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Coordinated control of AFS and DYC for

vehicle handling and stability based on

optimal guaranteed cost theoryXiujian Yang

a, Zengcai Wang

a& Weili Peng

a

aVehicular Institute of Mechanical Engineering Department ,

Shandong University , Jinan City, People's Republic of China

Published online: 31 Oct 2008.

To cite this article:Xiujian Yang , Zengcai Wang & Weili Peng (2009) Coordinated control of AFS

and DYC for vehicle handling and stability based on optimal guaranteed cost theory, Vehicle

System Dynamics: International Journal of Vehicle Mechanics and Mobility, 47:1, 57-79, DOI:10.1080/00423110701882264

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Vehicle System Dynamics

Vol. 47, No. 1, January 2009, 5779

Coordinated control of AFS and DYC for vehicle

handling and stability based on optimal guaranteed

cost theory

Xiujian Yang*, Zengcai Wang and Weili Peng

Vehicular Institute of Mechanical Engineering Department, Shandong University, Jinan City,

Peoples Republic of China

(Received 16 July 2007; final version received 21 December 2007)

Considering the uncertainty of tyre cornering stiffness due to the frequent variation of running con-ditions, a new coordination scheme is proposed based on optimal guaranteed cost control techniqueby coordinating active front steering and direct yaw moment control. A general procedure to developan optimal guaranteed cost coordination controller (OGCC) is presented, and the effect of uncertaintydeviation magnitude on the control system is discussed. An optimal coordination (OC) scheme basedon LQR is also presented. Many simulations are carried out on an 8-DOF nonlinear vehicle modelfor a slalom manoeuvre and a lane-change manoeuvre, respectively. The simulation results show thatthe OGCC scheme has superior stability and tracking performances at different running conditions

compared with the OC scheme.

Keywords: active front steering; direct yaw moment; vehicle stability control; optimal guaranteedcost control; coordinated control; vehicle dynamics

1. Introduction

In the past two decades, vehicle chassis control system as the important part of vehicle active

safety control has made great progress, such as four wheel steering (4WS), vehicle stability

control system (VDC/ESP/VSC), active front steering (AFS), etc. All the systems can improve

the handling or the stability performance obviously in a certain region. The vehicle stability

relies on the balance of the front and rear tyre cornering forces. In detail, when the front tyre

cannot provide the cornering force, the vehicle will lead to drift out and loss of steerability; and

when the rear tyre cornering force reaches saturation, the vehicle will lead to spin out and loss

of stability. When the lateral acceleration is small, the tyre cornering force is approximately

proportional to the tyre slip angle; but when the lateral acceleration increases to a certain

value, the proportional relationship will no longer exist because of the saturation property of

*Corresponding author. Email: [email protected]

ISSN 0042-3114 print/ISSN 1744-5159 online 2009 Taylor & FrancisDOI: 10.1080/00423110701882264http://www.informaworld.com


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58 X. Yanget al.

the tyre. Therefore, 4WS and AFS, which depend on the lateral tyre force greatly, are mainly

effective in the linear region of the tyre. Vehicle stability control systems such as VDC, ESP

or VSC mainly use the active yaw moment generating from the difference of the longitudinal

tyre forces by driveline or braking to keep the vehicle stable, which is also called direct yaw

moment control (DYC). 4WS and AFS can effectively improve the steerability performance

in the linear region of the tyre. However, DYC can keep the vehicle stable in critical situations

where the tyre cornering force reaches saturation [15]. Therefore, each individual chassis

control system has a certain operating region. The vehicle handling and stability performances

canbe enhanced in all driving conditions by coordinatingthe individual chassis control systems

exertingthe advantage of each subsystem.Along withthe developments ofAFS andDYC, some

researchers investigate the integration of steering and braking to enhance vehicle dynamics.

Nagaiet al. [6,7] propose a coordination scheme that is composed of a steering angle-based

feedforward controller and an optimal state feedback controller. Boada et al. [8] design a

control scheme by integrating front steering and front wheel braking using fuzzy logic control.

In [9], steering and braking are coordinated by rules designed beforehand based on a modelregulator to enhance the yaw dynamics. In [10], the vehicle lateral dynamics control is regarded

as a multi-input and multi-output system control problem and an integration of steering and

braking scheme is presented using feedback linearisation technique.

As an important parameter in the vehicle dynamic control system, tyre cornering stiffness is

affected by many aspects (e.g. vehicle weight, adhesion coefficient, etc.), which is a disadvan-

tage for a model-following based vehicle stability controller. From the open-public literature, it

is easy to find that most of the model-following based vehicle stability controllers are designed

using a certain constant for the tyre cornering stiffness parameter [7,11,12]. Though some

researchers considered the uncertainty of the parameter in the controller design, the robust-

ness or stability performance of the closed-loop system is the primary objective [1316]. Ono[3,13] and Mammar [14] design a robust steering controller based on Htheory to reduce the

influence of the tyre cornering stiffness uncertainty on the system performance. Since robust

performance is the design objective for a robust controller, some other performances of the

system may not be guaranteed. In [15], an uncertain TS fuzzy model is founded to handle

the tyre cornering stiffness uncertainty when designing a 4WS stability controller. Though

quadratic optimal control based vehicle stability controller considers the tracking error and

the control input simultaneously, which is more suitable for realistic application, it may lose

stability and cannot obtain the optimal performance when the tyre cornering stiffness varies

in a large range for the change of running conditions. Fortunately, optimal guaranteed cost

control theory that can obtain a relative optimal performance for a system with norm-bounded

time-varying parameter uncertainties provides a good means to solve the problem. For all

the norm-bounded time-varying parameter uncertainties, optimal guaranteed cost control can

not only keep the closed-loop system stable but maintain the given quadratic cost function

within a certain bound [16]. In the past, the solution of the optimal guaranteed cost problem

was difficult, but the situation has been changed since linear matrix inequality (LMI) tool-

box of Matlab appeared. Thus the solution of optimal guaranteed cost problem is equivalent

to the solution of a set of LMIs. In this paper, the coordination of AFS and DYC based on

optimal guaranteed cost control theory is presented to reduce the influence of the variation

of tyre cornering stiffness uncertainty on vehicle dynamic control for the change of driving

conditions.

The rest of the paper is organised as follows. In Section 2, an 8-DOF nonlinear vehiclemodel and tyre model are described briefly. Section 3 gives an analysis of control logic and

the distribution of brake forces. An optimal guaranteed cost coordination control scheme

(OGCC) for the upper controller is presented in Section 4 in detail. Some simulation results

are carried out in Section 5. Section 6 presents the conclusions of the paper.


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Vehicle System Dynamics 59

Figure 1. 8-DOF nonlinear vehicle model. (a)XYplane; (b)YZplane; (c)ZXplane.

2. 8-DOF nonlinear vehicle model

Nonlinear vehicle model (Figure 1ac) reflecting the actual vehicle characteristics is used to

test the control schemes proposed in the paper. The 8-DOF nonlinear vehicle model with front

wheel driving and front wheel steering includes longitudinal, lateral, roll, yaw dynamics and

four wheels rotational dynamics. The notations are described in Appendix 1.

2.1. Vehicle model

Equations (1)(4) represent the longitudinal, lateral, yaw and roll dynamics, respectively:

mt(Vx Vy )mshs =

4i=1

Fxi , (1a)

4

i=1Fxi =(Fxw1 + Fxw2) cos f+(Fxw3+ Fxw4)(Fyw1 + Fyw2) sin f, (1b)

mt (Vy +Vx )+mshs+ (lfmuflrmur) =4

i=1

Fyi

4i=1

Fyi =(Fxw1 + Fxw2) sin f+(Fyw1 + Fyw2) cos f+(Fyw3+ Fyw4),

(2)


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60 X. Yanget al.

Ixx+ ms(Vy +Vx )hscos =msghssin (Kf+Kr) (Cf+Cr), (3)

Izz =tw

2(Fx1+ Fx3 Fx2 Fx4)+lf(Fy1+ Fy2)lr(Fy3+ Fy4). (4a)

The longitudinalandlateral forces of the ith wheel in the vehicle coordinates have the following

relationships with the tyre forces:Fxi =Fxwicos i Fywisin i

Fyi =Fxwisin i +Fywicos i(i =1, 2, 3, 4). (4b)

For a front steering vehicle:

1 =2 =f, 3 =4 =0.

For the variation of the tyre normal force has significant effects on the vehicle handling andstability performance [17], the tyre normal force model includes the load transfers due to the

longitudinal and lateral accelerations:

F z1 =mtglr

2l

1

2Fl +

ay

tw

mslrshfroll

l+mufhuf

+

1

tw(Kf Cf), (5a)

F z2 =mtglr

2l

1

2Fl

ay

tw

mslrshfroll

l+mufhuf

1

tw(Kf Cf), (5b)

F z3 =mtglf

2l

+1

2

Fl +ay

twmslfshrroll

l

+murhur + 1tw

(Kr Cr), (5c)

F z4 =mtglf

2l+

1

2Fl

ay

tw

mslfshrroll

l+murhur

1

tw(Kr Cr), (5d)

where

Fl =(mufhuf+mshs+ murhur)ax

l.

2.2. Tyre model

Slip angle for each wheel is defined as

1 =farctan

Vy +lf

Vx +(tw/2)

, 2 =f arctan

Vy +lf

Vx (tw/2)

,

3 =arctan

Vy +lr

Vx +(tw/2)

, 4 =arctan

Vy +lr

Vx (tw/2)

. (6)

Longitudinal wheel slip ratio can be described as

i =Rwwi Vx

max(Rwwi , Vx ), (i =1, 2, 3, 4). (7)

Tyre model for the 8-DOF nonlinear vehicle model needs to express the interaction between

longitudinal andlateral tyre forces. Considering the situationwhere the combination of steering

and braking is referred in this paper, Dugoff tyre model [18] is selected here, which can be

defined as follows:


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longitudinal tyre force

Fxwi =Cx i

1if (S), (i =1, 2, 3, 4), (8a)

lateral tyre force

Fywi =Citan i

1if (S), (i =1, 2, 3, 4),

C1 =C2 =Cf, C3 =C4 =Cr, (8b)

where

S=F zi (1r Vx2

i +tan2 i )

2

C2i

2i +C

2i tan

2 i

(1i ),

f(S)=

1 S >1

S(2S) S


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62 X. Yanget al.

which can be described as

d =K

1+Tsf, (11)

whereK =

Vx / l

1+(mt/ l2)((lf/Cr)(lr/Cf))V2x.

The value of time constant Tcan be obtained by the following formula [7]:

T =IzVx

2Cflf(lf+lr)+mtlrV2x.

For the desired slip angle response, it is not uniform. A steady state desired slip angle (see

Equation (12)) is deduced based on a 2-DOF vehicle model in [23]. However, a zero slip angle

is selected for the desired response in [7,24].

ss =lr (lfmtV

2x/2Crl)

l+ (mtV2x(lrCrlfCf)/2CrCfl)fss. (12)

As mentioned above, the slip angle response to the front wheel steering angle is also a second

order problem. In this paper, for the consideration of convenience, a first order model is also

used in the controller reasoning, which can be formulated as

d =K

1+T sf, (13)

where K can be obtained from Equation (12), that is K =ss/fss; T is assumed to beequal toT.

The desired yaw rate response and slip angle response cannot always be obtained when the

tyre force goes beyond the adhesion limit of the tyre. Thus, the desired yaw rate and slip angle

both have an upper bound, which can be expressed as follows, respectively [23]:

d_bound =g

Vx, d_bound =tan

1(0.02g). (14)

The desired yaw rate and slip angle responses for controller design can be rewritten as

d =K

1+Tsf,

Kf

d_bound

d =d_boundsgn (Kf)

1+Ts,Kf > d_bound , (15)

d =

K

1+T sf,

K f d_boundd =

d_boundsgn (K f)

1+T s,K f > d_bound . (16)

3.2. Analysis of control scheme

Figure 2 shows the whole structure of the optimal guaranteed cost based coordination scheme

including an upper controller and a lower controller. In detail, the upper controller that is the

key part studied in this paper calculates the active steer angle and the corrective yaw moment

needed to track the desired yaw rate and the desired slip angle. Note that when the vehicle


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Figure 2. Block diagram of the coordination control scheme.

is in the linear region, only steering is used to follow the desired response; and when the

vehicle reaches the handling limit, steering and braking work together. Since vehicle stability

is directly related to the sideslip motion, sideslip angle is often bounded to keep the vehicle in

the linear region [2,25,26]. In this paper, we partition the stable and the unstable region by the

phaseplane method about slip angle, which is described in [25,26]. A stability bound defined

in [28] is used here, which is formulated as

2.4979 + 9.549


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64 X. Yanget al.

Figure 3. Yaw rate response comparisons for all cases.

Table 1. Control decision.

Status M f Braking wheel

(a)d 0, >0, d < FL(b)d >0, 0, d > + + + RR

(c)d 0, + + + FR(d)d 0, d < FL

(e)d


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4. Upper coordination controller design

4.1. Optimal guaranteed cost control for uncertain systems

Consider the following linear uncertain systems [29]:

x(t)= (A +A)x(t)+(B +B)u(t) x(0)= x0, (18)

wherex(t) R n is the system state vector,u(t) R n is the control input vector,AandB are

known constant real matrices of appropriate dimensions, A andB are real-valued matrix

functions representing time-varying parameter uncertainties of the system model. The param-

eter uncertainties considered here are assumed to be norm-bounded and have the following

form:

[A B] =DF(t)[E1 E2], (19)

whereD,E1 and E2 are known constant real matrices of appropriate dimensions and F(t)

Rij is an unknown matrix satisfying

FT(t)F(t) I .

Consider a quadratic cost function associated with system (18) as:

J =

0 [x

T

(t)Qx(t)+u

T

Ru(t)]dt, (20)

where Q and R are given positive-definite symmetric matrices. For system (18) with cost

function (20), if the state feedback control law u(t )= K xcan make the closed-loop system

asymptotically stable and the upper bound of the closed-loop system cost function value J

is no more than a positive value J, J is an upper bound of the cost function and u(t )

is a quadratically guaranteed cost controller. Especially, u(t)is an optimal guaranteed cost

controller ifu(t )= K xcan bring a minimum upper bound of the cost function. A guaranteed

cost controller can make the uncertain closed-loop system not only asymptotically stable but

robust with respect to parameter uncertainties. Theorem 1 gives the solution of the optimal

guaranteed cost problem for uncertain system (18) with cost function (20).By defining

= [1, 2, . . . , l ], k >0, k =1, 2, . . . , l ,

M =diag{1Ii1i1, 2Ii2i2, . . . , l Iil il },

N =diag{11 Ii1i1, 12 Ii2i2, . . . ,

1l Iil il },

Equation (19) can be denoted as

DF(t)[E1 E2] =D MF(t)[N E1 NE2].

THEOREM1 For system (18)and cost function (20), u(t)= WX1x(t)is an optimal state

feedback control law, if there exists a solution ( , W , X, M) for the following optimisation


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66 X. Yanget al.

problem:

min,W,X,M

Trace(M), s.t.

(i)

(AX+B W )

T

+AX+ BW (E1X+E2W )T

X WT

D ME1X+ E2W N

1 0 0 0

X 0 Q1 0 0

W 0 0 R1 0

MDT 0 0 0 M1

0.

4.2. Optimal guaranteed cost controller design

In this section, two optimal guaranteed cost controllers are designed. The first one is the

coordination of DYC and AFS based on optimal guaranteed cost theory and the other is an

optimal guaranteed cost AFS controller. When the vehicle is in the linear region, only the AFS

controller (the second one) is active; and when the vehicle enters the nonlinear region, the

coordination controller (the first one) begins to work.

Though the tyre cornering stiffness is affected by many aspects, the surface adhesion coeffi-

cient is the primary aspect. Therefore, the variation of tyre cornering stiffness is treated as the

variation of the surface adhesion coefficient in this paper. The actual tyre cornering stiffness

can be described as

Cf =Cf0 , Cr =Cr0 ,

whereCf0 , Cr0 are the nominal cornering stiffness of the front and rear tyres; Cf,Cr are the

actual cornering stiffness of the front and rear tyres. A 2-DOF vehicle model is selected to

design the controller and the actual response dynamic equation can be expressed as follows:

xac =A0xac+ B10u1+ B20u2 (21)

with

xac =

u1 =f, u2 =

fM

.

Choose d, d as the state variables and fas the system input. Then the desired response

dynamic equation can be derived from Equations (11)(14):

xd =Adxd + Bdu1. (22)

The error dynamic equation can be deduced by Equations (21) and (22) as

e = xac xd =A0(xac xd)+(A0 Ad)xd + (B10 Bd)f+B20u2. (23)

In Equation (23), let

A0 =

a011 a012a021 a022

, Ad =

ad11 ad12ad21 ad22

(24)

with Equations (11) and (13), then the term (A0 Ad)xd in Equation (23) can be expressed

as

(A0 Ad)xd =

a1 a2T

f = Af (25)

with a1 =(a011 ad11)K +(a012 ad12)K , a2 =(a021 ad21)K +(a022 ad22)K .


13/25


Then Equation (23) can be rewritten as

e = A0e+( A+B10 Bd)f+B20u2. (26)

For dynamic system (26), fcan be viewed as the reference input. When analysing the effectof the control input on the system, the reference input can be set to zero. With f =0,

Equation (26) can be rewritten as

x =A0x + B20u (27)

with

x =e =

r

, u=

fM

.

As mentioned above, the tyre cornering stiffness is not constant but varies with road adhe-

sion coefficient. Considering the variation, the uncertainty of tyre cornering stiffness can beexpressed as follows:

Cf =Cf0(1+ff), f 1

Cr =Cr0 (1+rr), r 1, (28)

wheref andr are the deviation magnitude of the cornering stiffness for the front and rear

tyre, respectively, from the nominal values Cf0,Cr0 and f,r are perturbations. Then similar

to the uncertain form of Equation (18), Equation (27) can be written as:

x =(A0 + A)x + (B0+ B)u, (29)

where

A0 =

2(Cf0 + Cr0 )

mtVx

2(Cf0 lfCr0 lr)

mtV2x1

2(Cf0 lfCr0 lr)

Iz

2(Cf0 l2f +Cr0 l

2r)

IzVx

B0 =

2Cf0

mt Vx0

2Cf0 lf

Iz

1

Iz

,

A= DFE1, B =DFE2,

D =

2Cf0 f

mt Vx

2Cr0 r

mt Vx

2Cf0 flf

Iz

2Cr0 rlr

IzVx

, F = f 00 r ,

E1 =

1lf

Vx

1lr

Vx

, E2 =

1 0

0 0

.

The reasoning for the design of optimal guaranteed cost based AFS controller is similar

to that of the coordination controller presented above. The uncertain equation is the same

as Equation (29) except that the control input matrix B0 and matrix E2 should be set toB0 = [2Cf0 /(mtV )2Cf0 lf/Iz]

T andE2 = [1 0]T, respectively.

When designing the guaranteed cost controller, the uncertainty deviation magnitudes fand r should be selected first. It is obvious that the choice of uncertainty deviation magnitude

affects the controller performance. From Figure 4, we can find that for both the coordination


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68 X. Yanget al.

Figure 4. Effect of uncertainty deviation magnitude on guaranteed cost.

controller (AFS+DYC) and the AFS controller, the guaranteed cost increases with the incre-

ment of uncertainty deviation magnitude, and furthermore the guaranteed cost increases more

rapidly when the deviation magnitude exceeds 0.7. In other words, the existence of system

uncertainty leads to the degradation of the system performance. The guaranteed cost can be

interpreted as the performance of the system with parametric uncertainties being guaranteed to

be not more than this bound. The bigger the deviation magnitude, the worse performance can

be guaranteed. Both the front tyre cornering stiffness deviation magnitude fand the rear one

rconform to this law. The selection offand r, which is like the selection of the front and

rear tyre cornering stiffness, affects the vehicle dynamic response greatly. It relies on the expe-

rience to a certain degree. The effects of deviation magnitude on the vehicle dynamic response

can be referred to Section 5 (Figure 7). Without loss of generality, we assume the front andrear tyre cornering stiffness have the same deviation magnitude here, that is f =r =.

For the OGCC

Kcorrd =

kf1 kf2kM1 kM2

, (30)

the active steer anglefand corrective yaw momentMare formulated as, respectively,

f =kf1+ kf2 , M =kM1+kM2

and the variations of the active steer angle gains kf1, kf2 and corrective yaw moment

gainskM1,kM2 versusthe uncertainty deviation magnitude are shown in Figure 5. We findthat the control gains increase with the increment of the deviation magnitude on the whole.

Figure 5. Variation of controller gains with the uncertainty deviation magnitude.


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The increment of the control gains means that much more control effort is needed that is not

desired. On the other hand, bigger deviation magnitude that means more model parameter

uncertainties being considered in the controller design has much advantage when the driv-

ing condition varies in a large range. In short, the selection of deviation magnitude is also

conflictive. By trade-off, we choose = 0.5 in this paper.

The parameters for controller design are listed in the following [2]:

mt =1704 kg, Cf0 =63,224 N/rad, Cr0 =84,680 N/rad, Iz =3048.1 k g m2,

lf =1.135, lr =1.555 m, Vx =33.33 m/s, = 0.8.

For the design of control laws, the following weights are selected for coordination scheme

and AFS, respectively:

Qgc = 80000 00 8000

, Rgc = 80000 00 0.0001

Qga =

80000 0

0 8000

, Rga =80000.

Then for performance index (20), optimal guaranteed cost control laws can be obtained by

solving a set of LMIs according to Theorem 1.

The optimal guaranteed cost controller for coordination of AFS and DYC is

ugc(t)=

0.6624 0.7342

5172.2567 5464.0043

x(t), (31)

whereugc = [f M]T, and the optimal guaranteed cost controller for AFS is

uga(t )=

0.6699 0.7610

x(t), (32)

whereuga =f.

Since the AFS controller is used with the coordinated controller in the OGCC scheme,

in the sequel, we will call the combination of the two controllers optimal guaranteed cost

coordinated control, i.e. OGCC, which will be compared with the optimal coordination (OC)

scheme based on LQR.

4.3. Optimal coordination controller design

There are two optimal controllers introduced in this section, both of which are designed based

on LQR. The first controller coordinates DYC and AFS simultaneously and the second one

controls AFS only. The two controllers are combined in the OC scheme.

For system (27), define performance index as

Joc =

0

[xToc(t)Qocxoc(t )+uTocRocuoc(t)]dt, (33)

where xoc = [ ]T, uoc = [f M]T. For the reason of comparison, the weights areselected as same as those used in the design of the OGCC scheme in Section 4.2, i.e.

Qoc =

80000 0

0 8000

, Roc =

80000 0

0 0.0001

,


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70 X. Yanget al.

then an LQR problem can be formulated. A minimal performance index can be obtained by

solving Riccati equation

P A+ATP PBR1BP +Q = 0. (34)

Then the analytical solution for the control input is formulated as

uoc(t )= R1BTPx(t). (35)

With the parameters used for OGCC design, the OC law and the optimal AFS control law are

calculated as, respectively:

uoc(t)=

0.0925 0.2558

1542.5643 1137.3698

x(t ), uoa(t)=

0.0972 0.2583

x(t).

Similarly, in the sequel, we will call the combination of the two controllers optimal coordinatedcontrol, i.e. OC, that will be compared with the OGCC scheme.

In addition, from Sections 4.2 and 4.3, we can find that the OGCC scheme and OC scheme

each have two controllers, i.e. the coordination (AFS+DYC) controller and theAFS controller.

It is noted that the AFS controllers are not derived from the coordinated controller directly but

designed all alone.

4.4. Response analysis

For system (27), taking the initial statex0 = [0.01 0.1]T and the adhesion coefficient =

0.2, the state response and control input comparisons between OGCC and OC are shown in

Figure 6. State and control input response comparisons for the closed-loop system.


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Figure 6. It is noted that the OGCC scheme presents faster response than the OC scheme,

but the control input is also much bigger than OC. If the control input dose not exceed the

saturation limit of the actuator, OGCC maybe a good method.

5. Simulation analysis

In this section, a number of simulations are carried out on an 8-DOF nonlinear vehicle model

platform presented in Section 2 to analyse and evaluate the OGCC scheme proposed in

Section 4. Two different manoeuvres are considered here. The first manoeuvre is related

to the sinusoidal with increasing amplitude steering input, which is often used in the vehicle

handling performance test, and we call this manoeuvre slalom in the sequel. The second one

is a single lane-change manoeuvre with a single sinusoidal steering input. In all simulations,

the initial longitudinal velocity is 120 km/h and the values of vehicle parameter are listed in

Table 2 [2].

In order to present the effects of uncertainty deviation magnitude on the control perfor-

mance, Figure 7 shows the vehicle response comparisons at different uncertainty deviation

Table 2. Value of vehicle model parameters in simulation.

mt 1704.7 kg hfroll 0.130 m Kf 65,312 Nm/rad

ms 1526.9 kg hrroll 0.110 m Kr 32,311 Nm/rad

muf 98.1 kg hs 0.445 m Cf 3823 Nm/rad/s

mur 79.1 kg huf 0.313 m Cr 2653 Nm/rad/s

lf 1.135 m hur 0.313 m Cx 50,000 N/unit slip

lr 1.555 m tw 1.535 m Cf 105,850 N/radlfs 1.115 m Izz 3048.1kgm

2 Cr 79,030 N/rad

lrs 1.675 m Ixx 744kgm2 g 9.81 m/s2

Figure 7. Slalom manoeuvre responses versus different uncertainty deviation magnitude ( = 0.8).


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72 X. Yanget al.

Figure 8. Steer angle for slalom manoeuvre with increasing magnitude.

magnitude (= 0.1, 0.3, 0.5, 0.7) for a slalom manoeuvre with the steering input signalshown in Figure 8. It is noted that with the increment of deviation magnitude, the slip angle

and yaw rate error both become smaller. The reason is likely that much tyre cornering stiffness

error exists between the control model and the real vehicle (nonlinear vehicle model), and

the error becomes smaller with the increment of deviation magnitude. It is also noted that the

corrective yaw moment and the active steer angle both become bigger with the increment of

deviation magnitude that is consistent with the solution shown in Figure 5.

Figures 911 show the response comparisons for a slalom manoeuvre with the steering

input shown in Figure 8 on a dry road with the adhesion coefficient of 0.9. Figure 9 shows the

response comparisons from different points of view, including lateral dynamics and longitu-

dinal dynamics. We can easily find that compared with the OC scheme, the OGCC schemepresents superior tracking performance to the reference response. The uncontrolled vehicle

will lose stability and even turn over. Figure 10 shows the variations of stability index of the

two schemes that can be used for analysis and evaluation combining with the control effort

comparisons shown in Figure 11. As stated before, when the stability index is below one,

only AFS system is active to enhance the handling performance; and when the stability index

exceeds one, the braking system begins to work with the active steering system to keep the

vehicle stable. It also observed that the control effort for the OGCC scheme is bigger than that

of the OC scheme, the phenomenon of which is consistent with the fact stated in Figure 5.

Similarly, Figures 1214 show the response comparisons for a slalom manoeuvre with the

steering input shown in Figure 8 on an icy road with the adhesion coefficient of 0.2. It is

observed that the OGCC scheme is still stable and presents satisfying tracking performance

to the drivers intent but the OC scheme is unstable. It can be explained that the running

condition has deviated greatly on the icy road from that on the dry road where the controller is

designed and the tyre cornering stiffness has changed greatly. Fortunately, the OGCC scheme

considers the uncertainty of the tyre cornering stiffness beforehand (with the uncertainty

deviation magnitude of 0.5); however, the OC scheme is not. From the comparison of stability

index shown in Figure 13, it is also easy to find that the OGCC scheme can achieve good

stability performance when performing the slalom manoeuvre on the icy road at high speed.

A familiar phenomenon can be found in the control effort comparisons shown in Figure 14,

that is more control effort is needed for the OGCC scheme. However, it is still satisfying for

its good stability and tracking performance, if the control effort does not exceed the actuatorslimit because stability is always the primary objective for a vehicle steering at high speed.

There are also some methods for the optimal guaranteed cost control theory to handle the

actuators saturation in the literature [30]. In fact, actuators saturation is a common problem

not only for optimal guaranteed cost control but for all the control methods.


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Figure 9. Slalom manoeuvre response comparisons on dry road ( = 0.9).

Figures 1518 show the response comparisons for a single lane-change manoeuvre with

the steering input shown in Figure 15 on a dry road with the adhesion coefficient of 0.8.

Single sinusoidal steering input is often used to imitate the single lane-change and roadblock

avoiding manoeuvres in vehicle dynamics test. Note that the condition in this test is the sameas that where the controller is designed. The tyre cornering stiffness uncertainty is small in

Figure 10. Comparisons of stability index for slalom manoeunvre on dry road ( = 0.9).


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74 X. Yanget al.

Figure 11. Control effort comparisons for slalom manoeuvre on dry road ( = 0.9).

Figure 12. Slalom manoeuvre response comparisons on icy road ( = 0.2).


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Figure 13. Comparison of stability index for slalom manoeunvre on icy road ( = 0.2).

Figure 14. Control effort comparisons for slalom manoeuvre on icy road (= 0.2).

Figure 15. Steer angle for single lane-change manoeuvre.


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76 X. Yanget al.

Figure 16. Single lane-change manoeuvre response comparisons on dry road ( = 0.8).

Figure 17. Comparisons of stability index for single lane-change manoeunvre on dry road ( = 0.8).

Figure 18. Control effort comparisons for single lane-change manoeuvre on dry road ( = 0.8).


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this condition, so small response difference between OC and OGCC is achieved though the

OGCC presents superior performance. The uncontrolled vehicle cannot track the reference

response. From the time histories of stability index, we can find that both OGCC and OC are

stable, so braking is not used (Figure 18).

6. Conclusions

Vehicle chassis coordinated control is one of the main trends of vehicle active safety control.

Since handling and stability can be effectively improved by AFS and DYC, respectively, in

order to exert the advantages of the two subsystems, a coordination scheme is selected here.

Unlike the conventional OC scheme that is conservative because of the frequent variation

of tyre cornering stiffness, an OGCC scheme is proposed in this paper, which considers the

uncertainty of tyre cornering stiffness beforehand.

A number of simulations are conducted on an 8-DOF nonlinear vehicle model for a slalommanoeuvre and a lane-change manoeuvre to illustrate the effects of the OGCC scheme by

comparing with the responses of the OC scheme and the passive vehicle. From the simulation

results, we can find that when the vehicle is on a dry road at high speed, the response difference

of the two coordination schemes is small but the difference becomes very large when the

vehicle is on an icy road at high speed, in which condition the OGCC scheme is still stable

and presenting good tracking performance to the drivers intent but the OC scheme will lose

stability. In other words, the change of running conditions has more influence on the OC

scheme. The problem for the OGCC scheme is the control effort. More control effort is needed

for OGCC compared with OC. However, OGCC scheme is still satisfying if the control effort

does not exceed the actuators saturation limit because to keep the vehicle stable is alwaysmore important. Therefore, the OGCC scheme can also be interpreted as the improvement of

control effect being realised by exerting the ability of the actuator, which is not made the best

use of for the OC scheme. The research in the future will consider the actuators saturation for

the OGCC scheme.

Open-loop evaluation is conducted in this paper only. The driver characteristic will be

included and the effectiveness of OGCC will be evaluated in the drivervehicleroad closed-

loop system in the future.

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Appendix 1. Description of vehicle model parameters

mt , ms total mass, sprung mass of the vehiclemuf, mur front, rear unsprung mass

l wheel base

lf, lr distance between centre of gravity (CG) and the front, rear axle

lfs, lrs distance between CG and the front, rear axle

hfroll, hrroll, hs height of front, rear roll centre, sprung mass CG to roll centre

huf, hur height of front, rear unsprung mass CG

tw wheel track widthax, ay vehicle longitudinal, lateral accelerationVx , Vy vehicle longitudinal velocity, lateral velocity , yaw rate aboutz axis, roll angle about x axisFxi , Fyi longitudinal, lateral force of thei th wheel in the vehicle coordinates,i =1, 2, 3, 4Fxwi , Fywi longitudinal, lateral tyre force,i = 1, 2, 3, 4

Izz, Ixx vehicle moment of inertia about yaw axis, roll axisKf, Kr front, rear suspension roll stiffnessCf, Cr front, rear suspension roll dampingf steer angle of front wheelCx , Cf, Cr longitudinal tyre stiffness, cornering stiffness of the front wheel, rear wheeli , i thei th wheel slip angle, slip ratio, i = 1, 2, 3, 4F zi normal force of thei th wheel,i =1, 2, 3, 4


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Rw, Jw, w wheel rolling radius, moment of inertia, angular speed

g gravity acceleration

friction coefficient between tyre and road

Tbi , Pbi active brake torque, pressure of thei th wheel,i =1, 2, 3, 4

Kb brake gain

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article 1Coordinated control of AFS and DYC for vehicle handling and stability based on optimal guaranteed cost theory