Towards implementation of PDE control for Stefan system: Input-to-state stability and sampled-data design

Abstract

The Stefan system is a representative model for a liquid–solid phase change which describes the dynamics of a material’s temperature profile and the liquid–solid interface position. Our previous work designed a boundary feedback control to stabilize the phase interface position modeled by the Stefan system. This paper resolves two issues our previous work did not study, that are, the robustness analysis under the unknown heat loss and the digital control action. First, we introduce the one-phase Stefan problem with a heat loss by modeling a 1-D diffusion Partial Differential Equation (PDE) dynamics of the liquid temperature and the interface position governed by an Ordinary Differential Equation (ODE) with a time-varying disturbance. We focus on the closed-loop system under the control law proposed in our previous work, and show an estimate of L2 norm in a sense of Input-to-State Stability (ISS) with respect to the unknown heat loss. Second, we consider the sampled-data control of the one-phase Stefan problem without the heat loss, by applying Zero-Order-Hold (ZOH) to the control law in our previous work. We prove that the closed-loop system under the sampled-data control law satisfies the global exponential stability in the spatial L2 norm. Analogous ISS result for the two-phase Stefan problem which incorporates the dynamics of the solid phase is also obtained. Numerical simulation verifies our theoretical results for showing the robust performance under the heat loss and the digital control implemented to vary at each sampling time.