The role of vector autoregressive modeling in predictor-based subspace identification

Subspace identification for closed loop systems has been recently studied by several authors. A class of new and consistent closed-loop subspace algorithms is based on identification of a predictor model, in a way similar as prediction error methods (PEM) do. Experimental evidence suggests that these methods have a behavior which is very close to PEM in certain examples. The asymptotical statistical properties of one of these methods have been studied recently allowing to show (i) its relation with CCA and (ii) that Cramér Rao lower bound is not reached in general. Very little however is known concerning their relative performance. In this paper we shall discuss the link between these “predictor based” methods; to this purpose we exploit the role which Vector AutoRegressive with eXogenous inputs models play in all these algorithms. The results of this paper provide a unifying framework under which all these algorithms can be viewed; also the link with VARX modeling have important implications as to computational complexity is concerned, leading to very computationally attractive implementations. We also hope that this framework, and in particular the relation with VARX modeling followed by model reduction will turn out to be useful in future developments of subspace identification, such as the quest for efficient procedures and the statistical analysis with finite-data.