A direct iterative learning control method for multi-agent systems under hybrid attacks

Abstract

The application relates to the technical field of intelligent control, in particular to a direct iterative learning control method for a multi-agent system under hybrid attacks, which comprises the following steps: S1, constructing a linear parameterized model of multi-agent consistency output under non-repeated initial values; S2, establishing an iterative disturbance observer and a parameter estimator for the multi-agent system under FDI attacks; S3, constructing a forward compensation strategy for the multi-agent system under DoS attacks; and S4, constructing a direct iterative learning control scheme for the multi-agent system based on the observer and the forward compensation. The application effectively reduces the complexity of the controller, improves the adaptability of the system to different attacks, can accurately realize the consistency tracking target of the multi-agent system in a complex attack environment, and has a wide application prospect.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of intelligent control, and more particularly relates to a control method for a multi-agent system under hybrid attack. BACKGROUND

[0002] In recent years, multi-agent systems (MASs) have been widely used in many fields such as robots, control systems, social networks, and communication networks as a promising paradigm. However, with the increasing popularity of communication networks, achieving consensus control of multi-agent systems in complex environments has gradually become a research focus. Consensus control aims to ensure that all agents follow a specific control protocol and eventually reach a synchronized state or accurately track a preset trajectory.

[0003] Application No. 202411352061.4 proposes a method for achieving mean-square consensus control of a multi-agent system under hybrid attack, which belongs to the technical field of intelligent control. The method includes establishing a model of the multi-agent system, designing a controller, constructing an error system between the follower agents and the leader, and establishing mean-square consensus conditions for the multi-agent system under deception attack and DOS attack. The method controls the multi-agent system to achieve leader-follower mean-square consensus. The method improves the control strategy by considering possible deception attacks and DOS attacks during the operation of the multi-agent system, thereby achieving leader-follower mean-square bounded consensus and improving control efficiency and accuracy. However, the technology has the following defects:

[0004] 1. Complex controller design: The controller design in the comparative document involves multiple random functions and pulse control. The generation of random functions requires precise algorithm support, and in actual applications, the stability of random functions is difficult to guarantee due to factors such as hardware performance and environmental interference. Pulse control has high requirements for trigger conditions and execution accuracy, and even slight deviations can affect control effectiveness. Complex design not only increases development difficulty, but also increases debugging and maintenance costs, limiting its widespread application in actual scenarios.

[0005] 2. High dependence on model: The method in the comparative document is based on specific deception attack and DOS attack models. However, actual network attack methods are constantly emerging, and new attack patterns are constantly emerging, making it difficult for existing models to cover all attack types. Once attacked by a model outside, the system's defense capability will be greatly reduced. At the same time, the error system in the comparative document requires information about the model of the multi-agent system. If the system structure changes or there are unknown parameters, the error system will not be able to accurately operate, resulting in insufficient universality and robustness of the method.

[0006] 3. Strict mean-square consensus conditions: In the comparative document, the error converges to:

[0007] It can be known that the mean square consensus condition established by the comparison file contains complex mathematical inequalities and a large number of parameter restrictions. These conditions have extremely high requirements for the matching degree of system parameters. In actual systems, due to factors such as differences in device performance, fluctuations in environmental variables, etc., it is difficult to meet such strict conditions. This makes it necessary to repeatedly adjust parameters during system design, increases the design cycle and debugging difficulty, and reduces the system deployment efficiency.

[0008] In view of the above, it becomes a problem faced by the present application to propose a control method of a multi-agent system which is simple to implement, can process mixed attacks and has strong applicability. SUMMARY

[0009] To solve the problems of complex controller design, high dependence on model and strict mean square consensus condition of the existing multi-agent system under attack, the present application proposes a direct iterative learning control method for a multi-agent system under mixed attack.

[0010] The object of the present application is achieved by the following technical solution: a direct iterative learning control method for a multi-agent system under mixed attack, comprising the following steps:

[0011] S1, constructing a linear parameterized model of multi-agent consensus output under non-repeated initial value, using state transition method to transfer the consensus output at all previous times to the initial state;

[0012] S2, establishing an observer and a parameter estimator for a multi-agent system under FDI attack, the observer iteratively estimates the total uncertainty, and introduces the system uncertainty caused by different initial values and FDI attack into the tail term of the equivalent linearization parameter at the same time; using difference calculation to remove the influence of error data of the consensus output after attack on observation, and introducing intermediate variable state function and adjustable parameters to adjust the observation value in real time and observe the tail term;

[0013] S3, constructing a forward compensation strategy for a multi-agent system under DoS attack, and adding compensation to the consensus output;

[0014] S4, constructing a direct iterative learning control scheme for a multi-agent system based on the observer and the forward compensation.

[0015] Further, the step S1 comprises:

[0016] Step S1-1, according to the topological structure information, constructing the consensus output of the multi-agent system as

[0017]

[0018] Wherein,

[0019] yk,j (t+1) represents the system output of the jth agent at the kth iteration t+1 time;

[0020] y d (t+1) represents the desired output at t+1 time;

[0021] N j represents the set of all neighbor agents of the jth agent;

[0022] a j,i is the adjacency matrix A=(a j,i )∈R N×N elements constructed for the multi-agent system topology;

[0023] a j,0 represents the communication state of agent j and virtual leader 0;

[0024] Step S1-2, construct the nonlinear dynamic relationship between the ideal consensus output and the input variable:

[0025] ε k,j (t+1)=f j (ε k,j (t),L,ε k,j (t-n ε ),u k,j (t),L,u k,j (t-n u )) (2)

[0026] where u k,j (t)∈R represents the system input; f j (g) represents the nonlinear function of agent j; n ε and n u represent the order of the system consensus output and input respectively;

[0027] Step S1-3, combined with the state transition iterative dynamic linearization method, get the linear parameterization model between the consensus output ε k,j (t+1) and the input vector u k,j (t):

[0028]

[0029] where Φ k,j (t)=[φ k,j (0),L,φ k,j (t)]∈R 1×(t+1) is the equivalent linearization parameter vector; is the unknown disturbance caused by the non-repeated initial value in the system; u k,j (t)=[u k,j(0), L, u k,j (t)] T ∈ R t+1 , Δ denotes the difference operator.

[0030] Further, the step S2 comprises:

[0031] Step S2-1, establishing the FDI attack model as follows:

[0032]

[0033] wherein,

[0034] is the consistency output after attack;

[0035] δ k,j (t+1) is the data injected by the attacker;

[0036] Further, the consistency output after attack is obtained from the data model (3) as follows:

[0037]

[0038] wherein, ξ k,j (t+1) = ζ k,j (t) + δ k,j (t+1) is the total uncertainty term;

[0039] Step S2-2, using the following iterative disturbance observer to estimate the total uncertainty of the multi-agent system with attack:

[0040]

[0041] wherein, is the estimated value of the total uncertainty term ξ k,j (t+1); θ k,j (t+1) is a state function; K ∈ (0, 1) is a constant; is the estimated value of Φ k,j (t).

[0042] The estimation algorithm of Φ k,j (t) is obtained by using the projection algorithm as follows:

[0043]

[0044] wherein, η ∈ (0, 2] is a step factor; μ > 0 is a weight factor; ||g|| represents the two norm operator of the vector.

[0045] Further, the step S3 comprises:

[0046] Step S3-1, the probability of DoS attack is described as:

[0047]

[0048] wherein, is a weight coefficient; beta k,j (t+1)=0 indicates that the DoS attack successfully occurs between the sensor and the controller of the agent j;

[0049] Step S3-2, the forward compensation scheme of the multi-agent system under DoS attack is:

[0050]

[0051] wherein, indicates the consistent output after compensation.

[0052] Further, the step S4 comprises:

[0053] Step S4-1, the following iterative learning control law is designed in combination with the observer and the compensation strategy:

[0054]

[0055] wherein, lambda>0 is a weight factor; 0<rho<=1 is a step factor;

[0056] Step S3-2, the direct iterative learning control scheme of the multi-agent system based on the observer and the compensation strategy:

[0057]

[0058] wherein, is the initial value of ; nu is a very small positive number.

[0059] Advantages of the present application:

[0060] 1, the present application is aimed at the mixed attack constituted by FDI attack and DoS attack, and the observer and the forward compensation strategy are adopted, instead of directly embedding the attack into the controller. The observer can monitor the uncertainty caused by the attack in real time, and the forward compensation can offset the influence of the attack in time, so as to ensure that the system control performance is not disturbed. This separated design enhances the adaptability of the system to different attack types, effectively improves the system security and stability. In addition, the control scheme of the present application does not involve any random or pulse function, and the controller design is realized by using certain parameters, only in combination with the parameter estimation algorithm and the observer algorithm, it can be ensured that all parameters are known exact control information, the controller design is simple and easy to realize.

[0061] 2、The application discards the dependence on system model information, and directly calculates the consistency output according to the multi-agent system output and topological relationship. This design simplifies the controller design process and avoids the control deviation caused by inaccurate model. It is especially suitable for scenes where the system structure is unknown and the parameters are difficult to obtain, such as emerging industrial control systems or complex network environments, greatly widening the application range.

[0062] 3、The application adopts a linear data model, uses multiple input information to describe the consistency output, and takes into account the system uncertainty. Through the synergistic effect of multiple parameters, the dynamic changes of the system can be captured more comprehensively, and compared with single parameter description, the complex system characteristics can be more accurately reflected, the control precision is effectively improved, and the system can maintain stable operation in the actual complex environment. In addition, only a small amount of parameters are needed to achieve good control performance. Reducing parameters means reducing parameter debugging complexity, engineers do not need to spend a lot of time on parameter optimization, can quickly deploy the system, realize the consistency control effect, improve the project implementation efficiency, and reduce the cost of manpower and time. BRIEF DESCRIPTION OF DRAWINGS

[0063] Figure 1 A flow chart of a direct iterative learning control method for a multi-agent system under hybrid attack according to the application;

[0064] Figure 2 A principle block diagram of a direct iterative learning control method for a multi-agent system under hybrid attack according to the application;

[0065] Figure 3 A communication topological graph between agents;

[0066] Figure 4 A random initial state curve graph of four agents;

[0067] Figure 5 An energy graph of channel injection attack between agents;

[0068] Figure 6 A DoS attack sequence graph of agents under different iterations;

[0069] Figure 7 An output comparison graph of the direct iterative learning control method according to the application;

[0070] Figure 8 A convergence performance graph of the direct iterative learning control method according to the application. DETAILED DESCRIPTION

[0071] With reference to the drawings of the embodiments of the present application, the technical solutions in the embodiments of the present application will be clearly and completely described, obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments of the present application, all other embodiments obtained by a person of ordinary skill in the art without creative labor fall within the protection scope of the present application.

[0072] Referring to Figure 1 , Figure 2 , the embodiment discloses a direct iterative learning control method for a multi-agent system under a hybrid attack, including the following steps:

[0073] I. Construct a linear parameterization model of the consistency output of the multi-agent under non-repeated initial values;

[0074] First, a consistency output model of the multi-agent system is established based on the topological structure information, and a linear parameterization model of the system under the condition of non-repeated initial values is constructed. Considering the influence of FDI attacks and DoS attacks on the normal operation of the system, an observer and a forward compensation strategy are respectively used to optimize the algorithm, and then a parameter estimator and a direct iterative learning control scheme are designed to achieve the given consistency goal.

[0075] The commonly used model is The position relationship between the agent and the leader is not considered. The model of the embodiment has a more comprehensive consistency description, and considers the position information of the neighbor agent and the leader agent, which can more accurately describe that all agents finally track the output trajectory of the leader, and then achieve the consistency control goal.

[0076] The following is a detailed design process:

[0077] Consider an unknown discrete-time multi-agent system, where y k,j (t)∈R represents the system output, u k,j (t)∈R represents the system input; k represents the iteration number, j∈{1,2,K,N} represents the number of agents, t∈{0,1,K,T} represents the discrete time, and T is a positive integer; N j represents the set of all neighbor agents of the jth agent; the adjacency matrix A=(a j,i )∈R N×N , and a j,j =0; a j,i =1 represents that the agent j and i are neighbor agents, otherwise, a j,i =0;

[0078] Based on the topological structure information, the consistency output model of the multi-agent system is constructed as

[0079]

[0080] where y d (t+1) denotes the desired output at time t+1, i.e., the output of the virtual leader; a j,0 denotes the communication status between agent j and the virtual leader 0; a j,0 = 1 means that agent j can directly receive the information of leader 0, otherwise, a j,0 = 0.

[0081] The ideal nonlinear dynamic relationship between the consensus output and the input variable is constructed as follows:

[0082] ε k,j (t+1) = f j (ε k,j (t), L, ε k,j (t-n ε ), u k,j (t), L, u k,j (t-n u ))(a2)

[0083] The ideal nonlinear relationship between the consensus output and the input is directly constructed in this embodiment, without any mathematical model of the system, thus, the dynamics of the agents can be linear or nonlinear, affine or non-affine, homogeneous or non-homogeneous. Here, f j (g) denotes the nonlinear function of agent j; n ε and n u denote the orders of the system consensus output and input, respectively, which are two unknown positive integers.

[0084] Assumption 1: The nonlinear function f j (g) satisfies the global Lipschitz condition, i.e.,

[0085] |f j (ε j,1 , u j,1 ) - f j (ε j,2 , u j,2 )| ≤ L ε |ε j,1 - ε j,2 | + L u |u j,1 - u j,2 |

[0086] where 0 < L ε < ∞ and 0 < L u < ∞ are two Lipschitz constants;

[0087] Further, combined with the state transition iterative dynamic linearization method, the consensus output εk,j (t+1) and the input vector u k,j (t) between them:

[0088]

[0089] where,

[0090] u k,j (t) = [u k,j (0), L, u k,j (t)] T ∈ R t+1 ;

[0091] is the unknown disturbance caused by the non-repetitive initial value in the system;

[0092] Φ k,j (t) = [φ k,j (0), L, φ k,j (t)] ∈ R 1×(t+1) represents the equivalent linearization parameter vector;

[0093] Δu k,j (t) = u k,j (t) - u k-1,j (t), Δε k,j (t+1) = ε k,j (t+1) - ε k-1,j (t+1), Δ denotes the difference

[0094] operator; hereinafter, the meaning of Δ will be the difference operator unless otherwise specified;

[0095] The specific derivation process of the linear parameterization model is as follows:

[0096] In each test, the relationship between the input and the consistent output sequence (2) can be represented by the following algebraic function:

[0097]

[0098] where, u k,j (t) = [u k,j (0), K, u k,j (t)] T ∈ R t+1 ; the function is a composite function of f j (·) and has similar properties to f j (·).

[0099] By iteratively differentiating the above algebraic function, the differential mean value theorem can be used to obtain

[0100]

[0101] wherein, denotes a certain point in the interval [ε k,j (0),ε k-1,j (0)], the optimal partial derivative value of ε k,j (0) ; denotes a certain point in the interval [u k,j (t),u k-1,j (t)], the optimal partial derivative value of u k,j (t) to u k,j (t).

[0102] Definition and Then the iterative dynamic linearization data model (3) is obtained, wherein Φ k,j (t) is bounded according to assumption 1.

[0103] The embodiment uses a state transition method to ingeniously transfer the consistency output of the multi-agent system at all previous times to the initial state. This operation breaks the strong dependence of the traditional model on parameters, makes the model more concise and efficient by reducing the influence of parameters. At the same time, considering multiple input factors to describe the actual consistency output, compared with the single input model, it can more comprehensively and accurately reflect the dynamic changes of the system. The model is constructed based on the non-repeated initial value condition, breaking the same initial value assumption condition of the traditional iterative learning control method, and is closer to the actual system running state, taking into account the system uncertainty, effectively improving the adaptability of the model to complex environments.

[0104] II. Establishing an iterative disturbance observer and parameter estimator of a multi-agent system under FDI attack;

[0105] The following FDI attack model is established:

[0106]

[0107] wherein,

[0108] is the consistency output after the attack;

[0109] δ k,j (t+1) is the data injected by the attacker;

[0110] Further, the consistency output after the attack is obtained from the data model (a3) as follows:

[0111]

[0112] wherein, is the total uncertainty term;

[0113] Further, the following iterative perturbation observer is designed to estimate the total uncertainty of the multi-agent system:

[0114]

[0115] wherein,

[0116] is the estimate of the total uncertainty term ξ k,j (t+1);

[0117] θ k,j (t+1) is a state function; K ∈ (0, 1) is a constant;

[0118] is the estimate of Φ k,j (t);

[0119] Further, the criterion function of the linearized parameter Φ k,j (t) is designed based on the FDI attack model as follows:

[0120]

[0121] wherein, μ > 0 is a weight factor.

[0122] By optimization method, the parameter estimation law obtained from the criterion function (a7) is:

[0123]

[0124] wherein, η ∈ (0, 2] is a step factor; ||g|| represents the two-norm operator of the vector.

[0125] This embodiment innovatively designs an iterative observer to realize the iterative estimation of the total uncertainty in the face of FDI attacks. The system uncertainty caused by different initial values and the impact caused by FDI attacks are integrated into the tail term of the equivalent linearization parameter. By deeply combining with the equivalent linearization model, the interference of the error data in the consistent output after the attack on the observation process is eliminated by using the difference calculation method. The intermediate variable state function and adjustable parameters are introduced to dynamically adjust the observation value in real time, ensuring that the tail term change can be accurately captured. The observer runs along the iteration axis, which is different from the traditional observer along the time axis, can make full use of the historical iteration information, get rid of the dependence on the mechanism model information, establish the iterative evolution relationship, provide accurate and reliable state information for the controller design, and effectively resist the damage of FDI attacks to the system.

[0126] III. Constructing a forward compensation strategy for a multi-agent system under DoS attack;

[0127] Considering a typical DoS attack that affects the normal exchange of consensus output information by locking the communication channel between the sensor-controller, the probability of DoS attack is described as:

[0128]

[0129] wherein,

[0130] is a weight factor;

[0131] β k,j (t+1) = 0 indicates that a successful DoS attack occurs between the sensor and the controller of agent j;

[0132] Further, a forward compensation scheme for multi-agent systems under DoS attack is designed:

[0133]

[0134] wherein, represents the compensated consensus output.

[0135] The embodiment fully considers the actual situation that the network channel between agents in the multi-agent system is subjected to DoS attack, and applies the compensation mechanism to the consensus output link. Unlike the common scheme of adding compensation to the controller to actuator link, the present strategy is more suitable for the attack scene of multi-agent systems. At the same time, the concept of virtual leader is introduced, and when the real leader information is blocked due to DoS attack, the forward data is used for effective compensation to ensure the continuity of the control signal. The strategy is simple and intuitive in description, can accurately respond to DoS attack, avoid the problems of loss of tracking signal and decline of control precision caused by attack, and significantly enhances the robustness of the system under DoS attack.

[0136] Four, a direct iterative learning control scheme for multi-agent systems based on observer and forward compensation is constructed.

[0137] The direct iterative learning control scheme for the multi-agent system is designed as follows by combining the observer and forward compensation strategy:

[0138]

[0139] wherein, is the initial value of ; v is a very small positive number; other symbols are the same as the definition of the same symbols in the foregoing weight.

[0140] If the multi-agent system satisfies assumption 1 and the controller parameter adjustment is within the allowed range, then the direct iterative learning control method based on observer and forward compensation can guarantee that the consensus output ε k,j(t+1) is convergent and bounded; further, the tracking error e k,j (t+1) is also convergent and bounded.

[0141] The embodiment combines the observer designed in the foregoing with the forward compensation strategy to form a complete direct iterative learning control scheme.The observer provides accurate system state information for the controller to help the controller eliminate the influence of attacks and system uncertainties; the forward compensation strategy ensures stable transmission of the control signal under DoS attacks.The scheme directly uses the consistency output after compensation and integrates the topological relationship of the agent into the control law to give full play to the collaborative advantages of the multi-agent system, realize accurate control of the consistency tracking target under attacks, and effectively improve the overall performance of the system in a mixed attack environment.

[0142] The application realizes full-time period control along the iteration axis, only involves control parameters of five known conditions of K in (0, 1), η in (0, 2], μ>0, λ>0 and 0<ρ≤1, and only needs to adjust the parameter value in a certain range to efficiently achieve the consistency control goal in the iteration process.

[0143] In order to verify the correctness of the method of the application, the method of the application is simulated as follows:

[0144] As shown in Figure 3 , a heterogeneous multi-agent system with four follower agents is considered, and the dynamic model is as follows:

[0145]

[0146] The desired trajectory is:

[0147] y d (t)=1.6sin(t / 75)+0.8cos(t / 20),t∈[0,200] (a12)

[0148] The initial state is

[0149] ε k,j (0)=0.5+s k,j (0) (a13)

[0150] Where s k,j (0) randomly changes with iterations in the range of [-0.025, 0.025], as shown in Figure 4 .

[0151] The energy function of the injected attack is:

[0152] h k,j (t)=ο k (t)(1+0.5sin(2tπ / 1000)) (a14)

[0153] Among them, ο k (t) = 0.05rand, such as Figure 5 The FDI attack shown changes over time and iterations; as... Figure 6 The probabilities of the four agents suffering a DoS attack are as follows:

[0154] The controller parameters are set to λ = 0.001, μ = 0.9, η = 0.2, ρ = 0.1, K = 0.4, ν = 10. -5 , u 1,j (t) = [0, L, 0] T , j = 1, 2, 3, 4; Applying the proposed direct iterative learning control method, the tracking performance of the system output at the 1st, 50th, 100th, and 200th iterations is as follows: Figure 7 As shown; the convergence of the tracking error is as follows Figure 8 As shown, the vertical axis represents the average tracking error (ATE), which is defined as follows: from Figure 7 and Figure 8 It can be seen that when the model information is unknown and only input / output data is available, the proposed direct iterative learning control method can effectively improve the control performance of the system and reduce the impact of attacks on the system.

[0155] The foregoing has provided a detailed description of one embodiment of the present invention, but this description is merely a preferred embodiment and should not be construed as limiting the scope of the invention. All equivalent variations and modifications made within the scope of the claims of this invention should still fall within the patent coverage of this invention.

Claims

1. A direct iterative learning control method for multi-agent systems under hybrid attacks, characterized in that, The method comprises the following steps: S1, constructing a linear parameterization model of the consensus output of the multi-agent system under non-repeated initial values, and using a state transition method to transfer the consensus output at all previous time points to the initial state; S2, establishing an observer and a parameter estimator for the multi-agent system under FDI attack, the observer iteratively estimates the total uncertainty, and introduces the system uncertainty caused by different initial values and the FDI attack into the tail term of the equivalent linearization parameter; differential calculation is used to remove the influence of the error data of the consensus output after the attack on the observation, and an intermediate variable state function and an adjustable parameter are introduced to adjust the observation value in real time and observe the tail term, specifically as follows: Step S2-1, the following FDI attack model is established: Further, the consensus output after the attack is obtained from the linear parameterization model as follows: for the kth iteration consensus output after the jth agent is attacked at time instant for the kth iteration consensus output of the jth agent at time injecting to the attacker at the kth iteration data of the jth agent at the moment Step S2-2, the following iterative disturbance observer is used to estimate the total uncertainty of the multi-agent system with attack: wherein is the total uncertainty term; is the equivalent linearized parameter vector; is the unknown disturbance caused by non-repeating initial conditions in the system; , denotes the difference operator, denotes the system input; S3, constructing a forward compensation strategy for the multi-agent system under DoS attack, and adding compensation to the consensus output; wherein is an estimate of the total uncertainty term ; is a state function; is a constant; is an estimate of the total uncertainty term ; The estimation algorithm for is as follows: wherein is a step factor; is a weight factor; denotes the two-norm operator of a vector; S4, constructing a direct iterative learning control scheme for the multi-agent system based on the observer and the forward compensation. The step S1 comprises:

2. The direct iterative learning control method for multi-agent systems under hybrid attacks according to claim 1, characterized in that, Step S1-1, constructing the consensus output of the multi-agent system as follows according to the topological structure information: Step S1-2, constructing a nonlinear dynamic correlation between the ideal consensus output and the input variable: The step S3 comprises: denotes the kth iteration the system output of the jth agent at time representing desired output at a time instant; represents the set of all neighbor agents of the th intelligent agent; Constructing an adjacency matrix for a multi-agent system topology of elements; representing an agent communication state with virtual leader 0; Step S3-1, the probability of DoS attack is described as follows: wherein, represents an agent nonlinear function, and respectively represent the orders of system consistency output and input; Step S1-3, get consistency output by combining state transition iterative dynamic linearization method linear parameterization model between input vector ​ 。 3. The direct iterative learning control method for multi-agent systems under hybrid attacks according to claim 1, characterized in that, Step S3-2, the forward compensation scheme for the multi-agent system under DoS attack is as follows: The step S4 comprises: wherein, is a weight coefficient; denotes a successful occurrence of a DoS attack between the sensor and the controller of agent j; Step S4-1, combining the observer and the compensation strategy to design the following iterative learning control law: wherein, represents the consistent output after compensation.

4. The direct iterative learning control method for multi-agent systems under hybrid attacks according to claim 3, characterized in that, Step S4-2, the direct iterative learning control scheme for the multi-agent system based on the observer and the compensation strategy is as follows: ​ wherein ; is a weight factor; is a step factor; ​ wherein, is the initial value of is a very small positive number.

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