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Vehicle System Dynamics: InternationalJournal of Vehicle Mechanics andMobilityPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/nvsd20

Coordinated control of AFS and DYC forvehicle handling and stability based onoptimal guaranteed cost theoryXiujian Yang a , Zengcai Wang a & Weili Peng aa Vehicular Institute of Mechanical Engineering Department ,Shandong University , Jinan City, People's Republic of ChinaPublished online: 31 Oct 2008.

To cite this article: Xiujian Yang , Zengcai Wang & Weili Peng (2009) Coordinated control of AFSand DYC for vehicle handling and stability based on optimal guaranteed cost theory, VehicleSystem Dynamics: International Journal of Vehicle Mechanics and Mobility, 47:1, 57-79, DOI:10.1080/00423110701882264

To link to this article: http://dx.doi.org/10.1080/00423110701882264

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Vehicle System DynamicsVol. 47, No. 1, January 2009, 57–79

Coordinated control of AFS and DYC for vehiclehandling 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,People’s 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 conditionscompared 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 activesafety control has made great progress, such as four wheel steering (4WS), vehicle stabilitycontrol system (VDC/ESP/VSC), active front steering (AFS), etc. All the systems can improvethe handling or the stability performance obviously in a certain region. The vehicle stabilityrelies on the balance of the front and rear tyre cornering forces. In detail, when the front tyrecannot provide the cornering force, the vehicle will lead to drift out and loss of steerability; andwhen the rear tyre cornering force reaches saturation, the vehicle will lead to spin out and lossof stability. When the lateral acceleration is small, the tyre cornering force is approximatelyproportional to the tyre slip angle; but when the lateral acceleration increases to a certainvalue, the proportional relationship will no longer exist because of the saturation property of

*Corresponding author. Email: 123yxjpmsl@mail.sdu.edu.cn

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

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58 X. Yang et al.

the tyre. Therefore, 4WS and AFS, which depend on the lateral tyre force greatly, are mainlyeffective in the linear region of the tyre. Vehicle stability control systems such as VDC, ESPor VSC mainly use the active yaw moment generating from the difference of the longitudinaltyre forces by driveline or braking to keep the vehicle stable, which is also called direct yawmoment control (DYC). 4WS and AFS can effectively improve the steerability performancein the linear region of the tyre. However, DYC can keep the vehicle stable in critical situationswhere the tyre cornering force reaches saturation [1–5]. Therefore, each individual chassiscontrol system has a certain operating region. The vehicle handling and stability performancescan be enhanced in all driving conditions by coordinating the individual chassis control systemsexerting the advantage of each subsystem.Along with the developments ofAFS and DYC, someresearchers investigate the integration of steering and braking to enhance vehicle dynamics.Nagai et al. [6,7] propose a coordination scheme that is composed of a steering angle-basedfeedforward controller and an optimal state feedback controller. Boada et al. [8] design acontrol 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 regardedas a multi-input and multi-output system control problem and an integration of steering andbraking scheme is presented using feedback linearisation technique.

As an important parameter in the vehicle dynamic control system, tyre cornering stiffness isaffected 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, itis easy to find that most of the model-following based vehicle stability controllers are designedusing a certain constant for the tyre cornering stiffness parameter [7,11,12]. Though someresearchers 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 [13–16]. Ono[3,13] and Mammar [14] design a robust steering controller based on H∞ theory to reduce theinfluence of the tyre cornering stiffness uncertainty on the system performance. Since robustperformance is the design objective for a robust controller, some other performances of thesystem may not be guaranteed. In [15], an uncertain T–S fuzzy model is founded to handlethe tyre cornering stiffness uncertainty when designing a 4WS stability controller. Thoughquadratic optimal control based vehicle stability controller considers the tracking error andthe control input simultaneously, which is more suitable for realistic application, it may losestability and cannot obtain the optimal performance when the tyre cornering stiffness variesin a large range for the change of running conditions. Fortunately, optimal guaranteed costcontrol theory that can obtain a relative optimal performance for a system with norm-boundedtime-varying parameter uncertainties provides a good means to solve the problem. For allthe norm-bounded time-varying parameter uncertainties, optimal guaranteed cost control cannot only keep the closed-loop system stable but maintain the given quadratic cost functionwithin a certain bound [16]. In the past, the solution of the optimal guaranteed cost problemwas 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 equivalentto the solution of a set of LMIs. In this paper, the coordination of AFS and DYC based onoptimal guaranteed cost control theory is presented to reduce the influence of the variationof tyre cornering stiffness uncertainty on vehicle dynamic control for the change of drivingconditions.

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 andthe 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 resultsare 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) X–Y plane; (b) Y–Z plane; (c) Z–X plane.

2. 8-DOF nonlinear vehicle model

Nonlinear vehicle model (Figure 1a–c) reflecting the actual vehicle characteristics is used totest the control schemes proposed in the paper. The 8-DOF nonlinear vehicle model with frontwheel driving and front wheel steering includes longitudinal, lateral, roll, yaw dynamics andfour 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 − Vyγ ) − mshsγ φ =4∑

i=1

Fxi, (1a)

4∑i=1

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

⎧⎪⎪⎨⎪⎪⎩

mt(Vy + Vxγ ) + mshsφ + (lfmuf − lrmur)γ =4∑

i=1Fyi

4∑i=1

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

(2)

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60 X. Yang et al.

Ixxφ + ms(Vy + Vxγ )hs cos φ = msghs sin φ − (Kφf + Kφr)φ − (Cφf + Cφr)φ, (3)

Izzγ = tw

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

The longitudinal and lateral forces of the ith wheel in the vehicle coordinates have the followingrelationships with the tyre forces:{

Fxi = Fxwi cos δi − Fywi sin δi

Fyi = Fxwi sin δi + Fywi cos δi

(i = 1, 2, 3, 4). (4b)

For a front steering vehicle:

δ1 = δ2 = δf , δ3 = δ4 = 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 thelongitudinal and lateral accelerations:

Fz1 = mtglr

2l− 1

2Fl + ay

tw

(mslrshfroll

l+ mufhuf

)+ 1

tw(−Kφfφ − Cφf φ), (5a)

Fz2 = mtglr

2l− 1

2Fl − ay

tw

(mslrshfroll

l+ mufhuf

)− 1

tw(−Kφfφ − Cφf φ), (5b)

Fz3 = mtglf

2l+ 1

2Fl + ay

tw

(mslfshrroll

l+ murhur

)+ 1

tw(−Kφrφ − Cφrφ), (5c)

Fz4 = mtglf

2l+ 1

2Fl − ay

tw

(mslfshrroll

l+ murhur

)− 1

tw(−Kφrφ − Cφrφ), (5d)

where

Fl = (mufhuf + mshs + murhur)ax

l.

2.2. Tyre model

Slip angle for each wheel is defined as

α1 = δf − arctan

(Vy + lfγ

Vx + (tw/2)γ

), α2 = δf − arctan

(Vy + lfγ

Vx − (tw/2)γ

),

α3 = arctan

( −Vy + lrγ

Vx + (tw/2)γ

), α4 = arctan

( −Vy + lrγ

Vx − (tw/2)γ

). (6)

Longitudinal wheel slip ratio can be described as

λi = Rwωwi − Vx

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

Tyre model for the 8-DOF nonlinear vehicle model needs to express the interaction betweenlongitudinal and lateral tyre forces. Considering the situation where the combination of steeringand braking is referred in this paper, Dugoff tyre model [18] is selected here, which can bedefined as follows:

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Vehicle System Dynamics 61

longitudinal tyre force

Fxwi = Cxλi

1 − λi

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

lateral tyre force

Fywi = Ci tan αi

1 − λi

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

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

where

S =μFzi(1 − εrVx

√λ2

i + tan2 αi)

2√

C2i λ

2i + C2

i tan2 αi

(1 − λi),

f (S) ={

1 S > 1

S(2 − S) S < 1.

The corrective yaw moment is only from brake torque regardless of the driveline withoutaffecting the validation of the proposed coordination scheme. The wheel rotational dynamicscan be expressed as

Jwωwi = −Tbi − RwFxwi , (i = 1, 2, 3, 4). (9)

The relationship between braking torque and braking pressure is defined as

Tbi = KbPbi , (i = 1, 2, 3, 4). (10)

3. Realisation of the control scheme

3.1. Desired state response

Model-following technique is often used in vehicle dynamic control systems. The yaw rateand slip angle of the reference model are the desired responses tracked by the actual vehicle.However, following both variables simultaneously is more effective than following only onevariable of the two [7,19,20], since the yaw response reflects more handling performance andthe slip response reflects more stability performance. So both yaw rate and slip angle arecontrolled in this paper. The two inputs for the coordination controller are the yaw rate errorand slip angle error and the two outputs are active steer angle and corrective yaw moment.Yawrate can be measured by a sensor but the slip angle is often estimated, which is not discussedhere. About slip angle estimation algorithms, the readers can refer to [1,21,22]; however, it isassumed that the slip angle can be obtained directly in this paper.

The reference model to produce the desired state responses (yaw rate and slip angle) is awell-known 2-DOF bicycle model. The responses of yaw rate and slip angle to the front wheelsteering input are second order system problem in fact. In the procedure of controller design,however, for the consideration of convenience, a first order yaw rate response model is chosen,

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62 X. Yang et al.

which can be described as

γd = Kγ

1 + Tγ sδf , (11)

where

Kγ = Vx/l

1 + (mt/l2)((lf/Cr) − (lr/Cf))V 2x

.

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

Tγ = IzVx

2Cf lf(lf + lr) + mtlrV 2x

.

For the desired slip angle response, it is not uniform. A steady state desired slip angle (seeEquation (12)) is deduced based on a 2-DOF vehicle model in [23]. However, a zero slip angleis selected for the desired response in [7,24].

βss = lr − (lfmtV2x /2Crl)

l + (mtV 2x (lrCr − lfCf)/2CrCf l)

δfss. (12)

As mentioned above, the slip angle response to the front wheel steering angle is also a secondorder problem. In this paper, for the consideration of convenience, a first order model is alsoused in the controller reasoning, which can be formulated as

βd = Kβ

1 + Tβsδf , (13)

where Kβ can be obtained from Equation (12), that is Kβ = βss/δfss; Tβ is assumed to beequal to Tγ .

The desired yaw rate response and slip angle response cannot always be obtained when thetyre force goes beyond the adhesion limit of the tyre. Thus, the desired yaw rate and slip angleboth have an upper bound, which can be expressed as follows, respectively [23]:

γd_bound = μg

Vx

, βd_bound = tan−1(0.02 μg). (14)

The desired yaw rate and slip angle responses for controller design can be rewritten as⎧⎪⎪⎨⎪⎪⎩

γd = Kγ

1 + Tγ sδf ,

∣∣Kγ δf

∣∣ ≤ γd_bound

γd = γd_bound sgn (Kγ δf)

1 + Tγ s,

∣∣Kγ δf

∣∣ > γd_bound

, (15)

⎧⎪⎪⎨⎪⎪⎩

βd = Kβ

1 + Tβsδf ,

∣∣Kβδf

∣∣ ≤ βd_bound

βd = βd_bound sgn (Kβδf)

1 + Tβs,

∣∣Kβδf

∣∣ > βd_bound

. (16)

3.2. Analysis of control scheme

Figure 2 shows the whole structure of the optimal guaranteed cost based coordination schemeincluding an upper controller and a lower controller. In detail, the upper controller that is thekey part studied in this paper calculates the active steer angle and the corrective yaw momentneeded to track the desired yaw rate and the desired slip angle. Note that when the vehicle

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Vehicle System Dynamics 63

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 thevehicle reaches the handling limit, steering and braking work together. Since vehicle stabilityis directly related to the sideslip motion, sideslip angle is often bounded to keep the vehicle inthe linear region [2,25,26]. In this paper, we partition the stable and the unstable region by thephase–plane method about slip angle, which is described in [25,26]. A stability bound definedin [28] is used here, which is formulated as∣∣2.4979β + 9.549β

∣∣ < 1, (17)

when the inequation (17) is satisfied, the vehicle is considered to be stable and AFS is used toenhance the handling performance; otherwise, it is considered to have exceeded the handlinglimit and DYC is also used with AFS to keep the vehicle stable. For a detailed descriptionof β − β phase–plane, refer to [2]. In this paper, we call the left term of inequation (17)stability index. The lower controller determines the distribution of brake torques on the fourwheels in order to realise the corrective yaw moment. There are many ways to generate thecorrective yaw moment by braking force either by an individual wheel braking or by multi-wheel braking. Superior performance can be obtained by multi-wheel braking compared withan individual wheel braking [25]; however, multi-wheel braking also induces some problemsthat are complex to handle, such as redundancy control, optimisation of the brake force and soon. Since the objective of this paper is to demonstrate the effectiveness of the OGCC scheme,single wheel braking is used here. It is more effective by braking the rear inward wheel tocorrect the case where the course trace becomes difficult due to the saturation of front wheelcornering force and by braking the front outward wheel to correct the case where the vehiclebecomes unstable with the sudden increase of side slip angle [25,27].

The relationship between the generated active steer angle, corrective yaw moment and yawrate response, slip angle response needs to be analysed first. The comparison of actual yawrate response with the desired response can be reduced to six cases shown in Figure 3. Theerror between the actual yaw rate response and the desired response is defined as

γ = γd − γ.

Active steer angle and corrective yaw moment are described by δf and M , respectively.For each case in Figure 3, the sign of γ , δf and M as well as the braking wheel aredefined in Table 1. It is easy to find from Table 1 that active steer angle δf and corrective

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64 X. Yang et al.

Figure 3. Yaw rate response comparisons for all cases.

Table 1. Control decision.

Status γ M δf Braking wheel

(a) γd ≥ 0, γ > 0, γd < γ − − − FL(b) γd > 0, γ ≥ 0, γd > γ + + + RR(c) γd ≥ 0, γ < 0, γd > γ + + + FR(d) γd < 0, γ > 0, γd < γ − − − FL(e) γd < 0, γ ≤ 0, γd < γ − − − RL(f) γd < 0, γ < 0, γd > γ + + + FR

yaw moment M both have the same sign with the yaw rate response error γ . The errorbetween the actual slip angle response and the desired response is defined as

β = βd − β.

Similarly, we can find that active steer angle δf and corrective yaw moment M both havethe opposite sign with the slip angle response error β. From the analysis mentioned above,it is concluded that there exist straightforward relationships between γ , β and δf , M .Therefore, the upper controller can be realised by a static state feedback control law that isgiven in detail in Section 4. The steady state braking pressure decided by the lower controlleris defined as

PFL = −2MRw

Kbtw cos δf, PFR = 2MRw

Kbtw cos δf, PRL = −2MRw

Kbtw, PRR = 2MRw

Kbtw,

Kb = AwμbRb,

where PFL, PFR, PRL and PRR are brake pressures at the front left, front right, rear left andrear right wheels, respectively; Aw is the brake area of the wheel; μb is the brake frictioncoefficient; Rb is the brake radius.

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Vehicle System Dynamics 65

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)

where x(t) ∈ Rn is the system state vector, u(t) ∈ Rn is the control input vector, A and B areknown constant real matrices of appropriate dimensions, A and B are real-valued matrixfunctions representing time-varying parameter uncertainties of the system model. The param-eter uncertainties considered here are assumed to be norm-bounded and have the followingform:

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

where D, E1 and E2 are known constant real matrices of appropriate dimensions and F(t) ∈Ri×j is an unknown matrix satisfying

F T(t)F (t) ≤ I.

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

J =∫ ∞

0[xT(t)Qx(t) + uTRu(t)]dt, (20)

where Q and R are given positive-definite symmetric matrices. For system (18) with costfunction (20), if the state feedback control law u∗(t) = Kx can make the closed-loop systemasymptotically 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 costcontroller if u∗(t) = Kx can bring a minimum upper bound of the cost function. A guaranteedcost controller can make the uncertain closed-loop system not only asymptotically stable butrobust with respect to parameter uncertainties. Theorem 1 gives the solution of the optimalguaranteed cost problem for uncertain system (18) with cost function (20).

By defining

ε = [ε1, ε2, . . . , εl], εk > 0, k = 1, 2, . . . , l,

M = diag{ε1Ii1×i1, ε2Ii2×i2, . . . , εlIil×il},N = diag{ε−1

1 Ii1×i1, ε−12 Ii2×i2, . . . , ε

−1l Iil×il},

Equation (19) can be denoted as

DF(t)[E1 E2] = DMF(t)[NE1 NE2].

THEOREM 1 For system (18) and cost function (20), u∗(t) = W X−1x(t) is an optimal statefeedback control law, if there exists a solution ( ˜ε, W , X, M) for the following optimisation

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66 X. Yang et al.

problem:

minε,W,X,M

Trace(M), s.t.

(i)

⎡⎢⎢⎢⎢⎣

(AX + BW)T + AX + BW (E1X + E2W)T X WT DM

E1X + E2W −N−1 0 0 0X 0 −Q−1 0 0W 0 0 −R−1 0

MDT 0 0 0 −M−1

⎤⎥⎥⎥⎥⎦ < 0,

(ii)

[M I

I X

]> 0.

4.2. Optimal guaranteed cost controller design

In this section, two optimal guaranteed cost controllers are designed. The first one is thecoordination of DYC and AFS based on optimal guaranteed cost theory and the other is anoptimal guaranteed cost AFS controller. When the vehicle is in the linear region, only the AFScontroller (the second one) is active; and when the vehicle enters the nonlinear region, thecoordination 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 thevariation of the surface adhesion coefficient in this paper. The actual tyre cornering stiffnesscan be described as

Cf = μCf0, Cr = μCr0,

where Cf0, Cr0 are the nominal cornering stiffness of the front and rear tyres; Cf , Cr are theactual cornering stiffness of the front and rear tyres. A 2-DOF vehicle model is selected todesign the controller and the actual response dynamic equation can be expressed as follows:

xac = A0xac + B10u1 + B20u2 (21)

with

xac =[γ

β

]u1 = δf , u2 =

[δf

M

].

Choose βd, γd as the state variables and δf as the system input. Then the desired responsedynamic 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)δf + B20u2. (23)

In Equation (23), let

A0 =[a011 a012

a021 a022

], Ad =

[ad11 ad12

ad21 ad22

](24)

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

(A0 − Ad)xd = [a1 a2

]Tδf = Aδf (25)

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

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Vehicle System Dynamics 67

Then Equation (23) can be rewritten as

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

For dynamic system (26), δf can 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 δf = 0,Equation (26) can be rewritten as

x = A0x + B20u (27)

with

x = e =[β

r

], u =

[δf

M

].

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 + fρf), ‖ρf‖ ≤ 1

Cr = Cr0(1 + rρr), ‖ρr‖ ≤ 1, (28)

where f and r are the deviation magnitude of the cornering stiffness for the front and reartyre, respectively, from the nominal values Cf0,Cr0 and ρf , ρr are perturbations. Then similarto 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(Cf0lf − Cr0lr)

mtV 2x

− 1

−2(Cf0lf − Cr0lr)

Iz

−2(Cf0l2f + Cr0l

2r )

IzVx

⎤⎥⎥⎥⎦ B0 =

⎡⎢⎢⎣

2Cf0

mtVx

0

2Cf0lf

Iz

1

Iz

⎤⎥⎥⎦ ,

A = DFE1, B = DFE2,

D =

⎡⎢⎢⎣

−2Cf0f

mtVx

2Cr0r

mtVx

−2Cf0f lf

Iz

−2Cr0rlr

IzVx

⎤⎥⎥⎦ , F =

[ρf 00 ρr

],

E1 =

⎡⎢⎢⎣

1lf

Vx

−1lr

Vx

⎤⎥⎥⎦ , E2 =

[−1 00 0

].

The reasoning for the design of optimal guaranteed cost based AFS controller is similarto that of the coordination controller presented above. The uncertain equation is the sameas Equation (29) except that the control input matrix B0 and matrix E2 should be set toB0 = [2Cf0/(mtV ) 2Cf0lf/Iz]T and E2 = [−1 0]T, respectively.

When designing the guaranteed cost controller, the uncertainty deviation magnitudes f

and r should be selected first. It is obvious that the choice of uncertainty deviation magnitudeaffects the controller performance. From Figure 4, we can find that for both the coordination

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68 X. Yang et 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 morerapidly when the deviation magnitude exceeds 0.7. In other words, the existence of systemuncertainty leads to the degradation of the system performance. The guaranteed cost can beinterpreted as the performance of the system with parametric uncertainties being guaranteed tobe not more than this bound. The bigger the deviation magnitude, the worse performance canbe guaranteed. Both the front tyre cornering stiffness deviation magnitude f and the rear oner conform to this law. The selection of f and r, which is like the selection of the front andrear 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 responsecan 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 =[kδf 1 kδf 2

kM1 kM2

], (30)

the active steer angle δf and corrective yaw moment M are formulated as, respectively,

δf = kδf 1β + kδf 2γ, M = kM1β + kM2γ

and the variations of the active steer angle gains kδf 1, kδf 2 and corrective yaw momentgains kM1, kM2 versus the 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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Vehicle System Dynamics 69

The increment of the control gains means that much more control effort is needed that is notdesired. On the other hand, bigger deviation magnitude that means more model parameteruncertainties 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 alsoconflictive. 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 kg m2,

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

For the design of control laws, the following weights are selected for coordination schemeand AFS, respectively:

Qgc =[

80000 00 8000

], Rgc =

[80000 0

0 0.0001

]

Qga =[

80000 00 8000

], Rga = 80000.

Then for performance index (20), optimal guaranteed cost control laws can be obtained bysolving a set of LMIs according to Theorem 1.

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

u∗gc(t) =

[ −0.6624 0.7342−5172.2567 5464.0043

]x(t), (31)

where u∗gc = [δf M]T, and the optimal guaranteed cost controller for AFS is

u∗ga(t) = [−0.6699 0.7610

]x(t), (32)

where u∗ga = δf .

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 costcoordinated 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 basedon LQR. The first controller coordinates DYC and AFS simultaneously and the second onecontrols AFS only. The two controllers are combined in the OC scheme.

For system (27), define performance index as

Joc =∫ ∞

0[xT

oc(t)Qocxoc(t) + uTocRocuoc(t)]dt, (33)

where xoc = [β γ ]T, uoc = [δf 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 00 8000

], Roc =

[80000 0

0 0.0001

],

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70 X. Yang et al.

then an LQR problem can be formulated. A minimal performance index can be obtained bysolving Riccati equation

PA + ATP − PBR−1BP + Q = 0. (34)

Then the analytical solution for the control input is formulated as

u∗oc(t) = −R−1BTPx(t). (35)

With the parameters used for OGCC design, the OC law and the optimal AFS control law arecalculated as, respectively:

u∗oc(t) =

[ −0.0925 0.2558−1542.5643 1137.3698

]x(t), u∗

oa(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 schemeeach 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 butdesigned all alone.

4.4. Response analysis

For system (27), taking the initial state x0 = [0.01 − 0.1]T and the adhesion coefficient μ =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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Vehicle System Dynamics 71

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 thesaturation 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 modelplatform presented in Section 2 to analyse and evaluate the OGCC scheme proposed inSection 4. Two different manoeuvres are considered here. The first manoeuvre is relatedto the sinusoidal with increasing amplitude steering input, which is often used in the vehiclehandling performance test, and we call this manoeuvre slalom in the sequel. The second oneis 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 inTable 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 Kφf 65,312 Nm/radms 1526.9 kg hrroll 0.110 m Kφr 32,311 Nm/radmuf 98.1 kg hs 0.445 m Cφf 3823 Nm/rad/smur 79.1 kg huf 0.313 m Cφr 2653 Nm/rad/slf 1.135 m hur 0.313 m Cx 50,000 N/unit sliplr 1.555 m tw 1.535 m Cf 105,850 N/radlfs 1.115 m Izz 3048.1 kg m2 Cr 79,030 N/radlrs 1.675 m Ixx 744 kg m2 g 9.81 m/s2

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

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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 angleand yaw rate error both become smaller. The reason is likely that much tyre cornering stiffnesserror exists between the control model and the real vehicle (nonlinear vehicle model), andthe error becomes smaller with the increment of deviation magnitude. It is also noted that thecorrective yaw moment and the active steer angle both become bigger with the increment ofdeviation magnitude that is consistent with the solution shown in Figure 5.

Figures 9–11 show the response comparisons for a slalom manoeuvre with the steeringinput shown in Figure 8 on a dry road with the adhesion coefficient of 0.9. Figure 9 shows theresponse 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 vehiclewill lose stability and even turn over. Figure 10 shows the variations of stability index of thetwo schemes that can be used for analysis and evaluation combining with the control effortcomparisons 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 indexexceeds one, the braking system begins to work with the active steering system to keep thevehicle stable. It also observed that the control effort for the OGCC scheme is bigger than thatof the OC scheme, the phenomenon of which is consistent with the fact stated in Figure 5.

Similarly, Figures 12–14 show the response comparisons for a slalom manoeuvre with thesteering input shown in Figure 8 on an icy road with the adhesion coefficient of 0.2. It isobserved that the OGCC scheme is still stable and presents satisfying tracking performanceto the driver’s intent but the OC scheme is unstable. It can be explained that the runningcondition has deviated greatly on the icy road from that on the dry road where the controller isdesigned and the tyre cornering stiffness has changed greatly. Fortunately, the OGCC schemeconsiders the uncertainty of the tyre cornering stiffness beforehand (with the uncertaintydeviation magnitude of 0.5); however, the OC scheme is not. From the comparison of stabilityindex shown in Figure 13, it is also easy to find that the OGCC scheme can achieve goodstability 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 forits good stability and tracking performance, if the control effort does not exceed the actuator’slimit 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 theactuator’s saturation in the literature [30]. In fact, actuator’s saturation is a common problemnot only for optimal guaranteed cost control but for all the control methods.

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Vehicle System Dynamics 73

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

Figures 15–18 show the response comparisons for a single lane-change manoeuvre withthe 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 roadblockavoiding 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 (μ = 0.9).

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74 X. Yang et al.

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

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

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Vehicle System Dynamics 75

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

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

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

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

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

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

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Vehicle System Dynamics 77

this condition, so small response difference between OC and OGCC is achieved though theOGCC presents superior performance. The uncontrolled vehicle cannot track the referenceresponse. From the time histories of stability index, we can find that both OGCC and OC arestable, 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, inorder 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 variationof tyre cornering stiffness, an OGCC scheme is proposed in this paper, which considers theuncertainty 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 bycomparing with the responses of the OC scheme and the passive vehicle. From the simulationresults, we can find that when the vehicle is on a dry road at high speed, the response differenceof the two coordination schemes is small but the difference becomes very large when thevehicle is on an icy road at high speed, in which condition the OGCC scheme is still stableand presenting good tracking performance to the driver’s intent but the OC scheme will losestability. In other words, the change of running conditions has more influence on the OCscheme. The problem for the OGCC scheme is the control effort. More control effort is neededfor OGCC compared with OC. However, OGCC scheme is still satisfying if the control effortdoes not exceed the actuator’s saturation limit because to keep the vehicle stable is alwaysmore important. Therefore, the OGCC scheme can also be interpreted as the improvement ofcontrol effect being realised by exerting the ability of the actuator, which is not made the bestuse of for the OC scheme. The research in the future will consider the actuator’s saturation forthe OGCC scheme.

Open-loop evaluation is conducted in this paper only. The driver characteristic will beincluded and the effectiveness of OGCC will be evaluated in the driver–vehicle–road 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 massl wheel baself , lr distance between centre of gravity (CG) and the front, rear axlelfs, lrs distance between CG and the front, rear axlehfroll, hrroll, hs height of front, rear roll centre, sprung mass CG to roll centrehuf , hur height of front, rear unsprung mass CGtw wheel track widthax, ay vehicle longitudinal, lateral accelerationVx, Vy vehicle longitudinal velocity, lateral velocityγ, φ yaw rate about z axis, roll angle about x axisFxi , Fyi longitudinal, lateral force of the ith wheel in the vehicle coordinates, i = 1, 2, 3, 4Fxwi , Fywi longitudinal, lateral tyre force, i = 1, 2, 3, 4Izz, Ixx vehicle moment of inertia about yaw axis, roll axisKφf , Kφr front, rear suspension roll stiffnessCφf , Cφr front, rear suspension roll dampingδf steer angle of front wheelCx, Cf , Cr longitudinal tyre stiffness, cornering stiffness of the front wheel, rear wheelαi, λi the ith wheel slip angle, slip ratio, i = 1, 2, 3, 4Fzi normal force of the ith wheel, i = 1, 2, 3, 4

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Vehicle System Dynamics 79

Rw, Jw, ωw wheel rolling radius, moment of inertia, angular speedg gravity accelerationμ friction coefficient between tyre and roadTbi , Pbi active brake torque, pressure of the ith wheel, i = 1, 2, 3, 4Kb brake gain

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