## Abstract

A novel technique for selecting the poles of or-thonormal basis functions (OBF) in Volterra models of any order is presented. It is well-known that the usual large number of parameters required to describe the Volterra kernels can be significantly reduced by representing each kernel using an appropriate basis of orthonormal functions. Such a representation results in the so-called OBF Volterra model, which has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a nonlinear static mapping given by the Volterra polynomial series. Aiming at optimizing the poles that fully param-eterize the orthonormal bases, the exact gradients of the outputs of the orthonormal filters with respect to their poles are computed analytically by using a back-propagation-through-time technique. The expressions relative to the Kautz basis and to generalized orthonormal bases of functions (GOBF) are addressed; the ones related to the Laguerre basis follow straightforwardly as a particular case. The main innovation here is that the dynamic nature of the OBF filters is fully considered in the gradient computations. These gradients provide exact search directions for optimizing the poles of a given orthonormal basis. 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. The Levenberg-Marquardt algorithm is adopted here as the optimization procedure. Unlike previous related work, the proposed approach relies solely on input-output data measured from the system to be modeled, i.e., no information about the Volterra kernels is required. Examples are presented to illustrate the application of this approach to the modeling of dynamic systems, including a real magnetic levitation system with nonlinear oscillatory behavior.

Originalsprog | Engelsk |
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Artikelnummer | 5308390 |

Tidsskrift | IEEE Transactions on Automatic Control |

Vol/bind | 54 |

Udgave nummer | 12 |

Sider (fra-til) | 2757-2772 |

ISSN | 0018-9286 |

DOI | |

Status | Udgivet - dec. 2009 |

Udgivet eksternt | Ja |

### Bibliografisk note

Funding Information:Manuscript received April 17, 2008; revised September 23, 2008, February 10, 2009, and July 08, 2009. First published November 03, 2009; current version published December 09, 2009. This work was supported by the Brazilian National Council for Scientific and Technological Development (CNPq), under Grants 140706/2005-4, 306229/2006-4, and 301063/2007-9, and also by the Research Foundation of the State of São Paulo (Fapesp), under Grant 06/50231-5. Recommended by Associate Editor J.-F. Zhang.