Extended Self-Organizing Maps with Local Linear Mappings for Function Approximation and System Identification

A new method for function approximation and system identification based on the SelfOrganizing Map is presented. The standard Self-Organizing Map (SOM) is extended with Local Linear Mappings to enable the original algorithm to learn input-output relationships with reasonable accuracy. To every node in the SOM along with the input weight two output weights are assigned: one that stores the output part of an input-output pair and one that stores the local gradient matrix (Jacobian) of the input-output mapping that is calculated from the training pairs. A training algorithm for the Jacobian matrices is derived based on a Least Squares approximation by utilising properties of the pseudoinverses of vectors. The method is tested in function approximation of a multivariable function, system identification of a highly nonlinear system and system identification of a hydraulic actuator based on input-output measurements.