Multiresolution methods for reduced-order models for dynamical systems

Reduced-order input/output models are derived for a class of nonlinear systems by utilizing wavelet approximations of kernels appearing in Volterra series representations. Although Volterra series representations of nonlinear system input/output have been understood from a theoretical standpoint for some time, their practical use has been limited as a result of the dimensionality of approximations of the higher-order, nonlinear terms. In general, wavelets and multiresolution analysis have shown considerable promise for the compression of signals, images, and, most importantly here, some integral operators. Unfortunately, causal Volterra series representations are expressed in terms of integrals that are restricted to products of half-spaces, and there is a significant difficulty in deriving wavelets that are appropriate for Volterra kernel representations that are restricted to semi-infinite domains. In addition, it is necessary to derive Volterra kernel expansions that are consistent with the method of sampling used to obtain the input and output data. This paper derives discrete approximations for truncated Volterra series representations in terms of a specific class of biorthogonal wavelets. When a zero-order hold is used for both the input and output signals, it is shown that a consistent approximation of the input/output system is achieved for a specific choice of biorthogonal wavelet families. This family is characterized by the fact that all of the wavelets are biorthogonal with respect to the characteristic function of the dyadic intervals used to define the zero-order hold. It is also simple to show that an arbitrary choice of wavelet systems will not, in general, provide a consistent approximation for arbitrary input/output mappings, Numerical studies of the derived methodologies are carried out by using experimental pitch/plunge response data from the TAMU Nonlinear Aeroelastic Testbed.