Analysis method of T-S fuzzy partial coupling network under control of containment pulse

Abstract

The invention discloses an analysis method of a T-S fuzzy partial coupling network under the control of a contained pulse, and belongs to the field of kinetic analysis. Firstly, a T-S fuzzy system is added to a part of the coupling network, and then a containment pulse controller is added to the network model to ensure that the network achieves global consistent stability. Secondly, constructing a comparison system through a recombination method and a pulse control principle; finally, the stability process of the network error model is analyzed through a step function method, the analysis process can be divided into two parts, the first part is a single-step step function, and the second part is a multi-step step function. According to the method, the monotonicity and continuity of the Lyapunov function do not need to be met, and the existing Lyapunov stability method is greatly expanded. Under the analysis of a step function, the method finally provides some new criteria to ensure the stability of the T-S fuzzy partial coupling network under the control of the containment pulse.

Description

Technical Field

[0001] The present invention belongs to the field of dynamic analysis, and specifically relates to the problems of the stability of T-S fuzzy systems, partially coupled network systems, impulsive control systems, and step function hypothesis testing networks. Background Art

[0002] In complex networks, the connections between nodes usually require coupling to transmit information. In the face of external interference, only some nodes can complete communication, which forms the problem of partially coupled networks. Therefore, it is very necessary to analyze the stability of partially coupled complex networks. In a partially coupled network, the interaction between nodes usually presents a non-linear relationship. This means that the coupling relationship between nodes cannot be simply described by linear weighted sum superposition, but may involve more complex non-linear function relationships. Therefore, the T-S fuzzy model can be used to more conveniently analyze the dynamic characteristics of partially coupled networks. Therefore, the T-S fuzzy partially coupled network has the advantage of solving the non-linear problem of nodes in the network and has more practical applications in the network.

[0003] In traditional stability analysis methods, the Lyapunov function must satisfy monotonicity and continuity within the impulsive interval, which usually leads to conservative stability conditions. However, in actual networks, there will inevitably be more chaotic complex network systems, or the network does not have the two properties of monotonicity and continuity within the impulsive interval. Therefore, in order to study the stability problems of a wider range of hybrid systems, the proposed step function method is more general and greatly expands the existing Lyapunov stability method. The present invention aims at the stability problem of the T-S fuzzy partially coupled network. Combining the impulsive control theory, a comparison system of the T-S fuzzy partially coupled network under pinning impulse is constructed by adding pinning impulses to the T-S fuzzy partially coupled network and selecting a reorganization method. Then, the single-step step function and multi-step step function in the step function are used to analyze the stability of the network. Different from the previous analysis methods, the step function method does not require whether the network satisfies monotonicity or continuity within the impulsive interval, thus solving the problems of a wider range of hybrid systems and finally obtaining a stability criterion. Summary of the Invention

[0004] The objective of the present invention is to address the stability issue of a T-S fuzzy partially coupled network under pinning pulse control. Combining with the traditional Lyapunov function method, an analysis method for a T-S fuzzy partially coupled network under pinning pulse control is proposed. Specifically, by adding a T-S fuzzy system to the partially coupled network and then incorporating a pinning pulse controller into the network model, the global uniform stability of the network is ensured. Secondly, a comparison system is constructed through the recombination method and the pulse control principle. Finally, the step function method is used to analyze the stability of the network error model. The specific analysis process can be divided into two parts. The first part is the single-step step function, and the second part is the multi-step step function. This method does not require the monotonicity and continuity of the Lyapunov function, greatly expanding the existing Lyapunov stability method.

[0005] The present invention specifically includes the following steps:

[0006] ① Establish a T-S fuzzy partially coupled network model

[0007] Add a T-S fuzzy system to the partially coupled network, and process the non-linear nodes in the network through the T-S fuzzy system to obtain a T-S fuzzy partially coupled network, which is specifically described as follows:

[0008] Rule τ: If α 1 (t) is is Then

[0009]

[0010] where, Φ i (t) = (Φ i1 (t), Φ i2 (t), …, Φ in (t)) T and Ψ i (t) = (Ψ i1 (t), Ψ i2 (t), …, Ψ in (t)) T represent the state vectors of the i-th node belonging to R n ; A τ and B τ are matrices belonging to R n×n ; F(·) is a non-linear function that satisfies R n →R n , and F(0) = 0; θ > 0 represents the coupling strength; D = diag{d 1 , d 2 , …, d n} is the inner coupling matrix; Q = q ij represents belonging to R N×NThe external coupling matrix, where N defines the number of nodes on the complex network under consideration. If there is an edge from node j to node i (j ≠ i), then q ij > 0; otherwise, q ij = 0; is the channel matrix, where is defined as follows: If the s-th layer channel of the edge from node j to node i is active, then Otherwise, is the controller to be designed. α(t) = (α 1 (t), α 2 (t), …, α μ (t)) T is the premise vector of the α j variable, represents the fuzzy set. In addition,

[0011]

[0012] where, Ω τ (α(t)) > 0 and is expressed as the level of the membership function, and there is

[0013] ② Design the pinning pulse controller

[0014]

[0015] where, σ(·) represents the Dirac function, g ι ∈ (0, 1) represents the pulse control gain, t ι represents the pulse instant sequence satisfying the condition 0 = t 0 < t 1 < t 2 < … < t ι < …, and lim ι→∞ t ι = +∞. φ ι is the number of controlled nodes.

[0016] ③ Method for analyzing the stability of the T - S fuzzy partially coupled network under pinning pulse control

[0017] A. Assume that there exists a positive definite function θ: R n → R + , then the step function Λ(t) can be defined as

[0018]

[0019] such that it satisfies

[0020] 1) For a function ∈ belonging to the ξ class, Λ(t 1 ) ≤ ∈(θ(ei (t 0 )));

[0021] 2) Λ(t) decreases with time t, and lim t→∞ Λ(t) = 0.

[0022] Then the origin of the network model can be classified as globally uniformly attractive stable (GUAS). Additionally, when t ∞ = ∞, the origin of the network model is classified as globally attractive stable (GAS).

[0023] Furthermore, the T-S fuzzy partially coupled network under pinning pulse control can be analyzed by a single-step step function: Let where is the largest eigenvalue of the matrix , is the largest eigenvalue of the matrix , If there exists a positive definite function θ: R n → R + , then the following conditions can all be satisfied:

[0024]

[0025] B. Assume that there exists a positive definite function θ: R n → R + , and there exists an integer Then the step function Λ(t) can be defined as

[0026]

[0027] where ι ≥ 1 and it satisfies

[0028] 1) For a function ∈ belonging to the ξ class, Λ(t 1 ) ≤ ∈(θ(e i (t 0 )));

[0029] 2) Λ(t) decreases with time t, and lim t→∞ Λ(t) = 0.

[0030] Then the origin of the network model can be classified as globally uniformly attractive stable (GUAS). Additionally, when t ∞ = ∞, the origin of the network model is classified as globally attractive stable (GAS).

[0031] Furthermore, the T-S fuzzy partially coupled network under pinning pulse control can be analyzed by a multi-step step function: Let where is the largest eigenvalue of the matrix ​ is the largest eigenvalue of the matrix . If there exists a positive definite function θ: R → R n → R + , then the following conditions can all be satisfied:

[0032]

[0033] The present invention aims at the stability problem of the T-S fuzzy partial coupled network under pinning pulse control. Under the analysis of the step function, through simple single-step step function analysis, the criteria that the network needs to meet to achieve stability are obtained, and further through multi-step step function analysis, the criteria that the network needs to meet to achieve stability are obtained, so that the research on the stability requirements of the network does not need to meet the traditional analysis method of Lyapunov function.

[0034] The present invention mainly uses the method of simulation experiment for verification, and all steps and conclusions are verified correctly on MATLAB. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 is the flowchart of the work of the present invention.

[0036] Figure 2 is the trajectory diagram of the T-S fuzzy partial coupled network in the simulation of the present invention.

[0037] Figure 3 is the trajectory diagram of the T-S fuzzy partial coupled network under pinning pulse control in the simulation of the present invention.

[0038] Figure 4 is the trajectory diagram of the T-S fuzzy partial coupled network under pinning pulse control under the analysis of single-step step function and multi-step step function in the simulation of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0039] The present invention will be further described below in conjunction with embodiments, but it should not be understood that the above-mentioned subject scope of the present invention is limited to the following embodiments. Without departing from the above-mentioned technical idea of the present invention, various substitutions and changes made according to the common general knowledge and conventional means in the art should all be included within the protection scope of the present invention.

[0040] Embodiment 1:

[0041] This embodiment discloses an analysis method for a T-S fuzzy partial coupled network under pinning pulse control, including the following steps:

[0042] ① Establish a T-S fuzzy partial coupled network model

[0043] Add a T-S fuzzy system to the partial coupling network. Process the non-linear nodes in the network through the T-S fuzzy system to obtain a T-S fuzzy partial coupling network, which is specifically described as follows:

[0044] Rule τ: If α 1 (t) is is Then

[0045]

[0046] where, Φ i (t) = (Φ i1 (t), Φ i2 (t), …, Φ in (t)) T and Ψ i (t) = (Ψ i1 (t), Ψ i2 (t), …, Ψ in (t)) T represent the state vectors of the i-th node belonging to R n ; A τ and B τ are matrices belonging to R n×n ; F(·) is a non-linear function, satisfying R n →R n , and F(0) = 0; θ > 0 represents the coupling strength; D = diag{d 1 , d 2 , …, d n} is the internal coupling matrix; Q = q ij represents the external coupling matrix belonging to R N×N . N defines the number of nodes on the considered complex network. If there is an edge from node j to node i (j ≠ i), then q ij > 0; otherwise, q ij = 0; is the channel matrix, where, is defined as follows: If the s-th layer channel of the edge from node j to node i is active, then Otherwise, u i (t) is the controller to be designed. α(t) = (α 1 (t), α 2 (t), …, α μ (t)) T is the premise vector of the α j variable, represents the fuzzy set. In addition,

[0047]

[0048] Among them, Ω τ (α(t)) > 0 and represented as the level of the membership function, there is h τ (α(t)) ≥ 0, τ = 1, 2, …, r. Further, in the dynamic networks (1) and (2), to solve the complexity increased due to the introduction of the channel matrix and will use the recombination method to handle the constraints within the channels, making and Therefore, it can be obtained that

[0049]

[0050] ② Design the pinning pulse controller

[0051]

[0052] Among them, σ(·) represents the Dirac function, g ι ∈(0, 1) represents the pulse control gain, t ι represents that the pulse instant sequence satisfies the condition 0 = t 0 < t 1 < t 2 < … < t ι < …, and lim ι→∞ t ι = +∞. φ ι is the number of control nodes. Then, after introducing the pinning pulse controller (4) into the nodes in Υ ι , using the property of σ(·), the expression of the network with pulse control is:

[0053]

[0054] Secondly, define e i (t) = Ψ i (t) - Φ i (t), then the corresponding error system is obtained from equation (4):

[0055]

[0056] Among them, in the formula Therefore, for i = 1, 2, …, N, when and only when t → ∞, ||e i (t)|| → 0, the drive - response network (6) can reach stability. The exponential set Υ ι is described as: the re - ordered e 1 (t ι ), e 2 (t ι ), …, e N(t ι ) At the pulse moment t ι , there is Then and Υ ι = φ ι .

[0057] Finally, define The error system can be described as:

[0058]

[0059] Therefore, for equation (7), the left - continuous system is discontinuous at t = t ι (ι ∈ N).

[0060] ③ Method for analyzing the stability of the T - S fuzzy partially - coupled network under pinning pulse control

[0061] A. Assume that there exists a positive - definite function θ: R n → R + , then the step function Λ(t) can be defined as

[0062]

[0063] such that it satisfies

[0064] 1) For a function ∈ belonging to the ξ class, Λ(t 1 ) ≤ ∈(θ(e i (t 0 )));

[0065] 2) Λ(t) decreases with time t, and lim t→∞ Λ(t) = 0.

[0066] Then the origin of (7) can be classified as globally uniformly attractive stable (GUAS). Additionally, when t ∞ = ∞, then the origin of (7) is classified as globally attractive stable (GAS).

[0067] Furthermore, the T - S fuzzy partially - coupled network under pinning pulse control can be analyzed by a single - step step function: Let where is the largest eigenvalue of the matrix , is the largest eigenvalue of the matrix , If there exists a positive - definite function θ: R n → R + , then the following conditions can all be satisfied:

[0068]

[0069] Proof: Define the following function:

[0070]

[0071] The corresponding step function is as follows:

[0072]

[0073] For \(t\in(t ι ,t ι+1 \), by calculating the derivative of \(\theta(e i (t))\) with respect to \(t\) along the trajectory of system (7), we can obtain

[0074]

[0075]

[0076] Expand and derive the following inequality:

[0077]

[0078] where In addition, there is

[0079]

[0080] Let Therefore, there is

[0081]

[0082] where \(d = \max 1≤s≤n \{|d s |\},\lambda N =\min 1≤s≤n \{\lambda sN \}, \lambda sN is 's eigenvalue. From inequalities (9)-(11), equation (12) is as follows:

[0083]

[0084]

[0085] For any \(\iota\in N\), let Since \(g ι \in(0,1)\), we get \(0\lt\psi ι \lt1\) and \((1 - \psi ι )(N - \varphi ι )=[\psi ι -(1 - g ι ) 2 \varphi ι. According to the selection of nodes in the set Υ t , it can be obtained that

[0086]

[0087] This means that

[0088]

[0089] Therefore, for any ι ∈ N, the second equation of Equation (7) can be deduced to

[0090]

[0091] Then, it will be proved that the conditions of Definition 3 are satisfied. From Equations (12) and (13), it can be seen that since It is observed that for t ∈ [t 0 , t 1 ,

[0092]

[0093] Therefore, for t ∈ [t 0 , t 1 ,

[0094]

[0095] This shows that the first condition in A is satisfied.

[0096] For t ∈ (t 1 , t 2 ,

[0097]

[0098] For t ∈ (t 2 , t 3 ,

[0099]

[0100] Similarly, for t ∈ (t ι-1 , t ι ,

[0101]

[0102] For t ∈ (t ι , t ι+1 ,

[0103]

[0104] Therefore, for t ∈ (t ι , t ι+1, ι ≥ 1,

[0105]

[0106] Thus, it can be obtained that for ι ≥ 1, there is

[0107]

[0108] In addition, for t ∈ (t ι , t ι+1 )

[0109]

[0110] Therefore, the second condition in A is satisfied. Thus, the proof is completed.

[0111] B. Assume that there exists a positive definite function θ: R n → R + , and there exist integers Then the step function Λ(t) can be defined as

[0112]

[0113] where ι ≥ 1 and it satisfies

[0114] 1) For a function ∈ belonging to the ξ class, Λ(t 1 ) ≤ ∈(θ(e i (t 0 )));

[0115] 2) Λ(t) decreases with time t, and lim t→∞ Λ(t) = 0.

[0116] Then the origin of (7) can be classified as globally uniformly attractive and stable (GUAS). In addition, when t ∞ = ∞, the origin of (7) is classified as globally attractive and stable (GAS).

[0117] Furthermore, the T-S fuzzy partially coupled network under pinning pulse control can be analyzed by a multi-step step function: Let where is the largest eigenvalue of the matrix , is the largest eigenvalue of the matrix , If there exists a positive definite function θ: R n → R + , then the following conditions can all be satisfied:

[0118]

[0119] Proof: Define the same function:

[0120]

[0121] Then the relevant step function is as follows:

[0122]

[0123] The proof process is the same as that of A and is omitted here. Directly obtain the following relevant inequalities:

[0124]

[0125] In the following, we will prove that the conditions of B are satisfied. is derived from inequalities (12) and (13). Therefore, for t ∈ [t 0 , t 1 ,

[0126]

[0127] For t ∈ (t 1 , t 2 ,

[0128]

[0129] Similarly, for

[0130]

[0131] Therefore, for

[0132]

[0133] where this means that the first condition in B is satisfied.

[0134] Similarly, for

[0135]

[0136] For

[0137]

[0138] For

[0139]

[0140] Therefore, for

[0141]

[0142] Then, there is

[0143]

[0144] In addition, for

[0145]

[0146] Therefore, the second condition of B is satisfied. Thus, the proof of Theorem 2 is completed.

[0147] The effects of the present invention can be further illustrated by the following simulation experiments:

[0148] In this simulation experiment, a T-S fuzzy partially coupled network consisting of five nodes is studied, where each node represents a three-dimensional system. The error system of this network is described as follows:

[0149]

[0150] Among them, let Ψ(t) = (Ψ 11 (t), Ψ 21 (t), Ψ 31 (t)) T , F(Ψ(t)) = (|Ψ 11 (t) + 1| - |Ψ 11 (t) - 1|, 0, 0) T And

[0151]

[0152] The coupling parameters of this model are defined as follows:

[0153]

[0154] And the channel matrix is selected:

[0155]

[0156] In addition, θ = 3, W = 2 are Lipschitz constants. The initial values in the system are randomly selected from a uniform distribution within the real number interval [-40, 40]. It is not difficult to find from the simulation results that the error system fails to reach stability, as Figure 2 shown.

[0157] To further illustrate Figure 3 the results, the first step of using the recombination method is to set Next, we collect the diagonal elements of the s-th layer from G ij (i, j = 1, 2, …, N) Rearrange into a new matrix

[0158]

[0159]

[0160] Through simple calculations, β can be obtained m = 0.1880, β M = 0.2116, and λ N = -5.7889, For simplicity, focus on the case of equidistant pulse intervals. Design the number of control nodes φ ι = 4, the pulse gain g ι = 0.25, through simple calculations, ψ can be obtained ι = 0.7120, b ι = 0.8010.

[0161] Furthermore, in Figure 4 , first consider the case of a single-step step function. Since λ > 0, let Γ = 0.1, and calculate the condition c) ζ = 1.4581 of the single-step step function. In addition, indicates that condition d) in the single-step step function is satisfied. Similarly, in the case of a multi-step step function, let Calculate the conditions of the multi-step step function and This shows that condition d) is satisfied in the multi-step step function.

Claims

1. An analysis method for TS fuzzy partial coupling network under pinning pulse control, characterized by: The following steps are involved: ① Establish TS fuzzy partial coupling network model Add TS fuzzy system to the partially coupled network, process the nonlinear nodes in the network through TS fuzzy system, and obtain TS fuzzy partially coupled network. The specific description is as follows: Rule τ: If α1(t) is …,α μ (t) Yes Then Among them, Φ i (t)=(Φ i1 (t),Φ i2 (t),…,Φ in (t)) T and i (t)=(Ψ i1 (t),Ψ i2 (t),…,Ψ in (t)) T Indicates that it belongs to R n The state vector of the i-th node; A τ and B τ It belongs to R n×n ; F(·) is a nonlinear function that satisfies R n →R n , and F(0)=0; θ>0 indicates the coupling strength; D=diag{d1,d2,…,d n } is the internal coupling matrix; Q = q ij Indicates that it belongs to R N×N The external coupling matrix of the complex network is q, where N defines the number of nodes on the complex network under consideration; if there is an edge from node j to node i (j≠i), then q ij > 0; otherwise, q ij =0; is the channel matrix, where It is defined as follows: If the s-th layer channel from node j to node i is active, then otherwise, u i (t) is the controller to be designed; α(t) = (α1(t), α2(t),…,α μ (t)) T is α j The premise vector of variables, represents a fuzzy set; in addition, Among them, Ω τ (α(t))>0 and Expressed as the level of the membership function, ② Design of pulse control controller Where σ(·) represents the Dirac function, g ι ∈(0,1) represents the pulse control gain, t ι Indicates that the pulse instant sequence satisfies the condition 0=t0<t1<t2<…<t ι <…, and lim ι→∞ t ι =+∞;φ ι is the number of control nodes; ③ Method for analyzing the stability of TS fuzzy partially coupled networks under pinned pulse control A. Assume there is a positively defined function :R n →R + , then the step function Λ(t) can be defined as Make it satisfy 1) For functions belonging to the class ξ , 2)Λ(t) decreases with time t, and lim t→∞ Λ(t)=0; Then the origin of the network model can be classified as globally uniformly attractive and stable (GUAS). In addition, when t ∞ =∞, the origin of the network model is classified as globally attractively stable (GAS); B. Assume there exists a positively defined function :R n →R + , and there exists an integer Then the step function Λ(t) can be defined as Among them, ι≥1 satisfies 1) For functions belonging to the class ξ , 2)Λ(t) decreases with time t, and lim t→∞ Λ(t)=0; Then the origin of the network model can be classified as globally uniformly attractive and stable (GUAS). In addition, when t ∞ =∞, the origin of the network model is classified as globally attractively stable (GAS).

2. The method for analyzing a TS fuzzy partial coupling network under pinning pulse control according to claim 1, characterized in that: set up in, is a matrix The maximum eigenvalue of is a matrix The maximum eigenvalue of If there exists a positive definite function :R n →R + , then the following conditions are met:

3. The method for analyzing a TS fuzzy partial coupling network under pinning pulse control according to claim 1, characterized in that: set up in, is a matrix The maximum eigenvalue of is a matrix The maximum eigenvalue of If there exists a positive definite function :R n →R + , then the following conditions are met:

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