Parameter-based conditions for closed-loop system identifiability of ARX models with routine operating data

Traditionally, closed-loop system identification in the absence of external excitation has focused on determining the identifiability of plant model based on the interplay between the orders of the different polynomials present. However, due to the presence of the controller, it is possible that the system may not be globally identifiable at a given complexity, but may be locally identifiable given certain restrictions or relationships between the individual parameters present in the system. In order to obtain parameter-specific solutions to the problem, many different approaches can be taken. In this paper, the focus will be primarily on an expectation-based analysis of the Fisher information matrix to determine parameter-based constraints on closed-loop identification. Additionally, a method for determining an analytical expression for the expectation operation will be presented. The proposed approach will be illustrated using a first-order autoregressive model with exogenous input controlled by a lead-lag controller. Monte Carlo simulations are used to validate the resulting constraints.