3,684 research outputs found
Nonconservative Lagrangian mechanics II: purely causal equations of motion
This work builds on the Volterra series formalism presented in [D. W.
Dreisigmeyer and P. M. Young, J. Phys. A \textbf{36}, 8297, (2003)] to model
nonconservative systems. Here we treat Lagrangians and actions as `time
dependent' Volterra series. We present a new family of kernels to be used in
these Volterra series that allow us to derive a single retarded equation of
motion using a variational principle
A kernel method for non-linear systems identification â infinite degree volterra series estimation
Volterra series expansions are widely used in analyzing
and solving the problems of non-linear dynamical
systems. However, the problem that the number of
terms to be determined increases exponentially with the
order of the expansion restricts its practical application.
In practice, Volterra series expansions are truncated
severely so that they may not give accurate representations
of the original system. To address this problem,
kernel methods are shown to be deserving of exploration.
In this report, we make use of an existing result
from the theory of approximation in reproducing kernel
Hilbert space (RKHS) that has not yet been exploited in
the systems identification field. An exponential kernel
method, based on an RKHS called a generalized Fock
space, is introduced, to model non-linear dynamical systems
and to specify the corresponding Volterra series
expansion. In this way a non-linear dynamical system
can be modelled using a finite memory length, infinite
degree Volterra series expansion, thus reducing the
source of approximation error solely to truncation in
time. We can also, in principle, recover any coefficient
in the Volterra series
Linear parameter estimation for multi-degree-of-freedom nonlinear systems using nonlinear output frequency-response functions
The Volterra series approach has been widely used for the analysis of nonlinear systems. Based on the Volterra series, a novel concept named Nonlinear Output Frequency Response Functions (NOFRFs) was proposed by the authors. This concept can be considered as an alternative extension of the classical frequency response function for linear systems to the nonlinear case. In this study, based on the NOFRFs, a novel algorithm is developed to estimate the linear stiffness and damping parameters of multi-degree-of-freedom (MDOF) nonlinear systems. The validity of this NOFRF based parameter estimation algorithm is demonstrated by numerical studies
Versatile surrogate models for IC buffers
In previous papers [1,2] the authors have investigated the use of Volterra series in the identification of IC buffer macro-models. While the approach benefited from some of the inherent qualities of Volterra series it preserved the two-state paradigm of earlier methods (see [3] and its references) and was thus limited in its versatility. In the current paper the authors tackle the challenge of going beyond an application or device-oriented approach and build versatile surrogate models that mimic the behavior of IC buffers over a wide frequency band and for a variety of loads thus achieving an unprecedented degree of generality. This requires the use of a more general system identification paradig
Compositional nonlinear audio signal processing with Volterra series
We develop a compositional theory of nonlinear audio signal processing based
on a categorification of the Volterra series. We augment the classical
definition of the Volterra series to be functorial with respect to a base
category whose objects are temperate distributions and whose morphisms are
certain linear transformations. This leads to formulae describing how the
outcomes of nonlinear transformations are affected if their input signals are
first linearly processed. We then consider how nonlinear audio systems change,
and introduce as a model thereof the notion of morphism of Volterra series. We
show how morphisms can be parameterized and used to generate indexed families
of Volterra series, which are well-suited to model nonstationary or
time-varying nonlinear phenomena. We describe how Volterra series and their
morphisms organize into a functor category, Volt, whose objects are Volterra
series and whose morphisms are natural transformations. We exhibit the
operations of sum, product, and series composition of Volterra series as
monoidal products on Volt and identify, for each in turn, its corresponding
universal property. We show, in particular, that the series composition of
Volterra series is associative. We then bridge between our framework and a
subject at the heart of audio signal processing: time-frequency analysis.
Specifically, we show that an equivalence between a certain class of
second-order Volterra series and the bilinear time-frequency distributions
(TFDs) can be extended to one between certain higher-order Volterra series and
the so-called polynomial TFDs. We end with prospects for future work, including
the incorporation of nonlinear system identification techniques and the
extension of our theory to the settings of compositional graph and topological
audio signal processing.Comment: Master's thesi
Volterra Series Truncation and Kernel Estimation of Nonlinear Systems in the Frequency Domain
The Volterra series model is a direct generalisation of the linear convolution integral and is capable of displaying the intrinsic features of a nonlinear system in a simple and easy to apply way. Nonlinear system analysis using Volterra series is normally based on the analysis of its frequency-domain kernels and a truncated description. But the estimation of Volterra kernels and the truncation of Volterra series are coupled with each other. In this paper, a novel complex-valued orthogonal least squares algorithm is developed. The new algorithm provides a powerful tool to determine which terms should be included in the Volterra series expansion and to estimate the kernels and thus solves the two problems all together. The estimated results are compared with those determined using the analytical expressions of the kernels to validate the method. To further evaluate the effectiveness of the method, the physical parameters of the system are also extracted from the measured kernels. Simulation studies demonstrates that the new approach not only can truncate the Volterra series expansion and estimate the kernels of a weakly nonlinear system, but also can indicate the applicability of the Volterra series analysis in a severely nonlinear system case
Projective stochastic equations and nonlinear long memory
A projective moving average is a Bernoulli shift
written as a backward martingale transform of the innovation sequence. We
introduce a new class of nonlinear stochastic equations for projective moving
averages, termed projective equations, involving a (nonlinear) kernel and a
linear combination of projections of on "intermediate" lagged innovation
subspaces with given coefficients . The class of such
equations include usual moving-average processes and the Volterra series of the
LARCH model. Solvability of projective equations is studied, including a nested
Volterra series representation of the solution . We show that under
natural conditions on , this solution exhibits
covariance and distributional long memory, with fractional Brownian motion as
the limit of the corresponding partial sums process
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