Unknown parameter complex network finite time synchronization control method based on bounded interference and mixed time delay

Abstract

The invention provides an unknown parameter complex network finite time synchronization control method based on bounded interference and mixed time delay, and relates to the technical field of synchronization control in automation. The method comprises the following steps: establishing a driving network model and a response network model of unknown parameters based on bounded interference and mixed time delay; determining a semi-global practical finite time synchronization condition according to the driving network model and the response network model; setting a synchronization error of the driving network and the response network, and establishing a synchronization error system; and designing a finite time state self-adaptive synchronous controller and an unknown parameter updating law according to the synchronization error, and acting the finite time state self-adaptive synchronous controller and the unknown parameter updating law on the response network to meet the condition of semi-global practical finite time synchronization, so that the finite time of the response network is synchronized to the driving network. According to the invention, the synchronization precision and the anti-disturbance capability are significantly improved, and the problem of synchronous oscillation and even instability caused by neglecting time lag or interference in a traditional method is avoided.

Description

Technical Field

[0001] The present invention relates to the technical field of synchronization control in automation, and in particular to a finite-time synchronization control method for a complex network with unknown parameters based on bounded interference and mixed time delay. Background Art

[0002] Synchronization is a widespread phenomenon in the real world. It occurs when two or more dynamical systems, under different initial conditions, interact with each other, causing their dynamical characteristics to converge. Researchers have primarily pursued two approaches to achieve synchronization in complex networks. One approach involves modifying the internal characteristics of complex networks, such as adjusting the network's coupling strength, optimizing the topology matrix, and adjusting the clustering coefficient. However, this control approach is complex to implement and lacks universal applicability, making it difficult to apply to similar complex networks. Another approach involves leveraging external control methods to achieve network synchronization. Specifically, based on the characteristics of complex networks, researchers have integrated multidisciplinary theoretical foundations such as control theory, nonlinear system dynamics, and matrix theory to design practical control strategies and high-performance controllers, thereby achieving synchronization in complex networks. This approach, with its relatively flexible control mechanism and strong universality, provides a more effective approach to solving the synchronization control problem in complex networks. It also opens up broad space for further research in this field and is of great significance for advancing the theory and technology of synchronization control in complex networks. Through the tireless efforts of researchers both domestically and internationally, the field of synchronization control has introduced numerous control methods to achieve synchronization in complex networks, achieving fruitful results and demonstrating broad application prospects in fields such as power systems, artificial intelligence, communications, healthcare, and aviation. However, many challenges remain to be addressed. Therefore, in-depth research on complex network synchronization is necessary, both from a theoretical and practical perspective, and holds significant theoretical and practical significance.

[0003] Synchronization in complex networks in the real world is often affected by multiple factors, and the states of the network and its nodes often change over time. Nodes are affected by time, including the hysteresis of their own dynamics and the time lags caused by the interaction and transmission between nodes. In addition to the influence of time, the parameters of the network node dynamics system also have a certain impact on the network's synchronization ability. In real complex networks, the parameters of the node dynamics system are often uncertain. To address node dynamics systems with unknown parameters, a driver-response network model is often used in research. The dynamics system parameters of the driver network are known, while the dynamics system of the response network contains unknown parameters. Furthermore, complex networks may be subject to interference from unknown factors, such as changes in the network's environment, unstable external input signals, or poorly understood dynamic mechanisms within the network. To effectively address these complex factors in practical research and applications, a comprehensive approach and strategy are needed to achieve synchronization control in complex networks.

[0004] In order to address the above problems, the present invention considers the influence of bounded interference and mixed time delay on the synchronization of complex networks with unknown parameters, and proposes a finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay.

[0005] The invention patent application number 202310707475.3 discloses a coupling synchronization control method and system for discrete memristor neural networks, including the following steps: Step S1: Establishing a discrete memristor neural network with random perturbations and mixed time delays; Step S2: Designing a pulse-based coupling synchronization controller based on the discrete memristor neural network with random perturbations and mixed time delays established in Step S1, and constructing a synchronization error system based on the designed coupling synchronization controller; Step S3: Selecting a corresponding Lyapunov function based on the synchronization error system constructed in Step S2 and combining it with the coupling synchronization controller to achieve coupling synchronization of the discrete memristor neural network; Step S4: Building a discrete memristor neural network model and using the discrete memristor neural network model to perform numerical simulation to verify the coupling synchronization effect between discrete memristor neural networks. The above invention has low control cost and high control accuracy. However, the random interference in the above invention cannot be directly applied to bounded interference.

[0006] In real-world complex networks, such as power grids, biological neural networks, and multi-robot systems, external disturbances and mixed time delays are an objective reality. However, many theoretical models overlook these key factors when constructing them, resulting in overly idealized models. Due to the lack of consideration for disturbances and time delays, theoretical models struggle to accurately characterize the dynamic characteristics of actual systems. In practical applications, the synchronization accuracy, speed, and robustness of such models often fall far short of theoretical expectations or design goals. Furthermore, the disturbances and unavoidable time delays present in real-world systems can easily undermine control effectiveness. This series of issues makes it difficult for many research results to be truly applied in complex network systems with disturbances and time delays in the real world. Summary of the Invention

[0007] To address the technical issues that the synchronization control of complex networks does not comprehensively consider bounded interference and mixed time delays, and the theoretical model has difficulty in accurately characterizing the dynamic characteristics of actual systems, the present invention proposes a finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delays, and proposes an adaptive control method and an unknown parameter update law to achieve finite-time synchronization of complex networks. By combining an adaptive control strategy with an unknown parameter update law, the present invention significantly improves synchronization accuracy and anti-disturbance capability, avoiding synchronization oscillations and even instability caused by ignoring time delays or interference in traditional methods. By designing an adaptive parameter update law, unknown node parameters in the network dynamic model are identified and corrected, maintaining synchronization robustness without relying on precise mathematical models.

[0008] In order to achieve the above object, the technical solution of the present invention is implemented as follows: a finite-time synchronization control method for a complex network with unknown parameters based on bounded interference and mixed time delay, the steps of which are as follows:

[0009] Step 1: Establish a driving network model and a response network model with unknown parameters based on bounded interference and mixed time delay;

[0010] Step 2: Determine the conditions for semi-global practical finite-time synchronization based on the driving network model and the response network model;

[0011] Step 3: According to the driving network model and the response network model established in step 1, the synchronization error of the driving network and the response network is set, and a synchronization error system is established;

[0012] Step 4: Design a finite-time state adaptive synchronization controller and an unknown parameter update law based on the synchronization error, apply the finite-time state adaptive synchronization controller and the unknown parameter update law to the response network, and satisfy the conditions of semi-global practical finite-time synchronization, so that the response network is synchronized with the driving network in a finite time.

[0013] Preferably, a time-varying and time-delayed complex dynamic network with N nodes is considered, where each node is a dynamic system, and the driving network model is:

[0014]

[0015] The response network model is:

[0016]

[0017] Among them, x i (t)=(x i1 (t),x i2 (t),…,x in (t)) T ∈R n is the state vector of the i-th node in the driving network at time t, n is the dimension of the node, x i1 (t),x i2 (t),…,x in (t) represents the state of each dimension for the i-th node in the driving network, Represents the state vector x i The derivative of (t), y i (t)=(y i1 (t),y i2 (t),…,y in (t)) T ∈R n is the state vector of the i-th node in the response network at time t, y i1 (t),y i2 (t),…,y in (t) represents the state of each dimension for the i-th node in the response network; Represents the state vector y i The derivative of (t); f(x i (t),x i (t-τ1)) and f(y i (t),y i (t-τ1)) are the function parts of the driving network and the response network that do not contain unknown parameters after separation, τ1 represents the node delay, and τ1 represents the coupling delay; g(x i (t))α i and They are the function parts of the driving network and the response network containing unknown parameters, α i is a known parameter in the node state equation driving the ith node of the network, is the unknown parameter in the node state equation of the i-th node in the response network, F(x i (t),x i (t-τ1),αi )=f(x i (t),x i (t-τ1))+g(x i (t))α i ,F(y i (t),y i (t-τ1),α i )=f(y i (t),y i (t-τ1))+g(y i (t))α i . F(x i (t),x i (t-τ1),α i ) and F(y i (t),y i (t-τ1),α i ) are all nonlinear smooth functions; E is the system matrix, W i (t) is the external interference component of the i-th node, and the external interference vector W(t)=(W1(t),W2(t),…,W N (t)) T , and satisfies ‖W(t)‖ 2 ≤δ, δ is a positive constant; u i (t) is the designed finite-time state adaptive synchronous controller, Γ=diag{γ1,γ2,…,γ n} is the internal coupling matrix, γ1,γ2,…,γ n Indicates the influence coefficient of the state variable on its own change, A=(a ij ) N×N ∈R n×n and B=(b ij ) N×N ∈R n×n They are the external coupling matrix of the network without time delay and the external coupling matrix with time delay.

[0018] Preferably, the external coupling matrix A and the external coupling matrix B satisfy the following condition: if node i is connected to node j, i≠j, then there is an element a ij ≠0, otherwise, a ij =0; if nodes i and j are connected, then b ij ≠0, otherwise, b ij =0; external coupling matrix A = (a ij ) N×N and B=(b ij ) N×N The following dissipation conditions are met:

[0019] Preferably, the semi-global practical finite-time synchronization is: if there exists a constant time t1>0 and a constant δ>0, so that the state vectors of the corresponding nodes of the response network and the driving network satisfy: The driving network and the response network are said to achieve semi-global practical finite-time synchronization.

[0020] Preferably, the synchronization error between the driving network and the response network is: i (t) = y i (t)-x i (t);

[0021] According to the driving network, response network and synchronization error, the synchronization error system is established as:

[0022]

[0023] in, Represents the node state vector in the error network.

[0024] Preferably, the synchronization error e is used i (t) Design the finite-time state adaptive synchronization controller as:

[0025]

[0026] Where i = 1, 2, ..., N, parameters k, p, η, λ i , μ is a positive constant, sign(e i (t)) is the synchronization error e i (t) is a sign function, and sign(e i (t))=(sign(e i1 (t)),sign(e i2 (t)),…,sign(e in (t))) T ,i=1,2,…,N,e i1 (t), e i2 (t), e in (t) represents the error value of different dimensions of the i-th node; M is the error value The upper bound of |||| represents the error value The 2-norm of |||| 2 represents the vector norm, T is the transpose operation, and the error value

[0027] Preferably, the unknown parameter update law is: λ i Is a positive number.

[0028] Preferably, the Lyapunov function is constructed according to the synchronization error as follows:

[0029]

[0030] Among them, η,λ i is a positive constant;

[0031] The derivative of the Lyapunov function V(t) satisfies the semi-global practical finite-time stability condition, and the condition for semi-global practical finite-time synchronization is obtained.

[0032] Preferably, the Lyapunov function is applied to the node state vector Derivative:

[0033]

[0034] Where I represents the unit matrix, p is a constant, is a non-zero constant;

[0035] The derivative of the Lyapunov function V(t) satisfies the conditions of semi-global practical finite-time stability, the synchronization error system is semi-global practical finite-time stable, and the driving network and the response network are finite-time synchronized.

[0036] Preferably, given any real-valued matrix G=(G ij ) N×N 、External coupling matrix A=(a ij ) n×N , B=(b ij ) n×N and the internal coupling matrix Γ=diag{γ1,γ2,…,γ i}, G′=diag{G 11 ,G 22 ,…,G NN}; with element a i , Represent the i-th column and j-th row of the external coupling matrix A respectively; i , Denote the i-th column and j-th row of the external coupling matrix B, respectively, and the following equation holds:

[0037]

[0038] Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention includes: establishing a complex dynamic network model of a nonlinear system with N identical nodes with unknown parameters under bounded interference and mixed time delay; establishing a desired trajectory system model; obtaining an error system model based on the complex dynamic network system model and the desired trajectory system model; and judging the synchronization state of the complex dynamic network model based on the convergence trend of the error system model; determining a controller based on the error system model; and achieving a stable desired trajectory for the complex dynamic network based on the semi-global practical finite-time stability theory. The method of the present invention can utilize adaptive control methods to solve the finite-time synchronization problem of complex networks, and can effectively save control resources, reduce control costs, and shorten stabilization time. The benefits of the present invention are specifically manifested in:

[0039] 1. This paper studies the characteristics of dynamic network models with mixed time delays and unknown parameters under bounded interference. By incorporating multiple realistic factors, it significantly improves the universality and practical fit.

[0040] 2. The present invention adopts a finite-time synchronization control method, which can shorten the stabilization time and improve the response speed and control efficiency of the system compared to the asymptotic synchronization control method. It has broad application prospects and important practical significance.

[0041] 3. Design an adaptive controller with an integral term, which can compensate and adjust the dynamic characteristics of the system in real time and accurately, and has stronger adaptability and robustness. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0043] Figure 1 Flowchart of the present invention.

[0044] Figure 2 This is a first error component diagram of the driving network and the response network under the action of the adaptive controller in Example 2 of the present invention.

[0045] Figure 3 This is a second error component diagram of the driving network and the response network under the action of the adaptive controller in Example 2 of the present invention.

[0046] Figure 4 This is a third error component diagram of the driving network and the response network under the action of the adaptive controller in Example 2 of the present invention.

[0047] Figure 5 This is an identification diagram of the unknown parameter a value in the response network under the action of the adaptive controller in Example 2 of the present invention.

[0048] Figure 6 This is an identification diagram of the unknown parameter b value in the response network under the action of the adaptive controller in Example 2 of the present invention.

[0049] Figure 7 This is an identification diagram of the unknown parameter c value in the response network under the action of the adaptive controller in Example 2 of the present invention. DETAILED DESCRIPTION

[0050] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without creative work are within the scope of protection of the present invention.

[0051] Example 1

[0052] like Figure 1 As shown, this embodiment provides a finite-time synchronization control method for a complex network with unknown parameters based on bounded interference and mixed time delay. The synchronization control method includes the following steps:

[0053] Step 1: Establish a driving network model and a response network model with unknown parameters based on bounded interference and mixed time delay.

[0054] In the study of complex network synchronization control, nonlinear complex dynamic network models are typically decomposed into a driver network model and a response network model, which together form a master-slave synchronization framework. Within this framework, the driver network model serves as an ideal reference source, defining the target synchronization trajectory of the system. The response network model, the actual controlled object, requires a controller to track the state of the driver network to achieve synchronization. These two models form a master-slave tracking relationship, with the goal of eliminating the state error between the response network and the driver network within a finite time.

[0055] Bounded disturbances refer to unpredictable disturbances of limited magnitude in the external environment (such as power load fluctuations and mechanical noise). These disturbances can contaminate node states, leading to decreased synchronization accuracy and even loss of synchronization. Hybrid delays, on the other hand, consist of node delays (internal computational delays at nodes) and coupling delays (signal transmission delays within the network topology). The combined effects of these two factors can distort information propagation, causing the controller to generate erroneous commands based on outdated data, leading to delayed synchronization convergence, oscillations, or even complete instability. These factors collectively undermine the effectiveness of theoretical control strategies and are a key obstacle to achieving robust finite-time synchronization in practical engineering.

[0056] The implementation method of step 1 is:

[0057] 1.1 Establish a driving network model based on bounded interference and mixed time delay: Consider the following complex dynamic network with N nodes and time delay, each node is a dynamic system.

[0058]

[0059] 1.2 Establish the unknown parameter response network model based on bounded interference and mixed time delay as follows:

[0060]

[0061] In the described driving network model and response network model, x i (t)=(x i1 (t),x i2 (t),…,x in (t)) T ∈R n is the state vector of the i-th node in the driving network at time t, n is the dimension of the node, x i1 (t),x i2 (t),…,x in (t) represents the state of each dimension for the i-th node in the driving network, Represents the state vector x i The derivative of (t), y i (t)=(y i1 (t),y i2 (t),…,y in (t)) T ∈R n is the state vector of the i-th node in the response network at time t, y i1 (t),y i2 (t),…,y in (t) represents the state of each dimension for the i-th node in the response network. Represents the state vector y i The derivative of (t). f(x i (t),x i (t-τ1)) and f(y i (t),y i (t-τ1)) are the function parts of the driving network and the response network that do not contain unknown parameters after separation, τ1 represents the node delay, and τ1 represents the coupling delay. g(x i (t))α i and They are the function parts of the driving network and the response network containing unknown parameters, α iis a known parameter in the node state equation driving the ith node of the network, is the unknown parameter in the node state equation of the i-th node in the response network, let F(x i (t),x i (t-τ1),α i )=f(x i (t),x i (t-τ1))+g(x i (t))α i ,F(y i (t),y i (t-τ1),α i )=f(y i (t),y i (t-τ1))+g(y i (t))α i . F(x i (t),x i (t-τ1),α i ) and F(y i (t),y i (t-τ1),α i ) are all nonlinear smooth functions. E is the system matrix, W i (y) is the external interference component of the i-th node, satisfying ‖W(t)‖ 2 ≤δ, δ is a positive constant, the external interference vector W(t)=(W1(t),W2(t),…,W N (t)) T ,u i (t) is the designed controller, Γ=diag{γ1,γ2,…,γ n} is the internal coupling matrix, γ1,γ2,…,γ n Indicates the coupling strength or influence coefficient of the state variable on its own change, A=(a ij ) N×N ∈R n ×n and B=(b ij ) N×N ∈R n×n They are the network's time-delay-free external coupling matrix and the time-delay external coupling matrix, which satisfy the following conditions: if nodes i and j (i≠j) are connected, then a ij ≠0, otherwise, a ij = 0. If nodes i and j (i≠j) are connected, then b ij ≠0, otherwise, b ij =0. A=(a ij ) N×N and B=(b ij ) N×NThe following dissipation conditions are met: N is the total number of nodes.

[0062] Step 2: Based on the unknown parameter driving network model and response network model with bounded interference and mixed time delay established in step 1, a semi-global practical finite-time synchronization definition is given:

[0063] If there are constants t1>0 and δ>0, so that the state vectors of the corresponding nodes of the response network and the driving network satisfy:

[0064]

[0065] The driving network and the response network are said to achieve semi-global practical finite-time synchronization. Where t1 represents time.

[0066] The synchronization problem between the driving network and the response network is transformed into an error stability problem between the two. If the error between the actual trajectory (the state of the node in the response network) and the expected trajectory (the state of the node in the driving network) meets this condition, it is said that the response network and the driving network achieve semi-global practical finite-time synchronization.

[0067] Step 3: According to the driving network model and response network model with unknown parameters based on bounded interference and mixed time delay established in step 1, the synchronization errors of the driving network and response network are set, and a synchronization error system is established.

[0068] The implementation method of step 3 is:

[0069] 3.1 Based on the unknown parameter driving network model and response network model based on bounded interference and mixed time delay established in step 1, the synchronization error of the driving network and response network is set to:

[0070] e i (t) = y i (t)-x i (t)

[0071] 3.2 Based on the driving network and response network, and the synchronization error set in step 3.1, a synchronization error network is established as follows:

[0072]

[0073] in, Represents the node state vector in the error network. Substitute into the derived V function.

[0074] Step 4: construct a structure based on the synchronization error set in step 3, and design a finite-time state adaptive synchronization controller and an unknown parameter update law. Apply the finite-time state adaptive synchronization controller and the unknown parameter update law to the response network so that the response network is synchronized with the drive network in a finite time.

[0075] The implementation method of step 4 is:

[0076] 4.1 Using the synchronization error e i (t) Design a finite-time state adaptive synchronous controller:

[0077]

[0078] Where i = 1, 2, ..., N, parameters k, p, η, λ i , μ is a positive constant, sign(e i (t)) is the synchronization error e i (t) is a sign function, and sign(e i (t))=(sign(e i1 (t)),sign(e i2 (t)),…,sign(e in (t))) T ,i=1,2,…,N,e i1 (t), e i2 (t), e in (t) represents the error value of different dimensions of the i-th node. M is The upper bound of |||| represents The 2-norm of The square of the absolute value of |||| 2 Represents the vector norm. T is the transposition operation, the error value

[0079] 4.2 Design unknown parameter update law:

[0080] 4.3 Based on the synchronization error set in step 3, construct the Lyapunov function V(t), and give its specific expression as follows:

[0081]

[0082] Among them, η,λ i Is a positive number.

[0083] It will be subsequently proved that the derivative of the above Lyapunov function V(t) satisfies the semi-global practical finite-time stability condition, and the semi-global practical finite-time synchronization condition is obtained.

[0084] 4.4 The finite time adaptive synchronization controller u i (t) and the designed unknown parameter update law Acting on the response network, the response network is synchronized with the drive network for a finite time.

[0085] Based on the theory of semi-global practical finite-time stability, the effectiveness of the synchronization method is proved. That is, based on the theory of semi-global practical finite-time stability, the complex dynamic network is led to a stable desired trajectory.

[0086] Example 2

[0087] A finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay is provided. This embodiment mainly includes two parts:

[0088] One is to theoretically prove the effectiveness of the finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay proposed in Example 1.

[0089] Secondly, numerical simulation is used to simulate and verify the synchronization performance of the driving network model and the response network model with unknown parameters based on bounded interference and mixed time delay in Example 1.

[0090] 1. Theoretical Proof

[0091] The following are the assumptions and lemmas used in the proof:

[0092] Assumption 1: For any x,y∈R n and t>0, there exists a positive number η>0, so that the vector function f has the following inequality

[0093]

[0094] Lemma 1: Consider the following nonlinear system V(x) is a smooth positive definite function that satisfies the following conditions:

[0095]

[0096] Among them, C>0,0<β<1,D>0, then this nonlinear system is semi-globally practical finite-time stable. It is the mathematical expression of nonlinear system, f(x) is the nonlinear function about x, is the derivative of V(x) with respect to x.

[0097] Lemma 2: Let x1, x2, …, x n ∈R nLet \(x\) be an arbitrary real vector and \(0 < q < 2\) be a real number. Then the following inequality holds:

[0098] \(\|x_1\| q +\|x_2\| q +\cdots+\|x n \| q \geq(\|x_1\| 2 +\|x_2\| 2 +\cdots+\|x n \| 2 ) q / 2 ,

[0099] When \(q = 1\), we have

[0100] \|x_1\|+\|x_2\|+\cdots+\|x n \|\geq(\|x_1\| 2 +\|x_2\| 2 +\cdots+\|x n \| 2 ) 1 / 2 .

[0101] Lemma 3: Suppose \(x\) and \(y\) are vectors with appropriate dimensions. Then the following inequality holds:

[0102] 2x T y\leq\varepsilon -1 x T M -1 x+\varepsilon y T My,

[0103] where \(\varepsilon\) is a scalar and \(M\in R n×n is a positive definite matrix.

[0104] Lemma 4: Given arbitrary real matrices \(G=(G ij ) N×N , A=(a ij ) n×N , B=(b ij ) n×N and diagonal matrix \(\Gamma = diag\{\gamma_1,\gamma_2,\cdots,\gamma n}\), \(G' = diag\{G 11 ,G 22 ,\cdots,G NN}\). Let \(a i , denote the \(i\)-th column and \(j\)-th row of matrix \(A\) respectively; let \(b i , denote the \(i\)-th column and \(j\)-th row of matrix \(B\) respectively. The following equation holds:

[0105]

[0106] Next, construct the Lyapunov function:

[0107]

[0108] The constructed Lyapunov function about the node state vector Derivative:

[0109]

[0110] Where I represents the unit matrix and p is a constant that is not zero. The first less than or equal to sign uses Assumption 1. The second less than or equal to sign uses Lemma The third one is less than or equal to as long as and From this we can get the sufficient condition for synchronization.

[0111] Therefore, according to Lemma 1, under the action of the finite-time adaptive synchronization controller and the unknown parameter update law, the response network synchronizes with the drive network in finite time. The synchronization problem between the drive network and the response network is transformed into the stability problem of the error network between the drive network and the response network. According to Lemma 1, as long as the derivative of the Lyapunov function V(t) satisfies the conditions, the error network can be said to be semi-globally practical finite-time stable, that is, the drive network and the response network are synchronized in finite time.

[0112] 2. Numerical Simulation

[0113] In this embodiment, a complex network model with five nodes and unknown parameters based on bounded interference and mixed time delay, where the node dynamics system is a Lorenz system, is taken as an example. The driving network is determined as follows:

[0114]

[0115] The response network is:

[0116]

[0117] Among them, t≥0, a=10, b=8 / 3, c=28, d=5, and there are 5 different trajectories in the figure representing the changes of 5 nodes over time. These are all uncertain parameters in the system.

[0118] The finite-time state adaptive synchronous controller is:

[0119]

[0120] The update law of unknown parameters is:

[0121] Here, represents the update law of the three uncertain parameters. i2 、y i1 、y i3 Indicates the value of the three dimensions of the ige node in the response network, e i1 (t), e i2 (t), e i3 (t) represents the error values ​​of the driving network and the response network in three dimensions.

[0122] The initial values ​​of the driving network and the response network are set as follows: where i = 1, 2, ... 5. In the numerical simulation, the following parameters are set: τ2 = 0.1, λ i =0.7, k=90.5, p=10, η=0.1, M=30, μ=1, sign function sign(e i (t)) is approximated by the tanh function; Γ is a 3×3 unit matrix. Select the initial condition: x i (0)=[4,3,1],y i (0)=[2+i,1+2*i,2*i],i=1,2,…5.

[0123] The topology matrix A, B and system matrix E are as follows:

[0124]

[0125] Numerical simulation experiments are carried out on the driving network and the response network. The specific simulation results are as follows: Figure 2 is a first error component diagram of the driving network and the response network under the action of the adaptive controller in specific embodiment 2 of the present invention; Figure 3 This is a second error component diagram of the driving network and the response network under the action of the adaptive controller in specific embodiment 2 of the present invention. Figure 4 This is a diagram of the third error component between the driving network and the response network under the action of the adaptive controller in specific embodiment 2 of the present invention. Figure 2-Figure 4 The trajectory shows that under the action of the adaptive controller, the error between the response network and the driving network fluctuates within a certain range, achieving synchronization of the response network with the driving network within a limited time, verifying the synchronization performance.

[0126] Figure 5 This is an identification diagram of the unknown parameter a value in the response network under the action of the adaptive controller in specific embodiment 2 of the present invention. Figure 6 This is an identification diagram of the unknown parameter b value in the response network under the action of the adaptive controller in specific embodiment 2 of the present invention. Figure 7This is an identification diagram of the unknown parameter c value in the response network under the action of the adaptive controller in specific embodiment 2 of the present invention. Figure 4-Figure 6 The trajectory shows that under the action of the unknown parameter update law, the response network can identify the true value of the unknown parameter, thus realizing the unknown parameter identification.

[0127] Finally, it should be noted that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art will be able to modify the technical solutions described in the aforementioned embodiments or substitute equivalents for some of the technical features. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention.

Claims

1. A finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay, characterized in that: The steps are as follows: Step 1: Establish a driving network model and a response network model with unknown parameters based on bounded interference and mixed time delay; Step 2: Determine the conditions for semi-global practical finite-time synchronization based on the driving network model and the response network model; Step 3: According to the driving network model and the response network model established in step 1, the synchronization error of the driving network and the response network is set, and a synchronization error system is established; Step 4: Design a finite-time state adaptive synchronization controller and an unknown parameter update law based on the synchronization error, apply the finite-time state adaptive synchronization controller and the unknown parameter update law to the response network, and satisfy the conditions of semi-global practical finite-time synchronization, so that the response network is synchronized with the driving network in a finite time.

2. The finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay according to claim 1 is characterized in that: Consider a time-varying and time-delayed complex dynamic network with N nodes, where each node is a dynamic system. The driving network model is: The response network model is: Among them, x i (t)=(x i1 (t),x i2 (t),…,x in (t)) T ∈R n is the state vector of the i-th node in the driving network at time t, n is the dimension of the node, x i1 (t),x i2 (t),…,x in (t) represents the state of each dimension for the i-th node in the driving network, Represents the state vector x i The derivative of (t), y i (t)=(y i1 (t),y i2 (t),…,y in (t)) T ∈R n is the state vector of the i-th node in the response network at time t, y i1 (t),y i2 (t),…,y in (t) represents the state of each dimension for the i-th node in the response network; Represents the state vector y i The derivative of (t); f(x i (t),x i (t-τ1)) and f(y i (t),y i (t-τ1)) are the function parts of the driving network and the response network that do not contain unknown parameters after separation, τ1 represents the node delay, and τ1 represents the coupling delay; g(x i (t))α i and They are the function parts of the driving network and the response network containing unknown parameters, α i is a known parameter in the node state equation driving the ith node of the network, is the unknown parameter in the node state equation of the i-th node in the response network, F(x i (t),x i (t-τ1),α i )=f(x i (t),x i (t-τ1))+g(x i (t))α i ,F(y i (t),y i (t-τ1),α i )=f(y i (t),y i (t-τ1))+g(y i (t))α i . F(x i (t),x i (t-τ1),α i ) and F(y i (t),y i (t-τ1),α i ) are all nonlinear smooth functions; E is the system matrix, W i (t) is the external interference component of the i-th node, and the external interference vector W(t)=(W1(t),W2(t),…,W N (t)) T , and satisfies ‖W(t)‖ 2 ≤δ, δ is a positive constant; u i (t) is the designed finite-time state adaptive synchronous controller, Γ=diag{γ1,γ2,…,γ n } is the internal coupling matrix, γ1,γ2,…,γ n Indicates the influence coefficient of the state variable on its own change, A=(a ij ) N×N ∈R n×n and B=(b ij ) N×N ∈R n×n They are the external coupling matrix of the network without time delay and the external coupling matrix with time delay.

3. The finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay according to claim 2 is characterized in that: The external coupling matrix A and the external coupling matrix B satisfy the following conditions: if node i is connected to node j, i≠j, then there is an element a ij ≠0, otherwise, a ij =0; If nodes i and j are connected, then b ij ≠0, otherwise, b ij =0; external coupling matrix A = (a ij ) N×N and B=(b ij ) N×N The following dissipation conditions are met:

4. The finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay according to claim 2 or 3, characterized in that: The semi-global practical finite-time synchronization is: if there exists a constant time t1>0 and a constant δ>0, so that the state vectors of the corresponding nodes of the response network and the driving network satisfy: The driving network and the response network are said to achieve semi-global practical finite-time synchronization.

5. The finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay according to claim 4 is characterized in that: The synchronization error between the driving network and the response network is: i (t) = y i (t)-x i (t); According to the driving network, response network and synchronization error, the synchronization error system is established as: in, Represents the node state vector in the error network.

6. The method for finite-time synchronization control of complex networks with unknown parameters based on bounded interference and mixed time delay according to claim 5 is characterized in that: Using the synchronization error e i (t) Design the finite-time state adaptive synchronization controller as: Where i = 1, 2, ..., N, parameters k, p, η, λ i , μ is a positive constant, sign(e i (t)) is the synchronization error e i (t) is a sign function, and sign(e i (t))=(sign(e i1 (t)),sign(e i2 (t)),…,sign(e in (t))) T ,i=1,2,…,N,e i1 (t), e i2 (t), e in (t) represents the error value of different dimensions of the i-th node; M is the error value The upper bound of || || represents the error value The 2-norm of |||| 2 represents the vector norm, T is the transpose operation, and the error value 7. The finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay according to claim 5 or 6, characterized in that: The unknown parameter update law is: λ i Is a positive number.

8. The finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay according to claim 7 is characterized in that: The Lyapunov function is constructed based on the synchronization error: Among them, η,λ i is a positive constant; The derivative of the Lyapunov function V(t) satisfies the semi-global practical finite-time stability condition, and the condition for semi-global practical finite-time synchronization is obtained.

9. The finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay according to claim 8 is characterized in that: For the Lyapunov function on the node state vector Derivative: Where I represents the unit matrix, p is a constant, is a non-zero constant; The derivative of the Lyapunov function V(t) satisfies the conditions of semi-global practical finite-time stability, the synchronization error system is semi-global practical finite-time stable, and the driving network and the response network are finite-time synchronized.

10. The finite-time synchronization control method for complex networks with unknown parameters based on bounded interference and mixed time delay according to claim 2, characterized in that: Given any real-valued matrix G = (G ij ) N×N 、External coupling matrix A=(a ij ) n×N , B=(b ij ) n×N and the internal coupling matrix Γ=diag{γ1,γ2,…,γ n }, G′=diag{G 11 ,G 22 ,…,G NN }; with element a i , Represent the i-th column and j-th row of the external coupling matrix A respectively; i , Denote the i-th column and j-th row of the external coupling matrix B, respectively, and the following equation holds:

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