Lagrangian quadratic programming approach for linear model predictive control

A Lagrangian approach to the solution of the quadratic programming problem resulting from the open-loop, optimal control law for linear model predictive control is presented. The Lagrangian is formed from the quadratic objective function, linear model, and constraint equations. By substituting the relationships determined from implicit differentiation of the model equations, the Lagrange multipliers for each of the linear model equality constraints are eliminated from the linear system obtained from partial differentiation of the Lagrangian. This technique results in a reduction in the dimension of the resulting linear system that must be solved.