Frequency delay‐dependent stability criterion for time‐delay systems thanks to Fourier–Legendre remainders

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 byℋ ∞ 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.