In the last five years, the slack matrices-based function has been widely investigated by constructing appropriate polynomials since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities (LMIs). However, the inherently complexitiy of slack matrices has not been considered until now, which is an open problem. In this paper, novel flexible polynomial-based functions (FPFs) are proposed by constructing flexible polynomials. Not only the inherently complex dimensions of the slack matrices but also higher-order time delays and zero components in the existing works are relaxed. Benefitting from FPFs, two representative stability criteria are, respectively, obtained for linear delay systems. According to the two stability conditions, all permutations and combinations of flexible vectors in FPFs can be freely chosen. An example is illustrated to demonstrate the effectiveness of the obtained results.

在过去的五年中，基于松弛矩阵的函数通过构造适当的多项式得到了广泛的研究，因为它是推导以线性矩阵不等式 (LMI) 表示的易处理稳定性条件的合适工具。然而，松弛矩阵固有的复杂性直到现在才被考虑，这是一个悬而未决的问题。在本文中，通过构造灵活多项式提出了新颖的基于灵活多项式的函数（FPF）。不仅松弛矩阵固有的复杂维度，而且现有工作中的高阶时间延迟和零分量都得到了放松。受益于 FPF，分别获得了线性延迟系统的两个代表性稳定性标准。根据两个稳定条件，可以自由选择 FPF 中灵活向量的所有排列和组合。举例说明所得结果的有效性。