149 research outputs found
Diffusive representations for fractional Laplacian: systems theory framework and numerical issues
Bridging the gap between an abstract definition of pseudo-differential operators, such as (-\Delta)^{\gamma} for - 1/2 < \gamma < 1/2, and a concrete way to represent them has proved difficult; deriving stable numerical schemes for such operators is not an easy task either. Thus, the framework of diffusive representations, as already developed for causal fractional integrals and derivatives,
is being applied to fractional Laplacian: it can be seen as an extension of the Wiener-ÂHopf factorization and splitting techniques to irrational transfer functions
Optimal control of fractional systems: a diffusive formulation
Optimal control of fractional linear systems on a finite horizon can be classically formulated using the adjoint system. But the adjoint of a causal fractional integral or derivative operator happens to be an anti-causal operator: hence, the adjoint equations are not easy to solve in the first place. Using an equivalent diffusive realization helps transform the original problem into a coupled system of PDEs, for which the adjoint system can be more easily derived and properly studied
Standard diffusive systems are well-posed linear systems
The class of well-posed linear systems as introduced by Salamon has become a well-understood class of systems, see e.g. the work of Weiss and the book of Staffans. Many partial partial differential equations with boundary control and point observation can be formulated as a well-posed linear system. In parallel to the development of well-posed linear systems, the class of diffusive systems has been developed. This class is used to model systems for which the impulse response has a long tail, i.e., decays slowly, or systems with a diffusive nature, like the Lokshin model in acoustics. Another class of models arethe fractional differential equations, i.e., a system which has fractional powers of s in its transfer functio
Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics
A fractional time derivative is introduced into the Burger's equation to
model losses of nonlinear waves. This term amounts to a time convolution
product, which greatly penalizes the numerical modeling. A diffusive
representation of the fractional derivative is adopted here, replacing this
nonlocal operator by a continuum of memory variables that satisfy local-in-time
ordinary differential equations. Then a quadrature formula yields a system of
local partial differential equations, well-suited to numerical integration. The
determination of the quadrature coefficients is crucial to ensure both the
well-posedness of the system and the computational efficiency of the diffusive
approximation. For this purpose, optimization with constraint is shown to be a
very efficient strategy. Strang splitting is used to solve successively the
hyperbolic part by a shock-capturing scheme, and the diffusive part exactly.
Numerical experiments are proposed to assess the efficiency of the numerical
modeling, and to illustrate the effect of the fractional attenuation on the
wave propagation.Comment: submitted to Siam SIA
Systems control theory applied to natural and synthetic musical sounds
Systems control theory is a far developped field which helps to study stability, estimation and control of dynamical systems. The physical behaviour of musical instruments, once described by dynamical systems, can then be controlled and numerically simulated for many purposes.
The aim of this paper is twofold: first, to provide the theoretical background on linear system theory, both in continuous and discrete time, mainly in the case of a finite number of degrees of freedom ; second, to give illustrative examples on wind instruments, such as the vocal tract represented as a waveguide, and a sliding flute
Damping models for PDEs: a port-Hamiltonian formulation
Some methodologies developped by practitioners to model damping phenomena for distributed parameter systems will be listed and revisited. We will try to put them in the framework of port-Hamiltonian systems, either linear or non-linear, either finite or infinite-dimensional. The most significant distinction occurs between static and dynamic damping models: for the latter, a particular attention will be paid to the notion of memory variables, such as those defined in mechanical engineering
Fractional equations and diffusive systems: an overview
The aim of this discussion is to give a broad view of the links between fractional differential equations (FDEs) or fractional partial differential equations (FPDEs) and so-called diffusive representations (DR). Many aspects will be investigated: theory and numerics, continuous time and discrete time, linear and nonlinear equations, causal and anti-causal operators, optimal diffusive representations, fractional Laplacian.
Many applications will be given, in acoustics, continuum mechanics, electromagnetism, identification, ..
State-space representation for digital waveguide networks of lossy flared acoustic pipes
This paper deals with digital waveguide modeling of wind instruments. It presents the application of state-space representations to the acoustic model of Webster-Lokshin. This acoustic model describes the propagation of longitudinal waves in axisymmetric acoustic pipes with a varying cross-section, visco-thermal losses at the walls, and without assuming planar or spherical waves. Moreover, three types of discontinuities of the shape can be taken into account (radius, slope and curvature), which can lead to a good fit of the original shape of pipe. The purpose of this work is to build low-cost digital simulations in the time domain, based on the Webster-Lokshin model. First, decomposing a resonator into independent elementary parts and isolating delay operators lead to a network of input/output systems and delays, of Kelly-Lochbaum network type. Second, for a systematic assembling of elements, their state-space representations are derived in discrete time. Then, standard tools of automatic control are used to reduce the complexity of digital simulations in time domain. In order to validate the method, simulations are presented and compared with measurements
Puzzles in pipes with negative curvature: from the Webster PDE to stable numerical simulation in real time
Minimal realizations of a class of delay-differential systems are derived for the digital simulation of waveguides, modelled by the Webster horn equation. Studying their stability is an interesting issue, since negative curvatures could lead to unstable systems. Spectral properties of Toeplitz matrix play a key role in this work
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