Optimization of spatially distributed systems with spatially local controllers and partial connectivity

This paper presents two different design approaches of networked distributed controllers for a class of partial differential equations. Such a work attempts to reduce controller complexity by opting to control a spatially distributed process by a network of interconnected control units. This approach reduces the need to use a centralized controller that must fuse sensory information by the distributed sensors and then dispense the controllers to the spatially distributed process via the spatially distributed actuators. In the first approach, the controller gains used to implement the local controllers by the controller units, along with the actuator and sensor location and the network connectivity are optimized by formulating the optimization problem into a minimization problem of an appropriately chosen performance measure of the closed-loop system. This takes the form of a parameterized operator Lyapunov equation whose optimal solution provides the optimal (proportional) controller gains, the optimal locations of the actuators and sensors, and the optimal network connectivity. The second approach considers the adaptation of the gains of the distributed controllers using Lyapunov-redesign methods to extract their adaptation. The well-posedness and convergence properties of the proposed controllers are summarized and numerical studies examining various aspects of the proposed work are presented.