ON THE ROLE OF FUTURE HORIZON IN CLOSED-LOOP SUBSPACE IDENTIFICATION

Abstract

Closed loop subspace identification has become a focus of interest with several recent developments. Notably are the innovation estimation approach (Qin and Ljung, 2003), the state space approach with ARX pre-estimates (SSARX, (Jansson, 2003)), and the whitening filter approach (Chiuso and Picci, 2004). All these approaches use an extended future horizon to form the projection or regression from which an observable subspace is extracted. Yet there are other methods such as OKID of (Phan and Longman, 1992) and that of (Ljung and McKelvey, 1996) that do not use an extended horizon in the projection or regression step. Instead, a single high order ARX model is used.A natural question is whether the future horizon is necessary and if so what role does it play in these steps. In this paper we investigate the role of the future horizon using the whitening filter approach of (Chiuso and Picci, 2004), which works for both open-loop and closed-loop data. We conclude that the role of future horizon in this algorithm is merely extending the order of a bank of already high order ARX models. The difference from a single ARX model is insignificant if the ARX order or past horizon is sufficiently high. The role of future horizon is mainly in the model reduction step where it serves to elevate the order of the Hankel matrix. We complement the analysis with simulations.