A Dissipativity Framework for Input-to-State Stability with Positivity of Dynamical Systems with Interior Equilibria

This paper proposes a module-based framework for analyzing and designing dynamical systems evolving on positive orthants and having stationary points in their interior. Dissipativity theory is a widely-used concept that allows one to assess a property of a system by aggregating dissipativity of its components. Popular storage functions and supply rates do apply to positive systems, but do not give information about the positivity. This paper employs non-vector space storage functions encompassing logarithmic functions to formulate input-to-state stability (ISS) with positivity guarantee, and shows how its dissipativity characterization can effectively address system interconnections. The developed supply rates and criteria are not only asymmetric to fit non-vector state spaces, but also carry equilibrium information which is inherent in positive systems exhibiting interior equilibria. Mathematical and practical examples illustrate that the proposed framework offers flexibility in applying to various system interconnections.