Syed Ali Ajwad
email: syed.ali.ajwad@univ-poitiers.fr

Tomas Ménard
email: tomas.menard@unicaen.fr

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Michael Defoort
email: michael.defoort@uphf.fr

Patrick Coirault
email: patrick.coirault@univ-poitiers.fr

Observer based leader-following consensus of second-order multi-agent systems with nonuniform sampled position data

Keywords: Second-order multi-agent systems, Continuous-discrete time observer, Leader-following consensus, Nonuniform and asynchronous sampling

This paper deals with the leader-following consensus problem of multi-agent systems with the consideration that each agent can only transmit its position state to the neighbors at irregular discrete sampling times. In the proposed algorithm, a continuous-discrete time observer is designed for the continuous estimation of both position and velocity from the discrete position information of the neighbors. These estimated states are then used for designing a continuous control law which solves the leader-following consensus problem. Moreover, the dynamics of the leader is not fixed and can be controlled through an external input. The stability analysis has been carried out by employing the Lyapunov approach which provides sufficient conditions to tune the parameters according to the maximum allowable sampling period. The developed algorithm has been simulated and then tested on an actual multi-robot system consisting of three differential drive wheeled robots. Both simulation and hardware results validate the effectiveness of the control algorithm.

Introduction

Development in technology has enabled advanced operations for autonomous robotic vehicles. The effectiveness of these robots can be further improved by utilizing them in a cooperative manner to accomplish much more complex tasks which cannot be realized by solo operation. Cooperative team work is not only limited to robotics but it also finds applications in the field of sensor networks [START_REF] Manfredi | Decentralized control algorithm for fast monitoring and efficient energy consumption in energy harvesting wireless sensor networks[END_REF], interferometers [START_REF] Chen | Satellite formation-containment flying control with collision avoidance[END_REF], electric power systems [START_REF] Yang | A decentralized coordination control method for parallel bidirectional power converters in a hybrid ac-dc microgrid[END_REF], distributed computation [START_REF] Boutin | Apollo: Scalable and coordinated scheduling for cloud-scale computing[END_REF], synchronization [START_REF] Jain | Synchronization of multi-agent systems with heterogeneous controllers[END_REF] etc.

Cooperative control of multi-agent systems (MAS) is classified into two categories, namely, centralized control and distributed control. In centralized control, all the agents communicate with a central control unit. On the other hand, distributed control scheme does not have any common control unit. All the agents make decisions by exchanging their information with neighbors. Study of distributed control of MAS has gained much popularity in the research community during the last decade because of its advantages over centralized control such as efficiency, scalability, flexibility, robustness and adaptability [START_REF] Ebegbulem | Distributed control of multi-agent systems over unknown communication networks using extremum seeking[END_REF].

Consensus is considered as a fundamental problem in distributed control in which all agents are required to reach an agreement on some value of interest [START_REF] Shan | Average-consensus tracking of multi-agent systems with additional interconnecting agents[END_REF]. For instance, in multi-robotics systems, it may require all robots to either reach a common position or follow the trajectory of a leader. The former case is defined as leaderless consensus while the latter is called leader-following consensus [START_REF] Ge | Event-based set-membership leader-following consensus of networked multi-agent systems subject to limited communication resources and unknown-but-bounded noise[END_REF]. The topic of leader-following consensus has received significant attention in the past years [START_REF] Wang | Leader-following consensus of multi-agent systems under directed communication topology via distributed adaptive nonlinear protocol[END_REF][START_REF] He | Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control[END_REF][START_REF] Madhevan | Tracking algorithm using leader follower approach for multi robots[END_REF][START_REF] Liu | Leader-following consensus for uncertain second-order nonlinear multi-agent systems[END_REF][START_REF] Ni | Leader-following consensus of multi-agent systems under fixed and switching topologies[END_REF][START_REF] Wu | Leader-following consensus of nonlinear discrete-time multi-agent systems with limited communication channel capacity[END_REF].

Various physical systems like robotic vehicles can be represented as linearized double integrator dynamic systems where position and velocity are the information states [START_REF] Djaidja | Leader-following consensus of double-integrator multi-agent systems with noisy measurements[END_REF][START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF]. Though these dynamics are continuous in nature, information exchange among the agents is in discrete time because of the digital means of communication [START_REF] Ge | A brief survey of recent advances in consensus of sampled-data multi-agent systems[END_REF].

Several control techniques, to achieve consensus in double integrator systems with sampled data, have been proposed by the research community. For instance, Pan and Qiao proposed a discrete time consensus algorithm for double integrator MAS in [START_REF] Pan | Consensus of double-integrator discrete-time multi-agent system based on second-order neighbors' information[END_REF] where communication delay was considered and conditions for consensus under such delays were derived using linear matrix inequalities. Several sufficient and necessary conditions on controller gain, communication topology and sampling time were proposed in [START_REF] Cao | Multi-vehicle coordination for double-integrator dynamics under fixed undirected/directed interaction in a sampled-data setting[END_REF][START_REF] Cao | Sampled-data discrete-time coordination algorithms for double-integrator dynamics under dynamic directed interaction[END_REF] for the convergence of sampled-data coordination protocols for agents with double integrator dynamics. In [START_REF] Casbeer | Discrete double integrator consensus[END_REF], average consensus was achieved asymptotically for a dynamic network provided that there always exists a directed spanning tree. Integral sliding mode control based consensus algorithm was proposed in [START_REF] Wang | Sliding-mode consensus algorithms for disturbed second-order multi-agent systems[END_REF] for both leaderless and leader tracking of second-order MAS with disturbance. The authors in [START_REF] Qin | A sufficient condition for convergence of sampled-data consensus for double-integrator dynamics with nonuniform and time-varying communication delays[END_REF] investigated a discrete-time consensus algorithm for second-order MAS. It was shown that agents achieve consensus in the presence of switching communication topology with non-uniform time delays and sampled data. Eichler and Werner discussed the optimization of convergence speed for only fixed communication topology [START_REF] Eichler | Optimal convergence speed of consensus under constrained damping for multi-agent systems with discrete-time double-integrator dynamics[END_REF]. A leader-following protocol for second-order MAS in a sampling setting with Markovian switching topology has been presented in [START_REF] Xie | Bounded consensus tracking for sampled-data second-order multi-agent systems with fixed and markovian switching topology[END_REF]. Wu et al. derived mean square consensus tracking with a virtual leader in [START_REF] Wu | Stochastic bounded consensus tracking of leader-follower multi-agent systems with measurement noises based on sampled-data with small sampling delay[END_REF]. In this paper, the tracking error not only depends on the sampling period and delay but also on the velocity and acceleration of the virtual leader. The problem of intermittent communication was addressed by Liu et al. in [START_REF] Liu | Consensus seeking of multi-agent systems with intermittent communication: a persistent-hold control strategy[END_REF]. The authors proposed a consensus algorithm based on persistent-hold techniques for first order systems.

However, the above mentioned techniques require information of both position and velocity states of the neighbors. In a practical scenario, the agents usually have limited on-board measurement resources that makes it difficult to measure their velocity along with their position. That is why, it is more affordable to measure and transmit only position information. To cope with this problem, available position information of the neighbors were used to estimate the velocity for consensus tracking problem in [START_REF] Xu | Observer-based consensus tracking for second-order leader-following nonlinear multi-agent systems with adaptive coupling parameter design[END_REF]. Chen and Li have proposed an observer to estimate the velocity via sampled data [START_REF] Chen | Observer-based consensus of second-order multi-agent system with fixed and stochastically switching topology via sampled data[END_REF]. Second-order consensus was achieved by Yu et al. by using only current and previous sampled position data [START_REF] Yu | Second-order consensus in multi-agent dynamical systems with sampled position data[END_REF]. It was shown that consensus can only be reached for specific sampling time intervals. Some necessary and sufficient conditions for sampling time period were established in [START_REF] Huang | Some necessary and sufficient conditions for consensus of second-order multi-agent systems with sampled position data[END_REF] for position based second-order consensus algorithms.

In the above cited literature, position information is supposed to be transmitted at regular constant intervals. However in practice, asynchronous and aperiodic transmission of data is inevitable. Furthermore, in many cases, asynchronous and aperiodic transmission is desired in order to avoid unnecessary consumption of the communication resources. For example, in event-triggered based consensus, local information is exchanged only when a specific event occurs. In [START_REF] Dimarogonas | Distributed event-triggered control for multi-agent systems[END_REF], Dimarogonas et al. proposed an event-triggering based distributed control for MAS. The next triggering time is calculated when a particular event takes place. An event-triggering based algorithm with a state-dependant triggering function was proposed in [START_REF] Li | A distributed event-triggered scheme for discrete-time multi-agent consensus with communication delays[END_REF]. An observer based event-triggering consensus protocol was proposed in [START_REF] Zhang | Observer-based output feedback event-triggered control for consensus of multi-agent systems[END_REF][START_REF] Ding | Observer-based event-triggering consensus control for multiagent systems with lossy sensors and cyber-attacks[END_REF][START_REF] Hu | An observer-based consensus tracking control and application to event-triggered tracking[END_REF] where the control law is updated using the estimated states. One drawback of such eventtriggering schemes is that it sometimes requires a continuous monitoring of the full state of the agents. Furthermore, it needs pre-knowledge of the event-triggering function or mechanism which dictates the sampling times for the data transmission.

In the current article, it is aimed at designing a leader-following consensus algorithm for second-order MAS which guarantees that each agent tracks the leader when only position information is transmitted with asynchronous and irregular sampling times. The proposed scheme is based on continuous-discrete time observers. Such observers estimate both the available and unavailable states in continuous time from information received in discrete time. Continuousdiscrete time observers have been studied widely in recent years, for example in [START_REF] Farza | Continuous-discrete time observers for a class of mimo nonlinear systems[END_REF][START_REF] Hernández-González | A cascade observer for a class of mimo non uniformly observable systems with delayed sampled outputs[END_REF][START_REF] Bouraoui | Observer design for a class of uncertain nonlinear systems with sampled outputsapplication to the estimation of kinetic rates in bioreactors[END_REF]. A leaderless consensus protocol for second-order MAS with asynchronous sampling and partial measurements has been introduced in [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF]. A continuous-discrete time observer has been proposed to reconstruct the position and velocity in continuous time from the available discrete position data. In the current paper, using the basic idea of [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF], a novel output-feedback controller is proposed to solve the problem of leader-following consensus problem. The leader is not assumed to be fixed. Its dynamics can be controlled by an external input that influences the stability of the whole system. Here, the novelty of this article lies in the development of a leader-following consensus protocol with an active leader in the presence of the following constraints: (i) an agent only transmits its position state to its neighbors (nor its velocity nor its input), (ii) the agents transmit their data with irregular sampling times, (iii) the sampling instants for each agent are totally independent which means that no synchronization is required for the transmission, (iv) the leader can communicate its position to only a small portion of the followers, (v) the communication topology among agents is directed. The designed controller ensures the practical consensus of MAS for the general case with an ultimate bound that can be made as small as desired by appropriately tuning the parameters. It is also shown that exponential consensus is obtained if the leader input is equal to zero. It is important to mention that the results of [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF] cannot be directly applied to the leader-following case due to the presence of aperiodic and asynchronous sampling period and the availability of position state only. Compared to [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF], the addition of an active leader changes the dynamics of overall system, resulting in new terms not only in the tracking error dynamics but also in the observer error dynamics. Consequently, the stability analysis can no longer be carried out with the same Lyapunov function as used in [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF]. Therefore, in this paper, a new stability proof is derived based on new Lyapunov functions. At last, another important contribution in this paper compared to [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF] is the experimental validation of the developed algorithm. Indeed, the algorithm is tested on a group of mobile robots using Robot Operating System (ROS) for linear movement. Both simulation and experiment results have validated the efficacy of this algorithm.

The remaining of the paper is organized as follows. Section 2 describes preliminary knowledge along with the problem formulation. Main results are presented in Section 3. Simulation and experimental results are discussed in Section 4 and 5 respectively while Section 6 concludes the article.

Preliminary knowledge and problem formulation

Notations

In this paper, the set of n × n real matrices is denoted R n×n . I n ∈ R n×n is the n-dimensional identity matrix. For any symmetric matrix A, λ min (A) and λ max (A) represent minimum and maximum eigenvalue of A respectively.

The symbol = means equal by definition while ⊗ represents the Kronecker product. D N i denotes the N × N matrix with all entries equal to zero except the i th diagonal entry which is 1. . 2 and . F represent the Euclidean and Frobenius norms respectively. If nothing is specified, then . denotes the Euclidean norm. diag(b 1 , . . . , b q ), with b i ∈ R m×m , i = 1, . . . , q, q, m ∈ N, is the diagonal by block matrix having b 1 , . . . , b q on its diagonal. 1 N ∈ R N represents the vector with all entries equal to 1.

Preliminary knowledge

A directed graph G is a pair (V, E), where V is a nonempty finite set of nodes and E ⊆ V ×V is a set of edges where an edge is an ordered pair of distinct nodes. For an edge (i, j), node j, called child node, can receive information from node i which is the parent node and i is a neighbor of j. A graph has a directed spanning tree if each node has one parent node except for one node, called the root, which has a directed path to all other nodes in the graph. The adjacency matrix A = (a ij ) ∈ R N ×N of G with N nodes is defined by a ij = 1 if (j, i) ∈ E and a ij = 0 otherwise. The Laplacian matrix L ∈ R N ×N is defined as l ii = j =i a ij , l ij = -a ij for i = j. L satisfies l ii 0, i = j and

N j=1 l ij = 0, i = 1 . . . N . Lemma 1. [41] Let M = (m ij ) 1≤i,j≤n ∈ R n×n
be a non-singular M-matrix, such that m ij ≤ 0, for all i = j, i, j = 1, . . . , n and all its eigenvalues have positive real parts. Then, there exists a diagonal matrix

Ω = diag(ω 1 , . . . , ω n ) > 0 such that ΩM + M T Ω > 0.

Problem formulation

Consider a group of N followers labeled from 1 to N and one leader labeled 0. The followers have the following second-order dynamics:

ṙi (t) = v i (t), i = 1, . . . , N vi (t) = u i (t) (1)
where r i , v i , u i ∈ R m represent respectively the position, velocity and control input of the i-th agent (m ∈ N). The dynamics of the leader is given by:

ṙ0 (t) = v 0 (t) v0 (t) = u 0 (t) (2)
where r 0 , v 0 , u 0 ∈ R m represent respectively the position, velocity and control input of the leader. It is assumed that the control input u 0 is bounded, that is, there exists δ 0 ≥ 0 such that u 0 (t) ≤ δ 0 for all t ≥ 0.

The communication connection between the followers is described by graph G and the corresponding adjacency and Laplacian matrices are A and L, respectively. The position of the leader is considered to be transmitted to a small portion of the followers. Let the diagonal matrix B = diag(b 1 , b 2 , . . . , b N ) be the interconnection relationship between the leader and followers, where b i = 1 if the information of the leader is accessible by the i th follower, otherwise b i = 0. The communication graph including the followers and the leader is denoted G. Let us define the following matrix

H = L + B Lemma 2. [42]
Matrix H is a nonsingular M-matrix if and only if the pinning joint communication topology G has a directed spanning tree.

Definition 1. The leader-following practical exponential consensus is achieved if

N i=1 e i (t) ≤ αe -βt + γ, ∀t ≥ 0 where e i = x i -x 0 with x i = r T i , v T i T , x 0 = r T 0 , v T 0 T , α, β > 0 and γ is a positive constant.
It is considered that each agent transmits only its position r j to its neighbor i at times t i,j k with k ∈ N, but not its velocity v j nor its input u j . The sampling instants t i,j k are supposed to verify 0 = t i,j 0 < t i,j 1 < • • • < t i,j k < . . . . Furthermore, one assumes that there exist constants τ m , τ M > 0, called respectively the minimum sampling period and the maximum sampling period, such that τ m < t i,j k+1 -t i,j k < τ M , for all k ∈ N and i = 1, . . . , N , j = 0, . . . , N . Figure 1 shows an example of sampling instants for data transmission between the neighbors. The problem under investigation is to design a distributed control law to solve the leader-following consensus problem under these communication constraints.

Observer based leader-following consensus

In this paper, the basic idea is to use classical continuous linear consensus controller for double integrator MAS with discrete position measurements only. In a classical consensus protocol, it is assumed that all states of the neighbors are available in continuous time. Whereas, the current article considers that only position data is transmitted to the neighbors at irregular and asynchronous time instants. The idea is to estimate the position and velocity states in continuous time from the available discrete position data by using a continuous-discrete time observer (as discussed in [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF] for the leaderless consensus problem). The proposed observer-based consensus protocol is given for t ≥ 0 by

u i (t) = -cλ 2 N j=1 a ij [r i,i (t) -ri,j (t)] -c2λ N j=1 a ij [v i,i (t) -vi,j (t)] -cλ 2 b i [r i,i (t) -ri,0 (t)] -c2λb i [v i,i (t) -vi,0 (t)] , i = 1, . . . , N (3)
where ri,j and vi,j are the estimated position and velocity of agent j by agent i. Their dynamics are given by ṙi,j (t

) = vi,j (t) -2θe -2θ(t-t i,j k ) ri,j (t i,j k ) -r j (t i,j k ) (4) vi,j (t) = -θ 2 e -2θ(t-t i,j k ) ri,j (t i,j k ) -r j (t i,j k ) (5)
for i = 1, . . . , N , j = 0, 1, . . . , N and t ∈ t i,j k , t i,j k+1 , k ∈ N, where c > 0 is the coupling strength, a ij is the (i, j)-th entry of the adjacency matrix A of the directed graph G while θ and λ > 0 are the observer and controller tuning parameter respectively. The initial conditions ri,j (0), vi,j (0) ∈ R m of the observers can be chosen arbitrarily. Before stating the main result, note that if G has a directed spanning tree, then according to Lemma 1, there exists a diagonal matrix Ω = diag(ω 1 , . . . , ω N ) such that H T Ω + ΩH > 0. Furthermore, the following notations will be used hereafter.

ω max = max{ω 1 , . . . , ω N }, (6)
ω min = min{ω 1 , . . . , ω N }, (7) ρ
= λ min (H T Ω + ΩH). (8) h max = max i,j |H ij | (9)
Theorem 1. Consider the MAS (1)-( 2) with the consensus protocol (3)-( 5) and assume that the communication topology G contains a directed spanning tree. If the control parameters θ, λ ≥ 1 and c > 0 satisfy the following

θ < ¯ τ M (10) c ≥ ω max ρ (11) λ < * θ ( 12
)
where ¯ is a positive constant, * ∈ (0, 1), ω max and ρ are given by ( 6) and (8), respectively, then the leader-following practical consensus problem is solved in the sense of Definition 1.

Remark 1. The conditions provided in Theorem 1 to achieve the leader-following consensus are only sufficient since they are obtained through Lyapunov based stability analysis which may lead to conservative bounds. Despite the limitation of Lyapunov based approaches, it provides some useful hints about the choice of the gains. For instance, it is clear from inequality (10) that the maximum sampling period directly influences the choice of θ. If the maximum sampling time is high, then θ must be chosen small enough and vice versa. It should be noted that θ and λ dictate the convergence speed of the observer and controller, respectively. In order to guarantee closed-loop stability, the controller dynamics must be slower than the observer dynamics which is represented by the fact * < 1. Therefore, for large maximum sampling period, the system convergence rate will be slow.

Before the proof of Theorem 1, we need some important results which are presented below.

Lemma 3. Let v 1 (t) and v 2 (t) be real valued functions verifying

d dt v 2 1 (t) + v 2 2 (t) ≤ -av 2 1 (t) -bv 2 2 (t) + c t t-δ v 2 2 (s)ds + k, (13)
for all t ≥ 0, where a, b, c, δ > 0 and k ≥ 0. There exists > 0, independent of a, b, c, k, and ᾱ ≥ 0 such that if δ < min b c , 1 σ , then v 1 (t) and v 2 (t) verify the following inequality

v 2 1 (t) + v 2 2 (t) ≤ ᾱe -σt + k σ , ∀t ≥ 0 ( 14
)
where σ is given by

σ = 1 2 min (a, b) (15)
PROOF. Let v = min a √ 2 , b √ 2 , ξ = 2 cδ b and κ = 1 -ξ. Since δ ∈ 0, min b c , 1 σ , one has 0 < 2 cδ b < 2 c b min b c , 1 σ ≤ 2 ⇒ 1 -2 < 1 -ξ =κ < 1 (16) 0 < vκδ < vδ < v min b c , 1 σ ≤ √ 2 (17)
Then > 0 can be chosen, independently of a, b, c, k such that for all δ ∈ 0, min b c ,

1 σ κ ∈ 1 √ 2 , 1 (18)
e vκδ ≤ 1 + 2vκδ (19)
Consider the following Lyapunov function

W (v t ) = v 2 1 (t) + v 2 2 (t) + c δ 0 t t-s e vκ(µ-t+s) v 2 2 (µ)dµds (20)
where

v t (s) = [v 1 (t + s), v 2 (t + s)] T , s ∈ [-δ, 0]. One has Ẇ (v t ) = d dt v 2 1 (t) + v 2 2 (t) + c δ 0 d dt t t-s e vκ(µ-t+s) v 2 2 (µ)dµds.
Applying Leibniz integration leads to

Ẇ (v t ) = d dt v 2 1 (t) + v 2 2 (t) -vκc δ 0 t t-s e vκ(µ-t+s) v 2 2 (µ)dµds + c δ 0 e vκs v 2 2 (t) -v 2 2 (t -s)ds ≤ -av 2 1 (t) -bv 2 2 (t) + c t t-δ v 2 2 (s)ds + k -vκc δ 0 t t-s e vκ(µ-t+s) v 2 2 (µ)dµds + c δ 0 e vκs v 2 2 (t)ds -c δ 0 v 2 2 (t -s)ds ≤ -av 2 1 (t) -bv 2 2 (t) + k + c e vκδ -1 vκ v 2 2 (t) -vκ W (v t ) -v 2 1 (t) -v 2 2 (t)
Since e vκδ -1 vκ ≤ 2δ and given the definition of v, the following inequalities are achieved

Ẇ (v t ) + vκW (v t ) ≤ (-a + vκ) v 2 1 (t) + (-b + 2cδ + vκ)v 2 2 (t) + k Ẇ (v t ) + vκW (v t ) ≤ -a 1 - 1 √ 2 v 2 1 (t) -b 1 -(1 -κ) - κ √ 2 v 2 2 (t) + k Ẇ (v t ) + vκW (v t ) ≤ -a √ 2 -1 √ 2 v 2 1 (t) -bκ √ 2 -1 √ 2 v 2 2 (t) + k Ẇ (v t ) ≤ -vκW (v t ) + k Ẇ (v t ) ≤ -σW (v t ) + k
In order to get an over-valuation of W , one uses the comparison Lemma 2.5 p85 [START_REF] Khalil | Noninear systems[END_REF]. The solution of the following ODE ż(t) = -σz(t) + k is given by

z(t) = z(0) - k σ e -σt + k σ Taking ᾱ = (W (v 0 ) -k σ ) ends the proof.
Remark 2. It is worth mentioning that the results of Lemma 2 of [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF], which deal with the exponential stability problem, cannot be directly applied to the current case where the problem of practical stability is considered. In the case of practical stability, the error dynamics does not converge exactly toward zero but stays in a ball of arbitrary small radius. Therefore, it is important to estimate the radius of this convergence ball and its dependency on the control parameters. Lemma 2 of [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF] does not provide such information. Therefore, Lemma 3 is needed in order to get an insight on the effect of tuning parameters on the final error bounds.

Proof of Theorem 1

The proof of Theorem 1 is divided into several steps. In the first step, system (1) along with the control law (3) and the observer (4)-( 5) are re-written in a more compact form. Equations for tracking and observer errors are also derived in this step. New coordinates for high-gain design are introduced in the second step, while, in step 3, variables of the state of tracking and observer errors are combined in new variables. Candidate Lyapunov functions are introduced in step 4 and inequalities involving their derivatives are derived. Then in step 5, it is shown that Lemma 3 can be applied if the conditions given in Theorem 1 are satisfied. Finally in the last step, an inequality involving the tracking error in original coordinates is obtained.

Step 1. The dynamics of the agents can be written as

ẋi = Ax i + Bu i i = 0, . . . , N r i = Cx i ( 21
)
where

A = 0 m I m 0 m 0 m , B = 0 m I m and C = I m 0 m . Denoting xi,j = (r T i,j , vT i,j ) T , system (4)
-( 5) can be written as ẋi,j (t) = Ax i,j (t) -θ∆ -1 θ K o e -2θ(t-κi,j (t)) (r i,j (κ i,j (t)) -r j (κ i,j (t))), i = 1 . . . N, j = 0 . . . N where κ i,j (t) = max t i,j k | t i,j k ≤ t, k ∈ N represents the last instant, among all the sampling times represented by t i,j k , when the measurements of agent j have been received by agent i, ∆ θ =

I m 0 m 0 m 1 θ I m , K o = P -1 C T = 7 2I m I m
T , with P the symmetric positive definite matrix solution of the equation P + A T P + P A = C T C (see [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] for more details). Defining the estimation error xi,j = xi,j -x j for j = 0, . . . , N and tracking error e i = x i -x 0 for i = 1, . . . , N gives ẋi,j (t) = Ax i,j (t) -θ∆ -1 θ K o e -2θ(t-κi,j (t)) (C xi,j (κ i,j (t))

-Cx j (κ i,j (t))) -Bu j (t) = (A -θ∆ -1 θ K o C)x i,j (t) -θ∆ -1 θ K o z i,j (t) -Bu j (t)
where z i,j (t) = e -2θ(t-κi,j (t)) C xi,j (κ i,j (t)) -C xi,j (t) and ėi (t) = Ae i (t) + Bu i (t) -Bu 0 (t) [START_REF] Bédoui | Robust nonlinear controllers for bioprocesses[END_REF] for more details).

u i = -cK c Γ λ N k=1 H ik e k -cK c Γ λ N k=1 H ik xi,k + b i cK c Γ λ xi,0 for i = 1, . . . , N and K c = B T Q = I m 2I m with Q the symmetric positive definite matrix solution of Q + QA + A T Q = QBB T Q and Γ λ = λ 2 I m 0 m 0 m λI m (see
Step 2. Consider the coordinates for high-gain design ēi = Γ λ e i and xi,j = ∆ θ xi,j using the equalities,

∆ θ A∆ -1 θ = θA, C∆ -1 θ = C, Γ λ AΓ -1 λ = λA, Γ λ B = λB, ∆ θ B = 1 θ B and B T ∆ -1 θ = θB T yields ėi (t) = λAē i (t) + λBu i (t) -λBu 0 (t) ẋi,j (t) = θ(A -K o C)x i,j (t) -θK o z i,j (t) - 1 θ Bu j (t) u i = -cK c N k=1 H ik ēk -cK c Γ λ ∆ -1 θ N k=1 H ik xi,k + b i cK c Γ λ ∆ -1 θ xi,0 Step 3. Denoting η c = [ē T 1 . . . ēT N ] T , η o i = [(x i,1 ) T . . . (x i,N ) T ] T , i = 1 . . . N and η o 0 = [(x 1,0 ) T . . . (x N,0 ) T ],
the tracking error can be written in compact form as

ηc = λ[I N ⊗ A]η c -cλ[H ⊗ (BK c )]η c -cλ N i=1 [(D N i H) ⊗ (BK c Γ λ ∆ -1 θ )]η o i +cλ[I N ⊗ (BK c Γ λ ∆ -1 θ )][B ⊗ I 2m ]η o 0 -λ[1 N ⊗ B]u 0
Step 4. Let us consider the following Lyapunov functions

Vc (η c ) = (η c ) T [Ω ⊗ Q]η c (22) V o (x i,j ) = (x i,j ) T P (x i,j ) (23)
Vo (η o ) = N i=1 N j=0 s ij V o (x i,j ) (24)
where s ij = 1 if agent i receives information from agent j and 0 otherwise for i = 1, . . . , N , j = 0, . . . , N and η o is the vector containing all the xi,j such that s ij = 1.

Step 4.1. The derivative of the Lyapunov function ( 22) can be written as

Vc (η c ) = λ(η c ) T [Ω ⊗ (A T Q + QA)]η c -cλ(η c ) T [(H T Ω) ⊗ ((BK c ) T Q)]η c -cλ(η c ) T [(ΩH) ⊗ (QBK c )]η c + 2cλ(η c ) T [Ω ⊗ (QBK c Γ λ ∆ -1 θ )][B ⊗ I 2m ]η o 0 -2cλ N i=1 (η c ) T [(ΩD N i H) ⊗ (QBK c Γ λ ∆ -1 θ )]η o i -2λ(η c ) T [(Ω1 N ) ⊗ (QB)]u 0
One has the following inequalities, whose derivations are detailed in Appendix B:

2cλ(η c ) T [Ω ⊗ (QBK c Γ λ ∆ -1 θ )][B ⊗ I 2m ]η o 0 -2cλ N i=1 (η c ) T [(ΩD N i H) ⊗ (QBK c Γ λ ∆ -1 θ )]η o i ≤ 2k 1 λ Γ λ ∆ -1 θ Vc (η c ) Vo (η o ) (25) -2λ(η c ) T [(Ω1 N ) ⊗ (QB)]u 0 ≤ 2λ k2 δ 0 V c (η c ). (26)
with

k 1 = c√ N + 1 √ ω max λ max (Q) λ min (P ) max{1, H } K c (27) k2 = √ N √ ω max λ max (Q) (28)
furthermore, if c ≥ ω max /ρ, then we have

λ(η c ) T [Ω⊗(AQ T +QA)]η c -cλ(η c ) T [(H T Ω)⊗((BK c ) T Q)]η c -cλ(η c ) T [(ΩH)⊗(QBK c )]η c ≤ -λ Vc (η c ) (29)
These inequalities lead to

Vc (η c ) ≤ -λ Vc (η c ) + 2k 1 λ Γ λ ∆ -1 θ Vc (η c ) Vo (η o ) + 2 k2 λδ 0 Vc (η c )
Step 4.2. For i, j such that s ij = 1, the derivative of ( 23) is given by

Vo (x i,j ) = θ(x i,j ) T [((A -K o C) T P + P (A -K o C))](x i,j ) -2θ(x i,j ) T P K o z i,j - 2 θ (x i,j ) T P Bu j
One has the following inequalities, whose derivations are detailed in Appendix B:

-2θ(x i,j (t)

) T P K o z i,j (t) ≤ 2θ 2 k3 V o (x i,j (t)) t t-τ M V o (x i,j (s))ds (30)
- 2 θ (x i,j ) T P Bu j ≤ 2 k4 θ V o (x i,j ) Vc (η c ) (31)
+ 2 k5 θ Γ λ ∆ -1 θ V o (x i,j ) N k=0 s jk V 0 (x j,k ) - 2 θ (x i,0 ) T P Bu 0 ≤ 2 k6 θ δ 0 V 0 (x i,0 ) (32)
where k3 = λ max (P )

λ min (P ) K o k4 = c K c h max √ N λ max (P ) λ min (Q) √ ω min k5 = c K c h max λ max (P ) λ min (P ) k6 = λ max (P )
Using inequalities ( 30), ( 31) and ( 32) together with the definition of P leads to

Vo (x i,j ) ≤ -θV o (x i,j ) + 2θ 2 k3 V o (x i,j ) t t-τ M V o (x i,j (s))ds + 2 k4 θ V o (x i,j ) Vc (η c ) +2 k5 θ Γ λ ∆ -1 θ V o (x i,j ) N k=0 s jk V o (x j,k ) ≤ -θV o (x i,j ) + 2θ 2 k3 V o (x i,j ) t t-τ M Vo (η o (s))ds + 2 k4 θ V o (x i,j ) Vc (η c ) +2 k5 θ Γ λ ∆ -1 θ V o (x i,j ) √ N + 1 Vo (η o )
for j = 1, . . . , N , while

Vo (x i,0 ) ≤ -θV o (x i,0 ) + 2θ 2 k3 V o (x i,j ) t t-τ M Vo (η o (s))ds + 2 k6 θ δ 0 V o (x i,0 )
Step 4.3. Letting λ = θ with ∈ (0, 1), then Γ λ ∆ -1 θ = λθ and using the above inequalities, one gets

Vo (η o ) ≤ -θ Vo (η o )+2θ 2 k 3 Vo (η o ) t t-τ M Vo (η o (s))ds+2λk 5 Vo (η o )+2 k 4 θ Vo (η o ) Vc (η c )+2 k 6 θ δ 0 Vo (η o )
where

k 3 = k3 √ N √ N + 1 (33) k 4 = k4 N (34) k 5 = k5 N √ N + 1 (35) k 6 = k6 √ N (36)
Moreover, we obtain

d dt Vc (η c ) = 1 2 Vc (η c ) Vc (η c ) ≤ - λ 2 Vc (η c ) + k 1 λ 2 θ Vo (η o ) + k2 λδ 0
and similarly

d dt Vo (η o ) ≤ - θ 2 Vo (η o ) + k 3 θ 2 t t-τ M Vo (η o (s))ds + k 4 θ Vc (η c ) + k 5 λ Vo (η o ) + k 6 θ δ 0 .
Step 5. We have

d dt Vc (η c ) + 3 2 θ 2 Vo (η o ) ≤ - θ 4 1 -4k 4 1 2
Vc (η c ) -

3 2 θ 3 4 1 -4k 1 1 2 -4k 5 Vo (η o ) - θ 4
Vc (η c ) -

3 2 θ 3 4 Vo (η o ) + k 3 3 2 θ 4 t t-τ M Vo (η o (s))ds + k2 θδ 0 + k 6 3 2 θδ 0 By choosing < * where * = min 1, 1 (4k4) 2 , 1 (8k1) 2 , 1 8k5 , we obtain d dt Vc (η c ) + 3 2 θ 2 Vo (η o ) ≤ - θ 4 Vc (η c ) - 3 2 θ 3 4 Vo (η o ) + k 3 3 2 θ 4 t t-τ M Vo (η o (s))ds + k 2 θδ 0
where

k 2 = max{ k2 , k 6 } (37)
Applying Lemma 3 with

a = θ 4 ( 38
) b = 3 2 θ 3 4 (39) c = k 3 3 2 θ 4 (40) k = k 2 θδ 0 ( 41
)
one obtains the existence of > 0 and ᾱ > 0 such that if

τ M < min b c , 1 σ (42)
with σ = 1 2 min(a, b) = λ 8 since λ ≥ 1, then Vc (η c ) + 3 2 θ 2 Vo (η o ) ≤ ᾱe -σt + k 2 θδ 0 σ Since min b c , 1 σ = min 1 4θk 3 , 8 λ ≥ min 1 4k 3 , 8 min 1 θ , 1 λ ≥ 1 4k 3 θ (43) then, if θ verifies τ M < ¯ θ (44)
with ¯ = 4k3 , the following inequality holds true

Vc (η c ) + 3 2 θ 2 Vo (η o ) ≤ ᾱe -λ 8 t + 8k 2 δ 0 ( 45
)
Step 6. One now comes back to the original coordinates of both observer and tracking errors. Since λ ≥ 1 and θ ≥ 1, one has

Vc (η c ) ≥ λl 1 N i=1 e i ( 46
)
Vo (η o ) ≥ l 2 θ N i=1 N j=0 xi,j (47)
where

l 1 = λ min (Q) √ ω min √ N ( 48
)
l 2 = λ min (P ) √ N √ N + 1 (49)
The derivation of ( 46) and ( 47) is provided in Appendix B. Using these inequalities, one obtains

Vc (η c ) + 3 2 θ 2 Vo (η o ) ≥ λl 1 N i=1 e i + 3 2 θl 2 N i=1 N j=0 xi,j
Over-valuation of the tracking error e i is given by

Vc (η c ) + 3 2 θ 2 Vo (η o ) ≥ λl 1 N i=1 e i
by using inequality [START_REF] Bédoui | Robust nonlinear controllers for bioprocesses[END_REF], it gives

N i=1 e i ≤ αe -λ 8 t + βδ 0 λ (50) with α = ᾱ λl 1 (51) β = 8k 2 l 1 (52)
This ends the proof.

Remark 3. It is clear from inequality (50) that the error will gradually converge and will enter in a ball centered at the origin. It means that practical consensus is achieved. The radius of the convergence ball of the tracking error is directly proportional to the bound of the leader input/acceleration δ 0 . The size of this ball can also be reduced by increasing the controller gain λ (while keeping in mind that the controller dynamics must remain slower than the observer dynamics in order to guarantee stability of the closed-loop system). Furthermore, if the leader is moving with constant velocity i.e. u 0 = 0, then the MAS achieves exponential consensus.

Remark 4. One can remark that the conditions on the control parameters given in Theorem 1 require some global information like H and N . Hence, each agent must have some global knowledge about the communication topology similarly to many existing works on consensus. Nevertheless, the tuning parameters θ, λ and c are chosen beforehand and then they remain constant for all t ≥ 0 as only fixed communication topology is considered. Once the gains are set, only local information is needed to achieve leader-following consensus.

Remark 5. In [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF], the problem of leaderless consensus has been discussed whereas the current paper deals with the problem of leader-following consensus. It is important to mention that the results of [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF] cannot be directly applied to the leader-following case. Indeed, the addition of an active leader is not straightforward because of aperiodic sampling periods and availability of partial data only. Both the tracking and estimation error dynamics are different from [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF] due to the presence of leader dynamics. Therefore, the stability analysis is different from the one presented in [START_REF] Menard | Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF].

Simulation results

Consider a multi-agent system with four followers, labeled from 1 to 4 and a leader denoted as 0. The communication topology of the system is shown in Figure 2. The adjacency and pinning matrices corresponding to this topology are

A =     0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0     , B =     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0    
Based on Theorem 1 and Remark 1, the gains for simulation purpose are chosen as c = 1, λ = 1.0 and θ = 10. Figure 3 depicts the estimation of both position and velocity of the leader by follower 1. Similarly, the state estimation of follower 1 by follower 2 is also shown in Figure 4. It can be seen that the observer estimates both position and velocity quite efficiently. The time intervals between transmission of position state information from follower 1 to follower 2 is presented in Figure 5. Figures 6 and7 show the leader-following consensus results for step and ramp position trajectories of the leader respectively. In case of non-zero leader input, only practical stability is achieved which is shown in Figure 8. It is worth noting that only position information is transmitted through the communication network, given in Figure 2, at nonuniform and asynchronous sampling instants. Velocities and inputs are completely unknown to the neighbors. Despite these constraints, the system achieved leader-following consensus.

Experimental results

The proposed algorithm has been implemented on a group of wheeled mobile robots. The group consists of 3 Mini-Lab robots by Enova ROBOTICS. Table 1 illustrates the specifications of Mini-Lab robot. It is equipped with various sensors including a camera for depth images and an odometer for position measurements. The open source control architecture of the robot allows the implementation of the developed algorithm using Robot Operating System (ROS). ROS is a multi-lingual framework where modules can be written in various languages like C++, Python and Lisp. Each process in ROS is separately designed and programmed and is referred to as a node. Nodes can be run on different computers since ROS provides a distributed computation environment. Moreover, data among the nodes can be shared by passing messages over a topic. A node can publish or subscribe to multiple topics at a time. A ROS master manages the naming and registration of the nodes in an overall ROS system and tracks the publishers and subscribers to the topics. All nodes can communicate to the ROS master for registration and then they can receive information of other registered nodes. In the current experimental setup, the robots are connected over a wireless network and each robot has its own ROS master. The ROS package rosmaster fkie has been used to establish and manage a ROS multi-master network that consists of three robots. The proposed consensus tracking algorithm has been simulated for Mini-Lab using ROS Gazebo simulator prior to the actual implementation. ENOVA ROBOTICS has provided URDF file of Mini-Lab which gives 3D visualization of the robot along with the kinematic and dynamic properties.

In the following experiment, robot motion is restricted to 1-D space (i.e. a robot is allowed to move only in a straight line), Hence, the nonholonomic constraints can be ignored and robots can be modelled as double integrator systems with position and velocity as states. Therefore, the proposed algorithm is implemented with the consideration that robots are moving only in the x-axis with some constant offset in their y-positions. The consensus is supposed to be achieved for the x-axis positions and the x-axis velocities only. The experimental setup is shown in Figure 9 while the communication topology for the experiments is illustrated in Figure 10. Keeping in mind the hardware limitation of the robots, the controller gains have been chosen in such a way that the maximum speed limit is never crossed. Experimental results of the leader-following consensus, for both step and ramp position trajectories of the leader, are shown in Figure 11. It is clear from these results that MAS achieved the leader-following consensus efficiently in spite of the presence of uncertainties and disturbances inherent for a real application. An example of the time period between information transmission from one robot to the other is depicted in Figure 12. This nonuniform transmission of data may be caused by multiple reasons including computation time and measurement delays.

Conclusion

In this article, an algorithm has been proposed to deal with the communication constraints in the leader-following consensus problem. These communication constraints include nonuniform and asynchronous sampling periods and availability of only the position state. An observer has been designed to estimate the states of the leader and followers through discrete position information of neighbors which is available at irregular time intervals. The effectiveness of the proposed law has been demonstrated through both simulation and implementation on a fleet of wheeled mobile robots. The design of consensus algorithms to further cope with the nonlinear model of the robot is planned as a future work.

d) The proof is straightforward and then not reported here. e) See [47, p.27], property IX. f) Denoting ρ(A) the spectral radius of matrix A, one has

A ⊗ B 2 2 = ρ (A ⊗ B) T (A ⊗ B) = ρ((A T A) ⊗ (B T B)) = ρ(A T A)ρ(B T B)) (by applying Lemma 4-e) = A 2
(A⊗B) = det([u 1 ⊗v 1 , . . . , u 1 ⊗ v m , u 2 ⊗ v 1 , . . . , u n ⊗ v m ]) = (det(A)) n (det(B)) m = 0 since (u i
) and (v j ) are basis of R n and R m respectively. k) Let x be a non zero vector R mn . Let (u i ) (resp. (v j )) be a basis of orthogonal eigenvectors of A (resp. B) of R n (resp. R m ), that is Au i = µ i u i , with µ i an eigenvalue of A (resp. Bv j = λ j v j , with λ j an eigenvalue of B).

Since, according to point j), (u i ⊗ v j ) 1≤i,1≤j≤m is a basis of R mn , there exist reals α ij such that

x = n i=1 m j=1 α ij u i ⊗ v j .
One has

x T (A ⊗ B)x = n i1,i2=1 m j1,j2=1 α i1j1 α i2j2 (u i1 ⊗ v j1 ) T (A ⊗ B)(u i2 ⊗ v j2 ) = n i1,i2=1 m j1,j2=1 α i1j1 α i2j2 ((u i1 ) T Au i2 ) ⊗ ((v j1 ) T Bu j2 ) = n i=1 m j=1 α 2 ij ((u i ) T Au i ) ⊗ ((v j ) T Bu j ) since (u i ) and (v j ) are orthogonal basis, = n i=1 m j=1 α 2 ij µ i λ j ((u i ) T u i ) ⊗ ((v j ) T u j )
Then, since B is semidefinite positive symmetric, one has λ j ≥ 0, j = 1, . . . , m. Furthermore

((u i ) T u i ) ⊗ ((v j ) T v j ) = ((u i ) T u i )((v j ) T v j ) ≥ 0, then n i=1 m j=1 α 2 ij µ i λ j ((u i ) T u i ) ⊗ ((v j ) T v j ) ≤ max i {µ i } n i=1 m j=1 α 2 ij λ j ((u i ) T u i ) ⊗ ((v j ) T v j ) = max i {µ i } n i=1 m j=1 α 2 ij ((u i ) T u i ) ⊗ ((v j ) T Bv j ) = max i {µ i }x T (I n ⊗ B)x
The same can be done for the other inequality.

0 ≤ 2cλ Γ λ ∆ -1 θ √ ω max λ max (Q) λ min (P ) BK c Vc (η c ) N i=0 s i0 V o (x i0 ) ≤ 2cλ Γ λ ∆ -1 θ √ ω max λ max (Q) λ min (P ) K c Vc (η c ) N i=0 s i0 V o (x i0 ) since B = 1 (B.3)
For the second term, we have -2θ(x i,j ) T P K o z i,j (t) ≤ 2θ λ max (P ) K o V o (x i,j ) z i,j (t) by using Lemma 4-i) (B.5) Since żi,j (t) = -2θe -2θ(t-κi,j (t)) C xi j (κ i,j (t)) -C ẋi,j (t) = -B T xi,j (t)

-2cλ N i=1 (η c ) T [(ΩD N i H) ⊗ (QBK c Γ λ ∆ -1 θ )]η o i = -2cλ N i=1 (η c ) T (Ω ⊗ Q)[(D N i H) ⊗ (BK c Γ λ ∆ -1 θ )]η o i ≤ 2cλ N i=1 Vc (η c ) λ max (Ω ⊗ Q) (D N i H) ⊗ (BK c Γ λ ∆ -1 θ ) 2 N j=1 s ij xi,j by
and z i,j (κ i,j (t)) = 0, ∀t ≥ 0, then we have z i,j (t) = - ((x i,j (s)) T P (x i,j (s)))ds by using Lemma 4-i)

Therefore, (B.5) becomes -2θ(x i,j ) T P K o z i,j (t) ≤ 2θ 2 λ max (P ) λ min (P )

K o = k3
V o (x i,j (t))

Figure 1 :

 1 Figure 1: Example of sampling instants for data transmission under a directed graph

Figure 2 :

 2 Figure 2: Communication topology.

Figure 3 :

 3 Figure 3: Estimation of the leader's states by follower 1.

Figure 4 :

 4 Figure 4: Estimation of follower 1's states by follower 2.

Figure 5 :

 5 Figure 5: Sampling period for transmission of position information from follower 1 to follower 2.

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6: Consensus tracking when the leader trajectory is a step function.

Figure 9 :

 9 Figure 9: Experimental Setup.

Figure 10 :Figure 11 :Figure 12 :

 101112 Figure 10: Communication topology for hardware experiments

1 θs 1 θs 1 θ

 111 Lemma 4-h) and i)≤ 2cλ ω max λ max (Q) H K c Γ λ ∆ -Vc ij xi,jby using Lemma 4-e) andD N i 2 = 1, B 2 = 1 ≤ 2cλ √ ω max λ max (Q) λ min (P ) H K c Γ λ ∆ -Vc (η c ) ij V (x i,j ) by using Lemma 4-i) (B.4)Finally, by using inequalities (B.3) and (B.4), one obtains2cλ(η c ) T [Ω ⊗ (QBK c Γ λ ∆ -1 θ )][b ⊗ I 2m ]η o 0 -2cλ N i=1 (η c ) T [(ΩD N i H) ⊗ (QBK c Γ λ ∆ -1 θ )]η o i ≤ 2cλ √ ω max λ max (Q) λ min (P ) max{1, H } K c Γ λ ∆ -Vc c ) Vo (η o ) by using Lemma 4-c).b) Derivation of inequality[START_REF] Wu | Stochastic bounded consensus tracking of leader-follower multi-agent systems with measurement noises based on sampled-data with small sampling delay[END_REF] One has-2λ(η c ) T [(Ω1 N ) ⊗ (QB)]u 0 = -2λ(η c ) T (Ω ⊗ Q)(1 N ⊗ B)u 0 ≤ 2 √ N √ ω max λ max (Q) = k2λδ 0 Vc (η c ) by using Lemma 4-g), h) and i) c) Derivation of inequality[START_REF] Chen | Observer-based consensus of second-order multi-agent system with fixed and stochastically switching topology via sampled data[END_REF] Using the equality(BK c ) T Q = QBK c = QBB T Q, one gets with point k) of Lemma 4 (H T Ω) ⊗ ((BK c ) T Q) + (ΩH) ⊗ (QBK c ) = [H T Ω + ΩH] ⊗ [QBB T Q] ≥ ρ(I N ⊗ [QBB T Q]) ≥ ρ ω max Ω ⊗ [QBB T Q] If c ≥ ωmax ρ , one then obtains λ(η c ) T [Ω ⊗ (A T Q + QA)]η c -cλ(η c ) T [(H T Ω) ⊗ ((BK c ) T Q)]η c -cλ(η c ) T [(ΩH) ⊗ (QBK c )]η c ≤ -λ Vc (η c ) d) Derivation of inequality (30) One has

  xi,j (s) ds from the fact that t -κ i,j (t) ≤ τ M , ∀t ≥ 0, andB T = 1 ≤ θ λ min (P ) t t-τ M

1 λΓ λ ∆ - 1 θV 0

 110 min (Q) √ ω minVc (η c ) by using Lemma 4-c)xj,k ≤ min (P ) V o (x j,k ) by using Lemma 4-i) So it leads to -2 θ (x i,j ) T P Bu j ≤ 2 θ V o (x i,j ) λ max (P ) u j ≤ 2 θ c K c h max √ N λ max (P ) λ min (Q) √ ω min = k4 V o (x i,j ) Vc (η c ) + 2 θ c K c h max λ max (P ) λ min (P ) = k5 V o (x i,j ) N k=0 s jk V o (x j,k ) f) Derivation of inequality (32) One has -2 θ (x i,0 ) T P Bu 0 ≤ 2 θ λ max (P ) (x i,0 ) T P (x i,0 ) (Bu 0 ) T (Bu 0 ) by using Lemma 4-h) (x i,0 )by using Lemma 4-i) and the fact that B = 1 and u 0 (t) ≤ δ 0 , ∀t ≥ 0 g) Derivation of inequalities (46) and (47) Qē i = Vc (η c )) and Lemma 4-c)Vc (η c ) ≥ λ λ min (Q) have λ min (P ) θ xi,j ≤ (x i,j ) T P xi,j

Table 1 :

 1 Specifications of Mini-Lab robot

	0	1	2

Appendix A. Lemma 4. We have the following results:

e) let µ i , i = 1, . . . , m and ν j , j = 1, . . . , n be respectively the eigenvalues of A ∈ R m×m and B ∈ R n×n , then the eigenvalues of A ⊗ B are µ i ν j with i = 1, . . . , m and j = 1, . . . , n;

Applying Lemma 4-b) gives the result.

Appendix B.

The detailed derivations of the inequalities, introduced in the proof of Theorem 1, are given in this Appendix.

a) Derivation of inequality ( 25) Inequality ( 25) is derived in two steps. An over-valuation of the two terms in the left hand-side of ( 25) are first obtained and then combined in order to obtain [START_REF] Xie | Bounded consensus tracking for sampled-data second-order multi-agent systems with fixed and markovian switching topology[END_REF].

For the first term, using Lemma 4-h)-i)-g) yields


Mihir Sahasrabudhe

Stergios Christodoulidis

Roberto Salgado

Stefan Michiels

Sherene Loi

Fabrice André

Nikos Paragios

Maria Vakalopoulou

Self-Supervised Nuclei Segmentation in Histopathological Images Using Attention

Keywords: Pathology, Whole Slide Images, Nuclei Segmentation, Deep Learning, Self-Supervision, Attention Models

Segmentation and accurate localization of nuclei in histopathological images is a very challenging problem, with most existing approaches adopting a supervised strategy. These methods usually rely on manual annotations that require a lot of time and effort from medical experts. In this study, we present a self-supervised approach for segmentation of nuclei for whole slide histopathology images. Our method works on the assumption that the size and texture of nuclei can determine the magnification at which a patch is extracted. We show that the identification of the magnification level for tiles can generate a preliminary self-supervision signal to locate nuclei. We further show that by appropriately constraining our model it is possible to retrieve meaningful segmentation maps as an auxiliary output to the primary magnification identification task. Our experiments show that with standard post-processing, our method can outperform other unsupervised nuclei segmentation approaches and report similar performance with supervised ones on the publicly available MoNuSeg dataset. Our code and models are available online to facilitate further research.

Introduction

Histology images are the gold standard in diagnosing a considerable number of diseases including almost all types of cancer. For example, the count of nuclei on whole-slide images (WSIs) can have diagnostic significance for numerous cancerous conditions [START_REF] Ruan | Predictive value of tumor-infiltrating lymphocytes to pathological complete response in neoadjuvant treated triple-negative breast cancers[END_REF]. The proliferation of digital pathology and highthroughput tissue imaging leads to the adoption in clinical practice of digitized https://github.com/msahasrabudhe/miccai2020_self_sup_nuclei_seg histopathological images that are utilized and archived every day. Such WSIs are acquired from glass histology slides using dedicated scanning devices after a staining process. In each WSI, thousands of nuclei from various types of cell can be identified. The detection of such nuclei is crucial for the identification of tissue structures, which can be further analyzed in a systematic manner and used for various clinical tasks. Presence, extent, size, shape, and other morphological characteristics of such structures are important indicators of the severity of different diseases [START_REF] Gleason | Histologic grading of prostate cancer: a perspective[END_REF]. Moreover, a quantitative analysis of digital pathology is important, to understand the underlying biological reasons for diseases [START_REF] Rubin | Rubin's pathology: clinicopathologic foundations of medicine[END_REF].

Manual segmentation or estimation of nuclei on a WSI is an extremely time consuming process which suffers from high inter-observer variability [START_REF] Andrion | Malignant mesothelioma of the pleura: interobserver variability[END_REF]. On the other hand, data-driven methods that perform well on a specific histopathological datasets report poor performance on other datasets due again to the high variability in acquisition parameters and biological properties of cells in different organs and diseases [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF]. To deal with this problem, datasets integrating different organs [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF][START_REF] Gamper | Pannuke: An open pan-cancer histology dataset for nuclei instance segmentation and classification[END_REF] based on images from The Cancer Genome Atlas (TCGA) provide pixelwise annotations for nuclei from variety of organs. Yet, these datasets provide access to only a limited range of annotations, making the generalization of these techniques ambiguous and emphasizing the need for novel segmentation algorithms without relying purely on manual annotations.

To this end, in this paper, we propose a self-supervised approach for nuclei segmentation without requiring annotations. The contributions of this paper are threefold: (i) we propose using scale classification as a self-supervision signal under the assumption that nuclei are a discriminative feature for this task; (ii) we employ a fully convolutional attention network based on dilated filters that generates segmentation maps for nuclei in the image space; and (iii) we investigate regularization constraints on the output of the attention network in order to generate semantically meaningful segmentation maps.

Related Work

Hematoxylin and eosin (H&E) staining is one of the most common and inexpensive staining schemes for WSI acquisition. A number of different tissue structures can be identified in H&E images such as glands, lumen (ducts within glands), adipose (fat), and stroma (connective tissue). The building blocks of such structures are a number of different cells. During the staining process, hematoxylin renders cell nuclei dark blueish purple and the epithelium light purple, while eosin renders stroma pink. A variety of standard image analysis methods are based on hematoxylin in order to extract nuclei [START_REF] Yi | Automatic extraction of cell nuclei from H&E-stained histopathological images[END_REF][START_REF] Boyle | The prognostic significance of the aberrant extremes of p53 immunophenotypes in breast cancer[END_REF] reporting very promising results, albeit evaluated mostly on single organs. A lot of research on the segmentation of nuclei in WSI images has been presented over the past few decades. Methodologies that integrate thresholding, clustering, watershed algorithms, active contours, and variants along with a variety of pre-and post-processing techniques have been extensively studied [START_REF] Gurcan | Histopathological image analysis: A review[END_REF]. A common problem among the aforementioned algorithmic approaches is the poor generalization across the wide spectrum of tissue morphologies introducing a lot of false positives.

To counter this, a number of learning-based approaches have been investigated in order to better tackle the variation over nuclei shape and color. One group of learning-based methods includes hand-engineered representations such as filter bank responses, geometric features, texture descriptors or other first order statistics paired with a classification algorithm [START_REF] Kong | Partitioning histopathological images: an integrated framework for supervised color-texture segmentation and cell splitting[END_REF][START_REF] Plissiti | Overlapping cell nuclei segmentation using a spatially adaptive active physical model[END_REF]. Recent success of deep learning-based methods and the introduction of publicly available datasets [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF][START_REF] Gamper | Pannuke: An open pan-cancer histology dataset for nuclei instance segmentation and classification[END_REF] formed a second learning-based group of supervised approaches. In particular, [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF] summarises some of these supervised approaches that are developed for multi-organ nuclei segmentation, most of them based on convolutional neural networks. Among them the best performing method proposes a multi-task scheme based on an FCN [START_REF] Long | Fully convolutional networks for semantic segmentation[END_REF] architecture using a ResNet [START_REF] He | Deep residual learning for image recognition[END_REF] backbone encoder with one branch to perform nuclei segmentation and a second one for contour segmentation. Yet, the emergence of self-supervised approaches in computer vision [17,[START_REF] Gidaris | Unsupervised representation learning by predicting image rotations[END_REF] has not successfully translated to applications in histopathology. In this paper, we proposed a self-supervised method for nuclei segmentation exploiting magnification level determination as a self-supervision signal.

Methodology

The main idea behind our approach is that given a patch extracted from a WSI viewed at a certain magnification, the level of magnification can be ascertained by looking at the size and texture of the nuclei in the patch. By extension, we further assume that the nuclei are enough to determine the level of magnification, and other artefacts in the image are not necessary for this task.

Contrary to several concurrent computer vision pipelines which propose to train and evaluate models by feeding images sampled at several scales in order for them to learn multi-scale features [START_REF] He | Deep residual learning for image recognition[END_REF] or models which specifically train for scale equivariance [START_REF] Worrall | Deep scale-spaces: Equivariance over scale[END_REF], we posit learning a scale-sensitive network which specifically trains for discriminative features for correct scale classification . Given a set of WSIs, we extract all tissue patches (tiles) from them at a fixed set of magnifications C. We consider only these tiles, with the "ground-truth" knowledge for each tile being at what magnification level it was extracted. Following our earlier reasoning, if nuclei in a given tile I ∈ R 3×H×W are enough to predict the level of magnification, we assume that there exists a corresponding attention map A, so that A I is also enough to determine the magnification, where represents element-wise multiplication, and A ∈ [0, 1] 1×H×W is a single channel attention image that focuses on the nuclei in the input tile (Figure 1).

We design a fully-convolutional feature extractor F to predict the attention map A from the patch I. Our feature extractor consists of several layers of convolution operations with a gradual increase in the dilation of the kernels so as to incorporate information from a large neighborhood around every pixel. This feature extractor F regresses a confidence map a = F (I) ∈ R 1×H×W , which is activated by a compressed and biased sigmoid function so that A = σ (a). In order to force the attention map to focus only on parts of the input patch, we apply a sparsity regularizer on A. This regularizer follows the idea and implementation of a concurrent work on unsupervised separation of nuclei and background [START_REF] Hou | Sparse autoencoder for unsupervised nucleus detection and representation in histopathology images[END_REF]. Sparsity is imposed by picking the η-th percentile value in the confidence map a for all images in the batch, and choosing a threshold τ equal to the average of this percentile over an entire training batch. Formally,

τ = 1 B B b=1 a (η) b , (1)
where a The "attended" image J = A I is now enough for magnification or scale classification. We train a scale classification network G, which we initialize as a ResNet-34 [START_REF] He | Deep residual learning for image recognition[END_REF], to predict the magnification level for each input tile J. The output of this network is scores for each magnification level, which is converted to probabilities using a softmax activation. The resulting model (Figure 1) is trainable in an end-to-end manner. We use negative log-likelihood to train the scale classification network G, and in turn the attention network F-

L scale (p, l) = -log pl ; pi = [softmax (ŝ)] i ; ŝ = G (J) ; 1 ≤ i ≤ N C , ( 2
)
where l is the scale ground-truth, and N C = |C| .

Note that the terms scale (in the context of computer vision) and magnification (in the context of histopathology) are semantically equivalent and used interchangeably.

Smoothness Regularization

We wish A to be semantically meaningful and smooth with blobs focusing on nuclei instead of having high frequency components. To this end, we incorporate a smoothness regularizer on the attention maps. The smoothness regularizer attempts simply to reduce the high frequency component that might appear in the attention map because of the compressed sigmoid. We employ a standard smoothness regularizer based on spatial gradients defined as

L smooth = 1 (H -1)(W -1) i,j A i+1,j -A i,j 1 + A i,j+1 -A i,j 1 . (3)

Transformation Equivariance

Equivariance is a commonly used constraint on feature extractors for imposing semantic consistency [START_REF] Thewlis | Unsupervised learning of object landmarks by factorized spatial embeddings[END_REF][START_REF] Cohen | Gauge equivariant convolutional networks and the icosahedral cnn[END_REF]. A feature extractor f is equivariant to a transformation g if g is replicated in the feature vector produced by f , i.e., f (g(x)) = g(f (x)) , for an image x. In the given context, we want the attention map obtained from F to be equivariant to a set T of certain rigid transforms. We impose equivariance to these transformations through a simple mean squared error loss on A. Formally, we define the equivariance constraint as

L equiv = 1 HW σ (t (F (I))) -σ (F (t (I))) 2 2 , (4)
for a transformation t ∈ T . We set T to include horizontal and vertical flips, matrix transpose, and rotations by 90, 180, and 270 degrees.Each training batch uses a random t ∈ T .

Training

The overall model is trained in an end-to-end fashion, with L scale being the guiding self-supervision loss. For models incorporating all constraints, i.e., smoothness, sparsity, and equivariance, the total loss is

L total = L scale + L smooth + L equiv . (5)
We refer to a model trained with all these components together as M proposed . We also test models without one of these losses to demonstrate how each loss contributes to the learning. More specifically, we define the following models:

1. M ¬smooth : does not include L smooth . 2. M ¬equiv : does not include L equiv .

3. M ¬sparse : does not include a sparsity regularizer on the attention map. In this case, the sigmoid is simply defined as σ(x) = 1 1+exp(-x) . 4. M ¬WSI : a model which does not sample images from WSIs, but instead from a set of pre-extracted patches (see Section 4.1).

We set the sparsity parameter η empirically in order to choose the 93-rd percentile value for sparsity regularization. This is equivalent to assuming that, on an average, 7% of the pixels in a tile represent nuclei.

Post Processing, Validation, and Model Selection

In order to retrieve the final instance segmentation from the attention image we employ a post processing pipeline that consists of 3 consequent steps. Firstly, two binary opening and closing morphological operations are sequentially performed using a coarse and a fine circular element (r = 2, r = 1). Next, the distance transform is calculated and smoothed using a Gaussian blur (σ = 1) on the new attention image and the local maxima are identified in a circular window (r = 7). Lastly, a marker driven watershed algorithm is applied using the inverse of the distance transform and the local maxima as markers.

As our model does not explicitly train for segmentation of nuclei, we require a validation set to determine which model is finally best-suited for our objective. To this end, we record the Dice score between the attention map and the ground truth on the validation set (see Section 4.1) at intermediate training epochs, and choose the epoch which performs the best. We noticed that, in general, performance increases initially on the validation, but flattens after ∼30 epochs.

Experimental Setup and Results

Dataset

For the purposes of this study we used the MoNuSeg database [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF]. This dataset contains thirty 1000 × 1000 annotated patches extracted from thirty WSIs from different patients suffering from different cancer types from The Cancer Genomic Atlas (TCGA). We downloaded the WSIs corresponding to patients included in the training split and extracted tiles of size 224 × 224 from three different magnifications, namely 10× , 20× , and 40× . For each extracted tile, we perform a simple thresholding in the HSV color space to determine whether the tile contains tissue or not. Tiles with less then 70% tissue cover are not used. Furthermore, a stain normalization step was performed using the color transfer approach described in [START_REF] Reinhard | Color transfer between images[END_REF]. Finally, a total of 1 125 737 tiles from the three aforementioned scales were selected and paired with the corresponding magnification level. The MoNuSeg train and test splits were employed, while the MoNuSeg train set was further split into training and validation as 19 and 11 examples, respectively. The annotations provided by MoNuSeg on the validation set were utilized for determining the four post processing parameters (Section 3.4) and for the final evaluation. For the model M ¬WSI , which does not use whole slide images, we use the MoNuSeg patches instead for training, using the same strategy to split training and validation. We further evaluated the performance of our model that was trained on the MoNuSeg training set on the TNBC [START_REF] Naylor | Segmentation of nuclei in histopathology images by deep regression of the distance map[END_REF] and CoNSeP [START_REF] Graham | Hover-net: Simultaneous segmentation and classification of nuclei in multitissue histology images[END_REF] datasets.

Implementation

We use the PyTorch [START_REF] Paszke | Automatic differentiation in pytorch[END_REF] library for our code. We use the Adam [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] optimizer in all our experiments, with an initial learning rate of 0.0002 , a weight decay of 0.0001 , and β 1 = 0.9 . We use a batch size of 32 , 100 minibatches per epoch, and randomly crop patches of size 160 × 160 from training images to use as inputs to our models. Furthermore, as there is a high imbalance among the number of tiles for each of the magnification level (images are about 4 times more in number for a one step increase in the magnification level), we force a per-batch sampling of images that is uniform over the magnification levels, i.e., each training batch is sampled so that images are divided equally over the magnification levels. This is important to prevent learning a biased model.

Results

To highlight the potentials of our method we compare its performance with supervised and unsupervised methods on the MoNuSeg testset presented in [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF]. In particular, in Table 1 we summarize the performance of three supervised methods (CNN2,CNN3 and Best Supervised) and two completely unsupervised methods (Fiji and CellProfiler) together with different variations of our proposed method. Our method outperforms the unsupervised methods, and it reports similar performance with CNN2 [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF] and CNN3 [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF] on the same dataset. While it reports lower performance than the best supervised method from [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF], our formulation is quite modular and able to adapt multi-task schemes similar to the one adapted by the winning method of [START_REF] Kumar | A dataset and a technique for generalized nuclear segmentation for computational pathology[END_REF].

On the TNBC and CoNSeP datasets, our method is strongly competitive among the unsupervised methods. We should emphasize that these results have been obtained without retraining on these datasets. The CoNSeP dataset consists mainly of colorectal adenocarcinoma which is under-represented in the training set of MoNuSeg, proving very good generalization of our method.

Moreover, from our ablation study (Table 1), it is clear that all components of the proposed model are essential. Sparsity is the most important as by removing it, the network regresses an attention map that is too smooth and not necessarily concentrating on nuclei, thus being semantically meaningless. Qualitatively, we observed that L smooth allows the network to focus on only on nuclei by removing attention over adjacent tissue regions, while L equiv further refines the attention maps by imposing geometric symmetry. Finally, in Figure 2 1. Quantitative results of the different benchmarked methods on three different public available datasets. AJI, AHD, and ADC stand for Aggregated Jaccard Index, Average Hausdorff Distance, and Average Dice Coefficient, respectively. Methods marked with † are supervised.

M proposed attention network together with the nuclei segmentation after the performed post-processing are summarised.

Conclusion

In this paper, we propose and investigate a self-supervised method for nuclei segmentation of multi-organ histopathological images. In particular, we propose the use of the scale classification as a guiding self-supervision signal to train an attention network. We propose regularizers in order to regress attention maps that are semantically meaningful. Promising results comparable with supervised methods tested on the publicly available MoNuSeg dataset indicate the potentials of our method. We show also via. experiments on TNBC and ConSeP that our model generalizes well on new datasets. In the future, we aim to investigate the integration of our results within a treatment selection strategy. Nuclei presence is often a strong bio-marker as it concerns emerging cancer treatments (immunotherapy). Therefore, the end-to-end integration coupling histopathology and treatment outcomes could lead to prognostic tools as it concerns treatment response. Parallelly, other domains in medical imaging share concept similarities with the proposed concept.

Fig. 1 .

 1 Fig. 1. A diagram of our approach. Each patch I is fed to the attention network F generating an attention map A. The "attended" image J is then given to the scale classification network G. Both networks are trained in an end-to-end fashion. s, p, and d for convolution blocks refer to stride, padding, and dilation.

  100 • HW -th largest value in the confidence map a b for the b-th image in the training batch of B images. The sigmoid is then defined as σ(x) = 1 1+exp(-r(x-τ )) . It is compressed in order to force sharp transitions in the activated attention map, the compression being determined by r. We use r = 20 in our experiments.

Fig. 2 .

 2 Fig. 2. Input, intermediate results and output of the post processing pipeline. From left to right: the input image; the attention map obtained from M proposed after the post-processing; the distance transform together with local maxima over-imposed in red; and the final result after the marker driven watershed.

  the segmentation map for one test image is presented. Results obtained from the

	Test dataset	Method	AJI [13]	AHD	ADC
		CNN2 [13] †	0.3482	8.6924	0.6928
		CNN3 [13] †	0.5083	7.6615	0.7623
		Best Supervised [13] †	0.691	-	-
	MoNuSeg test	CellProfiler [13] Fiji [13]	0.1232 0.2733	9.2771 8.9507	0.5974 0.6493
		M¬sparse	0.0312	13.1415	0.2283
		M ¬smooth	0.1929	8.8166	0.4789
		M¬WSI	0.3025	8.2853	0.6209
		M¬equiv	0.4938	8.0091	0.7136
		M proposed	0.5354	7.7502	0.7477
		U-Net[7] †	0.514	-	0.681
		SegNet+WS[7] †	0.559	-	0.758
	TNBC[15]	HoverNet[7] †	0.590	-	0.749
		CellProfiler	0.2080	-	0.4157
		M proposed	0.2656	-	0.5139
		SegNet[7] †	0.194	-	0.796
		U-Net[7] †	0.482	-	0.724
	CoNSeP[7]	CellProfiler[7]	0.202	-	0.434
		QuPath[7]	0.249	-	0.588
		M proposed	0.1980	-	0.587
	Table