Controlling 2D PDEs using mobile collocated actuators-sensors and their simultaneous guidance constrained over path-dependent reachability regions

Employing mobile actuators and sensors for control and estimation of spatially distributed processes offers a significant advantage over immobile actuators and sensors. In addition to the control performance improvement, one also comes across the economic advantages since fewer devices, if allowed to be repositioned within a spatial domain, must be employed. While simulation studies of mobile actuators report superb controller performance, they are far from reality as the mechanical constraints of the mobile platforms carrying actuators and sensors have to satisfy motional constraints. Terrain platforms cannot behave as point masses without inertia; instead they must satisfy constraints which are adequately represented as path-dependent reachability sets. When the control algorithm commands a mobile platform to reposition itself in a different spatial location within the spatial domain, this does not occur instantaneously and for the most part the motion is not omnidirectional. This constraint is combined with a computationally feasible and suboptimal control policy with mobile actuators to arrive at a numerically viable control and guidance scheme. The feasible control decision comes from a continuous-discrete control policy whereby the mobile platform carrying the actuator is repositioned at discrete times and dwells in a specific position for a certain time interval. Moving to a subsequent spatial location and computing its associated path over a physics-imposed time interval, a set of candidate positions and paths is derived using a path-dependent reachability set. Embedded into the path-dependent reachability sets that dictate the mobile actuator repositioning, a scheme is proposed to integrate collocated sensing measurements in order to minimize costly state estimation schemes. The proposed scheme is demonstrated with a 2D PDE having two sets of collocated actuator-sensor pairs onboard mobile platforms.