Encryption–decryption-based state estimation for nonlinear complex networks subject to coupled perturbation

Abstract

This paper discusses the encryption-decryption-based state estimation (EDBSE) issue for coupled perturbation complex networks (CPCNs) in the framework of the Kalman-type filtering scheme. A uniform distributed random variable is employed to characterize the coupled perturbation among different network units. A uniform-quantization-dependent encryption-decryption (UQDED) scheme is considered here to orchestrate the transmitted data. A novel EDBSE approach is developed such that the upper bounds of prediction error (PE) covariance (PEC) and estimation error (EE) covariance (EEC) can be derived by resolving Riccati-like difference equations and the estimation parameter (EP) is determined by minimizing the trace of the upper bound of EEC. Furthermore, a uniformly bounded condition is elaborated to evaluate the algorithm performance of EDBSE. Finally, an illustrative example is conducted to verify the validity of the introduced EDBSE method.

Keywords:

• State estimation
• complex networks
• coupled perturbation
• encryption–decryption scheme
• boundedness analysis

1. Introduction

Over the past few decades, the prosperous development of complex networks (CNs) has been witnessed, which are made up of large-scale coupled units with individual dynamics. For this reason, some complex systems in practice have been modelled and characterized by CNs, such as social networks, traffic networks, smart grids and so on. It plays a significant role to carry out the analysis and synthesis with respect to networked or dynamical systems recently, including but are not limited to synchronization (Wei et al., 2024; Zheng & Zhang, 2024), control (Ding & Sun, 2023; Pang et al., 2021, 2022; J. Tan et al., 2023; Zhu, 2019; Zhu & Huang, 2021) and state estimation (Caballero-Águila et al., 2020b; Feng et al., 2023; J. Hu et al., 2023; Song et al., 2024; H. Tan et al., 2024; K.-Y. Wang et al., 2024; L. Wang et al., 2022). Consequently, the design of state estimation algorithm for CNs has gained extensive research attention (C. Hu et al., 2023; J. Hu et al., 2022; F. Li et al., 2024; Sakthivel et al., 2024). For example, a Kalman-type state estimation problem was investigated in Jia et al. (2023) for a kind of quantized coupled CNs with imperfect measurements and amplify-forward-relay communication, where the estimator parameter (EP) was designed under the variance-constrained index to minimize the trace of the upper bound of estimation error (EE) covariance (EEC). A distributed state estimation issue was discussed in Peng et al. (2023) for a kind of CNs subject to nonlinear uncertainty. In addition, a sufficient principle regarding the stability of the upper bound of EEC was given. Although CNs have broad application prospects, they have not yet received sufficient study in the framework of Kalman-type filtering scheme, which stimulates us to carry out such an idea in this paper.

Recently, the state estimation for networked systems has appealed a great number of attention, see (Caballero-Águila et al., 2020a, 2020b, 2020a; Caballero-Águila & Linares-Pérez, 2023; García-Ligero et al., 2020; J. Hu et al., 2023; Tao et al., 2022) for more details. Actually, the coupled dynamic is one of the major focuses during the design of state estimation algorithm in the framework of CNs. The connection among different network units usually fluctuates due to the limited resource or changeable external environment. Consequently, a growing number of scholars have paid more attention on the state estimation for CNs with coupled dynamics. For instance, an $H_{\mathrm{\infty }}$ state estimation issue was investigated in Meng et al. (2023) for a class of CNs with missing packets, where an uncertain coupling was described by a parameter varying in a given interval and a sufficient criterion was presented to guarantee the pre-defined $H_{\mathrm{\infty }}$ performance. Taking the bandwidth-limited channel into account, L. Liu et al. (2020) emphasized the design of a state estimation algorithm for a kind of delayed CNs, where the phenomenon of coupled delay was considered and modelled due to the limited network resource. Furthermore, the sufficient condition regarding finite-time boundedness was established by resorting to algebraic inequalities. It should be pointed out that the varying environment may be caused by network attacks besides the resource-limited situation. In this scenario, this paper aims to addressing the state estimation problem for CNs suffering from coupled perturbation induced by opponent attacks.

On another frontier, in the networked environment, data-treatment has been frequently employed to alleviate the communication burden and avoid communication constraints (H. Gao et al., 2023; X. Wang et al., 2022; Y.-A. Wang et al., 2023; H. Yang et al., 2024; J. Yang et al., 2022). Especially, the data-security-treatment has gradually been a focus in the dynamical analyses of networked systems (Cheng et al., 2023; Luo et al., 2023; Pang et al., 2022, 2021; Sun et al., 2022; Wen et al., 2022; Wu et al., 2022; Zha et al., 2024; Zou et al., 2023). Specifically, the sliding mode consensus problem under privacy preservation and disturbance was explored by J. Liu et al. (2024) for a class of multi-agent systems, where an encryption–decryption strategy was proposed to enhance the data security. The secure state estimation method was proposed by Zou et al. (2023) for a class of multi-rate system, where an encryption–decryption scheme was considered to defend against eavesdroppers and uniform boundedness regarding EE was given. It is worth mentioning that the employment of encryption–decryption strategy can strengthen the data security while may resulting in data distortion described by decryption error, which enhances the handling difficulty of the design of state estimation in the framework of Kalman-type filtering. For example, the set-membership filtering problems for time-varying systems subject to encryption–decryption communication protocol were tackled in Y. Gao et al. (2022); J.-Y. Li et al. (2023) and Ma et al. (2021), where the upper bounds of decryption errors were described by resorting to uniform-type quantization scheme. It should be noted that the discussions regarding the state estimation for CNs with encryption–decryption strategy have been reported inadequately up to now, which motivates us to shorten such a gap.

Motivated by the aforementioned summary, this paper concentrates on the encryption–decryption-based state estimation (EDBSE) issue for coupled perturbation complex networks (CPCNs). The main encountered difficulties and challenges can be listed as (a) how to obtain an upper bound of EEC by resolving two coupled matrix equations in the simultaneous presence of decryption error, coupled perturbation and system noises; (b) how to determine EP in the framework of Kalman-type filtering so as to make the trace of the upper bound of EEC minimized at each instant; and (c) how to evaluate the developed estimation algorithm and give a boundedness analysis. The corresponding contributions can be summed up as (1) an attractive EDBSE method is proposed for CPCNs, where an upper bound of EEC with a recursive form involving encryption–decryption parameters and coupled perturbation information is obtained; (2) a selected EP is parameterized via optimizing variance-constrained index; and (3) a uniformly bounded condition is elaborated to evaluate the proposed estimation algorithm performance.

Notations

The notations used here are standard. For example, $\Vert \cdot \Vert$ denotes the Euclidean norm. Matrix $M^{\mathrm{T}}$ means the transpose of M and $M^{-1}$ denotes the inverse operation of M. The notation P>0 $(P\ge 0)$ signifies that P is symmetric and positive (semi-positive) definite. I and 0 denote the unit matrix and zero matrix with compatible dimensions.

2. Problem formulation and preliminaries

Consider the following CPCNs with S units (1) $\begin{array}{rl}{x}_{\kappa ,\theta +1}& =f\left(x_{\kappa ,\theta }\right)+{\omega }_{\mathrm{\kappa \kappa },\theta }\mathrm{\Pi }{x}_{\kappa ,\theta }\\ & +\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\zeta }_{\rho ,\theta }+B_{\kappa ,\theta }{u}_{\kappa ,\theta },\end{array}$(1) (2) $\begin{array}{rl}{y}_{\kappa ,\theta }& =C_{\kappa ,\theta }{x}_{\kappa ,\theta }+{\sigma }_{\kappa ,\theta },\end{array}$(2) where $x_{\kappa ,\theta }$ is the $n_x$-dimensional model state and $y_{\kappa ,\theta }$ denotes the $n_y$-dimensional sensor measurement. ${\mathcal{A}}_{\kappa ,\theta }=[{\omega }_{\mathrm{\kappa \gamma },\theta }{]}_{S\times S}$ and Π stand for the outer coupling matrix and the inner coupling matrix, respectively. $B_{\kappa ,\theta }$ and $C_{\kappa ,\theta }$ are well-dimensioned known system matrices. $u_{\kappa ,\theta }$ is a zero-mean system noise with variance $U_{\kappa ,\theta }>0$, and ${\sigma }_{\kappa ,\theta }$ is zero-mean noise from sensor measurement with variance ${\mathrm{\Sigma }}_{\kappa ,\theta }>0$. $f\left(x_{\kappa ,\theta }\right)$ is a nonlinear function satisfying $f^T\left(x_{\kappa ,\theta }\right)f\left(x_{\kappa ,\theta }\right)\le x_{\kappa ,\theta }^T\mathrm{\Phi }{x}_{\kappa ,\theta }$ with $f\left(0\right)=0$ and $\mathrm{\Phi }>0$ being a known scaling matrix. The coupled perturbation is described by ${\zeta }_{\rho ,\theta }$ with the form of (3) ${\zeta }_{\rho ,\theta }=x_{\rho ,\theta }+{\xi }_{\rho ,\theta },$(3) where ${\xi }_{\rho ,\theta }=[{\xi }_{\rho 1,\theta }{\xi }_{\rho 2,\theta }\dots {\xi }_{\rho n_x,\theta }{]}^T$ is an external perturbation usually caused by network attacks in the unit-to-unit channel, whose element ${\xi }_{\mathrm{\rho \varphi },\theta }$ is mutually independent regarding ϕ and uniformly distributed in $[-a,a]$ with mean 0 and variance ${\overrightarrow{\xi }}_{\mathrm{\rho \varphi },\theta }$ $(\varphi =1,2,\dots ,n_x)$.

Remark 2.1

Due to the uncertain network environment, the node-to-node communication may be contaminated by malicious attacks, which is specifically manifested in the occurrence of coupled perturbation. In this paper, a uniformly distributed random variable is employed to describe the phenomenon of coupled perturbation. Obviously, if a increases, the introduced estimation algorithm accuracy will decrease, which will be confirmed by the subsequent simulation example.

To enhance the security of the transmitted data, a uniform-quantization-dependent encryption–decryption (UQDED) strategy is employed. To begin with, the encryption procedure is described by (4) $\{\begin{array}{l}{\mathrm{\Im }}_{\kappa ,0}=0,\\ {\mathrm{\Im }}_{\kappa ,\theta }=z_{\kappa ,\theta }{\mathrm{\Theta }}_{\kappa ,\theta }+{\mathrm{\Im }}_{\kappa ,\theta -1},\\ {\displaystyle {\mathrm{\Theta }}_{\kappa ,\theta }=\mathcal{Q}\left(\frac{y_{\kappa ,\theta }-{\mathrm{\Im }}_{\kappa ,\theta -1}}{z_{\kappa ,\theta }}\right),}\end{array}$(4) where ${\mathrm{\Im }}_{\kappa ,\theta }$ is an auxiliary variable and ${\mathrm{\Theta }}_{\kappa ,\theta }$ denotes the generated codeword of encoder. $z_{\kappa ,\theta }>0$ is a scaling parameter. $\mathcal{Q}\left(\hslash \right)=\left[\mathfrak{Q}\right({\hslash }_1\left)\mathfrak{Q}\right({\hslash }_2)\cdots \mathfrak{Q}({\hslash }_{n_y}){]}^T$ functions as a quantization operation characterized by $\mathfrak{Q}\left({\hslash }_{\mathrm{\wp }}\right)=\{\begin{array}{ll}0,& -l/2\le {\hslash }_{\mathrm{\wp }}<l/2\\ {\displaystyle \mathrm{nl},}& \frac{(2n-1)l}{2}\le {\hslash }_{\mathrm{\wp }}<\frac{(2n+1)l}{2}\\ \mathrm{\Delta l},& {\hslash }_{\mathrm{\wp }}\ge (2\mathrm{\Delta }-1)l/2\\ -\mathfrak{Q}(-{\hslash }_{\mathrm{\wp }}),& {\hslash }_{\mathrm{\wp }}<-l/2\end{array}$where ${\hslash }_{\mathrm{\wp }}$ is the ℘-th element in $\hslash =[{\hslash }_1{\hslash }_2\cdots {\hslash }_{n_y}{]}^T$. l>0 is a scalar, $2\mathrm{\Delta }+1$ denotes quantization level, $n=1,2,\dots ,\mathrm{\Delta }-1$.

The decryption output is characterized by (5) $\{\begin{array}{l}{\overrightarrow{y}}_{\kappa ,0}=0,\\ {\overrightarrow{y}}_{\kappa ,\theta }=z_{\kappa ,\theta }{\mathrm{\Theta }}_{\kappa ,\theta }+{\overrightarrow{y}}_{\kappa ,\theta -1}.\end{array}$(5) Comparing (4(4) $\{\begin{array}{l}{\mathrm{\Im }}_{\kappa ,0}=0,\\ {\mathrm{\Im }}_{\kappa ,\theta }=z_{\kappa ,\theta }{\mathrm{\Theta }}_{\kappa ,\theta }+{\mathrm{\Im }}_{\kappa ,\theta -1},\\ {\displaystyle {\mathrm{\Theta }}_{\kappa ,\theta }=\mathcal{Q}\left(\frac{y_{\kappa ,\theta }-{\mathrm{\Im }}_{\kappa ,\theta -1}}{z_{\kappa ,\theta }}\right),}\end{array}$(4) ) and (5(5) $\{\begin{array}{l}{\overrightarrow{y}}_{\kappa ,0}=0,\\ {\overrightarrow{y}}_{\kappa ,\theta }=z_{\kappa ,\theta }{\mathrm{\Theta }}_{\kappa ,\theta }+{\overrightarrow{y}}_{\kappa ,\theta -1}.\end{array}$(5) ) yields ${\overrightarrow{y}}_{\kappa ,\theta }={\mathrm{\Im }}_{\kappa ,\theta }$. The decryption error is denoted by $y_{\kappa ,\theta }^e={\overrightarrow{y}}_{\kappa ,\theta }-y_{\kappa ,\theta }$. Obviously, we have (6) $\begin{array}{rl}{y}_{\kappa ,\theta }^e& =z_{\kappa ,\theta }{\mathrm{\Theta }}_{\kappa ,\theta }+{\overrightarrow{y}}_{\kappa ,\theta -1}-y_{\kappa ,\theta }\\ & =z_{\kappa ,\theta }[\mathcal{Q}\left(\frac{y_{\kappa ,\theta }-{\mathrm{\Im }}_{\kappa ,\theta -1}}{z_{\kappa ,\theta }}\right)-\frac{y_{\kappa ,\theta }-{\mathrm{\Im }}_{\kappa ,\theta -1}}{z_{\kappa ,\theta }}]\end{array}$(6) which signifies that (7) $||y_{\kappa ,\theta }^e|{|}^2\le \frac{n_y{z}_{\kappa ,\theta }^2{l}^2}{4}.$(7)

Remark 2.2

To guarantee the security and reliability, an encryption–decryption strategy is employed in the process of data transmission, where a uniform-type quantizer is used to process data exposed in the uncertain network environment. Moreover, the decryption error can be bound by the quantization density, scaling parameter and measurement output dimension. It can be seen from (7(7) $||y_{\kappa ,\theta }^e|{|}^2\le \frac{n_y{z}_{\kappa ,\theta }^2{l}^2}{4}.$(7) ) that the decryption error is proportional to $n_y$, $z_{\kappa ,\theta }^2$ and $l^2$. Actually, we can choose the smaller $z_{\kappa ,\theta }$ and l to satisfy the high-precision requirement in practice.

Considering UQDED and coupled perturbation, the following predictor-estimator with recursive form is constructed (8) $\begin{array}{rl}{\hat{x}}_{\kappa ,\theta +1|\theta }& =f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)+{\omega }_{\mathrm{\kappa \kappa },\theta }\mathrm{\Pi }{\hat{x}}_{\kappa ,\theta |\theta }\\ & +\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\hat{x}}_{\rho ,\theta |\theta },\end{array}$(8) (9) $\begin{array}{rl}{\hat{x}}_{\kappa ,\theta +1|\theta +1}& ={\hat{x}}_{\kappa ,\theta +1|\theta }+{\mathrm{\Xi }}_{\kappa ,\theta +1}({\overrightarrow{y}}_{\kappa ,\theta +1}\\ & -C_{\kappa ,\theta +1}{\hat{x}}_{\kappa ,\theta +1|\theta }),\end{array}$(9) where ${\hat{x}}_{\kappa ,\theta +1|\theta }$ and ${\hat{x}}_{\kappa ,\theta |\theta }$ stand for the prediction and estimation of system state, ${\mathrm{\Xi }}_{\kappa ,\theta +1}$ denotes the EP to be determined later.

3. Main results

We make a great effort to derive the upper bound regarding EEC and determine the EP to minimize the trace of such upper bound. First of all, the prediction error (PE) $x_{\kappa ,\theta +1}^p$ and EE $x_{\kappa ,\theta +1}^e$ are, respectively, presented by (10) $\begin{array}{rl}{x}_{\kappa ,\theta +1}^p& =x_{\kappa ,\theta +1}-{\hat{x}}_{\kappa ,\theta +1|\theta }\\ & =f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)+\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\\ & +\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\xi }_{\rho ,\theta }+B_{\kappa ,\theta }{u}_{\mathrm{\kappa \theta }}\end{array}$(10) and (11) $\begin{array}{rl}{x}_{\kappa ,\theta +1}^e& =x_{\kappa ,\theta +1}-{\hat{x}}_{\kappa ,\theta +1|\theta +1}\\ & =x_{\kappa ,\theta +1}^p-{\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e-{\mathrm{\Xi }}_{\kappa ,\theta +1}(y_{\kappa ,\theta +1}\\ & -C_{\kappa ,\theta +1}{\hat{x}}_{\kappa ,\theta +1|\theta })\\ & =x_{\kappa ,\theta +1}^p-{\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}{x}_{\kappa ,\theta +1}^p\\ & -{\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}\\ & =(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})x_{\kappa ,\theta +1}^p-{\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e\\ & -{\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}.\end{array}$(11) The following theorem introduces the covariance matrices regarding PE and EE, i.e. $X_{\kappa ,\theta +1}^p$ and $X_{\kappa ,\theta +1}^e$.

Theorem 3.1

$X_{\kappa ,\theta +1}^p$ and $X_{\kappa ,\theta +1}^e$ respectively satisfy the following recursions (12) $\begin{array}{rl}{X}_{\kappa ,\theta +1}^p& =\mathbb{E}\left\{x_{\kappa ,\theta +1}^p\right(x_{\kappa ,\theta +1}^p{)}^T\}\\ & =\mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)\left]\right[f\left(x_{\kappa ,\theta }\right)\\ & -f\left({\hat{x}}_{\kappa ,\theta |\theta }\right){]}^T\}+\mathbb{E}\{\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\\ & \times {\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\}+\mathbb{E}\{\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\\ & \times \mathrm{\Pi }{\xi }_{\rho ,\theta }{\left(\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\xi }_{\rho ,\theta }\right)}^T\}\\ & +\mathbb{E}\left\{B_{\kappa ,\theta }{u}_{\kappa ,\theta }\right(B_{\kappa ,\theta }{u}_{\kappa ,\theta }{)}^T\}+\mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)\\ & -f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)]{\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\\ & +\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)\left[f\right(x_{\kappa ,\theta })\\ & -f\left({\hat{x}}_{\kappa ,\theta |\theta }\right){]}^T\}\end{array}$(12) and (13) $\begin{array}{rl}{X}_{\kappa ,\theta +1}^e& =\mathbb{E}\left\{x_{\kappa ,\theta +1}^e\right(x_{\kappa ,\theta +1}^e{)}^T\}\\ & =\mathbb{E}\left\{\right(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})x_{\kappa ,\theta +1}^p\\ & \times \left[\right(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}\left)x_{\kappa ,\theta +1}^p{]}^T\right\}\\ & +\mathbb{E}\left\{\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e\left)\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e{)}^T\}\\ & +\mathbb{E}\left\{\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}\left)\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}{)}^T\}\\ & -\mathbb{E}\left\{\right(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})x_{\kappa ,\theta +1}^p\\ & \times ({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e{)}^T+({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e)\\ & \times \left[\right(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}\left)x_{\kappa ,\theta +1}^p{]}^T\right\}\\ & +\mathbb{E}\left\{\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e\left)\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}{)}^T\\ & +\left({\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}\right)\left({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e{)}^T\right\}.\end{array}$(13)

Proof.

The proof is omitted here for simplicity.

The following theorem gives the recursion with respect to the upper bound of EEC and parameterizes the EP to minimize the trace of the derived upper bound.

Theorem 3.2

For adjusted parameters $o_1>0$, $o_2>0$ and $o_3>0$, if the following matrix equations (14) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^p& =(1+o_1)\mathrm{tr}\left({\mathcal{X}}_{\kappa ,\theta }^e\mathrm{\Phi }\right)I+(1+o_1^{-1})\sum _{t=1}^s{\omega }_{\mathrm{\kappa t},\theta }\\ & \times \sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\mathcal{X}}_{\kappa ,\theta }^e{\mathrm{\Pi }}^T+\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }^2\mathrm{\Pi }\\ & \times {\mathrm{{\rm Y}}}_{\rho ,\theta }{\mathrm{\Pi }}^T+B_{\kappa ,\theta }{U}_{\kappa ,\theta }{B}_{\kappa ,\theta }^T\end{array}$(14) and (15) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^e& =(1+o_2)(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}){\mathcal{X}}_{\kappa ,\theta +1}^p\\ & \times (I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}{)}^T+(1+o_3^{-1}){\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times {\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T,\end{array}$(15) where ${\mathrm{{\rm Y}}}_{\rho ,\theta }=\left[\begin{array}{cccc}{\overrightarrow{\xi }}_{\rho 1,\theta }& 0& \cdots & 0\\ 0& {\overrightarrow{\xi }}_{\rho 2,\theta }& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& 0& \cdots & {\overrightarrow{\xi }}_{\rho n_x,\theta }\end{array}\right]$with initial condition ${\mathcal{X}}_{\kappa ,0}^e=X_{\kappa ,0}^e>0$ have solutions ${\mathcal{X}}_{\kappa ,\theta +1}^p>0$ and ${\mathcal{X}}_{\kappa ,\theta +1}^e>0$, then ${\mathcal{X}}_{\kappa ,\theta +1}$ is an upper bound of $X_{\kappa ,\theta +1}$, i.e. $X_{\kappa ,\theta +1}^e\le {\mathcal{X}}_{\kappa ,\theta +1}^e$.

Furthermore, if the EP is parameterized by (16) $\begin{array}{rl}& {\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & =(1+o_2){\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T\left\{\right(1+o_2)C_{\kappa ,\theta +1}\\ & \times {\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}I\\ & {+(1+o_3^{-1}){\mathrm{\Sigma }}_{\kappa ,\theta +1}\}}^{-1},\end{array}$(16) then the index $\mathrm{tr}\left({\mathcal{X}}_{\kappa ,\theta +1}^e\right)$ can be minimized at every instant.

Proof.

According to the inequality $\mathcal{P}{\mathcal{J}}^T+\mathcal{J}{\mathcal{P}}^T\le o\mathcal{P}{\mathcal{P}}^T+o^{-1}\mathcal{J}{\mathcal{J}}^T$ with o>0 being an adjusted parameter, $\mathcal{P}$ and $\mathcal{J}$ being well-dimensioned matrices, yields (17) $\begin{array}{rl}& \mathbb{E}\left\{\right(f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)){\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\\ & +\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)\left(f\right(x_{\kappa ,\theta })-f({\hat{x}}_{\kappa ,\theta |\theta }\left){)}^T\right\}\\ & \le o_1^{-1}\mathbb{E}\{\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }\\ & {\times x_{\rho ,\theta }^e)}^T\}+o_1\mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)]\\ & \times \left[f\right(x_{\kappa ,\theta })-f({\hat{x}}_{\kappa ,\theta |\theta }\left){]}^T\right\},\end{array}$(17) where $o_1>0$ is an adjusted parameter. Substitute (17(17) $\begin{array}{rl}& \mathbb{E}\left\{\right(f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)){\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\\ & +\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)\left(f\right(x_{\kappa ,\theta })-f({\hat{x}}_{\kappa ,\theta |\theta }\left){)}^T\right\}\\ & \le o_1^{-1}\mathbb{E}\{\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }\\ & {\times x_{\rho ,\theta }^e)}^T\}+o_1\mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)]\\ & \times \left[f\right(x_{\kappa ,\theta })-f({\hat{x}}_{\kappa ,\theta |\theta }\left){]}^T\right\},\end{array}$(17) ) into (12(12) $\begin{array}{rl}{X}_{\kappa ,\theta +1}^p& =\mathbb{E}\left\{x_{\kappa ,\theta +1}^p\right(x_{\kappa ,\theta +1}^p{)}^T\}\\ & =\mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)\left]\right[f\left(x_{\kappa ,\theta }\right)\\ & -f\left({\hat{x}}_{\kappa ,\theta |\theta }\right){]}^T\}+\mathbb{E}\{\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\\ & \times {\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\}+\mathbb{E}\{\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\\ & \times \mathrm{\Pi }{\xi }_{\rho ,\theta }{\left(\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\xi }_{\rho ,\theta }\right)}^T\}\\ & +\mathbb{E}\left\{B_{\kappa ,\theta }{u}_{\kappa ,\theta }\right(B_{\kappa ,\theta }{u}_{\kappa ,\theta }{)}^T\}+\mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)\\ & -f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)]{\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\\ & +\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)\left[f\right(x_{\kappa ,\theta })\\ & -f\left({\hat{x}}_{\kappa ,\theta |\theta }\right){]}^T\}\end{array}$(12) ) leading to (18) $\begin{array}{rl}& X_{\kappa ,\theta +1}^p\\ & \le (1+o_1)\mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)\left]\right[f\left(x_{\kappa ,\theta }\right)\\ & -f\left({\hat{x}}_{\kappa ,\theta |\theta }\right){]}^T\}+(1+o_1^{-1})\mathbb{E}\{(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\\ & \times \mathrm{\Pi }{x}_{\rho ,\theta }^e){\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\}+B_{\kappa ,\theta }{U}_{\kappa ,\theta }\\ & \times B_{\kappa ,\theta }^T+\mathbb{E}\{\left(\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\xi }_{\rho ,\theta }\right)\\ & \times {\left(\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\xi }_{\rho ,\theta }\right)}^T\}.\end{array}$(18) Handling the nonlinearity, one has (19) $\begin{array}{rl}& \mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)\left]\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right){]}^T\}\\ & \le \mathbb{E}\left\{\mathrm{tr}\right(\left(x_{\kappa ,\theta }^e{)}^T\mathrm{\Phi }{x}_{\kappa ,\theta }^e\right)I\}\\ & =\mathrm{tr}\left(X_{\kappa ,\theta }^e\mathrm{\Phi }\right)I,\end{array}$(19) where $\mathrm{\Phi }>0$ is a known scaling matrix defined before. In terms of the statistical characteristics of coupled perturbation, we have (20) $\begin{array}{rl}& \mathbb{E}\{\left(\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\xi }_{\rho ,\theta }\right)(\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\\ & {\times \mathrm{\Pi }{\xi }_{\rho ,\theta })}^T\}\\ & =\mathbb{E}\left\{\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }^2\mathrm{\Pi }{\xi }_{\rho ,\theta }{\xi }_{\rho ,\theta }^T{\mathrm{\Pi }}^T\right\}\\ & =\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }^2\mathrm{\Pi }{\mathrm{{\rm Y}}}_{\kappa ,\theta }{\mathrm{\Pi }}^T,\end{array}$(20) where ${\mathrm{{\rm Y}}}_{\kappa ,\theta }$ is defined in Theorem 3.1. Notice that (21) $\begin{array}{rl}& \mathbb{E}\left\{\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right){\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\right\}\\ & \le \sum _{t=1}^s{\omega }_{\mathrm{\kappa t},\theta }\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{X}_{\rho ,\theta }^e{\mathrm{\Pi }}^T.\end{array}$(21) Substitute (19(19) $\begin{array}{rl}& \mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)\left]\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right){]}^T\}\\ & \le \mathbb{E}\left\{\mathrm{tr}\right(\left(x_{\kappa ,\theta }^e{)}^T\mathrm{\Phi }{x}_{\kappa ,\theta }^e\right)I\}\\ & =\mathrm{tr}\left(X_{\kappa ,\theta }^e\mathrm{\Phi }\right)I,\end{array}$(19) )–(21(21) $\begin{array}{rl}& \mathbb{E}\left\{\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right){\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\right\}\\ & \le \sum _{t=1}^s{\omega }_{\mathrm{\kappa t},\theta }\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{X}_{\rho ,\theta }^e{\mathrm{\Pi }}^T.\end{array}$(21) ) into (18(18) $\begin{array}{rl}& X_{\kappa ,\theta +1}^p\\ & \le (1+o_1)\mathbb{E}\left\{\right[f\left(x_{\kappa ,\theta }\right)-f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)\left]\right[f\left(x_{\kappa ,\theta }\right)\\ & -f\left({\hat{x}}_{\kappa ,\theta |\theta }\right){]}^T\}+(1+o_1^{-1})\mathbb{E}\{(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\\ & \times \mathrm{\Pi }{x}_{\rho ,\theta }^e){\left(\sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{x}_{\rho ,\theta }^e\right)}^T\}+B_{\kappa ,\theta }{U}_{\kappa ,\theta }\\ & \times B_{\kappa ,\theta }^T+\mathbb{E}\{\left(\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\xi }_{\rho ,\theta }\right)\\ & \times {\left(\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\xi }_{\rho ,\theta }\right)}^T\}.\end{array}$(18) ) yielding (22) $\begin{array}{rl}& X_{\kappa ,\theta +1}^p\\ & \le (1+o_1)\mathrm{tr}\left(X_{\kappa ,\theta }^e\mathrm{\Phi }\right)I+(1+o_1^{-1})\sum _{t=1}^s{\omega }_{\mathrm{\kappa t},\theta }\\ & \times \sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{X}_{\kappa ,\theta }^e{\mathrm{\Pi }}^T+\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }^2\mathrm{\Pi }\\ & \times {\mathrm{{\rm Y}}}_{\kappa ,\theta }{\mathrm{\Pi }}^T+B_{\kappa ,\theta }{U}_{\kappa ,\theta }{B}_{\kappa ,\theta }^T.\end{array}$(22) Subsequently, the upper bound of EEC is derived. Recall inequality $\mathcal{P}{\mathcal{J}}^T+\mathcal{J}{\mathcal{P}}^T\le o\mathcal{P}{\mathcal{P}}^T+o^{-1}\mathcal{J}{\mathcal{J}}^T$ leading to (23) $\begin{array}{rl}& -\mathbb{E}\left\{\right[(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})x_{\kappa ,\theta +1}^p\left]\right({\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times y_{\kappa ,\theta +1}^e{)}^T+\left({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e\right)\left[\right(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times C_{\kappa ,\theta +1}\left)x_{\kappa ,\theta +1}^p{]}^T\right\}\\ & \le o_2(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})X_{\kappa ,\theta +1}^p(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times C_{\kappa ,\theta +1}{)}^T+o_2^{-1}{\mathrm{\Xi }}_{\kappa ,\theta +1}\mathbb{E}\left\{y_{\kappa ,\theta +1}^e\right(y_{\kappa ,\theta +1}^e{)}^T\}\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}^T\end{array}$(23) and (24) $\begin{array}{rl}& \mathbb{E}\left\{\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e\left)\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}{)}^T\\ & +\left({\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}\right)\left({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e{)}^T\right\}\\ & \le o_3{\mathrm{\Xi }}_{\kappa ,\theta +1}\mathbb{E}\left\{y_{\kappa ,\theta +1}^e\right(y_{\kappa ,\theta +1}^e{)}^T\}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T\\ & +o_3^{-1}{\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T,\end{array}$(24) where $o_2>0$ and $o_3>0$ are adjusted parameters. In terms of (23(23) $\begin{array}{rl}& -\mathbb{E}\left\{\right[(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})x_{\kappa ,\theta +1}^p\left]\right({\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times y_{\kappa ,\theta +1}^e{)}^T+\left({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e\right)\left[\right(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times C_{\kappa ,\theta +1}\left)x_{\kappa ,\theta +1}^p{]}^T\right\}\\ & \le o_2(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})X_{\kappa ,\theta +1}^p(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times C_{\kappa ,\theta +1}{)}^T+o_2^{-1}{\mathrm{\Xi }}_{\kappa ,\theta +1}\mathbb{E}\left\{y_{\kappa ,\theta +1}^e\right(y_{\kappa ,\theta +1}^e{)}^T\}\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}^T\end{array}$(23) ) and (24(24) $\begin{array}{rl}& \mathbb{E}\left\{\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e\left)\right({\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}{)}^T\\ & +\left({\mathrm{\Xi }}_{\kappa ,\theta +1}{\sigma }_{\kappa ,\theta +1}\right)\left({\mathrm{\Xi }}_{\kappa ,\theta +1}{y}_{\kappa ,\theta +1}^e{)}^T\right\}\\ & \le o_3{\mathrm{\Xi }}_{\kappa ,\theta +1}\mathbb{E}\left\{y_{\kappa ,\theta +1}^e\right(y_{\kappa ,\theta +1}^e{)}^T\}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T\\ & +o_3^{-1}{\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T,\end{array}$(24) ), one has (25) $\begin{array}{rl}& X_{\kappa ,\theta +1}^e\\ & \le (1+o_2)(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})X_{\kappa ,\theta +1}^p\\ & \times (I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}{)}^T+(1+o_2^{-1}+o_3){\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times \mathbb{E}\left\{y_{\kappa ,\theta +1}^e\right(y_{\kappa ,\theta +1}^e{)}^T\}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T+(1+o_3^{-1})\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T.\end{array}$(25) Note that (26) $\begin{array}{rl}\mathbb{E}\left\{y_{\kappa ,\theta +1}^e\right(y_{\kappa ,\theta +1}^e{)}^T\}& \le \mathbb{E}\left\{||y_{\kappa ,\theta +1}^e|{|}^2\right\}I\\ & \le \frac{n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}I.\end{array}$(26) Thus, we have (27) $\begin{array}{rl}& X_{\kappa ,\theta +1}^e\\ & \le (1+o_2)(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})X_{\kappa ,\theta +1}^p\\ & \times (I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}{)}^T+(1+o_3^{-1}){\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times {\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T.\end{array}$(27) Considering (14(14) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^p& =(1+o_1)\mathrm{tr}\left({\mathcal{X}}_{\kappa ,\theta }^e\mathrm{\Phi }\right)I+(1+o_1^{-1})\sum _{t=1}^s{\omega }_{\mathrm{\kappa t},\theta }\\ & \times \sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\mathcal{X}}_{\kappa ,\theta }^e{\mathrm{\Pi }}^T+\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }^2\mathrm{\Pi }\\ & \times {\mathrm{{\rm Y}}}_{\rho ,\theta }{\mathrm{\Pi }}^T+B_{\kappa ,\theta }{U}_{\kappa ,\theta }{B}_{\kappa ,\theta }^T\end{array}$(14) )–(15(15) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^e& =(1+o_2)(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}){\mathcal{X}}_{\kappa ,\theta +1}^p\\ & \times (I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}{)}^T+(1+o_3^{-1}){\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times {\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T,\end{array}$(15) ), (22(22) $\begin{array}{rl}& X_{\kappa ,\theta +1}^p\\ & \le (1+o_1)\mathrm{tr}\left(X_{\kappa ,\theta }^e\mathrm{\Phi }\right)I+(1+o_1^{-1})\sum _{t=1}^s{\omega }_{\mathrm{\kappa t},\theta }\\ & \times \sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{X}_{\kappa ,\theta }^e{\mathrm{\Pi }}^T+\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }^2\mathrm{\Pi }\\ & \times {\mathrm{{\rm Y}}}_{\kappa ,\theta }{\mathrm{\Pi }}^T+B_{\kappa ,\theta }{U}_{\kappa ,\theta }{B}_{\kappa ,\theta }^T.\end{array}$(22) ), (27(27) $\begin{array}{rl}& X_{\kappa ,\theta +1}^e\\ & \le (1+o_2)(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1})X_{\kappa ,\theta +1}^p\\ & \times (I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}{)}^T+(1+o_3^{-1}){\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times {\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T.\end{array}$(27) ) as well as mathematical induction, one has $X_{\kappa ,\theta +1}^e\le {\mathcal{X}}_{\kappa ,\theta +1}^e$.

Finally, determine the EP to minimize the trace of ${\mathcal{X}}_{\kappa ,\theta +1}^e$. Calculate $\frac{\mathrm{\partial }\mathrm{tr}\left({\mathcal{X}}_{\kappa ,\theta +1}^e\right)}{\mathrm{\partial }{\mathrm{\Xi }}_{\kappa ,\theta +1}}$ leading to (28) $\begin{array}{rl}& \frac{\mathrm{\partial }\mathrm{tr}\left({\mathcal{X}}_{\kappa ,\theta +1}^e\right)}{\mathrm{\partial }{\mathrm{\Xi }}_{\kappa ,\theta +1}}\\ & =-2(1+o_2)(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}){\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T\\ & +\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{2}{\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & +2(1+o_3^{-1}){\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Sigma }}_{\kappa ,\theta +1}.\end{array}$(28) Let $\frac{\mathrm{\partial }\mathrm{tr}\left({\mathcal{X}}_{\kappa ,\theta +1}^e\right)}{\mathrm{\partial }{\mathrm{\Xi }}_{\kappa ,\theta +1}}=0$ resulting in (29) $\begin{array}{rl}& {\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & =(1+o_2){\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T\left\{\right(1+o_2)C_{\kappa ,\theta +1}\\ & \times {\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}I\\ & {+(1+o_3^{-1}){\mathrm{\Sigma }}_{\kappa ,\theta +1}\}}^{-1},\end{array}$(29) which can minimize the trace of ${\mathcal{X}}_{\kappa ,\theta +1}^e$ and the proof is verified completely.

Remark 3.1

Taking nonlinearity, coupled perturbation and UQDED into account simultaneously, two recursive matrix equations are concretized in (14(14) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^p& =(1+o_1)\mathrm{tr}\left({\mathcal{X}}_{\kappa ,\theta }^e\mathrm{\Phi }\right)I+(1+o_1^{-1})\sum _{t=1}^s{\omega }_{\mathrm{\kappa t},\theta }\\ & \times \sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\mathcal{X}}_{\kappa ,\theta }^e{\mathrm{\Pi }}^T+\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }^2\mathrm{\Pi }\\ & \times {\mathrm{{\rm Y}}}_{\rho ,\theta }{\mathrm{\Pi }}^T+B_{\kappa ,\theta }{U}_{\kappa ,\theta }{B}_{\kappa ,\theta }^T\end{array}$(14) ) and (15(15) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^e& =(1+o_2)(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}){\mathcal{X}}_{\kappa ,\theta +1}^p\\ & \times (I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}{)}^T+(1+o_3^{-1}){\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times {\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T,\end{array}$(15) ). In the framework of the Kalman-type filtering strategy, the trace of (15(15) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^e& =(1+o_2)(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}){\mathcal{X}}_{\kappa ,\theta +1}^p\\ & \times (I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}{)}^T+(1+o_3^{-1}){\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times {\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T,\end{array}$(15) ) is minimized by selecting the EP in (16(16) $\begin{array}{rl}& {\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & =(1+o_2){\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T\left\{\right(1+o_2)C_{\kappa ,\theta +1}\\ & \times {\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}I\\ & {+(1+o_3^{-1}){\mathrm{\Sigma }}_{\kappa ,\theta +1}\}}^{-1},\end{array}$(16) ).

4. Boundedness analysis

This section is concerned with the performance discussion of the proposed EDBSE algorithm. A uniformly bounded condition regarding ${\mathcal{X}}_{\kappa ,\theta +1}^e$ is elaborated and testified in the following theorem. Before proceeding, the following assumption is presented necessarily.

Assumption 4.1

For positive scalars $\overline{c}$, $\overline{\phi }$, $\overline{\sigma }$, $\overline{\pi }$, $\overline{b}$, $\overline{u}$, $\overline{r}$, $\overline{z}$ and $\overline{l}$, the following inequalities $\begin{array}{rl}& C_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}^T\le \overline{c}I,\mathrm{\Phi }\le \overline{\phi }I,{\mathrm{\Sigma }}_{\kappa ,\theta +1}\le \overline{\sigma }I,\\ & \mathrm{\Pi }{\mathrm{\Pi }}^T\le \overline{\pi }I,B_{\kappa ,\theta +1}{B}_{\kappa ,\theta +1}^T\le \overline{b}I,U_{\kappa ,\theta +1}\le \overline{u}I,\\ & {\mathrm{{\rm Y}}}_{\rho ,\theta +1}\le \overline{r}I,z_{\kappa ,\theta +1}\le \overline{z},l\le \overline{l}\end{array}$hold for each θ, ρ and κ.

Theorem 4.1

Assume that ${\mathcal{X}}_{\kappa ,0}^e\le \overline{x}I$. If the following condition holds (30) $(1+o_2)\overline{c}{\overline{x}}_p-\frac{(1+o_2{)}^2\overline{c}({\overline{x}}_p{)}^2}{(1+o_2)\overline{c}{\overline{x}}_p+(1+o_3^{-1})\overline{\sigma }+\overline{m}}\le \overline{x}$(30) with $\overline{m}=\frac{(1+o_2^{-1}+o_3)n_y{\overline{z}}^2{\overline{l}}^2}{4}$ and ${\overline{x}}_p=(1+o_1)n_x\overline{x}\overline{\phi }+(1+o_1^{-1})S^2{\overline{\omega }}^2\overline{\pi }\overline{x}+(S-1){\overline{\omega }}^2\overline{\pi }\overline{r}+\overline{b}\overline{u}$, then ${\mathcal{X}}_{\kappa ,\theta +1}^e$ is said to be uniformly bounded and satisfies ${\mathcal{X}}_{\kappa ,\theta +1}^e\le \overline{x}I$ for each θ.

Proof.

The minimized ${\mathcal{X}}_{\kappa ,\theta +1}^e$ can be derived according to the selected EP in (16(16) $\begin{array}{rl}& {\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & =(1+o_2){\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T\left\{\right(1+o_2)C_{\kappa ,\theta +1}\\ & \times {\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}I\\ & {+(1+o_3^{-1}){\mathrm{\Sigma }}_{\kappa ,\theta +1}\}}^{-1},\end{array}$(16) ) satisfying (31) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^e& =(1+o_2)C_{\kappa ,\theta +1}{\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T\\ & -(1+o_2{)}^2{\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T\left\{\right(1+o_2)C_{\kappa ,\theta +1}\\ & \times {\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T+(1+o_3^{-1}){\mathrm{\Sigma }}_{\kappa ,\theta +1}\\ & {+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}I\}}^{-1}{C}_{\kappa ,\theta +1}\\ & \times {\mathcal{X}}_{\kappa ,\theta +1}^p.\end{array}$(31) With the help of induction, based on the initial condition ${\mathcal{X}}_{\kappa ,0}^e\le \overline{x}I$ and assumption ${\mathcal{X}}_{\kappa ,\theta }^e\le \overline{x}I$, the conclusion ${\mathcal{X}}_{\kappa ,\theta +1}^e\le \overline{x}I$ will be proved inductively. Taking the Assumption 4.1 into account, one has (32) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^p& \le (1+o_1)n_x\overline{x}\overline{\phi }I+(1+o_1^{-1})S^2{\overline{\omega }}^2\overline{\pi }\overline{x}I\\ & +(S-1){\overline{\omega }}^2\overline{\pi }\overline{r}I+\overline{b}\overline{u}I\\ & :={\overline{x}}_{p}I\end{array}$(32) and (33) $\begin{array}{rl}& C_{\kappa ,\theta +1}{\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T+{\mathrm{\Sigma }}_{\kappa ,\theta +1}+z_{\kappa ,\theta +1}^2{l}^{2}I\\ & \le \overline{c}{\overline{x}}_{p}I+\overline{\sigma }I+\overline{z}\overline{l}I.\end{array}$(33) Accordingly, we can obtain $\begin{array}{rl}& {\mathcal{X}}_{\kappa ,\theta +1}^e\\ & \le \left[\right(1+o_2)\overline{c}{\overline{x}}_p-\frac{(1+o_2{)}^2\overline{c}({\overline{x}}_p{)}^2}{(1+o_2)\overline{c}{\overline{x}}_p+(1+o_3^{-1})\overline{\sigma }+\overline{m}}]I.\end{array}$In terms of the condition in (30(30) $(1+o_2)\overline{c}{\overline{x}}_p-\frac{(1+o_2{)}^2\overline{c}({\overline{x}}_p{)}^2}{(1+o_2)\overline{c}{\overline{x}}_p+(1+o_3^{-1})\overline{\sigma }+\overline{m}}\le \overline{x}$(30) ), we can arrive at obviously (34) ${\mathcal{X}}_{\kappa ,\theta +1}^e\le \overline{x}I.$(34) To sum up, the proof of the boundedness analysis is completed.

Remark 4.1

Due to the comprehensive consideration including nonlinearity, coupled perturbation, UQDED and system noises, the constructed predictor-estimator should hold the ability to resist the uncertain factors just mentioned, thereby ensuring that the EE is limited within an acceptable range. To clarify the performance evaluation of the introduced EDBSE method, a uniformly bounded analysis is elaborated based on some reasonable parameter assumptions. In order to facilitate the clarification of the developed EDBSE algorithm, the specific implementation steps are revealed in Table 1.

Table 1. The EDBSE algorithm.

$\mathbf{Algorithm}\mathbf{:}\mathbf{EDBSE}\mathbf{algorithm}$
$\mathrm{Step}1:$ Initialize algorithm parameters.
$\mathrm{Step}2:$ Derive prediction by (8(8) $\begin{array}{rl}{\hat{x}}_{\kappa ,\theta +1|\theta }& =f\left({\hat{x}}_{\kappa ,\theta |\theta }\right)+{\omega }_{\mathrm{\kappa \kappa },\theta }\mathrm{\Pi }{\hat{x}}_{\kappa ,\theta |\theta }\\ & +\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\hat{x}}_{\rho ,\theta |\theta },\end{array}$(8) ).
$\mathrm{Step}3:$ Compute ${\mathcal{X}}_{\kappa ,\theta +1}^p$ via (14(14) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^p& =(1+o_1)\mathrm{tr}\left({\mathcal{X}}_{\kappa ,\theta }^e\mathrm{\Phi }\right)I+(1+o_1^{-1})\sum _{t=1}^s{\omega }_{\mathrm{\kappa t},\theta }\\ & \times \sum _{\rho =1}^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\mathcal{X}}_{\kappa ,\theta }^e{\mathrm{\Pi }}^T+\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }^2\mathrm{\Pi }\\ & \times {\mathrm{{\rm Y}}}_{\rho ,\theta }{\mathrm{\Pi }}^T+B_{\kappa ,\theta }{U}_{\kappa ,\theta }{B}_{\kappa ,\theta }^T\end{array}$(14) ).
$\mathrm{Step}4:$ Determine EP based on (16(16) $\begin{array}{rl}& {\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & =(1+o_2){\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T\left\{\right(1+o_2)C_{\kappa ,\theta +1}\\ & \times {\mathcal{X}}_{\kappa ,\theta +1}^p{C}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}I\\ & {+(1+o_3^{-1}){\mathrm{\Sigma }}_{\kappa ,\theta +1}\}}^{-1},\end{array}$(16) ).
$\mathrm{Step}5:$ Derive estimation by (9(9) $\begin{array}{rl}{\hat{x}}_{\kappa ,\theta +1|\theta +1}& ={\hat{x}}_{\kappa ,\theta +1|\theta }+{\mathrm{\Xi }}_{\kappa ,\theta +1}({\overrightarrow{y}}_{\kappa ,\theta +1}\\ & -C_{\kappa ,\theta +1}{\hat{x}}_{\kappa ,\theta +1|\theta }),\end{array}$(9) ).
$\mathrm{Step}6:$ Compute ${\mathcal{X}}_{\kappa ,\theta +1}^e$ via (15(15) $\begin{array}{rl}{\mathcal{X}}_{\kappa ,\theta +1}^e& =(1+o_2)(I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}){\mathcal{X}}_{\kappa ,\theta +1}^p\\ & \times (I-{\mathrm{\Xi }}_{\kappa ,\theta +1}{C}_{\kappa ,\theta +1}{)}^T+(1+o_3^{-1}){\mathrm{\Xi }}_{\kappa ,\theta +1}\\ & \times {\mathrm{\Sigma }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T+\frac{(1+o_2^{-1}+o_3)n_y{z}_{\kappa ,\theta +1}^2{l}^2}{4}\\ & \times {\mathrm{\Xi }}_{\kappa ,\theta +1}{\mathrm{\Xi }}_{\kappa ,\theta +1}^T,\end{array}$(15) ).
$\mathrm{Step}7:$ Set $\theta =\theta +1$ and return to Step 2.

5. Numerical simulation

This section is devoted to carrying out a numerical simulation to demonstrate the validity of the proposed EDBSE algorithm.

Consider the CPCNs in (1(1) $\begin{array}{rl}{x}_{\kappa ,\theta +1}& =f\left(x_{\kappa ,\theta }\right)+{\omega }_{\mathrm{\kappa \kappa },\theta }\mathrm{\Pi }{x}_{\kappa ,\theta }\\ & +\sum _{\rho =1,\rho \ne \kappa }^s{\omega }_{\mathrm{\kappa \rho },\theta }\mathrm{\Pi }{\zeta }_{\rho ,\theta }+B_{\kappa ,\theta }{u}_{\kappa ,\theta },\end{array}$(1) )–(3(3) ${\zeta }_{\rho ,\theta }=x_{\rho ,\theta }+{\xi }_{\rho ,\theta },$(3) ) with 3 units. The system matrices are given by $\begin{array}{rl}{B}_{1,\theta }& ={\left[\begin{array}{cc}0.8& 0.8\end{array}\right]}^T,\\ B_{2,\theta }& ={\left[\begin{array}{cc}0.04-0.1\sin \left(0.1\theta \right)& 0.04\end{array}\right]}^T,\\ B_{3,\theta }& ={\left[\begin{array}{cc}-0.1& -0.2\end{array}\right]}^T,\\ C_{1,\theta }& =\left[\begin{array}{cc}0.3& -0.4\end{array}\right],\\ C_{2,\theta }& =\left[\begin{array}{cc}0.5& -0.5\end{array}\right],\\ C_{3,\theta }& =\left[\begin{array}{cc}0.5-0.1\cos \left(0.1\theta \right)& 0.6\end{array}\right].\end{array}$The inner and outer coupling matrices are, respectively, given by $\mathrm{\Gamma }=\mathrm{diag}\{0.5,0.5\}$ and ${\mathcal{A}}_{\kappa ,\theta }=[{\omega }_{\mathrm{\kappa \gamma },\theta }{]}_{3\times 3}$ $(\kappa =1,2,3)$ with ${\omega }_{\mathrm{\kappa \kappa },s}=-0.8$ and ${\omega }_{\mathrm{\kappa \gamma },s}=0.4$ $(\kappa \ne \gamma )$. The system state $x_{\kappa ,\theta }$ is made up of $x_{\kappa ,\theta }^1$ and $x_{\kappa ,\theta }^2$. Set the initial values of EDBSE as ${\overline{x}}_{\kappa ,0}=\mathbb{E}\left\{x_{\kappa ,0}\right\}=[0.40.4{]}^T$, ${\hat{x}}_{\kappa ,0|0}=[{\hat{x}}_{\kappa ,\theta |\theta }^1{\hat{x}}_{\kappa ,\theta |\theta }^2{]}^T={\overline{x}}_{\kappa ,0}$ and ${\mathcal{X}}_{\kappa ,0}^e=2I$. The other selected parameters satisfy $o_1=0.4$, $o_2=0.5$, $o_3=0.8$, a = 2, $z_{\kappa ,\theta }=1$, l = 1, $\mathrm{\Delta }=20$, $\mathrm{\Phi }=5I$. The variances of $u_{\kappa ,\theta }$ and ${\sigma }_{\kappa ,\theta }$ are respectively chosen as $U_{\kappa ,\theta }=0.2$ and ${\mathrm{\Sigma }}_{\kappa ,\theta }=0.1$. Moreover, the nonlinear function satisfies $f\left(x_{\kappa ,\theta }\right)=\left[\begin{array}{c}\sin \left(x_{\kappa ,\theta }^1{x}_{\kappa ,\theta }^2\right)\\ \cos \left(x_{\kappa ,\theta }^1{x}_{\kappa ,\theta }^2\right)\end{array}\right].$The corresponding simulation results of EDBSE method are presented in Figures 1–7. Figure 1 denotes $x_{1,\theta }$ and its estimation ${\hat{x}}_{1,\theta |\theta }$. Figure 2 characterizes $x_{2,\theta }$ and its estimation ${\hat{x}}_{2,\theta |\theta }$. $x_{3,\theta }$ and its estimation ${\hat{x}}_{3,\theta |\theta }$ are plotted in Figure 3. From the results of Figures 1–3, it can be seen that the developed EDBSE method can estimate the state of CPCNs effectively. The index $\mathrm{MSE}\kappa$ denotes the mean square error of the κ unit and MSE is the sum of $\mathrm{MSE}\kappa$. The MSE and its upper bound are shown in Figure 4, which signifies that the MSE always keeps below its upper bound. To reveal the influence of coupled perturbation on the EDBSE algorithm (Case I: a = 2 and Case II: a = 0.2), comparison results regarding $\mathrm{MSE}\kappa$ are drew in Figures 5–7 respectively performed by Monte Carlo simulation with 100 iteration, i.e. $\mathrm{MSE}\kappa =\frac{1}{100}\sum _{h=1}^{100}{\left[\right(x_{\kappa ,\theta }^1-{\hat{x}}_{\kappa ,\theta |\theta }^1)+(x_{\kappa ,\theta }^2-{\hat{x}}_{\kappa ,\theta |\theta }^2\left)\right]}^{\left(h\right)}.$Obviously, the coupled perturbation with a larger fluctuation range can result in the reduction of estimation accuracy.

Figure 1. $x_{1,\theta }$ and ${\hat{x}}_{1,\theta |\theta }$.

Figure 2. $x_{2,\theta }$ and ${\hat{x}}_{2,\theta |\theta }$.

Figure 3. $x_{3,\theta }$ and ${\hat{x}}_{3,\theta |\theta }$.

Figure 4. log(MSE) and its upper bound.

Figure 5. log(MSE1) and its upper bound.

Figure 6. log(MSE2) and its upper bound.

Figure 7. log(MSE3) and its upper bound.

6. Conclusions

An EDBSE algorithm for CPCNs has been proposed in this paper. Firstly, a bounded nonlinearity has been described to embody the dynamical behaviour of CPCNs. Then, considering the unreliable interaction among different units, a uniform distributed random variable has been adopted to describe the phenomenon of coupled perturbation. Consequently, a UQDED strategy has been employed to pre-process data so as to enhance security and reliability. An effective EDBSE approach has been developed to guarantee the existence of the upper bound of EEC, where the EP has been designed to minimize the trace of such upper bound. Moreover, a sufficient condition has been given and testified to clarify the uniform boundedness of EE. Finally, a simulation example has been presented to verify the availability of the concerned EDBSE algorithm. The extended research directions could include the memory-based event-triggered state estimation, stochastic-communication-based state estimation and so on.

Acknowledgments

CRediT authorship contribution statement: Chaoqing Jia: Conceptualization, Methodology, Investigation, Writing-original draft, Writing-review $\mathrm{\&}$ editing, Supervision. Peixia Gao: Methodology, Writing-original draft, Writing-review $\mathrm{\&}$ editing, Investigation, Software, Validation. Aozhan Zhou: Writing-review $\mathrm{\&}$ editing, Software, Validation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from the corresponding author, Chaoqing Jia, upon reasonable request.

Additional information

Funding

This work was supported in part by the National College Students' Innovative Entrepreneurial Training Plan Program of China [grant number 202310214116], the National Natural Science Foundation of China [grant number 12301567], the Postdoctoral Science Foundation of Heilongjiang Province of China [grant number LBH-Z22199] and the Fundamental Research Foundation for Universities of Heilongjiang Province [grant number 2022-KYYWF-0141].