CN113315667B - State estimation method of time-lag complex network system under outlier detection - Google Patents

State estimation method of time-lag complex network system under outlier detection Download PDF

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CN113315667B
CN113315667B CN202110853538.7A CN202110853538A CN113315667B CN 113315667 B CN113315667 B CN 113315667B CN 202110853538 A CN202110853538 A CN 202110853538A CN 113315667 B CN113315667 B CN 113315667B
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邹磊
王子栋
白星振
赵忠义
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Shandong University of Science and Technology
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Abstract

The invention discloses a state estimation method of a time-lag complex network system under outlier detection. The method comprises the following steps: establishing a state equation of a time-lag complex network system, and converting the state equation into a time-lag-free system equation; establishing a detection method of the intermittent occurrence outliers based on the non-time-lag system equation, and detecting all outliers of the time-lag complex network system in the operation process; according to the detection result of the outlier, obtaining an estimator parameter by using a convex optimization technology; and constructing an estimator, substituting the estimator parameters into the estimator, and calculating the estimated value of the time-lag complex network system state. The invention considers the influence of intermittent occurrence outliers in a time-lag complex network system and establishes a detection method, which can effectively detect all the output interfered by outliers; in addition, the invention also considers the influence of bounded noise, and utilizes the output which is not influenced by the outlier to generate the estimated value of the system state based on the convex optimization technology, thereby better meeting the application requirements of the actual industry.

Description

State estimation method of time-lag complex network system under outlier detection
Technical Field
The invention belongs to the field of state estimation, and particularly relates to a state estimation method of a time-lag complex network system under outlier detection.
Background
The time-lapse complex network is formed by connecting a large number of nodes which are coupled with each other through a given topology, and can be used for simulating systems in many industries, such as a smart grid, an intelligent transportation network, the world wide web and the like. For time-lag complex network systems, many internal variables are generally unavailable due to the fact that coupling nodes are more, the size is larger and the like.
The importance of the state estimation technology for reconstructing the system state by using the output of the time-lag complex network system is self-evident in consideration of the important role of the state variables of the time-lag complex network system on monitoring the system operation. The core of the state estimation technology of the system state is to generate an estimated value of the state meeting the specific system performance by using model information and measurement information of the system.
In practice, a time-lag complex network system is usually interfered by external amplitude bounded noise, and if the processing is improper, the accuracy of state estimation is greatly influenced; on the other hand, outliers contained in the system measurements are typically large in amplitude (compared to noise) and occur intermittently, and if the outliers are not properly processed, they will be much more disruptive to the estimation than the noise.
Disclosure of Invention
The invention aims to provide a state estimation method of a time-lag complex network system under outlier detection, which fully considers the influence of outliers and bounded noise of external amplitude values to ensure the state estimation precision of the time-lag complex network system.
In order to achieve the purpose, the invention adopts the following technical scheme:
a state estimation method of a time-lag complex network system under outlier detection comprises the following steps:
step 1, establishing a state equation of a time-lag complex network system, and converting the state equation into a time-lag-free system equation;
step 2, establishing a detection method of the intermittent occurrence outliers based on the time-lag-free system equation obtained in the step 1, and detecting all outliers of the time-lag complex network system in the operation process;
step 3, according to the detection result of the outlier, calculating by using a convex optimization technology to obtain an estimator parameter;
step 4, calculating a state estimation value of the time-lag complex network system by using an estimator;
and (3) calculating a state estimation value of the time-lag complex network system by utilizing an estimator related to the detection result according to the detection result of the intermittent occurrence outlier obtained in the step (2) and the estimator parameter solved in the step (3).
The invention has the following advantages:
as described above, the present invention provides a state estimation method for a time-lapse complex network system under outlier detection, which considers the influence of outliers occurring intermittently, and uses a high-order keeper to establish a detection method, and can give a high-precision estimation at each time no matter whether outliers in system measurement are detected at these times, thereby satisfying the requirements of the actual industry.
Drawings
FIG. 1 is a block diagram illustrating a flow chart of a method for estimating a state of a time-lag complex network system under outlier detection according to an embodiment of the present invention;
FIG. 2 is a schematic diagram illustrating the existence time of the outliers contained in the measurement output of the first set of sensors compared with the corresponding outlier detection results in accordance with an embodiment of the present invention;
FIG. 3 is a schematic diagram illustrating the existence time of the outliers contained in the measurement output of the second set of sensors compared with the corresponding outlier detection results according to an embodiment of the present invention;
FIG. 4 is a schematic diagram illustrating the existence time of the outliers contained in the measurement output of the third group of sensors in comparison with the corresponding outlier detection results in accordance with an embodiment of the present invention;
FIG. 5 is a schematic diagram illustrating a comparison between the system state (first component) of the first node and the estimated system state (first component) obtained by the estimation method according to the present invention;
FIG. 6 is a schematic diagram illustrating a comparison between the system state (second component) of the first node and the estimated system state (second component) obtained by the estimation method according to the present invention;
FIG. 7 is a schematic diagram showing a comparison between the system state (first component) of the second node and the estimated system state (first component) obtained by the estimation method according to the present invention;
fig. 8 is a schematic diagram showing a comparison between the system state (second component) of the second node and the estimated system state (second component) obtained by the estimation method according to the present invention;
FIG. 9 is a schematic diagram illustrating a comparison between the system state (first component) of the third node and the estimated system state (first component) obtained by the estimation method according to the present invention;
fig. 10 is a schematic diagram showing a comparison between the system state (second component) of the third node and the system estimated state (second component) obtained by the estimation method according to the present invention;
fig. 11 is a schematic diagram illustrating a comparison between the system state (first component) of the fourth node and the estimated system state (first component) obtained by the estimation method according to the present invention;
fig. 12 is a schematic diagram showing a comparison between the system state (second component) of the fourth node and the system estimated state (second component) obtained by the estimation method according to the present invention;
FIG. 13 is a schematic diagram illustrating a comparison of real output values of three nodes with estimator input values after a high-order keeper proposed by the method of the present invention;
fig. 14 shows a comparison of estimated error values for an estimated signal based on the method of the invention and based on the more applied lunberg estimator method.
Detailed Description
The invention is described in further detail below with reference to the following figures and detailed description:
as shown in fig. 1, a state estimation method for a time-lag complex network system under outlier detection includes the following steps:
step 1, adopting a mechanism modeling method, considering the coupling among nodes in the complex network, the time-lag phenomenon in the nodes, the influence of process noise and measurement noise and the influence of wild values on measurement, so as to establish a state equation of a time-lag complex network system, and converting the state equation into a non-time-lag system equation.
The state equation of the time-lag complex network system is shown as formula (1);
Figure 100002_DEST_PATH_IMAGE001
（1）
wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE002
to represent complex network systems
Figure 902253DEST_PATH_IMAGE002
A node; k represents a sampling instant;
Figure 100002_DEST_PATH_IMAGE003
to represent complex network systems
Figure 928109DEST_PATH_IMAGE002
An n-dimensional state vector of each node at time k;
Figure 100002_DEST_PATH_IMAGE004
is shown as
Figure 197679DEST_PATH_IMAGE002
Q-dimensional signal to be estimated at time k for each node.
j denotes the jth node of the complex network system, j =1,2, …, N denotes the number of nodes of the complex network system.
xj,kAn n-dimensional state vector representing the jth node of the complex network system at time k.
Figure 100002_DEST_PATH_IMAGE005
To represent complex network systems
Figure 657479DEST_PATH_IMAGE002
Coupling between individual nodes and jth node if
Figure 386401DEST_PATH_IMAGE002
If there is a coupling between one node and the jth node, then
Figure 402898DEST_PATH_IMAGE005
>0, otherwise, the first step is to perform the following steps,
Figure 589291DEST_PATH_IMAGE005
=0。
Figure 100002_DEST_PATH_IMAGE006
is shown as
Figure 100002_DEST_PATH_IMAGE007
Maximum skew of individual nodes;
Figure 100002_DEST_PATH_IMAGE008
an adjacency matrix representing a complex network system when
Figure 43494DEST_PATH_IMAGE007
When = j, there are
Figure 100002_DEST_PATH_IMAGE009
。
Figure 100002_DEST_PATH_IMAGE010
Representing a known positive definite incoupling matrix,
Figure 100002_DEST_PATH_IMAGE011
are all known constants.
Wherein, the matrix
Figure 100002_DEST_PATH_IMAGE012
、
Figure 100002_DEST_PATH_IMAGE013
、
Figure 100002_DEST_PATH_IMAGE014
、
Figure 100002_DEST_PATH_IMAGE015
All represent a known constant matrix.
Matrix array
Figure 495597DEST_PATH_IMAGE012
A parameter matrix representing the system, reflecting the influence of the system state at the current moment on the system state at the next moment; matrix array
Figure 565053DEST_PATH_IMAGE013
A parameter matrix representing the system, reflecting the influence of the system state at the historical moment on the system state at the next moment;
Figure 917537DEST_PATH_IMAGE014
a coupling matrix that is process noise;
Figure 100002_DEST_PATH_IMAGE016
reflecting the relationship between the signal to be estimated and the system state.
Figure 100002_DEST_PATH_IMAGE017
Representing an unknown initial state of the complex network system;
wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE018
。
Figure 100002_DEST_PATH_IMAGE019
the bounded process noise, representing the dimension r, for any sampling instant k,
Figure 100002_DEST_PATH_IMAGE020
satisfies the inequality:
Figure 100002_DEST_PATH_IMAGE021
wherein, in the step (A),
Figure 100002_DEST_PATH_IMAGE022
representing a known normal, the operator | | · | | represents taking the euclidean norm for the vector "·".
If the system has measurements of d nodes, 1< d < N, the measurement equation of the system is shown in formula (2).
Figure 100002_DEST_PATH_IMAGE023
（2）
Wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE024
representing the system measurement output in m dimensions;
Figure 100002_DEST_PATH_IMAGE025
indicating being bounded
Figure 100002_DEST_PATH_IMAGE026
Dimensional measurement noise, for any time k, satisfies
Figure 100002_DEST_PATH_IMAGE027
，
Figure 100002_DEST_PATH_IMAGE028
Is a known normal number.
CiAnd DiAre all known constant matrices.
oi,kAs field value of intermittent occurrence, oi,kThe following model is used for description, namely, the formula (3) shows.
Figure 100002_DEST_PATH_IMAGE029
（3）
In the formula (I), the compound is shown in the specification,
Figure 100002_DEST_PATH_IMAGE030
representing a unit step function;
Figure 100002_DEST_PATH_IMAGE031
denotes the j (th)1The amplitude of the secondary occurrence outliers, for any time k, is satisfied
Figure 100002_DEST_PATH_IMAGE032
。
Wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE033
represents a known normal number;
Figure 100002_DEST_PATH_IMAGE034
denotes the j (th)1The occurrence time of the individual outliers is,
Figure 100002_DEST_PATH_IMAGE035
denotes the j (th)1The disappearance moment of the individual outliers;
Figure 100002_DEST_PATH_IMAGE036
and
Figure 352761DEST_PATH_IMAGE035
satisfies the inequality:
Figure 100002_DEST_PATH_IMAGE037
i.e. any two outliers do not occur simultaneously.
Order to
Figure 100002_DEST_PATH_IMAGE038
Denotes the j (th)1Duration intervals of individual outliers; order to
Figure 100002_DEST_PATH_IMAGE039
Denotes the j (th)11 time outlier disappearance and jth1The time interval between the occurrences of the secondary outliers can then be given by the following equation (4).
Figure 100002_DEST_PATH_IMAGE040
（4）
Definition of
Figure 100002_DEST_PATH_IMAGE041
Has an initial value of
Figure 100002_DEST_PATH_IMAGE042
，
Figure 100002_DEST_PATH_IMAGE043
Is initially of
Figure 100002_DEST_PATH_IMAGE044
Then there is
Figure 100002_DEST_PATH_IMAGE045
(ii) a For arbitrary j1The following formula (5) and formula (6) are given.
Figure 100002_DEST_PATH_IMAGE046
（5）
Wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE047
is a known normal number; the physical meaning of equation (5) is jth in the system measurement1The duration interval of each outlier is not more than
Figure 100002_DEST_PATH_IMAGE048
。
Figure 100002_DEST_PATH_IMAGE049
（6）
Wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE050
is a known normal number; the physical meaning of the formula (6) is j 11 time outlier disappearance and jth1The time interval between the occurrence of the sub-outliers is not shorter than
Figure 100002_DEST_PATH_IMAGE051
. Let the state vector of the complex network system be:
Figure 100002_DEST_PATH_IMAGE052
。
then there are:
Figure 100002_DEST_PATH_IMAGE053
（7）
in the formula (I), the compound is shown in the specification,
Figure 100002_DEST_PATH_IMAGE054
；
A = diag{A1,A2,…,AN}，B = diag{B1,B2,…,BN}，E = diag{E1,E2,…,EN}；
Figure 100002_DEST_PATH_IMAGE055
；
diag {. } represents a block diagonal matrix,') "
Figure 100002_DEST_PATH_IMAGE056
"represents the kronecker product of the matrix; 1i-1、1N-iA column vector representing all elements as 1; order to
Figure 100002_DEST_PATH_IMAGE057
。
The following non-skew system is obtained:
Figure 100002_DEST_PATH_IMAGE058
（8）
wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE059
，
Figure 100002_DEST_PATH_IMAGE060
，
Figure 100002_DEST_PATH_IMAGE061
；
Figure 100002_DEST_PATH_IMAGE062
to represent
Figure 100002_DEST_PATH_IMAGE063
I represents an identity matrix;
Figure 100002_DEST_PATH_IMAGE064
an initial value representing the state of the dead time system,
Figure 100002_DEST_PATH_IMAGE065
is divided into
Figure 100002_DEST_PATH_IMAGE066
Individual blocks, i.e.
Figure 100002_DEST_PATH_IMAGE067
。
Finally, by adopting an observable decomposition technology, the time-lag-free system (8) is transformed into:
Figure 100002_DEST_PATH_IMAGE068
（9）
in the formula (I), the compound is shown in the specification,
Figure 100002_DEST_PATH_IMAGE069
is ni,1Dimension vector, representing the system's inability to observe at time kThe sub-states of (1);
Figure 100002_DEST_PATH_IMAGE070
is ni,2A dimension vector representing a partial state that can be observed by the system at time k;
wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE071
。
Figure 100002_DEST_PATH_IMAGE072
is a known constant matrix representing subsystem parameters that cannot be observed.
Figure 100002_DEST_PATH_IMAGE073
Also known as a constant matrix, representing the subsystem parameters that can be observed.
And 2, establishing a detection method of the intermittent occurrence outliers based on the time-lag-free system equation obtained in the step 1, and detecting all outliers of the time-lag complex network system in the operation process.
Based on an energy-based decomposition theory in a linear system theory and a Karley-Hamilton theorem in a matrix theory, a detection function and a detection threshold of a field value are given, whether the system measurement is influenced by the field value is judged by whether the detection function value exceeds the detection threshold, and all field values of the time-lag complex network in the operation process are detected.
For the ith measurement of a complex network system, a time k and a natural number s are given, and a detection function f is giveni(k, s) is defined as:
Figure 100002_DEST_PATH_IMAGE074
（10）
wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE075
the calculation is iterated by the following formula.
Figure 100002_DEST_PATH_IMAGE076
。
Wherein j is3
Figure 100002_DEST_PATH_IMAGE077
{0,1,…,ni,2-1}，
Figure 100002_DEST_PATH_IMAGE078
Representing a matrix of constants
Figure 100002_DEST_PATH_IMAGE079
Characteristic polynomial of
Figure 100002_DEST_PATH_IMAGE080
The coefficient of (d) is calculated by the following formula.
Figure 100002_DEST_PATH_IMAGE081
。
Where det (-) represents the determinant of the matrix "·"; defining a detection threshold value of
Figure 100002_DEST_PATH_IMAGE082
Then, then
Figure 743028DEST_PATH_IMAGE082
Obtained by solving equation (11).
Figure 100002_DEST_PATH_IMAGE083
（11）
In the formula (I), the compound is shown in the specification,
Figure 100002_DEST_PATH_IMAGE084
，
Figure 100002_DEST_PATH_IMAGE085
，
Figure 100002_DEST_PATH_IMAGE086
，
Figure 100002_DEST_PATH_IMAGE087
，
Figure 100002_DEST_PATH_IMAGE088
as given by the formula (2),
Figure 100002_DEST_PATH_IMAGE089
；
Figure 100002_DEST_PATH_IMAGE090
to represent
Figure 100002_DEST_PATH_IMAGE091
Absolute value of (a).
Figure 100002_DEST_PATH_IMAGE092
。
Wherein r represents process noise
Figure 100002_DEST_PATH_IMAGE093
The dimension of (a);
Figure 100002_DEST_PATH_IMAGE094
represents ni,2A zero matrix of rows Nr columns,
Figure 100002_DEST_PATH_IMAGE095
represents ni,2A zero matrix of rows Nrs columns.
When s =0, there are
Figure 100002_DEST_PATH_IMAGE096
；
Figure 100002_DEST_PATH_IMAGE097
。
All matrix norms referred to above are Frobenius norms.
When in use
Figure 100002_DEST_PATH_IMAGE098
And is
Figure 100002_DEST_PATH_IMAGE099
Then, the following two time series are defined:
Figure 100002_DEST_PATH_IMAGE100
and
Figure 100002_DEST_PATH_IMAGE101
wherein:
Figure 100002_DEST_PATH_IMAGE102
（12）
for all j2Not less than 0, the occurrence time and disappearance time of the outlier are determined by
Figure 100002_DEST_PATH_IMAGE103
And
Figure 100002_DEST_PATH_IMAGE104
are given one by one.
Formula (12) gives a method of detecting a field value occurring intermittently according to the detection function in formula (10) and the detection threshold in formula (11); for the ith sensor node, let j2=0,1, …, having:
1) j th2Detection of occurrence time of the secondary outlier:
let equation (10) detect function fiS =0 in (k, s) to give fi(k,0) not earlier than time of day
Figure 100002_DEST_PATH_IMAGE105
And satisfies the detection condition
Figure 100002_DEST_PATH_IMAGE106
Is the j-th time point k2The time of appearance of the secondary outliers.
Particularly, when j2When =0, stipulate
Figure 100002_DEST_PATH_IMAGE107
。
2) J th2Detection of disappearance time of the secondary outlier:
satisfies the detection condition
Figure 100002_DEST_PATH_IMAGE108
Order of (1)
Figure 100002_DEST_PATH_IMAGE109
The smallest natural number s is the j2The disappearance of the secondary outliers.
According to the detection result of the outlier, a high-order keeper is adopted, and then the input of the estimator is represented as:
Figure 100002_DEST_PATH_IMAGE110
（13）
wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE111
indicating that no outliers are detected at time k, the estimator uses the received system measurements;
Figure 100002_DEST_PATH_IMAGE112
indicating that the outlier is detected at time k, the estimator input is taken according to a high order hold mechanism
Figure 100002_DEST_PATH_IMAGE113
。
Figure 100002_DEST_PATH_IMAGE114
（14）
Wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE115
representing the input signal of the estimator.
Equation (14) represents the time of day
Figure 100002_DEST_PATH_IMAGE116
Linearly combining if outliers are detected in the estimator input
Figure 100002_DEST_PATH_IMAGE117
As input to the estimator.
Figure 100002_DEST_PATH_IMAGE118
Wherein, in the step (A),
Figure 100002_DEST_PATH_IMAGE119
indicating the occurrence time of the outlier closest to the current time k.
Figure 100002_DEST_PATH_IMAGE120
。
Wherein, aggregate
Figure 100002_DEST_PATH_IMAGE121
The element (b) represents the occurrence time of all the outliers not later than the current time k;
Figure 100002_DEST_PATH_IMAGE122
representation collection
Figure 100002_DEST_PATH_IMAGE123
The maximum value of the occurrence time of the field value at the current time k.
The invention identifies the system output interfered by the outlier by establishing the outlier detection method, if the outlier is detected in the system measurement output at a certain time or a certain time, a high-order retainer is adopted at the moment, and the estimation precision of the estimator is effectively ensured.
And 3, calculating to obtain estimator parameters by utilizing a convex optimization technology according to the detection result of the system outlier.
The analysis of the estimation performance is based on the detection result of the system outlier in step 2, and the analysis result shows that when a specific linear matrix inequality has a solution, the estimation error index of the system is finally bounded.
The inventive method thus gives the estimator parameters by solving an optimization problem (the function to be optimized is the final bound) that satisfies the inequality constraints described previously, which as above can be achieved by means of convex optimization techniques.
Estimator parameters
Figure 100002_DEST_PATH_IMAGE124
Is given by solving the solution of the optimization problem (17) with the linear matrix inequality constraints shown in equations (15) and (16), where equation (15), equation (16), and equation (17) are as follows:
Figure 100002_DEST_PATH_IMAGE125
（15）
MTM<P 1 （16）
Figure 100002_DEST_PATH_IMAGE126
（17）
in the formula (I), the compound is shown in the specification,
Figure 100002_DEST_PATH_IMAGE127
， 0<λ<1 represents a given constant;P 1 = diag{P 1,1, P 1,2,…, P 1,Nand (c) the step of (c) in which,P 1,1, P 1,2,…, P 1,Nall are positive definite matrix variables to be solved;P 2the positive definite matrix variable to be solved.
Figure 100002_DEST_PATH_IMAGE128
，
C = [diag{C 1,C 2,…,C d } 0]，
Figure 100002_DEST_PATH_IMAGE129
；
Figure 100002_DEST_PATH_IMAGE130
，
Figure 100002_DEST_PATH_IMAGE131
Is a matrix variable to be solved;
Figure 100002_DEST_PATH_IMAGE132
，
Figure 100002_DEST_PATH_IMAGE133
，
Figure 100002_DEST_PATH_IMAGE134
，
Figure 100002_DEST_PATH_IMAGE135
，
Figure 100002_DEST_PATH_IMAGE136
，
Figure 100002_DEST_PATH_IMAGE137
，
Figure 100002_DEST_PATH_IMAGE138
；
wherein the content of the first and second substances,
Figure 100002_DEST_PATH_IMAGE139
and
Figure 100002_DEST_PATH_IMAGE140
for the scalar to be solved for,
Figure 100002_DEST_PATH_IMAGE141
。
Figure 100002_DEST_PATH_IMAGE142
to represent
Figure 100002_DEST_PATH_IMAGE143
And
Figure 100002_DEST_PATH_IMAGE144
wherein the parameters and
Figure 100002_DEST_PATH_IMAGE145
are given by equation (11).
The estimator parameter
Figure 926096DEST_PATH_IMAGE124
Comprises the following steps:
Figure 100002_DEST_PATH_IMAGE146
，i=1,2…,d（18）。
wherein the calculation formula (18) calculates the estimator parameters
Figure 87081DEST_PATH_IMAGE124
When i =1,2 …, d, only the solved matrix needs to be usedP 1The first d sub-blocks ofP 1,1，P 1,2，，P 1,dThat is (due to d)<N）。
And 4, calculating a state estimation value of the time-lag complex network system by using the estimator.
And (3) calculating a state estimation value of the time-lag complex network system by using a Luneberg type estimator related to the detection result according to the detection result of the intermittent occurrence outlier obtained in the step (2) and the estimator parameter calculated in the step (3).
Using the input of the estimator given in equation (13) and given in equation (18)Is estimated from the estimated parameters
Figure 262847DEST_PATH_IMAGE124
Calculating a state estimation value of the time-lag complex network system, wherein a specific calculation formula is shown as a formula (19);
Figure 100002_DEST_PATH_IMAGE147
（19）
in the formula (19)
Figure DEST_PATH_IMAGE148
Is 1,2, … …, N, N represents the number of nodes of the complex network, but for
Figure 814177DEST_PATH_IMAGE148
The estimated value of the state has different calculation modes.
In particular, for the first d nodes of the system, i.e. for
Figure 163118DEST_PATH_IMAGE148
When values are taken in the set {1,2, … …, d }, corresponding sensor measurement information exists
Figure DEST_PATH_IMAGE149
Can be used for the treatment of the diseases,
Figure DEST_PATH_IMAGE150
the value of (2) is calculated by adopting a first formula in a formula (19);
for the (d + 1) th to (N) th nodes of the system, no corresponding measurement information is available,
Figure 844855DEST_PATH_IMAGE150
is calculated by the second formula in formula (19), and at this time,
Figure 316156DEST_PATH_IMAGE148
values are taken from the set { d +1, d +2, … …, N }.
Wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE151
to represent
Figure DEST_PATH_IMAGE152
Is determined by the estimated value of (c),
Figure DEST_PATH_IMAGE153
to represent
Figure DEST_PATH_IMAGE154
An estimated value of (d);
Figure DEST_PATH_IMAGE155
is shown as
Figure 105514DEST_PATH_IMAGE148
At time k for a nodeqAn estimate of a signal to be estimated of the dimension;
under the interference of bounded noise and outlier, the index of the estimated value of the signal to be estimated calculated by the formula (19) is finally bounded, and the final bound is the minimum value calculated by the optimization problem (17), namely
Figure DEST_PATH_IMAGE156
。
The method of the invention considers the influence of intermittent occurrence outliers in a time-lag complex network system and establishes a detection method, which can effectively detect all the output interfered by outliers; the invention also considers the influence of bounded noise, and utilizes the output which is not influenced by the outlier to generate the estimated value of the system state based on the convex optimization technology, thereby better meeting the application requirements of the actual industry.
Because the complex network model can describe many practical systems, such as artificial neural networks, social relationship networks, power systems, information networks, and the like, the types of sensors used are different for different systems, such as electric meters for circuit systems; for moving objects, the sensors may be position sensors, velocity sensors, and the like.
When the method provided by the invention is applied, only the parameters of the corresponding system are needed to be used, and if the parameters related to the noise cannot be accurately obtained, the parameters which are relatively conservative (such as the upper bound of the amplitude of the noise is larger) can be used.
The state estimation method of the time-lag complex network system under outlier detection proposed by the invention is explained in combination with simulation verification in the following to verify the validity of the method proposed by the invention.
Taking the simulation step length as 150, a four-node complex network system is used, wherein three groups of sensors corresponding to the first three nodes of the system sample the system (considering the situation that part of nodes have no measurement output, the fourth node of the system does not use the corresponding sensor, namely no measurement output). The parameters of the used time-lapse complex network are:
Figure DEST_PATH_IMAGE157
，
Figure DEST_PATH_IMAGE158
，
Figure DEST_PATH_IMAGE159
，A4=-0.9I2，I2is a second order identity matrix;
Figure DEST_PATH_IMAGE160
，
Figure DEST_PATH_IMAGE161
，
Figure DEST_PATH_IMAGE162
，
Figure DEST_PATH_IMAGE163
；
Figure DEST_PATH_IMAGE164
，
Figure DEST_PATH_IMAGE165
，E3=E4=0；
C1=[0.6 0.9]，C2=[0.9 -0.6]， C3=[0.7 0.3]；
M1=I，M2=1.1I，M3=0.9I，M4=0.8I；
D1=0.3， D2=0.4，D3=0.4；
Ξ=0.165I；
Figure DEST_PATH_IMAGE166
，τ=2，
Figure DEST_PATH_IMAGE167
=0.5，
Figure DEST_PATH_IMAGE168
=0.3。
the process noise and measurement noise added are: omega1,k=
Figure DEST_PATH_IMAGE169
1sin(k)， ω2,k=
Figure DEST_PATH_IMAGE170
2sin(1.2k)， ω3,k=
Figure 278654DEST_PATH_IMAGE170
3cos(0.8k0.8)， ω4,k=
Figure 460237DEST_PATH_IMAGE169
4cos(0.9k0.9+1.5)；
υ1,k=
Figure DEST_PATH_IMAGE171
1sin(k0.9+1.5)，υ2,k=
Figure 213692DEST_PATH_IMAGE171
2sin(1.2k0.9+1.5)，υ3,k=3sin(0.8k0.8-1.2)。
The parameters related to the outliers are
Figure DEST_PATH_IMAGE172
=3（i=1，2，3），
Figure DEST_PATH_IMAGE173
=25（i=1，2，3）；
For a given value of i =1,2,3,
Figure DEST_PATH_IMAGE174
are 2.7618, 3.2159, and 2.861, respectively.
By using the state estimation method provided by the invention, the estimation value is generated by using MATLAB software and is compared with the real value of each state variable of the time-lag complex network.
Fig. 2 to 4 show graphs of occurrence time of the outliers contained in the measurement output of the three sets of sensors and corresponding outlier detection results, wherein the upper half of fig. 2 to 4 shows the actual occurrence time and amplitude of the outliers of the first, second, and third nodes, and the lower half of fig. 2 to 4 shows the adopted outlier detection results, and the outlier detected is 1, otherwise 0.
As can be seen from fig. 2 to 4, the outlier detection method of the present invention can accurately detect the appearance time and the disappearance time of the outlier.
Fig. 5 to 12 show the trajectories of the system states of the first, second, third and fourth total four nodes and the estimated state of the system based on the estimation method proposed by the present invention (the solid line represents the true value of the system, and the dotted line represents the estimated value of the system).
FIGS. 5-6 are graphs showing the effect of estimating two components of the state of the first node; FIGS. 7-8 are graphs of the estimated effect of two components of the state of the second node; FIGS. 9-10 are graphs of the estimated effect of two components of the state of the third node; FIGS. 11-12 are graphs showing the effect of estimating two components of the state of the fourth node. It can be easily seen from fig. 5 to 12 that:
the estimated value of the system state obtained by the method of the invention can better track the true value of the system state.
Fig. 13 represents the actual output values of the first, second, and third nodes and the estimator input values after the high-order hold proposed by the present invention is adopted, the solid line represents the actual output trajectory of the system, and the dotted line represents the output trajectory after the high-order hold is adopted.
As can be seen from fig. 13, when the measured signal obtained by the estimator is detected to contain outliers, the input of the estimator generated by the advanced keeper of the present invention is very close to the real output of the system.
When the field value is included in the measurement, the measurement information is unavailable, and a common processing mode aiming at the problem is to adopt a zero input mechanism (the input of the estimator is zero and only uses model information for prediction) or a zero-order keeping mechanism (the input of the estimator adopts the input at the last moment), wherein the difference between the input of the estimator and the real output of the system which is not influenced by the field value is larger, which will certainly influence the estimation performance.
Fig. 14 shows a comparison of estimated error values based on the method of the invention and based on the more applied lunberg estimator method. The solid line represents the estimated error value trajectory based on the method of the invention and the dashed line represents the estimated error value trajectory based on the lunberg observer. As can be seen from fig. 14, the estimation error of the estimator according to the present invention is much lower than that of the traditional roberg estimator, and the effect is better than that of the traditional roberg estimator.
It should be understood, however, that the description herein of specific embodiments is not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.

Claims (1)

1. A state estimation method of a time-lag complex network system under outlier detection is characterized by comprising the following steps:
step 1, establishing a state equation of a time-lag complex network system, and converting the state equation into a time-lag-free system equation;
the state equation of the time-lag complex network system is shown as formula (1);
Figure DEST_PATH_IMAGE001
（1）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE002
to represent complex network systems
Figure 122448DEST_PATH_IMAGE002
A node; k represents a sampling instant;
Figure DEST_PATH_IMAGE003
to represent complex network systems
Figure 164222DEST_PATH_IMAGE002
An n-dimensional state vector of each node at time k;
Figure DEST_PATH_IMAGE004
is shown as
Figure 649953DEST_PATH_IMAGE002
Q-dimensional signals to be estimated of each node at k time;
j represents the jth node of the complex network system, j =1,2, …, N represents the number of nodes of the complex network system;
xj,krepresenting an n-dimensional state vector of a jth node of a complex network system at time k；
Figure DEST_PATH_IMAGE005
To represent complex network systems
Figure 982846DEST_PATH_IMAGE002
Coupling between individual nodes and jth node if
Figure 797218DEST_PATH_IMAGE002
If there is a coupling between one node and the jth node, then
Figure 432730DEST_PATH_IMAGE005
>0, otherwise, the first step is to perform the following steps,
Figure DEST_PATH_IMAGE006
=0；
Figure DEST_PATH_IMAGE007
is shown as
Figure 450495DEST_PATH_IMAGE002
Maximum skew of individual nodes;
Figure DEST_PATH_IMAGE008
an adjacency matrix representing a complex network system when
Figure 587079DEST_PATH_IMAGE002
When = j, there are
Figure DEST_PATH_IMAGE009
；
Figure DEST_PATH_IMAGE010
Representing a known positive definite incoupling matrix,
Figure DEST_PATH_IMAGE011
are all known constants;
wherein, the matrix
Figure DEST_PATH_IMAGE012
、
Figure DEST_PATH_IMAGE013
、
Figure DEST_PATH_IMAGE014
、
Figure DEST_PATH_IMAGE015
All represent a known constant matrix;
matrix array
Figure 876197DEST_PATH_IMAGE012
A parameter matrix representing the system, reflecting the influence of the system state at the current moment on the system state at the next moment; matrix array
Figure 557976DEST_PATH_IMAGE013
A parameter matrix representing the system, reflecting the influence of the system state at the historical moment on the system state at the next moment;
Figure 436939DEST_PATH_IMAGE014
a coupling matrix that is process noise;
Figure DEST_PATH_IMAGE016
reflecting the relation between the signal to be estimated and the system state;
Figure DEST_PATH_IMAGE017
representing an unknown initial state of the complex network system;
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE018
；
the measurement of d nodes of the system can be obtained, and if d is more than 1 and less than N, the measurement equation of the system is shown as a formula (2);
Figure DEST_PATH_IMAGE019
（2）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE020
representing the system measurement output in m dimensions;
Figure DEST_PATH_IMAGE021
indicating being bounded
Figure DEST_PATH_IMAGE022
Dimensional measurement noise, for any time k, satisfies
Figure DEST_PATH_IMAGE023
，
Figure DEST_PATH_IMAGE024
Is a known normal number;
Ciand DiAre all known constant matrices;
oi,kas field value of intermittent occurrence, oi,kThe following model is used for description, namely the formula (3);
Figure DEST_PATH_IMAGE025
（3）
in the formula (I), the compound is shown in the specification,
Figure DEST_PATH_IMAGE026
representing a unit step function;
Figure DEST_PATH_IMAGE027
denotes the j (th)1The amplitude of the secondary occurrence outliers, for any time k, is satisfied
Figure DEST_PATH_IMAGE028
；
Wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE029
represents a known normal number;
Figure DEST_PATH_IMAGE030
denotes the j (th)1The occurrence time of the individual outliers is,
Figure DEST_PATH_IMAGE031
denotes the j (th)1The disappearance moment of the individual outliers;
Figure 108704DEST_PATH_IMAGE030
and
Figure 320505DEST_PATH_IMAGE031
satisfies the inequality:
Figure DEST_PATH_IMAGE032
that is, any two outliers will not occur simultaneously;
order to
Figure DEST_PATH_IMAGE033
Denotes the j (th)1Duration intervals of individual outliers; order to
Figure DEST_PATH_IMAGE034
Denotes the j (th)11 time outlier disappearance and jth1The time interval between occurrences of the secondary outliers, then the following equation (4) can be obtained;
Figure DEST_PATH_IMAGE035
（4）
definition of
Figure DEST_PATH_IMAGE036
Has an initial value of
Figure DEST_PATH_IMAGE037
，
Figure DEST_PATH_IMAGE038
Is initially of
Figure DEST_PATH_IMAGE039
Then there is
Figure DEST_PATH_IMAGE040
(ii) a For arbitrary j1The following formula (5) and formula (6);
Figure DEST_PATH_IMAGE041
（5）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE042
is a known normal number; the physical meaning of equation (5) is jth in the system measurement1The duration interval of each outlier is not more than
Figure DEST_PATH_IMAGE043
；
Figure DEST_PATH_IMAGE044
（6）
Wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE045
is a known normal number; the physical meaning of the formula (6) is j11 time outlier disappearance and jth1Time intervals between occurrences of sub-outliersNot shorter than
Figure DEST_PATH_IMAGE046
；
Let the state vector of the complex network system be:
Figure DEST_PATH_IMAGE047
，
then there are:
Figure DEST_PATH_IMAGE048
（7）
in the formula (I), the compound is shown in the specification,
Figure DEST_PATH_IMAGE049
；
A = diag{A1,A2,…,AN}，B = diag{B1,B2,…,BN}，E = diag{E1,E2,…,EN}；
Figure DEST_PATH_IMAGE050
；
diag {. } represents a block diagonal matrix,') "
Figure DEST_PATH_IMAGE051
"represents the kronecker product of the matrix; 1i-1、1N-iA column vector representing all elements as 1;
order to
Figure DEST_PATH_IMAGE052
The following non-lag system is obtained:
Figure DEST_PATH_IMAGE053
（8）
wherein:
Figure DEST_PATH_IMAGE054
，
Figure DEST_PATH_IMAGE055
，
Figure DEST_PATH_IMAGE056
；
Figure DEST_PATH_IMAGE057
to represent
Figure DEST_PATH_IMAGE058
I represents an identity matrix;
Figure DEST_PATH_IMAGE059
an initial value representing the state of the dead time system,
Figure DEST_PATH_IMAGE060
is divided into
Figure DEST_PATH_IMAGE061
Individual blocks, i.e.
Figure DEST_PATH_IMAGE062
；
Finally, by adopting an observable decomposition technology, the time-lag-free system (8) is transformed into:
Figure DEST_PATH_IMAGE063
（9）
in the formula (I), the compound is shown in the specification,
Figure DEST_PATH_IMAGE064
is ni,1A dimension vector representing a sub-state that the system cannot observe at time k;
Figure DEST_PATH_IMAGE065
is ni,2A dimension vector representing a partial state that can be observed by the system at time k;
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE066
；
Figure DEST_PATH_IMAGE067
a known constant matrix representing subsystem parameters that cannot be observed;
Figure DEST_PATH_IMAGE068
also known as a constant matrix, representing observable subsystem parameters;
step 2, establishing a detection method of the intermittent occurrence outliers based on the time-lag-free system equation obtained in the step 1, and detecting all outliers of the time-lag complex network system in the operation process;
for the ith measurement of a complex network system, a time k and a natural number s are given, and a detection function f is giveni(k, s) is defined as:
Figure DEST_PATH_IMAGE069
（10）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE070
the calculation is carried out iteratively by the following formula;
Figure DEST_PATH_IMAGE071
；
wherein j is3
Figure DEST_PATH_IMAGE072
{0,1,…,ni,2-1}，
Figure DEST_PATH_IMAGE073
Representing a matrix of constants
Figure DEST_PATH_IMAGE074
Characteristic polynomial of
Figure DEST_PATH_IMAGE075
The coefficient of (d) is calculated by the following formula;
Figure DEST_PATH_IMAGE076
；
where det (-) represents the determinant of the matrix "·"; defining a detection threshold value of
Figure DEST_PATH_IMAGE077
Then, then
Figure 827315DEST_PATH_IMAGE077
Solving by a formula (11);
Figure DEST_PATH_IMAGE078
（11）
in the formula (I), the compound is shown in the specification,
Figure DEST_PATH_IMAGE079
，
Figure DEST_PATH_IMAGE080
，
Figure DEST_PATH_IMAGE081
，
Figure DEST_PATH_IMAGE082
，
Figure DEST_PATH_IMAGE083
；
Figure DEST_PATH_IMAGE084
to represent
Figure DEST_PATH_IMAGE085
Absolute value of (d);
Figure DEST_PATH_IMAGE086
；
wherein r represents process noise
Figure DEST_PATH_IMAGE087
The dimension of (a);
Figure DEST_PATH_IMAGE088
represents ni,2A zero matrix of rows Nr columns,
Figure DEST_PATH_IMAGE089
represents ni,2A zero matrix of rows Nrs columns;
when s =0, there are
Figure DEST_PATH_IMAGE090
；
Figure DEST_PATH_IMAGE091
；
All the matrix norms referred to above are Frobenius norms;
when in use
Figure DEST_PATH_IMAGE092
And is
Figure DEST_PATH_IMAGE093
Then, the following two time series are defined:
Figure DEST_PATH_IMAGE094
and
Figure DEST_PATH_IMAGE095
wherein:
Figure DEST_PATH_IMAGE096
（12）
for all j2Not less than 0, the occurrence time and disappearance time of the outlier are determined by
Figure DEST_PATH_IMAGE097
And
Figure DEST_PATH_IMAGE098
giving out the raw materials one by one;
formula (12) gives a method of detecting a field value occurring intermittently according to the detection function in formula (10) and the detection threshold in formula (11); for the ith sensor node, let j2=0,1, …, having:
1) j th2Detection of occurrence time of the secondary outlier:
let equation (10) detect function fiS =0 in (k, s) to give fi(k,0) not earlier than time of day
Figure DEST_PATH_IMAGE099
And satisfies the detection condition
Figure DEST_PATH_IMAGE100
Is the j-th time point k2The appearance time of the secondary outlier;
2) j th2Detection of disappearance time of the secondary outlier:
satisfies the detection condition
Figure DEST_PATH_IMAGE101
Order of (1)
Figure DEST_PATH_IMAGE102
The smallest natural number s is the j2The disappearance moment of the secondary outlier;
according to the detection result of the outlier, a high-order keeper is adopted, and then the input of the estimator is represented as:
Figure DEST_PATH_IMAGE103
（13）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE104
indicating that no outliers are detected at time k, the estimator uses the received system measurements;
Figure DEST_PATH_IMAGE105
indicating that the outlier is detected at time k, the estimator input is taken according to a high order hold mechanism
Figure DEST_PATH_IMAGE106
；
Figure DEST_PATH_IMAGE107
（14）
Wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE108
representing an input signal of an estimator; equation (14) shows that at time k, if a outlier is detected in the estimator input, the linear combinations will be
Figure DEST_PATH_IMAGE109
As an input to the estimator;
Figure DEST_PATH_IMAGE110
wherein, in the step (A),
Figure DEST_PATH_IMAGE111
indicating the occurrence time of the outlier closest to the current time k;
Figure DEST_PATH_IMAGE112
；
wherein, aggregate
Figure DEST_PATH_IMAGE113
The element (b) represents the occurrence time of all the outliers not later than the current time k;
Figure DEST_PATH_IMAGE114
representation collection
Figure DEST_PATH_IMAGE115
Maximum value in occurrence time of field value not later than current time k;
step 3, according to the detection result of the outlier, calculating by using a convex optimization technology to obtain an estimator parameter;
estimator parameters
Figure DEST_PATH_IMAGE116
Is given by solving the solution of the optimization problem (17) with the linear matrix inequality constraints shown in equations (15) and (16), where equation (15), equation (16), and equation (17) are as follows:
Figure DEST_PATH_IMAGE117
（15）
MTM<P 1 （16）
Figure DEST_PATH_IMAGE118
（17）
in the formula (I), the compound is shown in the specification,
Figure DEST_PATH_IMAGE119
， 0<λ<1 represents a given constant;P 1 = diag{P 1,1, P 1,2,…, P 1,Nand (c) the step of (c) in which,P 1,1, P 1,2,…, P 1,Nall are positive definite matrix variables to be solved;P 2a positive definite matrix variable to be solved;
Figure DEST_PATH_IMAGE120
，
C = [diag{C 1,C 2,…,C d } 0]，
Figure DEST_PATH_IMAGE121
；
Figure DEST_PATH_IMAGE122
，
Figure DEST_PATH_IMAGE123
is a matrix variable to be solved;
Figure DEST_PATH_IMAGE124
，
Figure DEST_PATH_IMAGE125
，
Figure DEST_PATH_IMAGE126
，
Figure DEST_PATH_IMAGE127
，
Figure DEST_PATH_IMAGE128
，
Figure DEST_PATH_IMAGE129
，
Figure DEST_PATH_IMAGE130
；
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE131
and
Figure DEST_PATH_IMAGE132
for the scalar to be solved for,
Figure DEST_PATH_IMAGE133
；
Figure DEST_PATH_IMAGE134
to represent
Figure DEST_PATH_IMAGE135
And
Figure DEST_PATH_IMAGE136
wherein the parameters and
Figure DEST_PATH_IMAGE137
are given by equation (11);
the estimator parameter
Figure DEST_PATH_IMAGE138
Comprises the following steps:
Figure DEST_PATH_IMAGE139
，i=1,2…,d（18）；
step 4, calculating a state estimation value of the time-lag complex network system by using an estimator;
calculating a state estimation value of the time-lag complex network system by using an estimator related to the detection result according to the detection result of the intermittent occurrence outlier obtained in the step 2 and the estimator parameter solved in the step 3;
calculating a state estimation value of the time-lag complex network system by using the input of the estimator given in the formula (13) and the estimator parameter given in the formula (18), wherein the specific calculation formula is shown as a formula (19);
Figure DEST_PATH_IMAGE140
（19）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE141
to represent
Figure DEST_PATH_IMAGE142
Is determined by the estimated value of (c),
Figure DEST_PATH_IMAGE143
to represent
Figure DEST_PATH_IMAGE144
An estimated value of (d);
Figure DEST_PATH_IMAGE145
is shown as
Figure DEST_PATH_IMAGE146
At time k for a nodeqAn estimate of a signal to be estimated of the dimension;
under the interference of bounded noise and outlier, the index of the estimated value of the signal to be estimated calculated by the formula (19) is finally bounded, and the final bound is the minimum value calculated by the optimization problem (17), namely
Figure DEST_PATH_IMAGE147
。
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