Inner approximations of the region of attraction for polynomial dynamical systems

TL;DR

This paper extends a convex LP approach to polynomial dynamical systems to generate a hierarchy of polynomial inner approximations of the region of attraction, with convergence guarantees, complementing previous outer approximation methods.

Contribution

It introduces a modified hierarchy for inner approximations of the ROA, providing similar convergence guarantees as the outer approximation approach.

Findings

Hierarchy of polynomial inner approximations with convergence guarantees

Method compatible with standard semidefinite programming solvers

Extension of previous outer approximation framework

Abstract

In a previous work we developed a convex infinite dimensional linear programming (LP) approach to approximating the region of attraction (ROA) of polynomial dynamical systems subject to compact basic semialgebraic state constraints. Finite dimensional relaxations to the infinite-dimensional LP lead to a truncated moment problem in the primal and a polynomial sum-of-squares problem in the dual. This primal-dual linear matrix inequality (LMI) problem can be solved numerically with standard semidefinite programming solvers, producing a hierarchy of outer (i.e. exterior) approximations of the ROA by polynomial sublevel sets, with a guarantee of almost uniform and set-wise convergence. In this companion paper, we show that our approach is flexible enough to be modified so as to generate a hierarchy of polynomial inner (i.e.\,interior) approximations of the ROA with similar convergence guarantees.