Proportional Integral Derivative Decentralized Control vs Linear

Quadratic Tracking Regulator in Vehicle Overtaking within a Platoon

Alessandro Bozzi

, Roberto Sacile

and Enrico Zero

DIBRIS – Department on Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova,

Genova, Italy

Keywords:

Autonomous Vehicles, PID, Linear Quadratic Control, Platooning.

Abstract:

This paper introduces a comparison between a decentralized Proportional Integral Derivative (PID) controller

and a centralized Linear Quadratic Tracking (LQT) controller to automatise the exchange of two inner vehicles

inside a platoon moving on a straight path. Lomonossoff’s model is used to represent vehicle’s longitudinal

dynamics. A case study is presented to demonstrate the effectiveness of both controllers respectively on

nonlinear and linearized model.

1 INTRODUCTION

Autonomous vehicle (AV) is an important research

ﬁeld of the current century which consists in a car ac-

quiring data and information in real time about the

neighboring environment and driving without the hu-

man interaction for a speciﬁed period of time. AVs are

classiﬁed accordingly with the vehicle autonomy de-

gree in six levels, from level 0 where there is no driv-

ing automation to level 5 where there is a full driving

automation (SAE, 2014). Equipping cars and light ve-

hicles with this technology will likely reduce crashes,

energy consumption, pollution and congestions (An-

derson et al., 2014). One of the main causes in road

trafﬁc accidents is the human behavior. An applica-

tion of new technologies to monitor driver’s condi-

tion becomes essential to detect anomalous driver be-

havior and prevent near miss accidents has been per-

formed (Zero et al., 2019).

As autonomous vehicles supplant human drivers,

automation’s ability to communicate and cooperate

with people will become more important. Not all ve-

hicles are equipped with sensors for autonomous driv-

ing, so it is also important that the autonomous vehi-

cles interact with the human drivers of other vehicles.

A cooperative maneuver among one autonomous car

and two human-driven vehicles equipped with sensors

and actuators has been tested in (Alonso et al., 2011).

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

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

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

In order to perform this work, the system needs the

position, speed, and intentions of the cars involved

in the maneuver. The authors managed the speed of

human-driven vehicles to make each element arrive at

the intersection at the same time, in order to analyze

the behavior of the unmanned car.

In any urban environment, the vehicle will need

to react safely to each type of unexpected event such

as ill-behaved pedestrians, and pranksters (Koopman

and Wagner, 2017).

In several works, the aim is the management of

vehicle trajectory while driving, in particular the min-

imization of the gap between the planned and the real

trajectory. A novel dynamics controller that consists

of longitudinal and lateral controllers for autonomous

vehicle to simultaneously control it as closer as pos-

sible to the driving limits while following the desired

path has been proposed (Ni and Hu, 2017).

Recent works are not focused on the behavior

management of just one vehicle but they focus on

the behavior of more AVs, namely a platoon of ve-

hicles. This aspect requires more than one sensor em-

bedded in the vehicle. The management of trajectory

of an AVs platoon is fundamental because it is related

to another goal - the reduction of energy consump-

tion. Indeed, it is veriﬁed that AV technologies re-

duce fuel consumption by managing better accelera-

tion and deceleration than a human driver (Anderson

et al., 2014).

The management of a trajectory for a vehicle in a

platoon is more complex rather than on a single AV

due to the unexpected events during the transit which

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

Proportional Integral Derivative Decentralized Control vs Linear Quadratic Tracking Regulator in Vehicle Overtaking within a Platoon.

DOI: 10.5220/0011272200003271

In Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2022), pages 427-433

ISBN: 978-989-758-585-2; ISSN: 2184-2809

427

can affect differently each element of the platoon. In-

deed, vehicle approaching or detachment from neigh-

bors can frequently happen. To generate countermea-

sures for each element of the system, in order to re-

store the correct position, speed and interdistance and

guarantee passengers’ safety, a control system based

on a real-time robust trajectory has been tested (Bozzi

et al., 2021). In order to ensure the safety of ma-

neuvers to let an external vehicle be inserted into the

platoon or alternatively to let a vehicle of the platoon

leave it, a longitudinal Model Predictive Control has

been implemented (Grafﬁone et al., 2020a). The au-

tomation of the overtaking maneuver such as the en-

trance and exit from a platoon is considered to be one

of the hardest challenges in the development of au-

tonomous vehicles. In this direction a fuzzy controller

has been performed to reduce the human interaction

during this maneuver (Naranjo et al., 2008).

This paper proposes a comparison between the

performance of a PID controller on a nonlinear con-

tinuous model and of a Linear Quadratic Tracking

(LQT) controller on a linearized discrete model to

swap the central vehicles of a four-vehicle platoon

moving on a straight path. The former control sys-

tem is decentralized as each vehicle has its own PID

to handle the maneuver, similarly to (Stankovic et al.,

2000), while the latter is computed in a centralized

way, as usually in these cases the leader governs the

vehicle position giving the optimal speed and acceler-

ation to the followers (Grafﬁone et al., 2020b).

The remainder of this work is organized as fol-

lows: Section II reviews the nonlinear and linear

model tested and it shows the driving scenario. In

Section III the case study is analyzed and the results

related to the nonlinear and linear controller simula-

tion. In Section IV conclusions about the comparison

of the two models are reported and the further devel-

opments of this work are proposed.

2 MODELS AND METHODS

In the literature, the longitudinal models are largely

investigated. This happens because the vehicle’s dis-

placement can be often subdivided in longitudinal and

lateral motion and the two are assumed additive. This

paper tackles the problem in the same way but using

a different longitudinal representation for elements of

the platoon. Indeed, they have been modeled through

the Lomonossoff’s equations, mainly used for trains

(as in (Lu et al., 2011)) but easily adaptable for cars

modelization. It faithfully represents vehicle’s evo-

lution overtime and provide the possibility of tak-

ing into accounr vehicle’s parameters such as mass

Table 1: Davis constants.

0.06

0.023

W 2

and frictions, that have primary importance especially

when dealing with trucks.

The model chosen is nonlinear due to a quadratic

term function of the speed.

2.1 Nonlinear Model

The Lomonossoff’s equations are:











˙x(t) = v(t)

˙v(t) = f (t) − (C

v(t) +C

(t))

−W gsin α(t)

(1)

where:

• x[m] and v[m/s] are state variables, respectively

position and speed of the vehicle

• f [kN] is the control input, corresponding to the

tractive effort

• C

[kN], C

[

m/s

], C

[

] are the Davis con-

stants, related respectively to mechanical resis-

tance, viscous mechanical resistance and aerody-

namic resistance

• W [tonnes] is the vehicle’s tare mass

• W

[tonnes] is the vehicle’s effective mass, includ-

ing rotary allowance

• α is the slope angle of the position of the vehicle

They represent the dynamics of the longitudinal mo-

tion for the individual vehicle by prompting a tractive

effort, with a maximum value similarly to what hap-

pens in train modelization.

In the following, it is assumed that α = 0, i.e. ve-

hicles are traveling on a ﬂat road. Vehicles are consid-

ered homogeneous and their parameter estimation for

the case study, which deals with heavy-duty vehicles

(HDVs), is listed on Table 1. The choice of analyzing

HDVs is due to recent literature results, which reveal

that the improvement in their performance is signif-

icant if they are placed in ascending order based on

their braking capabilities (Alam et al., 2014). So, an

algorithm to swap inner vehicles of the platoon can

provide long-term efﬁciency in fuel consumption and

emissions.

2.2 Linear Model

The model presented in (1) can be linearized for each

planned instant t

around a working state/control cou-

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

428

ple ( ¯v,

f ), supposing no acceleration in that instant of

time.

The resulting linear approximation that represents

the evolution of the system overtime is:

δ ˙x = A

δx + B

δ f (2)

where:

δx = [x(t) − ¯x(t) v(t) − ¯v(t)]

, δ f =



f (t) −

f (t)



(3)

0 1

0 −

+2C

¯v(t)

, B



1/W



(4)

It has to be noted that the A

matrix is time-variant,

as it depends on the actual speed of the vehicle. In

other words, the matrix A

is computed at each sam-

pling instant in order to let the system work around

an operating point that varies overtime and follows

the trend of vehicle’s speed.

The control algorithm is applied to the discretized

system with a sample time, for the case study, of

100ms.

2.3 Driving Scenario

This paper analyzes a driving routine in which two

inner vehicles exchange their position while the pla-

toon is moving on a straight path. Initial positions of

platoon is represented in Fig. 1a.

This kind of maneuver can be seen as an overtak-

ing with space constraints, since the vehicle behind

has to:

• position itself on the fast lane, thus exiting from

the string formation (Fig. 1b)

• overcome the vehicle in front of it in the platoon

formation (that in the meanwhile has to favor the

overtaking with a slow deceleration)

• settle at the correct distance between its neighbor-

ing vehicles (Fig. 1c)

• come back to the platoon lane (Fig. 1d)

The maneuver, graphically represented in Fig. 1 has

to be performed keeping similar speed with respect

to the rest of platoon and in a reasonable time frame.

Moreover, for the whole time, vehicles involved in the

swap have to prevent getting too closer to their neigh-

bors, thus endangering passengers’ safety.

The role of each element can be summarize as fol-

lows:

• Vehicle #1 has to proceed at constant speed and

measure the distance to its follower, in order to

improve the reconstruction of the surroundings

• Vehicle #2 decelerates in order to favor the over-

taking of vehicle #3, while measuring the distance

from the leader and the last element of the platoon

• Vehicle #3 overtakes vehicle #2 and measure the

distance from it and from the leader, in order to

re-enter the string formation in the best position

possible

• Vehicle #4, similarly to vehicle #1, has to proceed

at constant speed and increase the knowledge of

the environment by providing its measurements

(a) (b)

Figure 1: Overtaking maneuver with position constraints.

3 CASE STUDY

In the following, a four-vehicle platoon is considered

(M = 4). Vehicle #2 and vehicle #3 are involved in

the swap, while the ﬁrst and the last vehicle of the pla-

toon are assumed to proceed around the regime speed

reg

= 22[m/s]. The four-vehicle platoon represents

the most reasonable choice whereas what it happens it

that two trucks exchange their position and they have

to pay attention to adjacent elements of the platoon.

In more numerous platoons, the four-vehicle subset

can be taken into account by the controller only for

the time needed to perform the maneuver, while main-

taining other elements at constant speed.

The maneuver must ensure the safety of the whole

system, so between adjacent elements must intervene

a minimum distance, computed as in (Bozzi et al.,

2021). According to this formulation, the minimum

and the recommended distances are:

min

[m] =

3 ∗ v

reg

[km/h]

= 23.76m

opt

[m] = (

reg

[km/h]

)

= 62.73m

(5)

Proportional Integral Derivative Decentralized Control vs Linear Quadratic Tracking Regulator in Vehicle Overtaking within a Platoon

429

However, these bounds might even be reduced con-

sidering the faster reaction time of unmanned vehi-

cles with respect to human-driven ones. Even if there

are other rules to compute the optimal inter-vehicle

distance, such as the one stated by the Responsibility

Sensitive Safety (RSS) widely used in literature (e.g.

in (Shalev-Shwartz et al., 2017) and (Gassmann et al.,

2019)), it is useful to start with the recommended dis-

tances stated by the trafﬁc regulations which represent

the minimum constraints to satisfy within the road

nowadays. Of course, assuming only unmanned ve-

hicles it can emerge the possibility of taking into ac-

count shorter inter-vehicle distances.

The case study considers an initial inter-vehicle

span of d = 30[m], close to the critical bound (d

min

)

and thus representing a risky situation to perform an

overtaking maneuver between inner element of the

platoon. In the ﬁrst 5 seconds of the simulation ve-

hicles move at regime and vehicle #3 changes lane in

order to proceed with the overtaking maneuver. Re-

entering the lane is assumed to be done in the last 5

seconds of simulations, without altering the longitu-

dinal displacement.

The inter-vehicle distance should remain un-

changed at the end of the driving routine. Moreover,

vehicles involved in the swap should not get too closer

to other elements of the platoon (i.e. the leader for ve-

hicle #3, the last element for vehicle #2).

The performance of a PID controller on the con-

tinuous nonlinear system are presented and then com-

pared to a control algorithm that deals with the linear

discrete approximation of the Lomonossoff’s model

and makes usage of a LQT problem to compute the

optimal control input.

3.1 Nonlinear System

The control of the nonlinear system is governed by a

PID that acts converting the desired value in position

in a corresponding value in tractive effort. Each ve-

hicle that needs to alter its speed from the regime has

its own PID. The only communication that occurs be-

tween vehicles involves the message with the actual

position, in order for the PID to formulate the proper

control action to pursue the desired value. In fact, the

desired position is expressed in terms of the actual po-

sition, the position of the leader or the position of the

last element of the platoon (respectively for vehicle

#3 and vehicle #2) and the interdistance between ad-

jacent elements, computed as a function of the regime

speed. At each sampling instant k, it can be expressed

as follows:

(

(k) = x

(k) − 2d

(k) = x

(k) + 2d = x

(k) − d

(6)

Table 2: PID coefﬁcients.

Proportional 2.754

Integral 0.484

Derivative 2.986

Filter coefﬁcient 9.300

0 10 20 30 40 50 60

Time[s]

Speed[m/s]

reg

Figure 2: Vehicles’ speed: nonlinear system.

Reference trajectory of vehicle #3 should be written

with respect to the position of the leader, since unex-

pected behavior can arise when dealing with a large

scale platooning, as demonstrated in (Pates et al.,

2017). In this paper, though, considering the small

number of vehicles involved in the platoon, both for-

mulations give the same results.

The parameters of the continuous time PID has been

tuned using the Matlab/Simulink tool and lineariz-

ing the plant near the equilibrium point obtained with

the regime speed. Thus, unexpected behaviour may

emerge at very different speeds from the initial one.

The value used for the simulation are listed in Ta-

ble 2. The ﬁlter coefﬁcient is needed to improve the

action of the derivative term, not implemented as a

pure derivative because of its sensitiveness to noise.

Rate limiter has been added to avoid abrupt changes

between consecutive sampling instants, while output

saturation is needed to maintain vehicle around its

equilibrium point and increase the overall realism of

the control input on the virtual environment. Fig. 2

shows the trend of the velocities. As expected, there is

a slight acceleration from the third vehicle, while the

other starts decelerating to favor the overall maneu-

ver. The behavior is almost specular as both vehicles

have the same PID gains and their desired positions

are symmetric with respect to the center of platoon.

Fig. 3 conﬁrms the effectiveness of the nonlinear con-

troller, as the distance between vehicles varies accord-

ing to the expectation and it settles on multiples of 30

meters (based on the pair of vehicles analyzed), en-

suring that the initial inter-vehicle distance is main-

tained.

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

430

0 10 20 30 40 50 60

Time[s]

-40

-30

-20

-10

Inter-distance[m]

Figure 3: Inter-vehicle distances: nonlinear system.

It can be stated that the objective is achieved

smoothly and fastly, without abrupt changes in accel-

eration and thus preserving passenger’s comfort.

3.2 Linear Controller

The linear controller is more complex to implement

for many factors: ﬁrst of all, it relies on time-variant

information from the system. Second, it operates on

a discretized model which represents an approxima-

tion of the actual system. As a matter of fact, the state

of the vehicle is used to determine the linearized sys-

tem around the working point

v(t

) and to provide the

state for the LQT problem to be solved. For this rea-

son, it is assumed that the trends of the velocities for

the nonlinear evolution is known, as it will be used as

reference signal to be followed by the tracking algo-

rithm.

The knowledge of velocity trends is a strong as-

sumption, but it is reasonable as the behavior depends

on the actual speed of the vehicles. Thus, it is possible

to have a set of different maneuvers based on the ini-

tial regime speed of the other vehicles of the platoon.

The LQT requires a cost function in order to prior-

itize the tracking of certain state variables and the cost

of the control action. For the case study, the overall

cost function for each sampling instant k is:

J =

∑

i=1

∑

k=1



(k) − x

(k))

+ β

(k) − v

reg

)

+ γ

(k)



∑

i=1



(K + 1) − x

(K + 1))

(K + 1) − v

reg

)



(7)

With α, β and γ gains to be tuned to prioritize the re-

spective elements of the summation and K the control

horizon for the LQT problem. The higher the K, the

smoother but less responsive the reaction of the sys-

tem. In the case study it has been chosen K = 10 not to

overburden the computational cost. x

are retrieved as

in (6) for vehicles involved in swap, and computed as

constant displacement between consecutive sampling

instants for outer vehicles. v

reg

is the regime speed,

in the case study equals to 22m/s for each element of

the platoon.

More in detail the ﬁrst term is the quadratic devi-

ation from the desired trajectory, which for inner ve-

hicles x

is computed as in (6) and for outer vehicles

is not considered (i.e. α

,α

= 0), the second term is

the deviation from the desired speed and their trend is

supposed to be known, and the last term regards the

minimization of the input which translates in the min-

imum tractive effort to be applied to accomplished the

goals. The second summation is needed to represent

the ﬁnal control instant for the state.

The equivalent of the cost function in matrix form

to be prompted to the LQT is:

Q =







+ α

0 −α

0 0 0 −α

d − 2α

0 β

0 0 0 0 0 0 0

−α

0 α

0 0 0 0 0 2α

0 0 0 β

0 0 0 0 −β

reg

−α

0 0 0 α

0 0 0 0 0 β

0 0 −β

reg

0 0 0 0 0 0 α

0 0

0 0 0 0 0 0 0 β

−α

d − 2α

d 0 2α

d −β

reg

−α

d −β

reg

0 0 0







R =







1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1







(8)

Note that Q ∈ R

2M+1

and R ∈ R

M+1

due to non

quadratic terms present in the cost function that re-

quire to increase the size of the matrix and the state

vector, as explained in (Boyd, 2008).

It is clear that the control technique is centralized,

thus the state of the system is composed by all vehi-

cles belonging to platoon, even the ones that have to

proceed at constant speed. This centralization ensures

the optimal behavior from the platoon’s point of view,

disregarding the individual element favoring the over-

all safety of the system.

In the case study, it has been decided to greatly

privilege the tracking of speeds over the control cost,

setting high β gains. It has to be noted that, if all

α gains are set to zero, the platoon control problem

translates into M control problems concerning the in-

dividual vehicle, as there are no constraints related to

inter-vehicle distance.

This turned out to be necessary since with unitary

gains the control technique is not able to perform the

swap between vehicles. This is shown in Fig.4 and

happens because, with unitary gains, for the control

algorithm is not convenient to perform the maneuver

due to the cost of the input.

On the other hand, if huge importance is given to

the maneuver (i.e. gains to track velocity trends are

Proportional Integral Derivative Decentralized Control vs Linear Quadratic Tracking Regulator in Vehicle Overtaking within a Platoon

431

0 10 20 30 40 50 60

Time[s]

-50

-40

-30

-20

-10

Distance[m]

- x

)

- x

)

=1000

- x

)

- x

)

=1000

Figure 4: Difference between actual and desired position

with α = 1 and α = 1000.

0 10 20 30 40 50 60

Time[s]

-40

-30

-20

-10

Inter-distance[m]

X 59.9

Y 24.2651

X 36

Y 23.3478

Figure 5: Inter-vehicle distances: linear controller.

signiﬁcantly higher than gains for the control cost) the

inter-vehicle distance evolves as shown in Fig.5, with

two markers to point out the behaviour in the middle

of the simulation and its ﬁnal value.

The overall objective is achieved, even if vehicles

experiment little reductions in their interdistances.

This does not affect safety of the whole system, since

each element preserves a distance greater than d

min

with the following vehicle (very close to the mini-

mum bound for the ones involved in the swap). This is

even more valuable considering the high regime speed

and the unfavorable initial conditions that should not

suggest such a maneuver. Increasing the initial span

produces safer results as vehicles have more space in

which to exchange. Moreover, the input prompt re-

ported in Fig. 6 highlights the difference between the

nonlinear case, in which the output reach its satura-

tion, and the linear evolution, closer to the regime and

desired value, improving the fuel consumption over-

time.

0 10 20 30 40 50 60

Time[s]

Tractive force[kN]

Figure 6: Control action for vehicles involved in the swap.

4 CONCLUSION

This paper analyzes the swap of inner elements of a

four-vehicle platoon moving on a straight path. The

work shows a comparison between PID controller on

nonlinear continuous system and LQT controller on

a discretized linear system. A case study for vehi-

cles proceeding at medium speed is presented and an-

alyzed. Even if it is possible, after a correct tuning

of PID and gains, to perform the exchange at higher

velocity, it is not recommended for safety reasons. In

addition, the beneﬁts produced by such driving rout-

ing are detectable mainly on heavy-duty vehicles, that

do not travel at higher speed.

To conclude, it can be asserted that the driving ma-

neuver can be ﬂuently completed by means of the de-

centralized controller. It acts on the nonlinear con-

tinuous system guaranteeing passengers’ comfort and

safety, although it requires an accurate tuning of the

PID parameters to work properly.

Even with the linear controller, though, vehicles

can successfully achieve the swap without endanger-

ing the whole system, also when initial conditions are

not favourable for this maneuver and suggest a more

cautious drive. There still is an offset in the steady

state interdistance, that may be reﬁned by low-level

controllers to be activated once the system is reaching

its regime. Moreover, the tuning can be done more

roughly by prioritizing the velocity constraints over

the position ones. For these reasons and considering

the linearity of the controller and its action on a sim-

pliﬁed model of the system, the results obtained on

the nonlinear model are satisfactory.

Its main drawback is the need for a priori informa-

tion that may not be available at the beginning of the

maneuver. For future improvements, we surely aim

to provide a method to dynamically generate velocity

trends to accomplish the driving routine.

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

432

It is also worth analyzing an overtaking maneu-

ver that involves more than two vehicles, to fasten the

reaching of the optimal string formation in bigger pla-

toons, though it may require too much time to be ac-

complished even in a free highway road. However, its

feasibility will be studied in further works.

Instead, there is no need to test the algorithm at

much higher speed (> 100km/h), since its advantages

are relevant especially for heavy-duty vehicles that

should not travel and even perform overtaking maneu-

ver at those velocities.

ACKNOWLEDGEMENT

This study is supported by the SysE2021 project

(2021-2023), “Centre d’excellence transfrontalier

pour la formation en ing

enierie de syst

emes” de-

veloped in the framework of the Interreg V-A

France-Italie (ALCOTRA) (2014 - 2020), Programme

de coop

eration transfrontali

ere europ

eenne entre la

France et l’Italie.

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