A new strategy for minimum usage of external yaw moment
in vehicle dynamic control system
M.Mirzaei
Faculty of Mechanical Engineering,Sahand University of Technology,Tabriz,Iran
a r t i c l e i n f o Article history:Received 20September 2008Received in revid form 9June 2009Accepted 11June 2009Keywords:Vehicle dynamics LQ optimal control Desired handling Direct yaw moment control Analytical solution
a b s t r a c t
Due to the loss of vehicle directional stability in emergency maneuvers,a new complete
desired model for vehicle handling bad on the linear two-degrees-of-freedom (2DOF)
model and tire/road conditions is prented to be tracked by the direct yaw moment con-
trol (DYC)system.In order to maintain the vehicle actual motions,yaw rate and side-slip
angle,clo to the propod desired respons without excessively large external yaw
moment,a complete linear quadratic (LQ)optimal problem is formulated and its analytical
solution is obtained.Here,the derived control law is evaluated and its different versions
are discusd.It is shown that the side-slip tracking by DYC is more effective than the
yaw rate control to stabilize vehicle motions in nonlinear regimes.Also,optimal property
of the control law provides the possibility of reducing the external yaw moment as low as
possible,at the cost of some admissible tracking errors.Simulation studies of vehicle han-
dling,with and without control,have been conducted using a full nonlinear vehicle
dynamic model.The results,obtained during various maneuvers,indicate that when the
遗失的世界第一部propod optimal controller is engaged with the model,improvements in the handling per-
formance through a reduced external yaw moment can be acquired.
Ó2009Elvier Ltd.All rights rerved.1.Introduction
Direct yaw moment control (DYC)system is the latest active safety technology introduced to control vehicle directional stability under emergency situations.In a turning maneuver with high lateral acceleration where tire forces are approaching to or at the physical he limit of road adhesion,the vehicle side-slip angle grows and the effectiveness of vehicle steering angle in generating yaw moment becomes significantly reduced becau of tire force saturation.This fact is first illustrated by the so-called b -method (Shibahata et al.,1993).The decrea of restoring yaw moment generated by tire lat-eral force when the side-slip angle increas is the basic cau of vehicle unstable motion called spin motion and adding yaw moment will recover the vehicle stability.2017六级答案
A practical approach to generate a required external yaw moment,independent of lateral forces and steering angle,is the transver distribution of the vehicle braking force between the left and right wheels.This strategy known as differential braking can be achieved using the main parts of common anti-lock braking system (van Zanten et al.,1998).
It is shown that DYC is the most effective method on vehicle motion control compared with the other conventional con-trol systems such as four wheel steering (4WS)(Abe,1999;Selby et al.,2001).The 4
WS control,which depends on the rela-tion between tire lateral force and the steer angle as a control command,is efficient in a range where the lateral acceleration is low.But,in high lateral accelerations,as mentioned before,the steer input los its direct effectiveness on tire lateral force and thus on the yaw moment.Therefore,the lateral dynamics parameters,yaw rate and side-slip angle,can no longer be controlled by the steer command.
在美国找工作0968-090X/$-e front matter Ó2009Elvier Ltd.All rights rerved.
doi:10.2009.06.002
E-mail address:mirzaei@sut.ac.ir
Transportation Rearch Part C 18(2010)213–224
Contents lists available at ScienceDirect稗草
Transportation Rearch Part C
journal homepage:www.el s e v i e r.c o m /l o c a t e /t r c
214M.Mirzaei/Transportation Rearch Part C18(2010)213–224
Despite the high efficiency of DYC in a wide range of operation,the external yaw moment,as the control input of DYC, should be kept as low as possible becau of its some undesirable effects.It slows down the vehicle becau a corrective yaw moment is applied to the vehicle through the brakes.This effect must be kept to a minimum so that the driver can feel supported rather than overruled.Further to this,tire life is also shortened becau of extra braking.One of the recent ap-pro
aches to limit the excessive u of external yaw moment is integrating and coordinating DYC and4WS(Selby et al.,2001).
The prent study investigates the other effective approach reducing the external yaw moment for stabilizing vehicle handling dynamics.First,a suitable desired model for vehicle handling is developed to be tracked by DYC system.Then,con-sidering some admissible tracking errors,an optimal yaw moment control law is developed to reduce the external yaw mo-ment as much as possible by adequately following the behavior of the desired model.In this way,the calculated external yaw moment can also remain below the maximum admissible value determined by the maximum difference in the longitudinal forces that can be generated by the left and right wheels during every cornering maneuver.
Basically,the yaw rate and side-slip angle are taken as the two controlled variables for vehicle handling.Depending on which variable is controlled,different control types of DYC have been propod.The yaw rate control by DYC has been stud-ied frequently(Esmailzadeh et al.,2003;Mokhiamar and Abe,2006;Mirzaei et al.,2008).Some rearchers have ud the side-slip control type of DYC(Abe,1999;Abe et al.,2001;Eslamian et al.,2007).However,both variables have been controlled simultaneously by DYC(Zheng et al.,2006;Park et al.,2001;Ghoneim et al.,2000).In t
he above studies,veral control theories with various reference models have been employed.
In both side-slip control and yaw rate control by DYC,the desired(reference)models of the variables should be estab-lished according with driver steering commands and tire/road conditions.The preci desired model not only results the enhanced performance but also prevents the u of extra control effort.Several rearchers have ud the steady-state behavior of linear vehicle model during a cornering maneuver as a desired model for the yaw rate(Esmailzadeh et al., 2003;Zheng et al.,2006;Bang et al.,2001).The desired side-slip angle has been considered as zero in their studies.van Zan-ten(2000)pointed out that on dry asphalt the physical limit in which the vehicle shows unstable motion,is reached at a slip angle of approximately±12°,while on icy roads this value is about±2°.The models do not include the transient respon for vehicle motions and therefore make large tracking errors at the beginning of maneuvers.In the other rearches,a linear 2DOF vehicle plane model(bicycle model)has been adopted as a desired model to be followed by the controller(Mokhiamar and Abe,2006;Ghoneim et al.,2000).Although the linear bicycle model shows a stable motion,it is unable to predict the tire/road conditions.In the prent study,a complete desired model for vehicle handling bad on the linear2DOF model and tire/road conditions is prented.
On the other hand,there are veral control methods for tracking the desired model by DYC in the literature.Ghoneim et al.(2000)introduced a feedback control using yaw rate with proportional-derivative(PD)structure.They further en-hanced the performance of PD yaw rate control with a full state feedback utilizing both yaw rate and side-slip angle.Another yaw rate control type of DYC has been prented using proportional-integral-derivative(PID)control method(Bang et al., 2001).Sliding mode control method has been also employed infinding the yaw moment control law(Abe,1999;Abe et al.,2001;Mokhiamar and Abe,2006).In the methods,the optimization is not ud as a main procedure infinding the control laws.A predictive optimal yaw stability controller bad on a linearized vehicle model which was discretized via a bilinear transformation has been developed by Anwar(2005).Some rearches have developed the well-known LQ the-ory to improve vehicle handling and stability(Park et al.,2001;Zheng et al.,2006).The studies u on-line numerical com-putations in optimization which are not suitable for implementation.Esmailzadeh et al.(2003)have prented an analytical solution for LQ problem,but their propod control law is bad on tracking only the reference yaw rate obtained by the steady-state behavior of vehicle during a cornering maneuver.Another yaw rate control by DYC has been developed by a predictive optimal approach(Mirzaei et al.,2008).The same approach has been employed for the side-slip control type of DYC(Eslamian et al.,2007).In the prent paper,a complete LQ optimal problem is formulated to tr
ack the propod new desired models for both yaw rate and side-slip angle.The derived control law is developed in an analytical clod form which is easy to solve and implement.Also,the different control types of DYC are examined and the effect of weighting fac-tors on the control system performance is further investigated.
2.Vehicle system dynamics
2.1.Vehicle model for simulation
In order to accurately predict the vehicle respon during various maneuvers,simulation studies have been conducted using a comprehensive vehicle dynamic model.In this respect,an eight-degrees-of-freedom vehicle model which has been previously developed and validated by experimental results(Smith and Starkey,1995)has been ud as the vehicle plant model.The longitudinal velocity,lateral velocity,yaw rate,roll rate and rotational speeds of four wheels constitute the de-grees of freedom for this model.Therefore,the saturation property of tire lateral force at high slip angle,the effects of normal load transfer on the balance of the front and rear tire lateral forces,the roll steer,the roll camber and other characteristics which influence vehicle stability in reality are considered for simulation.
Since,the nonlinear8DOF vehicle model described above is too complicated for u in control system design,a simpler vehicle model referred to as the design model has to be employed for the controller design.In this paper,according to the LQ
optimal control theory requirements,the conventional linear 2DOF bicycle model is lected as the design model.Fig.1illus-trates the structure of vehicle clod loop system.As it is considered,the control signal is nt to the comprehensive vehicle model and conquently the derived motions are nt back to the designed controller.
2.2.Desired model for vehicle handling
In order to compensate the loss of vehicle stability in emergency situations due to nonlinear characteristics of vehicle dynamics and tire forces,a linear 2DOF vehicle plane model (bicycle model)with appropriate limitations within physical constraints is propod as a desired model to be followed by the controller.The governing equations for the linear vehicle model,in the state space form,are expresd as (Wong,2001):_b _r "#¼a 11a 12a 21a 22 !b r !þe 1
e 2 !d ð1Þ
where
a 11¼À2C a f þC a r mu ;a 12¼2bC a r ÀaC a f mu 2翻译网址
À1;a 21¼2bC a r ÀaC a f I z ;a 22¼À2b 2
C a r þa 2C a f I z u ;e 1¼2C a f mu ;e 2¼2aC a f I z ð2ÞIn the above equations,the vehicle side-slip angle b and the yaw rate r are the two state variables.The front wheel steer-ing angle d is considered as the driver input.Other parameters denote the following:C a f and C a r are the cornering stiffness coefficients of the front and rear tires,respectively.a and b are the distances of the mass center to the front and rear axles,respectively.m is the mass of the vehicle and I z is the moment of inertia about the vertical z -axis.u is the forward velocity of the vehicle.
The transfer function from the steering input to the side-slip angle and yaw rate can be derived from the state equations:
b ðs Þd ðs Þ¼G B 1þT B s 1ÀðT A =D A Þs þð1=D A Þs 2
ð3Þr ðs Þ¼G R 1þT R s A A A ð4Þ
如何用电脑发传真
where G B ¼e 2a 12Àe 1a 22A ;T B ¼e 1212122
;
G R ¼e 1a 21Àe 2a 11A ;T R ¼e 2121211;T A ¼a 11þa 22;D A ¼a 11a 22Àa 12a 21ð5ÞAccording to Eqs.(2)–(5),T A is always negative and hence the linear vehicle model is stable if D A is positive.This condition implies
l þk us u 2>0
ð6Þ
in which l denotes the length of the wheelba and k us ¼m 2l b C a f Àa C a r ð7Þ
is referred to as the understeer coefficient.Eq.(6)is always satisfied when k us is positive and the vehicle shows the under-steer behavior.Generally,a more under-steering behavior with an incread k us results a more cure stability of vehicle par-ticularly at high speeds (Esmailzadeh et al.,2003).
M.Mirzaei /Transportation Rearch Part C 18(2010)213–224215
To increa the vehicle stability limit,an intentional modification of yaw rate respon from cond tofirst order model has been propod(Mokhiamar and Abe,2002).This modification is due to the fact that when the side-slip angle converges to zero,from Eq.(1),the yaw rate respon can be reduced to afirst-order lag.As a result,the following equation can be
propod for the yaw
by keeping its steady-state value
rðsÞdðsÞ¼G R
1
1þT r s
ð8Þ
where T r is the yaw rate time constant.
It should be noted that the linear vehicle model rely on the nominal values of vehicle parameters along with a high-coef-ficient of friction condition.In order to achieve the desired respons of side-slip angle and yaw rate compatible with all deriving conditions especially on slippery roads and during high speed maneuvers,an appropriate limitation must be com-bined with the linear vehicle model.
Referring to the vehicle dynamics model,the term of lateral acceleration is defined as:
a y¼_v yþurð9Þwhere v y is the lateral velocity.So,the steady-state value of yaw rate during a constant cornering is as follows:
r ss¼a y
ss
u
ð10Þ
trespass
Since the lateral acceleration of the vehicle in terms of g units cannot exceed the maximum road coefficient of friction,the steady-state value of yaw rate r ss during a constant cornering maneuver must be limited bad upon the tire/road condition.
开发英文j r ss j l g
ð11Þ
where l is the road coefficient of friction.The measured lateral acceleration a y can be taken instead of l g.
The steady-state value of vehicle yaw rate during a constant cornering maneuver can be obtained from Eq.(4): r ss¼G R dð12ÞSubstituting expressions in(2)and(5)into(12)yields
r ss¼
u
lþk us u2
dð13Þ
In the same manner,the steady-state value of vehicle side-slip angle is obtained from Eq.(3):
b ss¼G B dð14Þ
Using Eqs.(2)and(5),the steady-state value of vehicle side-slip angle can be rewritten in terms of the steady-state value of yaw rate:短裤的英语单词
b ss¼r ss f uð15Þwhere
f u¼b
u
À
amu
2lC a r
ð16Þ
is the ratio of the steady-state values of side-slip angle to yaw rate.The Eqs.(15)and(16)also state dependency of the two controlled variables.
Now,the constraint(11)can be easily applied to the linear vehicle model.In this way,combination of the following con-dition with Eq.(4)makes the desired yaw rate to be smoothly limited to a value compatible with tire/road conditions.
r ss¼
G R d if j G R d j<l g
u
l g
u
signðG R dÞotherwi
(
ð17Þ
Applying the condition(17)to Eq.(15)and then combining with Eq.(3)also leads to the desired side-slip angle limited by tire/road conditions.
Fig.2compares the respons of the linear and limited linear(desired)vehicle models during a J-turn maneuver at a con-stant speed of80km/h.First,it is suppod that the vehicle travels on aflat high-friction road(l=0.8)and then the same maneuver is repeated on a low-friction road(l=0.4).The front wheel steering angle commanded by driver is shown in Fig.2a.
The time respons of lateral acceleration,yaw rate and side-slip angle depicted in Fig.2b–d indicate that although the linear vehicle model shows a stable motion,it cannot predict the tire/road conditions alone.Fig.2b illustrates that the re-spons of lateral acceleration for the linear model during both driving conditions coincide and reach over1g(solid line). This means that the respons of linear model lack compatibility with the maximum coefficient of friction.In contrast, the desired respons of vehicle obtained by the limited linear model are smoothly restricted to the values compatible with 216M.Mirzaei/Transportation Rearch Part C18(2010)213–224
tire/road conditions.In this model,the lateral acceleration does not exceed the maximum road coeffici
ent of friction as shown in Fig.2b.
3.Control system design
The purpo of control system is to maintain the vehicle actual motions,yaw rate and side-slip angle,clo to their de-sired respons with a minimum external yaw moment.To achieve this aim,an optimal approach should be applied for development of the yaw moment control law.The linear quadratic(LQ)method is considered as a suitable tool for solving such problems(Bryson and Ho,1975;Kirk,1970).
3.1.Development of the control law
In order to formulate a complete LQ problem,a performance index that penalizes the tracking errors and control expen-diture isfirst considered in the following form:
J¼1
2
Z t f
w bðbÀb dÞ2þw rðrÀr dÞ2þw u M2
z
h i
dtð18Þ
where w b,w r and w u are weighting factors indicating the relative importance of the corresponding terms.M z is the external yaw moment which must be determined from the control law.The subscript d denotes the desired respon.Minimization of the performance index(18)must be sought in order to improve the tracking accuracy by using minimum external yaw moment.
Now,the linear2DOF vehicle model,described in Eqs.(1)and(2),is extended to design the optimal yaw moment con-troller.The matrix form of the equations is as follows:
_X¼AXþE dþBUð19Þwhere
X¼b
r
!
;A¼
a11a12
a21a22
!
;E¼
e1
e2
!
;B¼
1=I z
!
;U¼½M z ð20
Þ网上一对一辅导
Fig.2.Comparison of the respons of unmodified linear and modified linear(desired)vehicle models during a J-turn maneuver:(a)steering wheel angle, (b)lateral acceleration,(c)yaw rate,(d)side-slip angle.
M.Mirzaei/Transportation Rearch Part C18(2010)213–224217
