Event-triggered boundary estimation for 2 × 2 hyperbolic PDEs with disturbance

This paper addresses the event-triggered boundary estimation problem for 2 × 2 linear hyperbolic partial differential equations (PDEs) subject to disturbances. These disturbances enter the PDEs through the boundary and are represented as an ordinary differential equation (ODE). The scenario considered involves transmitting boundary measurements to the observer only when necessary, as determined by a dynamic event-triggering condition. This approach aims to conserve communication and computational resources. The event-triggered observer is designed using the backstepping method, and an event-triggering condition is proposed to determine the time instants when sampled measurements should be transmitted. Under this event-triggered estimation mechanism, a minimal dwell-time between consecutive triggering time instants is guaranteed. Furthermore, the well-posedness of the solutions and the exponential convergence of the estimation error are ensured. Finally, the obtained results are applied to the Saint-Venant equations to demonstrate the effectiveness of the event-triggered estimation method.