Set-membership filtering for complex networks with constraint communication channels

Abstract

The set-membership filtering is studied for a class of multi-rate sampling complex networks with communication capacity constraint. For reducing communication load, the weighted try-once-discard scheduling protocol is utilized to transmit the most needed measurement. To improve the filtering performance, a novel mixed compensation method is proposed to obtain a compensatory measurement that is closer to the actual value. Accordingly, a mixed compensation dependent filter is designed, and a filtering error system is obtained. Sufficient conditions are established to ensure that the filtering error system satisfies $P_{T_k}$-dependent constraint. Then, a new algorithm is designed to obtain the optimized ellipsoid by minimizing the constraint matrix. Finally, an illustrative example is given to demonstrate the validity of the developed filter.

Introduction

With the expansion of system scale and the increasing of its complexity, complex networks that can be used to describe many complex systems are penetrating into many different fields such as mathematics, life, engineering, and so on (Basaras et al., 2019, Chu and Iu, 2017, Liu and Ye, 2021, Yang, Hu et al., 2021, Zhang et al., 2021). In most literature concerning complex networks, the sampling rates for different components (such as plant, sensor, and so on) are assumed to be the same for analysis convenience (Li and Xiao, 2021, Vega et al., 2020). In practical, however, for different components, or even for the same component, such an assumption is often unrealistic since it is difficult to guarantee that the sampling rate is the same under different physical constraints. Furthermore, it is clear that a higher sampling rate gives more available information to improve system performance by sacrificing more resources, which is often undesired for resource-limited case (Shen, Wang, Shen and Han, 2020). Consequently, it is of practical importance to adopt the multi-rate (MR) sampling scheme, and the research of MR sampling systems has received considerable attention (Lin and Sun, 2019, Shen, Wang, Shen, Alsaadi, 2020, Tian and Sun, 2021). Nevertheless, to our best knowledge, the research in MR sampling complex networks has been rarely addressed, which motivates our further study.

Wireless sensor networks, which consist of plenty of sensor nodes and are widely distribute over a variety of environments, are usually used to collect and process information as well as communicate with filter (Liu, Zhao et al., 2020, Rao et al., 2020, Wang et al., 2021, Xu et al., 2021). For the distributed sensor networks, the energy of sensors is always limited and difficult to charge. Furthermore, the capacity of communication channel is constrained, which implies that the transmitted information would inevitably suffer from data collisions or packets dropouts when all system components try to access a shared communication channel simultaneously. In order to overcome this constraint and achieve the goal of improving the transmission efficiency and saving energy, a weighted try-once-discard (WTOD) protocol (Shen, Wang, Shen, & Dong, 2021) is introduced in this paper, which is a dynamic scheduling scheme in accordance with different weights. Compared with stochastic communication protocol (Song, Wang, Liu, & Wei, 2020) and round-robin protocol (Xu, Lv, Lin, Lu, & Quevedo, 2022), the WTOD protocol assigns the transmission right to the most needed sensor according to the given quadratic selection principle (Ju et al., 2020, Zou et al., 2016). However, once the WTOD protocol is employed to select one for transmission, the rest of data is discarded, which is undesirable for filter (Liu, Wang, Chen, & Wei, 2020b). Therefore, how to compensate for the other measurements except the assigned one becomes a challenging problem. Some compensation methods, including the hold on compensation (Zou et al., 2016) and the zero compensation (Ju et al., 2020), are frequently used. It should be pointed out that the hold on compensation method and zero compensation method cannot ensure which one is closer to the actual value. Therefore, how to design a mixed compensation method by combining these methods is an interesting issue.

Due to the large scale of complex networks, it is extremely difficult to directly acquire the full state information of complex networks (Liu et al., 2020a, Liu, Wang, Lu et al., 2021, Wang and Luo, 2021), and the measurement is also subject to the noise. Thus the filter is necessary to be designed by the available measurement of complex networks (Liu, Wang, Zhou, 2021, Liu et al., 2022). Up to now, various filtering techniques have been presented, which include Kalman filtering, robust filtering, set-membership filtering, and other approaches (Ierardi, Orihuela, & Jurado, 2021). Kalman filtering is good at solving systems with Gaussian noise (Rocha and Terra, 2021, Xia and Wang, 2015), and robust filtering is mainly used to handle energy bounded noise (Dong et al., 2014, Dong et al., 2016, Yang, Tu et al., 2021). The set-membership filtering method, dated back to the 1960s, has an advantage in dealing with the unknown but bounded noise, and such a method has been recognized in research communities in the past decades (Liu, Wang, Wang et al., 2021, Orihuela et al., 2017). It needs to be emphasized that the unknown but bounded noise only requires to know the upper and lower bounds rather than its distribution, which is more conformable to practical situation in MR sampling complex networks. Nevertheless, little attention has been paid to set-membership issue for MR sampling complex networks, which motivates our current investigation.

Motivated by the above considerations, this work investigates the set-membership filtering for MR sampling complex networks with WTOD protocol. Different from the existing ones (Ju et al., 2020, Zou et al., 2016), a new mixed compensation method and corresponding filter design method are proposed for better filtering performance. The main contributions are as follows.

• For complex networks, MR sampling and communication capacity constraint are considered simultaneously, the WTOD protocol is introduced to overcome the constrained communication channel, and the Pseudo measurement (PM) approach is utilized for modeling synchronization. Compared with the existing ones (Lin and Sun, 2019, Shen, Wang, Shen, Alsaadi, 2020, Tian and Sun, 2021), the set-membership filtering for the model that concentrates on MR sampling time-varying complex networks and constrained communication channel is meaningful.

• A new mixed compensation method is proposed to obtain the measurement that is closer to the actual value. An indication function ${\varrho }_{i,t_k^i}^l$ is introduced to describe the used compensation method, which can be adjusted by a chosen compensation weight coefficient ${\alpha }_{T_k}$. Based on the measurement under the mixed compensation, a mixed compensation dependent filter is designed.

• Due to the influence of compensation weight coefficient ${\alpha }_{T_k}$ on the compensated measurements, an algorithm is proposed by minimizing the constraint matrix $P_{T_k}$ to find an optimized ${\alpha }_{T_k}$ and the corresponding optimized ellipsoid. The mixed compensation method and the optimized algorithm can be directly used to the other communication protocols which need compensation, such as stochastic communication and the round-robin protocol, and so on (Ju et al., 2020, Shen et al., 2021, Song et al., 2020, Xu et al., 2022, Zou et al., 2016).

The organization of this article is as follows. Section 2 formulates the discrete time-varying complex networks with $N$ nodes, proposes the WTOD protocol and mixed compensation method, and designs a mixed compensation dependent filter. In Section 3, sufficient conditions of the $P_{T_k}$-dependent constraint are derived. Then a set of optimized ellipsoids and the filter gains are obtained. In Section 4, a numerical simulation is given, and conclusions are given in the end.

Notations: $\mathbb{N}$, ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ denote, respectively, the set of natural numbers, n-dimensional Euclidean space and $n\times m$ real matrices. The diagonal matrix is denoted as $\mathrm{diag}\left\{\cdot \right\}$. The identity matrix and the zero matrix with appropriate dimensions are represented by $I$ and $0$. The symbol $\ast$ in a matrix represents the symmetric term. For a matrix $X$, the matrix $X^T$ and symbol $\mathrm{trace}\left\{X\right\}$ denote its transpose and the trace, respectively. The Kronecker delta function $\delta (\sigma ,l)$ is a binary function that equals $1$ if $\sigma =l$ and equals $0$ otherwise. For a vector $\mathit{y}$, $\max \left\{\mathit{y}\right\}$ denotes its largest element. The symbol “$\arg {\max }_{x\in \phi }f\left(x\right)$” stands for the argument $x$ of the maximum $f\left(x\right)$, where $\phi$ is the set of values of $x$. The notation “$\mathrm{mod}(a,m)$” returns the remainder after division of $a$ by $m$, where $a$ is the dividend and $m$ is the divisor.