# Optimal Stationary State Estimation Over Multiple
Markovian Packet Drop Channels

Jiapeng Xu<EMAIL_ADDRESS>Guoxiang Gu<EMAIL_ADDRESS>Vijay Gupta
<EMAIL_ADDRESS>Yang Tang<EMAIL_ADDRESS>Key Laboratory of Advanced
Control and Optimization for Chemical Processes, Ministry of Education, East
China University of Science and Technology, Shanghai 200237, China.  School of
Electrical Engineering and Computer Science, Louisiana State University, Baton
Rouge, LA 70803, USA.  Department of Electrical Engineering, University of
Notre Dame, Notre Dame, IN 46556, USA.

###### Abstract

In this paper, we investigate the state estimation problem over multiple
Markovian packet drop channels. In this problem setup, a remote estimator
receives measurement data transmitted from multiple sensors over individual
channels. By the method of Markovian jump linear systems, an optimal
stationary estimator that minimizes the error variance in the steady state is
obtained, based on the mean-square (MS) stabilizing solution to the coupled
algebraic Riccati equations. An explicit necessary and sufficient condition is
derived for the existence of the MS stabilizing solution, which coincides with
that of the standard Kalman filter. More importantly, we provide a sufficient
condition under which the MS detectability with multiple Markovian packet drop
channels can be decoupled, and propose a locally optimal stationary estimator
but computationally more tractable. Analytic sufficient and necessary MS
detectability conditions are presented for the decoupled subsystems
subsequently. Finally, numerical simulations are conducted to illustrate the
results on the MS stabilizing solution, the MS detectability, and the
performance of the optimal and locally optimal stationary estimators.

###### keywords:

Cicero; Catiline; orations.

###### keywords:

State estimationstabilizing solution Markovian packet dropsMarkovian jump
linear systems.

††thanks: The material in this paper was not presented at any conference.

, , ,

##  1 Introduction

Networked control systems (NCSs) attract a great deal of attention from the
control community due to their numerous advantages over conventional control
systems. Much effort has been devoted to the study of control and estimation
for NCSs over various communication channels in recent years. In the case of
wireless channels, one major issue is the occurrence of data packet drops that
may destroy the feedback stability of estimators and controllers. Hence, a
large number of existing works are focused on the stability and stabilization
of dynamic systems over packet drop channels [1, 2, 3, 4, 5, 6, 7, 8].

From the stochastic point of view, the packet drop channel is commonly modeled
as either an independent and identically distributed (i.i.d.) random process
or a two-state Markov chain by taking the temporal correlation into
consideration. Under such two modeling methods, the stability of Kalman
filtering with intermittent measurements has been well studied. In [1], the
authors show that over an i.i.d. packet drop channel, there exists a critical
packet arrival rate below which the Kalman filter is unstable. Multi-sensor
and distributed scenarios are further studied in [9] and [5], respectively.
Compared with the i.i.d. case, the stability problem of Kalman filtering with
Markovian packet drops is more complicated, yet many interesting results are
obtained. Authors of [10] are the first to introduce the notion of peak
covariance to evaluate the estimation performance. Some improved results are
obtained in [11], in comparison with [10]. By showing the equivalent stability
property for the estimation error covariances at packet reception times and
each time instant, the stability for Kalman filtering has been sufficiently
studied in [12] through exploiting the system structure. In [13], a necessary
and sufficient condition is provided for diagonalizable systems with multiple
sensors. Similar to the existence of the critical packet arrival rate shown in
the i.i.d. case, the existence of the critical curve in terms of the failure-
recovery rate is proved in [14]. For the NCS without an acknowledgment signal
sent by the actuator to the estimator, authors of [15] derive the optimal and
an approximate optimal estimator, and show the same stability for both of
them.

Since NCSs over Markovian packet drop channels can be considered as a class of
Markov jump linear systems (MJLSs)[16], an alternative approach for studying
the state estimation with Markovian packet drops is design of optimal
stationary jump estimators [17, 18], instead of the time-varying Kalman filter
(TVKF) mentioned above. Although the TVKF is known to be the optimal linear
estimator, it does not converge in the steady state and its estimation gain
explicitly depends on the realization of packet drops such that it needs to be
computed online. In this paper, we consider the stationary state estimation
problem over multiple Markovian packet drop channels. Different from the TVKF
studied in the existing literature, e.g. [1, 10, 11, 12, 14], we are
interested in the optimal stationary linear state estimator that remained
unknown for which the estimator gains can be computed off-line, leading to a
reduction in computational burden for the estimator. Compared to [17], where a
jump estimator has been designed based on the last finite measurement loss
modes for a single Markovian packet drop channel, we consider a nontrivial
case of multiple Markovian packet drop channels. More importantly, we
investigate two fundamental stability issues, the existence of the MS
stabilizing solution to the corresponding coupled algebraic Riccati equations
(CAREs) and the MS detectability for the NCS with Markovian packet drops,
which have not been studied by [17].

It is well known that estimation and control are dual problems. Their optimal
solutions are associated with their respective Riccati equations. Therefore,
feedback control problems over packet drop or fading channels are also related
to the study in this paper. In [19], it is shown that for the optimal linear
quadratic Gaussian (LQG) control with i.i.d. packet drops, the separation
principle still holds under a TCP-like protocol, and there exists a critical
arrival probability for the control data. The case of multiple lossy channels
is further considered in [20], while the LQG control with Markovian packet
drops is studied in [21]. In the presence of i.i.d. fading channels, the
existence of the MS stabilizing solution to the modified algebraic Riccati
equation is studied recently in [22]. Compared to [22], the existence of the
MS stabilizing solution in this paper is more involved, due to the temporal
correlation of packet drops.

The contributions of this paper are summarized as follows.

  1. 1.

An optimal stationary state estimator is obtained for the NCS over multiple
Markovian packet drop channels, by making use of the MJLS method. This is a
nontrivial generalization from a single channel studied in [17], since our
case brings a challenging issue that the complexity of the optimal estimator
increases exponentially with respect to the number of channels.

  2. 2.

A necessary and sufficient condition is derived for the existence of the MS
stabilizing solution to the associated filtering CAREs. It is shown that, in
addition to the MS detectability, only the controllability of the eigenvalues
on the unit circle is required, which is weaker than the stabilizability
assumption in e.g.[1, 5, 12, 17].

  3. 3.

We provide some sufficient and necessary conditions for the MS detectability
and propose a locally optimal stationary estimator, through exploring the
system structure. Specifically, the MS detectability with multiple Markovian
packet drop channels can be decoupled and a locally optimal estimator,
lowering the complexity from $N=2^{m}$ to $2m$, is obtained, resulting in
significant reduction on the computational complexity. Moreover, some analytic
MS detectability conditions are derived for the decoupled subsystems.

The remainder of this paper is organized as follows. Section 2 describes the
problem considered in this paper and gives the optimal stationary estimator.
Section 3 is focused on the existence of the MS stabilizing solution, which is
hinged on the MS detectability and the controllability of the eigenvalues on
the unit circle. The MS detectability is studied, and a locally optimal
stationary estimator is proposed in Section 4. Numerical examples are provided
in Section 5 to illustrate the stability results and the performance of the
proposed estimators, followed by some concluding remarks in Section 6.

The notations in this paper are standard. $\mathbb{R}^{m\times n}$
($\mathbb{C}^{m\times n}$ respectively) denotes the set of $m\times n$ real
(complex) matrices, with $\mathbb{R}^{n}:=\mathbb{R}^{n\times 1}$
($\mathbb{C}^{n}:=\mathbb{C}^{n\times 1}$). $\mathbb{S}^{n}_{+}$ is the set of
$n\times n$ real positive semidefinite matrices. For a matrix or vector $X$,
denote by $X^{*},X^{\prime}$, and $\bar{X}$ the conjugate transpose, transpose
and conjugate of $X$, respectively. $\rho(\cdot)$ denotes the spectral radius
of a matrix or operator, $\rm{diag}\\{\cdot\\}$ the (block) diagonalization
operation, $\rm vec\\{\cdot\\}$ the vectorization operation, and
$\rm{tr}(\cdot)$ the trace of a square matrix. $\otimes$ represents the
Kronecker product. $I_{n}$ represents the identity matrix of dimension
$n\times n$ and $\mathbf{1}$ denotes the indicator function. Finally, the
expectation operator and the probability of a random event are denoted by
$\mathbb{E}[\cdot]$ and $\Pr\\{\cdot\\}$, respectively. Other notations will
be made clear as we proceed.

##  2 Optimal Stationary State Estimator

Consider a discrete-time shift-invariant system described by

$\displaystyle x(k+1)$ $\displaystyle=Ax(k)+w(k),\ \ \ \ x(0)=x_{0},$ (1a)
$\displaystyle y_{i}(k)$ $\displaystyle=C_{i}x(k)+v_{i}(k),\ \ \ 1\leq i\leq
m,$ (1b)

with $x(k)\in\mathbb{R}^{n}$ the system state, and $y_{i}(k)\in\mathbb{R}$ the
output measurement obtained by the $i$th sensor. $w(k)$ and $v(k)={\rm
vec}\\{v_{1}(k),\ldots,v_{m}(k)\\}$ are mutually independent white noises
having mean zero and covariances $Q\geq 0$ and $R>0$, respectively. The
initial state $x_{0}$ is independent of $w(k)$ and $v(k)$, with mean
$\overline{x}_{0}$ and covariance $\Pi_{0}$. Define $y(k):={\rm
vec}\\{y_{1}(k),\ldots,y_{m}(k)\\}$ and
${C:=[C^{\prime}_{1},\ldots,C^{\prime}_{m}]^{\prime}}$. We assume without loss
of generality that $R$ is a diagonal matrix.

######  Remark 1.

It is worth mentioning that dimension one of the output measurement from each
sensor does not pose constraints on the results in this paper from being
generalized to the case of arbitrary dimensions. Since our focus is on the
multiple Markovian packet drop processes, we consider the collective
measurement $y(k)\in\mathbb{R}^{m}$ for convenience. $\Box$

We assume that the collective measurement $y(k)$ is sent through unreliable
channels suffering from packet drops; see Fig. 1. The signal received at the
remote estimator is given by

$\displaystyle{y_{\rm r}(k)}=\Gamma(k)y(k),$ (2)

where $\Gamma(k)\in\mathbb{R}^{m\times m}$ represents the presence of the $m$
packet drop channels in the diagonal form:

$\displaystyle\Gamma(k)={\rm
diag}\\{\gamma_{1}(k),\gamma_{2}(k),\ldots,\gamma_{m}(k)\\}.$ (3)

Here, $\gamma_{i}(k)\in\\{0,1\\}$ for $1\leq i\leq m$. If $\gamma_{i}(k)=1$,
$y_{i}(k)$ arrives at the estimator; otherwise $y_{i}(k)$ is dropped.
Moreover, $\\{\gamma_{i}(k)\\}_{i=1}^{m}$ are independent of each other, and
each $\gamma_{i}(k)$ is modeled as a time-homogeneous two-state Markov chain
with the transition probability matrix (TPM)

$\displaystyle P_{i}=\left[\begin{array}[]{cc}1-q_{i}&q_{i}\\\
p_{i}&1-p_{i}\end{array}\right],\ \ i=1,\ldots,m,$ (6)

where $q_{i}=\Pr\\{\gamma_{i}(k+1)=1|\gamma_{i}(k)=0\\}$ is the recovery rate
and $p_{i}=\Pr\\{\gamma_{i}(k+1)=0|\gamma_{i}(k)=1\\}$ is the failure rate.
Denote $\pi_{i,l}(k):=\Pr\\{\gamma_{i}(k)={l-1}\\},~{}l\in\\{1,2\\}$. Assume
that $0<p_{i},q_{i}<1$. Then, there exists the limit probability distribution
$\pi_{i}=\left[\ \pi_{i,1}~{}\pi_{i,2}\ \right]$ with

$\pi_{i,1}=\frac{p_{i}}{p_{i}+q_{i}},\ \ \
\pi_{i,2}=\frac{q_{i}}{p_{i}+q_{i}}.$

Figure 1: State estimation over packet drop channels.

Different from the TVKF with Markovian packet drops studied in the existing
literature [10, 11, 12, 13, 14, 15], we are interested in the optimal
stationary linear state estimator for which the estimator gains can be
computed off-line. We will apply the filtering theory from MJLSs [16] to
derive such an estimator. It is noted that the NCS described in (1) and (2)
can be rewritten into the following jump form:

$\displaystyle x(k+1)$ $\displaystyle=Ax(k)+w(k),\ \ \ x(0)=x_{0},$ (7a)
$\displaystyle{y_{\rm r}}(k)$
$\displaystyle=H_{\theta(k)}x(k)+D_{\theta(k)}v(k),$ (7b)

where $\theta(k)\in\mathcal{N}:=\\{1,\ldots,2^{m}\\}$, specified by

$\displaystyle\theta(k)=1+\sum_{i=1}^{m}2^{i-1}\gamma_{i}(k),$ (8)

is the Markovian jump variable. It follows that $H_{\theta(k)}=\Gamma(k)C$ and
$D_{\theta(k)}=\Gamma(k)$. For simplicity, denote $N=2^{m}$ in the remainder
of the paper.

######  Lemma 1.

Given the way of computing $\theta(k)$ in (8), the TPM $P$ for the Markov
process $\\{\theta(k)\\}$ is given by

$\displaystyle P:=P_{m}\otimes P_{m-1}\otimes\cdots\otimes P_{1},\ \ \ 1\leq
i,j\leq 2^{m},$ (9)

where $P=\left[\ p_{ij}\ \right]$,
$p_{ij}=\Pr\\{\theta(k+1)=j|\theta(k)=i\\}$. Moreover, by defining
$\mu_{i}(k):=\Pr\\{\theta(k)=i\\}$ for each $i\in\mathcal{N}$, and
$\mu(k):=\left[\ \mu_{1}(k)\ \cdots\ \mu_{N}(k)\ \right]$, $\theta(k)$ has a
unique stationary distribution $\mu=\left[\ \mu_{1}\ \cdots\ \mu_{N}\
\right]$, i.e., $\lim_{k\to\infty}\mu_{i}(k)=\mu_{i}$ for $i\in\mathcal{N}$
and

$\displaystyle\mu=\pi_{m}\otimes\pi_{m-1}\otimes\ldots\otimes\pi_{1}.$ (10)

Proof. We will prove that (9) holds for any $m\geq 1$ by induction. Obviously,
(9) holds when $m=1$. Assume that for $m=l$ satisfying $l>1$, we have

$\displaystyle P^{\\{l\\}}:=\left[\ p_{ij}^{\\{l\\}}\
\right]=P_{l}\otimes\cdots\otimes P_{1},\ \ \ 1\leq i,j\leq 2^{l}.$ (11)

In this case, denote $\vartheta(k)$ as the Markovian jump variable such that
$p_{ij}^{\\{l\\}}=\Pr\\{\vartheta(k+1)=j|\vartheta(k)=i\\}$. For $m=l+1$,
suppose that sensor $l+1$ is the newly added sensor compared with the case
$m=l$. Define

$\displaystyle P^{\\{l+1\\}}:=\left[\ p_{rs}^{\\{l+1\\}}\ \right],\ \ \ 1\leq
r,s\leq 2^{l+1}$ (12)

as the corresponding TPM. Then by (8) and the assumption that
$\\{\gamma_{i}(k)\\}_{i=1}^{m}$ are independent,

$\displaystyle\Pr\\{\gamma_{l+1}(k+1),\vartheta(k+1)=j|\gamma_{l+1}(k),\vartheta(k)=i\\}$
$\displaystyle=$ $\displaystyle
p_{(2^{l}\gamma_{l+1}(k)+i)(2^{l}\gamma_{l+1}(k+1)+j)}^{\\{l+1\\}}$
$\displaystyle=$
$\displaystyle\Pr\\{\gamma_{l+1}(k+1)|\gamma_{l+1}(k)\\}\Pr\\{\vartheta(k+1)=j|\vartheta(k)=i\\},$
$\displaystyle~{}\gamma_{l+1}(k+1),\gamma_{l+1}(k)\in\\{0,1\\},\ 1\leq i,j\leq
2^{l}.$ (13)

Therefore, we conclude that

$\displaystyle P^{\\{l+1\\}}=P_{l+1}\otimes P^{\\{l\\}}=P_{l+1}\otimes
P_{l}\otimes\cdots\otimes P_{1}.$ (14)

Similarly, there holds

$\displaystyle\mu(k)=\pi_{m}(k)\otimes\pi_{m-1}(k)\otimes\ldots\otimes\pi_{1}(k).$
(15)

Thus,
$\mu=\lim_{k\to\infty}\mu(k)=\pi_{m}\otimes\pi_{m-1}\otimes\ldots\otimes\pi_{1}$.
The uniqueness follows from the limit probability distributions
$\\{\pi_{i}\\}_{i=1}^{m}$. The proof is thus complete.

Let $\hat{x}(k)$ be the state estimation for $x(k)$ in (7a). Now, consider a
dynamic Markovian jump linear estimator described by

$\displaystyle\hat{x}(k+1)=A\hat{x}(k)+L_{\theta(k)}[y_{\rm
r}(k)-H_{\theta(k)}\hat{x}(k)].$ (16)

At each time instant, the estimator gain $L_{\theta(k)}$ is chosen from a
finite set of pre-computed values, i.e.,
$L_{\theta(k)}\in\\{L_{i}\in\mathbb{R}^{n\times m}\\}_{i\in\mathcal{N}}$. This
implies that the estimator gain $L_{\theta(k)}$ depends only on $\theta(k)$
(rather than on all the past modes $\\{\theta(0),\ldots,\theta(k)\\}$,
corresponding to the TVKF), which is an important feature of the jump
estimator. We aim to find a set of optimal gains, denoted by
$\\{K_{i}\in\mathbb{R}^{n\times m}\\}_{i\in\mathcal{N}}$ such that with
$L_{i}=K_{i}$ for each $i\in\mathcal{N}$, the stationary estimation cost

$\displaystyle J(\infty)=\lim_{k\to\infty}\mathrm{E}[\|x(k)-\hat{x}(k)\|^{2}]$
(17)

is bounded and minimized.

######  Theorem 1.

For the system dynamics described in (7), assume that there exists the MS
stabilizing solution (see Definition 2 in Section 3) $Y=(Y_{1},\ldots,Y_{N})$
to the following CAREs

$\displaystyle Y_{j}=\sum_{i=1}^{N}p_{ij}\big{[}$ $\displaystyle
AY_{i}A^{\prime}-AY_{i}H^{\prime}_{i}(H_{i}Y_{i}H^{\prime}_{i}+\mu_{i}R)^{-1}H_{i}Y_{i}A^{\prime}$
$\displaystyle+\mu_{i}Q\big{]},\ \ \ j\in\mathcal{N}.$ (18)

Then the optimal stationary gains for the jump estimator in (16) are given by

$\displaystyle
K_{j}=K_{j}(Y):=AY_{j}H^{\prime}_{j}(H_{j}Y_{j}H^{\prime}_{j}+\mu_{j}R)^{-1}$
(19)

for $j\in\mathcal{N}$, and the stationary optimal cost is given by

$\lim_{k\to\infty}J(k)=\sum_{i=1}^{N}{\rm tr}(Y_{i}).$

Proof. The result can be obtained by using the filtering theory of MJLSs
(Chapter 5 in [16]). To be more clear, define the estimation error by
$e(k):=x(k)-\hat{x}(k)$ and set
$Y_{j}(k):=\mathrm{E}\left[e(k)e(k)^{\prime}\mathbf{1}_{\theta(k)=j}\right],j\in\mathcal{N}$
such that

$\displaystyle\mathrm{E}\left[e(k)e(k)^{\prime}\right]=\sum_{j=1}^{N}Y_{j}(k).$

From (7) and (16) we have

$\displaystyle e(k+1)=$
$\displaystyle(A-L_{\theta(k)}(k)H_{\theta(k)})e(k)+w(k)$ $\displaystyle-
L_{\theta(k)}(k)D_{\theta(k)}v(k)$ (20)

with $L_{\theta(k)}$ replaced by $L_{\theta(k)}(k)$. From Proposition 3.35 2)
in [16], we have

$\displaystyle Y_{j}(k+1)=$
$\displaystyle\sum_{i=1}^{N}p_{ij}[(A-L_{i}(k)H_{i})Y_{i}(k)(A-L_{i}(k)H_{i})^{\prime}$
$\displaystyle+\mu_{i}$
$\displaystyle(k)(Q+L_{i}(k)D_{i}RD^{\prime}_{i}L^{\prime}_{i}(k)],\
j\in\mathcal{N}.$ (21)

By solving $\partial\text{tr}({Y_{j}(k+1)})/\partial{L_{i}(k)}=0$ for all
$i\in\mathcal{N}$, it gives $L_{i}(k)=K_{i}(k)$ with

$\displaystyle
K_{i}(k)=AY_{i}(k)H_{i}^{\prime}(H_{i}Y_{i}(k)H^{\prime}_{i}+\mu_{i}(k)D_{i}RD^{\prime}_{i})^{\dagger},$
(22)

which minimizes $\text{tr}(Y_{j}(k+1))$. The Moore-Penrose inverse is used,
since matrix $H_{i}Y_{i}H^{\prime}_{i}+\mu_{i}(k)D_{i}RD^{\prime}_{i}$ in
general is positive semi-definite and the relation
${\mathcal{R}}\\{H_{i}Y_{i}(k)A^{\prime}\\}\subseteq{\mathcal{R}}\\{H_{i}Y_{i}H^{\prime}_{i}+\mu_{i}(k)D_{i}RD^{\prime}_{i}\\}$
holds for any $i\in\mathcal{N}$ (${\mathcal{R}}\\{\cdot\\}$ denotes the range
space). With $L_{i}(k)=K_{i}(k)$, (2) becomes

$\displaystyle Y_{j}(k+1)=$
$\displaystyle\sum_{i=1}^{N}p_{ij}\big{[}AY_{i}(k)A^{\prime}-AY_{i}(k)H^{\prime}_{i}(H_{i}Y_{i}(k)H^{\prime}_{i}$
$\displaystyle+$
$\displaystyle\mu_{i}(k)D_{i}RD_{i}^{\prime})^{\dagger}H_{i}Y_{i}(k)A^{\prime}+\mu_{i}(k)Q\big{]}.$
(23)

As $k\to\infty$, (2) converges to the CAREs (1), $Y_{j}(k+1)\to Y_{j}$, and
$K_{i}(k)\to K_{i}$, where $(H_{i}Y_{i}H^{\prime}_{i}+\mu_{i}R)^{-1}$ takes
place of $(H_{i}Y_{i}H^{\prime}_{i}+\mu_{i}D_{i}RD^{\prime}_{i})^{\dagger}$
without changing the values of $Y_{j}$ and $K_{i}$, following from the special
form of $\Gamma(k)$. Also, the stationary optimal cost is
$\sum_{i=1}^{N}\text{tr}(Y_{i})$.

######  Remark 2.

It is seen from Theorem 1 that there are $N=2^{m}$ CAREs and gains for the
optimal stationary estimator, which may cause a difficulty on implementation
when the number of sensors is large. We will deal with this issue in Section 4
by proposing a locally optimal stationary estimator. $\Box$

##  3 MS Stabilizing Solution

From Theorem 1, it is known that the optimal stationary estimator is based on
the existence of the MS stabilizing solution to the CAREs (1). Therefore, we
focus on the necessary and sufficient condition for this existence problem in
this section.

###  3.1 Preliminaries of MJLSs

Let $\mathbb{H}^{m,n}$ represent the linear space composed of all
${N}$-sequences of real matrices $V=(V_{1},\ldots,V_{N})$ with
$V_{i}\in\mathbb{R}^{m,n}$. In the case of $m=n$, we denote
$\mathbb{H}^{n}=\mathbb{H}^{n,n}$, and define

$\displaystyle{\mathbb{H}^{n*}}$
$\displaystyle{:=\\{V=(V_{1},\ldots,V_{N})\in\mathbb{H}^{n};V_{i}=V^{\prime}_{i}\
\forall i\in\mathcal{N}\\},}$ $\displaystyle\mathbb{H}_{+}^{n}$
$\displaystyle:=\\{V=(V_{1},\ldots,V_{N})\in\mathbb{H}^{n*};V_{i}\geq 0\
\forall i\in\mathcal{N}\\}.$

For $V=(V_{1},\ldots,V_{N})\in\mathbb{H}^{n*}$ and
$S=(S_{1},\ldots,S_{N})\in\mathbb{H}^{n*}$, write that $V\geq S$ (or $V>S$) if
$V-S=(V_{1}-S_{1},\ldots,V_{N}-S_{N})\in\mathbb{H}^{n}_{+}$ (or
$V_{i}-S_{i}>0$). It is known that $\mathbb{H}^{m,n}$ can be equipped with the
inner product:

$\displaystyle\langle V,S\rangle=\sum_{i=1}^{N}{\rm tr}(V^{\prime}_{i}S_{i}),$
(24)

where $V=(V_{1},\ldots,V_{N})$ and $S=(S_{1},\ldots,S_{N})$ are in
$\mathbb{H}^{m,n}$. For $V=(V_{1},\ldots,V_{N})\in\mathbb{H}^{n}$ and
$L=(L_{1},\ldots,L_{N})\in\mathbb{H}^{n,m}$, define the operators
$\widetilde{\mathcal{L}}(\cdot)=(\widetilde{\mathcal{L}}_{1}(\cdot),\ldots,\widetilde{\mathcal{L}}_{N}(\cdot))$
and
$\mathcal{L}(\cdot)=(\mathcal{L}_{1}(\cdot),\ldots,\mathcal{L}_{N}(\cdot))$ as

$\displaystyle\widetilde{\mathcal{L}}_{j}(V)$
$\displaystyle:=\sum_{i=1}^{{N}}p_{ij}(A+L_{i}H_{i})V_{i}(A+L_{i}H_{i})^{\prime},$
(25) $\displaystyle\mathcal{L}_{j}(V)$
$\displaystyle:=\sum_{i=1}^{{N}}p_{ij}AV_{i}A^{\prime},\ \ \ j\in\mathcal{N}.$
(26)

Their respective adjoint operators $\widetilde{\mathcal{L}}^{*}$ and
$\mathcal{L}^{*}$ are given by

$\displaystyle\widetilde{\mathcal{L}}_{i}^{*}(V)$
$\displaystyle=\sum_{j=1}^{N}p_{ij}(A+L_{i}H_{i})^{\prime}V_{j}(A+L_{i}H_{i}),$
(27) $\displaystyle\mathcal{L}_{i}^{*}(V)$
$\displaystyle=\sum_{j=1}^{N}p_{ij}A^{\prime}V_{j}A,\ \ \ i\in\mathcal{N},$
(28)

satisfying the following equalities

$\displaystyle\langle\widetilde{\mathcal{L}}(V),S\rangle$
$\displaystyle=\langle V,\widetilde{\mathcal{L}}^{*}(S)\rangle,$ (29a)
$\displaystyle\langle{\mathcal{L}}(V),S\rangle$ $\displaystyle=\langle
V,{\mathcal{L}}^{*}(S)\rangle.$ (29b)

Recall the inner product defined in (24). Denote

$\displaystyle\mathbf{A}$ $\displaystyle:=(\underbrace{A,\ldots,A}_{N}),\
\mathbf{Q}:=(\underbrace{Q,\ldots,Q}_{N}),$ $\displaystyle\mathbf{H}$
$\displaystyle:=(H_{1},\ldots,H_{N}),\ \mathbf{p}:=\\{p_{ij}\\},\
i,j\in\mathcal{N}.$

Next, we introduce several notions regarding MJLSs.

######  Definition 1 (Def. 1, [23]).

The system described in (7), or simply $(\mathbf{H},\mathbf{A},\mathbf{p})$ is
said to be MS detectable, if there exists
$L=(L_{1},\ldots,L_{N})\in\mathbb{H}^{n,m}$ such that
$\rho(\widetilde{\mathcal{L}})<1$.

######  Definition 2 (Def. 5.7, [16]).

$Y=(Y_{1},\ldots,Y_{N})\in{\mathbb{H}}_{+}^{n}$ is said to be the MS
stabilizing solution for the CAREs (1), if $\rho(\widetilde{\mathcal{L}})<1$
holds with $L_{i}=-K_{i}(Y)$ for $i\in\mathcal{N}$.

Following the definition of the observability for MJLSs in Theorem 3 of [24],
we introduce the following definition for the uncontrollable eigenvalue of the
MJLS in (7).

######  Definition 3.

A real number $\lambda\geq 0$ is said to be an uncontrollable eigenvalue for
the pair $(\mathbf{A},\mathbf{Q})$ if there exists an eigenvector
$V=(V_{1},\ldots,V_{N})\in{\mathbb{H}}_{+}^{n}\backslash\\{0\\}$ of
$\mathcal{L}^{*}$ such that

$\displaystyle{\rm(a)}\ \mathcal{L}^{*}(V)=\lambda V,\ \ \ {\rm(b)}\ QV_{i}=0\
\ \forall i\in\mathcal{N}.$ (30)

###  3.2 Existence of MS Stabilizing Solution

We begin with the notion of the maximal solution. For
$X=(X_{1},\ldots,X_{N})\in\mathbb{H}^{n*}$, denote
$\mathcal{P}_{j}(X)=\sum_{i=1}^{N}p_{ij}X_{j}$ and define the operators
$\mathcal{X}(\cdot)=(\mathcal{X}_{1}(\cdot),\ldots,\mathcal{X}_{N}(\cdot))$
and
$\mathcal{R}(\cdot)=(\mathcal{R}_{1}(\cdot),\ldots,\mathcal{R}_{N}(\cdot))$ as

$\displaystyle\mathcal{X}_{j}(X)$
$\displaystyle:=\left[\begin{array}[]{cc}A\mathcal{P}_{j}(X)A^{\prime}+\mu_{j}Q-X_{j}&A\mathcal{P}_{j}(X)H^{\prime}_{j}\\\
H_{j}\mathcal{P}_{j}(X)A^{\prime}&\mu_{j}R+H_{j}\mathcal{P}_{j}(X)H^{\prime}_{j}\end{array}\right],$
(33) $\displaystyle\mathcal{R}_{j}(X)$
$\displaystyle:=\mu_{j}R+H_{j}\mathcal{P}_{j}(X)H^{\prime}_{j},\ \ \
j\in\mathcal{N}.$ (34)

Then define the following set

$\displaystyle{\Omega:=\left\\{X\in\mathbb{H}^{n*}|\mathcal{X}(X)\geq
0,\mathcal{R}(X)>0\right\\}.}$ (35)

######  Definition 4.

A solution $Y^{+}=(Y_{1}^{+},\ldots,Y_{N}^{+})$ to the CAREs (1) is said to be
the maximal solution if $Y^{+}\geq Y$ for any $Y=(Y_{1},\ldots,Y_{N})$ with
$Y_{j}=\mathcal{P}_{j}(X)$, $j\in\mathcal{N}$, $X\in\Omega$.

The maximal solution can be numerically computed by solving the following
convex programming problem [25]:

$\displaystyle\max\left\\{{\rm tr}\left(\sum_{j=1}^{N}X_{j}\right):\ \
X=(X_{1},\ldots,X_{N})\in\Omega\right\\}.$ (36)

######  Lemma 2 ([25]).

Recall Definitions 1 and 2.

  * (i)

Suppose that $(\mathbf{H},\mathbf{A},\mathbf{p})$ is MS detectable. Then there
exists the maximal solution
$Y^{+}=(Y_{1}^{+},\ldots,Y_{N}^{+})\in\mathbb{H}_{+}^{n}$ to the CAREs (1).
Moreover, with $L_{i}=-K_{i}(Y^{+})$ for $1\leq i\leq N$, there holds
$\rho(\widetilde{\mathcal{L}}^{*})\leq 1$.

  * (ii)

There exists at most one MS stabilizing solution to the CAREs (1), which
coincides with the maximal solution.

Having the solution $X^{+}=(X_{1}^{+},\ldots,X_{N}^{+})$ for the convex
programming problem (36), the maximal solution is
$Y^{+}=(Y_{1}^{+},\ldots,Y_{N}^{+})$ with $Y_{j}^{+}=\mathcal{P}_{j}(X^{+})$.
By Lemma 2 (ii), $Y^{+}$ is also the MS stabilizing solution to the CAREs (1),
if the latter exists. The following result is important, which can be applied
directly to the main result in this section.

######  Theorem 2.

The CAREs in (1) admit the MS stabilizing solution, if and only if

  * 1)

$(\mathbf{H},\mathbf{A},\mathbf{p})$ is MS detectable.

  * 2)

$\lambda=1$ is not an uncontrollable eigenvalue for $(\mathbf{A},\mathbf{Q})$.

Proof. See Appendix.

We remark that a similar result to Theorem 2 for the control CAREs is given in
Corollary 14 of [26]. Nonetheless, a self-contained and independent proof for
the CAREs (1) is provided in Appendix, in which the equalities (29) in terms
of inner product play an important role. Before presenting the main result in
this section, we need the following technical lemma.

######  Lemma 3.

Let $\lambda>0$. If $A^{\prime}XA=\lambda X$ and $QX=0$ admit a solution
$X\in\mathbb{S}_{+}^{n}\backslash\\{0\\}$, then there exist a nonzero vector
$x_{0}\in\mathbb{C}^{n}$ and $\omega_{0}\in\mathbb{R}$ such that

$\displaystyle A^{\prime}x_{0}=\sqrt{\lambda}e^{j\omega_{0}}x_{0},\ \
Qx_{0}=0.$ (37)

That is, $\sqrt{\lambda}e^{j\omega_{0}}$ is an uncontrollable eigenvalue for
$(A,Q)$ according to the well-known Popov–Belevitch–Hautus (PBH) test.

Proof. The hypothesis on $X\in\mathbb{S}_{+}^{n}\backslash\\{0\\}$ implies
that $X=GG^{\prime}$ with $G\in\mathbb{R}^{n\times r}$ and $r={\rm
rank}\\{X\\}>0$. It follows that $\lambda GG^{\prime}=A^{\prime}GG^{\prime}A$
and $QGG^{\prime}=0$. Hence, there exists an orthogonal matrix
$U\in\mathbb{R}^{r\times r}$ such that

$\displaystyle\sqrt{\lambda}GU=A^{\prime}G,\ \ \ QGU=0.$ (38)

Since all the eigenvalues of an orthogonal matrix are on the unit circle,
there exists a nonzero vector $u_{0}\in\mathbb{C}^{r}$ such that
$Uu_{0}=e^{j\omega_{0}}u_{0}$ for some $\omega_{0}\in\mathbb{R}$. Multiplying
the two equalities in (38) by $u_{0}$ from right verifies the
eigenvalue–eigenvector equation and $Qx_{0}=0$ in (37) by taking
$x_{0}=Gu_{0}$.

######  Theorem 3.

The CAREs in (1) admit the MS stabilizing solution, if and only if

  * 1)

$(\mathbf{H},\mathbf{A},\mathbf{p})$ is MS detectable;

  * 2)

$(A,Q)$ does not have uncontrollable eigenvalues on the unit circle, i.e.,

$\displaystyle{\rm rank}\left\\{\left[\begin{array}[]{cc}\lambda
I_{n}-A&Q\end{array}\right]\right\\}=n,\ \forall|\lambda|=1,$ (40)

where $\lambda$ is an eigenvalue of $A$.

Proof. By Theorem 2, it suffices to show that $1$ is an uncontrollable
eigenvalue for $(\mathbf{A},\mathbf{Q})$, if and only if there exists some
eigenvalue on the unit circle that is uncontrollable for $(A,Q)$. We first
show the sufficiency. By the PBH test, there exist some $|\lambda|=1$ and
$0\neq v\in\mathbb{C}^{n}$ such that

$\displaystyle A^{\prime}v=\lambda v,\ \ \ Qv=0.$ (41)

Let $V_{i}:=vv^{*}+\bar{v}\bar{v}^{*}\in\mathbb{S}_{+}^{n}$ for all
$i\in\mathcal{N}$ such that $V=(V_{1},\ldots V_{N})\in{\mathbb{H}}_{+}^{n}$.
Then by (28),

$\displaystyle\mathcal{L}_{i}^{*}(V)=$
$\displaystyle\sum_{j=1}^{N}p_{ij}A^{\prime}V_{i}A=A^{\prime}(vv^{*}+\bar{v}\bar{v}^{*})A$
$\displaystyle=$ $\displaystyle
A^{\prime}v(A^{\prime}v)^{*}+A^{\prime}\bar{v}(A^{\prime}\bar{v})^{*}$
$\displaystyle=$
$\displaystyle\lambda\bar{\lambda}vv^{*}+\lambda\bar{\lambda}\bar{v}\bar{v}^{*}=V_{i}\
\ \forall\ i\in\mathcal{N}.$ (42)

In addition, it is easy to see that $QV_{i}=0$ $\forall i\in\mathcal{N}$.
Therefore, 1 is an uncontrollable eigenvalue for the pair
$(\mathbf{A},\mathbf{Q})$.

To show the necessity, assume that there exists
$V=(V_{1},\ldots,V_{N})\in{\mathbb{H}}_{+}^{n}\backslash\\{0\\}$ such that
$\mathcal{L}_{i}^{*}(V)=V_{i}$ and $QV_{i}=0$. Recall the limit probability
distribution $\mu$ in (10) of Lemma 1 where
$\mu=\left[\begin{array}[]{cccc}\mu_{1}&\mu_{2}&\cdots&\mu_{N}\end{array}\right]\neq
0$ is a positive row vector. Since $\mu P=\mu$, there holds

$\displaystyle\mu_{j}=\sum_{i=1}^{N}\mu_{i}p_{ij}\ \ \ \forall\
j\in\mathcal{N}.$

By the above equality, $V_{i}=\mathcal{L}_{i}^{*}(V)$ with
$\mathcal{L}_{i}^{*}(V)$ defined in (28), and $QV_{i}=0$ $\forall
i\in\mathcal{N}$, we have

$\displaystyle X:=$
$\displaystyle\sum_{i=1}^{N}\mu_{i}V_{i}=\sum_{i=1}^{N}\mu_{i}\mathcal{L}_{i}^{*}(V)=\sum_{i=1}^{N}\sum_{j=1}^{N}\mu_{i}p_{ij}A^{\prime}V_{j}A$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{N}A^{\prime}V_{j}A\left(\sum_{i=1}^{N}\mu_{i}p_{ij}\right)=\sum_{j=1}^{N}\mu_{j}A^{\prime}V_{j}A$
$\displaystyle=$ $\displaystyle
A^{\prime}\left(\sum_{j=1}^{N}\mu_{j}V_{j}\right)A=A^{\prime}XA,$
$\displaystyle QX=$ $\displaystyle\sum_{i=1}^{N}\mu_{i}QV_{i}=0.$

There thus hold $A^{\prime}XA=X$ and $QX=0$. An application of Lemma 3 with
$\lambda=1$ concludes that $(A,Q)$ has at least an uncontrollable eigenvalue
on the unit circle. The proof is complete.

Condition 2) in Theorem 3 shows that the controllability requirement is the
same as that arising in the study of the standard ARE [27], which is an
interesting result. In Theorem 3, the matrix rank condition (40) is simple to
check, while the MS detectability for $(\mathbf{H},\mathbf{A},\mathbf{p})$ is
not straightforward, which needs to be numerically verified by solving a
feasibility problem generally, in terms of $2^{m}$ linear matrix inequalities
(LMIs); see (49) in the next section. The other issue is the exponential
complexity of the optimal stationary estimator in Theorem 1. In the next
section, we will derive some sufficient and necessary conditions for the MS
detectability by exploring the system structure, which show directly how
system parameters influence the MS detectability, and propose a locally
optimal stationary estimator that has a linear complexity.

##  4 MS Detectability and Locally Optimal Stationary Estimator

###  4.1 MS detectability for $(\mathbf{H},\mathbf{A},\mathbf{p})$

Theorem 3.9 of [16] describes the MS stability of MJLSs, based on which we
also have the following definition for the MS detectability for system (7),
coinciding exactly with Definition 1.

######  Definition 5.

We say that $(\mathbf{H},\mathbf{A},\mathbf{p})$ is MS detectable, if there
exist $\\{X_{i}>0\\}_{i=1}^{N}$ and
$L=(L_{1},\ldots,L_{N})\in\mathbb{H}^{n,m}$ such that either of the following
two inequalities holds:

$\displaystyle X_{i}$
$\displaystyle>\sum_{j=1}^{N}p_{ij}(A+L_{j}H_{j})^{\prime}X_{j}(A+L_{j}H_{j})\
\ \forall i\in\mathcal{N},$ (43) $\displaystyle X_{j}$
$\displaystyle>\sum_{i=1}^{{N}}p_{ij}(A+L_{i}H_{i})X_{i}(A+L_{i}H_{i})^{\prime}\
\ \forall j\in\mathcal{N}.$ (44)

The pair $(C,A)$ is assumed to be detectable throughout the section, which is
clearly weaker than the MS detectability of
$(\mathbf{H},\mathbf{A},\mathbf{p})$ that relates to channel parameters of
packet drops. We first provide an analytic necessary condition for the MS
detectability.

######  Theorem 4.

The triple $(\mathbf{H},\mathbf{A},\mathbf{p})$ is MS detectable, only if

$\displaystyle\prod_{i=1}^{N}(1-q_{i})\rho^{2}(A)<1.$ (45)

Proof. By (44), the MS detectability for $(\mathbf{H},\mathbf{A},\mathbf{p})$
implies that there exist $\\{X_{i}>0\\}_{i=1}^{N}$ such that

$\displaystyle X_{1}$
$\displaystyle>\sum_{i=2}^{{N}}p_{ij}(A+L_{i}H_{i})X_{i}(A+L_{i}H_{i})^{\prime}$
(46) $\displaystyle\ \ \
+\prod_{i=1}^{N}(1-q_{i})AX_{1}A\geq\prod_{i=1}^{N}(1-q_{i})AX_{1}A^{\prime}.$

This implies that $A\sqrt{\prod_{i=1}^{N}(1-q_{i})}$ must be a Schur stability
matrix, leading to the inequality in (45).

The MS detectability for $(\mathbf{H},\mathbf{A},\mathbf{p})$ can be
numerically verified by solving the following feasibility problem of LMIs that
can be obtained by using the MS detectability condition in (43) and the Schur
complement repeatedly.

######  Proposition 1.

Denote $X_{\rm d}:={\rm diag}\\{X_{1},\ldots,X_{N}\\}$ and
$\Psi_{i}(X,\Omega)=\left[\ \Psi_{i1}\ \Psi_{i2}\ \cdots\ \Psi_{iN}\ \right]$
where $\Psi_{ij}=\sqrt{p_{ij}}(A^{\prime}X_{j}+H_{j}^{\prime}\Omega_{j})$ for
$1\leq j\leq N$. Then, $(\mathbf{H},\mathbf{A},\mathbf{p})$ is MS detectable,
if and only if there exist $\\{X_{i}>0\\}_{i=1}^{N}$ and
$\\{\Omega_{i}\in\mathbb{R}^{n\times m}\\}_{i=1}^{N}$ such that LMI

$\displaystyle\left[\begin{array}[]{cc}X_{i}&\Psi_{i}(X,\Omega)\\\
\Psi_{i}(X,\Omega)^{\prime}&X_{\rm d}\end{array}\right]>0$ (49)

holds for each $i\in\mathcal{N}$.

A significant challenge to checking the MS detectability from Proposition 1
lies in the complexity, due to $N=2^{m}$ LMIs in (49). We will focus on
lowering the complexity from $N=2^{m}$ to $2m$. This is indeed possible
through decoupling the MS detectability for
$(\mathbf{H},\mathbf{A},\mathbf{p})$ into that for $m$ subsystems,
respectively.

Denote the state estimation error by

$\displaystyle e(k):=x(k)-\hat{x}(k).$ (50)

Taking the difference between (1a) and (16) yields

$\displaystyle e(k+1)=(A-L_{\theta(k)}H_{\theta(k)})e(k)$ (51)

after removing the terms about noises. Then the existence of
$\\{L_{\theta(k)}\\}$ that achieve the MS stability for the error dynamics
described in (51) is equivalent to the MS detectability for
$(\mathbf{H},\mathbf{A},\mathbf{p})$. Without loss of generality, the pair
$(C,A)$ for system (1) is assumed to be of the following Wonham decomposition
form [28]:

$\displaystyle A=\left[\begin{array}[]{cccc}A_{1}&0&\cdots&0\\\
A_{21}&A_{2}&\ddots&\vdots\\\ \vdots&\ddots&\ddots&0\\\
A_{m1}&\cdots&A_{m(m-1)}&A_{m}\end{array}\right],\
C=\left[\begin{array}[]{cccc}c_{1}&0&\cdots&0\\\ 0&c_{2}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&0\\\ 0&\cdots&0&c_{m}\end{array}\right],$ (60)

where $A_{i}\in\mathbb{R}^{n_{i}\times n_{i}}$, $c_{i}\in\mathbb{R}^{1\times
n_{i}}$, $\sum_{i=1}^{m}n_{i}=n$, and each pair $(c_{i},A_{i})$ is detectable
under the detectability of $(C,A)$. Define

$\displaystyle\theta_{i}(k)$ $\displaystyle:=\gamma_{i}(k)+1,\ \ \
h_{i,\theta_{i}(k)}:=\gamma_{i}(k)c_{i},$ $\displaystyle\mathbf{A}_{i}$
$\displaystyle:=(A_{i},A_{i}),\ \mathbf{h}_{i}:=(h_{i,1},h_{i,2}),\
\mathbf{p}_{i}:=\\{p_{lj}^{(i)}\\},$

for $1\leq i\leq m$, where $p_{lj}^{(i)}$ is the $(l,j)$th element of the TPM
$P_{i}$ in (6). Let us introduce a particular $L_{\theta(k)}$ that is in the
block diagonal form, conformal to that of $C$:

$\displaystyle\bar{L}_{\theta(k)}:={\rm
diag}\\{\ell_{\theta_{1}(k)},\ldots,\ell_{\theta_{m}(k)}\\},\ \
\ell_{\theta_{i}(k)}\in\mathbb{R}^{n_{i}}.$ (61)

######  Theorem 5.

If $(\mathbf{h}_{i},\mathbf{A}_{i},\mathbf{p}_{i})$ is MS detectable for all
$1\leq i\leq m$, then $(\mathbf{H},\mathbf{A},\mathbf{p})$ is MS detectable.

Proof. For system (7), consider a similarity transform $x(k)=S\tilde{x}(k)$
with

$\displaystyle{S={\rm
diag}\\{I_{n_{1}},\epsilon^{-1}I_{n_{2}},\ldots,\epsilon^{1-m}I_{n_{m}}\\}},\
\ \epsilon>0.$ (62)

Then the MS detectability for $(\mathbf{H},\mathbf{A},\mathbf{p})$, i.e.,
system (7), is equivalent to that for

$\displaystyle\tilde{x}(k+1)$
$\displaystyle=\tilde{A}\tilde{x}(k)+S^{-1}w(k),$ (63a) $\displaystyle{y_{\rm
r}(k)}$ $\displaystyle=\tilde{H}_{\theta(k)}\tilde{x}(k)+D_{\theta(k)}v(k),$
(63b)

where $\tilde{A}=S^{-1}AS$ and
$\tilde{H}_{\theta(k)}=H_{\theta(k)}S=\Gamma_{\theta(k)}CS$ that is in the
block diagonal form. Define
$\tilde{L}_{\theta(k)}:=S^{-1}\bar{L}_{\theta(k)}$, where
$\bar{L}_{\theta(k)}$ is given by (61). It follows from (51) that

$\displaystyle\tilde{e}(k+1)=(\tilde{A}-\tilde{L}_{\theta(k)}\tilde{H}_{\theta(k)})\tilde{e}(k),$
(64)

where $\tilde{e}(k)=S^{-1}e(k)$. Specifically,
$\tilde{A}-\tilde{L}_{\theta(k)}\tilde{H}_{\theta(k)}$ has the following lower
triangular form

$\displaystyle\tilde{A}-\tilde{L}_{\theta(k)}\tilde{H}_{\theta(k)}{=S^{-1}(A-\bar{L}_{\theta(k)}H_{\theta(k)})S}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}\mathcal{A}_{\theta_{1}(k)}&0&\cdots&0\\\
{\epsilon A_{21}}&\mathcal{A}_{\theta_{2}(k)}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&0\\\ {\epsilon^{m-1}A_{m1}}&\cdots&{\epsilon
A_{m(m-1)}}&\mathcal{A}_{\theta_{m}(k)}\end{array}\right],$

where
$\mathcal{A}_{\theta_{i}(k)}=A_{i}-\ell_{\theta_{i}(k)}h_{i,\theta_{i}(k)}$.
Hence as $\epsilon\to 0$, estimation error dynamics (64) approaches to a
diagonal form, implying that with the block diagonal gain
$\bar{L}_{\theta(k)}$ in (61), the MS detectability for
$\\{\mathbf{h}_{i},\mathbf{A}_{i},\mathbf{p}_{i}\\}_{i=1}^{m}$ is equivalent
to the MS stability for the error dynamics in (64). The proof is thus
complete.

######  Remark 3.

The Wonham decomposition (60), which gives $m$ detectable subsystems
$(c_{i},A_{i})$ from the original detectable system $(C,A)$, plays a critical
role in lowering the complexity from $N=2^{m}$ to $2m$, without which the
complexity reduction is not possible. This decomposition is also a powerful
tool when dealing with networked stabilization for multi-input systems over
logarithmic quantization and i.i.d. fading channels [3, 29, 4]. $\Box$

Now we obtain a sufficient condition on decoupling the MS detectability, but
the complexity of the stationary optimal estimator remains exponential. In the
next subsection, we propose a locally optimal stationary estimator, which has
the complexity $2m$ as well.

###  4.2 A locally optimal stationary estimator

In this subsection, instead of the general estimator gain $L_{\theta(k)}$ in
estimator (16), we consider the block diagonal gain in (61) that reduces
estimator (16) to

$\displaystyle\hat{x}(k+1)=A\hat{x}(k)+\bar{L}_{\theta(k)}[y_{\rm
r}(k)-H_{\theta(k)}\hat{x}(k)].$ (65)

Write system state $x(k)$ as
$x(k)=[x_{1}(k)^{\prime},\ldots,x_{m}(k)^{\prime}]^{\prime}$ with
$x_{i}(k)\in\mathbb{R}^{n_{i}}$ and measurement $y_{\rm r}(k)$ as $y_{\rm
r}(k)=[y_{{\rm r},1}(k),\ldots,y_{{\rm r},m}(k)]^{\prime}$, and let
$\hat{x}_{i}(k)$ be the state estimation of $x_{i}(k)$. It is observed that
estimator (65) can be written as the following $m$ sub-estimators:

$\displaystyle\mspace{-6.0mu}\hat{x}_{1}(k+1)=A_{1}\hat{x}_{1}(k)+\ell_{\theta_{1}(k)}[y_{{\rm
r},1}(k)-h_{1,\theta_{1}(k)}\hat{x}_{1}(k)],$
$\displaystyle\mspace{-6.0mu}\hat{x}_{i}(k+1)=A_{i}\hat{x}_{i}(k)+\ell_{\theta_{i}(k)}[y_{{\rm
r},i}(k)-h_{i,\theta_{i}(k)}\hat{x}_{i}(k)]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{j=1}^{i-1}A_{ij}\hat{x}_{j},\
2\leq i\leq m.$ (66)

Let $\\{Q_{i}\\}_{i=1}^{m}$ and $\\{R_{i}\\}_{i=1}^{m}$ be the sub-matrices on
the diagonal of $Q$ and $R$, respectively. We have the following estimation
result that is similar to that in Theorem 1. Hence, its proof is omitted.

######  Lemma 4.

Assume that there exists the MS stabilizing solutions
$\\{Z_{i}=(Z_{i,1},Z_{i,2})\\}$ to the following CAREs

$\displaystyle Z_{i,r}=$
$\displaystyle\sum_{j=1}^{2}p_{jr}^{(i)}\big{\\{}A_{j}Z_{i,j}[{I_{n_{i}}}+h_{i,j}^{\prime}(\pi_{i,j}R_{i})^{-1}h_{i,j}Z_{i,j}]^{-1}A_{j}^{\prime}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\pi_{i,j}Q_{j}\big{\\}},\ \ \ 1\leq i\leq
m,\ 1\leq r\leq 2.$ (67)

Then for each sub-estimator in (4.2), the optimal stationary gains that
minimize

$\displaystyle
J_{i}(\infty)=\lim_{k\to\infty}\mathrm{E}[\|x_{i}(k)-\hat{x}_{i}(k)\|^{2}]$
(68)

for $i=1,\ldots,m$ are given by

$\displaystyle\ell_{\theta_{i}(k)}=K_{i}(Z_{i}):=A_{i}Z_{i,2}c_{i}^{\prime}(c_{i}Z_{i,2}c_{i}^{\prime}+\pi_{i,2}R_{i})^{-1}$
(69)

for both $\theta_{i}(k)=1,2$.

Note that we have intentionally allowed the value of $l_{\theta_{i}(k)}$ for
the case $\theta_{i}(k)=1$ to be the same as that for $\theta_{i}(k)=2$,
without any effect, such that only a gain is required for each sub-estimator.
Define $\Sigma(k):=\mathbb{E}[e(k)e(k)^{\prime}]$ and
$\Sigma_{i}(k):=\mathbb{E}[e(k)e(k)^{\prime}\mathbf{1}_{\theta(k)=i}]$. Notice
that

$\displaystyle\Sigma(k)=\sum_{i=1}^{N}\Sigma_{i}(k).$ (70)

In light of Lemma 4, we present the following locally optimal estimator.

######  Proposition 2.

For the system dynamics described in (7) with $(C,A)$ in the Wonham
decomposition form (60), a locally optimal stationary estimator of which sub-
estimator $i$ in (4.2) minimizes cost $J_{i}(\infty)$ in (68) is given by

$\displaystyle\hat{x}(k+1)=A\hat{x}(k)+\bar{K}[y_{\rm
r}(k)-H_{\theta(k)}\hat{x}(k)],$ (71)

where $\bar{K}={\rm diag}\\{K_{1}(Z_{1}),\ldots,K_{m}(Z_{m})\\}$. Moreover,
the corresponding stationary error covariance $\Sigma(\infty)$ is given by the
solution of following coupled Lyapunov equations

$\displaystyle\Sigma_{j}(\infty)=$
$\displaystyle\sum_{i=1}^{N}p_{ij}\big{\\{}[A-\bar{K}H_{i}]\Sigma_{i}(\infty)[A-\bar{K}H_{i}]^{\prime}$
$\displaystyle+$
$\displaystyle\mu_{i}[\bar{K}D_{i}RD_{i}^{\prime}\bar{K}^{\prime}+Q]\big{\\}},\
j\in\mathcal{N}.$ (72)

######  Remark 4.

Compared with the complexity of solving $2^{m}$ CAREs for the optimal
stationary estimator, the estimator in (71) only needs to solve $2m$ CAREs and
one estimator gain $\bar{K}$. This is achieved by restricting the estimator
gain to the block diagonal form (61) and the estimation cost to local costs
(68). As it is expected, the estimator in (71) has a performance loss compared
with the optimal one, which will be illustrated in the simulation.
Nonetheless, it follows from Theorems 3 and 5 that if
$(\mathbf{h}_{i},\mathbf{A}_{i},\mathbf{p}_{i})$ is MS detectable for $1\leq
i\leq m$ and the rank condition (40) for each $(A_{i},Q_{i})$ holds, the error
dynamics described in (51) with $L_{\theta(k)}=\bar{K}$ is MS stable, implying
that the covariance of the locally optimal stationary estimator (71) converges
to a finite value $\Sigma(\infty)$. $\Box$

######  Remark 5.

If $m>n$, it is not possible to obtain the Wonham decomposition form in (60),
but in the following form:

$\displaystyle A=\left[\begin{array}[]{cccc}A_{1}&0&\cdots&0\\\
A_{21}&A_{2}&\ddots&\vdots\\\ \vdots&\ddots&\ddots&0\\\
A_{n1}&\cdots&A_{n(n-1)}&A_{n}\end{array}\right],\
C=\left[\begin{array}[]{cccc}c_{1}&0&\cdots&0\\\ 0&c_{2}&\ddots&\vdots\\\
\vdots&\ddots&\ddots&0\\\ 0&\cdots&0&c_{n}\\\ &&C_{0}&\end{array}\right],$

where $C_{0}=[C_{n+1}^{\prime},\ldots,C_{m}^{\prime}]^{\prime}T$ with $T$ the
Wonham decomposition transform matrix. In this case, we may not obtain the
ideal locally optimal estimator (71) that admits the low complexity $2m$ and
uses measurement information from all $m$ sensors. One simple way is to only
use the measurements from the $n$ sensors and abandon the left $m-n$ sensors.
One can also expect a better method to fuse the measurements from both the $n$
and $m-n$ sensors, which deserves a further study in the future. $\Box$

Since the MS detectability for each subsystem
$(\mathbf{h}_{i},\mathbf{A}_{i},\mathbf{p}_{i})$ is the sufficient MS
detectability condition for $(\mathbf{H},\mathbf{A},\mathbf{p})$, and is also
required by the locally optimal stationary estimator, we will investigate the
MS detectability for $(\mathbf{h}_{i},\mathbf{A}_{i},\mathbf{p}_{i})$ by
providing some analytic MS detectability conditions in the next subsection.

###  4.3 MS detectability for $(\mathbf{h}_{i},\mathbf{A}_{i},\mathbf{p}_{i})$

For convenience, we will omit the subscript $i$ in this subsection, i.e.,
denote $A:=A_{i}$, $C:=c_{i}$, $\gamma(k):=\gamma_{i}(k)$,
$\theta(k):=1+\gamma(k)$ for any $i=1,\ldots,m$ with a slight abuse of
notation. Also, the TPM for the process of Markovian packet drops
$\\{\gamma(k)\\}$ is now given by

$\displaystyle P=\left[\ p_{ij}\ \right]=\left[\begin{array}[]{cc}1-q&q\\\
p&1-p\end{array}\right].$ (75)

We further denote $C_{\theta(k)}:=\theta(k)C$ and

$\displaystyle\mathbf{A}:=(A,A),\ \ \mathbf{C}:=(C_{1},C_{2}),\ \
\mathbf{p}:=\\{p_{ij}\\},\ \ i,j\in\\{1,2\\}.$

For simplicity, define the following operators

$\displaystyle\phi_{1}(L,X_{1},X_{2})$
$\displaystyle:=(1-q)A^{\prime}X_{1}A+qA_{L}^{\prime}X_{2}A_{L},$ (76a)
$\displaystyle\phi_{2}(L,X_{1},X_{2})$
$\displaystyle:=pA^{\prime}X_{1}A+(1-p)A_{L}^{\prime}X_{2}A_{L},$ (76b)
$\displaystyle\psi_{1}(L,X_{1},X_{2})$
$\displaystyle:=(1-q)AX_{1}A^{\prime}+pA_{L}X_{2}A_{L}^{\prime},$ (76c)
$\displaystyle\psi_{2}(L,X_{1},X_{2})$
$\displaystyle:=qAX_{1}A^{\prime}+(1-p)A_{L}X_{2}A_{L}^{\prime},$ (76d)
$\displaystyle g_{1}(X_{1},X_{2})$
$\displaystyle:=(1-q)AX_{1}A^{\prime}+pAX_{2}A^{\prime}$ (76e)
$\displaystyle~{}~{}~{}~{}~{}~{}-pAX_{2}C^{\prime}(CX_{2}C^{\prime})^{-1}CX_{2}A^{\prime},$
$\displaystyle g_{2}(X_{1},X_{2})$
$\displaystyle:=qAX_{1}A^{\prime}+(1-p)AX_{2}A^{\prime}$ (76f)
$\displaystyle~{}~{}~{}~{}~{}~{}-(1-p)AX_{2}C^{\prime}(CX_{2}C^{\prime})^{-1}CX_{2}A^{\prime},$

where $A_{L}=A+LC$. The next lemma is useful.

######  Lemma 5.

The following statements are equivalent.

  * a)

The triple $(\mathbf{p},\mathbf{C},\mathbf{A})$ is MS detectable.

  * b)

There exist $X_{1}>0,X_{2}>0$, and $L\in\mathbb{R}^{n_{i}\times m}$ such that
$X_{i}>\psi_{i}(L,X_{1},X_{2})$ for $i=1,2$.

  * c)

There exist $X_{1}>0$ and $X_{2}>0$ such that $X_{i}>g_{i}(X_{1},X_{2})$ for
$i=1,2$.

  * d)

There exist $X_{1}>0,X_{2}>0$, and $\Omega\in\mathbb{R}^{n_{i}\times m}$ such
that the following LMIs hold:

$\displaystyle\left[\begin{array}[]{ccc}X_{1}&\sqrt{1-q}A^{\prime}X_{1}&\sqrt{q}(A^{\prime}X_{2}+C^{\prime}\Omega^{\prime})\\\
&X_{1}&0\\\ &0&X_{2}\end{array}\right]$ $\displaystyle>0,$
$\displaystyle\left[\begin{array}[]{ccc}X_{2}&\sqrt{p}A^{\prime}X_{1}&\sqrt{1-p}(A^{\prime}X_{2}+C^{\prime}\Omega^{\prime})\\\
&X_{1}&0\\\ &0&X_{2}\end{array}\right]$ $\displaystyle>0.$

Proof. a) $\Leftrightarrow$ b): This is straightforward in accordance with
Definition 5.

c) $\Rightarrow$ b): For any $L\in\mathbb{R}^{n_{i}\times m}$ and $X>0$,

$\displaystyle(A+LC)X(A+LC)^{\prime}=$ $\displaystyle
AXA^{\prime}-AXC^{\prime}(CXC^{\prime})^{-1}CXA^{\prime}$ $\displaystyle\ \ \
+(L+L_{X})CXC^{\prime}(L+L_{X})^{\prime},$

where $L_{X}=AXC^{\prime}(CXC^{\prime})^{-1}$. As a result, we have

$\displaystyle\psi_{1}(-L_{X_{2}},X_{1},X_{2})$
$\displaystyle=g_{1}(X_{1},X_{2}),$
$\displaystyle\psi_{2}(-L_{X_{2}},X_{1},X_{2})$
$\displaystyle=g_{2}(X_{1},X_{2}).$

The above two equalities imply

$\displaystyle\psi_{i}(L,X_{1},X_{2})\geq g_{i}(X_{1},X_{2}),\ \ i=1,2,$ (77)

for any $X_{1}>0$, $X_{2}>0$, and $L\in\mathbb{R}^{n_{i}\times m}$. Therefore,
if $X_{1}>g_{1}(X_{1},X_{2})$ and $X_{2}>g_{2}(X_{1},X_{2})$ hold for some
$X_{1}>0$ and $X_{2}>0$, then $X_{1}>\psi_{1}(-L_{X_{2}},X_{1},X_{2})$ and
$X_{2}>\psi_{2}(-L_{X_{2}},X_{1},X_{2})$, proving the statement of c)
$\Rightarrow$ b).

b) $\Rightarrow$ c): This is clearly true by
$X_{1}>\psi_{1}(L,X_{1},X_{2})\geq g_{1}(X_{1},X_{2})$ and
$X_{2}>\psi_{2}(L,X_{1},X_{2})\geq g_{2}(X_{1},X_{2})$.

a) $\Leftrightarrow$ d): This is straightforward from Proposition 1.

######  Theorem 6.

The triple $(\mathbf{p},\mathbf{C},\mathbf{A})$ is MS detectable, if

$\displaystyle\min\\{q,1-p\\}>\lambda_{\rm
c}=1-\frac{1}{\prod_{i}\max\\{|\lambda_{i}(A)|^{2},1\\}},$ (78)

where $\lambda_{i}(A)$ is the $i$th eigenvalue of $A$.

Proof. Recall the hypothesis on the detectability of $(C,A)$, assumed
throughout this section. Hence, $(c_{i},A_{i})$ is detectable with ${\rm
rank}\\{c_{i}\\}=1$ for $1\leq i\leq m$. Then by Lemma 5.4 of [19], there
exists $X>0$ such that

$\displaystyle X>AXA^{\prime}-\lambda
AXC^{\prime}(CXC^{\prime})^{-1}CXA^{\prime},$ (79)

if and only if $\lambda>\lambda_{\rm c}$. Therefore, if (78) holds, there
exists $\bar{X}>0$ to (79) with $\lambda=\min\\{q,1-p\\}$ such that

$\displaystyle\bar{X}$
$\displaystyle>A\bar{X}A^{\prime}-qA\bar{X}C^{\prime}(C\bar{X}C^{\prime})^{-1}C\bar{X}A^{\prime},$
$\displaystyle\bar{X}$
$\displaystyle>A\bar{X}A^{\prime}-(1-p)A\bar{X}C^{\prime}(C\bar{X}C^{\prime})^{-1}C\bar{X}A^{\prime}.$

Let $\bar{X}=X_{1}=pX_{2}/q$. Then, the above inequalities reduce to

$\displaystyle X_{1}>g_{1}(X_{1},X_{2}),\ \ \ X_{2}>g_{2}(X_{1},X_{2}).$

Therefore, the sufficient condition in (78) holds, following from the
equivalence of a) and c) in Lemma 5.

When $m=n$, i.e., the order of $A_{i}$ is one for $1\leq i\leq n$, we have the
following analytic necessary and sufficient condition for the MS detectability
of $(\mathbf{p},\mathbf{C},\mathbf{A})$.

######  Theorem 7.

If $m=n$, then the triple $(\mathbf{p},\mathbf{C},\mathbf{A})$ is MS
detectable, if and only if

$\displaystyle q>1-\frac{1}{\rho(A)^{2}}.$ (80)

Proof. The argument for necessity is the same as that in Theorem 4. So we only
show the sufficiency. First, we observe that if there exists a matrix $L$ such
that $A_{L}=A+LC=0$, then the MS detectability condition in (44) becomes the
case that there exist $X_{1}>0,X_{2}>0$ such that

$\displaystyle X_{1}>(1-q)AX_{1}A^{\prime},\ X_{2}>pAX_{1}A^{\prime}.$ (81)

In this case, if $q>1-\rho(A)^{-2}$, we can always find some $X_{1}>0$ and
$X_{2}>0$, rendering inequalities in (81) true. Clearly, when $A$ and $C$ are
both scalars, the choice of $L=-A/C$ makes $A_{L}=0$, which completes the
proof.

######  Remark 6.

It is worth mentioning that the results in Section 3 and this subsection can
be applied to the dual optimal control problem studied in [21], of which focus
is on the convergence issue under assumptions of stabilizability and
detectability of the system. To be specific, consider the linear system
described by

$\displaystyle x({k+1)}=Ax(k)+Bu_{\rm r}(k),\ \ u_{\rm r}(k)=\gamma(k)u(k),$
(82)

where $u(k)\in\mathbb{R}^{m}$ is the control signal sent from the remote
controller via the Markovian packet drop channel. In accordance with the MJLS
theory in [16], the optimal controller is given by $u(k)=F(X)x(k)$, where
$F(X)$ is computed using the MS stabilizing solution to following control
CAREs:

$\displaystyle X_{i}$
$\displaystyle=A^{\prime}\mathcal{E}_{i}(X)A+W+A^{\prime}\mathcal{E}_{i}(X)B_{i}F_{i}(X),$
(83) $\displaystyle F(X)$
$\displaystyle=-(U+B^{\prime}\mathcal{E}_{i}(X)B)^{-1}B^{\prime}\mathcal{E}_{i}(X)A,\
i=1,2.$

Here, $\mathcal{E}_{i}(X)=\sum_{j=1}^{2}p_{ij}X_{j}$, $B_{1}=0,B_{2}=B$ and
$(W\geq 0,U>0)$ are the weighting matrices for the state and control signal,
respectively. According to the notion of MS stabilizability in Definition 2 of
[23], which is dual to the MS detectability, similar results can be obtained
for such an optimal control problem. $\Box$

##  5 Simulation Examples

###  5.1 Existence of the MS stabilizing solution

First, we illustrate the theoretical results about the MS stabilizing solution
and MS detectability by a numerical example. Consider a third-order system of
the form in 1 with $C=R=I_{3}$, and

$\displaystyle A$ $\displaystyle=\left[\begin{array}[]{ccc}1&0&0\\\ 1&1.2&0\\\
1&1.5&1.3\end{array}\right],\ \ Q=\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&0\end{array}\right].$ (90)

Note that $(C,A)$ is already in the form of Wonham decomposition. Clearly,
$\lambda=1.3$ is an uncontrollable eigenvalue for $(A,Q)$. Nonetheless,
according to the condition 2) in Theorem 3, we only require that $|\lambda|=1$
be controllable. This condition is satisfied in this example. Let the
parameters of the three channels be

$\displaystyle p_{1}=0.50,\ \ \ p_{2}=0.60,\ \ \ p_{3}=0.70,$ $\displaystyle
q_{1}=0.20,\ \ \ q_{2}=0.32,\ \ \ q_{3}=0.51.$

So, $q_{1}>0,q_{2}>1-1.2^{-2}=0.3056$, and $q_{3}>1-1.3^{-2}=0.4083$. Then by
Theorems 5 and 7, the system is MS detectable. Therefore, by Theorem 3, the MS
stabilizing solution to the CAREs (1) exists. By solving the convex
programming problem (36) using YALMIP [30], we can obtain the stabilizing
solution $Y$. Then with $\\{K_{i}(Y)\\}_{i=1}^{8}$ computed by (19), we have
$\rho(\widetilde{\mathcal{L}}^{*})=\rho(\mathcal{A})=0.9297<1$, according to
Remark 3.5 of [16], where

$\mathcal{A}=(P^{\prime}\otimes I_{9}){\rm
diag}\\{(A-K_{i}(Y)H_{i})\otimes(A-K_{i}(Y)H_{i})\\}_{i\in\mathcal{N}}.$

Thus, $Y$ is indeed the MS stabilizing solution from Definition 2.

###  5.2 Performance of the optimal and locally optimal stationary estimators

We will use a target tracking example [31] to show the estimation performance
of the optimal and locally optimal stationary estimators in Theorem 1 and
Proposition 2, respectively. For brevity, the two estimators will be
abbreviated as OS estimator and LOS estimator, respectively. The system
dynamics is described by [31]

$\displaystyle x(k+1)=\left[\begin{array}[]{ccc}1&0&0\\\ T&1&0\\\
T^{2}/2&T&1\end{array}\right]x(k)+w(k),$ (94)

where $T$ is the sampling period and $w(k)$ is the Gaussian noise with
covariance

$\displaystyle
Q=2\alpha\sigma_{m}^{2}\left[\begin{array}[]{ccc}T&T^{2}/2&T^{3}/6\\\
T^{2}/2&T^{3}/3&T^{4}/8\\\ T^{3}/6&T^{4}/8&T^{5}/20\end{array}\right],$ (98)

with $\alpha$ the reciprocal of the maneuver time constant and
$\sigma_{m}^{2}$ the variance of the target acceleration. The first, second
and third entries of $x(k)$ represent the acceleration, speed and position of
the target, respectively. Suppose that there are three sensors measuring the
target acceleration, speed and position, respectively. As a result, the
measurement model is given by

$\displaystyle y(k)=\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right]x(k)+v(k).$ (102)

The covariance of the Gaussian noise $v(k)$ is assumed to be $R=0.01I_{3}$.
The other system parameters are set to $T=1$s, $\alpha=0.01$, and
$\sigma_{m}^{2}=10$. In this example, the original $(C,A)$ in [31] is already
in the form of Wonham decomposition. By Theorems 5 and 7, the conditions
$q_{1}>0,q_{2}>0$ and $q_{3}>0$ are sufficient to guarantee the MS
detectability of the system.

Figure 2: The target position error variance of the OS and LOS estimators. The
theoretical value for OS estimator is the $(3,3)$th element in the matrix
$\sum_{i=1}^{N}Y_{i}$, and the theoretical value for LOS one is the $(3,3)$th
element in the matrix $\lim\limits_{k\to\infty}\sum_{i=1}^{N}\Sigma_{i}(k)$.
Figure 3: A realization of target position estimates of the OS and LOS
estimators. Figure 4: A sample path of packet drops.

Set the channel parameters as

$\displaystyle p_{1}=0.20,\ \ \ p_{2}=0.30,\ \ \ p_{3}=0.20,$ $\displaystyle
q_{1}=0.85,\ \ \ q_{2}=0.75,\ \ \ q_{3}=0.80.$

For both OS and LOS estimators, we perform a Monte Carlo simulation with
50,000 trials over the time horizon of $k\in[1,\ 50]$ to show the estimation
performance represented by the target position error variance. Fig. 2 shows
the empirical variances of the OS and LOS estimators, of which both are close
to their respective theoretical values. As expected, the performance of the
LOS estimator is inferior to that of the globally optimal one that has much
higher complexity. On the other hand, numerical results for the target
position tracking shown in Fig. 3, illustrate fairly good estimation
performance for both estimators. Fig. 4 shows the corresponding packet drop
sequences of sensor data for the tracking in Fig. 3.

##  6 Conclusion

This paper studies the stationary state estimation over multiple Markovian
packet drop channels. We have investigated the existence of the MS stabilizing
solution, which is pivotal to the proposed estimator. It is shown that the
known stabilizability condition in the existing literature is not necessary;
only the controllability of the eigenvalues on the unit circle is required. In
addition, a sufficient condition is derived showing that the MS detectability
with multiple Markovian packet drop channels can be decoupled. Based on the
decoupling method, a locally optimal stationary estimator with much lower
complexity is proposed. In fact, the exponential complexity of the original
optimal estimator is reduced to the linear complexity of the locally optimal
estimator. Some analytic sufficient and necessary MS detectability conditions
are also derived for the decoupled subsystems, each of which corresponds to
the scenario of single Markovian packet drop channel. The results in this
paper are potentially applicable to smart and optimal manufacturing in which a
network of sensors collaboratively sense the interested process states [32].

The authors would like to thank the Associate Editor and the anonymous
reviewers for their constructive suggestions that have improved the quality of
this work.

## Appendix

The following technical Lemma will be useful in the proof of Theorem 2.

######  Lemma 6.

The following two statements are true:

  * (i)

For any $A,B$ in $\mathbb{S}_{+}^{n}$, ${\rm tr}(AB)\geq 0$, and the equality
holds if and only if $AB=0$.

  * (ii)

For any $A\in\mathbb{C}^{m\times n}$, $AA^{*}=0$ if and only if $A=0$.

Proof of Theorem 2. _Necessity:_ It is obvious that the MS detectability for
$(\mathbf{H},\mathbf{A},\mathbf{p})$ is necessary. In order to prove that
condition 2) is also necessary, assume on the contrary that condition 2) does
not hold but the CAREs in (1) has the MS stabilizing solution $Y$, implying
that $\widetilde{\mathcal{L}}^{*}$ with $L_{i}=-K_{i}(Y)$ is a stable
operator. Since condition 2) is false, from Definition 3 there holds

$\displaystyle\mathcal{L}^{*}(V)=V,\ Q^{1/2}V_{i}=0,\ \forall
i\in\mathcal{N},$ (103)

where $V=(V_{1},\ldots,V_{N})\in{\mathbb{H}}_{+}^{n}\backslash\\{0\\}$. In
light of the optimal state estimator gains in (19) and the definition for
$\mathcal{L}_{j}(\cdot)$ in (26), CAREs (1) in Theorem 1 can be rewritten as

$\displaystyle
Y_{j}=\mathcal{L}_{j}(Y)-\sum_{i=1}^{N}p_{ij}\left[K_{i}(Y)H_{i}Y_{i}A^{\prime}-\mu_{i}Q\right].$
(104)

Multiplying both sides of the above CAREs by $V_{j}$ from right, and applying
the operations of trace and summation yield

$\displaystyle\sum_{j=1}^{N}{\rm tr}(Y_{j}V_{j})=$
$\displaystyle\sum_{j=1}^{N}{\rm
tr}\\{\mathcal{L}_{j}(Y)V_{j}\\}+\sum_{j=1}^{N}\sum_{i=1}^{N}p_{ij}\mu_{i}{\rm
tr}(QV_{j})$ $\displaystyle-\sum_{j=1}^{N}\sum_{i=1}^{N}p_{ij}{\rm
tr}\left(K_{i}(Y)H_{i}Y_{i}A^{\prime}V_{j}\right).$ (105)

By the inner product in (24) and the relationship in (29b), we have

$\sum_{j=1}^{N}{\rm tr}\\{\mathcal{L}_{j}(Y)V_{j}\\}=\sum_{j=1}^{N}{\rm
tr}\\{Y_{j}\mathcal{L}^{*}_{j}(V)\\}.$

Then combining (103), we conclude that equality (Appendix) is equivalent to

$\displaystyle\sum_{j=1}^{N}{\rm tr}(Y_{j}V_{j})$
$\displaystyle=\sum_{j=1}^{N}\left[{\rm
tr}(Y_{j}V_{j})-\sum_{i=1}^{N}p_{ij}{\rm
tr}\left\\{K_{i}(Y)H_{i}Y_{i}A^{\prime}V_{j}\right\\}\right]$

implying that $K_{i}(Y)H_{i}Y_{i}A^{\prime}V_{j}=0\ \forall i,j\in\mathcal{N}$
by Lemma 6 (i). Noticing the expression of $K_{i}(Y)$ in (19), we further
conclude $H_{i}Y_{i}A^{\prime}V_{j}=0$ by Lemma 6 (ii), and thus
$K_{i}(Y)^{\prime}V_{j}=0$ $\forall i,j\in\mathcal{N}$. Therefore, by setting
$L_{i}=-K_{i}(Y)$ in the operator $\widetilde{\mathcal{L}}^{*}$, we obtain

$\displaystyle\widetilde{\mathcal{L}}_{i}^{*}(V)$
$\displaystyle=\sum_{j=1}^{N}p_{ij}[A-K_{i}(Y)H_{i}]^{\prime}V_{j}[A-K_{i}(Y)H_{i}]$
$\displaystyle=\sum_{j=1}^{N}p_{ij}A^{\prime}V_{j}A=\mathcal{L}_{i}^{*}(V)=V_{i},$
(106)

which means that $\widetilde{\mathcal{L}}^{*}(V)$ with $L_{i}=-K_{i}(Y)$ is
unstable, contradicting the assumption on the MS stabilizing solution. This
concludes the necessity of condition 2).

_Sufficiency:_ It suffices to show that $\rho(\widetilde{\mathcal{L}}^{*})<1$
under conditions 1) and 2). Assume on the contrary,
$\rho(\widetilde{\mathcal{L}}^{*})\geq 1$. Setting $L_{i}=-K_{i}(Y^{+})$
implies that $\rho(\widetilde{\mathcal{L}}^{*})=1$, by Lemma 2 (i). Let
$(\lambda=1,V)$ be an eigenvalue–eigenvector pair for
$\widetilde{\mathcal{L}}^{*}$ such that $\widetilde{\mathcal{L}}^{*}(V)=V$.
Rewrite CAREs (1) as

$\displaystyle Y_{j}^{+}=$
$\displaystyle\sum_{i=1}^{N}p_{ij}\big{\\{}[A-K_{i}(Y^{+})H_{i}]Y_{i}^{+}[A-K_{i}(Y^{+})H_{i}]^{\prime}$
$\displaystyle+\mu_{i}[K_{i}(Y^{+})RK_{i}(Y^{+})^{\prime}+Q]\big{\\}}.$ (107)

The same manipulations as in (Appendix) and the adjoint relation
$\sum_{j=1}^{N}{\rm
tr}\\{\widetilde{\mathcal{L}}_{j}(Y^{+})V_{j}\\}=\sum_{j=1}^{N}{\rm
tr}\\{Y_{j}^{+}\widetilde{\mathcal{L}}^{*}_{j}(V)\\}$ lead to

$\displaystyle\sum_{j=1}^{N}{\rm tr}(Y_{j}^{+}V_{j})=$
$\displaystyle\sum_{j=1}^{N}{\rm
tr}\\{Y_{j}^{+}\widetilde{\mathcal{L}}^{*}_{j}(V)\\}$
$\displaystyle+\sum_{j=1}^{N}\sum_{i=1}^{N}$ $\displaystyle p_{ij}\mu_{i}{\rm
tr}\\{K_{i}(Y^{+})RK_{i}(Y^{+})^{\prime}V_{j}+QV_{j}\\}.$

Since $\widetilde{\mathcal{L}}^{*}(V)=V$, the above equation implies that
$QV_{j}=0$, and $K_{i}(Y^{+})^{\prime}V_{j}=0$, by again Lemma 6, further
leading to ${\mathcal{L}^{*}}(V)=V$. This contradicts condition 2) and
concludes the sufficiency proof. The proof is now complete.

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