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

TL;DR

The paper addresses designing fixed-time stable systems with a predefined Upper Bound on the Settling Time ($T_c$) by combining time-scale transformations with Lyapunov analysis. It develops a general framework that constructs both autonomous and non-autonomous systems whose settling time is explicitly governed by $T_c$, and identifies conditions under which $T_c$ is the least UBST. Central to the approach is a Lyapunov differential inequality $\dot{V}(x)\le -\frac{1}{T_c}\Psi(V(x),\\hat{t})\mathcal{H}(V(x))$, together with parameter transformations that map known settling-time properties of asymptotically stable systems to fixed-time behavior with predefined UBST. The method unifies existing fixed-time design techniques and enables the systematic generation of new vector fields, including autonomous, non-autonomous, and second-order fixed-time algorithms, with explicit $T_c$ as the guaranteed settling-time bound. These results reduce UBST conservativeness and have practical impact for real-time convergence requirements in control, estimation, and coordination tasks.

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.

Figures (2)

• Figure 1: Examples of autonomous fixed-time systems with $t_0=0$ and $T_c=1$ with $x\in\mathbb{R}$. From left to right: System \ref{['eq:autosys0']} with $\alpha = 1,\beta=2,p=0.5,q=2$ and $k=1$; System \ref{['eq:autosys2']}; System \ref{['eq:autosys1']} with $p=1/2$.
• Figure 2: Examples of non-autonomous fixed-time stable system \ref{['Eq:NonSystemDyn']} with $t_0=0$, $T_c=1$, $\lfloor x\rceil^{\frac{1}{2}}=|x|^{\frac{1}{2}}\hbox{sign}(x)$ and $x\in\mathbb{R}$. From left to right. $\Phi(\psi^{-1}(\hat{t}))^{-1}$ in \ref{['Syst:ExTBG']}; $\Phi(\psi^{-1}(\hat{t}))^{-1}$ in \ref{['Syst:Sec']}; and $\Phi(\psi^{-1}(\hat{t}))^{-1}$ in \ref{['Syst:poly_eta']} with $p=0.5$, $q=2$, $\alpha=1$ and $\beta=2$.

Theorems & Definitions (31)

• Remark 1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 3.2
• Lemma 3.3
• Lemma 3.5
• Lemma 3.7
• Lemma 3.8