A Halpern Fixed-Point Iteration-Based Approach to Harmonic Model Predictive Control Problem

Harmonic model predictive control (HMPC) is a variant of MPC that offers improved system performance compared to other MPC formulations within a shorter prediction horizon. However, achieving this benefit requires formulating the MPC as a quadratic second-order cone programming (QSOCP) problem, as opposed to the conventional quadratic programming problem commonly found in linear MPC formulations. This distinction arises from the need to incorporate a special class of second-order cone (SOC) intersections in the constraints. To handle the cone constraints in this problem efficiently, we introduced an explicit expression for the intersection of the SOC constraints and derived a closed-form solution projected onto it. By exploiting the inherent sparsity structure of the HMPC problem and using Cholesky decomposition, we reformulate it as a strongly QSOCP problem. This reformulation allows us to represent the objective function as a quadratic function using an identity matrix. We then use the Halpern fixed-point iteration-based scheme to solve this problem. Through numerical experiments, we demonstrated the superiority of our proposed method in terms of the computational time required to solve HMPC problems.