new comparisons of 4ws and brake-fas based on imc for vehicle … · 2016. 10. 20. · 1266 m....

Journal of Mechanical Science and Technology 25 (5) (2011) 1265~1277

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0309-z

Comparisons of 4WS and Brake-FAS based on IMC for vehicle stability control†

Men Jinlai*, Wu Bofu and Chen Jie School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai, China

(Manuscript Received September 9, 2010; Revised February 15, 2011; Accepted February 15, 2011)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract The paper proposes a multi-input-multi-output internal model control (MIMO IMC) based on combined brake and front wheel active

steering (Brake-FAS) for vehicle stability control and makes comparisons with the four wheel steering internal model control (4WS IMC). Brake control would change vehicle velocity which will make the vehicle control model nonlinear. To solve the nonlinearity in-volved in the Brake-FAS, an inverse system method is introduced to turn the nonlinear internal vehicle model into a pseudo-linear system, and then the design of main IMC controller and related filters is discussed in details. Comparisons of the Brake-FAS IMC and 4WS IMC were done on the basis of simulations which were composed of different combinations of driving maneuvers and road conditions in Simulink where an 11DOF vehicle model verified by CarSim7 was built.

Keywords: Brake-FAS; 4WS IMC; Inverse system; Pseudo-linear system ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

With the improvement of speed, vehicle active safety be-comes an unavoidable and indispensable matter which auto-makers have to deal with. In order to keep vehicle high-speed stability, more and more electronic control systems are being applied on vehicles, which could to the utmost guarantee ve-hicle’s stability especially in severe and emergency conditions such as driving on an icy road surface in winter or steering fiercely to avoid a collision with other vehicles at an ex-tremely high speed. According to related data [1], 30% acci-dents and 50% severe accidents in average are avoidable with the help of active safety equipment. Thus, the European Union has decided that all new manufactured cars must be equipped with ABS since July 2006, Australia and the United States also have issued laws or standards to guarantee all new vehi-cles to install ESP (Electric Stability Program) in 2011.

For vehicle active control, vehicle yaw rate and sideslip an-gle are usually chosen as control targets, and braking and/or active steering are usually applied to fulfill control strategies. In recent years, four-wheel-steering (4WS) control is widely studied [1-8] with the development of steer-by-wire technolo-gies. However, braking control is still indispensable and would be much more effective when combined with active steering [9-12]. In spite of the advantages of braking control, it

will introduce nonlinearity into vehicle stability control system which would make the controller design a hard work. In [13], a linearization strategy is proposed by supposing the velocity is constant. In [14], a fully integrated vehicle controller with steering and brakes is proposed, using the exact feedback lin-earization method. The feedback linearization control method amounts to canceling the nonlinearities in a nonlinear system so that the closed–loop dynamics is in a linear form.

Internal Model Control (IMC), which was first proposed by Horowits [15] and then developed by Garcia and Morari [16, 17], has been widely applied in industrial process control for its disturbance rejection and robustness even in face of inputs limitations. IMC was introduced by M. Canale et al. into the vehicle stability control system in [18, 19], but the controllers were designed mainly based on linear vehicle models which did not take the velocity changes into account, which makes sense only when there was no braking intervening in for brak-ing torque would have impact on the vehicle velocity and introduce nonlinearity into vehicle model. In order to solve the problem caused by nonlinear vehicle model, a new strategy for nonlinear system known as inverse system is adopted in Section 5.

The paper is organized into seven sections. The reference yaw rate and sideslip angle by considering limitations coming from the road surface friction coefficient are derived in Sec-tion 2 from a 2DOF vehicle model. In Section 3, estimations of vehicle velocity and sideslip angle are presented and the results are testified by CarSim7. Section 4 is mainly about the design of IMC controller for 4WS vehicle stability control. In

† This paper was recommended for publication in revised form by Associate Editor Jeong Sam Han

*Corresponding author. Tel.: +86 021 5499 5723, Fax.: +86 021 5499 5721 E-mail address: [email protected]

© KSME & Springer 2011

1266 M. Jinlai et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1265~1277

Section 5, Brake-FAS IMC is discussed in details and a brak-ing strategy is introduced together. Section 6 mainly describes the simulations based on a validated 11DOF Simulink model verified by CarSim7. In the end, conclusions and some future work are discussed.

2. Reference yaw rate and sideslip angle for vehicle stability control

The widely applied 2DOF bicycle model is introduced for the determination of reference yaw rate and sideslip angle, and lateral forces of tire are supposed in their linear regions,

yf yf fF K α=

(1)

yr yr rF K α= (2) where Kyf and Kyr are combined cornering stiffness of the front and rear tires respectively and they are negative for the sign definition of lateral forces and slip angles, αf and αr are slip angles of front and rear tires respectively.

Thus, the dynamics of the 2DOF model could be expressed as follows:

( ) /yf f yr rv u K K mγ α α+ = +

(3)

( ) /yf f f yr r r zK L K L Iγ α α= − (4)

where v, u are vehicle longitudinal and lateral velocities re-spectively, γ is vehicle yaw rate, m is vehicle mass, Iz is vehi-cle yaw moment of inertia, Lf is distance between the front axle and the center of gravity, Lr is distance between the rear axle and the center of gravity.

When the sideslip angle is small enough, vehicle longitudi-nal and lateral velocities could be simplified as:

cos( )v V Vβ= =

(5)

sin( )u V Vβ β= = (6) where V is the vehicle velocity, β is vehicle sideslip angle.

The tire slip angles could be derived from vehicle kinematics as follows:

tan( )f ff f f f f

v L V La L

u V Vγ β γ γα δ δ β δ

+ += − = − = + −

(7)

tan( )r rr r r

v L V La Lu V Vγ β γ γα δ β− −

= − = = − (8)

whereδf δr are front and rear wheel steering angles respec-tively andδr is zero for the 2DOF model.

By combining Eqs. (1)-(8), there are:

2yf yr yf f yr r yf

fK K K L K L K

mV mVmVβ β γ δ

+ −= + −

(9)

2 2

.yf f yr r yf f yr r yf ff

z z z

K L K L K L K L K LI I V I

γ β γ δ− +

= + − (10)

The steady state yaw rate of vehicle could be derived from

Eqs. (9) and (10) by setting 0β =i

and 0γ = ,

22

/

1 ( )s f

f r

yr yf

V LLm L VK KL

γ δ=+ −

(11) where L is vehicle wheelbase and L = Lf +Lr.

In addition, the vehicle lateral dynamics is limited by the road surface friction coefficient. The maximum lateral accel-eration of vehicle on certain road surface could be determined as:

maxya gµ= (12)

where aymax is vehicle maximum lateral acceleration, µ is road surface friction coefficient, g is the gravity acceleration.

As a result, the maximum of vehicle steady yaw rate is lim-ited as:

max| | .sg

Vµγ = (13)

The reference yaw rate of vehicle motion control could be

formulated as follows:

22

/ 1 ,11 ( )

( ),

f yf r r

yr yfref

f y

V L a gLm L sVK KL

g sign a gV

δ µτ

γ

µ δ µ

⎧ ≤⎪ +⎪ + −⎪= ⎨⎪⎪ >⎪⎩ (14)

Fig. 1. Reference vehicle model for determinations of the referenceyaw rate and sideslip angle, the coordinate system shown is the basisof following dynamic equations. C.O.G is short for vehicle Center-Of-Gravity.

M. Jinlai et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1265~1277 1267

where τr is time constant of the inertia and usually assigned with a value from 0.1s to 0.25s, s stands for the Laplace trans-formation.

The reference sideslip angle could be deduced from the ref-erence yaw rate as:

1refk

sβ

β

βτ

=+

(15) where kβ is a proportional coefficient which is normally very small and is set to zero for following simulations, τβ is a time constant for sideslip angle.

3. Estimation of sideslip angle and vehicle velocity

The strategy for estimation of vehicle sideslip angle is based on the lateral acceleration, longitudinal acceleration and the yaw rate which could be measured by on-board devices. Their interrelations are shown as follows:

xa V Vβγ= − (16) ya V Vβ γ= + (17)

where ax and ay are vehicle longitudinal and lateral accelera-tions respectively.

It could be rewritten into the form of state equations as:

xV V aβγ= + (18)

/ .ya Vβ γ= − (19) In the above equations, V and β are parameters to be identi-fied and the left are measurable, the main point of the pro-posed algorithm is to solve the equations. Obviously the equa-tions are nonlinear and the to-be-identified parameters are coupled with each other. To solve the nonlinear equations, one way is to linearize the equations and then to get its analytic solutions and the proposal presented is another way which relies on discretization of the equations and then comes to their numerical solutions.

( 1)

(( 1) ) ( ) ( ) ( ) (( 1) )k T

kT

x k T T x kT u kT k T Bdτ τ+

+ =Φ + Φ + −∫ (20)

where Φ (T) is the state transition matrix, T is sampling period.

Because the state transition matrix is hard to be solved out for nonlinearity of the equations, an approximate discretization is adopted.

(( 1) ) ( )( ) x k T x kTx kTT

+ −≈

(21)

The approximate discretization is of little difference with

the exact discretization when the sampling period is short enough. The results of the discretization are shown below and testified by simulations in CarSim7:

(( 1) ) ( ( ) ( ) ( ) ( )) ( )xV k T V kT kT kT a kT T V kTβ γ+ = + +

(22) (( 1) ) ( ( ) / ( ) ( )) ( ) .yk T a kT V kT kT T kTβ γ β+ = − + (23)

As the Fig. 2 shows, the errors of estimated parameters com-

pared with the outputs of CarSim7 are small enough to verify the high accuracy of proposed strategy.

4. 4WS based on IMC for vehicle stability control

Fig. 3 depicts the structure of IMC for 4WS vehicle lateral stability control. The IMC controller is composed of the main controller ( )C s and a filter ( )F s which will make up the model differences between the real vehicle and the internal model applied in the IMC control structure. ( )G s stands for the real vehicle which is substituted with a 11DOF vehicle model constructed in Matlab/Simulink based on and testified by CarSim7 with Pacejka5.2 (MF) tire model. ( )cG s is the internal 3DOF vehicle model with front and rear wheel steer-ing angles as its inputs, and the lateral forces involved in the internal model are still supposed to be linear with tire slip angles. ( , )ff Vδ is the function presented in Section 2 for determination of vehicle reference yaw rate and sideslip angle. µ is introduced to reflect the road surface conditions and to put limitations to the reference yaw rate. d is disturbances from outside just like measuring noise etc., and ( )D s is the transfer function of disturbances.

The internal model applied in IMC is a 3DOF bicycle model shown in Fig. 4 with front and rear wheel steering an-

Fig. 2. Velocity estimations (left) and vehicle sideslip angle estima-tions (right) compared with the that of CarSim7. The solid line (—) stands for that of CarSim7 and the dashed line (--) for estimations ofthe proposed strategy.

Fig. 3. Overall schematic of the proposed IMC.

Fig. 4. 3DOF vehicle model introduced for controller design.


gle as its inputs. The dynamic equations are as follows:

2yf yr yf f yr r yf yr

f rK K K L K L K K

mV mV mVmVβ β γ δ δ

+ −= + − −

(24)

2 2

.yf f yr r yf f yr r yf f yr rf r

z z z z

K L K L K L K L K L K LI I V I I

γ β γ δ δ− +

= + − + (25)

After Laplace transformation, the system transfer function

could be written as:

2 2

2 2

1 0 1 0

2 1 0 2 1 0

1 0 1 0

2 1 0 2 1 0

( )( )( )( )

f

r

a s a b s be s e s e e s e s e ss

ss c s c d s de s e s e e s e s e

δβδγ

+ +⎡ ⎤⎢ ⎥+ + + + ⎡ ⎤⎡ ⎤ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦⎢ ⎥+ + + +⎣ ⎦

(26)

where 1 z yfa I K V= − , 2

0 r yf yr f yfa LL K K mL K V= + , 1b = z yrI K V− , 2

0 f yf yr yr rb LL K K mK V L= − , 21 f yfc mL K V= − ,

0 yf yrc LK K V= , 21 yr rd mK L V= , 0 yf yrd LK K V= − ,

22 ze mI V= , 2 2

1 ( ) ( )f yf r yr yf yr ze mV L K L K K K I V= − + − + , 2 2

0 ( ) .yf yr f yf r yre L K K L K L K mV= + − According to the theories [16, 17] of IMC, the controller is

designed as the very inverse of the internal model only when the internal model is a minimum system which requires the zeros and poles of its transfer function should be in the LHP (Left-Half-Plane). For the transfer function matrix of the inter-nal vehicle model, the transfer function at the first line and first column has a zero point lying in the RHP (Right-Half-Plane), so the inverse is not applicable for it is unstable due to the RHP point. In order to solve the problem, the transfer matrix is re-vised into the minimum form as follows:

2 2

2 2

1 1 0

2 1 0 2 1 0

1 0 1 0

2 1 0 2 1 0

( )( ).

( )( )f

r

a b s be s e s e e s e s e ss

ss c s c d s de s e s e e s e s e

δβδγ

+⎡ ⎤⎢ ⎥+ + + + ⎡ ⎤⎡ ⎤ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦⎢ ⎥+ + + +⎣ ⎦

(27)

And then the IMC controller ( )C s is designed as the inverse of the revised transfer function matrix:

2 2

2 2

3 2 3 23 2 1 0 3 2 1 0

2 1 0 2 1 03 2 2

3 2 1 0 2 1 0

2 1 0 2 1 0

( ) ( )( ) ( )

f

r

aa s aa s aas aa bbs bbs bbs bbee s ees e ee s ees ees s

s scc s cc s ccs cc dd s dds ddee s ees ee ee s ees ee

δ βδ γ

⎡ ⎤+ + + + + +⎢ ⎥+ + + +⎡ ⎤ ⎡ ⎤⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥+ + + + + ⎣ ⎦⎣ ⎦ ⎢ ⎥

+ + + +⎢ ⎥⎣ ⎦

(28)

where 2 3

0 ( )f yf r yr yf yraa L K L K mLV L K K= − + 2 2

1

2 3 2

( ) ( ) /

(( ) / )f yf r yr z yr yf yf

r f yf r yr yf yr

aa mLV L K L K I K K K

L m V L K L K K L K

= − + − +

− − +

2 22

2 2

( ) /

( ( ) ( )) /z f yf r yf yf r

f yf r yr z yr yf yf

aa mI V L K L K K mL V

m L K L K I K K K

= + +

+ + +

2 33 /z r yfm I L Vaa K= −

2 20

2

( )(( )

) / /f yf r f yf r yr

yf yr yf

bb LL K mL V L K L K mV

L K K K V

= − −

+

2 2 21

2 2

(( )(

) (( ) )) /f yf r f yf r yr z yr

z yf z f yf r yr yf yr yf

bb LL K mL V mL K mL K I K

I K I L K L K mV L K K K

= − − − −

− − − +

2 2 2 22

2

(( ) ( (

) ( ))) / /f yf r z z f yf

r yr z yr yf yf

bb LL K mL V mV I I V m L K

L K I K K K V

= − +

+ + +

2 23 /z yfmV Ibb K−=

2 20

2

2

( )

( ) / /f yf r yr yf yr

r f yf yf

mV L K mV L K L Kc K

mV L L K K V

c

L

− −= +

−

2 2

2 2

2 2 2

1 (( )

( )

( )) /

z yr z yf f yf r yr

f yf r f r yf

z f yf r yr yf yr yf

I K I K mL K mL K

L K mV L L L K

I mV L K mV L K L K

c

K

c

K

− − − − −

− + −

+

=

− +

2

2

22

2

( ( )

( )) /z r f yf f r yf

z z yr z yf f yf r yr yf

mI V mV L L K L L K

VI I K I K mL K mL K

c

K

c −

+ − −

= − −

− −

2 23 /z yfmV Icc K−=

2 20 (( ) ) /f yf r yr yf yr z yrL K L K mV K K L Idd K− +=

2 21 ( ) /f yf r yr z yr z yf z yrmL K mL K I K I V Kdd K I− − −= −

2 22 /z yrd Vd mI K=

2 2 20 ( )yf z r r f yr f yf yree LK I V L L L K mV L L K K= − + − − +

2 2 2 31 ( )z yr f yf r z r fee I K mL K LV L I mV m L L V= − − + +

22 .z fmIe Ve L=

For the main IMC controller ( )C s , a first-order filter ( )F s

is applied and designed as follows:

11

( )1

1

sF s

s

β

γ

λ

λ

⎡ ⎤⎢ ⎥+⎢ ⎥= ⎢ ⎥⎢ ⎥

+⎢ ⎥⎣ ⎦

(29)

where βλ and γλ are time constants which could adjust the IMC controller’s performance.

5. Brake-FAS based on IMC for vehicle stability con-trol

5.1 Brake strategy design

To determine which wheel to brake in Brake-FAS vehicle stability control system during the steering maneuver, tire forces characteristics are analyzed first. As shown in Fig. 5, with the increase of braking torque the lateral forces are de-creasing, based on which signs and changing trends of yaw moment under longitudinal control are depicted in Table 1,


where it describes the left turn maneuvers. In Table 1, LF, RF, LR and RR stand for left-front, right-front, left-rear and right-rear wheels respectively; minus of longitudinal forces means that the forces are caused by braking torque on wheels; plus of lateral forces means that the forces have the same direction with the vehicle’s steering direction; plus of yaw moment means enhancement of yaw moment to the steering maneu-vers while minus for the reduction. A detailed description for table1 could be as follows: When the vehicle steers to the left,

(1) Lateral forces of front wheels namely left-front (LF) and right-front (RF) wheels will generate plus yaw moment with the same direction with steering direction, while lateral forces of rear wheels that are left-rear (LR) and right-rear (RR) wheels will produce minus yaw moment to weaken the steer-ing effect.

(2) Longitudinal forces due to the braking control on left side wheels namely left-front (LF) and left-rear (LR) wheels will result in plus yaw moment to enhance steering maneuvers, while braking control on right side wheels which implies right-front (RF) and right-rear (RR) wheels will cause minus yaw moment.

(3) Longitudinal control forces due to braking on wheels will decrease the lateral forces.

(4) Braking LF wheel will increase the longitudinal-force-caused yaw moment and decrease the lateral-force-caused yaw moment which are both plus and will come to an uncer-tainty of the total yaw moment; braking RF wheel will in-crease the minus yaw moment due to longitudinal forces and decrease the plus yaw moment due to lateral forces which will

jointly weaken the steering effect; braking LR wheel will in-crease plus yaw moment by longitudinal forces and decrease minus yaw moment by lateral forces which comes to enhance the steering maneuver together; braking RR wheel will in-crease yaw moment of longitudinal forces and decrease yaw moment of lateral forces which are both minus and would come to an uncertainty effect.

When steering to the right, there are similar results. Thus, the following conclusions could be drawn,

(1) When steering to the left and the vehicle is understeering which means the yaw moment is smaller than the desired, the LR wheel should be braked to generate total plus yaw moment to enhance the steering effect; and when the vehicle is over steering which means the yaw moment is larger than expected, the RF wheel should be applied on a braking torque to give birth to a total minus yaw moment to weaken the over-steering-ability.

(2) Similarly, when steering to the right and the vehicle is understeering, the RR wheel should be controlled with brak-ing torques, and when the vehicle is over steering, the LF wheel should be brake. 5.2 Dynamics of Brake-FAS controls

The vehicle model introduced for combined Brake-FAS control is of 4DOF (degree-of-freedom) depicted in Fig. 4. As discussed above, there is just one wheel needed control under different conditions, thus the model in Fig. 4 is of 4 degree of freedom which is composed of vehicle longitudinal and lateral kinematics, front wheel steering and one wheel braking. The related dynamic equations are shown as follows:

xFV Vm

βγ= +

(30)

2yf yr yf f yr r yf

fK K K L K L K

mV mVmVβ β γ δ

+ −= + −

(31) 2 2

,

2yf f yr r yf f yr r yf f x left

fz z z

K L K L K L K L K L F dI I V I

γ β γ δ− +

= + − − (32)

or 2 2

,

2yf f yr r yf f yr r yf f x right

fz z z

K L K L K L K L K L F dI I V I

γ β γ δ− +

= + − + (33)

where xF stands for braking force of wheels which is defined negative, left and right for left wheels and right wheels respectively.

For simplicity, the braking force is supposed to be linear with the slip ratios as follows:

x xF K λ=

(34)

where xK is longitudinal stiffness, and λ is slip ratio de-fined for braking process as:

w

w

V RV

ωλ −=

(35)

Table 1. Yaw moment changes with longitudinal control interference.

Sign and Changes of yaw moment due to longitudinal

and lateral forces Control Wheel

Sign and Trends of Longitudinal

Control Forces

Sign and Trends of Lateral

Forces Mz_Fx Mz_Fy ∑Mz

LF -↑ +↓ +↑ +↓ + ～

RF -↑ +↓ -↑ +↓ +↓

LR -↑ +↓ +↑ -↓ +↑

RR -↑ +↓ -↑ -↓ - ～

Fig. 5. Tire lateral forces accompanied with different slip ratios caused by braking torques.


where R stands for wheel radius, ω is wheel angular veloc-ity, wV is wheel linear speed which is approximately equal to the vehicle velocity V , thus tire slip ratio could be rewritten as:

.V RV

ωλ −=

(36)

By combining above equations and just considering the left

wheel braking situations, which are similar to the right side braking, there are:

xKV Vm

βγ λ= +

(37)

2yf yr yf f yr r yf

fK K K L K L K

mV mVmVβ β γ δ

+ −= + −

(38)

2 2

.2

yf f yr r yf f yr r yf f xf

z z z z

K L K L K L K L K L K dI I V I I

γ β γ δ λ− +

= + − − (39)

Obviously, the dynamic equations are nonlinear mainly due

to the coupled state parameter V . Let 1 2 3[ , , ] [ , , ]T Tx x x x V β γ= = , 1 2[ , ] [ , ]T T

fu u u δ λ= = , 1 2[ , ] [ , ]T Ty y y β γ= = , then above equations could be

changed into state-space forms:

1 1 2 2( ) ( ) ( )x f x g x u g x u= + +

(40)

( )y h x=

(41)

where

1 2 3

2 32

1 12 2

32

1

( ) yf yr yf f yr r

yf f yr r yf f yr r

z z

x x xK K K L K Lx xf x

m x m x

K L K L K L K L xxI I x

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥+ −⎢ ⎥= +⎢ ⎥⎢ ⎥

− +⎢ ⎥+⎢ ⎥⎣ ⎦

,

1

0

( ) yf

yf f

z

Kg x

mVK L

I

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= −⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

, 2 ( ) 0

2

x

x

z

Km

g xK d

I

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

, 0 1 0

( )0 0 1

h x⎡ ⎤

= ⎢ ⎥⎣ ⎦

The nonlinear dynamics involved are special known as af-fine nonlinear equations and they could be solved through exact linearization when the relative order equals to the sys-tem dimension. As to the specific equations here, the relative order is two and the system dimension is three, thus the exact linearization could not be applied. To solve the problem, the inverse system method is introduced. 5.3 Inverse system theory

Definition1: Suppose a nonlinear or linear system ∑ with ( )u t as its inputs and ( )y t as its outputs and the cause-and-

effect relationship between the inputs and outputs is described with a factor θ and then the system could be expressed with a definite initial conditions 0 0( )x t x= as:

0: ( ) [ , ( )] .y x uθ∑ ⋅ = ⋅

There is another system Π with an initial condition of

0 0( )x t x= commonly decided by 0x , and a cause-and-effect relationship factor of ˆ : d dy uθ → , where dy is a function with a derivative order of n and satisfies a certain initial condi-tion determined by 0x , if the following equation is satisfied:

ˆ .d d d dy u yθθ θ= =

Then, Π is defined as the inverse system of ∑ and ∑ is called the original system.

Definition2: αΠ is a third system with an initial condition of 0 0( )x t x= commonly determined by 0x , and a cause-and-effect relationship factor of ˆ : uαθ ϕ → . When

( ) ( )dy tαϕ = , if there is:

ˆ ˆ .dd

d y u ydt

α

α αθθ ϕ θθ θ= = =

Fig. 6. 4DOF vehicle model for brake-FAS control.

Fig. 7. Schematic of the inverse system control by constituting a pseudo-linear system.

Fig. 8. Pseudo-linear system composed of inverse system and original system.


Then, αΠ is defined as αth order integral inverse system of ∑ .

Definition3: The system composed of ∑ and its αth or-der integral inverse system αΠ as follows:

ˆ : d ydt

α

αθθ ϕ=

is defined as a pseudo-linear system which is characterized with a linear relationship between its inputs and outputs but maybe some nonlinearities internal.

The principle of inverse system theory is to solve out the inverse system Π with determined initial conditions and then the αth order integral inverse system αΠ to generate the pseudo-linear system:

ˆ : d ydt

α

αθθ ϕ=

and based on the linear relations of its inputs and outputs to derive a proper control strategy. The overall schematic of inverse system control is depicted in Fig. 5, where r stands for the reference inputs, rθ is a linear controller, x is state vector of systemθ .

As to the Brake-FAS vehicle stability control system with the state space expressed by Eqs. (40) and (41), its inverse system could be solved out as:

3

1 1 3 2 1 112 2

yf yr yf f yr r

yf yf yf yf

K K K L K Lm x mu x x x x yK K K x K

+ −=− + + −

(42)

32 1 3 2 1 1 2

1

2 4 4 2 2f yr yr r f z

x x x x x

mL K L K L L mLx Iu x x x x y yK d K d K d x K d K d

= − + + −

(43)

Let 1 1yϕ = and 2 2yϕ = , then the 1st order integral in-

verse system is obtained as:

31 1 3 2 1 1

12 2yf yr yf f yr r

yf yf yf yf

K K K L K Lm x mu x x x xK K K x K

ϕ+ −

=− + + −

(44)

32 1 3 2 1 1 2

1

2 4 4 2 2 .f yr yr r f z

x x x x x

mL K L K L L mLx Iu x x x xK d K d K d x K d K d

ϕ ϕ= − + + −

(45)

And then, the pseudo-linear system could be derived by con-necting the original system and the 1st order integral system in series. As the Fig. 8 shows that after the transformation the system could be changed into an integral system which is much simplified and easy to control. 5.4 IMC controller for Brake-FAS vehicle stability control

As discussed in Section 5.3, the pseudo-linear system is equivalent to an integral system with zero poles which will intend to become unstable with disturbances. Thus, in order to improve its performance a feed back is added as shown in Fig. 9.

The schematic of Brake-FAS IMC vehicle stability control is depicted in Fig. 10 which is of a little difference compared with the structure for 4WS IMC. As for the 4WS IMC, the internal model applied is a definite and linear 3DOF bicycle model, while for Brake-FAS IMC the internal model is a feed-back pseudo-linear system composed of a 4DOF nonlinear vehicle model and its 1st order integral inverse system which is much more complicated than that of 4WS IMC. And the main con-troller ( )C s is the inverse of the 3DOF bicycle model for 4WS IMC and the inverse of the feed-back pseudo-linear sys-tem for Brake-FAS IMC. Besides, the controller outputs are reference inputs for the pseudo-linear system and not applied directly on the actual vehicle ( )G s but go through the 1st order integral inverse system α̂θ to generate the control in-puts namely front wheel steering angle fδ and slip ratio λ which would be transformed into the braking torque bT by a nonlinear function, and then the fδ and bT will take effect on the actual vehicle.

The other components left in Fig. 10 are similar to that of Fig. 4. The IMC controller is composed of the main controller

( )C s and a filter ( )F s which will make up the model dif-ferences between the real vehicle and the internal model ap-plied in the IMC control structure. ( )G s stands for the real vehicle which is substituted with a 11DOF vehicle model constructed in Matlab/Simulink based on and testified by Car-Sim7 with Pacejka5.2 (MF) tire model. ( , )ff Vδ is the func-tion presented in Section 2 for determination of vehicle refer-ence yaw rate and sideslip angle. µ is introduced to reflect the road surface conditions and to put limitations to the refer-ence yaw rate. d is disturbances from outside just like measuring noises etc., and ( )D s is the transfer function of disturbances.

For the Brake-FAS IMC, the main controller ( )C s and the filter ( )F s are designed respectively as follows:

1

( )1

sC s

s+⎡ ⎤

= ⎢ ⎥+⎣ ⎦

(46)

11

( )1

1

sF s

s

β

γ

λ

λ

⎡ ⎤⎢ ⎥′ +⎢ ⎥= ⎢ ⎥⎢ ⎥′ +⎢ ⎥⎣ ⎦

(47)

where βλ′ and γλ′ are time constants just like that of 4WS

Fig. 9. Schematic of pseudo-linear system with feedback.


IMC which could adjust the IMC controller’s performance. The nonlinear transfer function from slip ratio λ into brak-

ing torque bT could be deduced from wheel dynamics. The wheel rotation dynamics could be described as follows:

x b

w

F R TI

ω −=

(48)

where ω is wheel angular velocity, xF is longitudinal force generated from braking torque, R is wheel radius, bT is braking torque and wI is wheel spin inertia moment.

From Eq. (36), the derivative of slip ratio could be expressed as:

2 .RV R VVω ωλ − +

=

(49)

By combining Eqs. (37), (48) and (49), the braking torque is

obtained as follows:

2( ) .w w w x w wb x

I I I K I IT V V K R VR R mR mR R

βγ βγλ λ λ λ= − − + + −

(50)

6. Simulations

6.1 Driver model

According to ISO/TR388 [20] and based on a reference tra-jectory [21], the reference yaw angle of vehicle during the dou-ble lane change is depicted in Fig. 12, and then a simple driver model [22, 23] is derived for the double lane change maneuver as:

* ( )

1d

f refd

ks

δ ψ ψτ

= −+

(51)

where refψ is the reference yaw angle of vehicle in the double lane change process, ψ is vehicle actual yaw angle, dk is

the driver gain with a value of 7.7 and dτ is driver time con-stant with a value of 0.08 in the following simulations.

The driver model is introduced in order to compare the con-trol results of 4WS IMC with that of Brake-FAS IMC, thus the simple driver model which could fulfill the purpose is chosen instead of a complex one. 6.2 Simulations and analysis

In order to test the proposed control strategy CarSim7 was introduced as a benchmark, and an 11DOF vehicle model was built in Matlab/Simulink and validated by CarSim7 as Fig. 14 shows. The vehicle parameters are listed in Table 2.

As shown in Fig. 14, the differences between the Mat-lab/Simulink model and CarSim7 are small enough to testify the 11DOF-model’s capability to substitute CarSim7 to some extent, and it means that simulations based on the Simulink model are credible.

Considering about the high velocity of vehicle, steering an-

Fig. 10. Overall structure of Brake-FAS IMC.

Fig. 11. Tire rotational dynamics.

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

50 70 90 110 130 150 170 190

X(m)

Yaw Angle(rad)

Ψref(rad)

Fig. 12. Reference yaw angles derived for the ISO double lane change.

Fig. 13. 11DOF Matlab/Simulink vehicle model introduced in follow-ing simulations.

Fig. 14. Yaw rates (left) and vehicle sideslip angles (right) of CarSim7 and the 11DOF Simulink model with the same front wheel steering angles. The solid line (—) stands for the yaw rate given out by Car-Sim7, and the dotted line (--) by 11DOF Simulink model.


gle limits were imposed on both of the front wheel steering and rear wheel steering with 7º and 2º, respectively.

To test the effectiveness of the Brake-FAS IMC based on pseudo-linear system and to make thorough and detailed com-parisons with the 4WS IMC based on a 3DOF bicycle model, several driving maneuvers were introduced such as double lane change, sine steering angle inputs and step steering inputs. Besides, different road conditions which were mainly in forms of low road surface friction coefficient and combination of low and high road surface frictions were applied accompanying with selected driving maneuvers. Another test introduced white noises into the measured vehicle yaw rate to verify the distur-bance rejection characteristics of the proposed control strate-gies.

6.2.1 Sine steering tests with magnitude of 5º and period of

2πs on low road surface with friction coefficient µ=0.4 and time constants λβ=0.25, λγ=0.5, λ´β=10, λ´γ=0.3

As depicted in Fig. 15, the reference yaw rate of Brake-FAS IMC was different from that of 4WS IMC, which was aroused by the changes of vehicle velocity caused by interference of braking torque under Brake-FAS IMC. When there was a brake torque applied on certain wheels, the vehicle would slow down reflected by the decrease of velocity which would come to the increase of reference yaw rate as shown in the determi-nation of reference yaw rate in section 2. Another phenomenon about the yaw rates was that the response rate of vehicle under 4WS IMC was slower than that of Brake-FAS IMC which well illustrated different characteristics of braking yaw moment control and steering yaw moment control. As discussed in sec-tion 5.1, the braking torque would cause both the longitudinal and lateral forces to contribute to control yaw moment with the same trend to increase or decrease the yaw moment jointly, while the 4WS IMC could only made use of the wheel steering angles to generate the control yaw moment.

In addition, the active front wheel steering angles of Brake-FAS IMC were larger than that of 4WS IMC as shown in Fig. 16, which also made greater contributions to the tracking rate. But the faster response rate of Brake-FAS IMC would be at the cost of bigger sideslip angles as depicted in Fig. 15 (upper right). In addition, the braking torque on certain wheels would arouse oscillations for both lateral and longitudinal acceleration, because the braking torque was made up of many oscillations as shown in Fig. 16. The lateral friction coefficients for Brake-FAS IMC fell into the nonlinear regions of tire characteristics

Table 2. Vehicle and tire parameters for simulations. Parameters Values Description

m 1530kg Vehicle mass

Iz 4192kg•m² Yaw inertia of vehicle

Lf 1.11m Distance between front axle and vehicle center of gravity

Lr 1.666m Distance between rear axle and vehicle center of gravity

L 2.776m Wheelbase

d 1.55m Track width

R 0.298m Wheel radius

Iw 0.9kg•m² Spin inertia of wheel

V 120km/h Initial velocity of vehicle

Kf -40849.6N/rad Cornering stiffness of single front wheel

Kr -31148.3N/rad Cornering stiffness of single rear wheel

Kxf 49573N Longitudinal stiffness of single front wheel

Kxr 32488N Longitudinal stiffness of single rear wheel

g 9.8m/s² Gravity acceleration

Fig. 15. Yaw rates (upper left), sideslip angles (upper right), lateral accelerations (lower left) and longitudinal accelerations (lower right) of Brake-FAS and 4WS IMC for vehicle stability control with sine front wheel steering angles on the road surface with friction coefficient µ=0.4. Upper left: the solid line (—) stands for reference yaw rate for Brake-FAS IMC control, dashed line (--) for yaw rates of vehicle under Brake-FAS IMC control, center line (-•-) for reference yaw rate for 4WS IMC control, and dotted line (••) for yaw rates of vehicle under 4WS IMC control. Upper right: solid line (—) stands for reference sideslip angle which is set to zero in the simulations, dashed line (--) for sideslip angles of vehicle under Brake-FAS IMC control, and dot-ted line (••) for that of vehicle under 4WS IMC control. Lower left: solid line (—) stands for lateral accelerations of vehicle under Brake-FAS IMC control, and dashed line (--) for that of vehicle under 4WS IMC control.

Fig. 16. Steering angles (upper left) for both Brake-FAS and 4WS IMC controls, Braking torques (upper right) for Brake-FAS IMC control, lateral coefficients (lower left) of tires for vehicle under Brake-FAS IMC and longitudinal accelerations (lower right) of tires for vehicle under 4WS IMC for vehicle stability control with sine front wheel steering angles on the road surface with friction coefficient µ=0.4. Upper left: the solid line (—) stands for original sine front wheel steer-ing angles, dashed line (--) for front wheel steering angles of vehicle under Brake-FAS IMC control, center line (-•-) for front wheel steering angles of vehicle under 4WS IMC control, and dotted line (••) for rear wheel steering angles of vehicle under 4WS IMC control.


while the 4WS IMC could still keep its tire lateral characteris-tics linear as in Fig. 16.

6.2.2 Step steering tests of 5º with a steering rate of 100º/s on

road surface with µ=0.4 and time constants λβ=0.25, λγ=0.5, λ´β=1000, λ´γ=1.5

As shown in Fig. 17, the response rate of Brake-FAS IMC was still faster than that of 4WS IMC as analyzed in section 6.2.1, and bigger sideslip angles would pay for the faster track-ing rate of Brake-FAS IMC. But the front wheel steering an-gles of Brake-FAS IMC were no bigger than that of 4WS IMC which was different from section 6.2.1 since the time constant λ´β was so big that it would make the front wheel steering an-gles very insensitive to steering maneuvers. Similarly, the os-cillations in braking torques were responsible for that in yaw rates of Brake-FAS IMC.

6.2.3 Double lane change with the driver model on road surface with µ=0.4 and time constants λβ=0.25, λγ=0.5, λ´β=20, λ´γ=0.3

The Brake-FAS IMC came to better yaw rate tracking and better trace following than 4WS IMC as shown in Fig. 18 and 19. However, the overshot of yaw rates for Brake-FAS IMC was a little bigger than that for 4WS IMC, which was the same

Fig. 17. Yaw rates (upper left), sideslip angles (upper right), steeringangles (lower left) and braking torques (lower right) of Brake-FAS and 4WS IMC for vehicle stability control with step front wheel steering angles on the road surface with friction coefficient µ=0.4. Upper left: the solid line (—) stands for reference yaw rate for Brake-FAS IMC control, dashed line (--) for yaw rates of vehicle under Brake-FAS IMC control, center line (-•-) for reference yaw rate for 4WS IMC control, and dotted line (••) for yaw rates of vehicle under 4WS IMC control. Upper right: solid line (—) stands for reference sideslip angle which isset to zero in the simulations, dashed line (--) for sideslip angles ofvehicle under Brake-FAS IMC control, and dotted line (••) for that ofvehicle under 4WS IMC control. Lower left: the solid line (—) stands for original step front wheel steering angles, dashed line (--) for front wheel steering angles of vehicle under Brake-FAS IMC control, centerline (-•-) for front wheel steering angles of vehicle under 4WS IMCcontrol, and dotted line (••) for rear wheel steering angles of vehicleunder 4WS IMC control.

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350

X(m)

Y(m

)

Y_IMC_Brk+FASY_IMC_4WS

Fig. 18. Trajectories of vehicle under control of Brake-FAS and 4WS.The solid line (—) stands for that of vehicle under Brake-FAS IMC control, and dashed line (--) for that of 4WS IMC control.

Fig. 19. Yaw rates (upper left), sideslip angles (upper right), lateral accelerations (lower left) and longitudinal accelerations (lower right) of Brake-FAS and 4WS IMC for vehicle stability control to fulfill double lane change with the same driver model on the road surface with friction coefficient µ=0.4. Upper left: the solid line (—) stands for reference yaw rate for Brake-FAS IMC control, dashed line (--) for yaw rates of vehicle under Brake-FAS IMC control, center line (-•-) for reference yaw rate for 4WS IMC control, and dotted line (••) for yaw rates of vehicle under 4WS IMC control. Upper right: solid line (—) stands for reference sideslip angle which is set to zero in the simula-tions, dashed line (--) for sideslip angles of vehicle under Brake-FAS IMC control, and dotted line (••) for that of vehicle under 4WS IMC control. Lower left: solid line (—) stands for lateral accelerations of vehicle under Brake-FAS IMC control, and dashed line (--) for that of vehicle under 4WS IMC control.

Fig. 20. Steering angles (upper left) for both Brake-FAS and 4WS IMC controls, Braking torques (upper right) for Brake-FAS IMC control, lateral coefficients (lower left) of tires for vehicle under Brake-FAS IMC and longitudinal accelerations (lower right) of tires for vehicle under 4WS IMC for vehicle stability control to fulfill double lane change with the same driver model on the road surface with friction coefficient µ=0.4. Upper left: the solid line (—) stands for original front wheel steering angles in Brake-FAS IMC determined by the driver model, dashed line (--) for front wheel steering angles of vehicle under Brake-FAS IMC control, center line (-•-) for original front wheel steering angles in 4WS IMC determined by driver model, double dot-ted line (-••-) for front wheel steering angles of vehicle under 4WS IMC control, and dotted line (••) for rear wheel steering angles of vehicle under 4WS IMC control.


with sideslip angles and the front wheel steering depicted in Fig. 20. All these facts could be explained for the interference of control braking torque which was introduced to achieve good reference tracking while also arouse oscillations in yaw rates, lateral and longitudinal accelerations and made the tire lateral characteristics fall out of its linear regions. By contrast, the 4WS IMC made the vehicle track well with the desired trajectories and yaw rates with little steering angles both for front and rear wheel which could maintain the tire linearity.

6.2.4 Double lane change with white noise interfered with

the yaw rate on road surface with µ=0.4 and time con-stants λβ=0.25, λγ=0.5, λ´β=20, λ´γ=0.45

Figs. 21 and 22 well show off the disturbance rejection for

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350

X(m)

Y(m

)


Fig. 21. Reference yaw angles derived for the ISO double lane change.

Fig. 22. Yaw rates (upper left), sideslip angles (upper right), steering angles (lower left) and braking torques (lower right) of Brake-FAS and 4WS IMC for vehicle stability control to fulfill double lane change with white noises interfered with the yaw rate on the road surface withfriction coefficient µ=0.4. Upper left: the solid line (—) stands for reference yaw rate for Brake-FAS IMC control, dashed line (--) for yaw rates of vehicle under Brake-FAS IMC control, center line (-•-) for reference yaw rate for 4WS IMC control, and dotted line (••) for yaw rates of vehicle under 4WS IMC control. Upper right: solid line (—) stands for reference sideslip angle which is set to zero in the simula-tions, dashed line (--) for sideslip angles of vehicle under Brake-FAS IMC control, and dotted line (••) for that of vehicle under 4WS IMCcontrol. Lower left: the solid line (—) stands for original front wheel steering angles in Brake-FAS IMC determined by the driver model, dashed line (--) for front wheel steering angles of vehicle under Brake-FAS IMC control, center line (-•-) for original front wheel steeringangles in 4WS IMC determined by driver model, double dotted line (-••-) for front wheel steering angles of vehicle under 4WS IMC control,and dotted line (••) for rear wheel steering angles of vehicle under4WS IMC control.

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10

Time(s)

Fric

tion

Coe

ff

Fig. 23. Changes of friction coefficient during double lane changing maneuvers with the proposed driver model.

Fig. 24. Yaw rates (upper left), sideslip angles (upper right), steering angles (lower left) and braking torques (lower right) of Brake-FAS and 4WS IMC for vehicle stability control to fulfill double lane change with road surface friction coefficient µ changes. Upper left: the solid line (—) stands for reference yaw rate for Brake-FAS IMC control, dashed line (--) for yaw rates of vehicle under Brake-FAS IMC control, center line (-•-) for reference yaw rate for 4WS IMC control, and dot-ted line (••) for yaw rates of vehicle under 4WS IMC control. Upper right: solid line (—) stands for reference sideslip angle which is set to zero in the simulations, dashed line (--) for sideslip angles of vehicle under Brake-FAS IMC control, and dotted line (••) for that of vehicle under 4WS IMC control. Lower left: the solid line (—) stands for original front wheel steering angles in Brake-FAS IMC determined by the driver model, dashed line (--) for front wheel steering angles of vehicle under Brake-FAS IMC control, center line (-•-) for original front wheel steering angles in 4WS IMC determined by driver model, double dotted line (-••-) for front wheel steering angles of vehicle under 4WS IMC control, and dotted line (••) for rear wheel steering angles of vehicle under 4WS IMC control.

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150 200 250 300 350X(m)

Y(m

)


Fig. 25. Trajectories of vehicle under control of Brake-FAS and 4WS with white noises interfered with vehicle yaw rates. The solid line (—) stands for that of vehicle under Brake-FAS IMC control, and dashed line (--) for that of 4WS IMC control.


both Brake-FAS IMC and 4WS IMC which had come to little changes in the control angles and braking torques, and the tracking efficiency for yaw rates and trajectories were almost the same with that without disturbances as in section 6.2.3.

6.2.5 Double lane change on road surface with µ changes

and time constants λβ=0.25, λγ=0.5, λ´β=25, λ´γ=0.38 As shown in Figs. 24 and 25, in the face of friction coeffi-

cient changes in Fig. 23 both the Brake-FAS and 4WS IMC could come to good trajectories tracking, but they could not any more maintain the efficiency of yaw rate following as in above simulations. In addition, the sideslip angles of Brake-FAS IMC were very big, the front wheel steering angles ar-rived at the limitations preset for safety consideration and espe-cially the braking torques were as great as 10,000 Nm which would be tough for the actual hydraulic executive valves, while for 4WS IMC all the control inputs namely the front and rear wheel steering angles could keep within its limits all the time.

7. Conclusions

Through the simulations several conclusions could be drawn as follows:

(1) Brake+FAS IMC could achieve a faster response rate but was easy to generate bigger sideslip angles, while 4WS IMC could keep small sideslip angles by adjusting the front and rear wheel steering angles. For the sake of stability which requires small sideslip angles 4WS IMC was better than Brake-FAS IMC.

(2) Fixed time constants λβ, λγ for 4WS IMC were suitable for almost all kinds of driving maneuvers and road surface conditions which should owe to the applied linear internal ve-hicle model. While time constants λ´β, λ´γ needed to be ad-justed according to different driving maneuvers and road sur-face conditions which was due to the nonlinearity of the inter-nal vehicle model.

(3) Both the two kinds of control showed good disturbances rejection as white noises interfered with the vehicle yaw rate which was a merit of IMC.

In order to achieve good efficiency, the time constants in Brake-FAS need to be renewed according to different condi-tions, and a further research about how to adjust the two pa-rameters should be done in the future.

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Men Jinlai received his B.S. and M.S. in Vehicle Engineering from Tongji University and Shanghai Jiaotong Uni-versity, China in 2005, 2007, respec-tively. Men Jinlai is currently a Ph.D. in Vehicle Engineering in Shanghai Jiaotong University, China. His research fields are vehicle active safety, hybrid

vehicle and flowmeter for measuring dynamic oil consump-tion of engines.

new comparisons of 4ws and brake-fas based on imc for vehicle … · 2016. 10. 20. · 1266 m....

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