A distributed arbitrary-time optimization method based on multi-agent system

Abstract

This invention discloses a distributed arbitrary-time optimization method for multi-agent systems, comprising the following steps: requiring the objective function to be a convex function and twice continuously differentiable; utilizing the gradient information of the objective function; and then combining the information of the agent nodes themselves, the reference model information, and the information of their neighboring agent nodes to design a control algorithm that minimizes the team's objective function at any given time. This invention achieves distributed arbitrary-time optimization of multi-agent systems, addressing the impact of initial values, some parameters, or constants of each agent on the settling time.

Description

Technical Field

[0001] This invention relates to the field of multi-agent control technology, and in particular to distributed arbitrary-time optimization of multi-agent systems. Background Technology

[0002] With the widespread application of distributed control strategies in various fields such as UAV formation, distributed sensor networks, and distributed optimal scheduling of power systems, distributed optimization of multi-agent systems has also attracted widespread attention. Typically, the cooperative minimization of the global objective function in a distributed optimization task within a multi-agent system can be represented as a single optimization problem. Distributed optimization, through the collaborative coordination among multiple agents, can better solve large-scale optimization problems, thus being more complex and widely applicable than traditional optimization. Therefore, the popularity of distributed optimization has been steadily increasing in recent years.

[0003] First, we define distributed optimization in multi-agent systems. An agent is a device or machine possessing autonomy, reactivity, initiative, sociality, and evolutionary potential. A multi-agent system is a network composed of multiple agents that communicate through the network's topology rules to collaboratively solve complex tasks. Therefore, distributed optimization in multi-agent systems involves each agent communicating through the network's topology rules and minimizing the team's objective function through cooperation.

[0004] As research deepens, distributed optimization algorithms for multi-agent systems are being applied more extensively in practical engineering, and the problems they address are becoming increasingly complex. Previously, the settling time of finite-time optimization relied on the initial values ​​of the multiple agents, and there were constraints on these initial values, requiring each agent's initial state to be the minimum of its local cost function. Under even stricter conditions, the settling time of fixed-time optimization algorithms is only affected by a few parameters or constants. Therefore, these algorithms all have significant limitations.

[0005] To date, most existing research on distributed optimization of multi-agent systems has significant limitations on the settling time, and it cannot be arbitrarily specified. In practical applications, the fewer constraints on the settling time, the better. Therefore, researching distributed arbitrary-time optimization of multi-agent systems is of practical significance. Summary of the Invention

[0006] The purpose of this invention is to provide a distributed arbitrary-time optimization method based on multi-agent systems to solve the problems mentioned in the background art.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A distributed, arbitrary-time optimization method based on multi-agent systems includes the following steps:

[0009] S1: Define the dynamic model of the research problem;

[0010] S2: Define the optimization objectives;

[0011] S3: Define the objective function and gradient information;

[0012] S4: Using the information of the agent node itself and the information of its neighboring agent nodes, design a distributed control algorithm. Under the Lyapunov stability theorem, the algorithm enables multiple agents to reach consensus at any pre-specified time and achieve the optimal solution to the optimization problem.

[0013] S5: The above dynamic model, control algorithm, objective function, and gradient information are programmed into each multi-agent system.

[0014] Compared with existing technologies, the technical highlights of this invention are: based on gradients and information, a novel distributed arbitrary-time optimization algorithm is designed. This algorithm enables each agent to minimize the global objective function within a predetermined time, without requiring each agent to minimize its local objective function in its initial state at the initial time. The steady-state time does not contain any parameters or constants. Attached Figure Description

[0015] Figure 1 This is a multi-agent communication topology diagram in this embodiment.

[0016] Figure 2 This is a state response diagram of multiple agents under the proposed method in this embodiment.

[0017] Figure 3 This is the control input diagram for the multiple agents in this embodiment. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of this invention clearer and easier to understand, the invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0019] A distributed, arbitrary-time optimization method based on multi-agent systems includes:

[0020] S1. Define the dynamic model of the research problem;

[0021] The multi-agent dynamics model contains n agents, and the agent dynamics model is as follows:

[0022]

[0023] Where x(t)=[x1(t),x2(t),…,xn (t)] T ∈R n u(t) = [u1(t), u2(t), ..., u n (t)] T ∈R n These represent the agent's position state and control input, respectively.

[0024] S2: Define the optimization objectives;

[0025] Optimization goal:

[0026]

[0027] Where f i (x i Let represent the objective function of agent i.

[0028] S3: Define the objective function and gradient information;

[0029] The objective function of each multi-agent is a convex function and is twice continuously differentiable.

[0030] S4: Design a distributed control algorithm using information from the agent node itself and information from its neighboring agent nodes;

[0031]

[0032] in g = [g1, g2, ..., g n ] T ,I n Represents an n-dimensional identity matrix, 1 n =[1,1,…,1] T λ2(L) is the second smallest non-zero eigenvalue of the Laplace matrix L, t f The settling time is arbitrarily specified in advance, as shown below:

[0033]

[0034] in As shown below:

[0035]

[0036] As shown below:

[0037]

[0038] As shown below:

[0039]

[0040] in H = [H1,H2,…,H] n ] T Let μ1 and μ2 represent the gradient and Hessian matrix of the objective function, respectively. There exist ε > 0, p1, p2,

[0041] S5: The above dynamic model, control algorithm, objective function, and gradient information are programmed into each multi-agent system.

[0042] In this embodiment, preferably, x i (t)∈R,u i (t)∈R.

[0043] The novel Laplace algorithm corresponding to the multi-agent communication topology in this embodiment is as follows:

[0044]

[0045] The objective function is selected as follows:

[0046]

[0047] The initial values ​​for each multi-agent are selected as follows:

[0048] x(0)=[-1.1,-1.2,-1.3,-1.4,-1.5] T

[0049] Preferably, the system control parameters are designed as follows:

[0050] eta1=eta2=μ1=μ2=μ3=3, p1=p2=p3=10.

[0051] Verification yields

[0052] t f Select 1s and 2s, according to Figure 2 It can be seen that the states of each agent can converge to the same state within 1 second and 2 seconds, respectively, and obtain the optimal value of -0.9474. And according to... Figure 3 This shows that the control of each agent will not tend to infinity and will gradually converge to 0. That is, the algorithm can achieve distributed arbitrary-time optimization of multi-agent systems.

[0053] The above embodiments are merely preferred embodiments of the present invention. It should be noted that those skilled in the art can make several modifications and improvements without departing from the concept of the present invention, and these should also be considered within the scope of protection of the present invention. These will not affect the effectiveness of the implementation of the present invention or the practicality of the patent.

Claims

1. A multi-agent system based distributed arbitrary-time optimization method, characterized in that: include: S1: Clearly define the dynamic model of the research problem; specifically: The multi-agent dynamics model contains n agents, and the agent dynamics model is as follows: Where, x(t)=[x1(t),x2(t),…,x n (t)] T ∈R n u(t) = [u1(t), u2(t), ..., u n (t)] T ∈R n These represent the agent's position state and control input, respectively. S2: Define the optimization goal; the optimization goal is specific to: where f i (x i ) represents the objective function of agent i; S3: Define the objective function and gradient information; specifically, the objective function of each multi-agent is a convex function and is twice continuously differentiable; the communication network of the agents is an undirected connected topology, and the system has feasible inputs to achieve the control objective; S4: Utilizing information from the agent nodes themselves and their neighboring agent nodes, design a distributed control algorithm. Under Lyapunov's stability theorem, ensure that multiple agents reach consensus at any pre-specified time and achieve the optimal solution to the optimization problem. The designed control algorithm is as follows: in g = [g1, g2, ..., g n ] T I n Represents an n-dimensional identity matrix, 1 n =[1,1,…,1] T λ2(L) is the second smallest non-zero eigenvalue of the Laplace matrix L, t f The settling time is arbitrarily specified in advance, as shown below: in As shown below: As shown below: As shown below: Where ▽f=[▽f1,▽f2,…,▽f n ] T H = [H1, H2, ..., H n ] T Let these represent the gradient and Hessian matrix of the objective function, respectively. There exists ε>

0. ; S5: The above dynamic model, control algorithm, objective function, and gradient information are programmed into each multi-agent system.

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