# Distributed Event-Triggered Leader-Follower Consensus of Nonlinear Multi-
Agent Systems

Mathieu Marchand1, Vincent Andrieu2, Sylvain Bertrand1 and Hélène Piet-
Lahanier1 *This work was funded by French grant ANR Delicio
(ANR-19-CE23-0006).1M. Marchand, S. Bertrand and H. Piet-Lahanier are with
Université Paris-Saclay, ONERA, Traitement de l’information et systèmes,
91123, Palaiseau, France (e-mail: {mathieu.marchand; sylvain.bertrand;
helene.piet-lahanier}@onera.fr).2V. Andrieu is with Universit́e Lyon, CNRS,
LAGEPP, Villeurbanne, France (e-mail: vincent.andrieu@gmail.com).

###### Abstract

We consider the distributed leader-follower consensus problem with event-
triggered communications. The system under consideration is a non-linear
input-affine multi agent system. The agents are assumed to have identical
dynamics structure with uncertain parameters and satisfying an incremental
stabilisability condition. A distributed control law is proposed which
achieves consensus based on two novel Communication Triggering Conditions
(CTCs): the first one to achieve an asymptotic consensus but without any
guarantees on Zeno behaviour and the second one to exclude Zeno behaviour but
with practical consensus.

##  I Introduction

As missions to be performed by autonomous systems become increasingly
challenging, there is a growing interest in the use of Multi-Agent Systems
(MAS) for many applications such as inspection/mapping/exploration with
unmanned aerial vehicles or mobile robots, see e. g. [1], [2]. The use of such
systems allows to achieve complex or highly demanding tasks with agents with
lower abilities through cooperation. MAS control may be either centralized or
distributed. In the centralized case, control laws are designed on the basis
of a global knowledge of the MAS which makes this approach very efficient.
However, it is at the cost of being very sensitive to potential deficiency or
loss of the central agent. In distributed control, each agent computes its
control input according to local information (i.e. from the agents located in
its neighbourhood). At the cost of potential discrepancy of resulting
performances, this strategy provides enhanced robustness to the loss of an
agent. Thus, distributed approaches have been increasingly employed,
especially for cooperative MAS. This cooperation requires the definition of a
communication strategy.

Development of distributed control to make a MAS reach a consensus, i.e. a
situation where the state vectors of the agents converge to an identical
value, has received significant interest in the last decades. This problem can
be split in two categories: leaderless consensus [1, 3] and leader-follower
consensus [4, 5, 6, 7]. In these works, consensus schemes are based on either
continuous or periodic exchanges of state values by agents within the same
neighbourhood.

Continuous communications are intractable in practice and periodic ones are
therefore preferred. Nevertheless they can be still energy consuming, and
bandwidth problems may arise when dealing with networks composed of large
number of agents.

In event-triggered communication [8], exchange of information between
connected agents is decided by evaluating a Communication Triggering Condition
(CTC) usually defined from a function of the current states and some
information error. Solutions to the consensus problem with event-triggered
communication have been proposed for single and double integrator MAS, see
e.g. [9, 10]. Then, the problem has also been studied for agents with general
linear dynamics, as in [11]. Several extensions have been suggeted to improve
the previous result, e.g. as in [12], resulting in reduced communications.

Regarding nonlinear systems, various articles proposed distributed control
laws and CTCs for MAS with Euler-Lagrange nonlinear dynamics, see e.g. [13,
14, 15]. Distributed event-triggered control has also been investigated for
agents with more general nonlinear dynamics. Solutions for reaching consensus
with event-triggered control in MAS with nonlinear fully-actuated input-affine
dynamics have been proposed in [16, 17, 18], or e.g. in [18] regarding the
finite-time event-triggered consensus problem.

For possibly fully actuated systems, an interesting approach has been
presented in [19] to solve an event-triggered MAS consensus problem with
nonlinear input-affine dynamics. The problem is split into two simpler
problems. First, the consensus problem of a virtual MAS with linear dynamics
with the same graph topology as the real one is solved, considering an event-
triggered mechanism. Then, the state of the virtual system can be used as
reference for the tracking problem of agents of the real MAS with nonlinear
dynamics. Nevertheless, the approach is restricted to agents described by
minimum phase dynamics.

In this paper, we consider the leader-follower consensus problem for nonlinear
input-affine MAS. Contrary to aforementionned works, the dynamics considered
here are neither assumed to be fully-actuated nor assumed to be described by
minimum phase system. Moreover, uncertainty on some parameters is considered
in the system description. Two novel distributed event-triggered control
schemes with associated CTCs are proposed.

The paper is organised as follows. In Section II, the problem is defined. Main
results are provided in Section III. In Section IV the efficiency of the
algorithms is analysed through an illustrative simulation example.

Notations. Given a matrix $M\in{\mathbb{R}}^{n\times m}$, we denote by
$m_{ij}$ the scalar element in row $i$ and column $j$ of $M$. The identity
matrix is denoted by $I$. For any matrix $A$, $A^{\top}$ denotes its transpose
matrix and, for square matrices, ${\texttt{He}}\\{A\\}:=A+A^{\top}$. A matrix
$P$ satisfies $P>0\;(\geq 0)$ if $P$ is symmetric and positive definite (semi-
positive definite). The Kronecker product is denoted by $\otimes$. The largest
and the smallest eigenvalue of $M$ are respectively denoted by
$\lambda_{\max}(M)$ and $\lambda_{\min}(M)$. The sets of positive real numbers
and non null positive real numbers are respectively denoted by
${\mathbb{R}}_{+}$ and ${\mathbb{R}}_{+}^{*}$ We also denote by $C^{1}$, the
set of differentiable functions with continuous derivative.

##  II Problem Statement

###  II-A System description

In this paper, we consider the problem of synchronising a MAS composed by a
leader agent with state $x_{1}$ in ${\mathbb{R}}^{n}$ and $N-1$ identical
followers with states $x_{2},\dots,x_{N}$ also in ${\mathbb{R}}^{n}$. The
leader’s dynamics are described by

$\dot{x}_{1}(t)=f(x_{1}(t),\,\theta),$ (1)

with $\theta\in\Omega\subset{\mathbb{R}}^{p}$ a vector of constant parameters,
possibly unknown, and $f:{\mathbb{R}}^{n}\times\Omega\mapsto{\mathbb{R}}^{n}$
a $C^{1}$ vector field. The followers’ dynamics are described by

$\dot{x}_{i}(t)=f(x_{i}(t),\,\theta)+B\,u_{i}(t),\quad\,i=2,\dots,N,$ (2)

where $x_{i}\in{\mathbb{R}}^{n}$ is the state of the agent $i$,
$u_{i}:t\mapsto{\mathbb{R}}^{m}$ is the control input and $B$ is in
${\mathbb{R}}^{n\times m}$.

The communication graph $\mathcal{G}$ between the agents is described by
$\left\\{\mathcal{V},\,\mathcal{E},A\right\\}$, where
$\mathcal{V}=\\{1,\,\dots\,,N\\}$ denotes the set of vertices,
$\mathcal{E}\subset\\{1,\dots,N\\}^{2}$ denotes the set of edges which
represents the communication capability among agents and
$A=\left(a_{ij}\right)_{(i,j)\in\\{1,\dots,N\\}^{2}}$ is the adjacency matrix
which entries $a_{ij}\geq 0$ represent the weights of the communication
capabilities. We have $a_{ij}=0$ if $(i,j)\notin\mathcal{E}$ and $a_{ij}>0$ if
$(i,j)\in\mathcal{E}$. If $(i,j)\in\mathcal{E}$, agent $j$ is a neighbour of
agent $i$. We introduce the Laplacian matrix of the graph denoted $L$ and
defined as

$l_{ij}=-a_{ij},\quad\text{for $i\neq j$},\hskip
19.91684ptl_{ij}=\sum_{k=1}^{N}a_{ik},\quad\text{for $i=j$}.$ (3)

where $l_{ij}$ is the $(i,\,j)$-th entry of $L$.

Our objective is to design an event-triggered distributed control law in order
to make the MAS asymptotically reach a consensus, i.e. to make the consensus
manifold $\mathcal{D}$ defined by

$\mathcal{D}=\left\\{\mathbf{x}=\text{col}(x_{1},\,\dots,\,x_{N})\in
R^{nN}\,|\,x_{1}=\dots=x_{N}\right\\}\\!,$ (4)

asymptotically stable along the solution of the complete dynamical system
(1)-(2). We denote for all $\mathbf{x}\in{\mathbb{R}}^{nN}$ its Euclidean
distance to the set $\mathcal{D}$ by $\left|\mathbf{x}\right|_{\mathcal{D}}$.

###  II-B Limitation of communications among the MAS

To reduce the communications, the agents can communicate only at some
particular time instants, that will be defined by the CTC. When an agent
communicates, it sends its current state value to all its neighbours. Hence,
for all $j$ in $\mathcal{V}$, there exists a discrete sequence $t_{j,p}$ in
${\mathbb{R}}_{+}$ which is the $p$-th instant of communication from agent $j$
to its neighbours.

###  II-C Structure of the distributed control law

Inspired by [11], to each agent $i$ is attached an estimate of the state of
its neighbours in the communication graph denoted $\hat{x}^{i}_{j}$ for all
$(i,j)$ in $\mathcal{E}$. For all $(i,j)$ in $\mathcal{E}$,

$\displaystyle{{\dot{\hat{x}}}}^{i}_{j}(t)$
$\displaystyle=f({\hat{x}}^{i}_{j}(t),\,\hat{\theta}),\forall
t\in\left[t_{j,p},t_{j,(p+1)}\right),$ (5a)
$\displaystyle{\hat{x}}_{j}^{i}(t_{j,p})$ $\displaystyle=x_{j}(t_{j,p}).$ (5b)

where $\hat{\theta}\in\Omega$ is an a-priori estimate of the vector of unknown
parameters $\theta$, assumed to be available to all the agents. We denote
(skipping the time dependency of the solution)

$\hat{\mathbf{x}}^{i}=\text{col}\\{\hat{x}_{j}^{i}\\},\,\forall\,(i,j)\in\mathcal{E},\quad\mathbf{x}=\text{col}(x_{1},\,\dots,\,x_{N}).$

The distributed control law is then defined as

$u_{i}(t)=\phi_{i}(\hat{\mathbf{x}}^{i}(t)),$ (6)

where $\phi_{i}:{\mathbb{R}}^{nN}\mapsto{\mathbb{R}}^{m}$ is a function in the
form111With a slight abuse of notations this function definition also
incorporates agents $j$ which are not neighbours of $i$ since their associate
weights $l_{ij}$ are $0$.

$\phi_{i}(\mathbf{\hat{x}}^{i})=-\kappa\sum_{j=1}^{N}l_{ij}\,\alpha(\hat{x}_{j}^{i}),$
(7)

with $\kappa\in\mathbb{R}$ and
$\alpha:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, a to-be-designed function.

###### Remark 1

To limit the complexity of the estimators and the amount of exchanged
information, it has been chosen not to use the control input $u_{i}$ in the
estimators (5). This corresponds to the open-loop estimation approach
described in [20].

The following section is dedicated to selection of the communication instants
$t_{j,p}$, the function $\alpha$ and the parameter $\kappa$ in order to obtain
exponential convergence of the trajectories to the consensus manifold
$\mathcal{D}$.

##  III Asymptotic Consensus

###  III-A Additional assumptions

To reach consensus in the MAS one needs to assume connectivity of the
communication graph and a stabilisability property (see for instance [1]).
These assumptions are detailed in this subsection.

####  III-A1 Graph connectivity

###### Assumption 1

The graph $\mathcal{G}$ is connected and undirected.

As a consequence, according to Assumption 1, the Laplacian matrix $L$
associated to the graph $\mathcal{G}$ can be partitioned as (see [21])

$L=\begin{pmatrix}l_{11}&L_{12}\\\ L_{21}&L_{22}\end{pmatrix}\\!,$ (8)

where
$L_{22}=L_{22}^{\top}\in{\mathbb{R}}^{{\left(N-1\right)}\times\left(N-1\right)}$
is a positive definite matrix satisfying:

$L_{22}\geq\mu\,I,$ (9)

with $\mu>0$ and $l_{11}\in{\mathbb{R}}$, $L_{21}$ and $L_{12}^{\top}$ in
${\mathbb{R}}^{N-1}$.

####  III-A2 Stabilisability of individual dynamics

The second assumption is related to a stabilisability property of each agents
dynamics.

###### Assumption 2

There exists a positive definite matrix $P\in\mathbb{R}^{n\times n}$ and
$q\in\mathbb{R}^{*}_{+}$ such that the following condition holds

$\frac{\partial f}{\partial x}(x,\,\theta)^{\top}P+P\,\frac{\partial
f}{\partial x}(x,\,\theta)-\rho\,P\,B\,B^{\top}P\leq-q\,P,$ (10)

with $\rho>0$ and for all $\theta\in\Omega$ and $x\in\mathbb{R}^{n}$.

In the particular case in which $f(x,\theta)=Fx$ for some matrix $F$ (i.e. in
the linear framework), equation (10) is a Riccati equation in the form

$F^{\top}P+P\,F-\rho\,P\,B\,B^{\top}P\leq-q\,P,$

which has been employed in many control design to achieve consensus (see for
instance [11]). Hence, (10) is a nonlinear extension of the Ricatti equation.
It has been employed to achieve consensus for instance in [22] when no
restriction of communication is imposed.

Obtaining a matrix $P$ which satisfies (10) is not an easy task. Indeed, it
depends on the coordinates in which is written the system dynamics. For
instance, in the particular case in which the system is feedback linearizable,
then this assumption is trivially satisfied. Also, in [23], the authors
suggest several methods to write equation (10) as a Linear Matrix Inequality
(LMI) which might be easily checked. It relies on the decomposition of $f$ as
a linear system to which is added an incremental sector bounded nonlinearity.
This type of assumption has also been employed in [24].

###  III-B Asymptotic event-triggered consensus

As in [11], aditionnaly to the estimators (5) of the states of its neighbours,
each agent of the MAS also maintains an estimate of its own state
corresponding to the current estimate employed by its neighbours to compute
their control law. Hence, we introduce for $i=1,\dots,N$,

$\displaystyle{{\dot{\hat{x}}}}^{i}_{i}(t)$
$\displaystyle=f({\hat{x}}^{i}_{i}(t),\,\hat{\theta}),\forall
t\in\left[t_{i,p},t_{i,(p+1)}\right),$ (11a)
$\displaystyle{\hat{x}}_{i}^{i}(t_{i,p})$ $\displaystyle=x_{i}(t_{i,p}).$
(11b)

where $\hat{\theta}\in\Omega$ is the same a priori estimate of the vector of
unknown parameters $\theta$ used in (5).

For $i=1,\dots,N$, let $e_{i}=x_{i}-\hat{x}_{i}^{i}$ be the error between
$x_{i}$ and $\hat{x}_{i}$, and $r_{i}=x_{i}-x_{1}$ the state error between the
agent $i$ and the leader (note that $r_{1}=0$).

Consider the following CTC defining the sequence of communication instants
$t_{i,\,p}$ for all $i=\\{1,...,N\\}$

$\displaystyle t_{i,\,p+1}$
$\displaystyle=\inf\\!\left\\{\\!s\\!>t_{i,\,p}\,|\,\delta_{i}(s)-\sigma_{i}\,w_{i}(s)^{\top}\Theta_{i}\,w_{i}(s)>0\right\\}\\!,$
(12)

with $0<\sigma_{i}<1$,

$w_{i}=\sum_{j=1}^{N}l_{ij}\left(\hat{x}_{j}^{i}-\hat{x}_{i}^{i}\right)\\!,$
(13)

and

$\displaystyle\delta_{i}$
$\displaystyle=e_{i}^{\top}S_{i}\,e_{i}+\left|w_{i}^{\top}R_{i}\,e_{i}\right|,$
(14)

with

$\displaystyle R_{i}$ $\displaystyle=2\,\kappa\,P\,B\,B^{\top}P,$ (15)
$\displaystyle\Theta_{i}$
$\displaystyle=2\,\kappa_{2}\,\varepsilon\left(1-2\,l_{ii}\,b_{i}\right)P\,B\,B^{\top}P,$
(16) $\displaystyle S_{i}$
$\displaystyle=\left[\left(2\,\kappa\,l_{ii}\,b_{i}+\frac{2\,\kappa\,l_{ii}}{b_{i}}+\kappa_{2}\,\varepsilon\left(\frac{4\,l_{ii}}{b_{i}}\right.\right.\right.$
$\displaystyle\hskip
42.67912pt\left.\left.\left.-N\,M_{i}\left(\frac{b_{i}}{2}+\frac{1}{2\,b_{i}}\right)\right)\right)P\,B\,B^{\top}P\right]\\!.$
(17)

where $0<b_{i}<\frac{1}{2\,{l_{ii}}}$, $M_{i}=\sum_{j=1}^{N}{l}_{ij}^{2}$ and
$\varepsilon\leq\frac{1}{\lambda_{\max}(L)}$, with
$\kappa=\kappa_{1}+\kappa_{2}$, $\kappa_{1}$ and $\kappa_{2}$ in
${\mathbb{R}}_{+}$.

Using the following theorem, one can define the distributed event-triggered
control law ensuring exponential asymptotic consensus of the MAS.

###### Theorem 1

Suppose Assumption 1 and 2 hold. Then with the distributed control law
$u_{i}=\phi_{i}(\mathbf{\hat{x}}^{i})$ for $i=2,\dots,N$, defined in (5)-(7)
with $\kappa=\kappa_{1}+\kappa_{2}$, for any $\kappa_{1}>\frac{\rho}{2\,\mu}$
and $\kappa_{2}>0$, $\alpha$ defined as

$\alpha(x)=B^{\top}P\,x,$ (18)

$\rho$ satisfying (10), $\mu$ satisfying (9) and $t_{i,\,p}$ defined according
to the CTC (12), there exist positive constants $k_{1}$ and $k_{2}>0$ such
that for any $\mathbf{x}(0)$ in ${\mathbb{R}}^{Nn}$ and for all $t$ in the
time domain of definition of the solution

$\left|\mathbf{x}(t)\right|_{\mathcal{D}}\leq
k_{1}\exp(-k_{2}\,t)\left|\mathbf{x}(0)\right|_{\mathcal{D}}.$ (19)

where $\mathbf{x}(t)$ denotes the solution of the system (1)-(2).

###### Proof:

The proof of Theorem 1 is provided in Appendix VI-A. ∎

###### Remark 2

One can notice that the computation of the CTC and the control law by agent
$i$ requires the value of $l_{ij}\,\hat{x}_{j}$ for all $j$. When $j$ is a
neighbour of $i$, then $i$ has access to $\hat{x}_{j}$. When $j$ is not a
neighbour of $i$ then $l_{ij}=0$, which implies $l_{ij}\,\hat{x}_{j}=0$.
Therefore the CTC (12) and the control law are fully distributed.

###### Remark 3

It has to be noticed that this theorem does not exclude the existence of Zeno
behaviour which implies that solutions may not be defined for all positive
times. In section III-C, we propose another CTC to exclude Zeno behaviour.
However the asymptotic convergence is lost and only a practical stabilisation
i.e. bounded consensus is obtained.

###  III-C Practical event-triggered Consensus excluding Zeno behaviour

In order to exclude Zeno behaviour, the former CTC (12) is modified by
introducing a small positive constant to guarantee a positive inter event
time. With this CTC, asymptotic consensus will not be ensured anymore, but the
deviation from the consensus can be bounded.

###### Assumption 3

The function $f$ is $k$-Lipschitz uniformly in $\theta$, i.e., for all $x_{i}$
and $x_{j}$ in ${\mathbb{R}}^{n}$

$\left|f(x_{i},\,\theta)-f(x_{j},\,\theta)\right|\leq
k\left|x_{i}-x_{j}\right|\\!.$ (20)

Moreover we assume the following property on $f$ holds

$\left|f(x_{i},\,\hat{\theta})-f(x_{i},\,\theta)\right|\leq\Delta,$ (21)

with $(\theta,\,\hat{\theta})$ in $\Omega^{2}$, for all
$x_{i}\in\mathbb{R}^{n}$, and $\Delta\in{\mathbb{R}}^{+}$.

The following new CTC is introduced to define the sequence of communication
instants $t_{i,\,p}$, for all $i=\\{1,\,\dots,\,N\\}$ as

$\displaystyle
t_{i,\,p+1}\\!=\\!\inf\\!\left\\{\\!s\\!>\\!t_{i,\,p}\,|\,\delta_{i}(s)\\!-\\!\sigma_{i}\,w_{i}(s)^{\\!\top}\Theta_{i}\,w_{i}(s)\\!-\\!\xi\\!>\\!0\right\\}\\!,$
(22)

with $0<\sigma_{i}<1$, $\xi>0$,

$w_{i}=\sum_{j=1}^{N}l_{ij}\left(\hat{x}_{i}^{i}-\hat{x}_{j}^{i}\right),$ (23)

with $\delta_{i}$ defined in (14) and $\Theta_{i}$ defined in (16).

Using the following theorem, one can define the distributed-event triggered
control law ensuring practical consensus of the MAS and absence of Zeno
behaviour.

###### Theorem 2

Suppose Assumptions 1, 2 and 3 hold. Then with the distributed control law
$u_{i}=\phi_{i}(\mathbf{\hat{x}}^{i})$ for $i=2,\dots,N$, defined in (5)-(7)
with $\kappa=\kappa_{1}+\kappa_{2}$, for any $\kappa_{1}>\frac{\rho}{2\,\mu}$
and $\kappa_{2}>0$, $\alpha$ defined in (18), $\rho$ satisfying (10), $\mu$
satisfying (9) and $t_{i,\,p}$ defined according to the CTC (22) then for all
$\mathbf{x}(0)$ in ${\mathbb{R}}^{Nn}$ the solution $\mathbf{x}(t)$ of the
system (2) is defined for all positive time instants and

$\lim_{t\rightarrow\infty}\left|\mathbf{x}(t)\right|_{\mathcal{D}}^{2}\leq\frac{N\,\xi}{q\,\lambda_{\min}(P)},$
(24)

with $q$ respecting (10). Moreover, there is no Zeno behaviour, and the inter
event-time of the solution is bounded by $\tau_{i}$

$\displaystyle\tau_{i}$
$\displaystyle=\frac{1}{k}\log\left(\sqrt{\frac{\xi}{c_{1}}+\frac{c_{2}^{2}}{4\,c_{1}^{2}}}+1-\frac{c_{2}}{2\,c_{1}}\right)\\!,$
(25)

with $k$ the Lipschitz constant of $f$, $c_{1}=\left|S_{i}\right|\nu^{2}$ and
$c_{2}=w_{i_{\max}}\left|R_{i}\right|\nu$, where
$w_{i_{\max}}=\max_{t}\left|w_{i}(t)\right|$ and
$\nu=\frac{\kappa\left|B\,B^{\top}P\right|\left[w_{i_{\max}}+\Delta\right]}{k}.$

###### Proof:

The proof of Theorem 2 is provided in Appendix VI-B ∎

##  IV Simulation examples

Consider a MAS of $N=5$ agents with a communication graph illustrated in
Figure 1.

$1$$2$$3$$5$$4$

Figure 1: Network Structure

The function $f$ and the matrix $B$ defining the dynamics of the agents are
chosen as

$f(z,\,\theta)\\!=\\!\begin{pmatrix}z_{2}+\theta\,\cos(z_{2})\\\
-z_{1}+\theta\cos(z_{2})+\theta^{2}\\!\cos(z_{1})\sin(z_{1})\end{pmatrix}\\!\\!,B=\begin{pmatrix}0\\\
-1\end{pmatrix}\\!,$ (26)

with $z=\left[z_{1}\,z_{2}\right]^{\top}$. For the simulations, the unknown
parameter is set to $\theta=0.5$ and the initial condition of the MAS is

$\mathbf{x}(0)=\begin{pmatrix}(0.95&0.63)^{\top}\\\ (-0.70&-0.73)^{\top}\\\
(-0.33&-0.54)^{\top}\\\ (-0.25&0.02)^{\top}\\\
(0.86&0.01)^{\top}\end{pmatrix}\\!,$

Using (26) with $\theta=0.5$, a matrix $P$ satisfying condition (10) with
constants $\rho=0.02$, $q=1$ is

$P=\begin{pmatrix}5&2\\\ 2&1\end{pmatrix}\\!,$ (27)

The other parameters used either in the CTC or control law are chosen as
$\kappa_{2}=5$, $\varepsilon=\frac{1}{\lambda_{\max}(L)}$,
$b_{i}=(5\,l_{ii})^{-1}$, $\kappa_{1}=0.1$, $\sigma_{1}=0.8$, and for all
$i\in\\{2,\,\dots,\,5\\}$, $\sigma_{i}=0.9$.

A sampling period of $10^{-2}s$ has been chosen for all the numerical
simulations. Different simulations have been performed to illustrate the
resulting performances of the proposed approach in terms of reduction of
communication. Analysis of the robustness to the a priori value $\hat{\theta}$
is also presented as well as a comparison of efficiency of the two proposed
CTCs (12) and (22).

Figure 2 presents the time evolution of the state of each agent of the MAS and
the communication instants, with the CTC (12) using $\hat{\theta}=0.40$, for
which (10) is satisfied with the chosen $P$. The simulation duration $T_{sim}$
is $10\,s$.

Figure 2: Time evolution of the components of state for each agent (top: first
component, middle: second component), and communication instants for each
agent (bottom). Case with the CTC (12), $\theta=0.50$ and $\hat{\theta}=0.40$.

It can be observed that the consensus is reached with limited communications.
The CTC is mostly triggered by the errors due to neglecting the control input
$u_{i}$ in the estimators. Thus, very few communications are triggered by the
leader (agent 1) which is not controlled.

The results presented in Figure 3 are obtained for $\hat{\theta}=0.35$
corresponding to a larger uncertainty but still satisfying (10). The CTC (12)
is also used in this example and other parameters’ values remain unchanged. As
can be seen, consensus is also reached in this case by the agents with non-
periodic communications.

Figure 3: Time evolution of the components of state for each agent (top: first
component, middle: second component), and communication instants for each
agent (bottom). Case with the CTC (12), $\theta=0.50$ and $\hat{\theta}=0.35$.

In both cases, non-periodic communications are obtained, as triggered by the
proposed CTC (12), and convergence to an asymptotic consensus is observed. In
order to compare the resulting performances, we introduce a performance index
$\Gamma$ to qualify the evolution of the consensus error and the reduction of
communications from the CTC. $\Gamma$ is defined as

$\Gamma=\frac{1}{N}\sum_{k=0}^{K}\left[\sum_{i=1}^{N}\left|r_{i}(k)\right|_{2}+\chi\,v_{i}(k)\right]$
(28)

where $K$ is the number of simulated instants, $v_{i}(k)$ is defined as

$\displaystyle 1$ $\displaystyle\text{if the CTC is verified},$ (29a)
$\displaystyle 0$ $\displaystyle\text{otherwise}.$ (29b)

and $\chi$ is a tuning parameter that allows to balance between consensus
accuracy and communication reduction. Comparisons of the performances for the
two values of $\hat{\theta}$ are presented in Table I, for $\chi$ equals to
$0.1$ and for two values of $T_{sim}$, respectively $10$ and $30$ seconds. As
expected, it can be observed that a larger uncertainty ($\hat{\theta}=0.35$)
results in some discrepancy of the performances, however limited. In order to
compare the efficiency of the two CTCs, simulations have been performed with
CTC (22) and the two values of $\hat{\theta}$. The initial condition and the
other parameters remain unchanged. The parameter $\xi$ has been selected equal
to 20. This value has been chosen to illustrate the bounded consensus in the
simulation. Note that better performances in terms of consensus accuracy could
be obtained for smaller values of $\xi$.

Figure 4 presents the corresponding time evolution of the state of each agent
of the MAS and the communication instants.

Figure 4: Time evolution of the components of state for each agent (top: first
component, middle: second component), and communication instants for each
agent (bottom). Case with the CTC (22), $\theta=0.50$ and $\hat{\theta}=0.40$.

As expected, with this CTC and value of $\xi$, the communications are
drastically reduced at the cost of losing asymptotic convergence to the
consensus. Bounded variations around the consensus manifold appear clearly on
the figure. The corresponding values of performance index $\Gamma$ for these
examples are also presented in Table I.

Values of $\Gamma$
---
| CTC (12) | CTC (22)
$T_{sim}$ | $\hat{\theta}=0.40$ | $\hat{\theta}=0.35$ | $\hat{\theta}=0.40$ | $\hat{\theta}=0.35$
10 s | $387$ | $393$ | $343$ | $395$
30 s | $727$ | $764$ | $843$ | $843$
Table I: Comparison of $\Gamma$

From Table I, it can be seen that, for a shorter simulation, the CTC (22) is
better than the CTC (12) according the performance index $\Gamma$. This result
is consistent since the CTC (22) triggers less communications than the CTC
(12). Nevertheless since the asymptotic consensus is not reached with the CTC
(22), for a longer simulation, the CTC (12) proves to be more efficient. In
order to explicit this phenomena, the quantity
$\frac{1}{N}\sum_{k=1}^{K}\sum_{i=1}^{N}\left|r_{i}(k)\right|_{2}$, which
represents the quality of the consensus and the quantity
$\sum_{i=1}^{K}\sum_{k=1}^{N}v_{i}(k)$ which represents the number of
communications in the MAS have been evaluated separately.

Tables II and III present the different values of these quantities for all the
simulated examples. Clearly, the CTC (12) allows to reach the consensus with
more accuracy than the CTC (22) at the cost of increasing the number of
communications.

Values of $\frac{1}{N}\sum_{k=1}^{K}\sum_{i=1}^{N}\left|r_{i}(k)\right|_{2}$
---
| CTC (12) | CTC (22)
$T_{sim}$ | $\hat{\theta}=0.40$ | $\hat{\theta}=0.35$ | $\hat{\theta}=0.40$ | $\hat{\theta}=0.35$
10 s | $311$ | $300$ | $420$ | $393$
30 s | $409$ | $388$ | $829$ | $830$
Table II: Comparison of
$\frac{1}{N}\sum_{k=1}^{K}\sum_{i=1}^{N}\left|r_{i}(k)\right|_{2}$ Values of
$\sum_{k=1}^{K}\sum_{i=1}^{N}v_{i}(k)$
---
| CTC (12) | CTC (22)
$T_{sim}$ | $\hat{\theta}=0.40$ | $\hat{\theta}=0.35$ | $\hat{\theta}=0.40$ | $\hat{\theta}=0.35$
10 s | $3800$ | $4650$ | $250$ | $200$
30 s | $15850$ | $18800$ | $700$ | $650$
Table III: Comparison of number of communications in the MAS

##  V Conclusion

In this paper, a new distributed event-triggered control law is proposed for
the leader-follower consensus problem of a MAS described with uncertain
control-affine nonlinear dynamics. It is based on two novel Communication
Triggering Conditions (CTCs): the first one allowing to achieve an asymptotic
consensus but without any guarantee on Zeno behaviour and the second one that
exclude Zeno behaviour but results only in a bounded consensus. Simulations
have been presented to compare the performances of the two approaches.
Robustness to uncertainty is also illustrated. Future work will consider
agents with more generic nonlinear dynamics and state perturbations and MAS
composed by heterogeneous agents.

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##  VI Appendix

###  VI-A Proof of the Theorem 1

###### Proof:

Since (5) and (11) follow the same dynamical system, it yields for all $(j,i)$
in $\mathcal{E}$

$\hat{x}_{j}^{i}(t)=\hat{x}_{j}^{j}(t).$

For $i=2,\dots,N$, the $r_{i}$’s dynamics are given by:

$\dot{r_{i}}=f(r_{i}+x_{1},\,\theta)-f(x_{1},\,\theta)-\kappa\sum_{j=1}^{N}l_{ij}\,B\,\alpha(x_{1}+r_{j}-e_{j}).$
(30)

Using the fact that for $i=2,\dots,N$:

$\sum_{j=1}^{N}l_{ij}\,B\,\alpha(x_{1})=B\,\alpha(x_{1})\sum_{j=1}^{N}l_{ij}=0,$
(31)

we can write:

$\displaystyle\dot{r_{i}}$
$\displaystyle=f(r_{i}+x_{1},\,\theta)-f(x_{1},\,\theta)$ $\displaystyle\hskip
42.67912pt-\kappa\sum_{j=1}^{N}l_{ij}\,B\left(\alpha(x_{1}+r_{j}-e_{j})-\alpha(x_{1})\right).$
(32)

Consider the following function which is defined on the time domain of
definition of the solution,

$V_{i}(t)=r_{i}(t)^{\top}P\,r_{i}(t)\ ,\ i=2,\dots,N.$ (33)

Using the relation (32), the time-derivative of $(V_{i})^{\prime}s$ can be
computed as (forgetting the time dependency of the state variables):

$\displaystyle\dot{V}_{i}$
$\displaystyle=2\,r_{i}^{\top}P\,\left(f(r_{i}+x_{1},\,\theta)-f(x_{1},\,\theta)\right)$
$\displaystyle\hskip
36.98866pt-2\,\kappa\,r_{i}^{\top}P\,B\sum_{j=1}^{N}l_{ij}\left(\alpha(r_{j}+x_{1})-\alpha(x_{1})\right)\\!.$
(34)

For fixed $x_{1},r_{i},\theta,e_{i}$, let us define
$F_{i}(s):\mathbb{R}\rightarrow\mathbb{R}^{n}$ and
$A_{i}(s):\mathbb{R}\rightarrow\mathbb{R}^{n}$ as:

$\displaystyle F_{i}(s)$ $\displaystyle=f(x_{1}+r_{i}\,s,\,\theta),$ (35)
$\displaystyle A_{i}(s)$
$\displaystyle=\alpha(x_{1}+\left(r_{i}-e_{i}\right)s).$ (36)

Note that
$\left(f_{i}(r_{i}+x_{1},\,\theta)-f_{i}(x_{1},\,\theta)\right)=\int_{0}^{1}\frac{d\,F_{i}(s)}{ds}ds$.
Hence

$\displaystyle\dot{V}_{i}$
$\displaystyle=\\!\int_{0}^{1}\\!\\!r_{i}^{\top}P\,\frac{dF_{i}(s)}{ds}\\!+\\!\frac{dF_{i}(s)}{ds}^{\\!\top}\\!\\!\\!P\,r_{i}\\!-\\!2\,\kappa\,r_{i}^{\top}\\!PB\\!\sum_{j=1}^{N}\\!l_{ij}\frac{dA_{j}(s)}{ds}ds.$

One can notice that:

$\displaystyle\frac{d\,F_{i}(s)}{ds}$ $\displaystyle=\frac{\partial
f}{\partial x_{i}}(x_{1}+r_{i}\,s,\,\theta)\,r_{i},$ (37)
$\displaystyle\frac{d\,A_{i}(s)}{ds}$
$\displaystyle=\frac{\partial\alpha}{\partial
x_{i}}(x_{1}+\left(r_{i}-e_{i}\right)s)\,\left(r_{i}-e_{i}\right).$ (38)

Therefore, for $i=2,\dots,N$,

$\displaystyle\dot{V}_{i}=$
$\displaystyle\int_{0}^{1}\\!r_{i}^{\top}\\!\\!\left(\\!P\,\frac{\partial
f}{\partial x_{i}}(x_{1}+r_{i}\,s,\,\theta)+\frac{\partial f}{\partial
x_{i}}(x_{1}+r_{i}\,s,\,\theta)^{\top}\\!P\\!\right)r_{i}\,ds$
$\displaystyle-2\,\kappa\,r_{i}^{\top}P\,B\sum_{j=1}^{N}l_{ij}B^{\top}P\,\left(r_{j}-e_{j}\right).$

Consider $v$ given as
$v=\left[(B^{\top}P\,r_{2})^{\top},...,(B^{\top}P\,r_{N})^{\top}\right]^{\top}$.
Define $V(t)=\sum_{i=2}^{N}V_{i}(t)$.

$\displaystyle\dot{V}=\int_{0}^{1}\sum_{i=2}^{N}r_{i}^{\top}{\texttt{He}}\left(P\,\frac{\partial
f}{\partial x_{i}}(x_{1}+r_{i}\,s,\,\theta)\right)\,r_{i}\,ds$
$\displaystyle+2\kappa\sum_{i=2}^{N}\sum_{j=1}^{N}l_{ij}\,r_{i}^{\top}PBB^{\top}\\!P\,e_{j}-2\kappa
v^{\top}(L_{22}\otimes I_{n})\,v.$ (39)

Using (9) and the condition (10) and selecting any
$\kappa_{1}\geq\frac{\rho}{2\,\mu}$, we have:

$\displaystyle\dot{V}\leq$
$\displaystyle\sum_{i=2}^{N}-q\,r_{i}^{\top}Pr_{i}+2\,\kappa\sum_{i=2}^{N}\sum_{j=1}^{N}l_{ij}\,r_{i}^{\top}P\,B\,B^{\top}P\,e_{j}$
$\displaystyle-2\,\kappa_{2}\sum_{i=2}^{N}\sum_{j=1}^{N}l_{ij}\,r_{i}^{\top}\,P\,B\,B^{\top}P\,r_{j}.$

We denote $\hat{x}=\text{col}(\hat{x}_{1},\,\dots,\,\hat{x}_{N})$,
$r=\text{col}(r_{1},\,\dots,\,r_{N})$ and
$e=\text{col}(e_{1},\,\dots,\,e_{N})$. Since $r_{1}=0$, one has:

$\displaystyle\sum_{i=2}^{N}\sum_{j=1}^{N}l_{ij}\,$ $\displaystyle
r_{i}^{\top}\,P\,B\,B^{\top}P\,r_{j}=r^{\top}\\!\left(L\otimes\left(P\,B\,B^{\top}P\right)\right)r,$
$\displaystyle\sum_{i=2}^{N}\sum_{j=1}^{N}l_{ij}\,$ $\displaystyle
r_{i}^{\top}\,P\,B\,B^{\top}P\,e_{j}=r^{\top}\\!\left(L\otimes\left(P\,B\,B^{\top}P\right)\right)e.$

Define $\varepsilon\leq\frac{1}{\lambda_{\max}(L)}$. Since $L$ is a symmetric
semi-definite postive matrix, one has

$-L\leq-\varepsilon\,L^{2}\leq-\varepsilon\,L^{\top}L.$ (40)

Moreover $P\,B\,B^{\top}P$ is a semi-definite positive matrix, so

$-L\otimes\left(P\,B\,B^{\top}P\right)\leq-\varepsilon\,\left(L^{\top}L\right)\otimes\left(P\,B\,B^{\top}P\right)\\!.$
(41)

Thus:

$\displaystyle-\\!\\!\sum_{i=2}^{N}\sum_{j=1}^{N}l_{ij}\,r_{i}^{\top}\\!PBB^{\top}\\!P\,r_{j}\leq-\varepsilon\,r^{\top}\\!\\!\left(\\!\left(L^{\top}\\!L\right)\otimes\left(PBB^{\top}\\!P\right)\\!\right)r,$
$\displaystyle\,\leq-\varepsilon\\!\sum_{i=1}^{N}\\!\left[\sum_{j=1}^{N}l_{ij}(r_{j}-r_{i})^{\top}\\!PBB^{\top}\\!P\sum_{k=1}^{N}l_{ik}(r_{k}-r_{i})\right]\\!\\!.$
(42)

One can notice that

$\displaystyle\left(L\otimes I_{N}\right)r$
$\displaystyle=\sum_{j=1}^{N}l_{ij}\,(r_{j}-r_{i}),$
$\displaystyle=\sum_{j=1}^{N}l_{ij}\left[(\hat{x}_{j}-\hat{x}_{i})+(e_{j}-e_{i})\right],$
$\displaystyle=\left(L\otimes I_{n}\right)\hat{x}+\left(L\otimes
I_{n}\right)e.$ (43)

Since $L$ is symmetric and $\sum^{N}_{j=1}l_{ij}=0$, we have

$\displaystyle r^{\top}\\!\\!$
$\displaystyle\left(L\\!\otimes\\!\left(PBB^{\top}\\!P\right)\\!\right)e=\sum_{i=1}^{N}\sum_{j=1}^{N}l_{ij}\,r_{i}^{\top}P\,B\,B^{\top}P\,e_{j},$
$\displaystyle\hskip
42.67912pt=\sum_{j=1}^{N}\sum_{i=1}^{N}l_{ji}\,r_{i}^{\top}P\,B\,B^{\top}P\,e_{j},$
$\displaystyle\hskip
42.67912pt=\sum_{i=1}^{N}\sum_{j=1}^{N}l_{ij}\,r_{j}^{\top}P\,B\,B^{\top}P\,e_{i},$
$\displaystyle\hskip
42.67912pt=\sum_{i=1}^{N}\sum_{j=1}^{N}l_{ij}\,(r_{j}-r_{i})^{\top}P\,B\,B^{\top}P\,e_{i},$
$\displaystyle\hskip 42.67912pt=\left(\left(L\otimes
I_{n}\right)r\right)^{\top}\\!\left(I_{N}\\!\otimes\\!\left(PBB^{\top}\\!P\right)\right)e.$
(44)

Therefore, using (42), (43) and (44), we have

$\displaystyle\dot{V}\leq\sum_{i=1}^{N}\left[-q\,r_{i}^{\top}P\,r_{i}+2\,\kappa\,\sum_{j=1}^{N}l_{ij}\left(\hat{x}_{j}-\hat{x}_{i}\right)^{\top}\\!PBB^{\top}\\!P\,e_{i}\right.$
$\displaystyle+2\,\kappa\\!\sum_{j=1}^{N}\\!l_{ij}\,e_{j}^{\top}PBB^{\top}\\!\\!P\,e_{i}-2\,\kappa_{2}\,\varepsilon\\!\sum_{j=1}^{N}\\!l_{ij}\,e_{j}^{\top}PBB^{\top}\\!\\!P\\!\sum_{k=1}^{N}\\!l_{ik}\,e_{k}$
$\displaystyle-2\,\kappa_{2}\,\varepsilon\left(\sum_{j=1}^{N}l_{ij}\left(\hat{x}_{j}-\hat{x}_{i}\right)\right)^{\\!\top}\\!\\!\\!PBB^{\top}\\!P\left(\sum_{k=1}^{N}l_{ik}\left(\hat{x}_{k}-\hat{x}_{i}\right)\right)$
$\displaystyle\left.-4\,\kappa_{2}\,\varepsilon\left(\sum_{j=1}^{N}l_{ij}\left(\hat{x}_{j}-\hat{x}_{i}\right)\right)^{\\!\top}\\!\\!\\!PBB^{\top}\\!P\sum_{k=1}^{N}l_{ik}\,e_{k}\right]\\!.$
(45)

Since $L$ is symmetric, we have

$\displaystyle\sum_{i=1}^{N}\sum_{j=1}^{N}\left|l_{ij}\right|e_{j}^{\top}P\,B\,B^{\top}P\,e_{j}=\sum_{i=1}^{N}2\,l_{ii}\,e_{i}^{\top}P\,B\,B^{\top}P\,e_{i}.$
(46)

Define $w_{i}=\sum_{j=1}^{N}l_{ij}\left(\hat{x}_{j}-\hat{x}_{i}\right)$. Note
that $w_{1}\neq 0$. Thus using the inequality
$\left|x^{\top}y\right|\leq\frac{b_{i}}{2}x^{\top}x+\frac{1}{2\,b_{i}}y^{\top}y$
with $b_{i}>0$, and (46), we have

$\displaystyle\dot{V}\leq\sum_{i=1}^{N}\\!\left[\vphantom{\sum_{j}^{N}}\left|2\,\kappa\,w_{i}^{\top}PBB^{\top}\\!P\,e_{i}\right|-2\,\kappa_{2}\,\varepsilon\,w_{i}^{\top}PBB^{\top}\\!P\,w_{i}\right.$
$\displaystyle+\kappa\left(2\,l_{ii}\,b_{i}\,e_{i}^{\top}P\,B\,B^{\top}P\,e_{i}+\sum_{j=1}^{N}\frac{\left|l_{ij}\right|}{b_{i}}\,e_{i}^{\top}P\,B\,B^{\top}\\!P\,e_{i}\right)$
$\displaystyle+2\,\kappa_{2}\,\varepsilon\\!\left(2\,l_{ii}\,b_{i}\,w_{i}^{\top}PBB^{\top}\\!P\,w_{i}+\sum_{j=1}^{N}\frac{\left|l_{ij}\right|}{b_{i}}\,e_{i}^{\top}PBB^{\top}\\!P\,e_{i}\right)$
$\displaystyle\left.+\kappa_{2}\,\varepsilon\,N\\!\left(\frac{b_{i}}{2}+\frac{1}{2\,b_{i}}\right)\\!\sum_{j=1}^{N}\\!l_{ij}^{\,2}\,e_{i}^{\top}PBB^{\top}\\!\\!P\,e_{i}-q\,r_{i}^{\top}P\,r_{i}\right]\\!\\!.$
(47)

Define $M_{i}=\sum_{j=1}^{N}{l}_{ij}^{\,2}$. Therefore, we obtain

$\displaystyle\dot{V}$
$\displaystyle\leq\sum_{i=1}^{N}\left[-q\,r_{i}^{\top}P\,r_{i}+\left|w_{i}^{\top}R_{i}\,e_{i}\right|-w_{i}^{\top}\Theta_{i}\,w_{i}+e_{i}^{\top}S_{i}\,e_{i}\right]\\!,$

with

$\displaystyle R_{i}$ $\displaystyle=2\,\kappa\,P\,B\,B^{\top}P,$ (48)
$\displaystyle\Theta_{i}$
$\displaystyle=2\,\kappa_{2}\,\varepsilon\left(1-2\,l_{ii}\,b_{i}\right)P\,B\,B^{\top}P,$
(49) $\displaystyle S_{i}$
$\displaystyle=\left[\left(2\,\kappa\,l_{ii}\,b_{i}+\frac{2\,\kappa\,l_{ii}}{b_{i}}+\kappa_{2}\,\varepsilon\left(\frac{4\,l_{ii}}{b_{i}}\right.\right.\right.$
$\displaystyle\hskip
42.67912pt\left.\left.\left.-N\,M_{i}\left(\frac{b_{i}}{2}+\frac{1}{2\,b_{i}}\right)\right)\right)P\,B\,B^{\top}\\!P\right]\\!.$
(50)

and if we choose $0\leq b_{i}\leq\frac{1}{2\,l_{ii}}$, then $\Theta_{i}\geq
0$. Define
$\delta_{i}=e_{i}^{\top}S_{i}\,e_{i}+\left|w_{i}^{\top}R_{i}\,e_{i}\right|$.
Therefore

$\displaystyle\dot{V}$
$\displaystyle\leq\sum_{i=1}^{N}\left[\vphantom{\frac{2}{b_{i}}}-q\,r_{i}^{\top}P\,r_{i}-w_{i}^{\top}\Theta_{i}\,w_{i}+\delta_{i}\right]\\!.$
(51)

If the CTC (12) is respected then the error is reset to zero, that is,
$e_{i}(t_{i,\,k})=0\implies\delta_{i}(t_{i,\,k})=0$. It yields,

$\displaystyle\delta_{i}\leq\sigma_{i}\,w_{i}^{\top}\Theta_{i}\,w_{i}.$ (52)

Thus

$\dot{V}\leq\sum_{i=1}^{N}\left[\vphantom{\frac{2}{b_{i}}}-q\,r_{i}^{\top}P\,r_{i}+\left(\sigma_{i}-1\right)w_{i}^{\top}\Theta_{i}\,w_{i}\right]\\!.$
(53)

Since $\sigma_{i}-1<0$, and $\Theta_{i}\geq 0,$, then
$\left(\sigma_{i}-1\right)w_{i}^{\top}\Theta_{i}\,w_{i}\leq 0$. Finally we
have

$\dot{V}\leq-q\,V.$ (54)

Hence, on the time domain of existence of the solution, one gets,

$V(t)\leq\exp(-qt)V(0)\ .$

The matrix $P$ being positive definite, the result may be easily obtained. ∎

###  VI-B Proof of the Theorem 2

###### Proof:

Starting from (53), we have

$\displaystyle\dot{V}$
$\displaystyle\leq\sum_{i=1}^{N}\left[\vphantom{\frac{2}{b_{i}}}-q\,r_{i}^{\top}P\,r_{i}-w_{i}^{\top}\Theta_{i}\,w_{i}+\delta_{i}\right]\\!.$

According to the CTC (22),
$\delta_{i}<\sigma_{i}\,w_{i}^{\top}\Theta\,w_{i}+\xi$. Hence, one has

$\displaystyle\dot{V}$
$\displaystyle\leq\sum_{i=1}^{N}\left[\vphantom{\frac{2}{b_{i}}}-q\,r_{i}^{\top}P\,r_{i}-\left(1-\sigma_{i}\right)w_{i}^{\top}\Theta_{i}\,w_{i}+\xi\right]\\!,$
$\displaystyle\dot{V}$ $\displaystyle\leq N\,\xi-q\,V.$ (55)

By integrating (55), we obtained

$\displaystyle V(t)$ $\displaystyle\leq
e^{-q\,t}\,V(0)+N\,\xi\int_{0}^{t}e^{-q\,\left(t-\tau\right)}d\tau,$
$\displaystyle\leq\left(V(0)-\frac{N\,\xi}{q\,}\right)e^{-q\,t}+\frac{N\,\xi}{q}.$
(56)

We recall that $V=\sum_{i=2}^{N}r_{i}^{\top}P\,r_{i}$, thus, as $P$ is
definite positive, we have

$\displaystyle\lambda_{\min}(P)\left|r_{i}(t)\right|^{2}$ $\displaystyle\leq
r_{i}(t)^{\top}P\,r_{i}(t),$
$\displaystyle\leq\left(V(0)-\frac{N\,\xi}{q}\right)e^{-q\,t}+\frac{N\,\xi}{q}.$

Therefore, we have
$\lim_{t\rightarrow\infty}\left|r_{i}(t)\right|^{2}\leq\frac{\left(N-1\right)\xi}{q\,\lambda_{\min}(P)}$.
Note that, we have

$\displaystyle\left|\mathbf{x}\right|_{\mathcal{D}}^{2}=\min_{z\in{\mathbb{R}}^{n}}\sum_{i=1}^{N}\left|z-x_{i}\right|^{2}\leq\sum_{i=1}^{N}\left|x_{1}-x_{i}\right|^{2}=\left|r\right|^{2}$
(57)

Thus
$\lim_{t\rightarrow\infty}\left|\mathbf{x}(t)\right|_{\mathcal{D}}^{2}\leq\frac{N\,\xi}{q\,\lambda_{\min}(P)}$.
In this second part, we want to show that the inter event time between two
communications can be bounded. Consider the error’s dynamics

$\displaystyle\frac{d}{dt}\left|e_{i}\right|$
$\displaystyle=\left(e_{i}^{\top}e_{i}\right)^{-\frac{1}{2}}e_{i}^{\top}\dot{e}_{i},$
$\displaystyle=\left(e_{i}^{\top}e_{i}\right)^{-\frac{1}{2}}e_{i}^{\top}\\!\\!\left(f(x_{i},\,\theta)-\kappa\,BB^{\top}\\!P\,w_{i}-f(\hat{x}_{i},\,\hat{\theta})\right),$
$\displaystyle\leq\left(e_{i}^{\top}e_{i}\right)^{-\frac{1}{2}}\left|e_{i}\right|\left|f(x_{i},\,\theta)-f(\hat{x}_{i},\,\theta)\right|$
$\displaystyle\hskip
56.9055pt+\left(e_{i}^{\top}e_{i}\right)^{-\frac{1}{2}}\left|e_{i}\right|\left(\left|\kappa\,B\,B^{\top}P\,w_{i}\right|+\Delta\right)\\!,$
$\displaystyle\leq
k\left|e_{i}\right|+\kappa\,|B\,B^{\top}P|\,(w_{i_{\max}}+\Delta).$ (58)

where $w_{i_{\max}}=\max_{t}\left|w_{i}(t)\right|$ and $k$ the Lipschitz
constant of $f$. Define $w=\text{col}(w_{1},\dots,\,w_{N})$. By definition and
using (43) we have $w=\left(L\otimes I\right)\left(r-e\right)$. According to
the first part of the proof, we show that $V$ is bounded so as
$\left|r\right|$, thus according to the CTC (22), $\left|e\right|$ is also
bounded. To conclude $w_{i_{\max}}$ exists and it is finite. By integrating
(58) with $\left|e_{i}(0)\right|=0$, one obtains

$\left|e_{i}\right|\leq\frac{\kappa\left|B\,B^{\top}P\right|\left(w_{i_{\max}}+\Delta\right)}{k}\left[e^{k\,t}-1\right]\\!.$
(59)

Define
$\nu=\frac{\kappa\left|BB^{\top}\\!P\right|\left(w_{i_{\max}}+\Delta\right)}{k}$.
Thus we can bound the evolution of $\delta_{i}$ by

$\displaystyle\delta_{i}$
$\displaystyle\leq\left|S_{i}\right|\nu^{2}\left[e^{k\,t}-1\right]^{2}+w_{i_{\max}}\left|R_{i}\right|\nu\left[e^{k\,t}-1\right]\\!,$
$\displaystyle\leq
c_{1}\,e^{2\,k\,t}-\left(2\,c_{1}-c_{2}\right)\,e^{k\,t}+c_{1}-c_{2},$ (60)

with $c_{1}=\left|S_{i}\right|\nu^{2}$ and
$c_{2}=w_{i_{\max}}\left|R_{i}\right|\nu$.

One can notice that the time it takes for $\bar{\delta}_{i}$ to grow from $0$
to $w_{i}^{\top}\Theta_{i}\,w_{i}+\xi$ is not smaller that the time need for
$\bar{\delta}_{i}$ to grow from $0$ to $\xi$. Define by $\tau_{i}$ the lower
bound between two events. It can be obtained by solving the following equation

$\displaystyle\xi$
$\displaystyle=c_{1}\,e^{2\,k\,\tau_{i}}-\left(2\,c_{1}-c_{2}\right)\,e^{k\,\tau_{i}}+c_{1}-c_{2},$
$\displaystyle\xi$
$\displaystyle=\left(\sqrt{c_{1}}\,e^{k\,\tau_{i}}-\left(\frac{1}{2\,\sqrt{c_{1}}}\left(2\,c_{1}-c_{2}\right)\right)\right)^{2}+c_{1}-c_{2}$
$\displaystyle\quad-\left(\frac{1}{2\,\sqrt{c_{1}}}\left(2\,c_{1}-c_{2}\right)\right)^{2}\\!\\!.$
(61)

One can notice that

$c_{1}-c_{2}-\left(\frac{1}{2\,\sqrt{c_{1}}}\left(2\,c_{1}-c_{2}\right)\right)^{2}=-\frac{c_{2}^{2}}{4\,c_{1}}.$
(62)

Thus, we have

$\displaystyle e^{k\,\tau_{i}}$
$\displaystyle=\left(\sqrt{\frac{\xi}{c_{1}}+\frac{c_{2}^{2}}{4\,c_{1}^{2}}}+\frac{1}{2\,c_{1}}\left(2\,c_{1}-c_{2}\right)\right)\\!.$
(63)

This implies

$\displaystyle\tau_{i}$
$\displaystyle=\frac{1}{k}\log\left(\sqrt{\frac{\xi}{c_{1}}+\frac{c_{2}^{2}}{4\,c_{1}^{2}}}+1-\frac{c_{2}}{2\,c_{1}}\right).$
(64)

Since
$\sqrt{\frac{\xi}{c_{1}}+\frac{c_{2}^{2}}{4\,c_{1}^{2}}}-\frac{c_{2}}{2\,c_{1}}>0$,
then $\tau_{i}>0$. ∎