Secure consensus control for multi-agent systems under communication constraints via adaptive sliding mode technique
Abstract
The consensus tracking problem is investigated for a class of multi-agent systems (MASs) under communication constraints. In particular, as a result of the impact of amplitude attenuation and random interference, communication among followers may inevitably suffer from the fading phenomenon. Meanwhile, the controllers may also be subject to malicious deception attacks, which will disrupt the correct operation of the MASs. Thus, the agents can only update their states based on fading information exchanged with their neighbors and the false control input under attacks. The consensus tracking error variables are first designed via the fading signal received from neighbors. Then, an online estimation strategy is introduced to estimate the unknown attacks, based on which the adaptive sliding mode controller is designed to attenuate the effect of the time-varying attacks on MASs. Convergence analysis of the MASs under the designed control strategy is provided by using the Lyapunov stability theory and adaptive sliding mode control method. Finally, the effectiveness of the theoretical results is verified via numerical simulations.
Keywords
1. INTRODUCTION
As typical autonomous cyber-physical systems, multi-agent systems (MASs) provide an effective means to coordinate spatially distributed and networked agents, where agents interact together to optimize decisions and achieve system objectives. In recent decades, the development of cluster control has motivated more and more research on the consensus problem for MASs, such as multi-UAVs (Unmanned Aerial Vehicles) control [1-3], underwater cooperative operations [4], robot formation control [5-8], wireless sensors collaboration [9], microgrids control [10] and so on. As a key issue in the research of MASs, the consensus problem has received extensive attention in the past few decades. For example, Hu et al. proposed a new consensus protocol for complex networks composed of multiple subnetworks to ensure convergence[11]. Yao et al. considered the finite-time consensus problem of MASs based on the finite-time Lyapunov stability theory[12]. Rehman et al. investigated the consensus problem of leader-following MASs in both fixed undirected topology and fixed directed topology and proposed two distributed control protocols[13]. Liu et al. studied the positive consensus problem of MASs with directed communication topologies where all agents have identical continuous-time positive linear dynamics[14].
To handle the consensus problem, various control methods have been proposed including fuzzy control [15], robust
However, a key feature of the aforementioned works is that the information can be transmitted accurately among agents. In practical MASs, a satisfying communication environment cannot be guaranteed under wireless transmission networks. As a result of the impact of amplitude attenuation and random interference, the wireless link communication among agents will suffer from the fading phenomenon, resulting in the distortion of the data. This unfavorable factor motivated some interesting research on consensus tracking of wireless MASs subject to channel fading. Oral et al.[26] considered link outages between agents and obtained the probability expression for MASs reaching consensus. Gu et al. designed a distributed SMC law to deal with the impact of the information fading phenomenon in communication channels[27]. Ding et al. investigated the finite-time consensus control for MASs with channel fading via SMC technique. [28]
Another adverse phenomenon in the wireless transmission network is the inevitable malicious attacks, thereby rendering the secure control of MASs fundamental significance [29]. Considering the different mechanisms and effects on the MASs consensus problem, cyber-attacks can be divided into various types, for example, deception attacks [30], replay attacks [31] and denial-of-service (DoS) attacks [32]. Among them, deception attacks may lead to erroneous information feedback by tampering with the real packets via injecting false data. Cui et al. investigated the consensus tracking problem of MASs, which may be subject to deception attacks randomly. Recently, SMC strategy combined with adaptive mechanism has shown promising performance for constrained systems, for example, Chen et al. constructed an adaptive sliding mode control law to deal with the effects of adversarial cyber injection attacks[33]. It is of great practical significance to investigate the consensus problem for MASs against deception attacks[34]. Meanwhile, it is challenging to design a feasible SMC law under unknown and time-varying deception attacks.
Inspired by the above discussion and based on the expanded research of ref. [28], this paper will be concerned with the secure consensus control problem for multi-agent systems with malicious attacks and channel fading via the adaptive sliding mode technique, and the main contributions are highlighted as follows: (1) Both the position error and the velocity error are used to reflect the consistency of MASs, then the consensus tracking problem of MASs can be transformed into the stability problem of the tracking error system; (2) Coping with the effect of the fading channel between followers, the incomplete fading information received by the agent is introduced into the controller design; (3) An online estimation strategy is employed to estimate the unknown and time-varying attacks, based on which, an adaptive sliding mode controller is designed to attenuate the effect of the attacks on MASs; and (4) The distributed adaptive SMC strategy is designed to ensure the mean square consistency of MASs, despite the communication constraints.
Notation:
2. PROBLEM FORMULATION
2.1. Graph theory
Graph theory is an important tool to study MASs, which is a graph composed of several nodes and edges connecting the node. Each agent can be represented as a node, and the information interaction between agents can be denoted as an edge in graph theory. A directed weighted graph is represented by
with
Lemma 1[35] The matrix
Definition 1 Consider a multi-agent system with
2.2. System model
Consider a second-order MASs consisting of a leader labeled as node
where
The leader's dynamic is of the following form:
with
Define the
with
The tracking errors can be rewritten in the compact form:
with
From the above definition, one can obtain the tracking error system as:
Now, the consensus tracking problem of MASs (3)-(4) converts to the stabilization problem of the tracking error system (7). The objective of this work is to achieve leader-follower consistency.
2.3. Fading channel
As stated in the Introduction, the transmission between followers may be inevitably suffered from the channel fading phenomenon. In this work, the network channel is considered as a continuous one with time-varying channel gain, the transmitted data will be modeled as the actually received information with random attenuation. Hence, introduce the following memoryless multiplicative fading model:
where
Assuming that fading occurs only in the channel between followers, the special case of channel fading from the leader to the followers is not considered in this work. Hence, based on the fading information (8), the tracking errors (5) are rewritten as:
It can be seen that the tracking errors (9) involve the expectation of the random variable
Define
where
Remark 1. There are two special cases considered in the channel fading model (8): when
2.4. Deception attacks
Among various cyber-attacks, the deception attack on controllers is a common form and usually satisfies the following assumptions: the hackers can steal the state information or measurement output of the agents to generate false data, which can then be injected into the controller. As shown in Figure 1, the hackers can attack the controller of agent
The compact form of expression (11) can be written as:
where
Remark 2. The deception attacks considered in this work focus on the controller, that is,
3. MAIN RESULTS
3.1. Adaptive SMC law
To cope with the impact of the deception attacks, the information about the attack is usually utilized to design the controller. For example, when the upper bounds
Design the sliding function as follows:
with
From (7), we can obtain the derivative of the sliding function:
Under these constraints considered in this work, the
The compact form of expression (16) as:
Then, construct the sliding mode controller as follow:
where the robust term
with
where
with
where the continuous function
with
According to Imbedded Convex Sets Assumption [37], we obtain:
and
and these two conditions will be used in the following derivation.
Remark 3. In some existing reliable control methods, the known bounds of attacks are usually utilized, which may inevitably yield larger conservativeness. To overcome this shortcoming, the online estimation mechanism for unknown attacks/faults was proposed in some related works [33, 38]. Inspired by these works, the online estimation mechanism of the attack is integrated with the SMC technique in this work.
3.2. Consistence and Reachability
Theorem 1. Consider the MASs (3)-(4) with channel fading (8) and deception attacks (12), under the proposed SMC law (19)-(20), the reachability of the sliding surface
Proof. Choose the Lyapunov function as follows:
where
Then, by the expressions (15) and (21), the derivative of
Taking mathematical expectation to the above expression (27), one has:
It can be easily verified from expressions (5) and (9) that
By the conditions
Then, it follows from (25) that:
Hence, the reachability of the sliding surface
Theorem 2. Considering the MASs (3)-(4) subject to deception attacks (12) and channel fading model (8), the consensus tracking for MASs (3)-(4) will be achieved under the proposed sliding surface (14) and the SMC law (19)-(20).
Proof. Select the Lyapunov function:
Its derivative is given as:
When the sliding surface
Combining the results of Theorem 1, the consensus tracking of MASs (3)-(4) can be ensured under the proposed sliding surface (14) and the SMC law (19)-(20).
4. SIMULATION
Consider the second-order MASs with one leader and 4 followers, where the communication topology between agents is shown in Figure 2. The blue arrows indicate that the followers receive the complete information from the leader, while the red arrows indicate that the information interaction between followers is over fading channel. Thereby, follower
Then, according to the leader and followers' topology, we can get the adjacency matrix
In this simulation, the initial state of the leader's position, speed, and control input are set as
Simulation results are shown in Figures 3-7. Among them, Figure 3 shows the tracking trajectories of MASs under the robust control term
Figure 3. The trajectories of the MASs under the robust control term
Figure 5. The position error
Figure 6. The velocity error
Figure 7. Sliding variables
Remark 4. As shown in Figures 5-7, agents 1 and 2 have better consensus tracking performance with smaller amplitude oscillating and smoother curves, that is, because they can receive accurate information from the leader. In contrast, agents 3 and 4 perform worse because they cannot obtain information from the leader, but only from neighbor agents over fading channel (as shown in Figure 2). Even so, the proposed adaptive SMC scheme can still guarantee consensus tracking of all followers, as shown in Figure 4.
5. CONCLUSION
This work considered the consensus control problem of MASs under deception attacks and fading channels. Due to the fading channels, the position and velocity errors cannot be calculated accurately. To solve this problem, the consensus tracking error variables have been designed based on the fading data received from neighbor agents. Meanwhile, the distributed adaptive SMC strategy via fading information has been proposed to deal with the time-varying and unknown deception attacks injected by the hacker. Utilizing the proposed scheme, consensus tracking can be achieved. Only malicious attacks and channel fading have been considered in this work. In practical applications, there may coexist multiple constraints, such as actuator/sensor faults, packet dropout, random noise[39, 40], etc. Under these constraints, how to design a feasible consensus control method is worthy to research in future work.
DECLARATIONS
Acknowledgments
Special thanks to the School of Electronic and Electrical Engineering, Shanghai University of Engineering Science (China) for providing technical support for this research.
Authors' contributions
Methodology, software, validation, data curation, visual conceptualization, ritingization, writing- original draft: Ding M
Conceptualization, riting-reviewing and editing, investigation: Chen B
Availability of data and materials
Not applicable.
Financial support and sponsorship
This work is supported in part by the NNSF of China (62173222, 61803255), Shanghai Science and Technology Innovation Action Plan (22S31903700, 21S31904200), and the National Key R & D Program of China (Grant No. 2020AAA0109301).
Conflicts of interest
Both authors declared that there are no conflicts of interest.
Ethical approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Copyright
© The Author(s) 2023.
REFERENCES
1. Liao W, Wei XH, Lai JZ, Sun H. Formation control for multi-UAVs systems based on Kullback-Leibler divergence. IEEE Trans Inst Meas Control 2020;42:598-603.
2. Dong X, Zhou Y, Ren Z, et al. Time-varying formation tracking for second-order multi-agent systems subjected to switching topologies with application to quadrotor formation flying. IEEE Trans Ind Electron 2016;64:5014-24.
3. Belkacem K, Munawar K, Muhammad SS. Distributed cooperative control of autonomous multi-agent UAV systems using smooth control. J Syst Eng Electron 2021;31:1297-307.
4. Ali S, Muhammad NM. Distributed observer for a team of autonomous underwater vehicles utilizing a beacon unit on the surface. In: 2017 IEEE 7th International Conference on Underwater System Technology: Theory and Applications 2017.
5. Lv YK, Zhang H, Wang ZP, Yan HC. Distributed localization for dynamic multiagent systems with randomly varying trajectory lengths. IEEE Trans Ind Electron 2022;69:9298-308.
6. Li X, Dong X, Li Q, Ren Z. Event-triggered time-varying formation control for general linear multi-agent systems. J Frankl Inst 2019;356:10179-95.
7. Chai XF, Wang Q, Diao Q, Yu Y, Sun CY. Sampled-data-based dynamic event-triggered formation control for nonlinear multi-agent systems. IEEE Trans Inst Meas Control 2022;14:2719-28.
8. Li Y, Jiao XY, Sun BQ, Yang JY. Multi-welfare-robot cooperation framework for multi-task assignment in healthcare facilities based on multi-agent system. In: 2021 IEEE International Conference on Intelligence and Safety for Robotics 2021.
9. Das R, Dwivedi M. Multi agent dynamic weight based cluster trust estimation for hierarchical wireless sensor networks. Peer-to-Peer Netw 2022;15:1505-20.
10. Abianeh AJ, Wan YH, Ferdowsi F, Mijatovic N, Dragicevic T. Vulnerability identification and remediation of FDI attacks in islanded DC microgrids using multiagent reinforcement learning. IEEE Trans Ind Electron 2022;37:6359-70.
11. Hu HX, Wen GH, Yu WW, Huang TW, Cao JD. Distributed stabilization of multiple heterogeneous agents in the strong-weak competition network: a switched system approach. IEEE Trans Cybern 2021;51:5328-41.
12. Yao DJ, Dou CX, Zhao N, Zhnag TJ. Finite-time consensus control for a class of multi-agent systems with dead-zone input. J Frankl Inst 2021;358:3512-29.
13. Rehman AU, Rehan M, Iqbal N, Waris MZ. Leaderless adaptive output feedback consensus approach for one-sided Lipschitz multi-agents. J Frankl Inst 2020;357:8800-22.
14. Liu JJR, Yang N, Kwok KW, Lam J. Positive consensus of directed multi-agent systems. IEEE Trans Automat Contr 2022;67:3641-6.
15. Chen CLP, Wen GX, Liu YJ, Liu Z. Observer-based adaptive backstepping consensus tracking control for high-order nonlinear semi-strict-feedback multiagent systems. IEEE Trans Cybern 2016;46:1591-601.
16. Sheng L, Wang Z, Zou L. Output-feedback
17. Yu D, Ji XY. Finite-time containment control of perturbed multi-agent systems based on sliding-mode control. Int J Syst Sci 2018;49:299-311.
18. Zou Y, Sun X, Li S, Liu Y. Event-triggered distributed predictive control for asynchronous coordination of multi-agent systems. Automatica 2019;99:92-8.
19. Yu WW, Chen GR, Cao M. Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems. Automatica 2010;46:1089-95.
20. Tao T, Roy S, Baldi S. Adaptive synchronization of uncertain complex networks under state-dependent a priori Interconnections". In: 2021 60th IEEE Conference on Decision and Control (CDC), Austin, TX, USA, 2021: 1777-82.
21. Tao T, Roy S, Baldi S. Adaptive single-stage control for uncertain nonholonomic Euler-Lagrange systems. In: 2022 IEEE 61st Conference on Decision and Control (CDC), Cancun, Mexico. 2022: 2708-13.
22. Zuo ZY. Nonsingular fixed-time consensus tracking for second-order multi-agent networks. Automatica 2015;54:305-9.
23. Wang J, Zhang XR, Zhou JL. Chen YQ, Distribution consensus of nonlinear stochastic multi-agent systems based on sliding-mode control with probability density function compensation. J Frankl Inst 2020;357:9308-29.
24. Cong YR, Feng ZG, Song HW, Wang SM. Containment control of singular heterogeneous multi-agent systems. J Frankl Inst 2018;55:4629-43.
25. Rahmani R, Toshani H, Mobayen S. Consensus tracking of multi-agent systems using constrained neural-optimiser-based sliding mode control. Int J Syst Sci 2020;51:2653-74.
26. Oral E, Schmeink A, Dartmann G, Ascheid G, Pusane AE. Consensus analysis of wireless multi-agent systems over fading channels. IEEE Wireless Commun Lett 2021;10:1528-31.
27. Gu XW, Jia TG, Niu YG. Consensus tracking for multi-agent systems subject to channel fading: a sliding mode control method. Int J Syst Sci 2020;51:2703-11.
28. Ding M, Chen B, Hu ZX and Zhang Y. Finite-time consensus control for multi-agent systems with channel fading via sliding mode technique. In: 2022 34th Chinese Control and Decision Conference (CCDC), Hefei, China. 2022: 3706-11.
29. He WL, Xu WY, Ge XH, Han QL, Du WL, Qian F. Secure control of multiagent systems against malicious attacks: a brief survey. IEEE Trans Industr Inform 2022;18:3595-608.
30. Zhao L, Yang GH. Cooperative adaptive fault-tolerant control for multi-agent systems with deception attacks. J Frankl Inst 2020;357:3419-33.
31. Tahoun AH, Arafa M. Cooperative control for cyber-physical multi-agent networked control systems with unknown false data-injection and replay cyber-attacks. ISA Trans 2021;110:1-14.
32. Shang Y, Liu CL, Cao KC. Event-triggered consensus control of second-order nonlinear multi-agent systems under denial-of-service attacks. IEEE Trans Inst Meas Control 2021;10:2272-81.
33. Chen B, Niu YG, Zou YY. Security control for Markov jump system with adversarial attacks and unknown transition rates via adaptive sliding mode technique. J Frankl Inst 2019;356:3333-52.
34. Cui Y, Liu YR, Zhang WB, Alsaadi FE. Sampled-based consensus for nonlinear multiagent systems with deception attacks: the decoupled method. IEEE Trans Syst Man Cybern Syst 2018;51:561-73.
35. Li W, Niu YG, Cao ZR. Event-triggered sliding mode control for multi-agent systems subject to channel fading. Int J Syst Sci 2021;53:1233-44.
36. Chen B, Niu YG. Dynamic event-triggered sliding mode security control for Markovian jump systems: Learning-based iteration optimization method. Int J Robust Nonlinear Control 2021;32:2500-17.
37. Pomet JB, Praly L. Adaptive nonlinear regulation: estimation from the Lyapunov equation. IEEE Trans. Automat. Contr 1992;37:729-40.
38. Zou Z, Ho DWC, Wang Y. Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation. Automatica 2010;46:569-76.
39. Zhang QC, Zhou YY. Recent advances in non-gaussian stochastic systems control theory and its applications. IJNDI 2022;1:111-9.
Cite This Article

How to Cite
Ding, M.; Chen, B. Secure consensus control for multi-agent systems under communication constraints via adaptive sliding mode technique. Complex Eng. Syst. 2023, 3, 7. http://dx.doi.org/10.20517/ces.2023.06
Download Citation
Export Citation File:
Type of Import
Tips on Downloading Citation
Citation Manager File Format
Type of Import
Direct Import: When the Direct Import option is selected (the default state), a dialogue box will give you the option to Save or Open the downloaded citation data. Choosing Open will either launch your citation manager or give you a choice of applications with which to use the metadata. The Save option saves the file locally for later use.
Indirect Import: When the Indirect Import option is selected, the metadata is displayed and may be copied and pasted as needed.
Comments
Comments must be written in English. Spam, offensive content, impersonation, and private information will not be permitted. If any comment is reported and identified as inappropriate content by OAE staff, the comment will be removed without notice. If you have any queries or need any help, please contact us at support@oaepublish.com.