Geometric Methods in Stochastic Realization and System Identiication

In this paper we discuss some recent advances in modeling and identiication of stationary processes. We point out that identiication of linear state-space models for stationary signals can be seen as stochastic realization of wide-sense stationary processes in an appropriate background Hilbert space. The geometric theory of stochastic realization developed in the last two decades plays an important role in this interpretation. Identiication of models with exogenous inputs in conditions of absence of feedback can also be formulated as a stochastic realization problem. We discuss procedures for constructing minimal state-space models in presence of inputs, based on a generalization of stochastic realization theory for time series and we discuss geometric procedures for identifying (generically) minimal state-space models with inputs. This approach leads to numerical linear algebraic algorithms which have been named \subspace methods" in the literature. It has important advantages over the traditional parametric optimization approach, since it attacks directly the dynamic model building problem by system theoretic methods and leads to procedures which are more transparent and more structured than those traditionally used and found in the literature.