Inverse-Design of Disturbance Observers for SISO Systems via Closed-Loop Transfer Function Matching: Application to SPMSM Back-EMF Estimation

This paper presents an inverse-design method for disturbance observers of single-input single-output (SISO) plants of relative degree one, based on closed-loop transfer function matching. Given a designer-specified target closed-loop transfer function, the proposed framework directly derives the corresponding observer feedback transfer function for single-input single-output plants of relative degree one. Thus, the observer structure, parameters, and discrete-time realization are obtained systematically, avoiding heuristic structure selection and complex iterative gain tuning. The proposed inverse-design method is applied to back-EMF estimation for a surface-mounted permanent-magnet synchronous motor (SPMSM) to validate its effectiveness. Motivated by the need to extract rotor position and speed information from the back-EMF for sensorless control, the target closed-loop transfer function is chosen as a second-order band-pass filter to preserve phase information while attenuating harmonic distortion. The resulting back-EMF observer achieves near-zero phase delay at the operating frequency and provides improved harmonic attenuation compared with a gain-optimized PI-type back-EMF observer. Simulation results under injected 5th-, 7th-, and 11th-order harmonics and ±30% motor parameter variations confirm robust harmonic rejection, stable magnitude response, and near-zero phase delay at the selected center frequency, with 44%, 52%, and 70% reductions in the corresponding harmonic components compared with the gain-optimized PI-type back-EMF observer. Experimental results further show near-zero phase delay at different operating speeds and angle estimation error bounded within ±5°. These results validate the proposed closed-loop transfer function matching approach for practical SPMSM back-EMF estimation.