154 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, ..
Digital waveguide simulation of convex acoustic pipes
This work deals with the physical modelling of acoustic pipes for real-time simulation, using the âDigital Waveguide Networkâ approach and the horn equation. With this approach, a piece of pipe is represented by a two-port system with a loop which involves two delays for wave propagation, and some subsystems without internal delay. A well-known form of this system is the âKelly-Lochbaumâ framework. It allows the reduction of the computation complexity and it gives a physically meaningful interpretation of the involving subsystems. In this paper, we focus this work on the simulation of pipes
with a convex profile, for which a curvature coefficient is constant and negative. In the literature, it has been shown that such pipes have trapped modes. With the formalism of automatic control, adapted for âWaveguidesâ, we meet some substates of the system which do not take effect on the outputs.
But, using the âKelly-Lochbaumâ framework with the horn equation, two problems occur: first, even if the outputs are bounded, some substates have their values which diverge; second, there is an infinite number of such substates. The system is then unstable and cannot be simulated as such. The solution of this problem is obtained with two steps. First, we show that there is a simple standard form compatible with the âWaveguideâ approach, for which there is an infinite number of solutions which preserve the input/output relations. Second, we look for one solution which guarantees the stability of the system and which makes easier the approximation in order to get a low-cost simulation
Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions
This paper proves the asymptotic stability of the multidimensional wave
equation posed on a bounded open Lipschitz set, coupled with various classes of
positive-real impedance boundary conditions, chosen for their physical
relevance: time-delayed, standard diffusive (which includes the
Riemann-Liouville fractional integral) and extended diffusive (which includes
the Caputo fractional derivative). The method of proof consists in formulating
an abstract Cauchy problem on an extended state space using a dissipative
realization of the impedance operator, be it finite or infinite-dimensional.
The asymptotic stability of the corresponding strongly continuous semigroup is
then obtained by verifying the sufficient spectral conditions derived by Arendt
and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and V\~u
(Studia Math., 88 (1988))
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