# Remote State Estimation of Multiple Systems over Multiple Markov Fading
Channels

Wanchun Liu  Daniel E. Quevedo  Karl Henrik Johansson  Branka Vucetic  and
Yonghui Li

###### Abstract

††W. Liu is with School of Electrical and Information Engineering, The
University of Sydney, Australia. Emails: {wanchun.liu, yonghui.li,
<EMAIL_ADDRESS>D. E. Quevedo is with the School of Electrical
Engineering and Robotics, Queensland University of Technology (QUT), Brisbane,
Australia. Email<EMAIL_ADDRESS>K. H. Johansson is with School of
Electrical Engineering and Computer Science, KTH Royal Institute of
Technology, Stockholm, Sweden. Email<EMAIL_ADDRESS>B. Vucetic and Y. Li are
with School of Electrical and Information Engineering, The University of
Sydney, Australia. Emails: {branka.vucetic<EMAIL_ADDRESS>
We consider remote state estimation of multiple discrete-time linear time-
invariant (LTI) systems over multiple wireless time-varying communication
channels. Each system state is measured by a sensor, and the measurements from
sensors are sent to a remote estimator over the shared wireless channels in a
scheduled manner. We answer the following open problem: what is the
fundamental requirement on the multi-sensor-multi-channel system to guarantee
the existence of a sensor scheduling policy that can stabilize the remote
estimation system? To tackle the problem, we propose a novel policy
construction method, and develop a new analytical approach by applying the
asymptotic theory of spectral radii of products of non-negative matrices. A
necessary and sufficient stability condition is derived in terms of the LTI
system parameters and the channel statistics, which is more effective than
existing sufficient conditions available in the literature. Explicit
scheduling policies with stability guarantees are presented as well. We
further extend the analytical framework to cover remote estimation with four
alternative network setups and obtain corresponding necessary and sufficient
stability conditions.

###### Index Terms:

Estimation, Kalman filtering, linear systems, stability, mean-square error,
Markov fading channel

##  I Introduction

###  I-A Motivation

Industry 4.0, also known as the Fourth Industrial Revolution, is the
automation of traditional manufacturing and industrial processes through
customized and flexible mass production [1]. Replacing the communication
cables with wireless devices in conventional factories will be a game-changer:
In particular, for automatic control, Industry 4.0 will make use of large-
scale, interconnected deployment of massive spatially distributed industrial
devices such as sensors, actuators, and controllers. Given their low-cost and
scalable deployment, wireless remote state estimation from ubiquitous sensors
will play a key role in many industrial control applications, such as smart
manufacturing, industrial automation, e-commerce warehouses, and smart grids
[2].

However, unlike wired communications, wireless communications are often
unreliable and have a limited spectrum for transmission [3]. Consequently,
when wireless sensors are deployed in a remote estimation system, scheduling
policies need to be designed to allow sensors to update the measurement data
over a limited number of frequency channels. The design of such transmission
schedules is especially challenging since, due to variability of the
environment, wireless channels are time-varying. The transition process of
associated channel fading states are commonly modeled as a Markov processes
[4, 5, 6], wherein different channel states lead to different packet drop
probabilities. Due to transmission scheduling and packet dropouts, a multi-
sensor remote estimator cannot correctly receive all the sensor measurements.
This degrades the estimation performance and can even lead to instability,
i.e., the expected estimation error covariances may become unbounded.

In this paper, we tackle the fundamental problem: what are necessary and
sufficient conditions on system parameters that ensure stochastic stability of
a multi-sensor remote estimator under multiple Markov fading channels?

###  I-B Related Works

Existing work related to multi-system remote estimation and control can be
divided into two categories: perfect (wired) and imperfect (wireless)
communication channels.

Perfect communication channels. Early research in stability analysis of multi-
control-loop transmission scheduling over single and multiple independent
communication channels involved periodic and aperiodic scheduling policies [7,
8, 9, 10, 11], assuming perfect communication channels. The scheduling of
sensor-controller transmissions of multiple control loops over shared
communication channels was considered in [7, 8]. In [7], a sufficient
condition was derived to guarantee the existence of a periodic scheduling
policy that can stabilize a number of control loops; in [8], a periodic
scheduling policy was investigated that achieves optimal control performance.
Transmission scheduling of a multi-sensor-multi-actuator system was considered
in [9], where sensors and actuators of a plant exchange information with a
remote controller via a shared communication medium. A sufficient condition
was derived to guarantee the existence of a periodic scheduling policy that
can stabilize the system. Different from [7, 8, 9], a dynamic scheduler of a
multi-plant system was proposed in [10], where the sensor-controller and
controller-actuator communications shared a single channel. The scheduler does
not rely on a predefined sequence but decides access to the communication
channel dynamically at each time-step. The decision depends on the current
plant states, namely, the system with the largest state error has a higher
priority to be scheduled. Optimal remote estimation for multi-hop network
scheduling was considered in [11].

Imperfect communication channels. In practice, wireless channels are not
error-free, leading to transmission errors and packet dropouts. In industrial
control environments comprising moving machines and mobile robots, the channel
quality is time-varying [12]. Unlike single systems for which remote
estimation and control that have been well investigated (e.g., [13, 14] for
independent and identically distributed packet dropout scenarios and [15] for
Markovian packet dropout), multi-system remote estimation and control over
wireless channels have not drawn as much research attention, until recent
efforts motivated by standardization and growing deployment of wireless
technology.

Considering a wireless control architecture with multiple control loops over
shared wireless fading channels, optimal dynamic transmission scheduling
policies were investigated in [16, 17] with different objective functions. The
scheduling decision in each step depends on both the wireless channel and
control loop states. In [18] and [19], sensor transmission scheduling over
single and multiple packet drop channels of remote estimators were
investigated, respectively, and some sufficient mean-square stability
conditions in terms of the system parameters and optimal transmission
scheduling policies were analyzed. The follow-up work [20] considered a time-
correlated Markov fading channel scenario, and derived a sufficient condition
to guarantee the existence of a deterministic and stationary scheduling policy
that can stabilize the remote estimator. A deep reinforcement learning method
was proposed as well to find the optimal scheduling policy. In [21],
scheduling policy design of a spatially distributed large control system with
many sensors and actuators based on the wireless network standard IEEE
802.15.4 was investigated.

###  I-C Contributions

In this paper, we consider a multi-sensor remote estimator with multiple
frequency channels, where individual sensors measure different physical
processes. We allow the channels to be time-varying and correlated in both
frequency and time, as is common in practical applications [3]. We derive a
necessary and sufficient mean-square stability condition in terms of the
physical process parameters and the fading channel statistics. Our result
establishes that there exists at least one sensor scheduling policy over the
frequency channels that can stabilize the remote estimator if and only if the
condition holds.

It is worth emphasizing that the stability condition depends on the essential
parameters of the physical processes of interest and the communication
channels, rather than on the specific scheduling policy employed. In fact, to
find merely sufficient conditions for stability, one can construct a specific
policy and analyze its properties (e.g., [20, 22]). Such a sufficient
condition, however, is commonly not tight and thus cannot be proved to be
necessary. The analysis of necessary conditions is challenging as there is an
uncountable number of scheduling policies.

To the best of our knowledge, necessary and sufficient stability conditions
have not been established before for remote estimation or control over
multiple systems and wireless communication channels. In the present work, we
consider a general correlated channel, wherein the channel state in each
frequency channel is a random variable and correlated with the other channel
states. To tackle the challenge, we develop a novel policy construction method
and a stochastic estimation-cycle based analytical approach. Moreover, we
develop an asymptotic theory for the spectral radius of a product of non-
negative matrices to prove our key result. We also apply the analytical
framework to four alternative remote estimation architectures with different
network setups and derive necessary and sufficient stability conditions for
them.

###  I-D Outline and Notations

The remainder of this paper is organized as follows: Section II presents the
model of the multi-sensor-multi-channel remote estimator of interest. Section
III presents and proves the main result of the paper. Section IV extends the
main result to four alternative network setups. Section V draws conclusions.

_Notations:_ Sets are denoted by calligraphic capital letters, e.g.,
$\mathcal{A}$. $\mathcal{A}\backslash\mathcal{B}$ denotes set subtraction.
Matrices and vectors are denoted by capital and lowercase upright bold
letters, e.g., $\mathbf{A}$ and $\mathbf{a}$, respectively. $|\mathcal{A}|$
denotes the cardinality of the set $\mathcal{A}$. $\mathsf{E}\left[A\right]$
is the expectation of the random variable $A$. $(\cdot)^{\top}$ is the matrix
transpose operator. $\|\mathbf{v}\|_{1}$ is the sum of the vector
$\mathbf{v}$’s elements.
$|\mathbf{v}|\triangleq\sqrt{\mathbf{v}^{\top}\mathbf{v}}$ is the Euclidean
norm of a vector $\mathbf{v}$. $\text{Tr}(\cdot)$ is the trace operator.
$\text{diag}\\{\mathbf{v}\\}$ denotes the diagonal matrix with the diagonal
elements taken from $\mathbf{v}$. $\mathbb{N}$ and $\mathbb{N}_{0}$ denote the
sets of positive and non-negative integers, respectively. $\mathbb{R}^{m}$
denotes the $m$-dimensional Euclidean space. $\left[u\right]_{B\times B}$
denotes the $B\times B$ matrix with identical elements $u$.
$[\mathbf{A}]_{j,k}$ denotes the element at the $j$th row and $k$th column of
a matrix $\mathbf{A}$. $\\{v\\}_{\mathbb{N}_{0}}$ denotes the semi-infinite
sequence $\\{v_{0},v_{1},\dots\\}$. $\rho(\mathbf{A})$ is the spectral radius
of $\mathbf{A}$, i.e., the largest absolute value of its eigenvalues.
$\vartheta(\mathbf{A})$ is the sum of all the elements of $\mathbf{A}$.

##  II System Model

We consider a remote estimator with $N$ sensors measuring $N$ independent
physical process, as illustrated in Fig. 1. The index set of the sensors is
denoted as $\mathcal{N}\triangleq\\{1,2,\dots,N\\}$. The sensors are connected
with a local gateway, which collects the measurements and forwards them to a
remote estimator. Connections between sensors and the gateway are reliable and
not scheduled, while the gateway to remote estimator communications are
wireless and scheduled due to bandwidth limitations.

Figure 1: The multi-sensor-multi-channel remote estimator with a single
gateway. Processes, sensors, gateway and remote estimator are denoted as P$1$,
P$2$, …, P$N$, S$1$, S$2$, …, S$N$, G and RE, respectively.

The discrete-time linear time-invariant (LTI) model of the measurement of each
process $n$ is given as [14, 23, 24]

$\displaystyle\mathbf{x}_{n}{(t+1)}$
$\displaystyle=\mathbf{A}_{n}\mathbf{x}_{n}(t)+\mathbf{w}_{n}(t),$ (1)
$\displaystyle\mathbf{y}_{n}(t)$
$\displaystyle=\mathbf{C}_{n}\mathbf{x}_{n}(t)+\mathbf{v}_{n}(t),$

where $\mathbf{x}_{n}(t)\in\mathbb{R}^{l_{n}}$ is the process state vector,
$\mathbf{A}_{n}\in\mathbb{R}^{{l_{n}}\times{l_{n}}}$ the state transition
matrix, $\mathbf{y}_{n}(t)\in\mathbb{R}^{r_{n}}$ the measurement vector of the
sensor attached to the process,
$\mathbf{C}_{n}\in\mathbb{R}^{{r_{n}}\times{l_{n}}}$ the measurement matrix,
$\mathbf{w}_{n}(t)\in\mathbb{R}^{l_{n}}$ and
$\mathbf{v}_{n}(t)\in\mathbb{R}^{r_{n}}$ are the process and measurement noise
vectors, respectively. We assume $\mathbf{w}_{n}(t)$ and $\mathbf{v}_{n}(t)$
are independent and identically distributed (i.i.d.) zero-mean Gaussian
processes with corresponding covariance matrices $\mathbf{W}_{n}$ and
$\mathbf{V}_{n}$, respectively. Without loss of generality, we assume that
$\rho^{2}(\mathbf{A}_{1})\geq\rho^{2}(\mathbf{A}_{2})\geq\dots\geq\rho^{2}(\mathbf{A}_{N})$.

###  II-A Local Estimation

Each sensor adopts a local Kalman filter (KF) to estimate its process before
sending to the gateway [23, 24, 15]. We have

$\displaystyle\mathbf{x}^{s}_{n}({t|t-1})$
$\displaystyle=\mathbf{A}_{n}\mathbf{x}^{s}_{n}({t-1})$ (2a)
$\displaystyle\mathbf{P}^{s}_{n}({t|t-1})$
$\displaystyle=\mathbf{A}_{n}\mathbf{P}^{s}_{n}({t-1})\mathbf{A}^{\top}_{n}+\mathbf{W}_{n}$
(2b) $\displaystyle\mathbf{K}_{n}(t)$
$\displaystyle=\mathbf{P}^{s}_{n}({t|t-1})\mathbf{C}^{\top}_{n}(\mathbf{C}_{n}\mathbf{P}^{s}_{n}({t|t-1})\mathbf{C}^{\top}_{n}+\mathbf{V}_{n})^{-1}$
(2c) $\displaystyle\mathbf{x}^{s}_{n}(t)$
$\displaystyle=\mathbf{x}^{s}_{n}({t|t-1})+\mathbf{K}_{n}(t)(\mathbf{y}_{n}({t})-\mathbf{C}_{n}\mathbf{x}^{s}_{n}({t|t-1}))$
(2d) $\displaystyle\mathbf{P}^{s}_{n}(t)$
$\displaystyle=(\mathbf{I}_{n}-\mathbf{K}_{n}({t})\mathbf{C}_{n})\mathbf{P}^{s}_{n}({t|t-1})$
(2e)

where $\mathbf{I}_{n}$ is the $l_{n}\times l_{n}$ identity matrix,
$\mathbf{x}^{s}_{n}({t|t-1})$ is the prior state estimate,
$\mathbf{x}^{s}_{n}(t)$ is the posterior state estimate at time $t$,
$\mathbf{K}_{n}(t)$ is the Kalman gain. The matrices
$\mathbf{P}^{s}_{n}({t|t-1})$ and $\mathbf{P}^{s}_{n}(t)$ represent the prior
and posterior error covariance at the sensor at time $t$, respectively. The
first two equations above present the prediction steps while the last three
equations correspond to the updating steps. In particular,
$\mathbf{x}^{s}_{n}(t)$ is the sensor $n$’s estimate of $\mathbf{x}_{n}(t)$ at
time $t$, i.e., the pre-filtered measurement of $\mathbf{y}_{t}$, with the
estimation error covariance $\mathbf{P}^{s}_{n}(t)$ defined as:

$\mathbf{P}^{s}_{n}(t)\triangleq\mathsf{E}\left[(\mathbf{x}^{s}_{n}(t)-\mathbf{x}_{n}(t))(\mathbf{x}^{s}_{n}(t)-\mathbf{x}_{n}(t))^{\top}\right].$

We focus on the _remote_ estimation stability and assume that the _local_ KFs
are stable and operate in steady state [23, 24, 15], i.e.,
$\mathbf{P}^{s}_{n}(t)=\mathbf{\bar{P}}_{n},\forall
t\in\mathbb{N},n\in\mathcal{N}$.

###  II-B Markov Channel

We assume that there exist only $M<N$ frequency channels that can be used for
transmission of sensor data. The channels are correlated in both time and
frequency domains as detailed below.

The $M$-channel (vector) state $\mathbf{h}(t)$ is modeled as an aperiodic
Markov chain with $\bar{M}$ irreducible channel states,
$\mathcal{S}\triangleq\\{\mathbf{h}_{1},\mathbf{h}_{2},\dots,\mathbf{h}_{\bar{M}}\\}$,
where
$\mathbf{h}_{i}\triangleq[h_{i,1},h_{i,2},\dots,h_{i,M}]^{\top}\in\\{0,1\\}^{M}$.
Here, $h_{i,j}=0$ or $1$ means successful (on) or failed (off) transmission in
the $j$th frequency channel at the $i$th channel state. Let
$\mathbf{M}\in\mathbb{R}^{\bar{M}}\times\mathbb{R}^{\bar{M}}$ denote the state
transition probability matrix, where

$[\mathbf{M}]_{i,j}\triangleq\mathsf{Prob}\left[\mathbf{h}(t)=\mathbf{h}_{j}|\mathbf{h}(t-1)=\mathbf{h}_{i}\right]=p_{i,j}.$
(3)

Let $\tilde{\mathcal{S}_{i}}\in\mathcal{S}$ denote the set of channel state
with an ‘on’ state in the $i$th frequency channel, i.e.,

$\tilde{\mathcal{S}_{i}}\triangleq\\{\mathbf{h}_{j}:h_{j,i}=1,j\in\mathcal{\bar{M}}\\},i\in\mathcal{M},$
(4)

where $\mathcal{M}\triangleq\\{1,2,\dots,M\\}$ and
$\mathcal{\bar{M}}\triangleq\\{1,2,\dots,\bar{M}\\}$.

We make the following assumption on the availability of the channel state.

######  Assumption 1 (Known Previous Channel State [20]).

At time $t\in\mathbb{N}$, the current channel state is unknown but the
previous channel state $\mathbf{h}(t-1)$ is available.

###  II-C Transmission Scheduling and Remote Estimation

In each time slot, the gateway collects $N$ packets carrying the sensor
estimates
$\\{\mathbf{\hat{x}}^{s}_{1}(t),\dots,\mathbf{\hat{x}}^{s}_{N}(t)\\}$. It
schedules at most $M$ of the packets and sends them through $M$ frequency
channels to the remote estimator, as illustrated in Fig. 1. Each frequency
channel can transmit at most one packet. The unscheduled packets are
discarded. We make the following assumption on transmission redundancy.

######  Assumption 2 (Disabled Redundant Transmissions).

In each time slot, each packet can take at most one frequency channel for
transmission.

Due to the transmission scheduling and the fading channels, packets carrying
the estimated $N$ process states may or may not arrive at the remote
estimator. Let $\gamma_{n}(t)=1$ denote successful detection of sensor $n$’s
packet at time $t$, $n\in\mathcal{N}$. If $\gamma_{n}(t)=0$, the packet is not
scheduled or is scheduled but with failed detection. It is also assumed that
each packet transmission has a unit delay that is equal to the sampling period
of the system. The optimal remote estimator in the sense of minimum mean-
square error (MMSE) is obtained as [14]

$\hat{\mathbf{x}}_{n}(t)=\begin{cases}\mathbf{A}_{n}\hat{\mathbf{x}}_{n}(t-1),&\gamma_{n}(t-1)=0,\\\
\mathbf{A}_{n}\hat{\mathbf{x}}_{n}^{s}(t-1),&\gamma_{n}(t-1)=1,\end{cases}$
(5)

and can be simplified as [15]

$\hat{\mathbf{x}}_{n}(t)=\mathbf{A}^{\phi_{n}(t)}_{n}\hat{\mathbf{x}}^{s}_{n}(t-(\phi_{n}(t))),$
(6)

where $\phi_{n}(t)\in\mathbb{N}$ is the time duration between the previous
successful transmission and the current time $t$, and can be regarded as the
_age-of-information_ (AoI) [26].

From the above it follows that the estimation error covariance of process $n$
is given as

$\displaystyle\mathbf{P}_{n}(t)$
$\displaystyle\triangleq\mathsf{E}\left[(\hat{\mathbf{x}}_{n}(t)-\mathbf{x}_{n}(t))(\hat{\mathbf{x}}_{n}(t)-\mathbf{x}_{n}(t))^{\top}\right]$
(7) $\displaystyle=\zeta^{\phi_{n}(t)}(\bar{\mathbf{P}}_{n})$ (8)

where (8) is obtained by substituting (6) and (1) into (7) and

$\zeta_{n}(\mathbf{X})\triangleq\mathbf{A}_{n}\mathbf{X}\mathbf{A}_{n}^{\top}+\mathbf{W}_{n}$
(9)
$\zeta_{n}^{1}(\cdot)\triangleq\zeta_{n}(\cdot),\quad\zeta_{n}^{m+1}(\cdot)\triangleq\zeta_{n}(\zeta_{n}^{m}(\cdot)),\quad
m\geq 1.$

Thus, the quality of the remote estimation error of process $n$ in time slot
$t$ can be quantified via $\text{Tr}\left(\mathbf{P}_{n}(t)\right)$. For ease
of exposition, we introduce the following function

$c_{n}(i)\triangleq\text{Tr}\left(\zeta^{i}_{n}(\bar{\mathbf{P}}_{n})\right),\forall
i\in\mathbb{N}$ (10)

and note that, using (8), we can write

$\text{Tr}\left(\mathbf{P}_{n}(t)\right)\triangleq c_{n}(\phi_{n}(t)).$ (11)

Therefore, the estimation quality of process $n$ at time $t$ is a function of
its AoI state $\phi_{n}(t)$.

Propositions 1 and 2 of [15] allow us to state the following property of
$c_{n}(\cdot)$:

######  Lemma 1.

For any $\epsilon>0$, there exists $N^{\prime}>0$, $\kappa>0$ and $\eta>0$
such that

$c_{n}(i)<\kappa\left(\rho^{2}(\mathbf{A}_{n})+\epsilon\right)^{i},\forall
i>N^{\prime},$

and

$c_{n}(i)\geq\eta(\rho(\mathbf{A}_{n}))^{2i},\forall i>N^{\prime}.$

In this work, we solely focus on deterministic stationary scheduling policies.
Let $\nu_{n}(t)\in\\{0,1,\dots,M\\}$, $n\in\mathcal{N}$, denote the selected
frequency channel for process $n$ at time $t$. The sensor is not scheduled for
transmission if $\nu_{n}(t)=0$. Since available system states include the
current AoI states and the previous channel states, scheduling polices
$\pi(\cdot)$ can be written as

$\bm{\nu}(t)=\pi(\bm{\phi}(t),\mathbf{h}(t-1)),$ (12)

where $\bm{\phi}(t)\triangleq[\phi_{1}(t),\dots,\phi_{N}(t)]\in\mathbb{N}^{N}$
and $\bm{\nu}(t)=[\nu_{1}(t),\dots,\nu_{N}(t)]$.

Note that $\mathbf{P}_{n}(t)$ is a countable stochastic process taking values
from a countably infinite set

$\\{\zeta^{1}_{n}(\bar{\mathbf{P}}_{n}),\zeta^{2}_{n}(\bar{\mathbf{P}}_{n}),\dots\\}.$

If $\rho(\mathbf{A}_{n})\geq 1$, then this process will grow during periods of
consecutive packet dropouts. Since, due to fading, periods of consecutive
packet dropouts have unbounded support, at best one can hope for some type of
stochastic stability. Our focus is on mean-square stability.

######  Definition 1 (Average Mean-Square Stability).

The $N$-sensor remote estimator over $M$ frequency channels described above is
average mean-square stabilizable, if there exists a deterministic stationary
policy (12) such that the sum average estimation mean-square error (MSE) $J$
is bounded, where

$J\triangleq\sum_{n=1}^{N}J_{n}$ (13)

and

$J_{n}=\limsup_{T\to\infty}\frac{1}{T}\sum_{t=1}^{T}\mathsf{E}\left[\text{Tr}\left(\mathbf{P}_{n}(t)\right)\right],n\in\mathcal{N}.$
(14)

##  III Key Results

In this section, we present and prove the necessary and sufficient condition
for stabilizing the multi-sensor-multi-channel remote estimator in terms of
the multi-process parameters and the multi-channel statistics. As will become
apparent establishing such a necessary and sufficient stability condition is
highly non-trivial as we consider transmission scheduling for multiple sensor
packets over multiple fading channels that are correlated in both time and
frequency domains.

###  III-A Stability Condition

The physical process and channel parameters jointly determine the stability of
the overall remote estimator. Our result is stated in terms of the
$\bar{M}\times\bar{M}$ packet error probability matrix
$\mathbf{E}(\mathbf{v})$ obtained from the channel state transition
probability matrix $\mathbf{M}$:

$\displaystyle\left[\mathbf{E}(\mathbf{v})\right]_{i,j}$
$\displaystyle\triangleq\mathsf{Prob}\left[\mathbf{h}(t)=\mathbf{h}_{j},\mathbf{h}_{j}\notin\tilde{\mathcal{S}}_{v_{i}}|\mathbf{h}(t-1)=\mathbf{h}_{i}\right]$
(15)
$\displaystyle=p_{i,j}\mathbbm{1}(\mathbf{h}_{j}\notin\tilde{\mathcal{S}}_{v_{i}}).$

######  Theorem 1.

Consider $\rho_{\max}\triangleq\rho(\mathbf{A}_{1})$, and

$\lambda_{\infty}\triangleq\liminf\limits_{L\rightarrow\infty}\lambda_{L}=\min_{L\in\mathbb{N}}\lambda_{L},$
(16)
$\lambda_{L}\triangleq\min_{\mathbf{v}_{l}\in\mathcal{M}^{\bar{M}}}\rho\left(\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{E}(\mathbf{v}_{L})\right)^{\frac{1}{L}}$
(17)

where
$\mathbf{v}_{l}\triangleq[v_{l,1},v_{l,2},\dots,v_{l,\bar{M}}]^{\top}\in\mathcal{M}^{\bar{M}}$
is a vector of frequency channel selection at the $\bar{M}$ different channel
conditions.

The remote estimator described in Section II under Assumptions 1 and 2 is
stable if and only if

$\rho^{2}_{\max}\lambda_{\infty}<1.$ (18)

Theorem 1 shows that the stability depends on the spectral radius of the most
unstable process and a complex function of the channel state transition
probability matrix. Provided the condition is satisfied, one can always find a
scheduling policy that stabilizes the remote estimator; if (18) does not hold,
then no stabilizing scheduling policy exists. Theorem 1 does not provide
direct insights on the structure of a suitable scheduling policy. However, we
will construct a policy with stability guarantees in the proof of the
sufficiency part.

We note that $\lambda_{\infty}$ can be treated as an inverse quality indicator
of the parallel correlated channels: smaller values of $\lambda_{\infty}$
indicate a better channel quality. It is worth emphasizing that even though
mathematically $\lambda_{\infty}$ can be obtained by calculating
$\liminf\limits_{L\rightarrow\infty}\lambda_{L}$, neither the intermediate
variables $L$ nor $\lambda_{L}$ have a formal physical meaning.

######  Remark 1 (Computations).

Although $\lambda_{\infty}$ in (16) may not be achieved with a finite $L$, one
can approximate $\lambda_{\infty}$ by finding the minimum value in
$\\{\lambda_{1},\dots,\lambda_{L}\\}$ in (17). Since
$\lambda_{\infty}\leq\min\\{\lambda_{1},\dots,\lambda_{L}\\},\forall
L\in\mathbb{N}$, we have the sufficient stability condition

$\rho^{2}_{\max}\min\\{\lambda_{1},\dots,\lambda_{L}\\}<1,$

which approximates the necessary and sufficient condition (18) when $L$ is
large. Actually, (17) can be treated as a general spectral radius optimization
problem of a product of non-negative matrices taken from a finite matrix set.
In Fig. 2, we randomly construct the matrix set with $8$ non-negative
$4$-by-$4$ matrices, and plot $\lambda_{L}$ as a function $L$ with $6$
different matrix set realizations. It can be observed that $\lambda_{L}$ may
and may not monotonically decrease with the increasing $L$, and a small $L$
(e.g., $L=2$) can almost achieve the minimum value obtained by a large $L$,
i.e., $\lambda_{2}\approx\min\\{\lambda_{1},\lambda_{2},\dots,\lambda_{5}\\}$
in different cases.

To the best of our knowledge, Theorem 1 is the first necessary and sufficient
stability condition for remote estimation over a multi-sensor-multi-channel
network in the literature. Before proving our result, we will first establish
a relationship to existing results by focusing on a special case. Under the
idealized assumption that the Markov channels at different frequencies are
independent, a sufficient condition for stability was proved in [20]. It
corresponds to Corollary 1 below.

Figure 2: $\lambda_{L}$ versus $L$ over $6$ randomly generated matrix sets.

######  Corollary 1.

Consider the special case that Markov channels at different frequencies are
independent, and assume that the $\\{0,1\\}$ channel state transition
probability matrix of frequency channel $m$ is

$\mathbf{M}^{(m)}\triangleq\begin{bmatrix}\alpha^{(m)}_{00}&&\alpha^{(m)}_{01}\\\
\alpha^{(m)}_{10}&&\alpha^{(m)}_{11}\\\ \end{bmatrix},m\in\mathcal{M}.$ (19)

The remote estimator described in Section II under Assumptions 1 and 2 is
stable if [20]

$\rho_{\max}^{2}\alpha^{(m^{\star})}_{00}<1,$ (20)

where $\alpha^{(m^{\star})}_{00}\triangleq\min_{m}\alpha^{(m)}_{00}$.

The sufficient stability condition (20) is more restrictive than Theorem 1.

###### Proof.

We only need to show that
$\alpha^{(m^{\star})}_{00}=\rho(\mathbf{E}(\mathbf{v}^{\prime}_{m^{\star}}))$,
where
$\mathbf{v}^{\prime}_{m^{\star}}\triangleq[m^{\star},\dots,m^{\star}]^{\top}$,
as
$\rho(\mathbf{E}(\mathbf{v}^{\prime}_{m^{\star}}))\geq\min_{L\in\mathbb{N}}\min_{\mathbf{v}_{l}\in\mathcal{M}^{\bar{M}}}\rho\left(\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{E}(\mathbf{v}_{L})\right)^{\frac{1}{L}}$.
Without loss of generality, we assume the first half of the channel states in
$\mathcal{S}$ have the off state in the $m^{\star}$th channel, i.e.,

$\mathbf{h}_{i}\notin\mathcal{\tilde{S}}_{m^{\star}},\forall
i\in\\{1,2,\dots,\bar{M}/2\\}.$ (21)

Then, from (15), $\mathbf{E}(\mathbf{v}^{\prime}_{m^{\star}})$ can be written
as a block lower triangular matrix

$\mathbf{E}(\mathbf{v}^{\prime}_{m^{\star}})=\begin{bmatrix}\mathbf{E}_{00}&\\!\\!\\!\framebox{}\\!\\!\\!&\mathbf{0}\\\
\hline\cr\mathbf{E}_{10}&\\!\\!\\!\framebox{}\\!\\!\\!&\mathbf{0}\\\
\end{bmatrix}$ (22)

and thus
$\rho(\mathbf{E}(\mathbf{v}^{\prime}_{m^{\star}}))=\rho(\mathbf{E}_{00})$.
Further, it is easy to see that

$\sum_{j=1}^{\bar{M}/2}[\mathbf{E}_{00}]_{i,j}=\sum_{j=1}^{\bar{M}/2}\mathsf{Prob}\left[\mathbf{h}_{j}|\mathbf{h}_{i}\right]=\alpha^{(m^{\star})}_{00},i=1,2,\dots,\bar{M}/2,$
(23)

where the second equality is due to that the frequency channel $m^{\star}$ is
independent to the other channels and the first $\bar{M}/2$ of the (vector)
channel states contains all the possible channel states of the rest of the
$M-1$ frequency channels. Using the Perron-Frobenius Theorem [27], we have
$\rho(\mathbf{E}_{00})=\alpha^{(m^{\star})}_{00}$, which completes the proof.
∎

######  Example 1.

We numerically compare the sufficient stability condition in [20] and the
necessary and sufficient condition in Theorem 1 for a two frequency channel
scenario with $\rho^{2}_{\max}=2$. We consider different channel state
transition matrices (19) for the second frequency channel with parameters: (a)
$\alpha^{(2)}_{00}=0.9,\alpha^{(2)}_{11}=0.9$, (b)
$\alpha^{(2)}_{00}=0.9,\alpha^{(2)}_{11}=0.1$, (c)
$\alpha^{(2)}_{00}=0.6,\alpha^{(2)}_{11}=0.9$, and (d)
$\alpha^{(2)}_{00}=0.6,\alpha^{(2)}_{11}=0.1$. A larger $\alpha^{(2)}_{00}$
and $\alpha^{(2)}_{11}$ lead to a longer channel state memory in the ‘off’ and
the ‘on’ state, respectively. Then the stability regions in terms of the
parameters of first frequency channel state transition matrices,
$\alpha^{(1)}_{00}$ and $\alpha^{(1)}_{11}$, are shown in Fig. 3.

It is clear that Theorem 1 has a larger stability region than [20] in case
(c), which corresponds to that the second frequency channel has a shorter
memory in the bad (‘off’) state and a longer memory in the good (‘on’) state.
Hence when the quality of the second frequency channel is pretty good, the
stability requirement on the first channel based on [20] is more restrictive
than Theorem 1. $\square$

Figure 3: Comparison between stability conditions of Theorem 1 (solid line)
and [20] (dashed line). The gray area denotes the parameter sets that satisfy
the stability conditions.

We will prove the necessary and sufficiency parts of Theorem 1 in the sequel.
Since the last equality in (16) can be easily obtained by using the property
that $\rho(\mathbf{D}^{L})^{\frac{1}{L}}=\rho(\mathbf{D})$ holds for any
square matrix $\mathbf{D}$ and positive integer $L$, we only need to prove the
condition $\rho^{2}_{\max}\liminf\limits_{L\rightarrow\infty}\lambda_{L}<1$.

###  III-B Proof of Necessity

####  III-B1 Policy Construction

To prove the necessity, we consider the scenario that only the estimate of the
process with the largest spectral radius is scheduled for transmission in each
time slot in a selected frequency channel, while the other sensors’ estimates
are perfectly known by the remote estimator and do not need any transmission.
In other words, only one packet is scheduled in each time slot. We recall that
process $1$ has the largest spectral radius, and we drop out the process index
$n$ in the following analysis.

The channel scheduling policy (12) is reduced to

$\nu(t)=\pi(\phi(t),\mathbf{h}(t-1))\in\mathcal{M},$ (24)

where $\nu(t)$ and $\phi(t)$ denote $\nu_{1}(t)$ and $\phi_{1}(t)$,
respectively, for ease of notation.

From (24), once the AoI $\phi$ is given, the channel selection rule given the
previous channel state information can be written as

$\mathbf{v}(\phi)=[v_{1}(\phi),\dots,v_{\bar{M}}(\phi)]\in\mathcal{M}^{\bar{M}},$
(25)

where $v_{i}(\phi)=\pi(\phi,\mathbf{h}_{i})$.

Given the channel selection vector $\mathbf{v}(\phi)$, we define the
successful transmission probability matrix
$\tilde{\mathbf{E}}(\mathbf{v}(\phi))\in\mathbb{R}^{\bar{M}}\times\mathbb{R}^{\bar{M}}$,
where

$[\tilde{\mathbf{E}}(\mathbf{v}(\phi))]_{i,j}\triangleq\mathsf{Prob}\left[\mathbf{h}(t)=\mathbf{h}_{j},\mathbf{h}_{j}\in\tilde{\mathcal{S}}_{v_{i}(\phi)}|\mathbf{h}(t-1)=\mathbf{h}_{i}\right]=p_{i,j}\mathbbm{1}(\mathbf{h}_{j}\in\tilde{\mathcal{S}}_{v_{i}(\phi)}).$
(26)

In other words, $[\tilde{\mathbf{E}}(\mathbf{v}(\phi))]_{i,j}$ is the
probability that the current channel state is $\mathbf{h}_{j}$ and the
transmission is successful in the selected $v_{i}(\phi)$th frequency channel
given that the previous channel state is $\mathbf{h}_{i}$. Accordingly, we
define the failed transmission probability matrix

${\mathbf{E}}(\mathbf{v}(\phi))\triangleq\mathbf{M}-\tilde{\mathbf{E}}(\mathbf{v}(\phi))\in\mathbb{R}^{\bar{M}}\times\mathbb{R}^{\bar{M}}.$
(27)

####  III-B2 Analysis of the Average Cost

Similar to [15], we consider an estimation cycle based analysis method. Each
estimation cycle starts after a successful transmission and ends at the next
one, and thus the AoI state $\phi$ at the beginning of each estimation cycle
is equal to $1$. $T_{k}$ is the sum of transmissions in the $k$th estimation
cycle. $C_{k}$ is the sum MSE in the $k$th estimation cycle and is a function
of $T_{k}$ as

$C_{k}=g(T_{k})\triangleq\sum_{j=1}^{T_{k}}c(j).$ (28)

The channel state before the $k$th cycle is denoted as
$\mathbf{b}_{k}\in\mathcal{S}$, and a successful transmission occurs at
$\mathbf{b}_{k}$. Similar to Lemma 1 in [15], we have the following Markovian
property of the pre-cycle channel states.

######  Lemma 2.

$\\{\mathbf{b}\\}_{\mathbb{N}_{0}}$ is a time-homogeneous ergodic Markov chain
with $\bar{M}_{1}\leq\bar{M}$ irreducible states of $\mathcal{S}$. The state
transition matrix of $\\{\mathbf{b}\\}_{\mathbb{N}_{0}}$ is
$\mathbf{G}^{\prime}$, which is the $\bar{M}_{1}$-by-$\bar{M}_{1}$ matrix
taken from the top-left corner of

$\mathbf{G}=\sum_{j=1}^{\infty}\mathbf{\tilde{\Xi}}(j),$ (29)

where

$\mathbf{\tilde{\Xi}}(j)=\mathbf{\mathbf{\Xi}}(j-1)\tilde{\mathbf{E}}(\mathbf{v}(j)),j=1,2,\dots.$
(30)

and

$\mathbf{\mathbf{\Xi}}(j)=\begin{cases}\mathbf{I},&j=0\\\
\prod_{i=1}^{j}\mathbf{E}(\mathbf{v}(i)),&j>0.\end{cases}$ (31)

The stationary distribution of $\\{\mathbf{b}\\}_{\mathbb{N}_{0}}$ is
$\bm{\beta}\triangleq[\beta_{1},\dots,\beta_{\bar{M}_{1}}]^{\top}$, which is
the unique null-space vector of $(\mathbf{I-G^{\prime}})^{\top}$ and
$\beta_{i}>0,\forall i\in\mathcal{\bar{M}}_{1}$, where
$\mathcal{\bar{M}}_{1}\triangleq\\{1,2,\dots,\bar{M}_{1}\\}$.

######  Remark 2.

Only the first $\bar{M}_{1}$ channel states can be a pre-cycle state, and thus
the last $(\bar{M}-\bar{M}_{1})$ columns of $\mathbf{G}$ are all zeros.

From (13) it follows that the average estimation MSE can be rewritten as

$\displaystyle
J=\limsup\limits_{K\rightarrow\infty}\frac{\frac{1}{K}(C_{1}+C_{2}+\dots+C_{K})}{\frac{1}{K}(T_{1}+T_{2}+\dots+T_{K})}=\frac{\mathsf{E}\left[C\right]}{\mathsf{E}\left[T\right]},$
(32)

where the last equality holds because the distributions of $T_{k}$ and $C_{k}$
depend on $\mathbf{b}_{k}$, which is ergodic, and hence the unconditional
distributions of $T_{k}$ and $C_{k}$ are also ergodic; time averages are equal
to the ensemble averages and we drop the time indexes. Then, we have

$\mathsf{E}\left[T\right]=\lim\limits_{K\rightarrow\infty}\frac{1}{K}\sum_{k=1}^{K}T_{k}=\sum_{m=1}^{\bar{M}}\beta_{m}\mathsf{E}\left[T|\mathbf{b}=\mathbf{h}_{m}\right],$
(33)

and

$\mathsf{E}\left[C\right]=\lim\limits_{K\rightarrow\infty}\frac{1}{K}\sum_{k=1}^{K}C_{k}=\sum_{m=1}^{\bar{M}}\beta_{m}\mathsf{E}\left[C|\mathbf{b}=\mathbf{h}_{m}\right],$
(34)

where $\beta_{m}$ is defined in Lemma 2 when $m\in\mathcal{\bar{M}}_{1}$, and
$\beta_{m}=0$ when
$m\in\mathcal{\bar{M}}_{0}\triangleq\mathcal{\bar{M}}\backslash\mathcal{\bar{M}}_{1}=\\{\bar{M}_{1}+1,\dots,\bar{M}\\}$.

From the definition of estimation cycle and the property of channel state
transition, the conditional probability of the length of an estimation cycle
is obtained as

$\displaystyle\mathsf{Prob}\left[T=i|\mathbf{b}=\mathbf{h}_{m}\right]=\vartheta\left(\mathbf{L}_{m}\mathbf{\tilde{\Xi}}(i)\right)\triangleq\sum_{k=1}^{\bar{M}}\left[\mathbf{\tilde{\Xi}}(i)\right]_{m,k},\forall
m\in\mathcal{\bar{M}}$ (35)

where $\mathbf{L}_{m}$ is an all-zero matrix except for the $m$th diagonal
element, which equals to $1$.

Taking (35) into (33) and into (34), then after some algebraic manipulations,
one can obtain

$\displaystyle\mathsf{E}\left[T\right]$
$\displaystyle=\sum_{m=1}^{\bar{M}}\beta_{m}\left(\sum_{i=1}^{\infty}i\vartheta\left(\mathbf{L}_{m}\mathbf{\tilde{\Xi}}(i)\right)\right),$
(36) $\displaystyle\mathsf{E}\left[C\right]$
$\displaystyle=\sum_{m=1}^{\bar{M}}\beta_{m}\left(\sum_{i=1}^{\infty}g(i)\vartheta\left(\mathbf{L}_{m}\mathbf{\tilde{\Xi}}(i)\right)\right).$
(37)

From the definition of $g(i)$ in (28) and the property of $c(i)$ in Lemma 1,
$g(i)$ grows exponentially fast with $i$. It is easy to verify the property
below.

######  Lemma 3.

$J<\infty$ if and only if $\mathsf{E}\left[C\right]<\infty$.

From Lemma 3, it follows that it suffices to only investigate conditions such
that $\mathsf{E}\left[C\right]<\infty$ in the following.

####  III-B3 Proof of Necessity

We define a set of channel selection vectors
$\mathcal{F}\subset\mathcal{M}^{\bar{M}}$, where for any
$\mathbf{v}\in\mathcal{F}$ we have $\mathbf{\tilde{E}}(\mathbf{v})=\mathbf{0}$
leading to zero chance of successful transmission in any of the frequency
channels. Thus, for any
$\mathbf{v}\in\mathcal{\tilde{F}}\triangleq\mathcal{M}^{\bar{M}}\backslash\mathcal{F}$,
we have $\mathbf{\tilde{E}}(\mathbf{v})\neq\mathbf{0}$. It is clear that if
$\tilde{\mathcal{F}}=\emptyset$, the packet dropout occurs all the time at all
frequency channels. From the definition (17), we can prove that
$\lambda_{\infty}=1$ as $\mathbf{E}(\mathbf{v})$ is a stochastic matrix for
any $\mathbf{v}\in\mathcal{M}^{\bar{M}}$. Thus, the necessary condition of
stability is straightforward as $\rho^{2}(\mathbf{A})<1$, which can be written
as $\rho^{2}(\mathbf{A})\lambda_{\infty}<1$. In what follows, we will focus on
the scenario with $\tilde{\mathcal{F}}\neq\emptyset$.

We categorize all possible scheduling policies into two types:

######  Definition 2 (Type-I and II Policies).

For a type-I policy, there exists $\bar{\phi}>0$ such that

$\mathbf{v}(\phi)\in\mathcal{F},\forall\phi>\bar{\phi},$ (38)

For a type-II policy, if $\mathbf{v(\phi)}\in\mathcal{\tilde{F}}$, one can
always find $\phi^{\prime}>\phi$ such that
$\mathbf{v(\phi^{\prime})}\in\mathcal{\tilde{F}}$.

A type-I policy has a strictly zero chance of successful transmission when the
AoI is larger than a threshold, while a type-II policy still has a non-zero
success probability when the AoI is arbitrarily large. Thus, to stabilize the
system, a type-I policy should guarantee successful transmission within the
first $\bar{\phi}$ attempts, while it is not necessary for a type-II policy.

We first prove the necessary condition for type I and then type II policies.
Before proceeding further, we define the following

$\lambda_{m,\infty}\triangleq\liminf\limits_{L\rightarrow\infty}\lambda_{m,L},m\in\mathcal{\bar{M}},$
(39)

and

$\lambda_{m,L}\triangleq\min_{\mathbf{v}_{l}\in\mathcal{M}^{\bar{M}}}\rho\left(\mathbf{L}_{m}\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{E}(\mathbf{v}_{L})\right)^{\frac{1}{L}}.$
(40)

Type-I Policy. Since the scheduling policy consistently chooses channel
selection vectors from $\mathcal{F}$ in the high AoI scenario that leads to
zero chance of success, it is clear that a necessary condition to stabilize
the system is that the transmission process has a zero probability to fail for
consecutive $\bar{\phi}$ times at the beginning of an estimation cycle. This
can be written as

$\max_{m\in\mathcal{\bar{M}}}\vartheta(\mathbf{L}_{m}\mathbf{E}(\mathbf{v}(1))\mathbf{E}(\mathbf{v}(2))\cdots\mathbf{E}(\mathbf{v}(\bar{\phi})))=\mathbf{0},$
(41)

and hence $\max_{m\in\mathcal{\bar{M}}}\lambda_{m,\infty}=0$. In the
following, we focus on the type-II policy.

Type-II Policy. We categorize the channel states $\mathbf{h}$ in two cases:
(i) the pre-cycle states $\mathbf{h}=\mathbf{h}_{m}$ with $\beta_{m}>0$, i.e.,
$m\in\mathcal{\bar{M}}_{1}$, and (ii) the non-pre-cycle states
$\mathbf{h}=\mathbf{h}_{m}$ with $\beta_{m}=0$, i.e.,
$m\in\mathcal{\bar{M}}_{0}$. In other words, an estimation cycle can and
cannot start after a case (i) and a case (ii) channel state, respectively. Due
to the ergodicity of the channel states, both the cases of channel states
occur with non-zero probabilities. Then, we analyze the necessary conditions
to (i) make the average sum MSE of an estimation cycle in (37) bounded that
starts after a pre-cycle state

$\sum_{j=1}^{\infty}g(j)\vartheta\left(\mathbf{L}_{m}\mathbf{\tilde{\Xi}}(j)\right)<\infty$
(42)

and to (ii) make the average sum MSE of an estimation cycle bounded that
contains a non-pre-cycle state.

(i) Assume that the channel state $\mathbf{h}_{m}$ is a pre-cycle state. Using
Perron–Frobenius Theorem [27], we have

$\vartheta\left(\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)\right)\geq\rho\left(\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)\right),$
(43)

thus, there exists an element in $\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)$,
e.g., the $m^{\prime}$th element of the $m$th row, such that

$\left[\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)\right]_{m,m^{\prime}}\geq\frac{1}{\bar{M}}\rho\left(\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)\right).$
(44)

From (39), there is a constant $\kappa_{0}>0$ such that
$\rho\left(\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)\right)\geq\kappa_{0}(\lambda_{m,\infty})^{i}$.
Let $\mathbf{F}(i)\in\mathbb{R}^{\bar{M}\times\bar{M}}$ denote a degraded
matrix of $\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)$, where
$[\mathbf{F}(i)]_{m,m^{\prime}}=[\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)]_{m,m^{\prime}}$
and the other elements of $\mathbf{F}(i)$ are zeros. From (44), it is clear
that

$\vartheta\left(\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)\right)\geq\vartheta\left(\mathbf{F}(i)\right)\geq\frac{\kappa_{0}}{\bar{M}}\left(\lambda_{m,\infty}\right)^{i}.$
(45)

Based on the property of type-II policy, we define an infinite sequence of AoI
variables $\\{\tilde{\phi}_{1},\tilde{\phi}_{2},\dots\\}$, where
$\tilde{\phi}_{i}<\tilde{\phi}_{j}$ if $i<j$, and
$\mathbf{v}(\phi)\in\tilde{\mathcal{F}}$ if and only if
$\phi\in\\{\tilde{\phi}\\}_{\mathbb{N}}$. Also, we define the operator $\lceil
x\rceil_{\mathcal{A}}$ as the smallest value in the set $\mathcal{A}$ that is
no smaller than $x$. Building on
$\\{\tilde{\phi}_{1},\tilde{\phi}_{2},\dots\\}$ and a constant
$L\in\mathbb{N}$, we construct a sequence of AoI as
$\\{\varphi_{1},\varphi_{2},\dots\\}$, where

$\varphi_{i}=\begin{cases}\lceil
L\rceil_{\\{\tilde{\phi}\\}_{\mathbb{N}}},&i=1\\\
\lceil\varphi_{i-1}+L\rceil_{\\{\tilde{\phi}\\}_{\mathbb{N}}},&i>1,\end{cases}$
(46)

and thus $\mathbf{v}(\varphi_{i})\in\mathcal{\tilde{F}},\
\varphi_{i+1}-\varphi_{i}\geq L,\forall i\in\mathbb{N}$.

Then, we introduce the following technical lemma.

######  Lemma 4.

There is a constant $L>0$ such that the transmission is successful with at
least a non-zero probability $\xi$ within the next $L$ steps for any channel
selection vector at the $L$th step $\mathbf{v}_{L}\in\mathcal{\tilde{F}}$, no
matter what the current channel state is and the first $(L-1)$-step channel
selection vectors are, i.e.,

$\displaystyle\xi$
$\displaystyle\triangleq\min_{\mathbf{v}_{1},\mathbf{v}_{2},\dots,\mathbf{v}_{L-1}\in\mathcal{M}^{\bar{M}},\mathbf{v}_{L}\in\mathcal{\tilde{F}}}\min_{m\in\mathcal{\bar{M}}}\max_{l=1,\dots,L}\vartheta(\mathbf{L}_{m}\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{E}(\mathbf{v}_{l-1})\mathbf{\tilde{E}}(\mathbf{v}_{l}))>0,$
(47)

where $\mathbf{E}(\mathbf{v}_{0})\triangleq\mathbf{I}$.

###### Proof.

See Appendix. ∎

From Lemma 4, we can find a constant $L$ to construct the AoI sequence
$\\{\varphi\\}_{\mathbb{N}}$ in (46). Then, using Lemma 1 and (45), we have

$\displaystyle\sum_{j=\varphi_{i}-L}^{\varphi_{i}}g(j)\vartheta\left(\mathbf{L}_{m}\mathbf{\tilde{\Xi}}(j)\right)\geq
g(\varphi_{i})\vartheta\left(\mathbf{L}_{m}\mathbf{\tilde{\Xi}}(\varphi_{i})\right)$
(48) $\displaystyle\geq
g(\varphi_{i}-L)\vartheta\left(\mathbf{L}_{m}\mathbf{{\Xi}}(\varphi_{i}-L)\mathbf{E}(\mathbf{v}(\varphi_{i}-L+1))\mathbf{E}(\mathbf{v}(\varphi_{i}-L+2))\cdots\mathbf{E}(\mathbf{v}(\varphi_{i}-1))\mathbf{\tilde{E}}(\mathbf{v}(\varphi_{i}))\right)$
$\displaystyle\geq
g(\varphi_{i}-L)\vartheta\left(\mathbf{L}_{m}\mathbf{{\Xi}}(\varphi_{i}-L)\right)\xi$
$\displaystyle\geq\frac{\kappa_{0}\xi}{\bar{M}}(\rho(\mathbf{A})^{2})^{\varphi_{i}-L}\left(\lambda_{m,\infty}\right)^{\varphi_{i}-L}.$

Now the average sum MSE per cycle in (42) is

$\displaystyle\sum_{j=1}^{\infty}g(j)\vartheta\left(\mathbf{L}_{m}\mathbf{\tilde{\Xi}}(j)\right)\geq\sum_{i=1}^{\infty}\sum_{j=\varphi_{i}-L}^{\varphi_{i}}g(j)\vartheta\left(\mathbf{L}_{m}\mathbf{\tilde{\Xi}}(j)\right)$
(49)
$\displaystyle\geq\sum_{i=1}^{\infty}\frac{\kappa_{0}\xi}{\bar{M}}(\rho(\mathbf{A})^{2})^{\varphi_{i}-L}\left(\lambda_{m,\infty}\right)^{\varphi_{i}-L}.$

To make the last sum in (49) bounded, we must have
$\lim\limits_{i\rightarrow\infty}(\rho(\mathbf{A})^{2})^{\varphi_{i}}\left(\lambda_{m,\infty}\right)^{\varphi_{i}}=0$,
i.e., $(\rho(\mathbf{A})^{2})\lambda_{m,\infty}<1$. Thus, by considering all
the pre-cycle channel states,

$\max_{m\in\mathcal{\bar{M}}_{1}}\rho(\mathbf{A})^{2}\lambda_{m,\infty}<1$

holds if $\mathsf{E}\left[C\right]$ in (37) is bounded.

(ii) Assume now that the channel state $\mathbf{h}_{m^{\prime}}$ is a non-pre-
cycle state. Since the channel state transition process is an ergodic Markov
chain, given a state $\mathbf{h}_{m}$ with $m\in\mathcal{\bar{M}}_{1}$, it
will take finite steps to arrive at
$\mathbf{h}_{m^{\prime}},m^{\prime}\in\mathcal{\bar{M}}_{0}$ with a positive
probability no matter what the channel scheduling policy is. Mathematically,
there exits $L^{\prime}\geq 1$ such that

$\tilde{\beta}_{m,m^{\prime}}\triangleq[\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(L^{\prime})]_{m,m^{\prime}}>0.$

Then, it is straightforward to have

$\vartheta\left(\mathbf{L}_{m}\mathbf{\mathbf{\Xi}}(i)\right)\geq\tilde{\beta}_{m,m^{\prime}}\vartheta\left(\mathbf{L}_{m^{\prime}}\mathbf{E}(\mathbf{v}_{L^{\prime}+1})\mathbf{E}(\mathbf{v}_{L^{\prime}+2})\cdots\mathbf{E}(\mathbf{v}_{i})\right),\forall
i\geq L^{\prime}.$ (50)

We see that the right-hand side of (50) has a similar format to the left-hand
side of (43) in case (i). Then by taking (50) into (42), following similar
steps as in case (i) and considering all the non-pre-cycle channel states, we
obtain that

$\max_{m^{\prime}\in\mathcal{\bar{M}}_{0}}\rho(\mathbf{A})^{2}\lambda_{m^{\prime},\infty}<1$

holds if $\mathsf{E}\left[C\right]$ in (37) is bounded.

From cases (i) and (ii), a necessary condition to bound
$\mathsf{E}\left[C\right]$ can be uniformly written as

$\max_{m\in\mathcal{\bar{M}}}\rho(\mathbf{A})^{2}\lambda_{m,\infty}<1.$

From the technical lemma below, it is easy to prove that
$\max_{m\in\mathcal{\bar{M}}}\lambda_{m,\infty}=\lambda_{\infty}$. Then, the
necessary condition of type-I and II policies can be jointly written as
$\rho(\mathbf{A})^{2}\lambda_{\infty}<1$, which completes the proof of the
necessity of Theorem 1.

######  Lemma 5.

Given a sequence of $N$-by-$N$ matrices
$\\{\mathbf{M}_{1},\mathbf{M}_{2},\dots\\}$, the following equation holds

$\max_{i\in\mathcal{N}}\liminf\limits_{L\rightarrow\infty}\rho\left(\mathbf{L}_{i}\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right)^{\frac{1}{L}}=\liminf\limits_{L\rightarrow\infty}\rho\left(\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right)^{\frac{1}{L}},$
(51)

where $\mathbf{L}_{i}\in\mathbb{R}^{N\times N}$ is a diagonal matrix with
$i$th diagonal element equals to $1$ and the other elements are zeros.

###### Proof.

See Appendix. ∎

###  III-C Proof of Sufficiency

####  III-C1 Policy Construction

We consider a _persistent serial scheduling_ policy that persistently
schedules the transmission of one sensor at a time until it is successful and
then schedules the next sensor and so on. Although it seems that such a policy
cannot take advantage of the parallel channels, we will show that the policy
stabilizes the remote estimator if condition (18) holds. The policy can be
written as

$\pi(\bm{\phi}(t),\mathbf{h}(t-1))=\begin{cases}\pi_{1}(\phi_{1}(t),\mathbf{h}(t-1)),&\text{if
}\phi_{n}(t)=1,n=N\\\ \pi_{n+1}(\phi_{n+1}(t),\mathbf{h}(t-1)),&\text{if
}\phi_{n}(t)=1,n<N\\\ \pi(\bm{\phi}(t-1),\mathbf{h}(t-2)),&\text{else }\\\
\end{cases}$ (52)

where the policy

$\pi_{n}(\phi_{n}(t),\mathbf{h}(t-1))=[\underbrace{0,\dots,0}_{n-1},\nu_{n}(t),\underbrace{0,\dots,0}_{N-n}],n\in\mathcal{N}$
(53)

denotes scheduling the $n$th sensor on the $\nu_{n}(t)$th frequency channel.
Without loss of generality, the initial scheduling policy is given by
$\pi(\bm{\phi}(1),\mathbf{h}(0))=\pi_{1}({\phi}_{1}(1),\mathbf{h}(0))$. With a
slight abuse of notation, we drop out the zeros in (53) so that the frequency
channel selection rule for sensor $n$ is rewritten as

$\pi_{n}(\phi_{n}(t),\mathbf{h}(t-1))=\nu_{n}(t)\in\mathcal{M},n\in\mathcal{N}.$
(54)

Once the AoI of sensor $n$, $\phi_{n}$, is given, the channel selection rule
with different previous channel states is denoted as

$\mathbf{v}_{n}(\phi_{n})\triangleq\left[\pi_{n}(\phi_{n},\mathbf{h}_{1}),\pi_{n}(\phi_{n},\mathbf{h}_{2}),\dots,\pi_{n}(\phi_{n},\mathbf{h}_{\bar{M}})\right]\in\mathcal{M}^{\bar{M}}.$
(55)

Moreover, we assume that the frequency channel selection rule
$\mathbf{v}_{n}(\phi_{n}),\forall n\in\mathcal{N}$, is a _periodic policy_ in
terms of $\phi_{n}$ with $L$ potential channel selection vectors taken from
the set
$\mathcal{V}\triangleq\\{\mathbf{\tilde{v}}_{n}(1),\mathbf{\tilde{v}}_{n}(2),\dots,\mathbf{\tilde{v}}_{n}(L)\\}$,
where $\mathbf{\tilde{v}}_{n}(L)\in\mathcal{M}^{\bar{M}}$. From the definition
of $\lambda_{\infty}$ in (16), for an arbitrarily small $\epsilon>0$, we can
find a constant $L$ and a length-$L$ channel-selection-vector set
$\mathcal{V}$ satisfying the condition that

$\rho\left(\mathbf{E}(\mathbf{\tilde{v}}_{n}(1))\mathbf{E}(\mathbf{\tilde{v}}_{n}(2))\cdots\mathbf{E}(\mathbf{\tilde{v}}_{n}(L))\right)^{\frac{1}{L}}\leq\lambda_{\infty}+\epsilon.$
(56)

Then, the periodic channel selection policy of sensor $n$ is defined as

$\mathbf{v}_{n}(\phi_{n})=\mathbf{\tilde{v}}_{n}(\phi_{n}\bmod L),$ (57)

where $(a\bmod b)$ denotes the remainder of the Euclidean division of $a$ by
$b$ if the remainder is non-zero, otherwise $(a\bmod b)=b$.

####  III-C2 Analysis of the Average Cost

To analyze the average cost of the average estimation MSE of the $n$th process
$J_{n}$, the estimation cycle starts after a successful transmission of
process $n$ and ends at the following one, as illustrated in Fig. 4.

Figure 4: The persistent serial scheduling policy with three sensors. Green,
blue and red intervals denotes the scheduling of sensor S1, S2 and S3,
respectively. The vertical axis denotes the AoI functions of the sensors in
terms of the time.

$T_{n,k}$ and $C_{n,k}$ denote the sum of transmissions and the sum MSE of the
$k$th estimation cycle:

$C_{n,k}=g(T_{n,k})\triangleq\sum_{j=1}^{T_{n,k}}c_{n}(j),$ (58)

and

$T_{n,k}=\sum_{i=1}^{N}T_{(n,i),k},$ (59)

where $T_{(n,i),k}$ is the time duration scheduled for the sensor $i$’s
transmission during the $k$th estimation cycle of process $n$.

Similar to the single-sensor case, it can be proved that the time average of
$\\{C_{n}\\}_{\mathbb{N}_{0}}$ is equal to its ensemble average, and $J_{n}$
is bounded if $\mathsf{E}\left[C_{n}\right]$ is. In the following, we drop out
the time index $k$ and will analyze $\mathsf{E}\left[C_{n}\right]$. For the
ease of notation but without loss of generality, we analyze
$\mathsf{E}\left[C_{N}\right]$ for the $N$th process:

$\displaystyle\mathsf{E}\left[C_{N}\right]$
$\displaystyle=\\!\\!\sum_{t_{1}=1}^{\infty}\dots\\!\\!\sum_{t_{N-1}=1}^{\infty}\\!\\!\mathsf{E}\left[C_{N}|T_{(N,1)}=t_{1},\dots,T_{(N,N-1)}=t_{N-1}\right]$
(60)
$\displaystyle\times\mathsf{Prob}\left[T_{(N,1)}=t_{1},\dots,T_{(N,N-1)}=t_{N-1}\right].$

Let $\mathbf{b}_{N}$ denote the channel state before an estimation cycle of
process $N$. From (35), it is easy to obtain the conditional probability

$\displaystyle\mathsf{Prob}\left[T_{(N,N)}=t_{N}|T_{(N,1)}=t_{1},\dots,T_{(N,N-1)}=t_{n-1}\right]$
(61)
$\displaystyle=\vartheta(\text{diag}\\{[\varsigma_{1},\dots,\varsigma_{\bar{M}}]\\}\mathbf{\tilde{\Xi}}(t_{N})),$

where

$\varsigma_{m}\triangleq\mathsf{Prob}\left[\mathbf{b}_{N}=\mathbf{h}_{m}|T_{(N,1)}=t_{1},\dots,T_{(N,N-1)}=t_{n-1}\right],m\in\mathcal{\bar{M}}.$
(62)

Recall that $\mathbf{\tilde{\Xi}}(\cdot)$ was defined in (30).

Thus, from (58) and (61), the conditional expectation is simplified as

$\displaystyle\mathsf{E}\left[C_{N}|T_{(N,1)}=t_{1},\dots,T_{(N,N-1)}=t_{N-1}\right]$
(63)
$\displaystyle=\sum_{t_{N}=1}^{\infty}g\left(\sum_{i=1}^{N}t_{i}\right)\mathsf{Prob}\left[T_{(N,N)}=t_{N}|T_{(N,1)}=t_{1},\dots,T_{(N,N-1)}=t_{n-1}\right]$
$\displaystyle=\sum_{t_{N}=1}^{\infty}g\left(\sum_{i=1}^{N}t_{i}\right)\vartheta(\text{diag}\\{[\varsigma_{1},\dots,\varsigma_{\bar{M}}]\\}\mathbf{\tilde{\Xi}}(t_{N})).$

####  III-C3 Proof of Sufficiency

From Lemma 1 and the monotonicity of $g(\cdot)$, it can be proved that for any
$\epsilon^{\prime}>0$ we can find $\kappa>0$ such that

$g\left(\sum_{i=1}^{N}t_{i}\right)\leq\kappa(\rho^{2}(\mathbf{A}_{N})+\epsilon^{\prime})^{\sum_{i=1}^{N}t_{i}}.$
(64)

Since $\varsigma_{m}\leq 1,\forall m\in\mathcal{\bar{M}}$, it is clear that

$\displaystyle\vartheta(\text{diag}\\{[\varsigma_{1},\dots,\varsigma_{\bar{M}}]\\}\mathbf{\tilde{\Xi}}(t_{N}))\leq\vartheta(\mathbf{\tilde{\Xi}}(t_{N}))$
(65) $\displaystyle\leq\vartheta(\mathbf{\Xi}(t_{N}-(t_{N}\bmod L))),\text{ if
}t_{N}>L,$

where the last inequality is due to the property that
$\vartheta(\mathbf{M}_{1}\mathbf{M}_{2})\leq\vartheta(\mathbf{M}_{1})$ if
$\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ are non-negative matrices and the sum of
each row of $\mathbf{M}_{2}$ is no higher than $1$.

Taking (64) and (65) into (63),

$\displaystyle\mathsf{E}\left[C_{N}|T_{(N,1)}=t_{1},\dots,T_{(N,N-1)}=t_{n-1}\right]$
(66) $\displaystyle\leq\gamma_{0}+\kappa
L\sum_{j=2}^{\infty}(\rho^{2}(\mathbf{A}_{N})+\epsilon^{\prime})^{\sum_{i=1}^{N-1}t_{i}}(\rho^{2}(\mathbf{A}_{N})+\epsilon^{\prime})^{jL}\vartheta(\mathbf{\Xi}((j-1)L))$
$\displaystyle\leq\gamma_{0}+\kappa^{\prime}\kappa
L(\rho^{2}(\mathbf{A}_{N})+\epsilon^{\prime})^{\sum_{i=1}^{N-1}t_{i}}\sum_{j=2}^{\infty}(\rho^{2}(\mathbf{A}_{N})+\epsilon^{\prime})^{jL}(\lambda_{\infty}+\epsilon)^{(j-1)L}$

where $\gamma_{0},\kappa$, and $\kappa^{\prime}$ are positive constant; the
first inequality is obtained by dividing the infinity sum into length-$L$
segment sums and applying the inequalities (64) and (65). The second
inequality is obtained by first rewriting $\mathbf{\Xi}((j-1)L)$ as
$\mathbf{E}(\mathbf{\tilde{v}}_{n}(1))\mathbf{E}(\mathbf{\tilde{v}}_{n}(2))\cdots\mathbf{E}(\mathbf{\tilde{v}}_{n}(L))$
to the power of $(j-1)L$ and then using (56) and Lemma 6 below.

######  Lemma 6 ([15]).

Given an $N$-by-$N$ matrix $\mathbf{M}$, for any $\epsilon>0$, we can find
$\kappa>0$ such that
$[\mathbf{M}^{k}]_{i,j}<\kappa(\rho(\mathbf{M})+\epsilon)^{k},\forall
k\in\mathbb{N},i,j=1,2,\dots,N.\hfill\square$

As a consequence, if $\rho^{2}(\mathbf{A}_{N})\lambda_{\infty}<1$, then we can
always find a scheduling policy providing an arbitrary small $\epsilon$ to
make $\mathsf{E}\left[C_{N}|T_{(N,1)}=t_{1},\dots,T_{(N,N-1)}=t_{n-1}\right]$
bounded. Since the condition does not rely on $\\{T_{(N,1)},T_{(N,2)}\dots
T_{(N,N-1)}\\}$, it is straightforward to show that the policy leads to a
bounded $\mathsf{E}\left[C_{N}\right]$.

By applying the above method to the other $N-1$ sensors, a sufficient
condition for stabilizing the $N$-sensor remote estimator is obtained as
$\rho^{2}(\mathbf{A}_{1})\lambda_{\infty}<1$, which completes the proof of
sufficiency of Theorem 1.

######  Remark 3.

The policy above with persistent sensor scheduling and periodic channel
selection is a stability-guaranteeing policy. Note that such a policy, which
does not utilize the parallel frequency channels, is only constructed for the
proof of the sufficiency of Theorem 1, and is not optimal. Once the stability
condition is satisfied, we can find the optimal policy in (12) by solving a
Markov decision process problem as discussed in [20].

##  IV Extensions

In this section, we will investigate four important extensions to the
framework in Section III. The policy construction rules will be discussed,
while the proof details of necessary and sufficient conditions are omitted due
to space considerations. Note that the constructed policies to prove the
necessary and sufficient conditions for the first two extension scenarios are
the same as that provided in the previous section.

###  IV-A Known Current Channel State

In the previous section, we assumed that the sensor only knows the channel
state in the previous time slot. Now we would like to also investigate the
scenario that the channel state of the current time slot is known prior to
transmission.

######  Assumption 3 (Known Current Channel State).

At time $t\in\mathbb{N}$, the current channel state $\mathbf{h}(t)$ is known
by the gateway prior to transmission.

Given a channel selection vector $\mathbf{v}\in\mathcal{M}^{\bar{M}}$ and the
current channel state information $\mathbf{h}(t)$, we define diagonal matrices

$[\mathbf{{V}}(\mathbf{v})]_{m,m}\triangleq\mathsf{Prob}\left[\mathbf{h}_{m}\notin\tilde{\mathcal{S}}_{v_{m}}|\mathbf{h}(t)=\mathbf{h}_{m}\right]=\mathbbm{1}(\mathbf{h}_{m}\notin\tilde{\mathcal{S}}_{v_{m}}),m\in\mathcal{\bar{M}}.$
(67)

and $\mathbf{\tilde{V}}(\mathbf{v})=\mathbf{I}-\mathbf{V}(\mathbf{v})$.
Different from Section III, the successful transmission probability matrix is
$\mathbf{\tilde{V}}(\mathbf{v})\mathbf{M}$, where

$[\mathbf{\tilde{V}}(\mathbf{v})\mathbf{M}]_{i,j}\triangleq\mathsf{Prob}\left[\mathbf{h}(t+1)=\mathbf{h}_{j},\mathbf{h}_{i}\in\tilde{\mathcal{S}}_{v_{i}}|\mathbf{h}(t)=\mathbf{h}_{i}\right]=p_{i,j}\mathbbm{1}(\mathbf{h}_{i}\in\tilde{\mathcal{S}}_{v_{i}}),$
(68)

and the failed transmission probability matrix is
$\mathbf{V}(\mathbf{v})\mathbf{M}$. Then, a necessary and sufficient condition
can be proved by simply replacing $\mathbf{E}(\mathbf{v})$ and
$\mathbf{\tilde{E}}(\mathbf{v})$ with $\mathbf{V}(\mathbf{v})\mathbf{M}$ and
$\mathbf{\tilde{V}}(\mathbf{v})\mathbf{M}$, respectively, in Theorem 1.

######  Theorem 2.

The remote estimator described in Section II under Assumptions 2 and 3 is
stable if and only if the following condition holds

$\rho^{2}_{\max}\lambda^{\prime}_{\infty}<1,$ (69)

where

$\lambda^{\prime}_{\infty}\triangleq\liminf\limits_{L\rightarrow\infty}\lambda^{\prime}_{L}=\lambda^{\prime}_{1},$
(70)
$\lambda^{\prime}_{L}\triangleq\min_{\mathbf{v}_{l}\in\mathcal{M}^{\bar{M}}}\rho\left(\mathbf{V}(\mathbf{v}_{1})\mathbf{M}\mathbf{V}(\mathbf{v}_{2})\mathbf{M}\cdots\mathbf{V}(\mathbf{v}_{L})\mathbf{M}\right)^{\frac{1}{L}},$
(71)

and $\mathbf{V}(\mathbf{v})$ is defined in (67).

Note that to achieve the minimum in (71), we have
$\mathbf{v}_{l}=\mathbf{v}^{\star},\forall l=1,\dots,L$, where
$\mathbf{V}(\mathbf{v}^{\star})\leq\mathbf{V}(\mathbf{v}),\forall\mathbf{v}\in\mathcal{M}^{\bar{M}}$,
element-wise. This is due to the property that the spectral radius of one non-
negative matrix is larger than the other if the former is larger element-wise
[28]. Thus, the last equality in (70) is obtained directly.

######  Remark 4.

The stability condition in Theorem 2 is less restrictive than that in Theorem
1. It follows from by comparing (71) and (17),

$\min_{\mathbf{v}_{l}\in\mathcal{M}^{\bar{M}}}\rho\left(\mathbf{V}(\mathbf{v}_{1})\mathbf{M}\mathbf{V}(\mathbf{v}_{2})\mathbf{M}\cdots\mathbf{V}(\mathbf{v}_{L})\mathbf{M}\right)^{\frac{1}{L}}\leq\min_{\mathbf{v}_{l}\in\mathcal{M}^{\bar{M}}}\rho\left(\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{E}(\mathbf{v}_{L})\right)^{\frac{1}{L}},\forall
L\in\mathbb{N}.$ (72)

Again, using the property that the spectral radius of one non-negative matrix
is larger than the other if the former is larger element-wise [28], it is
clear that (72) holds if the following inequality holds element-wise

$\min_{\mathbf{v}^{\prime}\in\mathcal{M}^{\bar{M}}}\mathbf{V}(\mathbf{v}^{\prime})\mathbf{M}\leq\mathbf{E}(\mathbf{v}),\forall\mathbf{v}\in\mathcal{M}^{\bar{M}}.$
(73)

Since
$[\mathbf{E}(\mathbf{v})]_{i,j}=p_{i,j}\mathbbm{1}(\mathbf{h}_{j}\notin\tilde{\mathcal{S}}_{v_{i}})$,
and
$[\mathbf{V}(\mathbf{v}^{\prime})\mathbf{M}]_{i,j}=p_{i,j}\mathbbm{1}(\mathbf{h}_{i}\notin\tilde{\mathcal{S}}_{v^{\prime}_{i}})$,
we have $\mathbf{V}(\mathbf{v}^{\prime})\mathbf{M}\leq\mathbf{E}(\mathbf{v})$
when each element of $\mathbf{v}^{\prime}$ is equal to
$\arg\min_{v^{\prime}\in\mathcal{M}}\min_{j}\mathbbm{1}(\mathbf{h}_{j}\notin\tilde{\mathcal{S}}_{v^{\prime}})$.

######  Remark 5.

When specialising to a single-sensor-single-channel scenario, Theorem 1
reduces to Theorem 1 of [15]. Further, Remark 4 shows that current channel
state information known scenario has a less restrictive stability condition
than the previous channel state information known one. In Fig. 5(a), we plot
$\lambda_{\infty}$ in Theorem 1 and $\lambda^{\prime}_{\infty}$ in Theorem 2
with two frequency channels and $25$ uniformly generated $4$-by-$4$ matrix
$\mathbf{M}$. The gap between them is significant in most of the cases. In
Fig. 5(b), we randomly generate next-state-highly-predictable matrix
$\mathbf{M}$, i.e., each row of $\mathbf{M}$ has a dominant element larger
than $0.9$. Comparing with Fig. 5(a), $\lambda_{\infty}$ and
$\lambda^{\prime}_{\infty}$ are quite close to each other. Intuitively, this
is because when the channel states are highly predictable, the current channel
states are “almost known” given the previous channel states.

Figure 5: Stability conditions in the known previous and current channel state
information scenarios, with (a) uniformly generated $\mathbf{M}$ and (b)
randomly generated next-state-highly-predictable $\mathbf{M}$.

###  IV-B Hidden State Markov Fading Channel

In Section II, a binary-level Markov channel was considered to explicitly
model the packet drop process. Now we investigate the scenario with multi-
level hidden Markov channel states [15], and only the previous channel states
are available.

Consider an $M$-channel-multi-level channel state
$\mathbf{h}(t)\triangleq[h_{1}(t),h_{2}(t),\dots,h_{M}(t)]$, where the $m$th
frequency channel $h_{m}(t)$ has $M_{m}$ channel conditions and

$h_{m}(t)\in\mathcal{H}_{m}\triangleq\\{h^{(m)}_{1},h^{(m)}_{2},\dots,h^{(m)}_{M_{m}}\\},\;m\in\mathcal{M}.$

The channel state $\mathbf{h}(t)$ forms an ergodic Markov chain with $\bar{M}$
irreducible states, where

$\mathbf{h}(t)\in\mathcal{S}=\\{\mathbf{h}_{1},\mathbf{h}_{2},\dots,\mathbf{h}_{\bar{M}}\\}=\mathcal{H}_{1}\times\mathcal{H}_{2}\times\dots\times\mathcal{H}_{M}$

and $\mathbf{h}_{i}=[h_{i,1},h_{i,2},\dots,h_{i,M}]$.

Let $\mathbf{M}\in\mathbb{R}^{\bar{M}}\times\mathbb{R}^{\bar{M}}$ denote the
state transition probability matrix, where

$[\mathbf{M}]_{i,j}\triangleq\mathsf{Prob}\left[\mathbf{h}(t)=\mathbf{h}_{j}|\mathbf{h}(t-1)=\mathbf{h}_{i}\right]=p_{i,j}.$
(74)

For ease of notation, given channel state
$\mathbf{h}_{j},j\in\mathcal{\bar{M}}$, by considering $M$ different channels
for transmission at $t$, the set of possible packet drop probabilities is
denoted as

$\mathcal{D}_{j}\triangleq\\{d_{j,1},d_{j,2},\dots,d_{j,M}\\},$ (75)

where $d_{j,m}$ is the packet drop probability in channel $m$ with channel
condition $h_{j,m}$.

By replacing the binary state channel model with the multi-state channel
model, we have the result below.

######  Theorem 3.

The remote estimator described in Section II under the multi-level hidden
Markov channels and Assumptions 1 and 2 is stable if and only if

$\rho^{2}_{\max}\lambda^{\prime}_{\infty}<1,$ (76)

where $\rho_{\max}\triangleq\rho(\mathbf{A}_{1})$, and

$\lambda^{\prime}_{\infty}\triangleq\liminf\limits_{L\rightarrow\infty}\lambda^{\prime}_{L},$
(77)

$\lambda^{\prime}_{L}\triangleq\min_{\mathbf{v}_{l}\in\mathcal{M}^{\bar{M}}}\rho\left(\mathbf{E}^{\prime}(\mathbf{v}_{1})\mathbf{E}^{\prime}(\mathbf{v}_{2})\cdots\mathbf{E}^{\prime}(\mathbf{v}_{L})\right)^{\frac{1}{L}},$
(78)

and $\mathbf{E}^{\prime}(\mathbf{v})$ is a $\bar{M}\times\bar{M}$ matrix
defined as

$[\mathbf{E}^{\prime}(\mathbf{v})]_{i,j}\triangleq p_{i,j}d_{j,{v_{i}}}.$ (79)

###  IV-C Redundant Transmission Among Multiple Frequency Channels

We consider the scenario with redundant transmissions as below.

######  Assumption 4 (Enabled Redundant Transmissions).

In each time slot, each packet can use multiple frequency channels for
transmission.

For the necessity part, we consider a policy that allocates all the $M$
channels for sensor $1$’s packet transmission. For the sufficiency part, we
consider a persistent scheduling policy that persistently schedules the
transmission of one sensor by allocating all the channels to it until it is
successful and then schedules the following one. Different from the scenario
in Section III, the constructed policies do not need to consider channel
selection for transmission, and a scheduled packet is successfully received as
long as the channel state $\mathbf{h}(t)\neq[0,\dots,0]$. The analysis for the
constructed policies is a degraded version of Section III, and leads to the
following result:

######  Theorem 4.

The remote estimator described in Section II under Assumptions 1 and 4 is
stable if and only if

$\rho^{2}_{\max}\lambda^{\prime}_{\infty}<1,$ (80)

where

$\lambda^{\prime}_{\infty}\triangleq\rho\left(\mathbf{M}^{\prime}\right),$
(81)

and $\mathbf{M}^{\prime}\in\mathbb{R}^{\bar{M}}\times\mathbb{R}^{\bar{M}}$

$\mathbf{M}^{\prime}=\mathbf{M}-\mathbf{\tilde{M}}^{\prime}$ (82)
$[\mathbf{\tilde{M}}^{\prime}]_{i,j}=\begin{cases}0&\text{ if
}\mathbf{h}_{j}=[0,\dots,0]\\\ [\mathbf{M}]_{i,j}&\text{
otherwise}\end{cases}$ (83)

######  Remark 6.

Considering the special case that the $M$ channels are independent, each
channel is time-uncorrelated and the packet error probability is $p$, it is
easy to show that the elements on the $j$th column of $\mathbf{M}^{\prime}$
are all equal to $p^{M}$, where $\mathbf{h}_{j}=[0,0,\dots,0]$, and the other
elements are all zeros. Thus, $\rho(\mathbf{M}^{\prime})=p^{M}$, and the
necessary and sufficient condition is $\rho^{2}_{\max}p^{M}<1$, which
corresponds to Theorem 3 in [29]. However, in [29], only sufficiency was
proved.

###  IV-D Multiple Wireless Sensors

We consider a gateway-free scenario: under the scheduling of the remote
estimator, each sensor can send its measurement to the remote estimator
wirelessly as illustrated in Fig. 6. Different sensors with different
locations have different radio propagation environment, thus they have
different channel conditions at the same frequency channels [16].

Figure 6: The multi-wireless-sensor-multi-channel remote estimator without
gateway.

######  Assumption 5 (Multiple Sensors Access Multiple Frequency Channels).

Each sensor is able to access any of the $M$ frequency channels for
transmission; however, in each time slot, at most one packet from one sensor
can be scheduled in each frequency channel. The $M$-channel state between
sensor $n$ and the remote estimator is $\mathbf{h}^{(n)}(t),k\in\mathcal{K}$.
$\mathbf{h}^{(n)}(t)$ is modeled as an aperiodic Markov chain with
$\bar{M}^{(n)}$ irreducible channel states,
$\mathcal{S}^{(n)}\triangleq\\{\mathbf{h}^{(n)}_{1},\mathbf{h}^{(n)}_{2},\dots,\mathbf{h}^{(n)}_{\bar{M}^{(n)}}\\}$,
where
$\mathbf{h}^{(n)}_{i}\triangleq(h^{(n)}_{i,1},h^{(n)}_{i,2},\dots,h^{(n)}_{i,M})\in\\{0,1\\}^{M}$.
Note that the channel states $\mathbf{h}^{(n)}(t)$ and
$\mathbf{h}^{(n^{\prime})}(t)$ are independent if $n\neq n^{\prime}$. Let
$\mathbf{M}^{(n)}\in\mathbb{R}^{\bar{M}^{(n)}}\times\mathbb{R}^{\bar{M}^{(n)}}$
denote the state transition probability matrix of sensor $n$, where

$[\mathbf{M}^{(n)}]_{i,j}\triangleq\mathsf{Prob}\left[\mathbf{h}^{(n)}(t)=\mathbf{h}^{(n)}_{j}|\mathbf{h}^{(n)}(t-1)=\mathbf{h}^{(n)}_{i}\right]=p^{(n)}_{i,j}.$
(84)

Let $\tilde{\mathcal{S}_{i}}^{(n)}\in\mathcal{S}^{(n)}$ denote the channel
state with successful transmission in the $i$th frequency channel, i.e.,

$\tilde{\mathcal{S}_{i}}^{(n)}\triangleq\\{\mathbf{h}^{(n)}_{j}:h^{(n)}_{j,i}=1,j\in\mathcal{\bar{M}}^{(n)}\\},i\in\mathcal{M}$
(85)

where $\mathcal{\bar{M}}^{(n)}\triangleq\\{1,2,\dots,{\bar{M}}^{(n)}\\}$. Now
we define the following

$\displaystyle\lambda^{(n)}_{\infty}\triangleq\liminf\limits_{L\rightarrow\infty}\lambda^{(n)}_{L}$
(86)
$\displaystyle\lambda^{(n)}_{L}\triangleq\min_{\mathbf{v}_{l}\in\mathcal{M}^{\bar{M}}}\rho\left(\mathbf{E}^{(n)}(\mathbf{v}_{1})\mathbf{E}^{(n)}(\mathbf{v}_{2})\cdots\mathbf{E}^{(n)}(\mathbf{v}_{L})\right)^{\frac{1}{L}},$

where $\mathbf{E}^{(n)}(\mathbf{v})$ is a $\bar{M}^{(n)}\times\bar{M}^{(n)}$
matrix defined as

$[\mathbf{E}^{(n)}(\mathbf{v})]_{i,j}\triangleq
p^{(n)}_{i,j}\mathbbm{1}(\mathbf{h}^{(n)}_{j}\notin\tilde{\mathcal{S}}^{(n)}_{v_{i}}).$
(87)

To derive the necessary condition, we consider the policy that only sensor
$n^{\star}$, where
$n^{\star}=\arg\max_{n\in\mathcal{N}}(\rho^{(n)}_{\max})^{2}\lambda^{(n)}_{\infty}$,
is scheduled for transmission, while the packets of the other sensors are not
scheduled and are assumed to be perfectly known by the remote estimator. To
derive the sufficiency condition, we consider the persistent scheduling policy
in Section III-C1 that schedules sensor $1$’s packet until it is successful
and then sensor $2$ and so on. We obtain the following result:

######  Theorem 5.

Consider the remote estimator described above under Assumptions 1, 2 and 5. A
necessary stability condition is

$\max_{n\in\mathcal{N}}(\rho(\mathbf{A}_{n}))^{2}\lambda^{(n)}_{\infty}<1.$
(88)

A sufficient stability condition is

$\rho_{\max}^{2}\max_{n\in\mathcal{N}}\lambda^{(n)}_{\infty}<1.$ (89)

######  Remark 7.

For the special case that $\rho(\mathbf{A}_{1})=\dots=\rho(\mathbf{A}_{N})$,
the necessary condition equals the sufficient condition. Unlike the other
extension scenarios, the necessary and sufficient stability condition is not
derived in the multi-wireless-sensor scenario. This is mainly because the
divergence in different sensor channels introduces additional dimensions in
scheduling policy design and cost function analysis. We will tackle the
challenge in our future work by constructing new policies that achieve a
tighter gap between the necessity and sufficiency.

##  V Conclusions

We have tackled the open problem: what is the fundamental requirement on the
multi-sensor-multi-channel system to guarantee the existence of a sensor
scheduling policy that can stabilize remote estimation? To solve the problem,
we have proposed novel policy construction methods, and have developed an
estimation-cycle based analytical approach. We have derived a necessary and
sufficient stability condition in terms of the LTI system parameters and the
channel statistics. Numerical results have shown that the condition is more
effective than existing sufficient conditions available in the literature.
Scheduling policies with stability guarantees have been derived as well. We
have further extended the stability analysis framework to some other remote
estimator architectures with different network setups and obtained necessary
and sufficient stability conditions also for these systems. For future work,
we will consider stability analysis of a multi-control-loop system over shared
wireless channels.

## Appendix

###  V-A Proof of Lemma 4

Due to the periodicity of the Markov channel states, given any current channel
state, there exists a constant $(L-1)$ such that it can reach any channel
state in the $(L-1)$th time slots with non-zero probabilities [30], i.e.,

$[\mathbf{L}_{m}\mathbf{M}^{L-1}]_{m,m^{\prime}}>0,\forall
m,m^{\prime}\in\mathcal{\bar{M}}.$ (90)

Given a current channel state $\mathbf{h}_{m}$ and a sequence of channel
selection vectors $\\{\mathbf{v}_{1},\mathbf{v}_{2},\dots,\mathbf{v}_{L}\\}$
for the next $L$ transmissions, if there is a non-zero probability $\xi_{m}$
that the transmissions can be successful within $(L-1)$ time slots, we
directly have

$\max_{l=1,\dots,L-1}\vartheta(\mathbf{L}_{m}\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{\tilde{E}}(\mathbf{v}_{l}))>0.$
(91)

Otherwise, we have

$\mathbf{L}_{m}\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{\tilde{E}}(\mathbf{v}_{l})=\mathbf{0},\forall
l=1,\dots,L-1.$

Since
$\mathbf{M}=\mathbf{E}(\mathbf{v}_{l})+\mathbf{\tilde{E}}(\mathbf{v}_{l}),\forall
l=1,\dots,L-1$, it is clear that

$\mathbf{L}_{m}\mathbf{M}^{L-1}=\mathbf{L}_{m}\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{E}(\mathbf{v}_{L}).$
(92)

By using (90) and the fact that
$\vartheta(\mathbf{\tilde{E}}(\mathbf{v}_{L}))\neq\mathbf{0}$, we have
$\vartheta(\mathbf{L}_{m}\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{E}(\mathbf{v}_{L-1})\mathbf{\tilde{E}}(\mathbf{v}_{L}))>0$.
This completes the proof of Lemma 4.

###  V-B Proof of Lemma 5

We define the optimal index $i^{\star}$ such that

$\displaystyle
i^{\star}\triangleq\arg\max_{i\in\mathcal{N}}\rho\left(\mathbf{L}_{i}\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right)$
(93)

By using Gelfand corollaries [31], we have

$\rho\left(\mathbf{L}_{i^{\star}}\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right)\leq\rho(\mathbf{L}_{i^{\star}})\rho(\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L})$
(94)

and thus

$\displaystyle\liminf\limits_{L\rightarrow\infty}\rho\left(\mathbf{L}_{i^{\star}}\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right)^{\frac{1}{L}}$
$\displaystyle\leq\liminf\limits_{L\rightarrow\infty}\Big{(}\rho(\mathbf{L}_{i^{\star}})\rho(\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L})\Big{)}^{\frac{1}{L}}$
(95)
$\displaystyle=\liminf\limits_{L\rightarrow\infty}\rho(\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L})^{\frac{1}{L}}.$

From the definition of $\mathbf{L}_{i}$, it is clear that

$\sum_{i=1}^{N}\mathbf{L}_{i}\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}=\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}.$
(96)

From the matrix norm property that the spectral radius of a sum of matrices is
no larger than the sum of the spectral radius of the individuals [31], we have

$\displaystyle\rho\left(\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right)\leq\sum_{i=1}^{N}\rho\left(\mathbf{L}_{i}\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right)$
(97) $\displaystyle\leq
N\rho\left(\mathbf{L}_{i^{\star}}\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right),$

and thus

$\liminf\limits_{L\rightarrow\infty}\rho\left(\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right)^{\frac{1}{L}}\leq\liminf\limits_{L\rightarrow\infty}\rho\left(\mathbf{L}_{i^{\star}}\mathbf{M}_{1}\mathbf{M}_{2}\cdots\mathbf{M}_{L}\right)^{\frac{1}{L}}.$
(98)

Using the inequalities (95) and (98), this completes the proof of Lemma 5.

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|  Wanchun Liu (M’18) received the B.S. and M.S.E. degrees in electronics and
information engineering from Beihang University, Beijing, China, and Ph.D.
from The Australian National University, Canberra, Australia. She is currently
a Research Fellow at The University of Sydney. Her research interest lies in
ultra-reliable low-latency communications and wireless networked control
systems for industrial Internet of Things (IIoT). She was a recipient of the
Chinese Government Award for Outstanding Students Abroad 2017. She is a co-
chair of the Australian Communication Theory Webinar in 2020-2021, and a Guest
Editor of the _IEEE Internet of Things Journal_ Special Issue on “Advanced
Communications, Control and Computing for Industrial Internet of Things” in
2021.
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|  Daniel E. Quevedo (S’97–M’05–SM’14–F’21) received Ingeniero Civil
Electrónico and M.Sc. degrees from Universidad Técnica Federico Santa María,
Valparaíso, Chile, in 2000, and in 2005 the Ph.D. degree from the University
of Newcastle, Australia. He is Professor of Cyberphysical Systems at the
School of Electrical Engineering and Robotics, Queensland University of
Technology (QUT), in Australia. Before joining QUT, he established and led the
Chair in Automatic Control at Paderborn University, Germany. In 2003 he
received the IEEE Conference on Decision and Control Best Student Paper Award
and was also a finalist in 2002\. He is co-recipient of the 2018 IEEE
Transactions on Automatic Control George S. Axelby Outstanding Paper Award.
Prof. Quevedo currently serves as Associate Editor for _IEEE Control Systems_
and in the Editorial Board of the _International Journal of Robust and
Nonlinear Control_. From 2015–2018 he was Chair of the IEEE Control Systems
Society _Technical Committee on Networks & Communication Systems_. His
research interests are in networked control systems, control of power
converters and cyberphysical systems security.
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|  Karl Henrik Johansson (SM’08-F’13) is Professor at the School of Electrical
Engineering and Computer Science, KTH Royal Institute of Technology, Sweden.
He received MSc and PhD degrees from Lund University. He has held visiting
positions at UC Berkeley, Caltech, NTU, HKUST Institute of Advanced Studies,
and NTNU. His research interests are in networked control systems, cyber-
physical systems, and applications in transportation, energy, and automation.
He is a member of the Swedish Research Council’s Scientific Council for
Natural Sciences and Engineering Sciences. He has served on the IEEE Control
Systems Society Board of Governors, the IFAC Executive Board, and is currently
Vice-President of the European Control Association Council. He has received
several best paper awards and other distinctions from IEEE, IFAC, and ACM. He
has been awarded Distinguished Professor with the Swedish Research Council and
Wallenberg Scholar with the Knut and Alice Wallenberg Foundation. He has
received the Future Research Leader Award from the Swedish Foundation for
Strategic Research and the triennial Young Author Prize from IFAC. He is
Fellow of the IEEE and the Royal Swedish Academy of Engineering Sciences, and
he is IEEE Control Systems Society Distinguished Lecturer.
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|  Branka Vucetic (F’03) received the BSEE, MSEE and PhD degrees in 1972, 1978
and 1982, respectively, in Electrical Engineering, from The University of
Belgrade, Belgrade. She is an ARC Laureate Fellow and Director of the Centre
of Excellence for IoT and Telecommunications at the University of Sydney. Her
current work is in the areas of wireless networks and Internet of Things. In
the area of wireless networks, she explores ultra-reliable, low-latency
techniques and transmission in millimetre wave frequency bands. In the area of
the Internet of things, Vucetic works on providing wireless connectivity for
mission critical applications. Branka Vucetic is a Fellow of IEEE, the
Australian Academy of Science and the Australian Academy of Technological
Sciences and Engineering.
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|  Yonghui Li (M’04-SM’09-F’19) received his PhD degree in November 2002 from
Beijing University of Aeronautics and Astronautics. From 1999 – 2003, he was
affiliated with Linkair Communication Inc, where he held a position of project
manager with responsibility for the design of physical layer solutions for the
LAS-CDMA system. Since 2003, he has been with the Centre of Excellence in
Telecommunications, the University of Sydney, Australia. He is now a Professor
in School of Electrical and Information Engineering, University of Sydney. He
is the recipient of the Australian Queen Elizabeth II Fellowship in 2008 and
the Australian Future Fellowship in 2012. His current research interests are
in the area of wireless communications, with a particular focus on MIMO,
millimeter wave communications, machine to machine communications, coding
techniques and cooperative communications. He holds a number of patents
granted and pending in these fields. He is now an editor for IEEE transactions
on communications and IEEE transactions on vehicular technology. He also
served as a guest editor for several special issues of IEEE journals, such as
IEEE JSAC special issue on Millimeter Wave Communications. He received the
best paper awards from IEEE International Conference on Communications (ICC)
2014, IEEE PIMRC 2017 and IEEE Wireless Days Conferences (WD) 2014.
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