Volterra-series-based nonlinear system modeling and its engineering applications: A state-of-the-art review

This paper reviews the major developments of modeling techniques applied
to nonlinear dynamics and chaos. Model representations, parameter
estimation techniques, data requirements, and model validation are some of
the key topics that are covered in this paper, which surveys slightly over two
decades since the pioneering papers on the subject appeared in the literature.

/pdf/modeling-nonlinear-dynamics-and-chaos-a-review-41oql20wfl.pdf

Modeling Nonlinear Dynamics and Chaos: A Review

Empirical time series often contain observational noise. We investigate the effect of this noise on the estimated parameters of models fitted to the data. For data of physiological tremor, i.e. a small amplitude oscillation of the outstretched hand of healthy subjects, we compare the results for a linear model that explicitly includes additional observational noise to one that ignores this noise. We discuss problems and possible solutions for nonlinear deterministic as well as nonlinear stochastic processes. Especially we discuss the state space model applicable for modeling noisy stochastic systems and Bock's algorithm capable for modeling noisy deterministic systems.

Modeling Noisy Time Series: Physiological Tremor

This paper addresses the development of a new algorithm for parameter estimation of ordinary differential equations. Here, we show that (1) the simultaneous approach combined with orthogonal cyclic reduction can be used to reduce the estimation problem to an optimization problem subject to a fixed number of equality constraints without the need for structural information to devise a stable embedding in the case of non-trivial dichotomy and (2) the Newton approximation of the Hessian information of the Lagrangian function of the estimation problem should be used in cases where hypothesized models are incorrect or only a limited amount of sample data is available. A new algorithm is proposed which includes the use of the sequential quadratic programming (SQP) Gauss–Newton approximation but also encompasses the SQP Newton approximation along with tests of when to use this approximation. This composite approach relaxes the restrictions on the SQP Gauss–Newton approximation that the hypothesized model should be correct and the sample data set large enough. This new algorithm has been tested on two standard problems.

/pdf/parameter-estimation-of-ordinary-differential-equations-2jqj2mlmdd.pdf

Parameter estimation of ordinary differential equations

Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system

Stochastic sensitivity analysis

Determining mixed linear-nonlinear coupled differential equations from multivariate discrete time series sequences

A formalism to characterise nonlinear dynamical hysteresis is described for multi-channel input-output physical systems that can have multi-valued solutions. The formalism presented is based on an extension to the Volterra functional representation of nonlinear dynamical input-output processes, an extension that overcomes the single-valued limitation of both the Taylor and Volterra series expansions. One important attribute of the formalism is that the coefficients can be physically significant and another is that the response function values can be determined directly from noisy data thereby offering potential insight into the underlying physics of the observed phenomena. The estimated response function values are the empirical coefficients required by a phenomenological theory of the system and can be used to predict likely behaviour, and to design and precisely control improved systems. The response function values that characterise the multi-channel nonlinear and dynamical hysteresis behaviour are estimated using the simultaneous moment equation method. The coefficients that characterise the hysteretic behaviour are obtained by solving a tractable system of simultaneous moment equations that are generated by operating on a suitably truncated Volterra functional expansion. These simultaneous moment equations enable the unknown constant coefficient values to be determined from noisy multi-channel input-output data. In order to demonstrate the attributes of the formalism under controlled conditions, a numerical example is presented which illustrates how to accurately estimate the coefficients of multi-channel nonlinear dynamical hysteresis phenomena corrupted by additive noise. The problems of time-dependent coefficients and the analysis of real data are to be considered elsewhere.

A.D. Irving

Papers

Stochastic sensitivity analysis

Determining mixed linear-nonlinear coupled differential equations from multivariate discrete time series sequences

Dynamical hysteresis in communications: a volterra functional approach

Mixed order response function estimation from multi-input non-linear systems

General nonlinear response of a single input system to stochastic excitations