I Introduction
Distributed coordination control of multiagent systems (MASs) has received considerable attention in the past decade [Ren2013TIE, Shi2017TIE]. Simple Proportional (P) and ProportionalDerivative (PD) coordination strategies have been proven effective for networks of singleintegrator [Murray2004TAC], doubleintegrator [Ren2008TAC], EulerLagrange (EL) [Ren2009IJC], and more general secondorder nonlinear [Ren2013Automatica] systems. Further, adaptive control and output feedback control have overcome system uncertainties [Ren2013Automatica2, Wang2013TAC] and the lack of velocity measurements [Duan2014TAC, Jiang2017TAC], respectively.
Timedelayed communications foil agents’ immediate access to their neighbour information and, thus, distort the interagent couplings and endanger synchronization. Prior work has mitigated time delays in various ways. Rendering the MAS inputoutput passive has led to output feedback synchronization of affine nonlinear networks with constant time delays in [Chopra2012TAC]. Sufficient damping injection has provably coordinated EL networks with timevarying delays in [Nuno2013TRO]. An Immersion and Invariance (I&I
) observer has facilitated globally exponentially convergent velocity estimation and output feedback consensus of such networks in
[Nuno2016TCNS]. The need for accurate system dynamics in the I&I observer design has been eliminated in [Nuno2017TCST]. Adaptive consensus of uncertain EL networks has been explored in [Nuno2011TAC, Wang2014TAC] for constant time delays, and in [Polushin2014TAC, Dixon2017Cybernetics, Jia2017Cybernetics] for timevarying delays with sufficiently small rates of change of the delays.Conventional MAS coordination strategies assume a connected communication graph. In contrast, gradientbased controls derived from potentials dependent on interagent distances can both synchronize and maintain the connectivity of a MAS. Existing research has offered several potentials for connectivitypreserving MAS synchronization. Unbounded controls, derived from potentials that continuously increase interagent attraction when adjacent systems approach their communication boundaries, have prevented disconnection among singleintegrator and nonholonomic agents in [Egerstedt2007TRO, Dimos2007TAC, Dimos2008TRO, Zavlanos2008TRO]. Bounded controls have achieved similar objectives in [Dimos2008ICRA, Amir2010TAC, Feng2013Automatica, Sun2013Automatica, Amir2013TAC]. A specific class of bounded potentials has led to PDlike, as well as adaptive PDlike, control with distancedependent P gains for doubleintegrator networks in [Su2010SCL], and for secondorder nonlinear networks in [Su2011Automatica], respectively.
Connectivitypreserving coordination of MASs with timevarying delays has been less explored to date, and is the focus herein. The paper proposes a class of potential functions of interagent distances whose negative gradients it adopts as control inputs for singleintegrator MASs. In the presence of communication delays, agents cannot access timely neighbour information and gradientbased controls suffer from disturbances that would lead to instability. A dynamic filter, that increases the degree of singleintegrator MASs from one to two, suppresses the delayinduced disturbances through sufficient damping injection. The injected damping limits velocities appropriately to maintain each agent in its neighbours’ communication zones although the neighbours move during the time it takes their information to reach the agent. Because the dynamic filter leads to a closedloop system that can be regarded as a doubleintegrator MAS, the proposed methodology is readily extended to connectivitypreserving coordination of EL networks with timevarying delays. Rigorous Lyapunov stability analysis and simulations validate the proposed designs.
Ii Preliminaries
Let a MAS have agents, and let each agent have a broadcasting area (communication zone) of radius centered around its position . At time , agent can receive the delayed position of agent only if agent is in the current communication zone of agent , i.e., , with the timevarying delay from to .
The communications of the MAS are described by its communication graph , that comprises two sets : the set of graph nodes , that includes all agents in the MAS; the set of graph edges , that collects all communication links in the system. An oriented communication link or graph edge exists if agent receives information from agent . Agent is a neighbour of agent , i.e., , if and only if , where is the set of all neighbours of agent . A graph is undirected if agents and are adjacent, or neighbours of each other, i.e., if and only if . A sequence of adjacent agents, forms a path in an undirected graph. Then, an undirected graph is connected if and only if there exists a path between each pair of agents in the system.
The adjacency matrix associated to an undirected graph is a symmetric matrix, with if and otherwise. The corresponding weighted Laplacian is also symmetric, with and for . Assume that the undirected communication graph contains oriented edges. For each pair of adjacent agents and , only one of the oriented links and is labelled as , , with weight . Particularly, if , then agents and are the tail and the head of . The incidence matrix of the graph is defined by if agent is the head of , if agent is the tail of , and otherwise. The graph Laplacian and the incidence matrix obey [Egerstedt2010Princeton]: , where is a diagonal matrix with on its diagonal.
The following assumption underlies the connectivitypreserving synchronization strategy designed in this paper:

[label=A.0]

The undirected communication graph of the MAS is initially connected and each pair of adjacent agents is strictly within their communication distance, i.e., for some .
Under assumption 1, the connectivity of an MAS with static communication graph is preserved if all initial communication links are maintained. In an MAS with dynamic graph, initially nonadjacent pairs of agents can start to receive each other’s delayed information once they enter in each other’s communication zone. A dynamic graph requires a mechanism to create new links depending on interagent distances and to simultaneously maintain connectivity. When communication delays prevent agents’ access to timely information of other agents, building bidirectional links dynamically is not trivial. For simplicity, this paper considers only MASs with static communication graph and aims to render the initial edge set invariant (static) through control. Thus, the connectivitypreserving coordination problem in this paper is the following:
Problem.
Let a MAS with timevarying communication delays have agents with broadcasting radius and obey assumption 1. Design distributed control laws that use only the agents’ information and their neighbours’ delayed positions to render the original edge set invariant and coordinate the MAS.
Iii ConnectivityPreserving Coordination
The dynamics of a MAS with singleintegrator dimensional agents are:
(1) 
where: indexes the agent; and , and are the agent’s position, velocity and control input. The potential used in this paper to quantify the energy stored in the interagent connections belongs to a class of functions with the following properties:

is positive definite and strictly increasing with respect to for .

.

for bounded and , where: maps the components of a vector to their absolute values ; are the gradients of with respect to , ; and are positive constants; and .

, where is increasing with respect to and is positively lowerbounded for .
The particular potential with the above properties selected in this paper is , with a parameter to be determined, and provides the following control laws:
(2) 
where is the set of neighbours of agent at time ; are the gradients of with respect to ; and with are positive diagonal gain matrices to be determined. The gradient of the selected potential with respect to being:
the designed controls (2) are proportional controls whose statedependent gains increase with on . In other words, the controls (2) stiffen the interagent couplings when the connections are threatened.
A property of the selected potential, needed in the connectivity preservation and coordination proofs, is the following:
Proposition.
For the function , proper selection of guarantees that .
Proof.
and , so implies that , i.e., , and further . This can be guaranteed by selecting if ; and arbitrarily otherwise. ∎
A property of timedelayed signals needed in the proofs is:
Proposition.
Given any vectors , and any variable time delays bounded by , they obey
for some and .
Proof.
The CauchySchwartz inequality yields
where . Then, can be upperbounded by
after applying the CauchySchwartz inequality again. Lastly, using and inverting the integration order leads to
∎
The Lyapunov function that quantifies the energy of the singleintegrator MAS in closed loop with the controls (2) is:
(3) 
Assumption 1 together with the first property of imply that . The selection and the second property of lead to . Then, because is continuous, rendering it nonincreasing is sufficient for connectivity preservation, i.e., for all and . If all initial connections are maintained, then for all and , and is positive definite with respect to and . Then, coordination can be shown by showing that as .
By continuity of , there exists such that on , . The derivative of along the closedloop dynamics (1) and (2) on are:
where the system dynamics and the initial undirected and connected interaction graph have been applied. From the third property of the potential , it follows that and can be upperbounded by:
Lemma 1 and time integration of the above inequality yield
(4)  
with positive constants and
the smallest eigenvalues of
.Letting , Equation (4) can further be rewritten as , where with if , if and , and otherwise. It then follows that if all column summations of are positive, i.e.,
(5) 
In other words, , and connectivity is maintained if the damping gains can be selected sufficiently large to satisfy (5).
The strict inequality (5) means that there exists with strictly positive components such that , and thus , . In turn, indicates that . Then, the derivative of the closedloop dynamics (1) and (2) imply that . Using Barbalat’s lemma, it follows that as , . The derivative of the controller dynamics (2) means , and thus Barbalat’s lemma indicates , i.e., . From , it follows that . Therefore, , , i.e., , where with for , for , and for . Then, the interaction graph being undirected and connected, and , which implies that , i.e., coordination is achieved.
The parameters and in the design criterion (5) can be specified for the potential noting that the connectivity preservation and coordination proof shows under the condition that . If there exist and that satisfy the third property of on , then s that observe (5) render the set invariant and maintain MAS connectivity.
Proposition.
For the potential function with , where , there exist and that guarantee the third property of on the set .
Proof.
On the set , and and . Because , and both exist by selecting . Then,
because , where and . ∎
The above analysis is summarized in the following theorem:
Theorem.
Because the singleintegrator MAS (1) in closed loop with the controllers (2) can be regarded as the doubleintegrator MAS , with the statedependent proportional plus damping controls , the above analysis readily applies to doubleintegrator MAS, whose connectivitypreserving coordination the controls can enforce.
In EL networks with timevarying delays, the agent dynamics are:
(6) 
with , and the joint positions, velocities and accelerations; and the matrices of inertia and of Coriolis and centrifugal effects; and and the gravity and control torques. The dynamics (6) have the following properties:

If agent has only revolute joints, its inertia matrix is symmetric, positive definite and uniformly bounded by ;

is skewsymmetric.
Taking the agent controls:
(7) 
the coordination with connectivity preservation of the EL network can be studied using the following Lyapunov function:
(8) 
with selected to satisfy:
(9)  
Proposition.
There exists a (sufficiently large) that satisfies condition (9) for any initial EL network configuration.
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