Semi-global consistency control method for heterogeneous nonlinear multi-agent systems

Abstract

This invention discloses a semi-global consistency control method for heterogeneous nonlinear multi-agent systems. The method includes considering nonlinear terms related to the absolute and relative velocities of the agents, and heterogeneously constructing first-order and second-order agents to obtain a heterogeneous nonlinear multi-agent system. Based on the heterogeneous nonlinear multi-agent system, the method obtains the state equations of the heterogeneous nonlinear multi-agent system. Based on the state equations, the method obtains initial constraints that satisfy the semi-global consistency of the heterogeneous nonlinear multi-agent system. Based on the initial constraints, the method constructs an open-loop transfer function for system controller design, thereby constructing a system controller to achieve semi-global consistency control of the heterogeneous nonlinear multi-agent system. This invention solves the problem that traditional methods, when dealing with such systems, struggle to ensure coordinated consistency among agents under complex operating conditions, and that the nonlinear factors of the system easily lead to instability or failure to converge to a consistent state.

Description

Technical Field

[0001] This invention relates to the field of multi-agent technology, and in particular to a semi-global consistency control method for heterogeneous nonlinear multi-agent systems. Background Technology

[0002] With the rapid development of technology, multi-agent systems are increasingly widely used in numerous fields, and their cooperative control problem has attracted increasing attention. However, agents in real-world systems often exhibit heterogeneity and nonlinearity, posing a significant challenge to consistency control. Traditional methods struggle to ensure agent coordination under complex conditions when dealing with such systems, as nonlinear factors can easily lead to instability or failure to converge to a consistent state. Therefore, an innovative control method is urgently needed to address the consistency control problem of heterogeneous nonlinear multi-agent systems. Summary of the Invention

[0003] This invention provides a semi-global consistency control method for heterogeneous nonlinear multi-agent systems to overcome the above-mentioned technical problems.

[0004] To achieve the above objectives, the technical solution of the present invention is as follows:

[0005] A semi-global consensus control method for heterogeneous nonlinear multi-agent systems specifically includes the following steps:

[0006] S1: Define a nonlinear term with respect to the absolute or relative velocity of the agent, and use heterogeneous first-order agents and second-order agents to obtain a heterogeneous nonlinear multi-agent system.

[0007] Based on the heterogeneous nonlinear multi-agent system, obtain the state equations of the heterogeneous nonlinear multi-agent system.

[0008] S2: Construct Lyapunov functions based on heterogeneous nonlinear multi-agent systems;

[0009] And differentiate the Lyapunov function to obtain the derivative of the Lyapunov function;

[0010] The derivatives of the Lyapunov functions are rewritten based on the state equations of the heterogeneous nonlinear multi-agent system to obtain initial constraints that satisfy the semi-global consistency of the system. These initial constraints include a first initial constraint corresponding to the absolute velocity of the agents and a second initial constraint corresponding to the relative velocity of the agents.

[0011] S3: Based on the initial constraints, construct the open-loop transfer function for system controller design according to the heterogeneous nonlinear multi-agent state equations;

[0012] S4: Based on the second-order closed-loop gain shaping algorithm, the initial first-order controller of the first-order agent and the initial second-order controller of the second-order agent in a heterogeneous nonlinear multi-agent system are obtained according to the open-loop transfer function.

[0013] S5: Considering the influence of uncertain constant disturbances in heterogeneous nonlinear multi-agent systems, the final system controller is obtained based on the initial first-order controller and the initial second-order controller.

[0014] And based on the final system controller, semi-global consistency control of heterogeneous nonlinear multi-agent systems is achieved.

[0015] Furthermore, S1 specifically includes the following steps:

[0016] S11: Define the number of agents as N, where N = {1,2,…,m,…,n};

[0017] The number of second-order agents is set to {1,2,...,m};

[0018] The number of first-order agents is {m+1,m+2,...,n};

[0019] S12: Define a nonlinear term with respect to the absolute or relative velocity of the agent, and use heterogeneous first-order agents and second-order agents to obtain a heterogeneous nonlinear multi-agent system.

[0020] The heterogeneous nonlinear multi-agent system includes:

[0021] Considering the nonlinear term with respect to the absolute velocity of the agent, the first heterogeneous nonlinear multi-agent system obtained by heterogeneous first-order agents and second-order agents;

[0022] And the expression for the first heterogeneous nonlinear multi-agent system is:

[0023]

[0024] In the formula: x i (t) represents the position of the i-th agent in a heterogeneous nonlinear multi-agent system; v i (t) represents the velocity of the i-th agent; u i (t) represents the control input of the i-th agent; x represents i (t),v i The first derivative of (t); x j (t) represents the position of the j-th agent; a ij Let a represent the decision variable that allows agents i and j to exchange information, and a ij =1 indicates that agents i and j can exchange information; a ij= 0 means that agent i and agent j cannot transmit information to each other; k1, k2 represent the proportionality coefficients in the control input; f(v i ) represents a set nonlinear function of the absolute velocity of the agent, and satisfies that for When |v i | < α, there is |f(v i )| < β|v i |; β < k1, k1; α, β represent design parameters;

[0025] Considering the nonlinear term of the relative velocity of the agent, the second heterogeneous nonlinear multi-agent system obtained by the heterogeneous first-order agent and the second-order agent;

[0026] And the expression of the second heterogeneous nonlinear multi-agent system is

[0027]

[0028] In the formula: represents a set nonlinear function of the relative velocity of the agent; and satisfies that for When , there is The expression of the heterogeneous nonlinear multi-agent state equation considering the absolute velocity of the agent is

[0029] S13: According to the heterogeneous nonlinear multi-agent system, obtain the heterogeneous nonlinear multi-agent state equation considering the absolute velocity or relative velocity of the agent;

[0030] The expression of the heterogeneous nonlinear multi-agent state equation considering the relative velocity of the agent is

[0031] The expression of the heterogeneous nonlinear multi-agent state equation considering the relative velocity of the agent is

[0032] <00​​​​​​​​​​​​​​​​​​​​​The expression of

[0040]

[0041] S23: According to the derivative of the Lyapunov function A method for obtaining the first initial constraint condition that satisfies the semi - global consensus of the first heterogeneous non - linear multi - agent system considering the absolute speed of the agent, specifically including

[0042] S231: Rewrite the derivative of the Lyapunov function according to the heterogeneous non - linear multi - agent state equation, that is, substitute into the derivative of the Lyapunov function and simplify to obtain

[0043]

[0044] And for the simplified derivative of the Lyapunov function, further simplification can obtain the simplified derivative of the Lyapunov function, whose expression is

[0045]

[0046] Substitute into the simplified derivative of the Lyapunov function for rewriting, and obtain the rewritten derivative of the Lyapunov function, whose expression is

[0047]

[0048] S232: Assume that when |v i | < α and satisfy |f(v i )| < β|v i |, β < k1, then v i f(v i ) - k i v i 2 [[ID=四十八]]<0; [[ID=四十九]] [[ID=五十]]

[0049] [[ID=五十一]]From the Lyapunov function V(t), it can be known that V(t) ≥ v [[ID=五十二]] i [[ID=五十三]] 2 [[ID=五十四]]; [[ID=五十五]] [[ID=五十六]]

[0050] [[ID=五十七]]When v [[ID=五十八]] i [[ID=五十九]]<α, then there is V(t) <α [[ID=六十]] 2 [[ID=六十一]]According to the rewritten derivative of the Lyapunov function, it can be known that [[ID=六十二]] [[ID=六十三]]And V(t) is monotonically decreasing, then V(t) ≤ V(0); [[ID=六十四]] [[ID=六十五]]

[0051] [[ID=六十六]]And when V(0) ≤ α [[ID=六十七]] 2 [[ID=六十八]]Then there is v [[ID=六十九]] i [[ID=七十]] 2 [[ID=七十一]]≤ V(t) ≤ V(0) ≤ α [[ID=七十二]] 2 [[ID=七十三]]; [[ID=七十四]] [[ID=七十五]]

[0052] Then, considering the absolute velocity of the agent, the first initial constraint condition for the semi-global consistency of the first heterogeneous nonlinear multi-agent system is obtained;

[0053] And the first initial constraint is x. i (0), i∈{1,2,...,n},v i (0), i∈{1,2,...,m},V(0)≤α 2 ;

[0054] S24: A method for obtaining the second initial constraint condition that satisfies the semi-global consistency of the second heterogeneous nonlinear multi-agent system, considering the relative velocities of the agents. Specifically:

[0055] S241: Will Substituting the derivative of the Lyapunov function and simplifying, we get

[0056]

[0057] S242: Assuming when hour, 0 < β < 1, so it exists at this time.

[0058] Make

[0059] And Substituting S241 yields

[0060] According to the Lyapunov stability criterion, if If true, then heterogeneous nonlinear multi-agent systems can achieve semi-global consistency:

[0061]

[0062] And because

[0063] Then we can know when hour, Established;

[0064] Furthermore, according to S241, it can be known that... That is, if V(t) is monotonically decreasing, then V(t)≤V(0);

[0065] Then, considering the relative velocities of the agents, the second initial constraint condition that satisfies the semi-global consistency of the second heterogeneous nonlinear multi-agent system is obtained.

[0066] And the second initial constraint is...

[0067] Furthermore, S3 specifically includes the following steps:

[0068] S31: Based on the first or second initial constraints, obtain the state-space equation of the linear time-invariant system according to the state equation of the heterogeneous nonlinear multi-agent system.

[0069] And the state-space equation of the linear time-invariant system is expressed as follows:

[0070]

[0071] In the formula: x(t) is x i u(t) is a simplified form of u(t), and represents the state vector of a heterogeneous nonlinear multi-agent system; u(t) represents u i y(t) is a simplified form of the heterogeneous nonlinear multi-agent system and represents the input of the system; y(t) represents the output of the system; A represents the state transition matrix; B represents the input matrix; C represents the output matrix; and D represents the feedforward matrix.

[0072] S32: By performing a Laplace transform on the state-space equations of a linear time-invariant system, we can obtain...

[0073] sX(s)=AX(s)+BU(s)

[0074] Y(s)=CX(s)+DU(s)

[0075] In the formula: X(s) represents the output after the Laplace transform of x(t); U(s) represents the output after the Laplace transform of u(t); Y(s) represents the output after the Laplace transform of y(t); s represents the Laplace operator;

[0076] S33: Multiply both sides of the equation sX(s)=AX(s)+BU(s) by (sI-A) -1 able to obtain

[0077] X(s)=(sA) -1 BU(s)

[0078] In the formula: I represents the identity matrix;

[0079] S34: Set X(s)=(sA) -1 Substituting BU(s) into Y(s) = CX(s) + DU(s), we can obtain

[0080] Y(s) = [C(sA)] -1 B+D]U(s)

[0081] S35: Construct the open-loop transfer function for system controller design based on step S34;

[0082] And the expression for the open-loop transfer function is:

[0083]

[0084] Among them, for second-order agents in heterogeneous nonlinear multi-agent systems,

[0085] set up C =

[10] , D = 0, then the open-loop transfer function of the second-order agent in the heterogeneous nonlinear multi-agent system is:

[0086] For a first-order agent in a heterogeneous nonlinear multi-agent system

[0087] Given A = 0, B = 1, C = 1, D = 0, the open-loop transfer function of a first-order agent in a heterogeneous nonlinear multi-agent system is:

[0088] Furthermore, S4 specifically includes the following steps:

[0089] S41: Obtain the model expression for the second-order closed-loop gain shaping algorithm as follows:

[0090]

[0091] In the formula: T1 represents the time constant; s represents the Laplace operator; G represents the open-loop transfer function; K represents the controller;

[0092] S42: Based on step S41, obtain the initial first-order controller of the first-order agent and the initial second-order controller of the second-order agent in the heterogeneous nonlinear multi-agent system according to the open-loop transfer function. Their expressions are as follows:

[0093]

[0094] In the formula: K1 represents the initial first-order controller of the first-order agent in the heterogeneous nonlinear multi-agent system; K2 represents the initial second-order controller of the second-order agent in the heterogeneous nonlinear multi-agent system.

[0095] Furthermore, the expression for obtaining the final system controller based on the initial first-order controller and the initial second-order controller, as described in S5, is as follows:

[0096]

[0097] Where: K c1 ,K c2 ε represents the control inputs of the first-order and second-order agents in a heterogeneous nonlinear multi-agent system; ε represents the static error term considering the uncertain constant disturbances in the heterogeneous nonlinear multi-agent system.

[0098] Beneficial Effects: This invention provides a semi-global consistency control method for heterogeneous nonlinear multi-agent systems. By fully considering the nonlinear terms related to the absolute or relative velocities of the agents, heterogeneous first-order and second-order agents are used to obtain a heterogeneous nonlinear multi-agent system, thereby obtaining initial constraints that satisfy the semi-global consistency of the heterogeneous nonlinear multi-agent system. Two heterogeneous nonlinear multi-agent systems are designed, introducing velocity-related nonlinear terms into the control input. The initial constraints for achieving semi-global consistency of the heterogeneous nonlinear multi-agent system are derived by constructing a Lyapunov function. Finally, by constructing an open-loop transfer function for system controller design and combining it with a second-order closed-loop gain shaping algorithm, a final system controller is constructed to achieve semi-global consistency control of the heterogeneous nonlinear multi-agent system. This solves the problem that traditional methods struggle to ensure coordination and consistency among agents under complex conditions when dealing with heterogeneous nonlinear multi-agent systems, leading to instability or failure to converge to a consistent state due to nonlinear factors. This significantly improves the control accuracy and efficiency of semi-global consistency control for heterogeneous nonlinear multi-agent systems and provides a reliable theoretical basis for the collaborative control of multi-agent systems in complex environments. Attached Figure Description

[0099] To more clearly illustrate the technical solutions in the embodiments of the present 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 the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0100] Figure 1 This is a flowchart of the semi-global consistency control method for heterogeneous nonlinear multi-agent systems according to the present invention;

[0101] Figure 2 This is an undirected connected graph of the heterogeneous nonlinear multi-agent system in this embodiment;

[0102] Figure 3 This is the flowchart of the first Simunlink simulation in this embodiment;

[0103] Figure 4 This is a flowchart of the first Simunlink simulation process in this embodiment, and a position curve diagram of the six agents in the undirected connected graph.

[0104] Figure 5 This is a flowchart of the first Simunlink simulation process in this embodiment, showing the velocity curves of the four second-order agents in the undirected connected graph.

[0105] Figure 6This is a flowchart of the first Simunlink simulation process in this embodiment, and a graph showing the position error curves of the six agents in the undirected connected graph.

[0106] Figure 7 This is a flowchart of the first Simunlink simulation process in this embodiment, showing the velocity error curves corresponding to the four second-order agents in the undirected connected graph.

[0107] Figure 8 This is the flowchart of the second Simunlink simulation in this embodiment;

[0108] Figure 9 This is a flowchart of the second Simunlink simulation process in this embodiment, and a position curve diagram of the six agents in the undirected connected graph.

[0109] Figure 10 This is a flowchart of the second Simunlink simulation process in this embodiment, showing the velocity curves of the four second-order agents in the undirected connected graph.

[0110] Figure 11 This is a flowchart of the second Simunlink simulation process in this embodiment, and a graph showing the position error curves of the six agents in the undirected connected graph.

[0111] Figure 12 This is a flowchart of the second Simunlink simulation process in this embodiment, showing the velocity error curves corresponding to the four second-order agents in the undirected connected graph. Detailed Implementation

[0112] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0113] This embodiment provides a semi-global consistency control method for heterogeneous nonlinear multi-agent systems, such as... Figure 1 As shown, the specific steps include:

[0114] S1: Define a nonlinear term with respect to the absolute or relative velocity of the agent, and use heterogeneous first-order agents and second-order agents to obtain a heterogeneous nonlinear multi-agent system.

[0115] Based on the heterogeneous nonlinear multi-agent system, obtain the state equations of the heterogeneous nonlinear multi-agent system.

[0116] In this embodiment, the definition of semi-global consistency in a multi-agent system is: regardless of the initial constraint value, the multi-agent system can eventually achieve consistency, that is, all agents reach the same goal or state. This consistency is called global consistency. The other type of consistency is called semi-global consistency, that is, consistency can be guaranteed under a certain range of initial constraints, but it cannot be guaranteed to achieve consistency under all initial constraints.

[0117] Specifically, the following steps are included:

[0118] S11: Define the number of agents as N, where N = {1,2,…,m,…,n};

[0119] The number of second-order agents is set to {1,2,...,m};

[0120] The number of first-order agents is {m+1,m+2,...,n};

[0121] S12: Define a nonlinear term with respect to the absolute or relative velocity of the agent, and use heterogeneous first-order agents and second-order agents to obtain a heterogeneous nonlinear multi-agent system.

[0122] The heterogeneous nonlinear multi-agent system includes:

[0123] Considering the nonlinear term with respect to the absolute velocity of the agent, the first heterogeneous nonlinear multi-agent system obtained by heterogeneous first-order agents and second-order agents;

[0124] And the expression for the first heterogeneous nonlinear multi-agent system is:

[0125]

[0126] In the formula: x i (t) represents the position of the i-th agent in a heterogeneous nonlinear multi-agent system; v i (t) represents the velocity of the i-th agent; u i (t) represents the control input of the i-th agent; x represents i (t),v i The first derivative of (t); x j (t) represents the position of the j-th agent; a ij Let a represent the decision variable that allows agents i and j to exchange information, and a ij =1 indicates that agents i and j can exchange information; a ij =0 indicates that agents i and j cannot exchange information; k1, k2 represent the proportional coefficients in the control input; f(v iLet ) represent a nonlinear function defined with respect to the absolute velocity of the agent, and satisfy the condition for . When |v i When |<α, we have |f(v) i )|<β|v i |;β<k1,k1;α,β represent design parameters;

[0127] Considering the nonlinear term regarding the relative velocity of the agents, a second heterogeneous nonlinear multi-agent system is obtained by heterogeneous first-order agents and second-order agents;

[0128] And the expression for the second heterogeneous nonlinear multi-agent system is:

[0129]

[0130] In the formula: Let represent a defined nonlinear function relating the relative velocities of the agents; and satisfy the following condition: when Sometimes, 0 < β < 1;

[0131] S13: Obtain the heterogeneous nonlinear multi-agent system state equations that consider the absolute or relative velocities of the agents.

[0132] The heterogeneous nonlinear multi-agent state equation considering the absolute velocity of the agents is expressed as follows:

[0133]

[0134] The heterogeneous nonlinear multi-agent state equation considering the relative velocities of the agents is expressed as follows:

[0135]

[0136] S2: Construct Lyapunov functions based on heterogeneous nonlinear multi-agent systems;

[0137] And differentiate the Lyapunov function to obtain the derivative of the Lyapunov function;

[0138] The derivative of the Lyapunov function is rewritten based on the state equation of the heterogeneous nonlinear multi-agent system to obtain the initial constraints that satisfy the semi-global consistency of the heterogeneous nonlinear multi-agent system; the initial constraints include a first initial constraint corresponding to the absolute velocity of the agent and a second initial constraint corresponding to the relative velocity of the agent.

[0139] Specifically, the following steps are included:

[0140] S21: Construct Lyapunov functions based on heterogeneous nonlinear multi-agent systems;

[0141] And the expression for the Lyapunov function V(t) is:

[0142]

[0143] S22: Find the derivative of the Lyapunov function;

[0144] And the derivative of the Lyapunov function The expression is

[0145]

[0146] S23: Based on the derivative of Lyapunov's function A method for obtaining the first initial constraints that satisfy the semi-global consistency of a first heterogeneous nonlinear multi-agent system, considering the absolute velocity of the agents, specifically includes...

[0147] S231: Rewrite the derivative of the Lyapunov function based on the heterogeneous nonlinear multi-agent state equation, that is... Substituting the derivative of the Lyapunov function and simplifying, we get

[0148]

[0149] Furthermore, by simplifying the derivative of the simplified Lyapunov function, we can obtain the simplified derivative of the Lyapunov function, which is expressed as follows:

[0150]

[0151] Will Substituting the simplified derivative of the Lyapunov function, we can rewrite it to obtain the derivative of the rewritten Lyapunov function, which is expressed as follows:

[0152]

[0153] S232: Assume that when |v i |<α and satisfy|f(v) i )|<β|v i |,β<k1, then v i f(v i )-k i v i 2 <0;

[0154] From the Lyapunov function V(t), we know that V(t) ≥ v i 2 ;

[0155] When v i When α < α, then V(t) < α. 2According to the derivative of the rewritten Lyapunov function, we can know that And since V(t) is monotonically decreasing, then V(t)≤V(0);

[0156] And when V(0)≤α 2 When, then v i 2 ≤V(t)≤V(0)≤α 2 ;

[0157] Then, considering the absolute velocity of the agent, the first initial constraint condition for the semi-global consistency of the first heterogeneous nonlinear multi-agent system is obtained;

[0158] And the first initial constraint is x. i (0), i∈{1,2,...,n},v i (0), i∈{1,2,...,m},V(0)≤α 2 ;

[0159] S24: A method for obtaining the second initial constraint condition that satisfies the semi-global consistency of the second heterogeneous nonlinear multi-agent system, considering the relative velocities of the agents. Specifically:

[0160] S241: Will Substituting the derivative of the Lyapunov function and simplifying, we get

[0161]

[0162] S242: Assuming when hour, 0 < β < 1, so it exists at this time.

[0163] Make

[0164] And Substituting S241 yields

[0165] According to the Lyapunov stability criterion, if If true, then heterogeneous nonlinear multi-agent systems can achieve semi-global consistency:

[0166]

[0167] And because

[0168] Then we can know when hour, Established;

[0169] Furthermore, according to S241, it can be known that... That is, if V(t) is monotonically decreasing, then V(t)≤V(0);

[0170] Then, considering the relative velocities of the agents, the second initial constraint condition that satisfies the semi-global consistency of the second heterogeneous nonlinear multi-agent system is obtained.

[0171] And the second initial constraint is...

[0172] S3: Based on the first or second initial constraints, construct the open-loop transfer function for system controller design according to the heterogeneous nonlinear multi-agent state equation;

[0173] Specifically, the following steps are included:

[0174] S31: Based on the initial constraints, obtain the state-space equation of the linear time-invariant system according to the state equation of the heterogeneous nonlinear multi-agent system.

[0175] And the state-space equation of the linear time-invariant system is expressed as follows:

[0176]

[0177] In the formula: x(t) is x i u(t) is a simplified form of u(t), and represents the state vector of a heterogeneous nonlinear multi-agent system; u(t) represents u i y(t) is a simplified form of the heterogeneous nonlinear multi-agent system and represents the input of the system; y(t) represents the output of the system; A represents the state transition matrix; B represents the input matrix; C represents the output matrix; and D represents the feedforward matrix.

[0178] S32: By performing a Laplace transform on the state-space equations of a linear time-invariant system, we can obtain...

[0179] sX(s)=AX(s)+BU(s)

[0180] Y(s)=CX(s)+DU(s)

[0181] In the formula: X(s) represents the output after the Laplace transform of x(t); U(s) represents the output after the Laplace transform of u(t); Y(s) represents the output after the Laplace transform of y(t); s represents the Laplace operator;

[0182] S33: Multiply both sides of the equation sX(s)=AX(s)+BU(s) by (sI-A) -1 able to obtain

[0183] X(s)=(sA) -1 BU(s)

[0184] In the formula: I represents the identity matrix;

[0185] S34: Set X(s)=(sA) -1 Substituting BU(s) into Y(s) = CX(s) + DU(s), we can obtain

[0186] Y(s) = [C(sA)] -1 B+D]U(s)

[0187] S35: Construct the open-loop transfer function for system controller design based on step S34;

[0188] And the expression for the open-loop transfer function is:

[0189]

[0190] Among them, for second-order agents in heterogeneous nonlinear multi-agent systems,

[0191] set up Then the open-loop transfer function of the second-order agent in the heterogeneous nonlinear multi-agent system is:

[0192] For a first-order agent in a heterogeneous nonlinear multi-agent system

[0193] Given A = 0, B = 1, C = 1, D = 0, the open-loop transfer function of a first-order agent in a heterogeneous nonlinear multi-agent system is:

[0194] S4: Based on the second-order closed-loop gain shaping algorithm, the initial first-order controller of the first-order agent and the initial second-order controller of the second-order agent in a heterogeneous nonlinear multi-agent system are obtained according to the open-loop transfer function.

[0195] Specifically, the following steps are included:

[0196] S41: Obtain the model expression for the second-order closed-loop gain shaping algorithm as follows:

[0197]

[0198] In the formula: T1 represents the time constant; s represents the Laplace operator; G represents the open-loop transfer function; K represents the controller;

[0199] S42: Based on step S41, obtain the initial first-order controller of the first-order agent and the initial second-order controller of the second-order agent in the heterogeneous nonlinear multi-agent system according to the open-loop transfer function. Their expressions are as follows:

[0200]

[0201] In the formula: K1 represents the initial first-order controller of the first-order agent in the heterogeneous nonlinear multi-agent system; K2 represents the initial second-order controller of the second-order agent in the heterogeneous nonlinear multi-agent system.

[0202] S5: Considering the influence of uncertain constant disturbances in heterogeneous nonlinear multi-agent systems, the final system controller is obtained based on the initial first-order controller and the initial second-order controller; and semi-global consistency control of the heterogeneous nonlinear multi-agent system is realized based on the final system controller.

[0203] Specifically, in this embodiment, the controller designed using a closed-loop gain control algorithm can eliminate the influence of static errors on the system. A very small constant term is added to the denominator of the transfer function to reproduce the influence of uncertain constant-value disturbances on motion. Therefore, the transfer functions of the first-order and second-order systems are extended as follows: and The final system controller is obtained from the initial first-order controller and the initial second-order controller, and its expression is:

[0204]

[0205]

[0206] Where: K c1 ,K c2 ε represents the control inputs of the first-order and second-order agents in a heterogeneous nonlinear multi-agent system; ε represents the static error term considering the uncertain constant disturbances in the heterogeneous nonlinear multi-agent system.

[0207] The beneficial effects of this embodiment are as follows: By fully considering the nonlinear terms related to the absolute or relative velocity of the agents, heterogeneous first-order and second-order agents are used to obtain a heterogeneous nonlinear multi-agent system, thereby obtaining the initial constraints that satisfy the semi-global consistency of the heterogeneous nonlinear multi-agent system; by designing two heterogeneous nonlinear multi-agent systems, namely, introducing nonlinear terms related to velocity into the control input, and deriving the initial constraints for achieving the semi-global consistency of the heterogeneous nonlinear multi-agent system by constructing a Lyapunov function, and by constructing an open-loop transfer function for system controller design, combined with a second-order closed-loop gain shaping algorithm, a final system controller for achieving the semi-global consistency control of the heterogeneous nonlinear multi-agent system is constructed; this solves the problem that traditional methods are difficult to ensure that agents achieve coordination and consistency under complex working conditions when dealing with heterogeneous nonlinear multi-agent systems, which makes the nonlinear factors of the system prone to instability or failure to converge to a consistent state, greatly improving the control accuracy and efficiency of the semi-global consistency control of the heterogeneous nonlinear multi-agent system, and providing a reliable theoretical basis for the cooperative control of multi-agent systems in complex environments.

[0208] This embodiment also includes the relevant basic theoretical knowledge:

[0209] Graph theory: Graph theory is a powerful tool for analyzing the consistency of multi-agent systems. The relationships between vertices and edges in a graph describe the communication topology between agents. The connections and information exchange relationships between agents can be modeled using both directed and undirected graphs; directed graphs can be used (m... p ,n p Let m be represented by ) p ={1,2,3,...,P} is a finite non-empty set of nodes. The set of ordered pairs of nodes with edges is also called the edge set; in a directed graph, an edge (i,j) represents that agent j can obtain information from i and will pass v i Called the parent node, it will be used as v j Child nodes; however, the reverse is not necessarily true. Unlike directed graphs, node pairs in undirected graphs are unordered, meaning (i,j) indicates that agents i and j can exchange information; adjacency matrix A ij It is a matrix representing the adjacency relationship between vertices. For an undirected graph, for all i ≠ j, a ij =a ji a ij Represents the adjacency matrix A ij The element in the i-th row and j-th column, when (j,i)∈n p This indicates that (i,j)∈n p For directed graphs, when (j,i)∈n p a ij When it is 1, At that time, a ij =0; if for all, or vertex v i If the in-degree of a vertex equals the out-degree of its vertex, then the graph is said to be balanced. For an undirected graph, A... ij It is time-symmetric, and the in-degree of each vertex equals its out-degree; therefore, every undirected graph is balanced. All information about the corresponding topological graph can be obtained from the Laplace matrix, for any vertex v in the graph. i (For agent i), the number of non-zero elements in the i-th row of the Laplace matrix is ​​the number of agents with which this agent has communication relationships.

[0210] Lyapunov stability criterion:

[0211] Let the state equations of a linear time-invariant system be... Since A is a non-singular matrix, the origin is the only equilibrium state. The necessary and sufficient condition for the asymptotic stability of a linear time-invariant system is that equation A... TFor any given symmetric positive definite matrix Q, P + PA = -Q has a positive definite symmetric solution P. During the calculation, a Lyapunov function V(x) = x can be first established. T Px>0, where P is A T The unique positive definite symmetric solution of P+PA=-Q is given by: Let x(t; 0, x0) denote the solution of the system state equation at time t=0 with x0 as the initial constraint. The derivative of V(x) is calculated as follows:

[0212]

[0213] x(t;0,x0)≠0, at which point the system is asymptotically stable;

[0214] Heterogeneous Multi-Agent Systems: Common linear heterogeneous multi-agent systems consist of first-order and second-order components, with n agents labeled from 1 to n. The second-order component contains m agents. The states of each second-order agent are given as follows: Where, x i ∈R、v i ∈R and u i Let R represent the agent's position, velocity, and control input, respectively, and let x be the initial constraint. i (0)=x i0 , v represents the initial position of the i-th agent; i0 This represents the initial velocity of the i-th agent; and the state of each first-order agent is as follows: Where, x i ∈R and u i ∈R represent the position input and control input of the agent, respectively, and the initial constraint is x. i (0)=x i0 ,v i (0)=v i0 Its control input is If the system output meets the following conditions It is then assumed that heterogeneous multi-agent systems can achieve consistency.

[0215] This embodiment also includes Simulink simulation experiments:

[0216] Suppose we have an undirected connected graph with six vertices, such as Figure 2 Vertices 1, 2, 3, and 4 represent second-order agents; vertices 5 and 6 represent first-order agents. Assume f(v) i ) = v i 2 The semi-global consistency condition is V(0)≤α. 2 ;

[0217] like Figure 3As shown, experiments are carried out in Simulink. The control input with non - linear terms related to the absolute speed is input into the controller designed by the closed - loop gain shaping algorithm. Then, the output of the controller is input into the state - space expression of the system, and finally, the position and speed are output.

[0218] Let \(k_1 = 10\). Because when \(|v|\lt\alpha\), \(|f(v)|\lt\beta|v|\), \(\beta\lt k_1\), so \(|f(v)| = v\lt\alpha|v|\). At this time, \(\alpha=\beta\), and \(\beta\lt k_1\), so take \(\alpha=\beta = 9\). To satisfy the initial constraint condition \(V(0)\leq\alpha\) i |, take the initial constraint conditions as \(x(0)=[4,2,0, - 1,1, - 2]\), \(v(0)=[1, - 1,2, - 2]\). From i )|<β|v i |,β<k1,所以|f(v i )|=v i 2 <α|v[[ID=...]] i |, it can be seen from 2 Figures 4 to 7 that when the initial constraint conditions are satisfied and as time approaches infinity, the position and speed errors between agents are both 0, and the heterogeneous non - linear multi - agent system can achieve semi - global consensus control. 所示能够得知在满足初始约束条件的情况下,在时间趋近于无穷时,智能体之间的位置和速度误差都为0,异质非线性多智能体系统能够实现半全局一致性控制;

[0219] 如 Figure 8 [[ID=...]] 所示,在simiulink中进行试验,将带有与相对速度相关的非线性项的控制输入输入到闭环增益成型算法设计的控制器中,再将控制器... As shown, experiments are carried out in Simulink. The control input with non - linear terms related to the relative speed is input into the controller designed by the closed - loop gain shaping algorithm. Then... 时, 0<β<1,所以 此时α=β,且0<β<1,所以取α=β=0.5。为满足初始约束条件 [[ID=...]] 取初始约束条件为x(0)=[8,5,2,-4,1,-5],v(0)=[1,-5,5,3];由 [[ID=...]] Figures 9 to 12 能够得知在满足初始约束条件的情况下,在时间趋近于无穷时,智能体之间的位置和速度误差都为0,异质非线性多智能体系统能够实现半全局一致性控制。 it can be seen that when the initial constraint conditions are satisfied and as time approaches infinity, the position and speed errors between agents are both 0, and the heterogeneous non - linear multi - agent system can achieve semi - global consensus control...

[0220] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements for some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A semi-global consistency control method for heterogeneous nonlinear multi-agent systems, characterized in that, Specifically, the following steps are included: S1: Define a nonlinear term with respect to the absolute or relative velocity of the agent, and use heterogeneous first-order agents and second-order agents to obtain a heterogeneous nonlinear multi-agent system. Based on the heterogeneous nonlinear multi-agent system, obtain the state equations of the heterogeneous nonlinear multi-agent system. S2: Construct Lyapunov functions based on heterogeneous nonlinear multi-agent systems; And differentiate the Lyapunov function to obtain the derivative of the Lyapunov function; The derivative of the Lyapunov function is rewritten based on the state equation of the heterogeneous nonlinear multi-agent system to obtain the initial constraints that satisfy the semi-global consistency of the heterogeneous nonlinear multi-agent system. The initial constraints include a first initial constraint corresponding to the absolute velocity of the agent and a second initial constraint corresponding to the relative velocity of the agent. S3: Based on the initial constraints, construct the open-loop transfer function for system controller design according to the heterogeneous nonlinear multi-agent state equations; S4: Based on the second-order closed-loop gain shaping algorithm, the initial first-order controller of the first-order agent and the initial second-order controller of the second-order agent in a heterogeneous nonlinear multi-agent system are obtained according to the open-loop transfer function. S5: Considering the influence of uncertain constant disturbances in heterogeneous nonlinear multi-agent systems, the final system controller is obtained based on the initial first-order controller and the initial second-order controller. And based on the final system controller, semi-global consistency control of heterogeneous nonlinear multi-agent systems is achieved.

2. The semi-global consistency control method for heterogeneous nonlinear multi-agent systems according to claim 1, characterized in that, S1 specifically includes the following steps: S11: Define the number of agents as... and ; And set the number of second-order agents to be ; The number of first-order agents is ; S12: Define a nonlinear term with respect to the absolute or relative velocity of the agent, and use heterogeneous first-order agents and second-order agents to obtain a heterogeneous nonlinear multi-agent system. The heterogeneous nonlinear multi-agent system includes: Considering the nonlinear term with respect to the absolute velocity of the agent, the first heterogeneous nonlinear multi-agent system obtained by heterogeneous first-order agents and second-order agents; And the expression for the first heterogeneous nonlinear multi-agent system is: ； ； In the formula: In a heterogeneous nonlinear multi-agent system, the first... The location of each agent; Indicates the first The speed of each intelligent agent; Indicates the first Control input for each intelligent agent; express The first derivative; Indicates the first The location of each agent; Represents intelligent agents With intelligent agents Decision variables that can exchange information with each other, and Represents intelligent agents With intelligent agents They can exchange information with each other; Represents intelligent agents With intelligent agents They cannot exchange information with each other; This represents the proportional coefficient in the control input; Let represent a defined nonlinear function relating to the absolute velocity of the agent, satisfying the condition for . ,when Sometimes, ; , ; Indicate design parameters; Considering the nonlinear term regarding the relative velocity of the agents, a second heterogeneous nonlinear multi-agent system is obtained by heterogeneous first-order agents and second-order agents; And the expression for the second heterogeneous nonlinear multi-agent system is: ； ； In the formula: Let represent a defined nonlinear function relating the relative velocities of the agents; and satisfy the following condition: ,when Sometimes, , ; S13: Obtain the heterogeneous nonlinear multi-agent system state equations that consider the absolute or relative velocities of the agents. The heterogeneous nonlinear multi-agent state equation considering the absolute velocity of the agents is expressed as follows: The heterogeneous nonlinear multi-agent state equation considering the relative velocities of the agents is expressed as follows:

3. The semi-global consistency control method for heterogeneous nonlinear multi-agent systems according to claim 2, characterized in that, S2 specifically includes the following steps: S21: Construct Lyapunov functions based on heterogeneous nonlinear multi-agent systems; And Lyapunov function The expression is S22: Find the derivative of the Lyapunov function; And the derivative of the Lyapunov function The expression is S23: Based on the derivative of Lyapunov's function A method for obtaining the first initial constraints that satisfy the semi-global consistency of a first heterogeneous nonlinear multi-agent system, considering the absolute velocity of the agents, specifically includes... S231: Rewrite the derivative of the Lyapunov function based on the heterogeneous nonlinear multi-agent state equation, that is... Substituting the derivative of the Lyapunov function and simplifying, we get Furthermore, by further simplifying the derivative of the simplified Lyapunov function, we can obtain the simplified derivative of the Lyapunov function, which is expressed as follows: Will Substituting the simplified derivative of the Lyapunov function and rewriting it, we obtain the derivative of the rewritten Lyapunov function, whose expression is: S232: Assuming when And satisfy , ,but ; Lyapunov function able to know ; when At that time, there is According to the derivative of the rewritten Lyapunov function, we can know that ,and If it is monotonically decreasing, then ; And when At that time, there is ; Then, considering the absolute velocity of the agent, the first initial constraint condition for the semi-global consistency of the first heterogeneous nonlinear multi-agent system is obtained; And the first initial constraint is , , ; S24: A method for obtaining the second initial constraint condition that considers the relative velocities of agents and satisfies the semi-global consistency of the second heterogeneous nonlinear multi-agent system, specifically: S241: Will Substituting the derivative of the Lyapunov function and simplifying, we get S242: Assuming when hour, , Therefore, it exists at this time. , Make ; And Substituting S241 yields ; According to the Lyapunov stability criterion, if If true, then heterogeneous nonlinear multi-agent systems can achieve semi-global consistency: And because Then we can know when hour, Established; Furthermore, according to S241, it can be known that... ,Right now If it is monotonically decreasing, then ; Then, considering the relative velocities of the agents, the second initial constraint condition that satisfies the semi-global consistency of the second heterogeneous nonlinear multi-agent system is obtained. And the second initial constraint is... , , .

4. The semi-global consistency control method for heterogeneous nonlinear multi-agent systems according to claim 3, characterized in that, S3 specifically includes the following steps: S31: Based on the first or second initial constraints, obtain the state-space equation of the linear time-invariant system according to the state equation of the heterogeneous nonlinear multi-agent system. And the state-space equation of the linear time-invariant system is expressed as follows: In the formula: for It is a simplified form of and represents the state vector of a heterogeneous nonlinear multi-agent system; express The abbreviation of , and represents the input of a heterogeneous nonlinear multi-agent system; This represents the output of a heterogeneous nonlinear multi-agent system. Represents the state transition matrix; Represents the input matrix; Indicates the output matrix; Represents the feedforward matrix; S32: By performing a Laplace transform on the state-space equations of a linear time-invariant system, we can obtain... In the formula: Indicates to The output after the Laplace transform; Indicates to The output after the Laplace transform; Indicates to The output after the Laplace transform; Represents the Laplace operator; S33: Yes Multiply both sides of the equation by able to obtain In the formula: Represents the identity matrix; S34: Will Substitution able to obtain S35: Construct the open-loop transfer function for system controller design based on step S34; And the expression for the open-loop transfer function is: ； Among them, for second-order agents in heterogeneous nonlinear multi-agent systems, set up Then the open-loop transfer function of the second-order agent in a heterogeneous nonlinear multi-agent system is: ; For a first-order agent in a heterogeneous nonlinear multi-agent system set up Then the open-loop transfer function of a first-order agent in a heterogeneous nonlinear multi-agent system is: .

5. The semi-global consistency control method for a heterogeneous nonlinear multi-agent system according to claim 3, characterized in that, S4 specifically includes the following steps: S41: Obtain the model expression for the second-order closed-loop gain shaping algorithm as follows: In the formula: Represents the time constant; Represents the Laplace operator; Represent the open-loop transfer function; Indicates controller; S42: Based on step S41, obtain the initial first-order controller of the first-order agent and the initial second-order controller of the second-order agent in the heterogeneous nonlinear multi-agent system according to the open-loop transfer function. Their expressions are as follows: In the formula: This represents the initial first-order controller of a first-order agent in a heterogeneous nonlinear multi-agent system. This represents the initial second-order controller of a second-order agent in a heterogeneous nonlinear multi-agent system.

6. The semi-global consistency control method for a heterogeneous nonlinear multi-agent system according to claim 5, characterized in that, The expression for obtaining the final system controller based on the initial first-order controller and the initial second-order controller, as described in S5, is: In the formula: This represents the control inputs of the first-order and second-order agents in a heterogeneous nonlinear multi-agent system. This represents the static error term considering uncertain constant disturbances in a heterogeneous nonlinear multi-agent system.

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