# Data-driven Polytopic Output Synchronization of Heterogeneous
Multi-agent Systems from Noisy Data

Yifei Li, Wenjie Liu, Jian Sun, , Gang Wang, ,
Lihua Xie, , and Jie Chen The work was supported in part by the National Key
R&D Program of China under Grant 2022ZD0119302, and the National Natural
Science Foundation of China under Grants 62173034, 61925303, 62088101,
U20B2073. _(Corresponding author: Gang Wang.)_ Yifei Li, Wenjie Liu, Jian Sun,
and Gang Wang are with the National Key Lab of Autonomous Intelligent Unmanned
Systems, Beijing Institute of Technology, Beijing 100081, China, and also with
the Beijing Institute of Technology Chongqing Innovation Center, Chongqing
401120, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>gangwang@bit.edu.cn).Lihua Xie is with the Centre for
Advanced Robotics Technology Innovation (CARTIN), School of Electrical and
Electronic Engineering, Nanyang Technological University, Singapore (e-mail:
elhxie@ntu.edu.sg). Jie Chen is with the Department of Control Science and
Engineering, Tongji University, Shanghai 201804, China, and also with the
National Key Lab of Autonomous Intelligent Unmanned Systems, Beijing Institute
of Technology, Beijing 100081, China (e-mail: chenjie@bit.edu.cn).

###### Abstract

This paper proposes a novel approach to addressing the output synchronization
problem in unknown heterogeneous multi-agent systems (MASs) using noisy data.
Unlike existing studies that focus on noiseless data, we introduce a
distributed data-driven controller that enables all heterogeneous followers to
synchronize with a leader’s trajectory. To handle the noise in the state-
input-output data, we develop a data-based polytopic representation for the
MAS. We tackle the issue of infeasibility in the set of output regulator
equations caused by the noise by seeking approximate solutions via constrained
fitting error minimization. This method utilizes measured data and a noise-
matrix polytope to ensure near-optimal output synchronization. Stability
conditions in the form of data-dependent semidefinite programs are derived,
providing stabilizing controller gains for each follower. The proposed
distributed data-driven control protocol achieves near-optimal output
synchronization by ensuring the convergence of the tracking error to a bounded
polytope, with the polytope size positively correlated with the noise bound.
Numerical tests validate the practical merits of the proposed data-driven
design and theory.

###### Index Terms:

Data-driven control, heterogeneous MAS, output synchronization, noisy data,
polytope

##  I Introduction

The field of distributed control, particularly consensus control of multi-
agent systems (MASs), has garnered significant attention in recent decades.
Numerous research efforts have focused on achieving state consensus in
homogeneous MASs, where all agents share identical dynamics, as evidenced in
e.g., [1, 2, 3, 4, 5, 6] and their associated references. However, in real-
world scenarios, the presence of inevitable variations between agents and
system uncertainties stemming from physical characteristics introduce
heterogeneity in the dynamics of MASs. Consequently, there is a growing need
to investigate the problem of output synchronization in heterogeneous MASs,
which consist of agents with different dimensions and dynamics and find
extensive applications.

A well-documented approach for addressing the output synchronization problem
is based on the internal model principle [7]. This principle has been widely
applied to design protocols for output synchronization in various settings,
including heterogeneous linear MASs [8, 9, 10], nonlinear MASs [11], and their
generalizations with additional considerations [12, 13, 14].

However, these protocols rely on knowledge of the system dynamics model of
each agent, rendering them inapplicable when first-principle models are
unavailable or system identification is time-consuming or inaccurate. To
overcome this challenge, several results have explored model-free approaches
based on reinforcement learning (RL) [15, 16, 17]. Nevertheless, model-free
RL-based approaches often require a large amount of training data, demanding
considerable computing resources. Alternatively, the fundamental lemma
introduced by _Willems et al._ in [18] provides an alternative avenue for
designing controllers for unknown systems by using pre-collected data, such as
input, output, and/or state data. This line of research has gained significant
attention due to its advantages in terms of theoretical certification and
computational tractability compared to other data-driven approaches [19, 20,
21]. Moreover, a data-based representation for linear time-invariant systems
has been proposed in [22], which enables the design of stabilizing controllers
using data-dependent linear matrix inequalities.

The fundamental lemma has lately found applications in diverse areas of data-
driven control design and analysis, including data-driven model predictive
control [23, 24, 25], data-enabled policy optimization [26], robust control
[27, 28], event-triggered control [29, 30, 31, 32], and distributed control of
network systems [33, 34, 35], among others.

The problem of data-driven control design and analysis for output
synchronization in unknown heterogeneous MASs remains unexplored. While some
progress has been made, such as the work presented in [34] where the process
noise during offline data collection was assumed to be measurable and
perfectly known, the general case of having unknown noise has not been
addressed. This limitation arises from the fact that process noise cannot be
accurately measured in practical scenarios. Consequently, there is a pressing
need to revisit the data-driven output synchronization problem in
heterogeneous MASs, considering the presence of unknown noise. Moreover,
accurate system identification is hindered by noisy data, necessitating the
development of robust data-based methods for controller design that can handle
uncertainty. The fundamental premise of such methods is to impose reasonable
constraints on the noise.

In the field of data-driven control, two primary approaches are commonly used
to model noise in the data acquisition phase, namely zonotopic constraints and
quadratic constraints. The former approach, utilizing data-driven zonotopic
reachability analysis [36], has led to the development of robust data-driven
predictive controllers as demonstrated in [37, 38]. On the other hand, the
latter approach, introduced as a general framework in [27] leveraging the
matrix _S_ -lemma, has been widely employed in designing controllers from data
subject to noise modeled by quadratic constraints. Notable applications
include robust event-triggered control [29], and distributed control [35].
Furthermore, the feasibility of the output regulator equations, a crucial
component for achieving output synchronization using the internal model
principle, is compromised by noisy data. To the best of our knowledge, no
data-driven methods have been reported for solving the output regulator
equations from noisy data.

Motivated by these observations, this paper aims to develop data-driven
polytopic controllers for output synchronization of unknown heterogeneous MASs
using noisy data obtained offline. The first step involves describing the
noise using polytopes, which serves as the basis for a novel data-based
polytopic representation of MASs. To address the infeasibility issue of output
regulator equations, we seek approximate solutions by tackling a norm
minimization problem that incorporates noise using matrix polytopes. Building
on the polytopic MAS representation, we derive sufficient conditions for
stabilizing feedback controllers in the form of data-dependent semidefinite
programs. Further, we consider a new synchronization measure, termed
$\Delta$-optimal output synchronization, which accounts for regulator equation
errors in the presence of noisy data and where $\Delta$ is proportional to the
size of noise. We demonstrate that the proposed data-driven control protocol
achieves $\Delta$-optimal output synchronization and exhibits robustness
against the noise in data collected offline. This is achieved by ensuring the
convergence of tracking error to a $\Delta$-bounded polytope. Notably, the
synchronization recovers exact output synchronization when the noise vanishes.

The contributions of this paper are summarized as follows:

  1. c1)

We propose a data-based polytopic representation for heterogeneous MASs based
on state-input-output data corrupted by bounded noise;

  2. c2)

We derive approximate solutions to the data-driven output synchronization
problem by minimizing the norm of fitting error matrix polytopes; and,

  3. c3)

We establish the near-optimality and robustness of the data-driven polytopic
design of distributed output synchronization controllers against noise.

_Notation._ We adopt the following notation conventions throughout the paper.
The set of non-negative integers (real numbers) is denoted by $\mathbb{N}$
($\mathbb{R}$). The sets of $n$-dimensional real vectors and $n\times m$ real
matrices are represented by $\mathbb{R}^{n}$ and $\mathbb{R}^{n\times m}$,
respectively. For a vector $x\in\mathbb{R}^{n}$, the notation $x>0$ indicates
that each entry of $x$ is positive. The symbol $(\cdot)^{\top}$ denotes the
transpose operation, while $\otimes$ represents the Kronecker product. The
identity matrix of appropriate dimensions is denoted by $I$, and the zero
matrix is denoted by $\mathbf{0}$. The Frobenius norm of a real matrix
$X\in\mathbb{R}^{n\times m}$ is denoted by $\|X\|_{F}$. A symmetric matrix $P$
is said to be positive definite (positive semi-definite) if $P\succ\mathbf{0}$
($P\succeq\mathbf{0}$). The expression ${\rm{diag}}_{i=1}^{N}\\{a_{i}\\}$
represents a diagonal matrix with $a_{1},a_{2},\ldots,a_{N}$ as its main
diagonal elements.

##  II Preliminaries and Problem Formulation

###  II-A Graph Theory

Consider a weighted graph ${\mathcal{G}}=({\mathcal{V}},{\mathcal{E}})$ that
represents the interactions among a set of agents. The graph consists of two
components: a nonempty set of nodes ${\mathcal{V}}=\\{v_{1},\ldots,v_{N}\\}$
and a set of edges ${\mathcal{E}}\subseteq{\mathcal{V}}\times{\mathcal{V}}$.
An element in ${\mathcal{E}}$, denoted as $(v_{i},v_{j})$, represents a link
from node $v_{j}$ to node $v_{i}$.

The adjacency matrix $\mathcal{A}=[a_{ij}]\in\mathbb{R}^{N\times N}$ is
defined such that $a_{ij}>0$ if $(v_{j},v_{i})\in\mathcal{E}$, and $a_{ij}=0$
otherwise. The in-degree of node $v_{i}$ in graph $\mathcal{G}$ is given by
$d_{i}=\sum_{j=1}^{N}a_{ij}$, and can be represented by the diagonal matrix
$D={\rm{diag}}_{i=1}^{N}\\{d_{i}\\}$. The Laplacian matrix
$\mathcal{L}=[l_{ij}]\in\mathbb{R}^{N\times N}$ associated with $\mathcal{G}$
is defined as $\mathcal{L}=D-\mathcal{A}$.

The neighbor set of node $v_{i}$ is denoted as
$\mathcal{N}_{i}=\\{j\in{\mathcal{V}}|(i,j)\in{\mathcal{E}}\\}$. An extended
graph is represented by
$\bar{\mathcal{G}}=(\bar{\mathcal{V}},\bar{\mathcal{E}})$, where
$\bar{\mathcal{V}}=\mathcal{V}\cup v_{0}$, and $v_{0}$ corresponds to the node
associated with the leader. The set $\bar{\mathcal{E}}$ includes all the arcs
in ${\mathcal{E}}$ as well as the arcs between $v_{0}$ and ${\mathcal{E}}$.

A graph $\bar{\mathcal{G}}$ is said to contain a directed spanning tree if
there exists a node, known as the root, from which every other node in
$\bar{\mathcal{V}}$ can be reached through a directed path. The pinning matrix
$G={\rm{diag}}_{i=1}^{N}\\{g_{i}\\}$ describes the accessibility of the leader
node $v_{0}$ to the remaining nodes $v_{i}\in\mathcal{V}$. The pinning gain
$g_{i}>0$ if $(v_{0},v_{i})\in\bar{\mathcal{E}}$, and $g_{i}=0$ otherwise.

###  II-B Output Synchronization of Discrete-time MASs

Consider a discrete-time heterogeneous leader-following MAS consisting of a
leader indexed by $0$ and $N$ followers indexed by $1,2,\ldots,N$. The
dynamics of follower $i\in\\{1,2,\ldots,N\\}$ is described by

$\begin{split}{x}_{i}(t+1)&=\bar{A}_{i}x_{i}(t)+\bar{B}_{i}u_{i}(t)\\\
y_{i}(t)&=\bar{C}_{i}x_{i}(t),\quad\forall t\in\mathbb{N}\end{split}$ (1)

where $x_{i}(t)\in\mathbb{R}^{n_{i}}$ represents the state,
$u_{i}(t)\in\mathbb{R}^{p_{i}}$ denotes the control input, and
$y_{i}(t)\in\mathbb{R}^{q}$ is the measurement output.

In this paper, the true system matrices $\bar{A}_{i}\in\mathbb{R}^{n_{i}\times
n_{i}}$, $\bar{B}_{i}\in\mathbb{R}^{n_{i}\times p_{i}}$, and
$\bar{C}_{i}\in\mathbb{R}^{q\times n_{i}}$ are assumed unknown.

The dynamics of the leader is given by

$\begin{split}{x}_{0}(t+1)&={S}x_{0}(t)\\\ {y}_{0}(t)&={H}x_{0}(t)\end{split}$
(2)

where $x_{0}(t)\in\mathbb{R}^{n_{0}}$ and $y_{0}(t)\in\mathbb{R}^{q}$
represent the state and output of the leader, respectively. The leader’s
system matrices $S$ and $H$ are assumed real, constant, and known. The pair
$(S,H)$ is assumed to be observable.

The dynamics and state dimensions are allowed to differ across agents, while
the output dimensions must be identical for synchronization. The objective is
to synchronize the outputs of all followers with that of the leader by
implementing a distributed feedback control protocol for the MAS described by
(1)-(2), such that $\lim_{t\to\infty}\|y_{i}(t)-y_{0}(t)\|=0$ holds for all
$i\in\\{1,2,\ldots,N\\}$. To address this problem, the following assumptions
are made.

######  Assumption 1 (Communication topology).

The graph $\bar{\mathcal{G}}$ contains a directed spanning tree with the
leader node as the root.

######  Assumption 2 (Stabilizability and detectability).

The pair $(\bar{A}_{i},\bar{B}_{i})$ is stabilizable, and
$(\bar{C}_{i},\bar{A}_{i})$ is detectable for all $i\in\\{1,2,\ldots,N\\}$.

######  Assumption 3 (Oscillating leader dynamics).

The leader dynamics $S$ has all its poles on the unit circle and non-repeated.

Regarding the assumptions, we have a remark.

######  Remark 1.

Assumptions 1–3 are standard for achieving output synchronization in linear
heterogeneous MASs and have been utilized in several existing results, such as
[15, 34]. It is important to note that the state of the leader is bounded and
does not converge to zero under Assumption 3.

Based on these assumptions, we consider a distributed feedback control
protocol for each follower in (1) as follows

$u_{i}(t)=K_{i}[x_{i}(t)-\Pi_{i}\eta_{i}(t)]+\Gamma_{i}\eta_{i}(t)$ (3)

where $K_{i}\in\mathbb{R}^{p_{i}\times n_{i}}$ is the feedback gain matrix to
be designed, and $\Pi_{i}\in\mathbb{R}^{n_{i}\times n_{0}}$ and
$\Gamma_{i}\in\mathbb{R}^{p_{i}\times n_{0}}$ are the solutions to output
regulator equations given by

$\begin{split}\bar{A}_{i}\Pi_{i}+\bar{B}_{i}\Gamma_{i}&=\Pi_{i}S\\\
\bar{C}_{i}\Pi_{i}&=H.\end{split}$ (4)

An observer $\eta_{i}(t)\in\mathbb{R}^{n_{0}}$ is employed to estimate the
state of the leader, governed by the following distributed observer

$\begin{split}\eta_{i}(t&+1)=S\eta_{i}(t)+(1+d_{i}+g_{i})^{-1}F\\\
&\times\Big{[}\sum_{j=1}^{N}a_{ij}(\eta_{j}(t)-\eta_{i}(t))+g_{i}(x_{0}(t)-\eta_{i}(t))\Big{]}\end{split}$
(5)

where $d_{i}$ and $g_{i}$ are the in-degree and pinning gain of node $i$,
respectively, and $F\in\mathbb{R}^{n_{0}\times n_{0}}$ is a gain matrix to be
designed.

Next, we define the observer’s estimation error
$\delta_{i}(t):=\eta_{i}(t)-x_{0}(t)$. It follows from (2) and (5) that the
dynamics of $\delta_{i}(t)$ satisfies

$\begin{split}\delta_{i}(t+1)&={S}\delta_{i}(t)+(1+d_{i}+g_{i})^{-1}F\\\
&\quad\times\Big{[}\sum_{j=1}^{N}a_{ij}(\delta_{j}(t)-\delta_{i}(t))-g_{i}\delta_{i}(t)\Big{]}.\end{split}$
(6)

For the entire system, (6) can be expressed in a compact form as follows

$\delta(t+1)=\big{[}I_{N}\otimes S-(I_{N}+D+G)^{-1}(\mathcal{L}+G)\otimes
F\big{]}\delta(t)$ (7)

where
$\delta(t)=[\delta_{1}^{\top}(t)~{}\delta_{2}^{\top}(t)~{}\cdots~{}\delta_{N}^{\top}(t)]^{\top}$.

Next, we introduce the virtual tracking error
$\xi_{i}(t):=x_{i}(t)-\Pi_{i}\eta_{i}(t)$. By substituting (1), (3), and (5)
into the definition of $\xi_{i}(t)$, we obtain its dynamics as follows

$\xi_{i}(t+1)=(\bar{A}_{i}+\bar{B}_{i}K_{i})\xi_{i}(t)+(1+d_{i}+g_{i})^{-1}\Pi_{i}Fz_{i}(t)$
(8)

where
$z_{i}(t)=\sum_{j=1}^{N}a_{ij}(\delta_{i}(t)-\delta_{j}(t))+g_{i}\delta_{i}(t)$.

When the true system matrices $(\bar{A}_{i},\bar{B}_{i},\bar{C}_{i})$ are
known, the next lemma provides necessary and sufficient conditions for
achieving output synchronization, see e.g., [15, Theorem 1].

######  Lemma 1 ([15, Theorem 1]).

Suppose Assumptions 1-3 hold. The output synchronization of the MAS (1)-(2) is
achieved for all initial conditions under the distributed feedback control
strategy (3)-(5), if and only if there exist matrices $F$ and $K_{i}$ such
that $I_{N}\otimes S-(I_{N}+D+G)^{-1}(\mathcal{L}+G)\otimes F$ and
$\bar{A}_{i}+\bar{B}_{i}K_{i}$ are Schur stable for all
$i\in\\{1,2,\ldots,N\\}$.

###  II-C Pre-collecting Noisy Data

In order to address the challenge of unknown system matrices for each
follower, we propose a distributed data-driven approach. Specifically, we
excite each follower with some control inputs, obtaining a set of data denoted
by
$\mathbb{D}_{i}=\\{(x_{i}(T),u_{i}(T),y_{i}(T)):T\in\\{0,1,\ldots,\rho\\}\\}$
for each follower $i$. The set $\mathbb{D}_{i}$ consists of state, input, and
output measurements, and is obtained through an open-loop experiment on the
following perturbed system

$\begin{split}x_{i}(T+1)&=\bar{A}_{i}x_{i}(T)+\bar{B}_{i}u_{i}(T)+w_{i}(T)\\\
y_{i}(T)&=\bar{C}_{i}x_{i}(T)+v_{i}(T)\end{split}$ (9)

where $w_{i}(T)\in\mathbb{R}^{n_{i}}$ and $v_{i}(T)\in\mathbb{R}^{q}$
represent unknown process and measurement noise, respectively. These noises
satisfy the following assumption.

######  Assumption 4 (Polytopic noise).

For every $T$ and $i$, the process noise $w_{i}(T)$ and measurement noise
$v_{i}(T)$ belong respectively to polytopic sets $\mathcal{P}_{w_{i}}$ and
$\mathcal{P}_{v_{i}}$ given by

$\begin{split}\mathcal{P}_{w_{i}}&=\Big{\\{}w_{i}\big{|}w_{i}=\sum_{k=1}^{\gamma_{w_{i}}}\beta^{(k)}_{w,i}\hat{w}_{i}^{(k)},\beta^{(k)}_{w,i}\geq
0,\sum_{k=1}^{\gamma_{w_{i}}}\beta^{(k)}_{w,i}=1\Big{\\}}\\\
\mathcal{P}_{v_{i}}&=\Big{\\{}v_{i}\big{|}v_{i}=\sum_{k=1}^{\gamma_{v_{i}}}\beta^{(k)}_{v,i}\hat{v}_{i}^{(k)},\beta^{(k)}_{v,i}\geq
0,\sum_{k=1}^{\gamma_{v_{i}}}\beta^{(k)}_{v,i}=1\Big{\\}}\end{split}$

where $\hat{w}_{i}^{(k)}$ and $\hat{v}_{i}^{(k)}$ represent the $k$-th vertex
of polytopes $\mathcal{P}_{w_{i}}$ and $\mathcal{P}_{v_{i}}$, respectively,
and $\gamma_{w_{i}}$ and $\gamma_{v_{i}}$ denote the number of vertices.

To store the collected data, we define the following matrices per agent

$\displaystyle X_{i+}$
$\displaystyle:=\left[x_{i}(1)\;x_{i}(2)\;\cdots\;x_{i}(\rho)\right]$
$\displaystyle X_{i}$
$\displaystyle:=\left[x_{i}(0)\;x_{i}(1)\;\cdots\;x_{i}(\rho-1)\right]$
$\displaystyle U_{i}$
$\displaystyle:=\left[u_{i}(0)\;u_{i}(1)\;\cdots\;u_{i}(\rho-1)\right]$
$\displaystyle Y_{i}$
$\displaystyle:=\left[y_{i}(0)\;y_{i}(1)\;\cdots\;y_{i}(\rho-1)\right].$

The unknown process noise of length $\rho$ is denoted as
$\\{w_{i}(T)\\}_{T=0}^{\rho-1}$. Consequently, the stacked matrix per agent,
$W_{i}=[w_{i}(0)\;w_{i}(1)\;\cdots\;w_{i}(\rho-1)]$, belongs to the matrix
polytope $\mathcal{M}_{W_{i}}$, given by

$\mathcal{M}_{W_{i}}=\Big{\\{}W_{i}\Big{|}W_{i}=\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}\hat{W}_{i}^{(k)},\beta_{W,i}^{(k)}\geq
0,\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}=1\Big{\\}}$ (10)

which results from the concatenation of multiple disturbance polytopes
$\mathcal{P}_{w_{i}}$ as follows

$\begin{split}\hat{W}_{i}^{(1+(k-1)\rho)}&=\big{[}\hat{w}_{i}^{(k)}\quad\mathbf{0}_{n_{i}\times(\rho-1)}\big{]}\\\
\hat{W}_{i}^{(m+(k-1)\rho)}&=\big{[}\mathbf{0}_{n_{i}\times(m-1)}\quad\hat{w}_{i}^{(k)}\quad\mathbf{0}_{n_{i}\times(\rho-m)}\big{]}\\\
\hat{W}_{i}^{(\rho+(k-1)\rho)}&=\big{[}\mathbf{0}_{n_{i}\times(\rho-1)}\quad\hat{w}_{i}^{(k)}\big{]}\end{split}$
(11)

for each $k\in\\{1,2,\ldots,\gamma_{w_{i}}\\}$, $m\in\\{2,3,\ldots,\rho-1\\}$,
and $i\in\\{1,2,\ldots,N\\}$.

Similarly, upon denoting the sequence of measurement noise
$\\{v_{i}(T)\\}_{T=0}^{\rho-1}$ as
$V_{i}=[v_{i}(0)\;v_{i}(1)\;\cdots\;v_{i}(\rho-1)]$, we can deduce that
$V_{i}$ belongs to the following matrix polytope

$\mathcal{M}_{V_{i}}=\Big{\\{}V_{i}\Big{|}V_{i}=\sum_{k=1}^{\gamma_{v_{i}}\rho}\beta_{V,i}^{(k)}\hat{V}_{i}^{(k)},\beta_{V,i}^{(k)}\geq
0,\sum_{k=1}^{\gamma_{v_{i}}\rho}\beta_{V,i}^{(k)}=1\Big{\\}}$ (12)

with the vertices $\hat{V}_{i}^{(k)}$ defined in the same way as
$\hat{W}_{i}^{(k)}$ in (11).

###  II-D Problem Statement

Having introduced the output synchronization of discrete-time heterogeneous
MASs and the pre-collected data, we now formally state the problem of
interest.

######  Problem 1 (Data-driven output synchronization).

Given the state-input-output measurements $\mathbb{D}_{i}$ for each follower
$i$, and under Assumptions 1-3, the objective is to design a distributed
controller of the form (3)-(5) such that approximate output synchronization of
the heterogeneous MAS (1)-(2) is achieved for any initial states.

The main challenge in addressing Problem 1 lies in solving the output
regulator equations in (4), designing the controller gain $K_{i}$, and
performing synchronization analysis without knowledge of the true system
matrices, but using only available data. To tackle this challenge, we propose
a data-driven polytopic reachability analysis technique in this paper,
inspired by the zonotopic reachability analysis presented in [36].

##  III Distributed Data-driven Output Synchronization of Heterogeneous MASs

This section addresses the challenging problem of output synchronization in
the unknown heterogeneous MAS (1)-(2). Due to the presence of noisy data,
achieving asymptotic output synchronization for the unknown heterogeneous MAS
is impractical, unlike the model-based scenario depicted in Lemma 1. Instead,
we propose achieving $\Delta$-optimal output synchronization by ensuring the
stability of reachable error trajectories, which will be formalized in the
following sections.

Our approach begins by introducing a data-based polytopic representation that
characterizes an open-loop MAS using noisy data $\mathbb{D}_{i}$. Next, we
address the output regulator equations by solving a data-dependent norm
minimization problem. Subsequently, a polytopic controller is designed
directly from the data. Leveraging this controller and the approximate
solution to the output regulator equations, we propose a data-driven output
synchronization algorithm and provide a proof of its UBB property, effectively
addressing Problem 1. See Fig. 1 for an illustration of the leader-following
heterogeneous MAS architecture.

Figure 1: Distributed data-driven output synchronization.

###  III-A Data-based Polytopic Representation of MASs

In this part, we assume the next rank condition on the richness of the data in
$\mathbb{D}_{i}$.

######  Assumption 5.

The data matrix $\left[\begin{matrix}U_{i}\\\ X_{i}\end{matrix}\right]$ has
full row rank for each $i\in\\{1,2,\ldots,N\\}$.

The verification of Assumption 5 can be easily performed for a given dataset
$\mathbb{D}_{i}$. Note that this rank condition can be satisfied if the
measurements $(u_{i}(T),w_{i}(T))$ are persistently exciting of order
$n_{i}+1$ [22]. We now proceed with representing the heterogeneous MAS using
data.

Since the actual realization of noise $(w_{i}(T),v_{i}(T))$ is unknown, there
generally exist many systems $(A_{i},B_{i},C_{i})$ that are consistent with
the data $(X_{i},X_{i+},U_{i},Y_{i})$. We denote this set by $\Sigma_{i}$ as
follows:

$\begin{split}\Sigma_{i}:=\big{\\{}(A_{i}&,\,B_{i},\,C_{i})|X_{i+}=A_{i}X_{i}+B_{i}U_{i}+W_{i},\\\
&Y_{i}=C_{i}X_{i}+V_{i},W_{i}\in\mathcal{M}_{W_{i}},V_{i}\in\mathcal{M}_{V_{i}}\big{\\}}.\end{split}$
(13)

As mentioned in Sec. II-C, we assume knowledge of the polytopes
$\mathcal{P}_{w_{i}}$ and $\mathcal{P}_{v_{i}}$, which bound the noise
$w_{i}(T)$ and $v_{i}(T)$, along with their associated matrix polytopes
$\mathcal{M}_{W_{i}}$ and $\mathcal{M}_{V_{i}}$, respectively. Our objective
now is to compute a set $\mathcal{M}_{i}$ that provides an overapproximation
of all possible $(A_{i},B_{i},C_{i})$ consistent with the state-input-output
data and given noise bounds. To achieve this, we construct a data-based
polytopic representation of $(A_{i},B_{i},C_{i})$ in the following lemma,
inspired by [37]. This representation yields a matrix polytope
$\mathcal{M}_{i}\supseteq\Sigma_{i}$.

######  Lemma 2 (Data-based polytopic representation of MASs).

Suppose Assumptions 4-5 hold. For each $i\in\\{1,2,\ldots,N\\}$, given state-
input-output data $(X_{i},X_{i+},U_{i},Y_{i})$ of the MAS (9), the matrix
polytope is defined as:

$\begin{split}\mathcal{M}_{i}=\Big{\\{}(\mathcal{M}_{{Z}_{i}},\mathcal{M}_{{C}_{i}})\Big{|}\mathcal{M}_{{Z}_{i}}&=(X_{i+}-\mathcal{M}_{W_{i}})\left[\begin{matrix}U_{i}\\\
X_{i}\end{matrix}\right]^{\dagger},\\\
\mathcal{M}_{{C}_{i}}&=(Y_{i}-\mathcal{M}_{V_{i}})X_{i}^{\dagger}\Big{\\}}.\end{split}$
(14)

This matrix polytope characterizes all matrices $(A_{i},B_{i},C_{i})$
consistent with the data $(X_{i},X_{i+},U_{i},Y_{i})$ and noise bounds, i.e.,
$\Sigma_{i}\subseteq\mathcal{M}_{i}$.

###### Proof.

Consider any $(A_{i},B_{i},C_{i})\in\Sigma_{i}$. From (13), we can find
$W_{i}\in\mathcal{M}_{W_{i}}$ and $V_{i}\in\mathcal{M}_{V_{i}}$ such that:

$\begin{split}A_{i}X_{i}+B_{i}U_{i}&=X_{i+}-W_{i},\\\
C_{i}X_{i}&=Y_{i}-V_{i}.\end{split}$ (15)

By using (10), we can express every $W_{i}\in\mathcal{M}_{W_{i}}$ uniquely as
$W_{i}=\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}\hat{W}_{i}^{(k)}$ with
coefficients $\\{\beta_{W,i}^{(k)}\\}$. Furthermore, multiplying
$\left[\begin{matrix}U_{i}\\\ X_{i}\end{matrix}\right]^{\dagger}$ from the
right on both sides of the first equation in (15) yields:

$Z_{i}:=[B_{i}~{}A_{i}]=\Big{(}X_{i+}-\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}\hat{W}_{i}^{(k)}\Big{)}\left[\begin{matrix}U_{i}\\\
X_{i}\end{matrix}\right]^{\dagger}.$ (16)

Similarly, it can be easily deduced that:

$C_{i}=\Big{(}Y_{i}-\sum_{k=1}^{\gamma_{v_{i}}\rho}\beta_{V,i}^{(k)}\hat{V}_{i}^{(k)}\Big{)}X_{i}^{\dagger}.$
(17)

Hence, for every $(A_{i},B_{i},C_{i})\in\Sigma_{i}$, there exist
$\beta_{W,i}^{(k)}$, $k=1,2,\ldots,\gamma_{w_{i}}\rho$, and
$\beta_{V,i}^{(k)}$, $k=1,2,\ldots,\gamma_{v_{i}}\rho$, such that
$\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}=1$ and
$\sum_{k=1}^{\gamma_{v_{i}}\rho}\beta_{v,i}^{(k)}=1$, satisfying (16) and
(17). Therefore, for every $(A_{i},B_{i},C_{i})\in\Sigma_{i}$, it also holds
that $(A_{i},B_{i},C_{i})\in\mathcal{M}_{\Sigma_{i}}$ for $i=1,2,\ldots,N$ as
defined in (14), which concludes the proof. ∎

It is worth noting that a prevalent approach in prior literature for modeling
unknown, yet bounded noise is to use the energy form, typically in terms of a
quadratic full-block bound; see, e.g., [22, 24, 27, 35, 29]. In contrast, we
propose a novel approach by describing unknown noise using polytopes. We then
develop a data-based polytopic representation of MASs in Lemma 2, which allows
us to characterize all possible system matrices. This representation paves the
way for addressing Problem 1 in the subsequent analysis. Notably, compared
with the quadratic matrix inequality-based representation in [27, 35, 29], our
proposed polytopic representation maintains the simplicity and compactness,
while providing a more precise characterization of the system and resulting in
less conservative data-based stability conditions.

###  III-B Solution to Output Regulation Equations with Noisy Data

In this subsection, we introduce a data-driven approach to solve the output
regulator equations by minimizing the norm of noise-matrix polytopes. This
approach yields approximate solutions $(\Pi_{i},\Gamma_{i})$ for each follower
directly from noisy data, without requiring exact knowledge of the true system
matrices $(\bar{A}_{i},\bar{B}_{i},\bar{C}_{i})$.

We point out that the presence of $w_{i}(T)$ and $v_{i}(T)$ in data
$\mathbb{D}_{i}$ prevents us from obtaining accurate solutions to the initial
output regulator equations (4). Thus, we define $\Delta_{i1}$ and
$\Delta_{i2}$ as the errors in the regulator equations caused by the noise-
corrupted data. As an intermediate step, we modify the output regulator
equations (4) for any $(A_{i},B_{i},C_{i})\in\mathcal{M}_{i}$ as follows:

$\begin{split}\Delta_{i1}&=A_{i}\Pi_{i}+B_{i}\Gamma_{i}-\Pi_{i}S,\\\
\Delta_{i2}&=C_{i}\Pi_{i}-H.\end{split}$ (18)

Next, we make a crucial observation that directly finding the solutions
$\Pi_{i}$ and $\Gamma_{i}$ from (18) is infeasible due to the presence of
unknown terms $\Delta_{i1}$ and $\Delta_{i2}$. Accordingly, we formulate the
problem of determining the gains $\Pi_{i}$ and $\Gamma_{i}$ for any
$(A_{i},B_{i},C_{i})\in\mathcal{M}_{i}$ as an optimization problem that
minimizes $\Delta_{i1}$ and $\Delta_{i2}$ with respect to a chosen norm.
According to the data-based polytopic representation in Lemma 2, we define the
following optimization problem:

$\underset{\Pi_{i},\Gamma_{i}}{\min}~{}\Big{\|}\mathcal{M}_{Z_{i}}\left[\begin{matrix}\Gamma_{i}\\\
\Pi_{i}\end{matrix}\right]-\Pi_{i}S\Big{\|}_{F}+\|\mathcal{M}_{C_{i}}\Pi_{i}-H\|_{F}.$
(19)

Overall, Problem (19) is convex and can be efficiently solved using off-the-
shelf solvers. Let $(\Pi_{i}^{*},\Gamma_{i}^{*})$ denote any optimal solution
of (19) and $(\Delta_{i1}^{*},\Delta_{i2}^{*})$ denote the resulting error of
output regulation equations in (18).

The following result provides upper bounds on the unknown regulator equation
errors $\Delta_{i1}^{*}$ and $\Delta_{i2}^{*}$, which serves as a solid basis
for the subsequent analysis and design of data-driven output synchronization.
For simplicity, we omit the subscript $F$ in the sequel, using $\|\cdot\|$ for
$\|\cdot\|_{F}$.

######  Lemma 3 (Bounded regulator equation errors).

Consider the MAS (1)-(2) with the relaxed output regulator equations (18).
Suppose that Assumptions 1-5 hold. For any
$(A_{i},B_{i},C_{i})\in\mathcal{M}_{i}$ and $i\in\\{1,2,\ldots,N\\}$, there
exist two bounded matrix polytopes $\mathcal{M}_{\Delta_{i1}}$ and
$\mathcal{M}_{\Delta_{i2}}$ such that the regulator equation errors
$\Delta_{i1}\in\mathcal{M}_{\Delta_{i1}}$ and
$\Delta_{i2}\in\mathcal{M}_{\Delta_{i2}}$.

###### Proof.

It follows from the data-based polytopic representation in Lemma 2 that the
true system matrices $(\bar{A}_{i},\bar{B}_{i},\bar{C}_{i})\in\mathcal{M}_{i}$
can be expressed by a unique set of $\bar{\beta}_{W,i}^{(k)}$ and
$\bar{\beta}_{V,i}^{(k)}$, respectively, as follows:

$\begin{split}[\bar{B}_{i}\;\,\bar{A}_{i}]&=\Big{(}X_{i+}-\sum_{k=1}^{\gamma_{w_{i}}\rho}\bar{\beta}_{W,i}^{(k)}\hat{W}_{i}^{(k)}\Big{)}\left[\begin{matrix}U_{i}\\\
X_{i}\end{matrix}\right]^{\dagger}\\\
\bar{C}_{i}&=\Big{(}Y_{i}-\sum_{k=1}^{\gamma_{v_{i}}\rho}\bar{\beta}_{V,i}^{(k)}\hat{V}_{i}^{(k)}\Big{)}X_{i}^{\dagger}.\end{split}$
(20)

When $\bar{\beta}_{W,i}^{(k)}$ and $\bar{\beta}_{V,i}^{(k)}$ are known, we
denote the solution of (4) associated with
$(\bar{A}_{i},\bar{B}_{i},\bar{C}_{i})$ by $(\Pi_{i}^{s},\Gamma_{i}^{s})$,
which also serves as a candidate solution for the optimization problem (19).
Nonetheless, it may not necessarily yield the optimal objective value. By
utilizing (4), (14), and (20), it can be shown that the first term of (19)
satisfies

$\begin{split}&\quad~{}\Big{\|}{M}_{Z_{i}}\left[\begin{matrix}\Gamma_{i}^{s}\\\
\Pi_{i}^{s}\end{matrix}\right]-\Pi_{i}^{s}S\Big{\|}\\\
&=\Big{\|}(X_{i+}-\mathcal{M}_{W_{i}})\left[\begin{matrix}U_{i}\\\
X_{i}\end{matrix}\right]^{\dagger}\left[\begin{matrix}\Gamma_{i}^{s}\\\
\Pi_{i}^{s}\end{matrix}\right]-[\bar{B}_{i}\;\,\bar{A}_{i}]\left[\begin{matrix}\Gamma_{i}^{s}\\\
\Pi_{i}^{s}\end{matrix}\right]\Big{\|}\\\
&=\Big{\|}\sum_{k=1}^{\gamma_{w_{i}}\rho}\big{[}(\bar{\beta}_{W,i}^{(k)}-{\beta}_{W,i}^{(k)})\hat{W}_{i}^{(k)}\big{]}\left[\begin{matrix}U_{i}\\\
X_{i}\end{matrix}\right]^{\dagger}\left[\begin{matrix}\Gamma_{i}^{s}\\\
\Pi_{i}^{s}\end{matrix}\right]\Big{\|}\\\ &\leq
2\gamma_{w_{i}}\rho\bar{W}_{i}\Big{\|}\left[\begin{matrix}U_{i}\\\
X_{i}\end{matrix}\right]^{\dagger}\left[\begin{matrix}\Gamma_{i}^{s}\\\
\Pi_{i}^{s}\end{matrix}\right]\Big{\|}\end{split}$ (21)

with $\bar{W}_{i}:=\max_{k\in[1,\gamma_{w_{i}}\rho]}\|\hat{W}_{i}^{(k)}\|$.

Similarly, for the second term of (19), we have

$\begin{split}\mathcal{\|}{M}_{C_{i}}\Pi_{i}^{s}-H\|&=\Big{\|}\sum_{k=1}^{\gamma_{v_{i}}\rho}\big{[}(\bar{\beta}_{V,i}^{(k)}-{\beta}_{V,i}^{(k)})\hat{V}_{i}^{(k)}\big{]}X_{i}^{\dagger}\Pi_{i}^{s}\Big{\|}\\\
&\leq 2\gamma_{v_{i}}\rho\bar{V}_{i}\|X_{i}^{\dagger}\Pi_{i}^{s}\|\end{split}$
(22)

with $\bar{V}_{i}:=\max_{k\in[1,\gamma_{v_{i}}\rho]}\|\hat{V}_{i}^{(k)}\|$.

By substituting the optimal solution $(\Pi_{i}^{*},\Gamma_{i}^{*})$ of problem
(19) into (18), it follows from (21) and (22) that the regulator equation
errors $\Delta_{i1}^{*}$ and $\Delta_{i2}^{*}$ adhere to

$\displaystyle\Delta_{i1}^{*}$
$\displaystyle\in\\!\sum_{k=1}^{\gamma_{w_{i}}\rho}\big{[}(\bar{\beta}_{W,i}^{(k)}-{\beta}_{W,i}^{(k)})\hat{W}_{i}^{(k)}\big{]}\left[\begin{matrix}U_{i}\\\
X_{i}\end{matrix}\right]^{\dagger}\left[\begin{matrix}\Gamma_{i}^{s}\\\
\Pi_{i}^{s}\end{matrix}\right]\\!\triangleq\\!\mathcal{M}_{\Delta_{i1}}$ (23a)
$\displaystyle\Delta_{i2}^{*}$
$\displaystyle\in\\!\sum_{k=1}^{\gamma_{v_{i}}\rho}\big{[}(\bar{\beta}_{V,i}^{(k)}-{\beta}_{V,i}^{(k)})\hat{V}_{i}^{(k)}\big{]}X_{i}^{\dagger}\Pi_{i}^{s}\triangleq\mathcal{M}_{\Delta_{i2}}$
(23b)

where $\mathcal{M}_{\Delta{i1}}$ and $\mathcal{M}_{\Delta_{i2}}$ are bounded
polytopes, implying the boundedness of the regulator equation errors
$\Delta_{i1}^{*}$ and $\Delta_{i2}^{*}$. ∎

It is worth emphasizing that (19) provides a data-driven method to compute the
gain matrices $\Pi_{i}$ and $\Gamma_{i}$ independently of system matrices.
Furthermore, Lemma 3 ensures the practicability of the obtained $\Pi_{i}$ and
$\Gamma_{i}$ for the subsequent analysis and design of data-driven output
synchronization by constraining the regulator equation errors $\Delta_{i1}$
and $\Delta_{i2}$ within the bounded matrix polytopes
$\mathcal{M}_{\Delta_{i1}}$ and $\mathcal{M}_{\Delta_{i2}}$.

######  Remark 2 (Relationship between noise and output regulator equations).

In fact, problem (19) addresses a variant of the output regulation problem
that incorporates unknown noise. This implies that the solutions $\Pi_{i}$ and
$\Gamma_{i}$ of (19) satisfy the relaxed equations (18) when the noise
$w_{i}(T)$ and $v_{i}(T)$ is not identically equal to zero. According to (23),
the size of the matrix polytopes $\mathcal{M}_{\Delta_{i1}}$ and
$\mathcal{M}_{\Delta_{i2}}$ depends on the upper bound of noise, indicating
that $\Delta_{i1}$ and $\Delta_{i2}$ increase with the noise levels $w_{i}(T)$
and $v_{i}(T)$. However, in the noise-free case where $w_{i}(T)=0$ and
$v_{i}(T)=0$, the solution of (19) achieves zero cost (i.e., $\Delta_{i1}=0$
and $\Delta_{i2}=0$) and satisfies (4).

###  III-C Controller Design and Output Synchronization Analysis

In the following, we focus on Problem 1, which involves learning stabilizing
controllers $K_{i}$ for each follower and analyzing the output synchronization
of the MAS (1)-(2) under the proposed distributed control protocol ((3), (5),
and (18)). To begin with, we reconstruct the dynamics of the virtual tracking
error $\xi_{i}(t)$ as follows:

$\begin{split}\xi_{i}(t+1)&=(\bar{A}_{i}+\bar{B}_{i}K_{i})\xi_{i}(t)+\tilde{\delta}_{i}(t)\end{split}$
(24)

where we utilize (1), (3), (5), (18), and define
$\tilde{\delta}_{i}(t):=\Delta_{i1}(x_{0}(t)+\delta_{i}(t))+\Pi_{i}(1+d_{i}+g_{i})^{-1}Fz_{i}(t)$.

We observe that $\xi_{i}(t)$ belongs to a well-defined polytope
$\mathcal{P}_{\xi_{i},t}$, i.e., $\xi_{i}(t)\in\mathcal{P}_{\xi_{i},t}$ for
$t\in\mathbb{N}$ and $i\in\\{1,2,\ldots,N\\}$. The next lemma provides the
definition and boundedness guarantee of the polytope
$\mathcal{P}_{\xi_{i},t}$.

######  Lemma 4 (Virtual tracking error polytope).

Under Assumptions 1-5, let $\mathcal{P}_{\xi_{i},0}=\xi_{i}(0)$. At time
$t\in\mathbb{N}$, the reachable set of the virtual tracking error is given by:

$\mathcal{P}_{\xi_{i},t}:=(\bar{A}_{i}+\bar{B}_{i}K_{i})^{t}\xi_{i}(0)+\sum_{\ell=0}^{t-1}(\bar{A}_{i}+\bar{B}_{i}K_{i})^{\ell}\mathcal{P}_{\tilde{\delta}_{i},t-\ell-1}$
(25)

where
$\mathcal{P}_{\tilde{\delta}_{i},t}:=\Delta_{i1}^{*}\mathcal{P}_{x_{0}}+\Delta_{i1}^{*}\delta_{i}(t)+\Pi_{i}(1+d_{i}+g_{i})^{-1}Fz_{i}(t)$,
and $\mathcal{P}_{x_{0}}$ represents a well-defined polytope. Moreover, if
there exist matrices $K_{i}$ and $F$ such that $\bar{A}_{i}+\bar{B}_{i}K_{i}$
and $I_{N}\otimes S-(I_{N}+D+G)^{-1}(\mathcal{L}+G)\otimes F$ are Schur
stable, then $\mathcal{P}_{\xi_{i},t}$ is a uniformly bounded set for any
$t\in\mathbb{N}$ and $i\in\\{1,2,\ldots,N\\}$.

###### Proof.

First, we invoke Assumption 3 to guarantee the boundedness of the leader’s
state $x_{0}(t)$ for any $t\in\mathbb{N}$. By constructing a well-defined
polytope $\mathcal{P}_{x_{0}}$, we ensure that
$x_{0}(t)\in\mathcal{P}_{x_{0}}$ holds for all $t\in\mathbb{N}$. Moreover,
considering the dynamics of the observer estimation error $\delta_{i}(t)$ in
(7), we observe that if $I_{N}\otimes S-(I_{N}+D+G)^{-1}(\mathcal{L}+G)\otimes
F$ is Schur stable, the observer convergence is assured, i.e.,
$\lim_{t\to\infty}\delta_{i}(t)=0$. As a result, $\delta_{i}(t)$ and,
consequently, $z_{i}(t)$ defined in (8), are uniformly bounded. Building upon
these findings, we deduce from (24) that $\tilde{\delta}_{i}(t)$ forms a
uniformly bounded sequence. Thus, there exists a bounded polytope
$\mathcal{P}_{\tilde{\delta}_{i},t}$ defined as
$\mathcal{P}_{\tilde{\delta}_{i},t}:=\Delta_{i1}^{*}\mathcal{P}_{x_{0}}+\Delta_{i1}^{*}\delta_{i}(t)+\Pi_{i}(1+d_{i}+g_{i})^{-1}Fz_{i}(t)$
such that $\tilde{\delta}_{i}(t)\in\mathcal{P}_{\tilde{\delta}_{i},t}$ holds
for $t\in\mathbb{N}$.

Furthermore, due to the Schur stability of $\bar{A}_{i}+\bar{B}_{i}K_{i}$ and
the boundedness of $\delta_{i}(0)$, there exists a uniformly bounded set,
defined as in (25), denoted by $\mathcal{P}_{\xi_{i},t}$. This ensures that
$\xi_{i}(t)\in\mathcal{P}_{\xi_{i},t}$ for any $t\in\mathbb{N}$ and
$i\in\\{1,2,\ldots,N\\}$, thereby completing the proof. ∎

######  Remark 3 (Design a stabilizing matrix $F$).

We would like to emphasize that there exist techniques for computing a
stabilizing matrix $F$ while ensuring the condition that $I_{N}\otimes S$
$-(I_{N}+D+G)^{-1}(\mathcal{L}+G)\otimes F$ is Schur stable. An effective
approach is to solve discrete-time Riccati equations, as described in [39],
which provides a matrix $F$ satisfying the desired stability condition.

At this stage, two challenges need to be addressed. First, the polytope
$\mathcal{P}_{\xi_{i},t}$ derived in Lemma 4 cannot be directly applied in
practical scenarios due to the unknown true system matrices $\bar{A}_{i}$ and
$\bar{B}_{i}$. Second, it is crucial to ensure the stability of
$\bar{A}_{i}+\bar{B}_{i}K_{i}$. To tackle the former, we aim to construct a
conservative approximation $\bar{\mathcal{P}}_{\xi_{i},t}$ of
$\mathcal{P}_{\xi_{i},t}$ such that
$\mathcal{P}_{\xi_{i},t}\subseteq\bar{\mathcal{P}}_{\xi_{i},t}$. The latter
issue can be addressed by designing a controller gain $K_{i}$ that stabilizes
all $(A_{i},B_{i})$ in $\Sigma_{i}$.

As previously mentioned, we proceed to construct an approximation
$\bar{\mathcal{P}}_{\xi_{i},t}$ of agent $i$. First, we observe from Lemma 4
that

$\begin{split}\mathcal{P}_{\xi_{i},t}&=(\bar{A}_{i}+\bar{B}_{i}K_{i})\mathcal{P}_{\xi_{i},t-1}+\mathcal{P}_{\tilde{\delta}_{i},t}.\end{split}$
(26)

To this end, we define the matrix polytope as

$\mathcal{M}_{Z_{i}}^{K}:=\mathcal{M}_{Z_{i}}\left[\begin{matrix}K_{i}\\\
I\end{matrix}\right].$ (27)

The approximation $\bar{\mathcal{P}}_{\xi_{i},t}$ is obtained through the
following lemma.

######  Lemma 5 (Virtual tracking error polytope approximation).

For $i\in\\{1,2,\ldots,N\\}$, let $\bar{\mathcal{P}}_{\xi_{i},t}$ be defined
as

$\begin{split}\bar{\mathcal{P}}_{\xi_{i},t}&:=\mathcal{M}_{Z_{i}}^{K}\bar{\mathcal{P}}_{\xi_{i},t-1}+\mathcal{P}_{\tilde{\delta}_{i},t}\end{split}$
(28)

with $\bar{\mathcal{P}}_{\xi_{i},0}=\mathcal{P}_{\xi_{i},0}$. Then, it holds
that $\mathcal{P}_{\xi_{i},t}\subseteq\bar{\mathcal{P}}_{\xi_{i},t}$ for any
$t\in\mathbb{N}$.

###### Proof.

The validity of (28) for $t=0$ is straightforward. For a fixed $t>0$,
leveraging the induction step, we have
$\mathcal{P}_{\xi_{i},t-1}\subseteq\bar{\mathcal{P}}_{\xi_{i},t-1}$ and
$\bar{A}_{i}+\bar{B}_{i}K_{i}=Z_{i}\left[\begin{matrix}K_{i}\\\
I\end{matrix}\right]\in\mathcal{M}_{Z_{i}}\left[\begin{matrix}K_{i}\\\
I\end{matrix}\right]$. Consequently, we obtain
$(\bar{A}_{i}+\bar{B}_{i}K_{i})\mathcal{P}_{\xi_{i},t-1}\subseteq\mathcal{M}_{Z_{i}}^{K}\bar{\mathcal{P}}_{\xi_{i},t-1}$,
which completes the proof.

∎

Drawing on the aforementioned lemma, we can build on the result in [38] and
establish the following lemma to ensure the stability of the virtual tracking
error polytope $\bar{\mathcal{P}}_{\xi_{i},t}$.

######  Lemma 6 (Stability of the virtual tracking error polytope).

Given Assumptions 1-5 and an initial value $\xi_{i}(0)$ for agent
$i\in\\{1,2,\ldots,N\\}$, there exists a polytope
$\bar{\mathcal{P}}_{i}\subset\mathbb{R}^{n_{i}}$ satisfying the following
properties: i) $\bar{\mathcal{P}}_{\xi_{i},t}\subset\bar{\mathcal{P}}_{i}$ for
all $t\in\mathbb{N}$; and ii) $\bar{\mathcal{P}}_{i}$ is an invariant set,
i.e., for $\xi_{i}(t)\in\bar{\mathcal{P}}_{i}$, it holds that
$\mathcal{M}_{Z_{i}}^{K}\xi_{i}(t)+\tilde{\delta}_{i}(t)\in\bar{\mathcal{P}}_{i}$
for all $\tilde{\delta}_{i}(t)\in\mathcal{P}_{\tilde{\delta}_{i},t}$ and
$t\in\mathbb{N}$.

###### Proof.

The proof consists of two steps. In the first step, we compute the reachable
set of $\bar{\mathcal{P}}_{\xi_{i},t}$. In the second step, we ensure the
stability of this reachable set by proving its invariance, which in turn
guarantees the stability of $\bar{\mathcal{P}}_{\xi_{i},t}$.

First, recalling Lemma 4, we refer to the bounded and compact set
$\mathcal{P}_{\tilde{\delta}i,t}$ as the disturbance set. Building on this
set, $\bar{\mathcal{P}}_{\xi_{i},t}$, and $\mathcal{M}_{Z_{i}}^{K}$ defined in
(27), the reachable set for the virtual tracking error $\xi_{i}(t)$ at time
$t$ can be formulated as

$\Xi_{i,t}=\left\\{Q_{i}\Xi_{i,t-1}+\mathcal{P}_{\tilde{\delta}_{i},t}:Q_{i}\in\mathcal{M}_{Z_{i}}^{K}\right\\}$
(29)

with $\xi_{i}(0)=\Xi_{i,0}$. Hence, it follows from Lemma 5 that
$\bar{\mathcal{P}}_{\xi_{i},t}\subset\Xi_{i,t}$.

Next, we prove the stability of the reachable set $\Xi_{i,t}$. Assumption 2
states that the system is stabilizable, implying the existence of a positive
definite symmetric matrix ${P}_{i}$ such that
$Q_{i}^{\top}{P}_{i}Q-{P}_{i}\prec 0$ for all
$Q_{i}\in\mathcal{M}_{Z_{i}}^{K}$. Therefore, for sufficiently small
$\mathcal{P}_{\tilde{\delta}_{i},t}$, we have $\Xi_{i,t+1}\subseteq\Xi_{i,t}$
for $t\in\mathbb{N}$, ensuring the stability of the reachable trajectories
$\Xi_{i,t}$. Consequently, according to Lemma 5, there exists an invariant set
$\bar{\mathcal{P}}_{i}$ satisfying
$\bar{\mathcal{P}}_{i}\supset\Xi_{i,t}\supset\bar{\mathcal{P}}_{\xi_{i},t}$
for all $t\in\mathbb{N}$, with $\bar{\mathcal{P}}_{i}:=\Xi_{i,0}$. The proof
is complete. ∎

Lemma 6 provides a stability guarantee for $\bar{\mathcal{P}}_{\xi_{i},t}$,
which also implies the stability of the virtual tracking error polytope
${\mathcal{P}}_{\xi_{i},t}$ since
$\mathcal{P}_{\xi_{i},t}\subseteq\bar{\mathcal{P}}_{\xi_{i},t}$. With this
result in hand, we proceed to the problem of identifying a gain matrix $K_{i}$
for follower $i$. This gain matrix ensures that $A_{i}+B_{i}K_{i}$ is Schur
stable for all $(A_{i},B_{i})\in\mathcal{M}_{Z_{i}}$. We derive a convex
program, specifically a SDP, in the next theorem. This program aims to search
for a stabilizing matrix $K_{i}$ based on $(X_{i},X_{i+},U_{i},Y_{i})$.

######  Theorem 1.

Consider the MAS (1)-(2) under the distributed data-driven feedback protocol
(3) and (5) over graph $\bar{\mathcal{G}}$. Let Assumptions 1-5 hold. Define
$\Omega_{i}=X_{i+}-\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}\hat{W}_{i}^{(k)}$.
Then, the following SDP is feasible, and the gain matrix
$K_{i}=U_{i}M_{i}(X_{i}M_{i})^{-1}$ with any $M_{i}\in\mathbb{R}^{\rho\times
n_{i}}$ satisfying (30) renders $A_{i}+B_{i}K_{i}$ Schur stable for all
$(A_{i},B_{i})\in\mathcal{M}_{Z_{i}}$

$\begin{split}X_{i}M_{i}-\Omega_{i}M_{i}(X_{i}M_{i})^{-1}(\Omega_{i}M_{i})^{\top}&\succ
0\\\ X_{i}M_{i}&\succ 0.\end{split}$ (30)

###### Proof.

Consider any matrix $K_{i}$ that makes $A_{i}+B_{i}K_{i}$ Schur stable. Based
on the proof of Lemma 6, for all $(A_{i},B_{i})\in\mathcal{M}_{Z_{i}}$, there
exists a matrix $P_{i}\succ 0$ such that

$(A_{i}+B_{i}K_{i})^{\top}P_{i}(A_{i}+B_{i}K_{i})-P_{i}\prec 0.$ (31)

According to Assumption 5, any vector of length $n_{i}+m_{i}$ can be expressed
as a linear combination of the data matrix. That is, there exists a matrix
$G_{i}\in\mathbb{R}^{\rho\times n_{i}}$ satisfying

$\left[\begin{matrix}K_{i}\\\
I\end{matrix}\right]=\left[\begin{matrix}U_{i}\\\
X_{i}\end{matrix}\right]G_{i}$ (32)

which implies

$\begin{split}A_{i}+B_{i}K_{i}&=Z_{i}\left[\begin{matrix}U_{i}\\\
X_{i}\end{matrix}\right]G_{i}\\\
&=\Big{(}X_{i+}-\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}\hat{W}_{i}^{(k)}\Big{)}G_{i}\end{split}$
(33)

with the second identity following from (16). Substituting (33) into (31), we
have

$\begin{split}\Big{[}\Big{(}X_{i+}&-\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}\hat{W}_{i}^{(k)}\Big{)}G_{i}\Big{]}^{\top}P_{i}\Big{[}\Big{(}X_{i+}\\\
&-\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}\hat{W}_{i}^{(k)}\Big{)}G_{i}\Big{]}-P_{i}\prec
0.\end{split}$ (34)

Let $G_{i}=M_{i}P_{i}$ and
$\Omega_{i}=X_{i+}-\sum_{k=1}^{\gamma_{w_{i}}\rho}\beta_{W,i}^{(k)}\hat{W}_{i}^{(k)}$.
By a Schur complement argument, condition (34) implies

$P_{i}^{-1}-\Omega_{i}M_{i}P_{i}(\Omega_{i}M_{i})^{\top}\succ 0.$ (35)

Recall from (32) that $X_{i}G_{i}=I$ and $K_{i}=U_{i}G_{i}$. By exploiting
$P_{i}=(X_{i}M_{i})^{-1}$, seeking stability is equivalent to finding a matrix
$M_{i}$ such that

$X_{i}M_{i}-\Omega_{i}M_{i}(X_{i}M_{i})^{-1}(\Omega_{i}M_{i})^{\top}\succ 0$

with the choice $K_{i}=U_{i}M_{i}(X_{i}M_{i})^{-1}$. Then, it can be concluded
that $A_{i}+B_{i}K_{i}$ is stable as $P_{i}$ is symmetric and $P_{i}\succ 0$
for all $(A_{i},\,B_{i})\in\mathcal{M}_{Z_{i}}$. ∎

Before presenting our stability result, we define the tracking error
$e_{i}(t)$ of agent $i$ as $e_{i}(t):=y_{i}(t)-y_{0}(t)$. Our objective is to
analyze the convergence of the tracking error, ensuring that the induced
closed-loop system exhibits desired stability properties. Leveraging previous
results, including the optimization problem (19), Lemmas 3-6, and Theorem 1,
we present our distributed data-driven output synchronization solution for
unknown heterogeneous MASs (1)-(2) in Algorithm 1, accompanied by the
stability guarantees detailed below.

Algorithm 1 Distributed data-driven output synchronization

1:Input: desired lifespan of the MAS $\mathcal{T}$; initial states of the
leader $x_{0}(0)\in\mathbb{R}^{n_{0}}$ and follower
$x_{i}(0)\in\mathbb{R}^{n_{i}}$; noise polytopes $\mathcal{P}_{w_{i}}$ and
$\mathcal{P}_{v_{i}}$; regulator equation error polytopes
$\mathcal{M}_{\Delta{i1}}$ and $\mathcal{M}_{\Delta{i2}}$; state-input-output
data $\mathbb{D}_{i}$ for each $i\in\\{1,2,\ldots,N\\}$.

2:Construct matrix polytopes $\mathcal{M}_{W_{i}}$ and $\mathcal{M}_{V{i}}$
via (10).

3:Compute data-based polytopic representation $\mathcal{M}_{i}$ using Lemma 2.

4:Solve Problem (19) to obtain feasible solutions $\Pi_{i}$ and $\Gamma_{i}$.

5:Design the controller gain matrix $K_{i}$ and observer gain matrix $F$ based
on Theorem 1.

6:while $t<\mathcal{T}$ do

7: for $i=1,2,\ldots,N$ do

8: Broadcast $\eta_{i}(t)$ and $x_{i}(t)$ to agent $j\in\mathcal{N}_{i}$;

9: Compute $z_{i}(t)$ from (8) using the updated state $\eta_{j}(t)$ and
$x_{j}(t)$, $j\in\mathcal{N}_{i}$;

10: Update the feedback control protocol (3), (5), and the dynamics (1) of
agent $i$.

11: end for

12:end while

######  Theorem 2.

Consider the MAS described by (1)-(2) and the graph $\bar{\mathcal{G}}$.
Suppose Assumptions 1-5 hold. Denoting the optimal solution of Problem (19) by
$(\Pi_{i}^{*},\Gamma_{i}^{*})$, the tracking errors $e_{i}(t)$ are ultimately
uniformly bounded (UUB) under the distributed data-driven feedback protocol
(3) and (5) for any initial state and all $i\in\\{1,2,\ldots,N\\}$, if the
following two conditions are satisfied:

  1. 1.

The controller gain $K_{i}$ is designed as in Theorem 1.

  2. 2.

Choose matrix $F$ such that $I_{N}\otimes
S-(I_{N}+D+G)^{-1}(\mathcal{L}+G)\otimes F$ is Schur stable.

###### Proof.

It can be observed that the solutions $\Pi_{i}$ and $\Gamma_{i}$ to the
optimization problem (19) satisfy (18) for all
$(A_{i},B_{i},C_{i})\in\mathcal{M}_{i}$. Considering the true system matrices
$(\bar{A}_{i},\bar{B}_{i},\bar{C}_{i})\in\mathcal{M}_{i}$, we deduce from (1),
(2), and (18) that the tracking error $e_{i}(t)$ can be expressed as

$e_{i}(t)=\bar{C}_{i}x_{i}(t)-(\bar{C}_{i}\Pi_{i}^{*}-\Delta_{i2}^{*})x_{0}(t).$
(36)

Moreover, based on (7), if we select a stabilizing matrix $F$ that satisfies
the condition $I_{N}\otimes S-(I_{N}+D+G)^{-1}(\mathcal{L}+G)\otimes F$ being
Schur stable, the observer’s state asymptotically converges to the leader’s
state, i.e., $\lim_{t\to\infty}\delta_{i}(t)=\eta_{i}(t)-x_{0}(t)=0$.
Therefore, as $t\rightarrow\infty$, we have

$\lim_{t\to\infty}e_{i}(t)=\lim_{t\to\infty}\bar{C}_{i}\xi_{i}(t)+\Delta_{i2}^{*}x_{0}(t),$
(37)

where the fact that $\xi_{i}(t):=x_{i}(t)-\Pi_{i}^{*}\eta_{i}(t)$ has been
used.

Furthermore, we can recursively obtain from (28) that for $t\in\mathbb{N}$,
the polytope $\bar{\mathcal{P}}_{\xi_{i},t}$ satisfies

$\displaystyle\bar{\mathcal{P}}_{\xi_{i},t}=(\mathcal{M}_{Z_{i}}^{K})^{t}\bar{\mathcal{P}}_{\xi_{i},0}+\sum_{\ell=0}^{t-1}(\mathcal{M}_{Z_{i}}^{K})^{\ell}\mathcal{P}_{\tilde{\delta}_{i},t-\ell-1}$
(38)

As $\lim_{t\to\infty}\delta_{i}(t)=0$, it follows that
$\lim_{t\to\infty}z_{i}(t)=0$. Consequently, as $t\rightarrow\infty$, the set
$\bar{\mathcal{P}}_{\xi_{i},t}$ converges to the following set:

$(\mathcal{M}_{Z_{i}}^{K})^{t}\bar{\mathcal{P}}_{\xi_{i},0}+\sum_{\ell=0}^{t-1}(\mathcal{M}_{Z_{i}}^{K})^{\ell}\Delta_{i1}^{*}\mathcal{P}_{x_{0}}.$
(39)

Based on Lemmas 2-6, it can be concluded that matrices $\bar{C}_{i}$,
$\Delta_{i1}$, and $\Delta_{i2}$, as well as the sequences $\xi_{i}(t)$ and
$x_{0}(t)$, can be bounded and constrained within compact polytopes,
respectively. Specifically, we have $\bar{C}_{i}\in\mathcal{M}_{C_{i}}$,
$\Delta_{i1}^{*}\in\mathcal{M}_{\Delta_{i1}}$,
$\Delta_{i2}^{*}\in\mathcal{M}_{\Delta_{i2}}$,
$\xi_{i}(t)\in\mathcal{P}_{\xi_{i},t}\subseteq\bar{\mathcal{P}}_{\xi_{i},t}$,
and $x_{0}(t)\in\mathcal{P}_{x_{0}}$ for all $t\in\mathbb{N}$. By combining
(37) and (39), we can compute the reachable set of the tracking error
$e_{i}(t)$ at time $t$ (as $t\rightarrow\infty$) as follows:

$\displaystyle\mathcal{P}_{e_{i},t}$
$\displaystyle:=\mathcal{M}_{C_{i}}(\mathcal{M}_{Z_{i}}^{K})^{t}\bar{\mathcal{P}}_{\xi_{i},0}$
$\displaystyle~{}\quad+\big{[}\mathcal{M}_{C_{i}}\sum_{\ell=0}^{t-1}(\mathcal{M}_{Z_{i}}^{K})^{\ell}\mathcal{M}_{\Delta_{i1}}+\mathcal{M}_{\Delta_{i2}}\big{]}\mathcal{P}_{x_{0}}.$
(40)

According to Theorem 1, the controller gain matrix
$K_{i}=U_{i}M_{i}(X_{i}M_{i})^{-1}$ renders ${A}_{i}+{B}_{i}K_{i}$ Schur
stable for all $(A_{i},\,B_{i})\in\mathcal{M}_{Z_{i}}$, which implies the
existence of $P_{i}:=(X_{i}M_{i})^{-1}$ such that any
$Q_{i}\in\mathcal{M}_{Z_{i}}^{K}$ is also Schur stable. This means that there
exist constants $\mu>0$ and
$\beta=\max_{Q_{i}\in\mathcal{M}_{Z_{i}}^{K}}\lambda_{\min}(Q_{i})\in(0,1)$
such that
$(\mathcal{M}_{Z_{i}}^{K})^{t}\bar{\mathcal{P}}_{\xi_{i},0}\subseteq\mu\beta^{t}\bar{\mathcal{P}}_{\xi_{i},0}$
holds. Consequently, (40) obeys

$\begin{split}\mathcal{P}_{e_{i},t}&\subseteq\mu\beta^{t}\mathcal{M}_{C_{i}}\bar{\mathcal{P}}_{\xi_{i},0}\\\
&\quad+\Big{[}\mu\mathcal{M}_{C_{i}}\sum_{\ell=0}^{t-1}(\mathcal{M}_{Z_{i}}^{K})^{\ell}\mathcal{M}_{\Delta_{i1}}+\mathcal{M}_{\Delta_{i2}}\Big{]}\mathcal{P}_{x_{0}}.\end{split}$
(41)

Hence, as $t\rightarrow\infty$, $e_{i}(t)$ converges to an adjustable bounded
set as (41).

Therefore, it can be concluded that the tracking error $e_{i}(t)$ is UUB for
all $i\in\\{1,2,\ldots,N\\}$ when considering the proposed distributed data-
driven feedback protocol given by (3) and (5). This property enables the
outputs of all followers to approximately synchronize with the output of the
leader.

∎

######  Remark 4 ($\Delta$-optimal output synchronization).

The proposed distributed data-driven feedback control protocol (3)–(5)
achieves $\Delta$-optimal output synchronization for the leader-following MAS
(1)–(2) with unknown system matrices. This result, presented in Theorem 2,
addresses Problem 1 effectively. Specifically, for any
$(A_{i},B_{i},C_{i})\in\Sigma_{i}$ with $i\in\\{1,2,\ldots,N\\}$, the tracking
error $e_{i}(t)$ converges to a bounded and compact reachable set as given in
(41). The size of this set is influenced by the size of noise polytopes
$\mathcal{M}_{\Delta_{i1}}$ and $\mathcal{M}_{\Delta_{i2}}$, as defined in
(23). Notably, the magnitude of tracking error $e_{i}(t)$ is positively
correlated with the vertices of noise polytopes $\mathcal{P}_{w_{i}}$ and
$\mathcal{P}_{v_{i}}$, corresponding to the noise levels $w_{i}(T)$ and
$v_{i}(T)$, respectively. It is interesting that when $w_{i}(T)=0$ and
$v_{i}(T)=0$, the regulator equation errors $\Delta_{i1}=0$ and
$\Delta_{i2}=0$, resulting in the virtual tracking error $\xi_{i}(t)$
asymptotically converge to zero, as shown in Lemma 6. In this case, the
tracking error $e_{i}(t)$ asymptotically converges to zero too.

Based on the above remark, we can derive the following corollary for the
noise-free case.

######  Corollary 1.

Consider the leader-following MAS (9) with $w_{i}(T)=0$ and $v_{i}(T)=0$.
Under the same conditions as in Theorem 2, the tracking error $e_{i}(t)$
asymptotically converges to zero for all $(A_{i},B_{i},C_{i})\in\Sigma_{i}$
and $i\in\\{1,2,\ldots,N\\}$.

######  Remark 5 (Solvability).

It is important to highlight that the solution to Problem (19) and the SDP
(30) only needs to consider the vertices of the matrix polytopes
$\mathcal{M}_{Z_{i}}$ and $\mathcal{M}_{C_{i}}$. This is due to the convexity
of the polytope, which implies that any point within the polytope can be
expressed as a convex combination of its vertices. Therefore, if the problem
is addressed at all vertices of the polytope, it covers the entire polytope,
including its interior.

######  Remark 6 (Comparison).

Several existing approaches have explored data-driven output synchronization
for unknown MASs, e.g., the behavioral approach in [34] and the model-free RL-
based approach in [15, 16, 17]. In comparison to these existing works, our
approach exhibits the following key distinctions. Firstly, the proposed data-
driven method is robust to unknown noisy data, eliminating the requirement for
accurately measurable noise as seen in [34]. Secondly, in contrast to a large
amount training data required for [16, 17], our approach achieves output
synchronization only using limited data, i,e., as long as Assumption 5 is
satisfied. Furthermore, we propose a static data-driven design method using
historical data, while ensuring system stability, which circumvents real-time
iterative computations using online data, as in [16, 17]. In this sense, the
proposed method significantly reduces the computational burden, making it more
efficient and suitable for real-world implementation.

##  IV Numerical Examples

In this section, we present a numerical example to demonstrate the
effectiveness of the proposed data-driven method. We consider a discrete-time
heterogeneous MAS consisting of seven agents, including one leader and six
followers. The dynamics of the leader are described by (2), where

$S=\left[\begin{matrix}0&1\\\ -1&0\end{matrix}\right],\quad
H=\left[\begin{matrix}1&0\end{matrix}\right].$

The true dynamics of the six followers are given by (1) with

$\displaystyle\bar{A}_{1}=2,\bar{B}_{1}=3,\bar{C}_{1}=1$
$\displaystyle\bar{A}_{2}=\left[\begin{matrix}0&1\\\
1&-1\end{matrix}\right],\bar{B}_{2}=\left[\begin{matrix}0\\\
1\end{matrix}\right],\bar{C}_{2}=\left[\begin{matrix}1&1\end{matrix}\right]$
$\displaystyle\bar{A}_{3}=\left[\begin{matrix}0&1\\\
1&-2\end{matrix}\right],\bar{B}_{3}=\left[\begin{matrix}1\\\
0\end{matrix}\right],\bar{C}_{3}=\left[\begin{matrix}0&1\end{matrix}\right]$
$\displaystyle\bar{A}_{4}=\left[\begin{matrix}0&1\\\
-1&-3\end{matrix}\right],\bar{B}_{4}=\left[\begin{matrix}1\\\
1\end{matrix}\right],\bar{C}_{4}=\left[\begin{matrix}-1&1\end{matrix}\right]$
$\displaystyle\bar{A}_{5}=\left[\begin{matrix}0&1&0\\\ 0&0&1\\\
0&0&-4\end{matrix}\right],\bar{B}_{5}=\left[\begin{matrix}0\\\ 0\\\
4\end{matrix}\right],\bar{C}_{5}=\left[\begin{matrix}1&0&0\end{matrix}\right]$
$\displaystyle\bar{A}_{6}=\left[\begin{matrix}0&1&0\\\ 0&0&1\\\
0&0&-5\end{matrix}\right],\bar{B}_{6}=\left[\begin{matrix}0\\\ 0\\\
5\end{matrix}\right],\bar{C}_{6}=\left[\begin{matrix}2&0&0\end{matrix}\right].$

The network topology of these agents is shown in Fig. 2, which represents the
communication topology $\bar{\mathcal{G}}$ between agents.

Figure 2: The communication topology $\bar{\mathcal{G}}$ between agents.

In the data-driven setting, the true system matrices
$(\bar{A}_{i},\bar{B}_{i},\bar{C}_{i})$ are assumed unknown. To collect data
for each follower, we run the open-loop system (9) offline and gather a set of
noisy data $\mathbb{D}_{i}$ with a length of $\rho=20$. The inputs are
uniformly distributed within a polytope $\mathcal{P}_{u_{i}}$ defined as

$\mathcal{P}_{u_{i}}=\Big{\\{}u_{i}\Big{|}u_{i}=\sum_{k=1}^{\gamma_{u_{i}}}\beta^{(k)}_{u,i}\hat{u}_{i}^{(k)},\beta^{(k)}_{u,i}\geq
0,\sum_{k=1}^{\gamma_{u_{i}}}\beta^{(k)}_{u,i}=1\Big{\\}}$

where $\gamma_{u_{i}}=2$ and the two vertices are $\hat{u}_{i}^{(1)}=1$ and
$\hat{u}_{i}^{(2)}=-1$ for $i\in\\{1,2,\ldots,6\\}$. Furthermore, the random
process noise $w_{i}(T)$ is sampled from the polytope $\mathcal{P}_{w_{i}}$
with four vertices: $\hat{w}_{i}^{(1)}=\bar{w}_{i}*[1\,1]^{\top}$,
$\hat{w}_{i}^{(2)}=\bar{w}_{i}*[1\,-1]^{\top}$,
$\hat{w}_{i}^{(3)}=\bar{w}_{i}*[-1\,1]^{\top}$, and
$\hat{w}_{i}^{(4)}=\bar{w}_{i}*[-1\,-1]^{\top}$. Similarly, the measurement
noise $v_{i}(T)$ is bounded by $\mathcal{P}_{v_{i}}$ with two vertices:
$\hat{v}_{i}^{(1)}=\bar{v}_{i}$ and $\hat{v}_{i}^{(2)}=-\bar{v}_{i}$. Here,
select $\bar{w}_{i}=\bar{v}_{i}=0.01$.

To solve the output regulation equations (18), we need to compute the
optimization problem (19) to determine the values of
$(\Pi_{i}^{*},\Gamma_{i}^{*})$.

$\begin{split}&\Pi_{1}^{*}=\left[\begin{matrix}1.0000&-0.0000\end{matrix}\right],\quad\,\Gamma_{1}^{*}=\left[\begin{matrix}-0.6672&0.3336\end{matrix}\right],\\\
&\Pi_{2}^{*}=\left[\begin{matrix}0.5013&-0.4995\\\
0.4987&0.4995\end{matrix}\right],\quad\Gamma_{2}^{*}=\left[\begin{matrix}-0.5014&1.4994\end{matrix}\right],\\\
&\Pi_{3}^{*}=\left[\begin{matrix}1.9914&0.9954\\\
0.9952&-0.0020\end{matrix}\right],\quad\,\Gamma_{3}^{*}=\left[\begin{matrix}-1.9907&1.9898\end{matrix}\right],\\\
&\Pi_{4}^{*}=\left[\begin{matrix}-0.7995&-0.1998\\\
0.1996&-0.2002\end{matrix}\right],\;\,\Gamma_{4}=\left[\begin{matrix}-0.0002&-0.6007\end{matrix}\right],\\\
&\Pi_{5}^{*}=\left[\begin{matrix}0.9999&0.0008\\\ -0.0010&1.0010\\\
-1.0001&-0.0011\end{matrix}\right],\;\Gamma_{5}^{*}=\left[\begin{matrix}-1.0002&-0.2515\end{matrix}\right]\\\
&\Pi_{6}^{*}=\left[\begin{matrix}0.5004&0.0001\\\ -0.0004&0.5004\\\
-0.4996&-0.0004\end{matrix}\right],\;\Gamma_{6}^{*}=\left[\begin{matrix}-0.4991&-0.1006\end{matrix}\right].\end{split}$

Then, by solving the SDP in Theorem 1, we obtained the stabilizing controller
gain $K_{i}$ for each follower $i$ in (3). The specific values of $K_{i}$ are
as follows:

$\begin{split}&K_{1}=-16.6608,\quad K_{2}=[-7.0686\;-1.8060],\\\
&K_{3}=[-13.3323\;-10.3354],K_{4}=[-13.5107\;-15.1270],\\\
&K_{5}=[-0.5139\;-0.9054\;-0.5120],\\\
&K_{6}=[-0.7749\;-1.3071\;-0.5160].\end{split}$

####  IV-1 Comparison with the model-based approach

The simulation of the heterogeneous MAS was carried out using the feedback
control protocol (3) and (5). The initial states of the leader, followers, and
observers were randomly selected. The tracking errors $e_{i}$ for
$i\in\\{1,2,\ldots,N\\}$ under the data-driven control (according to Theorem
2) and model-based control (according to Lemma 1) are shown in Fig. 3,
respectively.

Figure 3: Tracking error of each follower under data-driven control and model-
based control.

From this figure, it is evident that output synchronization is achieved under
both control paradigms. The results demonstrate that the proposed data-driven
polytopic method achieves comparable performance to the model-based approach
and exhibits excellent robustness to noisy and limited data, highlighting the
effectiveness of the data-based polytopic controller. Additionally, the
absence of model information during implementation further emphasizes the
superiority of the data-driven method.

####  IV-2 Comparison of different noise levels

The proposed data-driven approach was tested under different disturbance
levels to investigate their effect on system performance and tracking errors.
First, let us define the error between the solutions
$(\Pi_{i}^{s},\Gamma_{i}^{s})$ and $(\Pi_{i}^{*},\Gamma_{i}^{*})$ of the
output regulation equation computed by (4) and (19), respectively, as follows

$\phi_{1}=\max_{i\in\\{1,2,\ldots,6\\}}\|\Pi_{i}^{s}-\Pi_{i}^{*}\|_{2},\quad\phi_{2}=\max_{i\in\\{1,2,\ldots,6\\}}\|\Gamma_{i}^{s}-\Gamma_{i}^{*}\|_{2}.$

Table I tabulates $\phi_{1}$ and $\phi_{2}$ for five different noise levels
$\bar{w}_{i}$, $\bar{v}_{i}$. It can be observed that a larger noise level
results in a lower accuracy of $(\Pi_{i}^{*},\Gamma_{i}^{*})$ relative to the
exact solution $(\Pi_{i}^{s},\Gamma_{i}^{s})$.

TABLE I: The error between the regulation equation solutions computed by (4)
and (19) under different noise levels

Noise level | 0.001 | 0.005 | 0.01 | 0.05 | 0.1
---|---|---|---|---|---
$\phi_{1}$ | $8.5149\times 10^{-4}$ | 0.0051 | 0.0110 | 0.0587 | 0.1463
$\phi_{2}$ | $5.8310\times 10^{-4}$ | 0.0040 | 0.0138 | 0.0876 | 0.1748

Furthermore, taking follower $6$ as an example, Fig. 4 displays the tracking
error $e_{6}(t)$ (the orange solid line) and the bounds of
$\mathcal{P}_{e_{6},t}$ (the blue dashed line) under three different noise
levels. As expected, the tracking error $e_{6}(t)$ remains within
$\mathcal{P}_{e_{6},t}$ at each time step. Moreover, the error bound (41) is
on the order (precisely, several times) of the noise size. This can be
attributed to the fact that the escalating uncertainties caused by noise
contribute to the expansion of noise polytopes $\mathcal{M}_{\Delta_{i1}}$ and
$\mathcal{M}_{\Delta_{i2}}$. As a result, the polytope $\mathcal{M}_{i}$ of
allowable system matrices becomes larger, significantly augmenting the
conservatism of the obtained data-driven control solutions.

(a)

(b)

(c)

Figure 4: Tracking error $e_{6}(t)$ and bounds of $\mathcal{P}_{e_{6},t}$
under data-driven control with different noise levels: (a)
$\bar{w}_{i}(t)=\bar{v}_{i}(t)=0.001$; (b)
$\bar{w}_{i}(t)=\bar{v}_{i}(t)=0.01$; (c) $\bar{w}_{i}(t)=\bar{v}_{i}(t)=0.1$.

##  V Conclusions

In conclusion, this paper has presented a data-driven polytopic approach for
output synchronization of unknown heterogeneous MASs, utilizing data instead
of explicit knowledge of each follower’s dynamics model. The proposed method
offers a certified data-driven feedback control protocol that can handle
perturbed offline data and uncertainties in the system matrices. By means of a
unique data-based polytopic representation of the MASs, an approximate
solution of the output regulator equations and a stabilizing control gain are
obtained. The stability of the tracking error polytope is ensured, and
sufficient data-based conditions for near-optimal output synchronization are
provided. Future research directions could explore less conservative
approximations of the tracking error polytope and investigate output
synchronization under event-triggered control. These advancements would
further enhance the performance and applicability of the data-driven approach
in real-world scenarios.

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