# Resilient Output Containment Control of Heterogeneous Multi-agent Systems
against Composite Attacks: A Novel Digital Twin Approach

Xin Gong, _Graduate Student Member, IEEE_ , Yukang Cui, _Member, IEEE_ ,
Lingbo Cao  This work was partially supported by the National Natural Science
Foundation of China under Grant 61903258, 61973156, 61603180, Qatar National
Research Fund NPRP12C-0814-190012. X. Gong is with the Department of
Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong
(e-mail: gongxin@connect.hku.hk). Y. Cui and Lingbo. Cao are with the College
of Mechatronics and Control Engineering, Shenzhen University, Shenzhen,
518060, China (e-mail: cuiyukang,lingbcao@gmail.com).

###### Abstract

This work studies the distributed resilient output containment control of
heterogeneous multi-agent systems against composite attacks, including Denial-
of-Services (DoS) attacks, false-data injection (FDI) attacks, camouflage
attacks (CAs), and actuation attacks (AAs). Inspired by the concept of Digital
Twin, a twin layer (TL) with higher security and privacy is used to decouple
the above problem into the defense protocol against DoS attacks on the TL and
the defense protocol against AAs on the cyber-physical layer (CPL). First,
considering the existence of TL layer modeling error, we provide distributed
observers to estimate the leaders dynamics for each follower. Then,
distributed observers, which use the estimates of leaders dynamics by the
above observers, are provided to observe the trajectories of followers under
DoS attacks. On the CPL, we give the estimated solutions to the output regular
equations by estimating the leaders dynamics. Moreover, decentralized adaptive
attack-resilient control schemes are provided to deal with the unbounded AAs.
Then, the output containment error can achieve uniformly ultimately bounded
(UUB) convergence by the above control protocols and the boundary of output
containment error can be given explicitly. Finally, the simulations show the
effectiveness of these proposed control protocols.

###### Index Terms:

Composite attacks, Containment, Directed graphs, High-order Multi-agent
systems, Twin Layer

##  I Introduction

DISTRIBUTED cooperative control of multi-agent systems (MASs) have attracted
extensive attention over the past decade due to its broad applications in
satellite formation [scharnagl2019combining], mobile multirobot
[wang2017adaptive], and smart grids [ansari2016multi]. As one of the most
general phenomena in the area of cooperative control of MASs, consensus
problems can be categorized into two classes, that is, leaderless consensus
[fax2004information, ren2007information, olfati2007consensus] and
leader–follower consensus [hong2013distributed, zhao2016distributed,
su2011cooperative].  Their control scheme is to drive all agent reach a
specified ideal trajectory. In recent years, many scholars have introduced
multiple leaders on the consensus, whose control target tracks ideal
trajectories to drive all followers into the convex hull formed by the
leaders, which is also named the containment problem [wieland2011internal,
su2012cooperative, kim2010output, wang2010distributed,
lunze2012synchronization, yaghmaie2016output]. Among them, Haghshenas et al.
[haghshenas2015] and Zuo et al. [zuo2017output] solved the containment control
problem of heterogeneous linear multi-agent systems by output regular
equation. Zuo et al. [zuo2019] considered the containment control problem of
heterogeneous linear multi-agent systems against camouflage attacks (CAs). Zuo
et al. [zuo2020] presented adaptive compensator against unknown actuator
attacks (AAs). Lu et al. [lu2021] considered the containment control problem
against false-data injection (FDI) attacks. Ma et al. [ma2021observer] solved
the containment control problem against denial-of-service (DoS) by presenting
an event-triggered observer. However, the above works of containment control
only considered the case of free attack or only against a type of attack.
Considering the vulnerability of MASs, the situation is more complex in
reality, it may face multiple attacks at the same time. Therefore, this work
considers the multi-agent elastic output containment control problem to resist
composite attacks, which including DoS attacks, CAs, FDI attacks and AAs.

In recent years, digital twins [zhu2020, gehrmann2019digital] have been widely
used in many fields, such as industry [abburu2020cognitwin], aerospace
[liu2022intelligent], and medicine [voigt2021digital] . Digital twin
technology is the real-time mapping of things in virtual space. Inspired by
this technology, this work designs a double-layer control structure consisting
of TL and CPL. TL layer is the virtual mapping of CPL. It has the same
communication topology as CPL, and can interact with CPL in real time. It has
high privacy and security, which can effectively cope with the interference of
error information in communication network, such as CAs and FDI attacks.
Therefore, the main work of this paper is to design the corresponding control
framework to resist DoS attacks and AAs. Since TL layer is only affected by
DoS attacks, the main work of this paper can be divided into two part: the
design to resist DoS attacks on TL and the design to resist AAs on CPL.

Resilient control protocols of MASs against DoS attacks have been investigated
in [fengz2017, zhangd2019, yed2019, deng2021, yang2020]. Feng et al.
[fengz2017] considered the leader-follower consensus through its latest state
and the leader dynamics under the DoS attacks. Yang et al. [yang2020] designed
a dual-terminal event-triggered mechanism against DoS attacks for linear
leader-following MASs. However, the above works need to know the leader
dynamics for each follower, which is unrealistic in some cases. To deal with
this problem, Cai et al. [cai2017] and Chen et al. [chen2019] used an adaptive
distributed observer to estimate the leader dynamics and state for the MAS
with single leader. Deng et al. [deng2021] proposed a distributed resilient
learning algorithm to learn the unknown dynamic model. But they have not taken
into account the simultaneous existence of leader estimation errors and DoS
attacks. Herein, motivated by the above works, we consider the TL with leader
modeling errors, which is inevitable by the online modeling. This is an
interesting and challenged work, because the leader modeling errors will
further affect the state observer errors and output containment errors. To
deal with this problem, we establish a double-TL structure to deal with
modeling errors and DoS attacks. On the first TL, distributed estimators are
established to provide leaders dynamics for each follower. Then, on the second
TL, distributed state observers are used to estimate the state of containment.

Resilient control protocols of MASs against AAs have been studied in
[chen2019, zuo2020, lu2021]. A pioneering work [de2014resilient] presented a
distributed resilient adaptive control scheme against attacks in the form of
exogenous disturbances and interagent uncertainties. Xie et al.
[xie2017decentralized] dealt with the decentralized adaptive fault-olerant
control problem for a class of uncertain large-scale interconnected systems
with external disturbances and AAs. Moreover, Jin et al. [jin2017adaptive]
provided an adaptive controller to guarantee that the closed-loop dynamic
system against time-varying adversarial sensor and actuator attacks could
achieve uniformly ultimately bounded (UUB) performance. It should be noted
that the above works [de2014resilient, xie2017decentralized, jin2017adaptive]
could only deal with bounded AAs (or disturbances) which cannot maximize their
destructive capacity indefinitely. Recently, Zuo et al. [zuo2020] provided an
adaptive control scheme for MAS against unbounded AAs. Based on the above
works, we provide double-CPL to deal with output containment problem against
AAs. On the first CPL, we provide the output regulator equation solver to deal
with the output containment of heterogeneous MAS. On the second CPL, we use
distributed attack-resilient protocols with an adaptive compensation signal
against unbounded AAs. Moreover, compared with [zuo2020], our adaptive
compensation signal is featured by a soft-sign function, and the containment
error bound can be given explicitly.

The above discussions indicate that the problem of output containment of
heterogeneous MASs with the simultaneous existence of leader estimation and
composite attacks has not been solved or studied. In this work, we provide a
hierarchical control scheme to deal with the above problem. Moreover, our main
contributions are summarised as follows:

  1. 1.

Different from the previous works, our work considers the digital twin
modeling error, so that the TL needs to consider the coexistence of DoS
attacks and modeling error, and the CPL needs to consider the coexistence of
output adjustment error and AAs error of CPL, which makes the analysis more
complex and more practical. Compared with the existing work [deng2021] in
which the leader state estimator start to work need the necessary time to
finish the estimation of leaders dynamics, the TL in our work could work in
all time.

  2. 2.

The TL-CPL double-layer control structure is used, and a double-TL is
established in the control framework. Different from CPL, the TL has higher
security. It can effectively defend against many types of attacks, such as CAs
and FDI attacks. Moreover, owing to the existence of TL, the resilient control
scheme is decoupled into the defense against DoS attacks on the TL and the
defense against potentially unbounded AAs on the CPL

  3. 3.

Compared with [zuo2020], our work employs adaptive compensation signal against
AAs, which possesses the UUB convergence and chattering-free merit. Moreover,
the containment output error bound of the UUB performance is given explicitly.

Notations: In this work, $I_{r}$ is the identity matrix with compatible
dimension. $\bm{1}_{m}$ (or $\bm{0}_{m}$) denotes a column vector of size $m$
filled with $1$ (respectively, 0). Denote the index set of sequential integers
as $\textbf{I}[m,n]=\\{m,m+1,\ldots~{},n\\}$, where $m<n$ are two natural
numbers. Define that the set of real numbers and the set of natural numbers as
$\mathbb{R}$ and $\mathbb{N}$, respectively. For any matrix
$A\in\mathbb{R}^{M\times N}$, ${\rm vec}(A)={\rm
col}(A_{1},A_{2},\dots,A_{N})$ where $A_{i}\in\mathbb{R}^{M}$ is the $i$th
column of $A$. Define ${\rm{\rm blkdiag}}(A_{1},A_{2},\dots,A_{N})$ as a block
diagonal matrix whose principal diagonal elements equal to given matrices
$A_{1},A_{2},\dots,A_{N}$. $\sigma_{\rm min}(A)$, $\sigma_{\rm max}(A)$ and
$\sigma(A)$ denote the minimum singular value, maximum singular value and the
spectrum of matrix $A$, respectively. $\lambda_{\rm min}(A)$ and $\lambda_{\rm
max}(A)$ represent the minimum eigenvalue, maximum eigenvalue of $A$. Herein,
$||\cdot||$ denotes the Euclidean norm, and $\otimes$ denotes the Kronecker
product.

##  II Preliminaries

###  II-A Graph Theory

We use a directed graph to represent the interaction among agents. For a MAS
consisting of $n$ agents, the graph $\mathcal{G}$ can be expressed by a pair
$(\mathcal{V},\mathcal{E},\mathcal{A})$, where
$\mathcal{V}=\\{1,2,\ldots~{},N\\}$ is the node set,
$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}=\\{(v_{j},\ v_{i})\mid v_{i},\
v_{j}\in\mathcal{V}\\}$ is the edge set, and
$\mathcal{A}=[a_{ij}]\in\mathbb{R}^{N\times N}$ is the associated adjacency
matrix. The weight of the edge $(v_{j},\ v_{i})$ is denoted by $a_{ij}$ with
$a_{ij}>0$ if $(v_{j},\ v_{i})\in\mathcal{E}$, otherwise $a_{ij}=0$. The
neighbor set of node $v_{i}$ is represented by
$\mathcal{N}_{i}=\\{v_{j}\in\mathcal{V}\mid(v_{j},\ v_{i})\in\mathcal{E}\\}$.
Define the Laplacian matrix as
$L=\mathcal{D}-\mathcal{A}\in\mathbb{R}^{N\times N}$ with $\mathcal{D}={\rm
blkdiag}(d_{i})\in\mathbb{R}^{N\times N}$ where
$d_{i}=\sum_{j\in\mathcal{F}}a_{ij}$ is the weight in-degree of node $v_{i}$.
A directed is _strongly connected_ if there is a path from $v_{i}$ to $v_{j}$
for any pair of nodes $(v_{i},v_{j})$.

###  II-B Some Useful Lemmas and Definition

######  Lemma 1 (Bellman-Gronwall Lemma [lewis2003])

Assuming $\Phi:[T_{a},T_{b}]\rightarrow\mathbb{R}$ is a nonnegative continuous
function, $\alpha:[T_{a},T_{b}]\rightarrow\mathbb{R}$ is integrable,
$\kappa\geq 0$ is a constant, and there has

$\Phi(t)\leq\kappa+\int_{0}^{t}\alpha(\tau)\Phi(\tau)\,d\tau,t\in[T_{a},T_{b}],$
(1)

then we obtain that

$\Phi(t)\leq\kappa e^{\int_{0}^{t}\alpha(\tau)\,d\tau}$

for all $t\in[T_{a},T_{b}]$.

##  III System Setup and Problem Formulation

In this section, a new problem called resilient containment of MAS group
against composite attacks is proposed. First, the model of the MAS group is
formulated and some basic definitions of the composite attacks is given as
follows.

###  III-A MAS group Model

In the framework of containment control, we consider a group of $N+M$ MAS,
which can be divided into two groups:

1) $M$ leaders are the roots of directed graph $\mathcal{G}$, which have no
neighbors. Define the index set of leaders as
$\mathcal{L}=\textbf{I}[N+1,N+M]$.

2) $N$ followers who coordinate with their neighbors to achieve the
containment set of the above leaders. Define the index set of followers as
$\mathcal{F}=\textbf{I}[1,N]$.

Similar to many existing works [zuo2020, zuo2021, haghshenas2015], we consider
the following leaders dynamics :

$\begin{cases}\dot{x}_{k}=Sx_{k},\\\ y_{k}=Rx_{k},\end{cases}$ (2)

where $x_{k}\in\mathbb{R}^{q}$ and $y_{k}\in\mathbb{R}^{p}$ are system state
and reference output of the $k$th leader, respectively.

The dynamics of each follower is given by

$\begin{cases}\dot{x}_{i}=A_{i}x_{i}+B_{i}u_{i},\\\
y_{i}=C_{i}x_{i},\end{cases}$ (3)

where $x_{i}\in\mathbb{R}^{ni}$, $u_{i}\in\mathbb{R}^{mi}$ and
$y_{i}\in\mathbb{R}^{p}$ are system state, control input and output of the
$i$th follower, respectively. For convenience, the notation
${}^{\prime}(t)^{\prime}$ can be omitted in the following discussion. We make
the following assumptions about the agents and the communication network.

###### Assumption 1

There exists a directed path from at least one leader to each follower
$i\in\mathcal{F}$ in the graph $\mathcal{G}$.

###### Assumption 2

The real parts of the eigenvalues of $S$ are non-negative.

###### Assumption 3

The pair $(A_{i},B_{i})$ is stabilizable and the pair $(A_{i},C_{i})$ is
detectable for $i\in\mathcal{F}$.

###### Assumption 4

For all $\lambda\in\sigma(S)$, where $\sigma(S)$ represents the spectrum of
$S$,

${\rm rank}\left[\begin{array}[]{c|c}A_{i}-\lambda I_{n_{i}}&B_{i}\\\
\hline\cr C_{i}&0\\\ \end{array}\right]=n_{i}+p,i\in\mathcal{F}.$ (4)

###### Remark 1

Assumptions 1,3 and 4 are standard in the classic output regulation problem,
Assumption 2 is avoid the trivial case of stable $S$. The modes associated
with the eigenvalues of $S$ with negative real parts will exponentially decay
to zero, and do not affect the asymptotic behavior of the closed-loop system.
$\hfill\hfill\square$

###### Definition 1

For the $i$th follower, the system accomplishes containment if there exists a
series of $\alpha_{\cdot i}$, which satisfy
$\sum_{k\in\mathcal{L}}\alpha_{ki}=1$ to ensure following equation holds:

${\rm
lim}_{t\rightarrow\infty}(y_{i}(t)-\sum_{k\in\mathcal{L}}\alpha_{ki}y_{k}(t))=0,$
(5)

where $i\in\textbf{I}[1,N]$.

###  III-B Attack Descriptions

In this work, we consider the MASs consisting of cooperative agents with
potential malicious attackers. As shown in the Fig. 1, the attackers will use
four kinds of attacks to compromise the containment performance of the MASs:

Figure 1: Resilient MASs against Composite Attacks: A double-layer framework.

1) DoS attacks (Denial-of-Services attacks): The communication graphs among
agents (both in TL and CPL) denied by attackers;

2) AAs (actuation attacks): The motor inputs infiltrated by attackers to
destroy the input signal of the agent;

3) FDI attacks (fault-data injection attacks): The exchanging information
among agents distort by attackers;

4) CAs (camouflage attacks): The attacker mislead its downstream agents by
disguising as a leader.

To resist the composite attacks, we introduced a new layer named TL with the
same communication topology as CPL, which yet have greater security and less
physical meanings. Therefore, this TL could effectively against most of the
above attacks. With the introduction of TL, the resilient control scheme can
be decoupled to defend against DoS attacks on TL and defend against potential
unbounded AAs on CPL. The following two small subsections give the definitions
and essential constraints for the DoS attacks and AAs, respectively.

1) DoS attacks: DoS attacks refers to a type of attack where an adversary
presents some or all components of a control system. It can affect the
measurement and control channels simultaneously, resulting in the loss of data
availability. Assume that there exists a $l\in\mathbb{N}$, define
$\\{t_{l}\\}_{l\in\mathbb{N}}$ and $\\{\Delta_{*}\\}_{l\in\mathbb{N}}$ as the
start time and the duration time of the $l$th attack sequence of DoS attacks,
that is, the $l$th DoS attack time-interval is
$A_{l}=[t_{l},t_{l}+\Delta_{*})$ with $t_{l+1}>t_{l}+\Delta_{*}$ for all
$l\in\mathbb{N}$. Therefore, for all $t\geq\tau\in\mathbb{R}$, the set of time
instants where the communication network is under DoS attacks are represent by

$\Xi_{A}(\tau,t)=\cup A_{l}\cap[\tau,t],l\in\mathbb{N},$ (6)

and the set of time instants where the communication network is allowed are

$\Xi_{N}(\tau,t)=[\tau,t]/\Xi_{A}(\tau,t).$ (7)

######  Definition 2 (Attack Frequency [fengz2017])

For any $\delta_{2}>\delta_{1}\geq t_{0}$, let $N_{a}(t_{1},t_{2})$ represent
the number of DoS attacks in $[t_{1},t_{2})$. Therefore,
$F_{a}(t_{1},t_{2})=\frac{N_{a}(t_{1},t_{2})}{t_{2}-t_{1}}$ is defined as the
attack frequency at $[t_{1},t_{2})$ for all $t_{2}>t_{1}\geq t_{0}$.

######  Definition 3 (Attack Duration [fengz2017])

For any $t_{2}>t_{1}\geq t_{0}$, let $T_{a}(t_{1},t_{2})$ represent the total
time interval of DoS attacks on multi-agent systems during $[t_{1},t_{2})$.
The attack duration over $[t_{1},t_{2})$ is defined as: there exist constants
$\tau_{a}>1$ and $T_{0}>0$ such that

$T_{a}(t_{1},t_{2})\leq T_{0}+\frac{t_{2}-t_{1}}{\tau_{a}}.$ (8)

2) Unbounded Actuation Attacks: For each follower, the system input is under
unknown actuator faults, which is described as

$\bar{u}_{i}=u_{i}+\chi_{i},\forall i\in\mathcal{F},$ (9)

where $\chi_{i}$ denotes the unknown unbounded attack signal caused by
actuator faults. That is, the ture values of $u_{i}$ and $\chi_{i}$ are
unknown and we can only measure the damaged control input information
$\bar{u}_{i}$.

###### Assumption 5

The actuator attack signal $\chi_{i}$ is unbounded and its derivative
$\dot{\chi}_{i}$ is bounded by $\bar{d}$.

###### Remark 2

In contrast to the works [deng2021] and [chen2019], which only consider the
bounded AAs, this work tackles with unbounded AAs under the Assumption 5. In
the case when the derivative of the attack signal is unbounded, that is, the
attack signal increases at a extreme speed, the MAS can reject the signal by
removing the excessively large derivative values, which can be easily
detected. $\hfill\hfill\square$

###  III-C Problem Formulation

Under the condition that there exists no attack, the $i$th follower output
containment error is represented as:

$\xi_{i}=\sum_{j\in\mathcal{F}}a_{ij}(y_{j}-y_{i})+\sum_{k\in\mathcal{L}}g_{ik}(y_{k}-y_{i}),$
(10)

where $a_{ij}$ is the weight of edge $(v_{i},v_{j})$ in the graph
$\mathcal{G}_{f}$, and $g_{ik}$ is the weight of the path from $i$th leader to
$k$th follower.

The global form of (10) is written as

$\xi=-\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})(y-\underline{y}_{k}),$
(11)

where $\Psi_{k}=(\frac{1}{m}L_{f}+G_{ik})$ with $L_{f}$ is the Laplacian
matrix of communication digraph $\mathcal{G}_{f}$ and $G_{ik}={\rm
diag}(g_{ik})$,
$\xi=[\xi_{1}^{\mathrm{T}},\xi_{2}^{\mathrm{T}},\dots,\xi_{n}^{\mathrm{T}}]^{\mathrm{T}}$,
$y=[y_{1}^{\mathrm{T}},y_{2}^{\mathrm{T}},\dots,y_{n}^{\mathrm{T}}]^{\mathrm{T}}$,
$\underline{y}_{k}=(l_{n}\otimes y_{k})$.

######  Lemma 2 ([haghshenas2015])

Suppose Assumption 1 holds, then the matrices $\Psi_{k}$ and
$\bar{\Psi}_{L}=\sum_{k\in\mathcal{L}}\Psi_{k}$ are positive-definite and non-
singular. Therefore, both $(\Psi_{k})^{-1}$ and
$(\bar{\Psi}_{L})^{-1}=(\sum_{k\in\mathcal{L}}\Psi_{k})^{-1}$ are non-
singular.

The global output containment error [zuo2019] can be written as:

$e=y-(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{y}_{k},$
(12)

where
$e=[e_{i}^{\mathrm{T}},e_{2}^{\mathrm{T}},\dots,e_{n}^{\mathrm{T}}]^{\mathrm{T}}$
and $\xi=-\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})e$.

###### Lemma 3

Define that $\Theta={\rm diag}(v)$ where $v=(\bar{\Psi}_{L})^{-1}\bm{1}_{N}$.
Under Assumption 1 and Lemma 2, the matrix
$\Omega=\Theta\bar{\Psi}_{L}+\bar{\Psi}_{L}^{\mathrm{T}}\Theta>0$ is a
positive diagonal matrix.

######  Lemma 4 ([zuo2019, Lemma 1])

Under Assumption 1, the output containment control objective in (1) is
achieved if ${\rm lim}_{t\rightarrow\infty}e=0$.

######  Definition 4 ([khalil2002nonlinear])

The signal $x(t)$ is said to be UUB with the ultimate bound $b$, if there
exist positive constants $b$ and $c$, independent of $t_{0}\geq 0$, and for
every $a\in(0,c)$, there is $T_{1}=T_{1}(a,b)\geq 0$, independent of $t_{0}$,
such that

$||x(t)||\leq a\Rightarrow||x(t)||\leq b,\forall t\geq t_{0}+T_{1}.$ (13)

Consider the composite attacks in Subsection III-B, the local neighboring
relative output containment information is rewritten as

$\bar{\xi}_{i}=\sum_{j\in\mathcal{F}}d_{ij}(\bar{y}_{i,j}-\bar{y}_{i,i})+\sum_{k\in\mathcal{L}}d_{ik}(\bar{y}_{i,k}-\bar{y}_{i,i})+\sum_{l\in{\mathcal{C}}}c_{il}(y_{l}-\bar{y}_{i,i}),$
(14)

where $d_{ij}(t)$ and $d_{ik}(t)$ are the edge weights which influence by the
DoS attacks. For the denied communication link, $d_{ij}(t)=0$ and
$d_{ik}(t)=0$; for the normal communication link, $d_{ij}(t)=a_{ij}$ and
$d_{ik}(t)=g_{ik}$. And $\bar{y}_{i,j}$ (or $\bar{y}_{i,i}$) represent the
information flow of $y_{j}$ to the $i$th follower, compromised by the FDI
attacks. $\mathcal{C}$ is the index set of all camouflage attackers. $c_{il}$
present the edge weight for $l$th camouflage attacks to the $i$th follower.
From equation (14), it can see that the attackers could disrupt the
communication between agents and distort the actuation inputs of all
followers.

Based on the discussion aforementioned, the attack-resilient containment
control problem is introduced as follow.

Problem ACMCA (Attack-resilient Containment control of MASs against Composite
Attacks): The resilient containment control problem is to design the input
$u_{i}$ in (3) for each follower, such that output containment error $e$ in
(12) is UUB under the composite attacks, i.e., the trajectories of each
follower converges into a point in the dynamic convex hull spanned by
trajectories of multiple leaders.

##  IV Main Results

In this section, a double-layer resilient control scheme is used to solve the
Problem ACMCA. First, a distributed virtual resilient TL, which can resist the
FDI attacks, CAs, and AAs naturally, is proposed to against DoS attacks.
Considering the molding errors of TL, we use the distributed observers to
estimate the leaders dynamics for all agents under the effect of DoS attacks.
Next, distributed estimators are proposed to observe the containment
trajectories of followers under DoS attacks. Then, we use the estimation of
leaders dynamics in Theorem 1 to get the solution of output regulator
equation. Finally, adaptive controllers are proposed to resist unbounded AAs
in CPL. The specific control framework is shown in Figure. 2.

Figure 2:  The control framework of ACMCA.

###  IV-A Distributed Observers to Estimate Leaders Dynamics

In this section, we design distributed observers to remold the leaders
dynamics in the TL.

To facilitate the analysis, define the leaders dynamics in (2) as follows:

$\Upsilon=[S;R]\in\mathbb{R}^{(p+q)\times q}$ (15)

and its estimations can be defined as follows:

$\displaystyle\hat{\Upsilon}_{i}(t)=[\hat{S}_{i}(t);\hat{R}_{i}(t)]\in\mathbb{R}^{(p+q)\times
q},$ (16)

where $\hat{\Upsilon}_{i}(t)$ will be updated by (1), and it converge to
$\Upsilon$ exponentially.

###### Theorem 1

Consider a group of $M$ leaders and $N$ followers with dynamic in (2) and (3).
Suppose that Assumption 1 holds. The problem of the leader unknown dynamics
under the DoS attacks is solved by the following dynamic estimates
$\hat{\Upsilon}_{i}(t)$ :

$\displaystyle\dot{\hat{\Upsilon}}_{i}(t)=\mu_{1}(\sum_{j\in\mathcal{F}}d_{ij}(\hat{\Upsilon}_{j}(t)\\!-\\!\hat{\Upsilon}_{i}(t))$
$\displaystyle\\!+\\!\sum_{k\in\mathcal{L}}d_{ik}(\Upsilon\\!-\\!\hat{\Upsilon}_{i}(t))),$
$\displaystyle i=\textbf{I}[1,N],$ (17)

where $\mu_{1}>\frac{\sigma_{\rm max}(S)}{\lambda_{\rm
min}(\Omega\Theta^{-1}))(1-\frac{1}{\tau_{a}})}$ is the estimator gain with
$\bar{\Psi}_{L}=\sum_{k\in\mathcal{L}}\Psi_{k}$.

Proof. Form the (1), it can seen that we only used the relative neighborhood
information to estimate leaders dynamics, therefore, the leader observer will
suffer the influence of DoS attacks.

Step 1: Let $\tilde{\Upsilon}_{i}(t)=\Upsilon-\hat{\Upsilon}_{i}(t)$, then

$\displaystyle\dot{\tilde{\Upsilon}}_{i}(t)$
$\displaystyle=\dot{\Upsilon}-\dot{\hat{\Upsilon}}_{i}(t)$
$\displaystyle=-\mu_{1}(\sum_{j=1}^{N}d_{ij}(\hat{\Upsilon}_{j}(t)-\hat{\Upsilon}_{i}(t))$
$\displaystyle+\sum_{k\in\mathcal{L}}d_{ik}(\Upsilon-\hat{\Upsilon}_{i}(t))).$
(18)

The global estimation dynamic error of system (1) can be written as

$\displaystyle\dot{\tilde{\Upsilon}}(t)$
$\displaystyle=-\mu_{1}(\sum_{k\in\mathcal{L}}\Psi_{k}^{D}\otimes
I_{p+q}\tilde{\Upsilon}(t))$ $\displaystyle=-\mu_{1}(\bar{\Psi}_{L}^{D}\otimes
I_{p+q})\tilde{\Upsilon}(t),t\geq t_{0}.$ (19)

where $\bar{\Psi}^{D}_{L}=\sum_{k\in\mathcal{L}}\Psi_{k}^{D}$,
$\Psi_{k}^{D}(t)=\begin{cases}0,t\in\Xi_{A},\\\
\Psi_{k},t\in\Xi_{N},\end{cases}$, and
$\tilde{\Upsilon}(t)=[\tilde{\Upsilon}_{1}(t),\tilde{\Upsilon}_{1}(t),\dots,\tilde{\Upsilon}_{N}(t)]$.
Form (IV-A), we have ${\rm
vec}(\dot{\tilde{\Upsilon}})=-\mu_{1}(I_{q}\otimes\bar{\Psi}^{D}_{L}\otimes
I_{p+q}){\rm vec}(\tilde{\Upsilon})$. Consider the following Lyapunov function

$V_{1}(t)={\rm
vec}(\tilde{\Upsilon})^{\mathrm{T}}(\Theta\otimes\bar{P}_{1}){\rm
vec}(\tilde{\Upsilon}),$ (20)

The derivative of $V_{1}(t)$ can be written as follows

$\displaystyle\dot{V}_{1}$ $\displaystyle={\rm
vec}(\tilde{\Upsilon})^{\mathrm{T}}(-\mu_{1}((\bar{\Psi}^{D}_{L})^{\mathrm{T}}\Theta+\Theta\bar{\Psi}^{D}_{L})\otimes\bar{P}_{1}){\rm
vec}(\tilde{\Upsilon})$ (21) $\displaystyle\leq-\mu_{1}\lambda_{\rm
min}(\Omega^{D}\Theta^{-1})V_{1},$

where
$\Omega^{D}=(\bar{\Psi}^{D}_{L})^{\mathrm{T}}\Theta+\Theta\bar{\Psi}^{D}_{L}$.
Then, we have

$\displaystyle V_{1}(t)\leq e^{-\mu_{1}\lambda_{\rm
min}(\Omega^{D}\Theta^{-1})(t-t_{0})}V_{1}(t_{0}),$ (22) $\displaystyle{\rm
vec}(\tilde{\Upsilon})\leq\frac{V_{1}(t_{0})}{\lambda_{\rm
min}(\Theta\otimes\bar{P}_{1})}e^{-\mu_{1}\lambda_{\rm
min}(\Omega^{D}\Theta^{-1})(t-t_{0})}.$

From Definition 3, one has

$\displaystyle\bar{\Psi}_{L}^{D}(t-t_{0})$
$\displaystyle=\bar{\Psi}_{L}\left\lvert\Xi_{N}(t_{0},t)\right\rvert$ (23)
$\displaystyle=\bar{\Psi}_{L}|t-t_{0}-\Xi_{A}(t_{0},t)|$
$\displaystyle\geq\bar{\Psi}_{L}(t-t_{0}-(T_{0}+\frac{t-t_{0}}{\tau_{a}}))$
$\displaystyle\geq\bar{\Psi}_{L}((1-\frac{1}{\tau_{a}})(t-t_{0})+T_{0})$
$\displaystyle\geq(1-\frac{1}{\tau_{a}})\bar{\Psi}_{L}(t-t_{0}),t\geq t_{0},$

then, it easy obtain that $\lambda_{\rm
min}(\Omega^{D}\Theta^{-1})(t-t_{0})\geq(1-\frac{1}{\tau_{a}})\lambda_{\rm
min}(\Omega\Theta^{-1})(t-t_{0}).$

Therefore, we can conclude that

${\rm vec}(\tilde{\Upsilon})\leq c_{\Upsilon}e^{-\alpha_{\Psi}(t-t_{0})},$
(24)

where $c_{\Upsilon}=\frac{V_{1}(t_{0})}{\lambda_{\rm
min}(\Theta\otimes\bar{P}_{1})}$, and
$\alpha_{\Psi}=\mu_{1}(1-\frac{1}{\tau_{a}})\lambda_{\rm
min}(\Omega\Theta^{-1})$, which implies that ${\rm vec}(\tilde{\Upsilon})$ and
$\tilde{\Upsilon}$ converge to zero exponentially at the rate of
$\alpha_{\Psi}$ at least. Moreover, by $\alpha_{\Psi}>\sigma_{\rm max}(S)$,
$\tilde{S}_{i}x_{k}$ converges to zero exponentially for
$i\in\mathcal{F},k\in\mathcal{L}$. $\hfill\hfill\blacksquare$

###### Remark 3

In practice, there exist kinds of noises and attacks among agents, so, we
reconstruct the leaders dynamics for all followers by the observers (1). The
estimator gain $\mu_{1}$ is designed to guarantee that the convergence rate of
the estimated value $\tilde{S}$ is greater than the divergence rate of the
leader state. Moreover, it guarantees the possible of convergence of the
containment error. And compared with the existing work [chen2019], which only
used the observer in the normal communication for consensus, this work
consider the heterogeneous output containment with the DoS attacks.
$\hfill\hfill\square$

###  IV-B Distributed Resilient State Estimator Design

Based on the remodeled leaders dynamics, a distributed resilient estimator is
proposed to estimate the leader state $x_{k}$ under DoS attacks. Consider the
following distributed resilient TL:

$\dot{z}_{i}=\hat{S}_{i}z_{i}-\mu_{2}G(\sum_{j\in\mathcal{F}}d_{ij}(z_{i}-z_{j})+\sum_{k\in\mathcal{L}}d_{ik}(z_{i}-x_{k})),$
(25)

where $z_{i}$ is the local state of the virtual layer, $\mu_{2}>0$ is the
coupling gain that can be choose by the user, and $G$ is the observer gain to
be determined later. The global state of virtual resilient layer can be
written as

$\dot{z}=\hat{S}_{b}z-\mu_{2}(I_{N}\otimes
G)(\sum_{k\in\mathcal{L}}(\Psi_{k}^{D}\otimes I_{p})(z-\underline{x}_{k})),$
(26)

where $\hat{S}_{b}={\rm blkdiag}(\hat{S}_{i})$, $z=[z_{1},z_{2},\dots,z_{N}]$,
$\underline{x}_{k}=l_{n}\otimes x_{k}$ and $\mu_{2}>0$ is the estimator gain
designed in Theorem 2.

Define the global virtual resilient layer state estimation error

$\displaystyle\tilde{z}$
$\displaystyle=z-(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k},$
(27)

then, for the normal communication, we have

$\displaystyle\dot{\tilde{z}}$
$\displaystyle=\hat{S}_{b}z-\mu_{2}(I_{N}\otimes
G)(\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})(z-\underline{x}_{k}))$ (28)
$\displaystyle-(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})(I_{n}\otimes
S)\underline{x}_{k}$ $\displaystyle=\hat{S}_{b}z-(I_{n}\otimes
S)z+(I_{n}\otimes S)z$ $\displaystyle-(I_{n}\otimes
S)(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k}+M$
$\displaystyle-\mu_{2}(I_{N}\otimes G)\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes
I_{p})$ $\displaystyle(z-(\sum_{rr\in\mathcal{L}}(\Psi_{rr}\otimes
I_{p}))^{-1}(\sum_{kk\in\mathcal{L}}(\Psi_{kk}\otimes
I_{p})\underline{x}_{kk}$
$\displaystyle+(\sum_{rr\in\mathcal{L}}(\Psi_{rr}\otimes
I_{p})^{-1}(\sum_{kk\in\mathcal{L}}(\Psi_{kk}\otimes
I_{p})\underline{x}_{kk}-\underline{x}_{k}))$
$\displaystyle=\tilde{S}_{b}z+(I_{n}\otimes S)\tilde{z}-\mu_{2}(I_{N}\otimes
G)\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\tilde{z}$
$\displaystyle-\mu_{2}(I_{N}\otimes
G)(\sum_{kk\in\mathcal{L}}\\!(\Psi_{kk}\otimes
I_{p})\underline{x}_{kk}\\!-\\!\sum_{k\in\mathcal{L}}\\!(\Psi_{k}\otimes
I_{p})\underline{x}_{k})\\!+\\!M$ $\displaystyle=(I_{n}\otimes
S)\tilde{z}-\mu_{2}(I_{N}\otimes G)\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes
I_{p})\tilde{z}+\tilde{S}_{b}\tilde{z}+F_{2}^{x}(t),$

where $F_{2}^{x}(t)=\tilde{S}_{b}(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k}+M$
and $M=(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})(I_{n}\otimes
S)\underline{x}_{k}-(I_{n}\otimes S)(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k}$
and $\tilde{S}_{b}={\rm blkdiag}(\tilde{S}_{i})$ for $i\in\mathcal{F}$.

For the denied communication, we have

$\displaystyle\dot{\tilde{z}}$
$\displaystyle=\hat{S}_{b}z-(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})(I_{n}\otimes
S)\underline{x}_{k}$ (29) $\displaystyle=\hat{S}_{b}z-(I_{n}\otimes
S)z+(I_{n}\otimes S)z$ $\displaystyle-(I_{n}\otimes
S)(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k}+M$
$\displaystyle=\tilde{S}_{b}z+(I_{n}\otimes S)\tilde{z}+M$
$\displaystyle=(I_{n}\otimes S)\tilde{z}+\tilde{S}_{b}\tilde{z}+F_{2}^{x}(t).$

So, we can conclude (28) and (29) that

$\dot{\tilde{z}}=\begin{cases}\begin{aligned}
\hat{S}_{b}\tilde{z}-\mu_{2}(I_{N}\otimes
G)\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\tilde{z}&+F_{2}^{x}(t),\\\
t\in\Xi_{N}(t_{0},t),\end{aligned}\\\
\hat{S}_{b}\tilde{z}+F_{2}^{x}(t),t\in\Xi_{A}(t_{0},t).\end{cases}$ (30)

######  Lemma 5 ([haghshenas2015])

Under Lemma 2 and the Kronecker product property $(P\otimes Q)(Y\otimes
Z)=(PY)\otimes(QZ)$, it easy to show that
$(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})(I_{n}\otimes
S)\underline{x}_{k}=(I_{n}\otimes S)(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k}$.

Proof: Let

$\displaystyle M=\sum_{k\in\mathcal{L}}M_{k}$
$\displaystyle=\sum_{k\in\mathcal{L}}((\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}(\Psi_{k}\otimes I_{p})(I_{n}\otimes S)$ (31)
$\displaystyle-(I_{n}\otimes S)(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}(\Psi_{k}\otimes I_{p}))\underline{x}_{k}.$

By the Kronecker product property $(P\otimes Q)(Y\otimes Z)=(PY)\otimes(QZ)$,
we can obtain that

$\displaystyle(I_{N}\otimes S)(\sum_{r\in\mathcal{L}}\Psi_{r}\otimes
I_{p})^{-1}(\Psi_{k}\otimes I_{p})$ (32) $\displaystyle=(I_{N}\otimes
S)((\sum_{r\in\mathcal{L}}\Psi_{r})^{-1}\Psi_{k})\otimes I_{p})$
$\displaystyle=(I_{N}\times(\sum_{r\in\mathcal{L}}\Psi_{r})^{-1}\Psi_{k}))\otimes(S\times
I_{p})$ $\displaystyle=(\sum_{r\in\mathcal{L}}\Psi_{r}\otimes
I_{p})^{-1}(\Psi_{k}\otimes I_{p})(I_{N}\otimes S).$

We can show that $M_{k}=0$ and obtain that

$M=\sum_{k\in\mathcal{L}}M_{k}=0.$ (33)

This Proof is completed. $\hfill\hfill\blacksquare$

Then $F_{2}^{x}(t)=\tilde{S}_{b}(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k}$
and ${\rm lim}_{t\rightarrow\infty}F_{2}^{x}(t)=0$ is exponentially at the
same rate of $\tilde{S}_{b}$.

###### Theorem 2

Consider the MASs (2)-(3) suffered from DoS attacks, which satisfy Assumption
1, and the DoS attacks satisfying Definition 2 and Definition 3. There exist
scalars $\tilde{\alpha}_{1}>0$, $\tilde{\alpha}_{2}>0$ and $\mu_{2}>0$ and
positive symmetric define matrices $\bar{P}_{2}>0$ such that:

$\displaystyle\bar{P}_{2}S+S^{\mathrm{T}}\bar{P}_{2}-\bar{P}_{2}^{\mathrm{T}}\mu_{2}^{2}\bar{P}_{2}+\tilde{\alpha}_{1}\bar{P}_{2}=0$
(34)
$\displaystyle\bar{P}_{2}S+S^{\mathrm{T}}\bar{P}_{2}-\tilde{\alpha}_{2}\bar{P}_{2}\leq
0$ (35)
$\displaystyle\frac{1}{\tau_{a}}<\frac{\alpha_{1}}{\alpha_{1}+\alpha_{2}}$
(36)

where $\alpha_{1}=\tilde{\alpha}_{1}-k_{1}||\Theta\otimes\bar{P}_{2}||$,
$\alpha_{2}=\tilde{\alpha}_{2}+k_{1}||\Theta\otimes\bar{P}_{2}||$ with
$k_{1}>0$, and $G=\mu_{2}\lambda_{\rm max}(\Omega^{-1}\Theta)\bar{P}_{2}$.
Then it can be guaranteed that the estimation error $\tilde{z}$ exponentially
converges to zero under DoS attacks.

Proof. Nest, we will prove that the TL achieve containment under the DoS
attacks. For clarity, we first redefine the set $\Xi_{A}[t_{0},\infty)$ as
$\Xi_{A}[t_{0},\infty)=\bigcup_{k=0,1,2,\dots}[t_{2k+1},t_{2k+2})$, where
$t_{2k+1}$ and $t_{2k+2}$ indicate the time instants that attacks start and
end, respectively. Then, the set $\Xi_{N}[t_{0},\infty)$ can be redefined as
$\Xi_{N}[t_{0},\infty)=[t_{0},t_{1})\bigcup_{k=1,2,\dots}[t_{2k},t_{2k+1})$.

Consider the following Lyapunov function candidate:

$V_{2}(t)=\tilde{z}^{\mathrm{T}}(\Theta\otimes\bar{P}_{2})\tilde{z}.$ (37)

when the communication is normal, compute the time-derivative of $V_{1}(t)$
yields that:

$\displaystyle\dot{V}_{2}$
$\displaystyle=\tilde{z}^{\mathrm{T}}(\Theta\otimes(\bar{P}_{2}S+S^{\mathrm{T}}\bar{P}_{2}))\tilde{z}$
(38)
$\displaystyle-\tilde{z}^{\mathrm{T}}((\bar{\Psi}^{\mathrm{T}}_{L}\Theta+\Theta\bar{\Psi}_{L})\otimes\lambda_{\rm
max}(\Omega^{-1}\Theta)\mu_{2}^{2}\bar{P}_{2}^{2})\tilde{z}$
$\displaystyle+\tilde{z}^{\mathrm{T}}(\Theta\otimes(\bar{P}_{2}\tilde{S}_{i}+\tilde{S}_{i}^{\mathrm{T}}\bar{P}_{2}))\tilde{z}+2\tilde{z}^{\mathrm{T}}(\Theta\otimes\bar{P}_{2})F_{2}^{x}$
$\displaystyle\leq\tilde{z}^{\mathrm{T}}(\Theta\otimes(\bar{P}_{2}S+S^{\mathrm{T}}\bar{P}_{2})-\Theta\otimes\mu_{2}^{2}\bar{P}_{2}^{2})\tilde{z}$
$\displaystyle+\tilde{z}^{\mathrm{T}}((\Theta\otimes\bar{P}_{2})\tilde{S}_{b}+\tilde{S}_{b}^{\mathrm{T}}(\Theta\otimes\bar{P}_{2}))\tilde{z}+2\tilde{z}^{\mathrm{T}}(\Theta\otimes\bar{P}_{2})F_{2}^{x}.$

By Young’s inequality, there exists scalar $k_{1}>0$ yields that

$\displaystyle 2\tilde{z}^{\mathrm{T}}(\Theta\otimes\bar{P}_{2})F_{2}^{x}\leq
k_{1}\tilde{z}^{\mathrm{T}}(\Theta\otimes\bar{P}_{2})^{2}\tilde{z}+\frac{1}{k_{1}}||F_{2}^{x}||^{2},$
(39)

then

$\dot{V}_{2}\leq-\tilde{\alpha}_{1}V_{2}+2||\tilde{S}_{b}||V_{2}+k_{1}||\Theta\otimes\bar{P}_{2}||V_{2}+\frac{1}{k_{1}}||F_{2}^{x}||^{2}.$
(40)

With $\tilde{S}_{b}$ and $F_{2}^{x}$ converge to $0$ exponentially, we can
obtain that

$\dot{V}_{2}\leq-(\alpha_{1}+a_{s}e^{-b_{s}t})V_{2}+\varphi(t),$ (41)

where $a_{s}>0$, $b_{s}>0$, and $\varphi(t)=a_{f}e^{-b_{f}}$ with $a_{f}>0$,
and $b_{f}>0$.

When the communication is denied, the time-derivative of $V_{1}(t)$ yields
that

$\displaystyle\dot{V}_{2}$
$\displaystyle=\tilde{z}^{\mathrm{T}}(I_{N}\otimes(\bar{P}_{2}S+S^{\mathrm{T}}\bar{P}_{2}))\tilde{z}+\tilde{z}^{\mathrm{T}}((\Theta\otimes\bar{P}_{2})\tilde{S}_{b}$
(42)
$\displaystyle+\tilde{S}_{b}^{\mathrm{T}}(\Theta\otimes\bar{P}_{2}))\tilde{z}+2\tilde{z}^{\mathrm{T}}(\Theta\otimes\bar{P}_{2})F_{2}^{x}$
$\displaystyle\leq\tilde{\alpha}_{2}V_{2}+\frac{||\tilde{S}_{b}||^{2}}{k_{1}\sigma_{\rm
min}(\Theta\bar{P}_{2})}V_{2}+k_{1}||\Theta\otimes\bar{P}_{2}||V_{2}+\frac{1}{k_{1}}||F_{2}^{x}||^{2}$
$\displaystyle\leq(\alpha_{2}+a_{s}e^{-b_{s}t})V_{2}+\varphi(t).$

Solved the inequality (41) and (42), we can obtain the following inequality

$V(t)\leq\begin{cases}\begin{aligned}
&e^{\int_{t_{2k}}^{t}-\alpha_{1}+a_{s}e^{-b_{s}t}\,d\tau}V_{2}(t_{2k})\\\
&+\int_{t_{2k}}^{t}e^{\int_{\tau}^{t}-\alpha_{1}+a_{s}e^{-b_{s}s}\,ds}\varphi(\tau)\,d\tau,t\in[t_{2k},t_{2k+1}),\end{aligned}\\\
\begin{aligned}
&e^{\int_{t_{2k+1}}^{t}\alpha_{2}+a_{s}e^{-b_{s}t}\,d\tau}V_{2}(t_{2k+1})\\\
&+\int_{t_{2k+1}}^{t}e^{\int_{\tau}^{t}\alpha_{2}+a_{s}e^{-b_{s}s}\,ds}\varphi(\tau)\,d\tau,t\in[t_{2k+1},t_{2k+2}).\end{aligned}\end{cases}$
(43)

Let $\alpha=\begin{cases}-\alpha_{1},t\in[t_{2k},t_{2k+1})\\\
\alpha_{2},t\in[t_{2k+1},t_{2k+2})\\\ \end{cases}$, then we could reconstruct
the (43) as follows

$v(t)\leq
e^{\int_{t_{0}}^{t}\alpha+a_{s}e^{-b_{s}t}\,d\tau}V_{2}(t_{0})+\int_{t_{0}}^{t}e^{\int_{\tau}^{t}-\alpha_{1}+a_{s}e^{-b_{s}s}\,ds}\varphi(\tau)\,d\tau.$
(44)

For $t\in\Xi_{N}[t_{0},t)$, it yields from (44) that

$\displaystyle V_{2}(t)$ $\displaystyle\leq
e^{\int_{t_{0}}^{t}+a_{s}e^{-b_{s}t}\,d\tau}e^{\int_{t_{0}}^{t}\alpha\,d\tau}V_{2}(t_{0})$
(45)
$\displaystyle+\int_{t_{0}}^{t}a_{f}e^{-b_{f}+\int_{\tau}^{t}a_{s}e^{-b_{s}s}\,ds}e^{\int_{\tau}^{t}\alpha\,ds}\,d\tau$
$\displaystyle\leq
e^{-\frac{a_{s}}{b_{s}}(e^{-b_{s}t}-e^{-b_{s}t_{0}})}e^{-\alpha_{1}|\Xi_{N}(t_{0},t)|+\alpha_{2}|\Xi_{A}(t_{0},t)|}V_{2}(t_{0})$
$\displaystyle+\int_{t_{0}}^{t}a_{f}\\!e^{\\!-\\!b_{f}\tau\\!-\\!\frac{a_{s}}{b_{s}}(e^{\\!-\\!b_{s}t}\\!-\\!e^{-b_{s}\tau})}e^{\\!-\\!\alpha_{1}|\Xi_{N}(\tau,t)|\\!+\\!\alpha_{2}|\Xi_{A}(\tau,t)|}\,d\tau.$

Similarly, it can be easily conclude that (48) holds for
$t\in[t_{2k+1},t_{2k+2})$, too. With the Definition 3, we have

$\displaystyle-\alpha_{1}|\Xi_{N}(t_{0},t)+\alpha_{2}|\Xi_{A}(t_{0},t)|$ (46)
$\displaystyle=-\alpha_{1}(t-t_{0}-|\Xi_{A}(t_{0},t)|)+\alpha_{2}|\Xi_{A}(t_{0},t)|$
$\displaystyle\leq-\alpha_{1}(t-t_{0})+(\alpha_{1}+\alpha_{2})(T_{0}+\frac{t-t_{0}}{\tau_{a}})$
$\displaystyle\leq-\eta(t-t_{0})+(\alpha_{1}+\alpha_{2})T_{0},$

where $\eta=(\alpha_{1}-\frac{\alpha_{1}+\alpha_{2}}{\tau})$. Substitute (46)
into (48), we have

$\displaystyle V_{2}(t)$ $\displaystyle\leq
e^{-\frac{a_{s}}{b_{s}}(e^{-b_{s}t}-e^{-b_{s}t_{0}})+(\alpha_{1}+\alpha_{2})(T_{0})}V_{2}(t_{0})e^{-\eta(t-t_{0})}$
(47)
$\displaystyle+\frac{a_{f}}{-b_{f}-\eta}e^{-\frac{a_{s}}{b_{s}}(e^{-b_{s}t}-e^{-b_{s}t_{0}})+(\alpha_{1}+\alpha_{2})T_{0}}$
$\displaystyle(e^{-b_{f}t}-e^{-\eta t+(-b_{f}+\eta)t_{0}})$ $\displaystyle\leq
c_{1}e^{-\eta(t-t_{0})}+c_{2}e^{-b_{f}t}$

with
$c_{1}=e^{\frac{a_{s}}{b_{s}}e^{-b_{s}t_{0}}+(\alpha_{1}+\alpha_{2})T_{0}}(V_{2}(t_{0})-\frac{a_{f}}{-b_{f}+\eta}e^{-b_{f}t_{0}})$
and
$c_{2}=\frac{a_{f}}{-b_{f}+\eta}e^{\frac{a_{s}}{b_{s}}e^{-b_{s}t_{0}}+(\alpha_{1}+\alpha_{2})T_{0}}$.
From (36), we could get that $\eta>0$. Therefore, it obvious that $V_{2}(t)$
converges to $0$ exponentially.

Here, we have proof the $\tilde{z}$ converges to $0$ which is bounded. Then
the inequality (39) can be rewritten as
$2\tilde{z}^{\mathrm{T}}(\Theta\otimes\bar{P}_{2})F_{2}^{x}\leq\varphi^{*}(t)$,
which show that $k_{1}$ is not necessary and there exist
$\alpha_{1}=\tilde{\alpha}_{1}$ and $\alpha_{2}=\tilde{\alpha}_{2}$.

###### Remark 4

Compare with the existing work [deng2021] which the leader state estimator
start to work after finishing the leader dynamic estimation, this work does
not need to wait for the leader state estimator to estimate an accurate state
and the TL begin to work at the first time which is more applicable to the
actual situation. Moreover, Deng et al. [deng2021] only dealt with the problem
of single leader consensus, we consider the more general case of output
containment with multiple leaders. In addition, different from the existing
work [yang2020] which can only obtain the bounded state tracking error under
DoS attacks, the state tracking error of TL in our work converges to zero
exponentially which have more conservative result. $\hfill\hfill\square$

###  IV-C Distributed Output Regulator Equation Solvers

Our control protocol needs to use the solution of the regulator equation to
provide appropriate feedforward control to achieve output containment. The
solution of the regular equation needs to know the dynamic of the leader
system. Since the MAS is fragile, there are many disturbs in the information
transmission channel of CPL. Therefore, we cannot directly use the leaders
dynamics matrices $S$ and $R$ as [zuo2020] here. Since the leaders dynamics
were remolded in TL, we propose to use the remolding dynamic matrices
$\hat{S}_{i}$ and $\hat{R}_{i}$ to calculate the solution of the regulator
equations. Then, the output regulator equation can be solved by the following
Theorem.

###### Theorem 3

suppose the Assumptions 1 and 4 hold, the estimated solutions
$\hat{\Delta}_{i}$ to the output regulator equations in (49) are solved as
follows:

$\displaystyle\dot{\hat{\Delta}}_{i}=-\mu_{3}\hat{\Phi}^{\mathrm{T}}_{i}(\hat{\Phi}_{i}\hat{\Delta}_{i}-\hat{\mathcal{R}}_{i}),$
(48)

where $\hat{\Delta}_{i}={\rm vec}(\hat{Y}_{i}),\hat{\Phi}_{i}=(I_{q}\otimes
M_{i}-\hat{S}_{i}^{\mathrm{T}}\otimes N_{i})$, $\hat{R}_{i}={\rm
vec}(\mathcal{\hat{R}}^{\ast}_{i})$;
$\hat{Y_{i}}=[\hat{\Pi}_{i}^{\mathrm{T}}$,
$\hat{\Gamma}_{i}^{\mathrm{T}}]^{\mathrm{T}}$,
$\hat{R}^{\ast}_{i}=[0,\hat{R}_{i}^{\mathrm{T}}]^{\mathrm{T}}$,
$M_{i}=\left[\begin{array}[]{cc}A_{i}&B_{i}\\\ C_{i}&0\\\ \end{array}\right]$,
$N_{i}=\left[\begin{array}[]{cc}I_{n_{i}}&0\\\ 0&0\\\ \end{array}\right]$, and
$\mu_{3}>\frac{\sigma_{\rm max}(S)}{\lambda_{\rm
min}(\Phi_{i}^{\mathrm{T}}\Phi_{i})}$ with $\Phi_{i}=(I_{q}\otimes
M_{i}-S^{\mathrm{T}}\otimes N_{i})$.

Proof. In Theorem 1, we realize that leaders dynamics estimations are time-
varying, so the output regulator equation influenced by leaders dynamics
estimations are also time-varying. Next, we begin proof that the estimated
solution of regulator equation converges to the solution of standard regulator
equation exponentially.

First, we proof that $\hat{\Delta}_{i}$ converges to $\Delta_{i}$ at an
exponential rate with $\Delta$ is the solution of the standard output
regulator equation defined in (52).

From Assumption 4, we can get that the following output regulator equation:

$\begin{cases}A_{i}\Pi_{i}+B_{i}\Gamma_{i}=\Pi_{i}S\\\
C_{i}\Pi_{i}=R\end{cases}$ (49)

have a pair unique solution matrices $\Pi_{i}$ and $\Gamma_{i}$ for
$i=\textbf{I}[1,N]$.

Rewriting the standard output regulator equation (49) yields as follow:

$\left[\begin{array}[]{cc}A_{i}&B_{i}\\\ C_{i}&0\\\
\end{array}\right]\left[\begin{array}[]{cc}\Pi_{i}\\\ \Gamma_{i}\\\
\end{array}\right]I_{q}-\left[\begin{array}[]{cc}I_{n_{i}}&0\\\ 0&0\\\
\end{array}\right]\left[\begin{array}[]{cc}\Pi_{i}\\\ \Gamma_{i}\\\
\end{array}\right]S=\left[\begin{array}[]{cc}0\\\ R\end{array}\right].$ (50)

Reformulating the equation (50) as follow:

$M_{i}YI_{q}-N_{i}YS=R_{i}^{\ast},$ (51)

where $M_{i}=\left[\begin{array}[]{cc}A_{i}&B_{i}\\\ C_{i}&0\\\
\end{array}\right],N_{i}=\left[\begin{array}[]{cc}I_{n_{i}}&0\\\ 0&0\\\
\end{array}\right],Y_{i}=\left[\begin{array}[]{cc}\Pi_{i}\\\ \Gamma_{i}\\\
\end{array}\right],$ and $R_{i}^{\ast}=\left[\begin{array}[]{cc}0\\\
R\end{array}\right]$. By Theorem 1.9 of Huang et al. [huang2004], the standard
form of the linear equation in Equation (50) can be rewritten as:

$\Phi_{i}\Delta_{i}=\mathcal{R}_{i},$ (52)

where $\Phi_{i}=(I_{q}\otimes M_{i}-S^{\mathrm{T}}\otimes N_{i})$,
$\Delta_{i}={\rm vec}(\left[\begin{array}[]{cc}\Pi_{i}\\\ \Gamma_{i}\\\
\end{array}\right])$, $\mathcal{R}_{i}={\rm vec}(\mathcal{R}^{\ast})$,
$R^{\ast}=\left[\begin{array}[]{cc}0\\\ R\\\ \end{array}\right]$. Noting that

$\displaystyle\dot{\hat{\Delta}}_{i}$
$\displaystyle=-\mu_{3}\hat{\Phi}^{\mathrm{T}}_{i}(\hat{\Phi}_{i}\hat{\Delta}_{i}-\hat{\mathcal{R}}_{i})$
(53)
$\displaystyle=-\mu_{3}\hat{\Phi}^{\mathrm{T}}_{i}\hat{\Phi}_{i}\hat{\Delta}_{i}+\mu_{3}\hat{\Phi}_{i}\hat{\mathcal{R}_{i}}$
$\displaystyle=-\mu_{3}\Phi_{i}^{\mathrm{T}}\Phi_{i}\hat{\Delta}_{i}+\mu_{3}\Phi_{i}^{\mathrm{T}}\Phi_{i}\hat{\Delta}_{i}-\mu_{3}\hat{\Phi}_{i}^{\mathrm{T}}\hat{\Phi}_{i}\hat{\Delta}_{i}+\mu_{3}\hat{\Phi}_{i}^{\mathrm{T}}\hat{\mathcal{R}_{i}}$
$\displaystyle-\mu_{3}\Phi_{i}^{\mathrm{T}}\hat{\mathcal{R}_{i}}+\mu_{3}\Phi_{i}^{\mathrm{T}}\hat{\mathcal{R}_{i}}-\mu_{3}\Phi_{i}^{\mathrm{T}}\mathcal{R}_{i}+\mu_{3}\Phi_{i}^{\mathrm{T}}\mathcal{R}_{i}$
$\displaystyle=-\mu_{3}\Phi_{i}^{\mathrm{T}}\Phi_{i}\hat{\Delta}_{i}+\mu_{3}(\Phi_{i}^{\mathrm{T}}\Phi_{i}-\hat{\Phi}_{i}^{\mathrm{T}}\hat{\Phi}_{i})\hat{\Delta}_{i}$
$\displaystyle+\mu_{3}(\hat{\Phi}_{i}^{\mathrm{T}}-\Phi_{i}^{\mathrm{T}})\hat{\mathcal{R}_{i}}+\mu_{3}\Phi_{i}^{\mathrm{T}}(\hat{\mathcal{R}_{i}}-\mathcal{R}_{i})+\mu_{3}\Phi_{i}^{\mathrm{T}}\mathcal{R}_{i}$
$\displaystyle=-\mu_{3}\Phi_{i}^{\mathrm{T}}\Phi_{i}\hat{\Delta}_{i}+\mu_{3}\Phi_{i}^{\mathrm{T}}\mathcal{R}_{i}+d_{i}(t),$

where
$d_{i}(t)=-\mu_{3}(\hat{\Phi}_{i}^{\mathrm{T}}\hat{\Phi}_{i}-\Phi_{i}^{\mathrm{T}}\Phi_{i})\hat{\Delta}_{i}+\mu_{3}\tilde{\Phi}_{i}^{\mathrm{T}}\hat{\mathcal{R}}_{i}+\mu_{3}\Phi_{i}^{\mathrm{T}}\tilde{\mathcal{R}}_{i}$
with
$\tilde{\Phi}_{i}=\hat{\Phi}_{i}-\Phi_{i}=\tilde{S}_{i}^{\mathrm{T}}\otimes
N_{i}$ and $\tilde{\mathcal{R}}={\rm vec}(\left[\begin{array}[]{cc}0\\\
\tilde{R}_{i}\end{array}\right])$. It obvious that
$\lim_{t\rightarrow\infty}d_{i}(t)=0$ exponentially at rate of
$\alpha_{\Psi}$.

Let $\tilde{\Delta}_{i}=\Delta_{i}-\hat{\Delta}_{i}$, the time derivative of
$\tilde{\Delta}_{i}$ can be computed via

$\displaystyle\dot{\tilde{\Delta}}_{i}$
$\displaystyle=-\mu_{3}\Phi_{i}^{\mathrm{T}}\Phi_{i}\tilde{\Delta}_{i}-\mu_{3}\Phi_{i}^{\mathrm{T}}\Phi_{i}\Delta_{i}+\mu_{3}\Phi_{i}^{\mathrm{T}}\mathcal{R}_{i}+d_{i}(t)$
(54)
$\displaystyle=-\mu_{3}\Phi_{i}^{\mathrm{T}}\Phi_{i}\tilde{\Delta}_{i}+d_{i}(t),$

solve the equation (54), we have

$\displaystyle\tilde{\Delta}_{i}(t)=\tilde{\Delta}_{i}(t_{0})e^{-\mu_{3}\Phi_{i}^{\mathrm{T}}\Phi_{i}(t-t_{0})}+\int_{t_{0}}^{t}d_{i}(\tau)e^{-\mu_{3}\Phi_{i}^{\mathrm{T}}\Phi_{i}(t-\tau)}\,d\tau,$
(55)

since $\Phi_{i}^{\mathrm{T}}\Phi_{i}$ is positive, and $d_{i}(t)$ converge to
$0$ at rate of $\alpha_{\Psi}$, it can obtain that
$\lim_{t\rightarrow\infty}\tilde{\Delta}_{i}=0$ exponentially. Moreover, with
$\mu_{3}>\frac{\sigma_{\rm max}(S)}{\lambda_{\rm
min}(\Phi_{i}^{\mathrm{T}}\Phi_{i})}$, the exponential convergence rate of
$\lim_{t\rightarrow\infty}\tilde{\Delta}_{i}=0$ is bigger than $\sigma_{\rm
max}(S)$. $\hfill\hfill\blacksquare$

###### Lemma 6

The distributed leaders dynamics observers in (1) ensure $\dot{\hat{\Pi}}_{i}$
and $\dot{\hat{\Pi}}_{i}x_{k}$ converge to zero exponentially.

###### Remark 5

The rate of convergence of $\dot{\hat{\Pi}}_{i}$ is at least as
$\dot{\tilde{\Delta}}_{i}$, so, the exponential convergence rate of
$\dot{\hat{\Pi}}_{i}$ is bigger than $\sigma_{\rm max}(S)$ by
$\mu_{3}>\frac{\sigma_{\rm max}(S)}{\lambda_{\rm
min}(\Phi_{i}^{\mathrm{T}}\Phi_{i})}$. Then, it obvious that
$\dot{\hat{\Pi}}_{i}x_{k}$ converges to zero exponentially.
$\hfill\hfill\square$

###  IV-D Adaptive Distributed Resilient Controller Design

Define the following state tracking error :

$\epsilon_{i}=x_{i}-\hat{\Pi}_{i}z_{i},\\\ $ (56)

then considering the decentralized adaptive attack-resilient control protocols
as follows:

$\displaystyle u_{i}=\hat{\Gamma}_{i}z_{i}+K_{i}\epsilon_{i}-\hat{\chi}_{i}$
(57)
$\displaystyle\hat{\chi}_{i}=\frac{B_{i}^{\mathrm{T}}P_{i}\epsilon_{i}}{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert+\omega}\hat{\rho_{i}}$
(58)
$\displaystyle\dot{\hat{\rho}}_{i}=\begin{cases}\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert+2\omega,&\mbox{if}\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert\geq\bar{d},\\\
\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert+2\omega\frac{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert}{\bar{d}},&\mbox{
otherwise},\end{cases}$ (59)

where $\hat{\chi}_{i}$ is an adaptive compensation signal, $\hat{\rho}_{i}$ is
an adaptive updating parameter and the controller gain $K_{i}$ is designed as

$K_{i}=-R_{i}^{-1}B_{i}^{\mathrm{T}}P_{i},$ (60)

where $P_{i}$ is the solution to

$A_{i}^{\mathrm{T}}P_{i}+P_{i}A_{i}+Q_{i}-P_{i}B_{i}R_{i}^{-1}B_{i}^{\mathrm{T}}P_{i}=0.$
(61)

###### Theorem 4

Consider heterogeneous MAS consisting of $M$ leader (2) and $N$ followers (3)
with unbounded faults. Under Assumptions 1-5, the Problem ACMCA is solved by
designing the leaders dynamics observer (1), the distributed state estimator
(25), the estimated output solution regulator equation (48) and the
distributed controller consisting of (56)-(61).

Proof. In Theorems 1 and 2, we proved that the TL can resist DoS attacks in
which the frequency of DoS attack satisfies Theorem 2. In the following, we
will show that the state tracking error (56) is UUB against unbounded AAs.

The derivative of $\epsilon_{i}$ is presented as follows:

$\displaystyle\dot{\epsilon}_{i}$
$\displaystyle=A_{i}x_{i}+B_{i}u_{i}+B_{i}\chi_{i}-\dot{\hat{\Pi}}_{i}z_{i}$
(62)
$\displaystyle-\hat{\Pi}_{i}(\hat{S}_{i}z_{i}-\mu_{2}(\sum_{j\in\mathcal{F}}d_{ij}(z_{i}-z_{j})+\sum_{k\in\mathcal{L}}d_{ik}(z_{i}-x_{k})))$
$\displaystyle=(A_{i}+B_{i}K_{i})\epsilon_{i}-\dot{\hat{\Pi}}_{i}z_{i}$
$\displaystyle+\mu_{2}\hat{\Pi}_{i}(\sum_{j\in\mathcal{F}}d_{ij}(z_{i}-z_{j})+\sum_{k\in\mathcal{L}}d_{ik}(z_{i}-x_{k}))+B_{i}\tilde{\chi}_{i},$

where $\tilde{\chi}_{i}=\chi_{i}-\hat{\chi}_{i}$.

The global state tracking error of (62) is

$\displaystyle\dot{\epsilon}$ $\displaystyle={\rm
blkdiag}(A_{i}+B_{i}K_{i})\epsilon-{\rm blkdiag}(\dot{\hat{\Pi}}_{i})z$ (63)
$\displaystyle+\\!\mu_{2}{\rm
blkdiag}(\hat{\Pi}_{i})(\sum_{k\in\mathcal{L}}(\Psi_{k}^{D}\otimes
I_{p})(z\\!-\\!\underline{x}_{k}))\\!+\\!{\rm blkdiag}(B_{i})\tilde{\chi}$
$\displaystyle=\bar{A}_{b}\epsilon-\dot{\hat{\Pi}}_{b}(\tilde{z}+(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k})$
$\displaystyle+\mu_{2}\hat{\Pi}_{b}\sum_{k\in\mathcal{L}}(\Psi_{k}^{D}\otimes
I_{p})\tilde{z}+\mu_{2}\hat{\Pi}_{b}\sum_{k\in\mathcal{L}}(\Psi_{k}^{D}\otimes
I_{p})$ $\displaystyle((\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes
I_{p})\underline{x}_{k})-\underline{x}_{k}))+B_{b}\tilde{\chi}$
$\displaystyle=\bar{A}_{b}\epsilon-\dot{\hat{\Pi}}_{b}(\tilde{z}+(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k})$
$\displaystyle+\mu_{2}\hat{\Pi}_{b}\sum_{k\in\mathcal{L}}(\Psi_{k}^{D}\otimes
I_{p})\tilde{z}+B_{b}\tilde{\chi},$

where $\bar{A}_{b}={\rm blkdiag}(A_{i}+B_{i}K_{i})$, $\dot{\hat{\Pi}}_{b}={\rm
blkdiag}(\dot{\hat{\Pi}}_{i})$, $\hat{\Pi}_{b}={\rm blkdiag}(\hat{\Pi}_{i})$
for $i=\textbf{I}[1,N]$ and
$\epsilon=[\epsilon_{1}^{\mathrm{T}}\epsilon_{2}^{\mathrm{T}}\dots\epsilon_{N}^{\mathrm{T}}]^{\mathrm{T}}$,
$\tilde{\chi}=[\tilde{\chi}_{1}^{\mathrm{T}}\tilde{\chi}_{2}^{\mathrm{T}}\dots\tilde{\chi}_{N}^{\mathrm{T}}]^{\mathrm{T}}$.

Consider the following Lyapunov function candidate:

$V=\epsilon^{\mathrm{T}}P_{b}\epsilon,$ (64)

where $P_{b}={\rm blkdiag}(P_{i})$ and its time derivate is given as follows:

$\displaystyle\dot{V}$
$\displaystyle=2\epsilon^{\mathrm{T}}P_{b}\dot{\epsilon}_{i}$ (65)
$\displaystyle=-\epsilon^{\mathrm{T}}{\rm
blkdiag}(Q_{i})\epsilon+2\epsilon^{\mathrm{T}}P_{b}(\dot{\hat{\Pi}}_{b}+\mu_{2}\hat{\Pi}_{b}\sum_{k\in\mathcal{L}}(\Psi_{k}^{D}\otimes
I_{p}))\tilde{z}$
$\displaystyle\\!+\\!2\epsilon^{\mathrm{T}}P_{b}\dot{\hat{\Pi}}_{b}(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes
I_{p})\underline{x}_{k}\\!+\\!2\epsilon^{\mathrm{T}}P_{b}B_{b}\tilde{\chi}$
$\displaystyle\leq-\sigma_{min}(Q_{b})\left\lVert\epsilon\right\rVert^{2}+2\epsilon^{\mathrm{T}}P_{b}B_{b}\tilde{\chi}$
$\displaystyle+2\left\lVert\epsilon^{\mathrm{T}}\right\rVert\left\lVert
P_{b}\right\rVert\left\lVert\dot{\hat{\Pi}}_{b}+\mu_{2}\hat{\Pi}_{b}\sum_{k\in\mathcal{L}}(\Psi_{k}^{D}\otimes
I_{p})\right\rVert\left\lVert\tilde{z}\right\rVert$
$\displaystyle+2\left\lVert\epsilon^{\mathrm{T}}\right\rVert\left\lVert
P_{b}\right\rVert||\dot{\hat{\Pi}}_{b}(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{x}_{k}||,$

where $Q_{b}={\rm blkdiag}(Q_{i})$.

By $\dot{\hat{\Pi}}_{i}$ and $\dot{\hat{\Pi}}_{i}x_{k}$ converge to zero
exponentially in Lemma 5, there exist positive constants $V_{\Pi}$ and
$\alpha_{\Pi}$ such that the following holds:

$\left\lVert\dot{\hat{\Pi}}_{b}(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes
I_{p})\underline{x}_{k}\right\rVert\leq V_{\Pi}\exp(-\alpha_{\Pi}),$ (66)

use Young’s inequality, we have

$\displaystyle 2\left\lVert\epsilon^{\mathrm{T}}\right\rVert\left\lVert
P_{b}\right\rVert\left\lVert\dot{\hat{\Pi}}_{b}(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes
I_{p})\underline{x}_{k}\right\rVert$ (67)
$\displaystyle\leq\left\lVert\epsilon\right\rVert^{2}+\left\lVert
P_{b}\right\rVert^{2}\left\lVert\dot{\hat{\Pi}}_{b}(\sum_{r\in\mathcal{L}}(\Psi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes
I_{p})\underline{x}_{k}\right\rVert^{2}$
$\displaystyle\leq(\frac{1}{4}\sigma_{min}(Q_{b})-\frac{1}{2}\beta_{V1})\left\lVert\epsilon_{i}\right\rVert^{2}$
$\displaystyle+\frac{\left\lVert
P\right\rVert^{2}}{(\frac{1}{4}\sigma_{min}(Q_{b})-\frac{1}{2}\beta_{V1})}V_{\Pi}^{2}\exp(-2\alpha_{\Pi})$
$\displaystyle\leq(\frac{1}{4}\sigma_{min}(Q_{b})-\frac{1}{2}\beta_{V1})\left\lVert\epsilon_{i}\right\rVert^{2}+\beta_{v21}e^{-2\alpha_{v1}t}.$

From Lemma 6, we can obtain that $\dot{\hat{\Pi}}$ converges to 0
exponentially. By $\tilde{z}$ also converges to zero exponentially, we
similarly obtain that

$\displaystyle 2\left\lVert\epsilon^{\mathrm{T}}\right\rVert\left\lVert
P\right\rVert\left\lVert\dot{\hat{\Pi}}_{b}+\mu_{2}\hat{\Pi}_{b}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes
I_{p})\right\rVert\left\lVert\tilde{z}\right\rVert$ (68)
$\displaystyle\leq(\frac{1}{4}\sigma_{min}(Q_{b})-\frac{1}{2}\beta_{V1})\left\lVert\epsilon\right\rVert^{2}+\beta_{v22}e^{-2\alpha_{v2}t}.$

Next, noting that

$\displaystyle\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\tilde{\chi}_{i}$
$\displaystyle=\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\chi_{i}-\frac{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert^{2}}{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert+\omega}\hat{\rho_{i}}$
(69)
$\displaystyle\leq\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert\left\lVert\chi_{i}\right\rVert-\frac{\left\lVert\tilde{y}_{i}^{\mathrm{T}}P_{i}C_{i}B_{i}\right\rVert^{2}}{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert+\omega}\hat{\rho_{i}}$
$\displaystyle=\frac{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert^{2}(\left\lVert\chi_{i}\right\rVert-\hat{\rho}_{i})+\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert\left\lVert\chi_{i}\right\rVert\omega}{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert+\omega}$
$\displaystyle=\frac{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert^{2}(\frac{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert+\omega}{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert}||\chi_{i}||-\hat{\rho}_{i})}{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert+\omega}.$

Noting that $d\left\lVert\chi_{i}\right\rVert/dt$ is bounded, so, if
$\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert\geq\bar{d}\geq\frac{d||\chi_{i}||}{dt}$,
that is,
$\frac{\bar{d}+\omega}{\bar{d}}\frac{d||\chi_{i}||}{dt}-\dot{\hat{\rho}}\leq\bar{d}+\omega-\dot{\hat{\rho}}\leq-\omega<0$.
Then, there exists $t_{2}>0$ such that for all $t\geq t_{2}$, we have

$(\frac{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert+\omega}{\left\lVert\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\right\rVert}||\chi_{i}||-\hat{\rho}_{i})\leq(\frac{\bar{d}+\omega}{\kappa}||\chi_{i}||-\hat{\rho}_{i})<0.$
(70)

Thus, we can obtain that
$\epsilon_{i}^{\mathrm{T}}P_{i}B_{i}\tilde{\chi}_{i}<0$ and
$\epsilon^{\mathrm{T}}P_{b}B_{b}\tilde{\chi}<0$ over $t\in[t_{2},\infty)$.

From (66) to (70), it yields

$\dot{V}\leq-(\beta_{V1}+\frac{1}{2}\sigma_{min}(Q_{b}))\epsilon^{\mathrm{T}}\epsilon+\beta_{V2}e^{\alpha_{V1}t}.$
(71)

Solving (71) yields the following:

$\displaystyle V(t)$ $\displaystyle\leq
V(0)\\!-\\!\int_{0}^{t}\\!(\beta_{V1}\\!+\\!\frac{1}{2}\sigma_{min}(Q_{b}))\epsilon^{\mathrm{T}}\epsilon\,d\tau\\!+\\!\int_{0}^{t}\\!\beta_{V2}e^{\alpha_{V1}t}\,d\tau,$
(72)

then

$\epsilon^{\mathrm{T}}\epsilon\leq-\int_{0}^{t}\frac{1}{\sigma_{min}(P)}\beta_{V3}\epsilon^{\mathrm{T}}\epsilon\,d\tau+\bar{B}.$
(73)

where $\beta_{V3}=\beta_{V1}+\frac{1}{2}\sigma_{min}(Q_{b})$ and
$\bar{B}=V(0)-\int_{0}^{t}\beta_{V2}e^{\alpha_{V1}t}\,d\tau$ are define as a
bounded constant. Recalling Bellman-Gronwall Lemma, (73) is rewritten as
follows:

$\left\lVert\epsilon\right\rVert\leq\sqrt{\bar{B}}e^{-\frac{\beta_{V3}t}{2\sigma_{min}(P)}},$
(74)

conclude (70) and (74), we can that $\epsilon$ is bounded by $\bar{\epsilon}$,
where
$\bar{\epsilon}=[\bar{\epsilon}_{1}^{\mathrm{T}}\bar{\epsilon}_{2}^{\mathrm{T}}\dots\bar{\epsilon}_{N}^{\mathrm{T}}]^{\mathrm{T}}$
with $||\bar{\epsilon}_{i}||=\frac{\bar{d}}{\sigma_{\rm min}(P_{i}B_{i})}$.

Hence, similar to Lemma 5, the global output synchronization error satisfied
that

$\displaystyle e$ $\displaystyle=y-(\sum_{r\in\mathcal{L}}(\Phi_{r}\otimes
I_{p}))^{-1}\sum_{k\in\mathcal{L}}(\Psi_{k}\otimes I_{p})\underline{y}_{k}$
(75) $\displaystyle=y-(I_{N}\otimes R)(z-\tilde{z})$ $\displaystyle={\rm
blkdiag}(C_{i})x-{\rm blkdiag}(C_{i}\hat{\Pi}_{i})z+{\rm
blkdiag}(C_{i}\hat{\Pi}_{i})z$ $\displaystyle-(I_{N}\otimes R)z+(I_{N}\otimes
R)\tilde{z}$ $\displaystyle={\rm blkdiag}(C_{i})\epsilon-{\rm
blkdiag}(\tilde{R}_{i})z+(I_{N}\otimes R)\tilde{z}.$

Since $\epsilon$, $\tilde{R}_{i}$, $\tilde{z}$ converge to 0 exponentially
which have been proofed before, it is obviously that the global output
containment error $e$ is bounded by $\bar{e}$ with
$\bar{e}=[\bar{e}_{1}^{\mathrm{T}}\bar{e}_{2}^{\mathrm{T}}\dots\bar{e}_{i}^{\mathrm{T}}]^{\mathrm{T}}$
and $\bar{e}_{i}=\frac{\bar{d}||C_{i}||}{\sigma_{\rm min}(P_{i}B_{i})}$ for
$i=\textbf{I}[1,N]$. The whole proof is completed. $\hfill\hfill\blacksquare$

###### Remark 6

Compared with [chen2019] which only solved the tracking problem with a single
leader against bounded attacks, we consider more challenge work with multiple
leaders output containment problem against unbounded attacks. Moreover, the
boundary of the output containment error is given, that is,
$\frac{||C_{i}||\bar{d}}{\sigma_{\rm min}(P_{i}B_{i})}$. It’s helpful for
designers to pre-estimate of the controller. $\hfill\hfill\square$

##  V Numerical Simulation

Figure 3: Topology graph. Figure 4: The estimations of leaders dynamics in
Theorem 1: The shadow areas denote time intervals against DoS attacks.

(a) State estimation error of TL (X-axis).

(b) State estimation error of TL (Y-axis).

Figure 5: Performance of the TL: the shadow areas denote time intervals
against DoS attacks. Figure 6: The estimated solution of the output regulating
equation in Theorem 3: The shadow areas denote time intervals against DoS
attacks. Figure 7:  Output trajectories of the leaders and followers.

(a) Output containment error (X-axis).

(b) Output containment error (Y-axis).

Figure 8: Output containment errors. Figure 9: The estimation error between TL
and CPL: The blue shadow areas denote the UUB bound by $\bar{d}$.

In this section, two examples will be given to llustrate the effectiveness of
the above control protocol.

###  V-A Example 1

We consider a multi-agent system consisting of seven agents (three leaders
indexed by 5 7 and four followers indexed by 1 4) with the corresponding graph
show in Fig. 3. Set that $a_{ij}=1$ if there exist a path from node $j$ to
node $i$. Consider the leaders dynamics are described by

$S=\left[\begin{array}[]{cc}0.5&-0.4\\\ 0.8&0.5\\\
\end{array}\right],R=\left[\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right],$

and the dynamics of follows are given by

$A_{1}\\!=\\!\left[\begin{array}[]{cc}3&-2\\\ 1&-2\\\
\end{array}\right],B_{1}\\!=\\!\left[\begin{array}[]{cc}1.8&-1\\\ 2&3\\\
\end{array}\right],C_{1}\\!=\\!\left[\begin{array}[]{cc}-0.5&1\\\ 2&-1.5\\\
\end{array}\right],$ $A_{2}\\!=\\!\left[\begin{array}[]{cc}0.6&-1\\\ 1&-2\\\
\end{array}\right],B_{2}\\!=\\!\left[\begin{array}[]{cc}1&-2\\\ 1.9&4\\\
\end{array}\right],C_{2}\\!=\\!\left[\begin{array}[]{cc}-0.5&1\\\ 1.5&1.4\\\
\end{array}\right],$ $A_{3}=A_{4}=\left[\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
0&0&-2\\\ \end{array}\right],B_{3}=B_{4}=\left[\begin{array}[]{ccc}6&0\\\
0&1\\\ 1&0\\\ \end{array}\right],$
$C_{3}=C_{4}=\left[\begin{array}[]{ccc}0.5&-0.5&0.5\\\ -0.5&-0.5&0.5\\\
\end{array}\right].$

The DoS attack periods are given as $[0.5+2k,1.53+2k)s$ for $k\in\mathbb{N}$,
which satisfied Assumption 3. The MAS can againt the CAs and FDI attacks since
the information transmitted on the CPL is not used in the hierarchal control
scheme. The AAs are designed as $\chi_{1}=\chi_{2}=\chi_{3}=0.01\times[2t\quad
t]^{\mathrm{T}}$ and $\chi_{4}=-0.01\times[2t\quad t]^{\mathrm{T}}$.

Chosen the gains in (1), (25) and (48) as $\mu_{1}=2$, $\mu_{2}=0.5$ and
$\mu_{3}=6$, respectively. It can be obtain by Theorem that
$P2=\left[\begin{array}[]{cc}0.6038&0.0108\\\ 0.0108&0.1455\end{array}\right]$
and $G=\left[\begin{array}[]{cc}4.5191&0.5186\\\
0.5186&4.0746\end{array}\right]$. By employing the above parameters, the
trajectories of leaders dynamics estimations are shown in Fig. 4, the estimate
of leaders dynamics $||\hat{\Upsilon}_{i}||$ against DoS attacks remained
stable after $6s$ with the value equal to the value of leaders dynamics. The
state estimation errors of TL are given in Figs. 5 which can be seen the
performance of TL is stable under the DoS attacks. The trajectories of the
solution errors of output regulating equations are given in Fig. 6, which
validates Theorem 3. And to solve the AAs on the CPL, we design the controller
gain in (60) as

$K_{1}=\left[\begin{array}[]{cc}-1.71&0.07\\\ 2.31&-1.12\\\
\end{array}\right],K_{2}=\left[\begin{array}[]{cc}-0.62&-0.29\\\ 1.16&-0.56\\\
\end{array}\right],$
$K_{3}=K_{4}=\left[\begin{array}[]{ccc}-1.00&-0.19&-0.08\\\
0.02&-0.96&-0.30\\\ \end{array}\right].$

Now, we can solve the ACMCA problem by Theorem 4. The output trajectories of
all the agents are showed in the Fig. 7. It can be seen that followers
eventually enter the convex hull formed by the leaders. The output containment
errors of followers are shown in Figs. 8, which can be seen that the local
output errors $e_{i}(t)$ is UUB. The tracking error between the CPL and the TL
$\epsilon_{i}$ is depicted in Fig. 9, which show the tracking error is UUB by
the prescribed error bound. These results show that the output containment
problem subject to composite attacks is solved.

###  V-B Example 2

Figure 10: Communication Topology graph of UAV swarm : blue UAVs and green
ones represent leaders and followers, respectively.

We consider a practice example of UAVs with the corresponding graph show in
Fig. 10. Assume that all the edges have the weight of 1.

The dynamics of leader UAVs and follower UAVs can be approximately described
as the following second-order systems [wang2018optimal, dong2016time]:

$\begin{cases}\dot{p}_{k}(t)=v_{k}(t),\\\
\dot{v}_{k}(t)=\alpha_{p_{0}}p_{k}(t)+\alpha_{v_{0}}v_{k}(t),\\\
y_{k}(t)=p_{k}(t),\end{cases}$ (76)

and

$\begin{cases}\dot{p}_{i}(t)=v_{i}(t),\\\
\dot{v}_{i}(t)=\alpha_{p_{i}}p(t)+\alpha_{v_{i}}v_{i}(t)+u_{i}(t),\\\
y_{i}(t)=p_{i}(t),\end{cases}$ (77)

where $p_{k}(t)$ and $v_{k}(t)$ ($p_{i}(t)$ and $v_{i}(t)$) represent the
position and velocity of the $k$th leader UAVs ( $i$th follower UAVs),
respectively. The damping constants of leaders are selected as:
$\alpha_{p_{0}}=-0.6,\alpha_{v_{0}}=0$. The damping constants of followers are
choose as: $\alpha_{p_{1}}=-1,\alpha_{v_{1}}=-1$,
$\alpha_{p_{2}}=-0.5,\alpha_{v_{2}}=-1.2$,
$\alpha_{p_{3}}=-1.2,\alpha_{v_{3}}=-1$,
$\alpha_{p_{4}}=-0.8,\alpha_{v_{4}}=-0.5$, and
$\alpha_{p_{5}}=-0.4,\alpha_{v_{5}}=-1.2$. Select the gains as $\mu_{1}=5$,
$\mu_{2}=1$, and $\mu_{3}=7$, respectively. We choose the DoS attack periods
as $[0.2+2k,1.86+2k)s$ for $k\in\mathbb{N}$, and the AAs given as
$\chi_{1}=[0.01t\quad 0.02t\quad 0.02t]^{\mathrm{T}},\chi_{2}=[0.02t\quad
0.01t\quad 0.02t]^{\mathrm{T}},\chi_{3}=[0.02t\quad 0.02t\quad
0.02t]^{\mathrm{T}},\chi_{4}=[-0.01t\quad-0.02t\quad-0.02t]^{\mathrm{T}},\chi_{5}=[-0.02t\quad-0.01t\quad
0.02t]^{\mathrm{T}}$.

Figure 11: The leaders dynamics modeling of TL in Theorem 1: The shadow areas
denote time intervals against DoS attacks.

(a) Distributed estimation performance of the TL (X-axis).

(b) Distributed estimation performance of the TL (Y-axis).

(c) Distributed estimation performance of the TL (Z-axis).

Figure 12: Distributed estimation performance of TL: the shadow areas denote
time intervals against DoS attacks.

Choose the initial values of updating parameters in (1) and (48) to be zero.
Then select the initial leader’s state as $p_{6}=[1\quad 1\quad
1]^{\mathrm{T}},v_{6}=[4\quad 5\quad 4]^{\mathrm{T}}$, $p_{7}=[4\quad 1\quad
2]^{\mathrm{T}},v_{7}=[5\quad 6\quad 6]^{\mathrm{T}}$, and $p_{8}=[6\quad
5\quad 8]^{\mathrm{T}},v_{6}=[4\quad 6\quad 5]^{\mathrm{T}}$, and randomly
choose other initial values. According to Theorem 2 and Theorem 4, it yields
that the state observer gain of TL
$G=\left[\begin{array}[]{cc}2.8483&0.2594\\\ 0.2594&3.0611\end{array}\right]$
and the gain of CPL $K_{1}=[-0.4142\quad-0.6818]$,
$K_{2}=[-0.6180\quad-0.7173]$, $K_{3}=[-0.3620\quad-0.6505]$,
$K_{4}=[-0.4806\quad-0.9870]$, $K_{5}=[-0.6770\quad-0.7478]$, respectively.

Figure 13: The estimated solution of the output regulating equation in Theorem
3: The shadow areas denote time intervals against DoS attacks. Figure 14:
Trajectories of the UAV swarm during $0\sim 30s$: The hollow and solid blocks
represent the start and end points of all UAVs, respectively.

(a) The positions of all agents at $t=0$s.

(b) The positions of all agents at $t=10$s.

(c) The positions of all agents at $t=20$s.

(d) The positions of all agent at $t=30$s.

Figure 15: The positions of all agents at different atme nt.

(a) The position error ($e_{i}^{x}(t)$) of all follower UAVs (X-axis).

(b) The position error ($e_{i}^{y}(t)$) of all follower UAVs (Y-axis).

(c) The position error ($e_{i}^{z}(t)$) of all follower UAVs (Z-axis).

Figure 16: The performance of our control scheme during the output
containment. Figure 17: The estimation error between TL and CPL: The blue
shadow areas denote the UUB bound by $\bar{d}$.

From Fig. 11, it depicts that $\hat{\Upsilon}$ converges to $\Upsilon$ quickly
under DoS attacks, which implies that the molding error converges to $0$
quickly. The state estimator error of TL are shown in Figs. 12, which show
that the performance of TL is well under the DoS attacks. Fig. 13 verifies the
validity of Theorem 3. Fig. 14 shows the trajectories of UAVs reference in the
simulation within $t=30s$, where the initial positions and the final positions
of UAVs are marked by unfilled squares and filled squares, respectively. Figs.
15 show the positions of UAVs at four different instants, ($t=0s$, $t=10s$,
$t=20s$ and $t=30s$). It can be seen that, the trajectory of each follower
stays in a small neighborhood around the dynamic convex hull spanned by the
leaders at Fig. 15(b)-(d), that is, the output containment is achieved, which
can also valid by Figs. 16. Fig. 17 shows the validity and practicability of
our adaptive attack-resilient control protocols.

##  VI Conclusion

The distributed resilient output containment control of heterogeneous multi-
agent systems subject to composite Attacks has been solved in this work.
Inspired by the concept of hierarchical protocol, a TL, which is resilient to
most attacks (including the CAs, FDI attacks, and AAs), is provided to
decouples the defense strategy into two sub-problems. The first sub-problem is
the defense against DoS attacks on the TL and the second sub-problem is the
defense against unbounded AAs on the CPL. For the first problem, we used
double-TL to solve the modeling error and DoS attacks: the first TL use the
distributed observers of leaders dynamics to cope with the the modeling
errors. Then, the distributed estimators is provided to observe the
trajectories of containment against DoS attacks. For the second problem, the
double-CPL is provided to solve the output containment aginst AAs: the first
CPL use output regulation equation solver to deal with the output containment,
and adaptive distributed control scheme is provided to solve the unbounded AAs
on the CPL. We have prove the UUB convergence of the above adaptive controller
and given the error bound explicitly. Finally, the simulations illustrate the
effectiveness of these proposed methods.