Exact Search Directions for Optimization of Volterra Models Using Kautz Functions

A novel technique for selecting the poles of orthonormal basis functions (OBF) in Volterra models of any order is presented. In general, a large number of parameters are required to represent the Volterra kernels, although this difficulty can be overcome by describing each kernel using a basis of orthonormal functions, such as the Kautz basis. Aiming at optimizing the poles that fully parameterize the Kautz functions, the exact gradients of the outputs of these functions with respect to their poles are computed analytically. These gradients provide exact search directions for optimizing the Kautz poles. Such search directions can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into account the error of estimation of the system output. Unlike previous related work, the proposed approach relies solely on input-output data measured from the system to be modeled. This approach has been illustrated by means of an application to the modeling of linear and nonlinear systems, including a real magnetic levitation system with nonlinear oscillatory behavior.