Signed Small-Gain Criteria Amenable to Asymmetry with Respect to Equilibria in Establishing iISS of Networks

This paper proposes a framework of signed small-gain criteria which are automatically amenable to asymmetry of gain-type dissipation with respect to equilibria. In theory and practice, the asymmetry has been avoided via over-bounding by symmetrization, which often results in unreasonably demanding or negative answers in stability, stabilization, robustness and robustification of dynamical systems. Natural systems are composed in accordance with the principle of conservation, and employ mechanisms which are more versatile than linearity. Exact energy balance is exhibited through nonlinear mechanisms, but it is not symmetric with respect to equilibria, due to the nonlinearity. Stability and robustness embedded in such energy balance cannot be detected after symmetrization. The signed small-gain methodology proposed in this paper provides powerful tools that absorb the asymmetry automatically in establishing integral input-to-stability of dynamical networks.