Observer-based Periodic Event-triggered Boundary Control of the One-phase Stefan Problem

Abstract

The Stefan problem models the phenomenon of phase transition between a liquid and a solid as the time evolution of the temperature profile of a liquid-solid material and its moving interface. This paper provides a novel observer-based periodic event-triggered boundary control (PETBC) strategy for the one-phase Stefan problem using the position and velocity measurements of the moving interface. We propose a method to convert a specific class of continuous-time dynamic event-triggers that require continuous monitoring to periodic event-triggers that only require periodic evaluation. We achieve this result by finding an upper bound on the underlying continuous-time event-trigger between two successive periodic evaluations. We provide an explicit criterion for choosing a sampling period for periodically evaluating the event-trigger. The control input is updated only at events indicated by the periodic event-trigger and is applied in a zero-order hold fashion between two events. We establish the closed-loop system well-posedness along with certain model validity conditions under the proposed PETBC. Further, we prove that the exponential convergence to the setpoint under continuous-time event-triggered boundary control (CETBC) is preserved under the proposed PETBC. We provide simulation results to validate the theoretical developments.