A rotor initial position estimation method for sensorless field-oriented control of permanent magnet synchronous motor

Abstract

In this paper, a new technique to improve initial rotor position detection at standstill of a permanent magnet synchronous motor (PMSM) is presented. Sensorless field-oriented control (FOC) of a PMSM at low speed remains a difficult task. In order to estimate the position and rotor speed, we proposed a novel structure of a full-order sliding mode observer (FO-SMO) in a sensorless FOC. At standstill, we used a voltage pulse sequence applied to the windings in order to detect the initial rotor position. With this technique, we managed to minimize the error on the estimated rotor position to 3.75° (electrical) compared with others. The validity of the proposed approach with a 1.1-kW low-speed PMSM sensorless FOC has been proved by experimental results.

Keywords

Digital signal processor FO-SMO initial rotor position PMSM drive sensorless sigmoid function

Introduction

The permanent magnet synchronous motor (PMSM) presently attracts attention in an industrial domain thanks to several favourable characteristics such as low inertia, high efficiency and low weight. For sensorless field-oriented control (FOC) of a PMSM drive, accurate knowledge of the rotor position and speed is required by using either a rotor position encoder or an estimator. The use of these sensors leads to the emergence of some problems, for example the cost of the sensor, increased equipment complexity and reduction of the mechanical robustness (Acarnley and Watson, 2006). Scientific research has evolved to solve these problems; a sensorless vector control technique has been developed for variable speed PMSM drives (Delpoux and Floquet, 2014; Lee, 2015; Yang et al., 2016), instead of using a rotor position sensor. Consequently, many methods for rotor position and speed estimation have been researched.

Generally, sensorless vector control approaches can be classified into two categories: high frequency injection techniques using the rotor salient effect of the motor (Wu and Selmon, 1991) and estimation methods based on the observer (Tomei and Verelli, 2011). In order to estimate the rotor position and velocity, several studies have used the extended Kalman filter (Boussak, 2005), the model reference adaptive system (Khlaief et al., 2013), the non-linear observer (Khlaief et al., 2016) and the sliding mode observer (SMO) (Qiao et al., 2011; Saadaoui et al., 2015, 2016). The latter observer is widely used in the research domain due to its robustness towards parametric uncertainties. This approach is based on the technique of a sliding mode and two observers model have been studied; regarding the first model, many proposed methods are based on the back electromotive force (EMF) (Kim et al., 2011), and the second is based on the full-order SMO (FO-SMO) (Corradini et al., 2012; Kim et al., 2011; Saadaoui et al., 2015).

In Kim et al. (2011) and Qiao et al. (2011), the SMO has been proposed in order to estimate the rotor position and velocity. The authors present a new sliding mode observer of the PMSM.

Several studies on reducing the chattering phenomenon and the important efforts based on high-order sliding modes methods to overcome this difficulty have been published (Corradini et al., 2012). Saadaoui et al. (2015) propose a sensorless speed control for a PMSM based on FO-SMO. The simulation results show the effectiveness of the new SMO approach.

The FO-SMO technique is not able to detect the initial rotor position of the PMSM. At start-up, without the knowledge of the initial rotor position, the motor can rotate in the wrong direction, which is not allowed in some industrial applications.

Boussak (2005), Saadaoui et al. (2015) and Schmidt et al. (1997) used a method based on a voltage pulse. This technique consists of feeding the stator of the PMSM in various phases by a voltage pulse sequence. With this approach, we managed to minimize the error on the initial rotor position detection to 3.75° (electrical) without additional materials. The sensorless FOC approach with initial rotor position detection of the PMSM drive is applied in a Matlab-Simulink program and a dSpace DS1103 controller board.

This paper proposes a sensorless FOC of PMSM based on new FO-SMO with initial rotor position detection. The following section presents the PMSM model and then we propose a new FO-SMO. The principle of the proposed initial rotor position detection is developed, and experimental results of the sensorless FOC are presented and discussed. Finally, we draw conclusions relating to the contribution of this paper and present proposals for future work.

PMSM modelling

The PMSM model is presented as follows:

{\begin{matrix} \frac{d i_{α}}{dt} = - \frac{R_{s}}{L_{s}} i_{α} + \frac{n_{p} K_{e}}{L_{s}} Ω \sin (θ) + \frac{1}{L_{s}} V_{α} \\ \frac{d i_{β}}{dt} = - \frac{R_{s}}{L_{s}} i_{β} + \frac{n_{p} K_{e}}{L_{s}} Ω \cos (θ) + \frac{1}{L_{s}} V_{β} \\ \frac{d Ω}{dt} = \frac{n_{p} K_{e}}{J} (i_{β} \cos (θ) - i_{α} \sin (θ)) - \frac{f}{J} Ω \\ \frac{d θ}{dt} = Ω \end{matrix}

(1)

where: $n_{p}$ are pole pairs, ${i_{α,}}_{β}$ are stator currents in an $(α - β)$ stationary reference frame, $V_{α, β}$ are stator voltages in an $(α - β)$ stationary reference frame, $R_{s}$ is the stator resistance, $L_{s}$ the stator winding inductance, $K_{e}$ the back EMF constant, $Ω$ the rotor mechanical speed, $θ_{r}$ the angular position, $J$ the moment of inertia and $f$ the viscous friction coefficient.

Proposed SMO

In this next subsection, a sigmoid function replaces a sign function in order to eliminate the chattering phenomenon. Therefore, the use of a low pass filter is not necessary.

Sigmoid function

The observer is based on a stator current estimator as stator currents and voltages. Then, a new FO-SMO can be constructed as follows

{\begin{matrix} \frac{d {\hat{i}}_{α}}{dt} = - \frac{R_{s}}{L_{s}} {\hat{i}}_{α} + \frac{n_{p} K_{e}}{L_{s}} \hat{Ω} \sin (\hat{θ}) + \frac{1}{L_{s}} V_{α} + K_{1} F (i_{α} - {\hat{i}}_{α}) \\ \frac{d {\hat{i}}_{β}}{dt} = - \frac{R_{s}}{L_{s}} {\hat{i}}_{β} + \frac{n_{p} K_{e}}{L_{s}} \hat{Ω} \cos (\hat{θ}) + \frac{1}{L_{s}} V_{β} + K_{1} F (i_{β} - {\hat{i}}_{β}) \\ \frac{d \hat{Ω}}{dt} = \frac{n_{p} K_{e}}{J} ({\hat{i}}_{β} \cos (\hat{θ}) - {\hat{i}}_{α} \sin (\hat{θ})) \\ - \frac{f}{J} \hat{Ω} + K_{2} F (i_{α} - {\hat{i}}_{α}) + K_{2} F (i_{β} - {\hat{i}}_{β}) \\ \frac{d \hat{θ}}{dt} = \hat{Ω} + K_{3} F (i_{α} - {\hat{i}}_{α}) + K_{3} F (i_{β} - {\hat{i}}_{β}) \end{matrix}

(2)

with $K_{1}$ , $K_{2}$ and $K_{3}$ the constant gains and F(x) the commutation function.

The sigmoid function is given by this expression

F (x) = [\frac{2}{(1 + e^{- α x})}] - 1

(3)

where $α$ is an adjustable parameter.

Stability analysis

The sliding surface $S (x)$ can be defined according to the errors between the actual and estimated currents for each phase given as follows

S (x) = {[S_{α} (x) S_{β} (x)]}^{T} = [\begin{matrix} i_{α} - {\hat{i}}_{α} \\ i_{β} - {\hat{i}}_{β} \end{matrix}]

(4)

where ${\hat{i}}_{α, β}$ are the estimated currents and ${i_{α,}}_{β}$ are the measured currents.

For the PMSM sensorless FOC, we used the following Lyapunov function to obtain the sliding condition

\begin{matrix} V = \frac{1}{2} ({S_{α}}^{2} + {S_{β}}^{2} + {\bar{Ω}}^{2} + {\bar{θ}}^{2}) \\ = \frac{1}{2} ({\bar{i}}_{α}^{2} + {\bar{i}}_{β}^{2} + {\bar{Ω}}^{2} + {\bar{θ}}^{2}) \end{matrix}

(5)

with ${\bar{i}}_{α} = i_{α} - {\hat{i}}_{α}, {\bar{i}}_{β} = i_{β} - {\hat{i}}_{β}, \bar{Ω} = Ω - \hat{Ω} and \bar{θ} = θ - \hat{θ}$ .

We take the time derivative of (5), which can be found as

\overset{\cdot}{V} = {\bar{i}}_{α} {\overset{\cdot}{\bar{i}}}_{α} + {\bar{i}}_{β} {\overset{\cdot}{\bar{i}}}_{β} + \bar{Ω} \overset{\cdot}{\bar{Ω}} + \bar{θ} \overset{\cdot}{\bar{θ}}

(6)

The dynamic error equations are given by

{\begin{matrix} {\overset{\cdot}{\bar{i}}}_{α} = - \frac{R_{s}}{L_{s}} {\bar{i}}_{α} + \frac{n_{p} K_{e}}{L_{s}} (Ω \sin θ - \hat{Ω} \sin \hat{θ}) - K_{1} F ({\bar{i}}_{α}) \\ {\overset{\cdot}{\bar{i}}}_{β} = - \frac{R_{s}}{L_{s}} {\bar{i}}_{β} + \frac{n_{p} K_{e}}{L_{s}} (- Ω \cos θ + \hat{Ω} \cos \hat{θ}) - K_{1} F ({\bar{i}}_{β}) \\ \overset{\cdot}{\bar{Ω}} = \frac{n_{p} K_{e}}{J} (i_{β} \cos θ - i_{α} \sin θ) - ({\hat{i}}_{β} \cos \hat{θ} - {\hat{i}}_{α} \sin \hat{θ}) \\ - \frac{f}{J} \bar{Ω} - K_{2} F ({\bar{i}}_{α}) - K_{2} F ({\bar{i}}_{β}) \\ \overset{\cdot}{\bar{θ}} = Ω - \hat{Ω} - K_{3} F ({\bar{i}}_{α}) - K_{3} F ({\bar{i}}_{β}) \end{matrix}

(7)

Substituting (7) in (6) gives

\begin{matrix} \overset{\cdot}{V} = - a_{1} {\bar{i}}_{α}^{2} + a_{2} {\bar{i}}_{α} (Ω \sin θ - \hat{Ω} \sin \hat{θ}) - K_{1} | {\bar{i}}_{α} | \\ - a_{1} {\bar{i}}_{β}^{2} + a_{2} {\bar{i}}_{β} (Ω \cos θ - \hat{Ω} \cos \hat{θ}) - K_{1} | {\bar{i}}_{β} | - a_{4} {\bar{Ω}}^{2} \\ + a_{3} \bar{Ω} [(i_{β} \cos θ - i_{α} \sin \hat{θ}) - ({\hat{i}}_{β} \cos \hat{θ} - {\hat{i}}_{α} \sin \hat{θ})] \\ - \bar{Ω} K_{2} F ({\bar{i}}_{α}) - \bar{Ω} K_{2} F ({\bar{i}}_{β}) + \bar{θ} \bar{Ω} - \bar{θ} K_{3} F ({\bar{i}}_{α}) - \bar{θ} K_{3} F ({\bar{i}}_{β}) \end{matrix}

(8)

with

{\begin{matrix} a_{1} = \frac{R_{S}}{L_{S}}, a_{2} = \frac{n_{p} K_{e}}{L_{S}}, a_{3} = \frac{n_{p} K_{e}}{J}, a_{4} = \frac{f}{J} \\ \begin{matrix} | {\bar{i}}_{α} | = i_{α} F ({\bar{i}}_{α}) \\ | {\bar{i}}_{β} | = i_{β} F ({\bar{i}}_{β}) \end{matrix} \end{matrix}

(9)

To guarantee the convergence, the Lyapunov function derivative is forced in such a way that $\overset{\cdot}{V} < 0$ .

Knowing that

{\begin{matrix} - a_{1} {\bar{i}}_{α}^{2} < 0, - a_{1} {\bar{i}}_{β}^{2} < 0, - a_{4} {\bar{Ω}}^{2} < 0 \\ | Ω \sin θ - \hat{Ω} \sin \hat{θ} | < 2 Ω_{max} \\ | {\bar{i}}_{α} (Ω \sin θ - \hat{Ω} \sin \hat{θ}) | < 4 i_{max} Ω_{max} \end{matrix}

(10)

Such that

{\begin{matrix} | {\bar{i}}_{α} | < 2 i_{max} \\ | {\bar{i}}_{β} | < 2 i_{max} \end{matrix}

(11)

Therefore

{\begin{matrix} | {\bar{i}}_{α, β} (Ω \sin θ - \hat{Ω} \sin \hat{θ}) | < 4 i_{max} Ω_{max} \\ | {\bar{i}}_{β} (Ω \cos θ - \hat{Ω} \cos \hat{θ}) | < 4 i_{max} Ω_{max} \\ | \bar{Ω} [(i_{β} \cos θ - i_{α} \sin \hat{θ}) - ({\hat{i}}_{β} \cos \hat{θ} - {\hat{i}}_{α} \sin \hat{θ})] | < 8 i_{max} Ω_{max} \\ | \bar{θ} \bar{Ω} | < 4 θ_{max} Ω_{max} \\ | \bar{Ω} (F ({\bar{i}}_{α}) + F ({\bar{i}}_{β})) | < 4 Ω_{max} \end{matrix}

(12)

We have selected K₁, K₂ and K₃ so that

\begin{matrix} a_{2} {\bar{i}}_{α} (Ω \sin θ - \hat{Ω} \sin \hat{θ}) + a_{2} {\bar{i}}_{β} (Ω \cos θ - \hat{Ω} \cos \hat{θ}) \\ - K_{1} | {\bar{i}}_{α} | - K_{1} | {\bar{i}}_{β} | \\ + a_{3} \bar{Ω} [(i_{β} \cos θ - i_{α} \sin \hat{θ}) - ({\hat{i}}_{β} \cos \hat{θ} - {\hat{i}}_{α} \sin \hat{θ})] \\ - \bar{Ω} K_{2} F ({\bar{i}}_{α}) - \bar{Ω} K_{2} F ({\bar{i}}_{β}) \\ + \bar{θ} \bar{Ω} - \bar{θ} K_{3} F ({\bar{i}}_{α}) - \bar{θ} K_{3} F ({\bar{i}}_{β}) < 0 \end{matrix}

(13)

Therefore,

K_{1} > 2 a_{2} Ω_{max}, K_{2} > 2 a_{3} i_{max} and K_{3} > Ω_{max}

(14)

The structure of the sensorless FOC of PMSM by using the new FO-SMO is shown in Figure 1.

Figure 1.

Block diagram of the sensorless field-oriented control of permanent magnet synchronous motor: (a) overall block diagram of system; (b) block diagram of the synchronous motor observer.

Initial rotor position detection

To received good operation at the start-up of the sensorless FOC of the PMSM, we must know the initial position of the motor (Haque et al., 2003). Furthermore, we are obliged to add a new technique to detect the initial rotor position of the PMSM.

We apply a suitable sequence of voltage pulses to the winding of the PMSM. Evaluation of the different signs of the peak current leads to initial rotor position information (Boussak, 2005).

The proposed detection method

We used the DSP (dS1103) to beget these impulses to inverter legs (represented in Figure 2).

Figure 2.

Three possible switching sequences for three-phase permanent magnet synchronous motor.

The voltage equations can be presented as follows:

{\begin{matrix} V_{a} = \frac{d}{dt} (L_{sa} i_{a} + M_{ab} i_{b} + M_{ac} i_{c}) \\ V_{b} = \frac{d}{dt} (L_{ba} i_{a} + L_{sb} i_{b} + M_{bc} i_{c}) \\ V_{c} = \frac{d}{dt} (L_{ca} i_{a} + M_{cb} i_{b} + L_{sc} i_{c}) \end{matrix}

(15)

with

L_{sa}, L_{sb}, L_{sc} : self inductances phases;

M_{ab}, M_{ac}, M_{ba}, M_{bc}, M_{ca} and M_{bc} : mutual inductances phase .

Given that the system is supposed to be linear, the suppression of the current $i_{c}$ in Equation (15) presents as:

{\begin{matrix} V_{a} = \frac{d}{dt} {(L_{a} - M_{ac}) i_{a} + (L_{b} - M_{ac}) i_{b}} \\ V_{b} = \frac{d}{dt} {(M_{ba} - M_{bc}) i_{a} + (L_{b} - M_{bc}) i_{b}} \\ V_{c} = \frac{d}{dt} {(M_{ac} - L_{c}) i_{a} + (M_{cb} - L_{c}) i_{b}} \end{matrix}

(16)

From the connection configuration of the phase to the PMSM test vector $U_{1} (C_{1} {\bar{C}}_{2} {\bar{C}}_{3})$ , we can show the potential differences $V_{ab}$ and $V_{ac}$ :

{\begin{matrix} V_{a b} = \frac{d}{d t} {(L_{a} - M_{a c} - L_{b} + M_{c b}) i_{a} + (L_{a} - M_{a c} - L_{b} + M_{b c}) i_{b}} \\ V_{a c} = \frac{d}{d t} {(L_{a} - 2 M_{a c} + L_{c}) i_{a} + (L_{a b} - M_{a c} - L_{c} - M_{c b}) i_{b}} \\ M_{a c} = M_{c a} and M_{c b} = M_{b c} \end{matrix}

(17)

After all calculations are performed, the expression of the current of the phase a is represented by:

\begin{matrix} \frac{d}{dt} (i_{a}) & = \frac{4}{9} (\frac{L_{0} + L_{1} \cos (2 θ_{r 0})}{{L_{0}}^{2} - {L_{1}}^{2}}) v_{ab} \\ = \frac{4}{9} (\frac{L_{0} + L_{1} \cos (2 θ_{r 0})}{{L_{0}}^{2} - {L_{1}}^{2}}) v_{ac} \end{matrix}

(18)

where

L₀: component of the self inductance due to space fundamental air-gap flux;

L₁: component of the self inductance due to rotor position dependent flux.

We find that the coefficient-related voltages are represented by:

Y_{eq} = Y_{0} + Y_{1} \cos (2 θ_{r 0})

(19)

with

Y_{0} = \frac{4}{9} (\frac{L_{0}}{{L_{0}}^{2} - {L_{1}}^{2}})

and

Y_{1} = \frac{4}{9} (\frac{L_{2}}{{L_{0}}^{2} - {L_{1}}^{2}})

Applying a positive voltage pulse $U_{1} (C_{1} {\bar{C}}_{2} {\bar{C}}_{3})$ at the terminals of the phase motor generates a current in the form:

i_{a} = i_{0} + Δ i_{a} = i_{0} + Δ i_{0} \cos (2 θ_{r 0})

(20)

with $i_{0} = \frac{1}{3} (i_{a} + i_{b} + i_{c})$ the dc current component.

In the same way, the expressions of the currents that correspond to the voltage vector $U_{2} ({\bar{C}}_{1} C_{2} {\bar{C}}_{3})$ and $U_{3} ({\bar{C}}_{1} {\bar{C}}_{2} C_{3})$ are expressed by:

{\begin{matrix} i_{b} = i_{0} + Δ i_{0} \cos (2 θ_{r 0} - \frac{2 π}{3}) \\ i_{c} = i_{0} + Δ i_{0} \cos (2 θ_{r 0} + \frac{2 π}{3}) \end{matrix}

(21)

We can observe that the monotonous function lets us substitute the currents of the stator by their respective differences $Δ i_{a}, Δ i_{b} and Δ i_{c}$ . The expression of the position estimation can be given as follows:

{\begin{matrix} θ_{r 0} = \frac{\sqrt{3}}{2} \frac{(Δ i_{a} - Δ i_{b})}{(2 Δ i_{a} - Δ i_{b} - Δ i_{c})} \\ or \\ θ_{r 0} = \frac{\sqrt{3}}{2} \frac{(Δ i_{a} - Δ i_{b})}{(2 Δ i_{a} - Δ i_{b} - Δ i_{c})} + 180 \circ \end{matrix}

(22)

We can determine, while using (23), the values in Table 1 for the initial position detection.

Table 1.

Initial position estimation.

$θ_{r 0}$	Sign of the differences	Sign of the peaks	Sign of the peaks
15–30° or 195–210°	ΔI_a–ΔI_c>0	ΔI_b<0	ΔI_a>0
30–45° or 210–225°	ΔI_c–ΔI_a>0	ΔI_b<0	ΔI_a>0
45–60° or 225–240°	ΔI_a–ΔI_b>0	ΔI_c>0	ΔI_a<0
60–75° or 240–255°	ΔI_b–ΔI_a>0	ΔI_c>0	ΔI_a<0
75–90° or 255–270°	ΔI_c–ΔI_b>0	ΔI_a<0	ΔI_b>0
90–105° or 270–285°	ΔI_b–ΔI_c>0	ΔI_a<0	ΔI_b>0
105–120° or 285–300°	ΔI_c–ΔI_a>0	ΔI_b>0	ΔI_a<0
120–135° or 300–315°	ΔI_a–ΔI_c>0	ΔI_b>0	ΔI_a<0
135–150° or 315–330°	ΔI_b–ΔI_a>0	ΔI_c<0	ΔI_a>0
150–165° or 330–345°	ΔI_a–ΔI_b>0	ΔI_c<0	ΔI_a>0
165–180° or 345–360°	ΔI_b–ΔI_c>0	ΔI_a>0	ΔI_b<0
180–195° or 360–375°	ΔI_c–ΔI_b>0	ΔI_a>0	ΔI_b<0

We conclude that, according to the Table 1, it is possible to detect the initial position of the rotor with two different values. This problem is resolved in the next subsection. To overcome this problem, another short-length test signal is applied in order to receive a saturated magnetic circuit of the motor winding.

After all $Δ I_{a}, Δ I_{b}, Δ I_{c} and I_{0}$ calculations have been performed, we gain the experimental curve detection of the rotor position in the sector of 15° (electrical). Figure 3 displays the difference values of the current peaks in relation to $I_{0}$ in the function of the electrical position for the case of a non-saturated condition of the winding inductance.

Figure 3.

Experimental results of the peak currents versus rotor position.

The experimental results (Figure 3) confirm the feasibility of the proposed initial position estimation approach.

Discrimination of the uncertainty of the initial rotor position

This method consists of applying a pulse voltage to a saturated induction motor (Boussak, 2005).

Figure 4 gives the discrimination of the two values of the initial position. If the magnetic flux is in the same direction as that engendered by the current pulse, we obtain an additive flux and therefore a large peak current. In the other case, we obtain a descending field. Therefore, the initial rotor position corresponds to a current of the most important variation. When the north pole of the field is close to one axis of the three phase windings, the current peak is the most important in the considered phase. By considering the study previously developed to avoid the ambiguity on the accuracy concerning the rotor initial position estimation, we can produce the results in Table 2.

Figure 4.

Discrimination of the two values of the initial rotor position.

Table 2.

Discrimination of the uncertainty of the initial rotor position.

Voltage pulse	Peak value of the current	Location of the initial position
$U_{1} (C_{1} {\bar{C}}_{2} {\bar{C}}_{3})$	i_a>i_b, i_a>i_c	−60°<θ_r0<60°
$U_{2} ({\bar{C}}_{1} C_{2} {\bar{C}}_{3})$	i_b>i_a, i_b>i_c	60°<θ_r0<180°
$U_{3} ({\bar{C}}_{1} {\bar{C}}_{2} C_{3})$	i_c>i_b, i_c>i_a	180°<θ_r0<360°

Reduction of the error detection

The initial position $θ$ is situated in a sector of 15° (electrical). In order to minimize the initial position detection error, we can apply the proposed approach of Khlaief et al. (2012). To validate the variation of current $i_{d}$ of each test vector for a given rotor position, we apply the vectors $V_{1}, V_{3}, V_{5}, V_{7}, V_{9} and V_{11}$ . These vectors are functions of the configurations provided by Figure 5 and of the vector modulation principle for vectors $V_{2}, V_{4}, V_{6}, V_{8}, V_{10} and V_{12}$ with a short duration pulse.

Figure 5.

Application of the 12 test vectors for $θ_{r 0} = 27 \circ$ .

We can see in Figure 6 that the two vectors $V_{2}$ and $V_{8}$ produce an important current compared with others, which allows us to notice that the north pole of the magnet supported by the d-axis is close to the vector $V_{2}$ .

Figure 6.

Currents i_d representation for each voltage vector for $θ_{r 0} = 27 \circ$ .

After the verification that the nearest vector to the d-axis shows us the rotor position, it is possible to integrate this technique in a sector of 15° to minimize the initial position detection error, which is represented by Figure 7.

Figure 7.

Principle for the error reduction of initial position $θ_{r 0} = 19 \circ$

In the first sector and a real angle of the rotor equal to 19° (electrical), located between 15° and 30°, Figure 8 shows us that vector $V_{1}^{'}$ provides a maximum current $i_{d}$ .

Figure 8.

Current $i_{d}$ in function of the vector tests.

Therefore, we can note that the vector $V_{1}^{'}$ is the nearest to the axis supported by the north pole of the magnet. Thus, this novel approach leads us to improve the initial position detection with an uncertainty maintaining 3.75° (electrical).

In Figure 9, the experimental results of the rotor position estimation error ( $θ_{r 0} - {\hat{θ}}_{r 0}$ ) using the voltage pulse method was traced in terms of the measured position.

Figure 9.

Estimation error versus real rotor position.

According to this figure, we can conclude that the accuracy obtained during our tests is sufficient to reset the sensorless FOC algorithms, as the maximum error is about 3.75° (electrical). The obtained initial position estimation value is generally sufficient to eliminate random movement and reaches a stable start during start-up.

Experimental results and analysis

Experimental set-up

The experimental set-up is represented in Figure 10. It is based on the card from dSpace DS 1103, a PMSM, power inverter, driver IR 2130 in order to amplify the SVPWM signals generated by dSpace DS1103 card and active load used to change the load torque. The active load consists of an auto-start synchronous motor, a brake resistor and a load control drive. An incremental encoder with 4096 pulses per revolution is used only to obtain the position signal, which is solely used for comparison with the estimated speed and position.

Figure 10.

Picture of the experimental apparatus.

Experimental results

A sensorless FOC of PMSM drive has been implemented in the Matlab/Simulink toolbox, as indicated in Figure 1. The rated parameters of the PMSM and controller are given by Tables A1 and A2 in the Appendix.

In order to verify the dynamic performances of the proposed sensorless algorithm, many experimental tests have been carried out (Figures 11 and 12).

Figure 11.

Sensorless field-oriented control using the sigmoid function.

Figure 12.

Experimental results with speed reference of 500 rpm: (a) $Ω_{r}$ and ${\hat{Ω}}_{r}$ ; (b) zoomed $θ_{r}$ and ${\hat{θ}}_{r}$ at start-up; (c) phase current $i_{a}$ , $i_{b}$ and $i_{c}$ ; (d) estimation error; (e) measured load torque.

Figure 11 illustrates the experimental results of sensorless FOC using a sigmoid function at 500 rpm with initial rotor position equal to zero ( $θ_{r 0} = 0$ rad); Figure 11(a) shows the evolution of the measured and estimated speed, and Figure 11(b) shows the zoomed actual and estimated rotor position between [1s, 3s]. It should be noted that the estimated position tracks the measured position very well. Figure 11(c) shows the zoomed measured and estimated current of the stationary reference frame $i_{α}$ , and Figure 11(d) shows the zoomed measured and estimated current of the stationary reference frame $i_{β}$ . The speed error is very small and quickly converges to zero. Figure 11(e) seems that speed error reached 9 rpm during the transient and was about 0.08 rpm at steady state. The position estimation error is shown in Figure 11(f).

The application of sigmoid function does not require the use of a low pass filter, which allows the building of a time delay.

These results show that FO-SMO sensorless FOC based on sigmoid function can estimate correctly the rotor position and speed during the speed operation.

Example of initial rotor position detection

Figure 12 gives an example of initial rotor position estimation with the start-up of the PMSM drive. Before starting, Figure 12(a) and (b) give the waveforms speed at standstill when we applied the voltage pulse to the stator winding with initial rotor position $θ_{r 0} = 2.094$ .

The PMSM is accelerated to 500 rpm with a speed reference. Figure 12(b) shows the zoomed measured and estimated position at start-up.

To estimate the initial rotor position, two kinds of suitable sequence dc voltage rectangular pulses are applied from the inverter to the stator windings of the motor at standstill. One voltage pulse is applied with a short time duration T_sh, another with a long time duration T_L. The motor starts rotating with the estimated rotor position with the FO-SMO technique.

Before starting, the rotor did not move, but the appearance of a small vibration on the rotor is noteworthy. After monitoring the peak currents (Figure 12c), the detected initial rotor position is about 2.1 electrical radians (Figure 12b). The estimated and the actual rotor position are very close, as shown in Figure 12(b). In Figure 12(d), the experimental results show the rotor speed error. The estimated and the actual rotor speed are very close, and the estimation error using the FO-SMO method does not exceed 9 rpm during transients and 0.08 rpm at steady state (Figure 12d). In Figure 12(d), we can observe that the load increases to 2 Nm; this increase is due to the coupling of the MSAP with the auto-synchronous motor without application of the load torque. It can be observed that the estimated position tracks the actual rotor position very well.

These results confirm the feasibility of sensorless FOC of PMSM drive. We also note that experimental results show validity of the method applied for estimation of the initial rotor position of the PMSM.

Conclusion

In this work, the use of the FO-SM observer leads to estimate the rotor speed of the PMSM drive. To avoid the chattering phenomenon in the FOC, we used the proposed observer, which applies a novel switching function commonly called the sigmoid. The stability of FO-SMO has been proved by the use of a Lyapunov stability analysis.

The FO-SMO is not observable at standstill; for this reason, it is not capable of estimating the initial rotor position for the PMSM. Therefore, we applied a voltage pulses tests approach to detect the initial rotor position.

The sensorless FOC has been experimentally validated using a dSpace DS 1103. Experimental results show that the proposed FO-SMO for sensorless FOC for PMSM is able to estimate precisely the speed and rotor position. These results proved the effectiveness of the proposed sensorless FOC of PMSM drives with detection of initial rotor position at standstill.

In future, we will be interested in detecting the initial rotor position at standstill for salient-PMSM.

Footnotes

Appendix

Table A2.

Parameters of controller.

Pole	6
PI currents controllers	$\begin{matrix} K_{p_i_{d}} = 2200, K_{i_i_{d}} = 40, \\ K_{p_i_{q}} = 400, K_{i_i_{q}} = 40 \end{matrix}$
IP speed controller	$K_{p_Ω} = 15, K_{i_Ω} = 6$
SMO gains	$\begin{matrix} K_{1} = 10000 \\ K_{2} = 25, K_{3} = 8 \end{matrix}$

PI, proportional–integral; IP, interior permanent; SMO, sliding mode observer.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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