Propagation of initial condition uncertainty for linear dynamical systems: beyond the Gaussian assumption

Propagation of uncertainty in the initial conditions of a dynamical system is necessary in various control applications. While several generally applicable methods based on Monte Carlo simulation and surrogate modeling exist for this task, they can be computationally intensive or difficult to set up for complex initial distributions. Here, we propose an approach for studying the propagation of initial condition uncertainty, tailored to linear dynamical systems. Our approach uses a class of maximum entropy probability distributions to track the state probability density in time. For deterministic linear systems with initial state distributions belonging to this distribution class, our method results in set of ODEs that allow the exact calculation of the state distribution at any time point in time, generalizing the results known for Gaussian initial distributions. For systems perturbed by noise, we show that the state distribution can be efficiently approximated in a maximum entropy sense via the moment equations. Our results provide a powerful computational alternative to commonly used uncertainty propagation methods, and can be exploited in the construction of filtering and control methods for linear systems with uncertain initial conditions.