Asymptotic variance of closed-loop subspace identification methods

In this paper, the asymptotic properties of a version of the "innovation estimation" algorithm by Qin and Ljung as well as of a version of the "whitening filter" based algorithm introduced by Jansson are studied. Expressions for the asymptotic error as the sum of a "bias" term plus a "variance" term are given. The analysis is performed under rather mild assumptions on the spectrum of the joint input-output process; however, in order to avoid unnecessary complications, the asymptotic variance formulas are computed explicitly only for finite memory systems, i.e., of the ARX type. This assumption could be removed at the price of some technical complications; the simulation results confirm that when the past horizon is large enough (as compared to the predictor dynamics) the asymptotic expressions provide a good approximation of the asymptotic variance also for ARMAX systems.