Multilateral consistency optimization method based on partial differential multi-agent system

Abstract

The invention discloses a multilateral consistency optimization control method for a partial differential multi-agent system. The multilateral consistency optimization control method comprises the following steps: a) constructing a diffusion partial differential multi-agent system model; b) proposing definitions of groups of the symbol network and structure balance, grouping the system, regarding the agents with positive weight connection as one group, and connecting the groups by non-positive weight values; c) establishing an error expression according to the definition of the group; d) proposing definitions of multilateral consistency and multilateral consistency optimization; e) designing a novel proportional partial derivative (PPD) controller; f) giving out a partial differential system suitability analysis under the controller; and g) designing a relationship between a cost function and a system state to obtain a condition for solving a multilateral consistency optimization problem, so that each agent in the group reaches an optimal value related to the group. The method is suitable for solving the distributed optimization problem of a large-scale multi-agent system, has quick responsiveness and can solve various practical problems.

Description

Technical Field

[0001] The present invention relates to the field of multi-agent system control and optimization, and in particular to a multilateral consistency optimization method based on partial differential multi-agent system. Background Art

[0002] In recent decades, partial differential equations have attracted widespread attention due to their applications in various fields such as 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 pipes. At the same time, the stability of PDE systems has also been widely studied in theory.

[0003] The focus is now shifting to the application of multi-agent systems characterized by partial differential equations in large-scale and complex problems. Among them, consistency, as a basic dynamic behavior in multi-agent systems, has become a key area of ​​concern for researchers. However, in actual engineering applications, not all agents show cooperative relationships and may contain adversarial information, which is very challenging for the study of consistency in multi-agent systems. In order to solve this problem, the concept of bilateral consistency with adversarial information was proposed. Subsequently, researchers have become increasingly interested in bilateral consistency in partial differential equation multi-agent systems. At present, the optimization problem of partial differential multi-agent systems has been applied in many fields such as robot deep learning, transportation, autonomous driving, and gas transmission.

[0004] With the acceleration of globalization, cooperation and competition among countries are intertwined, so the problem of multilateral cooperation and confrontation is becoming increasingly prominent. It is mainly reflected in international security, international business, international trade and international economy. In this context, in-depth discussion of the common interests and differences of all parties in multilateral relations is conducive to promoting the stability and development of international relations. Therefore, it is necessary to distinguish the parties so that the information exchange within each group is cooperative, and the information exchange between different groups is non-cooperative, especially antagonistic, which is called the multilateral consistency of the group. Summary of the invention

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

[0006] Technical solution:

[0007] A multilateral consensus optimization method based on partial differential multi-agent system, comprising the following steps:

[0008] a) Construct a diffuse micro multi-agent system model;

[0009] b) Propose the definition of group and structural balance of signed networks, group the system, treat agents with positive weight connections as a group, and connect each group with non-positive weights;

[0010] c) Establish the error expression according to the definition of the group;

[0011] d) Propose the definition of multilateral consistency and multilateral consistency optimization;

[0012] e) Design a new proportional partial derivative PPD controller and give the system well-posedness analysis 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 related to its group.

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

[0015] Let x∈[0,l] be a position variable, which ranges from 0 to a positive constant l, and t∈[0,+∞) be a time variable. The model of the diffusion partial micro multi-agent system 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, θ∈R + is the diffusion coefficient, and Represents w i The first-order derivative of (x,t) with respect to t and x, To represent w i The second-order 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). To simplify the subsequent explanation, we will simply write w i (x,t)=w i .

[0019] Furthermore, the definition of group and structural balance in symbolic networks is 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, is the set of edges A=(aij ) N×N is the adjacency matrix, a ij For node v i and v j There exists a subset So that all the p The weights between nodes in are positive, that is, for {(v i ,v i )∈E p |G p =(V p ,E p ,A p )},a ij ≥0, then graph G p All nodes in are assigned to the same group, each group contains at least one node, and every pair of nodes in a group is connected.

[0021] Definition 2: (Structurally balanced) An undirected connected signed graph G = (V, E, A) is called structurally balanced if V can be partitioned into K non-empty subsets V1, V2, ..., V K , that is, for p≠q∈{1,2,…,K}, V1∪V2∪…∪V K =V, V p ∩V q =Φ, so that

[0022]

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

[0024] The symbol diagram must satisfy the following assumptions:

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

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

[0027] Let e i,p =w i -c p sgn(a ij ) j , then 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 is the constant to be designed, e i,p represents the state of the i-th node at x and t, and Represents e i,p The first-order derivative with respect to t and x is, To represent e i,p The second derivative with respect to x, u i is the control input, e i0,p (x) represents e i,p initial conditions.

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

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

[0033]

[0034] Where m p ∈R is the pth group G p The consistency value of .

[0035] Definition 4: (Multilateral Consistency Optimization) Under the assumptions 1 and 2, design the optimal controller u pi The following optimization goals are achieved:

[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 for v i ∈V p , design a new proportional partial derivative (PPD) controller:

[0040]

[0041] where k p ,k d ,kg ∈R + is the control gain to be designed, and ▽f i,p (e i,p ) is f i,p (e i,p )’s gradient.

[0042] Next, the system well-posedness analysis under the controller is given as follows:

[0043] If Assumption 1 holds, if the relationship between the value function and the system state and the multilateral consensus target 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 micro multi-agent system (2) has a controller u i The solution exists and is unique.

[0047] Let e ​​= (e 1,p ,e 2,p ,...e N,p ) T . According to condition (7), we can get

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

[0049] Multiply (9) by e T , and integrating from 0 to l gives in,

[0050]

[0051] Consider the following energy function:

[0052]

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

[0054]

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

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

[0057] The uniqueness of the solution of 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 , considering the energy function By taking the derivative, we can get

[0058]

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

[0060] In order to further solve the multilateral consistency optimization problem, the main method is to design the relationship between the cost function and the system state, and obtain the conditions for solving the multilateral consistency optimization problem so that each agent in the group reaches the optimal value related to its group, as follows:

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

[0062] For, let, consider the following function:

[0063]

[0064] Taking the time derivative of (15) we can obtain:

[0065]

[0066] in

[0067]

[0068]

[0069] Through (17)-(21) and condition (8), we can get This shows that The multilateral consistency problem mentioned in Definition 3 is solved.

[0070] Next, we solve 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, the multilateral consistency optimization problem in Definition 4 can be solved by the controller (3).

[0072] According to condition (7), we can get So the value function f i,p (e i,p ) is a convex function. According to the above multilateral consistency results, we can know According to conditions (7) and (8), we can get therefore Can be minimized In this way, 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 more effectively solve the optimization problem of large-scale multi-agent systems and can quickly respond to changes in errors. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 It is the communication topology diagram between intelligent agents;

[0075] Figure 2 The figure is a flow chart of the method of the present invention. DETAILED DESCRIPTION

[0076] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0077] Figure 2 The design flow chart of the present invention is composed of the following eight steps, each of which is described as follows:

[0078] Step 1: Construct a diffuse micro 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, θ∈R + is the diffusion coefficient, and Represents w i The first-order derivative of (x,t) with respect to t and x, To represent w i The second-order 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). To simplify the subsequent explanation, we will simply write w i (x,t)=w i .

[0082] Step 2: Propose the definition of group and structural balance of the signed network:

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

[0084] Definition 2: (Structurally balanced) An undirected connected signed graph G = (V, E, A) is called structurally balanced if V can be partitioned into K non-empty subsets V1, V2, ..., V K , that is, for p≠q∈{1,2,...,K}, V1∪V2∪…∪V K =V, V p ∩V q =Φ, so that

[0085]

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

[0087] Step 3: Based on Figure 1 The communication topology information between agents in the system is used to group the system. Agents with positive weight connections can be regarded as a group, and each group is connected by non-positive weights. Figure 1 In the example, 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, v8, v9∈V3.

[0088] The assumptions about the symbol graph are as follows:

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

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

[0091] Let e i,p =w i -c p sgn(a ij ) j , then according to 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 is the constant to be designed, e i,p represents the state of the i-th node at x and t, and Represents e i,p The first-order derivative with respect to t and x is, To represent e i,p The second derivative with respect to x, u i is the control input, e i0,p (x) represents e i,p initial conditions.

[0095] Step 5: Propose the definition of multilateral consistency and multilateral consistency optimization:

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

[0097]

[0098] Where m p ∈R is the pth group G p The consistency value of .

[0099] Definition 4: (Multilateral Consistency Optimization) Under the condition that Assumption 1 holds, design the optimal controller u pi The following optimization goals are achieved:

[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 + is the control gain to be designed, and ▽f i,p (e i,p ) is f i,p (e i,p )’s gradient.

[0107] Step 7: Give the system suitability analysis under the controller:

[0108] If Assumption 1 holds, if the relationship between the value function and the system state and the multilateral consensus target 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 ∈R is the value coefficient, then for x∈[0,l], t∈[0,+∞), the partial micro multi-agent system has a controller u i The solution exists and is unique.

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

[0113] If Assumption 1 holds, if the relationship between the value function and the system state and the multilateral consensus target value is as follows:

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

[0115]

[0116] Then the multilateral consistency optimization problem in Definition 4 can be solved by the controller u i Be realized.

Claims

1. A multilateral consensus optimization method based on partial differential multi-agent system, characterized in that: The following steps are involved: a) Construct a diffuse micro multi-agent system model; b) Propose the definition of group and structural balance of signed networks, group the system, treat agents with positive weight connections as a group, and connect each group with non-positive weights; c) Establish the error expression according to the definition of the group; d) Propose the definition of multilateral consistency and multilateral consistency optimization; e) Design a new proportional partial derivative PPD controller and give the system well-posedness analysis 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 related to its group.

2. The multilateral consistency optimization method based on partial differential multi-agent system according to claim 1 is characterized in that: Step a) is as follows: Let x∈[0,l] be a position variable, ranging from 0 to a positive constant l, and t∈[0,+∞) be a time variable; the diffusion partial micro multi-agent system model is constructed as follows: w i (x,0)=w i0 (x), where w i (x, t) represents the state of the i-th node at x and t, θ∈R + is the diffusion coefficient, and Represents w i The first-order derivative of (x,t) with respect to t and x, To represent w i The second derivative of (x,t) with respect to x, u i (x, t) is the control input, ω i0 (x) represents the initial condition of ω(x,t), and is abbreviated as w i (x,t)=w i .

3. The multilateral consistency optimization method based on partial differential multi-agent system according to claim 2 is characterized in that: In step b), the group and structural balance of the symbolic network are defined as follows: Group definition: For an undirected connected symbolic graph G = (V, E, A), where V = {v1, v2, ..., v N } is a point set, is the set of edges A=(a ij ) N×N is the adjacency matrix, a ij For node v i and v j There is a subset of weights between So that all the p The weights between the nodes in are positive, that is, {(v i ,v i )∈E p |G p =(V p ,E p ,A p )}, there is a ij ≥0, then graph G p All nodes in are assigned to the same group, each group contains at least one node, and every pair of nodes in the group is connected; Definition of structural balance: An undirected connected symbolic graph G = (V, E, A) is called structurally balanced if V can be divided into K non-empty subsets V1, V2, ..., V K , that is, for p≠q∈{1,2,...,K}, V1∪V2∪…∪V K =V, V p ∩V q =Φ, so that In addition, the negative topology between different subsets needs to be such that for v i ∈V p ,v j ∈V q ,(v i ,v j )∈E, satisfying Πa ij >0; The following assumptions are made about the symbolic graph: Assumption 1: The undirected signed graph G of all groups is structurally balanced.

4. The multilateral consistency optimization method based on partial differential multi-agent system according to claim 3 is characterized in that: Step c) is as follows: Let e i,p =w i -c p sgn(a ij ) j , according to the original partial differential multi-agent system, the error system is as follows: And i,p (x,0)=and i0,p (x). where c p is the constant to be designed, e i,p represents the state of the i-th node at x and t, and Represents e i,p The first-order derivative with respect to t and x is, To represent e i,p The second derivative with respect to x, u i is the control input, e i0,p (x) represents e i,p initial conditions. According to the above error system, the definition of multilateral consistency is as follows: Definition of multilateral consistency: Under assumption 1, if the following equation holds, then the multi-agent system is said to achieve multilateral consistency: Where m p ∈R is the pth group G p The consistency value of Multilateral consistency optimization definition: Under the condition that assumption 1 holds, design the optimal controller u pi The following optimization goals are achieved: where f i,p (e i,p ) is about e i,p The value function of .

5. The multilateral consistency optimization method based on partial differential multi-agent system according to claim 4 is characterized in that: In step e), a new proportional partial derivative PPD controller is designed, specifically: For v i ∈V p , design a new proportional partial derivative PPD controller: where k p ,k d ,k g ∈R + is the control gain to be designed, and ▽f i,p (e i,p ) is f i,p (e i,p )’s gradient.

6. The multilateral consistency optimization method based on partial differential multi-agent system according to claim 5 is characterized in that: In step e), the system suitability analysis under the controller is given, specifically: If Assumption 1 holds, if the relationship between the value function and the system state and the multilateral consensus target value is as follows: ▽f i,p (e i,p )=ξ p e i,p +r p , where ξ p ∈R + , ρ p ∈R is the value coefficient, then for x∈[0,l], t∈[0,+∞), the partial micro multi-agent system has a controller u i The solution exists and is unique.

7. The multilateral consistency optimization method based on partial differential multi-agent system according to claim 6 is characterized in that: Step d) is specifically: (1) Consider a system of partial differential equations. If Assumption 1 holds; if the relationship between the value function and the system state and the multilateral consistency target value is satisfied, then the multilateral consistency problem in the multilateral consistency definition is solved by the controller u i be resolved; (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 consistency target value is satisfied, then the multilateral consistency optimization problem in the multilateral consistency optimization definition is solved by the controller u i Be realized.

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