Local Input-to-State Lyapunov Function Based Small-Gain Theorem for Nonlinear Systems

In this paper, a Lyapunov formulation of the small-gain theorem for interconnected nonlinear systems is presented. The interconnected nonlinear system we considered can be decomposed into two subsystems and the subsystems admit locally Lipschitz continuous locally input-to-state practical stable (LISpS) Lyapunov functions. The small-gain condition is only hold locally in this paper. We proved that, as long as the local small-gain condition holds, the interconnected nonlinear system is LISpS and we can construct the LISpS Lyapunov function of the interconnected nonlinear system by the LISpS Lyapunov functions of the subsystems. The domain of attraction and the input restrictions are given. A numerical example is given to verify the effectiveness of our methodology.