SPRUJ26 User guide

SPRUJ26A September 2021 – April 2024

2.3.1.1 Design of ESMO for PMSM

The conventional PLL integrated into the SMO is shown in Figure 2-14.

Figure 2-14 Block Diagram of eSMO With PLL for a PMSM

The traditional reduced-order sliding mode observer is constructed, which mathematical model is shown in Figure 2-14 and the block diagram is shown in Figure 2-15.

Equation 11.

[\begin{matrix} {\dot{\hat{i}}}_{α} \\ {\dot{\hat{i}}}_{β} \end{matrix}] = \frac{1}{L_{d}} [\begin{matrix} {- R}_{s} & - {\hat{ω}}_{e} (L_{d} - L_{q}) \\ {\hat{ω}}_{e} (L_{d} - L_{q}) & {- R}_{s} \end{matrix}] [\begin{matrix} {\hat{i}}_{α} \\ {\hat{i}}_{β} \end{matrix}] + \frac{1}{L_{d}} [\begin{matrix} V_{α} - {\hat{e}}_{α} + z_{α} \\ V_{β} - {\hat{e}}_{β} + z_{β} \end{matrix}]

where $z_{α}$ and $z_{β}$ are sliding mode feedback components and are defined as:

Equation 12.

[\begin{matrix} z_{α} \\ z_{β} \end{matrix}] = [\begin{matrix} k_{α} s i g n ({\hat{i}}_{α} - i_{α}) \\ k_{β} s i g n ({\hat{i}}_{β} - i_{β}) \end{matrix}]

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

Figure 2-15 Block Diagram of Traditional Sliding Mode Observer

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

Equation 13.

[\begin{matrix} {\hat{e}}_{α} \\ {\hat{e}}_{β} \end{matrix}] = \frac{ω_{c}}{s + ω_{c}} [\begin{matrix} z_{α} \\ z_{β} \end{matrix}]

Where $ω_{c} = 2 π f_{c}$ 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

Equation 14.

{\hat{θ}}_{e} = - \tan^{- 1} (\frac{{\hat{e}}_{α}}{{\hat{e}}_{β}})

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 $ω_{c}$ and back EMF frequency $ω_{e}$ , which is defined as:

Equation 15.

∆ θ_{e} = - \tan^{- 1} (\frac{ω_{e}}{ω_{c}})

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

Equation 16.

{\hat{θ}}_{e} = - \tan^{- 1} (\frac{{\hat{e}}_{α}}{{\hat{e}}_{β}}) + ∆ θ_{e}

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 17 in α-β coordinates is given by Equation 17 as:

Equation 17.

[\begin{matrix} {\hat{\dot{i}}}_{α} (n + 1) \\ {\hat{\dot{i}}}_{β} (n + 1) \end{matrix}] = [\begin{matrix} F_{α} \\ F_{β} \end{matrix}] [\begin{matrix} {\hat{\dot{i}}}_{α} (n) \\ {\hat{\dot{i}}}_{β} (n) \end{matrix}] + [\begin{matrix} G_{α} \\ G_{β} \end{matrix}] [\begin{matrix} V_{α}^{*} (n) - {\hat{e}}_{α} (n) + z_{α} (n) \\ V_{β}^{*} (n) - {\hat{e}}_{β} (n) + z_{β} (n) \end{matrix}]

Where the matrix $[F]$ and $[G]$ are given by Equation 18 and Equation 19 as:

Equation 18.

[\begin{matrix} F_{α} \\ F_{β} \end{matrix}] = [\begin{matrix} e^{- \frac{R_{s}}{L_{d}}} \\ e^{- \frac{R_{s}}{L_{q}}} \end{matrix}]

Equation 19.

[\begin{matrix} G_{α} \\ G_{β} \end{matrix}] = \frac{1}{R_{s}} [\begin{matrix} {1 - e}^{- \frac{R_{s}}{L_{d}}} \\ 1 - e^{- \frac{R_{s}}{L_{q}}} \end{matrix}]

The time discrete form of Equation 13 is given by Equation 20 as:

Equation 20.

[\begin{matrix} {\hat{e}}_{α} (n + 1) \\ {\hat{e}}_{β} (n + 1) \end{matrix}] = [\begin{matrix} {\hat{e}}_{α} (n) \\ {\hat{e}}_{β} (n) \end{matrix}] + 2 π f_{c} [\begin{matrix} z_{α} (n) - {\hat{e}}_{α} (n) \\ z_{β} (n) - {\hat{e}}_{β} (n) \end{matrix}]