In this paper, a new integral inequality lemma along with an appropriate Lyapunov–Krasovskii functional (LKF) is proposed for fuzzy time-delay system to enhance the delay upper bound estimate. The proposed lemma is referred to hereafter as Extended Affine Wirtinger inequality. The novelty of the lemma is twofold, it has the ability to integrate uncertain delay information utilizing the convex combination of certain and uncertain delay intervals involved in the proposed LKF, and it is compatible to derive delay-range-dependent (DRD) stability conditions for continuous time Takagi–Sugeno (T–S) fuzzy time-delay system. One noteworthy advantage of the proposed stability condition in a linear matrix inequality (LMI) framework is that it requires less number of matrix variables compared to the existing integral inequalities of the similar type, thus reducing the computational burden too. The efficacy of the proposed DRD stability condition over existing conditions is validated numerically by solving three examples related to fuzzy time-delay system.

中文翻译：

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使用新的扩展仿射维特林不等式的可变时滞TS模糊系统的改进的时域相关稳定性条件

本文提出了一个新的积分不等式引理，以及一个适当的Lyapunov-Krasovskii泛函（LKF），用于模糊时滞系统，以提高延迟上限估计。所提出的引理在下文中称为扩展仿射维特林不等式。引理的新颖性是双重的，它具有利用所提出的LKF所涉及的确定和不确定延迟间隔的凸组合来集成不确定延迟信息的能力，并且兼容导出依赖于延迟范围的（DRD）稳定性条件。连续时间Takagi–Sugeno（TS）模糊时滞系统。线性矩阵不等式（LMI）框架中拟议的稳定性条件的一个值得注意的优势是，与类似类型的现有积分不等式相比，它需要较少数量的矩阵变量，因此也减轻了计算负担。通过解决与模糊时滞系统有关的三个实例，对所提出的DRD稳定性条件在现有条件下的有效性进行了数值验证。