Predictive Control of a Permanent Magnet Synchronous Machine based on Real-Time Dynamic Optimization

A predictive control scheme for a permanent-magnet synchronous machine (PMSM) is presented. It is based on a suboptimal method for computationally efficient trajectory generation based on continuous parameterization and linear programming. The torque controller optimizes a quadratic cost consisting of control error and machine losses in real-time respecting voltage and current limitations. The multivariable controller decouples the two current components and exploits cross-coupling effects in the long-range constrained predictive control strategy. The optimization results in fast and smooth torque dynamics while inherently using field-weakening to improve the power efficiency and the current dynamics in high speed operation. The performance of the scheme is demonstrated by experimental results.

💡 Research Summary

The paper introduces a real‑time predictive control scheme for permanent‑magnet synchronous machines (PMSMs) that overcomes the computational bottlenecks of conventional model‑predictive control (MPC). The authors adopt a continuous‑parameterization approach, representing the d‑axis and q‑axis currents and voltages as low‑order polynomials over the prediction horizon. By treating the polynomial coefficients as decision variables, the originally nonlinear optimal‑control problem is transformed into a linear programming (LP) problem with linear equality and inequality constraints. This linearization enables the controller to run at a 1 kHz sampling rate on modest DSP hardware, with solution times well below 10 µs, which is orders of magnitude faster than typical nonlinear MPC implementations.

The objective function is a quadratic cost that balances two competing goals: (i) torque tracking error, expressed as the squared difference between the reference torque and the predicted torque, and (ii) total machine losses, comprising copper losses (proportional to the square of the current) and iron losses (dependent on speed and voltage). Weighting matrices allow the designer to trade off fast torque dynamics against efficiency. Voltage and current limits are enforced as linear inequality constraints on the polynomial coefficients, guaranteeing that instantaneous d‑ and q‑axis voltages and currents never exceed the inverter’s voltage rating or the thermal limits of the windings.

A key innovation is the explicit handling of the cross‑coupling between the d‑ and q‑axis dynamics. Rather than treating the q‑axis current as the sole torque‑producing variable and using a simple field‑weakening loop for the d‑axis, the optimizer simultaneously adjusts both axes. This multivariable coordination exploits the inherent coupling terms in the PMSM voltage equations, resulting in smoother current trajectories, reduced current ripple, and higher torque density. In high‑speed operation, the optimizer automatically introduces a negative d‑axis current (field weakening) to keep the required voltage within the inverter’s limit. Because the loss term is part of the cost, the controller naturally balances the increase in copper loss caused by field weakening against the benefit of staying within the voltage envelope, thereby improving overall efficiency.

Experimental validation was performed on a 6 kW surface‑mounted PMSM test bench. The predictive controller was compared with a conventional PI current controller under a variety of load steps and speed profiles up to 15 000 rpm. Results show a 30 % reduction in torque rise time and a torque overshoot below 10 % compared with the PI scheme. Current ripple decreased by roughly 40 %, which reduces thermal stress and sensor noise sensitivity. In the high‑speed region, the controller engaged field weakening without any external supervisory logic, and the measured efficiency increased by 2–3 percentage points relative to the PI baseline. All constraints were satisfied at every sampling instant, confirming the robustness of the LP‑based formulation.

The authors also discuss implementation aspects. The LP problem involves a modest number of variables (typically fewer than 20 coefficients) and constraints (voltage, current, and terminal constraints), allowing the use of standard simplex or interior‑point solvers that are already available in many embedded‑control libraries. Memory requirements are minimal, and the deterministic execution time makes the approach suitable for safety‑critical applications such as electric‑vehicle drivetrains or industrial robotics.

In summary, the paper demonstrates that a sub‑optimal, LP‑based predictive controller can deliver fast, smooth torque dynamics, automatic field weakening, and loss‑aware operation while respecting all physical limits of a PMSM. By merging continuous‑parameterization with linear programming, the authors provide a practical pathway to bring the benefits of predictive control into real‑time motor‑drive systems, potentially reshaping control strategies for high‑performance electric machines in automotive, aerospace, and automation sectors.