Unified Lyapunov Conditions for Predefined-time and Finite-time Stability of Nonlinear Systems

Core Concepts

This paper presents unified sufficient Lyapunov conditions to guarantee the predefined-time and finite-time stability of autonomous nonlinear systems. The proposed Lyapunov theorem consolidates and generalizes existing results on predefined-time and finite-time stability.

Abstract

The paper presents a unified Lyapunov theorem that provides sufficient conditions for the predefined-time and finite-time stability of autonomous nonlinear systems. The key aspects are: The theorem establishes equivalence with existing Lyapunov-based theorems on predefined-time and finite-time stability. It relaxes the constraint on the Lyapunov function compared to prior work, allowing for a broader class of bounded monotonic functions. If the Lyapunov function is unbounded, the theorem degenerates to a finite-time stability result. The theorem is applied to design a nonsingular sliding mode controller for an Euler-Lagrange system, ensuring its predefined-time stability. Numerical simulations demonstrate the effectiveness of the proposed controller in achieving predefined-time convergence.

Stats

This system is asymptotically stable, due to the fact eV^p > 0 always holds. The time derivative of V is rewritten as dψ(V)/dt = -1/Tc, where ψ(V) = 2 - e^(-V^p) is a bounded increasing function with ψ(V) ∈ [1, 2). T = (ψ(V0) - ψT) Tc < Tc, where ψT = 1 is the minimum value of ψ(V).

Quotes

"Unlike literature [6], the strict constraint of the function ψ(V) ∈ [0, 1) being a K∞ function is required. Therefore, ψ(V) will decrease to the minimum ψT = 1 from its arbitrary initial value ψ(V0) < 2." "It is proved that T ≥ Tc is valid for any ψ(V0) decreases to the minimum of ψ(V). The Lyapunov candidate V also converges to zero simultaneously."

Key Insights Distilled From

Unified Predefined-time Stability Conditions of Nonlinear Systems with Lyapunov Analysis

by Bing Xiao,Ha... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.01164.pdf

Deeper Inquiries

How can the proposed Lyapunov theorem be extended to handle stochastic nonlinear systems

The proposed Lyapunov theorem can be extended to handle stochastic nonlinear systems by incorporating stochastic analysis techniques into the stability analysis framework. One approach is to introduce stochastic Lyapunov functions that can capture the probabilistic nature of the system dynamics. By considering the stochastic nature of the system uncertainties or disturbances, the Lyapunov theorem can be adapted to provide stability guarantees in the presence of random fluctuations. Additionally, techniques such as stochastic stability theory and stochastic control can be integrated into the analysis to account for the uncertainties in the system's behavior.

What are the potential limitations or drawbacks of the unified Lyapunov conditions in terms of conservatism or applicability

While the unified Lyapunov conditions offer a comprehensive framework for analyzing predefined-time/finite-time stability of autonomous systems, there are potential limitations and drawbacks to consider. One limitation is the conservatism inherent in the stability analysis, which may lead to overly conservative stability guarantees. The applicability of the unified conditions may also be limited in complex systems with high-dimensional state spaces or nonlinear dynamics that do not easily conform to the assumptions of the Lyapunov theorem. Furthermore, the selection of the Lyapunov function and the tuning of parameters in the stability conditions may require expert knowledge and careful consideration, making the approach challenging to implement in practice.

Can the predefined-time stability framework be combined with optimal control techniques to achieve desired performance objectives

The predefined-time stability framework can be effectively combined with optimal control techniques to achieve desired performance objectives in control systems. By incorporating optimal control principles such as minimizing a cost function or maximizing a performance metric, the predefined-time stability framework can be used to design controllers that not only ensure stability within a predefined time but also optimize system performance. This integration allows for the synthesis of controllers that meet both stability and performance requirements, leading to improved control system behavior. Additionally, optimal control techniques can be used to tune the control parameters in the predefined-time stability framework to achieve specific performance objectives, such as minimizing settling time or maximizing control efficiency.