Balanced model reduction of partially-observed Langevin equations: an averaging principle

We study balanced model reduction of partially-observed stochastic differential equations of Langevin type. Upon balancing, the Langevin equation turns into a singularly perturbed system of equations with slow and fast degrees of freedom. We prove that in the limit of vanishing small Hankel singular values (i.e., for infinite scale separation between fast and slow variables), its solution converges to the solution of a reduced-order Langevin equation. The approach is illustrated with several numerical examples, and we discuss the relation to model reduction of deterministic control systems having an underlying Hamiltonian structure.