A Dynamic Event-Based Recursive State Estimation for Mobile Robot Localization

Abstract

This paper deals with the recursive state estimation issue for mobile robot localization under a dynamic event-based mechanism. To enhance the utilization of communication resources, a dynamic event-based transmission protocol is utilized to reduce unnecessary measurement transmissions by introducing an auxiliary dynamical variable to adjust threshold parameters. The primary objective of this paper is to develop a dynamic event-based recursive state estimation scheme for the mobile robot localization problem in the presence of the impact of the dynamic event-based mechanism such that an upper bound on the estimation error covariance is firstly guaranteed by using mathematical induction and then is locally minimized by virtue of appropriately choosing the gain parameters. Furthermore, the boundedness analysis of the estimation error is conducted by establishing an evaluation criteria in the mean-squared sense. Finally, an experimental example is conducted to verify the feasibility of the proposed mobile robot localization strategy.

1. Introduction

Over the past few years, mobile robots (MRs) have not only attracted wide research interest from the academic community but also sparked strong development enthusiasm in the industry due to their remarkable practical utility and extensive application prospects in the areas of intelligent inspection, warehouse management and healthcare, etc. [1,2,3]. The swift and efficient implementation of MR Localization (MRL) is a pivotal research issue in the field of MRs, as it serves as the cornerstone for enabling robots to autonomously and precisely complete their assigned tasks within diverse environments. Generally, the process of MRL is fundamentally bifurcated into two key stages: environmental data collection and transmission via sensors and implementing localization by utilizing positioning system, which are highly susceptible to various inherent factors such as inevitable external disturbances, limited communication resources, and instability of the system structure [4,5]. In such a context, it can significantly undermine the decision-making process and execution of an MR, thereby posing a high risk of malfunctioning, particularly in complex and dynamic settings such as crowded warehouses, intricate factory layouts, or busy healthcare facilities. Therefore, it is crucial to develop an efficient strategy that is capable of enhancing the accuracy and robustness of localization in various operating environments for MRs, which has given rise to some initial progress, see, e.g., [6,7,8]. In Ref. [6], a robust decentralized cooperative localization approach has been presented by leveraging a multi-centralized framework and M-estimation technique, effectively addressing communication failures and biased measurements to enhance localization accuracy. A novel distributed finite memory estimation method has been introduced in Ref. [7] for robust multiple-robot localization in wireless sensor networks subject to missing data and errors.

On the other research front, with the rapid development of sensor technology, a series of state estimation strategies have been developed and applied in various applications, especially in signal processing, the internet of things, and industrial automation. Regarding different performance requirements, state estimation algorithms can be categorized into several types, including a conventional Kalman filtering scheme and its variants [9,10], an $H_{\infty }$ filtering scheme [11,12], set-membership filtering [13,14], etc. As a matter of fact, due to many unforeseen disturbances in industrial scenarios, accurately completing the state estimation with the above methods is quite challenging [15,16]. To tackle such a problem, recursive state estimation schemes have been proposed where the estimation error covariance (EEC) is minimized in a certain sense, and accordingly, some elegant work is available [17,18,19]. For example, in Ref. [17], the study focused on the challenging task of designing a distributed recursive state estimation scheme for a class of multi-rate nonlinear systems within the context of sensor networks. In Ref. [19], in the delayed state-saturated system under the effect of the energy harvesting sensors, the joint state and fault estimation problem has been addressed. Unfortunately, when it comes to the issue of MRL, a recursive state estimation scheme tailored to MRL has not been considered yet, mainly due to the complex structure and nonlinear characteristics of MRs, making it difficult to derive the upper bound of the EEC and minimize it in a recursive process. As such, there is a practical demand to establish a recursive state estimation scheme to deal with the MRL problem, which constitutes one of the main motivations behind this paper.

It is notable that in most state estimation/filtering issues, one underlying assumption is made that the communication resources are often unlimited and available. In engineering practice, the energy storage of sensors and the capacity of channels are constrained [20], whereas the continuously transmitting data to the remote estimator may lead to resource wastage and risks of measurement transmission failures. Therefore, a huge amount of communication protocols have been proposed to address the challenge of limited communication resources, where the most prominent one is the event-triggered mechanism (ETM), which transmits measurements only when a predefined triggering condition is satisfied [21,22]. In view of the disparity in the triggering threshold, ETMs can be categorized into static ETMs and dynamic ETMs (DETMs) [23]. Different from the static ETMs with fixed threshold parameters, the DETMs have the ability of dynamically adjusting the threshold parameters via an auxiliary dynamical variable, thereby reducing more unnecessary triggers and further enhancing resource utilization without significantly degrading the system’s performance. To date, some elegant results have been made available on the issue of the state estimation/filtering concerning DETMs [24,25,26]. For instance, the filtering problem based on a dynamic event-triggering mechanism has been tackled in Ref. [24] for the discrete time-varying systems confronted with complex environments. In Ref. [25], an efficient dynamic event-triggered communication protocol has been utilized to deal with the consensus issue for multi-agent systems.

On the basis of the above discussion, there is currently no research on the recursive state estimation scheme for the MRL problem that truly satisfies the practical application needs, not to mention considering a DETM in the localization process of MRs. Therefore, the ultimate goal of this paper is to narrow this gap by developing a dynamic event-based recursive state estimator for MRL. Specifically, this problem remains unexplored and poses additional challenges due to the following identified issues: (1) How to devise a reasonable model that seamlessly integrates DETM into the MRL process. (2) How to exploit an appropriate approach to tackle the impact of DETM on the estimation performance. (3) How to formulate a recursive estimator for the MRL problem based on a DETM. (4) How to conduct the analysis of the estimation performance for the developed dynamic event-based state estimation algorithm. Corresponding to the identified issues, the main contributions of this paper can be highlighted from the four aspects below: (1) a spatial state equation is modeled by capturing the dynamic characteristics of MRs, which is then used to design the integration of the DETM into the realm of MRL; (2) the DETM is employed, for the first time, for the MRL problem to further save communication resources by setting a dynamically adjusted threshold; (3) a dynamic event-based estimator is firstly constructed so that the bounded EEC is minimized by designing the estimator gain parameters; (4) the mean-square boundedness analysis of the estimation error is quantified by establishing an evaluation criteria; and (5) a dynamic event-based estimation algorithm is developed that is suitable for online computation and real-time usage. Finally, a simulation experiment is conducted to verify the validity of the developed estimation strategy.

Notation: The notations employed in this paper are fairly standard. ${\chi }^{-1}$ and ${\chi }^T$ below mean the inverse and the transposition of $\chi$, respectively. $F>\beth$ (respectively, $F\ge \beth$) represents that $F-\beth$ is a positive definite (positive semi-define) matrix, with $F$ and ℶ being symmetric matrices. $I_n$ represents the n-dimensional identity matrix. $\mathrm{tr}\left\{\mathcal{V}\right\}$ represents the trace of the matrix $\mathcal{V}$. $\mathbb{E}\left\{\mathcal{Q}\right\}$ stands for the mathematical expectation of the stochastic variable $\mathcal{Q}$. $\parallel \mathcal{W}\parallel$ denotes the norm of a matrix $\mathcal{W}$.

2. Problem Formulation

2.1. MR Model and Sensor Measurement Model

Consider an MR model as depicted in Figure 1, and then the continuous-time system describing the motion of MR is determined by [27]:

$$\left\{\begin{array}{cc}\hfill \dot{x}=& \nu cos\phi \hfill \\ \hfill \dot{y}=& \nu sin\phi \hfill \\ \hfill \dot{\phi }=& \omega \hfill \end{array}\right.$$

(1)

where $(x,y)$ represents the position coordinates of MR, and $\phi$ represents its orientation angle. $\nu$ and $\omega$ represent the displacement velocity and the angular velocity, respectively.

During the sampling period t, the MR’s displacement and angular velocities that can be accurately measured by an odometer are assumed to be constant. Then, the discrete-time system for the MR can be constructed as

$$\left\{\begin{array}{cc}\hfill x_{k+1}=& x_k+{\nu }_{k}tcos{\phi }_k\hfill \\ \hfill y_{k+1}=& y_k+{\nu }_{k}tsin{\phi }_k\hfill \\ \hfill {\phi }_{k+1}=& {\phi }_k+{\omega }_{k}t\hfill \end{array}\right.$$

(2)

where $(x_k,y_k,{\phi }_k)$ is the position coordinates and orientation angle of MR. ${\nu }_k$ and ${\omega }_k$ represent the corresponding displacement and angular velocities, respectively.

By setting ${\mathcal{X}}_k={\left[x_k{y}_k{\phi }_k\right]}^T$, ${\xi }_k={\left[{\nu }_{k}t{\omega }_{k}t\right]}^T$ and considering the presence of disturbances, the dynamic system model is determined by

$${\mathcal{X}}_{k+1}=g({\mathcal{X}}_k,{\xi }_k)+{ϵ}_k,$$

(3)

where $g({\mathcal{X}}_k,{\xi }_k)={\mathcal{X}}_k+{[{\nu }_{k}tcos{\phi }_k{\nu }_{k}tsin{\phi }_k{\omega }_{k}t]}^T$. ${ϵ}_k$ denotes Gaussian noise with $\mathbb{E}\left\{{ϵ}_k\right\}=0$ and covariance $R_k>0$.

In Figure 1, the landmark point $\mathcal{M}(x_{\mathcal{M}},y_{\mathcal{M}})$ is given, and $d_k$ represents the distance between MR $(x_k,y_k)$ and the landmark $\mathcal{M}(x_{\mathcal{M}},y_{\mathcal{M}})$. Specifically, the distance $d_k$ and the azimuth ${\psi }_k$ at k are determined by

$$\left\{\begin{array}{cc}\hfill d_k=& \sqrt{{(x_{\mathcal{M}}-x_k)}^2+{(y_{\mathcal{M}}-y_k)}^2},\hfill \\ \hfill {\psi }_k=& {\phi }_k-arctan\left({\displaystyle \frac{y_{\mathcal{M}}-y_k}{x_{\mathcal{M}}-x_k}}\right).\hfill \end{array}\right.$$

(4)

By noting Equation (4), the sensor measurement model of the MR system with measurement noise can be induced as follows:

$$z_k={\mu }_k+{\varpi }_k,$$

(5)

where ${\mu }_k={\left[d_k{\psi }_k\right]}^T$ denotes the ideal measurement, and ${\varpi }_k$ is Gaussian noise with $\mathbb{E}\left\{{\varpi }_k\right\}=0$ and covariance $Q_k>0$.

2.2. Dynamic Event-Triggering Strategy

In practical communication networks where resource constraint is a common phenomenon, the continuous transmitting of data to the remote estimator may lead to resource wastage. Therefore, a DETM is utilized to reduce such resource consumption. Denote the triggering sequences of the measurement transmission instants by $0=s_0\le s_1\le s_2\le \cdots \le s_l\le \cdots$, where the triggering sequences $s_l$ satisfy

$$\begin{array}{c}\hfill s_{l+1}=min\{k\in [0,N]|k>s_l,{\displaystyle \frac{1}{\vartheta }}{\eta }_k+\tau -\parallel {\varrho }_k\parallel \le 0\}\end{array}$$

(6)

where ${\varrho }_k\triangleq z_k^{s_l}-z_k$, $z_k^{s_l}$ stands for the transmission at the latest triggering squence. $\vartheta$ and $\tau$ are known positive scalars. ${\eta }_k$ represents the internal dynamic variable that is determined by

$$\begin{array}{c}\hfill {\eta }_{k+1}=\gamma {\eta }_k+\tau -\parallel {\varrho }_k\parallel \end{array}$$

(7)

where $\gamma$ denotes a given scalar, and the internal dynamical variable ${\eta }_k$ satisfies ${\eta }_0\ge 0$. It is worth mentioning that the parameters $\gamma$ and $\vartheta$ in the DETM are usually assumed to satisfy $\gamma \vartheta >1$, thereby ensuring ${\eta }_k\ge 0$.

The estimator for MRL with the measurements ${\overline{z}}_k$ received by the DETM is constructed as

$$\left\{\begin{array}{cc}\hfill {\widehat{\mathcal{X}}}_{k|k-1}=& g({\widehat{\mathcal{X}}}_{k-1|k-1},{\xi }_{k-1})\hfill \\ \hfill {\widehat{\mathcal{X}}}_{k|k}=& {\widehat{\mathcal{X}}}_{k|k-1}+K_k(z_k+{\varrho }_k-{\widehat{\mu }}_{k|k-1})\hfill \end{array}\right.$$

(8)

Here, ${\widehat{\mathcal{X}}}_{k|k-1}$ and ${\widehat{\mathcal{X}}}_{k|k}$, respectively, stand for the prediction and the estimate of the state, and $K_k$ denotes the estimator gain matrix.

Define the prediction error as $e_{k|k-1}={\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k-1}$, and the estimation error as $e_{k|k}={\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k}$. The prediction error covariance and the EEC are denoted by $\mathbb{E}\left\{e_{k|k-1}{e}_{k|k-1}^T\right\}$ and $\mathbb{E}\left\{e_{k|k}{e}_{k|k}^T\right\}$, respectively. This paper aims to devise an estimator that ensures the upper bound ${\Sigma }_{k|k}$ on the EEC, and then the gain matrix $K_k$ is recursively parameterized to minimize such an upper bound.

3. Main Results

In this section, based on the measurement transmitted through the DETM, we are going to find a solution for MRL. An upper bound on the EEC is obtained mathematically, and then is minimized through two Riccati-like difference equations. In addition, the lemmas below are introduced to promote the derivation.

Lemma 1.

For two real vectors $\mathcal{H}$ and $\mathcal{S}$, we can obtain the following relationship

$$\begin{array}{c}\hfill \mathcal{H}{\mathcal{S}}^T+\mathcal{S}{\mathcal{H}}^T\le o\mathcal{H}{\mathcal{H}}^T+o^{-1}\mathcal{S}{\mathcal{S}}^T,\end{array}$$

where $o>0$ is an arbitrary scalar.

Lemma 2

([28]). Given the matrices $\mathcal{M}$, $\mathcal{N}$, $\mathcal{H}$, and $\mathcal{S}$ with known appropriate dimensions and $\mathcal{S}{\mathcal{S}}^T\le I$, for a positive symmetric matrix Γ and a given positive scalar α satisfying ${\alpha }^{-1}I-\mathcal{H}\Gamma {\mathcal{H}}^T>0$, we can have the inequality below

$$\begin{array}{c}\hfill (\mathcal{M}+\mathcal{N}\mathcal{S}\mathcal{H})\Gamma {(\mathcal{M}+\mathcal{N}\mathcal{S}\mathcal{H})}^T\le \mathcal{M}{({\Gamma }^{-1}-\alpha {\mathcal{H}}^T\mathcal{H})}^{-1}{\mathcal{M}}^T+{\alpha }^{-1}\mathcal{N}{\mathcal{N}}^T.\end{array}$$

Lemma 3

([29]). Define ${\mathcal{M}}_k(·):{\mathbb{R}}^n\to {\mathbb{R}}^n$ and ${\mathcal{N}}_k(·):{\mathbb{R}}^n\to {\mathbb{R}}^n$ for any $k\ge 0$. Assume that $\mathcal{A}={\mathcal{A}}^T>0$, ${\mathcal{M}}_k\left(\mathcal{A}\right)={\mathcal{M}}_k^T\left(\mathcal{A}\right)$ and ${\mathcal{N}}_k\left(\mathcal{A}\right)={\mathcal{N}}_k^T\left(\mathcal{A}\right)$. If the following differences equations

$$\begin{array}{cc}\hfill & {\mathcal{M}}_k\left(\mathcal{A}\right)\le {\mathcal{M}}_k\left(\mathcal{B}\right),\forall \mathcal{A}={\mathcal{A}}^T\le \mathcal{B}={\mathcal{B}}^T,\hfill \\ \hfill & {\mathcal{M}}_k\left(\mathcal{B}\right)\le {\mathcal{N}}_k\left(\mathcal{B}\right)\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{S}}_{k+1}={\mathcal{M}}_k\left({\mathcal{S}}_k\right),{\mathcal{H}}_{k+1}={\mathcal{N}}_k\left({\mathcal{H}}_k\right),{\mathcal{S}}_0={\mathcal{H}}_0>0\end{array}$$

hold, one has ${\mathcal{S}}_k\le {\mathcal{H}}_k$.

Lemma 4.

For the known positive scalars ${\beta }_{i,k}(i=1,2,3)$, if there exists a matrix, ${\overline{\mathcal{Z}}}_k$ holds

$$\begin{array}{cc}\hfill & {\overline{\mathcal{Z}}}_{k+1}\triangleq {\Psi }_k\left({\overline{\mathcal{Z}}}_k\right)\hfill \\ \hfill =& ((1+{\beta }_{1,k})(1+{\beta }_{2,k}){\gamma }^2+{\displaystyle \frac{(1+{\beta }_{3,k})(1+{\beta }_{1,k}^{-1})}{{\vartheta }^2}}){\overline{\mathcal{Z}}}_k\hfill \\ \hfill & +((1+{\beta }_{1,k})(1+{\beta }_{2,k}^{-1})+(1+{\beta }_{1,k}^{-1})(1+{\beta }_{3,k}^{-1})){\tau }^2,\hfill \end{array}$$

with the initial value ${\overline{\mathcal{Z}}}_0={\eta }_0^2$; then we can obtain that ${\overline{\mathcal{Z}}}_k$ is the upper bound of ${\mathcal{Z}}_k$, where ${\mathcal{Z}}_k\triangleq \mathbb{E}\left\{{\eta }_k^2\right\}$.

Proof. 

Combining Equation (7) and Lemma 1, one has

$$\begin{array}{c}\hfill {\varrho }_k^T{\varrho }_k\le {({\displaystyle \frac{1}{\vartheta }}{\eta }_k+\tau )}^2\le {\displaystyle \frac{1+{\beta }_{3,k}}{{\vartheta }^2}}{\eta }_k^2+(1+{\beta }_{3,k}^{-1})){\tau }^2.\end{array}$$

(9)

As such, it is not difficult to give rise to

$$\begin{array}{cc}\hfill & {\mathcal{Z}}_{k+1}=\mathbb{E}\left\{{\eta }_{k+1}^2\right\}\hfill \\ \hfill =& \mathbb{E}\left\{\right(\gamma {\eta }_k+\tau -\parallel {\varrho }_k{\parallel )}^2\}\hfill \\ \hfill \le & (1+{\beta }_{1,k})(1+{\beta }_{2,k}){\gamma }^2\mathbb{E}\left\{{\eta }_k^2\right\}+(1+{\beta }_{1,k})(1+{\beta }_{2,k}^{-1}){\tau }^2+(1+{\beta }_{1,k}^{-1})\mathbb{E}\left\{{\varrho }_k^T{\varrho }_k\right\}\hfill \\ \hfill \le & ((1+{\beta }_{1,k})(1+{\beta }_{2,k}){\gamma }^2+{\displaystyle \frac{(1+{\beta }_{3,k})(1+{\beta }_{1,k}^{-1})}{{\vartheta }^2}}){\mathcal{Z}}_k\hfill \\ \hfill & +((1+{\beta }_{1,k})(1+{\beta }_{2,k}^{-1})+(1+{\beta }_{1,k}^{-1})(1+{\beta }_{3,k}^{-1})){\tau }^2\hfill \\ \hfill =& {\Psi }_k\left({\mathcal{Z}}_k\right).\hfill \end{array}$$

(10)

By noting Lemma 3, one has ${\mathcal{Z}}_k\le {\overline{\mathcal{Z}}}_k$, which ends the proof of this lemma.    □

In what follows, we need to linearize the nonlinear system Equation (3) by approximating the nonlinear function $g({\mathcal{X}}_k,{\xi }_k)$ around ${\widehat{\mathcal{X}}}_{k|k}$. It is obtained that the linearized form of Equation (3) is as follows:

$$\begin{array}{cc}\hfill {\mathcal{X}}_{k+1}=& g({\widehat{\mathcal{X}}}_{k|k},{\xi }_k)+{\Pi }_k({\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k})+L_k{\mathcal{G}}_k{H}_k({\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k})+{ϵ}_k,\hfill \end{array}$$

(11)

where

$$\begin{array}{c}\hfill {\Pi }_k={\displaystyle \frac{\partial g}{\partial {\mathcal{X}}_k}}{|}_{{\mathcal{X}}_k={\widehat{\mathcal{X}}}_{k|k}}=\left[\begin{array}{ccc}{\displaystyle \frac{\partial g_x}{\partial x_k}}& {\displaystyle \frac{\partial g_x}{\partial y_k}}& {\displaystyle \frac{\partial g_x}{\partial {\phi }_k}}\\ {\displaystyle \frac{\partial g_y}{\partial x_k}}& {\displaystyle \frac{\partial g_y}{\partial y_k}}& {\displaystyle \frac{\partial g_y}{\partial {\phi }_k}}\\ {\displaystyle \frac{\partial g_{\phi }}{\partial x_k}}& {\displaystyle \frac{\partial g_{\phi }}{\partial y_k}}& {\displaystyle \frac{\partial g_{\phi }}{\partial {\phi }_k}}\end{array}\right]{|}_{{\mathcal{X}}_k={\widehat{\mathcal{X}}}_{k|k}}=\left[\begin{array}{ccc}1& 0& -{\nu }_{k}tsin{\widehat{\phi }}_{k|k}\\ 0& 1& {\nu }_{k}tcos{\widehat{\phi }}_{k|k}\\ 0& 0& 1\end{array}\right],\end{array}$$

and $L_k$ denotes a scaling matrix tailored to the considered problem, $H_k$ is a matrix with known dimensions, and ${\mathcal{G}}_k$ is an unknown matrix with ${\mathcal{G}}_k{\mathcal{G}}_k^T\le I$.

Similarly, the measurement Equation (5) can be denoted as

$$\begin{array}{c}\hfill z_k={\widehat{\mu }}_{k|k-1}+{\mathsf{\Lambda }}_k({\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k-1})+G_k{\Delta }_k{F}_k({\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k-1})+{\varpi }_k,\end{array}$$

(12)

where

$$\begin{array}{c}\hfill {\mathsf{\Lambda }}_k={\displaystyle \frac{\partial \mu }{\partial {\mathcal{X}}_k}}{|}_{{\mathcal{X}}_k={\widehat{\mathcal{X}}}_{k|k-1}}=\left[\begin{array}{ccc}{\displaystyle \frac{\partial {\mu }_d}{\partial x_k}}& {\displaystyle \frac{\partial {\mu }_d}{\partial y_k}}& {\displaystyle \frac{\partial {\mu }_d}{\partial {\phi }_k}}\\ {\displaystyle \frac{\partial {\mu }_{\psi }}{\partial x_k}}& {\displaystyle \frac{\partial {\mu }_{\psi }}{\partial y_k}}& {\displaystyle \frac{\partial {\mu }_{\psi }}{\partial {\phi }_k}}\end{array}\right]{|}_{{\mathcal{X}}_k={\widehat{\mathcal{X}}}_{k|k-1}}=\left[\begin{array}{ccc}{\displaystyle \frac{-(x_{\mathcal{M}}-{\widehat{x}}_{k|k-1})}{{\widehat{d}}_{k|k-1}}}& {\displaystyle \frac{-(y_{\mathcal{M}}-{\widehat{y}}_{k|k-1})}{{\widehat{d}}_{k|k-1}}}& 0\\ {\displaystyle \frac{-(y_{\mathcal{M}}-{\widehat{y}}_{k|k-1})}{{\widehat{d}}_{k|k-1}}}& {\displaystyle \frac{x_{\mathcal{M}}-{\widehat{x}}_{k|k-1}}{{\widehat{d}}_{k|k-1}}}& 1\end{array}\right]\end{array}$$

with ${\widehat{d}}_{k|k-1}=\sqrt{{(x_{\mathcal{M}}-{\widehat{x}}_{k|k-1})}^2+{(y_{\mathcal{M}}-{\widehat{y}}_{k|k-1})}^2}$, $G_k$ denotes a scaling matrix tailored to the considered problem, $F_k$ is a matrix with known dimensions, and ${\Delta }_k$ is an unknown matrix with ${\Delta }_k{\Delta }_k^T\le I$.

Thus, by noting Equations (8), (11), and (12), the prediction error and the estimation error are determined by

$$\begin{array}{c}\hfill e_{k|k-1}={\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k-1}=({\Pi }_{k-1}+L_{k-1}{\mathcal{G}}_{k-1}{H}_{k-1})e_{k-1|k-1}+{ϵ}_{k-1}\end{array}$$

(13)

and

$$\begin{array}{cc}\hfill & e_{k|k}={\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k}\hfill \\ \hfill =& {\mathcal{X}}_k-[{\widehat{\mathcal{X}}}_{k|k-1}+K_k(z_k+{\varrho }_k-{\widehat{\mu }}_{k|k-1})]\hfill \\ \hfill =& [I-K_k({\mathsf{\Lambda }}_k+G_k{\Delta }_k{F}_k)]e_{k|k-1}-K_k{\varpi }_k-K_k{\varrho }_k.\hfill \end{array}$$

(14)

3.1. Design of DETM-Based Recursive Estimator

In the following theorem, we have completed the calculation of the prediction error covariance $P_{k|k-1}$ and EEC $P_{k|k}$. Meanwhile, the upper bound of the EEC is also derived.

Theorem 1.

Define $a_k$ and $b_k$, and ${\alpha }_{i,k}(i=1,2)$ are given positive scalars. If the symmetric matrices satisfy the following equations

$$\begin{array}{cc}\hfill {\Sigma }_{k|k-1}\triangleq & {\Pi }_{k-1}{({\Sigma }_{k-1|k-1}^{-1}-{\alpha }_{1,k-1}{H}_{k-1}^T{H}_{k-1})}^{-1}{\Pi }_{k-1}^T+{\alpha }_{1,k-1}^{-1}{L}_{k-1}{L}_{k-1}^T+R_{k-1}\hfill \end{array}$$

(15)

and

$$\begin{array}{cc}\hfill {\Sigma }_{k|k}\triangleq & (1+a_k)(I-K_k{\mathsf{\Lambda }}_k){({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{(I-K_k{\mathsf{\Lambda }}_k)}^T\hfill \\ \hfill & +(1+a_k){\alpha }_{2,k}^{-1}{K}_k{G}_k{G}_k^T{K}_k^T+(1+a_k^{-1}+b_k)K_k{\overline{\Psi }}_k\left({\overline{\mathcal{Z}}}_k\right)K_k^T\hfill \\ \hfill & +(1+b_k^{-1})K_k{Q}_k{K}_k^T\hfill \end{array}$$

(16)

with $P_{0|0}\le {\Sigma }_{0|0}$, where

$$\begin{array}{c}\hfill {\overline{\Psi }}_k\left({\overline{\mathcal{Z}}}_k\right)=[(1+{\beta }_{3,k}){\vartheta }^{-2}{\overline{\mathcal{Z}}}_k+(1+{\beta }_{3,k}^{-1}){\tau }^2]I,\end{array}$$

and the symmetric positive matrices ${\Sigma }_{k|k-1}$ and ${\Sigma }_{k|k}$satisfy that

$$\begin{array}{c}\hfill {\alpha }_{1,k-1}^{-1}I-H_{k-1}{\Sigma }_{k-1|k-1}{H}_{k-1}^T>0,{\alpha }_{2,k}^{-1}I-F_k{\Sigma }_{k|k-1}{F}_k^T>0,\end{array}$$

then we arrive at ${\Sigma }_{k|k}$ being an upper bound on $P_{k|k}$, i.e., $P_{k|k}\le {\Sigma }_{k|k}$.

Proof. The initial condition is given that $P_{0|0}\le {\Sigma }_{0|0}$, and we assume that $P_{k-1|k-1}\le {\Sigma }_{k-1|k-1}$. Based on the above conditions, we seek to establish that $P_{k|k}\le {\Sigma }_{k|k}$.

By noting Lemma 2, it is derived from Equation (13) that

$$\begin{array}{cc}\hfill & P_{k|k-1}=\mathbb{E}\left\{e_{k|k-1}{e}_{k|k-1}^T\right\}\hfill \\ \hfill =& \mathbb{E}\left\{({\Pi }_{k-1}+L_{k-1}{\mathcal{G}}_{k-1}{H}_{k-1})e_{k-1|k-1}{e}_{k-1|k-1}^T\right.\hfill \\ \hfill & \left.\times {({\Pi }_{k-1}+L_{k-1}{\mathcal{G}}_{k-1}{H}_{k-1})}^T+{ϵ}_{k-1}{ϵ}_{k-1}^T\right\}\hfill \\ \hfill =& ({\Pi }_{k-1}+L_{k-1}{\mathcal{G}}_{k-1}{H}_{k-1})P_{k-1|k-1}{({\Pi }_{k-1}+L_{k-1}{\mathcal{G}}_{k-1}{H}_{k-1})}^T+R_{k-1}\hfill \\ \hfill \le & {\Pi }_{k-1}{(P_{k-1|k-1}^{-1}-{\alpha }_{1,k-1}{H}_{k-1}^T{H}_{k-1})}^{-1}{\Pi }_{k-1}^T+{\alpha }_{1,k-1}^{-1}{L}_{k-1}{L}_{k-1}^T+R_{k-1}.\hfill \end{array}$$

(17)

Next, based on Lemma 3 and Equation (17), we can obtain that $P_{k|k-1}\le {\Sigma }_{k|k-1}$.

Subsequently, we are prepared to demonstrate that $P_{k|k}\le {\Sigma }_{k|k}$. Similarly, it can be derived from Equation (15) that

$$\begin{array}{cc}\hfill & P_{k|k}=\mathbb{E}\left\{e_{k|k}{e}_{k|k}^T\right\}\hfill \\ \hfill =& \mathbb{E}\left\{{\mathcal{U}}_1{\mathcal{U}}_1^T-{\mathcal{U}}_1{\mathcal{U}}_3^T+{\mathcal{U}}_2{\mathcal{U}}_2^T+{\mathcal{U}}_2{\mathcal{U}}_3^T-{\mathcal{U}}_3{\mathcal{U}}_1^T+{\mathcal{U}}_3{\mathcal{U}}_2^T+{\mathcal{U}}_3{\mathcal{U}}_3^T\right\},\hfill \end{array}$$

(18)

where

$$\begin{array}{cc}\hfill & {\mathcal{U}}_1=[I-K_k({\mathsf{\Lambda }}_k+G_k{\Delta }_k{F}_k)]e_{k|k-1},\hfill \\ \hfill & {\mathcal{U}}_2=K_k{\varpi }_k,{\mathcal{U}}_3=K_k{\varrho }_k.\hfill \end{array}$$

In the light of Equation (18), and Lemmas 1 and 2, the EEC can be further calculated so that

$$\begin{array}{cc}\hfill P_{k|k}\le & (1+a_k)[I-K_k({\mathsf{\Lambda }}_k+G_k{\Delta }_k{F}_k)]P_{k|k-1}{[I-K_k({\mathsf{\Lambda }}_k+G_k{\Delta }_k{F}_k)]}^T\hfill \\ \hfill & +(1+a_k^{-1}+b_k)K_k\mathbb{E}\left\{{\varrho }_k{\varrho }_k^T\right\}{K}_k^T+(1+b_k^{-1})K_k{Q}_k{K}_k^T\hfill \\ \hfill \le & (1+a_k)(I-K_k{\mathsf{\Lambda }}_k){(P_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{(I-K_k{\mathsf{\Lambda }}_k)}^T\hfill \\ \hfill & +(1+a_k){\alpha }_{2,k}^{-1}{K}_k{G}_k{G}_k^T{K}_k^T+(1+a_k^{-1}+b_k)K_k\mathbb{E}\left\{{\varrho }_k{\varrho }_k^T\right\}{K}_k^T\hfill \\ \hfill & +(1+b_k^{-1})K_k{Q}_k{K}_k^T.\hfill \end{array}$$

(19)

Then, according to the dynamic triggering condition Equation (6), one has

$$\begin{array}{cc}\hfill {\varrho }_k{\varrho }_k^T\le & {\varrho }_k^T{\varrho }_{k}I\hfill \\ \hfill \le & {({\displaystyle \frac{1}{\vartheta }}{\eta }_k+\tau )}^{2}I\hfill \\ \hfill \le & ({\displaystyle \frac{(1+{\beta }_{3,k})}{{\vartheta }^2}}{\eta }_k^2+(1+{\beta }_{3,k}^{-1}){\tau }^2)I\hfill \\ \hfill =& {\overline{\Psi }}_k\left({\mathcal{Z}}_k\right).\hfill \end{array}$$

(20)

By virtue of Lemma 4, we can arrive at $\mathbb{E}\left\{{\varrho }_k{\varrho }_k^T\right\}\le {\overline{\Psi }}_k\left({\overline{\mathcal{Z}}}_k\right)$. Thus, it is easy to derive from Equation (19) that

$$\begin{array}{cc}\hfill P_{k|k}\le & (1+a_k)(I-K_k{\mathsf{\Lambda }}_k){(P_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{(I-K_k{\mathsf{\Lambda }}_k)}^T\hfill \\ \hfill & +(1+a_k){\alpha }_{2,k}^{-1}{K}_k{G}_k{G}_k^T{K}_k^T+(1+a_k^{-1}+b_k)K_k{\overline{\Psi }}_k\left({\overline{\mathcal{Z}}}_k\right)K_k^T\hfill \\ \hfill & +(1+b_k^{-1})K_k{Q}_k{K}_k^T.\hfill \end{array}$$

(21)

Due to $P_{k|k-1}\le {\Sigma }_{k|k-1}$, it yields that

$$\begin{array}{cc}\hfill & P_{k|k}\le {\Sigma }_{k|k}.\hfill \end{array}$$

(22)

Now, we have completed this proof.    □

In Theorem 1, we have derived the upper bound ${\Sigma }_{k|k}$ for the EEC. Subsequently, such an upper bound will be minimized by selecting the desired estimator parameter $K_k$ at each iteration.

Theorem 2.

The minimal upper bound of the EEC can be derived by choosing the following estimator parameter $K_{k+1}$ below

$$K_k={\Omega }_k{\aleph }_k^{-1},$$

(23)

where

$$\begin{array}{cc}\hfill {\Omega }_k\triangleq & (1+a_k){({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{\mathsf{\Lambda }}_k^T,\hfill \\ \hfill {\aleph }_k\triangleq & (1+a_k){\mathsf{\Lambda }}_k{({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{\mathsf{\Lambda }}_k^T\hfill \\ \hfill & +(1+a_k){\alpha }_{2,k}^{-1}{G}_k{G}_k^T+(1+a_k^{-1}+b_k){\overline{\Psi }}_k\left({\overline{\mathcal{Z}}}_k\right)\hfill \\ \hfill & +(1+b_k^{-1})Q_k.\hfill \end{array}$$

(24)

Furthermore, the minimal upper bound ${\Sigma }_{k|k}$ is determined by

$$\begin{array}{c}\hfill {\Sigma }_{k|k}=-{\Omega }_k{\aleph }_k^{-1}{\Omega }_k^T+(1+a_k){({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}.\end{array}$$

(25)

Proof. It is not difficult to rewrite the upper bound in Equation (16) through the completing-square technique as follows:

$$\begin{array}{cc}\hfill {\Sigma }_{k|k}=& (1+a_k){({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}-(1+a_k)K_k{\mathsf{\Lambda }}_k{({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}\hfill \\ \hfill & -(1+a_k){({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{\mathsf{\Lambda }}_k^T{K}_k^T\hfill \\ \hfill & +(1+a_k)K_k{\mathsf{\Lambda }}_k{({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{\mathsf{\Lambda }}_k^T{K}_k^T+(1+a_k){\alpha }_{2,k}^{-1}{K}_k{G}_k{G}_k^T{K}_k^T\hfill \\ \hfill & +(1+a_k^{-1}+b_k)K_k{\overline{\Psi }}_k\left({\overline{\mathcal{Z}}}_k\right)K_k^T+(1+b_k^{-1})K_k{Q}_k{K}_k^T\hfill \\ \hfill =& K_k{\aleph }_k{K}_k^T-{\Omega }_k{K}_k^T-K_k^T{\Omega }_k+(1+a_k){({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}\hfill \\ \hfill =& (K_k-{\Omega }_k{\aleph }_k^{-1}){\aleph }_k{(K_k-{\Omega }_k{\aleph }_k^{-1})}^T-{\Omega }_k\hfill \\ \hfill & \times {\aleph }_k^{-1}{\Omega }_k^T+(1+a_k){({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}.\hfill \end{array}$$

(26)

Accordingly, the minimal upper bound ${\Sigma }_{k|k}$ in Equation (25) can be obtained directly by letting the gain $K_k={\Omega }_k{\aleph }_k^{-1}$. The proof is now complete.    □

3.2. Performance Analysis

In this section, we will analyze the boundedness of the estimation error for the designed recursive estimator by giving a sufficient condition in mean-squared sense. To begin with, let us make the following definition.

Definition 1

([30]). For real numbers $\overline{\varsigma }>0$, $\underline{\varsigma }>0$, $\upsilon >0$ and $0<\iota <1$, the stochastic process ${\zeta }_k$ is considered to satisfy

$$\mathbb{E}\left\{\parallel {\zeta }_k{\parallel }^2\right\}\le {\displaystyle \frac{\overline{\varsigma }}{\underline{\varsigma }}}\mathbb{E}\left\{\parallel {\zeta }_0{\parallel }^2\right\}{\iota }^k+{\displaystyle \frac{\upsilon \overline{\varsigma }}{1-\iota }}$$

(27)

for any $k\ge 0$.

Theorem 3.

Assume that $\overline{\lambda }$, $\underline{\lambda }$, $\overline{f}$, ${\overline{g}}_1$, ${\overline{g}}_2$, $\overline{h}$, $\overline{l}$, $\overline{\delta }$, $\overline{\pi }$, ${\overline{q}}_1$, ${\underline{q}}_1$, ${\overline{q}}_2$, ${\underline{q}}_2$, and $\overline{\varrho }$ are positive scalars, if the following inequalities

$$\begin{array}{cc}\hfill & \underline{\lambda }\le \parallel {\mathsf{\Lambda }}_k\parallel \le \overline{\lambda },\parallel F_k\parallel \le \overline{f},\parallel G_k\parallel \le {\overline{g}}_1,\parallel {\mathcal{G}}_k\parallel \le {\overline{g}}_2,\hfill \\ \hfill & \parallel H_k\parallel \le \overline{h},\parallel L_k\parallel \le \overline{l},\parallel {\Delta }_k\parallel \le \overline{\delta },\parallel {\Pi }_k\parallel \le \overline{\pi },\hfill \\ \hfill & {\underline{q}}_1\le \parallel R_k\parallel \le {\overline{q}}_1,\parallel Q_k\parallel \le {\overline{q}}_2,\mathrm{tr}\left\{\mathbb{E}\left\{{\varrho }_k^T{\varrho }_k\right\}\right\}\le \overline{\varrho },\hfill \\ \hfill & \rho ={(\overline{\pi }+\overline{l}{\overline{g}}_2\overline{h})}^2{(1+{\displaystyle \frac{\overline{\lambda }(\overline{\lambda }+{\overline{g}}_1\overline{\delta }\overline{f})}{{\underline{\lambda }}^2}})}^2<1,\hfill \end{array}$$

(28)

hold, we can obtain that the estimation error is exponentially mean-square bounded.

Proof. Substituting Equation (13) into Equation (14), one has

$$e_{k|k}=B_k{C}_{k-1}{e}_{k-1|k-1}+r_k$$

(29)

where

$$\begin{array}{cc}\hfill A_k=& {\mathsf{\Lambda }}_k+G_k{\Delta }_k{F}_k,C_{k-1}={\Pi }_{k-1}+L_{k-1}{\mathcal{G}}_{k-1}{H}_{k-1},\hfill \\ \hfill B_k=& I-K_k{A}_k,r_k=B_k{ϵ}_{k-1}-K_k{\varpi }_k-K_k{\varrho }_k\hfill \end{array}$$

According to Equations (23) and (24), we have

$$\begin{array}{cc}\hfill \parallel K_k\parallel =& \parallel (1+a_k){({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{\mathsf{\Lambda }}_k^T{\aleph }_k^{-1}\parallel \hfill \\ \hfill <& \parallel (1+a_k){({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{\mathsf{\Lambda }}_k^T\hfill \\ \hfill & {\left[(1+a_k){\mathsf{\Lambda }}_k{({\Sigma }_{k|k-1}^{-1}-{\alpha }_{2,k}{F}_k^T{F}_k)}^{-1}{\mathsf{\Lambda }}_k^T\right]}^{-1}\parallel \hfill \\ \hfill \le & {\displaystyle \frac{\overline{\lambda }}{{\underline{\lambda }}^2}}\triangleq \overline{k}.\hfill \end{array}$$

(30)

Along the similar line, we can also confirm that

$$\begin{array}{cc}\hfill & \parallel A_k\parallel =\parallel {\mathsf{\Lambda }}_k+G_k{\Delta }_k{F}_k\parallel \le \overline{\lambda }+{\overline{g}}_1\overline{\delta }\overline{f}\triangleq \overline{a},\hfill \\ \hfill & \parallel B_k\parallel =\parallel I-K_k{A}_k\parallel \le 1+{\displaystyle \frac{\overline{\lambda }\overline{a}}{{\underline{\lambda }}^2}}\triangleq \overline{b},\hfill \\ \hfill & \parallel C_{k-1}\parallel =\parallel {\Pi }_{k-1}+L_{k-1}{\mathcal{G}}_{k-1}{H}_{k-1}\parallel \le \overline{\pi }+\overline{l}{\overline{g}}_2\overline{h}\triangleq \overline{c}.\hfill \end{array}$$

(31)

Then, based on Lemma 1 and Equation (29), one has

$$\begin{array}{cc}\hfill & \mathbb{E}\left\{r_k^T{r}_k\right\}\hfill \\ \hfill \le & (1+{\sigma }_1)\mathbb{E}\left\{{ϵ}_{k-1}^T{B}_k^T{B}_k{ϵ}_{k-1}\right\}+(1+{\sigma }_2)\mathbb{E}\left\{{\varpi }_k^T{K}_k^T{K}_k{\varpi }_k\right\}\hfill \\ \hfill & +(1+{\sigma }_1^{-1}+{\sigma }_2^{-1})\mathbb{E}\left\{{\varrho }_{k-1}^T{K}_k^T{K}_k{\varrho }_{k-1}\right\}\hfill \\ \hfill \le & (1+{\sigma }_1){\overline{q}}_1{\overline{b}}^2+(1+{\sigma }_2){\overline{q}}_2{\overline{k}}^2+(1+{\sigma }_1^{-1}+{\sigma }_2^{-1})\overline{\varrho }{\overline{k}}^2\triangleq \overline{r},\hfill \end{array}$$

(32)

where ${\sigma }_i(i=1,2)$ are positive scalars.

Subsequently, the iterative matrix equation with respect to ${\mathsf{\Phi }}_k$ is determined by

$${\mathsf{\Phi }}_k=B_k{C}_{k-1}{\mathsf{\Phi }}_{k-1}{C}_{k-1}^T{B}_k^T+R_{k-1}+\epsilon I,$$

(33)

where ${\mathsf{\Phi }}_0=R_0+\epsilon I$ with a scalar ε. Then, by noting Equation (29), it is sufficient to observe that

$$\begin{array}{cc}\hfill \parallel {\mathsf{\Phi }}_k\parallel =& \parallel B_k{\parallel }^2\parallel C_{k-1}{\parallel }^2\parallel {\mathsf{\Phi }}_{k-1}\parallel +\parallel R_{k-1}\parallel +\parallel \pi I\parallel \hfill \\ \hfill \le & \rho \parallel {\mathsf{\Phi }}_{k-1}\parallel +{\overline{q}}_1+\epsilon \hfill \end{array}$$

(34)

where ρ has been defined in Equation (28). Furthermore, the inequality can be obtained directly as follows:

$$\begin{array}{cc}\hfill \parallel {\mathsf{\Phi }}_k\parallel \le & {\rho }^k\parallel {\mathsf{\Phi }}_0\parallel +({\overline{q}}_1+\epsilon )\sum _{i=0}^{k-1}{\rho }^i\hfill \\ \hfill <& \parallel {\mathsf{\Phi }}_0\parallel +({\overline{q}}_1+\epsilon )\sum _{i=0}^{\infty }{\rho }^i\hfill \\ \hfill =& \parallel {\mathsf{\Phi }}_0\parallel +{\displaystyle \frac{{\overline{q}}_1+\epsilon }{1-\rho }}.\hfill \end{array}$$

(35)

On the other hand, it can be easily observed that

$${\mathsf{\Phi }}_k\ge \epsilon I.$$

(36)

Combining Equation (35) and Equation (36), we can obtain that $\underline{\varphi }I\le {\mathsf{\Phi }}_k\le \overline{\varphi }I$ for any $k\ge 0$ where $\overline{\varphi }$ and $\underline{\varphi }$ are positive scalars.

Define $V_k\left(e_{k|k}\right)=e_{k|k}^T{\mathsf{\Phi }}_k^{-1}{e}_{k|k}$. With Lemma 1 and Equation (29) we have

$$\begin{array}{cc}\hfill & \mathbb{E}\left\{V_k\left(e_{k|k}\right)\right|e_{k-1|k-1}\}-(1+\kappa )V_{k-1}\left(e_{k-1|k-1}\right)\hfill \\ \hfill =& \mathbb{E}\left\{{[B_k{C}_{k-1}{e}_{k-1|k-1}+r_k]}^T{\mathsf{\Phi }}_k^{-1}[B_k{C}_{k-1}{e}_{k-1|k-1}+r_k]\right\}\hfill \\ \hfill & -(1+\kappa )e_{k-1|k-1}^T{\mathsf{\Phi }}_{k-1}^{-1}{e}_{k-1|k-1}\hfill \\ \hfill \le & (1+\kappa )\mathbb{E}\left\{e_{k-1|k-1}^T[C_{k-1}^T{B}_k^T{\mathsf{\Phi }}_k^{-1}{B}_k{C}_{k-1}-{\mathsf{\Phi }}_{k-1}^{-1}]e_{k-1|k-1}\right\}\hfill \\ \hfill & +(1+{\kappa }^{-1})\mathbb{E}\left\{r_k^T{\mathsf{\Phi }}_{k-1}^{-1}{r}_k\right\}\hfill \end{array}$$

(37)

where κ is a positive scalar.

Recalling the definition of matrix ${\mathsf{\Lambda }}_{k+1}$, one has

$$\begin{array}{cc}\hfill & C_{k-1}^T{B}_k^T{\mathsf{\Phi }}_k^{-1}{B}_k{C}_{k-1}-{\mathsf{\Phi }}_{k-1}^{-1}\hfill \\ \hfill =& C_{k-1}^T{B}_k^T{(B_k{C}_{k-1}{\mathsf{\Phi }}_{k-1}{C}_{k-1}^T{B}_k^T+R_{k-1}+\epsilon I)}^{-1}{B}_k{C}_{k-1}-{\mathsf{\Phi }}_{k-1}^{-1}\hfill \\ \hfill =& -{[{\mathsf{\Phi }}_{k-1}+{\mathsf{\Phi }}_{k-1}{C}_{k-1}^T{B}_k^T{(R_{k-1}+\epsilon I)}^{-1}{B}_k{C}_{k-1}{\mathsf{\Phi }}_{k-1}]}^{-1}\hfill \\ \hfill =& -{[I+C_{k-1}^T{B}_k^T{(R_{k-1}+\epsilon I)}^{-1}{B}_k{C}_{k-1}{\mathsf{\Phi }}_{k-1}]}^{-1}{\mathsf{\Phi }}_{k-1}^{-1}\hfill \\ \hfill \le & -{(1+{\displaystyle \frac{{\overline{c}}^2{\overline{b}}^2\overline{\varphi }}{{\underline{q}}_1+\epsilon }})}^{-1}{\mathsf{\Phi }}_k^{-1}.\hfill \end{array}$$

(38)

Let $\theta =(1+{\kappa }^{-1}){\displaystyle \frac{{\overline{r}}^2}{\underline{\varphi }}}$. By substituting Equation (38) into Equation (37), it yields that

$$\begin{array}{cc}\hfill & \mathbb{E}\left\{V_k\left(e_{k|k}\right)\right|e_{k-1|k-1}\}-(1+\phi )V_{k-1}\left(e_{k-1|k-1}\right)\hfill \\ \hfill \le & -(1+\kappa ){(1+{\displaystyle \frac{{\overline{c}}^2{\overline{b}}^2\overline{\varphi }}{{\underline{q}}_1+\epsilon }})}^{-1}{V}_k\left(e_{k|k}\right)+\theta .\hfill \end{array}$$

(39)

Furthermore, based on Equation (39), we can obtain that

$$\begin{array}{c}\hfill \mathbb{E}\left\{V_k\left(e_{k|k}\right)\right|e_{k-1|k-1}\}\le \zeta V_{k-1}\left(e_{k-1|k-1}\right)+\theta ,\end{array}$$

(40)

where $\zeta =(1+\kappa )[1-{(1+{\displaystyle \frac{{\overline{c}}^2{\overline{b}}^2\overline{\varphi }}{{\underline{q}}_1+\epsilon }})}^{-1}]$. It is obvious that $\zeta \in (0,1)$ for some $\kappa >0$, and one has

$$\begin{array}{cc}\hfill & \mathbb{E}\{\parallel e_{k|k}{\parallel }^2\}\hfill \\ \hfill \le & {\displaystyle \frac{\overline{\varphi }}{\underline{\varphi }}}\mathbb{E}\{\parallel e_{0|0}{\parallel }^2\}{\zeta }^k+\kappa \overline{\varphi }\sum _{i=0}^{k-1}{\zeta }^i\hfill \\ \hfill \le & {\displaystyle \frac{\overline{\varphi }}{\underline{\varphi }}}\mathbb{E}\{\parallel e_{0|0}{\parallel }^2\}{\zeta }^k+\kappa \overline{\varphi }\sum _{i=0}^{\infty }{\zeta }^i\hfill \\ \hfill \le & {\displaystyle \frac{\overline{\varphi }}{\underline{\varphi }}}\mathbb{E}\{\parallel e_{0|0}{\parallel }^2\}{\zeta }^k+{\displaystyle \frac{\kappa \overline{\varphi }}{1-\zeta }}.\hfill \end{array}$$

(41)

It is direct that the estimation error $e_{k|k}$ is exponentially bounded in a mean-squared sense, which ends this proof.    □

Remark 1.

Until now, the problem of the dynamic event-based recursive state estimation has been addressed in MRL in the presence of a DETM. According to the received measurements under a DETM, a dynamic event-based estimator has been designed, where an upper bound on the EEC is guaranteed and, subsequently, such an upper bound is minimized in light of the appropriate choice of the estimator gain parameter. It is noteworthy that the innovative dynamic event-triggered estimation methodology demonstrates commendable performance, even under extreme conditions that encompass a DETM and multiple noise, showcasing its robustness and adaptability. It is evident that the performance of the designed dynamic event-based recursive state estimation scheme is influenced by a diverse range of three pivotal factors: (1) the complex structure and nonlinear characteristics of an MRL system; (2) the presence of process and measurement noises; and (3) the impact of a DETM. In the pursuit of realizing the dynamic event-based recursive state estimation scheme, all three of these fundamental factors have been meticulously considered and comprehensively integrated, thereby guaranteeing a robust and reliable estimation performance.

Remark 2.

Due to the influence of the complex structure and nonlinear characteristics of an MRL system, coupled with the process and measurement noises, as well as the impact of a DETM, it is very difficult to compute the actual EECs for the MR state. Fortunately, there exists an alternative way to solve such a problem by deriving the upper bound on EECs, which is a viable solution to design the desire estimator. In comparison with the results of the existing literature, the main merits of the designed dynamic event-based recursive state estimation scheme are highlighted as: (1) a spatial state equation is formulated by abstracting the dynamic behaviors of MRs, serving as the foundation for integrating a DETM into the framework of MRL; (2) for the first time, a DETM is introduced to the MRL problem, leveraging a dynamically adjustable threshold to further optimize communication resource utilization; (3) a pioneering dynamic event-triggered estimator is devised, where the bounded EEC is minimized through strategic design of the estimator gain parameters; (4) an evaluation criterion is established to quantify the mean-square boundedness of the estimation error, providing a rigorous analysis of the estimator’s performance; and (5) a dynamic event-based estimation algorithm is developed, tailored for online computations and real-time applications, ensuring efficient and timely estimation. Based on the above discussion, the developed dynamic event-based recursive state estimation scheme is outlined in Algorithm 1.

Algorithm 1 Dynamic Event-Based Recursive State Estimation for MRL

1: Set relevant parameters of MRL system. Set the positive scalar $a_k$, $b_k$, ${\alpha }_{1,k}$, ${\alpha }_{2,k}$ and initial values ${\widehat{\mathcal{X}}}_0={\mathcal{X}}_0$, ${\Sigma }_{0|0}=P_{0|0}$, $k=0$, $\mathcal{T}=100$; 2: Calculate the predicted state ${\widehat{\mathcal{X}}}_{k|k-1}$ according to (8); 3: Obtain the prediction error $e_{k|k-1}$ based on (13); 4: Calculate the upper bound of the prediction error covariance ${\Sigma }_{k|k-1}$ via (15); 5: Compute the desire estimator gain $K_s$ via (23); 6: Calculate the estimated state ${\widehat{\mathcal{X}}}_{k|k}$ according to (8); 7: Obtain the estimation error $e_{k|k}$ based on (14); 8: Calculate the upper bound of the estimation error covariance ${\Sigma }_{k|k}$ via (16); 9: If $k\le \mathcal{T}$, set $k=k+1$ and return to step 2, otherwise, continue to step 10; 10: Stop.

4. Experimental Example

In this section, we endeavor to demonstrate the feasibility of the developed estimation scheme by conducting an experimental example. The parameters for the MRL problem with the impact of a DETM are listed as follows:

$$\begin{array}{cc}\hfill & L_k={\left[\begin{array}{ccc}0.01& 0.01& 0.01\end{array}\right]}^T, G_k={\left[\begin{array}{cc}0.01& 0.01\end{array}\right]}^T\hfill \\ \hfill & H_k=\left[\begin{array}{ccc}1& 1& 1\end{array}\right], F_k=\left[\begin{array}{ccc}1& 1& 1\end{array}\right]\hfill \\ \hfill & R_k=0.02I, Q_k=0.01I.\hfill \end{array}$$

Consider the MR system. We set the odometer time interval to $t=0.8$ s, the linear velocity to ${\nu }_k=0.8$, and the angular velocity to ${\omega }_k={\displaystyle \frac{\pi }{40}}$. The landmark M is positioned at $\mathcal{M}(x_{\mathcal{M}}=0$ m $,y_{\mathcal{M}}=0$ m). We chose the dynamic triggering thresholds and conditions as $\tau =0.1$, $\gamma =0.4$, and $\vartheta =5$. The initial state of the MR is selected as ${\mathcal{X}}_0={\left[\begin{array}{ccc}0.1& 0.3& 0.2\end{array}\right]}^T$. In addition, the positive scalars are set as $a_k=0.2$, $b_k=0.3$, ${\alpha }_{1,k}=50$, ${\alpha }_{2,k}=330$, and ${\beta }_{i,k}=1(i=1,2,3)$.

According to the above parameters, the simulation results are presented in Figure 2, Figure 3, Figure 4 and Figure 5. In Figure 2, the solid blue line stands for the actual trajectory of the MR and the red dashed line stands for its estimate. The actual orientation angle of the MR and its estimate are plotted in Figure 3, where the solid blue line stands for the actual orientation angle and the red dashed line stands for its estimate. It is obvious that our developed algorithm can accurately estimate the trajectory and orientation angle of the MR, thereby achieving the efficient localization of the MR. Figure 4 presents the logarithm of the upper bound ${\Sigma }_{k|k}$ and the logarithm of the mean square error (MSE) for the estimated state of the MR defined by

$$\begin{array}{c}\hfill MS{E}_k\triangleq {\displaystyle \frac{1}{300}}\sum _{m=1}^{300}({\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k}){({\mathcal{X}}_k-{\widehat{\mathcal{X}}}_{k|k})}^T\end{array}$$

(42)

It is notable that the logarithm of the MSE is consistently below the logarithm of the upper bound, validating our theoretical findings. Additionally, the dynamic event-based release instants and release interval are shown in Figure 5, where we can easily observe that the triggers are reduced to avoid unnecessary transmission, and the estimation performance is not significantly affected. By noting the above simulation results, we can confirm that the proposed estimation strategy is an effective solution to address the MRL problem under a DETM.

5. Conclusions

In this paper, we have discussed the dynamic event-based recursive state estimation issue for MRL subject to a DETM. A dynamic event-triggered transmission strategy is introduced to improve the utilization of communication resources, thereby addressing the adverse effects of insufficient sensor energy storage and bandwidth limitations. With the aid of the mathematical induction method and stochastic analysis techniques, we have developed a desired estimator framework for solving the MRL problem in the presence of the influence of a DETM. The upper bound of the EEC has been derived, and then such an upper bound is minimized by parameterizing the estimator gain at each iteration. Moreover, the mean-square boundedness analysis of the estimation error is given with an evaluation criteria. Finally, we have conducted an experimental example to test the feasibility of our proposed state estimation strategy. In future research endeavors, we will delve deeper into the expansive applications of estimation algorithms under various complex scenarios, such as sensor saturation, cyber-attacks, and channel fading, etc. Furthermore, we will be committed to extending the 2D robot model-based estimation algorithm presented herein, seamlessly transitioning it into complex 3D environments, thereby catering to the pressing demands of practical applications. This extension will not only enrich our theoretical understanding but also enhance the practicality and applicability of the algorithms in real-world settings.