Solving constrained LQR problems by eliminating the inputs from the QP

In this paper a new approach to formulate the constrained Linear Quadratic Regulator (LQR) problem as a Quadratic Programming (QP) problem is introduced. The new approach takes advantage of the (Moore-Penrose) generalized inverse to eliminate control inputs as decision variables, hence the optimization is performed only over the states belonging to the prediction horizon. This allows one to save on computation if an interior point method is used to solve the QP problem compared to using existing formulations, where the optimization is done over the states and inputs.