Global Observability of Sensorless IPMSMs

We show that the pair (speed, rotor position) of interior permanent magnet synchronous motors is not globally observable, and the number of all indistinguishable pairs is at most four, an invariant independent of any parameter. This result follows as a corollary by virtue of characterizing the observability condition as a special restricted form of the injectivity of a family of polynomials via immersion. The field-oriented control (FOC), which causes the collapse of at most two indistinguishable points easy to be told apart, joins forces with the knowledge of motor/generator mode to make IPMSMs observable, leading to a simple algorithm.