The transition and the input matrix of the discretized state-space equations for the IPM-SM, must be calculated in the (*d*, *q*) reference frame, which turns with the rotor, following the rotor magnetic anisotropy, to allow to:

- filter measurements made in the (
*α*,*β*) fixed frame: the two-phase currents and the position of the rotor, generally - predict an initial state-space during the control computation.

These two matrices *F* and *G* (cf. equations (3.124)) are calculated, from the exponential function of the diagonalized evolution matrix *D* multiplied by the sampling period *T* (cf. equation (3.69)), from the input matrix *B* of the continuous-time state-space equations, and from the transfer matrix and its inverse, calculated in Appendix B and C respectively:

(E.1)

In addition, the expression of *D*^{− 1} ⋅ (*e*^{D ⋅ T} − *I*) ⋅ *P*^{− 1} ⋅ *B*, was already calculated in (3.72).

We will start by calculating *e*^{D ⋅ T} ⋅ *P*^{− 1}, by simply creating the product of the two matrices *e*^{D ⋅ T} by *P*^{− 1}, and then we will multiply the result by the matrix *P* on the left.

(E.2)

To reduce the first row of the produced matrix, we then use equations (3.42) and (3.44) and the reduced variable *ζ*_{d1}, defined in (3.49). A new relation is thus obtained:

(E.3)

To reduce ...

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