Adaptive cooperative secure control of networked multiple unmanned systems under FDI attacks

With the expanding applications of multiple unmanned systems in various ﬁelds, more and more research attention has been paid to their security. The aim is to enhance the anti-interference ability, ensure their reliability and stability, and better serve human society. This article conducts adaptive cooperative secure tracking consensus of networked multiple unmanned systems subjected to false data injection attacks. From a practical perspective, each unmanned system is modeled using high-order unknown nonlinear discrete-time systems. To reduce the communication bandwidth between agents, a quantizer-based codec mechanism is constructed. This quantizer uses a uniform logarithmic quantizer, combining the advantages of both quantizers. Because the transmission information attached to the false data can aﬀect the accuracy of the decoder, a new adaptive law is added to the decoder to overcome this diﬃculty. A distributed controller is devised in the backstepping framework. Rigorous mathematical analysis shows that our proposed control algorithms ensure that all signals of the resultant systems remain bounded. Finally, simulation examples reveal the practical utility of the theoretical analysis.


Introduction
Owing to the rapid development of technology in artificial intelligence, countries around the world have developed unmanned systems.These systems do not require the physical presence of human operators on board and have increasingly autonomous functions, overcoming the limitations of human operators.These applications have gradually permeated various sectors, including national industrial production, social life, defense technology, and others such as self-driving vehicles and unmanned aerial vehicle reconnaissance [1,2].Since multiple systems offer more flexibility and robustness than single systems, extensive research has been conducted on multiple unmanned systems (MUSs).In particular, the distributed cooperative control of MUSs has garnered significant attention from scholars [3][4][5][6][7][8].In future control designs, MUSs could be equipped with tiny built-in microprocessors that gather data from adjacent agents and control actions based on pre-established rules, given the ubiquitous nature of nonlinear phenomena in the real world.Consequently, digital platforms are employed for the controllers, and the control policy is modified only at specific time intervals.Thus, the study of nonlinear discrete-time MUSs holds great importance, and promising results have been reported so far [9][10][11][12].With the advancement of networks, conserving network transmission bandwidth in MUSs has emerged as a prevalent trend.
Quantization is a straightforward yet effective method to alleviate the transmission load in communication channels within MUSs, yielding commendable outcomes [13][14][15].Concerning control strategies employing quantized data, Zhang et al. [16] addressed quantized control for linear systems with state observers.Similarly, Liu et al. [17,18] explored input-quantized state feedback control and state-quantized output feedback control for nonlinear systems, respectively.To enhance data transmission's security and efficiency in resource-constrained communication networks, the encoding-decoding technology based on quantization has captured scholars' attention.In [19], two encoding-decoding schemes were proposed, enabling the attainment of average consensus in linear systems.Notably, recent extensions [20][21][22] have surfaced; for detailed information, refer to [20], which introduced a control protocol utilizing symmetric compensation techniques alongside a distributed codec scheme relying on quantization to achieve consensus in discrete-time systems within jointly connected networks.Dong [21] pioneered consensus among multiple nonlinear systems with limited bandwidth, introducing a novel approach to designing a distributed controller for both inner and outer loops.In [22], a quantized data-driven method based on a codec mechanism was proposed for nonlinear non-affine systems.Moreover, Ren et al. [23] and Zhu et al. [24] utilized data-driven control methods and an encoding-decoding uniform quantization mechanism, effectively compressing communication data volume between agents.Thus, the development of a practical control scheme for MUSs to alleviate transmission load in communication channels is of paramount importance.
As networks rapidly expand, network security issues have taken on greater prominence.In networked environments, MUSs can achieve efficient cooperation while simultaneously facing increased security risks.This has spurred researchers to delve into the security concerns of MUSs.The communication networks of MUSs are susceptible to denial-of-service and false data injection (FDI) attacks.FDI attacks occur when an adversary intercepts communication between agents and deliberately injects inaccurate packets [25].In [26], an optimal FDI attack scheme based on historical and current residuals was proposed to maximally undermine performance from the attacker's perspective.Li et al. [27] examined security consensus in a multiple input multiple output system under FDI attacks, employing an event-triggering proposal.Similarly, Meng et al. [28] presented the adaptive resilient control problem for linear multiple systems subjected to sensor and actuator attacks.Zhang et al. [29] introduced a resilient observer-based control strategy triggered by events to address secure consensus in multiple-agent scenarios under FDI attacks following Bernoulli processes.In addressing unknown FDI attacks, Wang et al. [30] proposed an observerbased fully asynchronous event-triggered controller to achieve bipartite consensus among multiple agents.In the context of output consensus driven under FDI attacks, Huo et al. [31] developed state and outputfeedback cooperative control strategies.Hu et al. [32] tackled stochastic analysis and controller design problems for networked systems, considering false data injection attacks in the sensor-controller and controller-actuator channels.Handling quasi-consensus in stochastic nonlinear time-varying multi-agent systems with multi-modal FDI models was addressed in [33].However, due to exposure, the communication networks among agents are more vulnerable to attacks.To address this, Tahoun and Arafa [34] designed a distributed adaptive secure control scheme for multi-agent networked systems under unknown FDI and replay attacks.Consequently, ensuring the reliability and security of communication networks necessitates the exploration of secure control techniques for networked nonlinear MUSs vulnerable to FDI attacks.
As a consequence, secure cooperative control of MUSs under FDI attacks has garnered significant interest and importance.Drawing from prior research, quantization-based coding and decoding techniques have demonstrated the capability to achieve secure and efficient data transmission within resource-constrained digital communication networks.In light of this, the current study delves into the secure cooperative control challenge posed by discrete-time nonlinear MUSs in the presence of FDI attacks.The study also introduces a quantization-based codec scheme designed to facilitate efficient and secure data transmission in resource-constrained digital communication networks.Existing quantization-based coding and decoding techniques have predominantly been applied to linear multi-agent systems [19,20,23,24], or specifically to the controller-to-actuator component of an agent [35,36].However, there is a noticeable dearth of results for high-order unknown nonlinear multi-agent systems.This gap arises due to the complexities introduced by high-order nonlinearity and unknown variables, rendering the application of existing quantization-based codec techniques challenging.Moreover, in the presence of cyber attacks, these conventional methods fail to ensure system implementation or consistency.In scenarios involving network attacks, existing techniques also fall short of guaranteeing the system's ability to achieve coherence.Consequently, the adaptation of quantization-based codec techniques to high-order unknown nonlinear MUSs under FDI attacks represent a formidable challenge, which serves as the impetus behind this paper.Consequently, this paper proposes a codec-based neural network control approach for addressing the distributed quantized cooperative control problem within high-order nonlinear multi-agent systems.The framework takes into account transmission channels between agents that are potentially subject to FDI.
The remainder of this study is organized as follows: Section 2 presents the preliminaries and problem formulation.The design of secure consensus control and the stability analysis are detailed in Sections 3 and 4, respectively.Section 5 provides simulation examples, while Section 6 concludes the study.

System model
This study discusses the secure cooperative control of MUSs comprising N agents, wherein each agent is described by where is the plant control input, and y i (k) is the plant output.
Remark 1. System ( 1) is a unified structure of multiple MUSs, such as the motion of aircraft wings [37], robotic manipulators [38], and ship steering systems [39].Each variable in (1) depends on the specific characteristics of the actual system.Definition 2. The solution of the system (1) is uniformly ultimately bounded, if for all states x(0) ∈ D, and ∀k ≥ R( , x(0)), x(k) ≤ hold, wherein D ⊂ R is a compact set, N ( , x(0)) is a number, and > 0.

Communication graph
A directed graph is used to describe information communication among agents.G = (N , E, A) is introduced with a set of vertices N = {1, . . ., N }, directed channels E, and relevant adjacency matrices A = (a i,j ).If information is transmitted from vertex j to i, (j, i) is defined as a graph edge (a directed channel), and, a i,j is defined as the weight.If and only if (j, i) ∈ E do we obtain, a i,j > 0; else, we obtain a i,j = 0.The set of neighboring vertexes i is defined as A graph is considered connected if there exists a path between any two nodes, which is formed by a series of edges.For the purposes of this study, assume that the directed graph G is connected.

Encoding-decoding scheme
This subsection introduces the designed codec, intended for later use in the controller design.Its purpose is to employ the quantizer for the reduction of information transfer between agents and to address the issue of insufficient bandwidth.Precisely, in this scheme, each transmission channel encodes the sender's state value as a data point prior to transmission.Subsequently, the receiver employs a decoder to estimate the sender's state after receiving the data.The encoder E i corresponding to agent i is expressed as follows: where ξ i,r (k) (r = 1, . . ., n) are the internal states of E i , x i,r (k) is the input, and s a i,r (k) is the output and is transmitted to the neighbor.Here, q is a quantizer, which is expressed as follows where the specifies constant ζ th > 0 that serves as the threshold for switching between the logarithmic and uniform quantizers; The uniform quantizer error is characterized by a parameter where q l (µ) represents the output of a logarithmic quantizer, as expressed below with ζmin 1+ > 0 determining the dead zone size for q l (µ), The threshold ζ th is a pre-set positive constant for switching between the logarithmic and uniform.The quantization error |q ij (µ) − µ| is bounded by max{ − 2 1+ ζ th , h}.Remark 2. The logarithmic quantizer reduces quantization errors when the amplitude is small.However, as the signal amplitude increases, the quantization levels of the logarithmic quantizer become coarser.To address this issue, a logarithmic-uniform quantizer is employed.This quantizer aims to minimize the average communication rate across instances.
Assuming that the communication between agents (j, i) may be subject to an FDI attack, s i j,r (k) = s a j,r (k) + δ i j (k) is the signal received by agent i from neighbor agent j after the FDI attack.The decoder D i j in agent i is designed as follows where δi ) is the estimation of the false signal δ i j (k), with L i , K i being designed parameters and z i,1 being designed as (6), ξ i j,r (k) (r = 1, . . ., n) are the outputs of D i j .Especially, when the communication is no attack, one has s i j,r (k) = s a j,r (k) and δi j (k) = 0. Lemma 1. Considering an unknown nonlinear MUS (1), the encoding scheme (2) can estimate the state x i .
Proof.Denote e i,r (k) = x i,r (k) − ξ i,r (k), which can be rewritten as where ∆ i,r = q(µ) − µ represents the quantized error.Thus, the errors e i,r (k) are bounded by ∆i,r = max{ m−m 2 1+m ζ th , h}.Remark 3. Unlike observers that are utilized for state estimation or observation within systems, encoders and decoders primarily handle the encoding and decoding of signals.An encoder/decoder constitutes a system or algorithm used for the encoding and decoding of signals or data.The encoder transforms the input signal into a designated encoding format, while the decoder reverts the encoded signal back to its original form.Their advantage lies in conserving communication bandwidth.
3 Distributed state-feedback controller design This section outlines the structure of a distributed controller based on a codec scheme within a backstepping framework.The following error variables are established where α i,r−1 (r = 2, . . ., n) represent the intermediate variables to be designed.
Step 1.According to (6), z i,1 (k + 1) can be produced where ξ j,1 (k + 1) = ¯ j,1 (χ j,1 ), ¯ j,1 (χ j,1 (k)) is a nonlinear mapping, χ j,1 = [ξ j,1 (k), s i j (k), δi j (k)] T .The desired feedback control is established and approximated using a neural network (NN) [40] as follows where w i,1 and ϑ i,1 are weights of the output and hidden layer, φ i,1 is the function of the hidden layer, T is the input of NN, and approximation error |ε i,1 (k)| ≤ εi,1 .Noting Choose suitable positive parameters Ψ i,1 and mi,1 such that |∆ j i,1 (k + 1) + g i,1 e i,2 (k) + Ψ i,1 e i,1 (k)| ≤ mi,1 ∆i holds.Design α i,1 (k) and ŵi,1 (k + 1) as follows where l i,1 and k i,1 are positive parameters.Then, one can obtain where wi, where The desired feedback control is established and approximated it by an NN as follows where and , where Ψ i,Υ and mi,Υ are positive constants.Construct the virtual controller and adaptive law as follows where l i,Υ and k i,Υ are positive constants.Thus, one has where wi,Υ (k where α i,n−1 (k + 1) = ¯ i,n (α i,n−1 (k)), ¯ i,n (α i,n−1 (k)) is a nonlinear mapping.An ideal feedback control is defined and approximated by an NN as where and choosing positive constants Ψ i,n and mi,n , then, k)| ≤ mi,n ∆i holds.The actual controller and adaptive law are established as follows where parameters l i,n > 0 and k i,n > 0. Thus, holds, where wi,n (k) = ŵi,n (k) − w i,n .
The block diagram of the design process is shown in Figure 1.

Stability analysis
Theorem 1.For a multi-agent system composed of N agents utilizing the unknown nonlinear system (1), if the conditions are satisfied, the designed distributed control scheme (23) ensures that the consensus error and all the signals are semi-globally bounded.
Proof.Adopt the Lynapunov function The difference of ( 26) can be calculated as By choosing 0 < l i,β < 1 and 0 < L i < 1, the subsequent inequality is valid Similarly, Furthermore, ∆V i can be rewritten as Based on (30), ∆V i (k) ≤ 0 holds as or or where As mentioned by the extension theorem in [41], the previous analysis proves that the error z i,1 (k), . . ., z i,n (k), adaptive laws ŵi,1 (k), . . ., ŵi,n (k) and δi (k) are bounded.Further, it can be observed that u i is bounded.In addition, we derive the bounds of δi j .According to (30), ∆V i (k) ≤ 0 holds as follows Based on the standard Lyapunov extension theorem, δi j is bounded.By representing e j,κ (k) = x j,κ (k) − ξ j,κ (k) (κ = 1, . . ., n), the error is reworked as where s a j,κ (k) denotes the true signal.Thus, the errors e j,κ (k) are bounded.Furthermore, one obtains lim In accordance with the bounds of z i,1 (k) and δi j (k), consensus errors are uniformly bounded.In conclusion, the proof is finished.

Simulation results
In this section, two simulation cases are presented to illustrate the effectiveness of the control method.
Example 1.A system comprising six single-link manipulators is utilized, with each single-link manipulator being modeled according to [42] where M = 1 kg and L i = 0.1i + 0.5 m.System states are selected as the current angle θ and associated angular velocity ω.Here, let the moment of inertia G −1 i = (0.1i + 0.5) 2 kg • m 2 , the viscous friction f d = 2 kg • m 2 , and acceleration of gravity g = 9.81 m s −2 .The Euler method with a sampling interval ∆t = 0.1 s is used to discrete the systems as follows They are connected to each other through a network, the structure of which is shown in Figure 2. We assume that the false signals 5 sin(k) and 5 cos(k) are injected into the edges (v 1 , v 6 ) and (v 4 , v 3 ).
Choose the parameters as l 1 = [0.1,0.9, 0.1, 0.01, 0.1, 0.1] T , l 2 = [0.9,0.5, 0.5, 0.5, 0. In Figure 3, the outputs of all the agents are plotted.Figure 4 shows the consensus errors.The control input u i , and norms of the adaptive laws w i,1 and w i,2 are shown in Figures 5-7, respectively.From observation, they are bounded.Figures 8-9 illustrate the inputs and outputs of quantizers.Example 2. Next, a cooperative example consisting of six unmanned ships is given to further demonstrate the effectiveness of the designed controller.Each unmanned ship model is as follows [43] T with T i > 0, the Norrbin coefficient K i , gain φ i , the course angle ψ i , course angle rate r i = ψi , and command rudder angle ν i .We define the x i,1 = ψ i and x i,2 = r i , and use the Euler method with the sampling interval ∆t = 0.1s to discrete the systems as The communication network structure is displayed in Figure 10.Assume that false signals 0.5 sin(k) and 0.5 cos(k) are injected into edges (v 2 , v 1 ) and (v 5 , v 4 ), respectively.
Choose the parameters as l 1 = [0.9,0.9, 0.9, 0.01, 0.9, 0.9] T , l 2 = [0.9,0.5, 0.5, 0.5, 0.5, 0.5] T , Figures 11-15 show the simulation results.In Figure 11, the outputs of all the agents are plotted.Figure 12 draws the consensus errors.The control input u i , and norms of the adaptive laws w i,1 and w i,2 are given in Figures 13-15, separately.From observation, they are bounded.Remark 4. Existing quantization-based coding and decoding techniques are primarily suited for linear multi-agent systems [19,20] or the controller-to-actuator segment of an agent [35,36].They rarely address the complexities of high-order unknown nonlinear multi-agent systems.This scarcity of coverage arises from the inherent difficulty of applying existing quantization-based codec techniques to systems characterized by high-order nonlinearity and unknown variables.Moreover, in the presence of cyber attacks, these conventional techniques fail to ensure system implementation.Similarly, when facing network attacks, these methods are incapable of ensuring system consistency.Hence, the challenge lies in adapting quantization-based codec techniques to high-order unknown nonlinear MUSs within the context of FDI attacks.

Conclusion
This study focuses on dealing with the distributed cooperative control issue in unknown nonlinear discrete-time MUSs by utilizing limited communication resources within a backstepping framework.The communication between agents is facilitated using a quantizer-based codec mechanism, which addresses the limitations of communication bandwidth and ensures efficient transmission.Furthermore, FDI attacks are considered in the communication channels among agents, prompting the use of an adaptive method to counter the detected attacks.Employing the Lyapunov analysis technique, the developed control strategy guarantees the boundedness of all signals through appropriate parameter selection.Finally, examples are provided to illustrate the feasibility and effectiveness of the proposed scheme.In future work, the scope of the study will be extended to encompass the distributed cooperative control of nonlinear continuous-time MUSs using a finite-level quantizer-based codec mechanism.