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

TL;DR

<3-5 sentence high-level summary> This paper develops a Lyapunov-based framework to establish fixed-time input-to-state stability (FxT ISS) for singularly perturbed (multi-time-scale) systems by extending the classical composite Lyapunov approach. The authors prove that if both the reduced (slow) and boundary-layer (fast) subsystems are FxT ISS and interconnection conditions are satisfied, the full system attains FxT ISS for sufficiently large time-scale separation, with a constructive method to estimate the required separation. A stylized scalar example and a fixed-time gradient-based optimization scheme with time-varying costs illustrate the theory in practice. The results provide a practical toolkit for designing and analyzing FxT ISS in interconnected multi-time-scale systems, with implications for robust control and online optimization under slowly varying disturbances.

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.

Figures (3)

• Figure 1: Trajectories of system \ref{['ex']} with and without the disturbance $u(t)$, where we use $\varepsilon=0.01$. The theoretically computed settling time bound of $18.15$ obtained using \ref{['settime']} is conservative compared to the observed trajectories.
• Figure 2: A block diagram illustrating the interconnection \ref{['exint']}.
• Figure 3: Trajectories of the FxT time-varying feedback optimization example, with $\varepsilon_0=5,0.2, 0$ and varying initial conditions.

Theorems & Definitions (23)

• Lemma 1
• Lemma 2
• Definition 1
• Definition 2
• Remark 1
• Theorem 1
• proof
• Remark 2
• Corollary 1
• proof