EXPLICIT SOLUTION TO THE SINGULAR LQ REGULATION PROBLEM

In this note, an explicit solution to the multivariable discrete linear quadratic (LQ) regulation problem is obtained in the limiting singular case where the input weighting matrix tends to zero. Such a solution follows from a suitable spectral factorization of the input spectrum density matrix under the assumption that the system is stabilizable and detectable, and its transfer function matrix is a full rank. The suitable spectral factor is shown to be the product of the system minimum-phase image and its unitary interactor matrix. The unitary interactor matrix defined here is a special case of the nilpotent interactor matrix defined in [5].