CN111221311B - Complex network distributed pulse synchronization method and system based on parameter variational method - Google Patents
Complex network distributed pulse synchronization method and system based on parameter variational method Download PDFInfo
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Abstract
The invention relates to a complex network distributed pulse synchronization method and a system based on a parameter variational method, which comprises the following steps: considering a type of follower network and validating its leader node; acquiring state information of each node through a sensor and establishing an error network model, so that a state information vector and a tracking state information vector of each node in the network can be acquired; and returning the state information of each adjacent node and arranging a distributed negative feedback controller, wherein the distributed negative feedback controller is influenced by the pulse disturbance and judges whether the pulse plays a positive role or a negative role according to the disturbance type. The present invention has less limitation, and the discrete control method is used to design new pulse controller to produce effect only in some time, so that the present invention has effective control effect and low cost.
Description
Technical Field
The invention relates to the technical field of complex network synchronization, in particular to a complex network distributed pulse synchronization method and system based on a parameter variational method.
Background
Currently, network disciplines are becoming important research points in research fields and disciplines along with the cross fusion of multiple disciplines such as computers, biology, control and the like. Among them, the synchronization phenomenon of the network is more concerned by many researchers as a cluster behavior. In recent efforts published by scholars, controllers that continuously operate, such as negative feedback controllers, holdback controllers, and the like, have been widely used so far to achieve complex network synchronization. However, the continuous controller has a higher control cost due to its long-time operation, and the control cost cannot be reduced well, while the discrete controller has less related results to be applied to the continuous system.
The pulse controller controls by generating an excitation signal at a certain instant, and the intensity of the excitation signal has a great influence on the control effect of the controller. However, many studies are currently focused on controller designs where the pulsed excitation signal is beneficial to the synchronization effect, and less are studied for the synchronization effect of pulsed controllers when the excitation signal is counterproductive.
To date, it has been necessary to obtain global information for complex networks for designing distributed controllers. Therefore, it would be a challenge to design some distributed control protocols by using only local information of the system (i.e. dynamic information of the system nodes themselves and their adjacent nodes).
Disclosure of Invention
Therefore, the technical problem to be solved by the present invention is to overcome the problem of large limitation caused by less research on the synchronization effect of the pulse controller when the excitation signal is counterproductive in the prior art, thereby providing a complex network distributed pulse synchronization method and system based on the parameter variation method, which reduce the limitation.
In order to solve the technical problem, the invention discloses a complex network distributed pulse synchronization method based on a parameter variation method, which comprises the following steps: considering a type of follower network and validating its leader node; acquiring state information of each node through a sensor and establishing an error network model; returning the state information of each adjacent node and arranging a distributed negative feedback controller; the distributed negative feedback controller is influenced by pulse disturbance, whether the pulse plays a positive role or a negative role is judged according to the disturbance type, when the pulse plays the positive role, a new pulse controller is generated, the new pulse controller and the distributed negative feedback controller play a control effect on the network at the same time, and the network achieves synchronization under the dynamic balance of the distributed negative feedback controller and the new pulse controller; when the pulse is in counter reaction, new pulse disturbance is generated, the new pulse disturbance and the original random disturbance simultaneously have interference effect on the network, and the network achieves synchronization under the control effect of the distributed negative feedback controller.
In one embodiment of the present invention, the method for determining whether the pulse is positive or negative according to the disturbance type is: judging the size of the parameter, when delta is larger than 1, the pulse plays a counteraction role, and when delta is larger than 0 and smaller than or equal to 1, the pulse plays a positive role.
In one embodiment of the invention, when δ > 1, the synchronization rate of exponential synchronization is
Wherein λ is the equation
The special solution of (2); when delta is more than 0 and less than or equal to 1, the synchronization rate of the exponential synchronization is
Wherein λ' is the equation
The special solution of (1).
In one embodiment of the invention, the network refers to a complex network of a type having multiple time lags and being affected by random disturbances.
In one embodiment of the present invention, the sensors are disposed on the nodes, the state information vector and the tracking state information vector of each node in the complex network are obtained through the sensors, and an error network model is obtained by defining an error vector.
In one embodiment of the invention, the complex network refers to a network composed of a plurality of coupled nodes.
In one embodiment of the invention, the distributed negative feedback controller is a sufficient condition to achieve synchronization between the follower network and the leader system.
In one embodiment of the invention, after the distributed negative feedback controller is influenced by the pulse, when the pulse coefficient is smaller, the distributed negative feedback controller is considered to form a new pulse controller after being disturbed, and the pulse signal plays a positive role in synchronization, so that a new combined controller is formed; when the pulse coefficient is larger, the controller is considered to form noise after being disturbed, and the pulse signal plays a negative role in synchronization and forms interference with the original disturbance on network synchronization.
In one embodiment of the present invention, the new combined controller refers to a combination of the distributed negative feedback controller and the new pulse controller.
The invention also provides a complex network distributed pulse synchronization system based on the parameter variation method, which comprises a leader node module, a first-class follower network and a second-class follower network, wherein the leader node module is used for considering the first-class follower network and confirming the leader node; the error network model module is used for acquiring state information of each node through a sensor and establishing an error network model; the distributed negative feedback controller module is used for returning the state information of each adjacent node and arranging the distributed negative feedback controller; the distributed negative feedback controller is influenced by pulse disturbance, whether the pulse plays a positive role or a negative role is judged according to the disturbance type, when the pulse plays the positive role, a new pulse controller is generated, the new pulse controller and the distributed negative feedback controller play a control effect on the network at the same time, and the network achieves synchronization under the dynamic balance of the distributed negative feedback controller and the new pulse controller; when the pulse is in reaction, generating new pulse disturbance, and simultaneously having interference effect on the network with the original random disturbance, wherein the network achieves synchronization under the control effect of the distributed negative feedback controller.
Compared with the prior art, the technical scheme of the invention has the following advantages:
the invention uses a discrete control method, designs the pulse controller, and can effectively reduce the cost while ensuring the control effect by only generating the action at certain moments;
considering that the controller is disturbed due to a severe working environment in the production process, for a complex network formed by a Lur' e system with random interference, the invention discusses the synchronization condition of the complex network when the pulse effect plays a role in opposition, namely the pulse controller is regarded as disturbance at the moment, and obtains the corresponding index synchronization rate;
in consideration of the phenomena of signal delay and the like in the industrial process, the invention considers a Lur' e power network with multiple time-varying time lags, and the conclusion shows that the synchronization rate of a complex network can be accelerated by a smaller time lag upper bound;
an average pulse theorem is introduced to estimate a Cauchy matrix, and compared with the traditional method of giving upper and lower bounds of pulse intervals, the conservative property of the obtained synchronization condition can be effectively reduced;
in the multiple time lags existing in a complex network, the method expands the application of a variational formula to the parameters of a pulse system after the estimation of the Cauchy matrix, and obtains the index synchronization rate by some mathematical methods;
the invention obtains the time-lag independent synchronization conditions of some power networks, and can find that the synchronization rate of the complex network is closely related to the upper bound of the time lag.
Drawings
In order that the present disclosure may be more readily and clearly understood, reference is now made to the following detailed description of the embodiments of the present disclosure taken in conjunction with the accompanying drawings, in which
FIG. 1 is a flow chart of a complex network distributed pulse synchronization method based on a parameter variational method;
FIG. 2 is a schematic diagram of the pulse effect of the present invention;
FIG. 3 is a drawing of the present invention
And s 1 (t) an inter-error evolution curve;
FIG. 4 is a drawing of the present invention
And s 2 (t) an inter-error evolution curve;
FIG. 5 is a drawing of the present invention
And s 3 (t) an inter-error evolution curve;
FIG. 6 is an overall error evolution curve of the dynamic error network of the present invention.
Detailed Description
Example one
As shown in fig. 1, the present embodiment provides a complex network distributed pulse synchronization method based on a parameter variational method, including the following steps: step S1, considering a type of follower network and confirming a leader node of the follower network; s2, acquiring state information of each node through a sensor and establishing an error network model; s3, returning the state information of each adjacent node and arranging a distributed negative feedback controller; s4, the distributed negative feedback controller is influenced by pulse disturbance, whether the pulse plays a positive role or a negative role is judged according to the disturbance type, when the pulse plays the positive role, a new pulse controller is generated, the new pulse controller and the distributed negative feedback controller play a control effect on the network at the same time, and the network achieves synchronization under the dynamic balance of the distributed negative feedback controller and the new pulse controller; when the pulse is in reaction, generating new pulse disturbance, and simultaneously having interference effect on the network with the original random disturbance, wherein the network achieves synchronization under the control effect of the distributed negative feedback controller.
In the complex network distributed pulse synchronization method based on the parameter variation method, in step S1, a class of follower networks is considered and a leader node thereof is confirmed, so that the synchronization problem in the present invention can be regarded as a class of leader-follower problem; in the step S2, the state information of each node is obtained through the sensor and an error network model is established, so that a state information vector and a tracking state information vector of each node in the network can be obtained, which is beneficial to establishing the error network model, and the synchronization problem of each node in the network can be converted into an error network global stability problem through processing the error network model, so that the error network model is easier to process; in the step S3, state information of each adjacent node is returned and a distributed negative feedback controller is arranged, so that synchronization of the power network is realized; in the step S4, the distributed negative feedback controller is influenced by pulse disturbance, whether the pulse acts positively or negatively is judged according to the disturbance type, when the pulse acts positively, a new pulse controller is generated, the new pulse controller and the distributed negative feedback controller simultaneously play a control effect on the network, and the network achieves synchronization under the dynamic balance of the distributed negative feedback controller and the new pulse controller; when the pulse acts as a counteraction, a new pulse disturbance is generated, and simultaneously an interference effect is acted on the network with the original random disturbance, the network achieves synchronization under the control effect of the distributed negative feedback controller, the limitation is reduced because the invention discusses the performance of the controller when the pulse effect acts as a positive effect and a negative effect respectively, and the new pulse controller is designed because a discrete control method is used, and the cost can be effectively lowered while the control effect is ensured by only acting at certain moments.
The network refers to a complex network with multiple time lags and affected by random disturbance. Specifically, the complex network refers to a network model with independently operating systems which are connected together in some way so as to mutually influence each other. Since systems communicate with each other during industrial production, the situation of cooperative work is very common. Due to the complex working environment of the factory, the independent system nodes are often interfered by noise, which causes communication delay and the like. Based on the above situation, the present invention considers a class of complex networks with multiple time lags and affected by random disturbances:
wherein
Is the state vector of the ith Lur' e system of the complex network;
are all constant matrices; the normal number c is the coupling strength of the complex network; containing the element r i Matrix of > 0
Represents an internal coupling matrix; function(s)
Represents a non-linear function; h (t) and tau (t) respectively satisfy the time-varying time lag of h (t) being more than or equal to 0 and more than or equal to h and tau (t) being more than or equal to 0 and less than or equal to tau;
is an external coupling matrix determined by a complex network topology, the sum of each row of elements is 0 when the condition is satisfied, and
wherein w ij ＝w ji The condition that the number of the Lur ' e system is more than 0 represents that the connection exists between the ith Lur ' e system and the jth Lur ' e system; u. of i (t) denotes a controller, which will be described in detail in the following section;
representing an m-dimensional brownian motion;
is to satisfy the condition
The noise strength matrix of (2). In the case of the cj, the ratio,
all have C = [ C = 1 ，c 2 ，…，c m ] T ，G＝[g 1 ，g 2 ，…，g m ] T . Use of
Is shown in the scope
And has dimensions of
All continuous functions of wherein
For the random Lue' r power network (1), the initial conditions are all set to
As a kind of cluster behavior, the synchronization aims at making all systems in the complex network reach the same state, so a reference target is needed, in the present invention, the following isolated Lur' e system affected by random disturbance is considered as the reference target:
wherein
Furthermore, the solution s (t) of the isolated Lur 'e system (2) can be considered as a leader, and correspondingly, all Lur' e systems in the complex power network (1) can be considered as followers. Therefore, the problem of synchronization of the complex network (1) with the isolated node (2) can be regarded as a leader-follower problem. The sensors are arranged on the nodes, the state information vector and the tracking state information vector of each node in the complex network are obtained through the sensors, and an error network model is obtained through defining error vectors. In particular, by means of sensors arranged on each node, a state information vector x for each node in a complex network can be obtained i (t) and tracking the state information vector s (t) such that for i =1,2, …, N, by defining an error vector e i (t)＝x i (t)-s(t)，
The following error network model can thus be obtained:
φ(Ge i (t-h(t)))＝f2(Gx i (t-h(t)))-f2(Gs(t-h(t)))，
By processing the error network model, the synchronization problem of each node of the complex network can be converted into an error network global stability problem, and the error network global stability problem is easier to process.
In the present invention, to simulate a more realistic situation, consider a controller u designed i (t) in the case of a pulse, subject to a constant multiple of the pulse perturbation μ, the constant can be distributed over (0, + ∞), so that the pulse can have an adverse or beneficial effect on synchronization. The error Lur' e network of the required control can then be obtained:
in order to realize network synchronization between a random Lur 'e network (1) and a leader Lur' e system (2), the invention designs the following distributed strategy based on containment control by transmitting the state information of adjacent nodes and target synchronization nodes to each node:
wherein
Representing the set of all other Lur 'e systems that have connections to the ith Lur' e system.
Represents a control coupling matrix, satisfies the condition that the sum of each row element is 0, and
wherein l ij ＝l ji A > 0 indicates that a connection exists between the ith and jth Lur' e systems. Non-negative parameters k and epsilon i = i =1,2, …, N are control gains and at least ∈ i Is greater than 0. Hereinafter, xi = diag { epsilon 1 ，ε 2 ，…，ε N Denotes a control gain matrix.
The method is characterized in that an error control Lur 'e network (4) and a distributed negative feedback controller (5) are comprehensively considered, initial values are set, and for convenience of understanding, the impulse error Lur' e network with random disturbance and multiple time-varying time lag is expressed in a mathematical way as follows:
therein Ψ i (t) is the initial value of the error network (6). In the present invention, an error e is assumed i (t) at time
Is right-continuous, and
defining: if λ > 0 and M > 0 exist, let Ψ for the initial value i (t) the following equation holds for any error network:
As shown in FIG. 2, the distributed negative feedback controller of this embodiment considers the controller u i (t) cases affected by impulse disturbances, two cases are considered in the present invention: when the pulse coefficient is smaller, the distributed negative feedback controller is considered to form a new pulse controller after being disturbed, and the pulse signal plays a positive role in synchronization, so that a new combined controller is formed; when the pulse coefficient is larger, the controller is considered to form noise after being disturbed, and the pulse signal plays a negative role in synchronization and forms interference on network synchronization together with the original disturbance. Wherein the new combination controller refers to a combination of the distributed negative feedback controller and the new pulse controller.
The synchronization condition of a complex network (6) with random disturbances and multiple time lags will be discussed below. All mathematical expressions are based on comparative lemma and the extension of parameter variation method, and the invention provides sufficient conditions for realizing the synchronization between the follower network (1) and the leader system (2) through the designed distributed containment controller (5).
The following Lyapunov function was chosen:
wherein
By using the Lyapunov function, the error between all nodes in the complex network and the target synchronization node is discussed, and it is obvious that V (t) > 0 is provided because the initial states of the nodes in the complex network are not consistent, and thus the global error of the complex network is necessarily greater than 0, so in the following discussion, it will be explained that the function in equation (7) is monotonically decreasing, that is, the error of the complex network can be continuously reduced until the global synchronization is achieved.
For the
Based on Lur' e network (6) with the following results:
considering the above equation, it can be rewritten as:
thus, there are:
by scaling the part in the above equation, one can get:
through calculation, the original formula can be simplified into:
for the
By considering the noise as a superposition of non-linear functions, taking into account the complex network (6) model and by means of a linearization method, the following calculations can be made:
and is
a＝-λ max {A+A T +BB T +q 1 CTC+DDT+M T M-2αΓ}，
Combining equations (8) and (9), a pulse comparison system with a specific solution of γ (t) can be obtained, and by pulse comparison theorem it can be concluded that V (t) ≦ γ (t) for any t > 0. By extending the parametric variational approach, the following integral equation with time lags γ (t-h (t)) and γ (t- τ (t)) can be obtained:
wherein H (t, sigma) (t is more than or equal to sigma and more than or equal to 0) is a Cauchy matrix of the following linear pulse system:
the method for judging whether the pulse plays a positive role or a negative role according to the disturbance type comprises the following steps: judging the size of the parameter, when delta is larger than 1, the pulse plays a counteraction role, and when delta is larger than 0 and smaller than or equal to 1, the pulse plays a positive role. The following is a detailed description:
(1) if δ > 1, the right side of the Cauchy matrix H (t, σ) can be determined by considering the average pulse interval N ζ (t, σ) is calculated as follows:
the formula (11) can be substituted for the formula (10):
wherein
Defining the equation for the pulse as
The equation must have a special solution λ > 0.
First, for
Considering the parameters δ > 0, ε > 0, λ > 0, the following must be taken into account:
secondly, it can be found by a counter-method that for all t ∈ [0, + ∞), the following inequality still holds:
let ∈ → 0, then we can get:
by the definition of the complex network synchronization and the result of equation (15) in the foregoing, the synchronization between the follower network (1) and the leader system (2) can be achieved by the designed distributed negative feedback controller (5) explicitly. Further, as can be seen from the formula (15), when δ > 1, the synchronization rate of exponential synchronization is
Wherein λ is the equation
The special solution of (1).
(2) If 0 < δ ≦ 1, the left side of the Cauchy matrix H (t, σ) may be calculated as follows:
by similar proving methods, one can obtain:
similarly, for the equation
The special solution λ' > 0 also exists.
In addition, for
Delta is more than 0 and less than or equal to 1, epsilon is more than 0, and lambda' is more than 0, so that the product can be obtained
Correspondingly, it can be shown by a counter-test that equation (18) holds for all t ∈ [0, + ∞ ]. Let E → 0, get
Based on the above discussion, the synchronization condition between the follower network (1) and the leader system (2) is obtained, and the certification is completed.
And (4) conclusion:
for a pulse sequence ξ = { t = 1 ，t 2 …, assuming the average pulse spacing is less than N a If the normal values δ, k, μ, α, β are present with the matrix L, xi such that the equations (20) - (22) hold for δ > 1; and if the equations (20), (21), (23) hold for 0 < δ ≦ 1, then the solution for the error Lur' e network is exponentially stable, of which
a＝-λmax{A+A T +B T B+q 1 C T C+D T D+M T M-2αΓ}
Namely, the designed distributed containment controller (5) realizes the exponential synchronization between the coupled Lur 'e network (1) and the leader Lur' e system (2). In addition, for delta > 1, the synchronization rate for exponential synchronization is
Wherein λ is the equation
The special solution of (1); and for the case of 0 < delta.ltoreq.1, the synchronization rate of the exponential synchronization is
Wherein λ' is the equation
The special solution of (1).
The effectiveness of the invention will be illustrated by a specific numerical simulation example.
Firstly, establishing a complex network formed by coupling N Lur' e systems, wherein the specific model is as follows:
where C =1,0, g =0, 1, a =10, b =19.53, C =0.1636, p = -1.4325, ρ =0.5, q = -0.7831, and ∈ =0.2. In this model, time lag is assumed
While the non-linear function f1 ( Cx(t))＝0.5(|x 1 (t)+1|-|x 1 (t)-1|)，f 2 (Gy(t-h(t))＝sin(ρ x1 (t-h (t)). Finally, the brownian motion in this model is represented by
Secondly, determining the state model of the target Lur' e system as follows:
then, specific parameters meeting the specific model are calculated by using an LMI tool box;
and finally, building a Simulink model to obtain a simulation result, and as can be seen from the graphs in FIGS. 2-6, the states of all the nodes are synchronized under the proposed conditions.
Example two
Based on the same inventive concept, the present embodiment provides a complex network distributed pulse synchronization system based on a parameter variation method, and the principle of solving the problem is the same as that of the complex network distributed pulse synchronization system based on the parameter variation method, and repeated parts are not described again.
The complex network distributed pulse synchronization system based on the parameter variation method in the embodiment comprises:
the leader node module is used for considering a type of follower network and confirming a leader node of the follower network;
the error network model module is used for acquiring state information of each node through a sensor and establishing an error network model;
the distributed negative feedback controller module is used for returning the state information of each adjacent node and arranging the distributed negative feedback controller;
the distributed negative feedback controller is influenced by pulse disturbance, whether the pulse plays a positive role or a negative role is judged according to the disturbance type, when the pulse plays the positive role, a new pulse controller is generated, the new pulse controller and the distributed negative feedback controller play a control effect on the network at the same time, and the network achieves synchronization under the dynamic balance of the distributed negative feedback controller and the new pulse controller; when the pulse is in reaction, generating new pulse disturbance, and simultaneously having interference effect on the network with the original random disturbance, wherein the network achieves synchronization under the control effect of the distributed negative feedback controller.
As will be appreciated by one skilled in the art, embodiments of the present invention may be provided as a method, system, or computer program product. Accordingly, the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
It should be understood that the above examples are only for clarity of illustration and are not intended to limit the embodiments. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments. And obvious variations or modifications therefrom are within the scope of the invention.
Claims (7)
1. A complex network distributed pulse synchronization method based on a parameter variation method is characterized by comprising the following steps:
step S1: considering a type of follower network and validating its leader node;
step S2: acquiring state information of each node through a sensor and establishing an error network model;
and step S3: returning the state information of each adjacent node and arranging a distributed negative feedback controller;
and step S4: the distributed negative feedback controller is influenced by pulse disturbance, whether the pulse plays a positive role or a negative role is judged according to the disturbance type, when the pulse plays the positive role, a new pulse controller is generated, the new pulse controller and the distributed negative feedback controller play a control effect on the network at the same time, and the network achieves synchronization under the dynamic balance of the distributed negative feedback controller and the new pulse controller; when the pulse is in reaction, generating new pulse disturbance, and simultaneously playing an interference effect on the network with the original random disturbance, wherein the network achieves synchronization under the control effect of the distributed negative feedback controller;
wherein the network refers to a complex network which has multiple time lags and is influenced by random disturbance, the complex network refers to a network model which has independently operated systems and is connected together in some way so as to mutually influence, the complex network which has multiple time lags and is influenced by random disturbance,
wherein
Is the state vector of the ith Lur' e system of the complex network;
are all constant matrices; the normal number c is the coupling strength of the complex network; containing the element r i Matrix of > 0
Representing an internal couplingCombining the matrixes; function f 1 ，f 2 ：
Representing a non-linear function; h (t) and tau (t) respectively satisfy the time-varying time lag of h (t) being more than or equal to 0 and more than or equal to h and tau (t) being more than or equal to 0 and less than or equal to tau;
is an external coupling matrix determined by a complex network topology, the sum of each row of elements is 0 when the condition is satisfied, and
wherein w ij ＝w ji The condition that the number of the Lur ' e system is more than 0 represents that the connection exists between the ith Lur ' e system and the jth Lur ' e system; u. of i (t) denotes a controller, which will be described in detail in the following section;
representing an m-dimensional brownian motion;
is to satisfy the condition
For c, for the noise intensity matrix of j ，
All have C = [ C = 1 ，c 2 ，…，c m ] T ，G＝[g 1 ，g 2 ，…，g m ] T Use of
Is shown in the scope
And has dimensions of
All continuous functions of wherein
For the random Lur' r power network (1), the initial conditions are all set to
Consider as the reference target an isolated Lur' e system that is affected by random perturbations as follows:
wherein
In addition, the solution s (t) of the isolated Lur 'e system (2) can be regarded as a leader, and correspondingly, all Lur' e systems in the complex power network (1) can be regarded as followers, the synchronization problem of the complex network (1) and the isolated node (2) can be regarded as a leader-follower problem, the sensors are arranged on the nodes, the state information vector and the tracking state information vector of each node in the complex network are obtained through the sensors, an error network model is obtained through defining the error vectors, and particularly, the state information vector x of each node in the complex network can be obtained through the sensors arranged on the nodes i (t) and tracking the state information vector s (t) such that for i =1,2, …, N, by defining an error vector e i (t)＝x i (t)-s(t)，
Thereby can be provided withThe following error network model was obtained:
wherein
φ(Ge i (t-h(t)))＝f 2 (Gx i (t-h(t)))-f 2 (Gs(t-h(t)))，
by transmitting the state information of the adjacent node and the target synchronous node to each node, the invention also comprises a distributed strategy based on containment control:
wherein
Represents the set of all other Lur 'e systems connected with the ith Lur' e system,
represents a control coupling matrix, satisfies the condition that the sum of each row element is 0, and
wherein l ij ＝l ji The condition that the parameter k is more than 0 represents that the connection exists between the ith Lur 'e system and the jth Lur' e system, and the non-negative parameters k and epsilon i (i =1,2, …, N) are control gains and have at least ε i & gt 0, xi = diag { epsilon 1 ，ε 2 ，…，ε N Denotes a control gain matrix for controlling the gain of the signal,
the error control Lur 'e network (4) and the distributed negative feedback controller (5) are comprehensively considered, initial values are set, and for convenience of understanding, the pulse error Lur' e network with random disturbance and multiple time-varying time lags is mathematically expressed as follows:
therein Ψ i (t) is the initial value of the error network (6), assuming an error e i (t) at time
Is right-continuous, and
defining: if λ > 0 and M > 0 exist, let Ψ for the initial value i The arbitrary error network of (t) holds for the following equation:
2. The complex network distributed pulse synchronization method based on the parameter variational method according to claim 1, characterized in that: the method for judging whether the pulse plays a positive role or a negative role according to the disturbance type comprises the following steps: judging the size of the parameter, when the pulse coefficient delta is larger than 1, the pulse plays a counteraction role, and when the delta is larger than 0 and smaller than or equal to 1, the pulse plays a positive role.
3. The complex network distributed pulse synchronization method based on the parameter variational method as claimed in claim 2, characterized in that: when delta > 1, the synchronization rate of exponential synchronization is
Wherein λ is the equation
The special solution of (2); when delta is more than 0 and less than or equal to 1, the synchronization rate of the exponential synchronization is
Wherein λ' is the equation
The special solution of (1).
4. The complex network distributed pulse synchronization method based on the parameter variation method according to claim 1, characterized in that: the distributed negative feedback controller is a sufficient condition to achieve synchronization between the follower network and the leader system.
5. The complex network distributed pulse synchronization method based on the parameter variation method according to claim 1, characterized in that: after the distributed negative feedback controller is influenced by the pulse, when the pulse coefficient delta is smaller, namely delta is more than 0 and less than or equal to 1, the distributed negative feedback controller is considered to form a new pulse controller after being disturbed, and the pulse signal plays a positive role in synchronization, so that a new combined controller is formed; when the pulse coefficient is larger, namely delta is larger than 1, the controller is considered to form noise after being disturbed, and the pulse signal has a negative effect on synchronization and forms interference on network synchronization together with the original disturbance.
6. The complex network distributed pulse synchronization method based on the parameter variational method according to claim 5, characterized in that: the new combined controller refers to a combination of the distributed negative feedback controller and the new pulse controller.
7. A complex network distributed pulse synchronization system based on a parameter variation method is characterized by comprising the following steps:
the leader node module is used for considering a type of follower network and confirming a leader node of the follower network;
the error network model module is used for acquiring state information of each node through a sensor and establishing an error network model;
the distributed negative feedback controller module is used for returning the state information of each adjacent node and arranging the distributed negative feedback controller;
the distributed negative feedback controller is influenced by pulse disturbance, whether the pulse plays a positive role or a negative role is judged according to the disturbance type, when the pulse plays the positive role, a new pulse controller is generated, the new pulse controller and the distributed negative feedback controller play a control effect on the network at the same time, and the network achieves synchronization under the dynamic balance of the distributed negative feedback controller and the new pulse controller; when the pulse is in reaction, generating new pulse disturbance which has interference effect on the network with the original random disturbance, the network achieves synchronization under the control effect of the distributed negative feedback controller,
wherein the network refers to a complex network which has multiple time lags and is influenced by random disturbance, the complex network refers to a network model which has independently operated systems and is connected together in some way so as to mutually influence, the complex network which has multiple time lags and is influenced by random disturbance,
wherein
Is the state vector of the ith Lur' e system of the complex network;
are all constant matrices; the normal number c is the coupling strength of the complex network; containing the element r i Matrix of > 0
Represents an internal coupling matrix; function f 1 ，f 2 ：
Representing a non-linear function; h (t) and tau (t) respectively satisfy the time-varying time lag of h (t) being more than or equal to 0 and more than or equal to h and tau (t) being more than or equal to 0 and less than or equal to tau;
is an external coupling matrix determined by a complex network topology, the sum of each row of elements is 0 when the condition is satisfied, and
wherein w ij ＝w ji > 0 represents the ith and jth Lur' e systemsThere is a connection between the systems; u. of i (t) denotes a controller, which will be described in detail in the following section;
representing an m-dimensional brownian motion;
is that the condition is satisfied
For c, for the noise intensity matrix of j ，
All have C = [ C = 1 ，c 2 ，…，c m ] T ，G＝[g 1 ，g 2 ，…，g m ] T Use of
Is shown in the scope
And has dimensions of
All continuous functions of wherein
For the random Lue' r power network (1), the initial conditions are all set to
Consider as the reference target an isolated Lur' e system that is affected by random perturbations as follows:
wherein
In addition, the solution s (t) of the isolated Lur 'e system (2) can be regarded as a leader, and correspondingly, all Lur' e systems in the complex power network (1) can be regarded as followers, the synchronization problem of the complex network (1) and the isolated node (2) can be regarded as a leader-follower problem, the sensors are arranged on the nodes, the state information vector and the tracking state information vector of each node in the complex network are obtained through the sensors, an error network model is obtained through defining the error vectors, and particularly, the state information vector x of each node in the complex network can be obtained through the sensors arranged on the nodes i (t) and tracking the state information vector s (t) such that for i =1,2, …, N, by defining an error vector e i (t)＝x i (t)-s(t)，
The following error network model can thus be obtained:
wherein
φ(Ge i (t-h(t)))＝f 2 (Gx i (t-h(t)))-f 2 (Gs(t-h(t)))，
by transmitting the state information of the adjacent node and the target synchronous node to each node, the invention also comprises a distributed strategy based on containment control:
wherein
Represents the set of all other Lur 'e systems connected with the ith Lur' e system,
represents a control coupling matrix, satisfies the condition that the sum of each row element is 0, and
wherein l ij ＝l ji The condition that the parameter k is more than 0 represents that the connection exists between the ith Lur 'e system and the jth Lur' e system, and the non-negative parameters k and epsilon i (i =1,2, …, N) are control gains and there is at least ε i & gt 0, xi = diag { epsilon 1 ，ε 2 ，…，ε N Denotes a control gain matrix for controlling the gain of the signal,
the error control Lur 'e network (4) and the distributed negative feedback controller (5) are comprehensively considered, initial values are set, and for convenience of understanding, the pulse error Lur' e network with random disturbance and multiple time-varying time lags is mathematically expressed as follows:
therein Ψ i (t) is the initial value of the error network (6), assuming an error e i (t) at time
Is right-continuous, and
defining: if λ > 0 and M > 0 exist, let Ψ for the initial value i The arbitrary error network of (t) holds for the following equation:
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