Transverse Dynamics and Regions of Stability for Nonlinear Hybrid Limit Cycles

TL;DR

This paper introduces a new algorithm to estimate the stability regions of nonlinear hybrid system limit cycles using transverse dynamics and sum-of-squares programming, applicable to a broad class of systems.

Contribution

It presents a novel method for constructing transverse dynamics and Lyapunov functions to estimate stability regions of nonlinear hybrid limit cycles.

Findings

The transverse dynamics construction is valid for many nonlinear hybrid systems.

The method effectively computes inner estimates of attraction regions.

Stabilization of unstable limit cycles is addressed using transverse linearization.

Abstract

This paper presents an algorithm for computing inner estimates of the regions of attraction of limit cycles of a nonlinear hybrid system. The basic procedure is: (1) compute the dynamics of the system transverse to the limit cycle; (2) from the linearization of the transverse dynamics construct a quadratic candidate Lyapunov function; (3) search for a new Lyapunov function verifying maximal regions of orbital stability via iterated of sum-of-squares programs. The construction of the transverse dynamics is novel, and valid for a broad class of nonlinear hybrid systems. The problem of stabilization of unstable limit cycles will also be addressed, and a solution given based on stabilization of the transverse linearization.