  Convergent, contractive or incremental stability properties of nonlinear
systems have attracted interest for control tasks such as observer design,
output regulation and synchronization. The convergence property plays a central
role in the neuromorphic (brain-inspired) computing of reservoir computing,
which seeks to harness the information processing capability of nonlinear
systems. This paper presents a small-gain theorem for discrete-time
output-feedback interconnected systems to be uniformly input-to-output
convergent (UIOC) with outputs converging to a bounded reference output
uniquely determined by the input. A small-gain theorem for interconnected
time-varying discrete-time uniform input-to-output stable systems that could be
of separate interest is also presented as an intermediate result. Applications
of the UIOC small-gain theorem are illustrated in the design of observer-based
controllers and interconnected nonlinear classical and quantum dynamical
systems (as reservoir computers) for black-box system identification.
