Linear-quadratic discrete-time control and constant directions

The general linear-quadratic discrete-time minimization problem is studied, in which no restrictions are placed on the singularity or otherwise of certain matrices, or the appearance of cross-product terms in the performance indices. Constant directions are characterized in a number of ways, and an algorithm is presented for replacing a prescribed problem with constant directions by one of lower state and/or control dimension, lacking constant directions. The replacement is achieved using a series of state and input co-ordinate basis changes, and allows simplification of the calculations obtaining the optimal control law and performance index of the original problem.