Poincaré map, extending classical stability analysis to robustness considerations.

Findings

ISS of a periodic orbit is equivalent to ISS of the Poincaré map fixed point.

The results recover the classical exponential stability equivalence.

Applicable to systems with periodic solutions, including robotic locomotion.

Abstract

In this paper we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincar\'e 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 \emph{forced} Poincar\'e map. This result extends the classical Poincar\'e 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 Poincar\'e 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) Poincar\'e map and 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\'e 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.