This paper investigates the stability of a linear finite-dimensional system interconnected to a single delay operator. From robust approaches, to derive delay-dependent frequency tests, a characterization of the delay behavior is required. Based on approximation methods, one describes the transported signal by lumped parameters. More precisely, by the use of the first Fourier–Legendre polynomials coefficients, we split the delay block into a finite-dimensional part interconnected to a specific infinite-dimensional residual part. Two models are investigated with residuals related to two Fourier–Legendre remainders of the delayed transfer function. The main contribution is to highlight that the finite-dimensional models based on the first Legendre coefficients are proven to be related to Padé approximations and are recognized to be more and more accurate as the dimension increases. Interestingly, this modeling allows computing in an accurate manner the root locus of time-delay systems. Furthermore, as a by-product of this result, taking into account the infinite-dimensional remainders to keep track of the initial time-delay system, stability criteria are proposed by  analysis. Considering both infinite-dimensional remainders as bounded delay-free uncertainties, the small-gain theorem provides a new sufficient condition of stability for retarded time-delay systems, which can be implemented as a delay-dependent frequency-sweeping test. Our results are illustrated on several academic examples.

中文翻译：

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基于傅立叶-勒让德余数的时延系统的频率时延相关稳定性判据

本文研究了与单个延迟算子互连的线性有限维系统的稳定性。从稳健的方法中，为了得出延迟相关的频率测试，需要对延迟行为进行表征。基于近似方法，一种通过集总参数来描述传输的信号。更准确地说，通过使用第一个傅里叶-勒让德多项式系数，我们将延迟块分成与特定无限维残差部分互连的有限维部分。使用与延迟传递函数的两个傅立叶-勒让德余数相关的残差研究了两个模型。主要贡献是强调基于第一个勒让德系数的有限维模型被证明与 Padé 近似相关，并且随着维数的增加被认为越来越准确。有趣的是，这种建模允许以准确的方式计算时延系统的根轨迹。此外，作为这个结果的副产品，考虑到无限维余数来跟踪初始时滞系统，稳定性标准被提出： 分析。考虑到无限维余数作为有界无延迟不确定性，小增益定理为延迟时间延迟系统提供了一个新的稳定的充分条件，它可以作为延迟相关的扫频测试来实现。我们的结果在几个学术例子中得到了说明。