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

Hélie, Thomas

Matignon, Denis

Mignot, Rémi

English

cote interne IRCAM: Mignot09aNone / NoneNational audiencePuzzles in pipes with negative curvature: from the Webster PDE to stable numerical simulation in real tim

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Puzzles in pipes with negative curvature: from the Webster PDE to stable numerical simulation in real time

Hal-Diderot

Open Archive Toulouse Archive Ouverte

 To cite this document: Mignot, Rémi and Hélie, Thomas and Matignon, Denis Puzzles in pipes with negative curvature: from the Webster PDE to stable numerical simulation in real time. (2009) In: IFAC workshop on Control of Distributed Parameter Systems, CDPS 09, 20-24 Jul 2009, Toulouse, France. Open Archive Toulouse Archive Ouverte (OATAO)  OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.  This is an author-deposited version published in: http://oatao.univ-toulouse.fr/  Eprints ID: 8976 Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@inp-toulouse.fr  Puzzles in pipes with negative curvature: from theWebster PDE to stable numerical simulation in real time∗R. Mignot, T. He´lie†IRCAM & CNRS, UMR 9912: Analysis-Synthesis team.D. Matignon‡Universite´ de Toulouse; ISAE, Applied Mathematics training unit.AbstractMinimal realizations of a class of delay-differential systems are derived forthe digital simulation of waveguides, modelled by the Webster horn equation.Studying their stability is an interesting issue, since negative curvatures couldlead to unstable systems. Spectral properties of Toeplitz matrix play a key rolein this work.KeywordsWebster equation, delay-differential systems, BIBO-stability, numerical sim-ulation.1 IntroductionThe wave equation with space-varying coefficients that models propagation in hornsis the Webster PDE, which is known to be conservative, whatever the shape of thehorn; hence, stable numerical schemes can be derived for it, e.g. forward Euler onthe vector (pressure, flow) with a CFL condition. But from an input-ouptput pointof view, in the case of negative curvature, it is well-known that unstable subsystemsare to be found, a paradox that has recently been fully understood in [MED08]thanks to minimal realization of a delay system modelling a convergent cone. In thepresent work, the same methodology is applied to a pipe with negative (constant)curvature, the discretization of which now gives rise to a delay-differential system.∗This work is supported by the CONSONNES project, ANR-05-BLAN-0097-01†1, pl. Igor Stravinsky, 75004 Paris, France. e-mail: {mignot,helie}@ircam.fr‡10, av. E. Belin, F-31055 Toulouse Cedex 4, France. e-mail: denis.matignon@isae.fr12 1D propagation in a convex acoustic pipeConsider the conservative acoustic 1D propagation into a pipe with varying radiusR(ℓ) = R0 cos(ℓ) where ℓ ∈ [−L/2, L/2] (with L < π) denotes the curvilinear ab-scissa measuring the length of the shape from ℓ = 0 (see Fig. 1a). Inside this pipesymmetrical w.r.t ℓ = 0, the acoustic pressure p(ℓ, t) is governed by the followingWebster equation [?], for the adimentional celerity c = 1, ∂2ℓ p(ℓ, t) + 2ζ(ℓ)∂ℓp(ℓ, t)−∂2t p(ℓ, t) = 0, where ζ = r′/r. Denoting X(ℓ, t) = [p(ℓ, t), r(ℓ)2v(ℓ, t)]T where v isthe particle velocity for the adimensional massic density ρ = 1, the acoustic problemcan be described by (see [?]) ∂ℓX(ℓ, t) +(0 1/r(ℓ)2r(ℓ)2 0)∂tX(ℓ, t) =(00)orequivalently, and introducing the state variable φ± = r(p± v),∂ℓφ±(ℓ, t)± ∂tφ±(ℓ, t) = ζ(ℓ)φ∓(ℓ, t). (1)Consider the scattering matrixM(s) defined by [φˆ+(L/2, s), φˆ−(−L/2, s)]T =M(s)[φˆ+(−L/2, s), pˆ−(L/2, sin the Laplace domain for s ∈ C+0= {s ∈ C|ℜe(s) > 0}. For r(ℓ) = r0 cos(ℓ), itscomputation yields M(s) =(T (s) R(s)R(s) T (s))where a square root of s2 − 1 is to befound...3 Approximation with pieces of conical pipesFor all N ∈ N∗ and n ∈ [0, N ]N, define ℓNn =(− 12+ nN)L and approximate the shaper with the continuous piecewise affine model r˜ as follows: if 1 ≤ n ≤ N and ℓ ∈[ℓn−1, ℓn], then r˜N (ℓ) = r(ℓn−1)+ξNn (ℓ−ℓn−1) with slope ξNn =(r(ℓn)−r(ℓn−1))/ǫNwhere ǫN = L/N . In each conical piece of pipe, the propagation of ideal sphericaln = 1 n = 2 . . . . . . n = Nℓ0 ℓ1 ℓ2 ℓN−1 ℓNTp TpTpTpTp TpTpTpTpTpR3R2R1 R4Figure 1: Shape of the pipe with radius r and its piecewise afinne approximationr˜N ... A AFFINER !waves is assumed. Solving (1) and assuming the continuity of φ± (that is, of acousticpressure and flow), we find the structure in Fig. 1b in which reflections functionsare all given by RǫN (s) where Rǫ(s) = αǫ/(s − αǫ) and αǫ = (1 − cos ǫ)/ǫ ∼0ǫ/2(see e.g. [?]). This structure corresponds to the following global state-space-likerepresentation, with input U = []T , output YN = []T , and state XN = [?]T ,sXN (s) = AN (e−τns)XN (s) +BN (e−τns)U(s), (2)YN = CN (e−τns)XN (s), (3)2where AN (w) = αǫNWN (w), WN (w) is the N×N -symmetrical Toeplitz matrix suchthat [WN (w)]ij = w|i−j|, BN (w) is composed of the first and last columns of WN (w)and CN (w) = wBN (w)T .4 SimulationsReferences[1] L. Ahlfors. Complex Analysis. McGraw-Hill, 1953.[2] W. Rudin. Real and Complex Analysis. McGraw-Hill, 1987.3

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Puzzles in pipes with negative curvature: from the Webster PDE to stable numerical simulation in real time

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