Event-triggered Boundary Feedback Control Stabilization For Time Fractional Wave System

Caputo fractional derivative is widely used in practical engineering,which makes up for the deficiency of the definition of fractional RiemannLiouville differential.Meanwhile,event-triggered determines whether to trigger the event according to the current system status.Operations can be performed only when the system status meets the triggering conditions,which is more practical.Firstly,we reviews the development and research status of fractional calculus and event-triggered control in mathematics,physics and practice.Secondly,three kinds of fractional calculus definitions and their relations,Mittag-Leffler function with two parameters and event triggering control are introduced in detail.Then,event-triggered boundary feedback control for a time-varying fractional wave equation with unknown time-varying perturbation on the boundary is studied.The main ideas are: An appropriate controller is designed to reject the unknown disturbance,and an invertible integral transformation is adopted to transform the time-varying fractional wave system into a stable one.The state feedback and output feedback event-triggered control stabilization problems are further studied respectively.Before the study of control stabilization,the analytical expression of the unique mild solution of the more general fractional wave equation is calculated.For the first problem,an event-triggered boundary state feedback controller is designed to reject the disturbance,and the event-triggered control rules are given to make the time-varying fractional wave system stable;For the second problem,in the first step,an extended Luenberger observer is designed to evaluate the overall states of the system.In the second step,a boundary output feedback controller based on event triggering and event-triggered control rules are designed to stabilize the system.The stability of the above two problems is demonstrated by Mittag-Leffler function and Riesz basis method.The results show that the time-varying fractional wave system can be stabilized under both controllers of event triggering strategy.Finally,both control problems give positive lower bounds of event-triggered time interval,avoiding the Zeno phenomenon and reducing the waste of resources.