State Tracking of Linear Ensembles via Optimal Mass Transport

We consider the problems of tracking an ensemble of indistinguishable agents with linear dynamics based only on output measurements. In this setting, the dynamics of the agents can be modeled by distribution flows in the state space and the measurements correspond to distributions in the output space. In this letter, we formulate the corresponding state estimation problem using optimal mass transport theory with prior linear dynamics, and the optimal solution gives an estimate of the state trajectories of the ensemble. For general distributions of systems this can be formulated as a convex optimization problem which is computationally feasible when the number of state dimensions is low. In the case where the marginal distributions are Gaussian, the problem is reformulated as a semidefinite programming problem and can be efficiently solved for tracking systems with a large number of states.