# 1. Introduction

Electric Vehicles (EVs) have been gaining a lot of attention due to its environmental friend ness. EVs can be controlled precisely with the quick response of the motor. Moreover, using in-wheel motors, several types of motion controls can be intensely researched.

The four-wheel independent driving system of electric vehicle makes use of four electric motors which are controlled independently to drive four wheels of the vehicle respectively and there is not any mechanical transmission link [1 - 3].

The in-wheel motor arrangement has some significant advantages over the conventional electric vehicle consisting of a centrally situated single motor driving two or four wheels by axles :

- High controllability for all-wheel driving vehicle- High energy efficiency- Greater freedom in layout design

Due to the advantages of in-wheel motors, there is no doubt that more and more electric vehicles will adopt the topologies of the power train system to make full use of these merits. The work proposed by this paper consists of the use of an electric vehicle driven by four in-wheel motors.

The advantages provided to these electric vehicles can be summarized as follows [4]:

- Quick torque generation- Easy torque measurement- Possibility of independently equipped motors for each wheel

These advantages could be used to contribute much more to the anti-skid control in electric vehicles than they currently do. Several researches have been carried out on this aspect of EVs [4-5]; however it is still not well recognized. Recently, permanent magnet synchronous motors have been the most popular choice for in-wheel application [6], because of its high power density, efficiency, smooth torque, and easy control drive. Since magnetic field is excited by high-energy permanent magnets, power density is high, and hence, the overall weight and volume can be greatly reduced. The absence of rotor copper losses makes the efficiency of PM machines inherently higher than induction machines. Also, and for improving the dynamic performance of the permanent magnet synchronous motor drives for electric vehicle propulsion, the vector control techniques are usually adopted. In recent years, an innovative control method called direct torque control (DTC) has gained the attraction for electric propulsion system [6-7]. This control strategy can produce fast torque control and may not need heavy computation on-line, in contrast to vector control.

Moreover, the number of systems using several electrical machines and/or static converters is increasing in electromechanical applications. These systems are called Multi-machines Multi-converters Systems (MMS) and could be an electric vehicle. Two three-leg inverters are used to supply four three-phase PMSM motors connected to four driven-in-wheels. Several control methods have been proposed for the control of multi-machine single-converter systems based primarily on scalar techniques with slow responses. With the introduction of vector control, more effective and powerful techniques have been applied to improve the response time and accuracy of motor control. The paper first gives a review of existing control strategies for such systems, then, a novel algorithm based on DTC is proposed and used for the control of a multi machine system. The performance of the proposed method is investigated using computer simulations.

This paper is organized as follows, in section 1, a four wheel vehicle model is presented. A linear vehicle model with three degree of freedom is applied to estimate the driver’s intention, computing the desired motion of the vehicle [8-10].

In section 2, a speed sensor less control is proposed. In this case, speed estimation is based on sliding mode observer method. The proposed estimator does not require rotor position information and hence retains the sensorless nature of DTC of PMSM in-wheel motor.

In section 3, a new control method based on Direct Torque Control (DTC) of PMS motors is proposed for a multi-machine system. Similar to conventional DTC, the proposed method has two separate control loops. In the torque control loop, before selection of optimum voltage from the DTC look-up table, the system overall requirement is determined based on requirements of motor torque. Also, Switchable Master- Slave control is used in the flux control loop. The Method, which is simulated for a two-parallel PMS machine system, can be extended to a multi-machine system. Simulation results are also provided to investigate the performance of the proposed technique.

In section 4, an approach based on the ratio of wheel acceleration to motor torque for electric vehicle anti-skid control is proposed. Two topology of controllers, a rules-based controller and a fuzzy logic controller are designed for the vehicle anti-skid control. Simulation results show that both of the controllers can successfully prevent vehicle skidding.

Finally, an exhaustive series of Matlab/Simulink based simulation tests will be carried out and include in the paper to evaluate the performance of the proposed sensorless control system on a curved road.

# 2. Structure of the Proposed Electric Vehicle

A non linear model is shown in Fig. 2, which can be found by applying the fundamental principles of dynamics at the centre of gravity (CG). The vehicle dynamics are described by the longitudinal velocity, lateral vehicle and yaw rate as follows [11-13]:

**Fig. 2**Four wheels vehicle model

Where vx is the longitudinal velocity, vy is the lateral one, the yaw rate while, Jv the moment of inertia, Mv is the vehicle mass, l f and lr are the distance from CG to the front and rear axle, Ft1 , Ft2 , Ft3 , Ft4 are the traction forces of motors, Fres is the resistant force which includes the aerodynamic drag force Faero , climbing force Fc and rolling force Frr . They have the following expressions [12]:

Using a linear tire model, the front and rear cornering forces can be expressed as the product of the cornering stiffness (Cf , Cr ) and the sideslip angle (αf , αr )

The sideslip angles of the wheels can be expressed easily in terms of the longitudinal, lateral, angular velocities and the steering angle δ . The explicit expressions of the sideslip angles for the front and rear axles are represented by Eq. (7).

The longitudinal slip can be defined for the four wheels as :

Where Rω is the wheel radius, ω is the angular velocity of the in-wheel motor and ut is the linear speed at which the contact zone moves on the ground and can be written for the four wheels as [12]:

The interrelationships between the slip ratio λ and traction coefficient μ can be described by various formulas. In this paper, the widely adopted Magic Formula [13-15] is applied to describe the relationship between slip and traction force and to build a vehicle model for the simulations that follows, as shown in Eq. (7).

Coefficient sets of c1 , c2 , c3 and c4 are defined in [13].

The longitudinal force for the four in-wheel motors can be calculated by the following equation:

Moreover, the drive system model can be described by the following mechanical equations:

Where Tri is the resistive torque. Nf , Nr are the front and rear normal forces and have the following expressions:

# 3. Sensorless DTC based on Sliding Mode Observer

In the performance analysis of the sensorless control of a PMSM, the motor is generally modelled in stationary reference frame due to the fact that angular speed and position information are ready to be extracted in this reference frame [16]. In stationary ( α , β ) reference frame :

The motor speed changes slowly, implying that ≈ 0 , the model of these induced back EMF is :

Fig. 3 shows the proposed sliding mode observer (SMO).

**Fig. 3**Typical traction curves

Unlike the conventional sensorless control designs, the sliding mode observer proposed here uses only the motor electrical equation. The observer was first proposed in [17- 18] and its main equations are :

Where K1 is a constant observer gain, , are the observer mismatches. Assuming that the motor parameters are identical with those in the model, the mismatch dynamics is then :

The dynamics are distributed by the unknown induced emf components. However, since the back EMF components are bounded, they may be suppressed by discontinuous inputs with . Similarly to the sliding mode observer theory, we define the sliding hyper plane on the stator current errors S = x = [i α i β ]T . When the sliding mode occurs after a finite time interval and , the system behaviour can be examined by applying the equivalent control method mentioned above. Matching Eqs. (11) and (15) gives:

To extract eα and eβ from the corresponding equivalent control values in (17), a low pass filter is used with zα , zβ , as filter outputs:

Where Δμ(t) is the error determined by the distortions of both slow and fast components of the discontinuous filter inputs. For high performance applications, zα , zβ cannot be used directly, as the estimation of induced EMF components contain disturbance Δμ(t) as shown in Fig. 4. Eq. (19) is used to design a better filter and at the same time estimating the rotation speed. The observer designed to undertake this filtering task is:

**Fig. 4**Block diagram of proposed SMO

Where K2 is a constant observer gain. The observer has the structure of an extended Kalman filter and is expected to have high filtring properties. Assuming Δμ(t) = 0 for the rest of the analysis. At first, we assume m ≈ 0 , which means ueq constant. Then using (2) and (7), the mismatch in the back EMF equations are:

Where , , are the observer errors.

It can be shown using Lyapunov function that the estimates and tend to ℯα and ℯβ asymptotically. Thus the speed estimation error should, also, be equal to zero. To prove the convergence of the observer, a Lyapunov function is defined as:

The time derivative along the solution of (9) can be calculated as:

Or

**Fig. 5**Simulation results of DTC with proposed SMO

# 4. Proposed master slave control technique

The proposed method is based on the conventional DTC technique of permanent magnet synchronous motors [19- 20]. In the DTC method, the flux and electromagnetic torque are controlled by adjusting the magnitude and position of the stator flux respectively. This principle is used in the proposed method. The method is explained for a two-machine system and can be extended to a mulimahine system [21]. In the proposed method, a conventional single-machine DTC strategy is independently applied to each motor and the same principle is extended to make it applicable to a multimachine DTC method. In this proposed control strategy, there are two control loops; one for the stator flux control and one for the electromagnetic torque as can be seen in Fig. 6 but with different procedure for each loop. The procedure of each control loop will be explained thoroughly in the following section of the paper.

**Fig. 6**Block diagram of the proposed control method

## 4.1 Electromagnetic torque control loop

The new idea in the suggested control loop is to consider the motors torque requirements and system overall needs before selecting a voltage vector. This is done by designing a new look-up table in which a three level comparator is used in the torque control loop. The procedure is explained below and shown in Table 1 where, -1, 0, 1 are the outputs of torque error comparator.

If both motors require a reduction in torque, a vector is applied to decrease torque.If no motor requires a torque change, then a vector is applied such that the torque is kept constant.If both motors require an increase in torque, then a vector is applied to increase the torque.If one motor requires a decrease in torque but the other does not, then a vector is applied to decrease the torque.If one motor requires a increase in torque but the other does not, then a vector is applied to increase the torque.If one motor requires a decrease in torque but the other one requires an increase, then a vector is applied such that the torque is kept constant.

**Table 1.**Proposed Table in torque control loop

Fig. 7 shows the implementation of the method. In this figure, different possible kinds of torque errors in torque control loop are presented. The black points are the typical position for the electromagnetic torque of the two motors. For the conditions depicted in Fig. 7(a), a voltage vector is applied such that the torque does not vary. For the conditions shown in Fig. 7(b), a voltage vector is applied to increase the torque. Also, for the conditions presented in Fig. 7(c), a voltage vector is applied to decrease torque.

**Fig. 7**Different possible cases of torque errors: (a) the torque should be kept constant; (b) the torque should be increased; (c) the torque should be decreased

Finally, using the output of this table and the output of the stator flux control loop, the appropriate voltage vector is selected based on the conventional DTC switching look-up table.

## 4.2 Stator flux control loop

Before introducing the proposed idea, some issue must be explained regarding parallel PMS motors. As a result of applying one voltage vector, stator flux vector of all the parallel PMS motors will vary instantaneously in the same direction. Therefore:

Where , and are stator voltage vector, stator current vector and stator flux vector respectively. Rs is the stator resistance.

In the stator flux control loop, one should know that the flux of any machine can go beyond its rated value. According to (25) the stator flux of each permanent magnet synchronous motor surely depends on the applied voltage. In cases where parameters of the motors are different, or motors load are not the same, stator fluxes will be different, or motors load are not the same, stator fluxes will be different. From (25) it can be seen that the stator flux vector only depends on stator resistance among all other parameters. Therefore, when stator resistances are the same, one expects to see the same flux for both machines. This is valid only at steady state, and during transients, the difference between the fluxes may be observed. This difference will also increase as motor speed decreases. For this reason, in cases where motor loads or stator resistances are different, speed reference cannot go below a certain value for speed control application. Thus, as the stator flux of one of the motors decreases, its torque generation capability will also decrease.

For these conditions, the mean control strategy cannot be used since the flux of the one machine can be saturated while its average value is equal to the reference value. Therefore, the master-slave control technique can be used for the stator flux control loop. In this way, only the stator flux of one motor is controlled. But the motor with the bigger stator flux magnitude has to be selected as the master, and its stator flux is set to the reference value. To prevent flux saturation at different situations, the master motor may change. Therefore, in the proposed method, switchable master-slave technique is employed for stator flux control.

To accurately select the master motor and prevent flux saturation, an index is needed. The product of stator resistance and electromagnetic torque, i.e. RsTs is selected as the index. For each motor, this product is calculated. The motor with the smallest RsTs is chosen as the master motor. In cases where motor parameters are equal, the motor with the lower torque is selected as the master. The duration and amount of difference between indices are important in order to prevent frequent change variation of master motor during transients.

In the conventional DTC, the final step is the selection of voltage vector using a look-up table. The voltage is selected with respect to the section of the stator flux. The proposed method uses the stator flux of the master motor for flux sector selection.

In order to ensure the stability of the system composed of two PMSM connected in parallel on the same inverter which uses the sensorless DTC “master-slave” structure, different loads are applied to both machines as shown on Fig. 8(c). We can readily notice that whatever values of the torque provided by the two machines, the system is always stable.

**Fig. 8.**Speed, torque, stator flux and motor current of two motors with proposed method using load change

The master machine is the one that provides the highest torque and the difference in position between the two machines corresponds to the theoretical basis θ1 < θ2 when the highest torque is provided by PMSM1.For the speeds of the two machines (Fig. 8 (a)), whether master or slave, there is a close follow up of the reference speed imposed by the control strategy whatever is the value of the load applied. We notice that the difference between the two speeds has very satisfactory rates, which indicates the good perfection in swapping the master and slave motors according to the control behavior when disturbances occur. We notice fast response of electromagnetic torques of the two motors (master and slave) when we apply different loads as shown in Fig. 8(b), this confirms the fast and good management in master and slave under the conditions laid down in the algorithm of the control. We remark the fast response of electromagnetic torque of the two motors (master and slave)when applying different loads, Fig. 8 (b), which confirms the speed and good alternation in master and slave under the conditions laid down in the algorithm control

The phase currents of the two machines present good waveforms and confirm the responses of the motors as far as the changes in loads are concerned. Fig. 8 (i), (j), (m), (n) which represent the trajectories of the stator magnetic flux show good magnetic stability of both machines which ensures a good behavior that was imposed by the DTC control “master-slave” to the two machines against all disturbances. The simulation results of this proposed method offers better steady state response.

# 5. Fuzzy logic anti-skid control

The static slip/friction models are the most common road/tire friction models used in the simulation of vehicle longitudinal dynamics. As shown in Fig. 1, they are defined as one-to-one maps between traction coefficient μ and longitudinal slip ratio λ . The relationship between traction coefficient and slip ratio λ is initially linear (linear region), but later reaches a maximum value of traction coefficient (non linear region) beyond which there is a decline in traction coefficient (thermal region). In general, the peak traction coefficient occurs at about .1 or 10% of the slip ratio. If an excessive torque is applied to the driven wheel in slippery road conditions, the quick increase of slip ratio easily causes traction characteristic to enter the thermal region which may lead to the loss of traction and consequently to vehicle skidding.

**Fig. 1**Proposed structure of electric vehicle driving front and rear wheel

Compared with internal combustion engines, the torque generated by electric motors can be accurately estimated and controlled based on the motor current feedback. With this new accurate torque feedback characteristics, a unique anti-skid control method can be applied, which is specified for electric vehicles. The anti-skid control is based on regulating the ratio of wheel acceleration to drive motor torque.

According to Eqs. (26) and Eq. (27) of the longitudinal vehicle dynamic :

The relationship between the equivalent linear wheel acceleration and the vehicle acceleration can be represented by :

Namely,

Let α stands for the ratio of vehicle acceleration the wheel acceleration :

Since the vehicle acceleration is not easily measured, a new parameter Rat is introduced for the proposed anti-skid control. It is defined as the ratio between wheel acceleration to wheel drive/braking torque T . According to Eq. (29) to Eq. (30), Rat can be represented as :

Namely,

For acceleration, the relationship between the slip-ratio λ and α can be described as :

The anti-skid control strategy used in this paper is to indirectly regulate α within certain safe range; therefore α can be considered to be a bounded function over time t . suppose the upper bound and lower bound of α(t) are αL and αH respectively, based on Eq. (34) the range of λ can be represented as :

Namely, λ is within the range of [1−αH ,1−αL ] . Consequently the range of the ratio Rat can also be determined as :

where RatH and RatL are the upper and lower bounds of Rat, respectively. As explained above, since α is difficult to obtain, an alternative indirect regulation of α is proposed by controlling Rat to stay within [RatL , RatH].

For braking, similarly:

And

Clearly the proposed Rat based anti-skid control is valid for both cases of vehicle acceleration and braking. For the latter, T includes the braking torque jointly generated by electric drive motor and hydraulic brake system. The estimation of hydraulic braking torque is beyond the scope of this paper. For the sake of clarity, only the anti-skid control for acceleration case is discussed below, in which the Rat is the ratio of wheel equivalent linear acceleration to drive motor torque Tm. Namely,

Obviously the road/tire friction can not be infinite; therefore the value of available vehicle acceleration is also limited within a certain range. If wheel acceleration is largely deviated from vehicle acceleration, vehicle can be considered being skidding. Usually a slip ratio λ ranging from 0.1 to 0.3 is considered to be safe [30]. According to Eq. (35), the corresponding α is from 0.7 to 0.9 to avoid vehicle skid. If α is less than 0.7, vehicle can be considered to be skidding; while α higher than 0.9 will sacrifice vehicle acceleration performance. As shown in Eq. (33) and Fig. 9, Rat has a one-to-one relationship with α ; therefore, wheel slip level can be estimated and controlled using the new parameter Rat , which is convenient to obtain for electric vehicles.

**Fig. 9.**The relationship between α and Rat.

It is intently to notice that in Eq. (32) the denominator is directly related with the equivalent inertia felt by drive motor when the ratio α of vehicle acceleration to wheel acceleration is kept constant. The slip of wheel on a low μ road surface can be considered to bring a sudden decrease of vehicle’s equivalent inertia from its normal value. Therefore, the over acceleration of wheel can be prevented by regulating the permissible variation range of the equivalent inertia through Rat based anti-skid control.

For a good tradeoff between vehicle acceleration performance and skid prevention, regulating Rat within a permissible range is preferred to keeping a fixed value of Rat. Moreover, the real vehicle dynamics is highly nonlinear with many uncertainties included. Fuzzy controllers have been known to have satisfied control performance for nonlinear systems. Therefore, a fuzzy logic anti-skid controller is introduced to regulate Rat within the range determined by α ∈[0.7 0.9](see Eq. (36) and Eq. (37)).

As shown in the control block diagram of Fig. 10, Rat is the one controller input calculated by dividing feedback signals of the motor torque Tm into the wheel acceleration . Its time derivative is added as another input. The output of the controller is the variation of compensation torque ΔTc , which is integrated to give the compensation torque Tc for vehicle skid prevention purpose. The final motor torque command is adjusted using the following equation:

**Fig. 10.**The Rat based fuzzy anti-skid control diagram

Due to the capability of accurate torque control of electric motors, the generated motor torque Tm can be considered equal to motor torque command .

The relationship between α and Rat is shown in Fig. 9. If Rat is “very high”, it indicates a dangerous situation that may lead to a serious vehicle skid. A big increment of the compensation torque is needed to immediately decrease motor torque Tm . On the contrary, if Rat is “very low”, the vehicle’s acceleration performance will be adversely limited. Since compared with the case of a “very high” Rat this situation is not dangerous, a relatively small decrement of the output compensation torque can be applied. The basic rules of the control strategy are described as follows:

1. IF Rat IS very high THEN ΔTc IS greatly increased2. IF Rat IS high THEN ΔTc IS slightly increased3. IF Rat IS normal THEN ΔTc IS zero4. IF Rat IS low THEN ΔTc IS slightly decreased5. IF Rat IS very low THEN ΔTc IS greatly decreased

The relationship between Rat and the controller output ΔTc is shown in Fig. 11.

**Fig. 11**The input-output relationship of the basic rules

For the fuzzy logic controller, another fuzzy input variable, the time derivative of Rat , , is added. As shown in the fuzzy rules table, Table 2, if Rat is very high and is negative, the compensation torque ΔTc increases slightly; if Rat is very high and is zero or positive, ΔTc increases a lot. The other rules follow similar considerations. The membership functions for the two input variables, Rat and its time derivative , and the output variable ΔTc are shown in Fig. 12. Based on the above fuzzy control strategy, the increment of compensation torque ΔTc is big when it is necessary to prevent vehicle skid as soon as possible; while in order to avoid adverse effect on vehicle’s acceleration performance the decrement is relatively small when there is no immediate danger of vehicle skid.

**Table 2.**Fuzzy logic rules

**Fig. 12.**Membership function for fuzzy input and output variables

In order to verify the effectiveness of the fuzzy logic anti-skid control, a large constant 50N.m motor torque command from driver is applied. The control method used in this paper for the motors is the direct torque control (DTC). From 0sec to 10 sec , the vehicle is simulated running on a dry road surface. Then the vehicle enters a snowy road. However, the test consists to simulate the passage of front left wheel and rear left wheel of the electric vehicle from a dry road to a snowy road (at t=10sec). Figs. 13 (a) and (b) show the simulation results of the velocities of the wheels and vehicle and the slip ratio without anti-skid control, respectively. After entering the snowy road, the front left and front rear wheels become much faster than the vehicle velocity, which lead to a large slip ratio of over 0.7, namely the vehicle is in a serious skid (skid phenomenon). This phenomenon will lead to the instability of the vehicle for two reasons [22]:

**Fig. 13.**Simulation results for the case of snowy road.

The imbalance of the traction forces;

A reduction in the side forces necessary to maintain the vehicle on its trajectory.

In order to evaluate the effect of the anti-skid control in normal case, a vehicle driving on a dry road surface is simulated. As shown in Fig. 14 the vehicle’s acceleration performance is slightly affected under the fuzzy anti-skid control. However, the velocity curve in nearly linear, which means a good driving experience can still be guaranteed with improved safety.

**Fig. 14.**Simulation results for the case of dry road.

# 6. Simulation results

The simulation is carried out by applying a skid phenomenon to both outer and inner driven wheels successively at different times in the turns, Fig. 15.

**Fig. 15.**Vehicle trajectory

The vehicle speed starts from zero to the reference speed 30km / h . At this operating point, two turns, one to the left and one to the right are imposed to the vehicle chassis by the steering angle as shown in Fig. 16. The vehicle turns to the left at t =12s, and when the steering angle reaches its maximum value (7°) at t =15s, and still be maintained at the mentioned value about 10 seconds, when we apply a skid phenomenon between t =15s and t = 23s to both front and rear right wheels which are driven by motor 1 and motor 2 respectively, when the vehicle is moving at 30km / h. The skidding occurs when moving from a dry road to a slippery road which leads to a loss of adherence. After that, the vehicle is steered towards the right at the instant of t = 37s for a period of 16 seconds applying a skid phenomenon for both front and rear left wheels together, before being put back to its trajectory at the instant t = 53s. After that, the vehicle stays on a straight road till t = 60s. When the steering angle is equal to zero, the electric vehicle drives on the straight road.

**Fig. 16.**Steering angle

In Fig. 16, when δ = 0° the electric vehicle moves straight, at this moment ωm1,2 =ωm3,4. Then, if the vehicle turns to the left, which means that δ is at the range from 0° to 7°, ωm3,4 decreases as δ increases, on the contrary, ωm1,2 increases.

Fig. 17 shows the trajectory traced out by the vehicle. It can be seen clearly that the vehicle starts from rest and continues on its linear path until the steering angle starts to rise by imposing a left turn at first.

**Fig. 17.**Vehicle path

Fig. 18 shows the longitudinal velocity of the vehicle vx . We notice a dissipation of energy due to the lateral sliding. As for the lateral velocity vy and the yaw moment r , we can immediately recognize that their existence depends solely on the steering angle reference as shown in Figs. 19 and Fig. 20. We can clearly see that these two speeds occur only during cornering and they vanish when the vehicle is traveling on a straight road.

**Fig. 18.**Longitudinal velocity

**Fig. 19.**Lateral velocity

**Fig. 20.**Angular velocity

Fig. 21 (a) shows the estimated rotational speeds of the motors. During the first steering, the motors (M1 and M2) located outside of the turn’s curvature, rotate at higher speeds than motors (M3 and M4). On the other hand, we can notice that the motors (M3 and M4) rotate at higher speeds than the motors (M1 and M2) during the second steering as can be seen from Fig. 21 (b). The skidding occurs when moving on the left turn between t =17s and t = 23s. The fuzzy logic anti-skid control has a great effect to maintain permanently the speed of the vehicle and those of the front and rear right motors (M1 and M2) close to their profiles, during the loss of adherence. However, the fuzzy anti-skid control reduces significantly the speed errors which allows the re-adhesion of the skidding wheels.

**Fig. 21.**Rotational speed of motors

Fig. 22 depicts the linear speed of wheels. A good tracking of the longitudinal velocity of the vehicle can be observed. In fact, we can notice also a slight decrease when the driver applies a steering. However, similar responses could be noticed for the speeds of the 4 driving wheels of the vehicle.

**Fig. 22**Linear speed of wheels

The behaviour of the longitudinal slip of wheels is illustrated by Fig. 23. We should note that the longitudinal slips ( s1, s2, s3, ss4 ) of the four wheels are maintained in the adhesive region. Therefore, it is confirmed that the antiskid control maintain the slip ratio around its optimal value.

**Fig. 23**Longitudinal slip of wheels

The loss of adherence imposed on the outer driven wheels (1 and 2) results to a reduction in the load applied to these wheels. Consequently its speed increases during the transient time which induces a small variation of the slip on inner driven wheels (3 and 4), see Fig. 23. The effect of this variation, leads to a temporary increase in the traction forces, Fig. 26. However, the fuzzy anti-skid control establishes a self-regulation by reducing the increases the electromagnetic torques of motors 3 and 4 (Fig. 24) to compensate the load torque of motors 3 and 4, Fig. 25. Fig. 27 shows the phase current of drive motors.

**Fig. 24**Torque of motors

**Fig. 25**Resistive torques

**Fig. 26**Traction forces

**Fig. 27**Phase current of the motor

It can be observed from Fig. 28, that the adhesive coefficient of the outer driven wheels is bigger than those of the inner one. Therefore, the stability is maintained during the vehicle turn.

**Fig. 28**Adhesive coefficient of wheels

# 7. Conclusion

In this paper, a new method is proposed for a multimachine control and a fuzzy logic anti-skid control for high performance electric vehicle based on sensorless direct torque control of in-wheel permanent synchronous motors. The sensorless master slave control based on DTC was carried out on in-wheel motors based on sliding mode observer. However, the anti-skid control for electric vehicle based on regulating the ratio of wheel acceleration to drive motor torque shows a very stable behaviour of the electric vehicle during the various conditions of adherence.