In this article, inspired by the Halanay inequality, we study stability of sampled-data systems with packet losses by proposing a nonuniform sampling interval approach. First, a sampled-data controller with an exponential gain is put forward to reduce conservatism. We obtain the sufficient condition for linear sampled-data systems to be exponentially stable by extending the famous Halanay inequality to sampled-data systems. The obtained sufficient conditions indicate that the maximal-allowable bound of sampling intervals is determined by the constant terms in the Halanay inequality, and the decay rate is presented in the form of a Lambert function. Compared with some existing results on the stability of sampled-data systems by using the Gronwall–Bellman Lemma, the conservatism induced by the exponential term via the Gronwall–Bellman Lemma can be reduced to some extent. Considering the phenomenon of packet losses, a new lemma is further proposed to generalize the proposed Halanay-like inequality. The results derived by the new lemma permit that there exist some sampling intervals with the upper bound violating the desired condition of the Halanay-like inequality. This permits us to establish exponential stability in significant cases that do not satisfy the Halanay-like inequality needed in the previous results. Finally, the sampled-data local exponential stability is investigated for nonlinear systems with strong nonlinearity.

中文翻译：

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丢包情况下采样数据系统的稳定性：非均匀采样间隔方法


在本文中，受哈拉奈不等式的启发，我们通过提出一种非均匀采样间隔方法来研究具有丢包情况的采样数据系统的稳定性。首先，提出了具有指数增益的采样数据控制器以减少保守性。通过将著名的哈拉奈不等式推广到采样数据系统，我们得到了线性采样数据系统指数稳定的充分条件。获得的充分条件表明，采样间隔的最大允许界限由Halanay不等式中的常数项确定，并且衰减率以Lambert函数的形式表示。与现有的一些利用Gronwall-Bellman引理对采样数据系统稳定性的结果相比，通过Gronwall-Bellman引理可以在一定程度上减少指数项引起的保守性。考虑到丢包现象，进一步提出了一个新的引理来推广所提出的类哈拉奈不等式。新引理得出的结果允许存在一些采样间隔，其上限违反了类 Halanay 不等式的期望条件。这使我们能够在不满足先前结果中所需的哈拉奈式不等式的重要情况下建立指数稳定性。最后，研究了强非线性非线性系统的采样数据局部指数稳定性。