Stability of sampled-data systems with packet losses : a nonuniform sampling interval approach

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.