CN112532475B - State estimation method of multilayer complex dynamic network - Google Patents

State estimation method of multilayer complex dynamic network Download PDF

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CN112532475B
CN112532475B CN202011318630.5A CN202011318630A CN112532475B CN 112532475 B CN112532475 B CN 112532475B CN 202011318630 A CN202011318630 A CN 202011318630A CN 112532475 B CN112532475 B CN 112532475B
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蒋国平
李坤
王欣伟
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses a state estimation method of a multilayer complex dynamic network, which comprises the following steps: (1) establishing mathematical models of multilayer dynamic networks with different quantity of nodes and different types of nodes in each layer and non-one-to-one correspondence of nodes between layers; (2) establishing a state observer network with the same topological structure and node dynamics as the multilayer dynamic network; (3) establishing error dynamics of a state observer network and a multilayer dynamic network; the error is the difference value between the node state of the multilayer dynamic network and the node state of the state observer network; (4) designing control parameters of a state observer network according to the progressive stability of error dynamics; (5) and obtaining the state estimation value of the multilayer dynamic network. The invention realizes the state estimation of the multilayer dynamic network with different node numbers, different node types and non-one-to-one correspondence of interlayer nodes, and the error between the node state of the state observer network and the node state of the original multilayer dynamic network can be converged to zero within 1.5 seconds and is kept stable.

Description

State estimation method of multilayer complex dynamic network
Technical Field
The invention relates to a network state estimation method, in particular to a state estimation method of a multilayer complex dynamic network.
Background
The complex dynamic network is formed by coupling a plurality of nodes with each other and is used for describing various systems in the real world, such as a communication network, a power network, a cellular neural network, a social relationship network and the like. Since the discovery of the small world and the non-scalability of complex dynamic networks, complex dynamic network research has received increasing attention in various fields.
Due to the fact that the number of complex dynamic network nodes is large, the connection relation between the complex dynamic network nodes is complicated, and limited by a communication mechanism, a working environment, network bandwidth and the like, only part of state information of the network nodes can be measured generally, and all state information of the network nodes is difficult to measure. In order to better understand the dynamic behavior of the complex dynamic network, better monitor the state change of the network node, and timely discover network faults and emergencies, it is necessary to establish a state estimator of the complex dynamic network node to monitor the state change of the node.
With the continuous and deep research of complex dynamic networks, people gradually realize that most networks in real society and engineering do not exist independently, but are structurally or functionally related to other networks to form a multi-layer network. At present, the research on the state estimation of the complex dynamic network is mainly focused on a single-layer complex dynamic network, and the state estimation research of a multi-layer complex dynamic network is not yet involved. Therefore, the state estimation research of the multi-layer complex dynamic network is particularly important and not very slow.
Disclosure of Invention
The purpose of the invention is as follows: the invention aims to solve the defects in the prior art, provides a state estimation method for a multilayer complex dynamic network, and solves the problem of state estimation of the multilayer complex dynamic network, wherein nodes in each layer are different in number and type, and nodes in each layer are not in one-to-one correspondence.
The technical scheme is as follows: the state estimation method of the multilayer complex dynamic network comprises the following steps:
(1) establishing mathematical models of multilayer dynamic networks with different quantity of nodes and different types of nodes in each layer and non-one-to-one correspondence of nodes between layers;
(2) establishing a state observer network with the same topological structure and node dynamics as the multilayer dynamic network;
(3) establishing error dynamics of a state observer network and a multilayer dynamic network; the error is the difference value between the node state of the multilayer dynamic network and the node state of the state observer network;
(4) obtaining control parameters of a state observer network according to the progressive stability of error dynamics;
(5) and obtaining the state estimation value of the multilayer dynamic network.
The model expression of the multilayer dynamic network in the step (1) is as follows:
Figure BDA0002792112540000021
Figure BDA0002792112540000022
wherein N isKAnd NRRespectively representing the number of nodes in the K-th and R-th layers,
Figure BDA0002792112540000023
the state variable of the ith node of the K (K is more than or equal to 1 and less than or equal to M) layer in the M layer network is represented,
Figure BDA0002792112540000024
representing the output variable of the ith node of the K-th network, i is more than or equal to 1 and less than or equal to NK；fK:Rn→RnIs a kinetic equation of the K-th network nodes; c. CKThe in-layer coupling strength of the K-th layer network;
Figure BDA0002792112540000025
is an in-layer coupling matrix of the K-th layer network, if a connecting edge from a node i to a node j exists, then
Figure BDA0002792112540000026
Otherwise
Figure BDA0002792112540000027
Is the interlayer coupling strength of the ith node of the K-th layer and the jth node of the R (1. ltoreq. R. ltoreq.M) th layer, and
Figure BDA0002792112540000028
Γ is the intra-layer and inter-layer inline matrices of the network node; h is the output matrix of the node.
The expression of the state observer network in the step (2) is as follows:
Figure BDA0002792112540000029
Figure BDA00027921125400000210
wherein the content of the first and second substances,
Figure BDA00027921125400000211
representing a state observation value of an ith node in a K-th layer network;
Figure BDA00027921125400000212
representing an output observation value of an ith node in a K-th layer network; gKGain matrix for control parameters of the state observer, GK＝[GK1 … GKn]。
The state observation error in the step (3)
Figure BDA00027921125400000213
The error dynamics of the system is then:
Figure BDA00027921125400000214
the step (4) comprises the following steps:
(41) design the Lyapunov function as
Figure BDA00027921125400000215
P is a positive definite symmetric matrix;
(42) solving a first derivative of the Lyapunov function in the step (41);
Figure BDA00027921125400000216
(43) let step (42) be
Figure BDA00027921125400000217
Gain matrix G for obtaining control parameters of state observerK。
Has the advantages that: compared with the prior art, the method has the obvious advantages that the state estimation of the multilayer dynamic network with different node numbers, different node types and non-one-to-one correspondence of the nodes between the layers is realized, and the error between the node state of the realized state observer network and the node state of the original multilayer dynamic network can be converged to zero within 1.5 seconds and is kept stable.
Drawings
FIG. 1 is a schematic diagram of the invention;
FIG. 2 is a schematic diagram of a multi-layer complex dynamic network according to the present invention;
FIG. 3 is a first dimension state observation error graph of a first layer network node of the present invention;
FIG. 4 is a second dimension state observation error graph for a first tier network node of the present invention;
FIG. 5 is a third dimensional state observation error graph of a first level network node of the present invention;
FIG. 6 is a first dimension state observation error graph of a second layer network node of the present invention;
fig. 7 is a second-dimensional state observation error map of a second-layer network node of the present invention;
fig. 8 is a third dimensional state observation error map of a second layer network node of the present invention.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
As shown in fig. 1, the method for estimating the state of the multi-layer complex dynamic network according to the present invention includes the following steps:
the state estimation method of the multilayer complex dynamic network comprises the following steps:
(1) establishing mathematical models of multilayer dynamic networks with different quantity of nodes and different types of nodes in each layer and non-one-to-one correspondence of nodes between layers;
(2) establishing a state observer network with the same topological structure and node dynamics as the multilayer dynamic network;
(3) establishing error dynamics of a state observer network and a multilayer dynamic network; the error is the difference value between the node state of the multilayer dynamic network and the node state of the state observer network;
(4) obtaining control parameters of a state observer network according to the progressive stability of error dynamics;
(5) and obtaining the state estimation value of the multilayer dynamic network.
The model expression of the multilayer dynamic network in the step (1) is as follows:
Figure BDA0002792112540000031
Figure BDA0002792112540000032
wherein N isKAnd NRRespectively representing the number of nodes in the K-th and R-th layers,
Figure BDA0002792112540000033
the state variable of the ith node of the K (K is more than or equal to 1 and less than or equal to M) layer in the M layer network is represented,
Figure BDA0002792112540000034
representing the output variable of the ith node of the K-th network, i is more than or equal to 1 and less than or equal to NK；fK:Rn→RnIs a kinetic equation of the K-th network nodes; c. CKThe in-layer coupling strength of the K-th layer network;
Figure BDA0002792112540000035
is an in-layer coupling matrix of the K-th layer network, if a connecting edge from a node i to a node j exists, then
Figure BDA0002792112540000036
Otherwise
Figure BDA0002792112540000037
Is the interlayer coupling strength of the ith node of the K-th layer and the jth node of the R (1. ltoreq. R. ltoreq.M) th layer, and
Figure BDA0002792112540000038
Γ is the intra-layer and inter-layer inline matrices of the network node; h is the output matrix of the node.
The expression of the state observer network in the step (2) is as follows:
Figure BDA0002792112540000041
Figure BDA0002792112540000042
wherein the content of the first and second substances,
Figure BDA0002792112540000043
representing a state observation value of an ith node in a K-th layer network;
Figure BDA0002792112540000044
representing an output observation value of an ith node in a K-th layer network; gKGain matrix for control parameters of the state observer, GK＝[GK1 … GKn]。
The state observation error in the step (3)
Figure BDA0002792112540000045
The error dynamics of the system is then:
Figure BDA0002792112540000046
the step (4) comprises the following steps:
(41) design the Lyapunov function as
Figure BDA0002792112540000047
P is a positive definite symmetric matrix;
(42) solving a first derivative of the Lyapunov function in the step (41);
Figure BDA0002792112540000048
(43) let step (42) be
Figure BDA0002792112540000049
Gain matrix G for obtaining control parameters of state observerK。
The structure of the multi-layer complex dynamic network in the embodiment is shown in FIG. 2, and the state equation of the mathematical model is
Figure BDA0002792112540000051
Wherein the content of the first and second substances,
c1＝c2＝1，H＝[1 0 0]，
Figure BDA0002792112540000052
Figure BDA0002792112540000053
Figure BDA0002792112540000054
Figure BDA0002792112540000055
Figure BDA0002792112540000061
Figure BDA0002792112540000062
Figure BDA0002792112540000063
the state observer network is designed as follows:
Figure BDA0002792112540000064
let error
Figure BDA0002792112540000065
The error kinetics are then:
Figure BDA0002792112540000066
let the Lyapunov function be:
Figure BDA0002792112540000067
solving the first derivative of the Lyapunov function
Figure BDA0002792112540000068
The following linear matrix inequalities can be obtained:
Figure BDA0002792112540000069
wherein the content of the first and second substances,
Figure BDA00027921125400000610
Λ＝diag(P,…,P)。
solving the linear matrix inequality by using Matlab can obtain:
Figure BDA0002792112540000071
Figure BDA0002792112540000072
and the gain matrix is brought into a state observer network to complete the state estimation of the multilayer complex dynamic network.
Fig. 3 to 8 are diagrams of observation errors of each state of each layer of network node in the present embodiment, respectively, and it can be known that the observation errors of each state tend to zero within 1.5 seconds and remain stable.

Claims (3)

1. A state estimation method of a multilayer complex dynamic network is characterized in that: the method comprises the following steps:
(1) establishing mathematical models of multilayer dynamic networks with different node numbers, different node types and non-one-to-one correspondence of interlayer nodes; the model expression of the multilayer dynamic network in the step (1) is as follows:
Figure FDA0003458437320000011
Figure FDA0003458437320000012
wherein N isKAnd NRRespectively representing the number of nodes in the K-th and R-th layers,
Figure FDA0003458437320000013
the state variable of the ith node of the K (K is more than or equal to 1 and less than or equal to M) layer in the M layer network is represented,
Figure FDA0003458437320000014
representing the output variable of the ith node of the K-th network, i is more than or equal to 1 and less than or equal to NK；fK:Rn→RnIs a kinetic equation of the K-th network nodes; c. CKThe in-layer coupling strength of the K-th layer network;
Figure FDA0003458437320000015
is an in-layer coupling matrix of the K-th layer network, if a connecting edge from a node i to a node j exists, then
Figure FDA0003458437320000016
Otherwise
Figure FDA0003458437320000017
Figure FDA0003458437320000018
Is the interlayer coupling strength of the ith node of the K-th layer and the jth node of the R (1. ltoreq. R. ltoreq.M) th layer, and
Figure FDA0003458437320000019
Γ is the intra-layer and inter-layer inline matrices of the network node; h is an output matrix of the node;
(2) establishing a state observer network with the same topological structure and node dynamics as the multilayer dynamic network;
(3) establishing error dynamics of a state observer network and a multilayer dynamic network; the error is the difference value between the node state of the multilayer dynamic network and the node state of the state observer network;
(4) obtaining control parameters of a state observer network according to the progressive stability of error dynamics; the step (4) comprises the following steps:
(41) design the Lyapunov function as
Figure FDA00034584373200000110
P is a positive definite symmetric matrix;
(42) obtaining a first derivative of the Lyapunov function in step (41)
Figure FDA00034584373200000111
(43) Let step (42) be
Figure FDA00034584373200000112
Obtaining a control parameter gain matrix G of the state observerK；
(5) Will gain matrix GKAnd substituting the state estimation value into a state observer network to complete the state estimation of the multilayer complex dynamic network and obtain the state estimation value of the multilayer dynamic network.
2. The state estimation method of the multi-layer complex dynamic network according to claim 1, wherein: the expression of the state observer network in the step (2) is as follows:
Figure FDA00034584373200000113
Figure FDA00034584373200000114
wherein the content of the first and second substances,
Figure FDA0003458437320000021
representing a state observation value of an ith node in a K-th layer network;
Figure FDA0003458437320000022
representing an output observation value of an ith node in a K-th layer network; gKGain matrix for control parameters of the state observer, GK＝[GK1 ··· GKn]。
3. The state estimation method of the multi-layer complex dynamic network according to claim 2, wherein: the state observation error in the step (3)
Figure FDA0003458437320000023
The error dynamics of the system is then:
Figure FDA0003458437320000024
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