TIDUF77 Design guide

TIDUF77 June 2024 – December 2024 MSPM0G1507

2.4.2.2.1.2 Design of ESMO for the IPMSM

Figure 2-21 shows the conventional PLL integrated into the SMO.

TIDA-010273 Block Diagram of eSMO With PLL for a PMSM

Figure 2-21 Block Diagram of eSMO With PLL for a PMSM

The traditional reduced-order sliding-mode observer is constructed, with the mathematical model shown in Equation 21 and the block diagram shown in Figure 2-22.

Equation 21.

[\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 shown in Equation 22:

Equation 22.

[\begin{matrix} z_{α} \\ z_{β} \end{matrix}] = [\begin{matrix} k_{α} sign ({\hat{i}}_{α} - i_{α}) \\ k_{β} sign ({\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 provide the stable operation of the SMO, then k_α and k_β are large enough to hold

Equation 23.

k_{α} > \max (|e_{α}|)

and

Equation 24.

k_{β} > \max (|e_{β}|)

TIDA-010273 Block Diagram of Traditional Sliding-Mode Observer

Figure 2-22 Block Diagram of Traditional Sliding-Mode Observer

The estimated value of EEMF in α-β axes (Equation 25, Equation 26) can be obtained by low-pass filter from the discontinuous switching signals z_α and z_α:

Equation 25.

{\hat{e}}_{α}

Equation 26.

{\hat{e}}_{β}

Equation 27.

[\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

Equation 28.

ω_{c} = 2 {πf}_{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, as Equation 29 defines:

Equation 29.

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

Low-pass filters remove the high-frequency term of the sliding-mode function, which results in phase delay. The delay can be compensated by the relationship between the cut-off frequency ω_c and back EMF frequency ω_e, which is defined as shown in Equation 30:

Equation 30.

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

Then the estimated rotor position by using SMO method is found with Equation 31:

Equation 31.

{\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 21 in α-β coordinates is given by Equation 32 as:

Equation 32.

[\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 33 and Equation 34 as:

Equation 33.

[\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 34.

[\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 27 is given by Equation 35 as:

Equation 35.

[\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 {πf}_{c} [\begin{matrix} z_{α} (n) - {\hat{e}}_{α} (n) \\ z_{β} (n) - {\hat{e}}_{β} (n) \end{matrix}]