Bridging the Prediction Error Method and Subspace Identification: A Weighted Null Space Fitting Method
For control and system identification researchers, this work provides a computationally efficient method that achieves maximum likelihood efficiency without non-convex optimization, resolving a key theoretical gap.
The paper introduces a weighted null space fitting (WNSF) method for state-space model estimation that combines the numerical robustness of subspace identification methods (SIMs) with the asymptotic efficiency of the prediction error method (PEM). The method is proven consistent and asymptotically efficient, addressing a long-standing open problem in the field.
Subspace identification methods (SIMs) have proven to be very useful and numerically robust for building state-space models. While most SIMs are consistent, few if any can achieve the efficiency of the maximum likelihood estimate (MLE). Conversely, the prediction error method (PEM) with a quadratic criteria is equivalent to MLE, but it comes with non-convex optimization problems and requires good initialization points. This contribution proposes a weighted null space fitting (WNSF) approach for estimating state-space models, combining some key advantages of the two aforementioned mainstream approaches. It starts with a least-squares estimate of a high-order ARX model, and then a multi-step least-squares procedure reduces the model to a state-space model on canoncial form. It is demonstrated through statistical analysis that when a canonical parameterization is admissible, the proposed method is consistent and asymptotically efficient, thereby making progress on the long-standing open problem about the existence of an asymptotically efficient SIM. Numerical and practical examples are provided to illustrate that the proposed method performs favorable in comparison with SIMs.