ISIS project: Model Predictive Control for Systems Including Binary Variables

Background

Model Predictive Control (MPC) has proved to be a strong method to control large MIMO systems and has gained substantial interest in the industry, especially within the process and petrochemical fields. The main benefit is the possibility to handle constraints on various variables in the plant.

An extension to ordinary MPC is also to include the ability to use binary variables as control signals and as internal variables in the model description.

The central idea in MPC is to state the control problem as an optimization problem, and solve this optimization problem on-line repeatedly. When binary variables are used in MPC, the optimization problem to solve is changed from a Quadratic Program (QP) to a Mixed Integer Quadratic Program (MIQP), where the latter is known in general to be NP-hard.

The research is performed in collaboration with ABB Corporate Research.

As an example of linear MPC, let us consider a linear system

$\displaystyle x_{k+1}$ $\displaystyle =$ $\displaystyle Ax_{k}+Bu_k$  

A standard MPC controller is typically defined as
$\displaystyle u_k$ $\displaystyle =$ $\displaystyle u_{k\vert k}$  
$\displaystyle u_{(\cdot\vert k)}$ $\displaystyle =$ $\displaystyle \arg \min_{u_{(\cdot\vert k)}} J_k$  
$\displaystyle J_k$ $\displaystyle =$ $\displaystyle \sum_{j=0}^{N-1} \vert\vert x_{k+j\vert k}\vert\vert _{Q}^{2}+\vert\vert u_{k+j\vert k}\vert\vert _{R}^{2}$  
$\displaystyle u$ $\displaystyle \in$ $\displaystyle \mathcal{U},~ x \in \mathcal{X}$  

The optimization problem can be solved with quadratic programming if the control and state-constraints are linear. This is most often the case since amplitude and rate-constraints are the most common constraints in reality.

Research Area

As mentioned above, when binary signals are used in the MPC problem, an NP-hard problem has to be solved in each time instant. Our research is aiming at finding and exploring the structure in MIQP problems originating from MPC. The objective with this research is to speed up the solution of these optimization problems.

Publications

D. Axehill. A Preprocessing Algorithm for MIQP Solvers with Applications to MPC. Reglermöte 2004, 2004.
D. Axehill. A Preprocessing Algorithm for MIQP Solvers with Applications to MPC. Reglermöte 2004 and ISIS Workshop 2004, 2004.