LAUSR

Dear editor, Event-triggered control is a control strategy that is not necessarily periodic and takes resource constraints into consideration, such as limited communication bandwidth and restricted energy resources [1]. In event-triggered control, the execution of control tasks is decided by an event, instead of a fixed period of time (known as timetriggered control). The event is generated by a triggering condition involving the state or output variables. Thus, event-triggered control relates the execution of control actions to the operation of systems; moreover, it has the potential to significantly reduce the use of communication resources without excessive loss of control performance [2]. For event-triggered control, it is not only suitable to simply study stability and control performance, but also important to examine sampling behavior. The authors in [1,3] proposed some useful tools to provide a positive lower bound of interevent times. Ref. [3] reported on the sampling performance of several popular types of eventtriggered control systems, both in the absence and presence of external disturbances. The examination of sampling behavior is very important because the event-triggered control system can be formulated as a hybrid system model [4]. In fact, when a positive lower bound of inter-event times cannot be ensured, Zeno behavior [5], a particular phenomenon of hybrid dynamical systems, might occur. Zeno behavior leads to occurrence of an infinite number of discrete transitions (events) within a finite time interval. Hence, Zeno behavior is extremely undesirable in event-triggered control. Thus, the conditions, both necessary and sufficient, for Zeno behavior in event-triggered control systems are very important in the study of control systems. Generally, the necessary conditions for the existence of Zeno behavior can be used to eliminate it in event-triggered control systems, while sufficient conditions could assist the designer to avoid creating an event-triggered control system having Zeno behavior. Therefore, we investigated the conditions related to Zeno behavior in eventtriggered control systems, and the main contributions can be described as follows. First, some necessary and sufficient conditions for Zeno behavior are provided by limiting the triggering condition to a specific form, which can cover several types of triggering conditions in existing studies. This triggering condition is triggered when the difference, between the current and the previously sampled output signals, violates the threshold function that involves the output signals. Thus, the corresponding Zeno equilibrium [5] is created when outputs make the threshold function equal to zero. This property is quite crucial to validate the sufficiency of our results. Second, the innate contradiction between finitetime stability and the considered event-triggered control is demonstrated. For unstable plant, it has been proven that the finite-time stability cannot be achieved by the triggering condition if the threshold function is independent of time. Our results