A Smooth Construction of ISS Lyapunov Functions for Interconnected Systems

To establish input-to-state stability (ISS) of an interconnected nonlinear system, the small-gain framework makes use of nonlinear gain functions of components systems. Computing gain functions is usually hard without introducing Lyapunov functions to component systems. In the ISS formulation, Lyapunov functions of component systems naturally lead to a Lyapunov functions in the max-separable form for the interconnection. Since the maximization is not differentiable. technical modification or occurrence of artifical behavior is unavoidable in actual use of a max-separable Lyapunov function. This paper proposes a practical Lyapunov function by securing continuous differentiability and keeping it as simple and intuitive as the max-separable Lyapunov function.