Orthogonal least squares algorithm for the approximation of a map and its derivatives with a RBF network

Radial basis function networks (RBFNs) are used primarily to solve curve-fitting problems and for non-linear system modeling. Several algorithms are known for the approximation of a non-linear curve from a sparse data set by means of RBFNs. Regularization techniques allow to define constraints on the smoothness of the curve by using the gradient of the function in the training. However, procedures that permit to arbitrarily set the value of the derivatives for the data are rarely found in the literature. In this paper, the orthogonal least squares (OLS) algorithm for the identification of RBFNs is modified to provide the approximation of a non-linear single-input single-output map along with its derivatives, given a set of training data. The interest in the derivatives of non-linear functions concerns many identification and control tasks where the study of system stability and robustness is addressed. The effectiveness of the proposed algorithm is demonstrated with examples in the field of data interpolation and control of non-linear dynamical systems.