Invariant Funnels around Trajectories using Sum-of-Squares Programming

TL;DR

This paper introduces numerical methods utilizing Sum-of-Squares programming to compute and verify invariant funnels around trajectories of polynomial differential equations, enhancing safety and control in complex systems.

Contribution

It presents two novel methods for certifying invariant regions around trajectories, one exact and one sampling-based, both leveraging SOS programming for polynomial Lyapunov functions.

Findings

Exact method certifies invariance despite approximate trajectories.

Sampling method is faster and nearly as accurate.

Methods successfully applied to satellite trajectory stabilization.

Abstract

This paper presents numerical methods for computing regions of finite-time invariance (funnels) around solutions of polynomial differential equations. First, we present a method which exactly certifies sufficient conditions for invariance despite relying on approximate trajectories from numerical integration. Our second method relaxes the constraints of the first by sampling in time. In applications, this can recover almost identical funnels but is much faster to compute. In both cases, funnels are verified using Sum-of-Squares programming to search over a family of time-varying polynomial Lyapunov functions. Initial candidate Lyapunov functions are constructed using the linearization about the trajectory, and associated time-varying Lyapunov and Riccati differential equations. The methods are compared on stabilized trajectories of a six-state model of a satellite.