Global Stabilization of Polynomial Systems Using Equilibrium-Independent Dissipativity

This work presents a proof that the existence of a sum of squares (SOS) decomposition for a closed-loop (asymptotic) stability condition valid around the origin is equivalent, for input-affine polynomial systems, to the existence of an SOS certificate for a strict dissipation inequality subject to a certain equality constraint. In addition, a new sufficient condition for feedback asymptotic stabilization of nonlinear systems around nonzero equilibria is proposed using the notion of equilibrium-independent dissipativity (EID). Then, this new condition is applied to polynomial systems, specifically. It is proved in this article that the aforementioned SOS certificate for strict dissipativity exists, with a strongly convex energy function, if and only if a certain equilibrium-independent strict dissipativity (EISD) condition also has an SOS decomposition. Consequently, not only the origin, but any assignable equilibrium point can be globally stabilized by an appropriate state feedback. As the systems considered are polynomial, semidefinite programming (SDP) can be used to solve this global stabilization problem. A numerical example provides further evidence on the usefulness of the proposed strategy.