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MPC-Based Downshift Control of Automated Manual Transmissions

  • Xin Li
  • Jidong Lyu
  • Jinlong Hong
  • Jinghua Zhao
  • Bingzhao Gao
  • Hong ChenEmail author
Article

Abstract

Automated manual transmissions, which usually adopt synchronizers to complete the gear shift process, have many advantageous features. However, the torque interruption and the challenging control objectives during the gear shift process limit its industrial application, especially for the power-on gear downshift. This paper proposes a model predictive control (MPC) method to control the clutch engagement process and effectively shorten the torque interruption, consequently enhancing the gear downshift quality. During the control law deduction, the proposed MPC also accounts for time-domain constraints explicitly. After the control law was deduced, it was validated through simulations under two typical power-on gear downshift working scenarios. Both of the simulation results demonstrate that the controller proposed in this paper can shorten the torque interruption time during power-on gear downshifts while minimizing vehicle jerk for overall satisfactory drivability.

Keywords

Automated manual transmission Gear downshift Heavy-duty trucks Clutch Model predictive control 

Lit of Symbols

\(\omega_{\text{e}}\)

Engine speed

\(\omega_{\text{c}}\)

Clutch output speed

\(I_{\text{e}}\)

Moment of inertia of engine crankshaft

\(I_{\text{v,i}}\)

Equivalent moment of inertia from the clutch output shaft to the vehicle

\(T_{\text{e}}\)

Engine output torque

\(T_{\text{c}}\)

Clutch friction torque

\(T_{\text{v0}}\)

Converted driving resistance

\(F_{\text{c}}\)

Clutch engagement force

\(\mu_{d}\)

Friction coefficient of the clutch

\(R_{\text{c}}\)

Effective radius of the clutch plates

\(N\)

Number of the clutch friction surfaces

1 Introduction

Currently, increasingly stringent emission and fuel consumption requirements demand higher efficiency from the transmissions of conventional vehicles. Compared with other kinds of transmissions, the automated manual transmission (AMT), shown in Fig. 1, has the highest power transfer efficiency, due to its rigid gear ratio and small power loss. In addition, the AMT also has the advantages of small volume, simple structure, and lightweight [1]; thus, it is suitable for trucks due to drivability and fuel economy.
Fig. 1

Scheme of AMT

However, when gears shift in an AMT, torque is interrupted, which limits its application in vehicles, and this loss of power may influence the drivability of the vehicle greatly. Many studies have dealt with this problem of AMTs from various aspects [2]. A nonlinear model-based control design method was applied in [3] to deduce an engine torque and engine speed control law; by using this control law, the gears shift process can be completed without the use of a synchronizer. Similarly, in [4], the feed-forward, bang–bang, and PID control algorithms were combined to control the torque and speed of the AMT vehicle powertrain, thus realizing gear shift control without using the clutch. One study [5] applied a fading memory Kalman filter to estimate the unavailable state variables and unknown input of a seamless clutchless two-speed transmission and validated it with gear upshift and downshift experimental results.

During the gear shift process, the synchronizer speed should be adjusted to achieve high gear shift quality. In this process, if the driveline damping and speed synchronizers are not controlled well, the shift time may increase. However, the clutch should be released when the vehicle begins to conduct gear shift; under this condition, the moment of inertia to be synchronized is small, which is helpful for shortening the gear shift time. In addition, from this point of view, although the torque interruption in an AMT cannot be eliminated entirely, the clutch friction torque can be deemed as a compensation torque when the clutch is in the slipping state, which is helpful for reducing the torque interruption duration. It is worth noting that the clutch cannot be engaged too fast to prevent a large vehicle jerk; in [6], a model predictive control (MPC) was applied in tackling these multiple contradictory objectives during the gear upshift process, which included fast torque tracking, less friction loss, and small vehicle jerk.

Power-on gear downshift is more likely to occur when the desired driving torque is suddenly increased, or the driving resistance increases. In other words, the vehicle cannot maintain the current driving speed or power without decreasing the gear ratio. As a consequence, gear downshift is more likely to cause a large longitudinal vehicle jerk compared with gear upshift. In addition, during the power-on gear downshift process, the engine speed needs to increase, thus satisfying the speed synchronization requirement of the synchronizer and the target gear. This engine speed regulation process needs to occur rapidly to shorten the torque interruption. Moreover, to avoid large vehicle jerk, no-lurch condition [7] should be satisfied, that is, the acceleration of the engine speed and the clutch output speed should be equal at the clutch slip-stick transition. Hence, it is more difficult to control the gear downshift process, compared with gear upshift.

In this paper, the dynamics of the AMT are given in detail and then, taking all of the requirements during the gear downshift process into consideration, a MPC-based control law is deduced. Then, this control law is applied for clutch friction torque tracking control, thus shortening the torque interruption time and improving the gear downshift quality. During the deduction of the control law, the MPC takes the time-domain constraints into account explicitly. The proposed control scheme was validated on a complete AMESim powertrain model. The simulation results show a significant power-on gear downshift quality improvement, shortened torque interruption, and reduced vehicle jerk.

The remainder of this paper is organized as follows. Section 2 introduces the dynamic process of power-on gear downshift of the AMT in detail, and Sect. 3 summarizes the control problems during the gear downshift process. In Sect. 4, the controller is designed using MPC, and it is evaluated in a complete powertrain simulation model in Sect. 5. Concluding remarks are given in Sect. 6.

2 Dynamic Process of Power-On Gear Downshift

Five steps are taken when the vehicle downshifts: (1) reduce engine torque, (2) disengage the clutch, (3) move the shift actuator to neutral after the transferred torque decreases to zero, (4) engage the target gear after the speed synchronizes, and (5) restore engine torque. As shown in Fig. 2, these actions can be lumped into three phases. The first is the torque reduction phase, in which the engine torque is reduced, and subsequently, the driveline torque is also reduced to 0 to guarantee smoothness when the vehicle operates in the neutral gear. The second phase should be satisfied to engage the synchronizer with the target gear without large oscillation. In the third phase, when the clutch begins to engage, the engine torque increases to the torque determined by the driver.
Fig. 2

Dynamic process of power-on gear downshift process

In the first phase, the clutch is disengaged to cut off the torque to the driveline while the engine torque is reduced to minimize the friction torque loss of the clutch. After the clutch is released completely, the engine torque should be regulated increasing, due to the engine speed increase requirement subsequently in the second phase (note that the engine speed should be decreased during the gear upshift). The first phase ends at the moment when the clutch is released totally, and the shift actuator can be moved to neutral easily.

In the second phase, the sleeve speed of the synchronizer is synchronized with the target gear. The clutch is disengaged completely in this phase, because the moment of inertia of the whole vehicle, which is linked with the output of the clutch, is large enough and the speed of this part can be considered as constant. Conversely, the moment of inertia that needs to be synchronized (from clutch output plate to the synchronizer) is small, such as 0.05 kg/m2 for trucks and 0.005 kg/m2 for passenger cars. Thus, the speed synchronization of the synchronizer can be completed in a very short time. After the speed of the synchronizer is synchronized to and engaged with the target gear, the second phase ends.

After the synchronizer is engaged with the target gear, and the engine is regulated a speed a little higher than the clutch output shaft speed, the clutch is re-engaged to transfer the torque from the engine to the tires. In this phase, the engine torque should be controlled to minimize both the clutch slipping torque loss and torque recovery time. After the clutch is re-engaged completely, the gear downshift process is completed and the engine torque is controlled according to the driver’s torque command.

The above analysis shows that the engine torque and speed control, clutch engagement and release control, and shift and actuator position control do not occur in a simple sequence, but overlap. To guarantee the overall performance of the gear shift, they should be controlled coordinately. In fact, after the gear shift instruction is received externally or determined by the AMT, the transmission control unit (TCU) coordinates the motions of the transmission actuators, clutch actuator, and engine to complete the shift process. On the one hand, the TCU sends the relevant signals to the engine control unit (ECU) via CAN bus; through corresponding controller and actuators, the engine torque and speed can be controlled precisely. On the other hand, the TCU also controls the clutch and synchronizer actuators [8].

The relevant actions of the gear downshift process of an AMT are listed in Table 1.
Table 1

AMT power-on gear downshift process

Phase

Engine

Clutch

Gear/synchronizer

E1/C1

Reduce engine torque

Disengage clutch

Current gear

E1/C1/G1

Regulate engine torque/increase engine speed

Disengage clutch

Disengage current gear/neutral gear

E1/G2

Regulate engine torque

 

Engage target gear

E2/C2

Reduce engine torque/engine speed is not less than that of the clutch output shaft

Engage clutch

Target gear

E3/C2

Restore engine torque

Engage clutch

Target gear

3 Control Problem Description

During the first phase of gear downshift, when the clutch is controlled in a slipping state, the torque transferred to the AMT input shaft is determined by the torque transfer capability of the clutch. If the clutch is disengaged too fast, torsional vibration of the transmission may occur. To avoid this problem, the clutch should be released completely at the moment when the clutch torque reaches 0. Due to the cost and durability issues, torque sensor is seldom used in commercial vehicles. Thus, torque observers [9, 10] are designed to determine when the clutch should be disengaged completely. In this process, the clutch can be regulated by feed-forward control using the estimated torque. Meanwhile, in this phase, the engine speed also needs to be regulated to increase, and PID control or feed-forward control can be used.

In the second phase, the shift actuator can disengage the current gear and consequently engage the target gear. The torque interruption happens in this phase as the clutch is disengaged completely. To effectively shorten the torque interruption and reduce the clutch friction loss, the engine speed should be controlled increasing to a value slightly higher than that of the target gear. The engine speed can be also controlled in an open-loop method during this process.

After the target gear is engaged, the clutch needs to re-engage and re-build the torque path from the engine to the tires, that is, the torque recovery phase. A summary of the multiple and contradictory control objectives during the clutch re-engagement process is as follows:
  1. 1.

    The clutch should be controlled completely engaged as soon as possible, i.e., minimizing clutch re-engagement time and clutch friction losses;

     
  2. 2.

    The transmission output torque should be regulated to the value determined by the driver as soon as possible, i.e., tracking the reference torque trajectory in the shortest time;

     
  3. 3.

    Vehicle jerk should be controlled small enough, i.e., guaranteeing the drivability during the whole gear shift process.

     

The first and second requirements need a faster rate of clutch engagement and torque recovery from the engine to the tires to reduce the torque interruption time and minimize the clutch friction losses. The third requirement is proposed to guarantee smooth transitions during the whole gear shift process, which is evaluated by using the vehicle jerk (namely change rate of acceleration). It is worth noting that the peak vehicle jerk happens at the moment of clutch slipping-stick transition, that is, the moment when the clutch begins to disengage or the clutch is engaged completely. Thus, to reduce the largest vehicle jerk, the accelerations of the engine speed and clutch output speed should be equal as well. To satisfy the overall requirements, the cooperation of engine control and clutch control [11] is important, as it has a significant influence on the overall gear shift quality. To simplify the control scheme, the engine can be controlled by an open-loop method, while the clutch regulation can be realized using feedback control. As the clutch is in a high transient state in this phase, it has to be controlled very precisely to achieve high gear downshift quality.

As aforementioned, the control issue during the last phase is to guarantee the performance of the clutch re-engagement. Considering the overall control objectives, MPC is a potential approach for handling multivariable systems; besides, it can also consider the constraints explicitly [12, 13, 14]. The real-time application of MPC to fast dynamic systems is always challenging. However, thanks to the rapid improvement of technologies, this problem has been solved to some extent and real-time MPC is applied in many fields, such as aerospace [15]. Therefore, this study applied a MPC-based controller to improve the power-on gear downshift quality.

4 Controller Design

4.1 Control-Oriented Modeling

The powertrain of interest in this paper is a medium-duty truck with a 6-speed AMT. From the viewpoint of the controller design, the driveline can be simplified as a two-mass system as shown in Fig. 3.
Fig. 3

Simplified two-mass drivetrain model

From Newton’s second law, the dynamic equations of the drivetrain motion can be presented as
$$\dot{\omega }_{\text{e}} = \frac{1}{{I_{\text{e}} }}T_{\text{e}} - \frac{1}{{I_{\text{e}} }}T_{\text{c}} $$
(1a)
$$\dot{\omega }_{\text{c}} = \frac{1}{{I_{{{\text{v}},{\text{i}}}} }}T_{\text{c}} - \frac{1}{{I_{{{\text{v}},{\text{i}}}} }}T_{{{\text{v}}0}} $$
(1b)
and subsequently can be arranged as
$$\Delta \dot{\omega } = \left( {\frac{{C_{\text{v}} }}{{I_{{{\text{v}},i}} }} - \frac{{C_{\text{e}} }}{{I_{\text{e}} }}} \right)\omega_{\text{e}} - \frac{{C_{\text{v}} }}{{I_{{{\text{v}},i}} }}\omega - \left( {\frac{1}{{I_{\text{e}} }} + \frac{1}{{I_{{{\text{v}},i}} }}} \right)T_{\text{c}} + \frac{1}{{I_{\text{e}} }}T_{\text{e}} + \frac{1}{{I_{{{\text{v}},i}} }}T_{\text{v0}}$$
(1c)
where \(\Delta \omega = \omega_{\text{e}} - \omega_{\text{c}}\).

4.2 Controller Design

As discussed above, to satisfy the requirements, during the clutch re-engagement, the engine torque \(T_{\text{e}}\) is controlled with an open-loop method and recovered as a ramp before the clutch is re-engaged in the last phase. The clutch torque \(T_{\text{c}}\) is used to satisfy the multiple and conflicting control objectives straightforwardly.

From Eq. (1), the clutch slipping speed \(\Delta \omega = \omega_{\text{e}} - \omega_{\text{c}}\) and the clutch transfer torque \(T_{\text{c}}\) are chosen as the system states, namely \(x = \left[ {\Delta \omega T_{\text{c}} } \right]^{\rm{T}}\). Then, Eq. (1c) is re-arranged in the following state-space form:
$$\dot{x} = A_{\text{c}} x + B_{cu} u + B_{cd} d $$
(2a)
$$y = Cx $$
(2b)
with
$$A_{\text{c}} = \left[ {\begin{array}{*{20}c} 0 & { - \left( {\frac{1}{{I_{\text{e}} + I_{{{\text{v}},{\text{i}}}} }}} \right)} \\ 0 & 0 \\ \end{array} } \right] $$
(3a)
$$B_{cu} = \left[ \begin{aligned} 0 \hfill \\ 1 \hfill \\ \end{aligned} \right] $$
(3b)
$$B_{cd} = \left[ {\begin{array}{*{20}c} {\frac{1}{{I_{\text{e}} }}} & {\frac{1}{{I_{\text{v,i}} }}} \\ 0 & 0 \\ \end{array} } \right] $$
(3c)
$$C = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right] .$$
(3d)
The control variable chosen is the derivative of the clutch transferred torque, where
$$u = \dot{T}_{\text{c}}$$
(4)
and
$$d = \left[ {T_{\text{e}} \;T_{{{\text{v}}0}} } \right]^{\rm{T}}$$
(5)
are deemed as the measured disturbances.

It is worth noting that it is clutch torque \(T_{\text{c}}\), not its derivative \(\dot{T}_{\text{c}}\), that is practically applied in the vehicle. Another assumption is that the clutch friction torque \(x_{2} = T_{\text{c}}\) can be obtained by using the clutch torque characteristic curve, which is related to the thrust bearing displacement of the clutch.

Model (2) can be discretized with sampling period \(T_{\text{s}}\), which is
$$x\left( {k + 1} \right) = Ax\left( k \right) + B_{u} u\left( k \right) + B_{d} d\left( k \right) $$
(6a)
$$y\left( k \right) = Cx\left( k \right) $$
(6b)
where \(A = e^{{A_{\text{c}} T_{\text{s}} }}\), \(B_{u} = \int_{0}^{{T_{\text{s}} }} {e^{{A_{\text{c}} \tau }} } {\text{d}}\tau \cdot B_{cu}\), \(B_{d} = \int_{0}^{{T_{\text{s}} }} {e^{{A_{\text{c}} \tau }} } {\text{d}}\tau \cdot B_{cd}\).

It is obvious that the first requirement (1) mainly focuses on the clutch re-engagement time and can be represented quantitatively by \(||\Delta \omega - \Delta \omega_{\text{ref}} ||^{2}\), wherein \(\Delta \omega_{\text{ref}}\) is chosen as 0 rad/s to shorten the re-engagement time as soon as possible. For (2), \(||T_{\text{c}} - T_{\text{c,ref}} ||^{2}\) can be selected as the penalty function, where \(T_{\text{c,ref}}\) is the desired clutch transfer torque determined by the driver. Finally, regarding the requirement of (3), \(||\dot{T}_{\text{c}} ||^{2}\) needs to be small enough, because the transmission output torque has a close relation with \(T_{\text{c}}\), it is reasonable to believe that vehicle jerk is related to \(\dot{T}_{\text{c}}\), so this can be added into the objective function.

Based on the analysis above, and combined with the discretized model (6), the objective function in this paper is defined as
$$J = ||\varGamma_{y} \left( {Y\left( {k + 1|k} \right) - R_{e} \left( {k + 1} \right)} \right)||^{2} + ||\varGamma_{u} U\left( k \right)||^{2}$$
(7)
where \(Y\left( {k + 1|k} \right)\) is the output prediction sequence, \(U\left( k \right)\) is the control input sequence, and \(R_{\text{e}} \left( {k + 1} \right)\) is the reference sequence, with
$$Y\left( {k + 1|k} \right) = \left[ \begin{array}{c} y\left( {k + 1|k} \right) \\ y\left( {k + 2|k} \right) \\ \vdots \\ y\left( {k + p|k} \right) \\ \end{array} \right]_{2p \times 1} $$
(8a)
$$U\left( k \right) = \left[ \begin{array}{c} u\left( {k|k} \right) \\ u\left( {k + 1|k} \right) \\ \vdots \\ u\left( {k + m - 1|k} \right) \\ \end{array} \right]_{m \times 1} $$
(8b)
$$R_{\text{e}} \left( {k + 1} \right) = \left[ \begin{array}{c} r\left( {k + 1} \right) \\ r\left( {k + 2} \right) \\ \vdots \\ r\left( {k + p} \right) \\ \end{array} \right]_{2p \times 1} $$
(8c)
and
$$y\left( {k + i|k} \right) = \left[ \begin{array}{c} \Delta \omega \left( {k + i|k} \right) \\ T_{\text{c}} \left( {k + i|k} \right) \\ \end{array} \right]_{2 \times 1} $$
(9a)
$$r\left( {k + i} \right) = \left[ \begin{array}{c} \Delta \omega_{\text{ref}} \left( {k + i} \right) \\ T_{\text{c,ref}} \left( {k + i} \right) \\ \end{array} \right]_{2 \times 1} $$
(9b)
with \(i = 1 \sim p\). Parameters \(p\) and \(m\) are the prediction horizon and control horizon, respectively, and they satisfy \(m \le p\). Matrices \(\varGamma_{y}\) and \(\varGamma_{u}\) are weighting matrices of the output variable y and control variable u.
$$\varGamma_{y} = \left[ \begin{array}{cccc} \gamma_{y,1} & 0 & \cdots & 0 \hfill \\ 0 & \gamma_{y,2} & \cdots & 0 \hfill \\ \vdots & \vdots & \ddots & 0 \hfill \\ 0 & 0 & \cdots & \gamma_{y,p} \hfill \\ \end{array} \right]_{2p \times 2p} $$
(10a)
$$\varGamma_{u} = \left[ \begin{array}{cccc} \gamma_{u,1} & 0 & \cdots & 0 \hfill \\ 0 & \gamma_{u,2} & \cdots & 0 \hfill \\ \vdots & \vdots & \ddots & 0 \hfill \\ 0 & 0 & \cdots & \gamma_{u,m} \hfill \\ \end{array} \right]_{m \times m} $$
(10b)
and
$$\gamma_{y,i} = \left[ {\begin{array}{*{20}c} {\gamma_{\Delta \omega ,i} } & 0 \\ 0 & {\gamma_{{T_{\text{c}} ,i}} } \\ \end{array} } \right]_{2 \times 2} $$
(11)
Then, the control objective function during the clutch re-engagement process can be presented as
$$\hbox{min} J\left( {x\left( k \right),U\left( k \right),m,p} \right) $$
(12)
subject to (6) and
$$\begin{aligned} u_{\hbox{min} } \left( {k + i} \right) \le u\left( {k + i} \right) \le u_{\hbox{max} } \left( {k + i} \right), \hfill \\ \quad \quad \quad \quad \quad \quad \quad \;\;\,i = 0,1, \cdots m - 1 \hfill \\ \end{aligned} $$
(13)
The main purpose of the input constraint (13) is to limit the vehicle longitudinal jerk and to smooth the overall gear downshift.

It can be seen that both \(\varGamma_{y}\) and \(\varGamma_{u}\) influence the gear downshift quality. Relatively higher \(\varGamma_{y}\) results in a faster clutch re-engagement process, while relatively higher \(\varGamma_{u}\) leads to smoother vehicle driving.

According to the theory of MPC, it can be inferred that the output prediction sequences are
$$Y\left( {k + 1|k} \right) = S_{x} x\left( k \right) + S_{u} U\left( k \right) + S_{d} D\left( k \right) $$
(14)
and \(S_{x}\), \(S_{u}\), and \(S_{d}\) are calculated by
$$S_{x} = \left[ \begin{array}{c} CA \\ CA^{2} \\ \vdots \\ CA^{p} \\ \end{array} \right]_{2p \times 2} $$
(15a)
$$S_{u} = \left[ \begin{array}{cccc}CB_{u} & 0 & \cdots & 0 \hfill \\ CAB_{u} & CB_{u} & \cdots & 0 \hfill \\ \vdots & \vdots & \ddots & \vdots \hfill \\ CA^{p - 1} B_{u} & CA^{p - 2} B_{u} & \cdots & CA^{p - m} B_{u} \hfill \\ \end{array} \right]_{2p \times m} $$
(15b)
$$S_{d} = \left[ \begin{array}{cccc} CB_{d} & 0 & \cdots & 0 \hfill \\ CAB_{d} & CB_{d} & \cdots & 0 \hfill \\ \vdots & \vdots & \ddots & \vdots \hfill \\ CA^{p - 1} B_{d} & CA^{p - 2} B_{d} & \cdots & CA^{p - m} B_{d} \hfill \\ \end{array} \right]_{2p \times 2m} $$
(15c)
In addition, the signals \(T_{\text{e}}\) and \(T_{v}\) can be obtained through the CAN bus or calculated by using external environmental information in the present sample time. However, their future values are unpredictable, so it is assumed in this paper that their values are equal to those in the present sample time and keep constant in the control horizon, that is,
$$D_{k} = \left[ \begin{array}{c} d\left( {k|k} \right) \\ d\left( {k|k} \right) \\ \vdots \\ d\left( {k|k} \right) \\ \end{array} \right]_{2m \times 1}$$
(16)
and
$$d\left( {k|k} \right) = \left[ \begin{aligned} T_{e} \left( {k|k} \right) \hfill \\ T_{v} \left( {k|k} \right) \hfill \\ \end{aligned} \right]_{2 \times 1} $$
(17)
If the inequality constraint (13) is not considered in the derivation of the control law, the optimal control sequence \(U^{*} \left( k \right) \in R^{m \times 1}\) at time \(k\) can be solved as
$$U^{*} \left( k \right) = \left( {S_{u}^{\rm{T}} \varGamma_{y}^{\rm{T}} \varGamma_{y} S_{u} + \varGamma_{u}^{\rm{T}} \varGamma_{u} } \right)^{ - 1} S_{u}^{\rm{T}} \varGamma_{y}^{\rm{T}} \varGamma_{y} E_{p} \left( {k + 1|k} \right)$$
(18)
where
$$E_{p} \left( {k + 1|k} \right) = R_{e} \left( {k + 1} \right) - S_{x} x\left( k \right) - S_{d} D\left( k \right) $$
(19)
Furthermore, the optimization problem (12) subject to input constraint (13) is re-arranged in a quadratic programming (QP) form:
$$\begin{aligned} \hbox{min} \frac{1}{2}U\left( k \right)^{\rm{T}} HU\left( k \right) + G\left( {k + 1|k} \right)^{\rm{T}} U\left( k \right) \hfill \\ {\text{s}}.{\text{t}}.\;C_{u} U\left( k \right) \ge b\left( {k + 1|k} \right) \hfill \\ \end{aligned} $$
(20)
where
$$\begin{aligned} H &= 2\left( {S_{u}^{\rm{T}} \varGamma_{y}^{\rm{T}} \varGamma_{y} S_{u} + \varGamma_{u}^{\rm{T}} \varGamma_{u} } \right) \hfill \\ G\left( {k + 1|k} \right) &= - 2\left( {S_{u}^{\rm{T}} \varGamma_{y}^{\rm{T}} \varGamma_{y} E_{p} \left( {k + 1|k} \right)} \right) \hfill \\ C_{u} &= \left[ { - I_{m \times m} \;I_{m \times m} } \right]^{\rm{T}} \hfill \\ b\left( {k + 1|k} \right) &= \left[ \begin{array}{c} - u_{\hbox{max} } \\ \vdots \\ - u_{\hbox{max} } \\ u_{\hbox{min} } \\ \vdots \\ u_{\hbox{min} } \\ \end{array} \right]_{2m \times 1} . \hfill \\ \end{aligned}$$
It can be seen that \(H \ge 0\) means the existence of the solution of (20), denoted as \(U^{*} \left( k \right)\) in this paper. We can take the above input constraints into account to and obtain the control sequence by solving (20). Then, the first element in \(U^{*} \left( k \right)\) is deemed as \(u\left( k \right)\), and the desired clutch torque \(T_{\text{c,req}}\) can be obtained by integrating \(u\left( k \right)\) and applied to control the clutch re-engagement. Then, in the next time sampling interval, this procedure is repeated to update the control command.
The clutch clamp force \(F_{\text{c}}\) can be obtained by using the desired clutch torque \(T_{\text{c,req}}\)
$$F_{\text{c}} = \frac{{T_{\text{c,req}} }}{{\mu_{d} R_{\text{c}} N}} $$
(21)

It is worth noting that, in practice, the torque transfer characteristics of the clutch are more complex, and more details about this can be seen in Ref. [16].

The overall control scheme during the clutch re-engagement process is shown in Fig. 4 and summarized as follows:
  • S1 Minimize the objective function (1) by using MPC to obtain the control law (18) or solving (20) under input constraint;

  • S2 Determine the desired clutch torque \(T_{\text{c,req}}\) according to the control law solved in S1 and subsequently calculate the clutch clamp force \(F_{\text{c}}\) using (21);

  • S3 Recover the engine torque \(T_{\text{e}}\) to the desired value before re-engaging the clutch; and

  • S4 Finish the gear downshift control when the engine torque is recovered and the clutch is re-engaged completely.

Fig. 4

Control scheme of the torque recovery phase

5 Simulation Results

5.1 Simulation Model of the Powertrain

To validate the control method, co-simulations of MATLAB/Simulink and the AMESim were carried out, wherein the controller was built in Simulink and the complete powertrain plant model was built in AMESim. The plant model is built based on a medium-duty truck equipped with a 6.2-L engine and a 6-speed AMT. More information about the powertrain model can be seen in Ref. [6].

Moreover, the observer and the controller are discretized, and the sampling interval is chosen as 10 s, while the simulation time interval is less than 0.1 ms.

5.2 Simulation Results of Power-On Gear Downshift

Power-on gear downshift always occurs when the desired driving torque increases suddenly or the driving resistance torque increases while the vehicle cannot drive normally at the current gear ratio. In this section, two conditions are taken as examples to conduct the simulations: (a) the accelerator pedal is suddenly pushed, that is, the desired driving torque is suddenly increased; and (b) the vehicle begins to drive up a slope.

5.2.1 Simulation of Sudden Acceleration

Figure 5 shows the simulation results when the accelerator pedal was suddenly pushed. To output more power, the AMT downshifted. The phases shown in Fig. 2 correspond to the following times: 13.2–13.4 s represents the torque reduction phase, 13.4–13.5 s represents the speed synchronization phase, and 13.5–14 s is the torque recovery phase.
Fig. 5

Simulation results of power-on gear downshift from second to first gear (tip-in of gas pedal)

First, the vehicle was driven in second gear, and at 13.2 s the accelerator pedal was pushed suddenly to increase the desired torque of the vehicle. To satisfy this requirement, the TCU determined that a gear downshift was needed and coordinated the clutch actuator, shift actuator, and engine control simultaneously to complete this process successfully with short shift time and minimal jerk.

At 13.2 s, the moment when the TCU decided to conduct a gear downshift, the clutch, and engine open-loop control was activated, the engine torque was reduced, and the clutch was subsequently disengaged gradually by tracking the reference trajectory. After the clutch was totally released at 13.4 s, the TCU controlled the shift actuator to set the AMT in neutral to complete the torque reduction phase.

During the speed synchronization phase, from 13.4 to 13.5 s, the TCU continued to control the shift actuator to engage the synchronizer with the first gear. Meanwhile, the engine speed was increased to 230 rad/s, which was a little higher than the clutch output shaft speed of 200 rad/s, to satisfy the speed synchronization condition.

Finally, in the torque recovery phase, from 13.5 to 14 s, the clutch was controlled by the proposed controller. In this phase, the prediction horizon was \(p\) = 10 s, the control horizon was \(m\) = 2 s, the reference value of the clutch slipping speed was \(\Delta \omega_{\text{ref}}\) = 0 rad/s, the desired clutch torque was \(T_{\text{c,req}}\) = 400 N m, and the constraints of the control variable were \(u_{\hbox{min} }\) = − 1000 N m/s and \(u_{\hbox{max} }\) = 1000 N m/s. As shown in the third subplot of Fig. 5, the clutch friction torque could be increased from zero to the desired torque gradually without severe oscillation, which guarantees a smooth clutch re-engagement process and is helpful for improving the overall power-on gear downshift performance.

Some relevant evaluation indicators [8] related to the gear downshift quality are listed in Table 2.
Table 2

Evaluation indicators related to the shift quality from second to first gear

Evaluation metrics

Value

Total shift time

0.8 s

Torque interruption time

0.35 s

Peak jerk

17.5 m/s3

Friction loss

1984 J

The total gear downshift time of 0.8 s, wherein the torque interruption lasted 0.35 s, demonstrates that the clutch friction torque during the re-engagement process can be deemed as a compensation torque that contributes to the reduction of the torque interruption duration. Moreover, the peak jerk of the vehicle was 17.5 m/s3, which is within an acceptable range [17]; thus, the drivability of the vehicle can also be guaranteed.

More realistically, the clutch actuation delay and the gear selection time should be included, and they would increase the whole gear downshift process duration by 0.2 s. However, in this case, the gear downshift process is still short enough that it has little negative influence on the vehicle drivability.

5.2.2 Simulation of Increased Resistance Torque

Figure 6 shows the simulation results when the vehicle drove from a flat road onto a road with slope of 5°. Up to 15 s, the vehicle drove on the flat road in fourth gear. When it began to climb the slope at 15 s, the TCU determined that a gear downshift should be carried out immediately. The phases corresponding to those in Fig. 2 are as follows: 15–15.2 s represents the torque reduction phase, 15.2–15.3 s represents the speed synchronization phase, and 15.3–15.7 s is the controlled torque recovery phase. The relevant evaluation indicators [8] are also listed in Table 3.
Fig. 6

Simulation results of power-on gear downshift from fourth to third gear (driving uphill)

Table 3

Evaluation indicators related to the shift quality from fourth to third gear

Evaluation metrics

Value

Total shift time

0.8 s

Torque interruption time

0.3 s

Peak jerk

11 m/s3

Friction loss

1582 J

It can be seen that the torque interruption time lasted 0.3 s, which is very short and has little influence on the vehicle dynamic performance during the whole gear downshift process; in addition, during the torque recovery phase, the clutch friction increased smoothly without large oscillation or vibration, which guarantees the comfort of vehicle occupants.

Thus, from these simulations, it can be concluded that the proposed controller can shorten the torque interruption of an AMT during power-on gear downshifts while improving the drivability for the driver, which validates the effectiveness of the controller.

6 Conclusions and Future Work

The multiple and contradictory control objectives during the AMT power-on gear downshift process influence the overall vehicle drivability. Aiming at this problem, this study applied MPC to improve the overall power-on gear downshift performance. Our study can be summarized as follows:
  1. (1)

    The dynamic process of power-on gear downshift was introduced, and its mathematical state-space model was explained;

     
  2. (2)

    The multiple and contradictory control objectives, including minimizing clutch engagement time, minimizing vehicle jerk, and shortening the engine torque recovery time, were considered in the objective control function;

     
  3. (3)

    MPC was applied to deduce the clutch engagement control law during the deduction, and the constraints were taken into consideration to explicitly obtain a practical control law; and

     
  4. (4)

    Simulations using the proposed control method were conducted under two typical maneuvers of power-on downshifting showed that the torque interruption time is short enough to have little impact on the drivability of the vehicle.

     

The proposed control scheme was validated through simulations. To illustrate the practical power-on gear downshifting performance of the controller, tests in an actual vehicle need to be conducted in our future work.

Under practical conditions, more issues should be evaluated, such as the time delay due to the hydraulic drive of the actuators and the engine intake-to-power delay. These are the next problems to be solved.

Notes

Acknowledgements

This work was supported by the National Nature Science Foundation of China (61520106008), China Automobile Industry Innovation and Development Joint Fund (U1664257), and Jilin Province Department of Education “Thirteen Five” scientific and technological research projects (JJKH20170379KJ).

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Copyright information

© China Society of Automotive Engineers(China SAE) 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Automotive Simulation and ControlJilin UniversityChangchunChina
  2. 2.Computer CollegeJilin Normal UniversitySipingChina
  3. 3.Department of Control Science and EngineeringJilin UniversityChangchunChina

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