Sensor selection and static output feedback of parabolic PDEs via state feedback kernel partitioning using modification of Voronoi Tessellations

The major contribution of this work is on the reduced complexity of controller design for a class of distributed parameter systems. An optimal full state controller is first designed and the full state feedback operator is assumed to admit a kernel representation (integral representation). With the aid of the feedback kernel, the spatial domain of definition of the distributed parameter system is partitioned into cells with the property that the kernel over each cell has the same area and is equal to the area of the kernel over the entire spatial domain divided by the total number of sensors. This partition is subsequently used to place a single sensor in each cell. To approximate the full state feedback controller by a static output feedback controller, an optimization scheme is subsequently provided to obtain the scalar output feedback gain corresponding to each cell. Such a domain partition has similarities with the computational geometry method of Centroidal Voronoi Tessellations with the additional requirement that each cell centroid must also result in two subcells of equal area of the kernel. Once the sensor locations are obtained, then a least squares scheme, considered in weak form, minimizes the error between the full state controller and the approximate output feedback controller and provides the static output feedback gains. Extensive simulation studies of a diffusion partial differential equation are included to provide an insight on the specifics of this new domain partitioning with the simultaneous sensor placement and static output feedback gain computation.