An approach to realization of stochastic systems with exogenous input

This paper solves the stochastic realization problem for a linear discrete-time system with an exogenous input. The oblique projection of the future outputs on the space of the past observations along the space of the future inputs is factorized as a product of the extended observability matrix and the state vector. The state vector is chosen by using the canonical correlation analysis (CCA) of past and future. We then derive the state equations of the optimal predictor of the future outputs in terms of the state vector and the future inputs. These equations lead to a forward innovation model for a stochastic system with exogenous inputs.