Extended EMF Observer

Compute electrical position and mechanical speed of permanent magnet synchronous motor (PMSM)

Since R2023a

• 

Libraries:
Motor Control Blockset / Sensorless Estimators

Description

The Extended EMF Observer block computes the electrical position, θ_e (or sin θ_e and cos θ_e), in addition to the mechanical speed of the PMSM from the measured voltage and current in the stationary αβ reference frame.

The block includes a linear extended EMF state observer that estimates the extended back-EMF. It uses a lowpass filter to remove any noise in the estimated back-EMF value. In addition, the block includes a speed feedforward-based quadrature phase-locked loop (PLL) that acts as a speed observer. The speed observer uses the computed extended back-EMF to determine the motor speed. It calculates speed with zero error even during constant angular acceleration. The block then computes the position output from the estimated motor electrical speed.

Equations

The IPMSM state equation is defined as:

$$\begin{array}{l}\left[\begin{array}{c}\stackrel{·}{i}\\ \stackrel{·}{e}\end{array}\right]=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ 0& A_{22}\end{array}\right]\left[\begin{array}{c}i\\ e\end{array}\right]+\left[\begin{array}{c}{B}_1\\ 0\end{array}\right]v+\left[\begin{array}{c}0\\ W\end{array}\right]\\ i=C\left[\begin{array}{c}i\\ e\end{array}\right]\end{array}$$

These equations describe the reduced order PMSM state model:

$$\begin{array}{l}\stackrel{·}{i}=A_{11}i+A_{12}e+B_{1}v\\ \stackrel{·}{e}=A_{22}e\end{array}$$

Extended back-EMF, $$e=\{(L_d-L_q)({\omega }_{re}{i}_d-{\stackrel{·}{i}}_q)+{\omega }_{re}{K}_E\}\left[\begin{array}{c}-\sin {\theta }_{re}\\ \cos {\theta }_{re}\end{array}\right]$$

This equation describes the reduced dimensional state observer:

$$\stackrel{·}{\widehat{e}}=\left({\widehat{A}}_{22}-\xi A_{12}\right)\widehat{e}+\xi \left(\stackrel{·}{i}-{\widehat{A}}_{11}i-B_{1}v\right)$$

These equations describe the extended EMF observer parameters:

$$\begin{array}{l}{A}_{22}-\xi A_{12}={\omega }_{re}J-\left(\alpha I+\beta J\right)\left(\frac{-1}{L_d}\right)I\\ \xi =\left(\alpha I+\beta J\right)\end{array}$$

To ensure convergence, if you set $$\beta ={\omega }_{re}\times L_d$$ and set α to a negative value, then the extended EMF observer gain can be determined as:

$$\text{Extended EMF }\text{}\text{observer gain}=\frac{-\alpha }{L_d}$$

where:

• $$i=\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]$$, $$e=\left[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}\right]$$, and $$v=\left[\begin{array}{c}{v}_{\alpha }\\ v_{\beta }\end{array}\right]$$.

• $$W=\left(L_d-L_q\right)\left({\omega }_{re}\stackrel{·}{i_d}-\stackrel{··}{i_q}\right)\left[\begin{array}{c}-\sin {\theta }_{re}\\ \cos {\theta }_{re}\end{array}\right]$$.

• $$A=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ 0& A_{22}\end{array}\right]$$, $$B=\left[\begin{array}{c}{B}_1\\ 0\end{array}\right]$$, and $$C=\left[\begin{array}{cc}I& 0\end{array}\right]$$.

• $$A_{11}=-\left(\frac{R}{L_d}\right)I+\left({\omega }_{re}\frac{\left(L_d-L_q\right)}{L_d}\right)J$$, $$A_{12}=\left(\frac{-1}{L_d}\right)I$$, and $$A_{22}={\omega }_{re}J$$.

• $$B_1=\left(\frac{1}{L_d}\right)I$$, $$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$, and $$J=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$$.

  Note

  A and C are an observable pair.

• "$$\stackrel{·}{}$$" and "$$\stackrel{·}{}$$$$\stackrel{·}{}$$" are the first-order and second-order differential operators, respectively.

• v_α and v_β are the stator voltages along the α- and β-axes, respectively.

• i_α and i_β are the stator currents along the α- and β-axes, respectively.

• e_α and e_β are the extended back-EMF along the α- and β-axes, respectively.

• i_d and i_q are the stator currents along the d- and q-axes, respectively.

• R is the stator resistance of PMSM.

• L_d and L_q are the stator inductances of PMSM across d and q axes, respectively.

• K_E is the EMF constant.

• ω_re is the angular velocity at electrical angle.

• θ_re is the rotor position at electrical angle.

• $$\widehat{e},{\widehat{A}}_{22}\text{, and }{\widehat{A}}_{11}$$ are the estimated values.

• ξ is the feedback gain.

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.

Dependencies

To enable this port, set the Position output parameter to Electrical position.

Data Types: single | double | fixed point

Sin θ_e — Sine of electrical position of PMSM

scalar

Sine of the estimated electrical position of the rotor.

Dependencies

To enable this port, set the Position output parameter to Sine and Cosine electrical position.

Data Types: single | double | fixed point

Cos θ_e — Cosine of electrical position of PMSM

scalar

Cosine of the estimated electrical position of the rotor.

Dependencies

To enable this port, set the Position output parameter to Sine and Cosine electrical position.

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

Input units — Units of block input

SI unit (default) | Per-unit

Units of inputs to the block.

Position output — Type of position output

Electrical position (default) | Sine and Cosine electrical position

Type of positions that the block outputs.

• Electrical position — Select this value to enable the θe port.

• Sine and Cosine electrical position — Select this value to enable the Sin θe and Cos θe ports.

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

50e-6 (default) | scalar

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

Motor Parameters

Stator resistance (ohm) — Resistance

0.36 (default) | scalar

Stator phase winding resistance (in ohm).

Stator d-axis inductance (H) — D-axis inductance

0.2e-3 (default) | scalar

Stator phase winding inductance (in Henry) along the d-axis of the rotating d-q reference frame.

Stator d-axis inductance (H) — Q-axis inductance

0.2e-3 (default) | scalar

Stator phase winding inductance (in Henry) along the q-axis of the rotating d-q reference frame.

Maximum application speed (RPM) — Maximum supported mechanical speed

6000 (default) | scalar

Maximum mechanical speed (in RPM) that the block can support. If you set the speed to a value higher than this value, the block generates incorrect outputs.

Note

The block uses this parameter to filter out noise in the estimated speed signal.

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

4 (default) | scalar

Number of pole pairs available in the motor.

Base voltage (V) — Nominal voltage corresponding to one per unit

13.8564 (default) | scalar

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

Base current (A) — Nominal current corresponding to one per unit

21.4286 (default) | scalar

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

Observer Parameters

EEMF observer gain — Extended EMF observer gain

10 (default) | scalar

Gain of the linear extended EMF state observer.

EEMF filter cutoff frequency (Hz) — Cutoff frequency of EEMF filter

7500 (default) | scalar

Cutoff frequency of the lowpass filter that the block uses to remove noise in the estimated back-EMF value.

Proportional (P) — Proportional gain of speed observer

250 (default) | scalar

Proportional gain of the speed observer used by the block.

Integral (I) — Integral gain of the speed observer

200000 (default) | scalar

Integral gain of the speed observer used by the block.

Number of data points for lookup table — Number of data points used for speed computation

1024 (default) | scalar

Number of data points in the lookup table that the speed observer uses to compute speed from the extended back-EMF value.

Speed filter cutoff frequency (Hz) — Cutoff frequency of speed filter

15 (default) | scalar

Cutoff frequency of the lowpass filter that the block uses to remove noise in the estimated speed.

Datatypes

Position unit — Unit of position output

Degrees (default) | Radians | Per-unit

Unit of the position output.

Position data type — Data type of position output

single (default) | double | fixed point

Data type of the position output.

Speed unit — Unit of speed output

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

Unit of the speed output.

Speed data type — Data type of speed output

single (default) | double | fixed point

Data type of the speed output.

References

[1] Zhiqian Chen, M. Tomita, S. Doki and S. Okuma, "An extended electromotive force model for sensorless control of interior permanent-magnet synchronous motors," in IEEE Transactions on Industrial Electronics, vol. 50, no. 2, pp. 288-295, April 2003, doi: 10.1109/TIE.2003.809391.

[2] T. Ohnuma, S. Doki and S. Okuma, "Extended EMF observer for wide speed range sensorless control of salient-pole synchronous motor drives," The XIX International Conference on Electrical Machines - ICEM 2010, Rome, Italy, 2010, pp. 1-6, doi: 10.1109/ICELMACH.2010.5608087.

[3] G. Liu, H. Zhang and X. Song, "Position-Estimation Deviation-Suppression Technology of PMSM Combining Phase Self-Compensation SMO and Feed-Forward PLL," in IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 9, no. 1, pp. 335-344, Feb. 2021, doi: 10.1109/JESTPE.2020.2967508.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

Version History

Introduced in R2023a