Design of Low-Bandwidth Spatially Distributed Feedback

The paper considers a family of linear time-invariant and spatially invariant (LTSI) systems that are both distributed and localized. The spatial responses of the distributed plant are localized in spatial neighborhoods of each location. The feedback computations are also distributed and the information flow is localized in a spatial neighborhood of each location. The feedback is aimed at controlling spatial distributions of variables in the systems with a relatively low bandwidth in the time direction. Such systems have many important applications including industrial processes, imaging systems, signal and image processing, and others. We describe a new method for designing (tuning) a certain family of low-bandwidth controllers for such plants. We consider LTSI controllers with a fixed structure, which is a PID or a similar low-bandwidth feedback in time and local in spatial coordinates. Two spatial feedback filters, symmetric and with finite spatial response, modify the local PID control signal by mixing in the error and control signals at nearby nodes. These two filters provide loop- shaping and regularization of the spatial feedback loop. Like an ordinary PID controller, this controller structure is simple, but provides adequate performance in many practical settings. We cast a variety of specifications on the steady-state spatial response of the controller and its time response as a set of linear inequalities on the design variables, and so can carry out the design of the spatial filters using linear programming. The method handles steady-state limits on actuator signals, error signals, and several constraints related to robustness to plant and controller variation. The method allows handling the effects of boundary conditions and guaranteed closed-loop spatial or time decay. It does appear to work very well for low-bandwidth controllers, and so is applicable in a variety of practical situations.