Nonlinear small-gain theorems for discrete-time large-scale systems

In this work, we present generalizations of the nonlinear ISS small-gain theorem for discrete-time large-scale systems consisting of multiple subsystems, in a general setting of input-to-output stability. In other words, these discrete-time, large-scale small-gain theorems are established in terms of several types of gain functions, such as asymptotic gains, gains in decay estimations of the output functions, and Lyapunov gains. As a result, the set of cyclic small-gain conditions recovers the case of previously developed small-gain theorems either for an interconnected system comprised of two subsystems, or a network of ISS systems.