Employing mobile sensor density to approximate state feedback kernels in static output feedback control of PDEs

This work considers the replacement of a full-state feedback controller by a static output feedback controller employing a finite number of point sensors. This is achieved by the approximation of the feedback kernel associated with the full state feedback operator. The feedback kernel is partitioned into equiareal cells and an appropriately selected centroid within each cell serves as the sensor location. This allows one to approximate the inner product of the feedback kernel and the full state by the finite weighted sum of static output feedback measurements. By equating the feedback kernel with the density of a hypothetical sensor network, the problem of approximating the sensor density becomes that of partitioning the sensor density using the proposed computational-geometry based decomposition that is based on a modification of Centroidal Voronoi Tessellations. When the control is considered over a finite horizon and/or the actuator itself is repositioned within the spatial domain, the resulting feedback kernel is rendered time-varying. This requires its partitioning at each time leading to mobile sensors within the spatial domain. Two guidance policies are proposed: one uses the partitioning of the kernel method at each time to find the optimal sensors thus resulting in moving sensors. The other method uses the kernel partitioning only at the initial time and subsequently uses the sensor density as the initial condition for an advection PDE that represents the evolution of the sensor density. This advection PDE is solved for the velocity thereby providing the velocity of the density of the sensor network. Projecting the sensor density velocity onto the same partitioning used for the kernel provides the sensor velocities. A numerical example of an advection diffusion PDE is presented to provide an understanding of this computational geometry based partitioning of feedback kernels.