On the use of proper weighting in reference tracking optimal control with guaranteed DARE solvability

The paper suggests to apply a proper output weight to achieve guaranteed discrete algebraic Riccati equation (DARE) solvability of the LQ optimal control problem with output tracking. Usually a positive semi definite state weighting matrix (Q) is mentioned to be sufficient for solvable DARE. However, the literature also proposes more general rank conditions. The paper shows a sufficient condition to guarantee DARE solvability under output tracking problems, by the appropriate selection of a Q > 0 matrix. At output tracking quadratic problems Q results as a sparse matrix. Therefore, a special attention has to be carried on the output weighting strategy guaranteing DARE solvability for a class of linear and mostly integrating systems. Injecting additional weights, the closed system performance can be increased. The importance of the proper weighting strategy is illustrated with a numerical example where it has been pointed out that inappropriate weights leads to infeasible DARE.