An Implicit Function Approach to Lyapunov functions for Interconnections Containing Non-ISS Components

This paper addresses the problem of establishing internal and external stability of feedback interconnection of integral input-to-state stable systems. Construction of Lyapunov functions has been essential for establishing the stability when components are not input-to-state stable (ISS). Typical Lyapunov functions are in max-separable or sum-separable from. In contrast to the max construction, solutions to the sum construction have not been given intuitive interpretations. The max construction has limitations such as non-smoothness and incapability of guaranteeing stability in the presence of non-ISS components. This paper aims at constructing a Lyapunov function in the non-ISS case through simple geometrical observations. The approach leads to a novel construction mixing the max and sum separability. The new Lyapunov function gives much better invariant sets than the sum-separable ones known in the literature. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.