research article nonlinear analysis and intelligent control of...

Research ArticleNonlinear Analysis and Intelligent Control ofIntegrated Vehicle Dynamics

C Huang L Chen H B Jiang C C Yuan and T Xia

School of Automobile and Traffic Engineering Jiangsu University Zhenjiang 212013 China

Correspondence should be addressed to L Chen chenlong163com

Received 24 September 2013 Revised 2 December 2013 Accepted 2 December 2013 Published 21 January 2014

Academic Editor Hui Zhang

Copyright copy 2014 C Huang et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

With increasing and more stringent requirements for advanced vehicle integration including vehicle dynamics and controltraditional control and optimization strategies may not qualify for many applicationsThis is because among other factors they donot consider the nonlinear characteristics of practical systemsMoreover the vehicle wheel model has some inadequacies regardingthe sideslip angle road adhesion coefficient vertical load and velocity In this paper an adaptive neuralwheel network is introducedand the interaction between the lateral and vertical dynamics of the vehicle is analyzed By means of nonlinear analyses such asthe use of a bifurcation diagram and the Lyapunov exponent the vehicle is shown to exhibit complicated motions with increasingforward speed Furthermore electric power steering (EPS) and active suspension system (ASS) which are based on intelligentcontrol are used to reduce the nonlinear effect and a negotiation algorithm is designed to manage the interdependences andconflicts among handling stability driving smoothness and safety Further a rapid control prototype was built using the hardware-in-the-loop simulation platform dSPACE and used to conduct a real vehicle test The results of the test were consistent with thoseof the simulation thereby validating the proposed control

1 Introduction

When a car is travelling at a high speed a slight variationof the steering could result in a crash Moreover steeringinstability is primarily caused by the lateral force acting onthe steering wheels which is particularly affected by thesideslip angle [1] When a vehicle navigates a large radiuscurve the change in the sideslip angle is small and linearlyrelated to the change in the lateral force However for highlateral accelerations the wheel characteristic is nonlinearand the lateral force varies nonlinearly thereby reducing thesteering stability [2] Under these complex conditions andthe accompanying uncertainties there are variations in thevertical load and lateral force acting on the wheel as well asthe longitudinal force of the vehicle It is therefore necessaryto develop an effective nonlinear model and employ moreaccurate processes such as the use of a neural network modeland the sliding mode to achieve control requirements [3ndash5]

Currently vehicle dynamics is mostly studied by meansof nonlinear dynamics For example Wu and Sheng [6]established the phase plane of the sideslip angle and the rate

at which it changes which afforded a better method for thequantitative determination of the stability region Howeverthe method is limited by the simplified wheel model andcannot be theoretically used to determine the state andanalyze the trend of the vehicle outside the stable area Inagakiet al [7] also studied the phase plane of the turning kineticenergy and forward kinetic energy and proposed a methodfor interpreting experimental observations and the results ofquantitative analyses However the application of his methodis also limited

Shi et al [8] established the potential energy functionfor the major parameters of a vehicle such as the sideslipangle of the wheel steering wheel angle and vehicle speed Inaddition he conducted qualitative and quantitative analysesto determine the steering stability region of a vehicle Yang etal [9] used the nonlinear dynamical center manifold theoryto simplify a high-dimensional system to a one-dimensionalcenter manifold system and theoretically analyzed the phe-nomenon of bifurcated limit cycles He pointed out thatsaddle node bifurcation occurs with increasing speed andfront wheel angle

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 832864 15 pageshttpdxdoiorg1011552014832864

2 Mathematical Problems in Engineering

Liu et al [10] developed a nonlinear steering model of afrontwheel drive vehicle analyzed theHopf bifurcation of thesystem and confirmed that if the front wheel is periodicallyperturbed chaotic motion would be produced in the systemChang [11] created the bifurcation diagram of steer-by-wirevehicles for certain parameter ranges and used it to obtainthe periodic and chaotic motions of the system He proposeda chaotic feedback controller for steering the vehicle

Ono et al [12] analytically showed that the heading angleof a vehicle would be unstable if the heading speed exceeds acritical value Beyond the critical speed the vehicle may spinand possibly topple Furthermore Chang and Lin [13] usedthe bifurcation Lyapunov exponent to analyze complicatedmotions and offered detailed explanations of the topologychange that occurs in the solutions of the mathematicalmodel of a lateral system A feedback controller of a linearlateral system has also been proposed and implemented inthe steering system to enable the escape of a vehicle from theunstable domain thereby preventing spinning and improvingsafety [14]

The steering and suspension are two important subsys-tems of the chassis of a vehicle and both directly affectthe overall vehicle performance including handling stabilitydriving smoothness and safety The two systems are coupledand interact with each other The suspension causes signif-icant variations in the vertical force acting on the wheelwhereas the steering affects the lateral force which in turnaffects the overall lateral dynamics [15]

Studies have been conducted on the integrated control ofelectric power steering (EPS) and active suspension system(ASS) for complex nonlinear time-varying vehicle dynamicsThe lateral and vertical dynamics of the vehicle were eval-uated using different indexes control strategies and wheeldynamics According to the linear superposition principle ifthe lateral and vertical dynamics are separately controlled thedetermined comprehensive properties would not be accurateOver the past few years diverse intelligent control strategiesfor integrated control have been proposed March and Shim[16] developed an integrated control system for active frontwheel steering and normal force control and used fuzzy logicto improve the vehicle handling Yoshimura and Emoto [17]also used fuzzy logic and skyhook dampers to develop asteering and suspension system of a half car model subjectedto irregular excitation by the surface of the road Othertechniques have been applied to the design of vehicle chassiscontrol such as sliding mode control and adaptive control[18ndash20]More recently the119867

infinapproachwas used to produce

promising designs of a vehicle chassis controller [21ndash25]Chen et al [26] developed a full car dynamic model

that integrates EPS and ASS and used it to design a randomsuboptimal control strategy based on output feedback forintegrated EPS and ASS control Wang et al [27] determinedthe structural parameters and used them to obtain thecontroller parameters by means of a simulated annealingalgorithm Furthermore for simultaneous optimization themajor mechanical parameters of the EPS and ASS andsome of the controller parameters were used as designvariables and the comprehensive dynamic performance ofthe automobile was selected as the target function An EPS

ASS and rule-based central controller that supervised andcoordinated the subsystems were designed based on thecoupling relationships between the steering and suspensionsystems

In the present study a fuzzy control method was used toreduce the nonlinear effect and a coordination mechanismwas introduced tomanage the interdependences and conflictsamong handling stability driving smoothness and safetyMoreover computer simulations were used to validate thebifurcation and chaotic motions implied by the integratednonlinear differential equations that describe the vehicledynamics Finally to conduct a real vehicle test a rapidcontrol prototype was built using the hardware-in-the-loopsimulation platform dSPACE The results of the test wereconsistent with those of the simulation thereby validating theproposed control

2 Integrated Vehicle Dynamics

21 Vehicle Model To develop the vehicle dynamics modelbased on the steering working condition the effect of theground tangential force on the cornering properties of thewheel and the aerodynamics were ignored as shown inFigure 1

During the steering process of the vehicle the effect of theroll angle and yaw velocity on changes in the sideslip anglecannot be ignored The equations of motion of the vehiclewere therefore derived by taking the effect of the roll angleinto consideration and are as followslateral movement of the vehicle

119898V ( 120573 + 120574) minus 119898119904ℎ119904

120601 = 1198781+ 1198782+ 1198783+ 1198784 (1)

yaw movement of the vehicle

119868120574120574 = 119897119891(1198781+ 1198782) minus 119897119903(1198783+ 1198784) (2)

vertical movement of the vehicle

1198981199042= 11986521+ 11986522+ 11986523+ 11986524 (3)

pitching movement of the vehicle

119868120579

120579 = 119897119903(11986521+ 11986522) minus 119897119891(11986523+ 11986524) (4)

roll movement of the vehicle

119868120601

120601 = 119898119904V ( 120573 + 120574) ℎ

119904+ 119898119904119892ℎ119904120601 + (119865

21+ 11986523minus 11986522minus 11986524) 119889

(5)

vertical movement of the unsuspended mass

119898119897119894119897119894= 119896119897119894(119911119900119894minus 119911119897119894) minus 1198652119894 (119894 = 1 2 3 4) (6)

Mathematical Problems in Engineering 3

d

d

c21

z11

z01 z03

z04

z21

xy

z

zs

z22z24

k21

k24c22

lf lr

z02

z12z14

z13kaf

k11

k14

m12m14

m11 m13

S2 S3

S4

S1

k22

k23

hs

z23

c23

c24

k12

k13

kar

120575f

120575f

1205731

12057321205733

1205734

120574

120573

120601

120579

Figure 1 Vehicle dynamics model

Taking into consideration the effect of the stabilizer baron the inclination angle of the vehicle body the resultant forceof the suspension is given by the following

11986521=11989621(11991111minus 11991121)+11988821(11minus 21) minus

119896119886119891

2119889(120601 minus

11991111minus 11991112

2119889)


119896119886119891

2119889(120601 minus

11991111minus 11991112

2119889)

11986523=11989623(11991113minus 11991123)+11988823(13minus 23) minus119896119886119903

2119889(120601 minus

11991113minus 11991114

2119889)


2119889(120601 minus

11991113minus 11991114

2119889)

(7)

When the pitch angle 120579 and inclination angle 120601 are in theminor range the following approximations can be obtained

11991121= 119911119904minus 119897119891120579 minus 119889120601

11991122= 119911119904minus 119897119891120579 + 119889120601

11991123= 119911119904+ 119897119903120579 + 119889120601

11991124= 119911119904+ 119897119891120579 minus 119889120601

(8)

22 Wheel Model When a real vehicle is traveling on anonflat road the effect of the road on the wheel varieswhich makes it necessary to obtain a dynamic model that

is adaptable to environmental changes The adaptive neuralnetwork (ANN) is more appropriate and is as follows

119879 = 119865119910 = NN

[[[[[

[

120572

119875

V119865119911

119865119910(119899minus1)

]]]]]

]

(9)

ANN includes the input layer hidden layer and outputlayer The input layer contains five variables namely thesideslip angle of thewheel120572 the wheel pressure119875 the verticalload119865

119911 the velocity V and the lateral force of the previous step

119865119910(119899minus1)

The hidden layer is composed of several nonlineartransfer functions whereas the output layer is composed of asingle variable which is the lateral force acting on the wheel119865119910The input of the neuron in the hidden layer of the network

is given by

120579119895(119896) =

119898

sum

119894=1

119908119894119895119901119894(119896) 119895 = 1 2 8 119898 = 5 (10)

The relationship between the input and output of thehidden layer is given by the following sigmoid functionwhich is an expression of the output of the neuron in thehidden layer

120585119895(119896) = 119891 [120579

119895(119896)] 119895 = 1 2 8 (11)


Figure 2 Analytical tire contact system

The output layer consists of one neuron the inputfunction of which is as follows

120589119897(119896) =

119899

sum

119895=1

V119895119897120585119895(119896) 119897 = 1 119899 = 8 (12)

The relationship between the input and output of theoutput layer is given by the following sigmoid function

119868119897= 119891 [120589

119897(119896)] 119897 = 1 (13)

Let us assume that the inclination angles of the left- andright-side wheels are equal and that the inclination angles ofthe front and rear wheels can be expressed as

1205721= 1205722= 120575119891minus 120573 minus

119897119891120574

V+ 119864119891120601

1205723= 1205724=119897119903120574

Vminus 120573 + 119864

119903120601

(14)

For our experiments we used a 16565R13 Radial Tyre(Hankooktire) The equipments used for the experimentsincluded a Vehicle Tyre Road Rotating Test Stand (Figure 2)and a T-8050 Tyre RoadContacting Pressure Analysis System(US Tekscan Company)

Based on the characteristics of the Magic formula modela comparative analysis of the fitting performances of theadaptive and Magic formula models was conducted usingexperimental tire data for different loads and with theassumption of steady working conditions namely a wheelspeed of 20 kmh and inflation pressure of 250 kPa (Figure 3)As shown in the figure the adaptive model produces abetter fit of the experimental data In the case of the Magicformula because the trigonometric function is used as thebase function the fitting curve is only close to and crossesthe experimental data Moreover there are distortions in theapproximation of the lateral force

3 Nonlinear Analysis of Lateral andVertical Vehicle Dynamics

31 CoupledMechanism Figure 4 is a diagram of the interac-tion between the suspension and steering systemsThedouble

3000

2000

1000

0

minus1000

minus2000

minus3000

minus15 minus10 minus5 0 5 10 15

Slip angel (∘)La

tera

l for

ce (N

)TestANNMagic

Figure 3 Model fitting

lines in the figure indicate the basic and direct actions thesolid lines indicate the indirect effect of its functions and thedashed lines indicate the constraintsWhen an increase in thegain of one of either the suspension or steering system is usedto improve the performance of the system the indirect effecton the performance of the other system is also significantlyincreased For example if the roll stiffness of the suspensionis increased the roll angle would decrease resulting in aweakening of the under steer characteristics and an increasein the yaw velocity gain which in turn deteriorates the yawresponse characteristics

It can be seen from (1)ndash(6) that the steering of a vehicleaffects the roll of the body by laterally accelerating it whichin turn affects the motion of the suspension subsystemThe vertical force acting on the wheel changes significantlywhich affects the vertical motion The wheel model (see (9))shows that this causes additional changes in the lateral forceacting on the wheel which in turn affects the overall lateraldynamics This indicates that there are intercoupling andinteraction between the steering and suspension systems ofthe vehicle

32 Nonlinear Computation An analytical solution of thenonlinear dynamic system of a coupled neural network isextremely cumbersome which makes its actual applicationdifficult It is thus necessary to obtain an approximate solu-tion by a numerical method that is to convert the differentialdifferential equation to a differential equation In this studyMatlabSimulinkwas used to develop a simulationmodel thecalculation flow of which is shown in Figure 5 The model


Suspensioncontrol

Suspensiondisplacement

Suspensionforce

Vertical jitter

Pitching movement

Roll movement

Yawing motion

Vertical load

Tire forceSteering angle

Ride performance

Vehicle attitude

Handling stability

Lateral motion

Change of orientation

angle

Steering control

Slip angle

Figure 4 Interaction between the suspension and steering systems

Table 1 Vehicle parameters

Parameter ValueKerb weightkg 900Maximum total weightkg 1330Unsprung mass (frontrear)kg 3533Front axle weight (emptyfull)kg 540640Wheel basemm 2335Front axle-centred distancemm 955Rear axle-centred distancemm 1380Frontrear wheel treadmm 13601355Vehicle bodylengthwidthheightmm 340015751670

Wheel modelmm 16565R13

parameters were chosen with reference to those of a realvehicle and are given in Table 1

ThemaximumLyapunov exponentmethod and the phaseplane method are the major methods used to examinethe nonlinear system in this paper In mathematics theLyapunov exponent characterizes the rate of separation ofinfinitesimally close trajectories Quantitatively the rate atwhich two trajectories separated by 120575119885

0in a phase space

diverge (provided the divergence can be treated using itslinearized approximation) is given by

|120575119885 (119905)| asymp 119890120582119905 10038161003816100381610038161205751198850

1003816100381610038161003816 (15)

The rate of separation may be different for differentorientations of the initial separation vector Thus there is aspectrum of Lyapunov exponents the number of which isequal to the dimensionality of the phase space The largestexponent is commonly referred to as the maximal Lyapunov

Input excitation

Vehicle model Tire model

Output result

Nonlinear research

Iterative loop ofthe design

Figure 5 Calculation flowchart

exponent (MLE) because it determines the predictability fora dynamic system A positive MLE is usually considered asan indication that the system is chaotic (provided some otherconditions are met eg the compactness of the phase space)It should be noted that an arbitrary initial separation vectorwould typically have a component in the direction of theMLE and the exponential growth rate would obliterate theeffect of the other exponents over time

The maximal Lyapunov exponent is defined as follows

120582 = lim119905rarrinfin

lim1205751198850rarr0

1

119905ln |120575119885 (119905)|10038161003816100381610038161205751198850

1003816100381610038161003816

(16)

The limit 1205751198850ensures the validity of the linear approximation

at any timeWhen computing the maximal Lyapunov exponent the

time series should be reconstructed to determine the closestpoint 119881

1198951015840 for every point 119881

119895


The separation interval is defined as follows

120596 =max (120591

119894)

Δ119905 119894 = 1 2 119872 (17)

It is assume that 119889119895(0) is the distance between 119881

119895and its

closest point 1198811198951015840 that is

119889119895(0) =

10038171003817100381710038171003817119881119895minus 1198811198951015840

1003817100381710038171003817100381710038161003816100381610038161003816119895 minus 119895101584010038161003816100381610038161003816gt 120596 (18)

For every point 119881119895in the phase space the distance to its

closest point after the forward evolution in the 119894th step iscalculated using

119889119895(119894) =

10038171003817100381710038171003817119881119895+1minus 119881119895+1198941015840

10038171003817100381710038171003817 (19)

where 119894 = 1198690 1198690+1 119873

Assuming that the closest point of 119881119895diverges at the rate

of the Lyapunov exponent namely 119889119895(119894) = 119889

119895(0) times 119890

120582(119894Δ119905) thelogarithm on both sides is used to obtain ln 119889

119895(119894) = ln 119889

119895(0) +

120582(119894Δ119905) The least square method is then used to fit the curveof ln 119889

119895(119894) relative to 119894Δ119905 to obtain the maximal Lyapunov

exponent 1205821

1205821=[119894 times 119910 (119894)]

sum 1198942 (20)

where 119910(119894) = (sum119901119895=1

ln 119889119895(119894))(119901Δ119905) and 119901 is the number of

nonzero 119889119895(119894)

A phase plane is a visual display of certain characteristicsof certain types of differential equations However a coordi-nate plane is a plane whose axes represent two state variablessuch as (119909 119910) (119902 119901) or any other pair of variables It is a two-dimensional case of the general n-dimensional phase spaceand can be used to determine the stability or otherwise of thedynamics

33 Analysis of Results The yaw angle rate 120574 of the chassis isthe riding and visual evaluation index of the driver duringsteering It is therefore necessary to study the dynamicbehavior of 120574 to explain a variety of phenomena and evaluatethe stability and control of the steering process

In the bifurcation diagram (Figure 6) as the velocity Vincreases 120574 initially maintains a single stable period WhenV reaches 15 kmh 120574 suddenly becomes chaotic As V furtherincreases to 45 kmh 120574 suddenly changes to a period-4motion After another short period of stability 120574 enters thechaotic state again and finally converges to a period-1 motionwith further increase in V

Figure 7 shows that when V increases from zero 120582 beginsat a negative value and continuously fluctuates as the absolutevalue of V increases 120582 is zero when V = 15 kmh andsubsequently becomes positive which indicates that thesystem becomes chaoticThis shift corresponds to the changeof 120574 from a period-1 motion to the chaotic state (Figure 6)As V increases from 15 to 50 kmh 120582 fluctuates alternatingincreasing from and decreasing back to zero When V is 25or 40 kmh 120582 assumes a local maximum value These localmaximums correspond to regions in which the attractors are

0 10 20 30 40 50 60 70 80 90 100 110 120

02

019

018

017

016

015

014

013

012

011

Yaw

vel

ocity

120574(r

ads

)

Velocity (kmh)

Figure 6 Bifurcation diagram

Table 2 Description of control mode

Identification Rule Working condition1 119888

2and 1198883and 1198885

Assist2 119888

1and 1198885

Damping3 119888

2and 1198886

Aligning

densely distributed in Figure 6 As V approaches 50 kmh 120582changes rapidly and assumes a global minimum value Inthe bifurcation diagram in Figure 6 the attractors maintainthe four-period stable state after which the chaotic state isreentered When V = 65 kmh 120582 assumes another maximumvalue which is the global maximum and corresponds toa narrow strip of densely distributed chaotic attractors inFigure 6 Subsequently 120582 gradually decreases and the chaoticattractors correspondingly decrease in density and becomeincreasingly narrow in scope The attractors are narrowestat V = 80 kmh at which point 120582 is zero after which itincreases again When V = 95 kmh 120582 assumes another localmaximum However when V exceeds 95 kmh 120582 continu-ously decreases The Lyapunov exponent tends to negativeinfinity with continuous thinning of the chaotic attractorsand convergence to a stable fixed point in the bifurcationdiagram (Figure 6)

4 Intelligent Control for Chaotic State

41 EPS Based on Fuzzy Control The input signals andsystem state are specified by 119878 = 119904

1 1199042 1199043 1199044 where 119904

1is

the steering wheel torque 1199042is the velocity 119904

3is the wheel

steering angle and 1199044is the motor current The environment

is categorized into three working conditions namely assistdamping and aligning (see Tables 2 and 3 for the criteria fordetermining the working condition)

The schedule of the fuzzy rules is based on the workingcondition In the assist condition a seamless velocity assistmode is employed using a velocity range of 0ndash200 kmh andintervals of 20 kmh For each velocity the assist provided


Table 3 Conditions of EPS hybrid control system

Identification Logical expression1198881

119879119889lt 1198791198890

1198882

not1198881

1198883

119906vehicle lt 119906vehiclemax

1198884

not1198884

1198885

120579ℎ

120579ℎgt 0

1198886

not1198885

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

0 25 50 75 100 125

Velocity (kmh)

120582

Figure 7 Lyapunov exponent diagram

by the motor can be divided into three segments which areas follows When the steering wheel input torque is in therange of 0-1 Nm there is a boosting dead zone and no assistis provided by the motor When the input torque is in therange of 1ndash6Nm there is an assist zone When the torque isbeyond this range there will be a 6Nm assist saturated zoneThe four parameters used for designing linear assist are thesteering wheel input torque119879

1198890at the beginning of the assist

the maximum steering wheel input torque 119879119889max the motor

current maximum value 119868119905max and the coefficient 119896(V) The

characteristics of the designed linear assist and its functionexpressions are as shown in Figure 8

119868119905=

010038161003816100381610038161198791198891003816100381610038161003816 le 1198791198890

119896 (V) 119891 (119879119889minus 1198791198890) 119879

1198890le10038161003816100381610038161198791198891003816100381610038161003816 le 119879119889max

119868119905max

10038161003816100381610038161198791198891003816100381610038161003816 ge 119879119889max

(21)

Based on the requirements and objectives the aligningcondition is composed of two parts One is the aligningcontrol the major function of which is the provision ofnecessary assist for easy steering of the wheel back tothe central position In this process the aligning controlfunctions as a PI adjuster for adjusting the target steeringwheel position 120579 and the actual steering wheel variance 119890

ℎ

outputting the control voltage and allowing the motor tobring the steering wheel back to the central position

119881mr119897 = minus(119870119901119890ℎ + 119870119894 int 119890ℎdt) (22)

0kmh20kmh40kmh60kmh80kmh100 kmh120 kmh

4 62

Torque Td (Nm)

Curr

entI

t(A

) 3025

20

15

10

5

35

O

Figure 8 Characteristics of a linear assist

The other part is damping control the major functionof which is to bring the vehicle back to the central positionand avoid the shimmy events under the damping conditionDuring this process a certain control voltage is generatedbased on the angular velocity of the aligning process whichproduces a certain damping torque in the motorThe quickerthe steering wheel spins the higher the control voltagegenerated and the larger the damping torque is Converselythe slower the steering wheel spins the smaller the dampingtorque is The steering wheel alignment speed can thereforebe adjusted by adjusting the damping coefficient

119881mrℎ = minus119870119889 119890ℎ (23)

The alignment control and the damping control can beintegrated in one PID returning control algorithmMoreoverby adjusting the coefficient alignments that produce differenteffects can be obtained as expressed by the following

119881mr = minus(119870119901119890ℎ + 119870119894 int 119890ℎdt + 119870119889 119890ℎ) (24)

In the damping condition the operation status of thesystem is determined by the value of the deviation |119890

119889| and

timely adjustment is made to the structure of the controllerThe controlling rules are as follows

IF 119890119889gt +119890119889max THEN 119870

119889PWM119898 = +119870119889PWMmax

IF 119890119889le minus119890119889max THEN 119870

119889PWM119898 = minus119870119889PWMmax(25)

The input variable 119890 and the rate of change of the deviation119890119888 are determined by computing the deviation between theactual current in the current booster motor and the targetcurrent to carry out fuzzy query using the fuzzy rules andto query the fuzzy matrix tableThe parameters 119896

119901 119896119894 and 119896

119889

controlled by the PID are then adjusted for adaptation (seeTable 4 and Figure 9)

42 ASS Based on Fuzzy Control In the external state 119875 =1199011 1199012 1199013 1199014 1199015 1199011is the vertical acceleration

119904 1199012is the


0 2 4 6

0

02

04

06

08

1 NM NS ZO PS PMNB

minus6 minus4 minus2

e

PB

Mem

bers

hipFi

Figure 9 Membership function of variable 119890

Figure 10 Fuzzy controller interface in MatlabSimulink

Table 4 Fuzzy rule table

119890119888

119890NB NM NS ZO PS PM PB

NB PB PB PM PM PS ZO ZONM PB PB PS ZO NS ZO NSNS PM PM PM PS ZO NS NSZO PM PM PS ZO NS NM NMPS PS PS ZO NS NS NM NMPM PS PS ZO NS NM NM NBPB ZO ZO NM NM NM NB NB

pitching angular speed of the vehicle 120579 1199013is the vehicle body

roll angular speed 120579 1199014is the suspension dynamic deflection

119891119889 and 119901

5is the wheel vertical acceleration

119905

To rank the changing amplitudes of 1199011 1199012 1199013 1199014 and 119901

5

based on their values and achieve selective control (using thebelief rules) of

119904 120601 119891119889 and

119905by combining the operation

condition of the wheel angel with the velocity signal the

120579h Th

SYS120593 120573 120579 120596 Zi rij Ui Tm Fi

TYRE

EPS

ASS

120572 P Fz Zi

Fy Fz

120575f Th Tm

F1 F2 F3 F4

Zi Zs

IEPS IASS

w(t)

Φ 120579

Figure 11 Schematic diagram

Task reception

Task convey and allocation

Task execution and return

Step 1

Step 2

Step 3

Step 4

No

Yes

Task integration and accomplishment

Figure 12 State flow of the controlling strategy


Sensors powerSignal

Driver line

Sensors

Actuators

Actuators power line

Rapidpro

SC PU power line

SCXI

ControlCurrent back

Autobox power line

+12Vpower

Sensors powerSignal

Driver line

Sensors

AAActAA uators

Rapidpro

IISCXI

ControlkkCurrent back

Figure 13 Rapid control system

Figure 14 Entire control loop

following control rules are used together with (24)-(25)

Steeringrarr 120579 cap 120601 cap 119904rarr ldquostabilityrdquo

Medium and low speed rarr 119904cap 119891119889rarr ldquocomfortrdquo

High speed rarr 120601 cap 119905cap 119904rarr ldquosafetyrdquo

The fuzzy logic system can be described as follows

if 1199091(119905) = 119865

1119895 1199092(119905) = 119865

2119895 119909

119899(119905) = 119865

119899119895

then 119910119895(119905) = 119866

119895(119895 = 1 2 4)

(26)

In this formula 119909 = (1199091 1199092 119909

119899)119879 and 119910 are the

language variables 119865119894119895(119894 = 1 2 119899) and 119866

119895are the fuzzy

sets and 119897 is the number of rules The output is

119891 (119909) =

sum119898

119897=1119910119897[prod119899

119894=1120583119865119894119895(119909)]

sum119898

119897=1[prod119899

119894=1120583119865119894119895(119909)]

(27)

where 120583119865119894119895

is the membership function of the fuzzy set of 119865119894119895

119865119894=

119898

sum

119895=1

(min119891 (119909119895)) 119894 = 1 2 3 4 (28)

st If ldquostabilityrdquo rarr 1199091= 120579 119909

2= 120601 119909

3= 119904

If ldquocomfortrdquo rarr 1199091= 119904 1199092= 119891119889

If ldquosafetyrdquo rarr 1199091= 120601 119909

2= 119905 1199093= 119904

The fuzzification and defuzzification processes of thefuzzy control of this studywere designed using the fuzzy logictoolbox in MATLABSimulink (see Figure 10)

43 Negotiation Algorithm for Intelligent Control Figure 11shows a control strategy for vehicle integrated chassis controlBecause it is difficult to directly determine the interrelationsbetween EPS and ASS the controller was designed using fourparts as follows TYRE EPS- ASS and SYSThe inputs are theroad noise w(t) steering wheel angle 120579

ℎ and steering wheel

torque 119879ℎ and the outputs are the controlled current 119868ASS

powered by ASS and the controlled current 119868EPS powered byEPS

The logical process of the negotiation algorithm is asfollows (see Figure 12)

input road noise w(t) steering wheel angle 120579ℎ and

steering wheel torque 119879ℎ

output the controlled current 119868ASS powered by ASSand the controlled current 119868EPS powered by assistmotors

Step 1 (task reception) Categorize the operation conditionsinto steering control suspension control and coordinationcontrol based on the external input w(t) 120579

ℎ and 119879

ℎ For

steering control and suspension control the operation con-ditions can be determined within the capacity limitationsof the single EPS and ASS which would initiate the corre-sponding agent and at the same time notify that the agenthas been occupied by the operation conditions and shouldstop receiving new operation conditions until the completionand return of the current condition However if the currentoperation condition fails to accomplish the process by a singleattempt it means that it is beyond its capacity The operationcondition would thus be rejected and would return to therank for allocation to others

Step 2 (task convey and allocation) With the whole vehiclemodel after obtaining signals 120573 120596 120579 120593 119885

119894for the vehicle


(a) Speed sensors installed onthe vehicle

(b) Test car

(c) Gyroscope (d) Data acquisition box

Figure 15 Main parts of the test and control systems

body andmaking reference to the values for the velocity V andthe rotation angle 120575

119891for the front wheels the system would

make a timely definition for employment and the proportionof the vague weights Based on the planned working con-ditions allocation strategy the target is subcategorized intoseveral subtargets for the respective accomplishment

Step 3 (task execution and return) After receiving orders forthe subtargets the EPS receives signals for the steering wheeltorque 119879

ℎand velocity V and then uploads a set of solutions

for the steering wheel assist and orientation Regarding theASS after receiving the order for the subtargets it receivesthe signals 120579 Φ 119885

119904 and 119885

119894for the vehicle body and uploads

a set of solutions for the orientation of 119865119894powered by the

suspension

Step 4 (task integration and accomplishment) The controllercoprocesses all the subsystems collected during the sequencesof vague relational weights as follows

input the proposals for 119886EPS and 119886SAS at the negotia-tion of turn 119895(119909(119895) 119910(119895)) utility value 119881

119895minus1

output the proposals for 119886EPS and 119886SAS at the negotia-tion of turn 119895 + 1(119909(119895+1) 119910(119895+1))

(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)

) then the negoti-ation is a failure if 119881

119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)

)then the negotiation is a success and is followed by areturn to the negotiation results (119909(119895) 119910(119895))

(3) Compute the concession 119877119904 = 119881119895(119909(119895)

119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)

) minus 119877119904 and return to 119886EPS and 119886SAS for a newproposal (119909(119895+1) 119910(119895+1))

(4) Output the proposal

Using the above task integration the return results forthe entire operation condition can be obtained to output thecontrolled current 119868ASS powered by the suspension and thesteering force-controlled current 119868EPS

5 Road Test and Result

51 Test Device and System The tests were performed usingthe control system shown in Figure 13 It comprises a rapidcontrol module an actuator system and a sensor systemThe SC unit is used for communication between the sensorsystem and the main controller unit whereas the PU isused for communication between the actuators and the maincontroller unit The 140115051507 MicroAutoBox is used asthe main controller unit In the actuator and dSPACE systeman input power of 12V was obtained from the car batteryThevehicle state was determined using an arm position sensorgyroscopes a vehicle speed sensor and a data acquiring boxThe entire control loop shown in Figure 14 is closed by thedriver The main parts of the control and test systems areshown in Figure 15

52 Road Test To validate the control a step input road andvarious S-shaped operation conditions were used to perform


0 5 10 15 20 25 30

PassivePID controlIntelligent control

Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005

(a) Slip angle of vehicle body

Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )

(b) Yaw velocity of vehicle body

PID

0118012

0122012401260128

0130132013401360138

014

minus00265 minus0026 minus00255 minus0025 minus00245 minus0024 minus00235

Slip angle 120573 (rad)

Yaw

vel

ocity

120574(r

admiddotsminus

1 )

(c) Phase portrait without control

Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )

minus0015 minus0017 minus0019 minus0021 minus0023 minus0027 minus0029minus0025

(d) Phase portrait with intelligent control

Figure 16 Vehicle results for angle step steering input

joint simulations for different speeds of a passive and activevehicle respectively The test vehicle was equipped with theoriginal passive suspension system and then with the activesuspension system with chaos control

521 Angle Step Steering Input Operation Conditions Asimulation analysis of the angle step steering input operationconditions can be used to illustrate the improved status of theASS achieved by stabilizing the handling of the vehicle

522 Simulation of the S-Shaped Operation Condition Theroad surface was simulated using a vehicle speed of 30 plusmn2 kmhThe primary variables that weremeasured during thesimulation were the steering wheel turning angle steeringwheel torque and velocity of the vehicle

523 Angle Step Steering Input Operation Conditions on aPulse Road A simulation analysis of the angle step steeringinput operation conditions on a pulse road can be usedto illustrate the improved status achieved by stabilizing the

handling and ride comfort of the vehicle The operationcondition used for the simulation consisted of a velocity of120 kmh (33ms) and front wheel steering angle of 6∘ For adry asphalt road when the vehicle arrived at the 130m pointafter 4 s the front and rear axle wheels were excited by thepulse of the road with an amplitude of 5 cm

53 Discussion Figures 16(a) and 16(b) show that the intel-ligent control applied to ASS + EPS effectively reduced thefirst resonance of the slip angle and the yaw velocity Itproduced better results than a passive suspension system andtraditional PID control for ASS+EPS especially for angle stepsteering Although the performance of intelligent control andtraditional PID control in the steady state are very close theformer afforded steady operation and produced little noise

Meanwhile it can be seen from the phase portraits inFigures 16(c) and 16(d) that the vehicle lateral dynamicruns along elliptic orbits when the intelligent controller isapplied but exhibits chaotic behavior when the traditionalPID controller is applied Moreover the oscillation peaks


PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control

minus15 minus1 minus05 05 1 15minus2

minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)

Slip angle 120573 (rad)0


Intelligent control

minus15 minus1 minus05 0 05 1 15minus2

minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Figure 17 Vehicle results for S-shaped input

produced by the irregularities are also significantlyweakenedThis indicates that the proposed control method can beeffectively used to suppress chaotic behavior

Under the S-shaped operation condition the slip angleand yaw velocity when the intelligent controller is applied aresignificantly better than the others as shown in Figures 17(a)and 17(b) The parameters also exhibit greater linearity in thephase portrait for the intelligent controller (Figure 17(d)) butdisorder for the PID controller (Figure 17(c))This also showsthat the proposed control method can be effectively used tosuppress chaotic behavior

In Table 5 the intelligent control was proposed to makethe Lyapunov exponents of the closed-loop system negativeand thereby eliminate chaos from the related system Fromthese table and figures it can be concluded that the chaoticoscillators were controlled and the performance of the vehicleand its lateral dynamic were improved

It can be seen from Figures 18(a) and 18(b) that when thevehicle is unevenly excited by the pulse road the intelligentcontrol significantly improves the vertical acceleration ofthe vehicle and the yaw velocity speed amplitude of thevibration However the effects of the control in the verticaldirection controlled by traditional PID are slightly lower

Table 5 Lyapunov exponents for road test

Measurement points Lyapunovexponents

Slip angleStep input PID control 189

Intelligent control minus055

Snake input PID control 229Intelligent control minus023

Yaw velocityStep input PID control 318



(the acceleration amplitude in the vertical direction is about1ms2) This is because in the traditional integrated controlsystem vertical control and other performance indicatorsare considered in a single frame and the amplitude of thecontrolling force is obtained from the optimal weight actingagainst the generalized suspension force The same situationoccurs during the pitchingmovement of the vehicle as shownin Figure 18(c) It should be noted that during the transfer


Intelligent controlPID controlPassive

0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)

Figure 18 Results for step-pulse road

from the step angle to the input the entire vehicle has astable roll angle of about 2∘ and the process lasts for about 4 sThe uneven excitation of the pulse road significantly affectsthe roll movement when the vertical load on the wheelschanges the yaw and lateral forces whereas this is not thecase for traditional PID control The negotiation algorithmcan compensate for this by continuous negotiation among thedifferent indicators Moreover the uneven pulse excitation ofthe road surface had almost no effect on the manipulation ofthe vehicle which also reflects the robustness of the controlstrategy

6 Conclusion

In this paper we considered the coupling mechanism ofadvanced vehicle integration as well as adaptive neural net-work of the vehicle wheels By nonlinear analysis such as theuse of bifurcation diagram and the Lyapunov exponent it was

shown that the lateral dynamics of the vehicle was character-ized by complicated motions with increasing forward speedElectric power steering and active suspension system basedon intelligent control were used to reduce the nonlinearityand a negotiation algorithm was designed to manage theinterdependences and conflicts among handling stabilitydriving smoothness and safety The results of rapid controlprototyping confirmed the feasibility of the proposed controlmethod By comparing the time response diagrams phaseportraits and Lyapunov exponents for different operationconditions we observed that the slip angle and yaw velocity ofthe lateral dynamics entered a stable domain and the chaoticmotions were successfully suppressed It was also shown thatthe safety was significantly improved Under the angle stepsteering input operation conditions on the pulse road theproposed intelligent control significantly improved the ridecomfort and handling stability The uneven pulse excitationof the road surface had almost no effect on the manipulationof the vehicle Future work will focus on optimizing and


improving the robustness of the control and the informationconstraints and faults of the integrated vehicle dynamics

Nomenclature

119898119898119904 Qualities of the vehicle and the

suspension respectivelyV Velocity120573 120573 Slip angle and slip angular speed

respectivelyℎ119904 Height of the center of gravity of the

vehicle under roll condition120601 120601 120601 Roll angle roll angular speed and roll

angular acceleration of the vehiclerespectively

119878119894 Lateral force on four wheels119868120574 Rotary inertia of the vehicle120574 120574 Yaw rate and yaw acceleration respectively119871119891 119897119903 Distances from the front and rear wheels

to the center of the body respectively119911119904 119904 119904 Vertical displacement vertical velocity

and vertical acceleration of the vehiclebody respectively

1198652119894 Composite force exerted by the

suspension on the vehicle body119868120579 119868120593 Pitching rotary inertia and roll rotary

inertia of the vehicle respectively119889 12 of the track119898119897119894 Unsuspended mass

119896119897119894 Stiffness of the wheels

1199110119894 Displacement of the road

1199111119894 1119894 1119894 Vertical displacement vertical velocityand vertical acceleration of theunsuspended mass

119896119886119891 119896119886119903 Stiffness of the stabilizer bar angle of the

front and rear suspensions respectively1198962119894 Stiffness of the suspension

1198882119894 Damping coefficient of the suspension

1199112119894 2119894 2119894 Vertical displacement vertical velocityand vertical acceleration of the suspendedmass respectively

119896119886119891 119896119886119903 Stiffness of the stabilizer rods of the front

and rear suspensions respectively120579119895(119896) Input of the jth neuron of the hidden layer

119901119894(119896) Output of the input layer119908119894119895 Weight of the connection between the

input and hidden layers120585119895 Output of the 119895th neuron of the hidden

layerV119895119897 Weight of the connection between the

hidden and output layers119868119897 Output of the neuron of the output layer1205721 1205722 Inclination angles of the left-side and

right-side wheels of the front axlerespectively

1205723 1205724 Inclination angles of the left-side and

right-side wheels of the rear axlerespectively

120582 Lyapunov exponent119881mr119897 Returning control voltage of the motor120579ℎ Angle of the steering wheel

119890ℎ Deviation between the target and actual

steering angles119870119901 Proportionality coefficient

119870119894 Integral coefficient

119881mrℎ Returning control voltage of the motor119870119889 Integral coefficient

119881mr Aligning control voltage of the motor119890119889= 119868119889119898119905

minus 119868119889119898

Deviation between the target and actualdamping currents

119868119889119898119905

Armature current for the target dampingtorque

119890119889max Maximum deviation between the target

and actual damping currents119870119889PWM119898 PWM duty ratio corresponding to 119890

119889

119870119889PWMmax PWM duty ratio corresponding to 119890

119889max

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant No 50875112 51305167) thePhD Programs Foundation of the Ministry of Educationof China (Grant no 20093227110013 20103227120011) theJiangsu Municipal Natural Science Foundation (Grant noBK2010337) and the Natural Science Foundation of HigherEducation of Jiangsu (Grant no 09KJA580001)

References

[1] H Glaser ldquoElectronic stability program ESPrdquo in Audi PressPresentation pp 9ndash13 1996

[2] X Peng X Meng K Guo D Lu and Y Xie ldquoExperimentalstudy of cornering properties of a tire on an icy and a drypavementrdquo Chinese Journal of Mechanical Engineering vol 40no 7 pp 24ndash28 2004

[3] H B Pacejka E Bakker and L Nyborg ldquoTyre modeling for usein vehicle dynamics studiesrdquo SAE Paper 870421 1987

[4] Y L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013

[5] B L Tian W R Fan Q Zong J Wang and F Wang ldquoAdaptivehigh order sliding mode controller design for hypersonicvehicle with flexible body dynamicsrdquoMathematical Problems inEngineering vol 2013 Article ID 357685 11 pages 2013

[6] GWu and Y Sheng ldquoReview on the application of chaos theoryin automobile nonlinear systemrdquo Chinese Journal of MechanicalEngineering vol 46 no 10 pp 81ndash87 2010

[7] S Inagaki I Kshiro and M Yamamoto ldquoAnalysis on vehiclestability in critical cornering using phase-plane methodrdquo inProceedings of the International Symposium on Advanced VehicleControl Tsukuba-shi Japan 1994


[8] S Shi Z Mao H Xiang and X Wang ldquoNonlinear analysismethods of vehicle cornering stabilityrdquo Chinese Journal ofMechanical Engineering vol 43 no 10 pp 77ndash81 2007

[9] X Yang Z Wang W Peng and S Zhu ldquoAnalysis on lateraldynamic stability and bifurcation of vehicle periodic steeringunder critical conditionrdquo Journal of Highway and Transporta-tion Research andDevelopment vol 43 no 11 pp 145ndash149 2009

[10] Z Liu G Payre and P Bourassa ldquoNonlinear oscillations andchaotic motions in a road vehicle system with driver steeringcontrolrdquo Nonlinear Dynamics vol 9 no 3 pp 281ndash304 1996

[11] S-C Chang ldquoOn controlling a chaotic vehicle dynamic systemby using ditherrdquo International Journal of Automotive Technologyvol 8 no 4 pp 467ndash476 2007

[12] E Ono S Hosoe H D Tuan and S Doi ldquoBifurcation invehicle dynamics and robust front wheel steering controlrdquo IEEETransactions on Control Systems Technology vol 6 no 3 pp412ndash420 1998

[13] S Chang and H Lin ldquoStudy on controlling chaos of permanentmagnet synchronous motor in electric vehiclesrdquo InternationalJournal of Vehicle Design vol 58 no 2 pp 387ndash398 2012

[14] Z Zhang K T Chau and Z Wang ldquoAnalysis and control ofchaos for lateral dynamics of electric vehiclesrdquo in Proceedings ofthe International Conference on Electrical Machines and Systems(ICEMS rsquo11) pp 1ndash6 Beijing China 2011

[15] J D Lozoya-Santos R Morales-Menendez and R A RMendoza ldquoControl of an automotive semi-active suspensionrdquoMathematical Problems in Engineering vol 2012 Article ID218106 21 pages 2013

[16] C March and T Shim ldquoIntegrated control of suspension andfront steering to enhance vehicle handlingrdquo Proceedings of theInstitution of Mechanical Engineers D vol 221 no 4 pp 377ndash391 2007

[17] T Yoshimura and Y Emoto ldquoSteering and suspension system ofa half car model using fuzzy reasoning and skyhook dampersrdquoInternational Journal of Vehicle Design vol 31 no 2 pp 229ndash250 2003

[18] A Alleyne ldquoA comparison of alternative intervention strategiesfor unintended roadway departure (URD) controlrdquo VehicleSystem Dynamics vol 27 no 3 pp 157ndash186 1997

[19] T Gordon M Howell and F Brandao ldquoIntegrated controlmethodologies for road vehiclesrdquoVehicle System Dynamics vol40 no 1ndash3 pp 157ndash190 2003

[20] Q Qu and J Zu ldquoVariable structure model following controlof four-wheel-steering vehiclerdquo International Journal of VehicleDesign vol 37 no 4 pp 291ndash310 2005

[21] P Gaspar I Szaszi and J Bokor ldquoDesign of robust controllersfor active vehicle suspension using the mixed 120583 synthesisrdquoVehicle System Dynamics vol 40 no 4 pp 193ndash228 2003

[22] S-S You H-S Choi H-S Kim T-W Lim and S-K JeongldquoActive steering for intelligent vehicles using advanced controlsynthesisrdquo International Journal of Vehicle Design vol 42 no3-4 pp 244ndash262 2006

[23] H Zhang Y Shi and A S Mehr ldquoOnHinfinfiltering for discrete-

time Takagimdashsugeno fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 20 no 2 pp 396ndash401 2012

[24] H Zhang Y Shi and B Mu ldquoOptimal Hinfin-based linear-

quadratic regulator tracking control for discrete-time Takagimdashsugeno fuzzy fystems with preview actionsrdquo Journal of DynamicSystems Measurement and Control vol 135 Article ID 044501pp 1ndash5 2013

[25] H Zhang Y Shi and M Liu ldquoHinfin

step tracking control fornetworked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[26] W Chen H Xiao L Liu and J W Zu ldquoIntegrated controlof automotive electrical power steering and active suspensionsystems based on random sub-optimal controlrdquo InternationalJournal of Vehicle Design vol 42 no 3-4 pp 370ndash391 2006

[27] Q Wang W Jiang W Chen and J Zhao ldquoSimultaneous opti-mization of mechanical and control parameters for integratedcontrol systemof active suspension and electric power steeringrdquoChinese Journal ofMechanical Engineering vol 44 no 8 pp 67ndash72 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Liu et al [10] developed a nonlinear steering model of afrontwheel drive vehicle analyzed theHopf bifurcation of thesystem and confirmed that if the front wheel is periodicallyperturbed chaotic motion would be produced in the systemChang [11] created the bifurcation diagram of steer-by-wirevehicles for certain parameter ranges and used it to obtainthe periodic and chaotic motions of the system He proposeda chaotic feedback controller for steering the vehicle

Ono et al [12] analytically showed that the heading angleof a vehicle would be unstable if the heading speed exceeds acritical value Beyond the critical speed the vehicle may spinand possibly topple Furthermore Chang and Lin [13] usedthe bifurcation Lyapunov exponent to analyze complicatedmotions and offered detailed explanations of the topologychange that occurs in the solutions of the mathematicalmodel of a lateral system A feedback controller of a linearlateral system has also been proposed and implemented inthe steering system to enable the escape of a vehicle from theunstable domain thereby preventing spinning and improvingsafety [14]

The steering and suspension are two important subsys-tems of the chassis of a vehicle and both directly affectthe overall vehicle performance including handling stabilitydriving smoothness and safety The two systems are coupledand interact with each other The suspension causes signif-icant variations in the vertical force acting on the wheelwhereas the steering affects the lateral force which in turnaffects the overall lateral dynamics [15]

Studies have been conducted on the integrated control ofelectric power steering (EPS) and active suspension system(ASS) for complex nonlinear time-varying vehicle dynamicsThe lateral and vertical dynamics of the vehicle were eval-uated using different indexes control strategies and wheeldynamics According to the linear superposition principle ifthe lateral and vertical dynamics are separately controlled thedetermined comprehensive properties would not be accurateOver the past few years diverse intelligent control strategiesfor integrated control have been proposed March and Shim[16] developed an integrated control system for active frontwheel steering and normal force control and used fuzzy logicto improve the vehicle handling Yoshimura and Emoto [17]also used fuzzy logic and skyhook dampers to develop asteering and suspension system of a half car model subjectedto irregular excitation by the surface of the road Othertechniques have been applied to the design of vehicle chassiscontrol such as sliding mode control and adaptive control[18ndash20]More recently the119867

infinapproachwas used to produce

promising designs of a vehicle chassis controller [21ndash25]Chen et al [26] developed a full car dynamic model

that integrates EPS and ASS and used it to design a randomsuboptimal control strategy based on output feedback forintegrated EPS and ASS control Wang et al [27] determinedthe structural parameters and used them to obtain thecontroller parameters by means of a simulated annealingalgorithm Furthermore for simultaneous optimization themajor mechanical parameters of the EPS and ASS andsome of the controller parameters were used as designvariables and the comprehensive dynamic performance ofthe automobile was selected as the target function An EPS

ASS and rule-based central controller that supervised andcoordinated the subsystems were designed based on thecoupling relationships between the steering and suspensionsystems

In the present study a fuzzy control method was used toreduce the nonlinear effect and a coordination mechanismwas introduced tomanage the interdependences and conflictsamong handling stability driving smoothness and safetyMoreover computer simulations were used to validate thebifurcation and chaotic motions implied by the integratednonlinear differential equations that describe the vehicledynamics Finally to conduct a real vehicle test a rapidcontrol prototype was built using the hardware-in-the-loopsimulation platform dSPACE The results of the test wereconsistent with those of the simulation thereby validating theproposed control

2 Integrated Vehicle Dynamics

21 Vehicle Model To develop the vehicle dynamics modelbased on the steering working condition the effect of theground tangential force on the cornering properties of thewheel and the aerodynamics were ignored as shown inFigure 1

During the steering process of the vehicle the effect of theroll angle and yaw velocity on changes in the sideslip anglecannot be ignored The equations of motion of the vehiclewere therefore derived by taking the effect of the roll angleinto consideration and are as followslateral movement of the vehicle

119898V ( 120573 + 120574) minus 119898119904ℎ119904

120601 = 1198781+ 1198782+ 1198783+ 1198784 (1)

yaw movement of the vehicle

119868120574120574 = 119897119891(1198781+ 1198782) minus 119897119903(1198783+ 1198784) (2)

vertical movement of the vehicle

1198981199042= 11986521+ 11986522+ 11986523+ 11986524 (3)

pitching movement of the vehicle

119868120579

120579 = 119897119903(11986521+ 11986522) minus 119897119891(11986523+ 11986524) (4)

roll movement of the vehicle

119868120601

120601 = 119898119904V ( 120573 + 120574) ℎ

119904+ 119898119904119892ℎ119904120601 + (119865

21+ 11986523minus 11986522minus 11986524) 119889

(5)

vertical movement of the unsuspended mass

119898119897119894119897119894= 119896119897119894(119911119900119894minus 119911119897119894) minus 1198652119894 (119894 = 1 2 3 4) (6)


d

d

c21

z11

z01 z03

z04

z21

xy

z

zs

z22z24

k21

k24c22

lf lr

z02

z12z14

z13kaf

k11

k14

m12m14

m11 m13

S2 S3

S4

S1

k22

k23

hs

z23

c23

c24

k12

k13

kar

120575f

120575f

1205731

12057321205733

1205734

120574

120573

120601

120579




119896119886119891

2119889(120601 minus

11991111minus 11991112

2119889)


119896119886119891

2119889(120601 minus

11991111minus 11991112

2119889)


2119889(120601 minus

11991113minus 11991114

2119889)


2119889(120601 minus

11991113minus 11991114

2119889)

(7)


11991121= 119911119904minus 119897119891120579 minus 119889120601

11991122= 119911119904minus 119897119891120579 + 119889120601

11991123= 119911119904+ 119897119903120579 + 119889120601

11991124= 119911119904+ 119897119891120579 minus 119889120601

(8)



119879 = 119865119910 = NN

[[[[[

[

120572

119875

V119865119911

119865119910(119899minus1)

]]]]]

]

(9)



119865119910(119899minus1)


is given by

120579119895(119896) =

119898

sum

119894=1

119908119894119895119901119894(119896) 119895 = 1 2 8 119898 = 5 (10)


120585119895(119896) = 119891 [120579

119895(119896)] 119895 = 1 2 8 (11)




120589119897(119896) =

119899

sum

119895=1

V119895119897120585119895(119896) 119897 = 1 119899 = 8 (12)


119868119897= 119891 [120589

119897(119896)] 119897 = 1 (13)


1205721= 1205722= 120575119891minus 120573 minus

119897119891120574

V+ 119864119891120601

1205723= 1205724=119897119903120574

Vminus 120573 + 119864

119903120601

(14)





3000

2000

1000

0

minus1000

minus2000

minus3000


Slip angel (∘)La

tera

l for

ce (N

)TestANNMagic






Suspensioncontrol


Suspensionforce

Vertical jitter

Pitching movement

Roll movement

Yawing motion

Vertical load


Ride performance

Vehicle attitude

Handling stability

Lateral motion


angle

Steering control

Slip angle







0in a phase space


|120575119885 (119905)| asymp 119890120582119905 10038161003816100381610038161205751198850

1003816100381610038161003816 (15)


Input excitation


Output result

Nonlinear research






lim1205751198850rarr0

1

119905ln |120575119885 (119905)|10038161003816100381610038161205751198850

1003816100381610038161003816

(16)




1198951015840 for every point 119881

119895



120596 =max (120591

119894)

Δ119905 119894 = 1 2 119872 (17)


119895and its


119889119895(0) =

10038171003817100381710038171003817119881119895minus 1198811198951015840

1003817100381710038171003817100381710038161003816100381610038161003816119895 minus 119895101584010038161003816100381610038161003816gt 120596 (18)



119889119895(119894) =

10038171003817100381710038171003817119881119895+1minus 119881119895+1198941015840

10038171003817100381710038171003817 (19)

where 119894 = 1198690 1198690+1 119873



119895(0) times 119890


119895(119894) = ln 119889

119895(0) +



exponent 1205821

1205821=[119894 times 119910 (119894)]

sum 1198942 (20)

where 119910(119894) = (sum119901119895=1


nonzero 119889119895(119894)





0 10 20 30 40 50 60 70 80 90 100 110 120

02

019

018

017

016

015

014

013

012

011

Yaw

vel

ocity

120574(r

ads

)

Velocity (kmh)




2and 1198883and 1198885

Assist2 119888

1and 1198885

Damping3 119888

2and 1198886

Aligning




1 1199042 1199043 1199044 where 119904

1is


3is the wheel







119879119889lt 1198791198890

1198882

not1198881

1198883


1198884

not1198884

1198885

120579ℎ

120579ℎgt 0

1198886

not1198885

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

0 25 50 75 100 125

Velocity (kmh)

120582







119868119905=

010038161003816100381610038161198791198891003816100381610038161003816 le 1198791198890

119896 (V) 119891 (119879119889minus 1198791198890) 119879

1198890le10038161003816100381610038161198791198891003816100381610038161003816 le 119879119889max

119868119905max

10038161003816100381610038161198791198891003816100381610038161003816 ge 119879119889max

(21)


ℎ


119881mr119897 = minus(119870119901119890ℎ + 119870119894 int 119890ℎdt) (22)


4 62

Torque Td (Nm)

Curr

entI

t(A

) 3025

20

15

10

5

35

O



119881mrℎ = minus119870119889 119890ℎ (23)




119889| and


IF 119890119889gt +119890119889max THEN 119870

119889PWM119898 = +119870119889PWMmax


119889PWM119898 = minus119870119889PWMmax(25)


119901 119896119894 and 119896

119889



119904 1199012is the


0 2 4 6

0

02

04

06

08

1 NM NS ZO PS PMNB


e

PB

Mem

bers

hipFi




119890119888





119891119889 and 119901


119905


5


119904 120601 119891119889 and



120579h Th

SYS120593 120573 120579 120596 Zi rij Ui Tm Fi

TYRE

EPS

ASS

120572 P Fz Zi

Fy Fz

120575f Th Tm

F1 F2 F3 F4

Zi Zs

IEPS IASS

w(t)

Φ 120579


Task reception



Step 1

Step 2

Step 3

Step 4

No

Yes




Sensors powerSignal

Driver line

Sensors

Actuators


Rapidpro

SC PU power line

SCXI

ControlCurrent back

Autobox power line

+12Vpower

Sensors powerSignal

Driver line

Sensors

AAActAA uators

Rapidpro

IISCXI









if 1199091(119905) = 119865

1119895 1199092(119905) = 119865

2119895 119909

119899(119905) = 119865

119899119895

then 119910119895(119905) = 119866

119895(119895 = 1 2 4)

(26)

In this formula 119909 = (1199091 1199092 119909

119899)119879 and 119910 are the


119895are the fuzzy


119891 (119909) =

sum119898

119897=1119910119897[prod119899

119894=1120583119865119894119895(119909)]

sum119898

119897=1[prod119899

119894=1120583119865119894119895(119909)]

(27)

where 120583119865119894119895


119865119894=

119898

sum

119895=1

(min119891 (119909119895)) 119894 = 1 2 3 4 (28)


2= 120601 119909

3= 119904



2= 119905 1199093= 119904











ℎ and 119879

ℎ For






(b) Test car









119904 and 119885



suspension



119895minus1


(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)


119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)



119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)








0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










d

d

c21

z11

z01 z03

z04

z21

xy

z

zs

z22z24

k21

k24c22

lf lr

z02

z12z14

z13kaf

k11

k14

m12m14

m11 m13

S2 S3

S4

S1

k22

k23

hs

z23

c23

c24

k12

k13

kar

120575f

120575f

1205731

12057321205733

1205734

120574

120573

120601

120579




119896119886119891

2119889(120601 minus

11991111minus 11991112

2119889)


119896119886119891

2119889(120601 minus

11991111minus 11991112

2119889)


2119889(120601 minus

11991113minus 11991114

2119889)


2119889(120601 minus

11991113minus 11991114

2119889)

(7)


11991121= 119911119904minus 119897119891120579 minus 119889120601

11991122= 119911119904minus 119897119891120579 + 119889120601

11991123= 119911119904+ 119897119903120579 + 119889120601

11991124= 119911119904+ 119897119891120579 minus 119889120601

(8)



119879 = 119865119910 = NN

[[[[[

[

120572

119875

V119865119911

119865119910(119899minus1)

]]]]]

]

(9)



119865119910(119899minus1)


is given by

120579119895(119896) =

119898

sum

119894=1

119908119894119895119901119894(119896) 119895 = 1 2 8 119898 = 5 (10)


120585119895(119896) = 119891 [120579

119895(119896)] 119895 = 1 2 8 (11)




120589119897(119896) =

119899

sum

119895=1

V119895119897120585119895(119896) 119897 = 1 119899 = 8 (12)


119868119897= 119891 [120589

119897(119896)] 119897 = 1 (13)


1205721= 1205722= 120575119891minus 120573 minus

119897119891120574

V+ 119864119891120601

1205723= 1205724=119897119903120574

Vminus 120573 + 119864

119903120601

(14)





3000

2000

1000

0

minus1000

minus2000

minus3000


Slip angel (∘)La

tera

l for

ce (N

)TestANNMagic






Suspensioncontrol


Suspensionforce

Vertical jitter

Pitching movement

Roll movement

Yawing motion

Vertical load


Ride performance

Vehicle attitude

Handling stability

Lateral motion


angle

Steering control

Slip angle







0in a phase space


|120575119885 (119905)| asymp 119890120582119905 10038161003816100381610038161205751198850

1003816100381610038161003816 (15)


Input excitation


Output result

Nonlinear research






lim1205751198850rarr0

1

119905ln |120575119885 (119905)|10038161003816100381610038161205751198850

1003816100381610038161003816

(16)




1198951015840 for every point 119881

119895



120596 =max (120591

119894)

Δ119905 119894 = 1 2 119872 (17)


119895and its


119889119895(0) =

10038171003817100381710038171003817119881119895minus 1198811198951015840

1003817100381710038171003817100381710038161003816100381610038161003816119895 minus 119895101584010038161003816100381610038161003816gt 120596 (18)



119889119895(119894) =

10038171003817100381710038171003817119881119895+1minus 119881119895+1198941015840

10038171003817100381710038171003817 (19)

where 119894 = 1198690 1198690+1 119873



119895(0) times 119890


119895(119894) = ln 119889

119895(0) +



exponent 1205821

1205821=[119894 times 119910 (119894)]

sum 1198942 (20)

where 119910(119894) = (sum119901119895=1


nonzero 119889119895(119894)





0 10 20 30 40 50 60 70 80 90 100 110 120

02

019

018

017

016

015

014

013

012

011

Yaw

vel

ocity

120574(r

ads

)

Velocity (kmh)




2and 1198883and 1198885

Assist2 119888

1and 1198885

Damping3 119888

2and 1198886

Aligning




1 1199042 1199043 1199044 where 119904

1is


3is the wheel







119879119889lt 1198791198890

1198882

not1198881

1198883


1198884

not1198884

1198885

120579ℎ

120579ℎgt 0

1198886

not1198885

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

0 25 50 75 100 125

Velocity (kmh)

120582







119868119905=

010038161003816100381610038161198791198891003816100381610038161003816 le 1198791198890

119896 (V) 119891 (119879119889minus 1198791198890) 119879

1198890le10038161003816100381610038161198791198891003816100381610038161003816 le 119879119889max

119868119905max

10038161003816100381610038161198791198891003816100381610038161003816 ge 119879119889max

(21)


ℎ


119881mr119897 = minus(119870119901119890ℎ + 119870119894 int 119890ℎdt) (22)


4 62

Torque Td (Nm)

Curr

entI

t(A

) 3025

20

15

10

5

35

O



119881mrℎ = minus119870119889 119890ℎ (23)




119889| and


IF 119890119889gt +119890119889max THEN 119870

119889PWM119898 = +119870119889PWMmax


119889PWM119898 = minus119870119889PWMmax(25)


119901 119896119894 and 119896

119889



119904 1199012is the


0 2 4 6

0

02

04

06

08

1 NM NS ZO PS PMNB


e

PB

Mem

bers

hipFi




119890119888





119891119889 and 119901


119905


5


119904 120601 119891119889 and



120579h Th

SYS120593 120573 120579 120596 Zi rij Ui Tm Fi

TYRE

EPS

ASS

120572 P Fz Zi

Fy Fz

120575f Th Tm

F1 F2 F3 F4

Zi Zs

IEPS IASS

w(t)

Φ 120579


Task reception



Step 1

Step 2

Step 3

Step 4

No

Yes




Sensors powerSignal

Driver line

Sensors

Actuators


Rapidpro

SC PU power line

SCXI

ControlCurrent back

Autobox power line

+12Vpower

Sensors powerSignal

Driver line

Sensors

AAActAA uators

Rapidpro

IISCXI









if 1199091(119905) = 119865

1119895 1199092(119905) = 119865

2119895 119909

119899(119905) = 119865

119899119895

then 119910119895(119905) = 119866

119895(119895 = 1 2 4)

(26)

In this formula 119909 = (1199091 1199092 119909

119899)119879 and 119910 are the


119895are the fuzzy


119891 (119909) =

sum119898

119897=1119910119897[prod119899

119894=1120583119865119894119895(119909)]

sum119898

119897=1[prod119899

119894=1120583119865119894119895(119909)]

(27)

where 120583119865119894119895


119865119894=

119898

sum

119895=1

(min119891 (119909119895)) 119894 = 1 2 3 4 (28)


2= 120601 119909

3= 119904



2= 119905 1199093= 119904











ℎ and 119879

ℎ For






(b) Test car









119904 and 119885



suspension



119895minus1


(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)


119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)



119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)








0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120589119897(119896) =

119899

sum

119895=1

V119895119897120585119895(119896) 119897 = 1 119899 = 8 (12)


119868119897= 119891 [120589

119897(119896)] 119897 = 1 (13)


1205721= 1205722= 120575119891minus 120573 minus

119897119891120574

V+ 119864119891120601

1205723= 1205724=119897119903120574

Vminus 120573 + 119864

119903120601

(14)





3000

2000

1000

0

minus1000

minus2000

minus3000


Slip angel (∘)La

tera

l for

ce (N

)TestANNMagic






Suspensioncontrol


Suspensionforce

Vertical jitter

Pitching movement

Roll movement

Yawing motion

Vertical load


Ride performance

Vehicle attitude

Handling stability

Lateral motion


angle

Steering control

Slip angle







0in a phase space


|120575119885 (119905)| asymp 119890120582119905 10038161003816100381610038161205751198850

1003816100381610038161003816 (15)


Input excitation


Output result

Nonlinear research






lim1205751198850rarr0

1

119905ln |120575119885 (119905)|10038161003816100381610038161205751198850

1003816100381610038161003816

(16)




1198951015840 for every point 119881

119895



120596 =max (120591

119894)

Δ119905 119894 = 1 2 119872 (17)


119895and its


119889119895(0) =

10038171003817100381710038171003817119881119895minus 1198811198951015840

1003817100381710038171003817100381710038161003816100381610038161003816119895 minus 119895101584010038161003816100381610038161003816gt 120596 (18)



119889119895(119894) =

10038171003817100381710038171003817119881119895+1minus 119881119895+1198941015840

10038171003817100381710038171003817 (19)

where 119894 = 1198690 1198690+1 119873



119895(0) times 119890


119895(119894) = ln 119889

119895(0) +



exponent 1205821

1205821=[119894 times 119910 (119894)]

sum 1198942 (20)

where 119910(119894) = (sum119901119895=1


nonzero 119889119895(119894)





0 10 20 30 40 50 60 70 80 90 100 110 120

02

019

018

017

016

015

014

013

012

011

Yaw

vel

ocity

120574(r

ads

)

Velocity (kmh)




2and 1198883and 1198885

Assist2 119888

1and 1198885

Damping3 119888

2and 1198886

Aligning




1 1199042 1199043 1199044 where 119904

1is


3is the wheel







119879119889lt 1198791198890

1198882

not1198881

1198883


1198884

not1198884

1198885

120579ℎ

120579ℎgt 0

1198886

not1198885

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

0 25 50 75 100 125

Velocity (kmh)

120582







119868119905=

010038161003816100381610038161198791198891003816100381610038161003816 le 1198791198890

119896 (V) 119891 (119879119889minus 1198791198890) 119879

1198890le10038161003816100381610038161198791198891003816100381610038161003816 le 119879119889max

119868119905max

10038161003816100381610038161198791198891003816100381610038161003816 ge 119879119889max

(21)


ℎ


119881mr119897 = minus(119870119901119890ℎ + 119870119894 int 119890ℎdt) (22)


4 62

Torque Td (Nm)

Curr

entI

t(A

) 3025

20

15

10

5

35

O



119881mrℎ = minus119870119889 119890ℎ (23)




119889| and


IF 119890119889gt +119890119889max THEN 119870

119889PWM119898 = +119870119889PWMmax


119889PWM119898 = minus119870119889PWMmax(25)


119901 119896119894 and 119896

119889



119904 1199012is the


0 2 4 6

0

02

04

06

08

1 NM NS ZO PS PMNB


e

PB

Mem

bers

hipFi




119890119888





119891119889 and 119901


119905


5


119904 120601 119891119889 and



120579h Th

SYS120593 120573 120579 120596 Zi rij Ui Tm Fi

TYRE

EPS

ASS

120572 P Fz Zi

Fy Fz

120575f Th Tm

F1 F2 F3 F4

Zi Zs

IEPS IASS

w(t)

Φ 120579


Task reception



Step 1

Step 2

Step 3

Step 4

No

Yes




Sensors powerSignal

Driver line

Sensors

Actuators


Rapidpro

SC PU power line

SCXI

ControlCurrent back

Autobox power line

+12Vpower

Sensors powerSignal

Driver line

Sensors

AAActAA uators

Rapidpro

IISCXI









if 1199091(119905) = 119865

1119895 1199092(119905) = 119865

2119895 119909

119899(119905) = 119865

119899119895

then 119910119895(119905) = 119866

119895(119895 = 1 2 4)

(26)

In this formula 119909 = (1199091 1199092 119909

119899)119879 and 119910 are the


119895are the fuzzy


119891 (119909) =

sum119898

119897=1119910119897[prod119899

119894=1120583119865119894119895(119909)]

sum119898

119897=1[prod119899

119894=1120583119865119894119895(119909)]

(27)

where 120583119865119894119895


119865119894=

119898

sum

119895=1

(min119891 (119909119895)) 119894 = 1 2 3 4 (28)


2= 120601 119909

3= 119904



2= 119905 1199093= 119904











ℎ and 119879

ℎ For






(b) Test car









119904 and 119885



suspension



119895minus1


(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)


119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)



119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)








0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Suspensioncontrol


Suspensionforce

Vertical jitter

Pitching movement

Roll movement

Yawing motion

Vertical load


Ride performance

Vehicle attitude

Handling stability

Lateral motion


angle

Steering control

Slip angle







0in a phase space


|120575119885 (119905)| asymp 119890120582119905 10038161003816100381610038161205751198850

1003816100381610038161003816 (15)


Input excitation


Output result

Nonlinear research






lim1205751198850rarr0

1

119905ln |120575119885 (119905)|10038161003816100381610038161205751198850

1003816100381610038161003816

(16)




1198951015840 for every point 119881

119895



120596 =max (120591

119894)

Δ119905 119894 = 1 2 119872 (17)


119895and its


119889119895(0) =

10038171003817100381710038171003817119881119895minus 1198811198951015840

1003817100381710038171003817100381710038161003816100381610038161003816119895 minus 119895101584010038161003816100381610038161003816gt 120596 (18)



119889119895(119894) =

10038171003817100381710038171003817119881119895+1minus 119881119895+1198941015840

10038171003817100381710038171003817 (19)

where 119894 = 1198690 1198690+1 119873



119895(0) times 119890


119895(119894) = ln 119889

119895(0) +



exponent 1205821

1205821=[119894 times 119910 (119894)]

sum 1198942 (20)

where 119910(119894) = (sum119901119895=1


nonzero 119889119895(119894)





0 10 20 30 40 50 60 70 80 90 100 110 120

02

019

018

017

016

015

014

013

012

011

Yaw

vel

ocity

120574(r

ads

)

Velocity (kmh)




2and 1198883and 1198885

Assist2 119888

1and 1198885

Damping3 119888

2and 1198886

Aligning




1 1199042 1199043 1199044 where 119904

1is


3is the wheel







119879119889lt 1198791198890

1198882

not1198881

1198883


1198884

not1198884

1198885

120579ℎ

120579ℎgt 0

1198886

not1198885

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

0 25 50 75 100 125

Velocity (kmh)

120582







119868119905=

010038161003816100381610038161198791198891003816100381610038161003816 le 1198791198890

119896 (V) 119891 (119879119889minus 1198791198890) 119879

1198890le10038161003816100381610038161198791198891003816100381610038161003816 le 119879119889max

119868119905max

10038161003816100381610038161198791198891003816100381610038161003816 ge 119879119889max

(21)


ℎ


119881mr119897 = minus(119870119901119890ℎ + 119870119894 int 119890ℎdt) (22)


4 62

Torque Td (Nm)

Curr

entI

t(A

) 3025

20

15

10

5

35

O



119881mrℎ = minus119870119889 119890ℎ (23)




119889| and


IF 119890119889gt +119890119889max THEN 119870

119889PWM119898 = +119870119889PWMmax


119889PWM119898 = minus119870119889PWMmax(25)


119901 119896119894 and 119896

119889



119904 1199012is the


0 2 4 6

0

02

04

06

08

1 NM NS ZO PS PMNB


e

PB

Mem

bers

hipFi




119890119888





119891119889 and 119901


119905


5


119904 120601 119891119889 and



120579h Th

SYS120593 120573 120579 120596 Zi rij Ui Tm Fi

TYRE

EPS

ASS

120572 P Fz Zi

Fy Fz

120575f Th Tm

F1 F2 F3 F4

Zi Zs

IEPS IASS

w(t)

Φ 120579


Task reception



Step 1

Step 2

Step 3

Step 4

No

Yes




Sensors powerSignal

Driver line

Sensors

Actuators


Rapidpro

SC PU power line

SCXI

ControlCurrent back

Autobox power line

+12Vpower

Sensors powerSignal

Driver line

Sensors

AAActAA uators

Rapidpro

IISCXI









if 1199091(119905) = 119865

1119895 1199092(119905) = 119865

2119895 119909

119899(119905) = 119865

119899119895

then 119910119895(119905) = 119866

119895(119895 = 1 2 4)

(26)

In this formula 119909 = (1199091 1199092 119909

119899)119879 and 119910 are the


119895are the fuzzy


119891 (119909) =

sum119898

119897=1119910119897[prod119899

119894=1120583119865119894119895(119909)]

sum119898

119897=1[prod119899

119894=1120583119865119894119895(119909)]

(27)

where 120583119865119894119895


119865119894=

119898

sum

119895=1

(min119891 (119909119895)) 119894 = 1 2 3 4 (28)


2= 120601 119909

3= 119904



2= 119905 1199093= 119904











ℎ and 119879

ℎ For






(b) Test car









119904 and 119885



suspension



119895minus1


(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)


119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)



119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)








0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120596 =max (120591

119894)

Δ119905 119894 = 1 2 119872 (17)


119895and its


119889119895(0) =

10038171003817100381710038171003817119881119895minus 1198811198951015840

1003817100381710038171003817100381710038161003816100381610038161003816119895 minus 119895101584010038161003816100381610038161003816gt 120596 (18)



119889119895(119894) =

10038171003817100381710038171003817119881119895+1minus 119881119895+1198941015840

10038171003817100381710038171003817 (19)

where 119894 = 1198690 1198690+1 119873



119895(0) times 119890


119895(119894) = ln 119889

119895(0) +



exponent 1205821

1205821=[119894 times 119910 (119894)]

sum 1198942 (20)

where 119910(119894) = (sum119901119895=1


nonzero 119889119895(119894)





0 10 20 30 40 50 60 70 80 90 100 110 120

02

019

018

017

016

015

014

013

012

011

Yaw

vel

ocity

120574(r

ads

)

Velocity (kmh)




2and 1198883and 1198885

Assist2 119888

1and 1198885

Damping3 119888

2and 1198886

Aligning




1 1199042 1199043 1199044 where 119904

1is


3is the wheel







119879119889lt 1198791198890

1198882

not1198881

1198883


1198884

not1198884

1198885

120579ℎ

120579ℎgt 0

1198886

not1198885

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

0 25 50 75 100 125

Velocity (kmh)

120582







119868119905=

010038161003816100381610038161198791198891003816100381610038161003816 le 1198791198890

119896 (V) 119891 (119879119889minus 1198791198890) 119879

1198890le10038161003816100381610038161198791198891003816100381610038161003816 le 119879119889max

119868119905max

10038161003816100381610038161198791198891003816100381610038161003816 ge 119879119889max

(21)


ℎ


119881mr119897 = minus(119870119901119890ℎ + 119870119894 int 119890ℎdt) (22)


4 62

Torque Td (Nm)

Curr

entI

t(A

) 3025

20

15

10

5

35

O



119881mrℎ = minus119870119889 119890ℎ (23)




119889| and


IF 119890119889gt +119890119889max THEN 119870

119889PWM119898 = +119870119889PWMmax


119889PWM119898 = minus119870119889PWMmax(25)


119901 119896119894 and 119896

119889



119904 1199012is the


0 2 4 6

0

02

04

06

08

1 NM NS ZO PS PMNB


e

PB

Mem

bers

hipFi




119890119888





119891119889 and 119901


119905


5


119904 120601 119891119889 and



120579h Th

SYS120593 120573 120579 120596 Zi rij Ui Tm Fi

TYRE

EPS

ASS

120572 P Fz Zi

Fy Fz

120575f Th Tm

F1 F2 F3 F4

Zi Zs

IEPS IASS

w(t)

Φ 120579


Task reception



Step 1

Step 2

Step 3

Step 4

No

Yes




Sensors powerSignal

Driver line

Sensors

Actuators


Rapidpro

SC PU power line

SCXI

ControlCurrent back

Autobox power line

+12Vpower

Sensors powerSignal

Driver line

Sensors

AAActAA uators

Rapidpro

IISCXI









if 1199091(119905) = 119865

1119895 1199092(119905) = 119865

2119895 119909

119899(119905) = 119865

119899119895

then 119910119895(119905) = 119866

119895(119895 = 1 2 4)

(26)

In this formula 119909 = (1199091 1199092 119909

119899)119879 and 119910 are the


119895are the fuzzy


119891 (119909) =

sum119898

119897=1119910119897[prod119899

119894=1120583119865119894119895(119909)]

sum119898

119897=1[prod119899

119894=1120583119865119894119895(119909)]

(27)

where 120583119865119894119895


119865119894=

119898

sum

119895=1

(min119891 (119909119895)) 119894 = 1 2 3 4 (28)


2= 120601 119909

3= 119904



2= 119905 1199093= 119904











ℎ and 119879

ℎ For






(b) Test car









119904 and 119885



suspension



119895minus1


(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)


119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)



119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)








0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119879119889lt 1198791198890

1198882

not1198881

1198883


1198884

not1198884

1198885

120579ℎ

120579ℎgt 0

1198886

not1198885

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

0 25 50 75 100 125

Velocity (kmh)

120582







119868119905=

010038161003816100381610038161198791198891003816100381610038161003816 le 1198791198890

119896 (V) 119891 (119879119889minus 1198791198890) 119879

1198890le10038161003816100381610038161198791198891003816100381610038161003816 le 119879119889max

119868119905max

10038161003816100381610038161198791198891003816100381610038161003816 ge 119879119889max

(21)


ℎ


119881mr119897 = minus(119870119901119890ℎ + 119870119894 int 119890ℎdt) (22)


4 62

Torque Td (Nm)

Curr

entI

t(A

) 3025

20

15

10

5

35

O



119881mrℎ = minus119870119889 119890ℎ (23)




119889| and


IF 119890119889gt +119890119889max THEN 119870

119889PWM119898 = +119870119889PWMmax


119889PWM119898 = minus119870119889PWMmax(25)


119901 119896119894 and 119896

119889



119904 1199012is the


0 2 4 6

0

02

04

06

08

1 NM NS ZO PS PMNB


e

PB

Mem

bers

hipFi




119890119888





119891119889 and 119901


119905


5


119904 120601 119891119889 and



120579h Th

SYS120593 120573 120579 120596 Zi rij Ui Tm Fi

TYRE

EPS

ASS

120572 P Fz Zi

Fy Fz

120575f Th Tm

F1 F2 F3 F4

Zi Zs

IEPS IASS

w(t)

Φ 120579


Task reception



Step 1

Step 2

Step 3

Step 4

No

Yes




Sensors powerSignal

Driver line

Sensors

Actuators


Rapidpro

SC PU power line

SCXI

ControlCurrent back

Autobox power line

+12Vpower

Sensors powerSignal

Driver line

Sensors

AAActAA uators

Rapidpro

IISCXI









if 1199091(119905) = 119865

1119895 1199092(119905) = 119865

2119895 119909

119899(119905) = 119865

119899119895

then 119910119895(119905) = 119866

119895(119895 = 1 2 4)

(26)

In this formula 119909 = (1199091 1199092 119909

119899)119879 and 119910 are the


119895are the fuzzy


119891 (119909) =

sum119898

119897=1119910119897[prod119899

119894=1120583119865119894119895(119909)]

sum119898

119897=1[prod119899

119894=1120583119865119894119895(119909)]

(27)

where 120583119865119894119895


119865119894=

119898

sum

119895=1

(min119891 (119909119895)) 119894 = 1 2 3 4 (28)


2= 120601 119909

3= 119904



2= 119905 1199093= 119904











ℎ and 119879

ℎ For






(b) Test car









119904 and 119885



suspension



119895minus1


(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)


119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)



119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)








0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6

0

02

04

06

08

1 NM NS ZO PS PMNB


e

PB

Mem

bers

hipFi




119890119888





119891119889 and 119901


119905


5


119904 120601 119891119889 and



120579h Th

SYS120593 120573 120579 120596 Zi rij Ui Tm Fi

TYRE

EPS

ASS

120572 P Fz Zi

Fy Fz

120575f Th Tm

F1 F2 F3 F4

Zi Zs

IEPS IASS

w(t)

Φ 120579


Task reception



Step 1

Step 2

Step 3

Step 4

No

Yes




Sensors powerSignal

Driver line

Sensors

Actuators


Rapidpro

SC PU power line

SCXI

ControlCurrent back

Autobox power line

+12Vpower

Sensors powerSignal

Driver line

Sensors

AAActAA uators

Rapidpro

IISCXI









if 1199091(119905) = 119865

1119895 1199092(119905) = 119865

2119895 119909

119899(119905) = 119865

119899119895

then 119910119895(119905) = 119866

119895(119895 = 1 2 4)

(26)

In this formula 119909 = (1199091 1199092 119909

119899)119879 and 119910 are the


119895are the fuzzy


119891 (119909) =

sum119898

119897=1119910119897[prod119899

119894=1120583119865119894119895(119909)]

sum119898

119897=1[prod119899

119894=1120583119865119894119895(119909)]

(27)

where 120583119865119894119895


119865119894=

119898

sum

119895=1

(min119891 (119909119895)) 119894 = 1 2 3 4 (28)


2= 120601 119909

3= 119904



2= 119905 1199093= 119904











ℎ and 119879

ℎ For






(b) Test car









119904 and 119885



suspension



119895minus1


(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)


119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)



119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)








0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sensors powerSignal

Driver line

Sensors

Actuators


Rapidpro

SC PU power line

SCXI

ControlCurrent back

Autobox power line

+12Vpower

Sensors powerSignal

Driver line

Sensors

AAActAA uators

Rapidpro

IISCXI









if 1199091(119905) = 119865

1119895 1199092(119905) = 119865

2119895 119909

119899(119905) = 119865

119899119895

then 119910119895(119905) = 119866

119895(119895 = 1 2 4)

(26)

In this formula 119909 = (1199091 1199092 119909

119899)119879 and 119910 are the


119895are the fuzzy


119891 (119909) =

sum119898

119897=1119910119897[prod119899

119894=1120583119865119894119895(119909)]

sum119898

119897=1[prod119899

119894=1120583119865119894119895(119909)]

(27)

where 120583119865119894119895


119865119894=

119898

sum

119895=1

(min119891 (119909119895)) 119894 = 1 2 3 4 (28)


2= 120601 119909

3= 119904



2= 119905 1199093= 119904











ℎ and 119879

ℎ For






(b) Test car









119904 and 119885



suspension



119895minus1


(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)


119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)



119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)








0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(b) Test car









119904 and 119885



suspension



119895minus1


(1) Compute 119881119895(119909(119895)

119910(119895)

)

(2) If 119881119895(119909(119895)

119910(119895)

) le 119881119895minus1(119909(119895minus1)

119910(119895minus1)


119895(119909(119895)

119910(119895)

) gt 119881119895minus1(119909(119895minus1)

119910(119895minus1)



119910(119895)

) minus

119881119895minus1(119909(119895minus1)

119910(119895minus1)

)N to generate 119881119895+1

= 119881119895(119909(119895)

119910(119895)








0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30


Time t (s)

Slip

angl

e (de

g)

minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0

0005


Time t (s)


00

5 10 15 20 25 30

005

010

015

020

025

Yaw

vel

ocity

120574(r

admiddotsminus

1 )


PID

0118012

0122012401260128

0130132013401360138

014



Yaw

vel

ocity

120574(r

admiddotsminus

1 )


Intelligent control

006

008

010

012

014

016

018


Yaw

vel

ocity

120574(r

admiddotsminus

1 )












PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










PID controlPassive

0 5 10 15 20 25 30

Intelligent control

Slip

angl

e120573(r

ad)

minus20

minus15

minus10

minus5

0

5

10

15

20

Time t (s)

minus3timesminus10


0 5 10 15 20 25 30minus20

minus15

minus10

minus5

0

5

10

15

20

PID controlPassive

Intelligent control

Time t (s)

Yaw

vel

ocity

(r

ads

)

minus2timesminus10


PID control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



Intelligent control


minus1

0

1

2

3

Yaw

vel

ocity

(r

ads

)



















0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 5 10 15 20Time (s)

minus2

0

1

2

3

minus3

minus1

Vert

ical

acce

lera

tion

(ms2)

(a)

0 5 10 15 20

30

40

50

60

0

20

Time (s)

10


Yaw

velo

city

(deg

s)

(b)

0 5 10 15 20Time (s)

Pitc

h an

gel (

deg)

0

1

2

3

minus1

minus2

minus3


(c)



6 Conclusion





Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Nomenclature


































minus 119868119889119898


119868119889119898119905




119889


119889max



Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article nonlinear analysis and intelligent control of...

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