# Robust Distributed Kalman filtering with
Event-Triggered Communication

Davide Ghion, Mattia Zorzi D. Ghion is with RMT s.r.l., Padova, Italy; e-mail:
<EMAIL_ADDRESS>.M. Zorzi is with the Department of Information
Engineering, University of Padova, Padova, Italy; e-mail:
<EMAIL_ADDRESS>.

###### Abstract

We consider the problem of distributed Kalman filtering for sensor networks in
the case there are constraints in data transmission and there is model
uncertainty. More precisely, we propose two distributed filtering strategies
with event-triggered communication where the state estimators are computed
according to the least favorable model. The latter belongs to a ball about the
nominal model. We also show that both the methods are stable in the sense that
the mean-square of the state estimation error is bounded in all the nodes.

###### Index Terms:

Distributed robust Kalman filtering, sensor networks (SNs), Event-triggered
communication, model uncertainty.

##  I Introduction

Sensor networks are ubiquitous in many field, e.g. monitoring, security, data
analysis and so on. The typical scenario is that the sensors collect
measurements and from them it is required to estimate some variables of
interest, i.e. the state of a dynamical model. This task is performed in a
distributed fashion and it can be accomplished in different ways, see [1, 2,
3, 4, 5, 6, 7]; for instance, each sensor can update its estimate and then
share the latter with its neighbors.

Nowadays the devices at each node of the sensor network are usually low-cost
and battery-supplied. The latter feature guarantees that these devices are
easily adaptable according to the surrounding environment. On the other hand,
the data transmission represents the most energy consuming node task.
Therefore, it is fundamental to guarantee that the rate of the data
transmission across the network is below a certain threshold in order to avoid
unexpected battery discharges. Data communication is typically scheduled by
means of data-driven (or event-triggered) strategies, see [8, 9, 10], [11, 12,
13, 14]. In this paper we focus our attention to the distributed scheme
proposed in [15]: each node updates its estimate with the new measurement (if
available); then, the latter is compared with the one just propagated in time
through the state space model (i.e. the estimate that can be computed also by
the neighbors of that node in the case there is no transmission). If the
discrepancy between them is large, then the node sends out the updated
estimate to its neighbors. Finally, each node performs a fusion of its
estimate and the ones regarding its neighbors. The appealing property of this
transmission rule is that it allows to obtain a distributed algorithm which
enjoys nice stability properties (i.e., mean-square boundedness of the state
estimation error in all the nodes) under minimal requirements.

In many situations the actual model is different from the nominal one (i.e.
the one used in the estimation algorithm), [16, 17]. Indeed, the fact that the
devices placed at each node are low-cost can lead to model uncertainty. In
such a scenario, the performance of the estimator based on the nominal model
will be poor. One possible way to address this issue is to consider the robust
Kalman filter proposed in [18]: the idea is to consider a dynamic minimax game
where one player is the estimator which minimizes the state prediction error,
while the other one is the hostile player which selects the least favorable
model in a set of plausible models called ambiguity set. The latter is a ball
which is formed by placing an upper bound on the discrepancy between the
nominal state space model and the models inside. The radius of this ball is
called tolerance and defines the magnitude of the uncertainty in the nominal
model.

Distributed strategies which are robust to model uncertainty have been already
proposed in the literature, see for instance [19, 20, 21, 22, 23, 24, 25],
however, to the best of the authors’ knowledge, none of them consider the case
with event-triggered communication.

In this paper we shall extend and formalize the ideas in the preliminary
conference paper [26]. More precisely, we present two new distributed Kalman
filters with event-triggered communication under model uncertainty. The
difference between the two approaches is the way with which the tolerance at
each node collecting measurements is designed. In one approach the tolerance
is the same for all those nodes, while in the other the tolerance depends on
the local model corresponding to the node. Our approaches represent a robust
generalization of the distributed strategy proposed in [15]. We also show
that, under reasonable assumptions, both the methods are stable in the sense
that the mean-square of the state estimation error is bounded in all the
nodes. Finally, we also present a Monte Carlo study showing the effectiveness
of the proposed approaches in the case of model uncertainty.

The outline of the paper is as follows. In Section II we formulate the
distributed state estimation problem characterized by data transmission
constraints and model uncertainty. In Section III we introduce the robust
distributed approach with event-triggered communication and uniform tolerance
across the network. In Section IV we analyze the stability property of the
proposed method. In Section V we propose the other robust distributed approach
with event triggered communication. In Section VI we consider the Monte Carlo
study showing the strength of our methods. Finally, in Section VII we draw the
conclusions.

Notation. Given a matrix $A$, $A^{\top}$ denotes its transpose matrix. Given a
symmetric matrix $Q$, then $Q>0$ ($Q\geq 0$) means that $Q$ is positive
(semi)definite; $\sigma_{min}(Q)$ and $\sigma_{max}(Q)$ denote the maximum and
the minimum eigenvalue of $Q$, respectively. Given an index set $\mathcal{N}$
and a set of matrices $\\{C^{i},i\in\mathcal{N}\\}$ having the same number of
columns, then $\mathrm{col}(C_{i},i\in\mathcal{N})$ is the matrix obtained by
stacking $C^{i}$’s. Given a set of matrices $\\{R^{i},i\in\mathcal{N}\\}$,
then $\mathrm{diag}(R^{i},i\in\mathcal{N})$ is the block diagonal matrix whose
main blocks are $R^{i}$’s. Let $x$ be a random vector, then $\mathbb{E}[x]$
and $\mathrm{Var}[x]$ denote its expectation and variance, respectively.
Finally, $x\sim\mathrm{N}(m,R)$ means that $x$ is Gaussian distributed with
mean $m$ and covariance matrix $R$.

##  II Problem formulation

Consider a network of nodes described by the digraph
$(\mathcal{N},\mathcal{A},\mathcal{S})$ where: $\mathcal{N}=\\{1,...,N\\}$ is
the set of the nodes, $\mathcal{A}\subseteq\mathcal{N}\times\mathcal{N}$ is
the set of edges and $\mathcal{S}\subseteq\mathcal{N}$ is the subset of
sensors node. The latter are the only one that have the capabilities to
perform measurements, while the nodes in $\mathcal{N}\setminus\mathcal{S}$ are
used to increase the connectivity of the network. If $(j,i)\in\mathcal{A}$,
then it means that node $j$ can transmit data to node $i$; moreover, all the
possible self-loops belong to $\mathcal{A}$. For each node $i\in\mathcal{N}$,
the subset $\mathcal{N}_{i}:=\\{j:(j,i)\in\mathcal{A},\;j\neq
i\\}\subseteq\mathcal{N}$ denotes the set of its in-neighbors, i.e. the set
composed by the nodes that can send information to node $i$.

We attach to this network the following nominal state space model

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Bw_{t}$ (1) $\displaystyle
y^{i}_{t}$
$\displaystyle=C^{i}x_{t}+D^{i}v^{i}_{t}\;,\;\;\;\;\;i\in\mathcal{S}$ (2)

where $x_{t}\in\mathbb{R}^{n}$ is the state, $y^{i}_{t}\in\mathbb{R}^{p_{i}}$
is the output at the sensor node $i$. Furthermore, matrix $B$ and $D^{i}$ are
full row rank matrices, $w_{t}$ and $v^{i}_{t}$ are zero-mean normalized
Gaussian white noises. The initial state $x_{0}$ is with mean $x_{0|-1}$ and
with covariance matrix $V_{0|-1}$. Finally, we assume that $w_{t}$,
$v^{i}_{t}$’s and $x_{0}$ are independent. Notice that, the model (1)-(2) can
be written as

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Bw_{t}$ (3) $\displaystyle
y_{t}$ $\displaystyle=Cx_{t}+Dv_{t}$ (4)

where $C:=\mathrm{col}(C^{i},i\in\mathcal{S})$,
$D:=\mathrm{diag}(D^{i},i\in\mathcal{S})$,
$y_{t}:=\mathrm{col}(y_{t}^{i},i\in\mathcal{S})\in\mathbb{R}^{p}$ and
$v_{t}:=\mathrm{col}(v_{t}^{i},i\in\mathcal{S})\in\mathbb{R}^{p}$.

The nominal model (1)-(2) over the time interval $t=1\ldots N$ can be
equivalently described by the conditional probability densities
$\phi_{t}(z_{t}|x_{t})$, where
$z_{t}=[\,x_{t+1}^{\top}\;y_{t}^{\top}\,]^{\top}$, $t=1\ldots N$, and
$f(x_{0})$. Taking the robust framework proposed in [18, 27, 28], we assume
that the actual model is described by the conditional probability densities
$\tilde{\phi}_{t}(z_{t}|x_{t})$, $t=1\ldots N$, and $f(x_{0})$. Moreover, we
assume that $\tilde{\phi}_{t}$ belongs to the ambiguity set

$\mathcal{B}_{t}=\\{\,\tilde{\phi}_{t}\hbox{ s.t.
}\tilde{\mathbb{E}}[\log(\tilde{\phi}_{t}/\phi_{t})|Y_{t-1}]\leq b\,\\},$
$\displaystyle\tilde{\mathbb{E}}[\log(\tilde{\phi}_{t}/\phi_{t})|Y_{t-1}]:=$
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n+p}}\tilde{\phi}_{t}(z_{t}|x_{t})\check{f}_{t}(x_{t}|Y_{t-1})\log\left(\frac{\tilde{\phi}_{t}(z_{t}|x_{t})}{\phi_{t}(z_{t}|x_{t})}\right)\mathrm{d}z_{t}\mathrm{d}x_{t},$

$Y_{t}:=\\{y_{0}\ldots y_{t}\\}$, $\check{f}_{t}$ is the actual probability
density of $x_{t}$ given $Y_{t-1}$ and $b>0$ is called tolerance which
accounts for model uncertainty. In plain words, $\mathcal{B}_{t}$ is a ball
about the nominal density with with radius $b$. Then, a robust state estimator
is the one solving the following minimax game:

$\displaystyle
x_{t+1|t}=\underset{g_{t}\in\mathcal{G}}{\operatorname*{arg\,min}}\underset{\tilde{\phi}_{t}\in\mathcal{B}_{t}}{\max}\,\tilde{\mathbb{E}}[\|x_{t+1}-g_{t}(y_{t})\|^{2}|Y_{t-1}]$
(5)

where $x_{t+1|t}$ is the estimator of $x_{t+1}$ given $Y_{t}$; $\mathcal{G}$
is the set of estimators having finite second order moments for any
$\tilde{\phi}_{t}\in\mathcal{B}_{t}$;

$\displaystyle\tilde{\mathbb{E}}$
$\displaystyle\left[\|x_{t+1}-g_{t}(y_{t})\|^{2}|Y_{t-1}\right]:=$
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n+p}}\|x_{t+1}-g_{t}(y_{t})\|^{2}\tilde{\phi}_{t}(z_{t}|x_{t})\check{f}_{t}(x_{t}|Y_{t-1})dz_{t}dx_{t}$

and it is assumed that
$\check{f}_{t}(x_{t}|Y_{t-1})\sim\mathrm{N}(\hat{x}_{t|t-1},V_{t|t-1})$. The
basic idea behind this paradigm is that when we are looking for an estimator
that minimizes properly the selected loss function, a hostile player called
“nature” conspires to select the worst possible model in the ambiguity set
$\mathcal{B}_{t}$. In [18] it has been shown that the (centralized) robust
estimator solution to (5) admits a Kalman-like structure. It is not difficult
to show that such a filter can be written in the information form as follows.
Let $P_{t|t-1}$ and $V_{t|t-1}$ denote the pseudo-nominal and least favorable,
respectively, covariance matrix of the prediction error at time $t$; let
$x_{t|t}$ denote the estimator of $x_{t}$ given $Y_{t}$ and $P_{t|t}$ denotes
the covariance matrices of the corresponding estimation error at time $t$. We
define the corresponding matrices in the information form as
$\Omega_{t|t-1}=P_{t|t-1}^{-1}$, $\Psi_{t|t-1}=V_{t|t-1}^{-1}$,
$\Omega_{t|t}=P_{t|t}^{-1}$ and the information states as

$q_{t|t-1}=\Psi_{t|t-1}{x}_{t|t-1},\quad q_{t|t}=\Omega_{t|t}{x}_{t|t}.$

Then, it is not difficult to see that the robust estimator obeys to

Correction step:
$\displaystyle\begin{cases}\Omega_{t|t}=\Psi_{t|t-1}+C^{\top}R^{-1}C\\\
q_{t|t}=q_{t|t-1}+C^{\top}R^{-1}y_{t},\end{cases}$ Prediction step:
$\displaystyle\begin{cases}\Omega_{t+1|t}=\\\ \hskip
5.69046ptQ^{-1}-Q^{-1}A(A^{\top}Q^{-1}A+\Omega_{t|t})A^{\top}Q^{-1}\\\
\text{Find }\theta_{t}>0\text{ s.t. }\gamma(\Omega_{t+1|t},\theta_{t})=b\\\
\Psi_{t+1|t}=\Omega_{t+1|t}-\theta_{t}\mathbf{I}_{n}\\\
q_{t+1|t}=\Psi_{t+1|t}A\Omega^{-1}_{t|t}q_{t|t},\end{cases}$

where $R:=DD^{\top}$, $Q:=BB^{\top}$ are positive definite matrices and

$\displaystyle\gamma$ $\displaystyle(\Omega,\theta)$
$\displaystyle:=\frac{1}{2}\left\\{tr[(\mathbf{I}_{n}-\theta\Omega^{-1})^{-1}-\mathbf{I}_{n}]+\log\det(\mathbf{I}_{n}-\theta\Omega^{-1})\right\\}.$

The parameter $\theta_{t}>0$ is called risk sensitivity parameter. It is worth
noting that given $\Omega>0$ and $b>0$, the equation $\gamma(\Omega,\theta)=b$
always admits a unique solution $\theta>0$ such that
$\Omega-\theta\mathbf{I}_{n}>0$. Furthermore, in the special case where $b=0$,
i.e. there is no model uncertainty, we have that $\gamma(\Omega,\theta)=0$
implies that $\theta_{t}=0$ and thus the above equations degenerates in the
usual Kalman equations in the information form.

In what follows, we face the problem to solve the minimax game in (5) where
the minimizer works in a distributed way and under data transmission
constraints. More precisely, each node $i\in\mathcal{N}$ must estimate the
state $x_{t}$, with $t\in\mathbb{Z}_{+}=\\{1,...,N\\}$, taking into account
that: i) the actual model does not coincide with the nominal one (3)-(4); ii)
each node $i$ can selectively transmits only the most relevant data without
compromising the stability properties.

##  III Robust event-triggered strategy

Before to introduce our robust distributed estimation paradigm we consider the
following quite simple scenario. We assume that $\mathcal{N}=\mathcal{S}$,
i.e. all the nodes are sensor nodes, and
$\mathcal{A}=\\{\,(j,j),\;j\in\mathcal{N}\,\\}$, i.e. the nodes do not
communicate. In the presence of model uncertainty, at node $i\in\mathcal{S}$
we can consider the robust Kalman filter in the information form based on the
local model

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Bw_{t}$ $\displaystyle
y^{i}_{t}$ $\displaystyle=C^{i}x_{t}+D^{i}v^{i}_{t}\;;$

thus, we obtain the following algorithm:

Correction step:
$\displaystyle\begin{cases}\Omega_{t|t}^{i}=\Psi_{t|t-1}^{i}+(C^{i})^{\top}(R^{i})^{-1}C^{i}\\\
q_{t|t}^{i}=q_{t|t-1}^{i}+(C^{i})^{\top}(R^{i})^{-1}y_{t}^{i}\end{cases}$ (6)
Prediction step: $\displaystyle\begin{cases}\Omega_{t+1|t}^{i}=\\\ \hskip
5.69046ptQ^{-1}-Q^{-1}A(A^{\top}Q^{-1}A+\Omega_{t|t}^{i})A^{\top}Q^{-1}\\\
\text{Find }\theta_{t}^{i}>0\text{ s.t.
}\gamma(\Omega_{t+1|t}^{i},\theta_{t}^{i})=b\\\
\Psi_{t+1|t}^{i}=\Omega_{t+1|t}^{i}-\theta_{t}^{i}\mathbf{I}_{n}\\\
q_{t+1|t}^{i}=\Psi_{t+1|t}^{i}A(\Omega^{i}_{t|t})^{-1}q_{t|t}^{i}\end{cases}$
(7)

where $R^{i}:=D^{i}(D^{i})^{\top}$. It is worth noting that each node has its
own risk sensitivity parameter $\theta_{t}^{i}$.

Next, we consider the scenario in which: i) $\mathcal{S}\subseteq\mathcal{N}$;
ii) each node $i$ can transmit its local estimate ${q}^{i}_{t|t}$ and
information matrix $\Omega^{i}_{t|t}$ to all its out-neighbors
$\overline{\mathcal{N}}_{i}=\\{\,j\,:\,(i,j)\in\mathcal{A},\,i\neq j\,\\}$ if
necessary. In plain words, each node $i$ can decide at any time step whether
to transmit or not its data, i.e. $(q^{i}_{t|t},\Omega^{i}_{t|t})$, without
compromising the stability properties of the algorithm. More precisely, we
require that $\mathbb{E}[\|x_{t}-x^{i}_{t|t}\|^{2}]$, where the expectation
operator is under the global least favorable model solution to (5), does not
diverge for any $i\in\mathcal{N}$ as $t$ approaches infinity.

The estimation paradigm that we now present is a robust generalization of the
distributed state estimation algorithm with event-triggered communication
proposed in [15] and it is composed by four steps described below.

Correction. At time $t$, the predicted pair $(q^{i}_{t|t-1},\Psi^{i}_{t|t-1})$
is available at node $i\in\mathcal{N}$. If $i\in\mathcal{S}$, i.e. it is a
sensor node, then also the measurement $y_{t}^{i}$ is available and thus the
correction step coincides with (6). If $i\notin\mathcal{S}$, no measurement is
available at the node, then we can only propagate the prediction couple.
Therefore, the so called information pair is obtained as

$(q^{i}_{t|t},\Omega^{i}_{t|t})=\begin{cases}\text{use equations
(\ref{eq:RKFfilteringloc}),}&\text{if $i\in\mathcal{S}$}\\\
(q^{i}_{t|t-1},\Psi^{i}_{t|t-1}),&\text{if $i\in\mathcal{N\setminus
S}$}.\end{cases}$

Information exchange. Each node $i\in\mathcal{N}$ sends its information couple
$(q^{i}_{t|t},\Omega^{i}_{t|t})$ to all its out-neighbors according to the
binary variable $c^{i}_{t}$:

  * •

if $c^{i}_{t}=1$, then node $i$ transmits the information couple to all its
out-neighbors at time $t$;

  * •

if $c^{i}_{t}=0$, then node $i$ does not transmit the information couple to
all its out-neighbors at time $t$.

It remains to define the binary variable $c^{i}_{t}$. Let
$n^{i}_{t}\in\mathbb{N}$ be the number of time instants elapsed from the most
recent transmission of node $i$, i.e. the most recently transmitted data is
$(q^{i}_{t-n^{i}_{r}|r-n^{i}_{t}},\Omega^{i}_{t-n^{i}_{t}|t-n^{i}_{t}})$.
Then, all the out-neighbors of node $i$ propagate
$(q^{i}_{t-n^{i}_{r}|t-n^{i}_{t}},\Omega^{i}_{t-n^{i}_{t}|t-n^{i}_{t}})$ in
time through a prediction step which takes into account the fact that the
actual model does not coincide with the nominal one (see (21) in the
prediction step below). Let $(\bar{q}^{i}_{t},\bar{\Psi}^{i}_{t})$ denote this
propagated pair at time $t$. Then, the transmission rule $c_{t}^{i}$ computed
at node $i$ is defined as in [15]:

$c^{i}_{t}=\begin{cases}0,&\quad\text{if
}\|{x}^{i}_{t|t}-\bar{x}^{i}_{t}\|^{2}_{\Omega^{i}_{t|t}}\leq\alpha\\\ &\text{
and
}\frac{1}{1+\beta}\Omega^{i}_{t|t}\leq\bar{\Psi}^{i}_{t}\leq(1+\delta)\Omega^{i}_{t|t}\\\
1,&\quad\text{otherwise}\end{cases}$ (8)

where $\bar{x}^{i}_{t}=(\bar{\Psi}^{i}_{t})^{-1}\bar{q}^{i}_{t}$ represents
the robust state prediction based on the propagation of the most recent
transmitted pair
$(q^{i}_{t-n^{i}_{r}|t-n^{i}_{t}},\Omega^{i}_{t-n^{i}_{t}|t-n^{i}_{t}})$;
$x^{i}_{t|t}=({\Omega}^{i}_{t|t})^{-1}q^{i}_{t|t}$ is the state estimate at
node $i$. In plain words, the transmission rule in (8) checks the discrepancy
between $(q^{i}_{t|t},\Omega^{i}_{t|t})$ and
$(\bar{q}^{i}_{t},\bar{\Psi}^{i}_{t})$. If the latter is large, then it means
that the out-neighbors own a prediction corresponding to node $i$ which is bad
and thus node $i$ must transmit the data. The positive scalars $\alpha$,
$\beta$ and $\delta$ can be tuned by the user in order to reach a desired
behavior in terms of transmission rate and performance. More precisely,
$\alpha$ tunes the bound on the discrepancy between $x^{i}_{t|t}$ and
$\bar{x}^{i}_{t}$, while $\beta$ and $\delta$ tunes the allowed mismatch
between the covariance matrices ${\Omega}^{i}_{t|t}$ and $\bar{\Psi}^{i}_{t}$.
In [29] it has been shown that the transmission strategy in (8) guarantees the
following upper bound. If we model the propagated and the information pairs as
$\mathrm{N}(\bar{q}^{i}_{t},\bar{\Psi}^{i}_{t})$ and
$\mathrm{N}(q^{i}_{t|t},\Omega^{i}_{t|t})$, respectively, then Condition (8)
guarantees that

$D_{KL}(\mathbb{N}(q^{i}_{t|t},\Omega^{i}_{t|t}),\mathbb{N}(\bar{q}^{i}_{t},\bar{\Psi}^{i}_{t}))\leq\frac{1}{2}[\alpha+\beta
n+n\log(1+\delta)]$

where $n$ is the state dimension and $D_{KL}$ denotes the Kullback-Leibler
divergence.

Information fusion. In this step, any node merges its information with the
ones corresponding to its in-neighbors. Let $\Pi\in\mathbb{R}^{N\times N}$
denote the consensus matrix whose element in position $(i,j)$ is defined as:

$\pi_{i,j}=\begin{cases}(d_{i}+1)^{-1},&\text{ if }(j,i)\in\mathcal{A}\\\
0,&\text{otherwise}\end{cases}$

where $d_{i}$ denotes the degree of node $i$; in this way we have that
$\pi_{i,j}$ with $j\in\mathcal{N}_{i}$ represents the coefficients of a convex
combination. Then, the fusion step is performed through the following convex
combination of the pairs:

$q^{i,F}_{t|t}=\pi_{i,i}q^{i}_{t|t}+\sum_{j\in\mathcal{N}_{i}}\pi_{i,j}\left[c^{j}_{t}q^{j}_{t|t}+(1-c^{j}_{t})\tilde{q}^{j}_{t}\right]$
(9)
$\Omega^{i,F}_{t|t}=\pi_{i,i}\Omega^{i}_{t|t}+\sum_{j\in\mathcal{N}_{i}}\pi_{i,j}\left[c^{j}_{t}\Omega^{j}_{t|t}+(1-c^{j}_{t})\tilde{\Omega}^{j}_{t}\right]$
(10)

where

$\tilde{q}^{j}_{t}=\frac{1}{1+\delta}\bar{q}^{j}_{t}\;\;\;,\;\;\;\tilde{\Omega}^{j}_{t}=\frac{1}{1+\delta}\bar{\Psi}^{j}_{t}.$

In view of (9)-(10), we can see that in the fusion step we consider
$(q^{j}_{t|t},\Omega^{j}_{t|t})$, if node $j$ transmitted its information pair
at time $t$. If node $j$ does not transmit, then the aforementioned pair is
not available at node $i$. To account for this lack, given that at each
iteration the nodes can calculate the pair
$(\bar{q}^{i}_{t},\bar{\Psi}^{i}_{t})$, which is certainly less informative
than $(q^{i}_{t|t},\Omega^{i}_{t|t})$, we consider
$(\bar{q}^{i}_{t},\bar{\Psi}^{i}_{t})$ in (9)-(10) shrunk by the factor
$(1+\delta)^{-1}$ in order to decrease its importance in the fusion step, see
[15] for more details.

Prediction. Once each node $i\in\mathcal{N}$ has computed the fused
information couple $(q^{i,F}_{t|t}\Omega^{i,F}_{t|t})$, the latter is
propagated in time with the robust prediction step in (7) where
$(q^{i}_{t|t},\Omega_{t|t}^{i})$ is now replaced by
$(q^{i,F}_{t|t},\Omega_{t|t}^{i,F})$:

$\displaystyle\left.\begin{array}[]{ll}\Omega_{t+1|t}^{i}&=Q^{-1}-Q^{-1}A(A^{\top}Q^{-1}A+\Omega_{t|t}^{i,F})A^{\top}Q^{-1}\\\
\text{Find }&\theta_{t}^{i}>0\text{ s.t.
}\gamma(\Omega_{t+1|t}^{i},\theta_{t}^{i})=b\\\
\Psi_{t+1|t}^{i}&=\Omega_{t+1|t}^{i}-\theta_{t}^{i}\mathbf{I}_{n}\\\
q_{t+1|t}^{i}&=\Psi_{t+1|t}^{i}A(\Omega^{i,F}_{t|t})^{-1}q_{t|t}^{i,F}.\end{array}\right.$
(15)

In this step we compute the propagated pair
$(\bar{q}^{i}_{t+1},\bar{\Psi}^{i}_{t+1})$, i.e. the one used in the case node
$i$ does not transmit its information pair. Notice this operation is performed
by both node $i$ and its out-neighbors and it can be summarized as follows. At
nodes $i\,\cup\,\mathcal{N}_{i}$ we have the pair
$(\breve{q}^{i}_{t},\breve{\Omega}^{i}_{t})$ defined as

$\displaystyle\begin{cases}\breve{q}^{i}_{t}&=c^{i}_{t}q^{i}_{t|t}+(1-c^{i}_{t})\bar{q}^{i}_{t}\\\
\breve{\Omega}^{i}_{t}&=c^{i}_{t}\Omega^{i}_{t|t}+(1-c^{i}_{t})\bar{\Psi}^{i}_{t};\end{cases}$
(16)

then, it is propagated in time with the robust prediction step in (7) where
$(q^{i}_{t|t},\Omega_{t|t}^{i})$ is now replaced by
$(\breve{q}^{i}_{t},\breve{\Omega}_{t}^{i})$:

$\displaystyle\left.\begin{array}[]{ll}\bar{\Omega}_{t+1}^{i}&=Q^{-1}-Q^{-1}A(A^{\top}Q^{-1}A+\breve{\Omega}_{t}^{i})A^{\top}Q^{-1}\\\
\text{Find }&\bar{\theta}_{t}^{i}>0\text{ s.t.
}\gamma(\bar{\Omega}_{t+1}^{i},\bar{\theta}_{t}^{i})=b\\\
\bar{\Psi}_{t+1}^{i}&=\bar{\Omega}_{t+1}^{i}-\bar{\theta}_{t}^{i}\mathbf{I}_{n}\\\
\bar{q}_{t+1}^{i}&=\bar{\Psi}_{t+1}^{i}A(\breve{\Omega}^{i}_{t})^{-1}\breve{q}_{t}^{i}.\end{array}\right.$
(21)

The procedure is summarized in Algorithm 1. It is worth noting that each node
$i$ is characterized by two risk sensitivity parameters, i.e. $\theta^{i}_{t}$
and $\bar{\theta}^{i}_{t}$. In the case that $b=0$, i.e. there is no model
uncertainty, in Algorithm 1 we have: $\theta^{i}_{t}=0$,
$\bar{\theta}^{i}_{t}=0$ and thus $\Psi_{t}^{i}=\Omega_{t}^{i}$,
$\bar{\Psi}_{t}^{i}=\bar{\Omega}_{t}^{i}$, i.e. we recover the distributed
Kalman algorithm with event-triggered communication proposed in [15].

Initialization: Set $(q^{i}_{0|-1},\Psi^{i}_{0|-1})$ for any $i\in\mathcal{N}$
For each $t=0,1,\ldots$
For each node $i\in\mathcal{N}$

Correction:

$(q^{i}_{t|t},\Omega^{i}_{t|t})=\begin{cases}\text{use
(\ref{eq:RKFfilteringloc}),}&\text{if $i\in\mathcal{S}$}\\\
(q^{i}_{t|t-1},\Psi^{i}_{t|t-1}),&\text{if $i\in\mathcal{N\setminus
S}$}\end{cases}$

Information exchange:

-

if $t=0$ set $c^{i}_{t}=1$, otherwise determine $c_{t}^{i}$ according to (8)

-

if $c^{i}_{t}=1$ transmit $(q^{i}_{t|t},\Omega^{i}_{t|t})$ to the out-
neighbors

-

receive $(q^{j}_{t|t},\Omega^{j}_{t|t})$ from all the in-neighbors
$j\in\mathcal{N}_{i}$ for which $c_{t}^{j}=1$

Information fusion:

$\tilde{q}^{j}_{t}=\frac{1}{1+\delta}\bar{q}^{j}_{t}\;\;\;,\;\;\;\tilde{\Omega}^{j}_{t}=\frac{1}{1+\delta}\bar{\Psi}^{j}_{t}\text{
with }j\in\mathcal{N}_{i}$ $\displaystyle q^{i,F}_{t|t}$
$\displaystyle=\pi_{i,i}q^{i}_{t|t}+\sum_{j\in\mathcal{N}_{i}}\pi_{i,j}\left[c^{j}_{t}q^{j}_{t|t}+(1-c^{j}_{t})\tilde{q}^{j}_{t}\right]$
$\displaystyle\Omega^{i,F}_{t|t}$
$\displaystyle=\pi_{i,i}\Omega^{i}_{t|t}+\sum_{j\in\mathcal{N}_{i}}\pi_{i,j}\left[c^{j}_{t}\Omega^{j}_{t|t}+(1-c^{j}_{t})\tilde{\Omega}^{j}_{t}\right]$

Prediction step:

-

Compute $(q^{i}_{t+1|t},\Psi^{i}_{t+1|t})$ using (15)

-

Compute $(\bar{q}^{i}_{t+1|t},\bar{\Psi}^{i}_{t+1})$ using (16) and (21)

Algorithm 1 RDKF with event-triggered communication

##  IV Stability analysis

Recall that the nominal (global) model (3)-(4) is different from the actual
one. In this section we analyze the stability properties of Algorithm 1 under
the least favorable model which is given by the maximizer of the minimax
problem in (5), that is, the centralized problem. In doing that, we need the
following assumptions:

A1.

The tolerance $b$ defining the ambiguity set in (5) is taken sufficiently
small;

A2.

The transition matrix A is invertible;

A3.

The system is collectively observable, i.e., the pair $(A,C)$ is observable;

A4.

The network is strongly connected, that is, there exists a directed path
between any pair $i,j\in\mathcal{N}$.

It is worth noting that assumptions A3-A4 are the same made in [15], in
particular Assumption A2 automatically holds in sampled-data systems where
matrix $A$ is obtained by discretizing the corresponding continuous-time
matrix. Finally, Assumption A1 is necessary to guarantee that the noise
processes characterizing the global least favorable model have uniformly
bounded variance, see the next proposition.

###### Proposition IV.1

Under assumptions A1 and A3, the least favorable model solution to (5) takes
the form

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+B\tilde{w}_{t}$ (22)
$\displaystyle y_{t}$ $\displaystyle=Cx_{t}+D\tilde{v}_{t}$ (23)

where $\tilde{w}_{t}$ and
$\tilde{v}_{t}:=\mathrm{col}(\tilde{v}_{t}^{i},i\in\mathcal{S})$ are zero-mean
Gaussian colored noises such that

$\displaystyle\underline{\rho}I_{n+N}\leq\mathrm{Var}\left(\left[\begin{array}[]{c}\tilde{w}_{t}\\\
\tilde{v}_{t+1}\end{array}\right]\right)\leq\overline{\rho}I_{n+N}$

with $\overline{\rho}\geq\underline{\rho}\geq 0$. Moreover, $\tilde{w}_{t}$
and $\tilde{v}_{t}$ are correlated.

###### Proof:

The least favorable model (22)-(23) has been characterized in [18, Section V]
where

$\displaystyle\tilde{w}_{t}:=H_{1,t}\varepsilon_{t}+L_{1,t}\epsilon_{t},\quad\tilde{v}_{t}:=H_{2,t}\varepsilon_{t}+L_{2,t}\epsilon_{t};$

$\epsilon_{t}$ is zero-mean normalized Gaussian white noise,

$\displaystyle\varepsilon_{t+1}=[A+BH_{1,t}-$ $\displaystyle
G_{t}(C+DH_{2,t})]\varepsilon_{t}$
$\displaystyle+(BL_{1,t}-G_{t}DL_{2,t})\epsilon_{t};$

the definition of the matrices $H_{1,t}$, $H_{2,t}$, $L_{1,t}$, $L_{2,t}$ and
$G_{t}$ can be found in [18]. In [30] it was shown that, under Assumptions A1
and A3, $H_{1,t}\rightarrow H_{1}$, $H_{2,t}\rightarrow H_{2}$,
$L_{1,t}\rightarrow L_{1}$, $L_{2,t}\rightarrow L_{2}$ and $G_{t}\rightarrow
G$ as $t\rightarrow\infty$. Moreover, $\check{A}:=A+BH_{1}-G(C+DH_{2})$ is
Schur stable. Accordingly
$\mathbb{E}[\varepsilon_{t}\varepsilon_{t}^{\top}]\rightarrow E_{0}$, with
$E_{0}\geq 0$, and
$\mathbb{E}[\varepsilon_{t+1}\varepsilon_{t}^{\top}]\rightarrow\check{E}_{1}=\check{A}E_{0}$
as $t\rightarrow\infty$. Then,

$\displaystyle\mathrm{Var}$
$\displaystyle\left(\left[\begin{array}[]{c}\tilde{w}_{t}\\\
\tilde{v}_{t+1}\end{array}\right]\right)$ $\displaystyle\rightarrow
HEH^{\top}+LL^{\top}+HML^{\top}+LM^{\top}H^{\top}$ (26)

where

$\displaystyle H:=\left[\begin{array}[]{cc}H_{1}&0\\\
0&H_{2}\end{array}\right],\quad
E:=\left[\begin{array}[]{cc}E_{0}&E_{1}^{\top}\\\
E_{1}&E_{0}\end{array}\right],$ (31) $\displaystyle
L:=\left[\begin{array}[]{cc}L_{1}&0\\\ 0&L_{2}\end{array}\right],\quad
M:=\left[\begin{array}[]{cc}0&0\\\ BL_{1}-GDL_{2}&0\end{array}\right].$ (36)

Hence, we have

$\displaystyle\underline{\rho}$
$\displaystyle=\sigma_{min}(HEH^{\top}+LL^{\top}+HML^{\top}+LM^{\top}H^{\top}),$
$\displaystyle\overline{\rho}$
$\displaystyle=\sigma_{max}(HEH^{\top}+LL^{\top}+HML^{\top}+LM^{\top}H^{\top}).$

∎

Let $e_{t}^{i}=x_{t}-x^{i}_{t|t}$ denote the estimation error at node $i$ and
$e_{t}=\mathrm{col}(e_{t}^{i},i\in\mathcal{N})$ the collective estimation
error. In order to prove the stability of the estimation error, we consider
the Lyapunov function

$\displaystyle\mathcal{V}_{t}(e_{t})=\sum_{i\in\mathcal{N}}p_{i}\|e_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}$
(37)

where $p_{i}$’s, strictly positive, are the components of a vector $p$
satisfying the condition $p^{\top}=p^{\top}\Pi$. Notice that, the existence of
such a vector is guaranteed by the Perron-Frobenious theorem since, by
Assumption A4, $\Pi$ is a primitive matrix. The next proposition shows that
$\mathcal{V}_{t}(e_{t})$ is a well-defined Lyapunov function because
$\Omega_{t|t}^{i}$’s are uniformly bounded.

###### Proposition IV.2

Assume that A2-A4 hold. Consider the sequence $\Omega_{t|t}^{i}$,
$i\in\mathcal{N}$, generated by Algorithm 1 with $\Psi_{0|-1}^{i}>0$. Then,
there exist three positive constants $\underline{\omega}$, $\overline{\omega}$
and $\underline{\omega}_{F}$ such that
$\underline{\omega}I\leq\Omega_{t|t}^{i}\leq\overline{\omega}I$ and
$\underline{\omega}I\leq\Omega_{t|t}^{i,F}\leq\overline{\omega}_{F}I$.

###### Proof:

We start by showing the upper bounds. By (7) we have that
$\Omega_{t+1|t}^{i}\leq Q^{-1}$ and thus
$\Psi_{t+1|t}^{i}\leq\Omega_{t+1|t}^{i}\leq Q^{-1}$ which is a uniform upper
bound. By the correction step in Algorithm 1 we have that
$\Omega^{i}_{t|t}\leq
Q^{-1}+(C^{i})^{\top}(R^{i})^{-1}C^{i}\leq\overline{\omega}I$ where

$\displaystyle\overline{\omega}:=\underset{i\in\mathcal{N}}{\max}\,\sigma_{max}(Q^{-1}+(C^{i})^{\top}(R^{i})^{-1}C^{i}).$

It is also worth noting that

$\displaystyle\Omega_{t|t}^{i,F}\leq\overline{\omega}I$ (38)

where have exploited the definition in (10) and the facts that
$\Omega_{t|t}^{i}\leq\overline{\omega}I$ and
$\tilde{\Omega}_{t}^{i}\leq\overline{\omega}I$, by the transmission rule (8).
We now prove the uniform lower bound. First, notice that if
$\Psi_{0|-1}^{i}>0$ $\forall\,i\in\mathcal{N}$ then $\Omega_{t|t}^{i}>0$
$\forall\,i\in\mathcal{N}$. Indeed, by induction we have that if
$\Psi_{t|t-1}^{i}>0$, by the correction step in Algorithm 1 it follows that
$\Omega_{t|t}^{i}\geq\Psi_{t|t-1}^{i}>0$ $\forall\,i\in\mathcal{N}$; thus,
$\Omega_{t|t}^{i,F}>0$ because it is a convex combination of positive definite
matrices; then, it follows
$\Omega_{t+1|t}^{i}=(A(\Omega^{i,F}_{t|t})^{-1}A^{\top}+Q)^{-1}>0$ by (15);
finally, the fact that $\Psi_{t+1|t}^{i}>0$ is guaranteed by the choice of
$\theta_{t}^{i}$ as the solution of
$\gamma(\Omega_{t+1|t}^{i},\theta_{t}^{i})=b$, see [18, 31]. Let
$\mathbf{1}_{\mathcal{S}}(i)$ denote the indicator function taking value 1 if
$i\in\mathcal{S}$ and 0 otherwise. By Lemma .1 in Appendix, there exists a
constant $\mu>0$ such that

$\displaystyle\Omega_{t|t}^{i}$
$\displaystyle=\Psi_{t|t-1}^{i}+\mathbf{1}_{\mathcal{S}}(i)(C^{i})^{\top}(R^{i})^{-1}C^{i}$
$\displaystyle=\Omega_{t|t-1}^{i}-\theta_{t-1}^{i}I+\mathbf{1}_{\mathcal{S}}(i)(C^{i})^{\top}(R^{i})^{-1}C^{i}$
$\displaystyle\geq\mu\Omega_{t|t-1}^{i}+\mathbf{1}_{\mathcal{S}}(i)(C^{i})^{\top}(R^{i})^{-1}C^{i}.$

Moreover, in view of (38), by [4, Lemma 1 -fact (ii)] there exists a constant
$\upsilon>0$ such that

$\displaystyle\Omega_{t|t}^{i}$ $\displaystyle\geq\mu\upsilon
A^{-\top}\Omega_{t-1|t-1}^{i,F}A^{-1}+\mathbf{1}_{\mathcal{S}}(i)(C^{i})^{\top}(R^{i})^{-1}C^{i}.$
(39)

In view of (8) and (10), it is not difficult to see that

$\displaystyle\Omega_{t|t}^{i,F}\geq\frac{1}{(1+\beta)(1+\delta)}\sum_{j\in\mathcal{N}}\pi_{i,j}\Omega^{j}_{t|t}.$
(40)

Taking into account (39), we obtain

$\displaystyle\Omega_{t|t}^{i}$ $\displaystyle\geq\nu
A^{-\top}\sum_{j\in\mathcal{N}}\pi_{i,j}\Omega^{j}_{t|t}A^{-1}+\phi_{i}$

where $\nu:=\mu\upsilon((1+\beta)(1+\delta))^{-1}>0$ and
$\phi_{i}:=\mathbf{1}_{\mathcal{S}}(i)(C^{i})^{\top}(R^{i})^{-1}C^{i}$. The
last inequality is similar to the one obtained in the proof of Lemma 1 in
[15]. Accordingly, using the same reasonings it is possible to conclude that
there exists a constant $\underline{\omega}>0$ such that
$\Omega_{t|t}^{i}\geq\underline{\omega}I$. It is also worth noting we have, by
(40), that $\Omega_{t|t}^{i,F}\geq\underline{\omega}_{F}I$ with
$\underline{\omega}_{F}:=\underline{\omega}((1+\beta)(1+\delta))^{-1}$. ∎

###### Proposition IV.3

Assume that A2-A4 hold. Let $(q^{i}_{t|t},\Omega^{i}_{t|t})$ be the sequence
generated according to Algorithm 1 with $\Psi_{0|-1}^{i}>0$
$\forall\,i\in\mathcal{N}$. Then, under the least favorable model in
(22)-(23), we have

$\displaystyle\|e_{t+1}^{i}\|^{2}_{\Omega_{t+1|t+1}^{i}}$
$\displaystyle\leq\gamma^{2}\left(\pi_{i,i}\|e^{i}_{t}+\xi_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}\right.$
$\displaystyle\left.+\sum_{j\in\mathcal{N}_{i}}\pi_{i,j}\|e_{t}^{j}+\xi_{t}^{i}+\eta_{t}^{j}\|^{2}_{\Omega_{t|t}^{j}}\right),\quad\forall\,i\in\mathcal{N}$

where $0<\gamma<1$ is a constant and

$\displaystyle\xi_{t}^{i}$
$\displaystyle:=A^{-1}[B\tilde{w}_{t}-\mathbf{1}_{\mathcal{S}}(i)(\Psi_{t+1|t}^{i})^{-1}(C^{i})^{\top}(R^{i})^{-1}D^{i}\tilde{v}_{t+1}^{i}],$
$\displaystyle\eta_{t}^{j}$
$\displaystyle:=(1-c^{j}_{t})(\hat{x}^{j}_{t|t}-\bar{x}^{j}_{t}),\quad\bar{x}_{t}^{j}:=(\bar{\Psi}_{t}^{j})^{-1}\bar{q}_{t}^{j}.$

###### Proof:

Recall that $q_{t+1|t}^{i}=\Psi^{i}_{t+1|t}x_{t+1|t}^{i}$ and
$q_{t|t}^{i}=\Omega^{i}_{t|t}x_{t|t}^{i}$. Then, it is not difficult to see
that

$\displaystyle{x}_{t+1|t+1}^{i}=(\Omega_{t+1|t+1}^{i})^{-1}\left(\Psi_{t+1|t}^{i}{x}_{t+1|t}^{i}\right.$
$\displaystyle\hskip
5.69046pt\left.+\mathbf{1}_{\mathcal{S}}(i)(C^{i})^{\top}(R^{i})^{-1}C^{i}x_{t+1}+\mathbf{1}_{\mathcal{S}}(i)(C^{i})^{\top}(R^{i})^{-1}D^{i}\tilde{v}_{t+1}^{i}\right)$
$\displaystyle
x_{t+1}=(\Omega_{t+1|t+1}^{i})^{-1}\left(\Psi_{t+1|t}^{i}x_{t+1}\right.$
$\displaystyle\hskip
5.69046pt+\left.\mathbf{1}_{\mathcal{S}}(i)(C^{i})^{\top}(R^{i})^{-1}C^{i}x_{t+1}\right)$

which implies

$\displaystyle e_{t+1}^{i}$ $\displaystyle=x_{t+1}-{x}_{t+1|t+1}^{i}$
$\displaystyle=(\Omega_{t+1|t+1}^{i})^{-1}\Psi_{t+1|t}^{i}(x_{t+1}-{x}_{t+1|t}^{i}+\bar{v}_{t+1}^{i})$

where
$\bar{v}_{t+1}^{i}=-\mathbf{1}_{\mathcal{S}}(i)(\Psi_{t+1|t}^{i})^{-1}(C^{i})^{\top}(R^{i})^{-1}D^{i}\tilde{v}_{t+1}$.
Since $\Psi^{i}_{t+1|t}\leq\Omega^{i}_{t+1|t+1}$, we have

$\displaystyle\|$ $\displaystyle e_{t+1}^{i}\|_{\Omega_{t+1|t+1}^{i}}^{2}$
$\displaystyle=\|x_{t+1}-{x}_{t+1|t}^{i}+\bar{v}_{t+1}^{i}\|^{2}_{\Psi^{i}_{t+1|t}(\Omega_{t+1|t+1}^{i})^{-1}\Psi^{i}_{t+1|t}}$
$\displaystyle\leq\|x_{t+1}-{x}_{t+1|t}^{i}+\bar{v}_{t+1}^{i}\|_{\Psi_{t+1|t}^{i}}^{2}.$
(41)

By Lemma 1 - fact (iii) in [4] there exists a constant $0<\gamma\leq 1$ such
that

$\Psi_{t+1|t}^{i}=\Omega_{t+1|t}^{i}-\theta_{t}^{i}\leq\Omega_{t+1|t}^{i}\leq\gamma^{2}A^{-T}\Omega_{t|t}^{i,F}A^{-1}$

where we exploited the fact that
$\Omega^{i,F}_{t|t}\geq\underline{\omega}_{F}I$ by Proposition IV.2. Taking
into account (41), it follows that

$\displaystyle\|e_{t+1}^{i}\|_{\Omega_{t+1|t+1}^{i}}^{2}$
$\displaystyle\leq\gamma^{2}\|x_{t+1}-{x}_{t+1|t}^{i}+\bar{v}_{t+1}^{i}\|_{A^{-\top}\Omega_{t|t}^{i,F}A^{-1}}^{2}$
$\displaystyle=\gamma^{2}\|A(x_{t}-{x}^{i,F}_{t|t})+B\tilde{w}_{t}+\bar{v}^{i}_{t+1}\|_{A^{-\top}\Omega_{t|t}^{i,F}A^{-1}}^{2}$
$\displaystyle=\gamma^{2}\|x_{t}-{x}^{i,F}_{t|t}+\xi^{i}_{t}\|_{\Omega_{t|t}^{i,F}}^{2}$

where we exploited the fact that
${x}^{i,F}_{t|t}=(\Omega_{t|t}^{i,F})^{-1}q^{i,F}_{t|t}$ and
$x_{t+1|t}^{i}=Ax_{t|t}^{i,F}$. Then, using the same reasonings in the second
part of the proof of Lemma 2 in [15] it is not difficult to see that

$\displaystyle\|e_{t+1}^{i}\|_{\Omega_{t+1|t+1}^{i}}^{2}$
$\displaystyle\leq\gamma^{2}\left(\pi_{i,i}\|{e}^{i}_{t}+\xi^{i}_{t}\|_{\Omega_{t|t}^{i}}^{2}\right.$
$\displaystyle\left.+\sum_{j\in\mathcal{N}_{i}}\pi_{i,j}\|{e}^{j}_{t}+\xi^{i}_{t}+\eta_{t}^{j}\|_{\Omega_{t|t}^{j}}^{2}\right)$

where

$\displaystyle\eta_{t}^{j}:=x_{t|t}^{j}-\check{x}_{t|t}^{j},\quad\check{x}_{t|t}^{j}=(\check{\Omega}_{t|t}^{j})^{-1}\check{q}_{t|t}^{j}$
$\displaystyle\check{q}^{j}_{t|t}:=c_{t}^{j}q^{j}_{t|t}+(1-c_{t}^{j})\tilde{q}^{j}_{t},\quad\check{\Omega}^{j}_{t|t}:=c_{t}^{j}\Omega^{j}_{t|t}+(1-c_{t}^{j})\tilde{\Omega}^{j}_{t}$

which concludes the proof. ∎

In view of Proposition IV.1 and Proposition IV.2, it follows that
$\xi_{t}^{i}$, with $i\in\mathcal{N}$, is uniformly bounded in mean-square
under the assumptions A1-A4. Indeed, if $i\notin\mathcal{S}$, then

$\displaystyle\xi_{t}^{i}$
$\displaystyle=A^{-1}B\tilde{w}_{t}=A^{-1}\left[\begin{array}[]{cc}B&0\end{array}\right]\left[\begin{array}[]{c}\tilde{w}_{t}\\\
\tilde{v}_{t+1}\end{array}\right]$ (45)
$\displaystyle\mathrm{Var}(\xi_{t}^{i})$
$\displaystyle=A^{-1}\left[\begin{array}[]{cc}B&0\end{array}\right]\mathrm{Var}\left(\left[\begin{array}[]{c}\tilde{w}_{t}\\\
\tilde{v}_{t+1}\end{array}\right]\right)\left[\begin{array}[]{c}B^{\top}\\\
0\end{array}\right]A^{-\top}$ (51)
$\displaystyle\leq\overline{\rho}^{\top}A^{-1}BB^{\top}A^{-\top}\leq\overline{\rho}\sigma_{max}(A^{-1}BB^{\top}A^{-\top})I;$

if $i\in\mathcal{S}$, then

$\displaystyle\xi_{t}^{i}$
$\displaystyle=A^{-1}\left[\begin{array}[]{cc}B&H_{i}\end{array}\right]\left[\begin{array}[]{c}\tilde{w}_{t}\\\
\tilde{v}_{t+1}\end{array}\right]$ (55)

where

$\displaystyle
H_{i}:=[\,0\;\ldots\;\underbrace{-(\Psi^{i}_{t+1|t})^{-1}(C^{i})^{\top}(R^{i})^{-1}D^{i}}_{\hbox{$i+1$-th
block}}\;\ldots\;0\,]$

and thus

$\displaystyle\mathrm{Var}(\xi_{t}^{i})$
$\displaystyle=A^{-1}\left[\begin{array}[]{cc}B&H_{i}\end{array}\right]\mathrm{Var}\left(\left[\begin{array}[]{c}\tilde{w}_{t}\\\
\tilde{v}_{t+1}\end{array}\right]\right)\left[\begin{array}[]{c}B^{\top}\\\
H_{i}^{\top}\end{array}\right]A^{-\top}$ (61)
$\displaystyle\leq\overline{\rho}A^{-1}(BB^{\top}+H_{i}H_{i}^{\top})A^{-\top}$
$\displaystyle\leq\overline{\rho}\sigma_{max}(A^{-1}(BB^{\top}+H_{i}H_{i}^{\top})A^{-\top})I.$

We conclude that $\mathrm{Var}[\xi_{t}^{i}]\leq\rho_{\xi}^{2}$ where

$\displaystyle\rho_{\xi}^{2}:=\overline{\rho}\,\underset{i\in\mathcal{N}}{\max}\,\sigma_{max}(A^{-1}(BB^{\top}+H_{i}H_{i}^{\top})A^{-\top}).$

Finally, it is worth noting that
$\|\eta_{t}^{j}\|^{2}_{\Omega_{t|t}^{j}}\leq\alpha$, indeed if $c_{t}^{j}=0$,
i.e. there is not transmission from node $j$, then
$\|\eta_{t}^{j}\|^{2}_{\Omega_{t|t}^{j}}=\|x_{t|t}^{j}-\bar{x}_{t}^{j}\|^{2}_{\Omega_{t|t}^{j}}\leq\alpha$
by the transmission rule in (8); otherwise, if $c_{t}^{i}=1$ then
$\|\eta_{t}^{j}\|^{2}_{\Omega_{t|t}^{j}}=0\leq\alpha$.

###### Theorem IV.1

Assume that the hypotheses A1-A4 hold. Then, the estimation error $e_{t}^{i}$,
with $i\in\mathcal{N}$, is uniformly bounded in mean-square as
$t\rightarrow\infty$, i.e.

$\displaystyle\lim_{t\rightarrow\infty}\mathbb{E}[\|e_{t}^{i}\|^{2}]\leq\left(\frac{\sqrt{2}\gamma}{1-\gamma}\frac{\sum_{i\in\mathcal{N}}\sqrt{p_{i}}(\sqrt{\overline{\omega}}\rho_{\xi}+\sqrt{\alpha})}{\sqrt{\underline{\omega}}\min_{i\in\mathcal{N}}\sqrt{p_{i}}}\right)^{2}$

where $\underline{\omega},\overline{\omega},\alpha,\gamma$ are the constants
given in Proposition IV.2 and Proposition IV.3.

###### Proof:

Using the same reasonings in the first part of the proof of Theorem 1 in [15],
it is not difficult to prove that

$\displaystyle\sqrt{\mathbb{E}[\mathcal{V}_{t+1}(e_{t+1})]}\leq\gamma\left(\sqrt{\mathbb{E}[\mathcal{V}_{t}(e_{t})]}+\sqrt{\mathbb{E}[\mathcal{V}_{t}(\omega_{t})]}\right)$
(62)
$\displaystyle\sqrt{\mathcal{V}_{t}(\omega_{t})}\leq\sum_{i\in\mathcal{N}}\sqrt{p_{i}}\sqrt{\mathbb{E}[\|\omega_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}}$
(63)

where $0<\gamma<1$ is the constant given by Proposition IV.3,
$\omega_{t}:=\mathrm{col}(\omega_{t}^{i};i\in\mathcal{N})$,

$\displaystyle\omega_{t}^{i}:=\left\\{\begin{array}[]{cc}\bar{\omega}_{t}^{i},&\hbox{
if }e_{t}^{i}=0\\\
\|\bar{\omega}_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}/\|e_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}e_{t}^{i},&\hbox{otherwise}\end{array}\right.$
(66)

and $\bar{\omega}_{t}^{i}$ is the vector in the set
$\\{\xi_{t}^{i}\\}\cup\\{\xi_{t}^{j}+\eta_{t}^{i},\,j\in\overline{\mathcal{N}}_{i}\\}$
maximizing the weighted norm $\|\cdot\|_{\Omega_{t|t}^{i}}$. Notice that

$\displaystyle
0\leq\mathbb{E}[\|\xi_{t}^{j}-\eta_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}]=\mathbb{E}[\|\xi_{t}^{j}\|^{2}_{\Omega_{t|t}^{i}}+\|\eta_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}-2(\xi_{t}^{j})^{\top}\Omega_{t|t}^{i}\eta_{t}^{i}]$

and thus

$\displaystyle\mathbb{E}[2(\xi_{t}^{j})^{\top}\Omega_{t|t}^{i}\eta_{t}^{i}]\leq\mathbb{E}[\|\xi_{t}^{j}\|^{2}_{\Omega_{t|t}^{i}}+\|\eta_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}].$

Moreover,

$\displaystyle\mathbb{E}[\|\omega_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}]=\mathbb{E}[\|\bar{\omega}_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}]$
$\displaystyle\leq\max\\{\mathbb{E}[\|\xi_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}],\max_{j\in\mathcal{N}}\mathbb{E}[\|\xi_{t}^{j}\|^{2}_{\Omega_{t|t}^{i}}+\|\eta_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}+2(\xi_{t}^{j})^{\top}\Omega_{t|t}^{i}\eta_{t}^{i}]\\}$
$\displaystyle\leq\max\\{\mathbb{E}[\|\xi_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}],\max_{j\in\mathcal{N}}2\,\mathbb{E}[\|\xi_{t}^{j}\|^{2}_{\Omega_{t|t}^{i}}+\|\eta_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}]\\}$
$\displaystyle\leq
2\,\max_{j\in\mathcal{N}}\mathbb{E}[\|\xi_{t}^{j}\|^{2}_{\Omega_{t|t}^{i}}+\|\eta_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}].$

Taking into account (63), we obtain

$\displaystyle\sqrt{\mathcal{V}_{t}(\omega_{t})}$
$\displaystyle\leq\sqrt{2}\sum_{i\in\mathcal{N}}\sqrt{p_{i}}\max_{j\in\mathcal{N}}\sqrt{\mathbb{E}[\|\xi_{t}^{j}\|^{2}_{\Omega_{t|t}^{i}}+\|\eta_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}]}$
$\displaystyle\leq\sqrt{2}\sum_{i\in\mathcal{N}}\sqrt{p_{i}}\left(\max_{j\in\mathcal{N}}\sqrt{\mathbb{E}[\|\xi_{t}^{j}\|^{2}_{\Omega_{t|t}^{i}}}+\sqrt{\mathbb{E}[\|\eta_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}]}\right)$
$\displaystyle\leq\sqrt{2}\sum_{i\in\mathcal{N}}\sqrt{p_{i}}\left(\sqrt{\overline{\omega}}\rho_{\xi}+\sqrt{\alpha}\right)$

where we exploited the upper bounds for
$\|\xi_{t}^{i}\|^{2}_{\Omega_{t|t}^{j}}$ and
$\|\eta_{t}^{i}\|^{2}_{\Omega_{t|t}^{i}}$. Taking into account (62) we obtain

$\displaystyle\sqrt{\mathbb{E}[\mathcal{V}_{t+1}(e_{t+1})]}$
$\displaystyle\hskip
22.76228pt\leq\gamma\sqrt{\mathbb{E}[\mathcal{V}_{t}(e_{t})]}+\sqrt{2}\gamma\sum_{i\in\mathcal{N}}\sqrt{p_{i}}\left(\sqrt{\overline{\omega}}\rho_{\xi}+\sqrt{\alpha}\right)$

which implies

$\displaystyle\underset{t\rightarrow\infty}{\mathrm{lim\,\sup}}\,\sqrt{\mathbb{E}[\mathcal{V}_{t}(e_{t})]}\leq\frac{\sqrt{2}\gamma}{1-\gamma}\sum_{i\in\mathcal{N}}\sqrt{p_{i}}\left(\sqrt{\overline{\omega}}\rho_{\xi}+\sqrt{\alpha}\right).$

Finally, recalling that $\Omega_{t|t}^{i}\geq\underline{\omega}I$ by
Proposition IV.2, it is sufficient to note that

$\displaystyle\mathbb{E}[\|e_{t}\|^{2}]\leq\frac{\mathbb{E}[\mathcal{V}_{t}(e_{t})]}{\underline{\omega}\min_{i\in\mathcal{N}}p_{i}}$

to obtain the claim. ∎

##  V A robust strategy with local tolerances

In Section III we have proposed a distributed strategy with event-triggered
communication which provides a solution to the minimax problem in (5) which is
suboptimal. Indeed, at each sensor node the state prediction in (15) is the
solution to the (local) minimax problem

$\displaystyle
x_{t+1|t}^{i}=\underset{g_{t}\in\mathcal{G}_{i}}{\operatorname*{arg\,min}}\underset{\tilde{\phi}_{t}^{i}\in\mathcal{B}_{t}^{i}}{\max}\,\tilde{\mathbb{E}}_{i}[\|x_{t+1}-g_{t}(y_{t}^{i})\|^{2}|Y_{t-1}]$
(67)

where

$\displaystyle\tilde{\mathbb{E}}_{i}$
$\displaystyle\left[\|x_{t+1}-g_{t}(y_{t}^{i})\|^{2}|Y_{t-1}\right]:=$
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n+p_{i}}}\|x_{t+1}-g_{t}(y_{t})\|^{2}\tilde{\phi}_{t}^{i}(z_{t}^{i}|x_{t})\check{f}_{t}(x_{t}|Y_{t-1})dz_{t}^{i}dx_{t}$

with $z_{t}^{i}:=[\,x_{t+1}^{\top}\;(y_{t}^{i})^{\top}\,]^{\top}$;
$\mathcal{B}_{t}^{i}$ is the ambiguity set about the local and nominal model
in (1)-(2), whose nominal density is denoted by $\phi_{t}^{i}$, with tolerance
$b$; $\mathcal{G}_{i}$ is the set of estimators having finite second order
moments for any $\tilde{\phi}_{t}^{i}\in\mathcal{B}_{t}^{i}$. Thus, the local
least favorable transition probability density $\tilde{\phi}_{t}^{i}$ solution
to (67) does not necessarily agree with the global least favorable density
$\tilde{\phi}_{t}$ obtained solving the centralized problem in (5).

Let

$\displaystyle\bar{f}_{t}(z_{t}|Y_{t-1}):=$
$\displaystyle\int_{\mathbb{R}^{n}}\phi_{t}(z_{t}|x_{t})\check{f}_{t}(x_{t}|Y_{t-1})\mathrm{d}x_{t}$
$\displaystyle\tilde{f}_{t}(z_{t}|Y_{t-1}):=$
$\displaystyle\int_{\mathbb{R}^{n}}\tilde{\phi}_{t}(z_{t}|x_{t})\check{f}_{t}(x_{t}|Y_{t-1})\mathrm{d}x_{t}$

be the pseudo-nominal and the least favorable conditional probability
densities of $z_{t}$ given $Y_{t-1}$, respectively. In a similar way we define
the pseudo-nominal and the least favorable conditional probability densities
of $z_{t}^{i}$ given $Y_{t-1}$, respectively, as

$\displaystyle\bar{f}_{t}^{i}(z_{t}|Y_{t-1}):=$
$\displaystyle\int_{\mathbb{R}^{n}}\phi_{t}^{i}(z_{t}|x_{t})\check{f}_{t}(x_{t}|Y_{t-1})\mathrm{d}x_{t}$
$\displaystyle\tilde{f}_{t}^{i}(z_{t}|Y_{t-1}):=$
$\displaystyle\int_{\mathbb{R}^{n}}\tilde{\phi}_{t}^{i}(z_{t}|x_{t})\check{f}_{t}(x_{t}|Y_{t-1})\mathrm{d}x_{t}.$

In [18] it has been shown that the least favorable density
$\tilde{\phi}^{i}_{t}$ solution to (67) is such that the Kullback-Leibler
divergence between $\bar{f}_{t}^{i}$ and $\tilde{f}_{t}^{i}$, i.e.

$\displaystyle
D_{KL}(\tilde{f}_{t}^{i},\bar{f}_{t}^{i}):=\int_{\mathbb{R}^{n+p_{i}}}\tilde{f}_{t}^{i}(z_{t}^{i}|Y_{t-1})\log\left(\frac{\tilde{f}_{t}^{i}(z_{t}^{i}|Y_{t-1})}{\bar{f}_{t}^{i}(z_{t}^{i}|Y_{t-1})}\right)\mathrm{d}z_{t}^{i}$

is equal to the tolerance of the ambiguity set $\mathcal{B}_{t}^{i}$ used in
(67).

Drawing inspiration from [22], it is possible to mitigate the fact that the
local least favorable density $\tilde{\phi}_{t}^{i}$ is different from the one
obtained from the global least favorable density $\tilde{\phi}_{t}$ by
considering the ambiguity set $\mathcal{B}_{t}^{i}$ about the nominal density
$\phi^{i}_{t}$ and with “local” tolerance

$\displaystyle b_{t}^{i}$
$\displaystyle=D_{KL}(\tilde{f}^{i,g}_{t},\bar{f}^{i}_{t}):=$
$\displaystyle\int_{\mathbb{R}^{n+p_{i}}}\tilde{f}_{t}^{i,g}(z_{t}^{i}|Y_{t-1})\log\left(\frac{\tilde{f}_{t}^{i,g}(z_{t}^{i}|Y_{t-1})}{\bar{f}_{t}^{i}(z_{t}^{i}|Y_{t-1})}\right)\mathrm{d}z_{t}^{i}$
(68)

where $\tilde{f}_{t}^{i,g}$ is the least favorable density at the sensor node
$i$ obtained by marginalizing $\tilde{f}_{t}$ with respect to $y_{t}^{l}$ with
$l\neq i$. Although such a choice does not guarantee that
$\tilde{\phi}_{t}^{i}$, i.e. the least favorable density solution to (67) with
tolerance (V), leads to a density $\tilde{f}_{t}^{i}$ which coincides with
$\tilde{f}_{t}^{i,g}$, at least it is ensured that

$\displaystyle
D_{KL}(\tilde{f}^{i}_{t},\bar{f}_{t}^{i})=D_{KL}(\tilde{f}_{t}^{i,g},\bar{f}_{t}^{i}).$

In [30] it has been shown that if $b>0$ in (5) is taken sufficiently small,
then the least favorable model $\tilde{\phi}_{t}$ solution to (5) converges to
a stationary Gaussian model as $t\rightarrow\infty$. Moreover, the conditional
mean of $z_{t}$ given $Y_{t-1}$ under $\tilde{f}_{t}$ is the same of the one
under $\bar{f}_{t}$, see [22, Theorem 1]. Let $K$ and $\tilde{K}$ denote the
asymptotic covariance matrices of $z_{t}$ given $Y_{t-1}$ under the Gaussian
densities $\bar{f}_{t}$ and $\tilde{f}_{t}$, respectively. Then,
$b_{t}^{i}\rightarrow b^{i}$ as $t\rightarrow\infty$ where

$\displaystyle
b^{i}=\frac{1}{2}\left[\log\det(K_{{i}}\tilde{K}_{{i}}^{-1})+\operatorname{tr}(\tilde{K}_{{i}}K^{-1}_{{i}})-(n+p_{i})\right];$

$K_{i}$ is the matrix obtained from $K$ by deleting the rows and the columns
corresponding to $y_{t}^{l}$ with $l\neq i$; $\tilde{K}_{i}$ is obtained from
$\tilde{K}$ likewise.

In conclusion, we propose the following distributed filtering strategy with
event-triggered communication. The central unit can compute offline the
asymptotic tolerance $b^{i}$ for each sensor node $i\in\mathcal{S}$ from the
global least favorable density $\tilde{\phi}_{t}$ and send it to the
corresponding sensor node. After this offline step, the remaining part of the
algorithm is as Algorithm 1 with the exception of the second equation in (15)
and (21) which become, respectively,

Find $\displaystyle\theta_{t}^{i}>0\text{ s.t.
}\gamma(\Omega_{t+1|t}^{i},\theta_{t}^{i})=b^{i}$ Find
$\displaystyle\bar{\theta}_{t}^{i}>0\text{ s.t.
}\gamma(\bar{\Omega}_{t+1}^{i},\bar{\theta}_{t}^{i})=b^{i}.$

In regard to the stability properties of this algorithm with local tolerances,
it is not difficult to see that under Assumptions A1-A4

$\displaystyle\lim_{t\rightarrow\infty}\mathbb{E}[\|e_{t}^{i}\|^{2}]\leq\left(\frac{\sqrt{2}\gamma}{1-\gamma}\frac{\sum_{i\in\mathcal{N}}\sqrt{p_{i}}(\sqrt{\overline{\omega}}\rho_{\xi}+\sqrt{\alpha})}{\sqrt{\underline{\omega}}\min_{i\in\mathcal{N}}\sqrt{p_{i}}}\right)^{2}$

for any $i\in\mathcal{N}$. The proof follows the same ideas exploited in
Section IV. The unique difference is the derivation of the constant
$\underline{\omega}$. More precisely, in the proof of Proposition IV.2 the
constant $\mu$ is derived as follows. Since each sensor node $i$ has its own
tolerance $b^{i}$, then we have that
$\gamma(\Omega^{i}_{t+1|t},\theta_{t}^{i})=b^{i}$. By Lemma .1 in Appendix, it
follows that there exists $\mu_{i}>0$ such that
$\Omega_{t|t}^{i}\geq\mu_{i}\Omega_{t|t-1}^{i}$ $\forall\,i\in\mathcal{S}$.
Then, $\Omega_{t|t}^{i}\geq\mu\Omega_{t|t-1}^{i}$ $\forall\,i\in\mathcal{S}$
with $\mu:=\min_{i\in\mathcal{S}}\mu_{i}>0$.

##  VI Simulations

In this section we evaluate the performance of the proposed robust distributed
Kalman algorithms with event-triggered communication. We consider the problem
of tracking the position of a target by using noisy position measurements
obtained by a network of $N=100$ nodes where $20$ of them are sensor nodes.
The possible connections among the nodes has been randomly generated in such a
way the network is strongly connected. Moreover, the percentage of node
connections is equal to $4\%$. The model for the motion of the target is

$\displaystyle\dot{x}^{c}_{s}=\Phi x^{c}_{s}+\dot{w}^{c}_{s},\quad
s\in\mathbb{R}$ (69)

where

$\displaystyle\Phi=\left[\begin{array}[]{cc}0&0\\\
\mathbf{I}_{3}&0\end{array}\right],$ (72)

$x^{c}_{s}=[\,v_{x,s}\;v_{y,s}\;v_{z,s}\;p_{x,s}\;p_{y,s}\;p_{z,s}\;\,]^{\top}$
with $v$ denoting the velocity, $p$ the position and the subscripts $x,y,z$
denoting the three spatial dimensions; $w_{s}^{c}$ is a Wiener process with
zero mean and rate of variance equal to 0.1. We discretize (69) with sampling
time equal to $0.1$. The corresponding discrete time model is
$x_{t+1}=Ax_{t}+Bw_{t}$ where $x_{t}$ is the sampled version of $x^{c}_{s}$,
$A=\mathbf{I}_{6}+0.1\Phi$, $B=\sqrt{0.001}I$, $w_{t}$ is a zero-mean
normalized Gaussian white noise and thus $Bw_{t}$ is the sampled version of
$w_{s}^{c}$. We assume that every sensor measures the position of the target
in either two horizontal dimensions, or a combination of one horizontal
dimension and the vertical dimension; in plain words, one sensor does not have
measurements in all the three dimensions. Therefore, we obtain the nominal
discrete state-space model (1)-(2) where
$C^{i}=[\,0\;0\;0\;\mathrm{diag(1,1,0)}\,]$, in the case that the sensor
measures only the horizontal positions, or
$C^{i}=[\,0\;0\;0\;\mathrm{diag(1,0,1)}\,]$,
$C^{i}=[\,0\;0\;0\;\mathrm{diag(0,1,1)}\,]$, in the case that the sensor
measures one horizontal position and the vertical position. Moreover,
$R^{i}=D^{i}(D^{i})^{\top}=\sqrt{k}PR_{0}P^{\top}$ where
$R_{0}=0.5\cdot\mathrm{diag}(1,4,7)$ and $P$ is a permutation matrix randomly
chosen for every node. Finally, the initial state $x_{0}$ is a Gaussian random
vector with zero-mean and covariance matrix $V_{0|-1}=I$. Since the previous
model is just an idealization of the underlying physical system, we assume
that the actual state-space model belongs to the ambiguity set
$\mathcal{B}_{t}$ about the aforementioned nominal model and with tolerance
$b=0.05$. More precisely, we assume that the actual model is the least
favorable model solution to (i.e. the maximizer of) the centralized problem in
(5).

In the following we consider the distributed algorithms:

  * •

RDKF – the distributed robust Kalman filter with event-triggered communication
in Algorithm 1 with $b=0.05$. Here, the transmission rule (8) is with
$\alpha=10$, $\beta=0.2$ and $\delta=0.5$;

  * •

RDKFLOC – the distributed robust Kalman filter with event-triggered
communication of Section V where each sensor nodes $i$ has its own tolerance
$b^{i}$ computed from the global least favorable model; the transmission rule
(8) is with $\alpha=10$, $\beta=0.2$ and $\delta=0.5$;

  * •

DKF1 – the distributed Kalman filter with event-triggered communication
proposed in [15] and the transmission rule is with $\alpha=10$, $\beta=0.2$
and $\delta=0.5$;

  * •

DKF2 – is the same as DKF1 but the transmission rule is with $\alpha=0.01$,
$\beta=0.2$ and $\delta=0.5$.

It is worth noting that RDKF, RDKFLOC and DKF1 have the same parameters for
the transmission rule. As we will see later, taking the parameter $\alpha$ in
(8) the same for RDKF, RDKFLOC and DKF1 provides a transmission rate for DKF1
which is smaller than the robust versions. For this reason, we also consider
DKF2 where the parameter $\alpha$ has been decreased in oder to increase the
transmission rate. We also tried to increase the transmission rate by keeping
fixed $\alpha=10$ and changing $\beta,\delta$; however, we did not notice a
significant growth in terms of transmission rate.

Figure 1: Average RMSE across the network. Figure 2: Average RMSE at each node
over the time horizon $[1,250]$. Figure 3: Average transmission rate across
the network. Figure 4: Tolerance used in RDKF (blue line) versus the local
tolerances at each sensors node used in RDKFLOC (purple bars). Figure 5:
Average risk sensitivity parameters across the communication nodes of the
network (in logarithmic scale). Figure 6: Average risk sensitivity parameter
across the sensor nodes of the network (in logarithmic scale).

We consider a Monte Carlo study which is composed by 500 independent runs
(which correspond to different target paths generated by the least favorable
model 111For more details on how to generate a realization from the least
favorable model see [18, Section V].) over a time horizon of 250 seconds. For
each run, we estimate the state using the aforementioned distributed
algorithms and for each of them we consider the following performance indexes:

  * •

The average root mean square error across the network at time $t$:

$\mathrm{RMSE}_{t}=\frac{1}{N}\sum_{i=1}^{N}\|x_{t}-x_{t|t}^{i}\|^{2}.$

  * •

The average root mean square error at node $i$ over the time horizon:

$\mathrm{RMSE}_{i}=\frac{1}{250}\sum_{t=1}^{250}\|x_{t}-x_{t|t}^{i}\|^{2}.$

  * •

The transmission rate across the network at time $t$, i.e. the faction of
nodes that transmit their data at time $t$.

Figure 1 and Figure 2 show the two aforementioned root mean square errors
averaged over the runs, while Figure 3 shows the corresponding averaged
transmission rate. As we can see the robust filters outperform DKF1 and DKF2.
In particular, even in the case we increase the transmission rate, i.e. as in
DKF2, RDKF and RDKFLOC outperform the standard algorithm. Clearly, DKF2
outperforms DKF1 in the steady state because the latter is penalized by the
low transmission rate across the network. Finally, RDKFLOC is slightly better
than RDKF: it requires a lower transmission rate and it exhibits a better RMSE
in the nodes characterized by a large estimation error, see Figure 2. This is
due by the fact that the local filters of the sensor nodes in RDKF are too
conservative, indeed their tolerances are larger than the ones in RDKFLOC, see
Figure 4.

Let the average risk sensitivity parameters across the communication nodes of
RDKF be defined as:

$\displaystyle\theta_{t}^{c}$
$\displaystyle:=\frac{1}{|\mathcal{C}|}\sum_{i\in\mathcal{C}}\theta_{t}^{i},\quad\bar{\theta}_{t}^{c}:=\frac{1}{|\mathcal{C}|}\sum_{i\in\mathcal{C}}\bar{\theta}_{t}^{i}$

where $\mathcal{C}=\mathcal{N}\setminus\mathcal{S}$ and $|\mathcal{C}|$
denotes the cardinality of set $\mathcal{C}$. The average risk sensitivity
parameters across the communication nodes in RDKFLOC, denoted by
$\theta_{t}^{LOC,c}$ and $\bar{\theta}_{t}^{LOC,c}$, are defined likewise.
Figures 5 shows the aforementioned quantities averaged over the Monte Carlo
runs. We can notice that $\theta_{t}^{c}\geq\bar{\theta}_{t}^{c}$ and
$\theta_{t}^{LOC,c}\geq\bar{\theta}_{t}^{LOC,c}$, moreover we have checked
that $\theta_{t}^{i}\geq\bar{\theta}_{t}^{i}$ for many communication nodes
both in RDKF and RDKFLOC. Since the mapping
$\theta\mapsto\gamma(\Omega,\theta)$ is monotone increasing, the mapping
$\Omega\mapsto\gamma(\Omega,\theta)$ is monotone decreasing according to the
partial order of positive definite matrices, see [32, 33], and in view of the
fact that

$\gamma(\Omega_{t+1|t}^{i},\theta_{t}^{i})=\gamma(\bar{\Omega}^{i}_{t+1},\bar{\theta}_{t}^{i}),$

it follows that
$\Omega_{t+1|t}^{i}\geq\bar{\Omega}^{i}_{t+1}\geq\bar{\Psi}^{i}_{t+1}$ for
many nodes both in RDKF and RDKFLOC. The latter inequality means that the
transmission from node $i$ typically produces an increase of information in
the corresponding out-neighbors nodes, which is the expected scenario. Notice
that $\theta_{t}^{c}\geq\theta_{t}^{LOC,c}$ and
$\bar{\theta}_{t}^{c}\geq\bar{\theta}_{t}^{LOC,c}$ which is just a consequence
of the fact that RDKF is more conservative than RDKFLOC; indeed, recall that
$b^{i}\leq b$ $\forall\,i\in\mathcal{S}$ see Figure 4.

Let the average risk sensitivity parameters across the sensor nodes of RDKF be
defined as:

$\displaystyle\theta_{t}^{s}$
$\displaystyle:=\frac{1}{|\mathcal{S}|}\sum_{i\in\mathcal{S}}\theta_{t}^{i},\quad\bar{\theta}_{t}^{s}:=\frac{1}{|\mathcal{S}|}\sum_{i\in\mathcal{S}}\bar{\theta}_{t}^{i}.$

The average risk sensitivity parameters across the sensor nodes in RDKFLOC,
denoted by $\theta_{t}^{LOC,s}$ and $\bar{\theta}_{t}^{LOC,s}$, are defined
likewise. Figure 6 shows the aforementioned quantities averaged over the Monte
Carlo runs. The observations done before nodes hold also in this case.

##  VII Conclusion

In this paper we have considered the problem to estimate the state over a
sensor network under model uncertainty and communication constraints. We have
proposed two robust distributed strategies with event-triggered communication.
More precisely, the sensor nodes compute their state estimate by solving a
minimax game: one player (i.e. the estimator) aims to minimize the estimation
error, while the other player selects the model in the ambiguity set which
maximizes such error. The communication among nodes is governed by a data-
driven rule which essentially allows the data transmission only in the case
the latter provides a substantial increase of information in the nodes
receiving it. The difference between the two distributed strategies is the way
the ambiguity sets are formed at each sensor node. A stability analysis of the
algorithms has been carried out showing that it is guaranteed mean-square
boundedness of the state estimation error in all the nodes, under the global
least favorable model, provided that the network is strongly connected, the
system collectively observable and the tolerance of the ambiguity set
corresponding to the global model is sufficiently small. Finally, a numerical
experiment showed that the proposed strategies are effective in the case there
is model uncertainty.

###### Lemma .1

Let $\Omega>0$ and $b,\theta>0$ such that $\gamma(\Omega,\theta)=b$. Then,
there exists a constant $\mu>0$ such that

$\displaystyle\Omega-\theta I\geq\mu\Omega.$

###### Proof:

The constraint $\gamma(\Omega,\theta)=b$ can be written as

$\displaystyle\operatorname{tr}(X)-\log\det(X)-n=2b$ (73)

where $X:=(I-\theta\Omega^{-1})^{-1}>0$. Let $\lambda_{k}>0$ denote the $k$-th
eigenvalue of $X$, then (73) can be written as

$\displaystyle\sum_{k=1}^{n}\lambda_{k}-\log\lambda_{k}-1=2b$ (74)

and the terms $\lambda_{k}-\log\lambda_{k}-1$ in the summation are
nonnegative. Hence, condition (74) implies that

$\displaystyle f(\lambda_{k}):=\lambda_{k}-\log\lambda_{k}-1\leq 2b,\quad
k=1\ldots n.$ (75)

Notice that $f$ is continuous for $\lambda>0$ and it is not difficult to see
that

$\displaystyle\underset{\lambda\rightarrow
0^{+}}{\lim}f(\lambda)=\infty,\quad\underset{\lambda\rightarrow\infty}{\lim}f(\lambda)=\infty,\quad\underset{\lambda>0}{\mathrm{argmin}}f(\lambda)=1.$

Accordingly, there exists $\overline{\lambda}>1$, which only depends on $b$,
such that condition (75) is satisfied for
$1\leq\lambda_{k}\leq\overline{\lambda}$, see Figure 7.

Figure 7: Pictorial description of $\overline{\lambda}$.

Accordingly, the constraint in (73) implies that

$\displaystyle(I-\theta\Omega^{-1})^{-1}=X\leq\overline{\lambda}I$
$\displaystyle I-\theta\Omega^{-1}\geq\overline{\lambda}^{-1}I$
$\displaystyle\Omega-\theta I\geq\overline{\lambda}^{-1}\Omega$

and thus $\mu=\overline{\lambda}^{-1}$. ∎

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