Optimal tracking control method for unmanned aerial vehicle swarm under denial-of-service attack

Abstract

This invention discloses a zero-sum game optimal tracking control method for UAV swarms under denial-of-service attacks. It establishes a communication topology model of the UAV swarm and constructs a nonlinear dynamics model of the UAVs containing unknown external interference. Based on a zero-sum game strategy, it constructs a stabilizing controller and a cooperative tracking controller. It identifies whether a follower UAV is under a denial-of-service attack based on the communication topology state, switches to stabilizing control for attacked UAVs to suppress state divergence, and reconstructs the communication topology for unattacked UAVs while maintaining cooperative tracking control. Simultaneously, a memory self-triggering mechanism is introduced to determine the communication update time based on the combined effect of historical and current information. This invention can improve the anti-attack capability, topology recovery capability, and cooperative flight stability of UAV swarms under conditions of communication link obstruction and external disturbances, while reducing continuous detection and communication resource consumption.

Description

Technical Field

[0001] This invention relates to the cooperative control of multiple unmanned aerial vehicle (UAV) systems, specifically to an optimal tracking and control method for a UAV swarm in a zero-sum game under a denial-of-service attack. Background Technology

[0002] Unmanned aerial vehicles (UAVs) are aircraft capable of autonomous flight or remote control. They are typically operated via radio remote control equipment or onboard program control systems, but can also achieve fully or semi-autonomous flight through onboard computers. Compared to manned aircraft, UAVs offer significant advantages such as lower cost, higher safety, and greater maneuverability. In recent years, with the rapid development of control technology, communication technology, computer science, and new materials technology, the performance and application areas of UAVs have continuously expanded, making them a hot topic in aerospace and intelligent systems research. In the civilian sector, UAVs play a vital role in aerial surveying and mapping, agricultural and forestry plant protection, disaster relief, logistics and transportation, environmental monitoring, power line inspection, news reporting, and film and television production.

[0003] Despite continuous improvements in the performance and functionality of individual drones, their mission execution capabilities remain limited by factors such as platform payload, endurance, and reliability. In contrast, multi-drone cooperative control technology effectively overcomes the limitations of single-drone operation, achieving higher mission completion rates and system robustness. When an individual drone fails during mission execution due to attacker attacks or equipment malfunctions, other drones in the swarm can autonomously take over the mission, ensuring the continuity and stability of the overall mission. Furthermore, drone formation control is a crucial component of multi-drone systems, encompassing formation generation, maintenance, and dynamic reconfiguration. Depending on different mission requirements, the drone swarm can flexibly adjust its formation to achieve optimal cooperation, significantly improving system survivability, mission execution efficiency, and overall security. This cooperative mechanism is particularly important in complex environments, effectively leveraging the advantages of swarm intelligence.

[0004] During actual flight, the control performance of unmanned aerial vehicles (UAVs) is not only affected by onboard resource limitations and complex environmental factors, but also susceptible to various security threats due to the openness, sharing, and insufficient protection measures of communication networks. Attackers may launch malicious attacks such as interference, hijacking, and denial-of-service attacks on UAV systems through network or physical layer means, resulting in serious consequences such as communication interruption, control failure, or even system paralysis or crash. Such attacks threaten the reliability of mission execution.

[0005] Furthermore, in the collaborative control of multiple unmanned aerial vehicle (UAV) systems, if a follower UAV adopts a fixed-period sampling method and transmits the collected status information to the navigator UAV or other nearby follower UAVs in real time, a large amount of duplicate or redundant information will be generated when the sampled data changes little. This continuous high-frequency data transmission not only occupies limited communication bandwidth but may also lead to network congestion or even communication link paralysis, thereby affecting the system's real-time performance and stability. To effectively reduce communication load and improve network utilization, event-triggered control strategies have been widely researched and applied. This method achieves "on-demand communication" by transmitting information only when the system state meets specific trigger conditions, thereby significantly reducing the number of communications and energy consumption while ensuring system performance. However, it requires frequent detection of trigger conditions and has high system computational complexity. Summary of the Invention

[0006] Purpose of the invention: To address the above-mentioned shortcomings, this invention provides an optimal tracking and control method for drone swarms under denial-of-service attacks, which improves stability through a zero-sum game approach.

[0007] Technical Solution: To solve the above problems, this invention adopts a zero-sum game optimal tracking and control method for drone swarms under denial-of-service attacks, including the following steps:

[0008] A communication topology model for a drone swarm is established, comprising a leader drone and several follower drones. The communication topology model includes information transmission from the leader drone to the follower drones and information communication between the follower drones. A nonlinear model of the drones considering unknown external interference is constructed. Based on a zero-sum game strategy, a stabilization controller and a cooperative tracking controller are constructed in conjunction with the nonlinear model of the drones.

[0009] Based on the communication topology model of the drone swarm, it is determined whether the follower drone is unable to communicate with its neighbors. If not, the follower drone is controlled by a cooperative tracking controller. If so, it is determined that the follower drone is under denial-of-service attack, and the follower drone under denial-of-service attack is controlled by a stabilizing controller. The communication topology model of the follower drones that are not under denial-of-service attack is reconstructed, and the original cooperative tracking controller is used for control.

[0010] The stabilization controller is:

[0011] ;

[0012] ;

[0013] in, This represents the estimated value of the optimal virtual control signal. This represents the estimated value of the worst-case interference input signal. Indicates the first The position vector of the follower drone. This represents the estimated value of the ideal evaluation weights for the optimal cost function under a zero-sum game strategy. The Euclidean norm of the Gausky function, representing the optimal cost function under a zero-sum game strategy. , Represents design constants. This indicates the degree of interference suppression.

[0014] Furthermore, an evaluation neural network is used to estimate the ideal evaluation weights of the optimal cost function under a zero-sum game strategy. The update rule for the evaluation neural network is obtained using gradient descent.

[0015] ;

[0016] ;

[0017] in, Indicates the learning rate. Indicates intermediate variables. Represents the Bellman residual. Represents the number of nodes in the neural network. Indicates the first Storage time, Indicates the total number of storage operations. Indicate intermediate variables In the The variable value at the next storage time. Represents the Bellman residual In the The variable value for the storage time.

[0018] Furthermore, the cooperative tracking controller is:

[0019] ;

[0020] ;

[0021] in, The degree matrix represents the first The correlation between the follower drone and the drone. Indicates the relationship with the first The adjacent follower drone The derivative of the output signal of the follower drone, Indicates formation error. Indicates the first The connection between the follower drone and the navigator drone. , Represents design constants. Indicates the degree of interference suppression. This indicates the total number of follower drones. Indicates the first Follower drone and the first Communication between follower drones This indicates the number of follower drones affected by a denial-of-service attack. This represents the estimated value of the ideal evaluation weights for the optimal cost function under a zero-sum game strategy. Let Gausky function be the Euclidean norm of the optimal cost function under a zero-sum game strategy.

[0022] Furthermore, a memory self-triggering mechanism is introduced for follower drones that have not been subjected to denial-of-service attacks, and the follower drones are actively controlled based on the memory self-triggering mechanism;

[0023] The controller based on the memory self-triggering mechanism is:

[0024] ;

[0025] in, This represents the optimal control signal considering the memory self-triggering mechanism. This represents the worst-case interference signal considering the memory self-triggering mechanism. Indicates the first The reciprocal of the mass of the drone. This represents a coordinate transformation based on dynamic surface technology. Indicates the first The velocity vector of the drone Indicates a self-triggered variable. , , , Represents design constants. and It is an estimate of the ideal evaluation weights of the optimal cost function under a zero-sum game strategy. The Euclidean norm of the Gausky function, representing the optimal cost function under a zero-sum game strategy. This indicates that the optimal cost function under a zero-sum game strategy includes self-triggered variables. The Euclidean norm of the Gausky function, This represents a coordinate transformation based on dynamic surface technology. This indicates the optimal virtual control signal without considering the memory self-triggering mechanism.

[0026] Furthermore, the triggering condition for the memory self-triggering mechanism is as follows:

[0027]

[0028] in, Indicates the first The memory of the follower drone is a self-triggered variable. Indicates the trigger time. Indicates the first Follower drone Output signal at time, Indicates the first Follower drone Output signal at time, It is a positive integer. It is a positive integer. , , , , , For design constants, express Formation error at any moment, Representing variables exist Rate of change over time.

[0029] Furthermore, the formula for calculating the optimal virtual control signal based on the memory self-triggering mechanism is as follows:

[0030]

[0031] in, This represents the estimated value of the optimal virtual control signal based on the memory self-triggering mechanism. This represents the estimated value of the worst-case interference input signal based on the memory self-triggering mechanism. This indicates that the variable contains a memory-triggered variable. The Find the first derivative of the output signal of the follower drone. The optimal cost function under a zero-sum game strategy includes memory-triggered variables. The Euclidean norm of the Gausky function, This indicates the formation error considering the memory self-trigger mechanism.

[0032] Furthermore, the triggering condition for the virtual control memory self-triggering mechanism is as follows:

[0033] ;

[0034] in, Indicates the first Follower drone The memory of a moment is a self-triggered variable. Indicates the first Follower drone The output signal at time, Indicates the first Follower drone The output signal at time, It is a positive integer. It is a positive integer. , , , , , For design constants, express Formation error at any moment, Representing variables exist Rate of change over time.

[0035] Furthermore, an evaluation neural network is used to estimate the ideal evaluation weights of the optimal cost function under a zero-sum game strategy. The update rule for the evaluation neural network is obtained using gradient descent.

[0036] ;

[0037] ;

[0038] ;

[0039] ;

[0040] ;

[0041] in, , Indicates the learning rate. , Indicates intermediate variables. , Represents the Bellman residual. Represents the number of nodes in the neural network. Indicates the first Storage time, Indicates the total number of storage operations. Indicate intermediate variables In the The variable value at the next storage time. Represents the Bellman residual In the The variable value at the next storage time. Indicate intermediate variables In the The variable value at the next storage time. Represents the Bellman residual In the The variable value at the next storage time. , and For design constants, , and This represents the weight update law. This represents the derivative of the output signal of the Navigator drone. Let Gausky function be the Euclidean norm of the optimal cost function under a zero-sum game strategy.

[0042] Furthermore, the constructed nonlinear model of the UAV considering unknown external disturbances is as follows:

[0043] ;

[0044] in, Indicates the first The position vector of the follower drone. Indicates the first The velocity vector of the follower drone This indicates the parts of the system that are not modeled or ignored. and It is an external disturbance. Indicates the first The output signal of the follower drone, Indicates the first Input signals from a follower drone.

[0045] The present invention also employs a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the above method.

[0046] The present invention also employs a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the above method.

[0047] Beneficial effects: Compared with the prior art, the significant advantages of this invention are: it considers the control strategy of the drone swarm when it is subjected to denial-of-service attacks, switches the drones subjected to denial-of-service attacks to calm control, so as to enhance the system's defense capabilities against attacks such as network intrusion, signal interference and data tampering, and ensure that the swarm can still maintain stable operation and coordinated flight when attacked, thereby improving the security and robustness of multi-drone swarm systems in complex and adversarial environments.

[0048] By adopting a self-triggering control mechanism, the computational complexity of the system is reduced and the frequent detection of triggering conditions is avoided. This mechanism can predict the next triggering time at the time of the previous triggering, without the need for continuous monitoring of the system status. This reduces the computational burden, improves the utilization efficiency of communication resources, reduces communication frequency, and reduces network resource consumption, thus showing broad application prospects in distributed control and network-constrained systems. Attached Figure Description

[0049] Figure 1 This is a flowchart illustrating the control method of the present invention.

[0050] Figure 2 This is a schematic diagram illustrating the communication topology switching of the drone cluster under a denial-of-service attack according to the present invention. Detailed Implementation

[0051] like Figure 1 As shown in this embodiment, a zero-sum game optimal tracking and control method for a drone swarm under a denial-of-service attack includes the following steps:

[0052] A communication topology model for a drone swarm is established, comprising a leader drone and several follower drones. The communication topology model includes information transmission from the leader drone to the follower drones and information communication between the follower drones. A nonlinear model of the drones considering unknown external interference is constructed. Based on a zero-sum game strategy, a stabilization controller and a cooperative tracking controller are constructed in conjunction with the nonlinear model of the drones.

[0053] Based on the communication topology model of the drone swarm, it is determined whether the follower drone is unable to communicate with its neighbors. If not, the follower drone is controlled by a cooperative tracking controller. If so, it is determined that the follower drone is under denial-of-service attack, and the follower drone under denial-of-service attack is controlled by a stabilizing controller. The communication topology model of the follower drones that are not under denial-of-service attack is reconstructed, and the original cooperative tracking controller is used for control.

[0054] The specific control method in this embodiment includes the following steps:

[0055] Step 1: Establish the communication topology model of the drone swarm, specifically including: introducing a directed graph. This represents the information exchange between drones, where This represents the set of all drone nodes. Denotes the set of edges. Represents an adjacency matrix, with elements Indicates the first The drone and the first The connection between drones and For adjacent drones, if Then it means the first The drone is able to start from the first If drones are used to acquire information, This means that two adjacent drones cannot exchange information, and at the same time, it defines... Define the degree matrix and Laplace matrix ,in , representing the degree matrix of the first The correlation between the follower drone and the drone.

[0056] Augmented graph ,in and And nodes This indicates the navigator drone node. Similarly, if Then it means the first The drone can obtain information from the Navigator drone, if This indicates that two adjacent drones cannot exchange information.

[0057] Assumption 1: Augmented graph It is connected and has Navigator drone nodes. For the spanning tree of the root node, the node The information can be sent to all other nodes along a directed path, and each follower drone sends information to at least one neighbor.

[0058] Step 2: Construct a nonlinear model of the UAV considering unknown external disturbances, specifically including the following UAV position dynamic model:

[0059] (1);

[0060] In the formula , and They are the first The position vector and velocity vector of the drone. It is the first The quality of the drone It is gravitational acceleration. It is a combination of external forces. Represents the unknown diagonal aerodynamic matrix. This is the transformation matrix between the body coordinate system and the ground coordinate system, and its expression is as follows:

[0061] (2)

[0062] in, , and It represents Euler angles.

[0063] Considering the external interference that the UAV may experience during flight, and combining equations (1)-(2), the first... The nonlinear equations for a drone can be rewritten as:

[0064] (3);

[0065] in, , , , , , , , , This indicates the parts of the system that are not modeled or ignored. and It is an external disturbance. This represents positional disturbances, primarily corresponding to kinematic displacement deviations and measurement uncertainties. This indicates velocity-term disturbances, primarily corresponding to force and torque disturbances at the dynamic level. Is and These represent the output and input signals of the drone, respectively. The output signal of the Navigator drone is defined as follows: .

[0066] Assumption 2: Output signal of the Navigator drone and its first derivative and second derivative Both are bounded. That is, there exists a constant. satisfy: .

[0067] Assumption 3: Navigator Drone Node It is unaffected by denial-of-service attacks, and at least one follower drone can obtain information from the navigator drone.

[0068] Assumption 4: For the game group and and game group and Assume that there exists an equilibrium strategy in both sets of games that can guarantee the stability of the closed-loop drone swarm system.

[0069] Lemma 1: For a continuous unknown nonlinear function Radial basis function neural networks can effectively approximate it, and the expression is:

[0070] ;

[0071] in, This represents the approximation error and satisfies... , It is a constant. This represents the basis function vector of a radial basis function neural network. This represents the ideal weight vector.

[0072] Subsequently, the optimal solution of the ideal weight vector is defined as:

[0073] ;

[0074] in, Represents the number of nodes in the neural network. It is a weight vector.

[0075] Next, the basis functions are defined as Gaussian functions, whose expressions are:

[0076] ;

[0077] in, and Indicates design parameters.

[0078] Step 3: In a drone swarm system, drones rely on wireless communication networks for information exchange and collaborative decision-making to complete complex distributed control tasks. During system operation, critical information such as flight status, attitude angular velocity, and control commands needs to be frequently exchanged. However, this high dependence on network communication also brings potential security risks, especially in open communication environments where the system is vulnerable to malicious network-layer attacks. When a communication link encounters a denial-of-service attack, attackers may block the network channel with a large number of invalid requests, preventing some or all drones from receiving necessary control information, leading to service interruption. Such attacks not only cause drones to lose awareness of the leader or neighboring drones' status but may also cause serious consequences such as flight yaw and attitude instability. Because the consistency control strategy highly depends on real-time information transmission and synchronous updates, once communication is delayed or interrupted, the overall collaborative performance of the system will drop sharply, and may even lead to swarm disintegration or mission failure. Therefore, a control strategy with denial-of-service attack resistance can improve the reliability of the communication link and the system's anti-interference capability, thereby ensuring the smooth execution of swarm collaborative tasks and guaranteeing the safe and stable operation of the drone swarm system in complex network environments.

[0079] like Figure 2The diagram illustrates the interaction topology of a drone swarm under a denial-of-service (DoS) attack, including the connected graph before the attack and the paralyzed graph after the attack. This can be understood as follows: when a drone node suffers a DoS attack, all information of that node becomes unavailable. Typically, considering that DoS attacks are usually limited by finite attack resources and require continuous energy consumption, the communication mechanism should be able to quickly recover when the attack enters a dormant state. Based on this, a switching method combining calm control and cooperative control is introduced to avoid mission failure in the presence of a DoS attack, ensuring the stable operation and cooperative flight of the drone swarm.

[0080] To establish sufficient conditions for consistency control in a drone swarm system, the following definition is given:

[0081] Definition 1: For and If the inequality If true, then define The average stay time, of which , , , ,and Indicates in Number of switches within.

[0082] To conserve communication resources in drone swarms, a memory-based self-triggering mechanism is designed. This mechanism's triggering condition depends not only on current information but also on historical information. The main design concept is as follows:

[0083] (4);

[0084] In the formula , Indicates the trigger time. Indicates formation error. Indicates the first The relative positions of the follower drone and the navigator drone. Indicates the first The relative positions of the follower drone and the navigator drone. , , , , , and These are the constants to be designed.

[0085] Lemma 2: For consistency errors Consistency error based on memory event triggering mechanism ,in Under the premise that the triggering conditions are met, the following relationship holds:

[0086] ;

[0087] In the formula ,and .

[0088] Step 4: Design of attack-resistant control strategy for zero-sum game based on memory self-triggering mechanism.

[0089] First, a backstepping control technique is employed to design a zero-sum game control strategy for a drone swarm system under a denial-of-service attack through a two-step iterative design. Considering the impact of denial-of-service attacks on the communication and collaborative performance of the drone swarm system, a memory-based self-triggered collaborative control method based on the zero-sum game strategy is further designed. The design process of this method can be divided into two cases: the attack period and the non-attack period. Specifically, let the attack period be... Non-attack periods are During non-attack periods, the system communication topology remains connected, allowing for the direct construction of a cooperative zero-sum game control strategy and the design of a cooperative controller to ensure the stability and coordination of the closed-loop UAV swarm system. However, during attack periods, attacked nodes cannot exchange information with their neighbors. For these nodes, a stabilizing controller needs to be designed to suppress state divergence. Simultaneously, the communication topology model for unattacked nodes is reconstructed, and the cooperative controller's control is preserved, thus maintaining the overall stability and performance of the system even with localized communication disruptions. The following section will provide a detailed analysis of the attack period.

[0090] During the attack period Inside, assuming there are a total One drone was affected by a denial-of-service attack, the rest Unattacked drones employ a cooperative zero-sum game control strategy through topology reconfiguration to maintain the overall cooperative performance of the system. For attacked drones... Since each drone is unable to communicate with neighboring nodes, a separate stabilization controller needs to be designed to ensure its stability. Therefore, during this period, separate controllers will be assigned to unattacked drones. Design of a collaborative tracking and control method for unmanned aerial vehicles (UAVs), and for the protection of attacked UAVs. The design of the drone system employs a calming control method to maintain the overall stability and coordination of the cluster system even when communication is disrupted.

[0091] First, for those that have not been attacked We constructed a collaborative zero-sum game control framework for drones without considering the memory self-trigger mechanism, laying the foundation for subsequent mechanism design.

[0092] Step 1: Convert the interfering input signal With virtual control signals The system is modeled as having two independent participants. Game theory analysis is used to determine the balance between them, allowing for the simultaneous solution of the control law and disturbance law to maintain the stability of the closed-loop system. To achieve this, the following optimal cost function is defined:

[0093] (5);

[0094] In the formula express Formation error between drones and These represent the optimal virtual control signal and the worst-case disturbance input signal, respectively. and Indicates the constant to be designed. This indicates the degree of interference suppression and satisfies the condition. .

[0095] Based on the optimal cost function (5), the Hamilton-Jacobi-Isax equation is obtained using the Bellman optimality principle, and its expression is:

[0096] (6);

[0097] in ,and Subsequently, the following stability condition is used:

[0098] ;

[0099] The optimal virtual control law and the worst-case disturbance law are solved, and their specific expressions are as follows:

[0100] (7);

[0101] According to equation (7), the memory-triggered zero-sum game strategy can be obtained. The expression is as follows:

[0102] (8);

[0103] In the formula , and Indicates the presence of trigger variables The partial derivatives of the optimal virtual control law, worst-case disturbance law, and optimal cost function are given. Based on this, the Hamilton-Jacobi-Isax equation under the memory self-triggering mechanism is defined as follows:

[0104] (9);

[0105] To achieve consistent control in zero-sum games, the following will be... Decomposed into the following form:

[0106] (10);

[0107] In the formula, ,and This represents the constant to be designed.

[0108] Substituting equation (10) into equation (7), we get:

[0109] (11)

[0110] Then, a radial basis function neural network is introduced. To approximate, we have:

[0111] (12);

[0112] Since the zero-sum game strategy obtained in equation (12) still contains unknown ideal weight vectors, it cannot be directly applied. Therefore, an evaluation neural network with fewer learning parameters is introduced to approximate the solution for this strategy:

[0113] (13);

[0114] (14);

[0115] (15);

[0116] In the formula , It is the weight The estimated value. Furthermore... , and They represent , and The estimated value.

[0117] Based on equations (14)-(15), and further considering the role of the memory self-triggering mechanism in equation (4), the following zero-sum game strategy based on the memory self-triggering mechanism is obtained:

[0118] (16);

[0119] In the formula , This indicates the inclusion of memory-triggered variables. Gaussian function, It also includes memory-triggered variables. Subsequently, based on equations (9) and (16), the following approximate Hamilton-Jacobi-Isax equation under the memory self-triggering mechanism can be obtained:

[0120] (17);

[0121] In the formula This represents the Bellman residual. Next, the objective function is defined. Using gradient descent, we obtain the following evaluation rule for neural network updates:

[0122] (18);

[0123] ;

[0124] In the formula , Represents the number of nodes in the neural network. Indicates the learning rate. Indicates the first Storage time, and Representing variables respectively and In the The variable value for the storage time.

[0125] Construct the following Lyapunov function:

[0126] ;

[0127] in, and These are the constants to be designed. This represents the estimation error. Based on the design of the zero-sum game strategy described above, for... Differentiation yields:

[0128] ;

[0129] In the formula , and This represents a coordinate transformation based on dynamic surface technology. It is a first-order filter The exported filter variables, and , Indicates the constant to be designed. Representing variables In the The value at time. Then, introducing Young's inequality, we obtain the following inequality:

[0130] (19)

[0131] In the formula: ;

[0132] yes and The upper realm, and It is a constant. express The smallest eigenvalue.

[0133] Step 2: To achieve asymptotic stability of the closed-loop UAV swarm system and solve for the zero-sum game equilibrium strategy, the actual control law is designed using the backstepping method. Based on equation (1) and the above analysis, we have:

[0134] (20);

[0135] (twenty one);

[0136] In the formula .

[0137] Throughout the design process of the control method, the introduction of a memory-triggered zero-sum game strategy in the first step resulted in discontinuous control signals. Therefore, in the second step, the control signals before the memory-triggered mechanism were used during coordinate transformation. To ensure the effectiveness of the memory-triggered zero-sum game strategy designed in the first step, this step further introduces a memory-triggered strategy to compensate for the deviation caused by the use of pre-control signals during coordinate transformation. The specific expression is as follows:

[0138] (twenty two);

[0139] In the formula Indicates the trigger time. , , , , , and These are the constants to be designed.

[0140] Similar to the first step, we first design a zero-sum game control method that does not consider the memory self-triggering mechanism. The optimal value function is defined as follows:

[0141] (twenty three);

[0142] In the formula and These represent the optimal control law and the worst-case disturbance law, respectively. ,and Indicates the degree of interference suppression. , and These are the constants to be designed. Additionally, constants... and Meet the conditions Then, by taking the time derivative of the optimal value function, we can obtain the following Hamilton-Jacobi-Isax equation:

[0143] (twenty four);

[0144] In the formula Following the same approach as the first step, we solve...

[0145]

[0146] To derive the following zero-sum game control strategy:

[0147] (25);

[0148] In order to achieve zero-sum game control, Decomposed into the following form:

[0149] (26);

[0150] In the formula ;

[0151] and These are the constants to be designed.

[0152] Then, substituting equation (26) into equation (25) gives:

[0153] (27);

[0154] Because there is an unknown continuous nonlinear function in equation (27) and We need to introduce a radial basis function neural network to approximate it, that is... and However, the ideal weights obtained using neural networks... and It is unknown. Therefore, similar to the first step, an identification and evaluation neural network with fewer learning parameters is introduced, and a memory self-triggering mechanism (22) is considered to obtain the following zero-sum game control strategy: (28);

[0155] In the formula , , Describe the Euclidean norm of the Gaussian function. Indicates the presence of self-triggered variables. The Euclidean norm of the Gausky function, and These are the weights and The estimated value.

[0156] Subsequently, the following identification and evaluation law for neural network weight update was designed:

[0157] (29);

[0158] (30);

[0159] in:

[0160] ;

[0161] ;

[0162] ;

[0163] In the formula and These are the constants to be designed. Indicates the learning rate. It is the Bellman residual. and They represent and In the Data value at any given time.

[0164] Construct the following Lyapunov function:

[0165]

[0166] In the formula Indicates the constant to be designed. and This indicates the estimation error.

[0167] For the self-triggered strategy (22), when At that time, the following relationship holds:

[0168] (31);

[0169] In the formula It is a constant. Then, according to equation (31), we have:

[0170] (32);

[0171] (33);

[0172] Based on the above analysis and the characteristics of the Gaussian function, the following relationship holds:

[0173] (34);

[0174] (35);

[0175] (36);

[0176] In the formula , and Describe bounded time-varying parameters and satisfy the following conditions. , and .

[0177] Based on the above analysis, Differentiation yields:

[0178]

[0179] In the formula and Next, for equation (36), by using Young's inequality, we can obtain:

[0180] (37);

[0181] In the formula: ; yes The upper realm, and They represent The minimum and maximum eigenvalues.

[0182] Construct the following integrated Lyapunov function:

[0183]

[0184] Combining equations (19) and (37), we can obtain:

[0185] (38);

[0186] In the formula:

[0187] ;

[0188]

[0189]

[0190]

[0191]

[0192] Secondly, regarding those subjected to denial-of-service attacks The design concept for this drone is to implement an appropriate zero-sum game strategy, enabling it to achieve stable self-control and maintain system stability. The specific design concept is as follows:

[0193] Step 1: Define the optimal cost function as follows:

[0194] (39);

[0195] In the formula and Indicates the constant to be designed. This indicates the degree of interference suppression and satisfies the condition. .

[0196] Similarly, using the Bellman optimality principle, we obtain the following Hamilton-Jacobi-Isax equation:

[0197] (40);

[0198] In the formula Then, by solving:

[0199]

[0200] To obtain a control strategy for a zero-sum game. The specific expression is as follows:

[0201]

[0202] Then, in order to achieve zero-sum game-based calm control, Decomposed into the following form:

[0203] (41);

[0204] In the formula ,and Indicates the constant to be designed. Represents the weight vector. It is the Gaussian function. This represents the approximation error.

[0205] Subsequently, similar to the above analysis, an evaluation neural network with fewer learning parameters is introduced to derive the following zero-sum game strategy:

[0206] (42);

[0207] (43);

[0208] (44);

[0209] In the formula , It is the weight The estimated value.

[0210] The Bellman residual is defined as follows:

[0211] (45);

[0212] Next, define the objective function. Using gradient descent, we obtain the following evaluation rule for neural network updates:

[0213] (46);

[0214] in , This represents the learning rate.

[0215] Step 2: Solve based on the optimal value function, and simultaneously approximate the parameters using a radial basis function neural network and an evaluation neural network. This yields the following zero-sum game strategy and parameter update law:

[0216] (47)

[0217] (48);

[0218] (49);

[0219] in:

[0220] ;

[0221] ;

[0222] ;

[0223] In the formula ,and Indicates the degree of interference suppression. , and These are the constants to be designed. Additionally, constants... and Meet the conditions , It is the Bellman residual. and They represent and In the Data values ​​at any given time and Represent the Gaussian function respectively and Euclidean norm, and These are the weights and The estimated value, , , and These are the constants to be designed. This represents the learning rate.

[0224] Subsequently, the Lyapunov function is defined as follows:

[0225]

[0226] in , .right The time derivative can be obtained similarly. ,and and It is a constant.

[0227] when When all drones are unaffected by denial-of-service attacks, the system can achieve zero-sum game-based collaborative control based on a memory-triggered mechanism. Its design philosophy is similar to that of the aforementioned systems targeting... Maintaining a consistent collaborative control design for all drones will also yield similar results. This will not be elaborated upon here.

[0228] In summary, we can conclude that:

[0229]

[0230] in:

[0231] ;

[0232] .

[0233] Based on the above analysis and discussion, the following theorem is derived:

[0234] Theorem: Consider by A drone swarm system consisting of one follower drone and one navigator drone, wherein the first... The system model of the follower drone is shown in equation (3). If assumptions 1 to 4 are satisfied, and the following conditions are met during the attack period: the zero-sum game cooperative tracking control strategy of the unattacked drone satisfies equations (14)-(15), the zero-sum game calming control strategy of the attacked drone satisfies equations (43)-(44) and (47), the zero-sum game control strategy based on the memory self-triggering mechanism satisfies equation (28), and the parameter update law satisfies equations (18), (29)-(30), (46), and (48)-(49), if there are suitable parameters , , , as well as If we make them satisfy the given constraints, then we can draw the following conclusion: ;

[0235] The following conclusions can be drawn:

[0236] (1) All signals within the closed-loop system (3) are bounded;

[0237] (2) During the non-attack phase, all follower drones can accurately track the trajectory of the navigator drone, thereby achieving consistent control performance; during the attack phase, the attacked drones can maintain their own stability, i.e., calm control, while the unattacked drones can still track the trajectory of the navigator drone;

[0238] (3) Control strategies based on memory self-triggering mechanisms do not exhibit the Zeno phenomenon.

[0239] Proof: Consider all Lyapunov functions Exportable ,in and Therefore, we can conclude that in the absence of attack, the follower drone can effectively track the trajectory of the navigator drone; however, under attack, if the communication topology is disrupted, each drone can only maintain its own stable control; if the overall topology remains connected, all drones can still track and control the trajectory of the navigator drone. Thus, the theorem is proven.

Claims

1. A method for optimal tracking and control of a drone swarm in a zero-sum game under a denial-of-service attack, characterized in that, Includes the following steps: A communication topology model for a drone swarm is established, wherein the drone swarm includes a leader drone and several follower drones, and the communication topology model includes information transmission from the leader drone to the follower drones and information communication between the follower drones. And construct a nonlinear model of the UAV that considers unknown external disturbances; Based on a zero-sum game strategy, a stabilization controller and a cooperative tracking controller are constructed by combining a nonlinear model of an unmanned aerial vehicle. Based on the communication topology model of the drone swarm, it is determined whether the follower drone is unable to communicate with its neighbors. If not, the follower drone is controlled by a cooperative tracking controller. If so, it is determined that the follower drone is under denial-of-service attack, and the follower drone under denial-of-service attack is controlled by a stabilizing controller. The communication topology model of the follower drones that are not under denial-of-service attack is reconstructed, and the original cooperative tracking controller is used for control. The stabilization controller is: ； ； in, This represents the estimated value of the optimal virtual control signal. This represents the estimated value of the worst-case interference input signal. Indicates the first The position vector of the follower drone. This represents the estimated value of the ideal evaluation weights for the optimal cost function under a zero-sum game strategy. The Euclidean norm of the Gausky function, representing the optimal cost function under a zero-sum game strategy. , Represents design constants. This indicates the degree of interference suppression.

2. The optimal tracking and control method for a drone swarm under a zero-sum game scenario during a denial-of-service attack, as described in claim 1, is characterized in that... An evaluation neural network is used to estimate the ideal evaluation weights of the optimal cost function under a zero-sum game strategy. The update rule for the evaluation neural network is obtained using gradient descent. ； ； in, Indicates the learning rate. Indicates intermediate variables. Represents the Bellman residual. Represents the number of nodes in a neural network. Indicates the first Storage time, Indicates the total number of storage operations. Indicate intermediate variables In the The variable value at the next storage time. Represents the Bellman residual In the The variable value for the storage time.

3. The optimal tracking and control method for a drone swarm under a zero-sum game scenario during a denial-of-service attack, as described in claim 1, is characterized in that... The cooperative tracking controller is: ； ； in, The degree matrix represents the first The correlation between the follower drone and the drone. Indicates the relationship with the first The adjacent follower drone The derivative of the output signal of the follower drone, Indicates formation error. Indicates the first The connection between the follower drone and the navigator drone. , Represents design constants. Indicates the degree of interference suppression. This indicates the total number of follower drones. Indicates the first Follower drone and the first Communication between follower drones This indicates the number of follower drones affected by a denial-of-service attack. This represents the estimated value of the ideal evaluation weights for the optimal cost function under a zero-sum game strategy. Let Gausky function be the Euclidean norm of the optimal cost function under a zero-sum game strategy.

4. The optimal tracking and control method for a drone swarm under a zero-sum game scenario during a denial-of-service attack, as described in claim 3, is characterized in that... A memory self-triggering mechanism is introduced for follower drones that have not been subjected to denial-of-service attacks, and active control of follower drones is based on the memory self-triggering mechanism; The controller based on the memory self-triggering mechanism is: ； in, This represents the optimal control signal considering the memory self-triggering mechanism. This represents the worst-case interference signal considering the memory self-triggering mechanism. Indicates the first The reciprocal of the mass of the drone. This represents a coordinate transformation based on dynamic surface technology. Indicates the first The velocity vector of the drone Indicates a self-triggered variable. , , , Represents design constants. and It is an estimate of the ideal evaluation weights of the optimal cost function under a zero-sum game strategy. The Euclidean norm of the Gausky function, representing the optimal cost function under a zero-sum game strategy. This indicates that the optimal cost function under a zero-sum game strategy includes self-triggered variables. The Euclidean norm of the Gausky function, This represents a coordinate transformation based on dynamic surface technology. This indicates the optimal virtual control signal without considering the memory self-triggering mechanism.

5. The event-triggered heterogeneous multi-UAV system tracking and control method according to claim 4, characterized in that, The triggering condition for the memory self-triggering mechanism is as follows: in, Indicates the first The memory of the follower drone is a self-triggered variable. Indicates the trigger time. Indicates the first Follower drone Output signal at time, Indicates the first Follower drone Output signal at time, It is a positive integer. It is a positive integer. , , , , , For design constants, express Formation error at any moment, Representing variables exist Rate of change over time.

6. The optimal tracking and control method for a drone swarm under a zero-sum game scenario during a denial-of-service attack, as described in claim 5, is characterized in that... The formula for calculating the optimal virtual control signal based on the memory self-triggering mechanism is: in, This represents the estimated value of the optimal virtual control signal based on the memory self-triggering mechanism. This represents the estimated value of the worst-case interference input signal based on the memory self-triggering mechanism. This indicates that the variable contains a memory-triggered variable. The Find the first derivative of the output signal of the follower drone. The optimal cost function under a zero-sum game strategy includes memory-triggered variables. The Euclidean norm of the Gausky function, This indicates the formation error considering the memory self-trigger mechanism.

7. The optimal tracking and control method for a drone swarm under a zero-sum game scenario during a denial-of-service attack, as described in claim 6, is characterized in that... The triggering condition for the virtual control memory self-triggering mechanism is: ； in, Indicates the first Follower drone The memory of a moment is a self-triggered variable. Indicates the first Follower drone Output signal at time, Indicates the first Follower drone Output signal at time, It is a positive integer. It is a positive integer. , , , , , For design constants, express Formation error at any moment, Representing variables exist Rate of change over time.

8. The optimal tracking and control method for a drone swarm under a zero-sum game scenario during a denial-of-service attack, as described in claim 7, is characterized in that... An evaluation neural network is used to estimate the ideal evaluation weights of the optimal cost function under a zero-sum game strategy. The update rule for the evaluation neural network is obtained using gradient descent. ； ； ； ； ； in, , Indicates the learning rate. , Indicates intermediate variables. , Represents the Bellman residual. Represents the number of nodes in a neural network. Indicates the first Storage time, Indicates the total number of storage operations. Indicate intermediate variables In the The variable value at the next storage time. Represents the Bellman residual In the The variable value at the next storage time. Indicate intermediate variables In the The variable value at the next storage time. Represents the Bellman residual In the The variable value at the next storage time. , and For design constants, , and This represents the weight update law. This represents the derivative of the output signal of the Navigator drone. Let Gausky function be the Euclidean norm of the optimal cost function under a zero-sum game strategy.

9. The optimal tracking and control method for a drone swarm under a zero-sum game scenario during a denial-of-service attack, as described in claim 1, is characterized in that... The constructed nonlinear model of the UAV considering unknown external disturbances is as follows: ； in, Indicates the first The position vector of the follower drone. Indicates the first The velocity vector of the follower drone This indicates the parts of the system that are not modeled or ignored. and It is an external disturbance. Indicates the first The output signal of the follower drone, Indicates the first Input signals from a follower drone.

10. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 9.

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