The conventional PLL integrated into the SMO is shown in Figure 2-18.

The traditional reduced-order sliding mode observer is constructed, which mathematical model is shown in Equation 51 and the block diagram is shown in Figure 2-19.

where $z_{\alpha }$ and $z_{\beta }$ are sliding mode feedback components and are defined as:

Where $k_{\alpha }$ and $k_{\beta }$ are the constant sliding mode gain designed by Lyapunov stability analysis. If $k_{\alpha }$ and $k_{\beta }$ are positive and significant enough to guarantee the stable operation of the SMO, the $k_{\alpha }$ and $k_{\beta }$ should be large enough to hold $k_{\alpha }>\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|e_{\alpha }\right|\right)$ and $k_{\beta }>\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|e_{\beta }\right|\right)$ .

The estimated value of EEMF in α-β axes ( ${\widehat{e}}_{\alpha }$ , ${\widehat{e}}_{\beta }$ ) can be obtained by low-pass filter from the discontinuous switching signals $z_{\alpha }$ and $z_{\alpha }$ :

Where ${\omega }_c=2\pi f_c$ is the cutoff angular frequency of the LPF, which is usually selected according to the fundamental frequency of the stator current.

Therefore, the rotor position can be directly calculated from arc-tangent the back EMF, defined as follow

Low pass filter removes the high-frequency term of the sliding mode function, which leads to occur phase delay resulting. It can be compensated by the relationship between the cut-off frequency ${\omega }_c$ and back EMF frequency ${\omega }_e$ , which is defined as:

And then the estimated rotor position by using SMO method is:

In a digital control application, a time discrete equation of the SMO is needed. The Euler method is the appropriate way to transform to a time discrete observer. The time discrete system matrix of Equation 51 in α-β coordinates is given by Equation 57 as:

Where the matrix $\left[F\right]$ and $\left[G\right]$ are given by Equation 58 and Equation 59 as:

The time discrete form of Equation 53 is given by Equation 60 as: