Performance Comparison of Two Estimators for Two-Phase Permanent Magnet Synchronous Motor

مقارنة اداء مخمنين لمحرك تزامني ذو طورين وذات مغانيط دائمية

Dr. Amjad J. Humaidi . امجد جليل حميد

ISSN: 25219154 / eISSN 25219162 Year: 2011 Volume: 14 Issue: 1 Pages: 46-60
Publisher: Al-Nahrain University جامعة النهرين


This paper presents and compares the performance of two Kalman filter schemes, the discrete extended Kalman filter (EKF) and unscented Kalman filter (UKF) for estimating the states (winding currents, rotor speed and rotor angular position) of two-phase Permanent Magnet Synchronous Motor (PMSM). Estimating the states of the system is performed by propagating the mean and covariance of the state distribution. For linear systems, the general recursive Kalman filter algorithm based on MMSE (minimum mean squared error) is the straightforward estimation technique to be implemented. For nonlinear systems, extended Kalman filter (EKF) is considered to be the best nonlinear estimator. The EKF is based on linearizing the state and output equations at every sampling instant. Therefore, this estimator requires continuously computation of the Jacobian matrix. The unscented Kalman filter (UKF) is based on implementation of the unscented transformation (UT) to the nonlinear state distribution (motor model). The UT uses the intuition that it is easier to approximate a probability distribution than it is to approximate an arbitrary nonlinear function or transformation. Apply this intuition to motor model, a set or cloud of points are generated around each state of motor model with specified sample mean and sample covariance. The nonlinear function (PMSM model) is applied to each of these points in turn to yield a transformed sample, and the predicted mean and covariance are calculated from the transformed sample. Based on predicted mean and covariance the UKF recursive algorithm can be developed. The performance comparisons are based on standard deviation estimation errors of both estimators and the time computation effort required execute the algorithms of both filters. The simulated results show that the UKF gives best estimates at motor low speed, while its estimation performance degrade at high motor speed. On the other hand, the EKF shows bad estimation characteristics at low frequency and it yields good estimates at high source frequency. However,, the EKF algorithm keeps lower time computation effort over wide range of rotor speed than that required to execute the UKF software for the same range of source frequency. The PMSM motor model and the algorithms of both filters are built in Matlab package using S-function capability and scalar control strategy are used to account for constant stator magnetizing flux.

يقدم البحث نموذجين من مرشحات كالمان ((Kalman Filters؛ وهما مرشح كالمان المعدل (Extended Kalman Filter) ؛ و((unscented Kalman filter وذلك لتخمين متغيرات المنظومة (states) وتشمل تيارات ملفات المجال، وزاوية والسرعة الزاوية للجزء الدوار (rotor). يعتمد مرشح كالمان المعدل في تخمين المتغيرات على تحويل ألانموذج اللاخطي الى أنموذج خطي. لذلك من الضروري حساب مصفوفة جاكوبين (Jaccobian Matrix) في كل لحظة زمنية. بينما يعتمد ((unscented Kalman filter على توليد عينات من النقاط بطريقة معينة يمكن استخدامها لغرض حساب المتوسط الحسابي (mean) والتباين ((covariance للمتغيرالعشوائي ذو توزيع ((Gaussian، حيث يتم تمرير هذه النقاط بالانموذج اللاخطي للمنظومة وايجاد، مرة اخرى، المتوسط الحسابي (mean) والتباين ((covariance لناتج خرج هذه النقاط لدرجة دقة تصل الى حدين فقط من متوالية تيلر (Taylor series). اثبت النتائج ان (UKF) يتميز باداء عالي للتخمين في حالة السرع الواطئة للجزء الدوار، بينما يتفوق اداء (EKF) في حالة السرع العالية للمحرك التزامني. وكذلك فان الزمن المستغرق لتنفيذ خوارزمية (EKF) هي اقل من الزمن المطلوب لتنفيذ خوارزمية (UKF) ولجميع مديات السرع للمحرك التزامني


Two-phase Permanent Magnet Synchronous Motor --- Scalar control --- EKF --- UKF.