  In this brief, we consider the constrained optimization problem underpinning
model predictive control (MPC). We show that this problem can be decomposed
into an unconstrained optimization problem with the same cost function as the
original problem and a constrained optimization problem with a modified cost
function and dynamics that have been precompensated according to the solution
of the unconstrained problem. In the case of linear systems subject to a
quadratic cost, the unconstrained problem has the familiar LQR solution and the
constrained problem reduces to a minimum-norm projection. This implies that
solving linear MPC problems is equivalent to precompensating a system using LQR
and applying MPC to penalize only the control input. We propose to call this a
constraint-separation principle and discuss the utility of both constraint
separation and general decomposition in the design of MPC schemes and the
development of numerical solvers for MPC problems.
