Performance Improvement of Servo Control System Driven by Novel PMSM-DTC Based On Fixed Sector Division Criterion

Abstract

In order to improve the performance of the servo control system driven by a permanent magnet synchronous motor (PMSM) under novel direct torque control (NDTC), which, utilizing composite active vectors, fixed sector division criterion, is proposed in this paper. The precondition of the accurate compensations of torque and flux errors is that the sector where the stator flux linkage is located can be determined accurately. Consequently, the adaptive sector division criterion is adopted in NDTC. However, the computation burden is inevitably increased with the using of the adaptive part. On the other hand, the main errors can be compensated through SV-DTC (DTC-utilizing single active vector), while another active vector applied in NDTC can only supply the auxiliary error compensation. The relationships of the two active vectors’ characteristics in NDTC are analyzed in this paper based on the active factor. Furthermore, the fixed sector division criterion is proposed for NDTC (FS-NDTC), which can classify the complexity of the control system. Additionally, the switching table for the selections of the two active vectors is designed. The effectiveness of the proposed FS-NDTC is verified through the experimental results on a 100-W PMSM drive system.

1. Introduction

A permanent magnet synchronous motor (PMSM) driven by direct torque control (DTC) has a lot of merits, including high reliability and good control performance. Therefore, it has been applied in industrial application widely [1,2,3,4,5,6,7]. Field-oriented control (FOC) and direct torque control are two widely applied high-performance control strategies for the PMSM [8,9,10,11,12,13]. Unlike the decoupled-analyzing method in FOC, torque and flux linkage are controlled directly in DTC, therefore, the quickest dynamic response can be obtained in the PMSM driven by DTC [14,15,16]. However, as only six active vectors can be selected to compensate the errors of flux linkage and torque in conventional DTC, the PMSM suffers from some drawbacks, such as large torque and flux linkage ripples [17,18,19,20].

The errors of torque and flux in the conventional PMSM-DTC system are compensated by a single active vector comprehensively, thus, the torque error and the flux error cannot be compensated accurately [1]. In this case, the error compensational strategy used in a conventional PMSM-DTC system can be called single vector error compensational strategy (SV-DTC). Furthermore, the torque vector and the flux vector are introduced into the PMSM-DTC system, and the novel direct torque control (NDTC) scheme is proposed [2], which can improve the steady-state performance of the PMSM effectively. Despite that a better performance of the PMSM driven by NDTC can be obtained, the computation burden is inevitably increased, due to the complexity of the control system.

The aim of this paper is to simplify the complexity of the servo control system by using a fixed sector division criterion. The relationships between the two active vectors’ characteristics in NDTC are analyzed in this paper based on the active factor firstly. It should be noted that the main errors can be compensated by SV-DTC, and another active vector applied in NDTC can only supply the auxiliary error compensation. Therefore, the two active vectors applied in NDTC can be defined as master vector and slave vector, respectively.

Because the master vector can be selected based on SV-DTC, the precision of the slave vector selection is the key to determining the error compensational results. Therefore, the three-dimensional (3-D) diagram and the two-dimensional (2-D) contour map of the active factor are introduced to determine the suitable slave vector with the variation of the error rate and the sector location of the stator flux linkage. Furthermore, the fixed sector division criterion, including master sector and slave sector, is proposed for NDTC (FS-NDTC), which can classify the complexity of the control system effectively. Additionally, the switching table for the selections of the two active vectors is designed in this paper.

The rest of this paper comprises the following sections: The characteristics of the active factors are analyzed in Section 2. The suitable slave vectors with the variation of the error rate and the sector location of the stator flux linkage are described in Section 3. The characteristics of master vector and slave vector are also illustrated in Section 3. The fixed sector division criterion and the design of the switching table for master and slave vectors are given in Section 4. The description of the experimental setup and discussions on experimental results are given in Section 5. The conclusion is analyzed in Section 6.

2. Analysis of the Active Factors

2.1. Design of the Novel Switching Table Considering the Error Rate State

In the proposed novel direct torque control (NDTC) scheme in [2], torque active vector and flux active vector are applied in one control period, as shown in Figure 1. It is also shown in Figure 1 that the proposed fixed sector division criterion is used to replace the adaptive sector division criterion.

As the main errors can be compensated by the applied active vector in SV-DTC [1], the two applied active vectors in NDTC can be classified as master vector (the applied active vector in SV-DTC) and slave vector (another applied active vector in NDTC).

The error rate state of the control system can be defined as

$${\tau }_i=\{\begin{array}{ll}{\tau }_0<\tau <\frac{1}{{\tau }_0},& i=\mathrm{A}\\ \tau <{\tau }_0,& i=\mathrm{B}\\ \tau >\frac{1}{{\tau }_0},& i=\mathrm{C}\end{array}$$

(1)

where τ₀ is the reference value of error rate state.

The value of the error rate state is calculated as

$$\tau =\frac{r_{\mathrm{T}}}{r_{\mathrm{F}}}$$

(2)

where r_T and r_F are torque error rate and flux error rate, respectively, which can be obtained by

$$\left[\begin{array}{c}{r}_{\mathrm{T}}\\ r_{\mathrm{F}}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{C_{\mathrm{T}}}& \\ & \frac{1}{C_{\mathrm{F}}}\end{array}\right]\left[\begin{array}{c}{e}_{\mathrm{T}}\\ e_{\mathrm{F}}\end{array}\right]$$

(3)

where C_T and C_F are the max compensations of torque error and flux error supplied by the active vector, respectively.

Therefore, the switching table of the active vectors in SV-DTC can be modified based on the value of the error rate state, as shown in

Table 1. Vector switching table.

Sector	Error Rate State	Error Property	Vn	Error Property	Vn
S2+	τA/C	a	VN+1	b	VN+3
		c	VN	d	VN+4
	τB	a	VN+2	b	VN+2
		c	VN+5	d	VN+5
S1+	τA/B/C	a	VN+1	b	VN+3
		c	VN	d	VN+4
S0	τA/B	a	VN+1	b	VN+2
		c	VN+5	d	VN+4
	τC	a	VN	b	VN+3
		c	VN	d	VN+3
S1−	τA/B/C	a	VN	b	VN+2
		c	VN+5	d	VN+3
S2−	τA/C	a	VN	b	VN+2
		c	VN+5	d	VN+3
	τB	a	VN+1	b	VN+1
		c	VN+4	d	VN+4

.

The error property in Table 1 is defined according to the values of torque error property and flux error property, as shown in

Table 2. Error property compound modes.

Error Property	Torque Error Property	Flux Error Property
a	1	1
b	1	−1
c	−1	1
d	−1	−1

.

2.2. Analysis of the Characteristics for the Active Vectors

The active angle and the impact angle of the one applied active vector in NDTC are θ_s and δ_s, respectively, as shown in Figure 2.

The active factors of torque error and flux error supplied by the applied active vector in Figure 1 can be calculated as

$${\mu }_{\mathrm{T}}=\sin {\theta }_{\mathrm{s}}$$

(4)

$${\mu }_{\mathrm{F}}=\cos {\theta }_{\mathrm{s}}$$

(5)

In NDTC, the adaptive sector division criterion is adopted. Therefore, the active factors of the torque error and the flux error within different error rate states and different small sectors are shown in

Table 3. Relationships among active factors and motor variables.

Error Rate State	τA		τB		τC
	μT	μF	μT	μF	μT	μF
S2+	0.5	0.9	1	0	0.5	0.9
S1+	(0.9, 0.5)	(0.5, 0.9)	(0.9, 0.5)	(0.5, 0.9)	(0.9, 0.5)	(0.5, 0.9)
S0	0.9	0.5	0.9	0.5	0	1
S1−	(0.5, 0)	(0.9, 1)	(0.5, 0)	(0.9, 1)	(0.5, 0)	(0.9, 1)
S2−	0.5	0.9	1	0	0.5	0.9

.

With the using of different active vectors, the active factors, active angle, and the impact angle are binary periodic functions. In order to obtain the variation character of the active factors, the vector angle θ_v that is between the applied active vector and the sector vector is introduced to replace the active angle. Therefore, the three-dimensional (3-D) diagram of μ_T and μ_F can be obtained, as shown in Figure 3a and Figure 4a. Furthermore, the two-dimensional contour map of the torque active factor and flux active factor are obtained by describing the third dimension, as shown in Figure 3b and Figure 4b.

3. Analysis of the Suitable Slave Vectors with the Variation of the Variables

3.1. Slave Vectors Selection in Main Sector

In SV-DTC, the suitable active vectors are V₂ or V₁, which can be applied to compensate the errors, while the stator flux linkage is located in S₁₊ or S₁₋ small sector. The vector angles of V₂ and V₁ are 60° and 0°, respectively, as shown in Figure 5.

(1) error rate state is τ_B

(a) S₁₊ sector

Torque error is larger than flux error in the error rate state of τ_B. From Figure 5, it can be observed that the max value of torque active factor supplied by the master vector V₂ is 0.9 with the impact changing from −30° and 0°.

In order to compensate the large torque error effectively, the active vector V₃, which supplied the best torque active factor, should be selected as the slave vector, and its vector angle is 120°.

(b) S₁₋ sector

It is also shown in Figure 5 that the max value of torque active factor of the master vector V₁ is only 0.5, with the impact changing from 0° and 30°. Therefore, the active vector V₂, which supplied the best torque active factor, should be selected as the slave vector to compensate the large torque error effectively, and its vector angle is 60°.

(2) error rate state is τ_C

(a) S₁₊ sector

The flux error is larger than the torque error in the error rate state of τ_C. From Figure 6, it can be observed that the wanted flux active factor cannot be supplied by the master vector V₂ with the impact changing from −30° and 0°.

In order to meet the required large flux error compensations, the active vector V₁, which supplied the best flux active factor, should be selected as the slave vector to compensate the large flux error effectively, and its vector angle is 0°.

(b) S₁₋ sector

It is also shown in Figure 6 that master vector V₁ can supply larger flux error compensations. To avoid the offset of the error compensations, the active vector V₆, which supplied the positive flux active factor, should be selected as the slave vector, and its vector angle is 300°.

(3) error rate state is τ_A

(a) S₁₊ sector

The stator flux linkage is located in S₁₊, and the torque active factor and the flux active factor are equal, so the relationships of the variables can be described as

$${\mu }_{\mathrm{T}}=\sin {\theta }_{\mathrm{s}2}={\mu }_{\mathrm{F}}=\cos {\theta }_{\mathrm{s}2}$$

(6)

The impact angle can be obtained as

$${\delta }_{\mathrm{s}2}={\theta }_{\mathrm{s}2}-60°=-15°$$

(7)

Figure 7 shows that the value of the torque active factor supplied by the master vector V₂ is the balance value of 0.7, while the impact angle is −15°. In addition, the torque error compensation will become smaller with the impact angle changing from −15° to −30°. Therefore, the active vector V₃ should be selected as the slave vector, which can supply the decreased torque error compensation supplied by the master vector V₂.

According to the two-dimensional contour map of the torque active factor in Figure 7, it can be observed that the value of the torque active factor supplied by the master vector V₂ is the balance value of 0.7, while the impact angle is −15°. In addition, the torque error compensation will become smaller with the impact angle changing from −15° to −30°. Therefore, the active vector V₃ should be selected as the slave vector, which can supply the decreased torque error compensation supplied by the master vector V₂.

According to the two-dimensional contour map of the flux active factor in Figure 8, it can be observed that the value of flux active factor supplied by the master vector V₂ is the balance value of 0.7, while the impact angle is −15°. Consequently, the flux error compensation will become smaller with the impact angle changing from −15° to 0°. In this case, the active vector V₁ should be selected as the slave vector, which can supply the decreased flux error compensation supplied by the master vector V₂.

(b) S₁₋ sector

When the stator flux linkage is located in S₁₋ sector, the variation range of the impact angle is from 0° to 30°. The comparison result of the torque active factor and the flux active factor supplied by the master vector V₁ can be expressed as

$${\mu }_{\mathrm{T}}<{\mu }_{\mathrm{F}}$$

(8)

According to the two-dimensional contour maps of the torque active factor and flux active factor in Figure 7 and Figure 8, it can be observed that the value of the torque active factor supplied by V₁ is constantly lower than the balance value 0.7, while the value of the flux active factor is constantly greater than the balance value 0.7. Thus, V₂ is the most suitable active vector to be the slave vector, considering the torque error and the flux error compensational results synthetically.

3.2. Slave Vectors Selection in Other Sectors

(1) S₀ sector

(a) The active vector V₂ will be selected as the master vector while the error rate state is τ_A. According to the two-dimensional contour map of the flux active factor in Figure 9, it can be observed that the value of the flux active factor supplied by the master vector V₂ is constantly lower than the balance value of 0.7. Therefore, in order to compensate the motor errors, the active vector V₁ should be selected as the slave vector, which can supply the flux error compensation supplied by the master vector V₂.

(b) The active vector V₂ will be selected as the master vector while the error rate state is τ_B. According to the two-dimensional contour map of the torque active factor in Figure 10, it can be observed that the master vector V₂ can supply the large torque error compensation. The torque error is much greater than the flux error in the error rate state of τ_B, therefore, the active vector V₃ that can supply the greatest torque error compensation should be selected as the slave vector.

(c) The active vector V₁ will be selected as the master vector while the error rate state is τ_C. According to the two-dimensional contour map of the flux active factor in Figure 9, it can be observed that the master vector V₁ can supply the greatest flux error compensation. The flux error is much greater than the torque error in the error rate state of τ_C, therefore, the active vector V₂ that can supply the positive flux error compensation should be selected as the slave vector to avoid the offset of the flux error compensations supplied by different active vectors.

(2) S₂₊ sector

(a) The active vector V₂ will be selected as the master vector while the error rate state is τ_A. According to the two-dimensional contour map of the torque active factor in Figure 10, it can be observed that the value of the torque active factor supplied by the master vector V₂ is constantly lower than the balance value 0.7. Therefore, the active vector V₁ should be selected as the slave vector, which can supply the insatiable torque error compensation supplied by the master vector V₂, which can compensate the motor errors effectively.

(b) The active vector V₃ will be selected as the master vector while the error rate state is τ_B. According to the two-dimensional contour map of the torque active factor in Figure 10, it can be observed that the master vector V₃ can supply the greatest torque error compensation. The torque error is much greater than the flux error in the error rate state of τ_B, therefore, the active vector V₂ that can supply the positive torque error compensation should be selected as the slave vector to avoid the offset of the torque error compensations supplied by different active vectors.

(c) The active vector V₂ will be selected as the master vector while the error rate state is τ_C. According to the two-dimensional contour map of the flux active factor in Figure 9, it can be observed that the master vector V₂ can supply a large flux error compensation. The flux error is much greater than the torque error in the error rate state of τ_C, hence, the active vector V₁ that can supply much greater flux error compensation should be selected as the slave vector.

(3) S₂₋ sector

(a) The active vector V₁ is selected as the master vector while the error rate state is τ_A. According to the two-dimensional contour map of the torque active factor in Figure 10, it can be observed that the value of torque active factor supplied by V₁ is constantly lower than the balance value 0.7. Therefore, the active vector V₂ should be selected as the slave vector, which can supply much greater torque error compensation.

(b) The active vector V₂ will be selected as the master vector while the error rate state is τ_B. According to the two-dimensional contour map of torque active factor in Figure 10, it can be observed that the master vector V₂ can supply the greatest torque error compensation. The torque error is much greater than the flux error in the error rate state of τ_B, therefore, the active vector V₁ that can supply the positive torque error compensation should be selected as the slave vector to avoid the offset of the torque error compensations supplied by different active vectors.

(c) The active vector V₁ will be selected as the master vector while the error rate state is τ_C. According to the two-dimensional contour map of the flux active factor in Figure 9, it can be observed that the master vector V₁ can supply the large flux error compensation. The flux error is much greater than the torque error in the error rate state of τ_C, hence, the active vector V₆ that can also supply the large flux error compensation should be selected as the slave vector.

3.3. Analysisthe Characteristics of the Master and Slave Vectors

From the aforementioned analyses, it can be observed that the suitable master vector and slave vector are different with the variation of the error rate state and the sector where the stator flux linkage is located.

Table 4. Relationships among active factors and motor variables.

Error Rate State	τA		τB		τC
	Vm	Vs	Vm	Vs	Vm	Vs
S2+	V2	V3	V3	V2	V2	V1
S1++	V2	V3	V2	V3	V2	V1
S1+−	V2	V1	V2	V3	V2	V1
S0	V2	V1	V2	V3	V1	V2
S1−	V1	V2	V1	V2	V1	V6
S2−	V1	V2	V2	V1	V1	V6

shows the conclusions of the suitable master vector and slave vector for different variations of PMSM.

There exists no difference with the applied sequence variation of master vector and slave vector. Therefore, the characteristics of the suitable active vector in each small sector can be described as:

(1) The selected master and slave vectors are the same, while the stator flux linkage is located in S₂₊ or S₁₊₊ sector;

(2) The selected master and slave vectors are the same, while the stator flux linkage is located in S₀ or S₁₊₋ sector;

(3) The selected master and slave vectors are the same, while the stator flux linkage is located in S₂₋ or S₁₋ sector.

In short, the small sectors S₂₊, S₀ and S₂₋ are not necessary. Furthermore, the adaptive sector division criterion is not suitable for NDTC.

4. Design of Master and Slave Vectors’ Switching Table

4.1. Devision of Master and Slave Sectors

The negative sector and the positive sector in each sector in conventional DTC are defined as master sector and slave sector, respectively. The slave sector can be divided into the positive slave sector and the negative slave sector through the median line l_m+, according to the Equation (7).

The location of the master sector and the slave sector will exchange equally, therefore, the master sector should also be divided into the positive master sector and the negative master sector through the median line l_m−. The analysis results of the sector characteristic for sector 1 is shown in Figure 11.

According to the analysis results in Figure 10, the division criterion for the master and slave sector, which is suitable for NDTC, can be obtained in terms of the active vector that is closest to the error, as shown in Figure 12.

4.2. Master and Slave Vectors Switching Table

The slave vector is applied to compensate the errors that the master vector cannot compensate, therefore, the master vector switching table is the same in SV-DTC, and the slave vector switching table that considers the error rate state is shown in

Table 5. Slave vector switching table.

τi	Sector	Error Property	Vs	Error Property	Vs
τA	Ss+	a	VN+2	b	VN+2
		c	VN+5	d	VN+5
	Ss−	a	VN	b	VN+2
		c	VN+5	d	VN+5
	Sm+	a	VN+1	b	VN+1
		c	VN	d	VN+4
	Sm−	a	VN+1	b	VN+1
		c	VN+4	d	VN+4
τB	Ss	a	VN+2	b	VN+2
		c	VN+5	d	VN+5
	Sm	a	VN+1	b	VN+1
		c	VN+4	d	VN+4
τC	Ss	a	VN	b	VN+4
		c	VN+1	d	VN+3
	Sm	a	VN+5	b	VN+3
		c	VN	d	VN+2

.

5. Experimental Analysis

5.1. Experimental System Setup

Experimental studies are carried out on a 100-W PMSM drive system to validate the effectiveness of the fixed sector division criterion for NDTC (FS-NDTC). The experimental hardware setup is illustrated in Figure 13. The parameters of the PMSM are given as follows: R_s = 0.76 Ω; L_s = 0.00182 H; the number of pole pairs p = 4. The DC voltage is 36 V. The rated speed and torque are 750 rpm and 1.2 Nm, respectively. This study compares the steady-state and the dynamic response performance of the PMSM, driven by FS-NDTC and NDTC. The experiments are implemented in a TMS320F28335 DSP control system with a sampling period of 100 μs.

5.2. Steady-State Performance

The steady-state performances of FS-NDTC and NDTC are compared under the same operating conditions. The PMSM is operated at 500 rpm and the reference values of torque and flux linkage are 0.8 Nm and 0.3 Wb, respectively. The torque and flux linkage waveforms of the PMSM driven by NDTC and FS-NDTC are shown in Figure 14a,b, respectively.

From these experimental results, it can be found that the torque ripples and the flux linkage ripples of NDTC are 0.32 Nm and 0.035 Wb, respectively. The torque ripples and the flux linkage ripples of FS-NDTC are 0.32 Nm and 0.045 Wb, respectively. Therefore, although the proposed fixed sector division criterion classify the complexity of NDTC, the ideal steady-state performance in NDTC will not be affected with the using of a fixed sector division criterion.

5.3. Dynamic Performance

To validate the fast dynamic response of the proposed FS-NDTC, the speed and torque responses of the PMSM driven by the two control strategies are tested. In the speed tests, a step change from 200 to 400 rpm is applied on the speed reference when the torque is set as 0.5 Nm. The speed waveforms of the PMSM driven by NDTC and FS-NDTC are as shown in Figure 15a,b, respectively.

In the torque and flux linkage tests, a step change from 0 to 100 rpm is applied on the speed reference when the torque and the flux linkage are set as 0.5 Nm and 0.3 Wb, respectively. The torque and flux linkage waveforms of the PMSM driven by NDTC and FS-NDTC are as shown in Figure 16a,b, respectively.

From the above tests, it can be seen that the ripples of the rotor speed with the using of NDTC and FS-NDTC are 29 and 27 rpm, respectively. Moreover, the settling time of the rotor speed using the two different control strategies are 0.014 and 0.015 s. In the torque and flux linkage tests, it can be seen that the flux linkage can adjust to the settling value within an instant, and the settling times of the torque using the two different control strategies are 0.034 and 0.032 s.

Therefore, the main advantage of NDTC, i.e., the fast dynamic response, is maintained in FS-NDTC. The experimental results show that dynamic response owns higher priority than ripples in dynamic-state condition, hence, the adaptive sector division criterion can be abandoned.

6. Conclusions

The fixed sector division criterion is introduced to replace the adaptive sector division criterion used in NDTC in this paper. The proposed fixed sector division criterion can simplify the determination process of the sector where the stator flux linkage is located. To ensure the reasonability of the fixed sector division criterion, the active factors of the applied active vectors are used to obtain the characteristic relationships of the master vector and the slave vector in NDTC. Finally, the switching table for the slave vector is designed, which can ensure the required error compensational results, using the proposed fixed sector division criterion.

Experimental results clearly indicate that the FS-NDTC exhibits similar control performance of torque and flux linkage when compared to NDTC, and the transient response performance can also be retained in FS-NDTC. In short, the steady and dynamic performances of the PMSM driven by NDTC and FS-NDTC are similar. The proposed fixed sector division criterion can simplify the complexity of the servo control system without decreasing the operation performance of the PMSM.