Semyon Tsynkov
(Department of Mathematics,
North Carolina State University & Tel Aviv University)
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.
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