Selecting the past and future for subspace identification of nonlinear systems with feedback and additive noise

A major issue in subspace system identification of nonlinear systems is the selection of nonlinear basis functions of the past and future on which the data are projected prior to the generalized singular value decomposition (SVD) central to the subspace approach. Previous approaches have used linear combinations of nonlinear basis functions of the past that may be regarded as a nonlinear ARX (NARX) model of the process. However, the nonlinear functions of the future are problematic since it is necessary to preserve the causality that involves the process inputs. Most approaches have used only linear combinations of the future that avoids the causality difficulty. But this ignores the effect of future inputs on future outputs, and can be very inefficient in capturing the functional canonical variables describing the process behavior leading to low accuracy models. In this paper, a new approach is proposed where the effects of future inputs on future outputs are removed using the NARX process of the past. For the case of linear systems with feedback, Larimore (2004) has shown that the parameter estimates are asymptotically efficient. This result extends to the nonlinear case with additive Gaussian measurement noise but may include arbitrary process noise. Once this is done, then the corrected future is used to construct a reverse NAR (RNAR) process giving an efficient description of the nonlinear future that also satisfies causality, asymptotically. To implement this approach, the NARX basis function selection is implemented using fast and numerically stable order-recursive computation of a generalized inverse of the covariance structure among the basis functions. Also leaps and bounds methods using the Akaike information criterion (AIC) are used to eliminate most of the candidate subsets in the regression subset selection of basis functions. The generalized SVD central to the subspace method determines the process state as linear combinations of the basis functions o- - f the past. The nonlinear state equations are determined by nonlinear regression of the outputs and the state one step ahead on the state and inputs