Sliding Mode Observer

Compute electrical position and mechanical speed of rotor

Since R2020a

• 

Libraries:
mcbpositiondecoderlib / Archive

Description

The Sliding Mode Observer block computes the electrical position and mechanical speed of a PMSM by using the per unit voltage and current values along the α- and β-axes of the stationary αβ reference frame.

Equations

These equations describe the computation of the electrical position and mechanical speed by the block.

$\frac{d{i}_{\alpha \beta }}{dt}=\Phi i_{\alpha \beta }+\Gamma V_{\alpha \beta }-\Gamma e_{\alpha \beta }$

$i_{\alpha \beta } = {\left[\begin{array}{cc}{i}_{\alpha }& i_{\beta }\end{array}\right]}^T$

$V_{\alpha \beta } = {\left[\begin{array}{cc}{V}_{\alpha }& V_{\beta }\end{array}\right]}^T$

$e_{\alpha \beta } = {\left[\begin{array}{cc}{e}_{\alpha }& e_{\beta }\end{array}\right]}^T=\left[\begin{array}{c}-\psi {\omega }_e\sin {\theta }_e\\ \psi {\omega }_e\cos {\theta }_e\end{array}\right]$

$\Phi =\left[\begin{array}{cc}-\frac{R}{L}& 0\\ 0& -\frac{R}{L}\end{array}\right]$

$\Gamma =\left[\begin{array}{cc}\frac{1}{L}& 0\\ 0& \frac{1}{L}\end{array}\right]$

These equations describe the discrete-time sliding mode observer operation by using per-unit values:

${\widehat{i}}_{\alpha \beta (k+1)P.U}=A{\widehat{i}}_{\alpha \beta (k)P.U}+\frac{V_{rated}}{I_{rated}}B(v_{\alpha \beta (k)P.U}-{\vartheta }_{\alpha \beta (k)P.U})$

${\vartheta }_{\alpha \beta (k+1)P.U}={\vartheta }_{\alpha \beta (k)P.U}+2\pi f_0\times ({\rm Z}(I_{rated}({\widehat{i}}_{\alpha \beta (k)P.U}-i_{\alpha \beta (k)P.U}))-{\vartheta }_{\alpha \beta (k)P.U})$

$A=e^{\Phi T_s}$

$B=\underset{0}{\overset{T_s}{{\displaystyle \int }}}{e}^{\Phi \tau }d\tau$

$$f_0=\frac{F_0}{F_s}$$

$F_s=\frac{1}{T_s}$

where:

• $e_{\alpha },i_{\alpha }$ are the stator back EMF and current for the α axis.

• $e_{\beta },i_{\beta }$ are the stator back EMF and current for the β axis.

• $v_{\alpha },v_{\beta }$ are the stator supply voltages.

• $R$ is the stator resistance.

• $$L$$ is the stator inductance.

• $\psi$ is the flux linkage due to permanent magnet.

• ${\omega }_e$ is the electrical angular velocity.

• ${\theta }_e$ is the electrical position of the rotor.

• $t$ is the time.

• $T_s$ is the sampling period.

• $$k$$ is the sample count.

• $V_{rated}$ is the nominal voltage corresponding to 1 per-unit.

• $I_{rated}$ is the nominal current corresponding to 1 per-unit.

• $$Z$$ is the attraction function.

  

• $f_0$ is the cut-off frequency of the filter in cycles per sample.

• $$F_0$$ is the cut-off frequency in cycles per second.

• $$F_s$$ is the sample frequency in samples per second.

• ${\vartheta }_{\alpha \beta (k)}$ is the estimated back EMF.

Tuning

Use the Current observer gain and Sliding surface limit parameters to tune the block.

• To improve stability, increase the Sliding surface limit or reduce the Current observer gain.

• To reduce distortion, decrease the Current observer gain or increase the Sliding surface limit.

Examples

Sensorless Field-Oriented Control of PMSM

Implements the field-oriented control (FOC) technique to control the speed of a three-phase permanent magnet synchronous motor (PMSM). For details about FOC, see Field-Oriented Control (FOC).

Open Model

Ports

Input

V_α — α-axis voltage

scalar

Per-unit voltage component along the α-axis of the stationary αβ reference frame.

Data Types: single | double | fixed point

V_β — β-axis voltage

scalar

Per-unit voltage component along the β-axis of the stationary αβ reference frame.

Data Types: single | double | fixed point

I_α — α-axis current

scalar

Per-unit current component along the α-axis of the stationary αβ reference frame.

Data Types: single | double | fixed point

I_β — β-axis current

scalar

Per-unit current component along the β-axis of the stationary αβ reference frame.

Data Types: single | double | fixed point

Rst — Reset the block

scalar

The pulse (true value) that resets and restarts the processing of the block algorithm.

Data Types: single | double | fixed point

Output

θ_e — Electrical position of PMSM

scalar

The estimated electrical position of the rotor.

Data Types: single | double | fixed point

⍵_m — Mechanical speed of PMSM

scalar

The estimated mechanical speed of the rotor.

Data Types: single | double | fixed point

Parameters

Observer parameters

Current observer gain — Sliding mode observer gain for current

1.1 (default) | scalar

The attraction function gain.

Sliding surface limit — Maximum limit of sliding surface of SMO

0.15 (default) | scalar

The boundary layer limit of the attraction function's domain.

Position unit — Unit of position output

Radians (default) | Degrees | Per unit

Unit of the position output.

Position data type — Data type of position output

single (default) | double | fixdt(1,16) | fixdt(1,16,0) | fixdt(1,16,2^0,0) | <data type expression>

Data type of the position output.

Speed unit — Unit of speed output

RPM (default) | Degrees/sec | Radians/sec | Per unit

Unit of the speed output.

Speed data type — Data type of speed output

single (default) | double | fixdt(1,16) | fixdt(1,16,0) | fixdt(1,16,2^0,0) | <data type expression>

Data type of the speed output.

Discrete step size (s) — Sample time after which block executes again

50e-6 (default) | scalar

The fixed time interval (in seconds) between every two consecutive instances of block execution.

Motor parameters

Stator resistance (ohm) — Resistance

0.4836 (default) | scalar

Stator phase winding resistance (in ohm).

Stator inductance (H) — Inductance

1e-3 (default) | scalar

Stator phase winding inductance (in Henry).

Maximum application speed (RPM) — Maximum supported speed value

6000 (default) | scalar

Maximum value of speed (in RPM) that the block can support. For a speed beyond this value, the block generates incorrect outputs.

Number of pole pairs — Number of pole pairs available in motor

4 (default) | scalar

Number of pole pairs available in the motor.

Base voltage — Nominal voltage corresponding to one per unit

68 (default) | scalar

The maximum phase voltage applied to PMSM. For details, see Per-Unit System.

Base current — Nominal current corresponding to one per unit

10 (default) | scalar

The maximum measurable current supplied to PMSM. For details, see Per-Unit System.

Note

The Sliding Mode Observer block may occasionally display the warning message 'Wrap on overflow detected.'

References

[1] Y. Kung, N. V. Quynh, C. Huang and L. Huang, "Design and simulation of adaptive speed control for SMO-based sensorless PMSM drive," 2012 4th International Conference on Intelligent and Advanced Systems (ICIAS2012), Kuala Lumpur, 2012, pp. 439-444 (doi: 10.1109/ICIAS.2012.6306234)

[2] Zhang Yan and V. Utkin, "Sliding mode observers for electric machines-an overview," IEEE 2002 28th Annual Conference of the Industrial Electronics Society. IECON 02, Sevilla, 2002, pp. 1842-1847 vol.3. (doi: 10.1109/IECON.2002.1185251)

[3] T. Bernardes, V. F. Montagner, H. A. Gründling and H. Pinheiro, "Discrete-Time Sliding Mode Observer for Sensorless Vector Control of Permanent Magnet Synchronous Machine," in IEEE Transactions on Industrial Electronics, vol. 61, no. 4, pp. 1679-1691, April 2014 (doi: 10.1109/TIE.2013.2267700)

[4] Z. Guo and S. K. Panda, "Design of a sliding mode observer for sensorless control of SPMSM operating at medium and high speeds," 2015 IEEE Symposium on Sensorless Control for Electrical Drives (SLED), Sydney, NSW, 2015, pp. 1-6. (doi: 10.1109/SLED.2015.7339255)

Version History

Introduced in R2020a