An Affine Integral Inequality of an Arbitrary Degree for Stability Analysis of Linear Systems With Time-Varying Delays

This paper is concerned with the stability analysis problems of linear systems with time-varying delays using integral inequalities. To reduce the conservatism of stability criteria obtained with Lyapunov-Kraosvksii approach, there has been a growing tendency to utilize various integral quadratic functions in the construction of Lyapunov-Krasovskii functionals. Consequently, integral inequalities also have played key roles to derive stability criteria guaranteeing the negativity of the Lyapunov-Krasovskii functional's derivative. Recently, by utilizing first-degree or second-degree orthogonal polynomials, new free-matrix-based integral inequalities have been proposed for integral quadratic functions containing both a system state variable and its time derivative. This paper tries to generalize these inequalities and their stability criteria with a note on the relation among the existing integral inequalities and the proposed one. In this note, it is shown that increasing a degree of the proposed integral inequality only reduces or maintains the conservatism by deriving the hierarchical stability criteria. Four numerical examples including practical systems demonstrate the effectiveness of the proposed methods in terms of allowable upper delay bounds.