Closed-loop Subspace Identification Algorithms for 2-D Separable-in-Denominator Systems*

Identification for closed-loop two-dimensional (2-D) causal, recursive, and separable- in-denominator (CRSD) systems in the Roesser form is discussed in this study. For closed- loop 2-D CRSD systems, in the presence of feedback control, there exists correlation between the unknown disturbances and the future inputs, which offers the fundamental limitation for utilizing the standard open-loop 2-D CRSD subspace identification methods. In other words, the existing open-loop 2-D CRSD subspace algorithms may result in biased estimates of plant parameters under closed-loop conditions. In this paper, based on the instrument variable method and principal component analysis, a novel 2-D CRSD subspace identification algorithm that are applicable to both open-loop and closed-loop data are developed. Additionally, we discuss the whiteness external excitation case, explain why the white set-point signal may influence the estimated model results, and subsequently adopt modified instrument variables to improve the proposed subspace identification method. Several numerical examples have been conducted to validate consistency and efficiency of the proposed subspace identification approaches.