A Non–conservative Small-Gain Theorem for GAS Discrete–Time Systems

This paper makes use of the concept of a finite–time Lyapunov function to derive a non– conservative small-gain theorem for stability analysis of interconnected discrete–time nonlinear systems. Firstly, it is shown that the existence of a global finite–time Lyapunov function is equivalent to global asymptotic stability (GAS) of the overall interconnected system. Secondly, it is indicated that existence of Lyapunov–type functions for each subsystem, together with a small-gain condition implies GAS of the interconnected system. Thirdly, the main result of this paper establishes that GAS of the interconnected system always yields a set of Lyapunov–type functions that satisfy the small-gain condition for a rather general class of GAS nonlinear systems. A simple example demonstrates the non–conservatism of the proposed small-gain theorem.