Consistency analysis of some closed-loop subspace identification methods

It seems fair to say that current state-of-the-art subspace identification methods provide reliable results only when applied to plants operating in open loop. However feedback is present in a variety of practical situations (even though often one cannot directly recognize physical controllers which “close the loop”) and there is a need of reliable identification methods and algorithms which could be used with multivariable systems in the presence of feedback. Various attempts to extend existing subspace identification algorithms to work in the presence of feedback have been made in the last decades. Among early references, we quote [1, 21, 13, 7, 20], while more recent work is presented in [17, 12]. Yet, as discussed in [3, 4] there are fundamental issues related to stochastic realization theory in the presence of feedback which remain unclear. In particular stochastic realization with feedback is still not fully understood when unstable open-loop transfer functions are involved, which of course is a very interesting situation in the applications. On the other hand, even some of the best recently proposed methods seem to run occasionally into troubles with unstable open-loop transfer functions. Unfortunately, even when restricting to stable open loop plants (a rather stringent restriction to be sure), the existing algorithms turn out to provide biased estimates. We shall argue that this is so mainly because one has to neglect the effect of initial conditions. This is in contrast to the classical open-loop subspace methods (N4SID, MOESP, CCA) from the literature, which instead provide consistent estimates by taking properly into account the “transient” effects due to initial conditions. Of course the usual way out to the bias due to neglecting initial conditions is to regress on enough past data (i.e. to keep the past data horizon of the algorithm suitably large). Provided the zeros of the system are not too close to the unit circle, the bias error can generally be made negligible. However, as we shall argue in this paper, when feedback is present, there may be situations in which, even when the past horizon of the algorithm is chosen very large, the bias is still unacceptable. In this paper we shall analyze the bias in subspace identification with feedback. We shall in particular point out that mishandling of the initial conditions is an intrinsic difficulty related to feedback, which pops up whenever we restrict to identification of the system in the forward loop only. The difficulty is visible from the structure of the (transient) Kalman filter-like representation of the output process, which in general does involve also the dynamics of the input process u which we don’t want to model. This phenomenon can be circumvented in the feedback-free case , see [6], but seems to be very hard to bypass when there is feedback. We believe that a correct handling of the initial condition should be a main step towards a satisfactory theory of subspace identification with feedback. This paper has been partly inspired and motivated by strong connections existing between the recent subspace algorithms of [17] and [12], and some theoretical work that we have been carrying through in the last years, dealing with stochastic state space construction in the presence of feedback. This work is preliminarily exposed in the papers [3, 4]. We shall demonstrate that these new algorithms, which we regard as a significant step forward in subspace identification of feedback systems, can be interpreted as possible numerical implementations of some stochastic realization constructions described in [3, 4]. Consistency analysis of the algorithms, and an explicit computation of the bias are then possible by using the framework of stochastic systems and stochastic realization theory. The structure of the paper is as follows: Section 2 states the problem and sets up basic notations;