# On the design of new classes of fixed-time stable systems with predefined upper bound for the settling time

## TL;DR

This paper introduces a methodology for designing fixed-time stable systems with a predefined upper bound on settling time, improving over traditional methods by reducing conservativeness and providing a broad class of such systems.

## Contribution

It presents a novel construction procedure for fixed-time stable systems with predefined and minimal upper bounds on settling time using time-scale transformations and Lyapunov analysis.

## Key findings

- Generated new fixed-time stable algorithms with predefined upper bounds.
- Provided conditions for the least possible upper bound of settling time.
- Demonstrated effectiveness through illustrative examples.

## Abstract

This paper aims to provide a methodology for generating autonomous and non-autonomous systems with a fixed-time stable equilibrium point where an Upper Bound of the Settling Time (UBST) is set a priori as a parameter of the system. In addition, some conditions for such an upper bound to be the least one are provided. This construction procedure is a relevant contribution when compared with traditional methodologies for generating fixed-time algorithms satisfying time constraints since current estimates of an UBST may be too conservative. The proposed methodology is based on time-scale transformations and Lyapunov analysis. It allows the presentation of a broad class of fixed-time stable systems with predefined UBST, placing them under a common framework with existing methods using time-varying gains. To illustrate the effectiveness of our approach, we generate novel, autonomous and non-autonomous, fixed-time stable algorithms with predefined least UBST.