[Paper Review] Sampled-Data Control of the Stefan System

Shumon Koga, Iasson Karafyllis+1 more|arXiv (Cornell University)|May 31, 2019

Stability and Controllability of Differential Equations43 references4 citations

TL;DR

This paper presents the first sampled-data boundary feedback control design for the Stefan problem using Zero-Order-Hold (ZOH) to implement continuous-time control laws in a digital setting. It proves global exponential stability in the L2 norm and model validity under conditions linking control gain and sampling period, validated via numerical simulations for one-phase and two-phase Stefan systems.

ABSTRACT

This paper presents results for the sampled-data boundary feedback control to the Stefan problem. The Stefan problem represents a liquid-solid phase change phenomenon which describes the time evolution of a material's temperature profile and the interface position. First, we consider the sampled-data control for the one-phase Stefan problem by assuming that the solid phase temperature is maintained at the equilibrium melting temperature. We apply Zero-Order-Hold (ZOH) to the nominal continuous-time control law developed in [23] which is designed to drive the liquid-solid interface position to a desired setpoint. Provided that the control gain is bounded by the inverse of the upper diameter of the sampling schedule, we prove that the closed-loop system under the sampled-data control law satisfies some conditions required to validate the physical model, and the system's origin is globally exponentially stable in the spatial $L_2$ norm. Analogous results for the two-phase Stefan problem which incorporates the dynamics of both liquid and solid phases with moving interface position are obtained by applying the proposed procedure to the nominal control law for the two-phase problem developed in [30]. Numerical simulation illustrates the desired performance of the control law implemented to vary at each sampling time and keep constant during the period.

Motivation & Objective

• To address the lack of sampled-data control designs for the Stefan problem, a class of nonlinear PDEs with state-dependent moving boundaries.
• To extend continuous-time boundary feedback control laws to practical digital implementation using Zero-Order-Hold (ZOH).
• To establish sufficient conditions on control gain and sampling period ensuring closed-loop stability and physical model validity.
• To validate the theoretical results through numerical simulations of one-phase and two-phase Stefan systems.
• To lay the foundation for future observer-based output feedback and quantized control extensions.

Proposed method

• Application of Zero-Order-Hold (ZOH) to continuous-time boundary feedback control laws derived via backstepping for the one-phase and two-phase Stefan problems.
• Use of Lyapunov-based analysis to prove global exponential stability in the L2 norm for the closed-loop system.
• Derivation of conditions linking control gain and sampling period to ensure physical model validity, such as boundary temperature above melting point.
• Employment of the boundary immobilization method combined with finite difference semi-discretization for numerical simulation.
• Formulation of energy-like Lyapunov functions to estimate system norms and establish stability bounds.
• Incorporation of the moving interface dynamics via an ODE coupled with the parabolic PDE, ensuring consistency with thermodynamic constraints.

Experimental results

Research questions

• RQ1 Can ZOH-based sampled-data control stabilize the one-phase Stefan problem with moving boundary and maintain physical model validity?
• RQ2 What conditions on control gain and sampling period ensure global exponential stability in the L2 norm for the sampled-data closed-loop system?
• RQ3 How does the proposed sampled-data control perform in terms of interface convergence, control input behavior, and boundary temperature dynamics?
• RQ4 Can the theoretical results for the one-phase problem be extended to the two-phase Stefan problem with both liquid and solid phase dynamics?
• RQ5 What are the implications of ZOH implementation on the positivity of control input and boundary temperature in the presence of discontinuities at sampling times?

Key findings

• The closed-loop system remains globally exponentially stable in the L2 norm when the control gain is bounded by the inverse of the maximum sampling period.
• The interface position converges monotonically and smoothly to the setpoint without overshoot, with the control input remaining positive and piecewise constant between sampling times.
• The boundary temperature remains above the melting temperature at all times, with transient spikes at sampling instants due to abrupt changes in control input.
• Numerical simulations confirm the theoretical stability and model validity conditions, showing consistent performance over 2 hours of simulation.
• The stability result holds for both one-phase and two-phase Stefan problems, with the same ZOH-based control law applied to the nominal continuous-time control design.
• The analysis ensures that the physical model remains valid, including the requirement that the liquid phase temperature exceeds the melting point, which is maintained via control input positivity.