4.1 Introduction

In the previous chapter, we have given several reasons for performing the identification of a system in closed loop when the objective is the computation of a model aimed for control design. Before we pursue the reasoning made there and extend it in the next chapters to the validation of a given model or controller, or to the reduction of a high-order model or controller, we address some problems that arise in closed-loop identification when the controller contains unstable poles or nonminimum-phase zeroes. Such singularities are likely to be encountered with controllers designed by optimal techniques like LQG or synthesis, since the designer has little control on the poles and zeroes of these controllers, but it should be noted that even PI or PID controllers are unstable, since they contain an integrator.

We show that, with some of the commonly used closed-loop identification methods, the resulting model is not stabilised by the controller used during identification, even though the true system is stabilised by the same controller ^[1]. Furthermore, a model that is not stabilised by the present controller is intrinsically flawed for the design of a better controller, in that it deprives the control designer of some of their most important robust-stability tools for the design of this new controller.

The objective of this chapter is thus to answer the following questions:

How do the different classical closed-loop identification methods compare with respect to this nominal instability issue? Which method should I use in function...