A Multilateral Consensus Optimization Method Based on Partial Differential Multi-Agent Systems

Abstract

This invention discloses a multilateral consensus optimization control method for partial differential equation (PDE) multi-agent systems. The steps are as follows: a) Constructing a diffuse PDE multi-agent system model; b) Proposing the definition of groups and structural equilibrium in symbolic networks, grouping the system, where agents with positive weight connections can be considered as a group, and each group is connected by non-positive weight connections; c) Establishing an error expression based on the group definition; d) Proposing the definitions of multilateral consensus and multilateral consensus optimization; e) Designing a novel proportional partial derivative (PPD) controller; f) Providing a well-posed analysis of the PDE system under the controller; g) Designing the relationship between the cost function and the system state to obtain the conditions for solving the multilateral consensus optimization problem, such that each agent in the group reaches its optimal value related to its group. This method is applicable to solving distributed optimization problems in large-scale multi-agent systems, has fast response capabilities, and can solve various practical problems.

Description

Technical Field

[0001] This invention relates to the field of control and optimization of multi-agent systems, and specifically to a multilateral consensus optimization method for multi-agent systems based on partial differential equations. Background Technology

[0002] In recent decades, partial differential equations (PDEs) have garnered widespread attention due to their applications in various fields, including image segmentation, highway traffic monitoring, robot swarm coordination, and autonomous vehicle deployment. PDEs can be used to model a variety of systems, including service function chains, flexible structures, robots, axially moving beams, and drilling rigs. Furthermore, the stability of PDE systems has been extensively studied theoretically.

[0003] Current research focuses on the application of multi-agent systems characterized by partial differential equations (PDEs) to large-scale and complex problems. Consistency, as a fundamental dynamic behavior in PDEs, has become a key area of ​​research interest. However, in practical engineering applications, not all agents exhibit cooperative relationships; adversarial information may be present, posing a significant challenge to the study of consistency in PDEs. To address this issue, the concept of bilateral consistency with adversarial information was proposed. Subsequently, researchers have shown increasing interest in bilateral consistency in PDE PDE PDE systems. Currently, the optimization problem of PDE PDE systems has been applied in various fields, including deep learning in robotics, transportation, autonomous driving, and gas transport.

[0004] With the acceleration of globalization, cooperation and competition among countries are intertwined, making the issue of multilateral cooperation and confrontation increasingly prominent. This is mainly reflected in international security, international business, international trade, and the international economy. Against this backdrop, in-depth discussions on the common interests and differences of all parties in multilateral relations are conducive to promoting the stability and development of international relations. Therefore, it is necessary to distinguish between the parties, ensuring that information exchange within each group is cooperative, while information exchange between different groups is non-cooperative, particularly antagonistic; this is known as multilateral consensus within a group. Summary of the Invention

[0005] Purpose of the invention: One objective of this invention is to provide a PPD control method for multilateral consensus optimization of partial differential multi-agent systems, which is applicable to optimization tasks of large-scale multi-agent systems.

[0006] Technical solution:

[0007] A multilateral consensus optimization method based on partial differential equation multi-agent systems includes the following steps:

[0008] a) Construct a diffusion-partial differential multi-agent system model;

[0009] b) Propose the definition of groups and structural balance in symbolic networks, group the system, regard agents with positive weight connections as a group, and connect each group with non-positive weights.

[0010] c) Based on the definition of a group, establish the error expression;

[0011] d) Propose definitions of multilateral consensus and multilateral consensus optimization;

[0012] e) Design a new proportional-partial-derivative (PPD) controller and provide a well-stability analysis of the system under the controller;

[0013] d) Design the relationship between the cost function and the system state to obtain the conditions for solving the multilateral consensus optimization problem, so that each agent in the group reaches the optimal value associated with its group.

[0014] Furthermore, the partial differential multi-agent system model is constructed as follows:

[0015] Let x∈[0,l] be the position variable, ranging from 0 to the positive constant l, and t∈[0,+∞) be the time variable. The diffusion partial differential multi-agent system model is constructed as follows:

[0016]

[0017] w i (x,0)=w i0 (x),(1c)

[0018] Where w i (x,t) represents the state of the i-th node at x and t, where θ∈R + Where is the diffusion coefficient. and They represent w respectively i The first derivative of (x,t) with respect to t and x. To represent w i The second derivative of (x,t) with respect to x is u. i (x,t) is the control input, ω i0 (x) represents the initial condition of ω(x,t). For simplicity, we will use w as a shorthand. i (x,t)=w i .

[0019] Furthermore, definitions of group and structural balance in symbolic networks are given:

[0020] Definition 1: (Group) For an undirected connected symbolic graph G = (V, E, A), where V = {v1, v2, ..., v...} N} is a point set, The set of edges A = (aij ) N×N Let a be an adjacency matrix. ij For node v i and v j The weights between them. There exists a subset. Make all in graph G p The weights between nodes in the matrix are positive, that is, for {(v i ,v i )∈E p |G p =(V p E p A p )},a ij If ≥0, then graph G p All nodes are assigned to the same group, each group contains at least one node, and every pair of nodes within a group is connected.

[0021] Definition 2: (Structural equilibrium) An undirected connected symbolic graph G = (V, E, A) is said to be structurally balanced if V can be divided into K non-empty subsets V1, V2, ..., V K That is, for p ≠ q ∈ {1, 2, ..., K}, we have V1 ∪ V2 ∪ ... ∪ V K =V,V p ∩V q =Φ, such that

[0022]

[0023] In addition, the negative topology between different subsets needs to ensure that for v i ∈V p ,v j ∈V q , (v i ,v j )∈E, satisfying Πa ij >0.

[0024] Symbolic diagrams must satisfy the following assumptions:

[0025] Assumption 1: The undirected symbolic graph G of all groups is structurally balanced.

[0026] Furthermore, the definitions of the error system expression, multilateral consistency, and multilateral consistency optimization are given as follows:

[0027] Let e i,p =w i -c p sgn(a ij )w j Therefore, according to (1), the error system can be obtained as follows:

[0028]

[0029] e i,p (x,0)=e i0,p (x).(2c)

[0030] Where c p e is the constant to be designed. i,p This represents the state of the i-th node at x and t. and They represent e respectively i,p First derivative with respect to t and x To represent e i,p The second derivative with respect to x, u i To control the input, e i0,p (x) represents e i,p The initial conditions.

[0031] Based on the above error system, the definition of multilateral consistency is as follows:

[0032] Definition 3: (Multilateral Consistency) Under Assumptions 1 and 2, if the following equation holds, then a multi-agent system is said to have achieved multilateral consistency:

[0033]

[0034] Where m p ∈R is the p-th group G p Consistency value.

[0035] Definition 4: (Multilateral Consistent Optimization) Under the conditions that Assumptions 1 and 2 hold, design an optimal controller u. pi This will achieve the following optimization objectives:

[0036]

[0037] ste i,p =-m p (5)

[0038] Where f i,p (e i,p ) is about e i,p The value function of .

[0039] Further regarding v i ∈V p Design a novel proportional-partial-derivative (PPD) controller:

[0040]

[0041] Where k p ,k d ,kg ∈R + Let f be the control gain to be designed, and ▽f i,p (e i,p ) is f i,p (e i,p The gradient of ).

[0042] The following is a system stability analysis under the controller:

[0043] If assumption 1 holds, then the relationship between the value function, the system state, and the multilateral consensus objective value is as follows:

[0044] ▽f i,p (e i,p )=ξ p e i,p +ρ p (7)

[0045]

[0046] Where ξ p ∈R + , ρ p ∈R is the value coefficient, then for x∈[0,l], t∈[0,+∞), the partial differential multi-agent system (2) in controller u i The solution exists and is unique.

[0047] Let e ​​= (e 1,p ,e 2,p ,...e N,p ) T We can obtain the result from condition (7).

[0048] e t =θe xx -(k p L+k g ξ p )ek d Le x -k g ρ p (9)

[0049] Multiply equation (9) by e T Integrating from 0 to l yields... in,

[0050]

[0051] Consider the following energy function:

[0052]

[0053] Taking the time derivative of (12), and combining (2b), (10), and (11), we can obtain...

[0054]

[0055] Where ε1=min{1,2k p λ²(L)}, where λ²(L) is the second smallest eigenvalue of L. According to Gronwall's inequality, we have...

[0056] This indicates that a solution to system (2) exists.

[0057] The uniqueness of the solution to system (2) is given below. Assume that system (2) has two solutions, e1 and e2, and let σ = e1 - e2, then σ t =θσ xx -(k p L+k g ξ p )σ-k d Lσ x Consider the energy function Taking its derivative, we get...

[0058]

[0059] Where ε2=min{1,2k p λ2(L)+2k g ξ p According to Gronwall's inequality, we have This shows that the solution to system (2) is unique.

[0060] To further address the multilateral consensus optimization problem, the main approach involves designing the relationship between the cost function and the system state to derive the conditions for solving the problem, ensuring that each agent in the group reaches its optimal value relative to its group. Specifically:

[0061] Consider the partial differential equation system (2) under the condition that assumption 1 holds. If conditions (7) and (8) are satisfied, the multilateral consistency problem in definition 3 can be solved by the controller (3).

[0062] For example, let's consider the following function:

[0063]

[0064] Taking the time derivative of (15) yields the following:

[0065]

[0066] in

[0067]

[0068]

[0069] By combining (17)-(21) with condition (8), we can obtain This indicates The multilateral consistency problem mentioned in Definition 3 has been solved.

[0070] Next, we will address the multilateral consistency optimization problem mentioned in Definition 4.

[0071] Consider the partial differential equation system (2), if assumption 1 holds. If condition (7) is satisfied, then the multilateral consensus optimization problem in definition 4 can be solved by the controller (3).

[0072] Based on condition (7), we can obtain Therefore, the value function f i,p (e i,p ) is a convex function. Based on the above multilateral consistency results, we know that... Based on conditions (7) and (8), we can obtain therefore Minimize Thus, the multilateral consistency optimization problem proposed in Definition 4 is solved.

[0073] Beneficial effects: Compared with the prior art, the advantages of the present invention are: it can solve the optimization problem of large-scale multi-agent systems more effectively, and it can react quickly to changes in error. Attached Figure Description

[0074] Figure 1 A topology diagram of communication between intelligent agents;

[0075] Figure 2 This is a flowchart of the method of the present invention. Detailed Implementation

[0076] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0077] Figure 2 The design flowchart of this invention consists of the following eight steps, each described below:

[0078] Step 1: Construct a diffusion-partially differential multi-agent system model:

[0079]

[0080] w i (x,0)=w i0 (x).

[0081] Where w i (x,t) represents the state of the i-th node at x and t, where θ∈R + Where is the diffusion coefficient. and They represent w respectively i The first derivative of (x,t) with respect to t and x. To represent w i The second derivative of (x,t) with respect to x is u. i (x,t) is the control input, ω i0 (x) represents the initial condition of ω(x,t). For simplicity, we will use w as a shorthand. i (x,t)=w i .

[0082] Step 2: Propose the definitions of groups and structural balance in symbolic networks:

[0083] Definition 1: (Group) For an undirected connected symbolic graph G = (V, E, A), where V = {v1, v2, ..., v...} N} is a point set, The set of edges A = (a ij ) N×N Let a be an adjacency matrix. ij For node v i and v j The weights between them. There exists a subset. Make all in graph G p The weights between nodes in the matrix are positive, that is, for {(v i ,v i )∈E p |G p =(V p E p A p )},a ij If ≥0, then graph G p All nodes are assigned to the same group, each group contains at least one node, and every pair of nodes within a group is connected.

[0084] Definition 2: (Structural equilibrium) An undirected connected symbolic graph G = (V, E, A) is said to be structurally balanced if V can be divided into K non-empty subsets V1, V2, ..., V K That is, for p ≠ q ∈ {1, 2, ..., K}, we have V1 ∪ V2 ∪ … ∪ V K =V,V p ∩V q =Φ, such that

[0085]

[0086] In addition, the negative topology between different subsets needs to ensure that for v i ∈V p ,v j ∈V q , (v i ,v j )∈E, satisfying Πa ij >0.

[0087] Step 3: According to Figure 1 The system groups agents based on the topological information exchanged between them. Agents with positive weighted connections can be considered as a group, and each group is connected by non-positive weighted connections. Figure 1 In the table, nodes 1-4 belong to group 1, 5-7 belong to group 2, and 8 and 9 belong to group 3. That is, v1, v2, v3, v4 ∈ V1, v5, v6, v7 ∈ V2, and v8, v9 ∈ V3.

[0088] The following assumptions are made regarding the symbol diagram:

[0089] Assumption 1: The undirected symbolic graph G of all groups is structurally balanced.

[0090] Step 4: Based on the definition of a group, establish the error expression and give the error system.

[0091] Let e i,p =w i -c p sgn(a ij )w j Therefore, based on the original partial differential multi-agent system, the error system can be obtained as follows:

[0092]

[0093] e i,p (x,0)=e i0,p (x).

[0094] Where c p e is the constant to be designed. i,p This represents the state of the i-th node at x and t. and They represent e respectively i,p First derivative with respect to t and x To represent e i,p The second derivative with respect to x, u i To control the input, e i0,p (x) represents e i,p The initial conditions.

[0095] Step 5: Define multilateral consensus and its optimization.

[0096] Definition 3: (Multilateral Consistency) Under Assumption 1, if the following equation holds, then a multi-agent system is said to have achieved multilateral consistency:

[0097]

[0098] Where m p ∈R is the p-th group G p Consistency value.

[0099] Definition 4: (Multilateral Consistency Optimization) Under the condition that Assumption 1 holds, design an optimal controller u. pi This will achieve the following optimization objectives:

[0100]

[0101] ste i,p =-m p ,

[0102] Where f i,p (e i,p ) is about e i,p The value function of .

[0103] Step 6: Design a new proportional-partial-derivative (PPD) controller:

[0104] For v i ∈V p Design a PPD optimization controller as follows:

[0105]

[0106] Where k p ,k d ,k g ∈R + Let f be the control gain to be designed, and ▽f i,p (e i,p ) is f i,p (e i,p The gradient of ).

[0107] Step 7: Provide a system stability analysis under the controller:

[0108] If assumption 1 holds, then the relationship between the value function, the system state, and the multilateral consensus objective value is as follows:

[0109] ▽f i,p (e i,p )=ξ p e i,p +ρ p ,

[0110]

[0111] Where ξ p ∈R + , ρ p Let ∈R be the value coefficient. Then, for x∈[0,l], t∈[0,+∞), the partial differential multi-agent system in the controller u i The solution exists and is unique.

[0112] Step 8: Solve the multilateral consistency optimization problem so that each agent in the group reaches its optimal value relative to its group:

[0113] If assumption 1 holds, then the relationship between the value function, the system state, and the multilateral consensus objective value is as follows:

[0114] ▽f i,p (e i,p )=ξ p e i,p +ρ p ,

[0115]

[0116] Therefore, the multilateral consensus optimization problem in Definition 4 can be solved by controller u. i It was achieved.

Claims

1. A multilateral consensus optimization control method based on partial differential equation multi-agent systems, characterized in that, Includes the following steps: a) Construct a diffusion-partial differential multi-agent system model; b) Propose the definition of groups and structural balance in symbolic networks, group the system, regard agents with positive weight connections as a group, and connect each group with non-positive weights. c) Based on the definition of a group, establish the error expression; d) Propose definitions of multilateral consensus and multilateral consensus optimization; e) Design a new proportional-partial-derivative (PPD) controller and provide a well-posed analysis of the system under the controller; d) Design the relationship between the cost function and the system state to obtain the conditions for solving the multilateral consensus optimization problem, so that each agent in the group reaches the optimal value associated with its group; Step a) is as follows: set up It is a positional variable, and its range is from to normal number , It is a time variable; the diffusion partial differential multi-agent system model is constructed as follows: in Indicates the first Each node and The state of being, Where is the diffusion coefficient. and Represent about and The first derivative, To represent about The second derivative, To control the input, express The initial conditions, and briefly noted. ; In step b), the definitions of group and structural balance in symbolic networks are as follows: Group definition: For an undirected connected symbolic graph ,in, For point set, It is the set of edges It is an adjacency matrix. For nodes and There exists a subset of the weights between them. Make all in the graph The weights between nodes in the array are positive, meaning that for... ,have Then the diagram All nodes are assigned to the same group, each group contains at least one node, and every pair of nodes within a group is connected; Definition of structural equilibrium: Undirected connected symbolic graph It is called structurally balanced, if Can be divided into a nonempty subset That is to say, for ,have , , making In addition, the negative topology between different subsets needs to ensure that for , ,satisfy ; The following assumptions are made about the symbol diagram: Assumption 1: An undirected symbolic graph of all groups They are all structurally balanced; Step c) is as follows: make The error system obtained from the original partial differential multi-agent system is as follows: in For the constant to be designed, Indicates the first Each node and The state of being, and Represent about and The first derivative, To represent about The second derivative, To control the input, express Initial conditions; Based on the above error system, the definition of multilateral consistency is as follows: Multilateral Consistency Definition: Under Assumption 1, if the following equation holds, then a multi-agent system is said to have achieved multilateral consistency: in It is the first Individual groups Consistency value; Multilateral consensus optimization definition: Under the condition that Assumption 1 holds, design an optimal controller. This will achieve the following optimization objectives: in It is about The value function; In step e), a new proportional-partial-derivative (PPD) controller is designed, specifically as follows: for Design a new proportional-partial-derivative (PPD) controller: in The control gain to be designed, and yes The gradient.

2. The multilateral consensus optimization control method based on partial differential equation multi-agent systems according to claim 1, characterized in that, Step e) provides a system stability analysis under the controller, specifically as follows: If assumption 1 holds, then the relationship between the value function, the system state, and the multilateral consensus objective value is as follows: in , As the value coefficient, then for , Partial differential multi-agent systems in controller The solution exists and is unique.

3. The multilateral consensus optimization control method for partial differential multi-agent systems according to claim 2, characterized in that, Step d) specifically refers to: (1) Consider a partial differential equation system. If Assumption 1 holds, and if the relationship between the value function and the system state and the multilateral consensus objective value is satisfied, then the multilateral consensus problem in the definition of multilateral consensus is solved by the controller. The problem has been solved; (2) Consider a partial differential equation system. If Assumption 1 holds; if the relationship between the value function and the system state and the multilateral consensus objective value is satisfied, then the multilateral consensus optimization problem in the definition of multilateral consensus optimization is solved by the controller. It was achieved.

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