Progressive Parametrization in Subspace Identification Models with finite horizons

Traditional subspace identification (SID) framework uses Kalman filter or predictor to interpret the SID models. To achieve this the horizons f, p have to approach infinity to be consistent. In practice, however, the horizons f, p are finite. We argue that for finite f and p the Kalman filter framework does not apply. In this paper, we introduce a progressive parametrization framework to interpret the models used in each step of SID methods and discuss how the progressively parametrized models lead to the recursive state space models, when additional assumptions are made. Monte-Carlo simulation is conducted on a closed-loop example to demonstrate what each step of SID contributes to the model estimate using the methods of HOARX, SSARX of Jansson, and that of canonical variate analysis. We also state that the intermediate non-recursive models can be useful for the purpose of state estimation, fault detection, and control.