Event-Based State Estimation for Networked Singularly Perturbed Complex Networks

Ren, Zerong

doi:https://doi.org/10.1155/2022/6122921

Complexity

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Abstract Introduction Numerical Example Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Behavior Analysis, Complexity and Control of Networked Dynamical Systems 2022

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Research Article | Open Access

Volume 2022 | Article ID 6122921 | https://doi.org/10.1155/2022/6122921

Event-Based State Estimation for Networked Singularly Perturbed Complex Networks

Zerong Ren¹

Academic Editor: Yue Song

Received02 Mar 2022

Revised22 Jun 2022

Accepted27 Jun 2022

Published14 Jul 2022

Abstract

This paper deals with the multievent-triggering-based state estimation for a class of discrete-time networked singularly perturbed complex networks (SPCNs). A small singularly perturbed scalar is adopted to establish a discrete-time SPCNs model. To reduce the communication burdens, the data transmission between the sensor and the estimator is managed by a multievent generator function. Depending on the singularly-perturbed-based Lyapunov theory, a sufficient condition is constructed to guarantee that the estimation error is exponentially ultimately bounded in the mean square. Finally, the validity of the developed result is demonstrated by a simulation example.

1. Introduction

A complex network is a set of interconnected nodes coupled by certain network topology, each node of which can be considered as a class of dynamic subsystems. Owing to its complex inherent structure, most systems in real life can be regarded simply as complex networks, including, but not limited to social networks, biological networks, power grid networks, and Internet [1–3]. Consequently, considerable research interest has been stirred over the past few decades and there has been a host of meaningful published achievements of complex networks [4–9].

As far as we know, however, the complex networks with two-time scales receive little attention. However, the two-time scale case of many real-life complex networks [10–12] is continually encountered. For instance, the circuit state variables become faster than the mechanical state variables in electronic power grids, due to the difference in the time scalars on the circuit and the mechanical systems [10]. This can result in the appearance of diverse time-scale subsystems in hosts of electromechanical systems, named fast and slow dynamics. In most of the existing literature [13–15], a singularly perturbed approach is adopted to deal with the two-time scale phenomenon of these real-life systems. In other words, the fast-slow subsystem is distinguished by introducing a small singularly perturbed scalar. Hence, such complex networks can be regarded as SPCNs [16–23].

What is worth mentioning is that a host of the reporting efforts [16–20] merely focuses on the synchronization phenomenon of the SPCNs. However, in some real-world scenarios, the exact state of the SPCNs on account of various factors, like the high number of nodes, disturbances in all directions, and high dimensions, is unavailable [7]. Thus, what we should pay attention to is the state estimation of the SPCNs. On the other hand, we noticed that besides [22, 23], the discrete-time SPCNs get little research concerns. The two important reasons for considering the discrete-time SPCNs are that computational simulation and network communication. Therefore, it is very necessary to investigate the state estimation of the discrete-time SPCNs.

In addition, increasing attention is devoted to the event-triggered protocol (ETP), in which the current packet is released if the ETP-based triggering condition is satisfied [24–26]. Past years have witnessed an increasing interest in ETPs, including static ETPs, dynamic ETPs, and memory ETPs [27–34]. It is noted that in the above-referred ETPs, the triggering parameter is assumed to be the same for all dynamic outputs/states. Nevertheless, such an assumption is difficult to be satisfied, especially in multisensor networks, which contributes to the varying triggering parameters. In light of above-discussed phenomenon, in this work, in order to save resource consumption and solve the communication congestion, a multi-ETP is employed to deal with the large information communication among nodes of the discrete-time SPCNs, which, to some extent, promotes the current research. As such, a natural and interesting question is how to design a proper multi-ETP for discrete-time SPCNs.

Based on the aforementioned observations, we try our best to develop the multi-ETP-based estimator design issue for the discrete-time SPCNs. Then, the mean square exponential bounded and state estimations are studied by using the Lyapunov function dependent on a singular perturbed parameter. In the end, a numerical example is presented to prove the effectiveness of the state estimator design method. It is worth emphasizing that even though the discrete-time SPCNs are unstable, the result of this work is still efficient. The highlights of our contributions are outlined as follows: (1) A nonlinear discrete-time SPCNs model is developed, which includes nonlinearities of complex networks and multitime scales. (2) As the study progress, the multi-ETP-based state estimation problem for the discrete-time SPCNs with nonlinearities is considered.

Notation: refers to a set of all nonnegative integers. represents the transpose of the matrix . symbolizes the diagonal matrix. . denotes the -dimensional unit matrix. denotes the minimal/maximal eigenvalue. means Euclidean vector norm. signifies the mathematical expectation.

2. Problem Formulations

Consider a type of SPCNs composed of coupled nodes described bywhere , , , , , , , and refer to the slow state and the fast state of node , respectively. and mean the measurement outputs of node , is a singular perturbation parameter, and signifies the bound disturbance input belonging to , which meets . refers to an inner-coupling matrix with given dimensions. , , , , and are known matrices with suitable dimensions.

The network topology is devoted to reflect the outer coupling phenomenon of the SPCNs. and symbolize the sets of nodes and edges. For any , the out-coupled configuration matrix is symmetric if , which satisfieswhere implies a connection between nodes and ; otherwise, .

Assumption 1 (see [23]). The nonlinear sector-valued functions and of SPCNs (1) satisfy the following assumption:where , , , , , and are constant matrices.

In this paper, for the sake of saving the communication resources between the sensors and estimators, a multievent-triggered approach is presented to reduce transmission energy. The triggering instant series of node can be assumed as where the new transmitted instant can be formulated as

With , is the given parameter of the -th node, is a weighting matrix of the -th node to be determined, and with referring to the latest transmitted measurement of node . Hence, for ,

Remark 1. Note that different from the existing static ETP, the multievent-triggered protocol is studied in (2). The proposed triggering protocol can be seen as a generalized framework of ETP, which cover the existing static ETP as a special case (i.e., ).

Subsequently, based on the multievent-triggered approach, a state estimator is constructed aswhere with and representing the state estimations of and , respectively. means the estimator gain of the -th node to be judged.

Let

Combining (1) and (4), the dynamics of the estimation error can be built aswhere

In the sequel, one reschedules the order of dynamic estimation errors (8). Denote , with being row-switching elementary matrices and being invertible. Then, one haswhich yieldswhere

To facilitate the derivation of the main results, the following definition and lemma are introduced.

Definition 1 (see [35]). Estimation error dynamics (7) is exponentially ultimately bounded in mean square (EUBMS), if for any solution with initial state ,holds, where , , and imply the mean square asymptotic upper bound of (11).

Lemma 1 (see [23]). Combined with Assumption 1, the nonlinear functions and of estimation error dynamics (7) satisfy the conditions as follows:where

Lemma 2 (see [23]). For any matrices and and a scalar , , if and hold, it yields .

3. Main Results

In this section, a sufficient condition is presented to guarantee that the estimation error dynamics (7) is exponentially ultimately bounded in mean square and the desired state estimator will be designed.

Theorem 1. For given , estimation error dynamics (7) is EUBMS, if there exist scalars and and matrices , such thatwhere

Proof. Firstly, construct the following Lyapunov functional candidate:According to (5), another equivalent form of the event-triggering condition is as follows:Along the trajectory of (11), calculating the mathematical expectation of the difference of V(k), one gains thatDepending on Young’s inequality, the following inequality holds:Combining (19) and (21) and Lemma 1, we obtain thatwhereApplying the Schur complement lemma to (16), it is clear that . Consequently, one haswhich yieldsThen, it follows from (25) thatMoreover, it is easy to obtain that and ; combining (26), it yields thatConsequently, estimation error dynamics (7) is EUBMS, and is the mean square asymptotic upper bound of (7), which completes the proof.

Theorem 2. For with the upper bound and , estimation error dynamics (7) is EUBMS, if there exist scalars and and matrices , such thatwhere

Moreover, the estimator gain matrices are calculated as

Proof. From (28)–(30) and Lemma 2, it follows that for , the following conditions hold:Substituting (32) into (33) and noticing that , one hasRecalling (29), it is clear that with . Then, is invertible.
Premultiplying and postmultiplying (34) by , its transposes yields (16). Therefore, inequality (16) can be guaranteed if (28)–(30) hold. This completes the proof.

4. Numerical Example

Similar to [23], consider SPCNs (1) with three nodes and the following parameters:

The out-coupled configuration matrix of SPCNs (1) and its inner-coupling matrix, respectively, are selected as follows:

In light of Assumption 1, the nonlinear vector-valued functions are chosen asand . The event-triggered thresholds are set as , , and , and the weighting matrices are calculated as . Other parameters are given as follows: .

According to Theorem 2, the gain matrices of state estimator (6) can be obtained as

The bound disturbance input is considered as , and the initial conditions of (1) and (6) are given as , , , and .

The simulation results are presented in Figures 1–5. Figures 1–3 display the state trajectories and their estimations of three nodes, respectively. The event-based release instants and release intervals of three nodes are shown in Figure 4. Figure 5 plots the evolutions of the estimation error dynamics. It can be discovered from Figure 5 that estimation error dynamics (7) is EUBMS.

5. Conclusions

This paper has investigated the issue of the multievent-triggered state estimation for a novel class of discrete-time nonlinear SPCNs. A discrete-time SPCN model with nonlinearities has been modeled. To alleviate energy consumption, a multievent triggered protocol is applied to regulate the communication among nodes of the SPCNs. Finally, a simulation has demonstrated the rationality, superiority, and effectiveness of the proposed method.

Data Availability

The data used to support the findings of this study are available from the author upon request.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

This work was supported by the Project of Youth Science and Technology Talents of Guizhou Province (Grant Number: Qian Jiao He KY Zi[2020]095).

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Copyright

Copyright © 2022 Zerong Ren. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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