# Distributed State Estimation for Linear Systems

Lili Wang<EMAIL_ADDRESS>Ji Liu<EMAIL_ADDRESS>A. Stephen Morse
<EMAIL_ADDRESS>Brian. D. O. Anderson<EMAIL_ADDRESS>Samueli
School of Engineering, University of California, Irvine  Department of
Electrical and Computer Engineering, Stony Brook University  Department of
Electrical Engineering, Yale University  School of Engineering, Australian
National University

###### Abstract

This paper studies a distributed state estimation problem for both continuous-
and discrete-time linear systems. A simply structured distributed estimator is
first described for estimating the state of a continuous-time, jointly
observable, input free, multi-channel linear system whose sensed outputs are
distributed across a fixed multi-agent network. The estimator is then extended
to non-stationary networks whose graphs switch according to a switching signal
with a fixed dwell time or a variable but with fixed average dwell time, or
switch arbitrarily under appropriate assumptions. The estimator is guaranteed
to solve the problem, provided a network-widely shared gain is sufficiently
large. As an alternative to sharing a common gain across the network, a fully
distributed version of the estimator is thus studied in which each agent
adaptively adjusts a local gain though the practicality of this approach is
subject to a robustness issue common to adaptive control. A discrete-time
version of the distributed state estimation problem is also studied, and a
corresponding estimator is proposed for time-varying networks. For each
scenario, it is explained how to construct the estimator so that its state
estimation errors all converge to zero exponentially fast at a fixed but
arbitrarily chosen rate, provided the network’s graph is strongly connected
for all time. This is accomplished by appealing to the “split-spectrum”
approach and exploiting several well-known properties of invariant subspace.
The proposed estimators are inherently resilient to abrupt changes in the
number of agents and communication links in the inter-agent communication
graph upon which the algorithms depend, provided the network is redundantly
strongly connected and redundantly jointly observable.

###### keywords:

Distributed Estimation, Multi-Agent Systems

††thanks: This paper was not presented at any IFAC meeting. Portions of this
paper were presented at 2019 American Control Conference [1] and IEEE
Conference on Decision and Control [2]. Corresponding author: Lili Wang.

, , ,

##  1 Introduction

With the growing interest in sensor networks and multi-agent systems, the
problem of estimating the state of a dynamical system whose measured outputs
are distributed across a network has been under study in one form or another
for a number of years [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1].

Depending on the nature of the system to be estimated, the distributed
estimation problem has continuous- and discrete-time versions. In its simplest
form, the continuous-time version of the distributed state estimation problem
starts with a network of $m>1$ agents labeled $1,2,\ldots,m$ which are able to
receive information from their neighbors. Neighbor relations are characterized
by a directed graph $\mathbb{N}$, which may or may not depend on time, whose
vertices correspond to agents and whose arcs depict neighbor relations. Each
agent $i$ senses a signal $y_{i}\in{\rm
I\\!R}^{s_{i}},\;i\in\mathbf{m}\stackrel{{\scriptstyle\Delta}}{{=}}\\{1,2,\ldots,m\\}$
generated by a continuous-time system of the form
$\dot{x}=Ax,\;y_{i}(t)=C_{i}x,\;i\in\mathbf{m}$ and $x\in{\rm I\\!R}^{n}$. It
is typically assumed that $\mathbb{N}$ is strongly connected and that the
system is jointly observable. It is invariably assumed that each agent
receives certain real-time signals from its neighbors although what is
received can vary from one problem formulation to the next. In all
formulations, the goal is to devise local estimators, one for each agent,
whose outputs are all asymptotically correct estimates of $x$. The local
estimator dynamics for agent $i$ are typically assumed to depend only on the
pair $(C_{i},A)$ and certain properties of $\mathbb{N}$. The problem is
basically the same in discrete time, except that rather than the continuous-
time model just described, the discrete-time model
$x(\tau+1)=Ax(\tau),\;y_{i}(\tau)=C_{i}x(\tau),\;i\in\mathbf{m},\;x\in{\rm
I\\!R}^{n}$ is considered instead. More precise problem formulations will be
given later.

###  1.1 Background

Study of distributed state estimation for linear systems can date back to the
so-called distributed Kalman filter problem [17], which involves system and
measurement noise in the problem formulation and has been widely studied for
years [18, 19, 3].

For the problem just described, the existing literature can be classified into
two categories, namely continuous- and discrete-time estimators/observers,
except for our earlier work of [16, 20] in which a hybrid observer was
proposed for a continuous-time linear system.

Continuous-time distributed estimators have recently appeared in [13, 14, 11,
15, 21]. By recasting it as a classical decentralized control problem, it is
capable of estimating state exponentially fast at a pre-assigned rate assuming
$\mathbb{N}$ is a constant strongly connected graph [11]. The work of [13]
seeks to propose a distributed estimator for a continuous-time system at the
expense of certain design flexibility. This is done, in essence, by exploiting
the $A$-invariance of the unobservable spaces of the pairs $(C_{i},A)$; this
in turn enables one to “split” the estimators into two parts, one based on
conventional spectrum assignment techniques and the other based on consensus.
The idea has been further developed in [14, 15, 21, 22]. Specifically, only
constant, undirected, connected neighbor graphs are addressed in [13]. The
work of [14, 15] extends the result of [13] to the case when the neighbor
graph is constant, directed, strongly connected. Establishing correctness
requires one to choose gains to ensure that certain LMIs hold. In [21],
motivated by a distributed least squares problem, a modified algorithm is
proposed to deal with measurement noise. Very recently, a distributed adaptive
algorithm has been proposed in [22] which allows the strongly connected
neighbor graph to change over time and each agent to compute its own gain in a
local manner. All these existing continuous-time distributed estimators
require a somewhat complicated gain computation procedure, and partially
because of this, do not, at least not directly, admit discrete-time
counterparts.

Discrete-time distributed estimators have been recently studied in [23, 24,
25, 26, 10, 12, 27, 28, 29, 30, 31]. Notable among them is the paper [10]
which solves the discrete-time distributed estimation problem for jointly
observable, linear systems with constant, directed, strongly connected
neighbor graphs. This is achieved by recasting the estimation problem as a
classical decentralized control problem. Although the observer is limited to
discrete-time systems, it has been proved possible to make use of the ideas in
[10] to obtain a distributed observer for continuous-time systems [11], but
still for constant neighbor graphs. There are other discrete-time distributed
observers/estimators which do not admit continuous-time extensions. The
algorithm in [23] works for fixed graphs with a relatively complicated
topology design by studying the roles of each agent in the network. The
distributed observer proposed in [24] can track the system only if the so-
called “scalar tracking capacity” condition is satisfied. By expanding on
earlier work in [12], the paper [27] provides a procedure for constructing
such an observer which exploits in some detail the structure of $\mathbb{N}$
and its relation to the structure of the data matrices defining the system.
The resulting algorithm, which is tailored exclusively to discrete-time
systems, requires a network-wide initialization step and thus can deal with
state estimation under assumptions which are weaker than strong connectivity.
There are also work in which the estimators incorporate two time scales. An
agent obtains data from its neighbors at a rate $q$ times that at which it
obtains measurement data[32, 3, 33].

The above discussion leads to a conclusion that there is no unified approach
to distributed state estimation design which is capable of handling
continuous-, discrete-time linear systems, time-varying neighbor graphs
simultaneously. This is precisely what we want to achieve in this paper.

The vast majority of papers on the subject assume $\mathbb{N}$ to be a
constant graph and take the local estimators to be linear, constant-
coefficient, dynamical systems. It is typically assumed that agent $i$
receives the current states of each of its neighbors’ estimators and possibly
the measurements $y_{j},j\in\mathcal{N}_{i}$, where $\mathcal{N}_{i}$ is the
set of labels of agent $i$’s neighbors, including itself. Accordingly, for the
continuous-time version of the problem, agent $i$ is presumed to estimate $x$
using an $n_{i}$ dimensional time-invariant linear system of the general form

$\displaystyle\dot{z}_{i}$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathcal{N}_{i}}(H_{ij}z_{j}+K_{ij}y_{j})$ (1)
$\displaystyle x_{i}$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathcal{N}_{i}}(M_{ij}z_{j}+N_{ij}y_{j})$ (2)

where $x_{i}(t)$ is to be an asymptotically correct estimate of $x(t)$ in the
sense that each estimation error $x_{i}(t)-x(t)$ tend to zero as
$t\rightarrow\infty$ at a preassigned, but arbitrarily fast exponential
convergence rate. The distributed estimator design problem is to define the
$n_{i}$ and the matrices
$H_{ij},K_{ij},M_{ij},N_{ij},\;j\in\mathcal{N}_{i},\;i\in\mathbf{m}$, so that
this is so.

It is known [11] that this design problem is equivalent to the algebraic
problem of constructing matrices
$H_{ij},K_{ij},M_{ij},N_{ij},\;j\in\mathcal{N}_{i},\;i\in\mathbf{m}$ and
$V_{i},\;i\in\mathbf{m}$, so that the estimator design equations

$\displaystyle I$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathcal{N}_{i}}(M_{ij}V_{j}+N_{ij}C_{j}),\;\;\;i\in\mathbf{m}$
$\displaystyle V_{i}A$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathcal{N}_{i}}(H_{ij}V_{j}+K_{ij}C_{j}),\;\;\;i\in\mathbf{m}$

hold and, in addition, so that the state transition matrix of the block matrix
$H=[H_{ij}]_{nm\times nm}$ has a prescribed convergence rate. Here $H_{ij}=0$
for $j\not\in\mathcal{N}_{i},\;i\in\mathbf{m}$. This algebraic problem has
been addressed in [11] for the case when the only signals transmitted to agent
$i$ are the states $z_{j},\;j\in\mathcal{N}_{i}$ of the estimators of each of
its neighbors. The problem was originally studied for discrete-time systems in
[10] where it was demonstrated that the problem could be recast mathematically
as a classical decentralized stabilization problem of the type considered in
[34]. Prompted by this clue, it was subsequently explained in [11] how to
choose all the matrices
$H_{ij},K_{ij},M_{ij},N_{ij},\;j\in\mathcal{N}_{i},\;i\in\mathbf{m}$ and
$V_{i}\;i\in\mathbf{m}$, so that the estimator design equations hold while at
the same time freely adjusting the spectrum of $H$. This was accomplished by
choosing one of the $n_{i}$ to equal $n+m$ and all remaining $n_{i}$ all equal
to $n$.

Although the estimator design equations shown above apply to every kind of
time-invariant distributed state estimator for a continuous-time, linear,
time-invariant system with a constant neighbor graph, there is a different way
to construct an estimator for the distributed state estimation problem under
study which does not directly exploit the estimator design equations and
which, when compared with that estimator studied in [11], has the virtues of
being more simply structured and very much easier to analyze, which in turn
(as we shall show) makes it possible to deal with time-varying neighbor
graphs. The idea stems from research originally reported in [13] and
subsequently extended in [14] which develops an entirely different and simpler
approach to the continuous-time version of the estimation problem. There are
however two distinct limitations of the types of estimators discussed in [13,
14]. First, as they stand these estimators cannot deal with time-varying
neighbor graphs. Second, there does not appear to be a way to easily modify
these estimators to address the discrete-time state estimation problem; this
is because the continuous-time distributed estimators rely on a “high gain”
concept for which there is no discrete-time counterpart.

###  1.2 Contributions

The aims of this paper are to describe a simply structured, unified approach
to the distributed state estimation problem and to design and analyze the
corresponding distributed estimators for both continuous- and discrete-time
linear systems with possibly time-varying neighbor graphs. This approach is
motivated by the ideas in [13, 14] but much simpler than those in [13, 14]. It
is termed the “split-spectrum” approach because it “splits” the system
spectrum into disjoint subsets corresponding to unobservable and observable
subspaces. Thus several well-known properties of invariant subspace are
exploited in the analysis.

The paper first describes the split-spectrum based distributed estimator for
the case when the system dynamics are continuous with a stationary network in
§2.1, and with associated background analysis is given in §2.2. The estimator
is then extended to deal with non-stationary networks whose neighbor graphs
switch according to a switching signal with a fixed dwell time or a variable
dwell time with prescribed average, the ideas being, detailed in §2.3. In the
case when the interconnection among the agents can always be implemented by
doubly stochastic matrices {e.g., undirected graphs with the Metropolis
weights [35]}, it is shown in §2.4 that the estimator functions correctly even
if the neighbor graph switches arbitrarily, provided the graph is always
strongly connected. The estimators mentioned above all rely on the existence
of a sufficiently large, network-widely shared gain. A fully distributed
version of the estimator is then studied in §2.5 where each agent can
adaptively adjust a local gain. The adaptive estimator is subject to a
robustness issue common to adaptive control. The proposed estimators, except
for the adaptive one, are inherently resilient to abrupt changes in the number
of agents and communication links in the inter-agent communication graph upon
which the algorithms depend, an issue which is discussed in §2.6. Then the
split-spectrum based estimator design is extended to the case when the system
dynamics is discrete in §3 for both constant and time-varying neighbor graphs.
Simulation validation is provided in §4.

The material in this paper was partially presented in [1, 2], but this paper
presents a more comprehensive treatment of the work. Specifically, the paper
crafts continuous-time distributed estimators for two types of non-stationary
networks in §2.3–§2.4 and a fully distributed adaptive estimator in §2.5,
which were not included in [1, 2].

##  2 Continuous-Time Distributed Estimator

We are interested in a network of $m>0$ {possibly mobile} autonomous agents
labeled $1,2,\ldots,m$ which are able to receive information from their
“neighbors”, where by a neighbor of agent $i$ is meant any other agent within
agent $i$’s reception range. We write $\mathcal{N}_{i}(t)$ for the labels of
agent $i$’s neighbors at time $t\in[0,\infty)$ and always take agent $i$ to be
a neighbor of itself. Neighbor relations at time $t$ are characterized by a
directed graph $\mathbb{N}(t)$ with $m$ vertices and a set of arcs defined so
that there is an arc in $\mathbb{N}(t)$ from vertex $j$ to vertex $i$ whenever
agent $j$ is a neighbor of agent $i$ at time $t$. Since each agent $i$ is
always a neighbor of itself, $\mathbb{N}(t)$ has a self-arc at each of its
vertices. Each agent $i$ can sense a continuous-time signal $y_{i}\in{\rm
I\\!R}^{s_{i}},\;i\in\mathbf{m}\stackrel{{\scriptstyle\Delta}}{{=}}\\{1,2,\ldots,m\\}$,
where

$\displaystyle y_{i}$ $\displaystyle=$ $\displaystyle
C_{i}x,\;\;\;i\in\mathbf{m}$ (3) $\displaystyle\dot{x}$ $\displaystyle=$
$\displaystyle Ax$ (4)

and $x\in{\rm I\\!R}^{n}$. We assume throughout that $C_{i}\neq
0,\;i\in\mathbf{m}$, and that the system defined by (3) and (4) is  jointly
observable; i.e., with
$C=\begin{bmatrix}C_{1}^{\prime}&C_{2}^{\prime}&\cdots&C_{m}^{\prime}\end{bmatrix}^{\prime}$,
the matrix pair $(C,A)$ is observable. Joint observability is equivalent to
the requirement that

$\bigcap_{i\in\mathbf{m}}\mathcal{V}_{i}=0$

where $\mathcal{V}_{i}$ is the unobservable space of $(C_{i},A)$; i.e.
$\mathcal{V}_{i}=\ker\begin{bmatrix}C_{i}^{\prime}&(C_{i}A)^{\prime}&\cdots&(C_{i}A^{n-1})^{\prime}\end{bmatrix}^{\prime}$.
As is well known, $\mathcal{V}_{i}$ is the largest $A$-invariant subspace
contained in the kernel of $C_{i}$. Generalizing the results that follow to
the case when $(C,A)$ is only detectable is quite straightforward and can be
accomplished using well-known ideas. However, the commonly made assumption
that each pair $(C_{i},A),\;i\in\mathbf{m}$, is observable, or even just
detectable, is very restrictive, grossly simplifies the problem and is
unnecessary. The assumption $C_{i}\neq 0$ is not necessary provided the more
relaxed problem is properly formulated. The assumption is made for the sake of
simplicity. The problem of interest is to construct a suitably defined family
of linear estimators in such a way so that no matter what the estimators’
initial states are, each agent obtains an asymptotically correct estimate
$x_{i}$ of $x$ in the sense that the estimation error $x_{i}(t)-x(t)$
converges to zero as fast as $e^{-\lambda t}$ does, where $\lambda$ is an
arbitrarily chosen but fixed positive number.

This section proposes the estimator first, and then analyses the estimator
beginning with the condition that the neighbor graph $\mathbb{N}(t)$ is
constant. A time-varying neighbor graph $\mathbb{N}(t)$ is then considered in
which changes occur according to a switching signal. Later, a fully
distributed algorithm based on use of multiplicative adaptive gain control is
developed.

###  2.1 The Estimator

The estimator to be considered consists of $m$ private estimators of the form

$\dot{x}_{i}=(A+K_{i}C_{i})x_{i}-K_{i}y_{i}\\\
-gP_{i}\bigg{(}x_{i}-\frac{1}{m_{i}(t)}\sum_{j\in\mathcal{N}_{i}(t)}x_{j}\bigg{)},\;\;i\in\mathbf{m}$
(5)

where $m_{i}(t)$ is the number of labels in $\mathcal{N}_{i}(t)$, $g$ is a
suitably defined positive gain, each $K_{i}$ is a suitably defined matrix, and
for each $i\in\mathbf{m}$, $P_{i}$ is the orthogonal projection on the
unobservable space of $(C_{i},A)$.

The definitions of $K_{i}$ and $g$ begin with the specification of a desired
convergence rate bound $\lambda>0$. To begin with, each matrix $K_{i}$ is
defined as follows. For each fixed $i\in\mathbf{m}$, write $Q_{i}$ for any
full rank matrix whose kernel is the unobservable space of $(C_{i},A)$ and let
$\bar{C}_{i}$ and $\bar{A}_{i}$ be the unique solutions to
$\bar{C}_{i}Q_{i}=C_{i}$ and $Q_{i}A=\bar{A}_{i}Q_{i}$ respectively. Then the
matrix pair $(\bar{C}_{i},\bar{A}_{i})$ is observable. A matrix $\bar{K}_{i}$
can be chosen to ensure that the convergence of
$e^{(\bar{A}_{i}+\bar{K}_{i}\bar{C}_{i})t}$ to zero is as fast as the
convergence of $e^{-\lambda t}$ to zero is. There are several well-documented
ways to do this {e.g, spectrum assignment algorithms or Riccati equation
solvers}, since each pair $(\bar{C}_{i},\bar{A}_{i})$ is observable. Having
chosen such $\bar{K}_{i}$, $K_{i}$ is then chosen to be
$K_{i}=Q_{i}^{-1}\bar{K}_{i}$ where $Q_{i}^{-1}$ is a right inverse for
$Q_{i}$. The definition implies that

$Q_{i}(A+K_{i}C_{i})=(\bar{A}_{i}+\bar{K}_{i}\bar{C}_{i})Q_{i}$ (6)

and that $(A+K_{i}C_{i})\mathcal{V}_{i}\subset\mathcal{V}_{i}$. The latter, in
turn, implies that there is a unique matrix $A_{i}$ which satisfies

$(A+K_{i}C_{i})V_{i}=V_{i}A_{i}$ (7)

where $V_{i}$ is a basis matrix111For simplicity, we assume that the columns
of $V_{i}$ constitute an orthonormal basis for $\mathcal{V}_{i}$ in which case
$P_{i}=V_{i}V_{i}^{\prime}$. for $\mathcal{V}_{i}$. Prior to explaining how to
choose $g$, it will be helpful to explain what defining the $K_{i}$ in this
way accomplishes.

####  2.1.1 The Error Dynamics

First note from (5), that the state estimation error $e_{i}=x_{i}-x$ satisfies

$\dot{e}_{i}=(A+K_{i}C_{i})e_{i}-gP_{i}\left(e_{i}-\frac{1}{m_{i}(t)}\sum_{j\in\mathcal{N}_{i}(t)}e_{j}\right)$
(8)

Consequently the overall error vector $e=\begin{bmatrix}e_{1}^{\prime}\
\ldots\ e_{m}^{\prime}\end{bmatrix}^{\prime}$ satisfies

$\dot{e}=(\bar{A}-gP(I_{mn}-\bar{S}(t)))e$ (9)

where $\bar{A}=$ block diag
$\\{A+K_{1}C_{1},A+K_{2}C_{2},\ldots,A+K_{m}C_{m}\\}$, $P=$ block diag
$\\{P_{1},P_{2},\ldots,P_{m}\\}$, $\bar{S}(t)=S(t)\otimes I_{n}$ with
$S(t)=D_{\mathbb{N}(t)}^{-1}\mathcal{A}^{\prime}_{\mathbb{N}(t)}$. Here
$I_{k}$ is the $k\times k$ identity matrix, and $\mathcal{A}_{\mathbb{N}(t)}$
is the adjacency matrix of $\mathbb{N}(t)$ and $D_{\mathbb{N}(t)}$ is the
diagonal matrix whose $i$th diagonal entry is the in-degree of
$\mathbb{N}(t)$’s $i$th vertex. Note that $\mathbb{N}(t)$ is the graph222The
graph of an $n\times n$ matrix $M$ is that directed graph on $n$ vertices
possessing a directed arc from vertex $i$ to vertex $j$ if $m_{ij}\neq 0$ {p.
357, [36]}. of $S^{\prime}(t)$ and that the diagonal entries of
$S^{\prime}(t)$ are all positive because each agent is a neighbor of itself.
The matrix $S(t)$ is evidently a stochastic matrix.

Note that the subspace
$\mathcal{V}=\mathcal{V}_{1}\oplus\mathcal{V}_{2}\oplus\cdots\oplus\mathcal{V}_{m}$
is $\bar{A}$ \- invariant because
$(A+K_{i}C_{i})\mathcal{V}_{i}\subset\mathcal{V}_{i},\;i\in\mathbf{m}$. Next,
let $Q=$ block diag $\\{Q_{1},Q_{2},\ldots,Q_{m}\\}$ and $V=$ block diag
$\\{V_{1},V_{2},\ldots,V_{m}\\}$ where $V_{i}$ is a matrix whose columns form
an orthonormal basis for $\mathcal{V}_{i}$. Then $Q$ is a full rank matrix
whose kernel is $\mathcal{V}$ and $V$ is a basis matrix for $\mathcal{V}$
whose columns form an orthonormal set. It follows that $P=VV^{\prime}$, that
$QP=0$, and that

$\displaystyle Q\bar{A}$ $\displaystyle=$ $\displaystyle\bar{A}_{V}Q$ (10)
$\displaystyle\bar{A}V$ $\displaystyle=$ $\displaystyle V\tilde{A}$ (11)

where

$\bar{A}_{V}=\text{block
diag}\;\\{\bar{A}_{1}+\bar{K}_{1}\bar{C}_{1},\ldots,\bar{A}_{m}+\bar{K}_{m}\bar{C}_{m}\\},$
(12) $\tilde{A}=\text{block diag}\;\\{A_{1},A_{2},\ldots,A_{m}\\}.$

Let $V^{-1}$ be any left inverse for $V$ and let $Q^{-1}$ be that right
inverse for $Q$ for which $V^{-1}Q^{-1}=0$. Then

$\bar{A}-gP(I_{mn}-\bar{S}(t))=T\begin{bmatrix}\bar{A}_{V}&0\\\
\hat{A}_{V}(t)&A_{V}(t)\end{bmatrix}T^{-1}$ (13)

where $\hat{A}_{V}(t)=V^{-1}(\bar{A}-g(I_{mn}-\bar{S}(t)))Q^{-1}$ and
$A_{V}(t)=\tilde{A}-gV^{\prime}(I_{mn}-\bar{S}(t))V.$ Here
$T=\begin{bmatrix}Q^{-1}&V\end{bmatrix}$. It is easy to check that
$T^{-1}=\begin{bmatrix}Q^{\prime}&V\end{bmatrix}^{\prime}$.

Recall that the $\bar{K}_{i}$ have been already been chosen so that each
matrix exponential $e^{(\bar{A}_{i}+\bar{K}_{i}\bar{C}_{i})t}$ converges to
zero as fast as $e^{-\lambda t}$ does. Because of this and the fact that
$\hat{A}_{V}(t)$ is a bounded matrix, to ensure that for each fixed $\tau$,
the state transition matrix $\Phi(t,\tau)$ converges to zero as fast as
$e^{-\lambda t}$ does, it is enough to choose $g$ so that the state transition
matrix of $A_{V}(t)$ converges to zero at least as fast as $e^{-\lambda t}$
does. The requisite condition on $g$ is provided below for three different
neighbor graph connectivity assumptions.

###  2.2 Constant Neighbor Graph

This subsection focuses on the case when the neighbor graph $\mathbb{N}(t)$ is
a constant graph $\mathbb{N}$. According to (13), the spectrum of
$\bar{A}-gP(I_{mn}-\bar{S})$ is equivalent to the union of the spectrum of
$\bar{A}_{V}$ and $A_{V}$. Since the spectrum of
$\bar{A}_{i}+\bar{K}_{i}\bar{C}_{i},\;i\in\mathbf{m}$, is assignable with
$\bar{K}_{i}$, to show for $g$ sufficiently large that
$\bar{A}-gP(I_{mn}-\bar{S})$ is a continuous-time stability matrix with a
prescribed convergence rate as large as $\lambda$, it is enough to show that
for $g$ sufficiently large, the matrix
$A_{V}=\tilde{A}-gV^{\prime}(I_{mn}-\bar{S})V$ is a continuous-time stability
matrix with a prescribed convergence rate as large as $\lambda$. This proves
to be a simple consequence of the following proposition.

######  Proposition 1.

$-V^{\prime}(I_{mn}-\bar{S})V$ is a continuous-time stability matrix.

Proof of Proposition 1: Recall $\bar{S}=S\otimes I_{n}$. Since $S$ is a
stochastic matrix, $S^{\prime}$ must have a spectral radius of $1$ and an
eigenvalue at $1$. Moreover, since $\mathbb{N}$ is the graph of $S^{\prime}$
and $\mathbb{N}$ is strongly connected by assumption, $S^{\prime}$ is
irreducible {Theorem 6.2.24, [36]}. Thus by the Perron-Frobenius Theorem there
must be a positive vector $\pi$ such that $S^{\prime}\pi=\pi$. Without loss of
generality, assume $\pi$ is normalized so that the sum of its entries equals
$1$; i.e., $\pi$ is a Perron vector [36]. Let $\Pi$ be that diagonal matrix
whose diagonal entries are the entries of $\pi$. Then $\Pi\mathbf{1}=\pi$
where $\mathbf{1}$ is the $m$-vector of all $1$s. Let $L=2\Pi-\Pi
S-S^{\prime}\Pi$. Clearly $L$ is a symmetric matrix and $L\mathbf{1}=0$.

We claim that the geometric multiplicity of $L$’s eigenvalue at $0$ is one. To
establish this claim, note first that every nonzero entry of $S^{\prime}$ is a
nonzero entry of $\Pi S+S^{\prime}\Pi$ because $S^{\prime}$ is a nonnegative
matrix and $\Pi$ is a diagonal matrix whose diagonal entries are all positive.
Moreover since $2(I_{m}-\Pi)$ is a nonnegative matrix, every nonzero entry of
$S^{\prime}$ is also a nonzero entry of $2I_{m}-L=2(I_{m}-\Pi)+\Pi
S+S^{\prime}\Pi$. Thus the graph of $S^{\prime}$ must be a spanning subgraph
of the graph of $2I_{m}-L$ so the graph of $2I_{m}-L$ must be strongly
connected. Therefore $2I_{m}-L$ must be irreducible. Note that the row sums of
$2I_{m}-L$ all equal $2$. Since $2I_{m}-L$ is nonnegative, its infinity norm
is $2$ so its spectral radius is no greater than $2$ {Theorem 5.6.9, [36]}.
Moreover $2$ is an eigenvalue of $2I_{m}-L$. Thus by the Perron-Frobenius
Theorem, the geometric multiplicity of this eigenvalue is one. It follows that
the geometric multiplicity of the eigenvalue of $L$ at $0$ is also one.

We claim that $L$ is positive semi-definite. To establish this claim, note
that $L$ can also be written as $L=D-\widehat{A}$ where $D$ is a diagonal
matrix whose entries are the diagonal entries of $L$ and $\widehat{A}$ is the
nonnegative matrix $\widehat{A}=D-L$. As such, $L$ is the generalized
Laplacian [37] of that simple undirected graph $\mathbb{G}$ whose adjacency
matrix is the matrix which results when the nonzero entries $a_{ij}$ in
$\widehat{A}$ are replaced by ones. Since $L$ can also be written as

$L=\sum_{(i,j)\in\mathcal{E}}a_{ij}(u_{i}-u_{j})(u_{i}-u_{j})^{\prime}$

where $u_{i}$ is the $i$th unit vector and $\mathcal{E}$ is the edge set of
$\mathbb{G}$, $L$ is positive semi-definite as claimed.

To proceed, set

$H=\text{ block diag
}\\{\pi_{1}I_{n_{1}},\pi_{2}I_{n_{2}},\ldots,\pi_{m}I_{n_{m}}\\}$ (14)

where $n_{i}=\dim\mathcal{V}_{i}$ and note that $VH=(\Pi\otimes I_{n})V$.
Since $((S-I_{m})^{\prime}\otimes I_{n})(\Pi\otimes
I_{n})=((S-I_{m})^{\prime}\Pi)\otimes I_{n}$ it must be true that
$(V^{\prime}((S-I_{m})\otimes
I_{n})V)^{\prime}H=V^{\prime}(((S-I_{m})^{\prime}\Pi)\otimes I_{n})V$ and thus
that

$\displaystyle\;H(V^{\prime}(I_{mn}-\bar{S})V)+(V^{\prime}(I_{mn}-\bar{S})V)^{\prime}H$
$\displaystyle=V^{\prime}(L\otimes I_{n})V$ (15)

Observe that this is a Lyapunov equation for the positive definite function
$z^{\prime}Hz$. Therefore to show that $-V^{\prime}(I_{mn}-\bar{S})V$ is a
stability matrix, it is enough to show that $V^{\prime}(L\otimes I_{n})V$ is
positive definite.

Since $L$ is positive semi-definite, so must be $L\otimes I_{n}$. Therefore
$V^{\prime}(L\otimes I_{n})V$ is at least positive semi-definite. Suppose
$z^{\prime}V^{\prime}(L\otimes I_{n})Vz=0$ where $z=$ col
$\\{z_{1},z_{2},\ldots,z_{m}\\}$ and $z_{i}\in{\rm
I\\!R}^{\dim(\mathcal{V}_{i})}$. To show that $V^{\prime}(L\otimes I_{n})V$ is
positive definite, it is enough to show that $z=0$. Since $L$’s eigenvalue at
$0$ has multiplicity one, $\ker L=$ span $\\{\mathbf{1}\\}$; therefore
$\ker(L\otimes I_{n})=$ column span $\mathbf{1}\otimes I_{n}$. The hypothesis
$z^{\prime}V^{\prime}(L\otimes I_{n})Vz=0$ implies that $(L\otimes I_{n})Vz=0$
so $Vz\in\ker(L\otimes I_{n})$. Therefore
$V_{i}z_{i}=V_{j}z_{j},\;i,j\in\mathbf{m}$. But because of joint
observability, $\bigcap_{i\in\mathbf{m}}\mathcal{V}_{i}=0$ so
$V_{i}z_{i}=0,\;i\in\mathbf{m}$. Thus $z_{i}=0,\;i\in\mathbf{m}$ so $z=0$
implying that $V^{\prime}(L\otimes I_{n})V$ is positive definite. Therefore
$-V^{\prime}(I_{mn}-\bar{S})V$ is a continuous-time stability matrix as
claimed.

To show that $e^{(\tilde{A}-gV^{\prime}((I_{m}-S)\otimes I_{n})V)t}$ can be
made to converge to zero as fast as $e^{-\lambda t}$ does by choosing $g$
sufficiently large, we exploit (15). Note in particular that

$\displaystyle H(\lambda I+A_{V})+(\lambda I+A_{V})^{\prime}H$
$\displaystyle=$ $\displaystyle H(\lambda I+\tilde{A})+(\lambda
I+\tilde{A})^{\prime}H-gV^{\prime}(L\otimes I_{n})V$

Since $V^{\prime}(L\otimes I_{n})V$ is positive definite, by picking $g$
sufficiently large, $H(\lambda I+\tilde{A})+(\lambda
I+\tilde{A})^{\prime}H-gV^{\prime}(L\otimes I_{n})V$ will be negative definite
implying that $\lambda I+A_{V}$ is a stability matrix and thus that
$\tilde{A}-gV^{\prime}(I_{mn}-\bar{S})V$ is a stability matrix for which
$e^{(\tilde{A}-gV^{\prime}(I_{mn}-\bar{S})V)t}$ converges to zero as fast as
$e^{-\lambda t}$ does. In other words, any value of $g$ will have the desired
property provided

$g\geq\frac{\lambda_{\max}\left(H(\lambda I+\tilde{A})+(\lambda
I+\tilde{A})^{\prime}H\right)}{\lambda_{\min}\left(V^{\prime}(L\otimes
I_{n})V\right)}$ (16)

where $\lambda_{\max}(.)$ and $\lambda_{\min}(.)$ are the largest eigenvalue
and the smallest eigenvalue of a symmetric matrix respectively. We summarize:

######  Theorem 1.

For any given positive number $\lambda$, if the neighbor graph $\mathbb{N}$ is
fixed and strongly connected, and the system defined by (3) and (4) is jointly
observable, there are matrices $K_{i}$, $i\in\mathbf{m}$ such that for $g$
sufficiently large, each estimation error $x_{i}(t)-x(t)$ of the distributed
estimator defined by (5), converges to zero as $t\rightarrow\infty$ as fast as
$e^{-\lambda t}$ converges to zero.

###  2.3  Switching Neighbor Graph

In the sequel the problem is studied under the assumption that $\mathbb{N}(t)$
changes according to a switching signal with a fixed dwell time or a variable
but with fixed average dwell time. To characterize the assumed time dependence
of $\mathbb{N}(t)$, let
$\mathcal{G}=\\{\mathbb{G}_{1},\mathbb{G}_{2},\ldots,\mathbb{G}_{|\mathcal{G}|}\\}$
denote the set of all directed, strongly connected graphs on $m$ vertices
which have self-arcs at all vertices; here $|\mathcal{G}|$ is the number of
graphs in $\mathcal{G}$. In some situations, the switching signals always have
consecutive discontinuities separated by a value which is no less than a fixed
positive real number $\tau_{D}$. It is called a dwell time [38]. In certain
situations, the switching signals may occasionally have consecutive
discontinuities separated by less than $\tau_{D}$, but for which the average
interval between consecutive discontinuities is no less than $\tau_{D}$. This
leads to the concept of average dwell-time. With $\tau_{D}$ and $\delta$ fixed
define $\mathcal{S}_{\text{avg}}$ for the set of all piecewise-constant
switching signals $\sigma:[0,\infty)\rightarrow\\{1,2,\ldots,|\mathcal{G}|\\}$
satisfying $\delta_{\sigma}(t_{0},t)\leq\delta_{0}+\frac{t-t_{0}}{\tau_{D}}$.
Here $\delta_{\sigma}(t_{0},t)$ denotes the number of discontinuities of
$\sigma$ in the open interval $(t_{0},t)$. The constant $\tau_{D}$ is called
the average dwell-time and $\delta_{0}$ the chatter bound [39]. By the set of
all time-varying neighbor graphs with average dwell-time $\tau_{D}$ is meant
the set $\\{\mathbb{G}_{\sigma}:\sigma\in\mathcal{S}_{\text{avg}}\\}$. Note
that switching according to a dwell time is a special case of switching
according to an average dwell time. In the following, it is assumed that
$\mathbb{N}\in\\{\mathbb{G}_{\sigma}:\sigma\in\mathcal{S}_{\text{avg}}\\}$.

The problem to which this section is addressed is this. For fixed averaged
dwell-time $\tau_{D}$ and the chatter bound $\delta_{0}$, devise a procedure
for crafting $m$ local estimators, one for each agent, so that for each
neighbor graph
$\mathbb{N}\in\\{\mathbb{G}_{\sigma}:\sigma\in\mathcal{S}_{\text{avg}}\\}$,
all $m$ state estimation errors converge to zero exponentially fast at a
prescribed rate.

The estimator to be considered, is still the same as the estimator described
in (5), with the exception that $g$ which is chosen differently. According to
§2.1, it remains to be shown that with $g$ sufficiently large, the state
transition matrix of $A_{V}(t)$ converges to zero as fast as $e^{-\lambda t}$
does. To accomplish this, use will be made of the following result.

######  Lemma 1.

Let $M_{1},M_{2},\ldots,M_{|\mathcal{G}|}$ be a set of $n\times n$
exponentially stable real matrices. For any $n\times n$ real matrix $N$ and
positive number $\lambda$ there is a positive number $g^{*}$, depending on
$\tau_{D}$ for which, for each $\sigma\in\mathcal{S}_{\text{avg}}$ and $g\geq
g^{*}$, all solutions to

$\dot{x}=(N+gM_{\sigma})x$ (17)

converge to zero as fast as $e^{-\lambda t}$ does.

Proof of Lemma 1 333The symbols used in this proof such as $g$, $c$ and
$\lambda^{*}$ are generic and do not have the same meanings as the same
symbols do when used elsewhere in this paper. By hypothesis, each $M_{i}$ is
exponentially stable. Thus there are positive constants $c_{i}>1$ and
$\lambda_{i}$ such that

$\|e^{M_{i}t}\|\leq c_{i}e^{-\lambda_{i}t}$ (18)

for any $i\in\\{1,2,\ldots,|\mathcal{G}|\\}$. Here $\|\cdot\|$ is any given
submultiplicative norm on $\mathbb{R}^{n\times n}$. Let

$c=\max_{i\in\\{1,2,\ldots,|\mathcal{G}|\\}}c_{i},$

and

$\lambda^{*}=\min\limits_{i\in\\{1,2,\ldots,|\mathcal{G}|\\}}\lambda_{i}.$

Fix $\lambda>0$ and let $g$ be any gain satisfying

$g\geq\frac{\tau_{D}(\lambda+\|N\|c)+\ln c}{\tau_{D}\lambda^{*}}$ (19)

We claim that for any number $\tau$, and any switching signal
$\sigma\in\mathcal{S}_{\text{avg}}$, the transition matrix of $gM_{\sigma}$,
namely $\Phi_{\sigma}(t,\tau)$, converges to zero as fast as $e^{-\alpha t}$
does where

$\alpha=\lambda+\|N\|c$ (20)

To understand why this is so, by (18),

$\|\Phi_{\sigma}(t,\tau)\|\leq
c^{\delta_{\sigma}(\tau,t)}e^{-g\lambda^{*}(t-\tau)}$ (21)

where $\delta_{\sigma}(\tau,t)$ is the number of switching between $(\tau,t)$.
By (19), $e^{g\lambda^{*}}\geq c^{\frac{1}{\tau_{D}}}e^{\alpha}$. From this
and the fact that
$\delta_{\sigma}(\tau,t)\leq\delta_{0}+\frac{t-\tau}{\tau_{D}}$,

$\|\Phi_{\sigma}(t,\tau)\|\leq
c^{\delta_{0}}c^{\frac{t-\tau}{\tau_{D}}}e^{-g\lambda^{*}(t-\tau)}\leq
c^{\delta_{0}-\frac{\tau}{\tau_{D}}}e^{-\alpha(t-\tau)}.$

Thus, the claim is true.

In view of (17) and the variation of constants formula,

$x(t)=\Phi_{\sigma}(t,0)x(0)+\int_{0}^{t}\Phi_{\sigma}(t,\mu)Nx(\mu)d\mu$ (22)

As $\|\Phi_{\sigma}(t,\tau)\|\leq ce^{-\alpha t}$ for all $\tau$ and
$\sigma\in\mathcal{S}_{\text{avg}}$,

$\|x(t)\|\leq ce^{-\alpha
t}\|x(0)\|+\int_{0}^{t}ce^{-\alpha(t-\mu)}\|N\|\|x(\mu)\|d\mu$

By multiplying by $e^{\alpha t}$ on both sides, one obtains $e^{\alpha
t}\|x(t)\|\leq c\|x(0)\|+\int_{0}^{t}||N||ce^{\alpha\mu}\|x(\mu)\|d\mu$. From
this and the Bellman-Gronwall Lemma there follows

$e^{\alpha t}\|x(t)\|\leq c\|x(0)\|e^{\int_{0}^{t}\|N\|cd\mu}$

Since $\int_{0}^{t}\|N\|cd\mu=\|N\|ct$, it follows that $e^{\alpha
t}\|x(t)\|\leq c\|x(0)\|e^{\|N\|ct}$ and thus that

$\|x(t)\|\leq c\|x(0)\|e^{(\|N\|c-\alpha)t}$

From this and (20) it follows that

$\|x(t)\|\leq c\|x(0)\|e^{-\lambda t}$

This completes the proof.

Recall that $A_{V}(t)=\tilde{A}-gV^{\prime}(I_{mn}-\bar{S}(t))V$. By
Proposition 1, for any fixed time $\tau$, $-V^{\prime}(I_{mn}-\bar{S}(\tau))V$
is exponentially stable if the graph of $S(\tau)^{\prime}$ is strongly
connected. Note $\tilde{A}$ is fixed and bounded. According to Lemma 1, for
each $\sigma\in\mathcal{S}_{\text{avg}}$ there is a positive number $g$,
depending on $\tau_{D}$ so that the transition matrix of $A_{V}(t)$ converges
to zero at least as fast as $e^{-\lambda t}$ does. This is accomplished by
choosing $g$ sufficiently large. Based on the proof of Lemma 1, it is
sufficient to pick $g$ to satisfy

$g\geq\frac{\ln c+(\lambda+\|\tilde{A}\|c)\tau_{D}}{\lambda^{*}\tau_{D}}$ (23)

where $c$ and $\lambda^{*}$ are two positive numbers chosen so that for any
fixed $\tau$, $\|e^{-V^{\prime}(I_{mn}-\bar{S}(\tau))Vt}\|\leq
ce^{-\lambda^{*}t}$, and $c>1$. We summarise:

######  Theorem 2.

For any fixed positive numbers $\tau_{D}$ and $\lambda$, there exists a
positive number $g^{*}$ with the following property. For any value of $g\geq
g^{*}$, any neighbor graph
$\mathbb{N}\in\\{\mathbb{G}_{\sigma}:\sigma\in\mathcal{S}_{\text{avg}}\\}$, if
the system defined by (3) and (4) is jointly observable, there are matrices
$K_{i},\;i\in\mathbf{m}$ such that, each state estimation error
$e_{i}=x_{i}-x,\;i\in\mathbf{m}$ of the distributed estimator defined by
$\eqref{eq:split_continuous_observer}$ converges to zero as
$t\rightarrow\infty$ as fast as $e^{-\lambda t}$ does.

######  Remark 1.

It is worth mentioning that when the neighbor graph is fixed, the dwell time
$\tau_{D}$ or the average dwell time $\tau_{D}$ is infinity. Then the
condition (23) in Theorem 2 degenerates to the condition (16) for exponential
stability with the ensured convergence rate $\lambda$ under a fixed topology.

###  2.4  Arbitrary Switching Neighbor Graph

In the sequel the problem is studied under arbitrary switching. The estimator
to be considered is the same as (5), with the exception of $g$ which are
chosen differently.

######  Theorem 3.

For any fixed positive number $\lambda$, there exists a positive number
$g^{*}$ with the following property. For any value of $g\geq g^{*}$, any time-
varying neighbor graph $\mathbb{N}(t)$, if the system defined by (3) and (4)
is jointly observable, the neighbor graph $\mathbb{N}(t)$ is undirected and
connected, and the stochastic matrix $S(t)$ of graph $\mathbb{N}(t)$ is doubly
stochastic, there are matrices $K_{i},\;i\in\mathbf{m}$ such that each state
estimation error $x_{i}(t)-x(t)$ of the distributed observer defined by (5),
converges to zero as $t\rightarrow\infty$ as fast as $e^{-\lambda t}$
converges to zero.

Proof: Recall $\Phi_{V}(t,\tau)$ is the transition matrix of $A_{V}(t)$ for
any $t\geq\tau\geq 0$. If we can show that there exist a constant $c$ so that

$\|\Phi_{V}(t,\tau)\|\leq ce^{-\lambda(t-\tau)},\;\;\;\forall t\geq\tau\geq 0$

the remaining proof is exactly the same as the proof of Theorem 1 which is
omitted here.

It is left to show that $\|\Phi_{V}(t,\tau)\|\leq
ce^{-\lambda(t-\tau)},\;\;\;\forall t\geq\tau\geq 0$ by choosing $g$
sufficiently large. We explore matrix $A_{V}(t)$. Recall that
$A_{V}(t)=\tilde{A}-gV^{\prime}((I_{m}-S(t))\otimes I_{n})V$. In particular,

$\displaystyle(\lambda I+A_{V}(t))+(\lambda I+A_{V}(t))^{\prime}$
$\displaystyle=$ $\displaystyle(\lambda I+\tilde{A})+(\lambda
I+\tilde{A})^{\prime}$ $\displaystyle-
gV^{\prime}((2I_{m}-S(t)-S^{\prime}(t))\otimes I_{n})V$

Since each $S(t)$ is doubly stochastic, $2I_{m}-S(t)-S^{\prime}(t)$ has row
sum $0$, all its off-diagonal entries are non-positive, and all its diagonal
entries are positive. That is this matrix can be seen as a generalized
Laplacian matrix of a connected graph. By Proposition 1, for any $t$,
$-V^{\prime}((2I_{m}-S(t)-S^{\prime}(t))\otimes I_{n})V$ is negative definite.
Thus by picking $g$ sufficiently large, $(\lambda I+A_{V}(t))+(\lambda
I+A_{V}(t))^{\prime}$ will be negative definite for any time $t$.

Consider system

$\dot{\bar{z}}=A_{V}(t)\bar{z}$

Let $V=\bar{z}^{\prime}\bar{z}$. Then

$\dot{V}=\bar{z}^{\prime}(A_{V}(t)^{\prime}+A_{V}(t))\bar{z}\leq-2\lambda\bar{z}^{\prime}\bar{z}$

Therefore, $\Phi_{V}(t,\tau)$ converges to zero as fast as
$e^{-\lambda(t-\tau)}$ does, i.e.,

$\|\Phi_{V}(t,\tau)\|\leq ce^{-\lambda(t-\tau)},\;\;\;\forall t\geq\tau\geq 0$

This completes the proof.

###  2.5 Distributed Estimator with Adaptive Gains

Obviously it may be disadvantageous to share a gain across the whole network,
and [22] has examined how one might find differing gains for each estimator in
a distributed way. Here we aim to design a simple adaptive distributed
estimator. The estimator for each agent $i$ still have form (5) while each
agent $i$ has its own gain $g_{i}$ which is obtained by

$\dot{g}_{i}=|V_{i}^{\prime}\sum_{j\in\mathcal{N}_{i}}\frac{1}{m_{i}}(e_{j}-e_{i})|_{2}^{2},\;\;\;i\in\mathbf{m}$
(24)

where $|.|_{2}$ is the two norm of a matrix and $g_{i}(0)$ is nonnegative but
otherwise arbitrary. The above equation can be rewritten as

$\dot{g}_{i}=|V_{i}^{\prime}W_{i}e|_{2}^{2},\;\;\;i\in\mathbf{m}$ (25)

where
$W_{i}=\begin{bmatrix}W_{i1}&\ldots&W_{im}\end{bmatrix}\in\mathbb{R}^{n\times
nm}$. Here $W_{ij}\in\mathbb{R}^{n\times n}$ is $\frac{1}{m_{i}}I_{n}$ if
$j\neq i$ and $j\in\mathcal{N}_{i}$, and $W_{ij}$ is $-I_{n}$ if $j=i$.
Otherwise $W_{ij}$ is a $0$ matrix. Let
$\text{column}\\{W_{1},W_{2},\ldots,W_{m}\\}=W$.

Different from (9), the error model turns to

$\dot{e}=(\bar{A}-G(t)P(I_{mn}-\bar{S}(t))e$ (26)

where $G(t)=\text{ block diag }\\{g_{1}(t)I_{n},\ldots,g_{m}(t)I_{n}\\}$. Let
$\begin{bmatrix}Qe\\\ z\end{bmatrix}=Te$ where
$T=\begin{bmatrix}Q^{-1}&V\end{bmatrix}$ as defined earlier.

Based on (13) and (26) the dynamic of $z_{i}$ can be written in the following
form

$\dot{z}_{i}=\tilde{A}_{i}z_{i}-g_{i}(t)V_{i}^{\prime}M_{i}e+\hat{A}_{i}Q_{i}e_{i},\;\;i\in\mathbf{m}$
(27)

where $\tilde{A}_{i}=V_{i}^{\prime}(A+K_{i}C_{i})V_{i}$, and
$\hat{A}_{i}=V_{i}^{\prime}(A+K_{i}C_{i})Q_{i}^{\prime}$.

First, we want to show that all $g_{i}(t)$ are bounded. We prove this by
contradiction. Without generality, suppose that $g_{i}$ for
$i\in\mathcal{V}_{u}=\\{1,2,\ldots,m_{1}\\}$ are unbounded, and $g_{i}$ for
$i\in\mathcal{V}_{b}=\\{m_{1}+1,m_{1}+2,\ldots,m\\}$ are bounded where
$\mathcal{V}_{u}\cap\mathcal{V}_{b}=0$ and
$\mathcal{V}_{u}\cup\mathcal{V}_{b}=\mathbf{m}$.

Let $R=R_{1}+R_{2}+R_{3}+R_{4}$ where the individual $R_{i}$ involve new
positive parameters $p$, $\alpha_{0}$, $\alpha_{m_{1}+1},\dots,\alpha_{m}$.

$R_{1}=\frac{1}{2}\sum_{i=1}^{m_{1}}\pi_{i}\frac{p}{g_{i}(t)}|z_{i}|_{2}^{2},\;\;R_{2}=\frac{1}{2}\sum_{i=m_{1}+1}^{m}\pi_{i}\frac{g_{i}(0)}{g_{i}(t)}|z_{i}|_{2}^{2}$

$R_{3}=-\sum_{i=m_{1}+1}^{m}\alpha_{i}g_{i}(t)$, and
$R_{4}=-\alpha_{0}\int_{0}^{t}|Qe|_{2}^{2}dt$. The way to pick positive
parameters $p$, $\alpha_{0}$, and $\alpha_{i}$ for $i\in\mathcal{V}_{b}$ is
specified as follows.

Picking $p\geq 1$:

Let $\mathcal{W}_{1}$ be a positive matrix matrix chosen such that

$z^{\prime}\mathcal{W}_{1}z=\sum_{i=1}^{m}\pi_{i}|\tilde{A}_{i}|_{2}|z_{i}|_{2}^{2}$
(28)

According to (15),

$F=\frac{1}{2}(HV^{\prime}(\bar{S}-I_{mn})V+V^{\prime}(\bar{S}-I_{mn})^{\prime}VH)>0$
(29)

Pick $p$ so that

$\mathcal{W}_{2}=pF-\mathcal{W}_{1}>0$ (30)

Picking $\alpha_{0}$, and $\alpha_{i}$, $i\in\mathcal{V}_{b}$:

Using the Cauchy-Schwarz inequality, the following three inequalities can be
derived. For $\beta_{1}>0,\;\beta_{2}>0$ and $\lambda_{i}>0{\text{ for
}}i\in\mathcal{V}_{b}$, all for the moment otherwise arbitrary, write
$\hat{A}=\text{block diag }\\{\hat{A}_{1},\ldots,\hat{A}_{m}\\}$

$z^{\prime}H\hat{A}Qe\leq\frac{\beta_{1}}{2}|\hat{A}^{\prime}Hz|_{2}^{2}+\frac{1}{2\beta_{1}}|Qe|_{2}^{2},$
(31) $-pz^{\prime}HV^{\prime}(\bar{S}-I_{mn})Q^{\prime}Qe\leq\\\
\frac{\beta_{2}}{2}|Q(\bar{S}-I_{mn})^{\prime}VHz|_{2}^{2}+\frac{1}{2\beta_{2}}|Qe|_{2}^{2}$
(32) $-\pi_{i}(g_{i}(0)-p)z_{i}^{\prime}V_{i}^{\prime}W_{i}e\leq\\\
\frac{\lambda_{i}}{2}|\pi_{i}(g_{i}(0)-p)z_{i}|_{2}^{2}+\frac{1}{2\lambda_{i}}|V_{i}^{\prime}W_{i}e|_{2}^{2}.$
(33)

Let $\mathcal{W}_{3}$ be a symmetric matrix chosen such that

$\displaystyle z^{\prime}\mathcal{W}_{3}z$ $\displaystyle=$ $\displaystyle
z^{\prime}\mathcal{W}_{2}z-\frac{\beta_{1}}{2}|\hat{A}^{\prime}Hz|_{2}^{2}$
(34)
$\displaystyle-\frac{\beta_{2}}{2}|Q(\bar{S}-I_{mn})^{\prime}VHz|_{2}^{2}$
$\displaystyle-\sum_{i=m_{1}+1}^{m}\frac{\lambda_{i}}{2}|\pi_{i}(g_{i}(0)-p)z_{i}|_{2}^{2}$

Pick positive $\beta_{1}$, $\beta_{2}$, and $\lambda_{i}$ for
$i\in\mathcal{V}_{b}$ to be small enough so that $W_{3}$ is positive definite.
Pick $\alpha_{0}$ and $\alpha_{i}$, $i\in\mathcal{V}_{b}$ according to (35),
and (36).

$\alpha_{0}=\frac{1}{2\beta_{1}}+\frac{1}{2\beta_{2}}$ (35)
$\alpha_{i}=\frac{1}{2\lambda_{i}}\;\;\;\;\;i\in\mathcal{V}_{b}$ (36)

Now we consider the derivative of $R$, i.e.,
$\dot{R}=\dot{R}_{1}+\dot{R}_{2}+\dot{R}_{3}+\dot{R}_{4}$.

$\displaystyle\dot{R}_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i=1}^{m_{1}}\pi_{i}\frac{p}{g_{i}^{2}}|z_{i}|_{2}^{2}|V_{i}^{\prime}W_{i}e|_{2}^{2}+\sum_{i=1}^{m_{1}}\pi_{i}\frac{p}{g_{i}}z_{i}^{\prime}\tilde{A}_{i}z_{i}$
$\displaystyle+\sum_{i=1}^{m_{1}}\pi_{i}\frac{p}{g_{i}}z_{i}^{\prime}\hat{A}_{i}Q_{i}e_{i}-\sum_{i=1}^{m_{1}}\pi_{i}pz_{i}^{\prime}V_{i}^{\prime}W_{i}e$

Since for $i\in\mathcal{V}_{u}$ each $g_{i}(t)$ is unbounded, there is a time
$T$ for $t\geq T$, $p\leq g_{i}(t)$ for all $i\in\mathcal{V}_{u}$. That is
$\frac{p}{g_{i}(t)}\leq 1$. Hence

$\displaystyle\dot{R}_{1}\leq\sum_{i=1}^{m_{1}}\pi_{i}z_{i}^{\prime}\tilde{A}_{i}z_{i}+\sum_{i=1}^{m_{1}}\pi_{i}z_{i}^{\prime}\hat{A}_{i}Q_{i}e_{i}-\sum_{i=1}^{m_{1}}\pi_{i}pz_{i}^{\prime}V_{i}^{\prime}W_{i}e$
(37)

According to (25), $g_{i}(t)$ is a non-decreasing function. As a result,
$\frac{g_{i}(0)}{g_{i}(t)}\leq 1$. Similar to the derivation of the inequality
for $\dot{R}_{1}$, it can be shown that

$\displaystyle\dot{R}_{2}$ $\displaystyle\leq$
$\displaystyle\sum_{i=m_{1}+1}^{m}\pi_{i}z_{i}^{\prime}\tilde{A}_{i}z_{i}+\sum_{i=m_{1}+1}^{m}\pi_{i}z_{i}^{\prime}\hat{A}_{i}Q_{i}e_{i}$
(38)
$\displaystyle-\sum_{i=m_{1}+1}^{m}\pi_{i}g_{i}(0)z_{i}^{\prime}V_{i}^{\prime}W_{i}e$

By the submultiplicity of the matrix two norm,
$z_{i}^{\prime}\tilde{A}_{i}z_{i}^{\prime}\leq|\tilde{A}_{i}|_{2}|z_{i}|_{2}^{2}$.
From this, (37), (38), (28) and (14),

$\displaystyle\dot{R}_{1}+\dot{R}_{2}$ $\displaystyle\leq$ $\displaystyle
z^{\prime}\mathcal{W}_{1}z+\sum_{i=1}^{m}\pi_{i}z_{i}^{\prime}\hat{A}_{i}Q_{i}e_{i}-\sum_{i=1}^{m}\pi_{i}pz_{i}^{\prime}V_{i}^{\prime}W_{i}e$
$\displaystyle-\sum_{i=m_{1}+1}^{m}\pi_{i}(g_{i}(0)-p)z_{i}^{\prime}V_{i}^{\prime}W_{i}e$
$\displaystyle=$ $\displaystyle
z^{\prime}\mathcal{W}_{1}z+z^{\prime}H\hat{A}Qe-
pz^{\prime}HV^{\prime}(\bar{S}-I_{mn})e$
$\displaystyle-\sum_{i=m_{1}+1}^{m}\pi_{i}(g_{i}(0)-p)z_{i}^{\prime}V_{i}^{\prime}W_{i}e$

It can be observed that
$z^{\prime}HV^{\prime}(\bar{S}-I_{mn})e=z^{\prime}HV^{\prime}(\bar{S}-I_{mn})(VV^{\prime}e+Q^{\prime}Qe)=z^{\prime}HV^{\prime}(\bar{S}-I_{mn})Vz+z^{\prime}HV^{\prime}(\bar{S}-I_{mn})Q^{\prime}Qe$.
From this and (29), $z^{\prime}HV^{\prime}((S-I_{m})\otimes
I_{n})e=z^{\prime}Fz+z^{\prime}HV^{\prime}((S-I_{m})\otimes
I_{n})Q^{\prime}Qe$. Thus

$\displaystyle\dot{R}_{1}+\dot{R}_{2}$ $\displaystyle\leq$ $\displaystyle
z^{\prime}(\mathcal{W}_{1}-pF)z+z^{\prime}H\hat{A}Qe$ $\displaystyle-
pz^{\prime}HV^{\prime}(\bar{S}-I_{mn})Q^{\prime}Qe$
$\displaystyle-\sum_{i=m_{1}+1}^{m}\pi_{i}(g_{i}(0)-p)z_{i}^{\prime}V_{i}^{\prime}W_{i}e$
$\displaystyle=$
$\displaystyle-z^{\prime}\mathcal{W}_{2}z++z^{\prime}H\hat{A}Qe$
$\displaystyle-pz^{\prime}HV^{\prime}(\bar{S}-I_{mn})Q^{\prime}Qe$
$\displaystyle-\sum_{i=m_{1}+1}^{m}\pi_{i}(g_{i}(0)-p)z_{i}^{\prime}V_{i}^{\prime}W_{i}e$

From this, (31), (32), and (33),

$\displaystyle\dot{R}_{1}+\dot{R}_{2}$ $\displaystyle\leq$
$\displaystyle-z^{\prime}\mathcal{W}_{2}z+\frac{\beta_{1}}{2}|\hat{A}^{\prime}Hz|_{2}^{2}$
(39)
$\displaystyle+\frac{\beta_{2}}{2}|Q(\bar{S}-I_{mn})^{\prime}VHz|_{2}^{2}$
$\displaystyle+\sum_{i=m_{1}+1}^{m}\frac{\lambda_{i}}{2}|\pi_{i}(g_{i}(0)-p)z_{i}|_{2}^{2}+\frac{1}{2\beta_{1}}|Qe|_{2}^{2}$
$\displaystyle+\frac{1}{2\beta_{2}}|Qe|_{2}^{2}+\sum_{i=m_{1}+1}^{m}\frac{1}{2\lambda_{i}}|V_{i}^{\prime}W_{i}e|_{2}^{2}$

It is direct to get that

$\dot{R}_{3}+\dot{R}_{4}=-\sum_{i=m_{1}+1}^{m}\alpha_{i}|V_{i}^{\prime}W_{i}e|_{2}^{2}-\alpha_{0}|Qe|_{2}^{2}$
(40)

From (39), (40), (35), and (36),

$\displaystyle\dot{R}$ $\displaystyle\leq$
$\displaystyle-z^{\prime}\mathcal{W}_{2}z+\frac{\beta_{1}}{2}|\hat{A}^{\prime}Hz|_{2}^{2}+\frac{\beta_{2}}{2}|Q((S-I)\otimes
I)^{\prime}VHz|_{2}^{2}$
$\displaystyle+\sum_{i=m_{1}+1}^{m}\frac{\lambda_{i}}{2}|\pi_{i}(g_{i}(0)-p)z_{i}|_{2}^{2}$

According to (34),

$\dot{R}\leq-z^{\prime}\mathcal{W}_{3}z<0$ (41)

Since $Qe$ is exponentially convergent, the limit of $R_{4}(t)$ as $t$ goes to
infinity exists. Due to the assumption that for $i\in\mathcal{V}_{b}$, the
$g_{i}$ are bounded. Thus $R_{3}$ is bounded. Therefore, $R$ is lower bounded.
From this and (41) it follows that $z\in\mathcal{L}^{2}$. From this and the
fact that $Qe\in\mathcal{L}^{2}$, we conclude $e\in\mathcal{L}^{2}$. This with
the definition of $\dot{g}_{i}$ imply that all $g_{i}$ for $i\in\mathbf{m}$
are bounded. Thus by contradiction all $g_{i}$ are bounded.

Next, we want to show that $e$ converges to zero. According to Theorem 1, let
$G_{1}=g_{1}I$ be a matrix chosen so that $\bar{A}-G_{1}P(I_{mn}-\bar{S}(t))$
is a stable matrix. From (26), the dynamic of $e$ can be rewritten as

$\dot{e}=(\bar{A}-G_{1}P(I_{mn}-\bar{S}(t)))e+(G_{1}-G)P(I_{mn}-\bar{S}(t))e$
(42)

Since $(I_{mn}-\bar{S}(t))e\in\mathcal{L}^{2}$, $G_{1}-G$ is bounded, and
$\bar{A}-G_{1}P(I_{mn}-\bar{S}(t))$ is stable, thus
$\dot{e}\in\mathcal{L}^{2}$. Thus (42) is input-to-state stable which implies
that $e$ would converge to zero asymptotically.

For an adaptive control algorithm, we have to notice that there are
fundamental issues that can arise in practice that will undermine the approach
[40]: these include the need to work with models of plants which may be very
accurate but are virtually never exact; the inability to know, given an
unknown plant, whether a desired control objective is practical or
impractical, and the possibility of transient instability, or extremely large
signals occurring before convergence. Thus, for this paper, our preference is
to stick with a given $g$ instead of using an adaptive algorithm.

###  2.6 Resilience

The concept of a passively resilient algorithm is proposed in [20]. By a
passively resilient algorithm for a distributed process is meant an algorithm
which, by exploiting built-in network and data redundancies, is able to
continue to function correctly in the face of abrupt changes in the number of
vertices and arcs in the inter-agent communication graph upon which the
algorithm depends. All the proposed continuous-time distributed estimators,
except for the adaptive one, are inherently resilient to these abrupt changes
provided the network is redundantly strongly connected and redundantly jointly
observable, with a careful gain picking before the algorithm starts. Details
can be found in Section 5 of [20]. The same resilient property is also
possessed by the discrete-time distributed estimators to be developed in the
next section.

##  3 Discrete-Time Distributed Estimator

In this section, a discrete time version of the distributed estimator problem
is studied. The estimator which solves this problem in discrete time is
described. Several techniques are outlined for picking the number of switches
required between “event times” in order to achieve a prescribed convergence
rate.

We are interested in the same time-varying network as used for the continuous-
time linear system and which is characterized by the neighbor graph
$\mathbb{N}(t)$. Each agent $i$ can sense a discrete-time signal
$y_{i}(\tau)\in{\rm I\\!R}^{s_{i}}$ at event times $\tau T$,
$\tau=0,1,2,\ldots$ where $T$ is a positive constant; for $i\in\mathbf{m}$ and
$\tau=0,1,2,\ldots$

$y_{i}(\tau)=C_{i}x(\tau),\;\;\;\;\;x(\tau+1)=Ax(\tau)$ (43)

and $x\in{\rm I\\!R}^{n}$. We assume throughout that $\mathbb{N}(t)$ is
strongly connected, and the system defined by (43) is jointly observable.

Each agent $i$ is to estimate $x$ using a dynamical system whose output
$x_{i}(\tau)\in{\rm I\\!R}^{n}$ is to be an asymptotically correct estimate of
$x(\tau)$ in the sense that the estimation error $x_{i}(\tau)-x(\tau)$
converges to zero as $\tau\rightarrow\infty$ as fast as $\lambda^{\tau}$ does,
where $\lambda$ is an arbitrarily chosen but fixed positive number444For the
type of observer to be developed, finite-time convergence is not possible.
less than $1$. To accomplish this it is assumed that the information agent $i$
can receive from neighbor $j$ at event time $\tau T$ is $x_{j}(\tau)$. It is
further assumed that agent $i$ can also receive certain additional information
from its neighbors at a finite number of times between each successive pair of
event times; what this information is will be specified below.

###  3.1 The Estimator

In this section it will be assumed that each agent’s neighbors do not change
between event times. In other words, for $i\in\mathbf{m}$,

$\mathcal{N}_{i}(t)=\mathcal{N}_{i}(\tau T),\;\;\;\;t\in[\tau
T,(\tau+1)T),\;\;\;\;\tau=0,1,2,\ldots$

With this assumption, the estimator to be considered consists of $m$ private
estimators, one for each agent. The estimator for agent $i$ is of the form

$x_{i}(\tau+1)=(A+K_{i}C_{i})\bar{x}_{i}(\tau)-K_{i}y_{i}(\tau)$ (44)

where $\bar{x}_{i}(\tau)$ is an “averaged state” computed recursively during
the real time interval $[\tau T,\;(\tau+1)T)$ using the update equations

$\displaystyle z_{i}(0,\tau)$ $\displaystyle=$ $\displaystyle x_{i}(\tau)$
(45) $\displaystyle z_{i}(k,\tau)$ $\displaystyle=$
$\displaystyle(I-P_{i})z_{i}(k-1,\tau)+$ (46)
$\displaystyle\frac{1}{m_{i}(\tau)}P_{i}\\!\\!\\!\\!\sum_{j\in\mathcal{N}_{i}(\tau
T)}\\!\\!\\!\\!z_{j}(k-1,\tau),\;\;k\in\mathbf{q}$
$\displaystyle\bar{x}_{i}(t)$ $\displaystyle=$ $\displaystyle z_{i}(q,\tau)$
(47)

Here $m_{i}(\tau)$ is the number of labels in $\mathcal{N}_{i}(\tau T)$, $q$
is a suitably defined positive integer, $\mathbf{q}=\\{1,2,\ldots,q\\}$, and
$P_{i}$ is the orthogonal projection on the unobservable space of $(C_{i},A)$.
Each matrix $K_{i}$ is defined as explained below. Meanwhile, we note that the
estimators incorporate two time scales. An agent obtains data from its
neighbors at a rate q times that at which it obtains measurement data.

As desriced in §2.1, for fixed $i\in\mathbf{m}$, write $Q_{i}$ for any full
rank matrix whose kernel is the unobservable space of $(C_{i},A)$, and let
$\bar{C}_{i}$ and $\bar{A}_{i}$ be the unique solutions to
$\bar{C}_{i}Q_{i}=C_{i}$ and $Q_{i}A=\bar{A}_{i}Q_{i}$ respectively. Let
$\lambda$ be a positive value bounded by $1$. Then the matrix pair
$(\bar{C}_{i},\bar{A}_{i})$ is observable. Thus by using a standard spectrum
assignment algorithm, a matrix $\bar{K}_{i}$ can be chosen to ensure that the
convergence of $(\bar{A}_{i}+\bar{K}_{i}\bar{C}_{i})^{\tau}$ to zero as
$\tau\rightarrow\infty$ is as fast as the convergence to zero of
$\lambda^{\tau}$. Having chosen such $\bar{K}_{i}$, $K_{i}$ is then defined to
be $K_{i}=Q_{i}^{-1}\bar{K}_{i}$ where $Q_{i}^{-1}$ is a right inverse for
$Q_{i}$. To explain what needs to be considered in choosing $q$ it is
necessary to describe the structure of the “error model” of the overall
estimator. This will be done next.

####  3.1.1 The Error Dynamic

For $i\in\mathbf{m}$, write $e_{i}(\tau)$ for the state estimation error
$e_{i}(\tau)=x_{i}(\tau)-x(\tau)$. In view of (44),

$e_{i}(\tau+1)=(A+K_{i}C_{i})\bar{e}_{i}(\tau)$

where $\bar{e}_{i}(\tau)=\bar{x}_{i}(\tau)-x(\tau)$. Moreover if
$\epsilon_{i}(k,\tau)\stackrel{{\scriptstyle\Delta}}{{=}}z_{i}(k,\tau)-x(\tau),\;k\in\\{0,1,\ldots,q\\}$
then

$\displaystyle\epsilon_{i}(0,\tau)$ $\displaystyle=$ $\displaystyle
e_{i}(\tau)$ $\displaystyle\epsilon_{i}(k,\tau)$ $\displaystyle=$
$\displaystyle(I-P_{i})\epsilon_{i}(k-1,\tau)+$
$\displaystyle\frac{1}{m_{i}(\tau)}P_{i}\sum_{j\in\mathcal{N}_{i}(\tau
T)}\epsilon_{j}(k-1,\tau),\;\;k\in\mathbf{q}$ $\displaystyle\bar{e}_{i}(\tau)$
$\displaystyle=$ $\displaystyle\epsilon_{i}(q,\tau)$

because of (45) – (47). It is possible to combine these $m$ subsystems into a
single system. Paralleling §2.1.1 let $e=$ col
$\\{e_{1},e_{2},\ldots,e_{m}\\}$, define $\bar{A}=$ block diag
$\\{A+K_{1}C_{1},A+K_{2}C_{2},\ldots,A+K_{m}C_{m}\\}$, $P=$ block diag
$\\{P_{1},P_{2},\ldots,P_{m}\\}$ and write $S(\tau)$ for the stochastic matrix
$S(\tau)=D_{\mathbb{N}(\tau T)}^{-1}\mathcal{A}^{\prime}_{\mathbb{N}(\tau T)}$
where $\mathcal{A}_{\mathbb{N}(\tau T)}$ is the adjacency matrix of
$\mathbb{N}(\tau T)$ and $D_{\mathbb{N}(\tau T)}$ is the diagonal matrix whose
$i$th diagonal entry is the in-degree of $\mathbb{N}(\tau T)$’s $i$th vertex.
Let $\bar{e}(\tau)=$ col
$\\{\bar{e}_{1}(\tau),\bar{e}_{2}(\tau),\ldots,\bar{e}_{m}(\tau)\\}$ and
$\epsilon(k,\tau)=\text{col}\\{\epsilon_{1}(k,\tau),\epsilon_{2}(k,\tau),\ldots,\epsilon_{m}(k,\tau)\\}$.
Then

$e(\tau+1)=\bar{A}\bar{e}(\tau)$

and

$\displaystyle\epsilon(0,\tau)$ $\displaystyle=$ $\displaystyle e(\tau)$
$\displaystyle\epsilon(k,\tau)$ $\displaystyle=$
$\displaystyle(I_{mn}-P(I_{mn}-\bar{S}(\tau)))\epsilon(k-1,\tau),\;k\in\mathbf{q}$
$\displaystyle\bar{e}(\tau)$ $\displaystyle=$ $\displaystyle\epsilon(q,\tau)$

where $\bar{S}(\tau)=S(\tau)\otimes I_{n}$. Clearly
$\bar{e}(\tau)=(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}e(\tau)$, so

$e(\tau+1)=\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}e(\tau)$ (48)

Our immediate aim is now to explain why for $q$ sufficiently large, the time-
varying matrix $\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}$ appearing in (48)
is a discrete-time stability matrix for which the product

$\Phi(\tau)=\prod_{s=1}^{\tau}\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(s)))^{q}$ (49)

converges to zero as $\tau\rightarrow\infty$ as fast as $\lambda^{\tau}$ does.
As described in §2.1, note that the subspace
$\mathcal{V}=\mathcal{V}_{1}\oplus\mathcal{V}_{2}\oplus\cdots\oplus\mathcal{V}_{m}$
is $\bar{A}$ \- invariant because
$(A+K_{i}C_{i})\mathcal{V}_{i}\subset\mathcal{V}_{i},\;i\in\mathbf{m}$. Next,
let $Q=$ block diag $\\{Q_{1},Q_{2},\ldots,Q_{m}\\}$ and $V=$ block diag
$\\{V_{1},V_{2},\ldots,V_{m}\\}$ It follows that $P=VV^{\prime}$, (10) and
(11). Also as before, $(A+K_{i}C_{i})V_{i}=V_{i}A_{i}$. Moreover

$\displaystyle Q(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}$ $\displaystyle=$
$\displaystyle Q$ (50) $\displaystyle(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}V$
$\displaystyle=$ $\displaystyle V(V^{\prime}\bar{S}(\tau)V)^{q}$ (51)

Note that (50) holds because $QP=0$. To understand why (51) is true, note
first that
$(I_{mn}-P(I_{mn}-\bar{S}(\tau)))V=V(I_{\bar{n}}-V^{\prime}(I_{mn}-\bar{S}(\tau))V)$
because $P=VV^{\prime}$; here $\bar{n}=\dim(\mathcal{V})$. But
$I_{\bar{n}}-V^{\prime}(I_{mn}-\bar{S}(\tau))V=V^{\prime}\bar{S}(\tau)V$
because $V^{\prime}V=I_{\bar{n}}$. Thus (51) holds for $q=1$; it follows by
induction that (51) holds for any positive integer $q$.

Using (10), (11),(50), and (51), one obtains the equations

$\displaystyle Q\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}$ $\displaystyle=$
$\displaystyle\bar{A}_{V}Q$ (52)
$\displaystyle\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}V$ $\displaystyle=$
$\displaystyle VA_{V}(\tau)$ (53)

where

$A_{V}(\tau)=\tilde{A}(V^{\prime}\bar{S}(\tau)V)^{q}$ (54)

with $\tilde{A}=\text{block diag }\\{A_{1},A_{2},\ldots,A_{m}\\}$. These
equations imply that555The notation $\hat{A}_{V}$ and $A_{V}$ are different
from the two defined in §3.

$\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}=T\begin{bmatrix}\bar{A}_{V}&0\\\
\hat{A}_{V}(\tau)&A_{V}(\tau)\end{bmatrix}T^{-1}$ (55)

with $T=\begin{bmatrix}Q^{-1}&V\end{bmatrix}$ and
$\widehat{A}_{V}(\tau)=V^{-1}\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}Q^{-1}$.

Since the spectrum of each
$\bar{A}_{i}+\bar{K}_{i}\bar{C}_{i},\;i\in\mathbf{m}$, is assignable with
$\bar{K}_{i}$, and $\widehat{A}_{V}(\tau)$ is a bounded matrix, to show that
for suitably defined $\bar{K}_{i}$ and $q$ sufficiently large, the matrix
$\Phi(\tau)$ defined in (49) converges to zero as fast as $\lambda^{\tau}$
does, it is sufficient to show that for $q$ sufficiently large, $A_{V}(\tau)$
is a discrete-time stability matrix whose state-transition matrix converges to
zero as fast as $\lambda^{\tau}$ does. To accomplish this, use will be made of
the following results.

######  Lemma 2.

Let $M$ be an $m\times m$ row stochastic matrix whose transpose has a strongly
connected graph. There exists a diagonal matrix $\Pi_{M}$ whose diagonal
entries are positive for which the matrix $L_{M}=\Pi_{M}-M^{\prime}\Pi_{M}M$
is positive semi-definite; moreover $L_{M}\mathbf{1}=0$ where $\mathbf{1}$ is
the $m$-vector of $1$s. If, in addition, the diagonal entries of $M$ are all
positive, then the kernel of $L_{M}$ is one-dimensional.

Proof of Lemma 2: Since $M$ is a stochastic matrix, it must have a spectral
radius of $1$ and an eigenvalue at $1$ as must $M^{\prime}$. Moreover, since
the graph of $M^{\prime}$ is strongly connected, $M^{\prime}$ is irreducible
{Theorem 6.2.24, [36]}. Thus by the Perron-Frobenius Theorem there must be a
positive vector $\pi$ such that $M^{\prime}\pi=\pi$. Without loss of
generality, assume $\pi$ is normalized so that the sum of its entries equals
$1$; i.e., $\pi$ is a probability vector. Let $\Pi_{M}$ be that diagonal
matrix whose diagonal entries are the entries of $\pi$. Then
$\Pi_{M}\mathbf{1}=\pi$.

Since $M\mathbf{1}=\mathbf{1}$, $\Pi_{M}\mathbf{1}=\pi$, and
$M^{\prime}\pi=\pi$, it must be true that $M^{\prime}\Pi_{M}M\mathbf{1}=\pi$
and thus that $L_{M}\mathbf{1}=0$. Thus $L_{M}$ can also be written as
$L_{M}=D-\hat{A}$ where $D$ is a diagonal matrix whose diagonal entries are
the diagonal entries of $L_{M}$ and $\hat{A}$ is the nonnegative matrix
$\hat{A}=D-L_{M}$. Arguing as in the proof of Proposition 1, it can be shown
that $L_{M}$ is positive-semidefinite.

Now suppose that the diagonal entries of $M$ are all positive. Then the
diagonal entries of $M^{\prime}\Pi_{M}$ must also all be positive. It follows
that every arc in the graph of $M^{\prime}$ must be an arc in the graph of
$M^{\prime}\Pi_{M}M$ so the graph of $M^{\prime}\Pi_{M}M$ must be strongly
connected. Since $I-\Pi_{M}$ is a nonnegative matrix, the graph of
$M^{\prime}\Pi_{M}M$ must be a spanning subgraph of the graph of
$I-\Pi_{M}+M^{\prime}\Pi_{M}M$. Since $I-L_{M}=I-\Pi_{M}+M^{\prime}\Pi_{M}M$
and the graph of $M^{\prime}\Pi M$ is strongly connected, the graph of
$I-L_{M}$ must be strongly connected as well. But $I-L_{M}$ is a nonnegative
matrix so it must be irreducible. In addition, since
$(I-L_{M})\mathbf{1}=\mathbf{1}$, the row sums of $(I-L_{M})$ all equal one.
Therefore the infinity norm of $I-L_{M}$ is one so its spectral radius is no
greater than $1$. Moreover $1$ is an eigenvalue of $I-L_{M}$. Thus by the
Perron-Frobenius Theorem, the geometric multiplicity of this eigenvalue is
one. It follows that the geometric multiplicity of the eigenvalue of $L_{M}$
at $0$ is also one; ie, the dimension of the kernel of $L_{M}$ is one as
claimed.

######  Proposition 2.

For each fixed value of $\tau$,

$(V^{\prime}\bar{S}(\tau)V)^{\prime}R(\tau)(V^{\prime}\bar{S}(\tau)V)-R(\tau)<0$
(56)

where $R(\tau)$ is the positive definite matrix,
$R(\tau)=V^{\prime}(\Pi_{S(\tau)}\otimes I_{n})V$.

Note that (56) shows that for each fixed $\tau$, $x^{\prime}R(\tau)x$ a
discrete-time Lyapunov function for the equation
$w(k+1)=V^{\prime}\bar{S}(\tau)Vw(k)$. Thus for fixed $\tau$,
$V^{\prime}\bar{S}(\tau)V$ is a discrete-time stability matrix.

Proof of Proposition 2: Fix $\tau$ and write $S$ for $S(\tau)$ and $\bar{S}$
for $\bar{S}(\tau)$. Note that the graph of $S^{\prime}$, namely $\mathbb{N}$,
is strongly connected. In view of Lemma 2, the matrix
$L=\Pi_{S}-S^{\prime}\Pi_{S}S$ is positive semi-definite and $L\mathbf{1}=0$.
Moreover, since the diagonal entries of $S$ and thus $S^{\prime}$ are all
positive, the kernel of $L$ is one-dimensional.

Write $R$ for $R(\tau)$. To prove the proposition it is enough to show that
the matrix

$Q=R-(V^{\prime}\bar{S}^{\prime}V)R(V^{\prime}\bar{S}V)$ (57)

is positive definite.

To proceed, set $\bar{L}=L\otimes I_{n}$ in which case $\bar{L}$ is positive
semi-definite because $L$ is. Moreover,
$\bar{L}=\bar{\Pi}-\bar{S}^{\prime}\bar{\Pi}\bar{S}$ where
$\bar{\Pi}=\Pi_{S}\otimes I_{n}$. Note that that $VRV^{\prime}=P\bar{\Pi}P$
where $P$ is the orthogonal projection matrix $P=VV^{\prime}$. Clearly
$VRV^{\prime}=P\bar{\Pi}^{\frac{1}{2}}\bar{\Pi}^{\frac{1}{2}}P$. Note that
both $P$ and $\bar{\Pi}^{\frac{1}{2}}$ are block diagonal matrices with
corresponding diagonal blocks of the same size. Because of this and the fact
that each diagonal block in $\bar{\Pi}^{\frac{1}{2}}$ is a scalar times and
identity matrix, it must be true that $P$ and $\bar{\Pi}^{\frac{1}{2}}$
commute; thus $P\bar{\Pi}^{\frac{1}{2}}=\bar{\Pi}^{\frac{1}{2}}P$. From this
and the fact that $P$ is idempotent, it follows that
$VRV^{\prime}=\bar{\Pi}^{\frac{1}{2}}P\bar{\Pi}^{\frac{1}{2}}$. Clearly
$\bar{\Pi}^{\frac{1}{2}}P\bar{\Pi}^{\frac{1}{2}}\leq\bar{\Pi}^{\frac{1}{2}}\bar{\Pi}^{\frac{1}{2}}$
so $VRV^{\prime}\leq\bar{\Pi}$. It follows using (57) that $Q\geq
R-V^{\prime}\bar{S}^{\prime}\bar{\Pi}\bar{S}V=R+V^{\prime}\bar{L}V-V^{\prime}\bar{\Pi}V$.
Therefore

$Q\geq V^{\prime}\bar{L}V$ (58)

In view of this, to complete the proof it is enough to show that
$V^{\prime}\bar{L}V$ is positive definite. This can be shown by the same proof
of Proposition 1.

Therefore $Q$ is positive definite because of (58). From this and (57) it
follows that (56) is true.

###  3.2 Choosing q

In what follows it will be assumed that each $\bar{K}_{i}$ has been selected
so that the the matrix $\bar{A}_{V}$ defined by (12), is such that
$\bar{A}_{V}^{\tau}$ converges to zero as $\tau\rightarrow\infty$ as fast as
$\lambda^{\tau}$ does. This can be done using standard spectrum assignment
techniques to make the spectral radius of $\bar{A}_{V}$ at least as small as
$\lambda$. In view of (55), it is clear that to assign the convergence rate of
the state transition matrix of $\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^{q}$
it is necessary and sufficient to control the convergence rate of the state
transition matrix of $A_{V}(\tau)$. This can be accomplished by choosing $q$
sufficiently large. There are two different ways to do this, each utilizing a
different matrix norm. Both approaches will be explained next using the
abbreviated notation $B(\tau)=V^{\prime}\bar{S}(\tau)V$; note that with this
simplification, $A_{V}(\tau)=\tilde{A}B^{q}(\tau)$ because of (54).

We summarize:

######  Theorem 4.

For any given $\lambda$ with $|\lambda|<1$, if the neighbor graph $\mathbb{N}$
is fixed and strongly connected, and the system defined by (43) is jointly
observable, there are matrices $K_{i}$, $i\in\mathbf{m}$ such that for
sufficiently large $q$, each estimation error $x_{i}(\tau)-x(\tau)$ of the
distributed estimator defined by (44)-(47), converges to zero as
$\tau\rightarrow\infty$ as fast as $\lambda^{\tau}$ converges to zero.

####  3.2.1 Weighted Two-Norm

For each fixed $\tau$ and each appropriately-sized matrix $M$, write
$\|M\|_{R(\tau)}$ for the matrix norm induced by the vector norm
$\|Mx\|_{R(\tau)}\stackrel{{\scriptstyle\Delta}}{{=}}\sqrt{x^{\prime}R(\tau)x}$.
Note that $\|M\|_{R(\tau)}$ is the largest singular value of
$R^{\frac{1}{2}}(\tau)MR^{-\frac{1}{2}}(\tau)$. Note in addition that

$(R^{\frac{1}{2}}(\tau)B(\tau)R^{-\frac{1}{2}}(\tau))^{\prime}(R^{\frac{1}{2}}(\tau)B(\tau)R^{-\frac{1}{2}}(\tau))<I$

because of (56). This shows that the largest singular value of
$R^{\frac{1}{2}}(\tau)B(\tau)R^{-\frac{1}{2}}(\tau)$ is less than one.
Therefore

$\|B(\tau)\|_{R(\tau)}<1$ (59)

a) $\mathbb{N}$ is constant

In this case both $B(\tau)$ and $R(\tau)$ are constant, so it is sufficient so
choose choose $q$ so that $\|\tilde{A}B^{q}(\tau)\|_{R(\tau)}\leq\lambda$.
Since $\|\cdot\|_{R(\tau)}$ is submultiplicative, this can be done by choosing
$q$ so that

$\|B(\tau)\|^{q}_{R(\tau)}\leq\frac{\lambda}{\;\;\;\;\;\|\tilde{A}\|_{R(\tau)}}$
(60)

This can always be accomplished because of (59).

b) $\mathbb{N}$ changes with time

In this case it is not possible to use the weighted two-norm
$\|\cdot\|_{R(\tau)}$ because it is time-dependent. A simple fix, but perhaps
not the most efficient one, would be to use the standard two-norm
$|\cdot|_{2}$ instead since it does not depend on time. Using this approach,
the first step would be to first choose, for each fixed $\tau$, an integer
$p_{1}(\tau)$ large enough so that $|B^{p_{1}(\tau)}(\tau)|_{2}<1$. Such
values of $p_{1}(\tau)$ must exist because each $B(\tau)$ is a discrete-time
stability matrix or equivalently, a matrix with a spectral radius less than
$1$. Computing such a value amounts to looking at the largest singular value
of $B^{p_{1}(\tau)}(\tau)$ for successively largest values of $p_{1}(\tau)$
until that singular value is less than $1$. Having accomplished this, a number
$p$ can easily be computed so that $|B^{p}(\tau)|_{2}<1\;\forall\tau$ since
there are only a finite number of distinct strongly connected graphs on $m$
vertices and consequently only a finite number of distinct matrices $B(\tau)$
in the set $\mathcal{B}=\\{B(\tau):\tau\geq 0\\}$. Choosing $p$ to be the
maximum of $p_{1}(\tau)$ with respect to $\tau$ is thus a finite computation.
The next step would be to compute an integer $\bar{p}$ large enough so that
each $|\tilde{A}(B^{p}(\tau))^{\bar{p}}|_{2}\leq\lambda$. A value of $q$ with
the required property would then be $q=p\bar{p}$.

####  3.2.2 Mixed Matrix Norm

There is a different way to choose $q$ which does not make use of either Lemma
2 or Proposition 2. The approach exploits the “mixed matrix norm” introduced
in [41]. To define this norm requires several steps. To begin, let
$|\cdot|_{\infty}$ denote the standard induced infinity norm and write ${\rm
I\\!R}^{mn\times mn}$ for the vector space of all $m\times m$ block matrices
$M=[M_{ij}]$ whose $ij$th entry is a matrix $M_{ij}\in{\rm I\\!R}^{n\times
n}$. With $n_{i}=\dim\mathcal{V}_{i},\;i\in\mathbf{m},$ and
$\bar{n}=n_{1}+n_{2}+\cdots n_{m}$, write ${\rm I\\!R}^{mn\times\bar{n}}$ for
the vector space of all $m\times m$ block matrices $M=[M_{ij}]$ whose $ij$th
entry is a matrix $M_{ij}\in{\rm I\\!R}^{n\times n_{j}}$. Similarly write
${\rm I\\!R}^{\bar{n}\times mn}$ for the vector space of all $m\times m$ block
matrices $M=[M_{ij}]$ whose $ij$th entry is a matrix $M_{ij}\in{\rm
I\\!R}^{n_{i}\times n}$. Finally write ${\rm I\\!R}^{\bar{n}\times\bar{n}}$
for the vector space of all $m\times m$ block matrices $M=[M_{ij}]$ whose
$ij$th entry is a matrix $M_{ij}\in{\rm I\\!R}^{n_{i}\times n_{j}}$.

Note that $B\in{\rm I\\!R}^{mn\times mn}$, $\tilde{A}\in{\rm
I\\!R}^{\bar{n}\times\bar{n}}$, $V\in{\rm I\\!R}^{mn\times\bar{n}}$, and
$V^{\prime}\in{\rm I\\!R}^{\bar{n}\times mn}$. For $M$ in any one of these
four spaces, the mixed matrix norm  [41] of $M$, written $\|M\|$, is

$\|M\|=\|\langle M\rangle\|_{\infty}$ (61)

where $\langle M\rangle$ is the matrix in ${\rm I\\!R}^{m\times m}$ whose
$ij$th entry is $\|M_{ij}\|_{2}$. It is very easy to verify that $\|\cdot\|$
is in fact a norm. It is even sub-multiplicative whenever matrix
multiplication is defined. Note in addition that $\|V\|=1$ and
$\|V^{\prime}\|=1$ because the columns of each $V_{i}$ form an orthonormal
set.

Recall that $P=VV^{\prime}$ is an orthogonal projection matrix. Using this,
the definition of $B(\tau)$ and the fact that $PV=V$, it is easy to see that
for any integer $p>0$

$B^{p}(\tau)=V^{\prime}(P\bar{S}(\tau)P)^{p}V$

Thus

$\|B^{p}(\tau)\|\leq\|(P\bar{S}(\tau)P)^{p}\|$

Using this and the fact that the graph of $S^{\prime}$ is strongly connected,
one can conclude that

$\|(P\bar{S}(\tau)P)^{p}\|<1,\;\;\;p\geq(m-1)^{2}$

This is a direct consequence of Proposition 2 of [41]. Thus

$\|B^{p}(\tau)\|<1,\;\;\;p\geq(m-1)^{2}$ (62)

a) $\mathbb{N}$ is constant

In this case $B(\tau)$ is constant so it is sufficient to choose $q$ so that
$\|\tilde{A}B^{q}(\tau)\|\leq\lambda$. This can be done by choosing
$q=p\bar{p}$ where $p\geq(m-1)^{2}$ and $\bar{p}$ is such that

$\|B^{p}(\tau)\|^{\bar{p}}\leq\frac{\lambda}{\|\tilde{A}\|}$ (63)

This can always be accomplished because of (62).

b) $\mathbb{N}$ changes with time

Note that (62) holds for all $\tau$. Assuming $p$ is chosen so that
$p\geq(m-1)^{2}$ it is thus possible to find, for each $\tau$, a positive
integer $\bar{p}(\tau)$, for which

$\|B^{p}(\tau)\|^{\bar{p}(\tau)}\leq\frac{\lambda}{\|\tilde{A}\|}$ (64)

Having accomplished this, a number $\bar{p}$ can easily be computed so that

$\|B^{p}(\tau)\|^{\bar{p}}\leq\frac{\lambda}{\|\tilde{A}\|}$ (65)

holds for all $\tau$, since there there are only a finite number of distinct
strongly connected graphs on $m$ vertices and consequently only a finite
number of distinct matrices $B(\tau)$ in the set $\mathcal{B}$ defined
earlier. Choosing $\bar{p}$ to be the maximum of $\bar{p}(\tau)$ with respect
to $\tau$ is thus a finite computation. A value of $q$ with the required
property would then be $q=p\bar{p}$.

##  4 Simulations

This section provides simulations to illustrate the state estimation
performance for both continuous and discrete time systems. The neighbor graph
will switch back and forth between Fig. 1 (a) and Fig. 1 (b), which can be
seen as modelling a connection failure happening between agent $1$ and agent
$3$ randomly.

###  4.1 Continuous dynamics

Consider the three channel, four-dimensional, discrete-time system described
by the equations $\dot{x}=Ax,\;y_{i}=C_{i}x,\;i\in\\{1,\;2,\;3\\}$, where

$A=\begin{bmatrix}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\ 0&0&-2&0\end{bmatrix}$

and $C_{i}$ is the $i$th unit row vector in ${\rm I\\!R}^{1\times 4}$. Note
that $A$ is a matrix with eigenvalues at $\pm 1j$, and $\pm 1.4142j$. While
the system is jointly observable, no single pair $(C_{i},A)$ is observable.
The observer convergence rate is designed to be $\lambda=1$. The first step is
to design $K_{i}$ as stated in §2.1.

For agent 1:

$\displaystyle{A}_{1}=\begin{bmatrix}0&-1\\\
1&0\end{bmatrix},\;\;\;{Q}_{1}=\begin{bmatrix}0&1&0&0\\\
1&0&0&0\end{bmatrix},$ $\displaystyle V_{1}=\begin{bmatrix}0&0&1&0\\\
0&0&0&1\end{bmatrix}^{\prime},\;\;\;K_{1}=\begin{bmatrix}-5&-5&0&0\end{bmatrix}^{\prime}$

For agent 2:

$\displaystyle{A}_{2}=\begin{bmatrix}0&-1\\\
1&0\end{bmatrix},\;\;\;{Q}_{2}=\begin{bmatrix}-1&0&0&0\\\
0&1&0&0\end{bmatrix},$ $\displaystyle V_{2}=\begin{bmatrix}0&0&1&0\\\
0&0&0&1\end{bmatrix}^{\prime},\;\;\;K_{2}=\begin{bmatrix}5&-5&0&0\end{bmatrix}^{\prime}$

For agent 3:

$\displaystyle{A}_{3}=\begin{bmatrix}0&-2\\\
1&0\end{bmatrix},\;\;\;{Q}_{3}=\begin{bmatrix}0&0&0&1\\\
0&0&1&0\end{bmatrix},$ $\displaystyle V_{3}=\begin{bmatrix}1&0&0&0\\\
0&1&0&0\end{bmatrix}^{\prime},\;\;\;K_{3}=\begin{bmatrix}0&0&-5&-4\end{bmatrix}^{\prime}$

Figure 1: The neighbor graph

Two cases are considered. First, suppose the neighbor $\mathbb{N}(t)$ is fixed
as shown in Figure 1(a). With random chosen initial state values and $g=10$,
traces of this simulation are shown in Fig. 2 where $x_{i}^{1}$ and $x^{1}$
denote the first components of $x_{i}$ and $x$ respectively. Moreover, the
norm of the estimation error is plotted in Fig. 3 from which we can see that
it is exponentially convergent with the rate $\lambda=1$.

Figure 2: Trajectory of the performance

Figure 3: The trajectory of the norm of the error

Second, suppose the neighbor $\mathbb{N}(t)$ is time-varying and switching
back and forth between Figure 1(a) and Figure 1(b) according to the indicator
function in Fig. 4. That is when the function value is $1$, the neighbor graph
is Figure 1(a), and when the function value is $0$, the neighbor graph is
Figure 1(b). It is arranged that the average dwell time is $\tau_{D}=0.0369$
for this simulation. The corresponding solution trajectories for $x_{i}^{3}$
and $x^{1}$ are shown in Fig. 5 and the norm of the estimation error is shown
in Fig. 6.

Figure 4: The indicator function

Figure 5: Trajectory of the performance

Figure 6: The trajectory of the norm of the error

###  4.2 Discrete dynamics

The following simulations are intended to illustrate how to pick the interger
parameter $q$ of the observer. Consider the three channel, four-dimensional,
discrete-time system described by the equations
$x(\tau+1)=Ax(\tau),\;y_{i}=C_{i}x,\;i\in\\{1,\;2,\;3\\}$, where $A$ and
$C_{i}$ are the same as described in §4.1. The observer convergence rate is
designed to be $\lambda=0.5$. The first step is also to design $K_{i}$ as
stated in §3.1. For each agent $i$, matrices $A_{i}$, $Q_{i}$, and $V_{i}$
remain the same as stated in §4.1. For agent 1:

$K_{1}=\begin{bmatrix}0.7&0.88&0&0\end{bmatrix}^{\prime}$

For agent 2:

$K_{2}=\begin{bmatrix}-0.8&0.7&0&0\end{bmatrix}^{\prime}$

For agent 3:

$K_{3}=\begin{bmatrix}0&0&0.7&1.88\end{bmatrix}^{\prime}$

Consider the case when the neighbor graph $\mathbb{N}$ is constant which is
Fig. 1 (a). With random chosen initial state values and $q=6$, the norm of the
estimation error is plotted in Fig. 7 from which we can see that it is
exponentially convergent with the rate $\lambda=0.5$.

Figure 7: The trajectory of the norm of the error

Figure 8: The indicator function

Figure 9: The trajectory of the norm of the error

Consider the case when the neighbor graph is switching between Fig. 1 (a) and
(b) according to Fig. 8. The norm of the estimation error is shown in Fig. 9
with the same value of $q$. Even though the error in Fig. 9 is larger than the
one in Fig. 7, it still does not overshoot the same upper-bound as that for
the fixed neighbor graph case.

##  5 Concluding Remarks

The distributed state estimation problem is studied when the neighbor graph is
time-varying. It has been shown that, each agent can estimate the state
exponentially fast with a pre-assigned convergence rate for both continuous-
time and discrete-time systems.

The distributed state estimators developed in both §2 and §3 rely on an
especially useful observation about distributed estimator structure first
noted in [13] and subsequently exploited in [15] and [1]. That is to split the
system spectrum into disjoint subsets corresponding to unobservable and
observable subspaces. Just how much further this idea can be advanced remains
to be seen. Generalization on the constraint of strong connectivity for the
neighbor graph can be studied in future work. Certainly the synchronous
switching upon which the local estimators in §3 depend can be relaxed by
judicious application of the mixed matrix norm discussed here. This
generalization will be addressed in future work.

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