Regenerative Braking Control Method Based on Predictive Optimization for Four-Wheel Drive Pure Electric Vehicle

Regenerative braking is the key to achieve efficient use of energy and extend the driving range in pure electric vehicles. This study proposes a new predictive control method integrating adaptive cubic exponential prediction and dynamic programming to address the problem of efficient energy recovery during the regular braking process of four-wheel pure electric vehicles. The method considers the dynamic characteristics of an electro-hydraulic combined braking system. The adaptive cubic exponential prediction is adopted to predict the vehicle velocity and braking intensity. The dynamic programming is employed to optimize the motor braking torques and wheel cylinder pressures under the condition of braking regulations, road constraints, and vehicle constraints. To verify the effectiveness of the new predictive control method, the ideal and multi-stage braking force distribution methods are employed for comparison. The results confirm that, under gradual braking conditions, the energy recovery efficiency achieved via the proposed method is improved by 1.55% and 6.40% considering the ideal and multi-stage braking force distribution methods, respectively.


I. INTRODUCTION
As the development of zero-emission pure electric vehicles is beneficial to the global environment [1]- [3], they are being increasingly developed and deployed worldwide [4].
Some researchers have conducted in-depth studies on electric vehicles, including handling stability control [5], yaw stability control [6], and inertial estimation [7]. However, the short driving range of these vehicles hinders their promotion and application. Research indicates that 50% of the energy of pure electric vehicles is wasted in the form of heat by a conventional braking system during deceleration in urban conditions [8]. Therefore, studies on the regenerative braking energy recovery of pure electric vehicles are necessary [9]- [11].
The associate editor coordinating the review of this manuscript and approving it for publication was Wei Xu .
The existing studies aiming to improve the energy recovery of regenerative braking mainly focus on two groups: system design and control [12]- [15]. Regarding system design, supercapacitors [16] or flywheels [17] are added in some studies to increase energy recovery, which can improve energy conversion efficiency. Regarding system control, in [18], model predictive control was applied to distribute braking energy for battery and supercapacitors, and a satisfactory control effect was obtained. However, these approaches increase both the complexity of the system and the cost of the entire vehicle owing to the addition of supercapacitors or flywheels. Thus, the recovery of kinetic energy by the motor becomes the best choice in all types of vehicles, as a trade-off between performance and cost [19].
The energy recovery is closely related to the braking control method when only the motor recovers the braking kinetic energy. An adequate braking control method can improve the energy recovery efficiency, whereas an inadequate method may cause the energy recovery effect to deteriorate, and even threaten the braking safety of the vehicle. In recent years, an increasing number of studies has focused on the energy recovery by motor braking. The methods can be further divided into two aspects: rule-and optimizationbased control [20]- [23]. Regarding rule-based control, early research mainly considered the regulations of the Economic Commission for Europe (ECE) and ideal braking force distribution [24]. Although this can improve the energy recovery efficiency, it does not consider the driver's intention recognition. To address driver's intention recognition, fuzzy logic [25] was adopted and a better energy recovery effect was obtained. To approximate to the actual control situation, system dynamics and driver intention recognition [26], [27] were considered, but only in terms of system modeling and not in terms of method formulation. The above methods are all rulebased, and the energy recovery effect can be further improved through optimization. Fuzzy logic optimized by genetic algorithm [28] was adopted to obtain electro-hydraulic braking force distribution rules. Although this method did not consider the driver's intention recognition, other studies considered this aspect [29] [30], obtaining an improvement in energy recovery efficiency, However, the dynamic characteristics of the system were neglected, as well as the future state information of the vehicle.
With the development of vehicle electrification and intelligence, energy recovery is an important research focus, whether it is a manually driven vehicle or an intelligent vehicle [31]. This study mainly focuses on the energy recovery of manually driven vehicles. It is well recognized that the dynamic characteristics of the electro-hydraulic combined braking system have an important influence on the control of the vehicle. However, regarding the formulation of regenerative braking methods, the dynamic characteristics of the braking system are rarely considered in existing studies. Neglecting these characteristics may worsen the recovery efficiency. In addition, the future of vehicle state information such as vehicle velocity and braking intensity has also important influence for vehicle energy recovery. If the state of the vehicle is known in advance, it may help to improve the energy recovery. To solve the mentioned issues, a predictive control method (PCM) that centers on the future of vehicle state information and dynamic characteristics of the electro-hydraulic combined braking system is proposed for four-wheel pure electric vehicles to improve energy recovery efficiency. Three vital advantages of the PCM are presented in the study: • An adaptive cubic exponential prediction is integrated into PCM to solve the problem of obtaining future vehicle state information, such as vehicle velocity and braking intensity; • A dynamic programming is integrated into PCM to consider dynamic characteristics of the electro-hydraulic combined braking system and obtain the front and rear motor braking torques, and front and rear wheel cylinder pressures; • A predictive optimization framework of PCM that includes adaptive cubic exponential prediction and dynamic programming is presented for the first time. The remainder of this paper is organized as follows. Section 2 illustrates the system structure and model of the vehicle, and Section 3 describes the vehicle PCM. The verification and discussion are presented in Section 4, and conclusions are presented in Section 5.

II. SYSTEM STRUCTURE AND MODEL
A. VEHICLE SYSTEM STRUCTURE The power system of the four-wheel pure electric vehicle is a centralized arrangement of front and rear axle dual motors. The vehicle system, which includes the battery, front and rear motors, main reducers I and II, wheel cylinders, and braking unit are displayed in Figure 1. The battery and the motors are electrically connected. The motors, main reducers, and wheels are mechanically connected, and the wheel cylinders and the braking unit are connected by pipelines.

B. VEHICLE MODEL
Because only a longitudinal braking analysis is performed in this study, the vehicle model is simplified to a left-right symmetrical model (i.e., the left and right wheel parameters are explained using a single wheel as an example). The established vehicle models include the vehicle dynamics model, efficiency and dynamic models of the motors, internal resistance model of the battery, transmission system, tire, and wheel cylinder models.

1) VEHICLE DYNAMICS MODEL
The vehicle driving equation is important for simulating vehicle driving. According to Newton's second law, the driving equation during vehicle braking is established as where F b , F f , F w , and F i denote the vehicle braking force, rolling, wind, and ramp resistances, respectively. δ denotes the vehicle rotation mass conversion factor. m denotes the vehicle curb mass and a denotes the vehicle acceleration.
The vehicle braking intensity is a key parameter of the vehicle and is represented by where z denotes the braking intensity of the vehicle and g denotes the acceleration of gravity. The load on the front and rear axles for a vehicle changes with modifications in the braking intensity, which is closely related to the generation of tire force. The corresponding mathematical model is as follows: where Fz 1 and Fz 2 denote the front and rear axle loads, respectively. c and b denote the distance from the center of gravity to the front and rear axles, respectively. h g denotes the height of gravity center and L denotes the wheelbase.

2) MOTOR MODEL
The motor is an essential component for the energy recovery of a four-wheel pure electric vehicle. In this study, the motor model is divided into motor efficiency and dynamic models. The front and rear motors efficiency models are displayed in Figures 2(a) and (b), respectively. The transfer function and state space methods are commonly used for the modeling of motor dynamic characteristics. The transfer function initial state value is zero; it is difficult to realize the dynamic simulation of a nonzero initial state motor. The torque in the actual control process of the motor is primarily from one torque (nonzero) to another torque (nonzero). In addition, the state-space method can easily set the initial state value. Therefore, this study adopts the state space method to establish the dynamic models of the motors.
where x 11 , x 21 , and x 12 , x 22 denote the states of the front and rear motors, respectively.ẋ 11 ,ẋ 21 andẋ 12 ,ẋ 22 denote the differentials of the states for the front and rear motors, respectively. u 1 and u 2 denote the input and y 1 and y 2 denote the outputs of the front and rear motors, respectively. a 11 , a 21 , b 11 , b 21 , a 12 , a 22 , b 12 , b 22 are constants and reflect the dynamic characteristics of the front and rear motors. Note that these parameters can be obtained through the transfer function. x 1i and y i represent motor output torques. The power delivered by the motor to the battery in the process of braking is where P b denotes the power transferred from the motor to the battery during braking. P b1 and P b2 denote the power, T p1 and T p2 denote the actual braking torque, n p1 and n p2 denote the speed, and η m1 and η m2 denote the braking efficiency of the front and rear motors, respectively.

3) TRANSMISSION SYSTEM MODEL
The transmission system of the vehicle is important for vehicle torque transmission. The relationship between the torque and rotation speed of the motor and wheel ends is where η 1 and η 2 denote the transmission system efficiency of the front and rear axles, respectively. i 1 and i 2 denote the transmission ratio of main reducers I and II, respectively. w 1 and w 2 denote the angular speeds of the front and rear wheels, respectively. T md1 and T md2 denote the braking torque generated by the front motor at the front wheel end and the rear motor at the rear wheel end, respectively.

4) BATTERY MODEL
The battery model adopts the common internal resistance model, as indicated in Figure 3. In this model, the battery is simplified into a system composed of a voltage source and internal resistance. According to Kirchhoff's voltage law, where E is the battery open circuit voltage, U is the charging voltage, I is the charging current, and r is the battery charging internal resistance The amount of battery energy recovery is an evaluation index to measure the effect of braking control. It is where Q denotes the energy recovery of the battery.

5) WHEEL CYLINDER MODEL
The wheel cylinder is directly related to the generation of mechanical braking force. The relationship between the front and rear wheel hydraulic braking torques and wheel cylinder pressure can be obtained using where T phf and T phr denote the actual hydraulic braking torques of the front and rear wheels, respectively, p pw1 and p pw2 denote the actual pressures of the front and rear wheel cylinders, respectively, K f and K r denote the braking factors of the front and rear axle brakes, respectively, D f and D r denote the wheel cylinder diameters of the front and rear wheels, respectively. R f and R r denote the effective radius of the front and rear wheel brake discs, respectively. To more accurately reflect the dynamic characteristics of the hydraulic braking system, the wheel cylinder model is regarded as a first-order inertial link [32], and the dynamic characteristics can be described using the state space method as where x k1 and x k2 denote the states,ẋ k1 andẋ k2 denote the differential states, u k1 and u k2 denote the inputs, and y k1 and y k2 denote the outputs of the front and rear wheel cylinders, respectively. a k1 , b k1 , a k2 , and b k2 are constants related to the dynamic characteristics of each wheel cylinder. Note that these parameters can be obtained through the transfer function and that x ki and y ki represent wheel cylinder braking torques.

6) TIRE MODEL
The tire model in this study includes the wheel dynamics model and magic model. In the process of vehicle braking, the wheel dynamics model is described by (11) where jw 1 and jw 2 denote the inertia, T d1 and T d2 denote the braking torque, and ux 1 and ux 2 denote the adhesion coefficients of the front and rear wheels, respectively. R t denotes the wheel radius. The four wheels of the vehicle have the same radius.ẇ 1 andẇ 2 denote the angular acceleration of the front and rear wheels, respectively. f denotes the rolling resistance coefficient. i is equal to ''1'' or ''2'', representing the front or rear wheel, respectively.
As the magic tire model is a semi-empirical model that can reflect the interaction with the road surface, it is suitable for the dynamic simulation panel of the vehicle [33] and was adopted in this study. Using a single tire as an example, its form is where Y (x) denotes the output variable: longitudinal force, and s denotes the slip ratio. B, C, D, and E refer to the stiffness, shape, peak, and curvature factors, respectively.
The slip rate is an important parameter of the tire model; its expression is where s 1 and s 2 denote the front and rear wheel slip rates, respectively, and v x denotes the vehicle velocity. The motor and hydraulic braking torques are coupled at the wheel. The coupling model is The schematic diagram of the vehicle simulation model is displayed in Figure 4. The simulation model includes six parts: the wheel cylinder, motor, transmission system, battery, vehicle dynamics, and tire models. The red dotted enclosure represents the predictive vehicle model (the first four parts).  T m1 and T m2 are the required torques of the front and rear motors, respectively. p w1 and p w2 are the wheel required cylinder pressures of the front and rear wheels, respectively. Y (x) 1 and Y (x) 2 are the longitudinal forces generated by the front and rear wheels, respectively. The motor model and wheel cylinder model accept commands (p w1 , p w2 T m1 , T m2 ), generate braking torques (T p1 , T p2 ) and wheel cylinder pressures (T phf , T phr ), and transfer the relevant parameters to the transmission system model (T p1 , n p1 , T p2 , n p2 ), battery model (P b1 , P b2 ), and tire model (T phf , T phr , T md1 , T md2 ). The tire model also accepts the parameters (v x , Fz1, Fz2) from the vehicle dynamics model and transmits the longitudinal force (Y (x) 1 , Y (x) 2 ) to the vehicle dynamics model.

III. PCM DESIGN
Performing data prediction based on the historical data of a vehicle to provide data support for its operation is essential to improve the efficient energy recovery during the braking process of the vehicle. In this study, the adaptive cubic exponential prediction is used to predict the key parameters of the vehicle operation. Based on the dynamic programming, the braking torques and wheel cylinder pressures are optimized under the constraints of ECE braking regulations, vehicle constraints, and motor maximum braking torque to achieve efficient energy recovery. As the adaptive cubic exponential prediction has a better prediction effect on gradual data, the prediction of sudden change data is not ideal. Therefore, this study is suitable for gradual braking intensity conditions. Note that the PCM in this study is employed when the battery state of charge (SOC) does not exceed the specified upper limit.

A. ADAPTIVE CUBIC EXPONENTIAL PREDICTION
The adaptive cubic exponential prediction optimizes the smoothing coefficient based on the cubic exponential smoothing method, obtains the prediction data model with the smallest error, and predicts the future data [34], which has an acceptable prediction effect on time-varying and nonlinear data. The ability of the cubic exponential smoothing method to adapt to time-varying data is not sufficient. However, the adaptive cubic exponential prediction method overcomes this shortcoming, thus, it was selected. The model of the cubic exponential smoothing is where t = 2, 3, 4, . . . . S (1) k,t , S k,t , and S k,t are the first, second, and third exponential smoothing values of the k-th prediction t period data, respectively. X k,t is the k-th prediction t period of the actual data. λ k is the k-th prediction smoothing coefficient (0 < λ k < 1).
To forecast future data, let T be the number of periods predicted forward from time t, and let a k,t , b k,t , and c k,t be the prediction coefficients of the k-th prediction The t + T period predicted value of the k-th is where T is the prediction step, that is, the interval between the target prediction period and current period. T = 1,2,3. . . . Y k,t+T is the t + T period predicted value of the k-th prediction.
The prediction accuracy is the basis for evaluating the quality of the prediction. This study adopts the sum of squared errors as the basis for evaluation.
where ff represents the sum of the squared errors. Y k,i represents the i-th data prediction value of the k-th prediction. X k,i represents the i-th data actual value of the k-th prediction. N is the number of data samples. The adaptive cubic exponential prediction refers to the optimization of the smoothing coefficients using the carpet traversal search algorithm to obtain dynamic smoothing coefficients where ε k,t is a function of time t, which represents the dynamic smoothing coefficient at the k-th prediction. λ k is the smoothing coefficient after the k-th prediction optimization.
The prediction flow chart of the adaptive cubic exponential prediction is displayed in Figure 5. This prediction first collects the historical data (braking intensity z k and vehicle velocity v xk ) derived from the vehicle simulation model. The carpet traversal search method is used to optimize the smoothing coefficient of the cubic exponential smoothing with the smallest square error as the goal (refer to [34] for details). The dynamic smoothing coefficient is updated according to Equation (19). Subsequently, the one-step prediction data (braking intensity z k+1 and vehicle velocity v x(k+1) ) are output, followed by data collection for the next data prediction. For adaptive cubic exponential prediction, one-step prediction results are the most accurate [34], thus they are adopted. In this study, the data predicted by the adaptive cubic exponential prediction and current data are combined into vectors ([z k v xk ; z k+1 v x(k+1) ]) to provide parameters for the dynamic programming.

B. DYNAMIC PROGRAMMING
The dynamic programming is suitable for linear and nonlinear systems, and adopted to obtain the optimal motor braking torques and wheel cylinder pressures for the vehicle control.
The dynamic programming must determine parameters such as decision variables, state variables, and index functions. The required braking torques of the front and rear motors and the required cylinder pressures of the front and rear wheels are selected as the decision variables, that is, the decision variables are [T m1 , T m2 , p w1 , p w2 ] T . The decision variables must respect the legal requirements. The braking force distribution coefficient is closely related to the decision variable. The relationship among the braking force distribution coefficient, front and rear axle braking torques, and total braking torque is (20) where β is the braking force distribution coefficient. T f and T r are the braking torques at the front and rear wheel ends, respectively, and T t is the total braking torque provided by the wheel ends.
The braking torques of the front and rear axles are constrained by the maximum braking torque on the ground. where µ 0 is the road adhesion coefficient. The right end of the equation represents the maximum braking torques (wheel ends) that the ground can provide.
In the process of vehicle braking, the braking force distribution coefficient must respect the requirements of the ECE regulations.
Equation (22) is the upper limit required by the ECE when the braking intensity is between 0.1 and 0.6, to prevent the front axle from distributing an excessive braking force. Equation (23) is the lower limit required by the ECE when the braking intensity is between 0.3 and 0.45 to prevent the rear axle from distributing an excessive braking force. Equation (24) is the middle limit required by the ECE when the braking intensity is between 0.15 and 0.8. In the process of vehicle braking, the motor torques are also constrained by the vehicle transmission system itself. If the braking torques are overly large, the vehicle produces an abnormal noise. The vehicle transmission system constraints (referred to as the vehicle constraints) are where γ 1 and γ 2 are the maximum proportional coefficients of the vehicle front and rear axle transmission systems that do not emit abnormal noise, respectively.
The braking torque of the motor is also constrained by the motor itself.
where T mf max and T mrmax are the maximum braking torques of the front and rear motors, respectively.
The selection of the index function in the dynamic programming is extremely important for its optimization. The energy recovery by the battery is selected as the index function, which can be obtained by combining Equations (5) and (8): T p1 n p1 η m1 /9550 + T p2 n p2 η m2 /9550 − I 2 r dt (27) Front and rear motor efficiencies η m1 and η m2 , respectively, are related to the speeds (n p1 and n p2 ) and torques (T p1 and T p2 ) of the motors, which can be obtained by consulting the table (motor efficiency maps). The current I can be obtained by Equation (8).
The selection of the state variables should be combined with the control object. This study considers the dynamic characteristics of the front and rear motors and the braking wheel cylinders. The means of the interferogram call are used for the simulation test. Therefore, the motor braking torques and wheel cylinder pressures are used as state variables, that is, the state variables are [T m1 , T m2 , p w1 , p w2 ] T . Because the state variables are consistent with the decision variables, the decision process of the dynamic programming is the state transition process.
The schematic diagram of the dynamic programming optimization process is shown in Figure 6. This study adopts two-stage dynamic programming, as one-step forecast of adaptive cubic exponential prediction is the most accurate. The [z k v xk ] and [z k+1 v x(k+1) ] act on the stages I and II, respectively. The SOC is considered approximately constant  throughout the process. Q fn and Q gm represent the index function values of states F n and G m , respectively, which are obtained by the predictive vehicle model. When min (Q fi + Q gj ) (i = 1,2, . . . n; j = 1,2, . . . m), the stage I optimal decision variable vector is obtained [T m1 , T m2 , p w1 , p w2 ] T .

C. PCM FRAMEWORK
The PCM integrates adaptive cubic exponential prediction and dynamic programming. In this study, the operating condition information is set to the gradient braking intensity, which is obtained from the driver's model and simulated via a sine function [35]. z = abs(sin(t/2))/2 (28) A framework of the PCM is displayed in Figure 7. The PCM is indicated by the red dotted line. First, the model is initialized with the battery SOC, vehicle velocity v xk , braking intensity z k , front and rear wheel required cylinder pressures p w1 and p w2 , respectively, front and rear motor required torques T m1 and T m2 , respectively. The dynamic programming calls the predictive vehicle model to optimize T m1 , T m2 , p w1 , and p w2 under the constraint conditions (Equations (21), (22), (23), (24), (25), (26)) to maximize the battery energy recovery.
Then, the vehicle simulation model outputs v xk . Next, v xk is evaluated: if it is lower than kk (kk is a constant), the simulation terminates; otherwise, the v xk is output to the adaptive cubic exponential prediction. This prediction relies on v xk and z k to obtain the predicted vectors [z k+1 v x(k+1) ], and then sends them to the dynamic programming for the next optimal control. This is repeated until v xk becomes lower than kk. It is important to note that the adaptive cubic exponential prediction must collect information (several sampling points) before making a prediction. Only the stage I optimization is performed in dynamic programming at the first few sampling points.

IV. SIMULATION VERIFICATION AND DISCUSSION
The effectiveness of vehicle control can be verified via simulation. This study verifies the PCM using a simulation test. When the vehicle velocity is low, energy recovery is not performed to protect the motors. The main technical parameters of the vehicle are listed in Table 1. The PCM is compared with the multi-stage braking force distribution method (MBFDM), commonly used in engineering, and the ideal braking force distribution method (IBFDM). The initial vehicle velocity was set to 75 km/h, the battery SOC was set to 70%, and a road with acceptable conditions was used as the simulated road; the road adhesion coefficient was 0.82. The gradual braking conditions [35] change as indicated in Figure 8.
The braking force distribution coefficients of the MBFDM, IBFDM, and PCM are displayed in Figure 9. The constraints for the above methods were the same. In Figure 9, the shaded area is the feasible region of the braking force distribution coefficient prescribed by the ECE regulations. The braking force distribution coefficients of the three methods are within the feasible range. The blue dotted, pink, and red lines represent the curve of the braking force distribution coefficient used by the IBFDM, MBFDM, and PCM, respectively. Among these, the braking force distribution coefficients of the IBFDM and MBFDM are only related to the braking intensity, whereas that of the PCM is zero at the initial stage of braking, and then changes in the direction of the arrow, indicating that the braking force distribution coefficient is related to the braking intensity and also to the vehicle velocity.
The changes in the front and rear motor braking torques under the three control methods are indicated in Figures 10(a) and (b), respectively, including a comparison of the required and actual braking torques. PCMD, MBFDMD, and IBFDMD are the required braking torques and PCDA, MBFDMA, and IBFDMA are the actual braking torques of the PCM, MBFDM, and IBFDM, respectively. The braking torque changes of the three control methods under the same braking intensity are not exactly the same, and the maximum braking torque attains the vehicle constraint (refer to the limit Equation (25) for the transmission system not to emit an abnormal noise). The enlarged sections indicate that the actual braking torques of the front and rear motors are not exactly the same as the required braking torques. However, they are consistent with the required braking torques at the end of the control duration. The braking torque is higher than zero owing to overshoot during the torque change of the motor.
The changes in the front and rear wheel cylinder pressures under the three control methods are presented in Figures 11(a) and (b), respectively, including also a FIGURE 11. Wheel cylinder pressure changes of front and rear wheels.
comparison of the required and actual wheel cylinder pressures. PCMDp, MBFDMDp, and IBFDMDp are the wheel required cylinder pressures and PCMAp, MBFDMAp and IBFDMAp are the actual wheel cylinder pressures of the PCM, MBFDM, and IBFDM, respectively. The wheel cylinder pressure changes of the three control methods under the same braking intensity are not exactly the same. From the enlarged section in Figure 11, we can observe that the actual and required wheel cylinder pressures are not completely consistent. However, at the end of the control duration, they are consistent with each other.
The changes in the battery SOC of the three control methods are displayed in Figure 12. The SOC started to increase from 70%. The final SOC of the PCM, MBFDM, and IBFDM were 70.104%, 70.095%, and 70.102%, respectively. The SOC under the PCM increased the most.  The changes in vehicle velocity are displayed in Figure 13. Combining Figure 10 we can see that when the vehicle velocity is low, the motors are off.
The vehicle energy flow is displayed in Figure 14. The braking energy is converted into four parts of energy, which are wind resistance loss, rolling resistance loss, ramp resistance loss, and recoverable energy. Recoverable energy (red dotted part) is stored from the wheel through the reducers to the motors and even to the battery, but the battery is the key to improving energy conversion efficiency. During the energy conversion process, energy loss occurs at the wheels, main reducers I and II, front and rear motors, and battery. The effective energy of the components, shown in Figure 15, refers to the energy flowing out of the component or the energy stored. It can be seen from Figure 15 that under different control methods, the energy stored by the battery FIGURE 15. Effective energy of component. VOLUME 9, 2021 is different. The recovered energies via PCM, IBFDM, and MBFDM during the entire braking process were 281.63 kJ, 275.25 kJ, and 255.25 kJ, respectively. When compared with the IBFDM and MBFDM, PCM increased energy recovery efficiency by 1.55% and 6.40%, respectively.
The changes in the front and rear motor braking efficiencies under the three control methods are indicated in Figures 16(a) and (b), respectively. Among the three control methods, the front and rear motor efficiency of PCM is not always the most efficient. At the initial stage of braking, the front motor under PCM and rear motor under MBFDM are not working (when efficiency is equal to zero, it means that the motor is off). The enlarged sections show that the motor efficiencies are different under different control methods. The battery efficiency change, shown in Figure 17, indicates that, among the three control methods, battery efficiency of PCM is the smallest, and the MBFDM, which recovers the least energy, has the highest battery efficiency for a long time. Combining Figures 16 and 17, we can see that it is difficult to achieve maximum energy conversion efficiency for dual-motor-driven vehicles simply by aiming for maximum efficiency of a single component. Figure 18 shows the changes in the total efficiency of the vehicle (which includes the efficiency of the motors, battery, and transmission system) under the three control methods. The total efficiency of the vehicle under PCM is the largest, whereas that under MBFDM is the smallest for a long-term in the entire braking process. This indicates the superiority of PCM. In summary, simulations of the three control methods were conducted under a gradual braking condition, and the changes in braking force distribution coefficients are displayed in Figure 9. The changes in the front and rear motor torques, presented in Figures 10(a) and (b), respectively, indicate that the actual braking torques of the front and rear motors are not exactly the same as the required braking torques. The pressure changes of the front and rear wheel cylinders under the three methods, presented in Figures 11(a) and (b), respectively, indicate that there is no hydraulic pressure on the rear wheels to participate at the early stage of braking. As seen in Figure 12, the SOC under the PCM increased the most. The changes in vehicle velocity are displayed in Figure 13. The vehicle energy flow and effective energy of the components, displayed in Figures 14 and 15, respectively, indicate that the PCM provides a higher energy recovery. The changes in the front and rear motor braking efficiencies, displayed in Figures 16(a) and (b), respectively, indicate that the front and rear motor efficiencies of PCM are not always the best. The changes in battery and total vehicle efficiencies, shown in Figures 17 and 18, respectively, indicate that the total efficiency of the vehicle under PCM is the largest in the entire braking process.

V. CONCLUSION
Energy recovery is a key technology to improve the driving range of pure electric vehicles. To improve the energy recovery efficiency during vehicle braking, the vehicle longitudinal control method under regular braking conditions is studied, and a PCM based on adaptive cubic exponential prediction and dynamic programming is proposed. The predictive optimization framework of PCM is presented for the first time. The adaptive cubic exponential prediction uses the historical data of the vehicle, predicts future key parameters such as vehicle velocity and braking intensity, and provides parameter support for the dynamic programming. The dynamic programming optimizes the front and rear motor braking torques, and front and rear wheel cylinder pressures to provide control parameters for the vehicle. To verify the superiority of the proposed method, simulations were performed under gradual braking conditions. The results indicated that the energy recovery of the proposed PCM increased by 1.55% and 6.40%, compared with the IBFDM and MBFDM, respectively. Moreover, this study provides new ideas for regenerative braking energy recovery.