Blind subspace system identification with Riemannian optimization

Abstract

Subspace identification methods provide a reliable set of methods to recover system parameters of linear dynamical systems based on the observation of their inputs and outputs. However, in the common case where one does not have access to the inputs, the identification problem becomes harder, and is referred to as blind system identification. On the other hand, if the inputs can be assumed to lie on a known subspace, identification techniques based on low-rank matrix recovery can be applied. In this case, blind subspace system identification has been formulated as the problem of simultaneously recovering structured low-rank matrices associated with both the system and inputs. Notwithstanding, the convex relaxation approach to this problem, where the objective function is defined as a sum of the nuclear norms of two matrices, has been shown to be significantly sub-optimal as it typically favors one of the objective terms. In this work, we propose a method for the joint identification of system and inputs using optimization over Riemann manifolds. Riemannian optimization defines operators that allow low-rank matrix constraints to be incorporated in the search space, producing feasible solutions by construction. Our approach takes advantage of this capability and formulates blind subsystem identification as a low-rank matrix approximation problem over the product manifold of fixed-rank matrices.