Finite element modeling for heat transfer processes using the method of integro-differential relations with applications in control engineering

Control design of spatially distributed thermal systems is a task that is necessary for a large variety of engineering applications. Early lumping techniques, that are applicable in this case, follow the methodology of first discretizing the infinite-dimensional system model and subsequently using the discretized model for a finite-dimensional control design. For that purpose, the governing partial differential equations are commonly reduced to a finite-dimensional set of ordinary differential equations. In this paper, the corresponding task is solved by an optimization-based version of the Method of Integro-Differential Relations (MIDR). On the one hand, the MIDR allows one to quantify and systematically influence the approximation quality and, on the other hand, to use the resulting models directly for control and state observer design. In the following, the MIDR is applied to a fundamental problem of linear heat transfer in tube-like structures for which cylindrical coordinates are a suitable choice for modeling the dynamic behavior in all three space directions. Illustrative simulation results of fundamental control and state estimation approaches conclude this paper. These simulation results serve as the basis for a future experimental validation of the presented modeling techniques on a laboratory test rig that is currently being built up at the Chair of Mechatronics at the University of Rostock.