Control Strategy of Dual-Disc Electromagnetic–EMB Composite Braking System Based on Hybrid Systems

Abstract

In this study, to address the problems of the redundant safety and mass production of electro-mechanical braking (EMB) structures that are widely used in distributed drive electric vehicles (DDEV), we designed a compact dual-disc electromagnetic–EMB composite brake. The composite brake embeds an electromagnetic brake into the original friction disc, which realizes an organic combination of the friction and electromagnetic brakes. Electromagnetic braking has the advantages of no friction, a rapid response, and a high-speed braking effect, which can effectively improve the reliability and mechanical redundancy of composite braking systems. The braking system comprises regenerative, electromagnetic, and friction braking, which are typical hybrid systems. We designed a mode-switching control strategy for a composite braking system based on the hybrid control theory. MATLAB/Simulink were used to model each system and set different simulation conditions. The simulation results showed that, under different working conditions, the hybrid automata control strategy had a fast response speed, small overshoot error, and adapted to different road conditions. The feasibility of the redundant design of the electromagnetic–friction–regenerative composite braking structure and the rationality of the hybrid automata control strategy design were verified.

1. Introduction

In the reconstruction of intelligent chassis technology systems for electric vehicles, the development of a key component such as the braking system has attracted significant attention because it plays a key role in vehicle safety and performance. With revolutionary progress in braking technology, EMB systems have received extensive attention in recent years [1]. As a real brake-by-wire system, EMB has several advantages over traditional brake systems [2]. First, the EMB system has a simple structure and can quickly adapt to different types of vehicles, which not only helps optimize vehicle dynamics performance, but also provides greater flexibility for vehicle design. In addition, EMB has a rapid braking response, which is critical in emergency braking situations, potentially preventing collisions and improving road safety. Furthermore, EMB is beneficial for integrated control. It can be seamlessly integrated with other vehicle control systems, such as the anti-lock braking system (ABS), electronic stability program (ESP), and advanced driver assistance system (ADAS), to achieve more comprehensive and intelligent vehicle control [3]. However, the development of EMB systems remains a challenge. They completely cancel the mechanical connection between the brake pedal and brake actuator. When designing a brake system, it is essential to comply with the functional safety requirements of the ASIL D level and the UNECE-R13H brake system standards formulated by the Economic Commission for Europe (UNECE). Technical conditions, along with factors such as the manufacturing costs, regulations, and standards, are the main challenges that restrict the mass production of EMB systems, and redundant safety issues have become prominent [4,5,6]. Researchers are actively designing reasonable and effective EMB redundancy schemes [7].

The mainstream EMB redundancy scheme includes two aspects: the electronic control system and actuator redundancy. The redundancy of the electronic control system includes that of the battery system and electronic control unit (ECU). For automobile safety integrity level ASIL D compliance, a power supply system requires a redundant design with power distribution and diagnostic functions [8]. Many EMB system designs have adopted an X-type dual-power-supply redundant topology. Weiberl [9] proposed a support capacitor local energy memory for short-term power supply backup. The BYD automotive team invented heterogeneous power-supply redundancy using high- and low-voltage batteries [10]. ECU redundancy is commonly categorized into dual, triple, or quad units [11]. The main schemes of redundant actuators are the double-motor single-caliper configuration and double-motor double-deceleration configuration [12,13]. BOSCH ’s EMB patent plan, announced in 2022, uses a dual-motor scheme. The transmission mechanism on the same side as the actuating motor and braking actuator enables smaller axial dimensions to facilitate integration within the hub [14]. Wando’s EMB patent scheme [15] adopted a belt-driven brake caliper structure. The brake caliper was equipped with a parking mechanism. After clamping the caliper, the brake assembly realized parking through an internal mechanical structure, which improved the redundancy of the EMB.

In addition to the methods of redundant backup of hardware and software, the design of composite braking systems is a promising research direction to improve the redundant safety of braking systems. Significant research has been conducted by domestic and foreign enterprises and institutions of regenerative–friction composite braking and electromagnetic–friction composite braking [16,17,18]. Especially for electromagnetic–friction composite braking, the advantages of electromagnetic brakes include frictionless operation, fast response speed, and suitability for application scenarios that require fast braking action, which provides a new idea for EMB redundancy safety design [19]. Gay et al. [20] designed a braking system integrated with electromagnetic and friction braking. The brake caliper and electromagnetic brake shared a brake disc, which reduced the axial distance of the composite braking system and facilitated integrated installation. In the electromagnetic and friction composite braking system developed by Song et al. [21], the brake caliper and U-shaped iron core were, respectively, installed on both sides of the brake disc. Energizing the coil created a magnetic field encircling the brake disc, thus generating a braking torque. Hu [22] proposed coordinated control of an electromagnetic braking system and friction braking system based on speed, and they designed two fuzzy control strategies for a composite braking system to realize ABS control. Professor He Ren’s group studied the eddy current retarders of commercial vehicles, as well as the application of electromagnetic–friction composite braking in passenger cars. They focused on structural design and theoretical research, which created a foundation for the application of electromagnetic–friction composite braking [23,24]. Liu [25] and Wouterse et al. [26] noted a critical speed in the process of electromagnetic braking, as well as two linear intervals with different slopes on both sides of the critical speed. Through appropriate controller design, the electric brake could be effectively used for composite braking. Zhu [27] proposed a braking mode switching control strategy for an electromagnetic, regenerative, and friction composite braking system of a DDEV based on the hybrid control theory.

In summary, the unique structure and redundancy requirements of EMB systems have brought new challenges to the integration of electromagnetic brakes and the realization of composite braking. A regenerative–electromagnetic–EMB composite braking scheme still lacks mature research. Therefore, based on the research on EMB redundant safety design combined with the structural characteristics of EMB, this study proposes a design scheme for a dual-disc electromagnetic–EMB composite brake that can realize the coordinated control of electromagnetic braking and the EMB braking function. Subsequently, a model of each component of the composite braking system was established, including the vehicle longitudinal dynamics, EMB braking, electromagnetic braking, regenerative braking, and tire models. Based on the hybrid system theory, a hybrid system model was developed, and a control strategy flowchart was designed to realize the coordinated operation control of the composite braking system under different braking modes. Finally, a simulation environment was established, and various braking conditions were designed. The control strategy has a fast response speed, small overshoot error, and robustness to different road conditions, which can provide a reference for solving the redundancy safety problem of EMB systems and could improve the ride comfort of composite braking vehicles.

2. Research Objects and Control Structure

2.1. Dual-Disc Electromagnetic and EMB Composite Brake Structure

The electromagnetic–EMB composite braking system designed in this study combines an electronic mechanical braking system with an independent voltage-source electromagnetic braking system, as shown in Figure 1. The electromagnetic braking system comprises an electromagnetic actuator (3), a battery (2), cable (1), and electromagnetic brake coils (20). The electromagnetic actuator regulates the current within the electromagnetic brake coils by following instructions from the composite braking system controller, enabling precise torque control. The size of the integrated brake friction disc is the same as that of the original EMB brake disc. The outer and inner parts serve as the EMB brake disc, and the inner part as the electromagnetic brake disc, respectively. The structural features of the integrated braking system in Figure 1 are as follows:

➀

The two systems share a brake bracket (22). Parts, such as the EMB transmission mechanism (5, 6, 11, and 17) and friction plate (23), are fixed through this shared bracket. The wheel is connected to the composite brake disc (21).

➁

The composite brake disc is mechanically locked to maintain a constant disc distance, thereby maintaining the electromagnetic braking gap and enhancing the reliability of the brake system.

➂

During braking, the EMB clamps the outer side of the composite brake disc, whereas the inner side is subjected to an electromagnetic braking force. In the same service cycle, the friction braking time is reduced, the service life of the outer friction brake disc is prolonged, and metal dust pollution is also reduced.

➃

The heat diversion port of the composite brake disc was designed in the middle. This design can dissipate heat from the center of the disc, significantly improving the heat dissipation effect.

2.2. Control Structure of the Electromagnetic–EMB–Regenerative Composite Braking System

The control structure relationship of the composite braking system of the DDEV is shown in Figure 2. The control structure presents the control strategy of the composite braking system and its relationship with the parameters of the vehicle. This control strategy switches among the different braking modes based on input signals, including the braking strength z, battery state of charge (SOC), wheel speed n, vehicle speed v, and braking torques $T_{EMB}$, $T_e$, and $T_R$. For each mode, a dedicated control algorithm was designed. As shown in Figure 2, Brake Controllers 1 to 7 correspond to Braking Modes $m_1$ to $m_7$ in the brake mode judge controller, respectively. The composite brake controller precisely issues control commands ($reg$, $ele$, $fri$, $U_R$, $I_e$, and $F_{cl}$) to each actuator of the system. Meanwhile, the vehicle provides real-time feedback on the operational status and other control information to the controller, forming a closed-loop control cycle for the composite braking system. Hence, the control strategy can be continuously optimized and adjusted according to the actual situation to ensure efficient and stable operation of the composite braking system under different working conditions.

3. System Modeling

3.1. Longitudinal Vehicle Dynamic Model

This study adopted the 1/4 vehicle single-wheel model to simplify the analysis of the vehicle braking process, and the force analysis of the wheel that was conducted is shown in Figure 3. The model is shown in Equation (1) and is based on the following assumptions: ➀ the road is an ideal road without a slope and is flat; ➁ the vehicle steering angle input is zero; and ➂ the influence of tire deformation on the braking performance is ignored [28].

In Equation (1), $s_r$ is the wheel slip rate; m is the mass of the vehicle; $\theta$ is the wheel angle; x is the displacement of the car; $F_f$ is the frictional resistance; f is the rolling resistance coefficient; $F_Z$ is the normal force of the ground on the tire; I is the moment of inertia of the wheel; $T_r$ is the rolling resistance moment; R is the tire radius; $T_b$ is the total braking torque, $T_e$ is the torque of the electromagnetic brake; $T_{EMB}$ is the EMB brake torque; $T_R$ is the regenerative braking torque; and $T_B$ is the lock braking torque.

$$\left\{\begin{array}{c}{\displaystyle \frac{m}{4}}\ddot{x}=-F_f=-f{F}_Z=-f{\displaystyle \frac{mg}{4}}\hfill \\ I\ddot{\theta }=-T_b-T_r+R{F}_f\hfill \\ s_r=1-{\displaystyle \frac{R\dot{\theta }}{\dot{x}}}\hfill \\ T_b=T_e+T_R+T_{EMB}\le T_B\hfill \end{array}.\right.$$

(1)

3.2. EMB Braking Model

The driving motor type of the EMB actuator is a BLDC torque motor. Motor friction directly affects the clamping force. The friction torque $T_D$ is a model composed of static, Coulomb, and viscous friction [29]. The model of the motor is shown in Equation (2):

$$\left\{\begin{array}{c}{U}_a=I_a{R}_a+L_a{\dot{I}}_a+K_t{\dot{\theta }}_m\hfill \\ \\ J_{\mathrm{m}}{\ddot{\theta }}_m=K_t{I}_a-T_{\mathrm{D}}-T_{\mathrm{L}}\hfill \\ T_D=\left\{\begin{array}{c}{T}_w,{\omega }_{\mathrm{\Delta }}=0\mathrm{and}\left|T_w\right|<T_s\\ T_s·\mathrm{sgn}\left(T_w\right),{\omega }_{\mathrm{\Delta }}=0\mathrm{and}\left|T_w\right|\ge T_s\\ G{\omega }_{\mathrm{\Delta }}+T_k·\mathrm{sgn}\left({\omega }_{\mathrm{\Delta }}\right),{\omega }_{\mathrm{\Delta }}\ne 0\end{array}\right.\hfill \end{array},\right.$$

(2)

where $U_a$ is the armature voltage; $I_a$ is the armature current; $R_a$ is the armature resistance; $L_a$ is the armature inductance; $K_t$ is the torque constant; ${\theta }_m$ is the rotor angle; $J_m$ is the moment of inertia of the motor; $T_D$ is the friction torque; $T_L$ is the load torque; $T_w$ is the external torque; ${\omega }_{\mathrm{\Delta }}$ denotes the relative sliding velocity; $T_s$ is the maximum static friction moment; $T_k$ is Coulomb friction torque; G is the viscous friction coefficient.

The EMB transmission mechanism and braking force models are shown in Equation (3):

$$\left\{\begin{array}{c}{x}_{emb}={\displaystyle \frac{{\mathrm{P}}_h}{2\pi i_p}}{\theta }_m\hfill \\ F_{cl}=f(x_{emb}^3,x_{emb}^2,x_{emb})\hfill \\ T_L={\displaystyle \frac{F_{cl}·{\mathrm{P}}_h}{2\pi ·i_p·{\eta }_s·{\eta }_p}}\hfill \\ T_{EMB}=2F_{cl}·R_b·{\mu }_b\hfill \end{array}\right.$$

(3)

where $F_{cl}$ is the clamping force; $x_{emb}$ is the axial displacement of the ball screw; $T_L$ is the load torque; $P_h$ is the ball screw lead; $i_p$ denotes the reduction ratio of planetary gears; $R_b$ is the radius; ${\mu }_b$ is the friction coefficient; ${\eta }_s$ is the transmission efficiency of the ball screw; ${\eta }_p$ represents the efficiency of the planetary gear transmission; and the nonlinear relationship between $F_{cl}$ and $x_{emb}$ can be fitted by a cubic polynomial.

3.3. Electromagnetic Braking Model

The speed, material, radius, magnetic field strength, and other factors of the brake disc will affect the electromagnetic brake torque. With the excitation current unchanged, a peak will appear in the brake torque as the electromagnetic brake disc’s speed varies. The speed corresponding to the peak value is the critical speed $n_p$. When the speed is below $n_p$, the braking torque has an approximately linear positive correlation with the speed; when above $n_p$, it shows an approximately linear negative correlation [25]. The relationship characteristics are shown in Figure 4b. To facilitate the analysis, this study made the following assumptions for the electromagnetic braking system: regardless of the internal structure of the brake disc and the influence of the brake caliper, the brake disc was regarded as a metal disc; the resistivity ${\rho }_r$ and permeability ${\mu }_r$ of the brake disc material were not affected by temperature and hysteresis effect; and, during electromagnetic braking, the magnetic flux of the electromagnetic coil was only in the circular area of the corresponding yoke with a radius of $r_e$, and the effect of magnetic flux leakage was ignored [18]. Based on the above assumptions, $k_{\mathrm{e}}$, $\mathrm{\Delta }h$, and $l_{\mathrm{s}}$ are the conversion coefficients, the air gap change, and the air gap width, which are all constants.

$$\left\{\begin{array}{c}{P}_{\mathrm{s}}={\displaystyle \frac{1}{t}}\underset{0}{\overset{t}{\int }}pdt={\displaystyle \frac{S_p^2\mathrm{\Delta }h{B}_s^2{\omega }_n^2}{16\pi {\rho }_r}}\hfill \\ P_e=12P_s={\displaystyle \frac{3S_p^2\mathrm{\Delta }h{B}_s^2{\omega }_n^2}{4\pi {\rho }_r}}\hfill \\ T_e={\displaystyle \frac{3N^2{S}_p^2\mathrm{\Delta }h{B}_s^2{\omega }_{\mathrm{n}}}{4\pi {\rho }_r}}\hfill \\ B_s={\displaystyle \frac{8\pi {\rho }_r{\mu }_r{N}_s{I}_e}{8\pi {\rho }_r{l}_{\mathrm{s}}+\sqrt{2}{k}_{\mathrm{e}}{\mu }_r\mathrm{\Delta }h{S}_{\mathrm{p}}{\omega }_{\mathrm{n}}}}\hfill \end{array}.\right.$$

(4)

As shown in Figure 4a, the electromagnetic brake designed in this study has a total of 12 electromagnetic coils. In a single circular area, the magnetic flux of the excitation coil is $\mathrm{\Phi }=B_s·S_p$, and the electromagnetic induction intensity is $B_s$. When the rotation speed is ${\omega }_n$, the magnetic flux is as follows: $\mathrm{\Phi }=B_s·S_{p}cos\left({\omega }_{n}t\right)$. The magnetic field will generate an induced electromotive force $\epsilon$ and eddy current $i_s$ on the conductor, and the coil area would be $S_p=\pi r_e^2$. As such, when it comes to the effective power $P_s$ being generated by a single coil in time t, $N_s$ is the number of turns of one coil, $P_e$ is the total braking power, $I_e$ is the excitation current, and the electromagnetic braking torque is $T_e$, as shown in Equation (4).

3.4. Regenerative Braking Model

The regenerative braking system model mainly includes the model of the motor and the model of the battery. The DDEV is driven by four independent hub motors for driving/braking. The motor model is described by the motion equation, torque equation, and voltage equation [30].

$$\left\{\begin{array}{c}{U}_R=K_{\mathrm{Re}}{\omega }_R+R_R{i}_R+L_R{i}_R\hfill \\ J_R{\dot{\omega }}_R=T_R-T_{\mathrm{RL}}-{\omega }_R{f}_R\hfill \\ T_R=K_{\mathrm{Rt}}{i}_R\hfill \end{array}.\right.$$

(5)

In Equation (5), $T_R$ and $T_{\mathrm{RL}}$, respectively, represent the regenerative braking torque and load torque; ${\omega }_R$, $J_R$, $f_R$, $R_R$, $i_R$, and $L_R$ represent the angular velocity, moment of inertia, friction coefficient, armature resistance, current, and inductance of the hub motor, respectively; $K_{\mathrm{Rt}}$ and $K_{\mathrm{Re}}$ represent the torque coefficient and back electromotive force coefficient, respectively; and $U_R$ represents the input voltage of the hub motor.

3.5. Tire Model

The construction methods of the tire mechanical model are generally divided into two categories: the theoretical model and empirical model. This study selected the Magic Formula method, which can accurately and concisely describe the mechanical properties of tires [31]. The calculation equation of tire longitudinal force $F_{X1}$ is as follows:

$$\begin{array}{c}\hfill F_{X1}=Dsin\left(Carctan\left(B{X}_1-E\left(B{X}_1-arctan\left(B{X}_1\right)\right)\right)\right)+S_V\end{array}.$$

(6)

In Equation (6), $X_1$ is the independent variable of the longitudinal force combination $X_1=\left(s_r+S_h\right)$; $s_r$ is the longitudinal slip ratio of the wheel; B is the stiffness factor, which represents the slope at the origin of the characteristic curve; C is the shape factor; D is the peak factor; E is the curvature factor of the curve; $S_h$ is the horizontal offset; and $S_v$ is the vertical drift.

4. Hybrid System Theory and Coordinated Control Strategy Design

Hybrid systems generally refer to complex systems containing continuous and discrete event systems. The use of hybrid control theory can better manage hybrid systems. The composite braking system investigated in this study has both continuous variable systems under certain braking conditions and discrete variable systems caused by braking mode switching, which have the typical characteristics of a hybrid system [32].

4.1. Hybrid System Modeling of the Electromagnetic–EMB–Regenerative Composite Braking System

The modeling methods for hybrid systems mainly include aggregation and extension methods. In practical applications, the hierarchical structure, hybrid automata, hybrid Petri net, hybrid logic dynamic system, and hybrid switching system models are the most widely used. Moreover, the composite braking system exhibits both discrete dynamic events and continuous dynamic behavior. Considering the hybrid characteristics of the hybrid braking system, in this study, we used MATLAB/Stateflow to construct a hybrid automaton (HA) model of the hybrid control strategy of the composite braking system [33]. The model consists of 11 tuples, as shown in Equation (7).

$$\begin{array}{c}\hfill H\mathrm{A}=\left(\begin{array}{c}{M}_{\mathrm{H}},X_H,V_{\mathrm{H}},Y_H,Ini{t}_H,f_H,h_H,In{\nu }_H,E_{\mathrm{H}},G_{\mathrm{H}},R_H\end{array}\right)\end{array}.$$

(7)

Among the above, $M_H=\{m_1,m_2,m_3,m_4,m_5,m_6,m_7\}$ is the set of control modes of the composite braking system, and the specific meaning is shown in

Table 1. Braking torque control algorithms of the composite braking system.

Control Mode	Controlled Objects	Brake Controllers	Control Algorithms	Main Parameters	Parameter Value/Range
$m_1$	Regenerative	Controller 1	Fuzzy PID	$K_{Rp},K_{Ri},K_{Rd},N_{RF}$	$(0.200,2.500)$; $(0.005,0.300)$; $(0.005,0.050)$; 27
$m_2$	EMB friction	Controller 2	Triple closed-loop PID	$K_{EMBp},K_{EMBi},K_{EMBd}$	$(0.100,3.500)$; $(0.010,1.000)$; $(0.001,0.050)$
$m_3$	Electromagnetic	Controller 3	PID	$K_{ep},K_{ei},K_{ed}$	$(0.200,8.000)$; $(0.100,0.800)$; $(0.005,0.600)$
$m_4$	Regenerative and Electromagnetic	Controller 4	Fuzzy PID, PID	$m_1$ and $m_3$	$m_1$ and $m_3$
$m_5$	Regenerative and EMB friction	Controller 5	Fuzzy PID, Triple closed-loop PID	$m_1$ and $m_2$	$m_1$ and $m_2$
$m_6$	Electromagnetic and EMB friction	Controller 6	PID, Triple closed-loop PID	$m_2$ and $m_3$	$m_2$ and $m_3$
$m_7$	Electromagnetic, Regenerative, and EMB friction	Controller 7	Fuzzy PID, PID, Triple closed-loop PID	$m_1,m_2$ and $m_3$	$m_1,m_2$, and $m_3$

. Moreover, $X_H$ is a set of continuous variables; $V_H$ and $Y_H$ are the set of input and output variables, respectively; $Ini{t}_H$ is the initial state set; $f_H$ is the mapping vector field; $h_H$ is the output mapping; $In{v}_H$ assigns the invariant sets to $m_i$; $E_H$ is the discrete jump set; $G_H$ is a protective set; and $R_H$ is the reset. The switching of braking modes is shown in Figure 5, where $m_1$, $m_2$, and $m_3$ are represented by circular icons; $m_4$, $m_5$, and $m_6$ are represented by square icons; and $m_7$ is represented by a triangular icon. The single braking state must be converted into three braking combinations after two braking combinations. Based on the above settings, when the control mode is $m_i$, the behavior of a subsystem can be expressed as ${\dot{f}}_i\left(x\right)=f_i(x,m_i,t)$, where the subscript i represents the serial number of the braking mode. In Figure 5, $e_j$, respectively, represents the discrete events between two of the seven modes. After the discrete event $e_j$ occurs, the corresponding control mode changes and simultaneously satisfies ${\dot{f}}_i\left(x\right)=f_i(x,m_i,t)$, where the subscript j represents the serial number (1–18) of the discrete events. The transformation of the discrete state and the discontinuous jump of the continuous dynamic occur simultaneously at the moment when the discrete event occurs.

4.2. Mode Switching Control Strategy of the Composite Braking System

The hybrid system includes a composite braking system and a hybrid controller. The composite braking system is the control object of the hybrid control structure. The hybrid controller designs the corresponding control algorithms for the seven modes. Under different braking conditions, the corresponding control mode and algorithm for control coordination are adopted. As shown in Table 1, the braking torque distribution strategy of the composite braking system is not limited to a certain control algorithm but integrates a variety of control algorithms to achieve the control and performance requirements of the composite braking system [34]. In Table 1, $K_{Rp}$, $K_{Ri}$, $K_{Rd}$, and $N_{RF}$, respectively, represent the proportional gain, integral gain, differential gain, and the number of fuzzy rules of the fuzzy PID control algorithm in the regenerative braking mode. $K_{EMBp},$$K_{EMBi},$ and $K_{EMBd}$, respectively, represent the proportional gain, integral gain, and differential gain of the triple closed-loop PID control algorithm in the EMB friction braking mode. Correspondingly, $K_{ep},$$K_{ei},$ and $K_{ed}$ refer to the PID parameters in the electromagnetic braking mode. The value ranges of the above parameters are listed in the table. In combination with the input parameters of the composite braking control strategy (z, n, v, SOC, $T_{EMB}$, $T_e$, and $T_R$), the control algorithm is adjusted according to different braking modes. The corresponding output parameters ($reg$, $ele$, $fri$, $U_R$, $I_e$, and $F_{cl}$) are output so as to realize the control of the composite brake actuators.

To fully leverage the characteristics of each sub-braking system, a control mode switching strategy for the composite braking system is formulated, as depicted in Figure 6. Figure 6 shows the inputs are the braking strength z, the initial SOC of the battery, and the wheel speed n. When $z>{\phi }_p$, the composite braking system will switch to $m_2$, where ${\phi }_p$ is the peak adhesion coefficient of the road. When $SOC\ge 90\%$, the regenerative braking system is disabled to ensure the safety of the battery. Energy consumption is required because of electromagnetic braking. A condition is set to prevent regenerative braking from occurring frequently. Specifically, regenerative braking can only be activated when the SOC drops below 90%. When $n\le n_{min}$, the electromagnetic braking system is disabled, and $n_{min}$ is the minimum speed of electromagnetic braking. When $z\le {\phi }_p$ and $T_b\le T_R$, the composite braking system switches to $m_1$. When $T_R<T_b\le T_R+T_e$, the composite braking system switches to $m_4$. When $T_b>T_{R\_imax}+T_{e\_imax}$, the composite braking system switches to $m_7$.

5. Simulation Studies and Results Analysis

5.1. Simulation Setup

Based on the proposed mathematical model, a DDEV model was developed using the Virtual Vehicle Combiner function module in MATLAB/Simulink. The established model is shown in Figure 7. The model includes four hub motors and deceleration mechanisms. The driving power of the electromagnetic braking and EMB originates from the vehicle’s battery system. The battery system includes a high-voltage power battery for the driving motor and a low-voltage DC power supply for the electromagnetic–EMB composite braking system. In addition, the electrical energy generated by regenerative braking returns to the high-voltage battery. This study was simplified because the simulation time was short. The simulation model was set to monitor only the SOC of the high-voltage power battery.

To verify the rationality of the composite braking mode-switching control strategy described in Section 4.2, five different braking conditions were simulated and analyzed by setting different initial SOC values: ${\phi }_p$ and z. The corresponding parameters are listed in

Table 2. Parameters of the composite braking simulation conditions.

No.	SOC (%)	${\mathit{\phi }}_{\mathit{p}}$	z(g)
1	60	0.8	0.1→0.3@2 s
2	60	0.8	0.3→0.7@2 s
3	89.9	0.8	0.1→0.3@2 s
4	89.9	0.8	0.3→0.7@2 s
5	60	0.65	0.3→0.7@2 s

, where g is the gravitational acceleration, and the initial speed of the vehicle is 72 km/h [35].

5.2. Results and Discussion

In Figure 8 and Figure 9, the Working Condition Nos. 1–4 (Table 2) are represented by black, red, blue, and green curves, respectively. The regenerative braking torque is $T_R$, the electromagnetic braking torque is $T_e$, and the EMB friction braking torque is $T_{EMB}$, which are, respectively, represented by solid lines, dotted lines, and dashed lines. The suffixes f and r represent the front and rear wheels, respectively.

Figure 8a,b correspond to the change in the front and rear wheel composite braking torque under the Working Condition Nos. 1–4, respectively. With a change in the SOC, z, and n, the front and rear wheel brakes complete fast switching between the seven braking modes according to the composite braking control strategy. The switching speed of electromagnetic braking and regenerative braking is faster than EMB friction, and the comprehensive switching time is less than 200 ms.

Figure 8 shows that, when the braking strength z increases stepwise at 2 s (e.g., from 0.1 g to 0.3 g in Working Condition No. 1), the system rapidly switches from single regenerative braking $m_1$ to composite modes $m_4$ or $m_5$, as evidenced by the abrupt activation of $T_e$ and $T_{EMB}$ alongside $T_R$. This demonstrates the hybrid automaton’s ability to respond within 200 ms. The results are in line with the design goal of rapid mode switching of the composite braking control strategy. Notably, the torque rise time of electromagnetic braking $T_e$ and regenerative braking $T_R$ is significantly faster than that of EMB friction braking $T_{EMB}$, as shown by their steeper slopes in Figure 8. This highlights the advantage of electromagnetic braking in emergency scenarios requiring rapid deceleration. Therefore, the optimization of the response speed of EMB can be the focus of subsequent research on composite braking.

The curves in Figure 9a illustrate that the composite braking force can quickly follow the braking torque demand curve under all four conditions when the braking strength step changes. At approximately 1.2 s, 3.5 s, and 6.8 s, the maximum torque fluctuations corresponding to different braking modes are all less than 8% (e.g., at 3.5 s in Working Condition No. 2), indicating that the hybrid model composite braking control strategy causes the actual braking torque to better adhere to the demand torque. This confirms the hybrid control strategy’s accuracy in torque allocation, particularly during mode transitions between $m_4$ and $m_7$.

The speed curves in Figure 9b further verify the previous analysis. Under Working Condition Nos. 1–4, except for the moment when the deceleration step changed, the braking deceleration in other stages showed good linear characteristics, which is highly consistent with the control requirements of the braking system. The piecewise linear deceleration curve in Figure 9b indicates that the braking performance of the composite braking system is stable and the error is small under different working conditions. As shown in Figure 9c, it is evident that the simulation continued for approx. 1 s, the SOC of the battery reached 90%, and regenerative braking immediately was disabled. The braking system turned into mode $m_6$. This was reflected in the sudden decrease in $T_R$ and increase in $T_{EMB}$, illustrating the strategy’s adaptability to battery state constraints.

In the long downhill scenario, the battery SOC of the DDEV may reach a high value, which prevents the involvement of regenerative braking in the braking process. If a vehicle relies solely on EMB friction braking, long-distance braking can easily cause excessive wear of the friction plates, leading to thermal recession. However, owing to the electromagnetic brake module, it is possible to achieve a safe long-distance downhill movement of the vehicle without loss of the friction disc by reasonably controlling the speed of the vehicle. Simultaneously, long-term electromagnetic–EMB composite braking also helps to consume the remaining power of the battery system, which, in turn, promotes the recovery of the regenerative braking function and forms a virtuous cycle.

The curves in Figure 10a show that the slip rates $s_{rf}$ and $s_{rr}$ of the front and rear wheels fluctuated between 0.1 and 0.25 before the end of the braking process, which verifies the effect of the triple closed-loop PID controller on the EMB friction braking. As shown in Figure 10b, $T_e$, $T_R$, and $T_{EMB}$ are represented by green, orange, and blue curves, respectively, and the suffixes f and r represent the front and rear wheels, respectively. The curves indicate that, as ${\phi }_p$ changes at 2 s, regenerative and electromagnetic braking can be rapidly disabled. Subsequently, EMB friction braking was promptly engaged for the front and rear wheels. The braking torques $T_{EMBf}$ and $T_{EMBr}$ fluctuated steadily over time, demonstrating that the control strategy switches swiftly in variable-adhesion road scenarios. The adoption of EMB friction braking alone can ensure the control stability of emergency braking, which satisfies the functional requirements of the braking system.

6. Conclusions

In summary, in this study, we created a structural deign of composite braking, developed a simulation model, and conducted numerical simulations. The conclusions are as follows.

(1)

An innovative dual-disc electromagnetic–EMB composite brake structure was designed. The axial space of the EMB friction brake disc was skillfully utilized, and the structural integration of the composite braking system was improved to realize the structural redundancy design of the EMB brake system.

(2)

A hybrid automaton mathematical model of each component of the composite braking system was established, and a control strategy was designed. The strategy comprehensively considers the working conditions of different SOCs, braking strengths, and adhesion coefficients, thereby verifying the feasibility of the composite braking scheme. The scheme can call the corresponding controller according to different trigger conditions, and, to effectively realize vehicle braking, the feasibility of the redundancy design was verified.

(3)

The composite braking system improves the upper limit of the braking strength and accelerates the overall braking response rate owing to the introduction of an electromagnetic braking control structure. In addition to the redundant function, the composite braking system compensates for the shortcomings of the slow response of the EMB friction brake and improves its ability to resist heat fading.

In addition, this study had the following limitations, which could be used as a focus of future research.

(1)

We did not perform a real vehicle test and failed to fully verify the unknown complex situation of real vehicle braking. In terms of the control strategy, the switching logic of composite braking systems under emergency braking and with abnormal battery systems were not considered. This part of the study can be debugged and calibrated in combination with a real vehicle test.

(2)

The modeling and simulation involved idealized assumptions, such as the friction model of the electromagnetic brake model and the EMB model. In the simulation analysis, SOC was only observed for high-voltage batteries in the battery system. Owing to the short simulation time, the redundant low-voltage battery for the EMB and electromagnetic braking remained unchanged. Therefore, in a follow-up study, the SOC change in the entire battery system should be fully considered for the working–condition cycle.