Direct torque control of PMSM using sliding mode

Article

Direct torque control of PMSM using sliding mode backstepping control with extended state observer

Journal of Vibration and Control 1–14 ! The Author(s) 2016 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546316650097 jvc.sagepub.com

Bowen Ning, Shanmei Cheng and Yi Qin

Abstract Based on the nonlinear characteristics of permanent magnet synchronous motor (PMSM), a nonlinear speed and direct torque control (DTC) using sliding mode backstepping method for PMSM is presented in this paper. The sliding mode speed controller is implemented with exponential reaching law to improve the robustness of the system, and further a step-by-step recursive design for backstepping torque and flux controllers is presented. The system stability with proposed scheme is mathematically proved using Lyapunov stability criteria. At the same time, the load torque is observed with the extended state observer (ESO), and is fed-forward to the controller for rejecting the load disturbance and to mitigate the chattering affect due to the sliding mode controller. Finally, simulation test results are demonstrated to support the effectiveness and feasibility of the proposed strategy.

Keywords Permanent magnet synchronous motor, direct torque control, sliding mode control, backstepping control, extended state observer

1. Introduction Permanent magnet synchronous motor (PMSM) possesses considerable advantages such as small size, simple design, high efficiency and high-power density (Choi et al., 2012; Wei et al., 2014). It has been widely used in applications like servo systems, robots and new energy electric vehicles due to the rapid development of power converters and control theory (Ananthamoorthy and Baskaran, 2015). Direct torque control (DTC) is one of the advanced control techniques adopted to control torque and flux linkage in industrial drives (Rahman et al., 1998; Sivaprakasam and Manigandan, 2015). However, conventional DTC presents some weaknesses including large torque and flux ripples, high-current distortions and unfixed switching frequency. Many research activities have been carried out in recent decades to overcome the weaknesses of the conventional DTC (Zhang and Zhu, 2011; Zhu et al., 2012; Cho et al., 2015; Foo and Rahman, 2008). Some methods have been proposed to improve the DTC performance, among which the space vector modulation (SVM) based DTC method is shown as one of

the very effective schemes. In some literatures (Foo and Rahman, 2008; Tang et al., 2004; Inoue et al., 2010), several SVM based DTC methods which could reduce the flux and torque ripples while acquiring fixed switching frequency are discussed. In these PMSM systems, the proportional plus integral (PI) controllers are used because of their simplicity in understanding and practical implementation. However PI controllers are sensitive to uncertainties and disturbances, and appropriate PI parameters are not easy to obtain. In fact, PMSM is considered as a complex nonlinear system due to its notable characteristics such as strong coupling between different state variables. Key Laboratory of Education Ministry for Image Processing and Intelligent Control, School of Automation, Huazhong University of Science and Technology, Wuhan, China Received: 15 October 2015; accepted: 21 April 2016 Corresponding author: Shanmei Cheng, Key Laboratory of Education Ministry for Image Processing and Intelligent Control, School of Automation, Huazhong University of Science and Technology, Hongshan District, Wuhan 430074, China. Email: [email protected]

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Journal of Vibration and Control

Recently, attempts have been made to control PMSM using advanced control algorithms due to the rapid development of control theory and the high-performance requirement for the drive system. Backstepping control is a newly developed controller that has a recursive and systematic design for nonlinear system (Yu et al., 2010; Liu et al., 2010). This method outputs the final control signal through a number of recursive steps adopting the Lyapunov function to ensure the stability of the system. Recently, the backstepping control method that could take the advantage of the nonlinear characteristics of motor has attracted attentions of researchers. In Trabelsi et al. (2012) based on the nonlinear of the induction motor, an adaptive backstepping controller along with the rotor-flux sliding mode observer is designed to control motor. In some literatures (Rahman et al., 2003; Foo and Rahman, 2009; Morawiec, 2013), the adaptive backstepping method is proposed to design the PMSM system and the results show that the system has the desirable performance. The nonlinear sliding mode control has outstanding qualities such as robustness against the parameter variation and external disturbances, thus, the system can have the fast dynamic performance (Zhang et al., 2013; Wang et al., 2013). In this study, the sliding mode control is used as speed controller and the sliding mode backstepping DTC scheme is proposed. In the real system, the superior disturbance rejection property of the sliding mode control is realized by increasing the switching gain. This further amplified the inherent chattering problem of the sliding mode control. To solve the conflict between the disturbance rejection and the chattering effect in sliding mode controller, the disturbance is observed and an appropriate compensation component is designed. Since it is difficult to measure the disturbances accurately using a simple observer, an extended state observer (ESO), which has superior performance compared to the earlier one, is introduced to estimate the internal dynamics and external disturbances of the system (Yao et al., 2014; Li and Li, 2014). In this paper, a feed-forward compensation component is employed with ESO to weaken the chattering and further enhance the disturbance rejection of the system. This paper is arranged as follows. First, the dynamic model of the system is discussed and a speed controller is designed. The steady state error in speed can be eliminated by adopting appropriate sliding surface and the sliding mode speed controller is obtained with exponential reaching law. Then the backstepping torque and flux controllers are designed to obtain the complete control law. The linear ESO is employed to observe load disturbance and added as a feed-forward term. At last, the feasibility and effectiveness of the proposed control method are validated by means of simulation results.

2. Mathematical model of PMSM To simplify the analysis, it is assumed that the magnetic circuit is unsaturated, the magnetic field in the air gap is distributed sinusoidally in space, and hysteresis and eddy-current losses are neglected. With this understanding, the dynamic model of the PMSM in the stationary a-b reference frame is given by the following equations (Adhavan and Jagannathan, 2014). 8 Rs u E _ > < i ¼ L i þ L L , u E i_ ¼ RLs i þ L L , > : !_ re ¼ 1J B!re þ np ðTe TL Þ

ð1Þ

Where E, E are the back EMFs along a, b axis respectively and can be expressed as (

E ¼ !re

f ð sin Þ,

E ¼ !re

f

cos

ð2Þ

The stator flux linkage is given by the following equation. 8 _ > < ¼ u i Rs , _ ¼ u i Rs , > : ¼ 2 þ 2

ð3Þ

The electromagnetic torque equation is as follows 3 T e ¼ np i 2

i

ð4Þ

Where u, u, i and i are voltages and currents in a-b frame, Rs is the stator resistance, L is the stator inductance, !re and are the rotor electrical angular velocity and the rotor electrical angle, respectively. f is the permanent magnet flux, and denote the stator flux linkages along the a-axis and b-axis, respectively. is the square of stator Fux linkage, np is the number of pole pairs, B and J are viscous friction coefficient and the moment of inertia, respectively. TL and Te represent the load and electromagnetic torques, respectively.

3. Sliding mode backstepping control scheme 3.1. Exponential reaching law In order to enhance the dynamic performance of sliding mode control, the concept of reaching law is proposed and then developed (Gao and Hung, 1993).The expression of exponential reaching law is given by the


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following equation, where sgn(s) is the sign function (Fallaha et al., 2011).

The sliding surface is designed as follows s ¼ cx1 þ x2 , c 4 0

s_ ¼ "sgnðsÞ ks,

" 4 0,

k40

ð12Þ

ð5Þ Taking the time derivative of (12), yields

When s > 0, equation (5) can be described as follows s_ ¼ cx_ 1 þ x_ 2 s_ ¼ " ks

ð6Þ ¼ 2 þ !€ þ

Solution for (6) is given by (7). " " sðtÞ ¼ þ s0 þ ekt , k k

s0 ¼ sð0Þ

ð7Þ

When t is sufficiently large, the reaching speed will be faster than the exponential law. When s(t) ¼ 0, the following expression is obtained. " " ¼ s0 þ ekt k k

ð8Þ

The time required to reach the sliding surface can be obtained by solving (8) and is given by (9). t¼

1h " "i ln s0 þ ln k k k

ð9Þ

In other words, it can be deduced from (9) that the system can reach the sliding surface from initial state in finite time. Furthermore, the reaching speed can be regulated by adjusting the parameters " and k directly. The system has fast speed when it is far away from the sliding surface, and it slows down as it approaches the sliding surface. Reaching law method can ensure dynamic performance for sliding mode control and mitigate the chattering effect.

3.2. Controller design The sliding mode speed controller for PMSM is designed to track the reference speed precisely even with load disturbance. To achieve this control objective, the following are defined as state variables.

x1 ¼ ! !, x2 ¼ x_ 1 ¼ !_ !_

ð10Þ

Where !* is the motor speed reference, ! is the actual rotor speed of the motor, and ! ¼ !re/np. Taking the time derivative of (10) and substituting (1) into the resulting equation, the following equation can be obtained (

x_ 1 ¼ !_ 1J ½B! þ ðTe TL Þ, x_ 2 ¼ x_ 1 ¼ !€ 1 B!_ þ T_ e

ð11Þ

T_ e B !_ J J

ð13Þ

To improve the dynamic response of sliding mode speed controller, it should be designed with exponential reaching law as discussed earlier. From (5) and (13), the expression for required reference torque is obtained and is given below. Te ¼ J

Z B cx2 þ !€ þ !_ þ "sgnðsÞ þ ks dt J

ð14Þ

It can be inferred from this equation that the integral term can act as a filter and attenuate the chattering effect in the sliding mode control. Choosing the Lyapunov function as V1 ¼ 12 s2 , the time derivative of function V1 is as follows V_ 1 ¼ s_s ¼ sð"sgnðsÞ ksÞ

ð15Þ

Therefore, the negative semi-definite of function V_ 1 can be guaranteed by an appropriate choice of the parameters " > 0 and k > 0, which results in the opposite signs for s and s_. The error between the reference speed and the actual speed will converge to zero with the sliding surface and then stay there, so the speed is asymptotically tracking. The backstepping torque and flux controller are designed to achieve the satisfactory torque and flux tracking. At this point, the torque and flux tracking errors are defined as follows

eT ¼ Te Te , e ¼

ð16Þ

Where * denotes the square of reference flux linkage. Now the torque tracking error eT and the flux tracking error e constitute a new system, and the torque tracking error dynamics can be obtained as follows.

B e_T ¼ J cx2 þ !€ þ !_ þ "sgnðsÞ þ ks J

3 Rs E Rs E np i i þ ð17Þ þ 2 L L L L

i u u þ þ i L L

J


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Figure 1. Schematic diagram of the PMSM DTC system using SMBC with ESO.

Figure 2. Schematic diagram of conventional PMSM DTC system.

The time derivative of function V2 is

Table 1. PMSM parameters. Parameter

Value

Mechanical inertia(J) Magnet flux linkage( f) Stator resistance(Rs) Stator inductance(L) Number of pole pairs(np) Rated torque(TN) Rated voltage(UN) Rated speed(nN)

Unit

V_ 2 ¼ V1 V_ 1 þ eT e_T þ e e_

0.000828 0.09428 0.779 0.003026 4 4 220 3000

kgm Wb H Nm V r/min

For stabilizing the flux components, the flux tracking error dynamics can be defined as,

The time derivatives of the Lyapunov function should be a non-positive value, hence e_T and e_ can be selected as follows.

i Rs

þ2

i Rs

2

u

2

u

ð18Þ

Defining the Lyapunov function V2 for whole system as, V2 ¼

1 2 V1 þ e2T þ e2 2

ð19Þ

e_T ¼ kT eT , e_ ¼ k e

ð21Þ

Where kT and k are positive constants. At last, the torque and flux tracking errors can be stabilized by designing the final control voltage outputs u and u as follows 1 þ i =L

B J cx2 þ !€ þ !_ þ "sgnðsÞ þ ks þ kT eT J

2 Rs E Rs E i þ i þ þ 3np L L L L

2 i Rs þ 2 i Rs þ k e i ð22Þ L 2

u ¼ e_ ¼ 2

ð20Þ

2

i


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Figure 3. The performance of sliding mode backstepping DTC system. (a) phase trajectory of the system and (b) speed response of the system. Table 2. Comparisons of conventional DTC and SMBC DTC. Method

Trip(Nm)

Conventional DTC Proposed SMBC DTC

0.778 0.135

u ¼

rip(Wb)

0.003 0.00063

Settling time(s) 0.08 0.08

Proof: According to (24), the time derivative of the Lyapunov function V2 is negative or zero which means that the function V2 is negative semi-definite; that is to say, V2 is not increasing. Hence, V2 is bounded and 0 V2 ð1Þ V2 ð0Þ 5 1, and following expression holds good. 8 _ > < V2 sð"sgnðsÞ ksÞ V_ 2 kT e2T > : _ V2 k e2

1

=L i i ( "

B J cx2 þ !€ þ !_ þ "sgnðsÞ þ ks þ kT eT J

2 Rs E Rs E i þ i þ þ 3np L L L L

2 i Rs þ 2 i Rs þ k e i ð23Þ L 2

Substituting the final control voltage outputs in function V_ 2 , the time derivative of the Lyapunov function is described as V_ 2 ¼ sð"sgnðsÞ ksÞ kT e2T k e2 0

ð24Þ

Therefore the asymptotically tracking of speed, torque and flux can be guaranteed by the control voltages u and u. This ensures the PMSM dynamic system has the satisfactory response.

ð25Þ

By integrating (25) from 0 to 1 we get, R1 8 > < V2 ð0Þ V2 ð1Þ R0 sð"sgnðsÞ þ ksÞdt 1 V2 ð0Þ V2 ð1Þ 0 kT e2T dt > R : 1 V2 ð0Þ V2 ð1Þ 0 k e2 dt

ð26Þ

Since 0 V2 ð0Þ V2 ð1Þ 5 1, the right side of (26) exists within the boundary. In addition the signals s, eT, e are uniformly continuous, according to Barbalat’s Lemma (Marino and Tomei, 1995), one can infer that lim s ¼ 0, lim eT ¼ 0, lim e ¼ 0

t!1

t!1

t!1

ð27Þ

This means the PMSM system is asymptotic stable, and the tracking errors of the signals will be converged to zero asymptotically.

3.4. The torque observer design based on ESO 3.3. Stability analysis Theorem 1. For the PMSM system described in (1)–(4), the control voltages given in (22) and (23) can ensure that the tracking errors of speed, torque and flux are asymptotically stable.

Sliding mode control suppresses the effects of parameter variations and external disturbances by employing discontinuous switching value. If the discontinuity in switching value can be decreased by observing the disturbances and compensating, the chattering can be


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Figure 4. The stator Fux response of the system. (a) -axis stator Fux, (b) -axis stator Fux and (c) Stator Fux trajectory curve.

reduced effectively. In this paper, the load disturbance is observed by ESO and then employed in control design as feed-forward compensation; hence the sliding speed controller can reduce the chattering and has better disturbance rejection. ESO is introduced as a kind of favorable disturbance observation technique by Han and the detailed principle of ESO are presented in Han (2009). It can not only obtain the state of the control system but also estimate the internal uncertainties and external disturbances which are considered as ‘extended states’. For nonlinear uncertain systems of the form given in (28), _ . . . , yðn1Þ , t þ wðtÞ þ bu yðnÞ ¼ f y, y,

ð28Þ

_ . . . , yðn1Þ , t is an unknown function, u is Where f y, y, the control input of the system, w(t) is an unknown disturbance, y is the output of the system. The state equations of the system can be obtained by rewriting the original system equations as 8 > < x_ 1 ¼ x2 , x_ 2 ¼ x3 , . . . , x_ n1 ¼ xn x_ n ¼ fðx1 , x2 , . . . , xn1 , xn , tÞ þ wðtÞ þ bu > : y ¼ x1

ð29Þ

Then the n-order ESO can be developed as follows 8 e ¼ z1 y > > > > > z_1 ¼ z2 01 falðe, 1 , Þ > > > > < z_2 ¼ z3 02 falðe, 2 , Þ ð30Þ .. > > . > > > > > > > z_n ¼ znþ1 0n falðe, n , Þ þ b0 u : z_nþ1 ¼ 0ðnþ1Þ falðe, nþ1 , Þ where falðe, i Þ is nonlinear function given by, falðe, i , Þ ¼

jeji sgnðeÞ jej 4 jej e=1i

ð31Þ

Considering the kinematic equation of PMSM, its second-order ESO is described as 8 > < e ¼ z1 !, z_1 ¼ z2 01 falðe, 1 , Þ þ ðBz1 =JÞ þ Te =J, > : z_2 ¼ 02 falðe, 2 , Þ

ð32Þ

where 01 , 02 are constants of observer, 1 , 2 are nonlinear factors, is a filter factor, z1 can observe speed signal !, and z2 can observe load torque TL. The secondorder ESO as given by (32) includes the nonlinear function, which requires the design of several parameters,


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Figure 5. Start-up transient performance for (a) conventional DTC and (b) SMBC DTC.

and moreover, structure is also complex. This makes it difficult to adjust the parameters in simulation or practical application. If the disturbance of system is observed by a linear function, then the structure of ESO can be simplified effectively (Miklosovic and Gao, 2004). The simplified linear ESO expression is given as follows.

z_1 ¼ z2 1 ðz1 !Þ þ ðBz1 =JÞ þ Te =J, z_2 ¼ 2 ðz1 !Þ

Now there are only two parameters 1 and 2 , which simplify the design of ESO effectively hence, parameters can be adjusted easily. Defining e1 ¼ z1 !, e2 ¼ z2 ðTL =JÞ, the following expression can be obtained.

ð33Þ

e_1 ¼ z_1 !_ ¼ ð1 þ B=JÞe1 þ e2 , e_2 ¼ z_2 ¼ 2 e1


ð34Þ

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Figure 6. Steady-state performance in the case of 300 rpm reference speed for (a) conventional DTC and (b) SMBC DTC.

The characteristic equation of (34) can be obtained and is given by (35) gðsÞ ¼ s2 þ ð1 þ B=JÞs þ 2

ð35Þ

The necessary and sufficient condition for the stability of the system, with respect to the continuous system of (35), is that the roots of the characteristic equation should be negative. It means that ð1 þ B=JÞ 4 0, 2 4 0, hence the observer poles are designed appropriately to obtain the desired performance. Assigning the desired value of poles, say s1 ¼ s2 ¼ pð p 4 0Þ, the following expression can be obtained. gðsÞ ¼ s2 þ ð1 þ B=JÞs þ 2 ¼ s2 þ 2ps þ p2

ð36Þ

It can be observed from (36) that the parameter values 1 , 2 can be designed by configuring the value of p alone. This means that the number of parameters that need to be adjusted is further decreased. The estimated load torque based on the ESO is considered as a feed-forward compensation input and is introduced in Te as (37) Te ¼ J

Z

cx2 þ !€ þ

B !_ þ "sgnðsÞ þ ks dt þ ktl T^ L J ð37Þ

where ktl > 0 is the feed-forward compensation coefficient of the load torque. Since the sampling frequency is high, the load torque can be assumed as constant within a control cycle. Thus the torque variation is considered


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as T_ L ¼ 0, the stability of the system is not affected due to the additional load torque compensation term. In addition, when the load torque is changed, the controller will respond quickly because of the feed-forward signal. At the same time, it can been seen that the sliding mode controller can acquire better load disturbance refection without the use of larger values for reaching law parameters " and k, hence the chattering is attenuated by reducing the gain of sliding mode controller. The entire schematic diagram of the proposed PMSM DTC control system is illustrated in Figure 1.

4. Simulation results and analysis To confirm the feasibility and effectiveness of the proposed control strategy, a simulation model of the

sliding mode backstepping control (SMBC) with ESO for PMSM DTC systems is established in MATLAB. The block diagram of simulation is also shown in Figure 1. The simulation studies of conventional DTC are also carried out and results are discussed in this section. In conventional DTC scheme, the reference torque is generated from the standard PI based speed controller as shown in Figure 2. Then the hysteresis controllers are used to control the torque and flux. The schematic diagram of the conventional DTC is presented in Figure 2. The parameters of the PMSM used during this study are shown in Table 1. The parameters for the controller of the SMBC are: k ¼ 500, " ¼ 90, c ¼ 105, kT ¼ 7000, k ¼ 4000, the reference stator flux is 0.09428 Wb and the saturation limit of T* e is 12Nm. A fixed sampling frequency of 10 kHz


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is used in the simulation for the proposed control methods. The performance of the PMSM with the proposed SMBC is shown in Figure 3. The phase trajectory of the system using the proposed SMBC is shown in Figure 3(a). It can be observed from these results that the system converges to the sliding surface fast and then reaches origin along the sliding surface. The reference tracking performance is demonstrated in Figure 3(b), where the reference speed is given as 1500 rpm. The system starts without any load torque, a load disturbance torque of TL ¼ 2Nm is introduced at 0.2 s and then the load torque is changed from 2Nm to 0 at 0.3 s. It can be seen from the figure that the controller responds fast to the reference. The stability of the proposed controller is

evident from the short settling time, small overshoot and negligible speed steady state error, which proves the good dynamic and static performance of the SMBC DTC system. In addition, the actual speed has little fluctuation and tracks the reference speed quickly when the load torque changes, hence the SMBC DTC system also shows good performance by rejecting the load disturbance. The details will be discussed later the section and values are summarized in Table. 2. The stator flux along the -axis and -axis are presented in Figure 4(a) and Figure 4(b). It can be observed from these results that the stator flux responses are sinusoidal and smooth. Figure 4(c) shows the circular trajectory curve of the stator flux with the small ripple in the - plane.


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Figure 9. Response in case of sudden load disturbance for (a) conventional DTC and (b) SMBC DTC.

In conventional DTC, the parameters of the PI speed controller are selected as: Kp ¼ 0.05, Ki ¼ 3, the saturation limit of T* e is 12Nm. The hysteresis band for the torque controller is 0.01Nm, hysteresis band for the flux controller is 0.0005 Wb and the reference stator flux is 0.09428 Wb. The sampling and control frequencies are set as 20 kHz, which is higher than that of the SMBC DTC system. The start-up transient response from standstill to 1500 rpm without load torque for proposed SMBC and conventional DTC is shown in Figure 5. The various

responses viz., torque, stator flux, stator current and speed performances are shown and compared. The torque ripple and flux ripple are calculated by using (37). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 N > 2 P > 1 > Te ðiÞ Te > < Trip ¼ N i¼1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > N P > > 2 > rip ¼ N1 : s ðiÞ s


i¼1

ð37Þ

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Figure 10. Performance of the SMBC system with ESO. (a) response of the load torque observer, (b) torque reference response and (c) speed response.

The torque ripple of conventional DTC is 0.778Nm while that of SMBC DTC is 0.135Nm, which shows a reduction of about 82% in SMBC DTC case. The flux ripple of conventional DTC is 0.003 Wb while that of SMBC DTC is 0.00063 Wb, which shows a reduction of about 79% in SMBC DTC case. In addition, the average value of flux is constant in both cases. The torque and flux responses in both the cases show that the proposed SMBC method is superior to the conventional DTC. Furthermore, the stator current is smooth and has fewer ripples with the proposed control scheme, which verifies the effectiveness and advantages of the proposed SMBC method. The speed response for the SMBC DTC at start-up process is also very fast, the settling time is 0.08 s, which is same as the conventional DTC. The above comparison results for conventional DTC and the proposed SMBC DTC can be summarized in Table 2. The steady-state responses for conventional DTC and SMBC DTC at different speed viz., 300 rpm, 1500 rpm, 3000 rpm are presented in Figure 6 to Figure 8. The torque, flux and stator current response are presented from top to bottom in these figures. The torque ripple and flux ripple are high at all speeds for the conventional DTC and are small for the SMBC DTC case, which shows that the SMBC DTC has superior performance compared to conventional DTC. The performances of conventional DTC and SMBC DTC for sudden load disturbance are shown in

Figure 9. It can be observed from this figure that the speed response of SMBC DTC is as good as conventional DTC for the changes in load. The recovery time for both the cases is 0.05 s. However, the speed ripple in case of SMBC DTC is smaller than that of the conventional DTC because of its small torque and flux ripples. The performance of the load torque observer can be tuned by configuring the poles of the characteristic equation. When the expected poles are chosen from the negative real axis, the observation error will converge to zero gradually. In this section, the poles in ESO are selected as p ¼ 3000. Figure 5(a) shows the observed load torque with ESO. It can be observed that the load torque is predicted precisely and quickly. The performance of the proposed control scheme with and without ESO for a step change in load torque is shown in Figure 10(b) and (c). The load variation considered in this case is same as Figure 3(b), and the feed-forward compensation coefficient is chosen as ktl ¼ 1. The figures are enlarged to illustrate the interesting details about the torque reference response and the speed response. The system response of the proposed SMBC without ESO is shown in dashed line and with ESO is shown in solid line. It can be seen from this figure that these two methods show strong robustness when load variations are encountered. Since the load torque is predicted and fed-forward in the proposed SMBC with ESO method, the load variation can be reflected in the torque reference.


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The instantaneous speed fluctuation is about 30 rpm during load change and the recovery time is about 0.05 s for the SMBC without ESO, which is decreased to 12 rpm and 0.02 s in the SMBC with ESO. From above analysis, it can be concluded that the SMBC with ESO has the better reference tracking characteristics, less speed fluctuation and less recovery time compared to SMBC method.

5. Conclusion Considering the nonlinear characteristics of PMSM, a sliding mode backstepping DTC approach has been developed in this paper. The torque and flux ripples can be effectively reduced through space vector modulation. The speed sliding mode controller can effectively improve the response and robustness of the system. The linear ESO is used to observe the load torque variations and is added as a feed-forward component, hence the system performance is further enhanced as it becomes less sensitive to load disturbances. The validity of the proposed method has been verified by simulations. Declaration of conflicting interests The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The authors received no financial support for the research, authorship, and/or publication of this article.

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