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In this paper, the Hammerstein identification problem with correlated inputs is studied in a prediction error framework using separable least squares methods. Thus, the identification is recast as an optimization over the parameters used to describe the nonlinearity. A sufficient condition is derived that guarantees that the identification problem is quasiconvex with respect to the parameters that describe the nonlinearity. Simulations using both IID and correlated inputs are used to illustrate the result.

Ph.D. Programs Foundation of Ministry of Education of China [20100121120020]; Major Science and Technology Special Project of Fujian Province, China [2009HZ0003-1]

This note compares and contrasts the non-linear parameter varying (NLPV) and state-dependent parameter (SDP) model classes. It shows that, while they have similarities, the two-stage SDP modelling procedure, involving non-parametric identification, followed by parametric estimation, is quite different from the single stage NLPV procedure. In particular, the SDP procedure allows for the identification of the model structure and the nature of the non-linearities, prior to the estimation of the parameters that characterize this identified model structure. In contrast to NLPV modelling, therefore, SDP estimation opens up the ‘black box’ and reveals the inner nature of the non-linear system.

In this paper, the parametrization of state-space systems by data driven local coordinates as introduced by (McKelvey et al., 2003) is modified. This modification leads to an alternative analogous parametrization which can be used for a suitable concentrated likelihood-type criterion function, where the concentration step can be done by a generalized least squares step. An obvious consequence is the reduced number of parameters resulting in less computational burden, but, of course, the criterion function itself is changed by the concentration step. The resulting new parametrization is called slsDDLC, and its topological and geometrical properties are investigated in detail.

We consider the least squares approximation of gridded 2D data by tensor product splines with free knots. The smoothing functional to be minimized—a generalization of the univariate Schoenberg functional—is chosen in such a way that the solution of the bivariate problem separates into the solution of a sequence of univariate problems in case of fixed knots. The resulting optimization problem is a constrained separable least squares problem with tensor product structure. Based on some ideas developed by the authors for the univariate case, an efficient method for solving the specially structured 2D problem is proposed, analyzed and tested on hand of some examples from the literature.

The aim of this paper is to propose a novel identification aalgorithm based on separable least squares ideas, for a class of nonlinear, possibly parameter-varying, input/output models. These models are given in the form of a Linear Fractional Transformation (LFT) where the "forward" part is represented by a conventional linear regression and the "feedback" part is given by a nonlinear map which can take into account scheduling variables available for measurement. The nonlinear part of the model can be parameterised according to various paradigms, like, e.g., Neural Network (NN) or NARX.

Separable least squares are generally written in the form ‖y−A(q)c ‖2 = min where minimization has to be carried out with respect to the parameters q and c. The latter enter linearly into the expression of the objective function and this leads to considerable simplications. These simplifications are not possible if the functional to be minimized is of the form σ‖y−Aj(q)c‖2 = min In this communication we modify previous methods, so that they can deal with the more general case (2). We also show that the statistical properties of both linear and nonlinear estimates can be evaluated according to a procedure that enables us to keep the two classes of parameters separate. An important application is examined and its numerical solution worked out in detail.