Indirect methods for the identification of linear plant models on the basis of closed loop data are based on the use of (reconstructed) input signals that are uncorrelated with the noise. This generally requires exact (linear) controller knowledge. On the other hand, direct identification requires exact plant and noise modelling (system in the model set) in order to achieve accurate results, although the controller can be non-linear. In this paper, a generalized approach to closed loop identification is presented that includes both methods as special cases and which allows novel combined methods to be generated. Besides providing robustness with respect to inexact controller knowledge, the method does not rely on linearity of the controller nor on exact noise modelling. The generalization is obtained by balancing input-noise decorrelation against noise whitening in a user-chosen flexible fashion. To this end, a user-chosen virtual controller is used to parametrize the plant model, thereby generalizing the dual-Youla method to cases where knowledge of the controller is inexact. Asymptotic bias and variance results are presented for the method. Also, the benefits of the approach are demonstrated via simulation studies.
Agüero, J. C., Goodwin, G. C., & Hof, Van den, P. M. J. (2011). A virtual closed loop method for closed loop identification. Automatica, 47(8), 1626-1637. https://doi.org/10.1016/j.automatica.2011.04.014