[Paper Review] Observer-based Event-triggered Boundary Control of the One-phase Stefan Problem

Bhathiya Rathnayake, Mamadou Diagne|arXiv (Cornell University)|Oct 2, 2022

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper proposes an observer-based event-triggered boundary control strategy for the one-phase Stefan problem using infinite-dimensional backstepping design. By employing a dynamic event-triggering condition, the control input is updated only when necessary, ensuring uniform dwell-time and excluding Zeno behavior, while achieving global exponential stability and convergence to the setpoint via Lyapunov analysis.

ABSTRACT

This paper provides an observer-based event-triggered boundary control strategy for the one-phase Stefan problem using the position and velocity measurements of the moving interface. The infinite-dimensional backstepping approach is used to design the underlying observer and controller. For the event-triggered implementation of the continuous-time observer-based controller, a dynamic event triggering condition is proposed. The triggering condition determines the times at which the control input needs to be updated. In between events, the control input is applied in a extit{Zero-Order-Hold} fashion. It is shown that the dwell-time between two triggering instances is uniformly bounded below excluding extit{Zeno behavior}. Under the proposed event-triggered boundary control approach, the well-posedness of the closed-loop system along with certain model validity conditions is provided. Further, using Lyapunov approach, the global exponential convergence of the closed-loop system to the setpoint is proved. A simulation example is provided to illustrate the theoretical results.

Motivation & Objective

• To address the challenge of stabilizing the one-phase Stefan problem with moving boundaries using boundary control.
• To design an output-feedback control strategy using only position and velocity measurements of the moving interface.
• To reduce control updates through event-triggering while maintaining system stability and avoiding Zeno behavior.
• To ensure well-posedness and exponential convergence of the closed-loop system under model validity conditions.

Proposed method

• The infinite-dimensional backstepping method is used to design both the observer and the controller for the PDE-ODE cascade system.
• A dynamic event-triggering condition is proposed to determine when control updates are needed, based on the current state of the closed-loop system.
• Between events, the control input is held constant using a Zero-Order-Hold (ZOH) implementation.
• The observer estimates the full state using only boundary measurements of the moving interface.
• Lyapunov-based analysis is employed to prove global exponential stability and convergence to the setpoint.
• Model validity conditions are enforced to ensure physical consistency of the system.

Experimental results

Research questions

• RQ1 Can an event-triggered control strategy be designed for the one-phase Stefan problem that reduces control updates while maintaining stability?
• RQ2 How can observer-based output feedback be implemented for a PDE-ODE system with moving boundaries?
• RQ3 Does the proposed dynamic event-triggering condition prevent Zeno behavior and ensure a uniform dwell-time between updates?
• RQ4 Can global exponential stability of the closed-loop system be guaranteed under the proposed event-triggered observer-based control?

Key findings

• The proposed event-triggered control strategy ensures a uniformly bounded minimal dwell-time between control updates, excluding Zeno behavior.
• The closed-loop system is well-posed and satisfies all required model validity conditions.
• Global exponential convergence of the system to the setpoint is proven using a Lyapunov function, with decay rate governed by the parameters in the triggering condition.
• Simulation results confirm monotonic convergence of the interface position to the setpoint without overshoot, and rapid decay of temperature and state estimation errors.
• The control input updates are significantly reduced compared to continuous or periodic sampled-data control, as shown in Fig. 3.
• The system achieves exponential stability with the chosen parameters, as evidenced by the decay of $\|T - T_m\|$, $\|T - \hat{T}\|$, and $|X(t)|$ to zero over time.