CN112731807A - Balance point containment control method of complex dynamic saturated network model - Google Patents
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Abstract
The invention discloses a balance point containment control method of a complex dynamic saturated network model, belonging to the technical field of network synchronization. The strategy can enable the complex dynamic network to reach a balance point, adjust the coupling strength between the nodes and the feedback gain of the constraint nodes, thereby reducing the control cost; the feedback controller is added on a few nodes of the network, all the nodes of the complex network can reach a preset balance point, the coupling strength between the nodes and the feedback gain of the constraint node are adjusted, and therefore the control cost is reduced.
Description
Technical Field
The invention belongs to the technical field of network synchronization, and particularly relates to a balance point containment control method of a complex dynamic saturated network model.
Background
Recently, the research on complex networks has started to pay more attention to the dynamic behavior of networks with a large number of nodes and a complex connection structure.
Synchronization is one of important dynamic behaviors, and refers to that two or more dynamic systems are coupled with each other so that states of the dynamic systems evolved respectively under different initial conditions are gradually close to each other and finally reach the same state. The complex network synchronization behavior is a technical problem with very practical significance and theoretical value in a complex dynamic system.
With the increase of network size and the complexity of network topology, how to effectively control these complex networks has become one of the hot spots of research.
The containment control strategy is a simple method, which uses the connectivity of the network and adds feedback controllers on a few nodes to make all nodes in the network reach the same state. The same state may be time varying or may be the balance point of the isolated node function.
The feedback gain of the controller may increase as the size of the network increases. In practice, however, the coupling strength and feedback gain cannot be infinite. For example, input and feedback currents in the power system network are saturated due to material limitations. In the prior art, saturation of coupling strength between nodes has been studied, but saturation of feedback gain has not been studied.
Disclosure of Invention
The purpose of the invention is as follows: the invention aims to provide a balance point containment control method of a complex dynamic saturated network model, which can enable all nodes of a complex network to reach a preset balance point by only adding feedback controllers on a few nodes of the network.
The technical scheme is as follows: in order to achieve the purpose, the invention provides the following technical scheme:
a balance point restraining control method of a complex dynamic saturated network model specifically comprises the following steps:
step 1: acquiring a function condition of a coupling strength saturation threshold value between nodes of a complex network;
step 2: establishing a mathematical model of saturation function constraint;
and step 3: and optimizing the synchronization capacity of the complex saturated network by using a balance point containment control strategy.
Further, the establishment of the mathematical model comprises the following steps:
1) one network consists of N linearly coupled identical nodes, and each node is an N-dimensional dynamic system;
2) the saturation function network comprises N nodes, and the mathematical model of the ith node at the time t is as follows:
in the formula (I), N is any positive integer, f (-) is a vector value function and describes the dynamic behavior of the node, and xi(t)＝(xi1(t),xi2(t),...,xin(t))T∈RnIs the state vector of node i at time t; gamma is belonged to Rn×nIs a matrix of constants 0-1 that reflects the coupling of internal variables.
Further, in the mathematical model, a ═ aij]N×N＝ATRepresenting an inter-node coupling matrix;
wherein, when there is a connection between node i and node j and i ≠ j, wherein j ═ 1,2ij＝ajiOtherwise let a be 1ij＝aji＝0；
Wherein, when i ═ j,
further, the coupling moment between the nodesArray A has N eigenvalues, which are written as: 0 > lambda2(A)≥…≥λN(A)；ci(t) is a function of the coupling strength of the ith node; u. ofi(t) is the feedback of node i;
assuming that the number of selected holdback nodes is m; in which feedback control ui(t) is:
in the formula (II)
di(t) > 0 is the saturation feedback gain;
represents the equilibrium point of the function f (·).
Further, the step 3 comprises the following steps:
define matrix B ═ L-diag (h)1d1(t),h2d2(t),…,hmdm(t),0, …,0), the characteristic values are as follows: 0 > lambda1(B)≥…≥λN(B)；
The saturation function of the coupling strength between nodes in the network is:
the saturation function of the feedback gain is:
wherein gamma isc＞0，γd> 0 saturation function gain of saturation function of coupling strength between nodes and saturation function of feedback gain, thetac＞0,θd> 0 is a saturation function and saturation of the feedback gain, respectively, for the coupling strength between the nodesError gain of the function.
Further, when the saturation function satisfies ci(t) c and the feedback gain satisfies diWhen (t) ═ d, the coupling strength and feedback gain are fixed in the connectivity network.
Has the advantages that: compared with the prior art, the balance point containment control method of the complex dynamic saturated network model provided by the invention researches the balance containment control of the complex network through a class of saturation functions applied to coupling strength and feedback gain, and designs the distributed controllers installed on the fixed nodes on the basis. The strategy enables the complex dynamic network to reach a balance point, and adjusts the coupling strength between the nodes and the feedback gain of the constraint nodes, thereby reducing the control cost; the feedback controller is added on a few nodes of the network, all the nodes of the complex network reach a preset balance point, the coupling strength between the nodes and the feedback gain of the constraint node are adjusted, and therefore the control cost is reduced.
Drawings
Fig. 1 is a dynamic network with a coupling strength c-10 between nodes;
fig. 2 is a dynamic network with a coupling strength c-20 between nodes;
fig. 3 is a dynamic network with improved function saturation coupling strength and feedback gain.
Detailed Description
The invention will be further described with reference to the following drawings and specific embodiments.
A balance point restraining control method of a complex dynamic saturated network model researches balance point restraining control of a complex network and designs a distributed controller installed on a fixed node on the basis. The strategy can enable the complex dynamic network to reach a balance point, adjust the coupling strength between the nodes and restrain the feedback gain of the nodes, and therefore control cost is reduced.
Meanwhile, the balance constraint control problem of a complex dynamic network model with saturated coupling strength and saturated feedback is researched. Considering the saturation of the coupling strength and the feedback gain among the nodes in the actual engineering, a containment strategy for improving the saturation function is provided, namely, all the nodes of the complex network can reach a preset balance point by only adding feedback controllers on a few nodes of the network.
A balance point restraining control method of a complex dynamic saturated network model specifically comprises the following steps:
step 1: acquiring a function condition of a coupling strength saturation threshold value between nodes of a complex network;
step 2: establishing a mathematical model of saturation function constraint;
and step 3: and optimizing the synchronization capacity of the complex saturated network by using a balance point containment control strategy.
Consider a network consisting of N linearly coupled identical nodes (N being any positive integer) and each node being an N-dimensional dynamic system. The saturation function network comprises N nodes, and the mathematical model of the ith node at the time t is as follows:
in formula (I), f (-) is a given, non-linear, continuously differentiated, vector-valued function that describes the dynamic behavior of the node, xi(t)＝(xi1(t),xi2(t),...,xin(t))T∈RnIs the state vector of node i at time t; gamma is belonged to Rn×nIs a matrix of constants 0-1 reflecting the coupling of internal variables; a ═ aij]N×N＝ATRepresenting an inter-node coupling matrix; let a be if there is a connection (i ≠ j) between node i and node j (j ≠ 1, 2.., N)ij＝ajiOtherwise let a be 1ij＝ajiWhen i is equal to j, 0, when i is equal to j,
the coupling matrix a has N eigenvalues, which are written as: 0 > lambda2(A)≥…≥λN(A)。ci(t) is a function of the coupling strength of the ith node. u. ofi(t) is the feedback of node i. Assume that the number of selected holdback nodes is m.
In which feedback control ui(t) is:
in the formula (II)
di(t) > 0 is the saturation feedback gain;
the equilibrium point defining matrix B, L-diag (h), representing the function f (·)1d1(t),h2d2(t),…,hmdm(t),0, …,0), the characteristic values are as follows: 0 > lambda1(B)≥…≥λN(B)。
The saturation function of the coupling strength between nodes in the network is:
the saturation function of the feedback gain is:
wherein gamma isc＞0，γd> 0 saturation function gain of saturation function of coupling strength between nodes and saturation function of feedback gain, thetac＞0,θd> 0 is the error gain of the saturation function of the coupling strength between the nodes and the saturation function of the feedback gain, respectively.
When the saturation function satisfies ci(t) c and the feedback gain satisfies diWhen (t) ═ d. It is said that in a connectivity network, the coupling strength and feedback gain are fixed.
Examples
A classical BA scale-free network model was studied. The initial nodes of the network are 4 and are connected to each other. For each new node, 3 new edges are probabilistically generated between the new node and the existing network nodes, where the network size N is 50.
Isolated nodes in the network consist of a Lorentz system with typical chaotic characteristics. The mathematical description model of the Lorentz system is
Lorentz systems are a class of nonlinear dynamical systems that have a mixing characteristic under certain parameters. Since the lorentz system is bounded, equation (V) is established.
The lorentz system has three balance points: [8.4853,8.4853,27]T,[-8.4853,-8.4853,27]TAnd [0,0]TThe number of fixed nodes is set to m-5, assuming an initial state x of the systemi＝(xi1,xi2,...,xin)T∈Rn(i-1, …, N-3) obeys the random number between normal distributions (0 is mathematically expected, variance 5). Assume that the synchronization status of the system is
When the feedback gain d is 1 and 5, respectively, λ1(B) Is-3.062 and-2.96.
As shown in fig. 1, when the coupling strength is 10, the feedback gains d are 1 (fig. 1(a)) and 5 (fig. 1(b)), respectively, and the times at which the control network reaches the equilibrium point are 9 seconds and 4 seconds, respectively.
As shown in fig. 2, when the coupling strength is 20, the feedback gains d are 1 (fig. 2(a)) and 5 (fig. 2(b)), respectively, and the times at which the control network reaches the equilibrium point are 5 seconds and 3 seconds, respectively. When there is saturation in the coupling strength and feedback gain of the network, where γc＝20，γd＝5，θc＝0.5，θd＝0.5。
As shown in fig. 3, the controlled network can still reach the equilibrium point position in around 3 seconds.
Claims (6)
1. A balance point containment control method of a complex dynamic saturated network model is characterized by comprising the following steps: the method specifically comprises the following steps:
step 1: acquiring a function condition of a coupling strength saturation threshold value between nodes of a complex network;
step 2: establishing a mathematical model of saturation function constraint;
and step 3: and optimizing the synchronization capacity of the complex saturated network by using a balance point containment control strategy.
2. The balance point containment control method of the complex dynamic saturation network model according to claim 1, characterized in that: the establishment of the mathematical model comprises the following steps:
1) one network consists of N linearly coupled identical nodes, and each node is an N-dimensional dynamic system;
2) the saturation function network comprises N nodes, and the mathematical model of the ith node at the time t is as follows:
in the formula (I), N is any positive integer, f (-) is a vector value function and describes the dynamic behavior of the node, and xi(t)＝(xi1(t),xi2(t),...,xin(t))T∈RnIs the state vector of node i at time t; gamma is belonged to Rn×nIs a matrix of constants 0-1 that reflects the coupling of internal variables.
3. The balance point containment control method of the complex dynamic saturation network model according to claim 2, characterized in that: in the mathematical model, A ═ aij]N×N＝ATRepresenting an inter-node coupling matrix;
wherein, when there is a connection between node i and node j and i ≠ j, wherein j ═ 1,2ij＝ajiOtherwise let a be 1ij＝aji0; wherein, when i ═ j,
4. the balance point containment control method of the complex dynamic saturation network model according to claim 3, characterized in that: the coupling matrix A between the nodes has N eigenvalues, and the eigenvalues are written as: 0 > lambda2(A)≥…≥λN(A)；ci(t) is a function of the coupling strength of the ith node; u. ofi(t) is the feedback of node i;
assuming that the number of selected holdback nodes is m; in which feedback control ui(t) is:
in the formula (II)
di(t) > 0 is the saturation feedback gain;
represents the equilibrium point of the function f (·).
5. The balance point containment control method of the complex dynamic saturation network model according to claim 4, characterized in that: the step 3 comprises the following steps:
define matrix B ═ L-diag (h)1d1(t),h2d2(t),…,hmdm(t),0, …,0), the characteristic values are as follows: 0 > lambda1(B)≥…≥λN(B)；
The saturation function of the coupling strength between nodes in the network is:
the saturation function of the feedback gain is:
wherein gamma isc＞0，γd> 0 saturation function gain of saturation function of coupling strength between nodes and saturation function of feedback gain, thetac＞0,θd> 0 is the error gain of the saturation function of the coupling strength between the nodes and the saturation function of the feedback gain, respectively.
6. The balance point containment control method of the complex dynamic saturation network model according to claim 5, characterized in that: when the saturation function satisfies ci(t) c and the feedback gain satisfies diWhen (t) ═ d, the coupling strength and feedback gain are fixed in the connectivity network.
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| US20140176203A1 (en) * | 2012-10-26 | 2014-06-26 | California Institute Of Technology | Synchronization of nanomechanical oscillators |
| CN106249717A (en) * | 2016-08-29 | 2016-12-21 | 上海交通大学 | A kind of control method for coordinating based on the modeling of executor's saturated multi-agent system |
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| CN116846773A (en) * | 2023-06-15 | 2023-10-03 | 哈尔滨理工大学 | Complex network synchronous control method with bit rate constraint |
| CN116846773B (en) * | 2023-06-15 | 2024-04-05 | 哈尔滨理工大学 | A complex network synchronization control method with bit rate constraints |
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