Positive safety control method and system for singular perturbation multi-agent system under DoS attack

Abstract

The invention discloses a positive security control method and system for a singular perturbation multi-agent system under DoS attack, and belongs to the field of multi-agent system control. Firstly, a state space mathematical model of the singular perturbation multi-agent system under the DoS attack is established, and a distributed security consistency control protocol is designed by using neighbor information of intelligent individuals; the method comprises the following steps: according to a positive system control theory and a corresponding closed-loop control system expression, obtaining sufficient and necessary conditions for keeping the positive property of the closed-loop control system according to the positive system control theory, decoupling the closed-loop control system by using a network decoupling technology, and finally controlling the closed-loop control system according to a security consistency protocol, thereby realizing network attack resistance and security consistency. And a control gain matrix for realizing the targets and constraints which need to be met by attack signals are obtained. According to the method, a positive state keeping method and a safety control technology are combined, DoS attacks are resisted on the premise of ensuring that the system state is always positive, and the consistency of the system is ensured.

Description

Technical Field

[0001] The present invention relates to the field of control of multi-agent systems, and in particular to a positive safety control method and system for a singularly perturbed multi-agent system under DoS attacks. Background Art

[0002] There are already relevant papers reporting on positive safety consistency control methods for singularly perturbed multi-agent systems under DoS attacks. These papers focus on resolving the coupling of system states at different time scales caused by singular perturbation parameters in the system, and then designing consistency control protocols to achieve the system's consistency goals.

[0003] For example, the paper "Event-triggered Cooperative Control of Multi-timescale Multi-agent Systems" published by Su Zhixuan, Ma Lei, Yang Chunyu, et al. constructed a Lyapunov function with singular perturbation parameters to deal with singular perturbation parameters in a singular multi-agent system, and proposed an event-triggered consistency control strategy to achieve system consistency while saving communication resources.

[0004] For example, Zhang Yao of Shandong University published a paper titled "Consistency Control of Multi-Agent Systems Based on Singular Perturbations." He proposed a linear transformation method to simplify discrete and continuous singular perturbation multi-agent systems, and designed a distributed control protocol for the simplified system to achieve system consistency.

[0005] In addition, Wu Zhenyuan from Guangxi University published a paper titled "Research on Consistency of Dual-Time-Scale Multi-Agent Systems", in which he proposed a method based on fast-slow state decomposition to decouple singularly perturbed multi-agent systems, and then designed an adaptive consistency protocol to achieve system consistency.

[0006] While these research papers address the problem of consistent control for singularly perturbed systems, they focus on singularly perturbed multi-agent systems in ideal communication environments. They fail to consider the impact of DoS network attacks, a critical factor in today's complex communication environments, on the control design and analysis of system consistency. Furthermore, these papers fail to constrain the system state. In practical engineering, the states of many systems, such as the amount of data transmitted in wireless routing networks, the content of substances in chemical reactions, and the number of organisms in population systems, are non-negative. This makes the methods proposed in these papers unsuitable for practical engineering requirements. Summary of the Invention

[0007] The purpose of the present invention is to provide a positive safety control method and system for a singularly perturbed multi-agent system under DoS attacks, which combines a positive state maintenance method with safety control technology to resist DoS attacks and ensure the consistency of the system while ensuring that the system state is always positive.

[0008] The technical solution adopted by the present invention to achieve the above technical objectives is: a positive safety control method for a singularly perturbed multi-agent system under DoS attack, comprising the following steps:

[0009] 1) Assume that the multi-agent system consists of multiple singularly perturbed intelligent individuals, and all intelligent individuals exchange information through a communication network with undirected connectivity as the topology feature. Establish a state space mathematical model of the singularly perturbed multi-agent system under DoS attack.

[0010] 2) Design a distributed security consistency control protocol using the neighbor information of intelligent individuals, and provide an analytical expression for the control protocol and the corresponding closed-loop control system expression;

[0011] 3) Based on the positive system control theory, we obtain the necessary and sufficient conditions for the closed-loop control system to maintain positivity;

[0012] 4) Use network decoupling technology to decouple the closed-loop control system and simplify the system structure;

[0013] 5) The closed-loop control system is controlled according to the security consistency protocol to achieve resistance to network attacks and security consistency, and the control gain matrix to achieve these goals and the constraints that the attack signal needs to meet are obtained.

[0014] As an optimization scheme for the positive safety control method of the singularly perturbed multi-agent system under the above-mentioned DoS attack, the state space mathematical model established in step 1) is:

[0015] Where,

[0016] x i (t)∈R n is the state vector of the i-th agent, x i (t) is its derivative, R n Represents n-dimensional real space. Because the state is affected by singular perturbations, the state is divided into two categories: the state with slower changes and rapidly changing conditions n1+n2=n;u i (t)∈R m is the control input; singular matrix E ε In represents the n-order identity matrix, the parameter ε is the singular perturbation parameter, and satisfies 0<ε<<1; the matrix A is the Metzler matrix, and the matrix B is a known constant matrix; They are all known constant matrices; α(t) represents the DoS attack signal suffered by the system communication network, and its value is 0 or 1. When α(t) = 1, the system is not attacked and can transmit signals normally. When α(t) = 0, it means that it is suffering from a DoS attack.

[0017] As another optimization scheme for the positive security control method of the singularly perturbed multi-agent system under the DoS attack, the specific operation of designing the distributed security consistency control protocol using the neighbor information of the intelligent individuals in step 2) is as follows:

[0018] Based on the state space mathematical model of the singular perturbation multi-agent system under DoS attack, the coupling strength τ and the relative state information x between neighboring intelligent individuals are integrated. i (t)-x j (t), designed control protocol u i (t) is of the form:

[0019] Among them, the coupling strength τ>0, the matrix K∈R m×n is the feedback gain matrix to be determined, a ij is the element of the Laplace matrix L, a ij >0 means that intelligent individual i and its neighbor intelligent individual j can communicate directly, otherwise the two cannot communicate directly.

[0020] As another optimization scheme for the positive safety control method of the singularly perturbed multi-agent system under the above-mentioned DoS attack, the closed-loop control system in step 2) is:

[0021] Where,

[0022] x i (t)∈R n is the state vector of the ith agent, Its derivative, R n Represents n-dimensional real space. Because the state is affected by singular perturbations, the state is divided into two categories: the state with slower changes and rapidly changing conditions n1+n2=n; ε represents the singular perturbation parameter satisfying 0<ε<<1; They are all known constant matrices; α(t) represents the DoS attack signal suffered by the system communication network, and its value is 0 or 1. When α(t) = 1, the system is not attacked and can transmit signals normally. When α(t) = 0, it means that it is suffering from a DoS attack.

[0023] As another optimization scheme for the positive safety control method of the singularly perturbed multi-agent system under the above DoS attack, let The closed-loop control system is transformed into:

[0024] As another optimization scheme for the positive safety control method of the singularly perturbed multi-agent system under the above-mentioned DoS attack, the necessary and sufficient conditions for the closed-loop control system to maintain positivity in step 3) are:

[0025] As another optimization scheme for the positive safety control method of the singularly perturbed multi-agent system under the above-mentioned DoS attack, the necessary and sufficient conditions for the closed-loop control system to maintain positivity in step 3) are proved as follows:

[0026] Rewrite the closed-loop control system as:

[0027] When the attack occurs, α(t)=0, where Since the matrix A is a Metzler matrix, then It is also a Metzler matrix, and the system is a positive system;

[0028] When the attack does not occur, α(t)=1, the matrix Expand into the following block matrix form:

[0029] According to the definition of system positivity, a closed-loop system is positive if and only if the matrix is the Metzler matrix;

[0030] like is a Metzler matrix if and only if its main diagonal matrix blocks are Metzler matrices and the elements in the off-diagonal matrix blocks are non-negative, that is, the matrix is a Metzler matrix and

[0031] make And ρ max =max{ρ i}, obviously is a Metzler matrix if and only if is the Metzler matrix;

[0032] Therefore, the necessary and sufficient condition for a closed-loop system to be a positive system is that the matrix is a Metzler matrix and

[0033] As another optimization scheme for the positive safety control method of the singularly perturbed multi-agent system under the DoS attack, the specific operation of decoupling the closed-loop control system using the network decoupling technology in step 4) is as follows:

[0034] Set the consistency error to Combined with the closed-loop control system in step (2), the consistency error system is obtained as:

[0035] make The vector form of the consistency error system is expressed as the following dual-coupled system:

[0036] is the Kronecker product. In the dual-coupled system, there are state coupling parameters ε and network coupling parameters L. In order to facilitate the subsequent system performance analysis, it is necessary to decouple the system based on the network parameters and simplify the system structure.

[0037] Perform equivalent decomposition on the Laplace matrix L, that is, L = MJM -1 , J is the Jordan matrix, M is the orthogonal matrix;

[0038] make The double-coupled system is transformed into the following system:

[0039] And the component form of the system is as follows:

[0040] λ l (L) is the eigenvalue of the matrix L. According to the characteristics of the cascade system, the component form asymptotic stability of the system is equivalent to the asymptotic stability of the following subsystems:

[0041] At this point, the control and analysis of the safety consistency of the closed-loop control system is equivalent to the control and analysis of the asymptotic stability of the above subsystems.

[0042] As another optimization scheme for the positive safety control method of the singularly perturbed multi-agent system under the above-mentioned DoS attack, the specific operation of obtaining the control gain matrix to achieve these goals and the constraints that the attack signal needs to satisfy in step 5) is:

[0043] Assume that there are matrices X1, X2∈R n×n Is a positive definite matrix, matrix Y1∈R m×n , and the constant μ makes the following conditions hold: X2A T E ε +E ε AX2-η2E ε X2≤0; (4)

[0044] And the DoS attack signal of the closed-loop control system meets

[0045] The parameters η1 and η2 in the above conditions are given positive constants, τ d >1 is the attack length ratio, F d (0,t) is the attack frequency, parameter μ≥1, then the control gain is selected The positive safety consistency of the closed-loop control system can be guaranteed.

[0046] The positive safety control system for a singularly perturbed multi-agent system under DoS attacks includes the following modules:

[0047] Control system model building module: used to establish the state space mathematical model of the multi-agent system under DoS network attacks, construct the system's input, output, and attack signals, and can decouple the system according to the parameters of the system's communication network;

[0048] Control strategy formulation module: It has a positive consistency safety control protocol, which is composed of the interaction information, coupling strength, and feedback gain between neighboring intelligent agents;

[0049] Control strategy execution module: Uses positive system strategy, security control strategy and consistency strategy to control the multi-agent system to ensure the system state is positive, resist attacks and achieve consistency;

[0050] Hardware module: includes a memory and a processor, and computer instructions are embedded in the memory device. These instructions execute the above method on the processor.

[0051] Compared with the prior art, the present invention has the following beneficial effects:

[0052] The present invention designs a distributed security consistency control protocol for singular perturbation systems that suffer from DoS attacks, and provides necessary and sufficient conditions for ensuring that the state of the corresponding closed-loop system is always in the non-negative quadrant. Then, network decoupling technology is used to simplify the closed-loop singular perturbation multi-agent system with dual coupling characteristics of network and state, so that the consistency problem of the original singular perturbation multi-agent system is converted into a stability problem of the simplified system, reducing the coupling degree of the system and facilitating performance analysis; finally, the stability analysis of the error system is completed using the multi-Lyapunov function method, and sufficient conditions for the system to achieve stability are obtained, which include the value of the feedback gain matrix in the distributed control protocol, and the constraints that the attack frequency and residence time need to meet; compared with the existing security consistency control method for singular perturbation multi-agent systems, the present invention reduces the analysis difficulty of the system and improves the ability of the multi-agent system to resist network attacks; and combined with the positive state maintenance method and security control technology, DoS attacks are resisted and the consistency of the system is guaranteed under the premise of ensuring that the system state is always positive. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a flowchart of the present invention;

[0054] Figure 2 A topological diagram of the communication network in the simulation test of the present invention;

[0055] Figure 3 This is a DoS attack signal diagram in the simulation test of the present invention;

[0056] Figure 4 This is a trajectory diagram of the first component of the control system state in the simulation test of the present invention;

[0057] Figure 5 This is a trajectory diagram of the second component of the control system state in the simulation test of the present invention;

[0058] Figure 6 This is a control input signal diagram of the control system in the simulation test of the present invention. DETAILED DESCRIPTION

[0059] The technical solution of the present invention is described in detail below with reference to specific embodiments. Parts not described in the following embodiments of the present invention should be understood as existing technologies known or should be known to those skilled in the art.

[0060] Example 1

[0061] The positive safety control method for a singularly perturbed multi-agent system under DoS attack includes the following steps:

[0062] Assuming that the multi-agent system consists of multiple singularly perturbed intelligent individuals, and all intelligent individuals exchange information through a communication network with undirected connectivity as the topology feature, a state space mathematical model of the singularly perturbed multi-agent system under DoS attack is established.

[0063] In the specific operation, a multi-agent system is assumed, which consists of N singular perturbation intelligent individuals, and all intelligent agents can exchange information through a communication network with an undirected topological connection. The state space mathematical model of the i-th singular perturbation intelligent individual in the system is expressed as follows:

[0064] Where x(t)∈R n is the state vector of the ith agent, Its derivative, R n Represents n-dimensional real space. Because the state is affected by singular perturbations, the state is divided into two categories, one of which is the state with slower changes. and rapidly changing conditions n1+n2=n;u i (t)∈R m is the control input; singular matrix E ε In represents the n-order identity matrix, the parameter ε is the singular perturbation parameter, and satisfies 0<ε<<1; the matrix A is the Metzler matrix, and the matrix B is a known constant matrix; All are known constant matrices;

[0065] The open network environment makes the above system extremely vulnerable to malicious network attacks. Consider the system's communication network being attacked by a malicious DoS attack. The attacker launches a DoS attack to block the transmission channel of the communication network, thereby preventing the normal transmission of data. Usually, DoS attacks have limited energy. After an attack, the attack signal will go dormant to accumulate energy. Let the attack interval of the kth attack be [t k ,t k +△ k ), where t k is the moment when the attack starts, k >0 represents the duration of the kth DoS attack, k is a positive integer, and the sleep interval after this attack is expressed as [t k +△ k ,t k+1 ), then in the time interval [0, t), the cumulative attack time, cumulative sleep time and attack frequency can be recorded as T d (0,t),T s (0,t) and F d(0, t); For any normal signal y(t), its signal y1(t) after suffering a DoS attack can be described as follows

[0066] Function α(t) represents the DoS attack signal suffered by the system communication network. Its value is 0 or 1. When α(t) = 1, the system is not under attack and can transmit signals normally. When α(t) = 0, it means that the system is under DoS attack.

[0067] The attack frequency and attack sleep time of DoS attack signals are defined as follows:

[0068] For any given time t2>t1≥t0, the number of times the DoS attack occurs in the time interval [t1, t2) is recorded as N d (t1, t2), then the DoS attack frequency Fd(t1, t2) in the time interval [t1, t2) is defined as

[0069] For any given time t2>t1≥t0, the total duration of the DoS attack in the time interval [t1, t2) is recorded as T d (t1, t2), the DoS attack duration Td(t1, t2) in the time interval [t1, t2) is defined as

[0070] Here τ d >1 is the attack length ratio;

[0071] For a singularly perturbed multi-agent system, the state space model of the system after it suffers a DoS network attack is described as follows:

[0072] 2) Design a distributed security consistency control protocol using the neighbor information of intelligent individuals, and provide an analytical expression for the control protocol and the corresponding closed-loop control system expression;

[0073] For the singularly perturbed multi-agent system under DoS attack, the coupling strength τ and the relative state information x between neighboring agents are used in the control protocol design process. i (t)-x j (t) Taking this into consideration, the designed control protocol u i (t) is of the form:

[0074] Among them, the coupling strength τ>0, the matrix K∈R m×n is the feedback gain matrix to be determined, aij is the element of the Laplace matrix L, a ij >0 means that intelligent individual i and its neighbor intelligent individual j can communicate directly, otherwise the two cannot communicate directly;

[0075] The control protocol u i Substituting (t) into the system state space model after the DoS network attack in step 1), the mathematical model of the closed-loop control system is obtained as follows:

[0076] Afterwards, The closed-loop control system is transformed into:

[0077] 3) Based on the positive system control theory, we obtain the necessary and sufficient conditions for the closed-loop control system to maintain positivity;

[0078] Using positive system control theory, we can obtain the necessary and sufficient conditions for the closed-loop control system in step 2) to always remain in the non-negative quadrant, and determine the value range of the control gain K.

[0079] The necessary and sufficient conditions for the closed-loop control system to always be in a positive state are: is a Metzler matrix, and the matrix Here the parameter ρ max >0 indicates the maximum degree of the communication topology graph, and the symbol ≥ indicates that all elements of the matrix or vector are non-negative;

[0080] The specific proof process for the above necessary and sufficient conditions is as follows:

[0081] Rewrite the closed-loop control system as:

[0082] When the attack occurs, α(t)=0, where Since the matrix A is a Metzler matrix, then It is also a Metzler matrix, and the system is a positive system;

[0083] When the attack does not occur, α(t)=1, the matrix Expand into the following block matrix form:

[0084] According to the definition of system positivity, if a closed-loop system is a positive system if and only if the matrix A is a Metzler matrix;

[0085] like is a Metzler matrix if and only if its main diagonal matrix blocks are Metzler matrices and the elements in the off-diagonal matrix blocks are non-negative, that is, the matrix is a Metzler matrix and

[0086] make And ρ max =max{ρ i}, obviously is a Metzler matrix if and only if is the Metzler matrix;

[0087] Therefore, the necessary and sufficient condition for a closed-loop system to be a positive system is that the matrix is a Metzler matrix and

[0088] 4) Use network decoupling technology to decouple the closed-loop control system, simplify the system structure, and facilitate performance analysis;

[0089] In order to analyze the consistency performance of the closed-loop system, the consistency error is set to Combined with the closed-loop control system in step (2), the consistency error system is obtained as:

[0090] make The vector form of the consistency error system is expressed as the following dual-coupled system:

[0091] is the Kronecker product. In the dual-coupled system, there are state coupling parameters ε and network coupling parameters L. In order to facilitate the subsequent system performance analysis, it is necessary to decouple the system based on the network parameters and simplify the system structure.

[0092] Perform equivalent decomposition on the Laplace matrix L, that is, L = MJM -1 , J is the Jordan matrix, M is the orthogonal matrix;

[0093] make The double-coupled system is transformed into the following system:

[0094] And the component form of the system is as follows:

[0095] λ l (L) is the eigenvalue of the matrix L. According to the characteristics of the cascade system, the asymptotic stability of the component form of the system is equivalent to the asymptotic stability of the following subsystem:

[0096] At this point, the control and analysis of the safety consistency of the closed-loop control system is equivalent to the control and analysis of the asymptotic stability of the above subsystems;

[0097] 5) Control the closed-loop control system according to the security consistency protocol to achieve resistance to network attacks and security consistency, and obtain the control gain matrix to achieve these goals and the constraints that the attack signal needs to meet;

[0098] For the controlled subsystem in step 4), construct an applicable Lyapunov function and use scaling and iteration techniques to perform mathematical derivation (Lyapunov function is a method well known in the industry and will not be explained again). Analyze the performance of the controlled system and obtain the conditions for achieving consistency of the closed-loop control system and the constraints that the DoS attack signal needs to meet.

[0099] The mathematical expression of the sufficient condition for ensuring that the system achieves positive safety consistency is:

[0100] Assume that there are matrices X1, X2∈R n×n Is a positive definite matrix, matrix Y1∈R m×n , and the constant μ makes the following conditions hold: X2A T E ε +E ε AX2-η2E ε X2≤0; (4)

[0101] And the DoS attack signal of the closed-loop control system meets

[0102] The parameters η1 and η2 in the above conditions are given positive constants, τ d >1 is the attack length ratio, F d (0,t) is the attack frequency, parameter μ≥1, then the control gain is selected The positive safety consistency of the closed-loop control system can be guaranteed. Conditions (1)-(2) are necessary and sufficient conditions to ensure that the state of the closed-loop control system is positive, conditions (3)-(4) are sufficient conditions to ensure that the closed-loop control system achieves consistency, and condition (5) is the condition to ensure the safety performance of the system, that is, the frequency F of DoS attacks d (0,t) and the dwell time (reflected in the parameter attack length rate τ d Therefore, when conditions (1)-(4) are satisfied, the control gain solved by these conditions is A corresponding control protocol can be obtained, which can achieve the control target, and a DoS attack signal that meets the requirements can be constructed according to the attack frequency and residence time constraints of condition (5).

[0103] The above conditions (1)-(2) for ensuring system positivity have been described in detail in step 3); the analysis process for obtaining conditions (16)-(18) for ensuring system safety consistency is as follows:

[0104] For the controlled subsystem in step 4), the multi-Lyapunov function is constructed as:

[0105] Among them, P θ(t) is a symmetric positive definite matrix, the characteristic function When θ(t) = 1, the system is not under DoS attack, and α(t) = 1 in the subsystem; when θ(t) = 2, the system is under DoS attack, and α(t) = 0 in the subsystem;

[0106] Find the time derivative of the above multi-Lyapunov function along the subsystem solution and substitute conditions (3)-(4) to obtain:

[0107] Put the two ends of the first inequality in the above equation in the time interval t∈[t k-1 +Δ k-1 ,t k ) and integrate the two ends of the second inequality in the above equation at t∈[t k ,t k +Δ k ) and integrate to get:

[0108] The above equations can be processed using iteration and scaling techniques to obtain:

[0109] Combining condition (5) with the definition of asymptotic stability, it can be concluded that the subsystem is asymptotically stable, which also shows that the states xi(t) of all agents in the singularly perturbed multi-agent system under DoS attack are consistent.

[0110] Based on the above analysis, it is proved that the security consistency control method of the present invention can resist DoS network attacks and achieve system consistency.

[0111] Example 2

[0112] The positive safety control system for a singularly perturbed multi-agent system under DoS attacks includes the following modules:

[0113] Control system model building module: used to establish the state space mathematical model of the multi-agent system under DoS network attacks, construct the system's input, output, and attack signals, and can decouple the system according to the parameters of the system's communication network;

[0114] Control strategy formulation module: It has a positive consistency safety control protocol, which is composed of the interaction information, coupling strength, and feedback gain between neighboring intelligent agents;

[0115] Control strategy execution module: Uses positive system strategy, security control strategy and consistency strategy to control the multi-agent system to ensure the system state is positive, resist attacks and achieve consistency;

[0116] Hardware module: includes a memory and a processor, and computer instructions are embedded in the memory device. These instructions execute the method of the above-mentioned embodiment 1 on the processor.

[0117] Those skilled in the art will appreciate that the above embodiments may be method flows, executable computer programs, or systems; therefore, the present invention may be implemented in the form of hardware embodiments, software embodiments, or a combination of software and hardware embodiments. Furthermore, the present invention may also be implemented in the form of a computer program product comprising executable computer program code stored on a computer storage medium (e.g., a hard disk, magnetic disk, optical disk, optical storage device, etc.).

[0118] The present invention is described with reference to flowcharts of methods, systems, and computer program products according to embodiments of the present invention. It should be understood that each process or combination of processes depicted in the flowcharts can be implemented using computer program instructions. These computer program instructions can be loaded onto a processor of a general-purpose computer, a special-purpose computer, or a programmable data processing device for execution, thereby generating a machine. When the computer or programmable data processor executes these instructions, a device is generated to implement the functions described in one or more processes in the flowcharts.

[0119] Simulation test

[0120] In order to verify the effectiveness of the control method proposed in the present invention, the following simulation test is carried out:

[0121] A multi-agent system consisting of five second-order singular perturbation multi-agents was built using the Simulink platform of MATLAB software. These five agents are numbered 1 to 5. The system matrix of the state space mathematical model of each agent is: The topology of their communication network is an undirected connected graph, such as Figure 2 shown.

[0122] The simulation parameters are selected as follows: coupling strength τ = 0.7 (empirical value), η1 = 0.8 (empirical value), η2 = 0.3 (empirical value). Using the LMI toolbox in MATLAB to solve the sufficient conditions in step 5, the following feasible solution is obtained:

[0123] And calculate the parameter μ = 1.2, and the DoS attack frequency 0.5, attack length rate 4. Then within the simulation time of 10s, the system suffered a total of 5 DoS attacks, with a total duration of 2.5s. From this, we can get the DoS attack signal diagram as follows Figure 3 shown.

[0124] Given the initial value of each agent is x1(0) = [0.6; 1.2] T , x2(0)=[0.2;2.2] T , x3(0)=[0.28;2.5] T , x4(0)=[0.4;3] T and x5(0)=[0.1;1.8] T The simulation step size is selected as 0.01, and the following simulation result graph is obtained:

[0125] Figure 4 and Figure 5 The dynamic change trajectories of the first and second components of the multi-agent system state under DoS attack are described respectively. It can be seen that the system state eventually tends to the same value, that is, the system state achieves consistency.

[0126] Figure 6 It is the control input signal diagram of the control system. It can be found that when the attack occurs, the control signal is 0. When the system is not attacked, the control signal of the system is normal, and the control signal u eventually converges.

[0127] From the above simulation results, it can be concluded that the method of the present invention can ensure that the system achieves consistency while resisting network attacks, which shows the effectiveness of the technical solution of the present invention.

Claims

1. A positive safety control method for a singularly perturbed multi-agent system under DoS attacks, characterized by: The following steps are involved: 1) Assume that the multi-agent system consists of multiple singularly perturbed intelligent individuals, and all intelligent individuals exchange information through a communication network with undirected connectivity as the topology feature. Establish a state space mathematical model of the singularly perturbed multi-agent system under DoS attack. 2) Design a distributed security consistency control protocol using the neighbor information of intelligent individuals, and provide an analytical expression for the control protocol and the corresponding closed-loop control system expression; 3) Based on the positive system control theory, we obtain the necessary and sufficient conditions for the closed-loop control system to maintain positivity; 4) Use network decoupling technology to decouple the closed-loop control system and simplify the system structure; 5) The closed-loop control system is controlled according to the security consistency protocol to achieve resistance to network attacks and security consistency, and the control gain matrix to achieve these goals and the constraints that the attack signal needs to meet are obtained.

2. The positive safety control method for a singularly perturbed multi-agent system under DoS attack according to claim 1, characterized in that: The state space mathematical model established in step 1) is: Where, x i (t)∈R n is the state vector of the ith agent, Its derivative, R n Represents n-dimensional real space. Because the state is affected by singular perturbations, the state is divided into two categories: the state with slower changes and rapidly changing conditions n1+n2=n;u i (t)∈R m is the control input; singular matrix E ε I in n represents the n-order identity matrix, the parameter ε is the singular perturbation parameter, and satisfies 0<ε<<1; the matrix A is the Metzler matrix, and the matrix B is a known constant matrix; They are all known constant matrices; α(t) represents the DoS attack signal suffered by the system communication network, and its value is 0 or 1. When α(t) = 1, the system is not attacked and can transmit signals normally. When α(t) = 0, it means that it is suffering from a DoS attack.

3. The positive safety control method for a singularly perturbed multi-agent system under DoS attack according to claim 1, characterized in that: The specific operations of designing a distributed security consistency control protocol using the neighbor information of intelligent individuals in step 2) are as follows: Based on the state space mathematical model of the singular perturbation multi-agent system under DoS attack, the coupling strength τ and the relative state information x between neighboring intelligent individuals are integrated. i (t)-x j (t), designed control protocol u i (t) is of the form: Among them, the coupling strength τ>0, the matrix K∈R m×n is the feedback gain matrix to be determined, a ij is the element of the Laplace matrix L, a ij >0 means that intelligent individual i and its neighbor intelligent individual j can communicate directly, otherwise the two cannot communicate directly.

4. The positive safety control method for a singularly perturbed multi-agent system under DoS attack according to claim 3, characterized in that: The closed-loop control system in step 2) is: Where, x i (t)∈R n is the state vector of the ith agent, Its derivative, R n Represents n-dimensional real space. Because the state is affected by singular perturbations, the state is divided into two categories: the state with slower changes and rapidly changing conditions n1+n2=n; singular matrix E ε I in n represents the n-order identity matrix, the parameter ε is the singular perturbation parameter, and satisfies 0<ε<<1; the matrix A is the Metzler matrix, and the matrix B is a known constant matrix; They are all known constant matrices; α(t) represents the DoS attack signal suffered by the system communication network, and its value is 0 or 1. When α(t) = 1, the system is not attacked and can transmit signals normally. When α(t) = 0, it means that it is suffering from a DoS attack.

5. The positive safety control method for a singularly perturbed multi-agent system under DoS attack according to claim 4, characterized in that: make The closed-loop control system is transformed into:

6. The positive safety control method for a singularly perturbed multi-agent system under DoS attack according to claim 1, characterized in that: The necessary and sufficient conditions for the closed-loop control system to maintain positivity in step 3) are: matrix is a Metzler matrix, and the matrix Among them, ε represents the singular perturbation parameter satisfying 0<ε<<1, and the parameter ρ max >0 represents the maximum degree of the communication topology graph, and the symbol A≥0 indicates that all elements of the matrix or vector A are non-negative.

7. The positive safety control method for a singularly perturbed multi-agent system under DoS attack according to claim 6, characterized in that: The necessary and sufficient conditions for the closed-loop control system to maintain positivity in step 3) are proven as follows: Rewrite the closed-loop control system as: When the attack occurs, α(t)=0, where Since the matrix A is a Metzler matrix, then It is also a Metzler matrix, and the system is a positive system; When the attack does not occur, α(t) = 1, and the matrix A is expanded into the following block matrix form: According to the definition of system positivity, if a closed-loop system is a positive system if and only if the matrix A is a Metzler matrix; like is a Metzler matrix if and only if its main diagonal matrix blocks are Metzler matrices and the elements in the off-diagonal matrix blocks are non-negative, that is, the matrix is a Metzler matrix and make And ρ max =max{ρ i }, obviously is a Metzler matrix if and only if is the Metzler matrix; Therefore, the necessary and sufficient condition for a closed-loop system to be a positive system is that the matrix is a Metzler matrix and 8. The positive safety control method for a singularly perturbed multi-agent system under DoS attack according to claim 1, characterized in that: The specific operation of decoupling the closed-loop control system using the network decoupling technology in step 4) is: Set the consistency error to Combined with the closed-loop control system in step (2), the consistency error system is obtained as: make The vector form of the consistency error system is expressed as the following dual-coupled system: is the Kronecker product; in a dual-coupled system, there are state coupling parameters ε and network coupling parameters L. In order to facilitate subsequent system performance analysis, it is necessary to decouple the system based on the network parameters and simplify the system structure; Perform equivalent decomposition on the Laplace matrix L, that is, L = MJM -1 , J is the Jordan matrix, M is the orthogonal matrix; make The double-coupled system is transformed into the following system: And the component form of the system is as follows: λ l (L) is the eigenvalue of the matrix L. According to the characteristics of the cascade system, the component form asymptotic stability of the system is equivalent to the asymptotic stability of the following subsystems: At this point, the control and analysis of the safety consistency of the closed-loop control system is equivalent to the control and analysis of the asymptotic stability of the above subsystems.

9. The positive safety control method for a singularly perturbed multi-agent system under DoS attack according to claim 1, characterized in that: The specific operations for obtaining the control gain matrix to achieve these goals and the constraints that the attack signal needs to satisfy in step 5) are: Assume that there are matrices X1, X2∈R n×n Is a positive definite matrix, matrix Y1∈R m×n , and the constant μ makes the following conditions hold: X1A T E ε +E ε AX1-tl l (L)(Y1 T B T E ε +E ε BY1)+η1E ε X1≤0; (3) X2A T AND ε +E ε AX2-η2E ε X2≤0; (4) And the DoS attack signal of the closed-loop control system meets The parameters η1 and η2 in the above conditions are given positive constants, τ d >1 is the attack length ratio, F d (0,t) is the attack frequency, parameter μ≥1, then the control gain is selected The positive safety consistency of the closed-loop control system can be guaranteed.

10. A positive safety control system for a singularly perturbed multi-agent system under DoS attacks, characterized by: Includes the following modules: Control system model building module: used to establish the state space mathematical model of the multi-agent system under DoS network attacks, construct the system's input, output, and attack signals, and can decouple the system according to the parameters of the system's communication network; Control strategy formulation module: It has a positive consistency safety control protocol, which is composed of the interaction information, coupling strength, and feedback gain between neighboring intelligent agents; Control strategy execution module: Uses positive system strategy, security control strategy and consistency strategy to control the multi-agent system to ensure the system state is positive, resist attacks and achieve consistency; Hardware module: includes a memory and a processor, and computer instructions are embedded in the memory device, and these instructions execute the method according to any one of claims 1 to 9 on the processor.

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