A second-order system privacy protection average consensus method based on edge disturbance

Abstract

The application discloses a second-order system privacy protection average consistency method based on edge disturbance, which comprises the following steps: determining the communication topology graph and Laplace matrix of a multi-agent system, and representing the information interaction relationship between agents by the Laplace matrix; establishing a dynamic model of the multi-agent system; in the initial iteration, each agent generates a disturbance signal for each edge connected to it, and determines the type of attacker; after confirming the attack type, the agent adds a disturbance signal to the transmission of position information and speed information to the neighbor, and the disturbed information is transmitted to the neighbor; after the disturbance signal stage ends, the agent switches to a standard second-order dynamic average consistency algorithm, and updates its own state according to the neighbor feedback information. The application can guarantee privacy protection without sacrificing the convergence accuracy of the consistency algorithm, is suitable for a resource-limited environment, and can effectively resist privacy theft of internal and external attackers.

Description

Technical Field

[0001] The present invention belongs to the field of computer science and control, and specifically relates to a privacy-preserving average consistency method for a second-order system based on edge perturbation. Background Art

[0002] In recent years, technological advancements and the evolving times have led to increasingly complex control tasks. For example, unmanned aerial vehicle (UAV) systems, intelligent transportation systems, smart grid systems, and sensor networks all rely on large-scale data analysis. However, single agents or traditional centralized processing systems are increasingly limited and unable to meet these demands. Consequently, swarm systems (also known as multi-agent systems) have emerged.

[0003] Multi-agent systems consist of multiple agents with autonomy and independent decision-making capabilities. Each agent executes a distributed control protocol and completes assigned tasks through interaction, communication, and collaboration. With the widespread use of multi-agent systems in practical applications, distributed control protocols have gradually attracted the attention and in-depth research of researchers both domestically and internationally. Consistency control, as a key issue in distributed control, has been widely applied and studied.

[0004] The core goal of consensus control is to achieve consistency in state, information, or behavior among multiple agents within a bounded timeframe through information exchange between agents. Traditional consensus algorithms typically require multiple agents to communicate their true state values ​​to neighboring agents for distributed computation. However, this mechanism introduces a serious problem: the initial state messages of agents, which often contain sensitive or private data, are vulnerable to theft by malicious attackers, leading to information leakage and potentially serious consequences. For example, in certain military missions, when drone swarms employ traditional consensus algorithms to complete specific tasks, their location information cannot remain private. This information leakage presents two potential security risks. First, malicious agents within the network could easily obtain the initial states of their neighbors and launch attacks. Second, external attackers could compromise communication links and obtain information in transit, potentially leaking the initial states of the agents and potentially exploiting them to attack the entire network. However, existing privacy-preserving methods suffer from issues such as high computational and communication costs, limited applicability to first-order systems, a trade-off between privacy and accuracy, and insufficient protection against external attackers. Therefore, developing a privacy-preserving average consensus algorithm with high convergence accuracy, low computational cost, and applicability to second-order systems has become a necessary and urgent technical need. Summary of the Invention

[0005] In response to the above-mentioned problems existing in the existing technology, the present invention proposes a privacy-preserving average consistency method for second-order systems based on edge perturbation. It has low communication overhead and is suitable for resource-constrained environments. While ensuring privacy protection, it does not sacrifice the convergence accuracy of the consistency algorithm. It is suitable for strongly connected directed balanced graphs and can effectively resist privacy theft by internal and external attackers.

[0006] In order to achieve the above object, the technical solutions specifically adopted by the present invention are as follows:

[0007] A privacy-preserving average consistency method for second-order systems based on edge perturbations includes the following steps:

[0008] Step 1: Determine the communication topology and Laplace matrix of the multi-agent system, and use the Laplace matrix to represent the information interaction relationship between agents;

[0009] Step 2: Establish a dynamic model of the multi-agent system;

[0010] Step 3: At the initial iteration, each agent generates a perturbation signal for each edge connected to it to determine the attacker type; the perturbation information follows specific constraints;

[0011] Step 4: After confirming the attack type, the agent adds a disturbance signal to the position information and speed information transmitted to its neighbors and transmits the generated interference information to its neighbors;

[0012] Step 5: After the disturbance signal phase ends, the agent switches to the standard second-order dynamic average consensus algorithm and updates its own state based on the information fed back by its neighbors.

[0013] Furthermore, in step 1, the communication topology between agents is a directed graph Represents a node set containing and directed edge sets Laplace matrix L = [l ij ]∈R N×N The definition is as follows: When i≠j, l ij =-a ij ; When i = j, Re(μ) and Im(μ) represent the real part and imaginary part of the eigenvalue μ of the Laplace matrix L, respectively.

[0014] Furthermore, in step 2, a discrete-time second-order agent system consisting of N agents is modeled, and the dynamic model of each agent is expressed as follows:

[0015] x i [k+1]=x i [k]+v i [k]

[0016] v i [k+1]=v i [k]+u i [k],i=1,2,…,N

[0017] Where k = 1, 2, ... are positive integers, x i [k]∈R N and v i [k]∈R N They represent the position state and velocity state of the i-th agent at iteration k, u i [k]∈R N is the control input of the i-th agent.

[0018] Furthermore, in step 3, for the internal curious attacker, his ability to judge the initial state of the agent is limited to the information set he obtains. Information Set The definition is as follows:

[0019]

[0020] Among them, ∈1,∈2,∈3,∈4 are coupling coefficients, and are the initial position and velocity states of the injected disturbance information, and is the disturbance information, and are the incoming neighbor set and outgoing neighbor set of agent i respectively.

[0021] Randomly select an agent in the topology graph and name it agent 1, and randomly select it from the neighbor set Select an agent and name it Agent 2, and the remaining agents in the topology are named Consider two implementation methods. First, confirm the initial state of the two implementation methods. In the first implementation method, define Agent 1, Agent 2, and The initial positions and velocities of the agents in the set are x1[0], x2[0], x3[0], v1[0], v2[0], v3[0]; in the second implementation: define agent 1, agent 2, and V 3 The initial position and velocity states of the agents in the set are in,

[0022] Then, based on the relationship of the initial state, the disturbance information and disturbance coefficient are designed as follows:

[0023]

[0024]

[0025] in, i,j∈1,2,3 is the disturbance information injected into the position information transmitted by agent i to agent j, is the corresponding perturbation information matrix; i,j∈1,2,3 is the disturbance information injected into the velocity information transmitted by agent i to agent j, is the corresponding perturbation information matrix; is the coupling coefficient under the second implementation mode; d i ,i∈1,2,3 is the degree of the node, a ij ,i,j∈1,2,3 are elements in the adjacency matrix, A ij is the corresponding matrix.

[0026] Furthermore, in step 3, when targeting an external eavesdropper, the information set of the external eavesdropper is The definition is as follows:

[0027]

[0028] Among them, ∈1,∈3,∈4 are coupling coefficients, and are the initial position and velocity states of the injected disturbance information respectively;

[0029] Define v[k]=[v1[k],v2[k],v3[k]] T ,x[k]=[x1[k],x2[k],x3[k]] T ,The relationship between the coupling coefficient and the initial state under the two implementation methods is as follows:

[0030]

[0031] Among them, δ∈2∈(-∞,0)∪(0,∞);

[0032] According to the relationship of the initial state, the relationship of the disturbance information is designed:

[0033]

[0034] Furthermore, after confirming the attack type, when agent i transmits position information and speed information to its neighbors, it adds specific disturbance information to the position information and speed information respectively. and( or ) to generate the transmission information after interference.

[0035] Furthermore, in step 5, after the disturbance signal phase ends, i.e., k>0, all agents will switch to the standard second-order dynamic average consensus algorithm for state update. In this phase, each agent updates its own state according to the following formula based on the information fed back by its neighbors:

[0036] x i [k+1]=x i [k]+v i [k]

[0037]

[0038] Among them, ∈3 and ∈4 are coupling coefficients, satisfying ∈4>∈3>0 and

[0039] The present invention has the following characteristics and beneficial effects:

[0040] The present invention proposes a privacy-preserving average consensus algorithm for second-order discrete multi-agent systems based on edge perturbation, which mainly targets the privacy leakage problem existing in second-order systems and has the following advantages: 1) By injecting perturbation signals at the edges, the initial position information and velocity information of the agent are effectively protected from being obtained by internal honest but curious agents and external eavesdroppers, while accurately converging to the average consensus state; 2) The algorithm is applicable to strongly connected balanced graphs, overcoming the limitation of existing methods that are limited to bidirectional transmission topologies and expanding the scope of application; 3) The algorithm of the present invention is lightweight and easy to implement, reducing computational and communication overheads. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Other features, objects and advantages of the present invention will become more apparent upon reading the detailed description of non-limiting embodiments with reference to the following drawings:

[0042] Figure 1 The communication topology diagrams of the first and second embodiments of the present invention are shown;

[0043] Figure 2 This is a position state update diagram of the five intelligent agents for the internal curiosity in the first and second implementation modes in the first embodiment of the present invention;

[0044] Figure 3 This is a speed state update diagram for the five intelligent agents of the internal curiosity in the first and second implementation modes in the first embodiment of the present invention;

[0045] Figure 4 Schematic diagram of the transmission information including the location status for the internal curious person in the first and second implementation modes in the first embodiment of the present invention

[0046] Figure 5 Schematic diagram of transmission information including speed status for internal curiosity in the first and second implementation modes in embodiment 1 of the present invention

[0047] Figure 6 This is a position state update diagram of the five intelligent agents against external attackers in the first and second implementation modes in the second embodiment of the present invention;

[0048] Figure 7 This is a speed state update diagram of the five intelligent agents against external attackers in the first and second implementation modes in the second embodiment of the present invention;

[0049] Figure 8 Schematic diagram of transmission information including location status in the first and second implementations for an external attacker in the second embodiment of the present invention;

[0050] Figure 9 Schematic diagram of transmission information including speed status in the first and second implementation modes against external attackers in the second embodiment of the present invention;

[0051] Figure 10 This is a flow chart of the invention of Embodiment 1 and Embodiment 2 of the present invention. DETAILED DESCRIPTION

[0052] The present invention is described in detail below in conjunction with specific embodiments. The following examples will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other.

[0053] To address the problems of existing privacy protection methods, such as high computational and communication costs, limited application to first-order systems, the trade-off between privacy and accuracy, and insufficient protection against external attackers, this embodiment of the present invention provides a privacy-preserving average consensus method for second-order multi-agent systems based on edge perturbations. The specific steps are as follows:

[0054] Step 1: Determine the communication topology of the multi-agent system and the Laplace matrix L; Specifically: In a network consisting of N agents, the communication topology between agents is a directed graph Indicates that it contains N nodes. The communication topology contains the node set and directed edge sets There are directed edges ε in the network ij Represented as a node pair (v i ,v j ), which means that node v i You can send a jTransmit information, node i’s incoming neighbor set and outgoing neighbor set and Represent the in-neighbors and out-neighbors of node i respectively. The in-degree and out-degree of node i are defined as and If the in-degree and out-degree of all nodes in the graph are equal, then the graph is called a balanced graph. In a balanced graph, we define For a simple graph consisting of N agents Its adjacency matrix A=[a ij ]∈R N×N The definition is as follows: ii =0; if and only if ε ij When ∈ε, a ij =1, otherwise a ij = 0, where i, j = 1, 2, ..., N and i ≠ j. Laplace matrix L = [l ij ]∈R N×N The definition is as follows: When i≠j, l ij =-a ij ; When i = j, Re(μ) and Im(μ) represent the real part and imaginary part of the eigenvalue μ of the Laplace matrix L, respectively.

[0055] Step 2: Model the discrete-time second-order agent system consisting of N agents. The dynamic model of each agent can be expressed as follows:

[0056] x i [k+1]=x i [k]+v i [k]

[0057] v i [k+1]=v i [k]+u i [k],i=1,2,…,N

[0058] Where k = 1, 2, ... are positive integers, x i [k]∈R N and v i [k]∈R N They represent the position state and velocity state of the i-th agent at iteration k, u i [k]∈R N is the control input of the i-th agent.

[0059] Step 3: When k = 0, each agent generates disturbance information to determine the attacker type; specifically, for agent i, if there is a directed edge ε ij∈ε, then agent i can generate two perturbation information for each edge connected to agent j, which are and These perturbation signals are added to the information transmitted between agents to ensure privacy protection. and satisfy:

[0060] 1) If and 2) and in,

[0061] Step 3-1: For an internal curious attacker, his ability to determine the initial state of the agent is limited to the information set he obtains. If the attacker receives exactly the same information when facing two agents with different initial positions and speeds, he will not be able to distinguish the different initial positions and speeds based on the collected information set. The definition is as follows:

[0062]

[0063] Among them, ∈1,∈2,∈3,∈4 are coupling coefficients, and are the initial position and velocity states of the injected disturbance information, and For disturbance information.

[0064] Without loss of generality, we randomly select an agent in the topological graph and name it agent 1, because the topological graph is a strongly connected balanced graph, and the neighbor set of agent 1 is It must not be empty, we randomly enter the neighbor set Select an agent and name it Agent 2, and the remaining agents in the topology are named We define Agent 1, Agent 2, and The position and velocity information of the agents in the set are: x1[k],x2[k],x3[k]=[x4[k],…,x N [k]] T ,v1[k],v2[k],v3[k]=[v4[k],…,v N [k]] T .

[0065] Furthermore, we define the Laplace matrix L, the adjacency matrix A, and the perturbation information matrices P and Q as block matrices, as follows:

[0066]

[0067] Step 3-2: Consider two implementation methods. First, confirm the initial state of the two implementation methods. In the first implementation method, we define Agent 1, Agent 2, and The initial position and velocity states of the agents in the set are x1[0], x2[0], x3[0], v1[0], v2[0], v3[0], and the disturbance information is further defined. And coupling coefficients ∈1,∈2,∈3,∈4. In the second implementation: we define the initial states as The specific relationship is as follows:

[0068] Step 3-3: Based on the relationship of the initial state, we further design the disturbance information and coupling coefficient in the second implementation method, as follows:

[0069]

[0070] in i,j∈1,2,3 is the disturbance information injected into the position information transmitted by agent i to agent j, is the corresponding perturbation information matrix; i,j∈1,2,3 is the disturbance information injected into the velocity information transmitted by agent i to agent j, is the corresponding perturbation information matrix; is the coupling coefficient under the second implementation method.

[0071] Step 3-4: For external eavesdroppers, we also use the same idea from step 3-1. The information set of the external eavesdropper is defined as follows:

[0072]

[0073] Among them, ∈1,∈3,∈4 are coupling coefficients, and are the initial position and velocity states of the injected disturbance information respectively.

[0074] Step 3-5: Define v[k] = [v1[k], v2[k], v3[k]] T ,x[k]=[x1[k],x2[k],x3[k]] T ,The relationship between the coupling coefficient and the initial state under the two implementation methods is as follows:

[0075]

[0076] Among them, δ∈2∈(-∞,0)∪(0,∞).

[0077] Step 3-6: Based on the relationship of the initial state, we further design the relationship of the perturbation information:

[0078]

[0079] Step 4: After confirming the attack type, when agent i transmits position information and speed information to its neighbors, it adds specific disturbance information to the position information and speed information respectively. and( or ) to generate the transmission information after interference. For example, other implementations of perturbations also follow the same idea. The position information transmitted by agent i to its neighbor agent j is and speed information Defined as:

[0080]

[0081] The initial state after these disturbances and is passed to agent j to ensure that the initial state of the agent is protected from being obtained by internal or external attackers during the information transmission process. The rules for each agent to update are as follows:

[0082] x i [1] = x i [0]+v i [0]

[0083]

[0084] Among them, ∈1∈(-∞,0)∪(0,∞) and ∈2∈(-∞,0)∪(0,∞) are both constants.

[0085] Step 5: After the disturbance signal phase ends, i.e., k>0, all agents will switch to the standard second-order dynamic average consensus algorithm for state update. In this phase, each agent updates its own state according to the following formula based on the information fed back by its neighbors:

[0086] x i [k+1]=x i [k]+v i [k]

[0087]

[0088] Among them, ∈3 and ∈4 are coupling coefficients, satisfying ∈4>∈3>0 and

[0089] Example 1: Take five multi-agents as an example, where agent 2 is an internally honest but curious agent. The topology diagram is shown below:

[0090] In the first implementation, the specific parameters are as follows:

[0091] x1[0]=8,x2[0]=2,x3[0]=-1,x4[0]=-4,x5[0]=-10

[0092] v1[0]=4, v2[0]=-1, v3[0]=-3, v4[0]=2, v5[0]=3

[0093] ∈1=0.2,∈2=0.4,∈3=0.9,∈4=1

[0094]

[0095] In the second implementation, the corresponding initial state, coupling coefficient and disturbance information are as follows:

[0096]

[0097]

[0098] The states of the five agents and the position and speed information transmitted at the initial moment are as follows: Figure 2-Figure 5 shown. Figure 2 and Figure 3 It shows that the second-order dynamic average consistency can be accurately achieved in both implementation methods. Figure 4 It shows that the position information transmitted between all agents is exactly the same. Figure 5 This shows that, except for Agents 3 and 4, all other agents transmit the same speed information. Therefore, the information received by the internal curious agent 2 is completely consistent, making it impossible to distinguish the initial states of other agents.

[0099] Embodiment 2: The present invention provides another example, in which agent 2 acts as an external eavesdropper to ensure that the initial state is not affected by its attack.

[0100] In the second implementation, the information corresponding to the first implementation in Example 1 is as follows:

[0101]

[0102] The states of the five agents and the position and speed information transmitted at the initial moment are as follows: Figure 6-Figure 9 shown. Figure 6 and Figure 7 It shows that all agents can accurately achieve second-order dynamic average consistency in both implementation methods. Figure 8 and Figure 9 It is shown that all agents pass the same position information and velocity information in the first iteration under both implementations. Thus, the external eavesdropper 2 receives the same information and cannot determine the initial state of the other agents.

[0103] The above shows and describes the basic principles, main features and advantages of the present application. Those skilled in the art should understand that the present application is not limited to the above-mentioned embodiments, and the above-mentioned embodiments and descriptions in the specification are only preferred examples of the present application and are not intended to limit the present application. Without departing from the spirit and scope of the present application, various changes and improvements can be made to the present application, and these changes and improvements all fall within the scope of the claimed present application. The scope of protection of the present application is defined by the appended claims and their equivalents.

Claims

1. A privacy-preserving average consistency method for second-order systems based on edge perturbations, characterized by: The steps include: Step 1: Determine the communication topology and Laplace matrix of the multi-agent system, and use the Laplace matrix to represent the information interaction relationship between the agents; the communication topology between the agents is represented by a directed graph Represents a node set containing and directed edge sets The Laplace matrix L=[l ij ]∈R N×N The definition is as follows: When i≠j, l ij =-a ij ; When i = j, Re(μ) and Im(μ) represent the real and imaginary parts of the eigenvalue μ of the Laplace matrix L, respectively. Step 2: Establish a dynamic model of the multi-agent system; Modeling a discrete-time second-order agent system consisting of N agents, the dynamic model of each agent is expressed as follows: x i [k+1]=x i [k]+v i [k] v i [k+1]=v i [k]+u i [k],i=1,2,…,N Where k = 1, 2, ... are positive integers, x i [k]∈R N and v i [k]∈R N They represent the position state and velocity state of the i-th agent at iteration k, u i (k)∈R N is the control input of the i-th agent; Step 3: At the initial iteration, each agent generates a perturbation signal for each edge connected to it to determine the attacker type; For an internal curious attacker, its ability to judge the initial state of the agent is limited to the information set it obtains. Information Set The definition is as follows: Among them, ∈1,∈2,∈3,∈4 are coupling coefficients, and are the initial position and velocity states of the injected disturbance information, and is the disturbance information, and are the incoming neighbor set and outgoing neighbor set of agent i respectively; Randomly select an agent in the topology graph and name it agent 1, and randomly select it from the neighbor set Select an agent and name it Agent 2, and the remaining agents in the topology are named Consider two implementation methods. First, confirm the initial state of the two implementation methods. In the first implementation method, define agent 1, agent 2, and v 3 The initial position and velocity states of the agents in the set are x1[0], x2[0], x3[0], v1[0], v2[0], v3[0], and the disturbance information is further defined. and coupling coefficients ∈1,∈2,∈3,∈4; in the second implementation: define agent 1, agent 2 and ν 3 The initial position and velocity states of the agents in the set are in, Then, based on the relationship of the initial state, the disturbance information and disturbance coefficient are designed as follows: in, is the disturbance information injected into the position information transmitted by agent i to agent j, is the corresponding perturbation information matrix; is the disturbance information injected into the velocity information transmitted by agent i to agent j, is the corresponding perturbation information matrix; is the coupling coefficient under the second implementation mode; d i ,i∈1,2,3 is the degree of the node, a ij ,i,j∈1,2,3 are elements in the adjacency matrix, A ij is the corresponding matrix; When targeting an external eavesdropper, the external eavesdropper's information set The definition is as follows: Among them, ∈1,∈3,∈4 are coupling coefficients, and are the initial position and velocity states of the injected disturbance information respectively; Define v[k]=[v1[k],v2[k],v3[k]] T ,x[k]=[x1[k],x2[k],x3[k]] T ,The relationship between the coupling coefficient and the initial state under the two implementation methods is as follows: Among them, δ∈2∈(-∞,0)∪(0,∞); According to the relationship of the initial state, the relationship of the disturbance information is designed: Step 4: After confirming the attacker type, when the agent transmits position information and speed information to its neighbors, it adds perturbation information to the position information and speed information respectively to generate perturbed transmission information; Step 5: When the disturbance signal phase ends, the agent switches to the standard second-order dynamic average consensus algorithm and updates its own state based on the information fed back by its neighbors. Each agent updates its own state according to the following formula based on the information fed back by its neighbors: x i [k+1]=x i [k]+v i [k] Among them, ∈3 and ∈4 are coupling coefficients, satisfying ∈4>∈3>0 and 2. The edge perturbation-based privacy-preserving average consistency method for a second-order system according to claim 1, characterized in that: After confirming the attacker type, when agent i transmits position information and speed information to its neighbors, it adds specific perturbation information to the position information and speed information respectively. and or To generate transmission information after interference.

3. The edge perturbation-based privacy-preserving average consistency method for a second-order system according to claim 1, characterized in that: In step 5, when the disturbance signal phase ends, that is, k>0, all agents will switch to the standard second-order dynamic average consensus algorithm for state update.

Images