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Research ArticleStudy on Oil Pressure Characteristics and Trajectory TrackingControl in Shift Process of Wet-Clutch for Electric Vehicles

Junqiu Li,1 Yihe Wang,1 Jianwen Chen,2 and Zongping He2

1Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing Institute of Technology, Beijing 100081, China2JIANGLU Machinery & Electronics Group Co., Ltd., Xiangtan 411199, China

Correspondence should be addressed to Junqiu Li; [email protected]

Received 19 October 2015; Revised 22 January 2016; Accepted 11 February 2016

Academic Editor: Mitsuhiro Okayasu

Copyright © 2016 Junqiu Li et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Accurate control of oil pressure of wet-clutch is of great importance for improving shift quality. Based on dynamic models of two-gear planetary transmission and hydraulic control system, a trajectory tracking model of oil pressure was built by sliding modecontrol method. An experiment was designed to verify the validity of hydraulic control system, through which the relationshipbetween duty cycle of on-off valve and oil pressure of clutch was determined. The tracking effect was analyzed by simulation.Results showed that oil pressure could follow well the optimal trajectory and the shift quality was effectively improved.

1. Introduction

Electric vehicles are gaining popularity because of highenergy efficiency, low emission, and diversified energy source[1]. To achieve high-efficiency and high-power, the integra-tion of high speed motor and transmission box is becominga trend of electric driving system. Two-gear planetary trans-mission and wet-clutch can be well matched with high speedmotor, which has good application prospect and practicalvalue in electric vehicles.

Because of high motor speed and large transformationratio, the clutch works in a large range of speed differencein shift process, which greatly influences shift quality. It isrequired that vehicles should be smooth and steady in shiftprocess; meanwhile the friction elements of clutch shouldwork with low heat load. These two targets are both affectedby clutch toque, which is decided by oil pressure if thephysical structure of clutch is confirmed [2, 3]. In previousresearches, based on optimal control theory, with shiftingjerk and sliding friction work selected as control targets, theauthors have obtained optimal trajectory of oil pressure inshift process [4].

Pressure-flow modeling of wet-clutch is widely used toanalyze shift quality [5, 6]. In this paper, the hydraulic systemmodel is built bymathematical analysis method to analyze oilpressure characteristics in shift process.

Proportional control method is widely applied in clutchpressure control [7, 8]. However, complex road conditionsin vehicle driving make strict demand on initial pressureand pressure change rate, and it is hard to acquire preferableshift quality. Fuzzy control method is widely accepted, andthe combination of fuzzy adjustment of variable factors andproportion adjustment of coefficient can obtain better effect[9]. Due to the complex nonlinear characteristics of clutch,changeable liquid viscosity, and variable external load, thereis a great necessity to work out a strategy with strong robust-ness and preferable tracking quality. On position trackingcontrol, Zhang et al. [10] effectively overcame the delay ofelectromagnetic valve and realized accurate control of clutchengagement process by predictive control method. For thelack of theories on system stability and robustness, it is hardto obtain preferable control effect. Xue et al. [11] realizedoil pressure tracking by fuzzy adaptive PID control method.However, the relationship between parameter characteristicand control objective in fuzzification process is not clear, soit is hard to adjust parameters and obtain optimal controlparameters. With advantages of fast response, no need ofonline identification, insensitive to external disturbance,simple structure, and strong robustness, slidingmode controlis especially applicable for complex nonlinear system [12,13]. Aiming at nonlinear characteristics of oil pressure andunobservable characteristics of clutch toque, a sliding mode

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 4869891, 11 pageshttp://dx.doi.org/10.1155/2016/4869891

2 Mathematical Problems in Engineering

Drivingmotor

Hydrocylinder

Ring gear

Planet carrierSun gearDifferential box

Tire

Half shaft

Pump

Overflowvalve V3

High speed on-off valve V1

Lubricationvalve V2 Oil channel

ECU

Two-gear transmission

Hydraulic control system

Clutch C

Brake B𝜏

pc

p0

ps

ps: system pressurepc: shift pressurep0: oil drainage pressure

Figure 1: Schematic diagram of system structure.

controller is designed to track optimal trajectory of shiftingoil pressure in this paper, the accuracy and robustness ofcontrol system are well improved.

2. Structure and Dynamic Modeling ofTwo-Gear Transmission

The powertrain with a two-gear transmission and combinedclutch is considered here, as schematically shown in Figure 1.The transmission system consists of a drivingmotor and two-gear planetary transmission. The clutch 𝐶 and brake 𝐵makeup a combined clutch and they are structurally connected.By the control of high speed on-off valve V1 and lubricationvalve V2, oil pressure could act on cylinder piston directly tocomplete shift action.

To realize downshift process, the Electric Control Unit(ECU) will enable oil charging circuit by closing V2 andopening V1. The rising pressure of oil will firstly push pistonto get clutch 𝐶 disengaged, after a short free phase, alongwith the continuousmovement of piston, the friction disks ofbrake 𝐵 are pressed tightly by piston; then brake 𝐵 is engagedanddownshift process is completed.On 1st gear, the transmis-sion ratio is given as 𝑖

1= 1+𝑘, where 𝑘 is the ratio of the teeth

number of ring gear to that of sun gear. To realize upshift pro-cess, ECUwill enable oil discharging circuit by closingV1 andopening V2, under the effect of return spring and graduallyfalling oil pressure, brake 𝐵 is disengaged; after a short freephase, piston is pushed to press the friction disk of clutch 𝐶tightly to engage the clutch and upshift process is completed.At 2nd gear, the transmission ratio is given as 𝑖

2= 1.

2.1. Dynamic Modeling of Planetary Transmission. For sim-plification, the damping and elasticity of transmission shaft,

Tbr

Tm 𝜔t𝜔o

Tcl

J1 + Js

Jc + J2

Tf

Jr

m: motorf: resistance forcet: inputo: outputcl: clutchbr: brake

s: sun gearr: ring gearc: planet carrierp: planet gear1: driving motor2: vehicle translational mass

Figure 2: Dynamic model of planetary transmission.

bearings, and gear mesh are ignored, and all the compo-nents of powertrain are assumed as concentrated mass. Thesimplifications above will not produce too much influenceto dynamic analysis of shift process [14]. The planetarytransmission is modeled as shown in Figure 2, where 𝑇

represents the torque, 𝐽 the moment of inertia, and 𝜔 theangular velocity, and all subscripts of relevant parameters areshown in Figure 2.

The rotation angle of input shaft 𝜃𝑡and rotation angle

of output shaft 𝜃𝑜are selected as generalized coordinates,

expressed as 𝑞1= 𝜃𝑡, 𝑞2= 𝜃𝑜.

Mathematical Problems in Engineering 3

According to principle of virtual work, the virtual work oftransmission system is as follows:

∑𝛿𝑊 = 𝑇𝑚𝛿𝜃𝑡− 𝑇br𝛿𝜃𝑟 − 𝑇cl (𝛿𝜃𝑡 − 𝛿𝜃𝑟) − 𝑇𝑓𝛿𝜃o, (1)

where 𝛿𝜃𝑡, 𝛿𝜃𝑟, and 𝛿𝜃

𝑜are, respectively, the virtual rotation

angle of input shaft, ring gear, and output shaft.The motion law of planetary gear system is described as

[

𝜔𝑟

𝜔𝑝

] =[[

[

−1

𝑘

𝑘 + 1

𝑘

−2

𝑘 − 1

𝑘 + 1

𝑘 − 1

]]

]

[𝜔𝑡

𝜔𝑜

] = [𝑎11

𝑎12

𝑎21

𝑎22

][𝜔𝑡

𝜔𝑜

] . (2)

According to (2), the virtual rotation angle of ring gearcan be expressed as

𝛿𝜃𝑟= 𝑎11𝛿𝜃𝑡+ 𝑎12𝛿𝜃𝑜. (3)

By substituting in (3), (1) is presented as

∑𝛿𝑊 = [𝑇𝑚− 𝑎11𝑇br − (1 − 𝑎11) 𝑇cl] 𝛿𝜃𝑡

+ [−𝑎12𝑇br + 𝑎12𝑇cl − 𝑇𝑓] 𝛿𝜃𝑜.

(4)

Then the generalized forces are obtained:

𝑄1= 𝑇𝑚− 𝑎11𝑇br − 𝑎12𝑇cl,

𝑄2= −𝑎12𝑇br + 𝑎12𝑇cl − 𝑇𝑓.

(5)

According to kinetic energy theory, the kinetic energy ofwhole system is described as

𝐸 =1

2[(𝐽1+ 𝐽𝑠) 𝜔2

𝑡+ 𝐽𝑟𝜔2

𝑟+ 𝐽𝑝𝜔2

𝑝+ 𝑚𝑝(𝜔𝑐𝑟𝑐)2

+ (𝐽𝑐+ 𝐽2) 𝜔2

𝑐] ,

(6)

where 𝑟𝑐is the equivalent radius of planet carrier and 𝑚

𝑝is

total mass of planet carrier.According to Lagrange equation,

𝑑

𝑑𝑡(𝜕𝐸

𝜕��𝑖

) −𝜕𝐸

𝜕𝑞𝑖

= 𝑄𝑖, (𝑖 = 1, 2) . (7)

Synthesizing (5)–(7), the dynamic modeling of transmis-sion is presented as follows:

[𝐽11

𝐽12

𝐽21

𝐽22

][��𝑡

��𝑜

] = [

𝑇𝑚− 𝑎11𝑇br − 𝑎12𝑇cl

−𝑎12𝑇br + 𝑎12𝑇cl − 𝑇𝑓

] , (8)

where

𝐽11= (𝐽1+ 𝐽𝑠) + 𝐽𝑟𝑎2

11+ 𝐽𝑝𝑎2

12,

𝐽12= 𝐽21= 𝐽𝑟𝑎11𝑎12+ 𝐽𝑝𝑎21𝑎22,

𝐽22= 𝐽𝑟𝑎2

12+ 𝐽𝑝𝑎2

22+ 𝑚𝑝𝑟2

𝑐+ 𝐽𝑐+ 𝐽2.

(9)

On-offvalve PistonDuty

cycle

Compression Overflow valve

PumpHydraulic system

Clutch toque

Brake toque

Qp

Qc

Qy

Qs

xc

𝜏

ps

ps

pc

Tbr

Tcl

oil

Figure 3: Modeling schematic of hydraulic system.

2.2. Hydraulic System Modeling. By analytic method, themathematical model of hydraulic system is built. Oil tubeleakage, cylinder leakage, and route loss are ignored in themodel.

The modeling schematic is shown in Figure 3. Modelingprocess is divided into three steps:

(i) Build the relationship among oil pressure 𝑝𝑐, duty

cycle 𝜏, system pressure 𝑝𝑠, and piston position 𝑥

𝑐by

the flow model of high speed on-off valve.(ii) Build the relationship between 𝑝

𝑐and 𝑥

𝑐by the

motion differential equation of piston.(iii) Build the relationship between 𝑝

𝑐and 𝑝

𝑠by the

continuity equation of fluid flow.

Eventually a single input single outputmodel of hydraulicsystem is built up and the relationship between 𝜏 and 𝑝

𝑐is

determined; then the friction toque of combined clutch couldbe determined.

2.2.1. Flow Model of High Speed On-Off Valve. High speedon-off valve can only work at “on” or “off” state with highswitching frequency controlled by Pulse Width Modulation(PWM). The average output flow 𝑄

𝑐is decided by duty cycle

𝜏, presented as

𝑄𝑐= 𝜏𝐶𝑑𝐴𝑑√2

𝜌(√(𝑝

𝑠− 𝑝𝑐) + √𝑝

𝑐− 𝑝0)

− 𝐶𝑑𝐴𝑑√2 (𝑝𝑐− 𝑝0)

𝜌,

(10)

where 𝐶𝑑is the valve flow coefficient, 𝐴

𝑑is the valve

throttling area, and 𝜌 is the oil density.


Brak

e

Clut

ch

Fseal

Fc

Fnon

mc

Fspring

Figure 4: Force analysis of piston in cylinder.

Assume that oil temperature keeps constant, according tofluid mechanics, the flow into cylinder; in other words, theflow through on-off valve satisfies

𝑄𝑐=𝐴𝑐𝑥𝑐+ 𝑉0

𝛽𝑒

𝑑𝑝𝑐

𝑑𝑡+ 𝐴𝑐��𝑐, (11)

where𝑉0is the initial volume of hydrocylinder,𝛽

𝑒is oil elastic

modulus, and 𝐴𝑐is the sectional area of cylinder.

Synthesizing (10)-(11), the relationship among 𝑝𝑐, 𝜏, 𝑝𝑠,

and 𝑥𝑐could be expressed as

𝑑𝑝𝑐

𝑑𝑡=

𝛽𝑒

𝐴𝑐𝑥𝑐+ 𝑉0

[𝐾1(𝑝𝑐) 𝜏 − 𝐾

2(𝑝𝑐) − 𝐴𝑐��𝑐] , (12)

where

𝐾1(𝑝𝑐) = 𝐶𝑑𝐴𝑑√2

𝜌(√(𝑝

𝑠− 𝑝𝑐) + √𝑝

𝑐− 𝑝0) ,

𝐾2(𝑝𝑐) = 𝐶𝑑𝐴𝑑√2 (𝑝𝑐− 𝑝0)

𝜌.

(13)

2.2.2. Piston Motion Differential Equation. As shown inFigure 4, the forces of cylinder piston satisfy

𝐹𝑐− 𝐹seal − 𝐹spring − 𝐹non = 𝑚𝑐��𝑐 + 𝑐𝑐��𝑐, (14)

where 𝐹𝑐represents the static force of oil pressure, 𝐹seal the

friction resistance of seal ring, 𝐹spring the force of returnspring, and 𝐹non the spring reversion force caused by defor-mation of friction disks.𝑚

𝑐is the effectivemass of piston; 𝑐

𝑐is

the viscous friction coefficient between cylinder and piston.For further description,

𝐹𝑐= 𝑝𝑐𝐴𝑐,

𝐹seal = 2𝜋𝜇seal𝑏𝑐 (𝑅1𝑐 + 𝑅2𝑐) 𝑝𝑐,

𝐹spring = 𝑘𝑐 (𝑥𝑐0 + 𝑥𝑐) ,

(15)

where 𝜇seal is the friction coefficient between seal ring andcylinder, 𝑏

𝑐is the thickness of seal ring, 𝑅

1𝑐and 𝑅

2𝑐are,

respectively, inner and outer radius of piston, and 𝑥𝑐0

is

the initial compression amount of return spring. Consider thefollowing:

𝐹non =

{{{{

{{{{

{

𝑘non ⋅ (𝑥𝑐 − 0) 𝑥𝑐min ≤ 𝑥𝑐 ≤ 0

0 0 < 𝑥𝑐< ℎ𝑝

𝑘non ⋅ (𝑥𝑐 − ℎ𝑝) ℎ𝑝≤ 𝑥𝑐≤ 𝑥𝑐max,

(16)

where 𝑘non is the compression stiffness of friction disksdeformation. Here we define the minimum and maximumdisplacement of piston as 𝑥

𝑐min and 𝑥𝑐max. The two critical

positions where the piston is without interactionwith frictiondisks are defined as 0 and ℎ

𝑝, namely, the starting point and

ending point of free phase.Then the relationship between 𝑝

𝑐and 𝑥

𝑐is expressed as

𝑝𝑐𝐴𝑏− 𝐹non = 𝑚𝑐��𝑐 + 𝑐𝑐��𝑐 + 𝑘𝑐 (𝑥𝑐0 + 𝑥𝑐) , (17)

where

𝐴𝑏= 𝐴𝑐− 2𝜋 ⋅ 𝜇seal ⋅ 𝑏𝑐 ⋅ (𝑅1𝑐 + 𝑅2𝑐) . (18)

2.2.3. Continuity Equation of Fluid Flow. According to liquidcontinuity [15], in downshift process, the flow equation ofhydraulic system should be

𝑄𝑝= 𝑄𝑐+ 𝑄𝑦+ 𝑄𝑠, (19)

where𝑄𝑝is the total flow fromhydraulic pump,𝑄

𝑐is the flow

through high speed on-off valve, 𝑄𝑦is the flow caused by oil

compression in connecting tube, and 𝑄𝑠is the flow through

overflow valve. Consider the following:

𝑄𝑝= 𝑛𝑝𝑉𝑝𝜉, (20)

where 𝑛𝑝is the rotational speed of oil pump, 𝑉

𝑝is the

pump displacement, and 𝜉 is the volume efficiency of pump.Consider the following:

𝑄𝑦=𝑉𝑦

𝛽𝑒

⋅𝑑𝑝𝑠

𝑑𝑡, (21)

where 𝑉𝑦is the volume of connecting tube. Consider the

following:

𝑄𝑠

=

{{{

{{{

{

0 𝑝𝑠< 𝑝𝑘

𝐶𝑑𝑠𝜋2𝑑3

𝑠sin𝛼𝑠(𝑝𝑠− 𝑝𝑘)

4𝑘𝑠

√2 (𝑝𝑠− 𝑝0)

𝜌𝑝𝑠≥ 𝑝𝑘,

(22)

where 𝐶𝑑𝑠is the port flow coefficient of overflow valve, 𝑑

𝑠is

the mean port diameter of overflow valve, 𝛼𝑠is the core half

cone angle of overflow valve, and 𝑝𝑘is the opening pressure

of overflow valve.


Synthesizing (11), (19)–(22), the relationship between 𝑝𝑐

and 𝑝𝑠will be

𝑉𝑦

𝛽𝑒

𝑑𝑝𝑠

𝑑𝑡+𝐴𝑐𝑥𝑐+ 𝑉0

𝛽𝑒

𝑑𝑝𝑐

𝑑𝑡+ 𝐴𝑐��𝑐− 𝑛𝑝𝑉𝑃𝜉 = 0,

𝑝𝑠< 𝑝𝑘

𝑉𝑦

𝛽𝑒

𝑑𝑝𝑠

𝑑𝑡+𝐶𝑑𝑠𝜋2𝑑3

𝑠sin𝛼𝑠(𝑝𝑠− 𝑝𝑘)

4𝑘𝑠

√2 (𝑝𝑠− 𝑝0)

𝜌

+𝐴𝑐𝑥𝑐+ 𝑉0

𝛽𝑒

𝑑𝑝𝑐

𝑑𝑡+ 𝐴𝑐��𝑐− 𝑛𝑝𝑉𝑃𝜉 = 0, 𝑝

𝑠≥ 𝑝𝑘.

(23)

Till then, by (12), (17), and (23), the hydraulic systemmodel is built up.

The toque transferred by combined clutch is given by(24) [16]. Downshift process includes three phases: clutch 𝐶disengagement phase, free phase, and brake 𝐵 engagementphase. In clutch disengagement phase, 𝑇cl = 0 and 𝑇br = 0; infree phase, 𝑇cl = 𝑇br = 0; in brake engagement phase, 𝑇cl = 0and 𝑇br = 0. So the toque is transferred by combined clutchin the first and third phase only:

𝑇cl, 𝑇br = 𝜇𝑁2 (𝑅3

2− 𝑅3

1)

3 (𝑅2

2− 𝑅2

1)⋅ [𝑝𝑐𝐴𝑏− 𝑘𝑐(𝑥𝑐0+ 𝑥𝑐)] , (24)

where 𝜇 is the friction coefficient of friction disk and it isassumed as a constant for simplification, 𝑁 is the numberof friction pairs, and 𝑅

1, 𝑅2are, respectively, the inner and

outer radius of friction disk. According to (24), there is adirect corresponding relationship between clutch toque andoil pressure.

3. Sliding Mode Controller Design ofCombined Clutch

Combined clutch is the shifting actuatingmechanism to real-ize toque transfer of vehicle driving system. Toque transferand shift quality are decided by oil pressure characteristics ofclutch. In this paper, sliding mode control method is used torealize trajectory tracking of oil pressure in each phase of shiftprocess.

3.1. Principle of Sliding Mode Controller. To simplify theanalysis and design of sliding mode control system, a linearswitching surface is adopted:

𝑠 = 𝑐𝑒1+ 𝑒2. (25)

Here a proper value of 𝑐 can ensure the accessibilityand stability of the sliding mode control system. It canbe determined by Hurwitz judgement and pole assignmentmethod.

The exponential reaching law is selected, shown as

𝑠 = −𝜀 sgn (𝑠) − 𝑘𝑠, 𝑘 > 0, 𝜀 > 0. (26)

Account the differential of (25)

𝑠 = 𝑐 𝑒1+ 𝑒2. (27)

According to (26)-(27),

−𝜀 sgn (𝑠) − 𝑘𝑠 = 𝑐 𝑒1+ 𝑒2. (28)

Through the state space equations of shift process, therelationship between deviation variables and control vari-ables could be determined [17, 18]. By (28), the controlvariable could be calculated to track the optimal trajectoryof oil pressure.

3.2. Sliding Mode Controller in Downshift Process. The oilpressure of the combined clutch in this paper is unable to bemeasured directly, so the tracking variable should be anothermeasurable state variable rather than oil pressure itself.

3.2.1. In Clutch Disengagement Phase and Brake EngagementPhase. Motor speed is a measurable state variable linked tooil pressure in these two phases, so it is selected as trackingvariable for these two phases.

Define the state variables as

𝑥1= 𝜔𝑚,

𝑥2= ��𝑚.

(29)

According to (8), (12), and (24), the state space equationof tracking system is expressed as

��1= 𝑥2,

��2= 𝑏1

𝑑𝑇𝑚

𝑑𝑡+ 𝑏2[𝐾1(𝑝𝑐) 𝜏 − 𝐾

2(𝑝𝑐)] .

(30)

For clutch disengagement phase,

𝑏1=

𝐽22

𝐽11𝐽22− 𝐽12𝐽21

, (31)

𝑏2=−𝑧𝜇𝑅𝑒𝐴𝑏𝛽𝑒𝑎12(𝐽12+ 𝐽22)

𝑉0(𝐽11𝐽22− 𝐽12𝐽21)

. (32)

For brake engagement phase,

𝑏1=

𝐽22

𝐽11𝐽22− 𝐽12𝐽21

, (33)

𝑏2=𝑧𝜇𝑅𝑒𝐴𝑏𝛽𝑒(𝑎12𝐽12− 𝑎11𝐽22)

𝑉0(𝐽11𝐽22− 𝐽12𝐽21)

. (34)

Then define the reference variables as

𝑟1= 𝜔∗

𝑚

𝑟2= ��∗

𝑚.

(35)

Consequently, the deviation variables are expressed as

𝑒1= 𝑟1− 𝑥1,

𝑒2= 𝑟2− 𝑥2.

(36)


For a further comment,

𝑒1= 𝑒2,

𝑒2= 𝑟1− 𝑏1

𝑑𝑇𝑚

𝑑𝑡− 𝑏2[𝐾1(𝑝𝑐) 𝜏 − 𝐾

2(𝑝𝑐)] .

(37)

Substituting (37) into (28),

−𝜀 sgn (𝑠) − 𝑘𝑠 = 𝑐𝑒2+ 𝑟1− 𝑏1

𝑑𝑇𝑚

𝑑𝑡

− 𝑏2[𝐾1(𝑝𝑐) 𝜏 − 𝐾

2(𝑝𝑐)] .

(38)

Finally,

𝜏 =1

𝑏2𝐾1(𝑝𝑐)[𝑐𝑒2+ 𝑟1− 𝑏1

𝑑𝑇𝑚

𝑑𝑡+ 𝑏2𝐾2(𝑝𝑐)

+ 𝜀 sgn (𝑠) + 𝑘𝑠] .(39)

Till then, on the basis of sliding mode control theory, thetrajectory tracking of motor speed in clutch disengagementphase and brake engagement phase is realized.The trajectoryof duty cycle of high speed on-off valve is obtained.

Here the proof of tracking stability is given.Define the Lyapunov function as

𝑉 =1

2𝑠2. (40)

Then

�� = 𝑠 ⋅ 𝑠

= 𝑠

⋅ ⟨𝑐𝑒2+ 𝑟1− 𝑏1

𝑑𝑇𝑚

𝑑𝑡− 𝑏2[𝐾1(𝑝𝑐) 𝜏 − 𝐾

2(𝑝𝑐)]⟩ .

(41)


�� = 𝑠 ⋅ {𝑐𝑒2+ 𝑟1− 𝑏1

𝑑𝑇𝑚

𝑑𝑡− 𝑏2[𝐾1(𝑝𝑐)

⋅1

𝑏2𝐾1(𝑝𝑐)[𝑐𝑒2+ 𝑟1− 𝑏1

𝑑𝑇𝑚

𝑑𝑡+ 𝑏2𝐾2(𝑝𝑐)

+ 𝜀 sgn (𝑠) + 𝑘𝑠] − 𝐾2(𝑝𝑐)]} = 𝑠 ⋅ [−𝜀 sgn (𝑠)

− 𝑘𝑠] = −𝜀 ⋅ |𝑠| − 𝑘𝑠2= −(

𝜀

2

√𝑉 +𝑘

2𝑉) ≤ 0.

(42)

Proof is completed.

3.2.2. In Free Phase. Piston position is linked to oil pressurein this phase. To ensure the smoothness of piston at the pointof disengagement and engagement, piston movement shouldbe controlled reasonably. Consequently, the piston position isselected as tracking variable for free phase.

Let

��𝑐= 𝐴𝑡2+ 𝐵𝑡 + 𝐶,

��𝑐=1

3𝐴𝑡3+1

2𝐵𝑡2+ 𝐶𝑡 + 𝐷,

𝑥𝑐=1

12𝐴𝑡4+1

6𝐵𝑡3+1

2𝐶𝑡2+ 𝐷𝑡 + 𝐸,

(43)

where 𝑡 is the time of piston movement and 𝐴, 𝐵, 𝐶,𝐷, 𝐸

are undetermined constants. To reduce the piston shock atthe point of engagement and disengagement, the speed andacceleration of piston should be zero at these two points.Thus, the undetermined constants can be determined. Theoptimal trajectories of piston position 𝑥∗

𝑐, ��∗𝑐, and ��∗

𝑐can be

determined.Let

𝑥1= ��𝑐,

𝑥2= ��𝑐.

(44)

According to (12) and (17)

��1= 𝑥2,

��2= 𝑀1𝜏 −𝑀

2−𝑀3−𝑐𝑐

𝑚𝑥2−𝑘𝑐

𝑚𝑥1,

(45)

where

𝑀1=𝐴𝑏

𝑚

𝛽𝑒


𝐾1(𝑝𝑐) ,

𝑀2=𝐴𝑏

𝑚

𝛽𝑒


𝐾2(𝑝𝑐) ,

𝑀3=𝐴𝑏

𝑚

𝛽𝑒


𝐴𝑐��𝑐.

(46)

Then

𝑟1= ��∗

𝑐,

𝑟2= ��∗

𝑐.

(47)

By (45) and (47),

𝑒1= 𝑒2,

𝑒2= 𝑟2−𝑀1𝜏 +𝑀

2+𝑀3+𝑐𝑐

𝑚𝑥2+𝑘𝑐

𝑚𝑥1.

(48)


𝜏 =1

𝑀1

[𝜀 sgn (𝑠) + 𝑘𝑠 +𝑀2+𝑀3+𝑐𝑐

𝑚𝑥2+𝑘𝑐

𝑚𝑥1

+ 𝑟2+ 𝑐𝑒2] .

(49)

Here the proof of tracking stability is given.


0

0.2

0.4

0.6

0.8

1

Dut

y cy

cle

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10Time (s)

(a) Step-input condition

(A)

(B)

(C)

(D)

Experimental resultSimulation result

0

0.5

1

1.5

2

2.5

Oil

pres

sure

(MPa

)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10Time (s)

(b) Oil pressure

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Pisto

n po

sitio

n (m

m)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10Time (s)

(c) Piston position

Driving motor Two-gear transmission

High speed on-off valveLubrication valve

Oil pump motor

(d) Experiment platform

Figure 5: Results of simulation and experiment in oil charging process.

Define the Lyapunov function as

𝑉 =1

2𝑠2. (50)

Then

�� = 𝑠 ⋅ 𝑠

= 𝑠

⋅ (𝑐𝑒2+ 𝑟2−𝑀1𝜏 +𝑀

2+𝑀3+𝑐𝑐

𝑚𝑥2+𝑘𝑐

𝑚𝑥1) .

(51)


�� = 𝑠 ⋅ {𝑐𝑒2+ 𝑟2+𝑀2+𝑀3+𝑐𝑐

𝑚𝑥2+𝑘𝑐

𝑚𝑥1−𝑀1

⋅1

𝑀1

[𝜀 sgn (𝑠) + 𝑘𝑠 +𝑀2+𝑀3+𝑐𝑐

𝑚𝑥2+𝑘𝑐

𝑚𝑥1

+ 𝑟2+ 𝑐𝑒2]} = 𝑠 ⋅ [−𝜀 sgn (𝑠) − 𝑘𝑠] = −𝜀 ⋅ |𝑠| − 𝑘𝑠2

= −(𝜀

2

√𝑉 +𝑘

2𝑉) ≤ 0.

(52)

Proof is completed.


Sliding mode controlOptimal trajectory

0.02 0.04 0.06 0.08 0.1 0.12 0.140Time (s)

90

110

130

150M

otor

spee

d (r

ad/s

)

(a) Clutch disengagement phase

0

5

10

15

20

25

0.15 0.2 0.25 0.3 0.35 0.4 0.45Time (s)

Pisto

n ve

loci

ty (m

m/s

)


(b) Free phase


280

300

320

Mot

or sp

eed

(rad

/s)

0.47 0.49 0.51 0.53 0.55 0.57 0.590.45Time (s)

(c) Brake engagement phase

Figure 6: Tracking results of state variables.

4. Result Analysis of Simulation andExperiment

In this paper, the simulation models of hydraulic system andsliding mode controller of two-gear transmission are built inMATLAB/simulink. An experiment platform is built to verifythe model of hydraulic system. On this basis, the trajectorytracking of oil pressure in downshift process is analyzed bysimulation.

4.1. Model Verification of Hydraulic System. Parameters ofhydraulic system are shown in Table 1. A kind of step-input condition of duty cycle is designed to analyze the

Table 1: Parameters of hydraulic system.

𝑅1𝑐/𝑅2𝑐/(mm) 89/135 𝑅

1/𝑅2/(mm) 86/119

𝜇seal 0.1 𝑏𝑐/(mm) 7

𝑉𝑐0/(mm3) 1.13 × 105 𝑚

𝑐/(kg) 6.82

𝑘𝑐/(N/mm) 1844.5 𝑥

𝑐0/(mm) 4.2

𝑥𝑐max/(mm) 4 𝜇 0.13𝑧cl/𝑧br 6/8 𝑝

𝑠/(MPa) 2

𝜌/(Kg/m3) 875 𝑘non (N/m) 3.07 × 1010

characteristics in oil charging process of combined clutch.Results of simulation and experiment are shown in Figure 5.


Table 2: Relevant parameters in vehicle shift process.

Curb weight/Kg 15000 Tire radius/m 0.478Gear ratio (Low/High) 3.5/1 Teeth ratio 𝑘 2.5Throttle opening 50% Vehicle speed/km/h 25Load toque 𝑇

𝑓/Nm 760 Final gear ratio 𝑖

06.5

As shown in Figure 5(b), the simulation result of oilpressure is basically consistent with experimental result. Butthe rising trend of experiment is obviously slower thansimulation. This is mainly because of the neglect of oiltemperature changing, route resistance of oil tube, and soforth. Overall, this experiment shows the validity of hydraulicsystemmodel. With further analysis, the characteristics of oilcharging process are explained as follows:

(1) (A)∼(B) part is the clutch disengagement phase inwhich the initial tension of spring is eliminated. Thepiston can be viewed as static; namely, 𝑥

𝑐= 0. Clutch

𝐶 is in sliding friction condition in this phase.(2) (B)∼(C) part is the free phase in which the clutch

and brake are all in separating state.The pistonmovesto maximum position from original position. In thisstep-input work condition, high speed on-off valveworks at “on” state in the whole shift process; the oilflows into cylinder rapidly so that it only takes 0.21 sfor piston to move to maximum position.

(3) (C)∼(D) part is the brake engagement phase. Pistonhas reached maximum position and oil pressurekeeps rising to systemic pressure rapidly. The mutualfriction condition between the brake and ring gearturns into relative static condition gradually and thenshift process is completed.

According to experiment result, it takes 0.24 s for oilpressure to rise to stable pressure. It means that theminimumtime of completing the whole downshift process is no lessthan 0.24 s. Due to the high speed of piston movement in thewhole process, there is a violent shock of piston at the pointof engagement, as shown in Figure 5(c). This further showsthe importance for the control of piston movement in shiftprocess.

4.2. Simulation Analysis of Oil Pressure Tracking in DownshiftProcess. In the previous research, based on optimal controltheory, the optimal trajectory of oil pressure andpistonmove-ment have been obtained to be used as reference trajectory ofslide mode controller in this paper.The trajectory tracking ofoil pressure of three phases is, respectively, simulated. Rele-vant parameters in vehicle shift process are shown in Table 2and tracking results of state variables are shown in Figure 6.

As shown in Figure 6, the tracking error is relatively largeat the beginning of each phase but it comes to a steadystate rapidly. This can well verify the stability of the slidingmode controller of this paper. Generally speaking, the statevariables tracked in each phase can keep consistent withoptimal trajectory, which can reflect the tracking effect of oilpressure indirectly.

(a) (b) (c) (d)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70Time (s)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Dut

y ra

tio

Figure 7: Duty cycle.

(a) (b) (c) (d)


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Oil

pres

sure

(MPa

)

0.1 0.2 0.3 0.4 0.5 0.6 0.70Time (s)

Figure 8: Oil pressure.

On the basis of sliding mode controller, combined withthe model of hydraulic system, the duty cycle is workedout and shown in Figure 7. The shift process is divided intofour parts, (a), (b), (c), and (d), where (a), (b), and (c) are,respectively, clutch disengagement phase, free phase, andbrake engagement phase; (d) is the phase that oil pressurerises to system pressure rapidly by setting a full duty cycle toprovide sufficient toque reserve for combined clutch.

Figure 8 shows the tracking effect of oil pressure. At thebeginning of each phase, there is fluctuation within a narrowrange in oil pressure, which corresponds to Figure 6. In phase(d), compared with Figure 5(c), the oil pressure rises faster,which is because that system pressure 𝑝

𝑠is continuously low

in the work condition of Figure 5 due to the fast oil chargingprocess. However, there is enough time for the building-up ofsystem oil pressure in Figure 7, so the oil pressure can rise tosteady state rapidly.


(a) (b) (c) (d)


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Pisto

n po

sitio

n (m

m)

0.1 0.2 0.3 0.4 0.5 0.6 0.70Time (s)

Figure 9: Piston position.

(a) (b) (c)

Motor toqueClutch toqueBrake toque

0.1 0.2 0.3 0.4 0.5 0.60Time (s)

0

50

100

150

200

250

300

350

400

450

Toqu

e (N

m)

Figure 10: Transmission torque.

The tracking of piston position has good effect, as shownin Figure 9. To reduce the shock, the piston engages anddisengages in low speed at the beginning and ending of freephase. To reduce interrupting time of power in shift process,the piston moves fast in the middle of free phase.

As shown in Figures 10–13, according to kinetic equationsand basic parameters of complete vehicle (Table 2), the trans-fer torque of combined clutch, rotation speed of each compo-nent, friction work, and shift jerk are obtained. By the controlof oil pressure, the clutch is disengaged completely at the endof phase (a); the brake is engaged completely at the end ofphase (c). In downshift process, the power of ring gear is cutoff, so the ring gear gradually slows down and stops at the end

(a) (b) (c)

Transmission speedRing gear speedMotor speed

0

50

100

150

200

250

300

350

Spee

d (r

ad/s

)

0.1 0.2 0.3 0.4 0.5 0.60Time (s)

Figure 11: Rotation speed.

(a) (b) (c)

0

50

100

150

200

250

300

350

400

Fric

tion

wor

k (J

)

0.1 0.2 0.3 0.4 0.5 0.60Time (s)

Figure 12: Friction work.

of shift process. Consequently, the slide friction of brake turnsinto static friction at the end. In phase (b), the constant poweroutput of motor leads to the rapid rise of motor speed. In thisphase, friction work and shift jerk are all close to zero. Shiftjerk mainly exists in phase (a) and (c). Because the shift timeis relatively short, there is no obvious change in output speedof transmission. The friction work of the whole shift processis about 350 J; the peak shift jerk is about 3m/s3.

5. Conclusion

(1) The models of two-gear transmission and hydraulicsystem are, respectively, built in this paper. Therelationship between duty cycle and oil pressure is


(a) (b) (c)

−3

−2

−1

0

1

2

3

Shift

jerk

(m/s3)

0.1 0.2 0.3 0.4 0.5 0.60Time (s)

Figure 13: Shift jerk.

obtained. A step-input condition of duty cycle isdesigned to analyze the characteristic of oil chargingprocess. An experiment platform is built to verify thevalidity and effectiveness of models.

(2) Aiming at the nonlinear characteristics of shift pro-cess and the problem of being unable to measure oilpressure, based on sliding mode control method, atracking model of oil pressure is built. Combiningwith the model of hydraulic system, the trackingof oil pressure is simulated in three phases of shiftprocess, respectively. Result shows that it can well andsteadily track the optimal trajectory of oil pressureand improve shift quality effectively.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work is supported by National Natural Science Foun-dation of China. The authors highly appreciate the abovefinancial support.

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