An asymmetric small-gain technique to construct Lyapunov-Krasovskii functionals for nonlinear time-delay systems with static components

The standard ISS small-gain theorem and the recently-developed ISS small-gain theorem assume that system components are characterized in a symmetric way with respect to the equilibrium. Dissipative properties of a nonlinear system is usually asymmetric. Formulating them into symmetric properties sometimes causes crucial conservativeness. The purpose of this paper is to develop a technique to take the asymmetry into account in stability analysis of time-delay systems. The result is based on decomposition of a system into integral input-to-state stable dynamic components and static components characterized in an asymmetric way with respect to the equilibrium. A Lyapunov-Krasovskii functional establishing robustness with respect to disturbances in the presence of time delays is constructed, and its effectiveness is illustrated by a network flow control example. The proposed iISS methodology covers a broader class of systems than existing approaches based on operator norms or the ISS gain.