# Distributed Observer Design over Directed Switching Topologies

Haotian Xu, Shuai Liu, , Bohui Wang, , Jingcheng Wang This work is supported
by National Natural Science Foundation of China (No.61821004, 62133008,
61903290). The corresponding author: Shuai Liu.H. Xu and S. Liu are with
School of Control Science and Engineering, Shandong University, Jinan 250061,
China (email<EMAIL_ADDRESS>liushuai@sdu.edu.cn)B. Wang is with the
School of Aerospace Science and Technology, Xidian University, Xi’an 710071,
China (e-mail<EMAIL_ADDRESS>wshchen@126.com).J. Wang is with Department
of Automation, Shanghai Jiao Tong University, Key Laboratory of System Control
and Information Processing, Ministry of Education of China, Shanghai, 200240;
He is also with Shanghai Engineering Research Center of Intelligent Control
and Management, Shanghai 200240, China (e-mail<EMAIL_ADDRESS>
###### Abstract

The distributed observer design problem holds significant importance in cases
in which the output information of a system is decentralized across different
subsystems. Each subsystem has a local observer and access to one part of the
measurement outputs and information exchanged through communication networks.
This paper focuses on the design of distributed observer with jointly
connected directed switching networks. The problem presents challenges due to
passive switching modes and the open-loop unboundedness that results from
local observability. To overcome these challenges, we develop a network
transformation mapping method whereby each local observer can classify itself
into an independent subgraph based on independent judgment. Next, an
observable decomposition and reorganization method is developed for the
digraph case to ensure that each subgraph possesses independent dynamic
properties. Asymptotic omniscience is then proven using a developed recursive
proof method. This paper includes many previous results as special cases,
because most are only suitable for undirected switching topologies or fast-
switching cases. An adaptive coupling gain design is proposed to simplify the
calculation and verification of conditions that guarantee asymptotic
omniscience. Finally, simulation results with the power system show the
validity of the developed theory.

###### Index Terms:

Distributed observer, Continuous time system estimation, Consensus, Sensor
networks, Switching topologies

##  I Introduction

Distributed observer has received extensive attention in recent years [1, 2,
3, 4, 5] owing to its great potential for improving the performance of
distributed control, such as the distributed cooperative control of battery
energy storage systems in DC microgrids [6], distributed controller designed
for flexible structure [7] and distributed Nash equilibrium game for nonlinear
systems [8].

Research on distributed observer is divided into two categories. The first
focuses on estimating the states of an external system and the second on
estimating the states of an interconnected system in a distributed way [9].
Since the first kind of distributed observer is the dual version of leader-
following systems [10, 11, 12], it is regarded as a kind of multi-agent
system. The distributed observer mentioned later refers to the second one.

Consider the linear system $\dot{\chi}=A\chi,~{}y=C\chi$ in which $A,C$ have
compatible dimensions and
$\chi\in\mathbb{R}^{n},y=[y_{1}^{T},\ldots,y_{N}^{T}]^{T}\in\mathbb{R}^{m}$
stand for states and measurement outputs respectively. Distributed observer is
a cooperative observer network composed of multiple local observers (also
termed “agents”). Each local observer is assumed to have access to one part of
outputs $y_{i}$ and the information exchanged via a communication network. Let
$\hat{\chi}_{i}$ be the estimates of $\chi$ generated by the $i$th agent.
Then, distributed observer aims to achieve omniscience asymptotically, i.e.,
$\lim_{t\to\infty}\|\hat{\chi}_{i}(t)-\chi(t)\|=0$ for all $i=1,\ldots,N$.

Up to now, distributed observer has been widely studied, such as the earliest
distributed Luenberger observer [13], distributed minimum order observer [14],
distributed observer with event-triggering mechanism [2], communication delay
[15], network attack [16], adaptive coupling gain [4], and so on. Distributed-
observer-based control approaches have been studied in [17, 18, 19], and some
claim that the distributed control law can achieve a control performance
similar to the centralized control [7, 8, 20]. It also provides profound
application value for research on distributed observer.

In the past few years, the problem of the distributed observer under switching
topologies has attracted much attention because failure or occasional
interruption of the communication network is inevitable [21, 16, 22]. One of
the significant challenges described in [23] is that local state estimates of
several different local observers cannot achieve consensus under the jointly
connected topologies (which are disconnected at any time), even if they belong
to the same connected branch. It is significantly different from the multi-
agent problem [24]. Also, the leader-following consensus of multi-agent
systems under real-time disconnected switching topologies relies on a
marginally stable open-loop leader system (refer to [25]). However, its
approaches are unsuitable for distributed observer because the non-consistency
terms in distributed observer are generally unstable. Consequently, most
previous results avoid real-time disconnected switching topologies and turn to
the case in which augmented topologies contain strongly connected networks
[26] or switch in fast-switching mode [27, 17, 28].

Recently, some studies have investigated the real-time disconnected switching
topologies problem in distributed observer design without resorting to a fast-
switching approach. [29] achieves distributed observer with a jointly
connected network, but it assumes that the system matrix $A$ is marginally
stable. However, unlike the multi-agent problem, we usually do not assume that
the estimated system is open-loop marginally stable in the distributed
observer problem, because the stability of the observer-based control system
needs to be proved—and it is thus impossible to know in advance whether the
observed system is marginally stable. Two studies, [30] and [31], have
implemented the distributed observer design for a class of unstable $A$
(relying on some constraint conditions). However, their stability proof
depends on the approach developed by [32], and this approach is only
applicable to the case of undirected graphs and cannot be extended to directed
graphs.

Therefore, when the system matrix is unstable and the communication topology
is jointly connected, how to design and implement a distributed observer with
asymptotic omniscience is unclear. Studies by [23] have shed light on this
issue by requiring that the output matrix $C$ be known by all agents and
addressing a states decomposition and reorganization method. With this
approach, asymptotic omniscience under jointly connected networks can be
achieved without any system constraints.

The fly in the ointment is that [23] only focuses on undirected communication
networks. Transformation from an undirected graph to a directed graph will
pose significant challenges in the case in which networks are not strongly
connected at any time. This is significantly different from the fixed topology
scenario, such as from the undirected graph case [33] to the directed graph
case [13]. The reason is that unconnected undirected graphs always contain
connected branches, yet directed graphs do not in general include a
counterpart. Specifically, [23] uses the state reorganization and
decomposition method in the undirected graph to construct a group of novel
distributed observers that can realize omniscience asymptotically in each
connected branch. However, there is no counterpart in a directed graph the
corresponds to the connected branch of an undirected graph. In most cases,
there is not even a separate subgraph in a directed graph. Even if separate
subgraphs exist, their nodes cannot share information sufficiently because
they are not always strongly connected, which prevent use of the state
reorganization and decomposition method. This will undoubtedly force
researchers to develop new approaches and devise state decomposition and
reorganization methods in directed graph cases.

This paper overcomes these difficulties in three steps. First, we consider a
network transformation mapping such that the transformed topology contains an
independent subgraph. Moreover, network transformation mapping can be directly
calculated by each local observer using the topology information of the
current network. Second, the states decomposition and reorganization method
for the digraph case is developed with the assistance of network
transformation mapping. This allows the dynamics of each local observer to be
decomposed into several different observable subspaces. Then, different
observable subsystems of each local observer employ different transformed
networks. As a result, the system after state reorganization has relatively
independent dynamic performance and there is a lower triangular
interconnection relationship between each group of reorganized systems.
Finally, a recursive proof method of state omniscience is proposed. Through
the convergence of error dynamics about the first group of reorganized states
and the lower triangular interconnection relationship, the convergence of all
reorganized error dynamics can be proved iteratively. Then, the asymptotic
omniscience of the distributed observer is obtained by applying the state
decomposition and reorganization method in reverse.

The contributions of this paper are concluded as follows. 1) Asymptotic
omniscience is achieved based on the proposed distributed observer under
jointly connected switching directed networks. In comparison, methods in the
literature usually rely on the undirected graph case [23], fast switching
(e.g., [17, 28]), and real-time strongly connected networks [26]. 2) An
original proposed network transformation mapping is investigated such that the
transformed network contains an independent subgraph with spanning trees. This
solves the problem whereby there are generally no independent subgraphs in a
directed graph. 3) With the help of network transformation mapping, this paper
proposes a reconstructed states decomposition and reorganization method for
directed graph such that each independent subgraph can contain relatively
independent observer dynamics. Similar methods are only suited for undirected
graphs in the literature. 4) With the aid of the recursive proof method, fix
and adaptive coupling gains are proposed to guarantee asymptotic omniscience.

The rest of th paper is organized as follows. The problem is formulated in
Section II, and the method for designing distributed observer is presented in
Section III. Section IV demonstrates the stability of the error dynamics with
the states decomposition and reorganization method and recursive proof method.
Section V supplements these results in terms of adaptive coupling gain. Two
numerical simulations with respect to the power system are employed in Section
VI to verify the stated theories, and Section VII concludes the paper.

##  II Problem formulation

Notation used throughout the paper are as follows.

(N1) Denote $x\in\mathbb{R}^{n}$ and $A\in\mathbb{R}^{n\times n}$ the $n$
dimension column vector and $n\times n$ dimension matrix, respectively.
$x^{T},A^{T}$ stand for their transpositions. In particular, $I_{n}$ is the
$n$ dimension identity matrix. $\|\cdot\|$ denotes the vector or the matrix
norm induced by $2$-norm. Given arbitrary elements $a_{i},i=1,\ldots,m$,
$col\\{a_{1},\ldots,a_{m}\\}\in\mathbb{R}^{m}$ represents the stack of
$a_{i},i=1,\ldots,m$ in a column and
$diag\\{a_{1},\ldots,a_{m}\\}\in\mathbb{R}^{m\times m}$ is the block diagonal
matrix with all $a_{i},i=1,\ldots,m$ on its diagonal. Denote
$\bar{\lambda}(\cdot)$ and $\underline{\lambda}(\cdot)$ the maximum eigenvalue
and minimum eigenvalue of a symmetric matrix, respectively. $sym\\{A\\}$
stands for $A+A^{T}$ for $A\in\mathbb{R}^{n\times n}$. Operators $\wedge$ and
$\vee$ are defined as $a\wedge b=\min\\{a,b\\}$ and $a\vee b=\max\\{a,b\\}$
for $a,b\in\mathbb{R}$.

(N2) Set $\mathcal{G}=\\{\mathcal{V},\mathcal{E},\mathcal{A}\\}$ to be a
directed graph and $\mathcal{V}$ and $\mathcal{E}$ the set of its nodes and
arcs respectively, and denote by $\mathcal{A}$ its adjacency matrix. The
element $a_{ij}=1$ of $\mathcal{A}$ indicates an arc pointing from node $j$ to
$i$, and $a_{ij}=0$ otherwise. A directed path from $i_{1}$ to $i_{k}$
composed of a set of nodes $\\{i_{1},\ldots,i_{k}\\}$ is defined by
$a_{i_{p+1}i_{p}}=1$ for all $p\in\\{1,\ldots,k\\}$. We say $i_{k}$ is able to
obtain information from $i_{1}$ by $l$ hops if the path between $i_{k}$ and
$i_{1}$ includes $l$ arcs. An input-degree matrix
$D=diag\\{d_{1},\ldots,d_{N}\\}$ is defined by $d_{i}=\sum_{j=1}^{N}a_{ij}$,
and the matrix $\mathcal{L}=D-\mathcal{A}$ is the graph Laplacian of
$\mathcal{G}$. The directed graph is a strongly connected graph if there is a
directed path between any pair of nodes belonging to $\mathcal{V}$. The
network contains a spanning tree if there is a node $k$ and the directed path
pointing from $k$ to $i$ exists $\forall i\in\mathcal{V}$ and $i\neq k$, where
$k$ is said to be a root of the spanning tree.

Consider a linear system,

$\displaystyle\dot{\chi}=A\chi,$ (1) $\displaystyle y=C\chi,$ (2)

where $\chi\in\mathbb{R}^{n}$ and $y\in\mathbb{R}^{p}$ are the states and
measurement outputs respectively; system matrix $A$ and output matrix $C$ are
with compatible dimension. Assume that the pair $(C,A)$ is observable and
further assume that each of $N$ agents only has access to one part of $y$.
Also, they are capable of reconstructing complete output information $y$ by
collaborating. The original output vector and output matrix can be rewritten
as $y=col\\{y_{1},\ldots,y_{N}\\}$ and $C=col\\{C_{1},\ldots,C_{N}\\}$,
respectively. $y_{i}=C_{i}\chi\in\mathbb{R}^{p_{i}}$ stands for the
information obtained by the $i$th agent (or the “$i$th node” in the
communication network), where $p=\sum_{i=1}^{N}p_{i}$. The pair $(C_{i},A)$
does not need to be observable for all $i=1,\ldots,N$. In this paper, the
output matrix $C$ is assumed to be known for all agents. Hence,  there exists
a nonsingular matrix $T$ that leads to a lower triangular form:

$\displaystyle\dot{x}_{0,i}=A_{io}x_{0,i}+\sum_{l=1}^{i-1}\Upsilon_{il}x_{0,l},$
(3) $\displaystyle y=\Sigma x=col\\{\Sigma_{1},\ldots,\Sigma_{N}\\}x,$ (4)

where $v_{i}$ is the observability index of the $i$th observable subsystem;
$x_{0,i}\in\mathbb{R}^{v_{i}}$ is the element of
$x=col\\{x_{0,1},\ldots,x_{0,N}\\}$ with $x=T\chi$;
$A_{io}\in\mathbb{R}^{v_{i}\times v_{i}}$, $C_{io}\in\mathbb{R}^{p_{i}\times
v_{i}}$, $\Upsilon_{il}\in\mathbb{R}^{v_{l}\times v_{i}}$; and

$\displaystyle\Sigma_{i}=\begin{bmatrix}0_{p_{i}\times\bar{v}_{i-1}},C_{io},0_{p_{i}\times(n-\bar{v}_{i})}\end{bmatrix}$
(5)

are matrices transformed from $A$ and $C$ with
$\bar{v}_{i}=\sum_{j=1}^{i}v_{i}$. Moreover, $\Upsilon_{il}$ stands for the
coupling matrix between $x_{0,l}$ and $x_{0,i}$; $v_{i}$ and $v_{l}$ are the
dimensions of $A_{io}$ and $A_{lo}$, respectively. Note that the pair
$(C_{io},A_{io})$ is observable according to the property of observability
decomposition. To further help the reader understand the relationship between
$A$ and $A_{io}$, $\Upsilon_{il}$, we specifically show that

$\displaystyle TAT^{-1}=\begin{bmatrix}A_{1o}&&&\\\ \Upsilon_{21}&A_{2o}&&\\\
\vdots&\vdots&\ddots&\\\
\Upsilon_{N1}&\Upsilon_{N2}&\cdots&A_{No}\end{bmatrix}.$ (6)

The overall goal of this paper is to design a distributed observer for the
underlying system. The $i$th agent owns the outputs of the $i$th observable
subsystem and establishes the $i$th local observer, such that its local state
estimation $\hat{\chi}_{i}$ satisfies
$\lim_{t\to\infty}\|\hat{\chi}_{i}-\chi\|=0$, or
$\lim_{t\to\infty}\|\hat{x}_{i}-x\|=0$, where $\hat{x}_{i}=T\hat{\chi}_{i}$.
As stated in Section 1, this property is the so-called asymptotic omniscience
defined by [3] and the references therein.

The communication network among $N$ agents will be described as follows, and
each agent is regarded as a node in the network. Given a piecewise function
$\sigma(t)$ that maps time $t$ to an index set
$\mathcal{P}=\\{1,\ldots,\varpi\\}$. The network that appears in time interval
$[0,+\infty)$ is indexed by $\sigma(t)$. For example, a graph at time $t$ can
be characterized as
$\mathcal{G}^{\sigma}=\\{\mathcal{V}^{\sigma},\mathcal{E}^{\sigma}\\}$, where
$\mathcal{V}^{\sigma}$ and $\mathcal{E}^{\sigma}$ are its node set and arc
set, respectively. Let
$\mathcal{A}^{\sigma}=[\alpha_{ij}^{\sigma}]_{i,j=1}^{N}$ be the adjacency
matrix of $\mathcal{G}^{\sigma}$. We now state the following two assumptions.

###### Assumption 1

Consider a time series $t_{0},~{}t_{1},\cdots$, $t_{k},\cdots$ subject to
$t_{k+1}-t_{k}=\mathcal{T}$ for $k=0,1,\cdots$, and a subsequence in
$[t_{k},t_{k+1})$ defined as $t_{k}^{0},\cdots,t_{k}^{\varpi}$ with
$t_{k}^{j+1}-t_{k}^{j}=\tau=\mathcal{T}/\varpi,j=1,\cdots,\ell_{k}-1$, where
$\mathcal{T}>0$ is a given constant, $t_{0}$ is the initial time of the system
and $t_{j}$ is the time moment with respect to the switching. Then, the
interconnection topology is supposed to remain unchanged in each
$[t_{k}^{j},t_{k}^{j+1})$, and it is further supposed that
$\mathcal{G}^{\sigma(t)}=\mathcal{G}^{j}$ if $t\in[t_{k}^{j},t_{k}^{j+1})$ for
arbitrary $k=0,1,\cdots$.

###### Assumption 2

We assume that the union $\bigcup_{\sigma\in\mathcal{P}}\mathcal{G}^{\sigma}$
of all $\mathcal{G}^{\sigma},~{}\sigma\in\mathcal{P}$ is strongly connected.
Besides, $\forall i,j$, there exists at least one graph belonging to
$\mathcal{P}$ which contains a path from $i$ to $j$.

###### Remark 1

Both Assumption 1 and Assumption 2 are mild. Assumption 1 differs from the
traditional assumption about jointly connected switching topologies, which it
requires a fixed dwell time. This is in line with some practical situations.
For example, each node switches communication signals according to a certain
communication protocol if communication resources are limited. Assumption 2
has one more directed path condition than the traditional assumption. The
directed path condition holds when network resources are limited and some
links in the strongly connected graph need to be interrupted intermittently.
Furthermore, we will give some illustration concerning the assumption of
“knowing the output matrix $C$ for all agents”. Indeed, many interconnected
systems in which the output functions of each subsystem are the same, such as
the power system consisting of several power-generation areas, the droop
control systems of microgrids, and the coupled Van der Pol system. In these
systems, as long as the agent knows the form of its own output function, this
is equivalent to knowing the form of other subsystems’ output functions.
Therefore, from this perspective, assuming that matrix $C$ is known to all
agents does not compromise the distributed characteristics of the distributed
observer.

##  III Design of distributed observer

In this section, we first present dynamic equations for the distributed
observer in Subsection III-A. Network transformation mapping will be
introduced in Subsection III-B, which is directly related to the acquisition
of some key parameters in distributed observer. Also, network transformation
mapping is one of the key technologies this paper can use to achieve
distributed observer for open-loop unstable systems under joint connected
switching topology conditions.

###  III-A Dynamics of the local observer

Dynamics of the $i$th local observer take the form of

$\displaystyle\dot{\hat{x}}_{i,i}=$ $\displaystyle
A_{io}\hat{x}_{i,i}+H_{io}\left(y_{i}-C_{io}\hat{x}_{i,i}\right)$
$\displaystyle+\mathbbm{r}_{i,i}\sum_{j=1}^{N}\alpha_{ij}^{\sigma}(i)\left(\hat{x}_{j,i}-\hat{x}_{i,i}\right)+\sum_{l=1}^{i-1}\Upsilon_{il}\hat{x}_{i,l},$
(7) $\displaystyle\dot{\hat{x}}_{i,k}=$ $\displaystyle
A_{ko}\hat{x}_{i,k}+\mathbbm{r}_{i,k}\sum_{j=1}^{N}\alpha_{ij}^{\sigma}(k)\left(\hat{x}_{j,k}-\hat{x}_{i,k}\right)$
$\displaystyle+\sum_{l=1}^{k-1}\Upsilon_{kl}\hat{x}_{i,l},~{}~{}k=1,\ldots,N,~{}k\neq
i.$ (8)

Herein, coupling gains $\mathbbm{r}_{i,k}$ are pending parameters. $H_{io}$ is
the observer gain chosen so that $A_{io}-H_{io}C_{io}$ is a Hurwitz matrix.
$\alpha_{ij}^{\sigma}(k)$, which is supposed to be highlighted, is the image
of the network transformation. In the next section, we will introduce the
construction method of $\alpha_{ij}^{\sigma}(k)$ and its function.

First, we state two lemmas (see the proof of Lemma 1 in the Appendix) that
will be used in this paper.

###### Lemma 1

Consider a linear time-variant system

$\dot{x}(t)=A(t)x(t)+M\xi(t),$

where $x\in\mathbb{R}^{n}$, $A(t),~{}M\in\mathbb{R}^{n\times n}$, and
$\xi(t)\in\mathbb{R}^{n}$ satisfies $\lim_{t\to\infty}\xi(t)=0$, then
$\lim_{t\to\infty}x(t)=0$ if the solution $\mathbbm{x}(t)$ of
$\dot{x}(t)=A(t)x(t)$ satisfies $\mathbbm{x}(t)\leq a_{1}e^{-a_{2}(t-t_{0})}$,
where $a_{1},a_{2}$ are positive constants and $t_{0}>0$ is the initial time.

######  Lemma 2 ([34])

Consider a graph $\mathcal{G}=\\{\mathcal{V},\mathcal{E},\mathcal{A}\\}$.
Assume $\mathcal{G}$ contains a spanning tree rooted at
$i_{r}\notin\mathcal{V}$. Set $b_{i}=1$ if there is a path from $i_{r}$ to
$i\in\mathcal{V}$. Then, all eigenvalues of matrix
$\mathcal{H}=\mathcal{L}+\mathcal{B}$ locate at the open right half of the
plane, where $\mathcal{B}\triangleq diag\\{b_{i},~{}i\in\mathcal{V}\\}$.

Figure 1: Illustration of the mapping on an adjacency matrix

###  III-B Network transformation mapping

Generally speaking, there is no strongly connected branch in directed
switching topologies, which means that the state decomposition and
reorganization method based on connected branches proposed in [23] is not
suitable for directed graphs. Nevertheless, a directed graph always has a
subgraph composed of some nodes with a tree structure. Although this tree
structure does not have good network properties like that of spanning tree
owing to disturbance from nodes outside the tree, it offers a new sight to
developing state decomposition and reorganization methods for directed graph
cases. However, because the tree structure lack the excellent properties of a
spanning tree, it cannot guarantee the asymptotic omniscience of the
distributed observer. Therefore, this section is devoted to proposing a
network transformation mapping to construct a subgraph that contains a
spanning tree.

To this end, denote $l_{ik}^{\sigma}$ the element located at row $i$ and
column $k$ of the matrix
$\mathcal{A}^{\sigma}_{r}=I_{N}+\mathcal{A}^{\sigma}+\left(\mathcal{A}^{\sigma}\right)^{2}+\cdots+\left(\mathcal{A}^{\sigma}\right)^{N-1}$.
Then, let $\ell_{ik}^{\sigma}=l_{ik}^{\sigma}\wedge 1$ be the elements of
$\mathcal{A}^{\sigma}_{r}$’s structure matrix, i.e, $\ell_{ik}^{\sigma}=1$ if
$l_{ik}^{\sigma}>0$ and $\ell_{ik}^{\sigma}=0$ otherwise. Construct a node set
with respect to a node $k$ as
$\mathcal{V}_{k}^{\sigma}=\\{i\in\mathcal{V}:~{}\ell_{ik}^{\sigma}=1\\}\cup\\{k\\}$.
Hence, we can state the following.

###### Lemma 3

Network $\mathcal{G}^{\sigma}$ is transformed to $\mathcal{G}^{\sigma}(k)$
with respect to an arbitrary given node $k$ if the elements of its adjacency
matrix $\alpha_{ij}^{\sigma}$ are transformed by

$\displaystyle\alpha_{ij}^{\sigma}(k)=\delta_{ij}^{\sigma}\alpha_{ij}^{\sigma},$
(9)
$\displaystyle\delta_{ij}^{\sigma}=\left(\left(1-\ell_{ik}^{\sigma}\right)\wedge\left(1-\ell_{jk}^{\sigma}\right)\right)\vee\left(\ell_{ik}^{\sigma}\wedge\ell_{jk}^{\sigma}\right).$
(10)

Then, $\mathcal{G}^{\sigma}(k)$ is the transformed network with adjacency
matrix $[\alpha_{ij}^{\sigma}(k)]_{i,j=1}^{N}$, and contains an independent
subgraph
$\mathcal{G}^{\sigma}_{k}\triangleq\\{\mathcal{V}^{\sigma}_{k},\mathcal{E}^{\sigma}_{k}\\}$.
In addition, $\mathcal{G}^{\sigma}(k)$ has a spanning tree rooted at $k$.
Furthermore, all the nodes belonging to $\mathcal{V}_{k}^{\sigma}$ have no
access to the information from
$\mathcal{V}\backslash\mathcal{V}_{k}^{\sigma}$.

###### Proof:

Based on the knowledge about complex networks, node $i$ obtains the
information of node $j$ at time $t$ by $1$ hop if and only if
$\alpha_{ij}^{\sigma}=1$. Moreover, node $i$ can get node $j$’s information by
$2$ hops at $t$ if and only if
$\sum_{l=1}^{N}\alpha_{il}^{\sigma}\alpha_{lj}^{\sigma}>0$. Hence, matrix
$\mathcal{A}^{\sigma}\cdot\mathcal{A}^{\sigma}$ denotes the reachable matrix
within $2$ hops. Similarly, $\left(\mathcal{A}^{\sigma}\right)^{l}$ denotes
the reachable matrix with $l$ hops.

As a result, the element $\ell_{ij}^{\sigma}=1$ denotes that node $i$ obtains
node $j$’s information within $N-1$ hops. Then, we can construct
$\mathcal{G}^{\sigma}_{k}$ by defining
$\mathcal{V}^{\sigma}_{k}=\\{k\\}\cup\\{l:~{}\ell_{lk}^{\sigma}\neq 0\\}$ and
$\mathcal{E}^{\sigma}_{k}=\\{(i,j)\in\mathcal{E}^{\sigma}:~{}\alpha_{ij}^{\sigma}(k)=1\\}$.
Therefore, there exists a directed path from $k$ to $i$ for $\forall
i\in\mathcal{V}_{k}^{\sigma}$.

To show that $\mathcal{G}^{\sigma}_{k}$ contains a spanning tree, we need to
discuss the elements in $\mathcal{E}_{k}^{\sigma}$ in the following three
cases. 1) The case in which $i,j\in\mathcal{V}^{\sigma}_{K}$. The
aforementioned proof leads thereby to $\ell_{ik}^{\sigma}=1$ and
$\ell_{jk}^{\sigma}=1$. Consequently,

$\displaystyle\alpha_{ij}^{\sigma}(k)=\delta_{ij}^{\sigma}\alpha_{ij}^{\sigma}(0\wedge
0)\vee(1\wedge 1)=\alpha_{ij}^{\sigma}.$

2) The case in which $i,j\in\mathcal{V}\backslash\mathcal{V}_{k}^{\sigma}$. In
light of the definition of $\mathcal{V}^{\sigma}_{k}$, we know there are no
arcs from node $k$ pointing to
$\mathcal{V}\backslash\mathcal{V}^{\sigma}_{k}$. Hence,
$\ell_{ik}^{\sigma}=\ell_{jk}^{\sigma}=0$. This means that
$\alpha_{ij}(k)=(1\wedge 1)\vee(0\wedge
0)\alpha_{ij}^{\sigma}=\alpha_{ij}^{\sigma}$.

3) The case in which $i\in\mathcal{V}_{k}^{\sigma}$ and
$j\in\mathcal{V}\backslash\mathcal{V}_{k}^{\sigma}$. Since $l_{ik}^{\sigma}=1$
and $l_{jk}^{\sigma}=0$, we have $\alpha_{ij}(k)=(0\wedge 1)\vee(1\wedge
0)\alpha_{ij}^{\sigma}=0$.

The above proof indicates that the network transformation mapping leaves the
arcs between the pair belonging to $\mathcal{V}_{k}^{\sigma}$ and
$\mathcal{V}\backslash\mathcal{V}_{k}^{\sigma}$ as they are and removes the
arcs between $\mathcal{V}_{k}^{\sigma}$ and
$\mathcal{V}\backslash\mathcal{V}_{k}^{\sigma}$. In other words, there is no
path from $\mathcal{V}\backslash\mathcal{V}_{k}^{\sigma}$ to
$\mathcal{V}_{k}^{\sigma}$, and also no path from $\mathcal{V}_{k}^{\sigma}$
to $\mathcal{V}\backslash\mathcal{V}_{k}^{\sigma}$. Hence, $\mathcal{G}_{k}$
is an independent subgraph. Moreover, because there exists a directed path
from $k$ to $i$ $\forall i\in\mathcal{V}_{k}^{\sigma}$, $\mathcal{G}_{k}$
contains a spanning tree rooted at $k$. ∎

According to the definition of Lemma 3, coupling gain can be preliminary
designed as

$\displaystyle\mathbbm{r}_{i,k}=(1-\ell_{ik}^{\sigma})\gamma+\ell_{ik}^{\sigma}\gamma_{i,k},~{}k=1,\ldots,N,$
(11)

where $\gamma$ and $\gamma_{i,j}$ are positive parameters that need to be
designed later.

###### Remark 2

This remark employs an example to show the function of Lemma 3. In Figure 1,
the subfigures (a) and (c) are original directed networks. First, transform
(a) with respect to node $1$ by (9), (10). Then, (a) can be transformed into
subfigure (b), which contains an independent subgraph $\mathcal{G}_{1}$ with
$\mathcal{V}_{1}=\\{1,3,4,5,6,7\\}$. The arcs among $\mathcal{V}_{1}$ are
preserved and the arcs that point from other nodes to $\mathcal{V}_{1}$ are
removed. Furthermore, $\mathcal{G}_{1}$ contains a spanning tree rooted at
node $1$. To further verify the effectiveness of Lemma 3, we transform network
(c) with respect to node $8$ and thus obtain the transformation in (d). A
subgraph $\mathcal{G}_{8}$ with $\mathcal{V}_{8}=\\{2,7,8,10\\}$ does not
obtain any information from the nodes outside and includes a spanning tree
rooted at $8$.

###### Remark 3

As explained in the introduction, multiple local observers cannot achieve
consistency even if they are in the same connected branch (undirected graph),
which is caused by the strong coupling between observable subsystems.
Moreover, the network transformation mapping in this section can only obtain
directed independent subgraphs, whose network properties are not as good as
the connected branches of undirected graphs. Therefore, to analyze the
performance of a distributed observer in independent subgraphs, it is also
necessary to propose a state decomposition and reorganization method (Section
IV) for directed graph cases, in order to weaken the coupling relationship and
cause the independent subgraphs to have independent dynamics.

##  IV Asymptotic omniscience of distributed observer

The definition of asymptotic omniscience suggests that achieving asymptotic
omniscience for distributed observer is equivalent to the stability of its
error dynamics. To demonstrate the stability of the error dynamics of
distributed observer, this section first presents a state decomposition and
reorganization approach in the case of directed graph, and derives some
preliminary conclusions (in Subsection IV-A). Subsequently, based on these
preliminary conclusions, we will prove the stability of error dynamics in
Subsection IV-B.

###  IV-A State decomposition and reorganization

In this section, state decomposition and reorganization methods are presented
to assist in analyzing the error dynamic performance of distributed observer
(7), (8), bearing in mind that the states have been decomposed in (3), (4).
This section will first state the observer error dynamics based on state
decomposition, then reorganize them on the basis of observable subspace.

The error dynamics are governed by

$\displaystyle\dot{e}_{i,i}=$
$\displaystyle\left(A_{io}-H_{io}C_{io}\right)e_{i,i}+\sum_{l=1}^{i-1}\Upsilon_{il}e_{i,l},$
(12) $\displaystyle\dot{e}_{i,k}=$ $\displaystyle
A_{ko}e_{i,k}+\mathbbm{r}_{i,k}\sum_{j=1}^{N}\alpha_{ij}^{\sigma}(k)\left(e_{j,k}-e_{i,k}\right)$
$\displaystyle+\sum_{l=1}^{k-1}\Upsilon_{kl}e_{k,l},~{}~{}k=1,\ldots,N,~{}k\neq
i.$ (13)

Note that the transformed network with respect to node $i$ indicates there is
no arc pointing to node $i$, so $e_{i,i}$ does not contain a coupling term.
Therefore, error dynamics can be reorganized in the compact form by the
following rule. Denote
$\varepsilon_{i\star}=col\\{e_{j,i},~{}j\in\mathcal{V}_{i}^{\sigma}\backslash\\{i\\}\\}$,
$\varepsilon_{i\diamond}=col\\{e_{k,i},~{}k\in\mathcal{V}\backslash\mathcal{V}_{i}^{\sigma}\\}$,
and $\varepsilon_{ii}=1_{\pi_{i}(\sigma)}\otimes e_{i,i}$, where
$\pi_{i}(\sigma)=\left|\mathcal{V}_{i}^{\sigma}\right|-1$; and denote
$\xi_{i\star}=col\\{\xi_{k,i},~{}k\in\mathcal{V}_{i}^{\sigma}\backslash\\{i\\}\\}$
with
$\xi_{k,i}=\sum_{j=1}^{N}\alpha_{kj}^{\sigma}(i)\left(\hat{x}_{j,i}-\hat{x}_{k,i}\right)$.
To move on, we denote the Laplacian matrix of
$\mathcal{V}_{i}\backslash\\{i\\}$ as $\mathcal{L}_{\pi_{i}(\sigma)}$. Then,
one states the properties of the reorganized states.

###### Lemma 4

The stability of $col\left\\{\varepsilon_{ii},\varepsilon_{i\star}\right\\}$
is equivalent to the stability of system
$col\left\\{\varepsilon_{ii},\xi_{i\star}\right\\}$ with the form

$\displaystyle\dot{\varepsilon}_{ii}=$
$\displaystyle\left(I_{\pi_{i}(\sigma)}\otimes\left(A_{io}-H_{io}C_{io}\right)\right)\varepsilon_{ii}$
$\displaystyle+\sum_{l=0}^{i-1}\left(I_{\pi_{i}(\sigma)}\otimes\Upsilon_{il}\right)\left(1_{\pi_{i}(\sigma)}\otimes
e_{i,l}\right),$ (14)

and

$\displaystyle\dot{\xi}_{i\star}=$
$\displaystyle\left(I_{\pi_{i}(\sigma)}\otimes
A_{io}\right)\xi_{i\star}-\left(\mathcal{H}_{\pi_{i}(\sigma)}\Gamma_{i}^{\sigma}\otimes
I_{v_{i}}\right)\xi_{i\star}$ $\displaystyle+\left(B_{i\star}^{\sigma}\otimes
H_{io}C_{io}\right)\varepsilon_{ii}$
$\displaystyle-\sum_{l=0}^{i-1}\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\left(I_{\pi_{i}(\sigma)}\otimes\Upsilon_{il}\right)\left(1_{\pi_{i}(\sigma)}\otimes
e_{i,l}\right)$
$\displaystyle+\sum_{l=0}^{i-1}\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\Upsilon_{i\star,l}\varepsilon_{i\star,l}.$ (15)

###### Proof:

Based on the definition, it is easy to get (4). Furthermore, according to (11)
that $\ell_{ji}^{\sigma}=1$ when
$j\in\mathcal{V}_{i}^{\sigma}\backslash\\{i\\}$, we have

$\displaystyle\dot{\varepsilon}_{i\star}=$
$\displaystyle\left(I_{\pi_{i}(\sigma)}\otimes
A_{io}\right)\varepsilon_{i\star}-\left(\Gamma_{i}^{\sigma}\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\varepsilon_{i\star}$
$\displaystyle+\left(\Gamma_{i}^{\sigma}\otimes
I_{v_{i}}\right)\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\varepsilon_{ii}+\sum_{l=1}^{i-1}\Upsilon_{i\star,l}\varepsilon_{i\star,l},$
(16)

where
$\Gamma_{i}^{\sigma}=diag\left\\{\gamma_{j,i},~{}j\in\mathcal{V}_{i}^{\sigma}\backslash\\{i\\}\right\\}$,
and
$\mathcal{H}_{\pi_{i}(\sigma)}=\mathcal{L}_{\pi_{i}(\sigma)}+B_{i\star}^{\sigma}$
with
$B_{i\star}^{\sigma}=diag\\{\alpha_{ji}(i),~{}j\in\mathcal{V}_{i}^{\sigma}\backslash\\{i\\}\\}$.
Also,
$\Upsilon_{i\star,t}=diag\\{\Upsilon_{jt},~{}j\in\mathcal{V}_{i}^{\sigma}\backslash\\{i\\}\\}$
and
$\varepsilon_{i\star,t}=col\\{e_{j,t},~{}j\in\mathcal{V}_{i}^{\sigma}\backslash\\{i\\}\\}$.

From the notation of $\xi_{i\star}$, we know

$\displaystyle\xi_{i\star}=\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\varepsilon_{i\star}-\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\varepsilon_{ii}.$ (17)

Obviously, based on the stability of $\varepsilon_{ii}$, $\xi_{i\star}$
converges to zero if and only if $\varepsilon_{ii}$, $\varepsilon_{i\star}$
converges to zero. Then, in the time interval in which $\sigma$ is a constant,
we have

$\displaystyle\dot{\xi}_{i\star}=$
$\displaystyle\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\dot{\varepsilon}_{i\star}-\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\dot{\varepsilon}_{ii}$ $\displaystyle=$
$\displaystyle\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\left(I_{\pi_{i}(\sigma)}\otimes
A_{io}\right)\varepsilon_{i\star}$
$\displaystyle-\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\left(\Gamma_{i}^{\sigma}\otimes
I_{v_{i}}\right)\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\varepsilon_{i\star}$
$\displaystyle+\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\left(\Gamma_{i}^{\sigma}\otimes
I_{v_{i}}\right)\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\varepsilon_{ii}$
$\displaystyle-\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\left(I_{\pi_{i}(\sigma)}\otimes\left(A_{io}-H_{io}C_{io}\right)\right)\varepsilon_{ii}$
$\displaystyle-\sum_{l=1}^{i-1}\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\left(I_{\pi_{i}(\sigma)}\otimes\Upsilon_{il}\right)\left(1_{\pi_{i}(\sigma)}\otimes
e_{i,l}\right)$
$\displaystyle+\sum_{l=1}^{i-1}\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\Upsilon_{i\star,l}\varepsilon_{i\star,l}.$ (18)

According to Lemma 3, subgraph $\mathcal{V}_{i}^{\sigma}$ has a spanning tree
rooted at $i$. Therefore, $\mathcal{H}_{\pi_{i}(\sigma)}$ is full rank due to
the conclusion of Lemma 2. Then, we have

$\displaystyle\varepsilon_{i\star}=$
$\displaystyle\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)^{-1}\xi_{i\star}$
$\displaystyle+\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)^{-1}\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\varepsilon_{ii}.$ (19)

Subsequently, substituting (IV-A) into (IV-A) leads to

$\displaystyle\dot{\xi}_{i\star}=$
$\displaystyle\left(I_{\pi_{i}(\sigma)}\otimes
A_{io}\right)\xi_{i\star}+\left(B_{i\star}^{\sigma}\otimes
A_{io}\right)\varepsilon_{ii}$
$\displaystyle-\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\left(\Gamma_{i}^{\sigma}\otimes I_{v_{i}}\right)\xi_{i\star}$
$\displaystyle-\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\left(I_{\pi_{i}(\sigma)}\otimes\left(A_{io}-H_{io}C_{io}\right)\right)\varepsilon_{ii}$
$\displaystyle-\sum_{l=1}^{i-1}\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\left(I_{\pi_{i}(\sigma)}\otimes\Upsilon_{il}\right)\left(1_{\pi_{i}(\sigma)}\otimes
e_{i,l}\right)$
$\displaystyle+\sum_{l=1}^{i-1}\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\Upsilon_{i\star,l}\varepsilon_{i\star,l}$ $\displaystyle=$
$\displaystyle\left(I_{\pi_{i}(\sigma)}\otimes
A_{io}\right)\xi_{i\star}-\left(\mathcal{H}_{\pi_{i}(\sigma)}\Gamma_{i}^{\sigma}\otimes
I_{v_{i}}\right)\xi_{i\star}$ $\displaystyle+\left(B_{i\star}^{\sigma}\otimes
H_{io}C_{io}\right)\varepsilon_{ii}$
$\displaystyle-\sum_{l=1}^{i-1}\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\left(I_{\pi_{i}(\sigma)}\otimes\Upsilon_{il}\right)\left(1_{\pi_{i}(\sigma)}\otimes
e_{i,l}\right)$
$\displaystyle+\sum_{l=1}^{i-1}\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\Upsilon_{i\star,l}\varepsilon_{i\star,l}.$

Hence, (4) is obtained. ∎

###### Remark 4

This remark describes the significance of observability decomposition and
state reorganization methods. The error dynamics of each local observer
contains the dynamics of $N$ observable subsystems, and the convergence of the
error dynamics corresponding to different observable subsystems depends on
different network topology. Therefore, the communication conditions can meet
the requirements of all observable subsystems in all local observers at the
same time if and only if the network is strongly connected. However, in the
scenario of jointly connected switching topologies, the real-time disconnected
network and the strong coupling relationship between observable subsystems
will make it impossible to analyze the performance of the error dynamics of
any observable subsystem. Hence, the observability decomposition method is
employed to decompose the state estimation of each local observer, which leads
to a weak coupling relationship between all decomposed systems. Furthermore,
according to the communication network requirements of the decomposed
observable subsystems, the subsystems from each local observer with the same
communication requirements are reorganized into a group of new compact states
including $\varepsilon_{ii},\varepsilon_{i\star}$, and
$\varepsilon_{i\diamond}$. In this way, all state components in new compact
states can meet the communication conditions required for stability analysis.
Consequently, the network properties of each new compact state and the weak
coupling among them help us construct the recursive proof method in the next
section, so as to obtain the asymptotic omniscience of the distributed
observer under the switching topologies.

###### Remark 5

As stated in the previous remark, each new compact state has a complete
network property that can meet its own requirements after state decomposition
and reorganization. Herein, we elaborate on the specific meaning of the
complete network property. Denote $\Xi_{ij}$ the $j$th observable subsystem of
the $i$th local observer after observability decomposition. A subsystem
$\Xi_{ij}$ has a complete network property if all of its parents and children
nodes are descendants of $\Xi_{jj}$ when $\Xi_{ij}$ is a descendant node of
$\Xi_{jj}$ (note that $\Xi_{ij}$ is a descendant node of $\Xi_{jj}$ if and
only if $i$ is the descendant of $j$ because $i$ in $\Xi_{ij}$ labels the
local observer). It is impossible for a typical network topology to possess
such properties, and the key to ensuring that the communication network
maintains such properties during the switching process in this paper lies in
the network transformation mapping proposed in Section III. The network
transformation mapping with respect to node $j$ can ensure that the descendant
nodes of $j$ will not receive information from other non-descendant nodes.
Therefore, it can ensure that a node $i$ is either not a descendant of $j$ or
it and its parents and children are all descendants of $j$ (a more detailed
description can be found in Remark 2). In summary, the state decomposition and
reorganization method in this section and the network transformation mapping
in the previous section are important foundations for the asymptotic
omniscience proof in the next section.

###  IV-B Stability analysis of the error dynamics of distributed observer

By observing the form (4) of $\xi_{i\star}$, it is not difficult to find that
there is a lower triangular coupling relationship between
$\xi_{1\star},\ldots,\xi_{N\star}$. Namely, the dynamics of $\xi_{i\star}$ are
affected by $e_{i,t}$ with $t=1,\ldots,i-1$. Their stability is proved
recursively in light of this coupling relationship, which is demonstrated in
the following theorem.

###### Theorem 1

Distributed observer formed by (7), (8) achieves omniscience asymptotically
under the switching directed networks subject to Assumptions 1 and 2, if its
coupling gain satisfies (11) with a positive constant $\gamma$ and a group of
sufficiently large $\gamma_{i,k}$ governed by

$\displaystyle\underline{\gamma}_{i}^{2}$
$\displaystyle>\bar{\lambda}(\Xi_{i1})+\frac{1}{4}\bar{\lambda}^{2}(\Xi_{i2})+\wp,$
(20) $\displaystyle\wp$
$\displaystyle>\frac{\bar{\lambda}_{i}\bar{\lambda}(\Xi_{i\diamond})}{\underline{\lambda}(P_{iu})}\frac{m(T_{k,i}^{c})}{m(T_{k,i})},$
(21)

for all $T_{k,i}$ and $T_{k,i}^{c}$ with $i=1,\ldots,N$ and $k=1,\ldots,N$.
Here, $m(\cdot)$ is measure and $T_{k,i}$ is a subset of $[t_{j},t_{j+1})$ in
which $k\in\mathcal{V}_{i}^{\sigma}$, and $T_{k,i}^{c}$ is
$[t_{j},t_{j+1})\backslash T_{k,i}$;
$\underline{\gamma}_{i}=\min_{j=1,\ldots,N}\\{\gamma_{j,i}\\}$;
$\Xi_{i1}^{\sigma}=Q_{io}^{\sigma}\otimes A_{io}$,
$\Xi_{i2}^{\sigma}=Q_{io}^{\sigma}B_{i\star}^{\sigma}\otimes H_{io}C_{io}$,
$\bar{\lambda}(\Xi_{i1})=\max_{t\in\mathbb{R}}\\{\bar{\lambda}(\Xi_{i1}^{\sigma})\\}$,
$\bar{\lambda}(\Xi_{i2})=\max_{t\in\mathbb{R}}\\{\bar{\lambda}(\Xi_{i2}^{\sigma})\\}$;
$\wp>0$ is a given constant; $Q_{io}^{\sigma}$ satisfies
$Q_{io}^{\sigma}\mathcal{H}_{\pi_{i}(\sigma)}+\mathcal{H}_{\pi_{i}(\sigma)}^{T}Q_{io}^{\sigma}>2I_{\pi_{i}(\sigma)}$;
$\bar{\lambda}_{i}^{\sigma}=\max\left\\{\bar{\lambda}(P_{io}),\bar{\lambda}\left(Q_{io}^{\sigma}\right)\right\\}$
and $\bar{\lambda}_{i}=\max_{t\in\mathbb{R}}\\{\bar{\lambda}_{i}^{\sigma}\\}$;
$\Xi_{i\diamond}^{\sigma}=2P_{iu}^{\sigma}\otimes A_{1o}$ with symmetric
positive definite matrix $P_{iu}^{\sigma}$ being solved by
$P_{iu}^{\sigma}\mathcal{L}_{\bar{\pi}_{i}(\sigma)}+\mathcal{L}_{\bar{\pi}_{i}(\sigma)}^{T}P_{iu}^{\sigma}\geq
0$, and $\mathcal{L}_{\bar{\pi}_{1}(\sigma)}$ is the Laplacian matrix that
corresponds to the subgraph among
$\mathcal{V}\backslash\mathcal{V}_{i}^{\sigma}$.

###### Proof:

This theorem will be proved by mathematical induction.

1) The performance of $\varepsilon_{11}$ and $\xi_{1\star}$. We will show that
$\lim_{t\to\infty}\|\varepsilon_{11}\|=0$ and
$\lim_{t\to\infty}\|\xi_{1\star}\|=0$. According to (4), it has

$\displaystyle\dot{\varepsilon}_{11}=\left(I_{\pi_{1}(\sigma)}\otimes(A_{1o}-H_{1o}C_{1o})\right)\varepsilon_{11}.$

A symmetric positive definite matrix $P_{1o}$ can be found such that
$sym\left\\{P_{1o}(A_{1o}-H_{1o}C_{1o})\right\\}=-2(\bar{\gamma}_{1}+\wp)I_{v_{1}}$,
where $\bar{\gamma}_{1}=\max_{j=1,\ldots,N}\\{\gamma_{j,1}\\}$. Then, by
setting
$V_{1,\varepsilon}=\frac{1}{2}\varepsilon_{11}^{T}(I_{\pi_{1}(\sigma)}\otimes
P_{1o})\varepsilon_{11}$, it yields
$\dot{V}_{1,\varepsilon}\leq-(\bar{\gamma}_{1}+\wp)\varepsilon_{11}^{T}\varepsilon_{11}$.
Also, since $\mathcal{H}_{\pi_{1}(\sigma)}$ is a nonsingular M-matrix, there
exists positive diagonal matrix $Q_{1o}^{\sigma}$ that satisfies
$Q_{1o}^{\sigma}\mathcal{H}_{\pi_{1}(\sigma)}+\mathcal{H}_{\pi_{1}(\sigma)}^{T}Q_{1o}^{\sigma}>2I_{\pi_{1}(\sigma)}$.
To move on, the Lyapunov function with respect to $\xi_{1\star}$ can be chosen
as

$\displaystyle
V_{1,\xi}=\frac{1}{2}\sum_{j\in\mathcal{V}_{1}^{\sigma}\backslash\\{1\\}}q_{1o,j}^{\sigma}\gamma_{j,1}\xi_{j,1}^{T}\xi_{j,1},$

with $q_{1o,j}^{\sigma}$ being the elements of a symmetric positive definite
matrix
$Q_{1o}^{\sigma}=diag\\{q_{1o,j},~{}j\in\mathcal{V}_{1}^{\sigma}\backslash\\{1\\}\\}$.
It follows that

$\displaystyle\dot{V}_{1,\xi}=$
$\displaystyle\xi_{1\star}^{T}\left(\Gamma_{1}^{\sigma}Q_{1o}^{\sigma}\otimes
I_{v_{1}}\right)\dot{\xi}_{1\star}$ $\displaystyle=$
$\displaystyle\xi_{1\star}^{T}\left(\Gamma_{1}^{\sigma}Q_{1o}^{\sigma}\otimes
A_{1o}-\Gamma_{1}^{\sigma}Q_{1o}^{\sigma}\mathcal{H}_{\pi_{1}(\sigma)}\Gamma_{1}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$
$\displaystyle+\xi_{1\star}^{T}\left(\Gamma_{1}^{\sigma}Q_{1o}^{\sigma}B_{1\star}^{\sigma}\otimes
H_{1o}C_{1o}\right)\varepsilon_{11}$ $\displaystyle\leq$
$\displaystyle\bar{\lambda}\left(Q_{1o}^{\sigma}\otimes
A_{1o}\right)\xi_{1\star}^{T}\left(\Gamma_{1}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$
$\displaystyle-\xi_{1\star}^{T}\left(\Gamma_{1}^{\sigma}\Gamma_{1}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$
$\displaystyle+\bar{\lambda}\left(Q_{1o}^{\sigma}B_{1\star}^{\sigma}\otimes
H_{1o}C_{1o}\right)\xi_{1\star}^{T}\left(\Gamma_{1}^{\sigma}\otimes
I_{v_{1}}\right)\varepsilon_{11}$ $\displaystyle\leq$
$\displaystyle\left(\bar{\lambda}(Q_{1o}^{\sigma}\otimes
A_{1o})-\underline{\gamma}_{1}^{2}\right)\xi_{1\star}^{T}\left(\Gamma_{1}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$
$\displaystyle+\bar{\lambda}\left(Q_{1o}^{\sigma}B_{1\star}^{\sigma}\otimes
H_{1o}C_{1o}\right)\xi_{1\star}^{T}\left(\Gamma_{1}^{\sigma}\otimes
I_{v_{1}}\right)\varepsilon_{11},$

where $\underline{\gamma}_{1}=\min_{j=1,\ldots,N}\\{\gamma_{j,1}\\}$.

For ease of notation, we define $\wp_{1,1}=-\bar{\gamma}_{1}+\wp$,
$\wp_{1,2}=\bar{\lambda}\left(Q_{1o}^{\sigma}B_{1\star}^{\sigma}\otimes
H_{1o}C_{1o}\right)$, $\wp_{1,3}=\bar{\lambda}(Q_{1o}^{\sigma}\otimes
A_{1o})-\underline{\gamma}_{1}^{2}+\wp$, and
$\bar{\Gamma}_{1}^{\sigma}=\Gamma_{1}^{\sigma}\otimes I_{v_{1}}$. Let
$V_{1,1}=V_{1,\varepsilon}+V_{1,\xi}$ and hence deduce

$\displaystyle\dot{V}_{1,1}\leq$
$\displaystyle\wp_{1,3}\xi_{1\star}^{T}\bar{\Gamma}_{1}^{\sigma}\xi_{1\star}+\wp_{1,2}\xi_{1\star}^{T}\bar{\Gamma}_{1}^{\sigma}\varepsilon_{11}-\bar{\gamma}_{1}\varepsilon_{11}^{T}\varepsilon_{11}$
$\displaystyle-\wp\xi_{1\star}^{T}\bar{\Gamma}_{1}^{\sigma}\xi_{1\star}-\wp\varepsilon_{11}^{T}\varepsilon_{11}$
$\displaystyle=$
$\displaystyle\begin{bmatrix}\varepsilon_{11}^{T}&\xi_{1\star}^{T}\end{bmatrix}\begin{bmatrix}-\bar{\gamma}_{1}I_{\pi_{1}(\sigma)v_{1}}&\frac{1}{2}\wp_{1,2}\bar{\Gamma}_{1}^{\sigma}\\\
\frac{1}{2}\wp_{1,2}\bar{\Gamma}_{1}^{\sigma}&\wp_{1,3}\bar{\Gamma}_{1}^{\sigma}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\\
\xi_{1\star}\end{bmatrix}$
$\displaystyle-\begin{bmatrix}\varepsilon_{11}^{T}&\xi_{1\star}^{T}\end{bmatrix}\begin{bmatrix}\wp
I_{\pi_{1}(\sigma)v_{1}}&\\\
&\wp\bar{\Gamma}_{1}^{\sigma}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\\
\xi_{1\star}\end{bmatrix}.$ (22)

Based on (20), we know

$\displaystyle\underline{\gamma}_{1}^{2}$
$\displaystyle>\bar{\lambda}(\Xi_{11})+\frac{1}{4}\bar{\lambda}^{2}(\Xi_{12})+\wp$
$\displaystyle>\bar{\lambda}(\Xi_{11}^{\sigma})+\frac{1}{4}\bar{\lambda}^{2}(\Xi_{12}^{\sigma})+\wp,$
(23)

where $\Xi_{11}^{\sigma}=Q_{1o}^{\sigma}\otimes A_{1o}$,
$\Xi_{12}^{\sigma}=Q_{1o}^{\sigma}B_{1\star}^{\sigma}\otimes H_{1o}C_{1o}$,
$\bar{\lambda}(\Xi_{11})=\max_{t\in\mathbb{R}}\\{\bar{\lambda}(\Xi_{11}^{\sigma})\\}$,
$\bar{\lambda}(\Xi_{12})=\max_{t\in\mathbb{R}}\\{\bar{\lambda}(\Xi_{12}^{\sigma})\\}$.
Hence,

$\displaystyle 0$
$\displaystyle>\wp_{1,3}I_{\pi_{1}(\sigma)v_{1}}+\frac{1}{4}\wp_{1,2}^{2}I_{\pi_{1}(\sigma)v_{1}}$
$\displaystyle>\wp_{1,3}I_{\pi_{1}(\sigma)v_{1}}+\frac{1}{4\bar{\gamma}_{1}}\wp_{1,2}^{2}\left(\Gamma_{1}^{\sigma}\otimes
I_{v_{1}}\right).$

Since $\Gamma_{1}^{\sigma}\otimes I_{v_{1}}$ is a diagonal positive definite
matrix, we further obtain that

$\displaystyle\wp_{1,3}\left(\Gamma_{1}^{\sigma}\otimes
I_{v_{1}}\right)+\frac{1}{4\bar{\gamma}_{1}}\wp_{1,2}^{2}\left(\Gamma_{1}^{\sigma}\otimes
I_{v_{1}}\right)^{2}<0.$ (24)

In light of the Schur complement lemma, (24) is equivalent to

$\displaystyle\begin{bmatrix}-\bar{\gamma}_{1}I_{\pi_{1}(\sigma)v_{1}}&\frac{1}{2}\wp_{1,2}\bar{\Gamma}_{1}^{\sigma}\\\
\frac{1}{2}\wp_{1,2}\bar{\Gamma}_{1}^{\sigma}&\wp_{1,3}\bar{\Gamma}_{1}^{\sigma}\end{bmatrix}<0.$
(25)

Substituting (25) into (IV-B) yields

$\displaystyle\dot{V}_{1,1}$
$\displaystyle<-\wp\varepsilon_{11}^{T}\varepsilon_{11}-\wp\xi_{1\star}^{T}\bar{\Gamma}_{1}^{\sigma}\xi_{1\star}$
$\displaystyle<-\frac{\wp}{\bar{\lambda}(P_{1o})}V_{1,\varepsilon}-\frac{\wp}{\bar{\lambda}\left(Q_{1o}^{\sigma}\right)}V_{1,\xi}$
$\displaystyle<-(\bar{\lambda}_{1}^{\sigma})^{-1}\wp V_{1,1},$

where
$\bar{\lambda}_{1}^{\sigma}=\max\left\\{\bar{\lambda}(P_{1o}),\bar{\lambda}\left(Q_{1o}^{\sigma}\right)\right\\}$.
Since the range of $\sigma(t)$ is limited, there exists
$\bar{\lambda}_{1}=\max_{t\in\mathbb{R}}\\{\bar{\lambda}_{1}^{\sigma}\\}$.
Hence

$\displaystyle
V_{1,1}(t)\leq\exp\left\\{-\bar{\lambda}_{1}^{-1}\wp(t-t_{0})\right\\}V_{1,1}(t_{0}).$

Now the nodes that belong to $\mathcal{V}\backslash\mathcal{V}_{1}^{\sigma}$
should be considered. Note that $\ell_{ik}^{\sigma}=0$ in this scenario and
thus $\mathbbm{r}_{i,k}=\gamma$. Recall that the compact form of these nodes
is
$\varepsilon_{1\diamond}=col\\{e_{k,1},~{}k\in\mathcal{V}\backslash\mathcal{V}_{1}^{\sigma}\\}$.
By setting $\bar{\pi}_{1}(\sigma)=N-\pi_{1}(\sigma)$, we obtain

$\displaystyle\dot{\varepsilon}_{1\diamond}=\left(I_{\bar{\pi}_{1}(\sigma)}\otimes
A_{1o}\right)\varepsilon_{1\diamond}-\gamma\left(\mathcal{L}_{\bar{\pi}_{1}(\sigma)}\otimes
I_{v_{1}}\right)\varepsilon_{1\diamond}.$

Since $\mathcal{L}_{\bar{\pi}_{1}(\sigma)}$ is a singular M-matrix, there
exists a symmetric positive definite matrix $P_{1u}^{\sigma}$ in light of [35,
Chapter 4] such that
$P_{1u}^{\sigma}\mathcal{L}_{\bar{\pi}_{1}(\sigma)}+\mathcal{L}_{\bar{\pi}_{1}(\sigma)}^{T}P_{1u}^{\sigma}$
is a symmetric positive semidefinite matrix. Subsequently, the Lyapunov
candidate can be chosen as
$V_{1,\diamond}(t)=\varepsilon_{1\diamond}^{T}\left(P_{1u}^{\sigma}\otimes
I_{\bar{\pi}_{1}(\sigma)}\right)\varepsilon_{1\diamond}$, which leads to

$\displaystyle\dot{V}_{1,\diamond}(t)=$
$\displaystyle\varepsilon_{1\diamond}^{T}\left(P_{1u}^{\sigma}\otimes
A_{1o}\right)\varepsilon_{1\diamond}+\varepsilon_{1\diamond}^{T}\left(P_{1u}^{\sigma}\otimes
A_{1o}^{T}\right)\varepsilon_{1\diamond}$
$\displaystyle-\gamma\varepsilon_{1\diamond}^{T}\left(\left(P_{1u}^{\sigma}\mathcal{L}_{\bar{\pi}_{1}(\sigma)}+\mathcal{L}_{\bar{\pi}_{1}(\sigma)}^{T}P_{1u}^{\sigma}\right)\otimes
I_{v_{1}}\right)\varepsilon_{1\diamond}$ $\displaystyle\leq$ $\displaystyle
2\varepsilon_{1\diamond}^{T}\left(P_{1u}^{\sigma}\otimes
A_{1o}\right)\varepsilon_{1\diamond}\leq\frac{\bar{\lambda}(\Xi_{1\diamond}^{\sigma})}{\underline{\lambda}(P_{1u}^{\sigma})}V_{1,\diamond}(t),$

where $\Xi_{1\diamond}^{\sigma}=2P_{1u}^{\sigma}\otimes A_{1o}$. Also, because
the range of $\sigma$ is finite, we find
$\bar{\lambda}(\Xi_{1\diamond})=\max_{t\in\mathbb{R}}\\{\bar{\lambda}(\Xi_{1\diamond}^{\sigma})\\}$,
$\underline{\lambda}(P_{1u})=\min_{t\in\mathbb{R}}\\{\underline{\lambda}(P_{1u}^{\sigma})\\}$,
and

$\displaystyle
V_{1,\diamond}(t)\leq\exp\left\\{\frac{\bar{\lambda}(\Xi_{1\diamond})}{\underline{\lambda}(P_{1u})}(t-t_{0})\right\\}V_{1,\diamond}(t_{0}).$

Let $W_{i,1}$ be the Lyapunov function of $e_{i,1}$. It satisfies

$\displaystyle
W_{i,1}(t)=e_{1,1}^{T}P_{1o}e_{1,1}+\frac{1}{2}\underline{\lambda}(Q_{1o})\gamma_{i,1}\xi_{i,1}^{T}\xi_{i,1}$

if $t\in T_{i,1}$, and

$\displaystyle W_{i,1}(t)=\underline{\lambda}(P_{1u})e_{i,1}^{T}e_{i,1}$

if $t\in T_{i,1}^{c}$, where $T_{i,1}$ is a subset of $[t_{k},t_{k+1})$ in
which $i\in\mathcal{V}_{1}^{\sigma}$, and $T_{i,1}^{c}$ is
$[t_{k},t_{k+1})\backslash T_{i,1}$. Let $\beth_{i,1}^{k\chi}$ be the
convergence (divergence) rate of $W_{i,1}(t)$ when
$t\in[t_{k}^{\chi-1},t_{k}^{\chi})$. Noting that
$\sum_{i\in\mathcal{V}_{1}^{\sigma}\backslash\\{1\\}}W_{i,1}(t)\leq V_{1,1}$
if $[t_{k}^{\chi-1},t_{k}^{\chi})\subset T_{i,1}$, and
$\sum_{i\in\mathcal{V}\backslash\mathcal{V}_{1}^{\sigma}}W_{i,1}(t)\leq
V_{1,\diamond}$ if $[t_{k}^{\chi-1},t_{k}^{\chi})\subset T_{i,1}^{c}$.
Therefore, $\beth_{i,1}^{k\chi}\leq-\bar{\lambda}_{1}^{-1}\wp$ if
$[t_{k}^{\chi-1},t_{k}^{\chi})\subset T_{i,1}$ and
$\beth_{i,1}^{k\chi}\leq\frac{\bar{\lambda}(\Xi_{1\diamond})}{\underline{\lambda}(P_{1u})}$
otherwise. Then,

$\displaystyle W_{i,1}(t_{k+1})=W_{i,1}\left(t_{k}^{\ell}\right)$
$\displaystyle=$
$\displaystyle\exp\left\\{\beth_{i,1}^{k\ell}(t_{k}-t_{k}^{\ell-1})\right\\}W_{i,1}\left(t_{k}^{\ell-1}\right)$
$\displaystyle\leq$
$\displaystyle\exp\left\\{\beth_{i,1}^{k\ell}(t_{k}^{\ell}-t_{k}^{\ell-1})\right\\}\exp\left\\{\beth_{i,1}^{k(\ell-1)}(t_{k}^{\ell-1}-t_{k}^{\ell-2})\right\\}$
$\displaystyle\times W_{i,1}\left(t_{k}^{\ell-2}\right)$ $\displaystyle\leq$
$\displaystyle\prod_{\chi=1}^{\ell}\exp\left\\{\beth_{i,1}^{k\chi}(t_{k}^{\chi}-t_{k}^{\chi-1})\right\\}W_{i,1}\left(t_{k}^{0}\right).$

According to the value of $\beth_{i,1}^{k\chi}$, we know

$\displaystyle\prod_{\chi=1}^{\ell}\exp\left\\{\beth_{i,1}^{k\chi}(t_{k}^{\chi}-t_{k}^{\chi-1})\right\\}$
$\displaystyle=$
$\displaystyle\exp\left\\{\sum_{\chi=1}^{\ell}\beth_{i,1}^{k\chi}(t_{k}^{\chi}-t_{k}^{\chi-1})\right\\}$
$\displaystyle=$ $\displaystyle\exp\left\\{-\bar{\lambda}_{1}^{-1}\wp
m(T_{i,1})+\frac{\bar{\lambda}(\Xi_{1\diamond})}{\underline{\lambda}(P_{1u})}m(T_{i,1}^{c})\right\\}.$

Consequently,

$\displaystyle
W_{i,1}(t_{k+1})\leq\exp\left\\{\beth_{i,1}\right\\}W_{i,1}(t_{k})$

with

$\displaystyle\beth_{i,1}=-\bar{\lambda}_{1}^{-1}\wp
m(T_{i,1})+\frac{\bar{\lambda}(\Xi_{1\diamond})}{\underline{\lambda}(P_{1u})}m(T_{i,1}^{c}).$

(21) indicates that $W_{i,1}(t_{k+1})\leq W_{i,1}(t_{k})$ because it leads to
$\beth_{i,1}<0$. Therefore, the sufficient conditions (20) and (21) guarantee
the convergence of $\varepsilon_{11}$ and $\xi_{1\star}$.

2) Suppose the convergence of $\varepsilon_{kk}$ and $\xi_{k\star}$ are
fulfilled for $k=1,\ldots,i-1$. We will show that
$\lim_{t\to\infty}\|\varepsilon_{ii}\|=0$ and
$\lim_{t\to\infty}\|\xi_{i\star}\|=0$. From Lemma 4, we note

$\displaystyle\dot{\varepsilon}_{ii}=$
$\displaystyle\left(I_{\pi_{i}(\sigma)}\otimes\left(A_{io}-H_{io}C_{io}\right)\right)\varepsilon_{ii}$
$\displaystyle+\sum_{l=1}^{i-1}\left(I_{\pi_{i}(\sigma)}\otimes\Upsilon_{il}\right)\left(1_{\pi_{i}(\sigma)}\otimes
e_{i,l}\right)$ (26)

and

$\displaystyle\dot{\xi}_{i\star}=$
$\displaystyle\left(I_{\pi_{i}(\sigma)}\otimes
A_{io}\right)\xi_{i\star}-\left(\mathcal{H}_{\pi_{i}(\sigma)}\Gamma_{i}^{\sigma}\otimes
I_{v_{i}}\right)\xi_{i\star}$ $\displaystyle+\left(B_{i\star}^{\sigma}\otimes
H_{io}C_{io}\right)\varepsilon_{ii}$
$\displaystyle-\sum_{l=1}^{i-1}\left(B_{i\star}^{\sigma}\otimes
I_{v_{i}}\right)\left(I_{\pi_{i}(\sigma)}\otimes\Upsilon_{it}\right)\left(1_{\pi_{i}(\sigma)}\otimes
e_{i,l}\right)$
$\displaystyle+\sum_{l=1}^{i-1}\left(\mathcal{H}_{\pi_{i}(\sigma)}\otimes
I_{v_{i}}\right)\Upsilon_{i\star,l}\varepsilon_{i\star,l}.$ (27)

Since $\lim_{t\to\infty}\|e_{i,t}\|=0$,
$\lim_{t\to\infty}\|\varepsilon_{i\star,t}\|=0$ for $i=1,\ldots,t-1$, in light
of Lemma 1, the stability of $\varepsilon_{ii}$ and $\xi_{i\star}$ is
equivalent to the stability

$\displaystyle\dot{\epsilon}_{ii}=\left(I_{\pi_{i}(\sigma)}\otimes(A_{io}-H_{io}C_{io})\right)\epsilon_{ii},$
(28) $\displaystyle\dot{\zeta}_{i\star}=\left(I_{\pi_{i}(\sigma)}\otimes
A_{io}\right)\zeta_{i\star}-\left(\mathcal{H}_{\pi_{i}(\sigma)}\Gamma_{i}^{\sigma}\otimes
I_{v_{i}}\right)\zeta_{i\star}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\left(B_{i\star}^{\sigma}\otimes
H_{io}C_{io}\right)\epsilon_{ii},$ (29)

where the newly defined notation $\epsilon_{ii}$ and $\zeta_{i\star}$ are
employed to replace $\varepsilon_{ii}$ and $\xi_{i\star}$ to avoid confusion.
By repeating the same process in the stability proof of the first subsystem,
we can obtain that the states (28), (29) converge to zero if (20) and (21)
hold.

Therefore, the trajectories of (4) and (4) converge to zero exponentially; so
do $\varepsilon_{ii}$ and $\varepsilon_{i\star}$, owing to Lemma 4. This
implies the convergence of all $e_{i}$ with $i=1,\ldots,N$. ∎

###### Remark 6

We can see from the proof of Theorem 1 that the stability of the error
dynamics of the distributed observer under the switching topology designed in
this paper does not depend on the initial value, which is also consistent with
general understanding of linear system theory.

###### Remark 7

This section proves a sufficient condition for a distributed observer to
achieve asymptotic omniscience in the case of directed switching networks.
Distributed observer dynamics are decomposed using the network transformation
mapping proposed in Section III, and the decomposed observer dynamics can make
full use of the properties of the directed support tree.

##  V Achievement of distributed observer with adaptive coupling gain

Due to the complex calculation of the sufficient conditions (20), adaptive
coupling gain will be proposed in this section.

One of the main roles of the unique design of $\mathbbm{r}_{i,k}$ in (7), (8)
is that it helps us to convert the constant coupling gain into an adaptive
one. In this paper, the adaptive gain is only employed in the subsystems
$\Xi_{ik}$ (see the definition in Remark 5) that satisfy
$\ell_{i,k}^{\sigma}=1$, because it is unnecessary to adaptively adjust the
coupling gain in subsystems without a convergence trend. For this reason, the
adaptive gain is introduced as

$\displaystyle\dot{\gamma}_{i,k}=\omega_{i,k},~{}\gamma_{i,k}(0)>0,$ (30)
$\displaystyle\omega_{i,k}=\left\|\sum_{j=1}^{N}\alpha_{ij}^{\sigma(t)}(k)\left(\hat{x}_{j,k}-\hat{x}_{i,k}\right)\right\|^{2}.$
(31)

The corresponding Lyapunov function chosen is
$V_{i,1}^{a}=V_{i,\varepsilon}^{a}+V_{i,\xi}^{a}$, where

$\displaystyle V^{a}_{i,\varepsilon}=$
$\displaystyle\frac{1}{2}\varepsilon_{ii}^{T}(I_{\pi_{i}(\sigma)}\otimes
P_{io})\varepsilon_{ii},$ (32) $\displaystyle V^{a}_{i,\xi}=$
$\displaystyle\frac{1}{2}\sum_{j\in\mathcal{V}_{i}^{\sigma(t)}\backslash\\{i\\}}\left(q_{io,j}^{\sigma}(2\gamma_{j,i}+\omega_{j,i})\omega_{j,i}\right.$
$\displaystyle\left.+q_{io,j}^{\sigma}(\gamma_{j,i}-\gamma_{i}^{*})^{2}\right),$
(33)

with a pending decision constant $\gamma_{i}^{*}$ and positive constant
$q_{io,j}^{\sigma}$ that belongs to
$Q_{io}^{\sigma}=col\\{q_{io,j}^{\sigma},~{}j\in\mathcal{V}_{i}^{\sigma}\backslash\\{i\\}\\}$.
Then, we can directly show that $\varepsilon_{ii}$ and $\xi_{i\star}$ have a
convergence trend (see the corresponding proof in the appendix). However, with
the adaptive gain, we cannot estimate the convergence rate when the node
satisfies $\ell_{i,k}^{\sigma}=1$. Hence, we fail to guarantee that the error
dynamics of $\hat{x}_{i,k}$ converge to zero with $t\to\infty$.

Fortunately, the proof in Theorem 1 shows that the convergence rate of
$V_{k,k}$ depends on the coupling gain
$\gamma_{i,k},~{}i\in\mathcal{V}_{k}^{\sigma}$. This ensures that the error
dynamics can converge at least at the speed of $\wp$ when (20) is satisfied.
Actually, the overall error dynamics cannot converge if the error convergence
rate of subsystems on those nodes satisfying $\ell_{i,k}^{\sigma}$ is less
than $\wp$. It thus leads to the increasing value of $\gamma_{i,k}$, until the
coupling gain fulfills the requirement of (20). The above statement indicates
that the distributed observer can achieve omniscience asymptotically with
adaptive gain under the switching directed networks.

Figure 2: Augmented communication networks with $4$ agents TABLE I: Parameters of the power system |  Area 1 |  Area 2 |  Area 3 |  Area 4
---|---|---|---|---
$M_{i}$ |  $12$ |  $10$ |  $8$ |  $8$
$R_{i}$ |  $0.05$ |  $0.0625$ |  $0.08$ |  $0.08$
$D_{i}$ |  $0.7$ |  $0.9$ |  $0.9$ |  $0.7$
$T_{t_{i}}$ |  $0.65$ |  $0.4$ |  $0.3$ |  $0.6$
$T_{g_{i}}$ |  $0.1$ |  $0.1$ |  $0.1$ |  $0.1$

Figure 3: Error dynamics of distributed observer with network transformation
mapping
Figure 4: Error dynamics of distributed observer without network
transformation mapping
Figure 5: Time-varying trajectory of coupling gain

##  VI Simulation with a power system

To verify the effectiveness of the proposed method, a power system consisting
of four power-generation areas is considered. The dynamics of the $i$th area
are governed by

$\displaystyle\dot{\chi}_{i}=A_{ii}\chi_{i}+\sum_{j=1,~{}j\neq
i}^{4}A_{ij}\chi_{j},$ $\displaystyle y_{i}=C_{i}\chi_{i},$

where $\chi_{i}\in\mathbb{R}^{4}$ and $y_{i}\in\mathbb{R}^{2}$ are system
states and measurement outputs, respectively; $A_{ij}\in\mathbb{R}^{4\times
4}$ for $i,j=1,2,3,4$ are system matrices; and $C_{i}\in\mathbb{R}^{2\times
4}$ is the output matrix of the $i$th area. $\chi_{i}$ takes the form
$\chi_{i}=col\\{\Delta\theta_{i},\Delta w_{i},\Delta P_{m_{i}},\Delta
P_{v_{i}}\\}$. Herein, $\Delta\theta_{i}$ is the angular displacement of the
rotor relative to the stationary reference axis on the stator; $\Delta w_{i}$
is the change of rotation mass; $\Delta P_{m_{i}}$ is the change of mechanical
power; and $\Delta P_{v_{i}}$ is the change of steam value position.

Now, denote $M_{i}$ the inertia constant, $R_{i}$ the speed regulation rate,
$D_{i}$ the percentage of load change, $T_{t_{i}}$ the time constant of the
prime mover, $T_{g_{i}}$ the time constant of the governor, and $P_{ij}$ the
slope of the power angle curve between power-generation areas $i$ and $j$ at
the initial operating angle.

Then, $A_{ii}$, $A_{ij}$, $C_{i}$ are expressed as

$\displaystyle A_{ii}=\begin{bmatrix}0&1&0&0\\\
-\frac{\sum_{j=1}^{N}P_{ij}}{2M_{i}}&-\frac{D_{i}}{2M_{i}}&\frac{1}{2M_{i}}&0\\\
0&0&-\frac{1}{T_{t_{i}}}&0\\\
0&-\frac{1}{R_{i}T_{g_{i}}}&0&-\frac{1}{T_{g_{i}}}\end{bmatrix},$
$\displaystyle A_{ij}=\begin{bmatrix}0&0&0&0&\\\
\frac{P_{ij}}{2M_{i}}&0&0&0\\\ 0&0&0&0&\\\
0&0&0&0&\end{bmatrix},~{}C_{i}^{T}=\begin{bmatrix}1&0\\\ 0&0\\\ 0&0\\\
0&1\end{bmatrix}.$

Figure 6: Augmented communication networks with $8$ agents TABLE II: Parameters of the power system |  Area 1 |  Area 2 |  Area 3 |  Area 4 |  Area 5 |  Area 6 |  Area 7 |  Area 8
---|---|---|---|---|---|---|---|---
$M_{i}$ |  $12$ |  $10$ |  $8$ |  $8$ |  $12$ |  $10$ |  $8$ |  $8$
$R_{i}$ |  $0.05$ |  $0.0625$ |  $0.08$ |  $0.08$ |  $0.05$ |  $0.0625$ |  $0.08$ |  $0.08$
$D_{i}$ |  $0.7$ |  $0.9$ |  $0.9$ |  $0.7$ |  $0.7$ |  $0.9$ |  $0.9$ |  $0.7$
$T_{t_{i}}$ |  $0.65$ |  $0.4$ |  $0.3$ |  $0.6$ |  $0.65$ |  $0.4$ |  $0.3$ |  $0.6$
$T_{g_{i}}$ |  $0.1$ |  $0.1$ |  $0.1$ |  $0.1$ |  $0.1$ |  $0.1$ |  $0.1$ |  $0.1$

Figure 7: Error dynamics of distributed observer with network transformation
mapping
Figure 8: Error dynamics of distributed observer without network
transformation mapping

Specific values of the aforementioned parameters are shown in Table I. It is
assumed that four nodes locate in four areas, and each node has access to
$y_{i}=C_{i}\chi$ as well as output matrix $C$. The communication networks
subject to Assumption 1 and 2 among the four nodes are shown in Figure 2.
Note that the eigenvalues of $A=[A_{ij}]_{i,j=1}^{N}$ are

$\displaystyle\left\\{\underbrace{0,\ldots,0}_{4},\underbrace{-0.029,\ldots,-0.029}_{4},\right.$
$\displaystyle\quad\quad\left.\underbrace{-1.54,\ldots,-1.54}_{4},\underbrace{-10,\ldots,-10}_{4}\right\\}.$

Therefore, the considered open-loop system is polynominal unstable because it
contains multiple zero eigenvalues. This means that the results, which require
the condition of open loop stability [29] and the conditions of undirected
networks [30, 31], are not well suited for this simulation example.

Note that the distributed observer in this example over undirected switching
networks can achieve omniscience asymptotically with the approach proposed in
[23]. However, the method in [23] is no longer available when all topologies
in the augmented networks are replaced by directed networks. This paper
designs a distributed observer by (7) and (8). The initial values of
$\gamma_{i,k}$ in (11) are chosen to be $10$ as well as $\gamma=100$.
$\chi_{i}(0)$ are set as $[1,1,1,1]^{T}$ for all $i=1,2,3,4$. The initial
values of local observers are chosen randomly in the interval $[-3,3]$.
Observer gain $H_{io}$ is chosen so that the eigenvalues of
$A_{io}-H_{io}C_{io}$ locate at $\\{-12,-13,-14,-15\\}$.

To achieve asymptotic omniscience, we can design distributed observer by (7)
and (8) with network transformation mapping. The trajectories of this
distributed observer with the fix or adaptive coupling gain are shown in
Figure 3, which shows that the distributed observer can achieve omniscience
asymptotically under the jointly connected switching directed networks.
Comparison results can be seen in Figure 4, which shows that the error
dynamics cannot stay at zero without using this paper’s method (there is no
network transformation mapping when designing a distributed observer). This
indicates that the current method, including [29, 30, 31], are invalid for
this example. Moreover, Figure 4 also implies that Figure 3 is not achieved by
a fast switching method because Liu. K et al have proved that the distributed
observer with jointly connected switching topologies can achieve omniscience
asymptotically if the networks switch fast enough [18, 17]. Therefore, the
instability shown in Figure 4 indicates that the dwell time ($0.4$s) of each
network is longer than the upper limit required for fast switching.

Figure 5 shows the adaptive coupling gain. Note that the adaptive coupling
gain in this figure converges to a large value. This is because the error
dynamics under switching topology decrease to zero with fluctuation.
Subsequently, the large error during the fluctuation process will lead to a
large value of $\omega_{i,k}$ in equation (34). As a result, the adaptive
coupling gain increases to a huge value in a short period of time.
Nonetheless, since there are no issues in the observer, such as actuator
saturation, that pose a threat to system safety, the large coupling gain will
not affect system operation. Therefore, although the adaptive strategy may
lead to excessive coupling gain, we still adopt this strategy to avoid solving
complex inequalities (20) and (21).

To further demonstrate the effectiveness of our theory, we expand the number
of subsystems and nodes to $8$. The communication networks between them are
shown in Figure 6, and the parameters of the power system containing $8$ areas
are given in Table II. We can also calculate that the eigenvalues of
$A=[A_{ij}]_{i,j=1}^{N}$ are

$\displaystyle\left\\{\underbrace{0,\ldots,0}_{8},\underbrace{-0.029,\ldots,-0.029}_{8},\right.$
$\displaystyle\quad\quad\left.\underbrace{-1.54,\ldots,-1.54}_{8},\underbrace{-10,\ldots,-10}_{8}\right\\}.$

Note that the system is also polynomial unstable, so only the distributed
observer developed in this paper can achieve omniscience asymptotically for
the underlying system. Figure 7 shows the error dynamics of our distributed
observer. It indicates that that the error dynamics of the distributed
observer fluctuate and converge to $0$ gradually when network transformation
mapping exists. In addition, the unstable results shown in Figure 8 also
demonstrate the indispensability of network transformation mapping.

##  VII Conclusions

In this work, we have concerned with distributed observer under directed
switching topologies. To address the challenge whereby there are no strong
connected branches in the augmented communication networks, a network
transformation mapping has been creatively proposed to construct the spanning
tree so that the asymptotic omniscience of the distributed observer can be
achieved by a state decomposition and reorganization method. The distributed
observer with adaptive coupling gain has been further developed for saving the
tedious calculation required by the verification of sufficient conditions.
Simulation results have shown that both the fixed coupling gain and the
adaptive coupling gain can help the distributed observer achieve omniscience
asymptotically under the directed switching topologies.

## Appendix

### A. Proof of Lemma 1

According to linear system theory, the solution of a linear time-varying
system (1) is

$\displaystyle
x(t)=\Phi(t,t_{0})x(t_{0})+\int_{t_{0}}^{t}\Phi(t,\tau)M\xi(\tau){\rm d}\tau,$

where $\Phi(t,t_{0})$ is a state transition matrix of $\dot{x}(t)=A(t)x(t)$.
Since $\mathbbm{x}_{0}\leq a_{1}e^{-a_{2}(t-t_{0})}$, we have
$\|\Phi(t,t_{0})\|\leq a_{1}e^{-a_{2}(t-t_{0})}$. Also,
$\lim_{t\to\infty}\xi(t)=0$ indicates that there exists $T_{1}>0$ such that
$\|\xi(t)\|\leq\varepsilon(a_{2}/a_{1}-\|M\|)/(\|M\|c)\triangleq\varepsilon^{\prime}$
with $c=\sup_{t>t_{0}}\|\xi(t)\|$ $\forall t>T_{1}$ and
$\forall\varepsilon>0$. Furthermore, there exists $T_{2}>0$ such that
$e^{-a_{2}(t-t_{0})}\leq\varepsilon^{\prime}$ $\forall t>T_{2}$. Then, by
defining $x^{\prime}(t)=\int_{t_{0}}^{t}\Phi(t,\tau)M\xi(\tau){\rm d}\tau$, we
have

$\displaystyle\left\|x^{\prime}(T)\right\|\leq$
$\displaystyle\int_{t_{0}}^{T_{m}}\|\Phi(t,\tau)\|\|M\|\|\xi(\tau)\|{\rm
d}\tau$ $\displaystyle+\int_{T_{m}}^{T}\|\Phi(t,\tau)\|\|M\|\|\xi(\tau)\|{\rm
d}\tau$ $\displaystyle\leq$
$\displaystyle\int_{t_{0}}^{T_{m}}a_{1}e^{-a_{2}(t-\tau)}\|M\|c{\rm d}\tau$
$\displaystyle+\int_{T_{m}}^{T}a_{1}e^{-a_{2}(t-\tau)}\|M\|\varepsilon^{\prime}{\rm
d}\tau$ $\displaystyle\leq$
$\displaystyle\frac{a_{1}\|M\|c\varepsilon^{\prime}}{a_{2}}+\frac{a_{1}\|M\|\varepsilon^{\prime}}{a_{2}}=\varepsilon$

for arbitrary $T>T_{m}=\max\\{T_{1},T_{2}\\}$. This implies
$\lim_{t\to\infty}x^{\prime}(t)=0$. Thus, $\lim_{t\to\infty}x(t)=0$.

### B. Proof of the convergence with adaptive gain

In this appendix, we only prove that $\varepsilon_{11}$ and $\xi_{1\star}$ can
still converge under the condition of adaptive growth of coupling gain (a
similar proof is well suited for $\varepsilon_{i1}$ and $\xi_{i\star}$).
However, since the coupling gain is time-varying, the convergence rate is also
time-varying, so estimation of the convergence rate that corresponds to
different coupling gain values still needs to follow the method in Theorem 1.

We show the proof as follows. First, we consider the performance of
$\varepsilon_{11}$. Since its dynamics

$\displaystyle\dot{\varepsilon}_{11}=\left(I_{\pi_{1}(\sigma)}\otimes(A_{1o}-H_{1o}C_{1o})\right)\varepsilon_{11}$

are stable, a symmetric positive definite matrix $P_{1o}$ can be found such
that

$\displaystyle
P_{1o}(A_{1o}-H_{1o}C_{1o})+(A_{1o}-H_{1o}C_{1o})^{T}P_{1o}=-2\gamma_{1}^{*}I_{v_{1}}.$

Then, by setting
$V_{1,\varepsilon}^{a}=\frac{1}{2}\varepsilon_{11}^{T}(I_{\pi_{1}(\sigma)}\otimes
P_{1o})\varepsilon_{11}$, we have
$\dot{V}_{1,\varepsilon}^{a}\leq-\gamma_{1}^{*}\varepsilon_{11}^{T}\varepsilon_{11}$.
To move on, the Lyapunov function with respect to $\xi$ can be chosen as

$\displaystyle V_{1,\xi}^{a}=$
$\displaystyle\frac{1}{2}\sum_{j\in\mathcal{V}_{1}^{\sigma(t)}\backslash\\{1\\}}q_{1o,j}(2\gamma_{j,1}+\omega_{j,1})\omega_{j,1}$
$\displaystyle+\sum_{j\in\mathcal{V}_{1}^{\sigma(t)}\backslash\\{1\\}}q_{1o,j}(\gamma_{j,1}-\gamma_{1}^{*})^{2}.$

It follows that

$\displaystyle\dot{V}_{1,\xi}^{a}=$
$\displaystyle\sum_{j\in\mathcal{V}_{1}^{\sigma(t)}\backslash\\{1\\}}q_{1o,j}^{\sigma}(\gamma_{j,1}+q_{1o,j}^{\sigma}\omega_{j,1})\dot{\omega}_{j,1}$
$\displaystyle+\sum_{j\in\mathcal{V}_{1}^{\sigma(t)}\backslash\\{1\\}}q_{1o,j}^{\sigma}(\gamma_{j,1}+\omega_{j,1}-\gamma_{1}^{*})\dot{\gamma}_{j,1}$
$\displaystyle=$
$\displaystyle\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}Q_{1o}^{\sigma}\otimes
I_{v_{1}}\right)\dot{\xi}_{1\star}+\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}Q_{1o}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$
$\displaystyle-\gamma_{1}^{*}\xi_{1\star}^{T}\left(Q_{1o}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$ $\displaystyle=$
$\displaystyle\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}Q_{1o}^{\sigma}\otimes
A_{1o}\right)\xi_{1\star}$
$\displaystyle-\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}Q_{1o}^{\sigma}\mathcal{H}_{\pi_{1}(\sigma)}\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$
$\displaystyle+\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}Q_{1o}^{\sigma}B_{1\star}^{\sigma(t)}\otimes
H_{1o}C_{1o}\right)\varepsilon_{11}$
$\displaystyle+\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}Q_{1o}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}-\gamma_{1}^{*}\xi_{1\star}^{T}\left(Q_{1o}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$ $\displaystyle\leq$
$\displaystyle\bar{\lambda}(Q_{1o}^{\sigma}\otimes
A_{1o})\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$
$\displaystyle+\bar{\lambda}(Q_{1o}^{\sigma})\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$
$\displaystyle-\gamma_{1}^{*}\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}\Pi_{1}^{\sigma}\otimes
I_{v_{1}}Q_{1o}^{\sigma}\otimes I_{v_{1}}\right)\xi_{1\star}$
$\displaystyle+\bar{\lambda}\left(Q_{1o}^{\sigma}B_{1\star}^{\sigma(t)}\otimes
H_{1o}C_{1o}\right)\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)\varepsilon_{11}.$

Now, knowledge of mean value inequality leads to

$\displaystyle-\gamma_{1}^{*}\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}-\gamma_{1}^{*}\xi_{1\star}^{T}\left(Q_{1o}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}$ $\displaystyle=$
$\displaystyle-\gamma_{1}^{*}\sum_{j\in\mathcal{V}_{1}^{\sigma(t)}\backslash\\{1\\}}(\gamma_{j,1}+\omega_{j,1})^{2}\xi_{j,1}^{T}\xi_{j,1}-q_{1o,j}\xi_{j,1}^{T}\xi_{j,1}$
$\displaystyle\leq$
$\displaystyle-2\gamma_{1}^{*}\sum_{j\in\mathcal{V}_{1}^{\sigma(t)}\backslash\\{1\\}}\sqrt{q_{1o,j}}(\gamma_{j,1}+\omega_{j,1})\xi_{j,1}^{T}\xi_{j,1}$
$\displaystyle\leq$
$\displaystyle-2\underline{\lambda}(Q_{1o}^{\sigma})\gamma_{1}^{*}\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)\xi_{1\star}.$

Let $V_{1,1}^{a}=V_{1,\varepsilon}^{a}+V_{1,\xi}^{a}$,
$\wp_{1,1}=-2\underline{\lambda}(Q_{1o}^{\sigma})\gamma_{1}^{*}+\bar{\lambda}(Q_{1o}^{\sigma}\otimes
A_{1o})+\bar{\lambda}(Q_{1o}^{\sigma})$,
$\wp_{1,2}=\bar{\lambda}\left(Q_{1o}^{\sigma}B_{1\star}^{\sigma(t)}\otimes
H_{1o}C_{1o}\right)$ and yield

$\displaystyle\dot{V}_{1,1}^{a}\leq$
$\displaystyle\wp_{1,1}\xi_{1\star}^{T}\bar{\Pi}_{1}^{\sigma}\xi_{1\star}+\wp_{1,2}\xi_{1\star}^{T}\left(\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)\varepsilon_{11}-\gamma_{1}^{*}\varepsilon_{11}^{T}\varepsilon_{11}$
$\displaystyle=$
$\displaystyle\begin{bmatrix}\varepsilon_{11}^{T}&\xi_{1\star}^{T}\end{bmatrix}\begin{bmatrix}\wp_{1,1}\bar{\Pi}_{1}^{\sigma}&\frac{1}{2}\wp_{1,2}\bar{\Pi}_{1}^{\sigma}\\\
\frac{1}{2}\wp_{1,2}\bar{\Pi}_{1}^{\sigma}&-\gamma_{1}^{*}I_{\pi_{1}(\sigma)\times
v_{1}}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\\
\xi_{1\star}\end{bmatrix},$

where $\bar{\Pi}_{1}^{\sigma}=\Pi_{1}^{\sigma}\otimes I_{v_{1}}$. In light of
the Schur complement lemma, $\dot{V}_{1,1}^{a}<0$ if and only if

$\displaystyle\wp_{1,1}\left(\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)+\frac{1}{4\gamma_{1}^{*}}\wp_{1,2}^{2}\left(\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)^{2}<0.$

Since $\Pi_{1}^{\sigma}\otimes I_{v_{1}}>0$, there exists a large enough
$\gamma_{1}^{*}$ such that

$\displaystyle\wp_{1,1}+\frac{1}{4\gamma_{1}^{*}}\wp_{1,2}^{2}\left(\Pi_{1}^{\sigma}\otimes
I_{v_{1}}\right)<0.$ (34)

Therefore, $\varepsilon_{11}$ and $\xi_{1\star}$ have a convergence trend
because (34) leads to $\dot{V}_{1,1}<0$. Based on similar approaches, we can
also show that $\varepsilon_{ii}$ and $\xi_{i\star}$ have convergence trend.

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