On the stabilization of persistently excited linear systems

TL;DR

This paper investigates the stabilization of linear control systems with unknown time-varying excitation signals, establishing conditions under which such systems can be stabilized with linear feedback and exploring a bifurcation phenomenon related to the excitation parameters.

Contribution

It provides a stabilizability criterion for systems with persistent excitation signals and analyzes the impact of eigenvalues and excitation parameters on stabilization capabilities.

Findings

Systems with eigenvalues having non-positive real parts are stabilizable under persistent excitation.

Stabilizability depends on the ratio / of excitation parameters, showing a bifurcation phenomenon.

Stabilization is not always possible for arbitrary system matrices A.

Abstract

We consider control systems of the type $\dot x = A x +\alpha(t)bu$, where $u\in\R$, $(A,b)$ is a controllable pair and $\alpha$ is an unknown time-varying signal with values in $[0,1]$ satisfying a persistent excitation condition i.e., $\int_t^{t+T}\al(s)ds\geq \mu$ for every $t\geq 0$, with $0<\mu\leq T$ independent on $t$. We prove that such a system is stabilizable with a linear feedback depending only on the pair $(T,\mu)$ if the eigenvalues of $A$ have non-positive real part. We also show that stabilizability does not hold for arbitrary matrices $A$. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter $\mu/T$.