The finite-time stability (FTS) is crucial for the network systems with high requirements on the convergence rate. When the finite-time settling time is globally bounded, the fixed-time stability (FxTS) is introduced. This paper focuses on the FTS/FxTS of the network systems with nonlinear protocols over signed digraphs. Sufficient criteria are obtained for ensuring the FTS/FxTS of the investigated network systems with general weighted matrices from a unified framework. Particularly, the proposed distributed nonlinear protocol is fully distributed with flexible heterogeneous coefficients. The explicit bounds on the finite/fixed-time settling time are also derived analytically. Moreover, the FTS/FxTS of the perturbed network systems is guaranteed with the technique of the sliding mode controller. As an application, the obtained results are used to achieve finite/fixed-time network modulus consensus over signed digraphs. Finally, two numerical examples are presented to verify the analytical results.

中文翻译：

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有符号有向图上具有非线性协议的网络系统的有限时间稳定性

有限时间稳定性（FTS）对于收敛速度要求高的网络系统至关重要。当有限时间稳定时间全局有界时，就引入了固定时间稳定性 (FxTS)。本文重点介绍了在有符号有向图中具有非线性协议的网络系统的 FTS/FxTS。获得了足够的标准来确保所研究的网络系统的 FTS/FxTS 具有来自统一框架的一般加权矩阵。特别是，所提出的分布式非线性协议是完全分布式的，具有灵活的异构系数。有限/固定时间稳定时间的显式界限也是通过分析推导出来的。此外，通过滑模控制器技术保证扰动网络系统的FTS/FxTS。作为应用程序，获得的结果用于在有符号有向图中实现有限/固定时间网络模数共识。最后，给出了两个数值算例来验证分析结果。