On the SDP Relaxation of Direct Torque Finite Control Set Model Predictive Control
Luca M. Hartmann, Orcun Karaca, Tinus Dorfling, Tobias Geyer, Adam Kurpisz
TL;DR
The paper addresses long-horizon direct-torque FCS-MPC for a 3L-NPC inverter by formulating a semidefinite programming relaxation of the nonlinear, integer-valued c1 problem and solving it in parallel with a conventional node-limited branch-and-bound method. By lifting the decision variables to a matrix $\boldsymbol{\Theta}(k)$ and relaxing rank and integer constraints, the authors obtain a convex SDP whose solution yields a candidate input sequence after simple extraction. The proposed dual strategy—SDP-derived sequence plus early-stopped B&B—demonstrates improved torque transients in case studies, while maintaining computational feasibility (SDP solves in ~8–10 ms on a CPU). The work bridges SDP-based optimization with power-electronics MPC, enabling robust long-horizon control with practical runtimes and pointing to FPGA implementations and theoretical approximation guarantees as avenues for future work.
Abstract
This paper formulates a semidefinite programming relaxation for a long horizon direct-torque finite-control-set model predictive control problem. In parallel with this relaxation, a conventional branch-and-bound algorithm tailored for the original problem, but with an iteration limit to restrict its computational burden, is also solved. An input sequence candidate is extracted from the solution of the semidefinite program in the lifted space. This sequence is then compared with the so-called early-stopping branch-and-bound solution, and the best of the two is applied in a receding horizon fashion. In simulated case studies, the proposed approach exhibits significant improvements in torque transients, as the branch-and-bound alone struggles to find a meaningful solution due to the imposed limit.