Hierarchical extended least squares estimation approaches for a multi-input multi-output stochastic system with colored noise from observation data

•The hierarchical identification principle is used to study the problem of parameter identification.•A recursive extended least squares (RELS) algorithm and a decomposition-based recursive extended least squares (D-RELS) algorithm are derived.•The RELS algorithm and D-RELS algorithm can give accurate parameter estimates for the finite data length.•The D-RELS algorithm requires less computational cost than the RELS algorithm. This paper uses the hierarchical identification principle to decompose a multi-input multi-output equation-error moving average system into two subsystems, and then proposes a decomposition-based recursive extended least squares (D-RELS) algorithm (i.e., hierarchical extended least squares algorithm) to estimate the parameter matrices of these two subsystems. Before that, a recursive extended least squares (RELS) algorithm is presented as a comparison. By analyzing the estimation results and the calculation amount, these two algorithms can estimate the system parameters effectively but the D-RELS algorithm has less computational cost than the RELS algorithm. The proposed D-RELS algorithm is extended to multi-input multi-output equation-error autoregressive moving average systems. The numerical simulation results demonstrate the effectiveness of the proposed algorithms.