PI control method for nonlinear switching random complex network system

Abstract

The invention provides a PI control method for a nonlinear switching random complex network system, and relates to the technical field of control and information. For a switching complex network system described by a nonlinear stochastic differential equation, a synchronization target is realized by designing a PI control law, and a synchronization stability condition of the network system is analyzed based on a Lyapunov theory. Cooperative influence of node dynamics, an internal coupling matrix, global coupling strength, random noise, controller parameters and a network topology structure on system synchronization performance is researched from the perspective of theory and simulation. The nonlinear switching and randomness characteristics of an actual network system are fully considered, the provided PI control method can effectively regulate and control the nonlinear switching random complex network system, and the anti-interference capability of the system is improved. The method has high robustness and applicability, can be widely applied to the fields of communication, traffic, electric power, biology and the like, and provides innovative technical support and theoretical basis for improving system reliability and operation efficiency.

Description

Technical Field

[0001] The present invention relates to the field of control and information technology, and in particular to a PI control method for a nonlinear switching random complex network system. Background Art

[0002] With the development of modern science and technology, complex network theory has been widely used in many fields such as communication, transportation, electricity and biology. In practice, complex systems usually exhibit significant nonlinear characteristics, accompanied by random disturbances (such as environmental noise, signal fluctuations) and switching dynamics (caused by changes in topology or operating mode). This complexity makes the modeling, analysis and control of the system face huge challenges. In particular, how to achieve effective control of system stability and performance under the conditions of nonlinearity, random disturbances and dynamic switching is one of the core problems in the current complex network research.

[0003] PI control (proportional-integral control) is a classic control method that is widely used in various control systems due to its simplicity and efficiency. However, there are many difficulties in applying PI control to nonlinear switching random complex network systems: on the one hand, the switching characteristics and random disturbances of the network system make it difficult for traditional PI control to effectively maintain system stability; on the other hand, nonlinear characteristics and complex dynamic coupling between nodes increase the complexity of controller design. Therefore, there is an urgent need for a PI control method that adapts to the network switching characteristics and can handle random disturbances to provide a solution for improving the stability and performance of complex network systems. Summary of the invention

[0004] In view of the deficiencies of the prior art, the present invention provides a PI control method for a nonlinear switching random complex network system. The present invention aims at a switching complex network system described by a nonlinear stochastic differential equation, achieves the synchronization target by designing a reasonable and effective PI control law, analyzes the synchronization stability conditions of the network system based on the Lyapunov theory, and studies the synergistic effects of the node's own dynamics, internal coupling matrix, global coupling strength, random noise, controller parameters, and network topology on the synchronization performance of the entire system from a theoretical and simulation perspective. Compared with the prior art, the present invention fully considers the nonlinear switching and random characteristics of the actual network system, and the PI control method provided can achieve effective regulation of the nonlinear switching random complex network system, improve the anti-interference ability of the system, and has a wider range of applicability in practical applications.

[0005] A PI control method for a nonlinear switching random complex network system comprises the following steps:

[0006] Step 1: Construct a nonlinear switching random complex network system consisting of N nodes;

[0007] Construct a nonlinear switching random complex network system consisting of N nodes, where the dynamic equation of the i-th node is expressed as follows:

[0008]

[0009] Where i = 1, ..., N, N represents the number of nodes, represents the state vector of node i at time t, x in (t) represents the nth component of the state vector of node i, and the superscript T represents the transpose. represents a set of n-dimensional real vectors; the constant c>0 represents the global coupling strength, is the internal coupling matrix, where γ i >0,i=1,…,N, Represents a set of n×n dimensional real matrices; Laplacian matrix Represents the external coupling relationship between network nodes, represents a set of N×N dimensional real matrices, where the matrix element l ij The definition is as follows: If there is an edge connecting nodes i and j, then l ij = l ji = -1; otherwise, l ij = l ji =0(i≠j); In addition, the diagonal elements are defined as Satisfy the dissipative coupling condition Here we consider a connected unweighted undirected network, that is, L is a symmetric semi-positive definite matrix with one and only one zero characteristic root; and are respectively continuous differentiable nonlinear functions representing the node's own dynamics. The random variable δ(t) is used to describe the random events in the system. δ(t) = 1 means that the node's own dynamics function in system (1) is F(·), δ(t) = 0 means that the node's own dynamics function in system (1) is H(·), and δ(t) satisfies the following Bernoulli distribution:

[0010]

[0011] Among them, the symbol Prob represents probability, and the symbol represents the expected value, δ 0 ∈[0,1] represents the probability value of the nonlinear function F(·); g(x i (t))db(t) represents the noise added to the i-th node, where b(t) is the one-dimensional Brownian motion. is the noise transfer function, which is used to describe the diffusion mode of Brownian motion at each node;

[0012] Step 2: Design a PI controller to enable the nonlinear switched random complex network system to achieve the synchronization goal;

[0013] make is the left eigenvector corresponding to the zero eigenroot of the Laplacian matrix L, satisfying the condition η N represents the Nth component of η, represents an N-dimensional column vector whose elements are all 1, represents a set of N-dimensional real vectors, and the superscript T represents transposition; let where x i (t) represents the state vector of node i, then the node error vector is defined as

[0014] The PI controller is designed as follows:

[0015]

[0016] Where c represents the global coupling strength, Γ is the internal coupling matrix, and l ij is the element of the Laplacian matrix; K P and K I Represent the proportional control gain and integral control gain respectively, represents the integral, represents the error vector of the jth node;

[0017] Control all nodes in the entire network to a synchronous state Then the controlled nonlinear switching random complex network system is expressed as follows:

[0018]

[0019] Among them, x i (t) represents the state vector of node i at time t, c represents the global coupling strength, Γ is the internal coupling matrix, l ij are the elements of the Laplacian matrix L; F(·) and H(·) are nonlinear functions representing the node's own dynamics, and the random variable δ(t) is used to describe the random events in the system; b(t) is the one-dimensional Brownian motion, g(·) is the noise transfer function; K P and K I Represent the proportional control gain and integral control gain respectively, represents the integral, represents the error vector of the jth node;

[0020] Step 3: Based on Lyapunov stability theory, analyze the synchronization stability conditions of nonlinear switched random complex network systems;

[0021] Step 3.1: According to the defined node errors, the error system corresponding to system (4) is obtained through analysis:

[0022]

[0023] in, represents the integral, represents the error vector of the jth node;

[0024] represents the state vector of the jth node; the superscript T represents transposition, represents a set of nN-dimensional real vectors, represents the Kronecker product, I N and I n represent the N-dimensional and n-dimensional unit matrices respectively, represents an N-dimensional column vector whose elements are all 1; c represents the global coupling strength, Γ is the internal coupling matrix, L is the Laplacian matrix, and the random variable δ(t) is used to describe the random events in the system; K P and K I Respectively represent the proportional control gain and the integral control gain;

[0025] Step 3.2: Based on the Lyapunov stability theory, the parameter conditions of the PI controller that enables the network system to achieve synchronization are obtained; Assumption 1: The functions F(·) and H(·) both satisfy the Lipschitz condition, that is, there are constants p>0 and q>0, respectively, such that

[0026] ‖F(x 1 (t))-F(x 2 (t))‖≤p‖x 1 (t)-x 2 (t)‖ (6)

[0027] ‖H(x 1 (t))-H(x 2 (t))‖≤q‖x 1 (t)-x 2 (t)‖ (7)

[0028] Assumption 2: There exists a constant matrix Make

[0029]

[0030] Here, the symbol tr represents trace, ‖·‖ represents 2-norm, and superscript T represents transpose;

[0031] Under the condition that assumptions 1 and 2 are met, the following inequality holds true, and the nonlinear switched random complex network system achieves the global synchronization goal under the designed PI controller:

[0032]

[0033]

[0034] Where c represents the global coupling strength, γ = min{γ 1 ,…,γ n}, γ i , i=1,…,n are the diagonal elements of the inner coupling matrix Γ, the symbol min represents the minimum value, λ 2 is the smallest non-zero eigenvalue of the Laplacian matrix L, δ 0 ∈[0,1] represents the probability value, p and q are the Lipschitz coefficients in hypothesis 1, is the constant matrix in Assumption 2; and I n represent the N-dimensional and n-dimensional unit matrices respectively, represents an N-dimensional column vector whose elements are all 1, and η is the left eigenvector corresponding to the zero eigenroot of the Laplacian matrix L.

[0035] The beneficial effects of adopting the above technical solution are:

[0036] The present invention provides a PI control method for a nonlinear switching random complex network system, which can effectively solve the problem of dynamic behavior control of a nonlinear switching random complex network system, and has important theoretical significance and practical application value. By combining nonlinear dynamics, random interference characteristics and system switching laws, a reasonable and effective PI control law is designed to achieve the global synchronization goal. The method can adapt to the dynamic changes of network nodes and the influence of random disturbances, and effectively improve the stability and response performance of the system. At the same time, the present invention has strong robustness and applicability, and can be widely used in multiple fields such as communications, transportation, electricity, biology, etc., to provide innovative technical support and theoretical basis for improving system reliability and operation efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 A flow chart of a PI control method for a nonlinear switching random complex network system provided for the implementation of the present invention;

[0038] Figure 2 A synchronization error diagram of a nonlinear switched random complex network system controlled by PI in the implementation of the present invention;

[0039] Figure (a) shows the synchronization error diagram of the first component of the node state, Figure (b) shows the synchronization error diagram of the second component of the node state, and Figure (c) shows the synchronization error diagram of the third component of the node state. DETAILED DESCRIPTION

[0040] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and examples. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0041] A PI control method for nonlinear switched random complex network systems, such as Figure 1 As shown, the following steps are included:

[0042] Step 1: Construct a nonlinear switching random complex network system consisting of N nodes;

[0043] Construct a nonlinear switching random complex network system consisting of N nodes, where the dynamic equation of the i-th node is expressed as follows:

[0044]

[0045] Where i = 1, ..., N, N represents the number of nodes, represents the state vector of node i at time t, x in (t) represents the nth component of the state vector of node i, and the superscript T represents the transpose. represents a set of n-dimensional real vectors; the constant c>0 represents the global coupling strength, is the internal coupling matrix, where γ i >0,i=1,…,N, Represents a set of n×n dimensional real matrices; Laplacian matrix Represents the external coupling relationship between network nodes, represents a set of N×N dimensional real matrices, where the matrix element l ij The definition is as follows: If there is an edge connecting nodes i and j, then l ij = l ji = -1; otherwise, l ij = l ji =0(i≠j); In addition, the diagonal elements are defined as Satisfy the dissipative coupling condition Here we consider a connected unweighted undirected network, that is, L is a symmetric semi-positive definite matrix with one and only one zero characteristic root; and are respectively continuous differentiable nonlinear functions representing the node's own dynamics. The random variable δ(t) is used to describe the random events in the system. δ(t) = 1 means that the node's own dynamics function in system (1) is F(·), δ(t) = 0 means that the node's own dynamics function in system (1) is H(·), and δ(t) satisfies the following Bernoulli distribution:

[0046]

[0047] Among them, the symbol Prob represents probability, and the symbol represents the expected value, δ 0 ∈[0,1] represents the probability value of the nonlinear function F(·); g(x i (t))db(t) represents the noise added to the i-th node, where b(t) is the one-dimensional Brownian motion. is the noise transfer function, which is used to describe the diffusion mode of Brownian motion at each node;

[0048] In this embodiment, the node's own dynamics functions F(·) and H(·) are respectively selected from system and Lorenz system, that is, n = 3; other parameters are selected as follows: δ 0 =0.5, c=0.1, Γ=I 3 ; Select an unweighted undirected nearest neighbor coupling network with 20 nodes and node degree 4, that is, N = 20; the noise transfer function g(x i (t)) = x i (t), the noise is Brownian motion with mean 0 and variance 0.01;

[0049] Step 2: Design a PI controller to enable the nonlinear switched random complex network system to achieve the synchronization goal;

[0050] make is the left eigenvector corresponding to the zero eigenroot of the Laplacian matrix L, satisfying the condition η N represents the Nth component of η, represents an N-dimensional column vector whose elements are all 1, represents a set of N-dimensional real vectors, and the superscript T represents transposition; let where x i (t) represents the state vector of node i, then the node error vector is defined as

[0051] The PI controller is designed as follows:

[0052]

[0053] Where c represents the global coupling strength, Γ is the internal coupling matrix, and l ij is the element of the Laplacian matrix; K P and K I Represent the proportional control gain and integral control gain respectively, represents the integral, represents the error vector of the jth node;

[0054] Control all nodes in the entire network to a synchronous state Then the controlled nonlinear switching random complex network system is expressed as follows:

[0055]

[0056]

[0057] Among them, x i (t) represents the state vector of node i at time t, c represents the global coupling strength, Γ is the internal coupling matrix, l ij are the elements of the Laplacian matrix L; F(·) and H(·) are nonlinear functions representing the node's own dynamics, and the random variable δ(t) is used to describe the random events in the system; b(t) is the one-dimensional Brownian motion, g(·) is the noise transfer function; K P and K I Represent the proportional control gain and integral control gain respectively, represents the integral, represents the error vector of the jth node;

[0058] Step 3: Based on Lyapunov stability theory, analyze the synchronization stability conditions of nonlinear switched random complex network systems;

[0059] Step 3.1: According to the defined node errors, the error system corresponding to system (4) is obtained through analysis:

[0060]

[0061] in, represents the integral, represents the error vector of the jth node;

[0062] represents the state vector of the jth node; the superscript T represents transposition, represents a set of nN-dimensional real vectors, represents the Kronecker product, I N and I nDenote the identity matrices of dimensions \(N\) and \(n\) respectively. denotes an \(N -\)dimensional column vector with all elements equal to 1; \(c\) represents the global coupling strength, \(\Gamma\) is the internal coupling matrix, \(L\) is the Laplacian matrix, and the random variable \(\delta(t)\) is used to describe random events in the system; \(K\) P and \(K\) I represent the proportional control gain and the integral control gain respectively.

[0063] Step 3.2: Based on the Lyapunov stability theory, obtain the parameter conditions of the PI controller for the network system to achieve synchronization.

[0064] Assumption 1: The functions \(F(\cdot)\) and \(H(\cdot)\) both satisfy the Lipschitz condition, that is, there exist constants \(p>0\) and \(q>0\) respectively, such that

[0065] \(\|F(x 1 (t)) - F(x 2 (t))\| \leq p\|x 1 (t)-x 2 (t)\|(6)

[0066] \|H(x 1 (t)) - H(x 2 (t))\| \leq q\|x 1 (t)-x 2 (t)\|(7)

[0067] Assumption 2: There exists a constant matrix such that

[0068]

[0069] where the symbol \(tr\) represents the trace, \(\|\cdot\|\) represents the 2 - norm, and the superscript \(T\) represents the transpose.

[0070] When Assumptions 1 and 2 are satisfied and the following inequality holds, the nonlinear switched stochastic complex network system achieves the global synchronization goal under the designed PI controller:

[0071]

[0072] where \(c\) represents the global coupling strength, \(\gamma=\min\{\gamma 1 ,\cdots,\gamma n \}, \gamma i , i = 1,\cdots,n\) are the diagonal elements of the internal coupling matrix \(\Gamma\), the symbol \(\min\) represents taking the minimum value, \(\lambda 2 is the smallest non - zero eigenvalue of the Laplacian matrix \(L\), \(\delta 0 \in[0,1]\) represents the probability value, and \(p\) and \(q\) are the Lipschitz coefficients in Assumption 1. is the constant matrix in Assumption 2; and I n represent the N-dimensional and n-dimensional unit matrices respectively, represents an N-dimensional column vector whose elements are all 1, and η is the left eigenvector corresponding to the zero eigenroot of the Laplacian matrix L.

[0073] In this embodiment, the parameter value that satisfies assumptions 1 and 2 is p=q=5. λ 2 =0.4799,λ max (M T M)=1; the design proportional control gain and integral control gain are K P =21, K I =7, through Matlab simulation, the synchronization error diagram of the controlled network system can be obtained as follows Figure 2 As shown, Figure 2 (a) Figure 2 (b) and Figure 2 (c) represents the synchronization error diagram of the first component of the node state, the second component of the node state, and the third component of the node state. The vertical axis parameter represents the synchronization error from the first component to the third component of the node state. It can be seen from the figure that all node errors eventually tend to 0 over time; it can be seen that the network system achieves the global synchronization goal under the action of the designed PI controller, verifying the effectiveness of the control method proposed in the present invention.

[0074] The above description is only a preferred embodiment of the present disclosure and an explanation of the technical principles used. 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 a specific combination of the above-mentioned technical features, but should also cover other technical solutions formed by any combination of the above-mentioned technical features or their equivalent features without departing from the above-mentioned inventive concept. For example, the above-mentioned features are replaced with the technical features with similar functions disclosed in the embodiments of the present disclosure (but not limited to) to form a technical solution.

Claims

1. A PI control method for a nonlinear switching random complex network system, characterized in that: The following steps are involved: Step 1: Construct a nonlinear switching random complex network system consisting of N nodes; Construct a nonlinear switching random complex network system consisting of N nodes, where the dynamic equation of the i-th node is expressed as follows: Where i = 1, ..., N, N represents the number of nodes, represents the state vector of node i at time t, x in (t) represents the nth component of the state vector of node i, and the superscript T represents the transpose. represents a set of n-dimensional real vectors; the constant c>0 represents the global coupling strength, is the internal coupling matrix, where γ i >0,i=1,…,N, Represents a set of n×n dimensional real matrices; Laplacian matrix Represents the external coupling relationship between network nodes, represents a set of N×N dimensional real matrices, where the matrix element l ij The definition is as follows: If there is an edge connecting nodes i and j, then l ij = l ji = -1; otherwise, l ij = l ji =0(i≠j); In addition, the diagonal elements are defined as Satisfy the dissipative coupling condition Here we consider a connected unweighted undirected network, that is, L is a symmetric semi-positive definite matrix with one and only one zero characteristic root; and are respectively continuous differentiable nonlinear functions representing the node's own dynamics. The random variable δ(t) is used to describe the random events in the system. δ(t) = 1 means that the node's own dynamics function in system (1) is F(·), δ(t) = 0 means that the node's own dynamics function in system (1) is H(·), and δ(t) satisfies the following Bernoulli distribution: Among them, the symbol Prob represents probability, and the symbol represents the expected value, δ0∈[0,1] represents the probability value of the nonlinear function F(·); g(x i (t))db(t) represents the noise added to the i-th node, where b(t) is the one-dimensional Brownian motion. is the noise transfer function, which is used to describe the diffusion mode of Brownian motion at each node; Step 2: Design a PI controller to enable the nonlinear switched random complex network system to achieve the synchronization goal; Step 3: Based on Lyapunov stability theory, analyze the synchronization stability conditions of nonlinear switched random complex network systems.

2. The PI control method for a nonlinear switching stochastic complex network system according to claim 1, characterized in that: The step 2 is specifically as follows: is the left eigenvector corresponding to the zero eigenroot of the Laplacian matrix L, satisfying the condition η N represents the Nth component of η, represents an N-dimensional column vector whose elements are all 1, represents a set of N-dimensional real vectors, and the superscript T represents transposition; let where x i (t) represents the state vector of node i, then the node error vector is defined as The PI controller is designed as follows: Where c represents the global coupling strength, Γ is the internal coupling matrix, and l ij is the element of the Laplacian matrix; K P and K I Represent the proportional control gain and integral control gain respectively, represents the integral, ζ j (t) represents the error vector of the jth node; Control all nodes in the entire network to a synchronous state Then the controlled nonlinear switching random complex network system is expressed as follows: Among them, x i (t) represents the state vector of node i at time t, c represents the global coupling strength, Γ is the internal coupling matrix, l ij are the elements of the Laplacian matrix L; F(·) and H(·) are nonlinear functions representing the node's own dynamics, and the random variable δ(t) is used to describe the random events in the system; b(t) is the one-dimensional Brownian motion, g(·) is the noise transfer function; K P and K I Represent the proportional control gain and integral control gain respectively, represents the integral, ζ j (t) represents the error vector of the jth node.

3. The PI control method for a nonlinear switching stochastic complex network system according to claim 1, characterized in that: The step 3 specifically comprises the following steps: Step 3.1: According to the defined node errors, the error system corresponding to system (4) is obtained through analysis: in, represents the integral, ζ j (t), j = 1, ..., N represents the error vector of the jth node; x j (t), j = 1, ..., N represents the state vector of the jth node; the superscript T represents the transposition, represents a set of nN-dimensional real vectors, represents the Kronecker product, I N and I n represent the N-dimensional and n-dimensional unit matrices respectively, represents an N-dimensional column vector whose elements are all 1; c represents the global coupling strength, Γ is the internal coupling matrix, L is the Laplacian matrix, and the random variable δ(t) is used to describe the random events in the system; K P and K I Respectively represent the proportional control gain and the integral control gain; Step 3.2: Based on Lyapunov stability theory, obtain the parameter conditions of the PI controller that enables the network system to achieve synchronization.

4. The PI control method for a nonlinear switching random complex network system according to claim 3, characterized in that: The step 3.2 specifically includes the following steps: Assumption 1: Both functions F(·) and H(·) satisfy the Lipschitz condition, that is, there exist constants p>0 and q>0, respectively, such that ‖F(x1(t))-F(x2(t))‖≤p‖x1(t)-x2(t)‖ (6) ‖H(x1(t))-H(x2(t))‖≤q‖x1(t)-x2(t)‖ (7) Assumption 2: There exists a constant matrix Make Here, the symbol tr represents trace, ‖·‖ represents 2-norm, and superscript T represents transpose; Under the condition that assumptions 1 and 2 are met, the following inequality holds true, and the nonlinear switched random complex network system achieves the global synchronization goal under the designed PI controller: Where c represents the global coupling strength, γ = min{γ1,…,γ n }, γ i , i=1,…,n are the diagonal elements of the inner coupling matrix Γ, the symbol min indicates the minimum value, λ2 is the smallest non-zero eigenvalue of the Laplacian matrix L, δ0∈[0,1] indicates the probability value, p and q are the Lipschitz coefficients in hypothesis 1, is the constant matrix in Assumption 2; I N and I n represent the N-dimensional and n-dimensional unit matrices respectively, represents an N-dimensional column vector whose elements are all 1, and η is the left eigenvector corresponding to the zero eigenroot of the Laplacian matrix L.

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