Bounding the Settling Time of Finite-Time Stable Systems using Sum of Squares

TL;DR

This paper develops a Sum-of-Squares based method to verify finite-time stability and compute bounds on the settling time for systems with non-polynomial vector fields, extending traditional Lyapunov approaches.

Contribution

It reformulates the non-polynomial Lyapunov function verification as polynomial inequalities, enabling SoS methods to certify finite-time stability and estimate settling time.

Findings

Successfully certifies finite-time stability in numerical examples.

Provides accurate bounds on the settling time.

Extends SoS methods to non-polynomial systems.

Abstract

Finite-time stability (FTS) of a differential equation guarantees that solutions reach a given equilibrium point in finite time, where the time of convergence depends on the initial state of the system. For traditional stability notions such as exponential stability, the convex optimization framework of Sum-of-Squares (SoS) enables computation of polynomial Lyapunov functions to certify stability. However, finite-time stable systems are characterized by non-Lipschitz, non-polynomial vector fields, rendering standard SoS methods inapplicable. To this end, we show that computation of a non-polynomial Lyapunov function certifying finite-time stability can be reformulated as feasibility of a set of polynomial inequalities under a particular transformation. As a result, SoS can be utilized to verify FTS and obtain a bound on the settling time. Numerical examples are used to demonstrate the accuracy of the conditions in both certifying finite-time stability and bounding the settling time.