Adaptive secure consensus control method for multi-agent spatio-temporal dynamic system with unknown boundary nonlinearity under mixed attack

Abstract

The application discloses a method for adaptive secure consensus control of a multi-agent spatio-temporal dynamic system with unknown boundary nonlinearity under hybrid attacks, comprising the following steps: constructing a multi-agent spatio-temporal dynamic system model with unknown boundary nonlinearity and a virtual leader model; establishing a hybrid network attack model containing deception attacks and denial of service (Dos) attacks, and using an adaptive radial basis neural network to approximate the unknown boundary nonlinearity function; defining a consensus error signal to obtain a consensus state error system; designing a composite adaptive neural network secure boundary consensus control scheme, constructing a Lyapunov function for the error system, and obtaining a sufficient condition for the error system to realize mean square secure consensus control. The application effectively solves the problem that the traditional method is difficult to simultaneously cope with complex network attacks and unknown boundary nonlinear disturbances, and significantly improves the security and robustness of the system.

Description

Technical Field

[0001] This invention relates to the field of security control technology, and in particular to an adaptive security consistency control method for multi-agent spatiotemporal dynamic systems with unknown boundary nonlinearity under hybrid attacks. Background Technology

[0002] In key fields such as industrial automation, national defense and security, and intelligent equipment, multi-agent systems (MAS) achieve efficient execution of complex tasks through distributed cooperative control, becoming a core support for modern engineering technology. Consistency control, as the foundation of MAS cooperative operation, directly determines the stability and reliability of group behavior. Its safety performance is crucial to the success or failure of missions in scenarios such as continuous production line operation and UAV swarm collaborative operations; any failure in coordination can lead to significant economic losses or safety risks. For example, in distributed sensor network monitoring tasks, consistency imbalances among agents will lead to data fusion distortion, potentially causing critical facility warning failures in severe cases. In UAV swarm reconnaissance missions, interruptions in group coordination can even lead to catastrophic consequences such as equipment collisions. Therefore, the safe and consistent control of MAS systems has always been a research focus for both academia and industry.

[0003] As the application scenarios of multi-agent systems become increasingly complex, traditional modeling methods based on ordinary differential equations (ODEs) are gradually revealing their limitations. ODE models only characterize system dynamics through the time dimension, simplifying spatial distribution characteristics to a point mass model. This makes it difficult to represent spatiotemporal coupling behaviors such as the airspace collaboration of UAV swarms and the spatial diffusion of distributed sensor networks, thus limiting the accurate modeling and analysis of swarm intelligence in complex scenarios. To address this issue, modeling methods based on partial differential equations (PDEs) have achieved a breakthrough. By introducing spatial dimension variables, PDE models can accurately describe the continuous evolution of agents in both time and space, making them particularly suitable for multi-agent systems with significant spatial distribution characteristics.

[0004] Based on PDE modeling, boundary control strategies have become a research hotspot due to their advantages such as low deployment cost and strong anti-interference capability. Unlike intra-domain control, which requires the deployment of actuators across the entire domain, boundary control can achieve global coordination by adjusting the input at the spatial boundary of the system, making it more feasible in engineering practice. However, the complex disturbances and system uncertainties in the actual operating environment pose severe challenges to the secure and consistent control of multi-agent spatiotemporal dynamic systems: on the one hand, the threat of hybrid attacks is becoming increasingly prominent, including spoofing attacks that tamper with communication data and denial-of-service (DoS) attacks that undermine data availability by consuming resources or congesting the network; on the other hand, multi-agent spatiotemporal dynamic systems generally exhibit unknown nonlinear characteristics at the boundary, such as the spatial time-varying characteristics of joint friction in robotic arms and the nonlinear coupling of aerodynamic drag in UAVs. These unmodeled dynamics can severely reduce control accuracy and even lead to system instability.

[0005] While existing research has made progress in the consistency control of multi-agent spatiotemporal dynamic systems, it still has significant limitations: First, most results do not consider security protection mechanisms under hybrid attacks, making it difficult to cope with the risk of collaborative failure in complex attack scenarios; Second, there is a lack of effective means to handle unknown boundary nonlinearities, and traditional control methods are difficult to achieve high-precision adaptive compensation; Third, a boundary control framework that integrates adaptive neural networks has not yet been formed, and it is impossible to use the ability of neural networks to approximate unknown nonlinearities to solve the modeling uncertainty problem.

[0006] Against this backdrop, researching adaptive and secure consistency control methods for multi-agent spatiotemporal dynamic systems, which are subject to both hybrid attack interference and unknown boundary nonlinearity, has become an urgent need to overcome existing technological bottlenecks and ensure the reliable operation of multi-agent systems in key areas. Summary of the Invention

[0007] This invention proposes an adaptive security consensus control method for multi-agent spatiotemporal dynamic systems with unknown boundary nonlinearity under hybrid attacks, aiming to solve a problem that has not been considered to date in the consensus control of multi-agent spatiotemporal dynamic systems.

[0008] To achieve the above objectives, this invention provides an adaptive security and consistency control method for a multi-agent spatiotemporal dynamic system with unknown boundary nonlinearity under hybrid attacks, comprising:

[0009] Construct a nonlinear multi-agent spatiotemporal dynamic system model with unknown boundaries and a virtual leader model;

[0010] A hybrid network attack model incorporating spoofing attacks and denial-of-service (DoS) attacks is established, and an adaptive radial basis function is used to approximate the nonlinear function of the unknown boundary.

[0011] By using the multi-agent spatiotemporal dynamic system model, the virtual leader model, and the hybrid network attack model, a consistency error signal is defined, and a consistency state error system is obtained.

[0012] Design a composite adaptive neural network security boundary consistency control scheme. For the error system, construct a Lyapunov function to obtain sufficient conditions for the error system to achieve mean square security consistency control, thereby realizing adaptive security boundary consistency control of a multi-agent spatiotemporal dynamic system under hybrid attacks and unknown boundary nonlinearity.

[0013] Preferably, a nonlinear multi-agent spatiotemporal dynamic system model and a virtual leader model with unknown boundaries are constructed, including:

[0014] A nonlinear multi-agent spatiotemporal dynamic system model with unknown boundaries and a virtual leader model are constructed based on parabolic partial differential equations. The nonlinear multi-agent spatiotemporal dynamic system with unknown boundaries includes a mathematical model of follower agents, specifically:

[0015]

[0016] In the formula, Let represent the state of the i-th agent at time t, where 'a' is a spatial variable, and The time variable t > 0; Let u be the initial value for N follower agents; i (t) represents the control input at the boundary; A is a known positive definite matrix; B is a known constant real matrix; g(z) i (L,t) is an unknown nonlinear function.

[0017] Preferably, the virtual leader model is:

[0018]

[0019] In the formula, This represents the state of the leader agent at time t, point a; L is the upper boundary of the spatial domain. This represents the initial state of the leader agent.

[0020] Preferably, the hybrid network attack model including spoofing attacks and denial-of-service (DoS) attacks is as follows:

[0021]

[0022] In the formula, To account for the error signal after considering the effects of mixed attacks; ρ(e(L,t)) is a nonlinear function satisfying the inequality: ||ρ(e(L,t))||2≤||Ge(·,t)||2, where G is a constant matrix determined by the nonlinear function ρ(e(·,t)); α s (t)∈{0,1}(s=1,2) is a random variable that satisfies the Bernoulli distribution.

[0023] Preferably, the nonlinear function with an unknown boundary is approximated using an adaptive radial basis function neural network, including:

[0024]

[0025] In the formula, and h i (z i (L,t) represent the optimal weights and radial basis functions of the neural network, respectively; approximation error in Given a positive number, defined This represents an estimate of the optimal weights;

[0026] The radial basis function is a Gaussian function:

[0027]

[0028] In the formula, υ h and z c Let z represent the width and center of the radial basis functions, respectively. i The state of the agent.

[0029] Preferably, the consistency error signal is:

[0030] e i (a,t)=z i (a,t)-z l (a,t),

[0031] In the formula, e i (a,t) represents the consistency error signal; z i (a,t) represents the state of the i-th agent at time t at point a; z l (a,t) represents the state of the leader agent at point a at time t.

[0032] Preferably, the consistency state error system is as follows:

[0033]

[0034] In the formula, e(a,t) is the agent error signal matrix; I N A is the identity matrix; A is a known positive definite matrix; a is a spatial variable; B is a known constant real matrix; u(t) is the control input matrix; g(z(L,t)) is a matrix composed of unknown nonlinear functions; This represents the error signal of the Nth agent. This serves as the control input for the Nth agent. Let be the unknown nonlinear function corresponding to the Nth agent.

[0035] Preferably, the composite adaptive neural network security boundary consistency control scheme includes:

[0036]

[0037] In the formula, u s (t) represents the security boundary consistency control part that ensures mean square security consistency control under hybrid attacks; To control the gain; It is the communication weight matrix between agents; a ij b represents the coupling weight coefficients between agents; i The coupling weight coefficient between agent i and the leader; Let D be the Laplace matrix of the system, and let D = diag{d1,d2,...,d...} n}, It is the in-degree matrix of the system; B*=diag{b1,b2,...,b n}; This is the adaptive control section; This is an error signal that takes into account the impact of hybrid attacks. h(z(L,t)) is the matrix formed by the optimal weight estimation, and z is the matrix formed by the radial basis functions. j (L, t) represents the state of agent j at boundary L, z i (L, t) represents the state of agent i at boundary L, z l (L, t) represents the state of the leader at the boundary L.

[0038] Preferably, the adaptive control section The weight parameters of the neural network are estimated using an adaptive weight update law, specifically as follows:

[0039]

[0040] In the formula, P is a positive definite matrix; A is a known positive definite matrix; h is the derivative of the estimate of the optimal weights. i (z i (L,t)) represents the radial basis functions of the neural network; γ represents the agent's error at the boundary; i Let be a given positive real number; This is an estimate of the optimal weights.

[0041] Preferably, the sufficient condition for the error system to achieve mean-square safe and consistent control is obtained by solving linear matrix inequalities, specifically:

[0042]

[0043] In the formula, Ψ is a real symmetric matrix.

[0044] Compared with the prior art, the present invention has the following advantages and technical effects:

[0045] (1) The composite adaptive neural network security boundary consistency control method proposed in this invention effectively solves the problem that multi-agent spatiotemporal dynamic systems are unable to cope with complex network attacks and unknown boundary nonlinear disturbances at the same time.

[0046] (2) The boundary control method proposed in this invention significantly reduces hardware deployment costs while ensuring system robustness by deploying only a limited number of sensors and actuators at the spatial domain boundary. This method effectively solves the high cost problem caused by the deployment of actuators throughout the entire spatial domain in traditional methods and has significant application value in engineering practice. Attached Figure Description

[0047] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings:

[0048] Figure 1 This is a flowchart of an adaptive security consistency control method for a multi-agent spatiotemporal dynamic system with unknown boundary nonlinearity under hybrid attacks, according to an embodiment of the present invention.

[0049] Figure 2 This is a topology diagram of an embodiment of the present invention;

[0050] Figure 3 This is a schematic diagram illustrating the evolution of the spatiotemporal state error of a multi-agent system under open-loop control mode according to an embodiment of the present invention.

[0051] Figure 4 This is a schematic diagram illustrating the evolution of spatiotemporal state error in a multi-agent system under closed-loop control mode according to an embodiment of the present invention.

[0052] Figure 5 This is a schematic diagram of the weight coefficients of the adaptive neural network in an embodiment of the present invention. Detailed Implementation

[0053] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0054] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0055] This embodiment proposes an adaptive security and consistency control method for multi-agent spatiotemporal dynamic systems with unknown boundary nonlinearity under hybrid attacks, such as... Figure 1 ,include:

[0056] Construct a nonlinear multi-agent spatiotemporal dynamic system model with unknown boundaries and a virtual leader model;

[0057] A hybrid network attack model incorporating spoofing attacks and denial-of-service (DoS) attacks is established, and an adaptive radial basis function is used to approximate the nonlinear function of the unknown boundary.

[0058] By using the multi-agent spatiotemporal dynamic system model, the virtual leader model, and the hybrid network supply model, a consistency error signal is defined to obtain a consistency state error system.

[0059] Design a composite adaptive neural network security boundary consistency control scheme. For the error system, construct a Lyapunov function to obtain sufficient conditions for the error system to achieve mean square security consistency control, thereby realizing adaptive security boundary consistency control of a multi-agent spatiotemporal dynamic system under hybrid attacks and unknown boundary nonlinearity.

[0060] Furthermore, a nonlinear multi-agent spatiotemporal dynamic system model with unknown boundaries and a virtual leader model are constructed, including:

[0061] A multi-agent spatiotemporal dynamic system model is constructed based on parabolic partial differential equations. The multi-agent spatiotemporal dynamic system includes N agents, each labeled i = 1, 2, ..., N, and the leader agent is labeled l.

[0062] In a multi-agent spatiotemporal dynamic system with unknown boundaries and nonlinearity, a mathematical model including a follower agent is specifically defined as follows:

[0063]

[0064] In the formula, Let represent the state of the i-th agent at time t, where 'a' is a spatial variable, and The time variable t > 0; Let u be the initial value for N follower agents; i (t) represents the control input at the boundary; A is a known positive definite matrix; B is a known constant real matrix; g(z) i (L,t) is an unknown nonlinear function.

[0065] The virtual leader model is as follows:

[0066]

[0067] In the formula, This represents the state of the leader agent at time t, point a; L is the upper boundary of the spatial domain. This represents the initial state of the leader agent. This represents the initial state of the leader agent.

[0068] Furthermore, the hybrid network attack model, which includes spoofing attacks and denial-of-service (DoS) attacks, is as follows:

[0069]

[0070] In the formula, To account for the error signal after considering the effects of mixed attacks; ρ(e(L,t)) is a nonlinear function satisfying the inequality: ||ρ(e(L,t))||2≤||Ge(·,t)||2, where G is a constant matrix determined by the nonlinear function ρ(e(·,t)); α s (t)∈{0,1}(s=1,2) is a random variable that satisfies the Bernoulli distribution.

[0071] Specifically, α s (t)∈{0,1}(s=1,2) is a random variable that satisfies the following Bernoulli distribution:

[0072]

[0073] In the formula, For the expected value of a random variable, This represents the probability of being attacked.

[0074] There are three cases when using the Bernoulli distribution to represent a mixed random network attack model:

[0075] (1) If α1(t) = 0 and α2(t) = 1, the system will be subject to a deception attack, causing the controller to receive maliciously tampered data ρ(e(L,t));

[0076] (2) If α1(t) = 1 and α2(t) = 0, the system will be subject to a DoS attack and the controller will be unable to receive any transmitted data;

[0077] (3) If α1(t) = 0 and α2(t) = 0, the system will not be attacked by the network.

[0078] Furthermore, an adaptive radial basis function neural network is used to approximate the nonlinear function of the unknown boundary, including:

[0079]

[0080] In the formula, and h i (z i (L,t) represent the optimal weights and radial basis functions of the neural network, respectively; approximation error in Given a positive number, defined This represents an estimate of the optimal weights;

[0081] The radial basis function is a Gaussian function:

[0082]

[0083] In the formula, υ h and z c Let z represent the width and center of the radial basis functions, respectively. i The state of the agent.

[0084] Furthermore, the consistency error signal is:

[0085] e i (a,t)=z i (a,t)-z l (a,t),

[0086] In the formula, e i (a,t) represents the consistency error signal; z i (a,t) represents the state of the i-th agent at time t at point a; z l (a,t) represents the state of the leader agent at point a at time t.

[0087] Define systematic error e i (a,t)=z i (a,t)-z l (a,t) can transform a leader-follower multi-agent spatiotemporal dynamic system with unknown boundary nonlinearity into an error system with spatiotemporal dynamics:

[0088]

[0089]

[0090] In the formula, e(a,t) is the agent error signal matrix; I N A is the identity matrix; A is a known positive definite matrix; a is a spatial variable; B is a known constant real matrix; u(t) is the control input matrix; g(z(L,t)) is a matrix composed of unknown nonlinear functions; This represents the error signal of the Nth agent. This serves as the control input for the Nth agent. Let be the unknown nonlinear function corresponding to the Nth agent.

[0091] Specifically, for any given initial state, if the following equation holds, then the multi-agent spatiotemporal dynamic system is said to achieve consistent control:

[0092]

[0093] Furthermore, to address the security consistency control problem of multi-agent spatiotemporal dynamic systems with unknown boundary nonlinearity under hybrid attacks, the following boundary consistency control scheme based on a composite adaptive neural network is proposed, including:

[0094]

[0095] In the formula, u s (t) represents the security boundary consistency control part that ensures mean square security consistency control under hybrid attacks; To control the gain; It is the communication weight matrix between agents; a ij b represents the coupling weight coefficients between agents; i The coupling weight coefficient between agent i and the leader; Let D be the Laplace matrix of the system, and let D = diag{d1,d2,...,d...} n}, It is the in-degree matrix of the system; B*=diag{b1,b2,...,b n}; This is the adaptive control section; This is an error signal that takes into account the impact of hybrid attacks. h(z(L,t)) is the matrix formed by the optimal weight estimation, and h(z(L,t)) is the matrix formed by the radial basis functions. j (L,t) represents the state of agent j at boundary L, z i (L, t) represents the state of agent i at boundary L, z l (L, t) represents the leader's state at boundary L. This is an estimate of the optimal weights.

[0096] Adaptive control section The weight parameters of the neural network are estimated using an adaptive weight update law, specifically as follows:

[0097]

[0098] In the formula, P is a positive definite matrix; A is a known positive definite matrix; h is the derivative of the estimate of the optimal weights. i (z i (L,t)) represents the radial basis functions of the neural network; γ represents the agent's error at the boundary; i Let be a given positive real number; This is an estimate of the optimal weights.

[0099] Furthermore, consensus control was achieved in a multi-agent spatiotemporal dynamic system with unknown boundary nonlinearity under hybrid attacks. Given positive numbers θ, δ, and λ, the probability of an attack occurring... and gain matrix If a positive definite matrix exists This makes the following LMI true:

[0100] Formula 1:

[0101] in:

[0102]

[0103] In the formula, θ, δ, and λ are given positive numbers; The probability of an attack occurring. is the gain matrix. I is the identity matrix.

[0104] For error systems, the following Lyapunov function is constructed:

[0105]

[0106] Taking the derivative of the above equation and obtaining the expectation, we get...

[0107] By combining inequality techniques and Equation 1, mean-square consistency control can be achieved for multi-agent spatiotemporal dynamic systems with unknown boundary nonlinearity under hybrid attacks.

[0108] To more clearly illustrate the technical solution of the present invention, specific embodiments are provided below for description:

[0109] This embodiment provides an adaptive security and consistency control method for a multi-agent spatiotemporal dynamic system with unknown boundary nonlinearity under hybrid attacks, the method comprising:

[0110] Step 1: Construct a multi-agent spatiotemporal dynamic system model based on parabolic partial differential equations. The multi-agent spatiotemporal dynamic system includes N agents, each labeled i = 1, 2, ..., N, and the leader agent is labeled l.

[0111] The mathematical model of a single follower agent in a multi-agent spatiotemporal dynamic system with unknown boundaries and nonlinearity is as follows:

[0112]

[0113] in, Represents the state of the i-th agent at time t, at position a, and is a spatial variable. The initial values ​​of the N follower agents are expressed as follows: (Time variable t > 0) u i(t) represents the control input at the boundary, A is a known positive definite matrix, B is a known constant real matrix, and g(z) i (L,t) is an unknown nonlinear function.

[0114] The leader agent model in a multi-agent spatiotemporal dynamic system is as follows:

[0115]

[0116] in, This represents the state of the leader agent at time t, where a is the initial state of the leader agent.

[0117] Step Two: Hybrid network attacks include spoofing attacks and DoS attacks. The Bernoulli distribution is used to represent the hybrid random network attack model, and its mathematical expression is:

[0118]

[0119] in, It is the error signal after considering the influence of mixed attacks. ρ(e(L,t)) is a nonlinear function that satisfies the following inequality: ||ρ(e(L,t))||2≤||Ge(·,t)||2, where G is a constant matrix determined by the nonlinear function ρ(e(·,t)).

[0120] α s (t)∈{0,1}(s=1,2) is a random variable that satisfies the following Bernoulli distribution:

[0121]

[0122] There are three cases when using the Bernoulli distribution to represent a mixed random network attack model:

[0123] (1) If α1(t) = 0 and α2(t) = 1, the system will be subject to a deception attack, causing the controller to receive maliciously tampered data ρ(e(L,t));

[0124] (2) If α1(t) = 1 and α2(t) = 0, the system will be subject to a DoS attack and the controller will be unable to receive any transmitted data;

[0125] (3) If α1(t) = 0 and α2(t) = 0, the system will not be attacked by the network.

[0126] The following adaptive radial basis neural network is used to approximate the unknown nonlinear function at the boundary:

[0127]

[0128] in, and h i (z i (L,t) represent the optimal weights and radial basis functions of the neural network, respectively. Approximation error. Where is a given positive number. Definition in This represents an estimate of the optimal weights. The following Gaussian function is chosen:

[0129]

[0130] Among them, υ h and z c These represent the width and center of the radial basis function, respectively.

[0131] Step 3: Define the systematic error e i (a,t)=z i (a,t)-z l (a,t) can transform the leader-follower multi-agent spatiotemporal dynamic system with unknown boundary nonlinearity into the following error system with spatiotemporal dynamics:

[0132]

[0133] For any given initial state, if the following equation holds, then the multi-agent spatiotemporal dynamic system is said to achieve consistent control.

[0134]

[0135] Step 4: To address the security consistency control problem of multi-agent spatiotemporal dynamic systems with unknown boundary nonlinearity under hybrid attacks, the following composite adaptive neural network boundary consistency control scheme is proposed:

[0136]

[0137] In the formula, u s (t) represents the security boundary consistency control part, which guarantees mean square security consistency control under hybrid attacks. It is about controlling the gain. This is the communication weight matrix between agents. If there is communication between agent j and agent i, then a ij >0, otherwise a ij =0, b i Let b be the coupling weight coefficient between agent i and the leader. i When b > 0, data transmission is possible between agent i and the leader; when b i When B* = 0, data transmission is not possible between the leaders of agents i. B* = diag{b1, b2, ..., b n} Let D be the Laplace matrix of the system, and let D = diag{d1,d2,...,d...} n}, It is the in-degree matrix of the system.

[0138] For the adaptive control part, the weight parameters of the neural network are estimated using the following adaptive weight update law:

[0139]

[0140] Where P is a positive definite matrix.

[0141] Step 5: Consistent control was achieved in a multi-agent spatiotemporal dynamic system with unknown boundaries and nonlinearity under hybrid attacks. Given positive numbers θ, δ, and λ, the probability of the attack occurring is... and gain matrix If a positive definite matrix exists This makes the following LMI hold true: Formula 1: in:

[0142]

[0143] For error systems, the following Lyapunov function is constructed: Taking the derivative of the above equation and obtaining the expectation, we get...

[0144] Where, ξ(a,t)=[e T (L,t),e T (a,t),ρ T (e(L,t))] T , β=min{μ i ,δ},

[0145] Let N = P -1 , Using inequality techniques and Schulbu's theorem, and multiplying both sides of Φ by... Φ can be converted to Ψ.

[0146]

[0147] Combining inequality techniques and Formula 1, we can obtain That is, the exponent of V(t) converges to the set Furthermore, it is found that the state error between the leader and followers tends to zero within the set; therefore, the mean square safe consistency control is achieved in the multi-agent spatiotemporal dynamic system with unknown boundary nonlinearity under hybrid attacks, under the boundary control protocol of composite adaptive neural network.

[0148] Step Six: Select an example for simulation verification;

[0149] Consider a leader-follower multi-agent spatiotemporal dynamic system consisting of 5 nodes, with the following topology: Figure 2 As shown.

[0150] Choose the following matrix and parameters:

[0151]

[0152] ρ(e(a,t))=tanh(e(a,t)), F=diag{1,1,1,1}, θ=1, δ=0.5, λ=0.15.

[0153] Using the LMI toolbox in MATLAB, we can obtain:

[0154]

[0155] This makes Equation 1 valid. Therefore, multi-agent spatiotemporal systems can achieve mean-square safe and consistent control.

[0156] The initial state of the agent is selected as follows:

[0157]

[0158] Figure 3 and Figure 4 The evolution of spatiotemporal state errors in a multi-agent system under open-loop and closed-loop control modes is illustrated. Simulation data shows that the open-loop system without an adaptive neural network boundary consistency control protocol (…) Figure 3 In this system, the system state errors of each agent exhibit divergent characteristics over time, with the system error showing a divergent trend as time increases. In stark contrast, the closed-loop system employing the adaptive neural network boundary consistency control protocol designed in this embodiment... Figure 4 It exhibits significant error suppression capability; the state errors of all agents decay exponentially under control, eventually converging over time. The weight coefficients of the adaptive neural network ( Figure 5 The method eventually converges over time. Therefore, the simulation results further demonstrate the effectiveness of the proposed method.

[0159] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), Synchlink, DRAM (SLDRAM), RAMbus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and RAMbus dynamic RAM (RDRAM), etc.

[0160] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. An adaptive security and consistency control method for a multi-agent spatiotemporal dynamic system with unknown boundary nonlinearity under hybrid attacks, characterized in that, include: Construct a nonlinear multi-agent spatiotemporal dynamic system model with unknown boundaries and a virtual leader model; A hybrid network attack model incorporating spoofing attacks and denial-of-service (DoS) attacks is established, and an adaptive radial basis function is used to approximate the nonlinear function of the unknown boundary. By using the multi-agent spatiotemporal dynamic system model, the virtual leader model, and the hybrid network attack model, a consistency error signal is defined, and a consistency state error system is obtained. Design a composite adaptive neural network security boundary consistency control scheme. For the error system, construct a Lyapunov function to obtain sufficient conditions for the error system to achieve mean square security consistency control, thereby realizing adaptive security boundary consistency control of a multi-agent spatiotemporal dynamic system under hybrid attacks and unknown boundary nonlinearity. The hybrid network attack model, which includes spoofing attacks and denial-of-service (DoS) attacks, is as follows: ， In the formula, Error signal after taking into account the impact of hybrid attacks; It is a nonlinear function that satisfies the following inequalities: , It is composed of nonlinear functions A defined constant matrix; For random variables that satisfy the Bernoulli distribution; Approximating nonlinear functions with unknown boundaries using adaptive radial basis function neural networks includes: ， In the formula, and These are the optimal weights and radial basis functions of the neural network; approximation error. ,in Given a positive number, defined , This represents an estimate of the optimal weights; The radial basis function is a Gaussian function: ， In the formula, and Let represent the width and center of the radial basis functions, respectively. The state of the agent; The consistency error signal is: ， In the formula, This is a consistency error signal; For the first An intelligent agent in time The state of being; For the leader intelligent agent in time The state of being; The consistency state error system is as follows: ， ， , ， In the formula, The agent's error signal matrix; It is the identity matrix; It is a known positive definite matrix; a For spatial variables; A known constant real matrix; To control the input matrix; A matrix composed of unknown nonlinear functions; This represents the error signal of the Nth agent. This serves as the control input for the Nth agent. Let be the unknown nonlinear function corresponding to the Nth agent; The composite adaptive neural network security boundary consistency control scheme includes: ， ， ， In the formula, The security boundary consistency control part ensures mean square security consistency control under hybrid attacks; To control the gain; It is the communication weight matrix between intelligent agents; These are the coupling weight coefficients between agents; For intelligent agents Coupling weight coefficient between the leader and the leader; Let Laplace's matrix be the system's matrix. , It is the in-degree matrix of the system; , ; This is the adaptive control section; This is an error signal that takes into account the impact of hybrid attacks. The matrix formed by the optimal weight estimation. The matrix formed by the radial basis functions. For intelligent agents j At the border L The state of being, For intelligent agents i At the border L The state of being, For leaders at the boundary L The state at which it is located.

2. The method according to claim 1, characterized in that, Constructing a nonlinear multi-agent spatiotemporal dynamic system model with unknown boundaries and a virtual leader model, including: A nonlinear multi-agent spatiotemporal dynamic system model with unknown boundaries and a virtual leader model are constructed based on parabolic partial differential equations. The nonlinear multi-agent spatiotemporal dynamic system with unknown boundaries includes a mathematical model of follower agents, specifically: ， In the formula, Indicates the first An intelligent agent in time The state of being; a It is a spatial variable, and Time variable ; for Initial values ​​for each follower agent; For control input at the boundary; It is a known positive definite matrix; A known constant real matrix; It is an unknown nonlinear function.

3. The method according to claim 2, characterized in that, The virtual leader model is as follows: ， In the formula, This indicates that the leader agent is in time The state of being; L It is the upper boundary of the spatial domain; This represents the initial state of the leader agent.

4. The method according to claim 1, characterized in that, The adaptive control section The weight parameters of the neural network are estimated using an adaptive weight update law, specifically as follows: ， In the formula, It is a positive definite matrix; It is a known positive definite matrix; The derivative of the estimate of the optimal weights; For the radial basis functions of the neural network; This represents the agent's error at the boundary. Let be a given positive real number; This is an estimate of the optimal weights.

5. The method according to claim 1, characterized in that, The sufficient condition for the error system to achieve mean-square safe and consistent control is obtained by solving the linear matrix inequality, specifically: ， In the formula, It is a real symmetric matrix.

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