A novel MPTC sensorless control strategy for ANFTSMC with ESO to control PMSM

ZHANG Bin， WU Xiaoliang， YANG Jianfeng,2， YANG Ping， SUN Xuewei

(1. School of Automation and Electrical Engineering， Lanzhou Jiaotong University， Lanzhou 730070， China；2. Key Laboratory of Optoelectronic Technology and Intelligent Control of the Ministry of Education，Lanzhou Jiaotong University， Lanzhou 730070， China)

Abstract： Aiming at the problem that the traditional control strategy of permanent magnet synchronous motor (PMSM) for electric vehicles has low control performance, a novel adaptive non-singular fast terminal sliding mode control (ANFTSMC) model predictive torque control (MPTC) strategy is proposed. A new adaptive exponential approach rate is designed, and the traditional switching function sgn() is replaced by the hyperbolic tangent function tanh(). A new ANFTSMC with extended state observer (ESO) is constructed as the speed regulator of the system, and ESO can observe disturbances. This improved method weakens chattering and improves the robustness of the system. To realize sensorless control of the speed control system, an ESO speed observer based on tanh(Fal) is constructed. Compared with the traditional ESO based on Fal function, the observation error is smaller, and the observation accuracy is higher. Finally, aiming at the model predictive torque control strategy used, a new objective function construction method is proposed, which avoids the design of weight coefficient, and the traditional voltage vector selection method is improved and optimized, which reduces the calculation amount of the algorithm.

Key words： permanent magnet synchronous motors (PMSM)； adaptive fast non-singular terminal sliding mode control (ANFTSMC)； extended state observer (ESO)； model predictive torque control (MPTC)

0 Introduction

At present, permanent magnet synchronous motor (PMSM) is widely used in the drive control system of electric vehicles, and its high performance and stability control are the basis for realizing the stable operation of electric vehicles[1]. Model predictive torque control (MPTC) has the advantages of simplified structure and easy realization of nonlinear multi-objective system control, and has become a high-performance control method of PMSM in recent years[2]. However, PMSM faces various unknown internal parameter changes and external multi-source interference in the actual complex automobile running process. Besides, the traditional MPTC control algorithm takes stator flux linkage and electromagnetic torque as control variables, and uses the tedious enumeration method to traverse the optimal voltage vector with the minimum cost function to realize control. On the one hand, the calculation is huge. On the other hand, in order to control two variables with different dimensions at the same time, it is necessary to design the weight coefficient to unify, which limits its practicability and is difficult to apply in actual production and life.

To solve these defects, researchers first improved the speed regulator of PMSM control system. The traditional research and application widely used proportional-integral (PI) regulator with simple algorithm and easy implementation, but the limitation of PI greatly reduced the control performance of MPTC. Sliding mode control (SMC), which is independent of an accurate mathematical model and has good robustness[3], is a research hotspot for scholars at home and abroad. A sliding mode variable structure method combined with model reference adaptive method is used to design PMSM speed observer[4], which improves the observation accuracy and robustness of MPTC system. However, the existence of switching function in sliding mode control rate makes the output torque of the system fluctuate greatly, which is not suitable for practical industrial application. The sliding mode control rate is designed by using a piecewise exponential function[5]which effectively reduces chattering, but increases the complexity of system algorithm.

Considering the complexity of the traditional MPTC algorithm and how to adjust the weight coefficient, a new model predictive flux control is proposed[6], which reconstructs the design process of the weight coefficient, reduces the computational complexity of the algorithm, and weakens the torque ripple and stator flux ripple, but the controller still uses the traditional PI control which greatly reduces the rapidity of the speed control system. Stator current is taken as a control variable[7], without weight coefficient design. By deducing the relationship between current vector tracking error and voltage vector error, the voltage vector can be quickly selected, which significantly reduces the complexity of the algorithm. However, the disadvantage of large torque ripple in predictive current control has not been solved.

At the same time, the traditional PMSM control system of electric vehicles needs high-precision speed sensors to achieve accurate closed-loop control. However, the existence of sensors will increase the design cost, maintenance cost, volume and complexity of the system, and sudden sensor failures in practical applications will also affect the running status of electric vehicles, and even bring serious personal safety problems. Therefore, sensorless is also a hot research topic all the time. An improved flux observer with model reference self-adaptation is designed[8-9], which realizes sensorless control of permanent magnet linear motor. Although the fluctuation caused by thrust is reduced, the observation accuracy is still insufficient. A PMSM rotor position observation method is proposed based on compensation matrix[10]. Combined with high-frequency excitation model, the rotor speed tracking performance and observation accuracy are obviously improved, but the observation effect of this method is not ideal under the working condition of motor running at high speed. An ESO speed observer was designed based on inverse hyperbolic sine function for sensorless control[11-12], and the observation error was reduced, but it was not obvious.

To solve the above problems, a novel adaptive non-singular fast terminal sliding mode control (ANFTSMC) with ESO disturbance observation is designed in this paper. The ESO can observe the disturbance signal generated during the operation of the control system and introduce the observed disturbance term into the sliding mode approaching rate. The adaptive sliding mode exponential approaching rate can further improve the convergence speed of the system and reduce or even eliminate chattering. Secondly, aiming at the sensorless design, in order to improve the observation accuracy, this paper adopts ESO based on tanh(Fal) function to observe the speed and position of PMSM rotor. Finally, in the aspect of improving the traditional MPTC, by analyzing and deducing the relationship among stator flux linkage, electromagnetic torque and stator voltage vector, a new minimum cost function is constructed to eliminate the weight coefficient, and then a new simplified switch vector selection table is established according to the deadbeat voltage vector selection method, which reduces the calculation amount, increases the rapidity and realizes the practicability. Theoretical analysis proves that the improved speed control system is feasible, and the simulation results verifies the effectiveness of the designed scheme.

1 Mathematical model of three-phase PMSM

In this paper, the surface-mounted PMSM system is chosen as the control object, assuming that the rotor winding of the motor is undamped, the hysteresis loss and eddy current loss can be neglected, the stator phase winding is symmetrical and the spatial air gap magnetomotive force is sinusoidal. In thed-qcoordinate system, the stator current model of PMSM is expressed as[13]

(1)

The stator flux linkage model is

(2)

whereid,iq,ud,uqis the stator current and stator voltage ofdandqaxes ind-qcoordinate system;ψd,ψqis the stator flux linkage ofdandqaxes;Rsis the tator resistance;pis the extreme logarithm;ωris the rotor mechanical angular velocity;ψfis the flux linkage of permanent magnets;LdandLqare the inductances ofdaxis andqaxis of armature winding, respectively. Because the surface mounted PMSM is used in this paper, the relationship isLq=Ld.

The mechanical motion equation of PMSM is expressed as

(3)

whereJis the moment of inertia of PMSM;Teis electromagnetic torque;TLis load torque, andBis the damping coefficient.

The electromagnetic torque equation of PMSM is expressed as

(4)

2 Construction of general form of ESO

For a nonlinear control system, let its expression be

(5)

wheref(x1,x2,v(t)) is an unknown nonlinear function, in whichv(t) is an unknown nonlinear disturbance;u(t) is the input, andx1is the output of the system. The first-order state equation established for this nonlinear system is expressed as

(6)

At this time, letx2=f(x1,x2,v(t)) be a new expansion variable, and further expand the first-order equation of state Eq.(6) into a second-order equation of state. The new expression of the nonlinear system after expansion can be expressed as

(7)

At present, many scholars are generally based on nonlinear functions when constructing second-order or higher-order dilated observers[14]. In order to further increase the rotational speed tracking performance and discrimination accuracy of the observer, this paper chooses the hyperbolic tangent nonlinear function tanh(·) instead of the traditional function. A new function is designed to construct the general form of the improved second-order extended state observer, and this new function is defined as tanh(Fal)=(ε,α,σ)=αtanh(σε)， whereαis a non-linear factor and generally takes 0≤α≤1. Whenα<1, the system will have the excellent characteristics that the error decreases, the gain increases automatically, and the gain decreases automatically when the error increases, which can effectively suppress the observation disturbance and weaken the chattering.σis the filter factor, and the general value is 0.000 1≤σ≤1. Fig.1 shows the image comparison between the tanh(Fal) function and the conventional function.

Fig.1 Comparison of traditional Fal and tanh(Fal) function images

By observing Fig.1, it can be known that the value range of tanh(Fal) function is (-1,+1), so it also has the saturation characteristics of the traditional saturation function method. The domain is (-∞,+∞), and it is continuous and monotonically increasing in the domain, and the curve is smooth. Whenx=0, then tanh 0=0. Whenx≠0 and the value ofxis in the field ofu(0,δ)={x|0-δ

Based on the expanded system (7), the ESO established by using the tanh(Fal) function is expressed as

(8)

wherea1>0,a2>0,b2>0.When designing parameters, the state variablesx1(t) andx2(t) contained in the expansion system (7) can be accurately estimated, that isz1(t)→x1(t),z2(t)→x2(t) by selecting appropriate parametersa1anda2.

Using the established ESO, the nonlinear uncertain system under external disturbance can be linearized into an integrator series system with simple structure, and its structure is shown in Fig.2.

Fig.2 ESO structure diagram

3 Design of PMSM speed expansion observer

To realize sensorless control of PMSM, an ESO speed observer is designed, and the first-order state-space observation equation is constructed from the stator current equation and expressed as

(9)

whereedandeqare the back electromotive force of PMSM, and the equation corresponding to Eq.(1) is expressed as

(10)

whereωeis the electrical angular velocity. The expansion ofedandeqin Eq.(9) is a new state quantityx″, which can be expressed as

(11)

According to Eq.(8), the first-order state space equation can be extended to the second-order expansion equation.

(12)

Eq.(12) is observable, from which the second-order ESO of the system (12) can be established as

(13)

(14)

(15)

According to the above analysis and deduction process of PMSM speed and position information, the structural block diagram of ESO speed and position information observer can be obtained as shown in Fig.3.

Fig.3 Structural block diagram of speed and position information observer for ESO

4 Design of adaptive nonsingular fast terminal sliding mode speed regulator with ESO

4.1 Design of nonsingular terminal sliding mode controller

4.1.1 Establishment of mathematical model of controller

In order to facilitate the design of the controller, the variables are defined as

(16)

(17)

where Δa, Δb, and Δcare the uncertain factors when the motor parameters change, and they are defined asgwhich includes all disturbance quantities and is expressed as

g=Δaiq+Δbx4+ΔcTL.

(18)

Eqs.(16)-(18) are combined to obtain the rewritten mathematical model of PMSM for the convenience of observation and analysis, that is

(19)

4.1.2 Design of self-adaption exponential reaching law

At present, when designing sliding mode approaching rate, researchers usually choose exponential approaching rate based on traditional switching function[15], and its expression is

(20)

It can be seen from Eq.(20) that this approach rate includes two gainsλandε. Increasing the value ofλcan realize the rapid convergence of sliding mode control, while decreasing the value ofεwill weaken the influence of chattering. However, in practical application, the existence ofεsgnsin Eq.(20) cannot completely eliminate chattering.

Therefore, in this section, a new exponential approach rate is designed from two aspects of improving the switching function and increasing the adaptability, and the traditional discontinuous switching function is replaced by the hyperbolic tangent function. The image comparison between the traditional switching function and the hyperbolic tangent function is shown in Fig.4.

Fig.4 Comparison between traditional switching function and hyperbolic tangent function

In Fig.4, the image of sgnsis discontinuous, which is the main cause of chattering, while tanhsis continuous and smooth, which can eliminate chattering and has good switching characteristics.

The expression of self-adaptive exponential approaching rate based on hyperbolic tangent is

(21)

whereλ>0,k>0,δ>0. For the convenience of analysis, the sliding mode motion characteristic diagram is given as shown in Fig.5.

Fig.5 Sliding mode motion characteristic diagram

To sum up, the self-adaptive exponential approach rate designed in this paper can adapt to the changes of closed-loop feedback and disturbance on line according to different operating conditions of the PMSM speed control system.

4.1.3 Design of sliding mode speed controller

(22)

whereβ>0;pandqare both odd numbers and satisfy the condition of 0

Define the control rate as

u(t)=ueq(t)+usw(t)，

(23)

whereueq(t) andusw(t) are equivalent control part and nonlinear switching control part, respectively. The equivalent control partueq(t) is designed by the adaptive exponential approach rate function Eq.(21), sliding surface function Eq.(22) and the defined control rate function Eq.(23), that is

(24)

The nonlinear control partusw(t) is defined as

(25)

whereF=lgtanhx+ε(1-e-δ|s|)·tanhs.lg、β、h、m、εandδare all constants to be designed. In order to further improve the control quality of PMSM system and eliminate the steady-state error, this paper designs to add an integral term to the control rate function, andu(t) becomes

(26)

At the same time, because the internal parameters of the PMSM speed control system will inevitably change and there are various uncertain disturbances, which will increase the burden of adaptive control rate. Therefore, it is necessary to compensate for the disturbance term. ESO was designed by referring to the second section of this paper to observe and compensate for the disturbance.

(27)

whereβ1andβ2are the coefficients of ESO, and a large value ofβcan improve the tracking speed and accuracy of ESO;z4is the real-time estimation term of disturbance. To sum up, by combining the control rate function Eq.(26) and the anti-disturbance ESO observer Eq.(27), the adaptive nonsingular terminal sliding mode speed controller based on ESO can be finally obtained as

(28)

According to the above design ideas of PMSM speed controller, the structure block diagram of adaptive nonsingular terminal sliding mode controller based on ESO can be obtained as shown in Fig.6.

Fig.6 Structure block diagram of ANFTSMC based on ESO

4.2 Proof of controller stability

To prove whether the designed controller is stable, Lyapunov function is defined as

(29)

Combined with Eq.(22), the derivative of Eq.(29) is obtained as

(30)

5 Design of improved model predictive torque control

5.1 Traditional MPTC principle

The structure block diagram of the traditional MPTC control system is shown in Fig.7. Because the inverter used in this paper is a three-phase two-level inverter, which can provide two zero vectors and six non-zero voltage vectors, the traditional MPTC control strategy will select the optimal value from eight vectorsUi(i=1,2,…,8) as the output value of the inverter through the minimum cost function according to the PMSM system model and various constraints, and then act on the PMSM[16].

Fig.7 Structure diagram of traditional model predictive torque control system

Because the traditional MPTC takes stator flux linkage and electromagnetic torque as control variables, the minimum cost function is generally as

(31)

(32)

In the current research, there is still no scientific basis for how to quantify the weight coefficient accurately. Most researchers try to get the best coefficient through repeated experiments, which is also the defect of the traditional MPTC algorithm.

In the next step, the mathematical model of PMSM in Eq.(1) is discretized by using Euler formula, and the prediction model of stator current can be obtained as

(33)

According to Eq.(2), the estimated values of stator flux linkageψdandψqofdandqaxes ind-qrotating coordinate system are

(34)

The estimated value of stator flux linkageψsis further obtained by

(35)

The estimated value of electromagnetic torqueTeat this time is

(36)

To sum up, the flow chart of traditional MPTC algorithm for PMSM speed control system in a sampling period is shown in Fig.8.

Fig.8 MPTC algorithm flow char

5.2 Improved design of MPTC

As can be seen from the above section, it is difficult to determine the weight coefficient of the traditional MPTC when designing the minimum cost function. It can be seen from the MPTC algorithm flow chart in Fig.8 that the algorithm needs to calculate Eq.(33) several times, and then traverse the optimal voltage vector with the minimum cost function to realize control, which is cumbersome and requires a great amount of calculation. To avoid this defect, researchers all use high-performance digital processors. However, in production practice, high-performance digital processors will undoubtedly increase the cost, which greatly reduces the practicability of the algorithm. Therefore, this paper improves the traditional MPTC, redesigns the minimum cost function to avoid determining the weight coefficient, and improves the vector selection scheme, so that the best voltage vector can be quickly selected, and finally reduces the complexity of MPTC algorithm.

5.2.1 Design of removing weight coefficient of minimum cost function

In order to reconstruct the minimum cost function and eliminate the weight coefficient, the relationship between stator fluxψs, electromagnetic torqueTeand stator voltageusvector in traditional MPTC should be analyzed.

As for the stator flux linkage, there is another calculation method for the prediction calculation of the stator flux linkageψs(k+1) at the (k+1)th time besides Eq.(35), which is expressed as

ψs(k+1)=ψs(k)+(us-Rsis)Ts,

(37)

whereTsis the sampling period of the digital processor, andRsisis the stator resistance voltage drop, which can be ignored, soψs(k+1) can be expressed as

ψs(k+1)=ψs(k)+usTs.

(38)

It can be seen that the magnitude of stator voltage vectorusdetermines the change of stator flux linkage estimated valueψs(k+1) at the (k+1)th time. Therefore, the control of the magnitude of stator flux linkage is essentially the control of stator voltage vector.

For electromagnetic torque, the relationship between torque change and stator flux linkage load angle change is expressed as

(39)

where ΔTeis the increment of torque change, and Δφis the change of load angle. A simplified position diagram of load angle between stator flux linkageψsand rotor flux linkageψffrom (k)th to (k+1)th time in PMSM operation shown in Fig.9 is given for more intuitive analysis.

Fig.9 Change diagram of stator and rotor flux linkage load angle

It can be seen from Fig.9 that the stator flux linkage load angle has the relationship of

Δφ=Δφ(k+1)-Δφ=Δθs-Δθr.

(40)

In a sampling period of the control process, the mechanical constant of the rotor of PMSM is much higher than the electrical constant of the stator, so in research or industrial application, people usually equate the rotor flux linkage as a constant, ignoring its control. Therefore, the change of rotor flux linkage position angle can be directly ignored in Eq.(40), that is Δθr=0, and Δφ=Δθsat this time.

The geometric relationship of stator flux change is analyzed as shown in Fig.10.

Fig.10 Geometric diagram of stator flux change in a sampling period

The geometric relationships of Eq.(41) are shown in Fig.6.

(41)

Since the change value of Δθsis small, there is sin Δθs→Δθs, which can be obtained according to Eq.(40). The coupling type (39) can be obtained as

(42)

Because the time in a sampling period is extremely short, the flux linkage amplitude and load angle changes contained in Eq.(42) are approximately equivalent to constant values that remain unchanged. Therefore, according to Eq.(42), the primary factor that causesTeto change is only the basic voltage vectorus.

According to the above analysis of the relationship among stator flux linkageψs, electromagnetic torqueTeand stator voltage vectorus, the changes of PMSM flux linkage and torque are directly related to basic voltage vector during operation, so the control ofψsandTein traditional MPTC can be directly changed into the control of single variableus.

Therefore, the improved new minimum cost function is

(43)

(44)

Components Δψαand Δψβof the deviation between the target stator flux linkage and the current stator flux linkage on axesαandβare calculated by

(45)

According to Eq.(46), the value of the target voltage vector is predicted by

(46)

(47)

5.2.2 Improved design of voltage vector selection

Traditional MPTC is cumbersome in screening ideal basic voltage vector. First, it needs to calculate the minimum cost function seven times, and then finally screen out the optimal basic voltage vector according to the minimum result[18]. In order to realize fast voltage vector selection, this paper improves the traditional method by combining deadbeat vector selection method. The Fig.11 shows the voltage vector screening diagram of three-phase two-level inverter.

Fig.11 Voltage vector screening diagram

In Fig.11, according to the positions of the basic voltage vectorsV1-V6and the reference voltage vectors, the distances between the basic voltage vectorsV1-V2and the reference voltage vectors are smaller than the distances between other basic voltage vectors and the reference voltage quantities.

In this paper, combined with deadbeat voltage vector screening method[19], when selecting the most ideal basic voltage vector, only the two basic voltage vectors closest to the reference voltage vector are screened, and then the screened results are substituted into Eq.(43) for calculation.

Taking Fig.11 as an example,Vsis at the position of [0,π/3]. The nearest basic voltage vectorsV1andV2can be selected according to the deadbeat screening method. Based on this principle, a simplified and improved new voltage vector fast screening table is designed as shown in Table 1.

Table 1 New-style fast voltage vector screening table

6 Simulation analysis

To verify the performance of the improved MPTC sensorless control strategy for ANFTSMC with ESO to control PMSM, the structure diagram of PMSM speed control system is designed as shown in Fig.12, and the simulation model is built on the Simulink simulation platform in MATLAB. The related modules are designed byS-function as shown in Fig.13.

Fig.12 Improved MPTC sensorless control system structure block diagram of ANFTSMC control PMSM with ESO

Fig.13 Improved MPTC sensorless control system simulation design drawing of ANFTSMC control PMSM with ESO

For the convenience of analysis and comparison, the simulation model of traditional PMSM-MPTC control system based on sliding mode control (SMC) is also built.

The main parameters of PMSM used in the system are shown in Table 2.

Table 2 PMSM parameters

6.1 Comparison of control effects between improved and traditional MPTC

Firstly, the MPTC algorithm improved in this paper is combined with ANFTSMC speed controller to simulate PMSM speed response. For the convenience of analysis, it is compared with the traditional MPTC algorithm based on ANFTSMC speed controller. The experimental setting running time is 0.4 s and the given speed of PMSM is 1 000 r/min. The speed response comparison of the two control algorithms is shown in Fig.14.

Fig.14 Comparison of PMSM speed response before and after MPTC improvement

It can be seen from the speed response diagram that the speed rising time of the improved MPTC algorithm in this paper is 0.014 s, and that of the traditional MPTC algorithm is 0.212 s, which shows that the speed of the improved MPTC is obviously improved.

6.2 Comparison between new ANFTSMC and traditional SMC

To achieve fair comparison and analysis, the ESO observer based on tanh(Fal) function designed in this paper is used in PMSM control systems of the two controllers, and the improved MPTC control strategy is used. First, the reference speed of PMSM is set at 1 000 r/min, and it is allowed to run under no load when it runs 0 s-0.2 s. When it runs to 0.2 s, a load of 2 N·m is suddenly applied to the motor, which lasts until 0.4 s. The speed response of PMSM controlled by the two sliding mode controllers before and after improvement is obtained by the output of the rotational speed scope in the simulation model Fig.13 as shown in Fig.15.

Fig.15 Comparison of speed response of two kinds of sliding mode controllers

It can be seen from Fig.15 that both the ANFTSMC designed in this paper and the traditional SMC make the PMSM reach the speed of 1 000 r/min without overshoot, but ANFTSMC responds faster than the traditional SMC and rises to the given speed more quickly, which shows that the desired rapidity of the control system has been realized.

Both of them are stable under no-load operation during 0 s-0.2 s, and suddenly load disturbance starts at 0.2 s. Both of them have speed fluctuation and can return to the given speed again. However, from the enlarged details in Fig.15, it can be found that ANFTSMC has smaller fluctuation than traditional SMC, and it takes less time to restore stability after small fluctuation, which shows that ANFTSMC has strong anti-disturbance ability, and the main reason is that anti-disturbance ESO is added to various system disturbances in this paper when designing sliding mode controller.

In order to further verify the anti-disturbance ability of PMSM speed control system designed in this paper, the three-phase current response and torque response of ANFTSMC and traditional SMC are tested, respectively, as shown in Figs.16 and 17.

It can be seen from (a) and (b) in Fig.16 that compared with the traditional three-phase current output by SMC, the sinusoidal performance of the output current waveform based on ANFTSMC control is better and smoother, whether under the no-load period or the period after sudden loading.

(a) Three phase current response of ANFTSMC

With regard to the torque response of PMSM output, Fig.17(a) shows that the torque ripple based on ANFTSMC control is very small, especially there is almost no fluctuation at the moment when the load is suddenly applied in 0.2 s. While the torque ripple of traditional SMC output is larger under the same circumstances with obvious fluctuation when the load is suddenly applied in 0.2 s. This unfavorable phenomenon will increase the energy consumption and loss of PMSM and affect the practical industrial application.

(a) Torque response of ANFTSMC

Besides, the follow-up performance of the output speed of the speed controller is another index to evaluate the dynamic performance of the system. The sine signal of PMSM with a running period of 0.4 s and a given amplitude of 1 000 is simulated, and the given follow-up of the two speed controllers is shown in Fig.18. It can be seen that the new ANFTSMC output speed wheel can follow the given speed in the start-up phase or the full-speed phase well, while the follow-up performance of the traditional SMC in the start-up phase has a large deviation.

Fig.18 Comparison diagram of speed following performance of two sliding mode controllers

Compared with the traditional PMSM-MPTC speed control system based on SMC, the improved MPTC control system based on ANFTSMC designed in this paper has better dynamic and static performance with greatly improved rapidity, stronger disturbance resistance and robustness.

6.3 Comparison of accuracy of ESO observer

To compare and analyze the observer performance fairly, the improved MPTC control strategy based on ANFTSMC is adopted, and then the ESO observer based on tanh(Fal) function and traditionalFalfunction is simulated and compared. The waveform diagram of comparing the actual value of PMSM rotor position with the observed estimated value is obtained by the output of the rotor position comparison scope in the simulation model Fig.13, as shown in Fig.19.

(a) ESO rotor position observation diagram based on tanh(Fal) function

It can be seen from Fig.19 that the ESO based on the new tanh(Fal) function has higher observation accuracy compared with the traditional ESO based onFalfunction. The observed value is closer to the actual value, and the observation error is obviously reduced.

7 Conclusions

In this study, the PMSM control system for electric vehicles is taken as the research object. Firstly, the PMSM mathematical model considering various unknown disturbances is established, and a new ANFTSMC with ESO disturbance observation is further designed. The application of this controller improves the anti-disturbance ability of the system, reduces the output torque ripple and improves the control performance. At the same time, the new ESO based on tanh(Fal) function realizes sensorless control of PMSM for electric vehicles, which avoids the shortcoming of low fault tolerance of traditional mechanical sensors. Compared with traditional ESO, the observation accuracy has been improved obviously. Considering that the traditional MPTC has some problems, such as nonstandard and imprecise value of weight coefficient in the minimum cost function, complicated selection process of optimal voltage, which leads to a large amount of calculation and difficulty in application, this study deduces and analyzes the relationship among stator flux linkage, electromagnetic torque and stator voltage vector in PMSM, designs a new construction method of minimum cost function, avoids the adjustment of weight coefficient, and redesigns and optimizes the optimal voltage vector selection method of MPTC according to the voltage vector distribution of the three-phase two-level inverter used. Finally, the simulation results prove the effectiveness, rapidity and anti-disturbance performance of the PMSM control strategy designed by this research, which provides stable operating conditions for MPTC application in the PMSM control system of electric vehicles.