# Practical Output Consensus of Nonlinear Heterogeneous Multi-Agent Systems
with Limited Data Rate

Maopeng Ran<EMAIL_ADDRESS>Lihua Xie<EMAIL_ADDRESS>School of Electrical
and Electronic Engineering, Nanyang Technological University, Nanyang Avenue,
Singapore 639798, Singapore

###### Abstract

This paper investigates the consensus problem for nonlinear heterogeneous
multi-agent systems with limited communication data rate. Each agent is
modeled by a higher-order strict-feedback continuous-time system with unknown
nonlinearities and external disturbance, and only the first state variable
being measurable. Extended state observers (ESOs) are used to estimate the
unmeasurable agent states and the unknown nonlinear dynamics. An ESO-based
distributed output feedback protocol with dynamic encoding and decoding is
then presented. It is shown that, for a connected undirected network, the
proposed protocol guarantees practical output consensus, in which the steady-
state consensus error can be made arbitrarily small. The ESO-based protocol
also shapes the transient consensus performance, as it is capable of
recovering the consensus performance of a linear counterpart with fully
measurable states. Furthermore, we prove that for higher-order uncertain
nonlinear multi-agent systems, consensus can be achieved with merely one bit
information exchange between each pair of adjacent agents at each time step.
Finally, simulations on third-order pendulum systems are given, which verify
the theoretical results.

###### keywords:

Multi-agent systems; consensus; uncertain nonlinear systems; data rate;
quantization; extended state observer.

††thanks: This paper was not presented at any IFAC meeting. Corresponding
author Lihua Xie.

,

##  1 Introduction

Multi-agent systems have attracted great attention in recent years owing to
their potential applications in a wide variety of tasks, such as exploration,
surveillance, and cooperative manipulation. Consensus, which aims at all
agents autonomously achieving an agreement of common interest by using local
information available at the node and received from neighboring agents, is one
of the most fundamental problems in multi-agent systems.

In most cases, the agents communicate with their neighbours through digital
networks. For a multi-agent system with a digital communication network, the
information to be exchanged between each pair of adjacent agents will be first
quantized by the sender and then sent out to its neighbours. When the
neighbours receive the quantized information through the digital communication
channels, they use a decoding algorithm to reconstruct the information. That
is to say, only quantized information can be exchanged between agents.
Quantized consensus for first-order multi-agent systems was investigated in
Kashyap, Basar, and Srikant (2007), Carli, Bullo, and Zampieri (2010), Li, Fu,
Xie, and Zhang (2011), and Mu and Liu (2015). Specifically, Li et al. (2011)
studied the average consensus problem for undirected networks of first-order
discrete-time integrator systems under finite bit-rate communication. A static
uniform quantizer with an exponentially decaying scaling function was
proposed, and it was shown in Li et al. (2011) that no matter how many agents
there are, the consensus can be achieved with as few as one bit information
exchange between each pair of adjacent agents at each time step.

For higher-order multi-agent systems, the limited data rate problem becomes
more challenging. Li and Xie (2012) developed a quantized-observer based
encoding-decoding scheme for second-order integrator systems with partially
measurable states. It was shown in Li and Xie (2012) that for a connected
network, two bits information exchange between each pair of adjacent agents at
each time step suffices to guarantee the consensus of the agents. For multi-
agent systems with $r$th-order integrator dynamics, Qiu, Xie, and Hong (2017a)
showed that $r$ bits of information exchange between each pair of adjacent
agents at each time step are necessary to guarantee consensus. Qiu, Xie, and
Hong (2017b) further investigated the data rate problem for $2r$th-order
multi-agent systems with oscillator dynamics, and the sufficient number of
communication bits to guarantee consensus was proved to be an integer between
$r$ and $2r$, depending on the location of the poles of the agent state
matrix. You and Xie (2011) and Meng, Li, and Zhang (2017) considered the
limited data rate problem for general linear multi-agent systems with fully
and partially measurable states, respectively. However, the minimum data rate
required in You and Xie (2011) and Meng et al. (2017) remains an open problem.

The works mentioned above mainly focused on linear multi-agent systems. In a
recent work, Dong (2019) first visited the limited data rate problem for a
class of nonlinear multi-agent systems with fully measurable states. In the
case that there exist a known model of the agent nonlinearity and a known
upper bound of the modeling error, a multi-loop control structure was proposed
in Dong (2019) to guarantee consensus. In Ran and Xie (2019), the limited data
rate problem was considered for a class of simple nonlinear multi-agent
systems without external disturbances.

Based on the above observations, there are several important and interesting
problems that remain unsolved. Firstly, it still lacks a practical solution to
handle the limited data rate problem for general multi-agent systems with
nonlinearities, uncertainties, and external disturbances. It should be pointed
out that the nonlinearities, uncertainties, and external disturbances exist in
almost all practical control systems (Khalil, 2002; Isidori, 1989). Secondly,
whether the data rates required in Li and Xie (2012) and Qiu et al. (2017a;
2017b) are conservative, and how many bits are sufficient for general higher-
order uncertain nonlinear multi-agent systems with partially measurable
states? Thirdly, for the limited data rate problem, can a comparable transient
consensus performance for a multi-agent system with nonlinearities,
uncertainties, and external disturbances, to its simple linear counterpart be
achieved?

In this paper, we investigate the limited data rate problem for nonlinear
heterogeneous multi-agent systems with partially measurable states. Each agent
is described by a strict-feedback system with unknown nonlinear dynamics and
external disturbance. Extended state observer (ESO) (Han, 2009; Khalil, 2017)
is the main tool used in this paper. The unknown dynamics is first regarded as
an extended state of the agent, and then the agent state and the defined
extended state are estimated simultaneously by the ESO. An ESO-based protocol
with dynamic encoding and decoding is then presented. Comparing with the
existing literature, the main contributions of this paper are threefold:

  1. i)

We provide a more practical solution to the limited data rate problem for
multi-agent systems. The proposed protocol is output-feedback-based and is
capable of handling nonlinearities, uncertainties, and external disturbances.
In practical applications, the agent states may be not fully available for
feedback; and its dynamics is very likely to be perturbed by uncertainties and
external disturbances (Khalil, 2002; Isidori, 1989).

  2. ii)

We show that, with the application of the proposed ESO-based protocol, merely
_one bit_ information exchange between each pair of adjacent agents at each
time step suffices to guarantee the output consensus of general higher-order
uncertain nonlinear multi-agent systems with partially measurable states. This
indicates that the data rates required in Li and Xie (2012) and Qiu et al.
(2017a; 2017b) are still conservative, and from a theoretical viewpoint, the
approach developed in this paper achieves the lowest data rate required at
each time step via output feedback.

  3. iii)

The proposed ESO-based protocol is capable of shaping the transient consensus
performance of the uncertain nonlinear multi-agent systems with limited data
rate. More specifically, the agent trajectories of an uncertain nonlinear
multi-agent system under the ESO-based output feedback protocol can be made
arbitrarily close to the trajectories of its linear counterpart with fully
measurable states.

##  2 Problem Formulation

###  2.1 Notation

Let $\mathbb{R}$ and $\mathbb{R}^{n}$ denote the sets of real numbers and
$n$-dimension real vectors, respectively. $I_{n}$ and $\textbf{0}_{n}$
represent the identity matrix and the vector of zeros of dimension $n$,
respectively. Let $\textbf{1}_{N}$ be the $N$-dimensional column vector with
all components being 1, and define
$J_{N}=(1/N)\textbf{1}_{N}\textbf{1}^{\textrm{T}}_{N}$.
$\textrm{diag}\\{a_{1},\ldots,a_{n}\\}$ is the diagonal matrix with the $i$-th
diagonal component equal to $a_{i}$. For a given vector or matrix $A$,
$A^{\rm{T}}$, $\|A\|$, and $\|A\|_{\infty}$ represent its transpose, Euclidean
norm, and $\infty$-norm, respectively. For a given positive number $a$,
$\lfloor a\rfloor$ and $\lceil a\rceil$ represent the largest integer not
greater than $a$ and the smallest integer not less than $a$, respectively. Big
$O$-notation in terms of $\nu$ is denoted as $O(\nu)$ and it is assumed to
hold for a sufficiently small positive $\nu$. Let $\textrm{sat}(\cdot)$
represent the unity saturation function defined by
$\textrm{sat}(\nu)=\textrm{sign}(\nu)\cdot\min\\{1,|\nu|\\}$. Throughout this
paper, for clear presentation, the time variable $t$ of a signal will be
omitted except when the dependence of the signal on $t$ is crucial.

###  2.2 Communication Graph

The communications between agents are modeled as a graph
$\mathcal{G}=\\{\mathcal{V},\mathcal{E},\mathcal{A}\\}$, where
$\mathcal{V}=\\{1,2,\ldots,N\\}$ is the index set of $N$ agents with $i$
representing the $i$th agent, $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$
is the edge set of paired agents, and
$\mathcal{A}=[a_{ij}]\in\mathbb{R}^{N\times N}$ is the weighted adjacency
matrix with $a_{ij}=1$ or $0$ indicating whether or not there is a
communication channel from agent $j$ to $i$. If $a_{ij}=a_{ji}$ for any pair
of neighbouring agents, the associated communication graph is called an
undirected graph. The neighborhood of the $i$th agent is represented by
$\mathcal{N}_{i}=\\{j\in\mathcal{V}|(i,j)\in\mathcal{E}\\}$. The in-degree of
agent $i$ is denoted as $\textrm{deg}_{i}=\sum_{j=1}^{N}a_{ij}$, and
$d^{*}=\max_{1\leq i\leq N}\textrm{deg}_{i}$ is the degree of $\mathcal{G}$.
The Laplacian matrix of $\mathcal{G}$ is defined as
$\mathcal{L}=\mathcal{D}-\mathcal{A}$, where
$\mathcal{D}=\textrm{diag}\\{\textrm{deg}_{1},\ldots,\textrm{deg}_{N}\\}$. A
sequence of edges $(i_{1},i_{2})$, $(i_{2},i_{3})$, $\ldots$,
$(i_{k-1},i_{k})$ is called a path from agent $i_{1}$ to agent $i_{k}$. The
graph $\mathcal{G}$ is a connected graph if for any two agents
$i,j\in\mathcal{V}$, there exists a path from $i$ to $j$. For a connected
graph, its Laplacian matrix $\mathcal{L}$ is a symmetric positive semidefinite
matrix and its eigenvalues in an ascending order are denoted by
$0=\lambda_{1}(\mathcal{L})<\lambda_{2}(\mathcal{L})\leq\cdots\leq\lambda_{N}(\mathcal{L})$
(Fiedler, 1973).

###  2.3 Problem Statement

We consider a multi-agent system consisting of $N$ nonlinear agents described
by

$\left\\{\begin{aligned}
\dot{x}_{im}=&f_{im}(x_{i1},\ldots,x_{im})+x_{i,m+1},1\leq m\leq r-1,\\\
\dot{x}_{ir}=&f_{ir}(x_{i},z_{i},\omega_{i})+u_{i},\\\
\dot{z}_{i}=&f_{i0}(x_{i},z_{i},\omega_{i}),\\\
y_{i}=&x_{i1},~{}i=1,\ldots,N,\end{aligned}\right.$ (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 each agent, $r$ is the relative degree,
$y_{i}\in\mathbb{R}$ is the measured output, $u_{i}\in\mathbb{R}$ is the
control input, $\omega_{i}\in\mathbb{R}^{n_{\omega_{i}}}$ is the external
disturbance, $f_{i0}\in
C^{1}\left(\mathbb{R}^{r}\times\mathbb{R}^{n_{i}-r}\times\mathbb{R}^{n_{\omega_{i}}},\mathbb{R}^{n_{i}-r}\right)$,
$f_{im}\in C^{r+1-m}\left(\mathbb{R}^{m},\mathbb{R}\right)$, $1\leq m\leq
r-1$, and $f_{ir}\in C^{1}\left(\mathbb{R}^{r}\right.$
$\left.\times\mathbb{R}^{n_{i}-r}\times\mathbb{R}^{n_{\omega_{i}}},\mathbb{R}\right)$
are locally Lipschitz functions.
Remark 1. System (1) covers the plants considered in the existing data rate
literature as special cases. Specifically, the first-order integrator
considered in Li et al. (2011) and Mu and Liu (2015), the second-order
integrator considered in Li and Xie (2012), and the higher-order dynamics
considered in You and Xie (2011), Ment et al. (2017), Qiu et al. (2017a;
2017b), Dong (2019), and Ran and Xie (2019), directly or under a well-defined
state transformation, fall into the form of system (1). What is more, since
the functions $f_{i0}(\cdot),f_{i1}(\cdot),\ldots,f_{ir}(\cdot)$ can be
different for different agents, the multi-agent system described by (1) is
heterogeneous.

For system (1), we have the following assumptions:
Assumption A1. The external disturbance $\omega_{i}$ and its derivative
$\dot{\omega}_{i}$ are bounded.
Assumption A2. The zero dynamics $\dot{z}_{i}=f_{i0}(x_{i},z_{i},\omega_{i})$,
with input $(x_{i},\omega_{i})$, is bounded-input-bounded-state (BIBS) stable.
Assumption A3. The communication graph $\mathcal{G}$ among the $N$ agents is
undirected and connected.

In this paper, we will investigate the consensus of the multi-agent system (1)
with a digital network. Signals to be transmitted between each pair of
adjacent agents are first quantized and encoded at the transmitters, and then
decoded at the receivers. The finite-level uniform quantizer employed in this
paper is given by

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

where $K\in\\{1,2,\ldots,\\}$, and $2K+1$ is the quantization level. Similar
to Li et al. (2011), it is assumed that the agent sends out no signal if the
output of the quantizer is 0. Therefore, for a $(2K+1)$-level quantizer
$q(\cdot)$, it suffices to use $\lceil\log_{2}(2K)\rceil$ bits to represent
its output. In particular, letting $K=1$ yields a one-bit quantizer.

In this paper, we aim to solve the output consensus problem for the nonlinear
heterogeneous multi-agent system (1) with limited data rate. For simplicity
and to illustrate our design scheme, we first consider the simple case that
each agent has full knowledge of its dynamics (i.e., $x_{i}$, $z_{i}$,
$\omega_{i}$, and $f_{im}(\cdot)$, $0\leq m\leq r$) in Section 3. Then, in
Section 4, we move forward to the general case that $\omega_{i}$ and
$f_{im}(\cdot)$, $0\leq m\leq r$, are uncertain, and each agent only has
access to its first state variable.

##  3 Consensus Under Full Information

In this section, based on the full information of the agent dynamics, new
state variables
$\varrho_{i}=[\varrho_{i1},\ldots,\varrho_{ir}]^{\rm{T}}\in\mathbb{R}^{r}$,
$1\leq i\leq N$, are first introduced for subsequent consensus protocol
design. Let $\varrho_{i1}=y_{i}=x_{i1}\triangleq\phi_{i1}(x_{i1})$,
$\varrho_{im}=\dot{\varrho}_{i,m-1}=\sum_{j=1}^{m-2}(x_{i,j+1}+f_{ij}(x_{i1},\ldots,x_{ij}))\frac{\partial\phi_{i,m-1}}{\partial
x_{ij}}+x_{im}+f_{i,m-1}(x_{i1},\ldots,x_{i,m-1})\triangleq\phi_{im}(x_{i1},\ldots,x_{im})$,
$2\leq m\leq r$, and
$\dot{\varrho}_{ir}=f_{ir}(x_{i},z_{i},\omega_{i})+u_{i}+\sum_{j=1}^{r-1}(x_{i,j+1}+f_{ij}(x_{i1},\ldots,x_{ij}))\frac{\partial\phi_{ir}}{\partial
x_{ij}}\triangleq F_{i}(\varrho_{i},z_{i},\omega_{i})+u_{i}$, where $F_{i}\in
C^{1}(\mathbb{R}^{r}\times\mathbb{R}^{n_{i}-r}\times\mathbb{R}^{n_{\omega_{i}}},\mathbb{R})$.
As a result, the original system (1) is transformed into the following
canonical form (Isidori, 1989):

$\left\\{\begin{aligned}
\dot{\varrho}_{i}=&A\varrho_{i}+B[F_{i}(\varrho_{i},z_{i},\omega_{i})+u_{i}],\\\
\dot{z}_{i}=&Z_{i}(\varrho_{i},z_{i},\omega_{i}),\\\
y_{i}=&\varrho_{i1},~{}i=1,\ldots,N,\end{aligned}\right.$ (3)

where $Z_{i}\in
C^{1}(\mathbb{R}^{r}\times\mathbb{R}^{n_{i}-r}\times\mathbb{R}^{n_{\omega_{i}}},\mathbb{R}^{n_{i}-r})$
represents the zero dynamics with respect to the new state variable
$\varrho_{i}$, and matrices
$A=\left[\begin{array}[]{cc}\textbf{0}_{r-1}&I_{r-1}\\\
0&\textbf{0}^{\textrm{T}}_{r-1}\end{array}\right]$ and
$B=\left[\begin{array}[]{c}\textbf{0}_{r-1}\\\ 1\\\ \end{array}\right]$. Since
$\varrho_{i1}=x_{i1}$, the output consensus of system (3) is equivalent to the
output consensus of system (1).
Lemma 1. _Let_

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

_where $k_{1},k_{2},\ldots,k_{r-1}$ are chosen such that the polynomial
$k_{1}+k_{2}\lambda+\cdots+k_{r-1}\lambda^{r-2}+\lambda^{r-1}$ is Hurwitz._

  * •

_If_ $s_{i}$ is bounded and $\lim_{t\rightarrow\infty}s_{i}=\Lambda$ for some
constant $\Lambda$, then the multi-agent system (1) achieves the following
output consensus:

$\lim_{t\rightarrow\infty}(y_{i}-y_{j})=0,~{}1\leq i\neq j\leq N.$ (5)

_Furthermore, if the convergence of_ $s_{i}$ is exponential, the convergence
of (5) is also exponential.

  * •

_If_ $s_{i}$ is bounded and
$\lim_{t\rightarrow\infty}|s_{i}-\Lambda|=O(\varepsilon)$ for some constant
$\Lambda$ and small positive constant $\varepsilon$, then the multi-agent
system (1) achieves the following practical output consensus:

$\lim_{t\rightarrow\infty}|y_{i}-y_{j}|=O(\varepsilon),~{}1\leq i\neq j\leq
N.$ (6)

Proof. The proof is straightforward from Lemma 1 in Dong (2019) and the linear
control theory (Chen, 1998). $\blacksquare$

Lemma 1 indicates that the consensus of $s_{i}$, $1\leq i\leq N$, guarantees
the output consensus of the multi-agent system (1). Thus, in the sequel, the
protocol $u_{i}$ will be designed to handle the consensus of $s_{i}$.
Specifically, the information of $s_{i}$ will be transmitted between each pair
of agents. The dynamics of $s_{i}$ can be written as

$\dot{s}_{i}=k_{1}\varrho_{i2}+k_{2}\varrho_{i3}+\cdots+k_{r-1}\varrho_{ir}+F_{i}(\varrho_{i},z_{i},\omega_{i})+u_{i}.$
(7)

Since the communication network is a digital network, $s_{i}$ can only be
exchanged in discrete-time. Let $T$ be the sampling period and $k=0,1,\ldots$
the sampling index. Data to be transmitted at time $t=kT$ is then denoted by
$s_{i}(kT)$. What is more, we mention that the sampling period $T$ is also a
design parameter in this paper. We have the following lemma.
_Lemma 2. (Li et al. (2011)) If Assumption A3 holds and
$T\in(0,2/\lambda_{N}(\mathcal{L}))$, then $\rho_{h}<1$, where_

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

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

For agent $j$, the encoder $\Upsilon_{j}$ to encode the signal $s_{j}(t)$ is
given by

$\left\\{\begin{aligned} \xi_{j}(0)&=0,\\\
\xi_{j}((k+1)T)&=\beta(kT)\Delta_{j}((k+1)T)+\xi_{j}(kT),\\\
\Delta_{j}((k+1)T)&=q\left(\frac{s_{j}((k+1)T)-\xi_{j}(kT)}{\beta(kT)}\right),\\\
&\quad k=0,1,\ldots\end{aligned}\right.$ (8)

where $\xi_{j}$ is the internal state of $\Upsilon_{j}$, $\Delta_{j}$ is the
output of $\Upsilon_{j}$, and $\beta(\cdot)$ is a scaling function to be
specified latter. The agent $j$ broadcasts $\Delta_{j}$ to its neighbors
through the digital network. Let agent $i$ be a neighbor of agent $j$. Then
agent $i$ receives $\Delta_{j}$ and uses the following decoder $\Psi_{ji}$ to
decode $\Delta_{j}$ (i.e., to reconstruct $s_{j}$):

$\left\\{\begin{aligned} \widehat{s}_{ji}(0)&=0,\\\
\widehat{s}_{ji}((k+1)T)&=\beta(kT)\Delta_{j}((k+1)T)+\widehat{s}_{ji}(kT),\\\
&\quad k=0,1,\ldots\end{aligned}\right.$ (9)

where $\widehat{s}_{ji}$ is the output of $\Psi_{ji}$.

Then for every agent, based on the full-information assumption and the outputs
of its own encoder and decoders, the control protocol is given by

$\displaystyle u_{i}(t)=$
$\displaystyle-\theta_{i}(t)+\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{s}_{ji}(kT)-\xi_{i}(kT)\right),$
$\displaystyle t\in[kT,(k+1)T),~{}k=0,1,\ldots,$ (10)

where
$\theta_{i}(t)=k_{1}\varrho_{i2}+k_{2}\varrho_{i3}+\cdots+k_{r-1}\varrho_{ir}+F_{i}(\varrho_{i},z_{i},\omega_{i})$.

In the sequel, we will investigate the convergence of the closed-loop system
under the full-information-based protocol. Define a compact set
$\mathcal{X}\subseteq\mathbb{R}^{r}$ and let
$C_{s}\geq\max_{\varrho_{i}\in\mathcal{X},1\leq i\leq N}|s_{i}|$. Recall that
the state transformation from $x_{i}$ to $\varrho_{i}$ doesn’t change the BIBS
property of the zero dynamics
$\dot{z}_{i}=Z_{i}(\varrho_{i},z_{i},\omega_{i})$. Let
$z_{i}(0)\in\mathcal{Z}_{i}$ for some compact set
$\mathcal{Z}_{i}\subseteq\mathbb{R}^{n_{i}-r}$. By Assumption A2, there exists
a positive constant $c_{zi}$ such that $\sup_{t\in[0,\infty)}\|z_{i}\|\leq
c_{zi}$ for all $\varrho_{i}\in\mathcal{X}$. Note that any compact subset of
$\mathbb{R}^{r}\times\mathbb{R}^{n_{i}-r}$ can be put in the interior of
$\mathcal{X}\times\mathcal{Z}_{i}$.
Theorem 1. _Consider the closed-loop system composed of the multi-agent system
( 1), the encoder (8), the decoder (9), and the protocol (3). Suppose
Assumptions A1 to A3 are satisfied, and the initial conditions of the agents
$(\varrho_{i}(0),z_{i}(0))\in\mathcal{X}\times\mathcal{Z}_{i}$, $1\leq i\leq
N$. Let_

$\displaystyle T\in$ $\displaystyle\left(0,2/\lambda_{N}(\mathcal{L})\right),$
(11) $\displaystyle\gamma\in$ $\displaystyle(\rho_{h},1),$ (12) $\displaystyle
K\geq$ $\displaystyle\left\lfloor K_{1}(T,\gamma)-\frac{1}{2}\right\rfloor+1,$
(13) $\displaystyle\beta(kT)=$ $\displaystyle\beta_{0}\gamma^{k},$ (14)

_where_

$\displaystyle
K_{1}(T,\gamma)=\frac{\sqrt{N}T^{2}\lambda^{2}_{N}(\mathcal{L})}{2\gamma(\gamma-\rho_{h})}+\frac{1+2Td^{*}}{2\gamma},$
(15)
$\displaystyle\beta_{0}>\max\left\\{\frac{2T\lambda_{N}(\mathcal{L})C_{s}}{\gamma(K+\frac{1}{2})},\frac{C_{s}}{K+\frac{1}{2}},\right.$
$\displaystyle\qquad\qquad\quad\left.\frac{2C_{s}(\gamma-\rho_{h})(2\gamma+T\lambda_{N}(\mathcal{L}))}{T\lambda_{N}(\mathcal{L})}\right\\}.$
(16)

_Then the multi-agent system ( 1) achieves the output consensus (5)
exponentially._
Proof. Under the full-information-based protocol (3), one has

$s_{i}((k+1)T)=s_{i}(kT)+T\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{s}_{ji}(kT)-\xi_{i}(kT)\right).$

The system above falls into the same form of the first-order integrator system
considered in Li et al. (2011). Then according to Theorem 3.1 in Li et al.
(2011) and the first statement in Lemma 1, one can readily conclude that the
multi-agent system (1) achieves the output consensus (5) exponentially.
$\blacksquare$

Note that the protocol (3) is an “ideal” one since it relies on system full-
information, which is unavailable in the general case. However, it provides
some insights on the “ideal” consensus performance the protocol could achieve.
We will show in the next section that we can recover the “ideal” consensus
performance by using the device ESO. Towards that end, let the ideal
trajectories generated by the full-information-based protocol be superscripted
by the symbol $\star$. Then, the dynamics of $s_{i}^{\star}$ can be specified
by

$\left\\{\begin{aligned}
&s^{\star}_{i}((k+1)T)=s^{\star}_{i}(kT)+T\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{s}^{\star}_{ji}(kT)-\widehat{s}^{\star}_{ij}(kT)\right),\\\
&\qquad\qquad\qquad\quad k=0,1,\ldots,\\\
&s^{\star}_{i}(t)=s^{\star}_{i}(kT)+(t-kT)\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{s}^{\star}_{ji}(kT)-\widehat{s}^{\star}_{ij}(kT)\right),\\\
&\qquad\quad~{}kT<t<(k+1)T.\end{aligned}\right.\\\ $ (17)

The ideal trajectory of $y_{i}^{\star}$ can be regarded as the output of the
$r$th-order linear system

$\left\\{\begin{aligned}
\dot{\varrho}^{\star}_{i}=&A^{\star}\varrho^{\star}_{i}+B^{\star}u_{i}^{\star},\\\
y_{i}^{\star}=&\varrho^{\star}_{i1},~{}i=1,\ldots,N,\end{aligned}\right.$ (18)

where
$A^{\star}=\left[\begin{array}[]{c}\begin{array}[]{cc}\textbf{0}_{r-1}&I_{r-1}\end{array}\\\
\left[0,-k_{1},\ldots,-k_{r-1}\right]\end{array}\right]$, $B^{\star}=B$, with
full-state-feedback based input

$\displaystyle u_{i}^{\star}(t)=$
$\displaystyle\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{s}^{\star}_{ji}(kT)-\xi^{\star}_{i}(kT)\right),$
$\displaystyle t\in[kT,(k+1)T),~{}k=0,1,\ldots$ (19)

The system (18) is referred to as the linear counterpart of the uncertain
nonlinear multi-agent system (1).

##  4 Consensus Under Output Feedback

The control design in the previous section is generally not realizable. In
this section, we consider the case that $\omega_{i}$ and $f_{im}(\cdot)$,
$0\leq m\leq r$, are uncertain, i.e., the external disturbance and agent
dynamics are unknown to the designer. In addition, each agent only has access
to its first state variable. We propose to use ESOs to estimate the
unmeasurable states, and the unknown external disturbance and agent dynamics.
The ESO is built for each agent locally, and then ESO-based local encoder,
decoder, and protocol are designed to guarantee the output consensus of the
multi-agent system (1) with limited data rate.

###  4.1 ESO-Based Protocol Design

Let $\varrho_{i,r+1}\triangleq F_{i}(\varrho_{i},z_{i},\omega_{i})$ be the
extended state of agent $i$. The ESO for agent $i$ is then designed as

$\left\\{\begin{aligned}
\dot{\hat{\varrho}}_{im}&=\hat{\varrho}_{i,m+1}+\frac{l_{im}}{\varepsilon^{m}}(y_{i}-\hat{\varrho}_{i1}),1\leq
m\leq r-1,\\\
\dot{\hat{\varrho}}_{ir}&=\hat{\varrho}_{i,r+1}+\frac{l_{ir}}{\varepsilon^{r}}(y_{i}-\hat{\varrho}_{i1})+u_{i},\\\
\dot{\hat{\varrho}}_{i,r+1}&=\frac{l_{i,r+1}}{\varepsilon^{r+1}}(y_{i}-\hat{\varrho}_{i1}),\end{aligned}\right.$
(20)

where
$\hat{\varrho}_{i}=[\hat{\varrho}_{i1},\ldots,\hat{\varrho}_{ir}]^{\rm{T}}$
and $\hat{\varrho}_{i,r+1}$ are the estimates of the agent state $\varrho_{i}$
and the extended state $\varrho_{i,r+1}$, respectively,
$\hat{\varrho}_{i1}(0)$ to $\hat{\varrho}_{i,r+1}(0)$ are set as 0,
$\varepsilon<1$ is a small positive constant, and
$L_{i}=[l_{i1},l_{i2},\ldots,l_{i,r+1}]^{\rm{T}}\in\mathbb{R}^{r+1}$ is
selected such that the matrix
$E_{i}=\left[\begin{array}[]{cc}\begin{array}[]{c}-L_{i}\end{array}&\begin{array}[]{c}I_{r-1}\\\
\textbf{0}^{\rm{T}}_{r-1}\end{array}\end{array}\right]$ is Hurwitz. Since
$\varepsilon$ is a small positive constant, the observer (20) exhibits peaking
phenomenon during the initial period. To prevent the peaking from
destabilizing the closed-loop system, we employ the well-known saturation
technique (Khalil, 2017; Ran, Wang, & Dong, 2017; Ran, Wang, Dong, & Xie,
2020) to saturate the outputs of the observer. Recall that we aim to recover
the consensus performance under the full-information-based protocol. Let
$M_{m}\geq\sup_{1\leq i\leq N}|\varrho^{\star}_{im}|$, $1\leq m\leq r+1$. Then
the outputs of the ESO are bounded by

$\overline{\varrho}_{im}=M_{m}\textrm{sat}\left(\frac{\hat{\varrho}_{im}}{M_{m}}\right),~{}1\leq
m\leq r+1.$ (21)

Let
$\overline{\varrho}_{i}=[\overline{\varrho}_{i1},\ldots,\overline{\varrho}_{ir}]^{\rm{T}}$,
and the information to be transmitted between each pair of agents be

$\overline{s}_{i}=k_{1}\overline{\varrho}_{i1}+k_{2}\overline{\varrho}_{i2}+\cdots+k_{r-1}\overline{\varrho}_{i,r-1}+\overline{\varrho}_{ir}.$
(22)

Similar to the previous section, the encoder $\Upsilon_{j}$ for agent $j$ is
designed as

$\left\\{\begin{aligned} \xi_{j}(0)&=0,\\\
\xi_{j}((k+1)T)&=\beta(kT)\Delta_{j}((k+1)T)+\xi_{j}(kT),\\\
\Delta_{j}((k+1)T)&=q\left(\frac{\overline{s}_{j}((k+1)T)-\xi_{j}(kT)}{\beta(kT)}\right),\\\
&\quad k=0,1,\ldots\end{aligned}\right.$ (23)

The decoder $\Psi_{ji}$ for agent $i$ is as follows:

$\left\\{\begin{aligned} \widehat{\overline{s}}_{ji}(0)&=0,\\\
\widehat{\overline{s}}_{ji}((k+1)T)&=\beta(kT)\Delta_{j}((k+1)T)+\widehat{\overline{s}}_{ji}(kT),\\\
&\quad k=0,1,\ldots\end{aligned}\right.$ (24)

The ESO-based protocol is then given by

$\left\\{\begin{aligned}
u_{i}(t)=&-\overline{\theta}_{i}(t)+\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{\overline{s}}_{ji}(kT)-\xi_{i}(kT)\right),\\\
&t\in[kT,(k+1)T),~{}k=0,1,\ldots\end{aligned}\right.$ (25)

where
$\overline{\theta}_{i}(t)=k_{1}\overline{\varrho}_{i2}+k_{2}\overline{\varrho}_{i3}+\cdots+k_{r-1}\overline{\varrho}_{ir}+\overline{\varrho}_{i,r+1}$.

###  4.2 Convergence Analysis

For subsequent use, some notations are defined as
$S(kT)=[s_{1}(kT),s_{2}(kT),\ldots,s_{N}(kT)]^{\rm{T}}$,
$\overline{S}(kT)=[\overline{s}_{1}(kT),\overline{s}_{2}(kT),\ldots,\overline{s}_{N}(kT)]^{\rm{T}}$,
$\widehat{\overline{S}}(kT)=[\xi_{1}(kT),\xi_{2}(kT),$
$\ldots,\xi_{N}(kT)]^{\rm{T}}$,
$e(kT)=\overline{S}(kT)-\widehat{\overline{S}}(kT)$,
$\delta(kT)=S(kT)-J_{N}S(kT)$.

By (7) and (25), the evolution of $s_{i}(kT)$ can be given by

$\displaystyle
s_{i}((k+1)T)=s_{i}(kT)+T\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{\overline{s}}_{ji}(kT)-\xi_{i}(kT)\right)$
$\displaystyle\quad+\int_{kT}^{(k+1)T}\left(\sum_{m=1}^{r}k_{m}\left(\varrho_{i,m+1}(t)-\overline{\varrho}_{i,m+1}(t)\right)\right)\textrm{d}t.$
(26)

By the structures of the encoder $\Upsilon_{j}$ and decoder $\Psi_{ji}$, one
has $\widehat{\overline{s}}_{ji}(kT)=\xi_{j}(kT)$, $i\in\mathcal{N}_{j}$,
$1\leq j\leq N$. It follows from (4.2) that

$\displaystyle S((k+1)T)=$ $\displaystyle
S(kT)-T\mathcal{L}\widehat{\overline{S}}(kT)+\Pi_{1}(kT)$ $\displaystyle=$
$\displaystyle(I-T\mathcal{L})\overline{S}(kT)+T\mathcal{L}e(kT)$
$\displaystyle+S(kT)-\overline{S}(kT)+\Pi_{1}(kT),$ (27)

where $\Pi_{1}(kT)=[\pi_{11}(kT),\ldots,\pi_{1N}(kT)]^{\textrm{T}}$, with

$\displaystyle\pi_{1i}(kT)=\int_{kT}^{(k+1)T}\left(\sum_{m=1}^{r}k_{m}\left(\varrho_{i,m+1}(t)-\overline{\varrho}_{i,m+1}(t)\right)\right)\textrm{d}t.$

Bearing in mind that $\mathcal{L}J_{N}=J_{N}\mathcal{L}=0$, one has

$\displaystyle\overline{S}((k+1)T)-\widehat{\overline{S}}(kT)$
$\displaystyle=$ $\displaystyle
S((k+1)T)-\widehat{\overline{S}}(kT)+\overline{S}((k+1)T)-S((k+1)T)$
$\displaystyle=$ $\displaystyle(I+T\mathcal{L})e(kT)-T\mathcal{L}\delta(kT)$
$\displaystyle+(I+T\mathcal{L})(S(kT)-\overline{S}(kT))$
$\displaystyle+\Pi_{1}(kT)+\overline{S}((k+1)T)-S((k+1)T).$ (28)

It then follows from (4.2) and (4.2) that

$\displaystyle\delta((k+1)T)=$
$\displaystyle(I-T\mathcal{L})\delta(kT)+T\mathcal{L}e(kT)$
$\displaystyle+(I-J_{N})\left[T\mathcal{L}\left(S(kT)-\overline{S}(kT)\right)\right.$
$\displaystyle\left.+\Pi_{1}(kT)\right],$ (29) $\displaystyle e((k+1)T)=$
$\displaystyle\left[\overline{S}((k+1)T)-\widehat{\overline{S}}(kT)\right]$
$\displaystyle-\beta(kT)Q\left(\frac{\overline{S}((k+1)T)-\widehat{\overline{S}}(kT)}{\beta(kT)}\right),$
(30)

where
$Q([\nu_{1},\ldots,\nu_{N}]^{\rm{T}})=[q(\nu_{1}),\ldots,q(\nu_{N})]^{\rm{T}}$.
Note that in the consensus error dynamics (4.2), the second term,
$T\mathcal{L}e(kT)$, is caused by the quantization process; the third term,
$(I-J_{N})\left[S(kT)-\overline{S}(kT)+\Pi_{1}(kT)\right]$, is caused by the
ESO estimation process. Denote

$\alpha(kT)=\frac{1}{\beta(kT)}\delta(kT),~{}\zeta(kT)=\frac{1}{\beta(kT)}e(kT).$
(31)

Before presenting the convergence results of the closed-loop system, we first
state the following lemma, which shows the boundedness of $\alpha(kT)$.
Lemma 3. _Consider the closed-loop system composed of the multi-agent system (
1), the ESO (20), the encoder (23), the decoder (24), and the protocol (25).
Suppose Assumptions A1 to A3 are satisfied, and the initial conditions of the
agents $(\varrho_{i}(0),z_{i}(0))\in\mathcal{X}\times\mathcal{Z}_{i}$, $1\leq
i\leq N$. Let the sampling period $T$, the parameter $\gamma$, and the
quantization parameter $K$ be selected according to (11), (12), and (13),
respectively; let the scaling function be selected as_

$\beta(kT)=\max\\{\beta_{0}\gamma^{k},\sqrt{\varepsilon}\\},$ (32)

_where $\beta_{0}$ is given by (3). Then there exists
$\varepsilon^{{\dagger}}>0$ such that for all
$0<\varepsilon<\varepsilon^{{\dagger}}$ and $k=0,1,\ldots$,
$\|\alpha(kT)\|<\overline{\alpha}$, where
$\overline{\alpha}=\max\left\\{\frac{2\sqrt{N}C_{s}}{\beta_{0}},\frac{T\sqrt{N}\lambda_{N}(\mathcal{L})}{2\gamma(\gamma-\rho_{h})}\right\\}$.
Furthermore, the quantizer will never be saturated._
Proof. Lemma 3 will be proved recursively. First, consider the case $t=0$. By
the fact that $\|\nu\|_{\infty}\leq\|\nu\|\leq\sqrt{N}\|\nu\|_{\infty}$ for
any $N$ dimensional vector $\nu$, one has

$\|\alpha(0)\|=\frac{\|\delta(0)\|}{\beta_{0}}\leq\frac{\sqrt{N}\|\delta(0)\|_{\infty}}{\beta_{0}}\leq\frac{2\sqrt{N}C_{s}}{\beta_{0}}<\overline{\alpha}.$
(33)

What is more, for $\zeta(0)$, one has

$\|\zeta(0)\|_{\infty}=\frac{\|\overline{S}(0)-\widehat{\overline{S}}(0)\|_{\infty}}{\beta_{0}}=0.$
(34)

In the sequel, we consider the case $t=(\kappa+1)T$, $\kappa\geq 0$, with the
assumption that $\|\alpha(kT)\|<\overline{\alpha}$ and the quantizer is not
saturated for all $0\leq k\leq\kappa$. It follows from the definitions of
$\alpha(kT)$ and $S(kT)$, and the boundedness of the control signal $u_{i}$ in
the time interval $[0,(\kappa+1)T]$ that $\|\varrho_{i}\|$ is bounded by some
$\varepsilon$-independent positive constant in the time interval
$[0,(\kappa+1)T]$.

We define the scaled ESO estimation error $\eta_{i}=[\eta_{i1},\ldots,$
$\eta_{i,r+1}]^{\rm{T}}\in\mathbb{R}^{r+1}$ with
$\eta_{im}=\frac{1}{\varepsilon^{r+1-m}}(x_{im}-\hat{x}_{im})$, $1\leq m\leq
r+1$. According to (3) and (20), the error dynamics can be written as

$\left\\{\begin{aligned}
\varepsilon\dot{\eta}_{im}&=\eta_{i,m+1}-l_{im}\eta_{i1},~{}1\leq m\leq r,\\\
\varepsilon\dot{\eta}_{i,r+1}&=\varepsilon\dot{F}_{i}(\varrho_{i},z_{i},\omega_{i})-l_{i,r+1}\eta_{i1}.\end{aligned}\right.$
(35)

By Assumptions A1 and A2, and the boundedness of $\|\varrho_{i}\|$, one can
conclude that in the time interval $[0,(\kappa+1)T]$,
$|\dot{F}_{i}(\varrho_{i},z_{i},\omega_{i})|\leq N_{1i}$ for some
$\varepsilon$-independent positive constant $N_{1i}$. Recall that the matrix
$E_{i}$ is Hurwitz by design. Let $P_{i}\in\mathbb{R}^{(r+1)\times(r+1)}$ be
the unique positive definite matrix solution to the matrix equation
$P_{i}E_{i}+E_{i}^{\textrm{T}}P_{i}=-I_{r+1}$, and define the Lyapunov
function candidate $V_{i}:\mathbb{R}^{r+1}\rightarrow\mathbb{R}$ as
$V_{i}(\eta_{i})=\eta^{\textrm{T}}_{i}P_{i}\eta_{i}$. Then one has
$\sigma_{i1}\|\eta_{i}\|^{2}\leq
V_{i}(\eta_{i})\leq\sigma_{i2}\|\eta_{i}\|^{2}$ and $\left|\frac{\partial
V_{i}(\eta_{i})}{\partial\eta_{i,r+1}}\right|\leq 2\sigma_{i2}\|\eta_{i}\|$,
where $\sigma_{i1}$ and $\sigma_{i2}$ are the minimal and maximal eigenvalues
of the matrix $P_{i}$, respectively. Then by (35), the time derivative of
$V_{i}(\eta_{i})$ satisfies

$\displaystyle\frac{\textrm{d}V_{i}(\eta_{i})}{\textrm{d}t}=$
$\displaystyle\frac{1}{\varepsilon}\left(\sum_{j=1}^{r}(\eta_{i,j+1}-l_{ij}\eta_{i1})\frac{\partial
V_{i}(\eta_{i})}{\partial{\eta_{ij}}}\right.$
$\displaystyle\left.-l_{i,r+1}\eta_{i1}\frac{\partial
V_{i}(\eta_{i})}{\partial{\eta_{i,r+1}}}\right)+\dot{F}_{i}(\varrho_{i},z_{i},\omega_{i})\frac{\partial
V_{i}(\eta_{i})}{\partial{\eta_{i,r+1}}}$ $\displaystyle\leq$
$\displaystyle-\frac{1}{\varepsilon}\|\eta_{i}\|^{2}+2\sigma_{i2}N_{1i}\|\eta_{i}\|$
$\displaystyle\leq$
$\displaystyle-\frac{1}{\sigma_{i2}\varepsilon}V_{i}(\eta_{i})+\frac{2\sigma_{i2}N_{1i}}{\sqrt{\sigma_{i1}}}\sqrt{V_{i}(\eta_{i})}.$
(36)

It follows that

$\frac{\textrm{d}\sqrt{V_{i}(\eta_{i})}}{\textrm{d}t}\leq-\frac{1}{2\sigma_{i2}\varepsilon}\sqrt{V_{i}(\eta_{i})}+\frac{\sigma_{i2}N_{1i}}{\sqrt{\sigma_{i1}}}.$
(37)

Solving the inequality above and using $\sigma_{i1}\|\eta_{i}\|^{2}\leq
V_{i}(\eta_{i})\leq\sigma_{i2}\|\eta_{i}\|^{2}$ again, one has

$\displaystyle\|\eta_{i}(t)\|\leq$
$\displaystyle\sqrt{\frac{\sigma_{i2}}{\sigma_{i1}}}\|\eta_{i}(0)\|e^{-\frac{1}{2\sigma_{i2}\varepsilon}t}+\frac{2\sigma_{i2}^{2}N_{1i}\varepsilon}{\sigma_{i1}}.$
(38)

Note that the right hand side of the inequality above converges to 0 as
$\varepsilon\rightarrow 0$ uniformly for any $t\in(0,(\kappa+1)T]$. Therefore,
there exists $\varepsilon_{0}>0$ such that for any $t_{0}>0$ and
$\varepsilon\in(0,\varepsilon_{0})$, $\|\eta_{i}(t)\|=O(\varepsilon)$,
$\forall t\in[t_{0},(\kappa+1)T]$. What is more, one can select
$M_{m}\geq\sup_{1\leq i\leq N}|\varrho^{\star}_{im}|$, $1\leq m\leq r+1$, such
that the saturations for the outputs of the observer will not be invoked in
the time interval $[t_{0},(\kappa+1)T]$, i.e.,
$\overline{\varrho}_{im}=\hat{\varrho}_{im}$, $1\leq m\leq r+1$.

With the convergence of the observers, and the assumption that
$\|\alpha(kT)\|<\overline{\alpha}$ and the quantizer is not saturated for all
$0\leq k\leq\kappa$, we are ready to investigate the bound of
$\alpha((\kappa+1)T)$. Let positive integer $k^{*}$ satisfy
$\beta_{0}\gamma^{k^{*}-1}>\sqrt{\varepsilon}$ and
$\beta_{0}\gamma^{k^{*}}\leq\sqrt{\varepsilon}$, i.e.,
$\beta(kT)=\sqrt{\varepsilon}$ for all $k\geq k^{*}$. We consider three cases.

_Case 1):_ $\kappa+1<k^{*}$. In this case, by (4.2)-(31), one has

$\displaystyle\alpha((\kappa+1)T)=$
$\displaystyle\gamma^{-1}(I-T\mathcal{L})\alpha(\kappa
T)+\gamma^{-1}T\mathcal{L}\zeta(\kappa T)$ $\displaystyle+\frac{\Pi_{2}(\kappa
T)}{\beta_{0}\gamma^{\kappa+1}},$ (39) $\displaystyle\zeta((\kappa+1)T)=$
$\displaystyle\gamma^{-1}\left(\Delta(\kappa T)-Q(\Delta(\kappa T))\right),$
(40)

where

$\displaystyle\Pi_{2}(\kappa T)=$
$\displaystyle(I-J_{N})\left[T\mathcal{L}\left(S(\kappa T)-\overline{S}(\kappa
T)\right)+\Pi_{1}(\kappa T)\right],$ $\displaystyle\Delta(\kappa T)=$
$\displaystyle(I+T\mathcal{L})\zeta(\kappa T)-T\mathcal{L}\alpha(\kappa
T)+\frac{\Pi_{3}(\kappa T)}{\beta_{0}\gamma^{\kappa}},$
$\displaystyle\Pi_{3}(\kappa T)=$ $\displaystyle(I+T\mathcal{L})(S(\kappa
T)-\overline{S}(\kappa T))$ $\displaystyle+\Pi_{1}(\kappa
T)+\overline{S}((\kappa+1)T)-S((\kappa+1)T).$

Let $\max_{0\leq k\leq\kappa}\|\Pi_{2}(kT)\|/(\beta_{0}\gamma^{k+1})\triangleq
N_{2}(\varepsilon)$ and $\|\Pi_{3}(\kappa T)\|/(\beta_{0}\gamma^{\kappa})$
$\triangleq N_{3}(\varepsilon)$. Due to the convergence of the observers and
$\beta(kT)\geq\sqrt{\varepsilon}$, one has $N_{2}(\varepsilon)\rightarrow 0$
and $N_{3}(\varepsilon)\rightarrow 0$, as $\varepsilon\rightarrow 0$.

Let $J$ be a unitary matrix defined by $J=[\textbf{1}/\sqrt{N},$
$\psi_{2},\ldots,\psi_{N}]$, where
$\psi_{i}^{\textrm{T}}\mathcal{L}=\lambda_{i}(\mathcal{L})\psi_{i}^{\rm{T}}$,
$2\leq i\leq N$. Let $\widetilde{\alpha}(\kappa T)=J^{-1}\alpha(\kappa
T)=J^{\rm{T}}\alpha(\kappa T)$ and decompose $\widetilde{\alpha}(\kappa
T)=[\widetilde{\alpha}_{1}(\kappa T),\widetilde{\alpha}_{2}(\kappa
T)]^{\rm{T}}$ with a scalar $\widetilde{\alpha}_{1}(\kappa T)$. One can easily
verify that $\widetilde{\alpha}_{1}(\kappa T)=0$. Denote
$\widetilde{P}_{\gamma,T}=\gamma^{-1}\textrm{diag}\left\\{1-T\lambda_{2}(\mathcal{L}),\ldots,1-T\lambda_{N}(\mathcal{L})\right\\}$
and $\psi=[\psi_{2},\ldots,\psi_{N}]$. It follows from the definition of
$\widetilde{\alpha}_{2}(\kappa T)$ and (39) that

$\displaystyle\widetilde{\alpha}_{2}((\kappa+1)T)=$
$\displaystyle\widetilde{P}_{\gamma,T}\widetilde{\alpha}_{2}(\kappa
T)+\gamma^{-1}T\psi^{\rm{T}}\mathcal{L}\zeta(\kappa T)$
$\displaystyle+\frac{\psi^{\rm{T}}\Pi_{2}(\kappa
T)}{\beta_{0}\gamma^{\kappa+1}}.$ (41)

By (4.2), one can establish the evolution equation from
$\widetilde{\alpha}_{2}(0)$ to $\widetilde{\alpha}_{2}((\kappa+1)T)$. This
together with $\alpha(\kappa T)=\psi\widetilde{\alpha}_{2}(\kappa T)$ and
$\widetilde{\alpha}_{2}(\kappa T)=\psi^{\rm{T}}\alpha(\kappa T)$ leads to

$\displaystyle\alpha((\kappa+1)T)=$
$\displaystyle\psi[\widetilde{P}_{\gamma,T}]^{\kappa+1}\psi^{\rm{T}}\alpha(0)$
$\displaystyle+\gamma^{-1}T\psi[\widetilde{P}_{\gamma,T}]^{\kappa}\psi^{\rm{T}}\mathcal{L}\zeta(0)$
$\displaystyle+\gamma^{-1}T\psi\sum_{i=0}^{\kappa-1}[\widetilde{P}_{\gamma,T}]^{i}\psi^{\rm{T}}\mathcal{L}\zeta((\kappa-i)T)$
$\displaystyle+\sum_{i=0}^{\kappa}\frac{\psi[\widetilde{P}_{\gamma,T}]^{i}\psi^{\rm{T}}\Pi_{2}((\kappa-i)T)}{\beta_{0}\gamma^{\kappa+1-i}}.$
(42)

Note that $\|\psi\|=1$, $\gamma\in(\rho_{h},1)$,
$\|\mathcal{L}\|=\lambda_{N}(\mathcal{L})$,
$\|\widetilde{P}_{\gamma,T}\|\leq\rho_{h}/\gamma$, and
$\|\zeta(0)\|_{\infty}=0<{C_{s}}/{\beta_{0}}$ and $\sup_{1\leq t\leq\kappa
T}\|\zeta(t)\|_{\infty}\leq 1/(2\gamma)$ since the quantizer is unsaturated
before $t=(\kappa+1)T$. By some manipulations, the Euclidean norm of
$\alpha((\kappa+1)T)$ is upper bounded by

$\displaystyle\frac{2\sqrt{N}C_{s}}{\beta_{0}}\left(\frac{\rho_{h}}{\gamma}\right)^{\kappa}+\frac{\sqrt{N}C_{s}T\lambda_{N}(\mathcal{L})}{\beta_{0}\gamma}\left(\frac{\rho_{h}}{\gamma}\right)^{\kappa}$
$\displaystyle+\frac{\sqrt{N}T\lambda_{N}(\mathcal{L})}{2\gamma(\gamma-\rho_{h})}\left(1-\left(\frac{\rho_{h}}{\gamma}\right)^{\kappa}\right)+\frac{\gamma
N_{2}(\varepsilon)}{\beta_{0}(\gamma-\rho_{h})}.$ (43)

Considering that
$\beta_{0}>\frac{2C_{s}(\gamma-\rho_{h})(2\gamma+T\lambda_{N}(\mathcal{L}))}{T\lambda_{N}(\mathcal{L})}$,
one has

$\displaystyle\|\alpha(\kappa+1)T\|<\frac{\sqrt{N}T\lambda_{N}(\mathcal{L})}{2\gamma(\gamma-\rho_{h})}+\frac{\gamma
N_{2}(\varepsilon)}{\beta_{0}(\gamma-\rho_{h})}.$ (44)

Hence for sufficiently small $\varepsilon$,
$\|\alpha((\kappa+1)T)\|<\overline{\alpha}$ holds. On the other hand,

$\displaystyle\|\Delta(\kappa T)\|_{\infty}$ $\displaystyle\leq$
$\displaystyle\|(I+T\mathcal{L})\|_{\infty}\|\zeta(\kappa
T)\|_{\infty}+T\|\mathcal{L}\|\|\alpha(\kappa T)\|+\frac{\|\Pi_{3}(\kappa
T)\|}{\beta_{0}\gamma^{\kappa}}$ $\displaystyle<$
$\displaystyle\frac{1+2Td^{*}}{2\gamma}+\frac{\sqrt{N}T^{2}\lambda^{2}_{N}(\mathcal{L})}{2\gamma(\gamma-\rho_{h})}+N_{3}(\varepsilon)$
$\displaystyle=$ $\displaystyle K_{1}(T,\gamma)+N_{3}(\varepsilon).$ (45)

Similarly, one can select a sufficiently small $\varepsilon$ such that

$\|\Delta((\kappa+1)T)\|_{\infty}\leq K+\frac{1}{2}.$ (46)

Therefore, the quantizer is unsaturated when $t=(\kappa+1)T$.

_Case 2):_ $\kappa+1=k^{*}$. In this case, one has

$\displaystyle\alpha((\kappa+1)T)=$
$\displaystyle\frac{\beta_{0}\gamma^{\kappa}}{\sqrt{\varepsilon}}(I-T\mathcal{L})\alpha(\kappa
T)+\frac{\beta_{0}\gamma^{\kappa}}{\sqrt{\varepsilon}}T\mathcal{L}\zeta(\kappa
T)$ $\displaystyle+\frac{\Pi_{2}(\kappa T)}{\sqrt{\varepsilon}},$ (47)
$\displaystyle\zeta((\kappa+1)T)=$
$\displaystyle\frac{\beta_{0}\gamma^{\kappa}}{\sqrt{\varepsilon}}\left(\Delta(\kappa
T)-Q(\Delta(\kappa T))\right).$ (48)

Note that
$\frac{\beta_{0}\gamma^{\kappa}}{\sqrt{\varepsilon}}\leq\frac{1}{\gamma}$.
Then by conducting a similar analysis as in (4.2)-(46), one can conclude that
$\|\alpha((\kappa+1)T)\|<\overline{\alpha}$ and the quantizer is unsaturated
for $t=(\kappa+1)T$ in this case.

_Case 3):_ $\kappa+1>k^{*}$. In this case, one has

$\displaystyle\alpha((\kappa+1)T)=$
$\displaystyle(I-T\mathcal{L})\alpha(\kappa T)+T\mathcal{L}\zeta(\kappa
T)+\frac{\Pi_{2}(\kappa T)}{\sqrt{\varepsilon}},$ (49)
$\displaystyle\zeta((\kappa+1)T)=$ $\displaystyle\Delta(\kappa
T)-Q(\Delta(\kappa T)).$ (50)

Note that $1<\frac{1}{\gamma}$. Again similar to case 1), one can obtain that
$\|\alpha((\kappa+1)T)\|<\overline{\alpha}$ and the quantizer is unsaturated
when $t=(\kappa+1)T$.

Therefore, combining Cases 1) to 3) leads to
$\|\alpha((\kappa+1)T)\|<\overline{\alpha}$ and the quantizer is unsaturated
when $t=(\kappa+1)T$ if $\|\alpha(kT)\|<\overline{\alpha}$ and the quantizer
is unsaturated for all $0\leq k\leq\kappa$. Finally, we arrive at the
conclusion that $\|\alpha(kt)\|<\overline{\alpha}$ for all $k\geq 0$ and
quantizer will never be saturated. The proof of Lemma 3 is completed.
$\blacksquare$

Now, with Lemma 3, we are in a position to state the convergence results of
the closed-loop system.
Theorem 2. _Consider the closed-loop system composed of the multi-agent system
( 1), the ESO (20), the encoder (23), the decoder (24), and the protocol (25).
Suppose Assumptions A1 to A3 are satisfied, and the initial conditions of the
agents $(\varrho_{i}(0),z_{i}(0))\in\mathcal{X}\times\mathcal{Z}_{i}$, $1\leq
i\leq N$. Let the sampling period $T$, the parameter $\gamma$, the
quantization parameter $K$, and the scaling function $\beta(kT)$ be selected
according to (11), (12), (13), and (32), respectively. Then for any
$\sigma>0$, there exists $\varepsilon^{{\ddagger}}>0$, which is dependent on
$\sigma$, such that for all $0<\varepsilon<\varepsilon^{{\ddagger}}$,_

  * •

_the ESO (_ 20) recovers the agent state and the unknown dynamics, i.e., there
exists $\tau_{1}(\varepsilon)>0$ satisfying $\lim_{\varepsilon\rightarrow
0}\tau_{1}(\varepsilon)=0$, such that $\forall
t\in[\tau_{1}(\varepsilon),\infty)$,

$\left|\varrho_{im}(t)-\hat{\varrho}_{im}(t)\right|\leq\sigma,1\leq i\leq
N,~{}1\leq m\leq r+1.$ (51)

  * •

_the multi-agent system (_ 1) achieves practical output consensus, i.e.,

$\lim_{t\rightarrow\infty}\left|y_{i}(t)-y_{j}(t)\right|\leq\sigma,~{}1\leq
i\neq j\leq N.$ (52)

  * •

_the ESO-based protocol (_ 25) recovers the output consensus performance under
the full-information-based protocol, i.e., with same initial conditions,

$\sup_{t\in[0,\infty)}\left|y_{i}(t)-y^{\star}_{i}(t)\right|\leq\sigma,~{}1\leq
i\leq N.$ (53)

Proof. By Lemma 3, for any $\varepsilon\in(0,\varepsilon^{{\dagger}})$,
$\|\alpha(kT)\|<\overline{\alpha}$ for all $k\geq 0$. Then following a similar
line of the arguments as in (35) to (38), one can conclude that there exists
$\varepsilon^{{\ddagger}}_{1}\in(0,\varepsilon^{{\dagger}}]$ such that for all
$\varepsilon\in(0,\varepsilon^{{\ddagger}}_{1})$, the statement in the first
bullet holds.

By the definitions of $\alpha(kT)$ and $\delta(kT)$, and the selection of
$\beta(kT)$, for $k\geq k^{*}$, one has
$\|\delta(kT)\|=\beta(kT)\|\alpha(kT)\|<\overline{\alpha}\sqrt{\varepsilon}=O(\sqrt{\varepsilon})$.
Thus $|s_{i}(kT)-s_{j}(kT)|=O(\sqrt{\varepsilon})$, $1\leq i\neq j\leq N$, for
$k\geq k^{*}$. Furthermore, for any $t\in[kT,(k+1)T)$, one has

$\displaystyle
s_{i}(t)-s_{i}(kT)=(t-kT)\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{\overline{s}}_{ji}(kT)-\xi_{i}(kT)\right)$
$\displaystyle\qquad\qquad+\int_{kT}^{t}\sum_{m=1}^{r}k_{m}\varepsilon^{r+1-m}\eta_{i,m+1}(\nu)\textrm{d}\nu.$
(54)

Note that the right hand side of the equation above converges to
$O(\varepsilon)$. Thus one can conclude that
$\lim_{t\rightarrow\infty}|s_{i}-\Lambda|=O(\varepsilon)$ for some constant
$\Lambda$. By Lemma 1, the statement in the second bullet holds.

For $t\geq\tau_{1}(\varepsilon)$, considering the convergence of the observer,
the trajectory of $s_{i}(t)$ under the ESO-based protocol can be given by

$\left\\{\begin{aligned}
&s_{i}((k+1)T)=s_{i}(kT)+T\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{s}_{ji}(kT)-\widehat{s}_{ij}(kT)\right)\\\
&\qquad\qquad\qquad+O(\varepsilon),~{}k=0,1,\ldots,\\\
&s_{i}(t)=s_{i}(kT)+(t-kT)\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{s}_{ji}(kT)-\widehat{s}_{ij}(kT)\right)\\\
&\qquad\quad+O(\varepsilon),~{}t\in[\tau_{1}(\varepsilon),\infty)\cap(kT,(k+1)T),\end{aligned}\right.$
(55)

Since $\dot{s}_{i}$ and $\dot{s}_{i}^{\star}$ are bounded uniformly in
$\varepsilon$ and $s_{i}(0)=s_{i}^{\star}(0)$, one has

$s_{i}(t)-s^{\star}_{i}(t)=O(\tau_{1}(\varepsilon)),~{}\forall
t\in[0,\tau_{1}(\varepsilon)].$ (56)

Hence
$s_{i}(\tau_{1}(\varepsilon))-s^{\star}_{i}(\tau_{1}(\varepsilon))=O(\tau_{1}(\varepsilon))$,
this together with (17) and (55), the continuous dependence of the solutions
of differential equations on initial conditions and parameters (see Theorem
9.1 in Khalil (2002)), and the exponential convergence of the equation
$s_{i}((k+1)T)=s_{i}(kT)+T\sum_{j\in\mathcal{N}_{i}}a_{ij}\left(\widehat{s}_{ji}(kT)-\widehat{s}_{ij}(kT)\right)$
(i.e., the exponential convergence of $s_{i}^{\star}(t)$), leads to

$s_{i}(t)-s_{i}^{\star}(t)=O(\varepsilon)+O(\tau_{1}(\varepsilon)),~{}\forall
t\geq\tau_{1}(\varepsilon).$ (57)

Combining (56) and (57) gives

$s_{i}(t)-s_{i}^{\star}(t)=O(\varepsilon)+O(\tau_{1}(\varepsilon)),~{}\forall
t\geq 0.$ (58)

Since $y_{i}$ is uniquely decided by $s_{i}$, (58) indicates (53). Therefore
the statement in the third bullet holds, and this completes the proof of
Theorem 2. $\blacksquare$
Remark 2. The first bullet in Theorem 2 shows the convergence of the ESO. For
sufficiently small $\varepsilon$, the ESO estimation error and the transient
period $\tau_{1}(\varepsilon)$ can be both made arbitrarily small. The second
bullet shows that the proposed ESO-based protocol enables practical output
consensus for the multi-agent system (1). Also note that similar to Li et al.
(2011), the convergence rate of the practical consensus in this paper is also
decided by the scaling function $\beta(kT)$. Since
$\|\delta(kT)\|<\beta(kT)\overline{\alpha}$ and
$\beta(kT)=\max\\{\beta_{0}\gamma^{k},\sqrt{\varepsilon}\\}$, the practical
convergence needs at most $k^{*}$ steps. The third bullet shows the
performance recovery property of the proposed protocol as the trajectories of
$y_{i}$, $1\leq i\leq N$, can be made arbitrarily close to $y_{i}^{\star}$,
which are generated by the full-information-based protocol. Unlike the
existing consensus protocol for multi-agent systems which gives importance
mainly to the steady-state consensus error, the proposed protocol also shapes
the transient performance. Also note that $y_{i}^{\star}$ can be regarded as
the output of the linear system (18) with a full-state-feedback control
protocol $u_{i}^{\star}$. Thus the proposed ESO-based protocol is capable of
driving the trajectories of the outputs of a higher-order multi-agent system
with unknown nonlinear dynamics, external disturbances, and partially
measurable states, arbitrarily close to the outputs of its linear counterpart
with fully measurable states.
Remark 3. In this section, since the external disturbance $\omega_{i}$ and
functions $f_{im}(\cdot)$, $0\leq m\leq r$, are unknown, and each agent only
has access to its first state variable $x_{i1}$, an ESO is employed to
estimate these information. Then the quantizer uses the ESO estimated
information, and consequently the ESO estimation error will be injected into
the quantization process (see (4.2)-(30)). In this case, the scaling function
cannot decrease to 0. If $\beta(kT)\rightarrow 0$ as in (14), the ESO
estimation error will be enlarged to infinity during the quantization process,
which makes the quantizer saturated. To handle this problem, in Theorem 2, we
let the scaling function converge to $\sqrt{\varepsilon}$. Recall that the ESO
estimation error is of the order of $O(\varepsilon)$, letting
$\beta(kT)\rightarrow\sqrt{\varepsilon}$ guarantees that the ESO estimation
error will not make the quantizer saturated.

From Theorem 2, it can be observed that the number of the quantization levels,
$2K+1$, increases when the number of the agents $N$ increases. In particular,
$\lim_{N\rightarrow\infty}(2K+1)=\infty$. However, for a practical digital
network, the number of the quantization levels is limited. Our next theorem
shows that no matter how many agents there are, the proposed ESO-based
protocol guarantees consensus with a fixed number of quantization levels. The
lowest number of the quantization levels we achieve is 3 (i.e., $K=1$, a one-
bit quantizer).
Theorem 3. _Consider the closed-loop system composed of the multi-agent system
( 1), the ESO (20), the encoder (23), the decoder (24), and the protocol (25).
Suppose Assumptions A1 to A3 are satisfied, and the initial conditions of the
agents $(\varrho_{i}(0),z_{i}(0))\in\mathcal{X}\times\mathcal{Z}_{i}$, $1\leq
i\leq N$. For any given $K\geq 1$, let the scaling function $\beta(kT)$ be
selected according to (32), and_

$\displaystyle T\in$
$\displaystyle(0,\min\\{2/(\lambda_{2}(\mathcal{L})+\lambda_{N}(\mathcal{L})),T_{m}\\}),$
(59) $\displaystyle\gamma=$ $\displaystyle
1-(1-\epsilon_{0})T\lambda_{2}(\mathcal{L}),$ (60)

_where $\epsilon_{0}\in(0,1)$ and_

$\displaystyle T_{m}=$ $\displaystyle
2K\epsilon_{0}\lambda_{2}(\mathcal{L})\left[\sqrt{N}\lambda^{2}_{N}(\mathcal{L})+2\epsilon_{0}\lambda_{2}(\mathcal{L})d^{*}\right.$
$\displaystyle+\left.(2K+1)(1-\epsilon_{0})\epsilon_{0}\lambda_{2}^{2}(\mathcal{L})\right]^{-1}.$

_Then the three statements in Theorem 2 hold._
Proof. Note that (59) and (60) are sufficient conditions for (11) and (12),
respectively. What is more, it can be verified from the definition of $T_{m}$
that

$\frac{1}{2}<K_{1}(T,\gamma)<K+\frac{1}{2},$ (61)

which indicates that the condition (13) is also satisfied. Therefore,
following a similar line of the proof of Theorem 2, one can conclude that the
statements in Theorem 3 hold. $\blacksquare$
Remark 4. Theorem 3 shows that for a connected undirected network of $r$th-
order uncertain nonlinear agents with partially measurable states, no matter
how many agents there are, one can always design an ESO-based protocol to
guarantee consensus with only one bit information exchange between each pair
of adjacent agents at each time step. From a theoretical perspective, the
proposed approach achieves the lowest data rate via output feedback. For
comparison, in the literature, first-order integrator multi-agent systems
require at least one-bit data rate (Li et al., 2011); second-order integrator
multi-agent systems require at least two-bit data rate (Li & Xie, 2012);
$r$th-order integrator multi-agent systems require at least $r$-bit data rate
(Qiu et al., 2017a); and $2r$th-order multi-agent systems with oscillator
dynamics require at least $r$ to $2r$-bit data rate (Qiu et al., 2017b).

Finally, we provide some explanations on the parameters selection of the
proposed ESO-based protocol to end this section.
Remark 5. The saturation bounds $M_{m}$ satisfying $M_{m}\geq\sup_{1\leq i\leq
N}|\varrho^{\star}_{im}|$, $1\leq m\leq r+1$, are selected such that the
saturations will not be invoked in the steady state of the observer. The
values of $M_{1}$ to $M_{r}$ can be obtained by simulating the linear system
(18) with initial condition $\varrho^{\star}_{i}(0)\in\mathcal{X}$ and the
control protocol $u_{i}^{\star}(t)$. The calculation of $M_{r+1}$ might end up
with a conservative value since $F_{i}(\cdot)$ is unknown. The parameter
$C_{s}$ satisfying $C_{s}\geq\max_{\varrho_{i}\in\mathcal{X},1\leq i\leq
N}|s_{i}|$ is chosen to prevent the saturation of the quantizer in the initial
stage (Li et al., 2011).
Remark 6. The parameter $\varepsilon$ is important for the successful
implementation of the proposed ESO-based protocol. From (38) and the fact that
$\|\eta_{i}(0)\|\propto\frac{1}{\varepsilon^{r}}$, one has

$\displaystyle\|\eta_{i}(t)\|\leq$
$\displaystyle\sqrt{\frac{\sigma_{i2}}{\sigma_{i1}}}\|\eta_{i}(0)\|\varepsilon^{r+1}+\frac{2\sigma_{i2}^{2}N_{1i}\varepsilon}{\sigma_{i1}}=O(\varepsilon),$
(62)

for all $t\geq t_{\varepsilon}$, where
$t_{\varepsilon}\triangleq-2\sigma_{i2}(r+1)\varepsilon\ln\varepsilon$. Since
$N_{1i}$ is unknown due to the unknown external disturbance and agent
dynamics, it is difficult to accurately establish the bound of the ESO
estimation error. Therefore, inherent from the previous ESO results (Khalil,
2017), the calculation of the upper bound of $\varepsilon$ (i.e.,
$\varepsilon^{{\ddagger}}$) is not straightforward. What is more, for
practical control systems, the lower bound of $\varepsilon$ is also limited by
some implementation issues, such as noises. In practice, the value of
$\varepsilon$ can be selected by a simple trial and error procedure. Our
experiences and many ESO application examples in the literature (see Chen,
Yang, Guo, and Li (2016) and Sariyildiz, Oboe, and Ohnishi (2020), and
references therein) indicate that it is generally not difficult to select a
satisfactory value of $\varepsilon$.

##  5 Example

Figure 1: The digital communication network.

In this section, an application example with five pendulum systems is
investigated to illustrate the effectiveness of the proposed design scheme.
Each pendulum with the inclusion of motor dynamics is described by (Kwan,
1995)

$\left\\{\begin{aligned} \dot{x}_{i1}=&x_{i2},\\\
\dot{x}_{i2}=&x_{i3}-p_{i}\sin(x_{i1})-q_{i}\cos(x_{i1}),\\\
\dot{x}_{i3}=&-x_{i3}+u_{i}+\omega_{i},~{}1\leq i\leq 5,\end{aligned}\right.$
(63)

where $x_{i1}$ and $x_{i2}$ denote the angular position and rate of the $i$th
pendulum, respectively; $x_{i3}$ denotes the motor shaft angle; $u_{i}$ is the
input motor torque; $p_{i}$ and $q_{i}$ are system parameters; and
$\omega_{i}$ is the external disturbance. By defining $\varrho_{i1}=x_{i1}$,
$\varrho_{i2}=x_{i2}$, and
$\varrho_{i3}=x_{i3}-p_{i}\sin(x_{i1})-q_{i}\cos(x_{i1})$, the pendulum
systems can be rewritten as

$\left\\{\begin{aligned} \dot{\varrho}_{i1}=&\varrho_{i2},\\\
\dot{\varrho}_{i2}=&\varrho_{i3},\\\
\dot{\varrho}_{i3}=&-\varrho_{i3}-p_{i}\sin(\varrho_{i1})-q_{i}\cos(\varrho_{i1})-p_{i}\varrho_{i2}\cos(\varrho_{i1})\\\
&+q_{i}\varrho_{i2}\sin(\varrho_{i1})+\omega_{i}+u_{i},~{}1\leq i\leq
5.\end{aligned}\right.$

To make the pendulum systems heterogeneous, we let $p_{i}=10+i$ and
$q_{i}=2+0.2i$, $1\leq i\leq 5$. The external disturbances are numerically
taken as $\omega_{i}=\sin(2t)$.

(a) $\varepsilon=0.1$.

(b) $\varepsilon=0.05$.

(c) $\varepsilon=0.01$.

(d) $\varepsilon=0.001$.

Figure 2: Simulation results of the ESO-based protocol with different values
of $\varepsilon$. Figure 3: Response of the ESO for agent 1 with
$\varepsilon=0.01$.

Fig. 1 depicts the communication network among the five pendulums. Let
$k_{1}=k_{2}=4$. The gains $L_{i}$, $1\leq i\leq 5$, for the observers are set
as $L_{i}=[4~{}6~{}4~{}1]^{\rm{T}}$, which places the eigenvalues of the
matrix $E_{i}$ at $-1$. The initial conditions of the pendulum systems
$x_{im}(0)$, $1\leq i\leq 5$, $1\leq m\leq 3$, are assumed to be randomly
distributed in $[-4.5,4.5]$, while the initial conditions of the observers are
set as 0. The bounds are set as $M_{1}=5$, $M_{2}=5$, $M_{3}=15$, $M_{4}=25$,
and $C_{s}=40$.

First, we simulate the higher quantization level case. Select $T=0.05$,
$\gamma=0.93$, $K=10$, and $\beta_{0}=10$, which satisfy all the conditions in
Theorem 2. Fig. 2 shows the response of $y_{i}$, $1\leq i\leq 5$, with
different values of $\varepsilon$. One can see that the five agents achieve
practical output consensus, and smaller value of $\varepsilon$ leads to
smaller steady-state consensus error. Fig. 3 depicts the performance of the
ESO for agent 1 with $\varepsilon=0.01$. It can be observed that both the
agent state $\varrho_{1}$ and the unknown nonlinear dynamics
$\varrho_{14}\triangleq-\varrho_{13}-p_{1}\sin(\varrho_{11})-q_{1}\cos(\varrho_{11})-p_{1}\varrho_{12}\cos(\varrho_{11})+q_{1}\varrho_{12}\sin(\varrho_{11})+\omega_{1}$
are well-estimated by the ESO. Also note that the observer exhibits peaking
phenomenon during the short transient period.

Then, we investigate the performance of the proposed protocol with one-bit
quantizers (i.e., $K=1$). Let $\epsilon_{0}=0.5$, and by (59), the sampling
period needs to satisfy $0<T<0.0233$. According to Theorem 3, we select
$T=0.015$, $\gamma=0.9881$, and $\beta_{0}=30$. The trajectories of $y_{i}$,
$1\leq i\leq 5$, with $\varepsilon=0.01$, are shown in Fig. 4, which
illustrates the practical output consensus of the five agents.

Figure 4: Simulation results of the ESO-based protocol with one-bit
quantizers.

##  6 Conclusion

In this paper, we explored the output feedback quantized consensus problem for
uncertain nonlinear heterogeneous multi-agent systems. An ESO-based protocol
with dynamic encoding and decoding was proposed. The proposed protocol shapes
the transient consensus performance, and guarantees a steady-state consensus
error that can be made arbitrarily small by tuning the observer parameter. It
was also shown that, for a connected undirected network, merely one bit
information exchange between each pair of adjacent agents at each time step
suffices to guarantee consensus. The approach developed in this paper provides
a practical solution with some intriguing properties to the consensus problem
of multi-agent systems with limited data rate. Future research works will be
directed at the communication-constrained control problems for multi-agent
systems in the presence of network attacks (Feng et al., 2020).

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