A Lyapunov-Based Small-Gain Theorem for Fixed-Time Stability
Provides a theoretical tool for certifying fixed-time stability in interconnected systems, relevant for control engineers and researchers in nonlinear systems.
This paper develops a Lyapunov-based small-gain theorem for fixed-time stability (FxTS) in interconnected systems, filling a gap in stability analysis. The result is illustrated with examples, including fixed-time feedback optimization without time-scale separation.
This paper introduces a novel Lyapunov-based small-gain methodology for establishing fixed-time stability (FxTS) guarantees in interconnected dynamical systems. Specifically, we consider interconnections in which each subsystem admits an individual fixed-time input-to-state stability (ISS) Lyapunov function that certifies FxT-ISS. We then show that if a nonlinear small-gain condition is satisfied, then the entire interconnected system is FxTS. Our results are analogous to existing Lyapunov-based small-gain theorems developed for asymptotic and finite-time stability, thereby filling an important gap in the stability analysis of interconnected dynamical systems. The proposed theoretical tools are further illustrated through analytical and numerical examples, including the first result on fixed-time feedback optimization of dynamical systems without time-scale separation between the plant and the controller.