# Formation Control of Nonlinear Multi-Agent
Systems Using Three-Layer Neural Networks

Kiarash Aryankia${}^{\href https://orcid.org/0000-0002-4751-3925}$, and Rastko
R. Selmic${}^{\href https://orcid.org/0000-0001-9345-8077}$ This work was
supported by the Natural Sciences and Engineering Research Council of Canada
(NSERC) Discovery Grant #RGPIN-2018-05093. (Corresponding Author: Kiarash
Aryankia). The authors are with the Department of Electrical and Computer
Engineering, Concordia University, Montreal, QC, Canada (email:
<EMAIL_ADDRESS>rastko.selmic@concordia.ca).

###### Abstract

This paper considers a leader-following formation control problem for
heterogeneous, second-order, uncertain, input-affine, nonlinear multi-agent
systems modeled by a directed graph. A tunable, three-layer neural network
(NN) is proposed with an input layer, two hidden layers, and an output layer
to approximate an unknown nonlinearity. Unlike commonly used trial and error
efforts to select the number of neurons in a conventional NN, in this case an
a priori knowledge allows one to set up the number of neurons in each layer.
The NN weights tuning laws are derived using the Lyapunov theory. The leader-
following and formation control problems are addressed by a robust integral of
the sign of the error (RISE) feedback and a NN-based control. The RISE
feedback term compensates for unknown leader dynamics and the unknown, bounded
disturbance in the agent error dynamics. The NN-based term compensates for the
unknown nonlinearity in the dynamics of multi-agent systems, and semi-global
asymptotic tracking results are rigorously proven using the Lyapunov stability
theory. The results of the paper are compared with two previous results to
evaluate the efficiency and performance of the proposed method.

###### Index Terms:

Formation control, Multi-layer neural networks, Nonlinear multi-agent systems,
Second-order systems.

##  I Introduction

Mutli-agent systems have received significant attention from researchers and
have gained more attraction due to their various applications. One of the
significant applications of multi-agent systems is the formation control which
has been inspired by nature. In [1], different categories of formation control
problems were reviewed. The displacement control, one of the main categories
of formation control problems, includes leader-following consensus control
problem (also known as distributed cooperative tracking control problem) as a
particular case that was widely studied in the literature, e.g., [2, 3, 4, 5].
In the formation control problem of multi-agent systems, it is common to
consider single- and double-integrator dynamics [6, 4, 7] or nonlinear
dynamics [3, 2, 8, 9] to describe the dynamics of multi-agent systems. Most of
the literature that considers an unknown nonlinear dynamics, utilizes NNs or
fuzzy logic controllers to approximate the uncertain nonlinearity in the
system dynamics. There are two main reasons for this: first, the multi-layer
NNs and fuzzy logic controllers are universal approximators that can
approximate any continuous and smooth function over a compact set at a desired
accuracy [10, 3]; and second, a precise mathematical model for the dynamics of
a multi-agent system is not required, which eliminates an extra effort to
acquire an accurate system dynamics [11].

The literature on formation control mainly uses one-layer NNs, e.g., radial
basis function neural networks (RBFNNs) [12, 2] or two-layer NNs [8], to
compensate for the unknown nonlinearity (usually caused by uncertainty) of
system dynamics. These studies commonly use the universal approximation
property where the RBFNNs and the Gaussian activation functions are utilized.
In RBFNN, hidden layer neurons are usually uniformly distributed on a regular
lattice or they are randomly initialized. In [13], authors constructed an
RBFNN where a minimum distance of any two centers of the bell-shaped
activation functions is the same as the width of the activation function. An
RBFNN can approximate a nonlinearity if the input variables of the activation
function are located in certain neighborhoods of the RBF networks centers
[13]. Thus, one can select a large number of neurons (not a priori known) and
distribute activation functions over a regular lattice uniformly to cover the
compact set. A one-layer NN with a tunable, hidden layer, in the form of
linear-in-the-parameter can approximate a nonlinearity in the dynamics of a
nonlinear system at a desired accuracy. Still, the exact number of neurons for
control systems design is not clearly established. Although the self-
organizing receptive field method was introduced in [14] to optimize the
receptive fields locations in RBFNN, these results have not been used in the
formation control problem to approximate the unknown nonlinearity.

These limitations and restrictions motivated us to propose the three-layer NN
with two hidden layers, where the numbers of hidden neurons are clearly given
(see Remark 6). Here we use the sigmoid activation function.

To address the number of neurons in a NN, a self-structuring NN was proposed
in [11], whereby a neuron divides into two neurons by satisfying a specific
condition, but the output of these two neurons remains identical to the output
of a neuron without division. This work was extended to a nonlinear multi-
agent system [8], where authors designed an observer to estimate states of
leader for each agent and addressed cooperative tracking control problem of a
leader. Compared with existing results, we use two hidden layers instead of
one [11, 8, 15, 16, 17], and we developed tuning laws for all three layers.
The other novelty is that the number of neurons is fixed.

To avoid the chattering effect of the sliding mode controllers, the RISE
feedback controller has gained significant attention. Although a RISE feedback
control can potentially exceed the actuators limit due to the large integral
gain, it has been shown that it can be implementable in real-time
applications, e.g., [18]. The RISE feedback has been used in [19], to
compensate for a nonlinearity in a class of higher-order, multi-input multi-
output, nonlinear systems. These results were extended in [15], where authors
used the NN-based controller to address the tracking problem of an uncertain
system. In [15], the authors used the discontinuous projection algorithm,
which requires an a priori knowledge of the convex set and maximum and minimum
of NN’s weights [20]. In a similar study, [21], authors used two-layer NN
(input, hidden, and output layers) where only the weights between two outer
layers can be tuned. The tuning laws in [21] are derived from Lyapunov
stability theory. These results further extended in [16, 18], where the
authors used a two-layer NN, and a discontinuous projection algorithm has
tuned the NN weights matrices between any two consecutive layers.

In comparison to [19, 15, 20, 21, 16, 18], the novelty of our work is that we
utilize a three-layer NN comprised of an input layer, two hidden layers, and
an output layer. The NN tuning laws are derived using the Lyapunov theory.
Moreover, all NN weights matrices are tunable, and the tuning law for each of
them has been established.

Several of the NN-based formation control results that address the
displacement-based, leader-following problem established only uniform ultimate
bounded stability, due to NN approximation error and lack of communication
from leader to all agents, e.g., [22, 8, 23, 24, 3, 5]. We propose to use a
RISE feedback controller with two hidden layers NN which achieves semi-global,
asymptotic stability for the leader-following formation control for
heterogeneous, second-order, uncertain, input-affine, nonlinear multi-agent
systems.

The main contributions of the paper are as follows:

  1. 1.

We consider the leader to be connected to at least one of the other agents.
Unlike [17, 15, 25, 21, 16, 18], where the desired trajectory signals
(information from the leader) are fed into the first layer of the NN, the
novelty here is to construct the NN without these signals. Thus, effects of
the leader dynamics and unknown disturbances in the tracking error of each
agent are ameliorated by the RISE feedback controller.

  2. 2.

We determine the number of neurons in each layer and derived the tuning laws
for the NN weights matrices. Although the literature considered one- or two-
layer NN [2, 5, 17, 26, 24, 16, 18], we developed our results for a three-
layer NN.

  3. 3.

The semi-global, asymptotic, leader-following performance is achieved, which
distinguishes our work from [3, 23, 2, 5].

This paper is organized as follows. In Section II, the basic graph theory,
problem statement, and NN design with two hidden layers are given. In Section
III, the control design of RISE feedback and NN adaptive tuning laws for a
second-order, multi-agent system with disturbance and unknown nonlinearity in
the system dynamics are given, where the stability of the proposed method is
proven using the Lyapunov stability theory. In Section IV, the numerical
simulation results are presented to substantiate the theoretical results.
Then, results are compared with two different methods to illustrate the
performance of the proposed method. Finally, the conclusion is given in
Section V.

##  II Problem Formulation and Preliminaries

###  II-A Notations

The notations that are used in the paper are given here. Let
${\mathbb{R}^{n}}$ indicate $n$-dimensional Euclidean space and
$\mathbb{R}^{n\times m}$ denote the set of $n\times m$ real matrices. Let
${I}_{n}$ denote $n\times n$ identity matrix and
${\mathbf{1}}_{N}=[1,...,1]^{T}$ is an all-one vector of size $N$. With
$diag(a_{1},a_{2},...)$, we denote a block diagonal matrix with diagonal
elements of $a_{i}$. The notation vec(.) represents a stacked vector of a
matrix column-wise. For a square matrix $A$, $tr(A)$ is the sum of diagonal
elements of matrix $A$. The Kronecker product is represented by $\otimes$.
With $A^{T}$, we indicate the transpose of matrix A. For a matrix $A$, we
denote the minimum and maximum singular values by $\underline{\sigma}(A)$ and
$\bar{\sigma}(A)$, respectively. By $||.||$ we denote the 2-norm of a vector
or matrix and the Frobenius norm for an arbitrary matrix $A$ is defined as
$||A||_{F}=\sqrt{tr(A^{T}A)}$. For any vector $x\in\mathbb{R}^{n}$, the
Manhattan norm is $||x||_{1}=\sum_{i=1}^{n}|x_{i}|$. With a superscript ′ we
denote the partial derivative of a function with respect to the given
variable. The signum function is denoted by sgn(.). The notation
$\underset{\mu(H)=0}{\cap}$ denotes the intersection over all sets $H$ of
Lebesgue measure zero, and $\overline{co}$ denotes the convex closure.

###  II-B Graph Theory

We model interactions among agents by a directed graph $G=(V,E)$, where
$V=\\{v_{1},...,v_{N}\\}$ is a non-empty set of $N$ vertices and $E\subset
V\times V$ is a set of edges. We assume that the graph is simple, i.e., no
repeated edges and no self-loops. Adjacency, or connectivity matrix is
$A=[a_{ij}]$ with $a_{ij}>0$, if $(v_{j},v_{i})\in E$, and $a_{ij}=0$,
otherwise. Note that we consider $a_{ii}=0$.

###### Definition 1

The neighborhood $N_{i}$ of the vertex $v_{i}$ is a set of all the edges
adjacent to $v_{i}$

$\begin{array}[]{lr}N_{i}=\\{V_{j}\in V|\ (v_{j},v_{i})\in E\\}.\end{array}$
(1)

Let us define the in-degree matrix $D=diag(deg_{i})$, with
${deg}_{i}=\sum_{j\in N_{i}}a_{ij}$ as the sum of incoming edges weights to
the vertex $i$. The graph Laplacian is $L=D-A$, where a sum of all rows is
equal to zero.

The leader adjacency matrix is defined as $B=diag(b_{1},...b_{N})$, with
$b_{i}>0$, if the leader sends its information to the $i$-th agent, and
$b_{i}=0$, otherwise. It is assumed that the leader is connected to at least
one agent.

######  Lemma 1 ([27])

The directed graph $G$ is strongly connected if and only if its Laplacian
matrix is irreducible.

######  Definition 2 ([28, 3])

A square matrix $P\in\mathbb{R}^{N\times N}$ is said to be a non-singular
M-matrix, if it is positive definite, and the off-diagonal elements have non-
positive values.

######  Lemma 2 ([28])

Let the directed graph $G=(V,E)$ be strongly connected. If $\exists b_{i}\neq
0$, then $L+B$ is a non-singular M-matrix and the following inequality holds

$\Pi\triangleq P(L+B)+(L+B)^{T}P>0,$ (2)

where $P=diag(p_{i})\triangleq diag(1/q_{i})$ is a positive definite matrix
and $\mathbf{q}\triangleq(L+B)^{-1}{\mathbf{1}}_{N}$ with
$\mathbf{q}=[q_{1},...,q_{N}]^{T}$.

###  II-C Problem Formulation

The multi-agent system is considered to consist of $N$ agents where the
dynamics of each agent is given by

$\begin{array}[]{lr}\dot{p}_{i}=v_{i},\\\
\dot{v}_{i}=f_{i}(p_{i},v_{i})+g_{i}(p_{i},v_{i})u_{i}(t)+w_{i}(t),\ \ i=1,\
...\ ,N,\end{array}$ (3)

where the variables $p_{i}\in\mathbb{R}^{n}$ and $v_{i}\in\mathbb{R}^{n}$
denote the position and velocity states of each agent, respectively. The
functions $f_{i}(p_{i},v_{i})\in\mathbb{R}^{n}$ and
$g_{i}(p_{i},v_{i})\in\mathbb{R}^{n\times n}$ are unknown nonlinear
$\mathcal{C}^{2}$ functions. The control input is $u_{i}\in\mathbb{R}^{n}$ and
a disturbance of an each agent is $w_{i}\in\mathbb{R}^{n}$.

One can rewrite the dynamics of the multi-agent system in a stacked form as
follows:

$\begin{split}\dot{\mathbf{x}}_{1}&=\mathbf{x}_{2},\\\
\dot{\mathbf{x}}_{2}&=f(\mathbf{x_{1}},\mathbf{x_{2}})+g(\mathbf{x_{1}},\mathbf{x_{2}})u+w,\end{split}$
(4)

where the stacked vector
$\mathbf{x}_{1}=[{p^{T}_{1}},{p^{T}_{2}},...,{p^{T}_{N}}]^{T}\in\mathbb{R}^{nN}$
and the stacked vector
$\mathbf{x}_{2}=[{v^{T}_{1}},{v^{T}_{2}},...,{v^{T}_{N}}]^{T}\in\mathbb{R}^{nN}$.
Consider the overall states of a multi-agent system as a vector
$\mathbf{x}=[\mathbf{x}_{1}^{T},\mathbf{x}_{2}^{T}]^{T}$. The stacked vector
$f(\mathbf{x})=[f^{T}_{1}(p_{1},v_{1}),f^{T}_{2}(p_{2},v_{2}),...,f^{T}_{N}(p_{N},v_{N})]^{T}\in\mathbb{R}^{nN}$
consists of nonlinearities of all agents, the overall control input gain
matrix is
$g(\mathbf{x})=diag(g_{1}(p_{1},v_{1}),g_{2}(p_{2},v_{2}),...,g_{N}(p_{N},v_{N}))\in\mathbb{R}^{nN\times
nN}$, the stacked control input vector is
$u=[u_{1}^{T},u_{2}^{T},...,u_{N}^{T}]^{T}\in\mathbb{R}^{nN}$, and the stacked
disturbance vector is
$w=[w_{1}^{T},w_{2}^{T},...,w_{N}^{T}]^{T}\in\mathbb{R}^{nN}$.

The leader’s dynamics is given by

$\begin{split}&\dot{p}_{l}=v_{l},\\\
&\dot{v}_{l}=f_{l}(p_{l},v_{l}),\end{split}$ (5)

where $p_{l},v_{l}\in\mathbb{R}^{n}$ denote the position and velocity states
of the leader. Here we use some standard assumptions [26, 29, 25, 16, 15, 30,
31]

###### Assumption 1

The trajectory of the leader and its first three time-derivatives are bounded,
i.e. $p_{l},v_{l},f_{l},\dot{f}_{l}\in\mathcal{L}_{\infty}$. The bounds of
$p_{l},v_{l},f_{l},\dot{f}_{l}$ are considered to be unknown.

###### Assumption 2

The gain matrix $g_{i}(p_{i},v_{i})$ is a symmetric, positive definite matrix
and its inverse satisfies the following inequality

$\underline{g}||x||^{2}\leq x^{T}g_{i}^{-1}x\leq\bar{g}||x||^{2},$ (6)

where $x\in\mathbb{R}^{n}$ and constants $\underline{g}$ and $\bar{g}$ are
unknown positive constants.

###### Remark 1

Assumption 2 states a condition for the controllability of the system (3).
These bounds are considered for analytical purposes, and their exact values
are considered to be unknown for control design. We removed an a priori
knowledge about $\underline{g}$ in our control design. Its exact value is not
required and this consideration distinguishes our work from [2, 24, 26, 32].
This assumption is not hard to satisfy as the large class of nonlinear
systems, described by Euler–Lagrange formulation, meets this assumption [33,
15].

###### Assumption 3

The disturbance and the first two time-derivatives (i.e.
$w_{i},\dot{w}_{w},\ddot{w}_{i}$) are considered to be bounded with unknown
bounds.

Let us define the error vector for $i$-th agent as [9]

$\begin{split}&e_{i}=\sum_{j\in
N_{i}}a_{ij}(p_{i}-p_{j}-d_{i}+d_{j})+b_{i}(p_{i}-p_{l}-d_{i}),\end{split}$
(7)

where the constant vectors $d_{i}\in\mathbb{R}^{n},i\in\\{1,...,N\\}$
represents the desired relative position between agent $i$ and the leader.
Note that we define $d_{ij}=d_{i}-d_{j}$, representing the desired relative
position between agents $i$ and $j$. For the agent $i$, define $\delta_{i}$ as
$\delta_{i}\triangleq\dot{e}_{i}$

$\begin{split}\delta_{i}=\sum_{j\in
N_{i}}a_{ij}(v_{i}-v_{j})+b_{i}(v_{i}-v_{l}).\end{split}$ (8)

The control objective is to design a robust, distributed controller for each
agent using local information that ensures all agents achieve the desired
formation and maintain the leader’s velocity:

$\begin{split}&\text{lim}_{t\rightarrow\infty}||p_{i}-p_{l}-d_{i}||=0,\\\
&\text{lim}_{t\rightarrow\infty}||v_{i}-v_{l}||=0.\end{split}$ (9)

Figure 1: Three-layer neural network architecture.

###  II-D Neural Networks

Based on the universal approximation theorem, for a smooth nonlinear function
of $y(\mathcal{X})$ over a compact set of $\Omega_{\mathcal{X}}$ there exists
a three-layer NN [34], such that

${y}(\mathcal{X})=W^{T}\sigma_{1}(Z^{T}\sigma_{2}(V^{T}\mathcal{X}))+\epsilon,$
(10)

where $V$, $Z$, and $W$ are ideal NN weights matrices, $\sigma_{1}$ and
$\sigma_{2}$ are the activation functions, and $\epsilon$ is the approximation
error. Here we establish some standard assumptions to estimate the
nonlinearity [3, 33, 16].

###### Assumption 4

The unknown ideal neural network weights matrices $V$, $Z$ and $W$ are bounded
with a fixed bound such that

$\begin{split}||V||_{F}&\leq V_{m},\\\ ||Z||_{F}&\leq Z_{m},\\\ ||W||_{F}&\leq
W_{m}.\end{split}$ (11)

###### Assumption 5

For the vector $\mathcal{X}\in\mathcal{L}_{\infty}$, the approximation error
$\epsilon$, and its first and second time-derivatives are bounded with a fixed
bound.

In this paper, we use a three-layer NN, as shown in Fig. 1, to approximate an
unknown nonlinear function over a compact set $\Omega_{\mathcal{X}}$. The
first hidden layer has $m_{1}$ neurons and the second hidden layer has $m_{2}$
neurons. Considering
$\hat{y}=[\hat{y}_{1},...,\hat{y}_{n_{2}}]^{T}\in\mathbb{R}^{n_{2}}$, the
output of neural network (Fig. 1) can be written as

$\hat{y}=\hat{W}^{T}\sigma_{1}(\hat{Z}^{T}\sigma_{2}(\hat{V}^{T}\mathcal{X})),$
(12)

where $\mathcal{X}=[1,\bar{\mathcal{X}}^{T}]^{T}\in\mathbb{R}^{n_{1}+1}$ with
$\bar{\mathcal{X}}^{T}=[x_{1},...,x_{n_{1}}]^{T}\in\mathbb{R}^{n_{1}}$, the
neural network weights matrices $\hat{V}\in\mathbb{R}^{(n_{1}+1)\times
m_{1}}$, $\hat{Z}\in\mathbb{R}^{(m_{1}+1)\times m_{2}}$, and
$\hat{W}\in\mathbb{R}^{(m_{2}+1)\times n_{2}}$. We use
$\sigma_{1}(.)\in\mathbb{R}^{m_{2}+1}$ and
$\sigma_{2}(.)\in\mathbb{R}^{m_{1}+1}$ to denote the activation functions of
hidden layers. We consider the sigmoid function for both hidden layers, i.e.,
for a scalar $s_{i}$, one has $\sigma_{k}(s_{i})=\frac{1}{1+e^{-s_{i}}}$ for
$k\in\\{1,2\\}$. The activation function of the output layer is considered to
be linear. The ideal neural network weights matrices $V$, $Z$, and $W$ are
defined as follows:

$\small{\begin{array}[]{lr}V,Z,W=\arg\min_{\hat{V},\hat{Z},\hat{W}}\big{\\{}\sup_{\mathcal{X}\in\Omega_{\mathcal{X}}}||y(\mathcal{X})-\hat{y}(\mathcal{X})||\big{\\}}.\end{array}}$
(13)

Let us define $\tilde{y}=y-\hat{y}$,
$\tilde{\sigma}_{k}=\sigma_{k}-\hat{\sigma}_{k}$, $\tilde{V}=V-\hat{V}$,
$\tilde{Z}=Z-\hat{Z}$, and $\tilde{W}=W-\hat{W}$. Let us define
${\sigma}_{1}\triangleq\sigma_{1}({Z}^{T}{\sigma}_{2})$, with
${\sigma}_{2}\triangleq{\sigma}_{2}({V}^{T}{\mathcal{X}})$ and
$\hat{\sigma}_{1}\triangleq\sigma_{1}(\hat{Z}^{T}\hat{\sigma}_{2})$ with
$\hat{\sigma}_{2}\triangleq{\sigma}_{2}(\hat{V}^{T}{\mathcal{X}})$. Moreover,
we express
$\hat{\sigma}_{1}^{\prime}\triangleq\frac{d\sigma_{1}(s)}{ds}|_{s=\hat{Z}^{T}\hat{\sigma}_{2}}$
and
$\hat{\sigma}_{2}^{\prime}\triangleq\frac{d\sigma_{2}(s)}{ds}|_{s=\hat{V}^{T}\mathcal{X}}$.

###### Lemma 3

The estimation error satisfies the following

$\begin{split}y-\hat{y}=&\tilde{W}^{T}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}{\hat{\sigma}_{2}}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}^{T}{\mathcal{X}}\big{)}\\\
&+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}\tilde{Z}^{T}\big{(}\hat{\sigma}_{2}-\hat{\sigma}_{2}^{\prime}\hat{V}^{T}\mathcal{X}\big{)}\\\
&+\hat{W}^{T}\big{(}\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\tilde{V}^{T}\mathcal{X}\big{)}+\bar{\epsilon},\end{split}$
(14)

with

$\displaystyle\bar{\epsilon}=$
$\displaystyle\hat{W}^{T}\hat{\sigma}_{1}^{\prime}Z^{T}\textit{O}_{2}(.)+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}\tilde{Z}^{T}\hat{\sigma}_{2}^{\prime}V^{T}\mathcal{X}$
(15)
$\displaystyle+[\tilde{W}^{T}(\hat{\sigma}_{1}^{\prime}{Z}^{T}\sigma_{2})+\tilde{W}\hat{\sigma}_{i}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}^{T}\mathcal{X}+W^{T}\textit{O}_{1}(.)+\epsilon],$

where $\textit{O}_{1}(.)$ and $\textit{O}_{2}(.)$ are higher-order terms of
Taylor series expansion of ${\sigma}_{1}(Z^{T}\sigma_{2}(V^{T}{\mathcal{X}}))$
at $\hat{Z}^{T}\sigma_{2}(\hat{V}^{T}{\mathcal{X}})$ and
${\sigma}_{2}({V}^{T}{\mathcal{X}})$ at $\hat{V}^{T}{\mathcal{X}}$,
respectively.

###### Proof:

Following the procedure in [33, 11], from (10) and (12), one can write

$\begin{split}y-\hat{y}&={W}^{T}\sigma_{1}({Z}^{T}\sigma_{2}({V}^{T}\mathcal{X}))-\hat{W}^{T}\sigma_{1}(\hat{Z}^{T}\sigma_{2}(\hat{V}^{T}\mathcal{X}))+\epsilon.\end{split}$
(16)

By adding and subtracting
$W^{T}\hat{\sigma}_{1}+\hat{W}^{T}\tilde{\sigma}_{1}$, and rearranging (16),
one has

$\begin{split}y-\hat{y}&=\tilde{W}^{T}\hat{\sigma}_{1}+\tilde{W}^{T}\tilde{\sigma}_{1}+\hat{W}^{T}\tilde{\sigma}_{1}+\epsilon.\end{split}$
(17)

Taylor series expansions of
${\sigma}_{1}(Z^{T}\sigma_{2}(V^{T}{\mathcal{X}}))$ at
$\hat{Z}^{T}\sigma_{2}(\hat{V}^{T}{\mathcal{X}})$ and
${\sigma}_{2}({V}^{T}{\mathcal{X}})$ at $\hat{V}^{T}{\mathcal{X}}$ is

$\displaystyle{\sigma}_{1}(Z^{T}\sigma_{2}(V^{T}{\mathcal{X}}))=$
$\displaystyle{\sigma}_{1}(\hat{Z}^{T}\sigma_{2}(\hat{V}^{T}{\mathcal{X}}))$
(18)
$\displaystyle+\hat{\sigma}_{1}^{\prime}[Z^{T}\sigma_{2}(V^{T}{\mathcal{X}})-\hat{Z}^{T}\sigma_{2}(\hat{V}^{T}{\mathcal{X}})]+\textit{O}_{1}(.)$
$\displaystyle=$
$\displaystyle\hat{\sigma}_{1}+\hat{\sigma}_{1}^{\prime}[Z^{T}\sigma_{2}-\hat{Z}^{T}\hat{\sigma}_{2}]+\textit{O}_{1}(.),$
$\begin{split}\sigma_{2}(V^{T}{\mathcal{X}})=\hat{\sigma}_{2}+\hat{\sigma}_{2}^{\prime}\tilde{V}^{T}{\mathcal{X}}+\textit{O}_{2}(.).\end{split}$
(19)

Note that $\textit{O}_{1}(.)$ and $\textit{O}_{2}(.)$ are shortened notations
for
$\textit{O}_{1}(Z^{T}\sigma_{2}(V^{T}{\mathcal{X}})-\hat{Z}^{T}\sigma_{2}(\hat{V}^{T}{\mathcal{X}}))$
and $\textit{O}_{2}(\tilde{V}^{T}{\mathcal{X}})$, respectively. From (18) and
(19), it is evident that

$\begin{split}\tilde{\sigma}_{1}&=\hat{\sigma}_{1}^{\prime}[Z^{T}\sigma_{2}-\hat{Z}^{T}\hat{\sigma}_{2}]+\textit{O}_{1}(.),\\\
\tilde{\sigma}_{2}&=\hat{\sigma}_{2}^{\prime}\tilde{V}^{T}{\mathcal{X}}+\textit{O}_{2}(.).\end{split}$
(20)

Substituting (20) in (17), one has

$\begin{split}y-\hat{y}&=\tilde{W}^{T}\big{(}\hat{\sigma}_{1}+\hat{\sigma}_{1}^{\prime}[Z^{T}\sigma_{2}-\hat{Z}^{T}\hat{\sigma}_{2}]+\textit{O}_{1}(.)\big{)}+\hat{W}^{T}\tilde{\sigma}_{1}+\epsilon.\end{split}$
(21)

Rearranging (21), yields

$\begin{split}y-\hat{y}=&\tilde{W}^{T}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}+\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\tilde{\sigma}_{2}\big{)}+\hat{W}^{T}\tilde{\sigma}_{1}\\\
&+[\tilde{W}^{T}(\hat{\sigma}_{1}^{\prime}[\tilde{Z}^{T}\sigma_{2}+\hat{Z}^{T}\hat{\sigma}_{2}]+\textit{O}_{1}(.))+\epsilon].\end{split}$
(22)

From (20), (22) can be written as

$\displaystyle y-\hat{y}=$
$\displaystyle\tilde{W}^{T}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}^{T}\mathcal{X}\big{)}+\hat{W}^{T}\tilde{\sigma}_{1}$
(23)
$\displaystyle+[\tilde{W}^{T}(\hat{\sigma}_{1}^{\prime}[\tilde{Z}^{T}\sigma_{2}+\hat{Z}^{T}\hat{\sigma}_{2}+\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}V^{T}{\mathcal{X}}+\hat{Z}^{T}\textit{O}_{2}(.)]$
$\displaystyle+\textit{O}_{1}(.))+\epsilon].$

Following the same procedure for $\hat{W}^{T}\tilde{\sigma}_{1}$ and from
(20), one has

$\displaystyle y-\hat{y}=$
$\displaystyle\tilde{W}^{T}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}^{T}\mathcal{X}\big{)}$
(24)
$\displaystyle+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}[\tilde{Z}^{T}\tilde{\sigma}_{2}+\tilde{Z}^{T}\hat{\sigma}_{2}]+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}[\hat{Z}^{T}\tilde{\sigma}_{2}]+\hat{W}^{T}\textit{O}_{1}(.)$
$\displaystyle+[\tilde{W}^{T}(\hat{\sigma}_{1}^{\prime}[\tilde{Z}^{T}\sigma_{2}+\hat{Z}^{T}\hat{\sigma}_{2}+\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}V^{T}{\mathcal{X}}+\hat{Z}^{T}\textit{O}_{2}(.)]$
$\displaystyle+\textit{O}_{1}(.))+\epsilon].$

Substituting (20) in (24) leads to

$\displaystyle y-\hat{y}=$
$\displaystyle\tilde{W}^{T}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}^{T}\mathcal{X}\big{)}$
(25)
$\displaystyle+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}\tilde{Z}^{T}(\hat{\sigma}_{2}-\hat{\sigma}_{2}^{\prime}\hat{V}^{T}\mathcal{X})+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}\tilde{Z}^{T}\textit{O}_{2}(.)$
$\displaystyle+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}\tilde{Z}^{T}\hat{\sigma}_{2}^{\prime}V^{T}\mathcal{X}+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\tilde{V}^{T}\mathcal{X}$
$\displaystyle+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\textit{O}_{2}(.)+W^{T}\textit{O}_{1}(.)+\tilde{W}^{T}(\hat{\sigma}_{1}^{\prime}[\tilde{Z}^{T}\sigma_{2}$
$\displaystyle+\hat{Z}^{T}\hat{\sigma}_{2}+\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}V^{T}{\mathcal{X}}+\hat{Z}^{T}\textit{O}_{2}(.)])+\epsilon.$

From (19) and (25), one can derive $\bar{\epsilon}$ as (15). $\blacksquare$ ∎

The following lemma provides a bound for $\bar{\epsilon}$.

###### Lemma 4

The term $\bar{\epsilon}$ (15) satisfies the following inequality

$\begin{split}||\bar{\epsilon}||&\leq\Gamma\mu,\\\ \end{split}$ (26)

where $\Gamma$ is an unknown positive constant, and $\mu$ is defined as
follows:

$\begin{split}\mu=&||\hat{W}||_{F}+||\hat{W}||_{F}||\mathcal{X}||+||\hat{W}||_{F}||\hat{V}||_{F}||\mathcal{X}||\\\
&+||\hat{W}||_{F}||\hat{Z}||_{F}||\mathcal{X}||+||\hat{Z}||_{F}||\hat{V}||_{F}||\mathcal{X}||\\\
&+||\hat{W}||_{F}||\hat{Z}||_{F}||\hat{V}||_{F}||\mathcal{X}||+||\hat{Z}||_{F}+1.\end{split}$
(27)

###### Proof:

Following the procedure in [33, Lemma 4.3.1], [11], as well as using the fact
that $\sigma_{k}(.)$ and its derivative are bounded, and from (20), one can
show that

$\begin{split}||\textit{O}_{2}(.)||&\leq
c_{0}+c_{1}||\mathcal{X}||+c_{2}||\hat{V}||_{F}||\mathcal{X}||,\\\
||\textit{O}_{1}(.)||&\leq c_{3}+c_{4}||\hat{Z}||_{F},\\\ \end{split}$ (28)

where $c_{i}$, $i\in\\{0,..,4\\}$ are positive constants. Using Assumption 4,
and by substituting (28) in (15), one has

$\displaystyle||\bar{\epsilon}||\leq$
$\displaystyle||\hat{W}^{T}\hat{\sigma}_{1}^{\prime}Z^{T}||\
||\textit{O}_{2}(.)||+||\hat{W}^{T}\hat{\sigma}_{1}^{\prime}\tilde{Z}^{T}\hat{\sigma}_{2}^{\prime}V^{T}\mathcal{X}||$
(29)
$\displaystyle+||\tilde{W}^{T}\hat{\sigma}_{1}^{\prime}{Z}^{T}\sigma_{2}||+||\tilde{W}\hat{\sigma}_{i}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}^{T}\mathcal{X}||$
$\displaystyle+||W^{T}\textit{O}_{1}(.)+\epsilon||$ $\displaystyle\leq$
$\displaystyle
d_{0}||\hat{W}||_{F}(c_{0}+c_{1}||\mathcal{X}||+c_{2}||\hat{V}||_{F}||\mathcal{X}||)$
$\displaystyle+d_{1}||\hat{W}||_{F}\
||\mathcal{X}||+d_{2}||\hat{W}||_{F}||\hat{Z}||_{F}||\mathcal{X}||+d_{3}||\hat{W}||_{F}$
$\displaystyle+d_{4}||\hat{Z}||_{F}||\hat{V}||_{F}||\mathcal{X}||+d_{5}||\hat{W}||_{F}||\hat{Z}||_{F}||\hat{V}||_{F}||\mathcal{X}||$
$\displaystyle+d_{6}(c_{3}+c_{4}||\hat{Z}||_{F})+d_{7},$

where $d_{i}$, $i\in\\{1,..,7\\}$ are positive constants. Rearranging (29)
yields

$\displaystyle||\bar{\epsilon}||\leq$
$\displaystyle(d_{0}c_{0}+d_{3})||\hat{W}||_{F}+(c_{1}+d_{1})||\hat{W}||_{F}||\mathcal{X}||$
(30)
$\displaystyle+c_{2}||\hat{W}||_{F}||\hat{V}||_{F}||\mathcal{X}||+d_{2}||\hat{W}||_{F}||\hat{Z}||_{F}||\mathcal{X}||$
$\displaystyle+d_{4}||\hat{Z}||_{F}||\hat{V}||_{F}||\mathcal{X}||+d_{5}||\hat{W}||_{F}||\hat{Z}||_{F}||\hat{V}||_{F}||\mathcal{X}||$
$\displaystyle+d_{6}c_{3}+d_{7}+c_{4}||\hat{Z}||_{F}).$

From (30), one has

$\displaystyle||\bar{\epsilon}||\leq$
$\displaystyle\Gamma(||\hat{W}||_{F}+||\hat{W}||_{F}||\mathcal{X}||+||\hat{W}||_{F}||\hat{V}||_{F}||\mathcal{X}||$
(31)
$\displaystyle+||\hat{W}||_{F}||\hat{Z}||_{F}||\mathcal{X}||+||\hat{Z}||_{F}||\hat{V}||_{F}||\mathcal{X}||$
$\displaystyle+||\hat{W}||_{F}||\hat{Z}||_{F}||\hat{V}||_{F}||\mathcal{X}||+||\hat{Z}||_{F}+1),$

with

$\displaystyle\Gamma\triangleq
max\\{d_{0}c_{0}+d_{3},c_{1}+d_{1},c_{2},d_{2},d_{4},d_{5},d_{6}c_{3}+d_{7}+c_{4}\\}.$
(32)

From (32), it is straightforward to verify that selecting $\mu$ as (27)
implies (26). $\blacksquare$ ∎

###### Remark 2

Lemma 4 provides an upper bound for $\bar{\epsilon}$ using a three-layer
neural network. Note that this error is bounded as
$||\bar{\epsilon}||\leq\Gamma\mu$, where $\Gamma$ is considered to be an
unknown constant. Therefore, one can write it as
$||\bar{\epsilon}||\leq\bar{\epsilon}_{M}$, where $\bar{\epsilon}_{M}$
considered to be unknown. Note that we consider this bound for analytical
purposes, and its exact value is not required in our control design.

###### Remark 3

For any sufficiently smooth function over a compact set, the result of the
Stone-Weierstrass Theorem stipulates that the function can be approximated
using enough number of neurons, and the universal approximation property of NN
guarantees the boundedness of the approximation error [35, 15, 3, 34].

##  III Controller Design

Let us define
$\mathbf{\delta}=[\delta^{T}_{1},...,\delta_{N}^{T}]^{T}\in\mathbb{R}^{nN}$
and $\mathbf{e}=[e^{T}_{1},...,e_{N}^{T}]^{T}\in\mathbb{R}^{nN}$. Then, one
can write (7) and (8) as

$\begin{split}\mathbf{e}&=[(L+B)\otimes
I_{n}](\mathbf{x}_{1}-\mathbf{p}_{l}-d),\\\ \mathbf{\delta}&=[(L+B)\otimes
I_{n}](\mathbf{x}_{2}-\mathbf{v}_{l}),\end{split}$ (33)

where $\mathbf{p}_{l}=\mathbf{1}_{N}\otimes p_{l}$,
$d=[d_{1}^{T},...,d_{N}^{T}]^{T}$ and $\mathbf{v}_{l}=\mathbf{1}_{N}\otimes
v_{l}$. Let us define an auxiliary variable of
$\zeta_{i}=\delta_{i}+k_{1}e_{i}$ as in [3], where $k_{1}$ is a positive
constant gain. Then the stacked vector
$\zeta=[\zeta^{T}_{1},...,\zeta_{N}^{T}]^{T}\in\mathbb{R}^{nN}$ is defined as

$\begin{split}\zeta=\mathbf{\delta}+k_{1}\mathbf{e}.\end{split}$ (34)

One can derive the time-derivative of (34) as

$\dot{\mathbf{\zeta}}=[(L+B)\otimes
I_{n}]\big{(}f(\mathbf{x})+g(\mathbf{x})u+w-\mathbf{f}_{l}\big{)}+k_{1}\delta,$
(35)

where $\mathbf{f}_{l}=\mathbf{1}_{N}\otimes f_{l}$.

###  III-A RISE Feedback

Let us define the filtered error as

$r=\dot{\zeta}+k_{2}\zeta,$ (36)

where $k_{2}$ is a positive constant. The time-derivative of (36) is given by

$\dot{r}=\ddot{\zeta}+k_{2}\dot{\zeta}.$ (37)

From Assumption 2, Remark 1, Lemma 2, and (37), one can express the following
equation similar to [17] as

$\mathcal{G}(\mathbf{x})Q\dot{r}=\mathcal{G}(\mathbf{x})Q(\ddot{\zeta}+k_{2}\dot{\zeta}),$
(38)

where $\mathcal{G}(\mathbf{x})\triangleq g^{-1}(\mathbf{x})$ and
$Q\triangleq[(L+B)\otimes I_{n}]^{-1}$. By adding and subtracting
$\frac{1}{2}\dot{\mathcal{G}}(\mathbf{x})Qr+k_{2}\zeta$, and from (33)-(35),
(38) can be rewritten as

$\begin{split}\mathcal{G}(\mathbf{x})Q\dot{r}=&\mathcal{G}(\mathbf{x})\Big{(}(k_{2}+k_{1})f(\mathbf{x})+(k_{2}+k_{1})g(\mathbf{x})u\\\
&+(k_{2}+k_{1})w-(k_{2}+k_{1})\mathbf{f}_{l}+k_{1}k_{2}(\mathbf{x}_{2}-\mathbf{v_{l}})\\\
&+\dot{f}(\mathbf{x})+\dot{g}(\mathbf{x})u+{g}(\mathbf{x})\dot{u}+\dot{w}-\dot{\mathbf{f}}_{l}\Big{)}\\\
&+\frac{1}{2}\dot{\mathcal{G}}(\mathbf{x})Qr+k_{2}\zeta-\frac{1}{2}\dot{\mathcal{G}}(\mathbf{x})Qr-
k_{2}\zeta\\\ =&-\frac{1}{2}\dot{\mathcal{G}}(\mathbf{x})Qr-
k_{2}\zeta+\dot{u}+N_{1}+N_{2},\end{split}$ (39)

where

$\begin{split}\dot{\mathcal{G}}(\mathbf{x})&=\frac{\partial\mathcal{G}}{\partial{\mathbf{x}_{1}}}\mathbf{x}_{2}+\frac{\partial\mathcal{G}}{\partial{\mathbf{x}_{2}}}\big{(}f(\mathbf{x})+g(\mathbf{x})u+w\big{)},\\\
\dot{{f}}(\mathbf{x})&=\frac{\partial
f}{\partial{\mathbf{x}_{1}}}\mathbf{x}_{2}+\frac{\partial
f}{\partial{\mathbf{x}_{2}}}\big{(}f(\mathbf{x})+g(\mathbf{x})u+w\big{)},\\\
\dot{\mathbf{f}}_{l}&=\mathbf{1}_{N}\otimes\big{(}\frac{\partial
f_{l}}{\partial{{p}_{l}}}{v}_{l}+\frac{\partial
f_{l}}{\partial{{v}_{l}}}f_{l}\big{)}.\end{split}$ (40)

Using the fact that $\mathcal{G}(\mathbf{x})g(\mathbf{x})=I_{nN}$, and from
Assumption 2, $N_{1}$ and $N_{2}$ are given as follows:

$\begin{split}N_{1}=&(k_{1}+k_{2})(\mathcal{G}(\mathbf{x})f(\mathbf{x})+u)+\mathcal{G}(\mathbf{x})\frac{\partial
f}{\partial{\mathbf{x}_{1}}}\mathbf{x}_{2}\\\
&+\mathcal{G}(\mathbf{x})\frac{\partial
f}{\partial{\mathbf{x}_{2}}}\big{(}f(\mathbf{x})+g(\mathbf{x})u\big{)}+k_{2}\mathbf{x}_{2}+k_{2}k_{1}(\mathbf{x}_{1}-d)\\\
&+\frac{1}{2}\Big{(}\frac{\partial\mathcal{G}}{\partial{\mathbf{x}_{1}}}\mathbf{x}_{2}+\frac{\partial\mathcal{G}}{\partial{\mathbf{x}_{2}}}\big{(}f(\mathbf{x})+g(\mathbf{x})u\big{)}\Big{)}\big{(}f(\mathbf{x})-g(\mathbf{x})u\\\
&+(k_{1}+k_{2})\mathbf{x}_{2}+k_{1}k_{2}(\mathbf{x}_{1}-d)\big{)}+k_{1}k_{2}\mathcal{G}(\mathbf{x})\mathbf{x}_{2},\end{split}$
(41)
$\begin{split}N_{2}=&(k_{1}+k_{2})(\mathcal{G}(\mathbf{x})w-\mathcal{G}(\mathbf{x})\mathbf{f}_{l})+\mathcal{G}(\mathbf{x})(\dot{w}-\dot{\mathbf{f}}_{l})-k_{1}k_{2}\mathbf{p}_{l}\\\
&+\mathcal{G}(\mathbf{x})\frac{\partial
f}{\partial{\mathbf{x}_{2}}}w-k_{1}k_{2}\mathcal{G}(\mathbf{x}){\mathbf{v}}_{l}-\frac{\partial\mathcal{G}}{\partial{\mathbf{x}_{2}}}wg(\mathbf{x})u-k_{2}\mathbf{v}_{l}\\\
&+\frac{1}{2}\Big{(}\frac{\partial\mathcal{G}}{\partial{\mathbf{x}_{1}}}\mathbf{x}_{2}+\frac{\partial\mathcal{G}}{\partial{\mathbf{x}_{2}}}\big{(}f(\mathbf{x})+g(\mathbf{x})u\big{)}\Big{)}(w-\dot{\mathbf{v}}_{l}).\end{split}$
(42)

Note that $N_{1}=[N^{T}_{1_{1}},...,N^{T}_{1_{N}}]^{T}\in\mathbb{R}^{nN}$ and
$N_{2}=[N^{T}_{2_{1}},...,N^{T}_{2_{N}}]^{T}\in\mathbb{R}^{nN}$ consist of
unknown terms. The difference between $N_{1_{i}}$ and $N_{2_{i}}$ is that, for
each agent, we use NN to approximate the variable $N_{1_{i}}$ while we use a
time-varying robustifying term to compensate for $N_{2_{i}}$. The variable
$N_{1_{i}}$ is a function of available filtered error $\zeta_{i}$, states, and
agent control input. Note that feeding the agent control input into the input
layer of the NN causes the circular design problem. To avoid this problem,
similar to [11, 16], we construct the NN from the NN weights matrices, states
of the system dynamics, the robustifying term, and $\zeta_{i}$. This
consideration distinguishes our results from [17].

Considering (39), the NN-based control law is derived as

$\begin{split}\dot{u}=-(k_{3}+k_{4})\dot{\zeta}-k_{2}(k_{3}+k_{4}){\zeta}-\hat{N}_{1}-\kappa(t)\text{sgn}(\zeta(t)),\end{split}$
(43)

where $k_{3}$, $k_{4}$ are positive constants, $\hat{N}_{1}$ is the output of
NN, and $\kappa(t)=[diag(\kappa_{1},...,\kappa_{N})]\otimes
I_{n}\in\mathbb{R}^{nN\times nN}$ is the time-varying gain. The last term in
(43) is responsible for compensating the unknown term of $N_{2}$ in (39) and
the NN approximation error. For the agent $i$, let us define
$\hat{N}_{2_{i}}\triangleq\kappa_{i}\text{sgn}(\zeta_{i})$. The block diagram
of proposed control law is illustrated in Fig. 2.

We choose the following NN input vector for each agent:

$\begin{split}\bar{\mathcal{X}}_{i}=&[p^{T}_{i},v^{T}_{i},\zeta^{T}_{i},||\hat{V}_{i}||_{F},||\hat{Z}_{i}||_{F},||\hat{W}_{i}||_{F},\kappa_{i}(t)]^{T}\in\mathbb{R}^{3n+4},\end{split}$
(44)

and $\mathcal{X}_{i}=[1,\bar{\mathcal{X}}_{i}]^{T}\in\mathbb{R}^{3n+5}$. The
NN for $i$-th agent is described as (12), i.e.,
$y({\mathcal{X}_{i}})\triangleq N_{1_{i}}$ and
$\hat{y}({\mathcal{X}_{i}})\triangleq\hat{N}_{1_{i}}$. Let
$\tilde{V}=diag(\tilde{V}_{i})\in\mathbb{R}^{(3n+5)N\times m_{1}N}$,
$\tilde{Z}=diag(\tilde{Z}_{i})\in\mathbb{R}^{(m_{1}+1)N\times m_{2}N}$,
$\tilde{W}=diag(\tilde{W}_{i})\in\mathbb{R}^{(m_{2}+1)N\times nN}$,
$\hat{V}=diag(\hat{V}_{i})$, $\hat{Z}=diag(\hat{Z}_{i})$,
$\hat{W}=diag(\hat{W}_{i})$,
$\hat{\sigma}_{s}={[{\hat{\sigma}}^{T}_{1},...,{\hat{\sigma}}^{T}_{N}]}^{T}$,
and ${\hat{\sigma}^{\prime}}_{s}=[{\hat{\sigma}^{\prime
T}}_{1},...,{\hat{\sigma}^{\prime T}}_{N}]^{T},s\in\\{1,2\\}$. Let us define
$\mathcal{X}=[\mathcal{X}^{T}_{1},...,\mathcal{X}^{T}_{N}]^{T}$.

Define
$\hat{N}_{1}=\hat{W}^{T}\sigma_{1}(\hat{Z}^{T}\sigma_{2}(\hat{V}^{T}\mathcal{X}))$
and $\tilde{N}_{1}=[\tilde{N}^{T}_{1_{1}},...,\tilde{N}^{T}_{1_{N}}]^{T},$
with $\tilde{N}_{1_{i}}={N}_{1_{i}}-\hat{N}_{1_{i}}$ as

$\begin{split}\tilde{N}_{1_{i}}&=\tilde{W}_{i}^{T}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}{\hat{\sigma}_{2}}-\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}_{i}^{T}{\mathcal{X}_{i}}\big{)}\\\
&+\hat{W}_{i}^{T}\hat{\sigma}_{1}^{\prime}\tilde{Z}_{i}^{T}\big{(}\hat{\sigma}_{2}-\hat{\sigma}_{2}^{\prime}\hat{V}_{i}^{T}\mathcal{X}_{i}\big{)}\\\
&+\hat{W}_{i}^{T}\big{(}\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}\hat{\sigma}_{2}^{\prime}\tilde{V}_{i}^{T}\mathcal{X}_{i}\big{)}.\end{split}$
(45)

For the sake of the notation simplicity, the index of $i$ and the arguments of
$\sigma_{1}$ and $\sigma_{2}$ are dropped. From (36), (39), (43), Lemma 3, and
Remark 2, one has

$\begin{split}\mathcal{G}(\mathbf{x})Q\dot{r}=&-\frac{1}{2}\dot{\mathcal{G}}(\mathbf{x})Qr-
k_{2}\zeta-(k_{3}+k_{4}){r}-\kappa(t)\text{sgn}(\zeta)\\\
&+\tilde{N}_{1}+N_{2}+\epsilon,\end{split}$ (46)

where
$\epsilon=[\epsilon_{1}^{T},...,\epsilon^{T}_{N}]^{T}\in\mathbb{R}^{nN}$.

###### Remark 4

Let $\mathbf{x}\in\Omega_{\mathbf{x}}$, where set $\Omega_{\mathbf{x}}$ is a
compact set. Following the same procedure as in [17] and considering the
Assumptions 2 and 3, if the control input signal is bounded, i.e., $u\in
L_{\infty}$, the variable $\dot{\mathbf{x}}$ is bounded. As the
$\mathbf{p}_{l}$, $\mathbf{v}_{l}$ and $\mathbf{f}_{l}$ are bounded
(Assumption 1), signals $\mathbf{e},\delta$, and $\zeta$ remain bounded, as
well as $\dot{\zeta}$, i.e., $\dot{\zeta}\in\Omega_{\dot{\zeta}}$ (see (35)).
Moreover, considering $\kappa\in\Omega_{\kappa}$, $\zeta\in\Omega_{\zeta}$,
$Y\in\Omega_{Y}$ with
$Y=[||\hat{W}||_{F},||\hat{Z}||_{F},||\hat{V}||_{F}]^{T}$, and with the help
of (43), variable $\dot{u}$ is bounded, i.e., $\dot{u}\in L_{\infty}$. Then,
take the time-derivative of (44), consider Assumptions 1, 3, (35) and (36).
Then it follows that $\dot{\mathcal{X}}$ remains bounded in the compact set of
$\Omega_{\mathcal{X}}$ with
$\Omega_{\mathcal{X}}\triangleq\Omega_{\mathbf{x}}\times\Omega_{\mathbf{u}}\times\Omega_{\kappa}\times\Omega_{Y}$,
where $\Omega_{\mathbf{u}}$ is the compact set in which $u$ is bounded.

Let us define an agent’s NN weights matrices tuning laws as

$\begin{split}{\hat{W}}_{i}(t)=&\alpha_{i}\Big{(}k_{2}\int_{0}^{t}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}(s){\hat{\sigma}_{2}}\\\
&-\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}(s)\hat{\sigma}_{2}^{\prime}\hat{V}_{i}^{T}(s){\mathcal{X}_{i}(s)}\big{)}\zeta_{i}^{T}(s)ds,-\|\zeta_{i}\|_{1}{\hat{W}}_{i}\Big{)}\\\
{\hat{Z}}_{i}(t)=&\beta_{i}\Big{(}k_{2}\int_{0}^{t}\big{(}\hat{\sigma}_{2}-\hat{\sigma}_{2}^{\prime}\hat{V}_{i}^{T}(s)\mathcal{X}_{i}\big{)}\zeta_{i}^{T}(s)\hat{W}^{T}_{i}(s)\hat{\sigma}_{1}^{\prime}ds\\\
&-\|\zeta_{i}\|_{1}{\hat{Z}}_{i}\Big{)},\\\
{\hat{V}}_{i}(t)=&\gamma_{i}\big{(}k_{2}\int_{0}^{t}\mathcal{X}_{i}(s)\zeta^{T}_{i}(s)\hat{W}^{T}_{i}(s)\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}_{i}(s)\hat{\sigma}_{2}^{\prime}ds\\\
&-\|\zeta_{i}\|_{1}{\hat{V}}_{i}\big{)},\end{split}$ (47)

with $\alpha_{i},\beta_{i},\gamma_{i}$ being positive constants. Let us define
$\alpha=diag(\mathbf{1}_{m_{2}+1}^{T}\alpha_{1},...,\mathbf{1}_{m_{2}+1}^{T}\alpha_{N})\in\mathbb{R}^{(m_{2}+1)N\times(m_{2}+1)N}$,
$\beta=diag(\mathbf{1}_{m_{1}+1}^{T}\beta_{1},...,\mathbf{1}_{m_{1}+1}^{T}\beta_{N})\in\mathbb{R}^{(m_{1}+1)N\times(m_{1}+1)N}$
and
$\gamma=diag(\mathbf{1}_{3n+5}^{T}\gamma_{1},...,\mathbf{1}_{3n+5}^{T}\gamma_{N})\in\mathbb{R}^{(3n+5)N\times(3n+5)N}$.
Let the time-varying gain $\kappa_{i}(t)$ be defined as [17, 36]

$\begin{split}\dot{\kappa}_{i}(t)=r_{i}^{T}\text{sgn}(\zeta_{i}).\end{split}$
(48)

From (36) and by taking the integral from (48), one has

$\begin{split}{\kappa}_{i}(t)&=\int_{0}^{t}\big{(}\dot{\zeta}_{i}^{T}(s)\text{sgn}(\zeta_{i}(s))+k_{2}{\zeta}_{i}^{T}(s)\text{sgn}(\zeta_{i}(s))\big{)}ds\\\
&=\sum_{j=1}^{n}\int_{0}^{t}\frac{d\zeta_{i_{j}}(s)}{ds}\text{sgn}(\zeta_{i_{j}}(s))ds+\int_{0}^{t}k_{2}\|{\zeta}_{i}(s)\|_{1}ds\\\
&=\|{\zeta}_{i}(t)\|_{1}-\|{\zeta}_{i}(0)\|_{1}+k_{2}\int_{0}^{t}\|{\zeta}_{i}(s)\|_{1}ds.\end{split}$
(49)

Figure 2: The block diagram of proposed control law (43) for agent $i$. Note
that if $b_{i}\neq 0$, then the agent $i$ has the information of the leader.

###### Remark 5

The unknown term $N_{2}$ includes disturbance, its time-derivative, the
leader’s velocity, and its time-derivative. Following the similar procedure as
in [17, 19], considering the smooth functions for $f$ and $g$ in system
dynamics as well as Assumptions 1 and 3, one has $||N_{2}||\leq d_{M}$ and
$||\dot{N}_{2}||\leq d_{dM}$, where $d_{M}$ and $d_{dM}$ are considered to be
unknown.

Take the time-derivative of (15), recall Remark 4, and consider Assumption 5,
the boundedness of the activation function and its derivative, $\mathbf{x}$,
$\dot{\mathbf{x}}$, $Y$, and $\dot{\mathcal{X}}$ over the compact set of
$\Omega_{\mathcal{X}}$, then the boundedness of
$||\dot{\bar{\epsilon}}(\mathcal{X},\dot{\mathcal{X}})||$ follows – namely,
$||\dot{\bar{\epsilon}}(\mathcal{X},\dot{\mathcal{X}})||\leq\bar{\epsilon}_{dM}$.
We consider $\bar{\epsilon}_{M}$ and $\bar{\epsilon}_{dM}$ to be unknown while
authors in [21, 25, 15] considered these bounds to be known.

For the agent $i$, one has $||\bar{\epsilon}_{i}||\leq\bar{\epsilon}_{M_{i}}$,
$||\dot{\bar{\epsilon}}_{i}(\mathcal{X}_{i},\dot{\mathcal{X}_{i}})||\leq\bar{\epsilon}_{dM_{i}}$,
$||N_{2_{i}}||\leq d_{M_{i}}$, and $||\dot{N}_{2_{i}}||\leq d_{dM_{i}}$. Let
us define $\bar{N}_{2_{i}}\triangleq N_{2_{i}}+\bar{\epsilon}_{i}$. Note that
values of $\bar{\epsilon}_{M_{i}}$, $\bar{\epsilon}_{dM_{i}}$, $d_{M_{i}}$,
and $d_{{dM}_{i}}$ are considered to be unknown, and they only are used for
analytical purposes.

For the agent $i$, consider (45) and recall the fact that the activation
function and its derivative are bounded. Using the Assumption 4, for an
arbitrary set of $\Omega_{\mathbf{x}}$, the inputs of the NN are bounded and
their derivatives $(\mathcal{X}_{i},\dot{\mathcal{X}_{i}})$ are bounded.
Therefore, we can conclude that there exists upper bounds for $\tilde{N}_{1i}$
and $\dot{\tilde{N}}_{1i}$, namely $\|\tilde{N}_{1i}\|\leq N_{Mi}$ and
$\|\dot{\tilde{N}}_{1i}\|\leq N_{dMi}$, which are considered to be unknown.
This fact also can be found in [17, 36, 19, 15].

As continuous differentiability is a sufficient condition for being locally
Lipschitz [37, Corollary 4.1.1], [38, Theorem 8.4] can be written as follows:

######  Lemma 5 ([19])

Consider the close-loop system $\dot{\chi}=f(\chi,t)$, where
$f:D\times[0,\infty)$ with $D\in\mathbb{R}^{n}$ being a set containing
$\chi=0$. Let a continuously differentiable scalar function
$V:D\times[0,\infty)$ satisfy the following inequalities

$\begin{split}W_{1}(\chi)\leq&V(\chi,t)\leq W_{2}(\chi),\\\
\dot{V}(\chi,t)\leq&-W_{3}(\chi),\ \ \ \forall t\geq 0,\forall\chi\in
D,\end{split}$ (50)

where $W_{1}$, $W_{2}$ are continuous positive definite functions and $W_{3}$
is a uniformly continuous positive semi-definite function. If (50) holds,
select a closed ball $B(0,R)$ with the radius of $R>0$, and let
$\rho<min_{||\chi||=R}W_{1}(\chi)$. Solutions of $\dot{\chi}=f(\chi,t)$ with
$\chi(t_{0})\in\\{\chi\in D|\ W_{2}(\chi)\leq\rho\\}$ are bounded and

$W_{3}(\chi)\rightarrow 0\ \text{as}\ t\rightarrow\infty.$ (51)

###  III-B Stability Analysis

In this section, we prove that the proposed control law (43) guarantees the
stability of the leader-following formation control problem for the class of
heterogeneous, second-order, uncertain, nonlinear multi-agent systems (3).
Before we give the main theorem of the paper, the following lemma is stated,
which will be used to prove the theorem.

###### Lemma 6

Let an auxiliary function $L_{i}(t)\in R$ be defined as follows:

$\begin{split}L_{i}(t)\triangleq&r_{i}^{T}\big{(}\bar{N}_{2_{i}}-\kappa_{d_{i}}\text{sgn}(\zeta_{i})\big{)}+\dot{\zeta}^{T}_{i}\tilde{N}_{1_{i}}\\\
&+(W^{2}_{m}+V_{m}^{2}+Z^{2}_{m})\zeta^{T}_{i}\text{sgn}(\zeta_{i}).\end{split}$
(52)

Set
$\kappa_{d_{i}}=d_{M_{i}}+\bar{\epsilon}_{M_{i}}+\frac{1}{k_{2}}\max\\{d_{dM_{i}}+\bar{\epsilon}_{dM_{i}}+N_{dM_{i}}+(W^{2}_{m}+V^{2}_{m}+Z^{2}_{m}),k_{2}N_{M_{i}}\\}$.
Then, the following inequality holds

$\begin{split}M_{i}\triangleq&\
\kappa_{d_{i}}||\zeta_{i}(0)||_{1}-\zeta^{T}_{i}(0)(\bar{N}_{2_{i}}(0)+\tilde{N}_{1_{i}}(0))\\\
&-\int_{0}^{t}L_{i}(s)ds\geq 0.\end{split}$ (53)

###### Proof:

Similar to [36], from (36), and by integrating (52) one has

$\displaystyle\int_{0}^{t}L_{i}(s)ds=$
$\displaystyle\int_{0}^{t}\dot{\zeta_{i}}^{T}(s)\big{(}\bar{N}_{2_{i}}(s)+\tilde{N}_{1_{i}}(s)\big{)}ds$
(54)
$\displaystyle-\kappa_{d_{i}}\int_{0}^{t}\dot{\zeta_{i}}^{T}(s)\text{sgn}(\zeta_{i}(s))\big{)}ds$
$\displaystyle+k_{2}\int_{0}^{t}{\zeta}^{T}_{i}(s)\big{(}\bar{N}_{2_{i}}(s)-\kappa_{d_{i}}\text{sgn}(\zeta_{i}(s))$
$\displaystyle+\frac{(W^{2}_{m}+V_{m}^{2}+Z^{2}_{m})}{k_{2}}\text{sgn}(\zeta_{i}(s))\big{)}ds$
$\displaystyle=$
$\displaystyle\zeta_{i}^{T}(s)(\bar{N}_{2_{i}}(s)+\tilde{N}_{1_{i}}(s))\Big{|}_{0}^{t}+k_{2}\int_{0}^{t}{\zeta}^{T}_{i}(s)\big{(}\bar{N}_{2_{i}}(s)$
$\displaystyle-\kappa_{d_{i}}\text{sgn}(\zeta_{i}(s))+\frac{(W^{2}_{m}+V_{m}^{2}+Z^{2}_{m})}{k_{2}}\text{sgn}(\zeta_{i}(s))$
$\displaystyle-\frac{1}{k_{2}}\dot{\bar{N}}_{2_{i}}(s)-\frac{1}{k_{2}}\dot{\tilde{N}}_{1_{i}}(s)ds\big{)}-k_{d_{i}}\|\zeta_{i}(s)\|_{1}\Big{|}_{0}^{t}.$

Using Cauchy–Schwarz inequality, and the definition of $\kappa_{d_{i}}$, one
has

$\begin{split}\int_{0}^{t}L_{i}(s)\leq&\zeta_{i}^{T}(t)(\bar{N}_{2_{i}}(t)+\tilde{N}_{1_{i}}(t))-\zeta_{i}^{T}(0)(\bar{N}_{2_{i}}(0)+\tilde{N}_{1_{i}}(0))\\\
&-k_{d_{i}}\|\zeta_{i}(t)\|_{1}+k_{d_{i}}\|\zeta_{i}(0)\|_{1}\\\
\leq&\kappa_{d_{i}}||\zeta_{i}(0)||_{1}-\zeta^{T}_{i}(0)(\bar{N}_{2_{i}}(0)+\tilde{N}_{1_{i}}(0)).\end{split}$
(55)

Therefore, (53) is valid. This completes the proof. $\blacksquare$ ∎

###### Remark 6

We propose to set neurons numbers of the first hidden-layer and the second
hidden-layer to $m_{1}=3n+4$ and $m_{2}=2(3n+4)+2$, for each agent. From [39,
Theorem 2.4], one can find that a three-layer NN with $3n+4$ inputs, $3n+4$
neurons in the first hidden-layer, $2(3n+4)+2$ neurons in the second hidden-
layer, and the sigmoid activation function, can approximate $N_{1_{i}}$ with
an arbitrary accuracy.

###### Theorem 1

Let the multi-agent system (3) be modeled by a strongly connected directed
graph. For each agent, select the number of the hidden-layer neurons in (12)
as $m_{1}=3n+4$, $m_{2}=2(3n+4)+2$, and the NN input as (44). Under
Assumptions 1-5, choose $k_{1}>\frac{1}{2}$ and

$k_{2}>\frac{1+\bar{\sigma}(P(L+B))}{\underline{\sigma}(P(L+B))}.$ (56)

Set the control input as

$\begin{split}u_{i}(t)=&-(k_{3}+k_{4})\zeta_{i}(t)+(k_{3}+k_{4})\zeta(0)\\\
&-\int_{0}^{t}\Big{(}\hat{W}_{i}^{T}(s)\sigma_{1}\big{(}\hat{Z}_{i}^{T}(s)\sigma_{2}(\hat{V}_{i}^{T}(s)\mathcal{X}_{i}(s))\big{)}\\\
&+k_{2}(k_{3}+k_{4})\zeta_{i}(s)+\kappa_{i}(s)\text{sgn}(\zeta_{i}(s))\Big{)}ds,\end{split}$
(57)

where $k_{3}$ and $k_{4}$ satisfy the following conditions

$k_{3}>\frac{k_{2}}{2\underline{\sigma}(P(L+B))},\
k_{4}>\frac{\bar{\sigma}(P(L+B))}{2}.$ (58)

Set the time-varying gain $\kappa_{i}(t)$ as (49), and the neural networks
weights matrices tuning laws as

$\begin{split}\dot{\hat{W}}_{i}&=\alpha_{i}\Big{(}k_{2}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}{\hat{\sigma}_{2}}-\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}_{i}^{T}{\mathcal{X}_{i}}\big{)}\zeta_{i}^{T}-\|\zeta_{i}\|_{1}{\hat{W}}_{i}\Big{)},\\\
\dot{\hat{Z}}_{i}&=\beta_{i}\Big{(}k_{2}\big{(}\hat{\sigma}_{2}-\hat{\sigma}_{2}^{\prime}\hat{V}_{i}^{T}\mathcal{X}_{i}\big{)}\zeta_{i}^{T}\hat{W}_{i}^{T}\hat{\sigma}_{1}^{\prime}-\|\zeta_{i}\|_{1}{\hat{Z}}_{i}\Big{)},\\\
\dot{\hat{V}}_{i}&=\gamma_{i}\big{(}k_{2}\mathcal{X}_{i}\zeta_{i}^{T}\hat{W}_{i}^{T}\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}\hat{\sigma}_{2}^{\prime}-\|\zeta_{i}\|_{1}{\hat{V}}_{i}\big{)}.\end{split}$
(59)

Then, all the closed-loop signals are bounded, $\zeta\rightarrow 0$ as
$t\rightarrow\infty$, and agents achieve the desired formation and maintain
the leader’s velocity.

###### Proof:

Let us define the Lyapunov function candidate as

$\begin{split}V=&\frac{1}{2}\mathbf{e}^{T}\mathbf{e}+\frac{1}{4}\zeta^{T}\Pi\zeta+\frac{1}{2}r^{T}\mathcal{G}Qr+M+\frac{1}{2}\tilde{\kappa}^{T}\tilde{\kappa}+\bar{y},\end{split}$
(60)

where semi-positive term $M\triangleq\sum_{i=1}^{N}M_{i}$. The variable
$M_{i}$ is defined using Lemma 6 and $\bar{y}$ is given by

$\begin{split}\bar{y}\triangleq&\frac{1}{2}tr(\tilde{W}^{T}\alpha^{-1}\tilde{W})+\frac{1}{2}tr(\tilde{Z}^{T}\beta^{-1}\tilde{Z})+\frac{1}{2}tr(\tilde{V}^{T}\gamma^{-1}\tilde{V})\\\
=&\frac{1}{2}\text{vec}({\alpha^{\frac{-1}{2}}\tilde{W}})^{T}\text{vec}({\alpha^{\frac{-1}{2}}\tilde{W}})+\frac{1}{2}\text{vec}({\beta^{\frac{-1}{2}}\tilde{Z}})^{T}\text{vec}({\beta^{\frac{-1}{2}}\tilde{Z}})\\\
&+\frac{1}{2}\text{vec}({\gamma^{\frac{-1}{2}}\tilde{V}})^{T}\text{vec}({\gamma^{\frac{-1}{2}}\tilde{V}}).\end{split}$
(61)

Let us define $\bar{\kappa}=[\kappa_{1},...,\kappa_{N}]^{T}\in\mathbb{R}^{N}$,
${\kappa}_{d}=[{\kappa}_{d_{1}},...,\hat{\kappa}_{d_{N}}]^{T}\in\mathbb{R}^{N}$
and
$\tilde{\kappa}=[\tilde{\kappa}_{1},...,\tilde{\kappa}_{N}]\in\mathbb{R}^{N}$,
where $\tilde{\kappa}_{i}\triangleq\kappa_{i}-\kappa_{d_{i}}$. Moreover, let
$\chi=[\mathbf{e}^{T},\zeta^{T},r^{T},\sqrt{M},\text{vec}(\alpha^{\frac{-1}{2}}\hat{W})^{T},\text{vec}(\beta^{\frac{-1}{2}}\hat{Z})^{T}$,
$\text{vec}(\gamma^{\frac{-1}{2}}\hat{V})^{T},\kappa^{T}]^{T}$. It can be seen
that the following inequalities are valid for (60)

$L_{b}(\chi)\leq V\leq U_{b}(\chi),$ (62)

with

$L_{b}(\chi)=\underline{\eta}||\chi||^{2},\
U_{b}(\chi)=\bar{\eta}||\chi||^{2},$ (63)

where $\underline{\eta}\triangleq\
\text{min}\\{\frac{1}{2}\underline{g}\underline{\sigma}(Q),\frac{1}{2},\frac{1}{4}\underline{\sigma}(\Pi)\\}$,
$\bar{\eta}\triangleq\text{max}\\{\frac{1}{2}\bar{g}\bar{\sigma}(Q),1,\frac{1}{4}\bar{\sigma}(\Pi)\\}$.
To use Lemma 5, we should study the existence of the Filippov’s solution for
$\dot{\chi}=f(\chi,t)$, as $\dot{M}$ and $\dot{\kappa}$ are differential
equations with discontinuous right-hand side [40]. Using differential
inclusion [41, 42], an absolutely continuous Filipov’s solution exists for
$\dot{\chi}\in{K}{[\mathcal{F}}]({\chi},t)$ where an upper semi-continuous,
compact and convex set-valued map ${K}{[\mathcal{F}}]({\chi},t)$ is defined as
follows:

${K}{[\mathcal{F}}]({\chi},t)\triangleq\underset{R>0}{\cap}\underset{\mu(H)=0}{\cap}\overline{co}f(B(\chi,R)\backslash
H,t),$ (64)

where $B(\chi,R)$ is the closed ball with center $\chi$ and the radius $R$.
Using (64) and from [43, Theorem 2.2], the time-derivative of Lyapunov
function candidate exists almost everywhere:

$\dot{V}(\chi)\stackrel{{\scriptstyle
a.e.}}{{\in}}\underset{\Lambda\in\partial
V(\chi)}{\cap}\Lambda^{T}[{K}^{T}{[\mathcal{F}}]({\chi})]^{T}.$ (65)

As the Lyapunov function candidate (60) is continuously smooth, (65) can be
written as follows:

$\displaystyle\dot{V}(\chi)\stackrel{{\scriptstyle a.e.}}{{\in}}\triangledown
V\ {K}[$
$\displaystyle\dot{\mathbf{e}}^{T},\dot{\zeta}^{T},\dot{r}^{T},\dot{\mathcal{G}}^{T},\dot{M},\text{vec}{(\alpha^{-\frac{1}{2}}\dot{\hat{W}})}^{T},$
(66)
$\displaystyle\text{vec}{(\beta^{-\frac{1}{2}}\dot{\hat{Z}})}^{T},\text{vec}{(\gamma^{-\frac{1}{2}}\dot{\hat{V}})}^{T},\dot{\tilde{\kappa}}^{T}]^{T},$

where $\triangledown
V\triangleq[\mathbf{e}^{T},\frac{1}{2}\zeta^{T}\Pi,r^{T}\mathcal{G}Q,\frac{1}{2}\frac{\partial(r^{T}\mathcal{G}Qr)}{\partial\mathcal{G}},1,\text{vec}(\alpha^{-\frac{1}{2}}\hat{W})^{T},$
$\text{vec}(\beta^{-\frac{1}{2}}\hat{Z})^{T},\text{vec}(\gamma^{-\frac{1}{2}}\hat{V})^{T},\tilde{\kappa}^{T}]$.
Using (61) and (66), the time-derivative of (60) is given by

$\begin{split}\dot{V}\stackrel{{\scriptstyle
a.e.}}{{\in}}&\mathbf{e}^{T}\dot{\mathbf{e}}+\frac{1}{2}\zeta^{T}\Pi\dot{\zeta}+\frac{1}{2}r^{T}\dot{\mathcal{G}}Qr+r^{T}{\mathcal{G}}Q\dot{r}+\dot{M}+\tilde{\kappa}^{T}\dot{\tilde{\kappa}}\\\
&+tr(\tilde{W}^{T}\alpha^{-1}\dot{\tilde{W}})+tr(\tilde{Z}^{T}\beta^{-1}\dot{\tilde{Z}})+tr(\tilde{V}^{T}\gamma^{-1}\dot{\tilde{V}}).\end{split}$
(67)

Substituting (34), (36), (46), and (53) in (67) yields

$\displaystyle\dot{V}=$ $\displaystyle\mathbf{e}^{T}(\zeta-
k_{1}\mathbf{e})+\frac{1}{2}\zeta^{T}\Pi(r-k_{2}\zeta)+r^{T}\Big{(}-k_{2}\zeta-(k_{3}+k_{4}){r}$
(68)
$\displaystyle-\kappa(t)\text{Sgn}(\zeta)+\tilde{W}^{T}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}{\hat{\sigma}_{2}}-\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}^{T}{\mathcal{X}}\big{)}$
$\displaystyle+\hat{W}^{T}\hat{\sigma}_{1}^{\prime}\tilde{Z}^{T}\big{(}\hat{\sigma}_{2}-\hat{\sigma}_{2}^{\prime}\hat{V}^{T}\mathcal{X}\big{)}+\hat{W}^{T}\big{(}\hat{\sigma}_{1}^{\prime}\hat{Z}^{T}\hat{\sigma}_{2}^{\prime}\tilde{V}^{T}\mathcal{X}\big{)}$
$\displaystyle+N_{2}+\bar{\epsilon}_{M}\Big{)}+\tilde{\kappa}^{T}\dot{\tilde{\kappa}}-\sum_{i=1}^{N}\bigg{(}r_{i}^{T}\big{(}\bar{N}_{2_{i}}-\kappa_{d_{i}}\text{Sgn}(\zeta_{i})\big{)}$
$\displaystyle+\dot{\zeta}^{T}_{i}\tilde{N}_{1_{i}}+(W^{2}_{m}+V_{m}^{2}+Z^{2}_{m})\zeta^{T}_{i}\text{Sgn}(\zeta_{i})\bigg{)}$
$\displaystyle+tr(\tilde{W}^{T}\alpha^{-1}\dot{\tilde{W}})+tr(\tilde{Z}^{T}\beta^{-1}\dot{\tilde{Z}})+tr(\tilde{V}^{T}\gamma^{-1}\dot{\tilde{V}}),$

where by abusing notation,
$\text{Sgn}(\zeta_{i})=[\text{Sgn}(\zeta_{i1}),...,\text{Sgn}(\zeta_{in})]^{T}$.
The definition of $\text{Sgn}(x_{i})$, for $x_{i}\in R$ is given as in [42]

$\text{Sgn}(x_{i})=\begin{cases}-1&\text{if }x_{i}<0,\\\ [-1,1]&\text{if
}x_{i}=0,\\\ 1&\text{if }x_{i}>0.\end{cases}$ (69)

Reorganizing (68), from (36) and (45), one has

$\displaystyle\dot{V}\leq$ $\displaystyle\mathbf{e}^{T}(\zeta-
k_{1}\mathbf{e})+\frac{1}{2}\zeta^{T}\Pi(r-k_{2}\zeta)+r^{T}\big{(}-k_{2}\zeta$
(70)
$\displaystyle-(k_{3}+k_{4}){r}\big{)}+\tilde{\kappa}^{T}\dot{\tilde{\kappa}}+\sum_{i=1}^{N}(\kappa_{d_{i}}-\kappa_{i}(t))r_{i}^{T}\text{Sgn}(\zeta_{i})$
$\displaystyle-(W^{2}_{m}+V_{m}^{2}+Z^{2}_{m})\zeta^{T}_{i}\text{Sgn}(\zeta_{i})$
$\displaystyle+\sum_{i=1}^{N}\Big{(}tr(\tilde{W}_{i}^{T}\big{(}\alpha_{i}^{-1}\dot{\tilde{W}}_{i}+k_{2}\big{(}\hat{\sigma}_{1}-\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}{\hat{\sigma}_{2}}$
$\displaystyle-\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}\hat{\sigma}_{2}^{\prime}\hat{V}_{i}^{T}{\mathcal{X}_{i}}\big{)}\zeta_{i}^{T})\big{)}+tr\big{(}\tilde{Z}_{i}^{T}(\beta^{-1}\dot{\tilde{Z}}_{i}$
$\displaystyle+k_{2}\big{(}\hat{\sigma}_{2}-\hat{\sigma}_{2}^{\prime}\hat{V}_{i}^{T}\mathcal{X}_{i}\big{)}\zeta_{i}^{T}\hat{W}_{i}^{T}\hat{\sigma}_{1}^{\prime})\big{)}$
$\displaystyle+tr\big{(}\tilde{V}_{i}^{T}(\gamma^{-1}_{i}\dot{\tilde{V}}_{i}+k_{2}\mathcal{X}_{i}\zeta_{i}^{T}\hat{W}_{i}^{T}\hat{\sigma}_{1}^{\prime}\hat{Z}_{i}^{T}\hat{\sigma}_{2}^{\prime})\Big{)}.$

By substituting (48) and (59) in (70), one can get

$\displaystyle\dot{V}\leq$ $\displaystyle\mathbf{e}^{T}(\zeta-
k_{1}\mathbf{e})+\frac{1}{2}\zeta^{T}\Pi(r-k_{2}\zeta)+r^{T}(-k_{2}\zeta-(k_{3}+k_{4}){r})$
(71)
$\displaystyle+\sum_{i=1}^{N}\bigg{(}\|\zeta_{i}\|_{1}\Big{(}tr(\tilde{W}_{i}^{T}\hat{W}_{i})+tr(\tilde{Z}_{i}^{T}\hat{Z}_{i})+tr(\tilde{V}_{i}^{T}\hat{V}_{i})\Big{)}$
$\displaystyle-(W^{2}_{m}+V_{m}^{2}+Z^{2}_{m})\zeta^{T}_{i}\text{Sgn}(\zeta_{i})\bigg{)}.$

Applying Young’s inequality for $tr(\tilde{W}_{i}^{T}\hat{W}_{i})$, one has
$tr(\tilde{W}_{i}^{T}\hat{W}_{i})\leq\frac{W_{m}^{2}}{2}-\frac{\|\hat{W}_{i}\|_{F}^{2}}{2}$.
Following the same procedure for $tr(\tilde{Z}_{i}^{T}\hat{Z}_{i})$ and
$tr(\tilde{V}_{i}^{T}\hat{V}_{i})$, one can write

$\displaystyle\dot{V}\leq$ $\displaystyle\mathbf{e}^{T}(\zeta-
k_{1}\mathbf{e})+\frac{1}{2}\zeta^{T}\Pi(r-k_{2}\zeta)+r^{T}(-k_{2}\zeta-(k_{3}+k_{4}){r})$
(72)
$\displaystyle+\sum_{i=1}^{N}\big{(}\tilde{\kappa}_{i}^{T}\dot{\kappa}_{i}-\tilde{\kappa}^{T}_{i}r^{T}_{i}\text{Sgn}(\zeta_{i})\big{)}.$

From (2) and applying Cauchy–Schwarz inequality and Young’s inequality, the
following result can be obtained

$\begin{split}-k_{2}r^{T}\zeta\leq&\frac{k_{2}}{2}\underline{\sigma}(P(L+B))||\zeta||^{2}+k_{2}\frac{||r||^{2}}{2\underline{\sigma}(P(L+B))}.\end{split}$
(73)

Substituting (73) in (72) and from (2) one can write

$\begin{split}\dot{V}\leq&-(k_{1}-\frac{1}{2})||\mathbf{e}||^{2}-\big{(}\frac{k_{2}}{2}\underline{\sigma}(P(L+B))\\\
&-\frac{1+\bar{\sigma}(P(L+B))}{2}\big{)}||\zeta||^{2}-(k_{3}+k_{4}\\\
&-\frac{\bar{\sigma}(P(L+B))}{2}-\frac{k_{2}}{2\underline{\sigma}(P(L+B))})||r||^{2}).\end{split}$
(74)

From the given conditions for gains $k_{1},k_{2},k_{3},k_{4}$ in the Theorem
1, one has

$\begin{split}\dot{V}\leq&-\lambda_{1}||\mathbf{e}||^{2}-\lambda_{2}||\zeta||^{2}-\lambda_{3}||r||^{2},\end{split}$
(75)

where $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ are positive constants.
Consequently, (75) can be rewritten as

$\begin{split}\dot{V}\leq&-U(\chi),\end{split}$ (76)

where
$U(\chi)\triangleq-\underline{\lambda}||[\mathbf{e}^{T},\zeta^{T},r^{T}]^{T}||^{2}$,
with $\underline{\lambda}=\text{min}(\lambda_{1},\lambda_{2},\lambda_{3})$.
Note that in (76), $U(\chi)$ is a positive semi-definite function over
$\Omega_{\mathcal{X}}$. From (60) and (76), one has $V\in L_{\infty}$ over
$\Omega_{\mathcal{X}}$; hence,
$\mathbf{e},\zeta,r,\tilde{W},\tilde{Z},\tilde{V}$ and $\tilde{\kappa}$ are
bounded on set $\Omega_{\mathcal{X}}$. Consequently, from (34) $\delta\in
L_{\infty}$. From Assumption 1 and (33), one can conclude that $\mathbf{x}\in
L_{\infty}$. From (36), as $r\in L_{\infty}$ and $\zeta\in L_{\infty}$, one
can obtain $\dot{\zeta}\in L_{\infty}$. Using Lemma 2, Assumptions 1-3, and
(35), it can be deduced that $u\in L_{\infty}$. Based on Assumption 4, (43)
and (46) are also bounded. From the boundedness of $\mathbf{e},\delta,\zeta,r$
and the closed-loop terms, $U(\chi)$ is uniformly continuous, where Lemma 5
can be applied. Let $\Omega_{\mathcal{X}}=\\{\chi|\ ||\chi||\in B(0,R)\\}$,
and the initial conditions belong to the compact set
$\mathcal{S}\subset\Omega_{\mathcal{X}}$ as

$\mathcal{S}\triangleq\\{\chi\in\Omega_{\mathcal{X}}|U_{b}(\chi)\leq\underline{\eta}R^{2}\\},$
(77)

where the origin is within the set of $\mathcal{S}$. Then, the Lemma 5 is used
to conclude $||\mathbf{e}||^{2}\rightarrow 0,||\zeta||^{2}\rightarrow
0,||r||^{2}\rightarrow 0,\text{as}\ t\rightarrow\infty,\
\forall\chi(0)\in\mathcal{S}$. From (33) and (34), one can conclude that as
$||\mathbf{e}||\rightarrow 0$, and $||\delta||\rightarrow 0$, (9) holds.
Consequently, by increasing $R$, the semi-global asymptotic stability is
obtained [19]. This completes the proof. $\blacksquare$ ∎

##  IV Simulation Results

In order to study the performance of the proposed method, we consider a multi-
agent system with five two-link robot arms. The directed graph which models
the system is shown in Fig. 3. The dynamics of each agent is given by [2]

$\begin{array}[]{lr}\dot{p}_{i1}=p_{i2},\\\
M_{i}(p_{i1})\dot{p}_{i2}+V_{i}(p_{i1},p_{i2})p_{i2}+G_{i}(p_{i1})+w_{i}=\tau_{i},\end{array}$
(78)

for $i\in\\{1,...,5\\}$, where $p_{i1}=[p_{i11},p_{i12}]^{T}\in\mathbb{R}^{2}$
and $p_{i2}=[p_{i21},p_{i22}]^{T}\in\mathbb{R}^{2}$ are the joint position and
velocity states of $i$-th robot arm, respectively. In (78), the control input
is $\tau_{i}\triangleq u_{i}$. Expressions for the inertia matrix
$M_{i}(p_{i1})$, the Coriolis matrix $V_{i}(p_{i1},p_{i2})$, and the
gravitational vector $G_{i}(p_{i1})$ in (78) can be found in [32, 2]. In (78),
$w_{i}$ represents a bounded disturbance in the dynamics of each two-link
robot arm. The parameters of the each robot arm are $g=9.8m/s^{2}$,
$r_{i1}=1m$, $r_{i1}=1m$, $m_{i1}=0.8kg$, and $m_{i2}=1.7kg$,
$w_{i}=[-0.12cos(t),0.1sin(t)]^{T}$, $i\in\\{1,...,5\\}$. The leader dynamics
is given by

$\begin{array}[]{lr}\dot{p}_{l}=v_{l},\\\
\dot{v}_{l}=\begin{bmatrix}-p_{l_{1}}+0.2(1-p_{l_{1}}^{2})v_{l_{1}}\\\
-p_{l_{2}}+0.3(1-p_{l_{2}}^{2})v_{l_{2}}\end{bmatrix},\end{array}$ (79)

where $p_{l}=[p_{l_{1}},p_{l_{2}}]^{T}\in\mathbb{R}^{2}$ and
$v_{l}=[v_{l_{1}},v_{l_{2}}]^{T}\in\mathbb{R}^{2}$ are the position and
velocity states of the leader. The initial conditions for the five robots are
given as: $p_{11}(0)=[2.1,0]^{T}$, $p_{21}(0)=[0,2.5]^{T}$,
$p_{31}(0)=[-1.1,2]^{T}$, $p_{41}(0)=[-1.8,0.7]^{T}$,
$p_{51}(0)=[-1,-1.7]^{T}$, $p_{i2}(0)=[0,0]^{T},\forall i\in\\{1,...,5\\}$.
The desired displacement with respect to the leader for each agent is given
as: $d_{1}=[0,d]^{T}$,
$d_{2}=[-dsin(\frac{2\pi}{5}),dcos(\frac{2\pi}{5})]^{T}$,
$d_{3}=[-dsin(\frac{\pi}{5}),-dcos(\frac{\pi}{5})]^{T}$,
$d_{4}=[dsin(\frac{\pi}{5}),-dcos(\frac{\pi}{5})]^{T}$ and
$d_{5}=[dsin(\frac{2\pi}{5}),dcos(\frac{2\pi}{5})]^{T}$ with $d=1$. The leader
initial states are $p_{l}(0)=[1,-1]^{T}$ and $v_{l}(0)=[0,0]^{T}$,
respectively. All weights of edges in Fig. 3 are equal to $one$, as well as
the leader’s edge weight ($b_{1}=1$). Therefore, the adjacency matrix and
graph Laplacian are given by

$\small{A=\begin{bmatrix}0&0&0&0&1\\\ 1&0&0&0&0\\\ 1&1&0&0&0\\\ 0&0&1&0&1\\\
0&0&1&0&0\\\ \end{bmatrix},L=\begin{bmatrix}1&0&0&0&-1\\\ -1&1&0&0&0\\\
-1&-1&2&0&0\\\ 0&0&-1&2&-1\\\ 0&0&-1&0&1\\\ \end{bmatrix}.}$ (80)

We selected $10$ neurons for the input layer, $10$ neurons for the first
hidden layer, $22$ neurons for the second hidden layer, and $2$ neurons for
the output layer for each agent. The activation function for these layers is a
sigmoid function. The activation function for the output layer is linear. The
constants $\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$, $\forall i$, are chosen as
$\alpha_{i}=\frac{1}{20}$, $\beta_{i}=\frac{1}{20}$,
$\gamma_{i}=\frac{1}{20}$. The gains were selected as: $k_{1}=4$,
$k_{2}=37.5$, $k_{3}=380$, and $k_{4}=2$.

Figs. 4-10 show the simulation results for the multi-agent system (78). In
Fig. 4, the performance of the multi-agent system is shown where dash-dotted
lines indicate the multi-agent system achieving its desired formation. Fig. 5
illustrates trajectories of agents and the leader. The velocities of agents,
as well as the leader, are shown as Fig. 6, where velocities of agents become
identical to the leader velocity. The Frobenius norm of the NN weights
matrices of $\hat{V}_{i}$, $\hat{Z}_{i}$ and $\hat{W}_{i}$ are displayed as
Fig. 7. Fig. 8 demonstrates the time-varying gains $\kappa_{i}$. To avoid that
$\kappa_{i}(t)$ reaches high values, a dead-zone with size of $b=0.005$, has
been implemented, similar to [17, 36]. The error signals ${e}_{i}(t)$, their
time-derivatives $\delta_{i}(t)$, and the filtered errors $\zeta_{i}(t)$ are
shown in Fig. 9. The control input of each agent is shown in Fig. 10.

These results verify the performance of the proposed method based on RISE
feedback and the NN-based controller. To compare our proposed method with some
other existing works, we considered results on leader-following consensus
control of the nonlinear system with unknown nonlinearity [3, 2]. We included
the desired displacements and applied these methods similar to [44, p. 127].
We define the average of the control inputs cost function (ACI), $\nu(t)$ and
average of the formation errors cost function (AFE), $\vartheta(t)$, as
follows:

$\begin{split}\nu(t)=&\frac{1}{2N}\sum_{i=1}^{N}u_{i}^{T}(t)u_{i}(t),\\\
\vartheta(t)=&\frac{1}{2N}\sum_{i=1}^{N}||{e}_{i}(t)||_{1}.\end{split}$ (81)

We use these two functions as performance indices to evaluate and compare the
effectiveness of the proposed methods in comparison with [3, 2]. Fig. 11(a),
shows the semi-logarithmic graph of ACI function. However, the proposed method
(solid-line) energy consumption is almost the same as the other two methods.
Fig. 11(b) indicates that our proposed method settled faster and converged to
zeros in contrast with two other methods where the error remains bounded.

Figure 3: The directed graph of five agents with the leader as the root of the
spanning tree. Figure 4: Performance of the multi-agent system (78). Figure 5:
Evolution of agents and the leader position states (leader is the solid line).
Figure 6: Evolution of agents and the leader velocity states (leader is the
solid line). Figure 7: Frobenius Norm of NN weights matrices for agents.
Figure 8: Time-variant gain $\kappa_{i}$ of each agent. Figure 9: Error
signals of $\mathbf{e}$, $\delta$, and $\zeta$ for each agent. Figure 10:
Control inputs of each agent. Figure 11: (a) ACI function comparison between
proposed method and Method 1 [2] and Method 2 [3] ($y$-axis is in logarithmic
scale); (b) AFE function comparison between proposed method and Method 1 [2]
and Method 2 [3].

##  V Conclusion

This paper developed the leader-following formation control of the
heterogeneous, second-order, uncertain, input-affine, nonlinear multi-agent
systems modeled by a directed graph. The unknown nonlinearity in the dynamics
of a multi-agent system was approximated by a tunable, three-layer NN
consisting of an input layer, two hidden layers, and an output layer. The
proposed method can a priori set the number of neurons in each layer of NN.
The NN weights tuning laws were derived using the Lyapunov theory. A robust
integral of the sign of the error feedback control with an NN was developed to
guarantee semi-global asymptotic leader tracking. The boundedness of the
close-loop signals and asymptotic leader tracking formation were proven using
the Lyapunov stability theory. The paper results were compared with two
previous results, which showed the effectiveness of the proposed method.

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