Growth rates for persistently excited linear systems

TL;DR

This paper investigates the stabilization and destabilization of linear control systems with persistently exciting signals, extending previous results to higher dimensions and providing conditions for maximal convergence or divergence rates.

Contribution

It generalizes earlier findings from 2D systems to higher dimensions, establishing conditions under which maximal rates of convergence and divergence are equal.

Findings

Maximal convergence rate equals maximal divergence rate under certain conditions.

Results extend to general single-input systems with controllability assumption.

Provides precise bounds for stabilization and destabilization rates.

Abstract

We consider a family of linear control systems $\dot{x}=Ax+\alpha Bu$ where $\alpha$ belongs to a given class of persistently exciting signals. We seek maximal $\alpha$-uniform stabilisation and destabilisation by means of linear feedbacks $u=Kx$. We extend previous results obtained for bidimensional single-input linear control systems to the general case as follows: if the pair $(A,B)$ verifies a certain Lie bracket generating condition, then the maximal rate of convergence of $(A,B)$ is equal to the maximal rate of divergence of $(-A,-B)$. We also provide more precise results in the general single-input case, where the above result is obtained under the sole assumption of controllability of the pair $(A,B)$.