Tobit Kalman fusion filtering under dynamic event-triggering protocol with token bucket specification

2.1. System model

Consider the state-space model for a class of linear discrete time-varying systems and its measurements from $$ l $$ sensor nodes as follows:

where $$ x_{t}\in{\mathbb{R}}^{{n}_{x}} $$ and $$ z_{m, t}\in{\mathbb{R}^{n_y}} $$ are the state vector and measurement vector, respectively; $$ \omega_{t}\in{\mathbb{R}^{\omega}} $$ and $$ v_{m, t}\in{\mathbb{R}^{n_{y}}} $$ are white Gaussian noises with $$ \mathbb{E}\{\omega_{t}\}=0 $$, $$ \mathbb{E}\{v_{m, t}\}=0 $$, $$ \mathbb{D}\{\omega_{t}\}=Q_{t} $$, and $$ \mathbb{D}\{v_{m, t}\}=R_{m, t} $$. Furthermore, it is assumed that $$ {x}_{0} $$, $$ {\omega_{t}} $$ and $$ {v}_{m, t} $$ are mutually independent for different $$ t $$ and $$ m $$. $$ A_{t} $$, $$ B_{t} $$ and $$ C_{m, t} $$ are known time-varying matrices with compatible dimensions; the initial state $$ {x}_{0} $$ is a random variable with mean $$ \bar{x}_{0} $$ and variance $$ {P}_{0} $$. For convenience of later discussion, denote $$ z_{m, t}\triangleq\begin{bmatrix} z_{m, t}^{(1)}&z_{m, t}^{(2)}&\dots & z_{m, t}^{(n_y)} \end{bmatrix}^{T}\in\mathbb{R}^{n_y} $$.

The situation where one or multiple sensors cannot perform data measurement or acquisition properly for various reasons is referred to as "censored measurements". To address the issue of censored measurements and mitigate its adverse impact on system monitoring, the Tobit measurement model is used to formulate the one-side censored measurement:

where $$ y_{m, t}^{(j)} $$ is the censored measurement from the $$ m $$-th sensor and $$ \tau_{m}^{(j)} $$ is the one-side censoring threshold of $$ y_{m, t}^{(j)} $$.

In light of (2), define a series of Bernoulli random variables $$ \varrho_{m, t}^{(j)} (m=1, 2, \ldots, l; j=1, 2, \ldots, n_{y}) $$ to formulate $$ y_{m, t}^{(j)} $$ as follows:

which obeys the distribution law:

where $$ \bar{\varrho}_{m, t}^{(j)} $$ are known non-negative constants. Moreover, $$ \varrho_{m, t} $$ is uncorrelated with other random variables mentioned above.

The censoring probability $$ {\bar\varrho_{m, t}} $$ is approximated by

where $$ \hat{x}_{m, t|t-1} $$ represents prediction for $$ x_{t} $$ from node $$ m $$, and $$ \Phi( \cdot ) $$ is the cumulative distribution function of the standard normal distribution. From (5), it follows that $$ \bar{\varrho}_{m, t} $$ is related to the state $$ x_{t} $$, which is replaced with $$ \hat{x}_{m, t|t-1} $$ since $$ x_{t} $$ is unknown.

Similar to^[12], the measurement expectation and variance of $$ \bar{\varrho}_{m, t}^{(j)} $$ are given as follows:

where

Here, $$ \phi(\cdot ) $$ and $$ \Phi(\cdot) $$ are the probability density function and the cumulative distribution function of the standard normal distribution, respectively.

Denote $$ \varrho_{m, t}\triangleq\text{diag}\left\{\varrho_{m, t}^{(1)}, \varrho_{m, t}^{(2)}, \ldots, \varrho_{m, t}^{(n_{y})}\right\} $$, $$ y_{m, t}\triangleq\begin{bmatrix} y_m^{(1)}, y_m^{(2)}, \dots, y_m^{(n_y)} \end{bmatrix}^{T} $$, and $$ \tau_{m}\triangleq\begin{bmatrix} \tau_m^{(1)}, \tau_m^{(2)}, \dots, \tau_m^{(n_y)} \end{bmatrix}^{T} $$. Hence, (2) can be rewritten as:

Also, the following statistical information can be obtained from (6) and (7):

and

Remark 1.The censoring measurement is formulated as the Tobit model. The main challenge in dealing with the Tobit model arises from two aspects. Firstly, the Tobit model (2) represents a piecewise linear function, which is inherently nonlinear. To facilitate subsequent analysis, a random variable defined in (3) is introduced to reformulate the censoring measurement into a uniform form (8). Another issue involves determining the probability law and statistical property of the random variable (i.e., (5), (6), and (7)), where (5) is derived by using statistical information of normal distribution of measurement noise $$ v_{m, t} $$, and (6) and (7) are achieved by means of (5).

2.2. Information transmission protocols

To save precious energy and limited bandwidth, the DETP is used for node $$ m $$ to decide whether to transmit the current measurements to the filter. For this purpose, denote the event-triggering time sequence by $$ 0\le t_m^{1}<\cdots<t_m^{n}<\cdots $$ and define

where $$ \sigma_m $$ and $$ \chi_{m} $$ are given positive scalars, $$ y_{m, t_m^n} $$ represents the latest transmitted measurement until time $$ t $$, and the internal dynamic variable $$ \eta _{m, t} $$ satisfies:

where $$ \lambda_m $$ is a given positive scalar and $$ \eta_0\ge 0 $$ is the initial value. Moreover, $$ \lambda_{m}\chi_{m}>1 $$ and thus ensure $$ \eta_{m, t+1}\geq0 $$. Here, the inequality

is referred to as the event-triggering condition. For convenience, define the following indicator variable:

To cope with the measurement transmission in unexpected situations, such as the need to transfer multiple data at once, a token bucket triggering mechanism is introduced. First, the token bucket is employed to describe the available bandwidth resource, where the integer $$ g $$ and $$ c_{m} $$ are the token generating rate and transmission cost, respectively. For simplicity, $$ c_{m} $$ means the total cost for transmitting all elements of $$ y_{m, t} $$ at time instant $$ t $$ of node $$ m $$.

In order to unify the centralized utilization of the token bucket algorithm, the number of tokens is then evenly allocated to each sensor node, which is given as follows:

The internal token change of the token bucket is formulated as follows:

with initial values $$ \xi_{0}\geq 0 $$, where $$ b $$ is the token bucket capacity, and

After scheduled by the DETP with token bucket specification, the measurement received by the filter is formulated as

For the convenience of later analysis, (19) is rewritten as

Remark 2.Different from^[30], the introduction of the indicator variable $$ \alpha_{m, t} $$ is utilized to determine whether the conditions are met. This approach provides convenience for subsequent analysis, clearly defining the impact of the ETP. Building on this concept, another indicator variable $$ \beta_{m, t} $$ is implemented to assess whether there are sufficient tokens for information transmission.

The block diagram of FTKFF is depicted in Figure 1. As for node $$ m $$, the event trigger is used to determine whether the censored measurement $$ y_{m, t} $$ is transmitted or not, which is formulated by $$ \alpha_{m, t} $$. If the event-triggering condition (14) to be predetermined is met (i.e., $$ \alpha_{m, t}=1 $$), then the measurement to be transmitted is denoted as $$ y_{m, t_{m}^{n}} $$. Next, the token bucket $$ m $$ is employed to formulate the available communication resource, which is formulated by $$ \beta_{m, t} $$. If there exist sufficient tokens (i.e., $$ \beta_{m, t}=1 $$), then the filter can receive the measurement, i.e., $$ \bar{y}_{m, t} $$. Therefore, only when both token and event-triggering conditions are met can information transmission proceed. Such a procedure is expressed as (20).

Figure 1. A block diagram of FTKFF under DETP with token bucket.

2.3. Filter design

By using the available measurement $$ \bar{y}_{m, t} $$, construct a recursive filter

with the initial value $$ \hat{x}_{m, 0\mid0}=\bar{x}_{m, 0} $$, and for $$ t\in [t_{m}^n, t_{m}^{n+1})(n \ge 0, t\geq 1) $$, where $$ \hat{x}_{m, t|t-1} $$ and $$ \hat{x}_{m, t|t} $$ represent the prediction of one step and estimation from node $$ m $$, respectively; $$ \hat{y}_{m, t\mid t-1} $$ is the measurement estimate to be discussed later; and $$ K_{m, t} $$ is the filter gain to be designed.

The primary objective of this paper is to design a local filter of the form (21) and (22) for every node under the DETP with token bucket specifications for all censored measurements within the framework of TKF. Moreover, we are devoted to developing an FTKFF framework to address the multiple challenges induced by the censored measurements, the DETP and the token bucket specifications.

3.1. Filtering error and covariance matrix

In line with (5) and (6), the estimate of the measurement $$ \hat{y}_{m, t\mid t-1} $$ is derived as

where

Let $$ e_{m, t \mid t-1}\triangleq x_{t}-\hat{x}_{m, t|t-1} $$. From (1) and (21), one calculates

Subsequently, from (1), (8), (22) and (23), one has

where

By means of Eqs. (24) and (25), the covariance matrices of the prediction error and filtering error are, respectively, calculated as follows

and

where

Next, an upper bound for the filtering error covariance matrix $$ P_{m, t|t} $$ is provided to eliminate the uncertainties in Eq. (27).

Theorem 1.For the given positive scalars $$ \varepsilon _{j}(j=1, 2, ..., 14) $$, it is said that $$ \bar{P}_{m, t\mid t} $$ is an upper bound of the $$ {P}_{m, t\mid t} $$, namely,

with the initial value of $$ P_{0\mid 0}=\bar P_{0\mid 0}>0 $$. Moreover, $$ \bar P_{m, t\mid t} $$ satisfies the following recursions:

and

where

and

where $$ \bar{G}_{m, t} $$ is an upper bound of $$ G_{m, t}\triangleq\mathbb{E}\{\eta_{m, t}^{2}\} $$ with the initial condition $$ \bar{G}_{m, 0}=\eta_{m, 0}^{2} $$.

Here, $$ \circ $$ denotes the Hadamard product, which is defined as $$ [A\circ B]_{ij}=A_{ij}\cdot B_{ij} $$.

Proof. According to lemma 1 in^[36], it follows from Eq. (27) that

Following the same line, one has

Moreover, given the event-triggering condition (11), we have

In light of lemma 5 and the proof of lemma 4 in^[27], one derives

Substituting Eqs. (32) and (34) into (31), one obtains the following inequality

By using lemma 2 in^[27], one obtains that

The proof is now complete. □

Next, the filter gain $$ K_{m, t} $$ is designed such that $$ \text{tr}\{\bar{P}_{m, t|t}\} $$ is minimized at each time instant.

Theorem 2.Suppose that the positive scalars $$ \varepsilon_j(j=1, 2, ..., 14) $$ are given. The upper bound of the filtering error covariance $$ \bar{P}_{m, t\mid t} $$ given by Eq. (30) is minimized by the following filter gain

where

Proof. First, calculate the trace of $$ \bar{P}_{m, t\mid t} $$ obtained by Eq. (30) as follows

Next, one can derive the derivative of $$ \text{tr}\left \{\bar{P}_{m, t\mid t}\right \} $$ as follows

By letting (39) be zero, one acquires the filter gain as follows:

As such, one completes the proof of this theorem. □

Remark 3.Different from the approach to dealing with the DETP in^[6,27,37], an indicator variable is employed to formulate the impact of the DETP. From (37), it can be observed that the filter gain $$ K_{m, t} $$ is dependent on the indicator variable $$ \theta_{m, t} $$, which formulates the combined role of DETP with the token bucket specifications. Meanwhile, the filter gain can be calculated under two cases, i.e., $$ \theta_{m, t}=1 $$ (i.e., filter does not receive measurement) and $$ \theta_{m, t}=0 $$ (i.e., filter does receive measurement), respectively.

3.2. Boundedness analysis

In this section, the boundedness of $$ \bar{P}_{m, t|t} $$ will be discussed in the sense of mean square. Before carrying out the performance analysis, an assumption is provided to make some necessary constraints on the corresponding parameters.

Assumption 1.Suppose that the following inequalities hold

where $$ a $$, $$ b $$, $$ c $$, $$ \pi $$, $$ q $$, $$ r $$, $$ q $$, $$ o $$, $$ \rho $$, $$ \bar\tau $$, $$ \vartheta $$, $$ \xi $$, $$ \bar c $$, $$ \underline{c} $$, $$ \underline\varpi $$, $$ \bar\varpi $$, $$ \mu $$, and $$ \bar{\tau} $$ are all known positive real numbers.

Theorem 3.Suppose Assumption 1 holds, in line with the system (1) and censored measurement (2) under Assumption 1, if the following inequality holds:

where $$ w_{m, t} \triangleq\left \| I-K_{m, t}\bar{\varrho }_{m, t}C_{m, t} \right \|^{2} $$, then the filtering error (25) is mean-square bounded. Moreover, one has

where

Proof. In line with Assumption 1 and Eq. (26), it deduces that

By noting Eqs. (37) with (40), the upper bound of the filter gain $$ K_{m, t+1} $$ is given as

from which can be gained from Eq. (30):

By means of $$ w_{m, t} \triangleq\left \| I-K_{m, t}\bar{\varrho }_{m, t}C_{m, t} \right \|^{2} $$, one has

In the following, we can rewrite (47) as follows

which further implies

If (42) holds, i.e., $$ \kappa_1w_{m, t} a^2<1 $$, one then has (43), which implies that the convergence of the upper bound $$ \left\|\bar{P}_{m, t\mid t}\right\| $$. Meanwhile, it is observed that $$ \bar{P}_{m, t\mid t} $$ always serves as an upper bound for the true estimation error covariance $$ P_{m, t\mid t} $$. Consequently, the filtering error is mean-square bounded. Therefore, the proof is now complete. □

Remark 4.Theorem 3 shows the combined impact of the DETP with the token bucket specifications on the filtering error boundedness and the filtering accuracy. Noticing that $$ w_{m, t} $$ is dependent on filter gain $$ K_{m, t} $$ from sufficient condition (42) and $$ K_{m, t} $$ is dependent on $$ \theta_{m, t} $$ from (37), the DETP with the token bucket specifications will pose an impact on the bounded condition. In addition, as given in (43), $$ \theta_{m, t} $$ and $$ w_{m, t} $$ are involved in the upper bound of the filtering error covariance, showing that $$ \theta_{m, t} $$ would also pose effect on the filtering accuracy.

3.3. Federated Tobit Kalman fusion filtering

In line with federated filtering fusion, the state estimate and the error covariance are $$ \hat{x}_{m, t\mid t} $$ and $$ \bar{P}_{m, t\mid t} $$, respectively, produced by the $$ m $$th LTKF. Let $$ \hat{x}^g_{t\mid t} $$ and $$ \hat{P}^g_{t\mid t} $$ denote the global optimal estimate and covariance obtained at the fusion center under the federated fusion rule.

Theorem 4.The FTKFF scheme for systems (1)-(2) is expressed as

with the resulting updated values

where $$ \alpha_m $$ is the information weight of the $$ m $$-th local estimator and $$ {\sum_{m=1}^{l}} \alpha_m=1 $$; $$ \hat{x}_{m, t\mid t} $$, $$ \bar{P}_{m, t\mid t} $$, and $$ Q_{m, t} $$ are the local estimates, upper bounds, and process noise covariance to be initialized at time $$ t $$ for predictions at time $$ t+1 $$.

It is noted that the role of (51) is to allocate the fused result to the individual sub-filter. The detailed procedure of FTKFF is summarized in Algorithm 1.

In line with the boundedness analysis of Theorem 3, the same argument can be applied to deduce the boundedness.

According to Algorithm 1, one has the computational complexity of the scheme developed in this paper.

Remark 5.Now, we focus on the computational complexity of the scheme. In light of the system dimensions, such as $$ x_{t}\in\mathbb{R}^{n_{y}} $$, $$ y_{i, t}\in\mathbb{R}^{n_{y}} $$, $$ w_{t}\in\mathbb{R}^{w} $$ and the resulting matrix dimensions, such as $$ A_{t}\in{\mathbb{R}}^{n_{x}\times n_{x}} $$, $$ B_{t}\in\mathbb{R}^{n_{x}\times n_{w}} $$, $$ C_{i, t}\in\mathbb{R}^{n_{y}\times n_{x}} $$, one can obtain the computational complexity of the scheme developed in this paper at every fixed time instant. The FLOPS (Floating Point Operations Per Second) of main computational formulas from Algorithm 1 is $$ 7ln_{y}^{3}+(10l+1)n_{x}^{3}+36ln_{y}^{2}n_{x}+22ln_{y}n_{x}^{2}+(13l+6)n_{x}^{2}+18ln_{y}^{2}-17ln_{x}n_{y}-(2l+1)n_{x}-ln_{y}+2ln_{x}n_{w}^{2}-2ln_{x}n_{w}+2l+59 $$. The corresponding complexity is given as $$ O(7ln_{y}^{3}+(10l+1)n_{x}^{3}+36ln_{y}^{2}n_{x}+22ln_{y}n_{x}^{2}+2ln_{x}n_{w}^{2}) $$.

In the following, we will discuss the consistency of the federated fusion.

Theorem 5.If $$ P_{m, t\mid t}\le\bar{P}_{m, t\mid t} $$, then one has

Proof. First of all, one calculates

Noting such a fact:

one derives that

According to (49), one further has

which indicates that

By resorting to lemma 2 in^[41], one derives

Now, the proof is complete. □

Remark 6.The FTKFF scheme is proposed for a class of discrete time-varying systems under the schedule of the DETP with token bucket specifications. This paper embodies the following significant characteristics from two viewpoints: (1) a local Tobit Kalman filter (LTKFs) is elaborately designed based on an enhanced protocol model that gives combined consideration of the impacts incurred by the DETP and token bucket specifications, where two indicator variables are used to formulate their roles; (2) a federated fusion rule is chosen by productively integrating the local estimates from LTKFs.

Authors' contributions

Conceptualization and manuscript drafting: Chen X

Methodology and experiment: Zhang J

Manuscript edition: Song Y

Review and supervision: Han F

Availability of data and materials

Not applicable.

Financial support and sponsorship

This work was supported in part by the National Natural Science Foundation of China under Grants 62073070 and U21A2019.

Conflicts of interest

Han F is a Junior Editorial Board Member of the journal Complex Engineering Systems, while the other authors have declared that they have no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

© The Author(s) 2024.