Real-time Robust Trajectory Control for Vehicle Platoons: A Linear

Matrix Inequality-based Approach

Alessandro Bozzi

, Enrico Zero

, Roberto Sacile

and Chiara Bersani

DIBRIS – Department of Informatics, Bioengineering, Robotics and Systems Engineering,

University of Genova, Genova, Italy

Keywords:

Robust Control, Trajectory Planning, Linear Matrix Inequality, Vehicle Platooning.

Abstract:

This paper proposes a solution to dynamically adjust vehicle platoon trajectories. The goal of the control

algorithm is to keep the optimal interdistance between adjacent vehicles proceeding at cruising speed on a

straight road. After a proposal of the interdistance required between neighboring vehicles, a robust decentral-

ized controller based on a linear control law provides the speed proﬁle for each component of the platoon.

Its objective is to minimize the divergence in space in respect to the planned trajectories while assuring a safe

span between adjacent members of the platoon. The results on a limited instance demonstrate the effectiveness

of the proposed approach.

1 INTRODUCTION

Autonomous vehicles represent a topic of growing in-

terest in systems engineering. The goal to design un-

manned cars and to introduce them in the common

trafﬁc is one of the most ambitious projects in the

current century, studied all over the world. Since

the road is a complex environment, many parame-

ters have to be taken into account for the car’s au-

tonomous motion. Firstly, smart sensors, located on

the vehicles or on the infrastructures, have to acquire

speciﬁc data such as videos and images from the sur-

roundings in order to identify the elements useful to

manage the vehicle tracking. Other devices have to

be installed to transmit the information, in real time,

to the control center which analyses and evaluates the

data and sends the correct control commands to the

vehicles. Some vehicles can send or receive automat-

ically the information about their states from or to-

ward a centralized decision maker in order to manage

their own trajectory according to the reconstructed en-

vironment. When the surrounding is virtually rebuilt,

the automated system can provide its control law by

modifying the acceleration, the speed and/or the steer-

ing of the vehicle replacing the human intervention.

Control techniques can also be applied to vehicle pla-

https://orcid.org/0000-0002-2436-0946

https://orcid.org/0000-0002-9995-1724

https://orcid.org/0000-0003-4086-8747

https://orcid.org/0000-0002-5779-9605

tooning, which refers to a set of cooperative auto-

mated vehicles which travel on the same trajectory in

a string formation.

There are already several ﬁndings that show how

autonomous driving could improve the throughput of

the road, while reducing the fuel emissions, especially

for platooning (Alam et al., 2014). While dealing with

platoon, many factors have to be considered. In this

case, in fact, each vehicle does not only follow its

path, but also it interacts with the other members of

the platoon. The involved vehicles retrieve informa-

tion on the surroundings and have to proceed harmo-

niously with the rest of the system. The interdistance

between adjacent vehicles has to be constantly moni-

tored by means of safety zones, designed accordingly

to road trafﬁc rules and following principles intro-

duced in (Chen and Wang, 2007) and (Bersani et al.,

2015). The classiﬁcation of safer or less safe areas

leads the choice of the trajectory generator algorithm

to be used.

In literature, many works tackle the trajectory

planning problem for one vehicle. In (Alia et al.,

2015), the authors used clothoid tentacles and occu-

pancy grid to generate a feasible path for a vehicle

and even allowing it to avoid obstacles. To achieve

the same objective, in (Choi et al., 2008), the Bezier’s

curve method is adopted. However, the deﬁnition of

the trajectory for a vehicle included in a platoon ap-

pears more complex.

410

Bozzi, A., Zero, E., Sacile, R. and Bersani, C.

Real-time Robust Trajectory Control for Vehicle Platoons: A Linear Matrix Inequality-based Approach.

DOI: 10.5220/0010550704100415

In Proceedings of the 18th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2021), pages 410-415

ISBN: 978-989-758-522-7

In particular, unpredictable events during the tran-

sit due to external factors may cause vehicle ap-

proaching or detachment from the other members of

the platoon. In this case, a control system which gen-

erates countermeasures, for each element of the sys-

tem, to restore the correct position, speed and inter-

distance has to be introduced in order to guarantee

passengers’ safety. The approaches may be central-

ized or decentralized. In the ﬁrst case, usually the

leader regulates the vehicles’ position by providing

the optimal data for speed and acceleration to the fol-

lowers (Grafﬁone et al., 2020b), while in the second

case, each member of the platoon uses as reference

the distance and the speed of the preceding vehicle

(Ghasemi et al., 2013).

In (Liu et al., 2019), the authors proposed an opti-

mal platoon trajectory control method by optimizing

accelerations of the vehicles in order to satisfy con-

straints related to travel delay and fuel consumption.

In (Huang et al., 2019), a distributed control model

is presented where each vehicle is independently con-

trolled by itself and it shares only information about

its motion state in order to perform vehicle merging or

splitting. The same problem has also been tackled in

(Grafﬁone et al., 2020a) solved by a Model Predicted

Control model (MPC). In this latter case, the leader

coordinates the exchanged data with the vehicles in

order to perform the planned manoeuvres.

This paper presents a new algorithm to ensure the

correct maintenance of the distance between neigh-

boring vehicles, by means of a robust controller that

provide a linear control law. The ofﬂine procedure

should be activated when vehicles seem to have trou-

ble in keeping an optimal interdistance each other due

to external factors.

This work is organized as follows. Section 2 pro-

vides a deeper description of the safety zones clas-

siﬁcation and presents the proposed robust control

model. Section 3 shows a case study with simulation

and results, and ﬁnally Section 4 includes conclusions

and future developments.

2 MODEL AND METHOD

2.1 System Deﬁnition

The control of the platoon trajectory and the related

behaviour of the leader and the followers requires

an accurate model of the vehicles motion in order to

guarantee robustness and string stability.

To generate dynamically the set of trajectories, a

kinematic model of the vehicle has been used:

(t +1) = x

(t)+v

(t)∆t +w

(t) t = 0,..., T −1

(1)

where x

is the position of the i

vehicle, v

its veloc-

ity, w

the possible disturbance and ∆t the sampling

time. This simple approach ensures a fast modiﬁca-

tion of the conditions and paired with limiters it can

provide the bounding of the input respecting physical

constraints.

In matrix form, the aforementioned system of M

vehicles may be formalized as follows:

x(t + 1) = Ax(t) + ∆tBv(t) + w(t) t = 0,..., T − 1

(2)

with A and B identical matrices ∈ R

M×M

In addition, it is assumed that each vehicle can

access to the information about its own position and

on both the preceding and the following vehicle posi-

tions. So:

= C

(3)

and for the generic i

vehicle:





0 ... 0 1 0 0 0 ... 0

0 ... 0 0 1 0 0 ... 0

0 ... 0 0 0 1 0 ... 0





(4)

where C

∈ R

3×M

The ﬁrst and last vehicles of the platoon have

slightly different C

∈ R

2×M

matrix:



1 0 ... 0

0 1 ... 0





0 ... 1 0

0 ... 0 1



(5)

The speciﬁc structure of those matrices assures that

each vehicle monitors its own position and the posi-

tion of the vehicle which precedes and follows in the

string formation, as an example, by sensors such as

lidar and radar.

However, (Pates et al., 2017) and (Gao et al.,

2017) demonstrate that, to obtain better performances

of the control models regarding the scalability of the

system or in case of noisy measurements, it is recom-

mended to provide each vehicle also with information

coming from the leader’s position.

2.2 Safety Zones

In the management of the platoon trajectory, it is

signiﬁcant to monitor the interdistance between each

couple of consecutive vehicles, in order to apply a

proper controller able to handle speciﬁc goals accord-

ing to the different situations during the driving ses-

sions. When two vehicles get closer, a safety as-

pect must be taken into account. In the proposed

approach, to classify critical situations, the interdis-

tance between two adjacent vehicles may be repre-

sented as three colored states, green (optimal), yellow

Real-time Robust Trajectory Control for Vehicle Platoons: A Linear Matrix Inequality-based Approach

411

Figure 1: Safety zones division.

(sub-optimal) and red (critical). When a vehicle is too

close to the adjacent one, its position may be repre-

sented by a red zone. The state of the vehicle turns in

yellow, whenever a safer-distance is re-established ac-

cording to a predeﬁned threshold in space. The green

state represents a safe condition in respect to correct

interdistance among vehicles.

A schematic representation of safety zones is

shown in Fig.1 According to the state color, the ob-

jective of the control changes. The green zone is a

collision-free zone, in this case, the vehicle has to

keep a given speed, while maintaining the interdis-

tance. On the other hand, the red zone represents

collision-risk, as the distance with the preceding car

is not wide enough to guarantee safety. In this occur-

rence, the vehicle should perform a braking maneu-

ver and even stops itself in order to avoid a possible

collision, if for some external reason, the preceding

vehicle starts braking. The yellow intermediate zone

is by far the most interesting, since a vehicle must

pay attention to the preceding car because the inter-

distance is not small enough to induce an accident,

but not wide enough to allow a careless drive.

Hereinafter, the bounds of each zone are deﬁned

as follows (distance in m, velocity in km/h):

red

≤

3 ×V

green

≥





red

< d

yellow

< d

green

(6)

given accordingly to the reaction and the braking time

of a human driver.

More in details, these bounds have been assigned

accordingly to the rules of the Italian road trafﬁc,

which asserts as a recommended distance the sum of

the space traveled during the human reaction and the

braking time, while the minimum threshold is the re-

action space only. Nevertheless, these distances might

even be reduced, considering that autonomous ve-

hicles perform faster computation and input prompt

with respect to human being. In other words, the un-

manned vehicles’ reaction time is considerably lower.

Coherently with this reasoning, (Alam et al., 2014)

has shown that especially for heavy duty vehicles

(HDV) platooning the minimum distance to avoid col-

lision can be reduced.

However, in this paper it has been preferred to

maintain the canonical range. The proposed control

model will be applied considering the yellow inter-

mediate state of criticality.

2.3 Robust Control for Yellow Zone

The proposed robust control model deals with a situ-

ation in which vehicles are not optimally spaced (i.e.

they are in yellow zone) and while they are proceed-

ing with a given velocity. The behaviour in green and

red zone is assumed to be correctly performed by as-

signed controllers thus they are not considered in this

paper.

The robust controller’s purpose in the yellow zone

is to restore the desired interdistance minimizing the

speed variation, while keeping the planned trajectory,

despite disturbances that affect the system.

Thus, the trajectory of the i-th vehicle has to be

modiﬁed with respect to the desired reference path

a priori deﬁned x

(t), leading to apply the following

change of variable:

(

˜x

(t) = x

(t) − x

(t)

˜v

(t) = v

(t) − v

(t)

(7)

The problem can be deﬁned as minimax problem:

inf

sup

w6=0

J(v, w)

||w||

(8)

subject to (2) where the cost function J is designed as:

J(v,w) =

∑

i=1

T −1

∑

t=0

˜x

(t)+γ

˜v

(t)+

∑

i=2

T −1

∑

t=0

( ˜x

(t)− ˜x

i−1

(t))

(9)

In (9) α

,β

and γ

represent gains which give primary

importance respectively to tracking the trajectory,

restoring the optimal interdistance and minimizing

the difference from the desired velocity. The leader

vehicle is considered as an element of the platoon

(i.e. it is unmanned as the others) with the speciﬁc

feature that it does not have to care about keeping

some distance to a preceding vehicle.

Theorem 1. Let’s consider a time horizon of two in-

tervals, i.e. t = 0,...T and T = 1. The trajectory of

the platoon described by x(t) can be modiﬁed in real-

time, according to the problem deﬁned by (8) and (2)

by an optimal control law v(t) = Kx(t), which is lin-

ear, where K is the solution of the following linear

matrix inequality (LMI):

min

θ,K

θ (10)

s.t.

K =







1,1

1,2

0 0 0 ... 0 0

0 0 k

2,1

2,2

2,3

... 0 0

0 0 0 0 0 ... 0 0

... ... ... ... ... ... ... ...

0 0 0 0 0 ... k

M,M−1

M,M







(11)

ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics

412



− θI +

KC + C

KC −

−1



6 0

(12)

and θ > θ

inf

µ∈S

sup

06=x∈R



µ(Cx)





µ(Cx)





kxk



= θ

(13)

Proof. From the cost function of (9) we can deduce

the matrix Q

for the minimax problem, so parti-

tioned:





(14)

















+ β

−β

... 0 0

−β

+ β

... 0 0

... ... ... ... ...

0 0 ... α

M−1

+ β

M−1

+ β

−β

M−1

0 0 ... −β

M−1

+ β







= diag(γ

,γ

,...γ

)

= Q

= 0

(15)

Matrices thus obtained refer to a dynamic system. In

order to perform an ofﬂine computation of the trajec-

tory we need to transform the system from dynamic

to static. According to the available data on the state

and on the disturbance at initial time (i.e. x(0) and

w(0), hereinafter respectively x

and w

for sake of

notation) the following problem has to be solved:

inf

sup



+ v

+ x

(1)Q

x(1)

||x



(16)

By writing x(1) as a function of x(0) from (2) and

developing the computation, (16) may be rewritten as:

inf

sup



+ A

A)x

+ v

+ B

B) + x

+ v

||x(0),w(0)||



(17)

The numerator in (17) can be represented in the ma-

trix form as:









(18)

More in detail, the matrix

Q is composed as follows:

Q =







+ A

A A

A R + B



(19)

and it holds all the data needed to solve the LMI prob-

lem of Theorem 1.

Furthermore, Theorem 1 of (Gattami et al., 2012)

demonstrates that there exist linear decision µ

x) =

x, for i = 1, ...,N where the ﬁnite value θ

∗

of the

game represented by equation (13) is achieved.

Corollary 1.1. The control can be applied on wider

time horizon, i.e. T > 1, applying results in (Gattami

and Bernhardsson, 2007), Section VII.

Table 1: Initial speeds.

Vehicle speed [m/s]

1 18.79

2 17.55

3 17.67

4 15.59

5 14.66

3 CASE STUDY

The case study refers to a ﬁve-vehicle platoon (M=5)

which is moving along a rectilinear path. A decreas-

ing in the speed proﬁle of the vehicles which the pla-

toon consists of is considered.

The main goal of the control law is indeed to re-

store the correct distance, which is assumed to be at

least 29 meters according to (6) and provided that

platoon is willing to proceed at the cruising speed

of 15m/s, thus slightly reducing the velocity of the

ﬁrst vehicles. In other words, the robust control aims

at spacing the vehicles with the correct interdistance

with smaller as possible variation of their speed.

Initial speeds used for our simulation are listed

in Table 1. This happens because they are not op-

timally spaced as between two neighboring vehicles

intervenes a distance of 20 meters (caused by exter-

nal factors prior to the start of the simulation), and

thus platoon is trying to actuate some techniques to

increase the span.

In this scenario the noise is assumed to be additive

white Gaussian, with zero mean and known covari-

ance.

Gains are maintained unitary for demonstration

purposes, but in a real implementation they need to

be tuned empirically according to their individual im-

portance. In particular, it is reasonable to prioritize

the restoring of the optimal interdistance when ve-

hicles seem to face difﬁculties in exiting the yellow

zone due to external factors. This will cause them to

signiﬁcantly reduce their speed to obtain better stabil-

ity while resuming initial platoon shape.

Fig.2 shows the deviation of the vehicle position

with respect to the planned one. While in the very ﬁrst

instants, a huge difference appears between desired

and actual trajectories, at the end of the simulation

which lasts 50 seconds, the convergence is achieved.

The results demonstrate the effectiveness of the robust

controller which restores the initial and optimal state

for the vehicles in the string formation.

Fig.3 displays the interdistance between vehicles

during the simulation. Also in this case, the prede-

ﬁned values (29 meters) are recovered carrying out

the main objective of the control law.

Real-time Robust Trajectory Control for Vehicle Platoons: A Linear Matrix Inequality-based Approach

413

Figure 2: Divergence in vehicles’ position with respect to

the planned one.

Figure 3: Distance between neighboring vehicles.

Finally, Fig.4 displays the velocity trends, which

are coherent with the expectations: ﬁrst vehicles need

to slow down to ensure a correct repositioning of the

other members of the platoon reaching out the sta-

bility after about 50 seconds. After that, they can

proceed at cruising speed and, in cooperative way,

increase both the speed and interdistance with other

control techniques.

Figure 4: Trend of the velocities.

Table 2: Vehicles’ state at the end of the simulation.

Vehicle Velocity [m/s] Distance from

preceding car

[m]

1 14.92 /

2 15.00 29.78

3 15.08 29.41

4 15.01 28.98

5 14.99 29.02

Figure 5: Vehicles’ accelerations.

Table 2 sums up vehicles’ velocity and distance

with the preceding element of the platoon, validating

our control algorithm.

4 CONCLUSION

In this paper, a robust control approach has been used

to restore the optimal interdistance between neighbor-

ing vehicles in a platoon trajectory planning.

The interdistance that can occur between two ad-

jacent vehicles has been classiﬁed in three different

levels according to growing safe conditions. The pro-

posed control model is dedicated to manage the ve-

hicle’s position and speed in the sub-optimal yellow

condition which is very sensitive in the design of ve-

hicle’s trajectory. Here, in fact, vehicles may assume

many different behavior, from the most aggressive to

the most conservative one.

The proposed control low restores the optimal in-

terdistance between each vehicle of the platoon by

reducing the variation of the speed proﬁle. In other

words, the objective is to settle and maintain the pla-

toon cruising speed with vehicles optimally spaced.

Results show (see Fig.5) that the acceleration re-

quired to perform the maneuver is feasible (i.e. it os-

cillates between values in -1 and 1), allowing us to

have some conﬁdence when testing the trajectory on

a more realistic model.

In addition, the resistance to disturbances and the

linearity of the control law allow this algorithm to be

ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics

414

used when fast modiﬁcation of the planned trajectory

is required due to unforeseen approaching among ve-

hicles. For future improvements we aim to test out

the control algorithm on a large set of unmanned car

or on curvilinear trajectories. This involves the anal-

ysis of the curve parameters and the computation of

a control law both for the acceleration and the yaw

angle separately.

Moreover, we are interested in verifying if the pe-

riodic computation of gains could improve the overall

performance of the platoon. In our simulation, in fact,

gains are estimated before the activation of the robust

controller and maintained until the end of the experi-

ment. By periodically checking the gain optimality, it

may be possible to track better the trajectories proﬁle

during the evolution of the system.

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