@MASTERSTHESIS\{IMM2008-05623,
    author       = "J. Yang",
    title        = "Numerical Methods for Model Predictive Control",
    year         = "2008",
    school       = "Informatics and Mathematical Modelling, Technical University of Denmark, {DTU}",
    address      = "Richard Petersens Plads, Building 321, {DK-}2800 Kgs. Lyngby",
    type         = "",
    note         = "Supervised by Assoc. Prof. John Bagterp J{\o}rgensen, {IMM,} {DTU}.",
    url          = "http://www2.compute.dtu.dk/pubdb/pubs/5623-full.html",
    abstract     = "This thesis presents two numerical methods for the solutions of the unconstrained optimal control problem in model predictive control (MPC). The two methods are Control Vector Parameterization (CVP) and Dynamic Programming (DP). This thesis also presents a structured Interior-Point method for the solution of the constrained optimal control problem arising from {CVP}.
{CVP} formulates the unconstrained optimal control problem as a dense {QP} prob- lem by eliminating the states. In {DP,} the unconstrained optimal control problem is formulated as an extended optimal control problem. The extended optimal control problem is solved by {DP}. The constrained optimal control problem is formulated into an inequality constrained {QP}. Based on Mehrotra’s predictor- corrector method, the {QP} is solved by the Interior-Point method.

Each method discussed in this thesis is implemented in Matlab. The Matlab simulations verify the theoretical analysis of the computational time for the different methods. Based on the simulation results, we reach the following conclusion: The computational time for {CVP} is cubic in both the predictive horizon and the number of inputs. The computational time for {DP} is linear in the predictive horizon, cubic in both the number of inputs and states. The complexity is the same in terms of solving the constrained or unconstrained optimal control problem by {CVP}. Combining the effects of the predictive horizon, the number of inputs and the number of states, {CVP} is efficient for optimal control problems with relative short predictive horizons, while {DP} is efficient for optimal control problems with relative long predictive horizons.

The investigations of the different methods in this thesis may help others choose the efficient method to solve different optimal control problems. In 
addition, the {MPC} toolbox developed in this thesis will be useful for forecasting and comparing the results between the {CVP} method and the {DP} method."
}