Dissipative quadratizations of polynomial ODE systems
Yubo Cai, Gleb Pogudin
TL;DR
This paper tackles the problem of transforming polynomial ODEs into quadratic (or lower) lifted systems while preserving dissipativity at chosen equilibria. It proves that there exists a dissipativity-preserving quadratization for any finite set of dissipative equilibria and provides a constructive algorithm to find such quadratizations with minimal added variables. The method builds from inner-quadratic quadratizations and uses stabilizers with a gauging parameter $\lambda$ to ensure negative real parts of the Jacobian eigenvalues at the lifted equilibria. The authors demonstrate the approach with case studies in reachability analysis and chemical reaction networks and supply open-source code for reproducing the results.
Abstract
Quadratization refers to a transformation of an arbitrary system of polynomial ordinary differential equations to a system with at most quadratic right-hand side. Such a transformation unveils new variables and model structures that facilitate model analysis, simulation, and control and offers a convenient parameterization for data-driven approaches. Quadratization techniques have found applications in diverse fields, including systems theory, fluid mechanics, chemical reaction modeling, and mathematical analysis. In this study, we focus on quadratizations that preserve the stability properties of the original model, specifically dissipativity at given equilibria. This preservation is desirable in many applications of quadratization including reachability analysis and synthetic biology. We establish the existence of dissipativity-preserving quadratizations, develop an algorithm for their computation, and demonstrate it in several case studies.