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

Introduction: Localized Structures in Dissipative Media: From Optics to Plant Ecology

Localised structures in dissipative appears in various fields of natural science such as biology, chemistry, plant ecology, optics and laser physics. The proposed theme issue is to gather specialists from various fields of non-linear science toward a cross-fertilisation among active areas of research. This is a cross-disciplinary area of research dominated by the nonlinear optics due to potential applications for all-optical control of light, optical storage, and information processing. This theme issue contains contributions from 18 active groups involved in localized structures field and have all made significant contributions in recent years.Comment: 14 pages, 0 figure, submitted to Phi. Trasaction Royal Societ

Anticollocated backstepping observer design for a class of coupled reaction-diffusion PDEs

The state observation problem is tackled for a system ofncoupled reaction-diffusion PDEs, possessing the same diffusivity parameter and equipped with boundary sensing devices. Particularly, a backstepping-based observer is designed and the exponential stability of the error system is proven with an arbitrarily fast convergence rate. The transformation kernel matrix is derived in the explicit form by using the method of successive approximations, thereby yielding the observer gains in the explicit form, too. Simulation results support the effectiveness of the suggested design

Modeling Thermal Turbulence Using Implicit Large Eddy Simulation

A general description of a thermally coupled fluid flow is given by the incompressible Navier-Stokes equations coupled with the heat equation using Boussinesq approximation, whose mathematical structure is much well understood. A variational multiscale finite element approximation has been considered for the formulation of incompressible Navier-Stokes equation and heat equation. The complexity of these problems makes their numerical solution very difficult as the standard finite element method is unstable. In the incompressible Navier Stokes equations, two well known sources of numerical instabilities are the incompressibility constraint and the presence of the convective term. Many stabilization techniques used nowadays are based on scale separation, splitting the unknown into a coarse part induced by the discretization of the domain and a fine subgrid part. The modeling of the subgrid scale and its influence leads to a modified coarse scale problem providing stability. In convection-diffusion problem once global instabilities have been overcome by a stabilization method, there are still local oscillations near layers due to the lack of monotonicity of the method. Shock capturing techniques are often employed to deal with them. Proper choice of stabilization and shock capturing techniques can eliminate the local instabilities near layers of convection-diffusion equation. A very important issue of the formulation presented in this thermally coupled incompressible flow is the possibility to model turbulent flows. Some terms involving the velocity subgrid scale arise from the convective term in the Navier-Stokes equations which can be understood as the contribution from the Reynolds tensor of a LES approach and the contribution from the cross stress tensor. This opens the door of modeling thermal turbulence using LES automatically inherited by the formulation used in this work. Different classical benchmark problems are numerically solved in this thesis work for the convection-diffusion equation to show the capabilities of different combination of stabilization and shock capturing methods. In the case of thermally coupled incompressible flows some numerical and industrial examples are exhibited to check the performance of the different combination of stabilization and shock capturing methods and to compare them. The objective is to conclude which method works better to approximate the exact solution and eliminate instabilities and local oscillations