A pi control method for a nonlinear switching stochastic complex network system

Abstract

The application provides a PI control method for a nonlinear switching random complex network system, and relates to the technical field of control and information. The application aims at a switching complex network system described by a nonlinear random differential equation, realizes a synchronization target through the design of a PI control law, analyzes the synchronization stability condition of the network system based on Lyapunov theory, and studies the synergistic influence of the node self-dynamics, the inner coupling matrix, the global coupling strength, the random noise, the controller parameter and the network topology on the system synchronization performance from the aspects of theory and simulation. The application fully considers the nonlinear switching and randomness characteristics of the actual network system, and the provided PI control method can realize the effective regulation and control of the nonlinear switching random complex network system, and improves the anti-interference ability of the system. The application has strong robustness and applicability, can be widely applied to the fields of communication, transportation, power and biology, and provides innovative technical support and theoretical basis for improving the system reliability and operation efficiency.

Description

TECHNICAL FIELD

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

[0002] With the development of modern science and technology, complex network theory has been widely applied in communication, transportation, power and biology and other fields. The actual complex system usually shows significant nonlinear characteristics, accompanied by random disturbance (such as environmental noise, signal fluctuation) and switching dynamics (caused by changes in topology structure or operation mode). This complexity makes the modeling, analysis and control of the system face great challenges, especially how to realize effective control of system stability and performance under the coexistence of nonlinearity, random disturbance and dynamic switching is one of the core problems in current complex network research.

[0003] PI control (proportional-integral control) as a classical control method, due to its simplicity and high efficiency, is widely used in various control systems. However, the application of PI control to nonlinear switching random complex network systems faces many difficulties: on the one hand, the switching characteristics of network systems and random disturbances make it difficult for traditional PI control to effectively maintain system stability; on the other hand, the 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 switching characteristics of the network and can handle random disturbances, providing a solution for the stability and performance improvement of complex network systems. SUMMARY

[0004] In view of the shortcomings of the prior art, the present application provides a PI control method for a nonlinear switching random complex network system. The present application is aimed at a switching complex network system described by a nonlinear stochastic differential equation, and realizes the synchronization goal by designing a reasonable and effective PI control law. Based on Lyapunov theory, the synchronization stability conditions of the network system are analyzed, and the synergistic effects of node dynamics, internal coupling matrix, global coupling strength, random noise, controller parameters and network topology on the synchronization performance of the entire system are studied from the theoretical and simulation perspectives. Compared with the prior art, the present application fully considers the nonlinear switching and randomness characteristics of the actual network system, and the PI control method provided can realize effective regulation and control of the nonlinear switching random complex network system, improve the anti-interference ability of the system, and has a wider range of application in practical applications.

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

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

[0007] Construct a nonlinear switching stochastic 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, and N represents the number of nodes. Let x represent the state vector of node i at time t. in (t) represents the nth component of the state vector of node i, and the superscript T indicates transpose. Represents a set of n-dimensional real vectors; the constant c > 0 represents the global coupling strength. Let γ be the inner coupling matrix, where γ i >0, i=1,…,N, Represents the set of n×n dimensional real matrices; Laplacian matrix This indicates the external coupling relationship between network nodes. Let l represent the set of N×N dimensional real matrices, where l is the matrix element. ij Defined 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); Furthermore, diagonal elements are defined as Satisfying the dissipative coupling condition Here we consider a connected, unweighted, undirected network, where L is a symmetric positive semi-definite matrix with one and only one zero eigenvalue. and Let F(t) be a continuously differentiable nonlinear function representing the dynamics of the nodes themselves, and let δ(t) be a random variable used to describe random events in the system. Let δ(t) = 1 to indicate that the dynamics of the nodes in system (1) is F(·), and let δ(t) = 0 to indicate that the dynamics of the nodes in system (1) is H(·). Let δ(t) satisfy the following Bernoulli distribution:

[0010]

[0011] Wherein, the symbol Prob represents probability, and the symbol Let δ0∈[0,1] represent the expected value, and let g(x) represent the probability of the nonlinear function F(·). i (t))db(t) represents the noise added to the i-th node, where b(t) is a one-dimensional Brownian motion. This is the noise transfer function, used to describe how Brownian motion spreads at each node;

[0012] Step 2: Design a PI controller to achieve the synchronization goal of the nonlinear switching stochastic complex network system.

[0013] Let be the left eigenvector corresponding to the zero eigenvalue of the Laplacian matrix L, satisfying the condition η N denotes the Nth component of η, denotes an N-dimensional column vector with all elements being 1, denotes an N-dimensional real vector set, and the superscript T denotes the transpose; let where x i (t) denotes the state vector of node i, the node error vector is defined as

[0014] The PI controller is designed as follows:

[0015]

[0016] where c denotes the global coupling strength, Γ is the inner coupling matrix, l ij is the element of the Laplacian matrix; K P and K I denote the proportional control gain and the integral control gain, respectively, denotes the integral, denotes the error vector of the jth node;

[0017] Control all nodes of the entire network to the synchronization state The controlled nonlinear switching stochastic complex network system is represented as follows:

[0018]

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

[0020] Step 3: Based on Lyapunov stability theory, analyze the synchronization stability conditions of the nonlinear switching stochastic complex network system.

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

[0022]

[0023] where, denotes integral, denotes the error vector of the jth node; denotes the state vector of the jth node; the superscript T denotes transpose, denotes an nN-dimensional real vector set, denotes the Kronecker product, I N and I n denote the N-dimensional and n-dimensional identity matrices, respectively, denotes an N-dimensional column vector with all elements being 1; c denotes 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 event in the system; K P and K I denote the proportional control gain and the integral control gain, respectively;

[0024] Step 3.2: Based on 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, i.e., there exist constants p > 0 and q > 0 such that

[0025] ‖F(x1(t))-F(x2(t))‖≤p‖x1(t)-x2(t)‖ (6)

[0026] ‖H(x1(t))-H(x2(t))‖≤q‖x1(t)-x2(t)‖ (7)

[0027] Assumption 2: There exists a constant matrix such that

[0028]

[0029] where the symbol tr denotes trace, ‖·‖ denotes 2-norm, and the superscript T denotes transpose;

[0030] Under the conditions of satisfying Assumptions 1 and 2, the following inequality holds, and then the nonlinear switching stochastic complex network system achieves the global synchronization goal under the designed PI controller:

[0031]

[0032]

[0033] Where c represents the global coupling strength, and γ = min{γ1,…,γ} n}, γ i Let i = 1, ..., n be the diagonal elements of the inner coupling matrix Γ, min denotes the minimum value, λ2 is the smallest non-zero eigenvalue of the Laplacian matrix L, δ0 ∈ [0, 1] represents the probability value, and p and q are the Lipschitz coefficients in Hypothesis 1. This is the constant matrix in Assumption 2; and I n Let them represent the N-dimensional and n-dimensional identity matrices, respectively. Let η represent an N-dimensional column vector with all elements equal to 1, and let η be the left eigenvector corresponding to the zero eigenvalue of the Laplacian matrix L.

[0034] The beneficial effects of adopting the above technical solution are as follows:

[0035] This invention provides a PI control method for nonlinear switched stochastic complex network systems, which effectively solves the problem of dynamic behavior control in such systems and has significant theoretical and practical value. By combining nonlinear dynamics, random disturbance characteristics, and system switching laws, a reasonable and effective PI control law is designed to achieve global synchronization. This method can adapt to dynamic changes in network nodes and the influence of random disturbances, effectively improving the system's stability and response performance. Furthermore, this invention has strong robustness and applicability, and can be widely applied in multiple fields such as communications, transportation, power, and biology, providing innovative technical support and theoretical basis for improving system reliability and operational efficiency. Attached Figure Description

[0036] Figure 1 A flowchart of a PI control method for a nonlinear switched stochastic complex network system provided for the implementation of this invention;

[0037] Figure 2 This is a synchronization error diagram of a nonlinear switching stochastic complex network system controlled by PI in an embodiment of the present invention;

[0038] Figure (a) shows the synchronization error of the first component of the node state, Figure (b) shows the synchronization error of the second component of the node state, and Figure (c) shows the synchronization error of the third component of the node state. Detailed Implementation

[0039] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0040] A PI control method for nonlinear switched stochastic complex network systems, such as Figure 1As shown, comprising the following steps:

[0041] Step 1: constructing a nonlinear switching stochastic complex network system composed of N nodes;

[0042] Constructing a nonlinear switching stochastic complex network system composed of N nodes, wherein the dynamic equation of the i-th node is expressed as follows:

[0043]

[0044] Wherein, i = 1, …, N, N represents the number of nodes, x (t) represents the state vector of node i at time t, x in (t) represents the n-th component of the state vector of node i, and the superscript T represents transposition, x (t) represents the n-dimensional real vector set; constant c > 0 represents the global coupling strength, is the inner coupling matrix, wherein γ i > 0, i = 1, …, N, L represents an n × n-dimensional real matrix set; the Laplacian matrix represents the outer coupling relationship between network nodes, represents an N × N-dimensional real matrix set, and the matrix element l ij is defined as follows: if there is an edge connected between node i and node j, then l ij = l ji = -1; otherwise, l ij = l ji = 0 (i ≠ j); in addition, the diagonal element is defined as satisfying the dissipative coupling condition Here, a connected unweighted undirected network is considered, that is, L is a symmetric semi-positive definite matrix, and there is only one zero eigenvalue; and are respectively continuous differentiable nonlinear functions representing the node self-dynamics, and the random variable δ (t) is used to describe the random event in the system, δ (t) = 1 represents that the node self-dynamics function in system (1) is F (·), δ (t) = 0 represents that the node self-dynamics function in system (1) is H (·), and δ (t) satisfies the following Bernoulli distribution:

[0045]

[0046] Wherein, the symbol Prob represents probability, and the symbol represents expected value, and δ0 ∈ [0, 1] represents the probability value of the nonlinear function F (·) ; g (x i(t))db(t) represents the noise added to the ith node, where b(t) is a one-dimensional Brownian motion, is the noise transfer function, which describes the diffusion of Brownian motion at each node;

[0047] In this embodiment, the self-dynamics functions F(·) and H(·) of the nodes are respectively selected as the system and the Lorenz system, i.e., n = 3; other parameters are selected as follows: δ0= 0.5, c = 0.1, Γ = I3; a weightless and undirected nearest-neighbor coupling network containing 20 nodes and having a node degree of 4 is selected, i.e., N = 20; the noise transfer function g(x i (t)) = x i (t), and the noise is a Brownian motion with a mean of 0 and a variance of 0.01;

[0048] Step 2: Design a PI controller to enable the nonlinear switching random complex network system to achieve the synchronization target;

[0049] Let be the left eigenvector corresponding to the zero eigenvalue of the Laplacian matrix L, satisfying the condition η N denotes the Nth component of η, denotes an N-dimensional column vector with all elements being 1, denotes an N-dimensional real vector set, and the superscript T represents transposition; let where x i (t) represents the state vector of the node i, the node error vector is defined as

[0050] The PI controller is designed as follows:

[0051]

[0052] where c represents the global coupling strength, Γ is the inner coupling matrix, l ij is an element of the Laplacian matrix; K P and K I respectively represent the proportional control gain and the integral control gain, denotes integration, denotes the error vector of the jth node;

[0053] all nodes of the entire network are controlled to the synchronization state The controlled nonlinear switching random complex network system is represented as follows:

[0054]

[0055] where x i(t) denotes the state vector of node i at time t, c denotes the global coupling strength, Γ is the inner-coupling matrix, L is the Laplacian matrix, and ij is an element of the Laplacian matrix L; F(·) and H(·) are nonlinear functions representing the node's own dynamics, δ(t) is a random variable used to describe random events in the system; b(t) is a one-dimensional Brownian motion, g(·) is a noise transfer function; K P and K I denote the proportional control gain and the integral control gain, respectively, denotes the integral, denotes the error vector of the jth node;

[0056] Step 3: Based on Lyapunov stability theory, analyze the synchronization stability condition of the nonlinear switching stochastic complex network system;

[0057] Step 3.1: According to the defined node error, by analysis, the error system corresponding to system (4) is:

[0058]

[0059] where, denotes the integral, denotes the error vector of the jth node; denotes the state vector of the jth node; the superscript T denotes the transpose, denotes an nN-dimensional real vector set, denotes the Kronecker product, I N and I n denote the N-dimensional and n-dimensional identity matrices, respectively, denotes an N-dimensional column vector with all elements being 1; c denotes the global coupling strength, Γ is the inner-coupling matrix, L is the Laplacian matrix, and δ(t) is a random variable used to describe random events in the system; K P and K I denote the proportional control gain and the integral control gain, respectively;

[0060] Step 3.2: Based on Lyapunov stability theory, obtain the parameter condition of the PI controller that makes the network system achieve synchronization;

[0061] Assumption 1: The functions F(·) and H(·) both satisfy the Lipschitz condition, i.e., there exist constants p > 0 and q > 0 such that

[0062] ‖F(x1(t))-F(x2(t))‖≤p‖x1(t)-x2(t)‖ (6)

[0063] ‖H(x1(t))-H(x2(t))‖≤q‖x1(t)-x2(t)‖ (7)

[0064] Assumption 2: There exists a constant matrix. Make

[0065]

[0066] Where the symbol tr represents the trace, ‖·‖ represents the 2-norm, and the superscript T represents the transpose;

[0067] Under the conditions of Assumptions 1 and 2, the following inequality holds, and the nonlinear switching stochastic complex network system achieves the global synchronization objective under the designed PI controller:

[0068]

[0069] Where c represents the global coupling strength, and γ = min{γ1,…,γ} n}, γ i Let i = 1, ..., n be the diagonal elements of the inner coupling matrix Γ, min denotes the minimum value, λ2 is the smallest non-zero eigenvalue of the Laplacian matrix L, δ0 ∈ [0, 1] represents the probability value, and p and q are the Lipschitz coefficients in Hypothesis 1. This is the constant matrix in Assumption 2; and I n Let them represent the N-dimensional and n-dimensional identity matrices, respectively. Let η represent an N-dimensional column vector with all elements equal to 1, and let η be the left eigenvector corresponding to the zero eigenvalue of the Laplacian matrix L.

[0070] In this embodiment, the parameter values ​​that satisfy assumptions 1 and 2 are p = q = 5. λ² = 0.4799, λ max (M T M) = 1; the design proportional control gain and integral control gain are K and K respectively. P =21, K I =7, and the synchronization error diagram of the controlled network system can be obtained through Matlab simulation as follows: Figure 2 As shown, where Figure 2 (a) Figure 2 (b) and Figure 2 (c) The diagrams show the synchronization errors of the first, second, and third components of the node state, respectively. The vertical axis parameter represents the synchronization error from the first to the third component of the node state. As shown in the diagram, the errors of all nodes eventually tend to 0 over time. This indicates that the network system has achieved the global synchronization target under the action of the designed PI controller, verifying the effectiveness of the control method proposed in this invention.

[0071] The above description is merely that of the preferred embodiments of the present disclosure and the principles of the technology employed, and it is understood that the inventive scope of the embodiments of the present disclosure are not limited to the technical solutions formed by the specific combinations of the above technical features. It should also be understood that the inventive scope of the embodiments of the present disclosure includes other technical solutions formed by the combinations of the above technical features or equivalent features thereof without departing from the above inventive concept. For example, technical solutions formed by replacing the above features with the technical features with similar functions disclosed in the embodiments of the present disclosure (but not limited to) with each other.

Claims

1. A PI control method of a nonlinear switching stochastic complex network system, characterized in that, The method comprises the following steps: Step 1: constructing a nonlinear switched stochastic complex network system composed of N nodes; Constructing a nonlinear switched stochastic complex network system composed of N nodes, wherein the dynamic equation of the i-th node is expressed as follows: Where i = 1, ..., N, and N represents the number of nodes. Let x represent the state vector of node i at time t. in (t) represents the nth component of the state vector of node i, and the superscript T indicates transpose. Represents a set of n-dimensional real vectors; the constant c > 0 represents the global coupling strength. Let γ be the inner coupling matrix, where γ i >0, i=1,…,N, Represents the set of n×n dimensional real matrices; Laplacian matrix This indicates the external coupling relationship between network nodes. Let l represent the set of N×N dimensional real matrices, where l is the matrix element. ij Defined 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; furthermore, diagonal elements are defined as i = 1, ..., N, satisfying the dissipative coupling condition Here we consider a connected, unweighted, undirected network, where L is a symmetric positive semi-definite matrix with one and only one zero eigenvalue. and Let F(t) be a continuously differentiable nonlinear function representing the dynamics of the nodes themselves, and let δ(t) be a random variable used to describe random events in the system. Let δ(t) = 1 to indicate that the dynamics of the nodes in system (1) is F(·), and let δ(t) = 0 to indicate that the dynamics of the nodes in system (1) is H(·). Let δ(t) satisfy the following Bernoulli distribution: wherein the symbol Prob denotes a probability, the symbol denotes an expected value, δ0∈[0, 1] denotes a probability value of the occurrence of the non-linear function F(·); g(x i (t))db(t) denotes a noise added to the i-th node, wherein b(t) is a one-dimensional Brownian motion, is a noise transfer function, which is used to describe the diffusion manner of the Brownian motion at each node; Step 2: designing a PI controller to enable the nonlinear switched stochastic complex network system to achieve a synchronization target; Step 3: analyzing the synchronization stability condition of the nonlinear switched stochastic complex network system based on Lyapunov stability theory.

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

3. The PI control method of 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 error, the error system corresponding to the system (4) is obtained by analysis as follows: where denotes the integral, ζ j (t),j = 1,..., N denotes the error vector of the jth node; x j (t),j = 1,..., N denotes the state vector of the jth node; the superscript T denotes the transpose, denotes the set of nN-dimensional real vectors, denotes the Kronecker product, I N and I n denote the N-dimensional and n-dimensional identity matrices, respectively, denotes the N-dimensional column vector with all elements equal to 1; c denotes 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 denote the proportional and integral control gains, respectively; Step 3.2: based on Lyapunov stability theory, the parameter condition of the PI controller enabling the network system to achieve synchronization is obtained.

4. The PI control method of a nonlinear switching stochastic complex network system according to claim 3, characterized in that, The step 3.2 specifically comprises the following steps: Assumption 1: the functions F(·) and H(·) both satisfy the Lipschitz condition, that is, there exist constants p>0 and q>0, 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 such that Wherein, the symbol tr represents trace, ‖·‖ represents 2-norm, and the superscript T represents transpose; Under the conditions of satisfying the assumptions 1 and 2, if the following inequality exists, then the nonlinear switched stochastic complex network system achieves the global synchronization target under the designed PI controller: Where c represents the global coupling strength, and γ = min{γ1,…,γ} n }, γ i Let i = 1, ..., n be the diagonal elements of the inner coupling matrix Γ, min denotes the minimum value, λ2 is the smallest non-zero eigenvalue of the Laplacian matrix L, δ0 ∈ [0, 1] represents the probability value, and p and q are the Lipschitz coefficients in Hypothesis 1. This is the constant matrix in Assumption 2; I N and I n Let them represent the N-dimensional and n-dimensional identity matrices, respectively. Let η represent an N-dimensional column vector with all elements equal to 1, and let η be the left eigenvector corresponding to the zero eigenvalue of the Laplacian matrix L.

Images