Volterra Kernel Identification Using Triangular Wavelets

摘要

The Volterra series provides a convenient framework for the representation of nonlinear dynamical systems. One of the main drawbacks of this approach, however, is the large number of terms that are often needed to represent Wterra kernels. In this paper we present an approach whereby wavelets are used to obtain low-order estimates of first-order and second-order Volterra kernels. Several constructions of tensor-product wavelets have been employed for some Wterra kernel approximations. In this paper, a triangular wavelet basis is constructed for the representation of the triangular form of the second-order kernel. These wavelets are piecewise-constant, orthonormal, and are supported over the triangular domain over which the second-order kernel is defined. The well-known Haar wavelet is used concurrently for the identification of the first-order kernel. This kernel identification algorithm is demonstrated on a prototypical nonlinear oscillator. It is shown that accurate kernel estimates can be obtained in terms of a relatively small number of wavelet coefficients. It is also demonstrated that, for this particular system, the derived Wterra model is valid for input amplitudes below a specified bound. When the input amplitude exceeds this threshold, higher-order kernels are needed to adequately describe the system dynamics. Thus, the approach taken in this paper is applicable to a large class of nonlinear systems provided that the input excitation is sufficiently bounded.