Oblique Splitting Subspaces and Stochastic Realization with Inputs

Until recently stochastic realization theory has been primarily addressed to modeling of random processes in the absence of exogenous signals (inputs). However stochastic realization of models with exogenous inputs is also of interest. In particular it is of interest for the new class of “ subspace ” type identification algorithms. These algorithms can be formulated as stochastic realization algorithms in an appropriate data Hilbert space [23, 24] (as it is well-known subspace methods have important advantages over the traditional parametric optimization approach to identification). We discuss here procedures for constructing minimal state-space models in presence of inputs, based on a generalization of the idea of Markovian splitting subspace which is central in stochastic realization theory for random processes. In particular we discuss a geometric procedure for constructing the minimal state-space (the predictor space) of a process with inputs which leads to an interesting identification algorithm.