Volterra series

About: Volterra series is a research topic. Over the lifetime, 2731 publications have been published within this topic receiving 46199 citations.

Nonlinear image operators for the evaluation of local intrinsic dimensionality

Abstract: Local intrinsic dimensionality is shown to be an elementary structural property of multidimensional signals that cannot be evaluated using linear filters We derive a class of polynomial operators for the detection of intrinsically 2-D image features like curved edges and lines, junctions, line ends, etc Although it is a deterministic concept, intrinsic dimensionality is closely related to signal redundancy since it measures how many of the degrees of freedom provided by a signal domain are in fact used by an actual signal Furthermore, there is an intimate connection to multidimensional surface geometry and to the concept of 'Gaussian curvature' Nonlinear operators are inevitably required for the processing of intrinsic dimensionality since linear operators are, by the superposition principle, restricted to OR-combinations of their intrinsically 1-D eigenfunctions The essential new feature provided by polynomial operators is their potential to act on multiplicative relations between frequency components Therefore, such operators can provide the AND-combination of complex exponentials, which is required for the exploitation of intrinsic dimensionality Using frequency design methods, we obtain a generalized class of quadratic Volterra operators that are selective to intrinsically 2-D signals These operators can be adapted to the requirements of the signal processing task For example, one can control the "curvature tuning" by adjusting the width of the stopband for intrinsically 1-D signals, or the operators can be provided in isotropic and in orientation-selective versions We first derive the quadratic Volterra kernel involved in the computation of Gaussian curvature and then present examples of operators with other arrangements of stop and passbands Some of the resulting operators show a close relationship to the end-stopped and dot-responsive neurons of the mammalian visual cortex

Reduced-Order Modeling of Flutter and Limit-Cycle Oscillations Using the Sparse Volterra Series

Abstract: For the past two decades, the Volterra series reduced-order modeling approach has been successfully used for the purpose of flutter prediction, aeroelastic control design, and aeroelastic design optimization. The approach has been less successful, however, when applied to other important aeroelastic phenomena, such as aerodynamically induced limit-cycle oscillations. Similar to the Taylor series, the Volterra series is a polynomial-based approach capable of progressively approximating nonlinear behavior using quadratic, cubic, and higher-order functional expansions. Unlike the Taylor series, however, kernels of the Volterra series are multidimensional convolution integrals that are computationally expensive to identify. Thus, even though it is well known that aerodynamic nonlinearities are poorly approximated by quadratic Volterra series models, cubic and higher-order Volterra series truncations cannot be identified because their identification costs are too high. In this paper, a novel, sparse representation of the Volterra series is explored for which the identification costs are significantly lower than the identification costs of the full Volterra series. It is demonstrated that sparse Volterra reduced-order models are capable of efficiently modeling aerodynamically induced limit-cycle oscillations of the prototypical NACA 0012 benchmark model. These results demonstrate for the first time that Volterra series models are capable of modeling aerodynamically induced limitcycle oscillations.

Model Reduction of Bilinear Systems in the Loewner Framework

Abstract: The Loewner framework for model reduction is extended to the class of bilinear systems The main advantage of this framework over existing ones is that the Loewner pencil introduces a trade-off between accuracy and complexity Furthermore, through this framework, one can derive state-space models directly from input-output data without requiring initial system matrices The recently introduced methodology of Volterra series interpolation is also addressed Several numerical experiments illustrate the main features of this approach

Nonconservative Lagrangian mechanics: a generalized function approach

Abstract: We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. In particular, a formalism is developed that allows the inclusion of fractional derivatives. This is done within the Lagrangian framework by treating the action as a Volterra series. It is then possible to derive two equations of motion, one of these is an advanced equation and the other is retarded.

Nonlinear oscillation via Volterra series

Abstract: Using a novel approach, the amplitude and frequency of nearly sinusoidal nonlinear oscillators can be calculated by solving two algebraic nonlinear equations. These determining equations can be generated to within any desired accuracy using a recursive algorithm based on Volterra series. Our method inherits many desirable features of the harmonic balance method, the describing function method, and the averaging method. Our technique is analogous to, but is much simpler than, the classic approach due to Krylov, Bogoliubov, and Mitropolsky. Unlike conventional techniques, however, our approach imposes no severe restriction on either the degree of nonlinearity, or the amplitude of oscillation. Moreover, the accuracy of the solution can be determined by a constructive algorithm.