Department of Applied Mathematics & Physics, Kyoto Univiversity

Technical Report #99006 (April, 1999)

Realization of Stochastic Systems with Exogenous Inputs and Subspace Identification Methods
by Tohru Katayama and Giorgio Picci

This paper solves the stochastic realization problem for a
discrete-time stationary process 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 conditioned on the future inputs. We then derive the state
equations of the optimal predictor of the future outputs in terms of
the state vector and future inputs. These equations lead to a forward
innovation model for the output process in the presence of exogenous
inputs. The basic step of the realization procedure is a
factorization of the conditional covariance matrix of future outputs
and past data given future inputs. This factorization is based on CCA
and can be easily adapted to finite input-output data. We derive four
stochastic subspace identification algorithms which adapt the
realization procedure to finite input-output data. Numerical results
are also included.