Geometry of oblique splitting subspaces, minimality and Hankel operators To Anders (Professor Noci) as a token of esteem and friendship over many years

Stochastic realization theory provides a natural theoretical background for recent identification methods, called subspace methods, which have shown superior performance for multivariable state-space model-building. The basic steps of subspace algorithms are geometric operations on certain vector spaces generated by observed input-output time series which can be interpreted as “sample versions” of the abstract geometric operations of stochastic realization theory. The construction of the state space of a stochastic process is one such basic operation. In the presence of exogenous inputs the state should be constructed starting from input-output data observed on a finite interval. This and other related questions still seems to be not completely understood, especially in presence of feedback from the output process to the input, a situation frequently encountered in applications. This is the basic motivation for undertaking a first-principle analysis of the stochastic realization problem with inputs, as presented in this paper. It turns out that stochastic realization with inputs is by no means a trivial extension of the well-established theory for stationary processes (time-series) and there are fundamentally new concepts involved, e.g. in the construction of the state space under possible presence of feedback from the output process to the input. All these new concepts lead to a richer theory which (although far from being complete) substantially generalizes and puts what was known for the time series setting in a more general perspective.