Feedforward Control is Dual to Deconvolution

Bo Bernhardsson and Mikael Sternad

International Journal of Control, vol 57, pp 393-405, February 1993.

Paper available in Pdf

Outline:

Dualities between control and estimation problems are useful concepts in optimal control theory. By demonstrating a duality, the solution to one set of problems can be applied also to the dual set of problems. The paper discusses such a duality, which does not seem to have been noticed before.

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.

For 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. The result applies in continuous and discrete time, for the minimization of all transfer function norms which are invariant under transposition of transfer function matrices.

A derivation of a polynomial solution to the frequency-weighted discrete-time MIMO LQG feedforward control problems illustrates the use of the duality.

Related publications:

Earlier paper in IJC 1988, in which a relation, but not a duality, could be demonstrated.
Paper in IEEE Trans. AC 1995, where the duality is utilized in averaged H-2 robust design.
Book chapter 1993, incuding multivariable on LQ disturbance measurement feedforward design.
Paper in IEEE Trans. ASSP 1991, which includes multivariable deconvolution.