Notes on Stability of Time-Delay Systems: Bounding Inequalities and Augmented Lyapunov-Krasovskii Functionals
Notes on Stability of Time-Delay Systems: Bounding Inequalities and Augmented Lyapunov-Krasovskii Functionals
复制标题
DOI：
10.1109/tac.2016.2635381
复制
发表时间：
2017-10
影响因子：
6.8
通讯作者：
Chuan‐Ke Zhang;Yong He;Lin Jiang;Min Wu
中科院分区：
文献类型：
--
作者：
Chuan‐Ke Zhang;Yong He;Lin Jiang;Min Wu
边界不等式和Lyapunov-Krasovskii泛函（LKFs）对于时滞系统的稳定性分析具有重要意义。为了改进稳定性标准，人们对发展更紧密的不等式给予了很大的注意，而在讨论不平等的紧密性和标准的保守性之间的关系时，却没有考虑到LKFs的贡献。本照会与这个问题有关。首先，证明了当应用一个简单LKF时，基于wirtinger的不等式与Jensen不等式得到的稳定性判据是等价的，尽管基于wirtinger的不等式更为严格。这意味着更紧密的不平等并不总是导致更保守的标准。其次，为了实现基于wirtinger不等式的优点，需要一个适当的增广LKF，其导数中包含必要的积分向量。在此基础上，在LKF中引入两个时滞积型项，建立了新的稳定性判据。最后，给出了一个数值算例，验证了等效性陈述和所提出准则的有效性。
The bounding inequalities and the Lyapunov-Krasovskii functionals (LKFs) are important for the stability analysis of time-delay systems. Much attention has been paid to develop tighter inequalities for improving stability criteria, while the contribution of the LKFs has not been considered when discussing the relationship between the tightness of inequalities and the conservatism of criteria. This note is concerned with this issue. Firstly, it is proved that, when a simple LKF is applied, the stability criteria obtained by the Wirtinger-based inequality and the Jensen inequality are equivalent although the Wirtinger-based inequality is tighter. It means that the tighter inequality does not always lead to a less conservative criterion. Secondly, it is found that a suitable augmented LKF with necessary integral vectors in its derivative is required to achieve the advantage of the Wirtinger-based inequality. Based on this observation, two delay-product-type terms are introduced into the LKF to establish new stability criteria. Finally, a numerical example is given to verify the equivalence statements and to show the benefit of the proposed criteria.