Implementation of time-varying observers used in direct field orientation of motor drives by trapezoidal integration

The paper discusses the problem of implementing the state observers associated with direct field orientation (DFO) of motor drives using trapezoidal integration (Tustin method). Typically, the discrete-time equations of observers are obtained by emulating the continuous-time equations using the Euler method (forward rectangular rule). With Euler integration, the resulting equations are simple and the realtime implementation requires low computational effort. However, Euler-based observers become inaccurate if a small sampling time cannot used or if the motor drive operates at high frequency - this is because, as the sampling time increases, the Euler approximation of the integral starts losing more and more area from under the curve. The Tustin method (trapezoidal integration) offers an interesting alternative - it is theoretically a more accurate integration method, however, it is more complicated. The paper discusses the emulation procedure required to discretize continuous-time observers based on trapezoidal integration. The permanent magnet synchronous motor (PMSM) is used as an example of a time-varying plant - the paper develops a trapezoidal integration based observer for the PMSM and compares this with an Euler-based observer in terms of computational complexity and performance. The two observers are simulated comparatively in order to establish the conditions when trapezoidal integration outperforms the Euler method.