Denis Matignon

Observer-based controllers for fractional differential systems

The goal of this paper is to propose observer-based controllers, either in state-space form or in polynomial representation, for fractional differential systems. As for linear differential systems of integer order, polynomial representation allows us to take advantage of the Youla parametrization in order to asymptotically reject some perturbations. This is illustrated on a worked-out example.

Time-domain simulation of damped impacted plates. II. Numerical model and results

A time-domain model for the flexural vibrations of damped plates was presented in a companion paper [Part I, J. Acoust. Soc. Am. 109, 1422-1432 (2001)]. In this paper (Part II), the damped-plate model is extended to impact excitation, using Hertzs law of contact, and is solved numerically in order to synthesize sounds. The numerical method is based on the use of a finite-difference scheme of second order in time and fourth order in space. As a consequence of the damping terms, the stability and dispersion properties of this scheme are modified, compared to the undamped case. The numerical model is used for the time-domain simulation of vibrations and sounds produced by impact on isotropic and orthotropic plates made of various materials (aluminum, glass, carbon fiber and wood). The efficiency of the method is validated by comparisons with analytical and experimental data. The sounds produced show a high degree of similarity with real sounds and allow a clear recognition of each constitutive material of the plate without ambiguity.

DIFFUSIVE REPRESENTATIONS FOR THE ANALYSIS AND SIMULATION OF FLARED ACOUSTIC PIPES WITH VISCO-THERMAL LOSSES

Acoustic waves travelling in axisymmetric pipes with visco-thermal losses at the wall obey a Webster–Lokshin model. Their simulation may be achieved by concatenating scattering matrices of elementary transfer functions associated with nearly constant parameters (e.g. curvature). These functions are computed analytically and involve diffusive pseudo-differential operators, for which we have representation formula and input-output realizations, yielding direct numerical approximations of finite order. The method is based on some involved complex analysis.

Numerical simulations of xylophones. II. Time-domain modeling of the resonator and of the radiated sound pressure

This paper presents a time-domain modeling for the sound pressure radiated by a xylophone and, more generally, by mallet percussion instruments such as the marimba and vibraphone, using finite difference methods. The time-domain model used for the one-dimensional (1-D) flexural vibrations of a nonuniform bar has been described in a previous paper by Chaigne and Doutaut [J. Acoust. Soc. Am. 101, 539–557 (1997)] and is now extended to the modeling of the sound-pressure field radiated by the bar coupled with a 1-D tubular resonator. The bar is viewed as a linear array of equivalent oscillating spheres. A fraction of the bar field excites the tubular resonator which, in turn, radiates sound with a certain delay. In the present model, the open end of the resonator is represented by an equivalent pulsating sphere. The total sound field is obtained by summing the respective contributions of the bar and tube. Particular care is given for defining a valid approximation of the radiation impedance, both in continuous and discrete time domain, on the basis of Kreiss’s theory. The model is successful in reproducing the main features of real instruments: sharp attack, tuning of the bar, directivity, tone color, and aftersound due to the bar-resonator coupling.

Efficient solution of a wave equation with fractional-order dissipative terms

We consider a wave equation with fractional-order dissipative terms modeling visco-thermal losses on the lateral walls of a duct, namely the Webster-Lokshin model. Diffusive representations of fractional derivatives are used, first to prove existence and uniqueness results, then to design a numerical scheme which avoids the storage of the entire history of past data. Two schemes are proposed depending on the choice of a quadrature rule in the Laplace domain. The first one mimics the continuous energy balance but suffers from a loss of accuracy in long time simulation. The second one provides uniform control of the accuracy. However, even though the latter is more efficient and numerically stable under the standard CFL condition, no discrete energy balance has been yet found for it. Numerical results of comparisons with a closed-form solution are provided.

Simulation of fractionally damped mechanical systems by means of a Newmark-diffusive scheme

A Newmark-diffusive scheme is presented for the time-domain solution of dynamic systems containing fractional derivatives. This scheme combines a classical Newmark time-integration method used to solve second-order mechanical systems (obtained for example after finite element discretization), with a diffusive representation based on the transformation of the fractional operator into a diagonal system of linear differential equations, which can be seen as internal memory variables. The focus is given on the algorithm implementation into a finite element framework, the strategies for choosing diffusive parameters, and applications to beam structures with a fractional Zener model.

Fractional integro-differential boundary control of the Euler-Bernoulli beam

Absorbing boundary conditions are generally associated to long-range memory behaviors. In the case of the Euler-Bernoulli beam, they are naturally based on Abel-Volterra operators of order 1/2. Diffusive realizations of them are introduced and used for the construction of an original and efficient boundary dynamic feedback control.

Representations with poles and cuts for the time-domain simulation of fractional systems and irrational transfer functions

Fractional differential systems are infinite-dimensional systems which are difficult to study and simulate: they can be represented with poles and cuts. This representation applies to a wider class of irrational transfer functions, and is most useful for signal processing purposes, such as frequency-domain and time-domain simulations: the approximations in low dimension which give the most striking numerical results are obtained through an optimization procedure, the parameters of which are meaningful from a signal point of view. Ten such systems of increasing complexity are thoroughly investigated.

Diffusive Realisations of Fractional Integrodifferential Operators: Structural Analysis Under Approximation☆

Abstract Fractional integrals and derivatives are usually presented as non-standard convolutions, but they are also linked to classical diffusive operators on a Hilbert state space of infinite dimension. A variety of equivalent realisations are proposed, which can naturally be approximated by truncation in finite dimension. This process gives rise to convergent approximation algorithms from which some numerical simulations are derived.

Resonance modes in a one-dimensional medium with two purely resistive boundaries: Calculation methods, orthogonality, and completeness

Studying the problem of wave propagation in media with resistive boundaries can be made by searching for “resonance modes” or free oscillations regimes. In the present article, a simple case is investigated, which allows one to enlighten the respective interest of different, classical methods, some of them being rather delicate. This case is the one-dimensional propagation in a homogeneous medium having two purely resistive terminations, the calculation of the Green function being done without any approximation using three methods. The first one is the straightforward use of the closed-form solution in the frequency domain and the residue calculus. Then, the method of separation of variables (space and time) leads to a solution depending on the initial conditions. The question of the orthogonality and completeness of the complex-valued resonance modes is investigated, leading to the expression of a particular scalar product. The last method is the expansion in biorthogonal modes in the frequency domain, the modes having eigenfrequencies depending on the frequency. Results of the three methods generalize or∕and correct some results already existing in the literature, and exhibit the particular difficulty of the treatment of the constant mode.