Torsion Discriminance for Stability of Linear Time-Invariant Systems

TL;DR

This paper introduces a novel stability criterion for linear time-invariant systems based on the torsion of the state trajectory, linking geometric properties to system stability and asymptotic behavior.

Contribution

It proposes a new geometric approach to stability analysis using torsion, providing conditions involving the torsion's limit behavior for stability and asymptotic stability.

Findings

Stability is characterized by the torsion's limit behavior for initial states in sets of positive measure.

Asymptotic stability is linked to the torsion tending to infinity for certain initial conditions.

A relationship between trajectory curvature and stability is established when the system matrix is similar to a diagonal matrix.

Abstract

This paper proposes a new approach to describe the stability of linear time-invariant systems via the torsion $\tau(t)$ of the state trajectory. For a system $\dot{r}(t)=Ar(t)$ where $A$ is invertible, we show that (1) if there exists a measurable set $E_1$ with positive Lebesgue measure, such that $r(0)\in E_1$ implies that $\lim\limits_{t\to+\infty}\tau(t)\neq0$ or $\lim\limits_{t\to+\infty}\tau(t)$ does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set $E_2$ with positive Lebesgue measure, such that $r(0)\in E_2$ implies that $\lim\limits_{t\to+\infty}\tau(t)=+\infty$, then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the $i$th curvature $(i=1,2,\cdots)$ of the trajectory and the stability of the zero solution when $A$ is similar to a real diagonal matrix.