design of rollover prevention controller with linear matrix inequality-based trajectory sensitivity...

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Design of rollover prevention controllerwith linear matrix inequality-basedtrajectory sensitivity minimisationSeongjin Yim a & Youngjin Park ba BK21 Mechatronics Group , Chungnam National University , 220Kung-dong, Yuseong-gu, Daejeon, Koreab Department of Mechanical Engineering , KAIST , 373-1, Kusong-dong, YuSeong-gu, Daejeon, KoreaPublished online: 24 Feb 2011.

To cite this article: Seongjin Yim & Youngjin Park (2011) Design of rollover prevention controllerwith linear matrix inequality-based trajectory sensitivity minimisation, Vehicle SystemDynamics: International Journal of Vehicle Mechanics and Mobility, 49:8, 1225-1244, DOI:10.1080/00423114.2010.507275

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Vehicle System DynamicsVol. 49, No. 8, August 2011, 1225–1244

Design of rollover prevention controller with linear matrixinequality-based trajectory sensitivity minimisation

Seongjin Yima* and Youngjin Parkb

aBK21 Mechatronics Group, Chungnam National University, 220 Kung-dong, Yuseong-gu,Daejeon, Korea; bDepartment of Mechanical Engineering, KAIST, 373-1, Kusong-dong,

YuSeong-gu, Daejeon, Korea

(Received 7 September 2009; final version received 3 July 2010; first published 24 February 2011 )

This paper presents a method to design a rollover prevention controller for vehicle systems. The vehiclerollover can be prevented by a controller that minimises the lateral acceleration and the roll angle.Rollover prevention capability can be enhanced if the controlled vehicle system is robust to the variationof the height of the centre of gravity and the speed of the vehicle. For this purpose, a robust controller isdesigned with linear matrix inequality-based trajectory sensitivity minimisation. Differential brakingand active suspension are adopted as actuators that generate yaw and roll moments, respectively. Thenewly proposed method is shown to be effective in preventing rollover by the simulation on a non-linearmultibody dynamic simulation software, CarSim®.

Keywords: rollover prevention control; robust control; trajectory sensitivity minimisation; LMI;active suspension; differential braking

1. Introduction

In the early 2000s, a widespread supply of sports utility vehicles (SUVs) with high centre ofgravity (CG) increased the rollover accidents. For example, in USA, there were 2817 fatalitiesand 46,000 injuries in 2004 [1]. Most rollover accidents are fatal. For example, the portion ofthe rollover in all crashes is about 3%, of which 33% of all fatalities caused by the rollover[2]. For this reason, the rollover accidents should be prevented for the passenger safety.

The major factors that influence the rollover are the lateral acceleration ay , the height ofthe CG hs and the lateral force Fy , as shown in Figure 1. The untripped rollover occurred dueto the large lateral acceleration by the excessive steering at high speed. On the low frictionroad or at low speed, the rollover cannot occur as the lateral acceleration is small. Thus, it isnecessary to reduce the lateral acceleration and lateral force to prevent rollover.

Following the idea mentioned earlier, several control schemes have been proposed forrollover prevention. The most common scheme was to reduce the reference yaw rate throughdifferential braking or active steering for the purpose of generating under-steer characteristics

*Corresponding author. Email: [email protected]

ISSN 0042-3114 print/ISSN 1744-5159 online© 2011 Taylor & FrancisDOI: 10.1080/00423114.2010.507275http://www.informaworld.com

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1226 S. Yim and Y. Park

msay

hsFy

ms

Figure 1. Factors to influence the rollover.

of a vehicle [3–7]. However, these approaches did not take the variation of the height ofCG and the vehicle speed into account. The rollover prevention capability can be enhanced bydesigning a controller with the consideration of the variations of these factors. Gaspar et al. [8]used a two-degree-of-freedom (DOF) bicycle and one-DOF roll model with the longitudinalvelocity as a time-varying parameter and designed a gain-scheduled controller for the rolloverprevention. This paper did not consider the variation of the height of CG. Only an unmodelleddynamics was regarded as an unstructured uncertainty.

To enhance the rollover prevention capability, it is necessary to design a controller thatis robust to the variation of the height of CG and the vehicle speed. In this paper, theseparameters are treated as uncertainty. To design a robust controller, a trajectory sensitivityminimisation is adopted as a design methodology. A trajectory sensitivity is defined as aderivative of the state with respect to a particular parameter at its nominal value [9–12]. Forgiven nominal state-space models, it is easy to obtain the trajectory sensitivity model. Con-trary to the quadratic stabilisation [13] or polytopic uncertainty [14], the trajectory sensitivitymodel does not require information about the range of parameter variations. The reductionin the trajectory sensitivity makes the system robust to parameter variations. It was proventhat the minimum trajectory sensitivity is equivalent to the minimum eigenvalue sensitiv-ity [15]. For this reason, it is necessary to design a controller that minimises the trajectorysensitivity to make the controlled system be robust. In the research of [9], trajectory sensi-tivity is incorporated into linear quadratic (LQ) objective and minimised using LQ regulator(LQR). As the trajectory sensitivity is infinitesimal perturbation at nominal value, the con-troller designed by the trajectory sensitivity minimisation is far less conservative. However, thismethod cannot theoretically guarantee the robust stability for a specified range of parametervariations.

LQR is equivalent to H2 optimal control in a stochastic sense. In this paper, H2 opti-mal control is adopted for the trajectory sensitivity minimisation, and LMI is used to solvethe H2 optimal control as it can be easily solved by LMI [14]. The H2 optimal controllerwith trajectory sensitivity minimisation requires the state and trajectory sensitivity for feed-back. Because the trajectory sensitivity is a virtual state, it is impossible to measure thetrajectory sensitivity directly. To avoid introducing complicated trajectory sensitivity esti-mation procedure in control systems, it is better to design a controller based on the statevariables alone. As a result of this constraint, the controller should have a block-diagonalstructure, and this cannot be solved by standard LQR or H2 optimal control approach [11].To solve this problem, a state covariance matrix with block-diagonal structure for LMI isintroduced.

This paper is organised as follows. In Section 2, a three-DOF vehicle model that describesyaw and roll behaviours of vehicles and a design method of H2 controller that prevents rolloverare presented. In Section 3, a design method for robust rollover prevention controller is pro-posed utilising LMI-based trajectory sensitivity minimisation. Simulations are performed onlinear vehicle and non-linear models based on a commercial multibody dynamics software,Carsim® [16]. The simulation results are discussed in Section 4, and conclusions are drawnin Section 5.

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Vehicle System Dynamics 1227

2. Rollover prevention controller design

The rollover prevention controller consists of upper and lower level controllers [17]. The upperlevel controller that is designed through H2 optimal control based on a liner vehicle modelgenerates the control yaw/roll moment. The lower level controller distributes the controlyaw/roll moment to a brake and an active suspension, respectively. The actuators such as abrake or an active suspension are taken into account in the lower level controller.

2.1. Vehicle model

The vehicle model used in this paper is a three-DOF model, as shown in Figure 2. This modelconsists of a two-DOF bicycle and a one-DOF roll model, which describe the yaw and lateralmotion and the roll motion, respectively.

The equations of motion for this vehicle model are as follows [18].Lateral motion:

may − mshsφ = Fyf + Fyr. (1)

Yaw motion:

Izγ = lfFyf − lrFyr + Mγ . (2)

Roll motion:

Ixφ − mshsay = −Cφφ − Kφφ + msghsφ + Mφ. (3)

These equations of motion were different from Segel’s work, in that the coupling betweenyaw and roll motion is neglected [19]. In these equations, Mγ and Mφ are the controlyaw moment generated by active braking and the control roll moment generated by activesuspension, respectively. In Equations (1) and (3), the lateral acceleration ay is defined asfollows:

ay = vy + vx · γ. (4)

The relationship between Fy and α is given in Figure 3. In this figure, α represents the tyre slipangle that is defined as the difference between the direction of wheel velocity and the steeringangle. μ is the friction coefficient between the tyre and the road surface. Assuming that thelateral tyre forces Fyf and Fyr are linearly proportional to α and that tan−1 (θ) � θ , then the

φ

hs

ms, Ix

roll center

msay

msghs

zy

x

y

lf

lr

Fyr

Fyf

m, Iz

γ

δf

vy

vxV

(a) 2DOF Bicycle Model (b) 1DOF Roll Model

Mγ

Mφ

Figure 2. Three-DOF vehicle model.

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,

Figure 3. Relationship between Fy and α.

lateral tyre forces are given as Equation (5) [20]. As shown in Figure 3, Cf and Cr are validwithin the linear region where α is small. If α goes over the saturated region, Cf and Cr areno longer valid. Moreover, Fy varies according to the variation of μ. To capture these featuresof the lateral tyre forces, Cf and Cr should be regarded as uncertainty in the controller designprocedure. Regarding Cf and Cr as uncertainty has an effect on the reference yaw rate and thestate-space model for a vehicle, as seen later:

Fyf = −Cfαf , Fyr = −Crαr, (5)

where

αf = vy + lfγ

vx

− δf , αr = vy − lrγ

vx

.

The reference yaw rate γd generated by δf is modelled with a first-order system as follows:

γd =(

Kγ

τs + 1

)δf = Cf · Cr · (lf + lr) · vx

Cf · Cr · (lf + lr)2 + m · v2x · (lr · Cr − lf · Cf)

•(

δj

τ s + 1

), (6)

where τ is the time constant and Kγ is the steady-state yaw gain determined by the speed ofthe vehicle [17].

The state-space representation of Equation (6) is as follows:

γd = − 1

τγd + Kγ

τδf . (7)

The error of yaw rate eγ is defined as the difference between the actual yaw rate γ and thereference one γd:

eγ = γd − γ. (8)

The state x, control input u and disturbance w are defined as follows:

x �[vy γ φ φ γd

]T,

u �[Mγ Mφ

]T, (9)

w � δf .

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From these definitions and the equations of motion, the state-space equation of the vehiclemodel is obtained as follows:

x = Ax+B1w+B2u=E−1Aex+E−1Be1w+E−1Be2u

E =

⎡⎢⎢⎢⎢⎣

m 0 −mshs 0 00 Iz 0 0 0

−mshs 0 Ix 0 00 0 0 1 00 0 0 0 1

⎤⎥⎥⎥⎥⎦ ,

Ae =

⎡⎢⎢⎢⎢⎢⎣

a11 a12 0 0 0a21 a22 0 0 00 mshsvx −Cϕ msghs − Kφ 00 0 1 0 0

0 0 0 0 − 1

τ

⎤⎥⎥⎥⎥⎥⎦ , (10)

Be1 =

⎡⎢⎢⎢⎢⎢⎣

Cf

lfCf

00Kr

τ

⎤⎥⎥⎥⎥⎥⎦ , Be2 =

⎡⎢⎢⎢⎢⎣

0 01 00 10 00 0

⎤⎥⎥⎥⎥⎦ ,

where

a11 = −Cf + Cr

vx

, a12 = − lfCf − lrCr

vx

− mvx,

a21 = − lfCf − lrCr

vx

, a22 = − l2f Cf + l2

r Cr

vx

.

2.2. H2 optimal controller design for rollover prevention

The LQ cost functional is defined as follows:

J =∫ ∞

0

(q1e

2γ + q2a

2y + q3φ

2 + q4φ2 + q5M

2B + q6M

2AS

)dt. (11)

In Equation (11), qi is the weight of each objective. Through tuning the value of qi , it ispossible to emphasise each term in Equation (11). The weights qi of LQ cost functional givenin Equation (11) are set by the relation qi = 1/η2

i from Bryson’s rule, where ηi represents themaximum allowable value of each term [21].

In the cornering situation, a vehicle should follow the reference yaw rate γd with a smalllateral acceleration ay . To follow the reference yaw rate, the yaw rate error eγ is to be reduced.In the rollover situation, a lateral load transfer, which is calculated by the left and right verticaltyre forces, is a direct measure of the rollover danger. To prevent rollover, it is desirable toreduce the lateral load transfer. As the lateral load transfer is caused by the lateral acceleration,the penalty term on the lateral load transfer in LQ cost functional is equivalent to that on thelateral acceleration. Hence, the lateral acceleration ay should be reduced to prevent rollover.In addition, roll angle φ and roll rate φ are to be reduced. From these facts, the weights q2, q3

and q4 in Equation (11) are to be set to higher values. Equation (4) shows that the reductionin the lateral acceleration requires that of the yaw rate. This increases the yaw rate error.

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In other words, reduction in the yaw rate indicates that the yaw rate tracking performance isdeteriorated. From this fact, the rollover prevention can be achieved by reducing the lateralacceleration at the expense of the increase in the yaw rate error.

To design a controller that minimises the LQ cost functional shown in Equation (11) for thesystem represented by Equation (10), H2 optimal control is used. The cost functional of H2

control is shown as follows:

JLQ = JH2 =∫ ∞

0zT

2 z2dt, (12)

where

z2 = C2x + D22u =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 q1/21 0 0 −q

1/21

q1/22 a11 q

1/22 a12 0 0 0

0 0 0 q1/23 0

0 0 q1/24 0 0

0 0 0 0 0

0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

x +

⎡⎢⎢⎢⎢⎢⎢⎣

0 00 00 00 0

q1/25 00 q

1/26

⎤⎥⎥⎥⎥⎥⎥⎦

u.

Assuming that the full-state feedback controller u = −Kx is used, the H2 optimal controlproblem is to find K that minimises the cost functional JLQ. To solve this problem, the LMIoptimisation will be used [22]. The LMI optimisation has the extensibility to other controlmethods such as H2 and H∞ control and can handle several constraints such as pole placement,etc. The LMI is a convex optimisation, and therefore, the global optimum is guaranteed andfurthermore it is possible to obtain a solution in polynomial time with the interior-point method.The trajectory sensitivity minimisation proposed in this paper will be solved by the LMIoptimisation. For consistency, the H2 optimal controller is obtained by the following LMIoptimisation problem:

minX,L,W

trace(W),

X = XT > 0,

AX + XAT − B2L − LTBT2 + B1BT

1 < 0,[W (C2X − D22L)

(C2X − D22L)T X

]> 0,

(13)

where L = KX [14]. The LMI optimisation problem can be easily solved by the MATLABLMI Control Toolbox [14]. From the solutions X and L, the gain of the optimal controller Kis obtained as K = LX−1.

2.3. Distribution of the control yaw/roll moment

The control yaw moment Mγ and the control roll moment Mφ generated from the controllerare to be distributed to brake pressure and active suspension force of each wheel, respectively.According to the sign of Mγ , the braking input is applied to left or right wheels, as shown inFigure 4. For instance, if the sign of Mγ is positive, the braking input is applied to the leftwheels. According to the sign of Mφ , the active suspension is applied to left and right wheelsin the opposite direction, as shown in Figure 5. In Figures 4 and 5, the dark rotation arrowsrepresent the direction of the yaw and roll moments, Mγ and Mφ , respectively.

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+ M g g

( a ) Positive Yaw Moment ( b ) Negative Yaw Moment

x y

- M

Δ F x , fl

Δ F x , rl

Δ F x , fr

Δ F x , rr

Figure 4. Yaw moment distribution.

(a) Positive Roll Moment (b) Negative Roll Moment

z y

+ M - M

FAS,frFAS,fl FAS,fr FAS,fl

f f

Figure 5. Roll moment distribution.

Assuming that the control yaw moment is positive, the brake forces of left wheels can beobtained as follows:

Fx,fl = Fx,rl = Mγ

t. (14)

Hereafter, the subscripts fl, fr, rl and rr indicate the front left, front right, rear left and rearright wheel, respectively. From the following relation between the brake force and the brakepressure, the brake pressure can be obtained as follows:

Pfl = Prl = rt

KB

· Fx,fl. (15)

Assuming that the control roll moment is positive, the active suspension forces of the left andright wheels can be obtained as follows:

FAS,fl = FAS,rl = Mϕ

2t,

FAS,fr = FAS,rr = −Mϕ

2t. (16)

3. LMI-based trajectory sensitivity minimisation for robust rollover prevention

Let us consider the following linear time-invariant system:

x(t, p) = A(p)x(t) + B1(p)w(t) + B2(p)u(t), x ∈ �n, u ∈ �m, w ∈ �q,

z2(t, p) = C2(p)x(t) + D22(p)u(t), z2 ∈ �p, (17)

p = [p1 p2 · · · pr

], (18)

p0 = [p0

1 p02 · · · p0

r

], (19)

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where p is the vector of system parameters and p0 is the vector of nominal values of the param-eters. Let us assume that this system is controllable or stabilisable by a full-state feedback.This is a necessary condition for the existence of the solution of LQR.

The first-order trajectory sensitivity is defined as follows:

σi (t, p0) � ∂x(t)

∂pi

∣∣∣∣pi=p0

i

. (20)

In Equation (20), the subscript pi = p0i will be omitted hereinafter. With the above definition

of the trajectory sensitivity, the actual state can be written as follows:

x(t, p) = x(t, p0) + σ(t, p0) �p. (21)

In Equation (21), �p is the variation of the parameter vector p. According to Equation (21),the trajectory sensitivity indicates how much the parameter variation �p has the effect onthe state [12]. For this reason, the controlled system becomes robust against the parametervariation �p if the trajectory sensitivity is reduced by a certain control.

By differentiating the state equation in Equation (17) with respect to the particular parameterpi at its nominal value p0

i , the equation of trajectory sensitivity is easily obtained as follows:

σi = A(p0i )σ +

(∂A∂pi

)x +

(∂B1

∂pi

)w + B1(p

0i )

(∂w∂pi

)+

(∂B2

∂pi

)u + B2(p

0i )

(∂u∂pi

)

= Aσi + Apix + B1piw + B1wpi + B2piu + B2upi (22)

where

σi � ∂x∂pi

, wpi � ∂w∂pi

, upi � ∂u∂pi

,

A � A(p0i ), B1 � B1(p

0i ), B2 � B2(p

0i ),Api � ∂A

∂pi

, B1pi � ∂B1

∂pi

, B2pi � ∂B2

∂pi

.

As the disturbance w is independent of the parameter pi in Equation (22), let wpi = 0. UsingEquations (17) and (22), the following equation is obtained:

⎡⎢⎢⎢⎣

x

σ1...

σr

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣

A 0n×n · · · 0n×n

Ap1 A. . .

...... 0n×n

. . . 0n×n

Apr 0n×n · · · A

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎣

xσ1...

σr

⎤⎥⎥⎥⎦ +

⎡⎢⎢⎢⎣

B1

B1p1...

B1pr

⎤⎥⎥⎥⎦ w

+

⎡⎢⎢⎢⎢⎣

B2 0n×m · · · 0n×m

B2p1 B2. . .

...... 0n×m

. . . 0n×m

B2p2 0n×m · · · B2

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎣

uup1...

upr

⎤⎥⎥⎥⎦ . (23)

The state and sensitivity state are augmented into a single state ξ as defined in Equation (24),

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and then the augmented system is obtained as shown in Equation (25):

ξ �

⎡⎢⎢⎢⎣

xσ1...

σr

⎤⎥⎥⎥⎦ , ψ �

⎡⎢⎢⎢⎣

ww...

w

⎤⎥⎥⎥⎦ , ϕ �

⎡⎢⎢⎢⎣

uup1...

upr

⎤⎥⎥⎥⎦ , A �

⎡⎢⎢⎢⎢⎣

A 0n×n · · · 0n×n

Ap1 A. . .

...... 0n×n

. . . 0n×n

Apr 0n×n · · · A

⎤⎥⎥⎥⎥⎦ ,

B1 �

⎡⎢⎢⎢⎣

B1

B1p1...

B1pr

⎤⎥⎥⎥⎦ , B2 �

⎡⎢⎢⎢⎢⎣

B2 0n×m · · · 0n×m

B2p1 B2. . .

...... 0n×m

. . . 0n×m

B2p2 0n×m · · · B2

⎤⎥⎥⎥⎥⎦ , (24)

ξ = Aξ + B1ψ + B2ϕ. (25)

Even though the original system is completely controllable, the augmented system is notnecessarily. Nevertheless, if the original system that is open-loop unstable is stabilisable bya full-state feedback, then the augmented system is also stabilisable by a full-state feedbackas the equations of the trajectory sensitivity have the identical system matrix A, as shownin Equations (22) and (23). In other words, if all the RHP poles of the original system canbe moved to the left-half plane by a full-state feedback, then those of an augmented systemcome under. The original system (17) is not controllable since the reference yaw rate cannotbe regulated by the control input. However, with the values of the parameters given in Table 1,all the poles of the original system reside in the left-half plane. Thus, the original and theaugmented systems are stabilisable by a full-state feedback, and LQR or H2 optimal controlcan be applied to the augmented system.

The trajectory sensitivity can be minimised for each control framework for the augmentedsystem (25). The cost functional of H2 control with trajectory sensitivity minimisation is givenas follows:

JH2 = JH2 +∫ ∞

0

(r∑

i=1

σTi Siσi

)dt =

∫ ∞

0zT

2 z2dt, (26)

where

z2 �

⎡⎢⎢⎢⎢⎣

C2 0 · · · 0

0 S1/21

. . ....

.... . .

. . . 00 · · · 0 S1/2

r

⎤⎥⎥⎥⎥⎦ ξ +

⎡⎢⎢⎢⎢⎣

D22 0 · · · 0

0. . .

. . ....

.... . .

. . . 00 · · · 0 0

⎤⎥⎥⎥⎥⎦ ϕ = C2ξ + D22ϕ.

These problems are denoted as sensitivity-weighted H2 (SW-H2) control [23]. It is easy to solveSW-H2 control for the augmented system if the controller feeds the state and the trajectory

Table 1. Parameter values of vehicle model.

m 1146.6 hs 0.38ms 984.6 Cf 40,000Ix 442 Cr 50,000Iz 1302 Cφ 18,690lf 0.88 Kφ 6100lr 1.32

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sensitivity back such as ϕ = −Gξ. But, it is impossible to measure the trajectory sensitivitydirectly since the trajectory sensitivity is the virtual state obtained by differentiating the stateequation. To use the trajectory sensitivity for feedback, it is necessary to introduce the trajectorysensitivity estimation procedure. To avoid this procedure, it is desirable to design a controllerbased on the state variables alone with the form of the full-state feedback controller u = −Kx.Then, the following controller with the block-diagonal structure Equation (28) is obtained bydifferentiating u with respect to a particular parameter pi as shown in Equation (27):

upi = ∂u∂pi

= ∂u∂x

· ∂x∂pi

= −K∂x∂pi

= −Kσi , (27)

⎡⎢⎢⎢⎣

uup1...

upr

⎤⎥⎥⎥⎦ = −

⎡⎢⎢⎢⎢⎣

K 0m×n · · · 0m×n

0m×n K. . .

......

. . .. . . 0m×n

0m×n · · · 0m×n K

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎣

xσ1...

σr

⎤⎥⎥⎥⎦

= −Gξ, G �

⎡⎢⎢⎢⎢⎣

K 0m×n · · · 0m×n

0m×n K. . .

......

. . .. . . 0m×n

0m×n · · · 0m×n K

⎤⎥⎥⎥⎥⎦ . (28)

The H2 control with this block-diagonal structure is called the structurally constrainedsensitivity weighted H2 (SCSW-H2) control [11]. This method is similar to that of Luke et al.[24], in that the single control gain K tries to stabilise the multiple plants given in Equations (17)and (22). It is hard to obtain the optimal K for SCSW-H2. In the previous work of Flemingand Newmann [11], this problem was solved with the gradient search method. But, non-linearprogramming such as the gradient search has drawbacks in dependency on an initial condition,slow convergence and difficulty in proof of convexity.

To solve this problem, the LMI optimisation will be used. The LMI optimisation problemfor SW-H2 control with the system Equation (25) and the objective Equation (26) can beobtained as follows:

minX,G,W

trace(W),

X = XT > 0,(A − B2G

)X + X

(A − B2G

)T + B1BT1 < 0,⎡

⎣ W(C2 − D22G

)X

X(C2 − D22G

)TX

⎤⎦ > 0.

(29)

If the controller feeds only the state, then the controller has the block-diagonal structure,which is shown in Equation (28). It is hard to solve this problem due to the structure of G.Yimand Park [25] proposed a block-diagonal Lyapunov function to solve this problem. Followingthe previously proposed procedure, the state covariance matrix with a block-diagonal structure

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is defined as follows:

X �

⎡⎢⎢⎢⎢⎣

X0 0 · · · 0

0 X1. . .

......

. . .. . . 0

0 · · · 0 Xr

⎤⎥⎥⎥⎥⎦ , X = X

T> 0. (30)

To resolve the non-linearity between X and G, the new variable L is defined as follows:

L � GX =

⎡⎢⎢⎢⎢⎣

KX0 0 · · · 0

0 KX1. . .

......

. . .. . . 0

0 · · · 0 KXr

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣

L0 0 · · · 0

0 L1. . .

......

. . .. . . 0

0 · · · 0 Lr

⎤⎥⎥⎥⎥⎦ . (31)

Finally, the LMI optimisation problem for SCSW-H2 control is obtained as follows:

minX,L,W

trace(W

),

X = XT

> 0,

AX + XAT − B2L − L

TB

T2 + B1B

T1 < 0,⎡

⎣ W(C2X − D22L

)(C2X − D22L

)TX

⎤⎦ > 0.

(32)

This block-diagonal Lyapunov function is an approximation to the original problem. Hence,this LMI optimisation problem for SCSW-H2 gives the approximate solution to the originalproblem. This problem is also solved by MATLAB LMI Control Toolbox [14]. The controllerK is obtained as K = L0X−1

0 from the solutions X and L.To enhance the rollover prevention capability, the height of CG hs and vehicle speed vx are

to be regarded as uncertain parameters. In addition, the cornering stiffness of front and rearwheels, Cf and Cr, should also be regarded as pointed in Figure 3 as these can pose criticaleffect on the yaw dynamics of vehicles, as pointed in the research of Borner et al. [26]. Theweights Si for trajectory sensitivities in the cost functional (26) are computed as the maximumabsolute values of trajectory sensitivities through simulation for the closed-loop system withnominal H2 optimal gain K.

4. Simulation

The linear controllers are designed based on the linear vehicle model, shown in Equation (10).The nominal values of parameters in the linear vehicle model are referred from the SmallSUVmodel in CarSim�, as given in Table 1. The nominal H2 controller is designed with the LMIfor the nominal values of parameters. The values of ηi are given in Table 2. As explained inSection 2.2, ηi represents the maximum allowable value of each term in the LQ cost functional.For example, in Table 2, the desired values of the maximum lateral acceleration, the maximumroll angle and the maximum roll rate for rollover prevention are set to 7 m/s2, 5◦ and 10◦/s,respectively.

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Table 2. Weights in LQ cost functional.

η1 (m/s2) η2(◦) η3(

◦/s) η4 (rad/s) η5 (N m) η6 (N)

7 5 10 0.08 5000 2000

10-2

10-1

100

101

102

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (Hz)

Mag

nigu

de (d

B)

Nominal H2Proposed

Roll Moment Input

Yaw Moment Input

Figure 6. Singular value plots of sensitivity function for each controller.

4.1. Linear system analysis for the designed controllers

Figure 6 shows the singular value plots of the sensitivity function for the controllers. In figureshereinafter, Nominal H2 and Proposed denote the H2 controller designed with the nominalparameters, the nominal H2 controller, and the H2 controller designed by the proposed method,SCSW-H2 with the LMI, respectively. As shown in Figure 6, the sensitivity of roll momentinput for the proposed controller is larger than that of the nominal H2 one. In other words, theroll moment input has smaller contribution to the reduction in closed-loop sensitivity in theproposed controller. Contrary to this, the sensitivity of yaw moment input for the proposedcontroller is smaller than that of the nominal H2 one. This small sensitivity is caused by thefact that the proposed controller generates larger yaw moment input than the nominal H2 one.From this result, it can be concluded that the sensitivity reduction of the closed-loop systemis mainly caused by the increased yaw moment input in the proposed controller.

Based on the designed controllers, the Bode plots are drawn up, as shown in Figure 7. Inthese plots, the input is the steering angle, and the outputs are the lateral acceleration and theyaw rate error. As the frequency of the fishhook manoeuvre that is regarded as severe is near0.5 Hz, the frequency responses below 0.5 Hz should be checked [1]. As shown in Figure 7, thenominal H2 controller has reduced lateral acceleration and increased yaw rate error againstthe open-loop system. This results from the fact that large weights are set to q2 and q3 inEquation (11), as shown in Table 2. The reduction in the lateral acceleration indicates thatthe yaw rate is reduced. In other words, the nominal H2 controller makes the vehicle have anunder-steer characteristic. This increases the yaw rate error.As shown in Figure 7, the proposedcontroller has the minimum lateral acceleration and the maximum yaw rate error against theopen-loop system and the nominal H2 controller below the frequency 0.1 Hz. From this result,

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10-2

10-1

100

101

102

26

28

30

32

34

36

38

40Lateral Acceleration

Frequency (Hz)

Mag

nigu

de (d

B)

No ControlNominal H2Proposed

10-1

100

101

17

17.5

18

18.5

19

19.5

20Yaw Rate Error

Frequency (Hz)

Mag

nigu

de (d

B)


(a) (b)

Figure 7. Bode plots from the steering input to each output: (a) Bode plot from δf to ay and (b) Bode plot from δfto eγ .

it is concluded that the increased yaw moment input of the proposed controller increases theunder-steer characteristic of the controlled vehicle.

4.2. Rollover prevention control for CarSim vehicle model

With the designed controllers, the simulation was performed on the non-linear vehicle model,SmallSUV in CarSim�. In the simulation, the steering input is fixed fishhook manoeuvrewith 221◦, as shown in Figure 8 [1]. This vehicle model rolls over under the fixed fishhookmanoeuvre with 221◦ at 57 km/h. The initial speed of vehicle is set to 80 km/h, and the tyre–road friction coefficient is set to 1.1. The actuators of the brake and the active suspensionare modelled as a first-order system with the bandwidth of 16.6 Hz(= 1/0.06 s) and 12.5 Hz(= 1/0.08 s), respectively. To prevent the locking of a brake, the ABS provided in CarSimis used.

With the fixed nominal values of the system parameters, a rollover threshold is defined asthe pair of the minimum CG height and vehicle speed when the rollover occurs. In this paper,the rollover is defined by the onset of unstable roll. Figure 9 shows the rollover thresholds

A

-A

HandwheelAngle

T1

T2

A : 6.5*handwheel position at 0.3gT1 : command dwell time of 250msT2 : 3 seconds phase

-720deg/s

720deg/s

45deg/s

t

Figure 8. Fixed fishhook manoeuvre.

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75 80 85 90 95 100

500

550

600

650

700

750

800

veh

icle

C.G

. hei

gh

t (m

m)

vehicle speed (km/h)


Figure 9. Rollover thresholds for each controller.

0

20

40

60

80

100

120 Proposed

Nominal H2

No Control

Veh

icle

Sp

eed

(km

/h)

Figure 10. Rollover speeds for each controller.

for each controller. For a particular controller, the rollover cannot occur if the CG heightand vehicle speed are below the thresholds given in Figure 9. As shown in Figure 9, thecontrolled vehicle shows better performance in the rollover prevention than the uncontrolledone. For example, in Figure 9, the vehicle height of the rollover occurring at the same speed isincreased to the maximal extent of 25 and 38%, compared with the uncontrolled case for thenominal H2 controller and the proposed one, respectively. From these results, we concludethat the proposed controller considering the parameter uncertainties is superior to the nominalH2 one.

With fixed nominal values of the vehicle height, mass and inertia, a rollover speed is definedas the minimum vehicle speed when the rollover occurs. Figure 10 shows rollover speedsfor each controller. As shown in Figure 10, the rollover speed is increased to the extentof 53 and 119%, compared with the uncontrolled case for the nominal H2 controller and theproposed one, respectively. The proposed controller shows the best performance in the rolloverprevention.

Figures 11–13 show the responses of the CarSim SmallSUV model and control inputs underthe fixed fishhook manoeuvre with 221◦ at 80 km/h, respectively. In Figure 13(a) and (b), thelegends FL, FR, RL and RR represent the front left, front right, rear left and rear right wheels,respectively. As shown in Figure 11(c), the proposed controller shows poor yaw rate trackingperformance due to the larger braking input. Accordingly, the lateral acceleration and theroll angle are decreased, as shown in Figure 11(a) and (b). This is caused by the larger yawmoment and braking input of the proposed controller, as shown in Figures 12(a) and 13(b).

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0 2 4 6 8 10-2

-1

0

1

2

time

)g(

noitarelcc

AlaretaL


(a) Lateral acceleration (g)

0 2 4 6 8 10-10

-5

0

5

10

time

)ge

d(el

gn

Allo

R


(b) Roll angle (deg)

0 2 4 6 8 10-30

-20

-10

0

10

20

30

time

)s/ge

d(r

orrE

etaR

waY


(c) Yaw rate error (deg/s)

0 2 4 6 8 10-0.6

-0.4

-0.2

0

0.2

time

)g(

noitare lcc

Alani

duti

gn

oL


(d) Longitudinal acceleration (g)

0 2 4 6 8 1020

40

60

80

time

)s/m(

xV


(e) Longitudinal velocity (m/s)

Figure 11. Responses of CarSim SmallSUV model for each controller in Figure 4: (a) lateral acceleration (g);(b) roll angle (◦); (c) yaw rate error (◦/s); (d) longitudinal acceleration (g) and (e) longitudinal velocity (m/s).

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0 2 4 6 8 10-1

-0.5

0

0.5

1x 10

4

time

)m-

N(tu

pnIt

nem

oM

waY

Nominal H2Proposed

(a) Yaw moment input (N-m)

0 2 4 6 8 10-3000

-2000

-1000

0

1000

2000

3000

time

)m-

N(tu

pnIt

nem

oMll

oR

Nominal H2Proposed

(b) Roll moment input (N-m)

Figure 12. Control input in CarSim simulation for each controller: (a) yaw moment input (N m) and (b) roll momentinput (N m).

It can be known that the rollover prevention capability was determined by the braking inputfrom 0.5 to 1.5 s under the fixed fishhook manoeuvre, as shown in Figure 11(a) and (d). Thepeak value of the longitudinal acceleration appeared at 0.8 s for the nominal H2 and proposedcontrollers. The difference in the rollover prevention capability between the nominal H2 andproposed controllers was caused by the braking input after 1.7 s. The braking input of theproposed controller was larger than that of the nominal H2 controller after 1.7 s, as shown inFigure 13(a) and (b). As a result of this difference in the braking input, the longitudinal speedof the proposed controller was decreased further, compared with the nominal H2 controller,as shown in Figure 11(e).

As pointed in Figure 6, the yaw moment input of the proposed controller is larger thanthat of the nominal H2 controller, as shown in Figure 12(a). The active suspension inputsare vice versa. From these results, we know that the proposed controller makes the vehiclehave under-steer characteristic by the larger braking pressure, just like the implication of theBode plots in Figure 7. Figure 13 shows the applied brake pressures and the active suspensioninputs of each controller. As pointed in Figure 12(a) and (b), the brake input of the proposedcontroller is larger than that of the nominal H2 controller, and the active suspension inputsare vice versa. Figure 14 shows the trajectories of the vehicles with each controller. Thevehicle with the proposed controller has smaller braking distance and cornering radius thanthose of the nominal H2 controller due to the larger braking input generated from larger yawrate error.

Figure 15 shows the trajectory sensitivities for each controller with respect to the heightof CG. The trajectory sensitivities were computed by the numerical differentiation from thesimulation results on CarSim. As shown in Figure 15, the proposed controller reduced thetrajectory sensitivities compared with the nominal H2 one. This supposes that the robustnessof the controlled system can be enhanced by the proposed controller as the trajectory sen-sitivity indicates how much the parameter variation has the effect on the state, as shown inEquation (20) [12].

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0 2 4 6 8 100

2

4

6

8

10

time)a

PM(

erusser

Pekar

B

FLFRRLRR

(a) Applied brake pressure of Nominal H2 (MPa)

0 2 4 6 8 100

2

4

6

8

10

time

)aP

M(er

usserP

ekarB

FLFRRLRR

(b) Applied brake pressure of Proposed (MPa)

0 2 4 6 8 10-1000

-500

0

500

1000

time

)N(t

up

nIn

oisne

psu

Sevitc

A

Nominal H2Proposed

(c) Active suspension input (N)

Figure 13. Control input in CarSim simulation for each controller in Figure 4: (a) applied brake pressure of NominalH2 (MPa); (b) applied brake pressure of Proposed (MPa) and (c) active suspension input (N).

90 100 110 120 130 140 150 160-60

-50

-40

-30

-20

-10

0

10

X (m)

Y (

m)


Figure 14. Vehicle trajectories of each controller based on the CarSim vehicle model.

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0 2 4 6 8 10

-2

-1

0

1

2

x 10-4

time [sec]

elg

nAll

oR

Nominal H2Proposed

0 2 4 6 8 10-2

-1

0

1

2x 10

-3

time [sec]

noitarelec c

Alar et aL

Nominal H2Proposed

0 2 4 6 8 10-1

0

1

2x 10

-3

time [sec]

rorr

Eeta

Rwa

Y

Nominal H2Proposed

Figure 15. Trajectory sensitivities for each controller with respect to the vehicle height.

5. Conclusion

In this paper, the rollover prevention controller was proposed for the vehicle systems witha large CG height such as SUV and Van. The differential braking and the active suspensionwere adopted as an actuator. The H2 controller was designed on the basis of the nominalvalues of parameters to reduce the lateral acceleration and the roll angle. From the resultsof simulation, it was shown that the H2 controller makes the vehicle avoid the rollover byenhancing the under-steer characteristic. This controller presented good performance in therollover prevention, in that the vehicle height of rollover occurring at the same speed wasincreased to the maximal extent of 25% compared with the uncontrolled case through thesimulation on the non-linear CarSim� vehicle model.

As the key parameters to affect the vehicle rollover are the CG and the speed of vehicles, therollover prevention capability can be enhanced by the controller that is designed to be robustagainst the variation of these parameters. To design the robust controller, the trajectory sensi-tivity minimisation was adopted. The design problem of the robust controller was formulatedas the trajectory sensitivity minimisation and solved by LMI optimisation.

The controllers were designed on the basis of the linear vehicle model and were appliedto the non-linear CarSim� vehicle model. The proposed controller with consideration of the

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parameter uncertainties was superior to the nominal H2 controller. Through the simulation inCarSim�, the proposed controller is shown to improve the rollover prevention capability inthat the rollover threshold and the rollover speed were increased to the maximal extent of 25and 38%, compared with the nominal H2 controller, respectively. From these results, it canbe concluded that the controller robust to the height of CG and vehicle speed enhances therollover prevention capability.

If the longitudinal velocity and the height of CG can be accurately estimated online in thefuture work, a gain-scheduled controller will be an alternative to a single robust controller.Under the condition that controller gains are pre-computed for varying CG heights and lon-gitudinal velocities, the appropriate controller gains can be used with the help of the onlineparameter identification.Another future work is to use an output feedback instead of a full-statefeedback. If a static output feedback controller with the trajectory sensitivity minimisation canbe designed in the future work, the necessity of using an observer for full-state feedback willbe avoided.

Acknowledgements

This work was supported by the second stage BK21 Project and the Korea Science and Engineering Foundation(KOSEF) through the National Research Laboratory Program (R0A-2005-000-10112-0).

References

[1] National Highway Traffic Safety Administration (NHTSA). Testing the dynamic rollover resistance of two15-passenger vans with multiple load configurations, US Department of Transportation, 2004.

[2] National Highway Traffic Safety Administration. Motor vehicle traffic crash injury and fatality estimates,2002 early assessment, NCSA (National Center for Statistics and Analysis) Advanced Research and Analysis,2003.

[3] D. Odenthal, T. Bunte, and J.Ackermann, Nonlinear steering and braking control for vehicle rollover avoidance,European Control Conference, Karlsruhe, Germany, 1999.

[4] A.Y. Ungoren and H. Peng, Evaluation of vehicle dynamic control for rollover prevention, Int. J.Autom. Technol.5 (2004), pp. 115–122.

[5] J. Yoon, K. Yi, and D. Kim, Rollover index-based rollover mitigation system, Int. J. Autom. Technol. 7 (2006),pp. 821–826.

[6] B. Schofield and T. Hagglund, Optimal control allocation in vehicle dynamics control for rollover mitigation,American Control Conference, Westin Seattle Hotel, Seattle, WA, USA, 11–13 June 2008, pp. 3231–3236.

[7] B.C. Chen and H. Peng, Rollover prevention for sports utility vehicles with human-in-the-loop evaluations,Proceedings of AVEC2000, 22–24 August 2000, pp. 115–122.

[8] P. Gaspar, Z. Szabo, and J. Bokor, The design of integrated control system in heavy vehicles based on an LPVmethod, Proceedings of the 44th IEEE Conference on Decision, and European Control Conference, Seville,Spain, 2005, pp. 6722–6727.

[9] W.G. Tuel Jr, Optimal control of unknown systems, Ph.D. diss., Department of Electrical Engineering, RenssellaerPolytechnic Institute, Troy, NY, 1965.

[10] E. Kreindler, Closed-loop sensitivity reduction of linear optimal control systems, IEEE Trans. Autom. Control13 (1968), pp. 254–262.

[11] P.J. Fleming and M.M. Newmann, Design algorithms for a sensitivity constrained suboptimal regulator, Int. J.Control 25 (1977), pp. 956–978.

[12] P.M. Frank, Introduction to System Sensitivity Theory, Academic Press, New York, 1978.[13] I. Petersen, A stabilization algorithm for class of uncertain linear systems, Syst. Control Lett. 8 (1987),

pp. 351–357.[14] P. Gahinet, A. Nemirovski, A.J. Laub, and M. Chilali, LMI Control Toolbox User’s Guide, The MathWorks,

Natick, MA, 1995.[15] V. Guruprasada Rau, Minimum trajectory sensitivity of a linear system, Electron. Lett. 17 (1981), pp. 563–564.[16] Mechanical Simulation Corporation, CarSim User Manual Version 5, 2001.[17] R. Rajamani, Vehicle Dynamics and Control, Springer, New York, 2006.[18] M. Abe, Vehicle Dynamics and Control, Sankaido, Tokyo, 1992, pp. 147–172.[19] M. Segel, Theoretical prediction and experimental substantiation of the response of the automobile to steering

control, Proc. Autom. Div. Inst. Mech. Engrs 7 (1956), pp. 310–330.[20] S. Takano and M. Nagai, Dynamics Control of Large Vehicles for Rollover Prevention, Proceedings of the IEEE

International Vehicle Electronics Conference, 2001, pp. 85–89.

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[21] A.E. Bryson and Y.C. Ho, Applied Optimal Control, Hemisphere, New York, 1975.[22] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,

Vol. 15, SIAM, Philadelphia, 1994.[23] S.C.O. Grocott, J.P. How, and D.W. Miller, Experimental comparison of robust H2 control techniques for

uncertain structural systems, J. Guid. Control Dyn. 20 (1997), pp. 611–613.[24] R.A. Luke, P. Dorato, and T. Abdallah, Linear quadratic simultaneous performance design, American Control

Conference, Vol. 6, pp. 3602–3605, 4–6 June 1997.[25] S.J.Yim andY.J. Park, Controller design with trajectory sensitivity minimization and LMI, IEICE Trans. Fundam.

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the characteristic velocity, IFAC 15th Triennial World Congress, Barcelona, Spain, 2002.

Appendix 1. Nomenclature

ay lateral acceleration (m/s2)

Cf cornering stiffness of a front tyre (N/rad)

Cr cornering stiffness of a rear tyre (N/rad)

Cφ roll damping coefficient (N m s/rad)

FAS active suspension force applied at a wheel (N)

Fyf lateral tyre force of a front wheel (N)

Fyr lateral tyre force of a rear wheel (N)

g gravitational acceleration constant (9.81 m/s2)

h height of CG from ground (m)

hs height of CG from a roll centre (m)

Ix roll moment of inertia about roll axis (kg m2)

Iz yaw moment of inertia about yaw axis (kg m2)

K full-state feedback gain

KB brake pressure-force constant

lf distance from CG to a front axle (m)

lr distance from CG to a rear axle (m)

m vehicle total mass (kg)

ms sprung mass (kg)

Mγ control yaw moment (N m)

Mφ control roll moment (N m)

rt radius of a wheel (m)

t track width of a vehicle (m)

tf front track width (m)

vx longitudinal velocity of a vehicle (m/s)

vy lateral velocity of a vehicle (m/s)

V velocity of a vehicle (m/s)

W,W auxiliary matrices with LMI variables

X state-covariance matrix

γ yaw rate (rad/s)

γd reference yaw rate (rad/s)

δf front steering angle (rad)

Fx brake force applied to a wheel (N)

P brake pressure applied on a wheel (N)

φ roll angle (rad)

φ roll rate (rad/s)

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design of rollover prevention controller with linear matrix inequality-based trajectory sensitivity...

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