Relaxation of second-order matrix-valued multivariate polynomial inequalities: application to the stability analysis of linear systems with two additive time-varying delays

This study presents a relaxation method for second-order matrix-valued multivariate polynomial inequalities(SMMPI), with a specific application to the stability analysis of continuous-time linear systems with additive time-varying delays (ATDs). Initially, a general expression of SMMPI is decomposed into a first-order matrix-valued multivariate polynomial and an augmented multivariate vector. By utilizing the orthogonal complement of the augmented multivariate vector, a sufficient condition for the SMMPI is proposed with a newly introduced free matrix, which highly increases the freedom. Subsequently, the proposed relaxation method is employed for the stability analysis of linear systems with ATDs. Introducing a new Lyapunov-Krasovskii functional, the time derivative is formulated as a second-order matrix-valued multivariate polynomial on the two ATDs. The obtained stability criteria have superiority over existing studies since the degree of freedom is significantly increased due to the new relaxation technique. A practical example is presented to illustrate the effectiveness of the proposed technique.