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Volume 11 Issue 4
Apr.  2024

IEEE/CAA Journal of Automatica Sinica

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S. Wang, W. Ji, Y. Jiang, Y. Zhu, and  J. Sun,  “Relaxed stability criteria for time-delay systems: A novel quadratic function convex approximation approach,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 4, pp. 996–1006, Apr. 2024. doi: 10.1109/JAS.2023.123735
Citation: S. Wang, W. Ji, Y. Jiang, Y. Zhu, and  J. Sun,  “Relaxed stability criteria for time-delay systems: A novel quadratic function convex approximation approach,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 4, pp. 996–1006, Apr. 2024. doi: 10.1109/JAS.2023.123735

Relaxed Stability Criteria for Time-Delay Systems: A Novel Quadratic Function Convex Approximation Approach

doi: 10.1109/JAS.2023.123735
Funds:  This work was supported in part by the National Natural Science Foundation of China (62273058, U22A2045), the Key Science and Technology Projects of Jilin Province (20200401075GX), and the Youth Science and Technology Innovation and Entrepreneurship Outstanding Talents Project of Jilin Province (20230508043RC)
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  • This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays. By introducing two adjustable parameters and two free variables, a novel convex function greater than or equal to the quadratic function is constructed, regardless of the sign of the coefficient in the quadratic term. The developed lemma can also be degenerated into the existing quadratic function negative-determination (QFND) lemma and relaxed QFND lemma respectively, by setting two adjustable parameters and two free variables as some particular values. Moreover, for a linear system with time-varying delays, a relaxed stability criterion is established via our developed lemma, together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality. As a result, the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems. Finally, the superiority of our results is illustrated through three numerical examples.

     

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    Highlights

    • A novel quadratic convex function is constructed to approximate the quadratic function induced by LKF derivative, regardless of the sign of the coefficient in the quadratic term. As a result, the quadratic function convex approximation lemma is innovatively put forward, and other redundant conditions cannot be produced. When the two adjustable parameters and two free variables are set as particular values, our developed lemma can also be degenerated into the existing quadratic function negative determination (QFND) lemma and relaxed QFND lemma, respectively
    • To estimate the derivative of the constructed LKF, equivalent reciprocal convex combination (ERCC) technique in our previous work combining the B-L inequality is utilized to obtain the tighter lower bound. ERCC technique is able to solve the reciprocal convex combination problem equivalently and directly without the Schur complement and also the RCC condition is removed, which always exists in other results
    • The relaxed stability conditions for linear systems with time-varying delays can be obtained by combining our developed quadratic function convex approximation lemma, ERCC technique and the B-L inequality. The conservatism of stability conditions is reduced fundamentally, which is more general than the existing results. B-L inequality. The conservatism of stability conditions is reduced fundamentally, which is more general than the existing results

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