A methodology for designing fixed-time stable systems with a predefined upper-bound in their settling time

TL;DR

This paper introduces a new methodology for designing fixed-time stable systems with a user-defined upper-bound on settling time, ensuring bounded gains even as the system approaches equilibrium.

Contribution

The paper presents a general approach to redesign high-order systems into fixed-time stable systems with bounded gains and a specified settling time.

Findings

Successfully redesigns systems with bounded gains and fixed-time stability

Provides sufficient conditions for bounded time-varying gains

Demonstrates methodology with fixed-time online differentiators

Abstract

Algorithms having uniform convergence with respect to their initial condition (i.e., with fixed-time stability) are receiving increasing attention for solving control and observer design problems under time constraints. However, we still lack a general methodology to design these algorithms for high-order perturbed systems when we additionally need to impose a user-defined upper-bound on their settling time, especially for systems with perturbations. Here, we fill this gap by introducing a methodology to redesign a class of asymptotically, finite- and fixed-time stable systems into non-autonomous fixed-time stable systems with a user-defined upper-bound on their settling time. Our methodology redesigns a system by adding time-varying gains. However, contrary to existing methods where the time-varying gains tend to infinity as the origin is reached, we provide sufficient conditions to maintain bounded gains. We illustrate our methodology by building fixed-time online differentiators with user-defined upper-bound on their settling time and bounded gains.