Optimal controllers and adaptive controllers for multichannel feedforward control of stochastic disturbances

A time domain formulation for the multichannel feedforward control problem is used to derive an optimality condition for the least squares controller. It is also used to motivate an instantaneous steepest descent algorithm for the adaptation of the controller, which is known as the filtered error LMS algorithm. The convergence rate of this algorithm is limited by the correlation properties of each reference signal, their cross-correlation properties and the dynamics and coupling within the plant response. An expression is then derived for the transfer function of the optimum least squares controller. This suggests a new architecture for the adaptive controller whose convergence rate is not limited by the factors mentioned above. A set of white and uncorrelated reference signals are generated to drive a modified matrix of control filters, whose outputs are multiplied by the inverse of the minimum phase part of the plant response matrix before being fed to the physical plant. The error signals from the plant are then used to update this controller after being fed through the time-reverse transpose of the all-pass part of the plant matrix. The relationship is also discussed between this architecture and that using a Singular Value Decomposition of the plant response, which is used to control tonal disturbances.