Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000. Digital Object Identifier 10.1109/ACCESS.2023.DOI Gaussian Filtering with False Data Injection and Randomly Delayed Measurements SUMANTA KUMAR NANDA 1 , GUDDU KUMAR 2 , AMIT KUMAR NAIK 3 , MOHAMED ABDEL-HAFEZ 4 , VIMAL BHATIA 5,6 (Senior Member, IEEE), ONDREJ KREJCAR 6,7 , and ABHINOY KUMAR SINGH 8 (Member, IEEE) 1 Department of Electrical Engineering, Indian Institute of Technology Indore, Indore 453552, India (e-mail: phd1901102015@iiti.ac.in) 2 Department of Electrical Engineering, Indian Institute of Technology Indore, Indore 453552, India (e-mail: phd1901102010@iiti.ac.in) 3 Department of Electrical Engineering, Indian Institute of Technology Indore, Indore 453552, India (e-mail: phd1901102027@iiti.ac.in) 4 Electrical and Communication Engineering Department, College of Engineering, United Arab Emirates University, Al Ain P.O. Box 15551, United Arab Emirates (e-mail: mhafez@uaeu.ac.ae) 5 Department of Electrical Engineering, Indian Institute of Technology Indore, Indore 453552, India (e-mail: vbhatia@iiti.ac.in) 6 The faculty of Informatics and Management, University of Hradec Kralove, Czech Republic (e-mail: ondrej.krejcar@uhk.cz). 7 Malaysia-Japan International Institute of Technology, Universiti Teknologi Malaysia Kuala Lumpur, Jalan Sultan Yahya Petra, Kuala Lumpur 54100, Malaysia 8 Department of Electrical Engineering, Indian Institute of Technology Patna, Patna 801106, India (e-mail: abhinoy.singh@iitp.ac.in) Corresponding author: Mohamed Abdel-Hafez (e-mail:mhafez@uaeu.ac.ae). This work is supported by the Department of Science and Technology (DST), Government of India, through the INSPIRE Faculty Award, under Grant DST/INSPIRE/04/2018/000089 and the project (2023/2204) Grant Agency of Excellence, University of Hradec Kralove, Faculty of Informatics and Management, Czech Republic. This work is also partially funded by the Centre for Wireless Communications, University of Oulu, Finland, and the United Arab Emirates University, Al-Ain, UAE. ABSTRACT State estimation in cyber-physical systems is a challenging task involving integrating physical models and measurements to estimate dynamic states accurately in practical machine-to- machine and IoT deployments. However, integrating advanced wireless communication and intelligent measurements has increased vulnerability of external intrusion through a centralized server. This study addresses the challenge of Gaussian filtering for a specific type of stochastic nonlinear system vulnerable to cyber attacks and delayed measurements. These attacks occur randomly when data is transmitted from sensor nodes to remote filter nodes. To address this issue, a new cyber attack model is proposed that combines false data injection attacks and delayed measurement into a unified framework. The study also analyzes the stochastic stability of the proposed filter and establishes sufficient conditions to ensure that the filtering error remains bounded even in the presence of randomly occurring cyber attacks and delayed measurements. The proposed methodology is demonstrated and compared with other widely used approaches using simulated data to highlight its effectiveness and usefulness. INDEX TERMS Delay measurement, FDI, Gaussian filtering, nonlinear Bayesian filtering. I. INTRODUCTION F ILTERING is a recursive process for state estimation of dynamical systems from noisy measurements [1]. A popular nonlinear filtering method [2], Gaussian filter comprises of prediction and update steps, and is based on Bayesian approximation method. It approximates the unknown prior and posterior probability density functions (PDFs) as Gaussian, and characterizes them with mean and covariance [3]. The computation of mean and covariance involves intractable integrals, which are numerically approx- imated during the filtering process. Some popular Gaussian filters are extended Kalman filter (EKF) [4], unscented Kalman filter (UKF) [5], Cubature Kalman filter (CKF) [6], Gauss-Hermite filter (GHF) [7], and others. An alternate choice of Gaussian filtering is particle filtering, which is beyond the scope of this paper due to its huge computational complexity [8]. In this context, Gaussian filtering, a commonly used state estimation technique, often underperforms or fails to produce accurate results in the presence of irregularities [9]. VOLUME 4, 2016 1 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2023.3305288 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. For more information, see https://creativecommons.org/licenses/by-nc-nd/4