TIDUFE5 Design guide

TIDUFE5 July 2025

3.2.2.1.2 Design of ESMO for the IPMSM

Figure 3-16 shows the conventional PLL integrated into the SMO.

TIDA-010282 Block Diagram of eSMO With PLL for a
PMSM

Figure 3-16 Block Diagram of eSMO With PLL for a PMSM

The traditional reduced-order sliding mode observer is constructed (Equation 32 shows the mathematical model) and Figure 3-17 shows the block diagram.

Equation 32.

[\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 in Equation 33

Equation 33.

[\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 provide the stable operation of the SMO, the k_α and k_β are large enough to hold k_α > max(|e_α|) and k_β > max(|e_β|).

TIDA-010282 Block Diagram of Traditional Sliding Mode Observer

Figure 3-17 Block Diagram of Traditional Sliding Mode Observer

The estimated value of EEMF in α-β axes (ê_α, ê_β) can be obtained with the low-pass filter from the discontinuous switching signals z_α and z_α:

Equation 34.

[\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 = 2πf_c is the cutoff angular frequency of the LPF, which is typically 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 in Equation 35:

Equation 35.

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

The low-pass filter removes the high-frequency term of the sliding mode function, which leads to 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:

Equation 36.

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

Now the estimated rotor position is calculated by using the SMO method:

Equation 37.

{\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 32 in α-β coordinates is given by Equation 38 as:

Equation 38.

[\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 39 and Equation 40 as:

Equation 39.

[\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 40.

[\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 34 is given by Equation 41 as:

Equation 41.

[\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}]