Input-to-state stability with respect to two measurement functions: Discrete-time systems

Input-to-state stability (ISS) with respect to two measurement functions subsumes many ISS-type properties such as input-to-output stability (IOS) and state-independent input-to-output stability (SI-IOS). It also subsumes a version of incremental ISS. In this paper, we develop the notion of ISS with respect to two measurement functions for discrete-time systems and highlight the importance of commensurability of the two measurement functions and compactness of the zero set of the lower measurement function. Of particular interest, we show that the obvious extension of an implication-form ISS-Lyapunov function does not imply ISS with respect to two measurement functions.