Stability Analysis of Switched Time-Delay Systems With Unstable Subsystems: Slow–Fast Weighted Admissible Edge-Dependent Average Dwell Time Method

Abstract

This paper investigates the robust stability of a class of uncertain switched time-varying delay systems containing unstable subsystems. In the presence of unstable subsystems, a novel slow-fast weighted admissible edge-dependent average dwell time method is introduced for designing the switching signal. An enhanced multiple discontinuous Lyapunov-Krasovskii functional is constructed for analyzing the stability of the switched time-varying delay systems. By exploiting the Bessel-Legendre inequality together with the properties of convex functions and the designed switching signal, less conservative stability criteria are derived. Finally, a numerical example and a practical simulation demonstrate that the proposed method effectively reduces the conservatism of the derived criteria. Note to Practitioners—In many practical engineering systems, such as networked control systems, power electronic converters, and robotic systems, the system dynamics may switch among different operating modes and are often affected by delays arising from sensing, communication, and actuation. A more challenging situation occurs when some modes exhibit unstable dynamics under certain operating conditions. In this case, the main difficulty lies in coordinating the dwell time distribution among different admissible switching paths while simultaneously handling time-varying delays, uncertainties, and unstable modes. These coupled factors complicate the stability analysis and may render existing methods conservative or even inapplicable. Therefore, this paper investigates the stability analysis of switched time-delay systems with coexisting stable and unstable modes. To mitigate the adverse effects of unstable modes on the overall system stability, a slow-fast weighted admissible edge-dependent average dwell time switching strategy is developed. Furthermore, an enhanced Lyapunov-Krasovskii functional incorporating richer historical delay information is constructed to derive less conservative stability conditions. The results show that, under the proposed strategy, the dwell time range can be enlarged while maintaining system stability, and larger time delays can be tolerated, thereby significantly reducing the conservatism of existing analysis methods. These results provide theoretical guidance for the analysis and design of practical systems involving switching behaviors, time delays, and locally unstable dynamics.