Scalar Modules with Logarithmic Lyapunov Functions for Robust Positive Systems Exhibiting Interior Equilibria

This paper pursues the use of logarithmic functions to develop a dissipativity framework for module-based analysis and design of nonlinear dynamical systems evolving on positive orthants and having interior stationary points. For scalar module systems, the notion of asymmetric supply rate and asymmetric gain is formulated for establishing input-to-state stability (ISS) with positivity through the logarithmic storage functions. In contrast to systems without state space restrictions, targeting interior equilibria prevents asymptotic stability from implying ISS for additive perturbations. The tightness of the asymmetric supply rate is investigated by clarifying the involvement of the equilibria information.