Multi-objective parameter optimization strategy based on engine coordinated control for improving shifting quality

Abstract

In order to address the poor shifting quality issue of a certain model of heavy-duty vehicles, a multi-objective parameter optimization strategy based on engine coordinated control is proposed. This strategy aims to improve shifting quality by reducing the sliding friction work and impact during the shifting process. The Non-Dominated Sorting Genetic Algorithm II (NSGA-II) is employed to perform multi-objective optimization on the coordinated control parameters, which include external control torque of the engine, start of fuel cut-off timing, and duration of fuel cut-off. By comparing the performance of different parameter combinations in terms of sliding friction work and impact, the optimal parameter combination is determined. Through bench testing verification, it has been demonstrated that utilizing the optimized parameters for engine coordinated control during the torque phase of the shifting process can significantly enhance shifting quality. This strategy provides an effective solution for addressing shifting quality issues.

Introduction

The automatic transmission (AT) is currently the most widely used type of transmission, typically consisting of a torque converter, planetary gear set, wet clutches, hydraulic operating system, and electronic control system. ¹ The shifting process of an AT involves clutch-to-clutch shifting, where one clutch disengages while the other engages. Generally, the shifting process can be divided into two phases: the torque phase and the inertia phase. ² During the torque phase, the low-gear clutch begins to disengage while the high-gear clutch starts to engage gradually. In this phase, power is mainly transmitted by the low-gear clutch. In the inertia phase, the low-gear clutch continues to disengage while the high-gear clutch gradually engages. Power transmission is primarily carried out by the high-gear clutch in this phase. During the shifting process, if torque conversion is not well controlled, it can result in significant shifting shocks and increase clutch wear, thus reducing the lifespan of the clutch. Therefore, developing advanced shifting control strategies is of great significance for improving shifting quality. ³

To develop shifting control strategies, it is necessary to establish a control-oriented dynamic powertrain system model. Bond graph is a commonly used modeling method, which has been applied in the modeling of AT ⁴ and hybrid powertrains. ⁵ However, in practical AT systems, there are many nonlinear factors such as friction and inertia, which can affect the shifting process. Bond graph models have limitations in accurately describing these complex dynamic characteristics. An effective complementary method is to establish a dynamic model of the automatic transmission and vehicle powertrain system based on the Lagrangian method. This method can overcome the shortcomings of bond graphs in handling nonlinear characteristics and dynamic complexity, providing a more accurate and reliable foundation for the development of shifting control strategies.^6,7

In current research, controlling the clutch and engine is a key measure to improve shifting quality. By precisely controlling the timing and speed of the clutch engagement and disengagement during the shifting process, ⁸ or adjusting the torque output of the engine, ⁹ smoother shifting can be achieved. Furthermore, coordinated control of the clutch and engine during the shifting process can further enhance shifting quality.^10,11 In terms of coordinated control, Čorić et al. ¹² proposed an AT shifting control trajectory optimization method, representing the control trajectories of the clutch and engine as polynomial functions. They optimized the control trajectories of the clutch and engine in AT using pseudospectral collocation methods to achieve comfortable and efficient shifting. However, they did not analyze in depth the coordination relationship between the clutch and engine torques. Ranogajec et al., ¹³ based on the slip state of the clutch, developed six different open-loop control strategies. They quantitatively evaluated the shifting performance with different levels of complexity and applicability by optimizing the clutch and engine torque distribution. However, they did not make a comparison between the torque phase and inertia phase with different engine coordinated control strategies. Regarding coordinated control optimization, some researchers have used Linear Quadratic Regulator (LQR) to optimize control parameters such as engine torque and clutch torque during upshifting inertia phase of AT. ¹⁴ They utilized multi-objective optimization algorithms like genetic algorithms, ¹⁵ particle swarm optimization, ¹⁶ etc., to find the optimal combination of control parameters. They incorporated conflicting criteria with different weights in the cost function, including shifting duration, impact, sliding friction work, and other performance indices. ¹⁷ They even adjusted the weight values to personalize the control strategy according to user preferences and requirements. ¹⁸ Although these control strategies can improve shifting quality in certain scenarios, none of them have thoroughly analyzed the relationship between engine torque control in different phases (torque phase and inertia phase) and shifting quality.

This paper primarily focuses on improving shift quality through coordinated engine control. In order to conduct a dynamic analysis of the shifting process, the complex structure of the AT is modeled using the Lagrange method. By formulating six different coordinated control strategies, the relationship between coordinated control in different phases (torque phase and inertia phase) and shift quality is analyzed, and key parameters affecting shift quality are identified. Subsequently, two optimization strategies for coordinated control in different phases (torque phase and inertia phase) are developed, with the objective of minimizing sliding friction work and impact during the shifting process. The NSGA-II algorithm is employed to perform multi-objective optimization of the coordinated control parameters for the two strategies, resulting in a Pareto front of non-dominated solutions. Finally, a test bench experiment was conducted using the optimized coordinated control parameters to validate the proposed coordinated control strategies for their reliability.

Dynamic analysis of shift process

Powertrain structure

The powertrain system configuration studied in this research is shown in Figure 1. The torque output from the diesel engine is transmitted to the pump wheel of the torque converter, which then inputs to the turbine of the torque converter. The transmission receives the input from the torque converter and achieves changes in gear ratios through different combinations of engaging the shifting clutches. Finally, the power is transmitted to the driving wheels of the vehicle through a reduction mechanism. The transmission used is an electro-hydraulically controlled clutch-to-clutch planetary gear transmission. Its gear mechanism consists of two planetary gear sets and one compound gear set. During start-up, shifting from third to fourth gear or from fourth to third gear, it is necessary to control the coordination between four clutches to achieve gear changes. In other upshifting and downshifting processes, the main focus is on controlling the engagement and disengagement of two clutches to achieve power transfer and switching. Table 1 provides the engagement timing for gears 1–6 in this transmission.^19,20

Figure 1.

Power transmission system structure.

Table 1.

Clutch engagement timing table.

Gear	Ratio	Shift clutch
		TC1	TC2	TC3	TCL	TCH
First	G1	X			X
Second	G2		X		X
Third	G3			X	X
Fourth	G4	X				X
Fifth	G5		X			X
Sixth	G6			X		X

Transmission kinematics analysis

To obtain the speed characteristics of each planetary gear mechanism during the operation of the transmission, it is necessary to perform kinematic analysis of the transmission. By selecting the input shaft angular velocity and torque direction as the positive direction, and based on the speed characteristic equation of the planetary gear sets, ²¹ we can obtain:

$$(k_i+1){\omega }_{\mathrm{ci}}={\omega }_{\mathrm{si}}+k_i{\omega }_{\mathrm{ri}}$$

(1)

$${\omega }_{\mathrm{pi}}=\frac{2}{k_i-1}({\omega }_{\mathrm{si}}-{\omega }_{\mathrm{ci}})$$

(2)

Where $k$ is the structural parameter of the planetary array, $k=Z_{\mathrm{ri}}/Z_{\mathrm{si}}$ ; $i$ is the planet sequence number, taking 1, 2, 3, 4; ${\omega }_c$ is the angular velocity of the planet carrier; ${\omega }_s$ is the angular velocity of the solar wheel; ${\omega }_r$ is the angular velocity of the ring gear; ${\omega }_p$ is the angular velocity of the planetary gear.

Based on the structure of the compound planetary gear set, it can be seen that the transmission has three elements interconnected. To complete power transmission, one of the clutches in CL/CH/CR and one of the clutches in C1/C2/C3 need to be engaged. From the engagement timing table (Table 1) of the transmission, we can see that in first gear, the CL and C1 clutches are engaged; in second gear, the CL and C2 clutches are engaged. During the 1–2 shifting process, the CL clutch remains engaged, the C1 clutch gradually disengages, and the C2 clutch gradually engages. Therefore, the system has two degrees of freedom. Based on the equal angular velocities between the interconnected components, the 1 and 2 shifting process satisfies:

$${\omega }_i={\omega }_{s1}={\omega }_{s2}$$

(3)

$${\omega }_{r1}={\omega }_{c2}={\omega }_{s3}={\omega }_{r4}$$

(4)

$${\omega }_o={\omega }_{c3}={\omega }_{c4}$$

(5)

Select ${\omega }_i$ and ${\omega }_o$ as variables, and according to equations (1)–(5), the kinematic relationship between the planetary gear units of the transmission and the input/output shaft speed can be deduced, as shown in equation (6).

$$\left[\begin{array}{c}{\omega }_{s2}\hfill \\ \multicolumn{1}{c}{{\omega }_{c2}}\\ \multicolumn{1}{c}{{\omega }_{r2}}\\ \multicolumn{1}{c}{{\omega }_{r3}}\\ \multicolumn{1}{c}{{\omega }_{c3}}\\ \multicolumn{1}{c}{{\omega }_{s4}}\\ \multicolumn{1}{c}{{\omega }_{c1}}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}\hfill & a_{12}\hfill \\ \multicolumn{1}{c}{a_{21}}& a_{22}\hfill \\ \multicolumn{1}{c}{a_{31}}& a_{32}\hfill \\ \multicolumn{1}{c}{a_{41}}& a_{42}\hfill \\ \multicolumn{1}{c}{a_{51}}& a_{52}\hfill \\ \multicolumn{1}{c}{a_{61}}& a_{62}\hfill \\ \multicolumn{1}{c}{a_{71}}& a_{72}\hfill \end{array}\right]\left[\begin{array}{c}{\omega }_i\hfill \\ \multicolumn{1}{c}{{\omega }_o}\end{array}\right]$$

(6)

Where

$$a_{11}=a_{52}=1$$

(7)

$$a_{12}=a_{22}=a_{31}=a_{32}=a_{51}=a_{72}=0$$

(8)

$$a_{21}=\frac{1}{1+k_2}$$

(9)

$$a_{41}=-\frac{1}{(1+k_2)k_3}$$

(10)

$$a_{42}=\frac{(1+k_3)}{k_3}$$

(11)

$$a_{61}=-\frac{k_4}{1+k_2}$$

(12)

$$a_{62}=1+k_4$$

(13)

$$a_{71}=\frac{1+k_1+k_2}{(1+k_1)(1+k_2)}$$

(14)

Transmission dynamics analysis

This paper focuses on the analysis of a planetary gear transmission, disregarding clearances and friction between the planetary gears. The planetary gear system and shaft system are assumed to be rigid bodies, and the influence of system damping is neglected. The dynamics analysis of the 1–2 gear shifting process is carried out using Lagrange’s equations. ²²

The system has two degrees of freedom during the shifting process, and two generalized coordinates need to be defined. Therefore, the transmission input shaft angle and output shaft angle are selected as the generalized coordinate system, that is, $q_1={\phi }_i$ and $q_2={\phi }_o$ . Therefore, the virtual work of the system can be expressed by equation (15).

$$\delta W=T_i\delta {\phi }_i+T_{\mathrm{TC}1}\delta {\phi }_{r3}+T_{\mathrm{TC}2}\delta {\phi }_{s4}-T_o\delta {\phi }_o$$

(15)

Where $T_i$ is the transmission input shaft torque; $T_{\mathrm{TC}1}$ is C1 clutch torque; $T_{\mathrm{TC}2}$ is C2 clutch torque; $T_o$ is the torque of transmission output shaft; ${\phi }_i$ is the speed of transmission input shaft; ${\phi }_{r3}$ is the rotational speed of the No. 3 planetary gear ring; ${\phi }_{s4}$ is the rotational speed of No. 4 planetary sun gear; ${\phi }_o$ is the transmission output shaft speed.

According to the principle of virtual work for active forces shown in equation (16), the two generalized forces of the system can be expressed as equations (17) and (18).

$$\delta W=\sum _{\alpha }^s{Q}_{\alpha }\delta q_{\alpha }=0$$

(16)

$$Q_1=Q_{\phi i}=\frac{\sum \delta W}{\delta q_i}$$

(17)

$$Q_2=Q_{\phi o}=\frac{\sum \delta W}{\delta q_o}$$

(18)

The Lagrange function $L$ can be represented as:

$$L=T-V$$

(19)

Where $\alpha$ is the number of degrees of freedom; $T$ is the kinetic energy of the system; $V$ is the potential energy of the system.

Due to the fact that the potential energy of the system remains constant during power transmission in the gearbox, the kinetic energy of the system can be expressed as:

$$\begin{array}{c}\multicolumn{1}{c}{T=\frac{1}{2}(I_{s1s2}{\omega }_{s2}^2+I_{c2s3r4r1}{\omega }_{c2}^2+I_{r2}{\omega }_{r2}^2}\\ +I_{c3c4}{\omega }_{c3}^2+I_{r3}{\omega }_{r3}^2+I_{s4}{\omega }_{s4}^2+I_{c1}{\omega }_{c1}^2)\hfill \end{array}$$

(20)

Where $I_{s1s2}$ is the equivalent moment of inertia of $s_1$ / $s_2$ and the connected shaft; $I_{c2s3r4r1}$ is the equivalent moment of inertia of $c_2$ / $s_3$ / $r_4$ / $r_1$ and the connected shaft; $I_{r2}$ is the equivalent moment of inertia of $r_2$ and the connected shaft; $I_{c3c4}$ is the equivalent moment of inertia of $c_3$ / $c_4$ and the connected shaft; $I_{r3}$ is the equivalent moment of inertia of $r_3$ and the connected shaft; $I_{s4}$ is the equivalent moment of inertia of $s_4$ and the connected shaft; $I_{c1}$ is the equivalent moment of inertia of $c_1$ and the connected shaft.

The basic form of the Lagrange equation can be expressed as:

$$\frac{d}{\mathrm{dt}}\left(\frac{\partial L}{\partial {\stackrel{·}{q}}_{\alpha }}\right)-\frac{\partial L}{\partial q_{\alpha }}=Q_{\alpha }$$

(21)

According to the equations (6)–(21), it can be obtained that:

$$\left[\begin{array}{cc}{b}_{11}\hfill & b_{12}\hfill \\ \multicolumn{1}{c}{b_{21}}& b_{22}\hfill \end{array}\right]\left[\begin{array}{c}{\stackrel{··}{\phi }}_i\hfill \\ \multicolumn{1}{c}{{\stackrel{··}{\phi }}_o}\end{array}\right]=\left[\begin{array}{c}{L}_1\hfill \\ \multicolumn{1}{c}{L_2}\end{array}\right]$$

(22)

Where:

$$\begin{array}{c}\multicolumn{1}{c}{b_{11}=I_{s1s2}+a_{21}^2{I}_{c2s3r4r1}+a_{51}^2{I}_{c3c4}}\\ +a_{41}^2{I}_{r3}+a_{61}^2{I}_{s4}+a_{71}^2{I}_{c1}\hfill \end{array}$$

(23)

$$\begin{array}{c}\multicolumn{1}{c}{b_{12}=a_{21}{a}_{22}{I}_{c2s3r4r1}+a_{51}{a}_{52}{I}_{c3c4}}\\ +a_{41}{a}_{42}{I}_{r3}+a_{61}{a}_{62}{I}_{s4}+a_{71}{a}_{72}{I}_{c1}\hfill \end{array}$$

(24)

$$\begin{array}{c}\multicolumn{1}{c}{b_{21}=a_{21}{a}_{22}{I}_{c2s3r4r1}+a_{51}{a}_{52}{I}_{c3c4}}\\ +a_{41}{a}_{42}{I}_{r3}+a_{61}{a}_{62}{I}_{s4}+a_{71}{a}_{72}{I}_{c1}\hfill \end{array}$$

(25)

$$\begin{array}{c}\multicolumn{1}{c}{b_{22}=a_{22}^2{I}_{c2s3r4r1}+a_{52}^2{I}_{c3c4}}\\ +a_{42}^2{I}_{r3}+a_{62}^2{I}_{s4}+a_{72}^2{I}_{c1}\hfill \end{array}$$

(26)

$$L_1=T_i+a_{41}{T}_{\mathrm{TC}1}+a_{61}{T}_{\mathrm{TC}2}$$

(27)

$$L_2=a_{42}{T}_{\mathrm{TC}1}+a_{62}{T}_{\mathrm{TC}2}-T_o$$

(28)

It can be seen from equation (22) that, $b_{11}$ , $b_{12}$ , $b_{21}$ , and b₂₂ are constants. Therefore, during the shifting process, there is a fixed linear relationship between the angular acceleration of the transmission input shaft (turbine shaft), the angular acceleration of the transmission output shaft, the torque transmitted by the clutch, the torque on the transmission input shaft, and the torque on the transmission output shaft. Therefore, when identifying the rotational inertia model of the transmission, it is not necessary to identify the rotational inertia of each component individually. Instead, it is sufficient to identify $b_{11}$ , $b_{12}$ or $b_{21}$ , $b_{22}$ as equivalent rotational inertias. In this study, experimental data was collected during stable operation in gears 1 and 2, and an iterative algorithm using a variable forgetting factor was employed to obtain the numerical values of the equivalent rotational inertias.

According to the iterative algorithm with a variable forgetting factor, the optimization objective with forgetting factor can be obtained as follows:

$$J_{\mathrm{RLS}}=\sum _{k=1}^N{\lambda }^{N-k}{[z(k)-h^T(k)\theta ]}^2$$

(29)

Where $\lambda (0<\lambda <1)$ is the forgetting factor, which is used to adjust the weight of new and old data. The weight of earlier data is smaller, and the weight of newly collected data is larger; $h(k)$ is the sample input set; $z(k)$ is the sample output set; $\theta$ is the set of parameters to be identified.

The expression of recursive least squares method with forgetting factor is as follows. ²³

$$\begin{array}{c}\hat{\theta }\left(k\right)=\hat{\theta }(k-1)\hfill \\ +K\left(k\right)[z\left(k\right)-h^{\mathrm{T}}(k)\hat{\theta }(k-1)]\hfill \end{array}$$

(30)

$$K\left(k\right)=\frac{P(k-1)h\left(k\right)}{1+h^{\mathrm{T}}(k)P(k-1)h\left(k\right)}$$

(31)

$$P\left(k\right)=[I-K\left(k\right)h^T\left(k\right)]P(k-1)$$

(32)

According to equation (22), the sample set and parameter set of the system to be identified are as follows:

$$\begin{array}{c}{h}^T\left(k\right)=h_1^T\left(k\right)=h_2^T\left(k\right)\hfill \\ ={[{\stackrel{··}{\phi }}_i(k-1),{\stackrel{··}{\phi }}_o(k-1)]}^T\hfill \end{array}$$

(33)

$$\theta =\left[\begin{array}{c}{\theta }_1\hfill \\ \multicolumn{1}{c}{{\theta }_2}\end{array}\right]=\left[\begin{array}{cc}{b}_{11}\hfill & b_{12}\hfill \\ \multicolumn{1}{c}{b_{21}}& b_{22}\hfill \end{array}\right]$$

(34)

According to the power conservation of planetary gears, the torque corresponding relationship of each planetary row can be determined as follows:

$$T_{\mathrm{si}}:T_{\mathrm{ri}}:T_{\mathrm{ci}}=1:k_i:-(1+k_i)$$

(35)

Where $T_s$ is the torque of the sun gear; $T_{\mathrm{ri}}$ is the ring gear torque; $T_{\mathrm{ci}}$ is the planet carrier torque.

Therefore, the torque of clutches $T_{\mathrm{TC}1}$ and $T_{\mathrm{TC}2}$ can be deduced when the transmission is in first and second gear.

$$\{\begin{array}{c}{T}_{\mathrm{TC}1}=T_o/a_{42}{T}_{\mathrm{TC}2}=0(\mathrm{gear}=1)\hfill \\ \multicolumn{1}{c}{T_{\mathrm{TC}2}=T_o/a_{62}{T}_{\mathrm{TC}1}=0(\mathrm{gear}=2)}\end{array}$$

(36)

During the identification process, the test bench can collect the input shaft speed, output shaft speed, and torque of the transmission, so the sample set of $h(k)$ and $z(k)$ can be obtained. During identification, assume that the initial identification value is small enough $\theta =[0.0010.001,0.0010.001]$ , the covariance matrix $P_{\left(0\right)}={10}^6\times I$ , and the initial forgetting factor is 0.99. The values of the parameters to be identified are obtained through identification, as shown in Table 2.

Table 2.

Equivalent inertia identification results.

Parameter	$b_{11}$	$b_{12}$ / $b_{21}$	$b_{22}$
Value	1.455	−0.835	4.574

Powertrain modeling

The engine is modeled as a torque source element, where the actual output torque $T_e$ of the engine is determined by the effective torque $T_i$ and the control torque $\mathrm{\Delta }{T}_{\mathrm{ec}}$ of the engine. The dynamic equation describing the engine rotational dynamics is as follows ²⁴ :

$$I_e{\stackrel{·}{\omega }}_e=T_e-T_p$$

(37)

$$T_e=T_i-\mathrm{\Delta }{T}_{\mathrm{ec}}$$

(38)

Where $I_e$ is the total inertia from the engine output end to the hydraulic torque converter pump wheel; ${\omega }_e$ is the angular speed of the engine; $T_e$ is the actual output torque of the engine; $T_p$ is the torque of the hydraulic torque converter pump wheel; $T_i$ is the effective torque of the engine; $\mathrm{\Delta }{T}_{\mathrm{ec}}$ is the external control torque of the engine.

The hydraulic torque converter is a hydraulic coupling mechanism that connects the engine and the transmission. Its primary characteristic parameters, such as the torque ratio, torque converter efficiency, and pump-turbine torque coefficient, can characterize the performance of the hydraulic torque converter. The dynamic equation describing the dynamics of the hydraulic torque converter is as follows:

$$T_p={\lambda }_B\rho {\mathrm{gn}}_p^2{D}^5$$

(39)

$$T_t=K{T}_p$$

(40)

$$n_t=i_p{n}_p$$

(41)

Where ${\lambda }_B$ is the torque coefficient of pump wheel; $\rho$ is the oil density of the working medium at the working oil temperature; $g$ is the acceleration of gravity; $n_p$ and $n_t$ are the rotational speed of pump wheel and turbine respectively; $D$ is the effective working diameter; $T_t$ is turbine torque; $K$ is the torque coefficient; $i_p$ is the transmission ratio of the hydraulic torque converter.

Based on the analysis of the kinematics and dynamics of the transmission, the dynamic equation of the transmission can be represented by the following equation:

$${\omega }_i={\omega }_t$$

(42)

$$T_i=T_t$$

(43)

$${\stackrel{·}{\omega }}_i=\frac{T_i+a_{41}{T}_{\mathrm{TC}1}+a_{61}{T}_{\mathrm{TC}2}-b_{12}{\stackrel{·}{\omega }}_o}{b_{11}}$$

(44)

$$T_o=a_{42}{T}_{\mathrm{TC}1}+a_{62}{T}_{\mathrm{TC}2}-b_{21}{\stackrel{·}{\omega }}_i-b_{22}{\stackrel{·}{\omega }}_o$$

(45)

The Karnop model is used for clutch modeling. When the clutch is sliding, the dynamic friction torque $T_{\mathrm{fslip}}$ is transmitted. The static friction torque $T_{\mathrm{fstick}}$ is transmitted when the clutch is fully engaged. Its dynamic equation is as follows. ²⁵

$$T_{\mathrm{ci}}=\{\begin{array}{c}{T}_{\mathrm{fslip}}=f(w_{\mathrm{rel}},p(t))\left|{\omega }_{\mathrm{rel}}\right|\ge \epsilon \hfill \\ T_{\mathrm{fstick}}=\{\begin{array}{c}{T}_S\mathrm{sgn}(T_k)\left|{\omega }_{\mathrm{rel}}\right|\le \epsilon ,\left|T_k\right|\ge T_S\hfill \\ T_k\left|{\omega }_{\mathrm{rel}}\right|\le \epsilon ,\left|T_k\right|\le T_S\hfill \end{array}\hfill \end{array}$$

(46)

Where the sliding friction torque can be expressed as:

$$T_{\mathrm{fslip}}=N_f{A}_p{r}_{e}p(t)\mu (t)\mathrm{sgn}({\omega }_{\mathrm{rel}})$$

(47)

$$\mu (t)={\mu }_C+({\mu }_S-{\mu }_C)\exp [-({\omega }_{\mathrm{rel}}/{\omega }_S{)}^{\delta }]$$

(48)

In the above formula, $T_{\mathrm{ci}}$ is the clutch torque; $T_k$ is the input torque of the clutch; $T_S$ is the static friction torque of the clutch; $N_f$ is the number of friction pairs; ${\omega }_{\mathrm{rel}}$ is the relative angular speed of the clutch master and slave ends; $A_p$ is the acting area of clutch oil pressure; $r_e$ is the equivalent clutch radius; $p(t)$ is the clutch oil pressure that changes with time; $\mu (t)$ is the dynamic friction coefficient; ${\mu }_C$ is the Coulomb friction coefficient; ${\mu }_S$ is the static friction coefficient; ${\omega }_S$ is the Stricbeck speed; $\delta$ is the Streiber shape coefficient; For $\delta =1$ , the Tustin model is obtained; When $\delta =2$ , the Gaussian exponential model is obtained. ²⁶

The acting radius r of the clutch friction plate can be directly defined according to the following formula.

$$r_e=\frac{2(r_o^3-r_i^3)}{3(r_o^2-r_i^2)}$$

(49)

Where $r_i$ and $r_o$ are the inner circle radius and outer circle radius of the clutch friction plate respectively.

The road resistance $T_f$ is:

$$T_f=\mathrm{mg}\sin \theta +\mathrm{mg}{f}_0\cos \theta +C_d{\mathrm{Av}}_v^2/21.15$$

(50)

The output end of the transmission transmits power to the two half-shafts connected to the wheel through the main reducer. Assuming that the torque of the two half-shafts is the same, the dynamic equation can be expressed as follows:

$$I_v{\stackrel{·}{\omega }}_v=T_o{i}_0-T_f/{\eta }_v$$

(51)

The driving efficiency of tracked vehicles can be expressed as:

$${\eta }_v=0.95-0.003v_v$$

(52)

In the above formula, $m$ is the mass of the whole vehicle; $g$ is the acceleration of gravity; $\theta$ is the road slope; $C_d$ is the vehicle wind resistance coefficient; $A$ is the windward area of the vehicle; $v_v$ is the vehicle speed; $I_v$ is the equivalent moment of inertia of the whole vehicle; ${\omega }_v$ is the wheel angular velocity; $i_0$ is main reduction ratio; ${\eta }_v$ is the track efficiency.

Research on improving shift quality based on engine coordinated control

The shift quality is usually evaluated by the jerk and sliding friction work loss, ²⁷ such as equations (53) and (54).

$$j=\frac{d{\stackrel{··}{v}}_v}{\mathrm{dt}}$$

(53)

$$w_c={\int }_{t_0}^{t_f}{T}_{\mathrm{ci}}({\omega }_b-{\omega }_f)\mathrm{dt}$$

(54)

Where $T_{\mathrm{ci}}$ is the friction torque of the shift clutch; ${\omega }_b$ and ${\omega }_f$ are the angular velocities of the driving and driven ends of the shift clutch; $t_0$ and $t_f$ are the starting and ending time of the gear shift respectively.

Based on the developed dynamic powertrain system simulation model, the simulation of the 1–2 upshifting process without coordination control is conducted. In the simulation, the vehicle is set to start in first gear, with the engine torque $T_e=5000$  N·m stably outputted and no external input torque applied. At 3.5 s, the signal to shift to second gear is issued. The torque converter is in an unlocked state during the shifting process, and the oil pressure curves of clutch C1 and C2 are obtained through experiments. As a result, the dynamic simulation results of the 1–2 upshifting process without coordination control are obtained, as shown in Figure 2.

Figure 2.

Dynamic simulation results of 1 liter 2 shifting process without coordinated control.

From Figure 2, it can be seen that the simulation results of various parameters are consistent with the operating conditions. The entire shifting process from first to second gear can be divided into four stages: pre-shift, torque phase, inertia phase, and post-shift. In the beginning of the torque phase, the oil pressure of the C1 clutch rapidly decreases, while the C2 clutch starts to engage slowly. At this point, the C1 clutch is not completely disengaged, and the C2 clutch is not fully engaged yet. The resistance experienced by the transmission (mainly due to the friction of the C2 clutch) does not increase significantly, so the input shaft speed of the transmission continues to increase gradually. When the torque phase ends and the inertia phase begins, the oil pressure of the C1 clutch continues to decrease, while the oil pressure of the C2 clutch keeps increasing. The C2 clutch is in a state of sliding friction. Through dynamic analysis of the shifting process, it is known that at this stage, the output torque of the transmission is mainly determined by the sliding friction torque of the C2 clutch. When the inertia phase ends, the C2 clutch is fully engaged and the C1 clutch is completely disengaged. At this point, the input shaft speed of the transmission is equivalent to the product of the output shaft speed and the second gear transmission ratio, indicating that the transmission has entered the second gear stage.

Many researchers have pointed out that improving the shifting quality can be achieved by coordinating and controlling the engine torque (fuel supply or intake air) during the shifting process. However, current research has not delved into the relationship between engine torque control in different phases (torque phase and inertia phase) and shifting quality. Although optimizing the clutch oil pressure variation curve during the shifting process can improve the shifting quality, it is challenging to apply the optimized oil pressure curve to all upshifting and downshifting processes, leading to certain application limitations. Therefore, based on keeping the clutch oil pressure unchanged, this study formulates six different coordinated control strategies as shown in Table 3 to investigate the impact of engine torque coordination control during the shifting process on shifting quality.

Table 3.

Coordinated control strategy for shift process.

Coordination-control strategy		Start-time (s)	Duration (s)	Engine-torque (N·m)
Torque phase control	ST 1	3.5	0.5	2000
	ST 2	3.5	0.5	200
	ST 3	3.5	1	200
Inertial phase control	ST 4	4	0.5	2000
	ST 5	4	0.5	200
	ST 6	4	0.6	200

The simulation results of the six control strategies are obtained and shown in Figures 3 and 4. From the simulation results, it can be observed that all six formulated coordinated control strategies are more effective in reducing the peak output shaft torque, impact severity, and sliding friction work compared to the case without coordinated control. However, the degree of improvement in shifting quality varies among the different strategies.

Figure 3.

Variation of torque, impact, and sliding friction work loss of strategy 1/2/3 output shaft.

Figure 4.

Variation of torque, impact, and sliding friction work loss of output shaft of strategy 4/5/6.

The simulation results of six control strategies are obtained through simulation, as shown in Figures 3 and 4. From the simulation results, it can be seen that the six coordinated control strategies formulated can reduce the peak torque, impact, and sliding friction work loss of the transmission output shaft more than the non-coordinated control, but the improvement of the shift quality is different.

Strategies 1, 2, and 3 mainly involve coordinating control at the beginning of the torque phase. From the variation of the transmission output shaft torque in Figure 3, it can be seen that reducing the engine’s output torque at the start of the torque phase creates a “torque pit.” The smaller the torque coordinated by the engine control, the more pronounced the “torque dip” becomes. If the same engine torque is applied during the torque phase (Strategies 2 and 3), the shape of the “torque dip” is similar. If the coordination control time is the same (Strategies 1 and 2), the smaller the engine’s coordinated torque, the lower the peak value of the output shaft torque. When the engine’s coordinated torque is 200  $N·m$ , the longer the coordination control time, the lower the torque peak value. Although Strategy 3 has a lower torque peak value and the greatest reduction in impact degree, it also produces a larger negative impact degree compared to Strategy 2. Therefore, coordination control duration is also a key factor affecting shifting quality.

Strategies 4, 5, and 6 primarily involve coordinated control during the inertia phase. From the torque variation of the transmission output shaft shown in Figure 4, it can be observed that strategies 5 and 6 have lower torque peaks compared to strategy 4. Under the same coordinated control torque from the engine (strategies 5 and 6), a longer coordination control time results in a larger negative jolt (seen in Figure 4). Based on the exploration of these patterns, the following conclusions can be drawn:

• (1) Without changing the clutch oil pressure, improving shift quality can be achieved by adopting engine coordinated control strategies. The three parameters – coordination control start time, coordination control duration, and coordination control engine torque – determine the quality of the shift;

• (2) Under reasonable coordination control parameters (strategies 2 and 5), although coordinating control during the torque phase can reduce torque peaks, impact, and sliding friction work loss more effectively compared to coordinating control during the inertia phase, it may introduce an additional “torque pit” during the torque phase;

• (3) To achieve optimal shift quality, optimization of the coordination control start time, coordination control duration, and coordination control engine torque is necessary.

Multi-objective parameter optimization for improving shift quality

Multi-objective parameter optimization strategy based on NSGA-II

NSGA-II is one of the most popular multi-objective genetic algorithms. It reduces the complexity of non-dominated sorting genetic algorithms and has the advantages of fast execution speed and good convergence of solution sets, making it a benchmark for other multi-objective optimization algorithms. According to the aforementioned research, using engine coordinated control strategy during shifting process can enhance the shifting quality. However, the selection of three parameters, namely the external control torque $\mathrm{\Delta }{T}_{\mathrm{ec}}$ , start time of fuel cut-off $t_s$ , and duration of fuel cut-off $\mathrm{\Delta }{t}_0$ , is based on engineering experience. In order to simultaneously achieve optimal values for both friction work and impact, these three parameters need to be optimized. From the simulation results mentioned above, it can be observed that as the parameters approach their optimal values, both friction work and impact reach their optimal values simultaneously. Therefore, in this study, the NSGA-II algorithm is employed to solve this multi-objective optimization problem, with the reciprocals of the root mean square value of impact during 1–2 shifting process and the minimum clutch C2 friction work as optimization objectives, as shown in equations (55) and (56).

$$\min J_1=1/J_{\mathrm{RMS}}=1/\sqrt{\frac{1}{t_f}{\int }_0^{t_f}{\stackrel{··}{v}}_v^2\mathrm{dt}}$$

(55)

$$\min J_2=W_c={\int }_0^{t_f}{T}_{c2}({\omega }_b-{\omega }_f)\mathrm{dt}$$

(56)

To investigate the improvement effect of coordinating the three optimal parameters in different phases (torque phase and inertia phase) on shifting quality, two optimization strategies were developed as shown in Figure 5. The optimization parameters for both strategies are the same: external control torque $\mathrm{\Delta }{T}_{\mathrm{ec}}$ , start time of fuel cut-off $t_s$ , and duration of fuel cut-off $\mathrm{\Delta }{t}_0$ . When shifting from gear N to gear N + 1, the first strategy searches for the optimal moment to coordinate the engine torque during the torque phase, while the second strategy searches for the optimal moment to coordinate the engine torque during the inertia phase.

Figure 5.

Principle of two optimization strategies.

Therefore, the number of optimization variables is 3 and can be represented as follows:

$$X=\left[\begin{array}{ccc}{x}_1\hfill & x_2\hfill & x_3\hfill \end{array}\right]=\left[\begin{array}{ccc}{t}_s\hfill & \mathrm{\Delta }{t}_0\hfill & \mathrm{\Delta }{T}_{\mathrm{ec}}\hfill \end{array}\right]$$

(57)

Table 4 shows the constraints of the optimization parameters of Strategy I and II.

Table 4.

Constraints of the two strategies.

Optimize variables	Optimization strategy
	Strategy I	Strategy II
$t_s$ $\mathrm{\Delta }{t}_0$ $\mathrm{\Delta }{T}_{\mathrm{ec}}$	$\begin{array}{c}{t}_1\le x_1\le t_2\hfill \\ 0\le x_2\le 1.5\hfill \\ 0\le x_3\le 5000\hfill \end{array}$	$\begin{array}{c}{t}_2\le x_1\le 0.5\hfill \\ 0\le x_2\le 1\hfill \\ 0\le x_3\le 5000\hfill \end{array}$

In this study, the NSGA-II algorithm was implemented with a population size of 100, a maximum iteration count of 50, a crowding distance coefficient of 0.35, and a fitness function deviation of 1e-6. The optimization results for the two strategies are shown in Table 5. From Table 5, it can be observed that the root mean square values of impact for optimization strategies 1 and 2 are 2.71 and 4.19 m/s³, respectively. This represents a reduction of 48.2% and 19.9%, respectively, compared to the case without coordinated control. The Pareto front of the objective functions obtained through optimization is shown in Figure 6. From the Pareto optimal results, it can be seen that using engine coordinated control strategy improves the shifting quality. Strategy 1 has a broader range of non-dominated solutions compared to Strategy 2, and it also provides better non-dominated solutions than Strategy 2. Hence, the NSGA-II algorithm can obtain the optimal solutions for both optimization strategies.

Table 5.

Optimization results of coordinated-control strategy during gear shift.

Optimization strategy	$x_1$	$x_2$	$x_3$	$1/J_1$	$J_2$
Normal	–	–	–	5.23	46,218
Strategy I	3.51	0.7	52.32	2.71	21,574
Strategy II	4.0	0.53	0.3	4.19	38,243

Figure 6.

Pareto boundary of two optimization strategies.

Hardware in the loop simulation verification and analysis

To validate the effectiveness of the proposed NSGA-II algorithm-based multi-objective optimization strategy in improving shifting quality, hardware-in-the-loop simulations were conducted. The test bench setup is shown in Figure 7. The powertrain system model runs in real-time simulation, considering the engine’s effective torque and throttle pedal as a simple linear relationship. Specifically, 0%–100% throttle opening corresponds to 0–5000 N·m torque. Gear positions and throttle inputs are simulated using external devices. An integrated controller sends engine control torque signals, and the signals are communicated through a CAN bus. Real-time monitoring is carried out through a designed interface. During the experiment, the procedure involved engaging first gear using the gear lever and then pressing the throttle pedal to 100% opening. When the vehicle speed reached 6.2 km/h, second gear was engaged. For comparative analysis, three sets of shift tests were conducted: uncoordinated control, coordination control with Strategy 1 optimized by the NSGA-II algorithm, and coordination control with Strategy 2 optimized by the NSGA-II algorithm. After multiple tests, the results of the upshift dynamic process experiments were obtained, as shown in Figure 8.

Figure 7.

Hardware-in-the-loop simulation bench.

Figure 8.

Shift dynamic process of two optimization strategies.

From the experimental results, it can be seen that Optimization Strategy 1 has a coordination control time that is 0.5 s earlier and a longer duration of 0.17 s compared to Optimization Strategy 2. The output shaft torque peak value for Optimization Strategy 1 is 6875 N·m, which is a 29% decrease compared to no coordination control. The output shaft torque peak value for Optimization Strategy 2 is 8570 N·m, which is a 12% decrease compared to no coordination control. From the variation in turbine speed, it can be seen that both Optimization Strategy 1 and 2 reduce the engine output torque during the shifting process. This allows for a shorter synchronization time for the clutch, which is beneficial for reducing frictional losses. However, the decrease in turbine speed is more significant. The longer the engine power interruption time, the faster the turbine speed decreases during the inertia phase. Therefore, while using coordination control to improve shifting quality, there will be a loss in vehicle power. From the variation in vehicle speed, it can also be seen that at the sixth second, the vehicle speed without coordination control can reach 14.8 km/h, while the vehicle speed with Optimization Strategy 1 and 2 is 12.9 and 13.4 km/h respectively. This represents a power reduction of 12.8% and 9.5%. Therefore, to improve shifting quality while also considering power, it is necessary to carefully select the coordination control parameters.

Figures 9 and 10 show the variation in impact and sliding friction work loss. The peak values of impact and sliding friction work loss during the shifting process without coordination control are 10.25 m/s³ and 46,218 J, respectively. For Optimization Strategy 1, the peak values of impact and sliding friction work loss are 7.68 m/s³ and 21,574 J, representing a reduction of 25% and 53.3% compared to no coordination control. For Optimization Strategy 2, the peak values of impact and sliding friction work loss are 9.75 m/s³ and 38,243 J, representing a reduction of 4.8% and 17.3% compared to no coordination control. Therefore, when the engine torque is stably output during the shifting process, Strategy 1 optimization shows better results in improving shifting quality. In other words, using engine coordination control strategy during the torque phase has a more pronounced effect on enhancing shifting quality.

Figure 9.

Comparison of the impact of two optimization strategies.

Figure 10.

Comparison of sliding friction work loss of two optimization strategies.

In order to verify the reliability of the proposed optimization strategies, coordinated control of the gear shifting process was conducted for engine torques of 4000 and 3000 N·m respectively, and the simulation results are shown in Table 6. It can be observed that when the engine torque is stabilized at 4000 N·m, the coordinated control of gear shifting based on optimization strategy 1 reduces the peak impact and sliding friction work by 51.6% and 43.9% respectively compared to no coordinated control. Similarly, using optimization strategy 2 for coordinated control of gear shifting reduces the peak impact and sliding friction work by 11.3% and 6.1% respectively compared to no coordinated control. When the engine torque is stabilized at 3000 N·m, the control effect of optimization strategy 1 is also superior to that of optimization strategy 2. Therefore, the multi-objective parameter optimization strategy proposed in this paper, which is based on coordinated engine control to improve shift quality, demonstrates a certain level of reliability.

Table 6.

Comparison of shift quality improvement effects at different torque outputs of the engine.

Engine-torque (N·m)	Optimization strategy	The peak values of impact (m/s3)	Friction work loss (J)
4000	Normal	9.67	23,220
	Strategy I	4.68	13,030
	Strategy II	8.58	21,800
3000	Normal	4.58	9633
	Strategy I	2.08	6196
	Strategy II	3.58	8942

Conclusion

This paper proposes a multi-objective parameter optimization strategy based on coordinated engine control to improve shift quality. The strategy considers the external control torque of the engine, the start of fuel cut-off timing, and the duration of fuel cut-off as optimization parameters. The sliding friction work loss and impact during the shifting process are used as performance indicators. The NSGA-II algorithm is employed to achieve multi-objective optimization of the coordinated control parameters, resulting in Pareto fronts of non-dominated solutions for torque phase and inertia phase coordination control, respectively. Based on the results of parameter optimization, a comparison of the effects of coordinated control in different phases (torque phase and inertia phase) reveals that when the engine torque is stably output during the shifting process, using engine coordinated control in the torque phase yields more noticeable improvements in shift quality. When applying the proposed optimized strategy for shift coordination control, it is observed that the shift quality is improved regardless of the engine’s constant torque output. Therefore, this strategy demonstrates a certain level of reliability. However, it has also been observed that using engine coordinated control (fuel cut-off control) to improve shift quality may weaken vehicle power. If both shift quality and power performance need to be improved simultaneously, a trade-off in the selection of coordinated control parameters is required. The analysis in this paper primarily focuses on the shifting process under the condition of the engine’s constant torque stable output, without considering the transient characteristics of the engine. In future work, a more accurate engine model will be established to develop control strategies accordingly.