Nonlinear Optimal Control for the Nine-Phase Permanent Magnet Synchronous Motor

Multi-phase electric motors and in particular nine-phase permanent magnet synchronous motors (9-phase PMSMs) find use in electric actuation, traction and propulsion systems. They exhibit advantages comparing to three-phase motors because of achieving high power and torque rates under moderate variations of voltage and currents in their phases, while also exhibiting fault tolerance. In this article a novel nonlinear optimal control method is developed for the dynamic model of nine-phase PMSMs. First it is proven that the dynamic model of these motors is differentially flat. Next, to apply the proposed nonlinear optimal control, the state-space model of the nine-phase PMSM undergoes an approximate linearization process at each sampling instance. The linearisation procedure is based on first-order Taylor-series expansion and on the computation of the system's Jacobian matrices. It takes place at each sampling interval around a temporary operating point which is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. For the linearized model of the system an H-infinity feedback controller is designed. To compute the feedback gains of this controller an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. First it is demonstrated that the H-infinity tracking performance criterion is satisfied, which signifies robustness of the control scheme against model uncertainty and perturbations. Moreover, under mild assumptions it is also proven that the control loop is globally asymptotically stable. Additionally it is experimentally confirmed through simulation tests, that the nonlinear optimal control method achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs. Finally, to apply state estimation-based control without the need to measure the entire state vector of the nine-phase PMSM, the H-infinity Kalman Filter is used as a robust state estimator.