Finite volume and finite element models for real-time control and state estimation of two-dimensional heat transfer processes

Control design for distributed parameter systems usually makes use of either finite volume or finite element approximations of the governing partial differential equations (PDEs). The aim of using both finite volume and finite element models is to obtain a finite-dimensional state-space representation of the dynamics which can be used directly for the design of feedback or feedforward controllers as well as state observers. However, finite volume models only provide a coarse description of flow and storage variables since these are assumed to be piecewise homogeneous in the corresponding volume elements. In contrast, finite element models exploit parameterizable ansatz functions such as polynomials for each element so that smooth representations of the before-mentioned quantities become possible. However, classical finite element techniques do not provide reliable measures for the quantification of the achievable approximation quality. This drawback is removed by the method of integrodifferential relations (MIDR). Simulations and experiments for the observer-based control of a spatially two-dimensional heat transfer process with distributed control inputs are presented in this paper to visualize differences between finite element models relying on the MIDR and finite volume representations.