Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2014, Article ID 963725, 12 pages
http://dx.doi.org/10.1155/2014/963725
Research Article
Coordinated Control of Downshift Powertrain of Combined
Clutch Transmissions for Electric Vehicles
Junqiu Li and Han Wei
National Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology, Beijing 100081, China
Correspondence should be addressed to Junqiu Li; [email protected]
Received 6 March 2014; Accepted 11 May 2014; Published 1 June 2014
Academic Editor: Jun-Juh Yan
Copyright © 2014 J. Li and H. Wei. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
To improve the shift quality of electric vehicles equipped with two-gear automatic transmissions, the coordinated control of the
combined clutch and the motor is proposed. The dynamic model of shift process is built up, the dynamic characteristics of each
phase of downshift process are analyzed, and linear quadratic optimal control is used to optimize the shift process. As a result,
the optimal trajectories of the motor torque and oil pressure of the combined clutch are obtained. Compared to the clutch control
only, the simulation results indicate that shift quality is improved remarkably by employing the proposed coordinated control.
Specifically, the shift jerk and sliding friction work are decreased by 43% and 44%, respectively, with accelerator pedal angle 50%.
In contrast, the reduced percentages are 57% and 89% when accelerator pedal is not depressed.
1. Introduction
Electric vehicles (EVs) are paid wide attention to on account
of deteriorated environment and energy crisis in recent years.
The characteristics of wide working range, constant torque at
low speed, and constant power at high speed make the motor
suitable for vehicles, and transmissions can be removed in
theory. Even so, large EVs still need transmissions with
fewer gears to keep climbing performance and high speed
performance in balance. Electric buses used in the 2008
Olympic Games adopted an AMT with three gears, whose
synchronizers were worn badly [1]. The two-gear combined
clutch transmission studied in this paper uses planetary gear
set to change the gear ratio, but the torque converter is
removed; therefore it integrates the advantages of ATs and
AMTs [2, 3].
Shift process of conventional automatic transmissions is
mostly to engage the oncoming clutch during the process
of disengaging the off-going clutch [4–6]. Much different
from conventional automatic transmissions, the shift of twogear transmission is achieved by the combined clutch, which
includes a brake and a clutch. Both of them are structurally
connected. The shift action is executed by single hydraulic
system. Therefore, shift timing problem does not exist any
longer and only one hydraulic cylinder is in need, which
reduces the complexity of the shift structure and control.
Because of these differences, the dynamic model and control
strategy of shift process should be studied in detail.
Shift control focuses on the control of clutch pressure
in initial studies on automatic transmissions [7, 8]. With
the increasing demand for shift quality, the coordinated
powertrain control has emerged in recent years, the essence of
which is coordinated control of the engine and the shift clutch
to enhance shift quality and to extend the service life of the
clutch simultaneously. With respect to internal combustion
engine vehicles (ICEVs), Toyota applied Electric Controlled
Transmission-intelligence (ECT-i) to Lexus LS400 [9]. ECT-i
can control the automatic transmission A341E and engine
concertedly. Ibamoto et al. [10] suggested a way to use the estimated output torque and control of engine ignition advance
to optimize the gear shifts of an automatic transmission. A
control algorithm that combines speed and torque control of
the AMT vehicle powertrain to achieve shift control without
using the clutch is proposed [11, 12]. Anna and Govindswamy
[13] studied the influence of the engine torque control on
the shift quality based on different vehicles. Goetz et al. [14]
studied integrated powertrain control of gearshifts on twin
clutch transmissions, and a gearshift controller for twin clutch
transmissions is developed. The controller incorporates the
control of engine variables to achieve synchronization whilst
2
Mathematical Problems in Engineering
Hydraulic cylinder
Shift valve
Oil
Brake B
Driving shaft
Ring gear
Clutch C
Planet carrier
Differential
gear box
Driving
motor
Sun gear
Tire
Figure 1: Schematic graph of powertrain with combined clutch transmission.
the transfer of engine torque from clutch to clutch is managed
by a clutch slip control.
The significant difference between EVs and ICEVs is that
the driving motor replaces the engine. Good controllability
and sensitive response make the driving motor easier to
achieve coordinated control of powertrain. Hu et al. and
Zhang et al. [15, 16] presented a control strategy of the motor
torque and speed to perform the smooth gear shifts in AMTs
without releasing the clutch. Gu and Cheng [17] accomplished
the coordinated control of upshift power of twin clutch
transmission for EVs based on particle swarm optimization
(PSO). Zhu et al. [18] studied open-loop control method of
DCT shift process under pure electric vehicle system. Both
upshift and downshift algorithm were described.
Recently, optimal control theory, especially linear
quadratic regulator (LQR), has been widely applied in the
clutch shift control because of simplicity and engineering
advantages. Xue et al. [19] analyzed the oil pressure during
the clutch engagement process of CVTs under a variety of
work conditions adopting LQR algorithm. Qin and Chen
[20] realized the unification of optimal starting control
between DCTs and AMTs when the sliding friction work
was selected as the minimum object, and the shift jerk was
converted to one of constraints.
This paper studies the electric vehicle in which the
motor is installed in front of the two-speed combined clutch
transmission. On the basis of analysis of the shift process, a
dynamic model of the shift process is established, and the
dynamic characteristics of each phase are analyzed. Linear
quadratic optimal control is chosen as the control strategy
in which the motor torque and the oil pressure of combined
clutch are controlled concertedly to reduce the shift jerk
and friction work. As a result, the optimal trajectories of
the motor torque and oil pressure of the combined clutch
are obtained. The simulation results show that the proposed
control strategy can efficiently improve the shift quality.
2. Clutch System Description
We consider the powertrain in EVs with a two-gear combined
clutch transmission, as schematically shown in Figure 1.
The powertrain combines the motor characteristics of wide
working range, constant torque at low speed, and constant
power at high speed with the speed variation of transmission.
The combined clutch consists of a brake B and a clutch
C, which are structurally connected. Consequently, single
hydraulic system can afford to complete the shift action, with
oil pressure acting on the brake piston only. Efficient driving
and excellent shift quality can be implemented by designing
reasonable control strategy.
A planetary gear set is adopted as the shift gear and
combined clutch is used as the actuator. Power is transferred
from the sun gear to the planet carrier. When brake B is
engaged and clutch C is disengaged, the powertrain operates
on the 1st gear and the speed ratio is given out by 𝑖1 = 1 + 𝑘,
where 𝑘 is the ratio of the teeth number of the ring gear to
that of the sun gear. If clutch C is disengaged and brake B is
engaged, the vehicle is driven on the 2nd gear with a speed
ratio of 𝑖2 = 1.
The 2nd to 1st downshift is considered here, which is
divided into three phases: clutch C disengagement phase,
free phase, and brake B engagement phase. Specifically, firstly
hydraulic oil fills the brake cylinder, oil pressure moves the
brake piston, and clutch C is disengaged. Then there is a
free phase, where both the clutch C and the brake B are
disengaged. Lastly, along with the continuous movement of
the brake piston, the brake B starts to engage. When the brake
B is engaged fully, the downshift action is completed. During
the 1st to 2nd upshift, hydraulic system starts to unload and
return spring makes the brake B disengaged. After a free
phase, clutch C starts to engage until it is fully engaged.
3. Combined Clutch Transmission
Shift Dynamics Model
The vehicle powertrain may be treated as a multi-rigidbody system. For the purpose of simplification, damping,
and elasticity of transmission shaft, bearings and gear mesh
for reducing vibration and shock are ignored, and all parts
of the powertrain are assumed to exist in the form of
concentrated mass [21]. The dynamic model of combined
clutch transmission and force analysis of the planet gear are
Mathematical Problems in Engineering
3
where 𝜔𝑝(𝑎) is the absolute angular speed of the planet gear:
Jr + Jcl2 + Jbr1
Tc
Tcl
Tm
𝜔𝑝(𝑎) = 𝜔𝑝(𝑟) + 𝜔𝑝(𝑒) ,
Tr
Tbr
(2)
where 𝜔𝑝(𝑟) is the relative angular speed of the planet gear.
Ts
𝜔𝑝(𝑟) = (𝜔𝑐 − 𝜔𝑠 )(2/(𝑘 − 1)). 𝜔𝑝(𝑒) is the transport angular speed
of the planet gear, 𝜔𝑝(𝑒) = 𝜔𝑜 .
Based on the above equations and according to the
planetary gear train kinematics [23],
𝜔m
J1 + Js + Jcl1
𝜔o
Tf
𝜔𝑝(𝑎) = (𝜔𝑜 − 𝜔𝑚 )
Jc + J2
2
𝑘+1
2
+ 𝜔𝑜 =
𝜔 −
𝜔
𝑘−1
𝑘−1 𝑜 𝑘−1 𝑚
(3)
𝜔𝑚 + 𝑘𝜔𝑟 = (1 + 𝑘) 𝜔𝑜 .
Figure 2: Dynamic model of combined clutch transmission.
The principle of virtual work and Lagrange principle can
be arranged as follows:
Fc
𝜔(r)
p
∑ 𝛿𝜔𝑖 ⋅ 𝐽𝑖 𝜔̇ 𝑖 = ∑ 𝛿𝑊𝑖𝑍 + ∑ 𝛿𝜔𝑖 ⋅ 𝑀𝑖𝑒 ,
Fr
Tf
𝜔m
Fs
(4)
where 𝛿𝑊𝑖𝑍 is virtual work made by the force of constraint,
which is zero ignoring the slip of tooth mesh case, and 𝑀𝑖𝑒 is
the external torque. As a result, (4) can be transformed into
𝜔o
𝑛
∑ [𝛿𝜔𝑖 (𝐽𝑖 𝜔̇ 𝑖 − 𝑀𝑖𝑒 )] = 0,
(5)
𝑖=1
Tm
𝛿𝜔𝑠 [(𝐽1 + 𝐽𝑠 + 𝐽cl 1 ) 𝜔̇ 𝑠 + 𝑇cl − 𝑇𝑚 )
𝜔r
+ 𝛿𝜔𝑟 [(𝐽𝑟 + 𝐽cl 2 + 𝐽br1 ) 𝜔̇ 𝑟 − 𝑇cl + 𝑇br ] + 𝑁𝛿𝜔𝑝
(6)
× (𝐽𝑝 + 𝑚𝑝 𝑅𝑐2 ) 𝜔̇ 𝑝(𝑎) + 𝛿𝜔𝑐 [(𝐽𝑐 + 𝐽2 ) 𝜔̇ 𝑐 + 𝑇𝑅 ] = 0.
When (3) are considered, (6) is presented as follows:
Figure 3: Force analysis of the planet gear.
[(𝐽1 + 𝐽𝑠 + 𝐽cl 1 ) +
shown in Figures 2 and 3 individually, where 𝑇𝑚 is the motor
torque. 𝑇𝑓 is resistance torque. 𝑇cl is the clutch transmission
torque. 𝑇br is the brake transmission torque. 𝐽1 , 𝐽cl 1 , 𝐽cl 2 , 𝐽br1 ,
and 𝐽2 are the equivalent moment of inertia of the motor,
the driving and driven parts of the clutch, the driving parts
of the brake, and the vehicle translational mass, respectively.
In addition, 𝜔 means the angular velocity. 𝑇 represents the
torque. 𝐹 shows the force. Parameters relevant to sun gear,
ring gear, planet carrier, and planet gears are expressed in
subscripts 𝑠, 𝑟, 𝑐, and 𝑝.
The dynamic equation of every element in the planetary
mechanism during shift process can be presented as [22]
× (𝐽𝑝 + 𝑚𝑝 𝑅𝑐2 ) ] 𝜔̇ 𝑚 − [
= 𝑇𝑚 −
[𝐽𝑐 + 𝐽2 +
𝑇cl − 𝑇br − 𝑇𝑟 = (𝐽cl 2 + 𝐽br1 + 𝐽𝑟 ) 𝜔̇ 𝑟
𝑅 − 𝑅𝑠
(𝐹𝑟 − 𝐹𝑠 ) 𝑟
= 𝐽𝑝 𝜔̇ 𝑝(𝑎) ,
2
𝑘+1
(𝐽𝑟 + 𝐽cl 2 + 𝐽br1 )
𝑘2
+
2 (𝑘 + 1) 𝑁
(𝐽𝑝 + 𝑚𝑝 𝑅𝑐2 )] 𝜔̇ 𝑜
(𝑘 − 1)2
𝑘+1
1
𝑇 + 𝑇 ,
𝑘 cl 𝑘 br
(𝑘 + 1)2
(𝑘 + 1)2 𝑁
(𝐽𝑟 + 𝐽cl 2 + 𝐽br1 ) +
2
𝑘
(𝑘 − 1)2
× (𝐽𝑝 + 𝑚𝑝 𝑅𝑐2 ) ] 𝜔̇ 𝑜 − [
𝑇𝑚 − 𝑇cl − 𝑇𝑠 = (𝐽1 + 𝐽𝑠 + 𝐽cl 1 ) 𝜔̇ 𝑚
𝑇𝑐 − 𝑇𝑓 = (𝐽𝑐 + 𝐽2 ) 𝜔̇ 𝑜
1
4𝑁
(𝐽𝑟 + 𝐽cl 2 + 𝐽br1 ) +
2
𝑘
(𝑘 − 1)2
𝑘+1
(𝐽𝑟 + 𝐽cl 2 + 𝐽br1 )
𝑘2
+
(1)
=
2 (𝑘 + 1) 𝑁
(𝐽𝑝 + 𝑚𝑝 𝑅𝑐2 )] 𝜔̇ 𝑚
2
(𝑘 − 1)
𝑘+1
𝑘+1
𝑇cl −
𝑇 − 𝑇𝑓 .
𝑘
𝑘 br
(7)
4
Mathematical Problems in Engineering
Equation (7) is expressed in matrix form which means the
shift dynamic as follows:
𝑇𝑚
1
𝑘+1
0 ] [𝑇 ]
𝐽11 𝐽12 𝜔̇ 𝑚
[1 − 𝑘
cl ]
𝑘
][ ] = [ 𝑘 + 1
[
][
[𝑇br ]
𝑘+1
𝐽21 𝐽22 𝜔̇ 𝑜
−
−1
0
[
] [ 𝑇𝑓 ]
𝑘
𝑘
=[
𝑇 − 𝑎11 𝑇br
𝜔̇
𝐽 𝐽
].
[ 11 12 ] [ 𝑚 ] = [ 𝑚
−𝑎12 𝑇br − 𝑇𝑓
𝐽21 𝐽22 𝜔̇ 𝑜
4. Optimal Control for Downshift
where
+
1
(𝐽 + 𝐽 + 𝐽 )
𝑘2 𝑟 cl 2 br1
4𝑁
(𝐽𝑝 + 𝑚𝑝 𝑅𝑐2 )
(𝑘 − 1)2
𝐽12 = 𝐽21 = −
𝑘+1
(𝐽𝑟 + 𝐽cl 2 + 𝐽br1 )
𝑘2
−
𝐽22 = 𝐽𝑐 + 𝐽2 +
2 (𝑘 + 1) 𝑁
(𝐽𝑝 + 𝑚𝑝 𝑅𝑐2 )
(𝑘 − 1)2
(9)
(𝑘 + 1)2
(𝐽𝑟 + 𝐽cl 2 + 𝐽br1 )
𝑘2
(𝑘 + 1)2 𝑁
(𝐽𝑝 + 𝑚𝑝 𝑅𝑐2 )
2
(𝑘 − 1)
+
1
𝑎11 = − ,
𝑘
𝑎12 =
𝑘+1
.
𝑘
𝑁 is the number of the planet gear.
𝑇𝑓 depends on road drive resistance, composed of rolling
resistance 𝐹𝑓 , air resistance 𝐹𝑤 , and grade resistance 𝐹𝑖 , and
can be calculated from (10):
𝑇𝑓 =
𝐹𝑥 𝑟𝑤 (𝐹𝑓 + 𝐹𝑤 + 𝐹𝑖 ) 𝑟𝑤
=
𝑖0
𝑖0
(10)
𝐹𝑓 = 𝑚𝑔𝑓 cos 𝛼
(11)
𝐹𝑖 = 𝑚𝑔 sin 𝛼
(12)
𝐹𝑤 =
2
𝐶𝐷𝐴𝑢V
,
21.15
(13)
where 𝑚 is vehicle mass. 𝑔 is gravity acceleration. 𝑢V is vehicle
speed. 𝛼 is road ramp. 𝐶𝐷 is air resistance coefficient. 𝐴 is
windward area. 𝑟𝑤 is tire radius. 𝑖0 is final drive ratio. 𝑓 is
rolling resistance coefficient.
During clutch C disengagement phase, 𝑇cl ≠ 0, 𝑇br = 0,
(8) can be transformed into
𝑇 − 𝑎12 𝑇cl
𝜔̇
𝐽 𝐽
].
[ 11 12 ] [ 𝑚 ] = [ 𝑚
𝑎12 𝑇cl − 𝑇𝑓
𝐽21 𝐽22 𝜔̇ 𝑜
𝑇
𝐽11 𝐽12 𝜔̇ 𝑚
][ ] = [ 𝑚 ].
−𝑇𝑓
𝐽21 𝐽22 𝜔̇ 𝑜
According to previous studies [24, 25], there are two basic
requirements during the engagement of the clutch. One is
the comfortability, which means the engagement should be
smooth. The other requirement is longer clutch life, which
requires the dissipated energy caused by friction to be as small
as possible. The shift jerk defined as the vehicle acceleration
variation rate is to evaluate the comfortability or smoothness,
and the sliding friction work defined as the work produced
by friction between the clutch plates is used to evaluate
dissipated energy, which has positive correlation with the
shift time [26]. The shift jerk and sliding friction work
are chosen to establish the object function. The optimal
trajectories of oil pressure of the combined clutch and the
motor torque are obtained based on linear quadratic optimal
control.
4.1. Downshift Control Flow. Based on the dynamic analysis
of shift and the coordinated control idea of the motor torque
and the oil pressure of the combined clutch, the flow chart
of the shift control for EVs is proposed as shown in Figure 4
[27], where 𝑛𝑟 is the speed of ring gear and 𝑛0 is the specified
speed value at the end of the free phase. In this paper the
downshift coordinated control includes three phases: (1)
the coordinated control of the motor torque and the oil
pressure of transmission in clutch C disengagement phase;
(2) the control of the motor torque in the free phase; (3) the
coordinated control of the motor torque and the oil pressure
of transmission in brake B engagement phase. After shift, the
motor torque and the brake transmission torque should be
adjusted to power requirements.
4.2. Linear Quadratic Optimal Control. The control law
worked out by linear quadratic optimal control is the linear
function of state variables, which endows it with significance
in engineering. In the shift control, disturbance matrix is out
of consideration generally. However, it exists in this paper for
further accurate control model.
The linear time-varying system can be written into the
following state space model:
𝑋̇ (𝑡) = 𝐴𝑋 (𝑡) + 𝐵𝑈 (𝑡) + 𝑉,
(17)
(14)
During the free phase, 𝑇cl = 0, 𝑇br = 0, (8) can be
transformed into
[
(16)
(8)
𝑇𝑚 − 𝑎11 𝑇br − 𝑎12 𝑇cl
],
−𝑎12 𝑇br + 𝑎12 𝑇cl − 𝑇𝑓
𝐽11 = (𝐽1 + 𝐽𝑠 + 𝐽cl 1 ) +
During brake B engagement phase, 𝑇cl = 0, 𝑇br ≠ 0, (8)
can be transformed into
(15)
where 𝑉 is the disturbance matrix. We look for a control
which minimizes the performance index:
𝐽=
1 𝑡𝑓
∫ (𝑋(𝑡)𝑇 𝑄1 𝑋 (𝑡) + 𝑈(𝑡)𝑇 𝑄2 𝑈 (𝑡)) 𝑑𝑡.
2 0
(18)
Mathematical Problems in Engineering
5
Start
Free
phase
Control for the
Driving on the
motor torque
origin gear
No
nr < n0
Calculating the shifting speed
and throttle opening based on
the shifting schedule
Yes
No
Shifting
brake torque
Yes
Coordinated
control for the
motor torque
Decreasing the
clutch torque
No
No
No
nr is zero
No
The clutch
Yes
torque is zero
Brake B
engagement
phase
The brake is
Clutch C
disengagement
Coordinated
control for the
motor torque
Increasing the
engaged completely
Yes
phase
The clutch is
disengaged completely
End
Figure 4: Flow chart of downshift coordinated control.
The Hamiltonian dynamic equation governing the clutch
engagement can be constructed and presented in the following form:
𝐻=
1
(𝑋(𝑡)𝑇 𝑄1 𝑋 (𝑡) + 𝑈(𝑡)𝑇 𝑄2 𝑈 (𝑡))
2
Using the Hamiltonian approach and the adjoint equation, The following differential equations can be obtained
𝜕𝐻
= −𝑄𝑋 (𝑡) − 𝐴𝑇 𝜆 (𝑡) ,
𝜆̇ (𝑡) = −
𝜕𝑥
(19)
+ 𝜆𝑇 (𝐴𝑋 (𝑡) + 𝐵𝑈 (𝑡) + 𝑉) ,
where 𝜆 is lagrangian multiplier.
According to the maximum principle, (19) can be
acquired.
One has
𝜕𝐻
= 𝑄2 𝑈 (𝑡) + 𝐵𝑇 𝜆 (𝑡) = 0;
𝜕𝑢
(20)
𝑈 (𝑡) = −𝑄2 −1 𝐵𝑇 𝜆 (𝑡) .
(21)
equally,
(22)
𝑋̇ (𝑡) = 𝐴𝑋 (𝑡) − 𝐵𝑄2 −1 𝐵𝑇 𝜆 (𝑡) + 𝑉,
with the set of conditions
𝑋 (𝑡0 ) = 𝑥0 ,
(23)
𝑁 [𝑋 (𝑡𝑓 ) , 𝑡𝑓 ] = 0,
(24)
𝜆 (𝑡𝑓 ) =
𝜕𝑁𝑇
𝜕𝑋 (𝑡𝑓 )
V,
(25)
where 𝑡0 is initial time, 𝑡𝑓 is the terminal time, V is the
unknown constant, and 𝑁 is the terminal constraint.
6
Mathematical Problems in Engineering
Considering the effect of the perturbation matrix and the
terminal constraint [28], define
𝜆 (𝑡) = 𝑃 (𝑡) 𝑋 + 𝑀 (𝑡) V + ℎ (𝑡)
(26)
𝜓 = 𝐾 (𝑡) 𝑋 + 𝐿 (𝑡) V + 𝑟 (𝑡) ,
(27)
where 𝜓 is the terminal function. 𝑃(𝑡) is the matrix that solves
the Riccati equation. ℎ(𝑡) has been introduced in order to
compensate for the presence of the disturbance vector 𝑉.
𝑀(𝑡) is used to solve the problem of the unknown constant V.
By computing (26)-(27) at 𝑡𝑓 , the 𝑃(𝑡𝑓 ), 𝑀(𝑡𝑓 ), ℎ(𝑡𝑓 ), 𝐾(𝑡𝑓 ),
𝐿(𝑡𝑓 ), and 𝑟(𝑡𝑓 ) can be obtained.
By substituting (26)-(27) in (22), after simple algebraic
manipulations one obtains the following three matrix differential equations:
𝑃̇ (𝑡) + 𝑃 (𝑡) 𝐴 + 𝐴𝑇 𝑃 (𝑡) − 𝑃 (𝑡) 𝐵𝑄2 −1 𝐵𝑇 𝑃 (𝑡) + 𝑄1 = 0
𝑀̇ (𝑡) − 𝑃 (𝑡) 𝐵𝑄2 −1 𝐵𝑇 𝑀 (𝑡) + 𝐴𝑇 𝑀 (𝑡) = 0
Substitution of transformations in (33) into (14) results in
the equation of motion of the dynamic system in terms of the
state variables as follows:
𝑋̇ = 𝐴 1 𝑋 + 𝐵1 𝑈 + 𝑉1 ,
where
𝑇
𝑋 = [𝑥1 𝑥2 𝑥3 ] ,
𝑎12 (𝐽21 + 𝐽22 ) −𝑎12 2 (𝐽11 + 2𝐽12 + 𝐽22 )
0
]
[ 𝐽 𝐽 −𝐽 𝐽
𝐽11 𝐽22 − 𝐽12 𝐽21
],
𝐴1 = [
]
[0 11 22 0 12 21
0
0
0
0
] (35)
[
0 0
𝐵1 = [ 1 0 ] ,
[0 1]
(28)
These differential equations can be solved backward in
time from the above terminal conditions. Now, by differentiating (27) we have
[𝐾̇ (𝑡) + 𝐾 (𝑡) 𝐴 1 − 𝐾 (𝑡) 𝐵𝑄2 −1 𝐵𝑇 𝑃 (𝑡)] 𝑋 (𝑡)
(30)
that holds for any 𝑋(𝑡) and V. Therefore
𝐾̇ (𝑡) + 𝐾 (𝑡) 𝐴 − 𝐾 (𝑡) 𝐵𝑄2 −1 𝐵𝑇 𝑃 (𝑡) = 0
(31)
𝑟 ̇ (𝑡) − 𝐾 (𝑡) 𝐵𝑄2 −1 𝐵𝑇 ℎ (𝑡) + 𝐾 (𝑡) 𝑉 = 0,
which determine 𝐾(𝑡), 𝐿(𝑡), and 𝑟(𝑡). From (23), since V is a
constant, one obtains
V = 𝐿−1 (𝑡0 ) [𝜓 − 𝐾 (𝑡0 ) 𝑋 (𝑡0 ) + 𝑟 (𝑡0 )] .
Finally, the optimal control variables 𝑈 (𝑡) and state
variables 𝑋∗ (𝑡) can be obtained.
4.3. Optimal Control Model for Downshift
4.3.1. Clutch C Disengagement Phase. Considering the shift
jerk and sliding friction work of downshift process and
the coordinated control idea, the following state variables,
control variables, and transformations are introduced below:
𝑢1 =
𝑑𝑇𝑚
,
𝑑𝑡
𝑥2 = 𝑇𝑚 ,
𝑢2 =
𝑑𝑇cl
.
𝑑𝑡
𝑥3 = 𝑇cl ,
(33)
𝑡𝑚
𝑡𝑚
0
0
(37)
𝑊 = ∫ 𝑇cl (𝜔𝑚 − 𝜔𝑟 ) 𝑑𝑡 = ∫ 𝑥1 𝑥3 𝑑𝑡.
It is shown through experiments that the shift jerk is
inversely proportional to friction work and to time as well.
Therefore, the shift jerk may be reduced by increasing the
frictional time but cause short service life of the clutch at
the expense of the plate excessive wear. For a compromise of
the two evaluation criterions in contradictory, an objective
function is proposed for the optimal control and expressed
below:
𝐽=
(32)
∗
𝑥1 = 𝜔𝑚 − 𝜔𝑟 ,
𝑟𝑤
𝑑𝑎 𝑟𝑤 𝑑2 𝜔𝑜
=
=
2
𝑑𝑡
𝑖0 𝑑𝑡
𝑖0 (𝐽11 𝐽22 − 𝐽21 𝐽12 )
× [𝑎12 (𝐽11 + 𝐽21 ) 𝑢2 − 𝐽21 𝑢1 ] ,
+ 𝑟 ̇ − 𝐾𝐵𝑄2 −1 𝐵𝑇 ℎ + 𝐾𝑉 = 0,
(36)
where 𝑡𝑚 is the end time of clutch C disengagement.
In this phase, the shift jerk 𝑗 and the sliding friction work
𝑊 are expressed below:
(29)
and, by using (22)
𝐿̇ (𝑡) − 𝐾 (𝑡) 𝐵𝑄2 −1 𝐵𝑇 𝑀 (𝑡) = 0
𝑎12 (𝐽11 + 𝐽21 )
[ 𝐽 𝐽 − 𝐽 𝐽 𝑇𝑓 ]
]
𝑉1 = [
[ 11 22 0 12 21 ] .
0
[
]
𝑁 [𝑥 (𝑡𝑚 ) , 𝑡𝑚 ] = 𝑥3 (𝑡𝑚 ) = 0,
𝑗=
+ [𝐿 (𝑡) − 𝐾 (𝑡) 𝐵𝑄2 −1 𝐵𝑇 𝑀 (𝑡)] V
𝑇
𝑈 = [𝑢1 𝑢2 ] ,
When clutch C is disengaged completely, its torque must
be zero; that is, the terminal constraint is
ℎ̇ (𝑡) − 𝑃 (𝑡) 𝐵𝑄2 −1 𝐵𝑇 ℎ (𝑡) + 𝐴𝑇 ℎ (𝑡) + 𝑃 (𝑡) 𝑉 = 0.
𝐾̇ (𝑡) 𝑋 + 𝐾 (𝑡) 𝑋̇ + 𝐿̇ (𝑡) V + 𝑟 ̇ (𝑡) = 0,
(34)
1 𝑡𝑚
∫ (𝑊 + 𝜂𝑗2 ) 𝑑𝑡
2 0
=
1 𝑡𝑚
2
∫ {𝑥1 𝑥3 + 𝜂[𝑎12 (𝐽11 + 𝐽21 )𝑢2 − 𝐽21 𝑢1 ] } 𝑑𝑡
2 0
=
1 𝑡𝑚 𝑇
∫ (𝑥 𝑄1 𝑥 + 𝑢𝑇 𝑄2 𝑢) 𝑑𝑡,
2 0
(38)
where 𝜂(0 < 𝜂 < 1) is weight coefficient of the shift jerk and
the larger the 𝜂 is, the more the shift jerk is considered [29]:
0 0 0.5
𝑄1 = [ 0 0 0 ] ,
[0.5 0 0 ]
𝐽21 2
0
𝑄2 = 𝜂 [
2] .
2
−2𝑎12 (𝐽11 + 𝐽21 ) 𝐽21 𝑎12 (𝐽11 + 𝐽21 )
(39)
Mathematical Problems in Engineering
7
Obviously, the optimal coordinated control is equivalent
to seeking the optimal trajectories of the motor torque and
the clutch C transmission torque to minimize the value of the
object function.
By computing (26)-(27) at 𝑡𝑚 , one obtains
0 0 0
𝑃 (𝑡𝑚 ) = [0 0 0] ,
[0 0 0]
0
𝑀 (𝑡𝑚 ) = [0] ,
[1]
0
ℎ (𝑡𝑚 ) = [0] ,
[0]
𝐾 (𝑡𝑚 ) = [0 0 1] ,
(40)
𝐿 (𝑡𝑚 ) = 0,
𝑟 (𝑡𝑚 ) = 0.
4.3.2. Free Phase. During this phase, both the clutch C and
the brake B are disengaged. When the motor torque is
constant, the shift jerk and the sliding friction work will
keep being zero, which is an ideal state. However, in order to
accomplish the fast synchronization of the driving and driven
parts of the brake and to decrease the sliding friction work in
the brake B engagement phase, the motor torque is supposed
to be controlled under the condition that the shift jerk is less
than the recommended value, 10 m/s3 .
Define
𝑥2 = 𝑇𝑚 ,
𝑢=
𝑑𝑇𝑚
.
𝑑𝑡
𝑋̇ = 𝐴 2 𝑋 + 𝐵2 𝑈 + 𝑉2 ,
(42)
where
𝑇
𝑉2 = [
[
(𝑎11 𝐽12 − 𝑎12 𝐽11 ) 𝑇𝑓
𝐽11 𝐽22 − 𝐽12 𝐽21
0
(46)
0
𝐵2 = [ ] ,
1
4.3.3. Brake B Engagement Phase. Define
𝑥1 = 𝜔𝑟 ,
𝑢1 =
𝑑𝑇𝑚
,
𝑑𝑡
𝑢2 =
𝑥3 = 𝑇br ,
(48)
𝑑𝑇br
.
𝑑𝑡
The state equation (51) can be derived from (14):
𝑋̇ = 𝐴 3 𝑋 + 𝐵3 𝑈 + 𝑉3 ,
(49)
where
𝑇
𝑋 = [𝑥1 𝑥2 𝑥3 ] ,
𝑇
𝑈 = [𝑢1 𝑢2 ] ,
𝑎11 𝐽22 − 𝑎12 𝐽21 2𝑎11 𝑎12 𝐽12 − 𝑎11 2 𝐽22 − 𝑎12 2 𝐽11
0
[ 𝐽 𝐽 −𝐽 𝐽
]
𝐽11 𝐽22 − 𝐽12 𝐽21
],
𝐴3 = [
[0 11 22 0 12 21
]
0
0
0
0
[
]
𝑎11 𝐽12 − 𝑎12 𝐽11
[ 𝐽 𝐽 − 𝐽 𝐽 𝑇𝑓 ]
11 22
12 21
].
𝑉3 = [
[
]
0
0
[
]
(50)
At the end of the brake engagement phase, the speed
difference of the driving and driven parts of the brake B is
zero; that is, the terminal constraint is
]
At the end of free phase, the angular velocity of the ring
gear is required to be less than 30 rad/s; that is, the terminal
constraint is
(44)
where 𝑡𝑛 is the end time of free phase.
During this phase, the shift jerk is expressed below:
𝐽21 𝑟𝑤
𝑟 𝑑
𝐽21 𝑇𝑚
)=
𝑢.
𝑗= 𝑤 (
𝑖0 𝑑𝑡 𝐽11 𝐽22 − 𝐽21 𝐽12
𝑖0 (𝐽11 𝐽22 − 𝐽21 𝐽12 )
𝑥2 = 𝑇𝑚 ,
(43)
].
𝑁 [𝑥 (𝑡𝑛 ) , 𝑡𝑛 ] = 𝑥1 (𝑡𝑛 ) ≤ 30,
(47)
where 𝑚𝑐 is the equivalent piston mass. 𝑥0 is the initial
compression amount of the return spring. 𝑥𝑐 is piston
displacement. 𝑐 is the piston friction coefficient. 𝑘𝑠 is the
return spring stiffness. 𝑝 is the oil pressure.
According to (47), the oil pressure of combined clutch in
free phase can be acquired.
0 0
𝐵3 = [ 1 0 ] ,
[0 1]
𝑇
𝑈 = [𝑢1 ]
𝑎 𝐽 − 𝑎12 𝐽21
0 11 22
𝐴 2 = [ 𝐽11 𝐽22 − 𝐽12 𝐽21 ] ,
0
[0
]
1 𝑡𝑛
1 𝑡𝑛 2
∫ 𝑢 𝑑𝑡 = ∫ 𝑈𝑇 𝑄2 𝑈 𝑑𝑡,
2 0
2 0
where 𝑄1 = 0, 𝑄2 = [1].
The optimal torque of the motor control law in free phase
can be derived from solutions of the above differential matrix
equations.
During this phase, ignoring the frictional resistance in
sealing ring and splines, the equation of motion of the brake
piston is established as
(41)
The state equation has the following expression, resembling (15):
𝑋 = [𝑥1 𝑥2 ] ,
𝐽=
𝑚𝑐 𝑥𝑐̈ + 𝑐𝑥𝑐̇ + 𝑘𝑠 (𝑥0 + 𝑥𝑐 ) = 𝑝𝐴 𝑐 ,
The optimal trajectories of motor torque and clutch C
transmission torque in the clutch C disengagement phase
can be derived from solving the differential equations as
described above with different weight coefficient.
𝑥1 = 𝜔𝑟 ,
As a result, the object function can be given as follows:
(45)
N [x (𝑡𝑓 ) , 𝑡𝑓 ] = 𝑥1 (𝑡𝑓 ) = 0,
(51)
where 𝑡𝑓 is the end time of brake B engagement phase.
In this phase, the shift jerk 𝑗 and the sliding friction work
𝑊 are expressed below:
𝑗=
𝑟𝑤
× [(𝑎11 𝐽21 − 𝑎12 𝐽11 ) 𝑢2 − 𝐽21 𝑢1 ]
𝑖0 (𝐽11 𝐽22 − 𝐽21 𝐽12 )
𝑡𝑓
𝑡𝑓
0
0
𝑊 = ∫ 𝑇br 𝜔𝑟 𝑑𝑡 = ∫ 𝑥1 𝑥3 𝑑𝑡.
(52)
8
Mathematical Problems in Engineering
Similar to the clutch C disengagement phase, the objective function can be expressed as follows:
𝐽=
𝑡𝑓
1
∫ (𝑊 + 𝜂𝑗2 ) 𝑑𝑡
2 0
=
1 𝑡𝑓
2
∫ {𝑥1 𝑥3 + [(𝑎11 𝐽21 − 𝑎12 𝐽11 )𝑢2 − 𝐽21 𝑢1 ] } 𝑑𝑡
2 0
=
1 𝑡𝑓 𝑇
∫ (𝑋 𝑄1 𝑋 + 𝑈𝑇 𝑄2 𝑈) 𝑑𝑡,
2 0
(53)
where
0 0 0.5
𝑄1 = [ 0 0 0 ] ,
[0.5 0 0 ]
𝑄2 = 𝜂 [
Parameters
Value
Vehicle mass fully equipped 𝑚/kg
Air resistance coefficient 𝐶𝐷
Rolling resistance coefficient 𝑓
Vehicle frontal area 𝐴/m2
Tire radius 𝑟𝑤 /m
Final drive ratio 𝑖0
Transmission ratio (2 gears)
Ratio of the number of teeth of ring
gear and sun gear 𝑘
15000
0.6
0.0015
7.82
0.478
6.5
[3.5; 1]
Motor
(54)
2
0
𝐽21
2] .
−2𝐽21 (𝑎11 𝐽21 − 𝑎12 𝐽11 ) (𝑎11 𝐽21 − 𝑎12 𝐽11 )
The optimal trajectories of the motor torque and brake
B transmission torque in brake engagement phase can be
acquired with reference to the solving process in clutch
disengagement phase.
The correlation of the clutch transmission torque and the
oil pressure is given by (55) [30]. Apparently, the optimal
trajectory of the oil pressure can be transformed from the
optimal trajectory of the clutch/brake transmission torque:
𝑇 = 𝜇𝑍
Table 1: The vehicle parameters.
2 (𝑅2 3 − 𝑅1 3 ) 󵄨
󵄨󵄨𝐹𝑠 − 𝑝𝐴 𝑐 󵄨󵄨󵄨 ,
󵄨
3 (𝑅 2 − 𝑅 2 ) 󵄨
2
(55)
Rated/peak power/(kW)
Rated/peak torque/(Nm)
Plate number of the clutch/brake 𝑍
Equivalent piston mass 𝑚𝑐 /kg
Brake piston friction coefficient
𝑐/(kg/s)
Return spring stiffness 𝑘𝑠 /(N/mm)
Return spring initial compression
amount 𝑥0 /mm
Brake piston area 𝐴 𝑐 /mm2
Friction coefficient of clutch/brake
disc 𝜇
𝐽11 /𝐽12 /𝐽21 /𝐽22
Weight coefficient of the shift jerk 𝜂
2.5
Permanent magnet
synchronous motor
130/170
653/850
6/8
6.82
0.5
2520
4.2
32.371
0.13
0.63/−0.044/−0.044/81.2
0.5
1
where 𝑇 is the clutch/brake transmission torque. 𝐹𝑠 is the
return spring force. 𝜇 is the friction coefficient of clutch/brake
plate, treated as a constant in this paper. 𝑅1 and 𝑅2 are
clutch/brake plate inner and outer radius, respectively. 𝑍 is
the plate number of the clutch/brake. 𝐴 𝑐 is the brake piston
area.
In reality, the shift time is very short, which makes the
real-time online calculation of the differential matrix equation impractical. A solution to this problem is to calculate
the optimal trajectory under different working conditions
offline and then to conduct function fitting. At last, the
fitting coefficients are stored in the memory of vehicle control
system. We can get the optimal trajectory quickly through the
look-up table and interpolation way to get fit coefficients in
online control.
5. Simulation Results and Analysis
Based on MATLAB software, a simulation model is built up
to analyze the above discussed shift control strategy and to
compare the shift quality employing coordinated control with
that adopting oil pressure of combined clutch only. The focus
is put on uphill downshift in this paper, and the road ramp
studied is 5%. The shift vehicle speed is set as 25 km/h. The
vehicle parameters are shown in Table 1.
Figure 5 indicates simulation results of the coordinated
control of powertrain and the control of oil pressure of the
combined clutch only when the accelerator pedal angle is
50%, consisting of four graphs, which is similar to Figure 6,
with the accelerator pedal angle 0: (a) shows the optimal
trajectories of the motor torque, (b) depicts the hydraulic oil
pressure of the combined clutch, (c) shows the speeds of the
motor, the ring gear, and the transmission output shaft, and
(d) pictures the shift jerk in downshift. Table 2 represents the
maximum sliding friction work during downshift.
In the case where the coordinated control of powertrain
is employed, the motor torque is not determined by the
accelerator pedal any more, the signal of which just represents
power requirement from the driver. As can be seen from the
profiles of the motor torque in Figure 5, the motor torque
is kept roughly constant in clutch disengagement phase,
which means that the changes of the motor torque can be
neglected to simplify the control process, and at the end of
this phase, the clutch transmission torque goes to zero. In
free phase, the motor torque keeps increasing to regulate the
ring gear rotational speed to the specified range and then
to decrease the shift time and the sliding friction work. In
brake engagement phase, the motor torque is supposed to
reduce to make sure that the vehicle speed is almost constant.
At the point of synchronization of the driving and driven
parts of the brake, actual transition from stick to slip at the
brake is accomplished. Namely, the brake transmission torque
becomes static friction torque that does not depend on the oil
Mathematical Problems in Engineering
9
×106
2
800
1.8
1.6
1.4
600
Oil pressure (Pa)
Motor torque (Nm)
700
500
400
1.2
1
0.8
0.6
0.4
300
0.2
200
0
0.4
0.2
0.6
0.8
0
1
0
0.2
Time (s)
0.8
1
(b) The oil pressure curves
3500
10
3000
8
6
2500
4
Shift jerk (m/s3 )
Speed (rpm)
0.6
Time (s)
(a) The motor torque curves
2000
1500
nm
1000
0.2
0
−2
−6
nr
0
2
−4
no
500
0
0.4
−8
0.4
0.6
0.8
1
−10
0
0.2
0.4
Time (s)
Coordinated control
Clutch control only
(c) The speed curves
0.6
0.8
1
Time (s)
Coordinated control
Clutch control only
(d) The shift jerk curve
Figure 5: Downshift simulation results with the accelerator pedal angle 50%.
pressure, which leads to a big shift jerk. At the end of brake
engagement phase, the motor torque depends on motor speed
and accelerator pedal when combined clutch is controlled
only. The absence of motor torque control makes a further
bigger shift jerk.
The combined clutch is largely different from the conventional wet clutch in structure, which results in the tremendous
difference in oil pressure between two kinds of clutches.
From the profiles of the oil pressure in Figures 5 and 6,
the oil pressure curves of combined clutch consist of five
periods. Firstly, the combined clutch is filled with oil quickly,
completed in an instant, so that the oil pressure increases
rapidly, aiming to make the actual transmission torque of
the clutch equal to its dynamic friction torque to mitigate
the shift jerk at the beginning of the clutch disengagement.
Secondly, the oil pressure keeps increasing slowly to surpass
the initial value of return spring, to separate friction plates of
the clutch, and to decrease the clutch transmission torque to
zero. Thirdly, the increased oil pressure is used to eliminate
the surplus space of the brake until the friction plates are
connected. During this period, both the clutch and brake
transmission torque are zeros. In the following period,
10
Mathematical Problems in Engineering
×106
2
800
1.8
1.6
600
Oil pressure (Pa)
Motor torque (Nm)
700
500
400
300
1.2
1
0.8
0.6
0.4
200
0.2
100
0
1.4
0
0
0.2
0.4
0.6
0.8
1
0
0.2
3000
2500
2500
Speed (rpm)
3000
2000
0
nm
0.4
1500
500
nr
0.2
nm
2000
1000
no
0
1
(b) The oil pressure curves
3500
0.6
0.8
0
1
nr
0.2
0
0.4
Time (s)
no
0.6
0.8
Time (s)
Transmission output shaft speed
Ring gear speed
Motor speed
Transmission output shaft speed
Ring gear speed
Motor speed
(c) The speed curves under coordinated control
(d) The speed curves under clutch control only
5
0
Shift jerk (m/s3 )
Speed (rpm)
(a) The motor torque curves
3500
500
0.8
Coordinated control
Clutch control only
Coordinated control
1000
0.6
Time (s)
Time (s)
1500
0.4
−5
−10
−15
−20
0
0.2
0.4
0.6
0.8
1
Time (s)
Coordinated control
Clutch control only
(e) The shift jerk curves
Figure 6: Downshift simulation results with the accelerator pedal angle 0.
1
Mathematical Problems in Engineering
11
Table 2: The maximum sliding friction work.
Coordinated control
Accelerator pedal angle/%
Sliding friction work/KJ
50
1.34
0
1.53
the brake plates clearance has been eliminated. Oil pressure
is rising continuously until the driving and driven parts of
brake B reach the same speed. This is main stage to control
the shift jerk and sliding friction work. At the end of this
period, the actual transition is from slip to stick at the brake.
After that, the brake transmission torque has nothing to do
with the oil pressure. During the last period, in order to
produce torque reserve for the bake in case the brake slips,
the oil pressure is raised to the line pressure, which is the
main pressure in the hydraulic system. It must be stressed
that the rapid increase of oil pressure will not affect the
driving comfort since the driving and driven parts of the
brake already achieved synchronization.
With the accelerator pedal angle 0, the control of oil
pressure of the combined clutch only means there is power
interrupt during downshift. It is shown from Figure 6 that
there is little variation in the speed of the motor, the ring
gear, and the transmission output shaft during the clutch
C disengagement phase and free phase. By contrast, the
coordinated control strategy adjusts the three speeds through
increasing the motor torque in the two phases.
It is shown from Figures 5 and 6 that there are three
significant differences between the two accelerator pedal
angles when coordinated control of powertrain is employed:
the changing trend of combined clutch oil pressure during
brake B engagement phase, the speed variation during the
free phase, and the shift jerk at the end of brake B engagement
phase. Ultimately, it comes down to the initial value of motor
torque. In other words, the terminal value of motor torque
must be zero with combined clutch control only.
Compared to the combined clutch control only, the
downshift time is 0.45 s with coordinated control of powertrain, 0.2 s less than that in the control strategy without the
motor. In addition, the adjustment of the ring gear rotational
speed in free phase makes the sliding friction work down
from 2.4 KJ to 1.34 KJ when the accelerator pedal angle is 50%
and from 14.5 KJ to 1.53 KJ when the accelerator pedal angle
is 0. In other words, the sliding friction work is decreased
by 44% and 89% separately. During the brake engagement
phase when the accelerator pedal angle is 50%, the maximum
value of the shift jerk is almost equal to that in the control
strategy without the motor because the motor torque is
supposed to decrease to make sure that the vehicle speed is
almost constant. But the shift jerk is from 8.1 m/s3 down to
1.6 m/s3 , which improves the shift comfortability at the time
of synchronization of the driving and driven parts of the
brake B. In the whole shift process, the coordinated control
of powertrain makes the shift jerk change from 8.1 m/s3
to 4.6 m/s3 with the accelerator pedal angle 50% and from
13.1 m/s3 down to 5.6 m/s3 with the accelerator pedal angle 0,
Combined clutch control only
Accelerator pedal angle/%
Sliding friction work/KJ
50
2.4
0
14.5
which means the shift jerk is decreased by 43% and 57% under
different accelerator pedal angles.
6. Conclusions
(1) Based on two-gear combined clutch transmission, the
dynamic model of shift process is established by the
principle of virtual work and Lagrange principle. On
this basis, the shift process is divided into three phases
and each phase has different dynamics equation.
(2) The dynamic coordinated system for EVs regards
the motor and transmission as a whole. Through
coordinated control of the motor torque and the oil
pressure of the combined clutch, the shift jerk and
sliding friction work are reduced by a large margin
no matter how much the accelerator pedal angle is.
At the same time, the shift time can be shortened
by adjusting the speed difference of the driving and
driven parts of brake in free phase. In a word, an
effective method is provided to solve the problem of
automatic transmission shift control in this paper.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgment
This work is supported by Beijing Municipal Education
Commission, China. The authors highly appreciate the above
financial support.
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