Lyapunov analysis of nonlinear systems with rational vector field and Jacobian

This paper studies Lur'e type nonlinear systems where both the vector field and its Jacobian are rational with respect to the states and a sector bounded nonlinearities. Conditions to assess stability and compute induced L
₂
-gain bounds for such systems are cast in terms of rational inequalities. A numerical solution for these inequalities is formulated as a convex optimisation problem given by a sum-of-squares program. Examples are given for nonlinear systems with the arctangent and rational nonlinearities. The proposed method is compared to the Popov and Circle criteria in a numerical example and is shown to outperform both of these classical results.