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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)
11986522=11989622(11991112minus 11991122)+11988822(12minus 22) minus
119896119886119891
2119889(120601 minus
11991111minus 11991112
2119889)
11986523=11989623(11991113minus 11991123)+11988823(13minus 23) minus119896119886119903
2119889(120601 minus
11991113minus 11991114
2119889)
11986524=11989624(11991114minus 11991124)+11988824(14minus 24) minus119896119886119903
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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
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Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
11986522=11989622(11991112minus 11991122)+11988822(12minus 22) minus
119896119886119891
2119889(120601 minus
11991111minus 11991112
2119889)
11986523=11989623(11991113minus 11991123)+11988823(13minus 23) minus119896119886119903
2119889(120601 minus
11991113minus 11991114
2119889)
11986524=11989624(11991114minus 11991124)+11988824(14minus 24) minus119896119886119903
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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
11986522=11989622(11991112minus 11991122)+11988822(12minus 22) minus
119896119886119891
2119889(120601 minus
11991111minus 11991112
2119889)
11986523=11989623(11991113minus 11991123)+11988823(13minus 23) minus119896119886119903
2119889(120601 minus
11991113minus 11991114
2119889)
11986524=11989624(11991114minus 11991124)+11988824(14minus 24) minus119896119886119903
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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
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Mathematical Problems in Engineering
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4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
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
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
(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
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
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)
PassivePID controlIntelligent control
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
Slip angle 120573 (rad)
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
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
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
(a) Slip angle of vehicle body
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
(b) Yaw velocity of vehicle body
PID control
minus15 minus1 minus05 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)0
(c) Phase portrait without control
Intelligent control
minus15 minus1 minus05 0 05 1 15minus2
minus1
0
1
2
3
Yaw
vel
ocity
(r
ads
)
Slip angle 120573 (rad)
(d) Phase portrait with intelligent control
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
Intelligent control minus016
Snake input PID control 516Intelligent control minus166
(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
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
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
Intelligent controlPID controlPassive
Yaw
velo
city
(deg
s)
(b)
0 5 10 15 20Time (s)
Pitc
h an
gel (
deg)
0
1
2
3
minus1
minus2
minus3
Intelligent controlPID controlPassive
(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
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 15
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of