Stability analysis of linear systems with state jump - motivated by periodic motion control of passive walker

In this paper, we study the stability of periodic solutions of linear systems with state jump. Such a model arises, for example, when we consider the periodic motion of passive walkers [M. Garcia, et. al. The simplest walking model: stability, complexity, and scaling, 1998]. In our previous work [K. Hirata, H. Kokame and K. Konishi, On stability of simplified passive walker model and effect of feedback control, 2002], we derived such a model by simplifying the result of Garcia et.al.[T.McGeer, Passive dynamic walking, 1990] and analyzed its stability via the Poincare map. Also the effects of the feedback control strategies, OGY and DFC (delayed feedback control) methods, were examined in the same framework. However, the Poincare map was introduced in a rather ad-hoc manner there. In this paper, we refine the mathematical treatment of the Poincare map. After defining a special type of stability for the periodic solution considered here, we show that the stability via the Poincare map is equivalent to this specific definition. Also the effect of the data loss caused by the variation of the state jump interval in the OGY case is examined.