# Cooperative Predictive Cruise Control: A Bargaining Game Approach

Miguel F. Arevalo-Castiblanco  Jaime Pachon  Duvan Tellez-Castro  Eduardo
Mojica-Nava

(Received: date / Accepted: date)

###### Abstract

This paper considers a cooperative cruise control problem from a predictive
control perspective. Online decision-making is used to be executed during the
driving process based on the information obtained from the network. We
formalize a synchronization problem approach from a predictive control theory
using bargaining games to find an operating agreement between the vehicles.
Finally, we test these results in an emulation environment in a hardware-in-
the-loop system.

##  1 Introduction

In recent years, there has been an exponential increase in current vehicle
transportation issues such as accidents, jam traffic, and air contamination,
among others [1, 2]. The main companies in the automotive industry have
prioritized the development of vehicle control systems (e.g., 14% of vehicles
in the United States employed it recently). These control strategies mainly
regulate the speed of a motor vehicle based on the presence of relatives in
its vicinity [3]. For this reason, using cooperative control strategies may
reduce the transit time of vehicles on roads [4].

Recently, several cooperative control approaches have been proposed for
networked vehicles [5]. The first works focused on cruise control without
employing inter-vehicle communications to improve highway comfort and energy
expenditure. The rise of wireless communications allows a network of vehicles
to be seen as a cooperative study case [2]. The Cooperative Adaptive Cruise
Control (CCAC) concept was developed using synchronization laws based on a
reference acceleration profile [6]. The CCAC technique is based on string
stability, where vehicles maintain a predefined distance at a constant speed,
but it might not result in an optimal response [7].

On the other hand, predictive control strategies can find the optimal values
in terms of energy or performance [8]. In [9, 10], distributed predictive
control strategies for agent synchronization have been developed using inter-
agent communication and the construction of cost functions that involve
network information. A Model Predictive Control (MPC) problem involving
different agents where decisions depend on others can be considered as a game
[11]. Likewise, if these agents manage a common goal, the problem can be
described as a bargaining game, where a set of players with a target modify
their actions based on a disagreement between them [12]. Theoretically, the
point of disagreement is defined as the minimum satisfaction expected for the
negotiation. In practical implementation, cooperative cruise strategies can
present several drawbacks, and only a few authors have validated these
developments in highway vehicle platoons [13]. The main practical tests of the
theories explained are usually validated in dynamic emulation models, or robot
networks [14, 15].

The emulation of dynamic systems in hardware-in-the-loop enables the
validation of real-time methods in situations where the models are unavailable
(e.g., autonomous vehicle network). Emulation hardware based on embedded
systems is used as a dynamic environment simulation with predefined models
interaction and a visualization interface for validating desired signals [16].
For distributed networks, hardware emulation is commonly used in electrical
and communication systems to validate future installations. The validation of
these systems is mainly given by the need to prove the algorithm’s correct
operation in harsh application environments [17, 18]. In recent works, few
results have been based on the theory of bargaining games as a problem of
cooperative control in distributed systems, without considering vehicle
platoon emulation [19]. Other works have emulated multi-agent systems without
considering the application of predictive control techniques or game theory in
vehicles [20].

Identifying the research gap falls at the application level in the case of
cooperative cruise control, where the control strategy must solve a
distributed optimization problem according to the states of the agents, the
point of disagreement and the predefined cost functions for energy expenditure
presets for simulation and emulation. A technique that allows managing control
algorithms efficiently without increasing computational expense in the
presence of the network is a distributed bargaining methodology. The main
contribution of this paper then, is threefold: 1) First, the appropriation of
a networked vehicle control problem is to be worked from a predictive
perspective in a novel way. 2) Second, the inclusion of bargaining game theory
for the cooperative control problem in simulation. This development allows it
to be compared with centralized and decentralized predictive control
algorithms for symmetric and non-symmetric cases. 3) Finally, a validation of
the procedures developed in emulation with hardware-in-the-loop tools.

The rest of the paper is organized as follows: Section 2 presents the
cooperative cruise control and bargaining games as a tool for Distributed
Model Predictive Control (DMPC) background. Section 3 contextualizes the
bargaining problem for the solution of the cooperative cruise problem. In
Section 5, we show the study case application and its simulation. Section V
shows the emulation of the system in hardware-in-the-loop, and finally,
Section 6 sets out the conclusions of the work.

##  2 Background

This section presents the basic foundations of the cooperative cruise control
theory, and DMPC as a bargaining game.

###  2.1 Cooperative Cruise Control

The cooperative cruise control problem has been extensively studied in latest
years. Recent developments have focused on wireless vehicle-to-vehicle (V2V)
communication that has grown commercially. The wireless communication led to
the definition of the Grand Cooperative Driving Challenge (GCDC) to manage a
platoon of vehicles that have this technology. The main objective of
cooperative cruise control is synchronizing the vehicles on the road with the
traffic profile considered by an established reference. The traffic profile is
commonly constituted by the inter-vehicle distance and the speed on a highway,
reducing vehicle time and fuel consumption. Communication is usually
considered by the predecessor vehicle as a string stability case.

A simple description of the CACC setting (for longitudinal dynamics) is
considered. In this case, each vehicle is modeled through its physical and
mechanical parameters. The dynamics proposed in this case are linear as

$\displaystyle\dot{p}(t)$ $\displaystyle=v(t),$ (1) $\displaystyle\dot{v}(t)$
$\displaystyle=a_{1}p(t)+a_{2}v(t)+b(\tau(t)+f(p,v)),$ (2)

where variables are speed $v$ and position $p$, respectively. The parameters
$a_{1}$ and $a_{2}$ are transmission parameters, $b$ is related to
transmission efficiency, and $\tau$ has the dimension of acceleration, or the
force when it is multiplied by the vehicle mass. The term $f(p,v)$ is
associated with an input uncertainty. These parameters can be included in the
dynamics due to the approximation by an inversible steady-stable time-
invariant model without the presence of uncertainties [21]. This approach
considerably reduces the complexity of the model without losing performance,
and it has been used as an approximation in previous theory for validation of
this type of problem [22].

The controller must be able to regulate the speed of each vehicle and maintain
a distance from its neighbors. The graphic representation of vehicles platoon
is shown in Fig. 1, where the distance between each vehicle is defined as
$d(t)=p_{1}(t)-p_{2}(t)$, that is the difference of the position of vehicles 1
and 2 with its respective subscripts $p_{1}$ and $p_{2}$.

Figure 1: Vehicle Platoon Approach Diagram.

Let the states of the system be $x(t)=[d(t),v(t)]^{\top}$, where the distance
is position minus a predefined distance, and the control input $u(t)=\tau(t)$
with a possibly input matched uncertainty $f(p,v)$. It is possible to write
the system as a linear affine continuous model of the form

$\dot{x}(t)=Ax(t)+b(u(t)+f(x)),$ (3)

with

$A=\begin{bmatrix}0&1\\\ a_{1}&a_{2}\end{bmatrix},\quad b=\begin{bmatrix}0\\\
b\end{bmatrix}.$ (4)

This model considers the acceleration of neighboring vehicles in a $k-th$
instant of time. For implementation control prediction, it is necessary to use
the discrete dynamics of the system. Therefore (3) is modified as

$x(k+1)=A_{k}x(k)+b_{k}(u(k)+f(x_{k})),$ (5)

with $A_{k}=e^{AT}$ and $b_{k}=A-1(A_{k}-I)b_{k}$.

Note that onboard sensors for controller action measure distance and speed,
and the parameters of each vehicle can be different for a heterogeneous case.

###### Remark 1

For the managed approximations of the cooperative cruise control theory, the
considered model relates the position, inter-vehicular distance, and the speed
of the vehicles in the network, assuming, in this case, a constant
acceleration in an instant of time for each agent during development.

###  2.2 Distributed Model Predictive Control with Bargaining Games

In this section, we introduce the basic concepts of MPC from a bargaining game
perspective. It is contextualized as the negotiation method for solving a
distributed optimization problem. The block diagram representing the framework
is presented in Fig. 2, where it is observed that each vehicle, through its
dynamics, enters a bargaining algorithm along a prediction horizon. This
algorithm generates the optimal control action $u^{*}_{i}(k)$ together with
the game relationship variable, the disagreement point $\beta_{i}$. This block
diagram is used for non-symmetric bargaining cases. For symmetric cases, the
characteristics of each vehicle are the same. All cases are regulated by
regulatory aspects or physical restrictions of the vehicles to be considered.

Figure 2: Model predictive control with bargaining games.

A bargaining game is mathematically defined as the tuple
$G=(N,\Lambda_{i},\psi_{i}),\forall i\in N$, where $N$ is the number of
vehicles involved in the process, $\Lambda$ is the decision space of the
control law, and $\psi_{i}$ is defined as the local cost function of each
vehicle. It is assumed that the vehicles are in a negotiating position to
achieve a common objective, such as Nash’s notions [23], where each player is
sought to achieve the best benefit in common with their neighborhood according
to the application [24]. In the game, if it is impossible to reach an
agreement, the term disagreement point is used for the bargain between
vehicles [25].

Assuming the dynamics of each vehicle as in (3) and with its discrete
representation (5), the particular objective is to achieve energy-level
optimization in each vehicle’s operation. For this optimization problem, a
usually quadratic cost function is defined in a distributed way as

$L_{i}(\tilde{x}_{i}(k),\tilde{u}_{i}(k))=\sum_{t=0}^{N_{p}}\psi_{i}(\tilde{x}_{i}(k),\tilde{u}_{i}(k)),$
(6)

where $\tilde{x}_{i}(k)$ is the representation of vehicle $i$ states built
along the prediction horizon
$[x^{\top}_{i}(k),\ldots,x\top_{i}(k+N_{p})]^{\top}$, likewise
$\tilde{u}_{i}(k)=[u_{i}(k),\ldots,u_{i}(k+N_{u}),\ldots,u_{i}(k+N_{p})]$
considering the control horizon $N_{u}$ with $N_{u}\leq N_{p}$. The local cost
function is defined as

$\psi_{i}(\tilde{u}(k))=\tilde{u}^{\top}(k)Q_{uui}\tilde{u}(k)+2x^{\top}_{i}(k)Q_{xui}\tilde{u}(k)+x_{i}^{\top}(k)Q_{xxi}x_{i}(k),$
(7)

which is positive defined, convex, and where $Q_{uui},Q_{xui}$ and $Q_{xxi}$
are weighted positive definite matrices, i.e., $Q_{uui}\succ 0$.

For the control problem formulation, it is important to define a decision
space $\Lambda=\prod_{i=1}^{N}\Lambda_{i}$ for the whole system according to
the physical operating conditions. An MPC problem with communication between
agents is interpreted as a bargain so that it can be a bargaining game. For
the analysis and solution of this type of game, Nash proposes an axiomatic
methodology [26], which was used in continuous and static systems [25].

A continuous representation for a bargaining game is with the tuple
$(S,\beta_{d})$, where $S$ is the game decision space, which is a non-empty
closed subset of $\mathbb{R}^{N}$ and $\beta_{d}\in\text{int}(S)$ is the
interaction disagreement point. For implementation purposes, it is important
to consider the discrete dynamics of the game, so is then defined as
${(\Gamma(k),\beta_{d}(k))}^{\infty}_{k=0}$ with $\Gamma(k)$, a closed non-
empty $\mathbb{R}^{N}$ subset that contains the profit functions values of
each vehicle. The values of the states $x_{i}$, the set $\Gamma$, and the
point of disagreement $\beta$ vary dynamically.

The evolution value of the disagreement point varies as

$\beta_{i}(k+1)=\begin{cases}\beta_{i}(k)-\mu(\beta_{i}(k)-\psi_{i}(\tilde{u}(k)))&\mbox{if
}\beta_{i}(k)\geq\psi_{i}(\tilde{u}(k)),\\\
\beta_{i}(k)+(\psi_{i}(\tilde{u}(k))-\psi_{i}(k))&\mbox{if
}\beta_{i}(k)<\psi_{i}(\tilde{u}(k)),\end{cases}$ (8)

where $0\leq\mu\leq 1$ is an adjustment constant according to the definition
of the axioms of the negotiation processes raised from the work of John Nash
[27]. In this case, if a vehicle decides to cooperate in the road, the
disagreement point is reduced with a
$\mu\left[\beta_{i}(k)-\psi_{i}(\tilde{u}(k))\right]$ factor, otherwise, it is
increased by a $\left[\psi(\tilde{u}(k))-\beta_{i}(k)\right]$ factor. As a
complementary definition for the game, $\zeta_{i}$ is defined as the utopia
point available for the vehicle $i$ as $\zeta_{i}(\Gamma)=\max{\psi_{i}}\text{
, }(\psi_{i})_{i\in N}\in\Gamma$ , $\forall i\in N,$ $\Theta$ is defined as
the union of the cost functions $\psi_{i}$ of the game, where then the
discrete game can be interpreted as $\\{\Theta,\beta(k)\\}^{\infty}_{k=0}$.

###### Remark 2

The analysis of a bargaining game can be carried out symmetrical for a game
with similar characteristics between its players or non-symmetrical for a game
where these characteristics differ, i.e., synchronization for oscillator
systems with homogeneous characteristics or control of mechanical systems with
heterogeneous physical characteristics.

For the solution of a bargaining game, a non-symmetrical centralized scenario
is proposed based on [28] as

$\displaystyle\max_{\tilde{u}(k)}\sum_{i=1}^{N}(\lambda_{i}\log(\beta_{i}(k)-\psi_{i}(\tilde{u}_{i}(k)))),$
(9) s.t. $\displaystyle\beta_{i}(k)>\psi_{i}(\tilde{u}(k)),$ (10)
$\displaystyle\tilde{u}(k)\in\Lambda,$ (11)

where $\lambda_{i}$ is a weight variable usually defined as
$\lambda_{i}=\frac{1}{N}$, with $N$ as the number of vehicles involved in the
process. However, for a distributed control analysis, the solution to the
optimization problem is proposed as

$\displaystyle\max_{\tilde{u}_{i}(k)}\sum_{r=1}^{N}(\log(\beta_{r}(k)-\kappa_{r}(\tilde{u}_{i}(k),\tilde{u}_{-i}(k)))),$
(12) s.t.
$\displaystyle\beta_{r}(k)>\sigma_{r}(\tilde{u}_{i}(k),\tilde{u}_{-i}(k))$
(13) $\displaystyle\tilde{u}_{i}(k)\in\Lambda,$ (14)

with $\kappa_{r}(\tilde{u}_{i}(k),\tilde{u}_{-i}(k))$ a distributed cost
function usually defined quadratic and $\tilde{u}_{-i}(k)$ the set of the
remaining vehicles control actions except for the agent $i$.

The optimization problem (14) differs from problem (11) about it considers
$\tilde{u}_{-i}(k)$ fixed and only optimizes as a function of
$\tilde{u}_{i}(k)$, this means that optimization does not involve the
decisions of the entire network cooperatively. The solution to this problem
then arises as a negotiation model that depends on the context given by the
cooperative cruise control theory. This type of methodology does not use
iterative solutions as others commonly used in distributed optimization
problems [29], which reduces the computational cost in operation with great
benefits in high-impact applications such as vehicle platoon. Bargaining
methodology allows the solution to a distributed control problem by solving
only one local optimization with the information collected by its neighbors
and achieving an agreement based on the Nash theory of bargaining through the
defined disagreement point. In summary, the objective is to apply a
distributed control methodology for vehicles network on a highway. Based on
the communication of their states, a negotiation can be interpreted as the
solution to an optimization problem (14).

##  3 Cooperative Cruise Control as a Bargaining Game

Considering the definition of a DMPC as a bargaining game presented in Section
2, the cooperative cruise control problem is contextualized. In this scenario,
the vehicles on a highway synchronize their dynamics from a reference model in
vehicle distance and speed. The global cost function of the DMPC must be made
up of two terms: one term associated with the tracking error in the distance
between vehicles, and the other term with the speed of each one during its
transit on the road.

The cooperative cruise control model is taken from (5), where matrices
$A_{k},B_{k}$ are obtained from dynamic models and (4). The output is defined
in this case as the speed of each vehicle. The vehicle’s acceleration gives
the control action $u(k)$. An operative constraint is defined according to
comfort parameters of $32.6$ m/s [30].

Initially considering each vehicle independently, the local MPC problem is
formulated as

$\displaystyle\min_{\tilde{u}(k)}J\left(\tilde{u},x(k)\right)$ (15)
$\displaystyle\text{s.t}.$ $\displaystyle
x_{i}(k+N_{p}+1)=\bar{A}_{i}x_{i}(k+1)+\bar{b}_{i}u_{i}(k+1),$ (16)
$\displaystyle y_{i}(k+N_{p}+1)=C_{i}x_{i}(k+1)+d_{i}u_{i}(k+1),$ (17)
$\displaystyle x_{i}\in\mathcal{X},$ (18) $\displaystyle u_{i}\in\Lambda,$
(19)

with $N_{u}<\alpha<N_{p}-1$, $\Lambda$ is the decision space of the control
law, $\bar{A}_{i}$ and $\bar{b}_{i}$ are the state matrix and vector resulting
from the prediction along $N_{p}$. The characteristics of the software and
hardware determine any possible delay that may be found in the communication
for control operations.

Let $\kappa_{i}(\tilde{u}_{i}(k),\tilde{u}_{-i}(k))$ be the global cost
function of each vehicle defined as

$\displaystyle\kappa_{i}(\tilde{u}_{i}(k),\tilde{u}_{-i}(k))=$
$\displaystyle\lambda|\tilde{v}_{r}(k)-\tilde{y}_{v}(\tilde{u}_{i}(k),\tilde{u}_{-i}(k))|$
(20)
$\displaystyle+[\tilde{u}^{\top}_{i}(k),\tilde{u}^{\top}_{-i}(k)]\bar{H}_{i}[\tilde{u}^{\top}_{i}(k),\tilde{u}^{\top}_{-i}(k)]^{\top}+2\bar{F}[\tilde{u}^{\top}_{i}(k),\tilde{u}^{\top}_{-i}(k)]^{\top},$
(21)

where $H_{i}$ and $F_{i}$ are matrices obtained from $Q_{uu}$ and $Q_{ux}$,
respectively. The input restrictions and states are time-independent and may
differ for each vehicle. Therefore, the bargaining game for cooperative cruise
control is defined as
$G_{CCC}=\\{N,\\{\tilde{u}_{i}(k),\tilde{u}_{-i}(k)\\},\Lambda_{i}\\}$,
$\forall i\in N$. Each vehicle at the control level has the same objective, to
minimize the synchronization error to maintain the distance between vehicles
and the speed in a stable state. The solution of this game with discrete
characteristics of the form $\\{\Theta,\beta(k)\\}^{\infty}_{k=0}$ is solved
by the following algorithm

Result: Optimal control signals

1 Initialize $u_{i}$, $\beta_{i}$ ;

2 while  _$e_{ij} >\Delta$_ do

3 $x_{i}$ send to others $x_{i}(k)$,$\beta_{i}(k)$;

4 $x_{i}$ solve (14);

5 $x_{i}$ selects the first control action $\tilde{u}(k)$;

6 Each vehicle modifies $\beta_{d}$ according to (8);

7 $x_{i}$ sends the modification of $\tilde{u}(k)$.

8 end while

Algorithm 1 Distributed bargaining algorithm

Algorithm 1 methodologically explains that each vehicle sends its dynamics
information to its neighbors as long as the synchronization error $e_{ij}$ is
greater than a given constant $\Delta$. With this information, it is possible
to solve the optimization problem (14) in each agent, to subsequently modify
the values of the disagreement point until achieving convergence in the
network synchronization. Finally, the modification to the point of
disagreement is sent back to the neighbors. That process is one of the main
contributions of this work since it summarizes the control methodology used in
simulation and beyond with implementation for a cooperative cruise control
problem.

The bargaining game result is mathematically defined as the tuple
$\xi(\Theta,\beta(k))\\\ =\\{\psi_{1},\psi_{2},\ldots,\psi_{\mathcal{N}}\\}$
composed of the profit of each vehicle. If there is no cooperation on the
highway, its value in the tuple is replaced by the disagreement point.

###### Proposition 1

The proposed solution $\xi(\Theta,\beta(k))$ of a discrete bargaining game
${(\Theta,\beta(k))}^{\infty}_{k=0}$ in $k$ steptime is unique and depends on
optimization problem (16) and $\Theta$ that must be convex.

Proof: It follows by [19].

As we mentioned in Remark 1, we consider two cases, one with similar
characteristics of vehicles assimilating a symmetric game and another one with
vehicles with non-similar characteristics assimilating a non-symmetrical game.
However, we can note that non-symmetric bargaining games are the most commonly
found in real-life applications since each subsystem that makes up a game
usually has different characteristics than the others. The following lemma is
proposed, based on Algorithm 1.

###### Lemma 1

Consider a cooperative cruise control problem as a bargaining game
$\\{(\Theta,\beta(k))\\}_{k=0}^{\infty}$, then solution $\xi(\Theta,\beta(k))$
is the Nash bargaining solution at time $k$ computed by Algorithm 1.

Proof: According to the definition of the game and satisfying its axiomatic
analysis [Section 2, [25]], the solution of the cooperative cruise problem as
a game $\\{(\Theta,\beta(k))\\}_{k=0}^{\infty}$ is defined as the Nash
bargaining solution for every step time $k$ obtained through the negotiation
problem $\blacksquare$.

Explicitly, this theory allows a complete structuring of the network, if
required, through the transmission of the utility functions or system inputs
to benefit the solution of each local optimization problem. These algorithms
can make their own decisions separately, so their implementation does not need
an iterative process. That decision-making process considerably reduces the
computational burden that Lagrange multiplier-based solution methods can
present.

##  4 Simulation Results

For the mentioned methodology, an application field is proposed based on the
problem of an autonomous network of vehicles that is increasing nowadays,
where each vehicle must follow the same patterns (position and speed). The
most well-known technique for this problem is cooperative adaptive cruise
control, an extension of the adaptive cruise control, a problem worked at a
platoon level with onboard sensors. In this case, each agent is modeled as a
linear second-order system such as in (5) with the remarks considered. For the
experiment, the leading vehicle defines an acceleration profile that all
agents must follow through a fixed distance between each one. This means in
terms of synchronization $x_{i}-x_{j}\xrightarrow{}0$. It is important to
highlight four aspects of this methodology, the vehicle dynamics, the
distributed controller, the information communicated, and the graph
communication topology [31].

A numerical simulation is performed to validate the proposed control laws.
Figure 3 shows the simulation digraph, where the agent $0$ acts as the leader
node. The formation control idea in the platoon starts from the graph
representing the communications, intercommunicated vehicles handle a speed and
inter-vehicular distance contemplated by the reference vehicle.

Figure 3: Study case communication graph. Table 1: Players Coefficients and Initial Conditions. | $a_{1}$ | $a_{2}$ | $b_{1}$ | $x_{0}$
---|---|---|---|---
$A_{0}$ | -0.25 | -0.5 | 1 | $[2\hskip 2.84544pt1]^{\top}$
$A_{1}$ | -1.25 | 1 | 0.5 | $[1\hskip 2.84544pt2.1]^{\top}$
$A_{2}$ | -0.5 | 2.5 | 0.75 | $[1\hskip 2.84544pt-0.2]^{\top}$
$A_{3}$ | -0.75 | 2 | 1.5 | $[3\hskip 2.84544pt2.3]^{\top}$
$A_{4}$ | -1.5 | 2.5 | 1 | $[3\hskip 2.84544pt0.6]^{\top}$
$A_{5}$ | -1 | 2 | 1 | $[2\hskip 2.84544pt-0.5]^{\top}$
$A_{6}$ | -0.75 | 1 | 0.5 | $[1\hskip 2.84544pt0.4]^{\top}$

In Table 1 the simulation parameters used are presented, highlighting that
these are used only for simulation, not for control design. The results of
symmetric and non-symmetric games are shown in both simulation and
implementation. MATLAB software and the fmincon optimization problem-solving
command are used for the optimization problems. The cost functions and
decision variables are packed along the prediction horizon and using
Kronecker-like structures for all the procedures. External parameters of
communication or interaction between the agents are not considered for
simulation purposes.

###  4.1 Symmetric Game

For the simulation of the system, symmetric and non-symmetric cases are
presented. In the symmetric case, the cost function is defined as (21), making
a grouping according to the theory, and the local cost function is defined as
(7).

The following figures show the agents’ response when the solution to the
distributed optimization problem is obtained. For simulation purposes, the
agents dynamic is defined with the parameters $a_{1}=1$ and $a_{2}=b_{1}=-1$.
Fig. 6 shows the network response with the bargaining model performed.

Similarly, Fig.8 shows the cost function and control action values evolution,
respectively, where it is evident that they achieve synchronization based on
Nash Theory.

We make a comparison with a centralized and decentralized model predictive
control problem, reflected in Figure 6–6. Generally, a centralized problem is
solved for each agent without any information sent. In a decentralized way,
the information is sent, and with it, a single optimization problem per agent
is solved. In the figures, it is evidenced that although the response of
centralized and decentralized systems handles a faster response,
synchronization is also achieved by communicating information in a distributed
way, optimizing the processes of the applied network.

Figure 4: Symmetric bargaining simulation standard game result

Figure 5: Symmetric bargaining simulation centralized result

Figure 6: Symmetric bargaining simulation decentralized result

Figure 7: Symmetric bargaining Cost Function

Figure 8: Symmetric bargaining Control action

###  4.2 Non-Symmetric Game

For the game with non-symmetric characteristics, the same case of cooperative
cruise control is used. However, the parameters of each agent are taken from
Table 1. The cost function used is the same as in the symmetric case (7). Fig.
11–11 shows the output response of each system under the negotiation model
compared with a centralized and decentralized predictive control methodology,
where convergence is observed equally handling the information in a
distributed way.

Figure 9: Non-Symmetric bargaining simulation game result

Figure 10: Non-Symmetric bargaining simulation centralized result

Figure 11: Non-Symmetric bargaining simulation decentralized result

In this case, the point of disagreement is plotted since, under the concept of
symmetric and non-symmetric games, there may be a variation in said value
(even considering symmetric cases with different initial conditions). Then,
the response of the disagreement point is observed in Fig. 14 where the Nash
agreement achieved from the consensus of this value is evidenced.

Figure 12: Disagreement point Non-symmetric bargaining simulation result

Figure 13: Cost Function Non-symmetric bargaining simulation result.

Figure 14: Control action Non-symmetric bargaining simulation result.

Finally, the cost function response is presented in Fig 14, where it is
possible to validate the Nash theory through the convergence of these values
in all agents, as well as the application of the control action for each one
in the Fig. 14.

##  5 Implementation Results

Basic experiments are performed to apply the developed algorithms with real-
time simulation hardware. The validation is made through a temporal response
of the developed algorithms, along with physical considerations. We use the
National Instruments CompactRio controllers connected through an Ethernet
network. Four controllers of two types are used for development, a NI9045
CompactRio controller and three NI9063 CompactRio controllers. The photo of
the modules inside the DESYNC laboratory at Universidad Nacional de Colombia
is shown in Fig. 15. Similarly, as seen in Fig. 16, the inclusion of each
controller within Rack is observed. Two NI9063 controllers and their two power
supplies are observed. In the other cabinet, the remaining NI9063 controller
and the NI9045 controller are shown next to their power supplies and the
Ethernet communication switch. The communications graph is defined as in Fig.
17.

Figure 15: DESYNC lab implementation racks Figure 16: Block Rack Diagram for
Implementation Figure 17: Communication graph used in emulation case

Similarly, for the dynamic models implemented, the same dynamic (2) is used
with a Tustin discretization for system matrices with a sample time of 0.1s.
The simulation parameters are observed in Table 2.

Table 2: Agent’s Coefficients and Initial Conditions for emulation case. | $a_{1}$ | $a_{2}$ | $b_{1}$ | $x_{0}$
---|---|---|---|---
$A_{0}$ | -0.25 | -0.5 | 1 | $[1\hskip 2.84544pt2]^{\top}$
$A_{1}$ | -1.25 | 1 | 0.5 | $[1\hskip 2.84544pt4]^{\top}$
$A_{2}$ | -0.5 | 2.5 | 0.75 | $[-1\hskip 2.84544pt2]^{\top}$
$A_{3}$ | -0.75 | 2 | 1.5 | $[1\hskip 2.84544pt4]^{\top}$

For the implementation, communication is made between Labview and Simulink,
where the dynamics of the controllers are emulated. For the response of the
symmetric game, the same parameters of the simulation case are used ($a_{1}=1$
and $a_{2}=b_{1}=-1$). Fig. 18 shows the controller’s response implemented in
a symmetric game. In emulation, the system presents some fluctuations
initially, but their response also achieves an adequate bargain. In all those
cases, the fluctuations are derived from the fact that by embedding the
dynamics and control in each module separately and requiring communication
between them, the response does not handle the same synchronization interval
as in simulation, where the communication does not have delays.

Figure 18: Output synchronization of bargaining game theory implementation in
the symmetric case.

Similarly, the response of the agents’ cost function is observed in Fig. 20,
which maintains similarity with the response of the simulation case and also
reaches a correct bargain, as well as the application of the control action in
the Fig. 20.

Figure 19: Cost function of bargaining game theory implementation in a
symmetric case.

Figure 20: Control action of bargaining game theory implementation in a
symmetric case.

The response of the controller to non-symmetric cases is also validated,
according to the parameters of Table 2, Fig. 21 shows the response of the
agents’ output in this case, where it is evidenced that as in the previous
cases, the system achieves correct bargaining in a distributed scenario.

Figure 21: Output synchronization of bargaining game theory implementation in
a Non-symmetric case.

Figures 24, 24 and 24 show the response of the point of disagreement, the cost
function, and the control action in the non-symmetrical case of emulation, the
response as in the previous cases, shows a fluctuation in their behavior,
followed by a correct bargain in both cases.

Figure 22: Disagreement point of bargaining game theory implementation in a
non-symmetric case.

Figure 23: Cost function of bargaining game theory implementation in a non-
symmetric case.

Figure 24: Control action of bargaining game theory implementation in a non-
symmetric case.

In the same way, it is important to validate cases where a correct bargain is
not achieved. Considering that, a scenario of multiple agents in the context
of mechanical systems based on [32], presents a non-linearity in its systemic
base that makes it difficult for the bargain algorithm, as is observed in Fig.
25. It is validated that in those scenarios, when the value of the point of
disagreement diverges, synchronization of all the systems is not completely
achieved.

Figure 25: Output synchronization of bargaining game theory implementation in
a non-bargaining case.

##  6 Conclusion

This paper solves the problem of the cooperative cruise in vehicles from a
control perspective. Its logic is validated through predictive control in
output synchronization of symmetric and non-symmetric agents with linear
dynamics. The use of algorithms based on game theory allows conceptualizing
the term cooperation and its relation with control through the Nash theory of
bargaining, which is validated throughout the paper. A bargaining game
methodology is used as a model predictive control strategy for game theory
analysis. Its extension to dynamic cases is presented even in a discrete
perspective for implementation. In this methodology, the analysis of the
disagreement point is used dynamically for negotiation between agents,
guaranteeing that this can be applied for symmetric and non-symmetrical games.
The unique solution can be summarized as a convex optimization problem
methodology.

At the simulation level, under the context of cooperative cruise control,
symmetric and non-symmetric cases are defined, and it is possible to
demonstrate the synchronization and the agreement reached through the
bargaining algorithm. In the cases of non-symmetric games, the system tends to
have a slower response and agreement derived from its characteristics.
Similarly, in implementation, by emulating dynamic systems in hardware-in-the-
loop, the dynamics of the agents are synchronized, achieving an agreement,
highlighting that in non-symmetric games, the system presents fluctuations in
its distributed response.

In future work, it is possible to strengthen this theory by including security
parameters in cost functions and unstable models in an open loop, likewise,
with the inclusion of heterogeneity parameters to have a better response in
non-symmetric game implementation.

## Declarations

The development is partially financed by the project ”RESEARCH PROGRAM IN
EMERGING TECHNOLOGIES FOR SMART ELECTRIC MICROGRIDS WITH HIGH PENETRATION OF
RENEWABLE ENERGIES” from Minciencias, Bogota, Colombia.

The authors declare that they have no conflict of interest.

The authors declare the transparency of the information and data obtained in
the development of the article.

The authors confirm the availability of the codes without any inconvenience
requested by email.

This work does not contain any studies with human or animal participants
performed by any of the authors.

Informed consent was obtained from all individual participants included in the
study.

Publication consent was obtained from all individual participants included in
the study.

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