Optimization-based tuning of low-bandwidth control in spatially distributed systems

We describe a new method for tuning a certain family of low-bandwidth controllers for linear time-invariant and spatially-invariant (LTSI) plants. We consider LTSI controllers with a fixed structure, which is PID in time and local in spatial coordinates. Two spatial feedback filters, assumed to be symmetric and have finite spatial response, modify the local PID control signal based on the error and control signals, respectively, at nearby nodes. 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 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. While the method does not directly handle some important constraints involving the effects of boundary conditions, or 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.