A Closed Form Expression of Nonlinear Scalings for Lyapunov Functions of ISS Networks

Abstract

For networks consisting of input-to-state stable systems, it is well-known that the popular Lyapunov function called the max-separable function can be constructed by solely relying on component-wise inverse maps of one single path characterizing the monotonicity of dissipation inequalities of component systems. Numerical algorithms have been developed to compute such a path. This paper proposes a useful closed-form expression for the inverse maps of a path and its extension to generate a sufficient variety of paths. The solution not only gives nonlinear scalings of the max-separable Lyapunov function explicitly, but also substantiates the rounding-off technique to remove the non-differentiable nature from the max-separable function.