Available via license: CC BY 4.0

Content may be subject to copyright.

Citation: Domina, Á.; Tihanyi, V.

LTV-MPC Approach for Automated

Vehicle Path Following at the Limit of

Handling. Sensors 2022,22, 5807.

https://doi.org/10.3390/s22155807

Academic Editor: Felipe Jiménez

Received: 9 June 2022

Accepted: 27 July 2022

Published: 3 August 2022

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional afﬁl-

iations.

Copyright: © 2022 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

sensors

Article

LTV-MPC Approach for Automated Vehicle Path Following

at the Limit of Handling

Ádám Domina * and Viktor Tihanyi

Department of Automotive Technologies, Budapest University of Technology and Economics,

1111 Budapest, Hungary; tihanyi.viktor@kjk.bme.hu

*Correspondence: domina.adam@edu.bme.hu

Abstract:

In this paper, a linear time-varying model predictive controller (LTV-MPC) is proposed

for automated vehicle path-following applications. In the ﬁeld of path following, the application

of nonlinear MPCs is becoming more common; however, the major disadvantage of this algorithm

is the high computational cost. During this research, the authors propose two methods to reduce

the nonlinear terms: one is a novel method to deﬁne the path-following problem by transforming

the path according to the actual state of the vehicle, while the other one is the application of a

successive linearization technique to generate the state–space representation of the vehicle used for

state prediction by the MPC. Furthermore, the dynamic effect of the steering system is examined

as well by modeling the steering dynamics with a ﬁrst-order lag. Using the proposed method, the

necessary segment of the predeﬁned path is transformed, the linearized model of the vehicle is

calculated, and the optimal steering control vector is calculated for a ﬁnite horizon at every timestep.

The longitudinal dynamics of the vehicle are controlled separately from the lateral dynamics by a PI

cruise controller. The performance of the controller is evaluated and the effect of the steering model

is examined as well.

Keywords:

automated vehicle; model predictive controller; path following; successive linearization;

vehicle dynamics

1. Introduction

Collision avoidance systems, advanced driver-assistance systems, and the many other

types of automated vehicle functions are becoming more and more popular and have

become the most related topics in the ﬁeld of automotive research. The advantages of

automatization of different vehicle functions include the potential to improve road safety,

reduce pollutant emissions and traveling times, and eliminate human errors, which are

the primary cause of accidents. An automated vehicle can avoid collisions by using the

steering and the braking system conventionally; furthermore, a new alternative solution

for accident prevention is to force the vehicle into an unstable state, in which a control

software can drive the vehicle at a level as high as a professional driver, for example, in

case of a drift maneuver [

1

–

3

]. Each of these studies aims to drive a vehicle at the level of a

professional human driver. In [

1

], an LQ controller is used for steady-state drifting, while

in [

2

], an MPC is applied for drifting in varying road surface conditions, and in [

3

], the drift

is realized by a reinforcement learning algorithm.

According to the structure of the automated vehicles, the motion planning and motion

control modules are essential to achieve these goals. In this paper, the authors focus on the

motion control layer by proposing an LTV-MPC structure for path-following tasks. Based

on the existing research and the experimental tests, the authors identify three features to

be considered in a path-following controller at higher vehicle speed, the inclusion of the

vehicle dynamics in the control law, and the knowledge of the path ahead of the vehicle.

Furthermore, the inclusion of the steering dynamics in the control law is identiﬁed as the

Sensors 2022,22, 5807. https://doi.org/10.3390/s22155807 https://www.mdpi.com/journal/sensors

Sensors 2022,22, 5807 2 of 23

third required feature, which has a high effect on the performance of the path-following

controller. Numerous path-following solutions are summarized in [

4

–

6

], e.g., geometric

controllers, such as Pure Pursuit and Stanley, linear quadratic regulators, model predictive

controllers, neural network based controllers, etc. The authors decided to apply MPC for

the path-following tasks, as the identiﬁed features can be incorporated into an MPC and

the MPC handles constraints well.

Three main types of MPC methods are used in the ﬁeld of path tracking, the linear

parameter-varying MPC (LPV-MPC) [

7

–

9

] the linear time-varying MPC (LTV-MPC) [

10

–

13

],

and the nonlinear MPC (NLMPC) [

10

,

14

–

16

] solutions. The LPV-MPC applies a linear

vehicle model for state prediction and the structure of that model does not change over

time; however, a few of the variables, e.g., the velocity of the vehicle, can change. The

model is calculated at each timestep based on the current states. The main advantage

of the LPV-MPC method is the low computational cost; however, this technique reaches

the limit of applicability when the controlled system leaves the linear region, where the

system model is deﬁned. Thus, the LPV-MPC method is unable to handle nonlinear system

dynamics in a wide range, e.g., tire dynamics in the nonlinear region of the tires. The change

in given parameters can be determined by measurements or estimated by state estimators.

The LTV-MPC [

10

–

12

,

17

] can handle the nonlinear behavior of the system by lineariz-

ing the nonlinear system at every timestep and estimating the states accordingly. The

predicted states are still based on a linear system model; however, these are extremely

close to the real states of the system, since the controller calculates the predicted states for

a short horizon, e.g., a few hundred ms, in which range the prediction is acceptable. In

the LTV-MPC structure, the current vehicle state is the basis of the linearization and an

additional transformation in the system matrices and the control input is required. The

LTV-MPC method claims a bit more computational cost than the LPV-MPC; however, it

still can run on an ECU or a rapid prototyping unit in real-time. When using LPV-MPC

or LTV-MPC, usually, a quadratic cost function is formulated, the optimum of which is

computationally cheaper to ﬁnd.

Applying the NLMPC, the evolution matrices are calculated based on a nonlinear

system model without linearization. A nonlinear cost function is formulated, the optimum

of which is computationally expensive to ﬁnd, and might limit the applicability of the

NLMPC, as discussed in [6].

In this paper, the authors propose an LTV-MPC method for automated vehicle path-

following scenarios. The aim of this research is to apply a nonlinear vehicle model for state

prediction, handled by a successive linearization technique, where the vehicle model is

linearized at every timestep by Jacobi linearization. The applied vehicle model is coupled

with a Pacejka tire model, the parameters of which are identiﬁed based on measurement

results. The application of a nonlinear vehicle model coupled with the Pacejka model is

expected to be suitable for handling the high sideslip angles of the tires, where the tires

are operating in the nonlinear region–which the LPV-MPC is unable to handle–hence, the

vehicle should be able to perform the path-following task at the limit of handling.

In the existing research, if the path is deﬁned in a forward-looking way, it is per-

formed by deﬁning the reference path in the global coordinate system by a series of (x,y)

points

[10,12,18–21]

. While this solution is obvious, it incorporates high nonlinearity into

the plant model, since the motion of the vehicle needs to be transformed from the vehicle’s

coordinate system–i.e., the ego frame–to the global frame. A novel method for the deﬁnition

of the reference path is presented, by which the number of the nonlinear terms involved in

the entire control problem is successfully reduced. According to our proposal, the path is

transformed from the global coordinate system into the vehicle’s coordinate system and

the reference states, which need to be realized by the vehicle, are expressed according to

the state–space representation of the vehicle plant model used for state prediction by the

MPC. In the existing research, the reference path is either not deﬁned in a forward-looking

way [

7

,

22

], or the forward-looking property is limited to the lateral displacement of the

Sensors 2022,22, 5807 3 of 23

vehicle [

8

,

12

]. In this article, the authors deﬁne both the lateral displacement and the

heading angle as a reference series for a ﬁnite horizon, as presented in Section 3.

Some studies considered the effect of the actuator fault [

23

] and the robustness of

the controlled system [

24

]; in this article, the effect of the steering system is analyzed.

The effect of the consideration of the steering dynamics in the vehicle model applied for

state prediction is discussed in this paper as well. The steering system is modeled as a

ﬁrst-order lag. Moreover, the performance of the controller is evaluated with and without

the inclusion of the steering dynamics in the plant model by applying the control input to

the same vehicle model, which includes the steering dynamics.

The contribution of this paper is twofold, (1) The proposed LTV-MPC method for path

following includes a novel reference deﬁnition method that effectively reduces the nonlinear

terms of the path-following problem by transforming the path from the global frame to

the ego frame. By this transformation, the nonlinearity that needs to be managed during

the Jacobian linearization is signiﬁcantly reduced as the state vector has two elements less,

since the position of the vehicle in the global coordinate system is not considered in the

plant model. This approach removes two equations from the model while retaining the

lookahead principle of the reference deﬁnition. (2) The analysis of the effect of the steering

dynamics on the path-following problem are incorporated in an LTV-MPC.

The structure of this paper is as follows. Section 2presents the vehicle model applied

in the MPC for state prediction, regarding the steering dynamics and the vehicle model

used for testing the controller. In Section 3, the generation of the reference path is presented.

The structure of the proposed MPC and the derivation of the quadratic programming (QP)

cost function is presented in Section 4. Section 5describes the results, while Section 6

provides an analysis, in which the controller is tested on a sine wave path and the effect

of the application of linear and nonlinear vehicle models in the state prediction is com-

pared. Finally, Section 7discusses the concluding remarks and suggestions for further

research directions.

2. Vehicle Modelling

In this section, the applied tire and vehicle models are presented. The authors use

different vehicle models for state prediction and for simulation testing. Furthermore, a

Pacejka tire model is applied in both models, the parameters of which are identiﬁed from

measurement data.

2.1. Vehicle Model for Testing

A four-wheel vehicle model is used for testing the path-following controller in a

simulation environment. The roll and pitch dynamics are neglected in the vehicle model

since they have no signiﬁcant effect on the path-following problem. The model describes

the planar dynamics of the vehicle: the angular acceleration around the vertical axis (1), the

longitudinal (2), and the lateral acceleration (3).

.

r=1

Iz

Mz(1)

ax=1

mFV

x+Vy

.

r(2)

ay=1

mFV

y−Vx

.

r(3)

where ris the yaw rate, I

z

is the moment of inertia of the vehicle around the axis z,M

z

is the

resultant torque, which rotates the vehicle around axis z,a

x

is the longitudinal acceleration,

mis the mass of the vehicle, F

xV

is the resultant of the longitudinal forces, acting on the

vehicle, V

y

is the lateral velocity of the vehicle in the ego coordinate system, a

y

is the lateral

acceleration, F

yV

is the resultant of the lateral forces acting on the vehicle, and V

x

is the

longitudinal velocity in the ego frame.

Sensors 2022,22, 5807 4 of 23

The resultant torque and forces are calculated by (4), (5), and (6)

Mz=−b(FyRL +FyRR) + sf(−FxFLsin(δFL )+FxFRsin(δF R)) + sr(−FxRL +FxRR)

+aFyFL cos(δFL)+FyFR cos(δFR )(4)

FV

x=FxRL +FxRR +FyFLsin(δF L )+FyFRsin(δFR )+FxFL cos(δF L)+FxFRcos(δFR )(5)

FV

y=FyRL +FyRR +FxF L sin(δFL )+FxFRsin(δF R )+FyFLcos(δFL )+FyFR cos(δFR)(6)

where aand bare the distances between the center of gravity (C.G.) and the front and the

rear axle, respectively, s

f

and s

r

are the front and rear track of the vehicle, respectively,

δFL

and

δFR

are the front-left and front-right steering angles of the individual wheels, F

xij

and

F

yij

are the longitudinal and lateral forces at the individual wheels, where imarks the front

and the rear axles, hence, i=Ras rear or i=Fas front, and jmarks the left and the right

wheels, hence, j=Las left or j=Ras right. In this model, the authors consider a rear-wheel

drive vehicle, with equally distributed traction force on the rear wheels, hence, F

xRL

and

F

xRR

are equal. Furthermore, the authors assume a front-wheel steering vehicle; hence, the

steering angle is solely interpreted at the front wheels.

In Figure 1, the tire sideslip angles and the sideslip angle of the vehicle at C.G. are also

shown. The vehicle sideslip angle is calculated by (7).

β=tan−1Vy

Vx(7)

Figure 1. The four-wheel vehicle model applied for controller testing.

The sideslip angle of an individual wheel is calculated using the velocity vector of the

given wheel and the steering angle at the front axle. The velocity vector of the wheels is

described by (8), the position vector of the wheels is deﬁned by (9), where the origin is the

C.G. of the vehicle, and the tire sideslip angles are described by (10) and (11).

vi=

vxi

vyi

vzi

=

Vx

Vy

Vz

+

0

0

r

×Pi(8)

PFL =ha sf0iT,PFR =ha−sf0iT,PRL =−b sr0T,PRR =−b−sr0T(9)

αFL =tan−1vy FL

vxFL −δFL ,αFR =tan−1vyFR

vxFR −δFR (10)

αRL =tan−1vyRL

vxRL ,αRR =tan−1vyRR

vxRR (11)

Sensors 2022,22, 5807 5 of 23

where i=FL

∨

FR

∨

RL

∨

RR and

α

denotes the tire sideslip angle. The horizontal velocity

V

z

in (8) is considered zero as the model focuses solely on the planar dynamics of the

vehicle, as stated previously.

A Pacejka tire model [

20

] is applied to calculate the lateral forces at each wheel. During

this research, the vehicle moves at constant speed; thus, the longitudinal forces are small,

especially compared to the lateral forces and, hence, solely a lateral tire model is used in

this article. The Pacejka tire model is described by Equations (12) and (13) [25].

Y=DsinCtan−1(Bφ)+Sv(12)

φ=(1−E)(α+Sh)+(E/B)tan−1(B(α+Sh)) (13)

where Bis the stiffness factor, Cis the shape factor, Dis the peak factor, Eis the curvature

factor, S

v

and S

h

are the vertical and the horizontal shift, respectively, and the

α

sideslip

angle is deﬁned in degrees. The Pacejka tire model and the meaning of the factor variables

are described in more detail in [25].

The entire vehicle model, including the tire model, is ﬁtted to a test vehicle used for

automated vehicle function tests, such as automated drift, Moose test, and other path-

following tasks. Further details of the test vehicle setup can be found in [1].

To determine the characteristics of the tires, a ramp steer maneuver is taken by the test

vehicle and the necessary variables are measured with the data acquisition system. The

parameters in the Pacejka tire model are ﬁtted to the measurement results, as shown in

Figure 2.

Figure 2. The measured and identiﬁed tire characteristics.

The characteristics of the rear tires have a greater slope at small sideslip angles, where

the tire model is nearly linear. That means the rear tires have greater cornering stiffness

than the front tires. This meets the expectations of the authors, since the front wheels

of the vehicle are mounted with 245/35 R19 tires, while 265/35 R19 tires are applied at

the rear wheels—using the same sidewall height, the wider tire becomes stiffer. During

the identiﬁcation of the tire characteristics, the vertical load of the tires is assumed to

be constant.

The identiﬁed tire models describe the characteristics of the tires located on the same

axle, not the individual wheels; hence, the results in Figure 2contain the lateral forces

generated by the two front tire pairs and the two rear tire pairs, respectively. Thus, when

modeling one wheel, the value of the lateral force needs to be halved.

Sensors 2022,22, 5807 6 of 23

2.2. Vehicle Model for State Prediction

The states of the vehicle model applied in the MPC for state prediction are advisable

to choose in a way to correspond to the control purpose. In this case, the aim is to drive the

vehicle along a predeﬁned path as fast as the vehicle is still able to execute the maneuver.

The applied vehicle model shown in Figure 3is a dynamic bicycle model, coupled with

different Pacejka tire models at the front and the rear wheels. Since the bicycle model

applies one wheel per axle, the identiﬁed Pacejka models can be used without halving the

values of the lateral forces of the tire characteristics. The applied vehicle model describes the

dynamics of the vehicle with four equations, enhanced by a ﬁfth equation for the steering

dynamics. The steering dynamics are considered to take into account the dynamic lag of

the steering system, in which way the controller considers that the demanded steering

angle will only be realized with a time delay. The dynamics of the steering system are

modeled by a ﬁrst-order lag, which has one tuning parameter, the time constant, to identify

the measurements made. The time constant is determined by measurements conducted on

the test vehicle.

Figure 3. The vehicle model applied to the MPC for state prediction.

The state vector of the system is chosen as x = [

v Vyϕrδact

]

T

, where yis the lateral

displacement of the vehicle in the ego frame,

ϕ

is the heading angle of the vehicle, and

δact

is the actual steering angle. The vehicle dynamics are described by (14)–(17), while the

steering dynamics by (18).

Vy=.

y(14)

ay=.

Vy= Fy f cos(δ) + Fyr

m!−Vxr(15)

r=.

ϕ(16)

.

r=aFy f cos(δ)−bFyr

Iz(17)

.

δact =−1

Tst

δact +1

Tst

δdem (18)

where

δdem

is the steering angle required by the MPC and T

st

is the time constant of the

ﬁrst-order steering model. To calculate the F

yf

and F

yr

lateral forces, the identiﬁed Pacejka

models are applied by determining the parameters of (12) and (13). The sideslip angles are

calculated by (19) and (20).

vyF =.

vy+ar,vyR =.

vy−br (19)

αF=tan−1vyF

vxF −δF,αR=tan−1vyR

vxR (20)

where v

yF

and v

yR

are the velocity of the front and the rear wheels and

αF

and

αR

are

the sideslip angles of the front and the rear wheels, respectively. The tests are conducted

without the consideration of the steering dynamics, in which case, the state vector is chosen

as x = [v Vyϕr]T, and (18) is neglected during the state prediction.

Sensors 2022,22, 5807 7 of 23

The future states of a vehicle using a vehicle model can be predicted for a ﬁnite time

horizon. In this article, the authors apply successive linearization to continuously generate

the state–space representation of the vehicle. The successive linearization allows linearizing

the nonlinear vehicle model at the current operating point, calculating the state–space

representation accordingly, and making the state prediction based on the linearized system.

The linearization is conducted at every timestep; hence, the MPC can use the latest state of

the vehicle as a basis of the state prediction.

The linearization of the vehicle model is solved by Jacobian linearization and leads to

the continuous-time [Ac,Bc,Cc,Dc] state–space representation of the system (21).

Ac(i,j)=∂Fi

∂xj

,Bc(i,j)=∂Fi

∂uj

,Cc(i,j)=∂zi

∂xj

,Dc(i,j)=∂zi

∂uj

(21)

where Fis the system of nonlinear equations (14)–(18), xis the state vector, uis the control

input, which, in this case, is the demanded steering angle

δdem

, while C

c

and D

c

are

considered constant matrices (22).

Cc=10000

00100,Dc=0

0(22)

Furthermore, matrix B

c

remains constant during the linearization in the following

form (23).

Bc=h0000 1

Tst iT(23)

The partial derivatives of the matrices are evaluated at the desired operation point,

which is speciﬁed by the state vector x

o

, the time derivative of the state vector

.

xo

, and the

control vector u

o

and, thereby, the partial derivatives in (21) need to be calculated based

on the initial conditions

x=xo

,

.

x=.

xo

and

u=uo

. The evolution of the continuous-time

system can be described by (24)

.

xL=.

xo+Ac(xL−xo)+Bc(uL−uo)(24)

where x

L

and u

L

are the linearized state of the system and the control input, respectively.

The constant terms of (24) are incorporated into K

c

, with which the continuous-time

representation of the state–space system becomes (25) and (26).

.

xL=AcxL+BcuL+Kc,Kc=.

xo−Acxo−Bcuo(25)

yL=CcxL+DcuL(26)

where yLis the output of the system.

Since the MPC is a discrete-time-control technique, the state–space representation

needs to be discretized to get the discrete-time state–space representation of the system

[Ad, Bd, Cd, Dd] and Kd(27)–(30).

Ad=eAcTs(27)

Bd=A−1

ceAcTs−IBc(28)

Kd=A−1

ceAcTs−IKc(29)

Cd=Cc,Dd=Dc(30)

where T

s

is the discretization timestep, which is equal to the timestep value of the MPC.

Finally, the discrete-time representation of the system (31) can be used for state prediction

by the MPC.

x(k+1)=Adx(k) + Bcu(k) + Kd

y(k+1)=Cdx(k+1)+Ddu(k+1)(31)

Sensors 2022,22, 5807 8 of 23

3. Reference Path Deﬁnition

As previously introduced, in numerous studies, the control problem is to minimize the

error between the spatial reference path and the position of the vehicle. In this research, the

state–space representation of the vehicle contains the position of the vehicle in the global

frame, according to the reference deﬁnition, where the path is deﬁned by a series of (x,y)

points in the global frame. The controlled state is the spatial position of the vehicle—X

and Y coordinates in the global frame. The spatial path-following problem is possibly

supplemented with the tracking of the heading and the yaw rate. The problem with the

formulation when the X and Y coordinates are deﬁned as a reference is that it introduces

several nonlinear terms into the vehicle model applied for state prediction by the MPC

and increases the dimension of the state vector by two. Therefore, if the control problem is

deﬁned to follow spatial points, the vehicle needs to be transformed to the global frame in

the state–space representation—this transformation is responsible for higher nonlinearity.

Thus, if the increase in the dimension of state vector xby two—displacement in X and

Y directions—can be omitted, then the computational requirements of the MPC can be

reduced, which increases directly proportionally with the increase in the dimension of

state vector x, as presented in [

26

]. In this article, a novel method is proposed for the

reduction in nonlinear terms, which is to transform the path to the vehicle and describe the

path-following task in a way that retains the advantageous property of the MPC, which

is the knowledge of the path ahead of the vehicle—the knowledge of the reference for a

ﬁnite horizon. In this article, the lateral displacement and the heading of the vehicle are

deﬁned as a reference to be followed, in accordance with the states of the vehicle model

used for state prediction. The reference state variables are chosen as they clearly deﬁne the

relationship between the vehicle and the path, with respect to the lateral terms. To calculate

the reference for a ﬁnite horizon, ﬁrstly, the lateral and angular errors are deﬁned, as shown

in Figure 4. The lateral error eis the distance between the C.G. point of the vehicle and the

intersection point Mon the yaxis of the ego frame and the reference path.

Figure 4. The interpretation of the lateral and angular errors.

The angular error

α

is the angle between the heading of the vehicle and the tangent of

the path at the intersection point M. To generate the reference for the lateral displacement

state yand for the heading angle state

ϕ

, the vehicle, the reference path, the ego frame, and

the global frame need to be considered, as shown in Figure 5. In Figure 5, the lateral errors

and the angle errors are shown for a ﬁnite N

p

horizon, where N

p

is the prediction horizon of

the MPC. If the lateral error values from the ego coordinate system can be seen, these errors

can be interpreted as reference lateral displacements, which corresponds to y, the ﬁrst state

of the state–space representation. In a similar way, the angular errors can be interpreted

as

ϕ

reference heading angles. By applying this method for reference generation, the

Sensors 2022,22, 5807 9 of 23

transformation of the vehicle coordinates from the ego frame to the global frame becomes

avoidable, resulting in omitting several nonlinear terms from the vehicle model.

Figure 5.

The interpretation of the lateral error reference and orientation angle reference in both ego

frame and global frame.

As stated before, if the vehicle model is not transformed into the global frame, the path

needs to be transformed into the ego frame of the vehicle. The transformation is deﬁned by

Equations (32)–(34),

γ=α1+ϕ(32)

H=0

e1(33)

Pre f ,i=H+cos(γ)−sin(γ)

sin(γ)cos(γ)Pi(34)

where

γ

is the transformation angle, e

1

and

α1

are the lateral and angular errors at the M

point, respectively, His an offset vector, which shifts the entire reference trajectory to the

necessary lateral displacement, P

i

is the (x,y) coordinate of the ith path point described in

the global frame, and P

ref,i

is the coordinate of the reference path point in the ego frame.

After the transformation, the lateral error references and the angular error references can be

calculated; the lateral error reference is the y-direction component of P

ref,i

, which determines

the series of e

1

, e

2

,

. . .

e

Np

, while the angular reference can be calculated considering the

tangent of the transformed path at every point where the lateral displacements are deﬁned.

After the reference values are calculated, the reference vector can be generated by

(35) and (36).

t=ei

αi(35)

T=t1t2· · · tNpT=he1α1e2α2· · · eNpαNpiT(36)

While the original reference path is described by N

p

points, the reference lateral

displacements and the heading angles need to be calculated for every path point. The result

is a stacked matrix Twith a dimension of 2

×

N

p

. Using this method of reference deﬁnition,

several nonlinear terms are omitted from the vehicle model, the reference is coherent to

the states of the vehicle model, the reference lateral displacement is coupled with the ﬁrst

state in the state–space model, and the angle reference is coupled with the third state.

Furthermore, the presented reference generation method preserves the forward-looking

nature of the problem deﬁnition, which is necessary for the path-following task.

Sensors 2022,22, 5807 10 of 23

4. MPC Structure

As stated in Section 2, the authors apply a successive linearization technique, similar

to [

12

,

14

,

17

], to handle the nonlinear vehicle model. During the successive linearization,

the nonlinear vehicle model is linearized at every timestep based on the actual state vector

of the vehicle. The resulting state–space representation is applied to predict the future

vehicle states for a ﬁnite prediction horizon, the length of which is N

p

. Using an MPC

controller, the objective is to minimize the difference between the given reference states

in the system and the predicted states in the systems by calculating the optimal control

input vector as a result of an optimization process, while meeting a set of constraints. The

optimization process requires a cost function to minimize. In this section, the derivation

of the cost function is presented. The cost function depends on the tracking error and

the amount of the control input. The tracking error e(k) in the k-th timestep is deﬁned as

e(k) = y(k)−r(k)

, where y(k) is the current state of the system and r(k) is the given reference.

Then, the evolution of the error can be deﬁned for the k-th, k+ 1-th, and k+ 2-th timesteps,

etc., by (37).

e(k) = Cdx(k) + Ddu(k)−r(k)

e(k+1)=Cdx(k+1)+Ddu(k+1)−r(k+1)

=CdAdx(k) + CdBdu(k) + CdKd+Ddu(k+1)−r(k+1)

e(k+2)=Cdx(k+2)+Ddu(k+2)−r(k+2)=

CdA2

dx(k) + CdAdBdu(k) + CdBdud(k+1)+CdAdKd+CdKd+

Ddu(k+2)−r(k+2)

(37)

The evolution of the error needs to be calculated for the entire prediction horizon,

resulting in the error vector

e∈RNpNo

, where N

o

is the number of outputs in the system,

which is equal to the rows of matrix C

d

. In this case, N

o

= 2 corresponds to the lateral

displacement and the heading angle references.

e==

Px(k) + =

Hu +=

EKd−r(38)

e=

e(k)

e(k+1)

e(k+2)

.

.

.

ek+Np−1

,=

P=

Cd

CdAd

CdA2

d

.

.

.

CdANp−1

d

(39a)

=

H=

Dd0 0 . . .

CdBdDd0 . . .

CdAdBdCdBdDd. . .

.

.

..

.

..

.

....

CdANp−2

dBdCdANp−3

dBdCdANp−4

dBd. . .

(39b)

u=

u(k)

u(k+1)

u(k+2)

.

.

.

uk+Np−1

,=

E=

0

Cd

Cd(I+Ad)

.

.

.

CdI+∑Np−2

i=1Ai

d

(39c)

Sensors 2022,22, 5807 11 of 23

r=

r(k)

r(k+1)

r(k+2)

.

.

.

rNp−1

(39d)

where

u∈RNpNu

is the control input vector, i.e., the result of the optimization process, N

u

is the number of the control inputs, which is equal to the columns of matrix B

d

—in this

case—N

u

= 1 corresponding to the steering input,

=

P∈RNpNo×Nx

,N

x

is the dimension of

the state vector,

=

H∈RNpNo×NpNu

,

=

E∈RNpNo×Nx

,

r∈RNpNo

is the reference matrix, which

deﬁnes the reference for the N

p

horizon. In (39), P,H, and Eare the error evolution matrices.

The constant terms in (38) can be combined as

=

K==

EKd−r

, which leads to a more compact

form of e.

e==

Px(k) + =

Hu +=

K(40)

As the evolution of the tracking error is calculated, the cost function can be deﬁned as

J(k) = 1

2e(k)T=

Qe(k) + u(k)T=

Ru(k)(41)

where

=

Q∈RNpNo×NpNo

penalizes the deviation from the reference states and

=

R∈RNpNu×NpNu

penalizes the number of the control input. Both

=

Q

and

=

R

are diagonal square matrices, built

by using matrix

=

q

and scalar

r

, according to (42) and (43), where q

1

and q

2

are the weights

of the deviation from the reference lateral distance and the heading angle, respectively, and

ris the weight of the control input.

=

Q=

=

q0 . . . 0

0=

q. . . 0

.

.

..

.

....0

0 0 . . . =

q

,=

R=

r0 . . . 0

0r. . . 0

.

.

..

.

....0

0 0 . . . r

(42)

=

q=q10

0q2,r= [r](43)

In this article, the authors apply a quadratic cost function (41), the optimum of which

is found by the built-in numerical solver in the MATLAB software package, called quadprog.

Substituting (40) into (41) results in

J(k) = 1

2=

Px(k) + =

Hu +=

KT=

Q=

Px(k) + =

Hu +=

K+u(k)T=

Ru(k))(44)

The following task is to aggregate the terms that do not depend on the control input

u

and aggregate the quadratic and linear terms of (44), which leads to (45).

J(k) = 1

2u(k)T

=

G

z }| {

=

HT=

Q=

H+=

Ru(k) +

=

WT

z }| {

x(k)T=

PT+

=

KT=

Q=

Hu(k)(45)

A more compact form of the cost function is in (46).

J(k) = 1

2u(k)T=

Gu(k) +

=

WTu(k)(46)

Sensors 2022,22, 5807 12 of 23

When solving optimal control problems, the aim is to minimize the tracking errors

and the amount of control input. However, a steady-state error might occur while tracking,

since the cost of the steady-state error might be less than the cost of the higher value of the

control inputs. To avoid the steady-state error, the cost function is deﬁned by the increments

in the current control input, rather than deﬁning the individual control inputs. Applying

this solution, the effect of the control input values on the cost function is minimal. The

control input vector can be transformed as

u(k) = u(k−1)+ρu1

u(k+1)=u(k) + ρu2=u(k−1)+ρu1+ρu2

uk+Np−1=uk+Np−2+ρuNp=u(k−1)+ρu1+ρu2+. . . +ρuNp

(47)

which can be written as

u

z }| {

u(k)

u(k+1)

.

.

.

uk+Np−1

=

u(k−1)

z }| {

u(k−1)

u(k−1)

.

.

.

u(k−1)

+

=

D

z }| {

I0 . . .

I I . . .

.

.

..

.

....

I I . . .

ρu(k)

z }| {

ρu1

ρu2

.

.

.

ρuNp

(48)

where

=

D∈RNpNu×NpNu

is a lower diagonal matrix built of

I∈RNu×Nu

matrices and zero

matrices of the same dimensions. Performing the transformation, the evolution of the

tracking error can be written as

e==

Px(k) + =

Hu(k−1)+=

Dρu(k)+=

K(49)

where

u(k−1)

is constant for the entire prediction horizon, thus, it can be included in the

constant term

=

K

. The new const function, the solution of which is the optimal series of

control input increments, is given in the following form

JD(k) = 1

2ρu(k)T=

GDρu(k) +

=

WT

Dρu(k)(50)

where the matrices are deﬁned as =

GD=DT=

GD and =

WT

D=uT(k−1)=

GD.

5. Results

A reference path is deﬁned for testing the controller shown in Figure 6. The path

contains a double-lane-change section, which is considered an evasive maneuver [

27

], and

a U-turn section, which is built by a clothoid segment, marked with Aand Bin Figure 6. As

the vehicle drives along the U-turn section, the curvature of the path is the linear function

of the length of the arc. During the simulations, the vehicle is started from the left side of

the path, using a 0.2 m offset, which is applied to examine how the vehicle can ﬁnd the path.

The simulations are conducted at 50, 60, and 70 km/h, using vehicle models, excluding

and including the steering dynamics; hence, in total, six cases are examined. During the

simulations, the identiﬁed model of the steering system is applied to the vehicle model in

every case; hence, the consideration of the steering dynamics in the prediction model is

expected to lead to better results. The value of N

p

and N

c

is 10 and the sampling time is

0.05 s.

The results are summarized in Table 1, where eis the lateral error and

φ

is the orien-

tation error. As the velocity of the vehicle is increased, the average and maximal errors

are also increased; furthermore, the application of the steering dynamics can reduce the

errors signiﬁcantly.

Sensors 2022,22, 5807 13 of 23

Please note that the ﬁgures below sometimes do not show the difference between

the reference and the real steering angle, which is due to the accurate reference tracking.

As shown in Figures 7and 8, at 50 km/h, the consideration of the steering dynamics can

increase the accuracy of the controller and result in a smoother steering command, which

increases the stability of the vehicle and provides better ride comfort to the passengers.

Furthermore, smaller steering interventions result in smaller sideslip angles.

Figure 6. The reference path.

Figure 7. Simulation results at 50 km/h, without steering dynamics.

Sensors 2022,22, 5807 14 of 23

Table 1. Simulation results.

Speed [km/h] Steering Dynamics eavg [m] emax [m] φavg [deg] φmax [deg]

50 No 0.066 0.220 0.516 2.561

Yes 0.023 0.200 0.510 3.862

60 No 0.085 0.425 0.407 6.159

Yes 0.077 0.362 0.378 5.363

70 No 0.544 4.672 1.942 25.336

Yes 0.522 4.669 1.937 25.869

The result at 60 km/h is shown in Figures 9and 10. The results show similar behavior

to the 50 km/h cases; however, the errors are a little higher, although still in an acceptable

range. As shown in Figure 2, the tires become saturated when reaching about 4 deg and

2 deg of sideslip at the front and the rear wheels, respectively.

The tire sideslips reach about 7 deg at the front wheels and 5 deg at the rear wheels,

which means the tires operate in a saturated state when the vehicle drives at the ending

section of the U turn, resulting in a higher lateral error there.

The results at 70 km/h are shown in Figures 11–14. Figures 12 and 14 show the

same results as Figures 11 and 13, respectively, but in a smaller time window to make the

results more visible. The vehicle reaches and exceeds the friction limit at the U turn, which

leads to high lateral errors—the vehicle is physically not able to follow the path due to its

great curvature.

Figure 8. Simulation results at 50 km/h, including steering dynamics.

Sensors 2022,22, 5807 15 of 23

Figure 9. Simulation results at 60 km/h, without steering dynamics.

Figure 10. Simulation results at 60 km/h, including steering dynamics.

Sensors 2022,22, 5807 16 of 23

Figure 11. Simulation results at 70 km/h, without steering dynamics.

The importance of the 70 km/h case is to show whether the controller can handle

the vehicle at the friction limit or increase the steering angle to a high value, applying

unnecessarily large sideslip angles, which can not decrease the path-following errors.

According to the simulation results, the controller can handle the nonlinear characteristics

of the tires and operates stably, even when the sideslip angles reach high values. As shown

in Figures 11 and 13, the sideslip angles reach a maximal 8 deg and 5 deg at the front and

the rear wheels, respectively. The further increase in the sideslip would not generate a

larger lateral force. As the cost function penalizes the value of the steering intervention, the

steering angle does not increase. In this research, the authors do not apply constraints on

the sideslip angle; however, the controller does not generate too-large values due to the

inclusion of the nonlinear Pacejka tire model in the plant model.

In the double-lane-change section, the errors do not show a signiﬁcant increase com-

pared to the lower-speed scenarios; the controller could drive the vehicle faster in this

section, but the higher speed would result in a greater error at the U-turn. Please consider

that in Table 1, the high values of the average errors at 70 km/h are caused by the large

errors at the U-turn, which has a serious effect when calculating the average errors.

Overall, the inclusion of the steering dynamics in the plant model leads to a more

accurate path following, with a smoother steering intervention, while the complexity of

the model is not seriously increased. Furthermore, as shown by the fourth subgraphs in

Figures 7–14, the inclusion of the steering dynamics in the plant model results in a more

accurate steering demand tracking, where the tracking error is decreased in every case.

Sensors 2022,22, 5807 17 of 23

Figure 12. Simulation results at 70 km/h, without steering dynamics in 0–15 sec interval.

Figure 13. Simulation results at 70 km/h, including steering dynamics.

Sensors 2022,22, 5807 18 of 23

Figure 14. Simulation results at 70 km/h, including steering dynamics in 0–15 sec interval.

6. Evaluation of the Effectiveness of the Controller and Comparison of the

Linear/Nonlinear Plant Model

In this section, the effectiveness of the controller is evaluated by testing on a sine wave

path. Furthermore, the effect of the vehicle plant model is also evaluated by comparing

the results of a linear vehicle model and a nonlinear vehicle model. The reference path is

shown in Figure 15. The distance between the cones is 30 m, which means 60 m wavelength,

and the lateral peak value of the sine wave is 2.5 m in both directions. Each test scenario

presented in this section is conducted using the four-wheel vehicle model presented in

Section 2.1 and the plant model contains the steering dynamics in each test case.

Figure 15. Sine wave reference path.

Sensors 2022,22, 5807 19 of 23

The results of the scenarios conducted on the sine wave path are summarized in

Table 2. Please note that when the linear plant model is applied in the 70 km/h scenario,

the results are calculated solely for the time interval of 0–4.2 s, since at 4.2 s, the vehicle

left the path. In the ﬁrst scenario, the controller presented in Section 4is tested, using the

vehicle model presented in Section 2.2 for state prediction. The results at 70 km/h vehicle

velocity are shown in Figure 16. The controller drives the vehicle accurately and stably,

the maximum sideslip angles are 4 deg, and the small values in the lateral and the angular

errors also demonstrate the accuracy of the controller.

Table 2. Simulation results on the sine wave path.

Speed [km/h] eavg [m] emax [m] φavg [deg] φmax [deg]

Nonlinear plant model

70 0.098 0.192 0.689 2.414

Linear plant model 60 0.390 0.687 2.442 4.259

70 1.225 2.903 15.307 71.385

Figure 16.

Simulation results on the sine wave path at 70 km/h, including steering dynamics, using

the nonlinear vehicle model for state prediction.

In the next scenario, the nonlinear vehicle plant model is replaced by a linear one, in

which small angle assumptions are applied during the linearization of the model, and the

vehicle model is coupled with a linear tire model, while the model of the steering dynamics

remains unchanged. The resulting linear vehicle model is described by Equations (51)–(55),

Vy=.

y(51)

ay=.

Vy=−cfvy−cflfr

mVx

+cfδ

m+−crvy+crlrr

mVx

−Vxr(52)

r=.

ϕ(53)

Sensors 2022,22, 5807 20 of 23

.

r=

−lfcfvy−l2

fcfr

IzVx

+lfcfδ

Iz

+lrcrvy−l2

rcrr

IzVx(54)

.

δact =−1

Tst

δact +1

Tst

δdem (55)

where the lateral force is a linear function of the sideslip angle (56).

Fy f =−cαfαf,Fyr =−cαrαr(56)

In (56),

cαf

and

cαr

are the cornering stiffness of the front and rear tires, respectively,

which are identiﬁed from the measurement results (Figure 2), and the state vector remains

unchanged, x= [v Vyϕrδact]T.

The ﬁrst test using the linear plant model is conducted at 60 km/h and the results are

shown in Figure 17. Both the lateral and angular errors are large, while the controller is

unable to drive the vehicle along the path when the velocity is increased to 70 km/h, as

shown in Figure 18.

Figure 17.

Simulation results on the sine wave path at 60 km/h, including steering dynamics, using

the linear vehicle model for state prediction.

In the 70 km/h scenario, the vehicle is unable to follow the path anymore, resulting in

leaving the path. The reason for the poor performance of the linear-model-based control is

the neglection of the nonlinearities of the controlled vehicle in the state prediction. Both the

nonlinear terms in the vehicle model and the tire model are neglected, which results in a

less accurate state prediction; hence, in the 60 km/h scenario, performance degradation of

the controller is realized, and at 70 km/h, the controller is unable to drive the vehicle along

the path.

The results clearly show that the application of the nonlinear vehicle model coupled

with the nonlinear tire model for state prediction has a signiﬁcant advantage over the linear

vehicle model and linear tire model couple. Using the nonlinear model, the controller is

able to drive the vehicle stable at higher speeds and more accurately than with the linear

Sensors 2022,22, 5807 21 of 23

model. Furthermore, the effectiveness of the proposed LTV-MPC method is also conﬁrmed

in this section by testing the controller in the sine wave path.

Figure 18.

Simulation results on the sine wave path at 70 km/h, including steering dynamics, using

linear the vehicle model for state prediction.

7. Conclusions

In this paper, an LTV-MPC structure is proposed for path-following scenarios. The

controller uses a nonlinear vehicle model coupled with a Pacejka tire model for state

prediction. Furthermore, the dynamics of the steering system are also incorporated into the

prediction model. The reference deﬁnition is embedded into the LTV-MPC, corresponding

to the states in the plant model, which results in a simpliﬁed plant model, removing two

equations from the model, which are responsible for the coordinate transformation. The

presented reference deﬁnition method can reduce the nonlinear terms in the path-following

problem deﬁnition by transforming the path into the ego frame, resulting in a simpliﬁed

description of the vehicle dynamics, the nonlinearities of which are easier to handle. Using

this method, the transformation from the ego frame to the global frame is practically

conducted separately from the vehicle model.

During the simulation tests, the inclusion of the steering dynamics results in a more

accurate path-following performance with smaller errors and smoother steering command

demanded by the controller. Furthermore, the proposed reference deﬁnition is proven

to be effective, while the nonlinear terms in the plant model are reduced. The proposed

controller structure coupled with the plant model can handle the vehicle at sharp corners

when the tires operate in the nonlinear region. The vehicle drives stably during the tests,

while the errors remain low. According to the results, the controller is applicable for even

emergency scenarios, e.g., for a double-lane change. The effectiveness of the proposed

controller is tested in a sine wave path, where the controller is proven to be as accurate as

on the double-lane-change path. Furthermore, the effect of the linear and nonlinear plant

Sensors 2022,22, 5807 22 of 23

model is also analyzed and the advantage of applying the nonlinear plant model is clearly

proven on the sine wave path.

Regarding future research opportunities, the vehicle plant model and the model of

the steering system, as an actuator, could be further detailed, which could result in more

accurate state prediction. The dynamics of the steering system could be modeled as a func-

tion of the vehicle velocity, the yaw rate, and the lateral acceleration, and the self-aligning

torque of the tires could be incorporated into the plant model as well. Furthermore, the

proposed reference deﬁnition may reduce the computational requirements of an NLMPC,

as the plant model has fewer nonlinear terms.

Author Contributions:

Conceptualization, Á.D. and V.T.; Formal analysis, Á.D. and V.T.; Inves-

tigation, Á.D.; Methodology, Á.D.; Project administration, V.T.; Resources, V.T.; Software, Á.D.;

Supervision, V.T.; Validation, Á.D.; Writing—review & editing, Á.D. and V.T. All authors have read

and agreed to the published version of the manuscript.

Funding:

The research reported in this paper and carried out at the Budapest University of Technol-

ogy and Economics was supported by the National Research Development and Innovation Fund

(TKP2020 National Challenges Subprogram, Grant No. BME-NC) based on the charter of bolster

issued by the National Research Development and Innovation Ofﬁce under the auspices of the

Ministry for Innovation and Technology.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1.

Bárdos, Á.; Domina, Á.; Tihanyi, V.; Szalay, Z.; Palkovics, L. Implementation and experimental evaluation of a MIMO drift-

ing controller on a test vehicle. In Proceedings of the 2020 IEEE Intelligent Vehicles Symposium (IV), Las Vegas, NV, USA,

19 October–13 November 2020; pp. 1472–1478. [CrossRef]

2.

Czibere, S.; Domina, Á.; Bárdos, Á.; Szalay, Z. Model predictive controller design for vehicle motion control at handling limits in

multiple equilibria on varying road surfaces. Energies 2021,14, 6667. [CrossRef]

3.

Orgován, L.; Bécsi, T.; Aradi, S. Autonomous drifting using reinforcement learning. Period. Polytech. Transp. Eng.

2021

,49,

292–300. [CrossRef]

4.

Lu, Z.; Shyrokau, B.; Boulkroune, B.; van Aalst, S.; Happee, R. Performance benchmark of state-of-the-art lateral path-following

controllers. In Proceedings of the 2018 IEEE 15th International Workshop on Advanced Motion Control (AMC), Tokyo, Japan,

9–11 March 2018; pp. 541–546. [CrossRef]

5.

Viana, I.B.; Kanchwala, H.; Ahiska, K.; Aouf, N. A comparison of trajectory planning and control frameworks for cooperative

autonomous driving. J. Dyn. Syst. Meas. Control 2021,143, 071002. [CrossRef]

6.

Viana, Í.B.; Kanchwala, H.; Aouf, N. Cooperative trajectory planning for autonomous driving using nonlinear model predictive

control. In Proceedings of the 2019 IEEE International Conference on Connected Vehicles and Expo (ICCVE), Graz, Austria,

4–8 November 2019; pp. 1–6. [CrossRef]

7.

Kianfar, R.; Falcone, P.; Frederikson, J. A distributed model predictive control approach to active steering control of string stable

cooperative vehicle platoon. IFAC Proc. Vol. 2013,46, 750–755. [CrossRef]

8.

Wu, N.; Huang, W.; Song, Z.; Wu, X.; Zhang, Q.; Yao, S. Adaptive dynamic preview control for autonomous vehicle trajectory

following with DDP based path planner. In Proceedings of the 2015 IEEE Intelligent Vehicles Symposium (IV), Seoul, Korea,

28 June–1 July 2015; pp. 1012–1017.

9.

Alcalá, E.; Puig, V.; Quevedo, J.; Rosolia, U. Autonomous racing using Linear Parameter Varying-Model Predictive Control

(LPV-MPC). Control Eng. Pract. 2020,95, 104270. [CrossRef]

10.

Falcone, P.; Borelli, F.; Asgari, J.; Tseng, H.E.; Hrovat, D. Predictive active steering control for autonomous vehicle systems. IEEE

Trans. Control Syst. Technol. 2007,15, 566–580. [CrossRef]

11.

Katriniok, A.; Maschuw, J.P.; Christen, F.; Eckstein, L.; Abel, D. Optimal vehicle dynamics control for combined longitudinal

and lateral autonomous vehicle guidance. In Proceedings of the 2013 European Control Conference (ECC), Zurich, Switzerland,

17–19 July 2013; pp. 974–979.

12.

Xu, Y.; Tang, W.; Chen, B.; Qiu, L.; Yang, R. A model predictive control with preview-follower theory algorithm for trajectory

tracking control in autonomous vehicles. Symmetry 2021,13, 381. [CrossRef]

Sensors 2022,22, 5807 23 of 23

13.

Wang, Z.; Bai, Y.; Wang, J.; Wang, X. Vehicle path tracking LTV-MPC controller parameter selection considering CPU computational

load. J. Dyn. Syst. Meas. Control 2018,141, 051004. [CrossRef]

14.

Borrelli, F.; Falcone, P.; Keviczky, T.; Asgari, J.; Hrovat, D. MPC-based approach to active steering for autonomous vehicle systems.

Int. J. Veh. Auton. Syst. 2005,3, 265–291. [CrossRef]

15.

Yu, C.; Zheng, Y.; Shyrokau, B.; Ivanov, V. MPC-based path following design for automated vehicles with rear wheel steering. In

Proceedings of the IEEE International Conference on Mechatronics, ICM 2021, Piscataway, NJ, USA, 7–9 March 2021. [CrossRef]

16.

Chowdhri, N.; Ferranti, L.; Iribarren, F.S.; Shyrokau, B. Integrated nonlinear model predictive control for automated driving.

Control Eng. Pract. 2021,106, 104654. [CrossRef]

17.

Falcone, P.; Borrelli, F.; Tseng, H.E.; Asgari, J.; Hrovat, D. Linear timevarying model predictive control and its application to active

steering systems: Stability analysis and experimental validation. Int. J. Robust Nonlinear Control 2007,18, 862–875. [CrossRef]

18.

Falcone, P.; Tseng, H.E.; Borrelli, F.; Asgari, J.; Hrovat, D. MPC-based yaw and lateral stabilisation via active front steering and

braking. Veh. Syst. Dyn. 2008,46, 611–628. [CrossRef]

19.

Ducajú, J.M.S.; Llobregat, J.J.S.; Cuenca, Á.; Tomizuka, M. Autonomous ground vehicle lane-keeping LPV model-based control:

Dual-rate state estimation and comparison of different RealTime control strategies. Sensors 2021,21, 1531. [CrossRef] [PubMed]

20.

Zhang, W.; Wang, Z.; Drugge, L.; Nybacka, M. Evaluating model predictive path following and yaw stability controllers for over

actuated autonomous electric vehicles. IEEE Trans. Veh. Technol. 2020,69, 12807–12821. [CrossRef]

21.

Wei, S.; Zou, Y.; Zhang, X.; Zhang, T.; Li, X. An integrated longitudinal and lateral vehicle following control system with radar

and vehicle-to-vehicle communication. IEEE Trans. Veh. Technol. 2019,68, 1116–1127. [CrossRef]

22.

Ni, L.; Gupta, A.; Falcone, P.; Johannesson, L. Vehicle lateral motion control with performance and safety guarantees. IFAC-

PapersOnLine 2016,49, 285–290. [CrossRef]

23.

Moradi, M.; Fekih, A. A stability guaranteed robust fault tolerant control design for vehicle suspension systems subject to actuator

faults and disturbances. IEEE Trans. Control Syst. Technol. 2015,23, 1164–1171. [CrossRef]

24.

Jin, X.; Wang, J.; Yan, Z.; Xu, L.; Yin, G.; Chen, N. Robust Vibration Control for Active Suspension System of In-Wheel-Motor-

Driven Electric Vehicle Via µ-Synthesis Methodology. J. Dyn. Syst. Meas. Control 2022,144, 051007. [CrossRef]

25.

Pacejka, H.B.; Bakker, E.; Nyborg, L. Tyre Modelling for Use in Vehicle Dynamics Studies; SAE Technical Paper Series; SAE

International: Warrendale, PA, USA, 1987; p. 870421.

26.

Schwenzer, M.; Ay, M.; Bergs, T.; Abel, D. Review on model predictive control: An engineering perspective. Int. J. Adv. Manuf.

Technol. 2021,117, 1327–1349. [CrossRef]

27.

Rákos, O.; Aradi, S.; Bécsi, T. Lane Change Prediction Using Gaussian Classiﬁcation, Support Vector Classiﬁcation and Neural

Network Classiﬁers. Period. Polytech. Transp. Eng. 2020,48, 327–333. [CrossRef]