32bfe510f29fd4870c

Optimal yaw moment control law forimproved vehicle handling

E. Esmailzadeh *, A. Goodarzi, G.R. Vossoughi

Department of Mechanical Engineering, Sharif University of Technology,

P.O. Box 11365-9567, Tehran, Iran

Abstract

A new optimal control law for direct yaw moment control, to improve the vehicle handling,

is developed. Although, this can be considered as part of a multi-layer system, for the traction

control of a motorized wheels electric vehicle, but the results of this study are quite general and

can be applied to other types of vehicles. The dynamic model of the vehicle system is initially

developed and, using the well-known optimal control theory, an optimal controller is de-

signed. Two different versions of control laws are considered here and the performance of each

version of the control law is compared with the other one. The numerical simulation of the

vehicle handling with and without the use of the optimal yaw moment controller, and as-

suming a comprehensive non-linear vehicle dynamic model, has been carried out. Simulation

results obtained indicate that considerable improvements in the vehicle handling can be

achieved whenever the vehicle is governed by the optimal yaw moment control.

� 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Many researchers in the last decade have reported that direct yaw moment control

is one of the most effective methods of active chassis control, which could consid-

erably enhance the vehicle stability and controllability. They have showed that it is

the most important working principle behind the vehicle dynamic control (VDC)

systems. Direct yaw moment control is the advanced control system used to improve

the dynamic performances of road vehicles under various road conditions [1–3].

Mechatronics 13 (2003) 659–675

* Corresponding author. Address: Department of Mechanical Engineering, University of Victoria,

Victoria, BC, Canada V8W 3P6.

E-mail address: [email protected] (E. Esmailzadeh).

0957-4158/03/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0957-4158(02)00036-3

mail to: [email protected]

For VDC systems, the yaw moment control is considered as a way of controlling

the lateral motion of a vehicle during a severe driving maneuver. To achieve this goal

a yaw moment control strategy based on the vehicle dynamics state-feedbacks, as

well as an actuation system, is required.In accordance to the present available technology, the performance of VDC ac-

tuation systems is based on the individual control of wheel braking force known as

the differential braking. The transverse distribution of the vehicle braking force

between the wheels is the most common approach to generate the required yaw

moment. Differential braking can be achieved using the main parts of the common

anti-lock brake systems, which is based on a different and highly sophisticated

control strategy.

In spite of the conventional internal combustion engine (ICE) type vehicles, inmodern concepts of electric vehicles (EVs), known as motorized wheels EVs, the

VDC actuation system can work on the principle of controlling the vehicle traction

force rather than the braking force, as illustrated in Fig. 1. The power train of this

type of vehicles consists of two or four electric motors, which are integrated into each

one of the drive wheels and can be controlled independently [4–6]. In this configu-

ration, the traction force of each individual wheel can be independently controlled

by means of controlling the corresponding electric drive motor.

It must be noted that direct yaw moment control can become highly importantand crucial for EVs when compared with the ICE type vehicles. As a matter of a fact,

with the installation of heavy battery boxes at the rear of an electric car, the mass

center of the vehicle may now be shifted to an undesirable position, making the car

Fig. 1. Two possible ways of having external yaw moment generation.

660 E. Esmailzadeh et al. / Mechatronics 13 (2003) 659–675

to be inherently over-steered. This means that the vehicle must require additional

stabilizing external yaw moment in order to improve its overall dynamic road per-

formance.For the case of the conventional ICE type vehicles, which inherently exhibit an

under-steer behavior, the stabilizing external yaw moment is most appropriate even

when the car performs a critical road maneuver in the non-linear regime.

Regardless of the type of actuation systems, the yaw moment control strategy is

the fundamental part of a VDC system. The improvements that have been carried

out on the yaw moment control laws, are the fundamental design aspect of a VDC

system and, were reported by some researchers [7–10].

In this paper, two versions of a new optimal yaw moment control law are pre-sented. The control laws are developed as a part of the VDC system designed for a

motorized wheel EV, but in general, other types of vehicles can also make use of

these laws. The performances of the proposed control laws have been compared with

one of the previously developed control law. Finally, numerical simulations were

carried out in order to investigate the dynamic performance of the optimal controller

system.

2. Dynamic modeling of the system

A conventional linear two-degree of freedom model for vehicle handling, shown

in Fig. 2, is developed. The governing equations for the yaw and lateral motions of

the vehicle model, in the state-space form, are derived as [11,12]:

_XX ¼ AX þ BMz þ Ed ð1Þ

Fig. 2. Plan view of the vehicle dynamics model.

E. Esmailzadeh et al. / Mechatronics 13 (2003) 659–675 661

where

X ¼ vr

� �; A ¼ a11 a12

a21 a22

� �; B ¼ b1

b2

� �; E ¼ e1

e2

� �

and

a11 ¼ �2cr þ cf

mu; a12 ¼ 2

ðbcr � acfÞmu

� u; a21 ¼ 2ðbcr � acfÞ

Iu;

a22 ¼ �2ðb2cr þ a2cfÞ

Iu; b1 ¼ 0; b2 ¼

1

I; e1 ¼

2cf

m; e2 ¼

2acf

I

For the vehicle model, the lateral velocity v and the yaw rate r are considered as

the two state variables while the yaw moment Mz is the control input, which must be

determined from the control law. Moreover, the vehicle steering angle d is consideredas the external disturbance.

The cornering stiffness coefficients of the front and rear axles are cf and cr, re-

spectively. The mass of the vehicle is m and its mass moment of inertia about Z-axis

is I. The distances of the mass center from the front and rear axles are a and b,

respectively. The vehicle is traveling forward with the constant speed u.

3. Structure of the control law

The control law consists of the disturbance feed-forward signal, which is relatedto the input steering angle, and the two state variable feedback terms being those of

the yaw rate and the lateral velocity.

Mz ¼ Krr þ Kvvþ Kdd ð2Þ

In the above control law, according to the current state of technology, direct

measurements of the yaw rate r and the steering angle d are quite feasible. However,

due to the impracticality of direct measurement of the lateral velocity v, the online

estimation of this state variable is most desirable. As one of the few possible ap-proaches, this estimation can rely on the measurement of the respective longitudinal

and lateral accelerations ax and ay, respectively. This approach is based on the

integration and filtering of the following well-known equations of the longitudinal

and lateral accelerations:

_uu ¼ ax þ vr; and _vv ¼ ay � ur ð3Þ

In order to obtain further insights into the above-mentioned signals, an analytical

development of the feedback gain and the feed-forward gain needs to be fully ad-

dressed.


4. Optimal handling performance index

It has been stated before that improved steerability and reliable stability are thetwo important aspects of the optimum vehicle handling. One could hence define the

performance index for the optimum road handling of a vehicle in the following form

J ¼Z tf

to

1

2ðr

�� rdÞ2

�dt ð4Þ

where rd is the desired yaw rate of the vehicle concerned.

Minimization of the performance index must be sought in order to obtain the

optimum handling behavior. The desired value of the yaw rate can be determined byconsidering a vehicle moving with a constant forward speed while negotiating a

steady state cornering maneuver. Constant radius turn of a vehicle, being referred to

as the neutral steer behavior, could be achieved when the yaw rate is given by the

following relationship

rd ¼ uld ð5Þ

where u, l and d represent the vehicle longitudinal speed, the vehicle wheel base, andthe steering angle, respectively. The existence of the vehicle drift out (as the under-

steering behavior) and the vehicle spin (being the over-steering behavior) are judged

by whether the actual value of the yaw rate is greater or less than the calculated value

of rd.

It can be noted that in accordance to the above definition, the term (r) rd) in the

performance index is a measure of the vehicle steerability. Minimization of this term

leads the vehicle to a neutral steer behavior, and because of the great under-steer

tendency of most vehicles, therefore a stronger steerability is most needed. Although,this is a good behavior for the case of vehicles driving on low friction roads (specially

for the front-wheel drive vehicles), but in general, the design of controller, based on

the above definition of rd, can influence the vehicle stability particularly at high

speeds.

Therefore, the value of rd can be modified in terms of the longitudinal speed as

rd ¼ ulð1 þ kusu2Þ d ð6Þ

where kus is a positive constant and its increase lead the vehicle to an under-steering

behavior and a more severe stability. Eq. (6) could be rewritten in the following form

rd ¼ fud ð7Þwhere fu is a general function of the vehicle speed u.

Now let us consider an expression for the yaw rate differential term Dr as

Dr ¼ rd � r ð8Þwhere r is the actual yaw rate. A very good handling performance is characterized by

Dr � 0 ð9Þ


It is important to note that the control effort Mz must satisfy some physical

constrains due to both the actuation system and the road-tire performance limits. To

satisfy those limits the control effort Mz in the above performance index must be

modified as in the following form

J ¼Z tf

to

1

2ðr

�� rdÞ2 þ

1

2wM 2

z

�dt ð10Þ

where w is a weighting factor that indicates the relative importance of the corre-

sponding term.

By proper selection of the weighting factor in (10) and the appropriate design of

the control law, in order to minimize the performance index, one can not only

achieve an excellent handling behavior, but also satisfy the physical limits of thedynamic system.

5. Optimal controller design

To determine the values of the feedback and feed-forward control gains, which

are based on the defined performance index and the vehicle dynamic model, a LQR

problem has been formulated for which its analytical solution is obtained, [13].In that case, the performance index of Eq. (10) may be rewritten in the following

form

JðuÞ ¼ 1

2

Z 1

0

½UTRU þ ðXd � X ÞTQðXd � X Þdt ð11Þ

where

U ¼ ½Mz; R ¼ ½w; Q ¼ 0 0

0 1

� �; and Xd ¼ 0

fud

� �

The Hamiltonian function, in the expanded form, is therefore given by:

HðuÞ ¼ 12UTRU þ 1

2ðXd � X ÞTQðXd � X Þ þ PTðAX þ BU þ EW Þ ð12Þ

where

P ¼ p1

p2

� �; and W ¼ ½d

with the parameters p1 and p2 are the Lagrangian multipliers.

The co-state equations can be presented in the following form:

_PP ¼ � oHoX

¼ QðXd � X Þ � ATP ð13Þ

Moreover, the algebraic relationships that must be satisfied are given by:

oHoU

¼ RU � BTP ¼ 0 ð14Þ


and in a more simplified form:

U ¼ R�1BTP ð15Þ

Now let us write the matrix P in the following form:

P ¼ KX þ S ð16Þ

where K is a symmetric matrix that includes the state feedback gains, and S being the

feed-forward control effort vector.

Finally, combining Eqs. (15) and (16), the control law could be obtained as:

U ¼ R�1BTðKX þ SÞ ð17Þ

This equation can now be rewritten in the following form:

Mz ¼ � 1

Iwðk12vþ k22r þ s2Þ ð18Þ

In general, the system of Eqs. (1), (13) and (14) form a system of non-linear or-

dinary differential equations, which can be converted into a non-linear algebraic

system of equations by assuming that the solutions of the equations converge rapidly

to the constant values. Therefore, this would lead to:

_KK ¼ 0; and _SS ¼ 0 ð19Þ

This implies that the time varying gains are replaced by those with the constantvalue gains, which makes the control law quite simple and yet in a more practical

form. Using the above assumptions, the following system of algebraic equations

could then be formed:

ATK þ KAþ Q� KBR�1BTK ¼ 0 ð20Þ

and

ðAT þ KBR�1BTÞS � QXd þ EW ¼ 0 ð21Þ

Due to the symmetry of the matrix K, Eq. (20) can now be expanded into the

following three independent equations:

2a11k11 þ 2a21k12 �k2

12

I2w¼ 0 ð22Þ

a11k12 þ a21k22 þ a12k11 þ a22k12 �k12k22

I2w¼ 0 ð23Þ

and

1 þ 2a12k12 þ 2a22k22 �k2

22

I2w¼ 0 ð24Þ

The system of equations given above can therefore be solved analytically in order

to determine the corresponding values of the feedback gains


k22 ¼1

aTA

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT 2

A � 2DA þ a þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2

A þ aa211

qr !ð25Þ

where

TA ¼ a11 þ a22 ð26ÞDA ¼ a11a22 � a12a21 ð27Þ

and

a ¼ 1=I2w ð28ÞThe other values of the feedback gains can now be expressed as a function of k22

k12 ¼1

2a12

ak222

� 2a22k22 � 1

�ð29Þ

and

k11 ¼1

2a212

TA

�þ ð2DA þ 2a2

22 � aÞk22 � aða11 þ 3a22Þk222 þ a2k3

22

�ð30Þ

Using Eqs. (20), (21), (25), (29), (30) the feed-forward part of the controllers can

therefore be found as:

s2 ¼ kdd ð31Þ

where

kd ¼ a12e1k11 þ ða12e2 � a11e1Þk12 � a11e2k22 þ fua11

DA þ aða12k12 � k22a11Þð32Þ

According to Eqs. (2) and (18), the values of the feedback and feed-forward gainscan hence be expressed as:

Kr ¼ � 1

Iwk22; Kv ¼ � 1

Iwk12; and Kd ¼ � 1

Iwkd ð33Þ

where k22, k12, and kd have been defined by (25), (29) and (32), respectively.The variation of the optimal values of the feedback and feed-forward gains

with the vehicle velocity, for different values of the weighting factor w is shown in

Fig. 3. The vehicle considered here is a motorized wheel electric car with its speci-

fication data presented in Table 1. Due to the installation of a heavy battery box at

the rear of the car, its mass center has been shifted closer to the rear axle, which

makes the car inherently over-steer. As mentioned before, the implementation of

the yaw moment control is considered a way of improving the dynamic behavior

of vehicles from an over-steering behavior to an under-steering one.It can be seen from Fig. 3 that the yaw rate gain Kr is always negative and its

magnitude increases rapidly with the increase in vehicle speed u. On the other hand,

the magnitude of the yaw rate gain decreases when the value of w increased. The

lateral velocity gain Kv has positive values and the variation of its magnitude with u

and w are quite similar to Kr, but its magnitude is relatively smaller than Kr. The


variation of the steer angle gain Kd with the vehicle speed is completely different from

those of the other two gains. Its magnitude is relatively small and positive for vehicle

speeds of up to about 25 m/s.

For higher speeds and around the critical speed of 53 m/s, the steer angle gain

decreases sharply to a very large negative value. Those large negative values of the

steering angle gain provide the negative yaw moments, which in turn can force the

vehicle to become under-steer.

Fig. 3. Variations of the optimal values of gains with vehicle speed: (a) yaw rate gain, (b) lateral velocity

gain and (c) steer angle gain.

Table 1

Specifications data of the case study vehicle

Parameter Value

a 1.25 m

b 1.20 m

cf 15 000 N/rad

cr 15 000 N/rad

I 2500 kgm2

l 2.45 m

m 1300 kg

kus 0.001

w 2e)8


The values of the feedback and feed-forward gains found were then substituted

back into Eq. (2) to obtain the optimal control laws. In order to make the control

law quite practical and much simple to use the following approximate assumption

can be put forward

Kv � 0 ð34Þ

Substituting Eq. (34) into (2) would lead us to the semi-optimal control law

Mz ¼ Krr þ Kdd ð35Þ

6. Regulation of the control law

The remaining challenge in the process of controller design is to determine the

weighting factor w. It has been shown that the dynamic performance of the con-troller is extremely sensitive to the values of the weighting factor. When the value of

w approaches zero although the best yaw rate convergence is then achieved but the

yaw moment will approach infinity. For practical applications, the power of the

electric motors and the road surface conditions could limit the values of the yaw

moment. Therefore, the weighting factor w should be determined such that the yaw

moment would always remain below the maximum admissible value during every

cornering maneuver.

In other words, this statement can be formulated by the following relationship

MaxðkMzkÞ6Mmax ð36Þ

where Mmax is the maximum admissible value of the yaw moment, which can be

defined as

Mmax ¼T2ðDFtf þ DFtrÞmax ð37Þ

where DFt is the traction force difference between the left and the right wheels and T

is the wheel track. The traction force of every wheel is a function of many parameters

including the vehicle forward velocity, the motor control signal, the transmission

system characteristics, the dynamic normal load of the wheel, and the road fric-

tion coefficient. The value of Mmax can then be predicted according to the above-

mentioned parameters [14].

For a specific type of vehicle, Eq. (36) must first be verified with the given value ofMmax and a known value of the weighting factor, w. This implies that

½u; dðuÞ 2 D; and D ¼ fu6 umax; d6 dmaxðuÞg ð38Þ

where D is referred to as the control law domain of action. dmaxðuÞ is considered theultimate value of the steering angle for every speed, which increases the value of kMzkto the value of Mmax.

The controller domain of action for three specific values of the weighting factor,

based on the given value of Mmax, is illustrated in Fig. 4.


The influence of the values of the weighting factor on the performance of the

optimal control law is another important subject that needs to be investigated. The

steady state response of the vehicle to the step steering angle input, at a constant

vehicle speed, is one of the important performance measure when evaluating the

quality of the vehicle stability and its handling. To illustrate the effect of the

weighting factor regulation on the steady state performance of the optimal con-

troller, the yaw velocity gain can be considered as an appropriate measure. Fig. 5

shows the variations of the yaw velocity gain with vehicle speed for different valuesof the weighting factor.

It can be seen that due to the over-steer behavior of this vehicle, the yaw velocity

gain increases rapidly when the speed approach to its critical value. In contrast, the

vehicle that is governed by the optimal yaw moment control shows an under-steer

Fig. 4. Effect of weighting factor values on controller domains of action.

Fig. 5. Yaw velocity gain vs. speed for controlled and uncontrolled vehicles.


behavior with its gain value remains limited over the wide speed range of the vehicle.

However, it must be noted that in order to achieve a suitable road performance one

must sufficiently decrease the value of the weighting factor.

7. Comparison of optimal and semi-optimal control laws

In this section, the dynamic performance of both versions of the controller will

be compared in order to validate the approximation put forward. The variations

of the yaw velocity gain with the vehicle speed, for the optimal and the semi-optimal

control laws, are illustrated in Fig. 6. It can be seen that the values of the gain

for both versions of the controllers are very close to each other such that one may saythey are almost identical. One should always remember that the results presented

are valid only for the linear regime of the vehicle dynamics. However, for the non-

linear regime, where the tires exhibit a non-linear stiffness behavior, the lateral ve-

locity would increase rapidly and as shown before the yaw moment control should

not only based on the yaw rate signal but also on the lateral velocity (slip angle)

feedback.

In summary, one could put forward the following important statements:

• The optimal yaw moment control can certainly improve the vehicle handling

when its performance is compared to that of the uncontrolled vehicle.

• Although the performance of the optimal control law is much improved from var-

ious aspects but those obtained with the semi-control law is also acceptable for

this specific application. On the other hand, the optimal control law requires an

additional feedback control loop (lateral velocity) that makes the control system

even more complex and expensive.

Fig. 6. Yaw velocity gain vs. speed for optimal and semi-optimal control laws.


8. Comparison with other control laws

One of the well-known laws for the vehicle yaw moment control, reported in

recent literatures, was developed by Doniselli and others [15]

Mz ¼ hr rh

� uldi

ð39Þ

where hr is a negative constant coefficient.

Regardless of the structural similarity between the above-mentioned control law

and the semi-optimal control law (both are based on the yaw rate feedback and the

steer angle feed-forward), one can point out a major difference between them. The

optimal values of the gains Kr and Kd, were substituted by a constant term and alinear function of the vehicle speed, respectively. According to the values of the

gains, presented in Fig. 3, one is not permitted to proceed in this way, since such

simplifications indicate that superior performance cannot be achieved.

To illustrate the difference between the dynamic performance of the optimal

control law and a non-optimal control law, let us consider a case study vehicle that

once governed by the optimum control law and then with the semi-optimal control

laws. Once more in this study, the yaw velocity gain is considered as a measuring

parameter. It can be seen from Fig. 7, that although the non-optimal control law wasable to relatively decrease the yaw velocity gain, but its value is not an acceptable one

yet. On the other hand, the non-optimal law produces approximate neutral steering

characteristics, which is not regarded as completely safe, especially at high speeds.

9. Numerical simulation

The numerical simulation of the vehicle dynamic response during a standard road

maneuver and under different road conditions, when the vehicle is either uncontrolled

Fig. 7. Yaw velocity gain vs. speed for semi-optimal and non-optimal control laws.


or is under optimal control, has been carried out. The considered maneuver is a

constant speed cornering, which is performed either on a dry or snow-covered road.

The computer simulation model is based on the case study vehicle with its technical

specification data given in Table 1.In order to achieve a precise and reliable set of results, the computer simulation

model is assumed as a comprehensive eight degree-of-freedom dynamic model of a

four motorized wheels EV (Fig. 8). This model has been previously developed by the

authors for a detailed dynamic analysis and simulation of an electric car [16].

The longitudinal velocity, lateral velocity, yaw rate, roll rate, and the wheels

rotational speeds are the associated degrees of freedom of the model. The study

includes all the non-linearities that are present in vehicle systems, namely, the non-

linear behavior of tires, the longitudinal and the lateral normal load transfers of tires,the roll steer effect, and the camber angle changes due to the vehicle roll. It has been

assumed that a separately excited armature controlled DC motor drives each one of

the wheels through a fixed reduction gears. Armature terminal voltage, E of each

motor is controlled through a pulse width modulation controller with the set point e.The simulation results are obtained for a constant speed cornering road maneu-

ver, performed on a dry surface road (l ¼ 1), as illustrated in Fig. 9. The vehicle

maneuvers on a level road at a constant speed of 35 m/s and the steering angle input

is set at d ¼ 0:01 rad.

Fig. 8. Three-dimensional vehicle dynamic model with eight degree-of-freedom.


Results obtained from the computer simulation indicate that the vehicle, which

governed by the optimal controller have a superior performance when compared

with the uncontrolled vehicle. Therefore, inclusion of the optimal controller in ve-

hicles can considerably improve the road handling and safety of vehicles. The time

history diagrams of Fig. 9a and b illustrate that the steady state values of the lateral

acceleration and the yaw rate for the optimal controlled vehicle are considerably less

than those for the uncontrolled vehicle as well as for the neutral steering limit. This

means that lower sensitivity of the steering system is achieved at high speeds, whichexplains better safety of the vehicle. In other words, the uncontrolled vehicle shows

an inherent over-steer behavior, but a controlled vehicle, with an optimal yaw mo-

ment control, can exhibit an under-steer behavior. Furthermore, Fig. 9c indicates

that reduction in the vehicle slip angle is an important safety criterion, which could

certainly be achieved in the controlled vehicle.

The time responses of the desirable and achievable values of the optimal yaw

moment control are illustrated in Fig. 9d. It can be seen that there is quite a dif-

ference between the desirable value being calculated by the optimal yaw momentcontroller, and the achievable value generated by the electric motors. This is due to

Fig. 9. Simulation results of constant speed cornering maneuver on dry-surface roads: (a) lateral accel-

eration, (b) yaw rate, (c) slip angle and (d) yaw moment.


the performance of the electric motors torque controller being placed in the lowest

level of the control system [16].

The simulation results obtained for a constant-speed cornering maneuver, but

with vehicle travels on a flat snow-covered road (l ¼ 0:2), are presented in Fig. 10.

For the case of the uncontrolled vehicle, the yaw rate response and that of the slip

angle are both oscillatory and in general, one can say that the dynamic behavior of

the vehicle is quite unsafe. On the other hand, by using the optimal yaw moment

control, those oscillatory responses have been rapidly converged to their steady stateconstant values and therefore, the vehicle performs a safe and controllable behavior.

10. Conclusion

External yaw moment control is an effective way to improve vehicle handling. For

future cars, due to the fundamental changes in the vehicle power-train technology,

Fig. 10. Simulation results of constant speed cornering maneuver on snow-covered roads: (a) lateral

acceleration, (b) yaw rate, (c) slip angle and (d) yaw moment.


this idea will become reality in a much simple and applicable form. In the case of

modern EVs, with an independent drive on every wheel, the ability to generate in-

dependent traction force control will be quite a challenge. Therefore, an accurateactive yaw moment control could be made feasible.

Two versions of a new optimal yaw moment control strategy, based on the op-

timal control theory, have been presented. The structure of the semi-optimal version

is based on the yaw rate feedback and the steer angle feed-forward. However, for the

fully optimal control version one must know the lateral velocity, as well as an ad-

ditional feedback. On the other hand, the performance of the optimal control version

is slightly better than the semi-optimal one. It shows that the performance of the

optimal controller, with its design based on the yaw rate following performanceindex and the linear vehicle dynamics model, is not strongly affected by the lateral

velocity feedback. Therefore, it can be concluded that, due to a more simple struc-

ture and a satisfactory performance, the semi-optimal control law in the linear

vehicle dynamics regime can be more appropriate.

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