Stable Realization of a Delay System Modeling a Convergent Acoustic Cone

This paper deals with the physical modeling and the digital time simulation of acoustic pipes. We will study the simplified case of a single convergent cone. It is modeled by a linear system made of delays and a transfer function which represents the wave reflection at the entry of the cone. According to [1], the input/output relation of this system is causal and stable whereas the reflection function is unstable. In the continuous time-domain, a first state space representation of this delay system is done. Then, we use a change of state to separate the unobservable subspace and its orthogonal complement, which is observable. Whereas the unobservable part is unstable, it is proved that the observable part is stable, using the D-Subdivision method. Thus, isolating this latter observable subspace, to build the minimal realization, defines a stable system. Finally, a discrete-time version of this system is derived and is proved to be stable using the Jury criterion. The main contribution of this work is neither the minimal realization of the system nor the proofs of stability, but it is rather the solving of an old problem of acoustics which has heen achieved using standard tools of automatic control

On the singularities of fractional differential systems, using a mathematical limiting process based on physical grounds

Fractional systems are associated with irrational transfer functions for which nonunique analytic continuations are available (from some right-half Laplace plane to a maximal domain). They involve continuous sets of singularities, namely cuts, which link fixed branching points with an arbitrary path. In this paper, an academic example of the 1D heat equation and a realistic model of an acoustic pipe on bounded domains are considered. Both involve a transfer function with a unique analytic continuation and singularities of pole type. The set of singularities degenerates into uniquely defined cuts when the length of the physical domain becomes infinite. From a mathematical point of view, both the convergence in Hardy spaces of some right-half complex plane and the pointwise convergence are studied and proved

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

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

Digital waveguide simulation of convex acoustic pipes

This work deals with the physical modelling of acoustic pipes for real-time simulation, using the “Digital Waveguide Network” 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 “Kelly-Lochbaum” 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 “Waveguides”, we meet some substates of the system which do not take effect on the outputs. But, using the “Kelly-Lochbaum” 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 “Waveguide” 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

Digital waveguide modeling for wind instruments: building a state-space representation based on the Webster-Lokshin model

This paper deals with digital waveguide modeling of wind instruments. It presents the application of state-space representations for the refined 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). 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 Kelly-Lochbaum network of input/output systems and delays. 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 the time domain. The method is applied to a real trombone, and results of simulations are presented and compared with measurements. This method seems to be a promising approach in term of modularity, complexity of calculation and accuracy, for any acoustic resonators based on tubes

A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems

For conservative mechanical systems, the so-called Caughey series are known to define the class of damping matrices that preserve eigenspaces. In particular, for finite-dimensional systems, these matrices prove to be a polynomial of one reduced matrix, which depends on the mass and stiffness matrices. Damping is ensured whatever the eigenvalues of the conservative problem if and only if the polynomial is positive for positive scalar values. This paper first recasts this result in the port-Hamiltonian framework by introducing a port variable corresponding to internal energy dissipation (resistive element). Moreover, this formalism naturally allows to cope with systems including gyroscopic effects (gyrators). Second, generalizations to the infinite-dimensional case are considered. They consists of extending the previous polynomial class to rational functions and more general functions of operators (instead of matrices), once the appropriate functional framework has been defined. In this case, the resistive element is modelled by a given static operator, such as an elliptic PDE. These results are illustrated on several PDE examples: the Webster horn equation, the Bernoulli beam equation; the damping models under consideration are fluid, structural, rational and generalized fractional Laplacian or bi-Laplacian

Nonlinear damping models for linear conservative mechanical systems with preserved eigenspaces: a port-Hamiltonian formulation

This paper introduces linear and nonlinear damping models, which preserve the eigenspaces of conservative linear mechanical problems. After some recalls on the finite dimensional case and on Caughey's linear dampings, an extension to a nonlinear class is introduced. These results are recast in the port-Hamiltonian framework and generalized to infinite dimensional systems. They are applied to an Euler-Bernoulli beam, excited by a distributed force. Simulations yield sounds of xylophone, glockenspiel (etc) and some interpolations for nonlinear dampings

Numerical Simulation of Acoustic Waveguides for Webster-Lokshin Model Using Diffusive Representations

This paper deals with the numerical simulation of acoustic wave propagation in axisymmetric waveguides with varying cross-section using a Webster-Lokshin model. Splitting the pipe into pieces on which the model coefficients are nearly constant, analytical solutions are derived in the Laplace domain, enabling for the realization of the propagation by concatenating scattering matrices of transfer functions (§2). These functions involve standard differential and delay operators, as well as pseudo-differential operators of diffusive type, induced by both the viscothermal losses and the curvature. These operators are explicitly decomposed thanks to an asymptotic expansion, and the diffusive ones may be defined and classified (§3). Various equivalent diffusive realizations may be proposed, that are deeply linked to choices of cuts in the complex analysis of the transfer functions. Then, finite order approximations are given for their simulation (§4)

A MINIMAL PASSIVE MODEL OF THE OPERATIONAL AMPLIFIER : APPLICATION TO SALLEN-KEY ANALOG FILTERS

International audienceThis papers stems from the fact that, whereas there are passive models of transistors and tubes, a minimal passive model of the operational amplifier does not seem to exist. A new behavioural model is presented that is memoryless, fully described by its interaction ports, with a minimal number of equations, for which a passive power balance can be defined. The proposed model handles saturation, asymmetric power supply, and can be used with non-ideal voltage references. To illustrate the model in audio applications , the non-inverting voltage amplifier and a saturating Sallen-Key lowpass filter are considered