Brief paperAdaptive consensus for heterogeneous multi-agent systems under sensor and actuator attacks☆
Introduction
Over the past several decades, distributed control of multi-agent systems has attracted much attention due to its wide applications in various fields, such as robotics (Bullo et al., 2009), smart grid (Tan et al., 2014), sensor networks (Ogren et al., 2004) and so on. Most existing results on distributed consensus problem (Hu et al., 2016, Li, Chen et al., 2016, Li et al., 2013, Li, Su et al., 2016, Meng et al., 2020, Olfati-Saber and Murray, 2004) assume guaranteed availability of healthy local sensors, actuators of every agent and the intact communication topologies. However, as multi-agent systems are a significant subclass of cyber–physical systems that involve communications and collaborations between connected agents, they are prone to cyber–physical attacks. For example, the GPS sensors in a multi-vehicle system can be attacked and make the sensory data that the vehicles receive to be corrupted. The misleading data or false actuator input may severely affect the performance of the system and prohibit the accomplishment of system-level objectives. Therefore, design of resilient and secure architectures is of paramount importance for achieving desired coordinated goal of distributed networks under attacks.
Different methods have been proposed for detecting and mitigating deception attacks in multi-agent systems (Boem et al., 2017, Forti et al., 2018, Rahimian and Preciado, 2015, Sundaram and Hadjicostis, 2011, Teixeira et al., 2010), in which the sensor and/or actuator attacks were considered. There are some approaches to design mitigation techniques for addressing malicious attacks. The first one is to establish a monitor based on the discrepancy among agents and their neighbors to detect and identify attacks on neighbors, and then isolate the comprised agents (Pasqualetti et al., 2012, Sundaram and Hadjicostis, 2011). Although by this means, various attacks including sensor and actuator attacks as well as attacks on communication links can be counteracted, additional assumptions on the graph connectivity and the upper bound of the fraction of adversary agents are typically needed. Also the network connectivity may be harmed by rejecting the neighbor’s information. The second approach (Arabi et al., 2017, Hota and Sundaram, 2018, Jin and Haddad, 2019, Modares et al., 2018, Mustafa and Modares, 2019, Zeng and Chow, 2014, Zhu and Martínez, 2013) is to design resilient distributed control protocols to mitigate the effects of attacks instead of removing compromised agents. In Zeng and Chow (2014), a reputation-based resilient control algorithm was proposed for leader-following problem of multi-agent systems in the presence of misbehaving agents. Game-theory based resilient control architecture was designed in Hota and Sundaram (2018) to mitigate the effects of adversary information. Adaptive resilient architectures were applied to ensure that the intruded multi-agent systems under attacks achieve a cooperative goal with uniform ultimate boundedness in Arabi et al. (2017) and Jin and Haddad (2019). A novel resilient distributed algorithm was presented by adopting a receding-horizon control methodology for mitigation of replay attacks (Zhu & Martínez, 2013). In Mustafa and Modares (2019), a rigorous analysis of the effects of cyber attacks on discrete-time multi-agent systems was conducted and accordingly a mitigating approach for sensor and actuator attacks was proposed. Another way to protect nodes from attackers who try to steal or even alter exchanged information by hacking into the nodes or the communication channels is the digital signature based approach (Ruan et al., 2019).
However, the aforementioned references have all concentrated on homogeneous multi-agent systems under cyber attacks. In practice, it is not common that all agents in a multi-agent system have exactly the same dynamics. Usually individual agents in a multi-agent system have different dynamics and even different state dimensions. It is thus desirable to study the leader-following consensus problem for such multi-agent systems termed heterogeneous multi-agent systems in the presence of both sensor and actuator attacks.
In this paper, we apply and extend techniques from adaptive control theory to mitigate the effects of sensor and actuator attacks on leader-following consensus of heterogenous multi-agent systems. A novel adaptive cooperative controller is presented to foil the time-varying sensor and actuator attacks. The contributions of this paper are mainly as follows. (i) The proposed adaptive controller guarantees the leader-following consensus with cooperative uniform ultimate boundedness (UUB). This cooperative bound can be adjusted by appropriately choosing some free parameters in the designed adaptive controller, and particularly, the bound can be sufficiently small when there are only constant sensor attacks. (ii) Compared with (Pasqualetti et al., 2012, Sundaram and Hadjicostis, 2011), in which resilient function calculation and consensus were discussed under the constraints on the number of the malicious agents and the communication topologies, our results however do not need these assumptions. Instead, the only constraint we need is that there exist directed paths from the leader node to all the follower nodes in the communication topology (see Assumption 2). (iii) We present some results ensuring that the outputs of all the followers approach the output of the leader with UUB in heterogeneous multi-agent systems under both sensor and actuator attacks while only sensor attacks or actuator attacks were considered in most literature (e.g. Arabi et al. (2017) and Chen et al. (2019)). Modares et al. (2018) and Mustafa and Modares (2019) present a unified approach to study resilient consensus of homogeneous/heterogeneous multi-agent systems under both sensor and actuator attacks, while however only the intact agents are ensured to achieve consensus.
The rest of this paper is organized as follows. Section 2 reviews some preliminaries and formulates the studied problem. The main results of this paper are presented in Section 3. Section 4 gives an illustrative example to demonstrate the effectiveness of the obtained results, followed by a brief conclusion in Section 5.
Notations. and denote the sets of real numbers and real matrices, respectively. () represents an dimensional vector with all of its elements being 1 (0). is the identity matrix of dimension . For real symmetric matrices and , means that is positive (positive semi-, negative, negative semi-) definite. Denote by the Euclidean/induced norm for vectors/matrices. represents the eigenvalue of matrix . denotes the Kronecker product of matrices and . represents a diagonal matrix with , , on its diagonal.
Section snippets
Preliminaries and problem formulation
In this section, some fundamentals of algebraic graph theory and the studied problem are introduced.
In a directed graph , , and represent the vertex set, the directed edge set and the weighted adjacency matrix of , respectively. The weights are defined as , if and otherwise. A node with one edge incoming to node is called a neighbor of node . Denote by the set of the neighbors of node , then .
Main results
In this section, a novel distributed adaptive resilient control protocol is designed to mitigate the effects of sensor and actuator attacks, and achieve leader-following consensus with cooperative UUB.
Assumption 1 The sensor and actuator attacks, and , as well as their derivatives, and are bounded. Besides, the bounds are unknown.
To achieve leader-following consensus uniformly ultimately bounded for multi-agent system (1) with the leader (2) under sensor and actuator attacks, a
Example
In this section, we give a numerical example to illustrate the obtained results. Consider a heterogeneous multi-agent system with the dynamics matrices as (Wieland et al., 2011) where and for . The leader’s system matrices are assumed to be Consider and is chosen as , , , , , . The communication topology among the
Conclusion
In this paper, we addressed the adaptive distributed leader-following consensus for a kind of heterogeneous multi-agent systems under sensor and actuator attacks. Novel resilient distributed controllers were given by extending the adaptive methods and can be designed to ensure the leader following consensus with cooperative UUB for the studied multi-agent systems. Future research interest is to study resilient consensus of heterogeneous multi-agent systems under actuator dynamics and attacks
Min Meng received the B.S. and Ph.D. degrees from Shandong University, China, in 2010 and 2015, respectively. She had a position as a Research Associate at The University of Hong Kong, Hong Kong, China, from April to October in 2014, from July to September in 2016, and from January to March in 2017. From July 2015 to June 2016, she also worked at City University of Hong Kong, Hong Kong, China, as a Research Associate. Since July 2017, she has been a Research Fellow in the School of Electrical
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Min Meng received the B.S. and Ph.D. degrees from Shandong University, China, in 2010 and 2015, respectively. She had a position as a Research Associate at The University of Hong Kong, Hong Kong, China, from April to October in 2014, from July to September in 2016, and from January to March in 2017. From July 2015 to June 2016, she also worked at City University of Hong Kong, Hong Kong, China, as a Research Associate. Since July 2017, she has been a Research Fellow in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Her research interests include Boolean networks, multi-agent systems, distributed games and optimization, distributed secure control and estimation.
Gaoxi Xiao received the Ph.D. degree in computing from the Hong Kong Polytechnic University, Hong Kong, in 1998. He was a Postdoctoral Research Fellow with Polytechnic University, Brooklyn, New York, NY, USA, in 1999 and a Visiting Scientist at the University of Texas at Dallas during 1999–2001. He joined the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, in 2001, where he is currently an Associate Professor. His research interests include complex systems and networks, system resilience and system risk control, cyber–physical systems and smart grid, and cybersecurity. He serves as an Associate Editor for IEEE Transactions on Network Science and Engineering and an Academic Editor for PLOS ONE.
Beibei Li is currently an associate professor at the College of Cybersecurity, Sichuan University, P.R. China. He received his B.E. degree (awarded Outstanding Graduate) in communication engineering from Beijing University of Posts and Telecommunications, P.R. China, in 2014 and his Ph.D. degree (awarded Full Research Scholarship) from School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, in 2019. He was invited as a visiting researcher at University of New Brunswick, Canada, from March to August 2018 and at Zhejiang University, P.R. China, from February to April 2019. His research interests span several areas in security and privacy issues on cyber–physical systems with a focus on intrusion detection techniques, applied cryptography, and artificial intelligence. He is serving or has served as a Publication Co-Chair or a TPC member for several international conferences, including IEEE ICC, IEEE GLOBECOM, IEEE ICNC, PST, WCSP, EPEC, and ML4CS, etc.
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The work was partially supported by Ministry of Education, Singapore, under contract of MOE2016-T2-1-119, the China Postdoctoral Science Foundation (No. 2019TQ0217); the Provincial Key Research and Development Program of Sichuan, PR China (No. 20ZDYF3145). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Vijay Gupta under the direction of Editor Christos G. Cassandras.