Feedback control of sound radiation from a plate excited by a turbulent boundary layer

Abstract

In this paper it is shown that the problem of applying active control to a simply supported plate excited by a turbulent boundary layer can be presented in a form which allows the application of optimal control theory. Linear regulator theory and stochastic linear regulator theory are briefly summarized. The approach taken follows that of Baumann et al. who show that the optimal control theory can be applied to structural acoustic problems if the sound power radiated by the structure is estimated from the vibration signal by means of a ‘‘radiation filter.’’ A modal model for the vibration of an elastic structure is presented and the state space realization of the model is given. It is shown that a simple Corcos model of the turbulent boundary layer excitation can be modeled by using a number of ‘‘excitation filters.’’ The state space realization of these filters can be combined with the state space model of a vibrating plate to give a model to which optimal control theory can be applied. In similar fashion the sound power radiated by the plate can be estimated by means of a radiation filter matrix. A detailed account of the calculation of such a radiation filter matrix is given. The state space forms of the excitation filters and radiation filter matrix can be combined with the state space model of the plate to give a system state space realization which can be used in the solution of a linear quadratic regulator problem. Optimal control analysis is then applied to the state equations and optimal reductions in the far-field radiated power are obtained for various arrangements of control forces. These results give insight into the possibilities for obtaining useful reductions in turbulent boundary layer induced noise in aircraft by the application of feedback control.