Small-Gain Conditions and the Comparison Principle

The general input-to-state stability (ISS) small-gain condition for networks in a trajectory formulation is shown to be equivalent to the requirement that a discrete-time comparison system induced by the gain matrix of the network is ISS. This leads to a comparison principle, relating input-to-state stability of an artificial discrete-time system to the same stability property of a continuous-time nominal system. As a consequence, general small-gain conditions can now be verified by finding ISS Lyapunov functions.