-
Research ArticleDynamic Analysis and Control of the Clutch
Filling Process inClutch-to-Clutch Transmissions
Wei Guo,1 Yanfang Liu,1 Jing Zhang,2 and Xiangyang Xu1
1 School of Transportation Science & Engineering, Beijing
University of Aeronautics and Astronautics, Beijing 100191, China2
International Academy, Hefei University, Anhui 230601, China
Correspondence should be addressed to Yanfang Liu;
[email protected]
Received 20 February 2014; Revised 14 May 2014; Accepted 18 May
2014; Published 23 June 2014
Academic Editor: Yunhua Li
Copyright 2014 Wei Guo et al.This is an open access article
distributed under the Creative Commons Attribution License,
whichpermits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Clutch fill control in clutch-to-clutch transmissions influences
shift quality considerably. An oncoming clutch should be
appliedsynchronously with the release of an offgoing clutch to
shift gear smoothly; therefore, the gap between the piston and
clutch platesshould be eliminated when the torque capacity is near
zero at the end of the clutch fill phase. Open-loop control is
typicallyimplemented for the clutch fill because of the cost of
pressure sensor. Low control precision causes underfill or overfill
to occur,deteriorating shift quality. In this paper,
amathematicalmodel of an electrohydraulic clutch shift control
system is presented. Specialdynamic characteristic parameters for
optimal clutch fill control are subsequently proposed. An automatic
method for predictinginitial fill control parameters is proposed to
eliminate distinct discrepancies among transmissions caused by
manufacturing orassembling errors. To prevent underfill and
overfill, a fuzzy adaptive control method is proposed, in which
clutch fill controlparameters are adjusted self-adaptively and
continually. Road vehicle test results proved that applying the
fuzzy adaptive methodensures the consistency of shift quality even
after the transmissions status is changed.
1. Introduction
Automatic transmissions are used to transfer the power of
anengine smoothly and effectively to vehicle wheels at
optimaltransmission ratios according to performance requirementsand
economic demand. Figure 1 shows a two-dimensionalstructural
diagramof an 8-speed automatic transmissionwithfive shift elements,
namely, a brake 1 and four clutches 14, which are engaged or
separated by the electrohydrauliccontrol system of transmissions.
To shift from one gear toanother in an automatic transmission, one
clutch must bereleased and another must be applied synchronously,
whichis called a clutch-to-clutch shift [1, 2]. With the
developmentof six-speed transmissions or even more speeds
nowadays,considerable time and effort has been made to study
theclutch-to-clutch shift control technology [1, 3, 4].
Asyn-chronous clutch control would cause power interruption
oroverconstraint [5].Therefore, the gap between the piston
andclutch plates for the oncoming clutch should be eliminatedwhen
the clutch torque capacity is near zero at the end ofthe clutch
fill phase [6, 7]. Nowadays most clutch torque
models do not take hydraulic dynamic characteristics intoaccount
[8]. To optimize the engagement of clutches, clutchfill is usually
formulated as an optimization problem. Open-loop clutch pressure
control was proposed as a solution bymeans of dynamic programming
algorithm for cost reasons[6]. However, this controlmethod requires
precision trackingof the input pressure. Relative experiments were
proposedand the experimental results were used for optimal control
ofclutch fill process [9]. Low control precision causes underfillor
overfill to occur, deteriorating shift quality.
Most automatic transmissions use the electrohydraulicdriving
pattern, in which the shifting elements are controlledseparately.
Figure 2 shows a partial electrohydraulic controlsystem for a
single clutch and a torque convertor. The mainline pressure CV, is
provided by an oil pump, which ismechanically connected with the
engine output shaft. A flowregulation valve (indicated by number 1)
is used to dischargeredundant fluid flow, especially at a high
rotation speed, andto prevent the pressure and fluid flow in the
main hydrauliccircuit from exceeding the limitation. A
pressure-regulatingvalve (indicated by number 5) is used to
regulate the pressure
Hindawi Publishing CorporationMathematical Problems in
EngineeringVolume 2014, Article ID 293637, 14
pageshttp://dx.doi.org/10.1155/2014/293637
http://dx.doi.org/10.1155/2014/293637
-
2 Mathematical Problems in Engineering
B1C4
C3 C1
Outputleft
Outputright
C2
Torqueconvertor
Differential
Input
Figure 1: Schematic diagram of an 8-speed automatic
transmission.
1
2
345
6
7
8
9
10
Lubrication
12
11
13
(1) Flow control valve(2) Solenoid valve(3) Shift valve(4) Oil
duct(5) Pressure-regulating valve(6) Pump(7) Filter
(8) Tank(9) Solenoid valve(10) Control value(11) Disc spring(12)
Piston(13) Clutch friction disc
B
TC
PCV,P
PCV,A
PCU
PSV,V
PSV,K
Figure 2: A partial electrohydraulic control system for a
singleclutch.
in the main hydraulic circuit to satisfy the pressure demandsSV,
for all shift valves control; here 5 bar is preferred whichis
realized by the dimensions of the pressure-regulating valve.The
clutch piston chamber pressures are jointly controlledby a solenoid
valve (indicated by number 2) and a shiftvalve (indicated by number
3). By controlling the electriccurrent of the solenoid valve, the
pilot pressure SV, of theshift valve can be regulated. The pilot
pressure determinesthe position of the shift valve spool and
thereby determinesthe pressure and fluid flow in the clutch piston
chamber.When the clutch piston chamber pressure increases, once
thefrictional resistance and the return spring force are
overcome,the clutch piston starts to move rightward until the gap
iseliminated and then force is applied. To prevent overfill
orunderfill, the fill phase is controlled to end before a
presettime. To reduce the cost of transmission sensors, no
pressuresensors are assembled for clutches; consequently,
obtainingthe feedback pressure for closed-loop control is
difficult.Moreover, because of the strong nonlinear pressure
charac-teristics of solenoid valves and the response delay,
obtainingprecise target pressure by controlling the current is
difficult.Furthermore, the clutch pressure response differs based
onthe status of transmissions, such as the tolerance and wearof
parts. Therefore, building an open-loop clutch pressurecontrol
model for clutch fill phase based on the dynamicpressure
characteristics of electrohydraulic clutch controlsystems is
necessary to satisfy the requirements of clutch filltime andmaximal
fluid flow. Previous models have primarilydepended on car test
results, which cannot be adapted tocomplex working environments.
After transmissions run foran extensive period, wear inevitably
occurs.Therefore, a fuzzyadaptive fill control method, which was
proven to be capableof improving the shift quality effectively, is
proposed in thispaper.
2. Model of Hydraulic Clutch ShiftControl Unit
Figure 3 shows a clutch hydraulic control unit, in which
theclutch piston chamber pressure is controlled by a solenoidvalve
and a shift control valve mainly. The solenoid valvecontrols the
pilot pressure SV, at port of the shift valve tocontrol the
position of the valve spool and change the openingarea of port ,
thereby regulating the clutch piston chamberpressure CU. Port is
connected to the main line pressureCV, of this system, and port is
used for discharging theclutch cylinder. The pressure CV, at port
is controlledby a limp home valve, and there is no pressure in
normalmodes. Pressure response in failure modes is not
discussedhere. Orifice is used to control the total fluid flow
intothe solenoid valve and the shift valve. Here some symbolsand
subscripts are appointed in advance for computationalconvenience;
see Nomenclature.
2.1. Dynamic Model of a Solenoid Valve. Figure 4 showsthe
structural diagram of a normally open high-speed pro-portional
solenoid valve [7], mainly composed of a coil,armature, return
spring, and valve spool. The coil produces
-
Mathematical Problems in Engineering 3
V
APT
D
C
B
1
2
3PSV,V
PSV,C = 0
PSV,K = 5 bar
PCV,P PCV,A
PCU
(1) Clutch control solenoid valve(2) Shift valve(3) Clutch
Figure 3: Schematic diagram of a hydraulic clutch control
unit.
electromagnetic force and overcomes the resistance of thereturn
spring, and the armature drives the valve spool tomove in the axial
direction of the spool, thereby regulatingthe opening area of the
valve and, subsequently, the pressure.
According to the laws established by Newton, the dynam-ic
equation of the valve spool can be described as
SVSV + SVSV + SV (SV + 0SV) = SV. (1)
The electromagnetic subsystem of the solenoid valve canbe
simplified into a series connection of resistive and induc-tive
components [10]. The electromagnetic force is calculatedusing
SV = 2
SV, (2)
where the dynamic equation of the electric circuit is
writtenas
SV =(SV SVSV VSVSV)
SV. (3)
During the movement of the valve spool, the dynamicequation for
the hydraulic subsystem is expressed as
SV, = CdSVSV, (SV) 2
SV,. (4)
2.2. DynamicModel of a ShiftValve. Figure 5 shows the
struc-tural diagram of a shift control valve with double spools.
Forconvenience, the symbol
1
represents the pilot spool and
2
the main valve spool. When the electric control systemfails to
provide pilot pressure at port , the safety pressureat port still
can control the main spool
2
to ensure theclutch pressure which can be supplied from port.
There aresectional area differences in chambers II and III.
1 2 3
4 5
Flow out
Flow in
(1) Coil (2) Armature (3) Valve spool
(4) Spring(5) Strainer
K
T
Figure 4: Structural diagram of a variable-force solenoid
valve.
A
T PV C D
m1 m2
ds
I
P: InletV: Pilot portC: Limp home portA: Outlet
D: Pressure regulation portT: Discharging portIIII: Chambers
II III IV
Figure 5: Structure of the shift valve.
According to the laws established by Newton, the dynam-ic
equations of the valve spools are described as
CV,1
CV,1
= SV,CV,I CV,CV,II
2
1
, (5)
CV,2
CV,2
+ CVCV,2
+ CV (CV,2
+ 0CV,2
)
= CV,CV,II + 2
1
CV,CV,III CV,CV,IV,
(6)
where CV,III = CV,III CV,III is the sectional area differ-ence
of chamber III.
Because of the movement of the main spool, the relation-ship
between the pressure and fluid flow of the shift controlvalve at
all ports can be expressed as
CV, = sign (CV, CV,)CdCV
CV, (CV,2
)2
CV, CV,,
CV, = CdCVCV, (CV,2
)2
CV,,
-
4 Mathematical Problems in Engineering
CV, = sign (SV, SV,)Cd
2
SV, SV, SV,,
(7)
SV, =CV
0CV,I + CV,1
CV,I(CV, CV,
1
CV,I) , (8)
CV, =CV
0CV,IV CV,2
CV,IV(CV, + CV,
2
CV,IV) .
(9)
2.3. DynamicModel of the Clutch Fill Phase. Automatic
trans-mission fluid (ATF) flows from port of the shift valve
intothe clutch chamber through pipes in gear structure of
thetransmission. For convenience, the symbols and are theequivalent
length and radius of these, respectively;
is theequivalent centrifugal pressure in the piston chamber
causedby rotational speed of the clutch, calculated in [11].
Withthe function of the increasing inlet pressure, after the
returnspring force and friction force of the clutch are overcome,
thepiston starts tomove rightward until the gap is eliminated
andthen force is applied.
According to the laws established by Newton, thedynamic
equations of the clutch piston are described as
CUCU = CU (CU + ) seal CU (CU + 0CU)
CUCU (CU < ) ,
APP = CU (CU + ) seal
CU ( + 0CU) (CU = ) .
(10)
The fluid flow into the clutch piston chamber throughpipes and
the piston chamber pressure can be expressed as
CU =
4
8(CV, CU) +
leakCU , (11)
CU =CU
0CU + CUCU(CU CUCU) . (12)
The sealing force seal is defined as
seal
=
{{{{
{{{{
{
(seal (CU + ) + ) tanh(
)
(CU
> 0.1mm/s)
(CU + ) + (CU
0.1mm/s) .
(13)
The static viscous friction stick can be computed usingthe
Kanopp stick-slip model [12]. This viscous friction isgenerally
ignored in dynamic models [13]. However, in theclutch fill phase,
the pressure of ATF fluid is low and,thus, a large proportion of
viscous friction exists. From theperspective of numerical
computation, the friction of the Oseal ring is assumed to be equal
to stick when the velocity of
the clutch piston is under 0.1mm/s. When this occurs, stickis
the force used to balance the piston, and its acceleration is0.
Therefore, this viscous friction is the maximal limit value.Once
the friction exceeds this limitation, the piston starts tomove.
2.4. Clutch Pressure Characteristics. By combining (1)(12),the
clutch piston chamber pressure response can be simu-lated. Figure
6(a) shows the pressure response when applyingthe step current
command. The simulation results indicatedthat the clutch piston
chamber pressure is initially 310mA.Therefore, a calculated pilot
pressure of at least 0.6 bar isrequired to overcome the resistance
of the shift valve spool.When the small step control is applied,
the clutch pressureincreases with step to approximately 0.28 bar.
If the pressureis not sufficiently high to overcome the resistance
of theshift valve spool, then the clutch pressure does not
respondto the current step, the so-called loss of step. When
largestep current control is applied, the clutch pressure
overshootsor oscillates, but it rapidly stabilizes because of
systemdamping (II in Figure 6(a)); clearly, the stabilizing time
ofthe shift valve is shorter than that of the solenoid valve.Based
on Figure 6(a), the steady characteristics of the clutchpiston
chamber pressure, exhibited when the control currentincreases, can
be obtained, as illustrated in Figure 6(b), whichshows that
hysteresis exists.
3. Optimization of Clutch FillControl Parameters
3.1. Selection of Control Parameters. Theobjective of clutch
fillcontrol is to stably eliminate the gap between the piston
andclutch plates within a preset time.The ideal piston velocity
inthe clutch fill phase is shown in Figure 7. In the early
periodfrom0 to2, the piston velocity is nearly zero, and the
deadvolume of the clutch cylinders is first filled with ATF
fluid.Consequently, the piston velocity is low within a short
periodfrom 1 to 2, rapidly increases to the maximum within
theperiod from2 to3, ismaintained from3 to4, andfinallydecreases
from 4 to 5, and the clutch is engaged.
Figure 8 shows two methods for clutch fill pressurecontrol based
on the piston motion shown in Figure 7: trian-gle fill and square
fill. pre, FP, and FTP are the prefillpressure, fast fill pressure,
and stable fill torque pressure,respectively. FTP is the pressure
when the clutch transfers asmall torque. In practice, based on
characteristic curves, suchas pressure versus current curves, the
target pressure can betransformed into the control current of the
solenoid valve.Therefore, analysis of the target pressure and the
real pressureis necessary for optimizing clutch fill control.
According toFigures 7 and 8, the clutch fill phase is divided into
threestages.
(1) Prefill Stage. The prefill stage shown in Figure 8
ensuresthat the dead volume in the clutch cylinder (Figure 2) is
filledwithATFfluid and an initial amount of pressure. At this
stage,because of the sealing force, the piston velocity is
extremelylow.
-
Mathematical Problems in Engineering 5
10 30 50 70 900
200
400
600
800
1000
1200
0 20 40 60 800
10
20
Time (s)
Clut
ch p
ress
ure (
bar)
Con
trol c
urre
nt (m
A)
I
I
II
II
(a)
Control current (mA)
Clut
ch p
ress
ure (
bar)
200 400 600 800 10000
5
10
15
20
a b c d
DownUpper
I
I
(b)
Figure 6: Simulation results for the step response of the
system.
Time (s)
0T5T1T0 T2 T3 T4
Maximum
xC
U(m
m/s
)
Figure 7: Ideal piston velocity.
(2) Fast Fill Stage. This stage is the most crucial and
difficultstage, corresponding to the period from 2 to 3 in Figure
7.The triangle fill method can be used to increase the pressurepeak
substantially and enhance the piston velocity. However,applying
this method also increases the instability of thefill phase, and an
excessively high pressure peak increasesthe demand for systemic
fluid flow, thereby affecting themain pressure supply. Regarding
the square fill method, thepressure response is relatively slow but
stabler than thatproduced when applying the triangle fill
method.
(3) Stable Fill Stage. At this stage, the piston is driven
stablyto eliminate the gap between clutch plates, corresponding
tothe period from 3 to 5 in Figure 7. The control currentis
generally unvarying in this process. As the piston moves,the
elasticity of the wave spring gradually increases and, thus,the
pressure gradually increases until the piston stopsmovingwhen the
gap is completely eliminated.
Four parameters listed in Figure 8 were selected as clutchfill
control parameters: fast fill pressure FP, rapid fill timeII,
stable fill pressure FTP, and prefill pressure pre. Theseparameters
were used to determine the variation in the clutch
Pres
sure
(bar
)Fill
Time (s)0
Triangle fill
Square fill
A
Ppre
PFP
PFTP
TIIITII
TI
II III
II III
I PrefillFast fillStable fill
I
Figure 8: Design of clutch fill control pressure.
fill pressure and the movement of the piston; thus, the
clutchfill control was optimized by optimizing these
parameters.
3.2. Influence of Clutch System Parameters on Clutch Fill-ing.
The aforementioned equations indicate that numerousparameters
affect the clutch fill process. The consistencyof several
parameters, such as the number of coils in thesolenoid valve, the
magnetic resistance, and the sectional sizeof the valve core, is
maintained during assembly. However,the consistency of other
parameters, such as the springstiffness and preload force, main
supply pressure, ATF fluidtemperature, and sealing force, is
impossible to maintain,which causes differences in the oil fill
results. Therefore, theeffects of these parameters on the clutch
fill phase should be
-
6 Mathematical Problems in Engineering
analyzed and controlled to optimize the open-loop clutch
fillcontrol.
Based on the aforementioned (5)(12), the influence ofthese
parameters on filling the clutch under the same targetoil fill
pressure was simulated, as shown in Figure 9.
(1) ATF Fluid Temperature . Figure 9(a) shows the clutchpiston
chamber pressures at various temperatures. The tem-perature of the
ATF fluid directly affects its viscosity; thus,the resistance of
motion increases at low temperatures.In addition, the contraction
of components contributes tochanges in the tolerance clearance.
These factors slow clutchfill. As shown in Figure 9(a), when the
temperature is below20
C, the pressure response is extremely slow at the fast oilfill
stage and stable oil fill stage. As the temperature increases,the
speed of the oil fill response and the peak pressureclearly
increase. At high temperatures, such as 120C, theclutch piston
chamber pressure rapidly reaches the kiss-pointpressure, but this
process may cause overfill and shiftingimpact to occur. Therefore,
clutch fill control parametersshould be adjusted based on
temperature to meet the shiftquality requirements at various
temperatures.
(2) Main Oil Pressure CV,. Equation (8) indicates that themain
supply pressure affects the flow of the system and that itis
directly related to the rotation speed of the engine; thus, themain
supply pressure considerably influences the clutch fillperformance.
Figure 9(b) shows the simulated clutch pistonchamber pressure under
various main supply pressures.Underfill clearly occurs between 5
and 10 bar, whereas overfilloccurs at 20 bar. Therefore, the
influence of the main supplypressure should be considered; in other
words, the clutchfill intensity should be reduced to prevent
overfill at highpressures and increased to prevent underfill at low
pressures.
(3) Spring Stiffness CU and Preload 0CU. Tolerances existbecause
of both manufacture and assembly of return springsin clutches,
which are mainly reflected in the stiffness andpreload of the
return springs. These two parameters deter-mine the spring
resistance that occurswhen the pistonmoves.A strong resistance
causes the clutch to reach the kiss pointunder high pressure and
over a long time. Although this timecan be shortened when the
resistance is weak, shortening thetime may cause the piston to move
too fast and thereby affectthe shift quality. Figures 9(c) and 9(d)
show the simulationresults for clutch fill pressure at various
spring stiffness andpreload values. Clearly, increased spring
stiffness and preloadresult in a high maximal fill pressure and a
low pressurechange rate during the rapid fill stage.
(4) Static Frictional Resistance Coefficient
. The static fric-tional resistance coefficient affects the
maximal static frictionthatmust be overcomewhen the piston is
static ormoving at alow speed.This coefficient and the preload 0CU
jointly affectthe pressure when the piston starts to move.The
larger the
is, the higher the peak pressure is during the rapid fill
stage(Figure 9(e)). However, if
and 0CU are extremely small,then the acceleration of the clutch
piston becomes extremelyhigh, easily causing a piston movement bump
to occur. If thepiston speed reduces to a certain level, then
double-peak fill
occurs because of the static friction (Figure 9(e)), resulting
inunderfill; therefore, the clutch cannot reach the kiss point.
(5) Sealing Resistance Coefficient seal. Figure 9(f) showsthe
effects of the sealing resistance coefficient seal of theclutch
cylinder. The influence of the tolerance of seal onclutch fill
pressure is relatively small compared with that ofother parameters.
The resistance caused by seal influencesthe acceleration of the
piston. Thus, a high speed clearlyaffects resistance. Therefore,
seal can magnify the effects oftemperature, piston preload, piston
stiffness, andmain supplypressure on the speed of the piston.
3.3. Optimization of Clutch Fill Control Parameters. Since
themanufacturing tolerance, assembly tolerance, and
workingconditions also affect clutch fill pressure, clutch fill
controlshould cover all types of qualified tolerance and
normalworking conditions.
3.3.1. Prefill Pressure. Prefill pressure pre is included
toensure that the pipes in the gear structure of the
transmissionsbetween the shift valve and clutch piston are filled
with fluidand is not high to the extent that the resistance
required tomove the piston is overcome. Therefore, pre should not
betoo high to overfill when using clutches with low
resistance.Moreover, the minimum prefill pressure should avoid
thedropping downward of ATF oil of the rotating clutch. The oilflow
must be completed in the prefill time I. Furthermorethe prefill
time is generally limited to theminimum shift time.The constraint
condition on pre and I can be defined as
I max
0
(CV, leakCU ) = 0CU +
2
,
I max = 0.1 shift,
I = (1 I) I min + II max,
pre maxCU = (stick)min + (CU0CU)min,
pre min = ,
pre = (1 pre) pre min + pre pre max.
(14)
3.3.2. Fast Fill Pressure and Time. The fast fill stage is
theperiod when the piston accelerates after overcoming all typesof
resistance. The fast fill pressure and time affect the fillspeed
and stable fill pressure status. Figure 10(a) shows thesimulation
results for varying target pressures. Increasing thefast fill
pressure enhances the clutch pressure response speedand increases
the fill pressure at the initial stage. The time forthe clutch
pressure to reach the target value is shortened withincreasing the
fill pressure. However, pressure fluctuationoccurs during the third
and fourth fill process because thepiston speed is still extremely
high when the gap betweenclutch plates is eliminated. The ATF fluid
cannot absorb allof the remaining energy instantly; therefore, the
pressurefluctuation occurs inevitably.
-
Mathematical Problems in Engineering 7
00.80.60.40.20.0
4
3
2
1
Fill
pres
sure
(bar
)
= 110C
= 60C
= 60C
= 20C
= 20C
Time (s)
(a)
0 0.80.60.40.20.0
4
3
2
1
PCV,P = 20 bar
PCV,P = 15 barPCV,P = 15 bar
PCV,P = 10 barPCV,P = 5 bar
Time (s)
Fill
pres
sure
(bar
)
(b)
00.80.60.40.20.0
4
3
2
1
kCU = 80nm/mmkCU = 91nm/mmkCU = 91nm/mm
kCU = 103nm/mmkCU = 110nm/mm
Time (s)
Fill
pres
sure
(bar
)
(c)
0 0.80.60.40.20.0
4
3
2
1
Fill
pres
sure
(bar
)
Time (s)
x0CU = 6.5mmx0CU = 6mmx0CU = 6mm
x0CU = 5.5mmx0CU = 5mm
(d)
4
3
2
1
00.80.60.40.20.0
Time (s)
ks = 0.00346
ks = 0.00276
ks = 0.00276
ks = 0.00226
ks = 0.00186
Fill
pres
sure
(bar
)
(e)
00.80.60.40.20.0
4
3
2
1
Time (s)
kseal = 0.0021
kseal = 0.0018
kseal = 0.0018
kseal = 0.0015kseal = 0.0012
Fill
pres
sure
(bar
)
(f)
Figure 9: Oil fill results for various clutch parameters.
-
8 Mathematical Problems in Engineering
Once the shift valve suddenly opens wide, a large amountof ATF
oil flows to the piston cylinder, which increases theflow and
reduces the main line pressure (Figure 10(a)). So themain line
pressure and the systemflow can be used to evaluatethe effects of
fast fill pressure. Because the variation in mainline pressure also
affects other clutch pressures, it should notbe reduced
excessively.
The real fill pressure at the stable fill stage increases
withthe fast fill time (Figure 10(b)). The reason is that the
filledvolume and the maximal clutch piston speed increase at
thefast fill stage. Subsequently, the clutch piston eliminates
thegap and reaches the kiss point rapidly. So if a long fast fill
timeis required, overfill may occur and the shift quality may
beaffected.
The clutch fill is affected by the fast fill timeII, the fast
fillpressureFP, the stable fill timeIII, and the stable fill
pressureFTP combined. FTP affects the transferred torque
afterengagement, which has priority over clutch fill control. SoFTP
is treated as the input for optimizing other parameters.
The fast fill time II affects clutch fill significantly(Figure
9(b)), while the stable fill time III affects it slightly.The
difference of the control effect of III for differenttransmissions
could be compensated by adjusting the fast filltime II. So the
preferred strategy is to adjust II while IIIis usually set to be a
constant, which is determined by thevelocity and displacement
requirements of the clutch piston.The stable fill time III is
formulated as
III = [FTPCU seal CU (0.75 + 0CU)]
2CU. (15)
In order to avoid bad shifting performance, the percent-age of
the clutch fill time in the whole shift time is limitedstrictly,
such as within 47% used in this paper. So the fast filltime II is
constrained by
II II max
= 0.47shift
[FTPCU seal CU (0.75 + 0CU)]
2CU.
(16)
Meanwhile, the fast fill pressure FP should be applicablefor
different conditions such as different temperatures ofATF fluid and
main line pressures. Here the fast fill timeII is again limited as
II = 0.7II max. According to thecontrol requirement, the gap
between clutch plates should beeliminated with the clutch fill
time, which is formulated as
FP () + FTP () = 0.7II max
CU (FP)
+
III
0.7II max
CU (FTP) = .
(17)
In order to engage clutches smoothly, the energy of theclutch
piston in the end should be smaller than the extrusion
energy
of ATF fluid between clutch plates; otherwise thereis an impact.
So the clutch fill pressures are constrained by
0 FP FP CU FP + FTP FTP CU FTP
1
2CU
2
seal ,(18)
where FP and FTP are equivalent factors of the target
pressureand real pressure at fast fill phase and stable fill
phase,respectively.
In order to avoid that the main line pressure decreasesbelow the
safe level in fast fill phase, a safety fluid flow of ATFfluid is
required. So the maximum velocity of the piston isusually limited
by
CU (FP) CUmax pump, (19)
where pump is the flow of pump; is the safety factor forflow
loss.
So the fast fill pressure can be controlled as
FP = FP min (FP max 1, FP max 2) + (1 FP) FP min.(20)
Oversized FP would cause unstable clutch fill process; herean
optimal value2/3 is tested to be effective.
3.3.3. Stable Fill Pressure. The stable fill pressure
mainlyaffects the final stage of clutch engagement. At this
stage,because of increased spring resistance, the piston
deceleratesuntil the gap is eliminated. The simulation results for
variousstable fill pressures are shown in Figure 11. During period
in Figure 11, a reduced amount of ATF fluid is fed intothe piston
chamber because of a sudden current reduction.However, the piston
continues to move rapidly, causing thefill pressure to decrease
according to (12). During period in Figure 11, when the piston
speed decreases to a certainlevel, namely, CU > CUCU, the fill
pressure increases tothe target level gradually. In an ideal
situation, the clutchpressure should be onlyKP and the piston speed
should equalzero when the fill is nearly completed. However,
numerousfactors affect fill pressure; consequently, realizing the
idealsituation is nearly impossible. Nevertheless, the robustness
ofthe clutch fill control can be improved by reducing the fastfill
pressure and increasing the stable fill pressure. Clearly,reducing
the fast fill pressure reduces the maximal pistonspeed; increasing
the stable fill pressure ensures that the fillpressure reaches the
kiss-point pressure before the fill processis completed. An
appropriate stable fill pressure FTP valuecan be calculated by
FTP = KP +10
2
(1)2
. (21)
4. Automatic Test Method for DeterminingInitial Clutch Fill
Parameters
Because of distinct inconsistencies in the mass productionof
transmissions and in the open-loop control characteristics
-
Mathematical Problems in Engineering 9
0 10 20 30 40 50 600
1
2
3
4
5
6
0
0
2
4
6
8
10 20 30 40 50 60
Pum
p flo
w (L
/min
) Li
ne p
ress
ure (
bar)
Pisto
n sp
eed
(mm
/s)
Clut
ch p
ress
ure (
bar)
0 10 20 30 40 50 6011
12
13
14
15
16
0 10 20 30 40 50 6019
19.5
20
20.5
21
21.5
Time (s)
Time (s)Time (s)
Time (s)
2
Q1Q2
Q3Q4
PCV,P1PCV,P2
PCV,P3PCV,P4
PCU1PCU2
PCU3PCU4
xCU1xCU2
xCU3xCU4
(A) (B)
(C) (D)
(a) Fast fill pressure
0 0.2 0.4 0.6 10
1
2
3
4
0.8
Real pressure
Command pressure
Time (s)
Fill
pres
sure
(bar
)
(b) Fast fill time
Figure 10: Pressure response at various target pressures and
fast fill times.
-
10 Mathematical Problems in Engineering
of clutch fill pressure, obtaining the initial fill parameters
iscrucial to improving shift quality. The direct method is
tocalculate fill parameters based on obtained clutch
pressuresignals by sensors from the transmission end-of-line
(EOL)test according to previos equations.
In the EOL test, because the allowed testing time is short,only
one key control parameter can be tested while theother control
parameters remain unchanged. By analyzingthe effects of the
aforementioned parameters on fill pressure,the fast fill time can
be adjusted to alter the characteristicparameters, thus enabling
the clutch pressure to reach thekiss-point pressure KP within the
preset time. The clutchfill testing scheme is formulated as shown
in Figure 12.During testing, the fast fill time gradually increases
andwhether the fill requirement is satisfied can be determinedby
observing three conditions denoted as 1, 2, and 3,respectively, in
Figure 12. 2 is defined from start2 to end2.1 is defined from
start1 to end1. The condition 1 requiresthat, immediately after the
fast fill stage ends, the pressure isboth below KP and within the
range of 1 to 1, ensuringthat the piston moves smoothly. The
condition 2 requiresthat, after the fill stage ends, the clutch
pressure is exceedingKP, ensuring that the piston reaches the final
engaged pointwithin the preset time.The condition3 requires that,
duringthe fast fill stage, themaximal fill pressure cannot
exceed
3
.The ranges for1,2, and3 are related to themass toleranceranges
of the transmission components.
In the fill phase, the clutch pressure within the range of1 and
2 is associated with the fast fill time. This pressurecan be
increased by increasing the fast fill time; therefore, theoptimal
fast fill time can be obtained by applying the learningrules of the
EOL test shown in Table 1. Figure 13 shows theautomatically
measured EOL results. Beginning from 120ms,the fast fill time
increases in increments of 10ms. As the fastfill time increases,
the pressure at 1 and 2 increases untilit reaches the required
level at 170ms. Moreover, during eachfast fill stage, the main line
pressure of the system decreasessuddenly and causes flow deficiency
when a large amount ofATF fluid instantly enters the piston
cylinder.
5. Effects of Overfill and Underfill onShift Quality
Shift quality optimization is applied to ensure that the
torquefrom the offgoing clutch is transferred smoothly to
theoncoming clutch without flares or tie-up.
5.1. Effects of Underfill. Because numerous factors affectclutch
fill and an open-loop is used for pressure control,the piston
cannot be ensured to reach the kiss point exactlywhen the clutch
fill phase ends. Figure 14 shows how underfillinfluences clutch
shifting processes. When clutch fill is com-pleted, the clutch
pressure does not reach the kiss point at thebeginning of the
torque exchange stage. However, because ofunderfill, the pressure
of the oncoming clutch cannot remainequal to the target pressure at
the initial stage of torqueexchange, causing the engine load to
decrease and, thus,engine flare to occur. If engine flare continues
for a long time
4
3
2
1
00.80.60.40.20.0
0
2
4Commandpressure
Piston speed
A B
Time (s)
Fill
pres
sure
(bar
)
Pisto
n sp
eed
(mm
/s)
Pmin 2
4
PFTP = PKPPFTP = PKP + 10 kT2P
Figure 11: Fill results at various stable fill pressures.
G1G2
Time (s)
Fill
pres
sure
(bar
)
UpDownG3
Tend2Tstart2
Tstart1
Tend1
PFP
PFTPPKP
Ppre
PL1 PH1PH2
PH3
Figure 12: Fill end-of-line test method based on the
pressurefeedback.
Table 1: Learning rules of the fill end-of-line test.
1 2 3 Fill learning rule <
1
< KP < 3 Up1
1
< KP < 3 Up >
1
< 3
Down1
1
> 2
< 3
Down1
1
KP 2 < 3 CorrectOther conditions Fail
at a high rotational speed, the clutch friction plates generate
ahigh amount of heat. Therefore, underfill would shorten thelife of
clutches. Moreover, engine flare aggravates the slippingof clutches
and causes the line pressure to increase based onthe PI
(proportional-integral) control of the offgoing clutch.Although
clutch slipping can be reduced, it causes fluctuationin the
rotational speed (Figure 14).
5.2. Effects of Overfill. Overfillmeans that a certain amount
oftorque is transferred by clutches during the clutch fill
phase.Overfill occurs when the fill pressure is too high. Figure
15shows the pressure response of the clutch when overfilloccurs. At
the fill stage, after the fast fill period ends, thepressure is
still higher than FTP; consequently, the oncoming
-
Mathematical Problems in Engineering 11
0 2 4 6 8 10 12 140
1
2
3
4
0
5
10
15
20
Clut
ch p
ress
ure (
bar)
Time (s)
Line
pre
ssur
e (ba
r)
Ok120ms 130ms 140ms 150ms 160ms
170ms
PFTP
PKP
Figure 13: Automatically measured fill results.
12
10
8
6
4
2
05251.651.250.850.450
3000
2800
2600
2400
2200
Inpu
t spe
ed (r
pm)
Clut
ch p
ress
ure (
bar)
Target input speed
Current input speed
Offgoingclutch
Oncomingclutch
Time (s)
Line
pre
ssur
e (ba
r)
20
10
0
Figure 14: Effects of underfill.
clutch can transfer a small amount of torque. This
inevitablycauses a sudden increase in engine load, causing the
rotationspeed of the transmission input shaft to decrease
rapidly,resulting in negative slip of the offgoing clutch.
Typicallythe pressure is reduced through PI control to
compensatefor the negative slip. Negative slip that still exists
before thetorque phase considerably affects the torque exchange
phasebecause, at this moment, the slow decrease in the torque ofthe
offgoing clutch and the rapid increase in the torque of theoncoming
clutch cause a substantial shift impact. If a smallamount of
positive slip exists before the torque phase begins,then a buffer
zones is used for the torque exchange control;thus, the shift
impact can be avoided. In conclusion, overfillcauses transmission
shift impact to occur.
6. Fuzzy-Adaption-Based Correction Method
Since certain components, especially those in constantmotion
such as the clutch return spring and the seal ring,exhibit
performance decay after being used for a long time,the
characteristics of these components change as the work-ing time
increases.TheEOL test data represent only the initial
2600
2400
2200
2000
1800
Time (s)
Offgoingclutch
Oncomingclutch
Line pressure
Target input speed
Current input speedInpu
t spe
ed (r
pm)
Clut
ch p
ress
ure (
bar)
Line
pre
ssur
e (ba
r)
12
10
8
6
4
2
02.01.61.20.80.40.0
Figure 15: Effects of overfill.
characteristics of transmissions. Therefore, achieving
clutchfill control by using the adaption method is the
prerequisitefor ensuring shift quality throughout the entire life
cycle of atransmission. Figure 16 shows the proposed adaption
controlstrategy for the clutch fill process. The current clutch
slip isused to evaluate the current shift quality during the
shiftingprocess and to optimize the fill control parameters.
Becausethe fill control affects only the shift quality at the fill
andtorque exchange stages, the fill control parameters II andFTP
can be optimized by monitoring the clutch slip based on1 and2,
shown in Figure 16.
According to the input speed
, the output speed
, andthe ratio of current gear, the clutch slip
can be calculatedby
=
=
. (22)
Both the fast fill timeII and stable fill pressureFTP
affectclutch fill results. The stable fill pressure FTP exerts a
moreevident influence on the speed adjustment required for
theclutch to reach the kiss-point pressure thanII does but
easilyleads to excessively high pressure and so affects shift
quality.The fast fill time II determines the movement speed of
thepiston after the fast fill stage.The correction coefficients of
thecontrol parameters FTP and II can be obtained by using
anempirical fuzzy adaption control method, which is describedin
Figure 17. The input variable of fuzzy adaptive control isthe
clutch slip
. The clutch slip is processed fuzzily andthen fuzzy subsets 1
and 2 are obtained, respectively. Thecorrection factors
and
for fast fill pressure and filltime are then calculated. First,
the current clutch slip domainin region 1 is defined as 1 = {
1
, 2
, 3
, 20 rpm} andthe fuzzy subset of which is {NB, NM, NS, ZO}. The
currentclutch slip domain in region 2 is defined as 2 = {
4
, 5
,20 rpm, 40 rpm,
6
, 7
, 8
}, the fuzzy subset of which is {NB,NS, ZO, PS, PM, PB}. Based
on the following observations,the fuzzy control rules shown in
Tables 2 and 3 are used.
-
12 Mathematical Problems in Engineering
Time (s)
Clut
ch p
ress
ure (
bar)
Inpu
t spe
ed (r
pm)
nint
no
nc
k1
k2
k3
k4k5
k6
k7
k8
Tfill TtqW1 W2
TIIPFTP
Figure 16: Adaption control strategy for the clutch fill
process.
Table 2: Empirical fuzzy control rules for fast fill time.
2
NB NS ZO PS PM PB1
NB NB NB NB NB NB NBNM NB NB NM NS NS NSNS NM NS NS ZO ZO PSZO
ZO ZO ZO PS PM PB
Table 3: Empirical fuzzy control rules for stable fill
pressure.
2
NB NS ZO PS PM PB1
NB NM NM NM NM NS NSNM NM NM NS NS NS NSNS NM NS NS ZO ZO PSZO
NS NS ZO PS PS PM
(1) The control priority level for adjusting fill
controlparameters is higher when negative slip occurs in1and2 than
when positive slip occurs.
(2) In 1 and 2, the fill control parameters are notadjusted
within the slip range of 20 to 40 rpm.
(3) The priority for adjusting fast fill time II is higherthan
that for adjusting the stable fill pressure FTP.
(4) When a negative slip occurs in2, only the stable
fillpressure FTP must be adjusted.
Based on the rules shown in Tables 2 and 3, the
correctioncoefficients for fast fill timeII and stable fill
pressureFTP canbe obtained using
II () = II ( 1) + 20 ,
FTP () = FTP ( 1) + 0.2 .(23)
Figures 18 and 19 show the adaption result for the fastfill time
II and stable fill pressure FTP of clutch 4 duringthe 200,000 km
transmission durability test. The fill controlparameters should be
corrected when the transmission isbeing used to enable the fast
fill time and stable fill pressureto meet the shift quality
requirements. Figure 18 shows that ahigh main line pressure results
in a short fast fill time anda high clutch torque coefficient
results in a low stable fillpressure. In Figure 19, the horizontal
axis is the clutch torquecoefficient, namely, the ratio coefficient
of the clutch torqueto the input shaft torque of the transmission,
which is relatedto the mechanical structure of the transmission.
The clutchtorque coefficient varies when using different gears. The
testresults indicate that the transmission characteristics changeas
the mileage increases especially the first 8,000 km.
7. Conclusions
(1) A mathematical model of an electrohydraulic clutchshift
control system in an automatic transmission waspresented.
(2) By analyzing the effects of key model parameters onclutch
filling process, four special dynamic character-istic parameters
were chosen for optimal clutch fillcontrol.
(3) An automatic method was proposed for testing initialclutch
fill parameters.
(4) In order to prevent the underfill or overfill in clutchfill
phase, a fuzzy adaption control method was pro-posed. 200,000 km
road vehicle test results verifiedthat this method can effectively
prevent the clutchshift quality from declining through the natural
decayof the performance of components during the lifecycle of the
transmission.
Nomenclature
SV,CV,CU, SP: Solenoid valve, shift control valve,clutch, and
return spring, respectively
sub1,sub2 (optional): Mass of the spool sub2 of the part
sub1,and only one subscript is used if there isonly one spool
sub1,sub2: Fluid flow at port sub2 of the part sub1
leaksub1: Leakage of the part sub1
sub1,sub2: Pressure at port sub2 of the part sub1sub1,sub2
(optional): Electromagnetic force exerted on the
spool sub2 in the part sub1, and onlyone subscript is used if
there is only onespool
sub1,sub2 (optional): Spring force exerted on the spool sub2in
the part sub1, and only one subscriptis used if there is only one
spool
sub1sub2: Force exerted on the mass sub1 fromthe mass sub2
sub1: Stiffness of the return spring of the partsub1
sub1: Stamping coefficient of the part sub1
-
Mathematical Problems in Engineering 13
Torque phasefuzzy processing
Clutch slipcalculation
Clutch controlstate
Timer
Fill phasefuzzy processing
Fill controlparametercorrection
Transmission system Fill pressurecalculation
Timer
P2C conversion
Empirical fuzzy logic
I
niTfill
ns
ns
ic
KT
KP
PP PFTP(n)
TII(n)
F1
F2
Ttq
Figure 17: Fill fuzzy adaptive control system diagram.
5 10 15 200
50
100
150
200
250
300
Fast
fill t
ime (
ms)
Line pressure (bar)
0km2000 km4000 km8000 km
12000 km16000 km20000 km
Figure 18: Adapted fast fill time during the entire life cycle
of thetransmission.
0.2 0.6 1 1.4 1.81.2
1.4
1.6
1.8
2
Stab
le fi
ll pr
essu
re (b
ar)
Clutch torque factor
0km2000 km4000 km8000 km
12000 km16000 km20000 km
Figure 19: Adapted stable fill pressure during the entire life
cycle ofthe transmission.
sub1,sub2 (optional): Displacement of the mass sub2 of thepart
sub1
0sub1,sub2 (optional): Preload of the mass sub2 of the part
sub1sub1,sub2 (optional): Velocity of themass sub2 of the part
sub1sub1,sub2 (optional): Acceleration of the mass sub2 of the
part
sub1sub1,sub2 (optional): Sectional area of the chamber sub2 of
the
part sub10sub1,sub2 (optional): Initial volume in the chamber
sub2 of the
part sub1 sub1,sub2: Opening area of the port sub2 of the
part
sub1sub1: Voltage applied in the coil of the part
sub1 sub1: Inductance in the part sub1sub1: Electric resistance
of the part sub1sub1: Electric current in the coil of the part
sub1Vsub1: Back EMF coefficient of the part sub1sub1: Magnetic
force coefficient of the part
sub1sub1: Bulk modulus of the part sub1Cdsub1: Unitless
discharge coefficient of the part
sub1: Density of the automatic transmission
fluid (ATF)seal: Sealing force of the clutch piston
: Static frictional resistance coefficient ofthe clutch
piston
stick: Static viscous friction of the clutch pistonpre: Prefill
pressure of the clutch pistonFP: Fast fill pressure of the clutch
pistonFTP: Stable fill pressure of the clutch pistonKP: Kiss-point
pressure when the clutch
starts to transfer torqueAPP: Cylinder force exerted on clutch
plates2
: Torque-pressure characteristiccoefficient of the clutch
: Relative torque coefficient of the inputshaft in the current
gear
I: Prefill time of the clutch pistonII: Fast fill time of the
clutch pistonIII: Stable fill time of the clutch piston
-
14 Mathematical Problems in Engineering
shift: Required shift time: Temperature of ATF fluidvar1: Weight
factors of the variable var1
: Vertical height from port of the shiftvalue to the center of
clutch pistonchamber
: Maximum stroke of the clutch pistonper: Stroke of the clutch
piston in clutch fill
phase per: Damping coefficient of O-type seal ring.
Conflict of Interests
The authors declare that there is no conflict of
interestsregarding the publication of this paper.
Acknowledgment
The authors would like to acknowledge financial support ofthe
National Science Foundation of China (51105017) and theNational
Science and Technology Support Program of China(2011BAG09B00).
References
[1] Z. Zhao, H. Chen, and Y. Yang, Fuzzy determination of
targetshifting time and torque control of shifting phase for dry
dualclutch transmission, Mathematical Problems in Engineering,vol.
2014, Article ID 347490, 19 pages, 2014.
[2] B. Luo, S. Liu, and Y. Mo, Automatic clutch control
strategyresearch based on multi-mode control, in Proceedings of
theInternational Conference on Systems and Informatics (ICSAI
12),pp. 9094, May 2012.
[3] C. Lazar, C.-F. Caruntu, and A.-E. Balau, Modelling
andpredictive control of an electro-hydraulic actuated wet
clutchfor automatic transmission, in Proceedings of the IEEE
Interna-tional Symposium on Industrial Electronics (ISIE 10), pp.
256261, July 2010.
[4] C. J. Lee, F. Samie, andC.-K.Kao, Control of selectable
one-wayclutch inGMsix-speed automatic transmissions,
inProceedingsof the ASME Dynamic Systems and Control Conference
(DSCC09), vol. 2, pp. 605609, January 2009.
[5] Z. Sun and K. Hebbale, Challenges and opportunities in
auto-motive transmission control, in Proceedings of the
AmericanControl Conference, pp. 32843289, June 2005.
[6] X. Song,M.A.M. Zulkefli, Z. Sun, andH.-C.Miao,
Automotivetransmission clutch fill control using a customized
dynamicprogramming method, Journal of Dynamic Systems, Measure-ment
and Control, Transactions of the ASME, vol. 133, no. 5,Article ID
054503, 2011.
[7] Q. Liu, H. Bo, and B. Qin, Analysis of the transient
electromag-netic field of direct action solenoid valve, in
Proceedings of the20th International Conference on Nuclear
Engineering and theASME Power Conference, vol. 1, pp. 475479, ASME,
July 2012.
[8] S. Iqbal, F. Al-Bender, B. Pluymers, and W. Desmet,
Mathe-matical model and experimental evaluation of drag torque
indisengaged wet clutches, ISRN Tribology, vol. 2013, Article
ID206539, 16 pages, 2013.
[9] X. Song, M. A. M. Zulkefli, and Z. Sun, Automotive
transmis-sion clutch fill optimal control: an experimental
investigation,
in Proceedings of the American Control Conference, pp. 27482753,
Marriott Waterfront, Baltimore, Md, USA, July 2010.
[10] P.-L. Chen, X.-L. Yu, and L. Liu, Simulation and
experimentalstudy of electro-pneumatic valve used in air-powered
engine,Journal of Zhejiang University: Science A, vol. 10, no. 3,
pp. 377383, 2009.
[11] X.-Y. Song, Z.-X. Sun, X.-J. Yang, and G.-M. Zhu,
Modelling,control, and hardware-in-the-loop simulation of an
automatedmanual transmission, Proceedings of the Institution of
Mechani-cal Engineers, D: Journal of Automobile Engineering, vol.
224, no.2, pp. 143160, 2010.
[12] D. Karnopp, Computer simulation of stick-slip friction
inmechanical dynamic systems, Journal of Dynamic
Systems,Measurement and Control, Transactions of the ASME, vol.
107,no. 1, pp. 100103, 1985.
[13] L. Glielmo, L. Iannelli, V. Vacca, and F. Vasca, Gearshift
controlfor automatedmanual transmissions, IEEE/ASMETransactionson
Mechatronics, vol. 11, no. 1, pp. 1726, 2006.
-
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Probability and StatisticsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
CombinatoricsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical
Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
The Scientific World JournalHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014
Stochastic AnalysisInternational Journal of