Discrete time integration of observers with continuous feedback based on Tustin´s method with variable prewarping

The paper discusses the problem of integrating the equations of state observers associated with direct field orientation (DFO) of motor drives and presents a method to improve the integration accuracy when the drive operates at high frequency. In a typical implementation, observer equations are integrated based on the Euler method (forward rectangular rule). Euler method is simple; however, integration is accurate only at low/medium speed. As the speed (frequency) increases, the integration process becomes more and more inaccurate because the rectangular 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 because it has implications related to the switching frequency of the power converter. Another idea is to use a more accurate integration method, for example, trapezoidal integration (Tustin method). At high frequency, trapezoidal integration performs better than the Euler method and the resulting estimates are closer to the expected values (in both magnitude and phase). In DFO drives, this leads to a more accurate field orientation angle. The paper presents an improvement to trapezoidal integration - the state equations of observers are integrated using a discrete-time filter that is prewarped as a function of the drive´s operating frequency. The algorithm estimates the applied frequency, prewarps the continuous-time transfer function used in practical integration and obtains a discrete time filter with frequency dependent coefficients. It is shown that the method produces an improvement over trapezoidal integration in the high-speed region. The implications related to DFO are studied by considering a full-order observer for the PMSM - integration with prewarping is compared with