Feedback control of noise in a 2-D nonlinear structural acoustics model
The problem considered by the authors is complex. The cavity (it could be the interior of an aircraft) responds to vibration of one of its walls, which is regarded as a beam. The acoustic response of the cavity is coupled with the vibrations of the structure. Piezo-electric patches are bounded to the vibrating beam. These patches exert bending moments on the vibrating beam. Only a two-dimensional cross-section of the cavity is considered in this paper. The purpose of the control is to minimize the noise. While this project deals with the fast growing study of ``smar materials'', it has a more modest purpose of noise control, through action of piezo-electric ceramic actuators. Acoustics are modeled by a simple wave equation \(\phi_{tt}=\Delta\phi\), where \(\phi\) is the velocity potential. The Euler-Bernoulli beam with a Voigt-Kelvin damping is one of the equations of the interacting system. The moment exerted by a piezo-electric patch is assumed to be proportional to the voltage applied to it. The nonlinearities arise because of the surface action of the patches. The coupling between the beam and the aboustic response is also nonlinear. All this is considered in the abstract setting, where the system of state equations is written in a weak form in an appropriate Hilbert space. The well-posedness of similar system was considered in a previous paper by the same authors, and is again discussed here. Gains are first computed in the LQR setting then are inserted numerically into the nonlinear system. Several control inputs are considered, and a reasonable success in reducing the noise is claimed.
scientific article; zbMATH DE number 1010961