On the asymptotic properties of closed loop CCA-type Subspace Algorithms: equivalence results and choice of the future horizon

In this paper, we shall consider a class of subspace algorithms for identification of linear time invariant systems operating in "closed loop." In particular we study algorithms based on the so-called "state-sequence" approach; we first show that the ADAPTx algorithm by Larimore is asymptotically equivalent to a number of recently developed algorithms, which we call CCA-type algorithms. Based on this equivalence result, we then study the effect of the "future horizon," which is one of the principal "user choices" in subspace identification. It is well known that for the CCA algorithm the asymptotic variance of any system invariant is a non increasing function of the future horizon when input signals are white (or absent). In particular we extend this result, valid for white noise input signals to a slightly more general class of input signals, which include proportional (output or state) feedback controllers and LQG controllers, provided the reference input is white. The condition on the input will be expressed in terms of its state space, which we regard as a rather natural condition in this framework. For the situations not covered by the above result, we shall also describe a computational procedure, based on some recently derived asymptotic variance formulas, which allows to optimize the choice of the future horizon. Some simulation results are included.