Global stabilization of classes of linear control systems with bounds on the feedback and its successive derivatives

TL;DR

This paper develops methods for globally stabilizing specific classes of linear control systems using static feedback laws with bounded amplitude and derivatives, addressing integrator chains and skew-symmetric systems.

Contribution

It introduces novel feedback laws ensuring bounded feedback and derivatives for integrator chains and skew-symmetric systems, expanding stabilization techniques.

Findings

Nested saturation works for integrator chains but not for skew-symmetric systems.

New feedback law proposed for skew-symmetric systems.

Successful stabilization demonstrated on third-order integrator and harmonic oscillator.

Abstract

In this paper, we address the problem of globally stabilizing a linear time-invariant (LTI) system by means of a static feedback law whose amplitude and successive time derivatives, up to a prescribed order $p$, are bounded by arbitrary prescribed values. We solve this problem for two classes of LTI systems, namely integrator chains and skew-symmetric systems with single input. For the integrator chains, the solution we propose is based on the nested saturations introduced by A.R. Teel. We show that this construction fails for skew-symmetric systems and propose an alternative feedback law. We illustrate these findings by the stabilization of the third order integrator with prescribed bounds on the feedback and its first two derivatives, and similarly for the harmonic oscillator with prescribed bounds on the feedback and its first derivative.