WGMwgm QEqe EPep PMSpms BECbec DEde

[1] [1]This work was supported by the National Natural Science Foundation of
China (Grant nos. 61873136 and 62033007), Taishan Scholars Climbing Program of
Shandong Province of China and Taishan Scholars Project of Shandong Province
of China (No. ts20190930) [1] [cor1]Corresponding author

# Controllability and observability of linear multi-agent systems over matrix-
weighted signed networks

Lanhao Zhao  Institute of Complexity Science,College of Automation, Qingdao
University, Qingdao 266071, China  Zhijian Ji<EMAIL_ADDRESS>Shandong
Key Laboratory of Industrial Control Technology, Qingdao 266071, China
Yungang Liu  School of Control Science and Engineering, Shandong University,
Jinan, 250061, China  Chong Lin

###### Abstract

In this paper, the controllability and observability of linear multi-agent
systems over matrix-weighted signed networks are analyzed. Firstly, the
definition of equitable partition of matrix-weighted signed multi-agent system
is given, and the upper bound of controllable subspace and a necessary
condition of controllability are obtained by combining the restriction
conditions of the coefficient matrix and matrix weight for the case of fixed
and switching topologies, respectively. The influence of different selection
methods of coefficient matrices on the results is discussed. Secondly, for the
case of heterogeneous systems, the upper bound of controllable subspace and
the necessary condition of controllability are obtained when the dynamics of
individuals in the same cell are the same. Thirdly, sufficient conditions for
controllable and uncontrollable union graphs are obtained by taking advantage
of the concept of switched systems and equitable partitions, respectively.
Finally, a necessary condition of observability is obtained in terms of the
dual system and the constraints of the coefficient matrix, and the
relationship between the observability and the controllability of the matrix-
weighted signed multi-agent systems is discussed.

###### keywords:

Controllability Observability Multi-Agent SystemsMatrix-Weighted Equitable
Partition General Linear Systems

lemmaLemma rmkRemark pfProof exmExample deDefinition coCorollary
proProposition

##  1 Introduction

Recently, the cooperative control of multiple individuals, such as UAV
formation control and social network modeling, has attracted increasing
attention of researchers. The concept of multi-agent system (MAS) was
introduced to solve the problem of cooperative control. The so-called multi-
agent system can be regarded as an ordered combination of multiple systems.
Each individual updates their states through neighbor information, resulting
in collective behaviors such as clustering and separation. Especially, many
works of multi-agent systems had emerged, such as consensus Wang et al.
(2020); Liu et al. (2020a); Sun et al. (2020, 2019), controllability Lu et al.
(2020); Qu et al. (2020); Ji et al. (2021); Guan et al. (2019); Guan and Wang
(2019); Liu et al. (2020b); Lou et al. (2020), observability Liu et al. (2020,
2020, 2019), stabilisability Guan and Kong (2019), optimal control Sun et al.
(2018, 2020, 2021); Han et al. (2020); Qi et al. (2020, 2018) and model-free
adaptive control Hou and Xiong (2019).

Controllability and observability are fundamental topics in control theory.
The concept of controllability provides helps for the design of control law,
and observability describes an ability to use partial information to figure
out the whole system states. Kalman Kalman (1963) first proposed the concept
of controllability, and it was generalized to multi-agent systems by Tanner
Tanner (2004). The system is controllable if the leader’s state can be
adjusted through an external input signal to guide the follower to any desired
state. Most of the existing results of controllability and observability are
obtained through the structural characteristics of graphs, especially the
partition of graphs. Analyzing the controllability from the perspective of
graph theory, especially partition, can directly determine the controllability
of the system and save the calculation time. For example, the effects of
network structure on system controllability from the perspective of equitable
or distance partition (EP or DP) were discussed in Rahmani et al. (2009); Ji
and Egerstedt (2007); Martini et al. (2010), the results show that the upper
and lower bounds of controllable subspace can be directly determined according
to the number of cells and path length in the network, which provides
convenience for the determination of system controllability. While Zamani and
Lin (2009); Liu et al. (2011) discussed the structural controllability of
different networks from the perspective of structured systems. It is worth
noting that these works deal with the simplest cases. To make the
controllability results proposed in these studies more applicable to various
scenarios, some related work has appeared in recent years, such as Guan et
al.Guan et al. (2017) extended the concept of equitable partition to the case
of the directed graph. Ji et al discussed the case of delay or switching
topology Ji et al. (2010), signed graph Sun et al. (2017); Gao et al. (2018);
Guan and Wang (2018); She et al. (2018) was also discussed in recent years.

Most of the existing results on the controllability and observability of
multi-agent systems are mainly based on scalar-weighted networks. The concept
of matrix weighted network is introduced to describe more complex systems, for
example, opinion dynamics Ye et al. (2020) and formation control Zhao and
Zelazo (2015). Compared with the scalar-weighted networks, the matrix-weighted
networks can manifest the interaction between the different states of
individuals and the states of corresponding neighbors more precisely. It is
non-trivial to analyze its controllability although the matrix weight network
seems to be a natural extension of the scalar weight network. For instance,
additional complexity is introduced because the rank of weight matrix is not
fixed when describing the dimension of controllable subspace. Pan et al.
(2020) analyzed the controllability of matrix-weighted network by using the
method of almost equitable partition and distance partition and gave the upper
and lower bounds of controllable subspace. Notably, the weight matrix is
assumed to be positive definite or positive semi-definite in their work.
However, the weight matrix may also be negative definite and negative semi-
definite in more cases. The consensus problem under the assumption that the
weight matrix can be negative definite or negative semi-definite was discussed
in Su et al. (2020), however, this assumption has not been taken into account
in the controllability analysis from the perspective of graph theory.
Therefore, it is meaningful to consider the controllability and observability
of the matrix-weighted signed networks under the assumption.

In addition, In the above work, the network topology is usually fixed, and
each individual is described by a first-order integrator. In engineering
practice, the network topology may change and the individual dynamics in the
network are also different. This brings a challenge to analyze the
controllability of the system by using graph theory. Firstly, the concept of
equitable partition is extended to directed signed multi-agent systems with
switched topology in Tian et al. (2019), however, the concept of equitable
partition is not used to analyze the relationship between union graph
controllability and system controllability. Secondly, Zhang et al. (2014)
analyzes the controllability of multi-agent systems whose individual dynamics
is a general linear system and reveals the new characteristics of the system
compared with the first-order integrator system. Therefore, for matrix-
weighted networks, it is meaningful to analyze the characteristics of
individual dynamics when it is a general linear system. Finally, the
difference of individuals dynamics in the network is also a challenge to the
controllability analysis, especially for graphical methods. Specifically, most
of the existing results of system controllability from the perspective of
graph theory analyze the influence of some special structures based on the
same dynamics of each individual. However, if we use different colors to label
network nodes with different dynamics, the distribution law of nodes with
different colors on the graph is a generalized graphic feature, which needs to
be further studied in the conventional graphic feature. Therefore, it is
necessary to analyze how individual dynamics affect the results of
controllability and observability of the matrix-weighted signed networks.

In this paper, the controllability and observability of linear multi-agent
systems over matrix-weighted signed networks are analyzed. The main
contributions of this paper are as follows.

(1) The definition of equitable partition of matrix-weighted signed multi-
agent system is first given, and the upper bound of controllable subspace and
a necessary graphic condition of controllability are obtained by combining the
restriction conditions of the coefficient matrix and matrix weight for the
case of fixed and switching topologies, respectively. In particular, the
equivalence of the controllability results based on the concept of equitable
partition for first-order integrator systems and general linear systems under
appropriate input matrix selection(it called complete control input)
conditions is discussed.

(2) For the case of heterogeneous systems, the upper bound of controllable
subspace and the necessary condition of controllability are obtained when the
dynamics of individuals in the same cell are the same. To our knowledge, this
is the first time to use the concept of equitable partition (EP) to deal with
the controllability of heterogeneous systems whether for scalar or matrix
weighted networks. The meaning of ’equitable’ is extended by combining the
characteristics of graph theory with the distribution law of each individual
dynamics.

(3) Sufficient conditions for controllable and uncontrollable union graphs are
obtained by taking advantage of the concept of switched systems and equitable
partitions, respectively. As far as we know, this is the first time that the
sufficient condition of uncontrollable union graph is given by using the
concept of EP.

(4) A necessary condition of observability is obtained in terms of the dual
system and the constraints of the coefficient matrix, and the relationship
between the observability and the controllability of the matrix-weighted
signed multi-agent systems is discussed. The observability results in this
paper are more applicable compared with the existing graph conditions of
observability obtained under the conditions of the first-order integrator,
undirected and unweighted graph.

The rest of this work is organized as follows. The notations and some
preliminaries used in this paper are provided in Section 2. The model of
matrix-weighted signed networks under general linear dynamics is established
in Section 3. Section 4, 5, and 6 present the main results of controllability
and observability of matrix weighted signed networks in different cases,
respectively. Some numerical examples are shown in Section 7, and the
conclusions are summarized in Section 8.

##  2 Notations and Preliminaries

In this paper, $R$ stands for the set of real number, $I_{m}$ and $0_{m}$
denote identity matrix and zero matrix with dimension $m$, respectively. For a
symmetric matrix $H\in\mathbb{R}^{n\times n}\left(H^{T}=H\right)$, it is
denoted by $H\succ 0$ ($H\succeq 0$ ) when $H$ is positive definite (positive
semi-definite), and $H$ is called negative definite (negative semi-definite)
when $-H$ is positive definite (positive semi-definite). $|H|=H$ when $H\succ
0$ or $H\succeq 0$ and $|H|=-H$ when $H\prec 0$ or $H\preceq 0$. $im(P)$
represents the vector space generated by the columns of matrix $P$. Graph
theory: $G=\\{V,E,\mathcal{A}\\}$ is called as matrix-weighted signed graph,
in which $V=\left\\{v_{1},v_{2}\cdots v_{n}\right\\}$ represents the vertex
set. $\mathcal{A}=\left[\mathcal{A}_{ij}\right]\in R^{nd\times nd}$ is the
weighted adjacency matrix which $\mathcal{A}_{ij}$ is a symmetric matrix
belonging to $\mathbb{R}^{d\times d}$. $E=E_{+}+E_{-}$ represents the edge
set, where ${E}_{+}=\left\\{(j,i)\mid\mathcal{A}_{ij}\succ
0\cup\mathcal{A}_{ij}\succeq 0\right\\}$ and
${E}_{-}=\left\\{(j,i)\mid\mathcal{A}_{ij}\prec 0\cup\mathcal{A}_{ij}\preceq
0\right\\}$ denote the sets of positive and negative edges, respectively. The
set of neighbors of agent $v_{i}$ is represented by $N_{i}=N_{i+}+N_{i-}$.
$N_{i+}=\left\\{j\in{V}\mid(j,i)\in{E}_{+}\right\\}$ and
$N_{i-}=\left\\{j\in{V}\mid(j,i)\in{E}_{-}\right\\}$ represent the set of
positive and negative neighbors, respectively.
$d_{i}=\sum_{j\in{N}_{i}}\left|\mathcal{A}_{ij}\right|$ denotes the degree of
$i$ for matrix-weighted signed graph. Let $L=D-\mathcal{A}$ be the Laplacian
matrix of $G$, where $D=\operatorname{diag}\left(d_{1},\cdots,d_{n}\right)$.
The entries of $L$ can be writen as follows

$l_{ij}=\left\\{\begin{array}[]{ll}d_{i},&i=j\\\ -\mathcal{A}_{ij},&i\neq
j\end{array}\right.$

Graph partition: For the vertex set $V$ of a graph, its subset $V_{i}$ is
called a cell. It is called a trivial cell if the cell contains only one
vertex, otherwise it is a nontrivial cell. If any vertex in $V_{1}$ also
belongs to $V_{2}$, then $V_{1}$ is a sub-cell of $V_{2}$. We define
$\pi=\left\\{V_{1},V_{2},\cdots,V_{k}\right\\}$. Then $\pi$ is a partition of
graph when $V_{i}\cap V_{j}=\varnothing$ and $\bigcup_{i}V_{i}=V$ for
$0<i,j<k$ and $i\neq j$. The characteristic matrix is

$P_{ij}\triangleq\left\\{\begin{array}[]{ll}I_{d\times d},&i\in V_{j}\\\
0_{d\times d},&i\notin V_{j}\end{array}\right.$

The characteristic matrix of $\pi=\\{\\{1,2\\},\\{3,4,5\\}\\}$ is

$P(\pi)=\left[\begin{array}[]{ll}I_{d\times d}&0_{d\times d}\\\ I_{d\times
d}&0_{d\times d}\\\ 0_{d\times d}&I_{d\times d}\\\ 0_{d\times d}&I_{d\times
d}\\\ 0_{d\times d}&I_{d\times d}\end{array}\right].$

##  3 Matrix-Weighted Signed Networks with General Linear Dynamics

Consider a multiagent system with $n$ agents which state is denoted by the
$x_{i}(t)\in\mathbb{R}^{d}$. The leader and follower are distinguished
according to whether the agent receives external input signals. We assume that
the first $m$ $(m<n)$ individuals are named as leaders where
$V_{L}=\left\\{v_{1},\cdots,v_{m}\right\\}$ is the leader set, and the
follower set is represented by $V_{F}=\left\\{v_{m+1},\cdots,v_{n}\right\\}$.

For the matrix-weighted signed networks under general linear dynamics, all
followers are governed by the following dynamics

$\dot{x}_{i}(t)=Ax_{i}(t)+Bu_{i}(t).$

For every leader, its dynamics is

$\dot{x}_{i}(t)=Ax_{i}(t)+Bu_{i}(t)+Cy_{i}(t)$

where $A\in R^{d\times d},B\in R^{d\times p},C\in R^{d\times q}$, $x_{i}\in
R^{d}$ represents the state of the $ith$ agent, and $u_{i}\in R^{p}$ reflects
the influence that each individual receives from the others, $y_{l}\in R^{q}$
represents an external input signal. The update rules based on neighbors are
as follows

$u_{i}(k)=\sum_{j\in
N_{i}}K\left[\mathcal{A}_{ij}x_{j}(k)-\left|\mathcal{A}_{ij}\right|x_{i}(k)\right]$

where $K$ stands for the feedback gain, $\mathcal{A}_{ij}$ is the connection
weight between the individual $i$ and $j$. Denote
$x(k)=\left[x_{1}(t),\cdots,x_{n}(t)\right]^{T}$ as the aggregate state vector
and $y(k)=\left[y_{1}(t),\cdots,y_{l}(t)\right]^{T}$ as the control input
vector. Then the expression of multiagent system can be written as

$\dot{x}(t)=\left(\left[\begin{array}[]{cccc}A&0&\cdots&0\\\ 0&A&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A\end{array}\right]-\left[\begin{array}[]{cccc}BK&0&\cdots&0\\\
0&BK&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&BK\end{array}\right]L\right)x(t)-\left(M\left[\begin{array}[]{cccc}C&0&\cdots&0\\\
0&C&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right]\right)y(t)$

where matrix $M$ is used to distinguish the leaders from the followers.

$M_{il}=\left\\{\begin{array}[]{ll}I_{d\times d}&i=l\\\ 0_{d\times
d}&\text{otherwise}\end{array}\right.$

For convenience, it can be rewritten as the following form

$\dot{x}(t)=\tilde{L}x(t)+\tilde{M}y(t)\quad\quad\quad\quad(1)$

where

$\tilde{L}=\left(\left[\begin{array}[]{cccc}A&0&\cdots&0\\\ 0&A&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A\end{array}\right]-\left[\begin{array}[]{cccc}BK&0&\cdots&0\\\
0&BK&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&BK\end{array}\right]L\right),\tilde{M}=\left(M\left[\begin{array}[]{cccc}C&0&\cdots&0\\\
0&C&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right]\right).$

System (1) includes both the influence of weight matrix and individual
dynamics. Compared with the linear dynamics in scalar networks Zhang et al.
(2014), the Kronecker product in the previous model of scalar networks is
replaced by matrix product in the system (1). This makes some existing results
based on Kronecker product properties may no longer be applicable especially
when multiplication is needed. So this situation deserves further discussion
for the new features brought by this matrix-weighted network.

##  4 Controllability and Graph partition

In this section, the controllability of matrix-weighted signed networks is
discussed by using graph partition. Firstly, the definition of equitable
partition of matrix-weighted signed networks is given. Secondly, the
controllability of matrix-weighted signed networks in fixed topology,
switching topology and the heterogeneous system is analyzed by using the
concept of equitable partition.

Denote a matrix-weighted signed graph as $G$, and let
$\pi=\left\\{V_{1},V_{2},\ldots,V_{k}\right\\}$ be a partition of $G$. The
partition $\pi$ is said to be an equitable partition (EP) of $G$ if for any
$r,s\in V_{i},i,j=1,2,\ldots,k$

$\sum_{t_{1}\in V_{j},t_{1}\in{N}_{r+}}\mathcal{A}_{rt_{1}}=\sum_{t_{2}\in
V_{j},t_{2}\in{N}_{s+}}\mathcal{A}_{st_{2}}$

and for any $r,s\in V_{i},i,j=1,2,\ldots,k$

$\sum_{t_{1}\in V_{j},t_{1}\in N_{r-}}\mathcal{A}_{rt_{1}}=\sum_{t_{2}\in
V_{j},t_{2}\in\mathcal{N}_{s-}}\mathcal{A}_{st_{2}}$

where $\left(t_{1},r\right),\left(t_{2},s\right)\in E$. Compared with the
definition for scaler weighted network in Guan and Wang (2018), The sum of
scalar weights in previous definition is replaced by the sum of matrices. This
shows the difference between matrix weighted network and scalar weighted
network when using the concept of equitable partition to analyze
controllability. Compared with the case that all the matrix weights are
positive definite in Pan et al. (2020), Definition 1 applies in the case that
the matrix weights are negative definite or semi negative definite. Therefore,
this extension is necessary and meaningful.

Denote $d(v_{i},Q)=\sum_{v_{j}\in Q}\left|\mathcal{A}_{ij}\right|$ and
$d(V_{i},Q)=d(v,Q)$ for all $v\in V_{i}$, then we give the concept of quotient
graph For an equitable partition $\pi=$
$\left\\{V_{1},V_{2},\ldots,V_{s}\right\\}$ of a matrix-weighted network $G$,
the quotient graph of $G$ over $\pi$ is a matrix-weighted network denoted by
$G/\pi$ with the node set $V(G/\pi)=\left\\{v_{1},v_{2},\ldots,v_{s}\right\\}$
and the edge set is $E(G/\pi)=\left\\{\left(v_{i},v_{j}\right)\mid
d\left(V_{i},V_{j}\right)\neq 0_{d\times d}\right\\},$ where the weight of
edge $\left(V_{i},V_{j}\right)$ is $d\left(V_{i},V_{j}\right)$ for $i\neq
j\in\underline{s}$.

Denote $L_{\pi}$ as the Laplacian matrix of $G/\pi$

$\left(L_{\pi}\right)_{ij}=\left\\{\begin{array}[]{ll}\sum_{v_{j}\in
V(G/\pi)}d\left(V_{i},V_{j}\right),&i=j\\\ -d\left(V_{i},V_{j}\right),&i\neq
j\end{array}\right.$

######  Lemma 4.1.

$\pi=\left\\{V_{1},V_{2},\cdots,V_{s}\right\\}$ is an EP for matrix-weighted
signed graph $G$ and $P_{\pi}$ is the characteristic matrix. Then $L$
satisfies

$LP_{\pi}=P_{\pi}L_{\pi}.$

Furthermore $im\left(P_{\pi}\right)$ is $L-$ invariant.

The proof is similar to Lemma 1 in Pan et al. (2020), and thus is omitted.

Lemma 1 shows the relationship between the characteristic matrix and the
Laplacian matrix. For the first-order system, the upper bound of the
controllable subspace can be obtained by the expression of the controllable
subspace in the geometric theory of linear systems Pan et al. (2020). However,
how to deal with the unknown coefficient matrix is a challenge for general
linear dynamics and worth further discussion.

###  4.1 The case of fixed topology

We first discuss the case of fixed topologies. The so-called fixed topology
means that the network topology between individuals does not change in the
evolution process of multi-agent systems. For system (1), the system is
controllable if and only if the matrix
$\left[\tilde{M}\quad\tilde{L}\tilde{M}\quad\tilde{L}^{2}\tilde{M}\cdots\tilde{L}^{dn-1}\tilde{M}\right]$
has full row rank, and the controllable subspace of system (1) is

$\mathcal{W}=\left\langle\tilde{L}\mid\tilde{M}\right\rangle=im(\tilde{M})+\tilde{L}\times
im(\tilde{M})+\cdots+\tilde{L}^{dn-1}\times im(\tilde{M}).$

It is a minimal $\tilde{L}-$ invariant subspace containing
$\operatorname{im}(\tilde{M})$.

Denote

$\tilde{P}_{\pi}=P_{\pi}\cdot\left[\begin{array}[]{cccc}C&0&\cdots&0\\\
0&C&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right].$

######  Lemma 4.2.

$\pi=\left\\{V_{1},V_{2},\cdots,V_{s}\right\\}$ is an EP for matrix-weighted
graph $G$ and $P_{\pi}$ is the characteristic matrix. If the matrix

$Q=\left[\begin{array}[]{cccc}Q_{1}&0&\cdots&0\\\ 0&Q_{1}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&Q_{1}\end{array}\right]-\left[\begin{array}[]{cccc}Q_{11}&Q_{12}&\cdots&Q_{1\mathrm{~{}s}}\\\
Q_{21}&Q_{12}&\cdots&Q_{2s}\\\ \vdots&\vdots&\ddots&\vdots\\\
Q_{n1}&Q_{n2}&\cdots&Q_{ns}\end{array}\right]$

exists, then there are matrices $Q_{1},Q_{ij}$ that make $AC=CQ_{1}$ and
$BKL_{\pi ij}C=CQ_{ij},(0\leq i\leq n,0\leq j\leq s)$. It follows that

$\tilde{L}\tilde{P}_{\pi}=\tilde{P}_{\pi}Q.$

Furthermore $im\left(\tilde{P}_{\pi}\right)$ is $\tilde{L}-$ invariant.

$\tilde{L}=\left(\left[\begin{array}[]{cccc}A&0&\cdots&0\\\ 0&A&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A\end{array}\right]-\left[\begin{array}[]{cccc}BK&0&\cdots&0\\\
0&BK&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&BK\end{array}\right]L\right),\tilde{P}_{\pi}=P_{\pi}\cdot\left[\begin{array}[]{cccc}C&0&\cdots&0\\\
0&C&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right].$

According to Lemma 1

$\aligned\tilde{L}\tilde{P}_{\pi}&=\left[\begin{array}[]{cccc}A&0&\cdots&0\\\
0&A&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A\end{array}\right]\cdot
P_{\pi}\cdot\left[\begin{array}[]{cccc}C&0&\cdots&0\\\ 0&C&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right]-\left[\begin{array}[]{cccc}BK&0&\cdots&0\\\
0&BK&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&BK\end{array}\right]\cdot L\cdot
P_{\pi}\cdot\left[\begin{array}[]{cccc}C&0&\cdots&0\\\ 0&C&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&C\end{array}\right]\\\
=\left[\begin{array}[]{cccc}AP_{11}C&AP_{12}C&\cdots&AP_{1s}C\\\
AP_{21}C&AP_{22}C&\cdots&AP_{2s}C\\\ \vdots&\vdots&\ddots&\vdots\\\
AP_{n1}C&AP_{n2}C&\cdots&AP_{ns}C\end{array}\right]-\left[\begin{array}[]{cccc}BK&0&\cdots&0\\\
0&BK&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&BK\end{array}\right]\cdot P_{\pi}\cdot
L_{\pi}\cdot\left[\begin{array}[]{cccc}C&0&\cdots&0\\\ 0&C&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&C\end{array}\right]\\\
=\left[\begin{array}[]{cccc}AP_{11}C&AP_{12}C&\cdots&AP_{1s}C\\\
AP_{21}C&AP_{22}C&\cdots&AP_{2s}C\\\ \vdots&\vdots&\ddots&\vdots\\\
AP_{n1}C&AP_{n2}C&\cdots&AP_{ns}C\end{array}\right]-\left[\begin{array}[]{cccc}BKP_{11}&BKP_{12}&\cdots&BKP_{1s}\\\
BKP_{21}&BKP_{22}&\cdots&BKP_{2s}\\\ \vdots&\vdots&\ddots&\vdots\\\
BKP_{n1}&BKP_{n2}&\cdots&BKP_{ns}\end{array}\right]\cdot
L_{\pi}\cdot\left[\begin{array}[]{cccc}C&0&\cdots&0\\\ 0&C&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&C\end{array}\right]$

The specific expression of the characteristic matrix is

$P_{ij}\triangleq\left\\{\begin{array}[]{ll}I_{d\times d}&i\in V_{j}\\\
0_{d\times d}&i\notin V_{j}\end{array}\right.$

Therefore, if $AC=CQ_{1}$ holds, then

$\left[\begin{array}[]{cccc}AP_{11}C&AP_{12}C&\cdots&AP_{1s}C\\\
AP_{21}C&AP_{22}C&\cdots&AP_{2s}C\\\ \vdots&\vdots&\ddots&\vdots\\\
AP_{n1}C&AP_{n2}C&\cdots&AP_{ns}C\end{array}\right]=P_{\pi}\cdot\left[\begin{array}[]{cccc}C&0&\cdots&0\\\
0&C&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right]\cdot\left[\begin{array}[]{cccc}Q_{1}&0&\cdots&0\\\
0&Q_{1}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&Q_{1}\end{array}\right]$

And if $BKL_{\pi ij}C=CQ_{ij},(0\leq i\leq n,0\leq j\leq s)$ holds, then

$\left[\begin{array}[]{cccc}BKP_{11}&BKP_{12}&\cdots&BKP_{1s}\\\
BKP_{21}&BKP_{22}&\cdots&BKP_{2s}\\\ \vdots&\vdots&\ddots&\vdots\\\
BKP_{n1}&BKP_{n2}&\cdots&BKP_{ns}\end{array}\right]\cdot
L_{\pi}\cdot\left[\begin{array}[]{cccc}C&0&\cdots&0\\\ 0&C&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right]=P_{\pi}\cdot\left[\begin{array}[]{cccc}C&0&\cdots&0\\\
0&C&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right]\cdot\left[\begin{array}[]{cccc}Q_{11}&Q_{12}&\cdots&Q_{1s}\\\
Q_{21}&Q_{22}&\cdots&Q_{2s}\\\ \vdots&\vdots&\ddots&\vdots\\\
Q_{n1}&Q_{n2}&\cdots&Q_{ns}\end{array}\right]$

Denote $Q$ as

$Q=\left[\begin{array}[]{cccc}Q_{1}&0&\cdots&0\\\ 0&Q_{1}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&Q_{1}\end{array}\right]-\left[\begin{array}[]{cccc}Q_{11}&Q_{12}&\cdots&Q_{1\mathrm{~{}s}}\\\
Q_{21}&Q_{12}&\cdots&Q_{2s}\\\ \vdots&\vdots&\ddots&\vdots\\\
Q_{n1}&Q_{n2}&\cdots&Q_{ns}\end{array}\right]$

It can be derived that

$\tilde{L}\tilde{P}_{\pi}=\tilde{P}_{\pi}Q$

if $AC=CQ_{1}$ and $BKL_{\pi ij}C=CQ_{ij},(0\leq i\leq n,0\leq j\leq s)$.
Furthermore $im\left(\tilde{P}_{\pi}\right)$ is $\tilde{L}-$ invariant.

######  Theorem 4.3.

The controllable subspace $\mathcal{W}$ of the system (1) satisfies
$\mathcal{W}\subseteq im\left(\tilde{P}_{\pi}\right)$ if the matrix $Q$
exists.

Every column of $M$ is also a column of $P_{\pi_{L}^{*}},$ and then every
column of $\tilde{M}$ is also a column of $\tilde{P}_{\pi}$. It follows that
$im(\tilde{M})\subseteq im\left(\tilde{P}_{\pi_{L}^{*}}\right)$. By Lemma $2$,
$\operatorname{im}(\tilde{P})$ is $\tilde{L}-$ invariant. Then

$\aligned\mathcal{W}&=im(\tilde{M})+\tilde{L}\times
im(\tilde{M})+\cdots+\tilde{L}^{dn-1}\times im(\tilde{M})\\\ \subseteq
im\left(\tilde{P}_{\pi}\right)+\tilde{L}\times
im\left(\tilde{P}_{\pi}\right)+\cdots+\tilde{L}^{dn-1}\times
im\left(\tilde{P}_{\pi}\right)\\\ =im\left(\tilde{P}_{\pi}\right).$

Suppose that the matrix $Q$ exists, then every cell in the EP is trivial when
system (1) is controllable. If a certain cell is a nontrivial cell under
controllable system (1), the dimension of $im\left(\tilde{P}_{\pi}\right)$ is
less than $dn$. By Theorem $1,$ the dimension of the controllable subspace is
also less than $dn,$ and accordingly system (1) is uncontrollable. This is
contrary to the assumption. Therefore, when system (1) is controllable, each
cell is a trivial cell. The proof is complete. Theorem 1 and Corollary 1 are
similar to the results under the scalar weight network. For instance,
$\tilde{L}\tilde{P}_{\pi}=\tilde{P}_{\pi}Q$ holds for matrix weighted networks
base on Lemma 2. It is obtained that $LP=PL_{\pi}$ if matrices $A$ is chosen
as zero matrices and $B$, $K$ and $C$ are all chosen as identity matrices,
which is same as $LP=PL_{\pi}$ for scaler weighted network Guan and Wang
(2018). It also shows that the scalar weighted network is a special case of
matrix weight network. It is worth noting that Theorem 1 and Corollary 1 are
also different from the conclusion under the scalar weighted network. It
mainly in the following two points. The weight matrix can not be ignored and
does not meet the commutative law in matrix operation compared with scalar
weight. And the dimension of the weight matrix also needs to be considered
when estimating the upper bound of the controllable subspace of the system.

From Theorem 1 and Corollary 1, matrices $Q_{1}$ and $Q_{ij}$ need to be found
to make $AC=CQ_{1}$ and $BKL_{\pi ij}C=CQ_{ij},(0\leq i\leq n,0\leq j\leq s)$
hold. There is no such limitation for the similar results in Pan et al.
(2020), so it is necessary to further discuss this condition.

First, we discuss the simplest case that matrices $A$, $B$, $K$, $L_{\pi ij}$
and $C$ are all chosen as identity matrices, and matrices $Q_{1}$ and $Q_{ij}$
can be found to make $AC=CQ_{1}$ and $BKL_{\pi ij}C=CQ_{ij},(0\leq i\leq
n,0\leq j\leq s)$ hold for this case.

$Q_{1}=Q_{ij}=I_{d\times d},(0\leq i\leq n,0\leq j\leq s).$

Then the conditions of Theorem 1 and Corollary 1 both hold in this case.

Furthermore, the matrix $L_{\pi ij}$ is not always chosen as the identity
matrix in practice. So we discuss a more complicated case that matrices $A$,
$B$, $K$ and $C$ are all chosen as identity matrices. Note that $Q_{1}$ and
$Q_{ij}$ can still be found to make $AC=CQ_{1}$ and $BKL_{\pi
ij}C=CQ_{ij},(0\leq i\leq n,0\leq j\leq s)$ hold for this case.

$Q_{1}=I_{d\times d},Q_{ij}=L_{\pi ij},(0\leq i\leq n,0\leq j\leq s).$

Then the conditions of Theorem 1 and Corollary 1 both hold in this case. It
should be noted that the model considered in this case is the same to the one
in Pan et al. (2020).

In practical engineering, especially when individual dynamics is more complex,
matrices $A$, $B$, and $K$ are rarely selected as identity matrices, and only
matrix $C$ can be designed by our goals. At this point, if matrix $C$ is
chosen as an identity matrix, matrices $Q_{1}$ and $Q_{ij}$ can still be found
to make $AC=CQ_{1}$ and $BKL_{\pi ij}C=CQ_{ij},(0\leq i\leq n,0\leq j\leq s)$
hold for this case.

$Q_{1}=A,Q_{ij}=BKL_{\pi ij},(0\leq i\leq n,0\leq j\leq s).$

The conditions of Theorem 1 and Corollary 1 both hold in this case.

Furthermore, if matrix $C$ is choosen as an invertible matrix, the matrices
$Q_{1}$ and $Q_{ij}$ can still be found to make $AC=CQ_{1}$ and $BKL_{\pi
ij}C=CQ_{ij},(0\leq i\leq n,0\leq j\leq s)$ hold for this case.

$Q_{1}=C^{-1}AC,Q_{ij}=C^{-1}BKL_{\pi ij}C,(0\leq i\leq n,0\leq j\leq s).$

Then the conditions of Theorem 1 and Corollary 1 both hold in this case.

Based on the above discussion, it can be obtained that matrix $Q$ is always
exists when matrix $C$ is chosen as an invertible matrix. Therefore, this
situation is worth discussing. We take C as the identity matrix for the
convenience. When matrix C is the identity matrix, it means that the
components of the external control signal correspond to the leader’s state one
by one. Then the results of general linear system and first-order integrator
system are the same for the controllability analysis based on the concept of
equitable partition, which is not affected by system dynamics. This also
explains why there is no such limitation for the similar results in Pan et al.
(2020), and Theorem 1 is an extension of previous conclusions in Pan et al.
(2020).

Inspired by this situation, the concept of complete control input is
introduced

For leaders which are described as

$\dot{x}_{i}(t)=Ax_{i}(t)+Bu_{i}(t)+Cy_{i}(t),$

if $C$ is an invertible matrix, then the control input $y_{i}(t)$ is called
the complete control input.

According to Theorem 1, Corollary 1 and the Definition 3, the following result
can be obtained Every cell in the EP is trivial when system (1) is
controllable with a complete control input. In practical systems, matrix $C$
is sometimes not designable and not invertible. In this case, matrices $Q_{1}$
and $Q_{ij}$ need to be found to make $AC=CQ_{1}$ and $BKL_{\pi
ij}C=CQ_{ij},(0\leq i\leq n,0\leq j\leq s)$ hold. it reflects the influence of
the selection of coefficient matrix on controllability analysis.

###  4.2 The case of switching topology

In practical application, due to the influence of formation change, fault,
obstacle avoidance, and the other factors, the connection relationship between
individuals in multi-agent systems maybe change. The concept of switching
topologies is introduced to deal with this situation. For the case of
switching topology, denote $\left\\{G_{1},G_{2},\ldots,G_{t}\right\\}$ as the
set of switching topology. Function
$\sigma(x):R^{+}\rightarrow\\{1,2,\ldots,t\\}$ is the switching signal which
is used to determine the topological structure of the system at a certain time
point. For convenience, we only consider the situation of fixed leaders during
the switching process. Then the system can be written as

$\dot{x}(t)=\tilde{L}_{\sigma(x)}x(t)+\tilde{M}y(t)\quad\quad\quad\quad(2)$

where

$\tilde{L}_{\sigma(x)}=\left(\left[\begin{array}[]{cccc}A&0&\cdots&0\\\
0&A&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A\end{array}\right]-\left[\begin{array}[]{cccc}BK&0&\cdots&0\\\
0&BK&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&BK\end{array}\right]L_{\sigma(x)}\right),\tilde{M}=\left(M\left[\begin{array}[]{cccc}C&0&\cdots&0\\\
0&C&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right]\right).$

The switching sequence of the system (2) is

$\pi=\left\\{\left(i_{1},h_{1}\right),\left(i_{2},h_{2}\right),\ldots,\left(i_{M},h_{M}\right)\right\\}$

where $i_{m}\in\\{1,2,\ldots,t\\}$, and $h_{m}=t_{m}-t_{m-1}>$ 0 represents
the duration of the $m$-th topology. If for any initial state, there are one
switch sequence and external inputs that take the states to zeros, then the
system (2) is said to be controllable. The controllable state set is denoted
as $\mathcal{C}(\pi)$ and the subspace sequence Xie and L.Wang (2003) is

$\mathbb{W}_{1}=\sum_{\sigma(x)=1}^{t}\left\langle\tilde{L}_{\sigma(x)}\mid\tilde{M}\right\rangle,\quad\mathbb{W}_{i+1}=\sum_{\sigma(x)=1}^{t}\left\langle\tilde{L}_{\sigma(x)}\mid\mathbb{W}_{i}\right\rangle,\quad
i=1,2,\ldots$

we denote $\pi_{i}=\left\\{V_{1},V_{2},\cdots,V_{s}\right\\}$ as an EP for
matrix-weighted signed graph $G_{i}$ and
$\pi_{min}=\left\\{V_{1},V_{2},\cdots,V_{s}\right\\}$ is the least upper bound
for all $\pi_{i}$.

It can be seen that the expression of subspace is similar to that of fixed
topology, so we can get the following results by using the concept of EP.

######  Theorem 4.4.

For system (2), if the matrix $Q$ always exists for all $\tilde{L}_{i}$
corresponding to $G_{i}$, the dimension of its controllable state set
$\mathcal{C}$ satisfies

$\operatorname{dim}(\mathcal{C})\leq\operatorname{card}\left(\pi_{min}\right)\cdot
d$

Since $\mathcal{C}=\mathbb{W}_{N}$ Tian et al. (2019), the conclusion is
equivalent to $\operatorname{dim}\left(\mathbb{W}_{N}\right)\leq$
$\operatorname{card}\left(\pi_{min}\right)\cdot d$. According to Lemma 2, if
$Q$ always exists for all $\tilde{L}_{i}$ corresponding to $G_{i}$, so for any
$\pi_{i}$, $i\in\\{1,2,\ldots,t\\},\operatorname{im}\left(P_{i}\right)$ is
$\tilde{L}_{i}$ -invariant, where $P_{i}$ is the characteristic matrix of
$\pi_{i}$.

$\aligned\mathbb{W}_{1}&=\sum_{i=1}^{t}\sum_{m=0}^{N-1}\left(\tilde{L}_{i}\right)^{m}\operatorname{im}\left(\tilde{M}\right)\\\
\subseteq\sum_{i=1}^{t}\sum_{m=0}^{N-1}\left(\tilde{L}_{i}\right)^{m}\operatorname{im}\left(P_{i}\right)=\operatorname{im}\left(P_{i}\right)\\\
\mathbb{W}_{2}=\sum_{i=1}^{t}\sum_{m=0}^{N-1}\left(\tilde{L}_{i}\right)^{m}\mathbb{W}_{1}\subseteq\sum_{i=1}^{t}\sum_{m=0}^{N-1}\left(\tilde{L}_{i}\right)^{m}\operatorname{im}\left(P_{i}\right)=\operatorname{im}\left(P_{i}\right)\\\
\ldots\\\
\mathbb{W}_{N}=\sum_{i=1}^{t}\sum_{m=0}^{N-1}\left(\tilde{L}_{i}\right)^{m}\mathbb{W}_{N-1}\\\
\subseteq\sum_{i=1}^{t}\sum_{m=0}^{N-1}\left(\tilde{L}_{i}\right)^{m}\operatorname{im}\left(P_{i}\right)=\operatorname{im}\left(P_{i}\right)$

That is to say,
$\operatorname{dim}\left(\mathbb{W}_{N}\right)\leq\operatorname{dim}\left(\operatorname{im}\left(P_{i}\right)\right)=\operatorname{card}(P_{i})\cdot
d$, and then

$\operatorname{dim}\left(\mathbb{W}_{N}\right)\leq\operatorname{card}\left(\pi_{min}\right)\cdot
d.$

This completes the proof.

Theorem 2 shows that the dimension of the controllable subspace is limited by
the nontrivial cell contained in each subgraph. This reflects the influence of
switching topology on the controllability analysis based on the concept of
equitable partition.

###  4.3 The case of heterogeneous systems

In practical applications, the models of different individuals are often in
different due to the requirements of tasks or manufacturing reasons. We call
this situation as heterogeneous system, which brings challenges to
controllability analysis, especially from the perspective of the graphics
method. Here, we consider matrix weighted signed networks, in which each
individual can be described by general linear dynamics, and the coefficient
matrix can be different. All followers take the following dynamics

$\dot{x}_{i}(t)=A_{i}x_{i}(t)+B_{i}u_{i}(t)$

and for every leader, its dynamics is

$\dot{x}_{i}(t)=A_{i}x_{i}(t)+B_{i}u_{i}(t)+Cy_{i}(t)$

where $A_{i}\in R^{d\times d},B_{i}\in R^{d\times p},C\in R^{d\times q}$. Then
the dynamics of multi-agent system can be written as

$\dot{x}(t)=\left(\left[\begin{array}[]{cccc}A_{1}&0&\cdots&0\\\
0&A_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A_{n}\end{array}\right]-\left[\begin{array}[]{cccc}B_{1}KL_{11}&B_{1}KL_{12}&\cdots&B_{1}KL_{1n}\\\
B_{2}KL_{21}&B_{2}KL_{22}&\cdots&B_{2}KL_{2n}\\\
\vdots&\vdots&\ddots&\vdots\\\
B_{n}KL_{n1}&B_{n}KL_{n2}&\cdots&B_{n}KL_{nn}\end{array}\right]\right)x(t)-\left(M\left[\begin{array}[]{cccc}C&0&\cdots&0\\\
0&C&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right]\right)y(t)$

For convenience, it can be rewritten as the following form

$\dot{x}(t)=\tilde{L}^{{}^{\prime}}x(t)+\tilde{M}y(t)\quad\quad\quad\quad(3)$

where

$\tilde{L}^{{}^{\prime}}=\left(\left[\begin{array}[]{cccc}A_{1}&0&\cdots&0\\\
0&A_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A_{n}\end{array}\right]-\left[\begin{array}[]{cccc}B_{1}KL_{11}&B_{1}KL_{12}&\cdots&B_{1}KL_{1n}\\\
B_{2}KL_{21}&B_{2}KL_{22}&\cdots&B_{2}KL_{2n}\\\
\vdots&\vdots&\ddots&\vdots\\\
B_{n}KL_{n1}&B_{n}KL_{n2}&\cdots&B_{n}KL_{nn}\end{array}\right]\right).$

To deal with different coefficient matrices, we assume that the dynamics of
nodes in each nontrivial cell is the same, which means that
$A_{i}=A_{j},B_{i}=B_{j}$ if nodes $i,j$ belong to the same cell, and the
dynamics of nodes in different cells can be different. Then we can get one
result similar to Lemma 2.

######  Lemma 4.5.

$\pi=\left\\{V_{1},V_{2},\cdots,V_{k}\right\\}$ is an EP for matrix-weighted
graph $G$ and $P_{\pi}$ is the corresponding characteristic matrix. Suppose
$A_{i}=A_{j},B_{i}=B_{j}$ if nodes $i,j$ belong to same cell. When the matrix

$Q^{{}^{\prime}}=\left[\begin{array}[]{cccc}Q^{{}^{\prime}}_{1}&0&\cdots&0\\\
0&Q^{{}^{\prime}}_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&Q^{{}^{\prime}}_{s}\end{array}\right]-\left[\begin{array}[]{cccc}Q^{{}^{\prime}}_{11}&Q^{{}^{\prime}}_{12}&\cdots&Q^{{}^{\prime}}_{1s}\\\
Q^{{}^{\prime}}_{21}&Q^{{}^{\prime}}_{12}&\cdots&Q^{{}^{\prime}}_{2s}\\\
\vdots&\vdots&\ddots&\vdots\\\
Q^{{}^{\prime}}_{n1}&Q^{{}^{\prime}}_{n2}&\cdots&Q^{{}^{\prime}}_{ns}\end{array}\right]$

exists, which means that there are matrices
$Q^{{}^{\prime}}_{1},Q^{{}^{\prime}}_{ij}$ that make
$A_{i}C=CQ^{{}^{\prime}}_{i}$ and $B_{i}KL_{\pi
ij}C=CQ^{{}^{\prime}}_{ij},(0\leq i\leq n,0\leq j\leq s)$ holds, then there is

$\tilde{L}^{{}^{\prime}}\tilde{P}_{\pi}=\tilde{P}_{\pi}Q^{{}^{\prime}}.$

Furthermore $im\left(\tilde{P}_{\pi}\right)$ is $\tilde{L}^{{}^{\prime}}-$
invariant.

Similar to the proof of Theorem 1 and Corollary 1, we can get the following
conclusion by Lemma 3,

######  Theorem 4.6.

Suppose that $A_{i}=A_{j},B_{i}=B_{j}$ if nodes $i,j$ belong to the same cell
and the matrix $Q^{{}^{\prime}}$ exist, then every cell in the EP is trivial
when system (3) is controllable.

Theorem 3 extends the EP-based method to the controllability analysis of
heterogeneous systems for the first time. The results show that the difference
of node dynamic equations does not affect the relevant conclusions of EP if
the nodes belong to different cells. From the ’equitable’ defined by EP, two
points can be called equitable points which not only means that the received
signals are the same (the sum of the weight matrix is the same), but also
their own dynamics should be the same. This deepens the understanding of EP.

##  5 Union graph and its controllablity

In this section, the controllability of the system is discussed from the
perspective of the union graph, more specifically the relationship of
controllability between union graph and system (2) is discussed, and a new
criterion to identify the controllability of the union graph is obtained by
using the concept of EP. The union graph of a collection of graphs $G_{m}=$
$\left\\{V,E_{m},\mathcal{A}_{m}\right\\},m=1,\ldots,t,$ is denoted by
$\tilde{G}=\\{V,\tilde{E},\mathcal{A}\\},$ where
$\tilde{E}=\bigcup_{m=1}^{t}E_{m},$ and
$\tilde{\mathcal{A}}=\sum_{m=1}^{t}\mathcal{A}_{m}$.

Accordingly, union graphs can be described as

$\dot{x}(t)=\tilde{L}^{*}x(k)+\tilde{M}y(k)\quad\quad\quad\quad(4)$

where

$\tilde{L}^{*}=\left(t\left[\begin{array}[]{cccc}A&0&\cdots&0\\\
0&A&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A\end{array}\right]-\left[\begin{array}[]{cccc}BK&0&\cdots&0\\\
0&BK&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&BK\end{array}\right]L^{*}\right),\tilde{M}=\left(M\left[\begin{array}[]{cccc}C&0&\cdots&0\\\
0&C&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C\end{array}\right]\right)$

$L^{*}$ is the Laplacian matrix of the union graph.

In order to discuss the relationship between the controllability of system (2)
and (4), the following lemma is introduced

######  Lemma 5.7.

System (2) is controllable if and only if the matrix

$\begin{array}[]{c}\tilde{W}_{c}=\left[\tilde{M},\tilde{L}_{1}\tilde{M},\ldots,\tilde{L}_{t}\tilde{M},\tilde{L}_{1}^{2}\tilde{M},\ldots,\tilde{L}_{1}^{dN-1}\tilde{M}\right.,\left.\tilde{L}_{1}^{dN-1}\tilde{L}_{2}\tilde{M},\ldots,\tilde{L}_{t}^{dN-1}\tilde{M}\right]\end{array}$

is full row rank.

The proof is similar to Lemma 1 in Guan et al. (2019), thus there is omitted.

######  Theorem 5.8.

System (2) is controllable if the system (4) is controllable

From the Kalman rank criterion, Tthe following matrix is full row rank

$\begin{array}[]{c}\left[\tilde{M},\tilde{L}^{*}\tilde{M},\ldots,\tilde{L}^{*dN-1}\tilde{M}\right]\end{array}$

which is expanded by using the properties of the union graph

$\begin{array}[]{c}\left[\tilde{M},\left(\tilde{L}_{1}+\ldots+\tilde{L}_{t}\right)^{*}\tilde{M},\ldots,\left(\tilde{L}_{1}+\ldots+\tilde{L}_{t}\right)^{*dN-1}\tilde{M}\right]\end{array}$

Continue to expand the above matrix and add some column vectors, then

$\begin{array}[]{l}\left[\tilde{M},\tilde{L}_{1}\tilde{M}+\cdots+\tilde{L}_{t}\tilde{M},\tilde{L}_{2}\tilde{M},\ldots,\tilde{L}_{t}\tilde{M},\ldots,\tilde{L}_{1}^{dN-1}\tilde{M}\right.\left.+\tilde{L}_{1}^{dN-2}\tilde{L}_{2}\tilde{M}+\cdots+\tilde{L}_{t}^{dN-1}\tilde{M},\tilde{L}_{1}^{dN-1}\tilde{L}_{2}\tilde{M},\ldots,\tilde{L}_{t}^{dN-1}\tilde{M}\right]\end{array}$

Because the basic transformation of the matrix does not change the rank of the
matrix, the above matrix can be transformed into

$\begin{array}[]{c}\left[\tilde{M},\tilde{L}_{1}\tilde{M},\ldots,\tilde{L}_{t}\tilde{M},\tilde{L}_{1}^{2}\tilde{M},\ldots,\tilde{L}_{1}^{dN-1}\tilde{M}\right.,\left.\tilde{L}_{1}^{dN-1}\tilde{L}_{2}\tilde{M},\ldots,\tilde{L}_{t}^{dN-1}\tilde{M}\right]\end{array}$

which is full row rank. According to Lemma 4, system (2) is controllable.
Theorem 4 is only a sufficient condition, that is, when the system (4) is
uncontrollable, the system (2) can also be controllable. See example 4 in
Section 7 for more details.

From Theorem 4 and its examples, the controllability of system (2) cannot be
determined when system (4) is uncontrollable only by above results. Therefore,
we need to further analyze more specific cases. Inspired by Theorem 2, the
following results are obtained

######  Theorem 5.9.

Suppose the matrix $Q$ always exists for all $\tilde{L}_{i}$ corresponding to
$G_{i}$, then the union graph is uncontrollable if any subgraph in the
switching sequence contains nontrivial cells.

If a proposition holds, so does its converse negative proposition. The
converse negative proposition of Theorem 4 is that system (4) is
uncontrollable if system (2) is uncontrollable. Theorem 2 shows that the
dimension of the controllable subspace of system (2) is less than $dN$ if any
subgraph in the switching sequence contains nontrivial cells. It means that
the system (2) is uncontrollable, so we can get that the union graph is also
uncontrollable.

Theorem 4 gives a special class of graphs. For this kind of graph, when system
(4) is uncontrollable, system (2) is also uncontrollable. It avoids the
uncertainty of previous results. At the same time, a new method to identify
the controllability of the union graph is also given, which simplifies the
process of identifying the controllability of the union graphs.

##  6 Observability analysis

Compared with controllability, observability is also a topic worth discussing,
which is used to measure the ability of reconstructing the state of the whole
network. Our previous discussion is based on the leader-follower structure.
The leader is used to receive external input signals to control the whole
network. Similarly, we can directly measure the state of some nodes, and then
use these information to estimate the overall states of the network.
Therefore, we call these nodes that can read the states directly as output
points and the remaining nodes are called observed points. Accordingly, an
architecture of output nodes-observed nodes is proposed to analyze the
observability.

From the viewpoint of control, it is expected to find a controllable and
observable system. For multi-agent systems, this requires that the
controllability factor should be considered when selecting the output points.
Therefore, in this paper, each leader in the network discussed above is
selected as an output point, and the system can be described as

$\dot{x}(t)=\tilde{L}x(t)+\tilde{M}y(t)\quad\dot{y}(t)=\tilde{M}^{T}x(t)\quad\quad\quad\quad(5)$

where

$\tilde{L}=\left[\begin{array}[]{cccc}A&0&\cdots&0\\\ 0&A&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A\end{array}\right]-\left[\begin{array}[]{cccc}BK&0&\cdots&0\\\
0&BK&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&BK\end{array}\right]L$
$\tilde{M}=M\left[\begin{array}[]{cccc}C&0&\cdots&0\\\ 0&C&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&C\end{array}\right].$

The observability of system (5) is equivalent to the controllability of its
dual system. The dual system is represented as follows.

$\dot{x}(t)=\tilde{L}^{T}x(t)+\tilde{M}y(t),\quad\dot{y}(t)=\tilde{M}^{T}x(t)\quad\quad\quad\quad(6)$

where $\tilde{L}^{T}$ is

$\begin{array}[]{c}\left[\begin{array}[]{cccc}A^{T}&0&\cdots&0\\\
0&A^{T}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&A^{T}\end{array}\right]\\\
-L^{T}\left[\begin{array}[]{cccc}K^{T}B^{T}&0&\cdots&0\\\
0&K^{T}B^{T}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&K^{T}B^{T}\end{array}\right]\end{array}$

and $\tilde{M}$ is

$\tilde{M}=M\left[\begin{array}[]{cccc}C&0&\cdots&0\\\ 0&C&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&C\end{array}\right].$

And then we transform the observability problem of system (5) into the
controllability problem of system (6). It is important to note that system (6)
and system (1) are in different forms due to the existence of coefficient
matrix, so the controllability of system (6) needs to be discussed.

######  Lemma 6.10.

$\pi=\left\\{V_{1},V_{2},\cdots,V_{k}\right\\}$ is an EP for matrix-weighted
graph $G$ and $P_{\pi}$ is the characteristic matrix. If the matrix

$Q^{*}=\left[\begin{array}[]{cccc}Q_{1*}&0&\cdots&0\\\ 0&Q_{1*}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&Q_{1*}\end{array}\right]-\left[\begin{array}[]{cccc}Q_{11*}&Q_{12*}&\cdots&Q_{1s*}\\\
Q_{21*}&Q_{12*}&\cdots&Q_{2s*}\\\ \vdots&\vdots&\ddots&\vdots\\\
Q_{n1*}&Q_{n2*}&\cdots&Q_{ns*}\end{array}\right]$

exists, which means that there are matrices $Q_{1*},Q_{ij*}$ that make
$A^{T}C=CQ_{1*}$ and $L_{\pi ij}K^{T}B^{T}C=CQ_{ij*},(0\leq i\leq n,0\leq
j\leq s)$ holds, then there is

$\tilde{L}^{T}\tilde{P}_{\pi}=\tilde{P}_{\pi}Q^{*}$

Furthermore $im\left(\tilde{P}_{\pi}\right)$ is $\tilde{L}-$ invariant.

The proof process is similar to Lemma 2, and thus is omitted here. Although
Lemma 5 and Lemma 2 are similar in form, the restriction conditions of the
coefficient matrix in the two cases are different, which is determined by the
characteristics of the dual system.

######  Theorem 6.11.

Suppose that the matrix $Q^{*}$ exists, then system (5) is unobservable if the
matrix weighted network contains nontrivial cells

Similar to Theorem 1 and Corollary 1, we can get from Lemma 5 that the system
(6) is uncontrollable if it contains the nontrivial cell. According to the
properties of dual systems, system (5) is unobservable.

If we take matrix $A$ as zero matrices, and matrices $B$, $C$, and $K$ as
identity matrices, then both matrices $Q$ and $Q^{*}$ exist according to Lemma
2 and Lemma 5. We call this case a first-order system. Choose and only choose
all leaders as output points, then a result is obtained System (5) is
uncontrollable and unobservable if the network contains nontrivial cells under
the first-order system. Theorem 6 and Corollary 3 use the method of dual
systems to discuss the influence of the existence of nontrivial cell on
observability. Compared with the previous work of using EP to deal with
observability of undirected and unweighted network O’Clery et al. (2013), the
situation considered in this paper is more extensive, and the method proposed
in this paper is more general, which can be used to analyze the observability
of multiagent systems in many cases.

##  7 Examples

In this section, we present several examples to demonstrate the relevant
results in aboves.

Figure 1: A matrix weighted signed graph with fixed topology including
nontrival cell Figure 2: A matrix weighted signed graph with switching
topology Figure 3: The case of heterogeneous system Figure 4: The case of
union graph which is uncontrollable Figure 5: The case of subgraph including
nontrival cell Figure 6: Observability of signed matrix weighted network

As showing in Figure 1, this is a network of four nodes. The coefficient
matrix of the system is selected as

$A=K=\left(\begin{array}[]{ll}1&0\\\
0&1\end{array}\right),B=C=\left(\begin{array}[]{ll}2&0\\\
0&2\end{array}\right).$

The weights of the selected matrix are as follows

$\mathcal{A}_{12}=\mathcal{A}_{13}=\left(\begin{array}[]{ll}1&2\\\
2&1\end{array}\right),\mathcal{A}_{24}=\mathcal{A}_{34}=\left(\begin{array}[]{ll}2&1\\\
1&2\end{array}\right).$

Agent 1 is chosen as the signle leader. In view of the definition, the nodes
can be divided into three cells
$\pi=\left\\{\left\\{1\right\\},\left\\{2,3\right\\},\left\\{4\right\\}\right\\}$.
The correcponding characteristic matrix is

$P=\left[\begin{array}[]{lll}1&0&0\\\ 0&1&0\\\ 0&1&0\\\
0&0&1\end{array}\right]$

then we get

$L=\left(\begin{array}[]{cccc}\mathcal{A}_{12}+\mathcal{A}_{13}&-\mathcal{A}_{12}&-\mathcal{A}_{13}&0\\\
-\mathcal{A}_{12}&\mathcal{A}_{12}-\mathcal{A}_{24}&0&-\mathcal{A}_{24}\\\
-\mathcal{A}_{13}&0&\mathcal{A}_{13}-\mathcal{A}_{34}&-\mathcal{A}_{34}\\\
0&-\mathcal{A}_{24}&-\mathcal{A}_{34}&-\mathcal{A}_{24}-\mathcal{A}_{34}\end{array}\right).$

After calculation, $rank(\tilde{L},\tilde{M})=6$. The system is
uncontrollable. Theorem 1 is verified. The switching topology is shown in
Figure 2. The coefficient matrix of the system is the same as the choice in
Example 2, and the weights of the selected matrix are as follows

$\mathcal{A}_{12}=\mathcal{A}_{13}=\mathcal{A}_{14}=\left(\begin{array}[]{ll}1&2\\\
2&1\end{array}\right).$

Agent 1 is chosen as the leader. Then, by definition, the nodes can be divided
into two cells
$\pi=\left\\{\left\\{1\right\\},\left\\{2,3,4\right\\}\right\\}$ for (a). For
Fig. 2(b), the nodes can be divided into three cells
$\pi=\left\\{\left\\{1\right\\},\left\\{2,3\right\\},\left\\{4\right\\}\right\\}$
for (b).

$L_{a}=\left[\begin{array}[]{cccc}\mathcal{A}_{12}+\mathcal{A}_{13}+\mathcal{A}_{14}&-\mathcal{A}_{12}&-\mathcal{A}_{13}&-\mathcal{A}_{14}\\\
-\mathcal{A}_{12}&\mathcal{A}_{12}&0&0\\\
-\mathcal{A}_{13}&0&\mathcal{A}_{13}&0\\\
-\mathcal{A}_{14}&0&0&\mathcal{A}_{14}\end{array}\right]$
$L_{b}=\left[\begin{array}[]{cccc}\mathcal{A}_{12}+\mathcal{A}_{14}&-\mathcal{A}_{12}&0&-\mathcal{A}_{14}\\\
-\mathcal{A}_{12}&\mathcal{A}_{12}&0&0\\\ 0&0&0&0\\\
-\mathcal{A}_{14}&0&0&\mathcal{A}_{14}\end{array}\right].$

According to Theorem 2, the upper bound of the sequence of controllable
subspaces is 4. After calculation, $\operatorname{dim}(\mathcal{C})=4$ too.
Thus Theorem 2 is verified. As showing in Figure 3, the coefficient matrix of
the system is selected as

$A_{1}=\left(\begin{array}[]{ll}1&0\\\
0&1\end{array}\right),A_{2}=A_{2}=\left(\begin{array}[]{ll}2&1\\\
1&2\end{array}\right),A_{3}=\left(\begin{array}[]{ll}1&2\\\
2&1\end{array}\right).$

the other coefficient matrix and weight matrix of the system are as Example 2.
After calculation, $rank(\tilde{L}^{{}^{\prime}},\tilde{M})=6$, The system (3)
is uncontrollable. Theorem 3 is verified. The other switching topologies and
their union graph are shown in Figure 4 where (c) is an union graph of (a) and
(b). The coefficient matrix of the system is the same as the choice in Example
2, and the weight matrix is

$\mathcal{A}_{12}=\mathcal{A}_{13}=\left(\begin{array}[]{ll}1&2\\\
2&1\end{array}\right).$

Agent 1 is chosen as the leader. For graphs in Fig. 3(a) and Fig. 3(b), there
are

$L_{3a}=\left[\begin{array}[]{ccc}\mathcal{A}_{12}&-\mathcal{A}_{12}&0\\\
-\mathcal{A}_{12}&\mathcal{A}_{12}&0\\\ 0&0&0\end{array}\right]$
$L_{3b}=\left[\begin{array}[]{ccc}\mathcal{A}_{13}&0&-\mathcal{A}_{13}\\\
0&0&0\\\ -\mathcal{A}_{13}&0&\mathcal{A}_{13}\end{array}\right].$

After calculation, $\operatorname{dim}(\mathcal{C})=6$. The system is
controllable, but the union graph is uncontrollable. This example shows that
the controllability of the system needs to be discussed separately when the
union graph is uncontrollable. The case of subgraph including nontrivial cell
is shown in Figure 5, where (a), (b) are subgraphs and (c) is the union graph.
The coefficient matrix of the system is selected as Example 2, and the weight
matrices are as follows

$\mathcal{A}_{12}=\mathcal{A}_{13}=\mathcal{A}_{23}=\left(\begin{array}[]{ll}1&2\\\
2&1\end{array}\right).$

Agent 1 is chosen as leader, and for Figure 3(a) and 3(b)

$L_{4a}=\left[\begin{array}[]{ccc}\mathcal{A}_{12}+\mathcal{A}_{13}&-\mathcal{A}_{12}&-\mathcal{A}_{13}\\\
-\mathcal{A}_{12}&\mathcal{A}_{12}&0\\\
-\mathcal{A}_{13}&0&\mathcal{A}_{13}\end{array}\right]$
$L_{4b}=\left[\begin{array}[]{ccc}0&0&0\\\
0&\mathcal{A}_{23}&-\mathcal{A}_{23}\\\
0&-\mathcal{A}_{23}&\mathcal{A}_{23}\end{array}\right].$

After calculation, $\operatorname{dim}(\mathcal{C})=4$, the system is
uncontrollable, and the union graph is uncontrollable too. Theorem 5 is
verified. As showing in Figure 6, we choose node 1 as the output point and the
coefficient matrix and weight matrix of the system are as Example 2. After
calculation, $rank(\tilde{L}^{T},\tilde{M})=6$, The system (6) is
uncontrollable, so system (5) is unobservable. Theorem 6 is verified.

##  8 Conclusion

In this paper, the controllability and observability of linear multi-agent
systems over matrix-weighted signed networks were analyzed. Firstly, the
definition of equitable partition of the matrix-weighted signed multi-agent
system was given, and the upper bound of controllable subspace and a necessary
condition of controllability were obtained by combining the restriction
conditions of coefficient matrix and matrix weight for the case of fixed and
switching topologies, respectively. The influence of different selection
methods of coefficient matrices on the results was discussed. In particular,
the equivalence of the controllability results based on equitable partition
for first-order integrator systems and general linear systems under
appropriate input matrix selection(it called complete control input)
conditions was discussed. Secondly, for the case of heterogeneous systems, the
upper bound of controllable subspace and the necessary condition of
controllability were obtained when the dynamics of individuals in the same
cell are the same. Thirdly, sufficient conditions for controllable and
uncontrollable union graphs were obtained by taking advantage of the concept
of switched systems and equitable partitions, respectively. Finally, a
necessary condition of observability was obtained in terms of dual system and
the constraints of the coefficient matrix, and the relationship between the
observability and the controllability of the matrix-weighted signed multi-
agent systems was discussed. For future work, we are interested in discussing
the case that the weight matrix is asymmetric or other forms of weight (such
as vector weight), to further improve the relevant results proposed in this
paper.

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