Sensor motion for optimal estimation in distributed dynamic environments

We propose a methodology of optimal path design for sensors through a distributed environment represented by a two or three-dimensional field quantity. We consider time-limited scenarios where the sensors can only make a small number of measurements, but where some portion of a physics-based model is available for the field of interest such as fluid flows, temperatures or concentrations. Thus the highly underdetermined inverse problem can be augmented with dynamical models. We consider stochastic settings where the fields are subject to partially unknown disturbances and boundary conditions. The main idea is to recast the sensor path planning problem as a deterministic optimal control problem to minimize metrics related to the optimal estimation error covariance, thus converting the stochastic estimation problem to a deterministic operator-valued problem. In the specific case of linear field dynamics, the signal to be designed is the sensors' paths which are inputs to the optimal error covariance Riccati equation, resulting in a deterministic, nonlinear, optimal control problem where the trace of the error covariance operator is to be minimized. For sensing modalities, we consider point-wise sampling as well as the more unusual case of line-integral measurements. The latter is motivated by tomographic reconstruction scenarios with a small number of sensors.