This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays. By introducing two adjustable parameters and two free variables, a novel convex function greater than or equal to the quadratic function is constructed, regardless of the sign of the coefficient in the quadratic term. The developed lemma can also be degenerated into the existing quadratic function negative-determination (QFND) lemma and relaxed QFND lemma respectively, by setting two adjustable parameters and two free variables as some particular values. Moreover, for a linear system with time-varying delays, a relaxed stability criterion is established via our developed lemma, together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality. As a result, the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems. Finally, the superiority of our results is illustrated through three numerical examples.

中文翻译：

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时滞系统的放宽稳定性准则：一种新颖的二次函数凸逼近方法

本文提出了一种二次函数凸逼近方法来处理时变线性系统稳定性分析引起的二次函数负定问题。通过引入两个可调参数和两个自由变量，构造了一个大于或等于二次函数的新颖凸函数，而不管二次项中系数的符号如何。通过将两个可调参数和两个自由变量设置为一些特定值，所开发的引理还可以分别退化为现有的二次函数否定定理（QFND）引理和松弛QFND引理。此外，对于具有时变延迟的线性系统，通过我们开发的引理以及等效倒数组合技术和贝塞尔-勒让德不等式建立了宽松的稳定性准则。因此，在构建用于线性时变延迟系统稳定性分析的 Lyapunov-Krasovskii 泛函的背景下，可以通过所提出的方法减少保守性。最后，通过三个数值例子说明了我们结果的优越性。