Robust stability and H∞$$ {H}_{\infty } $$ performance evaluation for time‐varying‐delayed uncertain linear systems based on variable‐augmented‐based free‐weighting matrices

Abstract

The problem of the robust stability and performance evaluation of time‐varying‐delayed uncertain linear systems (TVDULSs) is studied in this article. At first, to obtain a modified robust delay‐dependent condition of TVDULSs, a novel augmented Lyapunov‐Krasovskii functional (LKF) is established, and a variable‐augmented‐based free‐weighting matrices (VAFWMs) technique is tailored to evaluate the functional derivative. Compared with the existing robust stability methods, the constructed functional takes fully time delay information of the system into consideration and provides more delay‐cross terms, where the corresponding matrices are used to relax the constraints of linear matrix inequalities in the stability condition. Also, the VAFWMs method fully considers the linear relation of some delayed integral terms in the functional derivative, which avoids the usage of a quadratic order or higher order terms of the time delay. Then, an improved performance analysis criterion of TVDULSs is obtained against the disturbance on the basis of the proposed robust stability condition. The admissible maximal delay upper bounds that the uncertain linear system can tolerate are provided by proposed criterion under a given performance level. At last, two numerical examples are presented to verify the effectiveness and the superiority of the methods obtained.