Controle Extremal com Funções Barreira para a Equação Diferencial Parcial de Stefan

Abstract

This paper presents the design and analysis of a safe extremum seeking controller for static maps with input passed through a partial differential equation (PDE) of the diffusion type defined on a time-varying spatial domain whose boundary position is governed by an ordinary differential equation (ODE) – the so-called Stefan model. For this purpose, we design a compensator of the PDE with a moving boundary and the probing signal, which is the result of solving the problem of generating a sinusoid at the distal end of a boundary-actuated diffusion equation while maintaining a measure of control barrier functions (CBFs) positive. In this context, it is possible to guarantee that for all trajectories with safe initial conditions, the states remain on the safety set and attain the extremum. This is the first effort to pursue an extension of extremum seeking from the heat PDE to the Stefan PDE with CBFs. Simulation results are presented to illustrate the effectiveness of the proposed design.