Optimization-based Trajectory Planning for Heat Conduction

Heating and cooling procedures are crucial tasks in several industrial processes, and they have to be steered exactly and energy-efficiently to ensure a high-quality production. Hence, the design of optimal (open-loop) control strategies for heat conduction problems is an active research field. Flatness-based control is a popular open-loop control approach to steer parabolic partial differential equations like the heat equation. The control input signal is computed by a power series including the derivatives of a specified reference signal. However, this method has some limitations regarding, e.g, expensive computation due to a high number of derivatives and a weak capability to handle input constraints or nonlinear expressions in the original equations. In this contribution, we present a procedure to derive a parametrized input signal from flatness-based control for a simplified model. This parametrized input function is adjusted such that it fits to our original complex model. In detail, we compute a prototype input function with flatness-based control for a one-dimensional heat equation and design a parametrized input signal which is based on this prototype. We explain how to find suitable parameters for a well tracking of the measured and mean temperature. Moreover, we show that our proposed input signal is also able to handle input constraints and nonlinear boundary conditions. Finally, we discuss how to decrease the maximum input value while ensuring a suitable open-loop dynamics.