THE BEHAVIOR OF A MULTIPLE CHANNEL ACTIVE CONTROL-SYSTEM

The convergence behavior of an adaptive feedforward active control system is studied. This adjusts the outputs of a number of secondary sources to minimize a cost function comprising a combination of the sum of mean-square signals from a number of error sensors (the control error) and the sum of the mean-square signals fed to the secondary sources (the control effect). A steepest descent algorithm is derived which performs this function, the behavior of which is analyzed in the frequency domain not just in terms of the convergence of the individual secondary source signals, but also in terms of the convergence of the cost function and the control effort. It is found useful to describe this convergence in terms of the eigenvalues of the relevant Hessian matrix, which determine the speed of convergence of the modes of the control system, and in terms of the primary field transformed using a singular value decomposition of the transfer matrix, which determines the initial excitation of each of the modes. This illustrates that some modes not only converge slowly but also require an excessive control effort for complete convergence. This ill-conditioned behavior can be controlled by the proper choice of the cost function minimized. Laboratory experiments using a 16-loud-speaker 32-microphone control system to control the harmonic sound in an enclosure are presented. The behavior of the practical system is accurately predicted from the theoretical analysis of the adaptive algorithm. The effect of errors in the assumed transfer matrix used by the steepest descent algorithm is also briefly discussed.