Distributed state estimation over binary sensor networks with energy harvester and dynamic event-triggering protocol: a scalable design

2.1. Topology of sensor network

As for a BSN, its topology is formulated as a digraph $$ \mathscr{G}=(\mathscr{V}, \mathscr{E}, \mathscr{A}) $$, where $$ \mathscr{V}=\{1, 2, \ldots, N\} $$, $$ \mathscr{E}\subseteq\mathscr{V}\times\mathscr{V} $$, and $$ \mathscr{A}=[a_{ij}] $$ with $$ a_{ij}\geq 0 $$ denote the node set, the edge set, and the adjacency matrix, respectively. For different sensors $$ i\in\mathscr{V} $$ and $$ j\in\mathscr{V} $$, an edge is expressed as an ordered pair $$ (i, j)\in\mathscr{E} $$. Moreover, if $$ (i, j)\in\mathscr{E} $$, which indicates that j can send message to i, then $$ a_{ij}=1 $$; otherwise $$ a_{ij}=0 $$. $$ \mathscr{N}_i=\{j|(i, j)\in\mathscr{E}, j\neq{i}, i, j\in\mathscr{V}\} $$ denotes the neighboring set of node i. If the in-degree and out-degree of node i are denoted as $$ p_i=\sum_{j}a_{ij} $$ and $$ q_{i}=\sum_{j}a_{ji} $$, respectively, then one further has $$ {\mathscr{N}}_i\triangleq \{j_{i_{1}}, j_{i_{2}}, \ldots, j_{i_{p_{i}}}\} $$, where $$ j_{i_{d}} $$ denotes the d-th neighboring node.

2.2. System formulations

Consider a state-space mode of discrete time-varying systems as follows:

where $$ \bar{x}_{u}\in\mathbb{R}^{n_{\bar{x}}} $$ and $$ \omega_{u}\in\mathbb{R}^{n_{\omega}} $$ are the system state and the external disturbance belonging to $$ l_{2}[0, n-1] $$, respectively; $$ \bar{A}_{u}\in\mathbb{R}^{n_{\bar{x}}\times n_{\bar{x}}} $$ and $$ \bar{B}_{u}\in\mathbb{R}^{n_{\bar{x}}\times n_{\omega}} $$ are known time-varying matrices; u is the sample time belonging to $$ \mathscr{H}\triangleq \{0, 1, \ldots, n-1\} $$. $$ \bar{x}_{0} $$ is the initial state of the system.

The measurement of binary sensor $$ i\in\mathscr{V} $$ is given as follows:

where $$ y_{i, u}\in\mathbb{R} $$ and $$ z_{i, u}\in\mathbb{R} $$ are the measurement output (that is directly accessible) and input (that is inaccessible) of the binary sensor i; $$ \tau_{i}\in\mathbb{R} $$ is a fixed threshold, $$ v_{i, u}\in \mathbb{R}^{n_{v}} $$ is the measurement noise belonging to $$ l_{2}[0, n-1] $$; $$ s\in\mathbb{R} $$ is the duration designated for data collection, and $$ \bar C_{i, u-q}\in\mathbb{R}^{1\times n_{\bar{x}}} $$ and $$ D_{i, u}\in\mathbb{R}^{1\times n_{v}} $$ are two known time-varying matrices.

Regarding the integral measurements, the sample-state augmentation method (i.e., the lifting method) has been utilized to address integral measurements^[18]. Subsequently, the lifting method has become a widely adopted approach for studying estimation problems involving integral measurements^[44,45]. For integral measurement (2), denote $$ x_u\triangleq\begin{bmatrix}\bar{x}_u^T&\bar{x}_{u-1}^T&\cdots&\bar{x}_{u-s}^T\end{bmatrix}^T $$. As such, (1) and (2) are rewritten as:

where

2.3. Distributed estimator for binary sensor

Construct a distributed estimator for binary sensor i using all the available information:

where $$ \hat{x}_{i, u} $$, $$ \bar{z}_{i, u} $$ and $$ \bar{y}_{i, u} $$ are the estimated state, measurement output, and measurement input, respectively; $$ K_{i, u} $$ and $$ G_{ij, u} $$ are the estimator gains to be calculated later; $$ \theta_{i, u} $$ expresses whether the information from the BMs is used or not in the estimator.

Now, introduce the principle of obtaining useful information for estimation purposes. From (2), (5) and (7), under the case of $$ \theta_{i, u}=1 $$, i.e., $$ y_{i, u}\neq \bar{y}_{i, u} $$, which means that $$ \bar{z}_{i, u}>\tau_{i} $$ and $$ z_{i, u}<\tau_{i} $$. As a consequence, $$ \tau_{i} $$ can be expressed as linear combination of $$ \bar{z}_{i, u} $$ and $$ z_{i, u} $$, i.e.,

where the uncertain scalar $$ \delta_{i, u} $$ belongs to $$ [-1, 1] $$.

By resorting to (8), one further has

According to (4), $$ \hat{x}_{j, u} (j\in\mathscr{N}_i) $$ will be transferred to node i. To conserve sensor power and reduce network congestion, the following dynamic ETP is used to schedule the information transmission between neighboring nodes.

2.4. Dynamic event-triggering protocol

This subsection uses a dynamic ETP to determine whether the neighbor’s information should be sent to node i at the current time. The event-triggering time sequence $$ 0\leq t_{j, 0}<t_{j, 1}<\cdots<t_{j, u}<\cdots $$is determined by the following dynamic ETP:

where

and $$ r_{j, u} $$ is an internal dynamic variable whose evolution is given as

where $$ \mu_{j} $$, $$ \lambda_{j} $$, and $$ \varsigma_{j} $$ are given positive scalars. Moreover, the constraint $$ \lambda_j\mu_j\geq1 $$ is satisfied. As such, for all $$ u\in[0, +\infty) $$, one has $$ r_{j, u}\geq0 $$. Here, $$ r_{j, u}\geq 0 $$ is a dynamic variable to adjust the fixed threshold $$ \varsigma_j $$ in the traditional event-triggering condition for reducing the frequency of information transmission of node j.

In addition, to describe the effect of the dynamic ETP, define the following indicative variable as follows:

2.5. Energy harvesting

The inherent constraints in energy storage units within the sensors often prove inadequate for sustained information transmission between neighboring nodes. Consequently, our focus shifts to EH techniques, where a sensor, armed with an energy harvester, has the ability to harvest energy from the surrounding environment and convert it into electrical energy for the purposes of information transmission. Following a similar idea as in^[47,48,46], the EH model is formulated for a neighboring node j of node i.

To begin with, the energy collected by the sensor node j is denoted by a random variable $$ h_{j, u} $$ at the time instant u. Moreover, assume that $$ h_{j, u} $$ is finite and satisfies the well-known binomial distribution, the distribution law of which is given as

where the integer b is the upper bound of the collected energy, and the scalar $$ p_{j} $$ satisfies the constraint $$ 0<p_{j}<1 $$. For convenience in expression, $$ \mathbb{P}(h_{j, u}=m) $$ is abbreviated as $$ \bar{p}_{j, u}^{m} $$ satisfying $$ 0 \leq \bar{p}_{j, u}^{m}\leq1 $$. In addition, $$ h_{j, u} $$ and other stochastic variables in this paper are mutually independent for different nodes j and time instant u.

Remark 1 Under certain assumptions, the energy harvested by sensor node j, which follows a binomial distribution, can be interpreted based on the principles of photovoltaic power generation. For simplicity, let b denote the total number of photons incident on a specific area of the material surface during the time interval from $$ n-1 $$ to n. Each photon has two possible outcomes upon incidence: it is either absorbed (defined as "success") or not absorbed (defined as "failure, " i.e., reflected or transmitted). Assume that the probability of absorption is a constant p, and correspondingly, the probability of non-absorption is $$ 1-p $$. When a photon is absorbed, it contributes one unit of stored energy. It is generally accepted that photons do not interact with each other under typical light intensities (thus, the bosonic bunching effect can be neglected) and the absorption of each photon can therefore be considered an independent event. Consequently, during the time interval from $$ u-1 $$ to u, given that b photons illuminate the surface, the probability that exactly m photons are absorbed follows a binomial distribution.

Next, the energy level of the sensor node j $$ (j\in\mathscr{N}_{i}) $$ at time instant u is denoted by $$ \kappa_{j, u}\in\{0, 1, 2, ..., s_{j}\} $$, where $$ s_{j}>0 $$ represents the maximum capacity that can be stored. At instant u, the sensor node j $$ (j\in\mathscr{N}_{i}) $$ will consume $$ c_{j} $$ units of energy to broadcast its information simultaneously to all its out-neighboring nodes i if $$ \kappa_{j, u}>0 $$. Moreover, any surplus energy generated beyond the battery’s storage limit will be automatically jettisoned. In light of the above formulations, the dynamics of the energy level $$ \kappa_{j, u} $$ is modeled as

where

Here, $$ \sigma_{j, u}=0 $$ means that the neighboring node j cannot transmit $$ \hat{x}_{j, u} $$ to node i due to a lack of enough energy.

Remark 2 As introduced in the Introduction, EH and ETPs have been investigated for distinct underlying motivations. Specifically, EH is primarily motivated by the need to model the impact of energy consumption during information transmission. In contrast, the ETP aims to conserve energy by reducing the frequency of information transmissions through the imposition of a predefined triggering condition. In this paper, building upon existing results, we aim to investigate the precise energy-saving effect achieved within a framework that integrates both ETPs and EH techniques.

According to the previous analysis, the following distributed estimator is constructed:

2.6. Problem statement

Denoting $$ e_{i, u}\triangleq x_{u}-\hat{x}_{i, u} $$, one has the estimation-error dynamic equation:

Letting $$ \xi_{i, u}\triangleq \begin{bmatrix} w_{u}^{T}&v_{i, u}^{T}\end{bmatrix}^{T} $$, one further has

where

In the following, the system dynamics Equation 18 is provided with the specified performance metric.

Definition 1 For any node $$ i\in{\mathscr{V}} $$, let positive constants $$ \gamma $$ and $$ \iota_{i} $$, weighted matrices $$ U_{i}>0, R_{i, u}>0, $$ and $$ T_{i, u}>0 $$ be given. Under energy constraints Equation 15 and dynamic ETPs Equation 10, system dynamics Equation 18 satisfies the $$ H_{\infty} $$-consensus performance constraint over the finite horizon $$ \mathscr{H} $$ in the mean-square sense if the subsequent inequality holds

The aim of this paper is to find the appropriate estimator gains, denoted as $$ K_{i, u} $$ and $$ G_{ij, u} $$ $$ (i\in \mathscr{V} $$ and $$ j\in \mathscr{N}_i) $$, so that the inequality constraint Equation 19 holds.

3.1. Probabilistic distribution law of energy level

As discussed earlier, the transmission of information between neighboring nodes can only be achieved if the energy level is positive (that is, $$ \kappa_{j, u}>0 $$) due to the introduction of the EH technique. Next, we present the probabilistic distribution law of the random variable $$ \sigma_{j, u} $$.

Denote $$ p_{j, u}^m\triangleq\mathbb{P}(\kappa_{j, u}=m) $$ for $$ m = 0, 1, \ldots, s_{j} $$, $$ \phi_{j, u}\triangleq\begin{bmatrix}p_{j, u}^0&p_{j, u}^1&\cdots&p_{j, u}^{s_j}\end{bmatrix}^T $$ and $$ \gamma^{2}_{j, u}\triangleq\mathbb{E}\{(\sigma_{j, u}-\bar{\sigma}_{j, u})^{2}\} $$.

Lemma 1. ^[43] The following conclusions hold

where

and $$ \phi_{j, u} $$ satisfies the following dynamic equation

where

3.2. Vector dissipation

This subsection provides the LPA method to derive sufficient local conditions for any node $$ i\in{\mathscr{V}} $$ to ensure that Equation 18 satisfies the desired performance criterion Equation 19. Vector dissipation is the basis of the LPA method. As such, let us recall the definition of vector dissipation.

Let $$ {\mathbf{V}}_{u}\triangleq\begin{bmatrix}V_{1, u}& \cdots&V_{N, u}\end{bmatrix}^{T} $$ and $$ {\mathbf{S}}_{u}\triangleq\begin{bmatrix}S_{1, u}& \cdots&S_{N, u}\end{bmatrix}^{T} $$.

Definition 2 It is said that the system dynamics Equation 18 is stochastic dissipative regarding the supply rate function $$ S_{i, u} $$, if there exist a storage function $$ V_{i, u}\geq 0 $$ (with $$ V_{0}=0 $$) and a dissipation matrix $$ W\in {\mathbb{R}}^{N\times N} $$ such that the following inequality holds

Furthermore, the system dynamics Equation 18 is vector dissipation regarding the vector supply rate function $$ \mathbf{S}_{u} $$, if Equation 22 holds for any node $$ i\in{\mathscr{V}} $$ and $$ u\in\mathscr{H} $$, i.e.,

Here, for any two vectors with the same dimensions, $$ {\mathbf{e}}\triangleq\begin{bmatrix}e_{1}&e_{2}&\cdots& e_{N}\end{bmatrix}^{T} $$ and $$ {\mathbf{f}}\triangleq\begin{bmatrix}f_{1}& f_{2}&\cdots& f_{N}\end{bmatrix}^{T}\in{\mathbb{R}}^{N} $$, $$ {\mathbf{e}}\ll {\mathbf{f}} $$ implies that $$ e_{i}< f_{i} $$ $$ (\forall i\in{\mathscr{V}}) $$.

Here, as the focus of the LPA method, Equation 23 refers to the vector dissipation inequality, which is constructed by seeking suitable local conditions of node $$ i\in\mathscr{V} $$ with its neighboring nodes. In addition, the dissipation matrix W needs to be nonsingular and the column sub-stochastic, which will be constructed later.

3.3. Scalable distributed state estimation scheme

For proceeding further, for any node $$ i\in\mathscr{V} $$, introduce an interval-valued function regarding $$ q_{i} $$:

and choose the supply rate function:

where $$ \vartheta_{i}\in{\mathbb{R}} $$ is the positive parameter to be given by the subsequent theorem.

With the help of the LPA method, a sufficient criterion regarding the performance constraint Equation 19 is established in the subsequent theorem for the system dynamics Equation 18 by means of the vector dissipation of system Equation 18 regarding the supply rate function $$ S_{i, u} $$.

Theorem 1. Consider any binary sensor $$ i\in\mathscr{V} $$ with its neighboring nodes $$ j (j\in\mathscr{N}_{i}) $$. Suppose that the parameters $$ \alpha_{i}\in{\mathscr{I}}_{q_{i}} $$, $$ \alpha_{j}\in{\mathscr{I}}_{q_{j}} $$, $$ \gamma>0 $$, $$ \varsigma_i>0 $$, $$ \varsigma_j>0 $$, $$ \varrho_{i}>0 $$, $$ \vartheta_{i}>0 $$, and $$ \bar{r}_{i}>0 $$ satisfying the following conditions:

Let $$ U_{i} $$ and $$ P_{i, 0} $$ be given positive definite matrices satisfying $$ 0<P_{i, 0}\leq \gamma^{2}U_{i} $$, and let $$ K_{i, u} $$ and $$ G_{ij, u} $$ be the given estimator gains. Then, the system dynamics Equation 18 achieves the performance constraint Equation 19, if there exists a set of positive definite matrices $$ P_{i, u+1}>0 $$ such that the following $$ \theta_{i, u} $$-dependent inequality holds for any $$ u\in{\mathscr{H}} $$:

where

Proof 1. For node i, choose the scalar storage function with the following form:

where

Calculate the mathematical expectation of the storage function along the dynamics of Equation 12 and Equation 19

where

In addition, we also obtain

According to Equation 15, we introduce a positive scalar $$ \vartheta_{i} $$, such that the following inequality holds:

As such, one further has

which implies that

Then, in light of Equation 29, one immediately has

where

Moreover, since $$ \alpha_{i}\in{\mathscr{I}}_{q_{i}} $$ for $$ i\in{\mathscr{V}} $$, one immediately has W is the desired dissipation matrix.

By means of the double expectation theorem, one obtains that

which can be rewritten in a compact form:

Next, a scalar form is derived by making the linear transformation to Equation 32:

which is further calculated in light of the property of dissipation matrix W as follows:

Furthermore, the above inequality can be rewritten as

Next, Equation 34 is summed regarding u from $$ 0 $$ to $$ n-1 $$ and one has

which indicates that

Noticing that $$ \mathbb{E}\{V_{i, N+1}\}\geq0 $$, one derives from Equation 36 and $$ S_{i, u} $$ that

and, therefore, one has Equation 19 according to $$ P_{i, 0}\leq\gamma^{2}U_{i} $$, $$ \frac{\varsigma_i}{\mu_i}+\vartheta_{i}\sum_{j\in\mathscr{N}_i}\varsigma_j\leq \gamma^{2}\varrho_{i} $$, and $$ r_{i, 0}\leq n\gamma^{2}\bar{r}_{i} $$.

Furthermore, denoting $$ \iota_{i}\triangleq \varrho_{i}+\bar{r}_{i} $$, the proof is complete according to Definition Definition 1.

In the following, the desired estimator gains are calculated in terms of recursive linear matrix inequalities by resorting to the Schur Complement Lemma.

Theorem 2. Consider any binary sensor $$ i\in\mathscr{V} $$ with its neighboring nodes $$ j (j\in\mathscr{N}_{i}) $$. Suppose that the parameters $$ \alpha_{i}\in{\mathscr{I}}_{q_{i}} $$, $$ \alpha_{j}\in{\mathscr{I}}_{q_{j}} $$, $$ \gamma>0 $$, $$ \varsigma_i>0 $$, $$ \varsigma_j>0 $$, $$ \varrho_{i}>0 $$, $$ \vartheta_{i}>0 $$, and $$ \bar{r}_{i}>0 $$ satisfying the following conditions:

Let $$ U_{i} $$ and $$ P_{i, 0} $$ be given positive definite matrices satisfying $$ 0<P_{i, 0}\leq \gamma^{2}U_{i} $$. Then, the system dynamics Equation 18 satisfies the $$ H_\infty $$-consensus constraint in the mean square sense if there exists a sequence of positive definition matrices $$ P_{i, u+1}>0 $$ such that the following $$ \theta_{i, u} $$-dependent inequality holds

where

Furthermore, if Equation 38 is feasible, then one calculates

Proof 2. By resorting to the Schur Complement Lemma, it follows that Equation 25holds if and only if

Next, the left-hand side of Equation 39 is partitioned as

where

Next, one acquires the following compact form

where

By means of S-Procedure, $$ \Pi_{i, u}<0 $$ holds with $$ F_{i, u}F_{i, u}^{T}\leq I $$ if and only if thereexists a constant $$ \mu_{i1, u}>0 $$ such that

By reusing Schur Complement Lemma, Equation 42 holds if and only if

where

After performing the congruent transformation $$ \operatorname{diag}\{I, P_{i, u+1}, \underbrace{P_{i, u+1}, \cdots, P_{i, u+1}}_{p_{i}}, I\} $$ to Equation 43 and introducing some new notations, one easily has Equation 38 following some simple algebraic operations. From Equation 38, one immediately calculates the estimator gains $$ K_{i, u} $$ and $$ G_{ij, u} $$. As a result, the proof is complete with the aid of Theorem 1.

Remark 3 colorblueIt should be noted that Theorems 1 and 2 provide the parameter constraints that are helpful in achieving the desirable performance constraint and reproducing the results and understanding the engineering implications of the model.

Remark 4 The scalable distributed $$ H_{\infty} $$-consensus state estimation problem is investigated for a class of discrete time-varying system over BSNs with energy harvesters under the schedule of the dynamic ETP. Compared with the existing results, this paper exhibits the following characteristics: (1) the combined impact of the dynamic ETP and the EH techniques is studied within the framework of the scalable distributed estimation; (2) useful information is extracted from the BM subject to integral measurement; (3) a binomial distribution is used to formulate the limited energy collected from the surrounding environment, which is more in line with engineering practice; and (4) the scalable design is achieved by means of the LPA method, which implies that the distributed estimation scheme is implemented on every node.