A multi-agent robust containment control method based on self-triggered mechanism and medium

Abstract

This invention relates to a robust inclusion control method for multi-agent systems based on a self-triggering mechanism, comprising: determining the number of agents acting as leaders and followers in a multi-agent system, and determining the Laplace matrix of the communication topology of the multi-agent system; determining the state-space equations of the followers and leaders; determining the control objective; defining the state tracking error at the triggering moment, and designing a self-triggering inclusion control protocol using the state tracking error at the triggering moment; substituting the self-triggering inclusion control protocol into the state-space equations, and augmenting all state-space equations into a global form of the tracking error system; establishing inequality conditions for solving the control protocol gain matrix K in the global form of the tracking error system, and establishing self-triggering auxiliary conditions; proving that the multi-agent tracking error system is asymptotically stable; and linearizing the inequality conditions for solving the control protocol gain matrix K, and deriving the inequality for calculating the triggering moment using the self-triggering auxiliary conditions.

Description

Technical Field

[0001] This invention relates to the field of multi-agent system technology, and in particular to a multi-agent robust inclusion control method and medium based on a self-triggering mechanism. Background Technology

[0002] A multi-agent system refers to a system with multiple dynamically evolving agents that accomplish large and complex tasks through limited local information exchange and coordination.

[0003] Multi-agent cooperative control utilizes information from neighboring agents to design suitable control protocols, enabling the system to complete specific tasks. This is a popular research area within multi-agent system theory. Multi-agent cooperative control problems can be categorized into consensus problems, grouping problems, and swarming problems, with consensus problems being the most fundamental and common.

[0004] When there is only one leader agent in a multi-agent system, the consistency problem can be called the tracking control problem. However, with the deepening of industrialization and the increasing complexity of control tasks, in some practical systems, a multi-agent system with only one leader may not be able to complete certain complex tasks, and multiple leaders are required.

[0005] For multi-agent systems, the transmission of large amounts of information between agents via shared networks is unavoidable. Agents are highly susceptible to external interference during data transmission, which can lead to performance degradation or even instability.

[0006] Meanwhile, the theoretical analysis contains a potential assumption: data from an agent can be fully received by its neighboring agents without delay or loss due to data collisions. However, in reality, due to limited communication bandwidth, this assumption is easily invalidated. Excessive data transmission between agents can lead to network congestion, thereby causing some undesirable communication problems.

[0007] To overcome the aforementioned obstacles, it is necessary to consider how to enhance the system's anti-interference capabilities and address bandwidth limitations during data transmission when multiple leading agents are involved. Summary of the Invention

[0008] To address at least some of the aforementioned problems in the prior art, this invention provides a multi-agent robust inclusion control method based on a self-triggering mechanism, comprising:

[0009] Determine the number of agents acting as leaders and followers in a multi-agent system, and determine the Laplace matrix for the communication topology of the multi-agent system;

[0010] Determine the state-space equations for followers and leaders;

[0011] Define control objectives;

[0012] Define the state tracking error at the trigger moment, and design a self-triggering control protocol using the state tracking error at the trigger moment;

[0013] By incorporating the self-triggering control protocol into the state-space equations of the follower and leader, and by augmenting all state-space equations into a global form, we obtain the tracking error system in a global form.

[0014] Establish inequality conditions for solving the control protocol gain matrix K in the global form of the tracking error system, and establish self-triggering auxiliary conditions;

[0015] Prove that the tracking error system of a multi-agent system is asymptotically stable; and

[0016] The inequality conditions for solving the control protocol gain matrix K are linearized, and the inequality for calculating the trigger time is derived using the self-trigger auxiliary conditions.

[0017] Furthermore, in the multi-agent system, the leader sends data to the followers but does not receive data, while the followers receive data from the leader and their neighboring followers and send data to their neighboring followers.

[0018] For the follower, the k-th triggering time is denoted as t. k (k = 1, 2, ...), therefore the triggering time sequence is t1, t2, ... t k For the control protocol of the i-th follower, it can only receive and send the state x at the trigger time. i (t k ).

[0019] Furthermore, determining the number of agents acting as leaders and followers in the multi-agent system, and determining the Laplace matrix of the communication topology of the multi-agent system, includes:

[0020] For a multi-agent system, select M followers and NM leaders, where the number of followers and leaders is greater than 1.

[0021] The Laplace matrix of the communication topology G of a multi-agent system is:

[0022]

[0023] In the formula, L1∈R M×M L2∈R M×(N-M) .

[0024] Furthermore, the state-space equations for determining the followers and the leader include:

[0025] The state-space equation of the follower is:

[0026]

[0027] In the formula, Let x represent the state derivative of the i-th follower. i Let u represent the state of the i-th follower. i ω represents the control input of the i-th follower. i Indicates belonging to External disturbances in space satisfy ||ω i (t)‖≤α‖x i (t)‖, where α represents a known nonnegative scalar, A, B1, and B2 represent known coefficient matrices, and t represents the time variable;

[0028] The state-space equation for the leader is:

[0029]

[0030] In the formula, Let represent the state derivative of the i-th leader agent, xi represent the state of the i-th leader agent, and A represent the coefficient matrix.

[0031] Furthermore, the determination of control objectives includes:

[0032] When there are no interference terms, the control objective is to drive all followers asymptotically to converge to the convex hull formed by the leader in any initial state, defined as:

[0033]

[0034] In the formula, z i Let Ω(t) represent the control output of follower i, and let Ω(t) represent the convex hull formed by the leader. Therefore, the mathematical expression for the control objective in the absence of disturbance is:

[0035]

[0036] When there are disturbances, define an expression for the ability of a multi-agent system to attenuate external disturbances:

[0037]

[0038] In the formula, T zω (s) represents the closed-loop transfer function from disturbance to output, and the control objectives include:

[0039]

[0040] In the formula, γ represents a given positive scalar.

[0041] Furthermore, the state tracking error at the trigger moment is defined, and a self-triggered control protocol is designed using the state tracking error at the trigger moment, including:

[0042] H was proposed ∞ Control protocols drive followers to asymptotically converge to the convex hull formed by the leader:

[0043]

[0044] In the formula, K is the control protocol gain matrix, a ij The adjoint matrix associated with the communication topology graph G The (i,j)th term represents the information transmission connection relationship between agent i and agent j;

[0045] In H ∞ Based on the control protocol, an H-mode under a self-triggering mechanism is proposed. ∞ Control Protocol:

[0046] For followers, the definition is:

[0047]

[0048]

[0049] In the formula,

[0050] Finally, the self-triggered containment control protocol is represented as: for t∈[t] k ,t k+1 ),

[0051]

[0052] Furthermore, the process of incorporating the self-triggered control protocol into the state-space equations and transforming the state-space equations into a global form includes:

[0053] Substituting the self-triggering control protocol into the state-space equations, we get:

[0054]

[0055] Define the global form of the state variable:

[0056] Define control output: in Represents the convex hull formed by the leaders;

[0057] Define multi-agent measurement error: Its global form:

[0058] Transforming the global variables, since L1 is a positive definite matrix, there exists an orthogonal matrix U∈R. M×M Make:

[0059]

[0060] In the formula, 0 < λ1 ≤ λ2 ≤ … ≤ λ M ,make

[0061] The state-space equations are extended to a global form for the tracking error system, which is expressed as:

[0062]

[0063] Furthermore, the inequality conditions for solving the control protocol gain matrix K in the global state-space equations are established, and the self-triggering auxiliary conditions are established, including:

[0064] Establish the inequality conditions for solving the control protocol gain matrix K in the global form of the tracking error system:

[0065]

[0066] In the formula, λ i It is the i-th eigenvalue of matrix L1;

[0067] Establish self-triggering auxiliary conditions:

[0068]

[0069] In the formula,

[0070] Furthermore, the proof of asymptotic stability of multi-agent systems includes:

[0071] Based on the globalized multi-agent tracking error system state Construct the Lyapunov function;

[0072] Differentiating the Lyapunov function with respect to time, if the derivative is negative definite, it indicates that the multi-agent system is asymptotically stable; and

[0073] Proof || T zω (s)‖ ∞ <γ.

[0074] Furthermore, the inequality conditions for solving the control protocol gain matrix K are linearized, and the inequality for calculating the trigger time is derived using the self-triggering auxiliary conditions, including:

[0075] The inequality conditions for solving the control protocol gain matrix K are linearized using the Schur complement to the following linear matrix inequalities:

[0076]

[0077] In the formula,

[0078] definition Multiply the linear matrix inequalities on the left and right by Γ and ΓT respectively, and denote them as follows: Therefore, the above linear matrix inequality can be redefined as follows:

[0079]

[0080] In the formula, Because 0 < λ1 ≤ λ2 ≤ … ≤ λ M If for i=1 and i=M, Ψ i If <0 is true, then Ψ i <0 for all Both are true;

[0081] The inequality for calculating the triggering time, derived using the self-triggering auxiliary condition, is as follows:

[0082]

[0083] In the formula,

[0084] Furthermore, it also includes proving that multi-agent systems do not exhibit Zeno behavior, including:

[0085] Based on the derived self-triggering condition, prove T k >0, meaning the Zeno behavior does not exist:

[0086] choose

[0087]

[0088] because So

[0089]

[0090] The present invention also provides a computer-readable storage medium having a computer program stored thereon, the computer program performing the steps according to the method described above when executed by a processor.

[0091] This invention has at least the following beneficial effects: The invention discloses a robust multi-agent inclusion control method based on a self-triggering mechanism. In the case of multiple leaders, it proposes a self-triggering mechanism that reduces data transmission frequency, where followers only transmit data at the trigger moment, thus solving the problem of congestion in data transmission between followers. Based on the self-triggering mechanism, a self-triggering inclusion control protocol is designed to ensure that followers asymptotically converge to the convex hull formed by the leaders, while simultaneously satisfying the H... ∞ Robust performance indicators. This control method can reduce the bandwidth burden on the intelligent agent during information transmission and effectively resist the adverse effects of external interference on the system. Attached Figure Description

[0092] To further illustrate the above and other advantages and features of the various embodiments of the present invention, a more specific description of the various embodiments of the present invention will be presented with reference to the accompanying drawings. It is to be understood that these drawings depict only typical embodiments of the invention and are therefore not intended to limit its scope.

[0093] Figure 1 A flowchart of a multi-agent robust inclusion control method based on a self-triggering mechanism according to the present invention is shown;

[0094] Figure 2 A communication topology diagram of a multi-agent system according to an embodiment of the present invention is shown;

[0095] Figure 3 A state trajectory diagram of each agent according to an embodiment of the present invention is shown;

[0096] Figure 4 The control input of one of the followers in a multi-agent system according to an embodiment of the present invention is shown;

[0097] Figure 5 H is shown in one embodiment according to the present invention. ∞ Performance index change curves; and

[0098] Figure 6 A trigger gap diagram is shown according to one embodiment of the present invention. Detailed Implementation

[0099] It should be noted that the components in the accompanying drawings may be shown exaggerated for illustrative purposes and may not be to scale.

[0100] In this invention, the various embodiments are merely intended to illustrate the solutions of the invention and should not be construed as limiting.

[0101] In this invention, unless otherwise specified, the quantifiers “a” and “one” do not exclude scenarios involving multiple elements.

[0102] It should also be noted that, in the embodiments of the present invention, only a portion of the parts or components may be shown for clarity and simplicity. However, those skilled in the art will understand that, under the teachings of the present invention, the required parts or components can be added as needed for specific scenarios.

[0103] It should also be noted that within the scope of this invention, the terms "same", "equal", and "equal to" do not mean that the two values ​​are absolutely equal, but allow for a certain reasonable error. In other words, the terms also cover "substantially the same", "substantially equal", and "substantially equal to".

[0104] It should also be noted that in the description of this invention, the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not explicitly or implicitly suggest that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0105] Furthermore, the numbering of the steps in the methods of the present invention does not limit the execution order of the method steps. Unless otherwise specified, the method steps may be executed in different orders.

[0106] Figure 1 A flowchart of a multi-agent robust containment control method based on a self-triggering mechanism according to the present invention is shown.

[0107] like Figure 1 As shown, a multi-agent robust control method based on a self-triggering mechanism includes:

[0108] Step 1: Determine the number of agents acting as leaders and followers in the multi-agent system, and determine the Laplace matrix for the communication topology of the multi-agent system.

[0109] Step 2: Determine the state-space equations for the followers and the leader.

[0110] Step 3: Determine the control objective.

[0111] Step 4: Define the state tracking error at the trigger moment, and design a self-triggering control protocol using the state tracking error at the trigger moment.

[0112] Step 5: Substitute the self-triggering control protocol into the state-space equations, and augment all state-space equations into a global form to obtain the tracking error system in the global form.

[0113] Step 6: Establish the inequality conditions for solving the control protocol gain matrix K in the global form of the tracking error system, and establish self-triggering auxiliary conditions.

[0114] Step 7: Prove that the multi-agent tracking error system is asymptotically stable.

[0115] Step 8: Linearize the inequality conditions for solving the control protocol gain matrix K and derive the inequality for calculating the trigger time using the self-trigger auxiliary conditions.

[0116] Step 9: Prove that Zeno behavior does not exist in multi-agent systems.

[0117] Steps 1 through 9 are described in detail below.

[0118] 1. Determine the number of agents acting as leaders and followers in a multi-agent system, and determine the Laplace matrix for the topology of the multi-agent system.

[0119] 1.1 Select M followers and NM leaders for a multi-agent system, let... These represent the sets of indices for followers and leaders, respectively.

[0120] 1.2 Since the leader has no neighbors, the Laplace matrix of the communication topology G in the multi-agent system is:

[0121]

[0122] In the formula, L1∈R M×M L2∈R M×(N-M) .

[0123] 2. Determine the state-space equations for followers and leaders.

[0124] For the follower, in the general linear multi-agent system model with introduced disturbances, the follower's state-space equation is:

[0125]

[0126] In the formula, Let xi represent the state derivative of the i-th follower, and u represent the state of the i-th follower. i ω represents the control input of the i-th follower. i Indicates belonging to External disturbances in space satisfy ||ω i (t)‖≤α‖x i(t)‖, where α represents a known nonnegative scalar, A, B1, and B2 represent known coefficient matrices, and t represents the time variable.

[0127] The state-space equation for the leader is:

[0128]

[0129] In the formula, Let x represent the state derivative of the i-th leader agent. i Let A represent the state of the i-th leader agent, and let A represent the coefficient matrix.

[0130] The state-space equations of the followers and the leader together form a multi-agent system model.

[0131] Followers and leaders choose different general linear system models to determine the state-space equations.

[0132] 3. Determine the control objectives.

[0133] Due to limited network bandwidth, data communication conflicts may occur during information transmission between agents. In the multi-agent system considered in this invention, the leader only sends data to followers but does not receive data, while followers receive data from the leader and their neighboring followers and send data to their neighboring followers. In this situation, data transmission between followers is prone to congestion. To avoid this problem, this invention designs a self-triggering mechanism to control the moment of data transmission for following agents. Followers only transmit data at the triggering moment. It should be noted that due to the independence between agents, the triggering moment for each agent may be different. For followers, the k-th triggering moment is denoted as t. k (k = 1, 2, ...), therefore the triggering time sequence is t1, t2, ... t k For the control protocol of the i-th follower, it can only receive and send the state x at the trigger time. i (t k ).

[0134] 3.1. When there are no disturbance terms, determine the system's control objective. The control objective can be expressed as: under any initial state, drive all followers to asymptotically converge to the convex hull formed by the leader. Defined as follows:

[0135]

[0136] In the formula z i Let Ω(t) represent the control output of follower i, and let Ω(t) represent the convex hull formed by the leader. Therefore, the mathematical expression for the control objective in the absence of disturbance is:

[0137]

[0138] Specifically, if for any initial state, all followers asymptotically converge to the convex hull formed by the leader, then we say that the control protocol u is valid. i (t) solves the control problem, which can be abstractly expressed in mathematical language as: for when and If the convex hull is represented, then the containment control problem can be solved.

[0139] Applying the above abstract expression to this multi-agent system, a specific mathematical expression is defined. First, the control output of the multi-agent system is defined as:

[0140]

[0141]

[0142] if Then the system's inclusion control problem can be solved.

[0143] 3.2 When there are disturbances, determine the control objective of the system and define the attenuation capability of the multi-agent system to external disturbances:

[0144]

[0145] In the formula, T zω (s) represents the closed-loop transfer function from the disturbance to the output, if it satisfies ||T|| zω (s)‖ ∞ If <γ, where γ is a given positive scalar, then the system satisfies H. ∞ Robust performance metrics.

[0146] Therefore, the control objective when there is disturbance can be expressed as:

[0147]

[0148] In the formula, γ represents a given positive scalar. If both of these conditions are met simultaneously, then the control protocol u i (t) can solve the multi-agent H problem ∞ It includes control issues.

[0149] 4. Define the state tracking error at the trigger moment, and design a self-triggering control protocol using the state tracking error at the trigger moment.

[0150] 4.1, Propose H ∞ Control protocols drive followers to asymptotically converge to the convex hull formed by the leader:

[0151]

[0152] In the formula, K is the control protocol gain matrix, which will be determined in subsequent steps, a ij The adjoint matrix associated with the communication topology graph G The (i,j)th term represents the information transmission connection relationship between agent i and agent j.

[0153] 4.2, in H ∞ Based on the control protocol, an H-mode under a self-triggering mechanism is proposed. ∞ Control Protocol:

[0154] For followers, define the state tracking error at the trigger moment:

[0155]

[0156]

[0157] In the formula,

[0158] Finally, the self-triggered containment control protocol can be expressed as: for t∈[t] k ,t k+1 ),

[0159]

[0160] 5. Incorporate the self-triggering control protocol into the state-space equations of the followers and leaders, and augment all state-space equations into a global tracking error system.

[0161] 5.1 First, substituting the self-triggering control protocol designed in the previous step into the state-space equation, we get:

[0162]

[0163] 5.2 Define the global form of the state variable: Therefore, the state-space equation in step 5.1 can be written in global form:

[0164]

[0165] 5.3 Define control output: in This represents the convex hull formed by the leaders.

[0166] Define measurement error ε i The global form of (t): Therefore, the state-space equation in step 5.2 can be redefined as follows:

[0167]

[0168] Transforming the global variables, since L1 is a positive definite matrix, there exists an orthogonal matrix U∈R. M×M Make:

[0169]

[0170] In the formula, 0 < λ1 ≤ λ2 ≤ … ≤ λ M ,make The state-space equation can then be generalized to the global form of the tracking error system as follows:

[0171]

[0172] 6. Establish inequality conditions for solving the control protocol gain matrix K in the global state-space equations, and establish self-triggering auxiliary conditions.

[0173] Establish the inequality conditions for solving the controller gain matrix K:

[0174]

[0175] In the formula, λ i It is the i-th eigenvalue of matrix L1.

[0176] Established self-triggering auxiliary conditions:

[0177]

[0178] In the formula,

[0179] For a given scalar Given β∈(0,1], if there exists a matrix P>0 and a matrix K satisfying the above inequality conditions, and the follower's state in the multi-agent system only uses the state at the trigger time, with the trigger time calculated by the above self-triggering auxiliary conditions, then the multi-agent system is asymptotically stable under the self-triggering mechanism while satisfying the perturbation suppression level.

[0180] 7. Prove that the tracking error system of a multi-agent system is asymptotically stable.

[0181] We prove the asymptotic stability of the system using the self-triggering auxiliary condition and the inequality condition for solving the controller gain matrix K. That is, under the self-triggering mechanism, the followers can asymptotically converge to the convex hull composed of the leader and satisfy H. ∞ Performance metrics.

[0182] 7.1. Using the control protocol in step 4 and the self-triggering auxiliary conditions in step 6, prove that the tracking error system is asymptotically stable, specifically including:

[0183] Based on the globalized multi-agent tracking error system state To construct a Lyapunov function, we must first ensure that the constructed Lyapunov function is positive definite. The Lyapunov function is:

[0184]

[0185] Differentiating the Lyapunov function with respect to time, we get:

[0186]

[0187] because have:

[0188]

[0189] If the derivative of the Lyapunov function is negative definite, it indicates that the system is stable.

[0190] 7.2 Proof ||T zω (s)‖ ∞ <γ, meaning the system satisfies H ∞ Anti-interference performance.

[0191] If both the inequality conditions and the self-triggering auxiliary conditions for solving the control protocol gain matrix K hold, then we have:

[0192]

[0193] If the self-triggered auxiliary condition is true, we can obtain:

[0194]

[0195] have Right now definition Under zero initial conditions

[0196]

[0197] It is easy to obtain V(T)>0, J T <0, making T→∞, we can obtain

[0198]

[0199] That is, ||T zω (s)‖ ∞ <γ means that the multi-agent system asymptotically converges under the self-triggering mechanism and satisfies H. ∞ Performance metrics.

[0200] 8. Linearize the inequality conditions for solving the control protocol gain matrix K, and derive the inequality for calculating the trigger time using the self-trigger auxiliary conditions.

[0201] 8.1 The inequality conditions for solving the control protocol gain matrix K mentioned above can be linearized into the following linear matrix inequality using the Schur complement:

[0202]

[0203] In the formula, definition Multiply the above linear matrix inequalities on the left and right by Γ and ΓT respectively, and denote them as follows: Therefore, the above linear matrix inequality can be redefined as follows:

[0204]

[0205] In the formula, Because 0 < λ1 ≤ λ2 ≤ … ≤ λ M If for i=1 and i=M, Ψ i If <0 is true, then Ψ i <0 for all Both are valid.

[0206] 8.2. Derive the inequality for calculating the triggering time using the self-triggering auxiliary condition:

[0207] definition:

[0208]

[0209] In the formula, Therefore:

[0210]

[0211]

[0212] The self-triggering auxiliary condition can be seen from the above two formulas. The establishment can be by Ensure that, that is, if for any δ∈(0,1], if If it is true, then the self-triggered auxiliary condition must also be true.

[0213] Next, it will be by Derive the inequality for calculating the trigger time:

[0214] The time derivative of f(t) satisfies:

[0215]

[0216] in, From t k Integrating up to t, we have:

[0217]

[0218] Substituting the derivative of |f(t)| with respect to time, we get:

[0219]

[0220] in:

[0221]

[0222] because have:

[0223]

[0224]

[0225] From t k Integrating up to t, we have:

[0226]

[0227] Right now

[0228]

[0229] We can get:

[0230]

[0231] Obviously, if T k If the above inequality is satisfied, then we can obtain That is, the self-triggering auxiliary condition is met.

[0232] In summary, for a given scalar β, δ ∈ (0, 1] and γ > 0, if there exists a positive definite matrix... sum matrix For i = 1 and i = M, the following inequalities are satisfied:

[0233]

[0234]

[0235] And trigger time sequence By t k+1 =t k +T k The decision is made, where the trigger interval Tk Satisfy the following:

[0236]

[0237] In the formula, So many agent systems achieve asymptotic stability under self-triggering mechanisms while simultaneously satisfying disturbance suppression levels. Meanwhile, the gain matrix of the controller is K = QX -1 The aforementioned trigger interval T k The inequality is the self-triggering condition derived from it.

[0238] Step 9: Prove that the system does not exhibit Zeno behavior.

[0239] Based on the derived self-triggering condition, prove T k >0, meaning the Zeno behavior does not exist:

[0240] choose

[0241]

[0242] because So

[0243]

[0244] Therefore, Zeno behavior can be ruled out.

[0245] The following numerical simulation of a multi-agent system is presented to verify the effectiveness of the self-triggering mechanism and control protocol proposed in this invention.

[0246] Figure 2 A communication topology diagram of a multi-agent system according to an embodiment of the present invention is shown; Figure 3 A state trajectory diagram of each agent according to an embodiment of the present invention is shown; Figure 4 The control input of one of the followers in a multi-agent system according to an embodiment of the present invention is shown; Figure 5 H is shown in one embodiment according to the present invention. ∞ Performance index change curves; and Figure 6 A trigger gap diagram is shown according to one embodiment of the present invention.

[0247] Consider a multi-agent system with 5 followers and 8 leaders, whose communication topology is as follows: Figure 2 As shown, numbers 1-5 represent followers, and numbers 6-13 represent leaders. Clearly, matrices L1 and L2 are respectively:

[0248]

[0249]

[0250] Let the external disturbance of the system be ω(t)=[η(t),1.5η(t),2η(t),-η(t),-1.5η(t)] T ,in

[0251] Other variables were selected as follows: Given α = 1, δ = 0.9, and β = 0.5, solve the linear matrix inequality Ψ. i The feedback gain matrix is ​​obtained when <0:

[0252]

[0253] The initial state of the agent is selected as follows:

[0254] x(0)=[8,2,3,5,6,10,10,30,30,10,10,30,30]

[0255] y(0)=[4,3,8,6,9,10,30,30,10,10,30,30,10]

[0256] z(0)=[6,2,3,5,8,10,10,10,10,30,30,30,30]

[0257] v x (0) = [5,5,3,2,1,10,10,10,10,10,10,10,10]

[0258] v y (0) = [4,2,2,1,3,10,10,10,10,10,10,10,10]

[0259] v z (0) = [4,4,2,5,1,10,10,10,10,10,10,10,10]

[0260] Choosing a sampling interval ΔT = 0.0001s, the state trajectories of each agent are as follows: Figure 3 As shown, circles represent followers, and squares represent leaders. Leaders are located outside the followers, and followers eventually converge asymptotically to the convex hull formed by the leader. The control input signal for the first follower is as follows: Figure 4 As shown, the control input signals of other followers are... Figure 4 Similar. The system's H ∞ Performance curves as follows Figure 5As shown, it can be seen that, under a given disturbance immunity level γ = 1.46, the system can satisfy H ∞ Performance. The agent's trigger time t k and trigger interval T k like Figure 6 As shown, the trigger rate can be calculated. Furthermore, the time interval between the two triggers by the agent is greater than 0, indicating that no Zeno behavior occurred during the entire control process.

[0261] Due to its advantages such as simplicity of implementation, low complexity, strong robustness, and good scalability, multi-agent consensus has been widely applied in practical engineering applications such as multi-UAV systems, sensor network systems, multi-unmanned vehicle systems, satellite swarm flight systems, and multi-unmanned surface vessel systems. Among these, multi-UAV systems have played a significant role in areas with repetitive hazards, such as forest fire monitoring, search and rescue, firefighting, and power equipment inspection. This solution primarily considers its application in the multi-UAV domain, and the specific application process is as follows:

[0262] Step 1: Establish the kinematic and dynamic models of the UAV.

[0263] A multi-UAV system consisting of N UAVs is configured. A point-mass flight model is used to describe the UAVs' motion. Relevant variables are defined in an inertial coordinate system (a, b, h). It is assumed that the aircraft's thrust is along the velocity vector direction, the aircraft always performs coordinated maneuvers, the Earth is flat, fuel consumption is negligible, and the center of mass is time-invariant. Under these assumptions, the kinematic model of the i-th (i = 1, 2, ..., M) UAV is:

[0264]

[0265]

[0266]

[0267]

[0268]

[0269]

[0270] Among them, a i b is the forward and backward displacement distance. i h is the lateral displacement distance. i V is the flight altitude. i For flight speed, φ i Let θ be the pitch angle. i Let μ be the heading angle. i For the angle of inclination, T iFor engine thrust, D i It is thrust, L i It's lift, m i It is mass, and g is the acceleration due to gravity.

[0271] Step 2: Establish the dynamic equations of the UAV

[0272] right and Taking the derivative, we get:

[0273]

[0274]

[0275]

[0276] definition For position vectors, It is a velocity vector. Furthermore, considering the impact of external disturbances on the following drone, the motion model in step one can be determined as the following dynamic equation:

[0277]

[0278] in, For the state vector, To control the input, ω i Indicates belonging to External disturbances in space satisfy ||ω i (t)‖≤α‖x i (t)‖, where α represents a known nonnegative scalar. This represents the set of indices that follow the drone, i.e.

[0279] Meanwhile, without considering interference, the dynamic equation of the pilot drone can be expressed as:

[0280]

[0281] in, This represents the set of subscripts for the navigation drone, i.e.

[0282] Thus, a practical UAV motion model can be transformed into a state-space expression of the dynamic equations of a multi-agent system in the control domain, indicating that the method proposed in this scheme can be applied to the UAV field.

[0283] The control method derived from the technical solution of this invention can be used in the field of unmanned aerial vehicles (UAVs) to achieve the following technical effects:

[0284] 1. The follower drone can calculate the next data transmission time according to the self-triggering scheme proposed in this invention at the current data transmission time. This not only avoids data conflicts and resource waste caused by real-time data transmission, but also avoids unnecessary hardware devices in existing event triggering schemes, saving hardware costs.

[0285] 2. Under the control scheme proposed in this invention, the follower drone has a certain robustness to external disturbances. In the presence of external disturbances, it can enter the convex hull formed by the leader drone, and finally achieve consistent tracking of the multi-leader drone system.

[0286] The embodiments may be provided as computer program products that may include one or more machine-readable media on which machine-executable instructions are stored, which, when executed by one or more machines such as a computer, computer network, or other electronic equipment, may cause one or more machines to perform operations according to the embodiments of the invention. Machine-readable media may include, but are not limited to, floppy disks, optical disks, CD-ROMs (compact disc read-only memory) and magneto-optical disks, ROMs (read-only memory), RAMs (random access memory), EPROMs (erasable programmable read-only memory), EEPROMs (electrically erasable programmable read-only memory), magnetic or optical cards, flash memory, or other types of media / machine-readable media suitable for storing machine-executable instructions.

[0287] Furthermore, various embodiments can be downloaded as computer program products, wherein the program can be transmitted from a remote computer (e.g., a server) to a requesting computer (e.g., a client) via a communication link (e.g., a modem and / or a network connection) using one or more data signals implemented and / or modulated by a carrier wave or other propagation medium. Therefore, the machine-readable medium used herein may include such a carrier wave, but this is not required.

[0288] While some embodiments of the present invention have been described in this application, those skilled in the art will understand that these embodiments are merely illustrative. Numerous variations, alternatives, and improvements will arise in those skilled in the art under the teachings of this invention without departing from its scope. The appended claims are intended to define the scope of the invention and thereby cover methods and structures within the scope of the claims themselves and their equivalents.

Claims

1. A robust multi-agent control method based on a self-triggering mechanism, characterized in that, include: Determine the number of agents acting as leaders and followers in a multi-agent system, and determine the Laplace matrix for the communication topology of the multi-agent system; Determine the state-space equations for followers and leaders; Define control objectives; Define the state tracking error at the trigger moment, and design a self-triggering control protocol using the state tracking error at the trigger moment; The self-triggering control protocol is incorporated into the state-space equations of the followers and leaders, and all state-space equations are augmented into a tracking error system in global form. Establish a solution for the control protocol gain matrix in the global form of the tracking error system. The inequality conditions are established, and self-triggering auxiliary conditions are created. Prove that the tracking error system of a multi-agent system is asymptotically stable; as well as Solving the control protocol gain matrix The inequality conditions are linearized, and the inequality for calculating the trigger time is derived using the self-triggering auxiliary condition; among them, the control protocol gain matrix will be solved. Linearization of the inequality conditions includes: solving for the control protocol gain matrix The inequality conditions are linearized into linear matrix inequalities using Schur complement; The state-space equations for determining followers and leaders include: The state-space equation of the follower is: In the formula, Indicates the first The state derivative of each follower Indicates the first The state of a follower Indicates the first A follower's control input, Indicates belonging to External disturbances in space, satisfying , Represents a known nonnegative scalar. , , Represents a known coefficient matrix. Represents a time variable. Represents the set of indices of the followers; The state-space equation for the leader is: In the formula, Indicates the first The state derivative of a leader agent. Indicates the first The state of a leading intelligent agent Represents the coefficient matrix. A set of indices representing the leader; The control objectives include: When there are no interference terms, the control objective is to drive all followers asymptotically to converge to the convex hull formed by the leader in any initial state, defined as: ， In the formula, Indicates follower The control output, This represents the convex hull formed by the leaders. express In the The components of the dimension, therefore, the mathematical expression for the control objective under undisturbed conditions is: ； When there are disturbances, define an expression for the ability of a multi-agent system to attenuate external disturbances: In the formula, , The closed-loop transfer function represents the control objective from disturbance to output, and includes: In the formula, Represents a given positive scalar.

2. The method according to claim 1, characterized in that, In the multi-agent system, the leader sends data to the followers but does not receive data, while the followers receive data from the leader and their neighboring followers and send data to their neighboring followers. For followers, the first Each trigger time is denoted as Therefore, the triggering time sequence is For the first The follower's control protocol can only accept and send the status at the trigger moment. .

3. The method according to claim 2, characterized in that, Determining the number of agents acting as leaders and followers in a multi-agent system, and determining the Laplace matrix for the communication topology of the multi-agent system, includes: Selecting for multi-agent systems One follower and There are 1 leader, where the number of followers and leaders is greater than 1; Multi-agent system communication topology diagram The Laplace matrix is: In the formula, , .

4. The method according to claim 3, characterized in that, The definition of the state tracking error at the trigger moment is used to design a self-triggered control protocol, which includes: propose Control protocols drive followers to asymptotically converge to the convex hull formed by the leader: In the formula, To control the protocol gain matrix, For communication topology diagram Related adjoint matrix The Item, representing an intelligent agent and intelligent agents The information transmission connection relationship between them; exist Based on the control protocol, a self-triggering mechanism is proposed. Control Protocol: For followers, define the state tracking error at the trigger moment: In the formula, ; Finally, the self-triggered control protocol is represented as follows: For , , 。 5. The method according to claim 4, characterized in that, The self-triggering control protocol is incorporated into the state-space equations of both the follower and leader, and all state-space equations are augmented to a global form for the tracking error system, including: Substituting the self-triggering control protocol into the state-space equations, we get: ； Define the global form of the state variable: , , This represents the global form of the follower's state variable. Represents the global form of the leader's state variables; Define control output: ,in This represents the convex hull formed by the leaders. This indicates the communication relationship between followers. It indicates the communication relationship between followers and leaders. I 6 represents a 6-dimensional identity matrix; Define multi-agent measurement error: Its global form is: ; Transformations are performed on global variables because If a matrix is ​​positive definite, then there exists an orthogonal matrix. Make: In the formula, ,make , , , , , ; The state-space equations are extended to a global form for the tracking error system, which is expressed as: ， express An identity matrix of dimension 1.

6. The method according to claim 5, characterized in that, This involves establishing the control protocol gain matrix in solving the global state-space equations. The inequality conditions are established, and self-triggering auxiliary conditions are created, including: Establish a solution for the control protocol gain matrix in the global form of the tracking error system. Inequality conditions: ， In the formula, It is a matrix The 1 eigenvalue, Represents the coefficient matrix. Represents a Lyapunov positive definite matrix; Establish self-triggering auxiliary conditions: In the formula, , , β represents the self-triggered auxiliary condition guarantee coefficient. This represents the trigger error weight matrix. This represents the trigger state weight matrix.

7. The method according to claim 6, characterized in that, Proofs demonstrating the asymptotic stability of multi-agent systems include: Based on the globalized multi-agent tracking error system state Construct the Lyapunov function; Differentiating the Lyapunov function with respect to time, if the derivative is negative definite, it indicates that the multi-agent system is asymptotically stable; and prove .

8. The method according to claim 7, characterized in that, This will involve solving the control protocol gain matrix. The inequality conditions are linearized, and the inequalities for calculating the trigger time are derived using self-triggering auxiliary conditions, including: Solve for the control protocol gain matrix The inequality conditions are linearized using the Schur complement to the following linear matrix inequalities: In the formula, ; definition , respectively and Perform left and right multiplications on linear matrix inequalities, and denote... Therefore, the above linear matrix inequality can be redefined as follows: In the formula, ,because If for and , If it is established, then For all Both are true; The inequality for calculating the triggering time, derived using the self-triggering auxiliary condition, is as follows: ， In the formula, , , , ， ， , , , ， An auxiliary positive scalar representing the triggered state of a follower. Denotes an auxiliary positive definite matrix. Represents the global matrix of topological eigenvalues. Represents the global matrix norm of the system. A positive scalar representing the leader's triggered state. This represents the self-trigger threshold adjustment coefficient. This represents the operator for finding the largest eigenvalue of a matrix. This represents the identity matrix formed by the largest eigenvalues ​​of the matrix.

9. The method according to claim 8, characterized in that, This also includes proving that multi-agent systems do not exhibit Zeno behavior, including: Based on the derived self-triggering condition, prove In other words, Zeno behavior does not exist: choose because ,So >0 An auxiliary positive scalar representing the triggered state of a follower. This represents the operator for finding the minimum eigenvalue of a matrix. This represents the trigger state weight matrix.

10. A computer-readable storage medium having a computer program stored thereon, the computer program performing the steps of the method according to any one of claims 1 to 9 when executed by a processor.

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