Closed-loop Subspace System Identiication 1

In this paper we present a general framework for closed-loop subspace system identi cation. This framework consists of two new projection theorems which allow the extraction of nonsteady state Kalman lter states and of system related matrices directly from closed-loop input output data. Three algorithms for the identi cation of the state space matrices are then presented. The similarities between the theorems and algorithms, and the corresponding open-loop theorems and algorithms in the literature are emphasized. The closed-loop theory is illustrated with a simulation example. r u y k k P(z) C(z) k vk wk Figure 1: Standard feedback setup. uk is the input signal, yk the output signal and rk the reference signal. vk (measurement noise) and wk (process noise) are the disturbances acting on the linear plant P (z). The linear controller is represented by C(z). In this paper we assume that uk and yk are measured and that a limited number of impulse response samples (Markov parameters) of the controller is known. The controller C(z) does not have to be explicitly given.