1.5.2 Reduced Order Luenberger Observer

The discrete implementation of the Reduced Order Luenberger Observer is shown in the previous figure, and is represented by the following equations.

$\{\begin{array}{c}{\hat{z}}_{\alpha }(n+1)=(1-h)z_{\alpha }\left(n\right)+(\frac{h{L}_s}{T_c}-R_s)\left[h{i}_{\alpha }\right(n)-\omega T_c{i}_{\beta }(n\left)\right]+\left[h{\nu }_{\alpha }\right(n)-\omega T_c{\nu }_{\beta }(n\left)\right]\\ {\hat{z}}_{\beta }(n+1)=(1-h)z_{\beta }\left(n\right)+(\frac{h{L}_s}{T_c}-R_s)\left[h{i}_{\beta }\right(n)+\omega T_c{i}_{\alpha }(n\left)\right]+\left[h{\nu }_{\beta }\right(n)+\omega T_c{\nu }_{\alpha }(n\left)\right]\end{array}$

The estimated BEMF is calculated using the following equation:

$\{\begin{array}{c}{\hat{e}}_{\alpha }\left(n\right)=z_{\alpha }\left(n\right)-\frac{h{L}_s}{T_c}{i}_{\alpha }\left(n\right)+\omega L_s{i}_{\beta }\left(n\right)\\ {\hat{e}}_{\beta }\left(n\right)=z_{\beta }\left(n\right)-\frac{h{L}_s}{T_c}{i}_{\beta }\left(n\right)-\omega L_s{i}_{\alpha }\left(n\right)\end{array}$

Where

T_c – Computational Step time of the observer. Typically it is the control loop period.

R_s – Per Phase Stator resistance of the motor.

L_s - Per Phase Synchronous inductance of the motor.

ω - Electrical speed of the motor in rad/sec.

The constants to be considered are as follows:

$0<h<1$(arbitrary, its value determines the system dynamics)

$k=h\frac{L_s}{T_c}$

$c_0\triangleq 1-h$

$c_1\triangleq k-R_s$

$c_2\triangleq \omega T_c$

$c_3\triangleq \omega L_s$

The variables c₂ and c₃, which are speed dependent, should be recomputed at every iteration.