A bipartite synchronization method for switched complex networks with dynamic coupling and symbolic graph

Abstract

The present invention discloses a binary synchronization method for a switching complex network with dynamic coupling and symbolic graph, which belongs to the technical field of complex network synchronization, and solves the technical problem of binary synchronization of a complex network with cooperative-competitive interaction relationship and system parameter switching when node coupling exhibits dynamic behavior. The technical scheme includes the following steps: firstly, a system model of a switching complex network with dynamic coupling and symbolic graph is established; then, a controller of the switching complex network is designed; then, an error system between the switching complex network and a target node is constructed; and a binary synchronization condition of the switching complex network with dynamic coupling and symbolic graph is established to obtain a controller gain. The beneficial effects of the present invention are: cooperation and competition relationship coexist between nodes, node coupling exhibits dynamic behavior, system parameters are switched, and a controller is designed to realize binary synchronization of a switching complex network with dynamic coupling and symbolic graph in the sense that the error system is asymptotically stable in mean square.

Description

Technical Field

[0001] The present invention relates to the technical field of complex network synchronization, and in particular to a switching complex network binary synchronization method with dynamic coupling and symbolic graph. Background Art

[0002] The synchronization of complex networks has attracted great attention due to its potential application prospects in complex power networks, vehicle grid structures, and public urban population mobility networks. Based on the different interactions between nodes in complex networks, complex networks can be roughly divided into the following two categories: complex networks with cooperative interactions, which are usually described by unsigned graphs; complex networks with cooperative-competitive interactions, which are usually described by signed graphs.

[0003] Complex networks with symbolic graphs are characterized by the coexistence of cooperative and competitive relationships, which may arise in applications in robotics and aerospace, including the control of multiple satellites that have to deal with aerial debris that appear as non-cooperators. At the same time, considering that the internal structure of nodes in complex networks will change in practical applications, the problem of bipartite synchronization of switched complex networks with signed graphs has attracted widespread attention recently, for example, applying intermittent control methods to the bipartite synchronization of delayed coupled systems over signed graphs with Markovian switching via intermittent control (DOI: 10.1016 / j.jfranklin.2020.12.031); Exponential bipartite synchronization of random signed networks with Markovian switching via impulsive control (DOI: 10.1002 / rnc.5195); Bipartite Containment Control for Delayed Multiagent Systems With Markovian Switching Topologies Under Impulsive Control. Attacks, DOI: 10.1109 / ACCESS.2023.3267092). However, existing research on binary synchronization of switched complex networks with symbolic graphs ignores a problem that in actual situations, since the nodes in the complex dynamic network exhibit dynamic behavior, the node coupling used to connect the complex dynamic network also has dynamic behavior. At present, there is no report on binary synchronization of complex networks with symbolic graphs and system parameter switching that takes into account dynamic coupling. In this case, the present invention proposes a binary synchronization method for switched complex networks with dynamic coupling and symbolic graphs. Summary of the invention

[0004] In order to solve the technical problem of binary synchronization of complex networks with cooperative-competitive interaction and node parameter switching when node coupling exhibits dynamic behavior, the present invention proposes a binary synchronization method for switching complex networks with dynamic coupling and symbolic graph.

[0005] The present invention is achieved by the following measures: A switching complex network binary synchronization method with dynamic coupling and symbolic graph comprises the following steps:

[0006] Step 1: Consider a switching complex network consisting of N nodes with dynamic coupling and symbolic graph, and establish the system model as follows:

[0007]

[0008] in, is the state variable of node i, is the coupled state variable, z i (t) is the coupling output, α>0 indicates the coupling strength, is a nonlinear odd function, u i (t) is the control input; the matrices A(σ(t)), D(σ(t)), A r , B r and C r are all known constant real number matrices; σ(t) is the switching signal, which is a homogeneous Markov process of continuous-time discrete states and takes values ​​in the finite state set R ={1,2,…,R}; for symbolic graphs describing complex network topology Adjacency Matrix Satisfies: If there is an edge between node j and node i, a ij ≠0, i≠j; and a ii =0, i = 1, 2, ..., N; and a ij >0,a ij <0 indicates that there is a cooperative or competitive relationship between node j and node i. The Laplace matrix of in

[0009]

[0010] The dynamic model of a given target node is shown in the following formula (2):

[0011]

[0012] in, is the state variable of the target node, z0(t) is the output;

[0013] Step 2: Design a controller for switching complex networks. The controller model is expressed by the following formula (3):

[0014] u i (t) = K i (σ(t))(x i (t)-si x0(t)) (3)

[0015] Among them, K i (σ(t)) is the controller gain to be designed, and when i belongs to the first bipartite non-empty node subset, s i =1, otherwise s i = -1;

[0016] make have definition have Then the switching complex network (1) with controller (3) is expressed as,

[0017]

[0018] Step 3: Construct the error system between the switching complex network (1) and the target node (2)

[0019] make The error system is

[0020]

[0021] definition

[0022] G(δ(t))=[g T (δ1(t)) g T (δ2(t)) ... g T (δ N (t))] T ,

[0023]

[0024] K(σ(t))=diag{K1(σ(t)),K2(σ(t)),…,K N (σ(t))},

[0025] Then the error system (5) is expressed in the form of the following Kronecker product,

[0026]

[0027] Step 4: Establish the bisection synchronization condition of the switching complex network (1) with dynamic coupling and symbolic graph, and obtain the controller gain K ip , bipartite synchronization of the switching complex network (1) is achieved in the sense that the error system (5) is asymptotically stable in the mean square;

[0028] If there exists a matrix P 1p >0,P 2p >0,…,P Np>0,Q>0,X 1p , X 2p , …X Np , p∈ R , scalar μ>0, so that the following inequality holds:

[0029]

[0030] in, P p =diag{P 1p ,P 2p ,…,P Np}, X p =diag{X 1p ,X 2p ,…,X Np},

[0031] Get the controller gain K ip for,

[0032]

[0033] Then the bipartite synchronization of the switching complex network (1) is achieved in the sense that the error system (5) is asymptotically stable in the mean square.

[0034] Further, the method for bipartite synchronization of a switching complex network with dynamic coupling and symbolic graph described in step 1 is a nonlinear odd function, which means that the nonlinear function f(.) satisfies the Lipschitz condition, and there exists a constant ρ>0, so that ‖f(η1)-f(η2)‖≤ρ‖η1-η2‖,

[0035] Furthermore, in a switching complex network bipartite synchronization method with dynamic coupling and symbolic graph, in step 1, the σ(t) is a switching signal, which is a homogeneous Markov process of continuous-time discrete states and takes values ​​in a finite state set R ={1,2,…,R}, where the transition probability matrix of the switching signal σ(t) is ∏=[π pq ] R×R , the transition probability from switching mode p to switching mode q is as follows,

[0036]

[0037] Among them, h>0 is the stay time,

[0038] Furthermore, in a switching complex network bisection synchronization method with dynamic coupling and symbolic graph, in step 1, the symbolic graph describing the complex network topology structure It refers to the symbol diagram Structurally balanced, with bipartite non-empty node subsets and And it satisfies: When node i and node j belong to the same subset, then a ij ≥0; otherwise a ij ≤0.

[0039] Compared with the prior art, the present invention has the following beneficial effects:

[0040] (1) The present invention comprehensively considers the joint influence of the internal system structure change of the complex network and the network cooperation-competition interaction relationship, adopts system parameter switching to deal with the system structure change problem, uses symbolic graphs to describe the network cooperation-competition interaction relationship, and combines the actual situation of dynamic coupling between nodes to design a controller based on dynamic coupling to solve the binary synchronization problem of the complex network with dynamic coupling and symbolic graph switching;

[0041] (2) In the sense of mean square asymptotic stability of the error system, this study proposes a bipartite synchronization method for switching complex networks with dynamic coupling and symbolic graphs, which has more research and application value than the existing research on bipartite synchronization of switching complex networks based on symbolic graphs and the research on synchronization of complex networks based on dynamic coupling and unsigned graphs (Non-fragile passivitysynchronization control for complex dynamical networks with dynamics behaviorlinks, DOI: 10.1109 / ACCESS.2021.3124137). BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 It is a schematic diagram of a switching complex network binary synchronization method with dynamic coupling and symbolic graph of the present invention.

[0043] Figure 2 It is a symbolic diagram describing the complex network topology structure in the numerical example of the present invention.

[0044] Figure 3 is the switching complex network state x with controller function in the numerical example of the present invention i (t) Trajectory diagram, where (a) is the state x of the switching complex network. i1 (t) is the trajectory diagram; (b) is the state x of the switching complex network i2 Trajectory diagram of (t), i=1,…,5.

[0045] Figure 4 is the state error δ of the switching complex network with controller function in the numerical example of the present invention i(t) trajectory diagram, where (a) is the state error δ of the switching complex network i1 (t) is the trajectory diagram; (b) is the state error δ of the switching complex network i2 Trajectory diagram of (t), i=1,…,5. DETAILED DESCRIPTION

[0046] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. Of course, the specific embodiments described here are only used to explain the present invention and are not used to limit the present invention.

[0047] As attached Figure 1 As shown, consider a switching complex network consisting of N nodes with dynamic coupling and symbolic graph, and establish the system model as follows:

[0048]

[0049] in, is the state variable of node i, is the coupled state variable, z i (t) is the coupling output, α>0 indicates the coupling strength, is a nonlinear odd function, u i (t) is the control input; the matrices A(σ(t)), D(σ(t)), A r , B r and C r are all known constant real number matrices; σ(t) is the switching signal, which is a homogeneous Markov process of continuous-time discrete states and takes values ​​in the finite state set R ={1,2,…,R}; the transition probability matrix of the switching signal σ(t) is ∏=[π pq ] R×R , the transition probability from switching mode p to switching mode q is as follows,

[0050]

[0051] Among them, h>0 is the stay time, Symbolic diagrams for describing complex network topology Adjacency Matrix Satisfies: If there is an edge between node j and node i, a ij ≠0, i≠j; and a ii =0, i=1,2,...,N. And a ij >0,a ij <0 indicates that there is a cooperative or competitive relationship between node j and node i; symbol diagram The Laplace matrix of in

[0052] Assumption 1. Symbolic graph Structurally balanced, with bipartite non-empty node subsets and And it satisfies: When node i and node j belong to the same subset, then a ij ≥0; otherwise a ij ≤0.

[0053] The dynamic model of the given target node is shown in the following formula (12):

[0054]

[0055] in, is the state variable of the target node, and z0(t) is the output.

[0056] Design a controller for switching complex networks. The controller model is expressed by the following formula (13):

[0057] u i (t) = K i (σ(t))(x i (t)-s i x0(t)) (13)

[0058] Among them, K i (σ(t)) is the controller gain to be designed, and when When i =1, otherwise s i =-1.

[0059] Assumption 2. The nonlinear function f(.) satisfies the Lipschitz condition, and there exists a constant ρ>0, so that the following inequality,

[0060]

[0061] make have definition have Then the switching complex network (10) with controller (13) is expressed as,

[0062]

[0063] Definition 1. A complex network can achieve bipartite synchronization if the following conditions are met:

[0064]

[0065] Among them, when When i=1, otherwise s i =-1.

[0066] Let the error signal The error system is

[0067]

[0068] definition

[0069] G(δ(t))=[g T (δ1(t)) g T (δ2(t)) ... g T (δ N (t))] T ,

[0070]

[0071] K(σ(t))=diag{K1(σ(t)),K2(σ(t)),…,K N (σ(t))},

[0072] Then the error system (16) is expressed in the form of the following Kronecker product,

[0073]

[0074] When σ(t)=p∈ R ,A(σ(t)),D(σ(t)),K i (σ(t)) and K(σ(t)) are represented by A p , D p , K ip and K p .

[0075] Theorem 1. Under Assumptions 1 and 2, if there exists a matrix P 1p >0,P 2p >0,…,P Np >0,Q>0,X 1p , X 2p , …X Np , p∈ R , the scalar μ>0, so that the following inequality holds:

[0076]

[0077] in,

[0078] Get the controller gain K ip for,

[0079]

[0080] Then the bipartite synchronization of the switching complex network (10) is achieved in the sense of mean square asymptotic stability of the error system (16).

[0081] Numerical examples.

[0082] Consider a switching complex network with 5 nodes, where each node is a single-link robotic arm system with the following parameters:

[0083]

[0084] f(x i (t)) = [0 sin(x i1 (t))] T , x i (t) = [x i1 (t) x i2 (t)] T ,

[0085] Among them, x i1 (t) and x i2 (t) represents the angular position and angular velocity of the robot arm, i = 1, 2, 3, 4, 5. g is the acceleration due to gravity, F is the damping coefficient, l is the arm length, F is the damping coefficient, J p and M p They are the moment of inertia and the mass of the effective load, p=1,2,3, and their values ​​are: g=9.81, F=2, l=0.5, M1=J1=1, M2=J2=5, M3=J3=10.

[0086] In addition, the coupling parameters are

[0087] in,

[0088] Switching signal with 3 switching modes, transition probability matrix

[0089] Symbolic diagrams describing complex network topology As attached Figure 2 shown.

[0090] Symbol diagram The Laplace matrix of Expressed as The 5 nodes are divided into 2 subsets, V1 = {1, 2, 3} and V2 = {4, 5}, so the symbol graph Assumption 1 is satisfied.

[0091] Assuming the coupling strength α = 0.2, according to Theorem 1, the gain matrix of the controller is as follows:

[0092]

[0093]

[0094]

[0095]

[0096]

[0097] Switching complex network state trajectory with controller action x i1 and x i2 See attached Figure 3 , the state trajectories of nodes 1, 2, and 3 tend to be consistent, and the state trajectories of nodes 4 and 5 tend to be consistent. The state trajectories of the two clusters are the same in size and opposite in direction. The state error trajectory of the switching complex network with controller action δ i1 (t) and δ i2 (t) See attached Figure 4 , the state error trajectory of each node is asymptotically stable. As can be seen from the attached figure, the switching complex network with dynamic coupling and symbolic graph under the action of the controller achieves bipartite synchronization.

[0098] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for bipartite synchronization of switched complex networks with dynamic coupling and symbolic graph, characterized in that: The steps include: Step 1: Consider a switching complex network consisting of N nodes with dynamic coupling and symbolic graph, and establish the system model as follows: in, is the state variable of node i, is the coupled state variable, z i (t) is the coupling output, α>0 indicates the coupling strength, is a nonlinear odd function, u i (t) is the control input; the matrices A(σ(t)), D(σ(t)), A r , B r and C r are all known constant real number matrices; σ(t) is the switching signal, which is a homogeneous Markov process of continuous-time discrete states and takes values ​​in the finite state set R ={1,2,...,R}; for symbolic graphs describing complex network topology Adjacency Matrix Satisfies: If there is an edge between node j and node i, a ij ≠0, i≠j; and a ii =0, i = 1, 2, ..., N; and a ij >0,a ij <0 indicates that there is a cooperative or competitive relationship between node j and node i. The Laplace matrix of in The dynamic model of a given target node is shown in the following formula (2): in, is the state variable of the target node, z0(t) is the output; Step 2: Design a controller for switching complex networks. The controller model is expressed by the following formula (3): you i (t)=K i (σ(t))(x i (t)-s i x0(t)) (3) Among them, K i (σ(t)) is the controller gain to be designed, and when i belongs to the first bipartite non-empty node subset, s i =1, otherwise s i = -1; make have definition have Then the switching complex network (1) with controller (3) is expressed as, Step 3: Construct the error system between the switching complex network (1) and the target node (2) make The error system is definition G(δ(t))=[g T (δ1(t)) g T (δ2(t))...g T (d N (t))] T , K(σ(t))=diag{K1(σ(t)),K2(σ(t)),…,K N (σ(t))}, Then the error system (5) is expressed in the form of the following Kronecker product, Step 4: Establish the bisection synchronization condition of the switching complex network (1) with dynamic coupling and symbolic graph, and obtain the controller gain K ip , bipartite synchronization of the switching complex network (1) is achieved in the sense that the error system (5) is asymptotically stable in the mean square; If there exists a matrix P 1p >0,P 2p >0,…,P Np >0,Q>0,X 1p , X 2p , …X Np , p∈ R , scalar μ>0, so that the following inequality holds: in, X p =diag{X 1p ,X 2p ,…,X Np }, Get the controller gain K ip for, R is a finite set of states, R ={1,2,…,R}; ρ is the Lipschitz constant, ρ>0; Then the bipartite synchronization of the switching complex network (1) is achieved in the sense that the error system (5) is asymptotically stable in the mean square.

2. A switching complex network bisection synchronization method with dynamic coupling and symbolic graph according to claim 1, characterized in that: In step 1, is a nonlinear odd function, which means that the nonlinear function f(.) satisfies the Lipschitz condition, and there exists a constant ρ>0, so that ‖f(η1)-f(η2)‖≤ρ‖η1-η2‖, 3. A method for switching complex network bisection synchronization with dynamic coupling and symbolic graph according to claim 1, characterized in that: In step 1, the σ(t) is the switching signal, which is a homogeneous Markov process of continuous-time discrete states and takes values ​​in the finite state set R ={1,2,…,R}, where the transition probability matrix of the switching signal σ(t) is Π=[π pq ] R×R , the transition probability from switching mode p to switching mode q is as follows, Among them, the function o(h) is a high-order infinitesimal with respect to the residence time h, satisfying h>0 is the stay time, π pq ≥0, p≠q, 4. A method for switching complex network bisection synchronization with dynamic coupling and symbolic graph according to claim 1, characterized in that: In step 1, the symbolic diagram for describing the complex network topology structure It refers to the symbol diagram Structurally balanced, with bipartite non-empty node subsets and And it satisfies: When node i and node j belong to the same subset, then a ij ≥0; otherwise a ij ≤0.

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