CN112363388B - Dynamic classification control method for complex network nodes based on connection relation observer - Google Patents

Dynamic classification control method for complex network nodes based on connection relation observer Download PDF

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CN112363388B
CN112363388B CN202011222682.2A CN202011222682A CN112363388B CN 112363388 B CN112363388 B CN 112363388B CN 202011222682 A CN202011222682 A CN 202011222682A CN 112363388 B CN112363388 B CN 112363388B
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connection relation
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王颀
熊建斌
岑健
胡俊敏
陈康
刘鸣慧
钱文博
刘文浩
李春林
余得正
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Guangdong Polytechnic Normal University
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Abstract

The invention discloses a complex network node dynamic classification control method based on a connection relation observer, which comprises the following steps: s1, aiming at an undirected complex network, proving the rationality of using a Riccati matrix differential equation as a connection relation subsystem model; s2, designing a coupling item form of a special connection relation subsystem so that the connection relation subsystem and the node subsystem are mutually coupled in the complex network; s3, designing a state observer of the connection relation aiming at the connection relation subsystem; s4, designing controllers aiming at the connection relation subsystem and the node subsystem by using information in the connection relation subsystem observer, so that the nodes of the complex network are dynamically classified. The invention designs the controller aiming at the connection relation subsystem by using the information in the observer, so that the complex network asymptotically tracks a known sortable network, thereby achieving the purpose of dynamic classification of the complex network.

Description

Dynamic classification control method for complex network nodes based on connection relation observer
Technical Field
The invention belongs to the technical field of automatic control, and particularly relates to a dynamic classification control method for complex network nodes based on a connection relation observer.
Background
There are many dynamic complex systems in real life, which can be described as dynamic complex networks from the mathematical graph point of view. Like many complex systems in reality, the nodes of a complex network can also be divided into many categories. Thus, in the past years, complex network nodes have been categorized as a hot topic and have emerged as a vast array of research efforts. These algorithms are all partitioned by the density of connection relationships between nodes.
However, these algorithms described above do not take into account the sign of the connection relationship and are therefore not applicable to sign networks. In real life, many social networks can be abstracted into symbol networks, and research on node classification methods suitable for the symbol networks has important significance. In the existing related research results, one type of algorithm is developed from the traditional unsigned network node classification algorithm, such as an FEC algorithm, a Laplacian algorithm and the like. However, these algorithms do not yield accurate classification results because they do not take full advantage of the information provided by the negative connection in the network, although they take into account the negative connection. Amelio et al propose genetic algorithm, which takes the weight sum of negative connection relations in the same class as an objective function, and achieves the aim of classifying the nodes of the symbol network by continuously optimizing the function; jiang et al propose an SSBM model to measure the blocking of the network, as a statistical probability model whose parameters reflect the probability that nodes belong to different classes and the centrality of each node in its class. The algorithms classify nodes according to the density of the connection relations, which means that the algorithms cannot strictly ensure that the connection relations between nodes in the same class are positive and the connection relations between nodes in different classes are non-positive. In other words, these algorithms are not aware of the important role of the connection relation symbols in node classification.
In fact, the sign of the connection relationship plays an important role in the sign network. For example, in a social network, friendly, cooperative relationships between individuals are often represented by positive connections, and hostile, counterrelationships between individuals are often represented by negative connections. In the neural network, the positive connection relationship and the negative connection relationship between neurons represent mutual promotion and inhibition between neurons, respectively. Therefore, in a symbol network, it is important to study how to classify nodes by symbols of connection relations.
Some scholars have conducted some studies. For example, wang et al have studied the full necessity that generalized symbol network nodes can be classified based on the concept of structural holes and hole masters (broaders). The document states that generalized symbol networks can be divided into several categories when they do not appear in a hole master (Broker) structure. The above algorithm is proposed for static symbol networks and is not applicable to dynamic networks.
For dynamic symbol networks Chen et al propose a node phase based DEC algorithm whose core idea is to gradually bring together two nodes connected by a positive connection and gradually separate two nodes connected by a negative connection by a change of node phase. However, this method is only applicable to the case where the state of the node is a one-dimensional variable, and if the state of the node is multidimensional, the above method is not applicable. Gao et al propose a complex network control strategy based on node adaptation, which ultimately divides the nodes of the network into two categories and does not multiclassify the nodes of the symbol network.
Therefore, aiming at the defects of the research results, the invention provides a complex network node dynamic classification control method based on a connection relation observer. First, a complex network is considered as a large system of coupled node subsystems and connection relationship subsystems. The state observer design is carried out on the state (connection relation weight) of the connection relation subsystem, and the controller aiming at the connection relation subsystem is designed by using the information in the observer, so that the complex network asymptotically tracks a known sortable network, and the purpose of dynamically classifying the complex network is achieved.
Disclosure of Invention
The invention aims to solve the defects in the prior art, and provides a complex network node dynamic classification control method based on a connection relation observer.
In order to achieve the above purpose, the present invention provides the following technical solutions:
a complex network node dynamic classification control method based on a connection relation observer comprises the following steps:
s1, aiming at an undirected complex network, proving that a Riccati matrix differential equation can be used as a result of approximate linearization of the undirected complex network under certain assumption conditions, thereby proving rationality of using the Riccati matrix differential equation as a connection relation subsystem model aiming at the undirected network;
s2, designing a coupling item form of a special connection relation subsystem so that the connection relation subsystem and the node subsystem are mutually coupled in the complex network;
s3, designing a state observer of the connection relation aiming at the connection relation subsystem;
s4, designing controllers aiming at the connection relation subsystem and the node subsystem by using information in the connection relation subsystem observer, so that the nodes of the complex network are dynamically classified.
Preferably, in the step S1, for the undirected complex dynamic network, it is proved that the Riccati matrix differential equation is the result of approximate linearization of the dynamic equation through a series of assumption conditions, so as to further explain the rationality of using the Riccati matrix differential equation as the connection subsystem model.
Preferably, in the step S2, the following shape is designed: Φ (z) =Λ (z) +Λ (z) T Coupling terms of the connection relation subsystem in the formula, wherein Λ (z) =j -1 Γ T ζ(z)J -1 And ζ (z) is less than or equal to eta (t)||z|| 2 ，η(t)＞0。
Preferably, in the step S3, a shape is designed as follows:
Figure BDA0002762595550000031
connection relation subsystem state observer and +.>
Figure BDA0002762595550000032
And the robust term in the formula is used for observing the unknown connection relation state.
Preferably, in the step S4, by using information of the connection relation subsystem state observer, a shape is designed as follows:
Figure BDA0002762595550000033
directed to a node subsystem
Figure BDA0002762595550000041
The formula aims at the control of the connection relation subsystem to realize the dynamic classification of the complex network nodes.
Compared with the prior art, the invention has the beneficial effects that: compared with the prior art, the method treats the complex network as a large system formed by coupling a node subsystem and a connection relationship subsystem. The state observer design is carried out on the state (connection relation weight) of the connection relation subsystem, and the controller aiming at the connection relation subsystem is designed by using the information in the observer, so that the complex network asymptotically tracks a known sortable network, and the purpose of dynamically classifying the complex network is achieved.
Detailed Description
The present invention will be described in further detail with reference to specific embodiments in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
A complex network node dynamic classification control method based on a connection relation observer comprises the following steps:
s1, aiming at an undirected complex network, proving that a Riccati matrix differential equation can be used as a result of approximate linearization of the undirected complex network under certain assumption conditions, thereby proving rationality of using the Riccati matrix differential equation as a connection relation subsystem model aiming at the undirected network;
s2, designing a coupling item form of a special connection relation subsystem so that the connection relation subsystem and the node subsystem are mutually coupled in the complex network;
s3, designing a state observer of the connection relation aiming at the connection relation subsystem;
s4, designing controllers aiming at the connection relation subsystem and the node subsystem by using information in the connection relation subsystem observer, so that the nodes of the complex network are dynamically classified.
Specifically, in the step S1, for the undirected complex dynamic network, a series of assumption conditions prove that the Riccati matrix differential equation is a result of approximate linearization of the dynamic equation, and further demonstrate the rationality of using the Riccati matrix differential equation as a connection relationship subsystem model.
Specifically, in the step S2, the following shapes are designed: Φ (z) =Λ (z) +Λ (z) T Coupling terms of the connection relation subsystem in the formula, wherein Λ (z) =j -1 Γ T ζ(z)J -1 And ζ (z) is less than or equal to eta (t) z 2 ，η(t)＞0。
Specifically, in the step S3, the following shapes are designed:
Figure BDA0002762595550000051
connection relation subsystem state observer and +.>
Figure BDA0002762595550000052
Robust term in the equation for observing unknown connection state (connectionThe join relation weight).
Specifically, in the step S4, by using the information of the connection relation subsystem state observer, the following shape is designed:
Figure BDA0002762595550000053
directed to a node subsystem
Figure BDA0002762595550000054
The formula aims at the control of the connection relation subsystem to realize the dynamic classification of the complex network nodes.
Considering a time-varying, undirected, complex dynamic network of all real connections made up of N nodes, when the coupling relationship of a node to its wiring is not considered, the dynamic equations of an isolated connection relationship subsystem can be generally expressed as:
Figure BDA0002762595550000055
wherein x is ij (t) represents a connection relationship between the node i and the node j at the time t; x= (X ij )∈R N×N Representing a state matrix of the connection relation subsystem, f ij (X) is about X ij N of (2) 2 A primitive smoothing function, i, j=1, 2, …, N, satisfies f ij (X)＝f ji (X)。
Definition 1 consider equation (1), if a constant matrix is present
Figure BDA0002762595550000061
For any i, j e {1,2, …, N } satisfies f ij (X * ) Identical to 0, then x=x * Is a balance matrix of the connection relation subsystem (1).
Note F (X) = (F) ij (X)) N×N . The dynamic equation (1) can be written in the form of a matrix as follows:
Figure BDA0002762595550000062
thus, by definition 1 and dynamic equation (2) It can be seen that if and only if F (X * ) X=x at≡0 * Is a balance matrix of the network connection relation subsystem (2).
Mapping matrix X into vector space by a straightening operation, then balance matrix x=x in definition 1 * Can be regarded as the equilibrium point (state) of the dynamic equation (2) in the sense of Lyapunov, and notice the Euclidean norm X-X * ||＝||vec(X)-vec(X * ) I, thus for balance matrix x=x * In other words, the concepts of stability, asymptotic stability, etc. in the sense of Lyapunov can naturally be extended into the equation of dynamic equation (2).
Definition 2 if the balance matrix x=x of the dynamic equation (2) * Is stable (asymptotically stable) in the sense of Lyapunov, then the network connection subsystem represented by the equation of dynamic equation (2) is said to be stable (asymptotically stable).
Considering the network connection relation subsystem (2), the balance matrix is
Figure BDA0002762595550000063
Obviously, dynamic equation (2) can be converted into differential equation in vector form using matrix-straightening mapping>
Figure BDA0002762595550000064
The differential equation can thus be discussed at the equilibrium point vec (X using the approximate linearization method in Lyapunov stability theory * ) The approximate linearization problem at that point. However, such approximate linearization results have two drawbacks, one is that the state dimension of the approximated linear system increases to N 2 For complex networks with a plurality of nodes, the calculated amount is increased sharply, and the intuitiveness of the connection relationship between the network nodes is destroyed (obviously vec (X) is not as intuitive as X). Therefore, we will discuss dynamic equation (2) in its balance matrix from some characteristics of the connection relationship between network nodes>
Figure BDA0002762595550000065
Linearization problem at that point.
The following research results show that under certain conditions, the approximate linearization dynamic equation of the complex network connection subsystem is a Riccati matrix differential equation.
Considering dynamic equation (1), note f ij (X)＝f ji (X) we need the following assumptions.
Let 1 consider the dynamic equation (1), f ij (X) may be expressed in the form:
f ij (X)＝δ ij (x i1 x i2 … x iN )+δ ji (x j1 x j2 … x jN ) (3)
wherein the smoothing function delta ij (·)＝δ ji (. Cndot.) and satisfy
Figure BDA0002762595550000071
(a) Assume that equation (3) in 1 illustrates that for a given i, j, the function f ij (X) is only about X ik 、x jk A function of (k=1, 2, …, N) (the number of independent variables is 2N at most, not N 2 And (c) a). (b) Due to x ij Representing the weight of the connection between the ith node and the jth node, it is assumed that equation (3) in 1 means that there is only "connection x related to the ith node" in the network ik (k=1, 2, …, N) "and" connection relation x related to the jth node jk (k=1, 2, …, N) "affects x ij Rate of change of (2)
Figure BDA0002762595550000072
If 1 is assumed to be true, δ ij (·)＝δ ji (. Cndot.) and thus is easy to see
Figure BDA0002762595550000073
Record->
Figure BDA0002762595550000074
Figure BDA0002762595550000075
There is->
Figure BDA0002762595550000076
The function (3) is set at +.>
Figure BDA0002762595550000077
The expansion of the position can be obtained:
Figure BDA0002762595550000078
Figure BDA0002762595550000081
wherein,,
Figure BDA0002762595550000082
respectively indicate about->
Figure BDA0002762595550000083
Figure BDA0002762595550000084
Is infinitely small.
In Taylor formula (4), the higher order term is ignored
Figure BDA0002762595550000085
The following approximation equation is available:
Figure BDA0002762595550000086
consider the following N x N order real matrices:
Figure BDA0002762595550000087
memory matrix
Figure BDA0002762595550000088
Wherein->
Figure BDA0002762595550000089
If 1 is assumed to be true, then dynamic equation (2) can be found in balance matrix x=x using equations (5) and (6) * A first approximation of:
Figure BDA00027625955500000810
assuming that the N matrices in equation (6) are all equal, i.e., there is a constant matrix P that satisfies P k ＝P，k＝1,2,…,N。
It can be easily seen that the presence of an N-ary real function δ (·) results in a smooth function δ in hypothesis 1 ij (·)＝δ ji (·) =δ (·), then assuming 5.6 holds, this means the connection relationship x in the network ik 、x jk (k=1, 2, …, N) vs x ij Rate of change
Figure BDA0002762595550000091
The manner of influence is the same.
Identity matrix easy to verify
Figure BDA0002762595550000092
Thus, when 1-2 is assumed to hold, the dynamic equation (7) can be reduced to:
Figure BDA0002762595550000093
namely:
Figure BDA0002762595550000094
the dynamic equation (8) is a Riccati-type matrix differential equation, which also illustrates from another perspective the rationality of using the Riccati differential equation to describe the dynamic change of social network connections.
Herein, consider a generalized symbol network comprising N nodes, the dynamic equation for each node i satisfying the following equation:
Figure BDA0002762595550000095
wherein z is i ＝[z i1 z i2 … z in ] T ∈R n Representing the node state vector of the i-th node. A is that i ∈R n×n ；f i (z i )＝[f i1 (z i1 ) f i2 (z i2 ) … f in (z in )] T ∈R n Representing a continuous nonlinear function vector. G j (z j )＝[G j1 (z j1 ) G j2 (z j2 ) … G jn (z jn )] T Representing an inter-coupling nonlinear function vector. c i Indicating the coupling strength. u (u) i Indicating the control action exerted on node i. X is x ij Representing the connection relation weight between node i and node j.
The lead-in vector z= [ z ] 1 T z 2 T … z N T ] T ；A＝diag(A 1 A 2 … A N )；f(z)＝[f 1 (z 1 ) T f 2 (z 2 ) T … f N (z N ) T ] T ∈R Nn ；c＝diag(c 1 Π c 2 Π … c N Π)，Π∈R n A full one vector representing n dimensions; x= (X ij (t)) N×N ；G(z)＝[G 1 (z 1 ) T G 2 (z 2 ) T … G N (z N ) T ] T ∈R Nn ；u＝[u 1 T u 2 T … u N T ] T 。
In connection with the definition of the Kronecker product, the node subsystem dynamic equation (9) can be rewritten as:
Figure BDA0002762595550000101
the connection relation subsystem is considered to satisfy the following Riccati matrix differential equation:
Figure BDA0002762595550000102
wherein P is E R N×N Is a real matrix, Θ is a known constant matrix, Φ (z) ∈R N×N The coupling relation is the coupling relation between the connection relation subsystem and the node subsystem, and U is the control input of the connection relation subsystem. Y εR p×N Is the state output of the connection relation subsystem, Γ e R p×N Is the output gain matrix.
Depending on the nature of the straightening operation and the Kronecker product, the equation (11) can be rewritten as:
Figure BDA0002762595550000103
wherein,,
Figure BDA0002762595550000104
representing an N-order identity matrix.
Suppose 3 assumes that (P, Γ) is fully stabile for the connection relation subsystem (11). I.e. there is a matrix W.epsilon.R N×p Let P + wΓ be a Hurwitz matrix.
If 3 is assumed to be true, then there is a positive definite matrix J εR N×N So that for any matrix Q > 0, the following Lyapunov equation is satisfied:
(P+WΓ) T J+J(P+WΓ)＝-Q (13)
the following equation holds if the equations 3 and (13) hold in the quotation 1:
Figure BDA0002762595550000105
Figure BDA0002762595550000111
wherein,,
Figure BDA0002762595550000112
and (3) proving: from the formula (13), we can obtain:
Figure BDA0002762595550000113
Figure BDA0002762595550000114
by using the nature of the Kronecker product, it is known from formulas (14) and (15):
Figure BDA0002762595550000115
Figure BDA0002762595550000116
multiplying both sides (16 a) and (16 b) simultaneously
Figure BDA0002762595550000117
And +.>
Figure BDA0002762595550000118
And notice that
Figure BDA0002762595550000119
The method can obtain the following steps:
Figure BDA00027625955500001110
Figure BDA00027625955500001111
and the quotation mark 1 is obtained.
Suppose 4 suppose that Φ (z) in the connection relation subsystem (11) satisfies the following equation:
Φ(z)＝Λ(z)+Λ(z) T (18)
wherein Λ (z) =j -1 Γ T ζ(z)J -1 And ζ (z) is less than or equal to eta (t) z 2 ，η(t)＞0。
Considering that the state of the connection subsystem is not measurable, before the controller is designed, the following state observer is designed to estimate the state of the connection subsystem:
Figure BDA00027625955500001112
wherein,,
Figure BDA0002762595550000121
the estimated value of the state matrix X of the connection relation subsystem at the time t is shown. />
Figure BDA0002762595550000122
Representing the output of the state observer, +.>
Figure BDA0002762595550000123
Representing a robust term.
In an observer system (5.52), robust terms
Figure BDA0002762595550000124
The method meets the following conditions:
Figure BDA0002762595550000125
wherein,,
Figure BDA0002762595550000126
using the Kronecker product and the correlation properties of the straightening operation, one can obtain:
Figure BDA0002762595550000127
2 if 3 is assumed to be true, then the error between the estimated state in (19) and the state of the connection subsystem
Figure BDA0002762595550000128
Is asymptotically stable.
And (3) proving: using the formulas (10) and (19), it is possible to obtain:
Figure BDA0002762595550000129
selecting positive definite function
Figure BDA00027625955500001210
Its track derivative is:
Figure BDA00027625955500001211
Figure BDA0002762595550000131
from (23), it can be seen that the estimation error E is bounded, and
Figure BDA0002762595550000132
and 2, obtaining the syndrome of the quotation mark.
Control target: let X be * ∈R N×N The network connection relation matrix may be classified for a known node. By using states in a state observer (19)
Figure BDA0002762595550000133
And the state z (t) of the node subsystem (10), designed for the connection subsystem (11)Control U and design control U for node subsystem such that state X (t) of connection relationship subsystem tracks known classifiable network connection relationship matrix X * And the state of the node subsystem is guaranteed to be bounded, so that node classification is realized.
In order to achieve the above control objective, the following node subsystem controller u is selected:
Figure BDA0002762595550000141
selecting a controller U of the following connection relation subsystem:
Figure BDA0002762595550000142
note e = X-X * The following steps are:
Figure BDA0002762595550000143
theorem 1 considers a generalized symbol network composed of a node subsystem shown in (10) and a connection relationship subsystem shown in formula (11). A state observer of form (17) is built for the connection relationship subsystem. If 3-4 is assumed to be true, under the action of controllers (24) and (25), the state X of the connection subsystem may asymptotically track the given node-sortable network connection matrix X * And the status of the node subsystem
Figure BDA0002762595550000144
And (3) proving: selecting a positive definite function
Figure BDA0002762595550000145
Its derivative with respect to time is:
Figure BDA0002762595550000146
as can be seen from the quotation 2,
Figure BDA0002762595550000147
then, as shown in the formula (27):
Figure BDA0002762595550000148
from equation (28), the error in the error system (26) equation
Figure BDA0002762595550000149
And the status of the node subsystem->
Figure BDA00027625955500001410
The syndrome is known.
To sum up: compared with the prior art, the method treats the complex network as a large system formed by coupling a node subsystem and a connection relationship subsystem. The state observer design is carried out on the state (connection relation weight) of the connection relation subsystem, and the controller aiming at the connection relation subsystem is designed by using the information in the observer, so that the complex network asymptotically tracks a known sortable network, and the purpose of dynamically classifying the complex network is achieved.
Finally, it should be noted that: the foregoing description is only illustrative of the preferred embodiments of the present invention, and although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that modifications may be made to the embodiments described, or equivalents may be substituted for elements thereof, and any modifications, equivalents, improvements or changes may be made without departing from the spirit and principles of the present invention.

Claims (1)

1. A complex network node dynamic classification control method based on a connection relation observer is characterized in that: the method comprises the following steps:
s1, aiming at an undirected complex network, proving that a Riccati matrix differential equation can be used as a result of approximate linearization of the undirected complex network under certain assumption conditions, thereby proving rationality of using the Riccati matrix differential equation as a connection relation subsystem model aiming at the undirected network;
aiming at an undirected complex dynamic network, a series of assumption conditions prove that the Riccati matrix differential equation is the result of approximate linearization of a dynamic equation, and further the rationality of using the Riccati matrix differential equation as a connection relation subsystem model is demonstrated;
s2, designing a coupling term form of a special connection relation subsystem, wherein phi (z) =Λ (z) +Λ (z) T Coupling terms of the connection relation subsystem in the formula, wherein Λ (z) =j -1 Γ T ζ(z)J -1 And ζ (z) is less than or equal to eta (t) z 2 η (t) > 0, so that the connection relation subsystem and the node subsystem in the complex network are coupled with each other;
s3, aiming at the connection relation subsystem,
Figure FDA0004236913300000011
connection relation subsystem state observer and +.>
Figure FDA0004236913300000012
The robust term in the formula is used for observing the unknown connection relation state and designing a state observation detector of the connection relation;
s4, using information in the connection relation subsystem observer, designing a controller aiming at the connection relation subsystem and the node subsystem,
Figure FDA0004236913300000013
directed to a node subsystem
Figure FDA0004236913300000014
Aiming at the control of the connection relation subsystem, the dynamic classification of the nodes of the complex network is realized, so that the nodes of the complex network are realizedDynamic classification;
considering a time-varying undirected complex dynamic network of all real connections consisting of N nodes, when the coupling relationship of a node to its connection is not considered, the dynamic equation of an isolated connection relationship subsystem is expressed as:
Figure FDA0004236913300000021
wherein x is ij (t) represents a connection relationship between the node i and the node j at the time t; x= (X ij )∈R N×N Representing a state matrix of the connection relation subsystem, f ij (X) is about X ij N of (2) 2 A primitive smoothing function, i, j=1, 2, …, N, satisfies f ij (X)＝f ji (X)；
Definition 1 consider equation (1), if a constant matrix is present
Figure FDA0004236913300000022
For any i, j e {1,2, …, N } satisfies f ij (X * ) Identical to 0, then x=x * Is the balance matrix of the dynamic equation (1);
note F (X) = (F) ij (X)) N×N The method comprises the steps of carrying out a first treatment on the surface of the The dynamic equation (1) is written in the form of a matrix as follows:
Figure FDA0004236913300000023
thus, as can be seen from definition 1 and dynamic equation (2), if and only if F (X * ) X=x at≡0 * Is the balance matrix of the dynamic equation (2);
if the balance matrix of the equation (2) is x=x * Is stable in the Lyapunov sense, and the network connection relation subsystem expressed by the dynamic equation (2) is said to be stable;
consider the dynamic equation (2), whose balance matrix is
Figure FDA0004236913300000024
Converting dynamic equation (2) into differential equation in vector form using matrix-straightening mapping>
Figure FDA0004236913300000025
Considering dynamic equation (1), note f ij (X)＝f ji (X), consider dynamic equation (1), f ij (X) may be expressed in the form:
f ij (X)＝δ ij (x i1 x i2 …x iN )+δ ji (x j1 x j2 …x jN ) (3)
wherein the smoothing function delta ij (·)＝δ ji (. Cndot.) and satisfy
Figure FDA0004236913300000026
Assume that equation (3) in 1 illustrates that for a given i, j, the function f ij (X) is only about X ik 、x jk K=1, 2, …, function of N due to x ij Representing the weight of the connection between the ith node and the jth node, equation (3) therefore means that there is only "connection x associated with the ith node" in the network ik K=1, 2, …, N, "and" connection relation x related to the jth node jk K=1, 2, …, N "affects x ij Rate of change of (2)
Figure FDA0004236913300000031
δ ij (·)＝δ ji (. Cndot.) and thus is easy to see
Figure FDA0004236913300000032
Recording device
Figure FDA0004236913300000033
There is->
Figure FDA0004236913300000034
The function (3) is set at +.>
Figure FDA0004236913300000035
The expansion of the position can be obtained:
Figure FDA0004236913300000036
wherein,,
Figure FDA0004236913300000037
respectively indicate about->
Figure FDA0004236913300000038
Figure FDA0004236913300000039
Is infinitely small;
in Taylor formula (4), the higher order term is ignored
Figure FDA0004236913300000041
The following approximation equation is obtained:
Figure FDA0004236913300000042
consider the following N x N order real matrices:
Figure FDA0004236913300000043
memory matrix
Figure FDA0004236913300000044
Wherein->
Figure FDA0004236913300000045
Obtained by using equations (5) and (6)Dynamic equation (2) is found in the balance matrix x=x * A first approximation of:
Figure FDA0004236913300000046
the N matrices in equation (6) are all equal, i.e. there is a constant matrix P that satisfies P k ＝P，k＝1,2,…,N；
The dynamic equation (7) is simplified as:
Figure FDA0004236913300000047
namely:
Figure FDA0004236913300000048
considering a generalized symbol network comprising N nodes, the dynamic equation for each node i satisfies the following equation:
Figure FDA0004236913300000049
wherein z is i ＝[z i1 z i2 …z in ] T ∈R n Node state vector representing the ith node, A i ∈R n×n ；f i (z i )＝[f i1 (z i1 ) f i2 (z i2 )…f in (z in )] T ∈R n Representing a continuous nonlinear function vector, G j (z j )＝[G j1 (z j1 ) G j2 (z j2 )…G jn (z jn )] T Representing an inter-coupled nonlinear function vector, c i Represents the coupling strength, u i Representing the control action exerted on node i, x ij Representing the weight of the connection relation between the node i and the node j;
introduction ofVector z= [ z ] 1 T z 2 T …z N T ] T ；A＝diag(A 1 A 2 …A N )；
Figure FDA0004236913300000051
c＝diag(c 1 Π c 2 Π…c N Π)，Π∈R n A full one vector representing n dimensions; x= (X ij (t)) N×N ；G(z)＝[G 1 (z 1 ) T G 2 (z 2 ) T …G N (z N ) T ] T ∈R Nn ；/>
Figure FDA0004236913300000052
In connection with the definition of the Kronecker product, node subsystem dynamics equation (9) is rewritten as:
Figure FDA0004236913300000053
the connection relation subsystem is considered to satisfy the following Riccati matrix differential equation:
Figure FDA0004236913300000054
wherein P is E R N×N Is a real matrix, Θ is a known constant matrix, Φ (z) ∈R N×N Is the coupling relation between the connection relation subsystem and the node subsystem, U is the control input of the connection relation subsystem, Y is E R p×N Is the state output of the connection relation subsystem, Γ e R p×N Is the matrix of the output gains which are,
based on the nature of the straightening operation and Kronecker product, the formula (11) is rewritten as:
Figure FDA0004236913300000055
wherein,,
Figure FDA0004236913300000056
I N represents an N-order identity matrix, and positive definite matrix J epsilon R N×N So that for any matrix Q > 0, the following Lyapunov equation is satisfied:
(P+WΓ) T J+J(P+WΓ)＝-Q (13)
Figure FDA0004236913300000057
Figure FDA0004236913300000061
wherein,,
Figure FDA0004236913300000062
and (3) proving: from the formula (13), it is obtained:
Figure FDA0004236913300000063
Figure FDA0004236913300000064
Figure FDA0004236913300000065
Figure FDA0004236913300000066
multiplying both sides (16 a) and (16 b) simultaneously
Figure FDA0004236913300000067
And +.>
Figure FDA0004236913300000068
And notice that
Figure FDA0004236913300000069
The method comprises the following steps:
Figure FDA00042369133000000610
Figure FDA00042369133000000611
；
suppose that Φ (z) in Riccati matrix differential equation (11) satisfies the following equation:
Φ(z)＝Λ(z)+Λ(z) T (18)
wherein Λ (z) =j -1 Γ T ζ(z)J -1 And ζ (z) is less than or equal to eta (t) z 2 ，η(t)＞0；
The following state observer is first designed to estimate the state of the connection relation subsystem:
Figure FDA00042369133000000612
wherein,,
Figure FDA00042369133000000613
representing an estimated value of a state matrix X of the connection relation subsystem at a time t; />
Figure FDA00042369133000000614
Representing the output of the state observer, +.>
Figure FDA00042369133000000615
Representing a robust term;
robust item
Figure FDA0004236913300000071
The method meets the following conditions:
Figure FDA0004236913300000072
wherein,,
Figure FDA0004236913300000073
using the Kronecker product and the correlation properties of the straightening operation, we get:
Figure FDA0004236913300000074
and (3) proving: using the formulas (10) and (19), we obtain:
Figure FDA0004236913300000075
selecting positive definite function
Figure FDA0004236913300000076
Its track derivative is:
Figure FDA0004236913300000077
Figure FDA0004236913300000081
from (23), it can be seen that the estimation error E is bounded, and
Figure FDA0004236913300000082
control target: let X be * ∈R N×N A network connection relation matrix can be classified for a known node; by using states in a state observer (19)
Figure FDA0004236913300000083
And the state z (t) of the node subsystem (10), designing a control U for the connection relation subsystem satisfying the Riccati matrix differential equation (11) and designing the control U for the node subsystem such that the state X (t) of the connection relation subsystem tracks a known classifiable network connection relation matrix X * The state of the node subsystem is guaranteed to be bounded, and node classification is further achieved;
in order to achieve the above control objective, the following node subsystem controller u is selected:
Figure FDA0004236913300000084
selecting a controller U of the following connection relation subsystem:
Figure FDA0004236913300000085
note e = X-X * The following steps are:
Figure FDA0004236913300000091
consider a generalized symbol network consisting of a node subsystem shown in (10) and a connection relationship subsystem satisfying the formula (11); constructing a state observer of form (17) for the connection relationship subsystem; under the action of the controllers (24) and (25), the state X of the connection relation subsystem asymptotically tracks the given node-sortable network connection matrix X * And the status of the node subsystem
Figure FDA0004236913300000092
And (3) proving: selecting a positive definite function
Figure FDA0004236913300000093
Its derivative with respect to time is:
Figure FDA0004236913300000094
Figure FDA0004236913300000095
then, as shown in the formula (27):
Figure FDA0004236913300000096
from equation (28), the error in the error system (26) equation
Figure FDA0004236913300000097
And the status of the node subsystem->
Figure FDA0004236913300000098
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Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN1725131A (en) * 2004-07-21 2006-01-25 韩京清 Three-parameter fastest self-anti-interference controller device and self-anti-interference control method
DE60129905D1 (en) * 2000-10-02 2007-09-27 Nissan Motor Driver assistance system for a vehicle
CN106773696A (en) * 2016-12-28 2017-05-31 广东技术师范学院 Based on the feedback of status linear control method that deviation is separate
CN107710687A (en) * 2015-06-29 2018-02-16 高通股份有限公司 The customized network business model abnormal for detecting application
CN108508749A (en) * 2018-05-07 2018-09-07 北京航空航天大学 A kind of anti-interference iterative learning control method of Space Manipulator System for arresting noncooperative target
CN109001974A (en) * 2018-08-01 2018-12-14 浙江工业大学 The big coupled movements system position synchronisation control means of networking based on broad sense extended state observer
CN111290277A (en) * 2020-02-26 2020-06-16 鹏城实验室 Distributed multi-agent cooperative fault detection method, storage medium and equipment

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
EP1986317B1 (en) * 2007-04-27 2009-10-21 ABB Oy Stator resistance adaptation in sensorless PMSM drives
AU2015362611B2 (en) * 2014-12-16 2018-05-17 Aspin Kemp & Associates Holding Corp. Anti-recoil control design using a hybrid riser tensioning system in deepwater drilling

Patent Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
DE60129905D1 (en) * 2000-10-02 2007-09-27 Nissan Motor Driver assistance system for a vehicle
CN1725131A (en) * 2004-07-21 2006-01-25 韩京清 Three-parameter fastest self-anti-interference controller device and self-anti-interference control method
CN107710687A (en) * 2015-06-29 2018-02-16 高通股份有限公司 The customized network business model abnormal for detecting application
CN106773696A (en) * 2016-12-28 2017-05-31 广东技术师范学院 Based on the feedback of status linear control method that deviation is separate
CN108508749A (en) * 2018-05-07 2018-09-07 北京航空航天大学 A kind of anti-interference iterative learning control method of Space Manipulator System for arresting noncooperative target
CN109001974A (en) * 2018-08-01 2018-12-14 浙江工业大学 The big coupled movements system position synchronisation control means of networking based on broad sense extended state observer
CN111290277A (en) * 2020-02-26 2020-06-16 鹏城实验室 Distributed multi-agent cooperative fault detection method, storage medium and equipment

Non-Patent Citations (4)

* Cited by examiner, † Cited by third party
Title
Robust State Observer Design for Dynamic Connection Relationships in Complex Dynamical Networks;Zi-lin Gao，等;《International Journal of Control, Automation and Systems》;第17卷(第2期);全文 *
Stabilization of Networked Control Systems With Sparse Observer-Controller Networks;Mohammad Razeghi-Jahromi，等;《IEEE Transactions on Automatic Control》;第60卷(第6期);全文 *
复杂动态网络结构平衡控制设计研究;高子林;《中国博士学位论文全文数据库基础科学辑》(第1期);第A002-63页 *
耦合发生概率不确定的复杂网络传感器故障估计;董墕华，等;《山东科技大学学报(自然科学版)》;第39卷(第2期);全文 *

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