Model-Free Adaptive Control of Second-Order Multi-Agent Systems via Backstepping Under Mixed Attacks

Authors: st lei han
dong  liu
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-14 13:25:14

Abstract: Abstract—This paper employs model-free adaptive control
methods to investigate second-order multi-agent systems under
mixed attacks. Based on backstepping techniques, second-order
multi-agent systems are equivalently transformed into two interconnected first-order subsystems to reduce structural complexity.
Additionally, considering the impacts of multi-source mixed
network attacks,compensation mechanism employing predictive
distributed output are designed, which mitigate the negative
consequences of such attacks. Furthermore, leveraging only
input-output data from each subsystem, a distributed model-free adaptive control scheme utilizing backstepping is developed
to achieve consensus. Finally, the effectiveness of the proposed
method is validated through two simulation examples.

Main paper: 
 i. introduction: 
 In recent years, multi-agent systems (MASs) have seen widespread application and have emerged as a significant research direction in the field of network intelligence. These systems have achieved numerous outstanding research results in various practical fields, such as remote unmanned aerial vehicles (UAVs) [1] , multirotor drone systems [2] , and multirobot systems [3] . Compared to traditional control systems, multi-agent systems offer several advantages, including distribution, autonomy, efficiency, and cost-effectiveness. These advantages make multi-agent systems particularly appealing for a wide range of applications, from industrial automation to environmental monitoring, and from defense systems to intelligent transportation networks.Typically, current research on cooperative control problems can be classified into three primary directions: leaderless cooperative control [4] , leader-follower cooperative control [5] , and containment control [6] . Leaderless cooperative control focuses on achieving consensus among agents without a designated leader, emphasizing decentralized decision-making and robustness to individual agent failures. Leader-follower cooperative control, on the other hand, involves a hierarchical structure where certain agents (leaders) guide the behavior of the rest (followers), facilitating coordinated actions and efficient task completion. Containment control aims at ensuring that certain agents (leaders) steer the group of agents (followers) to stay within a desired area or follow a specific trajectory, which is crucial in applications such as formation flying and swarm robotics.It is important to note that cybersecurity plays a crucial role in multi-agent systems. In general, network attacks include denial of service (DoS) attacks, false data injection (FDI) attacks, deception attacks (DA), and more. These attacks can have various unexpected impacts on the system, disrupting normal operations and potentially causing significant damage. Consequently, an escalating emphasis has been placed on researching service denial assaults within the academic community in recent times. The work referenced as [7] introduces a collaborative defense mechanism aimed at countering sporadic incidents of such attacks. In [8] , an event-triggered state observer was developed to treat attack signals as states. In [9] , compensation for attacks was achieved by inversely compensating for deception attacks, mitigating their adverse effects on the system.In [10] , predictive control algorithms are applied to multiagent systems with communication constraints. These predictive control algorithms enable multi-agent systems to achieve cooperative control even in the presence of network attacks. By anticipating future states and adjusting control inputs accordingly, predictive control enhances the system's resilience to disruptions and uncertainties. This is particularly important in scenarios where reliable communication cannot always be guaranteed, and the system must adapt dynamically to changing conditions and potential threats. The integration of predictive control algorithms thus represents a significant advancement in the capability of multi-agent systems to operate effectively in complex and adversarial environments, ensuring robust performance and maintaining desired operational goals despite the presence of network attacks.It is noteworthy that most existing studies on multi-agent systems under network attacks assume a fundamental condition that the system model is known. Nevertheless, within a multitude of intricate systems, acquiring an accurate model is challenging, which significantly limits the depth and scope of research on cooperative control. Nevertheless, the challenge has been tackled following the introduction of a data-driven control approach Hou et al. [11] , which was expanded upon in subsequent works [12] - [13] , This approach to control is independent of the underlying mathematical model of the system, operating exclusively with the input and output data. This approach has opened new possibilities for the study of many complex systems, allowing researchers to explore cooperative control issues in greater depth. To date, a substantial body of research has employed this method to further investigate cooperative control problems [14] - [15] . However, for more complex second-order systems, this method still struggles to provide an adequate and comprehensive solution.Backstepping is a pivotal control algorithm in the realm of nonlinear control. The specific design process, as described in reference [16] , involves breaking down the model of the controlled system into several subsystems to ensure stability. Then, iterative design is applied to the intermediate virtual control variables, followed by monitoring the performance of these virtual variables to achieve desired control objectives. Reference [17] introduces a backstepping strategy for tracking control utilizing state observers to enhance system performance. Reference [18] utilizes a backstepping method with command control to sidestep the complexity of intermediate virtual variables, simplifying the control process. Reference [19] proposes a backstepping approach that tackles the computational challenges found in traditional backstepping algorithms. Reference [20] combines backstepping with a model-free approach to solve control problems by dividing second-order systems into two interrelated subsystems. In summary, the combination of backstepping and model-free control effectively addresses tracking control problems under external disturbances, providing robust solutions in complex environments.Drawing on the findings mentioned earlier, this paper delves into the control strategies for second-order nonlinear multiagent systems, employing a backstepping approach. It considers mixed network attacks and attack compensation mechanisms. When juxtaposed with existing literature on multi-agent systems, the highlights of this article are:(1) This paper applies the backstepping method to secondorder multi-agent systems, converting the systems into two interconnected serial first-order subsystems. By creating virtual desired velocities, the development of controllers for each subsystem is achieved.(2)Considering that sensor-controller communication channels suffer from mixed network attacks, compensation mechanisms are formulated to combine predictive distributed output with actual distributed output to counteract the attacked actual distributed output.(3) The proposed distributed model-free adaptive control scheme based on backstepping utilizes only real-time measured input-output (I/O) data and information from neighboring nodes, which positions it as the data-driven consensus control method for multi-agent systems.The subsequent sections will provide an in-depth exposition of the core material of this paper. Section II will delve into the principles of graph theory and the formalization of the issue at hand. Section III presents the controller design for second-order systems considering mixed network attacks and convergence analysis. Section IV provides a numerical case to demonstrate the control effectiveness of the proposed control scheme. Section V concludes the paper.The real numbers are represented by the symbol R. For any matrix A ∈ R N ×N , the norm is indicated as A . The term Diag(•) is used to denote a matrix with entries only on its main diagonal, while I denotes the standard matrix of the same size with ones on the diagonal and zeros elsewhere. Graph theory serves as a crucial framework in multi-agent systems to model interaction topologies. Consider a weighted directed graph G = (V, E, A), where V = {1, 2, . . . , N } is the set of vertices, E ⊆ V × V is the set of edges, and A is the matrix that encodes the connections between nodes.. The vertices V also function as indices for the agents.When agent j receives a message from agent i, the edge (i, j) ∈ E exists. Here, agent j is referred to as the child of agent i, and agent i is the parent of agent j. The neighborhood of agent i is described asN i = {j ∈ V | (j, i) ∈ E}. The weighted adjacency matrix A = (a i,j ) ∈ R N ×N is defined such that a i,i = 0 and a i,j = 1 if (j, i) ∈ E; otherwise, a i,j = 0.The Laplacian matrix L of graph G is defined as L = D-A, where D = diag(d in,1 , d in,2 , . . . , d in,N ). The in-degree d in,i of vertex i is given by N j=1 a i,j . A graph is considered connected when it is possible to move from any node to another through a viable route.Let us examine the subsequent second-order nonlinear system in discrete time:EQUATIONin this case, Additionally, f p (•) and h p (•) denotes an unknown nonlinear function.For System 1 Assumption 1 [21] : The system (1) adheres to the generalized Lipschitz condition, indicating that if the variation in con-trol input ∆u p (k -1) = 0, then ∆x p (k) ≤ b 1 ∆u p (k), where ∆x p (k) = x p (k)-x p (k-1) and ∆u p (k) = u p (k)-u p (k-1).The constantb 1 is positive.Assumption 2 [21] : The derivatives of the function f p (•) concerning the system's control signal u p (k) are uninterrupted.Assumption 3: There exists an upper limit to the proportion of the variation in u q (k) relative to u p (k) is bounded, i.e., |∆u q (k)/∆u p (k)| < ς, with ς being a positive constant.Assumption 4 [22] : The attribute of directed graph N is its possession of a directed root-branch structure that spans all its vertices.Remark 1: Assumptions 1 and 2 are commonly found in the domain of regulatory systems and support the application of dynamic linearization techniques throughout the present work. Assumption 3 is employed in the following proof. Assumption 4 is crucial to achieve consensus in leader-follower coordination.Lemma 1 [23] : Assuming that system (1) adheres to Assumptions 1 and 2 are satisfied and |∆u p (k)| = 0 for all time k, then a pseudo-partial derivative (PPD) parameter Φ p1 (k) exists. Consequently, the formulation of system (1) can be restructured to reflect this:EQUATIONwhereΦ p1 (k)| ≤ b1Similarly, for System 2 Finally, for System 2 Assumption 5 [21] : The system (1) adheres to the generalized Lipschitz condition, indicating that if the variation in con-trol input ∆x p (k -1) = 0, then ∆y p (k) ≤ b 2 ∆x p (k), where ∆y p (k) = y p (k)-y p (k -1) and ∆x p (k) = x p (k)-x p (k -1).The constantb 2 is positive.Assumption 6 [21] : It is assumed that the derivatives of f p (•) concerning the control input u p (k) of the system are unbroken throughout their domain.Assumption 7: It is posited that the proportional change in x q (k) relative to x p (k) is confined within a limit, i.e., |∆x q (k)/∆x p (k)| < ζ 1 with ζ 1 being a positive constant.Assumption 8 [22] : The directed graph G is characterized by the presence of a directed spanning tree.Lemma 2 [23] : For system (2), if Assumptions 5 and 6 are satisfied and |∆x p (k)| = 0 for all time k, then a pseudo-partial derivative (PPD) parameter Φ p2 (k) exists. Consequently, system (2) can be expressed as:EQUATIONwhere Φ p2 (k)| ≤ b2 For the overall systemEQUATIONwhereΦ p (k) = Φ p1 (k -1)Φ p2 (k)The error term associated with the output of the pth agent, when distributed, is delineated as follows:EQUATIONBy substituting equation (1) into equation ( 5) and introducing a new nonlinear function F(•), equation (5) becomes:EQUATIONTheorem 1: For equation ( 6), if assumptions 1-3 hold true and |∆u p (k)| = 0 at all times k, it follows that there exists a pseudo-partial derivative (PPD) parameter Φ p (k).Consequently, the formulation of equation ( 6) is rephrased to:EQUATIONwhere∆ε p (k + 1) = ε p (k + 1) -ε p (k) and |Φ p (k)| ≤ b p ,with b p being a positive constant. Proof: Utilizing system (6), ∆ε p (k + 1) is calculated as∆ε p (k + 1) = F p [y p (k), u p (k), y q (k), u q (k)] -F p [y p (k -1), u p (k -1), y q (k -1), u q (k -1)] + F p [y p (k -1), u p (k -1), y q (k -1), u q (k -1)] -F p [y p (k -1), u p (k -1), y q (k), u q (k)]Then, utilizing the mean value theorem for differentiation on the differenceF p [y p (k), u p (k), y q (k), u q (k)] -F p [y p (k - 1), u p (k-1), y q (k), u q (k)] with respect to u p (k), it is possible to derive ∆ε p (k + 1) = ∂F * p ∂u p (k) ∆u p (k) + Ψ p (k)where∂F * p ∂up(k)denotes the partial derivative value of F p with respect tou p (k) in the interval [u p (k-1), u p (k)]. The ter- m Ψ p (k) is defined as F p [y p (k -1), u p (k -1), y q (k), u q (k)]- F p [y p (k -1), y q (k), u q (k), u p (k -1)].Examine the equation for data discrepancies Ψ p (k) = η p (k)∆u p (k) + η q (k)∆u q (k) with variable η p (k) and η q (k) for every predetermined moment k. According to Assumption 3, there exists a solution η * p (k) whereΨ p (k) = η * p (k)∆u p (k) is valid. . Letting Φ p (k) = ∂F * p ∂up(k) + η *p (k), the formula above can be derived. Additionally, in accordance with Assumptions 1 and 4, A number of constants exist with the property that cp > 0 and cq > 0 such that:|∆ε p (k + 1)| ≤c p a pp |∆u p (k)| + q∈Np bq a pq |∆u q (k)| ≤ cp a pp |∆u p (k)| + q∈Np cq a pq |∆u q (k)| ≤   cp a pp + q∈Np bq a pq   |∆u p (k)| = c p |∆u p (k)|where c p = cp a pp + q∈Np bq a pq |e|. Therefore, one can follow from the formula above that the PPD parameter Φ p (k) remains bounded, i.e., |Φ p (k)| ≤ c i .This section elaborates on the main content of this article, including the design of mixed network attacks, compensation mechanisms, controller design, and proof of bounded distributed output errors. The theoretical structure of the article is illustrated in Fig. 1 . In this section, we will detail the development of a composite cyber-assault strategy. The attack, consisting of DoS, FDI, and DA, constitutes a stochastic attack. ε pi (k) represents the output signals under different attacks, with εp (k) denoting the final output signal subjected to the mixed attack.In the scenario where i = 1, the system endures a specific form of DoS attack, and the corresponding formula, labeled as equation ( 5), can be articulated as follows:EQUATIONwhere l 1 (k) represents the success of the DoS attacks and follows a Bernoulli distribution. If l 1 (k) = 0, it shows that the assaults achieved their intended outcome, with the likelihood of P{l 1 (k) = 0} = l. Conversely, if l 1 (k) = 1, the offensives did not succeed, and the likelihood is P{l1 (k) = 1} = 1 -l.In the case where i = 2, the system experiences FDI attacks, and thus formula ( 5) is transformed to:EQUATIONwhere π p (k) denotes the gain parameter of the FDI attacks, oscillating randomly within a predefined boundary. The occurrence of an FDI attack is ascertained usingl 2 (k) * = exp{-ε p (k) -εp (k) } If l 2 (k)* is below a predefined positive threshold denoted by ν, the attack is deemed to occur, resulting in l 2 (k) = 0.In the scenario where i = 3, the system faces Deception Attacks (DA), and equation ( 5) is modified to:EQUATIONThe variable l 3 (k) determines the success of the DA, conforming to a Bernoulli distribution. A value of l 3 (k) = 0, signifies a successful attack, occurring with certainty P{l 3 (k) = 0} = l. Conversely, when l 3 (k) = 1, the offensives did not succeed, with the probability P{l 3 (k) = 1} = 1 -l for failure.. Ultimately, l(k) signifies the outcome of a mixed attack, with l(k) = 1 representing a successful attack and l(k) = 0 indicating a failed attack. Here, l(k) = l 1 (k)l 2 (k)l 3 (k).To conclude, the methodology presented advances a refined distributed output error formula that incorporates an element of predictive compensation:EQUATIONwhere εp (k) = (ε p (k -1) + Φp2 (k -1)∆x p (k -1))Remark 2: Within this framework, DoS attacks are deliberate interruptions targeting network protocol implementation, aimed at rendering the computer or network incapable of providing standard services or accessing resources. FDI and DA attacks involve attackers injecting false signals to replace authentic information, thereby preventing the system from achieving its intended goals.For System 1,due to the compression of the nonlinear dynamic characteristics of the system into Φ p1 (k), obtaining its dynamic model remains challenging, but numerical variations can be estimated. Therefore, the cost function for Φ p1 (k) is given in the following form:J( Φp1 (k)) =|∆x p (k + 1) -Φp1 (k)∆u p (k -1)| 2 + µ| Φp1 (k) -Φp1 (k -1)| 2Taking the partial derivative of Φp1 (k) from the above equation yields 0, withΦp1 (k) = Φp1 (k -1) + η∆u p (k -1) µ + |∆u p (k -1)| 2 (∆x p (k + 1) -Φp1 (k -1)∆u p (k -1))where η ∈ (0, 2] is the step-size factor, which enhances the algorithm's flexibility and generality, and µ is a positive constant.Likewise, in the case of System 2, the associated cost function is formulated in the subsequent manner:J( Φp2 (k)) =|∆ε p (k + 1) -Φp2 (k)∆x p (k -1)| 2 + µ| Φp2 (k) -Φp2 (k -1)| 2Taking the partial derivative of Φp2 (k) from the above equation yields 0, withΦp2 (k) = Φp2 (k -1) + η∆x p (k -1) µ + |∆x p (k -1)| 2 (∆ε p (k + 1) -Φp (k -1)∆x p (k -1))Based on this, a control protocol is formulated for System 1 and System 2,u p (k) =u p (k -1) + ρ Φp1 (k) λ + Φp1 (k) 2 (x p (k) -x p (k)) x p (k) = x p (k -1) + ρ Φp2 (k) λ + Φp2 (k) 2 εp (k)Taking into account the current position and velocity state at the present time, the virtual desired velocity state is designed as follows:xp (k) = xp (k -1) + ρ Φ2 p1 (k) λ+ Φ2 p1 (k) xp (k -1) + g g = -x p (k -1)Consequently, the full control structure can be delineated belowEQUATIONEQUATIONEQUATIONEQUATIONThe above MFAC consists of PPD estimation algorithm(12) (14), PPD reset algorithm, control algorithm(13) (15), and virtual estimation algorithm (16) , which is designed with compensated distributed output∆ε p (k).In this segment, we delve into the convergence properties of system (1) when it is subjected to stochastic cyber-intrusions, with the principal outcomes encapsulated in Theorem 2.Theorem 2: For system (1) experiencing random cyberattacks, provided assumptions 1-4 are fulfilled and the controller parameters are selected as λ > 0, 0 < η < 2, µ > 0, ρ ∈ (0, 1], The control methodologies outlined in equations (12) to (16) are instrumental in attaining this consensus.Proof: The proof is structured into four main parts. First, it is verified that the PPD parameter approximation error is bounded. Second, we show the boundedness of the error in virtual tracking speed. Third, the verification of the distributed output forecast error's bounded nature is addressed. Fourth, the argument concludes with the proof of the distributed output error's boundedness. The rationale behind these bounded attributes is rooted in the mathematical expectation principle.part-1: Let Φp1 (k) = Φp1 (k) -Φ p1 (k) denote the approximation error of the PPD parameter, as stated in (12) .EQUATION)From equation (20) , it is derived that| Φp1 (k)| ≤ 1 - η∆u p (k -1) 2 µ + |∆u p (k -1)| 2 | Φp1 (k -1)| + |Φ p1 (k -1) -Φ p1 ](k)|. (18)Since |∆u p (k)| ≤ b, by appropriately selecting η and µ such that 0 < η ≤ 1 and µ ≥ 0, there exists a constant q 1 ensuring that holds.0 < 1 - η∆u p (k -1) 2 µ + |∆u p (k -1)| 2 ≤ q 1 < 1 (19) Since |Φ p1 (k)| ≤ a, in light of Assumption 4, it follows that the difference |Φ p1 (k -1) -Φ p1 (k)| ≤ a.Drawing from equations ( 8) and ( 9), we deduce that| Φp1 (k)| ≤ q 1 | Φp1 (k -1)| + a ≤ • • • ≤ q k 1 | Φp1 (0)| + a(1 -q k 1 ) 1 -q 1 (20) which implies that Φp1 (k) is bounded. Consequently, Φ p1 (k) is also bounded as Φp1 (k) is bounded. part-II: Let Φp2 (k) = Φp2 (k) -Φ p2 (k)be the estimation discrepancy associated with the PPD parameter, as per equation ( 14)EQUATIONBy evaluating the absolute value and then employing the expectation operator across the entirety of the preceding equation, the result isEQUATIONObserve that for 0 < η * < 2, µ * > 0, 0 < l(k) < 1. Consequently, a positive value d is identified, fulfilling the condition that 0 < |1-η * ∆x 2 p (k-1)l(k) µ * +∆u 2 (k-1) | = d < 1. Also because of |Φ p2 (k)| ≤ b 2 , so | Φ p2 (k)| ≤ 2b 2 .Consequently, the formulation of equation ( 22) is articulated to be,EQUATIONBased on the inequality above, it is evident that Φp2 (k) is uniformly bounded. Therefore, based on Φ p2 (k) being bounded, Φp2 (k) is also bounded.part-III: Define the virtual desired velocity tracking errorx p (k) = xp (k)-x p (k). Utilizing equations ( 15) and ( 16), may be represented in the form ofEQUATIONLet a Lyapunov function be defined:EQUATIONwhen lim k→∞ a = 0, so lim a→0k→∞ Φp1 (k) = 0EQUATIONEQUATIONEvidently, with the control scheme delineated within this document, the condition ∆v(k) < 0 is reliably met. The divergence between the current velocity and the envisioned target velocity achieves asymptotic and sustained convergence, thereby finalizing the verification.part-IV: Let us denote the distributed output forecast discrepancy by εp (k) = εp (k)-ε p (k). Utilizing the formulations from equation (11) , this discrepancy is articulated in the subsequent manner:EQUATIONObtain the absolute magnitude and then employ the expected value operator on the expression designated as equation ( 27), and scale them to derive the result, thenEQUATIONDue to the boundedness of Φp1 (k) proven in the part-I, and from our findings in the part-II that x p (k -1) is also bounded, we can conclude that Φp1 (k)x p (k -1) is bounded.Let Φp1 (k)x p (k -1) < q, where q is a positive constant. Given that l ∈ (0, 1), the relational expression is subsequently reformulated to read:EQUATIONThis indicates that the value of εp (k) remains consistently confined within limits. part-V: Due to ( 14) and e εp (k + 1) = ε * p -ε p (k -1), e εp (k + 1) can be expressed as,EQUATION)e εp (k)+ ρ * Φp2 (k)Φ p2 (k) λ * + Φ2 p2 (k) εp (k) (32) let ρ * Φp2(k)Φp2(k) λ * + Φ2 p2 (k) = m(k)Employing an identical approach as outlined in part-I, equation ( 35) is subsequently reformulated in the following manner:EQUATIONBased on the evidence presented within the demonstrations of part I and II, εp (k) and Φp2 (k -1) are bounded, so let(1-D)lE{|ε p (k)|}+(1-D)E{| Φp2 (k-1)∆x p (k-1)|} < d * with d * > 0 being a constant.Consequently, equation ( 29) is articulatedin the subsequent format:E{|e εp (k + 1)|} ≤ DE{|e εp (k)|} + (1 -D)lE{|ε p (k)|} (1 -D)E{| Φp2 (k -1)∆x p (k -1)|} ≤ DE{|e εp (k)|} + d * ≤ D k E{|e εp (1)|} + d * 1 -D (34) which illustrate ε * p is bounded. Due to e εp (k + 1) is bounded, ε p (k -1) is also bounded.The conclusion of the demonstration has been reached,this crucial result ensures that the proposed control algorithm effectively maintains system stability and achieves its intended control objectives under the specified conditions. The meticulous application of Lyapunov functions and the detailed stepby-step analysis provide a solid theoretical foundation for our approach. This comprehensive proof not only validates the robustness and reliability of the proposed control strategy but also highlights its practical applicability in real-world scenarios where multi-agent systems are subject to mixed network attacks. Consequently, we can confidently assert that the control strategy developed in this paper is both sound and effective, offering significant potential for enhancing the performance and resilience of multi-agent systems in dynamic and potentially hostile environments.In this segment, we introduce a computational example to substantiate the efficacy of the suggested methodology.Example 1: Consider a MASs comprising a single leader and four subordinate agents, as depicted in the communication structure diagram in Fig. 1 .The dynamics of each constituent agent's model are delineated below:                       Agent1 : y 1 (k + 1) = y 1 (k)x 1 (k) 1 + y 2 1 (k) + x 1 (k)u 1 (k) 1 + x 2 1 (k) + u 1 (k) Agent2 : y 2 (k + 1) = y 2 (k)x 2 (k) 1 + y 2 2 (k) + x 2 (k)u 2 (k) 1 + x 3 2 (k) + 0.5u 2 (k) Agent3 : y 3 (k + 1) = y 3 (k)x 3 (k) 1 + y 2 3 (k) + x 3 (k)u 3 (k) 1 + x 2 3 (k) + 0.9u 3 (k) Agent4 : y 4 (k + 1) = y 4 (k)x 4 (k) 1 + y 2 4 (k) + x 4 (k)u 4 (k) 1 + x 5 4 (k) + 0.8u 4 (k)Furthermore, one can determine the Laplacian matrix from the interaction structure diagram displayed in Fig. 2 , as follows:L =     1 0 0 -1 0 1 -1 0 0 -1 2 -1 0 -1 -1 2    The trajectory of the leading agent is defined through the subsequent mathematical expression: Remark 4: In the course of the simulation, the predetermined path for the fictitious leader is designated at node O. In this scenario, only agents 2 and 4 directly receive information from the leader, while agents 1 and 3 receive information indirectly through agents 2 and 4.y d (k) = 2, 0 < k ≤ 200 0.5, 200 < k ≤ 400The aforementioned simulation results indicate that, under time-invariant signals, the proposed MFAC algorithm exhibits superior control performance compared to the control method described in reference [24] . Specifically, the proposed algorithm achieves smaller errors and higher stability. Furthermore, the compensation method introduced in this paper demonstrates a significant improvement over the scenario without compensation. The control effect with compensation is markedly better than that without compensation. In summary, the MFAC algorithm with compensation proposed in this paper shows excellent control performance under time-invariant signals, making it a robust and effective solution for managing system stability and accuracy.Example 2: A multi-agent systems with a time-varying signal is analyzed, as showcased by the connectivity diagram in Fig. 1 .The path followed by the leader is defined by the formula below:EQUATION))Here, the starting conditions and controller settings for the agents are: l1 = 0.45, l2 = 0.6, l3 = 0.55, l4 = 0. proposed algorithm exhibits smaller errors, faster convergence, and better tracking performance compared to the algorithm presented in [24] . Through clear comparisons across various aspects, the proposed method shows good control effects in simulations.The simulation results illustrated above indicate that, under time-varying signals, the proposed MFAC algorithm demonstrates markedly better control performance compared to the control strategy outlined in reference [24] . This improvement is reflected in significantly reduced error margins and enhanced system stability. Moreover, the introduction of the compensation method proposed in this study shows a clear and substantial improvement over the scenario without compensation. Specifically, the control effect achieved with compensation is considerably superior to that without compensation. Thus, it can be concluded that the MFAC algorithm, when coupled with the proposed compensation technique, delivers outstanding control performance under time-varying signal conditions. This highlights the algorithm's robustness and efficacy in managing dynamic systems with fluctuating signals, ensuring both precision and stability in control outcomes.The present study investigates the adaptive control challenge for second-order nonlinear multi-agent networks subjected to a combination of cyber threats. To facilitate ongoing scholarly work, the output discrepancies across agents within each subsystem are determined employing backstepping methodologies devoid of model dependencies, subsequently converting them into a linear form via dynamic linearization. This technique streamlines the control formulation, rendering it more tractable. Additionally, the paper delves into the orchestration of hybrid cyber-assaults, proposing a countermeasure mechanism designed to alleviate their detrimental effects. Such a mechanism bolsters the system's resilience to diverse cyberintrusions. The strategy's convergence is substantiated through the application of Lyapunov stability theory, establishing a robust theoretical foundation. In conclusion, empirical simulations corroborate the method's commendable control efficacy, preserving system steadiness and potency amidst adverse scenarios. 
 
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Reviews: review: title: title: This article is very interesting and a good one.
main_review: main_review: In this paper,  the model-free adaptive control methods were proposed to investigate the second-order multi-agent systems under
mixed attacks. The obtained result is valuable and can be accepted if the following problems can be clarified.
1. In this paper, the control problem of second-order multi-agent systems under mixed attacks is studied. Can the method proposed in this paper be extended to solve the control problem of high order multi-agent systems?
2. In the introduction, there is a lack of relevant research on the control methods of multi-agent systems under mixed attack.
3. Please explain why (1) is a second-order system?
title: title: Review Comments for Manuscript No. 138
main_review: main_review: 1. The manuscript contains several noticeable errors in details such as text formatting, figure text size, and reference formatting. The authors should correct these issues.

2. Given that the study focuses on second-order systems, what are the advantages of this work compared to previous studies on strict-feedback MAS systems based on the backstepping method? In other words, why can't previous research be directly extended to the current study?

3. Regarding the cost function introduced after equation (11), what is the basis for choosing its particular form?

4. The physical meaning or the actual significance of the state vector in equation (1) is not clearly defined. However, the authors mention in the text before equation (12), “Taking into account the current position and velocity state at the present time,” which is confusing.

5. The simulation section needs improvement in the following aspects: First, in the Example, the form of the agents provided by the authors does not match the second-order agents described in equation (1). Additionally, the specific dynamics of the network attacks are not presented.
title: title: Model-Free Adaptive Control of Second-Order Multi-Agent Systems via Backstepping Under Mixed Attacks
main_review: main_review: This paper employs model-free adaptive control methods to investigate second-order multi-agent systems under mixed attacks. This work is well organized. Below are some comments.
(1)	The contributions should be illustrated in a clearer manner. For example, what is the main improvement of the paper compared to the existing results. The authors should explain the unique contributions of this paper.
(2)	I think that adding a block diagram/scheme will help to understand how your method works. 
(3)	Is the computational complexity of the proposed algorithm too large?
(4)	The simulation results should be explained more carefully.
(5)    The quality of Figures should be improved.
(6)	The paper is well presented, spelled correctly. I recommend authors to carefully read the entire paper to find possible misspellings.