Realization of Stochastic Systems with Application to Subspace Identification Method

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This study is concerned with the realization algorithms for the stochastic system subjected to exogenous input by extending the stochastic realization algorithm for time-series due to Akaike (1974).1. Under the assumption that there is no feedback from the output to the input, we have derived and characterized the family of minimal state-space models of the stationary process with exogenous inputs. We have introduced a very natural block structure which is generically minimal. This model structure leads to subspace-based identification algo-rithms which have a simpler structure of those existing in the literarture. These results are published in Signal Processing (1996) and in the Preprints of 13th IFAC World Congress (1996).2. We have solved the stochastic realization problem for a linear discrete-time system with an exogenous input. 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 the 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.