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arXiv · 2412.16797

Fixed-Time Input-to-State Stability for Singularly Perturbed Systems via Composite Lyapunov Functions

Michael Tang·Miroslav Krstic·Jorge Poveda
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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.

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Michael Tang, Miroslav Krstic, Jorge Poveda. 2024-12-21. Fixed-Time Input-to-State Stability for Singularly Perturbed Systems via Composite Lyapunov Functions. https://arxiv.org/abs/2412.16797

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