Model Decomposition and Optimal Flux Control for Linear Distributed Heat Transfer Systems

An optimal control problem for heat transfer in a steel bar heated and cooled by Peltier elements attached to its lower surface is studied. The bar has a cuboid shape with insulated vertical sides, whereas the heat exchange takes place on the body's upper and lower surfaces. A thermodynamic model accounting for heat capacity of cooling units is considered and an experimental identification of unknown structural parameters is carried out. The problem is additionally to find the feedforward control law for the elements' heat powers leading the system in finite time to a stationary state and minimizing a quadratic cost function. Order reduction and decomposition techniques based on the Fourier method are used to optimize the control signals.