Distributed finite-time synchronization method for multi-layer heterogeneous networks with containment impulse control

Abstract

The application provides a kind of distributed finite time synchronization method of multi-layer heterogeneous network of restraining pulse control, and relates to the field of control and information technology.The application combines distributed restraining control and pulse control to realize the finite time synchronization of multi-layer heterogeneous network.On the one hand, the method of distributed control can reduce the dependence on global information of network and improve the robust performance of controller. On the other hand, the discontinuous characteristics of pulse control can reduce the control cost while achieving control effect. Based on Lyapunov stability theory, sufficient conditions for network to achieve finite time synchronization are obtained. Compared with homogeneous network, the heterogeneous network model can better describe the state of each node in the network. Analyzing the heterogeneous network is conducive to deepening the understanding of complex network synchronization control, and has profound theoretical value and rich practical value.

Description

Technical Field

[0001] This invention relates to the field of control and information technology, and in particular to a finite-time synchronization method for multilayer heterogeneous networks using distributed restraint pulse control. Background Technology

[0002] In real life, complex networks constantly impact our lives, such as the internet, urban transportation networks, and communication networks. Therefore, complex networks have gradually become a research hotspot, attracting the attention of numerous researchers. With the discovery of synchronization phenomena such as bridge resonance and firefly bioluminescence, the synchronization problem of complex networks has become an important research direction. Combining modern control methods, various methods for controlling the synchronization of complex networks have been proposed, with restraint control being one of them. This invention proposes a distributed hybrid restraint pulse control method based on the concept of restraint control. On the one hand, it utilizes distributed restraint control to control nodes in a multi-layered heterogeneous complex network system, achieving finite-time synchronization; on the other hand, by incorporating pulse control, it reduces control costs while mitigating the impact of nonlinear terms in the network system on the restraint control component of the controller.

[0003] Compared to homogeneous networks, each node in a heterogeneous network has a different dynamic function. In the real world, many complex systems contain nodes with various dynamic types. For example, nodes in an electrical network have different parameters, while the biochemical network controlling cell division in mammals contains a wide variety of matrices and enzymes. This makes the dynamic equations of heterogeneous networks more realistic, and their study has significant practical application value. Furthermore, multi-layered networks are widely present in the real world, such as social networks and brain networks, compared to single-layered networks. Therefore, multi-layered network models can more comprehensively and realistically characterize the characteristics of real-world networks. It is worth noting that most research on synchronization problems assumes that the synchronization time of a network system is infinite, while practical applications often require network systems to synchronize within a finite time. Therefore, rationally designing distributed restraint pulse control strategies to achieve finite-time synchronization of multi-layered heterogeneous networks has significant economic and social implications. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a finite-time synchronization method for multilayer heterogeneous networks using distributed restraint pulse control. By designing a suitable distributed control law, the multilayer heterogeneous network system is controlled to a target state. Based on Lyapunov stability theory, the conditions for network synchronization are analyzed, and the criteria for network synchronization are given.

[0005] A finite-time synchronization method for multilayer heterogeneous networks based on distributed restraint pulse control includes the following steps:

[0006] Step 1: Construct a controlled multilayer heterogeneous complex network system with W layers and N nodes in each layer; the dynamic equation of the i-th node in the network is expressed as:

[0007]

[0008] in, Let x represent the state variable of the i-th node, n represent the dimension of a single node, and x represent the state variable of the i-th node. ij (t), j = 1, 2, ..., n represents the j-th component of the state of the i-th node. f represents the first derivative of the state of the i-th node with respect to time t; i (·) represents the self-dynamic function of the i-th node, which is continuously differentiable; the constant c>0 represents the global coupling strength of the network; Γ is the internal coupling matrix, representing the coupling relationship between the state variables of each node; W represents the number of network layers, and N represents the number of network nodes; To characterize the topology of the w-th layer network, the Laplacian matrix must satisfy L w =D w -A w ;D w Let w be the degree matrix of the w-th layer network. To characterize the adjacency matrix of the w-th layer network topology, it is defined as: when there is an edge connecting node i and node j, otherwise Represents the set of N×N dimensional real matrices; u i (t) represents the controller added to the i-th node;

[0009] Step 2: Construct an error system model for a multi-layered heterogeneous complex network system;

[0010] Suppose there exists a leader node that satisfies... in The state variables of the leader node are f0(x0(t)), which is a smooth function; the synchronization error is defined as e. i (t)=x i If (t)-x0(t), i=1,2,…,N, then the error system model is:

[0011]

[0012] Step 3: Design a distributed restraint pulse controller;

[0013] The distributed restraint pulse controller consists of a distributed control term and a pulse feedback control term. The formula for the distributed restraint pulse controller is as follows:

[0014]

[0015] Where μ is a constant and satisfies μ∈(0,1); D and ξ are the diagonal gain matrix and gain constant, respectively; pulse sequence Satisfying 0≤t0≤t1≤…≤t k ≤…≤t ∞ , and lim k→∞ t k =∞, 0 <t k -t k-1 <τ max , where τ max This represents the maximum pulse interval, and δ(.) is the Dirac function;

[0016] Taking into account both the system error model and the distributed pulse restraint controller, the error system under the action of the distributed pulse restraint controller is obtained as follows:

[0017]

[0018] Among them, I n It is an n×n dimensional identity matrix, e(t) at t=t k Continuous to the right, and has h is a constant;

[0019] Step 4: Determine whether the multi-layered heterogeneous complex network has achieved synchronization;

[0020] Consider an error system with nodal dynamics satisfying the Lipschitz condition. When the navigator's state is bounded and the diagonal gain matrix D is negative definite, there exists a constant Ψ∈(0,1) such that... And a settling time T, which enables multilayer heterogeneous complex network systems to achieve finite-time synchronization when t≥T.

[0021] The expression for the settling time T is as follows:

[0022] when hour,

[0023]

[0024] when hour,

[0025]

[0026] Where Π1>0, Π2>0, Π3>0 are all constants, and the parameters ∈ (0,1) and θ∈ (0,1); parameters V(t) represents the Lyapunov function, where

[0027] The beneficial effects of adopting the above technical solution are as follows:

[0028] This invention provides a finite-time synchronization method for multilayer heterogeneous networks using distributed tethered pulse control. On one hand, the distributed control method reduces reliance on global network information, improving the robustness of the controller. On the other hand, leveraging the discontinuous nature of pulse control, it achieves the desired control effect while reducing control costs. Compared to homogeneous networks, heterogeneous network models better describe the dynamic characteristics of individual nodes in real-world networks. Analyzing heterogeneous networks deepens the understanding of synchronization behavior in complex real-world networks, possessing profound theoretical and practical value. The research object of this invention is a multilayer heterogeneous complex network model, which more accurately reflects the characteristics of real-world networks. The research results can be applied to real-world networks such as the Internet, transportation networks, and neural networks. Attached Figure Description

[0029] Figure 1 This is a flowchart of an embodiment of the present invention;

[0030] Figure 2 This is a diagram of the two-layer network structure in an embodiment of the present invention;

[0031] Figure 3 It is a state diagram of nodes in the network;

[0032] Figure 4 It is an error graph between network nodes and leader nodes. Detailed Implementation

[0033] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0034] A finite-time synchronization method for multilayer heterogeneous networks based on distributed tethered pulse control, such as... Figure 1 As shown, it includes the following steps:

[0035] Step 1: Construct a controlled multilayer heterogeneous complex network system with W layers and N nodes in each layer; the dynamic equation of the i-th node in the network is expressed as:

[0036]

[0037] in, Let x represent the state variable of the i-th node, n represent the dimension of a single node, and x represent the state variable of the i-th node. ij (t), j = 1, 2, ..., n represents the j-th component of the state of the i-th node. f represents the first derivative of the state of the i-th node with respect to time t;i (·) represents the self-dynamic function of the i-th node, which is continuously differentiable; the constant c>0 represents the global coupling strength of the network; Γ is the internal coupling matrix, representing the coupling relationship between the state variables of each node; W represents the number of network layers, and N represents the number of network nodes; To characterize the topology of the w-th layer network, the Laplacian matrix must satisfy L w =D w -A w ;D w Let w be the degree matrix of the w-th layer network. To characterize the adjacency matrix of the w-th layer network topology, it is defined as: when there is an edge connecting node i and node j, otherwise Represents the set of N×N dimensional real matrices; u i (t) represents the controller added to the i-th node;

[0038] Step 2: Construct an error system model for a multi-layered heterogeneous complex network system;

[0039] Suppose there exists a leader node that satisfies... in The state variables of the leader node are f0(x0(t)), which is a smooth function; the synchronization error is defined as e. i (t)=x i If (t)-x0(t), i=1,2,…,N, then the error system model is:

[0040]

[0041] Step 3: Design a distributed restraint pulse controller;

[0042] The distributed restraint pulse controller consists of a distributed control term and a pulse feedback control term. The formula for the distributed restraint pulse controller is as follows:

[0043]

[0044] Where μ is a constant and satisfies μ∈(0,1); D and ξ are the diagonal gain matrix and gain constant, respectively; pulse sequence Satisfying 0≤t0≤t1≤…≤t k ≤…≤t ∞ , and lim k→∞ t k =∞, 0 <t k -t k-1 <τ max , where τ max This represents the maximum pulse interval, and δ(.) is the Dirac function;

[0045] Taking into account both the system error model and the distributed pulse restraint controller, the error system under the action of the distributed pulse restraint controller is obtained as follows:

[0046]

[0047] Among them, I n It is an n×n dimensional identity matrix, e(t) at t=t k Continuous to the right, and has h is a constant;

[0048] Step 4: Determine whether the multi-layered heterogeneous complex network has achieved synchronization;

[0049] Consider an error system with nodal dynamics satisfying the Lipschitz condition. When the navigator's state is bounded and the diagonal gain matrix D is negative definite, there exists a constant Ψ∈(0,1) such that... And a settling time T, which enables multilayer heterogeneous complex network systems to achieve finite-time synchronization when t≥T.

[0050] The expression for the settling time T is as follows:

[0051] when hour,

[0052]

[0053] when hour,

[0054]

[0055] Where Π1>0, Π2>0, Π3>0 are all constants, and the parameters ∈ (0,1) and θ∈ (0,1); parameters V(t) represents the Lyapunov function, where

[0056] This embodiment analyzes the finite-time synchronization conditions between a multi-layer heterogeneous network system (1) and a leader node.

[0057] When t≠t k Design the following Lyapunov function:

[0058]

[0059] in, I is a diagonal positive definite matrix. N It is an N×N dimensional identity matrix. This represents the Kronecker product.

[0060] Taking the derivative of V(t) over time, the final analysis yields the following results:

[0061]

[0062] in, Π3=Nω1h 2 。 ||.|| F The Frobenius norm is represented here by ||.|| for simplicity. F Indicates that λ min and λ max Let represent the minimum and maximum eigenvalues ​​respectively, ω1>0 be a constant, h>0 be a constant, and P be an n×n diagonal positive definite matrix.

[0063] When t = t k At that time, the Lyapunov function will satisfy:

[0064]

[0065] Finally, it is found that there exists a stable time T such that when t≥T, the controlled multilayer heterogeneous complex network system (1) can achieve finite-time synchronization.

[0066] In the implementation example, a two-layer network with eight nodes in each layer is designed, as shown in the following structure. Figure 2 As shown. The self-dynamic function of node i is selected as follows.

[0067]

[0068] Where i = 1, 2, ..., 8.

[0069] Navigator node is

[0070] x0(t)=[0.1x 01 0.1x 02 0.1x 03 ] T (9)

[0071] The coupling strength is set to c = 0.1, and the initial values ​​of the network nodes are x1(t) = [-0.5, -0.8, -0.5]. T x²(t) = [0.5, -0.6, 0.2] T x3(t) = [0.3, 3, -1] T x4(t) = [-1.3, 1, 0.5] T x5(t) = [1.8, -0.8, 5] T x6(t) = [-5, -1, 0.2] Tx7(t) = [-2, 1, -2] T x8(t) = [0.4, 0.1, 0.6] T The initial value of the leader node is x0(t) = [0.1, 0.2, 0.1]. T Simulations were performed using MATLAB to obtain... Figure 3 and Figure 4 Using the initial conditions, the settling time was estimated to be T = 7.29. Figure 3 and Figure 4 As can be seen, when t≥T, the multilayer heterogeneous network system (1) can achieve synchronization, that is, achieve finite-time synchronization.

[0072] The above description is merely a preferred embodiment of this disclosure and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of the invention involved in the embodiments of this disclosure is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-described inventive concept. For example, technical solutions formed by substituting the above-described features with (but not limited to) technical features with similar functions disclosed in the embodiments of this disclosure.

Claims

1. A finite-time synchronization method for multilayer heterogeneous networks using distributed restraint pulse control, characterized in that, Includes the following steps: Step 1: Construct a controlled multilayer heterogeneous complex network system with W layers and N nodes in each layer; the dynamic equation of the i-th node in the network is expressed as: （1）； in, Let represent the state variables of the i-th node, and n represent the dimension of a single node. This represents the j-th component of the state of the i-th node. This represents the first derivative of the state of the i-th node with respect to time t; Indicates the first The self-dynamic function of each node is continuously differentiable; constant The global coupling strength of the network; It is the internal coupling matrix, representing the coupling relationship between the state variables of each node; Indicates the number of network layers. Indicates the number of network nodes; To depict the first The Laplacian matrix of the layer network topology satisfies ; For the first The degree matrix of the layer network, To depict the first The adjacency matrix of a layered network topology is defined as: when a node and nodes When there is an edge connecting them, ,otherwise ; express A set of 3D real matrices; Indicates adding to Controllers on each node; Step 2: Construct an error system model for a multi-layered heterogeneous complex network system; Step 3: Design a distributed restraint pulse controller; Step 4: Determine whether the multi-layered heterogeneous complex network has achieved synchronization; Consider an error system with nodal dynamics satisfying the Lipschitz condition. When the navigator's state is bounded, the diagonal gain matrix... When it is negative time, there exists a constant. make and stabilization time , making when At that time, multi-layered heterogeneous complex network systems can achieve synchronization in a finite amount of time. It is a constant and satisfies , yes 3D identity matrix.

2. The finite-time synchronization method for multilayer heterogeneous networks with distributed restraint pulse control according to claim 1, characterized in that, Step 2 specifically involves: assuming there exists a leader node, and satisfying... ,in These are the state variables of the leader node. It is a smooth function; the synchronization error is defined as... , Then the error system model is: （2）。 3. The finite-time synchronization method for multilayer heterogeneous networks with distributed restraint pulse control according to claim 1, characterized in that, In step 3, the distributed restraint pulse controller consists of a distributed control term and a pulse feedback control term. The formula for the distributed restraint pulse controller is as follows: （3）； in, It is a constant and satisfies ; and These are the diagonal gain matrix and the gain constant, respectively; pulse sequence. satisfy ,and , ,in Indicates the maximum pulse interval. It is a Dirac function; Taking into account both the system error model and the distributed pulse restraint controller, the error system under the action of the distributed pulse restraint controller is obtained as follows: （4）； in, yes 3D identity matrix exist Continuous to the right, and has , It is a constant.

4. The finite-time synchronization method for multilayer heterogeneous networks with distributed restraint pulse control according to claim 3, characterized in that, The settling time mentioned in step 4 The expression is as follows: when hour, ； when hour, ； in, All are constants, parameters ;parameter , ; , Represents the Lyapunov function. .

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