Multi-variable Volterra kernels identification using time-delay neural networks: application to unsteady aerodynamic loading

Abstract

In the last decades, the Volterra series theory has been used to construct reduced-order models of nonlinear systems in engineering and applied sciences. For the particular case of weakly nonlinear aerodynamic and aeroelastic systems, the Volterra series theory has been tested as an alternative to the high computing costs of CFD methods. The Volterra series model determination depends on identifying the kernels associated with the respective convolution integrals. The Volterra kernels identification has been tried in many ways, but the majority of them addresses only the direct kernels of single-input, single-output nonlinear systems. However, multiple-input, multiple-output relations are the most typical case for many dynamic systems. In this case, the so-called Volterra cross-kernels represent the internal couplings between multiple inputs. Not many generalizations of the single-input kernel identification methods to multi-input Volterra kernels are available in the literature. This work proposes a methodology for the identification of Volterra direct kernels and cross-kernels, which is based on time-delay neural networks and the relationship between the kernels functions and the internal parameters of the network. Expressions to derive the pth-order Volterra direct kernels and cross-kernels from the internal parameters of a trained time-delay neural network are derived. The method is checked with a two-degree-of-freedom, two-input, one-output nonlinear system to demonstrate its capabilities. The application to a mildly nonlinear unsteady aerodynamic loading due to pitching and heaving motions of an airfoil is also evaluated. The Volterra direct kernels and cross-kernels of up to third order are successfully identified using training datasets computed with CFD simulations of the Euler equations. Comparisons between CFD simulations and Volterra model predictions are presented, thereby ensuring the potential of the method to systematically extract kernels from neural networks.