Multiresolution methods for representation of Volterra series and dynamical systems

Recently, the authors have derived a wavelet and multiresolution based methodology for obtaining reduced order approximations of Volterra series. While Volterra series provide a succinct characterization of nonlinear system response, their use has been limited in practice due to the large number of terms required to approximate the higher order, nonlinear terms. We have shown in Ref. 9 that a consistent approximation of the Volterra input/output representation is achieved if two conditions are satisfied: (1) a zero order hold is used for the input and output sequences, and (2) a biorthogonal wavelet family is selected such that the generator is dual to the characteristic functions that define the zero order hold. It is not straightforward, however, to extend this approach to higher order terms. In this paper we derive a family of multiwavelets that reproduce piecewise polynomials that are of prescribed order, are supported on [0, infinity), and are consequently well suited for the approximation of Volterra operators. These multiwavelets are derived by using an intertwining method due to Donovan, Geronimo, and Hardin(5,6) We present applications of these bases in the identification of Volterra operators representing system dynamics at the 41(st) Structures, Structural Dynamics, ann Materials Conference.