IMPROVED CONTROL DESIGN FOR AUTONOMOUS VEHICLES

International Journal of Information Technology, Control and Automation (IJITCA) Vol.12, No.1/2/3, July 2022
DOI:10.5121/ijitca.2022.12301 1
IMPROVED CONTROL DESIGN FOR
AUTONOMOUS VEHICLES
Mohamed Ali Jemmali and Hussein T. Mouftah
School of Electrical Engineering and Computer Science,
University of Ottawa, Ontario, Ottawa, Canada
ABSTRACT
In this paper, the autonomous vehicle presented as a discrete-time Takagi-Sugeno fuzzy (T-S) model. We
used the discrete-time T-S model since it is ready for the implementation unlike the continuous T-S fuzzy
model. The main goal is to keep the autonomous vehicle in the centreline of the lane regardless the
external disturbances. These disturbances are the wind force and the unknown curvature; they are applied
to test if the autonomous vehicle moves from the centreline. To ensure that the autonomous vehicle remain
on the centreline we propose two discrete-time fuzzy lateral controllers called also steering controllers.
KEYWORDS
Takagi-Sugeno model, Steering control, lane keeping, observers.
1. INTRODUCTION
Nowadays, vehicles have become an integral part of our daily lives. Most people worldwide need
vehicles to move around, such as cars, buses, bicycles, taxis, etc. Also, those who are physically
challenged use special vehicles. Due to the constant use of vehicles globally, so many people die
every day because of vehicle collisions. Many studies declare that the main causes of these
accidents are inattention, drowsiness, and illness [1] [2]. The purpose of autonomous cars lateral
control is to maintain the vehicle in the lane under varied limits, and it is one of the most
significant safety solutions. Over the last two decades, lateral stability control for autonomous
vehicles has gotten a lot of attention from scientists and engineers, and numerous findings have
been published [3]–[4]. Therefore, fruitful results with regard to stability and stabilization have
been developed by researchers. By merging fuzzy logic and PID control, several authors propose
a novel lateral control system design [5]. Others worked on controlling saturation through robust
yaw control and stabilising lateral dynamics with concerns of parameter uncertainty. [6] A robust
yaw-moment controller architecture for increasing vehicle handling and stability has been
designed, taking into account parameter uncertainty and control saturation. However, the state
vector only has two controllable parameters: the yaw rate and the sideslip angle; in this instance,
there is insufficient information about the condition of the vehicles to allow for robust control.
According to Sun et al. [7], the proposed proposal lacks a mathematical model, implying that no
mathematical stability and stabilisation criteria exist to get access to system control. On the other
hand, several researchers used deep learning and reinforcement learning approaches to examine
vision-based autonomous driving [8]. Despite this, the model has no constraints, such as lateral
wind force applied to the cars, unknown road curvature, or steering physical saturation. This
means that the proposed solution is still far from being applicable in real-world driving
conditions. Jiang and Astolfi [9] investigated an asymptotic stabilisation issue for a class of
nonlinear under actuated systems. Its solution is used in the control of a vehicle's nonlinear lateral
dynamics, together with back stepping and forward control design approaches. This approach

2
demonstrated that a vehicle may track any conceivable reference at a constant speed using the
established controller, and the lateral deviation converges to zero. The challenges in this scenario
are that the longitudinal vehicle speed is constant, there is only one controllable internal variable,
and the stability is local. Based on the aforementioned rationale, lateral control of autonomous
cars requires a system that provides access to several vehicle states and swipes into a wide range
of longitudinal speed. As a result, the Takagi–Sugeno (T–S) fuzzy models [10] have been widely
acknowledged and used. The T-S is well-known as a valuable and popular paradigm for
approximating complicated nonlinear systems. The nonlinear systems were approximated by
fuzzy "blending" local linear models using a set of "IF-THEN" rules, which has piqued the
control community's curiosity. In this paper, we will focus on discrete-time T-S fuzzy control.
This type of control is essentially based on feedback control, more specifically the parallel
distributed control (PDC) law. In addition, the state vector will contain six internal variables to
allow more accessibility for the autonomous vehicle control. In most cases, the state vector could
be unreadable, noisy, or completely inaccessible. Based on this motivation, in our work we
developed an observer called Luenberger multi observers to ensure the reconstruction of the
system state vector for accurate automatic steering control. The contributions in this paper are
summarized as follows:
1) Developing an improved PDC control law. The purpose is to improve the convergence
time.
2) Using multi observer to compute the real values of the autonomous vehicle internal
parameters which are the state vector parameters. This state vector will indeed be used in
the control law equation for the fuzzy controllers design. The stability and stabilization
conditions will be based on the quadratic Lyapunov function. The Linear Matrix
Inequality (LMI) approach is used for the optimization purpose.
In this paper, we will compare between two fuzzy controllers applied to the autonomous vehicle
in terms of stabilization and convergence speed. This paper is presented as follows. The vehicle
modelling which is the part who is consecrated for the vehicle parameters and models. The third
part presents the control design for the autonomous vehicle, which focuses on the stabilization of
the autonomous vehicle. The final part is consecrated to show the results.
2. VEHICLE PARAMETERS AND MODELS
In this part, we show the different steps for the vehicle modelling. We start by introducing the
vehicle parameters given in Table 1:

3
Table 1. Definition of parameters.
Parameters Description Value
Bs Steering system damping 5.73
Cf Front cornering stiffness 57000 N/rad
Cr Rear cornering stiffness 59000 N/rad
Is Steering system moment of inertia 0.02 kgm2
Iz Vehicle yaw moment of inertia 2800 kgm2
Kp Manual steering column coefficients 0.5
lf Distance from the CG to the front axle 1.3 m
lr Distance from the CG to the rear axle 1.6 m
ls Look-ahead distance 5 m
lw Distance from the CG to the
impactcenter of the wind force
0.4 m
M Mass of the vehicle 2025 kg
Rs Steering gear ratio 16
σt Tire length contact 0.13 m
2.1. Road – Vehicle Model
Using the well-known bicycle model described in [11]. The acquired model of the vehicle lateral
dynamics is expressed as:
[𝛽̇
𝑟̇
] = [
𝑎11 𝑎12
𝑎21 𝑎22
] [
𝛽
𝑟
] 𝛿 + [
𝑒1
𝑒2
] 𝑓
𝑤. (1)
Where 𝛽 is the sideslip angle at the center of gravity (CG), and the yaw rate is r shown in Figure
1. In (1), the lateral wind force is fw, and the elements of the system matrices are written as
follows:
𝑎11 = −2
(𝐶𝑟+ 𝐶𝑓)
𝑚𝜗𝑥
; 𝑎12 = 2
(𝑙𝑟𝐶𝑟− 𝑙𝑓𝐶𝑓)
𝑚𝜗𝑥
2 − 1 ;
𝑎21 = 2
(𝑙𝑟𝐶𝑟− 𝑙𝑓𝐶𝑓)
𝐼𝑧
; 𝑎22 = −2
(𝑙𝑟
2𝐶𝑟+ 𝑙𝑓
2
𝐶𝑓)
𝐼𝑧𝜗𝑥
.
𝑏1 =
2𝐶𝑓
𝑚𝜗𝑥
; 𝑏2 =
2𝑙𝑓𝐶𝑓
𝐼𝑧
; 𝑒1 =
1
𝑚𝜗𝑥
; 𝑒2 =
𝑙𝑤
𝐼𝑧
.

4
Figure 1. Lateral vehicle behavior modeling.
2.2. Vehicle Position Model on the Road
The vehicle positioning dynamics on the road is described by [11]:
{
𝑦̇𝐿 = 𝜗𝑥𝛽 + 𝑙𝑠𝑟 + 𝜗𝑥𝜓𝐿
̇ .
𝜓𝐿
̇ = 𝑟 − 𝜗𝑥𝜌𝑟.
(2)
Where 𝘺𝐿 is the lateral deviation error from the centerline of the lane projected forward a look
ahead distance 𝑙𝑠 and 𝜓𝐿 is the heading error between the tangent to the road and the vehicle
orientation. The road curvature is indicated by 𝜌𝑟.
2.3. The Vehicle Steering Model
The electronic power steering system is presented as [3]:
𝛿̈ = 2
𝐾𝑝𝐶𝑓𝜎𝑡
𝑅𝑠
2𝐼𝑠
𝛽 + 2
𝑅𝑠
2𝐼𝑠
𝑙𝑓
𝜗𝑥
𝑟 ⋯ − 2
𝑅𝑠
2𝐼𝑠
𝛿 −
𝐵𝑠
𝐼𝑠
𝛿̇ +
1
𝑅𝑠𝐼𝑠
𝑇𝑠. (3)
Where 𝑇𝑠 is the steering torque, δ is the steering angle, 𝑙𝑠 is the inertia moment of the steering
column, 𝐵𝑠 is the damping factor of the column, 𝑅𝑠 is the reduction ratio of the column, 𝜎𝑡 is the
width of the tire contact finally, the manual steering column coefﬁcient is 𝐾𝑝.
2.4. The Autonomous Vehicle Model
Based on (1), (2) and (3) the autonomous vehicles model is:
𝑥̇(𝑡) = 𝐴𝑣𝑥(𝑡) + 𝐵𝑣𝑢𝑢(𝑡) + 𝐵𝑣𝑤𝑤(𝑡). (4)
Where 𝑥 = [𝛽 𝑟 𝜓𝐿 𝘺𝐿 𝛿 𝛿
̇ ]
𝑇
is the vehicle vector state, 𝑤 = [𝑓
𝑤 𝜌𝑟]𝑇
is the disturbance vector,
and u = Ts is the input vector. The control-based system matrices in (4) are expressed as:

5
𝐴𝑣 =
[
𝑎11 𝑎12 0
𝑎21 𝑎22 0
0 1 0
0 𝑏1 0
0 𝑏2 0
0 0 0
𝑣𝑥 𝑙𝑠 𝑣𝑥
0 0 0
𝑎61 𝑎62 0
0 0 0
0 0 1
0 𝑎65 𝑎66]
;𝐵𝑣𝑙𝑤 =
[
𝑒1 0
𝑒2 0
0 −𝑣𝑥
0 0
0 0
0 0 ]
.Bv =
[
0
0
0
0
0
1
RsIs]
.
We have
𝑎61 = 2
𝑅𝑠
2𝐼𝑠
, 𝑎62 = 2
𝑅𝑠
2𝐼𝑠
𝑙𝑓
𝑣𝑥
,𝑎65 = −2
𝑅𝑠
2𝐼𝑠
, 𝑎66 = −
𝐵𝑠
𝐼𝑠
.
As a result, the control analysis and development in this work will be based on the discrete-time
system described below (5).
x(k + 1) = A x(k) + Bd u(k) + Blww(k) (5)
Te = 0.01 second is the discretization time. A is the discrete-time matrix of Av, Bd is the
discrete-time matrix of Bv and Blw is the discrete-time matrix of 𝐵𝑣𝑙𝑤.
3. AUTONOMOUS VEHICLE STABILIZATION ARCHITECTURE
In this section, we present the control design steps for the aim purpose is to control the
autonomous vehicle presented in (5).
3.1. The Autonomous Vehicle T-S Model
The discrete-time (T-S) system is presented by fuzzy IF-THEN rules.
The Rule i is of the form:
IF E1(k) is 𝑄1i and… En(k)is Qni THEN
{
x(k + 1) = Aix(k) + Biu(k).
y(k) = Cix(k).
(6)
Where 𝐸1,… , 𝐸𝑛 are linear functions, 𝑄1i, … . 𝑄𝑛𝑖 are the fuzzy sets, and n is the number of the
rules. In addition, 𝑥𝑘 ∈ ℝ𝑛
is the state vector; 𝑢𝑘 ∈ ℝ𝑝
is the measurable output vector; 𝐴𝑖, 𝐵𝑖
and 𝐶𝑖 are the system matrices with appropriate dimension, In addition, the premise variables are
represented by the vector ꓩ(𝑘) = [ꓩ1(𝑘) … . ꓩ𝑞(𝑘)]. The autonomous vehicle model is:
x(k + 1) = ∑ ωi
n
i=1 (ꓩ(k))(Aix(k) + Biu(k) + 𝐵𝑖
𝑤
w(k)). (7)
{
∑ 𝜔i
n
i=1 (ꓩ(k)) = 1
ωi(ꓩ(k)) ≥ 0.
(8)
With
ωi(ꓩ) =
wi(ꓩ(t))
∑ wi(ꓩ(t))
n
i=1
; wi(ꓩ(t)) = ∏ Mji(ꓩ(t))
q
j=1
.
The proposed Lyapunov function is:

6
V(x(k)) = xk
T
S xk. (9)
The discrete-time T-S fuzzy system described in equation (7) is asymptotically stable if there
exists a common symmetric matrix S = ST
> 0 such that the following LMI are feasible [12]
[13]:
{
S > 0.
Ai
T
S Ai − S < 0.
(10)
3.2. Autonomous Vehicle Stabilization
The control law is:
u(k) = − ∑ ωi(ꓩ(k))[Gix(k)] − 2Gix(k + 1).
n
i=1 (11)
Using the discrete-time T-S system previously described in (7), a closed loop control given by the
new PDC law:
{
x(k + 1) = ∑ ∑ ωi(ꓩ(k))ωj(ꓩ(k))ϑijx(k).
n
j=1
n
i=1
ϑij = ( Ai − BiGj).
(12)
Hypothesis 1: The T-S system is locally controllable as described in [14].
The discrete time T-S models stabilization conditions for a closed-loop PDC controller are that
there exists a symmetric matrix S > 0 as well as gains Gi, ∀i ∈ In satisfying:
{
Cdiscrete(ϑii,S) < 0, ∀𝑖 ∈ In,
Cdiscrete(ϑij,S) ≤ 0, ∀i, j ∈ In
2
ωi(ꓩ(k))ωj(ꓩ(k)) ≠ 0.
(13)
And,
Cdiscrete(ϑij,S) = [
ϑij+ϑji
2
]
T
S [
ϑij+ϑji
2
] − S. (14)
The conditions are:
(Ai − BiGi)T
S(Ai − BiGi) − S < 0. (15)
Multiplying (15) in pre and post by S−1
, we come by the following inequality:
S−1(AiS−1
− BiGiS−1)T
S(AiS−1
− BiGiS−1
) > 0. (16)
It’s supposed that X = S−1
and Hi = GiS−1
, thus we get the following:
X(AiX − BiHi)T
S (AiX − BiHi) > 0. (17)
The inequality (17) can be presented in LMI form by the application of the Schur complement as
follows:
[
X ∗
AiX − BiHi X
] > 0 ∀ 𝑖 ∈ 𝐼𝑛. (18)

7
By application of the same approach and the same steps, the condition Cdiscrete(ϑij, S) ≤ 0 is
given by:
[
X ∗
Ai+Aj
2
X −
1
2
(BjHi + BiHj) X
] ≥ 0 ∀(i, j) ∈ In
2
, i < 𝑗. (19)
The discrete-time T-S system described in equation (7) are globally asymptotically stable via the
novel PDC control law, if there are symmetric matrices such as S = ST
> 0 and M = MT
≥ 0
which verifies [15]:
{
𝐶𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒(ϑii,S) + (r − 1)M < 0, ∀𝑖 ∈ In,
𝐶𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒(ϑij,S) − M ≤ 0, ∀i,j ∈ In
2
,i < 𝑗,
ωi(ꓩ(k))ωj(ꓩ(k)) ≠ 0.
(20)
𝐶𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒(ϑij,S) = [
ϑij+ϑji
2
]
T
S [
ϑij+ϑji
2
] − S. (21)
Using theorem announced in [13] helps to decrease the conservatism. Theorem 1: if there exist
matrices S = ST
> 0, Mij = Mij
T
and matrices Gi which verifies:
{
Cdiscrete(ϑii,S) + (r − 1)M < 0, ∀𝑖 ∈ In.
Cdiscrete(ϑij,S) − M ≤ 0, ∀i,j ∈ In
2
,i < 𝑗.
ωi(ꓩ(k))ωj(ꓩ(k)) ≠ 0.
(22)
M = [
M11 … . M1n
M1n … . Mnn
]. (23)
Applying the previous equations, the autonomous vehicle model presented in (7) is globally
asymptotically stable.
{
X = S−1
.
Yii = XMijX.
Gi = HiX−1
.
∀i ∈ In.
(24)
3.3. Multi Observers Design
To perform the control of the autonomous vehicle we need the entire state vector 𝑥(𝑘). To
achieve this goal we applied the following observer’s equation:
{
X
̂(k + 1) = ∑ ωi(ꓩ(k))((Aix
̂(k) + Biu(k)) + Ni(y(k) − y
̂(k))) .
𝑛
𝑖=1
y
̂(k) = ∑ ωi
n
i=1 (ꓩ(k))Cix
̂(k).
(25)
The error state vector is written as:
x
̃(k) = x(k) − x
̂(k). (26)

8
Knowing that, the dynamics of the state vector error is given by:
{
x
̂(k + 1) = ∑ ∑ ωi
n
j=1 (ꓩ(k))ωj(ꓩ(k))αijx
̃(k).
n
i=1
αij = Ai − HiCj , ∀(i,j) ∈ In
2
.
(27)
In fact, the design of Luenberger observers requests the computing of local gains Ni ∈ In to
guarantee the convergence to 0 of the state vector error dynamics. In addition, we should
guarantee that S = ST
> 0 and matrices Ni ∈ In valid the following conditions:
{
𝐶𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒(αii, S) < 0, ∀𝑖 ∈ In.
𝐶𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒(αij, S) ≤ 0, ∀i,j ∈ In
2
ωi(ꓩ(k))ωj(ꓩ(k)) ≠ 0.
. (28)
𝐶𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒(αij, S) = [
𝛼ij+αji
2
]
T
S [
αij+αji
2
] − S. (29)
The equations (28) and (29) can be written as LMIs applying the Schur complement:
[S
S ∗
Ai+Aj
2
−
1
2
(HjCj + HjCi) S
] ≥ 0 (30)
When i < 𝑗.
We improve the observers by using the following theorem. Theorem 2: the multiple Luenberger
observers are globally asymptotically stable if there exist symmetric matrices S > 0 , Mij and
Ni ∈ In that satisfy:
{
Cdiscrete(αii,S) + Mii < 0, ∀𝑖 ∈ In.
Cdiscrete(αij, S) + Mij ≤ 0, ∀i, j ∈ In
2
.
[
M11 M1n
M1n Mnn
].
ωi(ꓩ(k))ωj(ꓩ(k)) ≠ 0.
αij = Ai − NiCj , ∀(i,j) ∈ In
2
.
(31)
The LMI’s are given by:
{
S > 0.
[
S − Mii ∗
SAi − HiCi S
] > 0 ∀ 𝑖 ∈ In.
[
S − Mij ∗
S
Ai+Aj
2
−
1
2
(HiCj + HjCi) S
] ≥ 0 ∀ i < 𝑗. 𝑖, 𝑗 ∈ In
2
M = [
M11 … . M1n
M1n … . Mnn
].
Hi = SNi.
. (32)

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4. RESULTS
We applied the improved PDC control law design and compared the obtained results with the
classic PDC [16] to the discrete-Time T-S fuzzy model representing the autonomous vehicle
system to ensure the lateral control purpose under certain constraints. Equations (25) and (32)
give:
{
S = X−1
.
Gi = HiS.
Hi = SNi.
(33)
Based on (33), we get the following gains and matrices:
S = 𝑒−03
[
0.0513 0.0139 0.0693 0.0120 0.1275 0.0070
0.0139 0.0062 0.0235 0.0041 0.0400 0.0025
0.0693 0.0235 0.1267 0.0196 0.21158 0.0121
0.0120 0.0041 0.0196 0.0051 0.0368 0.0019
0.1275 0.0400 0.2158 0.0368 0.4979 0.0266
0.0070 0.0025 0.0121 0.0019 0.0266 0.0029]
G1 = [ 538.7210 88.1555 137.5593 21.1442 − 146.3325 − 56.7097].
G2 = [545.2576 58.9360 139.5243 21.1255 − 148.8607 − 56.7512].
For multi-observer gains are:
N1 = 𝑒03
[
0.0116 0.0007
0.0828 − 0.0020
0.0285 − 0.0003
0.2673 − 0.0020
−0.0196 − 0.0273
9.3447 7.3696 ]
,N2 = 𝑒04
[
0.0026 0.0000
0.0152 − 0.0001
0.0076 − 0.0000
0.0624 − 0.0001
0.0074 − 0.0023
−1.2496 0.6264]
.
We tested our controller with different scenarios. The first scenario is to assume that the
autonomous vehicle system will start far from the origin with different orientation. The initial
state vector 𝑥0 = [0;0.02; 0.04; 0; 0; 0.9], which is not the system equilibrium point
[0;0;0;0;0;0], example, the lane centreline. We can certainly observe in figure 4-9 that the first
controller applied to the autonomous vehicle converges all the state variables to zero, which
means that our autonomous vehicle reaches the centreline of the lane. Figures 10-15 converges to
zero with fastest convergence time as presented in Table 2. These results prove the robustness of
our controllers with the advantage of the improved controller in term of convergence speed. The
second scenario is to apply a disturbance mitigation. We subjected the autonomous vehicle to a
lateral wind force of 1500 Newton. Figure 2 and Figure 3 show a remarkable robust stabilization.
Our model showed robustness and effectiveness against the disturbances.

10
Figure 2. Stabilized output1
Figure 3. Stabilized output2.

11
Figure 4. Sideslip angle.
Figure 5. Yaw rate.

12
Figure 6. Lateral deviation error.
Figure 7. Heading error.

13
Figure 8. Steering angle.
Figure 9. Steering rate.
Figure 4 presents the sideslip angle convergence to equilibrium point in 10 seconds. Figure 5
shows that the yaw rate converges in 12 seconds. Figures 6-9 show that the lateral deviation
error, the heading error and the steering angle converges in 8 seconds. These results show the
effectiveness of our first controller.

14
Figure 10. Heading error.
Figure 11. Sideslip angle.

15
Figure 12. Yaw rate.
Figure 13. Lateral deviation error.

16
Figure 14. Steering angle.
Figure 15. Steering rate.
Figure 10 presents the heading error convergence to equilibrium point. Figure 12 shows that the
yaw rate converges with fastest convergence speed and better manoeuvrability of -
40(degree/second). Figures 11, 13, 14, and 15 show that the lateral deviation error, the sideslip
angle and the steering angle converges in 5 seconds. These results show the effectiveness of our
first controller.

17
Table 2. Convergence Speed comparison.
Internal state vector
variables
Convergence time first
controller (second)
Convergence time
Improved controller
(second)
Heading error 8 5
Sideslip angel 10 5
Yaw rate 12 6
Steering angle 8 6
Steering rate 8 5
Lateral deviation error 8 5
Table 2 shows that the improved fuzzy controller provides fastest convergence speed for all the
internal variable state vector compared to the classic steering controller.
5. CONCLUSIONS
In this paper, we propose an improved control design for an autonomous vehicle. The discrete-
time T-S system represents the autonomous vehicle model. The state vector is computed by using
the Luenberger observer. Furthermore, we applied an improved PDC control law to the T-S fuzzy
model. Feasible LMI conditions have been developed to guarantee lane keeping under certain
limits. The results show that the lateral control of the autonomous vehicle to keep it on the
centreline of the lane is well done under various constraints and scenarios. In future work, we
will improve the heading error percentage to ensure more safety when we want to go back to the
centreline of the lane.
ACKNOWLEDGEMENTS
The research work was supported by the Canada Research Chairs Fund Program and Natural
Sciences and Engineering Research Council of Canada under Discovery Grant Project RGPIN
/1056-2017.
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