Fixed-Time Input-to-State Stability for Singularly Perturbed Systems via Composite Lyapunov Functions
Abstract
We study singularly perturbed systems that exhibit input-to-state stability (ISS) with fixed-time properties in the presence of bounded disturbances. In these systems, solutions converge to the origin within a time frame independent of initial conditions when undisturbed, and to a vicinity of the origin when subjected to bounded disturbances. First, we extend the traditional composite Lyapunov method, commonly applied in singular perturbation theory to analyze asymptotic stability, to include fixed-time ISS. We demonstrate that if both the reduced system and the boundary layer system exhibit fixed-time ISS, and if certain interconnection conditions are met, the entire multi-time scale system retains this fixed-time ISS characteristic, provided the separation of time scales is sufficiently pronounced. Next, we illustrate our findings via analytical and numerical examples, including a novel application in fixed-time feedback optimization for dynamic plants with slowly varying cost functions.
Cite
@article{arxiv.2412.16797,
title = {Fixed-Time Input-to-State Stability for Singularly Perturbed Systems via Composite Lyapunov Functions},
author = {Michael Tang and Miroslav Krstic and Jorge Poveda},
journal= {arXiv preprint arXiv:2412.16797},
year = {2025}
}