# Event-triggered state estimation for complex networks under deception attacks: a partial-nodes-based approach

*Complex Eng Syst*2023;3:14.

## Abstract

This paper addresses the issue of state estimation for a kind of complex network (CN) with distributed delays and random interference through output measurements. In the data transmission, the deception attacks are taken into account by resorting to a sequence of Bernoulli random variables with a given probability. Considering the complexity of the network, the fact that only partial output measurements are available in practical environments presents a new challenge. Therefore, the partial-nodes-based (PNB) state estimation problem is proposed. For the sake of data collision avoidance and energy saving, a general event-triggered scheme is adopted in the design of the estimator. A novel estimator is constructed to consider both cyber attacks and resource limitations, filling the gap in previous results on PNB state estimation. By using the Lyapunov method and several stochastic analysis techniques, a few sufficient conditions are derived to guarantee the desired security and convergency performance for the overall estimation error. The estimator gains are obtained by solving a set of matrix inequalities with nonlinear constraints. At last, two examples and simulations are presented to further show the efficiency of the proposed method.

## Keywords

*,*deception attacks

*,*partial-nodes-based (PNB) estimation

*,*event-triggered scheme

*,*finite-distributed delays

## 1. INTRODUCTION

Over the past few decades, complex networks (CNs) have gained significant research interest due to their diverse applications in natural and artificial systems, including but not limited to sensor networks, biological networks, and social networks, among others ^{[1-6]}. Generally speaking, a CN is composed of numerous nodes that can be described as various types of topologies. In CNs, each individual node exhibits intricate and diverse dynamical behavior ^{[7,8]}, which results in plenty of dynamics, including synchronization, chaos, and so on. Furthermore, it is widely acknowledged that time delays are an inevitable factor in data transmission. This can lead to a decline in performance and introduce additional challenges during analysis. So far, a great deal of research effort has been devoted to dynamic analysis and control for CNs with delays ^{[9-16]}.

In practical engineering, state information plays a key role in the analysis and design of CNs. However, the full state information is usually unavailable because of the large size of CNs, complex coupling relation, and inaccuracy of models. To cope with this issue, one possible solution is to estimate the state by using some available measurement outputs ^{[17-23]}. For example, in ^{[17]}, a finite-time ^{[18,19]}, the state estimation problems have been examined for different classes of CNs subject to both discrete and distributed time delays. For the state estimation of CNs, it should be noted that the majority of the existing literature has assumed the outputs of all nodes are accessible ^{[24]}. However, when a CN possesses a huge number of nodes, it might be unreasonable (even impossible) to get all outputs. Additionally, the sensor failure can also result in some output not being obtained. Taking these problems into consideration, a partial-node-based (PNB) state estimation, which implements the state estimation only via partial measurements of CNs, has attracted more and more research attention ^{[25-29]}.

In the networked communication environment, the components are usually interconnected through a shared communication network. On the one hand, during information transmission, opponents or attackers may capture and manipulate interchanged information between components, which causes degraded network performance or even destabilization of the system ^{[30]}. A lot of research works have been done to focus on cyber attacks ^{[31-39]}. For instance, the state estimation issue has been investigated for large-scale systems subjected to deception attacks in ^{[33]}. In ^{[38]}, ^{[40-42]}. Compared with the traditional periodic triggered mechanism, the most distinguishing feature of ETMs is to transmit information only when certain triggered conditions are met, which allows a considerable reduction of the network resource occupancy. However, from the perspective of security levels against cyber attacks, the event-triggered PNB state estimation for CNs has rarely been investigated, which is the main motivation of this paper^{[43-46]}.

To sum up the above discussions, this paper aims to investigate the event-triggered PNB state estimation problem for CNs under deception attacks. There are two significant contributions of the current research: (1) By employing partial output measurements, an event-triggered state estimator is designed for CNs subjected to deception attacks; (2) With the aid of stochastic analysis techniques and the Lyapunov method, the gain matrices and the event-triggered parameters are co-designed (by resorting to solutions of matrices inequalities) to ensure the desired secure performance of closed-loop systems under the deception attacks and the aperiodic data updating. The rest of this paper is presented as follows. Section 2 gives the problem formulation. In Section 3, an event-triggered PNB state estimation scheme is put forward for CNs against deception attacks. Two numerical simulation examples are presented in Section 4 to further demonstrate the effectiveness of the proposed method. Finally, several conclusions are derived in Section 5.

*Notation:*

## 2. MODEL DESCRIPTION

Consider a class of discrete-time CNs with

where

where

Actually, the issue of data safety usually arises in networked environments since the data may be subject to malicious cyber attacks during the transmission. In this paper, we assume that the measurement from the output sensors is affected by deception attacks as follows:

where

in which

with

*Remark 1:* From (4), it is noticeably known that

For the purpose of saving limited communication resources, an ETM is introduced during the data transmission. For clarity, the triggering instant sequence for node

in which the event generator function

with

*Remark 2:* From the perspective of reducing the data transmission rate, it has been proven that the event-triggered scheme is an effective implementation approach under which the data transmission is permitted only if a prescribed condition is met. For clarity, let

Within the event-triggered PNB scheme, the state estimator is constructed as follows:

where

We denote by

The error system is obtained as follows:

By defining

where

*Remark 3:* It is obviously noted that the estimation error system (9) is a subsystem of the augmented system (10). That is to say, the evolution of errors can be derived by analyzing the dynamic of the augmented system (10).

The definition and lemmas presented in this context play a crucial role in the stability analysis of the augmented system (10) and the design of an appropriate estimator.

*Definition 1 ^{[36]}:* For given constants

*Remark 4:* The parameter

*Lemma 1 ^{[47]}:* For a matrix

*Lemma 2 ^{[36]}:* For constants

## 3. ANALYSIS AND RESULTS

In this section, by resorting to the stochastic analysis techniques, we shall provide the analysis result to guarantee that the augmented system (10) is

For the

*Theorem 1:* Let positive constants

in which

the constant

and

*Proof:* We choose the Lyapunov-Krasovskii functional

Here,

where

For the differences of

and

Furthermore, it can be inferred from Lemma 1 that

Combining with

Combining (15)-(19), we have

Bearing in mind (20) and (21), it stems from (22) that

Recalling (12) gives

In addition, taking (21) into consideration, we have

It is readily derived from (24) and (25) that for a real number

where

For any integer

where

Noting that

Since

it is observed that

By following from (12), we get

For analysis simplification, we denote

In what follows, a method is proposed for designing the gain matrix of an estimator for the augmented system (10).

*Theorem 2:* Let positive constants

where

In addition, the matrices

*Proof:* Recalling (12) and denoting

where

By applying the Schur Complement Lemma, (32) is valid only if the subsequent inequality is satisfied:

It is worth noting that

The model establishment and consensus analysis are finally completed with the help of (29) (an important property of the Laplacian matrix). The gain matrices are designed by resorting to the feasible solutions of (30), which obviously depends on the Laplacian matrix. Both of them show the influence of the network communication topology.

*Remark 5:* Theorem 2 proposes an easy-to-check approach for designing the PNB- and-ETM-based state estimator, which enables the error system to achieve the desired convergence and security performance even in the presence of random deception attacks. The gain matrix for the estimator is determined by considering feasible solutions to the inequalities (29) and (30). These solutions are heavily influenced by various factors, including network parameters, external disturbances, inherent nonlinearities, the coupling Laplacian matrix, delay bounds, the intensity and frequency of deception attacks, and the event-triggering threshold.

*Remark 6:* Theorem 2 proposes an effective method to design state estimators for CNs by using the output of partial nodes. It differs significantly from the state estimators proposed in [10,11,13,17]. Compared with the results of PNB state estimation for CNs in [27-29], our result stands out by considering both deception attacks and event-triggering mechanisms simultaneously. This notable feature makes Theorem 2 more practical and feasible for network environments that are constrained by limited resources and susceptible to network attacks.

## 4. EXAMPLES AND SIMULATIONS

In this section, two examples are presented to demonstrate the validity of our method.

**Example 1**

Consider a CN with

Here, we choose

and

It is not difficult to verify that conditions (2) and (3) can be met with

According to (29) and using the Matlab software (with the YALMIP toolbox), we obtain the gain matrices for the estimator as follows

Let initial values be

In order to explain the relationship between the number of event triggerings and the estimation accuracy, we provide the following Table 1 to record the triggering frequency and the estimation error. It is readily observed that the larger number of triggerings usually indicates better estimation accuracy. Generally speaking, the threshold parameters

Triggering frequencies and estimation errors with different threshold parameters

Values of | Triggering of Node1 | Triggering of Node2 | Triggering of Node3 | Total Error |

**Example 2**

Consider a CN with 5-node discrete-time Chua's circuits borrowed from ^{[29]} and let

and other parameters are set as the same as in Example 1.

Without any difficulties, we derive that conditions (2) and (3) are satisfied with

By using the Matlab toolbox, the estimator gain matrices are derived as follows

For the aim of simulation, we choose initial states to be

## 5. CONCLUSIONS

The event-triggered PNB state estimation problem has been studied in this paper for a class of discrete-time CNs under deception attacks. A novel state estimator has been designed based on the measurement outputs from the fraction of nodes. A general event-triggering scheme has also been taken into account in the design of the estimator such that the communication resources are saved dramatically. Sufficient conditions have been established (in the form of matrix inequalities) to ensure the existence of the desired state estimator. Finally, a numerical example and simulations are presented to further demonstrate the effectiveness of the proposed method. Several future research topics include the state estimation of complex systems subjected to complexity attacks or communication protocols.

## Declarations

### Authors' contributions

Made substantial contributions to the conception and design of the study and performed data analysis and interpretation: Zhou L, Li B

Performed data acquisition and provided administrative, technical, and simulation: Zhou L

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

This work was supported in part by the National Natural Science Foundation of China under Grants 62273066, in part by the Science and Technology Research Program of Chongqing Municipal Education Commission under Grants KJZD-M202100701, in part by the Group Building Scientific Innovation Project for universities in Chongqing under Grant CXQT21021, and in part by the Joint Training Base Construction Project for Graduate Students in Chongqing under Grant JDLHPYJD2021016.

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2023.

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## Cite This Article

## How to Cite

Zhou, L.; Li B. Event-triggered state estimation for complex networks under deception attacks: a partial-nodes-based approach. *Complex. Eng. Syst.* **2023**, *3*, 14. http://dx.doi.org/10.20517/ces.2023.16

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