Weighted Least-Squares PARSIM

TL;DR

The paper targets variance and bias issues in classical subspace identification methods for discrete-time LTI systems, especially under closed-loop data. It introduces PARSIMopt, an optimized PARSIM that integrates a weighted least-squares stage to estimate the range space of the extended observability matrix $\Gamma_f L_p$ within a maximum-likelihood framework, achieving the BLUE for that intermediate quantity. A key novelty is to derive a consistent estimate of the optimal weighting matrix from a high-order ARX model (SSARX-inspired), enabling row-wise WLS that reduces variance relative to ordinary PARSIM. Simulation results across several examples show that PARSIMopt often yields smaller estimation variance than PARSIM and competitive performance against PBSIDopt and PEM, suggesting improved robustness for both open- and closed-loop identification.

Abstract

Subspace identification methods (SIMs) have proven very powerful for estimating linear state-space models. To overcome the deficiencies of classical SIMs, a significant number of algorithms has appeared over the last two decades, where most of them involve a common intermediate step, that is to estimate the range space of the extended observability matrix. In this contribution, an optimized version of the parallel and parsimonious SIM (PARSIM), PARSIM\textsubscript{opt}, is proposed by using weighted least-squares. It not only inherits all the benefits of PARSIM but also attains the best linear unbiased estimator for the above intermediate step. Furthermore, inspired by SIMs based on the predictor form, consistent estimates of the optimal weighting matrix for weighted least-squares are derived. Essential similarities, differences and simulated comparisons of some key SIMs related to our method are also presented.

Figures (4)

• Figure 1: Errors of Markov parameters.
• Figure 2: FIT (Example 1).
• Figure 3: FIT (Example 2).
• Figure 4: Joint FIT distribution of random systems.