A Novel Coordinated Control Strategy for Parallel Hybrid Electric Vehicles during Clutch Slipping Process

Abstract

For parallel hybrid electric vehicles (HEVs), the clutch serves as a vital enabling actuator element during mode transitions. The expected drivability and smoothness of parallel HEVs are difficult to be achieve owing to the neglect of clutch-torque-induced disturbance and different response characteristics of power sources during clutch slipping. To address this issue, this paper proposes a novel control strategy to coordinate the engine and motor during the clutch slipping process. A sliding mode control strategy based on a group-preserving scheme was applied to control the motor. The vehicle dynamic equation was constructed by the sliding surface with the Lagrange function. The equation solutions obtained by introducing the Runge–Kutta method were used as motor control inputs. Meanwhile, an adaptive PI controller was designed to regulate engine speed for the reduction in the speed difference of the clutch. The hardware-in-the-loop simulations were conducted to validate the outstanding performance of the proposal strategy. The verification results indicate that the proposed strategy not only reduces the vehicle jerk and frictional losses effectively, but also improves vehicle driving comfort and reliability.

1. Introduction

More severe environmental pollution and energy crisis problems are forcing the development of clean, efficient vehicles with low emissions. Hybrid electric vehicles (HEVs) with artistic multi-power-source designs possess the merits of high efficiency, a negligible driving range limitation and easy braking energy regeneration, representing a promising development direction in the automobile industry [1,2,3]. To reduce fuel consumption and emissions, HEVs can operate in a suitable mode for different traffic conditions [4,5,6,7]. However, the frequent mode transitions will lead to the deterioration in the performance of HEVs due to the clutch-torque-induced disturbance and differences in dynamic characteristics between the engine and motor. A noticeable jerk occurs especially during the clutch slipping process, which may shock the powertrain [8,9]. Hence, the quality of the mode transition should be considered when designing the control strategy for HEVs.

Recently, lots of control strategies have been investigated and presented that deal with the mode transition problem for HEVs. Koprubasi et al. presented a coordinated control strategy for parallel HEV based on hybrid control theory [10]. Hybrid controllers were designed for several switched hybrid dynamical systems, and results demonstrate that a seamless transition can be achieved using the proposed controllers. In [11], during mode transitions, the model reference control (MRC) method was applied in a series-parallel HEV to coordinate the clutch torque, motor torque and engine torque. Compared to the conventional method, the MRC method could reduce torque interruption and achieved less vehicle jerk. In [12], taking various unmodeled dynamics and parameter uncertainties into consideration, engine-side and motor-side controllers were devised separately by using feed-forward and robust feedback control strategies. The simulation results demonstrate that the drivability performance for a parallel HEV was improved significantly. Additionally, Kim et al. utilized disturbance observers to devise a four-phase control algorithm for parallel HEVs with a clutch during mode transitions [13]. Yang et al. proposed a hierarchical control system based on the robust H_∞ control strategy to realize a smooth and fast mode transition for a plug-in HEV [14]. Based on the available research, it can be deduced that the essence of these control algorithms is the coordinated control for the engine and motor during mode transitions. The expected vehicle drivability and smoothness can be guaranteed by suppressing fluctuations in coupling torque on the driveshaft while satisfying the torque requirements of the vehicle [15]. However, the influence of clutch torque is not considered in the existing coordinated control strategies. The speed fluctuations on both sides of the clutch, as well as the variation in the friction coefficient with the temperature of the friction plate, will results in a significant nonlinear characteristic of the torque transferred by the clutch [16,17,18,19]. The torque ripple generated by the engine at the side of driving plate can also have an adverse effect during the process of clutch engagement. Despite the short duration of the slipping phase, the nonlinear variation in torque will intensify the mismatching torque in the vehicle powertrain, leading to an obvious unpleasant driving sensation [20,21,22,23]. Hence, further improvement in the drivability and smoothness of HEVs during mode transition remains a challenge.

Focusing on the mode transition process from the electric mode to the hybrid driving mode, this paper proposes a novel coordinated strategy to deal with deterioration issues in the drivability and smoothness for parallel HEVs. First, a novel sliding mode control (SMC) strategy based on the group-preserving (GP) scheme was devised to control the motor. The dynamic differential equations for the vehicle during the clutch slipping process were constructed using the sliding surface with the Lagrange function, and the solutions could be solved by using the Runge–Kutta method. Second, an adaptive PI control strategy was designed to adjust the engine speed to realize the trajectory tracking of the driven plate, in which the parameter of the PI controller was updated online by a fuzzy algorithm. Finally, hardware-in-the-loop (HIL) simulations were conducted to validate the effectiveness of the proposed strategy.

The paper is organized as follows: Following the Introduction, the vehicle dynamic model is described in Section 2. In Section 3, the SMC-GP strategy and adaptive PI controller are described, respectively. Next, we explain how the proposed coordinated control strategy was implemented in the HIL, and verification results are provided and described in Section 4. Finally, conclusions and future works following on from this study are presented in Section 5.

2. Parallel HEV System Model

The simplified topology of the studied parallel HEV is shown in Figure 1. The engine, motor and automated mechanical transmission (AMT) are located on the same axis without the configuration of an extra coupling device. Such a layout provides greater reliability in a compact space. For dealing with different operating conditions, the clutch can be engaged or disengaged to enable the mode transition. In the electric drive mode, the clutch is open and only the motor provides the demanded torque to drive the vehicle [24,25]. In the hybrid drive mode, the engine drives the vehicle together with the motor, and the clutch is locked. Therefore, the clutch serves as a vital enabling actuator element for mode transitions of the parallel HEV. The specification of the main components of the studied parallel HEV is listed in

Table 1. The main parameters of the studied parallel HEV.

Components	Description
Engine	Diesel type, peak torque: 800 Nm, maximum power: 162 kW
Motor	Permanent magnet, peak torque: 850 Nm, maximum power: 95 kW
Gearbox	AMT, gear ratios: 6.71, 3.77, 2.26, 1.44, 1, 0.77
Battery	Lithium iron phosphate, nominal voltage: 483 V, capacity: 110 Ah

.

This study focused on the impact of both clutch-torque-induced disturbance and different dynamic characteristics between the engine and motor on the performance of the parallel HEV. Using the lumped parameter method [26], the differential equation for vehicle dynamic performance during the clutch slipping process can be depicted as:

$$\{\begin{array}{l}{J}_e{\dot{\mathsf{\omega }}}_e+b_e{\mathsf{\omega }}_e=T_e+T_c\\ J_m{\dot{\mathsf{\omega }}}_m+b_m{\mathsf{\omega }}_m=T_m-T_c-\frac{1}{i_g{i}_0}\left[k_s{\theta }_s+b_s\left(\frac{{\mathsf{\omega }}_m}{i_g{i}_0}-{\mathsf{\omega }}_w\right)\right]\\ J_w{\dot{\mathsf{\omega }}}_w+b_w{\mathsf{\omega }}_w=k_s{\theta }_s+b_s\left(\frac{{\mathsf{\omega }}_m}{i_g{i}_0}-{\mathsf{\omega }}_w\right)-T_L\end{array}$$

(1)

where J_e is the combined inertia of the engine and accessories, J_m is the combined inertia of the motor rotor and transmission on the motor side, and J_w is the inertia of the wheels. ω_e, ω_m and ω_w are the engine speed, motor speed and wheel speed, respectively. b_m is the motor internal friction coefficient, b_e is the engine friction coefficient, and b_w is the damping coefficient of tires. i_g and i₀ are the gear ratios of the AMT and final drive, respectively. b_s and k_s are the equivalent damping coefficient and spring coefficient of the driveline, respectively. θ_s denotes the torsional displacement of the driveshaft. T_e and T_m are the engine torque and motor torque, respectively. T_c and T_L are the clutch torque and vehicle load torque, respectively. In addition, the vehicle load torque can be calculated by:

$$T_L=r_w\left(mg{f}_r\cos \theta +mg\sin \theta +\frac{C_{D}A{v}_a^2}{21.15}+\delta m\frac{d{v}_a}{dt}\right)$$

(2)

where g is the gravity acceleration, m is the gross mass, f_r is the rolling resistance coefficient, θ is the road slope angle, A is the frontal area, C_D is the air drag coefficient, v_a is the vehicle velocity, and δ is the modified coefficient of the rotating mass.

Additionally, the torque transferred from the clutch can be divided into three parts, which is closely related to the friction under the action of the normal force between the two sides of clutch [27,28,29]. The clutch torque is calculated by:

$$T_c=\{\begin{array}{c}0,\mathrm{unlock}\\ \frac{2N_c{u}_c\left(R^3-r^3\right)F_c\mathrm{sgn}\left(\Delta \omega \right)}{3\left(R^2-r^2\right)},\mathrm{slip}\\ T_i,\mathrm{lock}\end{array}$$

(3)

where T_i denotes the input torque of the driving plate, u_c denotes the clutch friction coefficient, N_c denotes the friction plate number, F_c denotes the normal force, R denotes the outer radius of the diaphragm spring, r denotes the inside radius of the diaphragm spring, and △ω denotes the speed difference. Here, sgn() is the sign function expressed as the following mathematical formula.

$$\mathrm{sgn}\left(\Delta \omega \right)=\{\begin{array}{c}1,\Delta \omega \ge 0\\ -1,\Delta \omega <0\end{array}$$

(4)

According to Equation (3), the clutch torque exhibits strong nonlinear characteristics, especially in the slipping process, which has a prominent influence on the drivability and smoothness. Moreover, the different dynamic characteristics between the motor and engine will further cause the deterioration of driving comfort. In this paper, a coordinated control strategy was devised to improve drivability performance of parallel HEVs during the clutch slipping process.

3. Control Strategy Description

Comprehensively considering the energy conversion efficiency, a parallel HEV should be started in the electric mode [30]. The motor can provide the required torque to drive the vehicle, and the engine is shutdown in this situation. When the instantaneous velocity is greater than the preset threshold V_{e_max}, the mode transition from the electric mode to hybrid driving mode is triggered. The starting motor will start the engine at idle speed, and then the clutch is locked. When the driving plate rotates synchronously with the driven plate, the vehicle enters into the hybrid driving mode. On the contrary, when the instantaneous velocity is less than the preset threshold V_{e_max}, the vehicle will enter the electric mode. The diagram of the above-mentioned mode transition is shown in Figure 2.

During the clutch slipping process, the disturbance torque is introduced to the HEV powertrain. If the output torque of the motor is still in accordance with the torque commands given by the energy management strategy (EMS), the torque demand of the driver may not be satisfied, which results in the deterioration of the drivability performance. Furthermore, the speed difference of the clutch may increase the torque disturbance. The driven plate speed is equal to the speed of the motor, which can be derived from the vehicle velocity. Then, the speed difference can be reduced by engine speed regulation. Considering the interaction between the two power sources during clutch slipping, the proposed coordinated control strategy, including an adaptive PI control strategy for engine control and an SMC-GP algorithm for motor control, is proposed in this research, as illustrated in Figure 3.

3.1. Control Strategy for the Motor

For the studied parallel HEV, the motor is a key component that can be controlled to compensate for the disturbance torque during clutch slipping. In this situation, the motor, as the sole power supplier for the parallel HEV, must also be able to achieve accurate tracking of the target velocity. Here, we consider a general nonlinear system, as follows:

$$\{\begin{array}{c}\dot{x}\left(t\right)=f\left(x\left(t\right),u\left(t\right),t\right),x\in {ℝ}^n\\ x\left(t\right)|{}_{t=t_0}=x_0,t_0\le t\le t_f\end{array}$$

(5)

where x(t) stands for the system state, u(t) denotes the control input, and t₀ and t_f are the initial time and terminal time, respectively. For a given general performance function J(u), the control input u(t) should satisfy the following condition:

$$\min \left\{J\left(u\right)=F\left(x\left(t_f\right),t_f\right)+{\displaystyle {\int }_{t_0}^{t_f}\left[L\left(x,u,t\right)\right]dt}\right\}$$

(6)

$$\mathrm{s}.\mathrm{t}.\left|u\left(t\right)\right|\le U,t\in \left[t_0,t_f\right]$$

where F(x(t_f), t_f) is the terminal index function, and L(x, u, t) is the Lagrange function. U is the maximum of control input. Since the control input is independent of the Lagrange function, the above equation can be transformed into:

$$\min \left\{J\left(u\right)=F\left(x\left(t_f\right),t_f\right)+{\displaystyle {\int }_{t_0}^{t_f}\left[L\left(x,t\right)+u^2\right]dt}\right\}$$

(7)

The appropriate Lagrange function with positive constants M₀ and α can be defined by:

$$L(x,t)=M_0\mathit{exp}(-\alpha t)$$

(8)

On the basis of this definition, we can further deduce and obtain the equation as follows.

$$\begin{array}{c}{\displaystyle {\int }_{t_0}^{t_f}L\left(x,t\right)}dt={\displaystyle {\int }_{t_0}^{t_f}{M}_0}\exp \left(-\alpha t\right)dt\\ =\frac{M_0}{\alpha }\left[\exp \left(-\alpha t_0\right)-\exp \left(-\alpha t_f\right)\right]\\ \le \frac{M_0}{\alpha }\left[1-\exp \left(-\alpha t_f\right)\right]\\ <\frac{M_0}{\alpha }\end{array}$$

(9)

According to Equations (6), (7) and (9), the following inequality can be derived by

$$F\left(x\left(t_f\right),t_f\right)+\left(t_f-t_0\right)U^2+{\displaystyle {\int }_{t_0}^{t_f}{M}_0}\exp \left(-\alpha t\right)dt\le \min \left\{J\left(u\right)\right\}$$

(10)

That is, if the control input u(t) is derived from Equation (8) and satisfies corresponding constraints, a smaller performance index than that of Equation (10) can be obtained. Considering the complexity of the motor control problem during the clutch slipping process, an SMC strategy was selected to solve this problem.

3.1.1. SMC Strategy

In terms of control theory, SMC is a nonlinear control method with strong immunity and robustness for parameter perturbation and external disturbance, and does not rely on an accurate control system model [31]. The control input is changed for a certain purpose that can force the system trajectory to move on a predetermined sliding surface. As shown in Figure 4, the system trajectory will gradually converge to the sliding surface S(x) = 0. Therefore, the SMC strategy is suitable for application in the motor control during the clutch slipping process, as it can avoid the phenomenon of torque interruption and unfavorable drivability.

Here, the motor torque is selected as the control variable. The motor speed is regarded as the system state variable, and its ideal trajectory x_d can be defined as:

x_d = ω_wd·i_g·i₀

(11)

where ω_wd is the expected wheel speed, and its value is dependent on the vehicle velocity. From Equation (8), the sliding surface with the Lagrange function can be designed as:

S= k(x − x_d) − L(x, t) = k(x-x_d) − M₀ exp(−αt)

(12)

where k denotes the constant with a positive value. When the system is in sliding mode, the equation $\dot{s}=0$ will be satisfied. The control law u₁ can be derived as follows:

$$u_1=J_m{\dot{x}}_d-\frac{J_m}{k}{M}_0\alpha \exp \left(-\alpha t\right)+b_{m}x-T_c-d+\frac{1}{i_0{i}_g}\left[J_w{\dot{\mathsf{\omega }}}_w+b_w{\mathsf{\omega }}_w+T_L\right]-\chi \mathrm{sgn}\left(s\right)$$

(13)

where d is the disturbance, and it contains the modeling error and parameter perturbation; χ is a positive constant. When the state trajectory reaches the sliding mode surface, the system will traverse between both sides of the sliding surface and approach the equilibrium point. During this process, the control variable will generate high-frequency oscillation [32]. In this study, sgn(s) is replaced by the continuous function θ(s) to restrain the chattering phenomenon.

$$\theta \left(s\right)=\frac{s}{\left|s\right|+\sigma }$$

(14)

where σ is the positive constant with a smaller value.

3.1.2. GP-SMC Strategy

In 2001, the group-preserving scheme was first proposed by Liu [33]. The differential equations representing a nonlinear system can be converted into a Lie algebra in the augmented space of Minkowski. By using a group-preserving scheme with Cayley transformation and Padé approximation, the augmented state points can be updated to lie on the cone [34]. The asymptotic behavior of the original dynamic systems is preserved. In this way, spurious solutions and ghost fixed points can be easily excluded. The differential equations are solved using the Runge–Kutta method, which not only maintains the characteristics of the Hamilton function, but also guarantees an accuracy of O (h⁴), where h is the step size.

In general, the torque required by the vehicle is determined by the EMS in accordance with the driver’s intentions. For the proposed HEV powertrain, the vehicle velocity is related to the motor speed and can be expressed as:

$$v_a=0.377\frac{R_w{\mathsf{\omega }}_m}{i_0{i}_g}$$

(15)

where R_w is the wheel radius. The precise control of the motor helps to satisfy the driver’s driving intentions, especially during the clutch slipping process. Hence, the sliding mode control strategy based on the group-preserving (GP-SMC) scheme ensures that the motor can accurately accomplish a specific trajectory tracking. According to Equation (11), the differential equations concerning the vehicle dynamic during the clutch slipping process are reconstructed as:

$$\{\begin{array}{l}{J}_m\dot{x}=u_1-b_{m}x-T_c-\frac{1}{i_0{i}_g}\left[k_s{\theta }_s+b_s\left(\frac{x}{i_0{i}_g}-{\omega }_w\right)\right]\\ S=k\left(x-x_d\right)-M_0\exp \left(-\alpha t\right)=0\end{array}$$

(16)

where $\left|u_1\right|\le T_{m\_\max }$, T_{m_max} stands for the maximum motor torque.

The Runge–Kutta method was proposed by C. Runge and M. W. Kutta in 1900, and belongs to a single-step iterative method [35]. It is able to suppress cumulative errors and has the advantage of high arithmetic accuracy. The four points located on the cone can be defined as:

$$\{\begin{array}{c}{x}_k^1=x_k\\ x_k^2=x_k+\tau {\xi }_1\\ x_k^3=x_k+\tau {\xi }_2\\ x_k^4=x_k+h{\xi }_3\end{array}$$

(17)

where h is the step size, and τ = h/2. Here, we define the variable $\xi \left(x,t\right)$ as follows.

$$\xi \left(x,t\right)=\frac{1}{J_m}\left[u_1-b_{m}x-T_c-\frac{1}{i_0{i}_g}\left(k_s{\theta }_s+b_s\left(\frac{x}{i_0{i}_g}-{\mathsf{\omega }}_w\right)\right)\right]$$

(18)

Then, the slope values for these aforementioned points can be calculated by

$$\{\begin{array}{c}{\xi }_1=\xi \left(x_k,t_k\right)\\ {\xi }_2=\xi \left(x_k+{\lambda }_{k,1}{\xi }_1,t_k+\tau \right)\\ {\xi }_3=\xi \left(x_k+{\lambda }_{k,2}{\xi }_2,t_k+\tau \right)\\ {\xi }_4=\xi \left(x_k+{\lambda }_{k,3}{\xi }_3,t_k+h\right)\end{array}$$

(19)

where λ_k,1, λ_k,2 and λ_k,3 are the regulation coefficients for the corresponding slope values. The detailed computational procedure is listed in Figure 5, and the solution’s x_k can be regarded as the control law for the motor.

3.2. Control Strategy for the Engine

To avoid the disturbance torque generated from the large speed difference of the clutch, the engine was operated in the mode of speed regulation in this study. The driving plate of the clutch should be maintained to follow the speed of the driven plate.

As far as we know, conventional PI controllers with fixed parameters are not suitable for regulating the engine speed because of their complicated dynamic characteristics [36,37,38]. The parameters of the controller must be adjusted according to the actual operating state of the engine. Here, a fuzzy algorithm was introduced and applied in the PI controller to form an adaptive method, rather than the method with fixed parameters. The engine speed control principle diagram is depicted in Figure 6.

To improve the dynamic response of the engine speed regulation, eliminate integral saturation phenomenon, and avoid overshoot and oscillation of the engine speed, the fuzzy algorithm was adopted to correct parameters of the PI controller. The PI controller can be expressed as:

$$u(t)=K_{\mathrm{p}}·e(t)+K_{\mathrm{i}}{\int }_t^{0}e(\tau )d\mathsf{\tau }$$

(20)

$$e\left(t\right)={\mathsf{\omega }}_{\mathrm{e}\_\mathrm{cmd}}-{\mathsf{\omega }}_{\mathrm{e}}\left(t\right)$$

(21)

where K_p denotes the proportional coefficient, K_i denotes the integral coefficient, ω_{e_cmd} denotes the target speed for the engine, and e(t) denotes the speed error at the time t.

The parameters K_p and K_i of the controller can be normalized by Equations (22) and (23); then, the normalized parameters ${K^{\prime }}_{\mathrm{p}}$ and ${K^{\prime }}_{\mathrm{i}}$ in the interval [0, 1] are generated, which can be represented by the fuzzy sets {NB, NS, ZO, PS, PB}. Figure 7 shows their corresponding membership functions.

$${K^{\prime }}_{\mathrm{p}}=\frac{K_{\mathrm{p}}-K_{\mathrm{p}\_\min }}{K_{\mathrm{p}\_\max }-K_{\mathrm{p}\_\min }}$$

(22)

$${K^{\prime }}_{\mathrm{i}}=\frac{K_{\mathrm{i}}-K_{\mathrm{i}\_\min }}{K_{\mathrm{i}\_\max }-K_{\mathrm{i}\_\min }}$$

(23)

where K_{p_max} is the maximum of the parameter K_p, K_{p_min} is the minimum of the parameter K_p, K_{i_max} is the maximum of the parameter K_i, and K_{i_min} is the minimum of the parameter K_i.

The speed error e and its derivative $\dot{e}$ are regarded as the inputs, which can be expressed by the fuzzy sets {NB, NM, NS, ZO, PS, PM, PB}. Their membership functions are shown in Figure 8. The fuzzy inference rules for parameters $K_{\mathrm{p}}^{\prime }$ and $K_{\mathrm{i}}^{\prime }$ are listed in

Table 2. Fuzzy inference rules for the parameter $K_{\mathrm{p}}^{\prime }$.

		$\dot{e}$
		NB	NM	NS	ZO	PS	PM	PB
e	NB	NB	NB	NB	NS	NS	ZO	ZO
	NM	NB	NB	NS	NS	ZO	ZO	PS
	NS	NS	NS	NS	ZO	ZO	PS	PS
	ZO	NS	ZO	ZO	ZO	PS	PS	PS
	PS	ZO	ZO	ZO	PS	PS	PS	PB
	PM	ZO	ZO	PS	PS	PS	PB	PB
	PB	ZO	PS	PS	PS	PB	PB	PB

and

Table 3. Fuzzy inference rules for the parameter $K_{\mathrm{i}}^{\prime }$.

		$\dot{e}$
		NB	NM	NS	ZO	PS	PM	PB
e	NB	ZO	ZO	ZO	PS	PB	PB	PB
	NM	ZO	ZO	PS	PS	PS	PB	PB
	NS	ZO	PS	PS	PS	PS	PS	PB
	ZO	NS	PS	ZO	ZO	ZO	PS	PS
	PS	NB	NS	NS	NS	NS	NS	ZO
	PM	NB	NB	NS	NS	NS	ZO	ZO
	PB	NB	NB	NB	NS	ZO	ZO	ZO

, respectively. Finally, accurate parameters for the PI controller were achieved by the defuzzification. In this study, the height method was applied to calculate PI controller parameters, which can be expressed as [39]:

$$y(\mathrm{x})=\frac{{\displaystyle \sum _{n=1}^M{y}^n{\displaystyle \prod _{i=1}^2{F}_i^n({\mathrm{X}}_i)}}}{{\displaystyle \sum _{n=1}^M{\displaystyle \prod _{i=1}^2{F}_i^n({\mathrm{X}}_i)}}}$$

(24)

where $F_i^n$ is fuzzy sets, yⁿ is fuzzy singletons, x_i is the input variable, and M stands for the number of inference rules.

4. Results and Discussion

In order to validate the performance of the proposed strategy, the conducted HIL simulations are described in this section. The verification platform included a real-time parallel HEV model platform and a real-time control platform. Both of the two development platforms can generate the C code and perform online calibration in the MATLAB/Simulink environment [40].

In this study, the development process of the verification platform could be divided into three steps. First, both the parallel HEV model and the control strategy were designed in the MATLAB/Simulink environment. Second, by using the automatic code generation technology, the real-time kernels of the parallel HEV model and control strategy were generated. Third, the real-time kernel of the parallel HEV was downloaded into the SCALEXIO LabBox, which exhibits a strong computing capacity. In addition, the other kernel was flashed into the MicroAutobox, which is an electronic controller [41,42]. All set parameters and verification results in SCALEXIO LabBox and MicroAutobox were monitored by the software Control Desk operating on a laptop [40,41,42,43]. The HIL test bench is depicted in Figure 9.

As shown here, the unified model of parallel HEV was constructed with the specifications listed in Table 1. The test cycle was taken from a stretch of typical driving cycles that are widely used by manufacturers, as shown in Figure 9. The whole testing process consisted of the mode transition from the electric mode to the hybrid driving mode. When the vehicle velocity reached about 20 km/h, the engine was dragged to the idle speed by the starting motor and immediately ignited with fuel injection. The clutch was then engaged. After the clutch lockup, the parallel HEV entered the hybrid driving mode. To prove the effectiveness of the proposed coordinated control strategy, the performances of three different cases are compared in this section. A detailed description of three test cases is listed in

Table 4. Description of three test cases.

Cases	Control Strategy
I	Without coordinated control
II	Motor control with SMC
III	Motor control with GP-SMC and engine control with Fuzzy-PI

.

The simulation results for the two power sources for case I are shown in Figure 10. Because the coordinated control strategy was not considered in case I, the control commands for the engine and motor came directly from the energy management strategy. It can be observed that the mode transition occurred around 13.62 s. At the initial stage of the clutch slipping process, the driven plate speed was immediately reduced by about 89 r/min due to the speed difference of the clutch. At this moment, the torque transmitted by the clutch is regarded as an interference torque, which corresponds to a negative torque (−98.7 Nm) on the output shaft of the engine. If this disturbance torque cannot be suppressed or compensated in time, it will lead to a noticeable jerk and vibration in the hybrid powertrain. This means that the coupling torque on the driveshaft does not correspond to the torque required by the powertrain, which results in a large deviation from the actual velocity to the target value. In addition, the variation in the friction plate coefficient with temperature interacts with the engine torque on the driving plate side, which exacerbates the speed fluctuations at both ends of the clutch. Until 15.69 s later, the clutch was completely engaged and in a locked state. With the gradual increase in engine torque, the motor torque decreased to 100 Nm, and the parallel HEV completed the mode transition.

For case II, the SMC method was used to regulate the motor torque, and the engine still followed control commands that came from the vehicle energy management strategy. It can be seen from Figure 11 that the speed of driven plate decreased by about 43 r/min. This still adversely affected the vehicle performance during the clutch slipping process. Although the control strategy for the motor had a positive effect on depressing the disturbance torque, the fluctuation in engine speed cannot be ignored in this case. Hence, the dynamic responses of the engine should be taken into consideration.

The proposed coordinated control strategy was verified and taken as case III, whose verification results are shown in Figure 12. Similarly, when the velocity was greater than 20 km/h, the engine was dragged to the idle speed by the starting motor. At the engine side, an adaptive PI control strategy based on fuzzy algorithm was adopted to adjust engine speed to synchronize with the driven plate speed and reduce the speed difference of the clutch. The motor was controlled to compensate for the disturbance torque based on the SMC-GP method. From Figure 12, it is seen that the proposed coordinated control strategy exhibited a better control performance in terms of the speed and torque output from both power sources among the parallel HEV.

As is well-known, vehicle jerk and clutch frictional loss are two important criteria that are widely used to quantitatively evaluate the control strategy performance for mode transitions [44,45,46]. Jerk is defined as the derivative of vehicle acceleration, as shown in Equation (25). Vehicle jerk with a large value denotes that the driveline torque occurs as an obvious fluctuation phenomenon, and it will result in the deterioration of the vehicle’s drivability and smoothness. The recommended value is less than 17.64 m/s³ in China and 10 m/s³ in Germany, respectively [47,48,49].

$$\left|J\right|=\frac{da}{dt}=\frac{d^{2}v}{d{t}^2}$$

(25)

where J is the vehicle jerk, a is the vehicle acceleration, and v is the vehicle velocity determined by the driving cycle.

Clutch frictional loss indicates the work done by friction during the clutch slipping process, which is closely related to the clutch lifespan. The clutch frictional loss W_f can be written as:

$$W_f={\displaystyle {\int }_{t_1}^{t_2}{T}_c}\cdot \Delta \omega dt$$

(26)

where t₁ and t₂ stand for initial time and end time of the clutch slipping process, respectively.

Figure 13 shows the vehicle jerks for the three cases, and their performance comparisons are summarized in

Table 5. Performance comparison for the three cases.

Control Strategy	Clutch Frictional Loss (kJ)	Maximum Vehicle Jerk (m/s3)	Maximum Vehicle Velocity Error (%)
Case I	6.074	18.81	7.37
Case II	4.983	8.65	3.95
Case III	1.897	4.03	1.27

. Due to the influence of disturbance torque and diversity of dynamic responses between the two power sources, the maximum vehicle jerk during the clutch slipping process reached up to 18.81 m/s³ for case I. In addition, vehicle jerk fluctuations further deteriorated the drivability and smoothness performance. In case II, the maximum vehicle jerk could be reduced to 7.65 m/s³ when the SMC method was used to control motor to compensate for the disturbance torque. However, the clutch frictional loss reached up to 4.983 kJ. The black dashed line represents the jerk for case III in Figure 13. It can be observed that by using the proposed strategy, the vehicle jerk was significantly reduced. The clutch frictional loss in case III was only 1.897 kJ, which avoids the unnecessary energy loss and prolongs the clutch lifespan. Additionally, the maximum vehicle velocity error was also restricted to an acceptable range. Hence, the performance comparison for the three cases validates the effectiveness of the proposed strategy in terms of the smoothness and drivability improvement.

5. Conclusions

This paper presents a novel coordinated control strategy to address the deterioration of vehicle smoothness and drivability during the clutch slipping process. The strategy was combined with the motor control and engine control, taking both torque disturbances and model uncertainty into account. Based on the GP-SMC method, a suitable controller for the motor was designed to reduce torque interruption. The dynamic differential equations of the vehicle were constructed by means of the sliding surface with the Lagrange function, and solutions with an accuracy of O (h⁴) were obtained by using the Runge–Kutta method. Meanwhile, an adaptive PI controller was constructed to adjust the engine speed to reduce the speed difference of the clutch during the slipping process. To verify the effectiveness of the proposed coordinated control strategy, a control model was formulated in the MATLAB/Simulink environment, and HIL simulations were carried out. The results demonstrate that the expected smoothness and drivability of a parallel HEV could be obtained. The vehicle jerk under the proposed coordinated control was limited to 4 m/s³, while the value without coordinated control was 18.81 m/s³. The clutch frictional losses were also effectively reduced, which will extend the clutch lifespan. Although the complexity of the proposed algorithm increased, it was still acceptable. Hence, the proposed coordinated control strategy helps to improve the performance of parallel HEVs and promote their application in the automotive industry. For future work, the system delay and parameter sensitivity studies will be involved to improve the robustness and performance of the coordinated control strategy.