Closed-loop identification revisited

Identification of systems operating in closed loop has long been of prime interest in industrial applications. The problem offers many possibilities, and also some fallacies, and a wide variety of approaches have been suggested, many quite recently. The purpose of the current contribution is to place most of these approaches in a coherent framework, thereby showing their connections and display similarities and differences in the asymptotic properties of the resulting estimates. The common framework is created by the basic prediction error method, and it is shown that most of the common methods correspond to different parameterizations of the dynamics and noise models. The so-called indirect methods, e.g., are indeed "direct" methods employing noise models that contain the regulator. The asymptotic properties of the estimates then follow from the general theory and take different forms as they are translated to the particular parameterizations. We also study a new projection approach to closed-loop identification with the advantage of allowing approximation of the open-loop dynamics in a given, and user-chosen frequency domain norm, even in the case of an unknown, nonlinear regulator. (C) 1999 Elsevier Science Ltd. All rights reserved.