Active control of the axisymmetric vibration modes of a tom-tom drum using a modal-based observer-regulator
Résumé
This paper deals with an application of active control of percussion instruments. Our setup consists of a tom-tom drum with a circular membrane, a cylindrical cavity and a circular rigid wall on which a loudspeaker is mounted. The current applied to the loudspeaker is controlled in order to modify the frequencies of the axisymmetric drum membrane modes.
A hybrid PDE-ODE model of the axisymmetric transverse vibration of the tom-tom membrane coupled to a cavity and a loudspeaker, and its recast version as an infinite-dimensional port-Hamiltonian system, are recalled. Furthermore, a truncated modal projection of the model is provided.
In a previous work, a control law for the loudspeaker current was designed that modifies the frequency of the first axisymmetric vibration mode of the drum membrane, using finite-time and passivity-based methods. In addition to a pressure sensor, a sensor for the transverse displacement of the drum membrane is needed to implement the control law.
In this contribution, a finite-time observer is first designed to estimate the modal transverse displacements, removing the need for a displacement sensor. A control law for the loudspeaker current is then proposed that modifies the frequencies of multiple drum membrane modes. Finally, the control law is tested numerically on a model containing a finite number of modes. Stability and robustness of the observer-regulator controller are discussed as well as spillover effects on observation and control.
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