Remote State Estimation with Smart Sensors over Markov Fading Channels

In the long-term evolution of wireless applications from conventional wireless sensor networks (WSNs) to the Internet-of-Things (IoT) and the Industry 4.0, remote estimation is a key component [1, 2, 3, 4]. Driven by Moore’s Law, the accelerated development and adoption of smart sensor technology enables low-cost sensors with high computational capability [5]. Thus, in a number of remote estimation applications, it is practical to use smart sensors (e.g., with Kalman filters) to pre-estimate the dynamic states, and then send the estimated states rather than the raw measurements to the remote estimator. In the presence of communication constraints, the smart sensors provide better estimation performance than the conventional sensors that purely send raw measurement data to the remote estimator [6].

Unlike wired communications, wireless communications are unreliable and the channel status varies with time due to multipath propagation and shadowing caused by obstacles affecting the wave propagation. The transition process of the fading channel states is usually modeled as a Markov process [7, 8, 9], and different channel states lead to different packet drop probabilities of transmissions. The presence of an unreliable wireless communication channel degrades the estimation performance, and in some cases even lead to instability. Whilst stability when using conventional sensors has been well investigated, see literature survey below, stability when using a smart sensor has been much less considered. In this paper, we tackle the fundamental problem: what are necessary and sufficient conditions on system parameters that ensure stochastic stability of a smart-sensor-based remote estimation system over a Markov fading channel?

The existing work on remote estimation can be divided into two categories based on the sensor’s computational capability.

In the conventional sensor scenario, the sensor sends raw measurements to the remote estimator. When considering a static wireless channel, where neither the transceivers nor the wireless environment are moving, the packet drop probability during the remote estimation process remains fixed, so the packet arrival process is a Bernoulli process. It was proven in [10] that there exists a critical packet drop probability, such that the mean estimation error covariance is bounded for all initial conditions and diverges for some initial condition if the packet drop probability is less or greater than the critical probability, respectively. This result was further extended to a scenario with random packet delays in [6]. By modeling the packet arrival process as a Markovian binary switching process, sufficient conditions for stability in the sense of peak covariance were obtained in [11, 12]. For situations where the number of consecutive packet dropouts constitutes a bounded Markov process, peak covariance stability was investigated in [13]. By modeling the sequence of packet dropouts as a stationary finite-order Markov process, a necessary and sufficient stability condition was obtained in the sense of mean estimation error covariance in [14]. In contrast to [11, 12, 13, 14], which directly model packet-dropouts as a Markovian process and abstract away the underlying wireless channel, Markovian fading channel states were explicitly considered in [15]. A sufficient condition for exponential stability was derived by using stochastic Lyapunov functions. With the same multi-state Markov channel model as [15], optimal transmit power allocation policy under different channel conditions was proposed in [16] to achieve the minimum remote estimation error.

More recently, closed-loop control systems over multi-state Markov channels were investigated in [17, 18, 19]. Under an ideal assumption of perfect sensor measurements, a necessary and sufficient stability condition was obtained in [17], where the sensor and the controller were co-located; a sufficient stability condition of a half-duplex control system was obtained in [18], where the controller applied a scheduling policy determining when to receive the sensor’s packet or to transmit a control packet to the actuator. In [19], sufficient stability conditions in terms of maximum allowable transmission interval of a nonlinear system was investigated. The work [20] focused on bit-rate limited error-free communication channels, where the number of bits to be transmitted in each time slot formed a Markov chain. By combining results from quantization theory with insights from Markov Jump Linear Systems, [20] examined how the quantization errors induced by finite-bit quantizers affect the control and estimation quality.

In the smart sensor scenario, an estimator (e.g., based on a Kalman filter) of the sensor side pre-processes the raw measurements, such that an estimate is transmitted to the remote estimator over the wireless channel. It has been rigorously proved in [6] that smart sensor scenario performs better than the conventional sensor scenario when taking into account the transmission delay and failures. However, unlike the conventional-sensor-based scenario, most of the theoretical research on smart-sensor-based remote estimation considered static channels and assumed independent and identically distributed (i.i.d.) packet dropouts [6, 21, 22, 23, 24, 25, 26, 27, 28]. In [6], a necessary and sufficient condition for remote estimation stability was derived in the mean-square sense. In [21], an optimal sensor power scheduling policy under a sum power constraint was obtained. In [22], the optimal transmission scheduling policy of two sensors each measuring the state of one of the two systems was obtained in a closed form, where the sensors shared a single wireless channel. This work was extended to a multi-sensor-multi-channel scenario in [23], where the optimal transmission schedule policy was obtained by solving a Markov decision process problem. In [24], an optimal event-triggered transmission policy of a multi-sensor-multi-channel remote estimation system was proposed with a combined design target: the estimation error and the energy consumption of sensor transmissions.

In addition, optimal smart sensor transmission scheduling policies for single and multiple wireless channel scenarios were investigated under the presence of jamming attacks in [25] and [26], respectively; an optimal transmission scheduling policy under the presence of an eavesdropper was proposed in [27] to minimize the remote estimation error at the dedicated receiver while keeping the eavesdropper’s estimation error as large as possible.

More recently, a remote estimation system with retransmissions was proposed in [28], where the smart sensor can decide whether to retransmit the unsuccessfully transmitted local estimate (with a longer latency) or to send a new estimate (with a lower reliability). The obtained optimal retransmission scheduling policy found the optimal balance between the transmission latency and reliability on the remote estimation performance.

In this paper, we investigate mean-square stability of smart-sensor-based remote estimation over an error-prone multi-state time-homogeneous Markov channel. The $M$-state fading-channel model under consideration introduces an unbounded Markov chain in the analysis of the remote estimation system, which presents some non-trivial challenges. The main contributions are summarized as below.

1. 1.

   We derive a necessary and sufficient condition on the stability of a remote state estimation system in terms of the system matrix $\mathbf{A}$, the packet drop probabilities in different channel states $\{d_{1},\dots,d_{M}\}$ and the matrix of the channel state transitions $\mathbf{M}$. The remote state estimation is mean-square stable if and only if $\rho^{2}(\mathbf{A})\rho(\mathbf{DM})<1$, where $\rho(\cdot)$ denotes the spectral radius, and $\mathbf{D}$ is the diagonal matrix generated by $\{d_{1},\dots,d_{M}\}$.

2. 2.

   We derive asymptotic upper and lower bounds of the estimation error function in terms of the number of consecutive packet dropouts $i$, which are in the same order of $(\rho(\mathbf{A})+\epsilon)^{i}$ and $\rho^{i}(\mathbf{A})$, respectively, where $\epsilon$ is an arbitrarily small positive number.

To obtain these results, we propose a novel estimation-cycle based analytical approach. Moreover, we further develop the asymptotic theory of matrix power, which provides new element-wise bounds of matrix powers.

The remainder of this paper is organized as follows: Section II presents the model of the remote estimation system using a Markov fading channel. Section III presents and discusses the main results of the paper. Section IV proposes a stochastic estimation-cycle based analysis approach and derives some element-wise bounds of matrix powers. They are used in Section V to prove the main results. Section VI numerically evaluates the performance of the remote estimation system, and verifies the theoretical results. Section VII 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. $\text{diag}\{v_{1},v_{2},...,v_{K}\}$ denotes the diagonal matrix with the diagonal elements $\{v_{1},v_{2},...,v_{K}\}$. $\mathbb{N}$ and $\mathbb{N}_{0}$ denote the sets of positive and non-negative integers, respectively. $\mathbb{R}^{m}$ denotes the $m$-dimensional Euclidean space. $\rho(\mathbf{A})$ is the spectral radius of $\mathbf{A}$, i.e., the largest absolute value of its eigenvalues. $\left[u\right]_{B\times B}$ 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 $\{v_{0},v_{1},\cdots\}$.

The discrete-time linear time-invariant (LTI) model is given as (see e.g., [6, 21, 29])

where $\mathbf{x}_{t}\in\mathbb{R}^{n}$ is the process state vector, $\mathbf{A}\in\mathbb{R}^{n\times n}$ is the state transition matrix, $\mathbf{y}_{t}\in\mathbb{R}^{m}$ is the measurement vector of the smart sensor attached to the process, $\mathbf{C}\in\mathbb{R}^{m\times n}$ is the measurement matrix, $\mathbf{w}_{t}\in\mathbb{R}^{n}$ and $\mathbf{v}_{t}\in\mathbb{R}^{m}$ are the process and measurement noise vectors, respectively. We assume $\mathbf{w}_{t}$ and $\mathbf{v}_{t}$ are independent and are identically distributed (i.i.d.) zero-mean Gaussian processes with corresponding covariance matrices $\mathbf{W}$ and $\mathbf{V}$, respectively. In this work, we focus on the stability condition of the remote estimation of the process $\mathbf{x}_{t}$ in the sense of average remote estimation mean-square error. Note that, if $\rho^{2}(\mathbf{A})<1$, then the covariance of $\mathbf{x}_{t}$ is always bounded, and stability will trivially be satisfied. Thus, as commonly done in this context, in the sequel we focus on the more interesting case with $\rho^{2}(\mathbf{A})\geq 1$ indicating that the plant state grows up exponentially fast.

Note that in practice, feedback control of open-loop unstable LTI systems with Gaussian noise always requires sensors with unbounded measurement range, and adaptive zooming-in/zooming-out measurement range have been widely adopted [30]. Beyond the idealized situation of LTI systems, system models with exponentially growing modes are also obtained through linearization at unstable equilibria, see for example the Pendubot system in [27] and references therein.

Since the sensor’s measurements are noisy, a smart sensor is used to estimate the state of the process, $\mathbf{x}_{t}$. For that purpose a Kalman filter [21, 29] is used, which gives the minimum estimation MSE, based on the current and previous raw measurements:

where $\mathbf{I}$ is the $m\times m$ identity matrix, $\mathbf{x}^{s}_{t|t-1}$ is the prior state estimate, $\mathbf{x}^{s}_{t|t}$ is the posterior state estimate at time $t$, $\mathbf{K}_{t}$ is the Kalman gain. The matrices $\mathbf{P}^{s}_{t|t-1}$ and $\mathbf{P}^{s}_{t|t}$ 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 [31]. Note that $\mathbf{x}^{s}_{t|t}$ is the output of the Kalman filter at time $t$, i.e., the pre-filtered measurement of $\mathbf{y}_{t}$, with the estimation error covariance $\mathbf{P}_{t|t}^{s}$.

Using Assumption 1, the local Kalman filter of system (1) is stable, i.e., the error covariance matrix $\mathbf{P}_{t|t}^{s}$ converges to a finite matrix $\bar{\mathbf{P}}_{0}$ as the time index $t$ goes to infinity [31]. In the rest of the paper, we assume that the local Kalman filter operates in the steady state [6, 21, 22, 23, 24, 29], i.e., $\mathbf{P}_{t|t}^{s}=\bar{\mathbf{P}}_{0}$. Further, to simplify notation, the sensor’s estimation $\mathbf{x}_{t|t}^{s}$ shall be denoted by $\hat{\mathbf{x}}_{t}^{s}$.

The main characteristic of the wireless fading channel is that the channel quality is a time-varying random process that changes over time in a correlated manner [7, 8, 9]. We consider a finite-state time-homogeneous Markov block-fading channel [7]. It is assumed that the channel power gain $h_{t}>0$ remains constant during the $t$th time slot but may change slot by slot. We assume that the Markov channel has $M$ states, i.e,

The transition probability from state $i$ to state $j$ is time-homogeneous and given by

where $\mathcal{M}\triangleq\{1,\cdots,M\}$. The matrix of channel state transition probability is given as

We assume that all the channel states are aperiodic and positive recurrent. Thus, the Markov chain induced by $\mathbf{M}$ is ergodic [32].

We assume that the channel state information is available at both the sensor and the remote estimator, which can be achieved by standard channel estimation and feedback techniques, see e.g. [33] and the references therein. Let $\gamma_{t}=1$ and $\gamma_{t}=0$ denote the successful and failed packet detection of the remote estimator during time slot $t$, respectively. The packet drop probability in channel state $b_{i}$ is

Note that the transmission is always perfect if $d_{i}=0,\forall i\in\mathcal{M}$, while no information-carrying packet is delivered to the remote estimator if $d_{i}=1,\forall i\in\mathcal{M}$. We define the packet drop probability matrix as

In practice, the sensor can send a known sequence of symbols (called a pilot or channel training sequence) to the receiver, which can then estimate the channel power gain [35]. For the packet error probability at a certain channel condition, accurate value can be obtained by Monte Carlo simulation, i.e., sending a large sequence of packets to the receiver and calculate the ratio of failed packets.

The smart sensor sends its local estimate $\hat{\mathbf{x}}^{s}_{t}$ to the remote estimator at every time slot. Each packet transmission has a unit delay that is equal to the sampling period of the system. For the considered fading model, packets may or may not arrive at the receiver due to the random packet dropouts. To account for packet transmission delays and the failures, the remote estimation of the current system states is based on the previously detected information packet. The optimal remote estimator in the sense of minimum mean-square error (MMSE) can be obtained as [6, 22]

Assuming a packet was successfully received at time $t^{\prime}$, and the following transmission consecutively failed for $\delta\geq 1$ times before the current time $t$, i.e., $t=t^{\prime}+\delta+1$, from (8), it can be obtained that $\hat{\mathbf{x}}_{t^{\prime}+1}=\mathbf{A}\hat{\mathbf{x}}^{s}_{t^{\prime}}$, $\hat{\mathbf{x}}_{t^{\prime}+2}=\mathbf{A}^{2}\hat{\mathbf{x}}^{s}_{t^{\prime}}$, and $\hat{\mathbf{x}}_{t}=\hat{\mathbf{x}}_{t^{\prime}+\delta+1}=\mathbf{A}^{\delta+1}\hat{\mathbf{x}}^{s}_{t^{\prime}}$. Thus, (8) can be uniformly written as

where $\delta_{t}\in\mathbb{N}_{0}$ is the number of consecutive packet dropouts before time slot $t$. In other words, $(\delta_{t}+1)$ can be treated as the age-of-information (AoI) of the remote estimator in time slot $t$ [36].

Then, the estimation error covariance is given as

where (11) is obtained by substituting (9) and (1) into (10) and with:

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

From (11), we can write

Since $\mathbf{P}_{t}$ is a countable stochastic process taking value from a countable infinity set

it will grow during periods of consecutive packet dropouts when $\rho(\mathbf{A})\geq 1$. Since periods of consecutive packet dropouts have unbounded support, at best one can hope for some type of stochastic stability. In the present work, our focus is on the mean-square stability.

Note that establishing necessary and sufficient stability conditions is non-trivial as we consider correlated fading-channel model in the remote estimation system which induces a countable (and unbounded) Markov chain in the analysis. Some of the existing works adopt stochastic Lyapunov functions to elucidate such situations (see e.g., [15]). These however, merely lead to sufficient conditions.

Theorem 1 shows that the stability condition depends on the system matrix $\mathbf{A}$, the packet drop probability matrix $\mathbf{D}$ and the matrix of the channel state transitions $\mathbf{M}$. It is important to note that the necessary and sufficient condition is determined by both the spectral radiuses of $\mathbf{A}$ and the product of two matrices $\mathbf{D}$ and $\mathbf{M}$. Since $\rho(\mathbf{A})$ measures how fast the dynamic process varies, $\rho\left(\mathbf{D}\mathbf{M}\right)$ can be treated as an effective measurement of the Markov channel quality.

A pair of asymptotic upper and lower bounds of the estimation error function are given below.

Propositions 1 and 2 show that when a large number of consecutive packet dropouts occur, i.e., $i\gg 1$, the remote estimation error is upper and lower bounded by exponential functions in terms of $i$.

Before analyzing the long-term average MSE of the remote estimation system and derive the stability condition, we need to introduce and analyze estimation cycle. To be specific, the $k$th estimation cycle starts after the $k$th successful transmission and ends at the $(k+1)$th successful transmission as illustrated in Fig. 2. In other words, the estimation process is divided by the estimation cycles.

The channel state at the beginning of estimation cycle $k$, i.e., a post-success channel state, is denoted by $S_{k}\in\mathcal{B}^{\prime}\subset\mathcal{B}$, where

is the set of post-success channel states and the cardinality of $\mathcal{B}^{\prime}$ is $M^{\prime}\leq M$. Without loss of generality, we assume that $\mathcal{B}^{\prime}$ contains the first $M^{\prime}$ elements of $\mathcal{B}$. In other words, none of the last $(M-M^{\prime})$ elements of $\mathcal{B}$ can be a post-success channel state, while the others can.

Then, we have the following property of $S_{k}$.

Let $T_{k}$ denote the sum number of transmissions in the $k$th estimation cycle. The sum MSE in the $k$th estimation cycle, say $C_{k}$, is given as

From (15) it directly follows that the average estimation MSE can be rewritten as

Since $C_{k}$ is determined by $T_{k}$, and the distribution of $T_{k}$ depends on $S_{k}$ and the distribution of $S_{k}$ is time-invariant, the unconditional distributions of $T_{k}$ and of $C_{k}$ are also time-invariant. We thus drop the time indexes of $T_{k}$, $C_{k}$ and $S_{k}$. Then, the time average of $\{\cdots,T_{k},T_{k+1},\cdots\}$ and $\{\cdots,C_{k},C_{k+1},\cdots\}$ can be translated to the following ensemble averages as

and

where $\beta_{m}$ is defined in Lemma 1 for $m\in\{1,\cdots,M^{\prime}\}$ and $\beta_{m}=0$ when $m>M^{\prime}$.

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

If we now replace (27) into (25) and into (26), then after some algebraic manipulations, one can obtain:

where

Taking (28) and (29) into (24), we have

Therefore, it turns out that the average estimation MSE $J$ depends on the estimation error function $c(i)$ and the function $(\mathbf{DM})^{i}\mathbf{(I-D)M}$, both of which involve matrix powers. In what follows, we will introduce and prove some technical lemmas about the element-wise upper and lower bounds of matrix powers, which are the key steps for analyzing the sufficient and necessary stability conditions of the remote estimation system.

We give an element-wise upper bound of matrix powers as below.

Thus, if $r(k)$ is asymptotically and periodically lower bounded by $\underline{r}(k)$ with a period $l$, then the sum of the function $r(k)$ with a consecutive of $l$ samples is lower bounded by $\underline{r}(k)$. When a direct lower bound of the function $r(k)$ is intractable or very loose, we can resort to finding a periodical lower bound $\underline{r}(k)$, which might introduce a tight lower bound of the average of $r(k)$ per $l$ samples, i.e., $\underline{r}(k)/l$. It is clear that the periodical lower bound is tighter if the period $l$ is smaller. In Lemma 3, we will show how to determine the period of a specific problem in details. Definition 2 will be used to capture the lower bound of the average sum MSE in (23) for analyzing the necessary stability condition.

Given the preceding definitions, we can obtain an element-wise lower bound of matrix powers as below.

Taking (12) into (13), we have

From (31) and Lemma 2, for any $\epsilon>0$, there exists $\kappa,\kappa^{\prime},N>0$ such that for all $i>N$ we have

where $\mathcal{N}\triangleq\{1,\cdots,n\}$. Recall that $\mathbf{A}$ is an $n$-by-$n$ matrix. Thus, for any $\epsilon^{\prime}>\epsilon$, we can find $N^{\prime}>N$ and $\kappa^{\prime\prime}>0$ such that $c(i)\leq\kappa^{\prime\prime}(\rho(\mathbf{A})+\epsilon^{\prime})^{2i},\forall i>N^{\prime}$. This completes the proof of Proposition 1.

From Lemma 3(ii), we note that there exists $\eta>0$ and $j,k\in\mathcal{N}$ such that $\left\lvert[\mathbf{A}^{i}\mathbf{W}]_{j,k}\right\rvert^{2}$ is asymptotically and periodically lower bounded by $\eta(\rho(\mathbf{A}))^{2i}$ with the period $l$, which is no larger than the dimension of the matrix $\mathbf{A}$. Then, from (31), when $i$ is sufficiently large, we have

This completes the proof of Proposition 2.

Besides the upper and lower bounds of the estimation error function, to obtain the necessary and sufficient stability condition, we need some additional properties of $(\mathbf{DM})^{i}$ and $(\mathbf{DM})^{i}(\mathbf{(I-D)M})$.

By using the properties in Lemmas 4 and 5, Theorem 1 can be proved as in Appendix D.

Assume a $z$-by-$z$ matrix $\mathbf{Z}$ has $A$ different eigenvalues $\{\lambda_{1},\lambda_{2},\cdots,\lambda_{A}\}$. Represent $\mathbf{Z}$ in its Jordan normal form of $\mathbf{Z}=\mathbf{U}\mathbf{J}\mathbf{U}^{-1}$, where $\mathbf{U}$ is a invertible matrix and

Let $u_{m}$ denote the size of Jordan block $\mathbf{J}_{m}$. Then, $\mathbf{U}$ and $\mathbf{U}^{-1}$ can be represented as

where $\mathbf{F}_{m}$ and $\mathbf{G}_{m}$ are $z$-by $u_{m}$ matrices. Since $\mathbf{U}$ is of full rank, $\mathbf{F}_{m}$ and $\mathbf{G}_{m}$ have full column rank of $u_{m},\forall m=1,\cdots,A$.

Then, we have