A Nash equilibrium search algorithm for distributed games under unbounded attacks

Abstract

A distributed game Nash equilibrium search algorithm for unbounded attacks. The present invention has the following beneficial effects: the algorithm effectively mitigates the negative impact of unbounded FDI attacks on the network search process and information exchange between users by accurately estimating neighboring dynamics and fine-tuning drone parameters. It also deploys a distributed observer to counter FDI attacks and optimizes the gradient function to ensure the convergence performance of drone targets to resist the impact of attacks. Finally, using the Lyapunov stability analysis method, the convergence of the proposed algorithm's distributed Nash equilibrium strategy against unbounded attacks is described in detail. This strategy guides all participants into the Nash equilibrium neighborhood by adjusting the control gain, thereby ensuring that distributed Nash equilibrium can be achieved even when actuators and sensors are simultaneously subjected to unbounded attacks.

Description

Technical Field

[0001] The present invention relates to the field of distributed algorithms, and in particular to a distributed game Nash equilibrium search algorithm under unbounded attacks. Background Art

[0002] In recent years, with the remarkable development of efficient embedded computing, agile and lightweight hardware solutions, and a variety of complementary telecommunication technologies, multi-unmanned aerial systems (MUSs) have achieved remarkable success in various application scenarios. Game theory is widely applied to the control and optimization problems of networked systems. Each participant (user) is modeled as an independent and selfish decision maker who strives to optimize their own but interdependent cost functions. Distributed Nash equilibrium search describes the optimal strategy configuration of multiple participants in a distributed environment. Each participant's goal is to maximize their own utility (or minimize their cost), and their decisions are influenced by the choices of other participants. The game reaches a Nash equilibrium at a certain stable state.

[0003] In the actual application scenarios of drones, attacks against drones are also increasing. They can be mainly divided into two categories: point attacks and edge attacks. Point attacks mainly target the drone itself. For example, by injecting false data into the drone's controller and sensors, the system stability is damaged and the drone cannot fly according to the planned trajectory. These data attacks can be further divided into unbounded and bounded data attacks. Actuator attacks are also a common form. Edge attacks mainly target multi-drone networks. For example, in denial of service (DoS) attacks, malicious actors interrupt the normal functions of drone equipment, making it unable to use communications or other equipment. Ultimately, the system cannot handle normal traffic, thereby affecting the service of other users. To deal with unbounded attacks, a distributed Nash equilibrium search algorithm under unbounded attacks is proposed. Summary of the Invention

[0004] The present invention addresses the shortcomings and deficiencies of the existing technology by providing a distributed game Nash equilibrium search algorithm for unbounded attacks. This algorithm effectively mitigates the negative impact of unbounded FDI attacks on the network search process and information exchange between users by accurately estimating neighboring dynamics and fine-tuning drone parameters. Furthermore, the algorithm deploys a distributed observer to counter FDI attacks and optimizes the gradient function to ensure the convergence of drone targets, thereby resisting the impact of attacks. Finally, using the Lyapunov stability analysis method, the proposed algorithm's convergence to the distributed Nash equilibrium strategy against unbounded attacks is described in detail. This strategy guides all participants into the Nash equilibrium neighborhood by adjusting control gains, thereby ensuring that a distributed Nash equilibrium can be achieved even when both actuators and sensors are simultaneously subjected to unbounded attacks.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a distributed game Nash equilibrium search algorithm for unbounded attacks, characterized in that the dynamic equation of the drone is described using the following equation: When the attack is not considered, the UAV's Nash equilibrium algorithm uses gradient descent and state feedback. At this time, the system input u is:

[0006]

[0007] Where k is the control parameter, represents the cost function of multiple drones, Represents the pseudo gradient function of the current UAV and the neighboring UAV, where It represents the dynamic estimation of the current UAV’s neighbor information status in the communication network under partial information game. i Is a coefficient matrix. Under the action of u, multiple drones can realize Nash equilibrium search. By exploring the multi-drone system receiving unbounded false data to inject attack sensors and actuators, when the actuator of the drone i system is attacked, Among them, u i Represents the input of the original UAV system, ρ i Represents the status of the executor. A value of 1 indicates that there is an attack on the executor by a malicious attacker, and a value of 0 indicates that there is no attack. Represents an unbounded false data injection attack launched by a malicious attacker on the actuator of the drone; when the sensor of drone i is attacked, in, represents the unbounded false sensor data injected into the i-th UAV by a malicious attacker, where N i represents the set of neighbors of the i-th UAV. Although the attacks of sensors are unbounded, it is assumed that their derivatives are bounded. This property provides a key potential solution. The extended state observer (ESO) method has been adjusted to handle scenarios involving unbounded attacks. ESO can estimate unknown elements of the system state, including unknown attacks caused by such attacks. By adjusting the control parameter r, ESO enables users to maintain their pursuit of Nash equilibrium in the presence of attacks and minimize the interference of attacks on the dynamic estimation of UAVs within the UAV communication link network in the graph. Based on the actuator attack observer and the sensor attack observer, the following distributed game Nash equilibrium search algorithm is proposed to resist unbounded FDI attacks:

[0008]

[0009] Furthermore, it is worth noting that the attacker may choose a more sophisticated approach to evade certain detection and identification methods focused on sensor measurements. The attacker may not launch an independent and relatively large false measurement attack, which is easy to detect, but rather a simultaneous injection and related sensor attack. If a sensor attack occurs, then ρ xi is equal to 1, otherwise, ρ xi =0; Traditional distributed Nash control algorithms are not sufficient to solve the problem of unbounded FDI attacks. To solve this problem, a distributed Nash equilibrium search algorithm is designed. The main goal of these algorithms is to ensure the stability of the entire user system and promote its distributed search for targets. Even in the presence of malicious, unbounded FDI attacks, the control algorithm can effectively maintain the system. In addition, it also ensures the stability of the drone control system and the distributed search for Nash equilibrium.

[0010] Furthermore, the observer design for the actuator attack: In order to solve the actuator attack, an observer is designed to compensate for the unknown and potentially unlimited FDI attacks. The main goal of the observer is to adjust the control input to effectively counter the impact of the attack and thus keep the system

[0011] Stability: in, represents the observation of the unbounded executor attack, is an auxiliary variable, r1 and r2 are control parameters.

[0012] Furthermore, the observer design of the sensor attack: For the sensor attack that affects the dynamic estimation between the drone and its neighbors in the partial information game, a specific observer is designed for the unbounded FDI attack. The observer aims to mitigate the unbounded sensor attack by ensuring accurate estimation between users.

[0013]

[0014] Impact of the attack: in, represents the observation of sensor attack, is an auxiliary variable.

[0015] By employing the above technical solution, the present invention achieves the following beneficial effects: the algorithm effectively mitigates the negative impact of unbounded FDI attacks on network search processes and information exchange between users by accurately estimating neighboring dynamics and fine-tuning drone parameters. Furthermore, the algorithm deploys a distributed observer to counter FDI attacks and optimizes the gradient function to ensure the convergence of drone targets, thereby resisting the impact of attacks. Finally, using the Lyapunov stability analysis method, the proposed algorithm's convergence to the distributed Nash equilibrium strategy against unbounded attacks is described in detail. This strategy guides all participants into the Nash equilibrium neighborhood by adjusting control gains, thereby ensuring that distributed Nash equilibrium is achieved even when actuators and sensors are simultaneously subjected to unbounded attacks. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0017] Figure 1 Schematic diagram of the drone node and malicious attacker in the present invention.

[0018] Figure 2 It is the overall design framework diagram for implementing verification in the present invention.

[0019] Figure 3 It is a Nash equilibrium search graph when the UAV control system in the present invention is not subjected to any attack.

[0020] Figure 4 Schematic diagram of the drone status under the unbounded attack in the present invention.

[0021] Figure 5 Schematic diagram of the speed of the UAV under the unbounded attack in the present invention. DETAILED DESCRIPTION

[0022] The technical solution adopted in this specific embodiment is characterized in that the dynamic equation of the drone is described using the following equation:

[0023]

[0024] When the attack is not considered, the UAV's Nash equilibrium algorithm uses gradient descent and state feedback. At this time, the system input u is:

[0025]

[0026] Where k is the control parameter, represents the cost function of multiple drones, Represents the pseudo gradient function of the current UAV and the neighboring UAV, where It represents the dynamic estimation of the current UAV’s neighbor information status in the communication network under partial information game. i It is a coefficient matrix. At this time, under the action of u, multiple drones can realize Nash equilibrium search;

[0027] By exploring the multi-UAV system receiving unbounded false data injection attack sensors and actuators, when the actuator of the UAV i system is attacked, Among them, u i Represents the input of the original UAV system, ρ i Represents the status of the executor. A value of 1 indicates that there is an attack on the executor by a malicious attacker, and a value of 0 indicates that there is no attack. It represents an unbounded false data injection attack on the actuator of the drone launched by a malicious attacker;

[0028] When the sensor of drone i is attacked, in, represents the unbounded false sensor data injected into the i-th UAV by a malicious attacker, where N i represents the set of neighbors of the i-th drone.

[0029] See Figure 1 As shown, it is worth noting that the attacker may choose a more sophisticated approach to evade certain detection and identification methods focused on sensor measurements. The attacker may not launch an independent and relatively large false measurement attack, which is easy to detect, but rather a simultaneous injection and related sensor attack. If a sensor attack occurs, then ρ xi is equal to 1, otherwise, ρ xi =0;

[0030] Traditional distributed Nash control algorithms are not sufficient to address the problem of unbounded FDI attacks. To address this problem, a distributed Nash equilibrium search algorithm was designed. The main goal of these algorithms is to ensure the stability of the entire user system and facilitate its distributed search for targets. Even in the presence of malicious, unbounded FDI attacks, the control algorithm can effectively maintain the system. In addition, it also ensures the stability of the drone control system and the distributed search for Nash equilibrium.

[0031] Observer Design for Executor Attacks: To address the executor attack, an observer is designed to compensate for unknown and potentially unlimited FDI attacks. The main goal of the observer is to adjust the control input to effectively counter the impact of the attack and thus maintain the stability of the system:

[0032]

[0033] in, represents the observation of the unbounded executor attack, is an auxiliary variable, r1 and r2 are control parameters.

[0034] Observer Design for Sensor Attacks: For sensor attacks that affect the dynamic estimation between drones and their neighbors in a partial information game, a specific observer is designed for unbounded FDI attacks. The observer aims to mitigate the impact of unbounded sensor attacks by ensuring accurate estimation between users:

[0035]

[0036] in, represents the observation of sensor attack, is an auxiliary variable.

[0037] Although the attacks on sensors are unbounded, their derivatives are assumed to be bounded. This property provides a key potential solution. The Extended State Observer (ESO) method has been adapted to handle scenarios involving unbounded attacks. ESO can estimate unknown elements of the system state, including unknown attacks caused by such attacks. By adjusting the control parameter r, ESO enables users to maintain their pursuit of Nash equilibrium in the presence of attacks and minimize the interference of attacks on the dynamic estimates of the UAVs within the UAV communication link network in the figure.

[0038] Based on the actuator attack observer and the sensor attack observer, the following distributed game Nash equilibrium search algorithm is proposed to resist unbounded FDI attacks:

[0039]

[0040] The following are the experimental verification data and steps of the present invention:

[0041] For experimental verification of the overall framework, see Figure 2 shown.

[0042] The designed control algorithm is verified by using drones. The cost of designing five drones is as follows:

[0043]

[0044] For the parameter f a =0.8,f b =1,fc =0.01, where malicious attacks include:

[0045]

[0046] And sensor attacks are: First, when the system is not under any attack, the Nash equilibrium search of the drone is as follows: Figure 3 As shown:

[0047] pass Figure 3 It can be seen that when not under any attack, the five drones searched for the corresponding Nash equilibrium according to the predetermined cost function design.

[0048] Next, under unbounded attack, using the control algorithm designed above, the drone can still search for the Nash equilibrium designed by the cost function, such as Figure 4-5 As shown;

[0049] pass Figure 4-5 It can be seen that even if the drone is attacked by malicious unbounded false data injection from a malicious attacker, using the designed control algorithm, the drone can still successfully search for the corresponding Nash equilibrium according to the predetermined cost function.

[0050] The above description is only used to illustrate the technical solution of the present invention and is not intended to limit it. Other modifications or equivalent substitutions made to the technical solution of the present invention by ordinary technicians in this field should be included in the scope of the claims of the present invention as long as they do not depart from the spirit and scope of the technical solution of the present invention.

Claims

1. A distributed game Nash equilibrium search algorithm for unbounded attacks, characterized by: The dynamic equation of the drone is described by the following equation: When the attack is not considered, the UAV's Nash equilibrium algorithm uses gradient descent and state feedback. At this time, the system input u is: Where k is the control parameter, represents the cost function of multiple drones, Represents the pseudo gradient function of the current UAV and the neighboring UAV, where It represents the dynamic estimation of the current UAV’s neighbor information status in the communication network under partial information game. i It is a coefficient matrix. At this time, under the action of u, multiple drones can realize Nash equilibrium search; By exploring the multi-UAV system receiving unbounded false data injection attack sensors and actuators, when the actuator of the UAV i system is attacked, Among them, u i Represents the input of the original UAV system, ρ i Represents the status of the executor. A value of 1 indicates that there is an attack on the executor by a malicious attacker, and a value of 0 indicates that there is no attack. It represents an unbounded false data injection attack on the actuator of the drone launched by a malicious attacker; When the sensor of drone i is attacked, in, represents the unbounded false sensor data injected into the i-th UAV by a malicious attacker, where N i represents the set of neighbors of the i-th drone; While the attacks on sensors are unbounded, their derivatives are assumed to be bounded. This property provides a key potential solution. The Extended State Observer (ESO) approach has been adapted to handle scenarios involving unbounded attacks. ESO can estimate unknown elements of the system state, including those caused by such attacks. By adjusting the control parameter r, ESO enables users to maintain their pursuit of Nash equilibrium in the presence of attacks and minimize the interference of attacks on the dynamic estimates between drones within a network of drone communication links. Based on the actuator attack observer and the sensor attack observer, the following distributed game Nash equilibrium search algorithm is proposed to resist unbounded FDI attacks:

2. The distributed game Nash equilibrium search algorithm for unbounded attacks according to claim 1, characterized in that: It is worth noting that the attacker may choose a more sophisticated approach to evade certain detection and identification methods focused on sensor measurements. The attacker may not launch an independent and relatively large false measurement attack, which is easy to detect, but rather a simultaneous injection and related sensor attack. If a sensor attack occurs, then ρ xi is equal to 1, otherwise, ρ xi =0; Traditional distributed Nash control algorithms are not sufficient to address the problem of unbounded FDI attacks. To address this problem, a distributed Nash equilibrium search algorithm was designed. The main goal of these algorithms is to ensure the stability of the entire user system and facilitate its distributed search for targets. Even in the presence of malicious, unbounded FDI attacks, the control algorithm can effectively maintain the system. In addition, it also ensures the stability of the drone control system and the distributed search for Nash equilibrium.

3. The distributed game Nash equilibrium search algorithm for unbounded attacks according to claim 1, characterized in that: The observer design of the actuator attack: To address the actuator attack, an observer is designed to compensate for unknown and potentially unlimited FDI attacks. The main goal of the observer is to adjust the control input to effectively counter the impact of the attack and thus maintain the stability of the system: in, represents the observation of the unbounded executor attack, is an auxiliary variable, r1 and r2 are control parameters.

4. The distributed game Nash equilibrium search algorithm for unbounded attacks according to claim 1, characterized in that: The observer design of the sensor attack: For sensor attacks that affect the dynamic estimation between drones and their neighbors in a partial information game, a specific observer is designed for unbounded FDI attacks. The observer aims to mitigate the impact of unbounded sensor attacks by ensuring accurate estimation between users: in, represents the observation of sensor attack, is an auxiliary variable.

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