VAZQUEZ TREJO Juan Antonio, GUENARD Adrien, ADAM-MEDINA Manuel,PONSART Jean-Christophe, CIARLETTA Laurent, ROTONDO Damiano, and THEILLIOL Didier

1.Research Center for Automatic Control of Nancy, University of Lorraine, Nancy F-54000, France;

2.Electronic Engineering Department, National Institute of Technology of Mexico, Cuernavaca Morelos 62490, Mexico;

3.Lorraine Research Laboratory in Computer Science and Its Applications, University of Lorraine, Nancy F-54000, France;

4.University of Stavanger, Department of Electrical and Computer Engineering, Stavanger 4021, Norway

Abstract: The main contribution of this paper is the design of an event-triggered formation control for leader-following consensus in second-order multi-agent systems (MASs) under communication faults.All the agents must follow the trajectories of a virtual leader despite communication faults considered as smooth time-varying delays dependent on the distance between the agents.Linear matrix inequalities (LMIs)-based conditions are obtained to synthesize a controller gain that guarantees stability of the synchronization error.Based on the closed-loop system, an event-triggered mechanism is designed to reduce the control law update and information exchange in order to reduce energy consumption.The proposed approach is implemented in a real platform of a fleet of unmanned aerial vehicles (UAVs) under communication faults.A comparison between a state-of-theart technique and the proposed technique has been provided,demonstrating the performance improvement brought by the proposed approach.

Keywords: event-triggered, leader-following consensus, communication fault, formation control, unmanned aerial vehicle(UAV), experimental result.

1.Introduction

Leader-following consensus for multi-agent systems(MASs) has attracted interest due to its applications in collective missions including, self-organization, clusters of satellites, formation flying, and sensor networks [1].Leader-following consensus is a particular problem in multi-agent systems where all the agent trajectories must converge to the trajectory of a leader [2].Several research works have increased the focus on considering multiplicative and additive noises [3], switching topologies [4], time delays [5], particle swarm optimization [6],and event-triggered mechanisms [7], among others.The information exchange through digital networks is a key point in leader-following consensus.However, delays [8],packet losses [9], communication faults [10], or bandwidth limitations [11] are challenges in real engineering applications [12].

An alternative control strategy is the event-triggered approach, which is often used for reducing the information exchange and the control law rate [13].The difference between event-triggered and time-triggered approaches are that the latter considers a periodic control law update, whereas, in the former, the update of the control law and the information exchange between the agents are determined by an event generator [14,15].Related works have increased the focus on event-triggered leaderfollowing consensus in the last decades considering sufficient conditions using the M-matrix theory and algebraic inequalities for second-order nonlinear time-delayed dynamic agents [16]; linear matrix inequalities (LMIs)-based conditions using the M-matrix theory for reaching bipartite consensus in nonlinear second-order agents [17];nonuniform delays in heterogeneous agents [8]; bounded delays in fractional-order agents [18]; sufficient conditions including dependent and independent fixed delays,and time-varying delays [19]; constant delays in linear agents [20].Nevertheless, the aforementioned works have not considered a degradation in the communication based on the distances between agents.In [10], communication faults are modeled as a modification in the weights of the adjacency matrix as a result of a malfunction in the exchange of information.The communication faults in this work are considered as a delay-dependent on the distances between agents.Unlike [21], where a timetriggered control is designed to tolerate smooth communication faults, the main contribution of this paper inspired by [7], is the design of an event-triggered strategy to solve the leader-following consensus problem in second-order MASs under communication faults.A synthesis of a robust control gain is obtained in order to tolerate faults in the exchange of information.Then, based on the closed-loop system, an event-triggered mechanism is used in order to reduce the information exchange between agents and the control update rate.The proposed technique has been implemented in a real platform comprising a fleet of unmanned aerial vehicles (UAVs)achieving a desired formation and following a virtual leader agent in spite of the degradation in the exchange of information.

This paper is organized as follows.Preliminaries and problem statement are provided in Section 2.The eventtriggered leader-following formation control design is described in Section 3.The experimental results are shown in Section 4.Finally, the main conclusions are presented in Section 5.

2.Preliminaries and problem statement

2.1 Notation and graph theory

Given a matrixX,XTdenotes its transpose,X>0(<0)denotes a positive (negative) definite matrix.‖·‖ denotes the Euclidean norm.For simplicity, the symbol * within a symmetric matrix represents the symmetric entries.The Hermitian part of a square matrixXis denoted by He{X}=X+XT.The symbol ⊗ denotes the Kronecker product, which for real matricesA,B,C, andDwith appropriate dimensions, satisfing the following properties[22]:

Lemma 1[23] For a given matrixthe following statements are equivalent:

2.2 Problem statement

Consider the second-order MAS

wherepi(t) ,vi(t),ui(t)∈Rnare the position, velocity,and acceleration input withi=1,2,···,N, in ann-dimensional Euclidean space.

Leader-following consensus is designed such that all the agents follow the trajectories of a virtual leader.In this case, the leader’s dynamic is considered as follows:

wherepr(t),vr(t)∈Rnare the position and velocity of the leader agent.The leader agent position can be manipulated through its velocity.Let us define the rigid desiredposition formation from the agentito its neighborsj, ashi,hj∈Rn.According to [1], the classical leader-following formation control is given by

where Niis the set ofi’s neighbors.

Assumption 1The graph G is an undirected graph.

Assumption 2All the agents receive information states from the virtual leader agent.

Lemma 2[24] The Laplacian matrix L associated with an undirected graph has at least one zero eigenvalue and all the nonzero eigenvalues are positive.The Laplacian matrix L has exactly one zero eigenvalue if and only if the graph is connected.

Using the consensus protocol (3), the MAS (1)achieves the desired formation if the following is satisfied:

Bandwidth limitations, delays, or packet losses are challenges in real engineering applications in MASs.Let us define τij(t) as the communication faults between the agentiand the agentj.Based on τij(t), the leader-following formation control under communication faults (3)becomes

A degradation of the communication between agents can be associated to their distance as considered in [25].Communication faults are considered dependent on the agent positions in link with the distance between them and described by the following function:

where β1, β2, and β3are positive constants, andtfis the time of fault occurrence.

Assumption 3The derivative of the communication faultijij(t)≤dτ<1,i≠j,j∈Niwheredτis a fixed scalar.

Note that, when τij(t)=0, the leader-following formation control problem can be solved using (3).Nevertheless, as reported in [24], the longest delay to reach the consensus is determined as, where λN(L) is the maximum eigenvalue of the Laplacian matrix, and the delay is considered constant with the same value for all agents in a fixed, undirected, and connected graph.

The problem under consideration in this paper is to design an event-triggered leader-following formation control such that all the agents follow the leader’s trajectories subject to communication faults considered as smooth delays dependent on the agent positions.

3.Event-triggered leader-following formation control design

In the following subsections, a time-triggered formation control design and the event-triggered mechanism are presented in order to develop a strategy such that all the agents tolerate communication faults while reducing the information exchange.

3.1 Time-triggered leader-following formation control design

Let us define the error between the agentiand the leader as follows:

Adding the control gainKc∈Rn×2nand the scalar α,the leader-following control (5) is modified in order to tolerate communication faults whenas follows:

whereKcis the control gain to be designed and α>0 must be a positive constant which represents the relationship between the leader and the followers.Based on (8),(7) becomes

The following theorem provides LMI-based conditions for the computation of the control gainKc.

Theorem 1Given the non-zero eigenvalues of the Laplacian matrix λi(L),i=2,3,···,N, scalarsα>0,µ1>0 , µ2>0 , andijij≤dτ<1, the leader-following cons ensus is quadratically stable under (9), if there exist symmetric matricesP1>0,P2>0 , and a matrixKcsuch that the following inequality

ProofLet us define the following candidate Lyapunov functional inspired by [26]

The time derivative ofValong any solution of the system (11) is given by

According to [26] and Assumption 3, (14) is negativedefinite when

thus,

Let us perform a spectral decomposition of the Laplacia n matrix L, such that L=TJT−1with an invertible matrixT∈RN×Nand a diagonal matrixJ=diag(λ1=0,λ2,···,λN) .By Lemma 2, eigenvalues of L form a base matrixT.Let us define the following change of coordinof eigenvectors which are used to construct the invertible ates:

Taking (17) into (16) leads to

By Lemma 2, it is obtained that ψ1(t)=0 and ψ1(t−τ)=0 due to λ1=0, then (18) is rewritten as follows:

Then, the following matrix is obtained:

If matrix Ωi<0,i=2,3,···,N, then<0; thus, the synchronization error between the leader and the followers is quadratically stable.Using Schur complement(Lemma 1) in (12), the following inequality is obtained:

Note that:

where µ1>0.By taking into account the following inequality:

and combining (23) and (24), the following is obtained:

Based on (25) and (22), Ωi<0 is recovered, thus, the LMI (12) corresponds to (20) and the synchronization error is quadratically stable under (9), thus completing the proof.□

Remark 1Theorem 1 guarantees the time-triggered leader-following formation control design which is continuously updated.

3.2 Event-triggered mechanism

The following section, an event-triggered mechanism is developed in order to reduce the information exchange and the control update rate.

The update of the control law action in event-triggered approaches depends on an event error.This event error is calculated based on the last and the current state values.When the magnitude of the event error exceeds a threshold, the control law value is updated, otherwise, the control law keeps the last calculated value.The control in(9) is modified in order to design an event-triggered mechanism as follows:

According to [7], if the leader-following consensus is quadratically stable, then, the following event function can be considered:

where parametersc1>0 ,c2>0 , 0

4.Example: fleet of UAVs under communication faults

In order to illustrate the effectiveness of the proposed strategy, real implementations in a fleet of UAVs are presented in this section.

The experimental platform is described and some experimental results are shown in the following section.A video corresponding to the results can be found at https://youtu.be/Lo_kuGY9Wq4.

4.1 Experimental platform description

The experimental platform used for this implementation consists of an Optitrack system to recognize the UAVs in a three-dimensional space by image processing using Prime 17W cameras; Motive 2.1.1 is the software to manipulate the Optitrack which uses the virtual-reality peripheral network (VRPN) protocol with communication to a virtual machine; Ubuntu 16.04 is installed in the virtual machine with ROS Kinetic to manipulate the UAVs in parallel with Motive; identical Bebop 2 parrots are the UAVs (see Fig.1).The code is developed in Python 2.7.The sample time is 0.02 s.

Fig.1 Bebop 2 parrot

According to [27], a fleet of UAVs can be described as a second-order MAS if an inner closed-loop control is considered for each UAV employing their angles with the following references:

The goal of the three UAVs is to form an isosceles triangle and follow the trajectories of a virtual agent with the following desired formationh1=[0,0]T,h2=[0,1.5]T,andh3=[0.75,1.3]T.

The LMI in Theorem 1 is solved with the following parameters: µ1=1 , µ2=10 , α=1 , anddτ=0.2 obtaining the control gainThis control gain is used in both the time-triggered and event-triggered approach.Table 1 shows the initial values of the UAVs.

Table 1 Initial conditions of the UAVs

The communication topology is described by the following Laplacian matrix:

The communication fault is implemented through an artificial function with the following parameters:β1=0.8 , β2=1 , β3=0.6 , andtf=10s.All the UAVs are affected by the communication fault.The event-function has the following parameters:c1=0.03,c2=3, andc3=0.1.

Three implementations have been carried out for comparing the performance of the classical formation control(5), the time-triggered robust approach (9), and the eventtriggered approach (27).In the case of the classical formation control, the experiment had to be stopped to avoid the UAVs to crash.

4.2 Experimental results

Fig.2 illustrates the obtained trajectories of the UAVs using the classical formation control.The UAVs should follow the trajectory of the virtual agent in black.However,due to the communication faults, they start to oscillate,and they cannot maintain the formation.

Fig.2 Trajectories of UAVs (classical approach)

Fig.3 shows the UAVs trajectories obtained using the time-triggered robust control.The UAVs present a decrease in the oscillations with respect to the previous case.Moreover, they maintain the desired formation.

Fig.3 Trajectories of UAVs (proposed time-triggered approach)

Fig.4 presents the UAVs trajectories when the eventtriggered mechanism is used.The UAVs present a better performance, maintaining the formation despite the communication faults.

Fig.4 Trajectories of UAVs (proposed event-triggered approach)

Fig.5 presents the UAVs velocities obtained using the classical formation control.After approximately 140 s,the UAVs start to show stronger oscillations.As mentioned earlier, in order to preserve the integrity of the UAVs, the experiment has to be stopped.

Fig.5 Velocities of UAVs (classical approach)

Fig.6 shows the UAVs’ velocities using the timetriggered robust control.The oscillations decrease when compared to the classical formation control.However,there is still a small offset between the leader velocities and the velocities of the UAVs.Also, an offset is observed, induced by the fact that the control gain is smaller than that in the classical formation control.

Fig.6 Velocities of UAVs (proposed time-triggered approach)

Fig.7 illustrates the UAVs’ velocities using the eventtriggered control.The oscillations are smaller compared to the other two approaches.However, the offset is still present due to the control gain.It is worth highlighting that the event-triggered control reduces the information exchange between the agents and the update rate of the control law.

Fig.7 Velocities of UAVs (proposed event-triggered approach)

Fig.8 presents the event-triggered control law.The time interval between 15 s and 17 s is zoomed to illustrate when the control law keeps the last value.

Fig.8 Consensus control law (event-triggered proposed approach)

In order to measure the performance of the consensus,let us define, where, and.Fig.9 illustrates the evaluation of the performance of the consensus using the classical formation control.The performance presents oscillations after 140 s due to the communication faults.

Fig.9 Evaluation of consensus’ performance (classical approach)

Fig.10 shows the evaluation of the performance of the consensus using the time-triggered robust control.Compared to Fig.9, the performance has been improved.

Fig.10 Evaluation of the consensus’ performance (proposed timetriggered approach)

Fig.11 presents the evaluation of the consensus performance using the event-triggered control.Compared to Fig.10, the performance is smaller than the threshold value 1 due to the desired formation.

Fig.11 Evaluation of consensus’ performance (proposed eventtriggered approach)

Fig.12 presents the profile of the events for the eventtriggered control.It is considered 1 for UAV1, 2 for UAV2, 3 for UAV3if an event occurs, respectively, and 0 if there is no event.A zoom is considered in some intervals in order to show when an event occurs.

Fig.12 Events for updating the control law

Fig.13 shows the total number of events in each UAV.“No event” means that the control law and the exchange of information are not updated.For example, UAV1has 1 907 of no events compared with 6556 events.In contrast with time-triggered, the update of the information and the control law has been reduced.

Fig.13 Total number of events in each UAV

In order to quantify the performance between the approaches, the root mean square (RMS) metric is used.In Table 2, the RMS value ofdijis presented for each combination of UAVs corresponding to the classical formation control, the time-triggered robust control, and the event-triggered control.It should be noted that the eventtriggered approach reduces the energy consumption.

Table 2 Comparison of the consensus RMS

5.Conclusions

This paper has presented an event-triggered formation for second-order MAS under communication faults.The controller gain has been calculated by using LMI tools and an event-triggered mechanism has been introduced to reduce the information exchange between agents.The proposed approach has been implemented in a real platform of a fleet of UAVs subject to communication faults.A comparison between a state-of-the-art technique and the proposed technique has been provided, demonstrating the performance improvement brought by the proposed approach.For future work, a measurement of the energy consumption can be implemented in the real platform in order to compare the performance between the approaches.