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:
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:
In addition, the expression of D− 1 ⋅ (eD ⋅ T − I) ⋅ P− 1 ⋅ B, was already calculated in (3.72).
We will start by calculating eD ⋅ T ⋅ P− 1, by simply creating the product of the two matrices eD ⋅ T by P− 1, and then we will multiply the result by the matrix P on the left.
To reduce ...