Variational integrators in linear optimal filtering

Discrete-time estimation and control techniques play a crucial role in digital control architectures. These methods rely on accurate approximations of continuous-time system behavior. For mechanical systems, this includes not only the system state, but also mechanical properties such as symplecticity or the long-term energy behavior. Additionally, we aim to preserve the Hamiltonian structure of optimally controlled or filtered systems. In this contribution, it is discussed how these requirements can be met when replacing the standard discretization schemes by variational integrators. We show that if one chooses a symplectic discretization scheme for a Kalman filtering problem, the discretization inherits the Hamiltonian structure of the continuous-time linear quadratic problem. Numerical experiments with this filter show better results than obtained with standard discretization.