Robust filtering algorithm against hybrid-attacks and randomly occurring nonlinearities: Application to a quadrotor UAV

Abstract

This paper addresses the filtering problem for a class of discrete-time cyber-physical system (CPS) in simultaneous presence of physical attack, denial of service attacks (DoS), false data injection (FDI) attacks and randomly occurring nonlinearities (RONs). The malicious attacks can be simultaneously launched at the physical and cyber space to assault/disturb the system's state or block the measurement transmission channels. Firstly, the physical attack is considered as an injected unknown information, which can modify the system's state; then, a unified framework is proposed under which the phenomena of DoS attacks and FDI attacks are depicted by two independent Bernoulli distributed random variables, respectively. Furthermore, the available random probability information of the hybrid attacks and RONs are taken into account in the developed state estimation algorithm. Subsequently, by employing the stochastic analysis and matrix techniques, the exponential boundedness of the state estimation error is discussed from the theoretical aspects. As a typical CPS, a quadrotor unmanned aerial vehicle(UAV) system is provided to demonstrate the effectiveness and practicability of the proposed state estimation method.

Introduction

In recent years, as the deep combination of physical space and cyber space, cyber-physical systems (CPS) are well known for their wide applications in many areas, such as aerospace engineering, transportation, smart grids and medical health care [1], [2], [3]. Different from classical control systems, the physical components of CPS distributed in different locations, such as sensors, controllers, and actuators, are connected by wired/wireless transmission to complete the monitoring and perception of the perceived objects in the network coverage area. For example, as a typical CPS, unmanned aerial vehicle (UAV) relies on the communication between ground station and flight control system to achieve autonomous flight.

However, the realization of CPS relies heavily on sensor networks and distributed control networks. Due to the characteristics of network communication, there are many challenging problems in the CPS, for example,

• Limited communication constraints problem: due to the limited bandwidth of the network and sharing network resources, each subsystem can only transmit information after it has obtained the permission. Therefore, there are inevitable queuing, resource competition and network congestion. It is necessary to analyze the delay, packet loss, fault and quantization error induced by communication network, see for instance [4], [5], [6].

• Secure state estimation problem: for decades, filtering/state estimation (SE) has been proven to be not only the foundation of real-time sensing and dynamic monitoring, but also the premise of realizing closed-loop control, which have received extensive of research attention [7], [8], [9], [10], [11], [12], [13], [14]. Due to the openness of the communication system network, the system itself is vulnerable to hackers' illegal intrusion, destruction, and modification of data transmitted in the network. In the key infrastructures such as hydropower station, nuclear power station and thermal power station, any form of damages in the physical space or the cyber space may bring unpredictable loss or disaster to the environment, economy and even our lives. Therefore, malicious attacks have become one of the main threats faced by CPS, which attracts increasing interests to the research and applications on the security of CPS [15], [16], [17], [18], [19].

At present, the cyber-attacks that are concerned by most researchers mainly include denial of service (DoS) attack and false data injection (FDI) attack [15]. In fact, the attacks occur in a random manner because the protection devices may prevent and intercept the adversary attacks. The impact of DoS attacks on network communication is usually modeled as network delay or packet loss with certain statistical characteristics, such as Bernoulli process or Markov jump process obeying random independent and identically distribution. In [5], a state estimation method based on event-triggered sampling was proposed for CPS with sensor saturation and DoS attack. The problem of resilient filtering design for CPS under the consideration of DoS attack was developed in [16]. By considering the effects of resource constraints, an event-triggered distributed state estimation method was constructed against DoS attacks in [20]. The authors in [21] addressed the problem of resilient and secure remote monitoring against FDI attacks. A new resilient filter was investigated to ensure the security of fuzzy-model-based CPS with cyber-attacks in [22]. By taking the randomly occurring deception attacks and bounded disturbances into account, the distributed $H_{\infty }$ filter was designed in [23].

It is worth mentioning that most of the above papers focus on only a particular type of attack in the cyber space. However, in practice, it is more common that the attackers launch cooperative or mixed cyber-attacks instead of a single attack. For example, considering both the DoS attacks and replay attacks, the problem of event-based security control for state-dependent uncertain systems are studied in [24]. The authors in [25] investigated the problem of finite-time $H_{\infty }$ filtering subject to DoS attacks, FDI attacks and replay attacks in the data transmission channel from sensor to filter. Moreover, as pointed out in the work of [26], [27], according to a smart switched attack strategy, the malicious attacks can be launched in both the physical space and cyber space by governing the physical attacks/faults and cyber attacks. In fact, it is of practical significance to address the randomly physical-cyber hybrid-attacks in the hope of improving the security of CPS. Nevertheless, the research of secure state estimation of CPS with physical-cyber hybrid-attacks has not been fully studied yet, which motivates our current investigation.

On another research frontier, from the viewpoint of control engineering, the system is affected by additive nonlinear disturbance due to the change of environmental conditions in many engineering systems. In other words, nonlinear disturbances may occur randomly in a certain type of probability, which contribute significantly to the system complexities and result in extra challenges when coping with the filtering design problem. Recently, recursive filtering method was developed for discrete-time systems with randomly occurring nonlinearities (RONs) and missing measurements as in [28]. In [29], an event-based state estimator was proposed for Markovian jump systems with RONs. The event-triggered fault-tolerant control for networked control systems with RONs and DoS attacks was studied in [30].

Inspired by the above discussions, the major objective of this article is to design a robust state estimator for a class of discrete-time CPS subject to hybrid-attacks and RONs. Compared with the existing results, we need to handle the following challenges: i) how to build the model of the system under both the physical space attacks and mixed cyber space attacks; ii) how to comprehensively reveal the potential effects from the attacks and RONs onto the filtering performance and explore the involved relationship; iii) how to clearly analysis the performance of filtering method for the addressed CPS with certain complexities. Accordingly, the main contributions of this article could be summarized as: 1) the considered system model is a more comprehensive and realistic CPS, where the physical space and cyber space suffers from the simultaneous sophisticated attacks and randomly occurring nonlinearities; 2) by employing the independent Bernoulli distributed sequences to characterize the randomly phenomena, the corresponding influences onto the filtering performance are discussed; 3) the exponential boundedness of the filtering error is discussed from the theoretical aspects; 4) application on a quadrotor UAV system is demonstrate the efficacy and applicability of the implemented robust filtering scheme.

The reminder of this article is organized as follows. In Section 2, the state estimation problem for a class of discrete-time CPS with hybrid-attacks and RONs are formulated. In Section 3, by employing the stochastic analysis and matrix techniques, the upper bound of the estimation error covariance and the appropriate filter gain are derived. Moreover, sufficient conditions are given to ensure the boundedness of the estimation error in the mean square. In Section 4, as a typical CPS, a quadrotor UAV model is chosen to demonstrate the effectiveness of proposed state estimation algorithm. Finally, some conclusions are given in Section 5.

Notations. The notations are standard mathematical notations in this paper. ${\mathbb{R}}^n$ represents the n-dimensional Euclidean space. ${\mathbb{R}}^{n\times m}$ stands for the set of all $n\times m$ matrices. $E\left\{x\right\}$ means the expectation of random variable x. $tr(\cdot )$ represents the trace of a matrix. $P^T$ and $P^{-1}$ represent the transpose and the inverse of matrix P. $\Vert \cdot \Vert$ stands for the Euclidean norm of a vector and its induced-norm of a matrix.