Design and Simulation of a Field-Oriented Control System for a Permanent Magnet Synchronous Motor Using State-Space Modeling

Abstract

This article presents the design and simulation of a high-performance Field-Oriented Control (FOC) system for a Permanent Magnet Synchronous Motor (PMSM) within a state-space modeling framework. The continuous-time d–q model of a surface-mounted PMSM is derived, its eigenvalues are analyzed at a representative operating point to confirm small-signal stability, and the resulting differential equations are integrated by a hand-coded fixed-step fourth-order Runge–Kutta (RK4) routine. The accuracy of the integrator is benchmarked against MATLAB's adaptive ode45 over an open-loop step, and the empirical convergence rate is shown to follow the theoretical fourth-order slope until floating-point precision becomes the dominant error source. A cascaded FOC controller — outer speed loop and two inner d- and q-axis current loops with cross-coupling feed-forward — is then designed by pole-placement at a 200 Hz current-loop bandwidth and a 40 Hz speed-loop bandwidth. The closed-loop drive achieves a 45 ms 10-90 % rise time, 6 % overshoot, sub-milliradian-per-second steady-state error, and a 6 rad/s mechanical-speed dip on a 1.5 N·m load step that is recovered in under 30 ms. A back-EMF based speed estimator is integrated into the same simulation, demonstrating a software replacement for the rotor-position sensor with a steady-state estimation error of about 0.07 % of the operating speed under nominal parameters. A parameter-sensitivity study further shows that this estimator is mildly sensitive to stator-resistance drift (±20 %) but highly sensitive to magnet-flux mismatch (a 10 % error in ψPM produces a persistent ≈9-11 rad/s bias), quantifying the trade-off between simplicity and robustness in back-EMF based sensorless drives. The work demonstrates how three core computational tools — state-space analysis, numerical integration of ordinary differential equations, and observer design — combine to produce a working motor drive.