An Energy Efficient Control Strategy for Electric Vehicle Driven by In-Wheel-Motors Based on Discrete Adaptive Sliding Mode Control

Abstract

This paper presents an energy-efficient control strategy for electric vehicles (EVs) driven by in-wheel-motors (IWMs) based on discrete adaptive sliding mode control (DASMC). The nonlinear vehicle model, tire model and IWM model are established at first to represent the operation mechanism of the whole system. Based on the modeling, two virtual control variables are used to represent the longitudinal and yaw control efforts to coordinate the vehicle motion control. Then DASMC method is applied to calculate the required total driving torque and yaw moment, which can improve the tracking performance as well as the system robustness. According to the vehicle nonlinear model, the additional yaw moment can be expressed as a function of longitudinal and lateral tire forces. For further control scheme development, a tire force estimator using an unscented Kalman filter is designed to estimate real-time tire forces. On these bases, energy efficient torque allocation method is developed to distribute the total driving torque and differential torque to each IWM, considering the motor energy consumption, the tire slip energy consumption, and the brake energy recovery. Simulation results of the proposed control strategy using the co-platform of Matlab/Simulink and CarSim® demonstrate that it can accomplish vehicle motion control in a coordinated and economic way.

1 Introduction

Electric vehicles (EVs) have been considered as a substitution for the traditional vehicle with an internal combustion engine for the advantages of clean energy sources and emissions [1,2,3]. EVs driven by in-wheel-motors (IWMs) have been considered as a promising architecture for their noticeable advantages compared with other kinds of EVs [4, 5]. Firstly, the elimination of the transmission mechanism can save the producing cost and make more space for drivers and passengers. Secondly, the application of motorized wheels can improve motor-drive operation efficiency [6,7,8].

Besides the merits mentioned above, another significant advantage of EV driven by IWMs is that comprehensive performance can be elevated because the independent driving approach makes it possible to accomplish integrated optimization and control, then obtain more flexible responses under different driving conditions [9, 10]. Some prior researches have been done in this area. Yang et al. [11] proposed a current distribution control for dual directly driven wheel motors for EVs. They considered the status difference between two IWMs caused by fabricating qualities and different aging rates and so on, and then developed an internal controller that serves to distribute the current, instead of torque, to the driving wheels, thereby enhancing the robustness and stability of the system. Demirci et al. [12] proposed a control method for IWM drive EV based on the direct yaw-moment control, giving the optimized wheel force distribution as well as the coordination control of the hydraulic braking and the motor torque, which improves the stability of the four-wheel-drive electric vehicle effectively. Wu et al. [13] presented a layered vehicle dynamic control system, which is composed of an adaptive optimal control allocation method using neural networks for optimal distribution of tire forces and the sliding mode yaw moment observer for robust control of yaw dynamics to solve the stability control problem. Hu et al. [10] utilized the front-wheel differential drive-assist steering to achieve path-following control for independently actuated electric autonomous ground vehicles. The differential torque between the left and right wheels can be utilized to actuate the front wheels as the sole steering power when the regular steering system fails, avoiding dangerous consequences.

Moreover, torque distribution, namely distributing the IWM torque properly to accomplish the given targets such as high dynamic demand, driving stability demand or good energy efficiency is also an important issue. A lot of reports on this subject and relevant studies have been published. Li et al. [14] presented an ideal force distribution control method for the EV based on the friction circle of tire force, making the front and rear wheels reach the adhesion limits at the same time in different conditions. Another optimal torque distribution for EV driven by IWMs was introduced by Zhang et al. [15]. The linear quadratic regulator via a weighted least square method was used to calculate the required longitudinal force of each wheel. Then the IWM torques were obtained by a tire slip ratio controller because they considered the longitudinal force of each wheel was decided by the tire slip ratio. To increase the cruising range of EVs equipped with front and rear in-wheel-motors, an optimal torque distribution algorithm for longitudinal motion by considering the transfer of weight between front and rear axles and motor losses was proposed by Wang et al. [16]. The EV was modeled as a linear-time-invariant system with generalized frequency variables and then the output power of IWM was modeled as a convex function of the distribution ratio, providing an added value for front-rear-independent-drive EVs. Li et al. [17] proposed a coordinated control algorithm of vehicle dynamics performance and energy consumption for an EV driven by four IWMs and steered by two steer-by-wire systems. In their study, multi controllers were designed for each subsystem at first and then a rule-based coordinated control scheme is developed according to vehicle driving states to accomplish the whole control target.

According to the abovementioned studies, an energy efficient control strategy for the electric vehicle driven by in-wheel-motors is designed in this paper. The overall control scheme is displayed in Figure 1. Different from previous research studies, we are aiming to obtain a good comprehensive vehicle performance in both dynamic motion control and energy efficiency. The overall control scheme consists of three parts. The first part is the vehicle motion controller using adaptive discrete sliding mode control (ADSMC) algorithm to calculate the longitudinal and yaw control efforts, guaranteeing good dynamics performance and stability. The second part is the lateral tire force estimation. Rather than linear tire force calculation, an unscented Kalman filter (UKF) is applied considering the tire nonlinear characteristics because the yaw moment is closely related to the tire lateral force, so precise estimation can contribute to further torque allocation. In the third part, the longitudinal and yaw control efforts will be distributed to each IWM considering the energy efficiency. More specifically, the optimal allocator is designed considering IWM energy consumption, tire slip energy consumption and brake energy recovery.

Figure 1

Energy efficient control strategy for EV driven by IWMs based on DASMC

The rest of this paper is organized as follows. The vehicle nonlinear model and IWM model are built in Section 2, followed by the details about proposed control for IWM driven EV based on optimal torque allocation are presented in Section 3. Then the co-simulation results based on Matlab/Simulink and CarSim® are presented in Section 4, together with the related analyses. Finally, the conclusion will be summarized in Section 5.

2 Modeling

The vehicle nonlinear dynamic model, tire model and IWM model are constructed for further analysis and subsequent control strategy design.

2.1 Vehicle Dynamic Model

In this study, only longitudinal, lateral and yaw dynamics are concerned. A schematic diagram of the vehicle three degree-of-freedom nonlinear model [18] with four independently driven IWMs is shown in Figure 2.

Figure 2

Schematic diagram of vehicle nonlinear model

The longitudinal, lateral and yaw motions of this model can be expressed as

$$\left\{ \begin{gathered} \dot{v}_{x} = v_{y} \gamma - F_{w} - F_{f} + \frac{1}{m}\sum F_{x} , \hfill \\ \dot{v}_{y} = - v_{x} \gamma + \frac{1}{m}\sum F_{y} , \hfill \\ \dot{\gamma } = \frac{1}{{I_{z} }}M_{z} , \hfill \\ \end{gathered} \right.$$

(1)

where v_x, v_y and γ are the longitudinal velocity, lateral velocity, and yaw rate of the vehicle, respectively; $m$ is the vehicle total mass; I_z is the vehicle yaw inertia; ΣF_x is the total longitudinal tire force; ΣF_y is the total lateral tire force; M_z is the total yaw moment provided by four wheels; F_w and F_f are the aerodynamic resistance and rolling resistance, respectively, which can be expressed as

$$F_{w} = \frac{1}{2}C_{d} \rho Av_{x}^{2} ,$$

(2)

$$F_{f} = fmg,$$

(3)

where C_d is the aerodynamic resistance coefficient; ρ is the air density; A is the windward area; f is the rolling resistance coefficient; g is the gravity acceleration.

According to Figure 2, ΣF_x, ΣF_y and M_z can be written as

$$\left\{ \begin{gathered} \sum F_{x} = (F_{xfl} + F_{xfr} )\cos \delta_{f} - (F_{yfl} + F_{yfr} )\sin \delta_{f} + F_{xrl} + F_{xrr} , \hfill \\ \sum F_{y} = (F_{yfl} + F_{yfr} )\cos \delta_{f} + (F_{xfl} + F_{xfr} )\sin \delta_{f} + F_{yrl} + F_{yrr} , \hfill \\ M_{z} = c[(F_{xfr} - F_{xfl} )\cos \delta_{f} + (F_{yfl} - F_{yfr} )\sin \delta_{f} - F_{xrl} + F_{xrr} ] \hfill \\ \begin{array}{*{20}c} {} & { + a[(F_{yfl} + F_{yfr} )\cos \delta_{f} + (F_{xfl} + F_{xfr} )\sin \delta_{f} ] - b(F_{yrl} + F_{yrr} ),} \\ \end{array} \hfill \\ \end{gathered} \right.$$

(4)

where δ_f is the steering wheel angle; F_xi is the longitudinal tire force of the ith wheel (i = f l, f r, rl, rr); F_yi (i = f l, f r, rl, rr) represents the lateral tire force of the ith wheel; a and b are the distances from the front and the rear axle to the center of gravity, respectively; c is half of the track width. The longitudinal tire force can be calculated by the wheel rotational dynamics equation, which is

$$\dot{\omega }_{i} = \frac{1}{{I_{w} }}T_{i} - \frac{{R_{w} }}{{I_{w} }}F_{xi} ,\begin{array}{*{20}c} {} & {\left( {i = fl,fr,rl,rr} \right)} \\ \end{array} ,$$

(5)

where T_i and ω_i are the driving torque and the angular speed of the ith wheel; I_w and R_w are the wheel moment of inertia and tire rolling radius, respectively. T_i is the ith IWM output torque.

2.2 Tire Model

The tire forces can be expressed by a Dugoff tire model [19] as follows

$$F_{xi} = \mu F_{zi} k_{x} \frac{{\lambda_{i} }}{{1 + \lambda_{i} }}f(L_{i} ),$$

(6)

$$F_{yi} = \mu F_{zi} k_{y} \frac{{\tan \alpha_{i} }}{{1 + \alpha_{i} }}f(L_{i} ),$$

(7)

$$f(L_{i} ) = \left\{ {\begin{array}{*{20}c} {(2 - L_{i} )L_{i} } \\ 1 \\ \end{array} } \right.\begin{array}{*{20}c} {} \\ {} \\ \end{array} \begin{array}{*{20}c} {} \\ {} \\ \end{array} \begin{array}{*{20}c} {L_{i} \le 1,} \\ {L_{i} > 1,} \\ \end{array}$$

(8)

$$L_{i} = \frac{{(1 - \lambda_{i} )(1 - \varepsilon v_{x} \sqrt {k_{x}^{2} \cdot \lambda_{i}^{2} + k_{y}^{2} \cdot \tan^{2} \alpha_{i} } )}}{{2\sqrt {k_{x}^{2} \cdot \lambda_{i}^{2} + k_{y}^{2} \cdot \tan^{2} \alpha_{i} } }},$$

(9)

where $k_{x}$ is the longitudinal stiffness; $k_{y}$ is the lateral stiffness; $\mu$ is the road adhesion coefficient; $\alpha_{i}$ is the sideslip angle; $\lambda_{i}^{{}}$ is the longitudinal slip rate; $F_{zi}$ is the vertical load of the tire. More specifically, $\alpha_{i}$, $\lambda_{i}^{{}}$ and $F_{zi}$ can be written as

$$\left\{ \begin{gathered} \alpha_{fl} = \alpha_{fr} = - \delta_{f} + \frac{{a\gamma + v_{y} }}{{v_{x} }}, \hfill \\ \alpha_{rl} = \alpha_{rr} = \frac{{v_{y} - b\gamma }}{{v_{x} }}, \hfill \\ \end{gathered} \right.$$

(10)

$$\lambda_{i} = \frac{{\omega_{i} R_{w} - v_{i} }}{{v_{i} }},$$

(11)

$$\left\{ \begin{gathered} F_{zfi} = (\frac{1}{2}mg + m\frac{{a_{y} h}}{c})\frac{b}{l} - \frac{1}{2}ma_{x} \frac{h}{l}, \hfill \\ F_{zfr} = (\frac{1}{2}mg - m\frac{{a_{y} h}}{c})\frac{b}{l} - \frac{1}{2}ma_{x} \frac{h}{l}, \hfill \\ F_{zrl} = (\frac{1}{2}mg + m\frac{{a_{y} h}}{c})\frac{a}{l} + \frac{1}{2}ma_{x} \frac{h}{l}, \hfill \\ F_{zrr} = (\frac{1}{2}mg - m\frac{{a_{y} h}}{c})\frac{a}{l} + \frac{1}{2}ma_{x} \frac{h}{l}. \hfill \\ \end{gathered} \right.$$

(12)

2.3 IWM Model

Four permanent magnet synchronous motors (PMSMs) are applied as the driving motor [20]. Based on power invariant requirement, the d-q equivalent circuit of a PMSM is displayed as Figure 3.

Figure 3

D-q equivalent circuit of a PMSM

In Figure 3, i_d and i_q denote the d- and q-axis components of armature current, respectively; u_d and u_q denote the d- and q-axis components of terminal voltage, respectively; R_a is the armature winding resistance per phase; R_c is the iron loss resistance; φ_f is the flux linkage generated by permanent magnets; L_d and L_q denote the d- and q-axis components of armature self-inductance, respectively; ω_e is the rotor angular velocity, ω_e=ω_pn; p_n is the number of pole pair.

The voltages of equivalent circuit are derived as

$$\left\{ \begin{gathered} u_{d} = R_{a} i_{d} - \omega_{e} L_{q} i_{qt} , \hfill \\ u_{q} = R_{a} i_{q} + \omega_{e} (L_{d} i_{dt} + \varphi_{f} ). \hfill \\ \end{gathered} \right.$$

(13)

The electromagnetic torque equation can be expressed as

$$T_{e} = p_{n} (\varphi_{f} i_{qt} + (L_{d} - L_{q} )i_{dt} i_{qt} ),$$

(14)

where $L_{d} = L_{q}$ and the d-axis current can be controlled to be zero. Subsequently,

$$T_{e} = p_{n} \varphi_{f} i_{qt} .$$

(15)

Then the input power to a PMSM can be calculated by the following equation:

$$\begin{gathered} P_{in} = u_{d} i_{d} + u_{q} i_{q} \hfill \\ \begin{array}{*{20}c} {} & { = (R_{a} } \\ \end{array} i_{d} - \omega_{e} L_{q} i_{qt} )i_{d} + (R_{a} i_{q} + \omega_{e} L_{d} i_{dt} )i_{q} \hfill \\ \begin{array}{*{20}c} {} & { = R_{a} } \\ \end{array} (i_{d}^{2} + i_{q}^{2} ) + \omega_{e}^{2} (L_{d}^{2} i_{dt}^{2} + L_{q}^{2} i_{qt}^{2} )/R_{c} + \omega_{e} \varphi_{f} i_{qt} . \hfill \\ \end{gathered}$$

(16)

3 Control Design for IWM Driven EV

The control method design is proposed in this section, including vehicle motion controller design, lateral tire force estimation and the optimal energy torque allocation.

3.1 Vehicle Motion Controller Design

Road interferences and parameter uncertainties are not considered in the modeling in Section 2. In order to preserve system stability as well as maintain good system robustness, the ADSMC is applied for vehicle motion control design [21].

Assuming $\sin \delta_{f} \approx \delta_{f} ,\cos \delta_{f} \approx 1$, because of the small magnitude of front wheel steering angle, and substituting Eqs. (2)–(5) into Eq. (1), then Eq. (1) can be derived as

$$\left\{ \begin{gathered} \dot{v}_{x} = v_{y} \gamma + \frac{1}{{mR_{w} }}u_{1} - F_{w} - F_{f} - d_{1} , \hfill \\ \dot{v}_{y} = - v_{x} \gamma + \frac{1}{m}(F_{yfl} + F_{yfr} + F_{yrl} + F_{yrr} ), \hfill \\ \dot{\gamma } = a(F_{yfl} + F_{yfr} ) - b(F_{yrl} + F_{yrr} ) + \frac{c}{{I_{z} R_{w} }}u_{2} + d_{2} , \hfill \\ \end{gathered} \right.$$

(17)

where $u_{1} = T_{fl} + T_{fr} + T_{rl} + T_{rr}$ and $u_{2} = - T_{fl} + T_{fr} - T_{rl} + T_{rr}$ are the virtual control inputs to this nonlinear system; d₁ and d₂ are considered as modeling errors, which can be expressed as

$$d_{1} = - \frac{{I_{w} }}{{mR_{w} }}(\dot{\omega }_{1} + \dot{\omega }_{2} + \dot{\omega }_{3} { + }\dot{\omega }_{4} ),$$

(18)

$$d_{2} = \frac{{I_{w} }}{{I_{z} R_{w} }}(\dot{\omega }_{1} - \dot{\omega }_{2} + \dot{\omega }_{3} - \dot{\omega }_{4} ).$$

(19)

In real industrial applications, the control process is discrete. Furthermore, the real vehicle system is a nonlinear system and the longitudinal and yaw motion control problem proposed in our study is nonlinear. For these reasons, the discrete sliding mode control algorithm was adopted here. To reduce the chattering effect of discrete sliding mode control, the adaptive reaching law was introduced in the discrete sliding mode control. In summary, the most significant advantage of DASMC is that it can deal with the discrete nonlinear control problem and reduce the chattering effect of sliding mode control in the meantime. The vehicle model can be discretized as

$$\left\{ \begin{aligned} v_{x} (k + 1) & = v_{x} (k) + T_{s} (v_{y} (k)\gamma (k) \\ & \quad + \frac{1}{{mR_{w} }}u_{1} (k) - F_{w} (k) - F_{f} (k) - d_{1} (k))), \\ v_{y} (k + 1) & = v_{y} (k) + T_{s} ( - v_{x} (k)\gamma (k) \\ & \quad + \frac{1}{m}(F_{yfl} (k) + F_{yfr} (k) + F_{yrl} (k) + F_{yrr} (k))), \\ \gamma (k + 1) & = \gamma (k) + T_{s} (a(F_{yfl} (k) + F_{yfr} (k)) \\ & \quad - b(F_{yrl} (k) + F_{yrr} (k)) + d_{2} (k) + \frac{c}{{I_{z} R_{w} }}u_{2} (k))), \\ \end{aligned} \right.$$

(20)

The reference longitudinal velocity and yaw rate are defined as $v_{x}^{*}$ and $\gamma^{*}$. Errors between the reference values and the actual values are given as $e_{1} = v_{x}^{*} - v_{x}$ and $e_{2} = v_{\gamma }^{*} - v_{\gamma }$. To realize $e_{1} \to 0$ and $e_{2} \to 0$, the sliding faces are defined as $S_{1} (k) = \rho_{1} e_{1} (k)$ and $S_{2} (k) = \rho_{2} e_{2} (k)$, where $\rho_{1}$ and $\rho_{2}$ are positive defined.

The Lyapunov function candidates are chosen as:

$$\left\{ \begin{gathered} V_{1} (k) = \frac{1}{2}S_{1}^{2} (k), \hfill \\ V_{2} (k) = \frac{1}{2}S_{2}^{2} (k), \hfill \\ \end{gathered} \right.$$

(21)

and then

$$\left\{ \begin{gathered} \Delta V_{1} = S_{1}^{2} (k + 1) - S_{1}^{2} (k), \hfill \\ \Delta V_{2} = S_{2}^{2} (k + 1) - S_{2}^{2} (k). \hfill \\ \end{gathered} \right.$$

(22)

According to the Lyapunov theorem of asymptotic stability, $S_{j} = 0,(j = 1,2)$ is the system asymptotic stability surface, which means any starting state will eventually reach the switching surface $S_{j} = 0,(j = 1,2)$. The reaching condition is taken as

$$S_{j}^{2} (k + 1) < S_{j}^{2} (k), \, (j = 1,2).$$

(23)

When sampling time t_s is small enough, the existing and reaching condition can be derived as

$$\left\{ \begin{gathered} [(S_{i} (k + 1) - S_{i} (k)]{\text{sgn}} (S_{i} (k)) < 0, \hfill \\ [(S_{i} (k + 1) + S_{i} (k)]{\text{sgn}} (S_{i} (k)) > 0, \hfill \\ \end{gathered} \right. \, (j = 1,2).$$

(24)

The exponential reaching law is chosen for this discretized sliding mode control problem, which is

$$\frac{{S_{i} (k + 1) - S_{i} (k)}}{{t_{s} }} = - \varepsilon_{i} {\text{sgn}} (S_{i} (k)) - q_{i} S_{i} (k),j = 1,2,$$

(25)

where $\varepsilon_{j}$>0, q_j>0, 1−q_jt_s>0.

For Eq. (25), it can be derived as

$$\begin{gathered} \left[ {S_{j} \left( {k + 1} \right) - S_{j} \left( k \right)} \right]{\text{sgn}} \left( {S_{j} \left( k \right)} \right) \hfill \\ = \left[ { - q_{j} t_{s} S_{j} \left( k \right) - \varepsilon t_{s} {\text{sgn}} \left( {S_{j} \left( k \right)} \right)} \right]{\text{sgn}} \left( {S_{j} \left( k \right)} \right) \hfill \\ = - q_{i} t_{s} \left| {S_{j} \left( k \right)} \right| - \varepsilon t_{s} < 0\begin{array}{*{20}c} {} & {,j = 1,2} \\ \end{array} . \hfill \\ \end{gathered}$$

(26)

Meanwhile, when sampling time t_s is small enough, $2 - q_{j} t_{s} \gg 0$, then

$$\begin{gathered} [(S_{j} (k + 1) + S_{j} (k)]{\text{sgn}} (S_{j} (k)) \hfill \\ = [(2 - q_{i} t_{s} )S_{j} (k) - \varepsilon t_{s} {\text{sgn}} (S_{j} (k))]{\text{sgn}} (S_{j} (k)\begin{array}{*{20}c} {} & {(j = 1,2).} \\ \end{array} \hfill \\ = (2 - q_{i} t_{s} )\left| {S_{j} (k)} \right| - \varepsilon t_{s} \left| {S_{j} (k)} \right| > 0, \hfill \\ \end{gathered}$$

(27)

Thus, the reaching condition Eq. (23) can be achieved. Substituting Eq. (20) into Eq. (25), the control laws can be derived as

$$\left\{ \begin{gathered} u_{1} (k) = mR_{w} /t_{s} (v_{x}^{*} (k + 1) - v_{x}^{*} (k) + q_{1} t_{s} S_{1} (k) + \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \varepsilon_{1} t_{s} {\text{sgn}} (S_{1} (k)) - t_{s} (v_{y} (k)\gamma (k) - F_{w} (k) - F_{f} (k))), \hfill \\ u_{2} (k) = I_{z} R_{w} /ct_{s} (\gamma^{*} (k + 1) - \gamma^{*} (k) - \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} t_{s} F(k) + \varepsilon_{2} t_{s} {\text{sgn}} (S_{2} (k)) + q_{2} t_{s} S_{2} (k)), \hfill \\ \end{gathered} \right.$$

(28)

where

$$F(k) = a(F_{yfl} (k) + F_{yfr} (k)) - b(F_{yrl} (k) + F_{yrr} (k)) + d_{2} (k).$$

(29)

From Eq. (25),

$$S_{j} (k + 1) = (1 - q_{j} t_{s} - \frac{{\varepsilon_{j} t_{s} }}{{\left| {S_{j} (k)} \right|}})S_{j} (k) = p_{j} S_{j} (k).$$

(30)

Then,

$$\left| {p_{j} } \right| = \frac{{\left| {S_{j} (k + 1)} \right|}}{{\left| {S_{j} (k)} \right|}},p_{j} = 1 - q_{j} t_{s} - \frac{{\varepsilon_{j} t_{s} }}{{\left| {S_{j} (k)} \right|}}.$$

(31)

Based on Eq. (31), there are three circumstances: ① when $\left| {S_{j} (k)} \right| > {{\varepsilon_{j} t_{s} } \mathord{\left/ {\vphantom {{\varepsilon_{j} t_{s} } {(2 - q_{j} t_{s} }}} \right. \kern-0pt} {(2 - q_{j} t_{s} }})$, there is $p_{j} > 1 - q_{j} t_{s} - {{\varepsilon_{j} t_{s} (2 - q_{j} t_{s} )} \mathord{\left/ {\vphantom {{\varepsilon_{j} t_{s} (2 - q_{j} t_{s} )} {\varepsilon_{j} t_{s} }}} \right. \kern-0pt} {\varepsilon_{j} t_{s} }} = - 1$, then $\left| {p_{j} } \right| < 1,\left| {S_{j} (k + 1)} \right| < \left| {S_{j} (k)} \right|,$ which means $\left| {S_{j} (k)} \right|$ is decreasing; ② when $\left| {S_{j} (k)} \right| < {{\varepsilon_{j} t_{s} } \mathord{\left/ {\vphantom {{\varepsilon_{j} t_{s} } {2 - q_{j} t_{s} }}} \right. \kern-0pt} {2 - q_{j} t_{s} }}$, there is $p_{j} < 1 - q_{j} t_{s} - {{\varepsilon_{j} t_{s} (2 - q_{j} t_{s} )} \mathord{\left/ {\vphantom {{\varepsilon_{j} t_{s} (2 - q_{j} t_{s} )} {\varepsilon_{j} t_{s} }}} \right. \kern-0pt} {\varepsilon_{j} t_{s} }} = - 1$, then $\left| {p_{j} } \right| > 1,\left| {S_{j} (k + 1)} \right| > \left| {S_{j} (k)} \right|,$ which means $\left| {S_{j} (k)} \right|$ is increasing; ③ when $\left| {S_{j} (k)} \right| = {{\varepsilon_{j} t_{s} } \mathord{\left/ {\vphantom {{\varepsilon_{j} t_{s} } {(2 - q_{j} t_{s} )}}} \right. \kern-0pt} {(2 - q_{j} t_{s} )}},$ there is $p_{j} = 1 - q_{j} t_{s} - {{\varepsilon_{j} t_{s} (2 - q_{j} t_{s} )} \mathord{\left/ {\vphantom {{\varepsilon_{j} t_{s} (2 - q_{j} t_{s} )} {\varepsilon_{j} t_{s} }}} \right. \kern-0pt} {\varepsilon_{j} t_{s} }} = - 1,$ which means $\left| {S_{j} (k)} \right|$ is chattering. Thus, it can be concluded that the sufficient condition of $\left| {S_{j} (k)} \right|$ decrease is $\left| {S_{j} (k)} \right| > {{\varepsilon_{j} t_{s} } \mathord{\left/ {\vphantom {{\varepsilon_{j} t_{s} } {(2 - q_{j} t_{s} }}} \right. \kern-0pt} {(2 - q_{j} t_{s} }})$. To achieve this, it is required that

$$\varepsilon_{j} < \frac{1}{{t_{s} }}(2 - t_{s} q_{j} )\left| {S_{j} (k)} \right|.$$

(32)

If we take $\varepsilon_{j} = \left| {S_{j} (k)} \right|/2$ and the sampling time meets the requirement of $t_{s} < 4/(1 + 2q_{j} )$, Eq. (32) can be guaranteed. The hyperbolic tangent function tanh(S_j(k)/σ_j) is used to replace the sign switching function sgn(S_j(k)) in Eq. (28) to avoid the chattering effects in practical implementation. Here, σ_j is the boundary layer thicknesses.

3.2 Lateral Tire Force Estimation

According to the control low designed in Eq. (28), tire forces are critical to accomplishing the control target. Tire forces are hard to measure directly by sensors in practice. Analytical estimation is a practical way to obtain real-time tire forces and tire-road adhesion information [22, 23]. Based on the Dugoff tire model introduced in Section 2, an unscented Kalman filter is employed to estimate the real-time tire forces. According to the Dugoff tire model, the factors that determine the tire forces are road adhesion coefficient, tire vertical load, tire lateral stiffness, tire sideslip angle, and wheel slip ratio. Among all these factors, the road adhesion coefficient is the only one that can not be measured or calculated directly. Here we define nominal longitudinal and lateral tire forces $F_{xi}^{0}$ and $F_{yi}^{0}$ to describe the computable part of them, which are expressed as

$$\left\{ \begin{gathered} F_{xi}^{0} = F_{zi} k_{x} \frac{{\lambda_{i} }}{{1 + \lambda_{i} }}f(L_{i} ), \hfill \\ F_{yi}^{0} = F_{zi} k_{y} \frac{{\tan \alpha_{i} }}{{1 + \alpha_{i} }}f(L_{i} ). \hfill \\ \end{gathered} \right.$$

(33)

Then the measurement equations used for estimating are derived as

$$\left\{ \begin{gathered} a_{x} = \frac{1}{m}(\mu_{fl} F_{xfl}^{0} + \mu_{fr} F_{xfr}^{0} + \mu_{rl} F_{xrl}^{0} + \mu_{rr} F_{xrr}^{0} ), \hfill \\ a_{y} = \frac{1}{m}(\mu_{fl} F_{yfl}^{0} + \mu_{fr} F_{yfr}^{0} + \mu_{rl} F_{yrl}^{0} + \mu_{rr} F_{yrr}^{0} ), \hfill \\ \dot{\gamma } = \frac{1}{{I_{z} }}(a(\mu_{fl} F_{yfl}^{0} + \mu_{fr} F_{yfr}^{0} ) - b(\mu_{rl} F_{yrl}^{0} + \mu_{rr} F_{yrr}^{0} ) \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + c( - \mu_{fl} F_{xfl}^{0} + \mu_{fr} F_{xfr}^{0} - \mu_{rl} F_{xrl}^{0} + \mu_{rr} F_{xrr}^{0} )). \hfill \\ \end{gathered} \right.$$

(34)

The details of UKF are displayed in Figure 4, where system states are the four tire road adhesion coefficients, namely ${\varvec{x}} = [\mu_{fl} ,\mu_{fr} ,\mu_{rl} ,\mu_{rr} ]^\text{T}$; the inputs to the system are eight nominal ire forces, namely ${\varvec{u}} = [F_{yfl}^{0} ,F_{yfr}^{0} ,,F_{yrl}^{0} ,F_{yrr}^{0} ,F_{xfl}^{0} ,F_{xfr}^{0} ,F_{xrl}^{0} ,F_{xrr}^{0} ]^\text{T}$; the system outputs are ${\varvec{z}} = [a_{x} ,a_{y} ,\dot{\gamma }]^\text{T}$.

Figure 4

Procedure of UKF

After obtaining the estimated road adhesion coefficients, the lateral tire forces can be calculated by the Dugoff model. The estimation results of the front left tire are shown in Figure 5.

Figure 5

Lateral front left lateral tire force estimation results

In Figure 5, the blue curve is the tire force obtained by the linear tire model, which can be expressed as tire slip angle times tire lateral stiffness; the black curve is the reference value output by CarSim®; the red curve is the estimation result based on the method aforementioned. The result obtained by the linear model is larger than the reference because the lateral stiffness will decrease with the increase of the tire slip angle. It can be shown in Figure 5 that the estimation value matches the reference curve pretty well so it can be applied in the whole control strategy.

3.3 Optimal Energy Efficiency Torque Allocation

According to the former definition, the two virtual control inputs of the vehicle motion control are combinations of four IWM torque outputs. The optimal torque allocation is designed to complete the following tasks: ① minimizing the power consumption; ② satisfying the control requirements.

The inverter power consumption and the mechanical friction power consumption are considered uncontrollable, so the total power consumption can be described as

$$P_{con} = P_{out} + P_{iron} + P_{copper} ,$$

(35)

where P_out is the motor mechanical output; P_iron is the motor iron loss; P_copper is the motor copper loss. More specifically,

$$\begin{gathered} P_{copper} = R_{a} (i_{d}^{2} + i_{q}^{2} ) \hfill \\ \begin{array}{*{20}c} {} & { = R_{a} } \\ \end{array} ((i_{dt} - \frac{{\omega_{e} L_{q} i_{qt} }}{{R_{i} }})^{2} + (i_{qt} + \frac{{\omega_{e} (\varphi_{f} + L_{d} i_{dt} )}}{{R_{i} }})^{2} ), \hfill \\ \end{gathered}$$

(36)

$$\begin{gathered} P_{iron} = R_{i} (i_{di}^{2} + i_{qi}^{2} ) \hfill \\ \begin{array}{*{20}c} {} & { = \frac{{\omega_{e}^{2} L_{d}^{2} i_{qt}^{2} }}{{R_{i} }}} \\ \end{array} + \frac{{\omega_{e}^{2} (\varphi_{f} + L_{d} i_{dt} )^{2} }}{{R_{i} }}, \hfill \\ \end{gathered}$$

(37)

$$P_{out} = T_{e} \omega ,$$

(38)

$$i_{qt} = \frac{{T_{e} }}{{p_{n} \varphi_{f} }}.$$

(39)

In Section 2, it is assumed that $L_{d} = L_{q}$. According to Ref [20],

$$i_{dt} = \frac{{\omega_{e}^{2} L_{d} (R_{a} + R_{i} )\varphi_{f} }}{{R_{a} R_{i}^{2} + \omega_{e}^{2} L_{d}^{2} (R_{a} + R_{i} )}}.$$

(40)

Then the object function of motor power consumption is derived as

$$J_{1} = \left\| {\sum\limits_{i = fl}^{rr} {P_{con,i} } } \right\|^{2} .$$

(41)

The tire slip energy is considered to be important dissipation energy from driving axles to wheels [24]. To make the most use of driving torque, an objective function is introduced to minimize the tire slip energy, which is

$$\min J_{2} = \left\| {P_{tloss} } \right\|^{2} = \sum\limits_{i = fl}^{rr} {(T_{i} (k)\omega_{i} (k) - F_{xi} (k)v_{x} (k))^{2} } .$$

(42)

The IWM can work in both driving and braking modes, while under certain conditions the IWM braking will not be enough. To fully use the braking recovery energy, a parallel braking energy recovery strategy based on feedback braking is applied where the mechanical braking system will start to work when the IWM braking force is saturated. This can be described as

$$T_{t} = \max \{ - 4T_{e\max } ,u_{1} \} ,$$

(43)

$$T_{bm} = u_{1} - \max \{ u_{1} , - 4T_{e\max } \} ,$$

(44)

where $T_{t}$ is total torque provided by IWMs; $T_{e\max }$ is IWM output saturation; $T_{bm}$ is braking torque provided by mechanical braking system.

IWM torque output $T_{e}$ complies with its physical saturation, which is

$$- T_{e\max } \le T_{e} \le T_{e\max } ,$$

(45)

$$T_{e\max } (\omega ) = \left\{ \begin{gathered} T_{\max } ,{\text{ if }}\omega \le {{P_{\max } } \mathord{\left/ {\vphantom {{P_{\max } } {T_{\max } ,}}} \right. \kern-0pt} {T_{\max } ,}} \hfill \\ {{P_{\max } } \mathord{\left/ {\vphantom {{P_{\max } } {\omega ,{\text{ else,}}}}} \right. \kern-0pt} {\omega ,{\text{ else,}}}} \hfill \\ \end{gathered} \right.$$

(46)

where P_max and T_max are max power and max torque output of the IWM; $\omega$ is IWM rotational speed.

The first task can be accomplished by distributing the IWM torque to meet the following constraints:

$$\left\{ \begin{gathered} \Sigma T(k) = T_{fl} (k) + T_{fr} (k) + T_{rl} (k) + T_{rr} (k), \hfill \\ \Delta T(k) = - T_{fl} (k) + T_{fr} (k) - T_{rl} (k) + T_{rr} (k), \hfill \\ \end{gathered} \right.$$

(47)

$$J_{3} (k) = (u_{1} (k) - \sum T(k))^{2} + (u_{2} (k) - \Delta T(k))^{2} ,$$

(48)

where J₃ is the motion control cost function, which is to convert the hard constraints in Eq. (47) into the soft constrain in Eq. (48).

Then the overall cost-function of the optimal allocation is defined as

$$J(k) = \xi_{1} J_{1} (k) + \xi_{2} J_{2} (k) + \xi_{3} J_{3} (k),$$

(49)

where ξ₁, ξ₂ and ξ₃ are weights.

In summary, the optimal torque allocation problem can be reformulated as: minimizing the cost function Eq. (49) under the constraints of Eqs. (43)–(45).

This problem can be solved by the Sequential Quadratic Programming (SQP) proposed in Ref. [25] which can be summarized as follows. For optimal problem:

$$\left\{ {\begin{array}{*{20}c} {\min } & {f(x),} \\ \text{s.t.} & \begin{gathered} h_{i} (x) = 0,i \in E = \{ 1,2, \ldots ,l\} , \hfill \\ g_{i} (x) \ge 0,i \in I = \{ 1,2, \ldots ,m\} , \hfill \\ \end{gathered} \\ \end{array} } \right.$$

(50)

The Lagrange function is defined as

$$L(x,\mu ,\lambda ) = f(x) - \sum\limits_{i \in E} {\mu_{i} h_{i} (x)} - \sum\limits_{i \in I} {\mu_{i} g_{i} (x)} .$$

(51)

And then ${\varvec{A}}^{{\varvec{E}}} {\mathbf{(}}{\varvec{x}}{\mathbf{)}} = \nabla h(x)^\text{T}$, ${\varvec{A}}^{{\varvec{I}}} {\mathbf{(}}{\varvec{x}}{\mathbf{)}} = \nabla g(x)^\text{T}$, ${\varvec{A}}{\mathbf{(}}{\varvec{x}}{\mathbf{)}} = [A^{E} ;A^{I} ]$, $W(x,\mu ,\lambda ) = \nabla_{xx}^{2} L(x,\mu ,\lambda )$.

Initialization step, given initial pair $(x_{0} ,\mu_{0} ,\lambda_{0} ) \in {\mathbb{R}}^{n} \times {\mathbb{R}}^{l} \times {\mathbb{R}}^{m}$ and symmetric positive definite matrix $B_{0} \in {\mathbb{R}}^{n \times n}$, calculate ${\varvec{A}}_{{\mathbf{0}}}^{{\varvec{E}}} {\mathbf{(}}{\varvec{x}}{\mathbf{)}} = \nabla h(x_{0} )^\text{T}$, ${\varvec{A}}_{{\mathbf{0}}}^{{\varvec{I}}} {\mathbf{(}}{\varvec{x}}{\mathbf{)}} = \nabla g(x_{0} )^\text{T}$, $A_{0} = [A_{0}^{E} ;A_{0}^{I} ]$. Choose parameter $\eta \in (0,1/2)$ and the allowable errors $0 \le \varepsilon_{1} ,\varepsilon_{2} \ll 1.$ Set $k: = 0$.

• Step 1, solve the subproblem

  $$\left\{ {\begin{array}{*{20}c} {\min } & {\frac{1}{2}d^\text{T} B_{k} d + \nabla f(x_{k} )^\text{T} d,} \\ \text{s.t.} & \begin{gathered} h(x_{k} ) + A_{k}^{E} d = 0, \hfill \\ g(x_{k} ) + A_{k}^{I} d \ge 0, \hfill \\ \end{gathered} \\ \end{array} } \right.$$

  (52)

  to get the optimal solution d_k.

• Step 2, if $\left\| {d_{k} } \right\|_{1} \le \varepsilon_{1}$ and $\left\| {h_{k} } \right\|_{1} + \left\| {(g_{k} )_{\_} } \right\|_{1} \le \varepsilon_{2}$, stop and get an approximate KT point of the original problem $(x_{k} ,\mu_{k} ,\lambda_{k} )$.

• Step 3, for a certain cost function $\phi (x,\sigma )$, choose a penalty parameter $\sigma_{k}$ to make d_k is in the falling direction of $\phi (x,\sigma )$ at the point of x_k.

• Step 4, Armijo searching. Make m_k be the minimum nonnegative integer m satisfying the following inequality:

  $$\phi (x_{k} + \rho^{m} d_{k} ,\sigma_{k} ) - \phi (x_{k} ,\sigma_{k} ) \le \eta \rho^{m} \phi^{^{\prime}} (x_{k} ,\sigma ;d_{k} ),$$

  (53)

  then choose $\alpha_{k} : = \rho^{{m_{k} }}$,$x_{k + 1} : = x_{k} + \alpha_{k} d_{k}$.

• Step 5, calculate

  ${\varvec{A}}_{{\user2{k + }{\mathbf{1}}}}^{{\varvec{E}}} = \nabla h(x_{k + 1} )^\text{T},$ ${\varvec{A}}_{{\user2{k + }{\mathbf{1}}}}^{{\varvec{I}}} = \nabla g(x_{k + 1} )^\text{T}$, ${\varvec{A}}_{{\user2{k + }{\mathbf{1}}}} = [A_{k + 1}^{E} ;A_{k + 1}^{I} ]$ and the least-squares multiplier

  $$\left[ \begin{gathered} \mu_{k + 1} \hfill \\ \lambda_{k + 1} \hfill \\ \end{gathered} \right] = [A_{k + 1} A_{k + 1}^\text{T} ]^{ - 1} A_{k + 1} \nabla f_{k + 1} .$$

  (54)

• Step 6, correct B_k to B_k+1. Set

  $$s_{k} = \alpha_{k} d_{k} ,y_{k} = \nabla_{x} L(x_{k + 1} ,\mu_{k + 1} ,\lambda_{k + 1} ) - \nabla_{x} L(x_{k} ,\mu_{k + 1} ,\lambda_{k + 1} ),$$

  (55)
  $$B_{k + 1} = B_{k} - \frac{{B_{k} s_{k} s_{k}^\text{T} B_{k} }}{{s_{k}^\text{T} B_{k} s_{k} }} + \frac{{z_{k} z_{k}^\text{T} }}{{s_{k}^\text{T} z_{k} }},$$

  (56)

  where

  $$z_{k} = \theta_{k} y_{k} + (1 - \theta_{k} )B_{k} s_{k} ,$$

  (57)
  $$\theta_{k} = \left\{ {\begin{array}{*{20}c} {1,} & {f \, s_{k}^\text{T} y_{k} \ge 0.2s_{k}^\text{T} B_{k} s_{k} ,} \\ {\frac{{0.8s_{k}^\text{T} B_{k} s_{k} }}{{s_{k}^\text{T} B_{k} s_{k} - s_{k}^\text{T} y_{k} }},} & {if \, s_{k}^\text{T} y_{k} \ge 0.2s_{k}^\text{T} B_{k} s_{k} {. }} \\ \end{array} } \right.$$

  (58)

• Step 7, update k as k+1, then switch to step 1.

4 Simulation Results and Analyses

Simulations are conducted to verify the effectiveness of the proposed control strategy based on the co-simulation platform of CarSim® and Matlab/Simulink. The model is established based on a high-fidelity full-vehicle model in CarSim® and all the powertrain components are replaced by four IWMs. Meanwhile, simulation results by the torque distribution method shown in Eq. (59) are used as comparisons to validate the improvement of integrated optimal torque allocation. The parameters of the vehicle used in simulations are shown in Table 1.

$$\left\{ \begin{gathered} T_{fld} = \frac{b}{2(a + b)}\sum T{ - }\frac{1}{4}\Delta T, \hfill \\ T_{frd} = \frac{b}{2(a + b)}\sum T + \frac{1}{4}\Delta T, \hfill \\ T_{rld} = \frac{a}{2(a + b)}\sum T{ - }\frac{1}{4}\Delta T, \hfill \\ T_{rrd} = \frac{a}{2(a + b)}\sum T + \frac{1}{4}\Delta T. \hfill \\ \end{gathered} \right.$$

(59)

Table 1 Vehicle parameters

Double line change and new European driving cycle (NEDC) are conducted in simulations to verify the motion control performance and the economic efficiency of the proposed control. The results are shown in the subsequent sections.

4.1 Double Line Change

The control target of this study is to coordinate the motion control of EVs driven by IWMs in an effective way. The first simulation is the double line change with longitudinal velocity increase. The vehicle is set to accelerate from 15 to 20 m/s in 40 s and the double line change track is shown below. The air density is set as 1.206 kg/m³ during the simulation. Two comparison simulations are also conducted; one is conducted by using the normal sliding mode control with proposed torque optimal allocation, and the other is conducted by using the DASMC and the allocation method based on tire load. Simulation results are displayed in Figures 6 and 7.

Figure 6

Simulation results of longitudinal velocity

Figure 7

Simulation results of yaw rate

From Figures 6 and 7, it can be concluded that the proposed control method can accomplish the longitudinal and yaw motion control successfully. Moreover, the ADSMC outperforms the DSMC in suppressing chattering according to Figure 7.

The load-based allocation (LBA) method in Eq. (50) is applied as a comparison to the optimal allocation (OA) method. The results are shown in Figure 8.

Figure 8

Simulation results of IWM torques

In Figure 8, it can be seen that the yaw motion control is accomplished by differential torques between left and right IWMS in both two allocation methods. While the allocation ratios between front and rear IWMs are different. To quantitatively illustrate the improvement of the proposed control strategy more specifically, Table 2 shows the performances under different control strategies.

Table 2 Performances under different control strategies

The evaluation results are obtained based on the aforementioned simulations. It can be concluded that with the ADSMC, the tracking performance has been improved compared with the DSMC for its better chattering suppression performance. Furthermore, the power consumption is also decreased by ADSMC according to Table 2, because the frequent torque changes will cause more power consumption. According to the second and third rows of Table 2, the power consumption by OA is decreased as expected but the tracking performances of yaw rate and longitudinal velocity are slightly worse than LBA. Combining with the Eqs. (48)–(50), it can be concluded that the OA sacrifices the tracking performance to obtain lower energy consumption.

4.2 NEDC

Although the above simulation results can verify the coordinated motion control of the proposed control strategy, they are still not enough to demonstrate energy efficiency in such a short time interval. To fully prove the control performance of the proposed strategy, the NEDC is conducted. The vehicle is set to drive in a single line without steering. Similarly, simulations using the abovementioned control strategy ADSMC+LBA are conducted as the comparisons. The simulation results are shown in Figures 9 and 10. The control performance is displayed in Table 3.

Figure 9

Simulation results of longitudinal velocity

Figure 10

Simulation results of total power consumption

Table 3 Performances under different allocation methods

In Figure 9, it can be seen that both control strategies can track the reference longitudinal velocity in the whole-time range. While in Figure 10, the torque outputs with OA meet the torque output limit requirement and also have fewer sharp changes. According to Table 3, the control strategy with LBA shows better tracking performance at the cost of more power consumption and less energy efficiency. While even the control strategy with OA shows a bigger tracking error, it is still effective for the maximum absolute tracking error is 0.4960 which is totally acceptable. This indicates that the proposed control strategy can generate a proper control input to the system in an energy-efficient way, precisely to track the reference outputs to accomplish vehicle-coordinated motion control.

Based on all the simulation results exhibited above, the effectiveness of such an energy-efficient control strategy for EVs driven by IWMs based on DASMC is demonstrated.

5 Conclusions

An energy-efficient control strategy for EVs driven by IWMs based on DASMC is proposed in this study. Models are established firstly to demonstrate the operation mechanism of the whole system and two virtual control variables are used to describe the longitudinal and yaw control efforts to complete the vehicle coordinate motion control. Then DASMC method is applied to calculate the required total driving torque and yaw moment. A tire force estimator using UKF is designed to estimate real-time lateral tire forces used in the control scheme. Based on all the abovementioned factors, energy efficient torque allocation method is developed to distribute the total driving torque and differential torque to each IWM. Simulation results of the proposed control strategy using the co-platform of Matlab/Simulink and CarSim® demonstrate that this study can accomplish the vehicle motion control in a coordinated and economic way and improve the tracking performance as well as the system robustness.