Showing papers on "Volterra series published in 1987"

Modeling of the Nonlinear Response of an Electrodynamic Loudspeaker by a Volterra Series Expansion

Abstract: Modelisation des distorsions non lineaires a basse frequence de la reponse d'un haut-parleur electrodynamique

Volterra and generating power series approaches to identifiability testing

Abstract: Keywords . identifiability, nonlinear models, generating power series, Volterra series, modeling.

Accurate Modeling of Nonlinear Systems using Volterra Series Submodels

Abstract: We investigate the problem of accurately modeling nonlinear systems (such as aircraft flight in high angle-of-attack/sideslip flight) using simple low-order Volterra submodels. First, we apply this technique to a simplified nonlinear stall/post-stall aircraft model for the case of a longitudinal limit cycle. Our simulation study demonstrates that the responses of the Volterra submodels accurately match the responses of the original nonlinear model, whereas the responses of a piecewise-linear model do not. Next, we apply the technique to a simplified high a nonlinear model of wing rock. Our simulation study demonstrates that the second-order Volterra approximation predicts the wing rock limit cycle, while a linear approximation does not. Third-, fourth- and fifth-order Volterra approximations are observed to give wing rock amplitudes that converge quadratically to the nonlinear value.

The observation space and realizations of finite Volterra series

Abstract: In this paper we extend the results on the theory of realizations of finite Volterra series by exploiting the structural properties of the observation space. In general, two distinct minimal realizations of a finite Volterra series may be constructed, one based on the properties of the observation space and the other based on the properties of the Lie algebra. Both these realizations display the canonical structure found earlier for such systems. The results given here also yield information on the observation algebra generated by functions in the observation space, just as previous work gave information on the Lie algebra. As an application, the structure found here is applied to the finite-dimensional filtering problem for these systems.

Modeling of the nonlinear drift oscillations of moored vessels subject to non-Gaussian random sea-wave excitation

Abstract: A quadratic system model based on Volterra series representation is utilized to model the nonlinear response of moored vessels subjected to random seas. The key idea is to represent the relationship between the incident sea wave (input) and corresponding sway response of the moored vessel (output) with a parallel combination of linear and quadratic transfer functions, and to estimate them by processing actual input and output data. Compared to previous approaches, we take the important step of removing the restriction that the random input must possess Gaussian statistics. The feasibility and validity of the approach is demonstrated by analyzing experimental data taken in model basin tests. We also describe some of the deleterious consequences of assuming Gaussian sea-wave excitation when in fact the excitation is non-Gaussian.

Analysis and Modeling of Intermodulation Distortion in Wide-Band Cable TV Channels

Abstract: This correspondence deals with the development of a practical approach for the assessment of intermodulation distortion in modern cable television (CATV) systems operating over the 400 MHz range. In particular, Volterra series expansions are used in establishing appropriate high-frequency models for the description of such wide-band CATV channels. Furthermore, experimental measurements made on a typical system are utilized to illustrate the development of Volterra models which may be used subsequently in the prediction of intermodulation distortion in such systems.

Finite Volterra-series realizations and input-output approximations of non-linear discrete-time systems

Abstract: It is shown that a separability property of a given finite family of stationary kernels turns out to be necessary and sufficient for the realization of the associated functional by means of a polynomial affine system (i.e. a system that is polynomial in the input and affine in the state). Moreover, the discrete Volterra kernels associated with the input-output map of a non-linear analytic discrete-lime system, initialized at an equilibrium point, are shown to possess a separability property. On this basis, we state an approximation result for the given input-output map by considering the first kernels of the discrete Volterra series. Two explicit constructions of the approximating polynomial affine systems are proposed.

Decision-directed nonlinear filter for image processing

Abstract: A complete quadratic Volterra filter working under the control of a decision algorithm is proposed for image restoration and enhancement. A particular application in the enhancement of images having reduced luminance dynamics is considered.

A variable-step adaptation algorithm for memory-oriented Volterra filters

Abstract: A recently proposed adaptation algorithm for digital non-linear adaptive filters based on the truncated discrete Volterra series is modified and improved in order to get a faster and more accurate convergence. Performance examples are presented.

Application of Volterra-Wiener Series for bounding the overall conductivity of hetergeneous media. I. General procedure

Abstract: A general method of placing optimal bounds on the overall conductivity of random heterogeneous media is proposed. It utilizes truncated Volterra-Wiener functional series, generated by the random conductivity field K(x), as classes of trial functions for the classical variational principles. The method simplifies, unifies and/or generalizes the earlier proposed variational techniques of Beran (3), Dederichs and Zeller (10), Hori (15), Kr6ner (17) and Prager (25). The general procedure is displayed in detail in the simplest case of interest, the construction of the optimal third-order bounds, that requires knowledge of the two- and three-point correlation functions for K(x). The evaluation of these bounds is reduced to the solution of an integrodifferential equation whose coefficients and kernels are expressed through the said correlation functions. A perturbation solution to this equation is given. For Miller's cell materials the equation is explicitly solved and the obtained optimal bounds are shown to coincide with those of Beran-Miller.

Harmonic and transient scattering from weakly nonlinear objects

Abstract: A mathematical model for scattering of electromagnetic waves from weakly nonlinear objects is developed. The constitutive relations are based on Volterra series, but additional, physically plausible, heuristic assumptions have to be introduced in order to solve the scattering problem. The general theory is discussed in connection with scattering from circular cylinders. These canonical problems demonstrate the new phenomena involved. It is shown that the first order effects of the nonlinear scattering problem involve modification of the linear scattering coefficients and production of new multipole terms at the fundamental frequency. In addition, part of the energy is transformed into harmonics. The corresponding problem of transient scattering is considered. The new effects of pole migration and pole creation are discussed. The present study contributes to understanding the theoretical aspects of nonlinear scattering, and may also provide a method for remote sensing of nonlinear targets.

Application of Volterra-Wiener series for bounding the overall conductivity of heterogeneous media. II. Suspensions of eui-sized spheres

Abstract: The variational procedure of Part I of this paper (this Journal, 47 (1987), pp. 831-849) is employed for bounding the overall conductivity K* of a suspension of equi-sized spheres. The procedure uses classes of trial functions in the form of truncated Volterra-Wiener series generated here by the random density function of the system of sphere centers. Integrodifferential equations are derived whose solutions yield bounds, upper and lower, on K* for given two- and three-point distribution functions f2, f3 of the system of spheres. The bounds are explicitly calculated to the order c2, where c is the volume fraction of the spheres; they include a statistical parameter m2 that can be simply evaluated by means of the two-point distribution function f2 and that reflects the tendency of the spheres to form clusters at higher c. The predictions of various approximate models of composite mechanics are checked against the c2-bounds. The check yields a number of curious results concerning the random constitution of suspensions of impenetrable spheres that can be described by means of these models. In this paper we develop the technique of the Volterra-Wiener series for bounding the overall conductivity in random suspensions of equi-sized spheres. In ? 6 we describe the statistics of such suspensions by means of the multipoint distribution functions f, for the system of sphere centers. In ? 7 the two- and three-point moments (correlation functions) M' and M3? of the random conductivity field K(X) of the suspension are explicitly expressed by f2 and f3. In ? 8 the variational procedure of (I, ? 2) is slightly modified and an integrodifferential equation is derived whose solution leads to an upper bound K(3) on the overall conductivity, that depends on f2 and f3 only. In ? 9 a virial solution to this equation is given and the bound K(3) is explicitly calculated to the order c2 for a wide class of suspensions, denoted here by A. The bound includes a positive scalar parameter m2 that depends on the two-point function f2. In ? 10 the lower bound to the order c2 is given for the same class of suspensions S; it includes again the parameter M2. The so-obtained c2-bounds are not optimal for f2 given, because additional statistical information is tacitly employed when restricting the analysis to the class A. The bounds, however, coincide to the order (K)3. In ? 11 a simple formula for the statistical parameter m2 is derived and it is calculated for certain kinds of sphere distribution. An interpretation of m2 is also proposed. In ? 12 certain known results and approximations, that have been proposed to evaluate K*, are compared with the bounds in order to establish the range of applicability of the former. In this way it appears possible to extract some nontrivial statistical information about the kind of random suspensions for which the reviewed models of mechanics of composite materials can be employed when predicting the overall conductivity.

Identification of MGB cells by Volterra kernels. III. A glance into the black box

Abstract: Neuronal systems can be described by their transfer functions, which can be represented by a Volterra series expansion. While the high level of abstraction which characterizes this representation enables a global description, it is problematic, to some extent, in the context of linking the formal representation of the system to its actual structure. The formal representation is unique, yet there are multiple physical realizations of this representation. Separating the system's output into its logical components (linear, cross-linear, and self nonlinear, in this study), and inspecting the relative contribution of these components, might provide a key towards a linkage between the formal and actual representations. Based on results drawn from identification of MGB cells of the squirrel monkey, it is shown that the relative contributions can be described in neurobiological terms such as excitation and inhibition and thus be attributed to actual subsystems.

On the Identification of Non-Linear Operators and its Application (Invited contribution)

Abstract: This paper seeks to extend the analytical theory of non-linear systems, study new possible approaches to the problems of identification in a wide sense and problems involving the approximation of inverse operator for differential equations.

Nonlinear feedback control of a class of nonlinear systems

Abstract: A nonlinear feedback controller is designed on the basis of a truncated Volterra series representation of the process model, the necessary parameters of which can be obtained via suitable experiments. A simple example demonstrates the increased range and improved performance of the nonlinear controller compared to two linear controllers.

Theoretical and Experimental Characterisation of a Travelling-Wave Frequency Doubler using Generalised Volterra Series Representation

Abstract: Applying the generalized Volterra series approach we have analyzed a four-port frequency doubler. With this new analysis technique it has become possible to determine the influence of several distinct circuit parameters on output power. Especially, the contribution of the reflection coefficients has been evaluated. Comparison with experimental results revealed good agreement with theoretical values.

Optical Polynomial Vector Processing

Abstract: Polynomial processing can be applied to a number of potential applications, including general nonlinear operations using the Volterra series, eigenvalue computations via solutions to polynomial characteristic equations, pattern recognition, and polynomial neural networks[1]. Since most of these applications use vector-valued inputs (i.e., multiple-variable inputs), an optical polynomial vector processor is investigated and implemented here.

Volterra’s series solutions of free boundary plasma equilibria

Abstract: In this paper the nonlinear problem of the free boundary equilibrium of a plasma inside a conducting circular shell has been solved analytically in the high beta limit. The unknown function describing the plasma shape has been expanded in a Volterra functional series, the functional analogous to the Taylor series. The hierarchy of the linear integral equations obtained from the expansion is, at least in principle, analytically solvable, so that the solution of each equation can be given in a closed form. The analytical computations have been carried out up to the fourth order and the results compared with numerical computations.

Nonlinear Stochastic Response of Marine Vehicles

Abstract: The dynamic behavior of marine vehicles in extreme sea states is a matter of great concern following some recent and dramatic accidents. The complex problem of its prediction can be approached through the study, yet of broader scope, of nonlinear dynamic systems driven by stochastic processes. Nonlinear statistical dynamics is a relatively new and difficult field. Although the diversity of techniques now available may seem fostering, the achievement of a unified and general theory for nonlinear response to stochastic process appears as a quite remote event. Second-order statistics contain the most important information to describe a random process. Both theoretical and empirical evidence showing the superiority of the method of equivalent linearization to predict second-order statistics are exhibited and exemplified. The rationale underlying the Wiener-Hermite functional model appears to further support this affirmation. However, higher-order statistics cannot be accurately predicted within the framework of this technique whenever deviation from normal behavior becomes significant. A new technique for predicting the response moments and cumulants of nonlinear systems is presented. This technique relies upon the construction of a series of linear systems aimed at the prediction of the response statistics of a given order. Such linear systems are successively defined by linearizing the original nonlinear system and matching the Volterra functional model response statistics of the desired order. The linear system for predicting second-order statistics coincide with the one obtained using the method of equivalent linearization. This technique is exemplified by a nonlinear system governed by the Duffing equation with linear plus cubic damping. Several innovative results related to the transfer functions and the response cumulant of Volterra series are exhibited and used in our model. Response probability distributions can be constructed from knowledge of these statistical moments. Particular attention is devoted to the distribution of maximum entropy and its justification as a method of inference in such underdetennined moment problems. Finally, several applications to the rigid body behavior of marine vehicles serve to assess the accuracy and the versatility of these techniques. Response distributions of maxima so predicted compare very well with exact solutions or time domain simulation estimates when no exact solution is available.

Global bilinearization of non-linear systems and the existence of Volterra series

Abstract: A global Volterra series is developed for an analytic system defined on an analytic manifold M. Local bilinearizations are pieced together using the theory of fibre bundles, giving rise to ‘twisted’ bilinear systems denned globally on M.

Orthogonal Functional Series Representation of a Nonlinear System Subject to a Non-White Gaussian Input

Abstract: In determination of kernels of a volterra series model for an unknown nonlinear system from input/output experiments, the amount of computation required is generally excessive. If the input is a white Gaussian signal, however, it is greatly reduced by using G-functionals, which is a set of orthogonal functionals on such input. The purpose of the present paper is an extension of the theory of G-functionals, and it is shown that, a set of orthogonal functionals on non-white Gaussian signals can be derived by applying Karhunen-Loeve expansion theorem, and by use of such an orthogonal set of functionals for a basis of functional series model of a nonlinear system, the same amount of computational reduction as for a white Gaussian input can be obtained for a nonwhite Gaussian input.