Heterogeneous nonlinear multi-agent system semi-global consistency control method

Abstract

The invention discloses a semi-global consistency control method for a heterogeneous nonlinear multi-agent system. The semi-global consistency control method comprises the following steps: considering nonlinear terms about absolute speed and relative speed of agents, a heterogeneous first-order agent and a heterogeneous second-order agent to obtain the heterogeneous nonlinear multi-agent system; obtaining a heterogeneous nonlinear multi-agent state equation according to a heterogeneous nonlinear multi-agent system; obtaining an initial constraint condition meeting the semi-global consistency of the heterogeneous nonlinear multi-agent system according to the heterogeneous nonlinear multi-agent state equation; and on the basis of an initial constraint condition, an open-loop transfer function for design of a system controller is constructed according to the heterogeneous nonlinear multi-agent state equation, so that the system controller is constructed to realize semi-global consistency control of the heterogeneous nonlinear multi-agent system. It is difficult to ensure that intelligent agents are coordinated and consistent under complex working conditions, and non-linear factors of a system easily cause instability or cannot be converged to a consistent state.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-agent systems, and particularly to a semi-global consensus control method for heterogeneous non-linear multi-agent systems. Background Art

[0002] With the rapid development of technology, multi-agent systems are increasingly widely used in many fields, and their cooperative control problems have received more and more attention. However, the agents in actual systems often have heterogeneity and non-linearity characteristics, which pose great challenges to consensus control. Traditional methods are difficult to ensure that agents achieve coordination under complex working conditions, and the non-linear factors of the system are likely to lead to instability or failure to converge to a consistent state. Therefore, there is an urgent need for an innovative control method to solve the consensus control problem of heterogeneous non-linear multi-agent systems. Summary of the Invention

[0003] The present invention provides a semi-global consensus control method for heterogeneous non-linear multi-agent systems to overcome the above technical problems.

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

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

[0006] S1: Define non-linear terms regarding the absolute velocity or relative velocity of agents, and heterogeneous first-order agents and second-order agents to obtain a heterogeneous non-linear multi-agent system;

[0007] According to the heterogeneous non-linear multi-agent system, obtain the state equation of the heterogeneous non-linear multi-agent system;

[0008] S2: Construct a Lyapunov function according to the heterogeneous non-linear multi-agent system;

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

[0010] Rewrite the derivative of the Lyapunov function according to the state equation of the heterogeneous non-linear multi-agent system to obtain the initial constraint conditions that satisfy the semi-global consensus of the heterogeneous non-linear multi-agent system; the initial constraint conditions include a first initial constraint condition corresponding to the absolute velocity of the agent and a second initial constraint condition corresponding to the relative velocity of the agent

[0011] S3: Based on the initial constraint conditions, construct an open-loop transfer function for system controller design according to the state equation of the heterogeneous non-linear multi-agent system;

[0012] S4: Based on the second-order closed-loop gain shaping algorithm, obtain the initial first-order controller of the first-order agents and the initial second-order controller of the second-order agents in the heterogeneous non-linear multi-agent system according to the open-loop transfer function;

[0013] S5: Considering the influence of uncertain constant disturbances in the heterogeneous non-linear multi-agent system, obtain the final system controller according to the initial first-order controller and the initial second-order controller;

[0014] And realize the semi-global consensus control of the heterogeneous non-linear multi-agent system according to the final system controller.

[0015] Furthermore, the specific steps of S1 are as follows:

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

[0017] And set the number of second-order agents as {1, 2, ..., m};

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

[0019] S12: Define the non-linear terms regarding the absolute velocity or relative velocity of the agents, and heterogeneous first-order agents and second-order agents to obtain the heterogeneous non-linear multi-agent system;

[0020] And the heterogeneous non-linear multi-agent system includes:

[0021] Considering the non-linear terms regarding the absolute velocity of the agents, the first heterogeneous non-linear multi-agent system obtained by heterogeneous first-order agents and second-order agents;

[0022] And the expression of the first heterogeneous non-linear multi-agent system is

[0023]

[0024] In the formula: x i (t) represents the position of the i-th agent in the heterogeneous non-linear 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; represents the first derivative of x i (t), v i (t); x j (t) represents the position of the j-th agent; a ij represents the decision variable that can transmit information between agent i and agent j, and a ij = 1 indicates that information can be transmitted between agent i and agent j; a ij= 0 means that agent i and agent j cannot transmit information to each other; k 1 , k 2 represents the proportional coefficient 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 < k 1 , k 1 ; 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 0 < β < 1;

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

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

[0031]

[0032] The heterogeneous nonlinear multi-agent state equation considering the relative velocity of the agent, its expression is

[0033]

[0034] Furthermore, the specific steps of the S2 are as follows:

[0035] S21: Construct a Lyapunov function according to the heterogeneous nonlinear multi-agent system;

[0036] And the expression of the Lyapunov function V(t) is

[0037]

[0038] S22: Take the derivative of the Lyapunov function to obtain the derivative of the Lyapunov function;

[0039] And the derivative of the Lyapunov function The expression of is

[0040]

[0041] S23: According to the derivative of the Lyapunov function Obtain a method for the first initial constraint condition that considers the absolute velocity of the agent and satisfies the semi - global consensus of the first - order heterogeneous non - linear multi - agent system, specifically including

[0042] S231: Rewrite the derivative of the Lyapunov function according to the state equation of the heterogeneous non - linear multi - agent, 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 simplify to 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 satisfies |f(v i )| < β|v i |, β < k 1 , then v i f(v i ) - k i v i 2 <0;

[0049] From the Lyapunov function V(t), it can be known that V(t) ≥ v i 2 ;

[0050] When v i <α, then there is V(t) < α 2 , According to the rewritten derivative of the Lyapunov function, it can be known that And V(t) is monotonically decreasing, then V(t) ≤ V(0);

[0051] And when V(0) ≤ α2 When there is v i 2 ≤V(t)≤V(0)≤α 2 ;

[0052] Furthermore, obtain the first initial constraint condition that takes into account the absolute velocity of the agents and satisfies the semi-global consensus of the first heterogeneous non-linear multi-agent system;

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

[0054] S24: The method for obtaining the second initial constraint condition that takes into account the relative velocity of the agents and satisfies the semi-global consensus of the second heterogeneous non-linear multi-agent system is specifically

[0055] S241: Substitute into the derivative of the Lyapunov function and simplify to obtain

[0056]

[0057] S242: Assume that when there is 0 < β < 1, so at this time there exists

[0058] such that

[0059] and substitute into S241 to obtain

[0060] From the Lyapunov stability criterion, it can be known that if holds, then the heterogeneous non-linear multi-agent system can achieve semi-global consensus:

[0061]

[0062] And since

[0063] it can be known that when there is holds;

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

[0065] Furthermore, obtain the second initial constraint condition that takes into account the relative velocity of the agents and satisfies the semi-global consensus of the second heterogeneous non-linear multi-agent system;

[0066] And the second initial constraint condition is

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

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

[0069] And the expression of the state space equation of the linear time-invariant system is

[0070]

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

[0072] S32: Perform Laplace transform on the state space equation of the linear time-invariant system, and we can get

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

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

[0075] In the formula: X(s) represents the output after Laplace transform of x(t); U(s) represents the output after Laplace transform of u(t); Y(s) represents the output after 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 , and we can get

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

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

[0079] S34: Substitute X(s) = (s - A) -1 BU(s) into Y(s) = CX(s) + DU(s), and we can get

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

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

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

[0083]

[0084] Among them, for the second-order agent in the heterogeneous nonlinear multi-agent system,

[0085] Set C = [1 0], D = 0, then the open-loop transfer function of the second-order agent in the heterogeneous nonlinear multi-agent system is

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

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

[0088] Further, the specific steps of S4 are as follows:

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

[0090]

[0091] In the formula: T 1 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, and their expressions are

[0093]

[0094] In the formula: K 1 represents the initial first-order controller of the first-order agent in the heterogeneous nonlinear multi-agent system; K 2 represents the initial second-order controller of the second-order agent in the heterogeneous nonlinear multi-agent system.

[0095] Further, in S5, obtaining the final system controller according to the initial first-order controller and the initial second-order controller, and its expression is

[0096]

[0097] In the formula: Kc1 , K c2 denote the control inputs of the first-order agents and the second-order agents in the heterogeneous non-linear multi-agent system; ε denotes the static error term considering the uncertain constant disturbances in the heterogeneous non-linear multi-agent system.

[0098] Beneficial effects: The present invention provides a semi-global consensus control method for heterogeneous non-linear multi-agent systems. By fully considering the non-linear terms regarding the absolute or relative velocities of the agents, heterogeneous first-order agents and second-order agents are obtained to form a heterogeneous non-linear multi-agent system, so as to obtain the initial constraint conditions for satisfying the semi-global consensus of the heterogeneous non-linear multi-agent system. By designing two heterogeneous non-linear multi-agent systems, i.e., introducing non-linear terms related to velocity into the control input, and deriving the initial constraint conditions for achieving the semi-global consensus of the heterogeneous non-linear multi-agent system by constructing a Lyapunov function, and by constructing an open-loop transfer function for system controller design and combining with the second-order closed-loop gain shaping algorithm, the final system controller for achieving the semi-global consensus control of the heterogeneous non-linear multi-agent system is constructed. It solves the problem that traditional methods are difficult to ensure the coordination of agents under complex working conditions in dealing with heterogeneous non-linear multi-agent systems, and the non-linear factors of the system are likely to cause instability or inability to converge to a consistent state, greatly improving the control accuracy and efficiency of the semi-global consensus control of the heterogeneous non-linear multi-agent system, and at the same time providing a reliable theoretical basis for the cooperative control of multi-agent systems in complex environments. Description of the Drawings

[0099] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, the following drawings are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0100] Figure 1 is the flow chart of the semi-global consensus control method for the heterogeneous non-linear multi-agent system of the present invention;

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

[0102] Figure 3 is the first simunlink simulation flow block diagram in this embodiment;

[0103] Figure 4 is the position curve diagram corresponding to 6 agents in the undirected connected graph based on the first simunlink simulation flow block diagram in this embodiment;

[0104] Figure 5 In this embodiment, it is the velocity curve diagram corresponding to 4 second-order agents in the undirected connected graph based on the first Simulink simulation flowchart;

[0105] Figure 6 In this embodiment, it is the position error curve diagram corresponding to 6 agents in the undirected connected graph based on the first Simulink simulation flowchart;

[0106] Figure 7 In this embodiment, it is the velocity error curve diagram corresponding to 4 second-order agents in the undirected connected graph based on the first Simulink simulation flowchart;

[0107] Figure 8 In this embodiment, it is the second Simulink simulation flowchart;

[0108] Figure 9 In this embodiment, it is the position curve diagram corresponding to 6 agents in the undirected connected graph based on the second Simulink simulation flowchart;

[0109] Figure 10 In this embodiment, it is the velocity curve diagram corresponding to 4 second-order agents in the undirected connected graph based on the second Simulink simulation flowchart;

[0110] Figure 11 In this embodiment, it is the position error curve diagram corresponding to 6 agents in the undirected connected graph based on the second Simulink simulation flowchart;

[0111] Figure 12 In this embodiment, it is the velocity error curve diagram corresponding to 4 second-order agents in the undirected connected graph based on the second Simulink simulation flowchart. Detailed implementation manners

[0112] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0113] This embodiment provides a semi-global consensus control method for a heterogeneous non-linear multi-agent system, as Figure 1 shown, which specifically includes the following steps:

[0114] S1: Define the non - linear terms regarding the absolute or relative velocity of agents, and heterogeneous first - order and second - order agents to obtain a heterogeneous non - linear multi - agent system;

[0115] Based on the heterogeneous non - linear multi - agent system, obtain the state equation of the heterogeneous non - linear multi - agent system;

[0116] In this embodiment, the definition of semi - global consensus of the multi - agent system is as follows: Regardless of the values of the initial constraint conditions of the multi - agent system, consensus can be achieved finally, that is, all agents reach the same goal or state. This kind of consensus is called global consensus, and another kind of consensus relative to it is semi - global consensus, that is, the initial constraint conditions within a certain range can ensure the achievement of consensus, but it cannot guarantee that consensus can be achieved under all initial constraint conditions;

[0117] Specifically, it includes the following steps:

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

[0119] And set the number of second - order agents as {1, 2, …, m};

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

[0121] S12: Define the non - linear terms regarding the absolute or relative velocity of agents, and heterogeneous first - order and second - order agents to obtain a heterogeneous non - linear multi - agent system;

[0122] And the heterogeneous non - linear multi - agent system includes:

[0123] Consider the non - linear terms regarding the absolute velocity of agents, and the first heterogeneous non - linear multi - agent system obtained by heterogeneous first - order and second - order agents;

[0124] And the expression of the first heterogeneous non - linear multi - agent system is

[0125]

[0126] In the formula: x i (t) represents the position of the i - th agent in the heterogeneous non - linear 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; Represents the first - order derivative of x i (t), v i (t); x j (t) represents the position of the j - th agent; a ijThe decision variable indicating that agent i and agent j can transmit information to each other, and a ij = 1 indicates that agent i and agent j can transmit information to each other; a ij = 0 indicates that agent i and agent j cannot transmit information to each other; k 1 , k 2 represents the proportionality coefficient in the control input; f(v i ) represents a set nonlinear function regarding the absolute velocity of the agent, and satisfies that for when |v i | < α, there is |f(v i )| < β|v i |; β < k 1 , k 1 ; α, β represent design parameters;

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

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

[0129]

[0130] In the formula: represents a set nonlinear function regarding the relative velocity of the agents; and satisfies that for when , there is 0 < β < 1;

[0131] S13: According to the heterogeneous nonlinear multi-agent system, the heterogeneous nonlinear multi-agent state equation considering the absolute velocity or relative velocity of the agents is obtained;

[0132] The heterogeneous nonlinear multi-agent state equation considering the absolute velocity of the agents, its expression is

[0133]

[0134] The heterogeneous nonlinear multi-agent state equation considering the relative velocity of the agents, its expression is

[0135]

[0136] S2: Construct a Lyapunov function according to the heterogeneous nonlinear multi-agent system;

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

[0138] Rewrite the derivative of the Lyapunov function according to the heterogeneous non - linear multi - agent state equation to obtain the initial constraint conditions for semi - global consensus of the heterogeneous non - linear multi - agent system; the initial constraint conditions include a first initial constraint condition corresponding to the absolute velocity of the agent and a second initial constraint condition corresponding to the relative velocity of the agent.

[0139] Specifically, it includes the following steps:

[0140] S21: Construct a Lyapunov function according to the heterogeneous non - linear multi - agent system;

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

[0142]

[0143] S22: Take the derivative of the Lyapunov function to obtain the derivative of the Lyapunov function;

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

[0145]

[0146] S23: According to the derivative of the Lyapunov function Obtain a method for the first initial constraint condition that considers the absolute velocity of the agent and satisfies the semi - global consensus of the first heterogeneous non - linear multi - agent system, specifically including

[0147] 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 get

[0148]

[0149] And for the simplified derivative of the Lyapunov function, further simplification can obtain the simplified derivative of the Lyapunov function, and its expression is

[0150]

[0151] Substitute Into the simplified derivative of the Lyapunov function for rewriting to obtain the rewritten derivative of the Lyapunov function, and its expression is

[0152]

[0153] S232: Assume that when |v i | < α and satisfy |f(v i )| < β|v i |, β < k 1 Then vi f(v i ) - k i v i 2 <0;

[0154] It can be known from the Lyapunov function V(t) that V(t) ≥ v i 2 ;

[0155] When v i < α, then V(t) < α 2 , it can be known from the derivative of the rewritten Lyapunov function that and V(t) is monotonically decreasing, then V(t) ≤ V(0);

[0156] and when V(0) ≤ α 2 at this time, then there is v i 2 ≤ V(t) ≤ V(0) ≤ α 2 ;

[0157] Furthermore, the first initial constraint condition for satisfying the semi - global consensus of the first heterogeneous non - linear multi - agent system considering the absolute velocity of the agents is obtained;

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

[0159] S24: The method for obtaining the second initial constraint condition for satisfying the semi - global consensus of the second heterogeneous non - linear multi - agent system considering the relative velocity of the agents is as follows

[0160] S241: Substitute into the derivative of the Lyapunov function and simplify to obtain

[0161]

[0162] S242: Assume that when at this time, 0 < β < 1, so there exists

[0163] such that

[0164] and substitute into S241 to obtain

[0165] It can be known from the Lyapunov stability criterion that if If it holds, the heterogeneous nonlinear multi-agent system can achieve semi-global consensus:

[0166]

[0167] And since

[0168] Then it can be known that when At this time, Holds;

[0169] Also according to S241, it can be known that That is, V(t) is monotonically decreasing, so V(t) ≤ V(0);

[0170] Furthermore, obtain the second initial constraint condition for satisfying the semi-global consensus of the second heterogeneous nonlinear multi-agent system by considering the relative velocity of the agents;

[0171] And the second initial constraint condition is

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

[0173] Specifically, it includes the following steps:

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

[0175] And the expression of the state-space equation of the linear time-invariant system is

[0176]

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

[0178] S32: Perform Laplace transform on the state-space equation of the linear time-invariant system, and it can be obtained that

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

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

[0181] Where: X(s) represents the output after Laplace transform of x(t); U(s) represents the output after Laplace transform of u(t); Y(s) represents the output after 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 , and we can get

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

[0184] Where: I represents the identity matrix;

[0185] S34: Substitute X(s) = (s - A) -1 BU(s) into Y(s) = CX(s) + DU(s), and we can get

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

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

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

[0189]

[0190] Among them, for the second-order agent in the heterogeneous non-linear multi-agent system,

[0191] Set Then the open-loop transfer function of the second-order agent in the heterogeneous non-linear multi-agent system is

[0192] For the first-order agent in the heterogeneous non-linear multi-agent system,

[0193] Set A = 0, B = 1, C = 1, D = 0, then the open-loop transfer function of the first-order agent in the heterogeneous non-linear multi-agent system is

[0194] S4: Based on the second-order closed-loop gain shaping algorithm, 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 non-linear multi-agent system according to the open-loop transfer function;

[0195] Specifically, it includes the following steps:

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

[0197]

[0198] Where: T 1 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, and their expressions are

[0200]

[0201] Where: K 1 represents the initial first-order controller of the first-order agent in the heterogeneous nonlinear multi-agent system; K 2 represents the initial second-order controller of the second-order agent in the heterogeneous nonlinear multi-agent system;

[0202] S5: Considering the influence of the uncertain constant disturbance in the heterogeneous nonlinear multi-agent system, obtain the final system controller according to the initial first-order controller and the initial second-order controller; and realize the semi-global consensus control of the heterogeneous nonlinear multi-agent system according to the final system controller;

[0203] Specifically, for this embodiment, the controller designed using the closed-loop gain control algorithm can eliminate the influence of the static error on the system. Add a very small constant term to the denominator of the transfer function to reproduce the influence of the uncertain constant disturbance on the motion. Then the transfer functions of the first-order and second-order systems are extended to and to obtain the final system controller according to the initial first-order controller and the initial second-order controller, and its expression is

[0204]

[0205]

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

[0207] Beneficial effects of this embodiment: By fully considering the non-linear terms regarding the absolute speed or relative speed of the agents, heterogeneous first-order agents and second-order agents are used to obtain a heterogeneous non-linear multi-agent system, so as to obtain the initial constraint conditions that satisfy the semi-global consensus of the heterogeneous non-linear multi-agent system; by designing two heterogeneous non-linear multi-agent systems, that is, introducing non-linear terms related to speed in the control input, and deriving the initial constraint conditions for achieving the semi-global consensus of the heterogeneous non-linear multi-agent system by constructing a Lyapunov function, and by constructing an open-loop transfer function for system controller design, combined with the second-order closed-loop gain shaping algorithm, the final system controller for achieving the semi-global consensus control of the heterogeneous non-linear multi-agent system is constructed; it solves the problem that traditional methods are difficult to ensure the coordination of agents under complex working conditions when dealing with heterogeneous non-linear multi-agent systems, and the non-linear factors of the system are likely to cause instability or inability to converge to a consistent state, greatly improving the control accuracy and efficiency of the semi-global consensus control of the heterogeneous non-linear multi-agent system, and at the same time providing a reliable theoretical basis for the cooperative control of multi-agent systems in complex environments.

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

[0209] Graph theory: Graph theory is a powerful tool for analyzing the consensus of multi-agent systems, and the relationship between points and edges in the graph describes the communication topology among multi-agents. The connection and information exchange relationships among agents can be modeled using directed graphs and undirected graphs; a directed graph can be represented by (m p , n p ), where m p = {1, 2, 3,..., P} is a finite non-empty node set, is the set of node pairs of the edges, which is also called the edge set; the edge (i, j) in the directed graph means that agent j can obtain information from i, and v i is called the parent node, and v j is called the child node; but the reverse is not necessarily true. Different from the directed graph, the node pairs in the undirected graph have no order, that is, (i, j) means that agents i and j can exchange information with each other; the adjacency matrix A ij is a matrix representing the adjacent relationship between vertices. For an undirected graph, for all i ≠ j, a ij = a ji , and a ij represents the element in the i-th row and j-th column of the adjacency matrix A ij . When (j, i) ∈ n p , it indicates that (i, j) ∈ n p ; for a directed graph, when (j, i) ∈ n p , a ij is 1, when When a ij is 0; if for all, or vertex v i the in-degree is equal to the out-degree of the vertex, then the graph is called balanced. For an undirected graph, A ij is symmetric, and the in-degree of each vertex is equal to the out-degree. Therefore, every undirected graph is balanced; all information of the corresponding topological graph can be obtained from the Laplacian matrix. For any vertex v i (the agent numbered i), the number of non-zero elements in the i-th row of the Laplacian matrix is the number of agents related to the communication relationship of this agent.

[0210] Lyapunov stability criterion:

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

[0212]

[0213] x(t; 0, x 0 ) ≠ 0. At this time, the system is asymptotically stable;

[0214] Heterogeneous multi-agent system: The common linear heterogeneous multi-agent system consists of two parts, first-order and second-order. The number of agents is n, labeled from 1 to n, and the number of agents in the second-order part is m. The state of each second-order agent is given as follows where, x i ∈R, v i ∈R, and u i ∈R are the position, velocity, and control input of the agent respectively. The initial constraint condition is x i (0) = x i0 , represents the initial position of the i-th agent; v i0 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 are the position input and control input of the agent respectively, and the initial constraint condition is x i (0) = x i0 , v i (0) = v i0 ; Its control input is If the system output satisfies the following conditions Then it is considered that the heterogeneous multi-agent system can achieve consensus.

[0215] This embodiment also includes a simulink simulation experiment:

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

[0217] As Figure 3 shown, conduct experiments in simiulink. Input the control input with the non-linear term related to the absolute speed into the controller designed by the closed-loop gain shaping algorithm, then input the output of the controller into the state-space expression of the system, and finally output the position and speed;

[0218] Let k 1 = 10, because when |v i | < α, |f(v i )| < β|v i |, β < k 1 , so |f(v i )| = v i 2 < α|v i |, at this time α = β, and β < k 1 , so take α = β = 9; To satisfy the initial constraint condition V(0) ≤ α 2 , take the initial constraint condition as x(0) = [4, 2, 0, -1, 1, -2], v(0) = [1, -1, 2, -2]. From Figures 4 to 7 shown, it can be known that when the initial constraint condition is satisfied, when the 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;

[0219] As Figure 8As 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, the output of the controller is input into the state-space expression of the system. Finally, the position and speed are output. Let k 1 = 0.0001. Because when At this time 0 < β < 1, so At this time, α = β, and 0 < β < 1, so take α = β = 0.5. To meet the initial constraint conditions Take the initial constraint conditions as x(0) = [8, 5, 2, -4, 1, -5], v(0) = [1, -5, 5, 3]; From Figures 9 to 12 It can be known that when the initial constraint conditions are met, when the 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, and are not intended 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 described in the foregoing embodiments, or perform equivalent replacements on 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 consensus control method for heterogeneous nonlinear multi-agent systems, characterized in that: The specific steps include: S1: Define nonlinear terms about the absolute or relative speed of the agents, and heterogeneously construct first-order agents and second-order agents to obtain heterogeneous nonlinear multi-agent systems; According to the heterogeneous nonlinear multi-agent system, a heterogeneous nonlinear multi-agent state equation is obtained; S2: Constructing Lyapunov functions based on heterogeneous nonlinear multi-agent systems; And differentiate the Lyapunov function to obtain the derivative of the Lyapunov function; Rewriting the derivative of the Lyapunov function according to the heterogeneous nonlinear multi-agent state equation to obtain initial constraints that satisfy the semi-global consistency of the heterogeneous nonlinear multi-agent system, wherein the initial constraints include a first initial constraint corresponding to the absolute speed of the agent and a second initial constraint corresponding to the relative speed of the agent; S3: Based on the initial constraints, an open-loop transfer function for system controller design is constructed 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 the heterogeneous nonlinear multi-agent system are obtained according to the open-loop transfer function; S5: Considering the influence of uncertain constant disturbance in heterogeneous nonlinear multi-agent system, the final system controller is obtained according to 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. A semi-global consensus control method for heterogeneous nonlinear multi-agent systems according to claim 1, characterized in that: The S1 specifically includes the following steps: S11: Define the number of agents as N and N = {1, 2, …, m, …, n}; And set the number of second-order agents to {1,2,...,m}; The number of first-order agents is {m+1,m+2,...,n}; S12: Define nonlinear terms about the absolute or relative speed of agents, and heterogeneously construct first-order agents and second-order agents to obtain heterogeneous nonlinear multi-agent systems; And the heterogeneous nonlinear multi-agent system includes: Considering the nonlinear term about the absolute speed of the agent, the first heterogeneous nonlinear multi-agent system obtained by heterogeneous first-order agents and second-order agents; And the expression of the first heterogeneous nonlinear multi-agent system is Where: x i (t) represents the position of the i-th agent in the heterogeneous nonlinear multi-agent system; v i (t) represents the speed of the ith agent; u i (t) represents the control input of the ith agent; Represents x i (t),v i The first derivative of (t); x j (t) represents the position of the jth agent; a ij represents the decision variable that can transfer information between agents i and j, and a ij =1 means that agents i and j can transmit information to each other; a ij = 0 means that agents i and j cannot transmit information to each other; k1, k2 represent the proportional coefficients in the control input; f(v i ) represents a nonlinear function about the absolute speed of the agent, and satisfies When |v i |<α, there is |f(v i )|<β|v i |; β<k1,k1; α,β represent design parameters; Considering the nonlinear terms about the relative speed of the agents, the second heterogeneous nonlinear multi-agent system obtained by heterogeneous first-order agents and second-order agents; And the expression of the second heterogeneous nonlinear multi-agent system is Where: Represents a nonlinear function of the relative speed of the agent; and satisfies when Sometimes, there is 0<β<1; S13: Based on the heterogeneous nonlinear multi-agent system, the heterogeneous nonlinear multi-agent state equation considering the absolute speed or relative speed of the agent is obtained; The heterogeneous nonlinear multi-agent state equation considering the absolute speed of the agent is expressed as The heterogeneous nonlinear multi-agent state equation considering the relative speed of the agents is expressed as 3. A semi-global consensus control method for heterogeneous nonlinear multi-agent systems according to claim 2, characterized in that: The S2 specifically includes the following steps: S21: Constructing Lyapunov functions based on heterogeneous nonlinear multi-agent systems; And the expression of Lyapunov function V(t) is S22: Deriving the Lyapunov function to obtain the derivative of the Lyapunov function; And the derivative of the Lyapunov function The expression is S23: According to the derivative of Lyapunov function A method for obtaining a first initial constraint condition satisfying a first heterogeneous nonlinear multi-agent system semi-global consistency by taking into account the absolute speed of the agent, specifically comprising: S231: Rewrite the derivative of the Lyapunov function according to the heterogeneous nonlinear multi-agent state equation, that is, Substituting the derivative of the Lyapunov function into the simplified form, we get And the simplified Lyapunov function derivative can be further simplified to obtain the simplified Lyapunov function derivative, whose expression is Will Substitute the simplified Lyapunov function derivative and rewrite it to obtain the rewritten Lyapunov function derivative, whose expression is: S232: Assume that when |v i |<α and satisfies |f(v i )|<β|v i |,β<k1, then v i f(v i )-k i v i 2 <0; From the Lyapunov function V(t), we can know that V(t)≥v i 2 ; When v i <α, then V(t)<α 2 , according to the rewritten Lyapunov function derivative, we can know And V(t) is monotonically decreasing, then V(t)≤V(0); And when V(0)≤α 2 When v i 2 ≤V(t)≤V(0)≤α 2 ; Then, the first initial constraint condition satisfying the semi-global consistency of the first heterogeneous nonlinear multi-agent system is obtained by considering the absolute speed of the agent; And the first initial constraint is x i (0),i∈{1,2,...,n},v i (0),i∈{1,2,...,m},V(0)≤α 2 ; S24: A method for obtaining a second initial constraint condition that considers the relative speed of the agents and satisfies the semi-global consistency of the second heterogeneous nonlinear multi-agent system is as follows: S241: Substituting the derivative of the Lyapunov function and simplifying it, we get S242: Assume that hour, 0<β<1, so there exists Make and will Substituting S241, we can get According to the Lyapunov stability criterion, if If it holds, then the heterogeneous nonlinear multi-agent system can achieve semi-global consistency: And because Then we can know when hour, Established; According to S241, we can know That is, V(t) decreases monotonically, so V(t)≤V(0); Then, the second initial constraint condition that considers the relative speed of the agents and satisfies the semi-global consistency of the second heterogeneous nonlinear multi-agent system is obtained; And the second initial constraint is x i (0),i∈{1,...,n},v i (0),i∈{1,...,m}, 4. A method for controlling semi-global consistency of heterogeneous nonlinear multi-agent systems according to claim 3, characterized in that: The S3 specifically includes the following steps: S31: Based on the first initial constraint condition or the second initial constraint condition, obtaining a state space equation of the linear steady-state system according to the heterogeneous nonlinear multi-agent state equation; And the state space equation of the linear time-invariant system is expressed as Where: x(t) is x i (t), and represents the state vector of a heterogeneous nonlinear multi-agent system; u(t) represents u i (t) is an abbreviation of the system, and represents the input of the heterogeneous nonlinear multi-agent system; y(t) represents the output of the heterogeneous nonlinear multi-agent system; A represents the state transfer matrix; B represents the input matrix; C represents the output matrix; D represents the feedforward matrix; S32: Laplace transform of the state space equation of the linear steady-state system can be obtained sX(s)=AX(s)+BU(s) Y(s)=CX(s)+DU(s) Where: X(s) represents the output after Laplace transformation of x(t); U(s) represents the output after Laplace transformation of u(t); Y(s) represents the output after Laplace transformation of y(t); s represents the Laplace operator; S33: Multiply both sides of the equation sX(s)=AX(s)+BU(s) by (sI-A) -1 , can get X(s)=(s-A) -1 BU(s) Where: I represents the unit matrix; S34: Set X(s)=(sA) -1 Substituting BU(s) into Y(s)=CX(s)+DU(s), we can get Y(s)=[C(s-A) -1 B+D]U(s) S35: constructing an open-loop transfer function for system controller design according to step S34; And the expression of the open-loop transfer function is Among them, for the second-order agent in a heterogeneous nonlinear multi-agent system, 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 For the first-order agent in a heterogeneous nonlinear multi-agent system, Assuming A=0, B=1, C=1, D=0, the open-loop transfer function of the first-order agent in the heterogeneous nonlinear multi-agent system is 5. The method for controlling semi-global consistency of heterogeneous nonlinear multi-agent systems according to claim 3, characterized in that: The S4 specifically comprises the following steps: S41: The model expression of the second-order closed-loop gain shaping algorithm is obtained as follows: Where: T1 represents the time constant; s represents the Laplace operator; G represents the open-loop transfer function; K represents the controller; S42: Based on step S41, 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 are obtained according to the open-loop transfer function, and the expression is: Where: 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.

6. A method for controlling semi-global consistency of heterogeneous nonlinear multi-agent systems according to claim 5, characterized in that: The final system controller is obtained according to the initial first-order controller and the initial second-order controller as described in S5, and its expression is: Where: K c1 ,K c2 Represents the control inputs of the first-order agents and second-order agents in a heterogeneous nonlinear multi-agent system; ε represents the static error term considering uncertain constant disturbance in heterogeneous nonlinear multi-agent systems.

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