Estimation of Position and Resistance of a Sensorless PMSM: A Nonlinear Luenberger Approach for a Nonobservable System

We show that the pair (rotor position, resistance) of a nonsalient permanent magnet synchronous motor (PMSM) is not observable in the sensorless setting, namely, when only the electrical signals (voltages and intensities) are measured. However, if the rotation speed ω and the current in rotating frame i
_d
are not constantly zero, we prove that there only exist a maximum of six indistinguishable solutions, the resistance being the root of a polynomial of degree six. Besides, in the particular case where ω, i
_d
, and i
_q
are constant, we show that the number of possible solutions is reduced to two, with two well-identified possible values for the resistance. Those two solutions can be dissociated if the sign of i
_q
(namely, the mode of use of the motor) is known. Inspired by the nonlinear Luenberger methodology, this property enables us to propose an estimation algorithm, which relies on a one-dimensional minimization of a criterion involving the voltages and intensities filtered at three distinct sufficiently large frequencies. Its performances are tested and illustrated in simulations on real data.