Variable Elimination in Model Predictive Control Based on K-SVD and QR Factorization

For linearly constrained least-squares problems that depend on a vector of parameters, this article proposes techniques for reducing the number of involved optimization variables. After first eliminating equality constraints in a numerically robust way by QR factorization, we propose a technique based on singular value decomposition (SVD) and unsupervised learning, that we call
$K$
-SVD, and neural classifiers to automatically partition the set of parameter vectors in
$K$
nonlinear regions in which the original problem is approximated by using a smaller set of variables. For the special case of parametric constrained least-squares problems that arise from model predictive control (MPC) formulations, we propose a novel and very efficient QR factorization method for eliminating equality constraints. Together with SVD or
$K$
-SVD, the method provides a numerically robust alternative to standard condensing and move blocking, and to other complexity reduction methods for MPC based on basis functions. We show the good performance of the proposed techniques in numerical tests and in a problem of linearized MPC of a nonlinear benchmark process.