On input-to-state stability of delay difference equations

Input-to-state stability (ISS) of delay difference equations (DDEs) subject to external disturbances is studied. Each delayed state of the DDE is considered as a subsystem of an interconnected system. Thus, it can be proven via a small-gain theorem for interconnected systems that a DDE is ISS if it admits an ISS-Lyapunov-Razumikhin function (ISS-LRF). As a by-product of this approach, an explicit construction of an ISS-Lyapunov-Krasovskii function is also obtained. Then, necessary conditions under which the Razumikhin method can be used to establish ISS are derived. An example, which establishes that not every DDE that is ISS admits an ISS-LRF, indicates the significance of the developed necessary conditions. Moreover, these conditions provide a non-trivial necessary condition for linear DDEs in particular.