Distributed Optimization Control Methods for Multi-Agent Systems with Security Constraints and Uncertainty

Abstract

This invention relates to the fields of control and information technology, specifically disclosing a distributed optimization control method for multi-agent systems with uncertain security constraints. The method includes the following steps: constructing a topology graph based on the network structure of the multi-agent system and determining the adjacency matrix of the topology graph; determining the state equation; determining the objective function to be optimized and the inequality constraints to be satisfied; establishing optimal solution search conditions based on the control Lyapunov function method and the control barrier function method based on the state equation, the objective function to be optimized, and the inequality constraints to be satisfied; determining the system security maintenance conditions based on the multi-agent system; calculating the optimal control input using a quadratic programming method; and controlling the multi-agent system based on the optimal control input. This invention can significantly improve the long-term reliability of the system and significantly reduce communication and computational overhead, resulting in a significant improvement in computational efficiency.

Description

Technical Field

[0001] This invention relates to the fields of control and information technology, and in particular to a distributed optimization control method for multi-agent systems with safety constraints and uncertainties. Background Technology

[0002] Multi-agent systems consist of multiple agents with local autonomous decision-making capabilities connected through interaction. Each agent can perceive the local environmental state and interact with other agents. Given the prevalent large-scale and complex systems in the real world, the core objective of multi-agent systems is to model or reconstruct them as a distributed, communicable, and coordinated collection of subsystems. Essentially, they form an organic whole through the collaborative interaction of multiple agents to solve large and complex problems that exceed the capabilities of a single agent.

[0003] In recent years, the widespread application of multi-agent network optimization technology in fields such as robot collaborative operation, industrial IoT, smart grids, and 5G communication has driven distributed optimization to become a core research direction in the field of control. This technology aims to efficiently solve the minimization problem of a global objective function through local communication and cooperation among agents—where each agent relies only on its own objective function and the state information of its neighboring nodes, eliminating the need for centralized control. Compared to centralized methods, distributed optimization methods have the advantages of low computational cost and low communication cost, and improve the robustness and adaptability of the system. For example, Chinese invention patent application CN117742141A discloses a distributed local Lipshitz algorithm based on a control barrier function, which can solve safety-critical nonlinear nonconvex optimization problems. During the calculation process, each node only utilizes the information of its neighboring nodes, saving communication and computational costs and significantly improving computational efficiency.

[0004] However, in practical optimization problems of multi-agent systems, uncertainties in system ontology, parameters, and perceived information lead to uncertainties in their dynamic models. For second-order robot formation systems, ensuring system safety is paramount, requiring the robot formation's trajectory to always remain within a safe zone. The challenges in designing distributed optimization control methods lie in: 1. the existence of unknown nonlinear dynamics in the system dynamics; 2. the strict constraint of boundedness for control inputs; 3. the requirement to maintain system safety at all times; and 4. ensuring the distributed nature of the algorithm, each node can only utilize its own and its neighbors' information. Summary of the Invention

[0005] This invention aims to solve the aforementioned problems. To this end, it provides a distributed optimization control method for safety-constrained uncertain multi-agent systems, specifically addressing the optimization problem of safety-critical, input-constrained uncertain second-order nonlinear multi-agent systems, particularly the optimization of multi-agent trajectories in a second-order robot formation system within a safe region. This invention designs an optimal solution search mechanism based on a control Lyapunov function, introduces a control barrier function to ensure system safety, considers input constraints, and finally solves for the control input through quadratic programming. This significantly improves the long-term reliability of the system, substantially reduces communication and computational overhead, and significantly enhances computational efficiency.

[0006] This invention provides a distributed optimization control method for multi-agent systems with uncertain safety constraints, and the technical solution adopted is as follows: including the following steps:

[0007] S1: Construct a topology graph based on the network structure of the multi-agent system and determine the adjacency matrix of the topology graph;

[0008] S2: Determine the state equations based on the network structure of the multi-agent system, where the state equations include an unknown nonlinear dynamic part;

[0009] S3: Based on the network structure and adjacency matrix of the multi-agent system, determine the objective function to be optimized and the inequality constraints to be satisfied. Among them, the inequality constraints serve as safety constraints, which need to avoid collisions between agents.

[0010] S4: Based on the state equation, the objective function to be optimized, and the inequality constraints to be satisfied, the optimal solution search conditions are established based on the control Lyapunov function method, and the system safety maintenance conditions are established based on the control barrier function method. In this process, the estimated value of the unknown nonlinear dynamic part and the upper bound of the estimation error are first determined, and then the optimal solution search conditions and system safety maintenance conditions are established.

[0011] S5: Determine the input constraints based on the multi-agent system;

[0012] S6: Based on the optimal solution search conditions, system security maintenance conditions, and input constraints, the optimal control input is calculated using the quadratic programming method.

[0013] S7: Control the multi-agent system based on the optimal control input.

[0014] Furthermore, the multi-agent system is a second-order robot formation system.

[0015] Furthermore, in step S2, the state equation is:

[0016]

[0017]

[0018] in, For the first The location of each agent. for The first derivative, For the first The speed of each intelligent agent for The first derivative, For the first Control input for an intelligent agent For the unknown nonlinear dynamics, , This is the robot's actual weight. It is unknown. This is the robot's nominal weight. To control the gain, .

[0019] Furthermore, in step S3, the objective function to be optimized is... The inequality constraints that need to be satisfied are:

[0020]

[0021]

[0022]

[0023] Among them, variables The stack of decision variables belonging to each agent. , For intelligent agents The local objective function, , For robots The neighborhood group, For robots and robots The ideal distance between them For nodes and nodes The weight values ​​of the edges between them. The total number of nodes. As constraints, Let inequality constraint function be used. , To maintain a safe distance between intelligent agents, For a set of nodes, To find the minimum value.

[0024] Furthermore, in step S4, the optimal solution search condition is:

[0025]

[0026] in, For intelligent agents Ideal speed, for The estimated value, To estimate the upper bound of the error, To control the convergence rate of the Lyapunov function, for The first derivative, For auxiliary control input, ;

[0027] The conditions for maintaining system security are:

[0028]

[0029] in, For function right The first-order partial derivative, For function right The second-order partial derivative, The coefficients of the first control barrier function are . The coefficients of the second control barrier function are given.

[0030] Furthermore, in step S5, the input constraints are:

[0031]

[0032] in, To control the gain The lower bound, To control input The upper boundary.

[0033] Furthermore, in step S6,

[0034]

[0035]

[0036]

[0037] in, As slack variables, To control input With slack variables integration, This represents the optimal value for a constrained quadratic programming optimization problem. This is the matrix transpose.

[0038] Furthermore, in step S1, the agents are treated as nodes in the topology graph, and the communication relationships between agents are treated as edges.

[0039] The above-described one or more technical solutions in the embodiments of the present invention have at least one of the following technical effects:

[0040] The distributed optimization control method for multi-agent systems with safety constraints and uncertainties provided by this invention has significant technical advantages, especially in second-order robot formation scenarios, addressing the uncertainties and safety control requirements of multi-agent systems.

[0041] 1. Precisely adapts to second-order robot formation systems, meeting core requirements for collaborative control.

[0042] This invention specifically targets second-order robot formation systems (such as UAV swarm formations). By constructing an undirected connected topology graph and combining it with an adjacency matrix to quantize the network structure, it can accurately describe the position, velocity, and state relationships of agents in a second-order system. Simultaneously, by incorporating ideal and safe distance constraints between agents into the objective function, distributed collaboration can be achieved without relying on a central node. This maintains the stability of the formation and effectively avoids collisions between agents. Compared to general multi-agent control methods, this invention offers more precise adaptability to second-order robot formations and can be directly applied to practical scenarios such as UAV swarm collaborative operations and ground robot formation driving, significantly reducing adaptation costs in engineering applications and improving the flexibility and reliability of formation control.

[0043] 2. Incorporating unknown nonlinear dynamics into the state equations significantly improves control robustness.

[0044] Unlike existing technologies that ignore or simplify nonlinear dynamics, the state equations of this invention explicitly include unknown nonlinear dynamic components. This design is highly compatible with real-world engineering scenarios: in practical applications, robots are prone to uncertainties in their dynamic parameters due to wear, load variations, environmental disturbances, etc. State equations containing unknown nonlinear terms can accurately capture these uncertainties, avoiding control deviations caused by unknown parameters. Furthermore, by determining the estimated value and upper bound of the estimation error of the unknown nonlinear dynamic components, and combining this with the control Lyapunov function to establish optimal solution search conditions, this invention can effectively suppress the influence of unknown nonlinearities on the position and velocity state of the agent. Even in scenarios with fluctuating parameters, it can still ensure the motion stability of second-order robot formations, significantly improving the robustness and practical adaptability of the system control.

[0045] 3. Introduce input constraints to achieve a balance between control performance and hardware safety.

[0046] This invention effectively avoids the problem of control input exceeding the capabilities of the agent's hardware by setting input constraints (clearly defining the lower bound of the control gain and the upper bound of the control input) and incorporating these constraints into the quadratic programming process for solving the optimal control input. On the one hand, the lower bound of the control gain ensures sufficient control force, preventing slow system convergence and formation maintenance failure due to insufficient gain. On the other hand, the upper bound of the control input limits the output of the agent's actuators to not exceed the hardware's rated load, avoiding equipment overload damage. Simultaneously, by introducing slack variables, this invention integrates the control input and slack variables and solves for the optimal value through quadratic programming. This ensures that, while satisfying the input constraints, the control input can still drive the system to converge to the optimal solution, achieving a two-way balance between control performance (optimal solution convergence) and hardware safety (input not exceeding limits). This further guarantees the long-term stable operation of the second-order robot formation system and reduces the risk of equipment failure.

[0047] In summary, this invention significantly improves the practicality, robustness, and safety of multi-agent system control by directional adaptation of second-order robot formation systems, accurate modeling of unknown nonlinear dynamics, and strict control of input constraints. It provides a reliable technical solution for engineering applications in scenarios such as smart grid scheduling, UAV collaborative formation, and industrial IoT node clusters.

[0048] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0049] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0050] Figure 1 This is a flowchart of the method provided by the present invention. Detailed Implementation

[0051] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention. The following embodiments are used to illustrate this invention but should not be used to limit the scope of this invention.

[0052] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., refer to specific features, structures, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0053] The following is combined Figure 1 The present invention will be further described in detail below, providing a distributed optimization control method for multi-agent systems with uncertain safety constraints:

[0054] In this embodiment, as Figure 1 As shown, a distributed optimization control method for multi-agent systems with uncertain safety constraints is provided, including the following steps:

[0055] S1: Construct a topology graph based on the network structure of the multi-agent system and determine the adjacency matrix of the topology graph.

[0056] In this embodiment, the multi-agent system is a second-order robot formation system, such as the formation of a drone swarm.

[0057] By treating agents as nodes in a topological graph and the communication relationships between agents as edges, a topological graph is constructed. This graph is undirected and connected. The network topology diagram of the nodes is as follows:

[0058]

[0059] in, For a set of nodes, , Let be the set of edges. .

[0060] The adjacency matrix of a topological graph is of the following form:

[0061]

[0062] Among them, the elements in the adjacency matrix For nodes and nodes The weight values ​​of the edges between them. , If the node and nodes There are edges connecting them (intelligent agents) and intelligent agents If they can communicate with each other, then they have If the node and nodes There are no edges connecting them (intelligent agents) and intelligent agents If they cannot communicate with each other, then there is For undirected graphs, there exists The degree matrix of this network For a diagonal matrix, the first... The diagonal element corresponds to the first The degree of each node is then its Laplace matrix. for .

[0063] S2: Determine the state equation based on the network structure of the multi-agent system.

[0064] Unlike existing technologies, the state equations constructed in this embodiment include an unknown nonlinear dynamic component, specifically:

[0065]

[0066]

[0067] in, For the first The location of each agent. , For the real number field, for The first derivative, For the first The speed of each intelligent agent , for The first derivative, For the first Control input for an intelligent agent . For the unknown nonlinear dynamics, , To control the gain, .

[0068] For a second-order robot formation system, the dynamic equations are:

[0069]

[0070]

[0071] but , ,in, This is the robot's actual weight. It is unknown. , This refers to the robot's nominal weight; each robot's nominal weight is known. .

[0072] S3: Based on the network structure and adjacency matrix of the multi-agent system, determine the objective function to be optimized and the inequality constraints to be satisfied.

[0073] Objective function to be optimized The inequality constraints that need to be satisfied are:

[0074]

[0075]

[0076]

[0077] Among them, variables The stack of decision variables belonging to each agent. , For intelligent agents The local objective function, , Let inequality constraint function be used. , The total number of nodes. As constraints, For a set of nodes, To find the minimum value, this general formula covers a wide range of real-world scenarios where multiple agents (each with local objectives and constraints) adjust and compute their own decisions through communication with neighbors, ultimately achieving consistency in decision variables. Local objective function. Used to quantify estimation accuracy, constraints Used for collision avoidance. By designing distributed controllers, these controllers can drive the system toward the optimal solution to the constraint problem, while ensuring feasibility and coordination among agents.

[0078] For second-order robot formation systems, , For robots The neighborhood group, For robots and robots The ideal distance between them , To maintain a safe distance between agents, Avoid collisions between intelligent agents.

[0079] S4: Based on the state equation, the objective function to be optimized, and the inequality constraints to be satisfied, establish the optimal solution search conditions based on the control Lyapunov function method, and establish the system security maintenance conditions based on the control barrier function method.

[0080] The estimated value is determined based on the unknown nonlinear dynamics, as well as the upper bound of the estimation error. Then, the optimal solution search conditions and system safety maintenance conditions are established.

[0081] The optimal solution search condition is:

[0082]

[0083] in, For intelligent agents Ideal speed, for The estimated value, To estimate the upper bound of the error, To control the convergence rate of the Lyapunov function, , for The first derivative, For auxiliary control input, .

[0084] The conditions for maintaining system security are:

[0085]

[0086] in, For function right The first-order partial derivative, For function right The second-order partial derivative, The coefficients of the first control barrier function are . The coefficients of the second control barrier function are... , .

[0087] S5: Determine the input constraints based on the multi-agent system.

[0088] The input constraints are:

[0089]

[0090] in, To control the gain The lower bound, To control input The upper boundary.

[0091] S6: Based on the optimal solution search conditions, system security maintenance conditions, and input constraints, the optimal control input is calculated using the quadratic programming method.

[0092] Design feedback control inputs using quadratic programming:

[0093]

[0094]

[0095]

[0096] in, As slack variables, , To control input With slack variables integration, This represents the optimal value for a constrained quadratic programming optimization problem. This is the matrix transpose.

[0097] S7: Control the multi-agent system based on the optimal control input.

[0098] Generally, this method requires setting the runtime when used. Overall, this method determines the dynamic equations of a second-order robot formation system based on its actual conditions, estimates the unknown nonlinear components, and determines the estimated values. Upper bound of estimation error Then set the input constraints, local objective function, and inequality constraint function; finally, set the parameters for the control Lyapunov function method and the control barrier function method. , and This method outputs the optimal value for a constrained quadratic programming optimization problem. Based on this optimal value, the multi-agent system is controlled over the runtime.

[0099] This method is particularly suitable for formation control of UAV swarms in uncertain scenarios, ensuring that the UAVs' trajectories always remain within a safe zone. This method requires only communication between neighboring nodes to compute the local objective function, combining high computational efficiency with low communication overhead; simultaneously, it guarantees system safety and satisfies control input constraints.

[0100] This method can solve the optimization problem of safety-critical, input-constrained, uncertain second-order nonlinear multi-agent systems. By constructing targeted dynamic equations based on the actual situation of the agent system and reasonably estimating the unknown nonlinear components, this method can also be applied to other multi-agent systems containing unmodeled dynamics. For example, in real-world industrial scenarios, multi-agent systems (such as smart grid scheduling, UAV collaborative formation, and industrial IoT node clusters) often exhibit unmodeled dynamics. These dynamics may originate from internal parameter drift, external environmental disturbances, or unknown coupling relationships in complex interactions, easily leading to problems such as decreased stability and response lag in traditional control methods. This method can guarantee safety specifications and satisfy actuator input constraints throughout the time domain, ensuring that the input signal is always within the safety threshold, avoiding mechanical wear and downtime caused by actuators operating beyond limits, and significantly improving the long-term reliability of the system. Compared to traditional centralized architectures that rely on global topology information and central node decisions, its distributed architecture only requires local topology information. Each agent node only needs to obtain its own and its neighboring nodes' local topology information, and the calculation process is entirely based on the real-time state data of neighboring nodes, which will reduce communication and computing overhead and significantly improve computing efficiency.

[0101] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for distributed optimization control of a security-constrained uncertain multi-agent system, characterized in that, The method comprises the following steps: S1: constructing a topology graph according to the network structure of the multi-agent system and determining an adjacency matrix of the topology graph; the multi-agent system is a second-order robot formation system, and the second-order robot formation system is a formation of a UAV group; S2: determining a state equation according to the network structure of the multi-agent system, wherein the state equation comprises an unknown nonlinear dynamics part; In step S2, the state equation is: wherein is the position of the th agent, is the first derivative of is the velocity of the th agent, is the first derivative of is the control input of the th agent, is the unknown nonlinear dynamics part, , is the real weight of the robot, is unknown, is the nominal weight of the robot, is the control gain, ; S3: determining an objective function to be optimized and an inequality constraint to be satisfied according to the network structure and the adjacency matrix of the multi-agent system, wherein the inequality constraint is a safety constraint, and the safety constraint needs to avoid collision between agents; S4: establishing an optimal solution search condition based on a control Lyapunov function method and a system safety maintenance condition based on a control barrier function method according to the state equation, the objective function to be optimized and the inequality constraint to be satisfied, wherein an estimated value of the unknown nonlinear dynamics part and an upper bound of an estimation error are determined first, and then the optimal solution search condition and the system safety maintenance condition are established; In step S4, the optimal solution search condition is: wherein is the ideal velocity of the agent , is the estimated value of , is an upper bound of the estimation error, is the convergence rate of the control Lyapunov function, is the first derivative of , is the auxiliary control input, ; The system safety maintenance condition is: wherein is a function of the first order partial derivative of with respect to is a function of the second order partial derivative of with respect to is a coefficient of the first control barrier function, is a coefficient of the second control barrier function; S5: determining an input constraint condition according to the multi-agent system; In step S5, the input constraint condition is: wherein is a lower bound for the control gain is a lower bound for the control gain is an upper bound for the control input is an upper bound for the control input S6: calculating an optimal control input by using a quadratic programming method according to the optimal solution search condition, the system safety maintenance condition and the input constraint condition; In step S6, wherein, is a slack variable, is a control input is an integral of the slack variable , is an optimal value corresponding to the constrained quadratic programming optimization problem, is a matrix transpose; S7: controlling the multi-agent system according to the optimal control input.

2. The method of claim 1, wherein, In step S3, the objective function to be optimized and the inequality constraints to be satisfied are: where the variables are the stack of decision variables of each agent, , is the local objective function of the agent , , is the set of neighbors of the robot , is the ideal distance between the robot and the robot , is the weight value of the edge between the node and the node , is the total number of nodes, is the constraint condition, is the inequality constraint function, , is the safety distance between the agents, is the set of nodes, is the minimum value.

3. The method of claim 1, wherein, In step S1, the agents are taken as nodes of the topology graph, and the communication relationship between the agents is taken as an edge.

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