Distributed filtering based on Cauchy-kernel-based maximum correntropy subject to randomly occurring cyber-attacks

Abstract

This paper is concerned with the distributed filtering issue under the Cauchy-kernel-based maximum correntropy for large-scale systems subject to randomly occurring cyber-attacks in non-Gaussian environments. The considered cyber-attacks are hybrid and consist of both denial-of-service attacks and deception attacks. The weighted Cauchy kernel-based maximum correntropy criterion instead of the traditional minimum variance is put forward to evaluate the filtering performance against non-Gaussian noises as well as cyber-attacks. Based on the matrix decomposition and the fixed-point iterative update rules, the desired filter gain related with a set of Riccati-type equations is obtained to achieve the optimal filtering performance. Then, an improved version only dependent on the local information and neighboring one-step prediction is developed to realize the distributed implementation. Furthermore, the convergence of the developed fixed-point iterative algorithm is addressed via the famous Banach fixed-point theorem. Finally, a standard IEEE 39-bus power system is utilized to show the merit of the proposed distributed filtering algorithm in the presence of cyber-attacks and non-Gaussian noises.

Introduction

Due to the influence of physical constraints and environmental noises, system states of modern large-scale industrial systems are not always available, which gives rise to an ever-increasing research topic about state estimation and signal processing, see e.g. Khan and Moura, 2008, Mao et al., 2021, Rostami and Lotfifard, 2018 and Zhao et al. (2021). Existing filtering algorithms derived by the minimum mean square error (MMSE) criterion, such as the famous Kalman filtering (KF) and its extended version (Theodor & Shaked, 1996), are generally applicable for large-scale industrial systems with Gaussian noises.

The Gaussian assumption is not always satisfied for actual systems and hence the filtering performance is inevitably degraded if MMSE-based algorithms are arbitrarily performed. As such, a number of filtering techniques have been developed to process non-Gaussian noises, see e.g. Battilotti, Cacace, D’Angelo, Germani and Sinopoli, 2019, Chen et al., 2017 and Wang, Zhang and Wang (2020). For instance, an adaptive filtering has been discussed in Talebi, Werner, Li, and Mandic (2019) to relax the Gaussian assumption to the generalized setting of $\alpha$-stable distributions, and a robust Kalman filter has been proposed in Huang, Zhang, Wu, Li, and Chambers (2017) to deal with the heavy-tailed noises via Student’s distribution. Furthermore, two Kalman-like approaches have been developed in Battilotti, Cacace, D’Angelo and Germani (2019) and Cacace, Conte, D’Angelo, and Germani (2019a) to handle the non-Gaussian noise by a state space augmentation as well as Kronecker powers of states and outputs. The developed approach has been further employed to solve a target tracking problem in Battilotti, Cacace, D’Angelo, Germani and Sinopoli (2019) with nonlinear measurements and an LQ non-Gaussian problem in Cacace, Conte, D’Angelo, and Germani (2019b) with packet loss.

Recent studies have shown that the robustness of the algorithms can be effectively improved by using the maximum correntropy criterion (MCC) instead of the above MMSE criterion for the dynamic system disturbed by non-Gaussian noises without any restriction on their distribution, see e.g. Song, Ding, Dong and Han, 2020, Song, Ding, Dong, Wei et al., 2020. For instance, KF and unscented KF based on MCC have been established in Chen et al. (2017) and Liu, Chen, Xu, Wu, and Honeine (2017) to effectively restrain non-Gaussian disturbances for the linear and nonlinear systems. At the same time, some available forms can be found in the reported literature (Wang, Zhang et al., 2020) and typical forms include, but are not limited to, KF-like ones or their variants (Chen et al., 2017, Liu et al., 2017, Song, Ding, Dong and Han, 2020), square-root ones (Kulikova, 2017), Chandrasekhar-type recursion ones (Kulikova, 2020) as well as Rauch-Tung-Striebel smoother (Wang, Zhang et al., 2020). It should be pointed out that the desired filter gain cannot be directly obtained in comparison with the analytical solution of KF or extended KF and thus a fixed-point iterative scheme has been put forward to achieve the engineering applications. Furthermore, the convergence of such an iterative scheme has been addressed in Chen, Wang, Zhao, Zheng, and Príncipe (2015) and Wang, Lyu, He, Zhou and Wang (2020) with the help of the contraction mapping theorem (also known as the Banach fixed-point theorem). On the other hand, the Gaussian kernel function is usually exploited to serve as the cost function in the above mentioned MCC-based filtering algorithms. However, it is not always the best selection because of both the difficulty in the selection of a proper kernel bandwidth affecting the expected filtering performance and the appearance of singular matrices resulting in the breakdown of the developed algorithms (Wang, Lyu et al., 2020). As such, it is of great importance to seek out other types of kernel functions to overcome the inherent shortages of Gaussian ones. Fortunately, a primary yet challengeable attempt has been brought by resorting to a Cauchy kernel function and an initial result has been reported in Wang, Lyu et al. (2020), which lights up the following research.

Distributed filtering plays a vital role in large-scale industrial systems due to the inherent coupling among subsystems, such as Ding et al., 2019b, Farina and Carli, 2018 and Xie, Mo, and Sinopoli (2011). It is worth noting that the operation of large-scale systems is more vulnerable and the security threats are more severe compared to traditional networked control systems due mainly to more networked devices. Usually, cyber-attacks will lead to incomplete data collection or information transmission (Kosut et al., 2011, Sinopoli et al., 2003). Generally speaking, according to their types of physical implementation, cyber-attacks can be roughly divided into denial of service attacks (Song, Ding, Dong, Wei et al., 2020), replay attacks (Zhu & Martinez, 2014), and deception attacks (Ding, Han, Wang, & Ge, 2019a). In the past few years, some interesting results have been reported to disclose the impact on system performance (including the stability and the robustness) and also to provide some strategies about their detection and identification, see e.g. Milosevic et al., 2020, Pasqualetti et al., 2013, Sinopoli et al., 2003 and Ye, Woodford, Roy, and Sundaram (2021). It should be pointed out that the occurrence of cyber-attacks could not be single and hybrid cyber-attacks should be able to cause stronger destructiveness than that caused by single attacks (Du et al., 2019). Then, the injected attack signal of deception attacks should be non-Gaussian and can be approximated by mixed-Gaussian noises (Song, Ding, Dong and Han, 2020). For this kind of complex scenario, the MCC filtering based on Cauchy kernel functions should possess better robustness and adaptivity than the traditional KF or extended KF. Furthermore, sufficient conditions of the convergence need to be found for the corresponding fixed-point iterative scheme. However, such an issue remains a technical challenge due probably to the inherent couplings among subsystem dynamics, non-Gaussian noises, as well as hybrid cyber-attacks, which motivates the investigation of this paper.

Summarizing the above discussions, the focus of this paper is on the distributed MCC filtering issue based on the Cauchy kernel function for large-scale systems subject to hybrid cyber-attacks and non-Gaussian noises. The following two essential challenges should be solved: (1) how to design a suitable distributed filtering algorithm to suppress the impact from both hybrid cyber-attacks and non-Gaussian noises; (2) how to design a suitable fixed-point iterative scheme to seek the desired filter gains. The paper attempts to deal with the above two challenges and make the following highlighted contributions: (1) A Cauchy-kernel-based maximum correntropy filter consisting of both the one-step-ahead prediction and one-step update is constructed to deal with the state estimation of large-scale systems subject to both hybrid cyber-attacks and non-Gaussian noises; (2) the investigated filter is achieved via a set of Riccati-type equations, which are solved by the fixed-point iterative algorithm; (3) a degraded version only dependent on the local information and neighboring one-step prediction is developed to realize the distributed implementation, and (4) a sufficient condition of the convergence of the developed fixed-point iterative scheme is established by utilizing the famous Banach fixed-point theorem.

Notation: The notation used throughout the paper is fairly standard unless where otherwise noted. ${\Vert \cdot \Vert }_p$ denotes the $l_p$-norm of a vector or an induced norm of a matrix. ${\lambda }_{min}\left(\cdot \right)$ and ${\lambda }_{max}\left(\cdot \right)$ describe respectively the minimum and the maximum eigenvalues of the matrix. ${\left[x\right]}_s$ or ${\left[A\right]}_s$ denote the $s$th element of the vector $x$ or the $s$th row of the matrix $A$.