A note on relationship between some bounding inequalities in stability analysis of time-delay systems

In this paper, an extension of the generalized free matrix based inequality is introduced in a unified form suitable for the estimation of integrals and sums of quadratic functions. The equivalences of several known variants are shown, including the free matrix based inequalities and its simplified form. Secondly, the relationship between the (simplified) free matrix based estimation and the combination of the Bessel-based inequality with different bounding inequalities affine in the length of the intervals are investigated.