Feedback kernel approximations and sensor selection for controlled 2D parabolic PDEs using computational geometry methods

This paper presents a computational geometry approach for the approximation of full state feedback controllers by static output feedback for a class of 2D parabolic partial differential equations. First, the controller is designed with the assumption that the full infinite dimensional state is available for measurement and then it is approximated by a static output feedback controller which requires only few sensor measurements to be realized. The problem of sensor selection is solved by using computational geometry methods. This is achieved by using the feedback kernel to decompose the spatial domain in cells with the property that the volume of the feedback kernel over each cell is the same and equal to a fraction of the volume of the kernel over the entire spatial domain. A sensor is allocated in each of these cells and the subsequent gain optimization is achieved by minimizing the difference of the full state feedback and the approximate static output feedback, in a variational sense. Extensive numerical studies are included to highlight the various aspects of the proposed tessellation-based controller approximation and sensor selection.