Order estimation for subspace methods

In this paper the question of estimating the order in the context of subspace methods is addressed. Three di erent approaches are presented and the asymptotic properties thereof derived. Two of these methods are based on the information contained in the estimated singular values, while the third method is based on the estimated innovation variance. The case with observed inputs is treated as well as the case without exogeneous inputs. The two methods based on the singular values are shown to be consistent under fairly mild assumptions, while the same result for the third approach is only obtained on a generic set. The former can be applied to Larimore type of procedures as well as to MOESP type of procedures, whereas the latter is only applied to Larimore type of algorithms. This has implications for the estimation of the order of systems, which are close to the exceptional set, as is shown in a numerical example. All the estimation methods involve the choice of a penalty term. SuÆcient conditions on the penalty term to guarantee consistency are derived. The e ects of di erent choices of the penalty term are investigated in a simulation study.