Optimization Problems in the Context of Active Control of Sound

Abstract

The problem of suppressing the unwanted time-harmonic noise on a predetermined region of interest is solved by active means, i.e., by introducing the additional sources of sound, called controls, that generate the appropriate annihilating signal (anti-sound). The general solution for controls has been obtained previously for both the continuous and discrete formulation of the problem. Next, the controls can be optimized using different criteria. Minimization of their overall absolute acoustic source strength is equivalent to minimization of multi-variable complex functions in the sense of L₁ with conical constraints. The global L₁ optimum appears to be a special layer of monopoles on the perimeter of the protected region. The use of quadratic cost functions, such as the L₂ norm of the control sources, leads to a versatile numerical procedure. It allows one to analyze sophisticated geometries in the case of a constrained minimization. The optima obtained in the sense of L₂ differ drastically from those obtained in the sense of L₁. Finally, minimization of power required for operation of an active control system necessarily involves interaction between the sources of sound and the surrounding acoustic field. This was not the case for either L₁ or L₂. It turns out that one can build a surface control system that would require no power input for operation and would even produce a net power gain while providing the exact noise cancellation. This, of course, comes at the expense of having the original sources of noise produce even more energy.

In the talk, we will introduce the mathematical formulation of the noise control problem, describe the general solution for controls, and then outline the three foregoing optimization formulations, with the focus on the optimal solutions in the sense of L₁.