Stability analysis of time-delay system without delay-derivative-dependent information via augmented double integral functional

Formulating a stability criterion of time-delay systems from a positive definite Lyapunov-Krasovskii functional (LKF) is a two-step process, including, firstly, the estimation of the derivative of the functional and, finally, the determination of negative-definiteness condition. With regard to the estimation step, the Bessel-Legendre inequality provides tight estimation results if the state variables in this inequality are utilized by the LKF, which often results in the non-convexity problem of the LKF derivative with respect to the time-varying delay. Therefore, the introduction of double and multiple integral state variables into the LKF is not straightforward for the systems with time-varying delay but unknown delay-derivative-dependent information. In this paper, to avoid the presence of high-degree time-delay-pendent terms in the LKF derivative, the augmented double integral functional, which changes the form of integral state variables, is constructed in the LKF. Then, a less conservative stability criterion without a heavy computational burden is proposed. Finally, an example illustrates the superiority of the result.