A Lyapunov-like Characterization of Predefined-Time Stability (1910.14604v1)

Abstract: This technical note studies Lyapunov-like conditions to ensure a class of dynamical systems to exhibit predefined-time stability. The origin of a dynamical system is predefined-time stable if it is fixed-time stable and an upper bound of the settling-time function can be arbitrarily chosen a priori through a suitable selection of the system parameters. We show that the studied Lyapunov-like conditions allow to demonstrate equivalence between previous Lyapunov theorems for predefined-time stability for autonomous systems. Moreover, the obtained Lyapunov-like theorem is extended for analyzing the property of predefined-time ultimate boundedness with predefined bound, which is useful when analyzing uncertain dynamical systems. Therefore, the proposed results constitute a general framework for analyzing predefined-time stability, and they also unify a broad class of systems which present the predefined-time stability property. On the other hand, the proposed framework is used to design robust controllers for affine control systems, which induce predefined-time stability (predefined-time ultimate boundedness of the solutions) w.r.t. to some desired manifold. A simulation example is presented to show the behavior of a developed controller, especially regarding the settling time estimation.

• The paper introduces a Lyapunov-like theorem providing a unified framework to characterize predefined-time stability for both autonomous and perturbed dynamical systems.
• It clarifies that predefined-time stability allows setting an explicit upper bound for convergence time a priori, distinct from the general bounded settling time of fixed-time stability.
• The theoretical results are applied to design robust controllers ensuring predefined-time ultimate boundedness or stability, validated through numerical simulations for affine control systems.

Lyapunov Characterization of Predefined-Time Stability

The paper by Esteban Jiménez-Rodríguez et al. discusses the concept of predefined-time stability in dynamical systems, leveraging Lyapunov-like conditions to guarantee such stability. This notion significantly extends the understanding of system stability by allowing practitioners to not only ensure fixed-time stability but also to specifically dictate an upper bound on the convergence time through careful selection of system parameters.

One of the key distinctions highlighted is between fixed-time stability and predefined-time stability. Fixed-time stability improves upon finite-time stability by ensuring bounded settling time, but predefined-time stability further refines this by allowing the upper bound of the settling-time to be chosen a priori. This feature is especially critical in real-world applications where time constraints are stringent and non-negotiable, such as in automated control systems operating under regulatory or physical constraints.

The authors present a Lyapunov-like theorem that consolidates various results from prior literature on predefined-time stability, thereby creating a unified framework for analysis. The theorem utilizes class $\mathcal{K}^1$ functions, characterized by specific properties such as strict monotonicity and mapping within a bounded range, to formulate conditions under which a system can be ensured predefined-time stability. One of the significant advantages of this approach is its general applicability to both autonomous systems and systems subjected to perturbations, thus broadening its practical scope.

This paper further explores the application of these theoretical findings through the design of robust controllers for affine control systems. By adhering to the proposed Lyapunov-like conditions, it is demonstrated how continuous controllers can be devised to achieve predefined-time ultimate boundedness, or discontinuous controllers for predefined-time stability. These are critically important in engineering domains where system robustness and reliability must be assured despite external disturbances or model uncertainties.

Numerical simulations serving as proof-of-concept highlight the efficacy of the design methodologies proposed, showing that the derived controllers are adept at ensuring the desired time-bound responses while maintaining stability and robustness in practice. The theoretical and practical insights provided pave the way for advancements in nonlinear control systems where predefined-time guarantees are crucial.

In conclusion, the framework introduced by this paper is an important contribution to control theory, offering new methodologies for precision in system design through predefined-time stability. Going forward, further exploration and refinement within specific types of nonlinear systems could solidify these techniques as standard in industries dealing with critical time-dependent processes.