Random grouping based resilient beamforming

Abstract

This paper considers the resilient beamforming problem for a network of collaborative sensors in presence of faulty nodes. The main objective is to synchronize the signal phases at an intended destination and maximize the signal power. Towards this objective, we first propose a random grouping distributed beamforming (RG-DB) algorithm in the fault-free setup. For the RG-DB algorithm, each sensor classifies itself into one of two groups according to a random mechanism over a time horizon, during which a group of sensors control the phases of their transmissions, while the other group of sensors do not update their phases. This paper shows global convergence towards phase synchronization with probability one by applying the RG-DB algorithm. On the basis of the RG-DB algorithm, a random grouping based resilient distributed beamforming (RG-RDB) algorithm is then developed to solve the resilient beamforming problem in presence of faulty sensors. It is shown that the RG-RDB algorithm asymptotically makes the probability of selecting a faulty node for collective signal transmitting converge to zero, and thus solves the resilient beamforming problem by a subset of properly functioning sensors. Simulations are provided validating the algorithms developed in this paper.

Introduction

Beamforming technologies utilize an array of sensors to form a transmit/receive beam along a specific direction in order to enhance the communication quality such that long range communication or signal power enhancement can be achieved (Veen and Buckley, 1988, Zhou et al., 2018). Optimal collaborative beamforming with $m$ autonomous and independent sensors could lead to $m^2$-fold received signal power at a target destination or $m$-fold increased communication range in free space propagation. Therefore, collaborative beamforming stands out as a key technique for long range communication or signal enhancement, especially in distributed autonomous systems.

In most existing beamforming studies, all the sensors are assumed to function properly in collaborative beamforming in order to synchronize the signal phases and maximize the signal power at the receiver end. To achieve this goal, several techniques, such as low-bit feedback synchronization (Bucklew and Sethares, 2008, Hou et al., 2012, Hurnanen et al., 2011, Kumar et al., 2015, Mudumbai et al., 2005, Yang et al., 2016), master–slave open-loop synchronization (Mudumbai et al., 2007, Tian et al., 2016), round-trip open-loop (Brown III & Poor, 2008), two way source synchronization (Preuss & Brown III, 2010), and distributed robust beamforming (Jung and Lee, 2019, Ruan and De Lamare, 2019), are developed. Despite its advantages, collaborative beamforming inevitably suffers in practice from internal sensor faults or external cyber-attacks. In particular, due to the broadcast nature of radio propagation, potential adversaries can easily create unfavorable scenarios such as using false data injection attacks (FDIAs), which involve adding false signals on top of existing ones in the controller of some sensors or communication links, or hijacking attacks that completely replace the existing signals. As a result, the compromised sensor(s) become interferers to destabilize the system owing to imbalance in the iterative protocols of beamforming algorithms. Moreover, due to the energy, memory and computational power limitations, designing appropriate detection and prevention mechanism for distributed sensor networks is a challenging task as it may entail an excessively large amount of information exchange among the nodes. This motivates our study on light-weight resilient beamforming algorithms, with which a detection mechanism for faulty sensors can be simply merged with collaborative beamforming algorithms without complex computation.

As a matter of fact, resilient algorithms and resilient systems against node failures or cyber-attacks have been actively studied in recent years for distributed autonomous systems, such as in the framework of cyber–physical systems (Pajic et al., 2017), smart grids (Abbasi et al., 2017), multi-vehicle systems (Dong and Hu, 2016, Li et al., 2019), and wireless sensor networks (WSNs) (Zholbaryssov et al., 2019). In beamforming applications, distributed transmit beamforming is a form of cooperative communication, in which two or more autonomous sensors simultaneously transmit a common signal and control the phases of their transmissions so that the signals constructively combine at an intended destination. However, the existence of faulty sensors deteriorates signal coherence and even system stability, and thus the signal power in the desired direction may be largely reduced as illustrated in Fig. 1. Interestingly, there have been some works investigating this issue from the diagnosis perspective. Relying on minimum domain knowledge, an agnostic diagnosis (AD) method is considered in Miao et al. (2013) to detect faulty nodes. Fuchs et al. (2016) propose an antenna array diagnosis strategy based on a small number of far-field measurements by assuming that the power pattern of a reference array is known a priori. In a probabilistic framework, the Bayesian compressive sensing approach is discussed in Salucci et al. (2018). Moreover, heuristic algorithms such as genetic algorithms (Khan et al., 2016) and machine learning techniques (Chen et al., 2019, Wen et al., 2017) are also adopted to diagnose faulty sensors in a network.

In a very closely related field of multi-agent consensus and synchronization, the fault-agent tolerance problem is studied in a distributed manner. By supposing the maximum number $\widehat{m}$ of malicious agents in a network is known a priori, each agent removes the $\widehat{m}$ largest values as well as the $\widehat{m}$ smallest values from the received data in its coordination protocol. Such a scheme is called Mean-Subsequence-Reduced (MSR) algorithm, with which two metrics, network connectivity (Dibaji and Ishii, 2017, Sundaram and Hadjicostis, 2011) and network robustness (Zhang & Sundaram, 2012), are analyzed for resilient consensus problems. MSR-type algorithms are also adopted in various applications such as clock synchronization in WSNs (Kikuya et al., 2017) and spacecraft control (Wang et al., 2019) in the presence of non-cooperative robots.

Nevertheless, the existing resilient strategies either rely heavily on centralized and complex computation with the need of exchanging all other nodes’ information, or require to know the relative states as in the resilient consensus literature. However, distributed transmit beamforming aims to synchronize the signal phases at the receiver end, for which neither the absolute phase nor the relative phase offset is available. This paper aims to develop a light-weight and distributed resilient beamforming algorithm by using the signal power feedback information from the receiver. Towards this objective, we firstly propose a random grouping distributed beamforming (RG-DB) algorithm in the fault-free setup. For the RG-DB algorithm, each sensor classifies itself into one of two groups according to a random mechanism over a time horizon, during which a group of sensors adjust their transmitting signal phases by applying an identical control protocol, while the other group of sensors do not. Then we show that global convergence towards phase synchronization can be assured with probability one for the RG-DB algorithm. On the basis of the RG-DB algorithm, a random grouping based resilient distributed beamforming (RG-RDB) algorithm is then developed to solve the resilient beamforming problem in presence of faulty sensors that are due to either internal malfunctioning or external cyber-attacks. The RG-RDB algorithm merges a probability-based detection mechanism with the RG-DB algorithm interlacedly in the process of forming a beam. It is shown that the RG-RDB algorithm asymptotically makes the probability of selecting a subgroup having faulty nodes for collective signal transmitting converge to zero, and thus solves the resilient beamforming problem with a subset of properly functioning sensors. In other words, our algorithm is robust against node malfunctioning and cyber-attacks. Simulations are carried out to demonstrate the effectiveness of our algorithms.

Notation and Organization: The following notations are used throughout the paper. The set of nonnegative integers and the set of integers are denoted by $\mathbb{N}$ and $\mathbb{Z}$, respectively. $\iota$ represents the imaginary unit. $\mathbb{P}\left[\cdot \right]$ represents the probability measurement. The remainder of this paper is organized as follows: We formulate the distributed beamforming problem in Section 2. Sections 3 Random-grouping distributed beamforming, 4 Random-grouping resilient distributed beamforming present our main results in solving the beamforming problem and the resilient beamforming problem. Simulation and conclusions are provided in Sections 5 Algorithm evaluation, 6 Conclusions.