1) A novel gearshift system is introduced comprising a 2 degree-of-freedom electromagnetic actuator to simplify structure, increase efficiency, and improve shift quality of automated manual transmissions. 2) The gearshift process is divided into non-synchronization and synchronization phases, with different control algorithms designed for each. Extended state observer based inverse system method is used for non-synchronization, while active disturbance rejection controller is used for synchronization. 3) Comparative simulations and experiments demonstrate the effectiveness of the proposed control method in achieving good gearshift performance for the novel system. The control strategy provides a new solution for automated manual transmission applications.

1. Gearshift control system development for direct-drive automated
manual transmission based on a novel electromagnetic actuator
Shusen Lin a,b
, Siqin Chang a,⇑
, Bo Li a
a
School of Mechanical Engineering, Nanjing University of Science and Technology, PR China
b
College of Engineering, Zhejiang Normal University, PR China
a r t i c l e i n f o
Article history:
Received 16 August 2013
Accepted 22 September 2014
Available online 12 October 2014
Keywords:
Gearshift system
Automated manual transmission
Control strategy
Electromagnetic actuator
a b s t r a c t
A novel gearshift system which comprises a 2 degree-of-freedom electromagnetic actuator is introduced
to simplify the structure of gearshift system of automated manual transmission (AMT), increase trans-
mission efficiency and improve shift quality. The working principle and characteristics of the actuator
are analyzed. The gearshift process is divided into the non-synchronization and the synchronization
phase. Extended state observer (ESO) based inverse system method (ISM) and active disturbance rejec-
tion controller (ADRC) are designed for the two processes respectively. ISM can eliminate the nonlinearity
of the actuator and ESO can estimate and compensate the uncertainties, parameter variations and
external disturbances. ADRC is adopted to improve the tracking accuracy of the synchronization process.
Comparative simulations and experimental results demonstrate the effectiveness of the proposed control
method, and good gearshift performance has been achieved. Combined with the new designed control
strategy, the novel gearshift system provides a new solution for AMT applications.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Nowadays, vehicle manufactures put more attention on
reducing vehicle fuel consumption due to high fuel price and con-
cerns on global warming. Transmission system is one of the crucial
systems which affect the vehicle fuel economy. Currently, two
main types of transmission used in passenger cars are manual
transmission (MT) and automatic transmission (AT). MT has a high
mechanical efficiency and the driver can select gears autono-
mously. AT is convenient to operate but has relatively higher fuel
consumption. Therefore, a new type of transmission named
automated manual transmission (AMT), which combines the
advantages of AT and MT, represents a promising solution and is
spreading in the recent years [1].
AMT shares a similar mechanical structure with MT, but it is
equipped with electro-mechanical or electro-hydraulic actuators
which are controlled by a transmission control unit (TCU). The fuel
economy of vehicles equipped with AMT is supposed to be
improved compared with vehicles with AT. However, a poorly
designed or controlled AMT system may not achieve desired
results. Torque interruption which leads to driving comfort reduc-
tion is the bottleneck of the AMT for its wide application. However,
the driving comfort can be improved by proper gearshift control
strategy. An integral control strategy aiming to improve gearshift
quality should take into account the reduction of gearshift time,
driveline oscillations, friction work of clutch and synchronizer ring.
Currently, hydraulic and electrohydraulic actuation technolo-
gies are preferred for the control of AMT due to its higher density
and the readily available source of hydraulic power [2]. However,
hydraulic systems may represent up to 50% of the total transmis-
sion loss. Electrohydraulic actuation provides an alternative solu-
tion and it is widely employed in AMT vehicles though the loss
associated with leakage and flow is still present. Besides, electro-
hydraulic systems are complex, having many solenoids valves
and hydraulic lines, which occupy a large volume and are relatively
expensive. Presently, electromechanical actuation system is con-
sidered as an alternative to electrohydraulic system, since it offers
the potential for improving the efficiency, dynamic response and
robustness. Recent investigations to adopt electromechanical actu-
ation for AMT control generally utilize DC motors in conjunction
with reduction gear or motion conversion device to achieve the
desired force and motion. Although this solution offers great
potential for efficiency improvement, the large gear reduction ratio
and low efficiency of the gearbox compromise the performance.
Turner et al. present a direct-drive electromechanical actuation
system for gearshift control of AMT [2,3]. The actuation system
comprises a high-force moving magnet linear actuator and a rotary
actuator. As the actuation system employs the direct-drive
technology, it does not suffer from significant hysteresis,
http://dx.doi.org/10.1016/j.mechatronics.2014.09.008
0957-4158/Ó 2014 Elsevier Ltd. All rights reserved.
⇑ Corresponding author.
E-mail address: changsiqin@hotmail.com (S. Chang).
Mechatronics 24 (2014) 1214–1222
Contents lists available at ScienceDirect
Mechatronics
journal homepage: www.elsevier.com/locate/mechatronics

2. compliance and backlash. Only part of the published research
focuses on the structure innovation of AMT. Galvagno presents a
kind of AMT equipped with an additional flywheel to reduce the
torque gap during the gearshift [4]. R.P.G. Heath proposed the zero-
shift hub which is housed within an existing synchronizer to pro-
vide an uninterrupted path to deliver torque from the engine to
wheels [5]. Sandooja [6] developed a double indexing synchronizer
is developed to amplify the synchronizer capacity so that smooth
gearshift and good shift feeling are achieved.
Current research on AMT is focused on the control of clutch,
gearshift, engine speed and torque [7,8], and many kinds of
intelligent control algorithms such as fuzzy control, optimal con-
trol, sliding mode control, are adopted to solve the nonlinearity of
an AMT system and achieve better gearshift performance [9–11].
However, in spite of the extensive literature on AMT control, the
control methodology is still not mature enough for the wide
application of the AMT system. Few papers about the gearshift
synchronization process when the clutch is disengaged have been
published. Eventhough poorly controlled gearshift synchroniza-
tion process will cause vibration of the driveline and abrasion
of the synchronizer ring. Literature [12] presents the detailed
analysis of the synchronization process, but control method is
not involved.
In this research, a novel gearshift system based on direct-drive
technology is proposed. A totally improved direct-drive electromag-
netic rotary-linear actuator (EMRLA) is developed and adopted as a
gearshift actuator. The gearshift synchronization process is divided
into two main phases, the non-synchronization phase and the
synchronization phase. Speed difference is synchronized during
the synchronization phase and gaps are eliminated during non-syn-
chronization phase. Obviously, the task and the drag force are differ-
ent in each phase. As a result, it is necessary to adopt different
control algorithm to achieve the desired performance. Direct-drive
technology eliminates transmission mechanisms such as reduction
gear and lead screws, which lead to a lower component counts,
improved robustness and dynamic response of the gearshift system.
However, this structural simplification makes the EMRLA easily
affected by model uncertainties and disturbances. As a result, the
requirements for the control method are relatively high. PID control
algorithm is applied in previous work [13]. However, the algorithm
is insensitive to the variation of target displacement but sensitive
enough to the nonlinearity and disturbances, therefore the control
is not precise enough since the gearshift process is nonlinear and
suffers from disturbances. Inverse system method (ISM) is
introduced to eliminate the nonlinearity of the actuator and gear-
shift process [14]. Additionally, extended state observer (ESO) is
used to estimate the uncertainties and disturbances, and compen-
sates these unfavorable factors. The active disturbance rejection
controller (ADRC) is employed during the synchronization phase
[15]. The ADRC is not predicated on precise plant model and is
extremely tolerant of uncertainties and nonlinearity. Known
and unknown disturbances occurring during synchronization
process are lumped together as total disturbance, which is
estimated and compensated by ADRC in real time. Simulation and
experimental results indicate the effectiveness of the proposed
control strategy.
Nomenclature
U voltage
I current
R resistance
L inductance
m moving mass of the linear part
E back electromotive force
T electromagnetic torque
F electromagnetic force
x displacement of the linear part
Fc friction force
Jt rotary inertia of the actuator
x rotary speed
Td resistance torque
v velocity of the sleeve
c viscous friction damping coefficient
S displacement of the sleeve
xi element of the state variable x
x state variable
y output variable
A state matrix
B input matrix
C output matrix
w state variables of the pseudo-linear system
wi element of the state variable w
r desired value
yf feedback value
n damping ratio
xn natural frequency
ts transition time
u input variable of the pseudo-linear system
s laplace variable
K state variable feedback matrix
ai element of K
u control input
b, b0 system parameter
e estimate errors
z1 estimates of the output y
z2 estimates of the derivative of y
z3 estimates of total disturbance
h sampling period
b01, b02, b03 observer gains
fal(e, a, d)
nonlinear function
u control input
u0 intermediate control input
Jc equivalent inertia of the input shaft of the AMT
Ts friction torque
Js equivalent inertia of the output shaft
TL load torque
ig gear ratio
id differential ratio respectively
xc clutch speed
xs out put shaft speed
Fs gearshift force
fs friction coefficient
Rc effective radius of the friction cone
a half cone angle
d, d0 parameters of function f(v1 À v, v2, r, h)
v desired signal
y0 parameter of function f(v1 À v, v2, r, h)
v1 transitional trajectory of v
v2 differential signal of v1
rg parameter determining the dynamic characteristics of
v1
a, a4 parameter of function f(v1 À v, v2, r, h)
b11, b12 controller parameters
a1, a2 controller parameters
S. Lin et al. / Mechatronics 24 (2014) 1214–1222 1215

3. 2. The novel gearshift system
Fig. 1 shows the novel gearshift system which employs the
EMRLA as the AMT gearshift actuator. It consists of EMRLA, trans-
mission, shift block, shift lever and displacement sensor. Differen-
tiating from existing AMT gearshift systems, direct-drive
technology is adopted so that the EMRLA acts directly on the shift
rail of AMT. The large reduction gear and motion conversion device
are eliminated, which simplifies the structure of the system and
improves the mechanical efficiency.
The novel gearshift system based on an EMRLA offers a number
of advantages,
(1) Simplified construction and lower component count which
result in improved robustness.
(2) Elimination of reduction gear and motion conversion linkage
which improves efficiency and reduces mechanical hystere-
sis, compliance and backlash.
(3) Adoption of the EMRLA which has high driving ability and
fast dynamic response is beneficial to the reduction of gear-
shift time and the improvement of shift quality.
3. The EMRLA
A prototype of the EMRLA has been developed for gearshift con-
trol. The EMRLA is illustrated schematically in Fig. 2. The actuator
includes a high-force linear part which controls the engagement of
gears and a high-torque rotary part which is coupled to the shaft of
the linear part directly. The rotary part is in charge of gear
selection. The linear part comprises the output shaft, coil, perma-
nent magnets, outer core and inner core. The rotary part comprises
armature, permanent magnets, coils, outer core and inner core.
Both the linear part and the rotary part act on the same output
shaft which is connected with a shift lever as shown in Fig. 1.
The motion of the linear part and the rotary part do not interfere
with each other so that the output shaft can rotate and move
linearly at the same time. In order to achieve high driving ability,
a high energy sintered NdFeB magnet, with a maximum operating
temperature of 180 °C, is selected for the permanent magnet
design.
Fig. 3 presents the working principle of the EMRLA. The direc-
tion of the electromagnetic field and magnetic-curve of the rotary
part are shown in Fig. 3(a). There is a small gap between the arma-
ture and the inner core so that the armature can rotate freely. The
magnetic-curve produced by symmetrical coils overlap on the
armature and drives it to rotate to the right. The armature will
rotate to the left when the coils are energized reversely.
Halbach magnetized topology is utilized to maximize the actu-
ating force of the linear part [16]. The air gap is full of radial mag-
netic field. According to the Fleming’s left-hand rule, the direction
of the electromagnetic force acting on part 1 is towards right. Both
the direction of magnetic field and of current are reverse from part
1 which means the direction of the electromagnetic force is exactly
the same as the part 1. As a result, the output shaft moves towards
right. The motion could be bidirectional since the direction of the
current is alterable. The produced force is nearly proportional to
the current so that accurate motion control is achievable.
The electromagnetic actuator is a coupling system with strongly
interactive subsystems, including electrical, magnetic and mechan-
ical subsystems. The mathematical model can be described as
UðtÞ ¼ E þ RIðtÞ þ L dIðtÞ
dt
Electrical subsystem
FðtÞ ¼ kmIðtÞ
TðtÞ ¼ ktIðtÞ
&
Magnetic subsystem
m d2
xðtÞ
dt2 ¼ FðtÞ À Fc
Jt
_xðtÞ ¼ TðtÞ À Td
(
Mechanical subsystem
8
>>>>>>><
>>>>>>>:
ð1Þ
where U is the voltage applied to the actuator, E is the back electro-
motive force (EMF), I is the current through the coil, R and L repre-
sent the resistance and inductance of the coil respectively, F and T
represent the produced force and torque respectively, km is the force
coefficient and kt is the torque coefficient, m is the moving mass of
the linear part, x denotes the displacement, Fc is the friction force, Jt
is the rotary inertia of the actuator, x is the rotary speed, Td is the
resistance torque. Note that the electrical subsystem for linear part
is the same as the rotary part.
The specifications of the EMRLA are shown in Table 1 [13].
The electrical time constant and the electromechanical time
constant of the electromagnetic actuator are relatively small due
to the low moving mass and rotary inertia, and as a result the
dynamic response of the actuator is definitely fast. Additionally,
the driving ability of the actuator is large enough to realize gear-
shift. The quick response of the EMRLA is conducive to increasing
the controllability of the actuator.
4. Gearshift controller design
This research focuses on the gearshift process after the disen-
gagement of the clutch. The gearshift process can be divided into
several phases according to different research purposes [12,17].
Before the synchronization process, the sleeve moves forward to
eliminate the gap between the synchronizer ring and friction cone.
The synchronization process starts when the friction torque
emerges, and the rotary speed difference decreases. When the
rotary speed difference disappears, the sleeve moves forward again
and finally finishes meshing with target gear. The gearshift process
is divided into two main phases as synchronization phase and
Fig. 1. The novel gearshift system.
Fig. 2. Structure of the EMRLA.
1216 S. Lin et al. / Mechatronics 24 (2014) 1214–1222

4. non-synchronization phase. The inverse system method (ISM) [18]
is easy to realize in engineering applications. During the non-syn-
chronization process, inverse system method is employed to
achieve linearization of the gearshift system. The ISM of the gear-
shift system can be established by using feedback method. Besides,
an extended state observer (ESO) is introduced to replace the state
observer of the ISM. Model uncertainties and disturbance are esti-
mated and compensated by ESO so that fast and precise gearshift
control is achievable. The active disturbance rejection controller
(ADRC) [19] is a new way of control design, which is independent
of an accurate model and is highly tolerant of uncertainties and
disturbance. Since the disturbance and vibration are immeasur-
able, ADRC is adopted to consider all of these as total disturbance
and compensated by an ESO. It will reduce force ripple which
contributes to the improvement of the shift quality. The scheme
of the controller is shown in Fig. 4.
4.1. ISM-ESO controller design for non-synchronization phase
For a given system, the inverse system of the original system
can be developed by using feedback method. A pseudo-linear sys-
tem is obtained when the inverse system and the original system
are combined. As a result, linear system theory is appropriate to
be applied to achieve high performance. Combined with Eq. (1),
the mathematical model of the gearshift system can be described
as
_I ¼ À R
L
I À km
m
v þ u
L
_v ¼ km
m
I À c
m
v
_S ¼ v
8
>><
>>:
ð2Þ
where v is the velocity of the sleeve, c is the viscous friction
damping coefficient, S is the displacement of the sleeve. As shown
in the Eq. (1), the input variable of the gearshift system is voltage
u, and the output variable is S. Therefore, the gearshift system is a
single-input single output (SISO) system. The state variables are
given as
x ¼ x1 x2 x3½ ŠT
¼ I v S½ ŠT
ð3Þ
According to the mathematical model of the gearshift system
described in Eq. (2), the state equation can be depicted as
_x1
_x2
_x3
2
6
4
3
7
5 ¼
À R
L
km 0
km
L
À c
m
0
0 1 0
2
6
4
3
7
5
x1
x2
x3
2
6
4
3
7
5 þ
1
L
0
0
2
6
4
3
7
5u ð4Þ
The system output equation is expressed as
y ¼ 0 0 1½ Š
x1
x2
x3
2
6
4
3
7
5 ð5Þ
The necessary and sufficient condition for the reversibility of
the gearshift system is the existence of the relative order of the
state equation a in a given neighborhood. The step of reversibility
analysis can be described as [18]:
(1) Computing the derivatives of the output equation y = h(x, u)
until the input variable u appears in equation y(a)
= ha(x, u).
(2) If the partial derivative of the equation y(a)
= ha(x, u) is not
equal to zero in the neighborhood of (x0, u0), there is a rela-
tive order a of the gearshift system, and if the relative order
is less than or equal to the dimension of the state vector x, it
is reasonable to conclude that the gearshift system is
reversible.
Fig. 3. Working principle of the actuator.
Table 1
Specifications of the actuator.
Parameter Linear Rotary
Moving mass and rotary inertia 0.671 kg 7.04 Â 10À6
kg m2
Resistance 1.6 X 0.56 X
Inductance 1.1 mH 0.29 mH
Electrical time constant 0.69 ms 0.51 ms
Electromechanical time constant 0.97 ms 0.68 ms
Maximum driving ability 1300 N 2.5 Nm
Fig. 4. Scheme of the controller.
S. Lin et al. / Mechatronics 24 (2014) 1214–1222 1217

5. According to the above theory, the equations
y ¼ x3
_y ¼ _x3 ¼ x2
€y ¼ _x2 ¼ km
m
I À c
m
_y
y
v
¼ €x2 ¼ À c
m
€y þ km
m
À R
L
x1 À km
L
_y þ u
L
À Á
8
>>>><
>>>>:
ð6Þ
are obtained.
It is obvious that there is not input variable u in expression y; _y
and €y, but the expression y
v
includes the input variable u. Hence, the
relative order of the system a is 3. The relative order is equal to the
dimension of the state vector so that the gearshift system is
reversible. From the expression y
v
, the expression of the inverse
system can be solved as
u ¼
mL
km
y
v
þ
cL
km
€y þ km _y þ Rx1 ð7Þ
By connecting the inverse system with the original system, the
pseudo-linear system is obtained as shown in Fig. 5.
The state variables of the pseudo-linear system is given as
w ¼ w1 w2 w3½ ŠT
¼ y _y €y½ Š
T
ð8Þ
The state space equation of the pseudo-linear system can be
described as
_w1
_w2
_w3
2
6
4
3
7
5 ¼
0 1 0
0 0 1
0 0 0
2
6
4
3
7
5
w1
w2
w3
2
6
4
3
7
5 þ
0
0
1
2
6
4
3
7
5u ¼ Aw þ Bu
y ¼ 1 0 0½ Š
w1
w2
w3
2
6
4
3
7
5 ¼ Cw
8
>>>>>>>><
>>>>>>>>:
ð9Þ
As a result, the pseudo-linear system has been developed and
linear system theory can be applied.
The pseudo-linear system has a ath-order integral attribute,
thus linearization of the nonlinear system has been achieved. In
this section, state feedback control is used to design a controller
for the pseudo-linear system according to the system control target
(see Fig. 6).
If the state variable feedback matrix is K ¼ a0 a1 a2½ Š, then
the state feedback controller can be described as
u ¼ r À yf
yf ¼ a0y þ a1 _y þ a2€y
(
ð10Þ
where r is the desired value and yf is the feedback value.
To seek a rapid response with a low overshoot, a desired system
characteristic equation can be chosen such as
s2
þ 2nxn þ x2
n
À Á
ðs þ nxnÞ ¼ 0 ð11Þ
where n is the damping ratio, xn is the natural frequency.
The dynamic response is decided by the variables n and xn. The
transition time can be calculated by equation ts % 4/(nxn). Accord-
ing to the transient response of the system, the transition time is
the smallest when the value of the damping ration is 0.707. The
transition time is decided as 20 ms on the basis of experiments.
Therefore, the natural frequency is figured out as 283. The desired
system characteristic equation is
s3
þ a2s2
þ a1s þ a0 ¼ 0 ð12Þ
By applying Ackermann’s formula, a2 = 600, a1 = 160,021,
a0 = 16,004,218 is obtained.
Consider the following nonlinear second-order equation
_x1 ¼ x2
_x2 ¼ fðx1; x2Þ þ bu
y ¼ x1
8
><
>:
ð13Þ
where y is the output to control, x1, x2 are state variables, u is the
control input, b is the system parameter, and f(x1,x2) denotes the
total disturbance which is nonlinear. The objective is to synthesize
a control input u so that the output y gets to the desired point yd as
quickly and accurately as possible in spite of the total disturbance.
Consider the function f(x1,x2) as a new variable x3, and expressed as
_x3 ¼ wðtÞ. The Eq. (13) is converted to
_x1 ¼ x2
_x2 ¼ fðx1; x2Þ þ bu
_x3 ¼ wðtÞ
y ¼ x1
8
>>><
>>>:
ð14Þ
The discrete-time form of the ESO for Eq. (14) can be written as
e ¼ z1ðkÞ À yðkÞ
z1ðk þ 1Þ ¼ z1ðkÞ þ h Á ðz2ðkÞ À b01 Á eÞ
z2ðk þ 1Þ ¼ z2ðkÞ þ h Á ðz3ðkÞ À b02 Á falðe; 0:5; dÞ þ b Á uðkÞÞ
z3ðk þ 1Þ ¼ z3ðkÞ À h Á b03 Á falðe; 0:5; dÞ
8
>>><
>>>:
ð15Þ
where z1, z2, and z3 are estimates of x1, x2 and f(x1,x2) respectively, h
is the sampling period. b01, b02, and b03 are observer gains which
can be selected as b01 % 1/h, b02 % 1/1.6h1.5
, b03 % 1/8.6h2.2
.
Non-linear function fal(e,a,d) is defined as
falðe; a; dÞ ¼
e Á daÀ1
; jej 6 d
jeja
Á sgnðeÞ; jej > d
(
ð16Þ
Parameters a and d satisfy conditions of a < 1 and d = k Á h,
where k is a positive integer.
Since z3 tracks f(x1,x2) well, the control input u can be designed
as
u ¼ ðu0 À z3Þ=b ð17Þ
to compensate the total disturbance. Therefore, the original nonlin-
ear system Eq. (13) is linearized as
_x1 ¼ x2
_x2 ¼ u0
y ¼ x1
8
><
>:
ð18Þ
Fig. 5. Pseudo-linear system.
Fig. 6. State feedback controller.
1218 S. Lin et al. / Mechatronics 24 (2014) 1214–1222

6. Finally, the designed ISM-ESO control method for non-synchroniza-
tion phase is illustrated in Fig. 7. The parameters are: h = 0.0001,
b01 % 10,000, b02 % 625,000, b03 % 73,400,000, b % 23.87.
Stability is the most important characteristics of the control
system. The stability of Eq. (13) under the ISM-ESO controller has
been proved in reference [20,21] by using Lyapunov stability
principle, and Additionally, Lyapunov’s second method for stability
is adopted to prove the stability of the ESO controller in reference
[15].
The designed controller for the non-synchronization phase has
been compared with a tuned PID controller by simulation. In our
previous work, an incremental PID controller was adopted to real-
ize displacement control during the gearshift process [13], and the
controller is
eðkÞ ¼ ydðkÞÀxðkÞ
DuðkÞ ¼ KpðeðkÞÀeðkÀ1ÞÞþKieðkÞþKdðeðkÞÀ2eðkÀ1ÞþeðkÀ2ÞÞ
&
ð19Þ
where yd is the desired displacement, x is the actual displacement, e
is the error between desired and actual values, Kp, Ki and Kd are con-
troller parameters, Du is the increment of the control variable. The
gearshift process was divided into four phases and three of them
except synchronization phase were controlled by PID method.
According to the tuning method described in reference [23], the
controller parameters were determined by trial and error, and the
PID gains for the corresponding phase when the target
displacement is 4 mm are Kp = 13,500, Ki = 0.5, Kd = 10.
Fig. 8 shows the system response with respect to various target
displacement. The target displacements are set to 4 mm and
2.5 mm. Obviously, it is demonstrated that the ISM-ESO control
converges more quickly than PID control does, which ensures a
shorter shift time. When the target displacement is 4 mm, the
transition time of ISM-ESO control and PID control are 17.7 ms
and 24.4 ms respectively. When the target displacement is
2.5 mm, the transition time of ISM-ESO control is 4.9 ms and less
than that of PID control. Besides, since the PID parameters are
tuned when the target displacement is 4 mm, there is an overshoot
while the target displacement changes to 2.5 mm, and the
overshoot is nearly 1.1%. An overshoot may be result in crash of
synchronizer ring and cone which is not allowed. For ISM-ESO
control, no overshoot occurs at either target displacement.
The system parameters such as the coil resistance R may vary
because of some factors. Fig. 9 shows the simulation results when
the value of R increased by 20%. No matter how the R varies, the
simulation results of ISM-ESO control is the same with the original
R. The simulation results of PID control are relatively bad compared
with ISM-ESO control. The variation of R causes steady-state error
with PID control, and the range is 1.7%.
The gearshift system suffers from various disturbances during
gearshift. To test the robustness performance of the ISM-ESO con-
trol, a large step external disturbance F = À100 N is added between
0.01 s and 0.015 s. It can be seen in Fig. 10 that the displacement
trajectory of ISM-ESO control is not influenced to a great extent.In
conclusion, the ISM-ESO control has excellent performance and it
is conducive to improving the shift quality.
4.2. ADRC controller design for synchronization phase
The mathematical model of the synchronization phase can be
described as
Jc
_xc ¼ À Ts
ig
Js
_xs ¼ Ts À TL
id
Ts ¼ FsÁfsÁRc
sin a
8
>><
>>:
ð20Þ
where Jc is the equivalent inertia of the input shaft of the AMT, Ts is
the friction torque Js is the equivalent inertia of the output shaft, TL
is the load torque, ig and id is the gear ratio and the differential ratio
respectively, xc and xs represent the clutch speed and the output
shaft speed respectively. Fs is the gearshift force, fs is the friction
coefficient, Rc is the effective radius of the friction cone, a is the half
cone angle.
As it is known, the shift force is the critical factor which affects
the gearshift time, and the degree of impact is mostly influenced by
the change rate of the shift force. As a result, it is necessary to
design shift force control strategy to achieve good shift quality.
Optimal control based on Pontryagin’s minimum principle is
adopted to optimize the change rule of the shift force during the
synchronization phase so that a compromise between synchroni-
zation time and shift quality is obtained, and the result is shown
in Fig. 11. The target of the shift force control is to track the optimal
trajectory precisely. The controller should be reliable when
disturbance and uncertainty occur. Besides, considering that the
optimal trajectory varies according to the different work condition
of the transmission, the controller should adjust itself to track
different trajectories well. The active disturbance rejection
controller (ADRC) is exactly the appropriate control method.
Fig. 7. ISM-ESO controller.
Fig. 8. Response curve for different desired displacement. Fig. 9. Response curves with parameter variation.
S. Lin et al. / Mechatronics 24 (2014) 1214–1222 1219

7. The shift force can be described as
Fs ¼ kmI ð21Þ
Hence, the shift force control is converted into the closed-loop
control of current. The back electromotive force is zero during
the synchronization since there is no displacement process. As a
result, the electrical equation in Eq. (1) can be rewritten as
_I ¼ À
R
L
I þ
u
L
ð22Þ
The basic topology of the ADRC is given in Fig. 12, which is com-
prised of a tracking differentiator (TD), an extended state observer
(ESO), and a nonlinear state error feedback (NLSEF) controller.
The TD is used for generating a transitional trajectory v1 of the
desired signal v to improve transition performance especially when
v is a constant. Differential signal of v1 is given as v2 simulta-
neously. The discrete-time form of TD is written as
g ¼ fðv1 À v; v2; rg; hÞ
_v1 ¼ v1 þ hv2
_v2 ¼ v2 þ hg
8
><
>:
ð23Þ
where h is the sampling period, f(v1 À v, v2, rg, h) is a kind of
time-optimal control function, and the function is given as
fðv1 À v; v2; rg; hÞ ¼
d ¼ rg Á h
d0 ¼ h Á d
y0 ¼ v1 À v þ h Á v2
a4 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d
2
þ 8rg Á jy0j
q
a ¼
v2 þ ða4 À dÞ Á sgnðy0Þ=2; jy0j > d0
v2ðkÞ þ y0=h; jy0j 6 d0
&
f ¼
ÀrgsgnðaÞ; jy0j > d0
Àrga=d; jy0j 6 d0
&
8
>>>>>>>>>>>>>><
>>>>>>>>>>>>>>:
ð24Þ
where rg is a parameter of the function f(v1 À v, v2, rg, h) which
determines the dynamic characteristics of v1, and the larger the
value of rg, the shorter the time taken by the transitional trajectory
v1 of a specific v.
The main role of the ESO is to estimate the total disturbance,
and its discrete-time form with the sampling period h is
e ¼ z1 À y
z1 ¼ z1 þ hðz2 À b11 Á e þ b0 Á uÞ
z2 ¼ z2 À h Á b12 Á falðe; a; dÞ
8
><
>:
ð25Þ
where z1, and z2 are estimates of the output y and the total
disturbance respectively. b11 and b12 are observer gains which can
be selected as b11 % 1/h, b12 % 1/1.6h1.5
[19].
Non-linear function fal(e, a, d) is defined as the same in Eq. (16).
The control input u can be designed as
u ¼ u0 À
z3
b0
ð26Þ
The NLSEF is designed to produce the intermediate variable u0,
and it can be described as
e1 ¼ y1 À z1
e2 ¼ y2 À z2
u0 ¼ b11 Á falðe1; a1; dÞ þ b12 Á falðe2; a2; dÞ
8
><
>:
ð27Þ
where b11 and b12 are controller parameters, a1 and a2 satisfy the
condition 0 < a1 < 1 < a2. The main parameters are: b11 = 10,000,
b12 = 625,000, b0 = 909.
Though the ADRC theory can guarantee the stability of the
ADRC according to the reference [19], switched systems based
multiple Lyapunov function method which was proposed in refer-
ence [22] was adopted to prove the stability of the ADRC controller.
Besides, reference [24] also demonstrated the stability of the ADRC.
The desired trajectory is shown in Fig. 12, and the tracking
errors of the ADRC and PID are compared in Fig. 13. Both the two
controllers achieve good tracking performance, but the error of
ADRC is smaller. In addition, a tiny steady-state error about 0.6%
is seen in the error profile of PID control.
Fig. 10. Response curve with 100 N disturbance.
Fig. 11. Optimal trajectory of synchronization process.
Fig. 12. Basic topology of the ADRC. Fig. 13. Tracking error of the two controls.
1220 S. Lin et al. / Mechatronics 24 (2014) 1214–1222

8. 5. Experimental validation
In order to verify the designed gearshift system and assess the
performance of the control strategy, a gearshift test bench is devel-
oped. Fig. 14 shows the arrangement and the main components of
the test bench. It is mainly made up of six parts: actuator, trans-
mission, sensors, variable-frequency motor, control system and
other assistant mechanisms. The engine input is represented by a
variable-frequency motor. The actuator mounted on the transmis-
sion connects with the shift rail through a lever. The test bench is
mounted on a big plate to avoid vibration.
The structure of the controller is shown in Fig. 15. LPC2294
microcontroller is used as main controller since it has many
available on-chip resources. Sensor signals are transmitted to A/D
ports after filtered and amplified by peripheral circuits. The con-
troller deals with the signals and transmits them to PC through
CAN-BUS. The pulse signal produced by speed sensor is captured
by capture module. Modularization method is adopted during
software design.
Fig. 16 shows the displacement response and shift force when
the target displacement is 4 mm. Additionally, a large step distur-
bance F = À50 N is added at t = 0.008 s and removed at t = 0.01 ms.
The response of ISM-ESO control is quicker than that of PID control.
The influence of added disturbance to displacement profile is more
apparent with PID control and the overdamping lasts a long time
compared with ISM-ESO control.
Fig. 14. Test bench.
Fig. 15. Hardware schematic of the controller.
Fig. 16. Comparasion of the experimental results of the two controllers.
(a) Displacement
(b) Shift force
Fig. 17. Gearshift results with the two controllers.
Fig. 18. Degree of impact curve of the two controllers during synchronization
process.
S. Lin et al. / Mechatronics 24 (2014) 1214–1222 1221

9. Fig. 17 shows the displacement and shift force variations of the
entire gearshift process. The ripple of displacement and shift force
with PID control is more apparent than those with ISM-ESO–ADRC
control. Force ripple will result in vibrations, reduction of the gear-
shift comfort and should be avoided. Fig. 18 shows the degree of
impact of the synchronization process with ADRC control and PID
control. The degree of impact is lower with ADRC control. Besides,
PID control tracks the desired trajectory worse than ADRC control
during the synchronization process. The error curves of synchroni-
zation process in Fig. 19 prove it. Table 2 compares the main
indexes of shift quality with two control methods. Both the maxi-
mum degree of impact and the synchronization time are smaller
with ADRC control. Although the friction work per unit grows a lit-
tle, they are under the permission value 1.2 J/mm2
[25]. It is veri-
fied that the ADRC control track the desired trajectory better and
has decent robustness.
6. Conclusions
A novel gearshift system based on a 2-DOF electromagnetic
actuator is introduced. The novel system eliminates reduction
gears and drives the shift lever directly which improves the
dynamic response and efficiency of the gearshift system. It has a
lower component count which simplifies the system and improves
the robustness. The characteristics of the actuator indicate that the
2-DOF electromagnetic actuator has powerful driving force and
torque. Besides, the dynamic response is fast which is in favor of
improving shift quality.
Since the gearshift process is nonlinear and suffers from uncer-
tainties and external disturbances, robust controller is necessary to
guarantee good gearshift performance. The gearshift process is
divided into synchronization phase and non-synchronization
phase so that more suitable and effective control strategy can be
designed according to the different purpose of each process.
Inverse system method (ISM) is introduced to eliminate the non-
linearity of the actuator. Extended state observer (ESO) is used to
estimate the uncertainties and disturbances, and compensates
these unfavorable factors. The simulation and experiments results
prove that the designed ISM-ESO controller can achieve quick
response, good precision and robustness with respect to various
target displacement during non-synchronization process. The
ADRC is not predicated on precise plant model and is extremely
tolerant of uncertainties and nonlinearity. Compared with PID
control, it tracks desired trajectory better and the tracking error
is relatively small which results in low degree of impact and
synchronization time. Consequently, this novel gearshift system
along with ISM-ESO–ADRC controller provides a new solution for
wide applications of AMT.
Acknowledgements
This work was supported by the National Natural Science Foun-
dation of China (Grant No. 51306090).
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Fig. 19. Tracking errors of synchronization process with the two controller.
Table 2
Comparison of indexes with two control methods.
Method Maximum degree of
impact j/(m/s3
)
Friction work per
unit WA/(J/mm2
)
Synchronization
time t/ms
ADRC 2.95 0.077 90
PID 4.12 0.065 98
1222 S. Lin et al. / Mechatronics 24 (2014) 1214–1222