Lyapunov formulation of the ISS cyclic-small-gain theorem for hybrid dynamical networks

This paper presents a Lyapunov-based cyclic-small-gain theorem for the hybrid dynamical networks composed of input-to-state stable (ISS) subsystems whose motions may be continuous, impulsive or piecewise constant on the time-line. On the one hand, it is shown that hybrid dynamic networks with interconnection gains less than the identity function are ISS by means of Lyapunov functions. Additionally, an ISS-Lyapunov function for the total network is constructed using the ISS-Lyapunov functions of the subsystems. On the other hand, a novel result of this paper shows that a hybrid dynamic network satisfying the cyclic-small-gain condition can be transformed into one with interconnection gains less than the identity. In sharp contrast with several previously known results, the impulses of the subsystems are time triggered and the impulsive times for different subsystems may be different.