$L_p$-stabilization of integrator chains subject to input saturation using Lyapunov-based homogeneous design

TL;DR

This paper develops Lyapunov-based homogeneous feedback controllers for n-th integrator chains that achieve global asymptotic stability and $L_p$-stabilization under input saturation, with robustness to disturbances.

Contribution

It introduces new finite-gain $L_p$-stabilizing feedback laws for integrator chains using homogeneous control techniques, enhancing stability and robustness.

Findings

Design of explicit state feedback laws for stabilization

Achieves $L_p$-stability with arbitrarily small gain

Provides robustness results under disturbances

Abstract

Consider the $n$-th integrator $\dot x=J_nx+\sigma(u)e_n$, where $x\in\mathbb{R}^n$, $u\in \mathbb{R}$, $J_n$ is the $n$-th Jordan block and $e_n=(0\ \cdots 0\ 1)^T\in\mathbb{R}^n$. We provide easily implementable state feedback laws $u=k(x)$ which not only render the closed-loop system globally asymptotically stable but also are finite-gain $L_p$-stabilizing with arbitrarily small gain. These $L_p$-stabilizing state feedbacks are built from homogeneous feedbacks appearing in finite-time stabilization of linear systems. We also provide additional $L_\infty$-stabilization results for the case of both internal and external disturbances of the $n$-th integrator, namely for the perturbed system $\dot x=J_nx+e_n\sigma (k(x)+d)+D$ where $d\in\mathbb{R}$ and $D\in\mathbb{R}^n$.