# PhD Project

## Sound synthesis by physical modeling preserving passivity, and input/output inversion

Sound synthesis by physical modeling endeavors to simulate existing or imaginary sound production or transformation systems which comply with the laws of physics. It may be, for example, musical instruments (percussion, strings, winds, voice), electro-acoustic systems and electronic circuits (amplifiers, guitars, effects pedals, analog synthesizers). The advantage of this type of synthesis is to recover not only the timbre but also all natural behaviors (attacks, transient distortions, etc.).

This thesis is part of the work of the Analysis/Synthesis team of IRCAM. The objective is to build automatic simulation methods (synthesis) and inversion methods (analysis) of physical systems and explicitly exploit their passivity/dissipativity. Indeed, whether simple or complex, sound production systems have a common property: off the excitation sources (generators), they are passive. Very few sound analysis/synthesis tools seek to ensure and reproduce this property.

Standard tools usually consist of applying numerical schemes (finite differences, finite elements, decomposition in digital waveguides, etc.) for which the conditions of stability is to be found. Specific (and very effective) methods of discretization exist, which guarantee the conservation of energy. The numerical scheme can then no longer be preliminary but a result of the original model. It is based on an appropriate variational formulation (Hamiltonian formulation), which encodes more structural information than the differential formulation (PDEs). In summary, rather than discretizing the differential problem, discretizing the Hamiltonian leads to ensure a physical principle, here, the conservation of energy. The research conducted in this thesis is motivated by the fact that dissipative systems (which are not Hamiltonian) can be approximated by similar discretization methods which explicitly preserve the dynamic of power dissipation (that is to say, respecting the intrinsic physical principle of dissipation).

To address this issue, we will use a method of systems theory that can well represent this property and where current research is very active: it is the "port-Hamiltonian systems" framework . Historically, the research of schemes preserving the original behavior of energy was introduced in analytical mechanics and motivated by the numerical study of the stability of celestial systems (solar system, for example). Research on so-called symplectic schemes (eg preserving symmetries, natural geometric invariants, etc.) are also very active. In all cases, rather than discretizing the differential formulation (PDEs), an adequate variational formulation of the problem is discretized, which "encodes more structural information" (the Hamiltonian for the conservation of energy). The "port-Hamiltonian systems" introduced by theory of automatic systems are a possible generalization. They can integrate dissipation, the consideration some inputs, and connect multiple systems together while maintaining the energy balance.

The problem of inversion is itself ill-posed in general, in the sense that there are an infinite number of possible causes for the same result. In addition, the solutions may be very sensitive to small changes in parameters. It becomes crucial to regularize the problem to determine an appropriate solution. Again, energy considerations may be useful for this.

Because the phenomena involved in musical instruments can be complex (fractional derivation for the dissipation in acoustic tubes, non-trivial geometry, etc.), they are good candidates to advance (theoretical and applied) research on "port-Hamiltonian systems". Conversely, sound analysis/synthesis has much to withdraw from this approach insofar as the quality of dissipations plays a crucial role in the sound realism. It therefore becomes necessary to extend the results obtained for the case of conservative (Hamiltonian) systems to the dissipative case. In addition, the problems of inversion and observation status from its measured, include information on the structure of dynamic, has a regularizing, and any constraint preserving a known invariant system power. Finally, energy is often used as a Lyapunov functional in automatic system stabilization. But it can also draw a natural distance to measure a difference between the states of two twin systems. This can be exploited for the construction of state observers. Physical models of valves (reed, lips, glottis) used in sound synthesis have not a well-posed energy balance. The approach considered in this thesis should help overcome this problem. It will also focus on an automatic code generator for real-time simulations.