Adaptive Second-Order Sliding Mode Wheel Slip Control for Electric Vehicles with In-Wheel Motors

Abstract

The influence of the external environment can reduce the braking performance of the electric vehicle (EV) with in-wheel motors (IWM). In this paper, an adaptive sliding mode wheel slip control method with a vehicle speed observer consideration is proposed, which enables the EV to accurately track the optimal slip ratio in various environments and improve braking performance. First, the braking system dynamics model is established by taking the EV with IWM as the study object. Second, a super-twisting sliding mode observer is used to estimate the vehicle speed, and a new adaptive second-order sliding mode controller is constructed to control the braking torque. Finally, co-simulation experiments are performed under different conditions based on Carsim and MATLAB/Simulink, and the proposed scheme is validated by comparison with three control methods. The experimental results show that the proposed scheme has better control performance, and both the safety and control quality of the EV is improved.

1. Introduction

Electric vehicles (EVs) not only benefit energy conservation and environmental protection but also have better performance than internal combustion engine vehicles. EVs have received widespread attention in recent years [1,2]. As one of the most important indicators for evaluating the safety factor level of vehicles, excellent braking performance is more conducive to ensuring the driver’s own safety. The braking system of EVs is different from that of internal combustion engine vehicles. According to the actuators of the braking system, there are three braking modes for EVs: hydraulic friction braking (HFB), hybrid braking, and electric motor braking (EMB). Based on these three braking modes, the study on EV braking systems is mainly divided into three parts: coordinated control of HFB and EMB, anti-lock braking control of HFB and EMB, and EMB.

The widely used braking method in EVs is the coordinated control of HFB and EMB. Yang et al. [3] proposed a torque coordination control strategy based on brake pedal travel and rate of change and utilized the electric motor braking torque as the compensating braking torque. It reduced torque fluctuation and vehicle judder during mode switching. Tang et al. [4] presented a coordinated control strategy for an electro-hydraulic hybrid braking system with high-efficiency energy recovery, which improved the braking performance while increasing the energy recovery efficiency. Xu W et al. [5] developed a braking torque distribution strategy based on model predictive control to optimize the braking torque of the front and rear wheels. It better solved the multi-objective and constraint problem of coordinated control of the braking system. However, when entering the emergency braking state, the coordinated control of EVs will lead to the reduction of braking stability.

To solve the problem of braking stability in coordinated control, some of the literature has conducted studies on wheel anti-lock systems with HFB and EMB. Zhang et al. [6] proposed a new switching strategy for electric motor torque control between sliding mode control compensation and dual closed-loop PID control compensation to improve braking stability and maneuverability. Qiu C et al. [7] presented a serial control strategy for the wheel braking system and conducted experiments in different road environments. S. Lupberger et al. [8] constructed a continuous nonlinear controller to implement wheel speed tracking control considering the effects of the drivetrain. It ensures consistent control performance for both regenerative and hybrid braking. In practical applications, a new braking system needs to be redesigned for antilock synergistic control. However, it will complicate the control strategy of the anti-lock braking system (ABS) and endanger the driving safety of EVs. This is because the time response characteristics and braking execution modes of HFB and EMB are essentially different [9].

EMB has a faster response and lower delay than the other two braking modes. Thus, EVs that only use EMB can accurately control the braking torque and improve the braking performance. Jiang B et al. [10] proposed a regenerative braking strategy based on an optimized distribution algorithm. It optimizes the distribution of braking force in response to changes in braking intensity, such that braking stability and maximum braking energy recovery are satisfied. Savitski et al. [11] conducted experiments on three different configurations of EVs. The excellent performance of regenerative braking was demonstrated through experiments on low-friction road surfaces. Chen J et al. [12] designed a regenerative braking controller based on longitudinal velocity observer to improve regenerative braking energy recovery by distributing braking torque between front and rear wheels. Most studies to date have focused on regenerative braking and energy recovery [13,14,15]. However, in practical application, pure regenerative braking makes it difficult to meet braking torque requirements. As a result, almost all existing EVs disengage regenerative braking when ABS is activated and decelerate the EVs by friction braking.

As EVs continue to evolve, the number of electric motors is increasing, and the driving methods are becoming more diverse. However, it has been experimentally verified that EMB causes EVs to lose braking stability as the number of electric motors increases. Therefore, for EVs with IWM, the first step is to solve the problem of wheel brake holding. The response characteristics and braking mode of the electric motor anti-lock braking system (emABS) are completely different from traditional ABS. Although there are many studies on traditional ABS [16,17,18], this cannot solve the wheel-locking problem of EMB.

The main problem of emABS is how to choose the appropriate algorithm and control target variable to match the high nonlinearity of EV dynamics. Feng X et al. [19] designed a discrete fuzzy adaptive PID control algorithm to control wheel slip ratio. This algorithm makes up for the disadvantage of the PID control algorithm that cannot be automatically adjusted. Aksjonov et al. [20] proposed an anti-lock regenerative braking controller based on fuzzy logic and applied it to EVs to improve control stability. However, the design of a fuzzy logic controller relies too much on a large number of simulation experiments or practical experience. Nguyen et al. [21] proposed a hierarchical linear quadratic regulator to control the wheel slip ratio, which can significantly reduce the design cost and the complexity of real-time calculation. However, in the actual vehicle running process, EV dynamics are highly nonlinear and model uncertain and are affected by uncertain disturbances such as road friction and wind resistance [22]. Therefore, a nonlinear controller with stronger robustness should be designed for the slip ratio control system of EVs.

Strong nonlinearity has become an important problem in the design of slip ratio controllers for EVs. Sliding mode control (SMC) is a nonlinear control algorithm. Because of its strong robustness to parameter perturbation and external disturbances, SMC has become the preferred scheme for designing slip ratio controllers of EVs [23,24]. Dzmitry S et al. [25] applied the continuous twisting control algorithm to the slip ratio controller, which had a good effect on shortening the braking distance and improving driving stability and comfort. However, some known parts of the system are regarded as uncertainties in the single-wheel model, which will lead to excessive control action and reduce the safety of the braking process. He L et al. [26] proposed an SMC method that comprehensively considered wheel deceleration and slip ratio, which could control the deceleration rate while controlling the slip ratio. However, this will lead to the slow response speed of the braking system, inaccurate slip ratio control, and complicate the whole braking process, making the braking system unable to control each stage accurately. Hongwei W et al. [27] presented an improved global SMC scheme. However, the vehicle speed is calculated by longitudinal speed, which cannot be measured directly. Currently, most studies do not take vehicle speed into account. The common method used in the past for longitudinal velocity measurement has been the use of sensors and accelerometers, both of which have the same disadvantage of requiring accurate sensing systems and infrastructure. The sensing system requires frequent updates due to the accumulation of integration errors. Moreover, high-precision sensors are expensive [28]; thus, it is necessary to estimate the vehicle speed. Based on the above problems, this paper proposes an adaptive second-order sliding mode slip ratio control scheme for the full EV equipped with individual IWM. The adaptive gains are constructed based on the revised barrier function. The vehicle speed is observed by a super-twisting observer. At the same time, it is compared with the traditional PI control method, super-twisting sliding mode control (STSMC) method, and first-order sliding mode control (FOSMC) method. Through the simulation experiments in three road environments, the driving safety and quality control effects of the proposed control scheme are verified.

The structure of this paper is as follows. In Section 2, the dynamic model of the braking system and the estimation of tire–road friction coefficient are described. Section 3 introduces the vehicle speed observer and the EV slip ratio controller and proves the stability. Three other slip ratio controllers are also introduced for comparative experiments. In Section 4, the proposed control system is simulated under three road conditions and compared with the other three methods based on Carsim and MATLAB/Simulink. Finally, in Section 5, the conclusions are given.

2. Vehicle Model

Here, the wheel slip ratio equation is obtained by analyzing the braking system dynamics model. This is followed by the estimation of the road friction coefficient in conjunction with a widely recognized road tire model.

2.1. Brake System Dynamics Modeling

Firstly, the dynamic model of vehicle braking is analyzed. Changes in tire rolling radius during driving, tire transient dynamics, wheel rolling resistance, and air resistance are ignored. The single-wheel model of the EV under braking conditions is established, as shown in Figure 1.

As can be seen in Figure 1, $F_z$ is the vertical force of the tire, $F_x$ is the reverse friction force formed by friction between the tire and the ground, $V$ is the vehicle speed, $\omega$ is the angular velocity of the wheel, $T_b$ is the braking torque of the wheel, $r_w$ is the radius of the wheel.

According to Figure 1, the following vehicle motion equations can be obtained:

$$\left\{\begin{array}{l}m\dot{V}=-\sum F_{xi}=-{\mu }_{\lambda i}\sum F_{zi}\hfill \\ J_{\omega i}{\dot{\omega }}_i=-T_b+r_w{F}_{xi}\hfill \\ \sum F_{xi}=F_{xfl}+F_{xfr}+F_{xrl}+F_{xrr}\hfill \end{array}\right.$$

(1)

where $m$ is the mass of the EV, ${\mu }_{\lambda i}$ is the road friction coefficient, $J_{\omega }$ is the moment of inertia of the wheels, $r$ is the radius of the tires, $i$ is the total number of wheels, $fl$ is the left front wheel, $fr$ is the right front wheel, $rl$ is the left rear wheel, and $rr$ is the right rear wheel.

For vehicle longitudinal motion and braking mode, the wheel slip ratio ${\lambda }_i$ can be expressed as:

$${\lambda }_i={\displaystyle \frac{V-r_w{\omega }_i}{V}}$$

(2)

Take the derivative of the slip ratio, then:

$${\dot{\lambda }}_i=-{\displaystyle \frac{r_w{\dot{\omega }}_i}{V}}+{\displaystyle \frac{r_w{\omega }_i}{V^2}}\dot{V}$$

(3)

From Equations (1)–(3), it can be deduced as:

$${\dot{\lambda }}_i=-{\displaystyle \frac{1}{V}}({\displaystyle \frac{1}{m}}(1-{\lambda }_i)+{\displaystyle \frac{r_w^2}{J_{\omega i}}})F_z\mu (\lambda )+{\displaystyle \frac{r_w}{J_{\omega i}V}}{T}_b$$

(4)

It can be observed from Equation (4) that the infinitely fast system dynamics will lead to the loss of the system controllability when $V\to 0$. Therefore, the slip ratio controller should be disabled when $V$ is low.

Since ${\displaystyle \frac{1}{m}}(1-\lambda )<<{\displaystyle \frac{r_{\omega }^2}{J_{\omega i}}}$, Equation (4) can be simplified as:

$${\dot{\lambda }}_i=-{\displaystyle \frac{1}{V}}{\displaystyle \frac{r_{\omega }^2}{J_{\omega i}}}{F}_{zi}\mu (\lambda )+{\displaystyle \frac{r_{\omega }}{J_{\omega i}V}}{T}_b$$

(5)

In order to avoid extra state estimation, the influence of load distribution caused by the change in the tire vertical force $F_z$ and the road friction coefficient $\mu (\lambda )$ is treated as uncertainty. Because the wheel rolling resistance and air resistance are ignored, then:

$${\dot{\lambda }}_i=-{\displaystyle \frac{1}{V}}{\displaystyle \frac{r_{\omega }^2}{J_{\omega i}}}{F}_{zi}\mu (\lambda )+{\displaystyle \frac{r_{\omega }}{J_{\omega i}V}}{T}_b+h(x)=A(x)+B(x)T_b+h(x)$$

(6)

where $A(x)=-{\displaystyle \frac{1}{V}}{\displaystyle \frac{r_{\omega }^2}{J_{\omega i}}}{F}_{zi}\mu (\lambda )$, $B(x)={\displaystyle \frac{r_{\omega }}{J_{\omega i}V}}$, $h(x)$ is the parameter uncertainty of the slip ratio control system of the EV. Considering the physical limitations of actual EV, the numerical range of the tire vertical force, the road friction coefficient, wheel rolling resistance and air resistance are limited, thus $h(x)$ is bounded.

2.2. Estimation of Tire–Road Friction Coefficient

According to Equation (1), the road braking force $F_{xi}$ received by the EV is related to the road friction coefficient $\mu ({\lambda }_i)$ and the tire vertical force $F_{zi}$. The relationship between the road friction coefficient $\mu ({\lambda }_i)$ and the wheel slip ratio ${\lambda }_i$ is usually described in the form of road tire. The Burckhardt model [29], which is widely recognized to study their relationship, is described as:

$$\mu ({\lambda }_i)=c_1(1-e^{-c_2{\lambda }_i})-c_3{\lambda }_i$$

(7)

It can be observed that there is a nonlinear relationship between the road friction coefficient $\mu ({\lambda }_i)$ and the slip ratio ${\lambda }_i$. In different road conditions, the parameters $c_1$, $c_2$ and $c_3$ in the formula are different, which can be obtained by test fitting. The typical road conditions are shown in

Table 1. Parameter setting of typical Burckhardt tire model.

Pavement	Dry Cement	Dry Asphalt	Wet Pebbles	Wet Asphalt	Snow	Ice
$c_1$	1.1973	1.28	0.4004	0.857	0.1946	0.05
$c_2$	25.168	23.99	33.708	33.822	94.13	306.39
$c_3$	0.5373	0.52	0.1204	0.347	0.0646	0.001
${\lambda }_{opt}$	0.16	0.17	0.14	0.13	0.06	0.03
${\mu }_{\max }$	1.09	1.17	0.34	0.8013	0.1907	0.05

.

The $\mu -\lambda$ relationship diagram based on Burckhardt model under typical road conditions is shown in Figure 2. It can be seen that with the increase in wheel slip ratio ${\lambda }_i$, the road friction coefficient $\mu ({\lambda }_i)$ first increases and then decreases. The peak road friction coefficient ${\mu }_{\max }$ can be obtained when the slip ratio ${\lambda }_i$ is between 0.03 and 0.2. The maximum road friction is obtained by maintaining the peak friction coefficient.

3. Vehicle Speed Observer and Slip Ratio Controller Design

This paper proposed an adaptive second-order sliding mode slip ratio control scheme for the full EV equipped with individual IWM. Considering the nonlinear characteristics of EV dynamics and the problem of vehicle speed measurement during driving, the vehicle speed is observed with a super-twisting sliding mode observer. The control block diagram is shown as Figure 3. The slip ratio control system of EV consists of vehicle speed observer, slip ratio control activation logic controller, slip ratio control activation logic controller, sliding mode controller based on slip ratio, electric motor control model and Carsim simulation vehicle dynamics model. Firstly, considering environmental factors such as wind speed, road friction and aerodynamic force, the simulation environment is designed in Carsim2019.0 software. Disconnect the vehicle power system and change to external input, so as to simulate the driving conditions of full EV equipped with individual IWM for each wheel which is closest to the actual situation. Secondly, the dynamic analysis of the braking system is carried out through the single-wheel model, and the longitudinal slip ratio is taken as the control target variable. The sliding ratio control system is modeled in MATLAB/SimulinkR2021b software, and a sliding mode observer is designed to estimate the vehicle speed. Next, the composite scheme will be presented in detail.

3.1. The Proposed Composite Scheme

In order to calculate the slip ratio, it is necessary to measure the vehicle speed $V$ and wheel angular velocity ${\omega }_i$. Because the vehicle speed $V$ is an unmeasurable quantity, a super-twisting observer is constructed to estimate it.

Assuming $\widehat{V}$ is the estimated value of vehicle speed, the sliding mode surface is chosen as:

$$S_{sto}=V-\widehat{V}$$

(8)

According to Equation (1), by applying the super-twisting sliding mode algorithm to the observer, it can be obtained:

$$\left\{\begin{array}{l}\dot{\widehat{V}}=-{\displaystyle \frac{\sum F_{zi}}{m}}\mu ({\lambda }_i)+K_1|S_{sto}{|}^{1/2}sign(S_{sto})+u_1\hfill \\ {\dot{u}}_1=K_{2}sign(S_{sto})\hfill \end{array}\right.$$

(9)

where $K_1$ and $K_2$ are the observation gains.

Description of system stability. If these two parameters are chosen as $K_1=1.5\sqrt{D_{sto}}$, $K_2=1.1D_{sto}$, $D_{sto}$ is a positive constant, then the second-order sliding mode can be established to achieve finite-time stability. Detailed analysis of the stability of similar systems can be found in [30].

The function of the slip ratio controller is to control the actual slip ratio ${\lambda }_i$ near the desired slip ratio ${\lambda }_i^{*}$ via the output brake torque $T_b$ at the control wheel end. In the proposed vehicle slip ratio control block diagram, the direct slip control principle is used by the IWM to provide the electric motor demanded torque $T_b$ to maintain the desired wheel slip ratio ${\lambda }_i$. When the wheel slip ratio is higher than the desired value, the slip ratio control will be activated separately for each wheel to keep the wheel slip ratio near the desired value. Therefore, the slip ratio error is defined as:

$${\lambda }_{ei}={\lambda }_i^{*}-{\lambda }_i$$

(10)

Define the sliding mode surface as:

$$s=a_1{\lambda }_{ei}$$

(11)

where $a_1$ is a positive constant.

Substitute Equation (6) into Equation (11), $\dot{s}$ is obtained as:

$$\dot{s}=-{\displaystyle \frac{a_1}{\widehat{V}}}{\displaystyle \frac{r_w^2}{J_{wi}}}{F}_{zi}\mu ({\lambda }_i)+{\displaystyle \frac{a_1{r}_w}{J_{wi}\widehat{V}}}{\widehat{T}}_b+h(x)$$

(12)

For Equation (12), it is necessary to design a suitable control law and then establish the sliding modes with respect to s, to realize the slip rate tracking and subsequently the braking torque control. Then $u(t)$ is designed as:

$$u(t)=u_c+u_d$$

(13)

where $u_c$ is the equivalent control, and the expression for $u_c$ is calculated from the undisturbed system, i.e., $h(x)=0$. When $\dot{s}=0$, $u_c$ is obtained as:

$$u_c=r_w{F}_{zi}\mu ({\lambda }_i)$$

(14)

$u_c$ acts as a continuous control section to ensure that the system moves on the sliding surface. When there are uncertainties in the system, such as external disturbance or parameter perturbation, they can be overcome by $u_d$.

Equation (14) is substituted into Equation (12) to obtain:

$$\dot{s}={\displaystyle \frac{a_1{r}_w}{J_{wi}\widehat{V}}}{u}_d+h(x)$$

(15)

Let $v={\displaystyle \frac{a_1{r}_w}{J_{wi}\widehat{V}}}{u}_d$, then:

$$\dot{s}=v+h(x)$$

(16)

where $v\in R$ is the control input, $h(x)\in R$ represents the disturbance, which could be formulated as $h(x)={\epsilon }_1+{\epsilon }_2$. Assume that $|{\epsilon }_1|\le Y|s{|}^{1/2}$, $|{\epsilon }_2|\le Zt$, $Y$ and $Z$ are unknown upper bounds of square root growth disturbance ${\epsilon }_1$ and Lipschitz disturbance ${\epsilon }_2$, respectively.

Assuming that $\sigma >0$ is a given and fixed value, a revised barrier function $L_b$ can be defined as a strictly increasing continuous even function on $[0,\delta ]$. $L_b$: $x\in [-\sigma ,\sigma ]\to L_b(x)\in [\overline{K},\infty ]$. Where $\underset{\left|s\right|\to \sigma }{\lim }{L}_b=+\infty$, and $L_b(x)$ has a unique minimum at $x=0$, i.e., $L_b(0)=\overline{K}>0$, the expression for the revised barrier function is [31]:

$$L_b={\displaystyle \frac{\overline{K}{\sigma }^{1/n}}{{\sigma }^{1/n}-|x{|}^{1/n}}}$$

(17)

where $\overline{K}$ is a positive constant, $n$ is a positive integer to be chosen and is the smallest value that can be taken. The advantage of the revised barrier function is that it can alleviate the problem of having to choose very small sampling steps when applying the barrier function-based methods in practice.

The adaptive control law based on the revised barrier function can be constructed:

$$v=-L|s{|}^{1/2}sign(s)-L_1|s{|}^{\alpha }sign(s)-{\displaystyle \frac{P}{2}}{\displaystyle {\int }_0^{t}sign(s)}dt$$

(18)

where $L$, $L_1$, $P$ are all positive real numbers and $\alpha >1$. $L$, $P$ are adaptive gain parameters and $L_1$ is fixed parameters.

The expression of adaptive gain $L(s,t)$ and $P(s,t)$ are:

$$L(s,t)=\left\{\begin{array}{l}{L}_a=kt+L_0,0<t\le \overline{t}\\ L_b,t>\overline{t}\end{array}\right.$$

(19)

$$P(s,t)=2\tau L+2(\rho +4{\tau }^2)L_1{{\tau }_1}^{\alpha -1/2}+P_0$$

(20)

where $\rho ,\tau ,k,L_0,{\tau }_1>\sigma /2$ are positive constants, $L_0$, $P_0$ are the initial values of $L$, $P$ when $t=0$, respectively, $\overline{t}$ is the time when $|s|$ first reaches $\sigma /2$. The designated positions of $\left|s\right|$ and $\left|y\right|$ are $\sigma$ and $\zeta$, respectively.

It can be seen that if $L_1$ is zero, this super-twisting-like controller can be simplified to the traditional super-twisting algorithm. $|s{|}^{\alpha }sign(s)$ as a higher-order term to accelerate convergence. Therefore, the dynamic equation of the closed-loop system can be deduced as:

$$\left\{\begin{array}{l}\dot{s}=-L|s{|}^{1/2}sign(s)-L_1|s{|}^{\alpha }sign(s)+s_1+{\epsilon }_1\\ {\dot{s}}_1=-{\displaystyle \frac{P}{2}}sign(s)+{\dot{\epsilon }}_2\\ s(0)=s_0\\ s_1(0)=0\end{array}\right.$$

(21)

The proof of the above stability conclusion is divided into two steps. First, it is necessary to show that there exists a finite time $\overline{t}$ such that $|s|\le \sigma /2$ in Equation (21). First assume that $|s(0)|>\sigma /2$, according to Equation (19), it can be seen that the adaptive gain $L(s,t)=kt+L_0$. Meanwhile, the right-hand side of Equation (21) is sublinear with respect to $(s,s_1)$. This dynamic interval is defined by $U(s(0))$, its form is $[0,T_0]$. Prove by contradiction that $T_0$ is the time $\overline{t}$ required.

It is assumed that $|s|>\sigma$ is on $U(s(0))$ and $s$ is positive. According to the second equation in Equation (21):

$$-\tau (kt+L_0)-(\rho +4{\tau }^2)L_1{{\tau }_1}^{\alpha -1/2}-Z\le {\dot{s}}_1\le -\tau (kt+L_0)-(\rho +4{\tau }^2)L_1{{\tau }_1}^{\alpha -1/2}+Z$$

(22)

It is easy to obtain that $U(s(0))=[0,\infty )$ and $s_1$ becomes and remains negative for a finite time. According to the first equation of Equation (21):

$$\dot{s}\le -(kt+L_0)|s{|}^{1/2}sign(s)-L_1|s{|}^{\alpha }sign(s)$$

(23)

As a result, the sliding mold surface $s$ will tend to zero in finite time, contrary to the previous assumption that $|s|>\sigma$ on $U(s(0))$.

After that, it needs to be proved that after $\overline{t}$, $|s|$ is strictly limited to $|s|\le \sigma$ and $|s_1|\le \zeta$. Since there must exist $\overline{t}$ such that $|s|\le \sigma /2$, the control gain $L$ executes strategy $L_b$ in Equation (19).

According to the assumptions in Equation (16) for perturbations ${\epsilon }_1$ and ${\epsilon }_2$, define:

$$\left\{\begin{array}{l}{\epsilon }_1={\xi }_1{\vartheta }_1\hfill \\ {\dot{\epsilon }}_2={\displaystyle \frac{{\xi }_2{\vartheta }_1}{2|{\vartheta }_1|}}\hfill \end{array}\right.$$

(24)

where ${\xi }_1$ and ${\xi }_2$ are bounded and unknown, they are satisfied with $|{\xi }_1|\le Y$ and $|{\xi }_2|\le 2Z$, ${\vartheta }_1$ is obtained from a newly defined state vector $\vartheta ={\left[{\vartheta }_1,s_1\right]}^T$ as ${\vartheta }_1=|s{|}^{1/2}sign(s)$.

Substituting Equation (21) into the state vector $\vartheta$, and the derivative of $\vartheta$ is given as:

$$\dot{\vartheta }={\displaystyle \frac{1}{2|{\vartheta }_1|}}(R\left({\vartheta }_1\right)\vartheta +\left[\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right]{\vartheta }_1)$$

(25)

where $R\left({\vartheta }_1\right)=\left[\begin{array}{cc}-L-L_1|{\vartheta }_1{|}^{2\alpha -1}& 1\\ -P& 0\end{array}\right]$. It can observed that if ${\vartheta }_1$ and $s_1$ converge to the origin in finite and fixed time $T$, then $sign(s)=sign({\vartheta }_1)$, and $s$ will converge to the origin in finite and fixed time $T$.

Consider a Lyapunov candidate function:

$$V(\vartheta )={\vartheta }^T\left[\begin{array}{cc}\rho +4{\tau }^2& -2\tau \\ -2\tau & 1\end{array}\right]\vartheta$$

(26)

where $\rho >0$, $\tau >0$.

The derivative of this Lyapunov candidate function is given by:

$$\dot{V}(\vartheta )=\left({\vartheta }^T\dot{\vartheta }+{\dot{\vartheta }}^T\vartheta \right)\left[\begin{array}{cc}\rho +4{\tau }^2& -2\tau \\ -2\tau & 1\end{array}\right]=-{\displaystyle \frac{1}{2|{\vartheta }_1|}}{\vartheta }^{T}A(\vartheta )\vartheta$$

where

$$A(\vartheta )=\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{12}& A_{22}\end{array}\right)$$

(27)

with

$$\left\{\begin{array}{l}{A}_{11}=2\rho L-2(\rho +4{\tau }^2){\xi }_1+4\xi {\rho }_2+4\tau (2\tau L-\left[\begin{array}{cc}\rho +4{\tau }^2& -2\tau \\ -2\tau & 1\end{array}\right])+4(\rho +4{\tau }^2)L_1|{\vartheta }_1{|}^{2\alpha -1}\hfill \\ A_{12}=P-\rho -4{\tau }^2+2\tau {\xi }_1-{\xi }_2-2\tau L-2(\rho +4{\tau }^2)L_1|{\vartheta }_1{|}^{2\alpha -1}\hfill \\ A_{22}=4\tau \hfill \end{array}\right.$$

To ensure the positive definiteness of the matrix $A(\vartheta )$, define:

$$P=2\tau L+2(\rho +4{\tau }^2)L_1{\tau }_1^{\alpha -1/2}$$

(28)

The matrix $A(\vartheta )$ is positive definite and has the smallest eigenvalue ${\rho }_{\min }(A(\vartheta ))>2\tau$, if $L>{\displaystyle \frac{Y(\rho +4{\tau }^2)-\tau (4Z+1)+{(2\tau Y-2Z-\rho -4{\tau }^2)}^2/12\tau }{\rho (1-\eta )}}$, with positive constant $\eta <1$, then $\dot{V}(\vartheta )\le -{\displaystyle \frac{\tau }{|{\vartheta }_1|}}||\vartheta |{|}^2$.

Since

$${\rho }_{\min }(\left[\begin{array}{cc}\rho +4{\tau }^2& -2\tau \\ -2\tau & 1\end{array}\right])||\vartheta |{|}^2\le V(\vartheta )\le {\rho }_{\max }(\left[\begin{array}{cc}\rho +4{\tau }^2& -2\tau \\ -2\tau & 1\end{array}\right])||\vartheta |{|}^2$$

(29)

and

$$|{\vartheta }_1|\le \sqrt{{\vartheta }_1^2+{\vartheta }_2^2}=||\vartheta ||\le {\displaystyle \frac{V^{1/2}(\vartheta )}{{\rho }_{\min }(\left[\begin{array}{cc}\rho +4{\tau }^2& -2\tau \\ -2\tau & 1\end{array}\right])}}$$

(30)

then

$$V(\vartheta )\le -{\displaystyle \frac{\tau \sqrt{{\rho }_{\min }(\left[\begin{array}{cc}\rho +4{\tau }^2& -2\tau \\ -2\tau & 1\end{array}\right])}}{{\rho }_{\max }(\left[\begin{array}{cc}\rho +4{\tau }^2& -2\tau \\ -2\tau & 1\end{array}\right])}}{V}^{1/2}(\vartheta )$$

(31)

When $|s|\in \left[0,\sigma \right]$, $|s(t_1)|<\sigma$ and $L_b\left(\left|s\right|\right)$ are monotonically increasing functions with respect to $|s|$. Then the following conditions exist for $s_{11}<\sigma$:

$$s_{11}=\left\{\begin{array}{l}\sigma {(1-{\displaystyle \frac{\overline{K}}{L^{*}}})}^n,\overline{K}<{\displaystyle \frac{Y(\rho +4{\tau }^2)-\tau (4Z+1)+{(2\tau Y-2Z-\rho -4{\tau }^2)}^2/12\tau }{\rho (1-\eta )}}\hfill \\ 0,\overline{K}\ge {\displaystyle \frac{Y(\rho +4{\tau }^2)-\tau (4Z+1)+{(2\tau Y-2Z-\rho -4{\tau }^2)}^2/12\tau }{\rho (1-\eta )}}\hfill \end{array}\right.$$

(32)

If $|s|>s_{11}$, then $L_b>L^{*}$ and Equation (37) hold. Then $\dot{V}(\vartheta )$ is negative when $s_{11}\le |s|<\sigma$. Therefore, $|s|$ will still be restricted to $|s|\le s_{11}<\sigma$ when $t>t_1$.

3.2. Three Comparative Control Schemes with PI, FOSMC and STSMC

The PI control law is designed considering the control error equation in Equation (10):

$$u_{pi}=K_p(\widehat{V})\left({\lambda }_{ei}+{\displaystyle \frac{{\displaystyle \int {\lambda }_{ei}dt}}{t_i(\widehat{V})}}\right)$$

(33)

where $K_p$ is the proportional control gain and $t_i$ is the tuning parameter of the integral part.

To develop the FOSMC, the sliding variable is defined to be equal to the control error:

$$S_{sw}={\lambda }_{ei}={\lambda }_i^{*}-{\lambda }_i$$

(34)

The traditional FOSMC law is defined as:

$$u_{sw}=-K_{sw}sign(S_{sw})$$

(35)

where the control gain $K_{sw}$ is a positive constant.

In order to avoid excessive chattering in FOSMC, the sign function $sign(S_{sw})$ can be replaced by the following approximate value [32]:

$$si\widehat{g}n(S_{sw})={\displaystyle \frac{S_{sw}}{\left|S_{sw}\right|+\psi }}$$

(36)

where $\psi >0$ is a value as small as possible. Higher values may result in a loss of control performance. It is characterized by steady-state errors in the presence of matching uncertainties [33].

In order to suppress the chattering phenomenon in FOSMC, the sliding mode surface of the STSMC is designed as:

$$S_{stc}=a_1{\lambda }_{ei}+{\dot{\lambda }}_{ei}$$

(37)

where $a_1$ is a positive real number.

Substituting Equation (6) into Equation (37) and taking derivative:

$$\begin{array}{ll}{\dot{S}}_{stc}& =a_{1}A(x)+a_{1}B(x){\widehat{T}}_b+\ddot{e}+h(x)\\ & =-{\displaystyle \frac{a_1}{\widehat{V}}}{\displaystyle \frac{r_w^2}{J_{wi}}}{F}_{zi}\mu ({\lambda }_i)+{\displaystyle \frac{a_1{r}_w}{J_{wi}\widehat{V}}}{\widehat{T}}_b-{\displaystyle \frac{r_w^2{\dot{F}}_{xi}}{J_{wi}\widehat{V}}}+{\displaystyle \frac{r_w^2{F}_{xi}\dot{V}}{J_{wi}{\widehat{V}}^2}}-{\displaystyle \frac{r_w\dot{V}}{J_{wi}{\widehat{V}}^2}}{\widehat{T}}_b+{\displaystyle \frac{r_w}{J_{wi}{\widehat{V}}^2}}{\dot{\widehat{T}}}_b+h(x)\end{array}$$

(38)

Since there are several smaller components on the right-hand side of Equation (38) that cannot be reliably estimated, they are considered as system perturbations, such that Equation (38) can be modified to:

$${\dot{S}}_{stc}=-{\displaystyle \frac{a_1}{V}}{\displaystyle \frac{r_w^2}{J_{wi}}}{F}_{zi}\mu ({\lambda }_i)+{\displaystyle \frac{r_w^2{F}_{xi}\dot{V}}{J_{wi}{V}^2}}+{\displaystyle \frac{a_1{r}_w}{J_{wi}V}}{\widehat{T}}_b+h(x)$$

(39)

To design $u(t)$:

$$u(t)=u_a+u_b$$

(40)

where $u_a$ is the equivalent control, When $\dot{s}=0$, $u_a$ is obtained as:

$$u_a=r_w{F}_{zi}\mu ({\lambda }_i)-{\displaystyle \frac{r_w{F}_{xi}\dot{\widehat{V}}}{a_{1}V}}$$

(41)

Equation (41) is substituted into Equation (39) to obtain:

$${\dot{S}}_{stc}={\displaystyle \frac{a_1{r}_w}{J_{wi}\widehat{V}}}{u}_b+h(x)$$

(42)

As a special higher-order sliding mode algorithm, STSMC includes the discontinuous switching control in the integral term, which makes the control continuous and can greatly reduce the control chattering.

Let $u_{swt}={\displaystyle \frac{a_1{r}_w}{J_{wi}\widehat{V}}}{u}_b$, then:

$${\dot{S}}_{stc}=u_{swt}+h(x)$$

(43)

$u_{swt}$ is designed as a super-twisting algorithm:

$$\left\{\begin{array}{l}{u}_{swt}=-k_1{\left|S_{stc}\right|}^{1/2}sign(S_{stc})+u_1\hfill \\ {\dot{u}}_1=-k_{2}sign(S_{stc})\hfill \end{array}\right.$$

(44)

where the control gains $k_1$, $k_2$ are positive constants.

4. Simulation Results

4.1. Simulation Model and Parameter Settings

Since it is necessary to use EV model, road model, driver model and electric motor control model to provide input and output variables for the slip ratio control strategy, Carsim and MATLAB/Simulink co-simulation is designed in this paper. The main parameters of the simulated EV are shown in

Table 2. The main parameters of EV.

Description	Value	Units
Gross vehicle weight	1170	kg
Maximum torque of single electric motor	500	N·m
Radius of tire	0.31	m
Track width	1480	mm
Moment of inertia of wheel	0.6	kg/m2
Distance from center of mass to front axle	1040	mm
Distance from center of mass to rear axle	1460	mm

.

The integration algorithm in Simulink is ode1 and the fixed step size is set to 0.0005 s. Considering the practical physical application of the slip ratio controller, a random wind speed is designed in Carsim to simulate the real environment. At the same time, the EV is designed to drive in a straight line, and the control simulation is performed under different road conditions, including dry asphalt, wet asphalt, and snow road with different road friction coefficients. To verify the performance of the proposed control strategy, it is compared with PI control, FOSMC and STSMC. The superiority of the proposed scheme is verified by comparing driving safety and control quality. The key parameters of the proposed scheme are assigned in

Table 3. The key parameters of the proposed scheme.

Description	${\mathit{a}}_1$	${\mathit{L}}_0$	$\mathit{k}$	$\overline{\mathit{K}}$	$\mathit{n}$	$\mathit{\sigma }$	${\mathit{L}}_1$	$\mathit{\alpha }$	${\mathit{\tau }}_1$	$\mathit{\rho }$	$\mathit{\tau }$
Front wheel controller	80	1000	1000	50	5	0.2	50	1.1	5	1	2.5
Rear wheel controller	50	800	1000	100	5	0.2	20	1.1	5	1	2

.

4.2. Simulation Analysis

(1)

Dry asphalt pavement condition

The high intensity braking simulation experiment is carried out on the dry asphalt pavement with the peak road friction coefficient of 1.17. The initial speed of EV is set to 120 km/s and the ideal slip ratio is set as 0.17. The driver model is set up so that the driver presses the brake pedal at the beginning of the simulation. Dynamic information is provided to the slip rate control system. The simulation results are shown in Figure 4.

As can be observed in Figure 4, the slip ratio under the four controllers tends to be stable after varying degrees of fluctuation, and the optimal slip ratio of the current road surface can be quickly tracked. Due to the influence of vertical load, the slip ratio control of the front and rear wheels is slightly different but remains at the optimal slip ratio. As shown in Figure 4a,c, the proposed scheme can accurately provide the required braking torque during deceleration. EV and wheel speeds are always maintained. Under this high road friction condition, the EV takes only 3.48 s to complete the braking process.

Table 4. Numerical evaluation of four control strategies on dry asphalt pavements.

Evaluation Metric	The Proposed Scheme	PI	STSMC	FOSMC
The braking distance	58.05 m	58.39 m	58.16 m	58.14 m
The peak slip ratio	0.211	0.309	0.218	0.28
The response speed	0.12 s	1.18 s	0.53 s	0.02 s

summarizes the numerical evaluation of four control strategies for EV on dry asphalt.

(2)

Wet asphalt pavement condition

The medium strength braking simulation experiment of EV is carried out on the wet asphalt pavement. The peak road friction coefficient is set to 0.8013, the initial speed of EV is set to 80 km/s and the ideal slip ratio is set as 0.13. The simulation results are shown in Figure 5.

From Figure 5, it can be concluded that each controller can achieve better slip ratio control under wet asphalt pavement. As can be seen in Figure 5a, the proposed scheme can control the target slip ratio extremely accurately with fast response and very small slip ratio peaks under this road condition. The whole deceleration process takes only 2.87 s due to the moderate braking intensity. From Figure 5e, the peak front wheel slip ratio of the EV based on STSMC is higher under wet asphalt compared to the first condition. This is because the STSMC is not self-adjusting and always handles different simulation conditions with fixed parameters. The numerical evaluation of four control strategies for EV on wet asphalt is summarized in

Table 5. Numerical evaluation of four control strategies on wet asphalt pavements.

Evaluation Metric	The Proposed Scheme	PI	STSMC	FOSMC
The braking distance	32.15 m	32.39 m	32.32 m	32.23 m
The peak slip ratio	0.133	0.298	0.265	0.579
The response speed	0.09 s	1.07 s	0.575 s	0.575 s

.

(3)

Snow road condition

Simulation experiments of low intensity braking are carried out on the snow road with the peak road friction coefficient of 0.1907. The initial speed of EV is 40 km/s and the ideal slip ratio is 0.06. The simulation results are shown in Figure 6.

As shown in Figure 6, the slip ratio response speed of the four controllers on snow is improved compared to the two previously mentioned road surfaces. From Figure 6a, it can be seen that the ideal wheel slip ratio can be accurately tracked under the proposed scheme. However, due to the low road friction coefficient of the snow road, the tire is not able to obtain sufficient utilization coefficient from the road. As a result, EV can only achieve relatively small braking torques. From Figure 6c, it can be seen that the whole deceleration process is long, about 6.29 s. As shown in Figure 6f, the FOSMC under snow road has the largest jitter amplitude and the highest peak slip ratio compared to the previous two conditions. This is due to the external characteristics and fast response characteristics of the electric motor. Under low road friction coefficient, the FOSMC is difficult to control the slip ratio, which can easily lead to wheel lock.

Table 6. Numerical evaluation of the four control strategies on snow road.

Evaluation Metric	The Proposed Scheme	PI	STSMC	FOSMC
The braking distance	34.71 m	35.1 m	34.84 m	36.31 m
The peak slip ratio	0.099	0.343	0.279	0.628
The response speed	0.04 s	1.13 s	0.332 s	0.02 s

summarizes the numerical evaluation of four control strategies on the snow road.

Through the analysis of the control effects of the above four methods under three simulation conditions, we can have the following discussions. The slip ratio response speed of EV under traditional PI control is the slowest and the first peak slip ratio is the highest. This is because the traditional PI control has to deal with the dynamic process of many changes with the same mode and parameters. Thus, it is difficult to solve the contradiction between stationarity, rapidity and accuracy. However, the traditional PI control also has the advantage of no steady-state error. When it reaches the desired slip rate, the slip ratio can be tracked smoothly and accurately enough. The slip ratio of EV under FOSMC has the fastest response speed, which can reach the desired slip ratio faster than other control methods. However, the whole control process is affected by considerable chattering. This is because the braking torque is provided by the electric motor, which is characterized by high frequency and high amplitude. The chattering is well suppressed under STSMC. This is because the STSMC places the sign function in the higher derivative of the sliding mode variable. However, this may result in poor control under some test conditions because fixed control parameters are used under different operating conditions.

Compared with STSMC, the proposed scheme has improved the control effects in terms of both braking distances, the first peak of slip ratio and response time. This is due to the fact that the proposed scheme introduces adaptive control gains that can be continuously optimized with the control process, resulting in better control results in terms of driving safety and control quality of EV.

5. Conclusions

In this paper, the adaptive second-order sliding mode slip ratio control scheme for the full EV equipped with individual IWM for each wheel is proposed, and the super-twisting observer is designed to observe the vehicle speed. The simulation results demonstrated that the proposed scheme could prevent the wheels from locking during driving and improve the control quality. The following are some summary comments:

(1)

The barrier function-based adaptive second-order sliding mode slip ratio controller not only provides the required braking performance, but also has a better effect in accurately tracking the slip ratio.

(2)

The super-twisting observer can accurately track the wheel speed, and its response speed is fast and its chattering is low.

(3)

Simulation results show that the proposed scheme can achieve better control results in terms of driving safety and control quality.

Future work will mainly focus on experimental testing of the proposed scheme. EV tests on snow and ice will be further evaluated as conditions permit.