# Quantized Consensus under Data-Rate Constraints and DoS Attacks: A Zooming-
In and Holding Approach

Maopeng Ran, Shuai Feng, Juncheng Li, and Lihua Xie, _Fellow, IEEE_ M. Ran is
with the School of Automation Science and Electrical Engineering, Beihang
University, Beijing 100191, China (email: ranmp@buaa.edu.cn).J. Li, and L. Xie
(corresponding author) are with the School of Electrical and Electronic
Engineering, Nanyang Technological University, Singapore 639798 (email:
<EMAIL_ADDRESS>elhxie@ntu.edu.sg). S. Feng is with the School of
Automation, Nanjing University of Science and Technology, Nanjing 210094,
China (email: s.feng@njust.edu.cn).

###### Abstract

This paper is concerned with the quantized consensus problem for uncertain
nonlinear multi-agent systems under data-rate constraints and Denial-of-
Service (DoS) attacks. The agents are modeled in strict-feedback form with
unknown nonlinear dynamics and external disturbance. Extended state observers
(ESOs) are leveraged to estimate agents’ total uncertainties along with their
states. To mitigate the effects of DoS attacks, a novel dynamic quantization
with zooming-in and holding capabilities is proposed. The idea is to zoom-in
and hold the variable to be quantized if the system is in the absence and
presence of DoS attacks, respectively. The control protocol is given in terms
of the outputs of the ESOs and the dynamic-quantization-based encoders and
decoders. We show that, for a connected undirected network, the developed
control protocol is capable of handling any DoS attacks inducing bounded
consecutive packet losses with merely 3-level quantization. The application of
the zooming-in and holding approach to known linear multi-agent systems is
also discussed.

###### Index Terms:

Multi-agent systems, quantized consensus, data-rate, Denial-of-Service
attacks, uncertain nonlinear systems.

##  I Introduction

CONSENSUS over digital networks is known as one of the most important issues
in cooperative control of multi-agent systems [1]. The success of consensus
relies on efficient and reliable information exchange between the agents.
According to the basic communication principles, the capacity of a real
digital network is limited, i.e., at each time step or time interval, only
limited data can be transmitted reliably between the agents. On the other
hand, the digital network is vulnerable to malicious cyber attacks, such as
Denial-of-Service (DoS) [2]. DoS attacks refer to destroying the information
availability, which may deteriorate the consensus performance and even lead to
instability. Thus, it is of theoretical significance and practical relevance
to investigate the consensus problem for multi-agent systems under data-rate
constraints and DoS attacks.

###  I-A Literature Review

Data-rate constraints for multi-agent systems can be modeled by introducing a
quantization-encoding-decoding process with a finite number of quantization
levels [6]. At each time step, the sender encodes the quantized information
and sends out the code via the digital network. When the neighbors receive the
code, they use a decoding algorithm to reconstruct the information. Primary
studies on the quantized consensus problem have been on first-order integrator
multi-agent systems [3, 4, 5, 6, 7]. In [3], the states of the agents were
assumed to be integer-valued. In [4], a random dither was added to the sensor
state before quantization to make the quantization error a “white” noise. In
[6], with the use of a finite-level uniform quantizer and an exponentially
decaying scaling function, it was shown that the average consensus can be
achieved with one-bit data rate for a connected undirected network. The idea
in [6] was further generalized to multi-agent systems with higher-order
dynamics [8, 9, 10, 11, 12].

Recent years have also witnessed a growing interest towards control systems
under DoS attacks. In [13], DoS attacks were characterized by their levels of
average frequency and duration under which input-to-state stability of the
closed-loop system can be preserved. Such a characterization of DoS attacks
has motivated many interesting results, e.g., [14, 15, 16, 17]. For multi-
agent systems under DoS attacks, Deng and Wen [18] developed a resilient
fault-tolerant approach to handle the secure cooperative problem in the
presence of actuator faults and DoS attacks. In [19] and [20], the event-
triggered strategy was introduced to linear and nonlinear multi-agent systems
under DoS attacks, respectively. We mention that the communication data-rate
was implicitly assumed to be infinite in [18, 19, 20], since the agents can
communicate exact information with their neighbors.

For the simultaneous consideration of data-rate constraints and DoS attacks, a
few recent works have been done for centralized systems [21, 22, 23, 25, 24],
and multi-agent systems [26]. In [21], a dynamic encoding scheme was
developed, and sufficient conditions on DoS frequency and duration for
exponential state convergence were obtained. In [22, 23], the trade-off
between system resilience against DoS attacks and data-rate was analyzed, in
which the data-rate conditions are dependent on the plant state matrix, and
DoS frequency and duration. In [24], the input-to-state stability (ISS) of
linear systems with quantized state measurements under external disturbances
and DoS attacks was studied. In [25], the scenario that both input
(controller-to-plant) and output (plant-to-controller) channels are subject to
data-rate constraints and DoS attacks was investigated. An interesting
property of the approach in [25] is that when only the output channel is
subject to DoS attacks, the widely adopted network acknowledgement signal is
not required. For multi-agent systems under data-rate constraints and DoS
attacks, Feng and Ishii [26] proposed a dynamic quantization approach with
_zooming-in and zooming-out_ capabilities. Specifically, the scaling function
used for dynamic quantization zooms-in and out the variable to be quantized in
the absence and presence of DoS, respectively. It should be pointed out that
the required data-rate for each paired agents in [26] increases to infinity if
the number of the agents $N\rightarrow\infty$. What is more, the tolerable
level of DoS attacks specified by their average frequency and duration in [26]
may be low, since the quantized control scheme needs to zoom-out under DoS for
compensating the diverging consensus error.

By investigation of the existing literature, there are two important issues
that are noteworthy. On one hand, it still lacks a solution to the quantized
consensus problem of general _uncertain nonlinear_ multi-agent systems under
data-rate constraints and DoS attacks. The approaches in [21, 22, 23, 25, 24,
26] are for certain linear systems. However, most practical control systems
are inherently uncertain and nonlinear [27, 28]. On the other hand, it is
highly desirable to develop new techniques to achieve lower data-rate and
higher tolerance to DoS attacks.

###  I-B Main Contribution

The theme of this paper is to investigate the quantized consensus problem for
uncertain nonlinear multi-agent systems under data-rate constraints and DoS
attacks. The agents are modeled in strict-feedback form with unknown nonlinear
dynamics and external disturbance. When DoS occurs, the information
availability between the agents can be destroyed, and the consensus becomes
difficult under DoS attacks. Therefore, an important technical difficulty we
need to overcome is to efficiently govern the multi-agent systems during the
attack periods, with the uncertain nonlinear agent dynamics well-handled.
Based on the intuitive consideration that it is expected to lead to a higher
tolerance to DoS attacks if the multi-agent system can hold its status instead
of having a diverging trend when it is attacked, we develop a new dynamic
quantization scheme which we refer to as _zooming-in and holding_. The new
scheme is significantly different from that in the state-of-the-art result
[26], as it zooms-in the variable to be quantized in the absence of DoS
attacks, and holds the variable in the presence of DoS attacks. What is more,
we employ an extended state observer (ESO) [30, 29] to estimate the system
unknown nonlinear dynamics and external disturbance, and compensate them in
real time in the control action. _In summary, the main contribution of this
paper is the development of an ESO-based dynamic encoding-decoding scheme with
zooming-in and holding capabilities for uncertain nonlinear multi-agent
systems under data-rate constraints and DoS attacks._

Compared with the existing literature, the advantages of the control scheme
developed in this paper include:

  1. 1.

It is a more practical solution to the quantized consensus problem under data-
rate constraints and DoS attacks. The developed scheme is capable of handling
uncertain nonlinear dynamics and external disturbance. More importantly, it is
tolerable to serious DoS attacks. We only require that the consecutive packet
losses induced by DoS attacks be bounded. This is a minimum requirement since
a multi-agent system becomes $N$ single systems if an attack is always active.

  2. 2.

It is capable of leading to better consensus performance. This is due to the
nature of the zooming-in and holding mechanism, in which the consensus error
keeps invariant during DoS attacks. On the contrary, the consensus error may
diverge during DoS attacks for the zooming-in and zooming-out mechanism [26].

  3. 3.

It achieves theoretically the lowest data-rate. We show that merely 3-level
quantization suffices to guarantee quantized consensus, which is independent
from the number of the agents and DoS attacks.

The remainder of this paper is organized as follows. Section II provides some
preliminaries and problem formulation. The proposed control scheme for
uncertain nonlinear multi-agent systems is presented in Section III. In
Section IV, for ease of comparison, we apply the zooming-in and holding
approach to known linear multi-agent systems. Two examples are provided in
Section V. Finally, Section VI concludes the paper.

##  II Preliminaries and Problem Formulation

###  II-A Notation

In this paper, we use $\mathbb{R}$, $\mathbb{R}^{n}$, and $\mathbb{R}^{n\times
m}$ to denote the set of real numbers, $n$-dimensional real column vectors,
and $n\times m$ real matrices, respectively. For a given vector or matrix $A$,
its transpose, Euclidean norm, and $\infty$-norm are denoted by
$A^{\textrm{T}}$, $\|A\|$, and $\|A\|_{\infty}$, respectively. For a given
positive number $\nu$, the maximum integer less than or equal to $\nu$ is
denoted by $\lfloor\nu\rfloor$; the minimum integer greater than or equal to
$\nu$ is denoted by $\lceil\nu\rceil$. $I_{n}$ denotes the identity matrix of
dimension $n$. $\textbf{1}_{n}$ and $\textbf{0}_{n}$ denote the
$n$-dimensional column vector with all ones and zeros, repectively. Let
$J_{n}=(1/n)\textbf{1}_{n}\textbf{1}^{\textrm{T}}_{n}$. Big $O$-notation in
terms of $\nu$ is denoted as $O(\nu)$. Let $\textrm{sat}(\cdot)$ be the unity
saturation function defined by
$\textrm{sat}(\nu)=\textrm{sign}(\nu)\cdot\min\\{1,|\nu|\\}$.

###  II-B Graph Theory

For a group of $N$ agents, the corresponding graph $\mathcal{G}$ can be
expressed by a triple $\\{\mathcal{V},\mathcal{E},\mathcal{A}\\}$, where
$\mathcal{V}=\\{1,2,\ldots,N\\}$ is the set of nodes,
$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ is the set of edges, and
$\mathcal{A}=[a_{ij}]\in\mathbb{R}^{N\times N}$ is the adjacency matrix with
$a_{ij}=1$ if $(i,j)\in\mathcal{E}$, and $a_{ij}=0$ otherwise. It is assumed
$a_{ii}=0$, i.e., no self-loop in $\mathcal{G}$. The neighborhood of agent $i$
is denoted by $\mathcal{N}_{i}=\\{j\in\mathcal{V}|(i,j)\in\mathcal{E}\\}$. A
sequence of edges $(i_{1},i_{2})$, $(i_{2},i_{3})$, $\ldots$,
$(i_{j-1},i_{j})$ is called a path from agent $i_{1}$ to agent $i_{j}$. If for
any two agents $i,j\in\mathcal{V}$, there exists a path from agent $i$ to
agent $j$, $\mathcal{G}$ is called a connected graph. The in-degree of agent
$i$ is represented by $\textrm{deg}_{i}=\sum_{j=1}^{N}a_{ij}$, and
$d^{*}=\max_{1\leq i\leq N}\textrm{deg}_{i}$ is the degree of $\mathcal{G}$.
Denote
$\mathcal{D}=\textrm{diag}\\{\textrm{deg}_{1},\ldots,\textrm{deg}_{N}\\}$ and
the Lpalacian matrix of $\mathcal{G}$ by
$\mathcal{L}=\mathcal{D}-\mathcal{A}$. If $\mathcal{A}$ is a symmetric matrix,
$\mathcal{G}$ is called an undirected graph, and the eigenvalues of the
Laplacian matrix $\mathcal{A}$ in an ascending order are denoted by
$0=\lambda_{1}<\lambda_{2}\leq\cdots\leq\lambda_{N}$ [32].

###  II-C Problem Formulation

Consider a multi-agent system consisting of $N$ agents described by the
following single-input-single-output uncertain nonlinear dynamics:

$\left\\{\begin{aligned}
\dot{x}_{ij}=&f_{ij}(x_{i1},\ldots,x_{ij})+x_{i,j+1},~{}1\leq j\leq r-1,\\\
\dot{x}_{ir}=&f_{ir}(x_{i},z_{i},\omega_{i})+u_{i},\\\
\dot{z}_{i}=&f_{i0}(x_{i},z_{i},\omega_{i}),\\\
y_{i}=&x_{i1},~{}i=1,\ldots,N,\end{aligned}\right.$ (1)

where $x_{i}=[x_{i1},x_{i2},\ldots,x_{ir}]^{\rm{T}}\in\mathbb{R}^{r}$ is the
agent state, $z_{i}\in\mathbb{R}^{n_{i}-r}$ is the state of the zero dynamics;
$n_{i}$ is the dimension of agent $i$, $r$ is the relative degree;
$y_{i}\in\mathbb{R}$, $u_{i}\in\mathbb{R}$, and
$\omega_{i}\in\mathbb{R}^{n_{\omega_{i}}}$ are the measured output, control
input, and external disturbance, respectively; and $f_{i0}\in
C^{1}(\mathbb{R}^{r}\times\mathbb{R}^{n_{i}-r}\times\mathbb{R}^{n_{\omega_{i}}},\mathbb{R})$,
$f_{ij}\in C^{r+1-j}(\mathbb{R}^{j},\mathbb{R})$, $1\leq j\leq r-1$, and
$f_{ir}\in
C^{1}(\mathbb{R}^{r}\times\mathbb{R}^{n_{i}-r}\times\mathbb{R}^{n_{\omega_{i}}},\mathbb{R})$
are unknown locally Lipschitz functions. The strict-feedback system (1)
represents a wide class of physical plants [27], such as the pendulum with
motor dynamics in [33] and the third-order phase-locked loop in [34].

_Assumption A1:_ The communication graph $\mathcal{G}$ among the $N$ agents is
undirected and connected.

_Assumption A2:_ The external disturbances $\omega_{i}$ and their derivatives
$\dot{\omega}_{i}$ are bounded.

_Assumption A3:_ The internal dynamics
$\dot{z}_{i}=f_{i0}(x_{i},z_{i},\omega_{i})$ are bounded-input-bounded-state
(BIBS) stable with respect to $(x_{i},\omega_{i})$.

For a real digital network connecting the multi-agent system (1), information
to be transmitted between the agents is to be first quantized and encoded at
the transmitter, and then decoded at the receiver. In control protocols to be
developed in the sequel, we use the following finite-level uniform quantizer:

$q(\nu)=\left\\{\begin{aligned} &0,\quad-1/2<\nu<1/2,\\\
&i,\quad\frac{2i-1}{2}\leq\nu<\frac{2i+1}{2},~{}i=1,2,\ldots,K-1,\\\
&K,\quad\nu\geq\frac{2K-1}{2},\\\
&-q(-\nu),\quad\nu\leq-1/2,\end{aligned}\right.$ (2)

where $K\in\\{1,2,\ldots\\}$. For a ($2K+1$)-level quantizer specified by (2),
the agent is required to be capable of transmitting
$\lceil\textrm{log}_{2}(2K+1)\rceil$ bits at each time step. For subsequent
use, let
$Q(\nu)=[q(\nu_{1}),\ldots,q(\nu_{\ell})]^{\textrm{T}}\in\mathbb{R}^{\ell}$
with $\nu=[\nu_{1},\ldots,\nu_{\ell}]^{\textrm{T}}\in\mathbb{R}^{\ell}$. On
the other hand, it is assumed that the transmission is acknowledgement-based
and free of delay. That is, the receivers send acknowledgments to the
transmitters immediately if they receive encoded signals successfully [21, 23,
26, 25, 24, 22]. From a communication point of view, this can be realized by
requiring that the acknowledgments are sent by a more powerful source.

In this paper, similar to [26], we refer to DoS attacks as the event for which
all the encoded signals cannot be received by the decoders and it affects all
the agents. The attacks are assumed to be launched by an adversarial entity
possibly having limited energy. Hence, here we characterize the attacker’s
action by its capability of inducing consecutive transmission failures by
launching DoS. In addition to DoS, we apply periodic transmission strategy,
i.e., the transmission attempts of each agent takes place periodically at
$t=kT$, in which $k$ is a non-negative integer and $T>0$ is the sampling
interval. Due to DoS attacks, not all the transmission attempts at $kT$
succeed. Then, we let
$\\{v_{\ell}T\\}=\\{v_{0}T,v_{1}T,v_{2}T,...\\}\subseteq\\{kT\\}$ with
$\ell=0,1,...$ represent the sequence of time instants when the transmissions
are successful.

_Assumption A4:_ The duration of the interval $[v_{\ell}T,v_{\ell+1}T)$
calculated as $v_{\ell+1}T-v_{\ell}T$ with $\ell=0,1,...$ is bounded.

_Remark 1 (DoS Model):_ Under DoS attacks, the interval between two
consecutive successful transmissions can be prolonged by DoS attacks.
Assumption A4 constrains the maximal number of consecutive packet drops
induced by DoS attacks. In [13], a DoS model was developed to describe an
attacker’s action in terms of the DoS _average frequency and duration_ , and
the level of DoS was specified by $\frac{1}{T_{D}}+\frac{T}{\tau_{D}}$, where
$T_{D}>1$ and $\tau_{D}>T$ are DoS model parameters. More specifically,
$\frac{1}{T_{D}}+\frac{T}{\tau_{D}}=0$ indicates no DoS attacks, and
$\frac{1}{T_{D}}+\frac{T}{\tau_{D}}=1$ the attack can be always active. Here
we mention that the Assumption A4 has some connections with the DoS model in
[13]. Based on the model in [13], the papers [15, 16] presented a result
concerning the upper bound of the intervals between two consecutive successful
transmissions in the presence of DoS attacks. In particular, the value of the
upper bound depends on the level of DoS attacks, that is, if
$\frac{1}{T_{D}}+\frac{T}{\tau_{D}}$ is high, then the upper bound of the
intervals between two consecutive successful transmissions can be long. In the
context of our paper, this means $v_{\ell+1}T-v_{\ell}T$ can be large. What is
more, the results of the paper [26] were developed based on the DoS model in
[13], in which the upper bound of consecutive packet losses is involved in the
design process. Here, since we will design a novel zooming-in and holding
quantization strategy, we only require that the number of consecutive packet
losses induced by DoS is upper bounded as in Assumption A4, and the exact
upper bound is not necessary to be known for design. Our result is also
applicable under other DoS models which can imply bounded consecutive packet
losses. ∎

In the following, we aim to solve the quantized consensus problem for
uncertain nonlinear multi-agent system (1) over a digital network under data-
rate constraints and DoS attacks. Our main results will be stated in Section
III. The special case that the multi-agent system (1) is known and linear will
be discussed in Section IV.

##  III Quantized Consensus of Uncertain Nonlinear Multi-Agent Systems

In this section, we present the ESO-based dynamic encoding-decoding scheme
with zooming-in and holding capabilities. The control protocol is based on the
outputs of the observers, encoders, and decoders. Conditions for quantized
consensus under data-rate constraints and DoS attacks will be derived.

###  III-A Protocol Design

To present the main results, we first conduct a state transformation to the
agent dynamics. Let
$\rho_{i}=[\rho_{i1},\ldots,\rho_{ir}]^{\rm{T}}\in\mathbb{R}^{r}$ with
$\rho_{i1}=y_{i}$ and $\rho_{ij}=\dot{\rho}_{i,j-1}$, $2\leq j\leq r$. Then by
an iterative procedure [35], the agent dynamics (1) can be transformed into
the following normal form [28, 27]:

$\left\\{\begin{aligned}
\dot{\rho}_{i}=&A_{\rho}\rho_{i}+B_{\rho}[F_{i}(\rho_{i},z_{i},\omega_{i})+u_{i}],\\\
\dot{z}_{i}=&F_{i0}(\rho_{i},z_{i},\omega_{i}),\\\
y_{i}=&\rho_{i1},~{}i=1,\ldots,N,\end{aligned}\right.$ (3)

where $F_{i}\in
C^{1}(\mathbb{R}^{r}\times\mathbb{R}^{n_{i}-r}\times\mathbb{R}^{n_{\omega_{i}}},\mathbb{R})$,
$F_{i0}\in
C^{1}(\mathbb{R}^{r}\times\mathbb{R}^{n_{i}-r}\times\mathbb{R}^{n_{\omega_{i}}},\mathbb{R}^{n_{i}-r})$,
and matrices $A_{\rho}\in\mathbb{R}^{r\times r}$ and
$B_{\rho}\in\mathbb{R}^{r\times 1}$ are given by

$A_{\rho}=\left[\begin{array}[]{cccc}0&1&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&1\\\ 0&0&\cdots&0\\\
\end{array}\right],~{}B_{\rho}=\left[\begin{array}[]{c}0\\\ \vdots\\\ 0\\\
1\\\ \end{array}\right].$

_Lemma 1: For $1\leq i\leq N$, let_

$h_{i}=k_{1}\rho_{i1}+k_{2}\rho_{i2}+\cdots+k_{r-1}\rho_{i,r-1}+\rho_{ir},$
(4)

_where $k_{1}$, $\ldots$, $k_{r-1}$ are selected such that the polynomial
$k_{1}+k_{2}\lambda+\cdots+k_{r-1}\lambda^{r-1}+\lambda^{r}$ is Hurwitz. If
$h_{i}$, $1\leq i\leq N$, are bounded and
$\lim_{t\rightarrow\infty}h_{i}=h^{*}+O(\nu)$ for some constant $h^{*}$ and
small positive constant $\nu$, then
$\lim_{t\rightarrow\infty}(y_{i}-y_{j})=O(\nu)$, $1\leq i\neq j\leq N$._

_Proof:_ Let
$\tilde{\rho}_{i}=[\rho_{i1},\ldots,\rho_{i,r-1}]^{\rm{T}}\in\mathbb{R}^{r-1}$.
From (4), one has

$\dot{\tilde{\rho}}_{i}=\tilde{A}\tilde{\rho}_{i}+\tilde{B}h_{i},$ (5)

where the matrices $\tilde{A}\in\mathbb{R}^{(r-1)\times(r-1)}$ and
$\tilde{B}\in\mathbb{R}^{(r-1)\times 1}$ are given by

$\displaystyle\tilde{A}=\left[\begin{array}[]{cccc}0&1&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&1\\\
-k_{1}&-k_{2}&\cdots&-k_{r-1}\\\
\end{array}\right],\tilde{B}=\left[\begin{array}[]{c}0\\\ \vdots\\\ 0\\\ 1\\\
\end{array}\right].$ (14)

Note that system (5) is a stable linear time-invariant system in controllable
canonical form with state $\tilde{\rho}_{i}$ and input $h_{i}$. Then by Lemma
1 in [11] and the superposition principle in linear control theory, one can
readily conclude that $\lim_{t\rightarrow\infty}(y_{i}-y_{j})=O(\nu)$, $1\leq
i\neq j\leq N$. ∎

According to Lemma 1, the consensus of the variables $h_{i}$, $1\leq i\leq N$,
implies the output consensus of the multi-agent system (1). In the sequel, the
control protocol $u_{i}$ will be designed to make $h_{i}$ achieve consensus.
Hence, an attempting in transmitting the information of $h_{i}$ will be made
between each paired agents. In [9, 26], for higher-order multi-agent systems,
the agent state in vector form was transmitted through the digital network.
Recall that $h_{i}$ is a linear combination of the agent states and the agents
are single-output. We will show that transmitting the 1-dimensional signal
$h_{i}$ rather than the $r$-dimensional state enables us to achieve the lowest
data rate.

Figure 1: Architecture of the proposed control scheme for uncertain nonlinear
multi-agent systems under data-rate constraints and DoS attacks.

Since for each agent, only the first state variable $y_{i}=\rho_{i1}$ is
available, the unmeasurable agent states and unknown agent dynamics will be
estimated online. Augmenting an extended integrator to the chain of
integrators in (3), the ESO for the extended system is designed as

$\left\\{\begin{aligned}
\dot{\hat{\rho}}_{i}=&A_{\rho}\hat{\rho}_{i}+B_{\rho}(\hat{\rho}_{i,r+1}+u_{i})+L(\varepsilon)(y_{i}-\hat{\rho}_{i1}),\\\
\dot{\hat{\rho}}_{i,r+1}=&\frac{l_{r+1}}{\varepsilon^{r+1}}(y_{i}-\hat{\rho}_{i1}),\end{aligned}\right.$
(15)

where
$\hat{\rho}_{i}=[\hat{\rho}_{i1},\ldots,\hat{\rho}_{ir}]^{\rm{T}}\in\mathbb{R}^{r}$
is the estimate of the agent state $\rho_{i}$, and $\hat{\rho}_{i,r+1}$ is the
estimate of the extended state $\rho_{i,r+1}\triangleq
F_{i}(\rho_{i},z_{i},\omega_{i})$, which represents the total effect of the
unknown nonlinear dynamics and external disturbance. For the observer (15),
the initial condition is set as $\textbf{0}_{r+1}$, $\varepsilon<1$ is a small
positive constant,
$L(\varepsilon)=\left[\frac{l_{1}}{\varepsilon},\ldots,\frac{l_{r}}{\varepsilon^{r}}\right]^{\rm{T}}$,
and $l_{1}$, $\ldots$, $l_{r+1}$ are selected such that the following matrix
is Hurwitz:

$E=\left[\begin{array}[]{ccccc}-l_{1}&1&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ -l_{r}&0&\cdots&1\\\ -l_{r+1}&0&\cdots&0\\\
\end{array}\right].$ (16)

In this paper, we assume that the output $y_{i}$ is free of measurement noise.
Otherwise, some other techniques, such as nonlinear-gain structure [31],
should be adopted to improve the robustness to measurement noise. What is
more, the peaking phenomenon is also known as an important feature of the ESO
(15). More specifically, $|\rho_{ij}-\hat{\rho}_{ij}|$ , $2\leq j\leq r+1$,
might peak to $O(1/\varepsilon^{j-1})$ value during the initial short
transient period. To overcome the peaking phenomenon, the saturation technique
[30] is employed to saturate the estimates $\hat{\rho}_{i1}$ to
$\hat{\rho}_{i,r+1}$ outside a compact set of interest. Let
$\overline{\rho}_{ij}=M_{j}\textrm{sat}\left(\frac{\hat{\rho}_{ij}}{M_{j}}\right),~{}1\leq
j\leq r+1$, where $M_{j}$ are bounds selected such that the saturations will
not be invoked during the steady period of the observer [30]. Correspondingly,
denote
$\overline{\rho}_{i}=[\overline{\rho}_{i1},\ldots,\overline{\rho}_{ir}]^{\rm{T}}$
and

$\overline{h}_{i}=k_{1}\overline{\rho}_{i1}+k_{2}\overline{\rho}_{i2}+\cdots+k_{r-1}\overline{\rho}_{i,r-1}+\overline{\rho}_{ir}.$
(17)

Based on the output of the ESO, the information of $\overline{h}_{i}$ will be
transmitted through the DoS affected digital network. Recall that $t=kT$ with
$k=0,1,\ldots$ denotes the periodic sampling instants at which the multi-agent
system (1) attempts to exchange information. The data to be transmitted at
time $t=kT$ is denoted by $\overline{h}_{i}(kT)$, and some of them may not be
successfully transmitted if DoS is present at $kT$. For the instants at which
DoS is not present, as mentioned before Assumption A4, we let
$v_{0}T,v_{1}T,v_{2}T,\ldots$ represent the sequence of time instants when the
transmissions are successful. Let $H_{v}\triangleq\\{v_{i};i\geq 0\\}$.
According to the acknowledgement signal, the encoder for agent $i$ is designed
as

$\left\\{\begin{aligned} \chi_{i}(0)=&0,\\\
\chi_{i}((k+1)T)=&\chi_{i}(kT)+\beta(kT)\Delta_{i}((k+1)T),\\\
&\textrm{if}~{}k+1\in H_{v},\\\
\chi_{i}((k+1)T)=&\chi_{i}(kT),~{}\textrm{if}~{}k+1\notin H_{v},\\\
\Delta_{i}((k+1)T)=&q\left(\frac{\overline{h}_{i}((k+1)T)-\chi_{i}(kT)}{\beta(kT)}\right),\end{aligned}\right.$
(18)

where $\chi_{i}$ is the internal state of the encoder and $\Delta_{i}$ is a
binary data which represents the output of the encoder. In (18), $\beta(kT)$
is the scaling function which is initialized with $\beta(0)=\beta_{0}>0$ and
updated by the following mechanism:

$\beta((k+1)T)=\max\\{\gamma\beta(kT),\sqrt{\varepsilon}\\},$ (19)

with

$\gamma=\left\\{\begin{aligned} \gamma_{1},~{}\textrm{if}~{}k+1\in H_{v},\\\
1,~{}~{}\textrm{if}~{}k+1\notin H_{v},\end{aligned}\right.$ (20)

where $0<\gamma_{1}<1$ will be specified latter. Note that here we let
$\beta(kT)\rightarrow\sqrt{\varepsilon}$ rather than 0 as in [26]. The
explanation is that in the quantizer employed in (18), we use the ESO
estimated information $\overline{h}_{i}((k+1)T)$ rather than the accurate
information $h_{i}((k+1)T)$. In the subsequent section, we will show that for
any time interval $[\tau_{1},\tau_{2}]\subseteq[0,\infty)$,
$\frac{\int_{\tau_{1}}^{\tau_{2}}\left(h_{i}(\tau)-\overline{h}_{i}(\tau)\right)\textrm{d}\tau}{\sqrt{\varepsilon}}\to
0$ as $\varepsilon\to 0$. Therefore, letting
$\beta(kT)\rightarrow\sqrt{\varepsilon}$ prevents the saturation of the
quantizer from the accumulated ESO estimation error. What is more, note that
in (18), the scaling function $\beta(kT)$ appears in the denominator of the
information to be quantized. This together with its update mechanism
(19)-(20), for $\beta(kT)>\sqrt{\varepsilon}$, the sensitivity of the
quantizer to the term $\left[\overline{h}_{i}((k+1)T)-\chi_{i}(kT)\right]$
increases if the system is free of DoS attacks (since $\gamma=\gamma_{1}<1$),
and keeps invariant if under DoS attacks (since $\gamma=1$).

If agent $i$ is a neighbor of agent $j$, $\Delta_{j}$ (which is the output of
the encoder for agent $j$) will be transmitted through the DoS affected
digital network. Agent $i$ uses the following decoder to reconstruct
$\overline{h}_{j}$:

$\left\\{\begin{aligned} \hat{\overline{h}}_{ji}(0)=&0,\\\
\hat{\overline{h}}_{ji}((k+1)T)=&\hat{\overline{h}}_{ji}(kT)+\beta(kT)\Delta_{j}((k+1)T),\\\
&\textrm{if}~{}k+1\in H_{v},\\\
\hat{\overline{h}}_{ji}((k+1)T)=&\hat{\overline{h}}_{ji}(kT),~{}\textrm{if}~{}k+1\notin
H_{v},\end{aligned}\right.$ (21)

where $\hat{\overline{h}}_{ji}$ is the estimate of $\overline{h}_{j}$ obtained
by agent $i$. Since the initial conditions of the encoders and decoders are
set as 0, we let $v_{0}=0$ henceforth.

With the aid of the ESO (15), encoder (18), and decoder (21), we propose the
following protocol:

$\displaystyle u_{i}(t)=$
$\displaystyle-\sum_{\ell=1}^{r-1}k_{\ell}\overline{\rho}_{i,\ell+1}(t)-\overline{\rho}_{i,r+1}(t)$
$\displaystyle+c\sum_{j\in\mathcal{N}^{\textrm{DoS}}_{i}(kT)}a_{ij}\left(\hat{\overline{h}}_{ji}(kT)-\chi_{i}(kT)\right),$
$\displaystyle t\in[kT,(k+1)T),$ (22)

where $c>0$, and $\mathcal{N}_{i}^{\textrm{DoS}}(kT)$ represents the set of
agents which are able to exchange information with agent $i$ through the DoS
affected digital network at $t=kT$. That is,
$\mathcal{N}_{i}^{\textrm{DoS}}(kT)=\mathcal{N}_{i}$ if $k\in H_{v}$, and
$\mathcal{N}_{i}^{\textrm{DoS}}(kT)=\emptyset$ otherwise. In the control
protocol (III-A), the term
$-\sum_{\ell=1}^{r-1}k_{\ell}\overline{\rho}_{i,\ell+1}(t)-\overline{\rho}_{i,r+1}(t)$
is to compensate for the agent dynamics in real time, and the term
$c\sum_{j\in\mathcal{N}^{\textrm{DoS}}_{i}(kT)}a_{ij}\left(\hat{\overline{h}}_{ji}(kT)-\chi_{i}(kT)\right)$
satisfies

$\displaystyle
c\sum_{j\in\mathcal{N}^{\textrm{DoS}}_{i}(kT)}a_{ij}\left(\hat{\overline{h}}_{ji}(kT)-\chi_{i}(kT)\right)$
$\displaystyle=$ $\displaystyle
c\sum_{j\in\mathcal{N}^{\textrm{DoS}}_{i}(kT)}a_{ij}\left(\overline{h}_{j}(kT)-\overline{h}_{i}(kT)\right)$
$\displaystyle-c\sum_{j\in\mathcal{N}^{\textrm{DoS}}_{i}(kT)}a_{ij}\left(\overline{h}_{j}(kT)-\hat{\overline{h}}_{ji}(kT)\right)$
$\displaystyle+c\sum_{j\in\mathcal{N}^{\textrm{DoS}}_{i}(kT)}a_{ji}\left(\overline{h}_{i}(kT)-\hat{\overline{h}}_{ij}(kT)\right),$

where the second term represents the weighted sum of the estimation errors for
agent $i$’s neighbors’ information $\overline{h}_{j}(kT)$; and the last term
represents the weighted sum of the estimation errors for
$\overline{h}_{i}(kT)$ by agent $i$’s neighbors. The last term plays an error-
compensation role in the protocol and is mainly based on the assumption that
$a_{ij}=a_{ji}$. This mechanism was first considered in [37], and also
employed in many previous quantized consensus results [5, 6, 8, 9, 10, 11].
The extension of the protocol (III-A) to direct communication graphs is not
straightforward, and some extra techniques are needed [7].

The proposed control scheme for uncertain nonlinear multi-agent systems under
data-rate constraints and DoS attacks is depicted in Fig. 1.

###  III-B Convergence Analysis

Before stating the convergence results, we first present the dynamics of the
multi-agent system in the absence and presence of DoS attacks. Denote
$H(kT)=[h_{1}(kT),\ldots,h_{N}(kT)]^{\rm{T}}$,
$\overline{H}(kT)=[\overline{h}_{1}(kT),\ldots,\overline{h}_{N}(kT)]^{\rm{T}}$,
$\widehat{\overline{H}}(kT)=[\chi_{1}(kT),\ldots,\chi_{N}(kT)]^{\rm{T}}$,
$\mu(kT)=\overline{H}(kT)-\widehat{\overline{H}}(kT)$, and
$\delta(kT)=H(kT)-J_{N}H(kT)$. Note that $\mu(kT)$ and $\delta(kT)$ represent
the compact quantization error and consensus error, respectively.

If the multi-agent system is _in the absence of DoS attack_ , i.e., $k\in
H_{v}$, then by (III-A) and the definition of $h_{i}$, one has

$\displaystyle\dot{h}_{i}(t)=$
$\displaystyle\sum_{\ell=1}^{r}k_{\ell}\rho_{i,\ell+1}(t)+u_{i}(t)$
$\displaystyle=$ $\displaystyle
c\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\hat{\overline{h}}_{ji}(kT)-\chi_{i}(kT)\right)$
$\displaystyle+\sum_{\ell=1}^{r}k_{\ell}\left(\rho_{i,\ell+1}(t)-\overline{\rho}_{i,\ell+1}(t)\right),t\in[kT,(k+1)T),$

where $k_{r}=1$. It follows that

$\displaystyle h_{i}((k+1)T)=h_{i}(kT)$
$\displaystyle\qquad+cT\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\hat{\overline{h}}_{ji}(kT)-\chi_{i}(kT)\right)+\varsigma_{1i}(kT),$
(24)

where

$\displaystyle\varsigma_{1i}(kT)=\int_{kT}^{(k+1)T}\left(\sum_{\ell=1}^{r}k_{\ell}\left(\rho_{i,\ell+1}(t)-\overline{\rho}_{i,\ell+1}(t)\right)\right)\textrm{d}t.$

According to the structures of the paired encoder (18) and decoder (21), one
has

$\hat{\overline{h}}_{ji}(kT)=\chi_{j}(kT),~{}i\in\mathcal{N}_{j},~{}j=1,2,\ldots,N.$
(25)

This together with (III-B) leads to

$\displaystyle H((k+1)T)=$ $\displaystyle
H(kT)-cT\mathcal{L}\widehat{\overline{H}}(kT)+\varsigma_{1}(kT)$
$\displaystyle=$
$\displaystyle(I-cT\mathcal{L})\overline{H}(kT)+cT\mathcal{L}\mu(kT)$
$\displaystyle+H(kT)-\overline{H}(kT)+\varsigma_{1}(kT),$ (26)

where
$\varsigma_{1}(kT)=[\varsigma_{11}(kT),\ldots,\varsigma_{1N}(kT)]^{\textrm{T}}$.
Since $\mathcal{L}J_{N}=J_{N}\mathcal{L}=0$, by (III-B), one gets

$\displaystyle\delta((k+1)T)=$ $\displaystyle(I-J_{N})H((k+1)T)$
$\displaystyle=$
$\displaystyle(I-cT\mathcal{L})\delta(kT)+cT\mathcal{L}\mu(kT)$
$\displaystyle+cT\mathcal{L}\left(H(kT)-\overline{H}(kT)\right)$
$\displaystyle+(I-J_{N})\varsigma_{1}(kT),$ (27)

and

$\displaystyle\overline{H}((k+1)T)-\widehat{\overline{H}}(kT)$
$\displaystyle=$ $\displaystyle
H((k+1)T)-\widehat{\overline{H}}(kT)+\overline{H}((k+1)T)-H((k+1)T)$
$\displaystyle=$
$\displaystyle(I+cT\mathcal{L})\mu(kT)-cT\mathcal{L}\delta(kT)$
$\displaystyle+(I+cT\mathcal{L})(H(kT)-\overline{H}(kT))$
$\displaystyle+\varsigma_{1}(kT)+\overline{H}((k+1)T)-H((k+1)T).$ (28)

If the multi-agent system is _in the presence of DoS attack_ , i.e., $k\notin
H_{v}$, then one has

$\dot{h}_{i}(t)=\sum_{\ell=1}^{r}k_{\ell}\left(\rho_{i,\ell+1}(t)-\overline{\rho}_{i,\ell+1}(t)\right),t\in[kT,(k+1)T).$
(29)

It follows that

$\displaystyle H((k+1)T)=H(kT)+\varsigma_{1}(kT).$ (30)

By (30), one has

$\displaystyle\delta((k+1)T)=\delta(kT)+(I-J_{N})\varsigma_{1}(kT),$ (31)

and

$\displaystyle\overline{H}((k+1)T)-\widehat{\overline{H}}(kT)$
$\displaystyle=$
$\displaystyle\mu(kT)+\varsigma_{1}(kT)+H(kT)-\overline{H}(kT)$
$\displaystyle+\overline{H}((k+1)T)-H((k+1)T).$ (32)

According to the update mechanisms of the encoders and decoders, the evolution
equations of $\mu(kT)$ are stated as follows:

$\displaystyle\mu((k+1)T)=$
$\displaystyle\left[\overline{H}((k+1)T)-\widehat{\overline{H}}(kT)\right]$
$\displaystyle-\beta(kT)Q\left(\frac{\overline{H}((k+1)T)-\widehat{\overline{H}}(kT)}{\beta(kT)}\right),$
$\displaystyle\textrm{if}~{}k+1\in H_{v},$ (33) $\displaystyle\mu((k+1)T)=$
$\displaystyle\overline{H}((k+1)T)-\widehat{\overline{H}}(kT),$
$\displaystyle\textrm{if}~{}k+1\notin H_{v},$ (34)

where $\left[\overline{H}((k+1)T)-\widehat{\overline{H}}(kT)\right]$ is given
by (III-B) if $k\in H_{v}$, and given by (III-B) if $k\notin H_{v}$.

The dynamics of the multi-agent system, in terms of the quantization error
$\mu(kT)$ and consensus error $\delta(kT)$, are given by (III-B), (31), (33),
and (34). The following remark explains the design philosophy of the proposed
zooming-in and holding approach in the context of quantized consensus of
multi-agent systems under DoS attacks.

_Remark 2 (Design Philosophy):_ When the transmissions are successful, the
agents are driven to achieve consensus. In this case, similar to [6, 7, 8, 9,
10, 11, 26], the scaling function $\beta(kT)$ zooms-in the information to be
quantized (i.e., the term
$\left[\overline{h}_{i}((k+1)T)-\chi_{i}(kT)\right]$) to decrease the
consensus error. Therefore, $\gamma_{1}$ is selected less than 1. When the
transmissions are unsuccessful, the consensus process is interrupted. The
strategy in this paper is to hold the previous status of the whole system. In
the consensus error dynamics (31) under DoS attacks, anticipating that
$\varsigma_{1}(kT)\rightarrow 0$ as $\varepsilon\rightarrow 0$, one gets
$\delta((k+1)T)=\delta(kT)$ if $k\notin H_{v}$. What is more, the status of
the encoders and decoders are also hold, as we let
$\chi_{i}(kT)=\chi_{i}((k-1)T)$,
$\hat{\overline{h}}_{ji}(kT)=\hat{\overline{h}}_{ji}((k-1)T)$, and
$\beta(kT)=\beta((k-1)T)$ if $k\notin H_{v}$. Such a zooming-in and holding
mechanism is expected to lead to higher tolerance to DoS attacks. We should
point out that the zooming-in and holding approach is fundamentally different
from the zooming-in and zooming-out approach in [26] (see Section IV for more
details). ∎

In the sequel, we will state the convergence results of the closed-loop
system. Let $\mathcal{X}^{{\dagger}}$ be a compact subset of $\mathbb{R}^{r}$,
and $\mathcal{X}$ slightly smaller than $\mathcal{X}^{{\dagger}}$ (i.e.,
$\mathcal{X}\subseteq\mathcal{X}^{{\dagger}}$ and their boundaries are
disjoint). Denote $C_{h}\geq\max_{1\leq i\leq N}|h_{i}(0)|$. Let
$\omega_{i}\in\mathcal{W}_{i}$ for some compact set
$\mathcal{W}_{i}\subseteq\mathbb{R}^{n_{\omega_{i}}}$, and let
$z_{i}(0)\in\mathcal{Z}_{i}$ for some compact set
$\mathcal{Z}_{i}\subseteq\mathbb{R}^{n_{i}-r}$. By Assumption A3, there exists
a positive constant $N_{zi}$ such that $\sup_{t\in[0,\infty)}\|z_{i}(t)\|\leq
N_{zi}$ for all $z_{i}(0)\in\mathcal{Z}_{i}$,
$\rho_{i}\in\mathcal{X}^{{\dagger}}$, and $\omega_{i}\in\mathcal{W}_{i}$. Note
that any compact subset of $\mathbb{R}^{r}\times\mathbb{R}^{n_{i}-r}$ can be
put in the interior of $\mathcal{X}\times\mathcal{Z}_{i}$. We first state the
following lemma.

_Lemma 2[6]: If Assumption A1 holds and $c\in(0,2/(T\lambda_{N}))$, then
$\rho_{h}<1$, where_

$\rho_{h}=\max_{2\leq i\leq N}\left|1-cT\lambda_{i}\right|.$ (35)

_Furthermore, if $c\in(0,2/(T(\lambda_{2}+\lambda_{N})))$, then
$\rho_{h}=1-cT\lambda_{2}$._

Now, we are in a position to state our first main result.

_Theorem 1: Consider the multi-agent system (1) with a digital communication
network subject to data-rate constraints and DoS attacks. Given the control
protocol (III-A) with ESO (15), encoder (18), and decoder (21). Suppose
Assumptions A1 to A4 are satisfied, and the initial conditions of the agents
$(\rho_{i}(0),z_{i}(0))\in\mathcal{X}\times\mathcal{Z}_{i}$, $1\leq i\leq N$.
Let_

$\displaystyle c$ $\displaystyle\in\left(0,2/(T\lambda_{N})\right),$ (36)
$\displaystyle\gamma_{1}$ $\displaystyle\in(\rho_{h},1),$ (37) $\displaystyle
K$ $\displaystyle\geq\left\lfloor
K_{1}(c,\gamma_{1})-\frac{1}{2}\right\rfloor+1,$ (38) $\displaystyle\beta_{0}$
$\displaystyle>\max\left\\{\frac{2cT\lambda_{N}C_{h}}{\gamma_{1}(K+\frac{1}{2})},\frac{4C_{h}\gamma_{1}(\gamma_{1}-\rho_{h})}{cT\lambda_{N}}\right\\},$
(39)

_where_

$\displaystyle
K_{1}(c,\gamma_{1})=\frac{1+2cTd^{*}}{2\gamma_{1}}+\frac{\sqrt{N}c^{2}T^{2}\lambda^{2}_{N}}{2\gamma_{1}(\gamma_{1}-\rho_{h})}.$
(40)

_Then for any $\sigma>0$, there exists $\varepsilon^{{\dagger}}>0$, which is
dependent on $\sigma$, such that
$\forall\varepsilon\in(0,\varepsilon^{{\dagger}})$:_

  * •

_the ESO (_ 15) achieves practical convergence, i.e., there exists
$\tau_{0}(\varepsilon)>0$ satisfying $\lim_{\varepsilon\rightarrow
0}\tau_{0}(\varepsilon)=0$, such that $\forall
t\in[\tau_{0}(\varepsilon),\infty)$,

$\left|\rho_{ij}(t)-\hat{\rho}_{ij}(t)\right|\leq\sigma,~{}1\leq i\leq
N,~{}1\leq j\leq r+1;$ (41)

  * •

_the quantizer will never be saturated, and the multi-agent system (_ 1)
achieves practical output consensus, i.e.,

$\lim_{t\rightarrow\infty}\left|y_{i}(t)-y_{j}(t)\right|\leq\sigma,~{}1\leq
i\neq j\leq N.$ (42)

_Proof:_ See the Appendix. ∎

As stated in Theorem 1, the DoS attacks can be arbitrarily serious as long as
their induced maximal number of consecutive packet losses is bounded. Also
note that $K\rightarrow\infty$ as $N\rightarrow\infty$. However, the
quantization level (or data-rate) of a real digital network is limited. Our
next theorem shows that the selection of $K$ can be made independent of the
number of the agents. In particular, the lowest quantization level we achieve
is 3, i.e., $K=1$. That is, the communication channel is required to be
capable of transmitting merely 2 bits at each time step, no matter how large
the number of the agents is.

_Theorem 2: Consider the multi-agent system (1) with a digital communication
network subject to data-rate constraints and DoS attacks. Given the control
protocol (III-A) with ESO (15), encoder (18), and decoder (21). Suppose
Assumptions A1 to A4 are satisfied, and the initial conditions of the agents
$(\rho_{i}(0),z_{i}(0))\in\mathcal{X}\times\mathcal{Z}_{i}$, $1\leq i\leq N$.
For any given $K\geq 1$, let $\beta_{0}$ be selected according to (39), and
let_

$\displaystyle c\in$
$\displaystyle(0,\min\\{2/(T(\lambda_{2}+\lambda_{N})),c_{m}\\}),$ (43)
$\displaystyle\gamma_{1}=$ $\displaystyle 1-(1-\epsilon_{0})cT\lambda_{2},$
(44)

_where $\epsilon_{0}\in(0,1)$ and
$c_{m}=2K\epsilon_{0}\lambda_{2}T^{-1}\left[\sqrt{N}\lambda^{2}_{N}+2\epsilon_{0}\lambda_{2}d^{*}\right.$
$\left.+(2K+1)(1-\epsilon_{0})\epsilon_{0}\lambda_{2}^{2}\right]^{-1}$. Then
the statements in Theorem 1 hold._

_Proof:_ By the definition of $c_{m}$, one can verify that
$K_{1}(c,\gamma_{1})$ specified by (40) satisfies
$\frac{1}{2}<K_{1}(c,\gamma_{1})<K+\frac{1}{2}$. This together with (43) and
$(\ref{eq98})$ yields that the conditions in (36)-(38) are all satisfied.
Then, similar to Theorem 1, one can complete the proof of Theorem 2. ∎

Theorem 2 indicates that the proposed control protocol is capable of handling
any DoS attacks inducing bounded consecutive packet losses for the uncertain
nonlinear multi-agent system (1) with merely 2 bits information exchange
between each pair of adjacent agents for each transmission attempt. Also note
that $\gamma_{1}\rightarrow 1$ as $N\rightarrow\infty$, which means that,
under limited data rate, the convergence rate becomes slow if the number of
the agents increases.

_Remark 3 (Parameters Selection):_ The saturation bounds $M_{1}$ to $M_{r+1}$
satisfying $M_{j}\geq\sup_{1\leq i\leq N}|\rho_{ij}|$, $1\leq j\leq r+1$, are
selected such that the saturations will not be invoked in the steady state of
the observer [30]. The parameter $C_{h}$ is to prevent the saturation of the
quantizer in the initial stage [6, 7, 8, 9, 10, 11, 26]. The selections of
$M_{j}$ and $C_{h}$ might end up with some conservative values due to the
uncertain nonlinear agent dynamics. For the parameter $\varepsilon$,
theoretically, smaller $\varepsilon$ leads to higher estimation and consensus
accuracy. Unfortunately, in case of uncertain dynamics, concrete and
constructive algorithms to calculate the upper bound of $\varepsilon$ (i.e.,
$\varepsilon^{{\dagger}}$) are very challenging to obtain and still missing in
the literature [30]. This is indeed a cost we pay for the uncertainties. In
this paper, the scope of the current work is to develop a control scheme with
the existence of certain parameters to handle the quantized consensus problem
under data-rate constraints and DoS attacks. In practice, such parameters can
be decided by a simple trial and error procedure [30], or by a simple line
search algorithm [36]. In fact, the ESO has been successfully adopted in many
control engineering problems in recent years [39, 38]. At last, we mention
that the sampling period $T$ is not a design parameter in Theorem 1. As $T$ is
always multiplied by the constant $c$, the effects of the sampling period on
the control design can be adjusted by the value of $c$. ∎

_Remark 4 (Comparison with[12]): _ In our previous work [12], a higher-order
ESO-based protocol was developed to handle the limited data rate problem for a
class of simpler nonlinear multi-agent systems without DoS attacks. In this
paper, we consider more general uncertain nonlinear systems, and our focus is
on the coexistence problem of data-rate constraints and DoS attacks. Due to
the communication interruption caused by DoS, the consensus error may diverge
during the DoS periods, while in [12] the communication network is always
reliable. Therefore, in this paper new encoding-decoding scheme is needed, and
the behavior of the system during the attack periods must be carefully
governed. Accordingly, we developed a novel dynamic quantization with zooming-
in and holding capabilities, and showed that it enables better transient
consensus performance and higher level of tolerable DoS attacks. These render
the contribution of this paper with respect to [12] substantial. ∎

##  IV Quantized Consensus of Known Linear Multi-Agent Systems

In this section, for the ease of comparison with state-of-the-art results, we
revisit the quantized consensus problem for a class of known discrete-time
linear multi-agent systems under data-rate constraints and DoS attacks.

###  IV-A Protocol Design

Without loss of generality, it is assumed that the model of agent $i$, $1\leq
i\leq N$, is already in the following controllable canonical form:

$x_{i}((k+1)T)=Ax_{i}(kT)+Bu_{i}(kT),$ (45)

where $x_{i}\in\mathbb{R}^{r}$ is the state, and matrices
$A\in\mathbb{R}^{r\times r}$ and $B\in\mathbb{R}^{r\times 1}$ are given by

$\displaystyle A=\left[\begin{array}[]{cccc}0&1&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&1\\\ -a_{1}&-a_{2}&\cdots&-a_{r}\\\
\end{array}\right],B=\left[\begin{array}[]{c}0\\\ \vdots\\\ 0\\\ 1\\\
\end{array}\right].$

Here we are interested in the case where $A$ has at least one eigenvalue on or
outside the unit circle. Otherwise, the multi-agent system (45) can achieve
state consensus by setting $u_{i}(kT)\equiv 0$. This implies that any
communication protocol among the agents and any pattern of DoS attacks do not
influence the trivial consensus problem if $A$ is a stable matrix.

Denote

$h_{i}(kT)=k_{1}x_{i1}(kT)+\cdots+k_{r-1}x_{i,r-1}(kT)+x_{ir}(kT).$ (46)

For the discrete-time case, $k_{1}$, $\ldots$, $k_{r-1}$ are selected such
that the eigenvalues of matrix $\tilde{A}$ specified by (14) are real and
distinct, and located inside the unit circle. For subsequent use, let
$\tilde{\lambda}_{1}$, $\ldots$, $\tilde{\lambda}_{r-1}$ be the eigenvalues of
matrix $\tilde{A}$, and let
$\tilde{\Lambda}=\textrm{diag}\\{\tilde{\lambda}_{1},\ldots,\tilde{\lambda}_{r-1}\\}$.
It follows that there exists an invertible matrix
$\Gamma\in\mathbb{R}^{(r-1)\times(r-1)}$ such that
$\Gamma\tilde{A}\Gamma^{-1}=\tilde{\Lambda}$. Let
$\tilde{\Lambda}^{{\dagger}}=\textrm{diag}\\{1/(1-\tilde{\lambda}_{1}),\ldots,1/(1-\tilde{\lambda}_{r-1})\\}$
and $\tilde{A}_{r-1}=[-k_{1},\ldots,-k_{r-1}]$ be the $(r-1)$th row of matrix
$\tilde{A}$.

_Lemma 3: If $h_{i}(kT)$, $1\leq i\leq N$, $k\geq 0$, are bounded and
$\lim_{k\rightarrow\infty}h_{i}(kT)=h^{*}$ for some constant $h^{*}$, then_

$\lim_{k\rightarrow\infty}\|x_{i}(kT)-x_{j}(kT)\|=0,~{}1\leq i\neq j\leq N.$
(47)

_Furthermore,_

$\lim_{k\rightarrow\infty}x_{i}(kT)=\left[\begin{aligned}
&~{}~{}~{}~{}\Gamma\tilde{\Lambda}^{{\dagger}}\Gamma^{-1}\tilde{B}\\\
&1+\tilde{A}_{r-1}\Gamma\tilde{\Lambda}^{{\dagger}}\Gamma^{-1}\tilde{B}\end{aligned}\right]h^{*},$
(48)

_where the matrices $\tilde{A}$ and $\tilde{B}$ are given by (14)._

_Proof:_ Let
$\tilde{x}_{i}(kT)=[x_{i1}(kT),\ldots,x_{i,r-1}(kT)]^{\rm{T}}\in\mathbb{R}^{r-1}$.
By (46), one has

$\tilde{x}_{i}((k+1)T)=\tilde{A}\tilde{x}_{i}(kT)+\tilde{B}h_{i}(kT).$ (49)

Note that the eigenvalues of matrix $\tilde{A}$ are locate inside the unit
circle, and $h_{i}(kT)$ is bounded and converges to $h^{*}$. From the linear
control theory [40], one can readily obtain (47). Furthermore, according to
the response of the discrete-time linear system (49) and the relation
$\tilde{A}^{k}=\Gamma\tilde{\Lambda}^{k}\Gamma^{-1}$, one has
$\lim_{k\rightarrow\infty}\tilde{x}_{i}(kT)=\Gamma\tilde{\Lambda}^{{\dagger}}\Gamma^{-1}\tilde{B}h^{*}$.
This together with (46) yields (48). ∎

By Lemma 3, the consensus of $h_{i}(kT)$ implies the consensus of $x_{i}(kT)$.
The dynamics of $h_{i}(kT)$ is given by

$\displaystyle h_{i}((k+1)T)=$
$\displaystyle\sum_{\ell=1}^{r-1}k_{\ell}x_{i,\ell+1}(kT)-\sum_{m=1}^{r}a_{m}x_{im}(kT)$
$\displaystyle+u_{i}(kT).$ (50)

Similar to Section III, the encoder for agent $i$ is designed as

$\left\\{\begin{aligned} \chi_{i}(0)=&0,\\\
\chi_{i}((k+1)T)=&\chi_{i}(kT)+\beta(kT)\Delta_{i}((k+1)T),\\\
&\textrm{if}~{}k+1\in H_{v},\\\
\chi_{i}((k+1)T)=&\chi_{i}(kT),~{}\textrm{if}~{}k+1\notin H_{v},\\\
\Delta_{i}((k+1)T)=&q\left(\frac{h_{i}((k+1)T)-\chi_{i}(kT)}{\beta(kT)}\right),\end{aligned}\right.$
(51)

where the scaling function is updated by

$\beta((k+1)T)=\gamma\beta(kT),$ (52)

with $\gamma$ given by (20). Note that in (51) we use the accurate value of
$h_{i}$, while the ESO estimated value $\overline{h}_{i}$ in Section III. That
is, no additional ESO estimation error will be injected into the quantization
process here. Therefore, in this section we let the scaling function converge
to 0 rather than $\sqrt{\varepsilon}$ . The decoder for agent $i$
corresponding to the edge $(i,j)$ is as follows:

$\left\\{\begin{aligned} \hat{h}_{ji}(0)=&0,\\\
\hat{h}_{ji}((k+1)T)=&\hat{h}_{ji}(kT)+\beta(kT)\Delta_{j}((k+1)T),\\\
&\textrm{if}~{}k+1\in H_{v},\\\
\hat{h}_{ji}((k+1)T)=&\hat{h}_{ji}(kT),~{}\textrm{if}~{}k+1\notin
H_{v},\end{aligned}\right.$ (53)

where $\hat{h}_{ji}$ is the estimate of $h_{j}$ obtained by agent $i$. Here we
remind that for the known linear multi-agent systems in this section, $h_{j}$
is specified by (46). The protocol for the known linear multi-agent system
(45) is then given by

$\displaystyle u_{i}(kT)=$
$\displaystyle-\sum_{\ell=1}^{r-1}k_{\ell}x_{i,\ell+1}(kT)+\sum_{m=1}^{r}a_{m}x_{ij}(kT)+h_{i}(kT)$
$\displaystyle+cT\sum_{j\in\mathcal{N}^{\textrm{DoS}}_{i}(kT)}a_{ij}\left(\hat{h}_{ji}(kT)-\chi_{i}(kT)\right),$
(54)

where $c>0$.

###  IV-B Convergence Analysis

First of all, denote $H(kT)$ and $\delta(kT)$ as in Section III-B, and let
$\widehat{H}(kT)=[\chi_{1}(kT),\ldots,\chi_{N}(kT)]^{\rm{T}}$ and
$\mu(kT)=H(kT)-\widehat{H}(kT)$ in this subsection.

If the multi-agent system (45) is _in the absence of DoS attack_ , i.e., $k\in
H_{v}$, then by (IV-A) and (IV-A), one has

$\displaystyle
h_{i}((k+1)T)=h_{i}(kT)+cT\sum_{j\in\mathcal{N}_{i}}a_{ij}(\hat{h}_{ji}(kT)-\chi_{i}(kT)).$
(55)

This together with $\hat{h}_{ji}(kT)=\chi_{j}(kT)$, $i\in\mathcal{N}_{j}$,
gives

$\displaystyle H((k+1)T)=$ $\displaystyle H(kT)-cT\mathcal{L}\widehat{H}(kT)$
$\displaystyle=$ $\displaystyle(I-cT\mathcal{L})H(kT)+cT\mathcal{L}\mu(kT).$
(56)

By $\mathcal{L}J_{N}=J_{N}\mathcal{L}=0$, one has

$\displaystyle\delta((k+1)T)=(I-cT\mathcal{L})\delta(kT)+cT\mathcal{L}\mu(kT),$
(57)

and

$\displaystyle
H((k+1)T)-\widehat{H}(kT)=(I+cT\mathcal{L})\mu(kT)-cT\mathcal{L}\delta(kT).$
(58)

If the multi-agent system (45) is _in the presence of DoS attack_ , i.e.,
$k\notin H_{v}$, then one has

$h_{i}((k+1)T)=h_{i}(kT).$ (59)

It follows that

$\displaystyle\delta((k+1)T)=\delta(kT),$ (60)

and

$\displaystyle H((k+1)T)-\widehat{H}(kT)=\mu(kT).$ (61)

The evolution equations of $\mu(kT)$ are given by

$\displaystyle\mu((k+1)T)=$
$\displaystyle\left[H((k+1)T)-\widehat{H}(kT)\right]$
$\displaystyle-\beta(kT)Q\left(\frac{H((k+1)T)-\widehat{H}(kT)}{\beta(kT)}\right),$
$\displaystyle\textrm{if}~{}k+1\in H_{v},$ (62) $\displaystyle\mu((k+1)T)=$
$\displaystyle\mu(kT),~{}\textrm{if}~{}k+1\notin H_{v},$ (63)

where $\left[H((k+1)T)-\widehat{H}(kT)\right]$ is given by (58) if $k\in
H_{v}$, and given by (61) if $k\notin H_{v}$.

The dynamics of the multi-agent system (45), in terms of the quantization
error $\mu(kT)$ and consensus error $\delta(kT)$, are represented by (57),
(60), (62), and (63). Now we are ready to present the result concerning the
state consensus of the known linear multi-agent system (45) with the zooming-
in and holding approach.

_Theorem 3: Consider the known linear multi-agent system (45) with a digital
network subject to data-rate constraints and DoS attacks. Given the control
protocol (IV-A) with encoder (51) and decoder (53). Suppose Assumptions A1 and
A4 are satisfied, and the initial conditions of the agents
$x_{i}(0)\in\mathcal{X}$, $1\leq i\leq N$. For any given $K\geq 1$, let the
parameter $c$ and $\gamma_{1}$ be selected as in Theorem 2, and the scaling
function be updated by (52) and (20), and_

$\displaystyle\beta_{0}$
$\displaystyle>\max\left\\{\frac{C_{h}}{K+\frac{1}{2}},\frac{2C_{h}(\gamma_{1}-\rho_{h})(cT\lambda_{N}+2\gamma_{1})}{cT\lambda_{N}}\right\\}.$
(64)

_Then the quantizer will never be saturated, and the multi-agent system ( 45)
achieves state consensus specified by (47). Furthermore, the final consensus
value is given by_

$\lim_{k\rightarrow\infty}x_{i}(kT)=\left[\begin{aligned}
&~{}~{}~{}~{}\Gamma\tilde{\Lambda}^{{\dagger}}\Gamma^{-1}\tilde{B}\\\
&1+\tilde{A}_{r-1}\Gamma\tilde{\Lambda}^{{\dagger}}\Gamma^{-1}\tilde{B}\end{aligned}\right]\frac{1}{N}\sum_{m=1}^{N}\sum_{j=1}^{r}k_{j}x_{mj}(0).$
(65)

_Proof:_ The proof is straightforward by following a similar technique line as
in the Proposition 2 in the proof of Theorem 1. Here we omit the details due
to space limitation. The initial condition of the scaling function $\beta_{0}$
is given by (64) rather than (39) is because that in this section
$\|\mu(0)\|_{\infty}=\|H(0)-\widehat{H}(0)\|_{\infty}\leq C_{h}$ while in
Section III
$\|\mu(0)\|_{\infty}=\|\overline{H}(0)-\widehat{\overline{H}}(0)\|_{\infty}=0$.
∎

###  IV-C Discussion

In this subsection, we first briefly introduce the approach in [26] for
comparison purpose. In [26], a control scheme which adopts a zooming-in and
zooming-out mechanism was proposed to tackle the quantized consensus problem
of known linear multi-agent systems under data-rate constraints and DoS
attacks. The basic idea of this mechanism is to update the encoder as an open-
loop system during DoS attacks. In this case, the scaling function needs to
zoom-out the information to be quantized to prevent the quantizer being
saturated. Specifically, for the multi-agent system (45), the encoder in [26]
is given by

$\left\\{\begin{aligned} \hat{x}_{i}(0)=&\,\textbf{0}_{r},\\\
\hat{x}_{i}((k+1)T)=&\,A\hat{x}_{i}(kT)+\beta(kT)\Delta_{i}((k+1)T),\\\
&\textrm{if}~{}k+1\in H_{v},\\\
\hat{x}_{i}((k+1)T)=&\,A\hat{x}_{i}(kT),~{}\textrm{if}~{}k+1\notin H_{v},\\\
\Delta_{i}((k+1)T)=&\,Q\left(\frac{x_{i}((k+1)T)-\hat{x}_{i}(kT)}{\beta(kT)}\right),\end{aligned}\right.$
(66)

where $\hat{x}_{i}$ is the estimate of the state of agent $i$ by agent $i$
itself and its neighbors. In (66), the scaling function $\beta(kT)$ is updated
by (52) with

$\gamma=\left\\{\begin{aligned} \gamma_{1},~{}\textrm{if}~{}k+1\in H_{v},\\\
\gamma_{2},~{}\textrm{if}~{}k+1\notin H_{v},\end{aligned}\right.$ (67)

where $0<\gamma_{1}<1$ and $\gamma_{2}>1$. Accordingly, the control protocol
in [26] is given by

$u_{i}(kT)=K_{c}\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\hat{x}_{j}(kT)-\hat{x}_{i}(kT)\right),$
(68)

where the feedback gain $K_{c}$ is selected such that the eigenvalues of the
matrix $\textrm{diag}\\{A-\lambda_{2}BK_{c},\ldots,A-\lambda_{N}BK_{c}\\}$
locate inside the unit circle.

In the sequel, several remarks are presented that provide intuitive
explanations on the properties and advantages of the approach developed in
this paper.

_Remark 5 (Design Complexity):_ Compared with [26], the control design in this
paper is relatively simpler. When the system is subject to DoS attacks,
$\mathcal{N}^{\textrm{DoS}}_{i}(kT)=\emptyset$, and hence by (IV-A) and
(IV-A), one gets (59), i.e., $h_{i}((k+1)T)=h_{i}(kT)$. This implies that the
variable $h_{i}(kT)$ keeps invariant under DoS attacks. Consequently, in (51),
we are able to make the state of the encoder invariant during DoS attacks, and
hence the parameter of the scaling function $\gamma=1$ if the transmission
fails. By contrast, in (66), the control protocol is designed as (68) in which
the state estimation under DoS in the encoder is calculated according to the
open-loop equation $\hat{x}_{i}((k+1)T)=A\hat{x}_{i}(kT)$. In this case, the
consensus error driven by the control protocol (68) may diverge during DoS
attacks [26]. Therefore, in [26], the scaling function $\beta(kT)$ must be
updated accordingly to make sure that the quantizer is unsaturated during DoS
attacks. That is, zoom-out the term
$\left[x_{i}((k+1)T)-\hat{x}_{i}(kT)\right]$ by introducing an extra parameter
$\gamma_{2}$. It should be mentioned that the calculation of $\gamma_{2}$ is
not an easy work. In fact, the value of $\gamma_{2}$ is decided by the system
matrices $A$ and $B$, the Laplacian matrix $\mathcal{L}$, and $\Omega$ which
represents the maximum number of consecutive packet losses induced by DoS
attacks. Also note that the information of $\Omega$ is not required for
controller design in this paper. ∎

_Remark 6 (Consensus Performance):_ The consensus performance can be evaluated
by the convergence speed, transient performance, and final consensus value.
For the convergence speed and transient performance, it is mainly decided by
the scaling function $\beta(kT)$. During DoS attacks, the consensus error in
[26] may diverge while in this paper it keeps invariant (see (60)). In this
sense, the approach developed in this paper is able to lead to faster
convergence speed and better transient performance under the comparable level
of DoS attacks. For the final consensus value, in [26], it diverges to
infinity when $A$ has at least one eigenvalue outside the unit circle. On the
contrary, the final consensus value in Theorem 3 is given in terms of the
initial condition of the multi-agent system. This is important for some
practical applications, such as distributed computation [41]. ∎

_Remark 7 (Required Data Rate):_ Compared with [26], the approach developed in
this paper is capable of achieving lower data rate. On one hand, we encode the
information of a linear combination of the agent state (i.e., $h_{i}$) rather
than the agent state $x_{i}$ itself as in (66). That is to say, the dimension
of the encoding-decoding channel between each pair of adjacent agents in this
paper is 1 while in [26] is $r$ which is the dimension of $x_{i}$. On the
other hand, we show that, no matter how many agents there are, our proposed
protocol guarantees consensus with a fixed number of quantization level. More
importantly, the lowest number of the quantization level we achieve is 3
(i.e., $K=1$, a two-bit quantizer). In [26], the required quantization level
is proportional to the value of $\sqrt{Nr}$. This limits its capability for a
large group of agents. ∎

_Remark 8 (DoS Attack Tolerance):_ For the zooming-in and zooming-out
mechanism developed in [26], as the consensus error may diverge during DoS
attacks, the level of tolerable DoS attacks is expected to be low. More
specifically, the successful transmissions must occupy sufficient proportion
of the total transmissions, to compensate for the increase of the consensus
error caused by those unsuccessful transmissions. In [26], the tolerable DoS
level is given by
$\frac{1}{T_{D}}+\frac{T}{\tau_{D}}<\frac{-\ln\gamma_{1}}{\ln\gamma_{2}-\ln\gamma_{1}}$.
Note that the term $\frac{-\ln\gamma_{1}}{\ln\gamma_{2}-\ln\gamma_{1}}$ can be
much smaller than 1 in case the resulting $\gamma_{2}$ is large, which implies
a lower tolerance to DoS. In this paper, we only require that the number of
consecutive packet losses induced by DoS attacks is upper bounded, that is,
$\frac{1}{T_{D}}+\frac{T}{\tau_{D}}<1$. As discussed in [15], the bound “$<1$”
is the best bound that one can achieve in the sense that if the bound is
violated, a DoS attacker may corrupt all the transmission attempts and
consensus is not possible by any controller design. From this perspective, the
control scheme in this paper largely increases the tolerance to DoS attacks
with respect to [26]. ∎

##  V Examples

In this section, we present two simulation examples: a nonlinear academic one,
demonstrating the full capabilities of the proposed control scheme, and a
linear comparative example with the recent paper [26].

###  V-A Nonlinear Example

Consider a group of five agents with the following dynamics:

$\left\\{\begin{aligned} \dot{x}_{i1}=&x_{i1}+x_{i2},\\\
\dot{x}_{i2}=&x_{i3},\\\
\dot{x}_{i3}=&-p_{i3}x_{i3}-p_{i1}x_{i2}-p_{i2}x_{i2}^{3}+u_{i}+z_{i}+\omega_{i},\\\
\dot{z}_{i}=&-(x^{2}_{i3}+\omega_{i}^{2})z_{i},\\\ y_{i}=&x_{i1},~{}1\leq
i\leq 5,\\\ \end{aligned}\right.$ (69)

where $p_{i1}$, $p_{i2}$, and $p_{i3}$ are system parameters, and $\omega_{i}$
is the external disturbance. System (69) can be regarded as a connection of a
nonlinear Duffing equation [42] with stable zero dynamics and a first-order
linear system. Let $\rho_{i1}=x_{i1}$, $\rho_{i2}=x_{i1}+x_{i2}$, and
$\rho_{i3}=x_{i1}+x_{i2}+x_{i3}$. Then system (69) can be written as

$\left\\{\begin{aligned} \dot{\rho}_{i1}=&\rho_{i2},\\\
\rho_{i2}=&\rho_{i3},\\\
\dot{\rho}_{i3}=&\rho_{i3}-p_{i3}(\rho_{i3}-\rho_{i2})-p_{i1}(\rho_{i2}-\rho_{i1})\\\
&-p_{i2}(\rho_{i2}-\rho_{i1})^{3}+u_{i}+z_{i}+\omega_{i},\\\
\dot{z}_{i}=&-\left((\rho_{i3}-\rho_{i2})^{2}+\omega_{i}^{2}\right)z_{i},\\\
y_{i}=&\rho_{i1},~{}1\leq i\leq 5.\\\ \end{aligned}\right.$ (70)

The digital communication network among the five agents is depicted in Fig. 2.
In this simulation study, we set $p_{i1}=-1.1-0.2i$, $p_{i2}=1+0.2i$,
$p_{i3}=0.4+0.1i$, and $\omega_{i}=\frac{1}{2i}\sin(it)$, $1\leq i\leq 5$. The
initial conditions of the agents $\rho_{ij}(0)$ and $z_{i}(0)$, $1\leq i\leq
5$, $1\leq j\leq 3$, are randomly located in $[0,5]$. For the proposed control
scheme, let $k_{1}=k_{2}=4$, and $T=0.05$; the bounds are selected as
$M_{1}=10$, $M_{2}=10$, $M_{3}=10$, $M_{4}=100$, and $C_{h}=45$; the ESOs are
designed with
$[l_{1}~{}l_{2}~{}l_{3}~{}l_{4}]^{\rm{T}}=[4~{}6~{}4~{}1]^{\rm{T}}$, $1\leq
i\leq 5$.

Figure 2: The digital communication network of the five agents.

(a) $\varepsilon=0.01$.

(b) $\varepsilon=0.001$.

Figure 3: Trajectories of $y_{i}$ with $K=10$. The DoS attacks (colored in
gray) are present for $\sim$90% of total time. Figure 4: Performance of the
ESO with $\varepsilon=0.01$.

####  V-A1 Simulation with High Quantization Level

First, we simulate the high quantization level case. According to Theorem 1,
let $c=1$, $\gamma_{1}=0.96$, $\beta_{0}=10$, and $K=10$. We investigate a
randomly generated sustained DoS. Here for simulation purpose, the maximal
duration of the DoS is limited by 9 seconds, i.e., the number of consecutive
packet losses is no larger than 180, and the DoS attacks are present for
$\sim$90% of total time. Fig. 3 shows the simulation results with
$\varepsilon=0.01$ and $\varepsilon=0.001$. From this figure, one can observe
that the multi-agent system with unknown nonlinear dynamics, data-rate
constraints, and DoS attacks achieved practical convergence. Also note that
smaller $\varepsilon$ leads to smaller residual consensus error. Fig. 4
depicts the response of the ESO for agent 1 with $\varepsilon=0.01$. One can
see that both the agent states $\rho_{1j}$, $1\leq j\leq 3$, and the extended
state
$\rho_{14}\triangleq\rho_{13}-p_{13}(\rho_{13}-\rho_{12})-p_{11}(\rho_{12}-\rho_{11})-p_{12}(\rho_{12}-\rho_{11})^{3}+z_{1}+\omega_{1}$
are well-estimated.

####  V-A2 Simulation with Low Quantization Level

Next, we investigate the performance of the proposed control protocol with
two-bit quantizers (i.e., $K=1$). According to the conditions in Theorem 2, we
select $\epsilon_{0}=0.9$, $K=1$, $c=0.5$, $\gamma_{1}=0.9975$, and
$\beta_{0}=35$. Here we simulate a more serious DoS scenario. The duration of
each DoS attack is randomly selected between 0 to 95 seconds, while the
duration without DoS attack is randomly selected between 0 to 5 seconds. That
is, the DoS attacks are present for $\sim$95% of total time. Fig. 5 shows the
trajectories of $y_{i}$, $1\leq i\leq 5$, with $K=1$ and $\sim$95% DoS
attacks. One can see that the agents achieved output consensus with serious
DoS attacks and merely two-bit information exchange between each pair of
adjacent agents at each time step. The simulation results are consistent with
the theoretical results in Theorem 2.

Figure 5: Trajectories of $y_{i}$ with $K=1$ and $\sim$95% DoS attacks.

###  V-B Comparative Example

We consider the diving consensus problem of five autonomous underwater
vehicles (AUVs). The discrete-time model of the $i$th AUV with sampling period
1 second is as follows [43] (we have transformed the system into controllable
canonical form):

$\displaystyle\left[\begin{array}[]{c}x_{i1}(k+1)\\\ x_{i2}(k+1)\\\
x_{i3}(k+1)\\\ \end{array}\right]=$
$\displaystyle\left[\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
0.216&-1.502&2.286\\\ \end{array}\right]\left[\begin{array}[]{c}x_{i1}(k)\\\
x_{i2}(k)\\\ x_{i3}(k)\\\ \end{array}\right]$ (80)
$\displaystyle+\left[\begin{array}[]{c}0\\\ 0\\\ 1\\\
\end{array}\right]u_{i}(k),~{}1\leq i\leq 5.$ (84)

The communication graph of the AUVs is also given by Fig. 2. The initial
conditions are set as $x_{1}(0)=[57.55~{}181.56~{}113.06]^{\textrm{T}}$,
$x_{2}(0)=[96.21~{}32.08$ $180.69]^{\textrm{T}}$,
$x_{3}(0)=[26.76~{}75.41~{}197.66]^{\textrm{T}}$, $x_{4}(0)=[71.21~{}118.11$
$79.59]^{\textrm{T}}$, and $x_{5}(0)=[108.91$ $50.44~{}19.41]^{\textrm{T}}$,
which are generated by the _rand_ function in _Matlab_ with
$\|x_{i}(0)\|_{\infty}\leq 200$.

####  V-B1 Simulation with Low-Level DoS Attacks

First, we simulate the low-level DoS attacks case. For the control protocol in
[26], let $K_{c}=[0.0720~{}-0.4426~{}0.4029]$, $\gamma_{1}=0.9$,
$\gamma_{2}=4.8$, $\beta_{0}=100$, and $K=1600$ (see Theorem 1 in [26] for the
procedure of the calculation of these parameters). For the control protocol in
this paper, according to Theorem 3, we select $k_{1}=-0.25$, $k_{2}=0$,
$C_{h}=150$, $\epsilon_{0}=0.5$, $c=0.2$, $\gamma_{1}=0.9$, $\beta_{0}=100$,
and $K=1600$. Note that the same quantization parameter $K$, initial value of
the scaling function $\beta_{0}$, and parameter $\gamma_{1}$ are adopted to
make a fair comparison. The low-level DoS attacks are present for $\sim 5\%$
of total time. Fig. 6 shows the responses of
$\delta_{i1}(kT)=x_{i1}(kT)-\frac{1}{5}\sum_{j=1}^{5}x_{j1}(kT)$, $1\leq i\leq
5$, under the two control protocols with transmissions failed at
$k=10,30,45,70,85,105,125,145,165,185$. From this figure, one can observe that
the control protocol in this paper achieves much better consensus performance.
The main reason leads to this improvement is that the consensus error diverges
during the DoS attacks under the control protocol in [26], while is held
unchanged in this paper.

####  V-B2 Simulation with High-Level DoS Attacks

We further simulate the high-level DoS attacks case, i.e., the DoS attacks are
present for $\sim 95\%$ of total time. For the control protocol in [26], the
consensus cannot be achieved if the level of DoS attacks increases to $\sim
0.07$. In contrast to [26], the control protocol in this paper can tolerate
arbitrarily serious DoS attacks as long as their induced maximal number of
consecutive packet losses is bounded (i.e., present less than $100\%$ of total
time). Fig. 7(a) illustrates the responses of $\delta_{i1}(kT)$ under the
proposed control protocol with same parameters as in the previous simulation.

Now we show the potentials of data rate reduction by the control scheme
developed in this paper. For the multi-agent system (80), the number of
quantization levels in [26] must satisfy $2K+1\geq 2711$, while in this paper
$2K+1\geq 3$. According to Theorem 3, we now select $K=1$,
$\gamma_{1}=0.9925$, and $\beta_{0}=60$. The responses of $\delta_{i1}(kT)$
with 3-level quantizers and $\sim 95\%$ DoS attacks are depicted in Fig. 7(b),
from which one can see that the consensus has been successfully achieved.

Finally, we mention that the final consensus value of our proposed control
protocol in the simulations is $x_{i}=[133.40~{}133.40~{}133.40]^{\rm{T}}$,
which is consistent with the theoretical computation (65), and independent
with the quantization parameter $K$ and DoS attacks. On the contrary, the
AUVs’ states under the control protocol in [26] trend to infinity, as the
state matrix in (80) is unstable.

(a) Control protocol in [26].

(b) Control protocol in this paper.

Figure 6: Responses of $\delta_{i1}(kT)$ under the control protocols in [26]
and this paper with $K=1600$ and $\sim 5\%$ DoS attacks.

(a) $K=1600$.

(b) $K=1$.

Figure 7: Responses of $\delta_{i1}(kT)$ under the proposed control protocol
with different quantization levels and $\sim 95\%$ DoS attacks.

##  VI Conclusions

This paper investigated the quantized consensus problem for uncertain
nonlinear multi-agent systems under data-rate constraints and DoS attacks. An
ESO-based dynamic encoding-decoding scheme with zooming-in and holding
capabilities was developed, which exhibits several intriguing properties such
as better consensus performance, lower data-rate requirement, and higher
tolerance to DoS attacks. In particular, it was shown that the quantized
consensus can be achieved under two-bit data rate and any DoS attacks inducing
bounded consecutive packet losses. Two numerical examples verified the
capabilities and superiorities of the developed control scheme.

In this paper, as a preliminary theoretical research, we made some necessary
assumptions. Future research efforts will be devoted towards extending the
developed control scheme to more challenging scenarios, such as directed
networks [7], with network transmission delays [44], unreliable or no
acknowledgement signal [24, 45], and DoS attacks that affect only a part of
the communication graph.

## Appendix: Proof of Theorem 1

To facilitate the analysis, let $\alpha(kT)=\delta(kT)/\beta(kT)$ and
$\zeta(kT)=\mu(kT)/\beta(kT)$. Before stating the proof of Theorem 1, we need
two additional propositions. In the following, Proposition 1 is to analyze the
convergence of the ESO, and to verify that the acculturated ESO estimation
error will not lead to quantizer saturation if we let
$\beta(kT)\rightarrow\sqrt{\varepsilon}$. Proposition 2 is to show the
boundedness of $\alpha(v_{i}T)$.

_Proposition 1:_ Consider the system (1) with ESO (15). Suppose Assumptions A2
and A3 are satisfied, and the agents’ states $(\rho_{i},z_{i})$ are bounded.
Let $M_{j}\geq\sup_{1\leq i\leq N}|\rho_{ij}|$, $1\leq j\leq r+1$. Then there
exists $\varepsilon^{*}>0$ such that
$\forall\varepsilon\in(0,\varepsilon^{*})$:

  * •

there exists $\tau_{0}(\varepsilon)>0$ satisfying
$\lim_{\varepsilon\rightarrow 0}\tau_{0}(\varepsilon)=0$, such that $\forall
t\in[\tau_{0}(\varepsilon),\infty)$,

$|\rho_{ij}(t)-\hat{\rho}_{ij}(t)|=O(\varepsilon^{r+2-j}),~{}1\leq j\leq r+1.$
(85)

  * •

for any time interval $[\tau_{1},\tau_{2}]\subseteq[0,\infty)$,

$\frac{\int_{\tau_{1}}^{\tau_{2}}\left(\rho_{ij}(\tau)-\overline{\rho}_{ij}(\tau)\right)\textrm{d}\tau}{\sqrt{\varepsilon}}\rightarrow
0~{}\textrm{as}~{}\varepsilon\rightarrow 0,~{}1\leq j\leq r+1.$ (86)

_Proof of Proposition 1:_ Consider the scaled ESO estimation error
$\eta_{i}=[\eta_{i1},\ldots,\eta_{i,r+1}]^{\textrm{T}}\in\mathbb{R}^{r+1}$
with $\eta_{ij}=\frac{\rho_{ij}-\hat{\rho}_{ij}}{\varepsilon^{r+1-j}}$, $1\leq
j\leq r+1$. By (3) and (15), one has

$\dot{\eta}_{i}=\frac{1}{\varepsilon}E\eta_{i}+\bar{B}\dot{F}_{i}(\rho_{i},z_{i},\omega_{i}),$
(87)

where $E$ is given by (16), $\bar{B}=[0,B_{\rho}^{\textrm{T}}]^{\textrm{T}}$,
and

$\displaystyle\dot{F}_{i}(\rho_{i},z_{i},\omega_{i})=$
$\displaystyle\sum_{j=1}^{r-1}\rho_{i,j+1}\frac{\partial
F_{i}}{\partial\rho_{ij}}+\left[F_{i}(\rho_{i},z_{i},\omega_{i})+u_{i}\right]\frac{\partial
F_{i}}{\partial\rho_{ir}}$
$\displaystyle+F_{i0}(\rho_{i},z_{i},\omega_{i})\frac{\partial F_{i}}{\partial
z_{i}}+\dot{\omega}_{i}\frac{\partial F_{i}}{\partial\omega_{i}}.$

By the boundedness of $\rho_{i}$, and Assumptions A2 and A3, there exists
$\varepsilon$-independent positive constant $\iota_{1i}$ such that
$|\dot{F}_{i}(\rho_{i},z_{i},\omega_{i})|\leq\iota_{1i}$. Recall that the
matrix $E$ is Hurwitz by design. One can select a Lyapunov function candidate
$V_{i}(\eta_{i})=\eta_{i}^{\rm{T}}P\eta_{i}$, where
$P\in\mathbb{R}^{(r+1)\times(r+1)}$ is the unique positive definite matrix
solution to the equation $PE+E^{\rm{T}}P=-I_{r+1}$. Let $\sigma_{1}$ and
$\sigma_{2}$ be the minimal and maximal eigenvalues of the matrix $P$,
respectively. It follows that $\sigma_{1}\|\eta_{i}\|^{2}\leq
V_{i}(\eta_{i})\leq\sigma_{2}\|\eta_{i}\|^{2}$. Then, by (87), one has

$\displaystyle\frac{\textrm{d}V_{i}(\eta_{i})}{\textrm{d}t}=$
$\displaystyle\frac{1}{\varepsilon}\left(E^{\textrm{T}}P+PE\right)\|\eta_{i}\|^{2}$
$\displaystyle+\left(\dot{F}^{\textrm{T}}_{i}(\rho_{i},z_{i},\omega_{i})\bar{B}^{\textrm{T}}P\eta_{i}+\eta_{i}^{\rm{T}}P\bar{B}\dot{F}_{i}(\rho_{i},z_{i},\omega_{i})\right)$
$\displaystyle\leq$
$\displaystyle-\frac{1}{\varepsilon}\|\eta_{i}\|^{2}+2\sigma_{2}\iota_{1i}\|\eta_{i}\|$
$\displaystyle\leq$
$\displaystyle-\frac{1}{\sigma_{2}\varepsilon}V_{i}(\eta_{i})+\frac{2\sigma_{2}\iota_{1i}}{\sqrt{\sigma_{1}}}\sqrt{V_{i}(\eta_{i})}.$
(88)

By
$\frac{\textrm{d}V_{i}(\eta_{i})}{\textrm{d}t}=2\sqrt{V_{i}(\eta_{i})}\frac{\textrm{d}\sqrt{V_{i}(\eta_{i})}}{\textrm{d}t}$,
the theory of ordinary differential inequality, and some straightforward
manipulations, one can obtain

$\displaystyle\|\eta_{i}\|\leq$
$\displaystyle\frac{\sqrt{V_{i}(\eta_{i})}}{\sqrt{\sigma_{1}}}$
$\displaystyle\leq$
$\displaystyle\left(\frac{\sqrt{V_{i}(\eta_{i}(0))}}{\sqrt{\sigma_{1}}}-\frac{2\sigma_{2}^{2}\iota_{1i}\varepsilon}{\sigma_{1}}\right)e^{-\frac{1}{2\sigma_{2}\varepsilon}t}+\frac{2\sigma_{2}^{2}\iota_{1i}\varepsilon}{\sigma_{1}}$
$\displaystyle\leq$
$\displaystyle\sqrt{\frac{\sigma_{2}}{\sigma_{1}}}\|\eta_{i}(0)\|e^{-\frac{1}{2\sigma_{2}\varepsilon}t}+\frac{2\sigma_{2}^{2}\iota_{1i}\varepsilon}{\sigma_{1}}.$
(89)

Note that the right hand side of (Appendix: Proof of Theorem 1) is of the
order of $O(\varepsilon)$ for all
$t\geq\tau_{0}(\varepsilon)\triangleq-2(r+1)\sigma_{2}\varepsilon\ln\varepsilon$.
Since $\tau_{0}(\varepsilon)\rightarrow 0$ uniformly as
$\varepsilon\rightarrow 0$, and
$|\rho_{ij}-\hat{\rho}_{ij}|=\varepsilon^{r+1-j}|\eta_{ij}|$, $1\leq j\leq
r+1$, the conclusion in the first bullet holds. What is more, the saturations
will not be invoked after the transient period of the observer, i.e.,
$\overline{\rho}_{ij}=\hat{\rho}_{ij}$, $t\geq\tau_{0}(\varepsilon)$.

For the second bullet, consider three cases.

_Case 1):_ $\tau_{2}\leq\tau_{0}(\varepsilon)$. In this case, one has

$\displaystyle\frac{\int_{\tau_{1}}^{\tau_{2}}\left(\rho_{ij}(\tau)-\overline{\rho}_{ij}(\tau)\right)\textrm{d}\tau}{\sqrt{\varepsilon}}$
$\displaystyle\leq$ $\displaystyle
2M_{j}\frac{\tau_{0}(\varepsilon)}{\sqrt{\varepsilon}}\rightarrow
0~{}\textrm{as}~{}\varepsilon\rightarrow 0,~{}1\leq j\leq r+1.$ (90)

_Case 2):_ $\tau_{1}\geq\tau_{0}(\varepsilon)$. In this case, we start with
$j=1$. By (87), one has
$\rho_{i1}-\hat{\rho}_{i1}=\frac{\varepsilon^{r+1}}{l_{r+1}}\left(\dot{F}_{i}(x_{i},z_{i},\omega_{i})-\dot{\eta}_{i,r+1}\right)$.
It follows that

$\displaystyle\frac{\int_{\tau_{1}}^{\tau_{2}}\left(\rho_{i1}(\tau)-\overline{\rho}_{i1}(\tau)\right)\textrm{d}\tau}{\sqrt{\varepsilon}}$
$\displaystyle=$
$\displaystyle\frac{\varepsilon^{r+\frac{1}{2}}}{l_{r+1}}\left[\left(F_{i}(x_{i},z_{i},\omega_{i})-\eta_{i,r+1}\right)|_{t=\tau_{2}}\right.$
$\displaystyle\left.-\left(F_{i}(x_{i},z_{i},\omega_{i})-\eta_{i,r+1}\right)|_{t=\tau_{1}}\right].$
(91)

Since $F_{i}(x_{i},z_{i},\omega_{i})$ is bounded, and
$\eta_{i,r+1}=O(\varepsilon)$, $\forall t\geq\tau_{0}(\varepsilon)$, the right
hand side of (Appendix: Proof of Theorem 1) is of the order of
$O(\varepsilon^{r+\frac{1}{2}})$ For $j=2$, by (87), one has
$\rho_{i2}-\hat{\rho}_{i2}=\varepsilon^{r}\dot{\eta}_{i1}+\frac{l_{i1}}{\varepsilon}(\rho_{i1}-\hat{\rho}_{i1})$.
It follows that

$\displaystyle\frac{\int_{\tau_{1}}^{\tau_{2}}\left(\rho_{i2}(\tau)-\overline{\rho}_{i2}(\tau)\right)\textrm{d}\tau}{\sqrt{\varepsilon}}$
$\displaystyle=$
$\displaystyle\varepsilon^{r-\frac{1}{2}}(\eta_{i1}(\tau_{2})-\eta_{i1}(\tau_{1}))+\frac{l_{i1}}{\varepsilon}\frac{\int_{\tau_{1}}^{\tau_{2}}(\rho_{i1}(\tau)-\hat{\rho}_{i1}(\tau))\textrm{d}\tau}{\sqrt{\varepsilon}}.$
(92)

By (Appendix: Proof of Theorem 1), the right hand side of (Appendix: Proof of
Theorem 1) is of the order of $O(\varepsilon^{r-\frac{1}{2}})$. Then by a
simple iterative procedure, one can prove that

$\frac{\int_{\tau_{1}}^{\tau_{2}}\left(\rho_{ij}(\tau)-\overline{\rho}_{ij}(\tau)\right)\textrm{d}\tau}{\sqrt{\varepsilon}}=O(\varepsilon^{r+\frac{3}{2}-j}),~{}1\leq
j\leq r+1.$ (93)

_Case 3):_ $\tau_{1}<\tau_{0}(\varepsilon)<\tau_{2}$. Note that
$\int_{\tau_{1}}^{\tau_{2}}\left(\cdot\right)\textrm{d}\tau=\int_{\tau_{1}}^{\tau_{0}(\varepsilon)}\left(\cdot\right)\textrm{d}\tau+\int_{\tau_{0}(\varepsilon)}^{\tau_{2}}\left(\cdot\right)\textrm{d}\tau$.
Then by Cases 1) and 2), one can conclude that
$\frac{\int_{\tau_{1}}^{\tau_{2}}\left(\rho_{ij}(\tau)-\overline{\rho}_{ij}(\tau)\right)\textrm{d}\tau}{\sqrt{\varepsilon}}\rightarrow
0$ as $\varepsilon\rightarrow 0$ in this case.

Finally, combining Cases 1) to 3) leads to the conclusion in (86). This
completes the proof of Proposition 1. ∎

_Proposition 2:_ There exists $\varepsilon^{{\ddagger}}>0$ such that
$\forall\varepsilon\in(0,\varepsilon^{{\ddagger}})$,

$\|\alpha(v_{i}T)\|<\overline{\alpha},~{}i=0,1,\ldots,$ (94)

where
$\overline{\alpha}=\max\left\\{\frac{2\sqrt{N}C_{h}}{\beta_{0}},\frac{cT\sqrt{N}\lambda_{N}}{2\gamma_{1}(\gamma_{1}-\rho_{h})}\right\\}$.
Furthermore, the quantizer will never be saturated.

_Proof of Proposition 2:_ Proposition 2 will be proved recursively. For
$t=v_{0}T=0$, by (39) and the fact that
$\|\nu\|_{\infty}\leq\|\nu\|\leq\sqrt{N}\|\nu\|_{\infty}$,
$\forall\nu\in\mathbb{R}^{N}$, one has

$\|\alpha(0)\|=\frac{\|\delta(0)\|}{\beta_{0}}\leq\frac{\sqrt{N}\|\delta(0)\|_{\infty}}{\beta_{0}}\leq\frac{2\sqrt{N}C_{h}}{\beta_{0}}<\overline{\alpha}.$
(95)

What is more, by $\overline{H}(0)=\widehat{\overline{H}}(0)=\textbf{0}_{N}$,
one has

$\|\zeta(0)\|_{\infty}=\frac{\|\overline{H}(0)-\widehat{\overline{H}}(0)\|_{\infty}}{\beta_{0}}=0.$
(96)

According to the update mechanism of the scaling function, $\beta(kT)$ will
converge to $\sqrt{\varepsilon}$ at some successful transmission time instant
and then keep this value afterwards. Let $v_{m^{*}}T$ denote the first time
instant when $\beta(kT)=\sqrt{\varepsilon}$. We consider three cases, that is
$v_{m}<v_{m^{*}}$, $v_{m}=v_{m^{*}}$, and $v_{m}>v_{m^{*}}$.

Figure 8: Sequences of the successful (colored in red) and unsuccessful
(colored in gray) transmissions.

_Case 1):_ $v_{m}<v_{m^{*}}$. Recall that $v_{0}T,\ldots,v_{m}T,\ldots$ denote
the successful transmission time instants (see Fig. 8). Let
$(v_{m-1}+1)T,\ldots,$ $(v_{m}-1)T$ be the unsuccessful transmission time
instants between two consecutive successful transmission time instants
$v_{m-1}T$ and $v_{m}T$ 111If $v_{m-1}+1>v_{m}-1$, this indicates that
$v_{m}=v_{m-1}+1$, and there is no attack for the consecutive sampling time
instants $v_{m-1}T$ and $v_{m}T$.. By (III-B), (34), and
$\beta((v_{m-1}+1)T)=\beta(v_{m-1}T)$, the evolution equations of $\alpha(kT)$
and $\zeta(kT)$ from $k=v_{m-1}$ to $k=v_{m-1}+1$ is given by

$\displaystyle\alpha((v_{m-1}+1)T)=$
$\displaystyle(I-cT\mathcal{L})\alpha(v_{m-1}T)+cT\mathcal{L}\zeta(v_{m-1}T)$
$\displaystyle+\frac{\varsigma_{2}(v_{m-1}T)}{\beta(v_{m-1}T)},$ (97)
$\displaystyle\zeta((v_{m-1}+1)T)=$
$\displaystyle(I+cT\mathcal{L})\zeta(v_{m-1}T)-cT\mathcal{L}\alpha(v_{m-1}T)$
$\displaystyle+\frac{\varsigma_{3}(v_{m-1}T)}{\beta(v_{m-1}T)},$ (98)

where

$\displaystyle\varsigma_{2}(v_{m-1}T)=$ $\displaystyle
cT\mathcal{L}\left[H(v_{m-1}T)-\overline{H}(v_{m-1}T)\right]$
$\displaystyle+(I-J_{N})\varsigma_{1}(v_{m-1}T),$
$\displaystyle\varsigma_{3}(v_{m-1}T)=$
$\displaystyle(I+cT\mathcal{L})\left[H(v_{m-1}T)-\overline{H}(v_{m-1}T)\right]$
$\displaystyle+\varsigma_{1}(v_{m-1}T)$
$\displaystyle+\overline{H}((v_{m-1}+1)T)-H((v_{m-1}+1)T).$

For the transmissions at time instants $(v_{m-1}+1)T,\ldots,$ $(v_{m}-2)T$, by
(31) and (34), one has

$\displaystyle\alpha((k+1)T)=$
$\displaystyle\alpha(kT)+\frac{(I-J_{N})\varsigma_{1}(kT)}{\beta(v_{m-1}T)},$
(99) $\displaystyle\zeta((k+1)T)=$
$\displaystyle\zeta(kT)+\frac{\varsigma_{4}(kT)}{\beta(v_{m-1}T)},$ (100)

where $v_{m-1}+1\leq k\leq v_{m}-2$, and

$\displaystyle\varsigma_{4}(kT)=$
$\displaystyle\varsigma_{1}(kT)+H(kT)-\overline{H}(kT)$
$\displaystyle+\overline{H}((k+1)T)-H((k+1)T).$

Applying a simple iteration to (99)-(100) yields

$\displaystyle\alpha((v_{m}-1)T)=$ $\displaystyle\alpha((v_{m-1}+1)T)$
$\displaystyle+(I-J_{N})\sum_{j=v_{m-1}+1}^{v_{m}-2}\frac{\varsigma_{1}(jT)}{\beta(v_{m-1}T)},$
(101) $\displaystyle\zeta((v_{m}-1)T)=$
$\displaystyle\zeta((v_{m-1}+1)T)+\sum_{j=v_{m-1}+1}^{v_{m}-2}\frac{\varsigma_{4}(jT)}{\beta(v_{m-1}T)}.$
(102)

For the transmission at time instant $(v_{m}-1)T$, by (31), (33), and
$\beta(v_{m}T)=\gamma_{1}\beta((v_{m}-1)T)$, one has

$\displaystyle\alpha(v_{m}T)=$
$\displaystyle\frac{1}{\gamma_{1}}\alpha((v_{m}-1)T)+\frac{(I-J_{N})\varsigma_{1}((v_{m}-1)T)}{\beta(v_{m}T)},$
(103) $\displaystyle\zeta(v_{m}T)=$
$\displaystyle\frac{1}{\gamma_{1}}\zeta((v_{m}-1)T)+\frac{\varsigma_{4}((v_{m}-1)T)}{\beta(v_{m}T)}$
$\displaystyle-\frac{1}{\gamma_{1}}Q\left(\zeta((v_{m}-1)T)+\frac{\varsigma_{4}((v_{m}-1)T)}{\beta(v_{m-1}T)}\right).$
(104)

Therefore, by (97)-(98), (Appendix: Proof of Theorem 1)-(102), and
(103)-(Appendix: Proof of Theorem 1), the equations that describe the
evolution of $\alpha(kT)$ and $\zeta(kT)$ from $k=v_{m-1}$ to $k=v_{m}$ are
expressed as

$\displaystyle\alpha(v_{m}T)=$
$\displaystyle\frac{I-cT\mathcal{L}}{\gamma_{1}}\alpha(v_{m-1}T)+\frac{cT\mathcal{L}}{\gamma_{1}}\zeta(v_{m-1}T)$
$\displaystyle+\frac{\varsigma_{2}(v_{m-1}T)}{\beta(v_{m}T)}+\frac{(I-J_{N})\sum_{j=v_{m-1}+1}^{v_{m}-1}\varsigma_{1}(jT)}{\beta(v_{m}T)},$
(105) $\displaystyle\zeta(v_{m}T)=$
$\displaystyle\frac{1}{\gamma_{1}}\left[\Psi(v_{m-1}T)-Q(\Psi(v_{m-1}T))\right],$
(106)

where

$\displaystyle\Psi(v_{m-1}T)=$
$\displaystyle(I+cT\mathcal{L})\zeta(v_{m-1}T)-cT\mathcal{L}\alpha(v_{m-1}T)$
$\displaystyle+\frac{\varsigma_{3}(v_{m-1}T)}{\beta(v_{m-1}T)}+\sum_{j=v_{m-1}+1}^{v_{m}-1}\frac{\varsigma_{4}(jT)}{\beta(v_{m-1}T)}.$
(107)

For $m=1$, according to Proposition 1, (39), (95), and (96), one can readily
verify that $\|\alpha(v_{1}T)\|<\overline{\alpha}$ and
$\|\Psi(v_{0}T)\|_{\infty}<K+\frac{1}{2}$ for sufficiently small
$\varepsilon$. In the following, we investigate the case that $t=v_{m}T$,
$m\geq 2$, with the assumption that $\|\alpha(v_{i}T)\|<\overline{\alpha}$ and
the quantizer is unsaturated for all $t=v_{i}T$, $1\leq i\leq m-1$.

To facilitate the analysis, let $U$ be a unitary matrix defined by
$U=[\textbf{1}_{N}/\sqrt{N},\psi_{2},\ldots,\psi_{N}]$, where
$\psi^{\textrm{T}}_{i}\mathcal{L}=\lambda_{i}\psi_{i}^{\textrm{T}}$, $2\leq
i\leq N$. Denote
$\widetilde{\alpha}(kT)=U^{-1}\alpha(kT)=U^{\textrm{T}}\alpha(kT)$, and
decompose
$\widetilde{\alpha}(kT)=[\widetilde{\alpha}_{1}(kT),\widetilde{\alpha}_{2}(kT)]^{\textrm{T}}$
with a scalar $\widetilde{\alpha}_{1}(kT)$. One can verify that
$\widetilde{\alpha}_{1}(kT)=0$. Let
$\mathcal{M}_{\gamma_{1},c}=\gamma_{1}^{-1}\textrm{diag}\left\\{1-cT\lambda_{2},\ldots,1-cT\lambda_{N}\right\\}$
and $\psi=[\psi_{2},\ldots,\psi_{N}]$. It follows from (Appendix: Proof of
Theorem 1) that

$\displaystyle\widetilde{\alpha}_{2}(v_{m}T)=$
$\displaystyle\mathcal{M}_{\gamma_{1},c}\widetilde{\alpha}_{2}(v_{m-1}T)+\frac{cT\psi^{\textrm{T}}\mathcal{L}}{\gamma_{1}}\zeta(v_{m-1}T)$
$\displaystyle+\frac{\psi^{\textrm{T}}\varsigma_{2}(v_{m-1}T)}{\beta(v_{m}T)}$
$\displaystyle+\frac{\psi^{\textrm{T}}(I-J_{N})\sum_{j=v_{m-1}+1}^{v_{m}-1}\varsigma_{1}(jT)}{\beta(v_{m}T)}.$
(108)

By (Appendix: Proof of Theorem 1), one can establish the evolution equation of
$\widetilde{\alpha}_{2}(kT)$ from $k=v_{0}$ to $k=v_{m}$. This together with
$\alpha(kT)=\psi\widetilde{\alpha}_{2}(kT)$ and
$\widetilde{\alpha}_{2}(kT)=\psi^{\textrm{T}}\alpha(kT)$ yields

$\displaystyle\alpha(v_{m}T)=\psi\left[\mathcal{M}_{\gamma_{1},c}\right]^{m}\psi^{\textrm{T}}\alpha(v_{0}T)$
$\displaystyle\qquad+\psi\left[\mathcal{M}_{\gamma_{1},c}\right]^{m-1}\frac{cT\psi^{\textrm{T}}\mathcal{L}}{\gamma_{1}}\zeta(v_{0}T)$
$\displaystyle\qquad+\psi\sum_{i=0}^{m-2}\left[\mathcal{M}_{\gamma_{1},c}\right]^{i}\frac{cT\psi^{\textrm{T}}\mathcal{L}}{\gamma_{1}}\zeta(v_{m-1-i}T)$
$\displaystyle\qquad+\psi\sum_{i=0}^{m-1}\left[\mathcal{M}_{\gamma_{1},c}\right]^{i}\frac{\psi^{\textrm{T}}\varsigma_{2}(v_{m-1-i}T)}{\beta(v_{m-i}T)}$
$\displaystyle\qquad+\psi\sum_{i=0}^{m-1}\left[\mathcal{M}_{\gamma_{1},c}\right]^{i}\frac{\psi^{\textrm{T}}(I-J_{N})\sum_{j=v_{m-1-i}+1}^{v_{m-i}-1}\varsigma_{1}(jT)}{\beta(v_{m-i}T)}.$
(109)

Next we investigate the five terms on the right hand side of (Appendix: Proof
of Theorem 1) to seek for the upper-bound of $\|\alpha(v_{m}T)\|$. Note that
$\|\mathcal{M}_{\gamma_{1},c}\|\leq\frac{\rho_{h}}{\gamma_{1}}$, $\|\psi\|=1$,
and $\|\mathcal{L}\|=\lambda_{N}$. By (95), the first term satisfies

$\displaystyle\left\|\psi\left[\mathcal{M}_{\gamma_{1},c}\right]^{m}\psi^{\textrm{T}}\alpha(v_{0}T)\right\|\leq$
$\displaystyle\left(\frac{\rho_{h}}{\gamma_{1}}\right)^{m}\frac{2\sqrt{N}C_{h}}{\beta_{0}}$
$\displaystyle<$
$\displaystyle\left(\frac{\rho_{h}}{\gamma_{1}}\right)^{m-1}\frac{2\sqrt{N}C_{h}}{\beta_{0}}.$
(110)

The second term in (Appendix: Proof of Theorem 1) vanishes due to (96). For
the third term, recall that we have assumed that the quantizer is unsaturated
for all $t=v_{i}T$, $1\leq i\leq m-1$. According to (106), one has
$\max_{1\leq i\leq m-1}\|\zeta(v_{i}T)\|_{\infty}\leq\frac{1}{2\gamma_{1}}$,
and hence

$\displaystyle\left\|\psi\sum_{i=0}^{m-2}\left[\mathcal{M}_{\gamma_{1},c}\right]^{i}\frac{cT\psi^{\textrm{T}}\mathcal{L}}{\gamma_{1}}\zeta(v_{m-1-i}T)\right\|$
$\displaystyle\leq$
$\displaystyle\sum_{i=0}^{m-2}\left(\frac{\rho_{h}}{\gamma_{1}}\right)^{i}\frac{cT\sqrt{N}\lambda_{N}}{2\gamma^{2}_{1}}=\frac{cT\sqrt{N}\lambda_{N}}{2\gamma_{1}(\gamma_{1}-\rho_{h})}\left[1-\left(\frac{\rho_{h}}{\gamma_{1}}\right)^{m-1}\right].$
(111)

For the fourth and fifth terms, by Proposition 1 and the fact that
$\frac{\rho_{h}}{\gamma_{1}}<1$, they converge to 0 as $\varepsilon\rightarrow
0$. This together with (39), (Appendix: Proof of Theorem 1), and (Appendix:
Proof of Theorem 1) yields $\|\alpha(v_{m}T)\|<\overline{\alpha}$. What is
more, by $\|\alpha(v_{m-1}T)\|<\overline{\alpha}$,
$\|\zeta(v_{m-1}T)\|_{\infty}\leq\frac{1}{2\gamma_{1}}$, (36), and (39), one
has

$\displaystyle\|(I+cT\mathcal{L})\zeta(v_{m-1}T)-cT\mathcal{L}\alpha(v_{m-1}T)\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|(I+cT\mathcal{L})\|_{\infty}\|\zeta(v_{m-1}T)\|_{\infty}+cT\|\mathcal{L}\|\|\alpha(v_{m-1}T)\|$
$\displaystyle<$
$\displaystyle\frac{1+2cTd^{*}}{2\gamma_{1}}+\frac{\sqrt{N}c^{2}T^{2}\lambda^{2}_{N}}{2\gamma_{1}(\gamma_{1}-\rho_{h})}=K_{1}(c,\gamma_{1}).$
(112)

Hence, in (107), by Proposition 1 and (Appendix: Proof of Theorem 1), one has
$\|\Psi(v_{m-1}T)\|\leq K+\frac{1}{2}$ for sufficiently small $\varepsilon$.
That is, the quantizer is unsaturated when $t=v_{m}T$. Therefore, by
induction, we conclude that $\|\alpha(v_{m}T)\|<\overline{\alpha}$ and the
quantizer is unsaturated when $t=v_{m}T$, $1\leq m<m^{*}$.

_Case 2):_ $v_{m}=v_{m^{*}}$. In this case,
$\beta(v_{m^{*}-1}T)\leq\frac{\sqrt{\varepsilon}}{\gamma_{1}}$ and
$\beta(v_{m^{*}}T)=\sqrt{\varepsilon}$. By following the similar line as in
(97)-(106), one can obtain

$\displaystyle\alpha(v_{m^{*}}T)=$
$\displaystyle\frac{\beta(v_{{m^{*}}-1}T)}{\sqrt{\varepsilon}}(I-cT\mathcal{L})\alpha(v_{m^{*}-1}T)$
$\displaystyle+\frac{\beta(v_{m^{*}-1}T)}{\sqrt{\varepsilon}}cT\mathcal{L}\zeta(v_{m^{*}-1}T)$
$\displaystyle+\frac{\varsigma_{2}(v_{m^{*}-1}T)}{\sqrt{\varepsilon}}+\frac{(I-J_{N})\sum_{j=v_{m^{*}-1}+1}^{v_{m^{*}}-1}\varsigma_{1}(jT)}{\sqrt{\varepsilon}},$
(113) $\displaystyle\zeta(v_{m^{*}}T)=$
$\displaystyle\frac{\beta(v_{m^{*}-1}T)}{\sqrt{\varepsilon}}\left[\Psi(v_{m^{*}-1}T)-Q(\Psi(v_{m^{*}-1}T))\right].$
(114)

_Case 3):_ $v_{m}>v_{m^{*}}$. In this case,
$\beta(v_{m}T)=\beta(v_{m-1}T)=\sqrt{\varepsilon}$ , and hence

$\displaystyle\alpha(v_{m}T)=$
$\displaystyle(I-cT\mathcal{L})\alpha(v_{m-1}T)+cT\mathcal{L}\zeta(v_{m-1}T)$
$\displaystyle+\frac{\varsigma_{2}(v_{m-1}T)}{\sqrt{\varepsilon}}+\frac{(I-J_{N})\sum_{j=v_{m-1}+1}^{v_{m}-1}\varsigma_{1}(jT)}{\sqrt{\varepsilon}},$
(115) $\displaystyle\zeta(v_{m}T)=$
$\displaystyle\Psi(v_{m-1}T)-Q(\Psi(v_{m-1}T)).$ (116)

Note that in Case 2)
$\frac{\beta(v_{{m^{*}}-1}T)}{\sqrt{\varepsilon}}\leq\frac{1}{\gamma_{1}}$,
and in Case 3) $1<\frac{1}{\gamma_{1}}$. By Proposition 1 and conducting a
similar analysis as in (Appendix: Proof of Theorem 1)-(Appendix: Proof of
Theorem 1), one can verify that $\|\alpha(v_{i}T)\|<\overline{\alpha}$ and the
quantizer is unsaturated when $t=v_{m}T$, $m\geq m^{*}$.

Finally, combining Cases 1) to 3) leads to (94). What is more, by Cases 1) to
3), (III-B) and (29), and the update mechanism of the encoder, one can
conclude that the quantizer will never be saturated. This completes the proof
of Proposition 2. ∎

The proof of Proposition 2 is inspired by [6]. However, due to the uncertain
nonlinear dynamics and the introduction of the DoS attacks, several new and
important technique issues arise. On one hand, since the DoS destroys the
information availability, the behavior of the system in the presence and
absence of DoS attacks needs to be carefully analyzed. In fact, the evolution
equations (Appendix: Proof of Theorem 1) and (103) clearly reveal the
properties of the proposed zooming-in and holding approach. On the other hand,
in this work, the agent uncertain nonlinear dynamics, the DoS affected
quantization process, and the consensus process are highly coupled. In
(Appendix: Proof of Theorem 1), we show that the evolution of the scaled
consensus error is governed by the accumulated ESO estimation error, the
accumulated error induced by quantization, and the behavior of the DoS
attacker.

Finally, we are ready to state the proof of Theorem 1.

_Proof of Theorem 1:_ By Propositions 1 and 2, one can readily conclude the
statement in the first bullet of Theorem 1. For the second bullet, in
Proposition 2, we have proved that the quantizer will never be saturated for
all transmissions. According to the definition of $\alpha(kT)$ and the update
mechanism of the scaling function $\beta(kT)$, for $m\geq m^{*}$, one has
$\|\delta(v_{m}T)\|=\|(I-J_{N})H(v_{m}T)\|=\beta(v_{m}T)\|\alpha(v_{m}T)\|<\overline{\alpha}\sqrt{\varepsilon}$.
Hence, $h_{i}(v_{m}T)-h_{j}(v_{m}T)\rightarrow 0$, $1\leq i\neq j\leq N$, as
$m\rightarrow\infty$ and $\varepsilon\rightarrow 0$. This together with
Proposition 1, (III-B), and (29) yields that $h_{i}(t)-h_{j}(t)\rightarrow 0$,
$1\leq i\neq j\leq N$, as $t\rightarrow\infty$ and $\varepsilon\rightarrow 0$.
Finally, according to Lemma 1, one can conclude that the multi-agent system
(1) achieves the practical output consensus specified by (42). This completes
the proof of Theorem 1. ∎

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