Linear-Quadratic MPC

Model predictive control (MPC) is a control strategy which relies on the system's model to predict and optimize its trajectory online, applying inputs from the beginning of the predicted horizon. There are many algorithms for trajectory optimization, each adapted for a specific use-case. When talking about nonlinear systems in general, main factors determining the applicability of a specific algorithm are if the system is represented in continuous or discrete time and if its dynamics are continuous or hybrid.

In this course we will limit ourselves to the specific case of linear systems¹ represented in discrete time with a quadratic objective (LQ control problem) and box constraints on the system's inputs ². This specific control problem can be transcribed into a QP problem with specialized solvers dedicated to finding its solution. We will go through two formulations of this problem:

• direct - both states ${\mathbf{x}}_k$ and inputs ${\mathbf{u}}_k$ are the decision variables,
• indirect - only inputs ${\mathbf{u}}_k$ are decision variables.