# Stabilization for a perturbed chain of integrators in prescribed time

## TL;DR

This paper develops a new feedback control strategy for stabilizing chains of integrators within a prescribed time, addressing robustness to uncertainties and noise, and simplifies existing approaches through a homogeneity framework.

## Contribution

It introduces a novel robust feedback law for prescribed-time stabilization of perturbed integrator chains, unifying and simplifying previous methods.

## Key findings

- Achieves prescribed-time stabilization with bounded gains.
- Provides robustness against measurement noise and unmatched uncertainties.
- Simplifies existing stabilization proofs using homogeneity concepts.

## Abstract

In this paper, we consider issues relative to prescribed time stabilisation of a chain of integrators of arbitrary length, either pure (i.e., where there is no disturbance) or perturbed. In a first part, we revisit the proportional navigation feedback (PNF) approach and we show that it can be appropriately recasted within the framework of time-varying homogeneity. As a first consequence, we first recover all previously obtained results on PNF with simpler arguments. We then apply sliding mode inspired feedbacks to achieve prescribed stabilisation with uniformly bounded gains. However, all these feedbacks are robust to matched uncertainties only. In a second part, we provide a feedback law yet inspired by sliding mode which not only stabilises the pure chain of integrators in prescribed time but also exhibits some robustness in the presence of measurement noise and unmatched uncertainties.