SLUUDM5 User guide

SLUUDM5 March 2026 MSPM0G1507 , MSPM0G1519 , MSPM0G3507 , MSPM0G3519

2.3.2 Enhanced Sliding Mode Observer

Model-based BEMF estimation methods are used to achieve position sensorless control of the IPMSM drive system when the motor runs at middle or high speeds. The model methods estimates the rotor position by the back-EMF or the flux linkage model. The sliding mode observer is an observer-design method based on sliding mode control. The structure of the system is not fixed but purposefully changed according to the current state of the system, forcing the system to move according to the predetermined sliding mode trajectory. Advantages include fast response, strong robustness, and insensitivity to both parameter changes and disturbances.

Figure 2-12 Block Diagram of Traditional Sliding Mode Observer

The traditional reduced-order sliding mode observer is constructed, which mathematical model is shown in Figure 2-4, the block diagram is shown in Figure 2-12.

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 $k_{β} > m a x (|e_{β}|)$ .

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.

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 11 in α-β coordinates is given by Equation 14 as:

Equation 14.

[\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 15 and Equation 16 as:

Equation 15.

[\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 16.

[\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 17 as:

Equation 17.

[\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}]