Slip Suppression of Electric Vehicles Using Sliding Mode Control Method

This paper presents a slip suppression controller using sliding mode control method for electric vehicles which aims to improve the control performance and the energy conservation by controlling the slip ratio of wheel. In this method, a robust sliding mode controller against the model uncertainties is designed to obtain the maximum driving force by suppressing the slip ratio. The numerical simulations for one wheel model under variations in mass of vehicle and road condition are performed and demonstrated to show the effectiveness of the proposed method.


Introduction
In recent years, as a powerful solution against the environment and energy problems, Electric Vehicles (EVs) have attracted great interests in the world [1]. Compared with Internal Combustion Engine Vehicles (ICEVs), on one hand, EV is more environmental-friendly vehicle. On the other hand, it has significant performance advantages from the standpoint of electric and control engineering as the torque response is very quick and the torque can be measured easily and accurately [2]. As a result, advanced motion control such as the traction control with wheel slip suppression can be realized more easily for EVs than ICEVs [3,4].
As known to all, it is important to have good energy conservation for EVs, so it is hoped that there will be a control method which can reduce the electric energy consumption. When the vehicle is starting off or accelerating under slippery road conditions, the driving wheels falling into slip easily causes unstable driving situation and a lot of energy waste. The traction control is to provide the maximum driving force for driving wheels when accelerating, especially in wet, icy or snowy road conditions. For improving acceleration performance and saving electric energy for EVs, an advanced traction control is expected. During acceleration, the driving force of wheel directly depends on the friction coefficient between road and tire, which is in accordance with the wheel slip and road conditions. For this reason, it becomes possible to influence driving force by controlling the wheel slip.
There have been several methods proposed for the slip control of EVs, such as the method based on Model Following Control (MFC) in [5] and Model Predictive PID method (MP-PID) in [6]. Both of these methods show good performances under the nominal conditions where the situations, for example, mass of vehicle, road condition, and so on, are not changed. To meet high performance to the variation happened in these conditions, it is significant to construct robust control systems against the situation changing. About this point, Sliding Mode Control (SMC) has been performed good robustness for the systems with uncertainties and nonlinearities. Thereby [7,8] introduced robust control methods based on SMC for anti-lock braking system. In SMC, system trajectory is forced to reach the desired geometrical locus called sliding surface, then the trajectory slides along it when the motion of the system is in sliding mode [9,10]. The control objective of SMC is to take the system trajectory to reach the sliding surface in finite time, then to behave in sliding mode. In original SMC, chattering phenomenon always occurs through switching of the control inputs due to the vari-able structure of the sliding mode controller [11]. For reducing such undesired chattering effect, normally one boundary layer around the sliding surface is introduced in [12]. In contrast, the control performance such as steady state accuracy and transient response performance gets degradation, and moreover, it leads to energy loss. To overcome these disadvantages of conventional SMC method, an integral term is introduced in the design of the sliding surface [13]. Moreover, in order to get better control performance and save more energy, a new robust control method based on SMC for slip suppression of EVs is proposed. The numerical simulation results show the effectiveness of the proposed method.

Preliminaries of SMC
Consider a single input nonlinear dynamical system [12] described by where is the error be- tween the output state and the desired state . The problem of tracking is equivalent to remaining on the surface s for all . When , that is to say, the system trajectory reaches the surface which represents the error is zero. Hence, describes the sliding surface where the error will converge to zero exponentially. When , the trajectory is restricted to the sliding surface, which represents that the motion of the system is in sliding mode.
it in sliding mode. Generally, uses a discontinuous oosing the control input u , it is necessary to co s 0  The SMC law contains two parts, the equivalent control input eq and the hitting control input , which is defined as follows, eq can be interpreted as the continuous control law which would maintain when the dynamics is exactly known. When the dynamics is not exactly known, such as the uncertainties occur in the system or the trajectory of system is off the sliding surface, ht acts to bring the trajectory back to the sliding surface and keeps function to implement the switching action on sliding surface.
For ch ht u nsider the sliding condition [12], which is defined as where 0   . From Equation (4), 2 s shows that the dista ystem based on SMC sh squared nce to the sliding surface, which decreases along all system trajectories. Particularly, once the system trajectories reach the surface, they will remain on the surface. In other words, the system satisfying the sliding condition makes the trajectories reach the surface in finite time, and once on the surface, they cannot leave it. Furthermore, Equation (4) also implies that some dynamic uncertainties can be tolerated by keeping the motion of the system in sliding mode.
In general, to design a control s ould go through the following two steps. Firstly, design a sliding surface s which is invariant of the controlled dynamics. Secondly, choose the control input u which drives the system trajectory to the sliding surface in sliding mode in finite time.

Problem Formulation
ppropriate for acceleration on heel rotating motion an w m d

Vehicle Dynamics
A vehicle model which is a the longitudinal direction is described here. For simplicity, one wheel model directly driven by an electric motor is used for the derivation of control law and numerical simulations. Although the one wheel model is quite simple, it still retains the essential dynamics of the system. In deriving the dynamic equations of the system, the lateral and vertical motions are neglected. The rolling resistance and air resistance are also ignored. A simple one wheel model is shown in Figure 1.
The dynamic equation for the w d longitudinal vehicle motion are given by where  is the angular velocit l r d y of wheel, r the radius of whee and V the vehicle body speed. Othe parameters are defined in Table 1.
The tire driving force F is given by The friction coefficient   , c   , which is the ratio be d the n tween the driving force an ormal tire force, depends on the road condition (represented by road surface condition coefficient c ) and the wheel slip (represented by slip ratio  ). The slip ratio is defined as  Figure 2 shows the relationship between f fic riction coefient  and slip ratio  on the road surface con-

ion of Reference Slip Ratio λ * 3.2. Derivat
Let us choose the function By using Equation (12), Equation (11)  as can be rewritten and solving Equation (14) gives To obtain the maximum driving force all the time, the p ratio needs to d accurately on any cases, but in fact the road surface on which vehicles travel is not at a constant condition always. Besides, the mass of vehicle varies when the weight of load such as the luggage and passengers changes. Consequently, we need to control the slip ratio under the condition of the often change of vehicle mass and road condition.

Evaluation of
In this paper, we estimate the energy cost of EVs by calculating the electric energy consumed by motor. At first, And substituting Equations (5), (6 tion (19), the following equations can be obtained, Equation ( tion f is not exactly known, but it can be estimated as ˆm i Consider the system 18), the nonlinear funcf . The estimation error on f is assumed to be bounded by a own function The uncertainty in f is due t Accordingly, by using Equation ca o the parameter M and c.
(20) the estimation of f n be defined as where M is the estimated value of M mated for c. From these definitions, the error in estimation can be given by The reference slip ratio λ * is and solving Equation (32) gives equivalent control Then the estimate of the uivalent control input can be obtained as For meeting the sliding condition (making the system trajectory in the sliding mode) despite the dynamics f, the hitting control input is defined as stem is guaranteed with an exponential convergence once the sliding surface is encountered, when the sliding condition is satisfied. So Equation (41) guarantees that the trajectory will converge to the sliding surface in finite time if the error is not zero. That is to say, slip ratio can be suppressed to the reference value in finite time whenever the uncertainties occur in the system.

Chattering Reduction
In design of sliding mode control system control law requires switching However, because the actuato other imperfections, the action can lead to chatter in a neighborhood of the sliding surface. To reduce the chattering, the hitting control mht T can be rewritten by using the saturation function   is a design parameter re width of the boundary layer around 0 s presenting the the sliding surface and the saturation function is defined as  1 s s s sat s Thus, by using Equations (37), (38) a trol law of the system by the proposed SMC can be rewritten as nd (42), the con- M  (kg) and

Results of Robustness to the Variation in
In order to verify the robustness of proposed SMC with ion, the v s made by assigning

Numerical Simulations
The numerical simulations have been done using MATe computer platform is lations as follows. The first phase, the tim LAB/Simulink software [15]. Th listed in Table 2.
The simulation conditions are described here. The simulation time is set to 10 (s) in all. There are three phases in the simu e is from 0 (s) to 2 (s) and the car travels on the dry asphalt. The second phase, from 2 (s) to 8 (s), the car travels on ice road. The last phase, the car runs on wet asphalt during 8 (s) to 10 (s). The width of the boundary layer  defined in Equation (42)    , which are determined by trial and error.
By using Equations (28), (30), (35) and (38), the control law of the conventiona can be der ed as l SMC iv Likewise, in the conventional SMC, the parameters 1   and 1   . The values of paramet simulations are listed in Table 3. produ ed by the driver. Here, th ers used in the As the input to the simulation of system, the torque is ced by the pressure on the accelerator pedal, which is decided on the car speed desir e car speed is desired to achieve 180 (km/h) in 15 (s) by a fixed acceleration after starting the car. The range of variation in mass of the car M and road condition coefficient c are imposed as

Mass and Road Condition
variation both in the mass of the car and road condit ariation in the mass of the car i the value of M to 1000 (kg), 1100 (kg), 1200 (kg), 1300 (kg) and 1400 (kg) respectively. Figure 3 shows that the responses of slip ratio with different masses can converge to the reference value under the variation in the road condition. It is known that when the mass gets the nominal value 1200 (kg), in the first 2 (s), the response is more accurately than the car with other masses. But after 2 (s), the performance drops down with the mass increasing.
Next, we compare the proposed SMC with the conventional SMC and the case no control method used in the system. Figures 4-8 show the responses of slip ratio under three different road conditions for five different masses respectively. The responses with proposed SMC can suppress the slip ratio to the reference value 0.13 accurately in a very short time whenever both of the mass and road condition are changing. In addition, the slip ratio with the conventional SMC does not converge to the reference value because of the steady state error. When the car starts off at 0 (s) or runs into an ice road at 2 (s), the slip ratio response using control method grows with the in-     creasing wheel speed as a result of too much torque generated. As the car travels from ice road to wet asphalt from 8 (s), the slip ratio decreases with the decreasing wheel speed, when the torque generated at that time cannot meet that required on the wet asphalt. The car without control is to make the slip ratio to 0, so at the first stage the response is converged to 0. However, when the car runs into the ice road at 2 (s), the wheel spins out of control resulting that the wheel speed increasing suddenly, which leads to a large slip ratio value. Therefore, we can see that the proposed SMC have a good performance against the variation in both of the mass of the car and road condition.

Results of Acceleration Performance
It is different from the simulation condition described in p e ice road because the car cannot get road.
revious that the simulations are executed under unhanging road condition and mass every time. Figure 9 c shows the time required for 100 meters by the car with different control method. The x-axis label indicates the cases of different road condition and mass, for example, DA1000 says that the car with the mass 1000 (kg) is driving on the dry asphalt, WA1200 shows that the case with the mass 1200 (kg) on the wet asphalt and IR1400 is the case with mass 1400 (kg) on the ice road. As shown in the bar graph, it takes the minimum time for the car with the proposed method for the 100 meters in every case. So we can see that the car with proposed SMC have gained the best acceleration. In other words, the results also indicate the car with the proposed SMC decreases the loss of driving force mostly. Moreover, the time required is long on th enough driving force to accelerate on the slippery

Results of Energy Conservation
To confirm the effectiveness of the proposed SMC for energy conservation, we compare it to the conventional SMC and no control method. As a numerical example, we calculate the energy consumed in the simulations executed in 5.1. Figure 10 shows the results of electric energy consumed by different mass cases. We can see that the proposed SMC consumes the minimum energy in every case. The car without control takes most energy because the spin of wheel on the ice road from 2 (s) to 8 (s) leads to much energy loss. As the mass increases, the amount of energy cost decreases because the car suppresses the spin of wheel by increment of mass to get more driving force. Conversely, the energy consumption with the proposed SMC and conventional SMC increases due to the rising cost of control as the mass increases. From this perspective, it also implies that an EV should be made more light to save more energy.

Conclu
This paper proposed an extended SMC method ad the integral term to the sliding function for improvin performance of the slip ratio control for EVs. The control objective focused on suppressing the slip ratio to the reference value within the specified variation in ma vehicle and road conditions which allowed the vehicle to get the maximum driving force and minimum energy cost during acceleration.
As numerical examples based on the one wheel model, the simulations using the proposed method were ex cuted, ed in ass of vehicle and road conditions was verified. Beparameter sions ding g the ss of eand the robustness to the uncertainties caus m sides, by comparing to conventional SMC and no control, the vehicle with proposed method performed best acceleration performance; moreover, the results also showed that the proposed method could reduce the energy cost when the vehicle travels on a slippery road.
In our research, the design  and the gain i K of integral item added in the sliding function was determined by trial and error, so it is necessary to develop a method to find the optimal value of  and gain i K . Moreover, although the effectiveness of the proposed method for traction was just verified in the acceleration situation in this paper, we need to verify and improve the method for overall driving condition including braking situation. Then, the method is expected to be implemented and be one of the advanced motion control of EVs.