A binary quadratic function negative‐determination lemma and its application to stability analysis of systems with two additive time‐varying delay components

Abstract

This paper concentrates on the stability problem of systems with two additive time-varying delay components. For the construction of Lyapunov–Krasovskii functional (LKF), in the case that the introduced augmented vector contains the double integral term of the state vector, a special form of the binary quadratic function with respect to two time-varying delays has often been introduced into the derivative of the LKF. In order to determine the negative definiteness of such function, by making full use of the idea of partial differential, the convex/concave property and the slope characteristic of tangent lines of the binary quadratic function, a binary quadratic function negative-determination lemma is proposed in the case of the pluses or minuses of quadratic coefficients are unknown. Then, the obtained stability criterion in the form of linear matrix inequality shows a greater advantage than the previous criteria since not only some advanced techniques are employed to treat some integral terms, but also the proposed lemma is employed to deal with quadratic terms in functional derivative. Finally, a typical example is given to verify the superiority of the derived criterion.