Optimization in the Context of Active Control of Sound

Abstract

A problem of eliminating 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 control sources are optimized using different criteria. Minimization of the overall absolute acoustic source strength is equivalent to minimization of multi-variable complex functions in the sense of L ₁ with conical constraints. The global L ₁ optimum appears to be a special layer of monopoles on the perimeter of the protected region. The use of quadratic cost functions, e.g., the L ₂ norm of the controls, leads to a versatile numerical procedure. It allows one to analyze sophisticated geometries in the case of a constrained minimization. Finally, minimization of power consumed by an active control system always involves interaction between the sources of sound and the surrounding acoustic field, which was not the case for either L ₁ or L ₂. One can, in fact, build a control system that would require no power input for operation and may 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.

Work supported by the C&I Program, NASA Langley Research Center; performed while the authors were in residence at ICASE, NASA Langley Research Center.