Integrated Chassis Control of Active Front Steering and Yaw … · 2016. 6. 8. · ResearchArticle Integrated Chassis Control of Active Front Steering and Yaw Stability Control Based

Research ArticleIntegrated Chassis Control of Active Front Steeringand Yaw Stability Control Based on Improved InverseNyquist Array Method

Bing Zhu12 Yizhou Chen1 and Jian Zhao1

1 State Key Laboratory of Automotive Simulation and Control Jilin University Changchun 130022 China2 Key Laboratory of Bionic Engineering of Ministry of Education Jilin University Changchun 130022 China

Correspondence should be addressed to Jian Zhao zhaojianjlueducn

Received 8 December 2013 Accepted 6 February 2014 Published 20 March 2014

Academic Editors F Berto and M Gobbi

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

An integrated chassis control (ICC) system with active front steering (AFS) and yaw stability control (YSC) is introduced in thispaper The proposed ICC algorithm uses the improved Inverse Nyquist Array (INA) method based on a 2-degree-of-freedom(DOF) planar vehicle reference model to decouple the plant dynamics under different frequency bands and the change ofvelocity and cornering stiffness were considered to calculate the analytical solution in the precompensator design so that theINA based algorithm runs well and fast on the nonlinear vehicle system The stability of the system is guaranteed by dynamiccompensator together with a proposed PI feedback controller After the response analysis of the system on frequency domain andtime domain simulations under step steering maneuver were carried out using a 2-DOF vehicle model and a 14-DOF vehiclemodel by MatlabSimulink The results show that the system is decoupled and the vehicle handling and stability performance aresignificantly improved by the proposed method

1 Introduction

Vehicle safety and stability has been one of the hottestresearch topics during last several decades Many activecontrol systems such as antilock brake system (ABS) trac-tion control system (TCS) electric stability control (ESC)and active front steering (AFS) were developed and widelyequipped on various vehicles for safer more stable andcomfortable driving experience However as the complexityof vehicle active control systems increases the potentialconflicts among each system become increasingly problemsand concerns [1ndash3]

The primary objective of the chassis control systems is toimprove vehicle performances by actively controlling vehiclemotions However since vehicle sprung mass has six degreesof freedom (DOF) with strong couplings among them it ishard to regulate individual motion state without affectingothers [4ndash6] While each chassis control system is designedfor specific motion control and performance improvement

they may negatively impact others with potential conflictIn ABS design for example trade-off has to be madebetween stability and braking distance [7] Cooperation andintegration of the individual chassis subsystem have to beconsidered for further development of vehicle safety research[8]

There have been plenty of attempts to integrate thestand-alone chassis control subsystems to name a few theintegrated chassis control (ICC) [9] unified chassis control(UCC) [10] vehicle dynamics management (VDM) [11] andso on He et al proposed a strategy to integrate active steeringand Variable Torque Distribution (VTD) systems using thephase plane method and a rule based integration schemeis employed to determine and allocate the control tasksbetween these two subsystems [12] Wang et al brought outan integrated control technology of vehicle chassis based onmultiagent system (MAS) for coordination control betweensemiactive suspension (SAS) and electric power steering(EPS) [13]

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 919847 14 pageshttpdxdoiorg1011552014919847

2 The Scientific World Journal

Both yaw stability control and active front steering con-trol of a vehicle play important roles in its stability Howeverthey have their own drawbacks as well Thus there were alot of research works focusing on the cooperation betweenAFS and ESC controls to maintain vehicle desired yaw rateand side slip angle in order to improve vehicle handlingand stability [14ndash19] Cho et al described a UCC systemthat consists of a supervisor and a coordinator to integrateAFS and ESC The supervisor determines the target yawrate and lateral velocity based on typical control modesand the Karush-Kuhn-Tucker (KKT) condition is used tocompute the optimized coordination of tire forces consider-ing constraint corresponding to the tire friction circle [20]Ding and Taheri designed an adaptive integrated algorithmby integrating the AFS and DYC controls based on directLyapunovmethod [21] Li and Yu designed a supervisory andservo-loop structure for the integration control of AFS andDYC [22] The approach was used to reduce the conflictingeffects of the two dynamically coupled subsystems Doumiatiet al investigated the coordination of active front steering andrear braking in a driver-assist system for vehicle yaw controlThe coordination of these actuators was achieved througha suitable gain scheduled LPV (Linear Parameter Varying)controller [23]

However most of prior research has adopted a supervi-sory control method to coordinate the control commandsand actuations in order to avoid control conflicts and tomaximize resource sharing In such cases AFS and YSCcontrol techniques are optimized individually in specifichandling regions and the maximum benefit could be gainedthrough the coordinatedintegrated use of both methods ofcorrective yaw motion generation in the control strategyWhile this approach may be effective to some degree inreducing the interferences among multiple controlled sub-systems and easing the conflicts among different controlobjectives it is not a true ldquointegrationrdquo per se besides ithas added an extra layer of command hierarchy on top ofthe stand-alone subsystems The coupling mechanism anddecoupling method for these two active control systems lackin-depth research Actually the integration of AFS and YSCfor example is a typical two-input and two-output systemwith strong coupling in vehicle lateral dynamics it is thusmore desirable to decouple the dynamics so as to reduce oreliminate the control interference

Furthermore though easily and highly efficiently imple-mented facing the nonlinear vehicle systems and the velocityand cornering stiffness variation the traditional controlsystem design methods lose their edge However as generalsolutions to the problems above themodern controlmethodstend to have a complex control process Besides consideringthe heavy calculation burden it is hard to achieve real-timecontrol with some optimization control methods Thereforethe improvement of classical controlmethods that fit the non-linear requirement is what engineers have been researching[24 25]

Inverse Nyquist Array (INA) method a multivariablefrequency method developed by Rosenbrock 1969 and fur-ther enhanced by Mac Farlance 1970 has been proved to bevery effective in decoupling linear systems properly in both

high and low frequency bands [26ndash28] This method is ofinterest because it enables the utilization of classical single-loop systems for multivariable control system designs Afterdecoupling the plant by INA precompensator the classicalcommon control method for single-loop systems could beadopted So it has obtained widespread applications in thefield of automatic control and industry [29ndash31]

However the vehicle is a complex nonlinear system asis well known As a linear-model-based control methodthe controller designed by INA method shows less robuststabilities and cannot cover the complexity of vehicle states

In this paper an improved INA based feedback ICCcontroller is designed for AFS and YSC integration Firsta 2-DOF reference model is adopted Based on this modelthe plant of vehicle dynamics is decoupled by the prec-ompensation INA method The precompensator is solvedwith the consideration of variation of vehicle velocity andthe cornering stiffness of both axles which are functions ofvehicle longitudinal and lateral accelerations of the targetvehicle Thus the parameters in the linear 2-DOF referencemodel will change and the nonlinear characteristics of actualvehicle are taken into account It means that the INAdecoupling performance is regulated automatically based ondifferent vehicle states as well as various system frequencybands Then the target yaw moment and the target frontsteering angle are achieved by the feedback PI controllerAs analytic solutions of the precompensator are describedexplicitly the execution efficiency of the ICC controller ispretty high Finally simulations are performed to validate theproposed method and the results are discussed

2 Structure of INA Based FeedbackIntegrated Controller

The structure of the proposed ICC system for AFS andYSC integration is composed of a reference model andan INA based integrated controller which is shown inFigure 1 where 120575119878119882 is steering wheel angle 120575119891 is front wheelsteer angle V

119909 V119910are longitudinal and lateral velocity 120583 is

adhesion coefficient 120573 is sideslip angle 120574 is yaw rate 119886119909 119886119910

are vehicle longitudinal lateral acceleration 120575119888is active front

wheel steering angle 119879119911is the active yaw moment

The 2-DOF reference vehiclemodel which considers bothaccuracy and simplicity is used for target inputs calculationThe side slip angle and yaw rate are described in the modelas shown in Figure 2

The vehicle state space equation is

= A119909 + B119906

119910 = C119909

(1)

where

119909 = [120573 120574]119879

119906 = [120575119891 119879119885]119879

The Scientific World Journal 3

TZ

+

+

120575SW

120573d

INA controller

Precompensator

120575c

Dynamicscompensator

PI controller

PI controller

Vehicle

Drivercommand

Sensorssignals

2-DOFreference

model minus

minus

Vehicle states

120574120573

120574d

120575f x 120583

x ax ay

Figure 1 ICC system configurations

lf lrFyf Fyr

120575f

120573f120574

Tz

120573r120573x

y

Figure 2 2-DOF reference model

A = [11988611 1198861211988621 11988622

] =

[[[[

[

minus119888119891 + 119888119903

119898 sdot V119909

minus1 +119888119903119897119903 minus 119888119891119897119891

119898 sdot V2119909

119888119903119897119903 minus 119888119891119897119891

119868119911

minus1198881199031198972

119903+ 1198881198911198972

119891

119868119911sdot V119909

]]]]

]

B = [11988711

11988712

11988721

11988722

] =

[[[[

[

119888119891

119898 sdot V119909

0

119888119891119897119891

119868119911

1

119868119911

]]]]

]

C = [11988811

11988812

11988821

11988822

] = [1 0

0 1]

(2)

where 119888119891 119888119903are front and rear axle cornering stiffness 119898 is

mass 119897119891 119897119903describe the distances from the vehicle cg (center

of gravity) to the front and rear axle respectively 119868119911is yaw

inertia of the vehicle 120573119891 120573119903are front and rear slip angle

119865119910119891 119865119910119903are lateral force of the front and rear axle

To maintain lateral stability both the yaw rate and sideslip angle should be restricted within a stable field Thedesired yaw rate can be obtained from steady-state yaw rategain of the reference model

120574119889=

V119909sdot 120575119891

(119897119891 + 119897119903) sdot [1 + (V119909V119888ℎ)2]

(3)

where

V2119888ℎ

=119888119891 sdot 119888119903 sdot (119897119891 + 119897119903)

2

119898(119888119903119897119903minus 119888119891119897119891)

(4)

In addition the desired yaw rate should be constrained by theroad friction coefficient

10038161003816100381610038161205741198891003816100381610038161003816 le 120583 sdot

119892

V119909 (5)

To maintain lateral stability it is important to sustaindriverrsquos control authority which can be achieved when thevehicle sideslip angle is small According to some literatures[15 32] the desired sideslip angle can be chosen as

120573119889= 0 (6)

Furthermore the same 2-DOF model is also used forthe INA controller design By Laplace transformation thestate equation (1) can be rewritten into the system TransferFunction Matrix (TFM) as

G (119904)

= C(119904I minus A)minus1B + D

= [

[

11989211 (119904) 119892

12 (119904)

11989221 (119904) 119892

22 (119904)

]

]

=

[[[[

[

(11988711119904 minus 1198871111988622

+ 1198872111988612)

(119904 minus 11988611)(119904 minus 119886

22)minus 1198862111988612

1198872211988612

(119904 minus 11988611)(119904 minus 119886

22)minus 1198862111988612

(11988721119904+ 1198871111988621minus 1198872111988611)

(119904 minus 11988611)(119904 minus 119886

22)minus 1198862111988612

11988722 (119904 minus 11988611)

(119904 minus 11988611)(119904minus 119886

22)minus 1198862111988612

]]]]

]

(7)

Obviously it is a typical two-input two-output systemand G(119904) can be decoupled or pseudodecoupled in overallfrequency bands by INA method The structure of the INAcontroller designed in this paper is shown in Figure 3 Anappropriate precompensator K

119901is designed to make the

system TFM diagonal dominance that is to decouple thecontrol plant Then the design methods for classical SISOsystems can be used consequently [28] In this research adynamic compensator matrix K

119888and a feedback gain matrix

4 The Scientific World Journal

minus

minus

+

+

Dynamiccompensator

f1

f2

Kc1(s)

Kc2(s)

Precompensator

Feedback gains

TZ

g11

g21

g12

g22

120573d

120574d

Yaw rate 120574

PlantTarget inputs Outputs

Side slip angle 120573120575f

G(s)

Kp

F(s)

Kc(s)

Figure 3 Structure of INA controller

F for the decoupled MIMO system are designed utilizing theclassical SISO feedback PI control methods

In Figure 3

K119888 (119904) = [1198701198881 (119904) 0

0 1198701198882 (119904)]

F (119904) = [1198911 (119904) 0

0 1198912 (119904)

]

(8)

In this work unity feedbacks should be adopted betweenreference inputs and actual inputs that is

F (119904) = I2 (9)

3 Design of Nonlinear Modified INA System

The INA controller design flow is demonstrated in Figure 4First the appropriate precompensator K

119901is designed to

decouple the vehicle system G(119904) Being compensated theinverse system transfer function Kminus1

119901(s) Gminus1(119904) should be

diagonal dominance which can be judged by Gershgorinrsquosbands and Diagonal Dominance Factors Then the PIdynamic compensator matrixK

119888is designedThe parameters

of the PI controller should be set by the analysis of bodegraphs of open-loop system and the step responses of thedecoupled close-loop system

A square matrixQ(119904) is said to be of diagonal dominanceon a contour 119863 if for each column (or row) the modulus ofthe diagonal element is larger than the sum of the modulus ofthe off-diagonal elements for each complex variable 119904 in 119863

1003816100381610038161003816119902119894119894 (119904)1003816100381610038161003816 gt

119898

sum

119895=119894

(119895 = 119894)

10038161003816100381610038161003816119902119894119895 (119904)

10038161003816100381610038161003816(119894 = 1 2 119898) (10)

According to Gershgorinrsquos theorem [27] for specific 119904Gershgorinrsquos circles corresponding to 119898 column (or row) ofQ(119904) have the centres of 119902119894119894(119904) and the radii of 119903119894(119904)

119903119894 (119904) =

119898

sum

119895=1

(119895 = 119894)

10038161003816100381610038161003816119902119894119895 (119904)

10038161003816100381610038161003816(119894 = 1 2 119898) (11)

As 119904 runs through the contour 119863 the correspondingGershgorinrsquos circles for the 119898 columns (or rows) will sweepout 119898 bands centered by the trajectories 119902

119894119894(119904) respectively

These bands are called Gershgorinrsquos bands of the matrixQ(119904)It is clear that Gershgorinrsquos bands will not include the originof the complex 119904-plane and Q(119904) will be nonsingular for anyvalue of 119904 on a contour 119863 ifQ(119904) is diagonal dominance

Gershgorinrsquos bands of the inverse TFM Gminus1(119904) can beachieved by selecting Nyquist 119863-contour

Using the parameters in Table 1 Gershgorinrsquos bands ofGminus1(119904) when the vehicle is driving at longitudinal speed100 kmh are drawn in Figure 5 It is seen that the originalpoint of the complex plane is not included in Gershgorinrsquosbands of 119892minus1

11(119904) which is corresponding to the input of 119879

119911 It

means there is weak coupling for the vehicle system from thedirect yaw moment input on the sprung mass However for119892minus1

22(119904) which is corresponding to the input of front steering

wheel angle 120575119891 the origin of 119904-plane is located inGershgorinrsquosbands and strong coupling will be shown It is concluded thatthe steering input affects both yaw rate and side slip anglewhile the direct yawmoment affects yaw rate only which canalso be dedicated from (1) clearly

In order to analyze the diagonal dominance of the plantthe Diagonal Dominance Factor (DDF) is introduced TheDDF is defined as

119891119889119889

=

1003816100381610038161003816119892119894119894 (119904)1003816100381610038161003816

sum119898

119895=1

10038161003816100381610038161003816119892119894119895 (119904)

10038161003816100381610038161003816

(119894 = 1 2 119898) (12)

The DDF of uncompensated Gminus1(119904) is shown in Figure 6It is clear that the diagonal dominance of 1st row of Gminus1(119904)is weaker at lower frequency and is stronger at higher

The Scientific World Journal 5

Table 1 Vehicle parameters for Gershgorinrsquos bands calculation

Parameters Symbol ValuesMass (kg) 119898 1530Front axle cornering stiffness (Nrad) 119888

11989175435

Rear axle cornering stiffness (Nrad) 119888119903

54594Distance from vehicle CG to the front axle (m) 119897

119891111

Distance from vehicle CG to the rear axle (m) 119897119903

167Vehicle moment of inertia about the 119911-axis (kgm2) 119868

1199114192

Start

INA with Gershgorinrsquos bands Precompensatordesign

The closed-loop TFM

Step responses of thecompensated system

Performance

End

Yes

Yes

No

No

Dynamic compensatordesign

Input G(s)

Calculate Gminus1(s)

dominanceGminus1(s) diagonal

Figure 4 INA controller design flow

frequency while it is the opposite for the 2nd row In orderto achieve perfect decoupling through overall frequencybands the precompensations should be applied from lowerto higher frequency fields In this paper the precompensatoris designed according to the decoupling analysis of TFM atfrequency point 0 and +infin

At frequency point 0 the precompensator matrix can beachieved easily by

K119897= Gminus1 (0) (13)

However when the frequency is near to +infin the systemcannot be decoupled directly by the method of (13) as

G (infin) = lim119904rarrinfin

G (119904) = 0 (14)

which means that G(infin) is irreversible Thus a differentmethod is necessary

Rewrite matrix Gminus1(119904) as

Gminus1 (119904) =1

119889 (119904)P (119904) =

1

119889 (119904)[1199011 (119904) 119901

2 (119904) sdot sdot sdot 119901119898 (119904)]

(15)

whereP(119904) is a polynomialmatrix119901119894(119904) are column vectors of

P(119904) and 119889(119904) is the least common denominator of elementsof Gminus1(119904)

Let 119903119894be the highest degrees of the elements of each 119901

119894(119904)

the precompensator matrix for high frequency is

Kℎ= [

1199011 (119904)

1199041199031

1199012 (119904)

1199041199032sdot sdot sdot

119901119898 (119904)

119904119903119898] as 119904 997888rarr infin (16)

Thus by integrating K119897and K

ℎ the system can be

compensated perfectly from frequency 0 to +infin The prec-ompensator is

K119901 =1

119904K119897 + Kℎ (17)

The compensated INA becomes

Qminus1 (119904) = Kminus1119901

(119904) sdot Gminus1 (119904) (18)

By designing the precompensator from (13) to (17) withspecific V

119909= 100 kmh the DDF of compensated Qminus1(119904) is

shown in Figure 7It is seen that both of the DDFs of 11990211(119904) and 11990222(119904)

are almost 1 when the V119909 in the plant is 100 kmh whichmeans that the system is decoupled perfectly at this velocityHowever the DDFs decrease as V119909 departing from 100 kmhespecially in lower and middle frequency fields As shown inFigure 7(a) the DDF of 11990211(119904) decreased to 066 at frequency= 001Hz and V119909 = 40 kmh while in Figure 7(b) the DDFof11990222(119904) decreased to 081 at about frequency = 281Hz

and V119909= 160 kmh It is clear that the performance of

the compensator was affected negatively by the nonlinearcharacter caused by the variation of vehicle velocity

Using (7) and (13) to (17) the precompensator can besolved analytically By substituting (7) to (13) and (15)-(16)respectively we get

Kl =

[[[[[

[

119888119891 + 119888119903

119888119891

119898V2119909+ 119888119891119886 + 119888119903119887

V119909119888119891

minus119888119903 (119886 + 119887)minus (119886119898V2

119909minus 1198881199031198872+ 119888119903119886119887)

V119909

]]]]]

]

Kℎ= [

[

V119909119898

119888119891

0

minusV119909119886119898 119868

119911

]

]

(19)

6 The Scientific World Journal

0 1 2 3

minus05

0

05

1

15

Re

Imgminus111

(a)

Re

Im

minus4 minus2 0 2 4times105

times105

minus4

minus2

0

2

4gminus122

(b)

Figure 5 Uncompensated INA with Gershgorinrsquos bands at V119909= 100 kmh frequency from 001Hz to 10Hz

10minus2 10minus1 100 101 102 1030

02

04

06

08

1

Frequency (Hz)

Dia

gona

l dom

inan

ce fa

ctor

gminus111

gminus122

Figure 6 Diagonal Dominance Factors for uncompensated INA V119909

= 100 kmh

V119909can be achieved from the actual vehicle or vehicle

model directly while the cornering stiffness of each axle alsochanges on vehicle driving

As is well known the tire cornering force can be describedas map of tire vertical load road friction coefficient and tireslip angle

119888 =119865lowast

119910

120572lowast (20)

where 119865lowast

119910and 120572

lowast are coordinate values of specific point onlinear section of tire cornering characteristic curve at givenvertical load 119865

119911 which is shown in Figure 8

The vertical load of each wheel is calculated by thefollowing equations and then the cornering force of each

wheel can be interpolated according to data chart inFigure 8

119865119911119891119897 = 119898119892 sdot119897119903

2 (119897119891+ 119897119903)

minus 119898119886119909 sdot

ℎ119892

2 (119897119891+ 119897119903)

minus119896119891Φ

119896119891Φ

+ 119896119903Φ

sdot (119898119886119910ℎ119892+ 119898119892ℎ

119903Φ

119889)

119865119911119891119903 = 119898119892 sdot119897119903

2 (119897119891+ 119897119903)

minus 119898119886119909 sdot

ℎ119892

2 (119897119891+ 119897119903)

+119896119891Φ

119896119891Φ + 119896119903Φ

sdot (119898119886119910ℎ119892+ 119898119892ℎ

119903Φ

119889)

119865119911119903119897

= 119898119892 sdot119897119903

2 (119897119891 + 119897119903)+ 119898119886119909sdot

ℎ119892

2 (119897119891 + 119897119903)

minus119896119891Φ

119896119891Φ + 119896119903Φ

sdot (119898119886119910ℎ119892+ 119898119892ℎ

119903Φ

119889)

119865119911119903119903

= 119898119892 sdot119897119903

2 (119897119891 + 119897119903)+ 119898119886119909sdot

ℎ119892

2 (119897119891 + 119897119903)

+119896119891Φ

119896119891Φ

+ 119896119903Φ

sdot (119898119886119910ℎ119892+ 119898119892ℎ

119903Φ

119889)

(21)

where 119865119911119891119897 119865119911119891119903 119865119911119903119897 and 119865119911119903119903 are vertical load on front leftfront right rear left and rear right tire respectively119889 is wheeltrack ℎ

119892is height of center of sprung mass ℎ

119903is the distance

between height of center of sprung mass and roll center 119896119891Φ

and 119896119903Φ

are roll stiffness of front and rear axle respectivelyΦ is roll angle

The Scientific World Journal 7

10minus2100

102

50

100

150

07

08

09

1

Frequency (Hz)

DD

F ofq11

(s)

Vehicle velocity (kmmiddoth minus1)

(a)

10minus2100

102

Frequency (Hz)

08

085

09

095

1

50

100

150

DD

F ofq22

(s)

Vehicle velocity (kmmiddoth minus1)

(b)

Figure 7 DDF for 11990211(119904) and 119902

22(119904) ofQminus1(119904) without the consideration of nonlinear characteristics

0 5 10 15 200

1000

2000

3000

4000

5000

6000

120572 (deg)

Fy

(N)

Fz = 2500NFz = 4100NFz = 5800N

120572lowast

Fylowast

Figure 8 Curves for tire cornering characteristic

10minus2100

102

50

100

150

1

1

1

Frequency (Hz)

DD

F ofq11

(s)

Vehicle velocity (kmmiddoth minus1)

(a)

10minus2100

102

50

100

150

1

1

1

Frequency (Hz)

DD

F ofq22

(s)

Vehicle velocity (kmmiddoth minus1)

(b)

Figure 9 DDF for 11990211(119904) and 119902

22(119904) ofQminus1(119904) with the consideration of nonlinear characteristics

Then

119888119891 =119865lowast

119910119891119897+ 119865lowast

119910119891119903

120572lowast= 119891 (119865

lowast

119911119891119897+ 119865lowast

119911119891119903)

119888119903=

119865lowast

119910119903119897+ 119865lowast

119910119903119903

120572lowast= 119891 (119865

lowast

119911119903119897+ 119865lowast

119911119903119903)

(22)

where 119891(sdot) is the mapping function according to data chartFigure 8Thus the cornering stiffness is depicted as functionsof vehicle longitudinal and lateral accelerations

With the consideration of nonlinear characteristics theDDF of precompensated Qminus1(119904) is shown in Figure 9 TheDDFs of both diagonal elements are so close to 1 that thetiny deviation cannot be displayed in the ticks of verti-cal axis in the figures In fact the system is decoupledcompletely by the analytical solution and the deviation

8 The Scientific World Journal

minus05 0 050

02

04

06

08

1

12

Re

Imgminus111

(a)

minus05 0 050

02

04

06

08

1

12

Re

Im

gminus122

(b)

Figure 10 Gershgorinrsquos bands for Qminus1(119904) at V119909= 40 kmh frequency from 001Hz to 10Hz

minus05 0 050

02

04

06

08

1

12

Re

Im

gminus111

(a)

minus05 0 050

02

04

06

08

1

12

Re

Im

gminus122

(b)

Figure 11 Gershgorinrsquos bands for Qminus1(119904) at V119909= 100 kmh frequency from 001Hz to 10Hz

is caused by computer precision It means that the out-puts of vehicle yaw rate and side slip from both controlinputs are almost completely decoupled in common vehiclespeed region

The effectiveness of proposed precompensator is alsoproved by Gershgorinrsquos bands at speeds 40 kmh 100 kmhand 160 kmh which are shown in Figures 10 11 and 12respectively And the detailed sections of Figure 12 are shownin Figure 13 All of Gershgorinrsquos circles are tiny and keep

origin of s-plane outside thus the system is perfect diagonaldominance

As the system has been decoupled completely the feed-back PI controller design method for SISO is used tocompensate the system dynamics characteristic The TFM ofthe PI controller is

Kc (119904) = KcP + KcI1

119904= [

1198961198881199011

0

0 1198961198881199012

] + [1198961198881198941 0

0 1198961198881198942]

1

119904 (23)

The Scientific World Journal 9

minus05 0 050

02

04

06

08

1

12

Re

Imgminus111

(a)

minus05 0 050

02

04

06

08

1

12

Re

Im

gminus122

(b)

Figure 12 Gershgorinrsquos bands for Qminus1(119904) at V119909= 160 kmh frequency from 001Hz to 10Hz

minus005 0 0050

002

004

006

Re

Im

gminus111

(a)

0

Re

Im

minus5 5times10minus11

00439

00439

00439

gminus111

(b)

Figure 13 Detailed section of 119892minus111

in Figure 12

Thus TFM for the INA based integrated controller is

U (119904)

Δ (119904)= KINA

= Kc (119904) sdot KP (119904) = (Kh + Kl1

119904) (KcP + KcI

1

119904)

=KhKcP119904

2+ (KlKcP + KhKcI) 119904 + KlKcI

1199042

(24)

For the convenience of programming rewrite the con-troller TFM into state function mode

x120585 = A120585x120585 + B120585Δ

u = C120585x120585+ D120585Δ

(25)

where

x120585 = (1199091205851 1199091205852 1199091205853 1199091205854)119879

Δ = x119889minus x x

119889= (120575119889 120574

119889)119879

A120585 = [0 0

KcI 0] B120585 = [

I2

KcP]

C120585= [KhKcI Kl] D

120585= KhKcP

(26)

x and u were defined in (1)

4 Analysis and Simulation

Applying the proposed controller which is described by (25)to the 2-DOF system which is described by (1) the bode

10 The Scientific World Journal

minus20

0

20

40

60

80

100

Mag

nitu

de (d

B)

go(1 1)

10minus2 100 102

Frequency (Hz)

minus270

minus180

minus90

0

Phas

e (de

g)M

agni

tude

(dB)

minus450

minus400

minus350

minus300

minus250go(1 2)

10minus2 100 102minus180

minus135

minus90

Phas

e (de

g)

Frequency (Hz)

minus50

0

50

100

Mag

nitu

de (d

B)

go(2 2)

10minus2 100 102minus180

minus135

minus90

Phas

e (de

g)

Frequency (Hz)10minus2 100 102

Frequency (Hz)

minus270

minus180

minus90

0

90

180

270

Phas

e (de

g)M

agni

tude

(dB)

minus300

minus250

minus200

minus150

minus100go(2 1)

Figure 14 Bode graphs of the open-loop system

The Scientific World Journal 11

0 2 4 6 8 10

A(g

o12)

04

02

0

minus02

minus04

minus06

minus08

minus1

t (s)0 2 4 6 8 10

08

085

09

095

1

105

11

115

A(g

o11)

t (s)

0 2 4 6 8 10

A(g

o21)

015

01

005

0

minus005

minus01

minus015

minus02

t (s)0 2 4 6 8 10

A(g

o22)

14

12

1

08

06

04

02

0

t (s)

Figure 15 Step responses of the close-loop system

0 1 2 3 4 5 6

0

20

40

60

Time (s)

Stee

ring

inpu

t (de

g)

Figure 16 Steering wheel inputs for step steering simulation

graphs of the open-loop system 119892119900(119904) with different V119909 = 4050 160 are shown in Figure 14 The open-loop responseof 119892119900(1 1) and 119892

119900(2 2) can be analyzed from the top left and

bottom right graphs respectivelyIt is seen that gains 119892119900(1 2) and 119892119900(2 1) are close to zero

which implies that the systemhas been decoupled completelyand only the responses of 119892

119900(1 1) and 119892

119900(2 2) should be

considered for the performance of the controlled systemThesame conclusion can be drawn from the fact that the gain linesand phase lines of different V

119909are almost coincident and are

not affected by the minor diagonal elements of system TFM

The unit step response of the close-loop system 119892119888(119904) is

shown in Figure 15 It is seen that the responses of 119892119888(1 1)

and 119892119888(2 2) converge to 1 and the responses of 119892119888(1 2) and119892119888(2 1) converge to 0 quickly By setting the PI controllerKc(119904) carefully the time domain response of the close-loopsystem can be regulated to a satisfied level

Simulations were carried out by MatlabSimulink A2-DOF vehicle model and a 14-DOF vehicle model wereprogrammed and the step steering maneuver is simulated byboth vehicle models respectively [33ndash35] At 2 s a step steer-ing inputwith amplitude of 60 deg was appliedwith 03 sThesteering wheel input is shown in Figure 16 Both simulationsare performed on a road with a friction coefficient of 05Thesimulations lasted to 6 s

The simulation results of 2-DOF model are shownin Figure 17 During the simulation the longitudinalvelocity of the vehicle was set as decreasing linearly atthe rate of 4 kmh from 120 kmh which is shown inFigure 17(a) The yaw rate and side slip angle of targetvalue controlled and uncontrolled values are shown inFigures 17(b) and 17(c) Figures 17(d) and 17(e) show thesteering wheel and active yaw moment from the controllerIt is seen that the steering wheel and active yaw moment

12 The Scientific World Journal

0 1 2 3 4 5 695

100

105

110

115

120

Time (s)Longitudinal velocity

Long

itudi

nal v

eloc

ity(k

mmiddothminus1)

(a)

0 1 2 3 4 5 6

Time (s)

minus01

0

01

02

03

04

TargetControlled

Uncontrolled

Yaw

rate

(radmiddotsminus

1)

(b)

0 1 2 3 4 5 6

Time (s)

TargetControlled

Uncontrolled

minus4

minus3

minus2

minus1

0

1

Side

slip

angl

e (de

g)

(c)

0 1 2 3 4 5 6

Time (s)

0

20

40

60

DriverController

Stee

ring

whe

el an

gle

(deg

)

(d)

0 1 2 3 4 5 6

Time (s)

minus1500

minus1000

minus500

0

500

Active yaw moment

Yaw

mom

ent (

Nmiddotm

)

(e)

Figure 17 Simulation results for 2-DOF vehicle model

control are integrated by the proposed controller and bothyaw rate and side slip angle of the vehicle are reduced andcloser to target values

While simulating with 14-DOF model the longitudinalvelocity of the vehicle is generated automatically which isshown in Figure 18(a) From Figures 18(b) to 18(e) the simi-lar behaviors of the controller and vehicle in spite of largerfluctuations are shown The ICC coordinated the AFS andYSC system effectively and the side slip angle and yaw ratewere regulated to a desired region

5 Conclusion

This paper presents a control algorithm for integration ofAFS and YSC systems based on a 2-DOF vehicle model Thecoupling characteristic of the typical two-input two-outputsystem is analyzed and it is decoupled and compensatedby the INA design method In the research we found that

although the traditional INA controller could decouple thesystem well in given circumstances the performance ofthe compensator was affected negatively by some kinds ofnonlinear factors Therefore the nonlinear characteristicscaused by the change of velocity and cornering stiffness wereconsidered which implies that the improved INAmethod canbe qualified for the nonlinear vehicle driving maneuversThecorrection of the precompensator ensures the effectiveness ofthe controller covering overall frequency bands and vehiclespeeds As the analytic format of the precompensator isproposed the decoupling algorithm runs fast

After being decoupled a PI feedback controller isdesigned using the traditional design method for SISO linearsystems The frequency response of the open-loop systemand the step response on time domain of the close-loopsystem were analyzed The results show that the stability andresponse characteristics of the system are guaranteed by thefeedback gain matrix and dynamic compensation matrix

The Scientific World Journal 13

0 1 2 3 4 5 6100

105

110

115

120

125

Time (s) ControlledUncontrolled

Long

itudi

nal v

eloc

ity(k

mmiddothminus1)

(a)

Time (s) 0 1 2 3 4 5 6

0

01

02

03

minus01

TargetControlled

Uncontrolled

Yaw

rate

(radmiddotsminus

1)

(b)

Time (s) 0 1 2 3 4 5 6

0

1

Side

slip

angl

e (de

g)

TargetControlled

Uncontrolled

minus4

minus3

minus2

minus1

(c)

Time (s) 10 2 3 4 5 6

0

20

40

60

Driver Controller

Stee

ring

whe

el an

gle

(deg

)

(d)

Time (s) 0 1 2 3 4 5 6

0

200

Active yaw moment

minus800

minus600

minus400

minus200

Yaw

mom

ent (

Nmiddotm

)

(e)

Figure 18 Simulation results for 14-DOF vehicle model

Finally simulations were carried out using a 2-DOFmodel and a 14-DOF model by MatlabSimulink The sim-ulation results of step steering indicate that the proposedICC control can reduce yaw and side slip of the targetvehicle significantly and improve its handling and stabilityconsequently

Conflict of Interests

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

Acknowledgments

This work is partially supported by National Natural Sci-ence Foundation (51105169 51175215 and 51205156) JilinProvince Science and TechnologyDevelopment Plan Projects(201101028 20140204010GX) and Science Foundation for

Chinese Postdoctoral (2011M500053 2012T50292) and Fun-damental Research Funds of Jilin University

References

[1] F Yu D-F Li and D A Crolla ldquoIntegrated vehicle dynamicscontrol-state-of-the art reviewrdquo in Proceedings of the IEEEVehicle Power and Propulsion Conference (VPPC rsquo08) pp 1ndash6Harbin China September 2008

[2] Y Kou H Peng and D Jung ldquoDevelopment and evaluation ofan integrated chassis control systemrdquo JSAE Review of Automo-tive Engineering vol 29 no 3 2008

[3] A Elmarakbi C Rengaraj A Wheately and M Elkady ldquoNewintegrated chassis control systems for vehicle handling perfor-mance enhancementrdquo International Journal of Dynamics andControl vol 1 no 4 pp 360ndash384 2013

[4] T W Chu R P Jones and L M T Whittaker ldquoA systemtheoretic analysis of automotive vehicle dynamics and controlrdquoVehicle System Dynamics vol 37 pp 83ndash95 2003

14 The Scientific World Journal

[5] E H M Lim and J K Hedrick ldquoLateral and longitudinalvehicle control coupling for automated vehicle operationrdquo inProceedings of the American Control Conference (ACC rsquo99) pp3676ndash3680 San Diego Calif USA June 1999

[6] M Abe N Ohkubo and Y Kano ldquoA direct yaw momentcontrol for improving limit performance of vehicle handlingmdashcomparison and cooperation with 4WSrdquo Vehicle SystemDynamics vol 25 pp 3ndash23 1996

[7] R G Hebden C Edwards and S K Spurgeon ldquoAn applicationof sliding mode control to vehicle steering in a split-mumanoeuvrerdquo in Proceedings of the American Control Conferencevol 5 pp 4359ndash4364 June 2003

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

[9] C Rengaraj and D Crolla ldquoIntegrated chassis control toimprove vehicle handling dynamics performancerdquo SAE Paper2011-01-0958 2011

[10] W Cho J Yoon J Kim J Hur and K Yi ldquoAn investigation intounified chassis control scheme for optimised vehicle stabilityand manoeuvrabilityrdquo Vehicle System Dynamics vol 46 no 1pp 87ndash105 2008

[11] A Trachtler ldquoIntegrated vehicle dynamics control using activebrake steering and suspension systemsrdquo International Journalof Vehicle Design vol 36 no 1 pp 1ndash12 2004

[12] JHeDCrollaM Levesley andWManning ldquoIntegrated activesteering and variable torque distribution control for improvingvehicle handling and stabilityrdquo SAE Paper 2004-01-1071 2004

[13] R C Wang L Chen and H B Jiang ldquoIntegrated controlof semi-active suspension and electric power steering basedon multi-agent systemrdquo International Journal of Bio-InspiredComputation vol 4 no 2 pp 73ndash78 2012

[14] R K M Ali S H Tabatabaei R Kazemi et al ldquoIntegratedcontrol of AFS and DYC in the vehicle yaw stability manage-ment system using fuzzy logic controlrdquo SAE Paper 2008-01-1262 2008

[15] M Nagai M Shino and F Gao ldquoStudy on integrated control ofactive front steer angle and direct yaw momentrdquo JSAE Reviewvol 23 no 3 pp 309ndash315 2002

[16] G Burgio and P Zegelaar ldquoIntegrated vehicle control usingsteering and brakesrdquo International Journal of Control vol 79no 5 pp 534ndash541 2006

[17] AGoodarzi andMAlirezaie ldquoIntegrated fuzzyoptimal vehicledynamic controlrdquo International Journal of Automotive Technol-ogy vol 10 no 5 pp 567ndash575 2009

[18] B Lee A Khajepour and K Behdinan ldquoVehicle stabilitythrough integrated active steering and differential brakingrdquoSAE 2006-01-1022 2006

[19] R Marino and S Scalzi ldquoIntegrated control of active steeringand electronic differentials in four wheel drive vehiclesrdquo SAE2009-01-0446 2009

[20] W Cho J Choi C Kim S Choi and K Yi ldquoUnified chassiscontrol for the improvement of agility maneuverability andlateral stabilityrdquo IEEE Transactions on Vehicular Technology vol61 no 3 pp 1008ndash1020 2012

[21] N Ding and S Taheri ldquoAn adaptive integrated algorithm foractive front steering and direct yaw moment control based ondirect Lyapunov methodrdquo Vehicle System Dynamics vol 48 no10 pp 1193ndash1213 2010

[22] D Li and F Yu ldquoA novel integrated vehicle chassis controllercoordinating direct yaw moment control and active steeringrdquoSAE Paper 2007-01-3642 2007

[23] M Doumiati O Sename L Dugard J J Martinez-MolinaP Gaspar and Z Szabo ldquoIntegrated vehicle dynamics controlvia coordination of active front steering and rear brakingrdquoEuropean Journal of Control vol 19 no 2 pp 121ndash143 2013

[24] R Marino S Scalzi and F Cinili ldquoNonlinear PI front and rearsteering control in four wheel steering vehiclesrdquo Vehicle SystemDynamics vol 45 no 12 pp 1149ndash1168 2007

[25] T Acarman ldquoNonlinear optimal integrated vehicle controlusing individual braking torque and steering angle with on-linecontrol allocation by using state-dependent Riccati equationtechniquerdquo Vehicle System Dynamics vol 47 no 2 pp 155ndash1772009

[26] J M Maciejowski Multivariable Feedback Design Addison-Wesley Wokingham UK 1989

[27] N Munro ldquoMultivariable systems design using the inverseNyquist arrayrdquo Computer-Aided Design vol 4 no 5 pp 222ndash227 1972

[28] H H Rosenbrock Computer-Aided Control System DesignAcademic Press New York NY USA 1974

[29] D H Owens ldquoA historical view of multivariable frequencydomain controlrdquo in Proceedings of the 15th Triennial WorldCongress of the International Federation of Automatic ControlBarcelona Spain July 2002

[30] Z Deng Y Wang L Liu and Q Zhu ldquoGuaranteed costdecoupling control of bank-to-turn vehiclerdquo IET ControlTheoryand Applications vol 4 no 9 pp 1594ndash1604 2010

[31] L Shamgah and A Nobakhti ldquoDecomposition via QPrdquo IEEETransactions on Control Systems Technology vol 20 no 6 pp1630ndash1637 2012

[32] X Shen and F Yu ldquoInvestigation on integrated vehicle chassiscontrol based on vertical and lateral tyre behaviour correlativ-ityrdquo Vehicle System Dynamics vol 44 no 1 pp 506ndash519 2006

[33] B Zhu J Zhao L T Guo W W Deng and L Q RenldquoDirect yaw-moment control through optimal control alloca-tionmethod for light vehiclerdquoAppliedMechanics andMaterialsvol 246-247 pp 847ndash852 2013

[34] C Ghike and T Shim ldquo14degree-of-freedom vehicle model forroll dynamics studyrdquo SAE 2006-01-1277 2006

[35] R N JazarVehicle Dynamics Theory and Application SpringerNew York NY USA 2007