Pure complex network containment control method with communication noise

Abstract

The invention provides a simple complex network containment control method with communication noise, and relates to the technical field of control and information. According to the method, a network dynamic model containing communication noise is constructed based on a pure complex network topology structure, and a robust containment control strategy is designed for the randomness and uncertainty of the communication noise. External control input is applied by selecting a small number of contained nodes, and the overall state of the network is guided to converge to an expected target state. According to the method, the stability of the network system can be effectively maintained in a noise interference environment, the control precision and the anti-interference capability are improved, and the method is widely applied to complex network systems such as multi-agent cooperation, unmanned system formation and distributed sensing networks.

Description

Technical Field

[0001] The present invention relates to the field of control and information technology, and particularly to a pinning control method for a simplicial complex network with communication noise. Background Art

[0002] Currently, complex network systems have been widely applied in fields such as communication, transportation, energy, and multi-agent cooperation. The information transmission between network nodes is inevitably interfered by communication noise in practical applications. The communication noise may come from factors such as environmental noise, sensor errors, or signal quantization errors. This kind of noise interference will cause the distortion of the transmission of network state information, affect the system control performance, and even lead to system instability. Therefore, studying the control method of complex networks in the communication noise environment has important theoretical research significance and engineering application value.

[0003] A simplicial complex network is a mathematical model that can describe the high-order interaction relationships between multiple nodes. Compared with the traditional graph network model, the simplicial complex network can more accurately depict the cooperation and competition relationships between multiple nodes. The pinning control method realizes the effective control of the entire network by applying a small amount of external control input to the pinned nodes. However, most of the existing pinning control methods assume an ideal communication environment and ignore the influence of communication noise. Therefore, designing a pinning control method that can still maintain the system stability and convergence under the interference of communication noise has important research value for improving the robustness and reliability of the simplicial complex network control system. Summary of the Invention

[0004] Aiming at the deficiencies of the prior art, the present invention provides a pinning control method for a simplicial complex network with communication noise; the present invention aims at a simplicial complex network system affected by communication noise described by a nonlinear stochastic differential equation, and realizes the synchronization goal by designing a pinning controller. Compared with the prior art, the present invention fully considers the high-order interaction of the actual network system and the influence of communication noise. The provided pinning control method can effectively regulate the simplicial complex network system affected by communication noise, improve the robustness of the system, reduce the control cost, and has a wider application range in practical applications.

[0005] A pinning control method for a simplicial complex network with communication noise includes the following steps:

[0006] Step 1: Construct a simplicial complex network system with communication noise consisting of N nodes;

[0007] Construct a simplicial complex network system with communication noise consisting of N nodes, where the dynamic equation of the i-th node is expressed as follows:

[0008]

[0009] Among them, i = 1, …, N represents the state vector of node i at time t, x im (t) represents the m-th component of the state vector of node i, represents the state vector of node j 1 , and the superscript T represents the transpose, represents the set of m-dimensional real vectors; is a continuously differentiable function describing the dynamics of the node itself, represents the set of positive real

[0010] numbers; H (d) , d = 1, …, D: represents the internal coupling function between the node state variables in the d-simplex, satisfying the condition H (d) (x, …, x) ≡ 0, where D represents the dimension of the simplicial complex network, and x represents the node state; the constant σ d > 0, d = 1, …, D represents the coupling strength; is defined as follows: when (i, j 1 , …, j d ) belongs to the d-simplex, otherwise b(t) is a one-dimensional Wiener process used to represent communication noise, and σ c represents the noise intensity;

[0011] Step 2: Design a pinning controller to achieve the synchronization goal for the simplicial complex network system with communication noise;

[0012] The pinning controller is designed as follows:

[0013] u i (t) = k i σ 1 H (1) (x i (t), x s (t)) (2)

[0014] where i = 1, …, N, the constant σ 1 > 0 represents the coupling strength, represents the coupling function between the node state variables, represents the set of positive real numbers; k i represents the control gain. When the control action is applied to node i, k i > 0, otherwise k i = 0; represents the synchronization state, and x sm (t) represents the m-th component of the synchronization state, and the superscript T represents the transpose, Denote the set of m-dimensional real vectors;

[0015] Control all nodes of the entire network to the synchronous state x s (t). The controlled simplicial complex network system is expressed as follows:

[0016]

[0017] where, i = 1, …, N represents the state vector of node i at time t, and x im (t) represents the m-th component of the state vector of node i, represents the state vector of node j 1 , and the superscript T represents the transpose, denotes the set of m-dimensional real vectors; is a continuously differentiable function describing the dynamics of the node itself, denotes the set of positive real numbers; H (d) , d = 1, …, represents the internal coupling function between the node state variables in the d-simplex, satisfying the condition H (d) (x, …, x) ≡ 0, where D represents the dimension of the simplicial complex network, and x represents the node state; the constant σ d > 0, d = 1, …, D represents the coupling strength; is defined as follows: when (i, j 1 , …, j d ) belongs to the d-simplex, otherwise b(t) is a one-dimensional Wiener process used to represent communication noise, and σ c represents the noise intensity; u i is the designed pinning controller;

[0018] Step 3: Analyze the stability criterion of the controlled simplicial complex network system with communication noise;

[0019] Step 3.1: Define the node error as e i (t) = x i (t) - x s (t), i = 1, …, N. Through linearization and dimension reduction, the modified error system is expressed as follows:

[0020] dξ i (t) = [J F - λ i J ω(1) ξ i (t)dt - σ c λ i J ω(1) ξi (t)db(t) (4)

[0021] where ξ i (t) represents the correction error of the i-th node, and J F is the Jacobian matrix of the function F(·) at the synchronous state; λ i is the matrix P D = σ 1 C + σ 2 L (2) + … + σ D L (D) The eigenvalues of, C = L (1) + K, K = diag{k 1 , …, k N} is the control gain matrix, and the symbol diag represents a diagonal matrix with diagonal elements k i , i = 1, …, N being the control gains; L (d) , d = 1, …, D being the generalized Laplacian matrix; σ d , d = 1, …, D representing the coupling strength; σ c is the noise intensity, b(t) is a one-dimensional Wiener process; J ω(1) is the Jacobian matrix of the function ω (1) at the synchronous state, and ω (1) satisfies the following conditions:

[0022] H (1) (x i (t), x j (t)) = ω (1) (x j (t)) - ω (1) (x i (t)) (5)

[0023] where H (1) represents the internal coupling function between the node state variables in the 1-simplex, and ω (1) represents the function related to H (1) ;

[0024] Let η = λ i , and the following extended master stability equation can be obtained:

[0025]

[0026] where y(t) is an auxiliary variable, and J F is the Jacobian matrix of the function F(·) at the synchronous state; λ i is the matrix P D = σ 1 C + σ 2 L (2)+…+σ D L (D) The eigenvalues of, C = L (1) +K, K = diag{k 1 ,…,k N} is the control gain matrix, and the symbol diag represents a diagonal matrix. The diagonal elements k i , i = 1,…, N are the control gains; L (d) , d = 1,…, D are the generalized Laplacian matrices; σ d , d = 1,…, D represents the coupling strength; σ c is the noise intensity, and b(t) is a one-dimensional Wiener process; J ω(1) is the Jacobian matrix of the function ω (1) at the synchronous state, and ω (1) satisfies condition (5);

[0027] Step 3.2: The controlled simplicial complex network system (3) with communication noise is locally exponentially stable in the stochastic sense if and only if the maximum Lyapunov exponent ∧ of the extended master stability equation (6) is less than 0, where the maximum Lyapunov exponent ∧ is a function related to η and σ c ;

[0028] The beneficial effects of adopting the above technical solutions are as follows:

[0029] A pinning control method for a simplicial complex network with communication noise provided by the present invention can, in an environment with communication noise interference, apply an external control input to the network system through pinning nodes to effectively guide the overall network state. This method fully considers the high-order interaction between network nodes and the randomness and uncertainty of communication noise, designs a robust pinning control strategy, and ensures that the simplicial complex network system can still converge to the desired target state under noise interference. Compared with traditional control methods, the present invention has stronger anti-interference ability and stability, can effectively improve the control accuracy and robustness of the network system, and is widely applicable to the practical applications of complex network systems such as multi-agent cooperation, unmanned system formation, and distributed sensor networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 is a flow chart of the pinning control method for a simplicial complex network with communication noise provided for the implementation of the present invention;

[0031] Figure 2 is a schematic diagram of the structure of the simplicial complex network in the implementation of the present invention;

[0032] Figure 3 is a graph showing the relationship between the maximum Lyapunov exponent and the parameter η under different communication noises in the implementation of the present invention;

[0033] Among them, (a) is the relationship diagram when relationship diagram;

[0034] Figure 4 is the comparison diagram of the present invention implementation without communication noise and with communication noise;

[0035] Among them, (a) is the comparison diagram of the relationship between the maximum Lyapunov exponent and the parameter η when the noise intensities are 0 and 0.5 respectively, (b) is the synchronization error diagram when the noise intensity is 0, and diagram (c) is the synchronization error diagram when the noise intensity is 0.5. Detailed implementation method

[0036] The following combines the drawings and embodiments to further describe in detail the detailed implementation method of the present invention. The following embodiments are used to illustrate the present invention, but are not used to limit the scope of the present invention.

[0037] A method for pinning control of a simplicial complex network with communication noise, as Figure 1 shown, includes the following steps:

[0038] Step 1: Construct a simplicial complex network system with communication noise composed of N nodes;

[0039] Construct a simplicial complex network system with communication noise composed of N nodes, where the dynamic equation of the i-th node is expressed as follows:

[0040]

[0041] Among them, i = 1,..., N represents the state vector of node i at time t, x im (t) represents the m-th component of the state vector of node i, represents the state vector of node j 1 , and the superscript T represents the transpose, represents the set of m-dimensional real vectors; is a continuously differentiable function describing the dynamics of the node itself, represents the set of positive real numbers; H (d) , d = 1,..., D: represents the internal coupling function between the node state variables in the d-simplex, satisfying the condition H (d) (x,..., x) ≡ 0, Among them, D represents the dimension of the simplicial complex network, and x represents the node state; the constant σ d > 0, d = 1,..., D represents the coupling strength; The definition of is as follows: when (i, j 1 ,..., jd ) When it belongs to a d - simplex, Otherwise b(t) is a one - dimensional Wiener process used to represent communication noise, and σ c represents the noise intensity;

[0042] For example Figure 2 As shown, in this embodiment, a network with three nodes is constructed, including 3 0 - simplices, 3 1 - simplices and 1 2 - simplex. All the above simplices form a simplicial complex, and this simplicial complex is a 2 - dimensional simplicial complex, that is, D = 2. The self - dynamics of the nodes is the Lorenz system, which is expressed as follows:

[0043]

[0044] Among them, a 1 = 10, a 2 = 8 / 3, a 3 = 28.

[0045] Step 2: Design a pinning controller to make the simplicial complex network system with communication noise achieve the synchronization goal;

[0046] The pinning controller is designed as follows:

[0047] u i (t)=k i σ 1 H (1) (x i (t),x s (t)) (2)

[0048] Among them, i = 1,…,N, the constant σ 1 > 0 represents the coupling strength, represents the coupling function between node state variables, represents the set of positive real numbers; k i represents the control gain. When the control action is applied to node i, k i > 0, otherwise k i = 0; represents the synchronization state, x sm (t) represents the m - th component of the synchronization state, and the superscript T represents the transpose, represents the set of m - dimensional real vectors;

[0049] All the nodes of the entire network are controlled to the synchronization state x s (t), and the controlled simplicial complex network system is expressed as follows:

[0050]

[0051] Among them, $i = 1,\ldots,N$ represents the state vector of node $i$ at time $t$, $x$ im $(t)$ represents the $m$-th component of the state vector of node $i$, represents the state vector of node $j$ 1 and the superscript $T$ represents the transpose, represents the set of $m$-dimensional real vectors; is a continuously differentiable function that describes the dynamics of the node itself, represents the set of positive real numbers; $H$ (d) , $d = 1,\ldots,$ represents the internal coupling function between the node state variables in the $d$-simplex, satisfying the condition $H$ (d) $(x,\ldots,x)\equiv0$, where $D$ represents the dimension of the simplicial complex network and $x$ represents the node state; the constant $\sigma$ d $> 0$, $d = 1,\ldots,D$ represents the coupling strength; is defined as follows: when $(i,j$ 1 ,\ldots,j$ d ) belongs to the $d$-simplex, otherwise $b(t)$ is a one-dimensional Wiener process used to represent communication noise, $\sigma$ c represents the noise intensity; $u$ i is the designed pinning controller; it should be noted that the pinning controller designed in the present invention only needs to apply a control action to a small number of nodes in the network, greatly reducing the control cost;

[0052] Step 3: Analyze the stability discrimination conditions of the controlled simplicial complex network system with communication noise;

[0053] Step 3.1: Define the node error as $e$ i $(t)=x$ i (t)-x$ s (t), $i = 1,\ldots,N$. Through linearization and dimension reduction, the modified error system can be expressed as follows:

[0054] $d\xi$ i (t)=[J$ F -\lambda$ i J$ ω(1) \xi$ i (t)dt-\sigma$ c $\lambda$ i J$ ω(1) $\xi$ i (t)db(t) (4)

[0055] where $\xi$ i (t) represents the modified error of the $i$-th node, $J$ Fis the Jacobian matrix of the function F(·) in the synchronous state; λ i is the matrix P D = σ 1 C + σ 2 L (2) + … + σ D L (D) is the eigenvalue of, C = L (1) + K, K = diag{k 1 , …, k N} is the control gain matrix, and the symbol diag represents a diagonal matrix, with the diagonal element k i , i = 1, …, N being the control gains; L (d) , d = 1, …, D being the generalized Laplacian matrix; σ d , d = 1, …, D representing the coupling strength; σ c is the noise intensity, and b(t) is a one-dimensional Wiener process; is the function ω (1) in the synchronous state of the Jacobian matrix, ω (1) satisfies the following conditions:

[0056] H (1) (x i (t), x j (t)) = ω (1) (x j (t)) - ω (1) (x i (t)) (5)

[0057] where, H (1) represents the internal coupling function between the node state variables in the 1-simplex, and ω (1) represents the function related to H (1) ;

[0058] Let η = λ i , and the following extended master stability equation can be obtained:

[0059]

[0060] where, y(t) is an auxiliary variable, and J F is the Jacobian matrix of the function F(·) in the synchronous state; λ i is the matrix P D = σ 1 C + σ 2 L (2) + … + σ D L (D) is the eigenvalue of, C = L (1) + K, K = diag{k 1 , …, k N} is the control gain matrix, and the symbol diag represents a diagonal matrix with diagonal elements \(k_{i}\), \(i = 1,\cdots,N\) being the control gains; \(L_{d}\), \(d = 1,\cdots,D\) are the generalized Laplacian matrices; \(\sigma_{d}\), \(d = 1,\cdots,D\) represent the coupling strengths; \(\sigma\) is the noise intensity, and \(b(t)\) is a one-dimensional Wiener process; i , \(i = 1,\cdots,N\) are the control gains; \(L_{d}\) (d) , \(d = 1,\cdots,D\) are the generalized Laplacian matrices; \(\sigma_{d}\) d , \(d = 1,\cdots,D\) represent the coupling strengths; \(\sigma_{d}\) c is the noise intensity, and \(b(t)\) is a one-dimensional Wiener process; is the Jacobian matrix of the function \(\omega\) (1) at the synchronous state, and \(\omega\) (1) satisfies condition (5);

[0061] Step 3.2: The controlled simplicial complex network system (3) with communication noise is locally exponentially stable in the stochastic sense if and only if the maximum Lyapunov exponent \(\Lambda\) of the extended master stability equation (6) is less than 0, where the maximum Lyapunov exponent \(\Lambda\) is a function related to \(\eta\) and \(\sigma_{d}\); c is related;

[0062] The advantage of the stability analysis method is that it transforms the stability problem of a high-dimensional (mN-dimensional) complex nonlinear system at the synchronous state into the stability problem of a low-dimensional (m-dimensional) simple linear error system at the origin, and the maximum Lyapunov exponent is computable and easy to operate in practice, making it more suitable for engineering applications.

[0063] To verify the effectiveness of the control method described in the present invention, in this example, first, the inner coupling function is set to obtain the relationship diagrams of the maximum Lyapunov exponent and the parameter \(\eta\) when the noise intensities are \(\sigma_{d}\) c = 0, 1, 2, 3, as shown in Figure 3 (a); secondly, the inner coupling function is set to obtain the relationship diagrams of the maximum Lyapunov exponent and the parameter \(\eta\) when the noise intensities are \(\sigma_{d}\) c = 0, 1, 2, 3, as shown in Figure 3 (b). It can be seen from the figures that the larger the noise intensity, the smaller the range where the maximum Lyapunov exponent is less than 0, and the more difficult it is to control. Figure 4 (a) is the relationship diagram of the maximum Lyapunov exponent and the parameter \(\eta\) when the inner coupling function is and the noise intensities are \(\sigma_{d}\) c = 0, 0.5. In the network shown in Figure 2 , the coupling strengths are \(\sigma_{d}\) 1 = 5, \(\sigma_{d}\) 2 = 10. One of the nodes is selected as the controlled node, and the control gain is taken as 6. The synchronization error diagrams when \(\sigma_{d}\) c = 0 and 0.5 are obtained respectively, as shown in Figure 4 (b) and 4(c). FromFigure 4 (c) It can be seen that under the action of the controller, the network system has achieved the synchronization goal, indicating that the pinning control method proposed in the present invention is effective.

[0064] The above description is only the preferred embodiment of the present disclosure and the explanation of the applied technical principles. Those skilled in the art should understand that the scope of the invention involved in the embodiments of the present disclosure is not limited to the technical solutions formed by the specific combination of the above technical features, but should also cover other technical solutions formed by any combination of the above technical features or their equivalent features without departing from the above inventive concept. For example, the technical solutions formed by mutually replacing the above features with the technical features (but not limited to) having similar functions disclosed in the embodiments of the present disclosure.

Claims

1. A simplex complex network pinning control method with communication noise, characterized in that: The following steps are involved: Step 1: Construct a simple complex network system consisting of N nodes with communication noise; Step 2: Design a pinning controller to enable the simplex complex network system with communication noise to achieve the synchronization goal; Step 3: Analyze the stability criteria of the controlled simplicial complex network system with communication noise.

2. A simplex complex network pinning control method with communication noise according to claim 1, characterized in that: The step 1 is specifically as follows: construct a simple complex network system with communication noise consisting of N nodes, wherein the dynamic equation of the i-th node is expressed as follows: in, represents the state vector of node i at time t, x im (t) represents the mth component of the state vector of node i, represents the state vector of node j1, and the superscript T represents transposition. represents a set of m-dimensional real vectors; F(·): is a continuously differentiable function describing the node's own dynamics, represents the set of positive real numbers; H (d) , d=1,…,D: represents the internal coupling function between node state variables in the d-simplex, satisfying the condition Where D represents the dimension of the simple complex network, x represents the node state, and the constant σ d >0, d=1,…, D represents the coupling strength; The definition of is as follows: when (i,j1,…,j d ) belongs to a d-simplex, otherwise b(t) is a one-dimensional Wiener process, used to represent communication noise, σ c Indicates the noise intensity.

3. The method for controlling a simplex complex network with communication noise according to claim 1, characterized in that: The control controller described in step 2 is designed as follows: u i (t)=k i σ1H (1) (x i (t),x s (t)) (2) Where i = 1, ..., N, the constant σ1>0 represents the coupling strength, H (1) : represents the coupling function between node state variables, represents the set of positive real numbers; k i represents the control gain. When the control action is applied to node i, k i > 0, otherwise k i =0; Indicates the synchronous state, x sm (t) represents the mth component of the synchronous state, and the superscript T represents the transposition. represents a set of m-dimensional real vectors; Control all nodes in the entire network to the synchronous state x s (t), the controlled simplicial complex network system is expressed as follows: in, represents the state vector of node i at time t, x im (t) represents the mth component of the state vector of node i, represents the state vector of node j1, and the superscript T represents transposition. represents a set of m-dimensional real vectors; F(·): is a continuously differentiable function describing the node's own dynamics, represents the set of positive real numbers; H (d) , d=1,…,D: represents the internal coupling function between node state variables in the d-simplex, satisfying the condition Where D represents the dimension of the simple complex network, x represents the node state, and the constant σ d >0, d=1,…, D represents the coupling strength; The definition of is as follows: when (i,j1,…,j d ) belongs to a d-simplex, otherwise b(t) is a one-dimensional Wiener process, used to represent communication noise, σ c Indicates the noise intensity; u i The designed containment controller.

4. The method for controlling a simplex complex network with communication noise according to claim 1, characterized in that: The step 3 specifically comprises the following steps: Step 3.1: Define the node error as e i (t) = x i (t)-x s (t), i = 1, ..., N, through linearization and dimensionality reduction, the corrected error system is expressed as follows: Among them, ξ i (t) represents the correction error of the i-th node, J F is the Jacobian matrix of the function F(·) in the synchronous state; λ i is the matrix P D =σ1C+σ2L (2) +…+σ D L (D) The characteristic root of C = L (1) +K,K=diag{k1,…,k N } is the control gain matrix, the symbol diag represents a diagonal matrix, and the diagonal element k i , i=1,…,N is the control gain; L (d) ,d=1,…,D is the generalized Laplacian matrix;σ d ,d=1,…,D represents the coupling strength; σ c is the noise intensity, b(t) is a one-dimensional Wiener process; is the function ω (1) The Jacobian matrix in the synchronous state, ω (1) The following conditions must be met: H (1) (x i (t),x j (t))=ω (1) (x j (t))-ω (1) (x i (t)) (5) Among them, H (1) represents the internal coupling function between node state variables in the 1-simplex, ω (1) Indicates and H (1) Related functions; Let η = λ i , we can get the following expanded main stability equation: Among them, y(t) is an auxiliary variable, J F is the Jacobian matrix of the function F(·) in the synchronous state; λ i is the matrix P D =σ1C+σ2L (2) +…+σ D L (D) The characteristic root of C = L (1) +K,K=diag{k1,…,k N } is the control gain matrix, the symbol diag represents a diagonal matrix, and the diagonal element k i , i=1,…,N is the control gain; L (d) ,d=1,…,D is the generalized Laplacian matrix;σ d ,d=1,…,D represents the coupling strength; σ c is the noise intensity, b(t) is a one-dimensional Wiener process; is the function ω (1) The Jacobian matrix in the synchronous state, ω (1) Satisfy condition (5); Step 3.2: The controlled simplicial complex network system (3) with communication noise is locally exponentially stable in the stochastic sense if and only if the maximum Lyapunov exponent ∧ of the extended master stability equation (6) is less than 0, where the maximum Lyapunov exponent ∧ is related to η and σ c Related functions.

Images