Influence of the discretization method on the integration accuracy of observers with continuous feedback

Abstract

The paper discusses the problem of integrating the equations of state observers associated with direct field orientation (DFO) of motor drives and studies the influence of the discretization method used on the accuracy of integration. In a typical implementation, discrete-time integration is done using Euler's discretization method (forward rectangular rule) - the method is simple and integration is accurate when the drive operates at low and medium speed. However, as the frequency increases, the integration becomes inaccurate because the Euler approximation starts losing more and more area from under the curve. Theoretically, the problem could be alleviated by increasing the sampling frequency; however, this cannot always be done. Another idea would be to adopt a more accurate (but more computationally intensive) integration method, for example, trapezoidal integration (Tustin method). The paper shows that, at high frequency, under ideal conditions, trapezoidal integration performs better than the Euler method. In a real implementation, however, conditions are non-ideal since the measured signals bring dc offsets and imperfections into the terms to be integrated - as a result, pure integration must be replaced with quasi-low pass filtering. Under these conditions, the paper compares the Euler, Tustin and backward rectangular methods from the point of view of integration accuracy. The implications related to direct field orientation of motor drives are studied by considering a full-order observer for the PMSM - this is discretized using the three methods considered and the results are compared. At high frequency, neither integration method gives perfect results; the Euler method yields a waveform that leads the expected one while the backward rectangular method yield a waveforms that lags it. The paper finds that, surprisingly, when quasi-low pass filtering is used, the Tustin method is not significantly more accurate than the other ones - the waveform obtained lags the expected one by an angle comparable with the lead angle of the Euler method. It is shown that the integration accuracy depends on the frequency, sampling time, filter bandwidth and on the integration method used. Accurate high frequency drive DFO control would require correction of the magnitude/phase of the estimates.