A sufficient negative‐definiteness condition for cubic functions and application to time‐delay systems

Abstract

AbstractThis article studies the delay‐dependent stability of systems with a time‐varying delay. To get a stability criterion described as linear matrix inequalities (LMIs) from a positive definite Lyapunov‐Krasovskii functional (LKF), the negative‐definiteness condition, guaranteeing the negative definiteness of the derivative of the LKF, is necessary. This article proposes a negative‐definiteness lemma for the cubic function with respect to the time‐varying delay, which achieves the negative‐definiteness requirement without introducing any decision variable and extending the size of the LMIs contained in a stability criterion. Then benefiting from this lemma, an LKF with more delay information is constructed and applied to the stability analysis. After that, a delay‐dependent stability criterion is derived based on the LKF and the negative‐definiteness lemma. Finally, the contribution of the proposed lemma and the stability criterion is demonstrated with two examples.