# Matrix-Scaled Consensus over Undirected Networks

Minh Hoang Trinh1, , Hoang Huy Vu2, Nhat-Minh Le-Phan23, Quyen Ngoc Nguyen2
∗AI Department, FPT University, An Thinh Phu New Urban Area, Nhon Binh Ward,
Quy Nhon City, Binh Dinh 55117, Vietnam. Corresponding author. Email:
<EMAIL_ADDRESS>
‡Department of Automation Engineering, School of Electrical and Electronic
Engineering, Hanoi University of Science and Technology (HUST), 1 Dai Co Viet
Str., Hai Ba Trung Dist., Hanoi 11615, Vietnam. Emails:
<EMAIL_ADDRESS>
♯Research and Development in Navigation, Guidance and Control Technologies
Center, Viettel Aerospace Institute, Hanoi 13151, Vietnam. Email:
<EMAIL_ADDRESS>

###### Abstract

In this paper, we propose matrix-scaled consensus algorithms for linear
dynamical agents interacting over an undirected network. These algorithms are
designed to make the state vectors of all agents to asymptotically agree up to
some matrix scaling weights. First, the matrix scaled Laplacian and its
algebraic properties are examined. Second, we propose matrix-scaled consensus
algorithms for networks of single integrators with or without constant
parametric uncertainties. Third, observer-based matrix-scaled consensus
algorithms for homogeneous or heterogeneous linear agents are designed. The
effectiveness of the proposed algorithms are asserted by rigorous mathematical
analysis and supported by numerical simulations.

###### Index Terms:

multi-agent systems, consensus and synchronization

##  I Introduction

Collective behaviors displayed in nature such as bird flocking, fish
schooling, synchronous fireflies, have inspired a lot of research works.
Though simple, the consensus algorithm has been widely used for modeling and
studying such striking phenomena [1]. It is interesting that a lot of
multiagent systems such as autonomous vehicle formations, electrical system,
sensor networks, or social networks can be coordinated by appropriately
applying consensus algorithms and its modifications [2, 3].

Consider a network in which the interactions between subsystems, or agents, is
modeled by a graph. In the consensus algorithm, each agent updates its state
based on the sum of the relative states with its nearby agents. If the
interaction graph is connected, the agents’ states asymptotically converge to
a common point in the space, and the system is said to asymptotically achieve
a consensus [1].

A (scalar) scaled consensus model was proposed in [4], in which each agent has
a nonzero scalar scaling gain and updates its state variable based on the sum
of differences in the scaled states. Under the scaled consensus model, the
scaled states of all agents agree to the same virtual consensus value, and the
state variable of each agent converges to a value differing from the virtual
consensus value by an inverse of the scaling gain. Thus, under the scalar
scaled consensus algorithm [4], and suppose that each agent has a state
vector, the state vectors of all agents asymptotically distributed along a
straight line through the origin. The scaled consensus system can describe a
cooperative network, where agents have different levels of consensus on a
single topic. Further studies of the scalar scaled consensus algorithm with
consideration to switching graphs, time delays, disturbance attenuation, or
different agents’ models can be found in the literature [5, 6, 7, 8, 9, 10,
11].

This paper proposes a multi-dimensional generalization of the scaled consensus
model in [4]. We firstly consider an undirected network of single integrator
agents as a basic setup. Each agent has a state vector and a possibly non-
symmetric positive or negative definite scaling matrix. The agent updates the
state variables based on the relative matrix-scaled state vectors. Several
algebraic properties of the matrix scaled Laplacian, which determines the
asymptotic behavior of the whole system, are derived. Specifically, it is
shown that the matrix-scaled states eventually agree to a virtual consensus
vector. The state vector of each agent asymptotically converges to a vector,
which can be obtained from the virtual consensus point by a linear
transformation determined by the inverse of its scaling matrix. Thus, all
agents with the same scaling matrix converge to a common point in the space,
and clustering behaviors usually happen. To accommodate more complicated
interactions, nonlinear matrix-scaled consensus algorithms and corresponding
Lyapunov-based analysis are also provided. It is worth noting that in the
proposed matrix-scaled consensus model, the scaling matrices are not limited
to rotation matrices such as the algorithms considered in [12, 13, 14, 15,
16]. In [17], the proposed model was interpreted as a multi-dimensional model
opinion dynamics, where the positive/negative definite scaling matrix weights
capture the private belief system of an individual on $d$ logically dependent
topics. Clustering happens as the private belief system of each individual is
usually different from each other and not perfectly aligned with a social
norm. Second, we study the matrix-scaled consensus algorithm for a system of
single integrators with parametric uncertainties. An adaptive scheme is
developed, which guarantees the system to eventually achieve matrix-scaled
consensus. If in addition, a persistently exciting condition is satisfied, the
estimate variables eventually converge to the precise parameters. Third, we
consider a network of homogeneous linear agents and propose an observer-based
matrix-scaled consensus algorithm. It is shown that the agents asymptotically
achieve matrix-scaled consensus with regard to a trajectory of a virtual
agent. If we specify the agents to be linear oscillators in 2D, and the
scaling matrices to be a composition of rotation and expansion/compression
matrices, the asymptotic biases in phase and magnitude of each agent’s
trajectory with a common oscillator can be determined by the scaling matrices.
Thus, the algorithms can be used for designing robot’s trajectories,
referenced voltages in circuits, or explaining the alternation of temperature
during four seasons. Lastly, as practical networks often consists of agents
with different sizes and capacities, we propose a matrix-scaled consensus
algorithm for a network of heterogeneous linear agents. The proposed algorithm
combines the corresponding matrix-scaled consensus algorithm for homogeneous
linear agents with a disturbance observer which compensates the differences
between each agent’s model and a pre-specified one. The main challenges and
differences in the analysis of our proposed algorithms with regard to the
existing consensus algorithms for general linear agents in the literature,
e.g., [18, 19, 20, 21, 22, 23, 24], are originated from the asymmetry of the
matrix scaled Laplacian and the multi-dimension of the problem.

The remainder of this paper is organized as follows. Section II provides
theoretical background, problem formulation, and several properties of the
matrix-scaled Laplacian. Matrix-scaled consensus algorithms for single
integrator agents with and without parametric uncertainties are proposed and
examined in Section III. Section IV studies matrix-scaled consensus for
networks of homogeneous and heterogeneous linear agents. Simulations are given
in Section V to support the theoretical results. Finally, Section VI concludes
the paper.

##  II Preliminaries

###  II-A Notations

In this paper, the sets of real, complex, and natural numbers are denoted by
$\mathbb{R},\mathbb{C}$ and $\mathbb{N}$, respectively. Scalars are denoted by
lowercase letters, while bold font normal and capital letters are used for
vectors and matrices, respectively. We denote
$\mathbf{1}_{d}\in\mathbb{R}^{d}$ the $d$-dimensional vector of all 1 an
$\mathbf{0}_{m\times n}\in\mathbb{R}^{m\times n}$ the matrix of all zeros. The
transpose of a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ is denoted by
$\mathbf{A}^{\top}$. The kernel, image, rank, and determinant of $\mathbf{A}$
are respectively denoted as ker$(\mathbf{A})$, im$(\mathbf{A})$,
rank($\mathbf{A}$), and $\text{det}(\mathbf{A})$. A diagonal matrix with
elements $a_{1},a_{2},\ldots,a_{n}$ is denoted by
$\text{diag}(a_{1},a_{2},\ldots,a_{n})$. The 2-norm ($\infty$-norm) of a
vector $\mathbf{x}=[x_{1},\ldots,x_{d}]^{\top}$ is denoted by
$\|\mathbf{x}\|=\sqrt{\sum_{i=1}^{d}x_{i}^{2}}$ (correspondingly,
$\|\mathbf{x}\|_{\infty}=\max_{i\in\\{1,\ldots,d\\}}|x_{i}|$). The 2-norm of a
real matrix $\mathbf{A}$, denoted by $\|\mathbf{A}\|$, is defined as
$\max_{\|\mathbf{x}\|=1}\|\mathbf{A}\mathbf{x}\|$. A matrix
$\mathbf{A}\in\mathbb{R}^{d\times d}$ is positive definite (negative definite)
if and only if $\forall\mathbf{x}\in\mathbb{R}^{d}$,
$\mathbf{x}\neq\mathbf{0}_{d}$, then $\mathbf{x}^{\top}\mathbf{A}\mathbf{x}>0$
(resp., $\mathbf{x}^{\top}\mathbf{A}\mathbf{x}<0$). For a real, symmetric
positive semidefinite matrix $\mathbf{A}\in\mathbb{R}^{d\times d}$, which can
be diagonalized as
$\mathbf{T}^{-1}\mathbf{A}\mathbf{T}=\text{diag}(\omega_{1},\ldots,\omega_{d})$,
we use $\sqrt{\mathbf{A}}$ to denote its square root
$\mathbf{T}\text{diag}(\sqrt{\omega_{1}},\ldots,\sqrt{\omega_{d}})\mathbf{T}^{-1}$.
Let $\mathbf{x}_{1},\ldots,\mathbf{x}_{n}\in\mathbb{R}^{d}$, the vectorization
operator is defined as
vec$(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})=[\mathbf{x}_{1}^{\top},\ldots,\mathbf{x}_{n}^{\top}]^{\top}\in\mathbb{R}^{dn}$.
Given matrices $\mathbf{A}_{1},\ldots,\mathbf{A}_{n}$, we use
blkdiag$(\mathbf{A}_{1},\ldots,\mathbf{A}_{n})$ as the block diagonal matrix
with $\mathbf{A}_{k}$, $k=1,\ldots,n$ in the main diagonal.

###  II-B Graph theory

Consider a undirected graph
$\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{W})$, where
$\mathcal{V}=\\{1,\ldots,n\\}$ is the set of $n$ vertices,
$\mathcal{E}=\\{e_{k}=(i,j),~{}k=1,\ldots,m\\}\subset\mathcal{V}^{2}$ is the
set of $m$ edges, and $\mathcal{W}=\\{w_{ij}>0\\}_{(i,j)\in\mathcal{E}}$ is
the set of positive scalar weights corresponding to each edge of
$\mathcal{G}$. By undirectedness, if $(i,j)\in\mathcal{E}$, then
$(j,i)\in\mathcal{E}$. Further, it is assumed that there is no self-loop (an
edge with the same end vertices $(i,i)$, $i\in\mathcal{V}$) in the graph. The
neighbor set of a vertex $i$ is defined as
$\mathcal{N}_{i}=\\{j\in\mathcal{N}_{i}|~{}(j,i)\in\mathcal{E}\\}$. A path is
a sequences of vertices connected by edges in $\mathcal{E}$, where each vertex
appears one time, except for possibly the starting and the ending vertices.
For examples, the path $\mathcal{P}=i_{1}i_{2}\ldots i_{p}$ from $i_{1}$ to
$i_{p}$ has $(i_{k},i_{k+1})\in\mathcal{E}$, $k=1,\ldots,p-1$. A cycle is a
path with the same starting and ending vertices. A graph is connected if there
exists a path between any two vertices in $\mathcal{V}$.

A spanning subgraph
$\mathcal{G}^{\prime}=(\mathcal{V}^{\prime},\mathcal{E}^{\prime},\mathcal{W}^{\prime})$
of $\mathcal{G}$ has $\mathcal{V}^{\prime}=\mathcal{V}$ and
$\mathcal{E}\subseteq\mathcal{E}^{\prime}$. A spanning tree of $\mathcal{G}$
is a connected spanning sub-graph of $\mathcal{G}$ with $n-1$ edges. Let
$\mathcal{T}$ be a spanning tree of $\mathcal{G}$ and consider an arbitrary
labeling and orientation of the edges in $\mathcal{E}$ so that $n-1$ edges of
$\mathcal{E}(\mathcal{T})$ are $e_{1},\ldots,e_{n-1}$.

Corresponding to this labeling and edge orientation, the incidence matrix of
$\mathcal{G}$ is defined as $\mathbf{H}=[h_{ki}]\in\mathbb{R}^{m\times n}$,
where $h_{ki}=-1$ if $e_{k}=(i,j)$, $h_{ki}=-1$ if $e_{k}=(i,j)$, $h_{ki}=1$
if $e_{k}=(j,i)$, and $h_{ki}=0$ in other cases. It is wellknown that if
$\mathcal{G}$ is a connected graph, we have rank$(\mathbf{H})=n-1$ and
$\mathbf{H}\mathbf{1}_{n}=\mathbf{0}_{m}$ [25].

Let $\mathbf{H}_{\mathcal{T}}\in\mathbb{R}^{(n-1)\times n}$, and
$\mathbf{H}_{\mathcal{C}}\in\mathbb{R}^{(m-n+1)\times n}$ denote the
corresponding incidence matrices of the subgraphs $\mathcal{T}$, and the
subgraph
$(\mathcal{V},\mathcal{E}\setminus\mathcal{E}(\mathcal{T}),\mathcal{W}\setminus\mathcal{W}(\mathcal{T}))$.
Then, we can express the incidence matrix as

$\displaystyle\mathbf{H}=\begin{bmatrix}\mathbf{H}_{\mathcal{T}}\\\
\mathbf{H}_{\mathcal{C}}\end{bmatrix}=\begin{bmatrix}\mathbf{I}_{n-1}\\\
\mathbf{T}\end{bmatrix}\mathbf{H}_{\mathcal{T}}=\mathbf{R}\mathbf{H}_{\mathcal{T}},$

where $\mathbf{H}_{\mathcal{T}}\in\mathbb{R}^{(n-1)\times n}$,
$\mathbf{H}_{\mathcal{C}}\in\mathbb{R}^{(m-n+1)\times n}$, and
$\mathbf{T}=\mathbf{H}_{\mathcal{C}}\mathbf{H}_{\mathcal{T}}(\mathbf{H}_{\mathcal{T}}\mathbf{H}_{\mathcal{T}}^{\top})^{-1}$
[26]. The graph Laplacian of $\mathcal{G}$ can be defined as
$\mathbf{L}=[l_{ij}]=\mathbf{H}^{\top}\mathbf{W}\mathbf{H}\in\mathbb{R}^{n\times
n},$ where
$\mathbf{W}=\text{diag}(\ldots,w_{ij},\ldots)=\text{diag}(w_{1},\ldots,w_{m})$.
The matrix $\mathbf{L}$ of a connected graph is symmetric positive
semidefinite, with spectrum
$0=\lambda_{1}<\lambda_{2}\leq\ldots\leq\lambda_{n}$, and
ker$(\mathbf{L})=\text{im}(\mathbf{1}_{n})$.

###  II-C Agents’ models and the group’s objective

In this paper, we consider an $n$-agent system interacted via an undirected
weighted graph $\mathcal{G}$. The dynamics of each agent is modeled by either

  1. (i)

single-integrator with parametric uncertainty:

$\displaystyle\dot{\mathbf{x}}_{i}=\mathbf{u}_{i}+\bm{\phi}_{i}(t,\mathbf{x}_{i})\bm{\theta}_{i},~{}i=1,\ldots,n,$
(1)

where $\mathbf{x}_{i},\mathbf{u}_{i}\in\mathbb{R}^{d}$ are the state variable
vector and the control input,
$\bm{\phi}_{i}(t,\mathbf{x}_{i}),~{}\dot{\bm{\phi}}_{i}(t,\mathbf{x}_{i})\in\mathbb{R}^{d\times
r}$ are the matrices of known bounded continuous functions, and
$\bm{\theta}_{i}\in\mathbb{R}^{r}$ is a vector of constant unknown parameters;
or

  2. (ii)

homogeneous linear agents

$\displaystyle\dot{\mathbf{x}}_{i}$
$\displaystyle=\mathbf{A}\mathbf{x}_{i}+\mathbf{B}\mathbf{u}_{i},$ (2a)
$\displaystyle\mathbf{y}_{i}$
$\displaystyle=\mathbf{C}\mathbf{x}_{i},~{}i=1,\ldots,n,$ (2b)

where $(\mathbf{A},\mathbf{B})$ is stabilizable and $(\mathbf{A},\mathbf{C})$
is observable.

  3. (iii)

heterogeneous linear agents

$\displaystyle\dot{\mathbf{x}}_{i}$
$\displaystyle=\mathbf{A}_{i}\mathbf{x}_{i}+\mathbf{B}_{i}\mathbf{u}_{i},$
(3a) $\displaystyle\mathbf{y}_{i}$
$\displaystyle=\mathbf{C}_{i}\mathbf{x}_{i},~{}i=1,\ldots,n,$ (3b)

where $(\mathbf{A}_{i},\mathbf{B}_{i})$ is stabilizable and
$(\mathbf{A}_{i},\mathbf{C}_{i})$ is observable, $\forall
i\in\\{1,\ldots,n\\}$.

We associate to each agent a scaling matrix
$\mathbf{S}_{i}\in\mathbb{R}^{d\times d}$ ($d\geq 2$) and a state vector
$\mathbf{x}_{i}\in\mathbb{R}^{d}$. The matrix $\mathbf{S}_{i}$ is either
positive definite or negative definite. For each scaling matrix, we define a
matrix signum function

$\displaystyle\text{sign}(\mathbf{S}_{i})=\left\\{{\begin{array}[]{*{20}{ll}}{1},&\text{if
$\mathbf{S}_{i}$ is positive definite},\\\ {-1},&\text{if $\mathbf{S}_{i}$ is
negative definite}.\end{array}}\right.$ (6)

and an absolute matrix function
$|\mathbf{S}_{i}|=\text{sign}(\mathbf{S}_{i})\mathbf{S}_{i}$. It is not hard
to see that
$\text{sign}(\mathbf{S}_{i})=\text{sign}(\mathbf{S}_{i}^{\top})=\text{sign}(\mathbf{S}_{i}^{-1})$,
and $|\mathbf{S}_{i}^{-1}|=|\mathbf{S}_{i}|^{-1}$. Let
$\mathbf{x}=\text{vec}(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$,
$\mathbf{S}=\text{blkdiag}(\mathbf{S}_{1},\ldots,\mathbf{S}_{n})$, and
$|\mathbf{S}|=\text{blkdiag}(|\mathbf{S}_{1}|,\ldots,|\mathbf{S}_{n}|)$, we
aim to design matrix-scaled consensus algorithms so that
$\mathbf{x}(t)\to\mathcal{C}_{S}$ as $t\to+\infty$, $\forall\mathbf{x}(0)$,
where

$\displaystyle\mathcal{C}_{S}=\\{\mathbf{x}\in\mathbb{R}^{dn}|~{}\mathbf{S}_{i}\mathbf{x}_{i}=\mathbf{S}_{j}\mathbf{x}_{j},~{}\forall
i,j=1,\ldots,n\\}.$ (7)

is the matrix-scaled consensus (MSC) set.

The following lemma will be used throughout this paper.

###### Lemma II.1

The matrix
$\bm{\Omega}=(\text{sign}(\mathbf{S})\mathbf{L}\otimes\mathbf{I}_{d})\mathbf{S}$
has $d$ zero eigenvalues and $dn-d$ eigenvalues with positive real parts. The
left and right kernels of $\bm{\Omega}$ are spanned by columns of
$(\text{sign}(\mathbf{S})\mathbf{1}_{n})\otimes\mathbf{I}_{d}$ and
$\mathbf{S}^{-1}(\mathbf{1}_{n}\otimes\mathbf{I}_{d})$, respectively.

###### Proof:

Since two matrices
$\mathbf{M}=\mathbf{S}\bm{\Omega}\mathbf{S}^{-1}=|\mathbf{S}|(\mathbf{L}\otimes\mathbf{I}_{d})$
and $\bm{\Omega}$ are similar, they have the same spectrum. Because
$|\mathbf{S}|$ is positive definite,
rank$(\mathbf{M})=\text{rank}(\mathbf{L}\otimes\mathbf{I}_{d})=dn-d$, and thus
$\mathbf{M}$ has $d$ zero eigenvalues. Next, let
$\bar{\mathbf{H}}=\mathbf{H}\otimes\mathbf{I}_{d}$,
$\bar{\mathbf{W}}=\mathbf{W}\otimes\mathbf{I}_{d}$,
$\bar{\mathbf{L}}=\mathbf{L}\otimes\mathbf{I}_{d}$,

$\displaystyle\mathbf{X}=\begin{bmatrix}\mathbf{I}_{dn}&s^{-1}\mathbf{|S|}\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}}\\\
\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}&\mathbf{I}_{dm}\end{bmatrix},~{}\mathbf{Y}=\begin{bmatrix}\mathbf{I}_{dn}&\mathbf{0}_{dn\times
dm}\\\
-\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}&\mathbf{I}_{dm}\end{bmatrix},$

for any $0\neq s\in\mathbb{C}$, we have

$\displaystyle\mathbf{X}\mathbf{Y}$
$\displaystyle=\begin{bmatrix}\mathbf{I}_{dn}-s^{-1}\mathbf{|S|}\bar{\mathbf{L}}&s^{-1}\mathbf{|S|}\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}}\\\
\mathbf{0}_{dm\times dn}&\mathbf{I}_{dm}\end{bmatrix},$
$\displaystyle\mathbf{Y}\mathbf{X}$
$\displaystyle=\begin{bmatrix}\mathbf{I}_{dn}&s^{-1}\mathbf{|S|}\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}}\\\
\mathbf{0}_{dm\times
dn}&\mathbf{I}_{dm}-s^{-1}\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}\mathbf{|S|}\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}}\end{bmatrix}.$

From the fact that
$\text{det}(\mathbf{X}\mathbf{Y})=\text{det}(\mathbf{Y}\mathbf{X})$, one has

$\displaystyle\text{det}(\mathbf{I}_{dn}-s^{-1}\mathbf{|S|}\bar{\mathbf{L}})=\text{det}(\mathbf{I}_{dm}-s^{-1}\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}\mathbf{|S|}\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}}),$
$\displaystyle
s^{d(m-n)}\text{det}(s\mathbf{I}_{dn}-\mathbf{|S|}\bar{\mathbf{L}})=\text{det}(s\mathbf{I}_{dm}-\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}\mathbf{|S|}\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}}).$

Thus, the nonzero eigenvalues of two matrices
$\mathbf{M}=\mathbf{|S|}\bar{\mathbf{L}}$ and
$\mathbf{N}=\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}\mathbf{|S|}\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}}$
are the same. Since $\mathbf{|S|}$ and $\sqrt{\overline{\mathbf{W}}}$ are
positive definite, $\forall\mathbf{y}\in\mathbb{R}^{dm}$,
$\mathbf{y}^{\top}\mathbf{N}\mathbf{y}=\frac{1}{2}\mathbf{y}^{\top}(\mathbf{N}+\mathbf{N}^{\top})\mathbf{y}=\frac{1}{2}\mathbf{y}^{\top}\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}(\mathbf{|S|}+\mathbf{|S|}^{\top})\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}}\mathbf{y}\geq
0$. Therefore, $\mathbf{N}$, and thus $\bm{\Omega}$, have $dn-d$ eigenvalues
with positive real parts.111It is noted that
ker$(\mathbf{N})$=ker$(\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}})$,
and ker$(\bar{\mathbf{H}}^{\top}\sqrt{\overline{\mathbf{W}}})$ is referred to
as the cycle space of $\mathcal{G}$. Finally, the claim on ker$(\bm{\Omega})$
follows by direct computation. ∎

###### Remark II.1

We will refer to $\bm{\Omega}$ as the matrix-scaled Laplacian since its
structure resembles a Laplacian with block matrix weights [27]. If we further
assume that $\mathbf{S}_{i}=\mathbf{S}_{i}^{\top},~{}\forall i=1,\ldots,n$,
based on [28][Thm. 3], all eigenvalues $\Lambda_{1},\ldots,\Lambda_{dn}$ of
$\bm{\Omega}$ are real and

$\displaystyle\begin{split}\Lambda_{1}&=\ldots=\Lambda_{d}=0,\\\
\Lambda_{k}&=\theta_{k}\lambda_{\lceil
k/d\rceil},~{}k=d+1,\ldots,dn,\end{split}$ (8)

where $\lceil x\rceil$ denotes the closest natural number that is greater than
or equal to $x$, $\lambda_{k}$ is the $k$-smallest eigenvalue of $\mathbf{L}$
and $\theta_{k}\in[p_{\min},p_{\max}]$, $p_{\min}$ and $p_{\max}$ are
respectively the smallest and the largest eigenvalue of all
$|\mathbf{S}_{i}|$.

For example, consider $\mathcal{G}=\mathcal{C}_{6}$ \- the cycle of length
six, and $\mathbf{S}_{i}$ are given as
$\mathbf{S}_{1}=\mathbf{S}_{2}=\begin{bmatrix}2&-\frac{\sqrt{3}}{4}\\\
-\frac{\sqrt{3}}{4}&\frac{7}{4}\end{bmatrix},~{}\mathbf{S}_{3}=\mathbf{S}_{4}=\begin{bmatrix}2&0\\\
0&1\end{bmatrix},~{}\mathbf{S}_{5}=\mathbf{S}_{6}=\begin{bmatrix}-3&0\\\
0&-1\end{bmatrix}.$ Then,
$0=\lambda_{1}<\lambda_{2}=\lambda_{3}=1<\lambda_{4}=\lambda_{5}=3<\lambda_{6}=4$,
$p_{\min}=0.5$, $p_{\max}=3$, and the eigenvalues of $\bm{\Omega}$ are
$\\{0,~{}0,~{}1.059,~{}1.264,~{}2.088,2.387,3.406,3.477,~{}5.051,~{}6.657,$
$~{}7.388,~{}10.222\\}$, which all belong to the interval $[0.5,~{}12]$.

##  III Matrix-scaled consensus algorithms for a network of single-integrator
agents

###  III-A Matrix-scaled consensus of single-integrators

We firstly consider the ideal case, where agents are modeled by single
integrators. The matrix-scaled consensus algorithm is proposed as follows

$\displaystyle\dot{\mathbf{x}}_{i}$
$\displaystyle=\mathbf{u}_{i}=-\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\mathbf{x}_{i}-\mathbf{S}_{j}\mathbf{x}_{j}),\,i=1,\ldots,n.$
(9)

The asymptotic behavior of the system (9) is summarized in the following
theorem.

###### Theorem III.1

Let the graph $\mathcal{G}$ be connected. Under the algorithm (9),
$\mathbf{x}(t)$ exponentially converges to a point in $\mathcal{C}_{S}$.

###### Proof:

The system can be expressed in the matrix form as follows:

$\displaystyle\dot{\mathbf{x}}=-(\text{sign}(\mathbf{S})\mathbf{L}\otimes\mathbf{I}_{d})\mathbf{S}\mathbf{x}=-\bm{\Omega}\mathbf{x}.$
(10)

Let $\mathbf{V}=[\mathbf{v}_{1},\ldots,\mathbf{v}_{dn}]$ and
$\mathbf{V}^{-1}=\mathbf{Z}^{\top}$ be orthogonal matrices (which can be
chosen with real entries) such that
$\mathbf{V}_{[1:d]}=\mathbf{S}^{-1}(\mathbf{1}_{n}\otimes\mathbf{P})$,
$\mathbf{Z}_{[1:d]}=(\text{sign}(\mathbf{S})\mathbf{1}_{n}\otimes\mathbf{I}_{d})$,
where $\mathbf{P}\in\mathbb{R}^{d\times d}$ has yet to be determined.222We use
$\mathbf{V}_{[k:r]},k<r,$ to denote a sub-matrix containing columns
$k,\ldots,r$ of $\mathbf{V}$. Since
$\mathbf{V}^{-1}\mathbf{V}=\mathbf{I}_{dn}$, we have
$(\mathbf{1}_{n}^{\top}\text{sign}(\mathbf{S})\otimes\mathbf{I}_{d})\mathbf{S}^{-1}(\mathbf{1}_{n}\otimes\mathbf{I}_{d})\mathbf{P}=(\mathbf{1}_{n}^{\top}\otimes\mathbf{I}_{d})|\mathbf{S}|^{-1}(\mathbf{1}_{n}\otimes\mathbf{I}_{d})\mathbf{P}=\mathbf{I}_{d},$
It follows that

$\mathbf{P}=\left(\sum_{i=1}^{n}|\mathbf{S}_{i}|^{-1}\right)^{-1}.$

Moreover,

$\displaystyle(\mathbf{1}_{n}^{\top}\text{sign}(\mathbf{S})\otimes\mathbf{I}_{d})\mathbf{V}_{[d+1:dn]}=\mathbf{0}_{d\times
d(n-1)}.$ (11)

Thus, we can express

$\displaystyle\mathbf{V}^{-1}\bm{\Omega}\mathbf{V}=\begin{bmatrix}\mathbf{0}_{d\times
d}&\mathbf{0}_{d\times d(n-1)}\\\ \mathbf{0}_{d(n-1)\times
d}&\bm{\Omega}^{\prime}\end{bmatrix},$ (12)

where
$\bm{\Omega}^{\prime}=\mathbf{Z}_{[d+1:d(n-1)]}^{\top}\bm{\Omega}\mathbf{V}_{[d+1:dn]}$
is Hurwitz.

From linear control theory, we have

$\displaystyle\mathbf{x}(t)$
$\displaystyle=\text{exp}(-\bm{\Omega}^{\prime}t)\mathbf{x}(0)$
$\displaystyle=\mathbf{V}\begin{bmatrix}\mathbf{I}_{d}&\mathbf{0}_{d\times
d(n-1)}\\\ \mathbf{0}_{d(n-1)\times
d}&\text{exp}(-\bm{\Omega}^{\prime}t)\end{bmatrix}\mathbf{V}^{-1}\mathbf{x}(0),$

and thus,

$\displaystyle\lim_{t\to+\infty}\mathbf{x}(t)$
$\displaystyle=\mathbf{V}\begin{bmatrix}\mathbf{I}_{d}&\mathbf{0}_{d\times
d(n-1)}\\\ \mathbf{0}_{d(n-1)\times d}&\mathbf{0}_{d(n-1)\times
d(n-1)}\end{bmatrix}\mathbf{V}^{-1}\mathbf{x}(0),$
$\displaystyle=\mathbf{S}^{-1}(\mathbf{1}_{n}\otimes\mathbf{I}_{d})\mathbf{P}(\mathbf{1}_{n}^{\top}\text{sign}(\mathbf{S})\otimes\mathbf{I}_{d})\mathbf{x}(0)$
$\displaystyle=\mathbf{S}^{-1}(\mathbf{1}_{n}\otimes\mathbf{x}^{a}),$ (13)

where
$\mathbf{x}^{a}:=\mathbf{P}\sum_{i=1}^{n}\text{sign}(\mathbf{S}_{i})\mathbf{x}_{i}(0)$
and the convergence rate is exponential. It follows that
$\mathbf{x}(t)\to\mathbf{S}^{-1}(\mathbf{1}_{n}\otimes\mathbf{x}^{a})\in\mathcal{C}_{S}$,
as $t\to+\infty$. ∎

###### Remark III.1

Note that the proof that
$\lim_{t\to+\infty}\mathbf{x}(t)=\mathbf{S}\mathbf{x}^{a}$ can also be shown
by considering the Lyapunov candidate function
$V=\mathbf{x}^{\top}\mathbf{S}^{\top}(\mathbf{L}\otimes\mathbf{I}_{d})\mathbf{S}\mathbf{x}$.

###### Remark III.2

Let $\mathbf{f}:\mathbb{R}^{N}\to\mathbb{R}^{N}\,(N\in\mathbb{N}_{+})$ be a
Lipschitz continuous function that satisfies
$\mathbf{y}^{\top}\mathbf{f}(\mathbf{y})>\mathbf{0},~{}\forall\mathbf{0}_{N}\neq\mathbf{y}\in\mathcal{D}\subseteq\mathbb{R}^{N},\,\mathbf{0}_{d}\in\mathcal{D},~{}\mathbf{f}(\mathbf{0}_{N})=\mathbf{0}_{N}.$
Consider the following nonlinear MSC algorithm

$\displaystyle{\mathbf{u}}_{i}=-\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}\mathbf{f}(\mathbf{S}_{i}\mathbf{x}_{i}-\mathbf{S}_{j}\mathbf{x}_{j}),~{}i=1,\ldots,n.$
(14)

The matrix-scaled consensus system with generalized interaction (14) can be
expressed in matrix form as

$\displaystyle\dot{\mathbf{x}}=-(\text{sign}(\mathbf{S}){\mathbf{H}}^{\top}\mathbf{W}\otimes\mathbf{I}_{d})\mathbf{f}((\mathbf{H}\otimes\mathbf{I}_{d})\mathbf{S}\mathbf{x}).$
(15)

It follows that

$\displaystyle(\mathbf{1}_{n}^{\top}\text{sign}(\mathbf{S})\otimes\mathbf{I}_{d})\dot{\mathbf{x}}=-(\mathbf{1}_{n}^{\top}\mathbf{H}^{\top}\mathbf{W}\otimes\mathbf{I}_{d})\mathbf{f}((\mathbf{H}\otimes\mathbf{I}_{d})\mathbf{S}\mathbf{x})=\mathbf{0}_{d}$

which implies that $\mathbf{x}^{a}$ is time-invariant under (14).

For stability analysis, consider the Lyapunov function

$V(\mathbf{x})=\int_{\mathbf{0}_{dn}}^{\mathbf{x}}\mathbf{f}(\bar{\mathbf{H}}\mathbf{S}\mathbf{y})^{\top}\bar{\mathbf{W}}\bar{\mathbf{H}}\mathbf{S}d\mathbf{y},$

which is positive semidefinite and continuously differentiable in
$\mathcal{D}$. Along any trajectory of (14), we have

$\displaystyle\dot{V}$
$\displaystyle=-\mathbf{f}(\bar{\mathbf{H}}\mathbf{S}\mathbf{x})^{\top}\bar{\mathbf{W}}\bar{\mathbf{H}}|\mathbf{S}|\bar{\mathbf{H}}^{\top}\bar{\mathbf{W}}\mathbf{f}(\bar{\mathbf{H}}\mathbf{S}\mathbf{x})$
$\displaystyle=-\mathbf{f}(\bar{\mathbf{H}}\mathbf{S}\mathbf{x})^{\top}\bar{\mathbf{W}}\bar{\mathbf{H}}(|\mathbf{S}|+|\mathbf{S}|^{\top})\bar{\mathbf{H}}^{\top}\bar{\mathbf{W}}\mathbf{f}(\bar{\mathbf{H}}\mathbf{S}\mathbf{x})\leq
0.$

Based on LaSalle invariance principle, each trajectory of the system
approaches the largest invariant set in
$\Omega=\\{\mathbf{x}|~{}\dot{V}=0\\}$. Solving $\dot{V}=0$ gives
$\bar{\mathbf{H}}^{\top}\bar{\mathbf{W}}\mathbf{f}(\bar{\mathbf{H}}\mathbf{S}\mathbf{x})=\mathbf{0}$,
which further implies that
$\mathbf{x}^{\top}\mathbf{S}^{\top}\bar{\mathbf{H}}^{\top}\bar{\mathbf{W}}\mathbf{f}(\bar{\mathbf{H}}\mathbf{S}\mathbf{x})={0}$,
or $\mathbf{S}\mathbf{x}\in\text{im}(\mathbf{1}_{n}\otimes\mathbf{I}_{d})$.
Combining with $\mathbf{x}^{a}(t)=\mathbf{x}^{a}(0)$ yields
$\mathbf{S}\mathbf{x}\equiv\mathbf{1}_{n}\otimes\mathbf{x}^{a}$. Thus,
$\mathbf{x}(t)\to\mathbf{S}^{-1}(\mathbf{1}_{n}\otimes\mathbf{x}^{a})\in\mathcal{C}_{S}$,
as $t\to+\infty$.

This result expands the classes of possible interaction functions. For
example, if $\|\dot{\mathbf{x}}\|_{\infty}=\|\mathbf{u}\|_{\infty}$ must be
upper bounded by a constant $\beta>0$ for all $t\geq 0$, we may choose
$\mathbf{f}(\cdot)=\frac{\beta}{\max_{i\in\\{1,\ldots,n\\}}|\mathcal{N}_{i}|}\tanh(\cdot)$.
Moreover, a larger class of nonlinearly output-coupled systems can be
considered, for examples, the Kuramoto oscillators model has
$\mathbf{f}(\cdot)=\sin(\cdot)$ and
$\mathcal{D}=\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.

###  III-B Adaptive matrix-scaled consensus of single-integrators with
parametric uncertainties

Let each agent be modeled by the single integrator with uncertain parameters
(1). The following adaptive matrix-scaled consensus law is designed based on
certainty equivalence principle

$\displaystyle\mathbf{u}_{i}$
$\displaystyle=-\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\mathbf{x}_{i}-\mathbf{S}_{j}\mathbf{x}_{j})-\bm{\phi}_{i}(t,\mathbf{x}_{i})\hat{\bm{\theta}}_{i},$
(16a) $\displaystyle\dot{\hat{\bm{\theta}}}_{i}$
$\displaystyle=\gamma_{i}\bm{\phi}_{i}(t,\mathbf{x}_{i})^{\top}\mathbf{S}_{i}^{\top}\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\mathbf{x}_{i}-\mathbf{S}_{j}\mathbf{x}_{j}),$
(16b)

where $\gamma_{i}>0$ are adaptive rates, $i=1,\ldots,n$. We can rewrite the
system (1) under the matrix-scaled consensus algorithm (16a)–(16b) in matrix
form as follows:

$\displaystyle\dot{\mathbf{x}}$
$\displaystyle=-\bm{\Omega}\mathbf{x}+\text{blkdiag}(\bm{\phi}_{1},\ldots,\bm{\phi}_{n})(\bm{\theta}-\hat{\bm{\theta}}),$
(17a) $\displaystyle\dot{\hat{\bm{\theta}}}$
$\displaystyle=\text{blkdiag}(\gamma_{1}\bm{\phi}_{1},\ldots,\gamma_{n}\bm{\phi}_{n})^{\top}\mathbf{S}^{\top}(\mathbf{L}\otimes\mathbf{I}_{d})\mathbf{S}\mathbf{x},$
(17b)

where
$\mathbf{x}=\text{vec}(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathbb{R}^{dn}$,
$\bm{\theta}=\text{vec}(\bm{\theta}_{1},\ldots,\bm{\theta}_{n})\in\mathbb{R}^{nr}$,
$\hat{\bm{\theta}}=\text{vec}(\hat{\bm{\theta}}_{1},\ldots,\hat{\bm{\theta}}_{n})\in\mathbb{R}^{nr}$,
and
$\bm{\phi}(t)=\text{blkdiag}(\bm{\phi}_{1},\ldots,\bm{\phi}_{n})\in\mathbb{R}^{dn\times
dr}$.

###### Theorem III.2

Suppose that the graph $\mathcal{G}$ is connected and $\bm{\phi}(t)$ is
uniformly bounded. Under the adaptive matrix-scaled algorithm (16a)–(16b), we
have

  1. (i)

$\mathbf{x}(t)\to\mathcal{C}_{S}$ and $\bm{\theta}(t)\to\bm{\theta}^{*}$, as
$t\to+\infty$, and

  2. (ii)

if additionally, $\bm{\phi}_{i}(t)$ is persistently exciting (PE), i.e., there
exist $\mu_{1}\geq\mu_{2}>0$, $T>0$ such that for any $t\geq 0$,
$\mu_{1}\mathbf{I}_{d}\geq\int_{t}^{t+T}\bm{\phi}_{i}(\tau)\bm{\phi}_{i}(\tau)^{\top}d\tau\geq\mu_{2}\mathbf{I}_{d},$
then $\hat{\bm{\theta}}_{i}(t)\to\bm{\theta}_{i},~{}\forall i=1,\ldots,n,$ as
$t\to+\infty$.

###### Proof:

(i) Consider the function
$V=\frac{1}{2}\mathbf{x}^{\top}\mathbf{S}^{\top}\bar{\mathbf{L}}\mathbf{S}\mathbf{x}+\sum_{i=1}^{n}\frac{1}{2\gamma_{i}}(\bm{\theta}_{i}-\hat{\bm{\theta}}_{i})^{\top}(\bm{\theta}_{i}-\hat{\bm{\theta}}_{i}).$
Since $\mathbf{S}$ is invertible and $\bar{\mathbf{L}}$ is symmetric,
$\mathbf{S}^{\top}\bar{\mathbf{L}}\mathbf{S}$ has the same number of positive,
negative and zero eigenvalues as $\bar{\mathbf{L}}$. It follows that
$\mathbf{S}^{\top}\bar{\mathbf{L}}\mathbf{S}$ is symmetric positive
semidefinite. As a result, $V$ is continuously differentiable, positive
definite and radially unbounded with regard to
$\text{vec}(\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}\mathbf{S}\mathbf{x},\bm{\theta}-\hat{\bm{\theta}})$.
Moreover,
$\frac{1}{4}\|\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}\mathbf{S}\mathbf{x}\|^{2}+\frac{1}{2\max_{i\in\mathcal{V}}\gamma_{i}}\|\bm{\theta}-\hat{\bm{\theta}}\|^{2}\leq
V\leq\|\sqrt{\overline{\mathbf{W}}}\bar{\mathbf{H}}\mathbf{S}\mathbf{x}\|^{2}+\frac{1}{2\min_{i\in\mathcal{V}}\gamma_{i}}\|\bm{\theta}-\hat{\bm{\theta}}\|^{2}$,
and along any trajectory of the system,

$\displaystyle\dot{V}(t)=-\mathbf{x}^{\top}\mathbf{S}^{\top}\bar{\mathbf{L}}\left(|\mathbf{S}|+|\mathbf{S}^{\top}|\right)\bar{\mathbf{L}}\mathbf{S}\mathbf{x}\leq
0,$

which implies that $\|\bar{\mathbf{L}}\mathbf{S}\mathbf{x}\|$,
$\|\bm{\theta}-\hat{\bm{\theta}}\|$ are uniformly bounded, and
$\lim_{t\to+\infty}V(t)$ exists and is finite. As $\bm{\theta}$ is a constant
vector, it follows that $\hat{\bm{\theta}}$ is uniformly bounded. Further,
$\|\dot{\mathbf{x}}(t)\|\leq\|\bar{\mathbf{L}}\mathbf{S}\mathbf{x}(t)\|+\|\text{blkdiag}(\bm{\phi}_{1},\ldots,\bm{\phi}_{n})\|\|\bm{\theta}-\hat{\bm{\theta}}\|$
is uniformly bounded. Therefore,

$\displaystyle\ddot{V}(t)$
$\displaystyle=-2\mathbf{x}^{\top}\mathbf{S}^{\top}\bar{\mathbf{L}}\left(|\mathbf{S}|+|\mathbf{S}|^{\top}\right)\bar{\mathbf{L}}\mathbf{S}\dot{\mathbf{x}}$
$\displaystyle\leq-2\lambda_{\min}\left(|\mathbf{S}|+|\mathbf{S}|^{\top}\right)\|\bar{\mathbf{L}}\mathbf{S}\mathbf{x}\|\|\bar{\mathbf{L}}\|\|\mathbf{S}\|\|\dot{\mathbf{x}}\|$

is also uniformly bounded. It follows from Barbalat’s lemma that
$\lim_{t\to+\infty}\dot{V}(t)=0$, or
$\lim_{t\to+\infty}\bar{\mathbf{L}}\mathbf{S}\mathbf{x}(t)=\mathbf{0}_{dn}$.
This implies that
$\mathbf{S}_{i}\mathbf{x}_{i}\to\mathbf{S}_{j}\mathbf{x}_{j}$ as
$t\to+\infty$, or $\mathbf{x}(t)$ approaches the set $\mathcal{C}_{S}$ as
$t\to+\infty$. As $\lim_{t\to+\infty}V(t)$ exists and
$\lim_{t\to+\infty}\mathbf{x}^{\top}\mathbf{S}^{\top}\bar{\mathbf{L}}\mathbf{S}\mathbf{x}(t)={0}$,
it follows that $\lim_{t\to+\infty}\bm{\theta}(t)=\bm{\theta}^{*}$ exists and
is finite.

(ii) Next, let $\tilde{\bm{\theta}}=\hat{\bm{\theta}}-\bm{\theta}$ and define

$\displaystyle\mathbf{y}=[\mathbf{y}_{\mathcal{T}}^{\top},\mathbf{y}_{\mathcal{C}}^{\top}]^{\top}$
$\displaystyle=\bar{\mathbf{H}}\mathbf{S}\mathbf{x}$
$\displaystyle=\left(\begin{bmatrix}\mathbf{I}_{n-1}\\\
\mathbf{T}\end{bmatrix}\otimes\mathbf{I}_{d}\right)(\mathbf{H}_{\mathcal{T}}\otimes\mathbf{I}_{d})\mathbf{S}\mathbf{x}$
$\displaystyle=\left(\mathbf{R}\otimes\mathbf{I}_{d}\right)\mathbf{y}_{\mathcal{T}}=\bar{\mathbf{R}}\mathbf{y}_{\mathcal{T}},$

it follows that

$\displaystyle\dot{\mathbf{y}}_{\mathcal{T}}$
$\displaystyle=\bm{\varphi}(\mathbf{y}_{\mathcal{T}})+\bar{\mathbf{H}}_{\mathcal{T}}\mathbf{S}\text{blkdiag}(\bm{\phi}_{1},\ldots,\bm{\phi}_{n})\tilde{\bm{\theta}},$
(18a) $\displaystyle\dot{\tilde{\bm{\theta}}}$ $\displaystyle=\bm{\psi}(t).$
(18b)

Observe that

$\lim_{t\to+\infty}\mathbf{y}_{\mathcal{T}}(t)=\mathbf{0}_{d(n-1)},$
$\lim_{t\to+\infty}\bm{\varphi}(\mathbf{y}_{\mathcal{T}}(t))=-\bar{\mathbf{H}}_{\mathcal{T}}|\mathbf{S}|\lim_{t\to+\infty}\bar{\mathbf{L}}\mathbf{S}\mathbf{x}(t)=\mathbf{0}_{d(n-1)},$
$\lim_{t\to+\infty}\bm{\psi}(t)=-\lim_{t\to+\infty}\text{blkdiag}(\gamma_{1}\bm{\phi}_{1},\ldots,\gamma_{n}\bm{\phi}_{n})\mathbf{S}^{\top}\bar{\mathbf{L}}\mathbf{S}\mathbf{x}=\mathbf{0}_{nr},$

and

$\displaystyle W(t)$
$\displaystyle=\int_{t}^{t+T}\bar{\mathbf{H}}_{\mathcal{T}}\mathbf{S}\bm{\phi}_{i}(\tau)\bm{\phi}_{i}(\tau)^{\top}\mathbf{S}^{\top}\bar{\mathbf{H}}_{\mathcal{T}}^{\top}d\tau$
$\displaystyle=\bar{\mathbf{H}}_{\mathcal{T}}\mathbf{S}\left(\int_{t}^{t+T}\bm{\phi}_{i}(\tau)\bm{\phi}_{i}(\tau)^{\top}d\tau\right)\mathbf{S}^{\top}\bar{\mathbf{H}}_{\mathcal{T}}^{\top}$

satisfies

$\displaystyle\mu_{1}\bar{\mathbf{H}}_{\mathcal{T}}\mathbf{S}\mathbf{S}^{\top}\bar{\mathbf{H}}_{\mathcal{T}}^{\top}\leq
W(t)\leq\mu_{2}\bar{\mathbf{H}}_{\mathcal{T}}\mathbf{S}\mathbf{S}^{\top}\bar{\mathbf{H}}_{\mathcal{T}}^{\top},$
(19a) $\displaystyle\mu_{1}\lambda_{\max}(\mathbf{L}_{e})\mathbf{I}_{dn}\leq
W(t)\leq\mu_{2}\lambda_{\max}(\mathbf{L}_{e})\mathbf{I}_{dn}.$ (19b)

As
$\mathbf{L}_{e}=\bar{\mathbf{H}}_{\mathcal{T}}\mathbf{S}\mathbf{S}^{\top}\bar{\mathbf{H}}_{\mathcal{T}}^{\top}=\mathbf{L}_{e}^{\top}$
is positive semidefinite,
rank$(\mathbf{L}_{e})$=rank$(\bar{\mathbf{H}}_{\mathcal{T}})=d(n-1)$, it
follows that $\mathbf{L}_{e}$ is positive definite. Since all conditions of
Lemma A.1 (in the Appendix A) are satisfied,
$\lim_{t\to+\infty}\tilde{\bm{\theta}}(t)=\mathbf{0}_{nr}$ or
$\lim_{t\to+\infty}\hat{\bm{\theta}}(t)={\bm{\theta}}$. ∎

##  IV Matrix-scaled consensus algorithms for a network of linear dynamical
agents

In this section, we firstly propose two matrix scaled consensus algorithms for
the simplified general linear models (2) and (3). The proposed algorithms and
stability analyses in Subsection IV-A prepare for Subsections IV-B and IV-C,
where the general linear models (2) and (3) will be considered.

###  IV-A Simplified linear agents

####  IV-A1 The agents’ dynamics are identical

Let the agents be modeled by (2) with $\mathbf{B}=\mathbf{I}_{d}$. We assume
that each agent has information on the state variable $\mathbf{x}_{i}$ and
exchanges the matrix-scaled state $\mathbf{S}_{i}\mathbf{x}_{i}$ with its
neighboring agents. The following matrix-scaled consensus algorithm is
proposed

$\displaystyle\mathbf{u}_{i}=-c\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\mathbf{x}_{i}-\mathbf{S}_{j}\mathbf{x}_{j}),~{}i=1,\ldots,n,$
(20)

where $c>0$ is a coupling gain. The $n$-agent system can be written in matrix
form as

$\displaystyle\dot{\mathbf{x}}=(\mathbf{I}_{n}\otimes\mathbf{A}-c\bm{\Omega})\mathbf{x}.$
(21)

For stability analysis of the system (21), the following lemma whose proof is
given in Appendix A will be used.

###### Lemma IV.1

Let $\bm{\Theta}$ be a Hurwitz matrix, $\bm{\Delta}_{\Theta}$ is a
perturbation matrix of the same dimension with
$\|\bm{\Delta}_{\Theta}\|=\delta_{\Theta}$, and
$\mathbf{Q}=\mathbf{Q}^{\top}>0$ is the unique solution to the Lyapunov
equation $\mathbf{Q}\bm{\Theta}+\bm{\Theta}^{\top}\mathbf{Q}=-\mathbf{I}$. For
any $c>2\delta_{\Theta}\lambda_{\max}(\mathbf{Q})$, the matrix
$c\bm{\Theta}+\bm{\Delta}_{\Theta}$ is Hurwitz.

Let $\mathbf{y}=\mathbf{V}^{-1}\mathbf{x}\in\mathbb{R}^{dn}$, and define
$\mathbf{y}_{[k:r]}=[y_{k},y_{k+1},\ldots,y_{r}]^{\top}$, for $k<r$, it
follows that

$\displaystyle\dot{\mathbf{y}}_{[1:d]}$
$\displaystyle=(\mathbf{1}_{n}^{\top}\text{sign}(\mathbf{S}))\otimes\mathbf{I}_{d})(\mathbf{I}_{n}\otimes\mathbf{A}-c\bm{\Omega})\mathbf{V}\mathbf{y}=\mathbf{A}\mathbf{y}_{[1:d]},$
$\displaystyle\dot{\mathbf{y}}_{[d+1:dn]}$
$\displaystyle=\underbrace{\left(\mathbf{Z}_{[d+1:dn]}^{\top}(\mathbf{I}_{n}\otimes\mathbf{A})\mathbf{V}_{[d+1:dn]}-c\bm{\Omega}^{\prime}\right)}_{:=\bm{\Theta}_{c}}\mathbf{y}_{[d+1:dn]}.$

We have the following theorem on the system (21):

###### Theorem IV.1

Suppose that the graph $\mathcal{G}$ is connected. Let
$\mathbf{Q}=\mathbf{Q}^{\top}>0$ be the unique solution of the Lyapunov
equation
$\mathbf{Q}{\bm{\Omega}^{\prime}}+(\bm{\Omega}^{\prime})^{\top}\mathbf{Q}=-\mathbf{I}_{d(n-1)}$
and $c>2\|\mathbf{A}\|\lambda_{\max}(\mathbf{Q})$. Then,
$\mathbf{x}(t)\to\mathcal{C}_{S}$, as $t\to+\infty$.

###### Proof:

We have
$\|\mathbf{Z}_{[d+1:dn]}^{\top}(\mathbf{I}_{n}\otimes\mathbf{A})\mathbf{V}_{[d+1:dn]}\|\leq\|\mathbf{V}^{-1}(\mathbf{I}_{n}\otimes\mathbf{A})\mathbf{V}\|=\|\mathbf{I}_{n}\otimes\mathbf{A}\|=\|\mathbf{A}\|$.
Based on Lemma IV.1, the matrix $\bm{\Theta}_{c}$ is Hurwitz. It follows that
$\mathbf{y}_{[d+1:dn]}$ globally exponentially converges to
$\mathbf{0}_{d(n-1)}$.

The solution of equation (21) is

$\displaystyle\mathbf{x}(t)$
$\displaystyle=\mathbf{V}\mathbf{y}(t)=\mathbf{V}\text{blkdiag}(\text{exp}(\mathbf{A}t),\text{exp}(\bm{\Theta}_{c}t))\mathbf{y}(0)$
$\displaystyle=\mathbf{S}^{-1}(\mathbf{1}_{n}\otimes\mathbf{P})\text{exp}(\mathbf{A}t)(\mathbf{1}_{n}^{\top}\text{sign}(\mathbf{S})\otimes\mathbf{I}_{d})\mathbf{x}(0)+\bm{\xi}(t),$

where

$\bm{\xi}(t)=\mathbf{V}_{[d+1:dn]}\text{exp}(\bm{\Theta}_{c}t)\mathbf{Z}_{[d+1:dn]}^{\top}\mathbf{x}(0)\to\mathbf{0}_{dn}$

exponentially fast. Thus,

$\lim_{t\to+\infty}(\mathbf{x}_{i}(t)-\mathbf{S}_{i}^{-1}\mathbf{P}\text{exp}(\mathbf{A}t)\mathbf{P}^{-1}\mathbf{x}^{a})=\mathbf{0}_{d},~{}\forall
i=1,\ldots,n,$

or $\mathbf{x}(t)\to\mathcal{C}_{S}$, as $t\to+\infty$. ∎

####  IV-A2 Simplified heterogeneous linear agents with unknown system matrix
and full control input

In this subsection, we design an adaptive matrix-scaled consensus law for a
system of heterogeneous linear dynamics with full control input

$\dot{\mathbf{x}}_{i}=\mathbf{A}_{i}\mathbf{x}_{i}+\mathbf{u}_{i},$ (22)

where $\mathbf{A}_{i}\in\mathbb{R}^{d\times d}$ is unknown to each agent $i$.
Let
$\mathbf{A}_{i}=\mathbf{A}+(\mathbf{A}_{i}-\mathbf{A})=\mathbf{A}+\bar{\mathbf{A}}_{i}$,
where $\mathbf{A}$ is a pre-selected matrix such that
$\mathbf{W}_{d+1:dn}^{\top}\otimes\mathbf{A}\mathbf{V}_{[d+1:dn]}-c\bm{\Omega}^{\prime}$
is Hurwitz.

Let the $k$-th row of the matrix $\bar{\mathbf{A}}_{i}$ be denoted by
$\bar{\mathbf{A}}_{ik}$, we have the representation [24]:

$\bar{\mathbf{A}}_{i}\mathbf{x}_{i}=\begin{bmatrix}\bar{\mathbf{A}}_{i1}^{\top}\mathbf{x}_{i}\\\
\vdots\\\
\bar{\mathbf{A}}_{in}^{\top}\mathbf{x}_{i}\end{bmatrix}=\begin{bmatrix}\mathbf{x}_{i}^{\top}\bar{\mathbf{A}}_{i1}\\\
\vdots\\\
\mathbf{x}_{i}^{\top}\bar{\mathbf{A}}_{id}\end{bmatrix}=(\mathbf{I}_{d}\otimes\mathbf{x}_{i}^{\top})\text{vec}(\bar{\mathbf{A}}_{i}).$

By setting
$\bm{\phi}_{i}:=\mathbf{I}_{d}\otimes\mathbf{x}_{i}^{\top}\in\mathbb{R}^{d\times
d^{2}}$ and $\bm{\theta}_{i}:=\text{vec}(\bar{\mathbf{A}}_{i})$, the linear
model (22) is transformed into the single-integrator with uncertain parameters
(1).

To make the system achieves matrix-scaled consensus, the effect of the
uncertainty should be compensated faster than the evolution of
$\|\mathbf{x}\|$ (exponential rate). The matrix-scaled consensus algorithm is
proposed as follows

$\displaystyle\dot{\mathbf{z}}_{i}$
$\displaystyle=\mathbf{A}\mathbf{x}_{i}-c\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\mathbf{x}_{i}-\mathbf{S}_{j}\mathbf{x}_{j}),$
(23a) $\displaystyle\mathbf{u}_{i}$
$\displaystyle=-c\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\mathbf{x}_{i}-\mathbf{S}_{j}\mathbf{x}_{j})+\beta_{1}\text{sgn}(\mathbf{e}_{i})$
$\displaystyle\qquad+\beta_{2}\text{diag}(\text{sgn}(\mathbf{e}_{i}))(\mathbf{I}_{d}\otimes|\mathbf{x}_{i}|^{\top})\mathbf{1}_{d^{2}},$
(23b)

where $|\mathbf{x}_{i}|:=[|x_{i1}|,\ldots,|x_{id}|]^{\top}$,
$\mathbf{e}_{i}={\mathbf{z}}_{i}-\mathbf{x}_{i},$ $\beta_{1},\beta_{2}>0$ are
control gains, and sgn$(\cdot)$ denotes the signum function, defined as
sgn$(x)=1$ if $x>0$, sgn$(x)=-1$ if $x<0$, and sgn$(x)=0$ if $x=0$. The
following theorem will be proved.

###### Theorem IV.2

Let $\mathcal{G}$ be connected and
$\beta_{2}\geq\max_{i}\|\text{vec}(\mathbf{\bar{A}}_{i})\|_{\infty}$. Under
the algorithm (23), $\mathbf{x}(t)\to\mathcal{C}_{S}$ as $t\to+\infty$.

###### Proof:

We have

$\displaystyle\dot{\mathbf{e}}_{i}$
$\displaystyle=\bar{\mathbf{A}}_{i}\mathbf{x}_{i}-\beta_{1}\text{sgn}(\mathbf{e}_{i})-\beta_{2}\text{sgn}(\mathbf{e}_{i})(|\mathbf{x}_{i}|^{\top}\otimes\mathbf{I}_{d})\mathbf{1}_{d^{2}}$
$\displaystyle=-\beta_{1}\text{sgn}(\mathbf{e}_{i})+\bm{\phi}_{i}(t)\bm{\theta}_{i}-\beta_{2}\text{diag}(\text{sgn}(\mathbf{e}_{i}))(\mathbf{I}_{d}\otimes|\mathbf{x}_{i}|^{\top})\mathbf{1}_{d^{2}}$

As the right-hand side of the above equation is discontinuous, the solution of
$\mathbf{e}_{i}$ is understood in Filippov sense. Using the Lyapunov function
$V=\frac{1}{2}\|\mathbf{e}_{i}\|^{2}$, we have

$\displaystyle\dot{V}$
$\displaystyle\in^{\text{a.e.}}\tilde{V}=\bigcap_{\bm{\nu}\in\partial
V}\bm{\nu}^{\top}K[\dot{\mathbf{e}}_{i}]$
$\displaystyle=\mathbf{e}_{i}^{\top}\Big{(}-\beta_{1}\text{sgn}(\mathbf{e}_{i})+\bm{\phi}_{i}(t)\bm{\theta}_{i}$
$\displaystyle\qquad\qquad-\beta_{2}\text{diag}(\text{sgn}(\mathbf{e}_{i}))(|\mathbf{x}_{i}|^{\top}\otimes\mathbf{I}_{d})\mathbf{1}_{d^{2}}\Big{)}$
$\displaystyle\leq-\beta\|\mathbf{e}_{i}\|_{1}-\beta_{2}\sum_{k=1}^{d}|\mathbf{e}_{ik}|\|\mathbf{x}_{i}\|_{1}+\sum_{k=1}^{d}|\mathbf{e}_{ik}|\|\mathbf{x}_{i}\|_{1}\|\bm{\theta}_{i}\|_{\infty}$
$\displaystyle\leq-\beta\|\mathbf{e}_{i}\|_{1}\leq-\epsilon V^{1/2},$ (24)

for $\epsilon=\beta\sqrt{2}$, and this implies that $V(t)\to 0$ in finite time
[29, 30]. Therefore, there exists $T>0$ such that
$\mathbf{e}_{i}=\mathbf{0}_{d}$ or $\mathbf{x}_{i}=\mathbf{z}_{i}$ for $t\geq
T$. For $t\geq T$, $\mathbf{x}_{i}$ dynamics becomes the simplified
homogeneous linear agent (21). Thus, the system eventually achieves a matrix-
scaled consensus. ∎

###  IV-B Homogeneous general linear agents

In this subsection, we consider the $n$-agent system with identical general
linear model (2). The following observer-based matrix-scaled consensus
algorithm is proposed

$\displaystyle\dot{\hat{\mathbf{x}}}_{i}=\mathbf{A}\hat{\mathbf{x}}_{i}+\mathbf{B}\mathbf{u}_{i}+\mathbf{H}(\mathbf{C}{\hat{\mathbf{x}}}_{i}-\mathbf{y}_{i}),$
(25a)
$\displaystyle\dot{\bm{\eta}}_{i}=\mathbf{A}{\bm{\eta}}_{i}+\mathbf{B}\mathbf{u}_{i}+\mathbf{H}(\mathbf{C}{\hat{\mathbf{x}}}_{i}-\mathbf{y}_{i})$
$\displaystyle\qquad+c\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}(\hat{\mathbf{x}}_{i}-\bm{\eta}_{i})+\mathbf{S}_{j}(\hat{\mathbf{x}}_{j}-\bm{\eta}_{j})),$
(25b) $\displaystyle\mathbf{u}_{i}=\mathbf{K}\bm{\eta}_{i},\,i=1,\ldots,n,$
(25c)

where the matrices $\mathbf{H}$ and $\mathbf{K}$ are designed so that
$\mathbf{A}+\mathbf{H}\mathbf{C}$ and $\mathbf{A}+\mathbf{B}\mathbf{K}$ are
Hurwitz, and $c>0$ is a coupling gain.

In the proposed algorithm, (25a) is a Luenberger observer used to estimate the
state $\mathbf{x}$. The matrix-scaled consensus algorithm is conducted via the
auxiliary state ${\bm{\zeta}}_{i}=\hat{\mathbf{x}}_{i}-\bm{\eta}_{i}$. It is
worth noting that
$c\text{sign}(\mathbf{S}_{i})(\mathbf{S}_{i}(\hat{\mathbf{x}}_{i}-\bm{\eta}_{i})+\mathbf{S}_{j}(\hat{\mathbf{x}}_{j}-\bm{\eta}_{j}))=c\text{sign}(\mathbf{S}_{i})(\mathbf{S}_{i}\bm{\zeta}_{i}-\mathbf{S}_{j}\bm{\zeta}_{j})$
acts as a proportional control input for the scaled consensus process. When
the coupling gain $c$ is sufficiently large, it becomes a high-gain controller
and the auxiliary variables are forced to synchronize in matrix scale to a
solution of the system
$\dot{\bm{\zeta}}_{0}=\mathbf{P}\mathbf{A}\mathbf{P}^{-1}{\bm{\zeta}}_{0}$.
Finally, the state variables $\mathbf{x}_{i}$ achieve a matrix-scaled
consensus by forcing $\mathbf{\bm{\eta}}_{i}$ converge to $\mathbf{0}_{d}$
under the control law (25c).

For stability analysis, from Eq. (25), we can write

$\displaystyle\dot{\tilde{\mathbf{x}}}_{i}$
$\displaystyle=(\mathbf{A}+\mathbf{H}\mathbf{C}){\tilde{\mathbf{x}}}_{i},$
(26a) $\displaystyle\dot{\bm{\zeta}}_{i}$
$\displaystyle=\mathbf{A}{\bm{\zeta}}_{i}-c\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\bm{\zeta}_{i}-\mathbf{S}_{j}\bm{\zeta}_{j}),$
(26b)

where $i=1,\ldots,n,$ and
$\tilde{\mathbf{x}}_{i}=\hat{\mathbf{x}}_{i}-\mathbf{x}_{i}$. Denoting
$\bm{{\tilde{\mathbf{x}}}}=\text{vec}({\tilde{\mathbf{x}}}_{1},\ldots,{\tilde{\mathbf{x}}}_{n})$,
${\bm{\zeta}}=\text{vec}({\bm{\zeta}}_{1},\ldots,{\bm{\zeta}}_{n})$, and
${\bm{\eta}}=\text{vec}({\bm{\eta}}_{1},\ldots,{\bm{\eta}}_{n})$, we will
prove a theorem regarding the following system

$\displaystyle\dot{\tilde{\mathbf{x}}}$
$\displaystyle=\mathbf{I}_{n}\otimes(\mathbf{A}+\mathbf{H}\mathbf{C})\tilde{\mathbf{x}},$
(27a) $\displaystyle\dot{\bm{\zeta}}$
$\displaystyle=(\mathbf{I}_{n}\otimes\mathbf{A}-c\bm{\Omega}){\bm{\zeta}},$
(27b) $\displaystyle\dot{\bm{\eta}}$
$\displaystyle=(\mathbf{I}_{n}\otimes(\mathbf{A}+\mathbf{B}\mathbf{K})){\bm{\eta}}+c(\mathbf{I}_{n}\otimes(\mathbf{H}\mathbf{C}))\tilde{\mathbf{x}}+\bm{\Omega}{\bm{\zeta}}.$
(27c)

###### Theorem IV.3

Suppose that $\mathcal{G}$ is connected, $\mathbf{A}+\mathbf{H}\mathbf{C}$ and
$\mathbf{A}+\mathbf{B}\mathbf{K}$ are Hurwitz, and the control gain $c$ is
chosen so that
$\bm{\Theta}_{c}=\mathbf{Z}_{[d+1:dn]}^{\top}(\mathbf{I}_{n}\otimes\mathbf{A})\mathbf{V}_{[d+1:dn]}-c\bm{\Omega}^{\prime}$
is Hurwitz. Under the algorithm (25), $\mathbf{x}(t)\to\mathcal{C}_{S}$ as
$t\to+\infty$. Each trajectory of $\mathbf{x}_{i}(t),~{}i=1,\ldots,n,$ differs
from a common solution of
$\dot{\bm{\zeta}}_{0}=\mathbf{P}\mathbf{A}\mathbf{P}^{-1}\bm{\zeta}_{0}$ by
the scaling matrix $\mathbf{S}_{i}^{-1}$.

###### Proof:

(i) Because $\mathbf{A}+\mathbf{H}\mathbf{C}$ is Hurwitz,
${\tilde{\mathbf{x}}}$ converges to $\mathbf{0}_{dn}$ exponentially fast.

(ii) It follows from Theorem IV.1 that $\bm{\zeta}(t)$ exponentially converges
to $\mathcal{C}_{S}$. More specifically,

$\lim_{t\to+\infty}(\bm{\zeta}_{i}(t)-\mathbf{S}_{i}^{-1}\mathbf{P}\text{exp}(\mathbf{A}t)\mathbf{P}^{-1}\bm{\zeta}^{a})=\mathbf{0}_{d},~{}\forall
i=1,\ldots,n,$

where

$\bm{\zeta}^{a}=(\mathbf{1}_{n}^{\top}\text{sign}(\mathbf{S})\otimes\mathbf{P})\bm{\zeta}(0)=\mathbf{P}\sum_{i=1}^{n}\text{sign}(\mathbf{S}_{i})\bm{\zeta}_{i}(0).$

(iii) Consider equation (27c) with $\tilde{\mathbf{x}}=\mathbf{0}_{dn}$ and
$\bm{\zeta}\in\mathcal{C}_{S}$, we have the unforced system

$\dot{\bm{\eta}}=(\mathbf{I}_{n}\otimes(\mathbf{A}+\mathbf{B}\mathbf{K}))\bm{\eta}.$

As $\mathbf{A}+\mathbf{B}\mathbf{K}$ is Hurwitz, the unforced system is
globally exponentially stable, and (27c) is input-to-state stable [31]. As the
external input
$(\mathbf{I}_{n}\otimes(\mathbf{H}\mathbf{C}))\tilde{\mathbf{x}}+\bm{\Omega}{\bm{\zeta}}$
converges to $\mathbf{0}_{dn}$ exponentially fast according to (i) and (ii),
${\bm{\eta}}\to\mathbf{0}_{dn}$ exponentially fast.

(iv) Finally, as
$\bm{\zeta}=\hat{\mathbf{x}}-\bm{\eta}=\mathbf{x}-\tilde{\mathbf{x}}-\bm{\eta}$,

$\lim_{t\to+\infty}\bm{\eta}=\lim_{t\to+\infty}\tilde{\mathbf{x}}=\mathbf{0}_{dn},$

it follows that

$\lim_{t\to+\infty}(\mathbf{x}_{i}(t)-\mathbf{S}_{i}^{-1}\mathbf{P}\text{exp}(\mathbf{A}t)\mathbf{P}^{-1}\bm{\zeta}^{a})=\mathbf{0}_{d},$

for all $i=1,\ldots,n.$

Thus, we conclude that $\mathbf{x}(t)\to\mathcal{C}_{S}$, as $t\to+\infty$,
and the convergence rate is exponential. ∎

###  IV-C Heterogeneous general linear agents

Finally, we design a matrix-scaled consensus algorithm for the system with $n$
heterogenenous linear agents (3). The main idea is combining the control
strategy in subsection IV-B and the disturbance observer in IV-A2. Consider
the algorithm

$\displaystyle\dot{\hat{\mathbf{x}}}_{i}=\mathbf{A}_{i}\hat{\mathbf{x}}_{i}+\mathbf{B}_{i}\mathbf{u}_{i}+\mathbf{H}_{i}(\mathbf{C}_{i}{\hat{\mathbf{x}}}_{i}-\mathbf{y}_{i}),$
(28a)
$\displaystyle\dot{\bm{\eta}}_{i}=\mathbf{A}_{i}{\bm{\eta}}_{i}+\mathbf{B}_{i}\mathbf{u}_{i}+\mathbf{H}_{i}(\mathbf{C}_{i}{\hat{\mathbf{x}}}_{i}-\mathbf{y}_{i})-\hat{\mathbf{u}}_{i},$
(28b)
$\displaystyle\dot{\mathbf{z}}_{i}=\mathbf{A}\bm{\zeta}_{i}-c\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\bm{\zeta}_{i}-\mathbf{S}_{j}\bm{\zeta}_{j}),$
(28c)
$\displaystyle\hat{\mathbf{u}}_{i}=-c\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\bm{\zeta}_{i}-\mathbf{S}_{j}\bm{\zeta}_{j})$
$\displaystyle\qquad+\beta_{1}\text{sgn}(\mathbf{e}_{i})+\beta_{2}\text{diag}(\text{sgn}(\mathbf{e}_{i}))(\mathbf{I}_{d}\otimes|\bm{\zeta}_{i}|^{\top})\mathbf{1}_{d^{2}},$
(28d)
$\displaystyle\mathbf{u}_{i}=\mathbf{K}_{i}\bm{\eta}_{i},~{}i=1,\ldots,n,$
(28e)

where $\bm{\zeta}_{i}=\hat{\mathbf{x}}_{i}-\bm{\eta}_{i}$,
$\mathbf{e}_{i}={\mathbf{z}}_{i}-\bm{\zeta}_{i}$,
$\mathbf{A}_{i}+\mathbf{B}_{i}\mathbf{K}_{i}$ and
$\mathbf{A}_{i}+\mathbf{H}_{i}\mathbf{C}_{i}$ are Hurwitz. We have the
following theorem:

###### Theorem IV.4

Suppose that $\mathcal{G}$ is connected,
$\mathbf{A}_{i}+\mathbf{H}_{i}\mathbf{C}_{i}$ and
$\mathbf{A}_{i}+\mathbf{B}_{i}\mathbf{K}_{i}$ are Hurwitz, $\forall
i=1,\ldots,n$, $\beta_{2}\geq\max_{i}\|\bar{\mathbf{A}}_{i}\|_{\infty}$, and
the control gain $c$ is chosen so that
$\bm{\Theta}_{c}=\mathbf{Z}_{[d+1:dn]}^{\top}(\mathbf{I}_{n}\otimes\mathbf{A})\mathbf{V}_{[d+1:dn]}-c\bm{\Omega}^{\prime}$
is Hurwitz. Under the matrix scaled consensus algorithm (28),
$\mathbf{x}(t)\to\mathcal{C}_{S}$ as $t\to+\infty$.

###### Proof:

We have

$\displaystyle\dot{\bm{\zeta}}_{i}$
$\displaystyle=\mathbf{A}_{i}{\bm{\zeta}}_{i}+\hat{\mathbf{u}}_{i},$ (29a)
$\displaystyle\dot{\mathbf{z}}_{i}$
$\displaystyle=\mathbf{A}\bm{\zeta}_{i}-c\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\bm{\zeta}_{i}-\mathbf{S}_{j}\bm{\zeta}_{j}),$
(29b) $\displaystyle\dot{\mathbf{e}}_{i}$
$\displaystyle=\bar{\mathbf{A}}_{i}\bm{\zeta}_{i}-\beta_{1}\text{sgn}(\mathbf{e}_{i})-\beta_{2}\text{diag}(\text{sgn}(\mathbf{e}_{i}))(\mathbf{I}_{d}\otimes|\bm{\zeta}_{i}|^{\top})\mathbf{1}_{d^{2}}.$
(29c)

By a similar analysis as in the proof of Thm. IV.2,
$\mathbf{e}_{i}=\mathbf{z}_{i}-\bm{\zeta}_{i}=\mathbf{0}_{d}$ for $t\geq T$.
Thus, for $t\geq T$, we may consider $\bm{\zeta}_{i}$ to evolve according to
the equation
$\dot{\bm{\zeta}}_{i}=\mathbf{A}\bm{\zeta}_{i}-c\text{sign}(\mathbf{S}_{i})\sum_{j\in\mathcal{N}_{i}}(\mathbf{S}_{i}\bm{\zeta}_{i}-\mathbf{S}_{j}\bm{\zeta}_{j}),~{}i=1,\ldots,n.$

Similar to the proof of Thm. IV.3, it follows that ${\bm{\zeta}}_{i}(t)$
asymptotically achieves matrix-scaled consensus to a solution of the system
$\dot{\bm{\zeta}}_{0}=\mathbf{P}\mathbf{A}\mathbf{P}^{-1}\bm{\zeta}_{0}$. As
$\mathbf{A}_{i}+\mathbf{B}_{i}\mathbf{K}_{i}$ and
$\mathbf{A}_{i}+\mathbf{H}_{i}\mathbf{C}_{i}$ are Hurwitz,
$\tilde{\mathbf{x}}_{i}=\mathbf{x}_{i}-\hat{\mathbf{x}}_{i}\to\mathbf{0}_{d}$
exponentially fast, and the unforced system
$\dot{\bm{\eta}}_{i}=(\mathbf{A}_{i}+\mathbf{B}_{i}\mathbf{K}_{i})\bm{\eta}_{i}$
is globally exponentially stable. Moreover, the system (29b) has the external
input
$\mathbf{H}_{i}\mathbf{C}_{i}\tilde{\mathbf{x}}_{i}-\hat{\mathbf{u}}_{i}$,
which vanishes to $\mathbf{0}_{d}$ exponentially fast. Thus,
$\bm{\eta}_{i}\to\mathbf{0}_{d}$, as $t\to+\infty$.

Finally, from
$\mathbf{x}_{i}-\bm{\zeta}_{i}=\tilde{\mathbf{x}}_{i}+\bm{\eta}_{i}$, it
follows that $\mathbf{x}_{i}$ asymptotically achieves matrix-scaled consensus
with regard to a solution of
$\dot{\bm{\zeta}}_{0}=\mathbf{P}\mathbf{A}\mathbf{P}^{-1}\bm{\zeta}_{0}$. ∎

##  V Simulation results

(a) $\mathbf{x}_{i}(t)$

(b) $x_{i1}(t)$

(c) $x_{i2}(t)$

(d) $\|\mathbf{u}\|_{\infty}$

(e) $\mathbf{x}_{i}(t)$

(f) $x_{i1}(t)$

(g) $x_{i2}(t)$

(h) $\|\mathbf{u}\|_{\infty}$

Figure 1: The agents converge to three clusters in 2D under the matrix-scaled
consensus algorithms (9) (Figs. (a)–(d)) and (14) (Figs. (e)–(h)).

In this section, we provide simulations of the theoretical results in Sections
III, IV. In all simulations, a six-agent system in the two dimensional space
($d=2$) is considered. The interaction between agents are captured by an
undirected cycle of six vertices. Denote the SO(2) rotation matrix of angle
$\theta$ (rad) by
$\mathbf{R}(\theta)=\begin{bmatrix}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\end{bmatrix}$. The scaling matrices are chosen as
$\mathbf{S}_{1}=\mathbf{S}_{2}=\mathbf{R}(\frac{\pi}{3})$ (positive definite),
$\mathbf{S}_{3}=\mathbf{S}_{4}=-\mathbf{I}_{2}$ (negative definite), and
$\mathbf{S}_{5}=\mathbf{S}_{6}=\mathbf{R}(\frac{5\pi}{3})$ (positive
definite).

(a) $\mathbf{x}_{i}(t)$

(b) $\bm{\theta}_{i}(t),\hat{\bm{\theta}}_{i}(t)$

(c) $x_{i1}(t)$

(d) $x_{i2}(t)$

Figure 2: The six-agent system under the adaptive algorithm (16a)–(16b).

###  V-A Matrix-scaled consensus of single integrators

Figures 1 (a)–(d) shows a simulation of the six-agent system under the matrix-
scaled consensus algorithm (9) for 20 seconds, where the initial condition
$\mathbf{x}(0)$ was randomly selected. The agents’ trajectories asymptotically
converge to three clusters in 2D, which are vertices of an equilateral
triangle.

Next, we consider the six-agent system under the nonlinear matrix-scaled
algorithm (14) with the same initial state $\mathbf{x}(0)$. We would like the
input of each agent to satisfy $\|\mathbf{u}_{i}(t)\|_{\infty}<\beta=1$. This
objective is achieved by selecting
$\mathbf{f}(\cdot)=(\max_{i}|N_{i}|)^{-1}\text{tanh}(\cdot)=0.5\text{tanh}(\cdot)$.
Simulation results are depicted in Figs. 1(e)–(h). Observe that
$\mathbf{x}(t)$ converges to the same point as the previous simulation,
$\|\mathbf{u}_{i}\|_{\infty}\leq 1,\forall t\geq 0$, the settling time (which
is defined as the first time $\mathbf{x}(t)$ enters without escaping
$B_{5\%}=\\{\|\mathbf{x}(t)-\lim_{t\to+\infty}\mathbf{x}(t)\|<0.05\\}$)
becomes larger (approx. 10 sec in comparison with 5 sec).

###  V-B Matrix-scaled consensus of single-integrators with uncertain
parameters

Next, we consider a system consisting of 6-single-integrator agents with
parametric uncertainties

$\displaystyle\bm{\phi}_{i}(t)=\begin{bmatrix}0.2\sin(t)&0.5-0.2\sin(\frac{it}{\pi})\\\
-0.2\sin(\frac{t}{i\pi})&0.1\cos(\frac{t}{\pi})\end{bmatrix},~{}i=1,\ldots,6,$

which satisfy the persistently exciting condition, and $\gamma_{i}>0$ are
randomly chosen between $[1,10]$. The constant unknown parameters are
$\bm{\theta}_{i}=[i-0.5,~{}i]^{\top}$, and the initial estimates
$\hat{\bm{\theta}}_{i}(0)$ are randomly generated. Simulation results depicted
in Fig. 2 show that $\mathbf{x}_{i}\to\mathcal{C}_{S}$,
$\hat{\bm{\theta}}_{i}(t)\approx\bm{\theta}_{i}$ at $t=200$ seconds (Fig. 2
(b)). Due to the unknown parameters, the convergence rate, however, is much
slower in comparison with the ideal case.

(a) $\mathbf{A}$ is Hurwitz.

(b) $\mathbf{A}$ has a pair of imaginary eigenvalues,
$\mathbf{A}+\mathbf{A}^{\top}\neq\mathbf{0}_{d\times d}$.

(c) $\mathbf{A}$ is unstable.

(d) $\mathbf{A}=-\mathbf{A}^{\top}$ and has a pair of imaginary eigenvalues.

(e) State variables $x_{i1}(t)$ corresponding to trajectories in (d).

(f) State variables $x_{i2}(t)$ corresponding to trajectories in (d).

Figure 3: The six-agent system modeled by the simplified linear model (21)
with different choices of the matrix $\mathbf{A}$.

###  V-C Matrix-scaled consensus of simplified linear dynamical agents

We simulate the matrix-scaled consensus algorithm for the six-agent system
with the simplified linear model (21) for different matrices $\mathbf{A}$ as
follows

  * •

$\mathbf{A}=0.5\begin{bmatrix}-1&1\\\ -1&0\end{bmatrix}$ has eigenvalues
$-0.25\pm\frac{\sqrt{3}}{4}\imath$,

  * •

$\mathbf{A}=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}$ is skew-symmetric, has
eigenvalues $\pm\imath$,

  * •

$\mathbf{A}=0.5\begin{bmatrix}0&1\\\ -0.5&0\end{bmatrix}$ has a pair of
imaginary eigenvalues, $\mathbf{A}$ is not skew-symmetric,

  * •

$\mathbf{A}=0.5\begin{bmatrix}1&1\\\ -1&0\end{bmatrix}$ has eigenvalues
$0.25\pm\frac{\sqrt{3}}{4}\imath$,

where $\imath^{2}=-1$. In this subsection, the coupling gain is chosen as
$c=2$.

Simulation results depicted in Fig. 3 show that the agents’ trajectories are
affected by both the unforced dynamics (determined by
$\mathbf{A}\mathbf{x}_{i}$) and the matrix-scaled consensus algorithm.
Clearly, if $\mathbf{A}$ is stable (unstable), all agents’ states converge to
0 (resp., grow unbounded). In case $\mathbf{A}$ is marginally stable, the
agents asymptotically reach a matrix scaled consensus with regard to a
trajectory of the system
$\dot{\mathbf{r}}=\mathbf{P}\mathbf{A}\mathbf{P}^{-1}\mathbf{r}$. In case
$\mathbf{A}$ is skew-symmetric, agents move on a circle centered at the
origin, and if $\mathbf{A}$ is not skew-symmetric, agents with the same
scaling matrix move on a same elliptical trajectory. Each trajectory of
$\mathbf{x}_{i}(t)$ differs from a common solution of
$\dot{\mathbf{r}}=\mathbf{P}\mathbf{A}\mathbf{P}^{-1}\mathbf{r}$ by the
scaling matrix $\mathbf{S}_{i}$.

###  V-D Matrix-scaled consensus of general homogeneous linear dynamical
agents

(a) $\|\hat{\mathbf{x}}_{i}(t)\|$

(b) $\eta_{i1}(t)$ (c) $\eta_{i2}(t)$

(d) $\mathbf{x}_{i}$

(e) ${x}_{i1}$

(f) ${x}_{i2}$

Figure 4: Simulations of the homogeneous linear agent system (30) under the
matrix-scaled consensus algorithm (25).

In this subsection, let the agent be modeled by a linear oscillator with a
control input

$\displaystyle\dot{x}_{i1}$ $\displaystyle=x_{i2},$ (30a)
$\displaystyle\dot{x}_{i2}$ $\displaystyle=-\omega^{2}x_{i1}+u_{i},$ (30b)
$\displaystyle y_{i}$ $\displaystyle=x_{i1},~{}\,i=1,\ldots,6,$ (30c)

and $\omega=1$. The corresponding system matrices are
$\mathbf{A}=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}$,
$\mathbf{B}=\begin{bmatrix}0\\\ 1\end{bmatrix}$, $\mathbf{C}=[1,~{}0]$,
$\mathbf{K}=[-3,-4]$, $\mathbf{H}=[-8,-15]^{\top}$, and $c=2$. Simulation
corresponding to a randomly generated initial condition is displayed in Fig.
4. The state estimate error
$\tilde{\mathbf{x}}_{i}=\hat{\mathbf{x}}_{i}-\mathbf{x}_{i}$ and the auxiliary
variable $\bm{\eta}$ vanish exponentially fast. Figures 4(d), (e), (f) depict
the trajectories of 6 agents; Each agent’s trajectory differ to a common
solution of $\dot{\mathbf{r}}=\mathbf{P}\mathbf{A}\mathbf{P}^{-1}\mathbf{r}$
by its scaling matrix.

###  V-E Matrix-scaled consensus of general heterogeneous linear dynamical
agents

Consider a system of six heterogeneous linear agents, where matrix
$\mathbf{A}$ is the same as in the previous subsection, $\bar{\mathbf{A}}_{i}$
are randomly generated with entries belong to $[-0.2,~{}0.2]$. The matrices
$\mathbf{B}_{i}=\mathbf{B}+\bm{\Delta}_{B}$, where
$\bm{\Delta}_{B}=\begin{bmatrix}0&\delta_{b_{i}}\end{bmatrix}^{\top}$, with
$\delta_{b_{i}}$ be randomly generated to take values on the interval
$[-0.5,0.5]$, and $\mathbf{C}_{i}=\mathbf{C},~{}~{}\forall i=1,\ldots,n$. The
matrices $\mathbf{K}_{i}$ and $\mathbf{H}_{i}$ are designed so that each
matrix $\mathbf{A}_{i}+\mathbf{B}_{i}\mathbf{K}_{i}$ have eigenvalues $-2,-2$
and each matrix $\mathbf{A}_{i}+\mathbf{H}_{i}\mathbf{C}_{i}$ have eigenvalues
$-4,-4$, for $i=1,\ldots,6$. The control gains are given as $c=2$,
$\beta_{1}=2$ and $\beta_{2}=5$.

Figures 5(a), (b), (c) show that $\mathbf{x}(t)\to\mathcal{C}_{S}$
asymptotically under the algorithm (29). The system eventually achieves
matrix-scaled consensus on a trajectory of
$\dot{\mathbf{r}}=\mathbf{P}\mathbf{A}\mathbf{P}^{-1}\mathbf{r}$. For
comparison, the same heterogeneous system is simulated without using the
control law (29). Without the signum terms, Fig. 5(d), (e), (f) show that the
system diverges. Thus, simulation results are consistent with the analysis.

(a) $\mathbf{x}_{i}$ (b) $x_{i1}$ (c) $x_{i2}$

(d) $\mathbf{x}_{i}$ (e) $x_{i1}$ (f) $x_{i2}$

Figure 5: Simulation of the heterogeneous linear systems when the agents: (i)
employ the algorithm (29): (a), (b), (c); and (ii) do not include the signum
term in $\hat{\mathbf{u}}$ in (29): (d), (e), (f).

##  VI Conclusions

In this paper, the matrix-scaled consensus model has been proposed and studied
in detail. Several matrix-scaled consensus algorithms were designed for linear
agents interacting over an undirected network. The asymptotic behaviors of the
proposed algorithms associate with the algebraic properties of the matrix-
scaled Laplacian
$(\text{sign}(\mathbf{S})\mathbf{L}\otimes\mathbf{I}_{d})\mathbf{S}$. Main
difficulties in the stability analysis comes from the asymmetry and multi-
dimensionality of the matrix-scaled Laplacian.

As the study of the matrix-scaled consensus algorithm in this paper also
focused on undirected graphs and linear agents, further research will consider
the matrix-scaled consensus algorithm with directed or signed graphs.

##  Appendix A Asymptotic behavior of an adaptive system

###### Lemma A.1

[32] Given the system in the following form

$\displaystyle\dot{\mathbf{x}}_{1}$
$\displaystyle=\mathbf{h}(t)\mathbf{x}_{2}+\bm{\varphi}(t),~{}\mathbf{x}_{1}\in\mathbb{R}^{n},~{}\mathbf{x}_{2}\in\mathbb{R}^{m},$
(31a) $\displaystyle\dot{\mathbf{x}}_{2}$ $\displaystyle=\bm{\psi}(t),$ (31b)

where the functions $\mathbf{h}:\mathbb{R}^{+}\to\mathbb{R}^{n\times m}$,
$\bm{\varphi}:\mathbb{R}^{+}\to\mathbb{R}^{n}$, and
$\bm{\psi}:\mathbb{R}^{+}\to\mathbb{R}^{m}$ are such that (i)
$\lim_{t\to+\infty}\|\mathbf{x}_{1}(t)\|=\lim_{t\to+\infty}\|\bm{\varphi}(t)\|=\lim_{t\to+\infty}\|\bm{\psi}(t)\|={0}$,
(ii) $\mathbf{h}(t)$, $\dot{\mathbf{h}}(t)$ are bounded and (iii) there exist
positive constants $T,\mu_{1},\mu_{2}$ and $l$ such that for all $t\geq 0$,
$\mu_{1}\mathbf{I}_{n}\geq\int_{t}^{t+T}\mathbf{h}(\tau)^{\top}\mathbf{h}(\tau)d\tau\geq\mu_{2}\mathbf{I}_{n}$,
then $\lim_{t\to+\infty}\|\mathbf{x}_{2}(t)\|=0$.

##  Appendix B Proof of Lemma IV.1

Let $\bm{\Theta}_{c}=c\mathbf{\Theta}+\bm{\Delta}_{\Theta}$, and consider the
dynamical system

$\displaystyle\dot{\mathbf{x}}=\bm{\Theta}_{c}\mathbf{x}.$

Consider the Lyapunov function $V=\mathbf{x}^{\top}\mathbf{Q}\mathbf{x}$, we
have,
$\dot{V}=\mathbf{x}^{\top}(\mathbf{Q}\bm{\Theta}_{c}+\bm{\Theta}_{c}^{\top}\mathbf{Q})\mathbf{x}=-c\|\mathbf{x}\|^{2}+2\mathbf{x}^{\top}\mathbf{Q}\bm{\Delta}_{\Theta}\mathbf{x}\leq-(c-2\|\mathbf{Q}\|\|\bm{\Delta}_{\Theta}\|)\|\mathbf{x}\|^{2}\leq-(c-2{\delta}_{\Theta}\lambda_{\max}(\mathbf{Q}))\lambda_{\min}^{-1}(\mathbf{Q})V.$
Thus, $\mathbf{x}(t)$ is globally exponentially stable. Equivalently, the
matrix $\bm{\Theta}_{c}$ is Hurwitz.

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