On the design of non-autonomous fixed-time controllers with a predefined upper bound of the settling time

Abstract

This paper aims to introduce a design methodology to stabilize a chain of integrators in a fixed-time with predefined Upper Bound for the Settling-Time (UBST). This approach is based on time-varying gains (time-base generator) that become singular as the time approaches the predefined convergence time. We present the conditions such that every nonzero trajectory convergence exactly at the predefined time with feedback laws that are simpler than in previous approaches. Additionally, we present results such that the origin is reached before the singularity occurs, making this approach realizable in practice. A significant contribution, since current results, based on time-varying gains, required the time-varying gain to tend to infinity as the time approaches the prescribed/predefined-time. Moreover, our approach guarantees fixed-time convergence with predefined UBST even in the presence of bounded disturbances (with known bound) and provides a methodology to re-design an autonomous fixed-time controller where the estimate of the UBST for the closed-loop system is very conservative into a less conservative one with predefined \textit{UBST} that can be set arbitrarily tight. We present numerical examples to illustrate the effectiveness of the approach together with comparisons with existing autonomous fixed-time controllers.