Feedforward control is dual to deconvolution

Abstract

A duality is demonstrated between optimal feedforward control and optimal deconvolution, or input estimation. These two problems are normally discussed separately in the literature, but have close similarities. Duality between them can be demonstrated if and only if one uses general problem formulations, with frequency-shaped weighting in the criteria. From one of the problems, the dual problem can then be obtained immediately from the block diagram, by reversing the directions of arrows, interchanging summation points and node points and transposing all transfer function matrices. This result applies for continuous and discrete time problems, as well as for minimization of J = ||G||, for any transfer function norms for which ||G^T|| = ||G||. A derivation of a polynomial solution to the frequency-weighted discrete-time MIMO LQG feedforward control problem illustrates the use of the duality.