Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design

This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a liquid-solid material and its moving interface. This physical process is mathematically formulated as a diffusion partial differential equation (PDE) evolving on a time-varying spatial domain described by an ordinary differential equation (ODE). The state-dependency of the moving interface makes the coupled PDE-ODE system a nonlinear and challenging problem. We propose a full-state feedback control law, an observer design, and the associated output-feedback control law via the backstepping method. The designed observer allows estimation of the temperature profile based on the available measurement of solid phase length. The associated output-feedback controller ensures the global exponential stability of the estimation errors, the H1- norm of the distributed temperature, and the moving interface to the desired setpoint under some explicitly given restrictions on the setpoint and observer gain. The exponential stability results are established considering Neumann and Dirichlet boundary actuations.Comment: 16 pages, 11 figures, submitted to IEEE Transactions on Automatic Contro

Delay-Adaptive Boundary Control of Coupled Hyperbolic PDE-ODE Cascade Systems

This paper presents a delay-adaptive boundary control scheme for a $2\times 2$ coupled linear hyperbolic PDE-ODE cascade system with an unknown and arbitrarily long input delay. To construct a nominal delay-compensated control law, assuming a known input delay, a three-step backstepping design is used. Based on the certainty equivalence principle, the nominal control action is fed with the estimate of the unknown delay, which is generated from a batch least-squares identifier that is updated by an event-triggering mechanism that evaluates the growth of the norm of the system states. As a result of the closed-loop system, the actuator and plant states can be regulated exponentially while avoiding Zeno occurrences. A finite-time exact identification of the unknown delay is also achieved except for the case that all initial states of the plant are zero. As far as we know, this is the first delay-adaptive control result for systems governed by heterodirectional hyperbolic PDEs. The effectiveness of the proposed design is demonstrated in the control application of a deep-sea construction vessel with cable-payload oscillations and subject to input delay